LMFDB - Embedded newform 619.2.a.a.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [619,2,Mod(1,619)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(619, 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("619.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	619.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:	$4.94273988512$
Analytic rank:	$1$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	619.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.37173 q^{2} +0.756788 q^{3} -0.118367 q^{4} -1.92672 q^{5} -1.03811 q^{6} -0.911742 q^{7} +2.90582 q^{8} -2.42727 q^{9} +O(q^{10})$	\(q-1.37173 q^{2} +0.756788 q^{3} -0.118367 q^{4} -1.92672 q^{5} -1.03811 q^{6} -0.911742 q^{7} +2.90582 q^{8} -2.42727 q^{9} +2.64293 q^{10} +6.12907 q^{11} -0.0895789 q^{12} +0.967423 q^{13} +1.25066 q^{14} -1.45812 q^{15} -3.74925 q^{16} +1.00296 q^{17} +3.32955 q^{18} -1.01753 q^{19} +0.228060 q^{20} -0.689995 q^{21} -8.40740 q^{22} -8.98480 q^{23} +2.19909 q^{24} -1.28776 q^{25} -1.32704 q^{26} -4.10729 q^{27} +0.107920 q^{28} -8.45808 q^{29} +2.00014 q^{30} -4.61585 q^{31} -0.668689 q^{32} +4.63840 q^{33} -1.37578 q^{34} +1.75667 q^{35} +0.287310 q^{36} +7.02728 q^{37} +1.39577 q^{38} +0.732134 q^{39} -5.59869 q^{40} -1.86832 q^{41} +0.946484 q^{42} -0.388205 q^{43} -0.725481 q^{44} +4.67667 q^{45} +12.3247 q^{46} -7.67180 q^{47} -2.83739 q^{48} -6.16873 q^{49} +1.76646 q^{50} +0.759024 q^{51} -0.114511 q^{52} -6.32814 q^{53} +5.63408 q^{54} -11.8090 q^{55} -2.64936 q^{56} -0.770055 q^{57} +11.6022 q^{58} -0.806411 q^{59} +0.172593 q^{60} +10.8431 q^{61} +6.33168 q^{62} +2.21305 q^{63} +8.41577 q^{64} -1.86395 q^{65} -6.36262 q^{66} +7.03425 q^{67} -0.118717 q^{68} -6.79959 q^{69} -2.40967 q^{70} -6.02825 q^{71} -7.05322 q^{72} +12.9346 q^{73} -9.63951 q^{74} -0.974562 q^{75} +0.120442 q^{76} -5.58813 q^{77} -1.00429 q^{78} -11.7356 q^{79} +7.22375 q^{80} +4.17347 q^{81} +2.56282 q^{82} -1.04211 q^{83} +0.0816728 q^{84} -1.93241 q^{85} +0.532511 q^{86} -6.40097 q^{87} +17.8100 q^{88} -4.36032 q^{89} -6.41511 q^{90} -0.882040 q^{91} +1.06351 q^{92} -3.49322 q^{93} +10.5236 q^{94} +1.96050 q^{95} -0.506055 q^{96} -16.8355 q^{97} +8.46180 q^{98} -14.8769 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 9 q^{2} - 5 q^{3} + 15 q^{4} - 21 q^{5} - 6 q^{6} - 4 q^{7} - 21 q^{8} + 6 q^{9}+O(q^{10}) $ 21 * q - 9 * q^2 - 5 * q^3 + 15 * q^4 - 21 * q^5 - 6 * q^6 - 4 * q^7 - 21 * q^8 + 6 * q^9	\( 21 q - 9 q^{2} - 5 q^{3} + 15 q^{4} - 21 q^{5} - 6 q^{6} - 4 q^{7} - 21 q^{8} + 6 q^{9} + q^{10} - 27 q^{11} - 8 q^{12} - 11 q^{13} - 19 q^{14} - 10 q^{15} + 11 q^{16} - 14 q^{17} - 14 q^{18} - 15 q^{19} - 25 q^{20} - 42 q^{21} + 12 q^{22} - 14 q^{23} - 8 q^{24} + 16 q^{25} - 11 q^{26} - 5 q^{27} + q^{28} - 78 q^{29} + q^{30} - 8 q^{31} - 41 q^{32} - 6 q^{33} + 7 q^{34} - 3 q^{35} - q^{36} - 23 q^{37} + 21 q^{38} - 4 q^{39} + 12 q^{40} - 59 q^{41} + 39 q^{42} + 2 q^{43} - 50 q^{44} - 36 q^{45} - 15 q^{46} - 12 q^{47} + 10 q^{48} + 17 q^{49} - 23 q^{50} - 8 q^{51} + 18 q^{52} - 36 q^{53} - 4 q^{54} + 23 q^{55} - 28 q^{56} - 24 q^{57} + 46 q^{58} - 17 q^{59} + 8 q^{60} - 22 q^{61} + 42 q^{62} - 6 q^{63} + 49 q^{64} - 53 q^{65} + 29 q^{66} + 15 q^{67} - 16 q^{68} - 30 q^{69} + 44 q^{70} - 56 q^{71} + 12 q^{72} - 2 q^{73} - 12 q^{74} + 2 q^{75} - 4 q^{76} - 47 q^{77} + 36 q^{78} + 5 q^{79} + 15 q^{80} - 19 q^{81} + 47 q^{82} - q^{83} - 20 q^{84} - 29 q^{85} - 23 q^{86} + 44 q^{87} + 61 q^{88} - 12 q^{89} + 91 q^{90} + 5 q^{91} + 35 q^{92} - 15 q^{93} + 34 q^{94} - 17 q^{95} + 14 q^{96} + 21 q^{97} + 24 q^{98} - 38 q^{99}+O(q^{100}) \) 21 * q - 9 * q^2 - 5 * q^3 + 15 * q^4 - 21 * q^5 - 6 * q^6 - 4 * q^7 - 21 * q^8 + 6 * q^9 + q^10 - 27 * q^11 - 8 * q^12 - 11 * q^13 - 19 * q^14 - 10 * q^15 + 11 * q^16 - 14 * q^17 - 14 * q^18 - 15 * q^19 - 25 * q^20 - 42 * q^21 + 12 * q^22 - 14 * q^23 - 8 * q^24 + 16 * q^25 - 11 * q^26 - 5 * q^27 + q^28 - 78 * q^29 + q^30 - 8 * q^31 - 41 * q^32 - 6 * q^33 + 7 * q^34 - 3 * q^35 - q^36 - 23 * q^37 + 21 * q^38 - 4 * q^39 + 12 * q^40 - 59 * q^41 + 39 * q^42 + 2 * q^43 - 50 * q^44 - 36 * q^45 - 15 * q^46 - 12 * q^47 + 10 * q^48 + 17 * q^49 - 23 * q^50 - 8 * q^51 + 18 * q^52 - 36 * q^53 - 4 * q^54 + 23 * q^55 - 28 * q^56 - 24 * q^57 + 46 * q^58 - 17 * q^59 + 8 * q^60 - 22 * q^61 + 42 * q^62 - 6 * q^63 + 49 * q^64 - 53 * q^65 + 29 * q^66 + 15 * q^67 - 16 * q^68 - 30 * q^69 + 44 * q^70 - 56 * q^71 + 12 * q^72 - 2 * q^73 - 12 * q^74 + 2 * q^75 - 4 * q^76 - 47 * q^77 + 36 * q^78 + 5 * q^79 + 15 * q^80 - 19 * q^81 + 47 * q^82 - q^83 - 20 * q^84 - 29 * q^85 - 23 * q^86 + 44 * q^87 + 61 * q^88 - 12 * q^89 + 91 * q^90 + 5 * q^91 + 35 * q^92 - 15 * q^93 + 34 * q^94 - 17 * q^95 + 14 * q^96 + 21 * q^97 + 24 * q^98 - 38 * 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 $

$2$	−1.37173	−0.969957	−0.484978	−	0.874526i	$-0.661173\pi$
$2$	−1.37173	−0.969957	−0.484978	+	0.874526i	$0.661173\pi$
$3$	0.756788	0.436932	0.218466	−	0.975845i	$-0.429895\pi$
$3$	0.756788	0.436932	0.218466	+	0.975845i	$0.429895\pi$
$4$	−0.118367	−0.0591836
$5$	−1.92672	−0.861654	−0.430827	−	0.902435i	$-0.641778\pi$
$5$	−1.92672	−0.861654	−0.430827	+	0.902435i	$0.641778\pi$
$6$	−1.03811	−0.423805
$7$	−0.911742	−0.344606	−0.172303	−	0.985044i	$-0.555121\pi$
$7$	−0.911742	−0.344606	−0.172303	+	0.985044i	$0.555121\pi$
$8$	2.90582	1.02736
$9$	−2.42727	−0.809091
$10$	2.64293	0.835767
$11$	6.12907	1.84798	0.923992	−	0.382413i	$-0.124907\pi$
$11$	6.12907	1.84798	0.923992	+	0.382413i	$0.124907\pi$
$12$	−0.0895789	−0.0258592
$13$	0.967423	0.268315	0.134157	−	0.990960i	$-0.457167\pi$
$13$	0.967423	0.268315	0.134157	+	0.990960i	$0.457167\pi$
$14$	1.25066	0.334253
$15$	−1.45812	−0.376484
$16$	−3.74925	−0.937314
$17$	1.00296	0.243252	0.121626	−	0.992576i	$-0.461189\pi$
$17$	1.00296	0.243252	0.121626	+	0.992576i	$0.461189\pi$
$18$	3.32955	0.784783
$19$	−1.01753	−0.233438	−0.116719	−	0.993165i	$-0.537238\pi$
$19$	−1.01753	−0.233438	−0.116719	+	0.993165i	$0.537238\pi$
$20$	0.228060	0.0509958
$21$	−0.689995	−0.150569
$22$	−8.40740	−1.79246
$23$	−8.98480	−1.87346	−0.936730	−	0.350053i	$-0.886164\pi$
$23$	−8.98480	−1.87346	−0.936730	+	0.350053i	$0.886164\pi$
$24$	2.19909	0.448887
$25$	−1.28776	−0.257552
$26$	−1.32704	−0.260254
$27$	−4.10729	−0.790449
$28$	0.107920	0.0203950
$29$	−8.45808	−1.57063	−0.785313	−	0.619098i	$-0.787498\pi$
$29$	−8.45808	−1.57063	−0.785313	+	0.619098i	$0.787498\pi$
$30$	2.00014	0.365173
$31$	−4.61585	−0.829030	−0.414515	−	0.910042i	$-0.636049\pi$
$31$	−4.61585	−0.829030	−0.414515	+	0.910042i	$0.636049\pi$
$32$	−0.668689	−0.118209
$33$	4.63840	0.807442
$34$	−1.37578	−0.235944
$35$	1.75667	0.296931
$36$	0.287310	0.0478849
$37$	7.02728	1.15528	0.577639	−	0.816292i	$-0.303974\pi$
$37$	7.02728	1.15528	0.577639	+	0.816292i	$0.303974\pi$
$38$	1.39577	0.226425
$39$	0.732134	0.117235
$40$	−5.59869	−0.885231
$41$	−1.86832	−0.291782	−0.145891	−	0.989301i	$-0.546605\pi$
$41$	−1.86832	−0.291782	−0.145891	+	0.989301i	$0.546605\pi$
$42$	0.946484	0.146046
$43$	−0.388205	−0.0592007	−0.0296003	−	0.999562i	$-0.509423\pi$
$43$	−0.388205	−0.0592007	−0.0296003	+	0.999562i	$0.509423\pi$
$44$	−0.725481	−0.109370
$45$	4.67667	0.697156
$46$	12.3247	1.81718
$47$	−7.67180	−1.11905	−0.559523	−	0.828815i	$-0.689016\pi$
$47$	−7.67180	−1.11905	−0.559523	+	0.828815i	$0.689016\pi$
$48$	−2.83739	−0.409542
$49$	−6.16873	−0.881247
$50$	1.76646	0.249815
$51$	0.759024	0.106285
$52$	−0.114511	−0.0158798
$53$	−6.32814	−0.869236	−0.434618	−	0.900615i	$-0.643117\pi$
$53$	−6.32814	−0.869236	−0.434618	+	0.900615i	$0.643117\pi$
$54$	5.63408	0.766701
$55$	−11.8090	−1.59232
$56$	−2.64936	−0.354035
$57$	−0.770055	−0.101996
$58$	11.6022	1.52344
$59$	−0.806411	−0.104986	−0.0524929	−	0.998621i	$-0.516717\pi$
$59$	−0.806411	−0.104986	−0.0524929	+	0.998621i	$0.516717\pi$
$60$	0.172593	0.0222817
$61$	10.8431	1.38831	0.694156	−	0.719824i	$-0.255778\pi$
$61$	10.8431	1.38831	0.694156	+	0.719824i	$0.255778\pi$
$62$	6.33168	0.804124
$63$	2.21305	0.278818
$64$	8.41577	1.05197
$65$	−1.86395	−0.231195
$66$	−6.36262	−0.783184
$67$	7.03425	0.859371	0.429685	−	0.902979i	$-0.358624\pi$
$67$	7.03425	0.859371	0.429685	+	0.902979i	$0.358624\pi$
$68$	−0.118717	−0.0143966
$69$	−6.79959	−0.818574
$70$	−2.40967	−0.288010
$71$	−6.02825	−0.715422	−0.357711	−	0.933832i	$-0.616443\pi$
$71$	−6.02825	−0.715422	−0.357711	+	0.933832i	$0.616443\pi$
$72$	−7.05322	−0.831229
$73$	12.9346	1.51388	0.756938	−	0.653486i	$-0.226694\pi$
$73$	12.9346	1.51388	0.756938	+	0.653486i	$0.226694\pi$
$74$	−9.63951	−1.12057
$75$	−0.974562	−0.112533
$76$	0.120442	0.0138157
$77$	−5.58813	−0.636826
$78$	−1.00429	−0.113713
$79$	−11.7356	−1.32036	−0.660179	−	0.751108i	$-0.729520\pi$
$79$	−11.7356	−1.32036	−0.660179	+	0.751108i	$0.729520\pi$
$80$	7.22375	0.807640
$81$	4.17347	0.463719
$82$	2.56282	0.283016
$83$	−1.04211	−0.114387	−0.0571934	−	0.998363i	$-0.518215\pi$
$83$	−1.04211	−0.114387	−0.0571934	+	0.998363i	$0.518215\pi$
$84$	0.0816728	0.00891123
$85$	−1.93241	−0.209599
$86$	0.532511	0.0574221
$87$	−6.40097	−0.686257
$88$	17.8100	1.89855
$89$	−4.36032	−0.462193	−0.231096	−	0.972931i	$-0.574231\pi$
$89$	−4.36032	−0.462193	−0.231096	+	0.972931i	$0.574231\pi$
$90$	−6.41511	−0.676212
$91$	−0.882040	−0.0924629
$92$	1.06351	0.110878
$93$	−3.49322	−0.362230
$94$	10.5236	1.08543
$95$	1.96050	0.201143
$96$	−0.506055	−0.0516491
$97$	−16.8355	−1.70938	−0.854691	−	0.519137i	$-0.826253\pi$
$97$	−16.8355	−1.70938	−0.854691	+	0.519137i	$0.826253\pi$
$98$	8.46180	0.854771
$99$	−14.8769	−1.49519
$100$	0.152429	0.0152429
$101$	−16.0473	−1.59676	−0.798382	−	0.602151i	$-0.794311\pi$
$101$	−16.0473	−1.59676	−0.798382	+	0.602151i	$0.794311\pi$
$102$	−1.04117	−0.103092
$103$	−12.5673	−1.23829	−0.619146	−	0.785276i	$-0.712521\pi$
$103$	−12.5673	−1.23829	−0.619146	+	0.785276i	$0.712521\pi$
$104$	2.81116	0.275657
$105$	1.32943	0.129739
$106$	8.68047	0.843122
$107$	−4.69419	−0.453804	−0.226902	−	0.973918i	$-0.572860\pi$
$107$	−4.69419	−0.453804	−0.226902	+	0.973918i	$0.572860\pi$
$108$	0.486169	0.0467816
$109$	13.0403	1.24903	0.624516	−	0.781012i	$-0.285296\pi$
$109$	13.0403	1.24903	0.624516	+	0.781012i	$0.285296\pi$
$110$	16.1987	1.54448
$111$	5.31816	0.504777
$112$	3.41835	0.323004
$113$	−10.0328	−0.943810	−0.471905	−	0.881650i	$-0.656433\pi$
$113$	−10.0328	−0.943810	−0.471905	+	0.881650i	$0.656433\pi$
$114$	1.05631	0.0989321
$115$	17.3112	1.61427
$116$	1.00116	0.0929554
$117$	−2.34820	−0.217091
$118$	1.10618	0.101832
$119$	−0.914436	−0.0838262
$120$	−4.23702	−0.386785
$121$	26.5655	2.41504
$122$	−14.8737	−1.34660
$123$	−1.41392	−0.127489
$124$	0.546365	0.0490650
$125$	12.1147	1.08358
$126$	−3.03569	−0.270441
$127$	13.5377	1.20128	0.600640	−	0.799520i	$-0.294913\pi$
$127$	13.5377	1.20128	0.600640	+	0.799520i	$0.294913\pi$
$128$	−10.2068	−0.902158
$129$	−0.293789	−0.0258667
$130$	2.55683	0.224249
$131$	16.6499	1.45471	0.727353	−	0.686263i	$-0.240750\pi$
$131$	16.6499	1.45471	0.727353	+	0.686263i	$0.240750\pi$
$132$	−0.549035	−0.0477874
$133$	0.927726	0.0804441
$134$	−9.64907	−0.833552
$135$	7.91359	0.681094
$136$	2.91441	0.249908
$137$	−4.22733	−0.361165	−0.180582	−	0.983560i	$-0.557798\pi$
$137$	−4.22733	−0.361165	−0.180582	+	0.983560i	$0.557798\pi$
$138$	9.32717	0.793981
$139$	−8.17614	−0.693491	−0.346746	−	0.937959i	$-0.612713\pi$
$139$	−8.17614	−0.693491	−0.346746	+	0.937959i	$0.612713\pi$
$140$	−0.207932	−0.0175735
$141$	−5.80592	−0.488947
$142$	8.26911	0.693928
$143$	5.92940	0.495841
$144$	9.10046	0.758372
$145$	16.2963	1.35334
$146$	−17.7427	−1.46840
$147$	−4.66842	−0.385045
$148$	−0.831800	−0.0683735
$149$	11.7037	0.958807	0.479404	−	0.877595i	$-0.340853\pi$
$149$	11.7037	0.958807	0.479404	+	0.877595i	$0.340853\pi$
$150$	1.33683	0.109152
$151$	1.98896	0.161859	0.0809295	−	0.996720i	$-0.474211\pi$
$151$	1.98896	0.161859	0.0809295	+	0.996720i	$0.474211\pi$
$152$	−2.95676	−0.239825
$153$	−2.43445	−0.196813
$154$	7.66538	0.617694
$155$	8.89343	0.714337
$156$	−0.0866607	−0.00693841
$157$	−10.1942	−0.813583	−0.406791	−	0.913521i	$-0.633352\pi$
$157$	−10.1942	−0.813583	−0.406791	+	0.913521i	$0.633352\pi$
$158$	16.0980	1.28069
$159$	−4.78906	−0.379797
$160$	1.28837	0.101855
$161$	8.19182	0.645606
$162$	−5.72485	−0.449787
$163$	15.6625	1.22678	0.613391	−	0.789779i	$-0.289805\pi$
$163$	15.6625	1.22678	0.613391	+	0.789779i	$0.289805\pi$
$164$	0.221148	0.0172687
$165$	−8.93689	−0.695736
$166$	1.42949	0.110950
$167$	10.0022	0.773996	0.386998	−	0.922081i	$-0.373512\pi$
$167$	10.0022	0.773996	0.386998	+	0.922081i	$0.373512\pi$
$168$	−2.00500	−0.154689
$169$	−12.0641	−0.928007
$170$	2.65074	0.203302
$171$	2.46983	0.188872
$172$	0.0459507	0.00350371
$173$	7.85019	0.596839	0.298420	−	0.954435i	$-0.403541\pi$
$173$	7.85019	0.596839	0.298420	+	0.954435i	$0.403541\pi$
$174$	8.78038	0.665639
$175$	1.17411	0.0887541
$176$	−22.9794	−1.73214
$177$	−0.610282	−0.0458716
$178$	5.98116	0.448307
$179$	5.77928	0.431964	0.215982	−	0.976397i	$-0.430705\pi$
$179$	5.77928	0.431964	0.215982	+	0.976397i	$0.430705\pi$
$180$	−0.553564	−0.0412602
$181$	−12.6243	−0.938354	−0.469177	−	0.883104i	$-0.655449\pi$
$181$	−12.6243	−0.938354	−0.469177	+	0.883104i	$0.655449\pi$
$182$	1.20992	0.0896851
$183$	8.20590	0.606598
$184$	−26.1082	−1.92472
$185$	−13.5396	−0.995450
$186$	4.79174	0.351347
$187$	6.14718	0.449526
$188$	0.908089	0.0662292
$189$	3.74479	0.272393
$190$	−2.68926	−0.195100
$191$	−14.0558	−1.01704	−0.508519	−	0.861051i	$-0.669807\pi$
$191$	−14.0558	−1.01704	−0.508519	+	0.861051i	$0.669807\pi$
$192$	6.36895	0.459639
$193$	12.0945	0.870579	0.435290	−	0.900291i	$-0.356646\pi$
$193$	12.0945	0.870579	0.435290	+	0.900291i	$0.356646\pi$
$194$	23.0936	1.65803
$195$	−1.41062	−0.101016
$196$	0.730175	0.0521554
$197$	3.77688	0.269092	0.134546	−	0.990907i	$-0.457042\pi$
$197$	3.77688	0.269092	0.134546	+	0.990907i	$0.457042\pi$
$198$	20.4071	1.45027
$199$	3.69337	0.261816	0.130908	−	0.991395i	$-0.458211\pi$
$199$	3.69337	0.261816	0.130908	+	0.991395i	$0.458211\pi$
$200$	−3.74200	−0.264600
$201$	5.32344	0.375486
$202$	22.0125	1.54879
$203$	7.71159	0.541247
$204$	−0.0898436	−0.00629031
$205$	3.59972	0.251415
$206$	17.2389	1.20109
$207$	21.8086	1.51580
$208$	−3.62712	−0.251495
$209$	−6.23652	−0.431389
$210$	−1.82361	−0.125841
$211$	−12.4484	−0.856984	−0.428492	−	0.903546i	$-0.640955\pi$
$211$	−12.4484	−0.856984	−0.428492	+	0.903546i	$0.640955\pi$
$212$	0.749044	0.0514445
$213$	−4.56211	−0.312590
$214$	6.43914	0.440171
$215$	0.747961	0.0510105
$216$	−11.9351	−0.812078
$217$	4.20846	0.285689
$218$	−17.8877	−1.21151
$219$	9.78872	0.661461
$220$	1.39780	0.0942394
$221$	0.970282	0.0652682
$222$	−7.29506	−0.489612
$223$	9.86730	0.660764	0.330382	−	0.943847i	$-0.392822\pi$
$223$	9.86730	0.660764	0.330382	+	0.943847i	$0.392822\pi$
$224$	0.609671	0.0407354
$225$	3.12575	0.208383
$226$	13.7623	0.915455
$227$	8.05983	0.534950	0.267475	−	0.963565i	$-0.413811\pi$
$227$	8.05983	0.534950	0.267475	+	0.963565i	$0.413811\pi$
$228$	0.0911493	0.00603651
$229$	−23.9920	−1.58544	−0.792718	−	0.609589i	$-0.791335\pi$
$229$	−23.9920	−1.58544	−0.792718	+	0.609589i	$0.791335\pi$
$230$	−23.7462	−1.56578
$231$	−4.22903	−0.278249
$232$	−24.5777	−1.61360
$233$	−7.60333	−0.498111	−0.249055	−	0.968489i	$-0.580120\pi$
$233$	−7.60333	−0.498111	−0.249055	+	0.968489i	$0.580120\pi$
$234$	3.22109	0.210569
$235$	14.7814	0.964231
$236$	0.0954527	0.00621344
$237$	−8.88136	−0.576906
$238$	1.25436	0.0813078
$239$	−19.1600	−1.23936	−0.619679	−	0.784855i	$-0.712737\pi$
$239$	−19.1600	−1.23936	−0.619679	+	0.784855i	$0.712737\pi$
$240$	5.46685	0.352883
$241$	−2.15446	−0.138781	−0.0693905	−	0.997590i	$-0.522105\pi$
$241$	−2.15446	−0.138781	−0.0693905	+	0.997590i	$0.522105\pi$
$242$	−36.4405	−2.34249
$243$	15.4803	0.993062
$244$	−1.28346	−0.0821653
$245$	11.8854	0.759330
$246$	1.93951	0.123659
$247$	−0.984384	−0.0626348
$248$	−13.4128	−0.851715
$249$	−0.788659	−0.0499792
$250$	−16.6181	−1.05102
$251$	11.9155	0.752100	0.376050	−	0.926599i	$-0.377282\pi$
$251$	11.9155	0.752100	0.376050	+	0.926599i	$0.377282\pi$
$252$	−0.261952	−0.0165014
$253$	−55.0684	−3.46212
$254$	−18.5701	−1.16519
$255$	−1.46242	−0.0915806
$256$	−2.83067	−0.176917
$257$	−7.31804	−0.456487	−0.228243	−	0.973604i	$-0.573298\pi$
$257$	−7.31804	−0.456487	−0.228243	+	0.973604i	$0.573298\pi$
$258$	0.402998	0.0250895
$259$	−6.40707	−0.398116
$260$	0.220631	0.0136829
$261$	20.5301	1.27078
$262$	−22.8391	−1.41100
$263$	−2.72161	−0.167822	−0.0839109	−	0.996473i	$-0.526741\pi$
$263$	−2.72161	−0.167822	−0.0839109	+	0.996473i	$0.526741\pi$
$264$	13.4784	0.829536
$265$	12.1925	0.748981
$266$	−1.27259	−0.0780273
$267$	−3.29983	−0.201947
$268$	−0.832625	−0.0508607
$269$	−29.2998	−1.78644	−0.893220	−	0.449620i	$-0.851559\pi$
$269$	−29.2998	−1.78644	−0.893220	+	0.449620i	$0.851559\pi$
$270$	−10.8553	−0.660631
$271$	−11.5040	−0.698817	−0.349409	−	0.936970i	$-0.613618\pi$
$271$	−11.5040	−0.698817	−0.349409	+	0.936970i	$0.613618\pi$
$272$	−3.76033	−0.228004
$273$	−0.667517	−0.0404000
$274$	5.79874	0.350314
$275$	−7.89278	−0.475952
$276$	0.804848	0.0484462
$277$	21.6245	1.29929	0.649644	−	0.760238i	$-0.274918\pi$
$277$	21.6245	1.29929	0.649644	+	0.760238i	$0.274918\pi$
$278$	11.2154	0.672656
$279$	11.2039	0.670761
$280$	5.10456	0.305056
$281$	−3.44413	−0.205459	−0.102730	−	0.994709i	$-0.532758\pi$
$281$	−3.44413	−0.205459	−0.102730	+	0.994709i	$0.532758\pi$
$282$	7.96413	0.474257
$283$	−4.15039	−0.246715	−0.123357	−	0.992362i	$-0.539366\pi$
$283$	−4.15039	−0.246715	−0.123357	+	0.992362i	$0.539366\pi$
$284$	0.713547	0.0423412
$285$	1.48368	0.0878856
$286$	−8.13352	−0.480945
$287$	1.70342	0.100550
$288$	1.62309	0.0956415
$289$	−15.9941	−0.940828
$290$	−22.3541	−1.31268
$291$	−12.7409	−0.746883
$292$	−1.53103	−0.0895967
$293$	20.8692	1.21919	0.609597	−	0.792712i	$-0.291331\pi$
$293$	20.8692	1.21919	0.609597	+	0.792712i	$0.291331\pi$
$294$	6.40379	0.373477
$295$	1.55373	0.0904614
$296$	20.4200	1.18689
$297$	−25.1739	−1.46074
$298$	−16.0543	−0.930002
$299$	−8.69210	−0.502677
$300$	0.115356	0.00666010
$301$	0.353943	0.0204009
$302$	−2.72830	−0.156996
$303$	−12.1444	−0.697677
$304$	3.81499	0.218804
$305$	−20.8915	−1.19624
$306$	3.33939	0.190900
$307$	32.0912	1.83154	0.915772	−	0.401699i	$-0.131580\pi$
$307$	32.0912	1.83154	0.915772	+	0.401699i	$0.131580\pi$
$308$	0.661451	0.0376897
$309$	−9.51077	−0.541049
$310$	−12.1993	−0.692876
$311$	26.5425	1.50509	0.752543	−	0.658543i	$-0.228827\pi$
$311$	26.5425	1.50509	0.752543	+	0.658543i	$0.228827\pi$
$312$	2.12745	0.120443
$313$	29.0898	1.64425	0.822126	−	0.569306i	$-0.192788\pi$
$313$	29.0898	1.64425	0.822126	+	0.569306i	$0.192788\pi$
$314$	13.9836	0.789140
$315$	−4.26391	−0.240244
$316$	1.38911	0.0781436
$317$	−9.36707	−0.526107	−0.263054	−	0.964781i	$-0.584730\pi$
$317$	−9.36707	−0.526107	−0.263054	+	0.964781i	$0.584730\pi$
$318$	6.56927	0.368387
$319$	−51.8402	−2.90249
$320$	−16.2148	−0.906435
$321$	−3.55250	−0.198281
$322$	−11.2369	−0.626210
$323$	−1.02054	−0.0567843
$324$	−0.494002	−0.0274445
$325$	−1.24581	−0.0691051
$326$	−21.4847	−1.18993
$327$	9.86873	0.545742
$328$	−5.42900	−0.299766
$329$	6.99470	0.385630
$330$	12.2590	0.674834
$331$	21.2872	1.17005	0.585027	−	0.811014i	$-0.301084\pi$
$331$	21.2872	1.17005	0.585027	+	0.811014i	$0.301084\pi$
$332$	0.123352	0.00676983
$333$	−17.0571	−0.934725
$334$	−13.7203	−0.750743
$335$	−13.5530	−0.740480
$336$	2.58697	0.141131
$337$	3.75023	0.204288	0.102144	−	0.994770i	$-0.467430\pi$
$337$	3.75023	0.204288	0.102144	+	0.994770i	$0.467430\pi$
$338$	16.5486	0.900127
$339$	−7.59273	−0.412380
$340$	0.228734	0.0124048
$341$	−28.2908	−1.53203
$342$	−3.38793	−0.183198
$343$	12.0065	0.648289
$344$	−1.12805	−0.0608206
$345$	13.1009	0.705327
$346$	−10.7683	−0.578908
$347$	−16.2263	−0.871076	−0.435538	−	0.900170i	$-0.643442\pi$
$347$	−16.2263	−0.871076	−0.435538	+	0.900170i	$0.643442\pi$
$348$	0.757666	0.0406151
$349$	22.4490	1.20167	0.600835	−	0.799373i	$-0.294835\pi$
$349$	22.4490	1.20167	0.600835	+	0.799373i	$0.294835\pi$
$350$	−1.61055	−0.0860876
$351$	−3.97349	−0.212089
$352$	−4.09844	−0.218447
$353$	−14.7302	−0.784008	−0.392004	−	0.919964i	$-0.628218\pi$
$353$	−14.7302	−0.784008	−0.392004	+	0.919964i	$0.628218\pi$
$354$	0.837140	0.0444935
$355$	11.6147	0.616446
$356$	0.516119	0.0273542
$357$	−0.692034	−0.0366263
$358$	−7.92759	−0.418986
$359$	12.8152	0.676358	0.338179	−	0.941082i	$-0.390189\pi$
$359$	12.8152	0.676358	0.338179	+	0.941082i	$0.390189\pi$
$360$	13.5896	0.716232
$361$	−17.9646	−0.945507
$362$	17.3170	0.910163
$363$	20.1044	1.05521
$364$	0.104405	0.00547229
$365$	−24.9213	−1.30444
$366$	−11.2562	−0.588373
$367$	19.2763	1.00621	0.503106	−	0.864225i	$-0.332190\pi$
$367$	19.2763	1.00621	0.503106	+	0.864225i	$0.332190\pi$
$368$	33.6863	1.75602
$369$	4.53492	0.236078
$370$	18.5726	0.965543
$371$	5.76963	0.299544
$372$	0.413482	0.0214381
$373$	−24.7132	−1.27960	−0.639801	−	0.768540i	$-0.720983\pi$
$373$	−24.7132	−1.27960	−0.639801	+	0.768540i	$0.720983\pi$
$374$	−8.43225	−0.436021
$375$	9.16828	0.473448
$376$	−22.2929	−1.14967
$377$	−8.18255	−0.421423
$378$	−5.13683	−0.264210
$379$	−25.0147	−1.28492	−0.642459	−	0.766320i	$-0.722086\pi$
$379$	−25.0147	−1.28492	−0.642459	+	0.766320i	$0.722086\pi$
$380$	−0.232058	−0.0119043
$381$	10.2452	0.524877
$382$	19.2806	0.986484
$383$	−16.4915	−0.842678	−0.421339	−	0.906903i	$-0.638440\pi$
$383$	−16.4915	−0.842678	−0.421339	+	0.906903i	$0.638440\pi$
$384$	−7.72434	−0.394181
$385$	10.7667	0.548724
$386$	−16.5903	−0.844424
$387$	0.942279	0.0478987
$388$	1.99277	0.101167
$389$	22.1223	1.12164	0.560822	−	0.827937i	$-0.310485\pi$
$389$	22.1223	1.12164	0.560822	+	0.827937i	$0.310485\pi$
$390$	1.93498	0.0979814
$391$	−9.01135	−0.455724
$392$	−17.9252	−0.905360
$393$	12.6004	0.635607
$394$	−5.18085	−0.261007
$395$	22.6112	1.13769
$396$	1.76094	0.0884905
$397$	19.9481	1.00116	0.500582	−	0.865689i	$-0.333119\pi$
$397$	19.9481	1.00116	0.500582	+	0.865689i	$0.333119\pi$
$398$	−5.06630	−0.253951
$399$	0.702092	0.0351486
$400$	4.82815	0.241407
$401$	−3.49462	−0.174513	−0.0872564	−	0.996186i	$-0.527810\pi$
$401$	−3.49462	−0.174513	−0.0872564	+	0.996186i	$0.527810\pi$
$402$	−7.30230	−0.364205
$403$	−4.46548	−0.222441
$404$	1.89947	0.0945023
$405$	−8.04109	−0.399565
$406$	−10.5782	−0.524987
$407$	43.0707	2.13493
$408$	2.20559	0.109193
$409$	16.7498	0.828224	0.414112	−	0.910226i	$-0.364092\pi$
$409$	16.7498	0.828224	0.414112	+	0.910226i	$0.364092\pi$
$410$	−4.93783	−0.243862
$411$	−3.19919	−0.157804
$412$	1.48755	0.0732866
$413$	0.735239	0.0361787
$414$	−29.9154	−1.47026
$415$	2.00786	0.0985619
$416$	−0.646905	−0.0317171
$417$	−6.18760	−0.303008
$418$	8.55480	0.418429
$419$	16.7530	0.818440	0.409220	−	0.912436i	$-0.365801\pi$
$419$	16.7530	0.818440	0.409220	+	0.912436i	$0.365801\pi$
$420$	−0.157360	−0.00767840
$421$	11.9935	0.584528	0.292264	−	0.956338i	$-0.405591\pi$
$421$	11.9935	0.584528	0.292264	+	0.956338i	$0.405591\pi$
$422$	17.0758	0.831238
$423$	18.6215	0.905410
$424$	−18.3884	−0.893021
$425$	−1.29157	−0.0626502
$426$	6.25796	0.303199
$427$	−9.88608	−0.478421
$428$	0.555638	0.0268578
$429$	4.48730	0.216649
$430$	−1.02600	−0.0494780
$431$	−20.1810	−0.972085	−0.486043	−	0.873935i	$-0.661560\pi$
$431$	−20.1810	−0.972085	−0.486043	+	0.873935i	$0.661560\pi$
$432$	15.3993	0.740899
$433$	14.8317	0.712768	0.356384	−	0.934340i	$-0.384009\pi$
$433$	14.8317	0.712768	0.356384	+	0.934340i	$0.384009\pi$
$434$	−5.77285	−0.277106
$435$	12.3329	0.591316
$436$	−1.54354	−0.0739223
$437$	9.14232	0.437336
$438$	−13.4274	−0.641588
$439$	−38.6880	−1.84648	−0.923239	−	0.384227i	$-0.874468\pi$
$439$	−38.6880	−1.84648	−0.923239	+	0.384227i	$0.874468\pi$
$440$	−34.3148	−1.63589
$441$	14.9732	0.713009
$442$	−1.33096	−0.0633074
$443$	−3.85111	−0.182972	−0.0914859	−	0.995806i	$-0.529162\pi$
$443$	−3.85111	−0.182972	−0.0914859	+	0.995806i	$0.529162\pi$
$444$	−0.629496	−0.0298746
$445$	8.40109	0.398250
$446$	−13.5352	−0.640912
$447$	8.85724	0.418933
$448$	−7.67301	−0.362515
$449$	−33.5702	−1.58428	−0.792139	−	0.610341i	$-0.791033\pi$
$449$	−33.5702	−1.58428	−0.792139	+	0.610341i	$0.791033\pi$
$450$	−4.28767	−0.202123
$451$	−11.4511	−0.539209
$452$	1.18756	0.0558581
$453$	1.50522	0.0707213
$454$	−11.0559	−0.518878
$455$	1.69944	0.0796711
$456$	−2.23764	−0.104787
$457$	−20.8319	−0.974476	−0.487238	−	0.873269i	$-0.661996\pi$
$457$	−20.8319	−0.974476	−0.487238	+	0.873269i	$0.661996\pi$
$458$	32.9104	1.53780
$459$	−4.11943	−0.192279
$460$	−2.04907	−0.0955386
$461$	13.0784	0.609123	0.304562	−	0.952493i	$-0.401490\pi$
$461$	13.0784	0.609123	0.304562	+	0.952493i	$0.401490\pi$
$462$	5.80107	0.269890
$463$	27.3653	1.27178	0.635888	−	0.771782i	$-0.280634\pi$
$463$	27.3653	1.27178	0.635888	+	0.771782i	$0.280634\pi$
$464$	31.7115	1.47217
$465$	6.73044	0.312117
$466$	10.4297	0.483146
$467$	21.4385	0.992054	0.496027	−	0.868307i	$-0.334792\pi$
$467$	21.4385	0.992054	0.496027	+	0.868307i	$0.334792\pi$
$468$	0.277950	0.0128482
$469$	−6.41342	−0.296144
$470$	−20.2760	−0.935262
$471$	−7.71482	−0.355480
$472$	−2.34329	−0.107858
$473$	−2.37933	−0.109402
$474$	12.1828	0.559574
$475$	1.31034	0.0601224
$476$	0.108239	0.00496114
$477$	15.3601	0.703291
$478$	26.2823	1.20212
$479$	−19.8654	−0.907672	−0.453836	−	0.891085i	$-0.649945\pi$
$479$	−19.8654	−0.907672	−0.453836	+	0.891085i	$0.649945\pi$
$480$	0.975026	0.0445036
$481$	6.79836	0.309978
$482$	2.95533	0.134612
$483$	6.19947	0.282085
$484$	−3.14448	−0.142931
$485$	32.4372	1.47290
$486$	−21.2347	−0.963228
$487$	36.4829	1.65320	0.826600	−	0.562790i	$-0.190272\pi$
$487$	36.4829	1.65320	0.826600	+	0.562790i	$0.190272\pi$
$488$	31.5080	1.42630
$489$	11.8532	0.536020
$490$	−16.3035	−0.736517
$491$	−26.3205	−1.18783	−0.593914	−	0.804529i	$-0.702418\pi$
$491$	−26.3205	−1.18783	−0.593914	+	0.804529i	$0.702418\pi$
$492$	0.167362	0.00754526
$493$	−8.48308	−0.382059
$494$	1.35030	0.0607531
$495$	28.6636	1.28833
$496$	17.3060	0.777062
$497$	5.49621	0.246539
$498$	1.08182	0.0484777
$499$	0.259097	0.0115988	0.00579938	−	0.999983i	$-0.498154\pi$
$499$	0.259097	0.0115988	0.00579938	+	0.999983i	$0.498154\pi$
$500$	−1.43399	−0.0641299
$501$	7.56957	0.338183
$502$	−16.3448	−0.729505
$503$	13.6264	0.607570	0.303785	−	0.952741i	$-0.401750\pi$
$503$	13.6264	0.607570	0.303785	+	0.952741i	$0.401750\pi$
$504$	6.43071	0.286447
$505$	30.9186	1.37586
$506$	75.5388	3.35811
$507$	−9.12996	−0.405476
$508$	−1.60242	−0.0710961
$509$	−2.16338	−0.0958899	−0.0479450	−	0.998850i	$-0.515267\pi$
$509$	−2.16338	−0.0958899	−0.0479450	+	0.998850i	$0.515267\pi$
$510$	2.00605	0.0888292
$511$	−11.7930	−0.521691
$512$	24.2964	1.07376
$513$	4.17930	0.184521
$514$	10.0383	0.442773
$515$	24.2136	1.06698
$516$	0.0347750	0.00153088
$517$	−47.0210	−2.06798
$518$	8.78874	0.386155
$519$	5.94093	0.260778
$520$	−5.41630	−0.237521
$521$	−32.9473	−1.44345	−0.721723	−	0.692182i	$-0.756649\pi$
$521$	−32.9473	−1.44345	−0.721723	+	0.692182i	$0.756649\pi$
$522$	−28.1616	−1.23260
$523$	19.3592	0.846519	0.423260	−	0.906008i	$-0.360886\pi$
$523$	19.3592	0.846519	0.423260	+	0.906008i	$0.360886\pi$
$524$	−1.97080	−0.0860948
$525$	0.888549	0.0387795
$526$	3.73331	0.162780
$527$	−4.62949	−0.201664
$528$	−17.3906	−0.756827
$529$	57.7266	2.50985
$530$	−16.7248	−0.726479
$531$	1.95738	0.0849430
$532$	−0.109812	−0.00476097
$533$	−1.80746	−0.0782896
$534$	4.52647	0.195879
$535$	9.04437	0.391022
$536$	20.4403	0.882885
$537$	4.37369	0.188739
$538$	40.1913	1.73277
$539$	−37.8085	−1.62853
$540$	−0.936710	−0.0403096
$541$	−35.6867	−1.53429	−0.767147	−	0.641472i	$-0.778324\pi$
$541$	−35.6867	−1.53429	−0.767147	+	0.641472i	$0.778324\pi$
$542$	15.7803	0.677823
$543$	−9.55389	−0.409997
$544$	−0.670665	−0.0287545
$545$	−25.1249	−1.07623
$546$	0.915651	0.0391862
$547$	4.70155	0.201024	0.100512	−	0.994936i	$-0.467952\pi$
$547$	4.70155	0.201024	0.100512	+	0.994936i	$0.467952\pi$
$548$	0.500377	0.0213750
$549$	−26.3191	−1.12327
$550$	10.8267	0.461653
$551$	8.60637	0.366644
$552$	−19.7584	−0.840972
$553$	10.6998	0.455003
$554$	−29.6629	−1.26025
$555$	−10.2466	−0.434944
$556$	0.967787	0.0410433
$557$	−8.81424	−0.373471	−0.186736	−	0.982410i	$-0.559791\pi$
$557$	−8.81424	−0.373471	−0.186736	+	0.982410i	$0.559791\pi$
$558$	−15.3687	−0.650609
$559$	−0.375558	−0.0158844
$560$	−6.58620	−0.278318
$561$	4.65211	0.196412
$562$	4.72440	0.199287
$563$	25.9906	1.09537	0.547687	−	0.836683i	$-0.315509\pi$
$563$	25.9906	1.09537	0.547687	+	0.836683i	$0.315509\pi$
$564$	0.687231	0.0289376
$565$	19.3304	0.813237
$566$	5.69319	0.239303
$567$	−3.80512	−0.159800
$568$	−17.5170	−0.734997
$569$	46.5218	1.95029	0.975147	−	0.221558i	$-0.0711141\pi$
$569$	46.5218	1.95029	0.975147	+	0.221558i	$0.0711141\pi$
$570$	−2.03520	−0.0852452
$571$	−14.8705	−0.622313	−0.311156	−	0.950359i	$-0.600716\pi$
$571$	−14.8705	−0.622313	−0.311156	+	0.950359i	$0.600716\pi$
$572$	−0.701847	−0.0293457
$573$	−10.6372	−0.444376
$574$	−2.33663	−0.0975291
$575$	11.5703	0.482514
$576$	−20.4274	−0.851140
$577$	11.0186	0.458711	0.229356	−	0.973343i	$-0.426338\pi$
$577$	11.0186	0.458711	0.229356	+	0.973343i	$0.426338\pi$
$578$	21.9395	0.912563
$579$	9.15295	0.380383
$580$	−1.92895	−0.0800954
$581$	0.950139	0.0394184
$582$	17.4770	0.724444
$583$	−38.7856	−1.60633
$584$	37.5855	1.55530
$585$	4.52432	0.187057
$586$	−28.6269	−1.18256
$587$	−26.0095	−1.07353	−0.536763	−	0.843733i	$-0.680353\pi$
$587$	−26.0095	−1.07353	−0.536763	+	0.843733i	$0.680353\pi$
$588$	0.552588	0.0227883
$589$	4.69677	0.193527
$590$	−2.13129	−0.0877437
$591$	2.85830	0.117575
$592$	−26.3471	−1.08286
$593$	−37.5846	−1.54341	−0.771706	−	0.635979i	$-0.780597\pi$
$593$	−37.5846	−1.54341	−0.771706	+	0.635979i	$0.780597\pi$
$594$	34.5317	1.41685
$595$	1.76186	0.0722292
$596$	−1.38534	−0.0567457
$597$	2.79510	0.114396
$598$	11.9232	0.487575
$599$	20.4734	0.836520	0.418260	−	0.908327i	$-0.362640\pi$
$599$	20.4734	0.836520	0.418260	+	0.908327i	$0.362640\pi$
$600$	−2.83190	−0.115612
$601$	6.39307	0.260779	0.130389	−	0.991463i	$-0.458377\pi$
$601$	6.39307	0.260779	0.130389	+	0.991463i	$0.458377\pi$
$602$	−0.485512	−0.0197880
$603$	−17.0740	−0.695309
$604$	−0.235427	−0.00957940
$605$	−51.1841	−2.08093
$606$	16.6588	0.676717
$607$	24.9896	1.01430	0.507148	−	0.861859i	$-0.330700\pi$
$607$	24.9896	1.01430	0.507148	+	0.861859i	$0.330700\pi$
$608$	0.680412	0.0275943
$609$	5.83604	0.236488
$610$	28.6574	1.16031
$611$	−7.42187	−0.300257
$612$	0.288159	0.0116481
$613$	0.675281	0.0272743	0.0136372	−	0.999907i	$-0.495659\pi$
$613$	0.675281	0.0272743	0.0136372	+	0.999907i	$0.495659\pi$
$614$	−44.0204	−1.77652
$615$	2.72423	0.109851
$616$	−16.2381	−0.654251
$617$	−25.9403	−1.04432	−0.522159	−	0.852848i	$-0.674873\pi$
$617$	−25.9403	−1.04432	−0.522159	+	0.852848i	$0.674873\pi$
$618$	13.0462	0.524794
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	−1.05269	−0.0422771
$621$	36.9032	1.48087
$622$	−36.4090	−1.45987
$623$	3.97548	0.159274
$624$	−2.74496	−0.109886
$625$	−16.9029	−0.676114
$626$	−39.9032	−1.59485
$627$	−4.71972	−0.188488
$628$	1.20665	0.0481508
$629$	7.04805	0.281024
$630$	5.84892	0.233027
$631$	−2.81069	−0.111892	−0.0559460	−	0.998434i	$-0.517817\pi$
$631$	−2.81069	−0.111892	−0.0559460	+	0.998434i	$0.517817\pi$
$632$	−34.1015	−1.35649
$633$	−9.42081	−0.374443
$634$	12.8491	0.510301
$635$	−26.0834	−1.03509
$636$	0.566867	0.0224777
$637$	−5.96777	−0.236452
$638$	71.1105	2.81529
$639$	14.6322	0.578841
$640$	19.6655	0.777348
$641$	−14.6818	−0.579897	−0.289948	−	0.957042i	$-0.593638\pi$
$641$	−14.6818	−0.579897	−0.289948	+	0.957042i	$0.593638\pi$
$642$	4.87306	0.192324
$643$	16.7477	0.660465	0.330233	−	0.943900i	$-0.392873\pi$
$643$	16.7477	0.660465	0.330233	+	0.943900i	$0.392873\pi$
$644$	−0.969643	−0.0382093
$645$	0.566048	0.0222881
$646$	1.39990	0.0550783
$647$	−22.1815	−0.872045	−0.436023	−	0.899936i	$-0.643613\pi$
$647$	−22.1815	−0.872045	−0.436023	+	0.899936i	$0.643613\pi$
$648$	12.1273	0.476407
$649$	−4.94255	−0.194012
$650$	1.70891	0.0670290
$651$	3.18491	0.124826
$652$	−1.85393	−0.0726054
$653$	23.0706	0.902820	0.451410	−	0.892317i	$-0.350921\pi$
$653$	23.0706	0.902820	0.451410	+	0.892317i	$0.350921\pi$
$654$	−13.5372	−0.529346
$655$	−32.0796	−1.25345
$656$	7.00480	0.273492
$657$	−31.3957	−1.22486
$658$	−9.59481	−0.374045
$659$	−6.59606	−0.256946	−0.128473	−	0.991713i	$-0.541008\pi$
$659$	−6.59606	−0.256946	−0.128473	+	0.991713i	$0.541008\pi$
$660$	1.05783	0.0411762
$661$	−45.0602	−1.75264	−0.876319	−	0.481731i	$-0.840008\pi$
$661$	−45.0602	−1.75264	−0.876319	+	0.481731i	$0.840008\pi$
$662$	−29.2003	−1.13490
$663$	0.734298	0.0285178
$664$	−3.02819	−0.117517
$665$	−1.78747	−0.0693149
$666$	23.3977	0.906643
$667$	75.9942	2.94251
$668$	−1.18394	−0.0458079
$669$	7.46746	0.288709
$670$	18.5910	0.718234
$671$	66.4579	2.56558
$672$	0.461392	0.0177986
$673$	20.5317	0.791439	0.395720	−	0.918371i	$-0.370495\pi$
$673$	20.5317	0.791439	0.395720	+	0.918371i	$0.370495\pi$
$674$	−5.14428	−0.198150
$675$	5.28922	0.203582
$676$	1.42799	0.0549228
$677$	−18.1940	−0.699252	−0.349626	−	0.936889i	$-0.613691\pi$
$677$	−18.1940	−0.699252	−0.349626	+	0.936889i	$0.613691\pi$
$678$	10.4151	0.399991
$679$	15.3496	0.589063
$680$	−5.61524	−0.215335
$681$	6.09958	0.233736
$682$	38.8073	1.48601
$683$	18.8152	0.719942	0.359971	−	0.932963i	$-0.382787\pi$
$683$	18.8152	0.719942	0.359971	+	0.932963i	$0.382787\pi$
$684$	−0.292347	−0.0111781
$685$	8.14486	0.311199
$686$	−16.4696	−0.628812
$687$	−18.1568	−0.692727
$688$	1.45548	0.0554896
$689$	−6.12198	−0.233229
$690$	−17.9708	−0.684137
$691$	−30.2806	−1.15193	−0.575965	−	0.817475i	$-0.695373\pi$
$691$	−30.2806	−1.15193	−0.575965	+	0.817475i	$0.695373\pi$
$692$	−0.929206	−0.0353231
$693$	13.5639	0.515250
$694$	22.2581	0.844906
$695$	15.7531	0.597549
$696$	−18.6001	−0.705034
$697$	−1.87384	−0.0709768
$698$	−30.7939	−1.16557
$699$	−5.75411	−0.217640
$700$	−0.138976	−0.00525279
$701$	−40.6515	−1.53538	−0.767692	−	0.640819i	$-0.778595\pi$
$701$	−40.6515	−1.53538	−0.767692	+	0.640819i	$0.778595\pi$
$702$	5.45054	0.205717
$703$	−7.15048	−0.269685
$704$	51.5808	1.94402
$705$	11.1864	0.421303
$706$	20.2058	0.760454
$707$	14.6310	0.550255
$708$	0.0722374	0.00271485
$709$	−38.8530	−1.45916	−0.729578	−	0.683898i	$-0.760283\pi$
$709$	−38.8530	−1.45916	−0.729578	+	0.683898i	$0.760283\pi$
$710$	−15.9322	−0.597926
$711$	28.4855	1.06829
$712$	−12.6703	−0.474839
$713$	41.4724	1.55316
$714$	0.949281	0.0355260
$715$	−11.4243	−0.427244
$716$	−0.684078	−0.0255652
$717$	−14.5001	−0.541515
$718$	−17.5789	−0.656038
$719$	42.1167	1.57069	0.785343	−	0.619061i	$-0.212487\pi$
$719$	42.1167	1.57069	0.785343	+	0.619061i	$0.212487\pi$
$720$	−17.5340	−0.653454
$721$	11.4581	0.426723
$722$	24.6426	0.917101
$723$	−1.63047	−0.0606378
$724$	1.49430	0.0555352
$725$	10.8920	0.404519
$726$	−27.5778	−1.02351
$727$	−36.1229	−1.33973	−0.669863	−	0.742485i	$-0.733647\pi$
$727$	−36.1229	−1.33973	−0.669863	+	0.742485i	$0.733647\pi$
$728$	−2.56305	−0.0949929
$729$	−0.805094	−0.0298183
$730$	34.1851	1.26525
$731$	−0.389352	−0.0144007
$732$	−0.971310	−0.0359006
$733$	−44.8688	−1.65727	−0.828634	−	0.559790i	$-0.810882\pi$
$733$	−44.8688	−1.65727	−0.828634	+	0.559790i	$0.810882\pi$
$734$	−26.4417	−0.975983
$735$	8.99472	0.331775
$736$	6.00803	0.221459
$737$	43.1134	1.58810
$738$	−6.22067	−0.228986
$739$	−23.7453	−0.873487	−0.436743	−	0.899586i	$-0.643868\pi$
$739$	−23.7453	−0.873487	−0.436743	+	0.899586i	$0.643868\pi$
$740$	1.60264	0.0589143
$741$	−0.744970	−0.0273671
$742$	−7.91435	−0.290545
$743$	−13.0923	−0.480310	−0.240155	−	0.970735i	$-0.577198\pi$
$743$	−13.0923	−0.480310	−0.240155	+	0.970735i	$0.577198\pi$
$744$	−10.1507	−0.372141
$745$	−22.5498	−0.826160
$746$	33.8998	1.24116
$747$	2.52949	0.0925493
$748$	−0.727625	−0.0266046
$749$	4.27989	0.156384
$750$	−12.5764	−0.459224
$751$	−1.82954	−0.0667609	−0.0333805	−	0.999443i	$-0.510627\pi$
$751$	−1.82954	−0.0667609	−0.0333805	+	0.999443i	$0.510627\pi$
$752$	28.7635	1.04890
$753$	9.01751	0.328616
$754$	11.2242	0.408762
$755$	−3.83216	−0.139466
$756$	−0.443261	−0.0161212
$757$	10.6859	0.388387	0.194193	−	0.980963i	$-0.437791\pi$
$757$	10.6859	0.388387	0.194193	+	0.980963i	$0.437791\pi$
$758$	34.3133	1.24632
$759$	−41.6751	−1.51271
$760$	5.69685	0.206646
$761$	27.9514	1.01324	0.506620	−	0.862170i	$-0.330895\pi$
$761$	27.9514	1.01324	0.506620	+	0.862170i	$0.330895\pi$
$762$	−14.0536	−0.509108
$763$	−11.8894	−0.430424
$764$	1.66374	0.0601920
$765$	4.69049	0.169585
$766$	22.6219	0.817361
$767$	−0.780141	−0.0281693
$768$	−2.14221	−0.0773005
$769$	23.4582	0.845925	0.422963	−	0.906147i	$-0.360990\pi$
$769$	23.4582	0.845925	0.422963	+	0.906147i	$0.360990\pi$
$770$	−14.7690	−0.532238
$771$	−5.53820	−0.199454
$772$	−1.43159	−0.0515240
$773$	−50.3166	−1.80976	−0.904882	−	0.425664i	$-0.860041\pi$
$773$	−50.3166	−1.80976	−0.904882	+	0.425664i	$0.860041\pi$
$774$	−1.29255	−0.0464597
$775$	5.94411	0.213519
$776$	−48.9208	−1.75616
$777$	−4.84879	−0.173949
$778$	−30.3457	−1.08795
$779$	1.90107	0.0681130
$780$	0.166971	0.00597851
$781$	−36.9476	−1.32209
$782$	12.3611	0.442032
$783$	34.7398	1.24150
$784$	23.1281	0.826005
$785$	19.6413	0.701027
$786$	−17.2843	−0.616512
$787$	−10.7717	−0.383971	−0.191985	−	0.981398i	$-0.561493\pi$
$787$	−10.7717	−0.383971	−0.191985	+	0.981398i	$0.561493\pi$
$788$	−0.447059	−0.0159258
$789$	−2.05968	−0.0733267
$790$	−31.0164	−1.10351
$791$	9.14735	0.325242
$792$	−43.2296	−1.53610
$793$	10.4898	0.372505
$794$	−27.3633	−0.971086
$795$	9.22715	0.327253
$796$	−0.437174	−0.0154952
$797$	−51.0743	−1.80915	−0.904573	−	0.426319i	$-0.859810\pi$
$797$	−51.0743	−1.80915	−0.904573	+	0.426319i	$0.859810\pi$
$798$	−0.963078	−0.0340926
$799$	−7.69447	−0.272211
$800$	0.861112	0.0304449
$801$	10.5837	0.373956
$802$	4.79366	0.169270
$803$	79.2768	2.79762
$804$	−0.630120	−0.0222226
$805$	−15.7833	−0.556289
$806$	6.12541	0.215758
$807$	−22.1737	−0.780552
$808$	−46.6305	−1.64046
$809$	8.59922	0.302332	0.151166	−	0.988508i	$-0.451697\pi$
$809$	8.59922	0.302332	0.151166	+	0.988508i	$0.451697\pi$
$810$	11.0302	0.387561
$811$	−13.8386	−0.485939	−0.242969	−	0.970034i	$-0.578121\pi$
$811$	−13.8386	−0.485939	−0.242969	+	0.970034i	$0.578121\pi$
$812$	−0.912799	−0.0320330
$813$	−8.70608	−0.305335
$814$	−59.0812	−2.07079
$815$	−30.1772	−1.05706
$816$	−2.84577	−0.0996220
$817$	0.395011	0.0138197
$818$	−22.9761	−0.803342
$819$	2.14095	0.0748109
$820$	−0.426089	−0.0148797
$821$	−28.1615	−0.982844	−0.491422	−	0.870922i	$-0.663523\pi$
$821$	−28.1615	−0.982844	−0.491422	+	0.870922i	$0.663523\pi$
$822$	4.38841	0.153063
$823$	8.87067	0.309212	0.154606	−	0.987976i	$-0.450589\pi$
$823$	8.87067	0.309212	0.154606	+	0.987976i	$0.450589\pi$
$824$	−36.5183	−1.27217
$825$	−5.97316	−0.207959
$826$	−1.00855	−0.0350918
$827$	15.9862	0.555896	0.277948	−	0.960596i	$-0.410346\pi$
$827$	15.9862	0.555896	0.277948	+	0.960596i	$0.410346\pi$
$828$	−2.58142	−0.0897105
$829$	−2.67656	−0.0929606	−0.0464803	−	0.998919i	$-0.514800\pi$
$829$	−2.67656	−0.0929606	−0.0464803	+	0.998919i	$0.514800\pi$
$830$	−2.75423	−0.0956008
$831$	16.3651	0.567700
$832$	8.14161	0.282259
$833$	−6.18696	−0.214365
$834$	8.48770	0.293905
$835$	−19.2715	−0.666917
$836$	0.738200	0.0255312
$837$	18.9586	0.655306
$838$	−22.9806	−0.793851
$839$	41.7172	1.44024	0.720118	−	0.693851i	$-0.244088\pi$
$839$	41.7172	1.44024	0.720118	+	0.693851i	$0.244088\pi$
$840$	3.86307	0.133289
$841$	42.5392	1.46687
$842$	−16.4518	−0.566967
$843$	−2.60647	−0.0897717
$844$	1.47348	0.0507194
$845$	23.2441	0.799621
$846$	−25.5437	−0.878209
$847$	−24.2208	−0.832238
$848$	23.7258	0.814747
$849$	−3.14096	−0.107798
$850$	1.77168	0.0607680
$851$	−63.1387	−2.16437
$852$	0.540004	0.0185002
$853$	40.7768	1.39617	0.698086	−	0.716014i	$-0.254035\pi$
$853$	40.7768	1.39617	0.698086	+	0.716014i	$0.254035\pi$
$854$	13.5610	0.464048
$855$	−4.75866	−0.162743
$856$	−13.6405	−0.466221
$857$	22.4786	0.767854	0.383927	−	0.923363i	$-0.374571\pi$
$857$	22.4786	0.767854	0.383927	+	0.923363i	$0.374571\pi$
$858$	−6.15535	−0.210140
$859$	54.3888	1.85572	0.927861	−	0.372927i	$-0.121646\pi$
$859$	54.3888	1.85572	0.927861	+	0.372927i	$0.121646\pi$
$860$	−0.0885341	−0.00301899
$861$	1.28913	0.0439335
$862$	27.6828	0.942881
$863$	40.5024	1.37872	0.689359	−	0.724420i	$-0.257892\pi$
$863$	40.5024	1.37872	0.689359	+	0.724420i	$0.257892\pi$
$864$	2.74650	0.0934379
$865$	−15.1251	−0.514269
$866$	−20.3451	−0.691355
$867$	−12.1041	−0.411078
$868$	−0.498144	−0.0169081
$869$	−71.9283	−2.44000
$870$	−16.9173	−0.573551
$871$	6.80510	0.230582
$872$	37.8927	1.28321
$873$	40.8642	1.38305
$874$	−12.5408	−0.424197
$875$	−11.0455	−0.373406
$876$	−1.15866	−0.0391476
$877$	−34.7447	−1.17325	−0.586623	−	0.809860i	$-0.699543\pi$
$877$	−34.7447	−1.17325	−0.586623	+	0.809860i	$0.699543\pi$
$878$	53.0693	1.79100
$879$	15.7936	0.532704
$880$	44.2749	1.49251
$881$	−34.0749	−1.14801	−0.574006	−	0.818851i	$-0.694612\pi$
$881$	−34.0749	−1.14801	−0.574006	+	0.818851i	$0.694612\pi$
$882$	−20.5391	−0.691588
$883$	51.4060	1.72995	0.864974	−	0.501816i	$-0.167335\pi$
$883$	51.4060	1.72995	0.864974	+	0.501816i	$0.167335\pi$
$884$	−0.114850	−0.00386281
$885$	1.17584	0.0395255
$886$	5.28267	0.177475
$887$	−0.959159	−0.0322054	−0.0161027	−	0.999870i	$-0.505126\pi$
$887$	−0.959159	−0.0322054	−0.0161027	+	0.999870i	$0.505126\pi$
$888$	15.4536	0.518589
$889$	−12.3429	−0.413968
$890$	−11.5240	−0.386285
$891$	25.5795	0.856944
$892$	−1.16797	−0.0391064
$893$	7.80629	0.261228
$894$	−12.1497	−0.406347
$895$	−11.1350	−0.372203
$896$	9.30592	0.310889
$897$	−6.57808	−0.219636
$898$	46.0492	1.53668
$899$	39.0412	1.30210
$900$	−0.369986	−0.0123329
$901$	−6.34684	−0.211444
$902$	15.7077	0.523010
$903$	0.267859	0.00891380
$904$	−29.1536	−0.969634
$905$	24.3234	0.808537
$906$	−2.06475	−0.0685966
$907$	−17.5927	−0.584156	−0.292078	−	0.956395i	$-0.594347\pi$
$907$	−17.5927	−0.584156	−0.292078	+	0.956395i	$0.594347\pi$
$908$	−0.954020	−0.0316603
$909$	38.9511	1.29193
$910$	−2.33117	−0.0772775
$911$	7.41811	0.245773	0.122886	−	0.992421i	$-0.460785\pi$
$911$	7.41811	0.245773	0.122886	+	0.992421i	$0.460785\pi$
$912$	2.88713	0.0956026
$913$	−6.38718	−0.211385
$914$	28.5757	0.945200
$915$	−15.8104	−0.522677
$916$	2.83987	0.0938318
$917$	−15.1804	−0.501301
$918$	5.65073	0.186502
$919$	11.0625	0.364917	0.182459	−	0.983214i	$-0.441594\pi$
$919$	11.0625	0.364917	0.182459	+	0.983214i	$0.441594\pi$
$920$	50.3031	1.65844
$921$	24.2862	0.800259
$922$	−17.9400	−0.590823
$923$	−5.83187	−0.191958
$924$	0.500578	0.0164678
$925$	−9.04946	−0.297545
$926$	−37.5378	−1.23357
$927$	30.5042	1.00189
$928$	5.65583	0.185662
$929$	18.7168	0.614078	0.307039	−	0.951697i	$-0.400662\pi$
$929$	18.7168	0.614078	0.307039	+	0.951697i	$0.400662\pi$
$930$	−9.23232	−0.302740
$931$	6.27687	0.205716
$932$	0.899985	0.0294800
$933$	20.0870	0.657620
$934$	−29.4077	−0.962249
$935$	−11.8439	−0.387336
$936$	−6.82344	−0.223031
$937$	−25.7508	−0.841242	−0.420621	−	0.907236i	$-0.638188\pi$
$937$	−25.7508	−0.841242	−0.420621	+	0.907236i	$0.638188\pi$
$938$	8.79746	0.287247
$939$	22.0148	0.718426
$940$	−1.74963	−0.0570667
$941$	28.7127	0.936008	0.468004	−	0.883726i	$-0.344973\pi$
$941$	28.7127	0.936008	0.468004	+	0.883726i	$0.344973\pi$
$942$	10.5826	0.344800
$943$	16.7865	0.546643
$944$	3.02344	0.0984046
$945$	−7.21515	−0.234709
$946$	3.26379	0.106115
$947$	−26.9875	−0.876975	−0.438488	−	0.898737i	$-0.644486\pi$
$947$	−26.9875	−0.876975	−0.438488	+	0.898737i	$0.644486\pi$
$948$	1.05126	0.0341434
$949$	12.5132	0.406196
$950$	−1.79743	−0.0583162
$951$	−7.08889	−0.229873
$952$	−2.65719	−0.0861199
$953$	−28.3468	−0.918241	−0.459121	−	0.888374i	$-0.651835\pi$
$953$	−28.3468	−0.918241	−0.459121	+	0.888374i	$0.651835\pi$
$954$	−21.0699	−0.682162
$955$	27.0815	0.876336
$956$	2.26792	0.0733497
$957$	−39.2320	−1.26819
$958$	27.2498	0.880402
$959$	3.85423	0.124460
$960$	−12.2712	−0.396050
$961$	−9.69396	−0.312709
$962$	−9.32548	−0.300666
$963$	11.3941	0.367169
$964$	0.255017	0.00821356
$965$	−23.3026	−0.750138
$966$	−8.50397	−0.273611
$967$	−17.6693	−0.568206	−0.284103	−	0.958794i	$-0.591696\pi$
$967$	−17.6693	−0.568206	−0.284103	+	0.958794i	$0.591696\pi$
$968$	77.1945	2.48112
$969$	−0.772331	−0.0248108
$970$	−44.4949	−1.42865
$971$	−7.39225	−0.237229	−0.118614	−	0.992940i	$-0.537845\pi$
$971$	−7.39225	−0.237229	−0.118614	+	0.992940i	$0.537845\pi$
$972$	−1.83236	−0.0587730
$973$	7.45453	0.238981
$974$	−50.0446	−1.60353
$975$	−0.942814	−0.0301942
$976$	−40.6534	−1.30128
$977$	40.2736	1.28847	0.644233	−	0.764829i	$-0.277177\pi$
$977$	40.2736	1.28847	0.644233	+	0.764829i	$0.277177\pi$
$978$	−16.2593	−0.519916
$979$	−26.7247	−0.854124
$980$	−1.40684	−0.0449399
$981$	−31.6523	−1.01058
$982$	36.1045	1.15214
$983$	31.7824	1.01370	0.506850	−	0.862034i	$-0.330810\pi$
$983$	31.7824	1.01370	0.506850	+	0.862034i	$0.330810\pi$
$984$	−4.10860	−0.130977
$985$	−7.27698	−0.231864
$986$	11.6365	0.370580
$987$	5.29350	0.168494
$988$	0.116519	0.00370696
$989$	3.48794	0.110910
$990$	−39.3186	−1.24963
$991$	−50.5923	−1.60712	−0.803559	−	0.595225i	$-0.797063\pi$
$991$	−50.5923	−1.60712	−0.803559	+	0.595225i	$0.797063\pi$
$992$	3.08656	0.0979985
$993$	16.1099	0.511233
$994$	−7.53929	−0.239132
$995$	−7.11608	−0.225595
$996$	0.0933514	0.00295795
$997$	34.5592	1.09450	0.547250	−	0.836969i	$-0.315675\pi$
$997$	34.5592	1.09450	0.547250	+	0.836969i	$0.315675\pi$
$998$	−0.355410	−0.0112503
$999$	−28.8631	−0.913188

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	619.2.a.a.1.8	✓	21
3.2	odd	2		5571.2.a.e.1.14		21
4.3	odd	2		9904.2.a.j.1.9		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
619.2.a.a.1.8	✓	21	1.1	even	1	trivial
5571.2.a.e.1.14		21	3.2	odd	2
9904.2.a.j.1.9		21	4.3	odd	2

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:	\( N \)	\(=\)	\( 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	619.a (trivial)

\(f(q)\)	\(=\)	\(q-1.37173 q^{2} +0.756788 q^{3} -0.118367 q^{4} -1.92672 q^{5} -1.03811 q^{6} -0.911742 q^{7} +2.90582 q^{8} -2.42727 q^{9} +O(q^{10})\)	\(q-1.37173 q^{2} +0.756788 q^{3} -0.118367 q^{4} -1.92672 q^{5} -1.03811 q^{6} -0.911742 q^{7} +2.90582 q^{8} -2.42727 q^{9} +2.64293 q^{10} +6.12907 q^{11} -0.0895789 q^{12} +0.967423 q^{13} +1.25066 q^{14} -1.45812 q^{15} -3.74925 q^{16} +1.00296 q^{17} +3.32955 q^{18} -1.01753 q^{19} +0.228060 q^{20} -0.689995 q^{21} -8.40740 q^{22} -8.98480 q^{23} +2.19909 q^{24} -1.28776 q^{25} -1.32704 q^{26} -4.10729 q^{27} +0.107920 q^{28} -8.45808 q^{29} +2.00014 q^{30} -4.61585 q^{31} -0.668689 q^{32} +4.63840 q^{33} -1.37578 q^{34} +1.75667 q^{35} +0.287310 q^{36} +7.02728 q^{37} +1.39577 q^{38} +0.732134 q^{39} -5.59869 q^{40} -1.86832 q^{41} +0.946484 q^{42} -0.388205 q^{43} -0.725481 q^{44} +4.67667 q^{45} +12.3247 q^{46} -7.67180 q^{47} -2.83739 q^{48} -6.16873 q^{49} +1.76646 q^{50} +0.759024 q^{51} -0.114511 q^{52} -6.32814 q^{53} +5.63408 q^{54} -11.8090 q^{55} -2.64936 q^{56} -0.770055 q^{57} +11.6022 q^{58} -0.806411 q^{59} +0.172593 q^{60} +10.8431 q^{61} +6.33168 q^{62} +2.21305 q^{63} +8.41577 q^{64} -1.86395 q^{65} -6.36262 q^{66} +7.03425 q^{67} -0.118717 q^{68} -6.79959 q^{69} -2.40967 q^{70} -6.02825 q^{71} -7.05322 q^{72} +12.9346 q^{73} -9.63951 q^{74} -0.974562 q^{75} +0.120442 q^{76} -5.58813 q^{77} -1.00429 q^{78} -11.7356 q^{79} +7.22375 q^{80} +4.17347 q^{81} +2.56282 q^{82} -1.04211 q^{83} +0.0816728 q^{84} -1.93241 q^{85} +0.532511 q^{86} -6.40097 q^{87} +17.8100 q^{88} -4.36032 q^{89} -6.41511 q^{90} -0.882040 q^{91} +1.06351 q^{92} -3.49322 q^{93} +10.5236 q^{94} +1.96050 q^{95} -0.506055 q^{96} -16.8355 q^{97} +8.46180 q^{98} -14.8769 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 9 q^{2} - 5 q^{3} + 15 q^{4} - 21 q^{5} - 6 q^{6} - 4 q^{7} - 21 q^{8} + 6 q^{9}+O(q^{10}) \) 21 * q - 9 * q^2 - 5 * q^3 + 15 * q^4 - 21 * q^5 - 6 * q^6 - 4 * q^7 - 21 * q^8 + 6 * q^9	\( 21 q - 9 q^{2} - 5 q^{3} + 15 q^{4} - 21 q^{5} - 6 q^{6} - 4 q^{7} - 21 q^{8} + 6 q^{9} + q^{10} - 27 q^{11} - 8 q^{12} - 11 q^{13} - 19 q^{14} - 10 q^{15} + 11 q^{16} - 14 q^{17} - 14 q^{18} - 15 q^{19} - 25 q^{20} - 42 q^{21} + 12 q^{22} - 14 q^{23} - 8 q^{24} + 16 q^{25} - 11 q^{26} - 5 q^{27} + q^{28} - 78 q^{29} + q^{30} - 8 q^{31} - 41 q^{32} - 6 q^{33} + 7 q^{34} - 3 q^{35} - q^{36} - 23 q^{37} + 21 q^{38} - 4 q^{39} + 12 q^{40} - 59 q^{41} + 39 q^{42} + 2 q^{43} - 50 q^{44} - 36 q^{45} - 15 q^{46} - 12 q^{47} + 10 q^{48} + 17 q^{49} - 23 q^{50} - 8 q^{51} + 18 q^{52} - 36 q^{53} - 4 q^{54} + 23 q^{55} - 28 q^{56} - 24 q^{57} + 46 q^{58} - 17 q^{59} + 8 q^{60} - 22 q^{61} + 42 q^{62} - 6 q^{63} + 49 q^{64} - 53 q^{65} + 29 q^{66} + 15 q^{67} - 16 q^{68} - 30 q^{69} + 44 q^{70} - 56 q^{71} + 12 q^{72} - 2 q^{73} - 12 q^{74} + 2 q^{75} - 4 q^{76} - 47 q^{77} + 36 q^{78} + 5 q^{79} + 15 q^{80} - 19 q^{81} + 47 q^{82} - q^{83} - 20 q^{84} - 29 q^{85} - 23 q^{86} + 44 q^{87} + 61 q^{88} - 12 q^{89} + 91 q^{90} + 5 q^{91} + 35 q^{92} - 15 q^{93} + 34 q^{94} - 17 q^{95} + 14 q^{96} + 21 q^{97} + 24 q^{98} - 38 q^{99}+O(q^{100}) \) 21 * q - 9 * q^2 - 5 * q^3 + 15 * q^4 - 21 * q^5 - 6 * q^6 - 4 * q^7 - 21 * q^8 + 6 * q^9 + q^10 - 27 * q^11 - 8 * q^12 - 11 * q^13 - 19 * q^14 - 10 * q^15 + 11 * q^16 - 14 * q^17 - 14 * q^18 - 15 * q^19 - 25 * q^20 - 42 * q^21 + 12 * q^22 - 14 * q^23 - 8 * q^24 + 16 * q^25 - 11 * q^26 - 5 * q^27 + q^28 - 78 * q^29 + q^30 - 8 * q^31 - 41 * q^32 - 6 * q^33 + 7 * q^34 - 3 * q^35 - q^36 - 23 * q^37 + 21 * q^38 - 4 * q^39 + 12 * q^40 - 59 * q^41 + 39 * q^42 + 2 * q^43 - 50 * q^44 - 36 * q^45 - 15 * q^46 - 12 * q^47 + 10 * q^48 + 17 * q^49 - 23 * q^50 - 8 * q^51 + 18 * q^52 - 36 * q^53 - 4 * q^54 + 23 * q^55 - 28 * q^56 - 24 * q^57 + 46 * q^58 - 17 * q^59 + 8 * q^60 - 22 * q^61 + 42 * q^62 - 6 * q^63 + 49 * q^64 - 53 * q^65 + 29 * q^66 + 15 * q^67 - 16 * q^68 - 30 * q^69 + 44 * q^70 - 56 * q^71 + 12 * q^72 - 2 * q^73 - 12 * q^74 + 2 * q^75 - 4 * q^76 - 47 * q^77 + 36 * q^78 + 5 * q^79 + 15 * q^80 - 19 * q^81 + 47 * q^82 - q^83 - 20 * q^84 - 29 * q^85 - 23 * q^86 + 44 * q^87 + 61 * q^88 - 12 * q^89 + 91 * q^90 + 5 * q^91 + 35 * q^92 - 15 * q^93 + 34 * q^94 - 17 * q^95 + 14 * q^96 + 21 * q^97 + 24 * q^98 - 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more