LMFDB - Embedded newform 6038.2.a.c.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.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:	$48.2136727404$
Analytic rank:	$1$
Dimension:	$57$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.71541 q^{3} +1.00000 q^{4} +1.06645 q^{5} +1.71541 q^{6} +4.85447 q^{7} -1.00000 q^{8} -0.0573586 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.71541 q^{3} +1.00000 q^{4} +1.06645 q^{5} +1.71541 q^{6} +4.85447 q^{7} -1.00000 q^{8} -0.0573586 q^{9} -1.06645 q^{10} -4.21829 q^{11} -1.71541 q^{12} -5.04666 q^{13} -4.85447 q^{14} -1.82940 q^{15} +1.00000 q^{16} +5.34566 q^{17} +0.0573586 q^{18} -0.676346 q^{19} +1.06645 q^{20} -8.32742 q^{21} +4.21829 q^{22} +0.753052 q^{23} +1.71541 q^{24} -3.86269 q^{25} +5.04666 q^{26} +5.24463 q^{27} +4.85447 q^{28} -0.420186 q^{29} +1.82940 q^{30} -1.39465 q^{31} -1.00000 q^{32} +7.23610 q^{33} -5.34566 q^{34} +5.17703 q^{35} -0.0573586 q^{36} +3.97072 q^{37} +0.676346 q^{38} +8.65710 q^{39} -1.06645 q^{40} -2.34182 q^{41} +8.32742 q^{42} -2.53027 q^{43} -4.21829 q^{44} -0.0611699 q^{45} -0.753052 q^{46} +4.58870 q^{47} -1.71541 q^{48} +16.5659 q^{49} +3.86269 q^{50} -9.17001 q^{51} -5.04666 q^{52} +9.68212 q^{53} -5.24463 q^{54} -4.49858 q^{55} -4.85447 q^{56} +1.16021 q^{57} +0.420186 q^{58} -1.30045 q^{59} -1.82940 q^{60} -5.12051 q^{61} +1.39465 q^{62} -0.278446 q^{63} +1.00000 q^{64} -5.38199 q^{65} -7.23610 q^{66} -13.3698 q^{67} +5.34566 q^{68} -1.29180 q^{69} -5.17703 q^{70} -10.3650 q^{71} +0.0573586 q^{72} -2.60917 q^{73} -3.97072 q^{74} +6.62611 q^{75} -0.676346 q^{76} -20.4775 q^{77} -8.65710 q^{78} +5.09822 q^{79} +1.06645 q^{80} -8.82463 q^{81} +2.34182 q^{82} +2.00273 q^{83} -8.32742 q^{84} +5.70086 q^{85} +2.53027 q^{86} +0.720793 q^{87} +4.21829 q^{88} -2.27995 q^{89} +0.0611699 q^{90} -24.4988 q^{91} +0.753052 q^{92} +2.39241 q^{93} -4.58870 q^{94} -0.721286 q^{95} +1.71541 q^{96} +2.01635 q^{97} -16.5659 q^{98} +0.241955 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9}+O(q^{10}) $ 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9	\( 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9} + 15 q^{10} + 13 q^{11} - 5 q^{12} - 43 q^{13} + 28 q^{14} - 10 q^{15} + 57 q^{16} - 50 q^{18} - 6 q^{19} - 15 q^{20} - 23 q^{21} - 13 q^{22} - q^{23} + 5 q^{24} + 20 q^{25} + 43 q^{26} - 20 q^{27} - 28 q^{28} - 4 q^{29} + 10 q^{30} - 34 q^{31} - 57 q^{32} - 43 q^{33} + 26 q^{35} + 50 q^{36} - 64 q^{37} + 6 q^{38} + 8 q^{39} + 15 q^{40} + 27 q^{41} + 23 q^{42} - 29 q^{43} + 13 q^{44} - 76 q^{45} + q^{46} - 25 q^{47} - 5 q^{48} + 7 q^{49} - 20 q^{50} + 27 q^{51} - 43 q^{52} - 34 q^{53} + 20 q^{54} - 36 q^{55} + 28 q^{56} - 33 q^{57} + 4 q^{58} + 19 q^{59} - 10 q^{60} - 58 q^{61} + 34 q^{62} - 65 q^{63} + 57 q^{64} + 17 q^{65} + 43 q^{66} - 84 q^{67} - 33 q^{69} - 26 q^{70} + 22 q^{71} - 50 q^{72} - 82 q^{73} + 64 q^{74} + 8 q^{75} - 6 q^{76} - 41 q^{77} - 8 q^{78} + 8 q^{79} - 15 q^{80} + 25 q^{81} - 27 q^{82} - 23 q^{83} - 23 q^{84} - 58 q^{85} + 29 q^{86} - 17 q^{87} - 13 q^{88} + 18 q^{89} + 76 q^{90} - 4 q^{91} - q^{92} - 60 q^{93} + 25 q^{94} + 36 q^{95} + 5 q^{96} - 156 q^{97} - 7 q^{98} + 25 q^{99}+O(q^{100}) \) 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9 + 15 * q^10 + 13 * q^11 - 5 * q^12 - 43 * q^13 + 28 * q^14 - 10 * q^15 + 57 * q^16 - 50 * q^18 - 6 * q^19 - 15 * q^20 - 23 * q^21 - 13 * q^22 - q^23 + 5 * q^24 + 20 * q^25 + 43 * q^26 - 20 * q^27 - 28 * q^28 - 4 * q^29 + 10 * q^30 - 34 * q^31 - 57 * q^32 - 43 * q^33 + 26 * q^35 + 50 * q^36 - 64 * q^37 + 6 * q^38 + 8 * q^39 + 15 * q^40 + 27 * q^41 + 23 * q^42 - 29 * q^43 + 13 * q^44 - 76 * q^45 + q^46 - 25 * q^47 - 5 * q^48 + 7 * q^49 - 20 * q^50 + 27 * q^51 - 43 * q^52 - 34 * q^53 + 20 * q^54 - 36 * q^55 + 28 * q^56 - 33 * q^57 + 4 * q^58 + 19 * q^59 - 10 * q^60 - 58 * q^61 + 34 * q^62 - 65 * q^63 + 57 * q^64 + 17 * q^65 + 43 * q^66 - 84 * q^67 - 33 * q^69 - 26 * q^70 + 22 * q^71 - 50 * q^72 - 82 * q^73 + 64 * q^74 + 8 * q^75 - 6 * q^76 - 41 * q^77 - 8 * q^78 + 8 * q^79 - 15 * q^80 + 25 * q^81 - 27 * q^82 - 23 * q^83 - 23 * q^84 - 58 * q^85 + 29 * q^86 - 17 * q^87 - 13 * q^88 + 18 * q^89 + 76 * q^90 - 4 * q^91 - q^92 - 60 * q^93 + 25 * q^94 + 36 * q^95 + 5 * q^96 - 156 * q^97 - 7 * q^98 + 25 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.71541	−0.990394	−0.495197	−	0.868781i	$-0.664904\pi$
$3$	−1.71541	−0.990394	−0.495197	+	0.868781i	$0.664904\pi$
$4$	1.00000	0.500000
$5$	1.06645	0.476929	0.238465	−	0.971151i	$-0.423356\pi$
$5$	1.06645	0.476929	0.238465	+	0.971151i	$0.423356\pi$
$6$	1.71541	0.700314
$7$	4.85447	1.83482	0.917408	−	0.397947i	$-0.130277\pi$
$7$	4.85447	1.83482	0.917408	+	0.397947i	$0.130277\pi$
$8$	−1.00000	−0.353553
$9$	−0.0573586	−0.0191195
$10$	−1.06645	−0.337240
$11$	−4.21829	−1.27186	−0.635931	−	0.771746i	$-0.719384\pi$
$11$	−4.21829	−1.27186	−0.635931	+	0.771746i	$0.719384\pi$
$12$	−1.71541	−0.495197
$13$	−5.04666	−1.39969	−0.699845	−	0.714294i	$-0.746748\pi$
$13$	−5.04666	−1.39969	−0.699845	+	0.714294i	$0.746748\pi$
$14$	−4.85447	−1.29741
$15$	−1.82940	−0.472348
$16$	1.00000	0.250000
$17$	5.34566	1.29651	0.648257	−	0.761422i	$-0.275498\pi$
$17$	5.34566	1.29651	0.648257	+	0.761422i	$0.275498\pi$
$18$	0.0573586	0.0135196
$19$	−0.676346	−0.155164	−0.0775822	−	0.996986i	$-0.524720\pi$
$19$	−0.676346	−0.155164	−0.0775822	+	0.996986i	$0.524720\pi$
$20$	1.06645	0.238465
$21$	−8.32742	−1.81719
$22$	4.21829	0.899342
$23$	0.753052	0.157022	0.0785111	−	0.996913i	$-0.474983\pi$
$23$	0.753052	0.157022	0.0785111	+	0.996913i	$0.474983\pi$
$24$	1.71541	0.350157
$25$	−3.86269	−0.772538
$26$	5.04666	0.989731
$27$	5.24463	1.00933
$28$	4.85447	0.917408
$29$	−0.420186	−0.0780267	−0.0390133	−	0.999239i	$-0.512421\pi$
$29$	−0.420186	−0.0780267	−0.0390133	+	0.999239i	$0.512421\pi$
$30$	1.82940	0.334000
$31$	−1.39465	−0.250487	−0.125244	−	0.992126i	$-0.539971\pi$
$31$	−1.39465	−0.250487	−0.125244	+	0.992126i	$0.539971\pi$
$32$	−1.00000	−0.176777
$33$	7.23610	1.25964
$34$	−5.34566	−0.916773
$35$	5.17703	0.875078
$36$	−0.0573586	−0.00955977
$37$	3.97072	0.652782	0.326391	−	0.945235i	$-0.394167\pi$
$37$	3.97072	0.652782	0.326391	+	0.945235i	$0.394167\pi$
$38$	0.676346	0.109718
$39$	8.65710	1.38625
$40$	−1.06645	−0.168620
$41$	−2.34182	−0.365731	−0.182866	−	0.983138i	$-0.558537\pi$
$41$	−2.34182	−0.365731	−0.182866	+	0.983138i	$0.558537\pi$
$42$	8.32742	1.28495
$43$	−2.53027	−0.385863	−0.192932	−	0.981212i	$-0.561800\pi$
$43$	−2.53027	−0.385863	−0.192932	+	0.981212i	$0.561800\pi$
$44$	−4.21829	−0.635931
$45$	−0.0611699	−0.00911867
$46$	−0.753052	−0.111031
$47$	4.58870	0.669331	0.334666	−	0.942337i	$-0.391377\pi$
$47$	4.58870	0.669331	0.334666	+	0.942337i	$0.391377\pi$
$48$	−1.71541	−0.247599
$49$	16.5659	2.36655
$50$	3.86269	0.546267
$51$	−9.17001	−1.28406
$52$	−5.04666	−0.699845
$53$	9.68212	1.32994	0.664971	−	0.746869i	$-0.268444\pi$
$53$	9.68212	1.32994	0.664971	+	0.746869i	$0.268444\pi$
$54$	−5.24463	−0.713704
$55$	−4.49858	−0.606588
$56$	−4.85447	−0.648706
$57$	1.16021	0.153674
$58$	0.420186	0.0551732
$59$	−1.30045	−0.169305	−0.0846523	−	0.996411i	$-0.526978\pi$
$59$	−1.30045	−0.169305	−0.0846523	+	0.996411i	$0.526978\pi$
$60$	−1.82940	−0.236174
$61$	−5.12051	−0.655614	−0.327807	−	0.944745i	$-0.606310\pi$
$61$	−5.12051	−0.655614	−0.327807	+	0.944745i	$0.606310\pi$
$62$	1.39465	0.177121
$63$	−0.278446	−0.0350809
$64$	1.00000	0.125000
$65$	−5.38199	−0.667554
$66$	−7.23610	−0.890703
$67$	−13.3698	−1.63338	−0.816692	−	0.577073i	$-0.804195\pi$
$67$	−13.3698	−1.63338	−0.816692	+	0.577073i	$0.804195\pi$
$68$	5.34566	0.648257
$69$	−1.29180	−0.155514
$70$	−5.17703	−0.618774
$71$	−10.3650	−1.23010	−0.615050	−	0.788488i	$-0.710864\pi$
$71$	−10.3650	−1.23010	−0.615050	+	0.788488i	$0.710864\pi$
$72$	0.0573586	0.00675978
$73$	−2.60917	−0.305380	−0.152690	−	0.988274i	$-0.548794\pi$
$73$	−2.60917	−0.305380	−0.152690	+	0.988274i	$0.548794\pi$
$74$	−3.97072	−0.461587
$75$	6.62611	0.765117
$76$	−0.676346	−0.0775822
$77$	−20.4775	−2.33363
$78$	−8.65710	−0.980224
$79$	5.09822	0.573594	0.286797	−	0.957991i	$-0.407409\pi$
$79$	5.09822	0.573594	0.286797	+	0.957991i	$0.407409\pi$
$80$	1.06645	0.119232
$81$	−8.82463	−0.980515
$82$	2.34182	0.258611
$83$	2.00273	0.219829	0.109914	−	0.993941i	$-0.464942\pi$
$83$	2.00273	0.219829	0.109914	+	0.993941i	$0.464942\pi$
$84$	−8.32742	−0.908596
$85$	5.70086	0.618345
$86$	2.53027	0.272846
$87$	0.720793	0.0772771
$88$	4.21829	0.449671
$89$	−2.27995	−0.241674	−0.120837	−	0.992672i	$-0.538558\pi$
$89$	−2.27995	−0.241674	−0.120837	+	0.992672i	$0.538558\pi$
$90$	0.0611699	0.00644787
$91$	−24.4988	−2.56818
$92$	0.753052	0.0785111
$93$	2.39241	0.248081
$94$	−4.58870	−0.473289
$95$	−0.721286	−0.0740024
$96$	1.71541	0.175079
$97$	2.01635	0.204730	0.102365	−	0.994747i	$-0.467359\pi$
$97$	2.01635	0.204730	0.102365	+	0.994747i	$0.467359\pi$
$98$	−16.5659	−1.67341
$99$	0.241955	0.0243174
$100$	−3.86269	−0.386269
$101$	−12.5909	−1.25284	−0.626419	−	0.779487i	$-0.715480\pi$
$101$	−12.5909	−1.25284	−0.626419	+	0.779487i	$0.715480\pi$
$102$	9.17001	0.907967
$103$	3.46163	0.341085	0.170542	−	0.985350i	$-0.445448\pi$
$103$	3.46163	0.341085	0.170542	+	0.985350i	$0.445448\pi$
$104$	5.04666	0.494865
$105$	−8.88075	−0.866672
$106$	−9.68212	−0.940411
$107$	5.76801	0.557614	0.278807	−	0.960347i	$-0.410061\pi$
$107$	5.76801	0.557614	0.278807	+	0.960347i	$0.410061\pi$
$108$	5.24463	0.504665
$109$	−6.09862	−0.584142	−0.292071	−	0.956397i	$-0.594344\pi$
$109$	−6.09862	−0.584142	−0.292071	+	0.956397i	$0.594344\pi$
$110$	4.49858	0.428923
$111$	−6.81142	−0.646512
$112$	4.85447	0.458704
$113$	6.82556	0.642094	0.321047	−	0.947063i	$-0.395965\pi$
$113$	6.82556	0.642094	0.321047	+	0.947063i	$0.395965\pi$
$114$	−1.16021	−0.108664
$115$	0.803090	0.0748885
$116$	−0.420186	−0.0390133
$117$	0.289469	0.0267614
$118$	1.30045	0.119716
$119$	25.9503	2.37886
$120$	1.82940	0.167000
$121$	6.79395	0.617632
$122$	5.12051	0.463589
$123$	4.01719	0.362218
$124$	−1.39465	−0.125244
$125$	−9.45159	−0.845376
$126$	0.278446	0.0248059
$127$	0.516288	0.0458132	0.0229066	−	0.999738i	$-0.492708\pi$
$127$	0.516288	0.0458132	0.0229066	+	0.999738i	$0.492708\pi$
$128$	−1.00000	−0.0883883
$129$	4.34046	0.382157
$130$	5.38199	0.472032
$131$	−5.82134	−0.508613	−0.254306	−	0.967124i	$-0.581847\pi$
$131$	−5.82134	−0.508613	−0.254306	+	0.967124i	$0.581847\pi$
$132$	7.23610	0.629822
$133$	−3.28330	−0.284698
$134$	13.3698	1.15498
$135$	5.59312	0.481379
$136$	−5.34566	−0.458387
$137$	−15.3616	−1.31243	−0.656214	−	0.754575i	$-0.727843\pi$
$137$	−15.3616	−1.31243	−0.656214	+	0.754575i	$0.727843\pi$
$138$	1.29180	0.109965
$139$	−5.49046	−0.465695	−0.232847	−	0.972513i	$-0.574804\pi$
$139$	−5.49046	−0.465695	−0.232847	+	0.972513i	$0.574804\pi$
$140$	5.17703	0.437539
$141$	−7.87152	−0.662901
$142$	10.3650	0.869811
$143$	21.2882	1.78021
$144$	−0.0573586	−0.00477989
$145$	−0.448106	−0.0372132
$146$	2.60917	0.215937
$147$	−28.4173	−2.34382
$148$	3.97072	0.326391
$149$	−19.3454	−1.58484	−0.792419	−	0.609977i	$-0.791179\pi$
$149$	−19.3454	−1.58484	−0.792419	+	0.609977i	$0.791179\pi$
$150$	−6.62611	−0.541020
$151$	−11.0070	−0.895736	−0.447868	−	0.894100i	$-0.647817\pi$
$151$	−11.0070	−0.895736	−0.447868	+	0.894100i	$0.647817\pi$
$152$	0.676346	0.0548589
$153$	−0.306620	−0.0247887
$154$	20.4775	1.65013
$155$	−1.48732	−0.119465
$156$	8.65710	0.693123
$157$	1.45541	0.116154	0.0580771	−	0.998312i	$-0.481503\pi$
$157$	1.45541	0.116154	0.0580771	+	0.998312i	$0.481503\pi$
$158$	−5.09822	−0.405592
$159$	−16.6088	−1.31717
$160$	−1.06645	−0.0843100
$161$	3.65567	0.288107
$162$	8.82463	0.693329
$163$	−10.5625	−0.827320	−0.413660	−	0.910431i	$-0.635750\pi$
$163$	−10.5625	−0.827320	−0.413660	+	0.910431i	$0.635750\pi$
$164$	−2.34182	−0.182866
$165$	7.71692	0.600761
$166$	−2.00273	−0.155442
$167$	20.2343	1.56578	0.782890	−	0.622161i	$-0.213745\pi$
$167$	20.2343	1.56578	0.782890	+	0.622161i	$0.213745\pi$
$168$	8.32742	0.642474
$169$	12.4687	0.959134
$170$	−5.70086	−0.437236
$171$	0.0387943	0.00296667
$172$	−2.53027	−0.192932
$173$	−6.68271	−0.508077	−0.254038	−	0.967194i	$-0.581759\pi$
$173$	−6.68271	−0.508077	−0.254038	+	0.967194i	$0.581759\pi$
$174$	−0.720793	−0.0546432
$175$	−18.7513	−1.41747
$176$	−4.21829	−0.317965
$177$	2.23081	0.167678
$178$	2.27995	0.170890
$179$	1.20167	0.0898170	0.0449085	−	0.998991i	$-0.485700\pi$
$179$	1.20167	0.0898170	0.0449085	+	0.998991i	$0.485700\pi$
$180$	−0.0611699	−0.00455933
$181$	12.9017	0.958973	0.479486	−	0.877549i	$-0.340823\pi$
$181$	12.9017	0.958973	0.479486	+	0.877549i	$0.340823\pi$
$182$	24.4988	1.81597
$183$	8.78378	0.649316
$184$	−0.753052	−0.0555157
$185$	4.23456	0.311331
$186$	−2.39241	−0.175420
$187$	−22.5495	−1.64899
$188$	4.58870	0.334666
$189$	25.4599	1.85194
$190$	0.721286	0.0523276
$191$	−3.79391	−0.274518	−0.137259	−	0.990535i	$-0.543829\pi$
$191$	−3.79391	−0.274518	−0.137259	+	0.990535i	$0.543829\pi$
$192$	−1.71541	−0.123799
$193$	4.03902	0.290735	0.145368	−	0.989378i	$-0.453564\pi$
$193$	4.03902	0.290735	0.145368	+	0.989378i	$0.453564\pi$
$194$	−2.01635	−0.144766
$195$	9.23233	0.661141
$196$	16.5659	1.18328
$197$	24.0088	1.71056	0.855278	−	0.518169i	$-0.173386\pi$
$197$	24.0088	1.71056	0.855278	+	0.518169i	$0.173386\pi$
$198$	−0.241955	−0.0171950
$199$	−13.6434	−0.967153	−0.483577	−	0.875302i	$-0.660663\pi$
$199$	−13.6434	−0.967153	−0.483577	+	0.875302i	$0.660663\pi$
$200$	3.86269	0.273134
$201$	22.9348	1.61769
$202$	12.5909	0.885890
$203$	−2.03978	−0.143165
$204$	−9.17001	−0.642029
$205$	−2.49743	−0.174428
$206$	−3.46163	−0.241183
$207$	−0.0431940	−0.00300219
$208$	−5.04666	−0.349923
$209$	2.85302	0.197348
$210$	8.88075	0.612830
$211$	17.8142	1.22638	0.613189	−	0.789936i	$-0.289886\pi$
$211$	17.8142	1.22638	0.613189	+	0.789936i	$0.289886\pi$
$212$	9.68212	0.664971
$213$	17.7803	1.21828
$214$	−5.76801	−0.394293
$215$	−2.69840	−0.184029
$216$	−5.24463	−0.356852
$217$	−6.77030	−0.459598
$218$	6.09862	0.413051
$219$	4.47581	0.302447
$220$	−4.49858	−0.303294
$221$	−26.9777	−1.81472
$222$	6.81142	0.457153
$223$	0.669557	0.0448369	0.0224184	−	0.999749i	$-0.492863\pi$
$223$	0.669557	0.0448369	0.0224184	+	0.999749i	$0.492863\pi$
$224$	−4.85447	−0.324353
$225$	0.221559	0.0147706
$226$	−6.82556	−0.454029
$227$	−2.56221	−0.170060	−0.0850301	−	0.996378i	$-0.527099\pi$
$227$	−2.56221	−0.170060	−0.0850301	+	0.996378i	$0.527099\pi$
$228$	1.16021	0.0768369
$229$	−17.6117	−1.16381	−0.581907	−	0.813255i	$-0.697693\pi$
$229$	−17.6117	−1.16381	−0.581907	+	0.813255i	$0.697693\pi$
$230$	−0.803090	−0.0529542
$231$	35.1274	2.31122
$232$	0.420186	0.0275866
$233$	−6.50828	−0.426372	−0.213186	−	0.977012i	$-0.568384\pi$
$233$	−6.50828	−0.426372	−0.213186	+	0.977012i	$0.568384\pi$
$234$	−0.289469	−0.0189232
$235$	4.89361	0.319224
$236$	−1.30045	−0.0846523
$237$	−8.74555	−0.568084
$238$	−25.9503	−1.68211
$239$	−16.0562	−1.03859	−0.519296	−	0.854594i	$-0.673806\pi$
$239$	−16.0562	−1.03859	−0.519296	+	0.854594i	$0.673806\pi$
$240$	−1.82940	−0.118087
$241$	2.88780	0.186020	0.0930099	−	0.995665i	$-0.470351\pi$
$241$	2.88780	0.186020	0.0930099	+	0.995665i	$0.470351\pi$
$242$	−6.79395	−0.436732
$243$	−0.596006	−0.0382338
$244$	−5.12051	−0.327807
$245$	17.6666	1.12868
$246$	−4.01719	−0.256127
$247$	3.41328	0.217182
$248$	1.39465	0.0885606
$249$	−3.43551	−0.217717
$250$	9.45159	0.597771
$251$	20.4942	1.29358	0.646791	−	0.762667i	$-0.276111\pi$
$251$	20.4942	1.29358	0.646791	+	0.762667i	$0.276111\pi$
$252$	−0.278446	−0.0175404
$253$	−3.17659	−0.199711
$254$	−0.516288	−0.0323948
$255$	−9.77933	−0.612405
$256$	1.00000	0.0625000
$257$	27.0589	1.68789	0.843943	−	0.536433i	$-0.180229\pi$
$257$	27.0589	1.68789	0.843943	+	0.536433i	$0.180229\pi$
$258$	−4.34046	−0.270225
$259$	19.2757	1.19774
$260$	−5.38199	−0.333777
$261$	0.0241013	0.00149183
$262$	5.82134	0.359644
$263$	13.8185	0.852083	0.426041	−	0.904704i	$-0.359908\pi$
$263$	13.8185	0.852083	0.426041	+	0.904704i	$0.359908\pi$
$264$	−7.23610	−0.445351
$265$	10.3255	0.634288
$266$	3.28330	0.201312
$267$	3.91106	0.239353
$268$	−13.3698	−0.816692
$269$	12.2615	0.747599	0.373799	−	0.927510i	$-0.378055\pi$
$269$	12.2615	0.747599	0.373799	+	0.927510i	$0.378055\pi$
$270$	−5.59312	−0.340386
$271$	−25.3154	−1.53780	−0.768900	−	0.639368i	$-0.779196\pi$
$271$	−25.3154	−1.53780	−0.768900	+	0.639368i	$0.779196\pi$
$272$	5.34566	0.324128
$273$	42.0256	2.54351
$274$	15.3616	0.928027
$275$	16.2939	0.982562
$276$	−1.29180	−0.0777570
$277$	−20.8507	−1.25280	−0.626400	−	0.779502i	$-0.715472\pi$
$277$	−20.8507	−1.25280	−0.626400	+	0.779502i	$0.715472\pi$
$278$	5.49046	0.329296
$279$	0.0799954	0.00478920
$280$	−5.17703	−0.309387
$281$	10.1921	0.608007	0.304003	−	0.952671i	$-0.401677\pi$
$281$	10.1921	0.608007	0.304003	+	0.952671i	$0.401677\pi$
$282$	7.87152	0.468742
$283$	19.4740	1.15761	0.578805	−	0.815466i	$-0.303519\pi$
$283$	19.4740	1.15761	0.578805	+	0.815466i	$0.303519\pi$
$284$	−10.3650	−0.615050
$285$	1.23730	0.0732916
$286$	−21.2882	−1.25880
$287$	−11.3683	−0.671050
$288$	0.0573586	0.00337989
$289$	11.5761	0.680946
$290$	0.448106	0.0263137
$291$	−3.45888	−0.202763
$292$	−2.60917	−0.152690
$293$	7.93518	0.463578	0.231789	−	0.972766i	$-0.425542\pi$
$293$	7.93518	0.463578	0.231789	+	0.972766i	$0.425542\pi$
$294$	28.4173	1.65733
$295$	−1.38686	−0.0807463
$296$	−3.97072	−0.230793
$297$	−22.1234	−1.28373
$298$	19.3454	1.12065
$299$	−3.80040	−0.219783
$300$	6.62611	0.382559
$301$	−12.2831	−0.707988
$302$	11.0070	0.633381
$303$	21.5985	1.24080
$304$	−0.676346	−0.0387911
$305$	−5.46075	−0.312681
$306$	0.306620	0.0175283
$307$	−30.0031	−1.71236	−0.856182	−	0.516674i	$-0.827170\pi$
$307$	−30.0031	−1.71236	−0.856182	+	0.516674i	$0.827170\pi$
$308$	−20.4775	−1.16682
$309$	−5.93813	−0.337808
$310$	1.48732	0.0844743
$311$	1.96136	0.111219	0.0556093	−	0.998453i	$-0.482290\pi$
$311$	1.96136	0.111219	0.0556093	+	0.998453i	$0.482290\pi$
$312$	−8.65710	−0.490112
$313$	−12.3828	−0.699917	−0.349958	−	0.936765i	$-0.613804\pi$
$313$	−12.3828	−0.699917	−0.349958	+	0.936765i	$0.613804\pi$
$314$	−1.45541	−0.0821335
$315$	−0.296947	−0.0167311
$316$	5.09822	0.286797
$317$	−18.8305	−1.05763	−0.528814	−	0.848738i	$-0.677363\pi$
$317$	−18.8305	−1.05763	−0.528814	+	0.848738i	$0.677363\pi$
$318$	16.6088	0.931377
$319$	1.77247	0.0992391
$320$	1.06645	0.0596162
$321$	−9.89451	−0.552258
$322$	−3.65567	−0.203722
$323$	−3.61551	−0.201173
$324$	−8.82463	−0.490257
$325$	19.4937	1.08131
$326$	10.5625	0.585003
$327$	10.4617	0.578531
$328$	2.34182	0.129306
$329$	22.2757	1.22810
$330$	−7.71692	−0.424802
$331$	−14.2211	−0.781660	−0.390830	−	0.920463i	$-0.627812\pi$
$331$	−14.2211	−0.781660	−0.390830	+	0.920463i	$0.627812\pi$
$332$	2.00273	0.109914
$333$	−0.227755	−0.0124809
$334$	−20.2343	−1.10717
$335$	−14.2582	−0.779009
$336$	−8.32742	−0.454298
$337$	−33.4181	−1.82040	−0.910200	−	0.414168i	$-0.864073\pi$
$337$	−33.4181	−1.82040	−0.910200	+	0.414168i	$0.864073\pi$
$338$	−12.4687	−0.678210
$339$	−11.7086	−0.635926
$340$	5.70086	0.309173
$341$	5.88305	0.318585
$342$	−0.0387943	−0.00209775
$343$	46.4372	2.50737
$344$	2.53027	0.136423
$345$	−1.37763	−0.0741691
$346$	6.68271	0.359265
$347$	−23.5597	−1.26475	−0.632375	−	0.774662i	$-0.717920\pi$
$347$	−23.5597	−1.26475	−0.632375	+	0.774662i	$0.717920\pi$
$348$	0.720793	0.0386386
$349$	−6.89121	−0.368878	−0.184439	−	0.982844i	$-0.559047\pi$
$349$	−6.89121	−0.368878	−0.184439	+	0.982844i	$0.559047\pi$
$350$	18.7513	1.00230
$351$	−26.4679	−1.41275
$352$	4.21829	0.224835
$353$	−0.682518	−0.0363268	−0.0181634	−	0.999835i	$-0.505782\pi$
$353$	−0.682518	−0.0363268	−0.0181634	+	0.999835i	$0.505782\pi$
$354$	−2.23081	−0.118566
$355$	−11.0537	−0.586670
$356$	−2.27995	−0.120837
$357$	−44.5155	−2.35601
$358$	−1.20167	−0.0635102
$359$	30.7878	1.62492	0.812458	−	0.583019i	$-0.198129\pi$
$359$	30.7878	1.62492	0.812458	+	0.583019i	$0.198129\pi$
$360$	0.0611699	0.00322394
$361$	−18.5426	−0.975924
$362$	−12.9017	−0.678096
$363$	−11.6544	−0.611699
$364$	−24.4988	−1.28409
$365$	−2.78254	−0.145645
$366$	−8.78378	−0.459136
$367$	−28.0131	−1.46227	−0.731137	−	0.682231i	$-0.761010\pi$
$367$	−28.0131	−1.46227	−0.731137	+	0.682231i	$0.761010\pi$
$368$	0.753052	0.0392556
$369$	0.134324	0.00699261
$370$	−4.23456	−0.220144
$371$	47.0016	2.44020
$372$	2.39241	0.124041
$373$	9.91937	0.513605	0.256803	−	0.966464i	$-0.417331\pi$
$373$	9.91937	0.513605	0.256803	+	0.966464i	$0.417331\pi$
$374$	22.5495	1.16601
$375$	16.2134	0.837255
$376$	−4.58870	−0.236644
$377$	2.12054	0.109213
$378$	−25.4599	−1.30952
$379$	−10.7029	−0.549772	−0.274886	−	0.961477i	$-0.588640\pi$
$379$	−10.7029	−0.549772	−0.274886	+	0.961477i	$0.588640\pi$
$380$	−0.721286	−0.0370012
$381$	−0.885647	−0.0453731
$382$	3.79391	0.194113
$383$	−14.6152	−0.746803	−0.373401	−	0.927670i	$-0.621809\pi$
$383$	−14.6152	−0.746803	−0.373401	+	0.927670i	$0.621809\pi$
$384$	1.71541	0.0875393
$385$	−21.8382	−1.11298
$386$	−4.03902	−0.205581
$387$	0.145133	0.00737753
$388$	2.01635	0.102365
$389$	−28.0460	−1.42199	−0.710995	−	0.703197i	$-0.751755\pi$
$389$	−28.0460	−1.42199	−0.710995	+	0.703197i	$0.751755\pi$
$390$	−9.23233	−0.467497
$391$	4.02556	0.203581
$392$	−16.5659	−0.836703
$393$	9.98600	0.503727
$394$	−24.0088	−1.20955
$395$	5.43698	0.273564
$396$	0.241955	0.0121587
$397$	−19.2393	−0.965595	−0.482797	−	0.875732i	$-0.660379\pi$
$397$	−19.2393	−0.965595	−0.482797	+	0.875732i	$0.660379\pi$
$398$	13.6434	0.683881
$399$	5.63221	0.281963
$400$	−3.86269	−0.193135
$401$	−30.6769	−1.53193	−0.765965	−	0.642882i	$-0.777738\pi$
$401$	−30.6769	−1.53193	−0.765965	+	0.642882i	$0.777738\pi$
$402$	−22.9348	−1.14388
$403$	7.03834	0.350605
$404$	−12.5909	−0.626419
$405$	−9.41100	−0.467636
$406$	2.03978	0.101233
$407$	−16.7496	−0.830249
$408$	9.17001	0.453983
$409$	2.90316	0.143552	0.0717760	−	0.997421i	$-0.477133\pi$
$409$	2.90316	0.143552	0.0717760	+	0.997421i	$0.477133\pi$
$410$	2.49743	0.123339
$411$	26.3515	1.29982
$412$	3.46163	0.170542
$413$	−6.31301	−0.310643
$414$	0.0431940	0.00212287
$415$	2.13581	0.104843
$416$	5.04666	0.247433
$417$	9.41841	0.461222
$418$	−2.85302	−0.139546
$419$	23.3752	1.14196	0.570978	−	0.820966i	$-0.306564\pi$
$419$	23.3752	1.14196	0.570978	+	0.820966i	$0.306564\pi$
$420$	−8.88075	−0.433336
$421$	−10.5146	−0.512448	−0.256224	−	0.966617i	$-0.582478\pi$
$421$	−10.5146	−0.512448	−0.256224	+	0.966617i	$0.582478\pi$
$422$	−17.8142	−0.867180
$423$	−0.263202	−0.0127973
$424$	−9.68212	−0.470205
$425$	−20.6486	−1.00161
$426$	−17.7803	−0.861456
$427$	−24.8573	−1.20293
$428$	5.76801	0.278807
$429$	−36.5181	−1.76311
$430$	2.69840	0.130128
$431$	28.1981	1.35826	0.679128	−	0.734020i	$-0.262358\pi$
$431$	28.1981	1.35826	0.679128	+	0.734020i	$0.262358\pi$
$432$	5.24463	0.252332
$433$	−17.3077	−0.831757	−0.415879	−	0.909420i	$-0.636526\pi$
$433$	−17.3077	−0.831757	−0.415879	+	0.909420i	$0.636526\pi$
$434$	6.77030	0.324985
$435$	0.768687	0.0368557
$436$	−6.09862	−0.292071
$437$	−0.509324	−0.0243643
$438$	−4.47581	−0.213862
$439$	−29.3119	−1.39898	−0.699490	−	0.714643i	$-0.746589\pi$
$439$	−29.3119	−1.39898	−0.699490	+	0.714643i	$0.746589\pi$
$440$	4.49858	0.214461
$441$	−0.950195	−0.0452474
$442$	26.9777	1.28320
$443$	6.42674	0.305344	0.152672	−	0.988277i	$-0.451212\pi$
$443$	6.42674	0.305344	0.152672	+	0.988277i	$0.451212\pi$
$444$	−6.81142	−0.323256
$445$	−2.43145	−0.115262
$446$	−0.669557	−0.0317045
$447$	33.1854	1.56962
$448$	4.85447	0.229352
$449$	−18.8315	−0.888715	−0.444358	−	0.895849i	$-0.646568\pi$
$449$	−18.8315	−0.888715	−0.444358	+	0.895849i	$0.646568\pi$
$450$	−0.221559	−0.0104444
$451$	9.87848	0.465159
$452$	6.82556	0.321047
$453$	18.8815	0.887132
$454$	2.56221	0.120251
$455$	−26.1267	−1.22484
$456$	−1.16021	−0.0543319
$457$	−15.4594	−0.723162	−0.361581	−	0.932341i	$-0.617763\pi$
$457$	−15.4594	−0.723162	−0.361581	+	0.932341i	$0.617763\pi$
$458$	17.6117	0.822941
$459$	28.0360	1.30861
$460$	0.803090	0.0374443
$461$	−25.4778	−1.18662	−0.593309	−	0.804975i	$-0.702179\pi$
$461$	−25.4778	−1.18662	−0.593309	+	0.804975i	$0.702179\pi$
$462$	−35.1274	−1.63428
$463$	4.48763	0.208558	0.104279	−	0.994548i	$-0.466747\pi$
$463$	4.48763	0.208558	0.104279	+	0.994548i	$0.466747\pi$
$464$	−0.420186	−0.0195067
$465$	2.55137	0.118317
$466$	6.50828	0.301490
$467$	−7.61059	−0.352176	−0.176088	−	0.984374i	$-0.556344\pi$
$467$	−7.61059	−0.352176	−0.176088	+	0.984374i	$0.556344\pi$
$468$	0.289469	0.0133807
$469$	−64.9034	−2.99696
$470$	−4.89361	−0.225725
$471$	−2.49663	−0.115038
$472$	1.30045	0.0598582
$473$	10.6734	0.490764
$474$	8.74555	0.401696
$475$	2.61252	0.119870
$476$	25.9503	1.18943
$477$	−0.555353	−0.0254279
$478$	16.0562	0.734396
$479$	−2.50044	−0.114248	−0.0571240	−	0.998367i	$-0.518193\pi$
$479$	−2.50044	−0.114248	−0.0571240	+	0.998367i	$0.518193\pi$
$480$	1.82940	0.0835001
$481$	−20.0389	−0.913693
$482$	−2.88780	−0.131536
$483$	−6.27098	−0.285340
$484$	6.79395	0.308816
$485$	2.15033	0.0976415
$486$	0.596006	0.0270354
$487$	−5.05372	−0.229006	−0.114503	−	0.993423i	$-0.536528\pi$
$487$	−5.05372	−0.229006	−0.114503	+	0.993423i	$0.536528\pi$
$488$	5.12051	0.231794
$489$	18.1191	0.819373
$490$	−17.6666	−0.798096
$491$	−25.8437	−1.16631	−0.583155	−	0.812361i	$-0.698182\pi$
$491$	−25.8437	−1.16631	−0.583155	+	0.812361i	$0.698182\pi$
$492$	4.01719	0.181109
$493$	−2.24617	−0.101163
$494$	−3.41328	−0.153571
$495$	0.258032	0.0115977
$496$	−1.39465	−0.0626218
$497$	−50.3166	−2.25701
$498$	3.43551	0.153949
$499$	−19.1158	−0.855742	−0.427871	−	0.903840i	$-0.640736\pi$
$499$	−19.1158	−0.855742	−0.427871	+	0.903840i	$0.640736\pi$
$500$	−9.45159	−0.422688
$501$	−34.7102	−1.55074
$502$	−20.4942	−0.914701
$503$	13.1739	0.587395	0.293697	−	0.955898i	$-0.405114\pi$
$503$	13.1739	0.587395	0.293697	+	0.955898i	$0.405114\pi$
$504$	0.278446	0.0124030
$505$	−13.4275	−0.597515
$506$	3.17659	0.141217
$507$	−21.3890	−0.949921
$508$	0.516288	0.0229066
$509$	−11.4676	−0.508292	−0.254146	−	0.967166i	$-0.581794\pi$
$509$	−11.4676	−0.508292	−0.254146	+	0.967166i	$0.581794\pi$
$510$	9.77933	0.433036
$511$	−12.6661	−0.560317
$512$	−1.00000	−0.0441942
$513$	−3.54718	−0.156612
$514$	−27.0589	−1.19352
$515$	3.69164	0.162673
$516$	4.34046	0.191078
$517$	−19.3565	−0.851296
$518$	−19.2757	−0.846927
$519$	11.4636	0.503196
$520$	5.38199	0.236016
$521$	26.1402	1.14522	0.572611	−	0.819827i	$-0.305930\pi$
$521$	26.1402	1.14522	0.572611	+	0.819827i	$0.305930\pi$
$522$	−0.0241013	−0.00105489
$523$	−4.93397	−0.215747	−0.107874	−	0.994165i	$-0.534404\pi$
$523$	−4.93397	−0.215747	−0.107874	+	0.994165i	$0.534404\pi$
$524$	−5.82134	−0.254306
$525$	32.1663	1.40385
$526$	−13.8185	−0.602514
$527$	−7.45534	−0.324760
$528$	7.23610	0.314911
$529$	−22.4329	−0.975344
$530$	−10.3255	−0.448509
$531$	0.0745922	0.00323702
$532$	−3.28330	−0.142349
$533$	11.8184	0.511911
$534$	−3.91106	−0.169248
$535$	6.15127	0.265943
$536$	13.3698	0.577489
$537$	−2.06136	−0.0889543
$538$	−12.2615	−0.528632
$539$	−69.8796	−3.00993
$540$	5.59312	0.240690
$541$	−29.5794	−1.27172	−0.635860	−	0.771805i	$-0.719354\pi$
$541$	−29.5794	−1.27172	−0.635860	+	0.771805i	$0.719354\pi$
$542$	25.3154	1.08739
$543$	−22.1317	−0.949761
$544$	−5.34566	−0.229193
$545$	−6.50386	−0.278595
$546$	−42.0256	−1.79853
$547$	11.7861	0.503936	0.251968	−	0.967736i	$-0.418922\pi$
$547$	11.7861	0.503936	0.251968	+	0.967736i	$0.418922\pi$
$548$	−15.3616	−0.656214
$549$	0.293705	0.0125350
$550$	−16.2939	−0.694776
$551$	0.284191	0.0121070
$552$	1.29180	0.0549825
$553$	24.7491	1.05244
$554$	20.8507	0.885863
$555$	−7.26402	−0.308340
$556$	−5.49046	−0.232847
$557$	−19.8784	−0.842276	−0.421138	−	0.906997i	$-0.638369\pi$
$557$	−19.8784	−0.842276	−0.421138	+	0.906997i	$0.638369\pi$
$558$	−0.0799954	−0.00338648
$559$	12.7694	0.540089
$560$	5.17703	0.218769
$561$	38.6818	1.63315
$562$	−10.1921	−0.429926
$563$	16.5885	0.699123	0.349562	−	0.936913i	$-0.386331\pi$
$563$	16.5885	0.699123	0.349562	+	0.936913i	$0.386331\pi$
$564$	−7.87152	−0.331451
$565$	7.27909	0.306234
$566$	−19.4740	−0.818553
$567$	−42.8389	−1.79907
$568$	10.3650	0.434906
$569$	19.3599	0.811607	0.405804	−	0.913960i	$-0.366992\pi$
$569$	19.3599	0.811607	0.405804	+	0.913960i	$0.366992\pi$
$570$	−1.23730	−0.0518250
$571$	38.8137	1.62430	0.812151	−	0.583447i	$-0.198296\pi$
$571$	38.8137	1.62430	0.812151	+	0.583447i	$0.198296\pi$
$572$	21.2882	0.890106
$573$	6.50812	0.271881
$574$	11.3683	0.474504
$575$	−2.90881	−0.121306
$576$	−0.0573586	−0.00238994
$577$	−13.8479	−0.576494	−0.288247	−	0.957556i	$-0.593072\pi$
$577$	−13.8479	−0.576494	−0.288247	+	0.957556i	$0.593072\pi$
$578$	−11.5761	−0.481502
$579$	−6.92859	−0.287942
$580$	−0.448106	−0.0186066
$581$	9.72220	0.403345
$582$	3.45888	0.143375
$583$	−40.8420	−1.69150
$584$	2.60917	0.107968
$585$	0.308703	0.0127633
$586$	−7.93518	−0.327799
$587$	−22.3606	−0.922921	−0.461460	−	0.887161i	$-0.652674\pi$
$587$	−22.3606	−0.922921	−0.461460	+	0.887161i	$0.652674\pi$
$588$	−28.4173	−1.17191
$589$	0.943268	0.0388667
$590$	1.38686	0.0570963
$591$	−41.1850	−1.69412
$592$	3.97072	0.163196
$593$	25.7391	1.05698	0.528490	−	0.848939i	$-0.322758\pi$
$593$	25.7391	1.05698	0.528490	+	0.848939i	$0.322758\pi$
$594$	22.1234	0.907733
$595$	27.6746	1.13455
$596$	−19.3454	−0.792419
$597$	23.4040	0.957863
$598$	3.80040	0.155410
$599$	22.8937	0.935411	0.467706	−	0.883884i	$-0.345081\pi$
$599$	22.8937	0.935411	0.467706	+	0.883884i	$0.345081\pi$
$600$	−6.62611	−0.270510
$601$	5.89293	0.240378	0.120189	−	0.992751i	$-0.461650\pi$
$601$	5.89293	0.240378	0.120189	+	0.992751i	$0.461650\pi$
$602$	12.2831	0.500623
$603$	0.766875	0.0312296
$604$	−11.0070	−0.447868
$605$	7.24538	0.294567
$606$	−21.5985	−0.877380
$607$	25.1247	1.01978	0.509889	−	0.860240i	$-0.329686\pi$
$607$	25.1247	1.01978	0.509889	+	0.860240i	$0.329686\pi$
$608$	0.676346	0.0274294
$609$	3.49907	0.141789
$610$	5.46075	0.221099
$611$	−23.1576	−0.936856
$612$	−0.306620	−0.0123944
$613$	−41.1397	−1.66162	−0.830809	−	0.556558i	$-0.812122\pi$
$613$	−41.1397	−1.66162	−0.830809	+	0.556558i	$0.812122\pi$
$614$	30.0031	1.21082
$615$	4.28412	0.172752
$616$	20.4775	0.825064
$617$	−15.0568	−0.606165	−0.303082	−	0.952964i	$-0.598016\pi$
$617$	−15.0568	−0.606165	−0.303082	+	0.952964i	$0.598016\pi$
$618$	5.93813	0.238866
$619$	−4.83852	−0.194477	−0.0972383	−	0.995261i	$-0.531001\pi$
$619$	−4.83852	−0.194477	−0.0972383	+	0.995261i	$0.531001\pi$
$620$	−1.48732	−0.0597323
$621$	3.94948	0.158487
$622$	−1.96136	−0.0786434
$623$	−11.0680	−0.443428
$624$	8.65710	0.346561
$625$	9.23385	0.369354
$626$	12.3828	0.494916
$627$	−4.89411	−0.195452
$628$	1.45541	0.0580771
$629$	21.2261	0.846341
$630$	0.296947	0.0118307
$631$	29.4899	1.17398	0.586988	−	0.809596i	$-0.300314\pi$
$631$	29.4899	1.17398	0.586988	+	0.809596i	$0.300314\pi$
$632$	−5.09822	−0.202796
$633$	−30.5586	−1.21460
$634$	18.8305	0.747856
$635$	0.550594	0.0218496
$636$	−16.6088	−0.658583
$637$	−83.6022	−3.31244
$638$	−1.77247	−0.0701726
$639$	0.594522	0.0235189
$640$	−1.06645	−0.0421550
$641$	12.7863	0.505028	0.252514	−	0.967593i	$-0.418743\pi$
$641$	12.7863	0.505028	0.252514	+	0.967593i	$0.418743\pi$
$642$	9.89451	0.390505
$643$	−23.1913	−0.914576	−0.457288	−	0.889319i	$-0.651179\pi$
$643$	−23.1913	−0.914576	−0.457288	+	0.889319i	$0.651179\pi$
$644$	3.65567	0.144054
$645$	4.62887	0.182262
$646$	3.61551	0.142251
$647$	17.9749	0.706667	0.353334	−	0.935497i	$-0.385048\pi$
$647$	17.9749	0.706667	0.353334	+	0.935497i	$0.385048\pi$
$648$	8.82463	0.346664
$649$	5.48568	0.215332
$650$	−19.4937	−0.764605
$651$	11.6139	0.455183
$652$	−10.5625	−0.413660
$653$	19.9684	0.781425	0.390713	−	0.920513i	$-0.372229\pi$
$653$	19.9684	0.781425	0.390713	+	0.920513i	$0.372229\pi$
$654$	−10.4617	−0.409083
$655$	−6.20815	−0.242572
$656$	−2.34182	−0.0914328
$657$	0.149659	0.00583873
$658$	−22.2757	−0.868398
$659$	−6.94174	−0.270412	−0.135206	−	0.990818i	$-0.543170\pi$
$659$	−6.94174	−0.270412	−0.135206	+	0.990818i	$0.543170\pi$
$660$	7.71692	0.300381
$661$	5.56753	0.216552	0.108276	−	0.994121i	$-0.465467\pi$
$661$	5.56753	0.216552	0.108276	+	0.994121i	$0.465467\pi$
$662$	14.2211	0.552717
$663$	46.2779	1.79729
$664$	−2.00273	−0.0777211
$665$	−3.50146	−0.135781
$666$	0.227755	0.00882533
$667$	−0.316422	−0.0122519
$668$	20.2343	0.782890
$669$	−1.14857	−0.0444062
$670$	14.2582	0.550843
$671$	21.5998	0.833850
$672$	8.32742	0.321237
$673$	6.69109	0.257922	0.128961	−	0.991650i	$-0.458836\pi$
$673$	6.69109	0.257922	0.128961	+	0.991650i	$0.458836\pi$
$674$	33.4181	1.28722
$675$	−20.2584	−0.779746
$676$	12.4687	0.479567
$677$	−11.1374	−0.428046	−0.214023	−	0.976829i	$-0.568657\pi$
$677$	−11.1374	−0.428046	−0.214023	+	0.976829i	$0.568657\pi$
$678$	11.7086	0.449668
$679$	9.78832	0.375641
$680$	−5.70086	−0.218618
$681$	4.39526	0.168427
$682$	−5.88305	−0.225274
$683$	−25.5392	−0.977230	−0.488615	−	0.872500i	$-0.662498\pi$
$683$	−25.5392	−0.977230	−0.488615	+	0.872500i	$0.662498\pi$
$684$	0.0387943	0.00148334
$685$	−16.3823	−0.625936
$686$	−46.4372	−1.77298
$687$	30.2114	1.15263
$688$	−2.53027	−0.0964658
$689$	−48.8623	−1.86151
$690$	1.37763	0.0524455
$691$	34.5098	1.31281	0.656407	−	0.754407i	$-0.272076\pi$
$691$	34.5098	1.31281	0.656407	+	0.754407i	$0.272076\pi$
$692$	−6.68271	−0.254038
$693$	1.17456	0.0446180
$694$	23.5597	0.894313
$695$	−5.85528	−0.222104
$696$	−0.720793	−0.0273216
$697$	−12.5186	−0.474175
$698$	6.89121	0.260836
$699$	11.1644	0.422276
$700$	−18.7513	−0.708733
$701$	−29.1673	−1.10163	−0.550817	−	0.834626i	$-0.685684\pi$
$701$	−29.1673	−1.10163	−0.550817	+	0.834626i	$0.685684\pi$
$702$	26.4679	0.998965
$703$	−2.68558	−0.101289
$704$	−4.21829	−0.158983
$705$	−8.39455	−0.316157
$706$	0.682518	0.0256869
$707$	−61.1219	−2.29873
$708$	2.23081	0.0838391
$709$	8.08103	0.303489	0.151745	−	0.988420i	$-0.451511\pi$
$709$	8.08103	0.303489	0.151745	+	0.988420i	$0.451511\pi$
$710$	11.0537	0.414839
$711$	−0.292427	−0.0109669
$712$	2.27995	0.0854448
$713$	−1.05025	−0.0393321
$714$	44.5155	1.66595
$715$	22.7028	0.849036
$716$	1.20167	0.0449085
$717$	27.5431	1.02862
$718$	−30.7878	−1.14899
$719$	−8.48912	−0.316591	−0.158295	−	0.987392i	$-0.550600\pi$
$719$	−8.48912	−0.316591	−0.158295	+	0.987392i	$0.550600\pi$
$720$	−0.0611699	−0.00227967
$721$	16.8044	0.625828
$722$	18.5426	0.690082
$723$	−4.95378	−0.184233
$724$	12.9017	0.479486
$725$	1.62305	0.0602786
$726$	11.6544	0.432536
$727$	23.8775	0.885566	0.442783	−	0.896629i	$-0.353991\pi$
$727$	23.8775	0.885566	0.442783	+	0.896629i	$0.353991\pi$
$728$	24.4988	0.907987
$729$	27.4963	1.01838
$730$	2.78254	0.102987
$731$	−13.5260	−0.500277
$732$	8.78378	0.324658
$733$	−4.68120	−0.172904	−0.0864520	−	0.996256i	$-0.527553\pi$
$733$	−4.68120	−0.172904	−0.0864520	+	0.996256i	$0.527553\pi$
$734$	28.0131	1.03398
$735$	−30.3055	−1.11784
$736$	−0.753052	−0.0277579
$737$	56.3978	2.07744
$738$	−0.134324	−0.00494452
$739$	14.7968	0.544310	0.272155	−	0.962253i	$-0.412264\pi$
$739$	14.7968	0.544310	0.272155	+	0.962253i	$0.412264\pi$
$740$	4.23456	0.155665
$741$	−5.85519	−0.215096
$742$	−47.0016	−1.72548
$743$	28.3088	1.03855	0.519274	−	0.854608i	$-0.326202\pi$
$743$	28.3088	1.03855	0.519274	+	0.854608i	$0.326202\pi$
$744$	−2.39241	−0.0877099
$745$	−20.6309	−0.755856
$746$	−9.91937	−0.363174
$747$	−0.114874	−0.00420302
$748$	−22.5495	−0.824493
$749$	28.0006	1.02312
$750$	−16.2134	−0.592029
$751$	45.7904	1.67092	0.835458	−	0.549555i	$-0.185203\pi$
$751$	45.7904	1.67092	0.835458	+	0.549555i	$0.185203\pi$
$752$	4.58870	0.167333
$753$	−35.1560	−1.28116
$754$	−2.12054	−0.0772254
$755$	−11.7384	−0.427203
$756$	25.4599	0.925968
$757$	−0.164561	−0.00598108	−0.00299054	−	0.999996i	$-0.500952\pi$
$757$	−0.164561	−0.00598108	−0.00299054	+	0.999996i	$0.500952\pi$
$758$	10.7029	0.388748
$759$	5.44916	0.197792
$760$	0.721286	0.0261638
$761$	16.2532	0.589179	0.294590	−	0.955624i	$-0.404817\pi$
$761$	16.2532	0.589179	0.294590	+	0.955624i	$0.404817\pi$
$762$	0.885647	0.0320836
$763$	−29.6056	−1.07179
$764$	−3.79391	−0.137259
$765$	−0.326993	−0.0118225
$766$	14.6152	0.528069
$767$	6.56294	0.236974
$768$	−1.71541	−0.0618996
$769$	−4.46011	−0.160836	−0.0804179	−	0.996761i	$-0.525625\pi$
$769$	−4.46011	−0.160836	−0.0804179	+	0.996761i	$0.525625\pi$
$770$	21.8382	0.786994
$771$	−46.4171	−1.67167
$772$	4.03902	0.145368
$773$	−17.9538	−0.645752	−0.322876	−	0.946441i	$-0.604650\pi$
$773$	−17.9538	−0.645752	−0.322876	+	0.946441i	$0.604650\pi$
$774$	−0.145133	−0.00521670
$775$	5.38712	0.193511
$776$	−2.01635	−0.0723828
$777$	−33.0658	−1.18623
$778$	28.0460	1.00550
$779$	1.58388	0.0567484
$780$	9.23233	0.330571
$781$	43.7225	1.56452
$782$	−4.02556	−0.143954
$783$	−2.20372	−0.0787546
$784$	16.5659	0.591638
$785$	1.55212	0.0553974
$786$	−9.98600	−0.356189
$787$	37.3990	1.33313	0.666565	−	0.745447i	$-0.267764\pi$
$787$	37.3990	1.33313	0.666565	+	0.745447i	$0.267764\pi$
$788$	24.0088	0.855278
$789$	−23.7044	−0.843898
$790$	−5.43698	−0.193439
$791$	33.1344	1.17813
$792$	−0.241955	−0.00859750
$793$	25.8414	0.917657
$794$	19.2393	0.682779
$795$	−17.7124	−0.628195
$796$	−13.6434	−0.483577
$797$	42.6720	1.51152	0.755761	−	0.654848i	$-0.227267\pi$
$797$	42.6720	1.51152	0.755761	+	0.654848i	$0.227267\pi$
$798$	−5.63221	−0.199378
$799$	24.5296	0.867796
$800$	3.86269	0.136567
$801$	0.130775	0.00462070
$802$	30.6769	1.08324
$803$	11.0062	0.388402
$804$	22.9348	0.808847
$805$	3.89857	0.137407
$806$	−7.03834	−0.247915
$807$	−21.0336	−0.740417
$808$	12.5909	0.442945
$809$	−21.4279	−0.753364	−0.376682	−	0.926343i	$-0.622935\pi$
$809$	−21.4279	−0.753364	−0.376682	+	0.926343i	$0.622935\pi$
$810$	9.41100	0.330669
$811$	−43.0765	−1.51262	−0.756311	−	0.654212i	$-0.773000\pi$
$811$	−43.0765	−1.51262	−0.756311	+	0.654212i	$0.773000\pi$
$812$	−2.03978	−0.0715823
$813$	43.4264	1.52303
$814$	16.7496	0.587074
$815$	−11.2644	−0.394573
$816$	−9.17001	−0.321015
$817$	1.71134	0.0598722
$818$	−2.90316	−0.101507
$819$	1.40522	0.0491023
$820$	−2.49743	−0.0872140
$821$	−37.0210	−1.29204	−0.646021	−	0.763319i	$-0.723568\pi$
$821$	−37.0210	−1.29204	−0.646021	+	0.763319i	$0.723568\pi$
$822$	−26.3515	−0.919113
$823$	−9.21511	−0.321219	−0.160609	−	0.987018i	$-0.551346\pi$
$823$	−9.21511	−0.321219	−0.160609	+	0.987018i	$0.551346\pi$
$824$	−3.46163	−0.120592
$825$	−27.9508	−0.973124
$826$	6.31301	0.219658
$827$	32.7345	1.13829	0.569145	−	0.822237i	$-0.307274\pi$
$827$	32.7345	1.13829	0.569145	+	0.822237i	$0.307274\pi$
$828$	−0.0431940	−0.00150110
$829$	−0.407045	−0.0141373	−0.00706863	−	0.999975i	$-0.502250\pi$
$829$	−0.407045	−0.0141373	−0.00706863	+	0.999975i	$0.502250\pi$
$830$	−2.13581	−0.0741350
$831$	35.7676	1.24076
$832$	−5.04666	−0.174961
$833$	88.5555	3.06827
$834$	−9.41841	−0.326133
$835$	21.5788	0.746766
$836$	2.85302	0.0986738
$837$	−7.31445	−0.252824
$838$	−23.3752	−0.807484
$839$	13.4307	0.463678	0.231839	−	0.972754i	$-0.425526\pi$
$839$	13.4307	0.463678	0.231839	+	0.972754i	$0.425526\pi$
$840$	8.88075	0.306415
$841$	−28.8234	−0.993912
$842$	10.5146	0.362355
$843$	−17.4836	−0.602166
$844$	17.8142	0.613189
$845$	13.2972	0.457439
$846$	0.263202	0.00904906
$847$	32.9810	1.13324
$848$	9.68212	0.332485
$849$	−33.4060	−1.14649
$850$	20.6486	0.708242
$851$	2.99016	0.102501
$852$	17.7803	0.609141
$853$	−45.9317	−1.57267	−0.786336	−	0.617800i	$-0.788024\pi$
$853$	−45.9317	−1.57267	−0.786336	+	0.617800i	$0.788024\pi$
$854$	24.8573	0.850601
$855$	0.0413720	0.00141489
$856$	−5.76801	−0.197146
$857$	15.4284	0.527023	0.263512	−	0.964656i	$-0.415119\pi$
$857$	15.4284	0.527023	0.263512	+	0.964656i	$0.415119\pi$
$858$	36.5181	1.24671
$859$	41.1838	1.40517	0.702586	−	0.711598i	$-0.252028\pi$
$859$	41.1838	1.40517	0.702586	+	0.711598i	$0.252028\pi$
$860$	−2.69840	−0.0920147
$861$	19.5013	0.664604
$862$	−28.1981	−0.960432
$863$	54.5160	1.85575	0.927874	−	0.372895i	$-0.121635\pi$
$863$	54.5160	1.85575	0.927874	+	0.372895i	$0.121635\pi$
$864$	−5.24463	−0.178426
$865$	−7.12675	−0.242317
$866$	17.3077	0.588141
$867$	−19.8578	−0.674405
$868$	−6.77030	−0.229799
$869$	−21.5058	−0.729533
$870$	−0.768687	−0.0260609
$871$	67.4729	2.28623
$872$	6.09862	0.206525
$873$	−0.115655	−0.00391433
$874$	0.509324	0.0172281
$875$	−45.8824	−1.55111
$876$	4.47581	0.151224
$877$	−10.0843	−0.340523	−0.170261	−	0.985399i	$-0.554461\pi$
$877$	−10.0843	−0.340523	−0.170261	+	0.985399i	$0.554461\pi$
$878$	29.3119	0.989228
$879$	−13.6121	−0.459125
$880$	−4.49858	−0.151647
$881$	−10.4814	−0.353128	−0.176564	−	0.984289i	$-0.556498\pi$
$881$	−10.4814	−0.353128	−0.176564	+	0.984289i	$0.556498\pi$
$882$	0.950195	0.0319947
$883$	−41.5097	−1.39691	−0.698456	−	0.715653i	$-0.746129\pi$
$883$	−41.5097	−1.39691	−0.698456	+	0.715653i	$0.746129\pi$
$884$	−26.9777	−0.907359
$885$	2.37904	0.0799707
$886$	−6.42674	−0.215911
$887$	55.1685	1.85238	0.926189	−	0.377060i	$-0.123065\pi$
$887$	55.1685	1.85238	0.926189	+	0.377060i	$0.123065\pi$
$888$	6.81142	0.228576
$889$	2.50630	0.0840588
$890$	2.43145	0.0815023
$891$	37.2248	1.24708
$892$	0.669557	0.0224184
$893$	−3.10355	−0.103856
$894$	−33.1854	−1.10989
$895$	1.28152	0.0428364
$896$	−4.85447	−0.162176
$897$	6.51925	0.217671
$898$	18.8315	0.628417
$899$	0.586015	0.0195447
$900$	0.221559	0.00738529
$901$	51.7573	1.72429
$902$	−9.87848	−0.328917
$903$	21.0706	0.701187
$904$	−6.82556	−0.227015
$905$	13.7589	0.457362
$906$	−18.8815	−0.627297
$907$	−30.2264	−1.00365	−0.501826	−	0.864968i	$-0.667338\pi$
$907$	−30.2264	−1.00365	−0.501826	+	0.864968i	$0.667338\pi$
$908$	−2.56221	−0.0850301
$909$	0.722194	0.0239537
$910$	26.1267	0.866092
$911$	−49.5086	−1.64029	−0.820146	−	0.572154i	$-0.806108\pi$
$911$	−49.5086	−1.64029	−0.820146	+	0.572154i	$0.806108\pi$
$912$	1.16021	0.0384185
$913$	−8.44810	−0.279591
$914$	15.4594	0.511353
$915$	9.36744	0.309678
$916$	−17.6117	−0.581907
$917$	−28.2595	−0.933211
$918$	−28.0360	−0.925327
$919$	25.0822	0.827385	0.413692	−	0.910417i	$-0.364239\pi$
$919$	25.0822	0.827385	0.413692	+	0.910417i	$0.364239\pi$
$920$	−0.803090	−0.0264771
$921$	51.4676	1.69592
$922$	25.4778	0.839065
$923$	52.3086	1.72176
$924$	35.1274	1.15561
$925$	−15.3377	−0.504299
$926$	−4.48763	−0.147473
$927$	−0.198554	−0.00652138
$928$	0.420186	0.0137933
$929$	5.47667	0.179684	0.0898419	−	0.995956i	$-0.471364\pi$
$929$	5.47667	0.179684	0.0898419	+	0.995956i	$0.471364\pi$
$930$	−2.55137	−0.0836628
$931$	−11.2043	−0.367205
$932$	−6.50828	−0.213186
$933$	−3.36454	−0.110150
$934$	7.61059	0.249026
$935$	−24.0479	−0.786449
$936$	−0.289469	−0.00946160
$937$	15.4936	0.506153	0.253077	−	0.967446i	$-0.418557\pi$
$937$	15.4936	0.506153	0.253077	+	0.967446i	$0.418557\pi$
$938$	64.9034	2.11917
$939$	21.2416	0.693193
$940$	4.89361	0.159612
$941$	−28.0298	−0.913744	−0.456872	−	0.889532i	$-0.651030\pi$
$941$	−28.0298	−0.913744	−0.456872	+	0.889532i	$0.651030\pi$
$942$	2.49663	0.0813445
$943$	−1.76351	−0.0574279
$944$	−1.30045	−0.0423261
$945$	27.1516	0.883242
$946$	−10.6734	−0.347023
$947$	−19.3105	−0.627508	−0.313754	−	0.949504i	$-0.601587\pi$
$947$	−19.3105	−0.627508	−0.313754	+	0.949504i	$0.601587\pi$
$948$	−8.74555	−0.284042
$949$	13.1676	0.427438
$950$	−2.61252	−0.0847612
$951$	32.3022	1.04747
$952$	−25.9503	−0.841055
$953$	4.18686	0.135626	0.0678129	−	0.997698i	$-0.478398\pi$
$953$	4.18686	0.135626	0.0678129	+	0.997698i	$0.478398\pi$
$954$	0.555353	0.0179802
$955$	−4.04600	−0.130925
$956$	−16.0562	−0.519296
$957$	−3.04051	−0.0982858
$958$	2.50044	0.0807855
$959$	−74.5723	−2.40807
$960$	−1.82940	−0.0590435
$961$	−29.0549	−0.937256
$962$	20.0389	0.646079
$963$	−0.330845	−0.0106613
$964$	2.88780	0.0930099
$965$	4.30740	0.138660
$966$	6.27098	0.201766
$967$	−21.9010	−0.704289	−0.352145	−	0.935946i	$-0.614547\pi$
$967$	−21.9010	−0.704289	−0.352145	+	0.935946i	$0.614547\pi$
$968$	−6.79395	−0.218366
$969$	6.20210	0.199240
$970$	−2.15033	−0.0690430
$971$	−48.3128	−1.55043	−0.775216	−	0.631697i	$-0.782359\pi$
$971$	−48.3128	−1.55043	−0.775216	+	0.631697i	$0.782359\pi$
$972$	−0.596006	−0.0191169
$973$	−26.6533	−0.854465
$974$	5.05372	0.161932
$975$	−33.4397	−1.07093
$976$	−5.12051	−0.163903
$977$	−33.1039	−1.05909	−0.529544	−	0.848283i	$-0.677637\pi$
$977$	−33.1039	−1.05909	−0.529544	+	0.848283i	$0.677637\pi$
$978$	−18.1191	−0.579384
$979$	9.61749	0.307376
$980$	17.6666	0.564339
$981$	0.349809	0.0111685
$982$	25.8437	0.824706
$983$	−49.5933	−1.58178	−0.790890	−	0.611958i	$-0.790382\pi$
$983$	−49.5933	−1.58178	−0.790890	+	0.611958i	$0.790382\pi$
$984$	−4.01719	−0.128063
$985$	25.6041	0.815814
$986$	2.24617	0.0715327
$987$	−38.2120	−1.21630
$988$	3.41328	0.108591
$989$	−1.90543	−0.0605891
$990$	−0.258032	−0.00820080
$991$	4.91020	0.155978	0.0779888	−	0.996954i	$-0.475150\pi$
$991$	4.91020	0.155978	0.0779888	+	0.996954i	$0.475150\pi$
$992$	1.39465	0.0442803
$993$	24.3950	0.774152
$994$	50.3166	1.59594
$995$	−14.5499	−0.461264
$996$	−3.43551	−0.108858
$997$	−45.2065	−1.43170	−0.715851	−	0.698253i	$-0.753961\pi$
$997$	−45.2065	−1.43170	−0.715851	+	0.698253i	$0.753961\pi$
$998$	19.1158	0.605101
$999$	20.8250	0.658873

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.c.1.15	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.c.1.15	✓	57	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.71541 q^{3} +1.00000 q^{4} +1.06645 q^{5} +1.71541 q^{6} +4.85447 q^{7} -1.00000 q^{8} -0.0573586 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.71541 q^{3} +1.00000 q^{4} +1.06645 q^{5} +1.71541 q^{6} +4.85447 q^{7} -1.00000 q^{8} -0.0573586 q^{9} -1.06645 q^{10} -4.21829 q^{11} -1.71541 q^{12} -5.04666 q^{13} -4.85447 q^{14} -1.82940 q^{15} +1.00000 q^{16} +5.34566 q^{17} +0.0573586 q^{18} -0.676346 q^{19} +1.06645 q^{20} -8.32742 q^{21} +4.21829 q^{22} +0.753052 q^{23} +1.71541 q^{24} -3.86269 q^{25} +5.04666 q^{26} +5.24463 q^{27} +4.85447 q^{28} -0.420186 q^{29} +1.82940 q^{30} -1.39465 q^{31} -1.00000 q^{32} +7.23610 q^{33} -5.34566 q^{34} +5.17703 q^{35} -0.0573586 q^{36} +3.97072 q^{37} +0.676346 q^{38} +8.65710 q^{39} -1.06645 q^{40} -2.34182 q^{41} +8.32742 q^{42} -2.53027 q^{43} -4.21829 q^{44} -0.0611699 q^{45} -0.753052 q^{46} +4.58870 q^{47} -1.71541 q^{48} +16.5659 q^{49} +3.86269 q^{50} -9.17001 q^{51} -5.04666 q^{52} +9.68212 q^{53} -5.24463 q^{54} -4.49858 q^{55} -4.85447 q^{56} +1.16021 q^{57} +0.420186 q^{58} -1.30045 q^{59} -1.82940 q^{60} -5.12051 q^{61} +1.39465 q^{62} -0.278446 q^{63} +1.00000 q^{64} -5.38199 q^{65} -7.23610 q^{66} -13.3698 q^{67} +5.34566 q^{68} -1.29180 q^{69} -5.17703 q^{70} -10.3650 q^{71} +0.0573586 q^{72} -2.60917 q^{73} -3.97072 q^{74} +6.62611 q^{75} -0.676346 q^{76} -20.4775 q^{77} -8.65710 q^{78} +5.09822 q^{79} +1.06645 q^{80} -8.82463 q^{81} +2.34182 q^{82} +2.00273 q^{83} -8.32742 q^{84} +5.70086 q^{85} +2.53027 q^{86} +0.720793 q^{87} +4.21829 q^{88} -2.27995 q^{89} +0.0611699 q^{90} -24.4988 q^{91} +0.753052 q^{92} +2.39241 q^{93} -4.58870 q^{94} -0.721286 q^{95} +1.71541 q^{96} +2.01635 q^{97} -16.5659 q^{98} +0.241955 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9}+O(q^{10}) \) 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9	\( 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9} + 15 q^{10} + 13 q^{11} - 5 q^{12} - 43 q^{13} + 28 q^{14} - 10 q^{15} + 57 q^{16} - 50 q^{18} - 6 q^{19} - 15 q^{20} - 23 q^{21} - 13 q^{22} - q^{23} + 5 q^{24} + 20 q^{25} + 43 q^{26} - 20 q^{27} - 28 q^{28} - 4 q^{29} + 10 q^{30} - 34 q^{31} - 57 q^{32} - 43 q^{33} + 26 q^{35} + 50 q^{36} - 64 q^{37} + 6 q^{38} + 8 q^{39} + 15 q^{40} + 27 q^{41} + 23 q^{42} - 29 q^{43} + 13 q^{44} - 76 q^{45} + q^{46} - 25 q^{47} - 5 q^{48} + 7 q^{49} - 20 q^{50} + 27 q^{51} - 43 q^{52} - 34 q^{53} + 20 q^{54} - 36 q^{55} + 28 q^{56} - 33 q^{57} + 4 q^{58} + 19 q^{59} - 10 q^{60} - 58 q^{61} + 34 q^{62} - 65 q^{63} + 57 q^{64} + 17 q^{65} + 43 q^{66} - 84 q^{67} - 33 q^{69} - 26 q^{70} + 22 q^{71} - 50 q^{72} - 82 q^{73} + 64 q^{74} + 8 q^{75} - 6 q^{76} - 41 q^{77} - 8 q^{78} + 8 q^{79} - 15 q^{80} + 25 q^{81} - 27 q^{82} - 23 q^{83} - 23 q^{84} - 58 q^{85} + 29 q^{86} - 17 q^{87} - 13 q^{88} + 18 q^{89} + 76 q^{90} - 4 q^{91} - q^{92} - 60 q^{93} + 25 q^{94} + 36 q^{95} + 5 q^{96} - 156 q^{97} - 7 q^{98} + 25 q^{99}+O(q^{100}) \) 57 * q - 57 * q^2 - 5 * q^3 + 57 * q^4 - 15 * q^5 + 5 * q^6 - 28 * q^7 - 57 * q^8 + 50 * q^9 + 15 * q^10 + 13 * q^11 - 5 * q^12 - 43 * q^13 + 28 * q^14 - 10 * q^15 + 57 * q^16 - 50 * q^18 - 6 * q^19 - 15 * q^20 - 23 * q^21 - 13 * q^22 - q^23 + 5 * q^24 + 20 * q^25 + 43 * q^26 - 20 * q^27 - 28 * q^28 - 4 * q^29 + 10 * q^30 - 34 * q^31 - 57 * q^32 - 43 * q^33 + 26 * q^35 + 50 * q^36 - 64 * q^37 + 6 * q^38 + 8 * q^39 + 15 * q^40 + 27 * q^41 + 23 * q^42 - 29 * q^43 + 13 * q^44 - 76 * q^45 + q^46 - 25 * q^47 - 5 * q^48 + 7 * q^49 - 20 * q^50 + 27 * q^51 - 43 * q^52 - 34 * q^53 + 20 * q^54 - 36 * q^55 + 28 * q^56 - 33 * q^57 + 4 * q^58 + 19 * q^59 - 10 * q^60 - 58 * q^61 + 34 * q^62 - 65 * q^63 + 57 * q^64 + 17 * q^65 + 43 * q^66 - 84 * q^67 - 33 * q^69 - 26 * q^70 + 22 * q^71 - 50 * q^72 - 82 * q^73 + 64 * q^74 + 8 * q^75 - 6 * q^76 - 41 * q^77 - 8 * q^78 + 8 * q^79 - 15 * q^80 + 25 * q^81 - 27 * q^82 - 23 * q^83 - 23 * q^84 - 58 * q^85 + 29 * q^86 - 17 * q^87 - 13 * q^88 + 18 * q^89 + 76 * q^90 - 4 * q^91 - q^92 - 60 * q^93 + 25 * q^94 + 36 * q^95 + 5 * q^96 - 156 * q^97 - 7 * q^98 + 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more