LMFDB - Embedded newform 4002.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$31.9561308889$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.56155 q^{5} -1.00000 q^{6} -1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.56155 q^{5} -1.00000 q^{6} -1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.56155 q^{10} -3.12311 q^{11} +1.00000 q^{12} -5.12311 q^{13} +1.56155 q^{14} -1.56155 q^{15} +1.00000 q^{16} +1.56155 q^{17} -1.00000 q^{18} -2.43845 q^{19} -1.56155 q^{20} -1.56155 q^{21} +3.12311 q^{22} -1.00000 q^{23} -1.00000 q^{24} -2.56155 q^{25} +5.12311 q^{26} +1.00000 q^{27} -1.56155 q^{28} -1.00000 q^{29} +1.56155 q^{30} -1.00000 q^{32} -3.12311 q^{33} -1.56155 q^{34} +2.43845 q^{35} +1.00000 q^{36} +4.43845 q^{37} +2.43845 q^{38} -5.12311 q^{39} +1.56155 q^{40} +4.43845 q^{41} +1.56155 q^{42} +2.43845 q^{43} -3.12311 q^{44} -1.56155 q^{45} +1.00000 q^{46} +8.68466 q^{47} +1.00000 q^{48} -4.56155 q^{49} +2.56155 q^{50} +1.56155 q^{51} -5.12311 q^{52} +7.12311 q^{53} -1.00000 q^{54} +4.87689 q^{55} +1.56155 q^{56} -2.43845 q^{57} +1.00000 q^{58} -8.68466 q^{59} -1.56155 q^{60} -2.00000 q^{61} -1.56155 q^{63} +1.00000 q^{64} +8.00000 q^{65} +3.12311 q^{66} -1.12311 q^{67} +1.56155 q^{68} -1.00000 q^{69} -2.43845 q^{70} +8.00000 q^{71} -1.00000 q^{72} +4.24621 q^{73} -4.43845 q^{74} -2.56155 q^{75} -2.43845 q^{76} +4.87689 q^{77} +5.12311 q^{78} +12.2462 q^{79} -1.56155 q^{80} +1.00000 q^{81} -4.43845 q^{82} +5.12311 q^{83} -1.56155 q^{84} -2.43845 q^{85} -2.43845 q^{86} -1.00000 q^{87} +3.12311 q^{88} +17.3693 q^{89} +1.56155 q^{90} +8.00000 q^{91} -1.00000 q^{92} -8.68466 q^{94} +3.80776 q^{95} -1.00000 q^{96} -8.00000 q^{97} +4.56155 q^{98} -3.12311 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9} - q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{13} - q^{14} + q^{15} + 2 q^{16} - q^{17} - 2 q^{18} - 9 q^{19} + q^{20} + q^{21} - 2 q^{22} - 2 q^{23} - 2 q^{24} - q^{25} + 2 q^{26} + 2 q^{27} + q^{28} - 2 q^{29} - q^{30} - 2 q^{32} + 2 q^{33} + q^{34} + 9 q^{35} + 2 q^{36} + 13 q^{37} + 9 q^{38} - 2 q^{39} - q^{40} + 13 q^{41} - q^{42} + 9 q^{43} + 2 q^{44} + q^{45} + 2 q^{46} + 5 q^{47} + 2 q^{48} - 5 q^{49} + q^{50} - q^{51} - 2 q^{52} + 6 q^{53} - 2 q^{54} + 18 q^{55} - q^{56} - 9 q^{57} + 2 q^{58} - 5 q^{59} + q^{60} - 4 q^{61} + q^{63} + 2 q^{64} + 16 q^{65} - 2 q^{66} + 6 q^{67} - q^{68} - 2 q^{69} - 9 q^{70} + 16 q^{71} - 2 q^{72} - 8 q^{73} - 13 q^{74} - q^{75} - 9 q^{76} + 18 q^{77} + 2 q^{78} + 8 q^{79} + q^{80} + 2 q^{81} - 13 q^{82} + 2 q^{83} + q^{84} - 9 q^{85} - 9 q^{86} - 2 q^{87} - 2 q^{88} + 10 q^{89} - q^{90} + 16 q^{91} - 2 q^{92} - 5 q^{94} - 13 q^{95} - 2 q^{96} - 16 q^{97} + 5 q^{98} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9 - q^10 + 2 * q^11 + 2 * q^12 - 2 * q^13 - q^14 + q^15 + 2 * q^16 - q^17 - 2 * q^18 - 9 * q^19 + q^20 + q^21 - 2 * q^22 - 2 * q^23 - 2 * q^24 - q^25 + 2 * q^26 + 2 * q^27 + q^28 - 2 * q^29 - q^30 - 2 * q^32 + 2 * q^33 + q^34 + 9 * q^35 + 2 * q^36 + 13 * q^37 + 9 * q^38 - 2 * q^39 - q^40 + 13 * q^41 - q^42 + 9 * q^43 + 2 * q^44 + q^45 + 2 * q^46 + 5 * q^47 + 2 * q^48 - 5 * q^49 + q^50 - q^51 - 2 * q^52 + 6 * q^53 - 2 * q^54 + 18 * q^55 - q^56 - 9 * q^57 + 2 * q^58 - 5 * q^59 + q^60 - 4 * q^61 + q^63 + 2 * q^64 + 16 * q^65 - 2 * q^66 + 6 * q^67 - q^68 - 2 * q^69 - 9 * q^70 + 16 * q^71 - 2 * q^72 - 8 * q^73 - 13 * q^74 - q^75 - 9 * q^76 + 18 * q^77 + 2 * q^78 + 8 * q^79 + q^80 + 2 * q^81 - 13 * q^82 + 2 * q^83 + q^84 - 9 * q^85 - 9 * q^86 - 2 * q^87 - 2 * q^88 + 10 * q^89 - q^90 + 16 * q^91 - 2 * q^92 - 5 * q^94 - 13 * q^95 - 2 * q^96 - 16 * q^97 + 5 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.56155	−0.698348	−0.349174	−	0.937058i	$-0.613538\pi$
$5$	−1.56155	−0.698348	−0.349174	+	0.937058i	$0.613538\pi$
$6$	−1.00000	−0.408248
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.56155	0.493806
$11$	−3.12311	−0.941652	−0.470826	−	0.882226i	$-0.656044\pi$
$11$	−3.12311	−0.941652	−0.470826	+	0.882226i	$0.656044\pi$
$12$	1.00000	0.288675
$13$	−5.12311	−1.42089	−0.710447	−	0.703751i	$-0.751507\pi$
$13$	−5.12311	−1.42089	−0.710447	+	0.703751i	$0.751507\pi$
$14$	1.56155	0.417343
$15$	−1.56155	−0.403191
$16$	1.00000	0.250000
$17$	1.56155	0.378732	0.189366	−	0.981907i	$-0.439357\pi$
$17$	1.56155	0.378732	0.189366	+	0.981907i	$0.439357\pi$
$18$	−1.00000	−0.235702
$19$	−2.43845	−0.559418	−0.279709	−	0.960085i	$-0.590238\pi$
$19$	−2.43845	−0.559418	−0.279709	+	0.960085i	$0.590238\pi$
$20$	−1.56155	−0.349174
$21$	−1.56155	−0.340759
$22$	3.12311	0.665848
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	−2.56155	−0.512311
$26$	5.12311	1.00472
$27$	1.00000	0.192450
$28$	−1.56155	−0.295106
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	1.56155	0.285099
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	−3.12311	−0.543663
$34$	−1.56155	−0.267804
$35$	2.43845	0.412173
$36$	1.00000	0.166667
$37$	4.43845	0.729676	0.364838	−	0.931071i	$-0.381124\pi$
$37$	4.43845	0.729676	0.364838	+	0.931071i	$0.381124\pi$
$38$	2.43845	0.395568
$39$	−5.12311	−0.820353
$40$	1.56155	0.246903
$41$	4.43845	0.693169	0.346584	−	0.938019i	$-0.387341\pi$
$41$	4.43845	0.693169	0.346584	+	0.938019i	$0.387341\pi$
$42$	1.56155	0.240953
$43$	2.43845	0.371860	0.185930	−	0.982563i	$-0.440470\pi$
$43$	2.43845	0.371860	0.185930	+	0.982563i	$0.440470\pi$
$44$	−3.12311	−0.470826
$45$	−1.56155	−0.232783
$46$	1.00000	0.147442
$47$	8.68466	1.26679	0.633394	−	0.773830i	$-0.281661\pi$
$47$	8.68466	1.26679	0.633394	+	0.773830i	$0.281661\pi$
$48$	1.00000	0.144338
$49$	−4.56155	−0.651650
$50$	2.56155	0.362258
$51$	1.56155	0.218661
$52$	−5.12311	−0.710447
$53$	7.12311	0.978434	0.489217	−	0.872162i	$-0.337283\pi$
$53$	7.12311	0.978434	0.489217	+	0.872162i	$0.337283\pi$
$54$	−1.00000	−0.136083
$55$	4.87689	0.657600
$56$	1.56155	0.208671
$57$	−2.43845	−0.322980
$58$	1.00000	0.131306
$59$	−8.68466	−1.13065	−0.565323	−	0.824870i	$-0.691249\pi$
$59$	−8.68466	−1.13065	−0.565323	+	0.824870i	$0.691249\pi$
$60$	−1.56155	−0.201596
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−1.56155	−0.196737
$64$	1.00000	0.125000
$65$	8.00000	0.992278
$66$	3.12311	0.384428
$67$	−1.12311	−0.137209	−0.0686046	−	0.997644i	$-0.521855\pi$
$67$	−1.12311	−0.137209	−0.0686046	+	0.997644i	$0.521855\pi$
$68$	1.56155	0.189366
$69$	−1.00000	−0.120386
$70$	−2.43845	−0.291450
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	4.24621	0.496981	0.248491	−	0.968634i	$-0.420065\pi$
$73$	4.24621	0.496981	0.248491	+	0.968634i	$0.420065\pi$
$74$	−4.43845	−0.515959
$75$	−2.56155	−0.295783
$76$	−2.43845	−0.279709
$77$	4.87689	0.555774
$78$	5.12311	0.580077
$79$	12.2462	1.37781	0.688903	−	0.724853i	$-0.258092\pi$
$79$	12.2462	1.37781	0.688903	+	0.724853i	$0.258092\pi$
$80$	−1.56155	−0.174587
$81$	1.00000	0.111111
$82$	−4.43845	−0.490144
$83$	5.12311	0.562334	0.281167	−	0.959659i	$-0.409279\pi$
$83$	5.12311	0.562334	0.281167	+	0.959659i	$0.409279\pi$
$84$	−1.56155	−0.170379
$85$	−2.43845	−0.264487
$86$	−2.43845	−0.262945
$87$	−1.00000	−0.107211
$88$	3.12311	0.332924
$89$	17.3693	1.84114	0.920572	−	0.390573i	$-0.127723\pi$
$89$	17.3693	1.84114	0.920572	+	0.390573i	$0.127723\pi$
$90$	1.56155	0.164602
$91$	8.00000	0.838628
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−8.68466	−0.895754
$95$	3.80776	0.390668
$96$	−1.00000	−0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	4.56155	0.460786
$99$	−3.12311	−0.313884
$100$	−2.56155	−0.256155
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	−1.56155	−0.154617
$103$	−6.43845	−0.634399	−0.317200	−	0.948359i	$-0.602742\pi$
$103$	−6.43845	−0.634399	−0.317200	+	0.948359i	$0.602742\pi$
$104$	5.12311	0.502362
$105$	2.43845	0.237968
$106$	−7.12311	−0.691857
$107$	−5.80776	−0.561458	−0.280729	−	0.959787i	$-0.590576\pi$
$107$	−5.80776	−0.561458	−0.280729	+	0.959787i	$0.590576\pi$
$108$	1.00000	0.0962250
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	−4.87689	−0.464994
$111$	4.43845	0.421279
$112$	−1.56155	−0.147553
$113$	−9.56155	−0.899475	−0.449738	−	0.893161i	$-0.648483\pi$
$113$	−9.56155	−0.899475	−0.449738	+	0.893161i	$0.648483\pi$
$114$	2.43845	0.228382
$115$	1.56155	0.145616
$116$	−1.00000	−0.0928477
$117$	−5.12311	−0.473631
$118$	8.68466	0.799488
$119$	−2.43845	−0.223532
$120$	1.56155	0.142550
$121$	−1.24621	−0.113292
$122$	2.00000	0.181071
$123$	4.43845	0.400201
$124$	0	0
$125$	11.8078	1.05612
$126$	1.56155	0.139114
$127$	−0.876894	−0.0778118	−0.0389059	−	0.999243i	$-0.512387\pi$
$127$	−0.876894	−0.0778118	−0.0389059	+	0.999243i	$0.512387\pi$
$128$	−1.00000	−0.0883883
$129$	2.43845	0.214693
$130$	−8.00000	−0.701646
$131$	18.2462	1.59418	0.797089	−	0.603861i	$-0.206372\pi$
$131$	18.2462	1.59418	0.797089	+	0.603861i	$0.206372\pi$
$132$	−3.12311	−0.271831
$133$	3.80776	0.330175
$134$	1.12311	0.0970215
$135$	−1.56155	−0.134397
$136$	−1.56155	−0.133902
$137$	7.12311	0.608568	0.304284	−	0.952581i	$-0.401583\pi$
$137$	7.12311	0.608568	0.304284	+	0.952581i	$0.401583\pi$
$138$	1.00000	0.0851257
$139$	3.12311	0.264898	0.132449	−	0.991190i	$-0.457716\pi$
$139$	3.12311	0.264898	0.132449	+	0.991190i	$0.457716\pi$
$140$	2.43845	0.206086
$141$	8.68466	0.731380
$142$	−8.00000	−0.671345
$143$	16.0000	1.33799
$144$	1.00000	0.0833333
$145$	1.56155	0.129680
$146$	−4.24621	−0.351419
$147$	−4.56155	−0.376231
$148$	4.43845	0.364838
$149$	7.31534	0.599296	0.299648	−	0.954050i	$-0.403131\pi$
$149$	7.31534	0.599296	0.299648	+	0.954050i	$0.403131\pi$
$150$	2.56155	0.209150
$151$	8.68466	0.706747	0.353374	−	0.935482i	$-0.385034\pi$
$151$	8.68466	0.706747	0.353374	+	0.935482i	$0.385034\pi$
$152$	2.43845	0.197784
$153$	1.56155	0.126244
$154$	−4.87689	−0.392991
$155$	0	0
$156$	−5.12311	−0.410177
$157$	−9.80776	−0.782745	−0.391372	−	0.920232i	$-0.628000\pi$
$157$	−9.80776	−0.782745	−0.391372	+	0.920232i	$0.628000\pi$
$158$	−12.2462	−0.974256
$159$	7.12311	0.564899
$160$	1.56155	0.123452
$161$	1.56155	0.123068
$162$	−1.00000	−0.0785674
$163$	4.68466	0.366931	0.183465	−	0.983026i	$-0.441268\pi$
$163$	4.68466	0.366931	0.183465	+	0.983026i	$0.441268\pi$
$164$	4.43845	0.346584
$165$	4.87689	0.379666
$166$	−5.12311	−0.397630
$167$	−9.36932	−0.725020	−0.362510	−	0.931980i	$-0.618080\pi$
$167$	−9.36932	−0.725020	−0.362510	+	0.931980i	$0.618080\pi$
$168$	1.56155	0.120476
$169$	13.2462	1.01894
$170$	2.43845	0.187020
$171$	−2.43845	−0.186473
$172$	2.43845	0.185930
$173$	10.6847	0.812340	0.406170	−	0.913798i	$-0.366864\pi$
$173$	10.6847	0.812340	0.406170	+	0.913798i	$0.366864\pi$
$174$	1.00000	0.0758098
$175$	4.00000	0.302372
$176$	−3.12311	−0.235413
$177$	−8.68466	−0.652779
$178$	−17.3693	−1.30189
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−1.56155	−0.116391
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	−8.00000	−0.592999
$183$	−2.00000	−0.147844
$184$	1.00000	0.0737210
$185$	−6.93087	−0.509568
$186$	0	0
$187$	−4.87689	−0.356634
$188$	8.68466	0.633394
$189$	−1.56155	−0.113586
$190$	−3.80776	−0.276244
$191$	18.6847	1.35197	0.675987	−	0.736913i	$-0.263717\pi$
$191$	18.6847	1.35197	0.675987	+	0.736913i	$0.263717\pi$
$192$	1.00000	0.0721688
$193$	17.1231	1.23255	0.616274	−	0.787532i	$-0.288641\pi$
$193$	17.1231	1.23255	0.616274	+	0.787532i	$0.288641\pi$
$194$	8.00000	0.574367
$195$	8.00000	0.572892
$196$	−4.56155	−0.325825
$197$	−22.6847	−1.61621	−0.808107	−	0.589035i	$-0.799508\pi$
$197$	−22.6847	−1.61621	−0.808107	+	0.589035i	$0.799508\pi$
$198$	3.12311	0.221949
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	2.56155	0.181129
$201$	−1.12311	−0.0792178
$202$	−2.00000	−0.140720
$203$	1.56155	0.109600
$204$	1.56155	0.109331
$205$	−6.93087	−0.484073
$206$	6.43845	0.448588
$207$	−1.00000	−0.0695048
$208$	−5.12311	−0.355223
$209$	7.61553	0.526777
$210$	−2.43845	−0.168269
$211$	−18.9309	−1.30325	−0.651627	−	0.758539i	$-0.725913\pi$
$211$	−18.9309	−1.30325	−0.651627	+	0.758539i	$0.725913\pi$
$212$	7.12311	0.489217
$213$	8.00000	0.548151
$214$	5.80776	0.397011
$215$	−3.80776	−0.259687
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	0	0
$219$	4.24621	0.286932
$220$	4.87689	0.328800
$221$	−8.00000	−0.538138
$222$	−4.43845	−0.297889
$223$	4.49242	0.300835	0.150417	−	0.988623i	$-0.451938\pi$
$223$	4.49242	0.300835	0.150417	+	0.988623i	$0.451938\pi$
$224$	1.56155	0.104336
$225$	−2.56155	−0.170770
$226$	9.56155	0.636025
$227$	20.9309	1.38923	0.694615	−	0.719381i	$-0.255575\pi$
$227$	20.9309	1.38923	0.694615	+	0.719381i	$0.255575\pi$
$228$	−2.43845	−0.161490
$229$	5.80776	0.383788	0.191894	−	0.981416i	$-0.438537\pi$
$229$	5.80776	0.383788	0.191894	+	0.981416i	$0.438537\pi$
$230$	−1.56155	−0.102966
$231$	4.87689	0.320876
$232$	1.00000	0.0656532
$233$	−0.630683	−0.0413174	−0.0206587	−	0.999787i	$-0.506576\pi$
$233$	−0.630683	−0.0413174	−0.0206587	+	0.999787i	$0.506576\pi$
$234$	5.12311	0.334908
$235$	−13.5616	−0.884658
$236$	−8.68466	−0.565323
$237$	12.2462	0.795477
$238$	2.43845	0.158061
$239$	11.1231	0.719494	0.359747	−	0.933050i	$-0.382863\pi$
$239$	11.1231	0.719494	0.359747	+	0.933050i	$0.382863\pi$
$240$	−1.56155	−0.100798
$241$	1.31534	0.0847286	0.0423643	−	0.999102i	$-0.486511\pi$
$241$	1.31534	0.0847286	0.0423643	+	0.999102i	$0.486511\pi$
$242$	1.24621	0.0801095
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	7.12311	0.455079
$246$	−4.43845	−0.282985
$247$	12.4924	0.794874
$248$	0	0
$249$	5.12311	0.324664
$250$	−11.8078	−0.746789
$251$	2.24621	0.141780	0.0708898	−	0.997484i	$-0.477416\pi$
$251$	2.24621	0.141780	0.0708898	+	0.997484i	$0.477416\pi$
$252$	−1.56155	−0.0983686
$253$	3.12311	0.196348
$254$	0.876894	0.0550212
$255$	−2.43845	−0.152701
$256$	1.00000	0.0625000
$257$	−3.36932	−0.210172	−0.105086	−	0.994463i	$-0.533512\pi$
$257$	−3.36932	−0.210172	−0.105086	+	0.994463i	$0.533512\pi$
$258$	−2.43845	−0.151811
$259$	−6.93087	−0.430663
$260$	8.00000	0.496139
$261$	−1.00000	−0.0618984
$262$	−18.2462	−1.12725
$263$	1.80776	0.111472	0.0557358	−	0.998446i	$-0.482250\pi$
$263$	1.80776	0.111472	0.0557358	+	0.998446i	$0.482250\pi$
$264$	3.12311	0.192214
$265$	−11.1231	−0.683287
$266$	−3.80776	−0.233469
$267$	17.3693	1.06298
$268$	−1.12311	−0.0686046
$269$	−24.2462	−1.47832	−0.739159	−	0.673531i	$-0.764777\pi$
$269$	−24.2462	−1.47832	−0.739159	+	0.673531i	$0.764777\pi$
$270$	1.56155	0.0950331
$271$	16.4924	1.00184	0.500922	−	0.865493i	$-0.332994\pi$
$271$	16.4924	1.00184	0.500922	+	0.865493i	$0.332994\pi$
$272$	1.56155	0.0946830
$273$	8.00000	0.484182
$274$	−7.12311	−0.430323
$275$	8.00000	0.482418
$276$	−1.00000	−0.0601929
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−3.12311	−0.187311
$279$	0	0
$280$	−2.43845	−0.145725
$281$	−2.49242	−0.148685	−0.0743427	−	0.997233i	$-0.523686\pi$
$281$	−2.49242	−0.148685	−0.0743427	+	0.997233i	$0.523686\pi$
$282$	−8.68466	−0.517164
$283$	−18.4924	−1.09926	−0.549630	−	0.835408i	$-0.685231\pi$
$283$	−18.4924	−1.09926	−0.549630	+	0.835408i	$0.685231\pi$
$284$	8.00000	0.474713
$285$	3.80776	0.225552
$286$	−16.0000	−0.946100
$287$	−6.93087	−0.409116
$288$	−1.00000	−0.0589256
$289$	−14.5616	−0.856562
$290$	−1.56155	−0.0916975
$291$	−8.00000	−0.468968
$292$	4.24621	0.248491
$293$	30.4924	1.78139	0.890693	−	0.454605i	$-0.150220\pi$
$293$	30.4924	1.78139	0.890693	+	0.454605i	$0.150220\pi$
$294$	4.56155	0.266035
$295$	13.5616	0.789584
$296$	−4.43845	−0.257980
$297$	−3.12311	−0.181221
$298$	−7.31534	−0.423766
$299$	5.12311	0.296277
$300$	−2.56155	−0.147891
$301$	−3.80776	−0.219476
$302$	−8.68466	−0.499746
$303$	2.00000	0.114897
$304$	−2.43845	−0.139855
$305$	3.12311	0.178829
$306$	−1.56155	−0.0892680
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	4.87689	0.277887
$309$	−6.43845	−0.366270
$310$	0	0
$311$	−22.0540	−1.25057	−0.625283	−	0.780398i	$-0.715016\pi$
$311$	−22.0540	−1.25057	−0.625283	+	0.780398i	$0.715016\pi$
$312$	5.12311	0.290039
$313$	18.6847	1.05612	0.528060	−	0.849207i	$-0.322920\pi$
$313$	18.6847	1.05612	0.528060	+	0.849207i	$0.322920\pi$
$314$	9.80776	0.553484
$315$	2.43845	0.137391
$316$	12.2462	0.688903
$317$	17.1231	0.961729	0.480865	−	0.876795i	$-0.340323\pi$
$317$	17.1231	0.961729	0.480865	+	0.876795i	$0.340323\pi$
$318$	−7.12311	−0.399444
$319$	3.12311	0.174860
$320$	−1.56155	−0.0872935
$321$	−5.80776	−0.324158
$322$	−1.56155	−0.0870219
$323$	−3.80776	−0.211870
$324$	1.00000	0.0555556
$325$	13.1231	0.727939
$326$	−4.68466	−0.259459
$327$	0	0
$328$	−4.43845	−0.245072
$329$	−13.5616	−0.747673
$330$	−4.87689	−0.268464
$331$	−25.5616	−1.40499	−0.702495	−	0.711689i	$-0.747931\pi$
$331$	−25.5616	−1.40499	−0.702495	+	0.711689i	$0.747931\pi$
$332$	5.12311	0.281167
$333$	4.43845	0.243225
$334$	9.36932	0.512666
$335$	1.75379	0.0958197
$336$	−1.56155	−0.0851897
$337$	−15.1231	−0.823808	−0.411904	−	0.911227i	$-0.635136\pi$
$337$	−15.1231	−0.823808	−0.411904	+	0.911227i	$0.635136\pi$
$338$	−13.2462	−0.720499
$339$	−9.56155	−0.519312
$340$	−2.43845	−0.132243
$341$	0	0
$342$	2.43845	0.131856
$343$	18.0540	0.974823
$344$	−2.43845	−0.131472
$345$	1.56155	0.0840712
$346$	−10.6847	−0.574411
$347$	30.4384	1.63402	0.817011	−	0.576622i	$-0.195630\pi$
$347$	30.4384	1.63402	0.817011	+	0.576622i	$0.195630\pi$
$348$	−1.00000	−0.0536056
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	−4.00000	−0.213809
$351$	−5.12311	−0.273451
$352$	3.12311	0.166462
$353$	−20.7386	−1.10381	−0.551903	−	0.833908i	$-0.686098\pi$
$353$	−20.7386	−1.10381	−0.551903	+	0.833908i	$0.686098\pi$
$354$	8.68466	0.461584
$355$	−12.4924	−0.663029
$356$	17.3693	0.920572
$357$	−2.43845	−0.129056
$358$	0	0
$359$	7.06913	0.373094	0.186547	−	0.982446i	$-0.440270\pi$
$359$	7.06913	0.373094	0.186547	+	0.982446i	$0.440270\pi$
$360$	1.56155	0.0823011
$361$	−13.0540	−0.687051
$362$	12.0000	0.630706
$363$	−1.24621	−0.0654091
$364$	8.00000	0.419314
$365$	−6.63068	−0.347066
$366$	2.00000	0.104542
$367$	18.4924	0.965297	0.482648	−	0.875814i	$-0.339675\pi$
$367$	18.4924	0.965297	0.482648	+	0.875814i	$0.339675\pi$
$368$	−1.00000	−0.0521286
$369$	4.43845	0.231056
$370$	6.93087	0.360319
$371$	−11.1231	−0.577483
$372$	0	0
$373$	1.75379	0.0908077	0.0454039	−	0.998969i	$-0.485543\pi$
$373$	1.75379	0.0908077	0.0454039	+	0.998969i	$0.485543\pi$
$374$	4.87689	0.252178
$375$	11.8078	0.609750
$376$	−8.68466	−0.447877
$377$	5.12311	0.263853
$378$	1.56155	0.0803176
$379$	1.75379	0.0900861	0.0450430	−	0.998985i	$-0.485658\pi$
$379$	1.75379	0.0900861	0.0450430	+	0.998985i	$0.485658\pi$
$380$	3.80776	0.195334
$381$	−0.876894	−0.0449247
$382$	−18.6847	−0.955990
$383$	−10.6307	−0.543203	−0.271601	−	0.962410i	$-0.587553\pi$
$383$	−10.6307	−0.543203	−0.271601	+	0.962410i	$0.587553\pi$
$384$	−1.00000	−0.0510310
$385$	−7.61553	−0.388123
$386$	−17.1231	−0.871543
$387$	2.43845	0.123953
$388$	−8.00000	−0.406138
$389$	−19.3693	−0.982063	−0.491032	−	0.871142i	$-0.663380\pi$
$389$	−19.3693	−0.982063	−0.491032	+	0.871142i	$0.663380\pi$
$390$	−8.00000	−0.405096
$391$	−1.56155	−0.0789711
$392$	4.56155	0.230393
$393$	18.2462	0.920400
$394$	22.6847	1.14284
$395$	−19.1231	−0.962188
$396$	−3.12311	−0.156942
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	16.0000	0.802008
$399$	3.80776	0.190627
$400$	−2.56155	−0.128078
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	1.12311	0.0560154
$403$	0	0
$404$	2.00000	0.0995037
$405$	−1.56155	−0.0775942
$406$	−1.56155	−0.0774986
$407$	−13.8617	−0.687101
$408$	−1.56155	−0.0773084
$409$	1.12311	0.0555340	0.0277670	−	0.999614i	$-0.491160\pi$
$409$	1.12311	0.0555340	0.0277670	+	0.999614i	$0.491160\pi$
$410$	6.93087	0.342291
$411$	7.12311	0.351357
$412$	−6.43845	−0.317200
$413$	13.5616	0.667320
$414$	1.00000	0.0491473
$415$	−8.00000	−0.392705
$416$	5.12311	0.251181
$417$	3.12311	0.152939
$418$	−7.61553	−0.372488
$419$	−12.9309	−0.631714	−0.315857	−	0.948807i	$-0.602292\pi$
$419$	−12.9309	−0.631714	−0.315857	+	0.948807i	$0.602292\pi$
$420$	2.43845	0.118984
$421$	20.7386	1.01074	0.505370	−	0.862903i	$-0.331356\pi$
$421$	20.7386	1.01074	0.505370	+	0.862903i	$0.331356\pi$
$422$	18.9309	0.921540
$423$	8.68466	0.422263
$424$	−7.12311	−0.345929
$425$	−4.00000	−0.194029
$426$	−8.00000	−0.387601
$427$	3.12311	0.151138
$428$	−5.80776	−0.280729
$429$	16.0000	0.772487
$430$	3.80776	0.183627
$431$	18.2462	0.878889	0.439445	−	0.898270i	$-0.355175\pi$
$431$	18.2462	0.878889	0.439445	+	0.898270i	$0.355175\pi$
$432$	1.00000	0.0481125
$433$	21.3693	1.02694	0.513472	−	0.858106i	$-0.328359\pi$
$433$	21.3693	1.02694	0.513472	+	0.858106i	$0.328359\pi$
$434$	0	0
$435$	1.56155	0.0748707
$436$	0	0
$437$	2.43845	0.116647
$438$	−4.24621	−0.202892
$439$	−8.68466	−0.414496	−0.207248	−	0.978288i	$-0.566451\pi$
$439$	−8.68466	−0.414496	−0.207248	+	0.978288i	$0.566451\pi$
$440$	−4.87689	−0.232497
$441$	−4.56155	−0.217217
$442$	8.00000	0.380521
$443$	−32.4924	−1.54376	−0.771881	−	0.635767i	$-0.780684\pi$
$443$	−32.4924	−1.54376	−0.771881	+	0.635767i	$0.780684\pi$
$444$	4.43845	0.210639
$445$	−27.1231	−1.28576
$446$	−4.49242	−0.212722
$447$	7.31534	0.346004
$448$	−1.56155	−0.0737764
$449$	29.4233	1.38857	0.694286	−	0.719700i	$-0.255721\pi$
$449$	29.4233	1.38857	0.694286	+	0.719700i	$0.255721\pi$
$450$	2.56155	0.120753
$451$	−13.8617	−0.652724
$452$	−9.56155	−0.449738
$453$	8.68466	0.408041
$454$	−20.9309	−0.982334
$455$	−12.4924	−0.585654
$456$	2.43845	0.114191
$457$	33.4233	1.56348	0.781738	−	0.623607i	$-0.214334\pi$
$457$	33.4233	1.56348	0.781738	+	0.623607i	$0.214334\pi$
$458$	−5.80776	−0.271379
$459$	1.56155	0.0728870
$460$	1.56155	0.0728078
$461$	15.3693	0.715820	0.357910	−	0.933756i	$-0.383489\pi$
$461$	15.3693	0.715820	0.357910	+	0.933756i	$0.383489\pi$
$462$	−4.87689	−0.226894
$463$	28.4924	1.32416	0.662078	−	0.749435i	$-0.269675\pi$
$463$	28.4924	1.32416	0.662078	+	0.749435i	$0.269675\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	0.630683	0.0292158
$467$	−3.50758	−0.162311	−0.0811557	−	0.996701i	$-0.525861\pi$
$467$	−3.50758	−0.162311	−0.0811557	+	0.996701i	$0.525861\pi$
$468$	−5.12311	−0.236816
$469$	1.75379	0.0809824
$470$	13.5616	0.625548
$471$	−9.80776	−0.451918
$472$	8.68466	0.399744
$473$	−7.61553	−0.350162
$474$	−12.2462	−0.562487
$475$	6.24621	0.286596
$476$	−2.43845	−0.111766
$477$	7.12311	0.326145
$478$	−11.1231	−0.508759
$479$	25.6155	1.17040	0.585202	−	0.810888i	$-0.301015\pi$
$479$	25.6155	1.17040	0.585202	+	0.810888i	$0.301015\pi$
$480$	1.56155	0.0712748
$481$	−22.7386	−1.03679
$482$	−1.31534	−0.0599122
$483$	1.56155	0.0710531
$484$	−1.24621	−0.0566460
$485$	12.4924	0.567252
$486$	−1.00000	−0.0453609
$487$	23.3153	1.05652	0.528259	−	0.849083i	$-0.322845\pi$
$487$	23.3153	1.05652	0.528259	+	0.849083i	$0.322845\pi$
$488$	2.00000	0.0905357
$489$	4.68466	0.211848
$490$	−7.12311	−0.321789
$491$	−11.6155	−0.524201	−0.262101	−	0.965041i	$-0.584415\pi$
$491$	−11.6155	−0.524201	−0.262101	+	0.965041i	$0.584415\pi$
$492$	4.43845	0.200101
$493$	−1.56155	−0.0703288
$494$	−12.4924	−0.562061
$495$	4.87689	0.219200
$496$	0	0
$497$	−12.4924	−0.560362
$498$	−5.12311	−0.229572
$499$	−27.6155	−1.23624	−0.618120	−	0.786083i	$-0.712106\pi$
$499$	−27.6155	−1.23624	−0.618120	+	0.786083i	$0.712106\pi$
$500$	11.8078	0.528059
$501$	−9.36932	−0.418590
$502$	−2.24621	−0.100253
$503$	36.4384	1.62471	0.812355	−	0.583163i	$-0.198185\pi$
$503$	36.4384	1.62471	0.812355	+	0.583163i	$0.198185\pi$
$504$	1.56155	0.0695571
$505$	−3.12311	−0.138976
$506$	−3.12311	−0.138839
$507$	13.2462	0.588285
$508$	−0.876894	−0.0389059
$509$	14.1922	0.629060	0.314530	−	0.949248i	$-0.398153\pi$
$509$	14.1922	0.629060	0.314530	+	0.949248i	$0.398153\pi$
$510$	2.43845	0.107976
$511$	−6.63068	−0.293324
$512$	−1.00000	−0.0441942
$513$	−2.43845	−0.107660
$514$	3.36932	0.148614
$515$	10.0540	0.443031
$516$	2.43845	0.107347
$517$	−27.1231	−1.19287
$518$	6.93087	0.304525
$519$	10.6847	0.469004
$520$	−8.00000	−0.350823
$521$	−3.75379	−0.164456	−0.0822282	−	0.996614i	$-0.526204\pi$
$521$	−3.75379	−0.164456	−0.0822282	+	0.996614i	$0.526204\pi$
$522$	1.00000	0.0437688
$523$	−34.4924	−1.50825	−0.754124	−	0.656732i	$-0.771938\pi$
$523$	−34.4924	−1.50825	−0.754124	+	0.656732i	$0.771938\pi$
$524$	18.2462	0.797089
$525$	4.00000	0.174574
$526$	−1.80776	−0.0788223
$527$	0	0
$528$	−3.12311	−0.135916
$529$	1.00000	0.0434783
$530$	11.1231	0.483157
$531$	−8.68466	−0.376882
$532$	3.80776	0.165088
$533$	−22.7386	−0.984920
$534$	−17.3693	−0.751644
$535$	9.06913	0.392093
$536$	1.12311	0.0485108
$537$	0	0
$538$	24.2462	1.04533
$539$	14.2462	0.613628
$540$	−1.56155	−0.0671985
$541$	−3.06913	−0.131952	−0.0659761	−	0.997821i	$-0.521016\pi$
$541$	−3.06913	−0.131952	−0.0659761	+	0.997821i	$0.521016\pi$
$542$	−16.4924	−0.708410
$543$	−12.0000	−0.514969
$544$	−1.56155	−0.0669510
$545$	0	0
$546$	−8.00000	−0.342368
$547$	23.1231	0.988673	0.494336	−	0.869271i	$-0.335411\pi$
$547$	23.1231	0.988673	0.494336	+	0.869271i	$0.335411\pi$
$548$	7.12311	0.304284
$549$	−2.00000	−0.0853579
$550$	−8.00000	−0.341121
$551$	2.43845	0.103881
$552$	1.00000	0.0425628
$553$	−19.1231	−0.813197
$554$	−26.0000	−1.10463
$555$	−6.93087	−0.294199
$556$	3.12311	0.132449
$557$	24.6847	1.04592	0.522961	−	0.852356i	$-0.324827\pi$
$557$	24.6847	1.04592	0.522961	+	0.852356i	$0.324827\pi$
$558$	0	0
$559$	−12.4924	−0.528373
$560$	2.43845	0.103043
$561$	−4.87689	−0.205903
$562$	2.49242	0.105136
$563$	10.6307	0.448030	0.224015	−	0.974586i	$-0.428084\pi$
$563$	10.6307	0.448030	0.224015	+	0.974586i	$0.428084\pi$
$564$	8.68466	0.365690
$565$	14.9309	0.628146
$566$	18.4924	0.777294
$567$	−1.56155	−0.0655791
$568$	−8.00000	−0.335673
$569$	−9.06913	−0.380198	−0.190099	−	0.981765i	$-0.560881\pi$
$569$	−9.06913	−0.380198	−0.190099	+	0.981765i	$0.560881\pi$
$570$	−3.80776	−0.159490
$571$	42.4924	1.77825	0.889126	−	0.457662i	$-0.151313\pi$
$571$	42.4924	1.77825	0.889126	+	0.457662i	$0.151313\pi$
$572$	16.0000	0.668994
$573$	18.6847	0.780563
$574$	6.93087	0.289289
$575$	2.56155	0.106824
$576$	1.00000	0.0416667
$577$	5.61553	0.233777	0.116889	−	0.993145i	$-0.462708\pi$
$577$	5.61553	0.233777	0.116889	+	0.993145i	$0.462708\pi$
$578$	14.5616	0.605681
$579$	17.1231	0.711612
$580$	1.56155	0.0648400
$581$	−8.00000	−0.331896
$582$	8.00000	0.331611
$583$	−22.2462	−0.921344
$584$	−4.24621	−0.175709
$585$	8.00000	0.330759
$586$	−30.4924	−1.25963
$587$	6.43845	0.265743	0.132872	−	0.991133i	$-0.457580\pi$
$587$	6.43845	0.265743	0.132872	+	0.991133i	$0.457580\pi$
$588$	−4.56155	−0.188115
$589$	0	0
$590$	−13.5616	−0.558320
$591$	−22.6847	−0.933122
$592$	4.43845	0.182419
$593$	−2.87689	−0.118140	−0.0590699	−	0.998254i	$-0.518813\pi$
$593$	−2.87689	−0.118140	−0.0590699	+	0.998254i	$0.518813\pi$
$594$	3.12311	0.128143
$595$	3.80776	0.156103
$596$	7.31534	0.299648
$597$	−16.0000	−0.654836
$598$	−5.12311	−0.209499
$599$	36.9848	1.51116	0.755580	−	0.655056i	$-0.227355\pi$
$599$	36.9848	1.51116	0.755580	+	0.655056i	$0.227355\pi$
$600$	2.56155	0.104575
$601$	7.36932	0.300601	0.150300	−	0.988640i	$-0.451976\pi$
$601$	7.36932	0.300601	0.150300	+	0.988640i	$0.451976\pi$
$602$	3.80776	0.155193
$603$	−1.12311	−0.0457364
$604$	8.68466	0.353374
$605$	1.94602	0.0791172
$606$	−2.00000	−0.0812444
$607$	−12.0000	−0.487065	−0.243532	−	0.969893i	$-0.578306\pi$
$607$	−12.0000	−0.487065	−0.243532	+	0.969893i	$0.578306\pi$
$608$	2.43845	0.0988921
$609$	1.56155	0.0632773
$610$	−3.12311	−0.126451
$611$	−44.4924	−1.79997
$612$	1.56155	0.0631220
$613$	39.6155	1.60006	0.800028	−	0.599963i	$-0.204818\pi$
$613$	39.6155	1.60006	0.800028	+	0.599963i	$0.204818\pi$
$614$	−4.00000	−0.161427
$615$	−6.93087	−0.279480
$616$	−4.87689	−0.196496
$617$	−20.3002	−0.817255	−0.408627	−	0.912701i	$-0.633992\pi$
$617$	−20.3002	−0.817255	−0.408627	+	0.912701i	$0.633992\pi$
$618$	6.43845	0.258992
$619$	12.6847	0.509839	0.254920	−	0.966962i	$-0.417951\pi$
$619$	12.6847	0.509839	0.254920	+	0.966962i	$0.417951\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	22.0540	0.884284
$623$	−27.1231	−1.08666
$624$	−5.12311	−0.205088
$625$	−5.63068	−0.225227
$626$	−18.6847	−0.746789
$627$	7.61553	0.304135
$628$	−9.80776	−0.391372
$629$	6.93087	0.276352
$630$	−2.43845	−0.0971501
$631$	1.56155	0.0621644	0.0310822	−	0.999517i	$-0.490105\pi$
$631$	1.56155	0.0621644	0.0310822	+	0.999517i	$0.490105\pi$
$632$	−12.2462	−0.487128
$633$	−18.9309	−0.752435
$634$	−17.1231	−0.680045
$635$	1.36932	0.0543397
$636$	7.12311	0.282450
$637$	23.3693	0.925926
$638$	−3.12311	−0.123645
$639$	8.00000	0.316475
$640$	1.56155	0.0617258
$641$	33.6695	1.32987	0.664933	−	0.746903i	$-0.268460\pi$
$641$	33.6695	1.32987	0.664933	+	0.746903i	$0.268460\pi$
$642$	5.80776	0.229214
$643$	25.1231	0.990759	0.495379	−	0.868677i	$-0.335029\pi$
$643$	25.1231	0.990759	0.495379	+	0.868677i	$0.335029\pi$
$644$	1.56155	0.0615338
$645$	−3.80776	−0.149931
$646$	3.80776	0.149814
$647$	−1.36932	−0.0538334	−0.0269167	−	0.999638i	$-0.508569\pi$
$647$	−1.36932	−0.0538334	−0.0269167	+	0.999638i	$0.508569\pi$
$648$	−1.00000	−0.0392837
$649$	27.1231	1.06468
$650$	−13.1231	−0.514731
$651$	0	0
$652$	4.68466	0.183465
$653$	23.7538	0.929558	0.464779	−	0.885427i	$-0.346134\pi$
$653$	23.7538	0.929558	0.464779	+	0.885427i	$0.346134\pi$
$654$	0	0
$655$	−28.4924	−1.11329
$656$	4.43845	0.173292
$657$	4.24621	0.165660
$658$	13.5616	0.528684
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	4.87689	0.189833
$661$	12.4924	0.485899	0.242949	−	0.970039i	$-0.421885\pi$
$661$	12.4924	0.485899	0.242949	+	0.970039i	$0.421885\pi$
$662$	25.5616	0.993478
$663$	−8.00000	−0.310694
$664$	−5.12311	−0.198815
$665$	−5.94602	−0.230577
$666$	−4.43845	−0.171986
$667$	1.00000	0.0387202
$668$	−9.36932	−0.362510
$669$	4.49242	0.173687
$670$	−1.75379	−0.0677548
$671$	6.24621	0.241132
$672$	1.56155	0.0602382
$673$	−29.8078	−1.14900	−0.574502	−	0.818503i	$-0.694804\pi$
$673$	−29.8078	−1.14900	−0.574502	+	0.818503i	$0.694804\pi$
$674$	15.1231	0.582520
$675$	−2.56155	−0.0985942
$676$	13.2462	0.509470
$677$	0.246211	0.00946267	0.00473133	−	0.999989i	$-0.498494\pi$
$677$	0.246211	0.00946267	0.00473133	+	0.999989i	$0.498494\pi$
$678$	9.56155	0.367209
$679$	12.4924	0.479415
$680$	2.43845	0.0935102
$681$	20.9309	0.802073
$682$	0	0
$683$	−9.17708	−0.351151	−0.175576	−	0.984466i	$-0.556179\pi$
$683$	−9.17708	−0.351151	−0.175576	+	0.984466i	$0.556179\pi$
$684$	−2.43845	−0.0932364
$685$	−11.1231	−0.424992
$686$	−18.0540	−0.689304
$687$	5.80776	0.221580
$688$	2.43845	0.0929649
$689$	−36.4924	−1.39025
$690$	−1.56155	−0.0594473
$691$	−4.49242	−0.170900	−0.0854499	−	0.996342i	$-0.527233\pi$
$691$	−4.49242	−0.170900	−0.0854499	+	0.996342i	$0.527233\pi$
$692$	10.6847	0.406170
$693$	4.87689	0.185258
$694$	−30.4384	−1.15543
$695$	−4.87689	−0.184991
$696$	1.00000	0.0379049
$697$	6.93087	0.262525
$698$	14.0000	0.529908
$699$	−0.630683	−0.0238546
$700$	4.00000	0.151186
$701$	−34.5464	−1.30480	−0.652400	−	0.757875i	$-0.726238\pi$
$701$	−34.5464	−1.30480	−0.652400	+	0.757875i	$0.726238\pi$
$702$	5.12311	0.193359
$703$	−10.8229	−0.408194
$704$	−3.12311	−0.117706
$705$	−13.5616	−0.510758
$706$	20.7386	0.780509
$707$	−3.12311	−0.117456
$708$	−8.68466	−0.326389
$709$	−39.6155	−1.48779	−0.743896	−	0.668295i	$-0.767024\pi$
$709$	−39.6155	−1.48779	−0.743896	+	0.668295i	$0.767024\pi$
$710$	12.4924	0.468832
$711$	12.2462	0.459269
$712$	−17.3693	−0.650943
$713$	0	0
$714$	2.43845	0.0912566
$715$	−24.9848	−0.934380
$716$	0	0
$717$	11.1231	0.415400
$718$	−7.06913	−0.263818
$719$	50.7386	1.89223	0.946116	−	0.323828i	$-0.104970\pi$
$719$	50.7386	1.89223	0.946116	+	0.323828i	$0.104970\pi$
$720$	−1.56155	−0.0581956
$721$	10.0540	0.374430
$722$	13.0540	0.485819
$723$	1.31534	0.0489181
$724$	−12.0000	−0.445976
$725$	2.56155	0.0951337
$726$	1.24621	0.0462512
$727$	28.7386	1.06586	0.532928	−	0.846160i	$-0.321091\pi$
$727$	28.7386	1.06586	0.532928	+	0.846160i	$0.321091\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	6.63068	0.245413
$731$	3.80776	0.140835
$732$	−2.00000	−0.0739221
$733$	−19.7538	−0.729623	−0.364811	−	0.931081i	$-0.618867\pi$
$733$	−19.7538	−0.729623	−0.364811	+	0.931081i	$0.618867\pi$
$734$	−18.4924	−0.682568
$735$	7.12311	0.262740
$736$	1.00000	0.0368605
$737$	3.50758	0.129203
$738$	−4.43845	−0.163381
$739$	−26.2462	−0.965482	−0.482741	−	0.875763i	$-0.660359\pi$
$739$	−26.2462	−0.965482	−0.482741	+	0.875763i	$0.660359\pi$
$740$	−6.93087	−0.254784
$741$	12.4924	0.458921
$742$	11.1231	0.408342
$743$	12.9309	0.474388	0.237194	−	0.971462i	$-0.423772\pi$
$743$	12.9309	0.474388	0.237194	+	0.971462i	$0.423772\pi$
$744$	0	0
$745$	−11.4233	−0.418517
$746$	−1.75379	−0.0642108
$747$	5.12311	0.187445
$748$	−4.87689	−0.178317
$749$	9.06913	0.331379
$750$	−11.8078	−0.431159
$751$	19.7538	0.720826	0.360413	−	0.932793i	$-0.382636\pi$
$751$	19.7538	0.720826	0.360413	+	0.932793i	$0.382636\pi$
$752$	8.68466	0.316697
$753$	2.24621	0.0818565
$754$	−5.12311	−0.186573
$755$	−13.5616	−0.493555
$756$	−1.56155	−0.0567931
$757$	−6.68466	−0.242958	−0.121479	−	0.992594i	$-0.538764\pi$
$757$	−6.68466	−0.242958	−0.121479	+	0.992594i	$0.538764\pi$
$758$	−1.75379	−0.0637005
$759$	3.12311	0.113362
$760$	−3.80776	−0.138122
$761$	−24.6307	−0.892862	−0.446431	−	0.894818i	$-0.647305\pi$
$761$	−24.6307	−0.892862	−0.446431	+	0.894818i	$0.647305\pi$
$762$	0.876894	0.0317665
$763$	0	0
$764$	18.6847	0.675987
$765$	−2.43845	−0.0881622
$766$	10.6307	0.384102
$767$	44.4924	1.60653
$768$	1.00000	0.0360844
$769$	4.38447	0.158108	0.0790540	−	0.996870i	$-0.474810\pi$
$769$	4.38447	0.158108	0.0790540	+	0.996870i	$0.474810\pi$
$770$	7.61553	0.274445
$771$	−3.36932	−0.121343
$772$	17.1231	0.616274
$773$	−15.7538	−0.566624	−0.283312	−	0.959028i	$-0.591433\pi$
$773$	−15.7538	−0.566624	−0.283312	+	0.959028i	$0.591433\pi$
$774$	−2.43845	−0.0876482
$775$	0	0
$776$	8.00000	0.287183
$777$	−6.93087	−0.248644
$778$	19.3693	0.694424
$779$	−10.8229	−0.387771
$780$	8.00000	0.286446
$781$	−24.9848	−0.894028
$782$	1.56155	0.0558410
$783$	−1.00000	−0.0357371
$784$	−4.56155	−0.162913
$785$	15.3153	0.546628
$786$	−18.2462	−0.650821
$787$	−14.8769	−0.530304	−0.265152	−	0.964207i	$-0.585422\pi$
$787$	−14.8769	−0.530304	−0.265152	+	0.964207i	$0.585422\pi$
$788$	−22.6847	−0.808107
$789$	1.80776	0.0643581
$790$	19.1231	0.680370
$791$	14.9309	0.530881
$792$	3.12311	0.110975
$793$	10.2462	0.363854
$794$	2.00000	0.0709773
$795$	−11.1231	−0.394496
$796$	−16.0000	−0.567105
$797$	30.1080	1.06648	0.533239	−	0.845965i	$-0.320975\pi$
$797$	30.1080	1.06648	0.533239	+	0.845965i	$0.320975\pi$
$798$	−3.80776	−0.134793
$799$	13.5616	0.479773
$800$	2.56155	0.0905646
$801$	17.3693	0.613715
$802$	10.0000	0.353112
$803$	−13.2614	−0.467983
$804$	−1.12311	−0.0396089
$805$	−2.43845	−0.0859440
$806$	0	0
$807$	−24.2462	−0.853507
$808$	−2.00000	−0.0703598
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	1.56155	0.0548674
$811$	−36.0000	−1.26413	−0.632065	−	0.774915i	$-0.717793\pi$
$811$	−36.0000	−1.26413	−0.632065	+	0.774915i	$0.717793\pi$
$812$	1.56155	0.0547998
$813$	16.4924	0.578415
$814$	13.8617	0.485854
$815$	−7.31534	−0.256245
$816$	1.56155	0.0546653
$817$	−5.94602	−0.208025
$818$	−1.12311	−0.0392685
$819$	8.00000	0.279543
$820$	−6.93087	−0.242036
$821$	−38.4924	−1.34339	−0.671697	−	0.740826i	$-0.734434\pi$
$821$	−38.4924	−1.34339	−0.671697	+	0.740826i	$0.734434\pi$
$822$	−7.12311	−0.248447
$823$	−31.1231	−1.08488	−0.542442	−	0.840093i	$-0.682500\pi$
$823$	−31.1231	−1.08488	−0.542442	+	0.840093i	$0.682500\pi$
$824$	6.43845	0.224294
$825$	8.00000	0.278524
$826$	−13.5616	−0.471867
$827$	9.75379	0.339172	0.169586	−	0.985515i	$-0.445757\pi$
$827$	9.75379	0.339172	0.169586	+	0.985515i	$0.445757\pi$
$828$	−1.00000	−0.0347524
$829$	−17.8078	−0.618489	−0.309245	−	0.950983i	$-0.600076\pi$
$829$	−17.8078	−0.618489	−0.309245	+	0.950983i	$0.600076\pi$
$830$	8.00000	0.277684
$831$	26.0000	0.901930
$832$	−5.12311	−0.177612
$833$	−7.12311	−0.246801
$834$	−3.12311	−0.108144
$835$	14.6307	0.506316
$836$	7.61553	0.263389
$837$	0	0
$838$	12.9309	0.446689
$839$	39.5616	1.36582	0.682908	−	0.730504i	$-0.260715\pi$
$839$	39.5616	1.36582	0.682908	+	0.730504i	$0.260715\pi$
$840$	−2.43845	−0.0841344
$841$	1.00000	0.0344828
$842$	−20.7386	−0.714701
$843$	−2.49242	−0.0858436
$844$	−18.9309	−0.651627
$845$	−20.6847	−0.711574
$846$	−8.68466	−0.298585
$847$	1.94602	0.0668662
$848$	7.12311	0.244608
$849$	−18.4924	−0.634658
$850$	4.00000	0.137199
$851$	−4.43845	−0.152148
$852$	8.00000	0.274075
$853$	−40.9309	−1.40145	−0.700723	−	0.713433i	$-0.747139\pi$
$853$	−40.9309	−1.40145	−0.700723	+	0.713433i	$0.747139\pi$
$854$	−3.12311	−0.106870
$855$	3.80776	0.130223
$856$	5.80776	0.198505
$857$	48.7386	1.66488	0.832440	−	0.554115i	$-0.186943\pi$
$857$	48.7386	1.66488	0.832440	+	0.554115i	$0.186943\pi$
$858$	−16.0000	−0.546231
$859$	−40.7926	−1.39183	−0.695913	−	0.718126i	$-0.745000\pi$
$859$	−40.7926	−1.39183	−0.695913	+	0.718126i	$0.745000\pi$
$860$	−3.80776	−0.129844
$861$	−6.93087	−0.236203
$862$	−18.2462	−0.621468
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	−1.00000	−0.0340207
$865$	−16.6847	−0.567295
$866$	−21.3693	−0.726159
$867$	−14.5616	−0.494536
$868$	0	0
$869$	−38.2462	−1.29741
$870$	−1.56155	−0.0529416
$871$	5.75379	0.194960
$872$	0	0
$873$	−8.00000	−0.270759
$874$	−2.43845	−0.0824817
$875$	−18.4384	−0.623333
$876$	4.24621	0.143466
$877$	3.36932	0.113774	0.0568869	−	0.998381i	$-0.481883\pi$
$877$	3.36932	0.113774	0.0568869	+	0.998381i	$0.481883\pi$
$878$	8.68466	0.293093
$879$	30.4924	1.02848
$880$	4.87689	0.164400
$881$	−20.8769	−0.703360	−0.351680	−	0.936120i	$-0.614390\pi$
$881$	−20.8769	−0.703360	−0.351680	+	0.936120i	$0.614390\pi$
$882$	4.56155	0.153595
$883$	−10.7386	−0.361384	−0.180692	−	0.983540i	$-0.557834\pi$
$883$	−10.7386	−0.361384	−0.180692	+	0.983540i	$0.557834\pi$
$884$	−8.00000	−0.269069
$885$	13.5616	0.455867
$886$	32.4924	1.09160
$887$	−46.7386	−1.56933	−0.784665	−	0.619920i	$-0.787165\pi$
$887$	−46.7386	−1.56933	−0.784665	+	0.619920i	$0.787165\pi$
$888$	−4.43845	−0.148945
$889$	1.36932	0.0459254
$890$	27.1231	0.909169
$891$	−3.12311	−0.104628
$892$	4.49242	0.150417
$893$	−21.1771	−0.708664
$894$	−7.31534	−0.244662
$895$	0	0
$896$	1.56155	0.0521678
$897$	5.12311	0.171056
$898$	−29.4233	−0.981868
$899$	0	0
$900$	−2.56155	−0.0853851
$901$	11.1231	0.370564
$902$	13.8617	0.461545
$903$	−3.80776	−0.126714
$904$	9.56155	0.318013
$905$	18.7386	0.622893
$906$	−8.68466	−0.288528
$907$	−14.2462	−0.473038	−0.236519	−	0.971627i	$-0.576007\pi$
$907$	−14.2462	−0.473038	−0.236519	+	0.971627i	$0.576007\pi$
$908$	20.9309	0.694615
$909$	2.00000	0.0663358
$910$	12.4924	0.414120
$911$	−26.8769	−0.890471	−0.445236	−	0.895413i	$-0.646880\pi$
$911$	−26.8769	−0.890471	−0.445236	+	0.895413i	$0.646880\pi$
$912$	−2.43845	−0.0807451
$913$	−16.0000	−0.529523
$914$	−33.4233	−1.10554
$915$	3.12311	0.103247
$916$	5.80776	0.191894
$917$	−28.4924	−0.940903
$918$	−1.56155	−0.0515389
$919$	−34.5464	−1.13958	−0.569790	−	0.821790i	$-0.692976\pi$
$919$	−34.5464	−1.13958	−0.569790	+	0.821790i	$0.692976\pi$
$920$	−1.56155	−0.0514829
$921$	4.00000	0.131804
$922$	−15.3693	−0.506161
$923$	−40.9848	−1.34903
$924$	4.87689	0.160438
$925$	−11.3693	−0.373821
$926$	−28.4924	−0.936319
$927$	−6.43845	−0.211466
$928$	1.00000	0.0328266
$929$	−26.4924	−0.869188	−0.434594	−	0.900626i	$-0.643108\pi$
$929$	−26.4924	−0.869188	−0.434594	+	0.900626i	$0.643108\pi$
$930$	0	0
$931$	11.1231	0.364545
$932$	−0.630683	−0.0206587
$933$	−22.0540	−0.722015
$934$	3.50758	0.114771
$935$	7.61553	0.249054
$936$	5.12311	0.167454
$937$	24.9309	0.814456	0.407228	−	0.913327i	$-0.366495\pi$
$937$	24.9309	0.814456	0.407228	+	0.913327i	$0.366495\pi$
$938$	−1.75379	−0.0572632
$939$	18.6847	0.609751
$940$	−13.5616	−0.442329
$941$	−18.6307	−0.607343	−0.303671	−	0.952777i	$-0.598213\pi$
$941$	−18.6307	−0.607343	−0.303671	+	0.952777i	$0.598213\pi$
$942$	9.80776	0.319554
$943$	−4.43845	−0.144536
$944$	−8.68466	−0.282662
$945$	2.43845	0.0793227
$946$	7.61553	0.247602
$947$	−27.2311	−0.884890	−0.442445	−	0.896796i	$-0.645889\pi$
$947$	−27.2311	−0.884890	−0.442445	+	0.896796i	$0.645889\pi$
$948$	12.2462	0.397738
$949$	−21.7538	−0.706158
$950$	−6.24621	−0.202654
$951$	17.1231	0.555255
$952$	2.43845	0.0790305
$953$	17.1231	0.554672	0.277336	−	0.960773i	$-0.410549\pi$
$953$	17.1231	0.554672	0.277336	+	0.960773i	$0.410549\pi$
$954$	−7.12311	−0.230619
$955$	−29.1771	−0.944148
$956$	11.1231	0.359747
$957$	3.12311	0.100956
$958$	−25.6155	−0.827600
$959$	−11.1231	−0.359184
$960$	−1.56155	−0.0503989
$961$	−31.0000	−1.00000
$962$	22.7386	0.733123
$963$	−5.80776	−0.187153
$964$	1.31534	0.0423643
$965$	−26.7386	−0.860747
$966$	−1.56155	−0.0502421
$967$	18.2462	0.586759	0.293379	−	0.955996i	$-0.405220\pi$
$967$	18.2462	0.586759	0.293379	+	0.955996i	$0.405220\pi$
$968$	1.24621	0.0400547
$969$	−3.80776	−0.122323
$970$	−12.4924	−0.401108
$971$	0.876894	0.0281409	0.0140704	−	0.999901i	$-0.495521\pi$
$971$	0.876894	0.0281409	0.0140704	+	0.999901i	$0.495521\pi$
$972$	1.00000	0.0320750
$973$	−4.87689	−0.156346
$974$	−23.3153	−0.747072
$975$	13.1231	0.420276
$976$	−2.00000	−0.0640184
$977$	−23.3693	−0.747651	−0.373825	−	0.927499i	$-0.621954\pi$
$977$	−23.3693	−0.747651	−0.373825	+	0.927499i	$0.621954\pi$
$978$	−4.68466	−0.149799
$979$	−54.2462	−1.73372
$980$	7.12311	0.227539
$981$	0	0
$982$	11.6155	0.370666
$983$	−13.1231	−0.418562	−0.209281	−	0.977856i	$-0.567112\pi$
$983$	−13.1231	−0.418562	−0.209281	+	0.977856i	$0.567112\pi$
$984$	−4.43845	−0.141493
$985$	35.4233	1.12868
$986$	1.56155	0.0497300
$987$	−13.5616	−0.431669
$988$	12.4924	0.397437
$989$	−2.43845	−0.0775381
$990$	−4.87689	−0.154998
$991$	3.80776	0.120958	0.0604788	−	0.998169i	$-0.480737\pi$
$991$	3.80776	0.120958	0.0604788	+	0.998169i	$0.480737\pi$
$992$	0	0
$993$	−25.5616	−0.811171
$994$	12.4924	0.396236
$995$	24.9848	0.792073
$996$	5.12311	0.162332
$997$	−23.6695	−0.749621	−0.374810	−	0.927102i	$-0.622292\pi$
$997$	−23.6695	−0.749621	−0.374810	+	0.927102i	$0.622292\pi$
$998$	27.6155	0.874154
$999$	4.43845	0.140426

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.u.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.u.1.1	✓	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.56155 q^{5} -1.00000 q^{6} -1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.56155 q^{5} -1.00000 q^{6} -1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.56155 q^{10} -3.12311 q^{11} +1.00000 q^{12} -5.12311 q^{13} +1.56155 q^{14} -1.56155 q^{15} +1.00000 q^{16} +1.56155 q^{17} -1.00000 q^{18} -2.43845 q^{19} -1.56155 q^{20} -1.56155 q^{21} +3.12311 q^{22} -1.00000 q^{23} -1.00000 q^{24} -2.56155 q^{25} +5.12311 q^{26} +1.00000 q^{27} -1.56155 q^{28} -1.00000 q^{29} +1.56155 q^{30} -1.00000 q^{32} -3.12311 q^{33} -1.56155 q^{34} +2.43845 q^{35} +1.00000 q^{36} +4.43845 q^{37} +2.43845 q^{38} -5.12311 q^{39} +1.56155 q^{40} +4.43845 q^{41} +1.56155 q^{42} +2.43845 q^{43} -3.12311 q^{44} -1.56155 q^{45} +1.00000 q^{46} +8.68466 q^{47} +1.00000 q^{48} -4.56155 q^{49} +2.56155 q^{50} +1.56155 q^{51} -5.12311 q^{52} +7.12311 q^{53} -1.00000 q^{54} +4.87689 q^{55} +1.56155 q^{56} -2.43845 q^{57} +1.00000 q^{58} -8.68466 q^{59} -1.56155 q^{60} -2.00000 q^{61} -1.56155 q^{63} +1.00000 q^{64} +8.00000 q^{65} +3.12311 q^{66} -1.12311 q^{67} +1.56155 q^{68} -1.00000 q^{69} -2.43845 q^{70} +8.00000 q^{71} -1.00000 q^{72} +4.24621 q^{73} -4.43845 q^{74} -2.56155 q^{75} -2.43845 q^{76} +4.87689 q^{77} +5.12311 q^{78} +12.2462 q^{79} -1.56155 q^{80} +1.00000 q^{81} -4.43845 q^{82} +5.12311 q^{83} -1.56155 q^{84} -2.43845 q^{85} -2.43845 q^{86} -1.00000 q^{87} +3.12311 q^{88} +17.3693 q^{89} +1.56155 q^{90} +8.00000 q^{91} -1.00000 q^{92} -8.68466 q^{94} +3.80776 q^{95} -1.00000 q^{96} -8.00000 q^{97} +4.56155 q^{98} -3.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9} - q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{13} - q^{14} + q^{15} + 2 q^{16} - q^{17} - 2 q^{18} - 9 q^{19} + q^{20} + q^{21} - 2 q^{22} - 2 q^{23} - 2 q^{24} - q^{25} + 2 q^{26} + 2 q^{27} + q^{28} - 2 q^{29} - q^{30} - 2 q^{32} + 2 q^{33} + q^{34} + 9 q^{35} + 2 q^{36} + 13 q^{37} + 9 q^{38} - 2 q^{39} - q^{40} + 13 q^{41} - q^{42} + 9 q^{43} + 2 q^{44} + q^{45} + 2 q^{46} + 5 q^{47} + 2 q^{48} - 5 q^{49} + q^{50} - q^{51} - 2 q^{52} + 6 q^{53} - 2 q^{54} + 18 q^{55} - q^{56} - 9 q^{57} + 2 q^{58} - 5 q^{59} + q^{60} - 4 q^{61} + q^{63} + 2 q^{64} + 16 q^{65} - 2 q^{66} + 6 q^{67} - q^{68} - 2 q^{69} - 9 q^{70} + 16 q^{71} - 2 q^{72} - 8 q^{73} - 13 q^{74} - q^{75} - 9 q^{76} + 18 q^{77} + 2 q^{78} + 8 q^{79} + q^{80} + 2 q^{81} - 13 q^{82} + 2 q^{83} + q^{84} - 9 q^{85} - 9 q^{86} - 2 q^{87} - 2 q^{88} + 10 q^{89} - q^{90} + 16 q^{91} - 2 q^{92} - 5 q^{94} - 13 q^{95} - 2 q^{96} - 16 q^{97} + 5 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9 - q^10 + 2 * q^11 + 2 * q^12 - 2 * q^13 - q^14 + q^15 + 2 * q^16 - q^17 - 2 * q^18 - 9 * q^19 + q^20 + q^21 - 2 * q^22 - 2 * q^23 - 2 * q^24 - q^25 + 2 * q^26 + 2 * q^27 + q^28 - 2 * q^29 - q^30 - 2 * q^32 + 2 * q^33 + q^34 + 9 * q^35 + 2 * q^36 + 13 * q^37 + 9 * q^38 - 2 * q^39 - q^40 + 13 * q^41 - q^42 + 9 * q^43 + 2 * q^44 + q^45 + 2 * q^46 + 5 * q^47 + 2 * q^48 - 5 * q^49 + q^50 - q^51 - 2 * q^52 + 6 * q^53 - 2 * q^54 + 18 * q^55 - q^56 - 9 * q^57 + 2 * q^58 - 5 * q^59 + q^60 - 4 * q^61 + q^63 + 2 * q^64 + 16 * q^65 - 2 * q^66 + 6 * q^67 - q^68 - 2 * q^69 - 9 * q^70 + 16 * q^71 - 2 * q^72 - 8 * q^73 - 13 * q^74 - q^75 - 9 * q^76 + 18 * q^77 + 2 * q^78 + 8 * q^79 + q^80 + 2 * q^81 - 13 * q^82 + 2 * q^83 + q^84 - 9 * q^85 - 9 * q^86 - 2 * q^87 - 2 * q^88 + 10 * q^89 - q^90 + 16 * q^91 - 2 * q^92 - 5 * q^94 - 13 * q^95 - 2 * q^96 - 16 * q^97 + 5 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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