LMFDB - Embedded newform 4002.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$31.9561308889$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2389280.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 11x^{3} + 7x^{2} + 26x - 18 $ x^5 - x^4 - 11x^3 + 7x^2 + 26*x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.11043$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.25808 q^{5} +1.00000 q^{6} +1.35501 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.25808 q^{5} +1.00000 q^{6} +1.35501 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.25808 q^{10} +1.14765 q^{11} -1.00000 q^{12} +0.110434 q^{13} -1.35501 q^{14} +2.25808 q^{15} +1.00000 q^{16} +7.57588 q^{17} -1.00000 q^{18} -7.57588 q^{19} -2.25808 q^{20} -1.35501 q^{21} -1.14765 q^{22} -1.00000 q^{23} +1.00000 q^{24} +0.0989234 q^{25} -0.110434 q^{26} -1.00000 q^{27} +1.35501 q^{28} -1.00000 q^{29} -2.25808 q^{30} -10.2445 q^{31} -1.00000 q^{32} -1.14765 q^{33} -7.57588 q^{34} -3.05972 q^{35} +1.00000 q^{36} +11.0433 q^{37} +7.57588 q^{38} -0.110434 q^{39} +2.25808 q^{40} -6.08053 q^{41} +1.35501 q^{42} -7.97009 q^{43} +1.14765 q^{44} -2.25808 q^{45} +1.00000 q^{46} +4.67200 q^{47} -1.00000 q^{48} -5.16395 q^{49} -0.0989234 q^{50} -7.57588 q^{51} +0.110434 q^{52} +10.7491 q^{53} +1.00000 q^{54} -2.59147 q^{55} -1.35501 q^{56} +7.57588 q^{57} +1.00000 q^{58} +11.8360 q^{59} +2.25808 q^{60} +14.6550 q^{61} +10.2445 q^{62} +1.35501 q^{63} +1.00000 q^{64} -0.249369 q^{65} +1.14765 q^{66} -5.32989 q^{67} +7.57588 q^{68} +1.00000 q^{69} +3.05972 q^{70} -1.59428 q^{71} -1.00000 q^{72} +10.5553 q^{73} -11.0433 q^{74} -0.0989234 q^{75} -7.57588 q^{76} +1.55507 q^{77} +0.110434 q^{78} +0.564367 q^{79} -2.25808 q^{80} +1.00000 q^{81} +6.08053 q^{82} -12.2575 q^{83} -1.35501 q^{84} -17.1069 q^{85} +7.97009 q^{86} +1.00000 q^{87} -1.14765 q^{88} -0.540660 q^{89} +2.25808 q^{90} +0.149639 q^{91} -1.00000 q^{92} +10.2445 q^{93} -4.67200 q^{94} +17.1069 q^{95} +1.00000 q^{96} -14.7491 q^{97} +5.16395 q^{98} +1.14765 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} - 4 q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 - 4 * q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} - 4 q^{7} - 5 q^{8} + 5 q^{9} - q^{10} + 5 q^{11} - 5 q^{12} - 11 q^{13} + 4 q^{14} - q^{15} + 5 q^{16} + 4 q^{17} - 5 q^{18} - 4 q^{19} + q^{20} + 4 q^{21} - 5 q^{22} - 5 q^{23} + 5 q^{24} + 12 q^{25} + 11 q^{26} - 5 q^{27} - 4 q^{28} - 5 q^{29} + q^{30} + 5 q^{31} - 5 q^{32} - 5 q^{33} - 4 q^{34} - 6 q^{35} + 5 q^{36} + 9 q^{37} + 4 q^{38} + 11 q^{39} - q^{40} + 5 q^{41} - 4 q^{42} - 16 q^{43} + 5 q^{44} + q^{45} + 5 q^{46} + 8 q^{47} - 5 q^{48} - 5 q^{49} - 12 q^{50} - 4 q^{51} - 11 q^{52} - 4 q^{53} + 5 q^{54} - 33 q^{55} + 4 q^{56} + 4 q^{57} + 5 q^{58} + 23 q^{59} - q^{60} + 7 q^{61} - 5 q^{62} - 4 q^{63} + 5 q^{64} - 5 q^{65} + 5 q^{66} + 5 q^{67} + 4 q^{68} + 5 q^{69} + 6 q^{70} - 21 q^{71} - 5 q^{72} - 8 q^{73} - 9 q^{74} - 12 q^{75} - 4 q^{76} - 11 q^{78} - 8 q^{79} + q^{80} + 5 q^{81} - 5 q^{82} - 18 q^{83} + 4 q^{84} - 10 q^{85} + 16 q^{86} + 5 q^{87} - 5 q^{88} + 32 q^{89} - q^{90} + 10 q^{91} - 5 q^{92} - 5 q^{93} - 8 q^{94} + 10 q^{95} + 5 q^{96} - 16 q^{97} + 5 q^{98} + 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 - 4 * q^7 - 5 * q^8 + 5 * q^9 - q^10 + 5 * q^11 - 5 * q^12 - 11 * q^13 + 4 * q^14 - q^15 + 5 * q^16 + 4 * q^17 - 5 * q^18 - 4 * q^19 + q^20 + 4 * q^21 - 5 * q^22 - 5 * q^23 + 5 * q^24 + 12 * q^25 + 11 * q^26 - 5 * q^27 - 4 * q^28 - 5 * q^29 + q^30 + 5 * q^31 - 5 * q^32 - 5 * q^33 - 4 * q^34 - 6 * q^35 + 5 * q^36 + 9 * q^37 + 4 * q^38 + 11 * q^39 - q^40 + 5 * q^41 - 4 * q^42 - 16 * q^43 + 5 * q^44 + q^45 + 5 * q^46 + 8 * q^47 - 5 * q^48 - 5 * q^49 - 12 * q^50 - 4 * q^51 - 11 * q^52 - 4 * q^53 + 5 * q^54 - 33 * q^55 + 4 * q^56 + 4 * q^57 + 5 * q^58 + 23 * q^59 - q^60 + 7 * q^61 - 5 * q^62 - 4 * q^63 + 5 * q^64 - 5 * q^65 + 5 * q^66 + 5 * q^67 + 4 * q^68 + 5 * q^69 + 6 * q^70 - 21 * q^71 - 5 * q^72 - 8 * q^73 - 9 * q^74 - 12 * q^75 - 4 * q^76 - 11 * q^78 - 8 * q^79 + q^80 + 5 * q^81 - 5 * q^82 - 18 * q^83 + 4 * q^84 - 10 * q^85 + 16 * q^86 + 5 * q^87 - 5 * q^88 + 32 * q^89 - q^90 + 10 * q^91 - 5 * q^92 - 5 * q^93 - 8 * q^94 + 10 * q^95 + 5 * q^96 - 16 * q^97 + 5 * q^98 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−2.25808	−1.00984	−0.504922	−	0.863165i	$-0.668479\pi$
$5$	−2.25808	−1.00984	−0.504922	+	0.863165i	$0.668479\pi$
$6$	1.00000	0.408248
$7$	1.35501	0.512145	0.256073	−	0.966658i	$-0.417571\pi$
$7$	1.35501	0.512145	0.256073	+	0.966658i	$0.417571\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.25808	0.714067
$11$	1.14765	0.346028	0.173014	−	0.984919i	$-0.444649\pi$
$11$	1.14765	0.346028	0.173014	+	0.984919i	$0.444649\pi$
$12$	−1.00000	−0.288675
$13$	0.110434	0.0306289	0.0153145	−	0.999883i	$-0.495125\pi$
$13$	0.110434	0.0306289	0.0153145	+	0.999883i	$0.495125\pi$
$14$	−1.35501	−0.362141
$15$	2.25808	0.583034
$16$	1.00000	0.250000
$17$	7.57588	1.83742	0.918710	−	0.394933i	$-0.129232\pi$
$17$	7.57588	1.83742	0.918710	+	0.394933i	$0.129232\pi$
$18$	−1.00000	−0.235702
$19$	−7.57588	−1.73803	−0.869013	−	0.494790i	$-0.835245\pi$
$19$	−7.57588	−1.73803	−0.869013	+	0.494790i	$0.835245\pi$
$20$	−2.25808	−0.504922
$21$	−1.35501	−0.295687
$22$	−1.14765	−0.244679
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	0.0989234	0.0197847
$26$	−0.110434	−0.0216579
$27$	−1.00000	−0.192450
$28$	1.35501	0.256073
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−2.25808	−0.412267
$31$	−10.2445	−1.83996	−0.919981	−	0.391963i	$-0.871796\pi$
$31$	−10.2445	−1.83996	−0.919981	+	0.391963i	$0.871796\pi$
$32$	−1.00000	−0.176777
$33$	−1.14765	−0.199779
$34$	−7.57588	−1.29925
$35$	−3.05972	−0.517187
$36$	1.00000	0.166667
$37$	11.0433	1.81551	0.907755	−	0.419501i	$-0.137795\pi$
$37$	11.0433	1.81551	0.907755	+	0.419501i	$0.137795\pi$
$38$	7.57588	1.22897
$39$	−0.110434	−0.0176836
$40$	2.25808	0.357034
$41$	−6.08053	−0.949619	−0.474809	−	0.880089i	$-0.657483\pi$
$41$	−6.08053	−0.949619	−0.474809	+	0.880089i	$0.657483\pi$
$42$	1.35501	0.209082
$43$	−7.97009	−1.21543	−0.607714	−	0.794156i	$-0.707913\pi$
$43$	−7.97009	−1.21543	−0.607714	+	0.794156i	$0.707913\pi$
$44$	1.14765	0.173014
$45$	−2.25808	−0.336615
$46$	1.00000	0.147442
$47$	4.67200	0.681481	0.340741	−	0.940157i	$-0.389322\pi$
$47$	4.67200	0.681481	0.340741	+	0.940157i	$0.389322\pi$
$48$	−1.00000	−0.144338
$49$	−5.16395	−0.737707
$50$	−0.0989234	−0.0139899
$51$	−7.57588	−1.06084
$52$	0.110434	0.0153145
$53$	10.7491	1.47651	0.738253	−	0.674524i	$-0.235651\pi$
$53$	10.7491	1.47651	0.738253	+	0.674524i	$0.235651\pi$
$54$	1.00000	0.136083
$55$	−2.59147	−0.349434
$56$	−1.35501	−0.181071
$57$	7.57588	1.00345
$58$	1.00000	0.131306
$59$	11.8360	1.54091	0.770455	−	0.637495i	$-0.220029\pi$
$59$	11.8360	1.54091	0.770455	+	0.637495i	$0.220029\pi$
$60$	2.25808	0.291517
$61$	14.6550	1.87638	0.938190	−	0.346121i	$-0.112501\pi$
$61$	14.6550	1.87638	0.938190	+	0.346121i	$0.112501\pi$
$62$	10.2445	1.30105
$63$	1.35501	0.170715
$64$	1.00000	0.125000
$65$	−0.249369	−0.0309304
$66$	1.14765	0.141265
$67$	−5.32989	−0.651150	−0.325575	−	0.945516i	$-0.605558\pi$
$67$	−5.32989	−0.651150	−0.325575	+	0.945516i	$0.605558\pi$
$68$	7.57588	0.918710
$69$	1.00000	0.120386
$70$	3.05972	0.365706
$71$	−1.59428	−0.189206	−0.0946028	−	0.995515i	$-0.530158\pi$
$71$	−1.59428	−0.189206	−0.0946028	+	0.995515i	$0.530158\pi$
$72$	−1.00000	−0.117851
$73$	10.5553	1.23540	0.617700	−	0.786414i	$-0.288065\pi$
$73$	10.5553	1.23540	0.617700	+	0.786414i	$0.288065\pi$
$74$	−11.0433	−1.28376
$75$	−0.0989234	−0.0114227
$76$	−7.57588	−0.869013
$77$	1.55507	0.177217
$78$	0.110434	0.0125042
$79$	0.564367	0.0634962	0.0317481	−	0.999496i	$-0.489893\pi$
$79$	0.564367	0.0634962	0.0317481	+	0.999496i	$0.489893\pi$
$80$	−2.25808	−0.252461
$81$	1.00000	0.111111
$82$	6.08053	0.671482
$83$	−12.2575	−1.34543	−0.672716	−	0.739901i	$-0.734872\pi$
$83$	−12.2575	−1.34543	−0.672716	+	0.739901i	$0.734872\pi$
$84$	−1.35501	−0.147844
$85$	−17.1069	−1.85551
$86$	7.97009	0.859437
$87$	1.00000	0.107211
$88$	−1.14765	−0.122339
$89$	−0.540660	−0.0573099	−0.0286549	−	0.999589i	$-0.509122\pi$
$89$	−0.540660	−0.0573099	−0.0286549	+	0.999589i	$0.509122\pi$
$90$	2.25808	0.238022
$91$	0.149639	0.0156865
$92$	−1.00000	−0.104257
$93$	10.2445	1.06230
$94$	−4.67200	−0.481880
$95$	17.1069	1.75513
$96$	1.00000	0.102062
$97$	−14.7491	−1.49755	−0.748773	−	0.662826i	$-0.769357\pi$
$97$	−14.7491	−1.49755	−0.748773	+	0.662826i	$0.769357\pi$
$98$	5.16395	0.521638
$99$	1.14765	0.115343
$100$	0.0989234	0.00989234
$101$	8.08673	0.804659	0.402330	−	0.915495i	$-0.368201\pi$
$101$	8.08673	0.804659	0.402330	+	0.915495i	$0.368201\pi$
$102$	7.57588	0.750124
$103$	−9.15385	−0.901955	−0.450978	−	0.892535i	$-0.648925\pi$
$103$	−9.15385	−0.901955	−0.450978	+	0.892535i	$0.648925\pi$
$104$	−0.110434	−0.0108290
$105$	3.05972	0.298598
$106$	−10.7491	−1.04405
$107$	18.5575	1.79402	0.897010	−	0.442011i	$-0.145735\pi$
$107$	18.5575	1.79402	0.897010	+	0.442011i	$0.145735\pi$
$108$	−1.00000	−0.0962250
$109$	−18.5798	−1.77962	−0.889809	−	0.456332i	$-0.849163\pi$
$109$	−18.5798	−1.77962	−0.889809	+	0.456332i	$0.849163\pi$
$110$	2.59147	0.247087
$111$	−11.0433	−1.04819
$112$	1.35501	0.128036
$113$	−3.63820	−0.342253	−0.171127	−	0.985249i	$-0.554741\pi$
$113$	−3.63820	−0.342253	−0.171127	+	0.985249i	$0.554741\pi$
$114$	−7.57588	−0.709546
$115$	2.25808	0.210567
$116$	−1.00000	−0.0928477
$117$	0.110434	0.0102096
$118$	−11.8360	−1.08959
$119$	10.2654	0.941026
$120$	−2.25808	−0.206134
$121$	−9.68291	−0.880265
$122$	−14.6550	−1.32680
$123$	6.08053	0.548263
$124$	−10.2445	−0.919981
$125$	11.0670	0.989864
$126$	−1.35501	−0.120714
$127$	11.4239	1.01371	0.506855	−	0.862031i	$-0.330808\pi$
$127$	11.4239	1.01371	0.506855	+	0.862031i	$0.330808\pi$
$128$	−1.00000	−0.0883883
$129$	7.97009	0.701727
$130$	0.249369	0.0218711
$131$	−5.98839	−0.523208	−0.261604	−	0.965175i	$-0.584251\pi$
$131$	−5.98839	−0.523208	−0.261604	+	0.965175i	$0.584251\pi$
$132$	−1.14765	−0.0998897
$133$	−10.2654	−0.890122
$134$	5.32989	0.460433
$135$	2.25808	0.194345
$136$	−7.57588	−0.649626
$137$	−13.0568	−1.11552	−0.557760	−	0.830002i	$-0.688339\pi$
$137$	−13.0568	−1.11552	−0.557760	+	0.830002i	$0.688339\pi$
$138$	−1.00000	−0.0851257
$139$	10.6747	0.905416	0.452708	−	0.891659i	$-0.350458\pi$
$139$	10.6747	0.905416	0.452708	+	0.891659i	$0.350458\pi$
$140$	−3.05972	−0.258593
$141$	−4.67200	−0.393453
$142$	1.59428	0.133789
$143$	0.126739	0.0105985
$144$	1.00000	0.0833333
$145$	2.25808	0.187523
$146$	−10.5553	−0.873560
$147$	5.16395	0.425915
$148$	11.0433	0.907755
$149$	−4.67930	−0.383343	−0.191672	−	0.981459i	$-0.561391\pi$
$149$	−4.67930	−0.383343	−0.191672	+	0.981459i	$0.561391\pi$
$150$	0.0989234	0.00807706
$151$	−6.30149	−0.512808	−0.256404	−	0.966570i	$-0.582538\pi$
$151$	−6.30149	−0.512808	−0.256404	+	0.966570i	$0.582538\pi$
$152$	7.57588	0.614485
$153$	7.57588	0.612473
$154$	−1.55507	−0.125311
$155$	23.1328	1.85807
$156$	−0.110434	−0.00884181
$157$	12.1563	0.970181	0.485090	−	0.874464i	$-0.338787\pi$
$157$	12.1563	0.970181	0.485090	+	0.874464i	$0.338787\pi$
$158$	−0.564367	−0.0448986
$159$	−10.7491	−0.852461
$160$	2.25808	0.178517
$161$	−1.35501	−0.106790
$162$	−1.00000	−0.0785674
$163$	−18.7471	−1.46839	−0.734194	−	0.678940i	$-0.762440\pi$
$163$	−18.7471	−1.46839	−0.734194	+	0.678940i	$0.762440\pi$
$164$	−6.08053	−0.474809
$165$	2.59147	0.201746
$166$	12.2575	0.951364
$167$	−4.56378	−0.353156	−0.176578	−	0.984287i	$-0.556503\pi$
$167$	−4.56378	−0.353156	−0.176578	+	0.984287i	$0.556503\pi$
$168$	1.35501	0.104541
$169$	−12.9878	−0.999062
$170$	17.1069	1.31204
$171$	−7.57588	−0.579342
$172$	−7.97009	−0.607714
$173$	−12.4746	−0.948429	−0.474215	−	0.880409i	$-0.657268\pi$
$173$	−12.4746	−0.948429	−0.474215	+	0.880409i	$0.657268\pi$
$174$	−1.00000	−0.0758098
$175$	0.134042	0.0101326
$176$	1.14765	0.0865070
$177$	−11.8360	−0.889645
$178$	0.540660	0.0405242
$179$	0.201835	0.0150858	0.00754291	−	0.999972i	$-0.497599\pi$
$179$	0.201835	0.0150858	0.00754291	+	0.999972i	$0.497599\pi$
$180$	−2.25808	−0.168307
$181$	20.4652	1.52117	0.760585	−	0.649239i	$-0.224912\pi$
$181$	20.4652	1.52117	0.760585	+	0.649239i	$0.224912\pi$
$182$	−0.149639	−0.0110920
$183$	−14.6550	−1.08333
$184$	1.00000	0.0737210
$185$	−24.9367	−1.83338
$186$	−10.2445	−0.751161
$187$	8.69442	0.635799
$188$	4.67200	0.340741
$189$	−1.35501	−0.0985624
$190$	−17.1069	−1.24107
$191$	−9.98650	−0.722597	−0.361299	−	0.932450i	$-0.617667\pi$
$191$	−9.98650	−0.722597	−0.361299	+	0.932450i	$0.617667\pi$
$192$	−1.00000	−0.0721688
$193$	−17.6204	−1.26834	−0.634172	−	0.773192i	$-0.718659\pi$
$193$	−17.6204	−1.26834	−0.634172	+	0.773192i	$0.718659\pi$
$194$	14.7491	1.05893
$195$	0.249369	0.0178577
$196$	−5.16395	−0.368854
$197$	−26.1914	−1.86606	−0.933032	−	0.359794i	$-0.882847\pi$
$197$	−26.1914	−1.86606	−0.933032	+	0.359794i	$0.882847\pi$
$198$	−1.14765	−0.0815596
$199$	−13.8582	−0.982378	−0.491189	−	0.871053i	$-0.663438\pi$
$199$	−13.8582	−0.982378	−0.491189	+	0.871053i	$0.663438\pi$
$200$	−0.0989234	−0.00699494
$201$	5.32989	0.375942
$202$	−8.08673	−0.568980
$203$	−1.35501	−0.0951030
$204$	−7.57588	−0.530418
$205$	13.7303	0.958967
$206$	9.15385	0.637779
$207$	−1.00000	−0.0695048
$208$	0.110434	0.00765723
$209$	−8.69442	−0.601406
$210$	−3.05972	−0.211141
$211$	−3.37450	−0.232310	−0.116155	−	0.993231i	$-0.537057\pi$
$211$	−3.37450	−0.232310	−0.116155	+	0.993231i	$0.537057\pi$
$212$	10.7491	0.738253
$213$	1.59428	0.109238
$214$	−18.5575	−1.26856
$215$	17.9971	1.22739
$216$	1.00000	0.0680414
$217$	−13.8814	−0.942328
$218$	18.5798	1.25838
$219$	−10.5553	−0.713259
$220$	−2.59147	−0.174717
$221$	0.836636	0.0562782
$222$	11.0433	0.741179
$223$	−19.4445	−1.30210	−0.651051	−	0.759034i	$-0.725672\pi$
$223$	−19.4445	−1.30210	−0.651051	+	0.759034i	$0.725672\pi$
$224$	−1.35501	−0.0905353
$225$	0.0989234	0.00659489
$226$	3.63820	0.242010
$227$	16.5696	1.09976	0.549881	−	0.835243i	$-0.314673\pi$
$227$	16.5696	1.09976	0.549881	+	0.835243i	$0.314673\pi$
$228$	7.57588	0.501725
$229$	10.2304	0.676043	0.338021	−	0.941138i	$-0.390242\pi$
$229$	10.2304	0.676043	0.338021	+	0.941138i	$0.390242\pi$
$230$	−2.25808	−0.148893
$231$	−1.55507	−0.102316
$232$	1.00000	0.0656532
$233$	−5.90919	−0.387124	−0.193562	−	0.981088i	$-0.562004\pi$
$233$	−5.90919	−0.387124	−0.193562	+	0.981088i	$0.562004\pi$
$234$	−0.110434	−0.00721931
$235$	−10.5497	−0.688190
$236$	11.8360	0.770455
$237$	−0.564367	−0.0366595
$238$	−10.2654	−0.665406
$239$	9.98382	0.645799	0.322900	−	0.946433i	$-0.395342\pi$
$239$	9.98382	0.645799	0.322900	+	0.946433i	$0.395342\pi$
$240$	2.25808	0.145758
$241$	1.92188	0.123799	0.0618997	−	0.998082i	$-0.480284\pi$
$241$	1.92188	0.123799	0.0618997	+	0.998082i	$0.480284\pi$
$242$	9.68291	0.622441
$243$	−1.00000	−0.0641500
$244$	14.6550	0.938190
$245$	11.6606	0.744969
$246$	−6.08053	−0.387680
$247$	−0.836636	−0.0532339
$248$	10.2445	0.650525
$249$	12.2575	0.776785
$250$	−11.0670	−0.699940
$251$	−9.12246	−0.575805	−0.287902	−	0.957660i	$-0.592958\pi$
$251$	−9.12246	−0.575805	−0.287902	+	0.957660i	$0.592958\pi$
$252$	1.35501	0.0853575
$253$	−1.14765	−0.0721518
$254$	−11.4239	−0.716801
$255$	17.1069	1.07128
$256$	1.00000	0.0625000
$257$	−3.66492	−0.228611	−0.114306	−	0.993446i	$-0.536464\pi$
$257$	−3.66492	−0.228611	−0.114306	+	0.993446i	$0.536464\pi$
$258$	−7.97009	−0.496196
$259$	14.9638	0.929805
$260$	−0.249369	−0.0154652
$261$	−1.00000	−0.0618984
$262$	5.98839	0.369964
$263$	−25.2675	−1.55806	−0.779030	−	0.626986i	$-0.784288\pi$
$263$	−25.2675	−1.55806	−0.779030	+	0.626986i	$0.784288\pi$
$264$	1.14765	0.0706327
$265$	−24.2724	−1.49104
$266$	10.2654	0.629411
$267$	0.540660	0.0330879
$268$	−5.32989	−0.325575
$269$	5.97491	0.364297	0.182148	−	0.983271i	$-0.441695\pi$
$269$	5.97491	0.364297	0.182148	+	0.983271i	$0.441695\pi$
$270$	−2.25808	−0.137422
$271$	14.8604	0.902702	0.451351	−	0.892346i	$-0.350942\pi$
$271$	14.8604	0.902702	0.451351	+	0.892346i	$0.350942\pi$
$272$	7.57588	0.459355
$273$	−0.149639	−0.00905658
$274$	13.0568	0.788791
$275$	0.113529	0.00684605
$276$	1.00000	0.0601929
$277$	−25.0143	−1.50296	−0.751482	−	0.659753i	$-0.770661\pi$
$277$	−25.0143	−1.50296	−0.751482	+	0.659753i	$0.770661\pi$
$278$	−10.6747	−0.640226
$279$	−10.2445	−0.613321
$280$	3.05972	0.182853
$281$	−3.08403	−0.183978	−0.0919888	−	0.995760i	$-0.529322\pi$
$281$	−3.08403	−0.183978	−0.0919888	+	0.995760i	$0.529322\pi$
$282$	4.67200	0.278214
$283$	−1.10603	−0.0657467	−0.0328733	−	0.999460i	$-0.510466\pi$
$283$	−1.10603	−0.0657467	−0.0328733	+	0.999460i	$0.510466\pi$
$284$	−1.59428	−0.0946028
$285$	−17.1069	−1.01333
$286$	−0.126739	−0.00749425
$287$	−8.23917	−0.486343
$288$	−1.00000	−0.0589256
$289$	40.3939	2.37611
$290$	−2.25808	−0.132599
$291$	14.7491	0.864609
$292$	10.5553	0.617700
$293$	16.4457	0.960769	0.480385	−	0.877058i	$-0.340497\pi$
$293$	16.4457	0.960769	0.480385	+	0.877058i	$0.340497\pi$
$294$	−5.16395	−0.301168
$295$	−26.7265	−1.55608
$296$	−11.0433	−0.641880
$297$	−1.14765	−0.0665931
$298$	4.67930	0.271065
$299$	−0.110434	−0.00638657
$300$	−0.0989234	−0.00571134
$301$	−10.7995	−0.622475
$302$	6.30149	0.362610
$303$	−8.08673	−0.464570
$304$	−7.57588	−0.434506
$305$	−33.0921	−1.89485
$306$	−7.57588	−0.433084
$307$	−14.6869	−0.838226	−0.419113	−	0.907934i	$-0.637659\pi$
$307$	−14.6869	−0.838226	−0.419113	+	0.907934i	$0.637659\pi$
$308$	1.55507	0.0886083
$309$	9.15385	0.520744
$310$	−23.1328	−1.31386
$311$	−31.6135	−1.79264	−0.896319	−	0.443410i	$-0.853768\pi$
$311$	−31.6135	−1.79264	−0.896319	+	0.443410i	$0.853768\pi$
$312$	0.110434	0.00625210
$313$	3.62659	0.204987	0.102494	−	0.994734i	$-0.467318\pi$
$313$	3.62659	0.204987	0.102494	+	0.994734i	$0.467318\pi$
$314$	−12.1563	−0.686021
$315$	−3.05972	−0.172396
$316$	0.564367	0.0317481
$317$	−0.492551	−0.0276644	−0.0138322	−	0.999904i	$-0.504403\pi$
$317$	−0.492551	−0.0276644	−0.0138322	+	0.999904i	$0.504403\pi$
$318$	10.7491	0.602781
$319$	−1.14765	−0.0642558
$320$	−2.25808	−0.126230
$321$	−18.5575	−1.03578
$322$	1.35501	0.0755117
$323$	−57.3939	−3.19348
$324$	1.00000	0.0555556
$325$	0.0109245	0.000605983 0
$326$	18.7471	1.03831
$327$	18.5798	1.02746
$328$	6.08053	0.335741
$329$	6.33060	0.349017
$330$	−2.59147	−0.142656
$331$	−4.70382	−0.258545	−0.129273	−	0.991609i	$-0.541264\pi$
$331$	−4.70382	−0.258545	−0.129273	+	0.991609i	$0.541264\pi$
$332$	−12.2575	−0.672716
$333$	11.0433	0.605170
$334$	4.56378	0.249719
$335$	12.0353	0.657560
$336$	−1.35501	−0.0739218
$337$	15.2000	0.827995	0.413998	−	0.910278i	$-0.364132\pi$
$337$	15.2000	0.827995	0.413998	+	0.910278i	$0.364132\pi$
$338$	12.9878	0.706443
$339$	3.63820	0.197600
$340$	−17.1069	−0.927754
$341$	−11.7570	−0.636679
$342$	7.57588	0.409657
$343$	−16.4823	−0.889959
$344$	7.97009	0.429719
$345$	−2.25808	−0.121571
$346$	12.4746	0.670641
$347$	−8.19096	−0.439714	−0.219857	−	0.975532i	$-0.570559\pi$
$347$	−8.19096	−0.439714	−0.219857	+	0.975532i	$0.570559\pi$
$348$	1.00000	0.0536056
$349$	−19.4050	−1.03873	−0.519364	−	0.854553i	$-0.673831\pi$
$349$	−19.4050	−1.03873	−0.519364	+	0.854553i	$0.673831\pi$
$350$	−0.134042	−0.00716485
$351$	−0.110434	−0.00589454
$352$	−1.14765	−0.0611697
$353$	−4.88219	−0.259853	−0.129926	−	0.991524i	$-0.541474\pi$
$353$	−4.88219	−0.259853	−0.129926	+	0.991524i	$0.541474\pi$
$354$	11.8360	0.629074
$355$	3.60000	0.191068
$356$	−0.540660	−0.0286549
$357$	−10.2654	−0.543302
$358$	−0.201835	−0.0106673
$359$	−12.6269	−0.666422	−0.333211	−	0.942852i	$-0.608132\pi$
$359$	−12.6269	−0.666422	−0.333211	+	0.942852i	$0.608132\pi$
$360$	2.25808	0.119011
$361$	38.3939	2.02073
$362$	−20.4652	−1.07563
$363$	9.68291	0.508221
$364$	0.149639	0.00784323
$365$	−23.8346	−1.24756
$366$	14.6550	0.766029
$367$	20.7431	1.08278	0.541391	−	0.840771i	$-0.317898\pi$
$367$	20.7431	1.08278	0.541391	+	0.840771i	$0.317898\pi$
$368$	−1.00000	−0.0521286
$369$	−6.08053	−0.316540
$370$	24.9367	1.29640
$371$	14.5652	0.756185
$372$	10.2445	0.531151
$373$	−12.0109	−0.621902	−0.310951	−	0.950426i	$-0.600648\pi$
$373$	−12.0109	−0.621902	−0.310951	+	0.950426i	$0.600648\pi$
$374$	−8.69442	−0.449578
$375$	−11.0670	−0.571499
$376$	−4.67200	−0.240940
$377$	−0.110434	−0.00568765
$378$	1.35501	0.0696941
$379$	32.5185	1.67036	0.835181	−	0.549975i	$-0.185363\pi$
$379$	32.5185	1.67036	0.835181	+	0.549975i	$0.185363\pi$
$380$	17.1069	0.877567
$381$	−11.4239	−0.585265
$382$	9.98650	0.510954
$383$	4.66181	0.238207	0.119104	−	0.992882i	$-0.461998\pi$
$383$	4.66181	0.238207	0.119104	+	0.992882i	$0.461998\pi$
$384$	1.00000	0.0510310
$385$	−3.51147	−0.178961
$386$	17.6204	0.896854
$387$	−7.97009	−0.405143
$388$	−14.7491	−0.748773
$389$	−23.1125	−1.17185	−0.585924	−	0.810366i	$-0.699268\pi$
$389$	−23.1125	−1.17185	−0.585924	+	0.810366i	$0.699268\pi$
$390$	−0.249369	−0.0126273
$391$	−7.57588	−0.383129
$392$	5.16395	0.260819
$393$	5.98839	0.302074
$394$	26.1914	1.31951
$395$	−1.27438	−0.0641213
$396$	1.14765	0.0576714
$397$	3.11584	0.156380	0.0781898	−	0.996938i	$-0.475086\pi$
$397$	3.11584	0.156380	0.0781898	+	0.996938i	$0.475086\pi$
$398$	13.8582	0.694646
$399$	10.2654	0.513912
$400$	0.0989234	0.00494617
$401$	−13.4720	−0.672762	−0.336381	−	0.941726i	$-0.609203\pi$
$401$	−13.4720	−0.672762	−0.336381	+	0.941726i	$0.609203\pi$
$402$	−5.32989	−0.265831
$403$	−1.13134	−0.0563561
$404$	8.08673	0.402330
$405$	−2.25808	−0.112205
$406$	1.35501	0.0672480
$407$	12.6738	0.628217
$408$	7.57588	0.375062
$409$	−31.2822	−1.54680	−0.773402	−	0.633916i	$-0.781446\pi$
$409$	−31.2822	−1.54680	−0.773402	+	0.633916i	$0.781446\pi$
$410$	−13.7303	−0.678092
$411$	13.0568	0.644045
$412$	−9.15385	−0.450978
$413$	16.0378	0.789170
$414$	1.00000	0.0491473
$415$	27.6783	1.35868
$416$	−0.110434	−0.00541448
$417$	−10.6747	−0.522742
$418$	8.69442	0.425258
$419$	0.645785	0.0315487	0.0157743	−	0.999876i	$-0.494979\pi$
$419$	0.645785	0.0315487	0.0157743	+	0.999876i	$0.494979\pi$
$420$	3.05972	0.149299
$421$	−10.0439	−0.489510	−0.244755	−	0.969585i	$-0.578707\pi$
$421$	−10.0439	−0.489510	−0.244755	+	0.969585i	$0.578707\pi$
$422$	3.37450	0.164268
$423$	4.67200	0.227160
$424$	−10.7491	−0.522024
$425$	0.749431	0.0363528
$426$	−1.59428	−0.0772429
$427$	19.8577	0.960979
$428$	18.5575	0.897010
$429$	−0.126739	−0.00611903
$430$	−17.9971	−0.867897
$431$	−25.1401	−1.21096	−0.605479	−	0.795861i	$-0.707018\pi$
$431$	−25.1401	−1.21096	−0.605479	+	0.795861i	$0.707018\pi$
$432$	−1.00000	−0.0481125
$433$	−10.2961	−0.494798	−0.247399	−	0.968914i	$-0.579576\pi$
$433$	−10.2961	−0.494798	−0.247399	+	0.968914i	$0.579576\pi$
$434$	13.8814	0.666327
$435$	−2.25808	−0.108267
$436$	−18.5798	−0.889809
$437$	7.57588	0.362403
$438$	10.5553	0.504350
$439$	−19.8410	−0.946958	−0.473479	−	0.880805i	$-0.657002\pi$
$439$	−19.8410	−0.946958	−0.473479	+	0.880805i	$0.657002\pi$
$440$	2.59147	0.123544
$441$	−5.16395	−0.245902
$442$	−0.836636	−0.0397947
$443$	−22.5283	−1.07035	−0.535175	−	0.844741i	$-0.679754\pi$
$443$	−22.5283	−1.07035	−0.535175	+	0.844741i	$0.679754\pi$
$444$	−11.0433	−0.524093
$445$	1.22085	0.0578740
$446$	19.4445	0.920726
$447$	4.67930	0.221323
$448$	1.35501	0.0640182
$449$	30.3746	1.43347	0.716733	−	0.697348i	$-0.245637\pi$
$449$	30.3746	1.43347	0.716733	+	0.697348i	$0.245637\pi$
$450$	−0.0989234	−0.00466329
$451$	−6.97829	−0.328595
$452$	−3.63820	−0.171127
$453$	6.30149	0.296070
$454$	−16.5696	−0.777649
$455$	−0.337897	−0.0158409
$456$	−7.57588	−0.354773
$457$	−20.9017	−0.977739	−0.488869	−	0.872357i	$-0.662590\pi$
$457$	−20.9017	−0.977739	−0.488869	+	0.872357i	$0.662590\pi$
$458$	−10.2304	−0.478035
$459$	−7.57588	−0.353612
$460$	2.25808	0.105284
$461$	16.6481	0.775378	0.387689	−	0.921790i	$-0.373273\pi$
$461$	16.6481	0.775378	0.387689	+	0.921790i	$0.373273\pi$
$462$	1.55507	0.0723484
$463$	−27.3847	−1.27268	−0.636338	−	0.771410i	$-0.719552\pi$
$463$	−27.3847	−1.27268	−0.636338	+	0.771410i	$0.719552\pi$
$464$	−1.00000	−0.0464238
$465$	−23.1328	−1.07276
$466$	5.90919	0.273738
$467$	−9.57035	−0.442863	−0.221431	−	0.975176i	$-0.571073\pi$
$467$	−9.57035	−0.442863	−0.221431	+	0.975176i	$0.571073\pi$
$468$	0.110434	0.00510482
$469$	−7.22206	−0.333484
$470$	10.5497	0.486624
$471$	−12.1563	−0.560134
$472$	−11.8360	−0.544794
$473$	−9.14684	−0.420572
$474$	0.564367	0.0259222
$475$	−0.749431	−0.0343863
$476$	10.2654	0.470513
$477$	10.7491	0.492169
$478$	−9.98382	−0.456649
$479$	4.62400	0.211276	0.105638	−	0.994405i	$-0.466312\pi$
$479$	4.62400	0.211276	0.105638	+	0.994405i	$0.466312\pi$
$480$	−2.25808	−0.103067
$481$	1.21956	0.0556071
$482$	−1.92188	−0.0875394
$483$	1.35501	0.0616550
$484$	−9.68291	−0.440132
$485$	33.3047	1.51229
$486$	1.00000	0.0453609
$487$	−13.3419	−0.604580	−0.302290	−	0.953216i	$-0.597751\pi$
$487$	−13.3419	−0.604580	−0.302290	+	0.953216i	$0.597751\pi$
$488$	−14.6550	−0.663400
$489$	18.7471	0.847774
$490$	−11.6606	−0.526773
$491$	23.0794	1.04156	0.520778	−	0.853692i	$-0.325642\pi$
$491$	23.0794	1.04156	0.520778	+	0.853692i	$0.325642\pi$
$492$	6.08053	0.274131
$493$	−7.57588	−0.341200
$494$	0.836636	0.0376420
$495$	−2.59147	−0.116478
$496$	−10.2445	−0.459991
$497$	−2.16026	−0.0969008
$498$	−12.2575	−0.549270
$499$	25.5016	1.14161	0.570803	−	0.821087i	$-0.306632\pi$
$499$	25.5016	1.14161	0.570803	+	0.821087i	$0.306632\pi$
$500$	11.0670	0.494932
$501$	4.56378	0.203895
$502$	9.12246	0.407155
$503$	−41.9313	−1.86962	−0.934812	−	0.355144i	$-0.884432\pi$
$503$	−41.9313	−1.86962	−0.934812	+	0.355144i	$0.884432\pi$
$504$	−1.35501	−0.0603569
$505$	−18.2605	−0.812580
$506$	1.14765	0.0510191
$507$	12.9878	0.576809
$508$	11.4239	0.506855
$509$	0.602691	0.0267138	0.0133569	−	0.999911i	$-0.495748\pi$
$509$	0.602691	0.0267138	0.0133569	+	0.999911i	$0.495748\pi$
$510$	−17.1069	−0.757508
$511$	14.3025	0.632704
$512$	−1.00000	−0.0441942
$513$	7.57588	0.334483
$514$	3.66492	0.161653
$515$	20.6701	0.910834
$516$	7.97009	0.350864
$517$	5.36180	0.235812
$518$	−14.9638	−0.657471
$519$	12.4746	0.547576
$520$	0.249369	0.0109356
$521$	18.6550	0.817291	0.408645	−	0.912693i	$-0.366001\pi$
$521$	18.6550	0.817291	0.408645	+	0.912693i	$0.366001\pi$
$522$	1.00000	0.0437688
$523$	32.1886	1.40751	0.703754	−	0.710444i	$-0.251506\pi$
$523$	32.1886	1.40751	0.703754	+	0.710444i	$0.251506\pi$
$524$	−5.98839	−0.261604
$525$	−0.134042	−0.00585008
$526$	25.2675	1.10172
$527$	−77.6109	−3.38078
$528$	−1.14765	−0.0499449
$529$	1.00000	0.0434783
$530$	24.2724	1.05432
$531$	11.8360	0.513637
$532$	−10.2654	−0.445061
$533$	−0.671498	−0.0290858
$534$	−0.540660	−0.0233966
$535$	−41.9043	−1.81168
$536$	5.32989	0.230216
$537$	−0.201835	−0.00870980
$538$	−5.97491	−0.257597
$539$	−5.92638	−0.255267
$540$	2.25808	0.0971723
$541$	−29.0796	−1.25023	−0.625115	−	0.780533i	$-0.714948\pi$
$541$	−29.0796	−1.25023	−0.625115	+	0.780533i	$0.714948\pi$
$542$	−14.8604	−0.638307
$543$	−20.4652	−0.878248
$544$	−7.57588	−0.324813
$545$	41.9546	1.79714
$546$	0.149639	0.00640397
$547$	−18.4158	−0.787401	−0.393700	−	0.919239i	$-0.628805\pi$
$547$	−18.4158	−0.787401	−0.393700	+	0.919239i	$0.628805\pi$
$548$	−13.0568	−0.557760
$549$	14.6550	0.625460
$550$	−0.113529	−0.00484089
$551$	7.57588	0.322743
$552$	−1.00000	−0.0425628
$553$	0.764722	0.0325193
$554$	25.0143	1.06276
$555$	24.9367	1.05850
$556$	10.6747	0.452708
$557$	−25.0965	−1.06337	−0.531687	−	0.846941i	$-0.678442\pi$
$557$	−25.0965	−1.06337	−0.531687	+	0.846941i	$0.678442\pi$
$558$	10.2445	0.433683
$559$	−0.880171	−0.0372272
$560$	−3.05972	−0.129297
$561$	−8.69442	−0.367079
$562$	3.08403	0.130092
$563$	16.4721	0.694215	0.347107	−	0.937825i	$-0.387164\pi$
$563$	16.4721	0.694215	0.347107	+	0.937825i	$0.387164\pi$
$564$	−4.67200	−0.196727
$565$	8.21535	0.345622
$566$	1.10603	0.0464899
$567$	1.35501	0.0569050
$568$	1.59428	0.0668943
$569$	4.34591	0.182190	0.0910950	−	0.995842i	$-0.470963\pi$
$569$	4.34591	0.182190	0.0910950	+	0.995842i	$0.470963\pi$
$570$	17.1069	0.716531
$571$	27.4216	1.14756	0.573780	−	0.819010i	$-0.305476\pi$
$571$	27.4216	1.14756	0.573780	+	0.819010i	$0.305476\pi$
$572$	0.126739	0.00529924
$573$	9.98650	0.417192
$574$	8.23917	0.343896
$575$	−0.0989234	−0.00412539
$576$	1.00000	0.0416667
$577$	27.4219	1.14159	0.570793	−	0.821094i	$-0.306636\pi$
$577$	27.4219	1.14159	0.570793	+	0.821094i	$0.306636\pi$
$578$	−40.3939	−1.68017
$579$	17.6204	0.732278
$580$	2.25808	0.0937617
$581$	−16.6090	−0.689057
$582$	−14.7491	−0.611371
$583$	12.3362	0.510913
$584$	−10.5553	−0.436780
$585$	−0.249369	−0.0103101
$586$	−16.4457	−0.679367
$587$	5.81682	0.240086	0.120043	−	0.992769i	$-0.461697\pi$
$587$	5.81682	0.240086	0.120043	+	0.992769i	$0.461697\pi$
$588$	5.16395	0.212958
$589$	77.6109	3.19790
$590$	26.7265	1.10031
$591$	26.1914	1.07737
$592$	11.0433	0.453877
$593$	−25.5867	−1.05072	−0.525360	−	0.850880i	$-0.676069\pi$
$593$	−25.5867	−1.05072	−0.525360	+	0.850880i	$0.676069\pi$
$594$	1.14765	0.0470885
$595$	−23.1801	−0.950289
$596$	−4.67930	−0.191672
$597$	13.8582	0.567176
$598$	0.110434	0.00451599
$599$	−18.4212	−0.752671	−0.376335	−	0.926483i	$-0.622816\pi$
$599$	−18.4212	−0.752671	−0.376335	+	0.926483i	$0.622816\pi$
$600$	0.0989234	0.00403853
$601$	2.34861	0.0958019	0.0479009	−	0.998852i	$-0.484747\pi$
$601$	2.34861	0.0958019	0.0479009	+	0.998852i	$0.484747\pi$
$602$	10.7995	0.440157
$603$	−5.32989	−0.217050
$604$	−6.30149	−0.256404
$605$	21.8648	0.888930
$606$	8.08673	0.328501
$607$	−25.6416	−1.04076	−0.520380	−	0.853935i	$-0.674210\pi$
$607$	−25.6416	−1.04076	−0.520380	+	0.853935i	$0.674210\pi$
$608$	7.57588	0.307242
$609$	1.35501	0.0549077
$610$	33.0921	1.33986
$611$	0.515949	0.0208730
$612$	7.57588	0.306237
$613$	−29.7235	−1.20052	−0.600261	−	0.799804i	$-0.704937\pi$
$613$	−29.7235	−1.20052	−0.600261	+	0.799804i	$0.704937\pi$
$614$	14.6869	0.592715
$615$	−13.7303	−0.553660
$616$	−1.55507	−0.0626556
$617$	11.4087	0.459298	0.229649	−	0.973274i	$-0.426242\pi$
$617$	11.4087	0.459298	0.229649	+	0.973274i	$0.426242\pi$
$618$	−9.15385	−0.368222
$619$	−15.8834	−0.638407	−0.319203	−	0.947686i	$-0.603415\pi$
$619$	−15.8834	−0.638407	−0.319203	+	0.947686i	$0.603415\pi$
$620$	23.1328	0.929037
$621$	1.00000	0.0401286
$622$	31.6135	1.26759
$623$	−0.732599	−0.0293510
$624$	−0.110434	−0.00442091
$625$	−25.4848	−1.01939
$626$	−3.62659	−0.144948
$627$	8.69442	0.347222
$628$	12.1563	0.485090
$629$	83.6628	3.33585
$630$	3.05972	0.121902
$631$	−2.67521	−0.106498	−0.0532491	−	0.998581i	$-0.516958\pi$
$631$	−2.67521	−0.106498	−0.0532491	+	0.998581i	$0.516958\pi$
$632$	−0.564367	−0.0224493
$633$	3.37450	0.134124
$634$	0.492551	0.0195617
$635$	−25.7961	−1.02369
$636$	−10.7491	−0.426231
$637$	−0.570277	−0.0225952
$638$	1.14765	0.0454357
$639$	−1.59428	−0.0630685
$640$	2.25808	0.0892584
$641$	−13.0040	−0.513628	−0.256814	−	0.966461i	$-0.582673\pi$
$641$	−13.0040	−0.513628	−0.256814	+	0.966461i	$0.582673\pi$
$642$	18.5575	0.732405
$643$	−7.29741	−0.287782	−0.143891	−	0.989594i	$-0.545961\pi$
$643$	−7.29741	−0.287782	−0.143891	+	0.989594i	$0.545961\pi$
$644$	−1.35501	−0.0533948
$645$	−17.9971	−0.708635
$646$	57.3939	2.25813
$647$	18.4398	0.724944	0.362472	−	0.931995i	$-0.381933\pi$
$647$	18.4398	0.724944	0.362472	+	0.931995i	$0.381933\pi$
$648$	−1.00000	−0.0392837
$649$	13.5835	0.533198
$650$	−0.0109245	−0.000428495 0
$651$	13.8814	0.544053
$652$	−18.7471	−0.734194
$653$	−10.2437	−0.400866	−0.200433	−	0.979707i	$-0.564235\pi$
$653$	−10.2437	−0.400866	−0.200433	+	0.979707i	$0.564235\pi$
$654$	−18.5798	−0.726526
$655$	13.5223	0.528359
$656$	−6.08053	−0.237405
$657$	10.5553	0.411800
$658$	−6.33060	−0.246793
$659$	30.5839	1.19138	0.595689	−	0.803215i	$-0.296879\pi$
$659$	30.5839	1.19138	0.595689	+	0.803215i	$0.296879\pi$
$660$	2.59147	0.100873
$661$	−35.0039	−1.36150	−0.680748	−	0.732518i	$-0.738345\pi$
$661$	−35.0039	−1.36150	−0.680748	+	0.732518i	$0.738345\pi$
$662$	4.70382	0.182819
$663$	−0.836636	−0.0324922
$664$	12.2575	0.475682
$665$	23.1801	0.898884
$666$	−11.0433	−0.427920
$667$	1.00000	0.0387202
$668$	−4.56378	−0.176578
$669$	19.4445	0.751769
$670$	−12.0353	−0.464965
$671$	16.8187	0.649280
$672$	1.35501	0.0522706
$673$	37.9343	1.46226	0.731129	−	0.682239i	$-0.238994\pi$
$673$	37.9343	1.46226	0.731129	+	0.682239i	$0.238994\pi$
$674$	−15.2000	−0.585481
$675$	−0.0989234	−0.00380756
$676$	−12.9878	−0.499531
$677$	−4.38990	−0.168717	−0.0843587	−	0.996435i	$-0.526884\pi$
$677$	−4.38990	−0.168717	−0.0843587	+	0.996435i	$0.526884\pi$
$678$	−3.63820	−0.139724
$679$	−19.9852	−0.766961
$680$	17.1069	0.656021
$681$	−16.5696	−0.634948
$682$	11.7570	0.450200
$683$	35.1668	1.34562	0.672809	−	0.739816i	$-0.265087\pi$
$683$	35.1668	1.34562	0.672809	+	0.739816i	$0.265087\pi$
$684$	−7.57588	−0.289671
$685$	29.4833	1.12650
$686$	16.4823	0.629296
$687$	−10.2304	−0.390314
$688$	−7.97009	−0.303857
$689$	1.18707	0.0452238
$690$	2.25808	0.0859636
$691$	−1.67259	−0.0636282	−0.0318141	−	0.999494i	$-0.510128\pi$
$691$	−1.67259	−0.0636282	−0.0318141	+	0.999494i	$0.510128\pi$
$692$	−12.4746	−0.474215
$693$	1.55507	0.0590722
$694$	8.19096	0.310925
$695$	−24.1043	−0.914329
$696$	−1.00000	−0.0379049
$697$	−46.0653	−1.74485
$698$	19.4050	0.734491
$699$	5.90919	0.223506
$700$	0.134042	0.00506631
$701$	−5.08522	−0.192066	−0.0960331	−	0.995378i	$-0.530615\pi$
$701$	−5.08522	−0.192066	−0.0960331	+	0.995378i	$0.530615\pi$
$702$	0.110434	0.00416807
$703$	−83.6628	−3.15540
$704$	1.14765	0.0432535
$705$	10.5497	0.397327
$706$	4.88219	0.183744
$707$	10.9576	0.412103
$708$	−11.8360	−0.444822
$709$	−23.1857	−0.870759	−0.435380	−	0.900247i	$-0.643386\pi$
$709$	−23.1857	−0.870759	−0.435380	+	0.900247i	$0.643386\pi$
$710$	−3.60000	−0.135106
$711$	0.564367	0.0211654
$712$	0.540660	0.0202621
$713$	10.2445	0.383659
$714$	10.2654	0.384172
$715$	−0.286187	−0.0107028
$716$	0.201835	0.00754291
$717$	−9.98382	−0.372852
$718$	12.6269	0.471231
$719$	−21.7349	−0.810577	−0.405288	−	0.914189i	$-0.632829\pi$
$719$	−21.7349	−0.810577	−0.405288	+	0.914189i	$0.632829\pi$
$720$	−2.25808	−0.0841537
$721$	−12.4035	−0.461932
$722$	−38.3939	−1.42887
$723$	−1.92188	−0.0714756
$724$	20.4652	0.760585
$725$	−0.0989234	−0.00367392
$726$	−9.68291	−0.359366
$727$	−51.2866	−1.90211	−0.951057	−	0.309016i	$-0.900000\pi$
$727$	−51.2866	−1.90211	−0.951057	+	0.309016i	$0.900000\pi$
$728$	−0.149639	−0.00554600
$729$	1.00000	0.0370370
$730$	23.8346	0.882159
$731$	−60.3804	−2.23325
$732$	−14.6550	−0.541664
$733$	7.61677	0.281332	0.140666	−	0.990057i	$-0.455076\pi$
$733$	7.61677	0.281332	0.140666	+	0.990057i	$0.455076\pi$
$734$	−20.7431	−0.765642
$735$	−11.6606	−0.430108
$736$	1.00000	0.0368605
$737$	−6.11683	−0.225316
$738$	6.08053	0.223827
$739$	−13.5053	−0.496800	−0.248400	−	0.968658i	$-0.579905\pi$
$739$	−13.5053	−0.496800	−0.248400	+	0.968658i	$0.579905\pi$
$740$	−24.9367	−0.916691
$741$	0.836636	0.0307346
$742$	−14.5652	−0.534704
$743$	−7.23288	−0.265349	−0.132674	−	0.991160i	$-0.542356\pi$
$743$	−7.23288	−0.265349	−0.132674	+	0.991160i	$0.542356\pi$
$744$	−10.2445	−0.375581
$745$	10.5662	0.387117
$746$	12.0109	0.439751
$747$	−12.2575	−0.448477
$748$	8.69442	0.317900
$749$	25.1456	0.918799
$750$	11.0670	0.404110
$751$	−35.5498	−1.29723	−0.648615	−	0.761117i	$-0.724651\pi$
$751$	−35.5498	−1.29723	−0.648615	+	0.761117i	$0.724651\pi$
$752$	4.67200	0.170370
$753$	9.12246	0.332441
$754$	0.110434	0.00402178
$755$	14.2293	0.517856
$756$	−1.35501	−0.0492812
$757$	43.6829	1.58768	0.793841	−	0.608125i	$-0.208078\pi$
$757$	43.6829	1.58768	0.793841	+	0.608125i	$0.208078\pi$
$758$	−32.5185	−1.18112
$759$	1.14765	0.0416569
$760$	−17.1069	−0.620534
$761$	39.5754	1.43461	0.717303	−	0.696761i	$-0.245376\pi$
$761$	39.5754	1.43461	0.717303	+	0.696761i	$0.245376\pi$
$762$	11.4239	0.413845
$763$	−25.1758	−0.911423
$764$	−9.98650	−0.361299
$765$	−17.1069	−0.618503
$766$	−4.66181	−0.168438
$767$	1.30709	0.0471964
$768$	−1.00000	−0.0360844
$769$	48.3069	1.74199	0.870995	−	0.491292i	$-0.163475\pi$
$769$	48.3069	1.74199	0.870995	+	0.491292i	$0.163475\pi$
$770$	3.51147	0.126545
$771$	3.66492	0.131989
$772$	−17.6204	−0.634172
$773$	50.8011	1.82719	0.913595	−	0.406626i	$-0.133295\pi$
$773$	50.8011	1.82719	0.913595	+	0.406626i	$0.133295\pi$
$774$	7.97009	0.286479
$775$	−1.01342	−0.0364031
$776$	14.7491	0.529463
$777$	−14.9638	−0.536823
$778$	23.1125	0.828622
$779$	46.0653	1.65046
$780$	0.249369	0.00892885
$781$	−1.82966	−0.0654705
$782$	7.57588	0.270913
$783$	1.00000	0.0357371
$784$	−5.16395	−0.184427
$785$	−27.4500	−0.979731
$786$	−5.98839	−0.213599
$787$	−33.6083	−1.19801	−0.599003	−	0.800747i	$-0.704436\pi$
$787$	−33.6083	−1.19801	−0.599003	+	0.800747i	$0.704436\pi$
$788$	−26.1914	−0.933032
$789$	25.2675	0.899547
$790$	1.27438	0.0453406
$791$	−4.92980	−0.175283
$792$	−1.14765	−0.0407798
$793$	1.61841	0.0574715
$794$	−3.11584	−0.110577
$795$	24.2724	0.860853
$796$	−13.8582	−0.491189
$797$	−6.20765	−0.219886	−0.109943	−	0.993938i	$-0.535067\pi$
$797$	−6.20765	−0.219886	−0.109943	+	0.993938i	$0.535067\pi$
$798$	−10.2654	−0.363391
$799$	35.3945	1.25217
$800$	−0.0989234	−0.00349747
$801$	−0.540660	−0.0191033
$802$	13.4720	0.475714
$803$	12.1137	0.427483
$804$	5.32989	0.187971
$805$	3.05972	0.107841
$806$	1.13134	0.0398498
$807$	−5.97491	−0.210327
$808$	−8.08673	−0.284490
$809$	−49.0931	−1.72602	−0.863011	−	0.505185i	$-0.831424\pi$
$809$	−49.0931	−1.72602	−0.863011	+	0.505185i	$0.831424\pi$
$810$	2.25808	0.0793408
$811$	8.67470	0.304610	0.152305	−	0.988334i	$-0.451330\pi$
$811$	8.67470	0.304610	0.152305	+	0.988334i	$0.451330\pi$
$812$	−1.35501	−0.0475515
$813$	−14.8604	−0.521175
$814$	−12.6738	−0.444217
$815$	42.3325	1.48284
$816$	−7.57588	−0.265209
$817$	60.3804	2.11244
$818$	31.2822	1.09376
$819$	0.149639	0.00522882
$820$	13.7303	0.479483
$821$	−18.8347	−0.657336	−0.328668	−	0.944445i	$-0.606600\pi$
$821$	−18.8347	−0.657336	−0.328668	+	0.944445i	$0.606600\pi$
$822$	−13.0568	−0.455409
$823$	−32.2170	−1.12301	−0.561506	−	0.827472i	$-0.689778\pi$
$823$	−32.2170	−1.12301	−0.561506	+	0.827472i	$0.689778\pi$
$824$	9.15385	0.318889
$825$	−0.113529	−0.00395257
$826$	−16.0378	−0.558027
$827$	−2.22976	−0.0775364	−0.0387682	−	0.999248i	$-0.512343\pi$
$827$	−2.22976	−0.0775364	−0.0387682	+	0.999248i	$0.512343\pi$
$828$	−1.00000	−0.0347524
$829$	−40.9953	−1.42382	−0.711912	−	0.702268i	$-0.752171\pi$
$829$	−40.9953	−1.42382	−0.711912	+	0.702268i	$0.752171\pi$
$830$	−27.6783	−0.960729
$831$	25.0143	0.867737
$832$	0.110434	0.00382862
$833$	−39.1215	−1.35548
$834$	10.6747	0.369635
$835$	10.3054	0.356632
$836$	−8.69442	−0.300703
$837$	10.2445	0.354101
$838$	−0.645785	−0.0223083
$839$	56.5131	1.95105	0.975524	−	0.219893i	$-0.0705708\pi$
$839$	56.5131	1.95105	0.975524	+	0.219893i	$0.0705708\pi$
$840$	−3.05972	−0.105570
$841$	1.00000	0.0344828
$842$	10.0439	0.346136
$843$	3.08403	0.106219
$844$	−3.37450	−0.116155
$845$	29.3275	1.00890
$846$	−4.67200	−0.160627
$847$	−13.1204	−0.450823
$848$	10.7491	0.369126
$849$	1.10603	0.0379589
$850$	−0.749431	−0.0257053
$851$	−11.0433	−0.378560
$852$	1.59428	0.0546190
$853$	−7.26696	−0.248816	−0.124408	−	0.992231i	$-0.539703\pi$
$853$	−7.26696	−0.248816	−0.124408	+	0.992231i	$0.539703\pi$
$854$	−19.8577	−0.679515
$855$	17.1069	0.585045
$856$	−18.5575	−0.634282
$857$	0.721321	0.0246398	0.0123199	−	0.999924i	$-0.496078\pi$
$857$	0.721321	0.0246398	0.0123199	+	0.999924i	$0.496078\pi$
$858$	0.126739	0.00432681
$859$	47.3132	1.61431	0.807153	−	0.590342i	$-0.201007\pi$
$859$	47.3132	1.61431	0.807153	+	0.590342i	$0.201007\pi$
$860$	17.9971	0.613696
$861$	8.23917	0.280790
$862$	25.1401	0.856277
$863$	−13.4842	−0.459009	−0.229504	−	0.973308i	$-0.573710\pi$
$863$	−13.4842	−0.459009	−0.229504	+	0.973308i	$0.573710\pi$
$864$	1.00000	0.0340207
$865$	28.1687	0.957766
$866$	10.2961	0.349875
$867$	−40.3939	−1.37185
$868$	−13.8814	−0.471164
$869$	0.647693	0.0219715
$870$	2.25808	0.0765561
$871$	−0.588603	−0.0199440
$872$	18.5798	0.629190
$873$	−14.7491	−0.499182
$874$	−7.57588	−0.256258
$875$	14.9959	0.506954
$876$	−10.5553	−0.356629
$877$	18.2360	0.615785	0.307892	−	0.951421i	$-0.400376\pi$
$877$	18.2360	0.615785	0.307892	+	0.951421i	$0.400376\pi$
$878$	19.8410	0.669600
$879$	−16.4457	−0.554701
$880$	−2.59147	−0.0873586
$881$	26.5759	0.895365	0.447683	−	0.894193i	$-0.352249\pi$
$881$	26.5759	0.895365	0.447683	+	0.894193i	$0.352249\pi$
$882$	5.16395	0.173879
$883$	−27.5865	−0.928360	−0.464180	−	0.885741i	$-0.653651\pi$
$883$	−27.5865	−0.928360	−0.464180	+	0.885741i	$0.653651\pi$
$884$	0.836636	0.0281391
$885$	26.7265	0.898402
$886$	22.5283	0.756852
$887$	55.9788	1.87958	0.939792	−	0.341747i	$-0.111019\pi$
$887$	55.9788	1.87958	0.939792	+	0.341747i	$0.111019\pi$
$888$	11.0433	0.370589
$889$	15.4795	0.519167
$890$	−1.22085	−0.0409231
$891$	1.14765	0.0384476
$892$	−19.4445	−0.651051
$893$	−35.3945	−1.18443
$894$	−4.67930	−0.156499
$895$	−0.455759	−0.0152343
$896$	−1.35501	−0.0452677
$897$	0.110434	0.00368729
$898$	−30.3746	−1.01361
$899$	10.2445	0.341672
$900$	0.0989234	0.00329745
$901$	81.4341	2.71296
$902$	6.97829	0.232352
$903$	10.7995	0.359386
$904$	3.63820	0.121005
$905$	−46.2122	−1.53614
$906$	−6.30149	−0.209353
$907$	−32.2407	−1.07053	−0.535267	−	0.844683i	$-0.679789\pi$
$907$	−32.2407	−1.07053	−0.535267	+	0.844683i	$0.679789\pi$
$908$	16.5696	0.549881
$909$	8.08673	0.268220
$910$	0.337897	0.0112012
$911$	28.9301	0.958496	0.479248	−	0.877679i	$-0.340909\pi$
$911$	28.9301	0.958496	0.479248	+	0.877679i	$0.340909\pi$
$912$	7.57588	0.250862
$913$	−14.0672	−0.465557
$914$	20.9017	0.691366
$915$	33.0921	1.09399
$916$	10.2304	0.338021
$917$	−8.11432	−0.267959
$918$	7.57588	0.250041
$919$	8.51557	0.280903	0.140451	−	0.990088i	$-0.455145\pi$
$919$	8.51557	0.280903	0.140451	+	0.990088i	$0.455145\pi$
$920$	−2.25808	−0.0744467
$921$	14.6869	0.483950
$922$	−16.6481	−0.548275
$923$	−0.176062	−0.00579517
$924$	−1.55507	−0.0511580
$925$	1.09244	0.0359193
$926$	27.3847	0.899918
$927$	−9.15385	−0.300652
$928$	1.00000	0.0328266
$929$	15.0772	0.494668	0.247334	−	0.968930i	$-0.420446\pi$
$929$	15.0772	0.494668	0.247334	+	0.968930i	$0.420446\pi$
$930$	23.1328	0.758556
$931$	39.1215	1.28215
$932$	−5.90919	−0.193562
$933$	31.6135	1.03498
$934$	9.57035	0.313151
$935$	−19.6327	−0.642058
$936$	−0.110434	−0.00360965
$937$	13.8465	0.452345	0.226172	−	0.974087i	$-0.427379\pi$
$937$	13.8465	0.452345	0.226172	+	0.974087i	$0.427379\pi$
$938$	7.22206	0.235808
$939$	−3.62659	−0.118349
$940$	−10.5497	−0.344095
$941$	21.6604	0.706109	0.353055	−	0.935603i	$-0.385143\pi$
$941$	21.6604	0.706109	0.353055	+	0.935603i	$0.385143\pi$
$942$	12.1563	0.396075
$943$	6.08053	0.198009
$944$	11.8360	0.385227
$945$	3.05972	0.0995326
$946$	9.14684	0.297389
$947$	−32.3477	−1.05116	−0.525580	−	0.850744i	$-0.676152\pi$
$947$	−32.3477	−1.05116	−0.525580	+	0.850744i	$0.676152\pi$
$948$	−0.564367	−0.0183298
$949$	1.16566	0.0378390
$950$	0.749431	0.0243148
$951$	0.492551	0.0159721
$952$	−10.2654	−0.332703
$953$	39.7210	1.28669	0.643345	−	0.765576i	$-0.277546\pi$
$953$	39.7210	1.28669	0.643345	+	0.765576i	$0.277546\pi$
$954$	−10.7491	−0.348016
$955$	22.5503	0.729711
$956$	9.98382	0.322900
$957$	1.14765	0.0370981
$958$	−4.62400	−0.149395
$959$	−17.6921	−0.571308
$960$	2.25808	0.0728792
$961$	73.9493	2.38546
$962$	−1.21956	−0.0393202
$963$	18.5575	0.598006
$964$	1.92188	0.0618997
$965$	39.7882	1.28083
$966$	−1.35501	−0.0435967
$967$	12.7415	0.409739	0.204870	−	0.978789i	$-0.434323\pi$
$967$	12.7415	0.409739	0.204870	+	0.978789i	$0.434323\pi$
$968$	9.68291	0.311221
$969$	57.3939	1.84376
$970$	−33.3047	−1.06935
$971$	−46.8168	−1.50242	−0.751212	−	0.660061i	$-0.770530\pi$
$971$	−46.8168	−1.50242	−0.751212	+	0.660061i	$0.770530\pi$
$972$	−1.00000	−0.0320750
$973$	14.4643	0.463705
$974$	13.3419	0.427503
$975$	−0.0109245	−0.000349865 0
$976$	14.6550	0.469095
$977$	−46.5547	−1.48942	−0.744709	−	0.667389i	$-0.767412\pi$
$977$	−46.5547	−1.48942	−0.744709	+	0.667389i	$0.767412\pi$
$978$	−18.7471	−0.599467
$979$	−0.620486	−0.0198308
$980$	11.6606	0.372485
$981$	−18.5798	−0.593206
$982$	−23.0794	−0.736492
$983$	24.1215	0.769357	0.384679	−	0.923051i	$-0.374312\pi$
$983$	24.1215	0.769357	0.384679	+	0.923051i	$0.374312\pi$
$984$	−6.08053	−0.193840
$985$	59.1424	1.88443
$986$	7.57588	0.241265
$987$	−6.33060	−0.201505
$988$	−0.836636	−0.0266169
$989$	7.97009	0.253434
$990$	2.59147	0.0823625
$991$	44.8597	1.42501	0.712507	−	0.701665i	$-0.247560\pi$
$991$	44.8597	1.42501	0.712507	+	0.701665i	$0.247560\pi$
$992$	10.2445	0.325262
$993$	4.70382	0.149271
$994$	2.16026	0.0685192
$995$	31.2928	0.992049
$996$	12.2575	0.388393
$997$	16.4425	0.520740	0.260370	−	0.965509i	$-0.416155\pi$
$997$	16.4425	0.520740	0.260370	+	0.965509i	$0.416155\pi$
$998$	−25.5016	−0.807238
$999$	−11.0433	−0.349395

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.be.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.be.1.2	✓	5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.25808 q^{5} +1.00000 q^{6} +1.35501 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.25808 q^{5} +1.00000 q^{6} +1.35501 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.25808 q^{10} +1.14765 q^{11} -1.00000 q^{12} +0.110434 q^{13} -1.35501 q^{14} +2.25808 q^{15} +1.00000 q^{16} +7.57588 q^{17} -1.00000 q^{18} -7.57588 q^{19} -2.25808 q^{20} -1.35501 q^{21} -1.14765 q^{22} -1.00000 q^{23} +1.00000 q^{24} +0.0989234 q^{25} -0.110434 q^{26} -1.00000 q^{27} +1.35501 q^{28} -1.00000 q^{29} -2.25808 q^{30} -10.2445 q^{31} -1.00000 q^{32} -1.14765 q^{33} -7.57588 q^{34} -3.05972 q^{35} +1.00000 q^{36} +11.0433 q^{37} +7.57588 q^{38} -0.110434 q^{39} +2.25808 q^{40} -6.08053 q^{41} +1.35501 q^{42} -7.97009 q^{43} +1.14765 q^{44} -2.25808 q^{45} +1.00000 q^{46} +4.67200 q^{47} -1.00000 q^{48} -5.16395 q^{49} -0.0989234 q^{50} -7.57588 q^{51} +0.110434 q^{52} +10.7491 q^{53} +1.00000 q^{54} -2.59147 q^{55} -1.35501 q^{56} +7.57588 q^{57} +1.00000 q^{58} +11.8360 q^{59} +2.25808 q^{60} +14.6550 q^{61} +10.2445 q^{62} +1.35501 q^{63} +1.00000 q^{64} -0.249369 q^{65} +1.14765 q^{66} -5.32989 q^{67} +7.57588 q^{68} +1.00000 q^{69} +3.05972 q^{70} -1.59428 q^{71} -1.00000 q^{72} +10.5553 q^{73} -11.0433 q^{74} -0.0989234 q^{75} -7.57588 q^{76} +1.55507 q^{77} +0.110434 q^{78} +0.564367 q^{79} -2.25808 q^{80} +1.00000 q^{81} +6.08053 q^{82} -12.2575 q^{83} -1.35501 q^{84} -17.1069 q^{85} +7.97009 q^{86} +1.00000 q^{87} -1.14765 q^{88} -0.540660 q^{89} +2.25808 q^{90} +0.149639 q^{91} -1.00000 q^{92} +10.2445 q^{93} -4.67200 q^{94} +17.1069 q^{95} +1.00000 q^{96} -14.7491 q^{97} +5.16395 q^{98} +1.14765 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} - 4 q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 - 4 * q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} + 5 q^{6} - 4 q^{7} - 5 q^{8} + 5 q^{9} - q^{10} + 5 q^{11} - 5 q^{12} - 11 q^{13} + 4 q^{14} - q^{15} + 5 q^{16} + 4 q^{17} - 5 q^{18} - 4 q^{19} + q^{20} + 4 q^{21} - 5 q^{22} - 5 q^{23} + 5 q^{24} + 12 q^{25} + 11 q^{26} - 5 q^{27} - 4 q^{28} - 5 q^{29} + q^{30} + 5 q^{31} - 5 q^{32} - 5 q^{33} - 4 q^{34} - 6 q^{35} + 5 q^{36} + 9 q^{37} + 4 q^{38} + 11 q^{39} - q^{40} + 5 q^{41} - 4 q^{42} - 16 q^{43} + 5 q^{44} + q^{45} + 5 q^{46} + 8 q^{47} - 5 q^{48} - 5 q^{49} - 12 q^{50} - 4 q^{51} - 11 q^{52} - 4 q^{53} + 5 q^{54} - 33 q^{55} + 4 q^{56} + 4 q^{57} + 5 q^{58} + 23 q^{59} - q^{60} + 7 q^{61} - 5 q^{62} - 4 q^{63} + 5 q^{64} - 5 q^{65} + 5 q^{66} + 5 q^{67} + 4 q^{68} + 5 q^{69} + 6 q^{70} - 21 q^{71} - 5 q^{72} - 8 q^{73} - 9 q^{74} - 12 q^{75} - 4 q^{76} - 11 q^{78} - 8 q^{79} + q^{80} + 5 q^{81} - 5 q^{82} - 18 q^{83} + 4 q^{84} - 10 q^{85} + 16 q^{86} + 5 q^{87} - 5 q^{88} + 32 q^{89} - q^{90} + 10 q^{91} - 5 q^{92} - 5 q^{93} - 8 q^{94} + 10 q^{95} + 5 q^{96} - 16 q^{97} + 5 q^{98} + 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 + 5 * q^6 - 4 * q^7 - 5 * q^8 + 5 * q^9 - q^10 + 5 * q^11 - 5 * q^12 - 11 * q^13 + 4 * q^14 - q^15 + 5 * q^16 + 4 * q^17 - 5 * q^18 - 4 * q^19 + q^20 + 4 * q^21 - 5 * q^22 - 5 * q^23 + 5 * q^24 + 12 * q^25 + 11 * q^26 - 5 * q^27 - 4 * q^28 - 5 * q^29 + q^30 + 5 * q^31 - 5 * q^32 - 5 * q^33 - 4 * q^34 - 6 * q^35 + 5 * q^36 + 9 * q^37 + 4 * q^38 + 11 * q^39 - q^40 + 5 * q^41 - 4 * q^42 - 16 * q^43 + 5 * q^44 + q^45 + 5 * q^46 + 8 * q^47 - 5 * q^48 - 5 * q^49 - 12 * q^50 - 4 * q^51 - 11 * q^52 - 4 * q^53 + 5 * q^54 - 33 * q^55 + 4 * q^56 + 4 * q^57 + 5 * q^58 + 23 * q^59 - q^60 + 7 * q^61 - 5 * q^62 - 4 * q^63 + 5 * q^64 - 5 * q^65 + 5 * q^66 + 5 * q^67 + 4 * q^68 + 5 * q^69 + 6 * q^70 - 21 * q^71 - 5 * q^72 - 8 * q^73 - 9 * q^74 - 12 * q^75 - 4 * q^76 - 11 * q^78 - 8 * q^79 + q^80 + 5 * q^81 - 5 * q^82 - 18 * q^83 + 4 * q^84 - 10 * q^85 + 16 * q^86 + 5 * q^87 - 5 * q^88 + 32 * q^89 - q^90 + 10 * q^91 - 5 * q^92 - 5 * q^93 - 8 * q^94 + 10 * q^95 + 5 * q^96 - 16 * q^97 + 5 * q^98 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more