LMFDB - Embedded newform 8038.2.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$0$
Dimension:	$92$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.14562 q^{3} +1.00000 q^{4} -0.303255 q^{5} -3.14562 q^{6} +2.95430 q^{7} +1.00000 q^{8} +6.89495 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.14562 q^{3} +1.00000 q^{4} -0.303255 q^{5} -3.14562 q^{6} +2.95430 q^{7} +1.00000 q^{8} +6.89495 q^{9} -0.303255 q^{10} -3.23487 q^{11} -3.14562 q^{12} -5.88424 q^{13} +2.95430 q^{14} +0.953926 q^{15} +1.00000 q^{16} +2.81804 q^{17} +6.89495 q^{18} +0.492091 q^{19} -0.303255 q^{20} -9.29312 q^{21} -3.23487 q^{22} -4.10890 q^{23} -3.14562 q^{24} -4.90804 q^{25} -5.88424 q^{26} -12.2520 q^{27} +2.95430 q^{28} -1.93576 q^{29} +0.953926 q^{30} +0.161869 q^{31} +1.00000 q^{32} +10.1757 q^{33} +2.81804 q^{34} -0.895907 q^{35} +6.89495 q^{36} +0.947971 q^{37} +0.492091 q^{38} +18.5096 q^{39} -0.303255 q^{40} +6.24482 q^{41} -9.29312 q^{42} +7.13145 q^{43} -3.23487 q^{44} -2.09093 q^{45} -4.10890 q^{46} +2.29621 q^{47} -3.14562 q^{48} +1.72790 q^{49} -4.90804 q^{50} -8.86449 q^{51} -5.88424 q^{52} +2.13939 q^{53} -12.2520 q^{54} +0.980990 q^{55} +2.95430 q^{56} -1.54793 q^{57} -1.93576 q^{58} +3.07078 q^{59} +0.953926 q^{60} -8.34305 q^{61} +0.161869 q^{62} +20.3698 q^{63} +1.00000 q^{64} +1.78442 q^{65} +10.1757 q^{66} -4.79832 q^{67} +2.81804 q^{68} +12.9251 q^{69} -0.895907 q^{70} -3.59151 q^{71} +6.89495 q^{72} +15.7187 q^{73} +0.947971 q^{74} +15.4388 q^{75} +0.492091 q^{76} -9.55677 q^{77} +18.5096 q^{78} +16.7051 q^{79} -0.303255 q^{80} +17.8555 q^{81} +6.24482 q^{82} -2.65124 q^{83} -9.29312 q^{84} -0.854585 q^{85} +7.13145 q^{86} +6.08917 q^{87} -3.23487 q^{88} +6.02523 q^{89} -2.09093 q^{90} -17.3838 q^{91} -4.10890 q^{92} -0.509180 q^{93} +2.29621 q^{94} -0.149229 q^{95} -3.14562 q^{96} -18.6929 q^{97} +1.72790 q^{98} -22.3042 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) $ 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * 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
$3$	−3.14562	−1.81613	−0.908063	−	0.418833i	$-0.862439\pi$
$3$	−3.14562	−1.81613	−0.908063	+	0.418833i	$0.862439\pi$
$4$	1.00000	0.500000
$5$	−0.303255	−0.135620	−0.0678099	−	0.997698i	$-0.521601\pi$
$5$	−0.303255	−0.135620	−0.0678099	+	0.997698i	$0.521601\pi$
$6$	−3.14562	−1.28420
$7$	2.95430	1.11662	0.558311	−	0.829632i	$-0.311450\pi$
$7$	2.95430	1.11662	0.558311	+	0.829632i	$0.311450\pi$
$8$	1.00000	0.353553
$9$	6.89495	2.29832
$10$	−0.303255	−0.0958977
$11$	−3.23487	−0.975349	−0.487675	−	0.873025i	$-0.662155\pi$
$11$	−3.23487	−0.975349	−0.487675	+	0.873025i	$0.662155\pi$
$12$	−3.14562	−0.908063
$13$	−5.88424	−1.63199	−0.815997	−	0.578056i	$-0.803811\pi$
$13$	−5.88424	−1.63199	−0.815997	+	0.578056i	$0.803811\pi$
$14$	2.95430	0.789570
$15$	0.953926	0.246303
$16$	1.00000	0.250000
$17$	2.81804	0.683475	0.341738	−	0.939795i	$-0.388985\pi$
$17$	2.81804	0.683475	0.341738	+	0.939795i	$0.388985\pi$
$18$	6.89495	1.62515
$19$	0.492091	0.112893	0.0564467	−	0.998406i	$-0.482023\pi$
$19$	0.492091	0.112893	0.0564467	+	0.998406i	$0.482023\pi$
$20$	−0.303255	−0.0678099
$21$	−9.29312	−2.02793
$22$	−3.23487	−0.689676
$23$	−4.10890	−0.856765	−0.428383	−	0.903597i	$-0.640916\pi$
$23$	−4.10890	−0.856765	−0.428383	+	0.903597i	$0.640916\pi$
$24$	−3.14562	−0.642098
$25$	−4.90804	−0.981607
$26$	−5.88424	−1.15399
$27$	−12.2520	−2.35791
$28$	2.95430	0.558311
$29$	−1.93576	−0.359461	−0.179731	−	0.983716i	$-0.557523\pi$
$29$	−1.93576	−0.359461	−0.179731	+	0.983716i	$0.557523\pi$
$30$	0.953926	0.174162
$31$	0.161869	0.0290726	0.0145363	−	0.999894i	$-0.495373\pi$
$31$	0.161869	0.0290726	0.0145363	+	0.999894i	$0.495373\pi$
$32$	1.00000	0.176777
$33$	10.1757	1.77136
$34$	2.81804	0.483290
$35$	−0.895907	−0.151436
$36$	6.89495	1.14916
$37$	0.947971	0.155846	0.0779228	−	0.996959i	$-0.475171\pi$
$37$	0.947971	0.155846	0.0779228	+	0.996959i	$0.475171\pi$
$38$	0.492091	0.0798278
$39$	18.5096	2.96391
$40$	−0.303255	−0.0479488
$41$	6.24482	0.975277	0.487638	−	0.873046i	$-0.337858\pi$
$41$	6.24482	0.975277	0.487638	+	0.873046i	$0.337858\pi$
$42$	−9.29312	−1.43396
$43$	7.13145	1.08754	0.543768	−	0.839236i	$-0.316997\pi$
$43$	7.13145	1.08754	0.543768	+	0.839236i	$0.316997\pi$
$44$	−3.23487	−0.487675
$45$	−2.09093	−0.311697
$46$	−4.10890	−0.605825
$47$	2.29621	0.334937	0.167469	−	0.985877i	$-0.446441\pi$
$47$	2.29621	0.334937	0.167469	+	0.985877i	$0.446441\pi$
$48$	−3.14562	−0.454032
$49$	1.72790	0.246843
$50$	−4.90804	−0.694101
$51$	−8.86449	−1.24128
$52$	−5.88424	−0.815997
$53$	2.13939	0.293867	0.146934	−	0.989146i	$-0.453060\pi$
$53$	2.13939	0.293867	0.146934	+	0.989146i	$0.453060\pi$
$54$	−12.2520	−1.66729
$55$	0.980990	0.132277
$56$	2.95430	0.394785
$57$	−1.54793	−0.205029
$58$	−1.93576	−0.254177
$59$	3.07078	0.399782	0.199891	−	0.979818i	$-0.435941\pi$
$59$	3.07078	0.399782	0.199891	+	0.979818i	$0.435941\pi$
$60$	0.953926	0.123151
$61$	−8.34305	−1.06822	−0.534109	−	0.845415i	$-0.679353\pi$
$61$	−8.34305	−1.06822	−0.534109	+	0.845415i	$0.679353\pi$
$62$	0.161869	0.0205574
$63$	20.3698	2.56635
$64$	1.00000	0.125000
$65$	1.78442	0.221331
$66$	10.1757	1.25254
$67$	−4.79832	−0.586209	−0.293104	−	0.956080i	$-0.594688\pi$
$67$	−4.79832	−0.586209	−0.293104	+	0.956080i	$0.594688\pi$
$68$	2.81804	0.341738
$69$	12.9251	1.55599
$70$	−0.895907	−0.107081
$71$	−3.59151	−0.426233	−0.213117	−	0.977027i	$-0.568361\pi$
$71$	−3.59151	−0.426233	−0.213117	+	0.977027i	$0.568361\pi$
$72$	6.89495	0.812577
$73$	15.7187	1.83973	0.919866	−	0.392232i	$-0.128297\pi$
$73$	15.7187	1.83973	0.919866	+	0.392232i	$0.128297\pi$
$74$	0.947971	0.110199
$75$	15.4388	1.78272
$76$	0.492091	0.0564467
$77$	−9.55677	−1.08910
$78$	18.5096	2.09580
$79$	16.7051	1.87947	0.939737	−	0.341899i	$-0.111070\pi$
$79$	16.7051	1.87947	0.939737	+	0.341899i	$0.111070\pi$
$80$	−0.303255	−0.0339049
$81$	17.8555	1.98394
$82$	6.24482	0.689625
$83$	−2.65124	−0.291012	−0.145506	−	0.989357i	$-0.546481\pi$
$83$	−2.65124	−0.291012	−0.145506	+	0.989357i	$0.546481\pi$
$84$	−9.29312	−1.01396
$85$	−0.854585	−0.0926927
$86$	7.13145	0.769004
$87$	6.08917	0.652827
$88$	−3.23487	−0.344838
$89$	6.02523	0.638673	0.319337	−	0.947641i	$-0.396540\pi$
$89$	6.02523	0.638673	0.319337	+	0.947641i	$0.396540\pi$
$90$	−2.09093	−0.220403
$91$	−17.3838	−1.82232
$92$	−4.10890	−0.428383
$93$	−0.509180	−0.0527995
$94$	2.29621	0.236836
$95$	−0.149229	−0.0153106
$96$	−3.14562	−0.321049
$97$	−18.6929	−1.89797	−0.948987	−	0.315315i	$-0.897890\pi$
$97$	−18.6929	−1.89797	−0.948987	+	0.315315i	$0.897890\pi$
$98$	1.72790	0.174544
$99$	−22.3042	−2.24166
$100$	−4.90804	−0.490804
$101$	−7.97273	−0.793316	−0.396658	−	0.917966i	$-0.629830\pi$
$101$	−7.97273	−0.793316	−0.396658	+	0.917966i	$0.629830\pi$
$102$	−8.86449	−0.877716
$103$	12.6737	1.24878	0.624389	−	0.781114i	$-0.285348\pi$
$103$	12.6737	1.24878	0.624389	+	0.781114i	$0.285348\pi$
$104$	−5.88424	−0.576997
$105$	2.81819	0.275027
$106$	2.13939	0.207796
$107$	−12.0298	−1.16297	−0.581484	−	0.813558i	$-0.697528\pi$
$107$	−12.0298	−1.16297	−0.581484	+	0.813558i	$0.697528\pi$
$108$	−12.2520	−1.17895
$109$	−3.33546	−0.319479	−0.159740	−	0.987159i	$-0.551065\pi$
$109$	−3.33546	−0.319479	−0.159740	+	0.987159i	$0.551065\pi$
$110$	0.980990	0.0935337
$111$	−2.98196	−0.283035
$112$	2.95430	0.279155
$113$	6.86291	0.645609	0.322804	−	0.946466i	$-0.395374\pi$
$113$	6.86291	0.645609	0.322804	+	0.946466i	$0.395374\pi$
$114$	−1.54793	−0.144977
$115$	1.24605	0.116194
$116$	−1.93576	−0.179731
$117$	−40.5715	−3.75084
$118$	3.07078	0.282688
$119$	8.32534	0.763183
$120$	0.953926	0.0870812
$121$	−0.535632	−0.0486938
$122$	−8.34305	−0.755345
$123$	−19.6439	−1.77123
$124$	0.161869	0.0145363
$125$	3.00466	0.268745
$126$	20.3698	1.81468
$127$	0.438624	0.0389215	0.0194608	−	0.999811i	$-0.493805\pi$
$127$	0.438624	0.0389215	0.0194608	+	0.999811i	$0.493805\pi$
$128$	1.00000	0.0883883
$129$	−22.4329	−1.97510
$130$	1.78442	0.156504
$131$	9.35273	0.817152	0.408576	−	0.912724i	$-0.366026\pi$
$131$	9.35273	0.817152	0.408576	+	0.912724i	$0.366026\pi$
$132$	10.1757	0.885679
$133$	1.45379	0.126059
$134$	−4.79832	−0.414512
$135$	3.71549	0.319779
$136$	2.81804	0.241645
$137$	15.6556	1.33755	0.668773	−	0.743467i	$-0.266820\pi$
$137$	15.6556	1.33755	0.668773	+	0.743467i	$0.266820\pi$
$138$	12.9251	1.10025
$139$	4.20241	0.356444	0.178222	−	0.983990i	$-0.442965\pi$
$139$	4.20241	0.356444	0.178222	+	0.983990i	$0.442965\pi$
$140$	−0.895907	−0.0757180
$141$	−7.22302	−0.608288
$142$	−3.59151	−0.301393
$143$	19.0347	1.59176
$144$	6.89495	0.574579
$145$	0.587028	0.0487500
$146$	15.7187	1.30089
$147$	−5.43532	−0.448297
$148$	0.947971	0.0779228
$149$	6.46644	0.529752	0.264876	−	0.964283i	$-0.414669\pi$
$149$	6.46644	0.529752	0.264876	+	0.964283i	$0.414669\pi$
$150$	15.4388	1.26058
$151$	−15.8082	−1.28645	−0.643226	−	0.765676i	$-0.722404\pi$
$151$	−15.8082	−1.28645	−0.643226	+	0.765676i	$0.722404\pi$
$152$	0.492091	0.0399139
$153$	19.4302	1.57084
$154$	−9.55677	−0.770107
$155$	−0.0490877	−0.00394282
$156$	18.5096	1.48195
$157$	−0.611020	−0.0487647	−0.0243823	−	0.999703i	$-0.507762\pi$
$157$	−0.611020	−0.0487647	−0.0243823	+	0.999703i	$0.507762\pi$
$158$	16.7051	1.32899
$159$	−6.72971	−0.533700
$160$	−0.303255	−0.0239744
$161$	−12.1389	−0.956682
$162$	17.8555	1.40286
$163$	−3.84770	−0.301375	−0.150687	−	0.988581i	$-0.548149\pi$
$163$	−3.84770	−0.301375	−0.150687	+	0.988581i	$0.548149\pi$
$164$	6.24482	0.487638
$165$	−3.08583	−0.240231
$166$	−2.65124	−0.205776
$167$	14.7140	1.13860	0.569300	−	0.822130i	$-0.307214\pi$
$167$	14.7140	1.13860	0.569300	+	0.822130i	$0.307214\pi$
$168$	−9.29312	−0.716980
$169$	21.6243	1.66340
$170$	−0.854585	−0.0655437
$171$	3.39294	0.259465
$172$	7.13145	0.543768
$173$	−11.4450	−0.870148	−0.435074	−	0.900395i	$-0.643278\pi$
$173$	−11.4450	−0.870148	−0.435074	+	0.900395i	$0.643278\pi$
$174$	6.08917	0.461618
$175$	−14.4998	−1.09608
$176$	−3.23487	−0.243837
$177$	−9.65952	−0.726054
$178$	6.02523	0.451610
$179$	−2.05051	−0.153263	−0.0766313	−	0.997059i	$-0.524416\pi$
$179$	−2.05051	−0.153263	−0.0766313	+	0.997059i	$0.524416\pi$
$180$	−2.09093	−0.155849
$181$	4.27130	0.317484	0.158742	−	0.987320i	$-0.449256\pi$
$181$	4.27130	0.317484	0.158742	+	0.987320i	$0.449256\pi$
$182$	−17.3838	−1.28857
$183$	26.2441	1.94002
$184$	−4.10890	−0.302912
$185$	−0.287477	−0.0211357
$186$	−0.509180	−0.0373349
$187$	−9.11599	−0.666627
$188$	2.29621	0.167469
$189$	−36.1962	−2.63289
$190$	−0.149229	−0.0108262
$191$	−25.1646	−1.82085	−0.910424	−	0.413676i	$-0.864245\pi$
$191$	−25.1646	−1.82085	−0.910424	+	0.413676i	$0.864245\pi$
$192$	−3.14562	−0.227016
$193$	−7.46923	−0.537647	−0.268823	−	0.963190i	$-0.586635\pi$
$193$	−7.46923	−0.537647	−0.268823	+	0.963190i	$0.586635\pi$
$194$	−18.6929	−1.34207
$195$	−5.61313	−0.401965
$196$	1.72790	0.123421
$197$	2.26873	0.161640	0.0808201	−	0.996729i	$-0.474246\pi$
$197$	2.26873	0.161640	0.0808201	+	0.996729i	$0.474246\pi$
$198$	−22.3042	−1.58509
$199$	22.7289	1.61121	0.805604	−	0.592454i	$-0.201841\pi$
$199$	22.7289	1.61121	0.805604	+	0.592454i	$0.201841\pi$
$200$	−4.90804	−0.347051
$201$	15.0937	1.06463
$202$	−7.97273	−0.560959
$203$	−5.71881	−0.401382
$204$	−8.86449	−0.620639
$205$	−1.89377	−0.132267
$206$	12.6737	0.883019
$207$	−28.3307	−1.96912
$208$	−5.88424	−0.407998
$209$	−1.59185	−0.110111
$210$	2.81819	0.194473
$211$	1.73787	0.119640	0.0598200	−	0.998209i	$-0.480947\pi$
$211$	1.73787	0.119640	0.0598200	+	0.998209i	$0.480947\pi$
$212$	2.13939	0.146934
$213$	11.2975	0.774094
$214$	−12.0298	−0.822342
$215$	−2.16265	−0.147491
$216$	−12.2520	−0.833646
$217$	0.478211	0.0324631
$218$	−3.33546	−0.225906
$219$	−49.4450	−3.34119
$220$	0.980990	0.0661383
$221$	−16.5820	−1.11543
$222$	−2.98196	−0.200136
$223$	24.6758	1.65241	0.826206	−	0.563368i	$-0.190495\pi$
$223$	24.6758	1.65241	0.826206	+	0.563368i	$0.190495\pi$
$224$	2.95430	0.197393
$225$	−33.8407	−2.25604
$226$	6.86291	0.456514
$227$	21.2903	1.41309	0.706544	−	0.707669i	$-0.250253\pi$
$227$	21.2903	1.41309	0.706544	+	0.707669i	$0.250253\pi$
$228$	−1.54793	−0.102514
$229$	−5.69009	−0.376012	−0.188006	−	0.982168i	$-0.560202\pi$
$229$	−5.69009	−0.376012	−0.188006	+	0.982168i	$0.560202\pi$
$230$	1.24605	0.0821618
$231$	30.0620	1.97794
$232$	−1.93576	−0.127089
$233$	−18.5605	−1.21594	−0.607970	−	0.793960i	$-0.708016\pi$
$233$	−18.5605	−1.21594	−0.607970	+	0.793960i	$0.708016\pi$
$234$	−40.5715	−2.65224
$235$	−0.696338	−0.0454241
$236$	3.07078	0.199891
$237$	−52.5480	−3.41336
$238$	8.32534	0.539652
$239$	11.1095	0.718612	0.359306	−	0.933220i	$-0.383013\pi$
$239$	11.1095	0.718612	0.359306	+	0.933220i	$0.383013\pi$
$240$	0.953926	0.0615757
$241$	19.4353	1.25194	0.625968	−	0.779849i	$-0.284704\pi$
$241$	19.4353	1.25194	0.625968	+	0.779849i	$0.284704\pi$
$242$	−0.535632	−0.0344317
$243$	−19.4104	−1.24518
$244$	−8.34305	−0.534109
$245$	−0.523994	−0.0334767
$246$	−19.6439	−1.25245
$247$	−2.89558	−0.184242
$248$	0.161869	0.0102787
$249$	8.33982	0.528514
$250$	3.00466	0.190032
$251$	18.0799	1.14119	0.570596	−	0.821231i	$-0.306712\pi$
$251$	18.0799	1.14119	0.570596	+	0.821231i	$0.306712\pi$
$252$	20.3698	1.28317
$253$	13.2918	0.835646
$254$	0.438624	0.0275217
$255$	2.68820	0.168342
$256$	1.00000	0.0625000
$257$	−2.63732	−0.164511	−0.0822557	−	0.996611i	$-0.526212\pi$
$257$	−2.63732	−0.164511	−0.0822557	+	0.996611i	$0.526212\pi$
$258$	−22.4329	−1.39661
$259$	2.80059	0.174020
$260$	1.78442	0.110665
$261$	−13.3469	−0.826155
$262$	9.35273	0.577814
$263$	27.0513	1.66805	0.834027	−	0.551724i	$-0.186030\pi$
$263$	27.0513	1.66805	0.834027	+	0.551724i	$0.186030\pi$
$264$	10.1757	0.626270
$265$	−0.648780	−0.0398542
$266$	1.45379	0.0891374
$267$	−18.9531	−1.15991
$268$	−4.79832	−0.293104
$269$	18.2931	1.11535	0.557675	−	0.830059i	$-0.311694\pi$
$269$	18.2931	1.11535	0.557675	+	0.830059i	$0.311694\pi$
$270$	3.71549	0.226118
$271$	−5.39037	−0.327442	−0.163721	−	0.986507i	$-0.552350\pi$
$271$	−5.39037	−0.327442	−0.163721	+	0.986507i	$0.552350\pi$
$272$	2.81804	0.170869
$273$	54.6829	3.30956
$274$	15.6556	0.945787
$275$	15.8768	0.957410
$276$	12.9251	0.777997
$277$	2.21699	0.133206	0.0666029	−	0.997780i	$-0.478784\pi$
$277$	2.21699	0.133206	0.0666029	+	0.997780i	$0.478784\pi$
$278$	4.20241	0.252044
$279$	1.11608	0.0668180
$280$	−0.895907	−0.0535407
$281$	−24.9985	−1.49129	−0.745644	−	0.666344i	$-0.767858\pi$
$281$	−24.9985	−1.49129	−0.745644	+	0.666344i	$0.767858\pi$
$282$	−7.22302	−0.430125
$283$	−9.15271	−0.544072	−0.272036	−	0.962287i	$-0.587697\pi$
$283$	−9.15271	−0.544072	−0.272036	+	0.962287i	$0.587697\pi$
$284$	−3.59151	−0.213117
$285$	0.469419	0.0278060
$286$	19.0347	1.12555
$287$	18.4491	1.08901
$288$	6.89495	0.406289
$289$	−9.05865	−0.532862
$290$	0.587028	0.0344715
$291$	58.8008	3.44696
$292$	15.7187	0.919866
$293$	23.8266	1.39197	0.695984	−	0.718058i	$-0.254969\pi$
$293$	23.8266	1.39197	0.695984	+	0.718058i	$0.254969\pi$
$294$	−5.43532	−0.316994
$295$	−0.931230	−0.0542183
$296$	0.947971	0.0550997
$297$	39.6337	2.29978
$298$	6.46644	0.374591
$299$	24.1778	1.39824
$300$	15.4388	0.891362
$301$	21.0684	1.21437
$302$	−15.8082	−0.909659
$303$	25.0792	1.44076
$304$	0.492091	0.0282234
$305$	2.53007	0.144872
$306$	19.4302	1.11075
$307$	6.45305	0.368295	0.184148	−	0.982899i	$-0.441048\pi$
$307$	6.45305	0.368295	0.184148	+	0.982899i	$0.441048\pi$
$308$	−9.55677	−0.544548
$309$	−39.8667	−2.26794
$310$	−0.0490877	−0.00278800
$311$	−23.8424	−1.35198	−0.675988	−	0.736912i	$-0.736283\pi$
$311$	−23.8424	−1.35198	−0.675988	+	0.736912i	$0.736283\pi$
$312$	18.5096	1.04790
$313$	13.2515	0.749018	0.374509	−	0.927223i	$-0.377811\pi$
$313$	13.2515	0.749018	0.374509	+	0.927223i	$0.377811\pi$
$314$	−0.611020	−0.0344818
$315$	−6.17723	−0.348048
$316$	16.7051	0.939737
$317$	18.4246	1.03483	0.517414	−	0.855735i	$-0.326895\pi$
$317$	18.4246	1.03483	0.517414	+	0.855735i	$0.326895\pi$
$318$	−6.72971	−0.377383
$319$	6.26192	0.350600
$320$	−0.303255	−0.0169525
$321$	37.8413	2.11210
$322$	−12.1389	−0.676477
$323$	1.38673	0.0771599
$324$	17.8555	0.991970
$325$	28.8801	1.60198
$326$	−3.84770	−0.213104
$327$	10.4921	0.580214
$328$	6.24482	0.344812
$329$	6.78370	0.373998
$330$	−3.08583	−0.169869
$331$	21.5572	1.18489	0.592446	−	0.805610i	$-0.298162\pi$
$331$	21.5572	1.18489	0.592446	+	0.805610i	$0.298162\pi$
$332$	−2.65124	−0.145506
$333$	6.53621	0.358182
$334$	14.7140	0.805112
$335$	1.45512	0.0795015
$336$	−9.29312	−0.506981
$337$	6.50351	0.354269	0.177134	−	0.984187i	$-0.443317\pi$
$337$	6.50351	0.354269	0.177134	+	0.984187i	$0.443317\pi$
$338$	21.6243	1.17620
$339$	−21.5881	−1.17251
$340$	−0.854585	−0.0463464
$341$	−0.523626	−0.0283560
$342$	3.39294	0.183469
$343$	−15.5754	−0.840991
$344$	7.13145	0.384502
$345$	−3.91959	−0.211024
$346$	−11.4450	−0.615288
$347$	2.36907	0.127178	0.0635892	−	0.997976i	$-0.479745\pi$
$347$	2.36907	0.127178	0.0635892	+	0.997976i	$0.479745\pi$
$348$	6.08917	0.326414
$349$	1.56942	0.0840090	0.0420045	−	0.999117i	$-0.486626\pi$
$349$	1.56942	0.0840090	0.0420045	+	0.999117i	$0.486626\pi$
$350$	−14.4998	−0.775048
$351$	72.0939	3.84809
$352$	−3.23487	−0.172419
$353$	3.28232	0.174700	0.0873502	−	0.996178i	$-0.472160\pi$
$353$	3.28232	0.174700	0.0873502	+	0.996178i	$0.472160\pi$
$354$	−9.65952	−0.513398
$355$	1.08914	0.0578057
$356$	6.02523	0.319337
$357$	−26.1884	−1.38604
$358$	−2.05051	−0.108373
$359$	11.4762	0.605689	0.302845	−	0.953040i	$-0.402064\pi$
$359$	11.4762	0.605689	0.302845	+	0.953040i	$0.402064\pi$
$360$	−2.09093	−0.110202
$361$	−18.7578	−0.987255
$362$	4.27130	0.224495
$363$	1.68490	0.0884341
$364$	−17.3838	−0.911159
$365$	−4.76677	−0.249504
$366$	26.2441	1.37180
$367$	19.4851	1.01712	0.508558	−	0.861028i	$-0.330179\pi$
$367$	19.4851	1.01712	0.508558	+	0.861028i	$0.330179\pi$
$368$	−4.10890	−0.214191
$369$	43.0577	2.24149
$370$	−0.287477	−0.0149452
$371$	6.32039	0.328139
$372$	−0.509180	−0.0263998
$373$	1.76946	0.0916190	0.0458095	−	0.998950i	$-0.485413\pi$
$373$	1.76946	0.0916190	0.0458095	+	0.998950i	$0.485413\pi$
$374$	−9.11599	−0.471376
$375$	−9.45154	−0.488075
$376$	2.29621	0.118418
$377$	11.3905	0.586638
$378$	−36.1962	−1.86173
$379$	33.7902	1.73568	0.867842	−	0.496840i	$-0.165506\pi$
$379$	33.7902	1.73568	0.867842	+	0.496840i	$0.165506\pi$
$380$	−0.149229	−0.00765530
$381$	−1.37974	−0.0706865
$382$	−25.1646	−1.28753
$383$	26.8780	1.37340	0.686701	−	0.726940i	$-0.259058\pi$
$383$	26.8780	1.37340	0.686701	+	0.726940i	$0.259058\pi$
$384$	−3.14562	−0.160524
$385$	2.89814	0.147703
$386$	−7.46923	−0.380174
$387$	49.1710	2.49950
$388$	−18.6929	−0.948987
$389$	−18.0575	−0.915552	−0.457776	−	0.889068i	$-0.651354\pi$
$389$	−18.0575	−0.915552	−0.457776	+	0.889068i	$0.651354\pi$
$390$	−5.61313	−0.284232
$391$	−11.5791	−0.585578
$392$	1.72790	0.0872720
$393$	−29.4202	−1.48405
$394$	2.26873	0.114297
$395$	−5.06591	−0.254894
$396$	−22.3042	−1.12083
$397$	18.7731	0.942195	0.471098	−	0.882081i	$-0.343858\pi$
$397$	18.7731	0.942195	0.471098	+	0.882081i	$0.343858\pi$
$398$	22.7289	1.13930
$399$	−4.57306	−0.228940
$400$	−4.90804	−0.245402
$401$	7.89163	0.394089	0.197045	−	0.980395i	$-0.436866\pi$
$401$	7.89163	0.394089	0.197045	+	0.980395i	$0.436866\pi$
$402$	15.0937	0.752806
$403$	−0.952478	−0.0474463
$404$	−7.97273	−0.396658
$405$	−5.41476	−0.269062
$406$	−5.71881	−0.283820
$407$	−3.06656	−0.152004
$408$	−8.86449	−0.438858
$409$	16.0069	0.791488	0.395744	−	0.918361i	$-0.370487\pi$
$409$	16.0069	0.791488	0.395744	+	0.918361i	$0.370487\pi$
$410$	−1.89377	−0.0935268
$411$	−49.2465	−2.42915
$412$	12.6737	0.624389
$413$	9.07201	0.446405
$414$	−28.3307	−1.39238
$415$	0.804003	0.0394670
$416$	−5.88424	−0.288499
$417$	−13.2192	−0.647348
$418$	−1.59185	−0.0778599
$419$	21.3585	1.04343	0.521715	−	0.853120i	$-0.325292\pi$
$419$	21.3585	1.04343	0.521715	+	0.853120i	$0.325292\pi$
$420$	2.81819	0.137513
$421$	−4.92380	−0.239972	−0.119986	−	0.992776i	$-0.538285\pi$
$421$	−4.92380	−0.239972	−0.119986	+	0.992776i	$0.538285\pi$
$422$	1.73787	0.0845982
$423$	15.8323	0.769791
$424$	2.13939	0.103898
$425$	−13.8310	−0.670904
$426$	11.2975	0.547367
$427$	−24.6479	−1.19280
$428$	−12.0298	−0.581484
$429$	−59.8761	−2.89085
$430$	−2.16265	−0.104292
$431$	4.66746	0.224824	0.112412	−	0.993662i	$-0.464142\pi$
$431$	4.66746	0.224824	0.112412	+	0.993662i	$0.464142\pi$
$432$	−12.2520	−0.589477
$433$	4.88242	0.234634	0.117317	−	0.993095i	$-0.462571\pi$
$433$	4.88242	0.234634	0.117317	+	0.993095i	$0.462571\pi$
$434$	0.478211	0.0229549
$435$	−1.84657	−0.0885363
$436$	−3.33546	−0.159740
$437$	−2.02196	−0.0967232
$438$	−49.4450	−2.36258
$439$	−36.1862	−1.72707	−0.863537	−	0.504286i	$-0.831756\pi$
$439$	−36.1862	−1.72707	−0.863537	+	0.504286i	$0.831756\pi$
$440$	0.980990	0.0467669
$441$	11.9138	0.567322
$442$	−16.5820	−0.788726
$443$	29.9610	1.42349	0.711746	−	0.702437i	$-0.247905\pi$
$443$	29.9610	1.42349	0.711746	+	0.702437i	$0.247905\pi$
$444$	−2.98196	−0.141518
$445$	−1.82718	−0.0866167
$446$	24.6758	1.16843
$447$	−20.3410	−0.962096
$448$	2.95430	0.139578
$449$	−3.36038	−0.158586	−0.0792930	−	0.996851i	$-0.525266\pi$
$449$	−3.36038	−0.158586	−0.0792930	+	0.996851i	$0.525266\pi$
$450$	−33.8407	−1.59526
$451$	−20.2012	−0.951236
$452$	6.86291	0.322804
$453$	49.7266	2.33636
$454$	21.2903	0.999205
$455$	5.27173	0.247142
$456$	−1.54793	−0.0724887
$457$	−4.78819	−0.223982	−0.111991	−	0.993709i	$-0.535723\pi$
$457$	−4.78819	−0.223982	−0.111991	+	0.993709i	$0.535723\pi$
$458$	−5.69009	−0.265880
$459$	−34.5267	−1.61157
$460$	1.24605	0.0580972
$461$	13.2480	0.617020	0.308510	−	0.951221i	$-0.400170\pi$
$461$	13.2480	0.617020	0.308510	+	0.951221i	$0.400170\pi$
$462$	30.0620	1.39861
$463$	−29.5741	−1.37443	−0.687213	−	0.726456i	$-0.741166\pi$
$463$	−29.5741	−1.37443	−0.687213	+	0.726456i	$0.741166\pi$
$464$	−1.93576	−0.0898653
$465$	0.154412	0.00716066
$466$	−18.5605	−0.859799
$467$	21.2078	0.981382	0.490691	−	0.871334i	$-0.336744\pi$
$467$	21.2078	0.981382	0.490691	+	0.871334i	$0.336744\pi$
$468$	−40.5715	−1.87542
$469$	−14.1757	−0.654573
$470$	−0.696338	−0.0321197
$471$	1.92204	0.0885628
$472$	3.07078	0.141344
$473$	−23.0693	−1.06073
$474$	−52.5480	−2.41361
$475$	−2.41520	−0.110817
$476$	8.32534	0.381591
$477$	14.7510	0.675400
$478$	11.1095	0.508136
$479$	16.9522	0.774568	0.387284	−	0.921960i	$-0.373413\pi$
$479$	16.9522	0.774568	0.387284	+	0.921960i	$0.373413\pi$
$480$	0.953926	0.0435406
$481$	−5.57809	−0.254339
$482$	19.4353	0.885252
$483$	38.1845	1.73746
$484$	−0.535632	−0.0243469
$485$	5.66871	0.257403
$486$	−19.4104	−0.880476
$487$	21.0786	0.955163	0.477582	−	0.878587i	$-0.341513\pi$
$487$	21.0786	0.955163	0.477582	+	0.878587i	$0.341513\pi$
$488$	−8.34305	−0.377672
$489$	12.1034	0.547335
$490$	−0.523994	−0.0236716
$491$	−2.50209	−0.112918	−0.0564589	−	0.998405i	$-0.517981\pi$
$491$	−2.50209	−0.112918	−0.0564589	+	0.998405i	$0.517981\pi$
$492$	−19.6439	−0.885613
$493$	−5.45504	−0.245683
$494$	−2.89558	−0.130278
$495$	6.76387	0.304014
$496$	0.161869	0.00726815
$497$	−10.6104	−0.475941
$498$	8.33982	0.373716
$499$	28.4643	1.27424	0.637119	−	0.770765i	$-0.280126\pi$
$499$	28.4643	1.27424	0.637119	+	0.770765i	$0.280126\pi$
$500$	3.00466	0.134373
$501$	−46.2846	−2.06784
$502$	18.0799	0.806944
$503$	−29.4044	−1.31108	−0.655539	−	0.755161i	$-0.727559\pi$
$503$	−29.4044	−1.31108	−0.655539	+	0.755161i	$0.727559\pi$
$504$	20.3698	0.907341
$505$	2.41777	0.107589
$506$	13.2918	0.590891
$507$	−68.0218	−3.02095
$508$	0.438624	0.0194608
$509$	6.88932	0.305364	0.152682	−	0.988275i	$-0.451209\pi$
$509$	6.88932	0.305364	0.152682	+	0.988275i	$0.451209\pi$
$510$	2.68820	0.119036
$511$	46.4377	2.05428
$512$	1.00000	0.0441942
$513$	−6.02912	−0.266192
$514$	−2.63732	−0.116327
$515$	−3.84336	−0.169359
$516$	−22.4329	−0.987551
$517$	−7.42794	−0.326681
$518$	2.80059	0.123051
$519$	36.0017	1.58030
$520$	1.78442	0.0782522
$521$	22.7701	0.997578	0.498789	−	0.866723i	$-0.333778\pi$
$521$	22.7701	0.997578	0.498789	+	0.866723i	$0.333778\pi$
$522$	−13.3469	−0.584180
$523$	6.85159	0.299599	0.149800	−	0.988716i	$-0.452137\pi$
$523$	6.85159	0.299599	0.149800	+	0.988716i	$0.452137\pi$
$524$	9.35273	0.408576
$525$	45.6110	1.99063
$526$	27.0513	1.17949
$527$	0.456155	0.0198704
$528$	10.1757	0.442839
$529$	−6.11692	−0.265953
$530$	−0.648780	−0.0281812
$531$	21.1729	0.918825
$532$	1.45379	0.0630296
$533$	−36.7460	−1.59165
$534$	−18.9531	−0.820181
$535$	3.64811	0.157721
$536$	−4.79832	−0.207256
$537$	6.45014	0.278344
$538$	18.2931	0.788672
$539$	−5.58952	−0.240758
$540$	3.71549	0.159889
$541$	5.36739	0.230762	0.115381	−	0.993321i	$-0.463191\pi$
$541$	5.36739	0.230762	0.115381	+	0.993321i	$0.463191\pi$
$542$	−5.39037	−0.231536
$543$	−13.4359	−0.576590
$544$	2.81804	0.120822
$545$	1.01149	0.0433277
$546$	54.6829	2.34021
$547$	7.58542	0.324329	0.162165	−	0.986764i	$-0.448152\pi$
$547$	7.58542	0.324329	0.162165	+	0.986764i	$0.448152\pi$
$548$	15.6556	0.668773
$549$	−57.5249	−2.45510
$550$	15.8768	0.676991
$551$	−0.952570	−0.0405808
$552$	12.9251	0.550127
$553$	49.3520	2.09866
$554$	2.21699	0.0941908
$555$	0.904295	0.0383852
$556$	4.20241	0.178222
$557$	−45.8425	−1.94241	−0.971204	−	0.238250i	$-0.923426\pi$
$557$	−45.8425	−1.94241	−0.971204	+	0.238250i	$0.923426\pi$
$558$	1.11608	0.0472475
$559$	−41.9631	−1.77485
$560$	−0.895907	−0.0378590
$561$	28.6755	1.21068
$562$	−24.9985	−1.05450
$563$	26.7212	1.12616	0.563082	−	0.826401i	$-0.309615\pi$
$563$	26.7212	1.12616	0.563082	+	0.826401i	$0.309615\pi$
$564$	−7.22302	−0.304144
$565$	−2.08121	−0.0875573
$566$	−9.15271	−0.384717
$567$	52.7504	2.21531
$568$	−3.59151	−0.150696
$569$	−14.6995	−0.616235	−0.308118	−	0.951348i	$-0.599699\pi$
$569$	−14.6995	−0.616235	−0.308118	+	0.951348i	$0.599699\pi$
$570$	0.469419	0.0196618
$571$	−8.33082	−0.348634	−0.174317	−	0.984690i	$-0.555772\pi$
$571$	−8.33082	−0.348634	−0.174317	+	0.984690i	$0.555772\pi$
$572$	19.0347	0.795882
$573$	79.1585	3.30689
$574$	18.4491	0.770050
$575$	20.1666	0.841007
$576$	6.89495	0.287289
$577$	−23.5027	−0.978431	−0.489216	−	0.872163i	$-0.662717\pi$
$577$	−23.5027	−0.978431	−0.489216	+	0.872163i	$0.662717\pi$
$578$	−9.05865	−0.376790
$579$	23.4954	0.976434
$580$	0.587028	0.0243750
$581$	−7.83258	−0.324950
$582$	58.8008	2.43737
$583$	−6.92063	−0.286623
$584$	15.7187	0.650444
$585$	12.3035	0.508688
$586$	23.8266	0.984269
$587$	16.9211	0.698408	0.349204	−	0.937047i	$-0.386452\pi$
$587$	16.9211	0.698408	0.349204	+	0.937047i	$0.386452\pi$
$588$	−5.43532	−0.224149
$589$	0.0796546	0.00328211
$590$	−0.931230	−0.0383381
$591$	−7.13656	−0.293559
$592$	0.947971	0.0389614
$593$	−37.1722	−1.52648	−0.763239	−	0.646117i	$-0.776392\pi$
$593$	−37.1722	−1.52648	−0.763239	+	0.646117i	$0.776392\pi$
$594$	39.6337	1.62619
$595$	−2.52470	−0.103503
$596$	6.46644	0.264876
$597$	−71.4966	−2.92616
$598$	24.1778	0.988702
$599$	15.9269	0.650757	0.325379	−	0.945584i	$-0.394508\pi$
$599$	15.9269	0.650757	0.325379	+	0.945584i	$0.394508\pi$
$600$	15.4388	0.630288
$601$	−26.4878	−1.08046	−0.540229	−	0.841518i	$-0.681663\pi$
$601$	−26.4878	−1.08046	−0.540229	+	0.841518i	$0.681663\pi$
$602$	21.0684	0.858686
$603$	−33.0842	−1.34729
$604$	−15.8082	−0.643226
$605$	0.162433	0.00660384
$606$	25.0792	1.01877
$607$	12.9074	0.523896	0.261948	−	0.965082i	$-0.415635\pi$
$607$	12.9074	0.523896	0.261948	+	0.965082i	$0.415635\pi$
$608$	0.492091	0.0199569
$609$	17.9892	0.728960
$610$	2.53007	0.102440
$611$	−13.5115	−0.546615
$612$	19.4302	0.785421
$613$	−41.9325	−1.69364	−0.846818	−	0.531882i	$-0.821485\pi$
$613$	−41.9325	−1.69364	−0.846818	+	0.531882i	$0.821485\pi$
$614$	6.45305	0.260424
$615$	5.95710	0.240213
$616$	−9.55677	−0.385053
$617$	26.2192	1.05555	0.527773	−	0.849385i	$-0.323027\pi$
$617$	26.2192	1.05555	0.527773	+	0.849385i	$0.323027\pi$
$618$	−39.8667	−1.60367
$619$	−26.9006	−1.08123	−0.540613	−	0.841271i	$-0.681808\pi$
$619$	−26.9006	−1.08123	−0.540613	+	0.841271i	$0.681808\pi$
$620$	−0.0490877	−0.00197141
$621$	50.3424	2.02017
$622$	−23.8424	−0.955992
$623$	17.8004	0.713156
$624$	18.5096	0.740977
$625$	23.6290	0.945160
$626$	13.2515	0.529636
$627$	5.00736	0.199975
$628$	−0.611020	−0.0243823
$629$	2.67142	0.106517
$630$	−6.17723	−0.246107
$631$	8.44045	0.336009	0.168004	−	0.985786i	$-0.446268\pi$
$631$	8.44045	0.336009	0.168004	+	0.985786i	$0.446268\pi$
$632$	16.7051	0.664494
$633$	−5.46669	−0.217281
$634$	18.4246	0.731734
$635$	−0.133015	−0.00527853
$636$	−6.72971	−0.266850
$637$	−10.1674	−0.402846
$638$	6.26192	0.247912
$639$	−24.7633	−0.979619
$640$	−0.303255	−0.0119872
$641$	22.4928	0.888411	0.444206	−	0.895925i	$-0.353486\pi$
$641$	22.4928	0.888411	0.444206	+	0.895925i	$0.353486\pi$
$642$	37.8413	1.49348
$643$	9.10849	0.359204	0.179602	−	0.983739i	$-0.442519\pi$
$643$	9.10849	0.359204	0.179602	+	0.983739i	$0.442519\pi$
$644$	−12.1389	−0.478341
$645$	6.80288	0.267863
$646$	1.38673	0.0545603
$647$	−5.88118	−0.231213	−0.115607	−	0.993295i	$-0.536881\pi$
$647$	−5.88118	−0.231213	−0.115607	+	0.993295i	$0.536881\pi$
$648$	17.8555	0.701429
$649$	−9.93357	−0.389927
$650$	28.8801	1.13277
$651$	−1.50427	−0.0589571
$652$	−3.84770	−0.150687
$653$	23.6791	0.926635	0.463317	−	0.886192i	$-0.346659\pi$
$653$	23.6791	0.926635	0.463317	+	0.886192i	$0.346659\pi$
$654$	10.4921	0.410274
$655$	−2.83626	−0.110822
$656$	6.24482	0.243819
$657$	108.379	4.22829
$658$	6.78370	0.264456
$659$	0.589169	0.0229508	0.0114754	−	0.999934i	$-0.496347\pi$
$659$	0.589169	0.0229508	0.0114754	+	0.999934i	$0.496347\pi$
$660$	−3.08583	−0.120116
$661$	22.7043	0.883095	0.441548	−	0.897238i	$-0.354430\pi$
$661$	22.7043	0.883095	0.441548	+	0.897238i	$0.354430\pi$
$662$	21.5572	0.837845
$663$	52.1608	2.02576
$664$	−2.65124	−0.102888
$665$	−0.440868	−0.0170961
$666$	6.53621	0.253273
$667$	7.95384	0.307974
$668$	14.7140	0.569300
$669$	−77.6207	−3.00099
$670$	1.45512	0.0562160
$671$	26.9887	1.04189
$672$	−9.29312	−0.358490
$673$	9.60811	0.370365	0.185183	−	0.982704i	$-0.440712\pi$
$673$	9.60811	0.370365	0.185183	+	0.982704i	$0.440712\pi$
$674$	6.50351	0.250506
$675$	60.1335	2.31454
$676$	21.6243	0.831702
$677$	−45.5799	−1.75178	−0.875889	−	0.482513i	$-0.839724\pi$
$677$	−45.5799	−1.75178	−0.875889	+	0.482513i	$0.839724\pi$
$678$	−21.5881	−0.829088
$679$	−55.2244	−2.11932
$680$	−0.854585	−0.0327718
$681$	−66.9714	−2.56635
$682$	−0.523626	−0.0200507
$683$	24.9012	0.952819	0.476409	−	0.879224i	$-0.341938\pi$
$683$	24.9012	0.952819	0.476409	+	0.879224i	$0.341938\pi$
$684$	3.39294	0.129732
$685$	−4.74763	−0.181398
$686$	−15.5754	−0.594671
$687$	17.8989	0.682885
$688$	7.13145	0.271884
$689$	−12.5887	−0.479590
$690$	−3.91959	−0.149216
$691$	−26.3994	−1.00428	−0.502140	−	0.864786i	$-0.667454\pi$
$691$	−26.3994	−1.00428	−0.502140	+	0.864786i	$0.667454\pi$
$692$	−11.4450	−0.435074
$693$	−65.8935	−2.50309
$694$	2.36907	0.0899287
$695$	−1.27440	−0.0483409
$696$	6.08917	0.230809
$697$	17.5982	0.666577
$698$	1.56942	0.0594033
$699$	58.3844	2.20830
$700$	−14.4998	−0.548042
$701$	43.0991	1.62783	0.813915	−	0.580984i	$-0.197332\pi$
$701$	43.0991	1.62783	0.813915	+	0.580984i	$0.197332\pi$
$702$	72.0939	2.72101
$703$	0.466489	0.0175939
$704$	−3.23487	−0.121919
$705$	2.19042	0.0824959
$706$	3.28232	0.123532
$707$	−23.5538	−0.885833
$708$	−9.65952	−0.363027
$709$	36.1419	1.35734	0.678668	−	0.734445i	$-0.262557\pi$
$709$	36.1419	1.35734	0.678668	+	0.734445i	$0.262557\pi$
$710$	1.08914	0.0408748
$711$	115.181	4.31962
$712$	6.02523	0.225805
$713$	−0.665106	−0.0249084
$714$	−26.1884	−0.980076
$715$	−5.77238	−0.215875
$716$	−2.05051	−0.0766313
$717$	−34.9462	−1.30509
$718$	11.4762	0.428287
$719$	−28.7823	−1.07340	−0.536698	−	0.843774i	$-0.680329\pi$
$719$	−28.7823	−1.07340	−0.536698	+	0.843774i	$0.680329\pi$
$720$	−2.09093	−0.0779243
$721$	37.4419	1.39441
$722$	−18.7578	−0.698095
$723$	−61.1360	−2.27367
$724$	4.27130	0.158742
$725$	9.50077	0.352850
$726$	1.68490	0.0625324
$727$	31.8566	1.18149	0.590747	−	0.806857i	$-0.298833\pi$
$727$	31.8566	1.18149	0.590747	+	0.806857i	$0.298833\pi$
$728$	−17.3838	−0.644287
$729$	7.49157	0.277466
$730$	−4.76677	−0.176426
$731$	20.0967	0.743304
$732$	26.2441	0.970010
$733$	5.46683	0.201922	0.100961	−	0.994890i	$-0.467808\pi$
$733$	5.46683	0.201922	0.100961	+	0.994890i	$0.467808\pi$
$734$	19.4851	0.719210
$735$	1.64829	0.0607980
$736$	−4.10890	−0.151456
$737$	15.5219	0.571758
$738$	43.0577	1.58498
$739$	25.1948	0.926806	0.463403	−	0.886148i	$-0.346628\pi$
$739$	25.1948	0.926806	0.463403	+	0.886148i	$0.346628\pi$
$740$	−0.287477	−0.0105679
$741$	9.10841	0.334606
$742$	6.32039	0.232029
$743$	−36.3378	−1.33311	−0.666553	−	0.745458i	$-0.732231\pi$
$743$	−36.3378	−1.33311	−0.666553	+	0.745458i	$0.732231\pi$
$744$	−0.509180	−0.0186675
$745$	−1.96098	−0.0718448
$746$	1.76946	0.0647844
$747$	−18.2802	−0.668837
$748$	−9.11599	−0.333313
$749$	−35.5397	−1.29859
$750$	−9.45154	−0.345121
$751$	−13.5363	−0.493948	−0.246974	−	0.969022i	$-0.579436\pi$
$751$	−13.5363	−0.493948	−0.246974	+	0.969022i	$0.579436\pi$
$752$	2.29621	0.0837343
$753$	−56.8725	−2.07255
$754$	11.3905	0.414816
$755$	4.79391	0.174468
$756$	−36.1962	−1.31644
$757$	31.1892	1.13359	0.566796	−	0.823858i	$-0.308183\pi$
$757$	31.1892	1.13359	0.566796	+	0.823858i	$0.308183\pi$
$758$	33.7902	1.22731
$759$	−41.8109	−1.51764
$760$	−0.149229	−0.00541311
$761$	−12.1048	−0.438799	−0.219400	−	0.975635i	$-0.570410\pi$
$761$	−12.1048	−0.438799	−0.219400	+	0.975635i	$0.570410\pi$
$762$	−1.37974	−0.0499829
$763$	−9.85395	−0.356737
$764$	−25.1646	−0.910424
$765$	−5.89232	−0.213037
$766$	26.8780	0.971141
$767$	−18.0692	−0.652441
$768$	−3.14562	−0.113508
$769$	9.12474	0.329047	0.164523	−	0.986373i	$-0.447391\pi$
$769$	9.12474	0.329047	0.164523	+	0.986373i	$0.447391\pi$
$770$	2.89814	0.104442
$771$	8.29601	0.298773
$772$	−7.46923	−0.268823
$773$	46.1234	1.65894	0.829472	−	0.558549i	$-0.188642\pi$
$773$	46.1234	1.65894	0.829472	+	0.558549i	$0.188642\pi$
$774$	49.1710	1.76741
$775$	−0.794461	−0.0285379
$776$	−18.6929	−0.671035
$777$	−8.80961	−0.316043
$778$	−18.0575	−0.647393
$779$	3.07302	0.110102
$780$	−5.61313	−0.200982
$781$	11.6180	0.415726
$782$	−11.5791	−0.414066
$783$	23.7170	0.847576
$784$	1.72790	0.0617107
$785$	0.185295	0.00661345
$786$	−29.4202	−1.04938
$787$	−30.7504	−1.09613	−0.548066	−	0.836435i	$-0.684636\pi$
$787$	−30.7504	−1.09613	−0.548066	+	0.836435i	$0.684636\pi$
$788$	2.26873	0.0808201
$789$	−85.0931	−3.02940
$790$	−5.06591	−0.180237
$791$	20.2751	0.720900
$792$	−22.3042	−0.792547
$793$	49.0925	1.74333
$794$	18.7731	0.666233
$795$	2.04082	0.0723803
$796$	22.7289	0.805604
$797$	13.1353	0.465275	0.232637	−	0.972564i	$-0.425264\pi$
$797$	13.1353	0.465275	0.232637	+	0.972564i	$0.425264\pi$
$798$	−4.57306	−0.161885
$799$	6.47082	0.228921
$800$	−4.90804	−0.173525
$801$	41.5437	1.46787
$802$	7.89163	0.278663
$803$	−50.8478	−1.79438
$804$	15.0937	0.532315
$805$	3.68119	0.129745
$806$	−0.952478	−0.0335496
$807$	−57.5432	−2.02562
$808$	−7.97273	−0.280480
$809$	5.29907	0.186305	0.0931526	−	0.995652i	$-0.470306\pi$
$809$	5.29907	0.186305	0.0931526	+	0.995652i	$0.470306\pi$
$810$	−5.41476	−0.190255
$811$	−14.4369	−0.506946	−0.253473	−	0.967342i	$-0.581573\pi$
$811$	−14.4369	−0.506946	−0.253473	+	0.967342i	$0.581573\pi$
$812$	−5.71881	−0.200691
$813$	16.9561	0.594675
$814$	−3.06656	−0.107483
$815$	1.16683	0.0408724
$816$	−8.86449	−0.310319
$817$	3.50932	0.122776
$818$	16.0069	0.559667
$819$	−119.860	−4.18826
$820$	−1.89377	−0.0661334
$821$	−2.40648	−0.0839867	−0.0419934	−	0.999118i	$-0.513371\pi$
$821$	−2.40648	−0.0839867	−0.0419934	+	0.999118i	$0.513371\pi$
$822$	−49.2465	−1.71767
$823$	−16.1579	−0.563230	−0.281615	−	0.959527i	$-0.590870\pi$
$823$	−16.1579	−0.563230	−0.281615	+	0.959527i	$0.590870\pi$
$824$	12.6737	0.441509
$825$	−49.9426	−1.73878
$826$	9.07201	0.315656
$827$	39.5402	1.37495	0.687474	−	0.726209i	$-0.258720\pi$
$827$	39.5402	1.37495	0.687474	+	0.726209i	$0.258720\pi$
$828$	−28.3307	−0.984559
$829$	29.4721	1.02361	0.511804	−	0.859103i	$-0.328978\pi$
$829$	29.4721	1.02361	0.511804	+	0.859103i	$0.328978\pi$
$830$	0.804003	0.0279074
$831$	−6.97381	−0.241919
$832$	−5.88424	−0.203999
$833$	4.86929	0.168711
$834$	−13.2192	−0.457744
$835$	−4.46208	−0.154417
$836$	−1.59185	−0.0550553
$837$	−1.98323	−0.0685505
$838$	21.3585	0.737817
$839$	38.8091	1.33984	0.669920	−	0.742433i	$-0.266328\pi$
$839$	38.8091	1.33984	0.669920	+	0.742433i	$0.266328\pi$
$840$	2.81819	0.0972367
$841$	−25.2528	−0.870788
$842$	−4.92380	−0.169685
$843$	78.6360	2.70837
$844$	1.73787	0.0598200
$845$	−6.55767	−0.225591
$846$	15.8323	0.544324
$847$	−1.58242	−0.0543725
$848$	2.13939	0.0734668
$849$	28.7910	0.988104
$850$	−13.8310	−0.474401
$851$	−3.89512	−0.133523
$852$	11.2975	0.387047
$853$	−0.242397	−0.00829951	−0.00414975	−	0.999991i	$-0.501321\pi$
$853$	−0.242397	−0.00829951	−0.00414975	+	0.999991i	$0.501321\pi$
$854$	−24.6479	−0.843434
$855$	−1.02893	−0.0351886
$856$	−12.0298	−0.411171
$857$	52.5023	1.79344	0.896722	−	0.442594i	$-0.145942\pi$
$857$	52.5023	1.79344	0.896722	+	0.442594i	$0.145942\pi$
$858$	−59.8761	−2.04414
$859$	−1.52244	−0.0519451	−0.0259725	−	0.999663i	$-0.508268\pi$
$859$	−1.52244	−0.0519451	−0.0259725	+	0.999663i	$0.508268\pi$
$860$	−2.16265	−0.0737457
$861$	−58.0339	−1.97779
$862$	4.66746	0.158974
$863$	−34.6031	−1.17790	−0.588951	−	0.808169i	$-0.700459\pi$
$863$	−34.6031	−1.17790	−0.588951	+	0.808169i	$0.700459\pi$
$864$	−12.2520	−0.416823
$865$	3.47076	0.118009
$866$	4.88242	0.165911
$867$	28.4951	0.967744
$868$	0.478211	0.0162315
$869$	−54.0389	−1.83314
$870$	−1.84657	−0.0626046
$871$	28.2345	0.956689
$872$	−3.33546	−0.112953
$873$	−128.886	−4.36214
$874$	−2.02196	−0.0683937
$875$	8.87668	0.300086
$876$	−49.4450	−1.67059
$877$	2.24602	0.0758427	0.0379214	−	0.999281i	$-0.487926\pi$
$877$	2.24602	0.0758427	0.0379214	+	0.999281i	$0.487926\pi$
$878$	−36.1862	−1.22123
$879$	−74.9496	−2.52799
$880$	0.980990	0.0330692
$881$	−37.4157	−1.26057	−0.630283	−	0.776366i	$-0.717061\pi$
$881$	−37.4157	−1.26057	−0.630283	+	0.776366i	$0.717061\pi$
$882$	11.9138	0.401157
$883$	31.2370	1.05121	0.525604	−	0.850729i	$-0.323839\pi$
$883$	31.2370	1.05121	0.525604	+	0.850729i	$0.323839\pi$
$884$	−16.5820	−0.557714
$885$	2.92930	0.0984673
$886$	29.9610	1.00656
$887$	10.6446	0.357412	0.178706	−	0.983902i	$-0.442809\pi$
$887$	10.6446	0.357412	0.178706	+	0.983902i	$0.442809\pi$
$888$	−2.98196	−0.100068
$889$	1.29583	0.0434606
$890$	−1.82718	−0.0612473
$891$	−57.7601	−1.93503
$892$	24.6758	0.826206
$893$	1.12995	0.0378122
$894$	−20.3410	−0.680305
$895$	0.621829	0.0207854
$896$	2.95430	0.0986963
$897$	−76.0541	−2.53937
$898$	−3.36038	−0.112137
$899$	−0.313340	−0.0104505
$900$	−33.8407	−1.12802
$901$	6.02888	0.200851
$902$	−20.2012	−0.672625
$903$	−66.2734	−2.20544
$904$	6.86291	0.228257
$905$	−1.29529	−0.0430570
$906$	49.7266	1.65206
$907$	−16.8847	−0.560649	−0.280324	−	0.959905i	$-0.590442\pi$
$907$	−16.8847	−0.560649	−0.280324	+	0.959905i	$0.590442\pi$
$908$	21.2903	0.706544
$909$	−54.9715	−1.82329
$910$	5.27173	0.174756
$911$	24.7243	0.819151	0.409576	−	0.912276i	$-0.365677\pi$
$911$	24.7243	0.819151	0.409576	+	0.912276i	$0.365677\pi$
$912$	−1.54793	−0.0512572
$913$	8.57642	0.283838
$914$	−4.78819	−0.158379
$915$	−7.95866	−0.263105
$916$	−5.69009	−0.188006
$917$	27.6308	0.912449
$918$	−34.5267	−1.13955
$919$	−51.0869	−1.68520	−0.842600	−	0.538540i	$-0.818976\pi$
$919$	−51.0869	−1.68520	−0.842600	+	0.538540i	$0.818976\pi$
$920$	1.24605	0.0410809
$921$	−20.2989	−0.668871
$922$	13.2480	0.436299
$923$	21.1333	0.695610
$924$	30.0620	0.988968
$925$	−4.65268	−0.152979
$926$	−29.5741	−0.971866
$927$	87.3845	2.87008
$928$	−1.93576	−0.0635444
$929$	−50.2108	−1.64736	−0.823681	−	0.567053i	$-0.808083\pi$
$929$	−50.2108	−1.64736	−0.823681	+	0.567053i	$0.808083\pi$
$930$	0.154412	0.00506335
$931$	0.850284	0.0278669
$932$	−18.5605	−0.607970
$933$	74.9991	2.45536
$934$	21.2078	0.693942
$935$	2.76447	0.0904078
$936$	−40.5715	−1.32612
$937$	−26.6170	−0.869539	−0.434769	−	0.900542i	$-0.643170\pi$
$937$	−26.6170	−0.869539	−0.434769	+	0.900542i	$0.643170\pi$
$938$	−14.1757	−0.462853
$939$	−41.6842	−1.36031
$940$	−0.696338	−0.0227120
$941$	−49.0234	−1.59812	−0.799059	−	0.601253i	$-0.794668\pi$
$941$	−49.0234	−1.59812	−0.799059	+	0.601253i	$0.794668\pi$
$942$	1.92204	0.0626234
$943$	−25.6594	−0.835584
$944$	3.07078	0.0999454
$945$	10.9767	0.357072
$946$	−23.0693	−0.750047
$947$	53.7593	1.74694	0.873471	−	0.486876i	$-0.161864\pi$
$947$	53.7593	1.74694	0.873471	+	0.486876i	$0.161864\pi$
$948$	−52.5480	−1.70668
$949$	−92.4924	−3.00243
$950$	−2.41520	−0.0783595
$951$	−57.9568	−1.87938
$952$	8.32534	0.269826
$953$	37.4620	1.21351	0.606757	−	0.794887i	$-0.292470\pi$
$953$	37.4620	1.21351	0.606757	+	0.794887i	$0.292470\pi$
$954$	14.7510	0.477580
$955$	7.63130	0.246943
$956$	11.1095	0.359306
$957$	−19.6976	−0.636734
$958$	16.9522	0.547702
$959$	46.2513	1.49353
$960$	0.953926	0.0307878
$961$	−30.9738	−0.999155
$962$	−5.57809	−0.179845
$963$	−82.9451	−2.67287
$964$	19.4353	0.625968
$965$	2.26508	0.0729155
$966$	38.1845	1.22857
$967$	−9.05105	−0.291062	−0.145531	−	0.989354i	$-0.546489\pi$
$967$	−9.05105	−0.291062	−0.145531	+	0.989354i	$0.546489\pi$
$968$	−0.535632	−0.0172159
$969$	−4.36214	−0.140132
$970$	5.66871	0.182011
$971$	9.43849	0.302895	0.151448	−	0.988465i	$-0.451606\pi$
$971$	9.43849	0.302895	0.151448	+	0.988465i	$0.451606\pi$
$972$	−19.4104	−0.622590
$973$	12.4152	0.398013
$974$	21.0786	0.675402
$975$	−90.8458	−2.90939
$976$	−8.34305	−0.267055
$977$	−1.51598	−0.0485005	−0.0242503	−	0.999706i	$-0.507720\pi$
$977$	−1.51598	−0.0485005	−0.0242503	+	0.999706i	$0.507720\pi$
$978$	12.1034	0.387024
$979$	−19.4908	−0.622929
$980$	−0.523994	−0.0167384
$981$	−22.9978	−0.734264
$982$	−2.50209	−0.0798450
$983$	−4.76318	−0.151922	−0.0759610	−	0.997111i	$-0.524202\pi$
$983$	−4.76318	−0.151922	−0.0759610	+	0.997111i	$0.524202\pi$
$984$	−19.6439	−0.626223
$985$	−0.688003	−0.0219216
$986$	−5.45504	−0.173724
$987$	−21.3390	−0.679227
$988$	−2.89558	−0.0921208
$989$	−29.3024	−0.931763
$990$	6.76387	0.214970
$991$	44.0745	1.40007	0.700036	−	0.714108i	$-0.253167\pi$
$991$	44.0745	1.40007	0.700036	+	0.714108i	$0.253167\pi$
$992$	0.161869	0.00513936
$993$	−67.8109	−2.15191
$994$	−10.6104	−0.336541
$995$	−6.89265	−0.218512
$996$	8.33982	0.264257
$997$	−10.6281	−0.336594	−0.168297	−	0.985736i	$-0.553827\pi$
$997$	−10.6281	−0.336594	−0.168297	+	0.985736i	$0.553827\pi$
$998$	28.4643	0.901023
$999$	−11.6146	−0.367469

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.d.1.3	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.d.1.3	✓	92	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.14562 q^{3} +1.00000 q^{4} -0.303255 q^{5} -3.14562 q^{6} +2.95430 q^{7} +1.00000 q^{8} +6.89495 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.14562 q^{3} +1.00000 q^{4} -0.303255 q^{5} -3.14562 q^{6} +2.95430 q^{7} +1.00000 q^{8} +6.89495 q^{9} -0.303255 q^{10} -3.23487 q^{11} -3.14562 q^{12} -5.88424 q^{13} +2.95430 q^{14} +0.953926 q^{15} +1.00000 q^{16} +2.81804 q^{17} +6.89495 q^{18} +0.492091 q^{19} -0.303255 q^{20} -9.29312 q^{21} -3.23487 q^{22} -4.10890 q^{23} -3.14562 q^{24} -4.90804 q^{25} -5.88424 q^{26} -12.2520 q^{27} +2.95430 q^{28} -1.93576 q^{29} +0.953926 q^{30} +0.161869 q^{31} +1.00000 q^{32} +10.1757 q^{33} +2.81804 q^{34} -0.895907 q^{35} +6.89495 q^{36} +0.947971 q^{37} +0.492091 q^{38} +18.5096 q^{39} -0.303255 q^{40} +6.24482 q^{41} -9.29312 q^{42} +7.13145 q^{43} -3.23487 q^{44} -2.09093 q^{45} -4.10890 q^{46} +2.29621 q^{47} -3.14562 q^{48} +1.72790 q^{49} -4.90804 q^{50} -8.86449 q^{51} -5.88424 q^{52} +2.13939 q^{53} -12.2520 q^{54} +0.980990 q^{55} +2.95430 q^{56} -1.54793 q^{57} -1.93576 q^{58} +3.07078 q^{59} +0.953926 q^{60} -8.34305 q^{61} +0.161869 q^{62} +20.3698 q^{63} +1.00000 q^{64} +1.78442 q^{65} +10.1757 q^{66} -4.79832 q^{67} +2.81804 q^{68} +12.9251 q^{69} -0.895907 q^{70} -3.59151 q^{71} +6.89495 q^{72} +15.7187 q^{73} +0.947971 q^{74} +15.4388 q^{75} +0.492091 q^{76} -9.55677 q^{77} +18.5096 q^{78} +16.7051 q^{79} -0.303255 q^{80} +17.8555 q^{81} +6.24482 q^{82} -2.65124 q^{83} -9.29312 q^{84} -0.854585 q^{85} +7.13145 q^{86} +6.08917 q^{87} -3.23487 q^{88} +6.02523 q^{89} -2.09093 q^{90} -17.3838 q^{91} -4.10890 q^{92} -0.509180 q^{93} +2.29621 q^{94} -0.149229 q^{95} -3.14562 q^{96} -18.6929 q^{97} +1.72790 q^{98} -22.3042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more