LMFDB - Embedded newform 6026.2.a.i.1.12

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:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-1.00000 q^{2} -0.665355 q^{3} +1.00000 q^{4} +0.670202 q^{5} +0.665355 q^{6} -3.39786 q^{7} -1.00000 q^{8} -2.55730 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.665355 q^{3} +1.00000 q^{4} +0.670202 q^{5} +0.665355 q^{6} -3.39786 q^{7} -1.00000 q^{8} -2.55730 q^{9} -0.670202 q^{10} -2.66536 q^{11} -0.665355 q^{12} +2.25288 q^{13} +3.39786 q^{14} -0.445922 q^{15} +1.00000 q^{16} +3.18309 q^{17} +2.55730 q^{18} +4.73220 q^{19} +0.670202 q^{20} +2.26078 q^{21} +2.66536 q^{22} +1.00000 q^{23} +0.665355 q^{24} -4.55083 q^{25} -2.25288 q^{26} +3.69758 q^{27} -3.39786 q^{28} -9.07696 q^{29} +0.445922 q^{30} +2.40671 q^{31} -1.00000 q^{32} +1.77341 q^{33} -3.18309 q^{34} -2.27725 q^{35} -2.55730 q^{36} +10.0140 q^{37} -4.73220 q^{38} -1.49897 q^{39} -0.670202 q^{40} +10.3894 q^{41} -2.26078 q^{42} -10.8219 q^{43} -2.66536 q^{44} -1.71391 q^{45} -1.00000 q^{46} +2.12628 q^{47} -0.665355 q^{48} +4.54547 q^{49} +4.55083 q^{50} -2.11789 q^{51} +2.25288 q^{52} +2.74037 q^{53} -3.69758 q^{54} -1.78633 q^{55} +3.39786 q^{56} -3.14859 q^{57} +9.07696 q^{58} +2.19678 q^{59} -0.445922 q^{60} +0.869457 q^{61} -2.40671 q^{62} +8.68936 q^{63} +1.00000 q^{64} +1.50989 q^{65} -1.77341 q^{66} -5.55015 q^{67} +3.18309 q^{68} -0.665355 q^{69} +2.27725 q^{70} +13.2632 q^{71} +2.55730 q^{72} +0.381071 q^{73} -10.0140 q^{74} +3.02792 q^{75} +4.73220 q^{76} +9.05653 q^{77} +1.49897 q^{78} -6.31574 q^{79} +0.670202 q^{80} +5.21171 q^{81} -10.3894 q^{82} -8.49811 q^{83} +2.26078 q^{84} +2.13331 q^{85} +10.8219 q^{86} +6.03940 q^{87} +2.66536 q^{88} +13.3914 q^{89} +1.71391 q^{90} -7.65499 q^{91} +1.00000 q^{92} -1.60131 q^{93} -2.12628 q^{94} +3.17153 q^{95} +0.665355 q^{96} +3.23551 q^{97} -4.54547 q^{98} +6.81614 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 25 q^{2} - 4 q^{3} + 25 q^{4} - 3 q^{5} + 4 q^{6} - 11 q^{7} - 25 q^{8} + 19 q^{9}+O(q^{10}) $ 25 * q - 25 * q^2 - 4 * q^3 + 25 * q^4 - 3 * q^5 + 4 * q^6 - 11 * q^7 - 25 * q^8 + 19 * q^9	\( 25 q - 25 q^{2} - 4 q^{3} + 25 q^{4} - 3 q^{5} + 4 q^{6} - 11 q^{7} - 25 q^{8} + 19 q^{9} + 3 q^{10} - 12 q^{11} - 4 q^{12} - 6 q^{13} + 11 q^{14} + 25 q^{16} + 8 q^{17} - 19 q^{18} - 23 q^{19} - 3 q^{20} - 16 q^{21} + 12 q^{22} + 25 q^{23} + 4 q^{24} + 4 q^{25} + 6 q^{26} - 13 q^{27} - 11 q^{28} - 7 q^{29} - 7 q^{31} - 25 q^{32} + 3 q^{33} - 8 q^{34} - 18 q^{35} + 19 q^{36} - 7 q^{37} + 23 q^{38} - 2 q^{39} + 3 q^{40} - 10 q^{41} + 16 q^{42} - 26 q^{43} - 12 q^{44} + 20 q^{45} - 25 q^{46} - 2 q^{47} - 4 q^{48} + 2 q^{49} - 4 q^{50} - 28 q^{51} - 6 q^{52} + 47 q^{53} + 13 q^{54} - 38 q^{55} + 11 q^{56} - 4 q^{57} + 7 q^{58} - 19 q^{59} - 26 q^{61} + 7 q^{62} - 15 q^{63} + 25 q^{64} + 13 q^{65} - 3 q^{66} - 34 q^{67} + 8 q^{68} - 4 q^{69} + 18 q^{70} - 10 q^{71} - 19 q^{72} - 22 q^{73} + 7 q^{74} - 8 q^{75} - 23 q^{76} + 28 q^{77} + 2 q^{78} - 21 q^{79} - 3 q^{80} - 27 q^{81} + 10 q^{82} - 16 q^{83} - 16 q^{84} - 42 q^{85} + 26 q^{86} - 17 q^{87} + 12 q^{88} + 27 q^{89} - 20 q^{90} - 26 q^{91} + 25 q^{92} - 27 q^{93} + 2 q^{94} + 4 q^{96} + 4 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) 25 * q - 25 * q^2 - 4 * q^3 + 25 * q^4 - 3 * q^5 + 4 * q^6 - 11 * q^7 - 25 * q^8 + 19 * q^9 + 3 * q^10 - 12 * q^11 - 4 * q^12 - 6 * q^13 + 11 * q^14 + 25 * q^16 + 8 * q^17 - 19 * q^18 - 23 * q^19 - 3 * q^20 - 16 * q^21 + 12 * q^22 + 25 * q^23 + 4 * q^24 + 4 * q^25 + 6 * q^26 - 13 * q^27 - 11 * q^28 - 7 * q^29 - 7 * q^31 - 25 * q^32 + 3 * q^33 - 8 * q^34 - 18 * q^35 + 19 * q^36 - 7 * q^37 + 23 * q^38 - 2 * q^39 + 3 * q^40 - 10 * q^41 + 16 * q^42 - 26 * q^43 - 12 * q^44 + 20 * q^45 - 25 * q^46 - 2 * q^47 - 4 * q^48 + 2 * q^49 - 4 * q^50 - 28 * q^51 - 6 * q^52 + 47 * q^53 + 13 * q^54 - 38 * q^55 + 11 * q^56 - 4 * q^57 + 7 * q^58 - 19 * q^59 - 26 * q^61 + 7 * q^62 - 15 * q^63 + 25 * q^64 + 13 * q^65 - 3 * q^66 - 34 * q^67 + 8 * q^68 - 4 * q^69 + 18 * q^70 - 10 * q^71 - 19 * q^72 - 22 * q^73 + 7 * q^74 - 8 * q^75 - 23 * q^76 + 28 * q^77 + 2 * q^78 - 21 * q^79 - 3 * q^80 - 27 * q^81 + 10 * q^82 - 16 * q^83 - 16 * q^84 - 42 * q^85 + 26 * q^86 - 17 * q^87 + 12 * q^88 + 27 * q^89 - 20 * q^90 - 26 * q^91 + 25 * q^92 - 27 * q^93 + 2 * q^94 + 4 * q^96 + 4 * q^97 - 2 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.665355	−0.384143	−0.192071	−	0.981381i	$-0.561521\pi$
$3$	−0.665355	−0.384143	−0.192071	+	0.981381i	$0.561521\pi$
$4$	1.00000	0.500000
$5$	0.670202	0.299723	0.149862	−	0.988707i	$-0.452117\pi$
$5$	0.670202	0.299723	0.149862	+	0.988707i	$0.452117\pi$
$6$	0.665355	0.271630
$7$	−3.39786	−1.28427	−0.642136	−	0.766591i	$-0.721951\pi$
$7$	−3.39786	−1.28427	−0.642136	+	0.766591i	$0.721951\pi$
$8$	−1.00000	−0.353553
$9$	−2.55730	−0.852434
$10$	−0.670202	−0.211936
$11$	−2.66536	−0.803637	−0.401818	−	0.915719i	$-0.631622\pi$
$11$	−2.66536	−0.803637	−0.401818	+	0.915719i	$0.631622\pi$
$12$	−0.665355	−0.192071
$13$	2.25288	0.624837	0.312419	−	0.949944i	$-0.398861\pi$
$13$	2.25288	0.624837	0.312419	+	0.949944i	$0.398861\pi$
$14$	3.39786	0.908117
$15$	−0.445922	−0.115137
$16$	1.00000	0.250000
$17$	3.18309	0.772013	0.386007	−	0.922496i	$-0.373854\pi$
$17$	3.18309	0.772013	0.386007	+	0.922496i	$0.373854\pi$
$18$	2.55730	0.602762
$19$	4.73220	1.08564	0.542821	−	0.839849i	$-0.317356\pi$
$19$	4.73220	1.08564	0.542821	+	0.839849i	$0.317356\pi$
$20$	0.670202	0.149862
$21$	2.26078	0.493343
$22$	2.66536	0.568257
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0.665355	0.135815
$25$	−4.55083	−0.910166
$26$	−2.25288	−0.441827
$27$	3.69758	0.711599
$28$	−3.39786	−0.642136
$29$	−9.07696	−1.68555	−0.842775	−	0.538267i	$-0.819079\pi$
$29$	−9.07696	−1.68555	−0.842775	+	0.538267i	$0.819079\pi$
$30$	0.445922	0.0814139
$31$	2.40671	0.432257	0.216129	−	0.976365i	$-0.430657\pi$
$31$	2.40671	0.432257	0.216129	+	0.976365i	$0.430657\pi$
$32$	−1.00000	−0.176777
$33$	1.77341	0.308711
$34$	−3.18309	−0.545896
$35$	−2.27725	−0.384926
$36$	−2.55730	−0.426217
$37$	10.0140	1.64629	0.823147	−	0.567828i	$-0.192216\pi$
$37$	10.0140	1.64629	0.823147	+	0.567828i	$0.192216\pi$
$38$	−4.73220	−0.767664
$39$	−1.49897	−0.240027
$40$	−0.670202	−0.105968
$41$	10.3894	1.62255	0.811274	−	0.584667i	$-0.198775\pi$
$41$	10.3894	1.62255	0.811274	+	0.584667i	$0.198775\pi$
$42$	−2.26078	−0.348846
$43$	−10.8219	−1.65032	−0.825160	−	0.564899i	$-0.808915\pi$
$43$	−10.8219	−1.65032	−0.825160	+	0.564899i	$0.808915\pi$
$44$	−2.66536	−0.401818
$45$	−1.71391	−0.255495
$46$	−1.00000	−0.147442
$47$	2.12628	0.310149	0.155075	−	0.987903i	$-0.450438\pi$
$47$	2.12628	0.310149	0.155075	+	0.987903i	$0.450438\pi$
$48$	−0.665355	−0.0960357
$49$	4.54547	0.649352
$50$	4.55083	0.643584
$51$	−2.11789	−0.296563
$52$	2.25288	0.312419
$53$	2.74037	0.376418	0.188209	−	0.982129i	$-0.439732\pi$
$53$	2.74037	0.376418	0.188209	+	0.982129i	$0.439732\pi$
$54$	−3.69758	−0.503177
$55$	−1.78633	−0.240869
$56$	3.39786	0.454058
$57$	−3.14859	−0.417041
$58$	9.07696	1.19186
$59$	2.19678	0.285997	0.142998	−	0.989723i	$-0.454326\pi$
$59$	2.19678	0.285997	0.142998	+	0.989723i	$0.454326\pi$
$60$	−0.445922	−0.0575683
$61$	0.869457	0.111323	0.0556613	−	0.998450i	$-0.482273\pi$
$61$	0.869457	0.111323	0.0556613	+	0.998450i	$0.482273\pi$
$62$	−2.40671	−0.305652
$63$	8.68936	1.09476
$64$	1.00000	0.125000
$65$	1.50989	0.187278
$66$	−1.77341	−0.218292
$67$	−5.55015	−0.678058	−0.339029	−	0.940776i	$-0.610099\pi$
$67$	−5.55015	−0.678058	−0.339029	+	0.940776i	$0.610099\pi$
$68$	3.18309	0.386007
$69$	−0.665355	−0.0800993
$70$	2.27725	0.272184
$71$	13.2632	1.57405	0.787024	−	0.616922i	$-0.211621\pi$
$71$	13.2632	1.57405	0.787024	+	0.616922i	$0.211621\pi$
$72$	2.55730	0.301381
$73$	0.381071	0.0446010	0.0223005	−	0.999751i	$-0.492901\pi$
$73$	0.381071	0.0446010	0.0223005	+	0.999751i	$0.492901\pi$
$74$	−10.0140	−1.16411
$75$	3.02792	0.349634
$76$	4.73220	0.542821
$77$	9.05653	1.03209
$78$	1.49897	0.169725
$79$	−6.31574	−0.710577	−0.355288	−	0.934757i	$-0.615617\pi$
$79$	−6.31574	−0.710577	−0.355288	+	0.934757i	$0.615617\pi$
$80$	0.670202	0.0749309
$81$	5.21171	0.579079
$82$	−10.3894	−1.14731
$83$	−8.49811	−0.932789	−0.466394	−	0.884577i	$-0.654447\pi$
$83$	−8.49811	−0.932789	−0.466394	+	0.884577i	$0.654447\pi$
$84$	2.26078	0.246672
$85$	2.13331	0.231390
$86$	10.8219	1.16695
$87$	6.03940	0.647492
$88$	2.66536	0.284128
$89$	13.3914	1.41948	0.709742	−	0.704462i	$-0.248812\pi$
$89$	13.3914	1.41948	0.709742	+	0.704462i	$0.248812\pi$
$90$	1.71391	0.180662
$91$	−7.65499	−0.802461
$92$	1.00000	0.104257
$93$	−1.60131	−0.166048
$94$	−2.12628	−0.219309
$95$	3.17153	0.325392
$96$	0.665355	0.0679075
$97$	3.23551	0.328516	0.164258	−	0.986417i	$-0.447477\pi$
$97$	3.23551	0.328516	0.164258	+	0.986417i	$0.447477\pi$
$98$	−4.54547	−0.459161
$99$	6.81614	0.685047
$100$	−4.55083	−0.455083
$101$	−11.9405	−1.18812	−0.594061	−	0.804420i	$-0.702476\pi$
$101$	−11.9405	−1.18812	−0.594061	+	0.804420i	$0.702476\pi$
$102$	2.11789	0.209702
$103$	4.34785	0.428406	0.214203	−	0.976789i	$-0.431285\pi$
$103$	4.34785	0.428406	0.214203	+	0.976789i	$0.431285\pi$
$104$	−2.25288	−0.220913
$105$	1.51518	0.147867
$106$	−2.74037	−0.266168
$107$	−9.12065	−0.881726	−0.440863	−	0.897574i	$-0.645328\pi$
$107$	−9.12065	−0.881726	−0.440863	+	0.897574i	$0.645328\pi$
$108$	3.69758	0.355800
$109$	6.95171	0.665853	0.332926	−	0.942953i	$-0.391964\pi$
$109$	6.95171	0.665853	0.332926	+	0.942953i	$0.391964\pi$
$110$	1.78633	0.170320
$111$	−6.66287	−0.632412
$112$	−3.39786	−0.321068
$113$	−1.13052	−0.106350	−0.0531752	−	0.998585i	$-0.516934\pi$
$113$	−1.13052	−0.106350	−0.0531752	+	0.998585i	$0.516934\pi$
$114$	3.14859	0.294893
$115$	0.670202	0.0624967
$116$	−9.07696	−0.842775
$117$	−5.76131	−0.532633
$118$	−2.19678	−0.202230
$119$	−10.8157	−0.991475
$120$	0.445922	0.0407069
$121$	−3.89585	−0.354168
$122$	−0.869457	−0.0787170
$123$	−6.91262	−0.623290
$124$	2.40671	0.216129
$125$	−6.40098	−0.572521
$126$	−8.68936	−0.774110
$127$	−2.61248	−0.231821	−0.115910	−	0.993260i	$-0.536979\pi$
$127$	−2.61248	−0.231821	−0.115910	+	0.993260i	$0.536979\pi$
$128$	−1.00000	−0.0883883
$129$	7.20039	0.633959
$130$	−1.50989	−0.132426
$131$	1.00000	0.0873704
$131$	1.00000	0.0873704
$132$	1.77341	0.154356
$133$	−16.0794	−1.39426
$134$	5.55015	0.479460
$135$	2.47812	0.213283
$136$	−3.18309	−0.272948
$137$	11.4391	0.977310	0.488655	−	0.872477i	$-0.337488\pi$
$137$	11.4391	0.977310	0.488655	+	0.872477i	$0.337488\pi$
$138$	0.665355	0.0566388
$139$	−0.0220343	−0.00186893	−0.000934464	−	1.00000i	$-0.500297\pi$
$139$	−0.0220343	−0.00186893	−0.000934464		1.00000i	$0.500297\pi$
$140$	−2.27725	−0.192463
$141$	−1.41473	−0.119142
$142$	−13.2632	−1.11302
$143$	−6.00475	−0.502142
$144$	−2.55730	−0.213109
$145$	−6.08340	−0.505199
$146$	−0.381071	−0.0315376
$147$	−3.02435	−0.249444
$148$	10.0140	0.823147
$149$	−12.7364	−1.04341	−0.521704	−	0.853126i	$-0.674704\pi$
$149$	−12.7364	−1.04341	−0.521704	+	0.853126i	$0.674704\pi$
$150$	−3.02792	−0.247228
$151$	7.42620	0.604335	0.302168	−	0.953255i	$-0.402290\pi$
$151$	7.42620	0.604335	0.302168	+	0.953255i	$0.402290\pi$
$152$	−4.73220	−0.383832
$153$	−8.14013	−0.658091
$154$	−9.05653	−0.729796
$155$	1.61298	0.129558
$156$	−1.49897	−0.120013
$157$	4.52277	0.360956	0.180478	−	0.983579i	$-0.442235\pi$
$157$	4.52277	0.360956	0.180478	+	0.983579i	$0.442235\pi$
$158$	6.31574	0.502454
$159$	−1.82332	−0.144598
$160$	−0.670202	−0.0529841
$161$	−3.39786	−0.267789
$162$	−5.21171	−0.409470
$163$	7.42122	0.581274	0.290637	−	0.956833i	$-0.406133\pi$
$163$	7.42122	0.581274	0.290637	+	0.956833i	$0.406133\pi$
$164$	10.3894	0.811274
$165$	1.18854	0.0925280
$166$	8.49811	0.659581
$167$	−4.55293	−0.352317	−0.176158	−	0.984362i	$-0.556367\pi$
$167$	−4.55293	−0.352317	−0.176158	+	0.984362i	$0.556367\pi$
$168$	−2.26078	−0.174423
$169$	−7.92452	−0.609578
$170$	−2.13331	−0.163618
$171$	−12.1017	−0.925438
$172$	−10.8219	−0.825160
$173$	−19.1710	−1.45754	−0.728772	−	0.684756i	$-0.759909\pi$
$173$	−19.1710	−1.45754	−0.728772	+	0.684756i	$0.759909\pi$
$174$	−6.03940	−0.457846
$175$	15.4631	1.16890
$176$	−2.66536	−0.200909
$177$	−1.46164	−0.109864
$178$	−13.3914	−1.00373
$179$	−2.38254	−0.178079	−0.0890397	−	0.996028i	$-0.528380\pi$
$179$	−2.38254	−0.178079	−0.0890397	+	0.996028i	$0.528380\pi$
$180$	−1.71391	−0.127747
$181$	−19.0100	−1.41300	−0.706501	−	0.707712i	$-0.749727\pi$
$181$	−19.0100	−1.41300	−0.706501	+	0.707712i	$0.749727\pi$
$182$	7.65499	0.567425
$183$	−0.578498	−0.0427638
$184$	−1.00000	−0.0737210
$185$	6.71141	0.493433
$186$	1.60131	0.117414
$187$	−8.48409	−0.620418
$188$	2.12628	0.155075
$189$	−12.5639	−0.913886
$190$	−3.17153	−0.230087
$191$	−7.20331	−0.521213	−0.260607	−	0.965445i	$-0.583923\pi$
$191$	−7.20331	−0.521213	−0.260607	+	0.965445i	$0.583923\pi$
$192$	−0.665355	−0.0480178
$193$	−14.5057	−1.04414	−0.522072	−	0.852901i	$-0.674841\pi$
$193$	−14.5057	−1.04414	−0.522072	+	0.852901i	$0.674841\pi$
$194$	−3.23551	−0.232296
$195$	−1.00461	−0.0719416
$196$	4.54547	0.324676
$197$	−1.16500	−0.0830028	−0.0415014	−	0.999138i	$-0.513214\pi$
$197$	−1.16500	−0.0830028	−0.0415014	+	0.999138i	$0.513214\pi$
$198$	−6.81614	−0.484402
$199$	−15.6513	−1.10949	−0.554745	−	0.832021i	$-0.687184\pi$
$199$	−15.6513	−1.10949	−0.554745	+	0.832021i	$0.687184\pi$
$200$	4.55083	0.321792
$201$	3.69282	0.260471
$202$	11.9405	0.840129
$203$	30.8423	2.16470
$204$	−2.11789	−0.148282
$205$	6.96298	0.486315
$206$	−4.34785	−0.302929
$207$	−2.55730	−0.177745
$208$	2.25288	0.156209
$209$	−12.6130	−0.872461
$210$	−1.51518	−0.104557
$211$	−20.4743	−1.40951	−0.704754	−	0.709452i	$-0.748942\pi$
$211$	−20.4743	−1.40951	−0.704754	+	0.709452i	$0.748942\pi$
$212$	2.74037	0.188209
$213$	−8.82471	−0.604659
$214$	9.12065	0.623474
$215$	−7.25284	−0.494640
$216$	−3.69758	−0.251588
$217$	−8.17765	−0.555135
$218$	−6.95171	−0.470829
$219$	−0.253547	−0.0171331
$220$	−1.78633	−0.120434
$221$	7.17114	0.482383
$222$	6.66287	0.447183
$223$	−15.6735	−1.04957	−0.524787	−	0.851233i	$-0.675855\pi$
$223$	−15.6735	−1.04957	−0.524787	+	0.851233i	$0.675855\pi$
$224$	3.39786	0.227029
$225$	11.6378	0.775857
$226$	1.13052	0.0752011
$227$	−2.08245	−0.138217	−0.0691084	−	0.997609i	$-0.522015\pi$
$227$	−2.08245	−0.138217	−0.0691084	+	0.997609i	$0.522015\pi$
$228$	−3.14859	−0.208521
$229$	−0.900940	−0.0595358	−0.0297679	−	0.999557i	$-0.509477\pi$
$229$	−0.900940	−0.0595358	−0.0297679	+	0.999557i	$0.509477\pi$
$230$	−0.670202	−0.0441918
$231$	−6.02581	−0.396469
$232$	9.07696	0.595932
$233$	10.1771	0.666721	0.333361	−	0.942799i	$-0.391817\pi$
$233$	10.1771	0.666721	0.333361	+	0.942799i	$0.391817\pi$
$234$	5.76131	0.376628
$235$	1.42503	0.0929590
$236$	2.19678	0.142998
$237$	4.20221	0.272963
$238$	10.8157	0.701078
$239$	−11.5389	−0.746391	−0.373196	−	0.927753i	$-0.621738\pi$
$239$	−11.5389	−0.746391	−0.373196	+	0.927753i	$0.621738\pi$
$240$	−0.445922	−0.0287841
$241$	−5.92932	−0.381941	−0.190971	−	0.981596i	$-0.561163\pi$
$241$	−5.92932	−0.381941	−0.190971	+	0.981596i	$0.561163\pi$
$242$	3.89585	0.250435
$243$	−14.5604	−0.934048
$244$	0.869457	0.0556613
$245$	3.04638	0.194626
$246$	6.91262	0.440732
$247$	10.6611	0.678350
$248$	−2.40671	−0.152826
$249$	5.65426	0.358324
$250$	6.40098	0.404834
$251$	17.2997	1.09195	0.545974	−	0.837802i	$-0.316160\pi$
$251$	17.2997	1.09195	0.545974	+	0.837802i	$0.316160\pi$
$252$	8.68936	0.547378
$253$	−2.66536	−0.167570
$254$	2.61248	0.163922
$255$	−1.41941	−0.0888870
$256$	1.00000	0.0625000
$257$	25.9083	1.61612	0.808059	−	0.589102i	$-0.200518\pi$
$257$	25.9083	1.61612	0.808059	+	0.589102i	$0.200518\pi$
$258$	−7.20039	−0.448276
$259$	−34.0262	−2.11429
$260$	1.50989	0.0936392
$261$	23.2125	1.43682
$262$	−1.00000	−0.0617802
$263$	−15.7746	−0.972704	−0.486352	−	0.873763i	$-0.661673\pi$
$263$	−15.7746	−0.972704	−0.486352	+	0.873763i	$0.661673\pi$
$264$	−1.77341	−0.109146
$265$	1.83660	0.112821
$266$	16.0794	0.985889
$267$	−8.91002	−0.545284
$268$	−5.55015	−0.339029
$269$	−6.11680	−0.372948	−0.186474	−	0.982460i	$-0.559706\pi$
$269$	−6.11680	−0.372948	−0.186474	+	0.982460i	$0.559706\pi$
$270$	−2.47812	−0.150814
$271$	−10.2324	−0.621575	−0.310788	−	0.950479i	$-0.600593\pi$
$271$	−10.2324	−0.621575	−0.310788	+	0.950479i	$0.600593\pi$
$272$	3.18309	0.193003
$273$	5.09328	0.308259
$274$	−11.4391	−0.691063
$275$	12.1296	0.731443
$276$	−0.665355	−0.0400497
$277$	23.8901	1.43542	0.717708	−	0.696344i	$-0.245191\pi$
$277$	23.8901	1.43542	0.717708	+	0.696344i	$0.245191\pi$
$278$	0.0220343	0.00132153
$279$	−6.15467	−0.368471
$280$	2.27725	0.136092
$281$	22.5490	1.34516	0.672579	−	0.740025i	$-0.265186\pi$
$281$	22.5490	1.34516	0.672579	+	0.740025i	$0.265186\pi$
$282$	1.41473	0.0842458
$283$	17.4271	1.03594	0.517968	−	0.855400i	$-0.326689\pi$
$283$	17.4271	1.03594	0.517968	+	0.855400i	$0.326689\pi$
$284$	13.2632	0.787024
$285$	−2.11019	−0.124997
$286$	6.00475	0.355068
$287$	−35.3016	−2.08379
$288$	2.55730	0.150691
$289$	−6.86792	−0.403995
$290$	6.08340	0.357229
$291$	−2.15276	−0.126197
$292$	0.381071	0.0223005
$293$	−3.30962	−0.193350	−0.0966751	−	0.995316i	$-0.530821\pi$
$293$	−3.30962	−0.193350	−0.0966751	+	0.995316i	$0.530821\pi$
$294$	3.02435	0.176384
$295$	1.47229	0.0857199
$296$	−10.0140	−0.582053
$297$	−9.85538	−0.571867
$298$	12.7364	0.737801
$299$	2.25288	0.130288
$300$	3.02792	0.174817
$301$	36.7712	2.11946
$302$	−7.42620	−0.427330
$303$	7.94465	0.456408
$304$	4.73220	0.271410
$305$	0.582712	0.0333660
$306$	8.14013	0.465340
$307$	−13.3594	−0.762461	−0.381230	−	0.924480i	$-0.624500\pi$
$307$	−13.3594	−0.762461	−0.381230	+	0.924480i	$0.624500\pi$
$308$	9.05653	0.516044
$309$	−2.89286	−0.164569
$310$	−1.61298	−0.0916110
$311$	−19.8616	−1.12625	−0.563124	−	0.826372i	$-0.690401\pi$
$311$	−19.8616	−1.12625	−0.563124	+	0.826372i	$0.690401\pi$
$312$	1.49897	0.0848623
$313$	−21.3082	−1.20441	−0.602206	−	0.798341i	$-0.705711\pi$
$313$	−21.3082	−1.20441	−0.602206	+	0.798341i	$0.705711\pi$
$314$	−4.52277	−0.255235
$315$	5.82363	0.328124
$316$	−6.31574	−0.355288
$317$	23.6504	1.32834	0.664169	−	0.747582i	$-0.268785\pi$
$317$	23.6504	1.32834	0.664169	+	0.747582i	$0.268785\pi$
$318$	1.82332	0.102246
$319$	24.1934	1.35457
$320$	0.670202	0.0374654
$321$	6.06846	0.338709
$322$	3.39786	0.189355
$323$	15.0630	0.838130
$324$	5.21171	0.289539
$325$	−10.2525	−0.568706
$326$	−7.42122	−0.411023
$327$	−4.62535	−0.255782
$328$	−10.3894	−0.573657
$329$	−7.22479	−0.398316
$330$	−1.18854	−0.0654272
$331$	31.6327	1.73869	0.869346	−	0.494203i	$-0.164540\pi$
$331$	31.6327	1.73869	0.869346	+	0.494203i	$0.164540\pi$
$332$	−8.49811	−0.466394
$333$	−25.6089	−1.40336
$334$	4.55293	0.249125
$335$	−3.71972	−0.203230
$336$	2.26078	0.123336
$337$	3.03332	0.165235	0.0826177	−	0.996581i	$-0.473672\pi$
$337$	3.03332	0.165235	0.0826177	+	0.996581i	$0.473672\pi$
$338$	7.92452	0.431037
$339$	0.752197	0.0408537
$340$	2.13331	0.115695
$341$	−6.41474	−0.347378
$342$	12.1017	0.654384
$343$	8.34017	0.450327
$344$	10.8219	0.583476
$345$	−0.445922	−0.0240076
$346$	19.1710	1.03064
$347$	−25.7632	−1.38304	−0.691520	−	0.722357i	$-0.743059\pi$
$347$	−25.7632	−1.38304	−0.691520	+	0.722357i	$0.743059\pi$
$348$	6.03940	0.323746
$349$	23.9822	1.28374	0.641869	−	0.766815i	$-0.278159\pi$
$349$	23.9822	1.28374	0.641869	+	0.766815i	$0.278159\pi$
$350$	−15.4631	−0.826537
$351$	8.33021	0.444634
$352$	2.66536	0.142064
$353$	5.56720	0.296312	0.148156	−	0.988964i	$-0.452666\pi$
$353$	5.56720	0.296312	0.148156	+	0.988964i	$0.452666\pi$
$354$	1.46164	0.0776852
$355$	8.88900	0.471779
$356$	13.3914	0.709742
$357$	7.19628	0.380868
$358$	2.38254	0.125921
$359$	−16.2677	−0.858576	−0.429288	−	0.903168i	$-0.641235\pi$
$359$	−16.2677	−0.858576	−0.429288	+	0.903168i	$0.641235\pi$
$360$	1.71391	0.0903310
$361$	3.39373	0.178618
$362$	19.0100	0.999143
$363$	2.59212	0.136051
$364$	−7.65499	−0.401230
$365$	0.255394	0.0133680
$366$	0.578498	0.0302386
$367$	−1.48940	−0.0777462	−0.0388731	−	0.999244i	$-0.512377\pi$
$367$	−1.48940	−0.0777462	−0.0388731	+	0.999244i	$0.512377\pi$
$368$	1.00000	0.0521286
$369$	−26.5688	−1.38311
$370$	−6.71141	−0.348910
$371$	−9.31139	−0.483423
$372$	−1.60131	−0.0830242
$373$	6.34579	0.328572	0.164286	−	0.986413i	$-0.447468\pi$
$373$	6.34579	0.328572	0.164286	+	0.986413i	$0.447468\pi$
$374$	8.48409	0.438702
$375$	4.25893	0.219930
$376$	−2.12628	−0.109654
$377$	−20.4493	−1.05319
$378$	12.5639	0.646215
$379$	−25.9821	−1.33461	−0.667305	−	0.744784i	$-0.732552\pi$
$379$	−25.9821	−1.33461	−0.667305	+	0.744784i	$0.732552\pi$
$380$	3.17153	0.162696
$381$	1.73823	0.0890522
$382$	7.20331	0.368553
$383$	−28.2584	−1.44393	−0.721967	−	0.691927i	$-0.756762\pi$
$383$	−28.2584	−1.44393	−0.721967	+	0.691927i	$0.756762\pi$
$384$	0.665355	0.0339537
$385$	6.06970	0.309341
$386$	14.5057	0.738322
$387$	27.6748	1.40679
$388$	3.23551	0.164258
$389$	37.0291	1.87745	0.938726	−	0.344664i	$-0.112007\pi$
$389$	37.0291	1.87745	0.938726	+	0.344664i	$0.112007\pi$
$390$	1.00461	0.0508704
$391$	3.18309	0.160976
$392$	−4.54547	−0.229581
$393$	−0.665355	−0.0335627
$394$	1.16500	0.0586919
$395$	−4.23282	−0.212977
$396$	6.81614	0.342524
$397$	13.3645	0.670745	0.335372	−	0.942086i	$-0.391138\pi$
$397$	13.3645	0.670745	0.335372	+	0.942086i	$0.391138\pi$
$398$	15.6513	0.784528
$399$	10.6985	0.535594
$400$	−4.55083	−0.227541
$401$	−11.3007	−0.564331	−0.282165	−	0.959366i	$-0.591053\pi$
$401$	−11.3007	−0.564331	−0.282165	+	0.959366i	$0.591053\pi$
$402$	−3.69282	−0.184181
$403$	5.42203	0.270090
$404$	−11.9405	−0.594061
$405$	3.49290	0.173563
$406$	−30.8423	−1.53068
$407$	−26.6910	−1.32302
$408$	2.11789	0.104851
$409$	−1.51058	−0.0746936	−0.0373468	−	0.999302i	$-0.511891\pi$
$409$	−1.51058	−0.0746936	−0.0373468	+	0.999302i	$0.511891\pi$
$410$	−6.96298	−0.343877
$411$	−7.61108	−0.375427
$412$	4.34785	0.214203
$413$	−7.46436	−0.367297
$414$	2.55730	0.125685
$415$	−5.69545	−0.279579
$416$	−2.25288	−0.110457
$417$	0.0146607	0.000717935 0
$418$	12.6130	0.616923
$419$	31.3682	1.53244	0.766218	−	0.642581i	$-0.222136\pi$
$419$	31.3682	1.53244	0.766218	+	0.642581i	$0.222136\pi$
$420$	1.51518	0.0739333
$421$	−33.4032	−1.62797	−0.813987	−	0.580884i	$-0.802707\pi$
$421$	−33.4032	−1.62797	−0.813987	+	0.580884i	$0.802707\pi$
$422$	20.4743	0.996672
$423$	−5.43753	−0.264382
$424$	−2.74037	−0.133084
$425$	−14.4857	−0.702660
$426$	8.82471	0.427559
$427$	−2.95430	−0.142968
$428$	−9.12065	−0.440863
$429$	3.99529	0.192894
$430$	7.25284	0.349763
$431$	−29.8622	−1.43841	−0.719206	−	0.694797i	$-0.755494\pi$
$431$	−29.8622	−1.43841	−0.719206	+	0.694797i	$0.755494\pi$
$432$	3.69758	0.177900
$433$	−25.3290	−1.21723	−0.608617	−	0.793464i	$-0.708276\pi$
$433$	−25.3290	−1.21723	−0.608617	+	0.793464i	$0.708276\pi$
$434$	8.17765	0.392540
$435$	4.04762	0.194068
$436$	6.95171	0.332926
$437$	4.73220	0.226372
$438$	0.253547	0.0121150
$439$	−5.73920	−0.273917	−0.136959	−	0.990577i	$-0.543733\pi$
$439$	−5.73920	−0.273917	−0.136959	+	0.990577i	$0.543733\pi$
$440$	1.78633	0.0851600
$441$	−11.6241	−0.553530
$442$	−7.17114	−0.341096
$443$	−14.6720	−0.697089	−0.348545	−	0.937292i	$-0.613324\pi$
$443$	−14.6720	−0.697089	−0.348545	+	0.937292i	$0.613324\pi$
$444$	−6.66287	−0.316206
$445$	8.97493	0.425453
$446$	15.6735	0.742161
$447$	8.47424	0.400818
$448$	−3.39786	−0.160534
$449$	−3.29391	−0.155449	−0.0777247	−	0.996975i	$-0.524766\pi$
$449$	−3.29391	−0.155449	−0.0777247	+	0.996975i	$0.524766\pi$
$450$	−11.6378	−0.548614
$451$	−27.6914	−1.30394
$452$	−1.13052	−0.0531752
$453$	−4.94106	−0.232151
$454$	2.08245	0.0977341
$455$	−5.13039	−0.240516
$456$	3.14859	0.147446
$457$	4.46826	0.209016	0.104508	−	0.994524i	$-0.466673\pi$
$457$	4.46826	0.209016	0.104508	+	0.994524i	$0.466673\pi$
$458$	0.900940	0.0420982
$459$	11.7697	0.549364
$460$	0.670202	0.0312483
$461$	−21.9946	−1.02439	−0.512195	−	0.858869i	$-0.671168\pi$
$461$	−21.9946	−1.02439	−0.512195	+	0.858869i	$0.671168\pi$
$462$	6.02581	0.280346
$463$	4.28775	0.199268	0.0996342	−	0.995024i	$-0.468233\pi$
$463$	4.28775	0.199268	0.0996342	+	0.995024i	$0.468233\pi$
$464$	−9.07696	−0.421387
$465$	−1.07320	−0.0497686
$466$	−10.1771	−0.471443
$467$	−39.0018	−1.80479	−0.902393	−	0.430913i	$-0.858192\pi$
$467$	−39.0018	−1.80479	−0.902393	+	0.430913i	$0.858192\pi$
$468$	−5.76131	−0.266316
$469$	18.8586	0.870811
$470$	−1.42503	−0.0657319
$471$	−3.00925	−0.138659
$472$	−2.19678	−0.101115
$473$	28.8442	1.32626
$474$	−4.20221	−0.193014
$475$	−21.5354	−0.988114
$476$	−10.8157	−0.495737
$477$	−7.00795	−0.320872
$478$	11.5389	0.527778
$479$	15.6867	0.716746	0.358373	−	0.933579i	$-0.383332\pi$
$479$	15.6867	0.716746	0.358373	+	0.933579i	$0.383332\pi$
$480$	0.445922	0.0203535
$481$	22.5604	1.02867
$482$	5.92932	0.270073
$483$	2.26078	0.102869
$484$	−3.89585	−0.177084
$485$	2.16844	0.0984639
$486$	14.5604	0.660472
$487$	−9.11804	−0.413178	−0.206589	−	0.978428i	$-0.566236\pi$
$487$	−9.11804	−0.413178	−0.206589	+	0.978428i	$0.566236\pi$
$488$	−0.869457	−0.0393585
$489$	−4.93774	−0.223292
$490$	−3.04638	−0.137621
$491$	−27.2290	−1.22883	−0.614413	−	0.788985i	$-0.710607\pi$
$491$	−27.2290	−1.22883	−0.614413	+	0.788985i	$0.710607\pi$
$492$	−6.91262	−0.311645
$493$	−28.8928	−1.30127
$494$	−10.6611	−0.479666
$495$	4.56819	0.205325
$496$	2.40671	0.108064
$497$	−45.0664	−2.02150
$498$	−5.65426	−0.253373
$499$	0.612110	0.0274018	0.0137009	−	0.999906i	$-0.495639\pi$
$499$	0.612110	0.0274018	0.0137009	+	0.999906i	$0.495639\pi$
$500$	−6.40098	−0.286261
$501$	3.02932	0.135340
$502$	−17.2997	−0.772123
$503$	−8.18722	−0.365050	−0.182525	−	0.983201i	$-0.558427\pi$
$503$	−8.18722	−0.365050	−0.182525	+	0.983201i	$0.558427\pi$
$504$	−8.68936	−0.387055
$505$	−8.00253	−0.356108
$506$	2.66536	0.118490
$507$	5.27261	0.234165
$508$	−2.61248	−0.115910
$509$	−12.1602	−0.538991	−0.269496	−	0.963002i	$-0.586857\pi$
$509$	−12.1602	−0.538991	−0.269496	+	0.963002i	$0.586857\pi$
$510$	1.41941	0.0628526
$511$	−1.29483	−0.0572797
$512$	−1.00000	−0.0441942
$513$	17.4977	0.772542
$514$	−25.9083	−1.14277
$515$	2.91394	0.128403
$516$	7.20039	0.316979
$517$	−5.66729	−0.249247
$518$	34.0262	1.49503
$519$	12.7555	0.559905
$520$	−1.50989	−0.0662129
$521$	−35.6658	−1.56255	−0.781273	−	0.624189i	$-0.785429\pi$
$521$	−35.6658	−1.56255	−0.781273	+	0.624189i	$0.785429\pi$
$522$	−23.2125	−1.01599
$523$	−21.5639	−0.942925	−0.471462	−	0.881886i	$-0.656274\pi$
$523$	−21.5639	−0.942925	−0.471462	+	0.881886i	$0.656274\pi$
$524$	1.00000	0.0436852
$525$	−10.2884	−0.449024
$526$	15.7746	0.687805
$527$	7.66077	0.333708
$528$	1.77341	0.0771778
$529$	1.00000	0.0434783
$530$	−1.83660	−0.0797768
$531$	−5.61784	−0.243793
$532$	−16.0794	−0.697129
$533$	23.4060	1.01383
$534$	8.91002	0.385574
$535$	−6.11267	−0.264274
$536$	5.55015	0.239730
$537$	1.58524	0.0684079
$538$	6.11680	0.263714
$539$	−12.1153	−0.521843
$540$	2.47812	0.106641
$541$	6.48935	0.278999	0.139500	−	0.990222i	$-0.455451\pi$
$541$	6.48935	0.278999	0.139500	+	0.990222i	$0.455451\pi$
$542$	10.2324	0.439520
$543$	12.6484	0.542794
$544$	−3.18309	−0.136474
$545$	4.65905	0.199572
$546$	−5.09328	−0.217972
$547$	−32.1280	−1.37369	−0.686847	−	0.726802i	$-0.741006\pi$
$547$	−32.1280	−1.37369	−0.686847	+	0.726802i	$0.741006\pi$
$548$	11.4391	0.488655
$549$	−2.22347	−0.0948952
$550$	−12.1296	−0.517208
$551$	−42.9540	−1.82990
$552$	0.665355	0.0283194
$553$	21.4600	0.912573
$554$	−23.8901	−1.01499
$555$	−4.46547	−0.189549
$556$	−0.0220343	−0.000934464 0
$557$	11.9182	0.504992	0.252496	−	0.967598i	$-0.418749\pi$
$557$	11.9182	0.504992	0.252496	+	0.967598i	$0.418749\pi$
$558$	6.15467	0.260548
$559$	−24.3804	−1.03118
$560$	−2.27725	−0.0962315
$561$	5.64493	0.238329
$562$	−22.5490	−0.951171
$563$	1.59951	0.0674115	0.0337058	−	0.999432i	$-0.489269\pi$
$563$	1.59951	0.0674115	0.0337058	+	0.999432i	$0.489269\pi$
$564$	−1.41473	−0.0595708
$565$	−0.757676	−0.0318757
$566$	−17.4271	−0.732517
$567$	−17.7087	−0.743694
$568$	−13.2632	−0.556510
$569$	29.7103	1.24552	0.622761	−	0.782412i	$-0.286011\pi$
$569$	29.7103	1.24552	0.622761	+	0.782412i	$0.286011\pi$
$570$	2.11019	0.0883863
$571$	−38.5184	−1.61194	−0.805972	−	0.591953i	$-0.798357\pi$
$571$	−38.5184	−1.61194	−0.805972	+	0.591953i	$0.798357\pi$
$572$	−6.00475	−0.251071
$573$	4.79276	0.200220
$574$	35.3016	1.47346
$575$	−4.55083	−0.189783
$576$	−2.55730	−0.106554
$577$	−16.1198	−0.671077	−0.335539	−	0.942026i	$-0.608918\pi$
$577$	−16.1198	−0.671077	−0.335539	+	0.942026i	$0.608918\pi$
$578$	6.86792	0.285668
$579$	9.65145	0.401101
$580$	−6.08340	−0.252599
$581$	28.8754	1.19795
$582$	2.15276	0.0892347
$583$	−7.30407	−0.302504
$584$	−0.381071	−0.0157688
$585$	−3.86124	−0.159643
$586$	3.30962	0.136719
$587$	−3.12478	−0.128974	−0.0644868	−	0.997919i	$-0.520541\pi$
$587$	−3.12478	−0.128974	−0.0644868	+	0.997919i	$0.520541\pi$
$588$	−3.02435	−0.124722
$589$	11.3890	0.469276
$590$	−1.47229	−0.0606131
$591$	0.775139	0.0318849
$592$	10.0140	0.411573
$593$	27.6055	1.13362	0.566811	−	0.823848i	$-0.308177\pi$
$593$	27.6055	1.13362	0.566811	+	0.823848i	$0.308177\pi$
$594$	9.85538	0.404371
$595$	−7.24871	−0.297168
$596$	−12.7364	−0.521704
$597$	10.4137	0.426202
$598$	−2.25288	−0.0921273
$599$	−9.21585	−0.376550	−0.188275	−	0.982116i	$-0.560290\pi$
$599$	−9.21585	−0.376550	−0.188275	+	0.982116i	$0.560290\pi$
$600$	−3.02792	−0.123614
$601$	−10.8477	−0.442487	−0.221243	−	0.975219i	$-0.571012\pi$
$601$	−10.8477	−0.442487	−0.221243	+	0.975219i	$0.571012\pi$
$602$	−36.7712	−1.49868
$603$	14.1934	0.578000
$604$	7.42620	0.302168
$605$	−2.61101	−0.106152
$606$	−7.94465	−0.322729
$607$	9.39050	0.381149	0.190574	−	0.981673i	$-0.438965\pi$
$607$	9.39050	0.381149	0.190574	+	0.981673i	$0.438965\pi$
$608$	−4.73220	−0.191916
$609$	−20.5210	−0.831555
$610$	−0.582712	−0.0235933
$611$	4.79025	0.193793
$612$	−8.14013	−0.329045
$613$	28.5696	1.15391	0.576957	−	0.816774i	$-0.304240\pi$
$613$	28.5696	1.15391	0.576957	+	0.816774i	$0.304240\pi$
$614$	13.3594	0.539141
$615$	−4.63285	−0.186815
$616$	−9.05653	−0.364898
$617$	32.4045	1.30455	0.652277	−	0.757980i	$-0.273814\pi$
$617$	32.4045	1.30455	0.652277	+	0.757980i	$0.273814\pi$
$618$	2.89286	0.116368
$619$	−39.0485	−1.56949	−0.784746	−	0.619817i	$-0.787207\pi$
$619$	−39.0485	−1.56949	−0.784746	+	0.619817i	$0.787207\pi$
$620$	1.61298	0.0647788
$621$	3.69758	0.148379
$622$	19.8616	0.796378
$623$	−45.5021	−1.82300
$624$	−1.49897	−0.0600067
$625$	18.4642	0.738568
$626$	21.3082	0.851648
$627$	8.39214	0.335150
$628$	4.52277	0.180478
$629$	31.8755	1.27096
$630$	−5.82363	−0.232019
$631$	0.580118	0.0230941	0.0115471	−	0.999933i	$-0.496324\pi$
$631$	0.580118	0.0230941	0.0115471	+	0.999933i	$0.496324\pi$
$632$	6.31574	0.251227
$633$	13.6227	0.541452
$634$	−23.6504	−0.939277
$635$	−1.75089	−0.0694820
$636$	−1.82332	−0.0722992
$637$	10.2404	0.405740
$638$	−24.1934	−0.957825
$639$	−33.9179	−1.34177
$640$	−0.670202	−0.0264921
$641$	−30.3791	−1.19990	−0.599952	−	0.800036i	$-0.704814\pi$
$641$	−30.3791	−1.19990	−0.599952	+	0.800036i	$0.704814\pi$
$642$	−6.06846	−0.239503
$643$	13.2016	0.520620	0.260310	−	0.965525i	$-0.416175\pi$
$643$	13.2016	0.520620	0.260310	+	0.965525i	$0.416175\pi$
$644$	−3.39786	−0.133895
$645$	4.82571	0.190012
$646$	−15.0630	−0.592647
$647$	35.2463	1.38567	0.692837	−	0.721094i	$-0.256361\pi$
$647$	35.2463	1.38567	0.692837	+	0.721094i	$0.256361\pi$
$648$	−5.21171	−0.204735
$649$	−5.85522	−0.229837
$650$	10.2525	0.402136
$651$	5.44104	0.213251
$652$	7.42122	0.290637
$653$	19.8954	0.778566	0.389283	−	0.921118i	$-0.372723\pi$
$653$	19.8954	0.778566	0.389283	+	0.921118i	$0.372723\pi$
$654$	4.62535	0.180866
$655$	0.670202	0.0261870
$656$	10.3894	0.405637
$657$	−0.974513	−0.0380194
$658$	7.22479	0.281652
$659$	23.2928	0.907360	0.453680	−	0.891165i	$-0.350111\pi$
$659$	23.2928	0.907360	0.453680	+	0.891165i	$0.350111\pi$
$660$	1.18854	0.0462640
$661$	−37.5728	−1.46141	−0.730706	−	0.682693i	$-0.760809\pi$
$661$	−37.5728	−1.46141	−0.730706	+	0.682693i	$0.760809\pi$
$662$	−31.6327	−1.22944
$663$	−4.77135	−0.185304
$664$	8.49811	0.329791
$665$	−10.7764	−0.417892
$666$	25.6089	0.992324
$667$	−9.07696	−0.351461
$668$	−4.55293	−0.176158
$669$	10.4284	0.403187
$670$	3.71972	0.143705
$671$	−2.31742	−0.0894629
$672$	−2.26078	−0.0872116
$673$	−29.8570	−1.15090	−0.575452	−	0.817836i	$-0.695174\pi$
$673$	−29.8570	−1.15090	−0.575452	+	0.817836i	$0.695174\pi$
$674$	−3.03332	−0.116839
$675$	−16.8270	−0.647673
$676$	−7.92452	−0.304789
$677$	−35.7525	−1.37408	−0.687040	−	0.726620i	$-0.741090\pi$
$677$	−35.7525	−1.37408	−0.687040	+	0.726620i	$0.741090\pi$
$678$	−0.752197	−0.0288879
$679$	−10.9938	−0.421903
$680$	−2.13331	−0.0818089
$681$	1.38557	0.0530950
$682$	6.41474	0.245633
$683$	14.0646	0.538166	0.269083	−	0.963117i	$-0.413279\pi$
$683$	14.0646	0.538166	0.269083	+	0.963117i	$0.413279\pi$
$684$	−12.1017	−0.462719
$685$	7.66652	0.292923
$686$	−8.34017	−0.318429
$687$	0.599445	0.0228702
$688$	−10.8219	−0.412580
$689$	6.17373	0.235200
$690$	0.445922	0.0169760
$691$	−16.8164	−0.639725	−0.319862	−	0.947464i	$-0.603637\pi$
$691$	−16.8164	−0.639725	−0.319862	+	0.947464i	$0.603637\pi$
$692$	−19.1710	−0.728772
$693$	−23.1603	−0.879787
$694$	25.7632	0.977957
$695$	−0.0147675	−0.000560161 0
$696$	−6.03940	−0.228923
$697$	33.0703	1.25263
$698$	−23.9822	−0.907739
$699$	−6.77135	−0.256116
$700$	15.4631	0.584450
$701$	−10.7148	−0.404693	−0.202347	−	0.979314i	$-0.564857\pi$
$701$	−10.7148	−0.404693	−0.202347	+	0.979314i	$0.564857\pi$
$702$	−8.33021	−0.314404
$703$	47.3883	1.78729
$704$	−2.66536	−0.100455
$705$	−0.948153	−0.0357095
$706$	−5.56720	−0.209524
$707$	40.5721	1.52587
$708$	−1.46164	−0.0549318
$709$	−26.3019	−0.987788	−0.493894	−	0.869522i	$-0.664427\pi$
$709$	−26.3019	−0.987788	−0.493894	+	0.869522i	$0.664427\pi$
$710$	−8.88900	−0.333598
$711$	16.1513	0.605720
$712$	−13.3914	−0.501863
$713$	2.40671	0.0901318
$714$	−7.19628	−0.269314
$715$	−4.02439	−0.150504
$716$	−2.38254	−0.0890397
$717$	7.67748	0.286721
$718$	16.2677	0.607105
$719$	46.0316	1.71669	0.858345	−	0.513074i	$-0.171493\pi$
$719$	46.0316	1.71669	0.858345	+	0.513074i	$0.171493\pi$
$720$	−1.71391	−0.0638736
$721$	−14.7734	−0.550190
$722$	−3.39373	−0.126302
$723$	3.94510	0.146720
$724$	−19.0100	−0.706501
$725$	41.3077	1.53413
$726$	−2.59212	−0.0962027
$727$	9.59565	0.355883	0.177941	−	0.984041i	$-0.443056\pi$
$727$	9.59565	0.355883	0.177941	+	0.984041i	$0.443056\pi$
$728$	7.65499	0.283713
$729$	−5.94731	−0.220271
$730$	−0.255394	−0.00945257
$731$	−34.4470	−1.27407
$732$	−0.578498	−0.0213819
$733$	19.9457	0.736710	0.368355	−	0.929685i	$-0.379921\pi$
$733$	19.9457	0.736710	0.368355	+	0.929685i	$0.379921\pi$
$734$	1.48940	0.0549749
$735$	−2.02692	−0.0747642
$736$	−1.00000	−0.0368605
$737$	14.7931	0.544913
$738$	26.5688	0.978010
$739$	13.3186	0.489934	0.244967	−	0.969531i	$-0.421223\pi$
$739$	13.3186	0.489934	0.244967	+	0.969531i	$0.421223\pi$
$740$	6.71141	0.246716
$741$	−7.09341	−0.260583
$742$	9.31139	0.341832
$743$	−3.66185	−0.134340	−0.0671702	−	0.997742i	$-0.521397\pi$
$743$	−3.66185	−0.134340	−0.0671702	+	0.997742i	$0.521397\pi$
$744$	1.60131	0.0587070
$745$	−8.53598	−0.312734
$746$	−6.34579	−0.232336
$747$	21.7322	0.795141
$748$	−8.48409	−0.310209
$749$	30.9907	1.13238
$750$	−4.25893	−0.155514
$751$	−14.0114	−0.511283	−0.255641	−	0.966772i	$-0.582287\pi$
$751$	−14.0114	−0.511283	−0.255641	+	0.966772i	$0.582287\pi$
$752$	2.12628	0.0775373
$753$	−11.5104	−0.419464
$754$	20.4493	0.744721
$755$	4.97705	0.181133
$756$	−12.5639	−0.456943
$757$	−9.44368	−0.343236	−0.171618	−	0.985164i	$-0.554900\pi$
$757$	−9.44368	−0.343236	−0.171618	+	0.985164i	$0.554900\pi$
$758$	25.9821	0.943712
$759$	1.77341	0.0643707
$760$	−3.17153	−0.115044
$761$	−34.1167	−1.23673	−0.618364	−	0.785892i	$-0.712204\pi$
$761$	−34.1167	−1.23673	−0.618364	+	0.785892i	$0.712204\pi$
$762$	−1.73823	−0.0629694
$763$	−23.6209	−0.855135
$764$	−7.20331	−0.260607
$765$	−5.45553	−0.197245
$766$	28.2584	1.02102
$767$	4.94909	0.178701
$768$	−0.665355	−0.0240089
$769$	−16.7212	−0.602984	−0.301492	−	0.953469i	$-0.597485\pi$
$769$	−16.7212	−0.602984	−0.301492	+	0.953469i	$0.597485\pi$
$770$	−6.06970	−0.218737
$771$	−17.2382	−0.620820
$772$	−14.5057	−0.522072
$773$	43.9077	1.57925	0.789624	−	0.613590i	$-0.210275\pi$
$773$	43.9077	1.57925	0.789624	+	0.613590i	$0.210275\pi$
$774$	−27.6748	−0.994751
$775$	−10.9525	−0.393426
$776$	−3.23551	−0.116148
$777$	22.6395	0.812188
$778$	−37.0291	−1.32756
$779$	49.1646	1.76150
$780$	−1.00461	−0.0359708
$781$	−35.3511	−1.26496
$782$	−3.18309	−0.113827
$783$	−33.5628	−1.19944
$784$	4.54547	0.162338
$785$	3.03117	0.108187
$786$	0.665355	0.0237324
$787$	17.7914	0.634194	0.317097	−	0.948393i	$-0.397292\pi$
$787$	17.7914	0.634194	0.317097	+	0.948393i	$0.397292\pi$
$788$	−1.16500	−0.0415014
$789$	10.4957	0.373657
$790$	4.23282	0.150597
$791$	3.84135	0.136583
$792$	−6.81614	−0.242201
$793$	1.95879	0.0695585
$794$	−13.3645	−0.474288
$795$	−1.22199	−0.0433395
$796$	−15.6513	−0.554745
$797$	3.81395	0.135097	0.0675484	−	0.997716i	$-0.478482\pi$
$797$	3.81395	0.135097	0.0675484	+	0.997716i	$0.478482\pi$
$798$	−10.6985	−0.378722
$799$	6.76813	0.239439
$800$	4.55083	0.160896
$801$	−34.2458	−1.21002
$802$	11.3007	0.399042
$803$	−1.01569	−0.0358430
$804$	3.69282	0.130236
$805$	−2.27725	−0.0802626
$806$	−5.42203	−0.190983
$807$	4.06984	0.143265
$808$	11.9405	0.420064
$809$	−31.2612	−1.09909	−0.549543	−	0.835466i	$-0.685198\pi$
$809$	−31.2612	−1.09909	−0.549543	+	0.835466i	$0.685198\pi$
$810$	−3.49290	−0.122728
$811$	−27.1488	−0.953323	−0.476661	−	0.879087i	$-0.658153\pi$
$811$	−27.1488	−0.953323	−0.476661	+	0.879087i	$0.658153\pi$
$812$	30.8423	1.08235
$813$	6.80819	0.238774
$814$	26.6910	0.935518
$815$	4.97371	0.174222
$816$	−2.11789	−0.0741408
$817$	−51.2113	−1.79166
$818$	1.51058	0.0528163
$819$	19.5761	0.684045
$820$	6.96298	0.243158
$821$	−19.1189	−0.667255	−0.333627	−	0.942705i	$-0.608273\pi$
$821$	−19.1189	−0.667255	−0.333627	+	0.942705i	$0.608273\pi$
$822$	7.61108	0.265467
$823$	−38.3991	−1.33851	−0.669255	−	0.743033i	$-0.733387\pi$
$823$	−38.3991	−1.33851	−0.669255	+	0.743033i	$0.733387\pi$
$824$	−4.34785	−0.151464
$825$	−8.07049	−0.280978
$826$	7.46436	0.259718
$827$	6.30986	0.219415	0.109708	−	0.993964i	$-0.465009\pi$
$827$	6.30986	0.219415	0.109708	+	0.993964i	$0.465009\pi$
$828$	−2.55730	−0.0888724
$829$	−3.56625	−0.123861	−0.0619304	−	0.998080i	$-0.519726\pi$
$829$	−3.56625	−0.123861	−0.0619304	+	0.998080i	$0.519726\pi$
$830$	5.69545	0.197692
$831$	−15.8954	−0.551405
$832$	2.25288	0.0781047
$833$	14.4686	0.501309
$834$	−0.0146607	−0.000507657 0
$835$	−3.05138	−0.105598
$836$	−12.6130	−0.436231
$837$	8.89898	0.307594
$838$	−31.3682	−1.08360
$839$	−23.7059	−0.818419	−0.409209	−	0.912441i	$-0.634195\pi$
$839$	−23.7059	−0.818419	−0.409209	+	0.912441i	$0.634195\pi$
$840$	−1.51518	−0.0522787
$841$	53.3912	1.84108
$842$	33.4032	1.15115
$843$	−15.0031	−0.516733
$844$	−20.4743	−0.704754
$845$	−5.31103	−0.182705
$846$	5.43753	0.186946
$847$	13.2376	0.454848
$848$	2.74037	0.0941046
$849$	−11.5952	−0.397947
$850$	14.4857	0.496856
$851$	10.0140	0.343276
$852$	−8.82471	−0.302330
$853$	−13.6342	−0.466827	−0.233414	−	0.972378i	$-0.574990\pi$
$853$	−13.6342	−0.466827	−0.233414	+	0.972378i	$0.574990\pi$
$854$	2.95430	0.101094
$855$	−8.11056	−0.277375
$856$	9.12065	0.311737
$857$	33.9476	1.15963	0.579815	−	0.814748i	$-0.303125\pi$
$857$	33.9476	1.15963	0.579815	+	0.814748i	$0.303125\pi$
$858$	−3.99529	−0.136397
$859$	−35.5735	−1.21375	−0.606876	−	0.794797i	$-0.707577\pi$
$859$	−35.5735	−1.21375	−0.606876	+	0.794797i	$0.707577\pi$
$860$	−7.25284	−0.247320
$861$	23.4881	0.800473
$862$	29.8622	1.01711
$863$	−0.895710	−0.0304903	−0.0152452	−	0.999884i	$-0.504853\pi$
$863$	−0.895710	−0.0304903	−0.0152452	+	0.999884i	$0.504853\pi$
$864$	−3.69758	−0.125794
$865$	−12.8484	−0.436860
$866$	25.3290	0.860715
$867$	4.56960	0.155192
$868$	−8.17765	−0.277568
$869$	16.8337	0.571046
$870$	−4.04762	−0.137227
$871$	−12.5038	−0.423676
$872$	−6.95171	−0.235414
$873$	−8.27417	−0.280038
$874$	−4.73220	−0.160069
$875$	21.7497	0.735273
$876$	−0.253547	−0.00856657
$877$	−34.8547	−1.17696	−0.588480	−	0.808512i	$-0.700273\pi$
$877$	−34.8547	−1.17696	−0.588480	+	0.808512i	$0.700273\pi$
$878$	5.73920	0.193689
$879$	2.20207	0.0742741
$880$	−1.78633	−0.0602172
$881$	−48.2420	−1.62531	−0.812657	−	0.582742i	$-0.801980\pi$
$881$	−48.2420	−1.62531	−0.812657	+	0.582742i	$0.801980\pi$
$882$	11.6241	0.391405
$883$	26.9038	0.905386	0.452693	−	0.891666i	$-0.350463\pi$
$883$	26.9038	0.905386	0.452693	+	0.891666i	$0.350463\pi$
$884$	7.17114	0.241191
$885$	−0.979593	−0.0329287
$886$	14.6720	0.492917
$887$	3.29742	0.110717	0.0553583	−	0.998467i	$-0.482370\pi$
$887$	3.29742	0.110717	0.0553583	+	0.998467i	$0.482370\pi$
$888$	6.66287	0.223591
$889$	8.87686	0.297720
$890$	−8.97493	−0.300840
$891$	−13.8911	−0.465369
$892$	−15.6735	−0.524787
$893$	10.0620	0.336711
$894$	−8.47424	−0.283421
$895$	−1.59678	−0.0533746
$896$	3.39786	0.113515
$897$	−1.49897	−0.0500490
$898$	3.29391	0.109919
$899$	−21.8456	−0.728591
$900$	11.6378	0.387928
$901$	8.72284	0.290600
$902$	27.6914	0.922024
$903$	−24.4659	−0.814175
$904$	1.13052	0.0376005
$905$	−12.7405	−0.423510
$906$	4.94106	0.164156
$907$	19.4964	0.647366	0.323683	−	0.946166i	$-0.395079\pi$
$907$	19.4964	0.647366	0.323683	+	0.946166i	$0.395079\pi$
$908$	−2.08245	−0.0691084
$909$	30.5354	1.01280
$910$	5.13039	0.170071
$911$	43.3798	1.43724	0.718618	−	0.695405i	$-0.244775\pi$
$911$	43.3798	1.43724	0.718618	+	0.695405i	$0.244775\pi$
$912$	−3.14859	−0.104260
$913$	22.6505	0.749623
$914$	−4.46826	−0.147797
$915$	−0.387710	−0.0128173
$916$	−0.900940	−0.0297679
$917$	−3.39786	−0.112207
$918$	−11.7697	−0.388459
$919$	−13.9411	−0.459874	−0.229937	−	0.973206i	$-0.573852\pi$
$919$	−13.9411	−0.459874	−0.229937	+	0.973206i	$0.573852\pi$
$920$	−0.670202	−0.0220959
$921$	8.88874	0.292894
$922$	21.9946	0.724353
$923$	29.8804	0.983524
$924$	−6.02581	−0.198234
$925$	−45.5721	−1.49840
$926$	−4.28775	−0.140904
$927$	−11.1188	−0.365188
$928$	9.07696	0.297966
$929$	41.2786	1.35431	0.677154	−	0.735842i	$-0.263213\pi$
$929$	41.2786	1.35431	0.677154	+	0.735842i	$0.263213\pi$
$930$	1.07320	0.0351917
$931$	21.5101	0.704964
$932$	10.1771	0.333361
$933$	13.2150	0.432640
$934$	39.0018	1.27618
$935$	−5.68605	−0.185954
$936$	5.76131	0.188314
$937$	−7.35757	−0.240361	−0.120181	−	0.992752i	$-0.538347\pi$
$937$	−7.35757	−0.240361	−0.120181	+	0.992752i	$0.538347\pi$
$938$	−18.8586	−0.615756
$939$	14.1775	0.462666
$940$	1.42503	0.0464795
$941$	10.3822	0.338451	0.169225	−	0.985577i	$-0.445873\pi$
$941$	10.3822	0.338451	0.169225	+	0.985577i	$0.445873\pi$
$942$	3.00925	0.0980466
$943$	10.3894	0.338324
$944$	2.19678	0.0714991
$945$	−8.42032	−0.273913
$946$	−28.8442	−0.937806
$947$	15.7911	0.513141	0.256570	−	0.966526i	$-0.417407\pi$
$947$	15.7911	0.513141	0.256570	+	0.966526i	$0.417407\pi$
$948$	4.20221	0.136481
$949$	0.858508	0.0278684
$950$	21.5354	0.698702
$951$	−15.7359	−0.510272
$952$	10.8157	0.350539
$953$	30.8900	1.00063	0.500313	−	0.865845i	$-0.333218\pi$
$953$	30.8900	1.00063	0.500313	+	0.865845i	$0.333218\pi$
$954$	7.00795	0.226891
$955$	−4.82767	−0.156220
$956$	−11.5389	−0.373196
$957$	−16.0972	−0.520348
$958$	−15.6867	−0.506816
$959$	−38.8686	−1.25513
$960$	−0.445922	−0.0143921
$961$	−25.2078	−0.813154
$962$	−22.5604	−0.727377
$963$	23.3243	0.751614
$964$	−5.92932	−0.190971
$965$	−9.72176	−0.312955
$966$	−2.26078	−0.0727395
$967$	37.0403	1.19114	0.595568	−	0.803305i	$-0.296927\pi$
$967$	37.0403	1.19114	0.595568	+	0.803305i	$0.296927\pi$
$968$	3.89585	0.125217
$969$	−10.0223	−0.321961
$970$	−2.16844	−0.0696245
$971$	−53.6435	−1.72150	−0.860750	−	0.509028i	$-0.830005\pi$
$971$	−53.6435	−1.72150	−0.860750	+	0.509028i	$0.830005\pi$
$972$	−14.5604	−0.467024
$973$	0.0748696	0.00240021
$974$	9.11804	0.292161
$975$	6.82154	0.218464
$976$	0.869457	0.0278307
$977$	−38.9609	−1.24647	−0.623235	−	0.782034i	$-0.714182\pi$
$977$	−38.9609	−1.24647	−0.623235	+	0.782034i	$0.714182\pi$
$978$	4.93774	0.157892
$979$	−35.6929	−1.14075
$980$	3.04638	0.0973130
$981$	−17.7776	−0.567596
$982$	27.2290	0.868911
$983$	48.9246	1.56045	0.780227	−	0.625497i	$-0.215104\pi$
$983$	48.9246	1.56045	0.780227	+	0.625497i	$0.215104\pi$
$984$	6.91262	0.220366
$985$	−0.780785	−0.0248779
$986$	28.8928	0.920134
$987$	4.80705	0.153010
$988$	10.6611	0.339175
$989$	−10.8219	−0.344116
$990$	−4.56819	−0.145187
$991$	14.4871	0.460199	0.230100	−	0.973167i	$-0.426095\pi$
$991$	14.4871	0.460199	0.230100	+	0.973167i	$0.426095\pi$
$992$	−2.40671	−0.0764130
$993$	−21.0470	−0.667906
$994$	45.0664	1.42942
$995$	−10.4895	−0.332540
$996$	5.65426	0.179162
$997$	21.6914	0.686972	0.343486	−	0.939158i	$-0.388392\pi$
$997$	21.6914	0.686972	0.343486	+	0.939158i	$0.388392\pi$
$998$	−0.612110	−0.0193760
$999$	37.0276	1.17150

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6026.2.a.i.1.12	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6026.2.a.i.1.12	✓	25	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.665355 q^{3} +1.00000 q^{4} +0.670202 q^{5} +0.665355 q^{6} -3.39786 q^{7} -1.00000 q^{8} -2.55730 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.665355 q^{3} +1.00000 q^{4} +0.670202 q^{5} +0.665355 q^{6} -3.39786 q^{7} -1.00000 q^{8} -2.55730 q^{9} -0.670202 q^{10} -2.66536 q^{11} -0.665355 q^{12} +2.25288 q^{13} +3.39786 q^{14} -0.445922 q^{15} +1.00000 q^{16} +3.18309 q^{17} +2.55730 q^{18} +4.73220 q^{19} +0.670202 q^{20} +2.26078 q^{21} +2.66536 q^{22} +1.00000 q^{23} +0.665355 q^{24} -4.55083 q^{25} -2.25288 q^{26} +3.69758 q^{27} -3.39786 q^{28} -9.07696 q^{29} +0.445922 q^{30} +2.40671 q^{31} -1.00000 q^{32} +1.77341 q^{33} -3.18309 q^{34} -2.27725 q^{35} -2.55730 q^{36} +10.0140 q^{37} -4.73220 q^{38} -1.49897 q^{39} -0.670202 q^{40} +10.3894 q^{41} -2.26078 q^{42} -10.8219 q^{43} -2.66536 q^{44} -1.71391 q^{45} -1.00000 q^{46} +2.12628 q^{47} -0.665355 q^{48} +4.54547 q^{49} +4.55083 q^{50} -2.11789 q^{51} +2.25288 q^{52} +2.74037 q^{53} -3.69758 q^{54} -1.78633 q^{55} +3.39786 q^{56} -3.14859 q^{57} +9.07696 q^{58} +2.19678 q^{59} -0.445922 q^{60} +0.869457 q^{61} -2.40671 q^{62} +8.68936 q^{63} +1.00000 q^{64} +1.50989 q^{65} -1.77341 q^{66} -5.55015 q^{67} +3.18309 q^{68} -0.665355 q^{69} +2.27725 q^{70} +13.2632 q^{71} +2.55730 q^{72} +0.381071 q^{73} -10.0140 q^{74} +3.02792 q^{75} +4.73220 q^{76} +9.05653 q^{77} +1.49897 q^{78} -6.31574 q^{79} +0.670202 q^{80} +5.21171 q^{81} -10.3894 q^{82} -8.49811 q^{83} +2.26078 q^{84} +2.13331 q^{85} +10.8219 q^{86} +6.03940 q^{87} +2.66536 q^{88} +13.3914 q^{89} +1.71391 q^{90} -7.65499 q^{91} +1.00000 q^{92} -1.60131 q^{93} -2.12628 q^{94} +3.17153 q^{95} +0.665355 q^{96} +3.23551 q^{97} -4.54547 q^{98} +6.81614 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 25 q^{2} - 4 q^{3} + 25 q^{4} - 3 q^{5} + 4 q^{6} - 11 q^{7} - 25 q^{8} + 19 q^{9}+O(q^{10}) \) 25 * q - 25 * q^2 - 4 * q^3 + 25 * q^4 - 3 * q^5 + 4 * q^6 - 11 * q^7 - 25 * q^8 + 19 * q^9	\( 25 q - 25 q^{2} - 4 q^{3} + 25 q^{4} - 3 q^{5} + 4 q^{6} - 11 q^{7} - 25 q^{8} + 19 q^{9} + 3 q^{10} - 12 q^{11} - 4 q^{12} - 6 q^{13} + 11 q^{14} + 25 q^{16} + 8 q^{17} - 19 q^{18} - 23 q^{19} - 3 q^{20} - 16 q^{21} + 12 q^{22} + 25 q^{23} + 4 q^{24} + 4 q^{25} + 6 q^{26} - 13 q^{27} - 11 q^{28} - 7 q^{29} - 7 q^{31} - 25 q^{32} + 3 q^{33} - 8 q^{34} - 18 q^{35} + 19 q^{36} - 7 q^{37} + 23 q^{38} - 2 q^{39} + 3 q^{40} - 10 q^{41} + 16 q^{42} - 26 q^{43} - 12 q^{44} + 20 q^{45} - 25 q^{46} - 2 q^{47} - 4 q^{48} + 2 q^{49} - 4 q^{50} - 28 q^{51} - 6 q^{52} + 47 q^{53} + 13 q^{54} - 38 q^{55} + 11 q^{56} - 4 q^{57} + 7 q^{58} - 19 q^{59} - 26 q^{61} + 7 q^{62} - 15 q^{63} + 25 q^{64} + 13 q^{65} - 3 q^{66} - 34 q^{67} + 8 q^{68} - 4 q^{69} + 18 q^{70} - 10 q^{71} - 19 q^{72} - 22 q^{73} + 7 q^{74} - 8 q^{75} - 23 q^{76} + 28 q^{77} + 2 q^{78} - 21 q^{79} - 3 q^{80} - 27 q^{81} + 10 q^{82} - 16 q^{83} - 16 q^{84} - 42 q^{85} + 26 q^{86} - 17 q^{87} + 12 q^{88} + 27 q^{89} - 20 q^{90} - 26 q^{91} + 25 q^{92} - 27 q^{93} + 2 q^{94} + 4 q^{96} + 4 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) 25 * q - 25 * q^2 - 4 * q^3 + 25 * q^4 - 3 * q^5 + 4 * q^6 - 11 * q^7 - 25 * q^8 + 19 * q^9 + 3 * q^10 - 12 * q^11 - 4 * q^12 - 6 * q^13 + 11 * q^14 + 25 * q^16 + 8 * q^17 - 19 * q^18 - 23 * q^19 - 3 * q^20 - 16 * q^21 + 12 * q^22 + 25 * q^23 + 4 * q^24 + 4 * q^25 + 6 * q^26 - 13 * q^27 - 11 * q^28 - 7 * q^29 - 7 * q^31 - 25 * q^32 + 3 * q^33 - 8 * q^34 - 18 * q^35 + 19 * q^36 - 7 * q^37 + 23 * q^38 - 2 * q^39 + 3 * q^40 - 10 * q^41 + 16 * q^42 - 26 * q^43 - 12 * q^44 + 20 * q^45 - 25 * q^46 - 2 * q^47 - 4 * q^48 + 2 * q^49 - 4 * q^50 - 28 * q^51 - 6 * q^52 + 47 * q^53 + 13 * q^54 - 38 * q^55 + 11 * q^56 - 4 * q^57 + 7 * q^58 - 19 * q^59 - 26 * q^61 + 7 * q^62 - 15 * q^63 + 25 * q^64 + 13 * q^65 - 3 * q^66 - 34 * q^67 + 8 * q^68 - 4 * q^69 + 18 * q^70 - 10 * q^71 - 19 * q^72 - 22 * q^73 + 7 * q^74 - 8 * q^75 - 23 * q^76 + 28 * q^77 + 2 * q^78 - 21 * q^79 - 3 * q^80 - 27 * q^81 + 10 * q^82 - 16 * q^83 - 16 * q^84 - 42 * q^85 + 26 * q^86 - 17 * q^87 + 12 * q^88 + 27 * q^89 - 20 * q^90 - 26 * q^91 + 25 * q^92 - 27 * q^93 + 2 * q^94 + 4 * q^96 + 4 * q^97 - 2 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more