LMFDB - Embedded newform 363.8.a.g.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [363,8,Mod(1,363)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("363.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	363.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$113.395764251$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{89}, \sqrt{449})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 269x^{2} + 8100 $ x^4 - 269*x^2 + 8100
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$15.3118$ of defining polynomial
Character	$\chi$	$=$	363.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+15.3118 q^{2} +27.0000 q^{3} +106.451 q^{4} -150.902 q^{5} +413.419 q^{6} +450.951 q^{7} -329.950 q^{8} +729.000 q^{9} -2310.59 q^{10} +2874.18 q^{12} +1834.43 q^{13} +6904.88 q^{14} -4074.37 q^{15} -18677.9 q^{16} +3650.74 q^{17} +11162.3 q^{18} -35707.3 q^{19} -16063.8 q^{20} +12175.7 q^{21} -48865.9 q^{23} -8908.66 q^{24} -55353.4 q^{25} +28088.4 q^{26} +19683.0 q^{27} +48004.3 q^{28} -59377.9 q^{29} -62385.9 q^{30} +216739. q^{31} -243759. q^{32} +55899.4 q^{34} -68049.7 q^{35} +77603.0 q^{36} -56889.3 q^{37} -546743. q^{38} +49529.6 q^{39} +49790.3 q^{40} -155773. q^{41} +186432. q^{42} -9452.73 q^{43} -110008. q^{45} -748225. q^{46} -247316. q^{47} -504303. q^{48} -620186. q^{49} -847561. q^{50} +98569.9 q^{51} +195277. q^{52} +742117. q^{53} +301382. q^{54} -148792. q^{56} -964098. q^{57} -909182. q^{58} -1.27162e6 q^{59} -433721. q^{60} -1.38324e6 q^{61} +3.31867e6 q^{62} +328744. q^{63} -1.34161e6 q^{64} -276820. q^{65} -948760. q^{67} +388626. q^{68} -1.31938e6 q^{69} -1.04196e6 q^{70} -570590. q^{71} -240534. q^{72} +3.99621e6 q^{73} -871078. q^{74} -1.49454e6 q^{75} -3.80109e6 q^{76} +758387. q^{78} -6.79990e6 q^{79} +2.81854e6 q^{80} +531441. q^{81} -2.38517e6 q^{82} -9.23584e6 q^{83} +1.29612e6 q^{84} -550906. q^{85} -144738. q^{86} -1.60320e6 q^{87} +6.65992e6 q^{89} -1.68442e6 q^{90} +827239. q^{91} -5.20184e6 q^{92} +5.85196e6 q^{93} -3.78685e6 q^{94} +5.38832e6 q^{95} -6.58148e6 q^{96} -1.60645e7 q^{97} -9.49616e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 108 q^{3} + 26 q^{4} + 196 q^{5} + 2916 q^{9} + 702 q^{12} + 2032 q^{14} + 5292 q^{15} - 28734 q^{16} - 78648 q^{20} - 55532 q^{23} - 143052 q^{25} + 9204 q^{26} + 78732 q^{27} - 7816 q^{31} + 390716 q^{34}+ \cdots - 4819600 q^{97}+O(q^{100}) $ 4 * q + 108 * q^3 + 26 * q^4 + 196 * q^5 + 2916 * q^9 + 702 * q^12 + 2032 * q^14 + 5292 * q^15 - 28734 * q^16 - 78648 * q^20 - 55532 * q^23 - 143052 * q^25 + 9204 * q^26 + 78732 * q^27 - 7816 * q^31 + 390716 * q^34 + 18954 * q^36 - 763296 * q^37 - 550172 * q^38 + 54864 * q^42 + 142884 * q^45 - 1141988 * q^47 - 775818 * q^48 - 879924 * q^49 - 1035180 * q^53 + 2310616 * q^56 - 221596 * q^58 - 3891864 * q^59 - 2123496 * q^60 - 1530318 * q^64 + 1389632 * q^67 - 1499364 * q^69 - 5015440 * q^70 - 8309020 * q^71 - 3862404 * q^75 + 248508 * q^78 + 7783064 * q^80 + 2125764 * q^81 - 10542572 * q^82 - 3688636 * q^86 - 5296744 * q^89 + 9661056 * q^91 - 14347308 * q^92 - 211032 * q^93 - 4819600 * q^97

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 $

$2$	15.3118	1.35338	0.676692	−	0.736266i	$-0.263413\pi$
$2$	15.3118	1.35338	0.676692	+	0.736266i	$0.263413\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	106.451	0.831650
$5$	−150.902	−0.539885	−0.269943	−	0.962876i	$-0.587005\pi$
$5$	−150.902	−0.539885	−0.269943	+	0.962876i	$0.587005\pi$
$6$	413.419	0.781377
$7$	450.951	0.496920	0.248460	−	0.968642i	$-0.420076\pi$
$7$	450.951	0.496920	0.248460	+	0.968642i	$0.420076\pi$
$8$	−329.950	−0.227842
$9$	729.000	0.333333
$10$	−2310.59	−0.730672
$11$	0	0
$11$	0	0
$12$	2874.18	0.480154
$13$	1834.43	0.231579	0.115790	−	0.993274i	$-0.463060\pi$
$13$	1834.43	0.231579	0.115790	+	0.993274i	$0.463060\pi$
$14$	6904.88	0.672524
$15$	−4074.37	−0.311703
$16$	−18677.9	−1.14001
$17$	3650.74	0.180223	0.0901114	−	0.995932i	$-0.471278\pi$
$17$	3650.74	0.180223	0.0901114	+	0.995932i	$0.471278\pi$
$18$	11162.3	0.451128
$19$	−35707.3	−1.19432	−0.597159	−	0.802123i	$-0.703704\pi$
$19$	−35707.3	−1.19432	−0.597159	+	0.802123i	$0.703704\pi$
$20$	−16063.8	−0.448996
$21$	12175.7	0.286897
$22$	0	0
$23$	−48865.9	−0.837450	−0.418725	−	0.908113i	$-0.637523\pi$
$23$	−48865.9	−0.837450	−0.418725	+	0.908113i	$0.637523\pi$
$24$	−8908.66	−0.131545
$25$	−55353.4	−0.708524
$26$	28088.4	0.313416
$27$	19683.0	0.192450
$28$	48004.3	0.413264
$29$	−59377.9	−0.452097	−0.226049	−	0.974116i	$-0.572581\pi$
$29$	−59377.9	−0.452097	−0.226049	+	0.974116i	$0.572581\pi$
$30$	−62385.9	−0.421854
$31$	216739.	1.30669	0.653344	−	0.757061i	$-0.273366\pi$
$31$	216739.	1.30669	0.653344	+	0.757061i	$0.273366\pi$
$32$	−243759.	−1.31503
$33$	0	0
$34$	55899.4	0.243911
$35$	−68049.7	−0.268280
$36$	77603.0	0.277217
$37$	−56889.3	−0.184640	−0.0923198	−	0.995729i	$-0.529428\pi$
$37$	−56889.3	−0.184640	−0.0923198	+	0.995729i	$0.529428\pi$
$38$	−546743.	−1.61637
$39$	49529.6	0.133702
$40$	49790.3	0.123008
$41$	−155773.	−0.352979	−0.176489	−	0.984303i	$-0.556474\pi$
$41$	−155773.	−0.352979	−0.176489	+	0.984303i	$0.556474\pi$
$42$	186432.	0.388282
$43$	−9452.73	−0.0181308	−0.00906541	−	0.999959i	$-0.502886\pi$
$43$	−9452.73	−0.0181308	−0.00906541	+	0.999959i	$0.502886\pi$
$44$	0	0
$45$	−110008.	−0.179962
$46$	−748225.	−1.13339
$47$	−247316.	−0.347463	−0.173732	−	0.984793i	$-0.555583\pi$
$47$	−247316.	−0.347463	−0.173732	+	0.984793i	$0.555583\pi$
$48$	−504303.	−0.658184
$49$	−620186.	−0.753070
$50$	−847561.	−0.958906
$51$	98569.9	0.104052
$52$	195277.	0.192593
$53$	742117.	0.684710	0.342355	−	0.939571i	$-0.388775\pi$
$53$	742117.	0.684710	0.342355	+	0.939571i	$0.388775\pi$
$54$	301382.	0.260459
$55$	0	0
$56$	−148792.	−0.113219
$57$	−964098.	−0.689539
$58$	−909182.	−0.611861
$59$	−1.27162e6	−0.806075	−0.403037	−	0.915183i	$-0.632046\pi$
$59$	−1.27162e6	−0.806075	−0.403037	+	0.915183i	$0.632046\pi$
$60$	−433721.	−0.259228
$61$	−1.38324e6	−0.780267	−0.390133	−	0.920758i	$-0.627571\pi$
$61$	−1.38324e6	−0.780267	−0.390133	+	0.920758i	$0.627571\pi$
$62$	3.31867e6	1.76845
$63$	328744.	0.165640
$64$	−1.34161e6	−0.639730
$65$	−276820.	−0.125026
$66$	0	0
$67$	−948760.	−0.385385	−0.192692	−	0.981259i	$-0.561722\pi$
$67$	−948760.	−0.385385	−0.192692	+	0.981259i	$0.561722\pi$
$68$	388626.	0.149882
$69$	−1.31938e6	−0.483502
$70$	−1.04196e6	−0.363086
$71$	−570590.	−0.189199	−0.0945997	−	0.995515i	$-0.530157\pi$
$71$	−570590.	−0.189199	−0.0945997	+	0.995515i	$0.530157\pi$
$72$	−240534.	−0.0759473
$73$	3.99621e6	1.20232	0.601158	−	0.799130i	$-0.294706\pi$
$73$	3.99621e6	1.20232	0.601158	+	0.799130i	$0.294706\pi$
$74$	−871078.	−0.249888
$75$	−1.49454e6	−0.409067
$76$	−3.80109e6	−0.993254
$77$	0	0
$78$	758387.	0.180951
$79$	−6.79990e6	−1.55170	−0.775850	−	0.630918i	$-0.782678\pi$
$79$	−6.79990e6	−1.55170	−0.775850	+	0.630918i	$0.782678\pi$
$80$	2.81854e6	0.615473
$81$	531441.	0.111111
$82$	−2.38517e6	−0.477716
$83$	−9.23584e6	−1.77298	−0.886488	−	0.462751i	$-0.846862\pi$
$83$	−9.23584e6	−1.77298	−0.886488	+	0.462751i	$0.846862\pi$
$84$	1.29612e6	0.238598
$85$	−550906.	−0.0972996
$86$	−144738.	−0.0245380
$87$	−1.60320e6	−0.261018
$88$	0	0
$89$	6.65992e6	1.00139	0.500696	−	0.865623i	$-0.333078\pi$
$89$	6.65992e6	1.00139	0.500696	+	0.865623i	$0.333078\pi$
$90$	−1.68442e6	−0.243557
$91$	827239.	0.115076
$92$	−5.20184e6	−0.696465
$93$	5.85196e6	0.754416
$94$	−3.78685e6	−0.470252
$95$	5.38832e6	0.644794
$96$	−6.58148e6	−0.759232
$97$	−1.60645e7	−1.78717	−0.893583	−	0.448898i	$-0.851817\pi$
$97$	−1.60645e7	−1.78717	−0.893583	+	0.448898i	$0.851817\pi$
$98$	−9.49616e6	−1.01919
$99$	0	0
$100$	−5.89244e6	−0.589244
$101$	−1.99673e7	−1.92839	−0.964197	−	0.265189i	$-0.914566\pi$
$101$	−1.99673e7	−1.92839	−0.964197	+	0.265189i	$0.914566\pi$
$102$	1.50928e6	0.140822
$103$	−5.91653e6	−0.533502	−0.266751	−	0.963765i	$-0.585950\pi$
$103$	−5.91653e6	−0.533502	−0.266751	+	0.963765i	$0.585950\pi$
$104$	−605271.	−0.0527634
$105$	−1.83734e6	−0.154891
$106$	1.13631e7	0.926676
$107$	700139.	0.0552511	0.0276256	−	0.999618i	$-0.491205\pi$
$107$	700139.	0.0552511	0.0276256	+	0.999618i	$0.491205\pi$
$108$	2.09528e6	0.160051
$109$	−1.48121e7	−1.09553	−0.547765	−	0.836632i	$-0.684521\pi$
$109$	−1.48121e7	−1.09553	−0.547765	+	0.836632i	$0.684521\pi$
$110$	0	0
$111$	−1.53601e6	−0.106602
$112$	−8.42282e6	−0.566493
$113$	2.94311e7	1.91881	0.959404	−	0.282035i	$-0.0910093\pi$
$113$	2.94311e7	1.91881	0.959404	+	0.282035i	$0.0910093\pi$
$114$	−1.47621e7	−0.933212
$115$	7.37399e6	0.452127
$116$	−6.32085e6	−0.375987
$117$	1.33730e6	0.0771930
$118$	−1.94708e7	−1.09093
$119$	1.64631e6	0.0895563
$120$	1.34434e6	0.0710190
$121$	0	0
$122$	−2.11799e7	−1.05600
$123$	−4.20587e6	−0.203792
$124$	2.30722e7	1.08671
$125$	2.01422e7	0.922407
$126$	5.03366e6	0.224175
$127$	3.04913e6	0.132088	0.0660440	−	0.997817i	$-0.478962\pi$
$127$	3.04913e6	0.132088	0.0660440	+	0.997817i	$0.478962\pi$
$128$	1.06586e7	0.449226
$129$	−255224.	−0.0104678
$130$	−4.23861e6	−0.169208
$131$	3.04858e7	1.18481	0.592404	−	0.805641i	$-0.298179\pi$
$131$	3.04858e7	1.18481	0.592404	+	0.805641i	$0.298179\pi$
$132$	0	0
$133$	−1.61023e7	−0.593480
$134$	−1.45272e7	−0.521574
$135$	−2.97021e6	−0.103901
$136$	−1.20456e6	−0.0410623
$137$	−3.86462e7	−1.28406	−0.642029	−	0.766680i	$-0.721907\pi$
$137$	−3.86462e7	−1.28406	−0.642029	+	0.766680i	$0.721907\pi$
$138$	−2.02021e7	−0.654364
$139$	2.41409e7	0.762432	0.381216	−	0.924486i	$-0.375505\pi$
$139$	2.41409e7	0.762432	0.381216	+	0.924486i	$0.375505\pi$
$140$	−7.24397e6	−0.223115
$141$	−6.67752e6	−0.200608
$142$	−8.73676e6	−0.256060
$143$	0	0
$144$	−1.36162e7	−0.380003
$145$	8.96027e6	0.244081
$146$	6.11892e7	1.62720
$147$	−1.67450e7	−0.434785
$148$	−6.05594e6	−0.153556
$149$	2.43925e6	0.0604095	0.0302047	−	0.999544i	$-0.490384\pi$
$149$	2.43925e6	0.0604095	0.0302047	+	0.999544i	$0.490384\pi$
$150$	−2.28841e7	−0.553624
$151$	7.09883e7	1.67791	0.838953	−	0.544205i	$-0.183168\pi$
$151$	7.09883e7	1.67791	0.838953	+	0.544205i	$0.183168\pi$
$152$	1.17816e7	0.272116
$153$	2.66139e6	0.0600742
$154$	0	0
$155$	−3.27065e7	−0.705461
$156$	5.27249e6	0.111194
$157$	−2.24019e7	−0.461994	−0.230997	−	0.972954i	$-0.574199\pi$
$157$	−2.24019e7	−0.461994	−0.230997	+	0.972954i	$0.574199\pi$
$158$	−1.04119e8	−2.10005
$159$	2.00372e7	0.395317
$160$	3.67838e7	0.709964
$161$	−2.20362e7	−0.416146
$162$	8.13732e6	0.150376
$163$	−3.27244e7	−0.591855	−0.295927	−	0.955210i	$-0.595629\pi$
$163$	−3.27244e7	−0.591855	−0.295927	+	0.955210i	$0.595629\pi$
$164$	−1.65822e7	−0.293555
$165$	0	0
$166$	−1.41417e8	−2.39952
$167$	6.41663e7	1.06610	0.533052	−	0.846083i	$-0.321045\pi$
$167$	6.41663e7	1.06610	0.533052	+	0.846083i	$0.321045\pi$
$168$	−4.01737e6	−0.0653672
$169$	−5.93834e7	−0.946371
$170$	−8.43536e6	−0.131684
$171$	−2.60306e7	−0.398106
$172$	−1.00625e6	−0.0150785
$173$	−1.22716e7	−0.180194	−0.0900970	−	0.995933i	$-0.528718\pi$
$173$	−1.22716e7	−0.180194	−0.0900970	+	0.995933i	$0.528718\pi$
$174$	−2.45479e7	−0.353258
$175$	−2.49617e7	−0.352080
$176$	0	0
$177$	−3.43337e7	−0.465388
$178$	1.01975e8	1.35527
$179$	8.15449e7	1.06270	0.531351	−	0.847152i	$-0.321685\pi$
$179$	8.15449e7	1.06270	0.531351	+	0.847152i	$0.321685\pi$
$180$	−1.17105e7	−0.149665
$181$	−4.20419e7	−0.526996	−0.263498	−	0.964660i	$-0.584876\pi$
$181$	−4.20419e7	−0.526996	−0.263498	+	0.964660i	$0.584876\pi$
$182$	1.26665e7	0.155743
$183$	−3.73475e7	−0.450487
$184$	1.61233e7	0.190806
$185$	8.58474e6	0.0996842
$186$	8.96041e7	1.02102
$187$	0	0
$188$	−2.63271e7	−0.288968
$189$	8.87608e6	0.0956323
$190$	8.25049e7	0.872654
$191$	−1.00554e7	−0.104420	−0.0522101	−	0.998636i	$-0.516627\pi$
$191$	−1.00554e7	−0.104420	−0.0522101	+	0.998636i	$0.516627\pi$
$192$	−3.62235e7	−0.369348
$193$	1.55057e8	1.55253	0.776265	−	0.630406i	$-0.217112\pi$
$193$	1.55057e8	1.55253	0.776265	+	0.630406i	$0.217112\pi$
$194$	−2.45976e8	−2.41872
$195$	−7.47414e6	−0.0721838
$196$	−6.60195e7	−0.626291
$197$	9.24245e7	0.861302	0.430651	−	0.902519i	$-0.358284\pi$
$197$	9.24245e7	0.861302	0.430651	+	0.902519i	$0.358284\pi$
$198$	0	0
$199$	3.90241e7	0.351033	0.175516	−	0.984477i	$-0.443841\pi$
$199$	3.90241e7	0.351033	0.175516	+	0.984477i	$0.443841\pi$
$200$	1.82639e7	0.161431
$201$	−2.56165e7	−0.222502
$202$	−3.05736e8	−2.60986
$203$	−2.67765e7	−0.224656
$204$	1.04929e7	0.0865346
$205$	2.35065e7	0.190568
$206$	−9.05927e7	−0.722034
$207$	−3.56233e7	−0.279150
$208$	−3.42633e7	−0.264002
$209$	0	0
$210$	−2.81330e7	−0.209628
$211$	1.64991e8	1.20913	0.604564	−	0.796557i	$-0.293347\pi$
$211$	1.64991e8	1.20913	0.604564	+	0.796557i	$0.293347\pi$
$212$	7.89992e7	0.569439
$213$	−1.54059e7	−0.109234
$214$	1.07204e7	0.0747761
$215$	1.42644e6	0.00978856
$216$	−6.49441e6	−0.0438482
$217$	9.77389e7	0.649319
$218$	−2.26800e8	−1.48267
$219$	1.07898e8	0.694157
$220$	0	0
$221$	6.69702e6	0.0417358
$222$	−2.35191e7	−0.144273
$223$	1.21207e8	0.731916	0.365958	−	0.930631i	$-0.380741\pi$
$223$	1.21207e8	0.731916	0.365958	+	0.930631i	$0.380741\pi$
$224$	−1.09923e8	−0.653464
$225$	−4.03527e7	−0.236175
$226$	4.50643e8	2.59689
$227$	−7.69802e7	−0.436806	−0.218403	−	0.975859i	$-0.570085\pi$
$227$	−7.69802e7	−0.436806	−0.218403	+	0.975859i	$0.570085\pi$
$228$	−1.02629e8	−0.573456
$229$	−1.19863e8	−0.659569	−0.329785	−	0.944056i	$-0.606976\pi$
$229$	−1.19863e8	−0.659569	−0.329785	+	0.944056i	$0.606976\pi$
$230$	1.12909e8	0.611901
$231$	0	0
$232$	1.95918e7	0.103007
$233$	1.57766e8	0.817085	0.408542	−	0.912739i	$-0.366037\pi$
$233$	1.57766e8	0.817085	0.408542	+	0.912739i	$0.366037\pi$
$234$	2.04765e7	0.104472
$235$	3.73205e7	0.187590
$236$	−1.35366e8	−0.670372
$237$	−1.83597e8	−0.895874
$238$	2.52079e7	0.121204
$239$	3.34956e7	0.158707	0.0793534	−	0.996847i	$-0.474714\pi$
$239$	3.34956e7	0.158707	0.0793534	+	0.996847i	$0.474714\pi$
$240$	7.61006e7	0.355344
$241$	−1.98167e8	−0.911950	−0.455975	−	0.889993i	$-0.650709\pi$
$241$	−1.98167e8	−0.911950	−0.455975	+	0.889993i	$0.650709\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	−1.47248e8	−0.648909
$245$	9.35876e7	0.406571
$246$	−6.43995e7	−0.275810
$247$	−6.55026e7	−0.276579
$248$	−7.15132e7	−0.297718
$249$	−2.49368e8	−1.02363
$250$	3.08414e8	1.24837
$251$	−4.41895e8	−1.76385	−0.881924	−	0.471391i	$-0.843752\pi$
$251$	−4.41895e8	−1.76385	−0.881924	+	0.471391i	$0.843752\pi$
$252$	3.49952e7	0.137755
$253$	0	0
$254$	4.66877e7	0.178766
$255$	−1.48744e7	−0.0561759
$256$	3.34929e8	1.24771
$257$	−9.51348e7	−0.349601	−0.174801	−	0.984604i	$-0.555928\pi$
$257$	−9.51348e7	−0.349601	−0.174801	+	0.984604i	$0.555928\pi$
$258$	−3.90793e6	−0.0141670
$259$	−2.56543e7	−0.0917512
$260$	−2.94678e7	−0.103978
$261$	−4.32865e7	−0.150699
$262$	4.66792e8	1.60350
$263$	3.85784e8	1.30767	0.653837	−	0.756636i	$-0.273158\pi$
$263$	3.85784e8	1.30767	0.653837	+	0.756636i	$0.273158\pi$
$264$	0	0
$265$	−1.11987e8	−0.369665
$266$	−2.46555e8	−0.803207
$267$	1.79818e8	0.578154
$268$	−1.00997e8	−0.320505
$269$	6.61942e7	0.207342	0.103671	−	0.994612i	$-0.466941\pi$
$269$	6.61942e7	0.207342	0.103671	+	0.994612i	$0.466941\pi$
$270$	−4.54793e7	−0.140618
$271$	3.88853e8	1.18684	0.593421	−	0.804893i	$-0.297777\pi$
$271$	3.88853e8	1.18684	0.593421	+	0.804893i	$0.297777\pi$
$272$	−6.81881e7	−0.205455
$273$	2.23354e7	0.0664393
$274$	−5.91743e8	−1.73783
$275$	0	0
$276$	−1.40450e8	−0.402104
$277$	4.40365e8	1.24490	0.622449	−	0.782661i	$-0.286138\pi$
$277$	4.40365e8	1.24490	0.622449	+	0.782661i	$0.286138\pi$
$278$	3.69640e8	1.03186
$279$	1.58003e8	0.435563
$280$	2.24530e7	0.0611254
$281$	−5.76060e8	−1.54880	−0.774401	−	0.632696i	$-0.781948\pi$
$281$	−5.76060e8	−1.54880	−0.774401	+	0.632696i	$0.781948\pi$
$282$	−1.02245e8	−0.271500
$283$	2.16399e8	0.567547	0.283774	−	0.958891i	$-0.408414\pi$
$283$	2.16399e8	0.567547	0.283774	+	0.958891i	$0.408414\pi$
$284$	−6.07400e7	−0.157348
$285$	1.45485e8	0.372272
$286$	0	0
$287$	−7.02461e7	−0.175402
$288$	−1.77700e8	−0.438343
$289$	−3.97011e8	−0.967520
$290$	1.37198e8	0.330335
$291$	−4.33740e8	−1.03182
$292$	4.25402e8	0.999906
$293$	5.49455e8	1.27613	0.638066	−	0.769982i	$-0.279735\pi$
$293$	5.49455e8	1.27613	0.638066	+	0.769982i	$0.279735\pi$
$294$	−2.56396e8	−0.588432
$295$	1.91891e8	0.435188
$296$	1.87707e7	0.0420686
$297$	0	0
$298$	3.73494e7	0.0817573
$299$	−8.96411e7	−0.193936
$300$	−1.59096e8	−0.340200
$301$	−4.26272e6	−0.00900957
$302$	1.08696e9	2.27085
$303$	−5.39118e8	−1.11336
$304$	6.66937e8	1.36153
$305$	2.08734e8	0.421254
$306$	4.07506e7	0.0813036
$307$	−8.42847e8	−1.66251	−0.831255	−	0.555891i	$-0.812377\pi$
$307$	−8.42847e8	−1.66251	−0.831255	+	0.555891i	$0.812377\pi$
$308$	0	0
$309$	−1.59746e8	−0.308018
$310$	−5.00795e8	−0.954760
$311$	−9.24191e8	−1.74221	−0.871105	−	0.491097i	$-0.836596\pi$
$311$	−9.24191e8	−1.74221	−0.871105	+	0.491097i	$0.836596\pi$
$312$	−1.63423e7	−0.0304630
$313$	−2.52164e8	−0.464812	−0.232406	−	0.972619i	$-0.574660\pi$
$313$	−2.52164e8	−0.464812	−0.232406	+	0.972619i	$0.574660\pi$
$314$	−3.43014e8	−0.625256
$315$	−4.96082e7	−0.0894266
$316$	−7.23857e8	−1.29047
$317$	2.09625e8	0.369603	0.184802	−	0.982776i	$-0.440836\pi$
$317$	2.09625e8	0.369603	0.184802	+	0.982776i	$0.440836\pi$
$318$	3.06805e8	0.535017
$319$	0	0
$320$	2.02453e8	0.345381
$321$	1.89038e7	0.0318993
$322$	−3.37413e8	−0.563205
$323$	−1.30358e8	−0.215243
$324$	5.65726e7	0.0924056
$325$	−1.01542e8	−0.164079
$326$	−5.01070e8	−0.801007
$327$	−3.99927e8	−0.632505
$328$	5.13974e7	0.0804234
$329$	−1.11527e8	−0.172662
$330$	0	0
$331$	−7.72480e8	−1.17082	−0.585409	−	0.810738i	$-0.699066\pi$
$331$	−7.72480e8	−1.17082	−0.585409	+	0.810738i	$0.699066\pi$
$332$	−9.83167e8	−1.47450
$333$	−4.14723e7	−0.0615465
$334$	9.82502e8	1.44285
$335$	1.43170e8	0.208063
$336$	−2.27416e8	−0.327065
$337$	1.26330e9	1.79806	0.899028	−	0.437891i	$-0.144275\pi$
$337$	1.26330e9	1.79806	0.899028	+	0.437891i	$0.144275\pi$
$338$	−9.09267e8	−1.28080
$339$	7.94639e8	1.10782
$340$	−5.86446e7	−0.0809192
$341$	0	0
$342$	−3.98576e8	−0.538790
$343$	−6.51052e8	−0.871136
$344$	3.11893e6	0.00413096
$345$	1.99098e8	0.261035
$346$	−1.87901e8	−0.243872
$347$	−1.01177e9	−1.29995	−0.649976	−	0.759955i	$-0.725221\pi$
$347$	−1.01177e9	−1.29995	−0.649976	+	0.759955i	$0.725221\pi$
$348$	−1.70663e8	−0.217076
$349$	1.10483e9	1.39125	0.695626	−	0.718404i	$-0.255127\pi$
$349$	1.10483e9	1.39125	0.695626	+	0.718404i	$0.255127\pi$
$350$	−3.82209e8	−0.476500
$351$	3.61071e7	0.0445674
$352$	0	0
$353$	5.60998e8	0.678813	0.339406	−	0.940640i	$-0.389774\pi$
$353$	5.60998e8	0.678813	0.339406	+	0.940640i	$0.389774\pi$
$354$	−5.25712e8	−0.629848
$355$	8.61034e7	0.102146
$356$	7.08957e8	0.832808
$357$	4.44503e7	0.0517054
$358$	1.24860e9	1.43824
$359$	−1.36456e8	−0.155655	−0.0778275	−	0.996967i	$-0.524798\pi$
$359$	−1.36456e8	−0.155655	−0.0778275	+	0.996967i	$0.524798\pi$
$360$	3.62971e7	0.0410028
$361$	3.81141e8	0.426393
$362$	−6.43737e8	−0.713229
$363$	0	0
$364$	8.80606e7	0.0957033
$365$	−6.03038e8	−0.649112
$366$	−5.71857e8	−0.609682
$367$	1.82044e7	0.0192240	0.00961202	−	0.999954i	$-0.496940\pi$
$367$	1.82044e7	0.0192240	0.00961202	+	0.999954i	$0.496940\pi$
$368$	9.12713e8	0.954700
$369$	−1.13559e8	−0.117660
$370$	1.31448e8	0.134911
$371$	3.34659e8	0.340246
$372$	6.22949e8	0.627411
$373$	−7.96286e8	−0.794489	−0.397245	−	0.917713i	$-0.630034\pi$
$373$	−7.96286e8	−0.794489	−0.397245	+	0.917713i	$0.630034\pi$
$374$	0	0
$375$	5.43840e8	0.532552
$376$	8.16019e7	0.0791667
$377$	−1.08925e8	−0.104696
$378$	1.35909e8	0.129427
$379$	2.81423e8	0.265536	0.132768	−	0.991147i	$-0.457614\pi$
$379$	2.81423e8	0.265536	0.132768	+	0.991147i	$0.457614\pi$
$380$	5.73594e8	0.536243
$381$	8.23265e7	0.0762610
$382$	−1.53967e8	−0.141321
$383$	−1.16560e9	−1.06012	−0.530059	−	0.847961i	$-0.677830\pi$
$383$	−1.16560e9	−1.06012	−0.530059	+	0.847961i	$0.677830\pi$
$384$	2.87782e8	0.259361
$385$	0	0
$386$	2.37420e9	2.10117
$387$	−6.89104e6	−0.00604361
$388$	−1.71008e9	−1.48630
$389$	1.00118e9	0.862359	0.431179	−	0.902266i	$-0.358098\pi$
$389$	1.00118e9	0.862359	0.431179	+	0.902266i	$0.358098\pi$
$390$	−1.14443e8	−0.0976925
$391$	−1.78397e8	−0.150927
$392$	2.04631e8	0.171581
$393$	8.23116e8	0.684049
$394$	1.41519e9	1.16567
$395$	1.02612e9	0.837739
$396$	0	0
$397$	1.28665e9	1.03203	0.516015	−	0.856579i	$-0.327415\pi$
$397$	1.28665e9	1.03203	0.516015	+	0.856579i	$0.327415\pi$
$398$	5.97530e8	0.475082
$399$	−4.34761e8	−0.342646
$400$	1.03389e9	0.807723
$401$	−3.76915e8	−0.291903	−0.145951	−	0.989292i	$-0.546624\pi$
$401$	−3.76915e8	−0.291903	−0.145951	+	0.989292i	$0.546624\pi$
$402$	−3.92235e8	−0.301131
$403$	3.97593e8	0.302601
$404$	−2.12555e9	−1.60375
$405$	−8.01958e7	−0.0599872
$406$	−4.09997e8	−0.304046
$407$	0	0
$408$	−3.25232e7	−0.0237073
$409$	1.53594e9	1.11005	0.555026	−	0.831833i	$-0.312708\pi$
$409$	1.53594e9	1.11005	0.555026	+	0.831833i	$0.312708\pi$
$410$	3.59927e8	0.257912
$411$	−1.04345e9	−0.741352
$412$	−6.29821e8	−0.443687
$413$	−5.73439e8	−0.400555
$414$	−5.45456e8	−0.377797
$415$	1.39371e9	0.957204
$416$	−4.47158e8	−0.304533
$417$	6.51804e8	0.440191
$418$	0	0
$419$	2.50477e9	1.66348	0.831742	−	0.555163i	$-0.187344\pi$
$419$	2.50477e9	1.66348	0.831742	+	0.555163i	$0.187344\pi$
$420$	−1.95587e8	−0.128816
$421$	−1.42255e9	−0.929141	−0.464571	−	0.885536i	$-0.653791\pi$
$421$	−1.42255e9	−0.929141	−0.464571	+	0.885536i	$0.653791\pi$
$422$	2.52631e9	1.63641
$423$	−1.80293e8	−0.115821
$424$	−2.44862e8	−0.156006
$425$	−2.02081e8	−0.127692
$426$	−2.35893e8	−0.147836
$427$	−6.23774e8	−0.387730
$428$	7.45307e7	0.0459496
$429$	0	0
$430$	2.18414e7	0.0132477
$431$	1.99720e9	1.20158	0.600789	−	0.799408i	$-0.294853\pi$
$431$	1.99720e9	1.20158	0.600789	+	0.799408i	$0.294853\pi$
$432$	−3.67637e8	−0.219395
$433$	4.77303e8	0.282544	0.141272	−	0.989971i	$-0.454881\pi$
$433$	4.77303e8	0.282544	0.141272	+	0.989971i	$0.454881\pi$
$434$	1.49656e9	0.878779
$435$	2.41927e8	0.140920
$436$	−1.57677e9	−0.911098
$437$	1.74487e9	1.00018
$438$	1.65211e9	0.939462
$439$	1.91725e9	1.08156	0.540782	−	0.841163i	$-0.318128\pi$
$439$	1.91725e9	1.08156	0.540782	+	0.841163i	$0.318128\pi$
$440$	0	0
$441$	−4.52115e8	−0.251023
$442$	1.02543e8	0.0564846
$443$	2.58245e9	1.41130	0.705650	−	0.708561i	$-0.250655\pi$
$443$	2.58245e9	1.41130	0.705650	+	0.708561i	$0.250655\pi$
$444$	−1.63510e8	−0.0886554
$445$	−1.00500e9	−0.540636
$446$	1.85590e9	0.990564
$447$	6.58599e7	0.0348774
$448$	−6.05002e8	−0.317895
$449$	1.51304e9	0.788840	0.394420	−	0.918930i	$-0.370945\pi$
$449$	1.51304e9	0.788840	0.394420	+	0.918930i	$0.370945\pi$
$450$	−6.17872e8	−0.319635
$451$	0	0
$452$	3.13297e9	1.59578
$453$	1.91668e9	0.968739
$454$	−1.17871e9	−0.591166
$455$	−1.24832e8	−0.0621280
$456$	3.18104e8	0.157106
$457$	6.06245e8	0.297126	0.148563	−	0.988903i	$-0.452535\pi$
$457$	6.06245e8	0.297126	0.148563	+	0.988903i	$0.452535\pi$
$458$	−1.83532e9	−0.892651
$459$	7.18575e7	0.0346839
$460$	7.84970e8	0.376011
$461$	9.32097e8	0.443106	0.221553	−	0.975148i	$-0.428887\pi$
$461$	9.32097e8	0.443106	0.221553	+	0.975148i	$0.428887\pi$
$462$	0	0
$463$	1.19425e9	0.559195	0.279597	−	0.960117i	$-0.409799\pi$
$463$	1.19425e9	0.559195	0.279597	+	0.960117i	$0.409799\pi$
$464$	1.10905e9	0.515394
$465$	−8.83075e8	−0.407298
$466$	2.41568e9	1.10583
$467$	−1.12016e9	−0.508945	−0.254473	−	0.967080i	$-0.581902\pi$
$467$	−1.12016e9	−0.508945	−0.254473	+	0.967080i	$0.581902\pi$
$468$	1.42357e8	0.0641976
$469$	−4.27845e8	−0.191505
$470$	5.71445e8	0.253882
$471$	−6.04852e8	−0.266732
$472$	4.19572e8	0.183658
$473$	0	0
$474$	−2.81120e9	−1.21246
$475$	1.97652e9	0.846202
$476$	1.75251e8	0.0744795
$477$	5.41003e8	0.228237
$478$	5.12878e8	0.214791
$479$	1.19835e9	0.498207	0.249103	−	0.968477i	$-0.419864\pi$
$479$	1.19835e9	0.498207	0.249103	+	0.968477i	$0.419864\pi$
$480$	9.93162e8	0.409898
$481$	−1.04359e8	−0.0427587
$482$	−3.03429e9	−1.23422
$483$	−5.94976e8	−0.240262
$484$	0	0
$485$	2.42417e9	0.964864
$486$	2.19708e8	0.0868197
$487$	−3.74046e9	−1.46749	−0.733743	−	0.679427i	$-0.762228\pi$
$487$	−3.74046e9	−1.46749	−0.733743	+	0.679427i	$0.762228\pi$
$488$	4.56400e8	0.177777
$489$	−8.83559e8	−0.341708
$490$	1.43299e9	0.550248
$491$	−2.80782e9	−1.07050	−0.535248	−	0.844695i	$-0.679782\pi$
$491$	−2.80782e9	−1.07050	−0.535248	+	0.844695i	$0.679782\pi$
$492$	−4.47720e8	−0.169484
$493$	−2.16773e8	−0.0814782
$494$	−1.00296e9	−0.374318
$495$	0	0
$496$	−4.04823e9	−1.48963
$497$	−2.57308e8	−0.0940171
$498$	−3.81827e9	−1.38536
$499$	−3.29137e9	−1.18584	−0.592918	−	0.805263i	$-0.702024\pi$
$499$	−3.29137e9	−1.18584	−0.592918	+	0.805263i	$0.702024\pi$
$500$	2.14417e9	0.767120
$501$	1.73249e9	0.615515
$502$	−6.76621e9	−2.38717
$503$	3.30197e9	1.15687	0.578436	−	0.815727i	$-0.303663\pi$
$503$	3.30197e9	1.15687	0.578436	+	0.815727i	$0.303663\pi$
$504$	−1.08469e8	−0.0377398
$505$	3.01312e9	1.04111
$506$	0	0
$507$	−1.60335e9	−0.546388
$508$	3.24584e8	0.109851
$509$	1.57111e9	0.528074	0.264037	−	0.964513i	$-0.414946\pi$
$509$	1.57111e9	0.528074	0.264037	+	0.964513i	$0.414946\pi$
$510$	−2.27755e8	−0.0760276
$511$	1.80210e9	0.597455
$512$	3.76406e9	1.23940
$513$	−7.02827e8	−0.229846
$514$	−1.45668e9	−0.473145
$515$	8.92818e8	0.288030
$516$	−2.71689e7	−0.00870558
$517$	0	0
$518$	−3.92814e8	−0.124175
$519$	−3.31334e8	−0.104035
$520$	9.13368e7	0.0284862
$521$	2.70490e8	0.0837952	0.0418976	−	0.999122i	$-0.486660\pi$
$521$	2.70490e8	0.0837952	0.0418976	+	0.999122i	$0.486660\pi$
$522$	−6.62794e8	−0.203954
$523$	−5.38264e9	−1.64528	−0.822639	−	0.568564i	$-0.807499\pi$
$523$	−5.38264e9	−1.64528	−0.822639	+	0.568564i	$0.807499\pi$
$524$	3.24525e9	0.985345
$525$	−6.73966e8	−0.203273
$526$	5.90705e9	1.76978
$527$	7.91259e8	0.235495
$528$	0	0
$529$	−1.01695e9	−0.298678
$530$	−1.71473e9	−0.500299
$531$	−9.27011e8	−0.268692
$532$	−1.71411e9	−0.493568
$533$	−2.85755e8	−0.0817425
$534$	2.75333e9	0.782464
$535$	−1.05653e8	−0.0298293
$536$	3.13044e8	0.0878068
$537$	2.20171e9	0.613551
$538$	1.01355e9	0.280613
$539$	0	0
$540$	−3.16183e8	−0.0864092
$541$	−2.09946e9	−0.570054	−0.285027	−	0.958519i	$-0.592003\pi$
$541$	−2.09946e9	−0.570054	−0.285027	+	0.958519i	$0.592003\pi$
$542$	5.95404e9	1.60625
$543$	−1.13513e9	−0.304261
$544$	−8.89899e8	−0.236998
$545$	2.23518e9	0.591460
$546$	3.41996e8	0.0899180
$547$	−4.24627e9	−1.10931	−0.554654	−	0.832081i	$-0.687149\pi$
$547$	−4.24627e9	−1.10931	−0.554654	+	0.832081i	$0.687149\pi$
$548$	−4.11394e9	−1.06789
$549$	−1.00838e9	−0.260089
$550$	0	0
$551$	2.12023e9	0.539947
$552$	4.35330e8	0.110162
$553$	−3.06642e9	−0.771071
$554$	6.74278e9	1.68483
$555$	2.31788e8	0.0575527
$556$	2.56983e9	0.634077
$557$	4.46910e9	1.09579	0.547894	−	0.836548i	$-0.315430\pi$
$557$	4.46910e9	1.09579	0.547894	+	0.836548i	$0.315430\pi$
$558$	2.41931e9	0.589484
$559$	−1.73404e7	−0.00419872
$560$	1.27102e9	0.305841
$561$	0	0
$562$	−8.82052e9	−2.09612
$563$	4.40973e9	1.04144	0.520719	−	0.853728i	$-0.325664\pi$
$563$	4.40973e9	1.04144	0.520719	+	0.853728i	$0.325664\pi$
$564$	−7.10830e8	−0.166836
$565$	−4.44122e9	−1.03594
$566$	3.31345e9	0.768110
$567$	2.39654e8	0.0552134
$568$	1.88266e8	0.0431076
$569$	−1.04105e9	−0.236906	−0.118453	−	0.992960i	$-0.537794\pi$
$569$	−1.04105e9	−0.236906	−0.118453	+	0.992960i	$0.537794\pi$
$570$	2.22763e9	0.503827
$571$	1.44683e9	0.325231	0.162616	−	0.986689i	$-0.448007\pi$
$571$	1.44683e9	0.325231	0.162616	+	0.986689i	$0.448007\pi$
$572$	0	0
$573$	−2.71497e8	−0.0602870
$574$	−1.07559e9	−0.237387
$575$	2.70490e9	0.593353
$576$	−9.78035e8	−0.213243
$577$	−1.29000e9	−0.279559	−0.139779	−	0.990183i	$-0.544639\pi$
$577$	−1.29000e9	−0.279559	−0.139779	+	0.990183i	$0.544639\pi$
$578$	−6.07895e9	−1.30943
$579$	4.18653e9	0.896354
$580$	9.53832e8	0.202990
$581$	−4.16492e9	−0.881028
$582$	−6.64134e9	−1.39645
$583$	0	0
$584$	−1.31855e9	−0.273938
$585$	−2.01802e8	−0.0416754
$586$	8.41315e9	1.72710
$587$	−1.99334e9	−0.406768	−0.203384	−	0.979099i	$-0.565194\pi$
$587$	−1.99334e9	−0.406768	−0.203384	+	0.979099i	$0.565194\pi$
$588$	−1.78253e9	−0.361589
$589$	−7.73918e9	−1.56060
$590$	2.93819e9	0.588977
$591$	2.49546e9	0.497273
$592$	1.06257e9	0.210491
$593$	−7.62570e9	−1.50172	−0.750858	−	0.660463i	$-0.770360\pi$
$593$	−7.62570e9	−1.50172	−0.750858	+	0.660463i	$0.770360\pi$
$594$	0	0
$595$	−2.48432e8	−0.0483501
$596$	2.59662e8	0.0502396
$597$	1.05365e9	0.202669
$598$	−1.37257e9	−0.262470
$599$	−2.09746e9	−0.398750	−0.199375	−	0.979923i	$-0.563891\pi$
$599$	−2.09746e9	−0.398750	−0.199375	+	0.979923i	$0.563891\pi$
$600$	4.93125e8	0.0932025
$601$	−3.19200e9	−0.599795	−0.299897	−	0.953971i	$-0.596952\pi$
$601$	−3.19200e9	−0.599795	−0.299897	+	0.953971i	$0.596952\pi$
$602$	−6.52699e7	−0.0121934
$603$	−6.91646e8	−0.128462
$604$	7.55679e9	1.39543
$605$	0	0
$606$	−8.25487e9	−1.50680
$607$	−7.69068e9	−1.39574	−0.697869	−	0.716225i	$-0.745868\pi$
$607$	−7.69068e9	−1.39574	−0.697869	+	0.716225i	$0.745868\pi$
$608$	8.70396e9	1.57056
$609$	−7.22967e8	−0.129705
$610$	3.19610e9	0.570119
$611$	−4.53683e8	−0.0804652
$612$	2.83308e8	0.0499608
$613$	−3.71584e9	−0.651546	−0.325773	−	0.945448i	$-0.605625\pi$
$613$	−3.71584e9	−0.651546	−0.325773	+	0.945448i	$0.605625\pi$
$614$	−1.29055e10	−2.25002
$615$	6.34676e8	0.110025
$616$	0	0
$617$	9.16061e9	1.57010	0.785048	−	0.619435i	$-0.212638\pi$
$617$	9.16061e9	1.57010	0.785048	+	0.619435i	$0.212638\pi$
$618$	−2.44600e9	−0.416867
$619$	4.17363e9	0.707289	0.353645	−	0.935380i	$-0.384942\pi$
$619$	4.17363e9	0.707289	0.353645	+	0.935380i	$0.384942\pi$
$620$	−3.48165e9	−0.586697
$621$	−9.61828e8	−0.161167
$622$	−1.41510e10	−2.35788
$623$	3.00330e9	0.497612
$624$	−9.25108e8	−0.152422
$625$	1.28498e9	0.210530
$626$	−3.86108e9	−0.629070
$627$	0	0
$628$	−2.38471e9	−0.384217
$629$	−2.07688e8	−0.0332763
$630$	−7.59591e8	−0.121029
$631$	−7.34966e9	−1.16457	−0.582283	−	0.812986i	$-0.697841\pi$
$631$	−7.34966e9	−1.16457	−0.582283	+	0.812986i	$0.697841\pi$
$632$	2.24363e9	0.353542
$633$	4.45476e9	0.698090
$634$	3.20974e9	0.500215
$635$	−4.60121e8	−0.0713123
$636$	2.13298e9	0.328766
$637$	−1.13769e9	−0.174395
$638$	0	0
$639$	−4.15960e8	−0.0630665
$640$	−1.60841e9	−0.242531
$641$	1.91130e8	0.0286633	0.0143317	−	0.999897i	$-0.495438\pi$
$641$	1.91130e8	0.0286633	0.0143317	+	0.999897i	$0.495438\pi$
$642$	2.89451e8	0.0431720
$643$	−7.93068e9	−1.17645	−0.588223	−	0.808699i	$-0.700172\pi$
$643$	−7.93068e9	−1.17645	−0.588223	+	0.808699i	$0.700172\pi$
$644$	−2.34578e9	−0.346088
$645$	3.85139e7	0.00565143
$646$	−1.99602e9	−0.291307
$647$	−8.81420e9	−1.27943	−0.639717	−	0.768611i	$-0.720948\pi$
$647$	−8.81420e9	−1.27943	−0.639717	+	0.768611i	$0.720948\pi$
$648$	−1.75349e8	−0.0253158
$649$	0	0
$650$	−1.55479e9	−0.222062
$651$	2.63895e9	0.374885
$652$	−3.48355e9	−0.492216
$653$	−3.86041e9	−0.542547	−0.271274	−	0.962502i	$-0.587445\pi$
$653$	−3.86041e9	−0.542547	−0.271274	+	0.962502i	$0.587445\pi$
$654$	−6.12360e9	−0.856022
$655$	−4.60038e9	−0.639660
$656$	2.90951e9	0.402399
$657$	2.91324e9	0.400772
$658$	−1.70768e9	−0.233678
$659$	−1.04947e10	−1.42846	−0.714232	−	0.699909i	$-0.753224\pi$
$659$	−1.04947e10	−1.42846	−0.714232	+	0.699909i	$0.753224\pi$
$660$	0	0
$661$	1.06674e10	1.43666	0.718331	−	0.695701i	$-0.244906\pi$
$661$	1.06674e10	1.43666	0.718331	+	0.695701i	$0.244906\pi$
$662$	−1.18281e10	−1.58457
$663$	1.80820e8	0.0240962
$664$	3.04737e9	0.403958
$665$	2.42987e9	0.320411
$666$	−6.35016e8	−0.0832961
$667$	2.90156e9	0.378609
$668$	6.83058e9	0.886626
$669$	3.27259e9	0.422572
$670$	2.19219e9	0.281590
$671$	0	0
$672$	−2.96793e9	−0.377278
$673$	−9.04643e9	−1.14400	−0.571998	−	0.820255i	$-0.693831\pi$
$673$	−9.04643e9	−1.14400	−0.571998	+	0.820255i	$0.693831\pi$
$674$	1.93435e10	2.43346
$675$	−1.08952e9	−0.136356
$676$	−6.32143e9	−0.787050
$677$	5.69920e9	0.705917	0.352959	−	0.935639i	$-0.385176\pi$
$677$	5.69920e9	0.705917	0.352959	+	0.935639i	$0.385176\pi$
$678$	1.21674e10	1.49931
$679$	−7.24429e9	−0.888079
$680$	1.81771e8	0.0221689
$681$	−2.07847e9	−0.252190
$682$	0	0
$683$	5.60667e8	0.0673337	0.0336668	−	0.999433i	$-0.489281\pi$
$683$	5.60667e8	0.0673337	0.0336668	+	0.999433i	$0.489281\pi$
$684$	−2.77099e9	−0.331085
$685$	5.83181e9	0.693244
$686$	−9.96877e9	−1.17898
$687$	−3.23630e9	−0.380802
$688$	1.76557e8	0.0206693
$689$	1.36136e9	0.158565
$690$	3.04854e9	0.353281
$691$	−1.83589e9	−0.211677	−0.105838	−	0.994383i	$-0.533753\pi$
$691$	−1.83589e9	−0.211677	−0.105838	+	0.994383i	$0.533753\pi$
$692$	−1.30633e9	−0.149858
$693$	0	0
$694$	−1.54920e10	−1.75934
$695$	−3.64292e9	−0.411626
$696$	5.28977e8	0.0594709
$697$	−5.68687e8	−0.0636148
$698$	1.69169e10	1.88290
$699$	4.25968e9	0.471744
$700$	−2.65721e9	−0.292807
$701$	−9.45593e9	−1.03679	−0.518396	−	0.855141i	$-0.673471\pi$
$701$	−9.45593e9	−1.03679	−0.518396	+	0.855141i	$0.673471\pi$
$702$	5.52864e8	0.0603169
$703$	2.03137e9	0.220518
$704$	0	0
$705$	1.00765e9	0.108305
$706$	8.58989e9	0.918695
$707$	−9.00430e9	−0.958258
$708$	−3.65487e9	−0.387040
$709$	−1.00459e10	−1.05859	−0.529296	−	0.848437i	$-0.677544\pi$
$709$	−1.00459e10	−1.05859	−0.529296	+	0.848437i	$0.677544\pi$
$710$	1.31840e9	0.138243
$711$	−4.95712e9	−0.517233
$712$	−2.19744e9	−0.228159
$713$	−1.05912e10	−1.09429
$714$	6.80614e8	0.0699772
$715$	0	0
$716$	8.68056e9	0.883796
$717$	9.04382e8	0.0916294
$718$	−2.08939e9	−0.210661
$719$	−1.44553e10	−1.45036	−0.725180	−	0.688560i	$-0.758243\pi$
$719$	−1.44553e10	−1.45036	−0.725180	+	0.688560i	$0.758243\pi$
$720$	2.05472e9	0.205158
$721$	−2.66807e9	−0.265108
$722$	5.83595e9	0.577074
$723$	−5.35050e9	−0.526515
$724$	−4.47541e9	−0.438277
$725$	3.28677e9	0.320322
$726$	0	0
$727$	−1.44214e10	−1.39199	−0.695997	−	0.718045i	$-0.745037\pi$
$727$	−1.44214e10	−1.39199	−0.695997	+	0.718045i	$0.745037\pi$
$728$	−2.72948e8	−0.0262192
$729$	3.87420e8	0.0370370
$730$	−9.23360e9	−0.878499
$731$	−3.45094e7	−0.00326759
$732$	−3.97568e9	−0.374648
$733$	−8.73020e8	−0.0818767	−0.0409383	−	0.999162i	$-0.513035\pi$
$733$	−8.73020e8	−0.0818767	−0.0409383	+	0.999162i	$0.513035\pi$
$734$	2.78742e8	0.0260175
$735$	2.52686e9	0.234734
$736$	1.19115e10	1.10127
$737$	0	0
$738$	−1.73879e9	−0.159239
$739$	−1.31231e10	−1.19613	−0.598067	−	0.801446i	$-0.704064\pi$
$739$	−1.31231e10	−1.19613	−0.598067	+	0.801446i	$0.704064\pi$
$740$	9.13856e8	0.0829024
$741$	−1.76857e9	−0.159683
$742$	5.12423e9	0.460484
$743$	1.12095e10	1.00260	0.501299	−	0.865274i	$-0.332856\pi$
$743$	1.12095e10	1.00260	0.501299	+	0.865274i	$0.332856\pi$
$744$	−1.93086e9	−0.171888
$745$	−3.68089e8	−0.0326142
$746$	−1.21926e10	−1.07525
$747$	−6.73293e9	−0.590992
$748$	0	0
$749$	3.15729e8	0.0274554
$750$	8.32717e9	0.720747
$751$	−1.12643e10	−0.970428	−0.485214	−	0.874396i	$-0.661258\pi$
$751$	−1.12643e10	−0.970428	−0.485214	+	0.874396i	$0.661258\pi$
$752$	4.61933e9	0.396111
$753$	−1.19312e10	−1.01836
$754$	−1.66783e9	−0.141694
$755$	−1.07123e10	−0.905876
$756$	9.44870e8	0.0795327
$757$	6.20629e9	0.519992	0.259996	−	0.965610i	$-0.416279\pi$
$757$	6.20629e9	0.519992	0.259996	+	0.965610i	$0.416279\pi$
$758$	4.30910e9	0.359372
$759$	0	0
$760$	−1.77788e9	−0.146911
$761$	3.97591e9	0.327032	0.163516	−	0.986541i	$-0.447716\pi$
$761$	3.97591e9	0.327032	0.163516	+	0.986541i	$0.447716\pi$
$762$	1.26057e9	0.103210
$763$	−6.67955e9	−0.544391
$764$	−1.07041e9	−0.0868411
$765$	−4.01610e8	−0.0324332
$766$	−1.78475e10	−1.43475
$767$	−2.33270e9	−0.186670
$768$	9.04307e9	0.720364
$769$	−8.58317e9	−0.680621	−0.340310	−	0.940313i	$-0.610532\pi$
$769$	−8.58317e9	−0.680621	−0.340310	+	0.940313i	$0.610532\pi$
$770$	0	0
$771$	−2.56864e9	−0.201843
$772$	1.65060e10	1.29116
$773$	−1.50463e10	−1.17166	−0.585830	−	0.810434i	$-0.699231\pi$
$773$	−1.50463e10	−1.17166	−0.585830	+	0.810434i	$0.699231\pi$
$774$	−1.05514e8	−0.00817933
$775$	−1.19973e10	−0.925820
$776$	5.30047e9	0.407191
$777$	−6.92667e8	−0.0529726
$778$	1.53298e10	1.16710
$779$	5.56224e9	0.421569
$780$	−7.95631e8	−0.0600317
$781$	0	0
$782$	−2.73158e9	−0.204263
$783$	−1.16874e9	−0.0870061
$784$	1.15838e10	0.858506
$785$	3.38050e9	0.249424
$786$	1.26034e10	0.925781
$787$	8.51167e9	0.622448	0.311224	−	0.950337i	$-0.399261\pi$
$787$	8.51167e9	0.622448	0.311224	+	0.950337i	$0.399261\pi$
$788$	9.83870e9	0.716302
$789$	1.04162e10	0.754985
$790$	1.57118e10	1.13378
$791$	1.32720e10	0.953495
$792$	0	0
$793$	−2.53746e9	−0.180693
$794$	1.97009e10	1.39673
$795$	−3.02366e9	−0.213426
$796$	4.15417e9	0.291936
$797$	−1.57969e10	−1.10527	−0.552634	−	0.833424i	$-0.686377\pi$
$797$	−1.57969e10	−1.10527	−0.552634	+	0.833424i	$0.686377\pi$
$798$	−6.65698e9	−0.463732
$799$	−9.02885e8	−0.0626208
$800$	1.34929e10	0.931729
$801$	4.85508e9	0.333797
$802$	−5.77124e9	−0.395056
$803$	0	0
$804$	−2.72691e9	−0.185044
$805$	3.32531e9	0.224671
$806$	6.08786e9	0.409536
$807$	1.78724e9	0.119709
$808$	6.58823e9	0.439369
$809$	2.52400e9	0.167598	0.0837991	−	0.996483i	$-0.473295\pi$
$809$	2.52400e9	0.167598	0.0837991	+	0.996483i	$0.473295\pi$
$810$	−1.22794e9	−0.0811858
$811$	−3.59559e9	−0.236699	−0.118350	−	0.992972i	$-0.537760\pi$
$811$	−3.59559e9	−0.236699	−0.118350	+	0.992972i	$0.537760\pi$
$812$	−2.85040e9	−0.186835
$813$	1.04990e10	0.685223
$814$	0	0
$815$	4.93820e9	0.319534
$816$	−1.84108e9	−0.118620
$817$	3.37532e8	0.0216540
$818$	2.35180e10	1.50233
$819$	6.03057e8	0.0383588
$820$	2.50230e9	0.158486
$821$	−2.36760e10	−1.49317	−0.746583	−	0.665292i	$-0.768307\pi$
$821$	−2.36760e10	−1.49317	−0.746583	+	0.665292i	$0.768307\pi$
$822$	−1.59771e10	−1.00333
$823$	−1.17405e10	−0.734151	−0.367076	−	0.930191i	$-0.619641\pi$
$823$	−1.17405e10	−0.734151	−0.367076	+	0.930191i	$0.619641\pi$
$824$	1.95216e9	0.121554
$825$	0	0
$826$	−8.78038e9	−0.542105
$827$	2.39590e9	0.147299	0.0736495	−	0.997284i	$-0.476535\pi$
$827$	2.39590e9	0.147299	0.0736495	+	0.997284i	$0.476535\pi$
$828$	−3.79214e9	−0.232155
$829$	1.54175e10	0.939883	0.469942	−	0.882698i	$-0.344275\pi$
$829$	1.54175e10	0.939883	0.469942	+	0.882698i	$0.344275\pi$
$830$	2.13402e10	1.29547
$831$	1.18899e10	0.718742
$832$	−2.46109e9	−0.148148
$833$	−2.26414e9	−0.135720
$834$	9.98029e9	0.595747
$835$	−9.68286e9	−0.575574
$836$	0	0
$837$	4.26608e9	0.251472
$838$	3.83525e10	2.25133
$839$	6.69748e9	0.391511	0.195756	−	0.980653i	$-0.437284\pi$
$839$	6.69748e9	0.391511	0.195756	+	0.980653i	$0.437284\pi$
$840$	6.06232e8	0.0352908
$841$	−1.37241e10	−0.795608
$842$	−2.17819e10	−1.25749
$843$	−1.55536e10	−0.894201
$844$	1.75635e10	1.00557
$845$	8.96110e9	0.510932
$846$	−2.76061e9	−0.156751
$847$	0	0
$848$	−1.38612e10	−0.780575
$849$	5.84276e9	0.327674
$850$	−3.09422e9	−0.172817
$851$	2.77995e9	0.154626
$852$	−1.63998e9	−0.0908448
$853$	1.16105e10	0.640516	0.320258	−	0.947330i	$-0.396230\pi$
$853$	1.16105e10	0.640516	0.320258	+	0.947330i	$0.396230\pi$
$854$	−9.55110e9	−0.524748
$855$	3.92809e9	0.214931
$856$	−2.31011e8	−0.0125885
$857$	2.52052e10	1.36791	0.683955	−	0.729524i	$-0.260258\pi$
$857$	2.52052e10	1.36791	0.683955	+	0.729524i	$0.260258\pi$
$858$	0	0
$859$	2.78224e10	1.49768	0.748839	−	0.662752i	$-0.230612\pi$
$859$	2.78224e10	1.49768	0.748839	+	0.662752i	$0.230612\pi$
$860$	1.51846e8	0.00814066
$861$	−1.89664e9	−0.101269
$862$	3.05808e10	1.62620
$863$	1.27828e10	0.677000	0.338500	−	0.940966i	$-0.390080\pi$
$863$	1.27828e10	0.677000	0.338500	+	0.940966i	$0.390080\pi$
$864$	−4.79790e9	−0.253077
$865$	1.85182e9	0.0972841
$866$	7.30836e9	0.382391
$867$	−1.07193e10	−0.558598
$868$	1.04044e10	0.540007
$869$	0	0
$870$	3.70434e9	0.190719
$871$	−1.74043e9	−0.0892470
$872$	4.88726e9	0.249608
$873$	−1.17110e10	−0.595722
$874$	2.67171e10	1.35363
$875$	9.08317e9	0.458363
$876$	1.14858e10	0.577296
$877$	−3.36909e9	−0.168661	−0.0843304	−	0.996438i	$-0.526875\pi$
$877$	−3.36909e9	−0.168661	−0.0843304	+	0.996438i	$0.526875\pi$
$878$	2.93565e10	1.46377
$879$	1.48353e10	0.736775
$880$	0	0
$881$	2.27946e10	1.12309	0.561547	−	0.827445i	$-0.310206\pi$
$881$	2.27946e10	1.12309	0.561547	+	0.827445i	$0.310206\pi$
$882$	−6.92270e9	−0.339731
$883$	−2.11019e10	−1.03147	−0.515737	−	0.856747i	$-0.672482\pi$
$883$	−2.11019e10	−1.03147	−0.515737	+	0.856747i	$0.672482\pi$
$884$	7.12906e8	0.0347096
$885$	5.18105e9	0.251256
$886$	3.95420e10	1.91003
$887$	−1.24838e10	−0.600642	−0.300321	−	0.953838i	$-0.597094\pi$
$887$	−1.24838e10	−0.600642	−0.300321	+	0.953838i	$0.597094\pi$
$888$	5.06808e8	0.0242883
$889$	1.37501e9	0.0656372
$890$	−1.53883e10	−0.731689
$891$	0	0
$892$	1.29027e10	0.608698
$893$	8.83098e9	0.414981
$894$	1.00843e9	0.0472026
$895$	−1.23053e10	−0.573737
$896$	4.80651e9	0.223230
$897$	−2.42031e9	−0.111969
$898$	2.31674e10	1.06760
$899$	−1.28695e10	−0.590750
$900$	−4.29559e9	−0.196415
$901$	2.70927e9	0.123400
$902$	0	0
$903$	−1.15093e8	−0.00520168
$904$	−9.71079e9	−0.437185
$905$	6.34423e9	0.284517
$906$	2.93479e10	1.31108
$907$	1.81146e10	0.806125	0.403063	−	0.915172i	$-0.367946\pi$
$907$	1.81146e10	0.806125	0.403063	+	0.915172i	$0.367946\pi$
$908$	−8.19464e9	−0.363270
$909$	−1.45562e10	−0.642798
$910$	−1.91141e9	−0.0840831
$911$	−1.35544e10	−0.593972	−0.296986	−	0.954882i	$-0.595981\pi$
$911$	−1.35544e10	−0.593972	−0.296986	+	0.954882i	$0.595981\pi$
$912$	1.80073e10	0.786080
$913$	0	0
$914$	9.28270e9	0.402126
$915$	5.63583e9	0.243211
$916$	−1.27595e10	−0.548531
$917$	1.37476e10	0.588755
$918$	1.10027e9	0.0469406
$919$	3.84145e10	1.63264	0.816320	−	0.577600i	$-0.196011\pi$
$919$	3.84145e10	1.63264	0.816320	+	0.577600i	$0.196011\pi$
$920$	−2.43305e9	−0.103013
$921$	−2.27569e10	−0.959851
$922$	1.42721e10	0.599694
$923$	−1.04671e9	−0.0438146
$924$	0	0
$925$	3.14902e9	0.130822
$926$	1.82862e10	0.756806
$927$	−4.31315e9	−0.177834
$928$	1.44739e10	0.594520
$929$	−3.68031e10	−1.50602	−0.753008	−	0.658011i	$-0.771398\pi$
$929$	−3.68031e10	−1.50602	−0.753008	+	0.658011i	$0.771398\pi$
$930$	−1.35215e10	−0.551231
$931$	2.21452e10	0.899405
$932$	1.67944e10	0.679529
$933$	−2.49532e10	−1.00587
$934$	−1.71517e10	−0.688799
$935$	0	0
$936$	−4.41242e8	−0.0175878
$937$	−3.96782e10	−1.57566	−0.787832	−	0.615890i	$-0.788796\pi$
$937$	−3.96782e10	−1.57566	−0.787832	+	0.615890i	$0.788796\pi$
$938$	−6.55107e9	−0.259181
$939$	−6.80843e9	−0.268360
$940$	3.97282e9	0.156010
$941$	1.04269e10	0.407936	0.203968	−	0.978978i	$-0.434616\pi$
$941$	1.04269e10	0.407936	0.203968	+	0.978978i	$0.434616\pi$
$942$	−9.26137e9	−0.360992
$943$	7.61199e9	0.295602
$944$	2.37512e10	0.918932
$945$	−1.33942e9	−0.0516305
$946$	0	0
$947$	2.79277e10	1.06859	0.534293	−	0.845299i	$-0.320578\pi$
$947$	2.79277e10	1.06859	0.534293	+	0.845299i	$0.320578\pi$
$948$	−1.95442e10	−0.745054
$949$	7.33077e9	0.278431
$950$	3.02641e10	1.14524
$951$	5.65988e9	0.213391
$952$	−5.43199e8	−0.0204047
$953$	−5.16025e9	−0.193128	−0.0965640	−	0.995327i	$-0.530785\pi$
$953$	−5.16025e9	−0.193128	−0.0965640	+	0.995327i	$0.530785\pi$
$954$	8.28373e9	0.308892
$955$	1.51739e9	0.0563749
$956$	3.56565e9	0.131989
$957$	0	0
$958$	1.83489e10	0.674266
$959$	−1.74276e10	−0.638075
$960$	5.46622e9	0.199406
$961$	1.94633e10	0.707432
$962$	−1.59793e9	−0.0578689
$963$	5.10402e8	0.0184170
$964$	−2.10951e10	−0.758424
$965$	−2.33984e10	−0.838188
$966$	−9.11016e9	−0.325167
$967$	−4.71056e10	−1.67525	−0.837626	−	0.546244i	$-0.816057\pi$
$967$	−4.71056e10	−1.67525	−0.837626	+	0.546244i	$0.816057\pi$
$968$	0	0
$969$	−3.51967e9	−0.124271
$970$	3.71183e10	1.30583
$971$	−3.39042e10	−1.18847	−0.594233	−	0.804293i	$-0.702544\pi$
$971$	−3.39042e10	−1.18847	−0.594233	+	0.804293i	$0.702544\pi$
$972$	1.52746e9	0.0533504
$973$	1.08864e10	0.378868
$974$	−5.72732e10	−1.98607
$975$	−2.74163e9	−0.0947313
$976$	2.58360e10	0.889510
$977$	1.13011e10	0.387693	0.193847	−	0.981032i	$-0.437904\pi$
$977$	1.13011e10	0.387693	0.193847	+	0.981032i	$0.437904\pi$
$978$	−1.35289e10	−0.462462
$979$	0	0
$980$	9.96251e9	0.338125
$981$	−1.07980e10	−0.365177
$982$	−4.29928e10	−1.44879
$983$	2.38598e9	0.0801179	0.0400589	−	0.999197i	$-0.487245\pi$
$983$	2.38598e9	0.0801179	0.0400589	+	0.999197i	$0.487245\pi$
$984$	1.38773e9	0.0464325
$985$	−1.39471e10	−0.465004
$986$	−3.31919e9	−0.110271
$987$	−3.01124e9	−0.0996862
$988$	−6.97283e9	−0.230017
$989$	4.61916e8	0.0151837
$990$	0	0
$991$	1.52156e10	0.496629	0.248315	−	0.968679i	$-0.420123\pi$
$991$	1.52156e10	0.496629	0.248315	+	0.968679i	$0.420123\pi$
$992$	−5.28321e10	−1.71833
$993$	−2.08570e10	−0.675972
$994$	−3.93986e9	−0.127241
$995$	−5.88884e9	−0.189517
$996$	−2.65455e10	−0.851301
$997$	6.01330e10	1.92167	0.960837	−	0.277114i	$-0.0893780\pi$
$997$	6.01330e10	1.92167	0.960837	+	0.277114i	$0.0893780\pi$
$998$	−5.03968e10	−1.60489
$999$	−1.11975e9	−0.0355339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.8.a.g.1.4	yes	4
11.10	odd	2	inner	363.8.a.g.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.8.a.g.1.1	✓	4	11.10	odd	2	inner
363.8.a.g.1.4	yes	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q+15.3118 q^{2} +27.0000 q^{3} +106.451 q^{4} -150.902 q^{5} +413.419 q^{6} +450.951 q^{7} -329.950 q^{8} +729.000 q^{9} -2310.59 q^{10} +2874.18 q^{12} +1834.43 q^{13} +6904.88 q^{14} -4074.37 q^{15} -18677.9 q^{16} +3650.74 q^{17} +11162.3 q^{18} -35707.3 q^{19} -16063.8 q^{20} +12175.7 q^{21} -48865.9 q^{23} -8908.66 q^{24} -55353.4 q^{25} +28088.4 q^{26} +19683.0 q^{27} +48004.3 q^{28} -59377.9 q^{29} -62385.9 q^{30} +216739. q^{31} -243759. q^{32} +55899.4 q^{34} -68049.7 q^{35} +77603.0 q^{36} -56889.3 q^{37} -546743. q^{38} +49529.6 q^{39} +49790.3 q^{40} -155773. q^{41} +186432. q^{42} -9452.73 q^{43} -110008. q^{45} -748225. q^{46} -247316. q^{47} -504303. q^{48} -620186. q^{49} -847561. q^{50} +98569.9 q^{51} +195277. q^{52} +742117. q^{53} +301382. q^{54} -148792. q^{56} -964098. q^{57} -909182. q^{58} -1.27162e6 q^{59} -433721. q^{60} -1.38324e6 q^{61} +3.31867e6 q^{62} +328744. q^{63} -1.34161e6 q^{64} -276820. q^{65} -948760. q^{67} +388626. q^{68} -1.31938e6 q^{69} -1.04196e6 q^{70} -570590. q^{71} -240534. q^{72} +3.99621e6 q^{73} -871078. q^{74} -1.49454e6 q^{75} -3.80109e6 q^{76} +758387. q^{78} -6.79990e6 q^{79} +2.81854e6 q^{80} +531441. q^{81} -2.38517e6 q^{82} -9.23584e6 q^{83} +1.29612e6 q^{84} -550906. q^{85} -144738. q^{86} -1.60320e6 q^{87} +6.65992e6 q^{89} -1.68442e6 q^{90} +827239. q^{91} -5.20184e6 q^{92} +5.85196e6 q^{93} -3.78685e6 q^{94} +5.38832e6 q^{95} -6.58148e6 q^{96} -1.60645e7 q^{97} -9.49616e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{3} + 26 q^{4} + 196 q^{5} + 2916 q^{9} + 702 q^{12} + 2032 q^{14} + 5292 q^{15} - 28734 q^{16} - 78648 q^{20} - 55532 q^{23} - 143052 q^{25} + 9204 q^{26} + 78732 q^{27} - 7816 q^{31} + 390716 q^{34}+ \cdots - 4819600 q^{97}+O(q^{100}) \) 4 * q + 108 * q^3 + 26 * q^4 + 196 * q^5 + 2916 * q^9 + 702 * q^12 + 2032 * q^14 + 5292 * q^15 - 28734 * q^16 - 78648 * q^20 - 55532 * q^23 - 143052 * q^25 + 9204 * q^26 + 78732 * q^27 - 7816 * q^31 + 390716 * q^34 + 18954 * q^36 - 763296 * q^37 - 550172 * q^38 + 54864 * q^42 + 142884 * q^45 - 1141988 * q^47 - 775818 * q^48 - 879924 * q^49 - 1035180 * q^53 + 2310616 * q^56 - 221596 * q^58 - 3891864 * q^59 - 2123496 * q^60 - 1530318 * q^64 + 1389632 * q^67 - 1499364 * q^69 - 5015440 * q^70 - 8309020 * q^71 - 3862404 * q^75 + 248508 * q^78 + 7783064 * q^80 + 2125764 * q^81 - 10542572 * q^82 - 3688636 * q^86 - 5296744 * q^89 + 9661056 * q^91 - 14347308 * q^92 - 211032 * q^93 - 4819600 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more