LMFDB - Embedded newform 546.8.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,8,Mod(1,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("546.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	546.a (trivial)

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:	yes
Analytic conductor:	$170.562223914$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 51426x^{2} + 704960x + 421951600 $ x^4 - x^3 - 51426x^2 + 704960x + 421951600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-212.586$ of defining polynomial
Character	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +16.5813 q^{5} +216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +16.5813 q^{5} +216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +132.650 q^{10} -2509.84 q^{11} +1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} +447.695 q^{15} +4096.00 q^{16} +8559.21 q^{17} +5832.00 q^{18} -22698.1 q^{19} +1061.20 q^{20} +9261.00 q^{21} -20078.7 q^{22} -86919.3 q^{23} +13824.0 q^{24} -77850.1 q^{25} -17576.0 q^{26} +19683.0 q^{27} +21952.0 q^{28} +164971. q^{29} +3581.56 q^{30} -267590. q^{31} +32768.0 q^{32} -67765.6 q^{33} +68473.6 q^{34} +5687.39 q^{35} +46656.0 q^{36} -354355. q^{37} -181585. q^{38} -59319.0 q^{39} +8489.63 q^{40} +251432. q^{41} +74088.0 q^{42} +313220. q^{43} -160630. q^{44} +12087.8 q^{45} -695354. q^{46} +74057.5 q^{47} +110592. q^{48} +117649. q^{49} -622800. q^{50} +231099. q^{51} -140608. q^{52} -1.21654e6 q^{53} +157464. q^{54} -41616.4 q^{55} +175616. q^{56} -612849. q^{57} +1.31977e6 q^{58} -1.83111e6 q^{59} +28652.5 q^{60} -800465. q^{61} -2.14072e6 q^{62} +250047. q^{63} +262144. q^{64} -36429.1 q^{65} -542125. q^{66} -1.55632e6 q^{67} +547789. q^{68} -2.34682e6 q^{69} +45499.1 q^{70} +4.62638e6 q^{71} +373248. q^{72} -3.02816e6 q^{73} -2.83484e6 q^{74} -2.10195e6 q^{75} -1.45268e6 q^{76} -860874. q^{77} -474552. q^{78} +1.81803e6 q^{79} +67917.0 q^{80} +531441. q^{81} +2.01146e6 q^{82} +3.51805e6 q^{83} +592704. q^{84} +141923. q^{85} +2.50576e6 q^{86} +4.45421e6 q^{87} -1.28504e6 q^{88} +5.02737e6 q^{89} +96702.2 q^{90} -753571. q^{91} -5.56284e6 q^{92} -7.22493e6 q^{93} +592460. q^{94} -376364. q^{95} +884736. q^{96} +4.70305e6 q^{97} +941192. q^{98} -1.82967e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} - 328 q^{5} + 864 q^{6} + 1372 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10}) $ 4 * q + 32 * q^2 + 108 * q^3 + 256 * q^4 - 328 * q^5 + 864 * q^6 + 1372 * q^7 + 2048 * q^8 + 2916 * q^9	\( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} - 328 q^{5} + 864 q^{6} + 1372 q^{7} + 2048 q^{8} + 2916 q^{9} - 2624 q^{10} - 7945 q^{11} + 6912 q^{12} - 8788 q^{13} + 10976 q^{14} - 8856 q^{15} + 16384 q^{16} - 27285 q^{17} + 23328 q^{18} - 34802 q^{19} - 20992 q^{20} + 37044 q^{21} - 63560 q^{22} - 22316 q^{23} + 55296 q^{24} - 64650 q^{25} - 70304 q^{26} + 78732 q^{27} + 87808 q^{28} - 158470 q^{29} - 70848 q^{30} - 274647 q^{31} + 131072 q^{32} - 214515 q^{33} - 218280 q^{34} - 112504 q^{35} + 186624 q^{36} - 319169 q^{37} - 278416 q^{38} - 237276 q^{39} - 167936 q^{40} - 711254 q^{41} + 296352 q^{42} - 893034 q^{43} - 508480 q^{44} - 239112 q^{45} - 178528 q^{46} - 563129 q^{47} + 442368 q^{48} + 470596 q^{49} - 517200 q^{50} - 736695 q^{51} - 562432 q^{52} - 2186067 q^{53} + 629856 q^{54} - 1242387 q^{55} + 702464 q^{56} - 939654 q^{57} - 1267760 q^{58} + 53200 q^{59} - 566784 q^{60} - 2419497 q^{61} - 2197176 q^{62} + 1000188 q^{63} + 1048576 q^{64} + 720616 q^{65} - 1716120 q^{66} - 4908916 q^{67} - 1746240 q^{68} - 602532 q^{69} - 900032 q^{70} - 5546448 q^{71} + 1492992 q^{72} + 6114536 q^{73} - 2553352 q^{74} - 1745550 q^{75} - 2227328 q^{76} - 2725135 q^{77} - 1898208 q^{78} + 6523461 q^{79} - 1343488 q^{80} + 2125764 q^{81} - 5690032 q^{82} + 5265065 q^{83} + 2370816 q^{84} - 10347427 q^{85} - 7144272 q^{86} - 4278690 q^{87} - 4067840 q^{88} - 4679151 q^{89} - 1912896 q^{90} - 3014284 q^{91} - 1428224 q^{92} - 7415469 q^{93} - 4505032 q^{94} + 13009536 q^{95} + 3538944 q^{96} + 12566137 q^{97} + 3764768 q^{98} - 5791905 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 + 108 * q^3 + 256 * q^4 - 328 * q^5 + 864 * q^6 + 1372 * q^7 + 2048 * q^8 + 2916 * q^9 - 2624 * q^10 - 7945 * q^11 + 6912 * q^12 - 8788 * q^13 + 10976 * q^14 - 8856 * q^15 + 16384 * q^16 - 27285 * q^17 + 23328 * q^18 - 34802 * q^19 - 20992 * q^20 + 37044 * q^21 - 63560 * q^22 - 22316 * q^23 + 55296 * q^24 - 64650 * q^25 - 70304 * q^26 + 78732 * q^27 + 87808 * q^28 - 158470 * q^29 - 70848 * q^30 - 274647 * q^31 + 131072 * q^32 - 214515 * q^33 - 218280 * q^34 - 112504 * q^35 + 186624 * q^36 - 319169 * q^37 - 278416 * q^38 - 237276 * q^39 - 167936 * q^40 - 711254 * q^41 + 296352 * q^42 - 893034 * q^43 - 508480 * q^44 - 239112 * q^45 - 178528 * q^46 - 563129 * q^47 + 442368 * q^48 + 470596 * q^49 - 517200 * q^50 - 736695 * q^51 - 562432 * q^52 - 2186067 * q^53 + 629856 * q^54 - 1242387 * q^55 + 702464 * q^56 - 939654 * q^57 - 1267760 * q^58 + 53200 * q^59 - 566784 * q^60 - 2419497 * q^61 - 2197176 * q^62 + 1000188 * q^63 + 1048576 * q^64 + 720616 * q^65 - 1716120 * q^66 - 4908916 * q^67 - 1746240 * q^68 - 602532 * q^69 - 900032 * q^70 - 5546448 * q^71 + 1492992 * q^72 + 6114536 * q^73 - 2553352 * q^74 - 1745550 * q^75 - 2227328 * q^76 - 2725135 * q^77 - 1898208 * q^78 + 6523461 * q^79 - 1343488 * q^80 + 2125764 * q^81 - 5690032 * q^82 + 5265065 * q^83 + 2370816 * q^84 - 10347427 * q^85 - 7144272 * q^86 - 4278690 * q^87 - 4067840 * q^88 - 4679151 * q^89 - 1912896 * q^90 - 3014284 * q^91 - 1428224 * q^92 - 7415469 * q^93 - 4505032 * q^94 + 13009536 * q^95 + 3538944 * q^96 + 12566137 * q^97 + 3764768 * q^98 - 5791905 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	64.0000	0.500000
$5$	16.5813	0.0593231	0.0296615	−	0.999560i	$-0.490557\pi$
$5$	16.5813	0.0593231	0.0296615	+	0.999560i	$0.490557\pi$
$6$	216.000	0.408248
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	132.650	0.0419477
$11$	−2509.84	−0.568553	−0.284277	−	0.958742i	$-0.591753\pi$
$11$	−2509.84	−0.568553	−0.284277	+	0.958742i	$0.591753\pi$
$12$	1728.00	0.288675
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	2744.00	0.267261
$15$	447.695	0.0342502
$16$	4096.00	0.250000
$17$	8559.21	0.422535	0.211267	−	0.977428i	$-0.432241\pi$
$17$	8559.21	0.422535	0.211267	+	0.977428i	$0.432241\pi$
$18$	5832.00	0.235702
$19$	−22698.1	−0.759193	−0.379597	−	0.925152i	$-0.623937\pi$
$19$	−22698.1	−0.759193	−0.379597	+	0.925152i	$0.623937\pi$
$20$	1061.20	0.0296615
$21$	9261.00	0.218218
$22$	−20078.7	−0.402028
$23$	−86919.3	−1.48960	−0.744799	−	0.667289i	$-0.767454\pi$
$23$	−86919.3	−1.48960	−0.744799	+	0.667289i	$0.767454\pi$
$24$	13824.0	0.204124
$25$	−77850.1	−0.996481
$26$	−17576.0	−0.196116
$27$	19683.0	0.192450
$28$	21952.0	0.188982
$29$	164971.	1.25607	0.628035	−	0.778185i	$-0.283859\pi$
$29$	164971.	1.25607	0.628035	+	0.778185i	$0.283859\pi$
$30$	3581.56	0.0242185
$31$	−267590.	−1.61326	−0.806630	−	0.591057i	$-0.798711\pi$
$31$	−267590.	−1.61326	−0.806630	+	0.591057i	$0.798711\pi$
$32$	32768.0	0.176777
$33$	−67765.6	−0.328254
$34$	68473.6	0.298777
$35$	5687.39	0.0224220
$36$	46656.0	0.166667
$37$	−354355.	−1.15009	−0.575046	−	0.818121i	$-0.695016\pi$
$37$	−354355.	−1.15009	−0.575046	+	0.818121i	$0.695016\pi$
$38$	−181585.	−0.536831
$39$	−59319.0	−0.160128
$40$	8489.63	0.0209739
$41$	251432.	0.569741	0.284870	−	0.958566i	$-0.408049\pi$
$41$	251432.	0.569741	0.284870	+	0.958566i	$0.408049\pi$
$42$	74088.0	0.154303
$43$	313220.	0.600772	0.300386	−	0.953818i	$-0.402884\pi$
$43$	313220.	0.600772	0.300386	+	0.953818i	$0.402884\pi$
$44$	−160630.	−0.284277
$45$	12087.8	0.0197744
$46$	−695354.	−1.05330
$47$	74057.5	0.104046	0.0520231	−	0.998646i	$-0.483433\pi$
$47$	74057.5	0.104046	0.0520231	+	0.998646i	$0.483433\pi$
$48$	110592.	0.144338
$49$	117649.	0.142857
$50$	−622800.	−0.704618
$51$	231099.	0.243950
$52$	−140608.	−0.138675
$53$	−1.21654e6	−1.12244	−0.561219	−	0.827667i	$-0.689668\pi$
$53$	−1.21654e6	−1.12244	−0.561219	+	0.827667i	$0.689668\pi$
$54$	157464.	0.136083
$55$	−41616.4	−0.0337283
$56$	175616.	0.133631
$57$	−612849.	−0.438320
$58$	1.31977e6	0.888176
$59$	−1.83111e6	−1.16073	−0.580365	−	0.814356i	$-0.697090\pi$
$59$	−1.83111e6	−1.16073	−0.580365	+	0.814356i	$0.697090\pi$
$60$	28652.5	0.0171251
$61$	−800465.	−0.451531	−0.225766	−	0.974182i	$-0.572488\pi$
$61$	−800465.	−0.451531	−0.225766	+	0.974182i	$0.572488\pi$
$62$	−2.14072e6	−1.14075
$63$	250047.	0.125988
$64$	262144.	0.125000
$65$	−36429.1	−0.0164533
$66$	−542125.	−0.232111
$67$	−1.55632e6	−0.632174	−0.316087	−	0.948730i	$-0.602369\pi$
$67$	−1.55632e6	−0.632174	−0.316087	+	0.948730i	$0.602369\pi$
$68$	547789.	0.211267
$69$	−2.34682e6	−0.860019
$70$	45499.1	0.0158548
$71$	4.62638e6	1.53404	0.767020	−	0.641623i	$-0.221739\pi$
$71$	4.62638e6	1.53404	0.767020	+	0.641623i	$0.221739\pi$
$72$	373248.	0.117851
$73$	−3.02816e6	−0.911063	−0.455532	−	0.890220i	$-0.650551\pi$
$73$	−3.02816e6	−0.911063	−0.455532	+	0.890220i	$0.650551\pi$
$74$	−2.83484e6	−0.813238
$75$	−2.10195e6	−0.575318
$76$	−1.45268e6	−0.379597
$77$	−860874.	−0.214893
$78$	−474552.	−0.113228
$79$	1.81803e6	0.414864	0.207432	−	0.978249i	$-0.433489\pi$
$79$	1.81803e6	0.414864	0.207432	+	0.978249i	$0.433489\pi$
$80$	67917.0	0.0148308
$81$	531441.	0.111111
$82$	2.01146e6	0.402868
$83$	3.51805e6	0.675350	0.337675	−	0.941263i	$-0.390360\pi$
$83$	3.51805e6	0.675350	0.337675	+	0.941263i	$0.390360\pi$
$84$	592704.	0.109109
$85$	141923.	0.0250660
$86$	2.50576e6	0.424810
$87$	4.45421e6	0.725193
$88$	−1.28504e6	−0.201014
$89$	5.02737e6	0.755920	0.377960	−	0.925822i	$-0.376626\pi$
$89$	5.02737e6	0.755920	0.377960	+	0.925822i	$0.376626\pi$
$90$	96702.2	0.0139826
$91$	−753571.	−0.104828
$92$	−5.56284e6	−0.744799
$93$	−7.22493e6	−0.931416
$94$	592460.	0.0735718
$95$	−376364.	−0.0450377
$96$	884736.	0.102062
$97$	4.70305e6	0.523213	0.261607	−	0.965175i	$-0.415748\pi$
$97$	4.70305e6	0.523213	0.261607	+	0.965175i	$0.415748\pi$
$98$	941192.	0.101015
$99$	−1.82967e6	−0.189518
$100$	−4.98240e6	−0.498240
$101$	−1.68846e7	−1.63067	−0.815337	−	0.578987i	$-0.803448\pi$
$101$	−1.68846e7	−1.63067	−0.815337	+	0.578987i	$0.803448\pi$
$102$	1.84879e6	0.172499
$103$	7.52907e6	0.678908	0.339454	−	0.940623i	$-0.389758\pi$
$103$	7.52907e6	0.678908	0.339454	+	0.940623i	$0.389758\pi$
$104$	−1.12486e6	−0.0980581
$105$	153559.	0.0129454
$106$	−9.73236e6	−0.793684
$107$	−8.44548e6	−0.666471	−0.333235	−	0.942844i	$-0.608140\pi$
$107$	−8.44548e6	−0.666471	−0.333235	+	0.942844i	$0.608140\pi$
$108$	1.25971e6	0.0962250
$109$	−2.66797e6	−0.197328	−0.0986640	−	0.995121i	$-0.531457\pi$
$109$	−2.66797e6	−0.197328	−0.0986640	+	0.995121i	$0.531457\pi$
$110$	−332931.	−0.0238495
$111$	−9.56759e6	−0.664006
$112$	1.40493e6	0.0944911
$113$	−2.02475e7	−1.32007	−0.660034	−	0.751236i	$-0.729458\pi$
$113$	−2.02475e7	−1.32007	−0.660034	+	0.751236i	$0.729458\pi$
$114$	−4.90279e6	−0.309939
$115$	−1.44124e6	−0.0883675
$116$	1.05581e7	0.628035
$117$	−1.60161e6	−0.0924500
$118$	−1.46488e7	−0.820761
$119$	2.93581e6	0.159703
$120$	229220.	0.0121093
$121$	−1.31879e7	−0.676747
$122$	−6.40372e6	−0.319281
$123$	6.78867e6	0.328940
$124$	−1.71258e7	−0.806630
$125$	−2.58627e6	−0.118437
$126$	2.00038e6	0.0890871
$127$	1.77333e7	0.768202	0.384101	−	0.923291i	$-0.374511\pi$
$127$	1.77333e7	0.768202	0.384101	+	0.923291i	$0.374511\pi$
$128$	2.09715e6	0.0883883
$129$	8.45694e6	0.346856
$130$	−291433.	−0.0116342
$131$	−3.13542e7	−1.21856	−0.609278	−	0.792957i	$-0.708541\pi$
$131$	−3.13542e7	−1.21856	−0.609278	+	0.792957i	$0.708541\pi$
$132$	−4.33700e6	−0.164127
$133$	−7.78545e6	−0.286948
$134$	−1.24506e7	−0.447015
$135$	326370.	0.0114167
$136$	4.38231e6	0.149389
$137$	2.57462e7	0.855444	0.427722	−	0.903910i	$-0.359316\pi$
$137$	2.57462e7	0.855444	0.427722	+	0.903910i	$0.359316\pi$
$138$	−1.87746e7	−0.608125
$139$	1.21198e7	0.382774	0.191387	−	0.981515i	$-0.438701\pi$
$139$	1.21198e7	0.382774	0.191387	+	0.981515i	$0.438701\pi$
$140$	363993.	0.0112110
$141$	1.99955e6	0.0600711
$142$	3.70110e7	1.08473
$143$	5.51411e6	0.157688
$144$	2.98598e6	0.0833333
$145$	2.73543e6	0.0745140
$146$	−2.42253e7	−0.644219
$147$	3.17652e6	0.0824786
$148$	−2.26787e7	−0.575046
$149$	−1.77842e7	−0.440435	−0.220217	−	0.975451i	$-0.570677\pi$
$149$	−1.77842e7	−0.440435	−0.220217	+	0.975451i	$0.570677\pi$
$150$	−1.68156e7	−0.406812
$151$	6.81394e6	0.161057	0.0805283	−	0.996752i	$-0.474339\pi$
$151$	6.81394e6	0.161057	0.0805283	+	0.996752i	$0.474339\pi$
$152$	−1.16214e7	−0.268415
$153$	6.23966e6	0.140845
$154$	−6.88699e6	−0.151952
$155$	−4.43699e6	−0.0957035
$156$	−3.79642e6	−0.0800641
$157$	−8.73364e7	−1.80114	−0.900568	−	0.434715i	$-0.856849\pi$
$157$	−8.73364e7	−1.80114	−0.900568	+	0.434715i	$0.856849\pi$
$158$	1.45442e7	0.293353
$159$	−3.28467e7	−0.648040
$160$	543336.	0.0104869
$161$	−2.98133e7	−0.563015
$162$	4.25153e6	0.0785674
$163$	−2.54389e6	−0.0460088	−0.0230044	−	0.999735i	$-0.507323\pi$
$163$	−2.54389e6	−0.0460088	−0.0230044	+	0.999735i	$0.507323\pi$
$164$	1.60917e7	0.284870
$165$	−1.12364e6	−0.0194731
$166$	2.81444e7	0.477544
$167$	−3.16358e7	−0.525620	−0.262810	−	0.964848i	$-0.584649\pi$
$167$	−3.16358e7	−0.525620	−0.262810	+	0.964848i	$0.584649\pi$
$168$	4.74163e6	0.0771517
$169$	4.82681e6	0.0769231
$170$	1.13538e6	0.0177244
$171$	−1.65469e7	−0.253064
$172$	2.00461e7	0.300386
$173$	7.65935e7	1.12468	0.562342	−	0.826905i	$-0.309900\pi$
$173$	7.65935e7	1.12468	0.562342	+	0.826905i	$0.309900\pi$
$174$	3.56337e7	0.512789
$175$	−2.67026e7	−0.376634
$176$	−1.02803e7	−0.142138
$177$	−4.94399e7	−0.670148
$178$	4.02189e7	0.534516
$179$	−8.74602e6	−0.113979	−0.0569895	−	0.998375i	$-0.518150\pi$
$179$	−8.74602e6	−0.113979	−0.0569895	+	0.998375i	$0.518150\pi$
$180$	773617.	0.00988718
$181$	6.59452e7	0.826624	0.413312	−	0.910589i	$-0.364372\pi$
$181$	6.59452e7	0.826624	0.413312	+	0.910589i	$0.364372\pi$
$182$	−6.02857e6	−0.0741249
$183$	−2.16126e7	−0.260692
$184$	−4.45027e7	−0.526652
$185$	−5.87567e6	−0.0682270
$186$	−5.77995e7	−0.658610
$187$	−2.14822e7	−0.240233
$188$	4.73968e6	0.0520231
$189$	6.75127e6	0.0727393
$190$	−3.01092e6	−0.0318464
$191$	−2.31749e7	−0.240658	−0.120329	−	0.992734i	$-0.538395\pi$
$191$	−2.31749e7	−0.240658	−0.120329	+	0.992734i	$0.538395\pi$
$192$	7.07789e6	0.0721688
$193$	−8.88453e7	−0.889577	−0.444789	−	0.895636i	$-0.646721\pi$
$193$	−8.88453e7	−0.889577	−0.444789	+	0.895636i	$0.646721\pi$
$194$	3.76244e7	0.369968
$195$	−983586.	−0.00949929
$196$	7.52954e6	0.0714286
$197$	8.28691e7	0.772255	0.386128	−	0.922445i	$-0.373812\pi$
$197$	8.28691e7	0.772255	0.386128	+	0.922445i	$0.373812\pi$
$198$	−1.46374e7	−0.134009
$199$	1.18684e7	0.106759	0.0533796	−	0.998574i	$-0.483001\pi$
$199$	1.18684e7	0.106759	0.0533796	+	0.998574i	$0.483001\pi$
$200$	−3.98592e7	−0.352309
$201$	−4.20206e7	−0.364986
$202$	−1.35077e8	−1.15306
$203$	5.65850e7	0.474750
$204$	1.47903e7	0.121975
$205$	4.16907e6	0.0337988
$206$	6.02326e7	0.480061
$207$	−6.33642e7	−0.496532
$208$	−8.99891e6	−0.0693375
$209$	5.69686e7	0.431642
$210$	1.22848e6	0.00915375
$211$	7.86816e7	0.576613	0.288306	−	0.957538i	$-0.406908\pi$
$211$	7.86816e7	0.576613	0.288306	+	0.957538i	$0.406908\pi$
$212$	−7.78589e7	−0.561219
$213$	1.24912e8	0.885678
$214$	−6.75638e7	−0.471266
$215$	5.19360e6	0.0356397
$216$	1.00777e7	0.0680414
$217$	−9.17834e7	−0.609755
$218$	−2.13438e7	−0.139532
$219$	−8.17603e7	−0.526003
$220$	−2.66345e6	−0.0168642
$221$	−1.88046e7	−0.117190
$222$	−7.65407e7	−0.469523
$223$	2.29987e8	1.38879	0.694396	−	0.719594i	$-0.255672\pi$
$223$	2.29987e8	1.38879	0.694396	+	0.719594i	$0.255672\pi$
$224$	1.12394e7	0.0668153
$225$	−5.67527e7	−0.332160
$226$	−1.61980e8	−0.933429
$227$	−1.59117e8	−0.902872	−0.451436	−	0.892303i	$-0.649088\pi$
$227$	−1.59117e8	−0.902872	−0.451436	+	0.892303i	$0.649088\pi$
$228$	−3.92224e7	−0.219160
$229$	−3.67700e7	−0.202334	−0.101167	−	0.994869i	$-0.532258\pi$
$229$	−3.67700e7	−0.202334	−0.101167	+	0.994869i	$0.532258\pi$
$230$	−1.15299e7	−0.0624852
$231$	−2.32436e7	−0.124068
$232$	8.44650e7	0.444088
$233$	−5.41266e7	−0.280327	−0.140164	−	0.990128i	$-0.544763\pi$
$233$	−5.41266e7	−0.280327	−0.140164	+	0.990128i	$0.544763\pi$
$234$	−1.28129e7	−0.0653720
$235$	1.22797e6	0.00617234
$236$	−1.17191e8	−0.580365
$237$	4.90868e7	0.239522
$238$	2.34865e7	0.112927
$239$	−3.41930e8	−1.62011	−0.810056	−	0.586353i	$-0.800563\pi$
$239$	−3.41930e8	−1.62011	−0.810056	+	0.586353i	$0.800563\pi$
$240$	1.83376e6	0.00856255
$241$	5.78043e7	0.266011	0.133006	−	0.991115i	$-0.457537\pi$
$241$	5.78043e7	0.266011	0.133006	+	0.991115i	$0.457537\pi$
$242$	−1.05503e8	−0.478533
$243$	1.43489e7	0.0641500
$244$	−5.12298e7	−0.225766
$245$	1.95077e6	0.00847473
$246$	5.43093e7	0.232596
$247$	4.98678e7	0.210562
$248$	−1.37006e8	−0.570373
$249$	9.49874e7	0.389913
$250$	−2.06902e7	−0.0837479
$251$	−1.49265e8	−0.595800	−0.297900	−	0.954597i	$-0.596286\pi$
$251$	−1.49265e8	−0.595800	−0.297900	+	0.954597i	$0.596286\pi$
$252$	1.60030e7	0.0629941
$253$	2.18153e8	0.846915
$254$	1.41866e8	0.543201
$255$	3.83192e6	0.0144719
$256$	1.67772e7	0.0625000
$257$	−2.10341e8	−0.772961	−0.386481	−	0.922298i	$-0.626309\pi$
$257$	−2.10341e8	−0.772961	−0.386481	+	0.922298i	$0.626309\pi$
$258$	6.76555e7	0.245264
$259$	−1.21544e8	−0.434694
$260$	−2.33146e6	−0.00822663
$261$	1.20264e8	0.418690
$262$	−2.50833e8	−0.861649
$263$	3.07925e8	1.04376	0.521878	−	0.853020i	$-0.325232\pi$
$263$	3.07925e8	1.04376	0.521878	+	0.853020i	$0.325232\pi$
$264$	−3.46960e7	−0.116055
$265$	−2.01719e7	−0.0665865
$266$	−6.22836e7	−0.202903
$267$	1.35739e8	0.436430
$268$	−9.96044e7	−0.316087
$269$	−2.28991e8	−0.717273	−0.358637	−	0.933477i	$-0.616758\pi$
$269$	−2.28991e8	−0.717273	−0.358637	+	0.933477i	$0.616758\pi$
$270$	2.61096e6	0.00807285
$271$	9.18776e7	0.280425	0.140213	−	0.990121i	$-0.455221\pi$
$271$	9.18776e7	0.280425	0.140213	+	0.990121i	$0.455221\pi$
$272$	3.50585e7	0.105634
$273$	−2.03464e7	−0.0605228
$274$	2.05970e8	0.604890
$275$	1.95391e8	0.566552
$276$	−1.50197e8	−0.430010
$277$	−4.34265e8	−1.22765	−0.613827	−	0.789441i	$-0.710371\pi$
$277$	−4.34265e8	−1.22765	−0.613827	+	0.789441i	$0.710371\pi$
$278$	9.69582e7	0.270662
$279$	−1.95073e8	−0.537753
$280$	2.91194e6	0.00792738
$281$	−5.23377e8	−1.40716	−0.703578	−	0.710618i	$-0.748415\pi$
$281$	−5.23377e8	−1.40716	−0.703578	+	0.710618i	$0.748415\pi$
$282$	1.59964e7	0.0424767
$283$	1.67486e8	0.439263	0.219632	−	0.975583i	$-0.429514\pi$
$283$	1.67486e8	0.439263	0.219632	+	0.975583i	$0.429514\pi$
$284$	2.96088e8	0.767020
$285$	−1.01618e7	−0.0260025
$286$	4.41129e7	0.111502
$287$	8.62412e7	0.215342
$288$	2.38879e7	0.0589256
$289$	−3.37079e8	−0.821465
$290$	2.18834e7	0.0526893
$291$	1.26982e8	0.302077
$292$	−1.93802e8	−0.455532
$293$	3.48648e8	0.809749	0.404875	−	0.914372i	$-0.367315\pi$
$293$	3.48648e8	0.809749	0.404875	+	0.914372i	$0.367315\pi$
$294$	2.54122e7	0.0583212
$295$	−3.03621e7	−0.0688581
$296$	−1.81430e8	−0.406619
$297$	−4.94011e7	−0.109418
$298$	−1.42273e8	−0.311434
$299$	1.90962e8	0.413140
$300$	−1.34525e8	−0.287659
$301$	1.07434e8	0.227071
$302$	5.45115e7	0.113884
$303$	−4.55885e8	−0.941470
$304$	−9.29715e7	−0.189798
$305$	−1.32728e7	−0.0267862
$306$	4.99173e7	0.0995923
$307$	−3.76596e8	−0.742834	−0.371417	−	0.928466i	$-0.621128\pi$
$307$	−3.76596e8	−0.742834	−0.371417	+	0.928466i	$0.621128\pi$
$308$	−5.50959e7	−0.107446
$309$	2.03285e8	0.391968
$310$	−3.54959e7	−0.0676726
$311$	−7.99894e8	−1.50790	−0.753948	−	0.656934i	$-0.771853\pi$
$311$	−7.99894e8	−1.50790	−0.753948	+	0.656934i	$0.771853\pi$
$312$	−3.03713e7	−0.0566139
$313$	−1.51472e8	−0.279208	−0.139604	−	0.990207i	$-0.544583\pi$
$313$	−1.51472e8	−0.279208	−0.139604	+	0.990207i	$0.544583\pi$
$314$	−6.98691e8	−1.27360
$315$	4.14611e6	0.00747401
$316$	1.16354e8	0.207432
$317$	9.82236e8	1.73184	0.865921	−	0.500181i	$-0.166733\pi$
$317$	9.82236e8	1.73184	0.865921	+	0.500181i	$0.166733\pi$
$318$	−2.62774e8	−0.458234
$319$	−4.14050e8	−0.714143
$320$	4.34669e6	0.00741538
$321$	−2.28028e8	−0.384787
$322$	−2.38507e8	−0.398112
$323$	−1.94278e8	−0.320785
$324$	3.40122e7	0.0555556
$325$	1.71037e8	0.276374
$326$	−2.03511e7	−0.0325332
$327$	−7.20353e7	−0.113927
$328$	1.28733e8	0.201434
$329$	2.54017e7	0.0393258
$330$	−8.98914e6	−0.0137695
$331$	1.11624e9	1.69184	0.845919	−	0.533311i	$-0.179053\pi$
$331$	1.11624e9	1.69184	0.845919	+	0.533311i	$0.179053\pi$
$332$	2.25155e8	0.337675
$333$	−2.58325e8	−0.383364
$334$	−2.53087e8	−0.371669
$335$	−2.58058e7	−0.0375025
$336$	3.79331e7	0.0545545
$337$	−7.22489e8	−1.02832	−0.514158	−	0.857696i	$-0.671895\pi$
$337$	−7.22489e8	−1.02832	−0.514158	+	0.857696i	$0.671895\pi$
$338$	3.86145e7	0.0543928
$339$	−5.46682e8	−0.762141
$340$	9.08306e6	0.0125330
$341$	6.71608e8	0.917224
$342$	−1.32375e8	−0.178944
$343$	4.03536e7	0.0539949
$344$	1.60369e8	0.212405
$345$	−3.89134e7	−0.0510190
$346$	6.12748e8	0.795272
$347$	6.40337e8	0.822727	0.411363	−	0.911471i	$-0.365053\pi$
$347$	6.40337e8	0.822727	0.411363	+	0.911471i	$0.365053\pi$
$348$	2.85070e8	0.362596
$349$	−1.56775e8	−0.197419	−0.0987096	−	0.995116i	$-0.531471\pi$
$349$	−1.56775e8	−0.197419	−0.0987096	+	0.995116i	$0.531471\pi$
$350$	−2.13621e8	−0.266321
$351$	−4.32436e7	−0.0533761
$352$	−8.22423e7	−0.100507
$353$	3.69124e8	0.446643	0.223321	−	0.974745i	$-0.428310\pi$
$353$	3.69124e8	0.446643	0.223321	+	0.974745i	$0.428310\pi$
$354$	−3.95519e8	−0.473866
$355$	7.67113e7	0.0910040
$356$	3.21751e8	0.377960
$357$	7.92668e7	0.0922046
$358$	−6.99682e7	−0.0805953
$359$	3.00065e8	0.342283	0.171141	−	0.985247i	$-0.445255\pi$
$359$	3.00065e8	0.342283	0.171141	+	0.985247i	$0.445255\pi$
$360$	6.18894e6	0.00699129
$361$	−3.78667e8	−0.423626
$362$	5.27561e8	0.584511
$363$	−3.56073e8	−0.390720
$364$	−4.82285e7	−0.0524142
$365$	−5.02108e7	−0.0540471
$366$	−1.72900e8	−0.184337
$367$	4.57641e8	0.483274	0.241637	−	0.970367i	$-0.422316\pi$
$367$	4.57641e8	0.483274	0.241637	+	0.970367i	$0.422316\pi$
$368$	−3.56021e8	−0.372399
$369$	1.83294e8	0.189914
$370$	−4.70054e7	−0.0482438
$371$	−4.17275e8	−0.424242
$372$	−4.62396e8	−0.465708
$373$	5.89107e7	0.0587778	0.0293889	−	0.999568i	$-0.490644\pi$
$373$	5.89107e7	0.0587778	0.0293889	+	0.999568i	$0.490644\pi$
$374$	−1.71858e8	−0.169871
$375$	−6.98293e7	−0.0683799
$376$	3.79174e7	0.0367859
$377$	−3.62441e8	−0.348371
$378$	5.40102e7	0.0514344
$379$	−1.74034e9	−1.64209	−0.821043	−	0.570867i	$-0.806607\pi$
$379$	−1.74034e9	−1.64209	−0.821043	+	0.570867i	$0.806607\pi$
$380$	−2.40873e7	−0.0225188
$381$	4.78798e8	0.443522
$382$	−1.85399e8	−0.170171
$383$	−1.00025e9	−0.909733	−0.454866	−	0.890560i	$-0.650313\pi$
$383$	−1.00025e9	−0.909733	−0.454866	+	0.890560i	$0.650313\pi$
$384$	5.66231e7	0.0510310
$385$	−1.42744e7	−0.0127481
$386$	−7.10762e8	−0.629026
$387$	2.28337e8	0.200257
$388$	3.00995e8	0.261607
$389$	−1.30448e8	−0.112360	−0.0561802	−	0.998421i	$-0.517892\pi$
$389$	−1.30448e8	−0.112360	−0.0561802	+	0.998421i	$0.517892\pi$
$390$	−7.86869e6	−0.00671702
$391$	−7.43960e8	−0.629406
$392$	6.02363e7	0.0505076
$393$	−8.46562e8	−0.703534
$394$	6.62953e8	0.546067
$395$	3.01453e7	0.0246110
$396$	−1.17099e8	−0.0947589
$397$	−1.56843e9	−1.25805	−0.629026	−	0.777384i	$-0.716546\pi$
$397$	−1.56843e9	−1.25805	−0.629026	+	0.777384i	$0.716546\pi$
$398$	9.49470e7	0.0754902
$399$	−2.10207e8	−0.165670
$400$	−3.18874e8	−0.249120
$401$	−2.09206e7	−0.0162020	−0.00810100	−	0.999967i	$-0.502579\pi$
$401$	−2.09206e7	−0.0162020	−0.00810100	+	0.999967i	$0.502579\pi$
$402$	−3.36165e8	−0.258084
$403$	5.87896e8	0.447438
$404$	−1.08062e9	−0.815337
$405$	8.81198e6	0.00659145
$406$	4.52680e8	0.335699
$407$	8.89373e8	0.653889
$408$	1.18322e8	0.0862495
$409$	1.64286e9	1.18732	0.593661	−	0.804715i	$-0.297682\pi$
$409$	1.64286e9	1.18732	0.593661	+	0.804715i	$0.297682\pi$
$410$	3.33526e7	0.0238993
$411$	6.95148e8	0.493891
$412$	4.81860e8	0.339454
$413$	−6.28069e8	−0.438715
$414$	−5.06913e8	−0.351101
$415$	5.83339e7	0.0400638
$416$	−7.19913e7	−0.0490290
$417$	3.27234e8	0.220995
$418$	4.55749e8	0.305217
$419$	1.49087e9	0.990130	0.495065	−	0.868856i	$-0.335144\pi$
$419$	1.49087e9	0.990130	0.495065	+	0.868856i	$0.335144\pi$
$420$	9.82781e6	0.00647268
$421$	−3.06178e8	−0.199980	−0.0999899	−	0.994988i	$-0.531881\pi$
$421$	−3.06178e8	−0.199980	−0.0999899	+	0.994988i	$0.531881\pi$
$422$	6.29453e8	0.407727
$423$	5.39879e7	0.0346821
$424$	−6.22871e8	−0.396842
$425$	−6.66335e8	−0.421048
$426$	9.99297e8	0.626269
$427$	−2.74559e8	−0.170663
$428$	−5.40511e8	−0.333235
$429$	1.48881e8	0.0910414
$430$	4.15488e7	0.0252011
$431$	8.51959e8	0.512564	0.256282	−	0.966602i	$-0.417502\pi$
$431$	8.51959e8	0.512564	0.256282	+	0.966602i	$0.417502\pi$
$432$	8.06216e7	0.0481125
$433$	2.05827e9	1.21841	0.609206	−	0.793012i	$-0.291488\pi$
$433$	2.05827e9	1.21841	0.609206	+	0.793012i	$0.291488\pi$
$434$	−7.34267e8	−0.431162
$435$	7.38566e7	0.0430207
$436$	−1.70750e8	−0.0986640
$437$	1.97290e9	1.13089
$438$	−6.54082e8	−0.371940
$439$	9.91114e8	0.559111	0.279555	−	0.960130i	$-0.409813\pi$
$439$	9.91114e8	0.559111	0.279555	+	0.960130i	$0.409813\pi$
$440$	−2.13076e7	−0.0119248
$441$	8.57661e7	0.0476190
$442$	−1.50437e8	−0.0828658
$443$	−3.04674e8	−0.166503	−0.0832515	−	0.996529i	$-0.526530\pi$
$443$	−3.04674e8	−0.166503	−0.0832515	+	0.996529i	$0.526530\pi$
$444$	−6.12326e8	−0.332003
$445$	8.33603e7	0.0448435
$446$	1.83990e9	0.982024
$447$	−4.80173e8	−0.254285
$448$	8.99154e7	0.0472456
$449$	2.84882e9	1.48526	0.742630	−	0.669702i	$-0.233578\pi$
$449$	2.84882e9	1.48526	0.742630	+	0.669702i	$0.233578\pi$
$450$	−4.54022e8	−0.234873
$451$	−6.31054e8	−0.323928
$452$	−1.29584e9	−0.660034
$453$	1.83976e8	0.0929861
$454$	−1.27294e9	−0.638427
$455$	−1.24952e7	−0.00621875
$456$	−3.13779e8	−0.154970
$457$	−1.01974e9	−0.499787	−0.249893	−	0.968273i	$-0.580396\pi$
$457$	−1.01974e9	−0.499787	−0.249893	+	0.968273i	$0.580396\pi$
$458$	−2.94160e8	−0.143072
$459$	1.68471e8	0.0813168
$460$	−9.22391e7	−0.0441837
$461$	−9.34514e8	−0.444255	−0.222128	−	0.975018i	$-0.571300\pi$
$461$	−9.34514e8	−0.444255	−0.222128	+	0.975018i	$0.571300\pi$
$462$	−1.85949e8	−0.0877297
$463$	−1.81395e9	−0.849362	−0.424681	−	0.905343i	$-0.639614\pi$
$463$	−1.81395e9	−0.849362	−0.424681	+	0.905343i	$0.639614\pi$
$464$	6.75720e8	0.314018
$465$	−1.19799e8	−0.0552544
$466$	−4.33013e8	−0.198221
$467$	4.33705e8	0.197054	0.0985270	−	0.995134i	$-0.468587\pi$
$467$	4.33705e8	0.197054	0.0985270	+	0.995134i	$0.468587\pi$
$468$	−1.02503e8	−0.0462250
$469$	−5.33817e8	−0.238939
$470$	9.82376e6	0.00436451
$471$	−2.35808e9	−1.03989
$472$	−9.37526e8	−0.410380
$473$	−7.86131e8	−0.341571
$474$	3.92694e8	0.169368
$475$	1.76705e9	0.756521
$476$	1.87892e8	0.0798515
$477$	−8.86861e8	−0.374146
$478$	−2.73544e9	−1.14559
$479$	−1.16602e9	−0.484764	−0.242382	−	0.970181i	$-0.577929\pi$
$479$	−1.16602e9	−0.484764	−0.242382	+	0.970181i	$0.577929\pi$
$480$	1.46701e7	0.00605464
$481$	7.78518e8	0.318978
$482$	4.62434e8	0.188099
$483$	−8.04960e8	−0.325057
$484$	−8.44025e8	−0.338374
$485$	7.79827e7	0.0310386
$486$	1.14791e8	0.0453609
$487$	2.23938e9	0.878571	0.439286	−	0.898347i	$-0.355232\pi$
$487$	2.23938e9	0.878571	0.439286	+	0.898347i	$0.355232\pi$
$488$	−4.09838e8	−0.159640
$489$	−6.86849e7	−0.0265632
$490$	1.56062e7	0.00599254
$491$	1.75130e9	0.667692	0.333846	−	0.942628i	$-0.391653\pi$
$491$	1.75130e9	0.667692	0.333846	+	0.942628i	$0.391653\pi$
$492$	4.34475e8	0.164470
$493$	1.41202e9	0.530733
$494$	3.98942e8	0.148890
$495$	−3.03383e7	−0.0112428
$496$	−1.09605e9	−0.403315
$497$	1.58685e9	0.579813
$498$	7.59899e8	0.275710
$499$	1.60594e9	0.578599	0.289300	−	0.957239i	$-0.406578\pi$
$499$	1.60594e9	0.578599	0.289300	+	0.957239i	$0.406578\pi$
$500$	−1.65521e8	−0.0592187
$501$	−8.54167e8	−0.303467
$502$	−1.19412e9	−0.421294
$503$	−1.57612e9	−0.552206	−0.276103	−	0.961128i	$-0.589043\pi$
$503$	−1.57612e9	−0.552206	−0.276103	+	0.961128i	$0.589043\pi$
$504$	1.28024e8	0.0445435
$505$	−2.79969e8	−0.0967366
$506$	1.74523e9	0.598859
$507$	1.30324e8	0.0444116
$508$	1.13493e9	0.384101
$509$	−1.26134e9	−0.423955	−0.211977	−	0.977275i	$-0.567990\pi$
$509$	−1.26134e9	−0.423955	−0.211977	+	0.977275i	$0.567990\pi$
$510$	3.06553e7	0.0102332
$511$	−1.03866e9	−0.344349
$512$	1.34218e8	0.0441942
$513$	−4.46767e8	−0.146107
$514$	−1.68273e9	−0.546566
$515$	1.24842e8	0.0402749
$516$	5.41244e8	0.173428
$517$	−1.85872e8	−0.0591558
$518$	−9.72350e8	−0.307375
$519$	2.06802e9	0.649337
$520$	−1.86517e7	−0.00581711
$521$	2.75893e9	0.854690	0.427345	−	0.904089i	$-0.359449\pi$
$521$	2.75893e9	0.854690	0.427345	+	0.904089i	$0.359449\pi$
$522$	9.62110e8	0.296059
$523$	4.05357e9	1.23903	0.619515	−	0.784984i	$-0.287329\pi$
$523$	4.05357e9	1.23903	0.619515	+	0.784984i	$0.287329\pi$
$524$	−2.00667e9	−0.609278
$525$	−7.20969e8	−0.217450
$526$	2.46340e9	0.738047
$527$	−2.29036e9	−0.681658
$528$	−2.77568e8	−0.0820636
$529$	4.15014e9	1.21890
$530$	−1.61375e8	−0.0470838
$531$	−1.33488e9	−0.386910
$532$	−4.98269e8	−0.143474
$533$	−5.52396e8	−0.158018
$534$	1.08591e9	0.308603
$535$	−1.40037e8	−0.0395371
$536$	−7.96835e8	−0.223507
$537$	−2.36143e8	−0.0658058
$538$	−1.83192e9	−0.507189
$539$	−2.95280e8	−0.0812219
$540$	2.08877e7	0.00570837
$541$	1.21412e9	0.329663	0.164832	−	0.986322i	$-0.447292\pi$
$541$	1.21412e9	0.329663	0.164832	+	0.986322i	$0.447292\pi$
$542$	7.35021e8	0.198291
$543$	1.78052e9	0.477252
$544$	2.80468e8	0.0746943
$545$	−4.42385e7	−0.0117061
$546$	−1.62771e8	−0.0427960
$547$	−6.80364e9	−1.77740	−0.888700	−	0.458488i	$-0.848391\pi$
$547$	−6.80364e9	−1.77740	−0.888700	+	0.458488i	$0.848391\pi$
$548$	1.64776e9	0.427722
$549$	−5.83539e8	−0.150510
$550$	1.56313e9	0.400613
$551$	−3.74453e9	−0.953600
$552$	−1.20157e9	−0.304063
$553$	6.23584e8	0.156804
$554$	−3.47412e9	−0.868082
$555$	−1.58643e8	−0.0393909
$556$	7.75666e8	0.191387
$557$	1.71522e9	0.420559	0.210280	−	0.977641i	$-0.432563\pi$
$557$	1.71522e9	0.420559	0.210280	+	0.977641i	$0.432563\pi$
$558$	−1.56059e9	−0.380249
$559$	−6.88145e8	−0.166624
$560$	2.32955e7	0.00560550
$561$	−5.80020e8	−0.138699
$562$	−4.18701e9	−0.995010
$563$	−1.45049e9	−0.342560	−0.171280	−	0.985222i	$-0.554790\pi$
$563$	−1.45049e9	−0.342560	−0.171280	+	0.985222i	$0.554790\pi$
$564$	1.27971e8	0.0300356
$565$	−3.35729e8	−0.0783105
$566$	1.33988e9	0.310606
$567$	1.82284e8	0.0419961
$568$	2.36870e9	0.542365
$569$	4.27183e9	0.972123	0.486061	−	0.873925i	$-0.338433\pi$
$569$	4.27183e9	0.972123	0.486061	+	0.873925i	$0.338433\pi$
$570$	−8.12947e7	−0.0183866
$571$	6.75621e9	1.51872	0.759359	−	0.650672i	$-0.225513\pi$
$571$	6.75621e9	1.51872	0.759359	+	0.650672i	$0.225513\pi$
$572$	3.52903e8	0.0788441
$573$	−6.25721e8	−0.138944
$574$	6.89930e8	0.152270
$575$	6.76667e9	1.48435
$576$	1.91103e8	0.0416667
$577$	2.72966e9	0.591552	0.295776	−	0.955257i	$-0.404422\pi$
$577$	2.72966e9	0.591552	0.295776	+	0.955257i	$0.404422\pi$
$578$	−2.69663e9	−0.580863
$579$	−2.39882e9	−0.513598
$580$	1.75068e8	0.0372570
$581$	1.20669e9	0.255258
$582$	1.01586e9	0.213601
$583$	3.05333e9	0.638166
$584$	−1.55042e9	−0.322109
$585$	−2.65568e7	−0.00548442
$586$	2.78918e9	0.572579
$587$	7.40097e9	1.51027	0.755136	−	0.655568i	$-0.227571\pi$
$587$	7.40097e9	1.51027	0.755136	+	0.655568i	$0.227571\pi$
$588$	2.03297e8	0.0412393
$589$	6.07379e9	1.22478
$590$	−2.42897e8	−0.0486900
$591$	2.23747e9	0.445862
$592$	−1.45144e9	−0.287523
$593$	4.82175e9	0.949540	0.474770	−	0.880110i	$-0.342531\pi$
$593$	4.82175e9	0.949540	0.474770	+	0.880110i	$0.342531\pi$
$594$	−3.95209e8	−0.0773703
$595$	4.86795e7	0.00947408
$596$	−1.13819e9	−0.220217
$597$	3.20446e8	0.0616375
$598$	1.52769e9	0.292134
$599$	−4.55444e9	−0.865847	−0.432923	−	0.901431i	$-0.642518\pi$
$599$	−4.55444e9	−0.865847	−0.432923	+	0.901431i	$0.642518\pi$
$600$	−1.07620e9	−0.203406
$601$	7.76790e8	0.145963	0.0729815	−	0.997333i	$-0.476749\pi$
$601$	7.76790e8	0.145963	0.0729815	+	0.997333i	$0.476749\pi$
$602$	8.59476e8	0.160563
$603$	−1.13456e9	−0.210725
$604$	4.36092e8	0.0805283
$605$	−2.18672e8	−0.0401467
$606$	−3.64708e9	−0.665720
$607$	9.52087e9	1.72789	0.863945	−	0.503586i	$-0.167986\pi$
$607$	9.52087e9	1.72789	0.863945	+	0.503586i	$0.167986\pi$
$608$	−7.43772e8	−0.134208
$609$	1.52779e9	0.274097
$610$	−1.06182e8	−0.0189407
$611$	−1.62704e8	−0.0288572
$612$	3.99338e8	0.0704224
$613$	2.69289e9	0.472179	0.236089	−	0.971731i	$-0.424134\pi$
$613$	2.69289e9	0.472179	0.236089	+	0.971731i	$0.424134\pi$
$614$	−3.01277e9	−0.525263
$615$	1.12565e8	0.0195137
$616$	−4.40768e8	−0.0759761
$617$	−5.04887e9	−0.865358	−0.432679	−	0.901548i	$-0.642432\pi$
$617$	−5.04887e9	−0.865358	−0.432679	+	0.901548i	$0.642432\pi$
$618$	1.62628e9	0.277163
$619$	−3.78824e9	−0.641979	−0.320989	−	0.947083i	$-0.604015\pi$
$619$	−3.78824e9	−0.641979	−0.320989	+	0.947083i	$0.604015\pi$
$620$	−2.83968e8	−0.0478518
$621$	−1.71083e9	−0.286673
$622$	−6.39915e9	−1.06624
$623$	1.72439e9	0.285711
$624$	−2.42971e8	−0.0400320
$625$	6.03915e9	0.989455
$626$	−1.21178e9	−0.197430
$627$	1.53815e9	0.249208
$628$	−5.58953e9	−0.900568
$629$	−3.03300e9	−0.485954
$630$	3.31688e7	0.00528492
$631$	1.11029e10	1.75928	0.879638	−	0.475644i	$-0.157785\pi$
$631$	1.11029e10	1.75928	0.879638	+	0.475644i	$0.157785\pi$
$632$	9.30831e8	0.146677
$633$	2.12440e9	0.332908
$634$	7.85788e9	1.22460
$635$	2.94041e8	0.0455721
$636$	−2.10219e9	−0.324020
$637$	−2.58475e8	−0.0396214
$638$	−3.31240e9	−0.504975
$639$	3.37263e9	0.511347
$640$	3.47735e7	0.00524347
$641$	8.68379e9	1.30229	0.651143	−	0.758955i	$-0.274290\pi$
$641$	8.68379e9	1.30229	0.651143	+	0.758955i	$0.274290\pi$
$642$	−1.82422e9	−0.272086
$643$	1.43389e9	0.212704	0.106352	−	0.994329i	$-0.466083\pi$
$643$	1.43389e9	0.212704	0.106352	+	0.994329i	$0.466083\pi$
$644$	−1.90805e9	−0.281507
$645$	1.40227e8	0.0205766
$646$	−1.55422e9	−0.226829
$647$	4.97077e9	0.721537	0.360768	−	0.932655i	$-0.382514\pi$
$647$	4.97077e9	0.721537	0.360768	+	0.932655i	$0.382514\pi$
$648$	2.72098e8	0.0392837
$649$	4.59578e9	0.659937
$650$	1.36829e9	0.195426
$651$	−2.47815e9	−0.352042
$652$	−1.62809e8	−0.0230044
$653$	3.28133e9	0.461162	0.230581	−	0.973053i	$-0.425937\pi$
$653$	3.28133e9	0.461162	0.230581	+	0.973053i	$0.425937\pi$
$654$	−5.76282e8	−0.0805588
$655$	−5.19893e8	−0.0722885
$656$	1.02987e9	0.142435
$657$	−2.20753e9	−0.303688
$658$	2.03214e8	0.0278075
$659$	−6.00056e9	−0.816757	−0.408378	−	0.912813i	$-0.633906\pi$
$659$	−6.00056e9	−0.816757	−0.408378	+	0.912813i	$0.633906\pi$
$660$	−7.19131e7	−0.00973653
$661$	1.67846e9	0.226051	0.113026	−	0.993592i	$-0.463946\pi$
$661$	1.67846e9	0.226051	0.113026	+	0.993592i	$0.463946\pi$
$662$	8.92990e9	1.19631
$663$	−5.07723e8	−0.0676597
$664$	1.80124e9	0.238772
$665$	−1.29093e8	−0.0170226
$666$	−2.06660e9	−0.271079
$667$	−1.43391e10	−1.87104
$668$	−2.02469e9	−0.262810
$669$	6.20966e9	0.801819
$670$	−2.06446e8	−0.0265183
$671$	2.00904e9	0.256720
$672$	3.03464e8	0.0385758
$673$	−4.34546e9	−0.549519	−0.274760	−	0.961513i	$-0.588598\pi$
$673$	−4.34546e9	−0.549519	−0.274760	+	0.961513i	$0.588598\pi$
$674$	−5.77991e9	−0.727129
$675$	−1.53232e9	−0.191773
$676$	3.08916e8	0.0384615
$677$	9.16603e9	1.13533	0.567663	−	0.823261i	$-0.307848\pi$
$677$	9.16603e9	1.13533	0.567663	+	0.823261i	$0.307848\pi$
$678$	−4.37345e9	−0.538915
$679$	1.61315e9	0.197756
$680$	7.26645e7	0.00886219
$681$	−4.29616e9	−0.521273
$682$	5.37286e9	0.648575
$683$	−4.83662e9	−0.580857	−0.290428	−	0.956897i	$-0.593798\pi$
$683$	−4.83662e9	−0.580857	−0.290428	+	0.956897i	$0.593798\pi$
$684$	−1.05900e9	−0.126532
$685$	4.26906e8	0.0507475
$686$	3.22829e8	0.0381802
$687$	−9.92791e8	−0.116818
$688$	1.28295e9	0.150193
$689$	2.67275e9	0.311308
$690$	−3.11307e8	−0.0360759
$691$	−1.69887e9	−0.195879	−0.0979396	−	0.995192i	$-0.531225\pi$
$691$	−1.69887e9	−0.195879	−0.0979396	+	0.995192i	$0.531225\pi$
$692$	4.90199e9	0.562342
$693$	−6.27577e8	−0.0716310
$694$	5.12270e9	0.581756
$695$	2.00962e8	0.0227074
$696$	2.28056e9	0.256394
$697$	2.15206e9	0.240735
$698$	−1.25420e9	−0.139596
$699$	−1.46142e9	−0.161847
$700$	−1.70896e9	−0.188317
$701$	−4.51570e9	−0.495121	−0.247561	−	0.968872i	$-0.579629\pi$
$701$	−4.51570e9	−0.495121	−0.247561	+	0.968872i	$0.579629\pi$
$702$	−3.45948e8	−0.0377426
$703$	8.04319e9	0.873142
$704$	−6.57939e8	−0.0710691
$705$	3.31552e7	0.00356360
$706$	2.95299e9	0.315824
$707$	−5.79143e9	−0.616337
$708$	−3.16415e9	−0.335074
$709$	1.60085e10	1.68689	0.843447	−	0.537213i	$-0.180523\pi$
$709$	1.60085e10	1.68689	0.843447	+	0.537213i	$0.180523\pi$
$710$	6.13691e8	0.0643495
$711$	1.32534e9	0.138288
$712$	2.57401e9	0.267258
$713$	2.32588e10	2.40311
$714$	6.34134e8	0.0651985
$715$	9.14312e7	0.00935455
$716$	−5.59745e8	−0.0569895
$717$	−9.23212e9	−0.935372
$718$	2.40052e9	0.242030
$719$	1.23167e10	1.23579	0.617894	−	0.786261i	$-0.287986\pi$
$719$	1.23167e10	1.23579	0.617894	+	0.786261i	$0.287986\pi$
$720$	4.95115e7	0.00494359
$721$	2.58247e9	0.256603
$722$	−3.02934e9	−0.299549
$723$	1.56072e9	0.153582
$724$	4.22049e9	0.413312
$725$	−1.28430e10	−1.25165
$726$	−2.84858e9	−0.276281
$727$	1.04990e10	1.01339	0.506697	−	0.862124i	$-0.330866\pi$
$727$	1.04990e10	1.01339	0.506697	+	0.862124i	$0.330866\pi$
$728$	−3.85828e8	−0.0370625
$729$	3.87420e8	0.0370370
$730$	−4.01686e8	−0.0382170
$731$	2.68091e9	0.253847
$732$	−1.38320e9	−0.130346
$733$	9.93793e9	0.932035	0.466017	−	0.884776i	$-0.345688\pi$
$733$	9.93793e9	0.932035	0.466017	+	0.884776i	$0.345688\pi$
$734$	3.66113e9	0.341727
$735$	5.26709e7	0.00489288
$736$	−2.84817e9	−0.263326
$737$	3.90611e9	0.359425
$738$	1.46635e9	0.134289
$739$	6.82065e9	0.621684	0.310842	−	0.950462i	$-0.399389\pi$
$739$	6.82065e9	0.621684	0.310842	+	0.950462i	$0.399389\pi$
$740$	−3.76043e8	−0.0341135
$741$	1.34643e9	0.121568
$742$	−3.33820e9	−0.299984
$743$	−2.80748e9	−0.251106	−0.125553	−	0.992087i	$-0.540070\pi$
$743$	−2.80748e9	−0.251106	−0.125553	+	0.992087i	$0.540070\pi$
$744$	−3.69917e9	−0.329305
$745$	−2.94885e8	−0.0261279
$746$	4.71285e8	0.0415622
$747$	2.56466e9	0.225117
$748$	−1.37486e9	−0.120117
$749$	−2.89680e9	−0.251902
$750$	−5.58634e8	−0.0483519
$751$	−1.11715e10	−0.962438	−0.481219	−	0.876600i	$-0.659806\pi$
$751$	−1.11715e10	−0.962438	−0.481219	+	0.876600i	$0.659806\pi$
$752$	3.03339e8	0.0260116
$753$	−4.03016e9	−0.343985
$754$	−2.89953e9	−0.246336
$755$	1.12984e8	0.00955438
$756$	4.32081e8	0.0363696
$757$	−4.68714e9	−0.392711	−0.196355	−	0.980533i	$-0.562911\pi$
$757$	−4.68714e9	−0.392711	−0.196355	+	0.980533i	$0.562911\pi$
$758$	−1.39227e10	−1.16113
$759$	5.89014e9	0.488967
$760$	−1.92699e8	−0.0159232
$761$	−1.36075e10	−1.11926	−0.559631	−	0.828742i	$-0.689057\pi$
$761$	−1.36075e10	−1.11926	−0.559631	+	0.828742i	$0.689057\pi$
$762$	3.83038e9	0.313617
$763$	−9.15115e8	−0.0745830
$764$	−1.48319e9	−0.120329
$765$	1.03462e8	0.00835535
$766$	−8.00202e9	−0.643278
$767$	4.02294e9	0.321929
$768$	4.52985e8	0.0360844
$769$	2.04755e10	1.62365	0.811827	−	0.583898i	$-0.198473\pi$
$769$	2.04755e10	1.62365	0.811827	+	0.583898i	$0.198473\pi$
$770$	−1.14195e8	−0.00901427
$771$	−5.67920e9	−0.446269
$772$	−5.68610e9	−0.444789
$773$	−6.90807e9	−0.537933	−0.268967	−	0.963150i	$-0.586682\pi$
$773$	−6.90807e9	−0.537933	−0.268967	+	0.963150i	$0.586682\pi$
$774$	1.82670e9	0.141603
$775$	2.08319e10	1.60758
$776$	2.40796e9	0.184984
$777$	−3.28168e9	−0.250971
$778$	−1.04358e9	−0.0794508
$779$	−5.70704e9	−0.432543
$780$	−6.29495e7	−0.00474965
$781$	−1.16114e10	−0.872183
$782$	−5.95168e9	−0.445057
$783$	3.24712e9	0.241731
$784$	4.81890e8	0.0357143
$785$	−1.44815e9	−0.106849
$786$	−6.77250e9	−0.497474
$787$	2.23442e10	1.63401	0.817003	−	0.576633i	$-0.195634\pi$
$787$	2.23442e10	1.63401	0.817003	+	0.576633i	$0.195634\pi$
$788$	5.30362e9	0.386128
$789$	8.31396e9	0.602613
$790$	2.41162e8	0.0174026
$791$	−6.94488e9	−0.498939
$792$	−9.36792e8	−0.0670046
$793$	1.75862e9	0.125232
$794$	−1.25474e10	−0.889577
$795$	−5.44641e8	−0.0384437
$796$	7.59576e8	0.0533796
$797$	−2.47116e10	−1.72901	−0.864504	−	0.502627i	$-0.832367\pi$
$797$	−2.47116e10	−1.72901	−0.864504	+	0.502627i	$0.832367\pi$
$798$	−1.68166e9	−0.117146
$799$	6.33873e8	0.0439631
$800$	−2.55099e9	−0.176155
$801$	3.66495e9	0.251973
$802$	−1.67365e8	−0.0114565
$803$	7.60018e9	0.517988
$804$	−2.68932e9	−0.182493
$805$	−4.94344e8	−0.0333998
$806$	4.70316e9	0.316386
$807$	−6.18275e9	−0.414118
$808$	−8.64494e9	−0.576530
$809$	−1.01049e10	−0.670982	−0.335491	−	0.942043i	$-0.608902\pi$
$809$	−1.01049e10	−0.670982	−0.335491	+	0.942043i	$0.608902\pi$
$810$	7.04959e7	0.00466086
$811$	2.01551e10	1.32682	0.663410	−	0.748256i	$-0.269108\pi$
$811$	2.01551e10	1.32682	0.663410	+	0.748256i	$0.269108\pi$
$812$	3.62144e9	0.237375
$813$	2.48070e9	0.161904
$814$	7.11499e9	0.462369
$815$	−4.21810e7	−0.00272939
$816$	9.46580e8	0.0609876
$817$	−7.10951e9	−0.456102
$818$	1.31429e10	0.839563
$819$	−5.49353e8	−0.0349428
$820$	2.66821e8	0.0168994
$821$	1.67800e9	0.105825	0.0529127	−	0.998599i	$-0.483149\pi$
$821$	1.67800e9	0.105825	0.0529127	+	0.998599i	$0.483149\pi$
$822$	5.56118e9	0.349233
$823$	3.01965e10	1.88824	0.944122	−	0.329598i	$-0.106913\pi$
$823$	3.01965e10	1.88824	0.944122	+	0.329598i	$0.106913\pi$
$824$	3.85488e9	0.240030
$825$	5.27556e9	0.327099
$826$	−5.02456e9	−0.310218
$827$	−2.09299e10	−1.28676	−0.643379	−	0.765547i	$-0.722468\pi$
$827$	−2.09299e10	−1.28676	−0.643379	+	0.765547i	$0.722468\pi$
$828$	−4.05531e9	−0.248266
$829$	−9.76043e9	−0.595015	−0.297507	−	0.954719i	$-0.596155\pi$
$829$	−9.76043e9	−0.595015	−0.297507	+	0.954719i	$0.596155\pi$
$830$	4.66671e8	0.0283294
$831$	−1.17252e10	−0.708786
$832$	−5.75930e8	−0.0346688
$833$	1.00698e9	0.0603621
$834$	2.61787e9	0.156267
$835$	−5.24563e8	−0.0311814
$836$	3.64599e9	0.215821
$837$	−5.26698e9	−0.310472
$838$	1.19270e10	0.700127
$839$	−1.19167e10	−0.696608	−0.348304	−	0.937382i	$-0.613242\pi$
$839$	−1.19167e10	−0.696608	−0.348304	+	0.937382i	$0.613242\pi$
$840$	7.86224e7	0.00457687
$841$	9.96548e9	0.577713
$842$	−2.44942e9	−0.141407
$843$	−1.41312e10	−0.812422
$844$	5.03562e9	0.288306
$845$	8.00348e7	0.00456331
$846$	4.31903e8	0.0245239
$847$	−4.52345e9	−0.255786
$848$	−4.98297e9	−0.280610
$849$	4.52211e9	0.253609
$850$	−5.33068e9	−0.297726
$851$	3.08003e10	1.71317
$852$	7.99438e9	0.442839
$853$	1.12333e10	0.619707	0.309854	−	0.950784i	$-0.399720\pi$
$853$	1.12333e10	0.619707	0.309854	+	0.950784i	$0.399720\pi$
$854$	−2.19648e9	−0.120677
$855$	−2.74370e8	−0.0150126
$856$	−4.32409e9	−0.235633
$857$	−2.71544e10	−1.47370	−0.736848	−	0.676058i	$-0.763687\pi$
$857$	−2.71544e10	−1.47370	−0.736848	+	0.676058i	$0.763687\pi$
$858$	1.19105e9	0.0643760
$859$	5.91416e9	0.318359	0.159179	−	0.987250i	$-0.449115\pi$
$859$	5.91416e9	0.318359	0.159179	+	0.987250i	$0.449115\pi$
$860$	3.32390e8	0.0178198
$861$	2.32851e9	0.124328
$862$	6.81567e9	0.362438
$863$	6.58372e9	0.348685	0.174343	−	0.984685i	$-0.444220\pi$
$863$	6.58372e9	0.348685	0.174343	+	0.984685i	$0.444220\pi$
$864$	6.44973e8	0.0340207
$865$	1.27002e9	0.0667197
$866$	1.64661e10	0.861548
$867$	−9.10112e9	−0.474273
$868$	−5.87414e9	−0.304877
$869$	−4.56296e9	−0.235872
$870$	5.90853e8	0.0304202
$871$	3.41923e9	0.175334
$872$	−1.36600e9	−0.0697660
$873$	3.42853e9	0.174404
$874$	1.57832e10	0.799661
$875$	−8.87091e8	−0.0447651
$876$	−5.23266e9	−0.263001
$877$	−2.73245e10	−1.36790	−0.683949	−	0.729530i	$-0.739739\pi$
$877$	−2.73245e10	−1.36790	−0.683949	+	0.729530i	$0.739739\pi$
$878$	7.92891e9	0.395351
$879$	9.41350e9	0.467509
$880$	−1.70461e8	−0.00843208
$881$	−3.26826e10	−1.61028	−0.805138	−	0.593087i	$-0.797909\pi$
$881$	−3.26826e10	−1.61028	−0.805138	+	0.593087i	$0.797909\pi$
$882$	6.86129e8	0.0336718
$883$	2.45161e10	1.19837	0.599183	−	0.800612i	$-0.295492\pi$
$883$	2.45161e10	1.19837	0.599183	+	0.800612i	$0.295492\pi$
$884$	−1.20349e9	−0.0585950
$885$	−8.19777e8	−0.0397553
$886$	−2.43739e9	−0.117735
$887$	−3.52691e10	−1.69692	−0.848460	−	0.529259i	$-0.822470\pi$
$887$	−3.52691e10	−1.69692	−0.848460	+	0.529259i	$0.822470\pi$
$888$	−4.89860e9	−0.234762
$889$	6.08251e9	0.290353
$890$	6.66882e8	0.0317091
$891$	−1.33383e9	−0.0631726
$892$	1.47192e10	0.694396
$893$	−1.68097e9	−0.0789912
$894$	−3.84138e9	−0.179807
$895$	−1.45020e8	−0.00676159
$896$	7.19323e8	0.0334077
$897$	5.15597e9	0.238526
$898$	2.27906e10	1.05024
$899$	−4.41446e10	−2.02637
$900$	−3.63217e9	−0.166080
$901$	−1.04127e10	−0.474269
$902$	−5.04843e9	−0.229052
$903$	2.90073e9	0.131099
$904$	−1.03667e10	−0.466714
$905$	1.09346e9	0.0490379
$906$	1.47181e9	0.0657511
$907$	6.49706e9	0.289129	0.144564	−	0.989495i	$-0.453822\pi$
$907$	6.49706e9	0.289129	0.144564	+	0.989495i	$0.453822\pi$
$908$	−1.01835e10	−0.451436
$909$	−1.23089e10	−0.543558
$910$	−9.99615e7	−0.00439732
$911$	−3.67254e10	−1.60936	−0.804678	−	0.593712i	$-0.797662\pi$
$911$	−3.67254e10	−1.60936	−0.804678	+	0.593712i	$0.797662\pi$
$912$	−2.51023e9	−0.109580
$913$	−8.82973e9	−0.383972
$914$	−8.15795e9	−0.353402
$915$	−3.58364e8	−0.0154650
$916$	−2.35328e9	−0.101167
$917$	−1.07545e10	−0.460571
$918$	1.34777e9	0.0574997
$919$	−9.71584e9	−0.412930	−0.206465	−	0.978454i	$-0.566196\pi$
$919$	−9.71584e9	−0.412930	−0.206465	+	0.978454i	$0.566196\pi$
$920$	−7.37913e8	−0.0312426
$921$	−1.01681e10	−0.428876
$922$	−7.47611e9	−0.314136
$923$	−1.01641e10	−0.425466
$924$	−1.48759e9	−0.0620342
$925$	2.75866e10	1.14604
$926$	−1.45116e10	−0.600589
$927$	5.48869e9	0.226303
$928$	5.40576e9	0.222044
$929$	3.18508e10	1.30336	0.651682	−	0.758493i	$-0.274064\pi$
$929$	3.18508e10	1.30336	0.651682	+	0.758493i	$0.274064\pi$
$930$	−9.58391e8	−0.0390708
$931$	−2.67041e9	−0.108456
$932$	−3.46410e9	−0.140164
$933$	−2.15971e10	−0.870584
$934$	3.46964e9	0.139338
$935$	−3.56203e8	−0.0142514
$936$	−8.20026e8	−0.0326860
$937$	−4.80225e9	−0.190703	−0.0953513	−	0.995444i	$-0.530397\pi$
$937$	−4.80225e9	−0.190703	−0.0953513	+	0.995444i	$0.530397\pi$
$938$	−4.27054e9	−0.168956
$939$	−4.08975e9	−0.161201
$940$	7.85901e7	0.00308617
$941$	−3.95668e9	−0.154799	−0.0773993	−	0.997000i	$-0.524662\pi$
$941$	−3.95668e9	−0.154799	−0.0773993	+	0.997000i	$0.524662\pi$
$942$	−1.88647e10	−0.735311
$943$	−2.18543e10	−0.848684
$944$	−7.50021e9	−0.290183
$945$	1.11945e8	0.00431512
$946$	−6.28905e9	−0.241527
$947$	−2.85265e10	−1.09150	−0.545750	−	0.837948i	$-0.683755\pi$
$947$	−2.85265e10	−1.09150	−0.545750	+	0.837948i	$0.683755\pi$
$948$	3.14155e9	0.119761
$949$	6.65286e9	0.252683
$950$	1.41364e10	0.534941
$951$	2.65204e10	0.999879
$952$	1.50313e9	0.0564636
$953$	−1.61784e10	−0.605494	−0.302747	−	0.953071i	$-0.597904\pi$
$953$	−1.61784e10	−0.605494	−0.302747	+	0.953071i	$0.597904\pi$
$954$	−7.09489e9	−0.264561
$955$	−3.84269e8	−0.0142766
$956$	−2.18835e10	−0.810056
$957$	−1.11793e10	−0.412311
$958$	−9.32814e9	−0.342780
$959$	8.83095e9	0.323327
$960$	1.17361e8	0.00428127
$961$	4.40919e10	1.60261
$962$	6.22814e9	0.225552
$963$	−6.15675e9	−0.222157
$964$	3.69947e9	0.133006
$965$	−1.47317e9	−0.0527725
$966$	−6.43968e9	−0.229850
$967$	−2.03897e10	−0.725134	−0.362567	−	0.931958i	$-0.618100\pi$
$967$	−2.03897e10	−0.725134	−0.362567	+	0.931958i	$0.618100\pi$
$968$	−6.75220e9	−0.239266
$969$	−5.24550e9	−0.185205
$970$	6.23862e8	0.0219476
$971$	−1.99059e10	−0.697775	−0.348888	−	0.937165i	$-0.613440\pi$
$971$	−1.99059e10	−0.697775	−0.348888	+	0.937165i	$0.613440\pi$
$972$	9.18330e8	0.0320750
$973$	4.15708e9	0.144675
$974$	1.79151e10	0.621244
$975$	4.61799e9	0.159565
$976$	−3.27870e9	−0.112883
$977$	1.92627e10	0.660823	0.330412	−	0.943837i	$-0.392812\pi$
$977$	1.92627e10	0.660823	0.330412	+	0.943837i	$0.392812\pi$
$978$	−5.49480e8	−0.0187830
$979$	−1.26179e10	−0.429780
$980$	1.24850e8	0.00423736
$981$	−1.94495e9	−0.0657760
$982$	1.40104e10	0.472130
$983$	−3.50791e9	−0.117791	−0.0588954	−	0.998264i	$-0.518758\pi$
$983$	−3.50791e9	−0.117791	−0.0588954	+	0.998264i	$0.518758\pi$
$984$	3.47580e9	0.116298
$985$	1.37408e9	0.0458126
$986$	1.12962e10	0.375285
$987$	6.85846e8	0.0227048
$988$	3.19154e9	0.105281
$989$	−2.72249e10	−0.894909
$990$	−2.42707e8	−0.00794984
$991$	−4.13848e9	−0.135077	−0.0675387	−	0.997717i	$-0.521515\pi$
$991$	−4.13848e9	−0.135077	−0.0675387	+	0.997717i	$0.521515\pi$
$992$	−8.76839e9	−0.285187
$993$	3.01384e10	0.976783
$994$	1.26948e10	0.409989
$995$	1.96793e8	0.00633328
$996$	6.07919e9	0.194957
$997$	4.67921e10	1.49534	0.747669	−	0.664071i	$-0.231173\pi$
$997$	4.67921e10	1.49534	0.747669	+	0.664071i	$0.231173\pi$
$998$	1.28475e10	0.409131
$999$	−6.97477e9	−0.221335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.8.a.g.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.8.a.g.1.3	✓	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +16.5813 q^{5} +216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +16.5813 q^{5} +216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +132.650 q^{10} -2509.84 q^{11} +1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} +447.695 q^{15} +4096.00 q^{16} +8559.21 q^{17} +5832.00 q^{18} -22698.1 q^{19} +1061.20 q^{20} +9261.00 q^{21} -20078.7 q^{22} -86919.3 q^{23} +13824.0 q^{24} -77850.1 q^{25} -17576.0 q^{26} +19683.0 q^{27} +21952.0 q^{28} +164971. q^{29} +3581.56 q^{30} -267590. q^{31} +32768.0 q^{32} -67765.6 q^{33} +68473.6 q^{34} +5687.39 q^{35} +46656.0 q^{36} -354355. q^{37} -181585. q^{38} -59319.0 q^{39} +8489.63 q^{40} +251432. q^{41} +74088.0 q^{42} +313220. q^{43} -160630. q^{44} +12087.8 q^{45} -695354. q^{46} +74057.5 q^{47} +110592. q^{48} +117649. q^{49} -622800. q^{50} +231099. q^{51} -140608. q^{52} -1.21654e6 q^{53} +157464. q^{54} -41616.4 q^{55} +175616. q^{56} -612849. q^{57} +1.31977e6 q^{58} -1.83111e6 q^{59} +28652.5 q^{60} -800465. q^{61} -2.14072e6 q^{62} +250047. q^{63} +262144. q^{64} -36429.1 q^{65} -542125. q^{66} -1.55632e6 q^{67} +547789. q^{68} -2.34682e6 q^{69} +45499.1 q^{70} +4.62638e6 q^{71} +373248. q^{72} -3.02816e6 q^{73} -2.83484e6 q^{74} -2.10195e6 q^{75} -1.45268e6 q^{76} -860874. q^{77} -474552. q^{78} +1.81803e6 q^{79} +67917.0 q^{80} +531441. q^{81} +2.01146e6 q^{82} +3.51805e6 q^{83} +592704. q^{84} +141923. q^{85} +2.50576e6 q^{86} +4.45421e6 q^{87} -1.28504e6 q^{88} +5.02737e6 q^{89} +96702.2 q^{90} -753571. q^{91} -5.56284e6 q^{92} -7.22493e6 q^{93} +592460. q^{94} -376364. q^{95} +884736. q^{96} +4.70305e6 q^{97} +941192. q^{98} -1.82967e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} - 328 q^{5} + 864 q^{6} + 1372 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) 4 * q + 32 * q^2 + 108 * q^3 + 256 * q^4 - 328 * q^5 + 864 * q^6 + 1372 * q^7 + 2048 * q^8 + 2916 * q^9	\( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} - 328 q^{5} + 864 q^{6} + 1372 q^{7} + 2048 q^{8} + 2916 q^{9} - 2624 q^{10} - 7945 q^{11} + 6912 q^{12} - 8788 q^{13} + 10976 q^{14} - 8856 q^{15} + 16384 q^{16} - 27285 q^{17} + 23328 q^{18} - 34802 q^{19} - 20992 q^{20} + 37044 q^{21} - 63560 q^{22} - 22316 q^{23} + 55296 q^{24} - 64650 q^{25} - 70304 q^{26} + 78732 q^{27} + 87808 q^{28} - 158470 q^{29} - 70848 q^{30} - 274647 q^{31} + 131072 q^{32} - 214515 q^{33} - 218280 q^{34} - 112504 q^{35} + 186624 q^{36} - 319169 q^{37} - 278416 q^{38} - 237276 q^{39} - 167936 q^{40} - 711254 q^{41} + 296352 q^{42} - 893034 q^{43} - 508480 q^{44} - 239112 q^{45} - 178528 q^{46} - 563129 q^{47} + 442368 q^{48} + 470596 q^{49} - 517200 q^{50} - 736695 q^{51} - 562432 q^{52} - 2186067 q^{53} + 629856 q^{54} - 1242387 q^{55} + 702464 q^{56} - 939654 q^{57} - 1267760 q^{58} + 53200 q^{59} - 566784 q^{60} - 2419497 q^{61} - 2197176 q^{62} + 1000188 q^{63} + 1048576 q^{64} + 720616 q^{65} - 1716120 q^{66} - 4908916 q^{67} - 1746240 q^{68} - 602532 q^{69} - 900032 q^{70} - 5546448 q^{71} + 1492992 q^{72} + 6114536 q^{73} - 2553352 q^{74} - 1745550 q^{75} - 2227328 q^{76} - 2725135 q^{77} - 1898208 q^{78} + 6523461 q^{79} - 1343488 q^{80} + 2125764 q^{81} - 5690032 q^{82} + 5265065 q^{83} + 2370816 q^{84} - 10347427 q^{85} - 7144272 q^{86} - 4278690 q^{87} - 4067840 q^{88} - 4679151 q^{89} - 1912896 q^{90} - 3014284 q^{91} - 1428224 q^{92} - 7415469 q^{93} - 4505032 q^{94} + 13009536 q^{95} + 3538944 q^{96} + 12566137 q^{97} + 3764768 q^{98} - 5791905 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 + 108 * q^3 + 256 * q^4 - 328 * q^5 + 864 * q^6 + 1372 * q^7 + 2048 * q^8 + 2916 * q^9 - 2624 * q^10 - 7945 * q^11 + 6912 * q^12 - 8788 * q^13 + 10976 * q^14 - 8856 * q^15 + 16384 * q^16 - 27285 * q^17 + 23328 * q^18 - 34802 * q^19 - 20992 * q^20 + 37044 * q^21 - 63560 * q^22 - 22316 * q^23 + 55296 * q^24 - 64650 * q^25 - 70304 * q^26 + 78732 * q^27 + 87808 * q^28 - 158470 * q^29 - 70848 * q^30 - 274647 * q^31 + 131072 * q^32 - 214515 * q^33 - 218280 * q^34 - 112504 * q^35 + 186624 * q^36 - 319169 * q^37 - 278416 * q^38 - 237276 * q^39 - 167936 * q^40 - 711254 * q^41 + 296352 * q^42 - 893034 * q^43 - 508480 * q^44 - 239112 * q^45 - 178528 * q^46 - 563129 * q^47 + 442368 * q^48 + 470596 * q^49 - 517200 * q^50 - 736695 * q^51 - 562432 * q^52 - 2186067 * q^53 + 629856 * q^54 - 1242387 * q^55 + 702464 * q^56 - 939654 * q^57 - 1267760 * q^58 + 53200 * q^59 - 566784 * q^60 - 2419497 * q^61 - 2197176 * q^62 + 1000188 * q^63 + 1048576 * q^64 + 720616 * q^65 - 1716120 * q^66 - 4908916 * q^67 - 1746240 * q^68 - 602532 * q^69 - 900032 * q^70 - 5546448 * q^71 + 1492992 * q^72 + 6114536 * q^73 - 2553352 * q^74 - 1745550 * q^75 - 2227328 * q^76 - 2725135 * q^77 - 1898208 * q^78 + 6523461 * q^79 - 1343488 * q^80 + 2125764 * q^81 - 5690032 * q^82 + 5265065 * q^83 + 2370816 * q^84 - 10347427 * q^85 - 7144272 * q^86 - 4278690 * q^87 - 4067840 * q^88 - 4679151 * q^89 - 1912896 * q^90 - 3014284 * q^91 - 1428224 * q^92 - 7415469 * q^93 - 4505032 * q^94 + 13009536 * q^95 + 3538944 * q^96 + 12566137 * q^97 + 3764768 * q^98 - 5791905 * q^99

Introduction

L-functions

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