LMFDB - Embedded newform 6019.2.a.e.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6019,2,Mod(1,6019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6019, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6019 = 13 \cdot 463 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6019.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:	$48.0619569766$
Analytic rank:	$0$
Dimension:	$130$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	6019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.51967 q^{2} -3.32656 q^{3} +4.34873 q^{4} -0.493062 q^{5} +8.38182 q^{6} -0.741150 q^{7} -5.91802 q^{8} +8.06597 q^{9} +O(q^{10})$	\(q-2.51967 q^{2} -3.32656 q^{3} +4.34873 q^{4} -0.493062 q^{5} +8.38182 q^{6} -0.741150 q^{7} -5.91802 q^{8} +8.06597 q^{9} +1.24235 q^{10} -5.43412 q^{11} -14.4663 q^{12} +1.00000 q^{13} +1.86745 q^{14} +1.64020 q^{15} +6.21398 q^{16} +2.04346 q^{17} -20.3236 q^{18} -3.88601 q^{19} -2.14419 q^{20} +2.46548 q^{21} +13.6922 q^{22} -2.90672 q^{23} +19.6866 q^{24} -4.75689 q^{25} -2.51967 q^{26} -16.8522 q^{27} -3.22306 q^{28} +5.42123 q^{29} -4.13276 q^{30} -3.23353 q^{31} -3.82114 q^{32} +18.0769 q^{33} -5.14883 q^{34} +0.365433 q^{35} +35.0767 q^{36} -9.23599 q^{37} +9.79146 q^{38} -3.32656 q^{39} +2.91795 q^{40} -9.92194 q^{41} -6.21218 q^{42} -6.04449 q^{43} -23.6315 q^{44} -3.97703 q^{45} +7.32397 q^{46} -0.0696159 q^{47} -20.6712 q^{48} -6.45070 q^{49} +11.9858 q^{50} -6.79767 q^{51} +4.34873 q^{52} +1.79448 q^{53} +42.4621 q^{54} +2.67936 q^{55} +4.38614 q^{56} +12.9270 q^{57} -13.6597 q^{58} +14.2594 q^{59} +7.13278 q^{60} +6.00394 q^{61} +8.14743 q^{62} -5.97809 q^{63} -2.79995 q^{64} -0.493062 q^{65} -45.5478 q^{66} -8.29507 q^{67} +8.88644 q^{68} +9.66936 q^{69} -0.920770 q^{70} +2.25027 q^{71} -47.7346 q^{72} -13.3587 q^{73} +23.2716 q^{74} +15.8241 q^{75} -16.8992 q^{76} +4.02749 q^{77} +8.38182 q^{78} -1.85915 q^{79} -3.06388 q^{80} +31.8620 q^{81} +25.0000 q^{82} +0.955094 q^{83} +10.7217 q^{84} -1.00755 q^{85} +15.2301 q^{86} -18.0340 q^{87} +32.1592 q^{88} -4.34620 q^{89} +10.0208 q^{90} -0.741150 q^{91} -12.6405 q^{92} +10.7565 q^{93} +0.175409 q^{94} +1.91605 q^{95} +12.7112 q^{96} -6.77944 q^{97} +16.2536 q^{98} -43.8314 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 130 q + 10 q^{2} + 11 q^{3} + 146 q^{4} + 40 q^{5} + 4 q^{6} + 8 q^{7} + 24 q^{8} + 181 q^{9}+O(q^{10}) $ 130 * q + 10 * q^2 + 11 * q^3 + 146 * q^4 + 40 * q^5 + 4 * q^6 + 8 * q^7 + 24 * q^8 + 181 * q^9	\( 130 q + 10 q^{2} + 11 q^{3} + 146 q^{4} + 40 q^{5} + 4 q^{6} + 8 q^{7} + 24 q^{8} + 181 q^{9} + 5 q^{10} + 43 q^{11} + 28 q^{12} + 130 q^{13} + 47 q^{14} + 29 q^{15} + 170 q^{16} + 85 q^{17} + 20 q^{18} + 3 q^{19} + 73 q^{20} + 62 q^{21} + 12 q^{22} + 62 q^{23} - 5 q^{24} + 178 q^{25} + 10 q^{26} + 35 q^{27} - q^{28} + 134 q^{29} + 24 q^{30} + 14 q^{31} + 48 q^{32} + 4 q^{33} + 4 q^{34} + 50 q^{35} + 244 q^{36} + 32 q^{37} + 76 q^{38} + 11 q^{39} - 6 q^{40} + 48 q^{41} + 9 q^{42} + 34 q^{43} + 123 q^{44} + 115 q^{45} + 5 q^{46} + 25 q^{47} + 35 q^{48} + 210 q^{49} + 24 q^{50} + 20 q^{51} + 146 q^{52} + 193 q^{53} - 39 q^{54} + 32 q^{55} + 122 q^{56} + 7 q^{57} - 4 q^{58} + 50 q^{59} + 42 q^{60} + 57 q^{61} + 51 q^{62} + 8 q^{63} + 172 q^{64} + 40 q^{65} - 4 q^{66} + 21 q^{67} + 132 q^{68} + 92 q^{69} - 46 q^{70} + 58 q^{71} - 26 q^{72} + 15 q^{73} + 120 q^{74} + 23 q^{75} - 65 q^{76} + 192 q^{77} + 4 q^{78} + 32 q^{79} + 66 q^{80} + 326 q^{81} + 11 q^{82} + 33 q^{83} + 5 q^{84} + 43 q^{85} + 105 q^{86} + 31 q^{87} - 17 q^{88} + 84 q^{89} - 73 q^{90} + 8 q^{91} + 161 q^{92} + 52 q^{93} + 4 q^{94} + 59 q^{95} - 77 q^{96} + 9 q^{97} - 61 q^{98} + 100 q^{99}+O(q^{100}) \) 130 * q + 10 * q^2 + 11 * q^3 + 146 * q^4 + 40 * q^5 + 4 * q^6 + 8 * q^7 + 24 * q^8 + 181 * q^9 + 5 * q^10 + 43 * q^11 + 28 * q^12 + 130 * q^13 + 47 * q^14 + 29 * q^15 + 170 * q^16 + 85 * q^17 + 20 * q^18 + 3 * q^19 + 73 * q^20 + 62 * q^21 + 12 * q^22 + 62 * q^23 - 5 * q^24 + 178 * q^25 + 10 * q^26 + 35 * q^27 - q^28 + 134 * q^29 + 24 * q^30 + 14 * q^31 + 48 * q^32 + 4 * q^33 + 4 * q^34 + 50 * q^35 + 244 * q^36 + 32 * q^37 + 76 * q^38 + 11 * q^39 - 6 * q^40 + 48 * q^41 + 9 * q^42 + 34 * q^43 + 123 * q^44 + 115 * q^45 + 5 * q^46 + 25 * q^47 + 35 * q^48 + 210 * q^49 + 24 * q^50 + 20 * q^51 + 146 * q^52 + 193 * q^53 - 39 * q^54 + 32 * q^55 + 122 * q^56 + 7 * q^57 - 4 * q^58 + 50 * q^59 + 42 * q^60 + 57 * q^61 + 51 * q^62 + 8 * q^63 + 172 * q^64 + 40 * q^65 - 4 * q^66 + 21 * q^67 + 132 * q^68 + 92 * q^69 - 46 * q^70 + 58 * q^71 - 26 * q^72 + 15 * q^73 + 120 * q^74 + 23 * q^75 - 65 * q^76 + 192 * q^77 + 4 * q^78 + 32 * q^79 + 66 * q^80 + 326 * q^81 + 11 * q^82 + 33 * q^83 + 5 * q^84 + 43 * q^85 + 105 * q^86 + 31 * q^87 - 17 * q^88 + 84 * q^89 - 73 * q^90 + 8 * q^91 + 161 * q^92 + 52 * q^93 + 4 * q^94 + 59 * q^95 - 77 * q^96 + 9 * q^97 - 61 * q^98 + 100 * 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 $

$2$	−2.51967	−1.78167	−0.890837	−	0.454322i	$-0.849881\pi$
$2$	−2.51967	−1.78167	−0.890837	+	0.454322i	$0.849881\pi$
$3$	−3.32656	−1.92059	−0.960294	−	0.278990i	$-0.910000\pi$
$3$	−3.32656	−1.92059	−0.960294	+	0.278990i	$0.910000\pi$
$4$	4.34873	2.17436
$5$	−0.493062	−0.220504	−0.110252	−	0.993904i	$-0.535166\pi$
$5$	−0.493062	−0.220504	−0.110252	+	0.993904i	$0.535166\pi$
$6$	8.38182	3.42186
$7$	−0.741150	−0.280128	−0.140064	−	0.990142i	$-0.544731\pi$
$7$	−0.741150	−0.280128	−0.140064	+	0.990142i	$0.544731\pi$
$8$	−5.91802	−2.09234
$9$	8.06597	2.68866
$10$	1.24235	0.392866
$11$	−5.43412	−1.63845	−0.819224	−	0.573474i	$-0.805595\pi$
$11$	−5.43412	−1.63845	−0.819224	+	0.573474i	$0.805595\pi$
$12$	−14.4663	−4.17606
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.86745	0.499097
$15$	1.64020	0.423497
$16$	6.21398	1.55350
$17$	2.04346	0.495611	0.247805	−	0.968810i	$-0.420291\pi$
$17$	2.04346	0.495611	0.247805	+	0.968810i	$0.420291\pi$
$18$	−20.3236	−4.79031
$19$	−3.88601	−0.891512	−0.445756	−	0.895154i	$-0.647065\pi$
$19$	−3.88601	−0.891512	−0.445756	+	0.895154i	$0.647065\pi$
$20$	−2.14419	−0.479456
$21$	2.46548	0.538011
$22$	13.6922	2.91918
$23$	−2.90672	−0.606093	−0.303046	−	0.952976i	$-0.598004\pi$
$23$	−2.90672	−0.606093	−0.303046	+	0.952976i	$0.598004\pi$
$24$	19.6866	4.01851
$25$	−4.75689	−0.951378
$26$	−2.51967	−0.494148
$27$	−16.8522	−3.24322
$28$	−3.22306	−0.609101
$29$	5.42123	1.00670	0.503348	−	0.864084i	$-0.332101\pi$
$29$	5.42123	1.00670	0.503348	+	0.864084i	$0.332101\pi$
$30$	−4.13276	−0.754535
$31$	−3.23353	−0.580759	−0.290380	−	0.956912i	$-0.593782\pi$
$31$	−3.23353	−0.580759	−0.290380	+	0.956912i	$0.593782\pi$
$32$	−3.82114	−0.675489
$33$	18.0769	3.14678
$34$	−5.14883	−0.883017
$35$	0.365433	0.0617694
$36$	35.0767	5.84612
$37$	−9.23599	−1.51839	−0.759194	−	0.650865i	$-0.774407\pi$
$37$	−9.23599	−1.51839	−0.759194	+	0.650865i	$0.774407\pi$
$38$	9.79146	1.58838
$39$	−3.32656	−0.532675
$40$	2.91795	0.461368
$41$	−9.92194	−1.54955	−0.774773	−	0.632239i	$-0.782136\pi$
$41$	−9.92194	−1.54955	−0.774773	+	0.632239i	$0.782136\pi$
$42$	−6.21218	−0.958561
$43$	−6.04449	−0.921776	−0.460888	−	0.887458i	$-0.652469\pi$
$43$	−6.04449	−0.921776	−0.460888	+	0.887458i	$0.652469\pi$
$44$	−23.6315	−3.56258
$45$	−3.97703	−0.592860
$46$	7.32397	1.07986
$47$	−0.0696159	−0.0101545	−0.00507726	−	0.999987i	$-0.501616\pi$
$47$	−0.0696159	−0.0101545	−0.00507726	+	0.999987i	$0.501616\pi$
$48$	−20.6712	−2.98363
$49$	−6.45070	−0.921528
$50$	11.9858	1.69505
$51$	−6.79767	−0.951864
$52$	4.34873	0.603060
$53$	1.79448	0.246490	0.123245	−	0.992376i	$-0.460670\pi$
$53$	1.79448	0.246490	0.123245	+	0.992376i	$0.460670\pi$
$54$	42.4621	5.77835
$55$	2.67936	0.361284
$56$	4.38614	0.586122
$57$	12.9270	1.71223
$58$	−13.6597	−1.79361
$59$	14.2594	1.85642	0.928209	−	0.372058i	$-0.121348\pi$
$59$	14.2594	1.85642	0.928209	+	0.372058i	$0.121348\pi$
$60$	7.13278	0.920838
$61$	6.00394	0.768726	0.384363	−	0.923182i	$-0.374421\pi$
$61$	6.00394	0.768726	0.384363	+	0.923182i	$0.374421\pi$
$62$	8.14743	1.03472
$63$	−5.97809	−0.753169
$64$	−2.79995	−0.349994
$65$	−0.493062	−0.0611568
$66$	−45.5478	−5.60654
$67$	−8.29507	−1.01340	−0.506702	−	0.862121i	$-0.669136\pi$
$67$	−8.29507	−1.01340	−0.506702	+	0.862121i	$0.669136\pi$
$68$	8.88644	1.07764
$69$	9.66936	1.16405
$70$	−0.920770	−0.110053
$71$	2.25027	0.267058	0.133529	−	0.991045i	$-0.457369\pi$
$71$	2.25027	0.267058	0.133529	+	0.991045i	$0.457369\pi$
$72$	−47.7346	−5.62557
$73$	−13.3587	−1.56351	−0.781757	−	0.623583i	$-0.785676\pi$
$73$	−13.3587	−1.56351	−0.781757	+	0.623583i	$0.785676\pi$
$74$	23.2716	2.70527
$75$	15.8241	1.82720
$76$	−16.8992	−1.93847
$77$	4.02749	0.458976
$78$	8.38182	0.949054
$79$	−1.85915	−0.209170	−0.104585	−	0.994516i	$-0.533351\pi$
$79$	−1.85915	−0.209170	−0.104585	+	0.994516i	$0.533351\pi$
$80$	−3.06388	−0.342552
$81$	31.8620	3.54022
$82$	25.0000	2.76079
$83$	0.955094	0.104835	0.0524176	−	0.998625i	$-0.483307\pi$
$83$	0.955094	0.104835	0.0524176	+	0.998625i	$0.483307\pi$
$84$	10.7217	1.16983
$85$	−1.00755	−0.109284
$86$	15.2301	1.64231
$87$	−18.0340	−1.93345
$88$	32.1592	3.42818
$89$	−4.34620	−0.460696	−0.230348	−	0.973108i	$-0.573986\pi$
$89$	−4.34620	−0.460696	−0.230348	+	0.973108i	$0.573986\pi$
$90$	10.0208	1.05628
$91$	−0.741150	−0.0776936
$92$	−12.6405	−1.31787
$93$	10.7565	1.11540
$94$	0.175409	0.0180921
$95$	1.91605	0.196582
$96$	12.7112	1.29734
$97$	−6.77944	−0.688348	−0.344174	−	0.938906i	$-0.611841\pi$
$97$	−6.77944	−0.688348	−0.344174	+	0.938906i	$0.611841\pi$
$98$	16.2536	1.64186
$99$	−43.8314	−4.40523
$100$	−20.6864	−2.06864
$101$	9.15152	0.910611	0.455305	−	0.890335i	$-0.349530\pi$
$101$	9.15152	0.910611	0.455305	+	0.890335i	$0.349530\pi$
$102$	17.1279	1.69591
$103$	−9.81769	−0.967366	−0.483683	−	0.875243i	$-0.660701\pi$
$103$	−9.81769	−0.967366	−0.483683	+	0.875243i	$0.660701\pi$
$104$	−5.91802	−0.580309
$105$	−1.21563	−0.118634
$106$	−4.52148	−0.439165
$107$	−1.94925	−0.188441	−0.0942206	−	0.995551i	$-0.530036\pi$
$107$	−1.94925	−0.188441	−0.0942206	+	0.995551i	$0.530036\pi$
$108$	−73.2858	−7.05193
$109$	18.4079	1.76316	0.881580	−	0.472034i	$-0.156480\pi$
$109$	18.4079	1.76316	0.881580	+	0.472034i	$0.156480\pi$
$110$	−6.75109	−0.643691
$111$	30.7240	2.91620
$112$	−4.60549	−0.435178
$113$	−13.0078	−1.22367	−0.611833	−	0.790987i	$-0.709568\pi$
$113$	−13.0078	−1.22367	−0.611833	+	0.790987i	$0.709568\pi$
$114$	−32.5718	−3.05063
$115$	1.43319	0.133646
$116$	23.5754	2.18892
$117$	8.06597	0.745699
$118$	−35.9290	−3.30753
$119$	−1.51451	−0.138835
$120$	−9.70672	−0.886099
$121$	18.5296	1.68451
$122$	−15.1279	−1.36962
$123$	33.0059	2.97604
$124$	−14.0617	−1.26278
$125$	4.81075	0.430287
$126$	15.0628	1.34190
$127$	−1.74965	−0.155256	−0.0776282	−	0.996982i	$-0.524735\pi$
$127$	−1.74965	−0.155256	−0.0776282	+	0.996982i	$0.524735\pi$
$128$	14.6972	1.29906
$129$	20.1073	1.77035
$130$	1.24235	0.108962
$131$	0.00153299	0.000133938 0	6.69688e−5	−	1.00000i	$-0.499979\pi$
$131$	0.00153299	0.000133938 0	6.69688e−5		1.00000i	$0.499979\pi$
$132$	78.6115	6.84225
$133$	2.88012	0.249738
$134$	20.9008	1.80556
$135$	8.30920	0.715142
$136$	−12.0932	−1.03698
$137$	1.28089	0.109434	0.0547171	−	0.998502i	$-0.482574\pi$
$137$	1.28089	0.109434	0.0547171	+	0.998502i	$0.482574\pi$
$138$	−24.3636	−2.07397
$139$	6.54182	0.554870	0.277435	−	0.960744i	$-0.410516\pi$
$139$	6.54182	0.554870	0.277435	+	0.960744i	$0.410516\pi$
$140$	1.58917	0.134309
$141$	0.231581	0.0195027
$142$	−5.66993	−0.475810
$143$	−5.43412	−0.454424
$144$	50.1218	4.17682
$145$	−2.67300	−0.221981
$146$	33.6594	2.78567
$147$	21.4586	1.76988
$148$	−40.1648	−3.30153
$149$	1.26348	0.103508	0.0517542	−	0.998660i	$-0.483519\pi$
$149$	1.26348	0.103508	0.0517542	+	0.998660i	$0.483519\pi$
$150$	−39.8714	−3.25548
$151$	−17.8796	−1.45502	−0.727510	−	0.686097i	$-0.759323\pi$
$151$	−17.8796	−1.45502	−0.727510	+	0.686097i	$0.759323\pi$
$152$	22.9975	1.86534
$153$	16.4825	1.33253
$154$	−10.1480	−0.817745
$155$	1.59433	0.128060
$156$	−14.4663	−1.15823
$157$	−16.6645	−1.32997	−0.664986	−	0.746855i	$-0.731563\pi$
$157$	−16.6645	−1.32997	−0.664986	+	0.746855i	$0.731563\pi$
$158$	4.68443	0.372674
$159$	−5.96942	−0.473406
$160$	1.88406	0.148948
$161$	2.15431	0.169784
$162$	−80.2817	−6.30752
$163$	−22.1401	−1.73415	−0.867074	−	0.498180i	$-0.834002\pi$
$163$	−22.1401	−1.73415	−0.867074	+	0.498180i	$0.834002\pi$
$164$	−43.1478	−3.36928
$165$	−8.91303	−0.693879
$166$	−2.40652	−0.186782
$167$	−18.4659	−1.42893	−0.714467	−	0.699670i	$-0.753331\pi$
$167$	−18.4659	−1.42893	−0.714467	+	0.699670i	$0.753331\pi$
$168$	−14.5907	−1.12570
$169$	1.00000	0.0769231
$170$	2.53869	0.194709
$171$	−31.3445	−2.39697
$172$	−26.2859	−2.00428
$173$	21.3102	1.62019	0.810093	−	0.586301i	$-0.199416\pi$
$173$	21.3102	1.62019	0.810093	+	0.586301i	$0.199416\pi$
$174$	45.4397	3.44478
$175$	3.52557	0.266508
$176$	−33.7675	−2.54532
$177$	−47.4348	−3.56542
$178$	10.9510	0.820810
$179$	−6.80010	−0.508264	−0.254132	−	0.967170i	$-0.581790\pi$
$179$	−6.80010	−0.508264	−0.254132	+	0.967170i	$0.581790\pi$
$180$	−17.2950	−1.28909
$181$	−9.47245	−0.704081	−0.352041	−	0.935985i	$-0.614512\pi$
$181$	−9.47245	−0.704081	−0.352041	+	0.935985i	$0.614512\pi$
$182$	1.86745	0.138425
$183$	−19.9724	−1.47641
$184$	17.2020	1.26815
$185$	4.55392	0.334811
$186$	−27.1029	−1.98728
$187$	−11.1044	−0.812033
$188$	−0.302741	−0.0220796
$189$	12.4900	0.908516
$190$	−4.82780	−0.350245
$191$	−17.9790	−1.30091	−0.650456	−	0.759544i	$-0.725422\pi$
$191$	−17.9790	−1.30091	−0.650456	+	0.759544i	$0.725422\pi$
$192$	9.31420	0.672194
$193$	−14.8408	−1.06826	−0.534131	−	0.845402i	$-0.679361\pi$
$193$	−14.8408	−1.06826	−0.534131	+	0.845402i	$0.679361\pi$
$194$	17.0819	1.22641
$195$	1.64020	0.117457
$196$	−28.0523	−2.00374
$197$	−19.9555	−1.42177	−0.710884	−	0.703309i	$-0.751705\pi$
$197$	−19.9555	−1.42177	−0.710884	+	0.703309i	$0.751705\pi$
$198$	110.441	7.84868
$199$	−7.62012	−0.540176	−0.270088	−	0.962836i	$-0.587053\pi$
$199$	−7.62012	−0.540176	−0.270088	+	0.962836i	$0.587053\pi$
$200$	28.1514	1.99060
$201$	27.5940	1.94633
$202$	−23.0588	−1.62241
$203$	−4.01794	−0.282004
$204$	−29.5612	−2.06970
$205$	4.89213	0.341681
$206$	24.7373	1.72353
$207$	−23.4455	−1.62958
$208$	6.21398	0.430862
$209$	21.1170	1.46070
$210$	3.06299	0.211366
$211$	−9.04646	−0.622784	−0.311392	−	0.950282i	$-0.600795\pi$
$211$	−9.04646	−0.622784	−0.311392	+	0.950282i	$0.600795\pi$
$212$	7.80369	0.535959
$213$	−7.48564	−0.512908
$214$	4.91146	0.335741
$215$	2.98031	0.203255
$216$	99.7319	6.78589
$217$	2.39653	0.162687
$218$	−46.3819	−3.14138
$219$	44.4384	3.00287
$220$	11.6518	0.785564
$221$	2.04346	0.137458
$222$	−77.4144	−5.19571
$223$	18.8630	1.26316	0.631579	−	0.775311i	$-0.282407\pi$
$223$	18.8630	1.26316	0.631579	+	0.775311i	$0.282407\pi$
$224$	2.83204	0.189224
$225$	−38.3689	−2.55793
$226$	32.7752	2.18018
$227$	−21.5536	−1.43056	−0.715282	−	0.698836i	$-0.753702\pi$
$227$	−21.5536	−1.43056	−0.715282	+	0.698836i	$0.753702\pi$
$228$	56.2162	3.72301
$229$	−8.92387	−0.589706	−0.294853	−	0.955543i	$-0.595271\pi$
$229$	−8.92387	−0.589706	−0.294853	+	0.955543i	$0.595271\pi$
$230$	−3.61117	−0.238114
$231$	−13.3977	−0.881503
$232$	−32.0829	−2.10635
$233$	23.9526	1.56919	0.784594	−	0.620009i	$-0.212871\pi$
$233$	23.9526	1.56919	0.784594	+	0.620009i	$0.212871\pi$
$234$	−20.3236	−1.32859
$235$	0.0343250	0.00223911
$236$	62.0104	4.03653
$237$	6.18456	0.401730
$238$	3.81606	0.247358
$239$	−13.3013	−0.860388	−0.430194	−	0.902736i	$-0.641555\pi$
$239$	−13.3013	−0.860388	−0.430194	+	0.902736i	$0.641555\pi$
$240$	10.1922	0.657902
$241$	−21.5095	−1.38555	−0.692775	−	0.721154i	$-0.743612\pi$
$241$	−21.5095	−1.38555	−0.692775	+	0.721154i	$0.743612\pi$
$242$	−46.6885	−3.00125
$243$	−55.4340	−3.55609
$244$	26.1095	1.67149
$245$	3.18059	0.203201
$246$	−83.1639	−5.30234
$247$	−3.88601	−0.247261
$248$	19.1361	1.21514
$249$	−3.17717	−0.201345
$250$	−12.1215	−0.766631
$251$	0.590886	0.0372964	0.0186482	−	0.999826i	$-0.494064\pi$
$251$	0.590886	0.0372964	0.0186482	+	0.999826i	$0.494064\pi$
$252$	−25.9971	−1.63766
$253$	15.7955	0.993051
$254$	4.40854	0.276616
$255$	3.35167	0.209890
$256$	−31.4323	−1.96452
$257$	−7.05027	−0.439784	−0.219892	−	0.975524i	$-0.570570\pi$
$257$	−7.05027	−0.439784	−0.219892	+	0.975524i	$0.570570\pi$
$258$	−50.6638	−3.15419
$259$	6.84525	0.425343
$260$	−2.14419	−0.132977
$261$	43.7275	2.70666
$262$	−0.00386262	−0.000238633 0
$263$	−30.1192	−1.85723	−0.928613	−	0.371049i	$-0.878998\pi$
$263$	−30.1192	−1.85723	−0.928613	+	0.371049i	$0.878998\pi$
$264$	−106.979	−6.58413
$265$	−0.884788	−0.0543521
$266$	−7.25694	−0.444952
$267$	14.4579	0.884807
$268$	−36.0730	−2.20351
$269$	−20.4200	−1.24503	−0.622516	−	0.782607i	$-0.713889\pi$
$269$	−20.4200	−1.24503	−0.622516	+	0.782607i	$0.713889\pi$
$270$	−20.9364	−1.27415
$271$	14.0991	0.856459	0.428230	−	0.903670i	$-0.359137\pi$
$271$	14.0991	0.856459	0.428230	+	0.903670i	$0.359137\pi$
$272$	12.6980	0.769930
$273$	2.46548	0.149217
$274$	−3.22743	−0.194976
$275$	25.8495	1.55878
$276$	42.0494	2.53108
$277$	−2.05845	−0.123680	−0.0618402	−	0.998086i	$-0.519697\pi$
$277$	−2.05845	−0.123680	−0.0618402	+	0.998086i	$0.519697\pi$
$278$	−16.4832	−0.988598
$279$	−26.0816	−1.56146
$280$	−2.16264	−0.129242
$281$	−3.17940	−0.189667	−0.0948336	−	0.995493i	$-0.530232\pi$
$281$	−3.17940	−0.189667	−0.0948336	+	0.995493i	$0.530232\pi$
$282$	−0.583508	−0.0347474
$283$	12.7244	0.756389	0.378194	−	0.925726i	$-0.376545\pi$
$283$	12.7244	0.756389	0.378194	+	0.925726i	$0.376545\pi$
$284$	9.78580	0.580680
$285$	−6.37383	−0.377553
$286$	13.6922	0.809635
$287$	7.35365	0.434072
$288$	−30.8212	−1.81616
$289$	−12.8243	−0.754370
$290$	6.73508	0.395497
$291$	22.5522	1.32203
$292$	−58.0932	−3.39965
$293$	−18.6337	−1.08859	−0.544297	−	0.838893i	$-0.683204\pi$
$293$	−18.6337	−1.08859	−0.544297	+	0.838893i	$0.683204\pi$
$294$	−54.0686	−3.15334
$295$	−7.03078	−0.409348
$296$	54.6588	3.17698
$297$	91.5771	5.31384
$298$	−3.18356	−0.184418
$299$	−2.90672	−0.168100
$300$	68.8145	3.97301
$301$	4.47987	0.258216
$302$	45.0507	2.59237
$303$	−30.4431	−1.74891
$304$	−24.1476	−1.38496
$305$	−2.96032	−0.169507
$306$	−41.5303	−2.37413
$307$	−16.5713	−0.945776	−0.472888	−	0.881122i	$-0.656789\pi$
$307$	−16.5713	−0.945776	−0.472888	+	0.881122i	$0.656789\pi$
$308$	17.5145	0.997980
$309$	32.6591	1.85791
$310$	−4.01719	−0.228161
$311$	9.05177	0.513279	0.256639	−	0.966507i	$-0.417385\pi$
$311$	9.05177	0.513279	0.256639	+	0.966507i	$0.417385\pi$
$312$	19.6866	1.11454
$313$	32.2247	1.82145	0.910723	−	0.413017i	$-0.135525\pi$
$313$	32.2247	1.82145	0.910723	+	0.413017i	$0.135525\pi$
$314$	41.9890	2.36958
$315$	2.94757	0.166077
$316$	−8.08493	−0.454813
$317$	−2.80029	−0.157280	−0.0786399	−	0.996903i	$-0.525058\pi$
$317$	−2.80029	−0.157280	−0.0786399	+	0.996903i	$0.525058\pi$
$318$	15.0410	0.843455
$319$	−29.4596	−1.64942
$320$	1.38055	0.0771751
$321$	6.48429	0.361918
$322$	−5.42816	−0.302499
$323$	−7.94090	−0.441843
$324$	138.559	7.69773
$325$	−4.75689	−0.263865
$326$	55.7857	3.08969
$327$	−61.2350	−3.38630
$328$	58.7182	3.24217
$329$	0.0515958	0.00284457
$330$	22.4579	1.23627
$331$	−13.1732	−0.724062	−0.362031	−	0.932166i	$-0.617917\pi$
$331$	−13.1732	−0.724062	−0.362031	+	0.932166i	$0.617917\pi$
$332$	4.15344	0.227950
$333$	−74.4973	−4.08242
$334$	46.5279	2.54589
$335$	4.08999	0.223460
$336$	15.3204	0.835798
$337$	24.8668	1.35458	0.677290	−	0.735716i	$-0.263154\pi$
$337$	24.8668	1.35458	0.677290	+	0.735716i	$0.263154\pi$
$338$	−2.51967	−0.137052
$339$	43.2710	2.35016
$340$	−4.38156	−0.237624
$341$	17.5714	0.951544
$342$	78.9777	4.27062
$343$	9.96898	0.538274
$344$	35.7714	1.92866
$345$	−4.76760	−0.256679
$346$	−53.6947	−2.88664
$347$	−10.9940	−0.590191	−0.295096	−	0.955468i	$-0.595351\pi$
$347$	−10.9940	−0.590191	−0.295096	+	0.955468i	$0.595351\pi$
$348$	−78.4250	−4.20402
$349$	−9.29020	−0.497293	−0.248647	−	0.968594i	$-0.579986\pi$
$349$	−9.29020	−0.497293	−0.248647	+	0.968594i	$0.579986\pi$
$350$	−8.88326	−0.474830
$351$	−16.8522	−0.899506
$352$	20.7645	1.10675
$353$	−31.9099	−1.69839	−0.849196	−	0.528078i	$-0.822913\pi$
$353$	−31.9099	−1.69839	−0.849196	+	0.528078i	$0.822913\pi$
$354$	119.520	6.35241
$355$	−1.10952	−0.0588873
$356$	−18.9004	−1.00172
$357$	5.03809	0.266644
$358$	17.1340	0.905560
$359$	18.3523	0.968599	0.484299	−	0.874902i	$-0.339075\pi$
$359$	18.3523	0.968599	0.484299	+	0.874902i	$0.339075\pi$
$360$	23.5361	1.24046
$361$	−3.89891	−0.205206
$362$	23.8674	1.25444
$363$	−61.6399	−3.23525
$364$	−3.22306	−0.168934
$365$	6.58665	0.344761
$366$	50.3239	2.63047
$367$	−19.3866	−1.01197	−0.505986	−	0.862542i	$-0.668871\pi$
$367$	−19.3866	−1.01197	−0.505986	+	0.862542i	$0.668871\pi$
$368$	−18.0623	−0.941563
$369$	−80.0301	−4.16620
$370$	−11.4744	−0.596524
$371$	−1.32998	−0.0690489
$372$	46.7772	2.42528
$373$	−6.63936	−0.343773	−0.171886	−	0.985117i	$-0.554986\pi$
$373$	−6.63936	−0.343773	−0.171886	+	0.985117i	$0.554986\pi$
$374$	27.9794	1.44678
$375$	−16.0032	−0.826404
$376$	0.411988	0.0212467
$377$	5.42123	0.279207
$378$	−31.4707	−1.61868
$379$	19.3307	0.992954	0.496477	−	0.868050i	$-0.334627\pi$
$379$	19.3307	0.992954	0.496477	+	0.868050i	$0.334627\pi$
$380$	8.33236	0.427441
$381$	5.82031	0.298184
$382$	45.3010	2.31780
$383$	−5.64391	−0.288390	−0.144195	−	0.989549i	$-0.546059\pi$
$383$	−5.64391	−0.288390	−0.144195	+	0.989549i	$0.546059\pi$
$384$	−48.8912	−2.49497
$385$	−1.98580	−0.101206
$386$	37.3938	1.90330
$387$	−48.7547	−2.47834
$388$	−29.4819	−1.49672
$389$	−2.74648	−0.139252	−0.0696261	−	0.997573i	$-0.522181\pi$
$389$	−2.74648	−0.139252	−0.0696261	+	0.997573i	$0.522181\pi$
$390$	−4.13276	−0.209270
$391$	−5.93975	−0.300386
$392$	38.1753	1.92815
$393$	−0.00509956	−0.000257239 0
$394$	50.2811	2.53313
$395$	0.916675	0.0461229
$396$	−190.611	−9.57857
$397$	−2.62569	−0.131780	−0.0658899	−	0.997827i	$-0.520989\pi$
$397$	−2.62569	−0.131780	−0.0658899	+	0.997827i	$0.520989\pi$
$398$	19.2002	0.962417
$399$	−9.58087	−0.479643
$400$	−29.5592	−1.47796
$401$	−0.302156	−0.0150889	−0.00754447	−	0.999972i	$-0.502402\pi$
$401$	−0.302156	−0.0150889	−0.00754447	+	0.999972i	$0.502402\pi$
$402$	−69.5278	−3.46773
$403$	−3.23353	−0.161074
$404$	39.7975	1.98000
$405$	−15.7099	−0.780633
$406$	10.1239	0.502440
$407$	50.1895	2.48780
$408$	40.2287	1.99162
$409$	−9.12867	−0.451384	−0.225692	−	0.974199i	$-0.572464\pi$
$409$	−9.12867	−0.451384	−0.225692	+	0.974199i	$0.572464\pi$
$410$	−12.3266	−0.608765
$411$	−4.26097	−0.210178
$412$	−42.6945	−2.10341
$413$	−10.5684	−0.520035
$414$	59.0749	2.90337
$415$	−0.470920	−0.0231166
$416$	−3.82114	−0.187347
$417$	−21.7617	−1.06568
$418$	−53.2079	−2.60249
$419$	−9.87514	−0.482432	−0.241216	−	0.970471i	$-0.577546\pi$
$419$	−9.87514	−0.482432	−0.241216	+	0.970471i	$0.577546\pi$
$420$	−5.28646	−0.257953
$421$	36.8715	1.79701	0.898503	−	0.438967i	$-0.144656\pi$
$421$	36.8715	1.79701	0.898503	+	0.438967i	$0.144656\pi$
$422$	22.7941	1.10960
$423$	−0.561520	−0.0273020
$424$	−10.6197	−0.515740
$425$	−9.72050	−0.471513
$426$	18.8613	0.913834
$427$	−4.44982	−0.215342
$428$	−8.47676	−0.409740
$429$	18.0769	0.872761
$430$	−7.50939	−0.362135
$431$	−7.30293	−0.351770	−0.175885	−	0.984411i	$-0.556279\pi$
$431$	−7.30293	−0.351770	−0.175885	+	0.984411i	$0.556279\pi$
$432$	−104.720	−5.03832
$433$	21.4486	1.03076	0.515378	−	0.856963i	$-0.327652\pi$
$433$	21.4486	1.03076	0.515378	+	0.856963i	$0.327652\pi$
$434$	−6.03846	−0.289855
$435$	8.89189	0.426333
$436$	80.0511	3.83375
$437$	11.2955	0.540339
$438$	−111.970	−5.35013
$439$	−9.93106	−0.473984	−0.236992	−	0.971512i	$-0.576161\pi$
$439$	−9.93106	−0.473984	−0.236992	+	0.971512i	$0.576161\pi$
$440$	−15.8565	−0.755928
$441$	−52.0311	−2.47767
$442$	−5.14883	−0.244905
$443$	30.4813	1.44821	0.724105	−	0.689689i	$-0.242253\pi$
$443$	30.4813	1.44821	0.724105	+	0.689689i	$0.242253\pi$
$444$	133.611	6.34087
$445$	2.14295	0.101585
$446$	−47.5285	−2.25054
$447$	−4.20304	−0.198797
$448$	2.07518	0.0980432
$449$	33.9033	1.59999	0.799997	−	0.600004i	$-0.204834\pi$
$449$	33.9033	1.59999	0.799997	+	0.600004i	$0.204834\pi$
$450$	96.6770	4.55740
$451$	53.9170	2.53885
$452$	−56.5672	−2.66070
$453$	59.4775	2.79450
$454$	54.3080	2.54880
$455$	0.365433	0.0171318
$456$	−76.5024	−3.58255
$457$	35.4389	1.65776	0.828880	−	0.559426i	$-0.188979\pi$
$457$	35.4389	1.65776	0.828880	+	0.559426i	$0.188979\pi$
$458$	22.4852	1.05066
$459$	−34.4368	−1.60737
$460$	6.23257	0.290595
$461$	−9.39047	−0.437358	−0.218679	−	0.975797i	$-0.570175\pi$
$461$	−9.39047	−0.437358	−0.218679	+	0.975797i	$0.570175\pi$
$462$	33.7577	1.57055
$463$	−1.00000	−0.0464739
$463$	−1.00000	−0.0464739
$464$	33.6874	1.56390
$465$	−5.30363	−0.245950
$466$	−60.3527	−2.79578
$467$	−2.47796	−0.114666	−0.0573332	−	0.998355i	$-0.518260\pi$
$467$	−2.47796	−0.114666	−0.0573332	+	0.998355i	$0.518260\pi$
$468$	35.0767	1.62142
$469$	6.14789	0.283883
$470$	−0.0864875	−0.00398937
$471$	55.4354	2.55433
$472$	−84.3875	−3.88425
$473$	32.8465	1.51028
$474$	−15.5830	−0.715752
$475$	18.4853	0.848165
$476$	−6.58618	−0.301877
$477$	14.4742	0.662728
$478$	33.5148	1.53293
$479$	−34.6431	−1.58288	−0.791442	−	0.611244i	$-0.790670\pi$
$479$	−34.6431	−1.58288	−0.791442	+	0.611244i	$0.790670\pi$
$480$	−6.26743	−0.286068
$481$	−9.23599	−0.421125
$482$	54.1968	2.46860
$483$	−7.16645	−0.326085
$484$	80.5803	3.66274
$485$	3.34268	0.151783
$486$	139.675	6.33580
$487$	−35.7567	−1.62029	−0.810146	−	0.586228i	$-0.800612\pi$
$487$	−35.7567	−1.62029	−0.810146	+	0.586228i	$0.800612\pi$
$488$	−35.5314	−1.60843
$489$	73.6503	3.33058
$490$	−8.01404	−0.362038
$491$	10.5553	0.476353	0.238177	−	0.971222i	$-0.423450\pi$
$491$	10.5553	0.476353	0.238177	+	0.971222i	$0.423450\pi$
$492$	143.534	6.47100
$493$	11.0780	0.498930
$494$	9.79146	0.440539
$495$	21.6116	0.971370
$496$	−20.0931	−0.902207
$497$	−1.66779	−0.0748104
$498$	8.00542	0.358731
$499$	−31.8487	−1.42574	−0.712871	−	0.701295i	$-0.752606\pi$
$499$	−31.8487	−1.42574	−0.712871	+	0.701295i	$0.752606\pi$
$500$	20.9207	0.935600
$501$	61.4278	2.74439
$502$	−1.48884	−0.0664500
$503$	−7.93009	−0.353585	−0.176793	−	0.984248i	$-0.556572\pi$
$503$	−7.93009	−0.353585	−0.176793	+	0.984248i	$0.556572\pi$
$504$	35.3785	1.57588
$505$	−4.51227	−0.200793
$506$	−39.7993	−1.76929
$507$	−3.32656	−0.147738
$508$	−7.60876	−0.337584
$509$	−3.29458	−0.146030	−0.0730149	−	0.997331i	$-0.523262\pi$
$509$	−3.29458	−0.146030	−0.0730149	+	0.997331i	$0.523262\pi$
$510$	−8.44511	−0.373956
$511$	9.90078	0.437985
$512$	49.8044	2.20106
$513$	65.4880	2.89137
$514$	17.7643	0.783551
$515$	4.84073	0.213308
$516$	87.4414	3.84939
$517$	0.378301	0.0166377
$518$	−17.2478	−0.757823
$519$	−70.8896	−3.11171
$520$	2.91795	0.127961
$521$	−35.9607	−1.57547	−0.787733	−	0.616017i	$-0.788745\pi$
$521$	−35.9607	−1.57547	−0.787733	+	0.616017i	$0.788745\pi$
$522$	−110.179	−4.82239
$523$	28.5470	1.24827	0.624136	−	0.781316i	$-0.285451\pi$
$523$	28.5470	1.24827	0.624136	+	0.781316i	$0.285451\pi$
$524$	0.00666654	0.000291229 0
$525$	−11.7280	−0.511852
$526$	75.8903	3.30897
$527$	−6.60758	−0.287831
$528$	112.330	4.88852
$529$	−14.5510	−0.632652
$530$	2.22937	0.0968377
$531$	115.016	4.99127
$532$	12.5248	0.543021
$533$	−9.92194	−0.429767
$534$	−36.4290	−1.57644
$535$	0.961101	0.0415520
$536$	49.0904	2.12038
$537$	22.6209	0.976165
$538$	51.4517	2.21824
$539$	35.0538	1.50988
$540$	36.1345	1.55498
$541$	22.1350	0.951656	0.475828	−	0.879538i	$-0.342148\pi$
$541$	22.1350	0.951656	0.475828	+	0.879538i	$0.342148\pi$
$542$	−35.5251	−1.52593
$543$	31.5106	1.35225
$544$	−7.80834	−0.334780
$545$	−9.07625	−0.388784
$546$	−6.21218	−0.265857
$547$	−33.4128	−1.42863	−0.714314	−	0.699825i	$-0.753261\pi$
$547$	−33.4128	−1.42863	−0.714314	+	0.699825i	$0.753261\pi$
$548$	5.57026	0.237950
$549$	48.4276	2.06684
$550$	−65.1322	−2.77724
$551$	−21.0669	−0.897482
$552$	−57.2235	−2.43559
$553$	1.37791	0.0585946
$554$	5.18662	0.220358
$555$	−15.1489	−0.643033
$556$	28.4486	1.20649
$557$	21.2907	0.902117	0.451059	−	0.892494i	$-0.351047\pi$
$557$	21.2907	0.902117	0.451059	+	0.892494i	$0.351047\pi$
$558$	65.7169	2.78202
$559$	−6.04449	−0.255655
$560$	2.27079	0.0959586
$561$	36.9393	1.55958
$562$	8.01104	0.337925
$563$	−21.6782	−0.913626	−0.456813	−	0.889563i	$-0.651009\pi$
$563$	−21.6782	−0.913626	−0.456813	+	0.889563i	$0.651009\pi$
$564$	1.00708	0.0424059
$565$	6.41363	0.269823
$566$	−32.0613	−1.34764
$567$	−23.6145	−0.991717
$568$	−13.3171	−0.558774
$569$	10.3745	0.434922	0.217461	−	0.976069i	$-0.430222\pi$
$569$	10.3745	0.434922	0.217461	+	0.976069i	$0.430222\pi$
$570$	16.0599	0.672677
$571$	8.95764	0.374866	0.187433	−	0.982277i	$-0.439983\pi$
$571$	8.95764	0.374866	0.187433	+	0.982277i	$0.439983\pi$
$572$	−23.6315	−0.988083
$573$	59.8080	2.49852
$574$	−18.5287	−0.773375
$575$	13.8269	0.576623
$576$	−22.5843	−0.941014
$577$	−30.3835	−1.26488	−0.632441	−	0.774609i	$-0.717947\pi$
$577$	−30.3835	−1.26488	−0.632441	+	0.774609i	$0.717947\pi$
$578$	32.3129	1.34404
$579$	49.3687	2.05169
$580$	−11.6242	−0.482667
$581$	−0.707867	−0.0293673
$582$	−56.8240	−2.35543
$583$	−9.75139	−0.403861
$584$	79.0568	3.27140
$585$	−3.97703	−0.164430
$586$	46.9508	1.93952
$587$	7.71511	0.318437	0.159218	−	0.987243i	$-0.449103\pi$
$587$	7.71511	0.318437	0.159218	+	0.987243i	$0.449103\pi$
$588$	93.3176	3.84835
$589$	12.5655	0.517754
$590$	17.7152	0.729325
$591$	66.3830	2.73063
$592$	−57.3923	−2.35881
$593$	4.20516	0.172685	0.0863427	−	0.996265i	$-0.472482\pi$
$593$	4.20516	0.172685	0.0863427	+	0.996265i	$0.472482\pi$
$594$	−230.744	−9.46753
$595$	0.746746	0.0306136
$596$	5.49454	0.225065
$597$	25.3487	1.03746
$598$	7.32397	0.299499
$599$	−35.7059	−1.45890	−0.729452	−	0.684032i	$-0.760225\pi$
$599$	−35.7059	−1.45890	−0.729452	+	0.684032i	$0.760225\pi$
$600$	−93.6471	−3.82313
$601$	26.8824	1.09655	0.548277	−	0.836297i	$-0.315284\pi$
$601$	26.8824	1.09655	0.548277	+	0.836297i	$0.315284\pi$
$602$	−11.2878	−0.460056
$603$	−66.9078	−2.72470
$604$	−77.7535	−3.16375
$605$	−9.13626	−0.371442
$606$	76.7064	3.11598
$607$	14.7223	0.597561	0.298781	−	0.954322i	$-0.403420\pi$
$607$	14.7223	0.597561	0.298781	+	0.954322i	$0.403420\pi$
$608$	14.8490	0.602207
$609$	13.3659	0.541614
$610$	7.45901	0.302007
$611$	−0.0696159	−0.00281636
$612$	71.6778	2.89740
$613$	7.86960	0.317850	0.158925	−	0.987291i	$-0.449197\pi$
$613$	7.86960	0.317850	0.158925	+	0.987291i	$0.449197\pi$
$614$	41.7543	1.68507
$615$	−16.2740	−0.656229
$616$	−23.8348	−0.960331
$617$	−36.1289	−1.45450	−0.727248	−	0.686375i	$-0.759201\pi$
$617$	−36.1289	−1.45450	−0.727248	+	0.686375i	$0.759201\pi$
$618$	−82.2901	−3.31019
$619$	28.3630	1.14001	0.570003	−	0.821642i	$-0.306942\pi$
$619$	28.3630	1.14001	0.570003	+	0.821642i	$0.306942\pi$
$620$	6.93331	0.278449
$621$	48.9847	1.96569
$622$	−22.8075	−0.914496
$623$	3.22118	0.129054
$624$	−20.6712	−0.827509
$625$	21.4124	0.856498
$626$	−81.1955	−3.24523
$627$	−70.2470	−2.80540
$628$	−72.4694	−2.89185
$629$	−18.8733	−0.752530
$630$	−7.42690	−0.295895
$631$	33.2151	1.32227	0.661136	−	0.750266i	$-0.270075\pi$
$631$	33.2151	1.32227	0.661136	+	0.750266i	$0.270075\pi$
$632$	11.0025	0.437655
$633$	30.0935	1.19611
$634$	7.05580	0.280222
$635$	0.862686	0.0342347
$636$	−25.9594	−1.02936
$637$	−6.45070	−0.255586
$638$	74.2284	2.93873
$639$	18.1506	0.718026
$640$	−7.24665	−0.286449
$641$	0.0896369	0.00354044	0.00177022	−	0.999998i	$-0.499437\pi$
$641$	0.0896369	0.00354044	0.00177022	+	0.999998i	$0.499437\pi$
$642$	−16.3383	−0.644820
$643$	39.6658	1.56427	0.782133	−	0.623112i	$-0.214132\pi$
$643$	39.6658	1.56427	0.782133	+	0.623112i	$0.214132\pi$
$644$	9.36853	0.369172
$645$	−9.91417	−0.390370
$646$	20.0084	0.787221
$647$	−6.99180	−0.274876	−0.137438	−	0.990510i	$-0.543887\pi$
$647$	−6.99180	−0.274876	−0.137438	+	0.990510i	$0.543887\pi$
$648$	−188.560	−7.40733
$649$	−77.4874	−3.04165
$650$	11.9858	0.470121
$651$	−7.97219	−0.312455
$652$	−96.2813	−3.77067
$653$	17.6434	0.690438	0.345219	−	0.938522i	$-0.387805\pi$
$653$	17.6434	0.690438	0.345219	+	0.938522i	$0.387805\pi$
$654$	154.292	6.03329
$655$	−0.000755857 0	−2.95338e−5 0
$656$	−61.6548	−2.40721
$657$	−107.751	−4.20375
$658$	−0.130004	−0.00506810
$659$	10.1362	0.394849	0.197424	−	0.980318i	$-0.436742\pi$
$659$	10.1362	0.394849	0.197424	+	0.980318i	$0.436742\pi$
$660$	−38.7604	−1.50874
$661$	13.0087	0.505981	0.252991	−	0.967469i	$-0.418586\pi$
$661$	13.0087	0.505981	0.252991	+	0.967469i	$0.418586\pi$
$662$	33.1920	1.29004
$663$	−6.79767	−0.264000
$664$	−5.65226	−0.219350
$665$	−1.42008	−0.0550682
$666$	187.708	7.27355
$667$	−15.7580	−0.610151
$668$	−80.3031	−3.10702
$669$	−62.7488	−2.42601
$670$	−10.3054	−0.398133
$671$	−32.6261	−1.25952
$672$	−9.42094	−0.363421
$673$	−27.4928	−1.05977	−0.529885	−	0.848069i	$-0.677765\pi$
$673$	−27.4928	−1.05977	−0.529885	+	0.848069i	$0.677765\pi$
$674$	−62.6561	−2.41342
$675$	80.1643	3.08552
$676$	4.34873	0.167259
$677$	13.7578	0.528757	0.264378	−	0.964419i	$-0.414833\pi$
$677$	13.7578	0.528757	0.264378	+	0.964419i	$0.414833\pi$
$678$	−109.029	−4.18722
$679$	5.02458	0.192826
$680$	5.96270	0.228659
$681$	71.6994	2.74753
$682$	−44.2741	−1.69534
$683$	37.3545	1.42933	0.714666	−	0.699466i	$-0.246579\pi$
$683$	37.3545	1.42933	0.714666	+	0.699466i	$0.246579\pi$
$684$	−136.309	−5.21189
$685$	−0.631560	−0.0241307
$686$	−25.1185	−0.959030
$687$	29.6858	1.13258
$688$	−37.5604	−1.43198
$689$	1.79448	0.0683641
$690$	12.0128	0.457318
$691$	−17.7565	−0.675488	−0.337744	−	0.941238i	$-0.609664\pi$
$691$	−17.7565	−0.675488	−0.337744	+	0.941238i	$0.609664\pi$
$692$	92.6724	3.52288
$693$	32.4857	1.23403
$694$	27.7013	1.05153
$695$	−3.22552	−0.122351
$696$	106.726	4.04542
$697$	−20.2751	−0.767972
$698$	23.4082	0.886015
$699$	−79.6798	−3.01377
$700$	15.3317	0.579485
$701$	−32.2598	−1.21844	−0.609218	−	0.793003i	$-0.708516\pi$
$701$	−32.2598	−1.21844	−0.609218	+	0.793003i	$0.708516\pi$
$702$	42.4621	1.60263
$703$	35.8912	1.35366
$704$	15.2153	0.573447
$705$	−0.114184	−0.00430041
$706$	80.4023	3.02598
$707$	−6.78265	−0.255088
$708$	−206.281	−7.75251
$709$	−34.6679	−1.30198	−0.650991	−	0.759086i	$-0.725646\pi$
$709$	−34.6679	−1.30198	−0.650991	+	0.759086i	$0.725646\pi$
$710$	2.79563	0.104918
$711$	−14.9958	−0.562388
$712$	25.7209	0.963930
$713$	9.39896	0.351994
$714$	−12.6943	−0.475073
$715$	2.67936	0.100202
$716$	−29.5718	−1.10515
$717$	44.2475	1.65245
$718$	−46.2418	−1.72573
$719$	−25.2363	−0.941155	−0.470578	−	0.882359i	$-0.655954\pi$
$719$	−25.2363	−0.941155	−0.470578	+	0.882359i	$0.655954\pi$
$720$	−24.7132	−0.921006
$721$	7.27638	0.270987
$722$	9.82397	0.365610
$723$	71.5526	2.66107
$724$	−41.1931	−1.53093
$725$	−25.7882	−0.957749
$726$	155.312	5.76417
$727$	−17.9124	−0.664333	−0.332166	−	0.943221i	$-0.607780\pi$
$727$	−17.9124	−0.664333	−0.332166	+	0.943221i	$0.607780\pi$
$728$	4.38614	0.162561
$729$	88.8183	3.28957
$730$	−16.5962	−0.614252
$731$	−12.3517	−0.456842
$732$	−86.8548	−3.21024
$733$	40.5794	1.49883	0.749417	−	0.662098i	$-0.230334\pi$
$733$	40.5794	1.49883	0.749417	+	0.662098i	$0.230334\pi$
$734$	48.8478	1.80300
$735$	−10.5804	−0.390265
$736$	11.1070	0.409409
$737$	45.0764	1.66041
$738$	201.649	7.42282
$739$	43.7444	1.60916	0.804582	−	0.593841i	$-0.202389\pi$
$739$	43.7444	1.60916	0.804582	+	0.593841i	$0.202389\pi$
$740$	19.8038	0.728000
$741$	12.9270	0.474886
$742$	3.35110	0.123023
$743$	−21.3328	−0.782626	−0.391313	−	0.920258i	$-0.627979\pi$
$743$	−21.3328	−0.782626	−0.391313	+	0.920258i	$0.627979\pi$
$744$	−63.6573	−2.33379
$745$	−0.622975	−0.0228240
$746$	16.7290	0.612491
$747$	7.70376	0.281866
$748$	−48.2899	−1.76566
$749$	1.44469	0.0527877
$750$	40.3228	1.47238
$751$	−27.9459	−1.01976	−0.509881	−	0.860245i	$-0.670310\pi$
$751$	−27.9459	−1.01976	−0.509881	+	0.860245i	$0.670310\pi$
$752$	−0.432592	−0.0157750
$753$	−1.96562	−0.0716310
$754$	−13.6597	−0.497457
$755$	8.81575	0.320838
$756$	54.3158	1.97545
$757$	1.60211	0.0582295	0.0291148	−	0.999576i	$-0.490731\pi$
$757$	1.60211	0.0582295	0.0291148	+	0.999576i	$0.490731\pi$
$758$	−48.7071	−1.76912
$759$	−52.5444	−1.90724
$760$	−11.3392	−0.411316
$761$	20.1000	0.728624	0.364312	−	0.931277i	$-0.381304\pi$
$761$	20.1000	0.728624	0.364312	+	0.931277i	$0.381304\pi$
$762$	−14.6653	−0.531266
$763$	−13.6430	−0.493911
$764$	−78.1857	−2.82866
$765$	−8.12688	−0.293828
$766$	14.2208	0.513818
$767$	14.2594	0.514878
$768$	104.561	3.77303
$769$	8.42882	0.303951	0.151975	−	0.988384i	$-0.451437\pi$
$769$	8.42882	0.303951	0.151975	+	0.988384i	$0.451437\pi$
$770$	5.00357	0.180316
$771$	23.4531	0.844643
$772$	−64.5385	−2.32279
$773$	−3.21397	−0.115598	−0.0577992	−	0.998328i	$-0.518408\pi$
$773$	−3.21397	−0.115598	−0.0577992	+	0.998328i	$0.518408\pi$
$774$	122.846	4.41560
$775$	15.3815	0.552522
$776$	40.1208	1.44025
$777$	−22.7711	−0.816909
$778$	6.92023	0.248102
$779$	38.5568	1.38144
$780$	7.13278	0.255394
$781$	−12.2282	−0.437560
$782$	14.9662	0.535190
$783$	−91.3598	−3.26493
$784$	−40.0845	−1.43159
$785$	8.21664	0.293264
$786$	0.0128492	0.000458316 0
$787$	1.93324	0.0689126	0.0344563	−	0.999406i	$-0.489030\pi$
$787$	1.93324	0.0689126	0.0344563	+	0.999406i	$0.489030\pi$
$788$	−86.7809	−3.09144
$789$	100.193	3.56697
$790$	−2.30972	−0.0821761
$791$	9.64070	0.342784
$792$	259.395	9.21721
$793$	6.00394	0.213206
$794$	6.61588	0.234789
$795$	2.94330	0.104388
$796$	−33.1378	−1.17454
$797$	−10.5544	−0.373857	−0.186928	−	0.982374i	$-0.559853\pi$
$797$	−10.5544	−0.373857	−0.186928	+	0.982374i	$0.559853\pi$
$798$	24.1406	0.854568
$799$	−0.142257	−0.00503269
$800$	18.1768	0.642646
$801$	−35.0563	−1.23865
$802$	0.761333	0.0268836
$803$	72.5926	2.56174
$804$	119.999	4.23203
$805$	−1.06221	−0.0374380
$806$	8.14743	0.286981
$807$	67.9284	2.39119
$808$	−54.1589	−1.90530
$809$	21.3672	0.751230	0.375615	−	0.926776i	$-0.377431\pi$
$809$	21.3672	0.751230	0.375615	+	0.926776i	$0.377431\pi$
$810$	39.5839	1.39083
$811$	−35.4183	−1.24370	−0.621852	−	0.783135i	$-0.713619\pi$
$811$	−35.4183	−1.24370	−0.621852	+	0.783135i	$0.713619\pi$
$812$	−17.4729	−0.613180
$813$	−46.9014	−1.64491
$814$	−126.461	−4.43245
$815$	10.9164	0.382387
$816$	−42.2406	−1.47872
$817$	23.4890	0.821775
$818$	23.0012	0.804219
$819$	−5.97809	−0.208892
$820$	21.2746	0.742940
$821$	43.6326	1.52279	0.761395	−	0.648288i	$-0.224515\pi$
$821$	43.6326	1.52279	0.761395	+	0.648288i	$0.224515\pi$
$822$	10.7362	0.374469
$823$	38.7241	1.34984	0.674918	−	0.737892i	$-0.264179\pi$
$823$	38.7241	1.34984	0.674918	+	0.737892i	$0.264179\pi$
$824$	58.1013	2.02405
$825$	−85.9898	−2.99378
$826$	26.6288	0.926534
$827$	42.2187	1.46809	0.734045	−	0.679101i	$-0.237630\pi$
$827$	42.2187	1.46809	0.734045	+	0.679101i	$0.237630\pi$
$828$	−101.958	−3.54329
$829$	−17.7273	−0.615696	−0.307848	−	0.951436i	$-0.599609\pi$
$829$	−17.7273	−0.615696	−0.307848	+	0.951436i	$0.599609\pi$
$830$	1.18656	0.0411862
$831$	6.84755	0.237539
$832$	−2.79995	−0.0970709
$833$	−13.1817	−0.456719
$834$	54.8324	1.89869
$835$	9.10483	0.315086
$836$	91.8323	3.17609
$837$	54.4922	1.88353
$838$	24.8821	0.859537
$839$	23.6283	0.815740	0.407870	−	0.913040i	$-0.366272\pi$
$839$	23.6283	0.815740	0.407870	+	0.913040i	$0.366272\pi$
$840$	7.19414	0.248221
$841$	0.389689	0.0134376
$842$	−92.9039	−3.20168
$843$	10.5765	0.364273
$844$	−39.3406	−1.35416
$845$	−0.493062	−0.0169619
$846$	1.41484	0.0486433
$847$	−13.7332	−0.471879
$848$	11.1508	0.382921
$849$	−42.3285	−1.45271
$850$	24.4924	0.840083
$851$	26.8464	0.920284
$852$	−32.5530	−1.11525
$853$	−43.8466	−1.50128	−0.750639	−	0.660713i	$-0.770254\pi$
$853$	−43.8466	−1.50128	−0.750639	+	0.660713i	$0.770254\pi$
$854$	11.2121	0.383669
$855$	15.4548	0.528542
$856$	11.5357	0.394282
$857$	8.50819	0.290634	0.145317	−	0.989385i	$-0.453580\pi$
$857$	8.50819	0.290634	0.145317	+	0.989385i	$0.453580\pi$
$858$	−45.5478	−1.55498
$859$	−34.2445	−1.16841	−0.584203	−	0.811608i	$-0.698593\pi$
$859$	−34.2445	−1.16841	−0.584203	+	0.811608i	$0.698593\pi$
$860$	12.9606	0.441951
$861$	−24.4623	−0.833673
$862$	18.4010	0.626739
$863$	30.3377	1.03271	0.516353	−	0.856376i	$-0.327289\pi$
$863$	30.3377	1.03271	0.516353	+	0.856376i	$0.327289\pi$
$864$	64.3948	2.19076
$865$	−10.5073	−0.357258
$866$	−54.0434	−1.83647
$867$	42.6607	1.44883
$868$	10.4219	0.353741
$869$	10.1028	0.342715
$870$	−22.4046	−0.759587
$871$	−8.29507	−0.281068
$872$	−108.938	−3.68912
$873$	−54.6828	−1.85073
$874$	−28.4610	−0.962708
$875$	−3.56549	−0.120535
$876$	193.250	6.52932
$877$	−6.28104	−0.212096	−0.106048	−	0.994361i	$-0.533820\pi$
$877$	−6.28104	−0.212096	−0.106048	+	0.994361i	$0.533820\pi$
$878$	25.0230	0.844484
$879$	61.9861	2.09074
$880$	16.6495	0.561254
$881$	−8.75871	−0.295088	−0.147544	−	0.989055i	$-0.547137\pi$
$881$	−8.75871	−0.295088	−0.147544	+	0.989055i	$0.547137\pi$
$882$	131.101	4.41441
$883$	34.2754	1.15346	0.576729	−	0.816935i	$-0.304329\pi$
$883$	34.2754	1.15346	0.576729	+	0.816935i	$0.304329\pi$
$884$	8.88644	0.298883
$885$	23.3883	0.786189
$886$	−76.8028	−2.58024
$887$	−6.20778	−0.208437	−0.104218	−	0.994554i	$-0.533234\pi$
$887$	−6.20778	−0.208437	−0.104218	+	0.994554i	$0.533234\pi$
$888$	−181.825	−6.10166
$889$	1.29675	0.0434917
$890$	−5.39951	−0.180992
$891$	−173.142	−5.80047
$892$	82.0300	2.74657
$893$	0.270528	0.00905288
$894$	10.5903	0.354192
$895$	3.35287	0.112074
$896$	−10.8929	−0.363905
$897$	9.66936	0.322851
$898$	−85.4250	−2.85067
$899$	−17.5297	−0.584648
$900$	−166.856	−5.56187
$901$	3.66693	0.122163
$902$	−135.853	−4.52341
$903$	−14.9025	−0.495926
$904$	76.9801	2.56032
$905$	4.67050	0.155253
$906$	−149.864	−4.97888
$907$	26.4543	0.878402	0.439201	−	0.898389i	$-0.355262\pi$
$907$	26.4543	0.878402	0.439201	+	0.898389i	$0.355262\pi$
$908$	−93.7309	−3.11057
$909$	73.8159	2.44832
$910$	−0.920770	−0.0305232
$911$	−8.83352	−0.292668	−0.146334	−	0.989235i	$-0.546747\pi$
$911$	−8.83352	−0.292668	−0.146334	+	0.989235i	$0.546747\pi$
$912$	80.3284	2.65994
$913$	−5.19009	−0.171767
$914$	−89.2942	−2.95359
$915$	9.84766	0.325553
$916$	−38.8075	−1.28224
$917$	−0.00113617	−3.75197e−5 0
$918$	86.7694	2.86382
$919$	−30.5965	−1.00928	−0.504642	−	0.863329i	$-0.668376\pi$
$919$	−30.5965	−1.00928	−0.504642	+	0.863329i	$0.668376\pi$
$920$	−8.48166	−0.279632
$921$	55.1255	1.81645
$922$	23.6609	0.779229
$923$	2.25027	0.0740684
$924$	−58.2629	−1.91671
$925$	43.9346	1.44456
$926$	2.51967	0.0828014
$927$	−79.1893	−2.60092
$928$	−20.7153	−0.680012
$929$	−27.4111	−0.899328	−0.449664	−	0.893198i	$-0.648456\pi$
$929$	−27.4111	−0.899328	−0.449664	+	0.893198i	$0.648456\pi$
$930$	13.3634	0.438203
$931$	25.0675	0.821554
$932$	104.163	3.41199
$933$	−30.1112	−0.985797
$934$	6.24365	0.204298
$935$	5.47515	0.179057
$936$	−47.7346	−1.56025
$937$	−30.1630	−0.985382	−0.492691	−	0.870204i	$-0.663987\pi$
$937$	−30.1630	−0.985382	−0.492691	+	0.870204i	$0.663987\pi$
$938$	−15.4906	−0.505787
$939$	−107.197	−3.49825
$940$	0.149270	0.00486865
$941$	19.6621	0.640965	0.320483	−	0.947254i	$-0.396155\pi$
$941$	19.6621	0.640965	0.320483	+	0.947254i	$0.396155\pi$
$942$	−139.679	−4.55098
$943$	28.8403	0.939169
$944$	88.6078	2.88394
$945$	−6.15836	−0.200332
$946$	−82.7622	−2.69083
$947$	−33.0402	−1.07366	−0.536832	−	0.843689i	$-0.680379\pi$
$947$	−33.0402	−1.07366	−0.536832	+	0.843689i	$0.680379\pi$
$948$	26.8950	0.873508
$949$	−13.3587	−0.433641
$950$	−46.5769	−1.51115
$951$	9.31532	0.302070
$952$	8.96288	0.290489
$953$	−19.7287	−0.639075	−0.319538	−	0.947574i	$-0.603528\pi$
$953$	−19.7287	−0.639075	−0.319538	+	0.947574i	$0.603528\pi$
$954$	−36.4702	−1.18076
$955$	8.86475	0.286857
$956$	−57.8437	−1.87080
$957$	97.9989	3.16786
$958$	87.2892	2.82019
$959$	−0.949334	−0.0306556
$960$	−4.59248	−0.148222
$961$	−20.5443	−0.662719
$962$	23.2716	0.750308
$963$	−15.7226	−0.506654
$964$	−93.5390	−3.01269
$965$	7.31742	0.235556
$966$	18.0571	0.580977
$967$	28.2219	0.907557	0.453778	−	0.891115i	$-0.350076\pi$
$967$	28.2219	0.907557	0.453778	+	0.891115i	$0.350076\pi$
$968$	−109.659	−3.52456
$969$	26.4158	0.848599
$970$	−8.42246	−0.270429
$971$	47.0217	1.50900	0.754499	−	0.656301i	$-0.227880\pi$
$971$	47.0217	1.50900	0.754499	+	0.656301i	$0.227880\pi$
$972$	−241.067	−7.73224
$973$	−4.84847	−0.155435
$974$	90.0951	2.88683
$975$	15.8241	0.506775
$976$	37.3084	1.19421
$977$	1.60079	0.0512137	0.0256068	−	0.999672i	$-0.491848\pi$
$977$	1.60079	0.0512137	0.0256068	+	0.999672i	$0.491848\pi$
$978$	−185.574	−5.93401
$979$	23.6177	0.754826
$980$	13.8315	0.441832
$981$	148.478	4.74053
$982$	−26.5958	−0.848706
$983$	−29.3402	−0.935809	−0.467904	−	0.883779i	$-0.654991\pi$
$983$	−29.3402	−0.935809	−0.467904	+	0.883779i	$0.654991\pi$
$984$	−195.329	−6.22688
$985$	9.83928	0.313506
$986$	−27.9130	−0.888930
$987$	−0.171636	−0.00546325
$988$	−16.8992	−0.537636
$989$	17.5696	0.558682
$990$	−54.4541	−1.73067
$991$	−55.4650	−1.76191	−0.880953	−	0.473205i	$-0.843097\pi$
$991$	−55.4650	−1.76191	−0.880953	+	0.473205i	$0.843097\pi$
$992$	12.3558	0.392297
$993$	43.8212	1.39062
$994$	4.20227	0.133288
$995$	3.75719	0.119111
$996$	−13.8167	−0.437798
$997$	−60.7035	−1.92250	−0.961249	−	0.275682i	$-0.911096\pi$
$997$	−60.7035	−1.92250	−0.961249	+	0.275682i	$0.911096\pi$
$998$	80.2481	2.54021
$999$	155.647	4.92446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6019.2.a.e.1.9	✓	130

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6019.2.a.e.1.9	✓	130	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:	\( N \)	\(=\)	\( 6019 = 13 \cdot 463 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6019.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51967 q^{2} -3.32656 q^{3} +4.34873 q^{4} -0.493062 q^{5} +8.38182 q^{6} -0.741150 q^{7} -5.91802 q^{8} +8.06597 q^{9} +O(q^{10})\)	\(q-2.51967 q^{2} -3.32656 q^{3} +4.34873 q^{4} -0.493062 q^{5} +8.38182 q^{6} -0.741150 q^{7} -5.91802 q^{8} +8.06597 q^{9} +1.24235 q^{10} -5.43412 q^{11} -14.4663 q^{12} +1.00000 q^{13} +1.86745 q^{14} +1.64020 q^{15} +6.21398 q^{16} +2.04346 q^{17} -20.3236 q^{18} -3.88601 q^{19} -2.14419 q^{20} +2.46548 q^{21} +13.6922 q^{22} -2.90672 q^{23} +19.6866 q^{24} -4.75689 q^{25} -2.51967 q^{26} -16.8522 q^{27} -3.22306 q^{28} +5.42123 q^{29} -4.13276 q^{30} -3.23353 q^{31} -3.82114 q^{32} +18.0769 q^{33} -5.14883 q^{34} +0.365433 q^{35} +35.0767 q^{36} -9.23599 q^{37} +9.79146 q^{38} -3.32656 q^{39} +2.91795 q^{40} -9.92194 q^{41} -6.21218 q^{42} -6.04449 q^{43} -23.6315 q^{44} -3.97703 q^{45} +7.32397 q^{46} -0.0696159 q^{47} -20.6712 q^{48} -6.45070 q^{49} +11.9858 q^{50} -6.79767 q^{51} +4.34873 q^{52} +1.79448 q^{53} +42.4621 q^{54} +2.67936 q^{55} +4.38614 q^{56} +12.9270 q^{57} -13.6597 q^{58} +14.2594 q^{59} +7.13278 q^{60} +6.00394 q^{61} +8.14743 q^{62} -5.97809 q^{63} -2.79995 q^{64} -0.493062 q^{65} -45.5478 q^{66} -8.29507 q^{67} +8.88644 q^{68} +9.66936 q^{69} -0.920770 q^{70} +2.25027 q^{71} -47.7346 q^{72} -13.3587 q^{73} +23.2716 q^{74} +15.8241 q^{75} -16.8992 q^{76} +4.02749 q^{77} +8.38182 q^{78} -1.85915 q^{79} -3.06388 q^{80} +31.8620 q^{81} +25.0000 q^{82} +0.955094 q^{83} +10.7217 q^{84} -1.00755 q^{85} +15.2301 q^{86} -18.0340 q^{87} +32.1592 q^{88} -4.34620 q^{89} +10.0208 q^{90} -0.741150 q^{91} -12.6405 q^{92} +10.7565 q^{93} +0.175409 q^{94} +1.91605 q^{95} +12.7112 q^{96} -6.77944 q^{97} +16.2536 q^{98} -43.8314 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 130 q + 10 q^{2} + 11 q^{3} + 146 q^{4} + 40 q^{5} + 4 q^{6} + 8 q^{7} + 24 q^{8} + 181 q^{9}+O(q^{10}) \) 130 * q + 10 * q^2 + 11 * q^3 + 146 * q^4 + 40 * q^5 + 4 * q^6 + 8 * q^7 + 24 * q^8 + 181 * q^9	\( 130 q + 10 q^{2} + 11 q^{3} + 146 q^{4} + 40 q^{5} + 4 q^{6} + 8 q^{7} + 24 q^{8} + 181 q^{9} + 5 q^{10} + 43 q^{11} + 28 q^{12} + 130 q^{13} + 47 q^{14} + 29 q^{15} + 170 q^{16} + 85 q^{17} + 20 q^{18} + 3 q^{19} + 73 q^{20} + 62 q^{21} + 12 q^{22} + 62 q^{23} - 5 q^{24} + 178 q^{25} + 10 q^{26} + 35 q^{27} - q^{28} + 134 q^{29} + 24 q^{30} + 14 q^{31} + 48 q^{32} + 4 q^{33} + 4 q^{34} + 50 q^{35} + 244 q^{36} + 32 q^{37} + 76 q^{38} + 11 q^{39} - 6 q^{40} + 48 q^{41} + 9 q^{42} + 34 q^{43} + 123 q^{44} + 115 q^{45} + 5 q^{46} + 25 q^{47} + 35 q^{48} + 210 q^{49} + 24 q^{50} + 20 q^{51} + 146 q^{52} + 193 q^{53} - 39 q^{54} + 32 q^{55} + 122 q^{56} + 7 q^{57} - 4 q^{58} + 50 q^{59} + 42 q^{60} + 57 q^{61} + 51 q^{62} + 8 q^{63} + 172 q^{64} + 40 q^{65} - 4 q^{66} + 21 q^{67} + 132 q^{68} + 92 q^{69} - 46 q^{70} + 58 q^{71} - 26 q^{72} + 15 q^{73} + 120 q^{74} + 23 q^{75} - 65 q^{76} + 192 q^{77} + 4 q^{78} + 32 q^{79} + 66 q^{80} + 326 q^{81} + 11 q^{82} + 33 q^{83} + 5 q^{84} + 43 q^{85} + 105 q^{86} + 31 q^{87} - 17 q^{88} + 84 q^{89} - 73 q^{90} + 8 q^{91} + 161 q^{92} + 52 q^{93} + 4 q^{94} + 59 q^{95} - 77 q^{96} + 9 q^{97} - 61 q^{98} + 100 q^{99}+O(q^{100}) \) 130 * q + 10 * q^2 + 11 * q^3 + 146 * q^4 + 40 * q^5 + 4 * q^6 + 8 * q^7 + 24 * q^8 + 181 * q^9 + 5 * q^10 + 43 * q^11 + 28 * q^12 + 130 * q^13 + 47 * q^14 + 29 * q^15 + 170 * q^16 + 85 * q^17 + 20 * q^18 + 3 * q^19 + 73 * q^20 + 62 * q^21 + 12 * q^22 + 62 * q^23 - 5 * q^24 + 178 * q^25 + 10 * q^26 + 35 * q^27 - q^28 + 134 * q^29 + 24 * q^30 + 14 * q^31 + 48 * q^32 + 4 * q^33 + 4 * q^34 + 50 * q^35 + 244 * q^36 + 32 * q^37 + 76 * q^38 + 11 * q^39 - 6 * q^40 + 48 * q^41 + 9 * q^42 + 34 * q^43 + 123 * q^44 + 115 * q^45 + 5 * q^46 + 25 * q^47 + 35 * q^48 + 210 * q^49 + 24 * q^50 + 20 * q^51 + 146 * q^52 + 193 * q^53 - 39 * q^54 + 32 * q^55 + 122 * q^56 + 7 * q^57 - 4 * q^58 + 50 * q^59 + 42 * q^60 + 57 * q^61 + 51 * q^62 + 8 * q^63 + 172 * q^64 + 40 * q^65 - 4 * q^66 + 21 * q^67 + 132 * q^68 + 92 * q^69 - 46 * q^70 + 58 * q^71 - 26 * q^72 + 15 * q^73 + 120 * q^74 + 23 * q^75 - 65 * q^76 + 192 * q^77 + 4 * q^78 + 32 * q^79 + 66 * q^80 + 326 * q^81 + 11 * q^82 + 33 * q^83 + 5 * q^84 + 43 * q^85 + 105 * q^86 + 31 * q^87 - 17 * q^88 + 84 * q^89 - 73 * q^90 + 8 * q^91 + 161 * q^92 + 52 * q^93 + 4 * q^94 + 59 * q^95 - 77 * q^96 + 9 * q^97 - 61 * q^98 + 100 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more