LMFDB - Embedded newform 6026.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6026,2,Mod(1,6026)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6026, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6026.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6026 = 2 \cdot 23 \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6026.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:	$48.1178522580$
Analytic rank:	$1$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	6026.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.37676 q^{3} +1.00000 q^{4} -2.07688 q^{5} -2.37676 q^{6} -0.108208 q^{7} +1.00000 q^{8} +2.64897 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.37676 q^{3} +1.00000 q^{4} -2.07688 q^{5} -2.37676 q^{6} -0.108208 q^{7} +1.00000 q^{8} +2.64897 q^{9} -2.07688 q^{10} -2.80503 q^{11} -2.37676 q^{12} +0.0828135 q^{13} -0.108208 q^{14} +4.93625 q^{15} +1.00000 q^{16} -0.866728 q^{17} +2.64897 q^{18} +2.95166 q^{19} -2.07688 q^{20} +0.257183 q^{21} -2.80503 q^{22} -1.00000 q^{23} -2.37676 q^{24} -0.686551 q^{25} +0.0828135 q^{26} +0.834320 q^{27} -0.108208 q^{28} +1.41210 q^{29} +4.93625 q^{30} +5.07911 q^{31} +1.00000 q^{32} +6.66686 q^{33} -0.866728 q^{34} +0.224735 q^{35} +2.64897 q^{36} +7.73693 q^{37} +2.95166 q^{38} -0.196827 q^{39} -2.07688 q^{40} +6.79434 q^{41} +0.257183 q^{42} +3.52653 q^{43} -2.80503 q^{44} -5.50160 q^{45} -1.00000 q^{46} -2.96463 q^{47} -2.37676 q^{48} -6.98829 q^{49} -0.686551 q^{50} +2.06000 q^{51} +0.0828135 q^{52} +5.25858 q^{53} +0.834320 q^{54} +5.82571 q^{55} -0.108208 q^{56} -7.01538 q^{57} +1.41210 q^{58} -2.25825 q^{59} +4.93625 q^{60} -11.0963 q^{61} +5.07911 q^{62} -0.286639 q^{63} +1.00000 q^{64} -0.171994 q^{65} +6.66686 q^{66} -5.53201 q^{67} -0.866728 q^{68} +2.37676 q^{69} +0.224735 q^{70} -3.89898 q^{71} +2.64897 q^{72} -9.27785 q^{73} +7.73693 q^{74} +1.63176 q^{75} +2.95166 q^{76} +0.303525 q^{77} -0.196827 q^{78} +7.76096 q^{79} -2.07688 q^{80} -9.92988 q^{81} +6.79434 q^{82} -3.12732 q^{83} +0.257183 q^{84} +1.80009 q^{85} +3.52653 q^{86} -3.35622 q^{87} -2.80503 q^{88} +15.6851 q^{89} -5.50160 q^{90} -0.00896105 q^{91} -1.00000 q^{92} -12.0718 q^{93} -2.96463 q^{94} -6.13026 q^{95} -2.37676 q^{96} -5.48412 q^{97} -6.98829 q^{98} -7.43042 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 21 q^{2} + 21 q^{4} - 13 q^{5} - 18 q^{7} + 21 q^{8} + 7 q^{9}+O(q^{10}) $ 21 * q + 21 * q^2 + 21 * q^4 - 13 * q^5 - 18 * q^7 + 21 * q^8 + 7 * q^9	\( 21 q + 21 q^{2} + 21 q^{4} - 13 q^{5} - 18 q^{7} + 21 q^{8} + 7 q^{9} - 13 q^{10} - 4 q^{11} - 4 q^{13} - 18 q^{14} - 16 q^{15} + 21 q^{16} - 12 q^{17} + 7 q^{18} - 18 q^{19} - 13 q^{20} - 24 q^{21} - 4 q^{22} - 21 q^{23} + 2 q^{25} - 4 q^{26} - 9 q^{27} - 18 q^{28} - 16 q^{29} - 16 q^{30} - 7 q^{31} + 21 q^{32} - 15 q^{33} - 12 q^{34} + 7 q^{36} - 44 q^{37} - 18 q^{38} - 14 q^{39} - 13 q^{40} - 23 q^{41} - 24 q^{42} - 18 q^{43} - 4 q^{44} - 36 q^{45} - 21 q^{46} + 2 q^{47} - 13 q^{49} + 2 q^{50} - 26 q^{51} - 4 q^{52} - 39 q^{53} - 9 q^{54} - 32 q^{55} - 18 q^{56} - 22 q^{57} - 16 q^{58} - 27 q^{59} - 16 q^{60} - 34 q^{61} - 7 q^{62} - 28 q^{63} + 21 q^{64} - 25 q^{65} - 15 q^{66} - 19 q^{67} - 12 q^{68} - 24 q^{71} + 7 q^{72} - 8 q^{73} - 44 q^{74} + 50 q^{75} - 18 q^{76} - 16 q^{77} - 14 q^{78} - 27 q^{79} - 13 q^{80} + 33 q^{81} - 23 q^{82} + 7 q^{83} - 24 q^{84} - 22 q^{85} - 18 q^{86} - 15 q^{87} - 4 q^{88} - 12 q^{89} - 36 q^{90} - 20 q^{91} - 21 q^{92} - 43 q^{93} + 2 q^{94} - 14 q^{95} - 52 q^{97} - 13 q^{98} - 30 q^{99}+O(q^{100}) \) 21 * q + 21 * q^2 + 21 * q^4 - 13 * q^5 - 18 * q^7 + 21 * q^8 + 7 * q^9 - 13 * q^10 - 4 * q^11 - 4 * q^13 - 18 * q^14 - 16 * q^15 + 21 * q^16 - 12 * q^17 + 7 * q^18 - 18 * q^19 - 13 * q^20 - 24 * q^21 - 4 * q^22 - 21 * q^23 + 2 * q^25 - 4 * q^26 - 9 * q^27 - 18 * q^28 - 16 * q^29 - 16 * q^30 - 7 * q^31 + 21 * q^32 - 15 * q^33 - 12 * q^34 + 7 * q^36 - 44 * q^37 - 18 * q^38 - 14 * q^39 - 13 * q^40 - 23 * q^41 - 24 * q^42 - 18 * q^43 - 4 * q^44 - 36 * q^45 - 21 * q^46 + 2 * q^47 - 13 * q^49 + 2 * q^50 - 26 * q^51 - 4 * q^52 - 39 * q^53 - 9 * q^54 - 32 * q^55 - 18 * q^56 - 22 * q^57 - 16 * q^58 - 27 * q^59 - 16 * q^60 - 34 * q^61 - 7 * q^62 - 28 * q^63 + 21 * q^64 - 25 * q^65 - 15 * q^66 - 19 * q^67 - 12 * q^68 - 24 * q^71 + 7 * q^72 - 8 * q^73 - 44 * q^74 + 50 * q^75 - 18 * q^76 - 16 * q^77 - 14 * q^78 - 27 * q^79 - 13 * q^80 + 33 * q^81 - 23 * q^82 + 7 * q^83 - 24 * q^84 - 22 * q^85 - 18 * q^86 - 15 * q^87 - 4 * q^88 - 12 * q^89 - 36 * q^90 - 20 * q^91 - 21 * q^92 - 43 * q^93 + 2 * q^94 - 14 * q^95 - 52 * q^97 - 13 * q^98 - 30 * 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
$3$	−2.37676	−1.37222	−0.686110	−	0.727498i	$-0.740683\pi$
$3$	−2.37676	−1.37222	−0.686110	+	0.727498i	$0.740683\pi$
$4$	1.00000	0.500000
$5$	−2.07688	−0.928811	−0.464405	−	0.885623i	$-0.653732\pi$
$5$	−2.07688	−0.928811	−0.464405	+	0.885623i	$0.653732\pi$
$6$	−2.37676	−0.970306
$7$	−0.108208	−0.0408987	−0.0204493	−	0.999791i	$-0.506510\pi$
$7$	−0.108208	−0.0408987	−0.0204493	+	0.999791i	$0.506510\pi$
$8$	1.00000	0.353553
$9$	2.64897	0.882989
$10$	−2.07688	−0.656769
$11$	−2.80503	−0.845747	−0.422874	−	0.906189i	$-0.638979\pi$
$11$	−2.80503	−0.845747	−0.422874	+	0.906189i	$0.638979\pi$
$12$	−2.37676	−0.686110
$13$	0.0828135	0.0229683	0.0114842	−	0.999934i	$-0.496344\pi$
$13$	0.0828135	0.0229683	0.0114842	+	0.999934i	$0.496344\pi$
$14$	−0.108208	−0.0289197
$15$	4.93625	1.27453
$16$	1.00000	0.250000
$17$	−0.866728	−0.210212	−0.105106	−	0.994461i	$-0.533518\pi$
$17$	−0.866728	−0.210212	−0.105106	+	0.994461i	$0.533518\pi$
$18$	2.64897	0.624367
$19$	2.95166	0.677158	0.338579	−	0.940938i	$-0.390054\pi$
$19$	2.95166	0.677158	0.338579	+	0.940938i	$0.390054\pi$
$20$	−2.07688	−0.464405
$21$	0.257183	0.0561220
$22$	−2.80503	−0.598033
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−2.37676	−0.485153
$25$	−0.686551	−0.137310
$26$	0.0828135	0.0162411
$27$	0.834320	0.160565
$28$	−0.108208	−0.0204493
$29$	1.41210	0.262221	0.131110	−	0.991368i	$-0.458146\pi$
$29$	1.41210	0.262221	0.131110	+	0.991368i	$0.458146\pi$
$30$	4.93625	0.901231
$31$	5.07911	0.912236	0.456118	−	0.889919i	$-0.349240\pi$
$31$	5.07911	0.912236	0.456118	+	0.889919i	$0.349240\pi$
$32$	1.00000	0.176777
$33$	6.66686	1.16055
$34$	−0.866728	−0.148643
$35$	0.224735	0.0379871
$36$	2.64897	0.441494
$37$	7.73693	1.27194	0.635972	−	0.771712i	$-0.280599\pi$
$37$	7.73693	1.27194	0.635972	+	0.771712i	$0.280599\pi$
$38$	2.95166	0.478823
$39$	−0.196827	−0.0315176
$40$	−2.07688	−0.328384
$41$	6.79434	1.06110	0.530549	−	0.847654i	$-0.321986\pi$
$41$	6.79434	1.06110	0.530549	+	0.847654i	$0.321986\pi$
$42$	0.257183	0.0396842
$43$	3.52653	0.537791	0.268896	−	0.963169i	$-0.413341\pi$
$43$	3.52653	0.537791	0.268896	+	0.963169i	$0.413341\pi$
$44$	−2.80503	−0.422874
$45$	−5.50160	−0.820130
$46$	−1.00000	−0.147442
$47$	−2.96463	−0.432436	−0.216218	−	0.976345i	$-0.569372\pi$
$47$	−2.96463	−0.432436	−0.216218	+	0.976345i	$0.569372\pi$
$48$	−2.37676	−0.343055
$49$	−6.98829	−0.998327
$50$	−0.686551	−0.0970929
$51$	2.06000	0.288458
$52$	0.0828135	0.0114842
$53$	5.25858	0.722321	0.361161	−	0.932504i	$-0.382381\pi$
$53$	5.25858	0.722321	0.361161	+	0.932504i	$0.382381\pi$
$54$	0.834320	0.113537
$55$	5.82571	0.785539
$56$	−0.108208	−0.0144599
$57$	−7.01538	−0.929210
$58$	1.41210	0.185418
$59$	−2.25825	−0.293999	−0.146999	−	0.989137i	$-0.546962\pi$
$59$	−2.25825	−0.293999	−0.146999	+	0.989137i	$0.546962\pi$
$60$	4.93625	0.637267
$61$	−11.0963	−1.42073	−0.710366	−	0.703833i	$-0.751470\pi$
$61$	−11.0963	−1.42073	−0.710366	+	0.703833i	$0.751470\pi$
$62$	5.07911	0.645048
$63$	−0.286639	−0.0361131
$64$	1.00000	0.125000
$65$	−0.171994	−0.0213332
$66$	6.66686	0.820634
$67$	−5.53201	−0.675843	−0.337921	−	0.941174i	$-0.609724\pi$
$67$	−5.53201	−0.675843	−0.337921	+	0.941174i	$0.609724\pi$
$68$	−0.866728	−0.105106
$69$	2.37676	0.286128
$70$	0.224735	0.0268609
$71$	−3.89898	−0.462724	−0.231362	−	0.972868i	$-0.574318\pi$
$71$	−3.89898	−0.462724	−0.231362	+	0.972868i	$0.574318\pi$
$72$	2.64897	0.312184
$73$	−9.27785	−1.08589	−0.542945	−	0.839768i	$-0.682691\pi$
$73$	−9.27785	−1.08589	−0.542945	+	0.839768i	$0.682691\pi$
$74$	7.73693	0.899400
$75$	1.63176	0.188420
$76$	2.95166	0.338579
$77$	0.303525	0.0345899
$78$	−0.196827	−0.0222863
$79$	7.76096	0.873176	0.436588	−	0.899662i	$-0.356187\pi$
$79$	7.76096	0.873176	0.436588	+	0.899662i	$0.356187\pi$
$80$	−2.07688	−0.232203
$81$	−9.92988	−1.10332
$82$	6.79434	0.750309
$83$	−3.12732	−0.343268	−0.171634	−	0.985161i	$-0.554905\pi$
$83$	−3.12732	−0.343268	−0.171634	+	0.985161i	$0.554905\pi$
$84$	0.257183	0.0280610
$85$	1.80009	0.195248
$86$	3.52653	0.380276
$87$	−3.35622	−0.359825
$88$	−2.80503	−0.299017
$89$	15.6851	1.66261	0.831307	−	0.555813i	$-0.187593\pi$
$89$	15.6851	1.66261	0.831307	+	0.555813i	$0.187593\pi$
$90$	−5.50160	−0.579919
$91$	−0.00896105	−0.000939374 0
$92$	−1.00000	−0.104257
$93$	−12.0718	−1.25179
$94$	−2.96463	−0.305778
$95$	−6.13026	−0.628952
$96$	−2.37676	−0.242577
$97$	−5.48412	−0.556828	−0.278414	−	0.960461i	$-0.589809\pi$
$97$	−5.48412	−0.556828	−0.278414	+	0.960461i	$0.589809\pi$
$98$	−6.98829	−0.705924
$99$	−7.43042	−0.746785
$100$	−0.686551	−0.0686551
$101$	7.30874	0.727246	0.363623	−	0.931546i	$-0.381540\pi$
$101$	7.30874	0.727246	0.363623	+	0.931546i	$0.381540\pi$
$102$	2.06000	0.203970
$103$	−2.07329	−0.204287	−0.102144	−	0.994770i	$-0.532570\pi$
$103$	−2.07329	−0.204287	−0.102144	+	0.994770i	$0.532570\pi$
$104$	0.0828135	0.00812053
$105$	−0.534140	−0.0521267
$106$	5.25858	0.510758
$107$	4.62492	0.447107	0.223554	−	0.974692i	$-0.428234\pi$
$107$	4.62492	0.447107	0.223554	+	0.974692i	$0.428234\pi$
$108$	0.834320	0.0802825
$109$	−9.23348	−0.884407	−0.442203	−	0.896915i	$-0.645803\pi$
$109$	−9.23348	−0.884407	−0.442203	+	0.896915i	$0.645803\pi$
$110$	5.82571	0.555460
$111$	−18.3888	−1.74539
$112$	−0.108208	−0.0102247
$113$	0.879993	0.0827828	0.0413914	−	0.999143i	$-0.486821\pi$
$113$	0.879993	0.0827828	0.0413914	+	0.999143i	$0.486821\pi$
$114$	−7.01538	−0.657051
$115$	2.07688	0.193670
$116$	1.41210	0.131110
$117$	0.219370	0.0202808
$118$	−2.25825	−0.207889
$119$	0.0937866	0.00859740
$120$	4.93625	0.450616
$121$	−3.13183	−0.284712
$122$	−11.0963	−1.00461
$123$	−16.1485	−1.45606
$124$	5.07911	0.456118
$125$	11.8103	1.05635
$126$	−0.286639	−0.0255358
$127$	−17.8117	−1.58053	−0.790265	−	0.612766i	$-0.790057\pi$
$127$	−17.8117	−1.58053	−0.790265	+	0.612766i	$0.790057\pi$
$128$	1.00000	0.0883883
$129$	−8.38171	−0.737968
$130$	−0.171994	−0.0150849
$131$	1.00000	0.0873704
$131$	1.00000	0.0873704
$132$	6.66686	0.580276
$133$	−0.319392	−0.0276948
$134$	−5.53201	−0.477893
$135$	−1.73279	−0.149135
$136$	−0.866728	−0.0743213
$137$	5.36384	0.458264	0.229132	−	0.973395i	$-0.426411\pi$
$137$	5.36384	0.458264	0.229132	+	0.973395i	$0.426411\pi$
$138$	2.37676	0.202323
$139$	4.82719	0.409437	0.204718	−	0.978821i	$-0.434372\pi$
$139$	4.82719	0.409437	0.204718	+	0.978821i	$0.434372\pi$
$140$	0.224735	0.0189936
$141$	7.04621	0.593398
$142$	−3.89898	−0.327195
$143$	−0.232294	−0.0194254
$144$	2.64897	0.220747
$145$	−2.93277	−0.243554
$146$	−9.27785	−0.767840
$147$	16.6095	1.36993
$148$	7.73693	0.635972
$149$	−5.27006	−0.431740	−0.215870	−	0.976422i	$-0.569259\pi$
$149$	−5.27006	−0.431740	−0.215870	+	0.976422i	$0.569259\pi$
$150$	1.63176	0.133233
$151$	6.09903	0.496332	0.248166	−	0.968717i	$-0.420172\pi$
$151$	6.09903	0.496332	0.248166	+	0.968717i	$0.420172\pi$
$152$	2.95166	0.239411
$153$	−2.29593	−0.185615
$154$	0.303525	0.0244588
$155$	−10.5487	−0.847294
$156$	−0.196827	−0.0157588
$157$	4.02522	0.321247	0.160624	−	0.987016i	$-0.448649\pi$
$157$	4.02522	0.321247	0.160624	+	0.987016i	$0.448649\pi$
$158$	7.76096	0.617429
$159$	−12.4984	−0.991184
$160$	−2.07688	−0.164192
$161$	0.108208	0.00852796
$162$	−9.92988	−0.780165
$163$	0.170959	0.0133906	0.00669529	−	0.999978i	$-0.497869\pi$
$163$	0.170959	0.0133906	0.00669529	+	0.999978i	$0.497869\pi$
$164$	6.79434	0.530549
$165$	−13.8463	−1.07793
$166$	−3.12732	−0.242727
$167$	19.6392	1.51973	0.759863	−	0.650083i	$-0.225266\pi$
$167$	19.6392	1.51973	0.759863	+	0.650083i	$0.225266\pi$
$168$	0.257183	0.0198421
$169$	−12.9931	−0.999472
$170$	1.80009	0.138061
$171$	7.81886	0.597923
$172$	3.52653	0.268896
$173$	24.5496	1.86647	0.933237	−	0.359262i	$-0.116971\pi$
$173$	24.5496	1.86647	0.933237	+	0.359262i	$0.116971\pi$
$174$	−3.35622	−0.254435
$175$	0.0742901	0.00561580
$176$	−2.80503	−0.211437
$177$	5.36731	0.403431
$178$	15.6851	1.17565
$179$	−0.847188	−0.0633218	−0.0316609	−	0.999499i	$-0.510080\pi$
$179$	−0.847188	−0.0633218	−0.0316609	+	0.999499i	$0.510080\pi$
$180$	−5.50160	−0.410065
$181$	−22.8920	−1.70155	−0.850773	−	0.525533i	$-0.823866\pi$
$181$	−22.8920	−1.70155	−0.850773	+	0.525533i	$0.823866\pi$
$182$	−0.00896105	−0.000664237 0
$183$	26.3731	1.94956
$184$	−1.00000	−0.0737210
$185$	−16.0687	−1.18139
$186$	−12.0718	−0.885148
$187$	2.43119	0.177786
$188$	−2.96463	−0.216218
$189$	−0.0902799	−0.00656689
$190$	−6.13026	−0.444736
$191$	−4.53788	−0.328350	−0.164175	−	0.986431i	$-0.552496\pi$
$191$	−4.53788	−0.328350	−0.164175	+	0.986431i	$0.552496\pi$
$192$	−2.37676	−0.171528
$193$	−22.9480	−1.65183	−0.825915	−	0.563794i	$-0.809341\pi$
$193$	−22.9480	−1.65183	−0.825915	+	0.563794i	$0.809341\pi$
$194$	−5.48412	−0.393737
$195$	0.408788	0.0292739
$196$	−6.98829	−0.499164
$197$	−13.2147	−0.941511	−0.470756	−	0.882264i	$-0.656019\pi$
$197$	−13.2147	−0.941511	−0.470756	+	0.882264i	$0.656019\pi$
$198$	−7.43042	−0.528057
$199$	7.49401	0.531236	0.265618	−	0.964078i	$-0.414424\pi$
$199$	7.49401	0.531236	0.265618	+	0.964078i	$0.414424\pi$
$200$	−0.686551	−0.0485465
$201$	13.1482	0.927405
$202$	7.30874	0.514241
$203$	−0.152800	−0.0107245
$204$	2.06000	0.144229
$205$	−14.1111	−0.985559
$206$	−2.07329	−0.144453
$207$	−2.64897	−0.184116
$208$	0.0828135	0.00574208
$209$	−8.27949	−0.572704
$210$	−0.534140	−0.0368591
$211$	−5.21316	−0.358889	−0.179444	−	0.983768i	$-0.557430\pi$
$211$	−5.21316	−0.358889	−0.179444	+	0.983768i	$0.557430\pi$
$212$	5.25858	0.361161
$213$	9.26693	0.634960
$214$	4.62492	0.316153
$215$	−7.32420	−0.499507
$216$	0.834320	0.0567683
$217$	−0.549599	−0.0373092
$218$	−9.23348	−0.625370
$219$	22.0512	1.49008
$220$	5.82571	0.392770
$221$	−0.0717767	−0.00482823
$222$	−18.3888	−1.23417
$223$	−7.92300	−0.530563	−0.265282	−	0.964171i	$-0.585465\pi$
$223$	−7.92300	−0.530563	−0.265282	+	0.964171i	$0.585465\pi$
$224$	−0.108208	−0.00722993
$225$	−1.81865	−0.121243
$226$	0.879993	0.0585363
$227$	−7.15155	−0.474665	−0.237332	−	0.971429i	$-0.576273\pi$
$227$	−7.15155	−0.474665	−0.237332	+	0.971429i	$0.576273\pi$
$228$	−7.01538	−0.464605
$229$	−29.9139	−1.97676	−0.988382	−	0.151992i	$-0.951431\pi$
$229$	−29.9139	−1.97676	−0.988382	+	0.151992i	$0.951431\pi$
$230$	2.07688	0.136946
$231$	−0.721405	−0.0474650
$232$	1.41210	0.0927091
$233$	−6.35931	−0.416612	−0.208306	−	0.978064i	$-0.566795\pi$
$233$	−6.35931	−0.416612	−0.208306	+	0.978064i	$0.566795\pi$
$234$	0.219370	0.0143407
$235$	6.15720	0.401651
$236$	−2.25825	−0.146999
$237$	−18.4459	−1.19819
$238$	0.0937866	0.00607928
$239$	−21.0616	−1.36236	−0.681181	−	0.732115i	$-0.738533\pi$
$239$	−21.0616	−1.36236	−0.681181	+	0.732115i	$0.738533\pi$
$240$	4.93625	0.318633
$241$	−17.5666	−1.13157	−0.565784	−	0.824554i	$-0.691426\pi$
$241$	−17.5666	−1.13157	−0.565784	+	0.824554i	$0.691426\pi$
$242$	−3.13183	−0.201322
$243$	21.0979	1.35343
$244$	−11.0963	−0.710366
$245$	14.5139	0.927257
$246$	−16.1485	−1.02959
$247$	0.244437	0.0155532
$248$	5.07911	0.322524
$249$	7.43288	0.471039
$250$	11.8103	0.746950
$251$	−20.9312	−1.32117	−0.660583	−	0.750753i	$-0.729691\pi$
$251$	−20.9312	−1.32117	−0.660583	+	0.750753i	$0.729691\pi$
$252$	−0.286639	−0.0180565
$253$	2.80503	0.176350
$254$	−17.8117	−1.11760
$255$	−4.27838	−0.267923
$256$	1.00000	0.0625000
$257$	17.6755	1.10257	0.551283	−	0.834318i	$-0.314138\pi$
$257$	17.6755	1.10257	0.551283	+	0.834318i	$0.314138\pi$
$258$	−8.38171	−0.521822
$259$	−0.837195	−0.0520208
$260$	−0.171994	−0.0106666
$261$	3.74061	0.231538
$262$	1.00000	0.0617802
$263$	1.45115	0.0894815	0.0447408	−	0.998999i	$-0.485754\pi$
$263$	1.45115	0.0894815	0.0447408	+	0.998999i	$0.485754\pi$
$264$	6.66686	0.410317
$265$	−10.9215	−0.670900
$266$	−0.319392	−0.0195832
$267$	−37.2796	−2.28147
$268$	−5.53201	−0.337921
$269$	−1.97362	−0.120334	−0.0601669	−	0.998188i	$-0.519163\pi$
$269$	−1.97362	−0.120334	−0.0601669	+	0.998188i	$0.519163\pi$
$270$	−1.73279	−0.105454
$271$	−27.1721	−1.65059	−0.825294	−	0.564704i	$-0.808990\pi$
$271$	−27.1721	−1.65059	−0.825294	+	0.564704i	$0.808990\pi$
$272$	−0.866728	−0.0525531
$273$	0.0212982	0.00128903
$274$	5.36384	0.324041
$275$	1.92579	0.116130
$276$	2.37676	0.143064
$277$	−1.56531	−0.0940501	−0.0470251	−	0.998894i	$-0.514974\pi$
$277$	−1.56531	−0.0940501	−0.0470251	+	0.998894i	$0.514974\pi$
$278$	4.82719	0.289515
$279$	13.4544	0.805494
$280$	0.224735	0.0134305
$281$	−27.8384	−1.66070	−0.830351	−	0.557241i	$-0.811860\pi$
$281$	−27.8384	−1.66070	−0.830351	+	0.557241i	$0.811860\pi$
$282$	7.04621	0.419595
$283$	−18.0119	−1.07070	−0.535348	−	0.844632i	$-0.679819\pi$
$283$	−18.0119	−1.07070	−0.535348	+	0.844632i	$0.679819\pi$
$284$	−3.89898	−0.231362
$285$	14.5701	0.863060
$286$	−0.232294	−0.0137358
$287$	−0.735200	−0.0433975
$288$	2.64897	0.156092
$289$	−16.2488	−0.955811
$290$	−2.93277	−0.172218
$291$	13.0344	0.764091
$292$	−9.27785	−0.542945
$293$	−30.2746	−1.76866	−0.884331	−	0.466861i	$-0.845385\pi$
$293$	−30.2746	−1.76866	−0.884331	+	0.466861i	$0.845385\pi$
$294$	16.6095	0.968683
$295$	4.69012	0.273069
$296$	7.73693	0.449700
$297$	−2.34029	−0.135797
$298$	−5.27006	−0.305286
$299$	−0.0828135	−0.00478923
$300$	1.63176	0.0942099
$301$	−0.381598	−0.0219949
$302$	6.09903	0.350960
$303$	−17.3711	−0.997942
$304$	2.95166	0.169289
$305$	23.0457	1.31959
$306$	−2.29593	−0.131250
$307$	20.4721	1.16840	0.584202	−	0.811609i	$-0.301408\pi$
$307$	20.4721	1.16840	0.584202	+	0.811609i	$0.301408\pi$
$308$	0.303525	0.0172950
$309$	4.92771	0.280327
$310$	−10.5487	−0.599128
$311$	9.84036	0.557996	0.278998	−	0.960292i	$-0.409998\pi$
$311$	9.84036	0.557996	0.278998	+	0.960292i	$0.409998\pi$
$312$	−0.196827	−0.0111432
$313$	−18.3946	−1.03972	−0.519862	−	0.854250i	$-0.674017\pi$
$313$	−18.3946	−1.03972	−0.519862	+	0.854250i	$0.674017\pi$
$314$	4.02522	0.227156
$315$	0.595315	0.0335422
$316$	7.76096	0.436588
$317$	−5.68861	−0.319504	−0.159752	−	0.987157i	$-0.551070\pi$
$317$	−5.68861	−0.319504	−0.159752	+	0.987157i	$0.551070\pi$
$318$	−12.4984	−0.700873
$319$	−3.96098	−0.221772
$320$	−2.07688	−0.116101
$321$	−10.9923	−0.613530
$322$	0.108208	0.00603018
$323$	−2.55829	−0.142347
$324$	−9.92988	−0.551660
$325$	−0.0568557	−0.00315378
$326$	0.170959	0.00946856
$327$	21.9457	1.21360
$328$	6.79434	0.375155
$329$	0.320796	0.0176861
$330$	−13.8463	−0.762214
$331$	3.31417	0.182163	0.0910815	−	0.995843i	$-0.470968\pi$
$331$	3.31417	0.182163	0.0910815	+	0.995843i	$0.470968\pi$
$332$	−3.12732	−0.171634
$333$	20.4949	1.12311
$334$	19.6392	1.07461
$335$	11.4893	0.627730
$336$	0.257183	0.0140305
$337$	−27.0540	−1.47373	−0.736863	−	0.676042i	$-0.763694\pi$
$337$	−27.0540	−1.47373	−0.736863	+	0.676042i	$0.763694\pi$
$338$	−12.9931	−0.706734
$339$	−2.09153	−0.113596
$340$	1.80009	0.0976238
$341$	−14.2470	−0.771521
$342$	7.81886	0.422795
$343$	1.51364	0.0817289
$344$	3.52653	0.190138
$345$	−4.93625	−0.265759
$346$	24.5496	1.31980
$347$	−3.64406	−0.195624	−0.0978118	−	0.995205i	$-0.531184\pi$
$347$	−3.64406	−0.195624	−0.0978118	+	0.995205i	$0.531184\pi$
$348$	−3.35622	−0.179912
$349$	15.3347	0.820846	0.410423	−	0.911895i	$-0.365381\pi$
$349$	15.3347	0.820846	0.410423	+	0.911895i	$0.365381\pi$
$350$	0.0742901	0.00397097
$351$	0.0690930	0.00368791
$352$	−2.80503	−0.149508
$353$	1.77558	0.0945044	0.0472522	−	0.998883i	$-0.484954\pi$
$353$	1.77558	0.0945044	0.0472522	+	0.998883i	$0.484954\pi$
$354$	5.36731	0.285269
$355$	8.09774	0.429783
$356$	15.6851	0.831307
$357$	−0.222908	−0.0117975
$358$	−0.847188	−0.0447753
$359$	12.1509	0.641302	0.320651	−	0.947197i	$-0.396098\pi$
$359$	12.1509	0.641302	0.320651	+	0.947197i	$0.396098\pi$
$360$	−5.50160	−0.289960
$361$	−10.2877	−0.541457
$362$	−22.8920	−1.20317
$363$	7.44360	0.390687
$364$	−0.00896105	−0.000469687 0
$365$	19.2690	1.00859
$366$	26.3731	1.37854
$367$	4.93751	0.257736	0.128868	−	0.991662i	$-0.458866\pi$
$367$	4.93751	0.257736	0.128868	+	0.991662i	$0.458866\pi$
$368$	−1.00000	−0.0521286
$369$	17.9980	0.936937
$370$	−16.0687	−0.835372
$371$	−0.569019	−0.0295420
$372$	−12.0718	−0.625894
$373$	0.698850	0.0361851	0.0180925	−	0.999836i	$-0.494241\pi$
$373$	0.698850	0.0361851	0.0180925	+	0.999836i	$0.494241\pi$
$374$	2.43119	0.125714
$375$	−28.0702	−1.44954
$376$	−2.96463	−0.152889
$377$	0.116941	0.00602277
$378$	−0.0902799	−0.00464349
$379$	−13.1842	−0.677224	−0.338612	−	0.940926i	$-0.609957\pi$
$379$	−13.1842	−0.677224	−0.338612	+	0.940926i	$0.609957\pi$
$380$	−6.13026	−0.314476
$381$	42.3340	2.16883
$382$	−4.53788	−0.232178
$383$	−12.3879	−0.632994	−0.316497	−	0.948594i	$-0.602507\pi$
$383$	−12.3879	−0.632994	−0.316497	+	0.948594i	$0.602507\pi$
$384$	−2.37676	−0.121288
$385$	−0.630387	−0.0321275
$386$	−22.9480	−1.16802
$387$	9.34167	0.474864
$388$	−5.48412	−0.278414
$389$	33.6328	1.70525	0.852626	−	0.522521i	$-0.175008\pi$
$389$	33.6328	1.70525	0.852626	+	0.522521i	$0.175008\pi$
$390$	0.408788	0.0206998
$391$	0.866728	0.0438323
$392$	−6.98829	−0.352962
$393$	−2.37676	−0.119891
$394$	−13.2147	−0.665749
$395$	−16.1186	−0.811015
$396$	−7.43042	−0.373393
$397$	21.5807	1.08311	0.541553	−	0.840667i	$-0.317836\pi$
$397$	21.5807	1.08311	0.541553	+	0.840667i	$0.317836\pi$
$398$	7.49401	0.375641
$399$	0.759118	0.0380034
$400$	−0.686551	−0.0343275
$401$	−10.5331	−0.525998	−0.262999	−	0.964796i	$-0.584712\pi$
$401$	−10.5331	−0.525998	−0.262999	+	0.964796i	$0.584712\pi$
$402$	13.1482	0.655774
$403$	0.420619	0.0209525
$404$	7.30874	0.363623
$405$	20.6232	1.02478
$406$	−0.152800	−0.00758335
$407$	−21.7023	−1.07574
$408$	2.06000	0.101985
$409$	24.1264	1.19298	0.596488	−	0.802622i	$-0.296562\pi$
$409$	24.1264	1.19298	0.596488	+	0.802622i	$0.296562\pi$
$410$	−14.1111	−0.696896
$411$	−12.7485	−0.628839
$412$	−2.07329	−0.102144
$413$	0.244360	0.0120242
$414$	−2.64897	−0.130190
$415$	6.49508	0.318831
$416$	0.0828135	0.00406026
$417$	−11.4730	−0.561837
$418$	−8.27949	−0.404963
$419$	22.2818	1.08854	0.544268	−	0.838912i	$-0.316808\pi$
$419$	22.2818	1.08854	0.544268	+	0.838912i	$0.316808\pi$
$420$	−0.534140	−0.0260634
$421$	2.71744	0.132440	0.0662199	−	0.997805i	$-0.478906\pi$
$421$	2.71744	0.132440	0.0662199	+	0.997805i	$0.478906\pi$
$422$	−5.21316	−0.253773
$423$	−7.85321	−0.381836
$424$	5.25858	0.255379
$425$	0.595053	0.0288643
$426$	9.26693	0.448984
$427$	1.20070	0.0581060
$428$	4.62492	0.223554
$429$	0.552106	0.0266559
$430$	−7.32420	−0.353204
$431$	22.0113	1.06025	0.530124	−	0.847920i	$-0.322146\pi$
$431$	22.0113	1.06025	0.530124	+	0.847920i	$0.322146\pi$
$432$	0.834320	0.0401413
$433$	7.80509	0.375089	0.187544	−	0.982256i	$-0.439947\pi$
$433$	7.80509	0.375089	0.187544	+	0.982256i	$0.439947\pi$
$434$	−0.549599	−0.0263816
$435$	6.97049	0.334209
$436$	−9.23348	−0.442203
$437$	−2.95166	−0.141197
$438$	22.0512	1.05365
$439$	1.37982	0.0658554	0.0329277	−	0.999458i	$-0.489517\pi$
$439$	1.37982	0.0658554	0.0329277	+	0.999458i	$0.489517\pi$
$440$	5.82571	0.277730
$441$	−18.5118	−0.881512
$442$	−0.0717767	−0.00341407
$443$	22.4419	1.06625	0.533123	−	0.846038i	$-0.321018\pi$
$443$	22.4419	1.06625	0.533123	+	0.846038i	$0.321018\pi$
$444$	−18.3888	−0.872693
$445$	−32.5761	−1.54425
$446$	−7.92300	−0.375165
$447$	12.5256	0.592443
$448$	−0.108208	−0.00511233
$449$	−12.3543	−0.583034	−0.291517	−	0.956566i	$-0.594160\pi$
$449$	−12.3543	−0.583034	−0.291517	+	0.956566i	$0.594160\pi$
$450$	−1.81865	−0.0857320
$451$	−19.0583	−0.897420
$452$	0.879993	0.0413914
$453$	−14.4959	−0.681077
$454$	−7.15155	−0.335639
$455$	0.0186111	0.000872501 0
$456$	−7.01538	−0.328525
$457$	18.0206	0.842967	0.421483	−	0.906836i	$-0.361510\pi$
$457$	18.0206	0.842967	0.421483	+	0.906836i	$0.361510\pi$
$458$	−29.9139	−1.39778
$459$	−0.723128	−0.0337527
$460$	2.07688	0.0968352
$461$	14.7184	0.685505	0.342752	−	0.939426i	$-0.388641\pi$
$461$	14.7184	0.685505	0.342752	+	0.939426i	$0.388641\pi$
$462$	−0.721405	−0.0335628
$463$	−7.62867	−0.354534	−0.177267	−	0.984163i	$-0.556726\pi$
$463$	−7.62867	−0.354534	−0.177267	+	0.984163i	$0.556726\pi$
$464$	1.41210	0.0655552
$465$	25.0718	1.16267
$466$	−6.35931	−0.294589
$467$	−6.80461	−0.314880	−0.157440	−	0.987529i	$-0.550324\pi$
$467$	−6.80461	−0.314880	−0.157440	+	0.987529i	$0.550324\pi$
$468$	0.219370	0.0101404
$469$	0.598606	0.0276410
$470$	6.15720	0.284010
$471$	−9.56696	−0.440822
$472$	−2.25825	−0.103944
$473$	−9.89202	−0.454835
$474$	−18.4459	−0.847248
$475$	−2.02647	−0.0929806
$476$	0.0937866	0.00429870
$477$	13.9298	0.637802
$478$	−21.0616	−0.963336
$479$	26.1954	1.19690	0.598449	−	0.801161i	$-0.295784\pi$
$479$	26.1954	1.19690	0.598449	+	0.801161i	$0.295784\pi$
$480$	4.93625	0.225308
$481$	0.640722	0.0292144
$482$	−17.5666	−0.800139
$483$	−0.257183	−0.0117022
$484$	−3.13183	−0.142356
$485$	11.3899	0.517188
$486$	21.0979	0.957021
$487$	−40.4641	−1.83360	−0.916801	−	0.399344i	$-0.869238\pi$
$487$	−40.4641	−1.83360	−0.916801	+	0.399344i	$0.869238\pi$
$488$	−11.0963	−0.502304
$489$	−0.406329	−0.0183748
$490$	14.5139	0.655670
$491$	12.6295	0.569960	0.284980	−	0.958534i	$-0.408013\pi$
$491$	12.6295	0.569960	0.284980	+	0.958534i	$0.408013\pi$
$492$	−16.1485	−0.728030
$493$	−1.22391	−0.0551221
$494$	0.244437	0.0109978
$495$	15.4321	0.693622
$496$	5.07911	0.228059
$497$	0.421900	0.0189248
$498$	7.43288	0.333075
$499$	35.4878	1.58865	0.794326	−	0.607492i	$-0.207824\pi$
$499$	35.4878	1.58865	0.794326	+	0.607492i	$0.207824\pi$
$500$	11.8103	0.528173
$501$	−46.6776	−2.08540
$502$	−20.9312	−0.934206
$503$	−34.2603	−1.52759	−0.763796	−	0.645457i	$-0.776667\pi$
$503$	−34.2603	−1.52759	−0.763796	+	0.645457i	$0.776667\pi$
$504$	−0.286639	−0.0127679
$505$	−15.1794	−0.675475
$506$	2.80503	0.124699
$507$	30.8815	1.37150
$508$	−17.8117	−0.790265
$509$	−40.4817	−1.79432	−0.897161	−	0.441704i	$-0.854374\pi$
$509$	−40.4817	−1.79432	−0.897161	+	0.441704i	$0.854374\pi$
$510$	−4.27838	−0.189450
$511$	1.00393	0.0444115
$512$	1.00000	0.0441942
$513$	2.46263	0.108728
$514$	17.6755	0.779632
$515$	4.30599	0.189744
$516$	−8.38171	−0.368984
$517$	8.31587	0.365732
$518$	−0.837195	−0.0367842
$519$	−58.3485	−2.56121
$520$	−0.171994	−0.00754244
$521$	−2.57010	−0.112598	−0.0562992	−	0.998414i	$-0.517930\pi$
$521$	−2.57010	−0.112598	−0.0562992	+	0.998414i	$0.517930\pi$
$522$	3.74061	0.163722
$523$	39.0175	1.70611	0.853057	−	0.521817i	$-0.174746\pi$
$523$	39.0175	1.70611	0.853057	+	0.521817i	$0.174746\pi$
$524$	1.00000	0.0436852
$525$	−0.176569	−0.00770612
$526$	1.45115	0.0632730
$527$	−4.40221	−0.191763
$528$	6.66686	0.290138
$529$	1.00000	0.0434783
$530$	−10.9215	−0.474398
$531$	−5.98203	−0.259598
$532$	−0.319392	−0.0138474
$533$	0.562663	0.0243716
$534$	−37.2796	−1.61325
$535$	−9.60542	−0.415278
$536$	−5.53201	−0.238946
$537$	2.01356	0.0868915
$538$	−1.97362	−0.0850889
$539$	19.6023	0.844332
$540$	−1.73279	−0.0745673
$541$	−7.38422	−0.317473	−0.158736	−	0.987321i	$-0.550742\pi$
$541$	−7.38422	−0.317473	−0.158736	+	0.987321i	$0.550742\pi$
$542$	−27.1721	−1.16714
$543$	54.4086	2.33490
$544$	−0.866728	−0.0371606
$545$	19.1769	0.821447
$546$	0.0212982	0.000911480 0
$547$	−9.73443	−0.416214	−0.208107	−	0.978106i	$-0.566730\pi$
$547$	−9.73443	−0.416214	−0.208107	+	0.978106i	$0.566730\pi$
$548$	5.36384	0.229132
$549$	−29.3936	−1.25449
$550$	1.92579	0.0821161
$551$	4.16805	0.177565
$552$	2.37676	0.101161
$553$	−0.839795	−0.0357117
$554$	−1.56531	−0.0665035
$555$	38.1914	1.62113
$556$	4.82719	0.204718
$557$	−45.2274	−1.91634	−0.958172	−	0.286192i	$-0.907610\pi$
$557$	−45.2274	−1.91634	−0.958172	+	0.286192i	$0.907610\pi$
$558$	13.4544	0.569570
$559$	0.292045	0.0123522
$560$	0.224735	0.00949678
$561$	−5.77835	−0.243962
$562$	−27.8384	−1.17429
$563$	−15.1679	−0.639251	−0.319625	−	0.947544i	$-0.603557\pi$
$563$	−15.1679	−0.639251	−0.319625	+	0.947544i	$0.603557\pi$
$564$	7.04621	0.296699
$565$	−1.82764	−0.0768896
$566$	−18.0119	−0.757096
$567$	1.07449	0.0451243
$568$	−3.89898	−0.163598
$569$	−33.6878	−1.41227	−0.706133	−	0.708080i	$-0.749562\pi$
$569$	−33.6878	−1.41227	−0.706133	+	0.708080i	$0.749562\pi$
$570$	14.5701	0.610276
$571$	−26.7184	−1.11813	−0.559066	−	0.829123i	$-0.688840\pi$
$571$	−26.7184	−1.11813	−0.559066	+	0.829123i	$0.688840\pi$
$572$	−0.232294	−0.00971270
$573$	10.7854	0.450568
$574$	−0.735200	−0.0306866
$575$	0.686551	0.0286311
$576$	2.64897	0.110374
$577$	−24.2497	−1.00953	−0.504764	−	0.863258i	$-0.668420\pi$
$577$	−24.2497	−1.00953	−0.504764	+	0.863258i	$0.668420\pi$
$578$	−16.2488	−0.675860
$579$	54.5417	2.26668
$580$	−2.93277	−0.121777
$581$	0.338400	0.0140392
$582$	13.0344	0.540294
$583$	−14.7505	−0.610901
$584$	−9.27785	−0.383920
$585$	−0.455606	−0.0188370
$586$	−30.2746	−1.25063
$587$	45.1103	1.86190	0.930950	−	0.365146i	$-0.118981\pi$
$587$	45.1103	1.86190	0.930950	+	0.365146i	$0.118981\pi$
$588$	16.6095	0.684963
$589$	14.9918	0.617727
$590$	4.69012	0.193089
$591$	31.4082	1.29196
$592$	7.73693	0.317986
$593$	8.54272	0.350807	0.175404	−	0.984497i	$-0.443877\pi$
$593$	8.54272	0.350807	0.175404	+	0.984497i	$0.443877\pi$
$594$	−2.34029	−0.0960233
$595$	−0.194784	−0.00798536
$596$	−5.27006	−0.215870
$597$	−17.8114	−0.728973
$598$	−0.0828135	−0.00338650
$599$	−20.4788	−0.836740	−0.418370	−	0.908277i	$-0.637398\pi$
$599$	−20.4788	−0.836740	−0.418370	+	0.908277i	$0.637398\pi$
$600$	1.63176	0.0666165
$601$	−14.5455	−0.593323	−0.296662	−	0.954983i	$-0.595873\pi$
$601$	−14.5455	−0.593323	−0.296662	+	0.954983i	$0.595873\pi$
$602$	−0.381598	−0.0155528
$603$	−14.6541	−0.596761
$604$	6.09903	0.248166
$605$	6.50445	0.264444
$606$	−17.3711	−0.705652
$607$	28.4925	1.15648	0.578238	−	0.815868i	$-0.303740\pi$
$607$	28.4925	1.15648	0.578238	+	0.815868i	$0.303740\pi$
$608$	2.95166	0.119706
$609$	0.363169	0.0147163
$610$	23.0457	0.933091
$611$	−0.245512	−0.00993233
$612$	−2.29593	−0.0928076
$613$	−9.32014	−0.376437	−0.188219	−	0.982127i	$-0.560271\pi$
$613$	−9.32014	−0.376437	−0.188219	+	0.982127i	$0.560271\pi$
$614$	20.4721	0.826186
$615$	33.5385	1.35240
$616$	0.303525	0.0122294
$617$	−25.4058	−1.02280	−0.511400	−	0.859343i	$-0.670873\pi$
$617$	−25.4058	−1.02280	−0.511400	+	0.859343i	$0.670873\pi$
$618$	4.92771	0.198221
$619$	−2.09258	−0.0841079	−0.0420540	−	0.999115i	$-0.513390\pi$
$619$	−2.09258	−0.0841079	−0.0420540	+	0.999115i	$0.513390\pi$
$620$	−10.5487	−0.423647
$621$	−0.834320	−0.0334801
$622$	9.84036	0.394562
$623$	−1.69725	−0.0679987
$624$	−0.196827	−0.00787940
$625$	−21.0959	−0.843836
$626$	−18.3946	−0.735196
$627$	19.6783	0.785876
$628$	4.02522	0.160624
$629$	−6.70581	−0.267378
$630$	0.595315	0.0237179
$631$	32.5273	1.29489	0.647445	−	0.762112i	$-0.275837\pi$
$631$	32.5273	1.29489	0.647445	+	0.762112i	$0.275837\pi$
$632$	7.76096	0.308714
$633$	12.3904	0.492475
$634$	−5.68861	−0.225924
$635$	36.9928	1.46801
$636$	−12.4984	−0.495592
$637$	−0.578725	−0.0229299
$638$	−3.96098	−0.156817
$639$	−10.3283	−0.408580
$640$	−2.07688	−0.0820961
$641$	7.68132	0.303394	0.151697	−	0.988427i	$-0.451526\pi$
$641$	7.68132	0.303394	0.151697	+	0.988427i	$0.451526\pi$
$642$	−10.9923	−0.433831
$643$	8.57923	0.338332	0.169166	−	0.985588i	$-0.445893\pi$
$643$	8.57923	0.338332	0.169166	+	0.985588i	$0.445893\pi$
$644$	0.108208	0.00426398
$645$	17.4078	0.685433
$646$	−2.55829	−0.100654
$647$	−0.294591	−0.0115816	−0.00579078	−	0.999983i	$-0.501843\pi$
$647$	−0.294591	−0.0115816	−0.00579078	+	0.999983i	$0.501843\pi$
$648$	−9.92988	−0.390082
$649$	6.33445	0.248649
$650$	−0.0568557	−0.00223006
$651$	1.30626	0.0511965
$652$	0.170959	0.00669529
$653$	25.8398	1.01119	0.505595	−	0.862771i	$-0.331273\pi$
$653$	25.8398	1.01119	0.505595	+	0.862771i	$0.331273\pi$
$654$	21.9457	0.858146
$655$	−2.07688	−0.0811506
$656$	6.79434	0.265274
$657$	−24.5767	−0.958829
$658$	0.320796	0.0125059
$659$	−39.2154	−1.52762	−0.763808	−	0.645444i	$-0.776673\pi$
$659$	−39.2154	−1.52762	−0.763808	+	0.645444i	$0.776673\pi$
$660$	−13.8463	−0.538966
$661$	4.31112	0.167683	0.0838417	−	0.996479i	$-0.473281\pi$
$661$	4.31112	0.167683	0.0838417	+	0.996479i	$0.473281\pi$
$662$	3.31417	0.128809
$663$	0.170596	0.00662539
$664$	−3.12732	−0.121364
$665$	0.663341	0.0257233
$666$	20.4949	0.794160
$667$	−1.41210	−0.0546768
$668$	19.6392	0.759863
$669$	18.8310	0.728050
$670$	11.4893	0.443872
$671$	31.1253	1.20158
$672$	0.257183	0.00992106
$673$	27.5376	1.06150	0.530749	−	0.847529i	$-0.321911\pi$
$673$	27.5376	1.06150	0.530749	+	0.847529i	$0.321911\pi$
$674$	−27.0540	−1.04208
$675$	−0.572803	−0.0220472
$676$	−12.9931	−0.499736
$677$	42.0987	1.61798	0.808992	−	0.587820i	$-0.200014\pi$
$677$	42.0987	1.61798	0.808992	+	0.587820i	$0.200014\pi$
$678$	−2.09153	−0.0803247
$679$	0.593424	0.0227735
$680$	1.80009	0.0690304
$681$	16.9975	0.651345
$682$	−14.2470	−0.545547
$683$	22.6940	0.868364	0.434182	−	0.900825i	$-0.357038\pi$
$683$	22.6940	0.868364	0.434182	+	0.900825i	$0.357038\pi$
$684$	7.81886	0.298961
$685$	−11.1401	−0.425640
$686$	1.51364	0.0577911
$687$	71.0979	2.71256
$688$	3.52653	0.134448
$689$	0.435481	0.0165905
$690$	−4.93625	−0.187920
$691$	44.2578	1.68365	0.841823	−	0.539753i	$-0.181482\pi$
$691$	44.2578	1.68365	0.841823	+	0.539753i	$0.181482\pi$
$692$	24.5496	0.933237
$693$	0.804028	0.0305425
$694$	−3.64406	−0.138327
$695$	−10.0255	−0.380289
$696$	−3.35622	−0.127217
$697$	−5.88884	−0.223056
$698$	15.3347	0.580426
$699$	15.1145	0.571684
$700$	0.0742901	0.00280790
$701$	43.5646	1.64541	0.822707	−	0.568466i	$-0.192463\pi$
$701$	43.5646	1.64541	0.822707	+	0.568466i	$0.192463\pi$
$702$	0.0690930	0.00260775
$703$	22.8368	0.861306
$704$	−2.80503	−0.105718
$705$	−14.6342	−0.551154
$706$	1.77558	0.0668247
$707$	−0.790861	−0.0297434
$708$	5.36731	0.201716
$709$	−30.0985	−1.13037	−0.565186	−	0.824964i	$-0.691195\pi$
$709$	−30.0985	−1.13037	−0.565186	+	0.824964i	$0.691195\pi$
$710$	8.09774	0.303903
$711$	20.5585	0.771005
$712$	15.6851	0.587823
$713$	−5.07911	−0.190214
$714$	−0.222908	−0.00834211
$715$	0.482448	0.0180425
$716$	−0.847188	−0.0316609
$717$	50.0583	1.86946
$718$	12.1509	0.453469
$719$	14.2391	0.531028	0.265514	−	0.964107i	$-0.414458\pi$
$719$	14.2391	0.531028	0.265514	+	0.964107i	$0.414458\pi$
$720$	−5.50160	−0.205032
$721$	0.224346	0.00835508
$722$	−10.2877	−0.382868
$723$	41.7516	1.55276
$724$	−22.8920	−0.850773
$725$	−0.969480	−0.0360056
$726$	7.44360	0.276258
$727$	−34.2489	−1.27022	−0.635110	−	0.772421i	$-0.719045\pi$
$727$	−34.2489	−1.27022	−0.635110	+	0.772421i	$0.719045\pi$
$728$	−0.00896105	−0.000332119 0
$729$	−20.3550	−0.753888
$730$	19.2690	0.713179
$731$	−3.05654	−0.113050
$732$	26.3731	0.974778
$733$	28.5145	1.05321	0.526604	−	0.850111i	$-0.323465\pi$
$733$	28.5145	1.05321	0.526604	+	0.850111i	$0.323465\pi$
$734$	4.93751	0.182247
$735$	−34.4959	−1.27240
$736$	−1.00000	−0.0368605
$737$	15.5174	0.571592
$738$	17.9980	0.662515
$739$	−5.40009	−0.198645	−0.0993227	−	0.995055i	$-0.531668\pi$
$739$	−5.40009	−0.198645	−0.0993227	+	0.995055i	$0.531668\pi$
$740$	−16.0687	−0.590697
$741$	−0.580968	−0.0213424
$742$	−0.569019	−0.0208893
$743$	14.7065	0.539531	0.269765	−	0.962926i	$-0.413054\pi$
$743$	14.7065	0.539531	0.269765	+	0.962926i	$0.413054\pi$
$744$	−12.0718	−0.442574
$745$	10.9453	0.401005
$746$	0.698850	0.0255867
$747$	−8.28417	−0.303102
$748$	2.43119	0.0888932
$749$	−0.500451	−0.0182861
$750$	−28.0702	−1.02498
$751$	−18.9705	−0.692242	−0.346121	−	0.938190i	$-0.612501\pi$
$751$	−18.9705	−0.692242	−0.346121	+	0.938190i	$0.612501\pi$
$752$	−2.96463	−0.108109
$753$	49.7484	1.81293
$754$	0.116941	0.00425874
$755$	−12.6670	−0.460999
$756$	−0.0902799	−0.00328345
$757$	−37.8141	−1.37438	−0.687188	−	0.726480i	$-0.741155\pi$
$757$	−37.8141	−1.37438	−0.687188	+	0.726480i	$0.741155\pi$
$758$	−13.1842	−0.478870
$759$	−6.66686	−0.241992
$760$	−6.13026	−0.222368
$761$	−22.6951	−0.822696	−0.411348	−	0.911478i	$-0.634942\pi$
$761$	−22.6951	−0.822696	−0.411348	+	0.911478i	$0.634942\pi$
$762$	42.3340	1.53360
$763$	0.999133	0.0361711
$764$	−4.53788	−0.164175
$765$	4.76839	0.172401
$766$	−12.3879	−0.447594
$767$	−0.187013	−0.00675266
$768$	−2.37676	−0.0857638
$769$	−0.881841	−0.0318000	−0.0159000	−	0.999874i	$-0.505061\pi$
$769$	−0.881841	−0.0318000	−0.0159000	+	0.999874i	$0.505061\pi$
$770$	−0.630387	−0.0227176
$771$	−42.0103	−1.51296
$772$	−22.9480	−0.825915
$773$	−17.0663	−0.613832	−0.306916	−	0.951737i	$-0.599297\pi$
$773$	−17.0663	−0.613832	−0.306916	+	0.951737i	$0.599297\pi$
$774$	9.34167	0.335779
$775$	−3.48707	−0.125259
$776$	−5.48412	−0.196868
$777$	1.98981	0.0713840
$778$	33.6328	1.20580
$779$	20.0546	0.718531
$780$	0.408788	0.0146369
$781$	10.9368	0.391348
$782$	0.866728	0.0309941
$783$	1.17815	0.0421035
$784$	−6.98829	−0.249582
$785$	−8.35991	−0.298378
$786$	−2.37676	−0.0847761
$787$	−39.1028	−1.39386	−0.696932	−	0.717138i	$-0.745452\pi$
$787$	−39.1028	−1.39386	−0.696932	+	0.717138i	$0.745452\pi$
$788$	−13.2147	−0.470756
$789$	−3.44902	−0.122788
$790$	−16.1186	−0.573474
$791$	−0.0952220	−0.00338570
$792$	−7.43042	−0.264028
$793$	−0.918920	−0.0326318
$794$	21.5807	0.765872
$795$	25.9576	0.920623
$796$	7.49401	0.265618
$797$	−43.1978	−1.53015	−0.765073	−	0.643943i	$-0.777297\pi$
$797$	−43.1978	−1.53015	−0.765073	+	0.643943i	$0.777297\pi$
$798$	0.759118	0.0268725
$799$	2.56953	0.0909034
$800$	−0.686551	−0.0242732
$801$	41.5492	1.46807
$802$	−10.5331	−0.371937
$803$	26.0246	0.918389
$804$	13.1482	0.463702
$805$	−0.224735	−0.00792086
$806$	0.420619	0.0148157
$807$	4.69082	0.165125
$808$	7.30874	0.257120
$809$	−33.9941	−1.19517	−0.597584	−	0.801806i	$-0.703873\pi$
$809$	−33.9941	−1.19517	−0.597584	+	0.801806i	$0.703873\pi$
$810$	20.6232	0.724626
$811$	23.6049	0.828881	0.414441	−	0.910076i	$-0.363977\pi$
$811$	23.6049	0.828881	0.414441	+	0.910076i	$0.363977\pi$
$812$	−0.152800	−0.00536224
$813$	64.5814	2.26497
$814$	−21.7023	−0.760665
$815$	−0.355063	−0.0124373
$816$	2.06000	0.0721144
$817$	10.4091	0.364170
$818$	24.1264	0.843561
$819$	−0.0237375	−0.000829457 0
$820$	−14.1111	−0.492780
$821$	41.5584	1.45040	0.725199	−	0.688540i	$-0.241748\pi$
$821$	41.5584	1.45040	0.725199	+	0.688540i	$0.241748\pi$
$822$	−12.7485	−0.444656
$823$	51.5354	1.79641	0.898205	−	0.439577i	$-0.144872\pi$
$823$	51.5354	1.79641	0.898205	+	0.439577i	$0.144872\pi$
$824$	−2.07329	−0.0722265
$825$	−4.57714	−0.159355
$826$	0.244360	0.00850237
$827$	−46.9275	−1.63183	−0.815914	−	0.578173i	$-0.803766\pi$
$827$	−46.9275	−1.63183	−0.815914	+	0.578173i	$0.803766\pi$
$828$	−2.64897	−0.0920580
$829$	−6.16737	−0.214202	−0.107101	−	0.994248i	$-0.534157\pi$
$829$	−6.16737	−0.214202	−0.107101	+	0.994248i	$0.534157\pi$
$830$	6.49508	0.225448
$831$	3.72035	0.129058
$832$	0.0828135	0.00287104
$833$	6.05695	0.209861
$834$	−11.4730	−0.397279
$835$	−40.7883	−1.41154
$836$	−8.27949	−0.286352
$837$	4.23761	0.146473
$838$	22.2818	0.769711
$839$	18.2686	0.630701	0.315351	−	0.948975i	$-0.397878\pi$
$839$	18.2686	0.630701	0.315351	+	0.948975i	$0.397878\pi$
$840$	−0.534140	−0.0184296
$841$	−27.0060	−0.931240
$842$	2.71744	0.0936490
$843$	66.1652	2.27885
$844$	−5.21316	−0.179444
$845$	26.9853	0.928321
$846$	−7.85321	−0.269999
$847$	0.338888	0.0116443
$848$	5.25858	0.180580
$849$	42.8099	1.46923
$850$	0.595053	0.0204101
$851$	−7.73693	−0.265219
$852$	9.26693	0.317480
$853$	19.1953	0.657236	0.328618	−	0.944463i	$-0.393417\pi$
$853$	19.1953	0.657236	0.328618	+	0.944463i	$0.393417\pi$
$854$	1.20070	0.0410871
$855$	−16.2389	−0.555357
$856$	4.62492	0.158076
$857$	28.7631	0.982528	0.491264	−	0.871011i	$-0.336535\pi$
$857$	28.7631	0.982528	0.491264	+	0.871011i	$0.336535\pi$
$858$	0.552106	0.0188486
$859$	4.64181	0.158377	0.0791883	−	0.996860i	$-0.474767\pi$
$859$	4.64181	0.158377	0.0791883	+	0.996860i	$0.474767\pi$
$860$	−7.32420	−0.249753
$861$	1.74739	0.0595509
$862$	22.0113	0.749708
$863$	0.506362	0.0172368	0.00861839	−	0.999963i	$-0.497257\pi$
$863$	0.506362	0.0172368	0.00861839	+	0.999963i	$0.497257\pi$
$864$	0.834320	0.0283842
$865$	−50.9867	−1.73360
$866$	7.80509	0.265228
$867$	38.6194	1.31158
$868$	−0.549599	−0.0186546
$869$	−21.7697	−0.738486
$870$	6.97049	0.236322
$871$	−0.458125	−0.0155230
$872$	−9.23348	−0.312685
$873$	−14.5273	−0.491673
$874$	−2.95166	−0.0998415
$875$	−1.27797	−0.0432031
$876$	22.0512	0.745040
$877$	11.6073	0.391952	0.195976	−	0.980609i	$-0.437213\pi$
$877$	11.6073	0.391952	0.195976	+	0.980609i	$0.437213\pi$
$878$	1.37982	0.0465668
$879$	71.9553	2.42699
$880$	5.82571	0.196385
$881$	−18.4197	−0.620574	−0.310287	−	0.950643i	$-0.600425\pi$
$881$	−18.4197	−0.620574	−0.310287	+	0.950643i	$0.600425\pi$
$882$	−18.5118	−0.623323
$883$	35.5192	1.19532	0.597659	−	0.801751i	$-0.296098\pi$
$883$	35.5192	1.19532	0.597659	+	0.801751i	$0.296098\pi$
$884$	−0.0717767	−0.00241411
$885$	−11.1473	−0.374712
$886$	22.4419	0.753949
$887$	−6.02857	−0.202419	−0.101210	−	0.994865i	$-0.532271\pi$
$887$	−6.02857	−0.202419	−0.101210	+	0.994865i	$0.532271\pi$
$888$	−18.3888	−0.617087
$889$	1.92736	0.0646415
$890$	−32.5761	−1.09195
$891$	27.8536	0.933129
$892$	−7.92300	−0.265282
$893$	−8.75059	−0.292827
$894$	12.5256	0.418920
$895$	1.75951	0.0588140
$896$	−0.108208	−0.00361496
$897$	0.196827	0.00657188
$898$	−12.3543	−0.412267
$899$	7.17223	0.239207
$900$	−1.81865	−0.0606217
$901$	−4.55776	−0.151841
$902$	−19.0583	−0.634572
$903$	0.906965	0.0301819
$904$	0.879993	0.0292681
$905$	47.5440	1.58041
$906$	−14.4959	−0.481595
$907$	45.5446	1.51228	0.756141	−	0.654408i	$-0.227082\pi$
$907$	45.5446	1.51228	0.756141	+	0.654408i	$0.227082\pi$
$908$	−7.15155	−0.237332
$909$	19.3606	0.642151
$910$	0.0186111	0.000616951 0
$911$	19.4291	0.643716	0.321858	−	0.946788i	$-0.395693\pi$
$911$	19.4291	0.643716	0.321858	+	0.946788i	$0.395693\pi$
$912$	−7.01538	−0.232302
$913$	8.77221	0.290318
$914$	18.0206	0.596067
$915$	−54.7739	−1.81077
$916$	−29.9139	−0.988382
$917$	−0.108208	−0.00357333
$918$	−0.723128	−0.0238668
$919$	−7.40313	−0.244207	−0.122103	−	0.992517i	$-0.538964\pi$
$919$	−7.40313	−0.244207	−0.122103	+	0.992517i	$0.538964\pi$
$920$	2.07688	0.0684729
$921$	−48.6571	−1.60331
$922$	14.7184	0.484725
$923$	−0.322888	−0.0106280
$924$	−0.721405	−0.0237325
$925$	−5.31180	−0.174651
$926$	−7.62867	−0.250694
$927$	−5.49208	−0.180384
$928$	1.41210	0.0463545
$929$	−3.51986	−0.115483	−0.0577414	−	0.998332i	$-0.518390\pi$
$929$	−3.51986	−0.115483	−0.0577414	+	0.998332i	$0.518390\pi$
$930$	25.0718	0.822135
$931$	−20.6271	−0.676025
$932$	−6.35931	−0.208306
$933$	−23.3881	−0.765693
$934$	−6.80461	−0.222654
$935$	−5.04931	−0.165130
$936$	0.219370	0.00717034
$937$	−39.9807	−1.30611	−0.653057	−	0.757309i	$-0.726514\pi$
$937$	−39.9807	−1.30611	−0.653057	+	0.757309i	$0.726514\pi$
$938$	0.598606	0.0195452
$939$	43.7194	1.42673
$940$	6.15720	0.200826
$941$	−49.4709	−1.61271	−0.806353	−	0.591435i	$-0.798562\pi$
$941$	−49.4709	−1.61271	−0.806353	+	0.591435i	$0.798562\pi$
$942$	−9.56696	−0.311708
$943$	−6.79434	−0.221254
$944$	−2.25825	−0.0734997
$945$	0.187501	0.00609940
$946$	−9.89202	−0.321617
$947$	−32.2252	−1.04718	−0.523590	−	0.851971i	$-0.675407\pi$
$947$	−32.2252	−1.04718	−0.523590	+	0.851971i	$0.675407\pi$
$948$	−18.4459	−0.599095
$949$	−0.768331	−0.0249411
$950$	−2.02647	−0.0657472
$951$	13.5204	0.438430
$952$	0.0937866	0.00303964
$953$	−8.89303	−0.288074	−0.144037	−	0.989572i	$-0.546008\pi$
$953$	−8.89303	−0.288074	−0.144037	+	0.989572i	$0.546008\pi$
$954$	13.9298	0.450994
$955$	9.42466	0.304975
$956$	−21.0616	−0.681181
$957$	9.41429	0.304321
$958$	26.1954	0.846334
$959$	−0.580409	−0.0187424
$960$	4.93625	0.159317
$961$	−5.20261	−0.167826
$962$	0.640722	0.0206577
$963$	12.2512	0.394791
$964$	−17.5666	−0.565784
$965$	47.6603	1.53424
$966$	−0.257183	−0.00827473
$967$	−29.1583	−0.937666	−0.468833	−	0.883287i	$-0.655325\pi$
$967$	−29.1583	−0.937666	−0.468833	+	0.883287i	$0.655325\pi$
$968$	−3.13183	−0.100661
$969$	6.08042	0.195331
$970$	11.3899	0.365707
$971$	−4.38660	−0.140773	−0.0703863	−	0.997520i	$-0.522423\pi$
$971$	−4.38660	−0.140773	−0.0703863	+	0.997520i	$0.522423\pi$
$972$	21.0979	0.676716
$973$	−0.522339	−0.0167454
$974$	−40.4641	−1.29655
$975$	0.135132	0.00432769
$976$	−11.0963	−0.355183
$977$	8.44715	0.270248	0.135124	−	0.990829i	$-0.456857\pi$
$977$	8.44715	0.270248	0.135124	+	0.990829i	$0.456857\pi$
$978$	−0.406329	−0.0129930
$979$	−43.9970	−1.40615
$980$	14.5139	0.463629
$981$	−24.4592	−0.780922
$982$	12.6295	0.403022
$983$	30.5099	0.973114	0.486557	−	0.873649i	$-0.338253\pi$
$983$	30.5099	0.973114	0.486557	+	0.873649i	$0.338253\pi$
$984$	−16.1485	−0.514795
$985$	27.4455	0.874486
$986$	−1.22391	−0.0389772
$987$	−0.762454	−0.0242692
$988$	0.244437	0.00777659
$989$	−3.52653	−0.112137
$990$	15.4321	0.490465
$991$	35.3271	1.12220	0.561102	−	0.827747i	$-0.310378\pi$
$991$	35.3271	1.12220	0.561102	+	0.827747i	$0.310378\pi$
$992$	5.07911	0.161262
$993$	−7.87696	−0.249968
$994$	0.421900	0.0133819
$995$	−15.5642	−0.493418
$996$	7.43288	0.235520
$997$	−2.32354	−0.0735874	−0.0367937	−	0.999323i	$-0.511714\pi$
$997$	−2.32354	−0.0735874	−0.0367937	+	0.999323i	$0.511714\pi$
$998$	35.4878	1.12335
$999$	6.45508	0.204230

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6026.2.a.g.1.3	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6026.2.a.g.1.3	✓	21	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:	\( N \)	\(=\)	\( 6026 = 2 \cdot 23 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6026.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.37676 q^{3} +1.00000 q^{4} -2.07688 q^{5} -2.37676 q^{6} -0.108208 q^{7} +1.00000 q^{8} +2.64897 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.37676 q^{3} +1.00000 q^{4} -2.07688 q^{5} -2.37676 q^{6} -0.108208 q^{7} +1.00000 q^{8} +2.64897 q^{9} -2.07688 q^{10} -2.80503 q^{11} -2.37676 q^{12} +0.0828135 q^{13} -0.108208 q^{14} +4.93625 q^{15} +1.00000 q^{16} -0.866728 q^{17} +2.64897 q^{18} +2.95166 q^{19} -2.07688 q^{20} +0.257183 q^{21} -2.80503 q^{22} -1.00000 q^{23} -2.37676 q^{24} -0.686551 q^{25} +0.0828135 q^{26} +0.834320 q^{27} -0.108208 q^{28} +1.41210 q^{29} +4.93625 q^{30} +5.07911 q^{31} +1.00000 q^{32} +6.66686 q^{33} -0.866728 q^{34} +0.224735 q^{35} +2.64897 q^{36} +7.73693 q^{37} +2.95166 q^{38} -0.196827 q^{39} -2.07688 q^{40} +6.79434 q^{41} +0.257183 q^{42} +3.52653 q^{43} -2.80503 q^{44} -5.50160 q^{45} -1.00000 q^{46} -2.96463 q^{47} -2.37676 q^{48} -6.98829 q^{49} -0.686551 q^{50} +2.06000 q^{51} +0.0828135 q^{52} +5.25858 q^{53} +0.834320 q^{54} +5.82571 q^{55} -0.108208 q^{56} -7.01538 q^{57} +1.41210 q^{58} -2.25825 q^{59} +4.93625 q^{60} -11.0963 q^{61} +5.07911 q^{62} -0.286639 q^{63} +1.00000 q^{64} -0.171994 q^{65} +6.66686 q^{66} -5.53201 q^{67} -0.866728 q^{68} +2.37676 q^{69} +0.224735 q^{70} -3.89898 q^{71} +2.64897 q^{72} -9.27785 q^{73} +7.73693 q^{74} +1.63176 q^{75} +2.95166 q^{76} +0.303525 q^{77} -0.196827 q^{78} +7.76096 q^{79} -2.07688 q^{80} -9.92988 q^{81} +6.79434 q^{82} -3.12732 q^{83} +0.257183 q^{84} +1.80009 q^{85} +3.52653 q^{86} -3.35622 q^{87} -2.80503 q^{88} +15.6851 q^{89} -5.50160 q^{90} -0.00896105 q^{91} -1.00000 q^{92} -12.0718 q^{93} -2.96463 q^{94} -6.13026 q^{95} -2.37676 q^{96} -5.48412 q^{97} -6.98829 q^{98} -7.43042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 21 q^{2} + 21 q^{4} - 13 q^{5} - 18 q^{7} + 21 q^{8} + 7 q^{9}+O(q^{10}) \) 21 * q + 21 * q^2 + 21 * q^4 - 13 * q^5 - 18 * q^7 + 21 * q^8 + 7 * q^9	\( 21 q + 21 q^{2} + 21 q^{4} - 13 q^{5} - 18 q^{7} + 21 q^{8} + 7 q^{9} - 13 q^{10} - 4 q^{11} - 4 q^{13} - 18 q^{14} - 16 q^{15} + 21 q^{16} - 12 q^{17} + 7 q^{18} - 18 q^{19} - 13 q^{20} - 24 q^{21} - 4 q^{22} - 21 q^{23} + 2 q^{25} - 4 q^{26} - 9 q^{27} - 18 q^{28} - 16 q^{29} - 16 q^{30} - 7 q^{31} + 21 q^{32} - 15 q^{33} - 12 q^{34} + 7 q^{36} - 44 q^{37} - 18 q^{38} - 14 q^{39} - 13 q^{40} - 23 q^{41} - 24 q^{42} - 18 q^{43} - 4 q^{44} - 36 q^{45} - 21 q^{46} + 2 q^{47} - 13 q^{49} + 2 q^{50} - 26 q^{51} - 4 q^{52} - 39 q^{53} - 9 q^{54} - 32 q^{55} - 18 q^{56} - 22 q^{57} - 16 q^{58} - 27 q^{59} - 16 q^{60} - 34 q^{61} - 7 q^{62} - 28 q^{63} + 21 q^{64} - 25 q^{65} - 15 q^{66} - 19 q^{67} - 12 q^{68} - 24 q^{71} + 7 q^{72} - 8 q^{73} - 44 q^{74} + 50 q^{75} - 18 q^{76} - 16 q^{77} - 14 q^{78} - 27 q^{79} - 13 q^{80} + 33 q^{81} - 23 q^{82} + 7 q^{83} - 24 q^{84} - 22 q^{85} - 18 q^{86} - 15 q^{87} - 4 q^{88} - 12 q^{89} - 36 q^{90} - 20 q^{91} - 21 q^{92} - 43 q^{93} + 2 q^{94} - 14 q^{95} - 52 q^{97} - 13 q^{98} - 30 q^{99}+O(q^{100}) \) 21 * q + 21 * q^2 + 21 * q^4 - 13 * q^5 - 18 * q^7 + 21 * q^8 + 7 * q^9 - 13 * q^10 - 4 * q^11 - 4 * q^13 - 18 * q^14 - 16 * q^15 + 21 * q^16 - 12 * q^17 + 7 * q^18 - 18 * q^19 - 13 * q^20 - 24 * q^21 - 4 * q^22 - 21 * q^23 + 2 * q^25 - 4 * q^26 - 9 * q^27 - 18 * q^28 - 16 * q^29 - 16 * q^30 - 7 * q^31 + 21 * q^32 - 15 * q^33 - 12 * q^34 + 7 * q^36 - 44 * q^37 - 18 * q^38 - 14 * q^39 - 13 * q^40 - 23 * q^41 - 24 * q^42 - 18 * q^43 - 4 * q^44 - 36 * q^45 - 21 * q^46 + 2 * q^47 - 13 * q^49 + 2 * q^50 - 26 * q^51 - 4 * q^52 - 39 * q^53 - 9 * q^54 - 32 * q^55 - 18 * q^56 - 22 * q^57 - 16 * q^58 - 27 * q^59 - 16 * q^60 - 34 * q^61 - 7 * q^62 - 28 * q^63 + 21 * q^64 - 25 * q^65 - 15 * q^66 - 19 * q^67 - 12 * q^68 - 24 * q^71 + 7 * q^72 - 8 * q^73 - 44 * q^74 + 50 * q^75 - 18 * q^76 - 16 * q^77 - 14 * q^78 - 27 * q^79 - 13 * q^80 + 33 * q^81 - 23 * q^82 + 7 * q^83 - 24 * q^84 - 22 * q^85 - 18 * q^86 - 15 * q^87 - 4 * q^88 - 12 * q^89 - 36 * q^90 - 20 * q^91 - 21 * q^92 - 43 * q^93 + 2 * q^94 - 14 * q^95 - 52 * q^97 - 13 * q^98 - 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more