LMFDB - Embedded newform 42.9.c.a.13.6

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			13.6
Root			$16.2382 - 28.1253i$ of defining polynomial
Character	$\chi$	$=$	42.13
Dual form			42.9.c.a.13.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.3137 q^{2} +46.7654i q^{3} +128.000 q^{4} +1031.06i q^{5} -529.090i q^{6} +(-283.934 + 2384.15i) q^{7} -1448.15 q^{8} -2187.00 q^{9} +O(q^{10})$	\(q-11.3137 q^{2} +46.7654i q^{3} +128.000 q^{4} +1031.06i q^{5} -529.090i q^{6} +(-283.934 + 2384.15i) q^{7} -1448.15 q^{8} -2187.00 q^{9} -11665.2i q^{10} +24553.7 q^{11} +5985.97i q^{12} +31617.8i q^{13} +(3212.35 - 26973.6i) q^{14} -48218.1 q^{15} +16384.0 q^{16} -71096.4i q^{17} +24743.1 q^{18} +51386.1i q^{19} +131976. i q^{20} +(-111496. - 13278.3i) q^{21} -277794. q^{22} -104766. q^{23} -67723.5i q^{24} -672470. q^{25} -357715. i q^{26} -102276. i q^{27} +(-36343.6 + 305172. i) q^{28} -1.08008e6 q^{29} +545526. q^{30} +573939. i q^{31} -185364. q^{32} +1.14826e6i q^{33} +804364. i q^{34} +(-2.45822e6 - 292755. i) q^{35} -279936. q^{36} +2.41419e6 q^{37} -581368. i q^{38} -1.47862e6 q^{39} -1.49314e6i q^{40} -2.61563e6i q^{41} +(1.26143e6 + 150227. i) q^{42} +614755. q^{43} +3.14288e6 q^{44} -2.25494e6i q^{45} +1.18529e6 q^{46} -915508. i q^{47} +766204. i q^{48} +(-5.60356e6 - 1.35388e6i) q^{49} +7.60813e6 q^{50} +3.32485e6 q^{51} +4.04708e6i q^{52} +1.19170e7 q^{53} +1.15712e6i q^{54} +2.53165e7i q^{55} +(411181. - 3.45262e6i) q^{56} -2.40309e6 q^{57} +1.22197e7 q^{58} -1.03094e7i q^{59} -6.17192e6 q^{60} -8.60724e6i q^{61} -6.49338e6i q^{62} +(620964. - 5.21414e6i) q^{63} +2.09715e6 q^{64} -3.26000e7 q^{65} -1.29911e7i q^{66} -5.17705e6 q^{67} -9.10034e6i q^{68} -4.89941e6i q^{69} +(2.78115e7 + 3.31214e6i) q^{70} +2.02484e7 q^{71} +3.16711e6 q^{72} +3.52423e7i q^{73} -2.73134e7 q^{74} -3.14483e7i q^{75} +6.57742e6i q^{76} +(-6.97164e6 + 5.85398e7i) q^{77} +1.67287e7 q^{78} -4.76943e7 q^{79} +1.68930e7i q^{80} +4.78297e6 q^{81} +2.95925e7i q^{82} -3.60186e7i q^{83} +(-1.42715e7 - 1.69962e6i) q^{84} +7.33050e7 q^{85} -6.95516e6 q^{86} -5.05104e7i q^{87} -3.55576e7 q^{88} -9.28933e7i q^{89} +2.55117e7i q^{90} +(-7.53817e7 - 8.97738e6i) q^{91} -1.34100e7 q^{92} -2.68405e7 q^{93} +1.03578e7i q^{94} -5.29824e7 q^{95} -8.66861e6i q^{96} +1.17197e8i q^{97} +(6.33971e7 + 1.53175e7i) q^{98} -5.36990e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$			1031.06i			1.64970i	0.565349	+	0.824852i	$0.308742\pi$
$5$			1031.06i			1.64970i	−0.565349	+	0.824852i	$0.691258\pi$
$6$		−	529.090i		−	0.408248i
$7$	−283.934	+	2384.15i	−0.118257	+	0.992983i
$7$	−283.934	+	2384.15i	−0.118257	+	0.992983i
$8$	−1448.15			−0.353553
$9$	−2187.00			−0.333333
$10$		−	11665.2i		−	1.16652i
$11$	24553.7			1.67705			0.838527	−	0.544861i	$-0.183417\pi$
$11$	24553.7			1.67705			0.838527	+	0.544861i	$0.183417\pi$
$12$			5985.97i			0.288675i
$13$			31617.8i			1.10703i	0.832840	+	0.553514i	$0.186713\pi$
$13$			31617.8i			1.10703i	−0.832840	+	0.553514i	$0.813287\pi$
$14$	3212.35	−	26973.6i	0.0836201	−	0.702145i
$15$	−48218.1			−0.952457
$16$	16384.0			0.250000
$17$		−	71096.4i		−	0.851240i	−0.904902	−	0.425620i	$-0.860056\pi$
$17$		−	71096.4i		−	0.851240i	0.904902	−	0.425620i	$-0.139944\pi$
$18$	24743.1			0.235702
$19$			51386.1i			0.394304i	0.980373	+	0.197152i	$0.0631693\pi$
$19$			51386.1i			0.394304i	−0.980373	+	0.197152i	$0.936831\pi$
$20$			131976.i			0.824852i
$21$	−111496.	−	13278.3i	−0.573299	−	0.0682755i
$22$	−277794.			−1.18586
$23$	−104766.			−0.374376			−0.187188	−	0.982324i	$-0.559937\pi$
$23$	−104766.			−0.374376			−0.187188	+	0.982324i	$0.559937\pi$
$24$		−	67723.5i		−	0.204124i
$25$	−672470.			−1.72152
$26$		−	357715.i		−	0.782787i
$27$		−	102276.i		−	0.192450i
$28$	−36343.6	+	305172.i	−0.0591283	+	0.496492i
$29$	−1.08008e6			−1.52709			−0.763545	−	0.645754i	$-0.776543\pi$
$29$	−1.08008e6			−1.52709			−0.763545	+	0.645754i	$0.776543\pi$
$30$	545526.			0.673489
$31$			573939.i			0.621468i	0.950497	+	0.310734i	$0.100575\pi$
$31$			573939.i			0.621468i	−0.950497	+	0.310734i	$0.899425\pi$
$32$	−185364.			−0.176777
$33$			1.14826e6i			0.968247i
$34$			804364.i			0.601917i
$35$	−2.45822e6	−	292755.i	−1.63813	−	0.195088i
$36$	−279936.			−0.166667
$37$	2.41419e6			1.28814			0.644072	−	0.764965i	$-0.277244\pi$
$37$	2.41419e6			1.28814			0.644072	+	0.764965i	$0.277244\pi$
$38$		−	581368.i		−	0.278815i
$39$	−1.47862e6			−0.639143
$40$		−	1.49314e6i		−	0.583258i
$41$		−	2.61563e6i		−	0.925637i	−0.886453	−	0.462818i	$-0.846838\pi$
$41$		−	2.61563e6i		−	0.925637i	0.886453	−	0.462818i	$-0.153162\pi$
$42$	1.26143e6	+	150227.i	0.405384	+	0.0482781i
$43$	614755.			0.179816			0.0899080	−	0.995950i	$-0.471343\pi$
$43$	614755.			0.179816			0.0899080	+	0.995950i	$0.471343\pi$
$44$	3.14288e6			0.838527
$45$		−	2.25494e6i		−	0.549901i
$46$	1.18529e6			0.264724
$47$		−	915508.i		−	0.187616i	−0.995590	−	0.0938082i	$-0.970096\pi$
$47$		−	915508.i		−	0.187616i	0.995590	−	0.0938082i	$-0.0299040\pi$
$48$			766204.i			0.144338i
$49$	−5.60356e6	−	1.35388e6i	−0.972031	−	0.234854i
$50$	7.60813e6			1.21730
$51$	3.32485e6			0.491464
$52$			4.04708e6i			0.553514i
$53$	1.19170e7			1.51029			0.755147	−	0.655555i	$-0.227565\pi$
$53$	1.19170e7			1.51029			0.755147	+	0.655555i	$0.227565\pi$
$54$			1.15712e6i			0.136083i
$55$			2.53165e7i			2.76664i
$56$	411181.	−	3.45262e6i	0.0418100	−	0.351073i
$57$	−2.40309e6			−0.227652
$58$	1.22197e7			1.07982
$59$		−	1.03094e7i		−	0.850793i	−0.905007	−	0.425397i	$-0.860135\pi$
$59$		−	1.03094e7i		−	0.850793i	0.905007	−	0.425397i	$-0.139865\pi$
$60$	−6.17192e6			−0.476229
$61$		−	8.60724e6i		−	0.621648i	−0.950467	−	0.310824i	$-0.899395\pi$
$61$		−	8.60724e6i		−	0.621648i	0.950467	−	0.310824i	$-0.100605\pi$
$62$		−	6.49338e6i		−	0.439444i
$63$	620964.	−	5.21414e6i	0.0394189	−	0.330994i
$64$	2.09715e6			0.125000
$65$	−3.26000e7			−1.82627
$66$		−	1.29911e7i		−	0.684654i
$67$	−5.17705e6			−0.256911			−0.128456	−	0.991715i	$-0.541002\pi$
$67$	−5.17705e6			−0.256911			−0.128456	+	0.991715i	$0.541002\pi$
$68$		−	9.10034e6i		−	0.425620i
$69$		−	4.89941e6i		−	0.216146i
$70$	2.78115e7	+	3.31214e6i	1.15833	+	0.137948i
$71$	2.02484e7			0.796813			0.398407	−	0.917209i	$-0.369563\pi$
$71$	2.02484e7			0.796813			0.398407	+	0.917209i	$0.369563\pi$
$72$	3.16711e6			0.117851
$73$			3.52423e7i			1.24100i	0.784205	+	0.620502i	$0.213071\pi$
$73$			3.52423e7i			1.24100i	−0.784205	+	0.620502i	$0.786929\pi$
$74$	−2.73134e7			−0.910856
$75$		−	3.14483e7i		−	0.993922i
$76$			6.57742e6i			0.197152i
$77$	−6.97164e6	+	5.85398e7i	−0.198323	+	1.66529i
$78$	1.67287e7			0.451942
$79$	−4.76943e7			−1.22450			−0.612249	−	0.790665i	$-0.709735\pi$
$79$	−4.76943e7			−1.22450			−0.612249	+	0.790665i	$0.709735\pi$
$80$			1.68930e7i			0.412426i
$81$	4.78297e6			0.111111
$82$			2.95925e7i			0.654524i
$83$		−	3.60186e7i		−	0.758952i	−0.925201	−	0.379476i	$-0.876104\pi$
$83$		−	3.60186e7i		−	0.758952i	0.925201	−	0.379476i	$-0.123896\pi$
$84$	−1.42715e7	−	1.69962e6i	−0.286650	−	0.0341377i
$85$	7.33050e7			1.40429
$86$	−6.95516e6			−0.127149
$87$		−	5.05104e7i		−	0.881666i
$88$	−3.55576e7			−0.592928
$89$		−	9.28933e7i		−	1.48055i	−0.672302	−	0.740277i	$-0.734694\pi$
$89$		−	9.28933e7i		−	1.48055i	0.672302	−	0.740277i	$-0.265306\pi$
$90$			2.55117e7i			0.388839i
$91$	−7.53817e7	−	8.97738e6i	−1.09926	−	0.130913i
$92$	−1.34100e7			−0.187188
$93$	−2.68405e7			−0.358805
$94$			1.03578e7i			0.132665i
$95$	−5.29824e7			−0.650485
$96$		−	8.66861e6i		−	0.102062i
$97$			1.17197e8i			1.32383i	0.749581	+	0.661913i	$0.230255\pi$
$97$			1.17197e8i			1.32383i	−0.749581	+	0.661913i	$0.769745\pi$
$98$	6.33971e7	+	1.53175e7i	0.687330	+	0.166067i
$99$	−5.36990e7			−0.559018
$100$	−8.60762e7			−0.860762
$101$			2.05639e8i			1.97615i	0.153977	+	0.988074i	$0.450792\pi$
$101$			2.05639e8i			1.97615i	−0.153977	+	0.988074i	$0.549208\pi$
$102$	−3.76164e7			−0.347517
$103$		−	4.74686e6i		−	0.0421752i	−0.999778	−	0.0210876i	$-0.993287\pi$
$103$		−	4.74686e6i		−	0.0421752i	0.999778	−	0.0210876i	$-0.00671290\pi$
$104$		−	4.57875e7i		−	0.391393i
$105$	1.36908e7	−	1.14959e8i	0.112634	−	0.945774i
$106$	−1.34825e8			−1.06794
$107$	2.06913e8			1.57853			0.789265	−	0.614053i	$-0.210462\pi$
$107$	2.06913e8			1.57853			0.789265	+	0.614053i	$0.210462\pi$
$108$		−	1.30913e7i		−	0.0962250i
$109$	2.29533e8			1.62607			0.813035	−	0.582215i	$-0.197814\pi$
$109$	2.29533e8			1.62607			0.813035	+	0.582215i	$0.197814\pi$
$110$		−	2.86423e8i		−	1.95631i
$111$			1.12901e8i			0.743711i
$112$	−4.65198e6	+	3.90620e7i	−0.0295642	+	0.248246i
$113$	−2.03634e8			−1.24893			−0.624463	−	0.781054i	$-0.714682\pi$
$113$	−2.03634e8			−1.24893			−0.624463	+	0.781054i	$0.714682\pi$
$114$	2.71879e7			0.160974
$115$		−	1.08020e8i		−	0.617610i
$116$	−1.38250e8			−0.763545
$117$		−	6.91481e7i		−	0.369009i
$118$			1.16637e8i			0.601602i
$119$	1.69505e8	+	2.01867e7i	0.845267	+	0.100665i
$120$	6.98273e7			0.336744
$121$	3.88527e8			1.81251
$122$			9.73798e7i			0.439572i
$123$	1.22321e8			0.534417
$124$			7.34642e7i			0.310734i
$125$		−	2.90600e8i		−	1.19030i
$126$	−7.02541e6	+	5.89913e7i	−0.0278734	+	0.234048i
$127$	−4.47985e7			−0.172206			−0.0861030	−	0.996286i	$-0.527441\pi$
$127$	−4.47985e7			−0.172206			−0.0861030	+	0.996286i	$0.527441\pi$
$128$	−2.37266e7			−0.0883883
$129$			2.87492e7i			0.103817i
$130$	3.68827e8			1.29137
$131$		−	5.29314e7i		−	0.179733i	−0.995954	−	0.0898665i	$-0.971356\pi$
$131$		−	5.29314e7i		−	0.179733i	0.995954	−	0.0898665i	$-0.0286440\pi$
$132$			1.46978e8i			0.484124i
$133$	−1.22512e8	−	1.45903e7i	−0.391537	−	0.0466291i
$134$	5.85717e7			0.181664
$135$	1.05453e8			0.317486
$136$			1.02959e8i			0.300959i
$137$	4.56941e8			1.29711			0.648556	−	0.761167i	$-0.275373\pi$
$137$	4.56941e8			1.29711			0.648556	+	0.761167i	$0.275373\pi$
$138$			5.54305e7i			0.152839i
$139$			3.83658e8i			1.02774i	0.857867	+	0.513872i	$0.171790\pi$
$139$			3.83658e8i			1.02774i	−0.857867	+	0.513872i	$0.828210\pi$
$140$	−3.14652e8	−	3.74726e7i	−0.819064	−	0.0975442i
$141$	4.28141e7			0.108320
$142$	−2.29084e8			−0.563432
$143$			7.76335e8i			1.85654i
$144$	−3.58318e7			−0.0833333
$145$		−	1.11363e9i		−	2.51925i
$146$		−	3.98721e8i		−	0.877522i
$147$	6.33149e7	−	2.62053e8i	0.135593	−	0.561202i
$148$	3.09016e8			0.644072
$149$	8.53722e7			0.173209			0.0866047	−	0.996243i	$-0.472398\pi$
$149$	8.53722e7			0.173209			0.0866047	+	0.996243i	$0.472398\pi$
$150$			3.55797e8i			0.702809i
$151$	−8.78726e8			−1.69023			−0.845115	−	0.534585i	$-0.820468\pi$
$151$	−8.78726e8			−1.69023			−0.845115	+	0.534585i	$0.820468\pi$
$152$		−	7.44150e7i		−	0.139408i
$153$			1.55488e8i			0.283747i
$154$	7.88752e7	−	6.62303e8i	0.140235	−	1.17753i
$155$	−5.91768e8			−1.02524
$156$	−1.89263e8			−0.319571
$157$		−	9.18274e7i		−	0.151138i	−0.997141	−	0.0755690i	$-0.975923\pi$
$157$		−	9.18274e7i		−	0.151138i	0.997141	−	0.0755690i	$-0.0240773\pi$
$158$	5.39599e8			0.865850
$159$			5.57301e8i			0.871969i
$160$		−	1.91122e8i		−	0.291629i
$161$	2.97466e7	−	2.49778e8i	0.0442725	−	0.371749i
$162$	−5.41131e7			−0.0785674
$163$	−1.27431e9			−1.80519			−0.902596	−	0.430489i	$-0.858341\pi$
$163$	−1.27431e9			−1.80519			−0.902596	+	0.430489i	$0.858341\pi$
$164$		−	3.34800e8i		−	0.462818i
$165$	−1.18394e9			−1.59732
$166$			4.07504e8i			0.536660i
$167$			1.40636e9i			1.80814i	0.427386	+	0.904069i	$0.359435\pi$
$167$			1.40636e9i			1.80814i	−0.427386	+	0.904069i	$0.640565\pi$
$168$	1.61463e8	+	1.92290e7i	0.202692	+	0.0241390i
$169$	−1.83955e8			−0.225510
$170$	−8.29351e8			−0.992986
$171$		−	1.12381e8i		−	0.131435i
$172$	7.86886e7			0.0899080
$173$			8.52430e8i			0.951643i	0.879542	+	0.475822i	$0.157849\pi$
$173$			8.52430e8i			0.951643i	−0.879542	+	0.475822i	$0.842151\pi$
$174$			5.71460e8i			0.623432i
$175$	1.90937e8	−	1.60327e9i	0.203581	−	1.70944i
$176$	4.02288e8			0.419263
$177$	4.82122e8			0.491206
$178$			1.05097e9i			1.04691i
$179$	−1.26358e9			−1.23081			−0.615406	−	0.788210i	$-0.711008\pi$
$179$	−1.26358e9			−1.23081			−0.615406	+	0.788210i	$0.711008\pi$
$180$		−	2.88632e8i		−	0.274951i
$181$		−	7.70795e8i		−	0.718166i	−0.933306	−	0.359083i	$-0.883090\pi$
$181$		−	7.70795e8i		−	0.718166i	0.933306	−	0.359083i	$-0.116910\pi$
$182$	8.52846e8	+	1.01567e8i	0.777294	+	0.0925697i
$183$	4.02521e8			0.358909
$184$	1.51717e8			0.132362
$185$			2.48919e9i			2.12506i
$186$	3.03665e8			0.253713
$187$		−	1.74568e9i		−	1.42757i
$188$		−	1.17185e8i		−	0.0938082i
$189$	2.43841e8	+	2.90396e7i	0.191100	+	0.0227585i
$190$	5.99428e8			0.459963
$191$	−1.22735e9			−0.922224			−0.461112	−	0.887342i	$-0.652549\pi$
$191$	−1.22735e9			−0.922224			−0.461112	+	0.887342i	$0.652549\pi$
$192$			9.80741e7i			0.0721688i
$193$	1.13405e9			0.817343			0.408671	−	0.912682i	$-0.365992\pi$
$193$	1.13405e9			0.817343			0.408671	+	0.912682i	$0.365992\pi$
$194$		−	1.32594e9i		−	0.936086i
$195$		−	1.52455e9i		−	1.05440i
$196$	−7.17256e8	−	1.73297e8i	−0.486015	−	0.117427i
$197$	1.25901e9			0.835918			0.417959	−	0.908466i	$-0.362745\pi$
$197$	1.25901e9			0.835918			0.417959	+	0.908466i	$0.362745\pi$
$198$	6.07535e8			0.395285
$199$			2.23693e9i			1.42640i	0.700963	+	0.713198i	$0.252754\pi$
$199$			2.23693e9i			1.42640i	−0.700963	+	0.713198i	$0.747246\pi$
$200$	9.73840e8			0.608650
$201$		−	2.42107e8i		−	0.148328i
$202$		−	2.32654e9i		−	1.39735i
$203$	3.06672e8	−	2.57508e9i	0.180589	−	1.51637i
$204$	4.25581e8			0.245732
$205$	2.69688e9			1.52703
$206$			5.37046e7i			0.0298224i
$207$	2.29123e8			0.124792
$208$			5.18026e8i			0.276757i
$209$			1.26172e9i			0.661269i
$210$	−1.54893e8	+	1.30062e9i	−0.0796445	+	0.668763i
$211$	5.48896e8			0.276924			0.138462	−	0.990368i	$-0.455784\pi$
$211$	5.48896e8			0.276924			0.138462	+	0.990368i	$0.455784\pi$
$212$	1.52537e9			0.755147
$213$			9.46922e8i			0.460040i
$214$	−2.34095e9			−1.11619
$215$			6.33852e8i			0.296643i
$216$			1.48111e8i			0.0680414i
$217$	−1.36836e9	−	1.62961e8i	−0.617107	−	0.0734927i
$218$	−2.59687e9			−1.14980
$219$	−1.64812e9			−0.716494
$220$			3.24051e9i			1.38332i
$221$	2.24791e9			0.942346
$222$		−	1.27732e9i		−	0.525883i
$223$			7.03622e8i			0.284525i	0.989829	+	0.142262i	$0.0454377\pi$
$223$			7.03622e8i			0.284525i	−0.989829	+	0.142262i	$0.954562\pi$
$224$	5.26311e7	−	4.41936e8i	0.0209050	−	0.175536i
$225$	1.47069e9			0.573841
$226$	2.30386e9			0.883124
$227$		−	5.76372e8i		−	0.217070i	−0.994093	−	0.108535i	$-0.965384\pi$
$227$		−	5.76372e8i		−	0.217070i	0.994093	−	0.108535i	$-0.0346159\pi$
$228$	−3.07596e8			−0.113826
$229$		−	4.42318e8i		−	0.160839i	−0.996761	−	0.0804197i	$-0.974374\pi$
$229$		−	4.42318e8i		−	0.160839i	0.996761	−	0.0804197i	$-0.0256261\pi$
$230$			1.22211e9i			0.436716i
$231$	−2.73764e9	−	3.26032e8i	−0.961453	−	0.114502i
$232$	1.56413e9			0.539908
$233$	3.64885e9			1.23803			0.619016	−	0.785378i	$-0.287532\pi$
$233$	3.64885e9			1.23803			0.619016	+	0.785378i	$0.287532\pi$
$234$			7.82322e8i			0.260929i
$235$	9.43948e8			0.309511
$236$		−	1.31960e9i		−	0.425397i
$237$		−	2.23044e9i		−	0.706964i
$238$	−1.91773e9	−	2.28386e8i	−0.597694	−	0.0711807i
$239$	−1.28372e9			−0.393441			−0.196720	−	0.980460i	$-0.563029\pi$
$239$	−1.28372e9			−0.393441			−0.196720	+	0.980460i	$0.563029\pi$
$240$	−7.90006e8			−0.238114
$241$			2.19462e8i			0.0650566i	0.999471	+	0.0325283i	$0.0103559\pi$
$241$			2.19462e8i			0.0650566i	−0.999471	+	0.0325283i	$0.989644\pi$
$242$	−4.39568e9			−1.28164
$243$			2.23677e8i			0.0641500i
$244$		−	1.10173e9i		−	0.310824i
$245$	1.39594e9	−	5.77764e9i	0.387439	−	1.60356i
$246$	−1.38390e9			−0.377890
$247$	−1.62472e9			−0.436506
$248$		−	8.31152e8i		−	0.219722i
$249$	1.68442e9			0.438181
$250$			3.28777e9i			0.841669i
$251$			1.37174e9i			0.345601i	0.984957	+	0.172801i	$0.0552817\pi$
$251$			1.37174e9i			0.345601i	−0.984957	+	0.172801i	$0.944718\pi$
$252$	7.94834e7	−	6.67410e8i	0.0197094	−	0.165497i
$253$	−2.57239e9			−0.627849
$254$	5.06837e8			0.121768
$255$			3.42814e9i			0.810769i
$256$	2.68435e8			0.0625000
$257$		−	9.55370e8i		−	0.218998i	−0.993987	−	0.109499i	$-0.965075\pi$
$257$		−	9.55370e8i		−	0.218998i	0.993987	−	0.109499i	$-0.0349246\pi$
$258$		−	3.25261e8i		−	0.0734096i
$259$	−6.85471e8	+	5.75580e9i	−0.152332	+	1.27911i
$260$	−4.17280e9			−0.913134
$261$	2.36214e9			0.509030
$262$			5.98850e8i			0.127090i
$263$	2.98852e9			0.624644			0.312322	−	0.949976i	$-0.398893\pi$
$263$	2.98852e9			0.624644			0.312322	+	0.949976i	$0.398893\pi$
$264$		−	1.66286e9i		−	0.342327i
$265$			1.22872e10i			2.49154i
$266$	1.38607e9	+	1.65070e8i	0.276859	+	0.0329717i
$267$	4.34419e9			0.854798
$268$	−6.62663e8			−0.128456
$269$		−	7.69939e9i		−	1.47044i	−0.677829	−	0.735220i	$-0.737079\pi$
$269$		−	7.69939e9i		−	1.47044i	0.677829	−	0.735220i	$-0.262921\pi$
$270$	−1.19307e9			−0.224496
$271$		−	4.92911e9i		−	0.913885i	−0.889496	−	0.456942i	$-0.848945\pi$
$271$		−	4.92911e9i		−	0.913885i	0.889496	−	0.456942i	$-0.151055\pi$
$272$		−	1.16484e9i		−	0.212810i
$273$	4.19830e8	−	3.52525e9i	0.0755828	−	0.634658i
$274$	−5.16970e9			−0.917197
$275$	−1.65116e10			−2.88709
$276$		−	6.27125e8i		−	0.108073i
$277$	1.66469e9			0.282758			0.141379	−	0.989956i	$-0.454846\pi$
$277$	1.66469e9			0.282758			0.141379	+	0.989956i	$0.454846\pi$
$278$		−	4.34060e9i		−	0.726725i
$279$		−	1.25520e9i		−	0.207156i
$280$	3.55988e9	+	4.23954e8i	0.579166	+	0.0689742i
$281$	−7.51436e9			−1.20522			−0.602611	−	0.798035i	$-0.705873\pi$
$281$	−7.51436e9			−1.20522			−0.602611	+	0.798035i	$0.705873\pi$
$282$	−4.84386e8			−0.0765940
$283$			4.20309e9i			0.655274i	0.944804	+	0.327637i	$0.106252\pi$
$283$			4.20309e9i			0.655274i	−0.944804	+	0.327637i	$0.893748\pi$
$284$	2.59179e9			0.398407
$285$		−	2.47774e9i		−	0.375558i
$286$		−	8.78323e9i		−	1.31277i
$287$	6.23606e9	+	7.42666e8i	0.919142	+	0.109463i
$288$	4.05391e8			0.0589256
$289$	1.92106e9			0.275391
$290$			1.25993e10i			1.78138i
$291$	−5.48078e9			−0.764311
$292$			4.51102e9i			0.620502i
$293$			6.95941e9i			0.944282i	0.881523	+	0.472141i	$0.156519\pi$
$293$			6.95941e9i			0.944282i	−0.881523	+	0.472141i	$0.843481\pi$
$294$	−7.16326e8	+	2.96479e9i	−0.0958786	+	0.396830i
$295$	1.06296e10			1.40356
$296$	−3.49612e9			−0.455428
$297$		−	2.51125e9i		−	0.322749i
$298$	−9.65876e8			−0.122477
$299$		−	3.31247e9i		−	0.414445i
$300$		−	4.02538e9i		−	0.496961i
$301$	−1.74550e8	+	1.46567e9i	−0.0212644	+	0.178554i
$302$	9.94165e9			1.19517
$303$	−9.61678e9			−1.14093
$304$			8.41910e8i			0.0985761i
$305$	8.87463e9			1.02554
$306$		−	1.75914e9i		−	0.200639i
$307$			4.95393e9i			0.557695i	0.960335	+	0.278847i	$0.0899524\pi$
$307$			4.95393e9i			0.557695i	−0.960335	+	0.278847i	$0.910048\pi$
$308$	−8.92370e8	+	7.49310e9i	−0.0991613	+	0.832643i
$309$	2.21989e8			0.0243499
$310$	6.69509e9			0.724953
$311$		−	1.30232e10i		−	1.39211i	−0.717987	−	0.696057i	$-0.754936\pi$
$311$		−	1.30232e10i		−	1.39211i	0.717987	−	0.696057i	$-0.245064\pi$
$312$	2.14127e9			0.225971
$313$		−	1.43472e9i		−	0.149483i	−0.997203	−	0.0747413i	$-0.976187\pi$
$313$		−	1.43472e9i		−	0.149483i	0.997203	−	0.0747413i	$-0.0238131\pi$
$314$			1.03891e9i			0.106871i
$315$	5.37612e9	+	6.40254e8i	0.546043	+	0.0650295i
$316$	−6.10487e9			−0.612249
$317$	4.37110e9			0.432866			0.216433	−	0.976297i	$-0.430558\pi$
$317$	4.37110e9			0.432866			0.216433	+	0.976297i	$0.430558\pi$
$318$		−	6.30514e9i		−	0.616575i
$319$	−2.65200e10			−2.56101
$320$			2.16230e9i			0.206213i
$321$			9.67637e9i			0.911365i
$322$	−3.36544e8	+	2.82591e9i	−0.0313054	+	0.262867i
$323$	3.65337e9			0.335647
$324$	6.12220e8			0.0555556
$325$		−	2.12620e10i		−	1.90577i
$326$	1.44171e10			1.27646
$327$			1.07342e10i			0.938812i
$328$			3.78783e9i			0.327262i
$329$	2.18271e9	+	2.59944e8i	0.186300	+	0.0221869i
$330$	1.33947e10			1.12948
$331$	2.16460e10			1.80329			0.901644	−	0.432478i	$-0.142361\pi$
$331$	2.16460e10			1.80329			0.901644	+	0.432478i	$0.142361\pi$
$332$		−	4.61038e9i		−	0.379476i
$333$	−5.27983e9			−0.429382
$334$		−	1.59112e10i		−	1.27855i
$335$		−	5.33788e9i		−	0.423828i
$336$	−1.82675e9	−	2.17551e8i	−0.143325	−	0.0170689i
$337$	−7.19440e9			−0.557795			−0.278898	−	0.960321i	$-0.589969\pi$
$337$	−7.19440e9			−0.557795			−0.278898	+	0.960321i	$0.589969\pi$
$338$	2.08121e9			0.159459
$339$		−	9.52302e9i		−	0.721068i
$340$	9.38304e9			0.702147
$341$			1.40923e10i			1.04223i
$342$			1.27145e9i			0.0929384i
$343$	4.81891e9	−	1.29753e10i	0.348155	−	0.937437i
$344$	−8.90260e8			−0.0635745
$345$	5.05161e9			0.356577
$346$		−	9.64414e9i		−	0.672914i
$347$	4.72996e9			0.326242			0.163121	−	0.986606i	$-0.447844\pi$
$347$	4.72996e9			0.326242			0.163121	+	0.986606i	$0.447844\pi$
$348$		−	6.46534e9i		−	0.440833i
$349$			1.38112e9i			0.0930954i	0.998916	+	0.0465477i	$0.0148220\pi$
$349$			1.38112e9i			0.0930954i	−0.998916	+	0.0465477i	$0.985178\pi$
$350$	−2.16021e9	+	1.81389e10i	−0.143954	+	1.20876i
$351$	3.23374e9			0.213048
$352$	−4.55137e9			−0.296464
$353$			1.56401e10i			1.00726i	0.863920	+	0.503630i	$0.168002\pi$
$353$			1.56401e10i			1.00726i	−0.863920	+	0.503630i	$0.831998\pi$
$354$	−5.45458e9			−0.347335
$355$			2.08774e10i			1.31451i
$356$		−	1.18903e10i		−	0.740277i
$357$	−9.44038e8	+	7.92695e9i	−0.0581188	+	0.488015i
$358$	1.42958e10			0.870316
$359$	−9.36220e9			−0.563638			−0.281819	−	0.959468i	$-0.590938\pi$
$359$	−9.36220e9			−0.563638			−0.281819	+	0.959468i	$0.590938\pi$
$360$			3.26550e9i			0.194419i
$361$	1.43430e10			0.844524
$362$			8.72055e9i			0.507820i
$363$			1.81696e10i			1.04645i
$364$	−9.64885e9	−	1.14910e9i	−0.549630	−	0.0654567i
$365$	−3.63371e10			−2.04729
$366$	−4.55400e9			−0.253787
$367$		−	1.46661e10i		−	0.808443i	−0.914661	−	0.404221i	$-0.867543\pi$
$367$		−	1.46661e10i		−	0.808443i	0.914661	−	0.404221i	$-0.132457\pi$
$368$	−1.71648e9			−0.0935941
$369$			5.72038e9i			0.308546i
$370$		−	2.81619e10i		−	1.50264i
$371$	−3.38363e9	+	2.84118e10i	−0.178602	+	1.49970i
$372$	−3.43558e9			−0.179402
$373$	1.23977e10			0.640483			0.320241	−	0.947336i	$-0.396236\pi$
$373$	1.23977e10			0.640483			0.320241	+	0.947336i	$0.396236\pi$
$374$			1.97501e10i			1.00945i
$375$	1.35900e10			0.687220
$376$			1.32580e9i			0.0663324i
$377$		−	3.41498e10i		−	1.69053i
$378$	−2.75875e9	−	3.28546e8i	−0.135128	−	0.0160927i
$379$	−2.54231e10			−1.23217			−0.616087	−	0.787678i	$-0.711283\pi$
$379$	−2.54231e10			−1.23217			−0.616087	+	0.787678i	$0.711283\pi$
$380$	−6.78175e9			−0.325243
$381$		−	2.09502e9i		−	0.0994232i
$382$	1.38859e10			0.652111
$383$		−	2.27212e10i		−	1.05593i	−0.849265	−	0.527966i	$-0.822955\pi$
$383$		−	2.27212e10i		−	1.05593i	0.849265	−	0.527966i	$-0.177045\pi$
$384$		−	1.10958e9i		−	0.0510310i
$385$	−6.03584e10	−	7.18822e9i	−2.74723	−	0.327174i
$386$	−1.28304e10			−0.577949
$387$	−1.34447e9			−0.0599387
$388$			1.50013e10i			0.661913i
$389$	−3.46448e10			−1.51300			−0.756501	−	0.653992i	$-0.773093\pi$
$389$	−3.46448e10			−1.51300			−0.756501	+	0.653992i	$0.773093\pi$
$390$			1.72483e10i			0.745571i
$391$			7.44848e9i			0.318684i
$392$	8.11483e9	+	1.96063e9i	0.343665	+	0.0830333i
$393$	2.47535e9			0.103769
$394$	−1.42441e10			−0.591084
$395$		−	4.91759e10i		−	2.02006i
$396$	−6.87347e9			−0.279509
$397$			8.03155e9i			0.323324i	0.986846	+	0.161662i	$0.0516854\pi$
$397$			8.03155e9i			0.323324i	−0.986846	+	0.161662i	$0.948315\pi$
$398$		−	2.53080e10i		−	1.00861i
$399$	6.82320e8	−	5.72933e9i	0.0269213	−	0.226054i
$400$	−1.10177e10			−0.430381
$401$	2.09591e10			0.810580			0.405290	−	0.914188i	$-0.367170\pi$
$401$	2.09591e10			0.810580			0.405290	+	0.914188i	$0.367170\pi$
$402$			2.73913e9i			0.104884i
$403$	−1.81467e10			−0.687982
$404$			2.63218e10i			0.988074i
$405$			4.93155e9i			0.183300i
$406$	−3.46960e9	+	2.91337e10i	−0.127695	+	1.07224i
$407$	5.92774e10			2.16029
$408$	−4.81490e9			−0.173759
$409$		−	4.09668e10i		−	1.46399i	−0.681309	−	0.731996i	$-0.738589\pi$
$409$		−	4.09668e10i		−	1.46399i	0.681309	−	0.731996i	$-0.261411\pi$
$410$	−3.05117e10			−1.07977
$411$			2.13690e10i			0.748888i
$412$		−	6.07598e8i		−	0.0210876i
$413$	2.45791e10	+	2.92718e9i	0.844824	+	0.100612i
$414$	−2.59223e9			−0.0882414
$415$	3.71375e10			1.25205
$416$		−	5.86080e9i		−	0.195697i
$417$	−1.79419e10			−0.593369
$418$		−	1.42747e10i		−	0.467588i
$419$			3.22061e10i			1.04492i	0.852665	+	0.522458i	$0.174985\pi$
$419$			3.22061e10i			1.04492i	−0.852665	+	0.522458i	$0.825015\pi$
$420$	1.75242e9	−	1.47148e10i	0.0563172	−	0.472887i
$421$	1.01648e10			0.323572			0.161786	−	0.986826i	$-0.448275\pi$
$421$	1.01648e10			0.323572			0.161786	+	0.986826i	$0.448275\pi$
$422$	−6.21005e9			−0.195815
$423$			2.00222e9i			0.0625388i
$424$	−1.72576e10			−0.533970
$425$			4.78102e10i			1.46543i
$426$		−	1.07132e10i		−	0.325298i
$427$	2.05210e10	+	2.44389e9i	0.617286	+	0.0735140i
$428$	2.64849e10			0.789265
$429$	−3.63056e10			−1.07188
$430$		−	7.17122e9i		−	0.209758i
$431$	2.41142e10			0.698819			0.349409	−	0.936970i	$-0.386382\pi$
$431$	2.41142e10			0.698819			0.349409	+	0.936970i	$0.386382\pi$
$432$		−	1.67569e9i		−	0.0481125i
$433$			1.83741e9i			0.0522702i	0.999658	+	0.0261351i	$0.00832000\pi$
$433$			1.83741e9i			0.0522702i	−0.999658	+	0.0261351i	$0.991680\pi$
$434$	1.54812e10	+	1.84369e9i	0.436361	+	0.0519672i
$435$	5.20795e10			1.45449
$436$	2.93802e10			0.813035
$437$		−	5.38351e9i		−	0.147618i
$438$	1.86464e10			0.506638
$439$			7.04819e10i			1.89766i	0.315780	+	0.948832i	$0.397734\pi$
$439$			7.04819e10i			1.89766i	−0.315780	+	0.948832i	$0.602266\pi$
$440$		−	3.66622e10i		−	0.978155i
$441$	1.22550e10	+	2.96095e9i	0.324010	+	0.0782845i
$442$	−2.54322e10			−0.666339
$443$	−2.54470e10			−0.660726			−0.330363	−	0.943854i	$-0.607171\pi$
$443$	−2.54470e10			−0.660726			−0.330363	+	0.943854i	$0.607171\pi$
$444$			1.44513e10i			0.371855i
$445$	9.57790e10			2.44248
$446$		−	7.96058e9i		−	0.201189i
$447$			3.99246e9i			0.100002i
$448$	−5.95453e8	+	4.99993e9i	−0.0147821	+	0.124123i
$449$	1.14378e10			0.281421			0.140711	−	0.990051i	$-0.455061\pi$
$449$	1.14378e10			0.281421			0.140711	+	0.990051i	$0.455061\pi$
$450$	−1.66390e10			−0.405767
$451$		−	6.42234e10i		−	1.55234i
$452$	−2.60652e10			−0.624463
$453$		−	4.10939e10i		−	0.975854i
$454$			6.52091e9i			0.153492i
$455$	9.25626e9	−	7.77234e10i	0.215968	−	1.81345i
$456$	3.48005e9			0.0804870
$457$	−8.74536e8			−0.0200499			−0.0100250	−	0.999950i	$-0.503191\pi$
$457$	−8.74536e8			−0.0200499			−0.0100250	+	0.999950i	$0.503191\pi$
$458$			5.00426e9i			0.113731i
$459$	−7.27145e9			−0.163821
$460$		−	1.38266e10i		−	0.308805i
$461$			5.02332e10i			1.11221i	0.831111	+	0.556106i	$0.187705\pi$
$461$			5.02332e10i			1.11221i	−0.831111	+	0.556106i	$0.812295\pi$
$462$	3.09728e10	+	3.68863e9i	0.679850	+	0.0809649i
$463$	−4.52534e9			−0.0984753			−0.0492376	−	0.998787i	$-0.515679\pi$
$463$	−4.52534e9			−0.0984753			−0.0492376	+	0.998787i	$0.515679\pi$
$464$	−1.76961e10			−0.381773
$465$		−	2.76743e10i		−	0.591922i
$466$	−4.12820e10			−0.875421
$467$			8.11249e9i			0.170564i	0.996357	+	0.0852819i	$0.0271791\pi$
$467$			8.11249e9i			0.170564i	−0.996357	+	0.0852819i	$0.972821\pi$
$468$		−	8.85096e9i		−	0.184505i
$469$	1.46994e9	−	1.23429e10i	0.0303815	−	0.255109i
$470$	−1.06796e10			−0.218858
$471$	4.29434e9			0.0872595
$472$			1.49296e10i			0.300801i
$473$	1.50945e10			0.301561
$474$			2.52346e10i			0.499899i
$475$		−	3.45556e10i		−	0.678804i
$476$	2.16966e10	+	2.58390e9i	0.422633	+	0.0503324i
$477$	−2.60624e10			−0.503432
$478$	1.45237e10			0.278205
$479$			1.58429e10i			0.300948i	0.988614	+	0.150474i	$0.0480800\pi$
$479$			1.58429e10i			0.300948i	−0.988614	+	0.150474i	$0.951920\pi$
$480$	8.93790e9			0.168372
$481$			7.63314e10i			1.42601i
$482$		−	2.48293e9i		−	0.0460019i
$483$	1.16809e10	+	1.39111e9i	0.214630	+	0.0255607i
$484$	4.97315e10			0.906254
$485$	−1.20838e11			−2.18392
$486$		−	2.53062e9i		−	0.0453609i
$487$	7.07446e10			1.25770			0.628851	−	0.777526i	$-0.283526\pi$
$487$	7.07446e10			1.25770			0.628851	+	0.777526i	$0.283526\pi$
$488$			1.24646e10i			0.219786i
$489$		−	5.95934e10i		−	1.04223i
$490$	−1.57933e10	+	6.53665e10i	−0.273961	+	1.13389i
$491$	1.62645e10			0.279843			0.139921	−	0.990163i	$-0.455315\pi$
$491$	1.62645e10			0.279843			0.139921	+	0.990163i	$0.455315\pi$
$492$	1.56571e10			0.267208
$493$			7.67899e10i			1.29992i
$494$	1.83816e10			0.308656
$495$		−	5.53672e10i		−	0.922214i
$496$			9.40341e9i			0.155367i
$497$	−5.74920e9	+	4.82752e10i	−0.0942285	+	0.791222i
$498$	−1.90571e10			−0.309841
$499$	4.26709e10			0.688224			0.344112	−	0.938929i	$-0.388180\pi$
$499$	4.26709e10			0.688224			0.344112	+	0.938929i	$0.388180\pi$
$500$		−	3.71969e10i		−	0.595150i
$501$	−6.57691e10			−1.04393
$502$		−	1.55194e10i		−	0.244377i
$503$			2.40376e10i			0.375507i	0.982216	+	0.187754i	$0.0601207\pi$
$503$			2.40376e10i			0.375507i	−0.982216	+	0.187754i	$0.939879\pi$
$504$	−8.99252e8	+	7.55088e9i	−0.0139367	+	0.117024i
$505$	−2.12027e11			−3.26006
$506$	2.91033e10			0.443956
$507$		−	8.60273e9i		−	0.130198i
$508$	−5.73421e9			−0.0861030
$509$			1.90476e10i			0.283772i	0.989883	+	0.141886i	$0.0453166\pi$
$509$			1.90476e10i			0.283772i	−0.989883	+	0.141886i	$0.954683\pi$
$510$		−	3.87849e10i		−	0.573300i
$511$	−8.40231e10	−	1.00065e10i	−1.23230	−	0.146757i
$512$	−3.03700e9			−0.0441942
$513$	5.25556e9			0.0758839
$514$			1.08088e10i			0.154855i
$515$	4.89432e9			0.0695767
$516$			3.67990e9i			0.0519084i
$517$		−	2.24791e10i		−	0.314643i
$518$	7.75522e9	−	6.51194e10i	0.107715	−	0.904464i
$519$	−3.98642e10			−0.549432
$520$	4.72099e10			0.645683
$521$		−	7.19207e10i		−	0.976119i	−0.872810	−	0.488060i	$-0.837705\pi$
$521$		−	7.19207e10i		−	0.976119i	0.872810	−	0.488060i	$-0.162295\pi$
$522$	−2.67246e10			−0.359939
$523$		−	2.64873e10i		−	0.354023i	−0.984209	−	0.177011i	$-0.943357\pi$
$523$		−	2.64873e10i		−	0.354023i	0.984209	−	0.177011i	$-0.0566430\pi$
$524$		−	6.77521e9i		−	0.0898665i
$525$	7.49776e10	+	8.92925e9i	0.986948	+	0.117538i
$526$	−3.38112e10			−0.441690
$527$	4.08050e10			0.529018
$528$			1.88132e10i			0.242062i
$529$	−6.73351e10			−0.859842
$530$		−	1.39013e11i		−	1.76178i
$531$			2.25466e10i			0.283598i
$532$	−1.56816e10	−	1.86756e9i	−0.195769	−	0.0233145i
$533$	8.27004e10			1.02471
$534$	−4.91489e10			−0.604434
$535$			2.13341e11i			2.60411i
$536$	7.49717e9			0.0908319
$537$		−	5.90919e10i		−	0.710610i
$538$			8.71086e10i			1.03976i
$539$	−1.37588e11	−	3.32429e10i	−1.63015	−	0.393862i
$540$	1.34980e10			0.158743
$541$	4.62837e10			0.540305			0.270152	−	0.962818i	$-0.412926\pi$
$541$	4.62837e10			0.540305			0.270152	+	0.962818i	$0.412926\pi$
$542$			5.57665e10i			0.646214i
$543$	3.60465e10			0.414633
$544$			1.31787e10i			0.150479i
$545$			2.36663e11i			2.68253i
$546$	−4.74984e9	+	3.98837e10i	−0.0534451	+	0.448771i
$547$	−5.51464e10			−0.615982			−0.307991	−	0.951389i	$-0.599657\pi$
$547$	−5.51464e10			−0.615982			−0.307991	+	0.951389i	$0.599657\pi$
$548$	5.84884e10			0.648556
$549$			1.88240e10i			0.207216i
$550$	1.86808e11			2.04148
$551$		−	5.55012e10i		−	0.602138i
$552$			7.09511e9i			0.0764193i
$553$	1.35420e10	−	1.13710e11i	0.144805	−	1.21591i
$554$	−1.88338e10			−0.199940
$555$	−1.16408e11			−1.22690
$556$			4.91083e10i			0.513872i
$557$	−8.27853e10			−0.860067			−0.430033	−	0.902813i	$-0.641498\pi$
$557$	−8.27853e10			−0.860067			−0.430033	+	0.902813i	$0.641498\pi$
$558$			1.42010e10i			0.146481i
$559$			1.94372e10i			0.199061i
$560$	−4.02754e10	−	4.79649e9i	−0.409532	−	0.0487721i
$561$	8.16375e10			0.824211
$562$	8.50153e10			0.852220
$563$		−	1.45386e11i		−	1.44707i	−0.690289	−	0.723534i	$-0.742517\pi$
$563$		−	1.45386e11i		−	1.44707i	0.690289	−	0.723534i	$-0.257483\pi$
$564$	5.48020e9			0.0541602
$565$		−	2.09960e11i		−	2.06036i
$566$		−	4.75526e10i		−	0.463349i
$567$	−1.35805e9	+	1.14033e10i	−0.0131396	+	0.110331i
$568$	−2.93228e10			−0.281716
$569$	−5.20268e10			−0.496338			−0.248169	−	0.968717i	$-0.579829\pi$
$569$	−5.20268e10			−0.496338			−0.248169	+	0.968717i	$0.579829\pi$
$570$			2.80325e10i			0.265559i
$571$	−8.81801e10			−0.829518			−0.414759	−	0.909931i	$-0.636134\pi$
$571$	−8.81801e10			−0.829518			−0.414759	+	0.909931i	$0.636134\pi$
$572$			9.93709e10i			0.928272i
$573$		−	5.73977e10i		−	0.532446i
$574$	−7.05529e10	−	8.40231e9i	−0.649931	−	0.0774018i
$575$	7.04519e10			0.644498
$576$	−4.58647e9			−0.0416667
$577$		−	1.65148e10i		−	0.148994i	−0.997221	−	0.0744971i	$-0.976265\pi$
$577$		−	1.65148e10i		−	0.148994i	0.997221	−	0.0744971i	$-0.0237352\pi$
$578$	−2.17343e10			−0.194731
$579$			5.30344e10i			0.471893i
$580$		−	1.42545e11i		−	1.25962i
$581$	8.58738e10	+	1.02269e10i	0.753627	+	0.0897511i
$582$	6.20079e10			0.540450
$583$	2.92606e11			2.53284
$584$		−	5.10363e10i		−	0.438761i
$585$	7.12962e10			0.608756
$586$		−	7.87367e10i		−	0.667708i
$587$		−	2.17973e10i		−	0.183590i	−0.995778	−	0.0917951i	$-0.970740\pi$
$587$		−	2.17973e10i		−	0.183590i	0.995778	−	0.0917951i	$-0.0292605\pi$
$588$	8.10431e9	−	3.35428e10i	0.0677964	−	0.280601i
$589$	−2.94925e10			−0.245047
$590$	−1.20261e11			−0.992465
$591$			5.88780e10i			0.482618i
$592$	3.95541e10			0.322036
$593$		−	1.22622e11i		−	0.991630i	−0.868428	−	0.495815i	$-0.834869\pi$
$593$		−	1.22622e11i		−	0.991630i	0.868428	−	0.495815i	$-0.165131\pi$
$594$			2.84116e10i			0.228218i
$595$	−2.08138e10	+	1.74770e11i	−0.166067	+	1.39444i
$596$	1.09276e10			0.0866047
$597$	−1.04611e11			−0.823530
$598$			3.74763e10i			0.293057i
$599$	−1.47279e11			−1.14402			−0.572012	−	0.820245i	$-0.693837\pi$
$599$	−1.47279e11			−1.14402			−0.572012	+	0.820245i	$0.693837\pi$
$600$			4.55420e10i			0.351404i
$601$			5.16095e10i			0.395578i	0.980245	+	0.197789i	$0.0633760\pi$
$601$			5.16095e10i			0.395578i	−0.980245	+	0.197789i	$0.936624\pi$
$602$	1.97481e9	−	1.65822e10i	0.0150362	−	0.126257i
$603$	1.13222e10			0.0856371
$604$	−1.12477e11			−0.845115
$605$			4.00597e11i			2.99010i
$606$	1.08801e11			0.806759
$607$		−	1.05721e11i		−	0.778763i	−0.921077	−	0.389381i	$-0.872689\pi$
$607$		−	1.05721e11i		−	0.778763i	0.921077	−	0.389381i	$-0.127311\pi$
$608$		−	9.52513e9i		−	0.0697038i
$609$	1.20425e11	+	1.43416e10i	0.875479	+	0.104263i
$610$	−1.00405e11			−0.725163
$611$	2.89464e10			0.207696
$612$			1.99024e10i			0.141873i
$613$	5.21843e10			0.369571			0.184786	−	0.982779i	$-0.440841\pi$
$613$	5.21843e10			0.369571			0.184786	+	0.982779i	$0.440841\pi$
$614$		−	5.60474e10i		−	0.394350i
$615$			1.26121e11i			0.881629i
$616$	1.00960e10	−	8.47748e10i	0.0701176	−	0.588767i
$617$	2.55103e11			1.76025			0.880126	−	0.474741i	$-0.157458\pi$
$617$	2.55103e11			1.76025			0.880126	+	0.474741i	$0.157458\pi$
$618$	−2.51152e9			−0.0172180
$619$		−	2.88087e10i		−	0.196228i	−0.995175	−	0.0981140i	$-0.968719\pi$
$619$		−	2.88087e10i		−	0.196228i	0.995175	−	0.0981140i	$-0.0312810\pi$
$620$	−7.57463e10			−0.512619
$621$			1.07150e10i			0.0720488i
$622$			1.47340e11i			0.984373i
$623$	2.21472e11	+	2.63756e10i	1.47016	+	0.175085i
$624$	−2.42257e10			−0.159786
$625$	3.69444e10			0.242119
$626$			1.62320e10i			0.105700i
$627$	−5.90049e10			−0.381784
$628$		−	1.17539e10i		−	0.0755690i
$629$		−	1.71640e11i		−	1.09652i
$630$	−6.08238e10	−	7.24365e9i	−0.386110	−	0.0459828i
$631$	5.63711e10			0.355581			0.177791	−	0.984068i	$-0.443105\pi$
$631$	5.63711e10			0.355581			0.177791	+	0.984068i	$0.443105\pi$
$632$	6.90687e10			0.432925
$633$			2.56693e10i			0.159882i
$634$	−4.94534e10			−0.306083
$635$		−	4.61901e10i		−	0.284089i
$636$			7.13345e10i			0.435985i
$637$	4.28069e10	−	1.77172e11i	0.259989	−	1.07606i
$638$	3.00040e11			1.81091
$639$	−4.42832e10			−0.265604
$640$		−	2.44636e10i		−	0.145815i
$641$	9.61983e10			0.569817			0.284908	−	0.958555i	$-0.408037\pi$
$641$	9.61983e10			0.569817			0.284908	+	0.958555i	$0.408037\pi$
$642$		−	1.09476e11i		−	0.644432i
$643$		−	1.69382e11i		−	0.990887i	−0.868640	−	0.495443i	$-0.835006\pi$
$643$		−	1.69382e11i		−	0.990887i	0.868640	−	0.495443i	$-0.164994\pi$
$644$	3.80757e9	−	3.19716e10i	0.0221362	−	0.185875i
$645$	−2.96423e10			−0.171267
$646$	−4.13331e10			−0.237339
$647$			2.07631e11i			1.18488i	0.805614	+	0.592441i	$0.201836\pi$
$647$			2.07631e11i			1.18488i	−0.805614	+	0.592441i	$0.798164\pi$
$648$	−6.92648e9			−0.0392837
$649$		−	2.53134e11i		−	1.42683i
$650$			2.40552e11i			1.34759i
$651$	7.62092e9	−	6.39917e10i	0.0424310	−	0.356287i
$652$	−1.63111e11			−0.902596
$653$	−1.44044e11			−0.792212			−0.396106	−	0.918205i	$-0.629639\pi$
$653$	−1.44044e11			−0.792212			−0.396106	+	0.918205i	$0.629639\pi$
$654$		−	1.21444e11i		−	0.663840i
$655$	5.45757e10			0.296506
$656$		−	4.28544e10i		−	0.231409i
$657$		−	7.70750e10i		−	0.413668i
$658$	−2.46945e10	−	2.94093e9i	−0.131734	−	0.0156885i
$659$	−6.70039e10			−0.355270			−0.177635	−	0.984096i	$-0.556845\pi$
$659$	−6.70039e10			−0.355270			−0.177635	+	0.984096i	$0.556845\pi$
$660$	−1.51544e11			−0.798661
$661$		−	2.35273e11i		−	1.23244i	−0.787574	−	0.616220i	$-0.788663\pi$
$661$		−	2.35273e11i		−	1.23244i	0.787574	−	0.616220i	$-0.211337\pi$
$662$	−2.44896e11			−1.27512
$663$			1.05124e11i			0.544064i
$664$			5.21605e10i			0.268330i
$665$	1.50435e10	−	1.26318e11i	0.0769242	−	0.645921i
$666$	5.97345e10			0.303619
$667$	1.13156e11			0.571707
$668$			1.80015e11i			0.904069i
$669$	−3.29052e10			−0.164270
$670$			6.03912e10i			0.299691i
$671$		−	2.11340e11i		−	1.04254i
$672$	2.06673e10	+	2.46131e9i	0.101346	+	0.0120695i
$673$	1.68943e11			0.823533			0.411767	−	0.911289i	$-0.364912\pi$
$673$	1.68943e11			0.823533			0.411767	+	0.911289i	$0.364912\pi$
$674$	8.13953e10			0.394421
$675$			6.87774e10i			0.331307i
$676$	−2.35463e10			−0.112755
$677$			1.61601e11i			0.769288i	0.923065	+	0.384644i	$0.125676\pi$
$677$			1.61601e11i			0.769288i	−0.923065	+	0.384644i	$0.874324\pi$
$678$			1.07741e11i			0.509872i
$679$	−2.79416e11	−	3.32763e10i	−1.31454	−	0.156551i
$680$	−1.06157e11			−0.496493
$681$	2.69543e10			0.125325
$682$		−	1.59437e11i		−	0.736971i
$683$	−1.21402e11			−0.557884			−0.278942	−	0.960308i	$-0.589984\pi$
$683$	−1.21402e11			−0.557884			−0.278942	+	0.960308i	$0.589984\pi$
$684$		−	1.43848e10i		−	0.0657174i
$685$			4.71136e11i			2.13985i
$686$	−5.45197e10	+	1.46799e11i	−0.246183	+	0.662868i
$687$	2.06852e10			0.0928607
$688$	1.00721e10			0.0449540
$689$			3.76788e11i			1.67194i
$690$	−5.71525e10			−0.252138
$691$			1.08343e11i			0.475214i	0.971361	+	0.237607i	$0.0763631\pi$
$691$			1.08343e11i			0.475214i	−0.971361	+	0.237607i	$0.923637\pi$
$692$			1.09111e11i			0.475822i
$693$	1.52470e10	−	1.28027e11i	0.0661076	−	0.555095i
$694$	−5.35134e10			−0.230688
$695$	−3.95577e11			−1.69547
$696$			7.31469e10i			0.311716i
$697$	−1.85962e11			−0.787939
$698$		−	1.56255e10i		−	0.0658284i
$699$			1.70640e11i			0.714778i
$700$	2.44400e10	−	2.05219e11i	0.101791	−	0.854722i
$701$	1.43665e11			0.594946			0.297473	−	0.954730i	$-0.403856\pi$
$701$	1.43665e11			0.594946			0.297473	+	0.954730i	$0.403856\pi$
$702$	−3.65856e10			−0.150647
$703$			1.24056e11i			0.507921i
$704$	5.14929e10			0.209632
$705$			4.41441e10i			0.178696i
$706$		−	1.76948e11i		−	0.712240i
$707$	−4.90274e11	−	5.83879e10i	−1.96228	−	0.233693i
$708$	6.17116e10			0.245603
$709$	1.63535e11			0.647179			0.323590	−	0.946198i	$-0.395110\pi$
$709$	1.63535e11			0.647179			0.323590	+	0.946198i	$0.395110\pi$
$710$		−	2.36201e11i		−	0.929496i
$711$	1.04307e11			0.408166
$712$			1.34524e11i			0.523455i
$713$		−	6.01292e10i		−	0.232663i
$714$	1.06806e10	−	8.96832e10i	0.0410962	−	0.345079i
$715$	−8.00452e11			−3.06275
$716$	−1.61739e11			−0.615406
$717$		−	6.00337e10i		−	0.227153i
$718$	1.05921e11			0.398552
$719$		−	2.93936e11i		−	1.09986i	−0.835211	−	0.549930i	$-0.814654\pi$
$719$		−	2.93936e11i		−	1.09986i	0.835211	−	0.549930i	$-0.185346\pi$
$720$		−	3.69449e10i		−	0.137475i
$721$	1.13172e10	+	1.34780e9i	0.0418793	+	0.00498750i
$722$	−1.62273e11			−0.597169
$723$	−1.02632e10			−0.0375604
$724$		−	9.86618e10i		−	0.359083i
$725$	7.26323e11			2.62892
$726$		−	2.05566e11i		−	0.739953i
$727$		−	3.49590e11i		−	1.25147i	−0.780035	−	0.625736i	$-0.784799\pi$
$727$		−	3.49590e11i		−	1.25147i	0.780035	−	0.625736i	$-0.215201\pi$
$728$	1.09164e11	+	1.30006e10i	0.388647	+	0.0462848i
$729$	−1.04604e10			−0.0370370
$730$	4.11108e11			1.44765
$731$		−	4.37069e10i		−	0.153067i
$732$	5.15227e10			0.179454
$733$			4.83713e11i			1.67561i	0.545972	+	0.837803i	$0.316161\pi$
$733$			4.83713e11i			1.67561i	−0.545972	+	0.837803i	$0.683839\pi$
$734$			1.65928e11i			0.571655i
$735$	2.70193e11	+	6.52818e10i	0.925817	+	0.223688i
$736$	1.94198e10			0.0661810
$737$	−1.27116e11			−0.430854
$738$		−	6.47187e10i		−	0.218175i
$739$	−4.98515e10			−0.167148			−0.0835738	−	0.996502i	$-0.526633\pi$
$739$	−4.98515e10			−0.167148			−0.0835738	+	0.996502i	$0.526633\pi$
$740$			3.18616e11i			1.06253i
$741$		−	7.59805e10i		−	0.252017i
$742$	3.82814e10	−	3.21443e11i	0.126291	−	1.06045i
$743$	−1.83401e10			−0.0601791			−0.0300896	−	0.999547i	$-0.509579\pi$
$743$	−1.83401e10			−0.0601791			−0.0300896	+	0.999547i	$0.509579\pi$
$744$	3.88691e10			0.126857
$745$			8.80243e10i			0.285744i
$746$	−1.40264e11			−0.452890
$747$			7.87727e10i			0.252984i
$748$		−	2.23447e11i		−	0.713787i
$749$	−5.87497e10	+	4.93312e11i	−0.186672	+	1.56745i
$750$	−1.53754e11			−0.485938
$751$	5.68895e11			1.78843			0.894216	−	0.447637i	$-0.147734\pi$
$751$	5.68895e11			1.78843			0.894216	+	0.447637i	$0.147734\pi$
$752$		−	1.49997e10i		−	0.0469041i
$753$	−6.41497e10			−0.199533
$754$			3.86361e11i			1.19539i
$755$		−	9.06023e11i		−	2.78838i
$756$	3.12117e10	+	3.71707e9i	0.0955498	+	0.0113792i
$757$	−2.94565e11			−0.897012			−0.448506	−	0.893780i	$-0.648044\pi$
$757$	−2.94565e11			−0.897012			−0.448506	+	0.893780i	$0.648044\pi$
$758$	2.87630e11			0.871279
$759$		−	1.20299e11i		−	0.362489i
$760$	7.67267e10			0.229981
$761$		−	2.31247e11i		−	0.689505i	−0.938694	−	0.344752i	$-0.887963\pi$
$761$		−	2.31247e11i		−	0.689505i	0.938694	−	0.344752i	$-0.112037\pi$
$762$			2.37024e10i			0.0703028i
$763$	−6.51723e10	+	5.47242e11i	−0.192294	+	1.61466i
$764$	−1.57101e11			−0.461112
$765$	−1.60318e11			−0.468098
$766$			2.57061e11i			0.746657i
$767$	3.25960e11			0.941852
$768$			1.25535e10i			0.0360844i
$769$		−	1.70996e11i		−	0.488967i	−0.969653	−	0.244484i	$-0.921382\pi$
$769$		−	1.70996e11i		−	0.488967i	0.969653	−	0.244484i	$-0.0786184\pi$
$770$	6.82877e11	+	8.13254e10i	1.94258	+	0.231347i
$771$	4.46782e10			0.126438
$772$	1.45159e11			0.408671
$773$		−	1.53425e11i		−	0.429714i	−0.976646	−	0.214857i	$-0.931072\pi$
$773$		−	1.53425e11i		−	0.429714i	0.976646	−	0.214857i	$-0.0689285\pi$
$774$	1.52109e10			0.0423830
$775$		−	3.85957e11i		−	1.06987i
$776$		−	1.69720e11i		−	0.468043i
$777$	−2.69172e11	−	3.20563e10i	−0.738492	−	0.0879487i
$778$	3.91961e11			1.06985
$779$	1.34407e11			0.364982
$780$		−	1.95143e11i		−	0.527198i
$781$	4.97173e11			1.33630
$782$		−	8.42699e10i		−	0.225344i
$783$			1.10466e11i			0.293889i
$784$	−9.18088e10	−	2.21820e10i	−0.243008	−	0.0587134i
$785$	9.46800e10			0.249333
$786$	−2.80054e10			−0.0733757
$787$			7.16437e11i			1.86758i	0.357821	+	0.933790i	$0.383520\pi$
$787$			7.16437e11i			1.86758i	−0.357821	+	0.933790i	$0.616480\pi$
$788$	1.61153e11			0.417959
$789$			1.39759e11i			0.360639i
$790$			5.56362e11i			1.42840i
$791$	5.78187e10	−	4.85495e11i	0.147694	−	1.24016i
$792$	7.77645e10			0.197643
$793$	2.72142e11			0.688182
$794$		−	9.08667e10i		−	0.228624i
$795$	−5.74613e11			−1.43849
$796$			2.86327e11i			0.713198i
$797$		−	7.19098e11i		−	1.78219i	−0.453814	−	0.891096i	$-0.649937\pi$
$797$		−	7.19098e11i		−	1.78219i	0.453814	−	0.891096i	$-0.350063\pi$
$798$	−7.71956e9	+	6.48200e10i	−0.0190362	+	0.159844i
$799$	−6.50893e10			−0.159706
$800$	1.24652e11			0.304325
$801$			2.03158e11i			0.493518i
$802$	−2.37126e11			−0.573167
$803$			8.65331e11i			2.08123i
$804$		−	3.09897e10i		−	0.0741639i
$805$	2.57537e11	+	3.06707e10i	0.613276	+	0.0730365i
$806$	2.05306e11			0.486477
$807$	3.60065e11			0.848958
$808$		−	2.97797e11i		−	0.698674i
$809$	−5.00346e11			−1.16809			−0.584045	−	0.811721i	$-0.698531\pi$
$809$	−5.00346e11			−1.16809			−0.584045	+	0.811721i	$0.698531\pi$
$810$		−	5.57941e10i		−	0.129613i
$811$			7.74013e11i			1.78922i	0.446843	+	0.894612i	$0.352548\pi$
$811$			7.74013e11i			1.78922i	−0.446843	+	0.894612i	$0.647452\pi$
$812$	3.92540e10	−	3.29610e11i	0.0902943	−	0.758187i
$813$	2.30512e11			0.527632
$814$	−6.70647e11			−1.52755
$815$		−	1.31389e12i		−	2.97803i
$816$	5.44743e10			0.122866
$817$			3.15899e10i			0.0709022i
$818$			4.63487e11i			1.03520i
$819$	1.64860e11	+	1.96335e10i	0.366420	+	0.0436378i
$820$	3.45201e11			0.763513
$821$	−3.31259e11			−0.729114			−0.364557	−	0.931181i	$-0.618780\pi$
$821$	−3.31259e11			−0.729114			−0.364557	+	0.931181i	$0.618780\pi$
$822$		−	2.41763e11i		−	0.529544i
$823$	3.37179e11			0.734955			0.367477	−	0.930032i	$-0.380222\pi$
$823$	3.37179e11			0.734955			0.367477	+	0.930032i	$0.380222\pi$
$824$			6.87419e9i			0.0149112i
$825$		−	7.72173e11i		−	1.66686i
$826$	−2.78081e11	−	3.31173e10i	−0.597380	−	0.0711434i
$827$	−1.45029e11			−0.310050			−0.155025	−	0.987911i	$-0.549546\pi$
$827$	−1.45029e11			−0.310050			−0.155025	+	0.987911i	$0.549546\pi$
$828$	2.93277e10			0.0623961
$829$		−	1.35201e11i		−	0.286260i	−0.989704	−	0.143130i	$-0.954283\pi$
$829$		−	1.35201e11i		−	0.286260i	0.989704	−	0.143130i	$-0.0457167\pi$
$830$	−4.20163e11			−0.885331
$831$			7.78500e10i			0.163250i
$832$			6.63074e10i			0.138378i
$833$	−9.62563e10	+	3.98393e11i	−0.199917	+	0.827431i
$834$	2.02990e11			0.419575
$835$	−1.45005e12			−2.98289
$836$			1.61500e11i			0.330635i
$837$	5.87001e10			0.119602
$838$		−	3.64370e11i		−	0.738868i
$839$			8.50354e11i			1.71614i	0.513535	+	0.858069i	$0.328336\pi$
$839$			8.50354e11i			1.71614i	−0.513535	+	0.858069i	$0.671664\pi$
$840$	−1.98264e10	+	1.66479e11i	−0.0398223	+	0.334381i
$841$	6.66331e11			1.33200
$842$	−1.15002e11			−0.228800
$843$		−	3.51412e11i		−	0.695835i
$844$	7.02586e10			0.138462
$845$		−	1.89670e11i		−	0.372024i
$846$		−	2.26525e10i		−	0.0442216i
$847$	−1.10316e11	+	9.26308e11i	−0.214341	+	1.79979i
$848$	1.95247e11			0.377574
$849$	−1.96559e11			−0.378323
$850$		−	5.40911e11i		−	1.03621i
$851$	−2.52925e11			−0.482251
$852$			1.21206e11i			0.230020i
$853$		−	7.38212e11i		−	1.39439i	−0.716880	−	0.697196i	$-0.754431\pi$
$853$		−	7.38212e11i		−	1.39439i	0.716880	−	0.697196i	$-0.245569\pi$
$854$	−2.32168e11	−	2.76495e10i	−0.436487	−	0.0519823i
$855$	1.15873e11			0.216828
$856$	−2.99642e11			−0.558095
$857$			3.72967e11i			0.691428i	0.938340	+	0.345714i	$0.112363\pi$
$857$			3.72967e11i			0.691428i	−0.938340	+	0.345714i	$0.887637\pi$
$858$	4.10751e11			0.757931
$859$			4.70695e11i			0.864503i	0.901753	+	0.432251i	$0.142281\pi$
$859$			4.70695e11i			0.864503i	−0.901753	+	0.432251i	$0.857719\pi$
$860$			8.11331e10i			0.148322i
$861$	−3.47311e10	+	2.91631e11i	−0.0631983	+	0.530667i
$862$	−2.72822e11			−0.494139
$863$	−2.34915e11			−0.423515			−0.211757	−	0.977322i	$-0.567919\pi$
$863$	−2.34915e11			−0.423515			−0.211757	+	0.977322i	$0.567919\pi$
$864$			1.89582e10i			0.0340207i
$865$	−8.78911e11			−1.56993
$866$		−	2.07879e10i		−	0.0369606i
$867$			8.98391e10i			0.158997i
$868$	−1.75150e11	−	2.08590e10i	−0.308554	−	0.0367463i
$869$	−1.17107e12			−2.05355
$870$	−5.89213e11			−1.02848
$871$		−	1.63687e11i		−	0.284408i
$872$	−3.32399e11			−0.574902
$873$		−	2.56311e11i		−	0.441275i
$874$			6.09075e10i			0.104382i
$875$	6.92836e11	+	8.25114e10i	1.18195	+	0.140761i
$876$	−2.10959e11			−0.358247
$877$	7.92962e11			1.34046			0.670231	−	0.742153i	$-0.266195\pi$
$877$	7.92962e11			1.34046			0.670231	+	0.742153i	$0.266195\pi$
$878$		−	7.97412e11i		−	1.34185i
$879$	−3.25459e11			−0.545181
$880$			4.14785e11i			0.691660i
$881$			3.24550e11i			0.538739i	0.963037	+	0.269369i	$0.0868152\pi$
$881$			3.24550e11i			0.538739i	−0.963037	+	0.269369i	$0.913185\pi$
$882$	−1.38649e11	−	3.34993e10i	−0.229110	−	0.0553555i
$883$	1.00865e12			1.65919			0.829597	−	0.558363i	$-0.188570\pi$
$883$	1.00865e12			1.65919			0.829597	+	0.558363i	$0.188570\pi$
$884$	2.87733e11			0.471173
$885$			4.97099e11i			0.810344i
$886$	2.87900e11			0.467204
$887$			4.45222e11i			0.719254i	0.933096	+	0.359627i	$0.117096\pi$
$887$			4.45222e11i			0.719254i	−0.933096	+	0.359627i	$0.882904\pi$
$888$		−	1.63497e11i		−	0.262941i
$889$	1.27198e10	−	1.06806e11i	0.0203645	−	0.170998i
$890$	−1.08362e12			−1.72709
$891$	1.17440e11			0.186339
$892$			9.00636e10i			0.142262i
$893$	4.70444e10			0.0739779
$894$		−	4.51695e10i		−	0.0707124i
$895$		−	1.30284e12i		−	2.03048i
$896$	6.73678e9	−	5.65677e10i	0.0104525	−	0.0877681i
$897$	1.54909e11			0.239280
$898$	−1.29404e11			−0.198995
$899$		−	6.19901e11i		−	0.949038i
$900$	1.88249e11			0.286921
$901$		−	8.47252e11i		−	1.28562i
$902$			7.26605e11i			1.09767i
$903$	−6.85426e10	−	8.16289e9i	−0.103088	−	0.0122770i
$904$	2.94894e11			0.441562
$905$	7.94740e11			1.18476
$906$			4.64925e11i			0.690033i
$907$	−4.47745e10			−0.0661609			−0.0330804	−	0.999453i	$-0.510532\pi$
$907$	−4.47745e10			−0.0661609			−0.0330804	+	0.999453i	$0.510532\pi$
$908$		−	7.37756e10i		−	0.108535i
$909$		−	4.49732e11i		−	0.658716i
$910$	−1.04723e11	+	8.79340e11i	−0.152713	+	1.28230i
$911$	1.02342e12			1.48586			0.742932	−	0.669367i	$-0.233435\pi$
$911$	1.02342e12			1.48586			0.742932	+	0.669367i	$0.233435\pi$
$912$	−3.93722e10			−0.0569129
$913$		−	8.84391e11i		−	1.27280i
$914$	9.89424e9			0.0141774
$915$			4.15025e11i			0.592093i
$916$		−	5.66167e10i		−	0.0804197i
$917$	1.26196e11	+	1.50290e10i	0.178472	+	0.0212546i
$918$	8.22670e10			0.115839
$919$	1.44541e11			0.202641			0.101321	−	0.994854i	$-0.467693\pi$
$919$	1.44541e11			0.202641			0.101321	+	0.994854i	$0.467693\pi$
$920$			1.56430e11i			0.218358i
$921$	−2.31673e11			−0.321985
$922$		−	5.68324e11i		−	0.786453i
$923$			6.40209e11i			0.882094i
$924$	−3.50418e11	−	4.17320e10i	−0.480727	−	0.0572508i
$925$	−1.62347e12			−2.21757
$926$	5.11984e10			0.0696325
$927$			1.03814e10i			0.0140584i
$928$	2.00208e11			0.269954
$929$			6.99041e11i			0.938513i	0.883062	+	0.469256i	$0.155478\pi$
$929$			6.99041e11i			0.938513i	−0.883062	+	0.469256i	$0.844522\pi$
$930$			3.13098e11i			0.418552i
$931$	6.95709e10	−	2.87945e11i	0.0926038	−	0.383276i
$932$	4.67052e11			0.619016
$933$	6.09033e11			0.803737
$934$		−	9.17824e10i		−	0.120607i
$935$	1.79991e12			2.35508
$936$			1.00137e11i			0.130464i
$937$			1.03807e12i			1.34669i	0.739328	+	0.673346i	$0.235143\pi$
$937$			1.03807e12i			1.34669i	−0.739328	+	0.673346i	$0.764857\pi$
$938$	−1.66305e10	+	1.39644e11i	−0.0214829	+	0.180389i
$939$	6.70954e10			0.0863039
$940$	1.20825e11			0.154756
$941$			3.80107e11i			0.484783i	0.970179	+	0.242391i	$0.0779318\pi$
$941$			3.80107e11i			0.484783i	−0.970179	+	0.242391i	$0.922068\pi$
$942$	−4.85849e10			−0.0617018
$943$			2.74029e11i			0.346536i
$944$		−	1.68909e11i		−	0.212698i
$945$	−2.99417e10	+	2.51416e11i	−0.0375448	+	0.315258i
$946$	−1.70775e11			−0.213236
$947$	−1.10581e11			−0.137494			−0.0687468	−	0.997634i	$-0.521900\pi$
$947$	−1.10581e11			−0.137494			−0.0687468	+	0.997634i	$0.521900\pi$
$948$		−	2.85496e11i		−	0.353482i
$949$	−1.11429e12			−1.37383
$950$			3.90952e11i			0.479987i
$951$			2.04416e11i			0.249915i
$952$	−2.45469e11	−	2.92335e10i	−0.298847	−	0.0355904i
$953$	6.60043e11			0.800204			0.400102	−	0.916471i	$-0.368975\pi$
$953$	6.60043e11			0.800204			0.400102	+	0.916471i	$0.368975\pi$
$954$	2.94862e11			0.355980
$955$		−	1.26548e12i		−	1.52140i
$956$	−1.64316e11			−0.196720
$957$		−	1.24022e12i		−	1.47860i
$958$		−	1.79242e11i		−	0.212803i
$959$	−1.29741e11	+	1.08942e12i	−0.153392	+	1.28801i
$960$	−1.01121e11			−0.119057
$961$	5.23485e11			0.613778
$962$		−	8.63591e11i		−	1.00834i
$963$	−4.52519e11			−0.526177
$964$			2.80911e10i			0.0325283i
$965$			1.16928e12i			1.34837i
$966$	−1.32155e11	−	1.57386e10i	−0.151766	−	0.0180742i
$967$	1.60957e12			1.84078			0.920392	−	0.390998i	$-0.127870\pi$
$967$	1.60957e12			1.84078			0.920392	+	0.390998i	$0.127870\pi$
$968$	−5.62647e11			−0.640818
$969$			1.70851e11i			0.193786i
$970$	1.36713e12			1.54427
$971$		−	9.87433e11i		−	1.11079i	−0.831588	−	0.555394i	$-0.812568\pi$
$971$		−	9.87433e11i		−	1.11079i	0.831588	−	0.555394i	$-0.187432\pi$
$972$			2.86307e10i			0.0320750i
$973$	−9.14700e11	−	1.08934e11i	−1.02053	−	0.121538i
$974$	−8.00384e11			−0.889330
$975$	9.94327e11			1.10030
$976$		−	1.41021e11i		−	0.155412i
$977$	9.24662e11			1.01486			0.507429	−	0.861694i	$-0.330596\pi$
$977$	9.24662e11			1.01486			0.507429	+	0.861694i	$0.330596\pi$
$978$			6.74222e11i			0.736966i
$979$		−	2.28088e12i		−	2.48297i
$980$	1.78681e11	−	7.39538e11i	0.193719	−	0.801781i
$981$	−5.01989e11			−0.542023
$982$	−1.84012e11			−0.197879
$983$		−	9.84563e11i		−	1.05446i	−0.849723	−	0.527229i	$-0.823231\pi$
$983$		−	9.84563e11i		−	1.05446i	0.849723	−	0.527229i	$-0.176769\pi$
$984$	−1.77139e11			−0.188945
$985$			1.29812e12i			1.37902i
$986$		−	8.68779e11i		−	0.919182i
$987$	−1.21564e10	+	1.02075e11i	−0.0128096	+	0.107560i
$988$	−2.07964e11			−0.218253
$989$	−6.44053e10			−0.0673188
$990$			6.26408e11i			0.652104i
$991$	1.67557e11			0.173727			0.0868636	−	0.996220i	$-0.472316\pi$
$991$	1.67557e11			0.173727			0.0868636	+	0.996220i	$0.472316\pi$
$992$		−	1.06387e11i		−	0.109861i
$993$			1.01228e12i			1.04113i
$994$	6.50448e10	−	5.46171e11i	0.0666296	−	0.559479i
$995$	−2.30642e12			−2.35313
$996$	2.15606e11			0.219091
$997$		−	1.67030e9i		−	0.00169049i	−1.00000		0.000845246i	$-0.999731\pi$
$997$		−	1.67030e9i		−	0.00169049i	1.00000		0.000845246i	$-0.000269050\pi$
$998$	−4.82766e11			−0.486648
$999$		−	2.46913e11i		−	0.247904i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.9.c.a.13.6	yes	12
3.2	odd	2		126.9.c.c.55.8		12
4.3	odd	2		336.9.f.c.97.6		12
7.6	odd	2	inner	42.9.c.a.13.1	✓	12
21.20	even	2		126.9.c.c.55.11		12
28.27	even	2		336.9.f.c.97.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.9.c.a.13.1	✓	12	7.6	odd	2	inner
42.9.c.a.13.6	yes	12	1.1	even	1	trivial
126.9.c.c.55.8		12	3.2	odd	2
126.9.c.c.55.11		12	21.20	even	2
336.9.f.c.97.6		12	4.3	odd	2
336.9.f.c.97.7		12	28.27	even	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)

\(f(q)\)	\(=\)	\(q-11.3137 q^{2} +46.7654i q^{3} +128.000 q^{4} +1031.06i q^{5} -529.090i q^{6} +(-283.934 + 2384.15i) q^{7} -1448.15 q^{8} -2187.00 q^{9} +O(q^{10})\)	\(q-11.3137 q^{2} +46.7654i q^{3} +128.000 q^{4} +1031.06i q^{5} -529.090i q^{6} +(-283.934 + 2384.15i) q^{7} -1448.15 q^{8} -2187.00 q^{9} -11665.2i q^{10} +24553.7 q^{11} +5985.97i q^{12} +31617.8i q^{13} +(3212.35 - 26973.6i) q^{14} -48218.1 q^{15} +16384.0 q^{16} -71096.4i q^{17} +24743.1 q^{18} +51386.1i q^{19} +131976. i q^{20} +(-111496. - 13278.3i) q^{21} -277794. q^{22} -104766. q^{23} -67723.5i q^{24} -672470. q^{25} -357715. i q^{26} -102276. i q^{27} +(-36343.6 + 305172. i) q^{28} -1.08008e6 q^{29} +545526. q^{30} +573939. i q^{31} -185364. q^{32} +1.14826e6i q^{33} +804364. i q^{34} +(-2.45822e6 - 292755. i) q^{35} -279936. q^{36} +2.41419e6 q^{37} -581368. i q^{38} -1.47862e6 q^{39} -1.49314e6i q^{40} -2.61563e6i q^{41} +(1.26143e6 + 150227. i) q^{42} +614755. q^{43} +3.14288e6 q^{44} -2.25494e6i q^{45} +1.18529e6 q^{46} -915508. i q^{47} +766204. i q^{48} +(-5.60356e6 - 1.35388e6i) q^{49} +7.60813e6 q^{50} +3.32485e6 q^{51} +4.04708e6i q^{52} +1.19170e7 q^{53} +1.15712e6i q^{54} +2.53165e7i q^{55} +(411181. - 3.45262e6i) q^{56} -2.40309e6 q^{57} +1.22197e7 q^{58} -1.03094e7i q^{59} -6.17192e6 q^{60} -8.60724e6i q^{61} -6.49338e6i q^{62} +(620964. - 5.21414e6i) q^{63} +2.09715e6 q^{64} -3.26000e7 q^{65} -1.29911e7i q^{66} -5.17705e6 q^{67} -9.10034e6i q^{68} -4.89941e6i q^{69} +(2.78115e7 + 3.31214e6i) q^{70} +2.02484e7 q^{71} +3.16711e6 q^{72} +3.52423e7i q^{73} -2.73134e7 q^{74} -3.14483e7i q^{75} +6.57742e6i q^{76} +(-6.97164e6 + 5.85398e7i) q^{77} +1.67287e7 q^{78} -4.76943e7 q^{79} +1.68930e7i q^{80} +4.78297e6 q^{81} +2.95925e7i q^{82} -3.60186e7i q^{83} +(-1.42715e7 - 1.69962e6i) q^{84} +7.33050e7 q^{85} -6.95516e6 q^{86} -5.05104e7i q^{87} -3.55576e7 q^{88} -9.28933e7i q^{89} +2.55117e7i q^{90} +(-7.53817e7 - 8.97738e6i) q^{91} -1.34100e7 q^{92} -2.68405e7 q^{93} +1.03578e7i q^{94} -5.29824e7 q^{95} -8.66861e6i q^{96} +1.17197e8i q^{97} +(6.33971e7 + 1.53175e7i) q^{98} -5.36990e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 1536 q^{4} + 6420 q^{7} - 26244 q^{9}+O(q^{10}) \) 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

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