LMFDB - Embedded newform 8007.2.a.h.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30899 q^{2} +1.00000 q^{3} -0.286535 q^{4} +2.04407 q^{5} -1.30899 q^{6} -3.68915 q^{7} +2.99306 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.30899 q^{2} +1.00000 q^{3} -0.286535 q^{4} +2.04407 q^{5} -1.30899 q^{6} -3.68915 q^{7} +2.99306 q^{8} +1.00000 q^{9} -2.67568 q^{10} +3.09092 q^{11} -0.286535 q^{12} +4.30580 q^{13} +4.82908 q^{14} +2.04407 q^{15} -3.34483 q^{16} -1.00000 q^{17} -1.30899 q^{18} +4.59778 q^{19} -0.585698 q^{20} -3.68915 q^{21} -4.04600 q^{22} -2.49224 q^{23} +2.99306 q^{24} -0.821759 q^{25} -5.63627 q^{26} +1.00000 q^{27} +1.05707 q^{28} +3.86562 q^{29} -2.67568 q^{30} -6.56450 q^{31} -1.60776 q^{32} +3.09092 q^{33} +1.30899 q^{34} -7.54091 q^{35} -0.286535 q^{36} -2.44395 q^{37} -6.01846 q^{38} +4.30580 q^{39} +6.11804 q^{40} +8.37766 q^{41} +4.82908 q^{42} +5.52134 q^{43} -0.885656 q^{44} +2.04407 q^{45} +3.26233 q^{46} +1.30939 q^{47} -3.34483 q^{48} +6.60986 q^{49} +1.07568 q^{50} -1.00000 q^{51} -1.23376 q^{52} -2.61847 q^{53} -1.30899 q^{54} +6.31808 q^{55} -11.0419 q^{56} +4.59778 q^{57} -5.06007 q^{58} -4.88771 q^{59} -0.585698 q^{60} +4.11653 q^{61} +8.59289 q^{62} -3.68915 q^{63} +8.79420 q^{64} +8.80138 q^{65} -4.04600 q^{66} +11.7210 q^{67} +0.286535 q^{68} -2.49224 q^{69} +9.87100 q^{70} +1.92049 q^{71} +2.99306 q^{72} +5.38262 q^{73} +3.19911 q^{74} -0.821759 q^{75} -1.31742 q^{76} -11.4029 q^{77} -5.63627 q^{78} +6.42663 q^{79} -6.83708 q^{80} +1.00000 q^{81} -10.9663 q^{82} +5.83721 q^{83} +1.05707 q^{84} -2.04407 q^{85} -7.22740 q^{86} +3.86562 q^{87} +9.25132 q^{88} +0.588751 q^{89} -2.67568 q^{90} -15.8848 q^{91} +0.714114 q^{92} -6.56450 q^{93} -1.71398 q^{94} +9.39820 q^{95} -1.60776 q^{96} +3.77663 q^{97} -8.65226 q^{98} +3.09092 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 7 q^{2} + 56 q^{3} + 61 q^{4} + 17 q^{5} + 7 q^{6} + 5 q^{7} + 18 q^{8} + 56 q^{9}+O(q^{10}) $ 56 * q + 7 * q^2 + 56 * q^3 + 61 * q^4 + 17 * q^5 + 7 * q^6 + 5 * q^7 + 18 * q^8 + 56 * q^9	\( 56 q + 7 q^{2} + 56 q^{3} + 61 q^{4} + 17 q^{5} + 7 q^{6} + 5 q^{7} + 18 q^{8} + 56 q^{9} - 2 q^{10} + 35 q^{11} + 61 q^{12} + 8 q^{13} + 36 q^{14} + 17 q^{15} + 71 q^{16} - 56 q^{17} + 7 q^{18} - 2 q^{19} + 58 q^{20} + 5 q^{21} + 27 q^{22} + 40 q^{23} + 18 q^{24} + 85 q^{25} + 15 q^{26} + 56 q^{27} - 4 q^{28} + 41 q^{29} - 2 q^{30} + q^{31} + 43 q^{32} + 35 q^{33} - 7 q^{34} + 57 q^{35} + 61 q^{36} + 34 q^{37} + 52 q^{38} + 8 q^{39} + 14 q^{40} + 49 q^{41} + 36 q^{42} + 27 q^{43} + 66 q^{44} + 17 q^{45} + 10 q^{46} + 43 q^{47} + 71 q^{48} + 51 q^{49} + 30 q^{50} - 56 q^{51} - 7 q^{52} + 73 q^{53} + 7 q^{54} + 15 q^{55} + 118 q^{56} - 2 q^{57} - q^{58} + 53 q^{59} + 58 q^{60} + 15 q^{61} + 16 q^{62} + 5 q^{63} + 124 q^{64} + 107 q^{65} + 27 q^{66} + 20 q^{67} - 61 q^{68} + 40 q^{69} + 16 q^{70} + 56 q^{71} + 18 q^{72} + 49 q^{73} + 28 q^{74} + 85 q^{75} - 38 q^{76} + 50 q^{77} + 15 q^{78} - 4 q^{79} + 74 q^{80} + 56 q^{81} + 59 q^{82} + 35 q^{83} - 4 q^{84} - 17 q^{85} + 38 q^{86} + 41 q^{87} + 64 q^{88} + 66 q^{89} - 2 q^{90} + 5 q^{91} + 96 q^{92} + q^{93} - 12 q^{94} + 70 q^{95} + 43 q^{96} + 60 q^{97} + 26 q^{98} + 35 q^{99}+O(q^{100}) \) 56 * q + 7 * q^2 + 56 * q^3 + 61 * q^4 + 17 * q^5 + 7 * q^6 + 5 * q^7 + 18 * q^8 + 56 * q^9 - 2 * q^10 + 35 * q^11 + 61 * q^12 + 8 * q^13 + 36 * q^14 + 17 * q^15 + 71 * q^16 - 56 * q^17 + 7 * q^18 - 2 * q^19 + 58 * q^20 + 5 * q^21 + 27 * q^22 + 40 * q^23 + 18 * q^24 + 85 * q^25 + 15 * q^26 + 56 * q^27 - 4 * q^28 + 41 * q^29 - 2 * q^30 + q^31 + 43 * q^32 + 35 * q^33 - 7 * q^34 + 57 * q^35 + 61 * q^36 + 34 * q^37 + 52 * q^38 + 8 * q^39 + 14 * q^40 + 49 * q^41 + 36 * q^42 + 27 * q^43 + 66 * q^44 + 17 * q^45 + 10 * q^46 + 43 * q^47 + 71 * q^48 + 51 * q^49 + 30 * q^50 - 56 * q^51 - 7 * q^52 + 73 * q^53 + 7 * q^54 + 15 * q^55 + 118 * q^56 - 2 * q^57 - q^58 + 53 * q^59 + 58 * q^60 + 15 * q^61 + 16 * q^62 + 5 * q^63 + 124 * q^64 + 107 * q^65 + 27 * q^66 + 20 * q^67 - 61 * q^68 + 40 * q^69 + 16 * q^70 + 56 * q^71 + 18 * q^72 + 49 * q^73 + 28 * q^74 + 85 * q^75 - 38 * q^76 + 50 * q^77 + 15 * q^78 - 4 * q^79 + 74 * q^80 + 56 * q^81 + 59 * q^82 + 35 * q^83 - 4 * q^84 - 17 * q^85 + 38 * q^86 + 41 * q^87 + 64 * q^88 + 66 * q^89 - 2 * q^90 + 5 * q^91 + 96 * q^92 + q^93 - 12 * q^94 + 70 * q^95 + 43 * q^96 + 60 * q^97 + 26 * q^98 + 35 * 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 $

$2$	−1.30899	−0.925599	−0.462799	−	0.886463i	$-0.653155\pi$
$2$	−1.30899	−0.925599	−0.462799	+	0.886463i	$0.653155\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.286535	−0.143267
$5$	2.04407	0.914138	0.457069	−	0.889431i	$-0.348899\pi$
$5$	2.04407	0.914138	0.457069	+	0.889431i	$0.348899\pi$
$6$	−1.30899	−0.534395
$7$	−3.68915	−1.39437	−0.697185	−	0.716892i	$-0.745564\pi$
$7$	−3.68915	−1.39437	−0.697185	+	0.716892i	$0.745564\pi$
$8$	2.99306	1.05821
$9$	1.00000	0.333333
$10$	−2.67568	−0.846125
$11$	3.09092	0.931948	0.465974	−	0.884798i	$-0.345704\pi$
$11$	3.09092	0.931948	0.465974	+	0.884798i	$0.345704\pi$
$12$	−0.286535	−0.0827154
$13$	4.30580	1.19421	0.597107	−	0.802162i	$-0.296317\pi$
$13$	4.30580	1.19421	0.597107	+	0.802162i	$0.296317\pi$
$14$	4.82908	1.29063
$15$	2.04407	0.527778
$16$	−3.34483	−0.836207
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.30899	−0.308533
$19$	4.59778	1.05480	0.527401	−	0.849616i	$-0.323167\pi$
$19$	4.59778	1.05480	0.527401	+	0.849616i	$0.323167\pi$
$20$	−0.585698	−0.130966
$21$	−3.68915	−0.805039
$22$	−4.04600	−0.862610
$23$	−2.49224	−0.519669	−0.259834	−	0.965653i	$-0.583668\pi$
$23$	−2.49224	−0.519669	−0.259834	+	0.965653i	$0.583668\pi$
$24$	2.99306	0.610956
$25$	−0.821759	−0.164352
$26$	−5.63627	−1.10536
$27$	1.00000	0.192450
$28$	1.05707	0.199767
$29$	3.86562	0.717827	0.358913	−	0.933371i	$-0.383147\pi$
$29$	3.86562	0.717827	0.358913	+	0.933371i	$0.383147\pi$
$30$	−2.67568	−0.488510
$31$	−6.56450	−1.17902	−0.589509	−	0.807761i	$-0.700679\pi$
$31$	−6.56450	−1.17902	−0.589509	+	0.807761i	$0.700679\pi$
$32$	−1.60776	−0.284214
$33$	3.09092	0.538061
$34$	1.30899	0.224491
$35$	−7.54091	−1.27465
$36$	−0.286535	−0.0477558
$37$	−2.44395	−0.401782	−0.200891	−	0.979614i	$-0.564384\pi$
$37$	−2.44395	−0.401782	−0.200891	+	0.979614i	$0.564384\pi$
$38$	−6.01846	−0.976323
$39$	4.30580	0.689480
$40$	6.11804	0.967347
$41$	8.37766	1.30837	0.654185	−	0.756334i	$-0.273012\pi$
$41$	8.37766	1.30837	0.654185	+	0.756334i	$0.273012\pi$
$42$	4.82908	0.745143
$43$	5.52134	0.841997	0.420998	−	0.907061i	$-0.361680\pi$
$43$	5.52134	0.841997	0.420998	+	0.907061i	$0.361680\pi$
$44$	−0.885656	−0.133518
$45$	2.04407	0.304713
$46$	3.26233	0.481005
$47$	1.30939	0.190994	0.0954971	−	0.995430i	$-0.469556\pi$
$47$	1.30939	0.190994	0.0954971	+	0.995430i	$0.469556\pi$
$48$	−3.34483	−0.482784
$49$	6.60986	0.944265
$50$	1.07568	0.152124
$51$	−1.00000	−0.140028
$52$	−1.23376	−0.171092
$53$	−2.61847	−0.359675	−0.179837	−	0.983696i	$-0.557557\pi$
$53$	−2.61847	−0.359675	−0.179837	+	0.983696i	$0.557557\pi$
$54$	−1.30899	−0.178132
$55$	6.31808	0.851929
$56$	−11.0419	−1.47553
$57$	4.59778	0.608990
$58$	−5.06007	−0.664419
$59$	−4.88771	−0.636326	−0.318163	−	0.948036i	$-0.603066\pi$
$59$	−4.88771	−0.636326	−0.318163	+	0.948036i	$0.603066\pi$
$60$	−0.585698	−0.0756133
$61$	4.11653	0.527068	0.263534	−	0.964650i	$-0.415112\pi$
$61$	4.11653	0.527068	0.263534	+	0.964650i	$0.415112\pi$
$62$	8.59289	1.09130
$63$	−3.68915	−0.464790
$64$	8.79420	1.09928
$65$	8.80138	1.09168
$66$	−4.04600	−0.498028
$67$	11.7210	1.43195	0.715973	−	0.698128i	$-0.245983\pi$
$67$	11.7210	1.43195	0.715973	+	0.698128i	$0.245983\pi$
$68$	0.286535	0.0347474
$69$	−2.49224	−0.300031
$70$	9.87100	1.17981
$71$	1.92049	0.227920	0.113960	−	0.993485i	$-0.463646\pi$
$71$	1.92049	0.227920	0.113960	+	0.993485i	$0.463646\pi$
$72$	2.99306	0.352736
$73$	5.38262	0.629988	0.314994	−	0.949094i	$-0.397998\pi$
$73$	5.38262	0.629988	0.314994	+	0.949094i	$0.397998\pi$
$74$	3.19911	0.371889
$75$	−0.821759	−0.0948885
$76$	−1.31742	−0.151119
$77$	−11.4029	−1.29948
$78$	−5.63627	−0.638181
$79$	6.42663	0.723052	0.361526	−	0.932362i	$-0.382256\pi$
$79$	6.42663	0.723052	0.361526	+	0.932362i	$0.382256\pi$
$80$	−6.83708	−0.764409
$81$	1.00000	0.111111
$82$	−10.9663	−1.21103
$83$	5.83721	0.640717	0.320358	−	0.947296i	$-0.396197\pi$
$83$	5.83721	0.640717	0.320358	+	0.947296i	$0.396197\pi$
$84$	1.05707	0.115336
$85$	−2.04407	−0.221711
$86$	−7.22740	−0.779351
$87$	3.86562	0.414437
$88$	9.25132	0.986194
$89$	0.588751	0.0624075	0.0312038	−	0.999513i	$-0.490066\pi$
$89$	0.588751	0.0624075	0.0312038	+	0.999513i	$0.490066\pi$
$90$	−2.67568	−0.282042
$91$	−15.8848	−1.66518
$92$	0.714114	0.0744515
$93$	−6.56450	−0.680707
$94$	−1.71398	−0.176784
$95$	9.39820	0.964235
$96$	−1.60776	−0.164091
$97$	3.77663	0.383459	0.191730	−	0.981448i	$-0.438590\pi$
$97$	3.77663	0.383459	0.191730	+	0.981448i	$0.438590\pi$
$98$	−8.65226	−0.874011
$99$	3.09092	0.310649
$100$	0.235462	0.0235462
$101$	−7.39257	−0.735588	−0.367794	−	0.929907i	$-0.619887\pi$
$101$	−7.39257	−0.735588	−0.367794	+	0.929907i	$0.619887\pi$
$102$	1.30899	0.129610
$103$	13.4017	1.32051	0.660254	−	0.751042i	$-0.270449\pi$
$103$	13.4017	1.32051	0.660254	+	0.751042i	$0.270449\pi$
$104$	12.8875	1.26373
$105$	−7.54091	−0.735917
$106$	3.42757	0.332915
$107$	4.46311	0.431465	0.215732	−	0.976453i	$-0.430786\pi$
$107$	4.46311	0.431465	0.215732	+	0.976453i	$0.430786\pi$
$108$	−0.286535	−0.0275718
$109$	16.1688	1.54869	0.774347	−	0.632762i	$-0.218079\pi$
$109$	16.1688	1.54869	0.774347	+	0.632762i	$0.218079\pi$
$110$	−8.27032	−0.788545
$111$	−2.44395	−0.231969
$112$	12.3396	1.16598
$113$	−16.2616	−1.52976	−0.764881	−	0.644171i	$-0.777202\pi$
$113$	−16.2616	−1.52976	−0.764881	+	0.644171i	$0.777202\pi$
$114$	−6.01846	−0.563681
$115$	−5.09433	−0.475049
$116$	−1.10763	−0.102841
$117$	4.30580	0.398071
$118$	6.39799	0.588983
$119$	3.68915	0.338184
$120$	6.11804	0.558498
$121$	−1.44620	−0.131472
$122$	−5.38852	−0.487853
$123$	8.37766	0.755388
$124$	1.88096	0.168915
$125$	−11.9001	−1.06438
$126$	4.82908	0.430209
$127$	−1.82699	−0.162119	−0.0810595	−	0.996709i	$-0.525830\pi$
$127$	−1.82699	−0.162119	−0.0810595	+	0.996709i	$0.525830\pi$
$128$	−8.29604	−0.733274
$129$	5.52134	0.486127
$130$	−11.5209	−1.01045
$131$	−5.35907	−0.468224	−0.234112	−	0.972210i	$-0.575218\pi$
$131$	−5.35907	−0.468224	−0.234112	+	0.972210i	$0.575218\pi$
$132$	−0.885656	−0.0770865
$133$	−16.9619	−1.47078
$134$	−15.3427	−1.32541
$135$	2.04407	0.175926
$136$	−2.99306	−0.256653
$137$	−12.3630	−1.05625	−0.528123	−	0.849168i	$-0.677104\pi$
$137$	−12.3630	−1.05625	−0.528123	+	0.849168i	$0.677104\pi$
$138$	3.26233	0.277708
$139$	−8.10455	−0.687419	−0.343709	−	0.939076i	$-0.611683\pi$
$139$	−8.10455	−0.687419	−0.343709	+	0.939076i	$0.611683\pi$
$140$	2.16073	0.182615
$141$	1.30939	0.110271
$142$	−2.51391	−0.210963
$143$	13.3089	1.11295
$144$	−3.34483	−0.278736
$145$	7.90161	0.656193
$146$	−7.04582	−0.583116
$147$	6.60986	0.545172
$148$	0.700275	0.0575623
$149$	−6.04787	−0.495461	−0.247731	−	0.968829i	$-0.579685\pi$
$149$	−6.04787	−0.495461	−0.247731	+	0.968829i	$0.579685\pi$
$150$	1.07568	0.0878287
$151$	−20.9686	−1.70640	−0.853201	−	0.521582i	$-0.825342\pi$
$151$	−20.9686	−1.70640	−0.853201	+	0.521582i	$0.825342\pi$
$152$	13.7614	1.11620
$153$	−1.00000	−0.0808452
$154$	14.9263	1.20280
$155$	−13.4183	−1.07779
$156$	−1.23376	−0.0987799
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−8.41242	−0.669256
$159$	−2.61847	−0.207658
$160$	−3.28638	−0.259811
$161$	9.19427	0.724610
$162$	−1.30899	−0.102844
$163$	−20.1229	−1.57615	−0.788073	−	0.615582i	$-0.788921\pi$
$163$	−20.1229	−1.57615	−0.788073	+	0.615582i	$0.788921\pi$
$164$	−2.40049	−0.187447
$165$	6.31808	0.491862
$166$	−7.64087	−0.593047
$167$	7.68807	0.594921	0.297460	−	0.954734i	$-0.403860\pi$
$167$	7.68807	0.594921	0.297460	+	0.954734i	$0.403860\pi$
$168$	−11.0419	−0.851898
$169$	5.53991	0.426147
$170$	2.67568	0.205215
$171$	4.59778	0.351601
$172$	−1.58205	−0.120631
$173$	4.78755	0.363991	0.181995	−	0.983299i	$-0.441744\pi$
$173$	4.78755	0.363991	0.181995	+	0.983299i	$0.441744\pi$
$174$	−5.06007	−0.383603
$175$	3.03160	0.229167
$176$	−10.3386	−0.779302
$177$	−4.88771	−0.367383
$178$	−0.770672	−0.0577643
$179$	−5.25320	−0.392643	−0.196321	−	0.980540i	$-0.562900\pi$
$179$	−5.25320	−0.392643	−0.196321	+	0.980540i	$0.562900\pi$
$180$	−0.585698	−0.0436553
$181$	3.80371	0.282727	0.141364	−	0.989958i	$-0.454851\pi$
$181$	3.80371	0.282727	0.141364	+	0.989958i	$0.454851\pi$
$182$	20.7931	1.54128
$183$	4.11653	0.304303
$184$	−7.45944	−0.549917
$185$	−4.99561	−0.367284
$186$	8.59289	0.630061
$187$	−3.09092	−0.226031
$188$	−0.375186	−0.0273632
$189$	−3.68915	−0.268346
$190$	−12.3022	−0.892494
$191$	7.28846	0.527374	0.263687	−	0.964608i	$-0.415061\pi$
$191$	7.28846	0.527374	0.263687	+	0.964608i	$0.415061\pi$
$192$	8.79420	0.634667
$193$	19.9531	1.43625	0.718126	−	0.695913i	$-0.245000\pi$
$193$	19.9531	1.43625	0.718126	+	0.695913i	$0.245000\pi$
$194$	−4.94359	−0.354929
$195$	8.80138	0.630280
$196$	−1.89395	−0.135282
$197$	−6.34426	−0.452010	−0.226005	−	0.974126i	$-0.572566\pi$
$197$	−6.34426	−0.452010	−0.226005	+	0.974126i	$0.572566\pi$
$198$	−4.04600	−0.287537
$199$	−2.78841	−0.197665	−0.0988327	−	0.995104i	$-0.531511\pi$
$199$	−2.78841	−0.197665	−0.0988327	+	0.995104i	$0.531511\pi$
$200$	−2.45957	−0.173918
$201$	11.7210	0.826735
$202$	9.67683	0.680859
$203$	−14.2608	−1.00092
$204$	0.286535	0.0200614
$205$	17.1246	1.19603
$206$	−17.5427	−1.22226
$207$	−2.49224	−0.173223
$208$	−14.4022	−0.998610
$209$	14.2114	0.983021
$210$	9.87100	0.681164
$211$	−10.8859	−0.749415	−0.374708	−	0.927143i	$-0.622257\pi$
$211$	−10.8859	−0.749415	−0.374708	+	0.927143i	$0.622257\pi$
$212$	0.750283	0.0515296
$213$	1.92049	0.131590
$214$	−5.84218	−0.399363
$215$	11.2860	0.769701
$216$	2.99306	0.203652
$217$	24.2174	1.64399
$218$	−21.1649	−1.43347
$219$	5.38262	0.363724
$220$	−1.81035	−0.122054
$221$	−4.30580	−0.289639
$222$	3.19911	0.214710
$223$	−3.74454	−0.250753	−0.125376	−	0.992109i	$-0.540014\pi$
$223$	−3.74454	−0.250753	−0.125376	+	0.992109i	$0.540014\pi$
$224$	5.93127	0.396300
$225$	−0.821759	−0.0547839
$226$	21.2863	1.41595
$227$	−4.90108	−0.325296	−0.162648	−	0.986684i	$-0.552003\pi$
$227$	−4.90108	−0.325296	−0.162648	+	0.986684i	$0.552003\pi$
$228$	−1.31742	−0.0872484
$229$	16.8967	1.11657	0.558284	−	0.829650i	$-0.311460\pi$
$229$	16.8967	1.11657	0.558284	+	0.829650i	$0.311460\pi$
$230$	6.66845	0.439705
$231$	−11.4029	−0.750255
$232$	11.5700	0.759609
$233$	24.9830	1.63669	0.818347	−	0.574724i	$-0.194891\pi$
$233$	24.9830	1.63669	0.818347	+	0.574724i	$0.194891\pi$
$234$	−5.63627	−0.368454
$235$	2.67649	0.174595
$236$	1.40050	0.0911647
$237$	6.42663	0.417454
$238$	−4.82908	−0.313023
$239$	−8.07926	−0.522604	−0.261302	−	0.965257i	$-0.584152\pi$
$239$	−8.07926	−0.522604	−0.261302	+	0.965257i	$0.584152\pi$
$240$	−6.83708	−0.441332
$241$	−13.0622	−0.841412	−0.420706	−	0.907197i	$-0.638218\pi$
$241$	−13.0622	−0.841412	−0.420706	+	0.907197i	$0.638218\pi$
$242$	1.89306	0.121691
$243$	1.00000	0.0641500
$244$	−1.17953	−0.0755116
$245$	13.5110	0.863189
$246$	−10.9663	−0.699186
$247$	19.7971	1.25966
$248$	−19.6479	−1.24765
$249$	5.83721	0.369918
$250$	15.5772	0.985187
$251$	−12.1708	−0.768213	−0.384107	−	0.923289i	$-0.625491\pi$
$251$	−12.1708	−0.768213	−0.384107	+	0.923289i	$0.625491\pi$
$252$	1.05707	0.0665892
$253$	−7.70333	−0.484305
$254$	2.39152	0.150057
$255$	−2.04407	−0.128005
$256$	−6.72894	−0.420559
$257$	24.4142	1.52292	0.761459	−	0.648213i	$-0.224484\pi$
$257$	24.4142	1.52292	0.761459	+	0.648213i	$0.224484\pi$
$258$	−7.22740	−0.449958
$259$	9.01610	0.560233
$260$	−2.52190	−0.156401
$261$	3.86562	0.239276
$262$	7.01499	0.433387
$263$	14.9708	0.923138	0.461569	−	0.887104i	$-0.347287\pi$
$263$	14.9708	0.923138	0.461569	+	0.887104i	$0.347287\pi$
$264$	9.25132	0.569379
$265$	−5.35235	−0.328793
$266$	22.2030	1.36136
$267$	0.588751	0.0360310
$268$	−3.35847	−0.205151
$269$	7.92705	0.483321	0.241661	−	0.970361i	$-0.422308\pi$
$269$	7.92705	0.483321	0.241661	+	0.970361i	$0.422308\pi$
$270$	−2.67568	−0.162837
$271$	8.67981	0.527261	0.263630	−	0.964624i	$-0.415080\pi$
$271$	8.67981	0.527261	0.263630	+	0.964624i	$0.415080\pi$
$272$	3.34483	0.202810
$273$	−15.8848	−0.961389
$274$	16.1832	0.977660
$275$	−2.53999	−0.153167
$276$	0.714114	0.0429846
$277$	−5.73057	−0.344317	−0.172158	−	0.985069i	$-0.555074\pi$
$277$	−5.73057	−0.344317	−0.172158	+	0.985069i	$0.555074\pi$
$278$	10.6088	0.636274
$279$	−6.56450	−0.393006
$280$	−22.5704	−1.34884
$281$	26.2967	1.56873	0.784365	−	0.620300i	$-0.212989\pi$
$281$	26.2967	1.56873	0.784365	+	0.620300i	$0.212989\pi$
$282$	−1.71398	−0.102066
$283$	−8.14342	−0.484076	−0.242038	−	0.970267i	$-0.577816\pi$
$283$	−8.14342	−0.484076	−0.242038	+	0.970267i	$0.577816\pi$
$284$	−0.550287	−0.0326535
$285$	9.39820	0.556701
$286$	−17.4213	−1.03014
$287$	−30.9065	−1.82435
$288$	−1.60776	−0.0947381
$289$	1.00000	0.0588235
$290$	−10.3432	−0.607371
$291$	3.77663	0.221390
$292$	−1.54231	−0.0902566
$293$	27.7434	1.62078	0.810392	−	0.585888i	$-0.199254\pi$
$293$	27.7434	1.62078	0.810392	+	0.585888i	$0.199254\pi$
$294$	−8.65226	−0.504610
$295$	−9.99085	−0.581690
$296$	−7.31488	−0.425169
$297$	3.09092	0.179354
$298$	7.91663	0.458598
$299$	−10.7311	−0.620596
$300$	0.235462	0.0135944
$301$	−20.3691	−1.17405
$302$	27.4478	1.57944
$303$	−7.39257	−0.424692
$304$	−15.3788	−0.882033
$305$	8.41450	0.481813
$306$	1.30899	0.0748302
$307$	0.596712	0.0340562	0.0170281	−	0.999855i	$-0.494580\pi$
$307$	0.596712	0.0340562	0.0170281	+	0.999855i	$0.494580\pi$
$308$	3.26732	0.186173
$309$	13.4017	0.762396
$310$	17.5645	0.997597
$311$	17.2434	0.977785	0.488893	−	0.872344i	$-0.337401\pi$
$311$	17.2434	0.977785	0.488893	+	0.872344i	$0.337401\pi$
$312$	12.8875	0.729612
$313$	−10.2327	−0.578384	−0.289192	−	0.957271i	$-0.593387\pi$
$313$	−10.2327	−0.578384	−0.289192	+	0.957271i	$0.593387\pi$
$314$	1.30899	0.0738708
$315$	−7.54091	−0.424882
$316$	−1.84145	−0.103590
$317$	15.7484	0.884520	0.442260	−	0.896887i	$-0.354177\pi$
$317$	15.7484	0.884520	0.442260	+	0.896887i	$0.354177\pi$
$318$	3.42757	0.192208
$319$	11.9483	0.668977
$320$	17.9760	1.00489
$321$	4.46311	0.249106
$322$	−12.0352	−0.670698
$323$	−4.59778	−0.255827
$324$	−0.286535	−0.0159186
$325$	−3.53833	−0.196271
$326$	26.3407	1.45888
$327$	16.1688	0.894138
$328$	25.0748	1.38453
$329$	−4.83054	−0.266316
$330$	−8.27032	−0.455266
$331$	4.86737	0.267535	0.133767	−	0.991013i	$-0.457293\pi$
$331$	4.86737	0.267535	0.133767	+	0.991013i	$0.457293\pi$
$332$	−1.67256	−0.0917938
$333$	−2.44395	−0.133927
$334$	−10.0636	−0.550658
$335$	23.9586	1.30900
$336$	12.3396	0.673180
$337$	27.7367	1.51092	0.755458	−	0.655197i	$-0.227414\pi$
$337$	27.7367	1.51092	0.755458	+	0.655197i	$0.227414\pi$
$338$	−7.25171	−0.394441
$339$	−16.2616	−0.883209
$340$	0.585698	0.0317639
$341$	−20.2904	−1.09878
$342$	−6.01846	−0.325441
$343$	1.43930	0.0777147
$344$	16.5257	0.891006
$345$	−5.09433	−0.274270
$346$	−6.26688	−0.336909
$347$	34.2907	1.84082	0.920410	−	0.390956i	$-0.127855\pi$
$347$	34.2907	1.84082	0.920410	+	0.390956i	$0.127855\pi$
$348$	−1.10763	−0.0593753
$349$	2.38522	0.127678	0.0638391	−	0.997960i	$-0.479666\pi$
$349$	2.38522	0.127678	0.0638391	+	0.997960i	$0.479666\pi$
$350$	−3.96834	−0.212117
$351$	4.30580	0.229827
$352$	−4.96946	−0.264873
$353$	2.25712	0.120135	0.0600673	−	0.998194i	$-0.480868\pi$
$353$	2.25712	0.120135	0.0600673	+	0.998194i	$0.480868\pi$
$354$	6.39799	0.340049
$355$	3.92563	0.208351
$356$	−0.168698	−0.00894095
$357$	3.68915	0.195251
$358$	6.87641	0.363430
$359$	−9.77296	−0.515797	−0.257899	−	0.966172i	$-0.583030\pi$
$359$	−9.77296	−0.515797	−0.257899	+	0.966172i	$0.583030\pi$
$360$	6.11804	0.322449
$361$	2.13955	0.112608
$362$	−4.97903	−0.261692
$363$	−1.44620	−0.0759056
$364$	4.55153	0.238565
$365$	11.0025	0.575896
$366$	−5.38852	−0.281662
$367$	−28.8747	−1.50725	−0.753624	−	0.657305i	$-0.771696\pi$
$367$	−28.8747	−1.50725	−0.753624	+	0.657305i	$0.771696\pi$
$368$	8.33613	0.434551
$369$	8.37766	0.436124
$370$	6.53922	0.339958
$371$	9.65995	0.501520
$372$	1.88096	0.0975230
$373$	8.88774	0.460190	0.230095	−	0.973168i	$-0.426096\pi$
$373$	8.88774	0.460190	0.230095	+	0.973168i	$0.426096\pi$
$374$	4.04600	0.209214
$375$	−11.9001	−0.614519
$376$	3.91908	0.202111
$377$	16.6446	0.857239
$378$	4.82908	0.248381
$379$	16.3495	0.839817	0.419909	−	0.907566i	$-0.362062\pi$
$379$	16.3495	0.839817	0.419909	+	0.907566i	$0.362062\pi$
$380$	−2.69291	−0.138143
$381$	−1.82699	−0.0935994
$382$	−9.54055	−0.488137
$383$	13.0656	0.667620	0.333810	−	0.942640i	$-0.391666\pi$
$383$	13.0656	0.667620	0.333810	+	0.942640i	$0.391666\pi$
$384$	−8.29604	−0.423356
$385$	−23.3084	−1.18790
$386$	−26.1184	−1.32939
$387$	5.52134	0.280666
$388$	−1.08214	−0.0549371
$389$	−22.2340	−1.12731	−0.563654	−	0.826011i	$-0.690605\pi$
$389$	−22.2340	−1.12731	−0.563654	+	0.826011i	$0.690605\pi$
$390$	−11.5209	−0.583386
$391$	2.49224	0.126038
$392$	19.7837	0.999228
$393$	−5.35907	−0.270329
$394$	8.30459	0.418379
$395$	13.1365	0.660970
$396$	−0.885656	−0.0445059
$397$	14.1343	0.709379	0.354690	−	0.934984i	$-0.384587\pi$
$397$	14.1343	0.709379	0.354690	+	0.934984i	$0.384587\pi$
$398$	3.65002	0.182959
$399$	−16.9619	−0.849157
$400$	2.74864	0.137432
$401$	−7.19843	−0.359472	−0.179736	−	0.983715i	$-0.557524\pi$
$401$	−7.19843	−0.359472	−0.179736	+	0.983715i	$0.557524\pi$
$402$	−15.3427	−0.765224
$403$	−28.2654	−1.40800
$404$	2.11823	0.105386
$405$	2.04407	0.101571
$406$	18.6674	0.926446
$407$	−7.55405	−0.374440
$408$	−2.99306	−0.148179
$409$	17.5401	0.867304	0.433652	−	0.901081i	$-0.357225\pi$
$409$	17.5401	0.867304	0.433652	+	0.901081i	$0.357225\pi$
$410$	−22.4160	−1.10705
$411$	−12.3630	−0.609824
$412$	−3.84005	−0.189186
$413$	18.0315	0.887274
$414$	3.26233	0.160335
$415$	11.9317	0.585704
$416$	−6.92269	−0.339413
$417$	−8.10455	−0.396881
$418$	−18.6026	−0.909883
$419$	13.2065	0.645182	0.322591	−	0.946539i	$-0.395446\pi$
$419$	13.2065	0.645182	0.322591	+	0.946539i	$0.395446\pi$
$420$	2.16073	0.105433
$421$	3.92337	0.191213	0.0956067	−	0.995419i	$-0.469521\pi$
$421$	3.92337	0.191213	0.0956067	+	0.995419i	$0.469521\pi$
$422$	14.2496	0.693657
$423$	1.30939	0.0636647
$424$	−7.83725	−0.380610
$425$	0.821759	0.0398612
$426$	−2.51391	−0.121799
$427$	−15.1865	−0.734927
$428$	−1.27883	−0.0618148
$429$	13.3089	0.642559
$430$	−14.7734	−0.712434
$431$	−18.3422	−0.883515	−0.441757	−	0.897135i	$-0.645645\pi$
$431$	−18.3422	−0.883515	−0.441757	+	0.897135i	$0.645645\pi$
$432$	−3.34483	−0.160928
$433$	8.52648	0.409757	0.204878	−	0.978787i	$-0.434320\pi$
$433$	8.52648	0.409757	0.204878	+	0.978787i	$0.434320\pi$
$434$	−31.7005	−1.52167
$435$	7.90161	0.378853
$436$	−4.63293	−0.221877
$437$	−11.4588	−0.548148
$438$	−7.04582	−0.336662
$439$	4.99929	0.238603	0.119302	−	0.992858i	$-0.461934\pi$
$439$	4.99929	0.238603	0.119302	+	0.992858i	$0.461934\pi$
$440$	18.9104	0.901517
$441$	6.60986	0.314755
$442$	5.63627	0.268090
$443$	7.29511	0.346601	0.173301	−	0.984869i	$-0.444557\pi$
$443$	7.29511	0.346601	0.173301	+	0.984869i	$0.444557\pi$
$444$	0.700275	0.0332336
$445$	1.20345	0.0570491
$446$	4.90158	0.232096
$447$	−6.04787	−0.286055
$448$	−32.4432	−1.53280
$449$	8.78345	0.414516	0.207258	−	0.978286i	$-0.433546\pi$
$449$	8.78345	0.414516	0.207258	+	0.978286i	$0.433546\pi$
$450$	1.07568	0.0507079
$451$	25.8947	1.21933
$452$	4.65951	0.219165
$453$	−20.9686	−0.985192
$454$	6.41548	0.301094
$455$	−32.4696	−1.52220
$456$	13.7614	0.644438
$457$	19.2768	0.901732	0.450866	−	0.892592i	$-0.351115\pi$
$457$	19.2768	0.901732	0.450866	+	0.892592i	$0.351115\pi$
$458$	−22.1177	−1.03349
$459$	−1.00000	−0.0466760
$460$	1.45970	0.0680590
$461$	−23.0014	−1.07128	−0.535641	−	0.844446i	$-0.679930\pi$
$461$	−23.0014	−1.07128	−0.535641	+	0.844446i	$0.679930\pi$
$462$	14.9263	0.694435
$463$	−16.8468	−0.782937	−0.391469	−	0.920191i	$-0.628033\pi$
$463$	−16.8468	−0.782937	−0.391469	+	0.920191i	$0.628033\pi$
$464$	−12.9298	−0.600252
$465$	−13.4183	−0.622260
$466$	−32.7027	−1.51492
$467$	12.5409	0.580326	0.290163	−	0.956977i	$-0.406290\pi$
$467$	12.5409	0.580326	0.290163	+	0.956977i	$0.406290\pi$
$468$	−1.23376	−0.0570306
$469$	−43.2405	−1.99666
$470$	−3.50351	−0.161605
$471$	−1.00000	−0.0460776
$472$	−14.6292	−0.673364
$473$	17.0660	0.784697
$474$	−8.41242	−0.386395
$475$	−3.77826	−0.173359
$476$	−1.05707	−0.0484507
$477$	−2.61847	−0.119892
$478$	10.5757	0.483721
$479$	−9.88104	−0.451476	−0.225738	−	0.974188i	$-0.572479\pi$
$479$	−9.88104	−0.451476	−0.225738	+	0.974188i	$0.572479\pi$
$480$	−3.28638	−0.150002
$481$	−10.5231	−0.479814
$482$	17.0984	0.778810
$483$	9.19427	0.418354
$484$	0.414385	0.0188357
$485$	7.71972	0.350534
$486$	−1.30899	−0.0593772
$487$	−21.3010	−0.965241	−0.482621	−	0.875829i	$-0.660315\pi$
$487$	−21.3010	−0.965241	−0.482621	+	0.875829i	$0.660315\pi$
$488$	12.3210	0.557747
$489$	−20.1229	−0.909988
$490$	−17.6859	−0.798966
$491$	15.9475	0.719699	0.359849	−	0.933010i	$-0.382828\pi$
$491$	15.9475	0.719699	0.359849	+	0.933010i	$0.382828\pi$
$492$	−2.40049	−0.108222
$493$	−3.86562	−0.174099
$494$	−25.9143	−1.16594
$495$	6.31808	0.283976
$496$	21.9571	0.985904
$497$	−7.08499	−0.317805
$498$	−7.64087	−0.342396
$499$	40.3651	1.80699	0.903496	−	0.428596i	$-0.140992\pi$
$499$	40.3651	1.80699	0.903496	+	0.428596i	$0.140992\pi$
$500$	3.40979	0.152491
$501$	7.68807	0.343478
$502$	15.9315	0.711057
$503$	3.08166	0.137405	0.0687023	−	0.997637i	$-0.478114\pi$
$503$	3.08166	0.137405	0.0687023	+	0.997637i	$0.478114\pi$
$504$	−11.0419	−0.491844
$505$	−15.1110	−0.672429
$506$	10.0836	0.448272
$507$	5.53991	0.246036
$508$	0.523495	0.0232263
$509$	17.1287	0.759218	0.379609	−	0.925147i	$-0.376059\pi$
$509$	17.1287	0.759218	0.379609	+	0.925147i	$0.376059\pi$
$510$	2.67568	0.118481
$511$	−19.8573	−0.878436
$512$	25.4002	1.12254
$513$	4.59778	0.202997
$514$	−31.9581	−1.40961
$515$	27.3941	1.20713
$516$	−1.58205	−0.0696461
$517$	4.04723	0.177997
$518$	−11.8020	−0.518551
$519$	4.78755	0.210150
$520$	26.3430	1.15522
$521$	−13.7580	−0.602751	−0.301375	−	0.953506i	$-0.597446\pi$
$521$	−13.7580	−0.602751	−0.301375	+	0.953506i	$0.597446\pi$
$522$	−5.06007	−0.221473
$523$	−24.1972	−1.05807	−0.529035	−	0.848600i	$-0.677446\pi$
$523$	−24.1972	−1.05807	−0.529035	+	0.848600i	$0.677446\pi$
$524$	1.53556	0.0670812
$525$	3.03160	0.132310
$526$	−19.5967	−0.854455
$527$	6.56450	0.285954
$528$	−10.3386	−0.449930
$529$	−16.7887	−0.729944
$530$	7.00620	0.304330
$531$	−4.88771	−0.212109
$532$	4.86017	0.210715
$533$	36.0725	1.56247
$534$	−0.770672	−0.0333502
$535$	9.12292	0.394418
$536$	35.0816	1.51530
$537$	−5.25320	−0.226692
$538$	−10.3765	−0.447361
$539$	20.4306	0.880006
$540$	−0.585698	−0.0252044
$541$	−19.0045	−0.817066	−0.408533	−	0.912744i	$-0.633960\pi$
$541$	−19.0045	−0.817066	−0.408533	+	0.912744i	$0.633960\pi$
$542$	−11.3618	−0.488032
$543$	3.80371	0.163233
$544$	1.60776	0.0689321
$545$	33.0503	1.41572
$546$	20.7931	0.889861
$547$	30.3808	1.29899	0.649495	−	0.760366i	$-0.274980\pi$
$547$	30.3808	1.29899	0.649495	+	0.760366i	$0.274980\pi$
$548$	3.54244	0.151326
$549$	4.11653	0.175689
$550$	3.32484	0.141771
$551$	17.7732	0.757165
$552$	−7.45944	−0.317495
$553$	−23.7088	−1.00820
$554$	7.50128	0.318699
$555$	−4.99561	−0.212052
$556$	2.32223	0.0984846
$557$	−6.07574	−0.257438	−0.128719	−	0.991681i	$-0.541086\pi$
$557$	−6.07574	−0.257438	−0.128719	+	0.991681i	$0.541086\pi$
$558$	8.59289	0.363766
$559$	23.7738	1.00552
$560$	25.2230	1.06587
$561$	−3.09092	−0.130499
$562$	−34.4222	−1.45201
$563$	−20.6982	−0.872323	−0.436162	−	0.899868i	$-0.643662\pi$
$563$	−20.6982	−0.872323	−0.436162	+	0.899868i	$0.643662\pi$
$564$	−0.375186	−0.0157982
$565$	−33.2399	−1.39841
$566$	10.6597	0.448060
$567$	−3.68915	−0.154930
$568$	5.74814	0.241187
$569$	45.3936	1.90300	0.951499	−	0.307653i	$-0.0995436\pi$
$569$	45.3936	1.90300	0.951499	+	0.307653i	$0.0995436\pi$
$570$	−12.3022	−0.515282
$571$	10.1013	0.422725	0.211362	−	0.977408i	$-0.432210\pi$
$571$	10.1013	0.422725	0.211362	+	0.977408i	$0.432210\pi$
$572$	−3.81346	−0.159449
$573$	7.28846	0.304480
$574$	40.4564	1.68862
$575$	2.04802	0.0854085
$576$	8.79420	0.366425
$577$	22.3054	0.928586	0.464293	−	0.885682i	$-0.346308\pi$
$577$	22.3054	0.928586	0.464293	+	0.885682i	$0.346308\pi$
$578$	−1.30899	−0.0544470
$579$	19.9531	0.829221
$580$	−2.26408	−0.0940109
$581$	−21.5344	−0.893396
$582$	−4.94359	−0.204918
$583$	−8.09350	−0.335198
$584$	16.1105	0.666657
$585$	8.80138	0.363892
$586$	−36.3159	−1.50020
$587$	12.0699	0.498179	0.249090	−	0.968480i	$-0.419869\pi$
$587$	12.0699	0.498179	0.249090	+	0.968480i	$0.419869\pi$
$588$	−1.89395	−0.0781053
$589$	−30.1821	−1.24363
$590$	13.0780	0.538411
$591$	−6.34426	−0.260968
$592$	8.17458	0.335973
$593$	3.70113	0.151987	0.0759936	−	0.997108i	$-0.475787\pi$
$593$	3.70113	0.151987	0.0759936	+	0.997108i	$0.475787\pi$
$594$	−4.04600	−0.166009
$595$	7.54091	0.309147
$596$	1.73292	0.0709833
$597$	−2.78841	−0.114122
$598$	14.0470	0.574423
$599$	−24.6255	−1.00617	−0.503085	−	0.864237i	$-0.667802\pi$
$599$	−24.6255	−1.00617	−0.503085	+	0.864237i	$0.667802\pi$
$600$	−2.45957	−0.100412
$601$	−20.0215	−0.816696	−0.408348	−	0.912826i	$-0.633895\pi$
$601$	−20.0215	−0.816696	−0.408348	+	0.912826i	$0.633895\pi$
$602$	26.6630	1.08670
$603$	11.7210	0.477316
$604$	6.00823	0.244472
$605$	−2.95613	−0.120184
$606$	9.67683	0.393094
$607$	4.48477	0.182031	0.0910155	−	0.995849i	$-0.470989\pi$
$607$	4.48477	0.182031	0.0910155	+	0.995849i	$0.470989\pi$
$608$	−7.39212	−0.299790
$609$	−14.2608	−0.577879
$610$	−11.0145	−0.445965
$611$	5.63797	0.228088
$612$	0.286535	0.0115825
$613$	−9.82137	−0.396681	−0.198341	−	0.980133i	$-0.563555\pi$
$613$	−9.82137	−0.396681	−0.198341	+	0.980133i	$0.563555\pi$
$614$	−0.781093	−0.0315224
$615$	17.1246	0.690529
$616$	−34.1295	−1.37512
$617$	10.6688	0.429510	0.214755	−	0.976668i	$-0.431105\pi$
$617$	10.6688	0.429510	0.214755	+	0.976668i	$0.431105\pi$
$618$	−17.5427	−0.705672
$619$	0.210428	0.00845782	0.00422891	−	0.999991i	$-0.498654\pi$
$619$	0.210428	0.00845782	0.00422891	+	0.999991i	$0.498654\pi$
$620$	3.84481	0.154411
$621$	−2.49224	−0.100010
$622$	−22.5715	−0.905037
$623$	−2.17199	−0.0870191
$624$	−14.4022	−0.576548
$625$	−20.2159	−0.808637
$626$	13.3945	0.535351
$627$	14.2114	0.567548
$628$	0.286535	0.0114340
$629$	2.44395	0.0974465
$630$	9.87100	0.393270
$631$	−32.4933	−1.29354	−0.646769	−	0.762686i	$-0.723880\pi$
$631$	−32.4933	−1.29354	−0.646769	+	0.762686i	$0.723880\pi$
$632$	19.2353	0.765139
$633$	−10.8859	−0.432675
$634$	−20.6146	−0.818710
$635$	−3.73450	−0.148199
$636$	0.750283	0.0297507
$637$	28.4607	1.12765
$638$	−15.6403	−0.619204
$639$	1.92049	0.0759734
$640$	−16.9577	−0.670313
$641$	49.7412	1.96466	0.982331	−	0.187154i	$-0.0599264\pi$
$641$	49.7412	1.96466	0.982331	+	0.187154i	$0.0599264\pi$
$642$	−5.84218	−0.230572
$643$	7.89894	0.311504	0.155752	−	0.987796i	$-0.450220\pi$
$643$	7.89894	0.311504	0.155752	+	0.987796i	$0.450220\pi$
$644$	−2.63448	−0.103813
$645$	11.2860	0.444387
$646$	6.01846	0.236793
$647$	−19.9601	−0.784712	−0.392356	−	0.919814i	$-0.628340\pi$
$647$	−19.9601	−0.784712	−0.392356	+	0.919814i	$0.628340\pi$
$648$	2.99306	0.117579
$649$	−15.1075	−0.593023
$650$	4.63165	0.181668
$651$	24.2174	0.949157
$652$	5.76590	0.225810
$653$	7.35525	0.287833	0.143917	−	0.989590i	$-0.454030\pi$
$653$	7.35525	0.287833	0.143917	+	0.989590i	$0.454030\pi$
$654$	−21.1649	−0.827613
$655$	−10.9543	−0.428021
$656$	−28.0218	−1.09407
$657$	5.38262	0.209996
$658$	6.32315	0.246502
$659$	−28.6410	−1.11569	−0.557847	−	0.829944i	$-0.688372\pi$
$659$	−28.6410	−1.11569	−0.557847	+	0.829944i	$0.688372\pi$
$660$	−1.81035	−0.0704677
$661$	3.73766	0.145378	0.0726891	−	0.997355i	$-0.476842\pi$
$661$	3.73766	0.145378	0.0726891	+	0.997355i	$0.476842\pi$
$662$	−6.37135	−0.247630
$663$	−4.30580	−0.167223
$664$	17.4711	0.678011
$665$	−34.6714	−1.34450
$666$	3.19911	0.123963
$667$	−9.63406	−0.373032
$668$	−2.20290	−0.0852327
$669$	−3.74454	−0.144772
$670$	−31.3616	−1.21161
$671$	12.7239	0.491200
$672$	5.93127	0.228804
$673$	−32.9316	−1.26942	−0.634709	−	0.772751i	$-0.718880\pi$
$673$	−32.9316	−1.26942	−0.634709	+	0.772751i	$0.718880\pi$
$674$	−36.3072	−1.39850
$675$	−0.821759	−0.0316295
$676$	−1.58738	−0.0610529
$677$	28.2146	1.08437	0.542187	−	0.840258i	$-0.317597\pi$
$677$	28.2146	1.08437	0.542187	+	0.840258i	$0.317597\pi$
$678$	21.2863	0.817497
$679$	−13.9326	−0.534683
$680$	−6.11804	−0.234616
$681$	−4.90108	−0.187810
$682$	26.5600	1.01703
$683$	23.7537	0.908908	0.454454	−	0.890770i	$-0.349834\pi$
$683$	23.7537	0.908908	0.454454	+	0.890770i	$0.349834\pi$
$684$	−1.31742	−0.0503729
$685$	−25.2710	−0.965555
$686$	−1.88403	−0.0719326
$687$	16.8967	0.644651
$688$	−18.4679	−0.704084
$689$	−11.2746	−0.429529
$690$	6.66845	0.253864
$691$	3.54640	0.134911	0.0674557	−	0.997722i	$-0.478512\pi$
$691$	3.54640	0.134911	0.0674557	+	0.997722i	$0.478512\pi$
$692$	−1.37180	−0.0521480
$693$	−11.4029	−0.433160
$694$	−44.8863	−1.70386
$695$	−16.5663	−0.628395
$696$	11.5700	0.438560
$697$	−8.37766	−0.317327
$698$	−3.12225	−0.118179
$699$	24.9830	0.944946
$700$	−0.868657	−0.0328321
$701$	−4.18093	−0.157911	−0.0789557	−	0.996878i	$-0.525159\pi$
$701$	−4.18093	−0.157911	−0.0789557	+	0.996878i	$0.525159\pi$
$702$	−5.63627	−0.212727
$703$	−11.2367	−0.423801
$704$	27.1822	1.02447
$705$	2.67649	0.100803
$706$	−2.95456	−0.111196
$707$	27.2723	1.02568
$708$	1.40050	0.0526340
$709$	−1.89557	−0.0711895	−0.0355948	−	0.999366i	$-0.511333\pi$
$709$	−1.89557	−0.0711895	−0.0355948	+	0.999366i	$0.511333\pi$
$710$	−5.13862	−0.192849
$711$	6.42663	0.241017
$712$	1.76217	0.0660400
$713$	16.3603	0.612699
$714$	−4.82908	−0.180724
$715$	27.2044	1.01739
$716$	1.50522	0.0562529
$717$	−8.07926	−0.301725
$718$	12.7927	0.477421
$719$	−20.6209	−0.769029	−0.384514	−	0.923119i	$-0.625631\pi$
$719$	−20.6209	−0.769029	−0.384514	+	0.923119i	$0.625631\pi$
$720$	−6.83708	−0.254803
$721$	−49.4409	−1.84128
$722$	−2.80066	−0.104230
$723$	−13.0622	−0.485790
$724$	−1.08989	−0.0405056
$725$	−3.17660	−0.117976
$726$	1.89306	0.0702582
$727$	5.29859	0.196514	0.0982569	−	0.995161i	$-0.468673\pi$
$727$	5.29859	0.196514	0.0982569	+	0.995161i	$0.468673\pi$
$728$	−47.5440	−1.76210
$729$	1.00000	0.0370370
$730$	−14.4022	−0.533048
$731$	−5.52134	−0.204214
$732$	−1.17953	−0.0435966
$733$	28.2354	1.04290	0.521449	−	0.853282i	$-0.325392\pi$
$733$	28.2354	1.04290	0.521449	+	0.853282i	$0.325392\pi$
$734$	37.7969	1.39511
$735$	13.5110	0.498362
$736$	4.00693	0.147697
$737$	36.2287	1.33450
$738$	−10.9663	−0.403675
$739$	0.363693	0.0133787	0.00668933	−	0.999978i	$-0.497871\pi$
$739$	0.363693	0.0133787	0.00668933	+	0.999978i	$0.497871\pi$
$740$	1.43141	0.0526198
$741$	19.7971	0.727265
$742$	−12.6448	−0.464206
$743$	26.7444	0.981156	0.490578	−	0.871397i	$-0.336786\pi$
$743$	26.7444	0.981156	0.490578	+	0.871397i	$0.336786\pi$
$744$	−19.6479	−0.720329
$745$	−12.3623	−0.452920
$746$	−11.6340	−0.425951
$747$	5.83721	0.213572
$748$	0.885656	0.0323828
$749$	−16.4651	−0.601621
$750$	15.5772	0.568798
$751$	−7.11789	−0.259735	−0.129868	−	0.991531i	$-0.541455\pi$
$751$	−7.11789	−0.259735	−0.129868	+	0.991531i	$0.541455\pi$
$752$	−4.37969	−0.159711
$753$	−12.1708	−0.443528
$754$	−21.7876	−0.793459
$755$	−42.8614	−1.55989
$756$	1.05707	0.0384453
$757$	−3.82980	−0.139197	−0.0695983	−	0.997575i	$-0.522172\pi$
$757$	−3.82980	−0.139197	−0.0695983	+	0.997575i	$0.522172\pi$
$758$	−21.4014	−0.777334
$759$	−7.70333	−0.279613
$760$	28.1294	1.02036
$761$	38.1989	1.38471	0.692355	−	0.721557i	$-0.256573\pi$
$761$	38.1989	1.38471	0.692355	+	0.721557i	$0.256573\pi$
$762$	2.39152	0.0866355
$763$	−59.6493	−2.15945
$764$	−2.08839	−0.0755555
$765$	−2.04407	−0.0739037
$766$	−17.1028	−0.617948
$767$	−21.0455	−0.759910
$768$	−6.72894	−0.242810
$769$	9.19977	0.331752	0.165876	−	0.986147i	$-0.446955\pi$
$769$	9.19977	0.331752	0.165876	+	0.986147i	$0.446955\pi$
$770$	30.5105	1.09952
$771$	24.4142	0.879257
$772$	−5.71724	−0.205768
$773$	−30.9120	−1.11183	−0.555913	−	0.831240i	$-0.687631\pi$
$773$	−30.9120	−1.11183	−0.555913	+	0.831240i	$0.687631\pi$
$774$	−7.22740	−0.259784
$775$	5.39444	0.193774
$776$	11.3037	0.405779
$777$	9.01610	0.323451
$778$	29.1042	1.04344
$779$	38.5186	1.38007
$780$	−2.52190	−0.0902984
$781$	5.93609	0.212410
$782$	−3.26233	−0.116661
$783$	3.86562	0.138146
$784$	−22.1088	−0.789601
$785$	−2.04407	−0.0729562
$786$	7.01499	0.250216
$787$	14.3386	0.511117	0.255558	−	0.966794i	$-0.417741\pi$
$787$	14.3386	0.511117	0.255558	+	0.966794i	$0.417741\pi$
$788$	1.81785	0.0647582
$789$	14.9708	0.532974
$790$	−17.1956	−0.611793
$791$	59.9915	2.13305
$792$	9.25132	0.328731
$793$	17.7250	0.629432
$794$	−18.5017	−0.656600
$795$	−5.35235	−0.189828
$796$	0.798977	0.0283190
$797$	10.6472	0.377144	0.188572	−	0.982059i	$-0.439614\pi$
$797$	10.6472	0.377144	0.188572	+	0.982059i	$0.439614\pi$
$798$	22.2030	0.785979
$799$	−1.30939	−0.0463229
$800$	1.32119	0.0467111
$801$	0.588751	0.0208025
$802$	9.42270	0.332727
$803$	16.6373	0.587116
$804$	−3.35847	−0.118444
$805$	18.7938	0.662394
$806$	36.9993	1.30324
$807$	7.92705	0.279046
$808$	−22.1264	−0.778404
$809$	10.9604	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$809$	10.9604	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$810$	−2.67568	−0.0940139
$811$	16.9662	0.595765	0.297882	−	0.954603i	$-0.403720\pi$
$811$	16.9662	0.595765	0.297882	+	0.954603i	$0.403720\pi$
$812$	4.08623	0.143398
$813$	8.67981	0.304414
$814$	9.88821	0.346581
$815$	−41.1327	−1.44081
$816$	3.34483	0.117092
$817$	25.3859	0.888140
$818$	−22.9599	−0.802775
$819$	−15.8848	−0.555058
$820$	−4.90678	−0.171352
$821$	10.8704	0.379379	0.189689	−	0.981844i	$-0.439252\pi$
$821$	10.8704	0.379379	0.189689	+	0.981844i	$0.439252\pi$
$822$	16.1832	0.564452
$823$	−9.67396	−0.337213	−0.168607	−	0.985683i	$-0.553927\pi$
$823$	−9.67396	−0.337213	−0.168607	+	0.985683i	$0.553927\pi$
$824$	40.1121	1.39737
$825$	−2.53999	−0.0884312
$826$	−23.6032	−0.821259
$827$	−30.4256	−1.05800	−0.529000	−	0.848622i	$-0.677433\pi$
$827$	−30.4256	−1.05800	−0.529000	+	0.848622i	$0.677433\pi$
$828$	0.714114	0.0248172
$829$	25.0629	0.870472	0.435236	−	0.900316i	$-0.356665\pi$
$829$	25.0629	0.870472	0.435236	+	0.900316i	$0.356665\pi$
$830$	−15.6185	−0.542126
$831$	−5.73057	−0.198791
$832$	37.8661	1.31277
$833$	−6.60986	−0.229018
$834$	10.6088	0.367353
$835$	15.7150	0.543840
$836$	−4.07205	−0.140835
$837$	−6.56450	−0.226902
$838$	−17.2873	−0.597179
$839$	41.4378	1.43059	0.715296	−	0.698822i	$-0.246292\pi$
$839$	41.4378	1.43059	0.715296	+	0.698822i	$0.246292\pi$
$840$	−22.5704	−0.778752
$841$	−14.0570	−0.484725
$842$	−5.13567	−0.176987
$843$	26.2967	0.905707
$844$	3.11918	0.107367
$845$	11.3240	0.389557
$846$	−1.71398	−0.0589280
$847$	5.33524	0.183321
$848$	8.75834	0.300763
$849$	−8.14342	−0.279481
$850$	−1.07568	−0.0368954
$851$	6.09091	0.208794
$852$	−0.550287	−0.0188525
$853$	31.9800	1.09497	0.547487	−	0.836814i	$-0.315585\pi$
$853$	31.9800	1.09497	0.547487	+	0.836814i	$0.315585\pi$
$854$	19.8791	0.680248
$855$	9.39820	0.321412
$856$	13.3583	0.456579
$857$	54.1364	1.84927	0.924633	−	0.380860i	$-0.124372\pi$
$857$	54.1364	1.84927	0.924633	+	0.380860i	$0.124372\pi$
$858$	−17.4213	−0.594752
$859$	−0.0952164	−0.00324874	−0.00162437	−	0.999999i	$-0.500517\pi$
$859$	−0.0952164	−0.00324874	−0.00162437	+	0.999999i	$0.500517\pi$
$860$	−3.23384	−0.110273
$861$	−30.9065	−1.05329
$862$	24.0099	0.817780
$863$	−33.2838	−1.13300	−0.566498	−	0.824063i	$-0.691702\pi$
$863$	−33.2838	−1.13300	−0.566498	+	0.824063i	$0.691702\pi$
$864$	−1.60776	−0.0546971
$865$	9.78611	0.332738
$866$	−11.1611	−0.379270
$867$	1.00000	0.0339618
$868$	−6.93914	−0.235530
$869$	19.8642	0.673847
$870$	−10.3432	−0.350666
$871$	50.4682	1.71005
$872$	48.3943	1.63884
$873$	3.77663	0.127820
$874$	14.9995	0.507365
$875$	43.9013	1.48414
$876$	−1.54231	−0.0521097
$877$	36.5944	1.23571	0.617853	−	0.786294i	$-0.288003\pi$
$877$	36.5944	1.23571	0.617853	+	0.786294i	$0.288003\pi$
$878$	−6.54404	−0.220851
$879$	27.7434	0.935760
$880$	−21.1329	−0.712389
$881$	34.0947	1.14868	0.574339	−	0.818617i	$-0.305259\pi$
$881$	34.0947	1.14868	0.574339	+	0.818617i	$0.305259\pi$
$882$	−8.65226	−0.291337
$883$	−37.4687	−1.26092	−0.630460	−	0.776222i	$-0.717134\pi$
$883$	−37.4687	−1.26092	−0.630460	+	0.776222i	$0.717134\pi$
$884$	1.23376	0.0414958
$885$	−9.99085	−0.335839
$886$	−9.54926	−0.320814
$887$	−5.73546	−0.192578	−0.0962889	−	0.995353i	$-0.530697\pi$
$887$	−5.73546	−0.192578	−0.0962889	+	0.995353i	$0.530697\pi$
$888$	−7.31488	−0.245471
$889$	6.74004	0.226054
$890$	−1.57531	−0.0528045
$891$	3.09092	0.103550
$892$	1.07294	0.0359247
$893$	6.02029	0.201461
$894$	7.91663	0.264772
$895$	−10.7379	−0.358930
$896$	30.6054	1.02245
$897$	−10.7311	−0.358301
$898$	−11.4975	−0.383676
$899$	−25.3758	−0.846331
$900$	0.235462	0.00784874
$901$	2.61847	0.0872340
$902$	−33.8960	−1.12861
$903$	−20.3691	−0.677840
$904$	−48.6719	−1.61880
$905$	7.77506	0.258452
$906$	27.4478	0.911892
$907$	−5.62191	−0.186672	−0.0933362	−	0.995635i	$-0.529753\pi$
$907$	−5.62191	−0.186672	−0.0933362	+	0.995635i	$0.529753\pi$
$908$	1.40433	0.0466043
$909$	−7.39257	−0.245196
$910$	42.5026	1.40895
$911$	−55.2857	−1.83170	−0.915848	−	0.401526i	$-0.868480\pi$
$911$	−55.2857	−1.83170	−0.915848	+	0.401526i	$0.868480\pi$
$912$	−15.3788	−0.509242
$913$	18.0424	0.597115
$914$	−25.2333	−0.834642
$915$	8.41450	0.278175
$916$	−4.84150	−0.159968
$917$	19.7704	0.652877
$918$	1.30899	0.0432032
$919$	−38.0891	−1.25644	−0.628221	−	0.778035i	$-0.716217\pi$
$919$	−38.0891	−1.25644	−0.628221	+	0.778035i	$0.716217\pi$
$920$	−15.2476	−0.502700
$921$	0.596712	0.0196623
$922$	30.1087	0.991577
$923$	8.26925	0.272186
$924$	3.26732	0.107487
$925$	2.00833	0.0660336
$926$	22.0524	0.724686
$927$	13.4017	0.440169
$928$	−6.21498	−0.204017
$929$	−6.99270	−0.229423	−0.114711	−	0.993399i	$-0.536594\pi$
$929$	−6.99270	−0.229423	−0.114711	+	0.993399i	$0.536594\pi$
$930$	17.5645	0.575963
$931$	30.3906	0.996013
$932$	−7.15851	−0.234485
$933$	17.2434	0.564524
$934$	−16.4160	−0.537149
$935$	−6.31808	−0.206623
$936$	12.8875	0.421242
$937$	4.12262	0.134680	0.0673401	−	0.997730i	$-0.478549\pi$
$937$	4.12262	0.134680	0.0673401	+	0.997730i	$0.478549\pi$
$938$	56.6016	1.84811
$939$	−10.2327	−0.333930
$940$	−0.766907	−0.0250138
$941$	41.2397	1.34438	0.672188	−	0.740381i	$-0.265355\pi$
$941$	41.2397	1.34438	0.672188	+	0.740381i	$0.265355\pi$
$942$	1.30899	0.0426493
$943$	−20.8792	−0.679920
$944$	16.3486	0.532101
$945$	−7.54091	−0.245306
$946$	−22.3393	−0.726315
$947$	48.0954	1.56289	0.781446	−	0.623973i	$-0.214482\pi$
$947$	48.0954	1.56289	0.781446	+	0.623973i	$0.214482\pi$
$948$	−1.84145	−0.0598076
$949$	23.1765	0.752340
$950$	4.94572	0.160461
$951$	15.7484	0.510678
$952$	11.0419	0.357869
$953$	14.7944	0.479239	0.239620	−	0.970867i	$-0.422977\pi$
$953$	14.7944	0.479239	0.239620	+	0.970867i	$0.422977\pi$
$954$	3.42757	0.110972
$955$	14.8982	0.482093
$956$	2.31499	0.0748720
$957$	11.9483	0.386234
$958$	12.9342	0.417886
$959$	45.6092	1.47280
$960$	17.9760	0.580173
$961$	12.0927	0.390086
$962$	13.7747	0.444115
$963$	4.46311	0.143822
$964$	3.74278	0.120547
$965$	40.7855	1.31293
$966$	−12.0352	−0.387228
$967$	29.4953	0.948505	0.474252	−	0.880389i	$-0.342718\pi$
$967$	29.4953	0.948505	0.474252	+	0.880389i	$0.342718\pi$
$968$	−4.32855	−0.139125
$969$	−4.59778	−0.147702
$970$	−10.1051	−0.324454
$971$	−24.6886	−0.792296	−0.396148	−	0.918187i	$-0.629653\pi$
$971$	−24.6886	−0.792296	−0.396148	+	0.918187i	$0.629653\pi$
$972$	−0.286535	−0.00919060
$973$	29.8989	0.958515
$974$	27.8829	0.893426
$975$	−3.53833	−0.113317
$976$	−13.7691	−0.440738
$977$	61.7294	1.97490	0.987449	−	0.157936i	$-0.0504841\pi$
$977$	61.7294	1.97490	0.987449	+	0.157936i	$0.0504841\pi$
$978$	26.3407	0.842284
$979$	1.81978	0.0581606
$980$	−3.87138	−0.123667
$981$	16.1688	0.516231
$982$	−20.8751	−0.666152
$983$	22.5866	0.720401	0.360200	−	0.932875i	$-0.382708\pi$
$983$	22.5866	0.720401	0.360200	+	0.932875i	$0.382708\pi$
$984$	25.0748	0.799357
$985$	−12.9681	−0.413199
$986$	5.06007	0.161145
$987$	−4.83054	−0.153758
$988$	−5.67255	−0.180468
$989$	−13.7605	−0.437559
$990$	−8.27032	−0.262848
$991$	42.5808	1.35262	0.676312	−	0.736615i	$-0.263577\pi$
$991$	42.5808	1.35262	0.676312	+	0.736615i	$0.263577\pi$
$992$	10.5541	0.335094
$993$	4.86737	0.154461
$994$	9.27420	0.294160
$995$	−5.69973	−0.180693
$996$	−1.67256	−0.0529972
$997$	−55.2349	−1.74931	−0.874653	−	0.484749i	$-0.838911\pi$
$997$	−55.2349	−1.74931	−0.874653	+	0.484749i	$0.838911\pi$
$998$	−52.8377	−1.67255
$999$	−2.44395	−0.0773230

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.h.1.15	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.h.1.15	✓	56	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-1.30899 q^{2} +1.00000 q^{3} -0.286535 q^{4} +2.04407 q^{5} -1.30899 q^{6} -3.68915 q^{7} +2.99306 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.30899 q^{2} +1.00000 q^{3} -0.286535 q^{4} +2.04407 q^{5} -1.30899 q^{6} -3.68915 q^{7} +2.99306 q^{8} +1.00000 q^{9} -2.67568 q^{10} +3.09092 q^{11} -0.286535 q^{12} +4.30580 q^{13} +4.82908 q^{14} +2.04407 q^{15} -3.34483 q^{16} -1.00000 q^{17} -1.30899 q^{18} +4.59778 q^{19} -0.585698 q^{20} -3.68915 q^{21} -4.04600 q^{22} -2.49224 q^{23} +2.99306 q^{24} -0.821759 q^{25} -5.63627 q^{26} +1.00000 q^{27} +1.05707 q^{28} +3.86562 q^{29} -2.67568 q^{30} -6.56450 q^{31} -1.60776 q^{32} +3.09092 q^{33} +1.30899 q^{34} -7.54091 q^{35} -0.286535 q^{36} -2.44395 q^{37} -6.01846 q^{38} +4.30580 q^{39} +6.11804 q^{40} +8.37766 q^{41} +4.82908 q^{42} +5.52134 q^{43} -0.885656 q^{44} +2.04407 q^{45} +3.26233 q^{46} +1.30939 q^{47} -3.34483 q^{48} +6.60986 q^{49} +1.07568 q^{50} -1.00000 q^{51} -1.23376 q^{52} -2.61847 q^{53} -1.30899 q^{54} +6.31808 q^{55} -11.0419 q^{56} +4.59778 q^{57} -5.06007 q^{58} -4.88771 q^{59} -0.585698 q^{60} +4.11653 q^{61} +8.59289 q^{62} -3.68915 q^{63} +8.79420 q^{64} +8.80138 q^{65} -4.04600 q^{66} +11.7210 q^{67} +0.286535 q^{68} -2.49224 q^{69} +9.87100 q^{70} +1.92049 q^{71} +2.99306 q^{72} +5.38262 q^{73} +3.19911 q^{74} -0.821759 q^{75} -1.31742 q^{76} -11.4029 q^{77} -5.63627 q^{78} +6.42663 q^{79} -6.83708 q^{80} +1.00000 q^{81} -10.9663 q^{82} +5.83721 q^{83} +1.05707 q^{84} -2.04407 q^{85} -7.22740 q^{86} +3.86562 q^{87} +9.25132 q^{88} +0.588751 q^{89} -2.67568 q^{90} -15.8848 q^{91} +0.714114 q^{92} -6.56450 q^{93} -1.71398 q^{94} +9.39820 q^{95} -1.60776 q^{96} +3.77663 q^{97} -8.65226 q^{98} +3.09092 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 7 q^{2} + 56 q^{3} + 61 q^{4} + 17 q^{5} + 7 q^{6} + 5 q^{7} + 18 q^{8} + 56 q^{9}+O(q^{10}) \) 56 * q + 7 * q^2 + 56 * q^3 + 61 * q^4 + 17 * q^5 + 7 * q^6 + 5 * q^7 + 18 * q^8 + 56 * q^9	\( 56 q + 7 q^{2} + 56 q^{3} + 61 q^{4} + 17 q^{5} + 7 q^{6} + 5 q^{7} + 18 q^{8} + 56 q^{9} - 2 q^{10} + 35 q^{11} + 61 q^{12} + 8 q^{13} + 36 q^{14} + 17 q^{15} + 71 q^{16} - 56 q^{17} + 7 q^{18} - 2 q^{19} + 58 q^{20} + 5 q^{21} + 27 q^{22} + 40 q^{23} + 18 q^{24} + 85 q^{25} + 15 q^{26} + 56 q^{27} - 4 q^{28} + 41 q^{29} - 2 q^{30} + q^{31} + 43 q^{32} + 35 q^{33} - 7 q^{34} + 57 q^{35} + 61 q^{36} + 34 q^{37} + 52 q^{38} + 8 q^{39} + 14 q^{40} + 49 q^{41} + 36 q^{42} + 27 q^{43} + 66 q^{44} + 17 q^{45} + 10 q^{46} + 43 q^{47} + 71 q^{48} + 51 q^{49} + 30 q^{50} - 56 q^{51} - 7 q^{52} + 73 q^{53} + 7 q^{54} + 15 q^{55} + 118 q^{56} - 2 q^{57} - q^{58} + 53 q^{59} + 58 q^{60} + 15 q^{61} + 16 q^{62} + 5 q^{63} + 124 q^{64} + 107 q^{65} + 27 q^{66} + 20 q^{67} - 61 q^{68} + 40 q^{69} + 16 q^{70} + 56 q^{71} + 18 q^{72} + 49 q^{73} + 28 q^{74} + 85 q^{75} - 38 q^{76} + 50 q^{77} + 15 q^{78} - 4 q^{79} + 74 q^{80} + 56 q^{81} + 59 q^{82} + 35 q^{83} - 4 q^{84} - 17 q^{85} + 38 q^{86} + 41 q^{87} + 64 q^{88} + 66 q^{89} - 2 q^{90} + 5 q^{91} + 96 q^{92} + q^{93} - 12 q^{94} + 70 q^{95} + 43 q^{96} + 60 q^{97} + 26 q^{98} + 35 q^{99}+O(q^{100}) \) 56 * q + 7 * q^2 + 56 * q^3 + 61 * q^4 + 17 * q^5 + 7 * q^6 + 5 * q^7 + 18 * q^8 + 56 * q^9 - 2 * q^10 + 35 * q^11 + 61 * q^12 + 8 * q^13 + 36 * q^14 + 17 * q^15 + 71 * q^16 - 56 * q^17 + 7 * q^18 - 2 * q^19 + 58 * q^20 + 5 * q^21 + 27 * q^22 + 40 * q^23 + 18 * q^24 + 85 * q^25 + 15 * q^26 + 56 * q^27 - 4 * q^28 + 41 * q^29 - 2 * q^30 + q^31 + 43 * q^32 + 35 * q^33 - 7 * q^34 + 57 * q^35 + 61 * q^36 + 34 * q^37 + 52 * q^38 + 8 * q^39 + 14 * q^40 + 49 * q^41 + 36 * q^42 + 27 * q^43 + 66 * q^44 + 17 * q^45 + 10 * q^46 + 43 * q^47 + 71 * q^48 + 51 * q^49 + 30 * q^50 - 56 * q^51 - 7 * q^52 + 73 * q^53 + 7 * q^54 + 15 * q^55 + 118 * q^56 - 2 * q^57 - q^58 + 53 * q^59 + 58 * q^60 + 15 * q^61 + 16 * q^62 + 5 * q^63 + 124 * q^64 + 107 * q^65 + 27 * q^66 + 20 * q^67 - 61 * q^68 + 40 * q^69 + 16 * q^70 + 56 * q^71 + 18 * q^72 + 49 * q^73 + 28 * q^74 + 85 * q^75 - 38 * q^76 + 50 * q^77 + 15 * q^78 - 4 * q^79 + 74 * q^80 + 56 * q^81 + 59 * q^82 + 35 * q^83 - 4 * q^84 - 17 * q^85 + 38 * q^86 + 41 * q^87 + 64 * q^88 + 66 * q^89 - 2 * q^90 + 5 * q^91 + 96 * q^92 + q^93 - 12 * q^94 + 70 * q^95 + 43 * q^96 + 60 * q^97 + 26 * q^98 + 35 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more