LMFDB - Embedded newform 42.9.c.a.13.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [42,9,Mod(13,42)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(42, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("42.13");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 42 = 2 \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	42.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.1099016226$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 7731 x^{10} + 218714 x^{9} + 46944238 x^{8} + 954612102 x^{7} + \cdots + 37\!\cdots\!84 $ x^12 - 2x^11 + 7731x^10 + 218714x^9 + 46944238x^8 + 954612102x^7 + 95335059229x^6 + 576178351086x^5 + 122405480888457x^4 + 247993465027288x^3 + 91263792176643256x^2 - 730882789543557696*x + 37671313072244229184
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{18}\cdot 7^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			13.9
Root			$42.6889 + 73.9393i$ of defining polynomial
Character	$\chi$	$=$	42.13
Dual form			42.9.c.a.13.10

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.3137 q^{2} -46.7654i q^{3} +128.000 q^{4} +730.548i q^{5} -529.090i q^{6} +(-2148.48 + 1071.83i) q^{7} +1448.15 q^{8} -2187.00 q^{9} +O(q^{10})$	\(q+11.3137 q^{2} -46.7654i q^{3} +128.000 q^{4} +730.548i q^{5} -529.090i q^{6} +(-2148.48 + 1071.83i) q^{7} +1448.15 q^{8} -2187.00 q^{9} +8265.21i q^{10} -18934.9 q^{11} -5985.97i q^{12} -413.152i q^{13} +(-24307.3 + 12126.4i) q^{14} +34164.3 q^{15} +16384.0 q^{16} +149161. i q^{17} -24743.1 q^{18} +50809.9i q^{19} +93510.1i q^{20} +(50124.6 + 100475. i) q^{21} -214223. q^{22} +39352.9 q^{23} -67723.5i q^{24} -143075. q^{25} -4674.28i q^{26} +102276. i q^{27} +(-275006. + 137194. i) q^{28} -729438. q^{29} +386525. q^{30} -41630.7i q^{31} +185364. q^{32} +885496. i q^{33} +1.68757e6i q^{34} +(-783024. - 1.56957e6i) q^{35} -279936. q^{36} +2.63614e6 q^{37} +574848. i q^{38} -19321.2 q^{39} +1.05795e6i q^{40} +893911. i q^{41} +(567095. + 1.13674e6i) q^{42} -5.03478e6 q^{43} -2.42366e6 q^{44} -1.59771e6i q^{45} +445227. q^{46} +598476. i q^{47} -766204. i q^{48} +(3.46716e6 - 4.60562e6i) q^{49} -1.61871e6 q^{50} +6.97558e6 q^{51} -52883.4i q^{52} +1.11110e7 q^{53} +1.15712e6i q^{54} -1.38328e7i q^{55} +(-3.11134e6 + 1.55218e6i) q^{56} +2.37614e6 q^{57} -8.25265e6 q^{58} -1.74861e7i q^{59} +4.37304e6 q^{60} -1.53019e7i q^{61} -470997. i q^{62} +(4.69873e6 - 2.34410e6i) q^{63} +2.09715e6 q^{64} +301827. q^{65} +1.00182e7i q^{66} -2.20545e7 q^{67} +1.90926e7i q^{68} -1.84035e6i q^{69} +(-8.85891e6 - 1.77577e7i) q^{70} +2.40517e7 q^{71} -3.16711e6 q^{72} +3.14141e7i q^{73} +2.98245e7 q^{74} +6.69097e6i q^{75} +6.50367e6i q^{76} +(4.06812e7 - 2.02950e7i) q^{77} -218594. q^{78} -3.99613e7 q^{79} +1.19693e7i q^{80} +4.78297e6 q^{81} +1.01135e7i q^{82} +4.93228e7i q^{83} +(6.41595e6 + 1.28607e7i) q^{84} -1.08969e8 q^{85} -5.69620e7 q^{86} +3.41125e7i q^{87} -2.74206e7 q^{88} +9.82960e7i q^{89} -1.80760e7i q^{90} +(442829. + 887650. i) q^{91} +5.03717e6 q^{92} -1.94687e6 q^{93} +6.77099e6i q^{94} -3.71191e7 q^{95} -8.66861e6i q^{96} -1.46329e8i q^{97} +(3.92264e7 - 5.21067e7i) q^{98} +4.14105e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 1536 q^{4} + 6420 q^{7} - 26244 q^{9}+O(q^{10}) $ 12 * q + 1536 * q^4 + 6420 * q^7 - 26244 * q^9	$ 12 q + 1536 q^{4} + 6420 q^{7} - 26244 q^{9} + 4344 q^{11} - 12288 q^{14} + 59616 q^{15} + 196608 q^{16} - 224856 q^{21} - 508416 q^{22} + 499800 q^{23} - 3001476 q^{25} + 821760 q^{28} - 1278408 q^{29} + 705024 q^{30} + 2028912 q^{35} - 3359232 q^{36} + 7068648 q^{37} - 5473008 q^{39} + 1513728 q^{42} - 11388024 q^{43} + 556032 q^{44} + 8171520 q^{46} - 12346788 q^{49} + 30019584 q^{50} + 16727472 q^{51} + 19714968 q^{53} - 1572864 q^{56} - 10386144 q^{57} - 17696256 q^{58} + 7630848 q^{60} - 14040540 q^{63} + 25165824 q^{64} - 93770592 q^{65} - 9394008 q^{67} + 11218944 q^{70} + 5393208 q^{71} + 58512384 q^{74} + 24982968 q^{77} + 32638464 q^{78} + 134560968 q^{79} + 57395628 q^{81} - 28781568 q^{84} - 102074640 q^{85} - 282934272 q^{86} - 65077248 q^{88} - 96105408 q^{91} + 63974400 q^{92} + 202339296 q^{93} - 378351840 q^{95} + 387747840 q^{98} - 9500328 q^{99}+O(q^{100}) $ 12 * q + 1536 * q^4 + 6420 * q^7 - 26244 * q^9 + 4344 * q^11 - 12288 * q^14 + 59616 * q^15 + 196608 * q^16 - 224856 * q^21 - 508416 * q^22 + 499800 * q^23 - 3001476 * q^25 + 821760 * q^28 - 1278408 * q^29 + 705024 * q^30 + 2028912 * q^35 - 3359232 * q^36 + 7068648 * q^37 - 5473008 * q^39 + 1513728 * q^42 - 11388024 * q^43 + 556032 * q^44 + 8171520 * q^46 - 12346788 * q^49 + 30019584 * q^50 + 16727472 * q^51 + 19714968 * q^53 - 1572864 * q^56 - 10386144 * q^57 - 17696256 * q^58 + 7630848 * q^60 - 14040540 * q^63 + 25165824 * q^64 - 93770592 * q^65 - 9394008 * q^67 + 11218944 * q^70 + 5393208 * q^71 + 58512384 * q^74 + 24982968 * q^77 + 32638464 * q^78 + 134560968 * q^79 + 57395628 * q^81 - 28781568 * q^84 - 102074640 * q^85 - 282934272 * q^86 - 65077248 * q^88 - 96105408 * q^91 + 63974400 * q^92 + 202339296 * q^93 - 378351840 * q^95 + 387747840 * q^98 - 9500328 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/42\mathbb{Z}\right)^\times$.

$n$	$29$	$31$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	11.3137			0.707107
$2$	11.3137			0.707107
$3$		−	46.7654i		−	0.577350i
$3$		−	46.7654i		−	0.577350i
$4$	128.000			0.500000
$5$			730.548i			1.16888i	0.811438	+	0.584438i	$0.198685\pi$
$5$			730.548i			1.16888i	−0.811438	+	0.584438i	$0.801315\pi$
$6$		−	529.090i		−	0.408248i
$7$	−2148.48	+	1071.83i	−0.894828	+	0.446410i
$7$	−2148.48	+	1071.83i	−0.894828	+	0.446410i
$8$	1448.15			0.353553
$9$	−2187.00			−0.333333
$10$			8265.21i			0.826521i
$11$	−18934.9			−1.29328			−0.646638	−	0.762797i	$-0.723826\pi$
$11$	−18934.9			−1.29328			−0.646638	+	0.762797i	$0.723826\pi$
$12$		−	5985.97i		−	0.288675i
$13$		−	413.152i		−	0.0144656i	−0.999974	−	0.00723280i	$-0.997698\pi$
$13$		−	413.152i		−	0.0144656i	0.999974	−	0.00723280i	$-0.00230229\pi$
$14$	−24307.3	+	12126.4i	−0.632739	+	0.315660i
$15$	34164.3			0.674851
$16$	16384.0			0.250000
$17$			149161.i			1.78591i	0.450143	+	0.892957i	$0.351373\pi$
$17$			149161.i			1.78591i	−0.450143	+	0.892957i	$0.648627\pi$
$18$	−24743.1			−0.235702
$19$			50809.9i			0.389883i	0.980815	+	0.194941i	$0.0624517\pi$
$19$			50809.9i			0.389883i	−0.980815	+	0.194941i	$0.937548\pi$
$20$			93510.1i			0.584438i
$21$	50124.6	+	100475.i	0.257735	+	0.516629i
$22$	−214223.			−0.914485
$23$	39352.9			0.140626			0.0703130	−	0.997525i	$-0.477600\pi$
$23$	39352.9			0.140626			0.0703130	+	0.997525i	$0.477600\pi$
$24$		−	67723.5i		−	0.204124i
$25$	−143075.			−0.366273
$26$		−	4674.28i		−	0.0102287i
$27$			102276.i			0.192450i
$28$	−275006.	+	137194.i	−0.447414	+	0.223205i
$29$	−729438.			−1.03133			−0.515664	−	0.856791i	$-0.672455\pi$
$29$	−729438.			−1.03133			−0.515664	+	0.856791i	$0.672455\pi$
$30$	386525.			0.477192
$31$		−	41630.7i		−	0.0450782i	−0.999746	−	0.0225391i	$-0.992825\pi$
$31$		−	41630.7i		−	0.0450782i	0.999746	−	0.0225391i	$-0.00717503\pi$
$32$	185364.			0.176777
$33$			885496.i			0.746673i
$34$			1.68757e6i			1.26283i
$35$	−783024.	−	1.56957e6i	−0.521799	−	1.04594i
$36$	−279936.			−0.166667
$37$	2.63614e6			1.40657			0.703286	−	0.710907i	$-0.251715\pi$
$37$	2.63614e6			1.40657			0.703286	+	0.710907i	$0.251715\pi$
$38$			574848.i			0.275689i
$39$	−19321.2			−0.00835171
$40$			1.05795e6i			0.413260i
$41$			893911.i			0.316344i	0.987412	+	0.158172i	$0.0505600\pi$
$41$			893911.i			0.316344i	−0.987412	+	0.158172i	$0.949440\pi$
$42$	567095.	+	1.13674e6i	0.182246	+	0.365312i
$43$	−5.03478e6			−1.47267			−0.736336	−	0.676616i	$-0.763446\pi$
$43$	−5.03478e6			−1.47267			−0.736336	+	0.676616i	$0.763446\pi$
$44$	−2.42366e6			−0.646638
$45$		−	1.59771e6i		−	0.389626i
$46$	445227.			0.0994376
$47$			598476.i			0.122647i	0.998118	+	0.0613233i	$0.0195321\pi$
$47$			598476.i			0.122647i	−0.998118	+	0.0613233i	$0.980468\pi$
$48$		−	766204.i		−	0.144338i
$49$	3.46716e6	−	4.60562e6i	0.601436	−	0.798921i
$50$	−1.61871e6			−0.258994
$51$	6.97558e6			1.03110
$52$		−	52883.4i		−	0.00723280i
$53$	1.11110e7			1.40815			0.704075	−	0.710125i	$-0.251362\pi$
$53$	1.11110e7			1.40815			0.704075	+	0.710125i	$0.251362\pi$
$54$			1.15712e6i			0.136083i
$55$		−	1.38328e7i		−	1.51168i
$56$	−3.11134e6	+	1.55218e6i	−0.316370	+	0.157830i
$57$	2.37614e6			0.225099
$58$	−8.25265e6			−0.729259
$59$		−	1.74861e7i		−	1.44307i	−0.692381	−	0.721533i	$-0.743438\pi$
$59$		−	1.74861e7i		−	1.44307i	0.692381	−	0.721533i	$-0.256562\pi$
$60$	4.37304e6			0.337426
$61$		−	1.53019e7i		−	1.10516i	−0.833459	−	0.552581i	$-0.813643\pi$
$61$		−	1.53019e7i		−	1.10516i	0.833459	−	0.552581i	$-0.186357\pi$
$62$		−	470997.i		−	0.0318751i
$63$	4.69873e6	−	2.34410e6i	0.298276	−	0.148803i
$64$	2.09715e6			0.125000
$65$	301827.			0.0169085
$66$			1.00182e7i			0.527978i
$67$	−2.20545e7			−1.09445			−0.547227	−	0.836984i	$-0.684317\pi$
$67$	−2.20545e7			−1.09445			−0.547227	+	0.836984i	$0.684317\pi$
$68$			1.90926e7i			0.892957i
$69$		−	1.84035e6i		−	0.0811904i
$70$	−8.85891e6	−	1.77577e7i	−0.368967	−	0.739594i
$71$	2.40517e7			0.946481			0.473240	−	0.880933i	$-0.343084\pi$
$71$	2.40517e7			0.946481			0.473240	+	0.880933i	$0.343084\pi$
$72$	−3.16711e6			−0.117851
$73$			3.14141e7i			1.10620i	0.833116	+	0.553099i	$0.186555\pi$
$73$			3.14141e7i			1.10620i	−0.833116	+	0.553099i	$0.813445\pi$
$74$	2.98245e7			0.994597
$75$			6.69097e6i			0.211468i
$76$			6.50367e6i			0.194941i
$77$	4.06812e7	−	2.02950e7i	1.15726	−	0.577332i
$78$	−218594.			−0.00590555
$79$	−3.99613e7			−1.02596			−0.512981	−	0.858400i	$-0.671459\pi$
$79$	−3.99613e7			−1.02596			−0.512981	+	0.858400i	$0.671459\pi$
$80$			1.19693e7i			0.292219i
$81$	4.78297e6			0.111111
$82$			1.01135e7i			0.223689i
$83$			4.93228e7i			1.03929i	0.854383	+	0.519644i	$0.173935\pi$
$83$			4.93228e7i			1.03929i	−0.854383	+	0.519644i	$0.826065\pi$
$84$	6.41595e6	+	1.28607e7i	0.128868	+	0.258315i
$85$	−1.08969e8			−2.08751
$86$	−5.69620e7			−1.04134
$87$			3.41125e7i			0.595437i
$88$	−2.74206e7			−0.457242
$89$			9.82960e7i			1.56666i	0.621603	+	0.783332i	$0.286482\pi$
$89$			9.82960e7i			1.56666i	−0.621603	+	0.783332i	$0.713518\pi$
$90$		−	1.80760e7i		−	0.275507i
$91$	442829.	+	887650.i	0.00645759	+	0.0129442i
$92$	5.03717e6			0.0703130
$93$	−1.94687e6			−0.0260259
$94$			6.77099e6i			0.0867243i
$95$	−3.71191e7			−0.455725
$96$		−	8.66861e6i		−	0.102062i
$97$		−	1.46329e8i		−	1.65288i	−0.563022	−	0.826442i	$-0.690361\pi$
$97$		−	1.46329e8i		−	1.65288i	0.563022	−	0.826442i	$-0.309639\pi$
$98$	3.92264e7	−	5.21067e7i	0.425279	−	0.564923i
$99$	4.14105e7			0.431092
$100$	−1.83136e7			−0.183136
$101$			1.68457e8i			1.61884i	0.587231	+	0.809419i	$0.300218\pi$
$101$			1.68457e8i			1.61884i	−0.587231	+	0.809419i	$0.699782\pi$
$102$	7.89197e7			0.729096
$103$			4.04837e7i			0.359693i	0.983695	+	0.179846i	$0.0575600\pi$
$103$			4.04837e7i			0.359693i	−0.983695	+	0.179846i	$0.942440\pi$
$104$		−	598308.i		−	0.00511436i
$105$	−7.34015e7	+	3.66184e7i	−0.603876	+	0.301261i
$106$	1.25706e8			0.995713
$107$	2.43115e8			1.85471			0.927355	−	0.374182i	$-0.122076\pi$
$107$	2.43115e8			1.85471			0.927355	+	0.374182i	$0.122076\pi$
$108$			1.30913e7i			0.0962250i
$109$	−1.01523e8			−0.719213			−0.359607	−	0.933104i	$-0.617089\pi$
$109$	−1.01523e8			−0.719213			−0.359607	+	0.933104i	$0.617089\pi$
$110$		−	1.56501e8i		−	1.06892i
$111$		−	1.23280e8i		−	0.812085i
$112$	−3.52007e7	+	1.75609e7i	−0.223707	+	0.111603i
$113$	5.88119e6			0.0360705			0.0180352	−	0.999837i	$-0.494259\pi$
$113$	5.88119e6			0.0360705			0.0180352	+	0.999837i	$0.494259\pi$
$114$	2.68830e7			0.159169
$115$			2.87492e7i			0.164374i
$116$	−9.33681e7			−0.515664
$117$			903563.i			0.00482186i
$118$		−	1.97833e8i		−	1.02040i
$119$	−1.59876e8	−	3.20470e8i	−0.797250	−	1.59809i
$120$	4.94753e7			0.238596
$121$	1.44170e8			0.672564
$122$		−	1.73121e8i		−	0.781467i
$123$	4.18041e7			0.182641
$124$		−	5.32873e6i		−	0.0225391i
$125$			1.80847e8i			0.740749i
$126$	5.31601e7	−	2.65204e7i	0.210913	−	0.105220i
$127$	−4.06352e7			−0.156202			−0.0781012	−	0.996945i	$-0.524886\pi$
$127$	−4.06352e7			−0.156202			−0.0781012	+	0.996945i	$0.524886\pi$
$128$	2.37266e7			0.0883883
$129$			2.35453e8i			0.850248i
$130$	3.41479e6			0.0119561
$131$		−	1.24579e8i		−	0.423019i	−0.977376	−	0.211509i	$-0.932162\pi$
$131$		−	1.24579e8i		−	0.423019i	0.977376	−	0.211509i	$-0.0678379\pi$
$132$			1.13343e8i			0.373337i
$133$	−5.44596e7	−	1.09164e8i	−0.174048	−	0.348878i
$134$	−2.49518e8			−0.773897
$135$	−7.47174e7			−0.224950
$136$			2.16009e8i			0.631416i
$137$	3.50293e8			0.994373			0.497187	−	0.867644i	$-0.334366\pi$
$137$	3.50293e8			0.994373			0.497187	+	0.867644i	$0.334366\pi$
$138$		−	2.08212e7i		−	0.0574103i
$139$			4.00011e8i			1.07155i	0.844361	+	0.535775i	$0.179980\pi$
$139$			4.00011e8i			1.07155i	−0.844361	+	0.535775i	$0.820020\pi$
$140$	−1.00227e8	−	2.00905e8i	−0.260899	−	0.522972i
$141$	2.79880e7			0.0708101
$142$	2.72113e8			0.669263
$143$			7.82297e6i			0.0187080i
$144$	−3.58318e7			−0.0833333
$145$		−	5.32890e8i		−	1.20549i
$146$			3.55410e8i			0.782200i
$147$	−2.15384e8	−	1.62143e8i	−0.461257	−	0.347239i
$148$	3.37426e8			0.703286
$149$	−5.02630e8			−1.01977			−0.509886	−	0.860242i	$-0.670313\pi$
$149$	−5.02630e8			−1.01977			−0.509886	+	0.860242i	$0.670313\pi$
$150$			7.56997e7i			0.149530i
$151$	1.18014e8			0.226999			0.113500	−	0.993538i	$-0.463794\pi$
$151$	1.18014e8			0.226999			0.113500	+	0.993538i	$0.463794\pi$
$152$			7.35806e7i			0.137844i
$153$		−	3.26216e8i		−	0.595304i
$154$	4.60255e8	−	2.29611e8i	0.818307	−	0.408235i
$155$	3.04132e7			0.0526909
$156$	−2.47311e6			−0.00417586
$157$			1.13972e9i			1.87585i	0.346839	+	0.937925i	$0.387255\pi$
$157$			1.13972e9i			1.87585i	−0.346839	+	0.937925i	$0.612745\pi$
$158$	−4.52110e8			−0.725464
$159$		−	5.19609e8i		−	0.812996i
$160$			1.35417e8i			0.206630i
$161$	−8.45490e7	+	4.21797e7i	−0.125836	+	0.0627769i
$162$	5.41131e7			0.0785674
$163$	3.07878e8			0.436142			0.218071	−	0.975933i	$-0.430024\pi$
$163$	3.07878e8			0.436142			0.218071	+	0.975933i	$0.430024\pi$
$164$			1.14421e8i			0.158172i
$165$	−6.46897e8			−0.872769
$166$			5.58024e8i			0.734887i
$167$			2.48768e8i			0.319837i	0.987130	+	0.159918i	$0.0511231\pi$
$167$			2.48768e8i			0.319837i	−0.987130	+	0.159918i	$0.948877\pi$
$168$	7.25882e7	+	1.45503e8i	0.0911231	+	0.182656i
$169$	8.15560e8			0.999791
$170$	−1.23285e9			−1.47609
$171$		−	1.11121e8i		−	0.129961i
$172$	−6.44451e8			−0.736336
$173$		−	4.62514e8i		−	0.516346i	−0.966099	−	0.258173i	$-0.916880\pi$
$173$		−	4.62514e8i		−	0.516346i	0.966099	−	0.258173i	$-0.0831204\pi$
$174$			3.85938e8i			0.421038i
$175$	3.07395e8	−	1.53353e8i	0.327751	−	0.163508i
$176$	−3.10229e8			−0.323319
$177$	−8.17746e8			−0.833154
$178$			1.11209e9i			1.10780i
$179$	1.81214e9			1.76514			0.882571	−	0.470178i	$-0.155810\pi$
$179$	1.81214e9			1.76514			0.882571	+	0.470178i	$0.155810\pi$
$180$		−	2.04507e8i		−	0.194813i
$181$			1.02962e9i			0.959317i	0.877455	+	0.479659i	$0.159239\pi$
$181$			1.02962e9i			0.959317i	−0.877455	+	0.479659i	$0.840761\pi$
$182$	5.01004e6	+	1.00426e7i	0.00456621	+	0.00915295i
$183$	−7.15598e8			−0.638065
$184$	5.69891e7			0.0497188
$185$			1.92583e9i			1.64411i
$186$	−2.20264e7			−0.0184031
$187$		−	2.82435e9i		−	2.30968i
$188$			7.66050e7i			0.0613233i
$189$	−1.09622e8	−	2.19738e8i	−0.0859117	−	0.172210i
$190$	−4.19954e8			−0.322246
$191$	8.67724e8			0.652001			0.326001	−	0.945370i	$-0.394299\pi$
$191$	8.67724e8			0.652001			0.326001	+	0.945370i	$0.394299\pi$
$192$		−	9.80741e7i		−	0.0721688i
$193$	−1.06473e9			−0.767377			−0.383689	−	0.923462i	$-0.625346\pi$
$193$	−1.06473e9			−0.767377			−0.383689	+	0.923462i	$0.625346\pi$
$194$		−	1.65552e9i		−	1.16877i
$195$		−	1.41151e7i		−	0.00976212i
$196$	4.43796e8	−	5.89520e8i	0.300718	−	0.399461i
$197$	−7.06660e8			−0.469187			−0.234593	−	0.972094i	$-0.575376\pi$
$197$	−7.06660e8			−0.469187			−0.234593	+	0.972094i	$0.575376\pi$
$198$	4.68507e8			0.304828
$199$			5.84320e7i			0.0372596i	0.999826	+	0.0186298i	$0.00593039\pi$
$199$			5.84320e7i			0.0372596i	−0.999826	+	0.0186298i	$0.994070\pi$
$200$	−2.07195e8			−0.129497
$201$			1.03139e9i			0.631884i
$202$			1.90587e9i			1.14469i
$203$	1.56719e9	−	7.81835e8i	0.922861	−	0.460395i
$204$	8.92875e8			0.515549
$205$	−6.53045e8			−0.369767
$206$			4.58021e8i			0.254341i
$207$	−8.60648e7			−0.0468753
$208$		−	6.76908e6i		−	0.00361640i
$209$		−	9.62078e8i		−	0.504226i
$210$	−8.30443e8	+	4.14290e8i	−0.427005	+	0.213023i
$211$	6.23994e8			0.314811			0.157406	−	0.987534i	$-0.449687\pi$
$211$	6.23994e8			0.314811			0.157406	+	0.987534i	$0.449687\pi$
$212$	1.42221e9			0.704075
$213$		−	1.12478e9i		−	0.546451i
$214$	2.75053e9			1.31148
$215$		−	3.67814e9i		−	1.72137i
$216$			1.48111e8i			0.0680414i
$217$	4.46211e7	+	8.94428e7i	0.0201234	+	0.0403373i
$218$	−1.14860e9			−0.508560
$219$	1.46909e9			0.638664
$220$		−	1.77060e9i		−	0.755840i
$221$	6.16263e7			0.0258343
$222$		−	1.39476e9i		−	0.574231i
$223$		−	2.09615e9i		−	0.847622i	−0.905751	−	0.423811i	$-0.860692\pi$
$223$		−	2.09615e9i		−	0.847622i	0.905751	−	0.423811i	$-0.139308\pi$
$224$	−3.98251e8	+	1.98679e8i	−0.158185	+	0.0789150i
$225$	3.12906e8			0.122091
$226$	6.65381e7			0.0255057
$227$			4.96086e9i			1.86833i	0.356841	+	0.934165i	$0.383854\pi$
$227$			4.96086e9i			1.86833i	−0.356841	+	0.934165i	$0.616146\pi$
$228$	3.04146e8			0.112549
$229$		−	2.81546e9i		−	1.02378i	−0.859051	−	0.511890i	$-0.828945\pi$
$229$		−	2.81546e9i		−	1.02378i	0.859051	−	0.511890i	$-0.171055\pi$
$230$			3.25260e8i			0.116230i
$231$	−9.49102e8	−	1.90247e9i	−0.333323	−	0.668145i
$232$	−1.05634e9			−0.364629
$233$	−6.73147e8			−0.228395			−0.114197	−	0.993458i	$-0.536430\pi$
$233$	−6.73147e8			−0.228395			−0.114197	+	0.993458i	$0.536430\pi$
$234$			1.02226e7i			0.00340957i
$235$	−4.37216e8			−0.143359
$236$		−	2.23823e9i		−	0.721533i
$237$			1.86880e9i			0.592339i
$238$	−1.80879e9	−	3.62571e9i	−0.563741	−	1.13002i
$239$	−4.59513e9			−1.40833			−0.704167	−	0.710034i	$-0.748679\pi$
$239$	−4.59513e9			−1.40833			−0.704167	+	0.710034i	$0.748679\pi$
$240$	5.59749e8			0.168713
$241$			6.01644e9i			1.78349i	0.452536	+	0.891746i	$0.350519\pi$
$241$			6.01644e9i			1.78349i	−0.452536	+	0.891746i	$0.649481\pi$
$242$	1.63110e9			0.475574
$243$		−	2.23677e8i		−	0.0641500i
$244$		−	1.95864e9i		−	0.552581i
$245$	3.36463e9	+	2.53292e9i	0.933841	+	0.703004i
$246$	4.72959e8			0.129147
$247$	2.09922e7			0.00563988
$248$		−	6.02877e7i		−	0.0159376i
$249$	2.30660e9			0.600033
$250$			2.04605e9i			0.523789i
$251$		−	4.19396e9i		−	1.05664i	−0.849044	−	0.528322i	$-0.822821\pi$
$251$		−	4.19396e9i		−	1.05664i	0.849044	−	0.528322i	$-0.177179\pi$
$252$	6.01438e8	−	3.00044e8i	0.149138	−	0.0744017i
$253$	−7.45142e8			−0.181868
$254$	−4.59735e8			−0.110452
$255$			5.09600e9i			1.20523i
$256$	2.68435e8			0.0625000
$257$			5.00422e9i			1.14711i	0.819168	+	0.573554i	$0.194436\pi$
$257$			5.00422e9i			1.14711i	−0.819168	+	0.573554i	$0.805564\pi$
$258$			2.66385e9i			0.601216i
$259$	−5.66371e9	+	2.82550e9i	−1.25864	+	0.627908i
$260$	3.86339e7			0.00845425
$261$	1.59528e9			0.343776
$262$		−	1.40945e9i		−	0.299120i
$263$	−3.25545e9			−0.680437			−0.340218	−	0.940346i	$-0.610501\pi$
$263$	−3.25545e9			−0.680437			−0.340218	+	0.940346i	$0.610501\pi$
$264$			1.28233e9i			0.263989i
$265$			8.11711e9i			1.64595i
$266$	−6.16141e8	−	1.23505e9i	−0.123070	−	0.246694i
$267$	4.59685e9			0.904514
$268$	−2.82298e9			−0.547227
$269$			4.01159e9i			0.766138i	0.923720	+	0.383069i	$0.125133\pi$
$269$			4.01159e9i			0.766138i	−0.923720	+	0.383069i	$0.874867\pi$
$270$	−8.45331e8			−0.159064
$271$			3.34655e8i			0.0620469i	0.999519	+	0.0310234i	$0.00987665\pi$
$271$			3.34655e8i			0.0620469i	−0.999519	+	0.0310234i	$0.990123\pi$
$272$			2.44386e9i			0.446478i
$273$	4.15113e7	−	2.07091e7i	0.00747335	−	0.00372829i
$274$	3.96312e9			0.703128
$275$	2.70911e9			0.473692
$276$		−	2.35565e8i		−	0.0405952i
$277$	−1.02056e10			−1.73348			−0.866740	−	0.498760i	$-0.833789\pi$
$277$	−1.02056e10			−1.73348			−0.866740	+	0.498760i	$0.833789\pi$
$278$			4.52560e9i			0.757700i
$279$			9.10463e7i			0.0150261i
$280$	−1.13394e9	−	2.27298e9i	−0.184484	−	0.369797i
$281$	2.62903e8			0.0421667			0.0210834	−	0.999778i	$-0.493288\pi$
$281$	2.62903e8			0.0421667			0.0210834	+	0.999778i	$0.493288\pi$
$282$	3.16648e8			0.0500703
$283$		−	9.14877e9i		−	1.42632i	−0.701002	−	0.713160i	$-0.747263\pi$
$283$		−	9.14877e9i		−	1.42632i	0.701002	−	0.713160i	$-0.252737\pi$
$284$	3.07861e9			0.473240
$285$			1.73589e9i			0.263113i
$286$			8.85068e7i			0.0132286i
$287$	−9.58122e8	−	1.92055e9i	−0.141219	−	0.283073i
$288$	−4.05391e8			−0.0589256
$289$	−1.52733e10			−2.18949
$290$		−	6.02896e9i		−	0.852413i
$291$	−6.84311e9			−0.954293
$292$			4.02100e9i			0.553099i
$293$		−	4.68541e9i		−	0.635736i	−0.948135	−	0.317868i	$-0.897033\pi$
$293$		−	4.68541e9i		−	0.635736i	0.948135	−	0.317868i	$-0.102967\pi$
$294$	−2.43679e9	−	1.83444e9i	−0.326158	−	0.245535i
$295$	1.27745e10			1.68677
$296$	3.81754e9			0.497298
$297$		−	1.93658e9i		−	0.248891i
$298$	−5.68661e9			−0.721088
$299$		−	1.62587e7i		−	0.00203424i
$300$			8.56444e8i			0.105734i
$301$	1.08171e10	−	5.39643e9i	1.31779	−	0.657416i
$302$	1.33517e9			0.160513
$303$	7.87795e9			0.934637
$304$			8.32469e8i			0.0974707i
$305$	1.11788e10			1.29180
$306$		−	3.69071e9i		−	0.420944i
$307$		−	1.42482e10i		−	1.60401i	−0.597316	−	0.802006i	$-0.703766\pi$
$307$		−	1.42482e10i		−	1.60401i	0.597316	−	0.802006i	$-0.296234\pi$
$308$	5.20720e9	−	2.59776e9i	0.578630	−	0.288666i
$309$	1.89324e9			0.207669
$310$	3.44086e8			0.0372581
$311$		−	1.68607e9i		−	0.180233i	−0.995931	−	0.0901167i	$-0.971276\pi$
$311$		−	1.68607e9i		−	0.180233i	0.995931	−	0.0901167i	$-0.0287240\pi$
$312$	−2.79801e7			−0.00295278
$313$		−	4.79688e8i		−	0.0499783i	−0.999688	−	0.0249891i	$-0.992045\pi$
$313$		−	4.79688e8i		−	0.0499783i	0.999688	−	0.0249891i	$-0.00795511\pi$
$314$			1.28944e10i			1.32643i
$315$	1.71247e9	+	3.43265e9i	0.173933	+	0.348648i
$316$	−5.11504e9			−0.512981
$317$	−1.02510e10			−1.01514			−0.507572	−	0.861609i	$-0.669457\pi$
$317$	−1.02510e10			−1.01514			−0.507572	+	0.861609i	$0.669457\pi$
$318$		−	5.87871e9i		−	0.574875i
$319$	1.38118e10			1.33379
$320$			1.53207e9i			0.146110i
$321$		−	1.13694e10i		−	1.07082i
$322$	−9.56563e8	+	4.77209e8i	−0.0889796	+	0.0443900i
$323$	−7.57887e9			−0.696297
$324$	6.12220e8			0.0555556
$325$			5.91118e7i			0.00529835i
$326$	3.48324e9			0.308399
$327$			4.74775e9i			0.415238i
$328$			1.29452e9i			0.111844i
$329$	−6.41466e8	−	1.28582e9i	−0.0547507	−	0.109748i
$330$	−7.31881e9			−0.617141
$331$	5.33386e9			0.444354			0.222177	−	0.975006i	$-0.428684\pi$
$331$	5.33386e9			0.444354			0.222177	+	0.975006i	$0.428684\pi$
$332$			6.31332e9i			0.519644i
$333$	−5.76524e9			−0.468857
$334$			2.81449e9i			0.226159i
$335$		−	1.61119e10i		−	1.27928i
$336$	8.21241e8	+	1.64618e9i	0.0644338	+	0.129157i
$337$	1.06112e10			0.822705			0.411352	−	0.911476i	$-0.365057\pi$
$337$	1.06112e10			0.822705			0.411352	+	0.911476i	$0.365057\pi$
$338$	9.22701e9			0.706959
$339$		−	2.75036e8i		−	0.0208253i
$340$	−1.39481e10			−1.04376
$341$			7.88271e8i			0.0582986i
$342$		−	1.25719e9i		−	0.0918962i
$343$	−2.51267e9	+	1.36113e10i	−0.181535	+	0.983385i
$344$	−7.29113e9			−0.520668
$345$	1.34447e9			0.0949016
$346$		−	5.23275e9i		−	0.365111i
$347$	7.71650e8			0.0532234			0.0266117	−	0.999646i	$-0.491528\pi$
$347$	7.71650e8			0.0532234			0.0266117	+	0.999646i	$0.491528\pi$
$348$			4.36639e9i			0.297719i
$349$		−	2.17495e10i		−	1.46604i	−0.680205	−	0.733022i	$-0.738109\pi$
$349$		−	2.17495e10i		−	1.46604i	0.680205	−	0.733022i	$-0.261891\pi$
$350$	3.47777e9	−	1.73499e9i	0.231755	−	0.115618i
$351$	4.22555e7			0.00278390
$352$	−3.50984e9			−0.228621
$353$		−	2.18953e10i		−	1.41010i	−0.709155	−	0.705052i	$-0.750923\pi$
$353$		−	2.18953e10i		−	1.41010i	0.709155	−	0.705052i	$-0.249077\pi$
$354$	−9.25174e9			−0.589129
$355$			1.75709e10i			1.10632i
$356$			1.25819e10i			0.783332i
$357$	−1.49869e10	+	7.47665e9i	−0.922655	+	0.460293i
$358$	2.05020e10			1.24814
$359$	1.01314e10			0.609948			0.304974	−	0.952361i	$-0.401352\pi$
$359$	1.01314e10			0.609948			0.304974	+	0.952361i	$0.401352\pi$
$360$		−	2.31373e9i		−	0.137753i
$361$	1.44019e10			0.847991
$362$			1.16488e10i			0.678340i
$363$		−	6.74216e9i		−	0.388305i
$364$	5.66821e7	+	1.13619e8i	0.00322880	+	0.00647211i
$365$	−2.29495e10			−1.29301
$366$	−8.09607e9			−0.451180
$367$			2.56118e10i			1.41181i	0.708308	+	0.705903i	$0.249459\pi$
$367$			2.56118e10i			1.41181i	−0.708308	+	0.705903i	$0.750541\pi$
$368$	6.44758e8			0.0351565
$369$		−	1.95498e9i		−	0.105448i
$370$			2.17883e10i			1.16256i
$371$	−2.38718e10	+	1.19091e10i	−1.26005	+	0.628613i
$372$	−2.49200e8			−0.0130130
$373$	7.12600e9			0.368138			0.184069	−	0.982913i	$-0.441073\pi$
$373$	7.12600e9			0.368138			0.184069	+	0.982913i	$0.441073\pi$
$374$		−	3.19538e10i		−	1.63319i
$375$	8.45737e9			0.427672
$376$			8.66686e8i			0.0433621i
$377$			3.01369e8i			0.0149188i
$378$	−1.24024e9	−	2.48605e9i	−0.0607488	−	0.121771i
$379$	−7.68192e9			−0.372317			−0.186159	−	0.982520i	$-0.559604\pi$
$379$	−7.68192e9			−0.372317			−0.186159	+	0.982520i	$0.559604\pi$
$380$	−4.75124e9			−0.227862
$381$			1.90032e9i			0.0901835i
$382$	9.81718e9			0.461034
$383$			1.69808e10i			0.789159i	0.918862	+	0.394579i	$0.129110\pi$
$383$			1.69808e10i			0.789159i	−0.918862	+	0.394579i	$0.870890\pi$
$384$		−	1.10958e9i		−	0.0510310i
$385$	1.48265e10	+	2.97196e10i	0.674830	+	1.35269i
$386$	−1.20460e10			−0.542618
$387$	1.10111e10			0.490891
$388$		−	1.87301e10i		−	0.826442i
$389$	4.69924e8			0.0205224			0.0102612	−	0.999947i	$-0.496734\pi$
$389$	4.69924e8			0.0205224			0.0102612	+	0.999947i	$0.496734\pi$
$390$		−	1.59694e8i		−	0.00690286i
$391$			5.86993e9i			0.251146i
$392$	5.02098e9	−	6.66965e9i	0.212640	−	0.282461i
$393$	−5.82598e9			−0.244230
$394$	−7.99495e9			−0.331765
$395$		−	2.91936e10i		−	1.19922i
$396$	5.30055e9			0.215546
$397$			2.55715e9i			0.102942i	0.998674	+	0.0514712i	$0.0163910\pi$
$397$			2.55715e9i			0.102942i	−0.998674	+	0.0514712i	$0.983609\pi$
$398$			6.61082e8i			0.0263465i
$399$	−5.10510e9	+	2.54683e9i	−0.201425	+	0.100486i
$400$	−2.34415e9			−0.0915682
$401$	−3.60516e8			−0.0139427			−0.00697135	−	0.999976i	$-0.502219\pi$
$401$	−3.60516e8			−0.0139427			−0.00697135	+	0.999976i	$0.502219\pi$
$402$			1.16688e10i			0.446809i
$403$	−1.71998e7			−0.000652083
$404$			2.15625e10i			0.809419i
$405$			3.49419e9i			0.129875i
$406$	1.77307e10	−	8.84545e9i	0.652561	−	0.325549i
$407$	−4.99150e10			−1.81909
$408$	1.01017e10			0.364548
$409$		−	9.02972e9i		−	0.322687i	−0.986898	−	0.161343i	$-0.948417\pi$
$409$		−	9.02972e9i		−	0.322687i	0.986898	−	0.161343i	$-0.0515827\pi$
$410$	−7.38836e9			−0.261465
$411$		−	1.63816e10i		−	0.574102i
$412$			5.18192e9i			0.179846i
$413$	1.87422e10	+	3.75687e10i	0.644199	+	1.29130i
$414$	−9.73712e8			−0.0331459
$415$	−3.60327e10			−1.21480
$416$		−	7.65834e7i		−	0.00255718i
$417$	1.87066e10			0.618659
$418$		−	1.08847e10i		−	0.356542i
$419$			2.79138e10i			0.905655i	0.891598	+	0.452828i	$0.149585\pi$
$419$			2.79138e10i			0.905655i	−0.891598	+	0.452828i	$0.850415\pi$
$420$	−9.39539e9	+	4.68716e9i	−0.301938	+	0.150630i
$421$	4.05608e10			1.29116			0.645578	−	0.763694i	$-0.276617\pi$
$421$	4.05608e10			1.29116			0.645578	+	0.763694i	$0.276617\pi$
$422$	7.05968e9			0.222605
$423$		−	1.30887e9i		−	0.0408822i
$424$	1.60904e10			0.497856
$425$		−	2.13413e10i		−	0.654131i
$426$		−	1.27255e10i		−	0.386399i
$427$	1.64010e10	+	3.28758e10i	0.493355	+	0.988930i
$428$	3.11187e10			0.927355
$429$	3.65844e8			0.0108011
$430$		−	4.16135e10i		−	1.21719i
$431$	−2.67748e10			−0.775920			−0.387960	−	0.921676i	$-0.626820\pi$
$431$	−2.67748e10			−0.775920			−0.387960	+	0.921676i	$0.626820\pi$
$432$			1.67569e9i			0.0481125i
$433$		−	5.19153e10i		−	1.47688i	−0.674322	−	0.738438i	$-0.735564\pi$
$433$		−	5.19153e10i		−	1.47688i	0.674322	−	0.738438i	$-0.264436\pi$
$434$	5.04830e8	+	1.01193e9i	0.0142294	+	0.0285227i
$435$	−2.49208e10			−0.695993
$436$	−1.29949e10			−0.359607
$437$			1.99952e9i			0.0548276i
$438$	1.66209e10			0.451603
$439$			1.70186e10i			0.458210i	0.973402	+	0.229105i	$0.0735800\pi$
$439$			1.70186e10i			0.458210i	−0.973402	+	0.229105i	$0.926420\pi$
$440$		−	2.00321e10i		−	0.534460i
$441$	−7.58267e9	+	1.00725e10i	−0.200479	+	0.266307i
$442$	6.97221e8			0.0182676
$443$	2.19639e9			0.0570289			0.0285144	−	0.999593i	$-0.490922\pi$
$443$	2.19639e9			0.0570289			0.0285144	+	0.999593i	$0.490922\pi$
$444$		−	1.57799e10i		−	0.406042i
$445$	−7.18100e10			−1.83124
$446$		−	2.37152e10i		−	0.599359i
$447$			2.35057e10i			0.588766i
$448$	−4.50570e9	+	2.24779e9i	−0.111854	+	0.0558013i
$449$	3.92988e10			0.966927			0.483463	−	0.875365i	$-0.339379\pi$
$449$	3.92988e10			0.966927			0.483463	+	0.875365i	$0.339379\pi$
$450$	3.54012e9			0.0863313
$451$		−	1.69261e10i		−	0.409120i
$452$	7.52793e8			0.0180352
$453$		−	5.51896e9i		−	0.131058i
$454$			5.61257e10i			1.32111i
$455$	−6.48471e8	+	3.23508e8i	−0.0151302	+	0.00754813i
$456$	3.44102e9			0.0795845
$457$	−2.58374e10			−0.592357			−0.296178	−	0.955133i	$-0.595712\pi$
$457$	−2.58374e10			−0.592357			−0.296178	+	0.955133i	$0.595712\pi$
$458$		−	3.18532e10i		−	0.723922i
$459$	−1.52556e10			−0.343699
$460$			3.67990e9i			0.0821872i
$461$		−	4.35497e10i		−	0.964231i	−0.876108	−	0.482116i	$-0.839869\pi$
$461$		−	4.35497e10i		−	0.964231i	0.876108	−	0.482116i	$-0.160131\pi$
$462$	−1.07379e10	−	2.15240e10i	−0.235695	−	0.472450i
$463$	5.55873e10			1.20963			0.604814	−	0.796367i	$-0.293248\pi$
$463$	5.55873e10			1.20963			0.604814	+	0.796367i	$0.293248\pi$
$464$	−1.19511e10			−0.257832
$465$		−	1.42228e9i		−	0.0304211i
$466$	−7.61579e9			−0.161500
$467$		−	2.76361e10i		−	0.581044i	−0.956868	−	0.290522i	$-0.906171\pi$
$467$		−	2.76361e10i		−	0.581044i	0.956868	−	0.290522i	$-0.0938289\pi$
$468$			1.15656e8i			0.00241093i
$469$	4.73837e10	−	2.36387e10i	0.979349	−	0.488576i
$470$	−4.94653e9			−0.101370
$471$	5.32992e10			1.08302
$472$		−	2.53226e10i		−	0.510201i
$473$	9.53328e10			1.90457
$474$			2.11431e10i			0.418847i
$475$		−	7.26964e9i		−	0.142803i
$476$	−2.04641e10	−	4.10202e10i	−0.398625	−	0.799043i
$477$	−2.42997e10			−0.469384
$478$	−5.19879e10			−0.995843
$479$			6.13666e10i			1.16571i	0.812577	+	0.582854i	$0.198064\pi$
$479$			6.13666e10i			1.16571i	−0.812577	+	0.582854i	$0.801936\pi$
$480$	6.33283e9			0.119298
$481$		−	1.08913e9i		−	0.0203469i
$482$			6.80682e10i			1.26112i
$483$	1.97255e9	+	3.95397e9i	0.0362443	+	0.0726515i
$484$	1.84538e10			0.336282
$485$	1.06900e11			1.93202
$486$		−	2.53062e9i		−	0.0453609i
$487$	−7.66791e10			−1.36321			−0.681603	−	0.731722i	$-0.738717\pi$
$487$	−7.66791e10			−1.36321			−0.681603	+	0.731722i	$0.738717\pi$
$488$		−	2.21595e10i		−	0.390733i
$489$		−	1.43980e10i		−	0.251807i
$490$	3.80664e10	+	2.86568e10i	0.660325	+	0.497099i
$491$	8.02970e10			1.38157			0.690786	−	0.723059i	$-0.257265\pi$
$491$	8.02970e10			1.38157			0.690786	+	0.723059i	$0.257265\pi$
$492$	5.35092e9			0.0913205
$493$		−	1.08804e11i		−	1.84186i
$494$	2.37500e8			0.00398800
$495$			3.02524e10i			0.503894i
$496$		−	6.82077e8i		−	0.0112696i
$497$	−5.16746e10	+	2.57793e10i	−0.846938	+	0.422519i
$498$	2.60962e10			0.424287
$499$	−7.51380e10			−1.21187			−0.605937	−	0.795513i	$-0.707202\pi$
$499$	−7.51380e10			−1.21187			−0.605937	+	0.795513i	$0.707202\pi$
$500$			2.31484e10i			0.370375i
$501$	1.16337e10			0.184658
$502$		−	4.74492e10i		−	0.747161i
$503$		−	1.11123e9i		−	0.0173593i	−0.999962	−	0.00867965i	$-0.997237\pi$
$503$		−	1.11123e9i		−	0.0173593i	0.999962	−	0.00867965i	$-0.00276285\pi$
$504$	6.80449e9	−	3.39461e9i	0.105457	−	0.0526100i
$505$	−1.23066e11			−1.89222
$506$	−8.43032e9			−0.128600
$507$		−	3.81400e10i		−	0.577229i
$508$	−5.20131e9			−0.0781012
$509$			3.54760e10i			0.528523i	0.964451	+	0.264262i	$0.0851282\pi$
$509$			3.54760e10i			0.528523i	−0.964451	+	0.264262i	$0.914872\pi$
$510$			5.76546e10i			0.852223i
$511$	−3.36706e10	−	6.74926e10i	−0.493818	−	0.989857i
$512$	3.03700e9			0.0441942
$513$	−5.19663e9			−0.0750330
$514$			5.66163e10i			0.811127i
$515$	−2.95753e10			−0.420436
$516$			3.01380e10i			0.425124i
$517$		−	1.13321e10i		−	0.158616i
$518$	−6.40775e10	+	3.19669e10i	−0.889993	+	0.443998i
$519$	−2.16296e10			−0.298112
$520$	4.37092e8			0.00597806
$521$			1.14561e11i			1.55485i	0.628978	+	0.777423i	$0.283474\pi$
$521$			1.14561e11i			1.55485i	−0.628978	+	0.777423i	$0.716526\pi$
$522$	1.80486e10			0.243086
$523$			1.66287e9i			0.0222255i	0.999938	+	0.0111127i	$0.00353736\pi$
$523$			1.66287e9i			0.0222255i	−0.999938	+	0.0111127i	$0.996463\pi$
$524$		−	1.59461e10i		−	0.211509i
$525$	−7.17159e9	−	1.43754e10i	−0.0944014	−	0.189227i
$526$	−3.68312e10			−0.481141
$527$	6.20968e9			0.0805058
$528$			1.45080e10i			0.186668i
$529$	−7.67623e10			−0.980224
$530$			9.18346e10i			1.16387i
$531$			3.82422e10i			0.481022i
$532$	−6.97083e9	−	1.39730e10i	−0.0870238	−	0.174439i
$533$	3.69321e8			0.00457610
$534$	5.20074e10			0.639588
$535$			1.77607e11i			2.16793i
$536$	−3.19383e10			−0.386948
$537$		−	8.47454e10i		−	1.01911i
$538$			4.53859e10i			0.541741i
$539$	−6.56501e10	+	8.72068e10i	−0.777822	+	1.03323i
$540$	−9.56383e9			−0.112475
$541$	−7.44499e10			−0.869110			−0.434555	−	0.900645i	$-0.643094\pi$
$541$	−7.44499e10			−0.869110			−0.434555	+	0.900645i	$0.643094\pi$
$542$			3.78619e9i			0.0438738i
$543$	4.81505e10			0.553862
$544$			2.76491e10i			0.315708i
$545$		−	7.41673e10i		−	0.840671i
$546$	4.69646e8	−	2.34296e8i	0.00528446	−	0.00263630i
$547$	1.70204e11			1.90117			0.950586	−	0.310461i	$-0.100483\pi$
$547$	1.70204e11			1.90117			0.950586	+	0.310461i	$0.100483\pi$
$548$	4.48375e10			0.497187
$549$			3.34652e10i			0.368387i
$550$	3.06501e10			0.334951
$551$		−	3.70627e10i		−	0.402097i
$552$		−	2.66512e9i		−	0.0287052i
$553$	8.58561e10	−	4.28317e10i	0.918059	−	0.458000i
$554$	−1.15463e11			−1.22576
$555$	9.00621e10			0.949227
$556$			5.12014e10i			0.535775i
$557$	1.60997e11			1.67262			0.836310	−	0.548258i	$-0.184709\pi$
$557$	1.60997e11			1.67262			0.836310	+	0.548258i	$0.184709\pi$
$558$			1.03007e9i			0.0106250i
$559$			2.08013e9i			0.0213031i
$560$	−1.28291e10	−	2.57158e10i	−0.130450	−	0.261486i
$561$	−1.32082e11			−1.33349
$562$	2.97441e9			0.0298164
$563$		−	5.79247e10i		−	0.576541i	−0.957549	−	0.288271i	$-0.906920\pi$
$563$		−	5.79247e10i		−	0.576541i	0.957549	−	0.288271i	$-0.0930803\pi$
$564$	3.58246e9			0.0354050
$565$			4.29649e9i			0.0421619i
$566$		−	1.03506e11i		−	1.00856i
$567$	−1.02761e10	+	5.12654e9i	−0.0994254	+	0.0496012i
$568$	3.48305e10			0.334631
$569$	7.67792e10			0.732478			0.366239	−	0.930521i	$-0.380645\pi$
$569$	7.67792e10			0.732478			0.366239	+	0.930521i	$0.380645\pi$
$570$			1.96393e10i			0.186049i
$571$	1.02087e11			0.960342			0.480171	−	0.877175i	$-0.340575\pi$
$571$	1.02087e11			0.960342			0.480171	+	0.877175i	$0.340575\pi$
$572$			1.00134e9i			0.00935401i
$573$		−	4.05795e10i		−	0.376433i
$574$	−1.08399e10	−	2.17286e10i	−0.0998570	−	0.200163i
$575$	−5.63043e9			−0.0515074
$576$	−4.58647e9			−0.0416667
$577$			5.63596e10i			0.508469i	0.967143	+	0.254235i	$0.0818236\pi$
$577$			5.63596e10i			0.508469i	−0.967143	+	0.254235i	$0.918176\pi$
$578$	−1.72798e11			−1.54820
$579$			4.97924e10i			0.443046i
$580$		−	6.82099e10i		−	0.602747i
$581$	−5.28657e10	−	1.05969e11i	−0.463949	−	0.929984i
$582$	−7.74210e10			−0.674787
$583$	−2.10385e11			−1.82113
$584$			4.54924e10i			0.391100i
$585$	−6.60096e8			−0.00563617
$586$		−	5.30094e10i		−	0.449534i
$587$		−	1.81558e11i		−	1.52919i	−0.644509	−	0.764597i	$-0.722938\pi$
$587$		−	1.81558e11i		−	1.52919i	0.644509	−	0.764597i	$-0.277062\pi$
$588$	−2.75691e10	−	2.07543e10i	−0.230629	−	0.173619i
$589$	2.11525e9			0.0175752
$590$	1.44527e11			1.19272
$591$			3.30472e10i			0.270885i
$592$	4.31906e10			0.351643
$593$		−	6.43013e10i		−	0.519997i	−0.965609	−	0.259998i	$-0.916278\pi$
$593$		−	6.43013e10i		−	0.519997i	0.965609	−	0.259998i	$-0.0837221\pi$
$594$		−	2.19099e10i		−	0.175993i
$595$	2.34119e11	−	1.16797e11i	1.86797	−	0.931887i
$596$	−6.43367e10			−0.509886
$597$	2.73259e9			0.0215118
$598$		−	1.83946e8i		−	0.00143842i
$599$	−1.92687e11			−1.49674			−0.748368	−	0.663284i	$-0.769162\pi$
$599$	−1.92687e11			−1.49674			−0.748368	+	0.663284i	$0.769162\pi$
$600$			9.68956e9i			0.0747651i
$601$			1.33087e11i			1.02009i	0.860148	+	0.510045i	$0.170371\pi$
$601$			1.33087e11i			1.02009i	−0.860148	+	0.510045i	$0.829629\pi$
$602$	1.22382e11	−	6.10536e10i	0.931818	−	0.464864i
$603$	4.82332e10			0.364818
$604$	1.51058e10			0.113500
$605$			1.05323e11i			0.786144i
$606$	8.91289e10			0.660888
$607$		−	2.32672e11i		−	1.71391i	−0.515390	−	0.856956i	$-0.672353\pi$
$607$		−	2.32672e11i		−	1.71391i	0.515390	−	0.856956i	$-0.327647\pi$
$608$			9.41832e9i			0.0689222i
$609$	−3.65628e10	−	7.32900e10i	−0.265809	−	0.532814i
$610$	1.26473e11			0.913439
$611$	2.47262e8			0.00177416
$612$		−	4.17556e10i		−	0.297652i
$613$	1.86783e11			1.32280			0.661401	−	0.750033i	$-0.269962\pi$
$613$	1.86783e11			1.32280			0.661401	+	0.750033i	$0.269962\pi$
$614$		−	1.61200e11i		−	1.13421i
$615$			3.05399e10i			0.213485i
$616$	5.89127e10	−	2.93903e10i	0.409153	−	0.204118i
$617$	6.44286e10			0.444568			0.222284	−	0.974982i	$-0.428649\pi$
$617$	6.44286e10			0.444568			0.222284	+	0.974982i	$0.428649\pi$
$618$	2.14195e10			0.146844
$619$			1.30375e11i			0.888040i	0.896017	+	0.444020i	$0.146448\pi$
$619$			1.30375e11i			0.888040i	−0.896017	+	0.444020i	$0.853552\pi$
$620$	3.89289e9			0.0263454
$621$			4.02485e9i			0.0270635i
$622$		−	1.90757e10i		−	0.127444i
$623$	−1.05357e11	−	2.11187e11i	−0.699375	−	1.40190i
$624$	−3.16559e8			−0.00208793
$625$	−1.88006e11			−1.23212
$626$		−	5.42705e9i		−	0.0353400i
$627$	−4.49920e10			−0.291115
$628$			1.45884e11i			0.937925i
$629$			3.93210e11i			2.51202i
$630$	1.93744e10	+	3.88360e10i	0.122989	+	0.246531i
$631$	1.94784e11			1.22867			0.614335	−	0.789045i	$-0.289424\pi$
$631$	1.94784e11			1.22867			0.614335	+	0.789045i	$0.289424\pi$
$632$	−5.78701e10			−0.362732
$633$		−	2.91813e10i		−	0.181756i
$634$	−1.15976e11			−0.717815
$635$		−	2.96860e10i		−	0.182581i
$636$		−	6.65100e10i		−	0.406498i
$637$	−1.90282e9	−	1.43246e9i	−0.0115569	−	0.00870012i
$638$	1.56263e11			0.943133
$639$	−5.26010e10			−0.315494
$640$			1.73334e10i			0.103315i
$641$	−1.39487e11			−0.826229			−0.413115	−	0.910679i	$-0.635559\pi$
$641$	−1.39487e11			−0.826229			−0.413115	+	0.910679i	$0.635559\pi$
$642$		−	1.28630e11i		−	0.757182i
$643$		−	1.85805e11i		−	1.08696i	−0.839422	−	0.543480i	$-0.817106\pi$
$643$		−	1.85805e11i		−	1.08696i	0.839422	−	0.543480i	$-0.182894\pi$
$644$	−1.08223e10	+	5.39900e9i	−0.0629180	+	0.0313884i
$645$	−1.72010e11			−0.993835
$646$	−8.57451e10			−0.492356
$647$			1.54360e11i			0.880882i	0.897781	+	0.440441i	$0.145178\pi$
$647$			1.54360e11i			0.880882i	−0.897781	+	0.440441i	$0.854822\pi$
$648$	6.92648e9			0.0392837
$649$			3.31098e11i			1.86628i
$650$			6.68774e8i			0.00374650i
$651$	4.18282e9	−	2.08672e9i	0.0232887	−	0.0116182i
$652$	3.94083e10			0.218071
$653$	5.06084e10			0.278336			0.139168	−	0.990269i	$-0.455557\pi$
$653$	5.06084e10			0.278336			0.139168	+	0.990269i	$0.455557\pi$
$654$			5.37147e10i			0.293617i
$655$	9.10110e10			0.494457
$656$			1.46458e10i			0.0790859i
$657$		−	6.87026e10i		−	0.368733i
$658$	−7.25736e9	−	1.45474e10i	−0.0387146	−	0.0776033i
$659$	−1.76214e11			−0.934328			−0.467164	−	0.884171i	$-0.654724\pi$
$659$	−1.76214e11			−0.934328			−0.467164	+	0.884171i	$0.654724\pi$
$660$	−8.28028e10			−0.436385
$661$		−	1.22851e10i		−	0.0643538i	−0.999482	−	0.0321769i	$-0.989756\pi$
$661$		−	1.22851e10i		−	0.0643538i	0.999482	−	0.0321769i	$-0.0102440\pi$
$662$	6.03457e10			0.314206
$663$		−	2.88197e9i		−	0.0149154i
$664$			7.14271e10i			0.367443i
$665$	7.97497e10	−	3.97854e10i	0.407795	−	0.203440i
$666$	−6.52263e10			−0.331532
$667$	−2.87055e10			−0.145031
$668$			3.18423e10i			0.159918i
$669$	−9.80271e10			−0.489375
$670$		−	1.82285e11i		−	0.904590i
$671$			2.89739e11i			1.42928i
$672$	9.29128e9	+	1.86244e10i	0.0455616	+	0.0913280i
$673$	−9.02088e10			−0.439733			−0.219866	−	0.975530i	$-0.570562\pi$
$673$	−9.02088e10			−0.439733			−0.219866	+	0.975530i	$0.570562\pi$
$674$	1.20052e11			0.581740
$675$		−	1.46331e10i		−	0.0704892i
$676$	1.04392e11			0.499895
$677$			1.15253e11i			0.548653i	0.961637	+	0.274326i	$0.0884549\pi$
$677$			1.15253e11i			0.548653i	−0.961637	+	0.274326i	$0.911545\pi$
$678$		−	3.11168e9i		−	0.0147257i
$679$	1.56840e11	+	3.14385e11i	0.737865	+	1.47905i
$680$	−1.57805e11			−0.738047
$681$	2.31997e11			1.07868
$682$			8.91827e9i			0.0412233i
$683$	1.42847e11			0.656429			0.328215	−	0.944603i	$-0.393553\pi$
$683$	1.42847e11			0.656429			0.328215	+	0.944603i	$0.393553\pi$
$684$		−	1.42235e10i		−	0.0649804i
$685$			2.55906e11i			1.16230i
$686$	−2.84277e10	+	1.53994e11i	−0.128364	+	0.695358i
$687$	−1.31666e11			−0.591080
$688$	−8.24898e10			−0.368168
$689$		−	4.59052e9i		−	0.0203697i
$690$	1.52109e10			0.0671056
$691$		−	1.48850e11i		−	0.652886i	−0.945217	−	0.326443i	$-0.894150\pi$
$691$		−	1.48850e11i		−	0.652886i	0.945217	−	0.326443i	$-0.105850\pi$
$692$		−	5.92018e10i		−	0.258173i
$693$	−8.89698e10	+	4.43851e10i	−0.385753	+	0.192444i
$694$	8.73022e9			0.0376346
$695$	−2.92227e11			−1.25251
$696$			4.94001e10i			0.210519i
$697$	−1.33337e11			−0.564962
$698$		−	2.46067e11i		−	1.03665i
$699$			3.14800e10i			0.131864i
$700$	3.93465e10	−	1.96291e10i	0.163876	−	0.0817540i
$701$	1.05397e11			0.436473			0.218237	−	0.975896i	$-0.429970\pi$
$701$	1.05397e11			0.436473			0.218237	+	0.975896i	$0.429970\pi$
$702$	4.78066e8			0.00196852
$703$			1.33942e11i			0.548398i
$704$	−3.97093e10			−0.161660
$705$			2.04466e10i			0.0827682i
$706$		−	2.47717e11i		−	0.997095i
$707$	−1.80557e11	−	3.61927e11i	−0.722666	−	1.44858i
$708$	−1.04671e11			−0.416577
$709$	−1.60649e11			−0.635759			−0.317880	−	0.948131i	$-0.602971\pi$
$709$	−1.60649e11			−0.635759			−0.317880	+	0.948131i	$0.602971\pi$
$710$			1.98792e11i			0.782286i
$711$	8.73953e10			0.341987
$712$			1.42348e11i			0.553899i
$713$		−	1.63829e9i		−	0.00633917i
$714$	−1.69558e11	+	8.45886e10i	−0.652416	+	0.325476i
$715$	−5.71506e9			−0.0218674
$716$	2.31954e11			0.882571
$717$			2.14893e11i			0.813102i
$718$	1.14624e11			0.431298
$719$		−	6.92297e10i		−	0.259046i	−0.991576	−	0.129523i	$-0.958655\pi$
$719$		−	6.92297e10i		−	0.259046i	0.991576	−	0.129523i	$-0.0413446\pi$
$720$		−	2.61769e10i		−	0.0974064i
$721$	−4.33917e10	−	8.69786e10i	−0.160571	−	0.321863i
$722$	1.62939e11			0.599621
$723$	2.81361e11			1.02970
$724$			1.31791e11i			0.479659i
$725$	1.04365e11			0.377747
$726$		−	7.62789e10i		−	0.274573i
$727$			2.53794e11i			0.908538i	0.890864	+	0.454269i	$0.150100\pi$
$727$			2.53794e11i			0.908538i	−0.890864	+	0.454269i	$0.849900\pi$
$728$	6.41285e8	+	1.28545e9i	0.00228310	+	0.00457647i
$729$	−1.04604e10			−0.0370370
$730$	−2.59644e11			−0.914295
$731$		−	7.50993e11i		−	2.63007i
$732$	−9.15966e10			−0.319033
$733$		−	5.05247e11i		−	1.75020i	−0.483942	−	0.875100i	$-0.660795\pi$
$733$		−	5.05247e11i		−	1.75020i	0.483942	−	0.875100i	$-0.339205\pi$
$734$			2.89764e11i			0.998298i
$735$	1.18453e11	−	1.57348e11i	0.405880	−	0.539153i
$736$	7.29460e9			0.0248594
$737$	4.17599e11			1.41543
$738$		−	2.21181e10i		−	0.0745629i
$739$	−2.64955e11			−0.888369			−0.444185	−	0.895935i	$-0.646507\pi$
$739$	−2.64955e11			−0.888369			−0.444185	+	0.895935i	$0.646507\pi$
$740$			2.46506e11i			0.822055i
$741$		−	9.81708e8i		−	0.00325619i
$742$	−2.70078e11	+	1.34736e11i	−0.890992	+	0.444497i
$743$	2.15165e11			0.706018			0.353009	−	0.935620i	$-0.385158\pi$
$743$	2.15165e11			0.706018			0.353009	+	0.935620i	$0.385158\pi$
$744$	−2.81937e9			−0.00920155
$745$		−	3.67195e11i		−	1.19199i
$746$	8.06214e10			0.260313
$747$		−	1.07869e11i		−	0.346429i
$748$		−	3.61517e11i		−	1.15484i
$749$	−5.22328e11	+	2.60578e11i	−1.65965	+	0.827962i
$750$	9.56843e10			0.302410
$751$	4.34188e11			1.36495			0.682477	−	0.730907i	$-0.260903\pi$
$751$	4.34188e11			1.36495			0.682477	+	0.730907i	$0.260903\pi$
$752$			9.80544e9i			0.0306617i
$753$	−1.96132e11			−0.610054
$754$			3.40960e9i			0.0105492i
$755$			8.62147e10i			0.265334i
$756$	−1.40317e10	−	2.81265e10i	−0.0429559	−	0.0861049i
$757$	4.22834e11			1.28762			0.643808	−	0.765187i	$-0.277353\pi$
$757$	4.22834e11			1.28762			0.643808	+	0.765187i	$0.277353\pi$
$758$	−8.69110e10			−0.263268
$759$			3.48468e10i			0.105002i
$760$	−5.37542e10			−0.161123
$761$			2.29847e11i			0.685331i	0.939457	+	0.342666i	$0.111330\pi$
$761$			2.29847e11i			0.685331i	−0.939457	+	0.342666i	$0.888670\pi$
$762$			2.14997e10i			0.0637693i
$763$	2.18120e11	−	1.08815e11i	0.643572	−	0.321064i
$764$	1.11069e11			0.326001
$765$	2.38316e11			0.695838
$766$			1.92116e11i			0.558019i
$767$	−7.22443e9			−0.0208748
$768$		−	1.25535e10i		−	0.0360844i
$769$			5.05645e11i			1.44591i	0.690897	+	0.722953i	$0.257216\pi$
$769$			5.05645e11i			1.44591i	−0.690897	+	0.722953i	$0.742784\pi$
$770$	1.67742e11	+	3.36239e11i	0.477177	+	0.956500i
$771$	2.34024e11			0.662283
$772$	−1.36285e11			−0.383689
$773$		−	3.10061e11i		−	0.868418i	−0.900812	−	0.434209i	$-0.857028\pi$
$773$		−	3.10061e11i		−	0.868418i	0.900812	−	0.434209i	$-0.142972\pi$
$774$	1.24576e11			0.347112
$775$			5.95632e9i			0.0165109i
$776$		−	2.11907e11i		−	0.584383i
$777$	1.32136e11	+	2.64865e11i	0.362523	+	0.726676i
$778$	5.31658e9			0.0145116
$779$	−4.54195e10			−0.123337
$780$		−	1.80673e9i		−	0.00488106i
$781$	−4.55415e11			−1.22406
$782$			6.64107e10i			0.177587i
$783$		−	7.46039e10i		−	0.198479i
$784$	5.68059e10	−	7.54585e10i	0.150359	−	0.199730i
$785$	−8.32617e11			−2.19264
$786$	−6.59135e10			−0.172697
$787$		−	4.74703e11i		−	1.23744i	−0.785613	−	0.618718i	$-0.787652\pi$
$787$		−	4.74703e11i		−	1.23744i	0.785613	−	0.618718i	$-0.212348\pi$
$788$	−9.04525e10			−0.234593
$789$			1.52242e11i			0.392850i
$790$		−	3.30288e11i		−	0.847978i
$791$	−1.26356e10	+	6.30365e9i	−0.0322769	+	0.0161022i
$792$	5.99689e10			0.152414
$793$	−6.32200e9			−0.0159868
$794$			2.89308e10i			0.0727912i
$795$	3.79600e11			0.950292
$796$			7.47929e9i			0.0186298i
$797$		−	3.92298e11i		−	0.972260i	−0.873887	−	0.486130i	$-0.838408\pi$
$797$		−	3.92298e11i		−	0.972260i	0.873887	−	0.486130i	$-0.161592\pi$
$798$	−5.77577e10	+	2.88140e10i	−0.142429	+	0.0710547i
$799$	−8.92695e10			−0.219036
$800$	−2.65210e10			−0.0647485
$801$		−	2.14973e11i		−	0.522221i
$802$	−4.07877e9			−0.00985897
$803$		−	5.94821e11i		−	1.43062i
$804$			1.32017e11i			0.315942i
$805$	−3.08143e10	−	6.17671e10i	−0.0733784	−	0.147087i
$806$	−1.94593e8			−0.000461092
$807$	1.87603e11			0.442330
$808$			2.43952e11i			0.572346i
$809$	7.15124e11			1.66950			0.834751	−	0.550627i	$-0.185611\pi$
$809$	7.15124e11			1.66950			0.834751	+	0.550627i	$0.185611\pi$
$810$			3.95322e10i			0.0918356i
$811$		−	6.06590e11i		−	1.40221i	−0.713059	−	0.701104i	$-0.752691\pi$
$811$		−	6.06590e11i		−	1.40221i	0.713059	−	0.701104i	$-0.247309\pi$
$812$	2.00600e11	−	1.00075e11i	0.461431	−	0.230198i
$813$	1.56503e10			0.0358228
$814$	−5.64724e11			−1.28629
$815$			2.24919e11i			0.509796i
$816$	1.14288e11			0.257774
$817$		−	2.55816e11i		−	0.574170i
$818$		−	1.02160e11i		−	0.228174i
$819$	−9.68467e8	−	1.94129e9i	−0.00215253	−	0.00431474i
$820$	−8.35898e10			−0.184883
$821$	−1.49088e11			−0.328147			−0.164074	−	0.986448i	$-0.552463\pi$
$821$	−1.49088e11			−0.328147			−0.164074	+	0.986448i	$0.552463\pi$
$822$		−	1.85337e11i		−	0.405951i
$823$	−5.43817e11			−1.18537			−0.592684	−	0.805435i	$-0.701932\pi$
$823$	−5.43817e11			−1.18537			−0.592684	+	0.805435i	$0.701932\pi$
$824$			5.86267e10i			0.127171i
$825$		−	1.26693e11i		−	0.273486i
$826$	2.12044e11	+	4.25041e11i	0.455518	+	0.913084i
$827$	−1.25114e11			−0.267475			−0.133737	−	0.991017i	$-0.542698\pi$
$827$	−1.25114e11			−0.267475			−0.133737	+	0.991017i	$0.542698\pi$
$828$	−1.10163e10			−0.0234377
$829$		−	4.93095e11i		−	1.04403i	−0.852937	−	0.522014i	$-0.825181\pi$
$829$		−	4.93095e11i		−	1.04403i	0.852937	−	0.522014i	$-0.174819\pi$
$830$	−4.07663e11			−0.858992
$831$			4.77268e11i			1.00083i
$832$		−	8.66442e8i		−	0.00180820i
$833$	6.86981e11	+	5.17165e11i	1.42680	+	1.07411i
$834$	2.11642e11			0.437458
$835$	−1.81737e11			−0.373850
$836$		−	1.23146e11i		−	0.252113i
$837$	4.25781e9			0.00867531
$838$			3.15809e11i			0.640395i
$839$			2.36468e11i			0.477227i	0.971115	+	0.238614i	$0.0766930\pi$
$839$			2.36468e11i			0.477227i	−0.971115	+	0.238614i	$0.923307\pi$
$840$	−1.06297e11	+	5.30291e10i	−0.213502	+	0.106512i
$841$	3.18338e10			0.0636363
$842$	4.58894e11			0.912985
$843$		−	1.22947e10i		−	0.0243450i
$844$	7.98712e10			0.157406
$845$			5.95806e11i			1.16863i
$846$		−	1.48082e10i		−	0.0289081i
$847$	−3.09747e11	+	1.54526e11i	−0.601829	+	0.300239i
$848$	1.82042e11			0.352038
$849$	−4.27845e11			−0.823486
$850$		−	2.41449e11i		−	0.462541i
$851$	1.03740e11			0.197801
$852$		−	1.43972e11i		−	0.273225i
$853$			1.33766e11i			0.252668i	0.991988	+	0.126334i	$0.0403211\pi$
$853$			1.33766e11i			0.252668i	−0.991988	+	0.126334i	$0.959679\pi$
$854$	1.85557e11	+	3.71948e11i	0.348855	+	0.699279i
$855$	8.11794e10			0.151908
$856$	3.52068e11			0.655739
$857$			6.59143e9i			0.0122196i	0.999981	+	0.00610979i	$0.00194482\pi$
$857$			6.59143e9i			0.0122196i	−0.999981	+	0.00610979i	$0.998055\pi$
$858$	4.13905e9			0.00763751
$859$			8.20431e11i			1.50685i	0.657535	+	0.753424i	$0.271599\pi$
$859$			8.20431e11i			1.50685i	−0.657535	+	0.753424i	$0.728401\pi$
$860$		−	4.70803e11i		−	0.860686i
$861$	−8.98154e10	+	4.48069e10i	−0.163432	+	0.0815329i
$862$	−3.02922e11			−0.548658
$863$	−4.82843e11			−0.870488			−0.435244	−	0.900313i	$-0.643338\pi$
$863$	−4.82843e11			−0.870488			−0.435244	+	0.900313i	$0.643338\pi$
$864$			1.89582e10i			0.0340207i
$865$	3.37889e11			0.603544
$866$		−	5.87355e11i		−	1.04431i
$867$			7.14263e11i			1.26410i
$868$	5.71150e9	+	1.14487e10i	0.0100617	+	0.0201686i
$869$	7.56661e11			1.32685
$870$	−2.81946e11			−0.492141
$871$			9.11185e9i			0.0158319i
$872$	−1.47021e11			−0.254280
$873$			3.20021e11i			0.550961i
$874$			2.26220e10i			0.0387690i
$875$	−1.93837e11	−	3.88547e11i	−0.330678	−	0.662843i
$876$	1.88044e11			0.319332
$877$	−5.13038e11			−0.867263			−0.433632	−	0.901090i	$-0.642768\pi$
$877$	−5.13038e11			−0.867263			−0.433632	+	0.901090i	$0.642768\pi$
$878$			1.92543e11i			0.324004i
$879$	−2.19115e11			−0.367043
$880$		−	2.26637e11i		−	0.377920i
$881$		−	1.61370e11i		−	0.267867i	−0.990990	−	0.133933i	$-0.957239\pi$
$881$		−	1.61370e11i		−	0.267867i	0.990990	−	0.133933i	$-0.0427608\pi$
$882$	−8.57881e10	+	1.13957e11i	−0.141760	+	0.188308i
$883$	2.31400e11			0.380645			0.190323	−	0.981722i	$-0.439047\pi$
$883$	2.31400e11			0.380645			0.190323	+	0.981722i	$0.439047\pi$
$884$	7.88816e9			0.0129171
$885$		−	5.97403e11i		−	0.973854i
$886$	2.48493e10			0.0403255
$887$			1.16861e12i			1.88789i	0.330107	+	0.943944i	$0.392915\pi$
$887$			1.16861e12i			1.88789i	−0.330107	+	0.943944i	$0.607085\pi$
$888$		−	1.78529e11i		−	0.287115i
$889$	8.73040e10	−	4.35541e10i	0.139774	−	0.0697303i
$890$	−8.12437e11			−1.29488
$891$	−9.05648e10			−0.143697
$892$		−	2.68307e11i		−	0.423811i
$893$	−3.04085e10			−0.0478178
$894$			2.65936e11i			0.416321i
$895$			1.32386e12i			2.06323i
$896$	−5.09761e10	+	2.54309e10i	−0.0790924	+	0.0394575i
$897$	−7.60345e8			−0.00117447
$898$	4.44615e11			0.683721
$899$			3.03670e10i			0.0464904i
$900$	4.00519e10			0.0610455
$901$			1.65733e12i			2.51484i
$902$		−	1.91497e11i		−	0.289291i
$903$	−2.52366e11	−	5.05867e11i	−0.379560	−	0.760826i
$904$	8.51688e9			0.0127528
$905$	−7.52186e11			−1.12132
$906$		−	6.24399e10i		−	0.0926721i
$907$	1.17623e11			0.173805			0.0869025	−	0.996217i	$-0.472303\pi$
$907$	1.17623e11			0.173805			0.0869025	+	0.996217i	$0.472303\pi$
$908$			6.34990e11i			0.934165i
$909$		−	3.68415e11i		−	0.539613i
$910$	−7.33661e9	+	3.66007e9i	−0.0106987	+	0.00533733i
$911$	−1.32063e12			−1.91738			−0.958692	−	0.284448i	$-0.908190\pi$
$911$	−1.32063e12			−1.91738			−0.958692	+	0.284448i	$0.908190\pi$
$912$	3.89307e10			0.0562747
$913$		−	9.33921e11i		−	1.34409i
$914$	−2.92316e11			−0.418859
$915$		−	5.22779e11i		−	0.745819i
$916$		−	3.60378e11i		−	0.511890i
$917$	1.33528e11	+	2.67656e11i	0.188840	+	0.378529i
$918$	−1.72597e11			−0.243032
$919$	6.04281e11			0.847182			0.423591	−	0.905854i	$-0.360769\pi$
$919$	6.04281e11			0.847182			0.423591	+	0.905854i	$0.360769\pi$
$920$			4.16333e10i			0.0581151i
$921$	−6.66324e11			−0.926077
$922$		−	4.92708e11i		−	0.681814i
$923$		−	9.93699e9i		−	0.0136914i
$924$	−1.21485e11	−	2.43516e11i	−0.166661	−	0.334072i
$925$	−3.77167e11			−0.515189
$926$	6.28899e11			0.855336
$927$		−	8.85379e10i		−	0.119898i
$928$	−1.35211e11			−0.182315
$929$			4.42402e11i			0.593956i	0.954884	+	0.296978i	$0.0959789\pi$
$929$			4.42402e11i			0.593956i	−0.954884	+	0.296978i	$0.904021\pi$
$930$		−	1.60913e10i		−	0.0215110i
$931$	2.34011e11	+	1.76166e11i	0.311486	+	0.234489i
$932$	−8.61629e10			−0.114197
$933$	−7.88499e10			−0.104058
$934$		−	3.12667e11i		−	0.410860i
$935$	2.06332e12			2.69973
$936$			1.30850e9i			0.00170479i
$937$		−	7.46164e11i		−	0.968001i	−0.875068	−	0.484000i	$-0.839183\pi$
$937$		−	7.46164e11i		−	0.968001i	0.875068	−	0.484000i	$-0.160817\pi$
$938$	5.36085e11	−	2.67441e11i	0.692505	−	0.345475i
$939$	−2.24328e10			−0.0288550
$940$	−5.59636e10			−0.0716794
$941$		−	7.73947e11i		−	0.987081i	−0.869723	−	0.493541i	$-0.835702\pi$
$941$		−	7.73947e11i		−	0.987081i	0.869723	−	0.493541i	$-0.164298\pi$
$942$	6.03012e11			0.765812
$943$			3.51780e10i			0.0444861i
$944$		−	2.86493e11i		−	0.360766i
$945$	1.60529e11	−	8.00845e10i	0.201292	−	0.100420i
$946$	1.07857e12			1.34674
$947$	2.19027e9			0.00272331			0.00136166	−	0.999999i	$-0.499567\pi$
$947$	2.19027e9			0.00272331			0.00136166	+	0.999999i	$0.499567\pi$
$948$			2.39207e11i			0.296170i
$949$	1.29788e10			0.0160018
$950$		−	8.22466e10i		−	0.100977i
$951$			4.79390e11i			0.586094i
$952$	−2.31525e11	−	4.64091e11i	−0.281871	−	0.565009i
$953$	6.30930e11			0.764908			0.382454	−	0.923974i	$-0.375079\pi$
$953$	6.30930e11			0.764908			0.382454	+	0.923974i	$0.375079\pi$
$954$	−2.74920e11			−0.331904
$955$			6.33914e11i			0.762109i
$956$	−5.88176e11			−0.704167
$957$		−	6.45915e11i		−	0.770065i
$958$			6.94284e11i			0.824280i
$959$	−7.52599e11	+	3.75455e11i	−0.889793	+	0.443899i
$960$	7.16478e10			0.0843564
$961$	8.51158e11			0.997968
$962$		−	1.23221e10i		−	0.0143874i
$963$	−5.31692e11			−0.618237
$964$			7.70104e11i			0.891746i
$965$		−	7.77834e11i		−	0.896970i
$966$	2.23168e10	+	4.47340e10i	0.0256286	+	0.0513724i
$967$	−5.13723e11			−0.587521			−0.293760	−	0.955879i	$-0.594907\pi$
$967$	−5.13723e11			−0.587521			−0.293760	+	0.955879i	$0.594907\pi$
$968$	2.08780e11			0.237787
$969$			3.54429e11i			0.402007i
$970$	1.20944e12			1.36614
$971$		−	6.10968e11i		−	0.687293i	−0.939099	−	0.343646i	$-0.888338\pi$
$971$		−	6.10968e11i		−	0.687293i	0.939099	−	0.343646i	$-0.111662\pi$
$972$		−	2.86307e10i		−	0.0320750i
$973$	−4.28744e11	−	8.59416e11i	−0.478351	−	0.958853i
$974$	−8.67525e11			−0.963932
$975$	2.76439e9			0.00305901
$976$		−	2.50706e11i		−	0.276290i
$977$	7.82066e11			0.858352			0.429176	−	0.903221i	$-0.358804\pi$
$977$	7.82066e11			0.858352			0.429176	+	0.903221i	$0.358804\pi$
$978$		−	1.62895e11i		−	0.178054i
$979$		−	1.86122e12i		−	2.02613i
$980$	4.30672e11	+	3.24214e11i	0.466920	+	0.351502i
$981$	2.22030e11			0.239738
$982$	9.08457e11			0.976919
$983$		−	2.26482e11i		−	0.242561i	−0.992618	−	0.121280i	$-0.961300\pi$
$983$		−	2.26482e11i		−	0.242561i	0.992618	−	0.121280i	$-0.0387000\pi$
$984$	6.05388e10			0.0645734
$985$		−	5.16249e11i		−	0.548421i
$986$		−	1.23098e12i		−	1.30239i
$987$	−6.01317e10	+	2.99984e10i	−0.0633629	+	0.0316104i
$988$	2.68700e9			0.00281994
$989$	−1.98133e11			−0.207096
$990$			3.42267e11i			0.356307i
$991$	−8.67243e11			−0.899179			−0.449590	−	0.893235i	$-0.648430\pi$
$991$	−8.67243e11			−0.899179			−0.449590	+	0.893235i	$0.648430\pi$
$992$		−	7.71682e9i		−	0.00796878i
$993$		−	2.49440e11i		−	0.256548i
$994$	−5.84631e11	+	2.91660e11i	−0.598875	+	0.298766i
$995$	−4.26874e10			−0.0435519
$996$	2.95245e11			0.300016
$997$		−	1.83000e11i		−	0.185212i	−0.995703	−	0.0926061i	$-0.970480\pi$
$997$		−	1.83000e11i		−	0.185212i	0.995703	−	0.0926061i	$-0.0295197\pi$
$998$	−8.50089e11			−0.856924
$999$			2.69614e11i			0.270695i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.9.c.a.13.9	✓	12
3.2	odd	2		126.9.c.c.55.1		12
4.3	odd	2		336.9.f.c.97.11		12
7.6	odd	2	inner	42.9.c.a.13.10	yes	12
21.20	even	2		126.9.c.c.55.6		12
28.27	even	2		336.9.f.c.97.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.9.c.a.13.9	✓	12	1.1	even	1	trivial
42.9.c.a.13.10	yes	12	7.6	odd	2	inner
126.9.c.c.55.1		12	3.2	odd	2
126.9.c.c.55.6		12	21.20	even	2
336.9.f.c.97.2		12	28.27	even	2
336.9.f.c.97.11		12	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	42.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+11.3137 q^{2} -46.7654i q^{3} +128.000 q^{4} +730.548i q^{5} -529.090i q^{6} +(-2148.48 + 1071.83i) q^{7} +1448.15 q^{8} -2187.00 q^{9} +O(q^{10})\)	\(q+11.3137 q^{2} -46.7654i q^{3} +128.000 q^{4} +730.548i q^{5} -529.090i q^{6} +(-2148.48 + 1071.83i) q^{7} +1448.15 q^{8} -2187.00 q^{9} +8265.21i q^{10} -18934.9 q^{11} -5985.97i q^{12} -413.152i q^{13} +(-24307.3 + 12126.4i) q^{14} +34164.3 q^{15} +16384.0 q^{16} +149161. i q^{17} -24743.1 q^{18} +50809.9i q^{19} +93510.1i q^{20} +(50124.6 + 100475. i) q^{21} -214223. q^{22} +39352.9 q^{23} -67723.5i q^{24} -143075. q^{25} -4674.28i q^{26} +102276. i q^{27} +(-275006. + 137194. i) q^{28} -729438. q^{29} +386525. q^{30} -41630.7i q^{31} +185364. q^{32} +885496. i q^{33} +1.68757e6i q^{34} +(-783024. - 1.56957e6i) q^{35} -279936. q^{36} +2.63614e6 q^{37} +574848. i q^{38} -19321.2 q^{39} +1.05795e6i q^{40} +893911. i q^{41} +(567095. + 1.13674e6i) q^{42} -5.03478e6 q^{43} -2.42366e6 q^{44} -1.59771e6i q^{45} +445227. q^{46} +598476. i q^{47} -766204. i q^{48} +(3.46716e6 - 4.60562e6i) q^{49} -1.61871e6 q^{50} +6.97558e6 q^{51} -52883.4i q^{52} +1.11110e7 q^{53} +1.15712e6i q^{54} -1.38328e7i q^{55} +(-3.11134e6 + 1.55218e6i) q^{56} +2.37614e6 q^{57} -8.25265e6 q^{58} -1.74861e7i q^{59} +4.37304e6 q^{60} -1.53019e7i q^{61} -470997. i q^{62} +(4.69873e6 - 2.34410e6i) q^{63} +2.09715e6 q^{64} +301827. q^{65} +1.00182e7i q^{66} -2.20545e7 q^{67} +1.90926e7i q^{68} -1.84035e6i q^{69} +(-8.85891e6 - 1.77577e7i) q^{70} +2.40517e7 q^{71} -3.16711e6 q^{72} +3.14141e7i q^{73} +2.98245e7 q^{74} +6.69097e6i q^{75} +6.50367e6i q^{76} +(4.06812e7 - 2.02950e7i) q^{77} -218594. q^{78} -3.99613e7 q^{79} +1.19693e7i q^{80} +4.78297e6 q^{81} +1.01135e7i q^{82} +4.93228e7i q^{83} +(6.41595e6 + 1.28607e7i) q^{84} -1.08969e8 q^{85} -5.69620e7 q^{86} +3.41125e7i q^{87} -2.74206e7 q^{88} +9.82960e7i q^{89} -1.80760e7i q^{90} +(442829. + 887650. i) q^{91} +5.03717e6 q^{92} -1.94687e6 q^{93} +6.77099e6i q^{94} -3.71191e7 q^{95} -8.66861e6i q^{96} -1.46329e8i q^{97} +(3.92264e7 - 5.21067e7i) q^{98} +4.14105e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 1536 q^{4} + 6420 q^{7} - 26244 q^{9}+O(q^{10}) \) 12 * q + 1536 * q^4 + 6420 * q^7 - 26244 * q^9	\( 12 q + 1536 q^{4} + 6420 q^{7} - 26244 q^{9} + 4344 q^{11} - 12288 q^{14} + 59616 q^{15} + 196608 q^{16} - 224856 q^{21} - 508416 q^{22} + 499800 q^{23} - 3001476 q^{25} + 821760 q^{28} - 1278408 q^{29} + 705024 q^{30} + 2028912 q^{35} - 3359232 q^{36} + 7068648 q^{37} - 5473008 q^{39} + 1513728 q^{42} - 11388024 q^{43} + 556032 q^{44} + 8171520 q^{46} - 12346788 q^{49} + 30019584 q^{50} + 16727472 q^{51} + 19714968 q^{53} - 1572864 q^{56} - 10386144 q^{57} - 17696256 q^{58} + 7630848 q^{60} - 14040540 q^{63} + 25165824 q^{64} - 93770592 q^{65} - 9394008 q^{67} + 11218944 q^{70} + 5393208 q^{71} + 58512384 q^{74} + 24982968 q^{77} + 32638464 q^{78} + 134560968 q^{79} + 57395628 q^{81} - 28781568 q^{84} - 102074640 q^{85} - 282934272 q^{86} - 65077248 q^{88} - 96105408 q^{91} + 63974400 q^{92} + 202339296 q^{93} - 378351840 q^{95} + 387747840 q^{98} - 9500328 q^{99}+O(q^{100}) \) 12 * q + 1536 * q^4 + 6420 * q^7 - 26244 * q^9 + 4344 * q^11 - 12288 * q^14 + 59616 * q^15 + 196608 * q^16 - 224856 * q^21 - 508416 * q^22 + 499800 * q^23 - 3001476 * q^25 + 821760 * q^28 - 1278408 * q^29 + 705024 * q^30 + 2028912 * q^35 - 3359232 * q^36 + 7068648 * q^37 - 5473008 * q^39 + 1513728 * q^42 - 11388024 * q^43 + 556032 * q^44 + 8171520 * q^46 - 12346788 * q^49 + 30019584 * q^50 + 16727472 * q^51 + 19714968 * q^53 - 1572864 * q^56 - 10386144 * q^57 - 17696256 * q^58 + 7630848 * q^60 - 14040540 * q^63 + 25165824 * q^64 - 93770592 * q^65 - 9394008 * q^67 + 11218944 * q^70 + 5393208 * q^71 + 58512384 * q^74 + 24982968 * q^77 + 32638464 * q^78 + 134560968 * q^79 + 57395628 * q^81 - 28781568 * q^84 - 102074640 * q^85 - 282934272 * q^86 - 65077248 * q^88 - 96105408 * q^91 + 63974400 * q^92 + 202339296 * q^93 - 378351840 * q^95 + 387747840 * q^98 - 9500328 * q^99

Introduction

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