LMFDB - Embedded newform 4029.2.a.f.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4029,2,Mod(1,4029)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4029, 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("4029.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4029 = 3 \cdot 17 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4029.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:	$32.1717269744$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	4029.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.220817 q^{2} -1.00000 q^{3} -1.95124 q^{4} -3.79014 q^{5} -0.220817 q^{6} -3.98824 q^{7} -0.872503 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.220817 q^{2} -1.00000 q^{3} -1.95124 q^{4} -3.79014 q^{5} -0.220817 q^{6} -3.98824 q^{7} -0.872503 q^{8} +1.00000 q^{9} -0.836928 q^{10} -3.30218 q^{11} +1.95124 q^{12} +0.531345 q^{13} -0.880673 q^{14} +3.79014 q^{15} +3.70982 q^{16} +1.00000 q^{17} +0.220817 q^{18} +3.84760 q^{19} +7.39546 q^{20} +3.98824 q^{21} -0.729179 q^{22} +3.07235 q^{23} +0.872503 q^{24} +9.36513 q^{25} +0.117330 q^{26} -1.00000 q^{27} +7.78201 q^{28} -0.237786 q^{29} +0.836928 q^{30} +2.26472 q^{31} +2.56420 q^{32} +3.30218 q^{33} +0.220817 q^{34} +15.1160 q^{35} -1.95124 q^{36} -7.01645 q^{37} +0.849617 q^{38} -0.531345 q^{39} +3.30690 q^{40} +5.30414 q^{41} +0.880673 q^{42} -8.66151 q^{43} +6.44334 q^{44} -3.79014 q^{45} +0.678429 q^{46} +4.19422 q^{47} -3.70982 q^{48} +8.90607 q^{49} +2.06798 q^{50} -1.00000 q^{51} -1.03678 q^{52} +9.19560 q^{53} -0.220817 q^{54} +12.5157 q^{55} +3.47975 q^{56} -3.84760 q^{57} -0.0525073 q^{58} -3.05737 q^{59} -7.39546 q^{60} -9.10515 q^{61} +0.500089 q^{62} -3.98824 q^{63} -6.85341 q^{64} -2.01387 q^{65} +0.729179 q^{66} -10.4701 q^{67} -1.95124 q^{68} -3.07235 q^{69} +3.33787 q^{70} +7.20875 q^{71} -0.872503 q^{72} +8.40467 q^{73} -1.54935 q^{74} -9.36513 q^{75} -7.50759 q^{76} +13.1699 q^{77} -0.117330 q^{78} +1.00000 q^{79} -14.0607 q^{80} +1.00000 q^{81} +1.17125 q^{82} -0.683764 q^{83} -7.78201 q^{84} -3.79014 q^{85} -1.91261 q^{86} +0.237786 q^{87} +2.88116 q^{88} -7.82109 q^{89} -0.836928 q^{90} -2.11913 q^{91} -5.99489 q^{92} -2.26472 q^{93} +0.926157 q^{94} -14.5829 q^{95} -2.56420 q^{96} +9.23134 q^{97} +1.96661 q^{98} -3.30218 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + q^{2} - 22 q^{3} + 19 q^{4} + q^{5} - q^{6} - 15 q^{7} + 15 q^{8} + 22 q^{9}+O(q^{10}) $ 22 * q + q^2 - 22 * q^3 + 19 * q^4 + q^5 - q^6 - 15 * q^7 + 15 * q^8 + 22 * q^9	\( 22 q + q^{2} - 22 q^{3} + 19 q^{4} + q^{5} - q^{6} - 15 q^{7} + 15 q^{8} + 22 q^{9} - 13 q^{10} - 23 q^{11} - 19 q^{12} - 18 q^{13} - 9 q^{14} - q^{15} + 21 q^{16} + 22 q^{17} + q^{18} - 30 q^{19} - 7 q^{20} + 15 q^{21} + 4 q^{22} - 3 q^{23} - 15 q^{24} + 19 q^{25} - 7 q^{26} - 22 q^{27} - 25 q^{28} - 7 q^{29} + 13 q^{30} - 10 q^{31} + 31 q^{32} + 23 q^{33} + q^{34} - 11 q^{35} + 19 q^{36} - q^{37} - 29 q^{38} + 18 q^{39} - 59 q^{40} + 9 q^{42} - 43 q^{43} - 80 q^{44} + q^{45} - 43 q^{46} + 2 q^{47} - 21 q^{48} + 43 q^{49} + 25 q^{50} - 22 q^{51} - 5 q^{52} - q^{53} - q^{54} - 19 q^{55} - 8 q^{56} + 30 q^{57} - 43 q^{58} - 28 q^{59} + 7 q^{60} - 29 q^{61} - 3 q^{62} - 15 q^{63} + 23 q^{64} + 19 q^{65} - 4 q^{66} - 16 q^{67} + 19 q^{68} + 3 q^{69} - 5 q^{70} - q^{71} + 15 q^{72} - 19 q^{73} - 24 q^{74} - 19 q^{75} - 72 q^{76} + 24 q^{77} + 7 q^{78} + 22 q^{79} - 82 q^{80} + 22 q^{81} - 81 q^{82} - 29 q^{83} + 25 q^{84} + q^{85} - 42 q^{86} + 7 q^{87} - 43 q^{88} - 28 q^{89} - 13 q^{90} - 96 q^{91} - 11 q^{92} + 10 q^{93} - 63 q^{94} - 23 q^{95} - 31 q^{96} - 51 q^{97} + 12 q^{98} - 23 q^{99}+O(q^{100}) \) 22 * q + q^2 - 22 * q^3 + 19 * q^4 + q^5 - q^6 - 15 * q^7 + 15 * q^8 + 22 * q^9 - 13 * q^10 - 23 * q^11 - 19 * q^12 - 18 * q^13 - 9 * q^14 - q^15 + 21 * q^16 + 22 * q^17 + q^18 - 30 * q^19 - 7 * q^20 + 15 * q^21 + 4 * q^22 - 3 * q^23 - 15 * q^24 + 19 * q^25 - 7 * q^26 - 22 * q^27 - 25 * q^28 - 7 * q^29 + 13 * q^30 - 10 * q^31 + 31 * q^32 + 23 * q^33 + q^34 - 11 * q^35 + 19 * q^36 - q^37 - 29 * q^38 + 18 * q^39 - 59 * q^40 + 9 * q^42 - 43 * q^43 - 80 * q^44 + q^45 - 43 * q^46 + 2 * q^47 - 21 * q^48 + 43 * q^49 + 25 * q^50 - 22 * q^51 - 5 * q^52 - q^53 - q^54 - 19 * q^55 - 8 * q^56 + 30 * q^57 - 43 * q^58 - 28 * q^59 + 7 * q^60 - 29 * q^61 - 3 * q^62 - 15 * q^63 + 23 * q^64 + 19 * q^65 - 4 * q^66 - 16 * q^67 + 19 * q^68 + 3 * q^69 - 5 * q^70 - q^71 + 15 * q^72 - 19 * q^73 - 24 * q^74 - 19 * q^75 - 72 * q^76 + 24 * q^77 + 7 * q^78 + 22 * q^79 - 82 * q^80 + 22 * q^81 - 81 * q^82 - 29 * q^83 + 25 * q^84 + q^85 - 42 * q^86 + 7 * q^87 - 43 * q^88 - 28 * q^89 - 13 * q^90 - 96 * q^91 - 11 * q^92 + 10 * q^93 - 63 * q^94 - 23 * q^95 - 31 * q^96 - 51 * q^97 + 12 * q^98 - 23 * 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 $

$2$	0.220817	0.156142	0.0780708	−	0.996948i	$-0.475124\pi$
$2$	0.220817	0.156142	0.0780708	+	0.996948i	$0.475124\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.95124	−0.975620
$5$	−3.79014	−1.69500	−0.847500	−	0.530795i	$-0.821893\pi$
$5$	−3.79014	−1.69500	−0.847500	+	0.530795i	$0.821893\pi$
$6$	−0.220817	−0.0901483
$7$	−3.98824	−1.50741	−0.753707	−	0.657211i	$-0.771736\pi$
$7$	−3.98824	−1.50741	−0.753707	+	0.657211i	$0.771736\pi$
$8$	−0.872503	−0.308476
$9$	1.00000	0.333333
$10$	−0.836928	−0.264660
$11$	−3.30218	−0.995644	−0.497822	−	0.867279i	$-0.665867\pi$
$11$	−3.30218	−0.995644	−0.497822	+	0.867279i	$0.665867\pi$
$12$	1.95124	0.563274
$13$	0.531345	0.147369	0.0736843	−	0.997282i	$-0.476524\pi$
$13$	0.531345	0.147369	0.0736843	+	0.997282i	$0.476524\pi$
$14$	−0.880673	−0.235370
$15$	3.79014	0.978609
$16$	3.70982	0.927454
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0.220817	0.0520472
$19$	3.84760	0.882700	0.441350	−	0.897335i	$-0.354500\pi$
$19$	3.84760	0.882700	0.441350	+	0.897335i	$0.354500\pi$
$20$	7.39546	1.65368
$21$	3.98824	0.870306
$22$	−0.729179	−0.155461
$23$	3.07235	0.640629	0.320315	−	0.947311i	$-0.396211\pi$
$23$	3.07235	0.640629	0.320315	+	0.947311i	$0.396211\pi$
$24$	0.872503	0.178099
$25$	9.36513	1.87303
$26$	0.117330	0.0230103
$27$	−1.00000	−0.192450
$28$	7.78201	1.47066
$29$	−0.237786	−0.0441557	−0.0220779	−	0.999756i	$-0.507028\pi$
$29$	−0.237786	−0.0441557	−0.0220779	+	0.999756i	$0.507028\pi$
$30$	0.836928	0.152801
$31$	2.26472	0.406755	0.203377	−	0.979100i	$-0.434808\pi$
$31$	2.26472	0.406755	0.203377	+	0.979100i	$0.434808\pi$
$32$	2.56420	0.453290
$33$	3.30218	0.574835
$34$	0.220817	0.0378699
$35$	15.1160	2.55507
$36$	−1.95124	−0.325207
$37$	−7.01645	−1.15350	−0.576749	−	0.816922i	$-0.695679\pi$
$37$	−7.01645	−1.15350	−0.576749	+	0.816922i	$0.695679\pi$
$38$	0.849617	0.137826
$39$	−0.531345	−0.0850833
$40$	3.30690	0.522867
$41$	5.30414	0.828367	0.414183	−	0.910193i	$-0.364067\pi$
$41$	5.30414	0.828367	0.414183	+	0.910193i	$0.364067\pi$
$42$	0.880673	0.135891
$43$	−8.66151	−1.32087	−0.660434	−	0.750884i	$-0.729628\pi$
$43$	−8.66151	−1.32087	−0.660434	+	0.750884i	$0.729628\pi$
$44$	6.44334	0.971370
$45$	−3.79014	−0.565000
$46$	0.678429	0.100029
$47$	4.19422	0.611790	0.305895	−	0.952065i	$-0.401044\pi$
$47$	4.19422	0.611790	0.305895	+	0.952065i	$0.401044\pi$
$48$	−3.70982	−0.535466
$49$	8.90607	1.27230
$50$	2.06798	0.292457
$51$	−1.00000	−0.140028
$52$	−1.03678	−0.143776
$53$	9.19560	1.26311	0.631556	−	0.775330i	$-0.282417\pi$
$53$	9.19560	1.26311	0.631556	+	0.775330i	$0.282417\pi$
$54$	−0.220817	−0.0300494
$55$	12.5157	1.68762
$56$	3.47975	0.465001
$57$	−3.84760	−0.509627
$58$	−0.0525073	−0.00689455
$59$	−3.05737	−0.398036	−0.199018	−	0.979996i	$-0.563775\pi$
$59$	−3.05737	−0.398036	−0.199018	+	0.979996i	$0.563775\pi$
$60$	−7.39546	−0.954750
$61$	−9.10515	−1.16580	−0.582898	−	0.812546i	$-0.698081\pi$
$61$	−9.10515	−1.16580	−0.582898	+	0.812546i	$0.698081\pi$
$62$	0.500089	0.0635113
$63$	−3.98824	−0.502471
$64$	−6.85341	−0.856676
$65$	−2.01387	−0.249790
$66$	0.729179	0.0897557
$67$	−10.4701	−1.27912	−0.639561	−	0.768740i	$-0.720884\pi$
$67$	−10.4701	−1.27912	−0.639561	+	0.768740i	$0.720884\pi$
$68$	−1.95124	−0.236623
$69$	−3.07235	−0.369868
$70$	3.33787	0.398952
$71$	7.20875	0.855522	0.427761	−	0.903892i	$-0.359303\pi$
$71$	7.20875	0.855522	0.427761	+	0.903892i	$0.359303\pi$
$72$	−0.872503	−0.102825
$73$	8.40467	0.983693	0.491846	−	0.870682i	$-0.336322\pi$
$73$	8.40467	0.983693	0.491846	+	0.870682i	$0.336322\pi$
$74$	−1.54935	−0.180109
$75$	−9.36513	−1.08139
$76$	−7.50759	−0.861180
$77$	13.1699	1.50085
$78$	−0.117330	−0.0132850
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	−14.0607	−1.57203
$81$	1.00000	0.111111
$82$	1.17125	0.129342
$83$	−0.683764	−0.0750528	−0.0375264	−	0.999296i	$-0.511948\pi$
$83$	−0.683764	−0.0750528	−0.0375264	+	0.999296i	$0.511948\pi$
$84$	−7.78201	−0.849087
$85$	−3.79014	−0.411098
$86$	−1.91261	−0.206242
$87$	0.237786	0.0254933
$88$	2.88116	0.307133
$89$	−7.82109	−0.829034	−0.414517	−	0.910042i	$-0.636049\pi$
$89$	−7.82109	−0.829034	−0.414517	+	0.910042i	$0.636049\pi$
$90$	−0.836928	−0.0882200
$91$	−2.11913	−0.222145
$92$	−5.99489	−0.625011
$93$	−2.26472	−0.234840
$94$	0.926157	0.0955258
$95$	−14.5829	−1.49618
$96$	−2.56420	−0.261707
$97$	9.23134	0.937300	0.468650	−	0.883384i	$-0.344741\pi$
$97$	9.23134	0.937300	0.468650	+	0.883384i	$0.344741\pi$
$98$	1.96661	0.198658
$99$	−3.30218	−0.331881
$100$	−18.2736	−1.82736
$101$	5.24686	0.522082	0.261041	−	0.965328i	$-0.415934\pi$
$101$	5.24686	0.522082	0.261041	+	0.965328i	$0.415934\pi$
$102$	−0.220817	−0.0218642
$103$	16.9598	1.67110	0.835549	−	0.549416i	$-0.185150\pi$
$103$	16.9598	1.67110	0.835549	+	0.549416i	$0.185150\pi$
$104$	−0.463600	−0.0454597
$105$	−15.1160	−1.47517
$106$	2.03055	0.197224
$107$	−7.74033	−0.748286	−0.374143	−	0.927371i	$-0.622063\pi$
$107$	−7.74033	−0.748286	−0.374143	+	0.927371i	$0.622063\pi$
$108$	1.95124	0.187758
$109$	−10.0605	−0.963618	−0.481809	−	0.876276i	$-0.660020\pi$
$109$	−10.0605	−0.963618	−0.481809	+	0.876276i	$0.660020\pi$
$110$	2.76369	0.263507
$111$	7.01645	0.665972
$112$	−14.7956	−1.39806
$113$	−4.81919	−0.453352	−0.226676	−	0.973970i	$-0.572786\pi$
$113$	−4.81919	−0.453352	−0.226676	+	0.973970i	$0.572786\pi$
$114$	−0.849617	−0.0795740
$115$	−11.6446	−1.08587
$116$	0.463977	0.0430792
$117$	0.531345	0.0491228
$118$	−0.675121	−0.0621499
$119$	−3.98824	−0.365601
$120$	−3.30690	−0.301878
$121$	−0.0956188	−0.00869262
$122$	−2.01058	−0.182029
$123$	−5.30414	−0.478258
$124$	−4.41900	−0.396838
$125$	−16.5444	−1.47978
$126$	−0.880673	−0.0784566
$127$	0.276862	0.0245675	0.0122838	−	0.999925i	$-0.496090\pi$
$127$	0.276862	0.0245675	0.0122838	+	0.999925i	$0.496090\pi$
$128$	−6.64175	−0.587053
$129$	8.66151	0.762603
$130$	−0.444697	−0.0390025
$131$	−7.17489	−0.626873	−0.313436	−	0.949609i	$-0.601480\pi$
$131$	−7.17489	−0.626873	−0.313436	+	0.949609i	$0.601480\pi$
$132$	−6.44334	−0.560821
$133$	−15.3452	−1.33059
$134$	−2.31197	−0.199724
$135$	3.79014	0.326203
$136$	−0.872503	−0.0748165
$137$	18.3482	1.56759	0.783796	−	0.621018i	$-0.213281\pi$
$137$	18.3482	1.56759	0.783796	+	0.621018i	$0.213281\pi$
$138$	−0.678429	−0.0577517
$139$	−4.50301	−0.381940	−0.190970	−	0.981596i	$-0.561163\pi$
$139$	−4.50301	−0.381940	−0.190970	+	0.981596i	$0.561163\pi$
$140$	−29.4949	−2.49277
$141$	−4.19422	−0.353217
$142$	1.59182	0.133582
$143$	−1.75460	−0.146727
$144$	3.70982	0.309151
$145$	0.901241	0.0748440
$146$	1.85590	0.153595
$147$	−8.90607	−0.734560
$148$	13.6908	1.12537
$149$	9.59099	0.785724	0.392862	−	0.919597i	$-0.371485\pi$
$149$	9.59099	0.785724	0.392862	+	0.919597i	$0.371485\pi$
$150$	−2.06798	−0.168850
$151$	11.3043	0.919928	0.459964	−	0.887938i	$-0.347862\pi$
$151$	11.3043	0.919928	0.459964	+	0.887938i	$0.347862\pi$
$152$	−3.35704	−0.272292
$153$	1.00000	0.0808452
$154$	2.90814	0.234345
$155$	−8.58358	−0.689450
$156$	1.03678	0.0830089
$157$	3.34788	0.267190	0.133595	−	0.991036i	$-0.457348\pi$
$157$	3.34788	0.267190	0.133595	+	0.991036i	$0.457348\pi$
$158$	0.220817	0.0175673
$159$	−9.19560	−0.729258
$160$	−9.71866	−0.768327
$161$	−12.2533	−0.965693
$162$	0.220817	0.0173491
$163$	21.2770	1.66655	0.833273	−	0.552862i	$-0.186464\pi$
$163$	21.2770	1.66655	0.833273	+	0.552862i	$0.186464\pi$
$164$	−10.3496	−0.808171
$165$	−12.5157	−0.974346
$166$	−0.150987	−0.0117189
$167$	14.3890	1.11345	0.556726	−	0.830696i	$-0.312057\pi$
$167$	14.3890	1.11345	0.556726	+	0.830696i	$0.312057\pi$
$168$	−3.47975	−0.268469
$169$	−12.7177	−0.978283
$170$	−0.836928	−0.0641895
$171$	3.84760	0.294233
$172$	16.9007	1.28866
$173$	−9.78842	−0.744200	−0.372100	−	0.928193i	$-0.621362\pi$
$173$	−9.78842	−0.744200	−0.372100	+	0.928193i	$0.621362\pi$
$174$	0.0525073	0.00398057
$175$	−37.3504	−2.82342
$176$	−12.2505	−0.923414
$177$	3.05737	0.229806
$178$	−1.72703	−0.129447
$179$	−3.00748	−0.224789	−0.112395	−	0.993664i	$-0.535852\pi$
$179$	−3.00748	−0.224789	−0.112395	+	0.993664i	$0.535852\pi$
$180$	7.39546	0.551225
$181$	21.9820	1.63391	0.816956	−	0.576701i	$-0.195660\pi$
$181$	21.9820	1.63391	0.816956	+	0.576701i	$0.195660\pi$
$182$	−0.467941	−0.0346861
$183$	9.10515	0.673072
$184$	−2.68063	−0.197619
$185$	26.5933	1.95518
$186$	−0.500089	−0.0366683
$187$	−3.30218	−0.241479
$188$	−8.18393	−0.596874
$189$	3.98824	0.290102
$190$	−3.22017	−0.233615
$191$	−4.37193	−0.316342	−0.158171	−	0.987412i	$-0.550560\pi$
$191$	−4.37193	−0.316342	−0.158171	+	0.987412i	$0.550560\pi$
$192$	6.85341	0.494602
$193$	−1.50887	−0.108611	−0.0543054	−	0.998524i	$-0.517294\pi$
$193$	−1.50887	−0.108611	−0.0543054	+	0.998524i	$0.517294\pi$
$194$	2.03844	0.146351
$195$	2.01387	0.144216
$196$	−17.3779	−1.24128
$197$	5.46718	0.389520	0.194760	−	0.980851i	$-0.437607\pi$
$197$	5.46718	0.389520	0.194760	+	0.980851i	$0.437607\pi$
$198$	−0.729179	−0.0518205
$199$	0.604992	0.0428868	0.0214434	−	0.999770i	$-0.493174\pi$
$199$	0.604992	0.0428868	0.0214434	+	0.999770i	$0.493174\pi$
$200$	−8.17110	−0.577784
$201$	10.4701	0.738502
$202$	1.15860	0.0815187
$203$	0.948348	0.0665610
$204$	1.95124	0.136614
$205$	−20.1034	−1.40408
$206$	3.74502	0.260928
$207$	3.07235	0.213543
$208$	1.97119	0.136678
$209$	−12.7055	−0.878855
$210$	−3.33787	−0.230335
$211$	−4.72652	−0.325387	−0.162694	−	0.986677i	$-0.552018\pi$
$211$	−4.72652	−0.325387	−0.162694	+	0.986677i	$0.552018\pi$
$212$	−17.9428	−1.23232
$213$	−7.20875	−0.493936
$214$	−1.70920	−0.116839
$215$	32.8283	2.23887
$216$	0.872503	0.0593663
$217$	−9.03223	−0.613148
$218$	−2.22153	−0.150461
$219$	−8.40467	−0.567935
$220$	−24.4211	−1.64647
$221$	0.531345	0.0357421
$222$	1.54935	0.103986
$223$	5.24469	0.351210	0.175605	−	0.984461i	$-0.443812\pi$
$223$	5.24469	0.351210	0.175605	+	0.984461i	$0.443812\pi$
$224$	−10.2266	−0.683296
$225$	9.36513	0.624342
$226$	−1.06416	−0.0707870
$227$	−7.99497	−0.530645	−0.265323	−	0.964160i	$-0.585478\pi$
$227$	−7.99497	−0.530645	−0.265323	+	0.964160i	$0.585478\pi$
$228$	7.50759	0.497202
$229$	−5.67467	−0.374993	−0.187496	−	0.982265i	$-0.560037\pi$
$229$	−5.67467	−0.374993	−0.187496	+	0.982265i	$0.560037\pi$
$230$	−2.57134	−0.169549
$231$	−13.1699	−0.866515
$232$	0.207469	0.0136210
$233$	25.2059	1.65129	0.825645	−	0.564190i	$-0.190811\pi$
$233$	25.2059	1.65129	0.825645	+	0.564190i	$0.190811\pi$
$234$	0.117330	0.00767012
$235$	−15.8967	−1.03698
$236$	5.96566	0.388331
$237$	−1.00000	−0.0649570
$238$	−0.880673	−0.0570856
$239$	−2.86388	−0.185249	−0.0926247	−	0.995701i	$-0.529526\pi$
$239$	−2.86388	−0.185249	−0.0926247	+	0.995701i	$0.529526\pi$
$240$	14.0607	0.907615
$241$	−25.3131	−1.63056	−0.815280	−	0.579067i	$-0.803417\pi$
$241$	−25.3131	−1.63056	−0.815280	+	0.579067i	$0.803417\pi$
$242$	−0.0211143	−0.00135728
$243$	−1.00000	−0.0641500
$244$	17.7663	1.13737
$245$	−33.7552	−2.15654
$246$	−1.17125	−0.0746759
$247$	2.04440	0.130082
$248$	−1.97597	−0.125474
$249$	0.683764	0.0433318
$250$	−3.65330	−0.231055
$251$	8.09170	0.510743	0.255372	−	0.966843i	$-0.417802\pi$
$251$	8.09170	0.510743	0.255372	+	0.966843i	$0.417802\pi$
$252$	7.78201	0.490221
$253$	−10.1454	−0.637839
$254$	0.0611359	0.00383601
$255$	3.79014	0.237348
$256$	12.2402	0.765013
$257$	20.5050	1.27906	0.639532	−	0.768765i	$-0.279128\pi$
$257$	20.5050	1.27906	0.639532	+	0.768765i	$0.279128\pi$
$258$	1.91261	0.119074
$259$	27.9833	1.73880
$260$	3.92954	0.243700
$261$	−0.237786	−0.0147186
$262$	−1.58434	−0.0978809
$263$	−5.60359	−0.345532	−0.172766	−	0.984963i	$-0.555270\pi$
$263$	−5.60359	−0.345532	−0.172766	+	0.984963i	$0.555270\pi$
$264$	−2.88116	−0.177323
$265$	−34.8526	−2.14098
$266$	−3.38848	−0.207761
$267$	7.82109	0.478643
$268$	20.4296	1.24794
$269$	10.5131	0.640995	0.320497	−	0.947249i	$-0.396150\pi$
$269$	10.5131	0.640995	0.320497	+	0.947249i	$0.396150\pi$
$270$	0.836928	0.0509338
$271$	−25.1167	−1.52573	−0.762864	−	0.646559i	$-0.776207\pi$
$271$	−25.1167	−1.52573	−0.762864	+	0.646559i	$0.776207\pi$
$272$	3.70982	0.224941
$273$	2.11913	0.128256
$274$	4.05160	0.244766
$275$	−30.9253	−1.86487
$276$	5.99489	0.360850
$277$	−11.0027	−0.661089	−0.330545	−	0.943790i	$-0.607232\pi$
$277$	−11.0027	−0.661089	−0.330545	+	0.943790i	$0.607232\pi$
$278$	−0.994343	−0.0596367
$279$	2.26472	0.135585
$280$	−13.1887	−0.788177
$281$	11.5593	0.689568	0.344784	−	0.938682i	$-0.387952\pi$
$281$	11.5593	0.689568	0.344784	+	0.938682i	$0.387952\pi$
$282$	−0.926157	−0.0551518
$283$	−30.2843	−1.80022	−0.900108	−	0.435666i	$-0.856513\pi$
$283$	−30.2843	−1.80022	−0.900108	+	0.435666i	$0.856513\pi$
$284$	−14.0660	−0.834664
$285$	14.5829	0.863818
$286$	−0.387445	−0.0229101
$287$	−21.1542	−1.24869
$288$	2.56420	0.151097
$289$	1.00000	0.0588235
$290$	0.199010	0.0116863
$291$	−9.23134	−0.541150
$292$	−16.3995	−0.959710
$293$	13.2625	0.774804	0.387402	−	0.921911i	$-0.373372\pi$
$293$	13.2625	0.774804	0.387402	+	0.921911i	$0.373372\pi$
$294$	−1.96661	−0.114695
$295$	11.5878	0.674671
$296$	6.12187	0.355827
$297$	3.30218	0.191612
$298$	2.11786	0.122684
$299$	1.63248	0.0944086
$300$	18.2736	1.05503
$301$	34.5442	1.99109
$302$	2.49618	0.143639
$303$	−5.24686	−0.301424
$304$	14.2739	0.818664
$305$	34.5098	1.97602
$306$	0.220817	0.0126233
$307$	7.38733	0.421617	0.210809	−	0.977527i	$-0.432390\pi$
$307$	7.38733	0.421617	0.210809	+	0.977527i	$0.432390\pi$
$308$	−25.6976	−1.46426
$309$	−16.9598	−0.964809
$310$	−1.89540	−0.107652
$311$	−28.7553	−1.63056	−0.815282	−	0.579064i	$-0.803418\pi$
$311$	−28.7553	−1.63056	−0.815282	+	0.579064i	$0.803418\pi$
$312$	0.463600	0.0262462
$313$	−3.82591	−0.216253	−0.108127	−	0.994137i	$-0.534485\pi$
$313$	−3.82591	−0.216253	−0.108127	+	0.994137i	$0.534485\pi$
$314$	0.739269	0.0417194
$315$	15.1160	0.851689
$316$	−1.95124	−0.109766
$317$	11.4724	0.644354	0.322177	−	0.946679i	$-0.395585\pi$
$317$	11.4724	0.644354	0.322177	+	0.946679i	$0.395585\pi$
$318$	−2.03055	−0.113867
$319$	0.785212	0.0439634
$320$	25.9754	1.45207
$321$	7.74033	0.432023
$322$	−2.70574	−0.150785
$323$	3.84760	0.214086
$324$	−1.95124	−0.108402
$325$	4.97611	0.276025
$326$	4.69834	0.260217
$327$	10.0605	0.556345
$328$	−4.62787	−0.255532
$329$	−16.7276	−0.922220
$330$	−2.76369	−0.152136
$331$	13.6491	0.750222	0.375111	−	0.926980i	$-0.377605\pi$
$331$	13.6491	0.750222	0.375111	+	0.926980i	$0.377605\pi$
$332$	1.33419	0.0732230
$333$	−7.01645	−0.384499
$334$	3.17734	0.173856
$335$	39.6830	2.16811
$336$	14.7956	0.807168
$337$	−14.0476	−0.765224	−0.382612	−	0.923909i	$-0.624975\pi$
$337$	−14.0476	−0.765224	−0.382612	+	0.923909i	$0.624975\pi$
$338$	−2.80828	−0.152751
$339$	4.81919	0.261743
$340$	7.39546	0.401075
$341$	−7.47849	−0.404983
$342$	0.849617	0.0459420
$343$	−7.60185	−0.410462
$344$	7.55719	0.407456
$345$	11.6446	0.626926
$346$	−2.16145	−0.116201
$347$	−15.7903	−0.847666	−0.423833	−	0.905740i	$-0.639316\pi$
$347$	−15.7903	−0.847666	−0.423833	+	0.905740i	$0.639316\pi$
$348$	−0.463977	−0.0248718
$349$	−7.55392	−0.404352	−0.202176	−	0.979349i	$-0.564801\pi$
$349$	−7.55392	−0.404352	−0.202176	+	0.979349i	$0.564801\pi$
$350$	−8.24762	−0.440854
$351$	−0.531345	−0.0283611
$352$	−8.46744	−0.451316
$353$	14.3700	0.764838	0.382419	−	0.923989i	$-0.375091\pi$
$353$	14.3700	0.764838	0.382419	+	0.923989i	$0.375091\pi$
$354$	0.675121	0.0358823
$355$	−27.3222	−1.45011
$356$	15.2608	0.808822
$357$	3.98824	0.211080
$358$	−0.664103	−0.0350989
$359$	6.63223	0.350036	0.175018	−	0.984565i	$-0.444002\pi$
$359$	6.63223	0.350036	0.175018	+	0.984565i	$0.444002\pi$
$360$	3.30690	0.174289
$361$	−4.19597	−0.220841
$362$	4.85401	0.255121
$363$	0.0956188	0.00501869
$364$	4.13493	0.216729
$365$	−31.8549	−1.66736
$366$	2.01058	0.105095
$367$	−37.2812	−1.94606	−0.973031	−	0.230676i	$-0.925906\pi$
$367$	−37.2812	−1.94606	−0.973031	+	0.230676i	$0.925906\pi$
$368$	11.3979	0.594154
$369$	5.30414	0.276122
$370$	5.87226	0.305284
$371$	−36.6743	−1.90403
$372$	4.41900	0.229115
$373$	30.2400	1.56577	0.782885	−	0.622167i	$-0.213747\pi$
$373$	30.2400	1.56577	0.782885	+	0.622167i	$0.213747\pi$
$374$	−0.729179	−0.0377049
$375$	16.5444	0.854351
$376$	−3.65947	−0.188723
$377$	−0.126346	−0.00650717
$378$	0.880673	0.0452969
$379$	−27.8518	−1.43065	−0.715324	−	0.698793i	$-0.753721\pi$
$379$	−27.8518	−1.43065	−0.715324	+	0.698793i	$0.753721\pi$
$380$	28.4548	1.45970
$381$	−0.276862	−0.0141841
$382$	−0.965398	−0.0493941
$383$	−19.4345	−0.993057	−0.496528	−	0.868021i	$-0.665392\pi$
$383$	−19.4345	−0.993057	−0.496528	+	0.868021i	$0.665392\pi$
$384$	6.64175	0.338935
$385$	−49.9156	−2.54394
$386$	−0.333185	−0.0169586
$387$	−8.66151	−0.440289
$388$	−18.0125	−0.914449
$389$	−9.68047	−0.490819	−0.245410	−	0.969419i	$-0.578922\pi$
$389$	−9.68047	−0.490819	−0.245410	+	0.969419i	$0.578922\pi$
$390$	0.444697	0.0225181
$391$	3.07235	0.155375
$392$	−7.77057	−0.392473
$393$	7.17489	0.361925
$394$	1.20725	0.0608203
$395$	−3.79014	−0.190702
$396$	6.44334	0.323790
$397$	8.94257	0.448815	0.224407	−	0.974495i	$-0.427955\pi$
$397$	8.94257	0.448815	0.224407	+	0.974495i	$0.427955\pi$
$398$	0.133593	0.00669640
$399$	15.3452	0.768219
$400$	34.7429	1.73715
$401$	−3.51241	−0.175401	−0.0877007	−	0.996147i	$-0.527952\pi$
$401$	−3.51241	−0.175401	−0.0877007	+	0.996147i	$0.527952\pi$
$402$	2.31197	0.115311
$403$	1.20334	0.0599429
$404$	−10.2379	−0.509354
$405$	−3.79014	−0.188333
$406$	0.209412	0.0103929
$407$	23.1696	1.14847
$408$	0.872503	0.0431953
$409$	−0.0543466	−0.00268727	−0.00134363	−	0.999999i	$-0.500428\pi$
$409$	−0.0543466	−0.00268727	−0.00134363	+	0.999999i	$0.500428\pi$
$410$	−4.43918	−0.219236
$411$	−18.3482	−0.905050
$412$	−33.0926	−1.63036
$413$	12.1935	0.600004
$414$	0.678429	0.0333429
$415$	2.59156	0.127215
$416$	1.36247	0.0668007
$417$	4.50301	0.220513
$418$	−2.80559	−0.137226
$419$	17.7182	0.865590	0.432795	−	0.901492i	$-0.357527\pi$
$419$	17.7182	0.865590	0.432795	+	0.901492i	$0.357527\pi$
$420$	29.4949	1.43920
$421$	−9.83293	−0.479228	−0.239614	−	0.970868i	$-0.577021\pi$
$421$	−9.83293	−0.479228	−0.239614	+	0.970868i	$0.577021\pi$
$422$	−1.04370	−0.0508064
$423$	4.19422	0.203930
$424$	−8.02318	−0.389640
$425$	9.36513	0.454276
$426$	−1.59182	−0.0771239
$427$	36.3135	1.75734
$428$	15.1032	0.730043
$429$	1.75460	0.0847127
$430$	7.24906	0.349581
$431$	−17.6583	−0.850570	−0.425285	−	0.905060i	$-0.639826\pi$
$431$	−17.6583	−0.850570	−0.425285	+	0.905060i	$0.639826\pi$
$432$	−3.70982	−0.178489
$433$	−15.1622	−0.728649	−0.364325	−	0.931272i	$-0.618700\pi$
$433$	−15.1622	−0.728649	−0.364325	+	0.931272i	$0.618700\pi$
$434$	−1.99447	−0.0957378
$435$	−0.901241	−0.0432112
$436$	19.6304	0.940124
$437$	11.8212	0.565484
$438$	−1.85590	−0.0886783
$439$	−9.11027	−0.434809	−0.217405	−	0.976082i	$-0.569759\pi$
$439$	−9.11027	−0.434809	−0.217405	+	0.976082i	$0.569759\pi$
$440$	−10.9200	−0.520590
$441$	8.90607	0.424098
$442$	0.117330	0.00558083
$443$	−0.939933	−0.0446576	−0.0223288	−	0.999751i	$-0.507108\pi$
$443$	−0.939933	−0.0446576	−0.0223288	+	0.999751i	$0.507108\pi$
$444$	−13.6908	−0.649735
$445$	29.6430	1.40521
$446$	1.15812	0.0548385
$447$	−9.59099	−0.453638
$448$	27.3331	1.29137
$449$	20.6971	0.976759	0.488379	−	0.872631i	$-0.337588\pi$
$449$	20.6971	0.976759	0.488379	+	0.872631i	$0.337588\pi$
$450$	2.06798	0.0974857
$451$	−17.5152	−0.824759
$452$	9.40340	0.442299
$453$	−11.3043	−0.531121
$454$	−1.76543	−0.0828557
$455$	8.03180	0.376536
$456$	3.35704	0.157208
$457$	−28.1998	−1.31913	−0.659566	−	0.751647i	$-0.729260\pi$
$457$	−28.1998	−1.31913	−0.659566	+	0.751647i	$0.729260\pi$
$458$	−1.25307	−0.0585519
$459$	−1.00000	−0.0466760
$460$	22.7215	1.05939
$461$	3.50436	0.163214	0.0816072	−	0.996665i	$-0.473995\pi$
$461$	3.50436	0.163214	0.0816072	+	0.996665i	$0.473995\pi$
$462$	−2.90814	−0.135299
$463$	41.1642	1.91306	0.956532	−	0.291629i	$-0.0941972\pi$
$463$	41.1642	1.91306	0.956532	+	0.291629i	$0.0941972\pi$
$464$	−0.882142	−0.0409524
$465$	8.58358	0.398054
$466$	5.56589	0.257835
$467$	16.5020	0.763620	0.381810	−	0.924241i	$-0.375301\pi$
$467$	16.5020	0.763620	0.381810	+	0.924241i	$0.375301\pi$
$468$	−1.03678	−0.0479252
$469$	41.7572	1.92817
$470$	−3.51026	−0.161916
$471$	−3.34788	−0.154262
$472$	2.66756	0.122785
$473$	28.6018	1.31511
$474$	−0.220817	−0.0101425
$475$	36.0333	1.65332
$476$	7.78201	0.356688
$477$	9.19560	0.421037
$478$	−0.632396	−0.0289251
$479$	−21.4126	−0.978368	−0.489184	−	0.872181i	$-0.662705\pi$
$479$	−21.4126	−0.978368	−0.489184	+	0.872181i	$0.662705\pi$
$480$	9.71866	0.443594
$481$	−3.72815	−0.169989
$482$	−5.58957	−0.254598
$483$	12.2533	0.557543
$484$	0.186575	0.00848069
$485$	−34.9880	−1.58872
$486$	−0.220817	−0.0100165
$487$	−1.58896	−0.0720024	−0.0360012	−	0.999352i	$-0.511462\pi$
$487$	−1.58896	−0.0720024	−0.0360012	+	0.999352i	$0.511462\pi$
$488$	7.94427	0.359620
$489$	−21.2770	−0.962180
$490$	−7.45374	−0.336726
$491$	−2.28991	−0.103342	−0.0516711	−	0.998664i	$-0.516455\pi$
$491$	−2.28991	−0.103342	−0.0516711	+	0.998664i	$0.516455\pi$
$492$	10.3496	0.466598
$493$	−0.237786	−0.0107093
$494$	0.451440	0.0203112
$495$	12.5157	0.562539
$496$	8.40168	0.377246
$497$	−28.7502	−1.28962
$498$	0.150987	0.00676589
$499$	37.6406	1.68503	0.842513	−	0.538677i	$-0.181076\pi$
$499$	37.6406	1.68503	0.842513	+	0.538677i	$0.181076\pi$
$500$	32.2822	1.44370
$501$	−14.3890	−0.642852
$502$	1.78679	0.0797482
$503$	31.3737	1.39888	0.699442	−	0.714690i	$-0.253432\pi$
$503$	31.3737	1.39888	0.699442	+	0.714690i	$0.253432\pi$
$504$	3.47975	0.155000
$505$	−19.8863	−0.884929
$506$	−2.24029	−0.0995931
$507$	12.7177	0.564812
$508$	−0.540224	−0.0239686
$509$	23.9734	1.06260	0.531302	−	0.847182i	$-0.321703\pi$
$509$	23.9734	1.06260	0.531302	+	0.847182i	$0.321703\pi$
$510$	0.836928	0.0370598
$511$	−33.5199	−1.48283
$512$	15.9863	0.706503
$513$	−3.84760	−0.169876
$514$	4.52785	0.199715
$515$	−64.2799	−2.83251
$516$	−16.9007	−0.744011
$517$	−13.8501	−0.609125
$518$	6.17920	0.271498
$519$	9.78842	0.429664
$520$	1.75711	0.0770542
$521$	0.743064	0.0325542	0.0162771	−	0.999868i	$-0.494819\pi$
$521$	0.743064	0.0325542	0.0162771	+	0.999868i	$0.494819\pi$
$522$	−0.0525073	−0.00229818
$523$	−28.7566	−1.25744	−0.628718	−	0.777633i	$-0.716420\pi$
$523$	−28.7566	−1.25744	−0.628718	+	0.777633i	$0.716420\pi$
$524$	13.9999	0.611590
$525$	37.3504	1.63011
$526$	−1.23737	−0.0539519
$527$	2.26472	0.0986526
$528$	12.2505	0.533133
$529$	−13.5607	−0.589594
$530$	−7.69605	−0.334295
$531$	−3.05737	−0.132679
$532$	29.9421	1.29815
$533$	2.81833	0.122075
$534$	1.72703	0.0747360
$535$	29.3369	1.26835
$536$	9.13517	0.394579
$537$	3.00748	0.129782
$538$	2.32147	0.100086
$539$	−29.4094	−1.26675
$540$	−7.39546	−0.318250
$541$	43.4358	1.86745	0.933725	−	0.357991i	$-0.116538\pi$
$541$	43.4358	1.86745	0.933725	+	0.357991i	$0.116538\pi$
$542$	−5.54620	−0.238229
$543$	−21.9820	−0.943339
$544$	2.56420	0.109939
$545$	38.1305	1.63333
$546$	0.467941	0.0200260
$547$	−40.9703	−1.75176	−0.875881	−	0.482526i	$-0.839719\pi$
$547$	−40.9703	−1.75176	−0.875881	+	0.482526i	$0.839719\pi$
$548$	−35.8017	−1.52937
$549$	−9.10515	−0.388598
$550$	−6.82885	−0.291183
$551$	−0.914905	−0.0389763
$552$	2.68063	0.114095
$553$	−3.98824	−0.169597
$554$	−2.42959	−0.103223
$555$	−26.5933	−1.12882
$556$	8.78645	0.372629
$557$	24.5464	1.04006	0.520031	−	0.854147i	$-0.325920\pi$
$557$	24.5464	1.04006	0.520031	+	0.854147i	$0.325920\pi$
$558$	0.500089	0.0211704
$559$	−4.60225	−0.194654
$560$	56.0775	2.36971
$561$	3.30218	0.139418
$562$	2.55249	0.107670
$563$	−13.2567	−0.558703	−0.279352	−	0.960189i	$-0.590120\pi$
$563$	−13.2567	−0.558703	−0.279352	+	0.960189i	$0.590120\pi$
$564$	8.18393	0.344605
$565$	18.2654	0.768431
$566$	−6.68731	−0.281089
$567$	−3.98824	−0.167490
$568$	−6.28966	−0.263908
$569$	−42.1699	−1.76785	−0.883927	−	0.467624i	$-0.845110\pi$
$569$	−42.1699	−1.76785	−0.883927	+	0.467624i	$0.845110\pi$
$570$	3.22017	0.134878
$571$	−36.3594	−1.52159	−0.760796	−	0.648991i	$-0.775191\pi$
$571$	−36.3594	−1.52159	−0.760796	+	0.648991i	$0.775191\pi$
$572$	3.42364	0.143149
$573$	4.37193	0.182640
$574$	−4.67121	−0.194973
$575$	28.7730	1.19992
$576$	−6.85341	−0.285559
$577$	−26.6084	−1.10772	−0.553860	−	0.832610i	$-0.686846\pi$
$577$	−26.6084	−1.10772	−0.553860	+	0.832610i	$0.686846\pi$
$578$	0.220817	0.00918479
$579$	1.50887	0.0627064
$580$	−1.75854	−0.0730193
$581$	2.72702	0.113136
$582$	−2.03844	−0.0844960
$583$	−30.3655	−1.25761
$584$	−7.33310	−0.303446
$585$	−2.01387	−0.0832632
$586$	2.92859	0.120979
$587$	−33.1488	−1.36820	−0.684099	−	0.729389i	$-0.739805\pi$
$587$	−33.1488	−1.36820	−0.684099	+	0.729389i	$0.739805\pi$
$588$	17.3779	0.716651
$589$	8.71372	0.359043
$590$	2.55880	0.105344
$591$	−5.46718	−0.224890
$592$	−26.0297	−1.06982
$593$	27.9810	1.14904	0.574521	−	0.818490i	$-0.305188\pi$
$593$	27.9810	1.14904	0.574521	+	0.818490i	$0.305188\pi$
$594$	0.729179	0.0299186
$595$	15.1160	0.619695
$596$	−18.7143	−0.766568
$597$	−0.604992	−0.0247607
$598$	0.360480	0.0147411
$599$	−9.38607	−0.383504	−0.191752	−	0.981443i	$-0.561417\pi$
$599$	−9.38607	−0.383504	−0.191752	+	0.981443i	$0.561417\pi$
$600$	8.17110	0.333584
$601$	−42.7674	−1.74452	−0.872260	−	0.489043i	$-0.837346\pi$
$601$	−42.7674	−1.74452	−0.872260	+	0.489043i	$0.837346\pi$
$602$	7.62796	0.310892
$603$	−10.4701	−0.426374
$604$	−22.0573	−0.897500
$605$	0.362408	0.0147340
$606$	−1.15860	−0.0470648
$607$	−5.27631	−0.214159	−0.107079	−	0.994250i	$-0.534150\pi$
$607$	−5.27631	−0.214159	−0.107079	+	0.994250i	$0.534150\pi$
$608$	9.86601	0.400119
$609$	−0.948348	−0.0384290
$610$	7.62036	0.308539
$611$	2.22858	0.0901586
$612$	−1.95124	−0.0788742
$613$	−26.5735	−1.07329	−0.536646	−	0.843808i	$-0.680309\pi$
$613$	−26.5735	−1.07329	−0.536646	+	0.843808i	$0.680309\pi$
$614$	1.63125	0.0658320
$615$	20.1034	0.810647
$616$	−11.4908	−0.462976
$617$	−3.48150	−0.140160	−0.0700799	−	0.997541i	$-0.522325\pi$
$617$	−3.48150	−0.140160	−0.0700799	+	0.997541i	$0.522325\pi$
$618$	−3.74502	−0.150647
$619$	15.9692	0.641858	0.320929	−	0.947103i	$-0.396005\pi$
$619$	15.9692	0.641858	0.320929	+	0.947103i	$0.396005\pi$
$620$	16.7486	0.672641
$621$	−3.07235	−0.123289
$622$	−6.34968	−0.254599
$623$	31.1924	1.24970
$624$	−1.97119	−0.0789108
$625$	15.8800	0.635201
$626$	−0.844828	−0.0337661
$627$	12.7055	0.507407
$628$	−6.53251	−0.260675
$629$	−7.01645	−0.279764
$630$	3.33787	0.132984
$631$	35.6288	1.41836	0.709180	−	0.705028i	$-0.249065\pi$
$631$	35.6288	1.41836	0.709180	+	0.705028i	$0.249065\pi$
$632$	−0.872503	−0.0347063
$633$	4.72652	0.187862
$634$	2.53331	0.100610
$635$	−1.04934	−0.0416419
$636$	17.9428	0.711479
$637$	4.73219	0.187496
$638$	0.173388	0.00686451
$639$	7.20875	0.285174
$640$	25.1731	0.995055
$641$	42.8299	1.69168	0.845839	−	0.533438i	$-0.179100\pi$
$641$	42.8299	1.69168	0.845839	+	0.533438i	$0.179100\pi$
$642$	1.70920	0.0674567
$643$	−30.2662	−1.19358	−0.596791	−	0.802396i	$-0.703558\pi$
$643$	−30.2662	−1.19358	−0.596791	+	0.802396i	$0.703558\pi$
$644$	23.9091	0.942150
$645$	−32.8283	−1.29261
$646$	0.849617	0.0334277
$647$	−29.7954	−1.17138	−0.585689	−	0.810536i	$-0.699176\pi$
$647$	−29.7954	−1.17138	−0.585689	+	0.810536i	$0.699176\pi$
$648$	−0.872503	−0.0342751
$649$	10.0960	0.396302
$650$	1.09881	0.0430990
$651$	9.03223	0.354001
$652$	−41.5166	−1.62591
$653$	−11.7167	−0.458512	−0.229256	−	0.973366i	$-0.573629\pi$
$653$	−11.7167	−0.458512	−0.229256	+	0.973366i	$0.573629\pi$
$654$	2.22153	0.0868685
$655$	27.1938	1.06255
$656$	19.6774	0.768272
$657$	8.40467	0.327898
$658$	−3.69374	−0.143997
$659$	35.0881	1.36684	0.683419	−	0.730026i	$-0.260492\pi$
$659$	35.0881	1.36684	0.683419	+	0.730026i	$0.260492\pi$
$660$	24.4211	0.950592
$661$	19.2793	0.749879	0.374939	−	0.927049i	$-0.377664\pi$
$661$	19.2793	0.749879	0.374939	+	0.927049i	$0.377664\pi$
$662$	3.01396	0.117141
$663$	−0.531345	−0.0206357
$664$	0.596586	0.0231520
$665$	58.1602	2.25536
$666$	−1.54935	−0.0600363
$667$	−0.730562	−0.0282875
$668$	−28.0764	−1.08631
$669$	−5.24469	−0.202771
$670$	8.76270	0.338533
$671$	30.0668	1.16072
$672$	10.2266	0.394501
$673$	11.0711	0.426761	0.213380	−	0.976969i	$-0.431553\pi$
$673$	11.0711	0.426761	0.213380	+	0.976969i	$0.431553\pi$
$674$	−3.10196	−0.119483
$675$	−9.36513	−0.360464
$676$	24.8152	0.954432
$677$	9.63412	0.370269	0.185135	−	0.982713i	$-0.440728\pi$
$677$	9.63412	0.370269	0.185135	+	0.982713i	$0.440728\pi$
$678$	1.06416	0.0408689
$679$	−36.8168	−1.41290
$680$	3.30690	0.126814
$681$	7.99497	0.306368
$682$	−1.65138	−0.0632347
$683$	1.77907	0.0680741	0.0340370	−	0.999421i	$-0.489164\pi$
$683$	1.77907	0.0680741	0.0340370	+	0.999421i	$0.489164\pi$
$684$	−7.50759	−0.287060
$685$	−69.5422	−2.65707
$686$	−1.67862	−0.0640901
$687$	5.67467	0.216502
$688$	−32.1326	−1.22504
$689$	4.88603	0.186143
$690$	2.57134	0.0978891
$691$	−13.8838	−0.528164	−0.264082	−	0.964500i	$-0.585069\pi$
$691$	−13.8838	−0.528164	−0.264082	+	0.964500i	$0.585069\pi$
$692$	19.0996	0.726056
$693$	13.1699	0.500282
$694$	−3.48677	−0.132356
$695$	17.0670	0.647389
$696$	−0.207469	−0.00786409
$697$	5.30414	0.200909
$698$	−1.66804	−0.0631362
$699$	−25.2059	−0.953373
$700$	72.8796	2.75459
$701$	−19.7699	−0.746701	−0.373350	−	0.927690i	$-0.621791\pi$
$701$	−19.7699	−0.746701	−0.373350	+	0.927690i	$0.621791\pi$
$702$	−0.117330	−0.00442834
$703$	−26.9965	−1.01819
$704$	22.6312	0.852945
$705$	15.8967	0.598703
$706$	3.17315	0.119423
$707$	−20.9257	−0.786994
$708$	−5.96566	−0.224203
$709$	6.58904	0.247456	0.123728	−	0.992316i	$-0.460515\pi$
$709$	6.58904	0.247456	0.123728	+	0.992316i	$0.460515\pi$
$710$	−6.03321	−0.226422
$711$	1.00000	0.0375029
$712$	6.82392	0.255737
$713$	6.95800	0.260579
$714$	0.880673	0.0329584
$715$	6.65016	0.248702
$716$	5.86831	0.219309
$717$	2.86388	0.106954
$718$	1.46451	0.0546551
$719$	−19.0540	−0.710595	−0.355298	−	0.934753i	$-0.615620\pi$
$719$	−19.0540	−0.710595	−0.355298	+	0.934753i	$0.615620\pi$
$720$	−14.0607	−0.524012
$721$	−67.6397	−2.51904
$722$	−0.926543	−0.0344824
$723$	25.3131	0.941404
$724$	−42.8922	−1.59408
$725$	−2.22690	−0.0827049
$726$	0.0211143	0.000783625 0
$727$	−16.8288	−0.624145	−0.312072	−	0.950058i	$-0.601023\pi$
$727$	−16.8288	−0.624145	−0.312072	+	0.950058i	$0.601023\pi$
$728$	1.84895	0.0685266
$729$	1.00000	0.0370370
$730$	−7.03411	−0.260344
$731$	−8.66151	−0.320357
$732$	−17.7663	−0.656663
$733$	−15.8786	−0.586488	−0.293244	−	0.956038i	$-0.594735\pi$
$733$	−15.8786	−0.586488	−0.293244	+	0.956038i	$0.594735\pi$
$734$	−8.23233	−0.303861
$735$	33.7552	1.24508
$736$	7.87811	0.290391
$737$	34.5740	1.27355
$738$	1.17125	0.0431142
$739$	44.8738	1.65071	0.825355	−	0.564614i	$-0.190975\pi$
$739$	44.8738	1.65071	0.825355	+	0.564614i	$0.190975\pi$
$740$	−51.8899	−1.90751
$741$	−2.04440	−0.0751030
$742$	−8.09831	−0.297298
$743$	39.5168	1.44973	0.724865	−	0.688891i	$-0.241902\pi$
$743$	39.5168	1.44973	0.724865	+	0.688891i	$0.241902\pi$
$744$	1.97597	0.0724426
$745$	−36.3511	−1.33180
$746$	6.67753	0.244482
$747$	−0.683764	−0.0250176
$748$	6.44334	0.235592
$749$	30.8703	1.12798
$750$	3.65330	0.133400
$751$	27.1773	0.991712	0.495856	−	0.868405i	$-0.334854\pi$
$751$	27.1773	0.991712	0.495856	+	0.868405i	$0.334854\pi$
$752$	15.5598	0.567407
$753$	−8.09170	−0.294878
$754$	−0.0278995	−0.00101604
$755$	−42.8447	−1.55928
$756$	−7.78201	−0.283029
$757$	16.4403	0.597535	0.298767	−	0.954326i	$-0.403425\pi$
$757$	16.4403	0.597535	0.298767	+	0.954326i	$0.403425\pi$
$758$	−6.15015	−0.223384
$759$	10.1454	0.368256
$760$	12.7236	0.461535
$761$	−38.4468	−1.39369	−0.696847	−	0.717219i	$-0.745415\pi$
$761$	−38.4468	−1.39369	−0.696847	+	0.717219i	$0.745415\pi$
$762$	−0.0611359	−0.00221472
$763$	40.1236	1.45257
$764$	8.53068	0.308629
$765$	−3.79014	−0.137033
$766$	−4.29148	−0.155057
$767$	−1.62452	−0.0586579
$768$	−12.2402	−0.441681
$769$	15.4682	0.557798	0.278899	−	0.960320i	$-0.410030\pi$
$769$	15.4682	0.557798	0.278899	+	0.960320i	$0.410030\pi$
$770$	−11.0222	−0.397214
$771$	−20.5050	−0.738468
$772$	2.94416	0.105963
$773$	18.7785	0.675415	0.337708	−	0.941251i	$-0.390348\pi$
$773$	18.7785	0.675415	0.337708	+	0.941251i	$0.390348\pi$
$774$	−1.91261	−0.0687474
$775$	21.2094	0.761863
$776$	−8.05436	−0.289135
$777$	−27.9833	−1.00390
$778$	−2.13762	−0.0766373
$779$	20.4082	0.731200
$780$	−3.92954	−0.140700
$781$	−23.8046	−0.851795
$782$	0.678429	0.0242606
$783$	0.237786	0.00849778
$784$	33.0399	1.18000
$785$	−12.6889	−0.452886
$786$	1.58434	0.0565116
$787$	−18.5493	−0.661210	−0.330605	−	0.943769i	$-0.607253\pi$
$787$	−18.5493	−0.661210	−0.330605	+	0.943769i	$0.607253\pi$
$788$	−10.6678	−0.380024
$789$	5.60359	0.199493
$790$	−0.836928	−0.0297766
$791$	19.2201	0.683388
$792$	2.88116	0.102378
$793$	−4.83798	−0.171802
$794$	1.97468	0.0700786
$795$	34.8526	1.23609
$796$	−1.18049	−0.0418412
$797$	−36.7610	−1.30214	−0.651071	−	0.759017i	$-0.725680\pi$
$797$	−36.7610	−1.30214	−0.651071	+	0.759017i	$0.725680\pi$
$798$	3.38848	0.119951
$799$	4.19422	0.148381
$800$	24.0140	0.849025
$801$	−7.82109	−0.276345
$802$	−0.775602	−0.0273874
$803$	−27.7537	−0.979408
$804$	−20.4296	−0.720497
$805$	46.4416	1.63685
$806$	0.265720	0.00935957
$807$	−10.5131	−0.370079
$808$	−4.57790	−0.161050
$809$	−38.9655	−1.36995	−0.684977	−	0.728564i	$-0.740188\pi$
$809$	−38.9655	−1.36995	−0.684977	+	0.728564i	$0.740188\pi$
$810$	−0.836928	−0.0294067
$811$	−24.6889	−0.866946	−0.433473	−	0.901167i	$-0.642712\pi$
$811$	−24.6889	−0.866946	−0.433473	+	0.901167i	$0.642712\pi$
$812$	−1.85045	−0.0649382
$813$	25.1167	0.880880
$814$	5.11624	0.179324
$815$	−80.6428	−2.82479
$816$	−3.70982	−0.129870
$817$	−33.3260	−1.16593
$818$	−0.0120007	−0.000419594 0
$819$	−2.11913	−0.0740484
$820$	39.2266	1.36985
$821$	−32.3845	−1.13023	−0.565114	−	0.825013i	$-0.691168\pi$
$821$	−32.3845	−1.13023	−0.565114	+	0.825013i	$0.691168\pi$
$822$	−4.05160	−0.141316
$823$	−10.6531	−0.371343	−0.185671	−	0.982612i	$-0.559446\pi$
$823$	−10.6531	−0.371343	−0.185671	+	0.982612i	$0.559446\pi$
$824$	−14.7975	−0.515494
$825$	30.9253	1.07668
$826$	2.69254	0.0936856
$827$	40.3174	1.40197	0.700987	−	0.713174i	$-0.252743\pi$
$827$	40.3174	1.40197	0.700987	+	0.713174i	$0.252743\pi$
$828$	−5.99489	−0.208337
$829$	−6.24535	−0.216910	−0.108455	−	0.994101i	$-0.534590\pi$
$829$	−6.24535	−0.216910	−0.108455	+	0.994101i	$0.534590\pi$
$830$	0.572261	0.0198635
$831$	11.0027	0.381680
$832$	−3.64152	−0.126247
$833$	8.90607	0.308577
$834$	0.994343	0.0344313
$835$	−54.5362	−1.88730
$836$	24.7914	0.857429
$837$	−2.26472	−0.0782800
$838$	3.91248	0.135155
$839$	33.7878	1.16648	0.583242	−	0.812298i	$-0.301784\pi$
$839$	33.7878	1.16648	0.583242	+	0.812298i	$0.301784\pi$
$840$	13.1887	0.455054
$841$	−28.9435	−0.998050
$842$	−2.17128	−0.0748274
$843$	−11.5593	−0.398122
$844$	9.22257	0.317454
$845$	48.2017	1.65819
$846$	0.926157	0.0318419
$847$	0.381351	0.0131034
$848$	34.1140	1.17148
$849$	30.2843	1.03936
$850$	2.06798	0.0709313
$851$	−21.5570	−0.738964
$852$	14.0660	0.481893
$853$	27.1455	0.929445	0.464723	−	0.885456i	$-0.346154\pi$
$853$	27.1455	0.929445	0.464723	+	0.885456i	$0.346154\pi$
$854$	8.01866	0.274393
$855$	−14.5829	−0.498726
$856$	6.75346	0.230828
$857$	−42.5769	−1.45440	−0.727200	−	0.686426i	$-0.759179\pi$
$857$	−42.5769	−1.45440	−0.727200	+	0.686426i	$0.759179\pi$
$858$	0.387445	0.0132272
$859$	13.6077	0.464288	0.232144	−	0.972681i	$-0.425426\pi$
$859$	13.6077	0.464288	0.232144	+	0.972681i	$0.425426\pi$
$860$	−64.0559	−2.18429
$861$	21.1542	0.720932
$862$	−3.89926	−0.132809
$863$	−46.2225	−1.57343	−0.786716	−	0.617315i	$-0.788220\pi$
$863$	−46.2225	−1.57343	−0.786716	+	0.617315i	$0.788220\pi$
$864$	−2.56420	−0.0872358
$865$	37.0995	1.26142
$866$	−3.34808	−0.113772
$867$	−1.00000	−0.0339618
$868$	17.6240	0.598199
$869$	−3.30218	−0.112019
$870$	−0.199010	−0.00674706
$871$	−5.56322	−0.188502
$872$	8.77778	0.297253
$873$	9.23134	0.312433
$874$	2.61032	0.0882955
$875$	65.9832	2.23064
$876$	16.3995	0.554089
$877$	4.66961	0.157682	0.0788408	−	0.996887i	$-0.474878\pi$
$877$	4.66961	0.157682	0.0788408	+	0.996887i	$0.474878\pi$
$878$	−2.01171	−0.0678918
$879$	−13.2625	−0.447334
$880$	46.4310	1.56519
$881$	−14.7489	−0.496905	−0.248452	−	0.968644i	$-0.579922\pi$
$881$	−14.7489	−0.496905	−0.248452	+	0.968644i	$0.579922\pi$
$882$	1.96661	0.0662194
$883$	−12.4240	−0.418100	−0.209050	−	0.977905i	$-0.567037\pi$
$883$	−12.4240	−0.418100	−0.209050	+	0.977905i	$0.567037\pi$
$884$	−1.03678	−0.0348707
$885$	−11.5878	−0.389521
$886$	−0.207554	−0.00697290
$887$	25.7699	0.865267	0.432634	−	0.901570i	$-0.357584\pi$
$887$	25.7699	0.865267	0.432634	+	0.901570i	$0.357584\pi$
$888$	−6.12187	−0.205437
$889$	−1.10419	−0.0370334
$890$	6.54569	0.219412
$891$	−3.30218	−0.110627
$892$	−10.2336	−0.342648
$893$	16.1377	0.540027
$894$	−2.11786	−0.0708317
$895$	11.3987	0.381018
$896$	26.4889	0.884932
$897$	−1.63248	−0.0545068
$898$	4.57029	0.152513
$899$	−0.538518	−0.0179606
$900$	−18.2736	−0.609120
$901$	9.19560	0.306350
$902$	−3.86766	−0.128779
$903$	−34.5442	−1.14956
$904$	4.20476	0.139848
$905$	−83.3149	−2.76948
$906$	−2.49618	−0.0829300
$907$	49.0820	1.62974	0.814870	−	0.579644i	$-0.196808\pi$
$907$	49.0820	1.62974	0.814870	+	0.579644i	$0.196808\pi$
$908$	15.6001	0.517708
$909$	5.24686	0.174027
$910$	1.77356	0.0587930
$911$	−6.76367	−0.224090	−0.112045	−	0.993703i	$-0.535740\pi$
$911$	−6.76367	−0.224090	−0.112045	+	0.993703i	$0.535740\pi$
$912$	−14.2739	−0.472656
$913$	2.25791	0.0747259
$914$	−6.22701	−0.205971
$915$	−34.5098	−1.14086
$916$	11.0726	0.365850
$917$	28.6152	0.944957
$918$	−0.220817	−0.00728806
$919$	−6.36750	−0.210044	−0.105022	−	0.994470i	$-0.533491\pi$
$919$	−6.36750	−0.210044	−0.105022	+	0.994470i	$0.533491\pi$
$920$	10.1600	0.334964
$921$	−7.38733	−0.243421
$922$	0.773824	0.0254845
$923$	3.83033	0.126077
$924$	25.6976	0.845389
$925$	−65.7100	−2.16053
$926$	9.08978	0.298709
$927$	16.9598	0.557033
$928$	−0.609730	−0.0200154
$929$	−27.4767	−0.901481	−0.450741	−	0.892655i	$-0.648840\pi$
$929$	−27.4767	−0.901481	−0.450741	+	0.892655i	$0.648840\pi$
$930$	1.89540	0.0621527
$931$	34.2670	1.12306
$932$	−49.1827	−1.61103
$933$	28.7553	0.941407
$934$	3.64392	0.119233
$935$	12.5157	0.409307
$936$	−0.463600	−0.0151532
$937$	48.2679	1.57684	0.788422	−	0.615135i	$-0.210899\pi$
$937$	48.2679	1.57684	0.788422	+	0.615135i	$0.210899\pi$
$938$	9.22071	0.301067
$939$	3.82591	0.124854
$940$	31.0182	1.01170
$941$	−44.3429	−1.44554	−0.722769	−	0.691090i	$-0.757131\pi$
$941$	−44.3429	−1.44554	−0.722769	+	0.691090i	$0.757131\pi$
$942$	−0.739269	−0.0240867
$943$	16.2962	0.530676
$944$	−11.3423	−0.369160
$945$	−15.1160	−0.491723
$946$	6.31578	0.205344
$947$	36.1218	1.17380	0.586901	−	0.809659i	$-0.300348\pi$
$947$	36.1218	1.17380	0.586901	+	0.809659i	$0.300348\pi$
$948$	1.95124	0.0633733
$949$	4.46578	0.144965
$950$	7.95678	0.258152
$951$	−11.4724	−0.372018
$952$	3.47975	0.112779
$953$	26.5522	0.860110	0.430055	−	0.902803i	$-0.358494\pi$
$953$	26.5522	0.860110	0.430055	+	0.902803i	$0.358494\pi$
$954$	2.03055	0.0657414
$955$	16.5702	0.536199
$956$	5.58813	0.180733
$957$	−0.785212	−0.0253823
$958$	−4.72828	−0.152764
$959$	−73.1770	−2.36301
$960$	−25.9754	−0.838351
$961$	−25.8711	−0.834550
$962$	−0.823242	−0.0265424
$963$	−7.74033	−0.249429
$964$	49.3919	1.59081
$965$	5.71882	0.184095
$966$	2.70574	0.0870556
$967$	45.9715	1.47834	0.739172	−	0.673516i	$-0.235217\pi$
$967$	45.9715	1.47834	0.739172	+	0.673516i	$0.235217\pi$
$968$	0.0834277	0.00268147
$969$	−3.84760	−0.123603
$970$	−7.72596	−0.248066
$971$	30.5637	0.980835	0.490417	−	0.871488i	$-0.336844\pi$
$971$	30.5637	0.980835	0.490417	+	0.871488i	$0.336844\pi$
$972$	1.95124	0.0625860
$973$	17.9591	0.575742
$974$	−0.350869	−0.0112426
$975$	−4.97611	−0.159363
$976$	−33.7784	−1.08122
$977$	−5.05977	−0.161876	−0.0809382	−	0.996719i	$-0.525792\pi$
$977$	−5.05977	−0.161876	−0.0809382	+	0.996719i	$0.525792\pi$
$978$	−4.69834	−0.150236
$979$	25.8266	0.825423
$980$	65.8645	2.10396
$981$	−10.0605	−0.321206
$982$	−0.505652	−0.0161360
$983$	9.89839	0.315710	0.157855	−	0.987462i	$-0.449542\pi$
$983$	9.89839	0.315710	0.157855	+	0.987462i	$0.449542\pi$
$984$	4.62787	0.147531
$985$	−20.7213	−0.660237
$986$	−0.0525073	−0.00167217
$987$	16.7276	0.532444
$988$	−3.98912	−0.126911
$989$	−26.6112	−0.846186
$990$	2.76369	0.0878357
$991$	−41.1991	−1.30873	−0.654366	−	0.756178i	$-0.727064\pi$
$991$	−41.1991	−1.30873	−0.654366	+	0.756178i	$0.727064\pi$
$992$	5.80718	0.184378
$993$	−13.6491	−0.433141
$994$	−6.34856	−0.201364
$995$	−2.29300	−0.0726931
$996$	−1.33419	−0.0422753
$997$	8.76429	0.277568	0.138784	−	0.990323i	$-0.455681\pi$
$997$	8.76429	0.277568	0.138784	+	0.990323i	$0.455681\pi$
$998$	8.31170	0.263102
$999$	7.01645	0.221991

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.f.1.12	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.f.1.12	✓	22	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:	\( N \)	\(=\)	\( 4029 = 3 \cdot 17 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4029.a (trivial)

\(f(q)\)	\(=\)	\(q+0.220817 q^{2} -1.00000 q^{3} -1.95124 q^{4} -3.79014 q^{5} -0.220817 q^{6} -3.98824 q^{7} -0.872503 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.220817 q^{2} -1.00000 q^{3} -1.95124 q^{4} -3.79014 q^{5} -0.220817 q^{6} -3.98824 q^{7} -0.872503 q^{8} +1.00000 q^{9} -0.836928 q^{10} -3.30218 q^{11} +1.95124 q^{12} +0.531345 q^{13} -0.880673 q^{14} +3.79014 q^{15} +3.70982 q^{16} +1.00000 q^{17} +0.220817 q^{18} +3.84760 q^{19} +7.39546 q^{20} +3.98824 q^{21} -0.729179 q^{22} +3.07235 q^{23} +0.872503 q^{24} +9.36513 q^{25} +0.117330 q^{26} -1.00000 q^{27} +7.78201 q^{28} -0.237786 q^{29} +0.836928 q^{30} +2.26472 q^{31} +2.56420 q^{32} +3.30218 q^{33} +0.220817 q^{34} +15.1160 q^{35} -1.95124 q^{36} -7.01645 q^{37} +0.849617 q^{38} -0.531345 q^{39} +3.30690 q^{40} +5.30414 q^{41} +0.880673 q^{42} -8.66151 q^{43} +6.44334 q^{44} -3.79014 q^{45} +0.678429 q^{46} +4.19422 q^{47} -3.70982 q^{48} +8.90607 q^{49} +2.06798 q^{50} -1.00000 q^{51} -1.03678 q^{52} +9.19560 q^{53} -0.220817 q^{54} +12.5157 q^{55} +3.47975 q^{56} -3.84760 q^{57} -0.0525073 q^{58} -3.05737 q^{59} -7.39546 q^{60} -9.10515 q^{61} +0.500089 q^{62} -3.98824 q^{63} -6.85341 q^{64} -2.01387 q^{65} +0.729179 q^{66} -10.4701 q^{67} -1.95124 q^{68} -3.07235 q^{69} +3.33787 q^{70} +7.20875 q^{71} -0.872503 q^{72} +8.40467 q^{73} -1.54935 q^{74} -9.36513 q^{75} -7.50759 q^{76} +13.1699 q^{77} -0.117330 q^{78} +1.00000 q^{79} -14.0607 q^{80} +1.00000 q^{81} +1.17125 q^{82} -0.683764 q^{83} -7.78201 q^{84} -3.79014 q^{85} -1.91261 q^{86} +0.237786 q^{87} +2.88116 q^{88} -7.82109 q^{89} -0.836928 q^{90} -2.11913 q^{91} -5.99489 q^{92} -2.26472 q^{93} +0.926157 q^{94} -14.5829 q^{95} -2.56420 q^{96} +9.23134 q^{97} +1.96661 q^{98} -3.30218 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + q^{2} - 22 q^{3} + 19 q^{4} + q^{5} - q^{6} - 15 q^{7} + 15 q^{8} + 22 q^{9}+O(q^{10}) \) 22 * q + q^2 - 22 * q^3 + 19 * q^4 + q^5 - q^6 - 15 * q^7 + 15 * q^8 + 22 * q^9	\( 22 q + q^{2} - 22 q^{3} + 19 q^{4} + q^{5} - q^{6} - 15 q^{7} + 15 q^{8} + 22 q^{9} - 13 q^{10} - 23 q^{11} - 19 q^{12} - 18 q^{13} - 9 q^{14} - q^{15} + 21 q^{16} + 22 q^{17} + q^{18} - 30 q^{19} - 7 q^{20} + 15 q^{21} + 4 q^{22} - 3 q^{23} - 15 q^{24} + 19 q^{25} - 7 q^{26} - 22 q^{27} - 25 q^{28} - 7 q^{29} + 13 q^{30} - 10 q^{31} + 31 q^{32} + 23 q^{33} + q^{34} - 11 q^{35} + 19 q^{36} - q^{37} - 29 q^{38} + 18 q^{39} - 59 q^{40} + 9 q^{42} - 43 q^{43} - 80 q^{44} + q^{45} - 43 q^{46} + 2 q^{47} - 21 q^{48} + 43 q^{49} + 25 q^{50} - 22 q^{51} - 5 q^{52} - q^{53} - q^{54} - 19 q^{55} - 8 q^{56} + 30 q^{57} - 43 q^{58} - 28 q^{59} + 7 q^{60} - 29 q^{61} - 3 q^{62} - 15 q^{63} + 23 q^{64} + 19 q^{65} - 4 q^{66} - 16 q^{67} + 19 q^{68} + 3 q^{69} - 5 q^{70} - q^{71} + 15 q^{72} - 19 q^{73} - 24 q^{74} - 19 q^{75} - 72 q^{76} + 24 q^{77} + 7 q^{78} + 22 q^{79} - 82 q^{80} + 22 q^{81} - 81 q^{82} - 29 q^{83} + 25 q^{84} + q^{85} - 42 q^{86} + 7 q^{87} - 43 q^{88} - 28 q^{89} - 13 q^{90} - 96 q^{91} - 11 q^{92} + 10 q^{93} - 63 q^{94} - 23 q^{95} - 31 q^{96} - 51 q^{97} + 12 q^{98} - 23 q^{99}+O(q^{100}) \) 22 * q + q^2 - 22 * q^3 + 19 * q^4 + q^5 - q^6 - 15 * q^7 + 15 * q^8 + 22 * q^9 - 13 * q^10 - 23 * q^11 - 19 * q^12 - 18 * q^13 - 9 * q^14 - q^15 + 21 * q^16 + 22 * q^17 + q^18 - 30 * q^19 - 7 * q^20 + 15 * q^21 + 4 * q^22 - 3 * q^23 - 15 * q^24 + 19 * q^25 - 7 * q^26 - 22 * q^27 - 25 * q^28 - 7 * q^29 + 13 * q^30 - 10 * q^31 + 31 * q^32 + 23 * q^33 + q^34 - 11 * q^35 + 19 * q^36 - q^37 - 29 * q^38 + 18 * q^39 - 59 * q^40 + 9 * q^42 - 43 * q^43 - 80 * q^44 + q^45 - 43 * q^46 + 2 * q^47 - 21 * q^48 + 43 * q^49 + 25 * q^50 - 22 * q^51 - 5 * q^52 - q^53 - q^54 - 19 * q^55 - 8 * q^56 + 30 * q^57 - 43 * q^58 - 28 * q^59 + 7 * q^60 - 29 * q^61 - 3 * q^62 - 15 * q^63 + 23 * q^64 + 19 * q^65 - 4 * q^66 - 16 * q^67 + 19 * q^68 + 3 * q^69 - 5 * q^70 - q^71 + 15 * q^72 - 19 * q^73 - 24 * q^74 - 19 * q^75 - 72 * q^76 + 24 * q^77 + 7 * q^78 + 22 * q^79 - 82 * q^80 + 22 * q^81 - 81 * q^82 - 29 * q^83 + 25 * q^84 + q^85 - 42 * q^86 + 7 * q^87 - 43 * q^88 - 28 * q^89 - 13 * q^90 - 96 * q^91 - 11 * q^92 + 10 * q^93 - 63 * q^94 - 23 * q^95 - 31 * q^96 - 51 * q^97 + 12 * q^98 - 23 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more