LMFDB - Embedded newform 98.3.b.a.97.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [98,3,Mod(97,98)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(98, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("98.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 98 = 2 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	98.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$2.67030659073$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.1
Root			$-1.84776i$ of defining polynomial
Character	$\chi$	$=$	98.97
Dual form			98.3.b.a.97.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -2.16478i q^{3} +2.00000 q^{4} +8.15640i q^{5} +3.06147i q^{6} -2.82843 q^{8} +4.31371 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -2.16478i q^{3} +2.00000 q^{4} +8.15640i q^{5} +3.06147i q^{6} -2.82843 q^{8} +4.31371 q^{9} -11.5349i q^{10} +17.6569 q^{11} -4.32957i q^{12} +1.21371i q^{13} +17.6569 q^{15} +4.00000 q^{16} +16.6298i q^{17} -6.10051 q^{18} +0.371418i q^{19} +16.3128i q^{20} -24.9706 q^{22} +3.02944 q^{23} +6.12293i q^{24} -41.5269 q^{25} -1.71644i q^{26} -28.8213i q^{27} +2.38478 q^{29} -24.9706 q^{30} +1.26810i q^{31} -5.65685 q^{32} -38.2233i q^{33} -23.5181i q^{34} +8.62742 q^{36} -27.7574 q^{37} -0.525265i q^{38} +2.62742 q^{39} -23.0698i q^{40} -28.7988i q^{41} -55.3137 q^{43} +35.3137 q^{44} +35.1843i q^{45} -4.28427 q^{46} +58.3855i q^{47} -8.65914i q^{48} +58.7279 q^{50} +36.0000 q^{51} +2.42742i q^{52} -1.37258 q^{53} +40.7595i q^{54} +144.016i q^{55} +0.804041 q^{57} -3.37258 q^{58} -98.7735i q^{59} +35.3137 q^{60} -82.2975i q^{61} -1.79337i q^{62} +8.00000 q^{64} -9.89949 q^{65} +54.0559i q^{66} +97.9411 q^{67} +33.2597i q^{68} -6.55808i q^{69} +64.5685 q^{71} -12.2010 q^{72} -91.9940i q^{73} +39.2548 q^{74} +89.8968i q^{75} +0.742837i q^{76} -3.71573 q^{78} -19.7157 q^{79} +32.6256i q^{80} -23.5685 q^{81} +40.7276i q^{82} -43.4495i q^{83} -135.640 q^{85} +78.2254 q^{86} -5.16253i q^{87} -49.9411 q^{88} +7.86191i q^{89} -49.7582i q^{90} +6.05887 q^{92} +2.74517 q^{93} -82.5695i q^{94} -3.02944 q^{95} +12.2459i q^{96} -47.2126i q^{97} +76.1665 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 28 q^{9}+O(q^{10}) $ 4 * q + 8 * q^4 - 28 * q^9	$ 4 q + 8 q^{4} - 28 q^{9} + 48 q^{11} + 48 q^{15} + 16 q^{16} - 64 q^{18} - 32 q^{22} + 80 q^{23} - 36 q^{25} - 64 q^{29} - 32 q^{30} - 56 q^{36} - 128 q^{37} - 80 q^{39} - 176 q^{43} + 96 q^{44} + 96 q^{46} + 184 q^{50} + 144 q^{51} - 96 q^{53} + 320 q^{57} - 104 q^{58} + 96 q^{60} + 32 q^{64} + 256 q^{67} + 32 q^{71} - 128 q^{72} - 24 q^{74} - 128 q^{78} - 192 q^{79} + 132 q^{81} - 288 q^{85} + 64 q^{86} - 64 q^{88} + 160 q^{92} + 192 q^{93} - 80 q^{95} - 80 q^{99}+O(q^{100}) $ 4 * q + 8 * q^4 - 28 * q^9 + 48 * q^11 + 48 * q^15 + 16 * q^16 - 64 * q^18 - 32 * q^22 + 80 * q^23 - 36 * q^25 - 64 * q^29 - 32 * q^30 - 56 * q^36 - 128 * q^37 - 80 * q^39 - 176 * q^43 + 96 * q^44 + 96 * q^46 + 184 * q^50 + 144 * q^51 - 96 * q^53 + 320 * q^57 - 104 * q^58 + 96 * q^60 + 32 * q^64 + 256 * q^67 + 32 * q^71 - 128 * q^72 - 24 * q^74 - 128 * q^78 - 192 * q^79 + 132 * q^81 - 288 * q^85 + 64 * q^86 - 64 * q^88 + 160 * q^92 + 192 * q^93 - 80 * q^95 - 80 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/98\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

Coefficient data

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

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

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

$2$	−1.41421	−0.707107
$2$	−1.41421	−0.707107
$3$	− 2.16478i	− 0.721595i	−0.932644	−	0.360797i	$-0.882505\pi$
$3$	− 2.16478i	− 0.721595i	0.932644	−	0.360797i	$-0.117495\pi$
$4$	2.00000	0.500000
$5$	8.15640i	1.63128i	0.578559	+	0.815640i	$0.303615\pi$
$5$	8.15640i	1.63128i	−0.578559	+	0.815640i	$0.696385\pi$
$6$	3.06147i	0.510245i
$7$	0	0
$7$	0	0
$8$	−2.82843	−0.353553
$9$	4.31371	0.479301
$10$	− 11.5349i	− 1.15349i
$11$	17.6569	1.60517	0.802584	−	0.596539i	$-0.203458\pi$
$11$	17.6569	1.60517	0.802584	+	0.596539i	$0.203458\pi$
$12$	− 4.32957i	− 0.360797i
$13$	1.21371i	0.0933622i	0.998910	+	0.0466811i	$0.0148645\pi$
$13$	1.21371i	0.0933622i	−0.998910	+	0.0466811i	$0.985136\pi$
$14$	0	0
$15$	17.6569	1.17712
$16$	4.00000	0.250000
$17$	16.6298i	0.978225i	0.872221	+	0.489113i	$0.162679\pi$
$17$	16.6298i	0.978225i	−0.872221	+	0.489113i	$0.837321\pi$
$18$	−6.10051	−0.338917
$19$	0.371418i	0.0195483i	0.999952	+	0.00977417i	$0.00311126\pi$
$19$	0.371418i	0.0195483i	−0.999952	+	0.00977417i	$0.996889\pi$
$20$	16.3128i	0.815640i
$21$	0	0
$22$	−24.9706	−1.13503
$23$	3.02944	0.131715	0.0658573	−	0.997829i	$-0.479022\pi$
$23$	3.02944	0.131715	0.0658573	+	0.997829i	$0.479022\pi$
$24$	6.12293i	0.255122i
$25$	−41.5269	−1.66108
$26$	− 1.71644i	− 0.0660170i
$27$	− 28.8213i	− 1.06746i
$28$	0	0
$29$	2.38478	0.0822337	0.0411168	−	0.999154i	$-0.486908\pi$
$29$	2.38478	0.0822337	0.0411168	+	0.999154i	$0.486908\pi$
$30$	−24.9706	−0.832352
$31$	1.26810i	0.0409065i	0.999791	+	0.0204532i	$0.00651092\pi$
$31$	1.26810i	0.0409065i	−0.999791	+	0.0204532i	$0.993489\pi$
$32$	−5.65685	−0.176777
$33$	− 38.2233i	− 1.15828i
$34$	− 23.5181i	− 0.691710i
$35$	0	0
$36$	8.62742	0.239650
$37$	−27.7574	−0.750199	−0.375099	−	0.926985i	$-0.622391\pi$
$37$	−27.7574	−0.750199	−0.375099	+	0.926985i	$0.622391\pi$
$38$	− 0.525265i	− 0.0138228i
$39$	2.62742	0.0673697
$40$	− 23.0698i	− 0.576745i
$41$	− 28.7988i	− 0.702409i	−0.936299	−	0.351205i	$-0.885772\pi$
$41$	− 28.7988i	− 0.702409i	0.936299	−	0.351205i	$-0.114228\pi$
$42$	0	0
$43$	−55.3137	−1.28637	−0.643183	−	0.765713i	$-0.722386\pi$
$43$	−55.3137	−1.28637	−0.643183	+	0.765713i	$0.722386\pi$
$44$	35.3137	0.802584
$45$	35.1843i	0.781874i
$46$	−4.28427	−0.0931363
$47$	58.3855i	1.24224i	0.783714	+	0.621122i	$0.213323\pi$
$47$	58.3855i	1.24224i	−0.783714	+	0.621122i	$0.786677\pi$
$48$	− 8.65914i	− 0.180399i
$49$	0	0
$50$	58.7279	1.17456
$51$	36.0000	0.705882
$52$	2.42742i	0.0466811i
$53$	−1.37258	−0.0258978	−0.0129489	−	0.999916i	$-0.504122\pi$
$53$	−1.37258	−0.0258978	−0.0129489	+	0.999916i	$0.504122\pi$
$54$	40.7595i	0.754805i
$55$	144.016i	2.61848i
$56$	0	0
$57$	0.804041	0.0141060
$58$	−3.37258	−0.0581480
$59$	− 98.7735i	− 1.67413i	−0.547105	−	0.837064i	$-0.684270\pi$
$59$	− 98.7735i	− 1.67413i	0.547105	−	0.837064i	$-0.315730\pi$
$60$	35.3137	0.588562
$61$	− 82.2975i	− 1.34914i	−0.738211	−	0.674570i	$-0.764329\pi$
$61$	− 82.2975i	− 1.34914i	0.738211	−	0.674570i	$-0.235671\pi$
$62$	− 1.79337i	− 0.0289253i
$63$	0	0
$64$	8.00000	0.125000
$65$	−9.89949	−0.152300
$66$	54.0559i	0.819029i
$67$	97.9411	1.46181	0.730904	−	0.682480i	$-0.239099\pi$
$67$	97.9411	1.46181	0.730904	+	0.682480i	$0.239099\pi$
$68$	33.2597i	0.489113i
$69$	− 6.55808i	− 0.0950446i
$70$	0	0
$71$	64.5685	0.909416	0.454708	−	0.890641i	$-0.349744\pi$
$71$	64.5685	0.909416	0.454708	+	0.890641i	$0.349744\pi$
$72$	−12.2010	−0.169458
$73$	− 91.9940i	− 1.26019i	−0.776517	−	0.630096i	$-0.783016\pi$
$73$	− 91.9940i	− 1.26019i	0.776517	−	0.630096i	$-0.216984\pi$
$74$	39.2548	0.530471
$75$	89.8968i	1.19862i
$76$	0.742837i	0.00977417i
$77$	0	0
$78$	−3.71573	−0.0476375
$79$	−19.7157	−0.249566	−0.124783	−	0.992184i	$-0.539823\pi$
$79$	−19.7157	−0.249566	−0.124783	+	0.992184i	$0.539823\pi$
$80$	32.6256i	0.407820i
$81$	−23.5685	−0.290970
$82$	40.7276i	0.496678i
$83$	− 43.4495i	− 0.523488i	−0.965137	−	0.261744i	$-0.915702\pi$
$83$	− 43.4495i	− 0.523488i	0.965137	−	0.261744i	$-0.0842977\pi$
$84$	0	0
$85$	−135.640	−1.59576
$86$	78.2254	0.909598
$87$	− 5.16253i	− 0.0593394i
$88$	−49.9411	−0.567513
$89$	7.86191i	0.0883360i	0.999024	+	0.0441680i	$0.0140637\pi$
$89$	7.86191i	0.0883360i	−0.999024	+	0.0441680i	$0.985936\pi$
$90$	− 49.7582i	− 0.552869i
$91$	0	0
$92$	6.05887	0.0658573
$93$	2.74517	0.0295179
$94$	− 82.5695i	− 0.878399i
$95$	−3.02944	−0.0318888
$96$	12.2459i	0.127561i
$97$	− 47.2126i	− 0.486728i	−0.969935	−	0.243364i	$-0.921749\pi$
$97$	− 47.2126i	− 0.486728i	0.969935	−	0.243364i	$-0.0782510\pi$
$98$	0	0
$99$	76.1665	0.769359
$100$	−83.0538	−0.830538
$101$	196.532i	1.94586i	0.231092	+	0.972932i	$0.425770\pi$
$101$	196.532i	1.94586i	−0.231092	+	0.972932i	$0.574230\pi$
$102$	−50.9117	−0.499134
$103$	− 119.090i	− 1.15621i	−0.815963	−	0.578105i	$-0.803793\pi$
$103$	− 119.090i	− 1.15621i	0.815963	−	0.578105i	$-0.196207\pi$
$104$	− 3.43289i	− 0.0330085i
$105$	0	0
$106$	1.94113	0.0183125
$107$	−147.480	−1.37832	−0.689160	−	0.724609i	$-0.742020\pi$
$107$	−147.480	−1.37832	−0.689160	+	0.724609i	$0.742020\pi$
$108$	− 57.6426i	− 0.533728i
$109$	−130.385	−1.19619	−0.598095	−	0.801425i	$-0.704076\pi$
$109$	−130.385	−1.19619	−0.598095	+	0.801425i	$0.704076\pi$
$110$	− 203.670i	− 1.85155i
$111$	60.0887i	0.541340i
$112$	0	0
$113$	−1.94113	−0.0171781	−0.00858905	−	0.999963i	$-0.502734\pi$
$113$	−1.94113	−0.0171781	−0.00858905	+	0.999963i	$0.502734\pi$
$114$	−1.13708	−0.00997443
$115$	24.7093i	0.214864i
$116$	4.76955	0.0411168
$117$	5.23558i	0.0447486i
$118$	139.687i	1.18379i
$119$	0	0
$120$	−49.9411	−0.416176
$121$	190.765	1.57657
$122$	116.386i	0.953986i
$123$	−62.3431	−0.506855
$124$	2.53620i	0.0204532i
$125$	− 134.800i	− 1.07840i
$126$	0	0
$127$	22.3431	0.175930	0.0879651	−	0.996124i	$-0.471964\pi$
$127$	22.3431	0.175930	0.0879651	+	0.996124i	$0.471964\pi$
$128$	−11.3137	−0.0883883
$129$	119.742i	0.928235i
$130$	14.0000	0.107692
$131$	36.3662i	0.277605i	0.990320	+	0.138802i	$0.0443252\pi$
$131$	36.3662i	0.277605i	−0.990320	+	0.138802i	$0.955675\pi$
$132$	− 76.4466i	− 0.579141i
$133$	0	0
$134$	−138.510	−1.03365
$135$	235.078	1.74132
$136$	− 47.0363i	− 0.345855i
$137$	−75.9239	−0.554189	−0.277094	−	0.960843i	$-0.589371\pi$
$137$	−75.9239	−0.554189	−0.277094	+	0.960843i	$0.589371\pi$
$138$	9.27452i	0.0672067i
$139$	− 109.751i	− 0.789578i	−0.918772	−	0.394789i	$-0.870818\pi$
$139$	− 109.751i	− 0.789578i	0.918772	−	0.394789i	$-0.129182\pi$
$140$	0	0
$141$	126.392	0.896397
$142$	−91.3137	−0.643054
$143$	21.4303i	0.149862i
$144$	17.2548	0.119825
$145$	19.4512i	0.134146i
$146$	130.099i	0.891090i
$147$	0	0
$148$	−55.5147	−0.375099
$149$	−184.000	−1.23490	−0.617450	−	0.786610i	$-0.711834\pi$
$149$	−184.000	−1.23490	−0.617450	+	0.786610i	$0.711834\pi$
$150$	− 127.133i	− 0.847555i
$151$	216.284	1.43235	0.716173	−	0.697923i	$-0.245892\pi$
$151$	216.284	1.43235	0.716173	+	0.697923i	$0.245892\pi$
$152$	− 1.05053i	− 0.00691138i
$153$	71.7362i	0.468864i
$154$	0	0
$155$	−10.3431	−0.0667300
$156$	5.25483	0.0336848
$157$	− 182.570i	− 1.16287i	−0.813594	−	0.581433i	$-0.802492\pi$
$157$	− 182.570i	− 1.16287i	0.813594	−	0.581433i	$-0.197508\pi$
$158$	27.8823	0.176470
$159$	2.97135i	0.0186877i
$160$	− 46.1396i	− 0.288372i
$161$	0	0
$162$	33.3310	0.205747
$163$	−159.078	−0.975940	−0.487970	−	0.872860i	$-0.662262\pi$
$163$	−159.078	−0.975940	−0.487970	+	0.872860i	$0.662262\pi$
$164$	− 57.5976i	− 0.351205i
$165$	311.765	1.88948
$166$	61.4469i	0.370162i
$167$	− 74.3455i	− 0.445183i	−0.974912	−	0.222591i	$-0.928548\pi$
$167$	− 74.3455i	− 0.445183i	0.974912	−	0.222591i	$-0.0714516\pi$
$168$	0	0
$169$	167.527	0.991284
$170$	191.823	1.12837
$171$	1.60219i	0.00936953i
$172$	−110.627	−0.643183
$173$	188.648i	1.09045i	0.838290	+	0.545225i	$0.183556\pi$
$173$	188.648i	1.09045i	−0.838290	+	0.545225i	$0.816444\pi$
$174$	7.30092i	0.0419593i
$175$	0	0
$176$	70.6274	0.401292
$177$	−213.823	−1.20804
$178$	− 11.1184i	− 0.0624630i
$179$	−209.872	−1.17247	−0.586235	−	0.810141i	$-0.699390\pi$
$179$	−209.872	−1.17247	−0.586235	+	0.810141i	$0.699390\pi$
$180$	70.3687i	0.390937i
$181$	315.609i	1.74369i	0.489778	+	0.871847i	$0.337078\pi$
$181$	315.609i	1.74369i	−0.489778	+	0.871847i	$0.662922\pi$
$182$	0	0
$183$	−178.156	−0.973532
$184$	−8.56854	−0.0465682
$185$	− 226.400i	− 1.22378i
$186$	−3.88225	−0.0208723
$187$	293.631i	1.57022i
$188$	116.771i	0.621122i
$189$	0	0
$190$	4.28427	0.0225488
$191$	326.794	1.71096	0.855482	−	0.517833i	$-0.173261\pi$
$191$	326.794	1.71096	0.855482	+	0.517833i	$0.173261\pi$
$192$	− 17.3183i	− 0.0901994i
$193$	108.451	0.561921	0.280961	−	0.959719i	$-0.409347\pi$
$193$	108.451	0.561921	0.280961	+	0.959719i	$0.409347\pi$
$194$	66.7688i	0.344169i
$195$	21.4303i	0.109899i
$196$	0	0
$197$	40.5097	0.205633	0.102816	−	0.994700i	$-0.467215\pi$
$197$	40.5097	0.205633	0.102816	+	0.994700i	$0.467215\pi$
$198$	−107.716	−0.544019
$199$	245.390i	1.23311i	0.787310	+	0.616557i	$0.211473\pi$
$199$	245.390i	1.23311i	−0.787310	+	0.616557i	$0.788527\pi$
$200$	117.456	0.587279
$201$	− 212.021i	− 1.05483i
$202$	− 277.939i	− 1.37593i
$203$	0	0
$204$	72.0000	0.352941
$205$	234.894	1.14583
$206$	168.418i	0.817563i
$207$	13.0681	0.0631310
$208$	4.85483i	0.0233405i
$209$	6.55808i	0.0313784i
$210$	0	0
$211$	−36.2843	−0.171963	−0.0859817	−	0.996297i	$-0.527403\pi$
$211$	−36.2843	−0.171963	−0.0859817	+	0.996297i	$0.527403\pi$
$212$	−2.74517	−0.0129489
$213$	− 139.777i	− 0.656230i
$214$	208.569	0.974619
$215$	− 451.161i	− 2.09842i
$216$	81.5190i	0.377403i
$217$	0	0
$218$	184.392	0.845834
$219$	−199.147	−0.909348
$220$	288.033i	1.30924i
$221$	−20.1838	−0.0913293
$222$	− 84.9783i	− 0.382785i
$223$	− 88.9100i	− 0.398700i	−0.979928	−	0.199350i	$-0.936117\pi$
$223$	− 88.9100i	− 0.398700i	0.979928	−	0.199350i	$-0.0638830\pi$
$224$	0	0
$225$	−179.135	−0.796156
$226$	2.74517	0.0121468
$227$	− 320.287i	− 1.41096i	−0.708732	−	0.705478i	$-0.750732\pi$
$227$	− 320.287i	− 1.41096i	0.708732	−	0.705478i	$-0.249268\pi$
$228$	1.60808	0.00705299
$229$	− 2.90376i	− 0.0126802i	−0.999980	−	0.00634008i	$-0.997982\pi$
$229$	− 2.90376i	− 0.0126802i	0.999980	−	0.00634008i	$-0.00201812\pi$
$230$	− 34.9442i	− 0.151931i
$231$	0	0
$232$	−6.74517	−0.0290740
$233$	−16.6102	−0.0712883	−0.0356441	−	0.999365i	$-0.511348\pi$
$233$	−16.6102	−0.0712883	−0.0356441	+	0.999365i	$0.511348\pi$
$234$	− 7.40423i	− 0.0316420i
$235$	−476.215	−2.02645
$236$	− 197.547i	− 0.837064i
$237$	42.6803i	0.180086i
$238$	0	0
$239$	−397.470	−1.66305	−0.831527	−	0.555484i	$-0.812533\pi$
$239$	−397.470	−1.66305	−0.831527	+	0.555484i	$0.812533\pi$
$240$	70.6274	0.294281
$241$	302.018i	1.25319i	0.779347	+	0.626593i	$0.215551\pi$
$241$	302.018i	1.25319i	−0.779347	+	0.626593i	$0.784449\pi$
$242$	−269.782	−1.11480
$243$	− 208.371i	− 0.857494i
$244$	− 164.595i	− 0.674570i
$245$	0	0
$246$	88.1665	0.358400
$247$	−0.450793	−0.00182507
$248$	− 3.58673i	− 0.0144626i
$249$	−94.0589	−0.377746
$250$	190.636i	0.762545i
$251$	− 17.9974i	− 0.0717027i	−0.999357	−	0.0358514i	$-0.988586\pi$
$251$	− 17.9974i	− 0.0717027i	0.999357	−	0.0358514i	$-0.0114143\pi$
$252$	0	0
$253$	53.4903	0.211424
$254$	−31.5980	−0.124401
$255$	293.631i	1.15149i
$256$	16.0000	0.0625000
$257$	445.668i	1.73412i	0.498206	+	0.867059i	$0.333992\pi$
$257$	445.668i	1.73412i	−0.498206	+	0.867059i	$0.666008\pi$
$258$	− 169.341i	− 0.656361i
$259$	0	0
$260$	−19.7990	−0.0761500
$261$	10.2872	0.0394147
$262$	− 51.4296i	− 0.196296i
$263$	−33.5391	−0.127525	−0.0637626	−	0.997965i	$-0.520310\pi$
$263$	−33.5391	−0.127525	−0.0637626	+	0.997965i	$0.520310\pi$
$264$	108.112i	0.409514i
$265$	− 11.1953i	− 0.0422466i
$266$	0	0
$267$	17.0193	0.0637428
$268$	195.882	0.730904
$269$	− 131.716i	− 0.489651i	−0.969567	−	0.244826i	$-0.921269\pi$
$269$	− 131.716i	− 0.489651i	0.969567	−	0.244826i	$-0.0787307\pi$
$270$	−332.451	−1.23130
$271$	− 454.132i	− 1.67576i	−0.545851	−	0.837882i	$-0.683793\pi$
$271$	− 454.132i	− 1.67576i	0.545851	−	0.837882i	$-0.316207\pi$
$272$	66.5193i	0.244556i
$273$	0	0
$274$	107.373	0.391871
$275$	−733.235	−2.66631
$276$	− 13.1162i	− 0.0475223i
$277$	53.7645	0.194096	0.0970478	−	0.995280i	$-0.469060\pi$
$277$	53.7645	0.194096	0.0970478	+	0.995280i	$0.469060\pi$
$278$	155.212i	0.558316i
$279$	5.47022i	0.0196065i
$280$	0	0
$281$	−162.218	−0.577289	−0.288645	−	0.957436i	$-0.593205\pi$
$281$	−162.218	−0.577289	−0.288645	+	0.957436i	$0.593205\pi$
$282$	−178.745	−0.633848
$283$	225.037i	0.795182i	0.917563	+	0.397591i	$0.130154\pi$
$283$	225.037i	0.795182i	−0.917563	+	0.397591i	$0.869846\pi$
$284$	129.137	0.454708
$285$	6.55808i	0.0230108i
$286$	− 30.3070i	− 0.105968i
$287$	0	0
$288$	−24.4020	−0.0847292
$289$	12.4487	0.0430751
$290$	− 27.5081i	− 0.0948557i
$291$	−102.205	−0.351221
$292$	− 183.988i	− 0.630096i
$293$	− 92.6226i	− 0.316118i	−0.987430	−	0.158059i	$-0.949476\pi$
$293$	− 92.6226i	− 0.316118i	0.987430	−	0.158059i	$-0.0505236\pi$
$294$	0	0
$295$	805.637	2.73097
$296$	78.5097	0.265235
$297$	− 508.894i	− 1.71345i
$298$	260.215	0.873206
$299$	3.67685i	0.0122972i
$300$	179.794i	0.599312i
$301$	0	0
$302$	−305.872	−1.01282
$303$	425.450	1.40413
$304$	1.48567i	0.00488708i
$305$	671.252	2.20083
$306$	− 101.450i	− 0.331537i
$307$	− 430.230i	− 1.40140i	−0.713457	−	0.700700i	$-0.752871\pi$
$307$	− 430.230i	− 1.40140i	0.713457	−	0.700700i	$-0.247129\pi$
$308$	0	0
$309$	−257.803	−0.834314
$310$	14.6274	0.0471852
$311$	− 59.0008i	− 0.189713i	−0.995491	−	0.0948567i	$-0.969761\pi$
$311$	− 59.0008i	− 0.189713i	0.995491	−	0.0948567i	$-0.0302393\pi$
$312$	−7.43146	−0.0238188
$313$	313.725i	1.00232i	0.865356	+	0.501158i	$0.167093\pi$
$313$	313.725i	1.00232i	−0.865356	+	0.501158i	$0.832907\pi$
$314$	258.193i	0.822270i
$315$	0	0
$316$	−39.4315	−0.124783
$317$	470.548	1.48438	0.742190	−	0.670190i	$-0.233787\pi$
$317$	470.548	1.48438	0.742190	+	0.670190i	$0.233787\pi$
$318$	− 4.20212i	− 0.0132142i
$319$	42.1076	0.131999
$320$	65.2512i	0.203910i
$321$	319.263i	0.994588i
$322$	0	0
$323$	−6.17662	−0.0191227
$324$	−47.1371	−0.145485
$325$	− 50.4016i	− 0.155082i
$326$	224.971	0.690094
$327$	282.255i	0.863165i
$328$	81.4552i	0.248339i
$329$	0	0
$330$	−440.902	−1.33607
$331$	308.666	0.932526	0.466263	−	0.884646i	$-0.345600\pi$
$331$	308.666	0.932526	0.466263	+	0.884646i	$0.345600\pi$
$332$	− 86.8991i	− 0.261744i
$333$	−119.737	−0.359571
$334$	105.140i	0.314792i
$335$	798.847i	2.38462i
$336$	0	0
$337$	189.865	0.563398	0.281699	−	0.959503i	$-0.409102\pi$
$337$	189.865	0.563398	0.281699	+	0.959503i	$0.409102\pi$
$338$	−236.919	−0.700943
$339$	4.20212i	0.0123956i
$340$	−271.279	−0.797880
$341$	22.3907i	0.0656618i
$342$	− 2.26584i	− 0.00662526i
$343$	0	0
$344$	156.451	0.454799
$345$	53.4903	0.155044
$346$	− 266.788i	− 0.771064i
$347$	288.950	0.832710	0.416355	−	0.909202i	$-0.363307\pi$
$347$	288.950	0.832710	0.416355	+	0.909202i	$0.363307\pi$
$348$	− 10.3251i	− 0.0296697i
$349$	− 339.050i	− 0.971489i	−0.874101	−	0.485745i	$-0.838548\pi$
$349$	− 339.050i	− 0.971489i	0.874101	−	0.485745i	$-0.161452\pi$
$350$	0	0
$351$	34.9807	0.0996600
$352$	−99.8823	−0.283756
$353$	498.029i	1.41085i	0.708787	+	0.705423i	$0.249243\pi$
$353$	498.029i	1.41085i	−0.708787	+	0.705423i	$0.750757\pi$
$354$	302.392	0.854214
$355$	526.647i	1.48351i
$356$	15.7238i	0.0441680i
$357$	0	0
$358$	296.804	0.829062
$359$	−117.941	−0.328527	−0.164263	−	0.986417i	$-0.552525\pi$
$359$	−117.941	−0.328527	−0.164263	+	0.986417i	$0.552525\pi$
$360$	− 99.5164i	− 0.276434i
$361$	360.862	0.999618
$362$	− 446.338i	− 1.23298i
$363$	− 412.964i	− 1.13764i
$364$	0	0
$365$	750.340	2.05573
$366$	251.951	0.688391
$367$	175.066i	0.477020i	0.971140	+	0.238510i	$0.0766589\pi$
$367$	175.066i	0.477020i	−0.971140	+	0.238510i	$0.923341\pi$
$368$	12.1177	0.0329287
$369$	− 124.230i	− 0.336665i
$370$	320.178i	0.865347i
$371$	0	0
$372$	5.49033	0.0147590
$373$	−526.156	−1.41061	−0.705304	−	0.708905i	$-0.749189\pi$
$373$	−526.156	−1.41061	−0.705304	+	0.708905i	$0.749189\pi$
$374$	− 415.256i	− 1.11031i
$375$	−291.813	−0.778169
$376$	− 165.139i	− 0.439199i
$377$	2.89442i	0.00767751i
$378$	0	0
$379$	−383.245	−1.01120	−0.505600	−	0.862768i	$-0.668729\pi$
$379$	−383.245	−1.01120	−0.505600	+	0.862768i	$0.668729\pi$
$380$	−6.05887	−0.0159444
$381$	− 48.3681i	− 0.126950i
$382$	−462.156	−1.20983
$383$	0.127451i	0 0.000332769i	1.00000		0.000166385i	$5.29619e-5\pi$
$383$	0.127451i	0 0.000332769i	−1.00000		0.000166385i	$0.999947\pi$
$384$	24.4917i	0.0637806i
$385$	0	0
$386$	−153.373	−0.397338
$387$	−238.607	−0.616556
$388$	− 94.4253i	− 0.243364i
$389$	−372.909	−0.958634	−0.479317	−	0.877642i	$-0.659116\pi$
$389$	−372.909	−0.958634	−0.479317	+	0.877642i	$0.659116\pi$
$390$	− 30.3070i	− 0.0777102i
$391$	50.3790i	0.128847i
$392$	0	0
$393$	78.7250	0.200318
$394$	−57.2893	−0.145404
$395$	− 160.809i	− 0.407112i
$396$	152.333	0.384679
$397$	− 72.5164i	− 0.182661i	−0.995821	−	0.0913305i	$-0.970888\pi$
$397$	− 72.5164i	− 0.182661i	0.995821	−	0.0913305i	$-0.0291120\pi$
$398$	− 347.034i	− 0.871944i
$399$	0	0
$400$	−166.108	−0.415269
$401$	−12.3116	−0.0307023	−0.0153511	−	0.999882i	$-0.504887\pi$
$401$	−12.3116	−0.0307023	−0.0153511	+	0.999882i	$0.504887\pi$
$402$	299.844i	0.745880i
$403$	−1.53911	−0.00381912
$404$	393.064i	0.972932i
$405$	− 192.235i	− 0.474653i
$406$	0	0
$407$	−490.108	−1.20420
$408$	−101.823	−0.249567
$409$	185.279i	0.453004i	0.974011	+	0.226502i	$0.0727290\pi$
$409$	185.279i	0.453004i	−0.974011	+	0.226502i	$0.927271\pi$
$410$	−332.191	−0.810222
$411$	164.359i	0.399900i
$412$	− 238.179i	− 0.578105i
$413$	0	0
$414$	−18.4811	−0.0446403
$415$	354.392	0.853956
$416$	− 6.86577i	− 0.0165043i
$417$	−237.588	−0.569755
$418$	− 9.27452i	− 0.0221879i
$419$	− 491.676i	− 1.17345i	−0.809785	−	0.586726i	$-0.800417\pi$
$419$	− 491.676i	− 1.17345i	0.809785	−	0.586726i	$-0.199583\pi$
$420$	0	0
$421$	−471.373	−1.11965	−0.559825	−	0.828611i	$-0.689132\pi$
$421$	−471.373	−1.11965	−0.559825	+	0.828611i	$0.689132\pi$
$422$	51.3137	0.121596
$423$	251.858i	0.595409i
$424$	3.88225	0.00915625
$425$	− 690.586i	− 1.62491i
$426$	197.674i	0.464025i
$427$	0	0
$428$	−294.960	−0.689160
$429$	46.3919	0.108140
$430$	638.038i	1.48381i
$431$	−457.255	−1.06092	−0.530458	−	0.847711i	$-0.677980\pi$
$431$	−457.255	−1.06092	−0.530458	+	0.847711i	$0.677980\pi$
$432$	− 115.285i	− 0.266864i
$433$	− 248.941i	− 0.574921i	−0.957792	−	0.287461i	$-0.907189\pi$
$433$	− 248.941i	− 0.574921i	0.957792	−	0.287461i	$-0.0928110\pi$
$434$	0	0
$435$	42.1076	0.0967992
$436$	−260.770	−0.598095
$437$	1.12519i	0.00257480i
$438$	281.637	0.643006
$439$	− 430.001i	− 0.979501i	−0.871863	−	0.489751i	$-0.837088\pi$
$439$	− 430.001i	− 0.979501i	0.871863	−	0.489751i	$-0.162912\pi$
$440$	− 407.340i	− 0.925773i
$441$	0	0
$442$	28.5442	0.0645795
$443$	380.353	0.858585	0.429293	−	0.903165i	$-0.358763\pi$
$443$	380.353	0.858585	0.429293	+	0.903165i	$0.358763\pi$
$444$	120.177i	0.270670i
$445$	−64.1249	−0.144101
$446$	125.738i	0.281923i
$447$	398.320i	0.891097i
$448$	0	0
$449$	326.627	0.727455	0.363728	−	0.931505i	$-0.381504\pi$
$449$	326.627	0.727455	0.363728	+	0.931505i	$0.381504\pi$
$450$	253.335	0.562967
$451$	− 508.496i	− 1.12749i
$452$	−3.88225	−0.00858905
$453$	− 468.209i	− 1.03357i
$454$	452.954i	0.997697i
$455$	0	0
$456$	−2.27417	−0.00498721
$457$	−621.117	−1.35912	−0.679559	−	0.733621i	$-0.737829\pi$
$457$	−621.117	−1.35912	−0.679559	+	0.733621i	$0.737829\pi$
$458$	4.10653i	0.00896622i
$459$	479.294	1.04421
$460$	49.4186i	0.107432i
$461$	170.939i	0.370801i	0.982663	+	0.185401i	$0.0593583\pi$
$461$	170.939i	0.370801i	−0.982663	+	0.185401i	$0.940642\pi$
$462$	0	0
$463$	491.813	1.06223	0.531116	−	0.847299i	$-0.321773\pi$
$463$	491.813	1.06223	0.531116	+	0.847299i	$0.321773\pi$
$464$	9.53911	0.0205584
$465$	22.3907i	0.0481520i
$466$	23.4903	0.0504084
$467$	− 300.342i	− 0.643132i	−0.946887	−	0.321566i	$-0.895791\pi$
$467$	− 300.342i	− 0.643132i	0.946887	−	0.321566i	$-0.104209\pi$
$468$	10.4712i	0.0223743i
$469$	0	0
$470$	673.470	1.43292
$471$	−395.225	−0.839118
$472$	279.374i	0.591893i
$473$	−976.666	−2.06483
$474$	− 60.3591i	− 0.127340i
$475$	− 15.4239i	− 0.0324713i
$476$	0	0
$477$	−5.92092	−0.0124128
$478$	562.108	1.17596
$479$	161.823i	0.337834i	0.985630	+	0.168917i	$0.0540270\pi$
$479$	161.823i	0.337834i	−0.985630	+	0.168917i	$0.945973\pi$
$480$	−99.8823	−0.208088
$481$	− 33.6893i	− 0.0700402i
$482$	− 427.118i	− 0.886136i
$483$	0	0
$484$	381.529	0.788283
$485$	385.085	0.793990
$486$	294.681i	0.606340i
$487$	770.392	1.58191	0.790957	−	0.611872i	$-0.209583\pi$
$487$	770.392	1.58191	0.790957	+	0.611872i	$0.209583\pi$
$488$	232.773i	0.476993i
$489$	344.370i	0.704233i
$490$	0	0
$491$	−363.578	−0.740484	−0.370242	−	0.928935i	$-0.620725\pi$
$491$	−363.578	−0.740484	−0.370242	+	0.928935i	$0.620725\pi$
$492$	−124.686	−0.253427
$493$	39.6584i	0.0804431i
$494$	0.637518	0.00129052
$495$	621.245i	1.25504i
$496$	5.07241i	0.0102266i
$497$	0	0
$498$	133.019	0.267107
$499$	224.402	0.449703	0.224852	−	0.974393i	$-0.427810\pi$
$499$	224.402	0.449703	0.224852	+	0.974393i	$0.427810\pi$
$500$	− 269.600i	− 0.539201i
$501$	−160.942	−0.321242
$502$	25.4521i	0.0507015i
$503$	790.766i	1.57210i	0.618163	+	0.786050i	$0.287877\pi$
$503$	790.766i	1.57210i	−0.618163	+	0.786050i	$0.712123\pi$
$504$	0	0
$505$	−1603.00	−3.17425
$506$	−75.6468	−0.149500
$507$	− 362.660i	− 0.715305i
$508$	44.6863	0.0879651
$509$	409.968i	0.805438i	0.915324	+	0.402719i	$0.131935\pi$
$509$	409.968i	0.805438i	−0.915324	+	0.402719i	$0.868065\pi$
$510$	− 415.256i	− 0.814228i
$511$	0	0
$512$	−22.6274	−0.0441942
$513$	10.7048	0.0208670
$514$	− 630.270i	− 1.22621i
$515$	971.342	1.88610
$516$	239.485i	0.464117i
$517$	1030.90i	1.99401i
$518$	0	0
$519$	408.382	0.786863
$520$	28.0000	0.0538462
$521$	420.351i	0.806816i	0.915020	+	0.403408i	$0.132174\pi$
$521$	420.351i	0.806816i	−0.915020	+	0.403408i	$0.867826\pi$
$522$	−14.5483	−0.0278704
$523$	319.672i	0.611227i	0.952156	+	0.305613i	$0.0988615\pi$
$523$	319.672i	0.611227i	−0.952156	+	0.305613i	$0.901138\pi$
$524$	72.7324i	0.138802i
$525$	0	0
$526$	47.4315	0.0901739
$527$	−21.0883	−0.0400158
$528$	− 152.893i	− 0.289570i
$529$	−519.823	−0.982651
$530$	15.8326i	0.0298728i
$531$	− 426.080i	− 0.802411i
$532$	0	0
$533$	34.9533	0.0655785
$534$	−24.0690	−0.0450730
$535$	− 1202.91i	− 2.24843i
$536$	−277.019	−0.516827
$537$	454.328i	0.846048i
$538$	186.275i	0.346236i
$539$	0	0
$540$	470.156	0.870660
$541$	−361.608	−0.668407	−0.334203	−	0.942501i	$-0.608467\pi$
$541$	−361.608	−0.668407	−0.334203	+	0.942501i	$0.608467\pi$
$542$	642.240i	1.18494i
$543$	683.225	1.25824
$544$	− 94.0725i	− 0.172927i
$545$	− 1063.47i	− 1.95132i
$546$	0	0
$547$	93.4701	0.170878	0.0854389	−	0.996343i	$-0.472771\pi$
$547$	93.4701	0.170878	0.0854389	+	0.996343i	$0.472771\pi$
$548$	−151.848	−0.277094
$549$	− 355.008i	− 0.646644i
$550$	1036.95	1.88536
$551$	0.885750i	0.00160753i
$552$	18.5490i	0.0336033i
$553$	0	0
$554$	−76.0345	−0.137246
$555$	−490.108	−0.883077
$556$	− 219.503i	− 0.394789i
$557$	628.627	1.12860	0.564298	−	0.825572i	$-0.309147\pi$
$557$	628.627	1.12860	0.564298	+	0.825572i	$0.309147\pi$
$558$	− 7.73606i	− 0.0138639i
$559$	− 67.1347i	− 0.120098i
$560$	0	0
$561$	635.647	1.13306
$562$	229.411	0.408205
$563$	− 282.424i	− 0.501642i	−0.968034	−	0.250821i	$-0.919300\pi$
$563$	− 282.424i	− 0.501642i	0.968034	−	0.250821i	$-0.0807005\pi$
$564$	252.784	0.448198
$565$	− 15.8326i	− 0.0280223i
$566$	− 318.250i	− 0.562279i
$567$	0	0
$568$	−182.627	−0.321527
$569$	625.664	1.09959	0.549793	−	0.835301i	$-0.314707\pi$
$569$	625.664	1.09959	0.549793	+	0.835301i	$0.314707\pi$
$570$	− 9.27452i	− 0.0162711i
$571$	−661.401	−1.15832	−0.579160	−	0.815214i	$-0.696619\pi$
$571$	−661.401	−1.15832	−0.579160	+	0.815214i	$0.696619\pi$
$572$	42.8605i	0.0749310i
$573$	− 707.438i	− 1.23462i
$574$	0	0
$575$	−125.803	−0.218788
$576$	34.5097	0.0599126
$577$	− 334.988i	− 0.580569i	−0.956940	−	0.290285i	$-0.906250\pi$
$577$	− 334.988i	− 0.580569i	0.956940	−	0.290285i	$-0.0937499\pi$
$578$	−17.6051	−0.0304587
$579$	− 234.773i	− 0.405479i
$580$	38.9024i	0.0670731i
$581$	0	0
$582$	144.540	0.248350
$583$	−24.2355	−0.0415703
$584$	260.198i	0.445545i
$585$	−42.7035	−0.0729975
$586$	130.988i	0.223529i
$587$	1083.88i	1.84648i	0.384227	+	0.923239i	$0.374468\pi$
$587$	1083.88i	1.84648i	−0.384227	+	0.923239i	$0.625532\pi$
$588$	0	0
$589$	−0.470996	−0.000799654 0
$590$	−1139.34	−1.93109
$591$	− 87.6947i	− 0.148384i
$592$	−111.029	−0.187550
$593$	− 145.827i	− 0.245914i	−0.992412	−	0.122957i	$-0.960762\pi$
$593$	− 145.827i	− 0.245914i	0.992412	−	0.122957i	$-0.0392377\pi$
$594$	719.684i	1.21159i
$595$	0	0
$596$	−368.000	−0.617450
$597$	531.216	0.889809
$598$	− 5.19986i	− 0.00869541i
$599$	−154.246	−0.257505	−0.128753	−	0.991677i	$-0.541097\pi$
$599$	−154.246	−0.257505	−0.128753	+	0.991677i	$0.541097\pi$
$600$	− 254.267i	− 0.423778i
$601$	361.658i	0.601760i	0.953662	+	0.300880i	$0.0972804\pi$
$601$	361.658i	0.601760i	−0.953662	+	0.300880i	$0.902720\pi$
$602$	0	0
$603$	422.489	0.700646
$604$	432.569	0.716173
$605$	1555.95i	2.57182i
$606$	−601.677	−0.992866
$607$	361.943i	0.596282i	0.954522	+	0.298141i	$0.0963666\pi$
$607$	361.943i	0.596282i	−0.954522	+	0.298141i	$0.903633\pi$
$608$	− 2.10106i	− 0.00345569i
$609$	0	0
$610$	−949.294	−1.55622
$611$	−70.8629	−0.115979
$612$	143.472i	0.234432i
$613$	−513.360	−0.837454	−0.418727	−	0.908112i	$-0.637524\pi$
$613$	−513.360	−0.837454	−0.418727	+	0.908112i	$0.637524\pi$
$614$	608.436i	0.990939i
$615$	− 508.496i	− 0.826822i
$616$	0	0
$617$	613.767	0.994761	0.497380	−	0.867533i	$-0.334295\pi$
$617$	613.767	0.994761	0.497380	+	0.867533i	$0.334295\pi$
$618$	364.589	0.589949
$619$	454.594i	0.734400i	0.930142	+	0.367200i	$0.119684\pi$
$619$	454.594i	0.734400i	−0.930142	+	0.367200i	$0.880316\pi$
$620$	−20.6863	−0.0333650
$621$	− 87.3123i	− 0.140600i
$622$	83.4398i	0.134148i
$623$	0	0
$624$	10.5097	0.0168424
$625$	61.3116	0.0980986
$626$	− 443.674i	− 0.708745i
$627$	14.1968	0.0226425
$628$	− 365.140i	− 0.581433i
$629$	− 461.600i	− 0.733864i
$630$	0	0
$631$	40.7351	0.0645564	0.0322782	−	0.999479i	$-0.489724\pi$
$631$	40.7351	0.0645564	0.0322782	+	0.999479i	$0.489724\pi$
$632$	55.7645	0.0882350
$633$	78.5476i	0.124088i
$634$	−665.456	−1.04961
$635$	182.240i	0.286992i
$636$	5.94269i	0.00934386i
$637$	0	0
$638$	−59.5492	−0.0933373
$639$	278.530	0.435884
$640$	− 92.2792i	− 0.144186i
$641$	153.650	0.239703	0.119852	−	0.992792i	$-0.461758\pi$
$641$	153.650	0.239703	0.119852	+	0.992792i	$0.461758\pi$
$642$	− 451.506i	− 0.703280i
$643$	899.887i	1.39951i	0.714382	+	0.699756i	$0.246708\pi$
$643$	899.887i	1.39951i	−0.714382	+	0.699756i	$0.753292\pi$
$644$	0	0
$645$	−976.666	−1.51421
$646$	8.73506	0.0135218
$647$	709.014i	1.09585i	0.836528	+	0.547924i	$0.184582\pi$
$647$	709.014i	1.09585i	−0.836528	+	0.547924i	$0.815418\pi$
$648$	66.6619	0.102873
$649$	− 1744.03i	− 2.68726i
$650$	71.2786i	0.109659i
$651$	0	0
$652$	−318.156	−0.487970
$653$	−848.222	−1.29896	−0.649481	−	0.760378i	$-0.725014\pi$
$653$	−848.222	−1.29896	−0.649481	+	0.760378i	$0.725014\pi$
$654$	− 399.169i	− 0.610350i
$655$	−296.617	−0.452851
$656$	− 115.195i	− 0.175602i
$657$	− 396.835i	− 0.604011i
$658$	0	0
$659$	437.803	0.664345	0.332172	−	0.943219i	$-0.392218\pi$
$659$	437.803	0.664345	0.332172	+	0.943219i	$0.392218\pi$
$660$	623.529	0.944741
$661$	− 333.242i	− 0.504149i	−0.967708	−	0.252074i	$-0.918887\pi$
$661$	− 333.242i	− 0.504149i	0.967708	−	0.252074i	$-0.0811128\pi$
$662$	−436.520	−0.659395
$663$	43.6935i	0.0659027i
$664$	122.894i	0.185081i
$665$	0	0
$666$	169.334	0.254255
$667$	7.22453	0.0108314
$668$	− 148.691i	− 0.222591i
$669$	−192.471	−0.287700
$670$	− 1129.74i	− 1.68618i
$671$	− 1453.12i	− 2.16560i
$672$	0	0
$673$	366.253	0.544209	0.272105	−	0.962268i	$-0.412280\pi$
$673$	366.253	0.544209	0.272105	+	0.962268i	$0.412280\pi$
$674$	−268.510	−0.398382
$675$	1196.86i	1.77313i
$676$	335.054	0.495642
$677$	− 942.533i	− 1.39222i	−0.717935	−	0.696110i	$-0.754912\pi$
$677$	− 942.533i	− 1.39222i	0.717935	−	0.696110i	$-0.245088\pi$
$678$	− 5.94269i	− 0.00876503i
$679$	0	0
$680$	383.647	0.564186
$681$	−693.352	−1.01814
$682$	− 31.6652i	− 0.0464299i
$683$	−215.500	−0.315520	−0.157760	−	0.987477i	$-0.550427\pi$
$683$	−215.500	−0.315520	−0.157760	+	0.987477i	$0.550427\pi$
$684$	3.20438i	0.00468477i
$685$	− 619.266i	− 0.904038i
$686$	0	0
$687$	−6.28600	−0.00914993
$688$	−221.255	−0.321591
$689$	− 1.66592i	− 0.00241787i
$690$	−75.6468	−0.109633
$691$	916.732i	1.32668i	0.748320	+	0.663338i	$0.230861\pi$
$691$	916.732i	1.32668i	−0.748320	+	0.663338i	$0.769139\pi$
$692$	377.296i	0.545225i
$693$	0	0
$694$	−408.638	−0.588815
$695$	895.176	1.28802
$696$	14.6018i	0.0209796i
$697$	478.919	0.687115
$698$	479.489i	0.686947i
$699$	35.9574i	0.0514413i
$700$	0	0
$701$	541.894	0.773029	0.386515	−	0.922283i	$-0.373679\pi$
$701$	541.894	0.773029	0.386515	+	0.922283i	$0.373679\pi$
$702$	−49.4701	−0.0704703
$703$	− 10.3096i	− 0.0146651i
$704$	141.255	0.200646
$705$	1030.90i	1.46227i
$706$	− 704.319i	− 0.997619i
$707$	0	0
$708$	−427.647	−0.604021
$709$	−401.497	−0.566287	−0.283143	−	0.959078i	$-0.591377\pi$
$709$	−401.497	−0.566287	−0.283143	+	0.959078i	$0.591377\pi$
$710$	− 744.791i	− 1.04900i
$711$	−85.0479	−0.119617
$712$	− 22.2368i	− 0.0312315i
$713$	3.84163i	0.00538799i
$714$	0	0
$715$	−174.794	−0.244467
$716$	−419.744	−0.586235
$717$	860.437i	1.20005i
$718$	166.794	0.232304
$719$	306.327i	0.426046i	0.977047	+	0.213023i	$0.0683309\pi$
$719$	306.327i	0.426046i	−0.977047	+	0.213023i	$0.931669\pi$
$720$	140.737i	0.195469i
$721$	0	0
$722$	−510.336	−0.706837
$723$	653.803	0.904292
$724$	631.217i	0.871847i
$725$	−99.0324	−0.136596
$726$	584.019i	0.804434i
$727$	− 520.704i	− 0.716237i	−0.933676	−	0.358119i	$-0.883418\pi$
$727$	− 520.704i	− 0.716237i	0.933676	−	0.358119i	$-0.116582\pi$
$728$	0	0
$729$	−663.195	−0.909733
$730$	−1061.14	−1.45362
$731$	− 919.858i	− 1.25836i
$732$	−356.313	−0.486766
$733$	359.775i	0.490825i	0.969419	+	0.245412i	$0.0789234\pi$
$733$	359.775i	0.490825i	−0.969419	+	0.245412i	$0.921077\pi$
$734$	− 247.581i	− 0.337304i
$735$	0	0
$736$	−17.1371	−0.0232841
$737$	1729.33	2.34645
$738$	175.687i	0.238058i
$739$	−857.588	−1.16047	−0.580235	−	0.814449i	$-0.697039\pi$
$739$	−857.588	−1.16047	−0.580235	+	0.814449i	$0.697039\pi$
$740$	− 452.800i	− 0.611892i
$741$	0.975871i	0.00131696i
$742$	0	0
$743$	−863.038	−1.16156	−0.580779	−	0.814061i	$-0.697252\pi$
$743$	−863.038	−1.16156	−0.580779	+	0.814061i	$0.697252\pi$
$744$	−7.76450	−0.0104362
$745$	− 1500.78i	− 2.01447i
$746$	744.098	0.997450
$747$	− 187.429i	− 0.250908i
$748$	587.261i	0.785108i
$749$	0	0
$750$	412.686	0.550248
$751$	−1486.04	−1.97875	−0.989373	−	0.145398i	$-0.953554\pi$
$751$	−1486.04	−1.97875	−0.989373	+	0.145398i	$0.953554\pi$
$752$	233.542i	0.310561i
$753$	−38.9605	−0.0517403
$754$	− 4.09333i	− 0.00542882i
$755$	1764.10i	2.33656i
$756$	0	0
$757$	480.693	0.634998	0.317499	−	0.948259i	$-0.397157\pi$
$757$	480.693	0.634998	0.317499	+	0.948259i	$0.397157\pi$
$758$	541.990	0.715026
$759$	− 115.795i	− 0.152563i
$760$	8.56854	0.0112744
$761$	1222.02i	1.60581i	0.596107	+	0.802905i	$0.296713\pi$
$761$	1222.02i	1.60581i	−0.596107	+	0.802905i	$0.703287\pi$
$762$	68.4028i	0.0897675i
$763$	0	0
$764$	653.588	0.855482
$765$	−585.110	−0.764849
$766$	− 0.180242i	0 0.000235303i
$767$	119.882	0.156300
$768$	− 34.6366i	− 0.0450997i
$769$	290.136i	0.377289i	0.982045	+	0.188645i	$0.0604094\pi$
$769$	290.136i	0.377289i	−0.982045	+	0.188645i	$0.939591\pi$
$770$	0	0
$771$	964.775	1.25133
$772$	216.902	0.280961
$773$	− 1131.59i	− 1.46390i	−0.681359	−	0.731949i	$-0.738611\pi$
$773$	− 1131.59i	− 1.46390i	0.681359	−	0.731949i	$-0.261389\pi$
$774$	337.442	0.435971
$775$	− 52.6603i	− 0.0679488i
$776$	133.538i	0.172084i
$777$	0	0
$778$	527.373	0.677857
$779$	10.6964	0.0137309
$780$	42.8605i	0.0549494i
$781$	1140.08	1.45977
$782$	− 71.2467i	− 0.0911083i
$783$	− 68.7324i	− 0.0877808i
$784$	0	0
$785$	1489.11	1.89696
$786$	−111.334	−0.141646
$787$	667.719i	0.848435i	0.905560	+	0.424218i	$0.139451\pi$
$787$	667.719i	0.848435i	−0.905560	+	0.424218i	$0.860549\pi$
$788$	81.0193	0.102816
$789$	72.6049i	0.0920215i
$790$	227.419i	0.287872i
$791$	0	0
$792$	−215.431	−0.272009
$793$	99.8852	0.125959
$794$	102.554i	0.129161i
$795$	−24.2355	−0.0304849
$796$	490.780i	0.616557i
$797$	779.700i	0.978294i	0.872201	+	0.489147i	$0.162692\pi$
$797$	779.700i	0.978294i	−0.872201	+	0.489147i	$0.837308\pi$
$798$	0	0
$799$	−970.940	−1.21519
$800$	234.912	0.293640
$801$	33.9140i	0.0423396i
$802$	17.4113	0.0217098
$803$	− 1624.32i	− 2.02282i
$804$	− 424.043i	− 0.527416i
$805$	0	0
$806$	2.17662	0.00270053
$807$	−285.137	−0.353330
$808$	− 555.877i	− 0.687967i
$809$	1379.82	1.70559	0.852796	−	0.522245i	$-0.174905\pi$
$809$	1379.82	1.70559	0.852796	+	0.522245i	$0.174905\pi$
$810$	271.861i	0.335630i
$811$	− 736.976i	− 0.908725i	−0.890817	−	0.454363i	$-0.849867\pi$
$811$	− 736.976i	− 0.908725i	0.890817	−	0.454363i	$-0.150133\pi$
$812$	0	0
$813$	−983.098	−1.20922
$814$	693.117	0.851495
$815$	− 1297.51i	− 1.59203i
$816$	144.000	0.176471
$817$	− 20.5445i	− 0.0251463i
$818$	− 262.024i	− 0.320322i
$819$	0	0
$820$	469.789	0.572913
$821$	4.15642	0.00506263	0.00253132	−	0.999997i	$-0.499194\pi$
$821$	4.15642	0.00506263	0.00253132	+	0.999997i	$0.499194\pi$
$822$	− 232.438i	− 0.282772i
$823$	1228.08	1.49220	0.746098	−	0.665836i	$-0.231925\pi$
$823$	1228.08	1.49220	0.746098	+	0.665836i	$0.231925\pi$
$824$	336.836i	0.408782i
$825$	1587.29i	1.92399i
$826$	0	0
$827$	21.2548	0.0257011	0.0128506	−	0.999917i	$-0.495909\pi$
$827$	21.2548	0.0257011	0.0128506	+	0.999917i	$0.495909\pi$
$828$	26.1362	0.0315655
$829$	1223.92i	1.47639i	0.674590	+	0.738193i	$0.264321\pi$
$829$	1223.92i	1.47639i	−0.674590	+	0.738193i	$0.735679\pi$
$830$	−501.186	−0.603838
$831$	− 116.389i	− 0.140058i
$832$	9.70967i	0.0116703i
$833$	0	0
$834$	336.000	0.402878
$835$	606.392	0.726218
$836$	13.1162i	0.0156892i
$837$	36.5483	0.0436659
$838$	695.335i	0.829756i
$839$	− 73.5125i	− 0.0876192i	−0.999040	−	0.0438096i	$-0.986050\pi$
$839$	− 73.5125i	− 0.0876192i	0.999040	−	0.0438096i	$-0.0139495\pi$
$840$	0	0
$841$	−835.313	−0.993238
$842$	666.621	0.791712
$843$	351.168i	0.416569i
$844$	−72.5685	−0.0859817
$845$	1366.42i	1.61706i
$846$	− 356.181i	− 0.421017i
$847$	0	0
$848$	−5.49033	−0.00647445
$849$	487.156	0.573799
$850$	976.635i	1.14898i
$851$	−84.0892	−0.0988122
$852$	− 279.554i	− 0.328115i
$853$	704.025i	0.825352i	0.910878	+	0.412676i	$0.135406\pi$
$853$	704.025i	0.825352i	−0.910878	+	0.412676i	$0.864594\pi$
$854$	0	0
$855$	−13.0681	−0.0152843
$856$	417.137	0.487310
$857$	161.299i	0.188213i	0.995562	+	0.0941067i	$0.0299995\pi$
$857$	161.299i	0.188213i	−0.995562	+	0.0941067i	$0.970001\pi$
$858$	−65.6081	−0.0764663
$859$	− 902.603i	− 1.05076i	−0.850868	−	0.525380i	$-0.823923\pi$
$859$	− 902.603i	− 1.05076i	0.850868	−	0.525380i	$-0.176077\pi$
$860$	− 902.322i	− 1.04921i
$861$	0	0
$862$	646.656	0.750181
$863$	−802.705	−0.930133	−0.465066	−	0.885276i	$-0.653970\pi$
$863$	−802.705	−0.930133	−0.465066	+	0.885276i	$0.653970\pi$
$864$	163.038i	0.188701i
$865$	−1538.69	−1.77883
$866$	352.056i	0.406531i
$867$	− 26.9488i	− 0.0310828i
$868$	0	0
$869$	−348.118	−0.400596
$870$	−59.5492	−0.0684474
$871$	118.872i	0.136478i
$872$	368.784	0.422917
$873$	− 203.662i	− 0.233289i
$874$	− 1.59126i	− 0.00182066i
$875$	0	0
$876$	−398.294	−0.454674
$877$	1462.85	1.66802	0.834008	−	0.551753i	$-0.186041\pi$
$877$	1462.85	1.66802	0.834008	+	0.551753i	$0.186041\pi$
$878$	608.113i	0.692612i
$879$	−200.508	−0.228109
$880$	576.066i	0.654620i
$881$	− 851.416i	− 0.966420i	−0.875505	−	0.483210i	$-0.839471\pi$
$881$	− 851.416i	− 0.966420i	0.875505	−	0.483210i	$-0.160529\pi$
$882$	0	0
$883$	−306.303	−0.346889	−0.173444	−	0.984844i	$-0.555490\pi$
$883$	−306.303	−0.346889	−0.173444	+	0.984844i	$0.555490\pi$
$884$	−40.3675	−0.0456646
$885$	− 1744.03i	− 1.97065i
$886$	−537.901	−0.607111
$887$	− 560.594i	− 0.632011i	−0.948757	−	0.316005i	$-0.897658\pi$
$887$	− 560.594i	− 0.632011i	0.948757	−	0.316005i	$-0.102342\pi$
$888$	− 169.957i	− 0.191392i
$889$	0	0
$890$	90.6863	0.101895
$891$	−416.146	−0.467055
$892$	− 177.820i	− 0.199350i
$893$	−21.6854	−0.0242838
$894$	− 563.310i	− 0.630101i
$895$	− 1711.80i	− 1.91263i
$896$	0	0
$897$	7.95959	0.00887357
$898$	−461.921	−0.514389
$899$	3.02414i	0.00336389i
$900$	−358.270	−0.398078
$901$	− 22.8258i	− 0.0253339i
$902$	719.122i	0.797252i
$903$	0	0
$904$	5.49033	0.00607338
$905$	−2574.23	−2.84445
$906$	662.147i	0.730847i
$907$	1.70563	0.00188052	0.000940258	−	1.00000i	$-0.499701\pi$
$907$	1.70563	0.00188052	0.000940258		1.00000i	$0.499701\pi$
$908$	− 640.574i	− 0.705478i
$909$	847.783i	0.932654i
$910$	0	0
$911$	−629.539	−0.691042	−0.345521	−	0.938411i	$-0.612298\pi$
$911$	−629.539	−0.691042	−0.345521	+	0.938411i	$0.612298\pi$
$912$	3.21616	0.00352649
$913$	− 767.182i	− 0.840287i
$914$	878.392	0.961041
$915$	− 1453.12i	− 1.58810i
$916$	− 5.80751i	− 0.00634008i
$917$	0	0
$918$	−677.823	−0.738370
$919$	−562.940	−0.612557	−0.306279	−	0.951942i	$-0.599084\pi$
$919$	−562.940	−0.612557	−0.306279	+	0.951942i	$0.599084\pi$
$920$	− 69.8885i	− 0.0759657i
$921$	−931.354	−1.01124
$922$	− 241.745i	− 0.262196i
$923$	78.3674i	0.0849051i
$924$	0	0
$925$	1152.68	1.24614
$926$	−695.529	−0.751111
$927$	− 513.718i	− 0.554172i
$928$	−13.4903	−0.0145370
$929$	184.016i	0.198080i	0.995083	+	0.0990398i	$0.0315771\pi$
$929$	184.016i	0.198080i	−0.995083	+	0.0990398i	$0.968423\pi$
$930$	− 31.6652i	− 0.0340486i
$931$	0	0
$932$	−33.2203	−0.0356441
$933$	−127.724	−0.136896
$934$	424.748i	0.454763i
$935$	−2394.97	−2.56146
$936$	− 14.8085i	− 0.0158210i
$937$	− 200.616i	− 0.214104i	−0.994253	−	0.107052i	$-0.965859\pi$
$937$	− 200.616i	− 0.214104i	0.994253	−	0.107052i	$-0.0341412\pi$
$938$	0	0
$939$	679.147	0.723266
$940$	−952.431	−1.01322
$941$	715.285i	0.760133i	0.924959	+	0.380066i	$0.124099\pi$
$941$	715.285i	0.760133i	−0.924959	+	0.380066i	$0.875901\pi$
$942$	558.932	0.593346
$943$	− 87.2441i	− 0.0925176i
$944$	− 395.094i	− 0.418532i
$945$	0	0
$946$	1381.21	1.46006
$947$	865.588	0.914032	0.457016	−	0.889459i	$-0.348918\pi$
$947$	865.588	0.914032	0.457016	+	0.889459i	$0.348918\pi$
$948$	85.3606i	0.0900428i
$949$	111.654	0.117654
$950$	21.8126i	0.0229607i
$951$	− 1018.64i	− 1.07112i
$952$	0	0
$953$	−1499.57	−1.57352	−0.786762	−	0.617257i	$-0.788244\pi$
$953$	−1499.57	−1.57352	−0.786762	+	0.617257i	$0.788244\pi$
$954$	8.37345	0.00877720
$955$	2665.46i	2.79106i
$956$	−794.940	−0.831527
$957$	− 91.1540i	− 0.0952497i
$958$	− 228.852i	− 0.238885i
$959$	0	0
$960$	141.255	0.147140
$961$	959.392	0.998327
$962$	47.6439i	0.0495259i
$963$	−636.187	−0.660630
$964$	604.035i	0.626593i
$965$	884.568i	0.916651i
$966$	0	0
$967$	606.969	0.627682	0.313841	−	0.949475i	$-0.398384\pi$
$967$	606.969	0.627682	0.313841	+	0.949475i	$0.398384\pi$
$968$	−539.563	−0.557400
$969$	13.3711i	0.0137988i
$970$	−544.593	−0.561436
$971$	406.226i	0.418358i	0.977877	+	0.209179i	$0.0670791\pi$
$971$	406.226i	0.418358i	−0.977877	+	0.209179i	$0.932921\pi$
$972$	− 416.742i	− 0.428747i
$973$	0	0
$974$	−1089.50	−1.11858
$975$	−109.109	−0.111906
$976$	− 329.190i	− 0.337285i
$977$	283.591	0.290267	0.145133	−	0.989412i	$-0.453639\pi$
$977$	283.591	0.290267	0.145133	+	0.989412i	$0.453639\pi$
$978$	− 487.013i	− 0.497968i
$979$	138.817i	0.141794i
$980$	0	0
$981$	−562.442	−0.573335
$982$	514.177	0.523601
$983$	956.266i	0.972803i	0.873735	+	0.486402i	$0.161691\pi$
$983$	956.266i	0.972803i	−0.873735	+	0.486402i	$0.838309\pi$
$984$	176.333	0.179200
$985$	330.413i	0.335445i
$986$	− 56.0855i	− 0.0568818i
$987$	0	0
$988$	−0.901587	−0.000912537 0
$989$	−167.569	−0.169433
$990$	− 878.573i	− 0.887447i
$991$	−945.568	−0.954155	−0.477078	−	0.878861i	$-0.658304\pi$
$991$	−945.568	−0.954155	−0.477078	+	0.878861i	$0.658304\pi$
$992$	− 7.17346i	− 0.00723131i
$993$	− 668.196i	− 0.672906i
$994$	0	0
$995$	−2001.50	−2.01156
$996$	−188.118	−0.188873
$997$	− 940.358i	− 0.943187i	−0.881816	−	0.471594i	$-0.843679\pi$
$997$	− 940.358i	− 0.943187i	0.881816	−	0.471594i	$-0.156321\pi$
$998$	−317.352	−0.317988
$999$	800.003i	0.800804i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.3.b.a.97.1	✓	4
3.2	odd	2		882.3.c.a.685.3		4
4.3	odd	2		784.3.c.b.97.3		4
7.2	even	3		98.3.d.b.31.4		8
7.3	odd	6		98.3.d.b.19.4		8
7.4	even	3		98.3.d.b.19.3		8
7.5	odd	6		98.3.d.b.31.3		8
7.6	odd	2	inner	98.3.b.a.97.2	yes	4
21.2	odd	6		882.3.n.j.325.1		8
21.5	even	6		882.3.n.j.325.2		8
21.11	odd	6		882.3.n.j.19.2		8
21.17	even	6		882.3.n.j.19.1		8
21.20	even	2		882.3.c.a.685.4		4
28.3	even	6		784.3.s.j.705.2		8
28.11	odd	6		784.3.s.j.705.3		8
28.19	even	6		784.3.s.j.129.3		8
28.23	odd	6		784.3.s.j.129.2		8
28.27	even	2		784.3.c.b.97.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.3.b.a.97.1	✓	4	1.1	even	1	trivial
98.3.b.a.97.2	yes	4	7.6	odd	2	inner
98.3.d.b.19.3		8	7.4	even	3
98.3.d.b.19.4		8	7.3	odd	6
98.3.d.b.31.3		8	7.5	odd	6
98.3.d.b.31.4		8	7.2	even	3
784.3.c.b.97.2		4	28.27	even	2
784.3.c.b.97.3		4	4.3	odd	2
784.3.s.j.129.2		8	28.23	odd	6
784.3.s.j.129.3		8	28.19	even	6
784.3.s.j.705.2		8	28.3	even	6
784.3.s.j.705.3		8	28.11	odd	6
882.3.c.a.685.3		4	3.2	odd	2
882.3.c.a.685.4		4	21.20	even	2
882.3.n.j.19.1		8	21.17	even	6
882.3.n.j.19.2		8	21.11	odd	6
882.3.n.j.325.1		8	21.2	odd	6
882.3.n.j.325.2		8	21.5	even	6

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:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	98.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -2.16478i q^{3} +2.00000 q^{4} +8.15640i q^{5} +3.06147i q^{6} -2.82843 q^{8} +4.31371 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -2.16478i q^{3} +2.00000 q^{4} +8.15640i q^{5} +3.06147i q^{6} -2.82843 q^{8} +4.31371 q^{9} -11.5349i q^{10} +17.6569 q^{11} -4.32957i q^{12} +1.21371i q^{13} +17.6569 q^{15} +4.00000 q^{16} +16.6298i q^{17} -6.10051 q^{18} +0.371418i q^{19} +16.3128i q^{20} -24.9706 q^{22} +3.02944 q^{23} +6.12293i q^{24} -41.5269 q^{25} -1.71644i q^{26} -28.8213i q^{27} +2.38478 q^{29} -24.9706 q^{30} +1.26810i q^{31} -5.65685 q^{32} -38.2233i q^{33} -23.5181i q^{34} +8.62742 q^{36} -27.7574 q^{37} -0.525265i q^{38} +2.62742 q^{39} -23.0698i q^{40} -28.7988i q^{41} -55.3137 q^{43} +35.3137 q^{44} +35.1843i q^{45} -4.28427 q^{46} +58.3855i q^{47} -8.65914i q^{48} +58.7279 q^{50} +36.0000 q^{51} +2.42742i q^{52} -1.37258 q^{53} +40.7595i q^{54} +144.016i q^{55} +0.804041 q^{57} -3.37258 q^{58} -98.7735i q^{59} +35.3137 q^{60} -82.2975i q^{61} -1.79337i q^{62} +8.00000 q^{64} -9.89949 q^{65} +54.0559i q^{66} +97.9411 q^{67} +33.2597i q^{68} -6.55808i q^{69} +64.5685 q^{71} -12.2010 q^{72} -91.9940i q^{73} +39.2548 q^{74} +89.8968i q^{75} +0.742837i q^{76} -3.71573 q^{78} -19.7157 q^{79} +32.6256i q^{80} -23.5685 q^{81} +40.7276i q^{82} -43.4495i q^{83} -135.640 q^{85} +78.2254 q^{86} -5.16253i q^{87} -49.9411 q^{88} +7.86191i q^{89} -49.7582i q^{90} +6.05887 q^{92} +2.74517 q^{93} -82.5695i q^{94} -3.02944 q^{95} +12.2459i q^{96} -47.2126i q^{97} +76.1665 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 28 q^{9}+O(q^{10}) \) 4 * q + 8 * q^4 - 28 * q^9	\( 4 q + 8 q^{4} - 28 q^{9} + 48 q^{11} + 48 q^{15} + 16 q^{16} - 64 q^{18} - 32 q^{22} + 80 q^{23} - 36 q^{25} - 64 q^{29} - 32 q^{30} - 56 q^{36} - 128 q^{37} - 80 q^{39} - 176 q^{43} + 96 q^{44} + 96 q^{46} + 184 q^{50} + 144 q^{51} - 96 q^{53} + 320 q^{57} - 104 q^{58} + 96 q^{60} + 32 q^{64} + 256 q^{67} + 32 q^{71} - 128 q^{72} - 24 q^{74} - 128 q^{78} - 192 q^{79} + 132 q^{81} - 288 q^{85} + 64 q^{86} - 64 q^{88} + 160 q^{92} + 192 q^{93} - 80 q^{95} - 80 q^{99}+O(q^{100}) \) 4 * q + 8 * q^4 - 28 * q^9 + 48 * q^11 + 48 * q^15 + 16 * q^16 - 64 * q^18 - 32 * q^22 + 80 * q^23 - 36 * q^25 - 64 * q^29 - 32 * q^30 - 56 * q^36 - 128 * q^37 - 80 * q^39 - 176 * q^43 + 96 * q^44 + 96 * q^46 + 184 * q^50 + 144 * q^51 - 96 * q^53 + 320 * q^57 - 104 * q^58 + 96 * q^60 + 32 * q^64 + 256 * q^67 + 32 * q^71 - 128 * q^72 - 24 * q^74 - 128 * q^78 - 192 * q^79 + 132 * q^81 - 288 * q^85 + 64 * q^86 - 64 * q^88 + 160 * q^92 + 192 * q^93 - 80 * q^95 - 80 * q^99

Introduction

L-functions

Modular forms

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Representations

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Database

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