LMFDB - Embedded newform 2013.4.a.f.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2013,4,Mod(1,2013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2013, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2013.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2013 = 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2013.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:	$118.770844842$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	2013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.740360 q^{2} +3.00000 q^{3} -7.45187 q^{4} +7.63307 q^{5} -2.22108 q^{6} +1.52940 q^{7} +11.4399 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-0.740360 q^{2} +3.00000 q^{3} -7.45187 q^{4} +7.63307 q^{5} -2.22108 q^{6} +1.52940 q^{7} +11.4399 q^{8} +9.00000 q^{9} -5.65122 q^{10} +11.0000 q^{11} -22.3556 q^{12} +52.0801 q^{13} -1.13231 q^{14} +22.8992 q^{15} +51.1453 q^{16} +118.989 q^{17} -6.66324 q^{18} +37.2197 q^{19} -56.8806 q^{20} +4.58821 q^{21} -8.14396 q^{22} +143.774 q^{23} +34.3198 q^{24} -66.7362 q^{25} -38.5581 q^{26} +27.0000 q^{27} -11.3969 q^{28} -49.9148 q^{29} -16.9537 q^{30} +79.9974 q^{31} -129.385 q^{32} +33.0000 q^{33} -88.0950 q^{34} +11.6740 q^{35} -67.0668 q^{36} +365.840 q^{37} -27.5560 q^{38} +156.240 q^{39} +87.3219 q^{40} -251.441 q^{41} -3.39692 q^{42} -163.179 q^{43} -81.9705 q^{44} +68.6976 q^{45} -106.444 q^{46} +579.089 q^{47} +153.436 q^{48} -340.661 q^{49} +49.4088 q^{50} +356.968 q^{51} -388.094 q^{52} -540.387 q^{53} -19.9897 q^{54} +83.9638 q^{55} +17.4963 q^{56} +111.659 q^{57} +36.9549 q^{58} -204.827 q^{59} -170.642 q^{60} -61.0000 q^{61} -59.2268 q^{62} +13.7646 q^{63} -313.370 q^{64} +397.531 q^{65} -24.4319 q^{66} -533.898 q^{67} -886.694 q^{68} +431.321 q^{69} -8.64299 q^{70} -558.497 q^{71} +102.959 q^{72} +601.803 q^{73} -270.853 q^{74} -200.209 q^{75} -277.356 q^{76} +16.8234 q^{77} -115.674 q^{78} +606.675 q^{79} +390.395 q^{80} +81.0000 q^{81} +186.157 q^{82} +357.092 q^{83} -34.1907 q^{84} +908.255 q^{85} +120.811 q^{86} -149.744 q^{87} +125.839 q^{88} +526.464 q^{89} -50.8610 q^{90} +79.6515 q^{91} -1071.38 q^{92} +239.992 q^{93} -428.735 q^{94} +284.101 q^{95} -388.156 q^{96} +354.815 q^{97} +252.212 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9}+O(q^{10}) $ 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9	\( 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9} + 99 q^{10} + 418 q^{11} + 510 q^{12} + 209 q^{13} + 128 q^{14} + 105 q^{15} + 798 q^{16} + 512 q^{17} + 126 q^{18} + 487 q^{19} + 328 q^{20} + 315 q^{21} + 154 q^{22} + 417 q^{23} + 441 q^{24} + 925 q^{25} + 177 q^{26} + 1026 q^{27} + 902 q^{28} + 626 q^{29} + 297 q^{30} + 300 q^{31} + 1625 q^{32} + 1254 q^{33} - 180 q^{34} + 1086 q^{35} + 1530 q^{36} + 554 q^{37} + 845 q^{38} + 627 q^{39} + 329 q^{40} + 1378 q^{41} + 384 q^{42} + 1979 q^{43} + 1870 q^{44} + 315 q^{45} + 937 q^{46} + 1345 q^{47} + 2394 q^{48} + 2635 q^{49} + 800 q^{50} + 1536 q^{51} + 2006 q^{52} + 1497 q^{53} + 378 q^{54} + 385 q^{55} + 415 q^{56} + 1461 q^{57} + 1241 q^{58} + 2827 q^{59} + 984 q^{60} - 2318 q^{61} + 509 q^{62} + 945 q^{63} + 1003 q^{64} + 2810 q^{65} + 462 q^{66} + 369 q^{67} + 3936 q^{68} + 1251 q^{69} + 922 q^{70} + 965 q^{71} + 1323 q^{72} + 3081 q^{73} + 722 q^{74} + 2775 q^{75} + 2210 q^{76} + 1155 q^{77} + 531 q^{78} + 3795 q^{79} + 3793 q^{80} + 3078 q^{81} - 1678 q^{82} + 3869 q^{83} + 2706 q^{84} + 3553 q^{85} + 3305 q^{86} + 1878 q^{87} + 1617 q^{88} + 2849 q^{89} + 891 q^{90} + 1252 q^{91} + 4519 q^{92} + 900 q^{93} + 340 q^{94} + 1504 q^{95} + 4875 q^{96} + 2562 q^{97} + 6164 q^{98} + 3762 q^{99}+O(q^{100}) \) 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9 + 99 * q^10 + 418 * q^11 + 510 * q^12 + 209 * q^13 + 128 * q^14 + 105 * q^15 + 798 * q^16 + 512 * q^17 + 126 * q^18 + 487 * q^19 + 328 * q^20 + 315 * q^21 + 154 * q^22 + 417 * q^23 + 441 * q^24 + 925 * q^25 + 177 * q^26 + 1026 * q^27 + 902 * q^28 + 626 * q^29 + 297 * q^30 + 300 * q^31 + 1625 * q^32 + 1254 * q^33 - 180 * q^34 + 1086 * q^35 + 1530 * q^36 + 554 * q^37 + 845 * q^38 + 627 * q^39 + 329 * q^40 + 1378 * q^41 + 384 * q^42 + 1979 * q^43 + 1870 * q^44 + 315 * q^45 + 937 * q^46 + 1345 * q^47 + 2394 * q^48 + 2635 * q^49 + 800 * q^50 + 1536 * q^51 + 2006 * q^52 + 1497 * q^53 + 378 * q^54 + 385 * q^55 + 415 * q^56 + 1461 * q^57 + 1241 * q^58 + 2827 * q^59 + 984 * q^60 - 2318 * q^61 + 509 * q^62 + 945 * q^63 + 1003 * q^64 + 2810 * q^65 + 462 * q^66 + 369 * q^67 + 3936 * q^68 + 1251 * q^69 + 922 * q^70 + 965 * q^71 + 1323 * q^72 + 3081 * q^73 + 722 * q^74 + 2775 * q^75 + 2210 * q^76 + 1155 * q^77 + 531 * q^78 + 3795 * q^79 + 3793 * q^80 + 3078 * q^81 - 1678 * q^82 + 3869 * q^83 + 2706 * q^84 + 3553 * q^85 + 3305 * q^86 + 1878 * q^87 + 1617 * q^88 + 2849 * q^89 + 891 * q^90 + 1252 * q^91 + 4519 * q^92 + 900 * q^93 + 340 * q^94 + 1504 * q^95 + 4875 * q^96 + 2562 * q^97 + 6164 * q^98 + 3762 * 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 $

$2$	−0.740360	−0.261757	−0.130878	−	0.991398i	$-0.541780\pi$
$2$	−0.740360	−0.261757	−0.130878	+	0.991398i	$0.541780\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−7.45187	−0.931483
$5$	7.63307	0.682723	0.341361	−	0.939932i	$-0.389112\pi$
$5$	7.63307	0.682723	0.341361	+	0.939932i	$0.389112\pi$
$6$	−2.22108	−0.151125
$7$	1.52940	0.0825799	0.0412900	−	0.999147i	$-0.486853\pi$
$7$	1.52940	0.0825799	0.0412900	+	0.999147i	$0.486853\pi$
$8$	11.4399	0.505579
$9$	9.00000	0.333333
$10$	−5.65122	−0.178707
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−22.3556	−0.537792
$13$	52.0801	1.11111	0.555555	−	0.831480i	$-0.312506\pi$
$13$	52.0801	1.11111	0.555555	+	0.831480i	$0.312506\pi$
$14$	−1.13231	−0.0216159
$15$	22.8992	0.394170
$16$	51.1453	0.799145
$17$	118.989	1.69760	0.848800	−	0.528715i	$-0.177326\pi$
$17$	118.989	1.69760	0.848800	+	0.528715i	$0.177326\pi$
$18$	−6.66324	−0.0872523
$19$	37.2197	0.449410	0.224705	−	0.974427i	$-0.427858\pi$
$19$	37.2197	0.449410	0.224705	+	0.974427i	$0.427858\pi$
$20$	−56.8806	−0.635945
$21$	4.58821	0.0476776
$22$	−8.14396	−0.0789226
$23$	143.774	1.30343	0.651715	−	0.758464i	$-0.274050\pi$
$23$	143.774	1.30343	0.651715	+	0.758464i	$0.274050\pi$
$24$	34.3198	0.291896
$25$	−66.7362	−0.533890
$26$	−38.5581	−0.290841
$27$	27.0000	0.192450
$28$	−11.3969	−0.0769218
$29$	−49.9148	−0.319619	−0.159809	−	0.987148i	$-0.551088\pi$
$29$	−49.9148	−0.319619	−0.159809	+	0.987148i	$0.551088\pi$
$30$	−16.9537	−0.103177
$31$	79.9974	0.463482	0.231741	−	0.972777i	$-0.425558\pi$
$31$	79.9974	0.463482	0.231741	+	0.972777i	$0.425558\pi$
$32$	−129.385	−0.714760
$33$	33.0000	0.174078
$34$	−88.0950	−0.444358
$35$	11.6740	0.0563792
$36$	−67.0668	−0.310494
$37$	365.840	1.62550	0.812752	−	0.582609i	$-0.197968\pi$
$37$	365.840	1.62550	0.812752	+	0.582609i	$0.197968\pi$
$38$	−27.5560	−0.117636
$39$	156.240	0.641500
$40$	87.3219	0.345170
$41$	−251.441	−0.957769	−0.478885	−	0.877878i	$-0.658959\pi$
$41$	−251.441	−0.957769	−0.478885	+	0.877878i	$0.658959\pi$
$42$	−3.39692	−0.0124799
$43$	−163.179	−0.578711	−0.289355	−	0.957222i	$-0.593441\pi$
$43$	−163.179	−0.578711	−0.289355	+	0.957222i	$0.593441\pi$
$44$	−81.9705	−0.280853
$45$	68.6976	0.227574
$46$	−106.444	−0.341182
$47$	579.089	1.79721	0.898605	−	0.438759i	$-0.144582\pi$
$47$	579.089	1.79721	0.898605	+	0.438759i	$0.144582\pi$
$48$	153.436	0.461386
$49$	−340.661	−0.993181
$50$	49.4088	0.139749
$51$	356.968	0.980109
$52$	−388.094	−1.03498
$53$	−540.387	−1.40052	−0.700262	−	0.713886i	$-0.746934\pi$
$53$	−540.387	−1.40052	−0.700262	+	0.713886i	$0.746934\pi$
$54$	−19.9897	−0.0503751
$55$	83.9638	0.205849
$56$	17.4963	0.0417507
$57$	111.659	0.259467
$58$	36.9549	0.0836624
$59$	−204.827	−0.451969	−0.225984	−	0.974131i	$-0.572560\pi$
$59$	−204.827	−0.451969	−0.225984	+	0.974131i	$0.572560\pi$
$60$	−170.642	−0.367163
$61$	−61.0000	−0.128037
$61$	−61.0000	−0.128037
$62$	−59.2268	−0.121320
$63$	13.7646	0.0275266
$64$	−313.370	−0.612051
$65$	397.531	0.758580
$66$	−24.4319	−0.0455660
$67$	−533.898	−0.973524	−0.486762	−	0.873535i	$-0.661822\pi$
$67$	−533.898	−0.973524	−0.486762	+	0.873535i	$0.661822\pi$
$68$	−886.694	−1.58129
$69$	431.321	0.752536
$70$	−8.64299	−0.0147576
$71$	−558.497	−0.933540	−0.466770	−	0.884379i	$-0.654582\pi$
$71$	−558.497	−0.933540	−0.466770	+	0.884379i	$0.654582\pi$
$72$	102.959	0.168526
$73$	601.803	0.964873	0.482436	−	0.875931i	$-0.339752\pi$
$73$	601.803	0.964873	0.482436	+	0.875931i	$0.339752\pi$
$74$	−270.853	−0.425487
$75$	−200.209	−0.308241
$76$	−277.356	−0.418618
$77$	16.8234	0.0248988
$78$	−115.674	−0.167917
$79$	606.675	0.864003	0.432001	−	0.901873i	$-0.357808\pi$
$79$	606.675	0.864003	0.432001	+	0.901873i	$0.357808\pi$
$80$	390.395	0.545594
$81$	81.0000	0.111111
$82$	186.157	0.250703
$83$	357.092	0.472240	0.236120	−	0.971724i	$-0.424124\pi$
$83$	357.092	0.472240	0.236120	+	0.971724i	$0.424124\pi$
$84$	−34.1907	−0.0444108
$85$	908.255	1.15899
$86$	120.811	0.151481
$87$	−149.744	−0.184532
$88$	125.839	0.152438
$89$	526.464	0.627024	0.313512	−	0.949584i	$-0.398494\pi$
$89$	526.464	0.627024	0.313512	+	0.949584i	$0.398494\pi$
$90$	−50.8610	−0.0595691
$91$	79.6515	0.0917554
$92$	−1071.38	−1.21412
$93$	239.992	0.267592
$94$	−428.735	−0.470432
$95$	284.101	0.306822
$96$	−388.156	−0.412667
$97$	354.815	0.371402	0.185701	−	0.982606i	$-0.440544\pi$
$97$	354.815	0.371402	0.185701	+	0.982606i	$0.440544\pi$
$98$	252.212	0.259972
$99$	99.0000	0.100504
$100$	497.309	0.497309
$101$	−196.309	−0.193400	−0.0967002	−	0.995314i	$-0.530829\pi$
$101$	−196.309	−0.193400	−0.0967002	+	0.995314i	$0.530829\pi$
$102$	−264.285	−0.256550
$103$	445.082	0.425779	0.212889	−	0.977076i	$-0.431713\pi$
$103$	445.082	0.425779	0.212889	+	0.977076i	$0.431713\pi$
$104$	595.794	0.561754
$105$	35.0221	0.0325505
$106$	400.081	0.366597
$107$	−804.565	−0.726918	−0.363459	−	0.931610i	$-0.618404\pi$
$107$	−804.565	−0.726918	−0.363459	+	0.931610i	$0.618404\pi$
$108$	−201.200	−0.179264
$109$	989.826	0.869800	0.434900	−	0.900479i	$-0.356784\pi$
$109$	989.826	0.869800	0.434900	+	0.900479i	$0.356784\pi$
$110$	−62.1634	−0.0538823
$111$	1097.52	0.938486
$112$	78.2217	0.0659933
$113$	284.388	0.236752	0.118376	−	0.992969i	$-0.462231\pi$
$113$	284.388	0.236752	0.118376	+	0.992969i	$0.462231\pi$
$114$	−82.6680	−0.0679172
$115$	1097.43	0.889881
$116$	371.958	0.297720
$117$	468.721	0.370370
$118$	151.645	0.118306
$119$	181.983	0.140188
$120$	261.966	0.199284
$121$	121.000	0.0909091
$122$	45.1620	0.0335145
$123$	−754.324	−0.552968
$124$	−596.130	−0.431726
$125$	−1463.54	−1.04722
$126$	−10.1908	−0.00720529
$127$	−1963.85	−1.37215	−0.686076	−	0.727530i	$-0.740668\pi$
$127$	−1963.85	−1.37215	−0.686076	+	0.727530i	$0.740668\pi$
$128$	1267.09	0.874969
$129$	−489.537	−0.334119
$130$	−294.316	−0.198563
$131$	−1149.88	−0.766910	−0.383455	−	0.923560i	$-0.625266\pi$
$131$	−1149.88	−0.766910	−0.383455	+	0.923560i	$0.625266\pi$
$132$	−245.912	−0.162150
$133$	56.9239	0.0371122
$134$	395.277	0.254826
$135$	206.093	0.131390
$136$	1361.23	0.858270
$137$	−32.8825	−0.0205061	−0.0102531	−	0.999947i	$-0.503264\pi$
$137$	−32.8825	−0.0205061	−0.0102531	+	0.999947i	$0.503264\pi$
$138$	−319.333	−0.196981
$139$	−1181.14	−0.720740	−0.360370	−	0.932809i	$-0.617350\pi$
$139$	−1181.14	−0.720740	−0.360370	+	0.932809i	$0.617350\pi$
$140$	−86.9933	−0.0525163
$141$	1737.27	1.03762
$142$	413.489	0.244361
$143$	572.882	0.335012
$144$	460.307	0.266382
$145$	−381.003	−0.218211
$146$	−445.551	−0.252562
$147$	−1021.98	−0.573413
$148$	−2726.19	−1.51413
$149$	1626.20	0.894118	0.447059	−	0.894504i	$-0.352471\pi$
$149$	1626.20	0.894118	0.447059	+	0.894504i	$0.352471\pi$
$150$	148.226	0.0806843
$151$	2052.40	1.10611	0.553053	−	0.833146i	$-0.313463\pi$
$151$	2052.40	1.10611	0.553053	+	0.833146i	$0.313463\pi$
$152$	425.791	0.227212
$153$	1070.91	0.565866
$154$	−12.4554	−0.00651743
$155$	610.626	0.316430
$156$	−1164.28	−0.597546
$157$	2823.58	1.43533	0.717664	−	0.696390i	$-0.245211\pi$
$157$	2823.58	1.43533	0.717664	+	0.696390i	$0.245211\pi$
$158$	−449.158	−0.226159
$159$	−1621.16	−0.808593
$160$	−987.608	−0.487983
$161$	219.888	0.107637
$162$	−59.9692	−0.0290841
$163$	729.652	0.350618	0.175309	−	0.984513i	$-0.443907\pi$
$163$	729.652	0.350618	0.175309	+	0.984513i	$0.443907\pi$
$164$	1873.71	0.892146
$165$	251.891	0.118847
$166$	−264.376	−0.123612
$167$	−1570.12	−0.727544	−0.363772	−	0.931488i	$-0.618511\pi$
$167$	−1570.12	−0.727544	−0.363772	+	0.931488i	$0.618511\pi$
$168$	52.4888	0.0241048
$169$	515.341	0.234566
$170$	−672.436	−0.303373
$171$	334.977	0.149803
$172$	1215.99	0.539059
$173$	234.971	0.103263	0.0516315	−	0.998666i	$-0.483558\pi$
$173$	234.971	0.103263	0.0516315	+	0.998666i	$0.483558\pi$
$174$	110.865	0.0483025
$175$	−102.067	−0.0440886
$176$	562.598	0.240951
$177$	−614.480	−0.260944
$178$	−389.773	−0.164128
$179$	−4554.05	−1.90160	−0.950798	−	0.309812i	$-0.899734\pi$
$179$	−4554.05	−1.90160	−0.950798	+	0.309812i	$0.899734\pi$
$180$	−511.926	−0.211982
$181$	−2755.89	−1.13173	−0.565867	−	0.824497i	$-0.691458\pi$
$181$	−2755.89	−1.13173	−0.565867	+	0.824497i	$0.691458\pi$
$182$	−58.9708	−0.0240176
$183$	−183.000	−0.0739221
$184$	1644.76	0.658987
$185$	2792.48	1.10977
$186$	−177.681	−0.0700439
$187$	1308.88	0.511845
$188$	−4315.30	−1.67407
$189$	41.2939	0.0158925
$190$	−210.337	−0.0803128
$191$	2815.38	1.06656	0.533282	−	0.845937i	$-0.320958\pi$
$191$	2815.38	1.06656	0.533282	+	0.845937i	$0.320958\pi$
$192$	−940.111	−0.353368
$193$	−3826.48	−1.42713	−0.713564	−	0.700590i	$-0.752920\pi$
$193$	−3826.48	−1.42713	−0.713564	+	0.700590i	$0.752920\pi$
$194$	−262.691	−0.0972170
$195$	1192.59	0.437966
$196$	2538.56	0.925131
$197$	4537.82	1.64115	0.820575	−	0.571539i	$-0.193653\pi$
$197$	4537.82	1.64115	0.820575	+	0.571539i	$0.193653\pi$
$198$	−73.2956	−0.0263075
$199$	4214.84	1.50142	0.750708	−	0.660634i	$-0.229712\pi$
$199$	4214.84	1.50142	0.750708	+	0.660634i	$0.229712\pi$
$200$	−763.459	−0.269923
$201$	−1601.70	−0.562064
$202$	145.339	0.0506239
$203$	−76.3398	−0.0263941
$204$	−2660.08	−0.912956
$205$	−1919.27	−0.653891
$206$	−329.521	−0.111450
$207$	1293.96	0.434477
$208$	2663.65	0.887938
$209$	409.417	0.135502
$210$	−25.9290	−0.00852033
$211$	4853.86	1.58367	0.791833	−	0.610737i	$-0.209127\pi$
$211$	4853.86	1.58367	0.791833	+	0.610737i	$0.209127\pi$
$212$	4026.89	1.30457
$213$	−1675.49	−0.538980
$214$	595.667	0.190276
$215$	−1245.56	−0.395099
$216$	308.878	0.0972987
$217$	122.348	0.0382743
$218$	−732.828	−0.227676
$219$	1805.41	0.557070
$220$	−625.687	−0.191745
$221$	6196.99	1.88622
$222$	−812.559	−0.245655
$223$	1115.71	0.335038	0.167519	−	0.985869i	$-0.446425\pi$
$223$	1115.71	0.335038	0.167519	+	0.985869i	$0.446425\pi$
$224$	−197.882	−0.0590249
$225$	−600.626	−0.177963
$226$	−210.550	−0.0619715
$227$	212.102	0.0620164	0.0310082	−	0.999519i	$-0.490128\pi$
$227$	212.102	0.0620164	0.0310082	+	0.999519i	$0.490128\pi$
$228$	−832.069	−0.241689
$229$	4734.13	1.36611	0.683057	−	0.730365i	$-0.260650\pi$
$229$	4734.13	1.36611	0.683057	+	0.730365i	$0.260650\pi$
$230$	−812.497	−0.232932
$231$	50.4703	0.0143753
$232$	−571.023	−0.161593
$233$	−4306.61	−1.21088	−0.605440	−	0.795891i	$-0.707003\pi$
$233$	−4306.61	−1.21088	−0.605440	+	0.795891i	$0.707003\pi$
$234$	−347.022	−0.0969469
$235$	4420.23	1.22700
$236$	1526.34	0.421001
$237$	1820.02	0.498832
$238$	−134.733	−0.0366951
$239$	−4617.33	−1.24967	−0.624833	−	0.780759i	$-0.714833\pi$
$239$	−4617.33	−1.24967	−0.624833	+	0.780759i	$0.714833\pi$
$240$	1171.19	0.314999
$241$	−2912.95	−0.778587	−0.389294	−	0.921114i	$-0.627281\pi$
$241$	−2912.95	−0.778587	−0.389294	+	0.921114i	$0.627281\pi$
$242$	−89.5836	−0.0237961
$243$	243.000	0.0641500
$244$	454.564	0.119264
$245$	−2600.29	−0.678067
$246$	558.471	0.144743
$247$	1938.41	0.499344
$248$	915.165	0.234327
$249$	1071.27	0.272648
$250$	1083.54	0.274117
$251$	−10.2120	−0.00256803	−0.00128401	−	0.999999i	$-0.500409\pi$
$251$	−10.2120	−0.00256803	−0.00128401	+	0.999999i	$0.500409\pi$
$252$	−102.572	−0.0256406
$253$	1581.51	0.392999
$254$	1453.95	0.359170
$255$	2724.77	0.669143
$256$	1568.86	0.383022
$257$	−2139.67	−0.519335	−0.259667	−	0.965698i	$-0.583613\pi$
$257$	−2139.67	−0.519335	−0.259667	+	0.965698i	$0.583613\pi$
$258$	362.434	0.0874578
$259$	559.516	0.134234
$260$	−2962.35	−0.706605
$261$	−449.233	−0.106540
$262$	851.323	0.200744
$263$	−7241.99	−1.69795	−0.848974	−	0.528434i	$-0.822779\pi$
$263$	−7241.99	−1.69795	−0.848974	+	0.528434i	$0.822779\pi$
$264$	377.518	0.0880100
$265$	−4124.81	−0.956170
$266$	−42.1442	−0.00971438
$267$	1579.39	0.362012
$268$	3978.54	0.906821
$269$	4360.94	0.988443	0.494222	−	0.869336i	$-0.335453\pi$
$269$	4360.94	0.988443	0.494222	+	0.869336i	$0.335453\pi$
$270$	−152.583	−0.0343922
$271$	−1382.39	−0.309869	−0.154934	−	0.987925i	$-0.549517\pi$
$271$	−1382.39	−0.309869	−0.154934	+	0.987925i	$0.549517\pi$
$272$	6085.75	1.35663
$273$	238.954	0.0529750
$274$	24.3449	0.00536762
$275$	−734.098	−0.160974
$276$	−3214.15	−0.700975
$277$	2227.98	0.483272	0.241636	−	0.970367i	$-0.422316\pi$
$277$	2227.98	0.483272	0.241636	+	0.970367i	$0.422316\pi$
$278$	874.467	0.188659
$279$	719.976	0.154494
$280$	133.550	0.0285041
$281$	−1857.16	−0.394266	−0.197133	−	0.980377i	$-0.563163\pi$
$281$	−1857.16	−0.394266	−0.197133	+	0.980377i	$0.563163\pi$
$282$	−1286.20	−0.271604
$283$	8773.98	1.84296	0.921482	−	0.388421i	$-0.126979\pi$
$283$	8773.98	1.84296	0.921482	+	0.388421i	$0.126979\pi$
$284$	4161.84	0.869577
$285$	852.302	0.177144
$286$	−424.139	−0.0876917
$287$	−384.555	−0.0790925
$288$	−1164.47	−0.238253
$289$	9245.50	1.88184
$290$	282.080	0.0571182
$291$	1064.44	0.214429
$292$	−4484.56	−0.898763
$293$	−622.435	−0.124106	−0.0620530	−	0.998073i	$-0.519765\pi$
$293$	−622.435	−0.124106	−0.0620530	+	0.998073i	$0.519765\pi$
$294$	756.635	0.150095
$295$	−1563.46	−0.308569
$296$	4185.19	0.821821
$297$	297.000	0.0580259
$298$	−1203.97	−0.234042
$299$	7487.76	1.44825
$300$	1491.93	0.287122
$301$	−249.566	−0.0477899
$302$	−1519.52	−0.289531
$303$	−588.926	−0.111660
$304$	1903.61	0.359143
$305$	−465.617	−0.0874137
$306$	−792.855	−0.148119
$307$	−9653.19	−1.79458	−0.897291	−	0.441439i	$-0.854468\pi$
$307$	−9653.19	−1.79458	−0.897291	+	0.441439i	$0.854468\pi$
$308$	−125.366	−0.0231928
$309$	1335.25	0.245824
$310$	−452.083	−0.0828277
$311$	8954.05	1.63260	0.816298	−	0.577630i	$-0.196022\pi$
$311$	8954.05	1.63260	0.816298	+	0.577630i	$0.196022\pi$
$312$	1787.38	0.324329
$313$	−485.758	−0.0877210	−0.0438605	−	0.999038i	$-0.513966\pi$
$313$	−485.758	−0.0877210	−0.0438605	+	0.999038i	$0.513966\pi$
$314$	−2090.47	−0.375707
$315$	105.066	0.0187931
$316$	−4520.86	−0.804804
$317$	157.598	0.0279230	0.0139615	−	0.999903i	$-0.495556\pi$
$317$	157.598	0.0279230	0.0139615	+	0.999903i	$0.495556\pi$
$318$	1200.24	0.211655
$319$	−549.063	−0.0963687
$320$	−2391.98	−0.417861
$321$	−2413.69	−0.419686
$322$	−162.796	−0.0281748
$323$	4428.75	0.762918
$324$	−603.601	−0.103498
$325$	−3475.63	−0.593210
$326$	−540.205	−0.0917767
$327$	2969.48	0.502179
$328$	−2876.47	−0.484228
$329$	885.660	0.148413
$330$	−186.490	−0.0311089
$331$	5520.57	0.916731	0.458365	−	0.888764i	$-0.348435\pi$
$331$	5520.57	0.916731	0.458365	+	0.888764i	$0.348435\pi$
$332$	−2661.00	−0.439883
$333$	3292.56	0.541835
$334$	1162.46	0.190440
$335$	−4075.29	−0.664647
$336$	234.665	0.0381013
$337$	−3327.68	−0.537894	−0.268947	−	0.963155i	$-0.586676\pi$
$337$	−3327.68	−0.537894	−0.268947	+	0.963155i	$0.586676\pi$
$338$	−381.538	−0.0613992
$339$	853.165	0.136689
$340$	−6768.20	−1.07958
$341$	879.971	0.139745
$342$	−248.004	−0.0392120
$343$	−1045.59	−0.164597
$344$	−1866.76	−0.292584
$345$	3292.30	0.513773
$346$	−173.963	−0.0270298
$347$	8215.00	1.27091	0.635453	−	0.772140i	$-0.280813\pi$
$347$	8215.00	1.27091	0.635453	+	0.772140i	$0.280813\pi$
$348$	1115.88	0.171889
$349$	−6616.61	−1.01484	−0.507420	−	0.861699i	$-0.669401\pi$
$349$	−6616.61	−1.01484	−0.507420	+	0.861699i	$0.669401\pi$
$350$	75.5660	0.0115405
$351$	1406.16	0.213833
$352$	−1423.24	−0.215508
$353$	3811.77	0.574731	0.287365	−	0.957821i	$-0.407221\pi$
$353$	3811.77	0.574731	0.287365	+	0.957821i	$0.407221\pi$
$354$	454.936	0.0683039
$355$	−4263.04	−0.637349
$356$	−3923.14	−0.584062
$357$	545.948	0.0809374
$358$	3371.64	0.497756
$359$	7151.07	1.05131	0.525653	−	0.850699i	$-0.323821\pi$
$359$	7151.07	1.05131	0.525653	+	0.850699i	$0.323821\pi$
$360$	785.897	0.115057
$361$	−5473.69	−0.798031
$362$	2040.35	0.296239
$363$	363.000	0.0524864
$364$	−593.552	−0.0854686
$365$	4593.61	0.658741
$366$	135.486	0.0193496
$367$	3186.22	0.453186	0.226593	−	0.973990i	$-0.427241\pi$
$367$	3186.22	0.453186	0.226593	+	0.973990i	$0.427241\pi$
$368$	7353.34	1.04163
$369$	−2262.97	−0.319256
$370$	−2067.44	−0.290490
$371$	−826.468	−0.115655
$372$	−1788.39	−0.249257
$373$	3824.42	0.530887	0.265444	−	0.964126i	$-0.414482\pi$
$373$	3824.42	0.530887	0.265444	+	0.964126i	$0.414482\pi$
$374$	−969.046	−0.133979
$375$	−4390.61	−0.604613
$376$	6624.75	0.908631
$377$	−2599.57	−0.355132
$378$	−30.5723	−0.00415997
$379$	−13240.4	−1.79450	−0.897250	−	0.441522i	$-0.854439\pi$
$379$	−13240.4	−1.79450	−0.897250	+	0.441522i	$0.854439\pi$
$380$	−2117.08	−0.285800
$381$	−5891.54	−0.792212
$382$	−2084.39	−0.279180
$383$	137.554	0.0183517	0.00917584	−	0.999958i	$-0.497079\pi$
$383$	137.554	0.0183517	0.00917584	+	0.999958i	$0.497079\pi$
$384$	3801.27	0.505164
$385$	128.414	0.0169990
$386$	2832.97	0.373561
$387$	−1468.61	−0.192904
$388$	−2644.03	−0.345955
$389$	8926.67	1.16350	0.581748	−	0.813369i	$-0.302369\pi$
$389$	8926.67	1.16350	0.581748	+	0.813369i	$0.302369\pi$
$390$	−882.949	−0.114641
$391$	17107.6	2.21270
$392$	−3897.14	−0.502131
$393$	−3449.63	−0.442776
$394$	−3359.62	−0.429582
$395$	4630.79	0.589874
$396$	−737.735	−0.0936176
$397$	−15379.5	−1.94427	−0.972133	−	0.234431i	$-0.924677\pi$
$397$	−15379.5	−1.94427	−0.972133	+	0.234431i	$0.924677\pi$
$398$	−3120.50	−0.393006
$399$	170.772	0.0214268
$400$	−3413.24	−0.426655
$401$	−3373.50	−0.420111	−0.210056	−	0.977689i	$-0.567364\pi$
$401$	−3373.50	−0.420111	−0.210056	+	0.977689i	$0.567364\pi$
$402$	1185.83	0.147124
$403$	4166.27	0.514980
$404$	1462.87	0.180149
$405$	618.279	0.0758581
$406$	56.5189	0.00690884
$407$	4024.24	0.490108
$408$	4083.70	0.495523
$409$	2731.34	0.330211	0.165105	−	0.986276i	$-0.447204\pi$
$409$	2731.34	0.330211	0.165105	+	0.986276i	$0.447204\pi$
$410$	1420.95	0.171160
$411$	−98.6474	−0.0118392
$412$	−3316.69	−0.396606
$413$	−313.262	−0.0373236
$414$	−957.999	−0.113727
$415$	2725.71	0.322409
$416$	−6738.41	−0.794178
$417$	−3543.41	−0.416119
$418$	−303.116	−0.0354686
$419$	−1115.51	−0.130062	−0.0650312	−	0.997883i	$-0.520715\pi$
$419$	−1115.51	−0.130062	−0.0650312	+	0.997883i	$0.520715\pi$
$420$	−260.980	−0.0303203
$421$	−11994.7	−1.38857	−0.694283	−	0.719702i	$-0.744278\pi$
$421$	−11994.7	−1.38857	−0.694283	+	0.719702i	$0.744278\pi$
$422$	−3593.60	−0.414535
$423$	5211.80	0.599070
$424$	−6181.99	−0.708076
$425$	−7940.91	−0.906331
$426$	1240.47	0.141082
$427$	−93.2935	−0.0105733
$428$	5995.51	0.677112
$429$	1718.64	0.193419
$430$	922.160	0.103420
$431$	5007.19	0.559600	0.279800	−	0.960058i	$-0.409732\pi$
$431$	5007.19	0.559600	0.279800	+	0.960058i	$0.409732\pi$
$432$	1380.92	0.153795
$433$	7648.80	0.848910	0.424455	−	0.905449i	$-0.360466\pi$
$433$	7648.80	0.848910	0.424455	+	0.905449i	$0.360466\pi$
$434$	−90.5816	−0.0100186
$435$	−1143.01	−0.125984
$436$	−7376.05	−0.810204
$437$	5351.22	0.585774
$438$	−1336.65	−0.145817
$439$	9468.81	1.02943	0.514717	−	0.857360i	$-0.327897\pi$
$439$	9468.81	1.02943	0.514717	+	0.857360i	$0.327897\pi$
$440$	960.541	0.104073
$441$	−3065.95	−0.331060
$442$	−4588.00	−0.493731
$443$	15463.8	1.65849	0.829243	−	0.558889i	$-0.188772\pi$
$443$	15463.8	1.65849	0.829243	+	0.558889i	$0.188772\pi$
$444$	−8178.57	−0.874184
$445$	4018.54	0.428083
$446$	−826.026	−0.0876984
$447$	4878.61	0.516219
$448$	−479.269	−0.0505432
$449$	7204.46	0.757237	0.378619	−	0.925553i	$-0.376399\pi$
$449$	7204.46	0.757237	0.378619	+	0.925553i	$0.376399\pi$
$450$	444.679	0.0465831
$451$	−2765.85	−0.288778
$452$	−2119.22	−0.220531
$453$	6157.21	0.638611
$454$	−157.032	−0.0162332
$455$	607.985	0.0626435
$456$	1277.37	0.131181
$457$	5949.21	0.608955	0.304478	−	0.952520i	$-0.401518\pi$
$457$	5949.21	0.608955	0.304478	+	0.952520i	$0.401518\pi$
$458$	−3504.96	−0.357590
$459$	3212.72	0.326703
$460$	−8177.94	−0.828910
$461$	−6134.81	−0.619797	−0.309899	−	0.950770i	$-0.600295\pi$
$461$	−6134.81	−0.619797	−0.309899	+	0.950770i	$0.600295\pi$
$462$	−37.3662	−0.00376284
$463$	315.185	0.0316369	0.0158184	−	0.999875i	$-0.494965\pi$
$463$	315.185	0.0316369	0.0158184	+	0.999875i	$0.494965\pi$
$464$	−2552.91	−0.255422
$465$	1831.88	0.182691
$466$	3188.44	0.316956
$467$	−3405.33	−0.337430	−0.168715	−	0.985665i	$-0.553962\pi$
$467$	−3405.33	−0.337430	−0.168715	+	0.985665i	$0.553962\pi$
$468$	−3492.85	−0.344994
$469$	−816.545	−0.0803935
$470$	−3272.56	−0.321174
$471$	8470.75	0.828687
$472$	−2343.21	−0.228506
$473$	−1794.97	−0.174488
$474$	−1347.47	−0.130573
$475$	−2483.90	−0.239935
$476$	−1356.11	−0.130582
$477$	−4863.48	−0.466842
$478$	3418.48	0.327108
$479$	16246.2	1.54970	0.774852	−	0.632143i	$-0.217824\pi$
$479$	16246.2	1.54970	0.774852	+	0.632143i	$0.217824\pi$
$480$	−2962.83	−0.281737
$481$	19053.0	1.80612
$482$	2156.63	0.203800
$483$	659.663	0.0621444
$484$	−901.676	−0.0846803
$485$	2708.33	0.253564
$486$	−179.907	−0.0167917
$487$	−14351.2	−1.33535	−0.667673	−	0.744455i	$-0.732710\pi$
$487$	−14351.2	−1.33535	−0.667673	+	0.744455i	$0.732710\pi$
$488$	−697.837	−0.0647327
$489$	2188.96	0.202430
$490$	1925.15	0.177489
$491$	1480.06	0.136037	0.0680184	−	0.997684i	$-0.478332\pi$
$491$	1480.06	0.136037	0.0680184	+	0.997684i	$0.478332\pi$
$492$	5621.12	0.515081
$493$	−5939.34	−0.542585
$494$	−1435.12	−0.130707
$495$	755.674	0.0686162
$496$	4091.49	0.370389
$497$	−854.166	−0.0770917
$498$	−793.129	−0.0713674
$499$	13978.7	1.25405	0.627025	−	0.778999i	$-0.284272\pi$
$499$	13978.7	1.25405	0.627025	+	0.778999i	$0.284272\pi$
$500$	10906.1	0.975469
$501$	−4710.37	−0.420048
$502$	7.56054	0.000672198 0
$503$	12564.0	1.11372	0.556859	−	0.830607i	$-0.312006\pi$
$503$	12564.0	1.11372	0.556859	+	0.830607i	$0.312006\pi$
$504$	157.466	0.0139169
$505$	−1498.44	−0.132039
$506$	−1170.89	−0.102870
$507$	1546.02	0.135427
$508$	14634.3	1.27814
$509$	17193.3	1.49721	0.748603	−	0.663018i	$-0.230725\pi$
$509$	17193.3	1.49721	0.748603	+	0.663018i	$0.230725\pi$
$510$	−2017.31	−0.175153
$511$	920.399	0.0796792
$512$	−11298.2	−0.975228
$513$	1004.93	0.0864890
$514$	1584.13	0.135939
$515$	3397.34	0.290689
$516$	3647.96	0.311226
$517$	6369.98	0.541879
$518$	−414.243	−0.0351367
$519$	704.912	0.0596189
$520$	4547.74	0.383522
$521$	−2553.44	−0.214719	−0.107359	−	0.994220i	$-0.534240\pi$
$521$	−2553.44	−0.214719	−0.107359	+	0.994220i	$0.534240\pi$
$522$	332.594	0.0278875
$523$	−10700.8	−0.894671	−0.447335	−	0.894366i	$-0.647627\pi$
$523$	−10700.8	−0.894671	−0.447335	+	0.894366i	$0.647627\pi$
$524$	8568.73	0.714364
$525$	−306.200	−0.0254546
$526$	5361.68	0.444449
$527$	9518.84	0.786807
$528$	1687.79	0.139113
$529$	8503.88	0.698930
$530$	3053.84	0.250284
$531$	−1843.44	−0.150656
$532$	−424.189	−0.0345694
$533$	−13095.1	−1.06419
$534$	−1169.32	−0.0947592
$535$	−6141.30	−0.496283
$536$	−6107.77	−0.492193
$537$	−13662.1	−1.09789
$538$	−3228.67	−0.258732
$539$	−3747.27	−0.299455
$540$	−1535.78	−0.122388
$541$	−15574.8	−1.23773	−0.618867	−	0.785496i	$-0.712408\pi$
$541$	−15574.8	−1.23773	−0.618867	+	0.785496i	$0.712408\pi$
$542$	1023.47	0.0811102
$543$	−8267.67	−0.653407
$544$	−15395.5	−1.21338
$545$	7555.41	0.593832
$546$	−176.912	−0.0138666
$547$	9229.82	0.721460	0.360730	−	0.932670i	$-0.382528\pi$
$547$	9229.82	0.721460	0.360730	+	0.932670i	$0.382528\pi$
$548$	245.036	0.0191011
$549$	−549.000	−0.0426790
$550$	543.497	0.0421360
$551$	−1857.81	−0.143640
$552$	4934.29	0.380466
$553$	927.849	0.0713493
$554$	−1649.51	−0.126500
$555$	8377.44	0.640725
$556$	8801.68	0.671357
$557$	21353.6	1.62438	0.812190	−	0.583392i	$-0.198275\pi$
$557$	21353.6	1.62438	0.812190	+	0.583392i	$0.198275\pi$
$558$	−533.042	−0.0404399
$559$	−8498.38	−0.643011
$560$	597.071	0.0450551
$561$	3926.65	0.295514
$562$	1374.97	0.103202
$563$	−6433.61	−0.481606	−0.240803	−	0.970574i	$-0.577411\pi$
$563$	−6433.61	−0.481606	−0.240803	+	0.970574i	$0.577411\pi$
$564$	−12945.9	−0.966525
$565$	2170.76	0.161636
$566$	−6495.90	−0.482408
$567$	123.882	0.00917555
$568$	−6389.17	−0.471978
$569$	2306.68	0.169949	0.0849747	−	0.996383i	$-0.472919\pi$
$569$	2306.68	0.169949	0.0849747	+	0.996383i	$0.472919\pi$
$570$	−631.010	−0.0463686
$571$	−19564.1	−1.43386	−0.716929	−	0.697146i	$-0.754453\pi$
$571$	−19564.1	−1.43386	−0.716929	+	0.697146i	$0.754453\pi$
$572$	−4269.04	−0.312058
$573$	8446.14	0.615781
$574$	284.709	0.0207030
$575$	−9594.92	−0.695888
$576$	−2820.33	−0.204017
$577$	−19780.4	−1.42716	−0.713579	−	0.700574i	$-0.752927\pi$
$577$	−19780.4	−1.42716	−0.713579	+	0.700574i	$0.752927\pi$
$578$	−6845.00	−0.492585
$579$	−11479.4	−0.823953
$580$	2839.19	0.203260
$581$	546.137	0.0389975
$582$	−788.072	−0.0561282
$583$	−5944.25	−0.422274
$584$	6884.59	0.487819
$585$	3577.78	0.252860
$586$	460.826	0.0324856
$587$	−12707.5	−0.893514	−0.446757	−	0.894655i	$-0.647421\pi$
$587$	−12707.5	−0.893514	−0.446757	+	0.894655i	$0.647421\pi$
$588$	7615.68	0.534125
$589$	2977.48	0.208293
$590$	1157.52	0.0807701
$591$	13613.5	0.947519
$592$	18711.0	1.29901
$593$	14366.9	0.994904	0.497452	−	0.867492i	$-0.334269\pi$
$593$	14366.9	0.994904	0.497452	+	0.867492i	$0.334269\pi$
$594$	−219.887	−0.0151887
$595$	1389.09	0.0957093
$596$	−12118.2	−0.832856
$597$	12644.5	0.866843
$598$	−5543.63	−0.379090
$599$	−16369.6	−1.11660	−0.558300	−	0.829639i	$-0.688546\pi$
$599$	−16369.6	−1.11660	−0.558300	+	0.829639i	$0.688546\pi$
$600$	−2290.38	−0.155840
$601$	−9543.32	−0.647721	−0.323860	−	0.946105i	$-0.604981\pi$
$601$	−9543.32	−0.647721	−0.323860	+	0.946105i	$0.604981\pi$
$602$	184.769	0.0125093
$603$	−4805.09	−0.324508
$604$	−15294.2	−1.03032
$605$	923.602	0.0620657
$606$	436.017	0.0292277
$607$	11551.8	0.772444	0.386222	−	0.922406i	$-0.373780\pi$
$607$	11551.8	0.772444	0.386222	+	0.922406i	$0.373780\pi$
$608$	−4815.69	−0.321220
$609$	−229.019	−0.0152386
$610$	344.724	0.0228811
$611$	30159.1	1.99690
$612$	−7980.24	−0.527095
$613$	12590.3	0.829555	0.414777	−	0.909923i	$-0.363859\pi$
$613$	12590.3	0.829555	0.414777	+	0.909923i	$0.363859\pi$
$614$	7146.84	0.469744
$615$	−5757.81	−0.377524
$616$	192.459	0.0125883
$617$	−17913.0	−1.16880	−0.584401	−	0.811465i	$-0.698670\pi$
$617$	−17913.0	−1.16880	−0.584401	+	0.811465i	$0.698670\pi$
$618$	−988.562	−0.0643460
$619$	−26801.1	−1.74027	−0.870134	−	0.492815i	$-0.835968\pi$
$619$	−26801.1	−1.74027	−0.870134	+	0.492815i	$0.835968\pi$
$620$	−4550.30	−0.294749
$621$	3881.89	0.250845
$622$	−6629.22	−0.427343
$623$	805.176	0.0517796
$624$	7990.96	0.512651
$625$	−2829.25	−0.181072
$626$	359.636	0.0229616
$627$	1228.25	0.0782322
$628$	−21041.0	−1.33698
$629$	43531.1	2.75946
$630$	−77.7869	−0.00491921
$631$	−5603.59	−0.353527	−0.176763	−	0.984253i	$-0.556563\pi$
$631$	−5603.59	−0.353527	−0.176763	+	0.984253i	$0.556563\pi$
$632$	6940.32	0.436822
$633$	14561.6	0.914330
$634$	−116.680	−0.00730905
$635$	−14990.2	−0.936799
$636$	12080.7	0.753191
$637$	−17741.7	−1.10353
$638$	406.504	0.0252252
$639$	−5026.47	−0.311180
$640$	9671.79	0.597361
$641$	−14340.2	−0.883624	−0.441812	−	0.897108i	$-0.645664\pi$
$641$	−14340.2	−0.883624	−0.441812	+	0.897108i	$0.645664\pi$
$642$	1787.00	0.109856
$643$	11112.3	0.681533	0.340767	−	0.940148i	$-0.389313\pi$
$643$	11112.3	0.681533	0.340767	+	0.940148i	$0.389313\pi$
$644$	−1638.57	−0.100262
$645$	−3736.67	−0.228110
$646$	−3278.87	−0.199699
$647$	−13981.0	−0.849538	−0.424769	−	0.905302i	$-0.639645\pi$
$647$	−13981.0	−0.849538	−0.424769	+	0.905302i	$0.639645\pi$
$648$	926.635	0.0561754
$649$	−2253.09	−0.136274
$650$	2573.22	0.155277
$651$	367.044	0.0220977
$652$	−5437.27	−0.326595
$653$	30734.1	1.84183	0.920916	−	0.389760i	$-0.127442\pi$
$653$	30734.1	1.84183	0.920916	+	0.389760i	$0.127442\pi$
$654$	−2198.48	−0.131449
$655$	−8777.09	−0.523587
$656$	−12860.0	−0.765396
$657$	5416.23	0.321624
$658$	−655.707	−0.0388482
$659$	27667.9	1.63549	0.817745	−	0.575580i	$-0.195224\pi$
$659$	27667.9	1.63549	0.817745	+	0.575580i	$0.195224\pi$
$660$	−1877.06	−0.110704
$661$	−13692.7	−0.805725	−0.402862	−	0.915261i	$-0.631985\pi$
$661$	−13692.7	−0.805725	−0.402862	+	0.915261i	$0.631985\pi$
$662$	−4087.21	−0.239960
$663$	18591.0	1.08901
$664$	4085.11	0.238754
$665$	434.504	0.0253374
$666$	−2437.68	−0.141829
$667$	−7176.44	−0.416601
$668$	11700.4	0.677695
$669$	3347.13	0.193434
$670$	3017.18	0.173976
$671$	−671.000	−0.0386046
$672$	−593.647	−0.0340780
$673$	−10659.7	−0.610554	−0.305277	−	0.952264i	$-0.598749\pi$
$673$	−10659.7	−0.610554	−0.305277	+	0.952264i	$0.598749\pi$
$674$	2463.68	0.140797
$675$	−1801.88	−0.102747
$676$	−3840.25	−0.218494
$677$	481.057	0.0273095	0.0136547	−	0.999907i	$-0.495653\pi$
$677$	481.057	0.0273095	0.0136547	+	0.999907i	$0.495653\pi$
$678$	−631.649	−0.0357793
$679$	542.654	0.0306703
$680$	10390.4	0.585961
$681$	636.307	0.0358052
$682$	−651.495	−0.0365792
$683$	−20190.0	−1.13111	−0.565556	−	0.824710i	$-0.691339\pi$
$683$	−20190.0	−1.13111	−0.565556	+	0.824710i	$0.691339\pi$
$684$	−2496.21	−0.139539
$685$	−250.994	−0.0140000
$686$	774.115	0.0430843
$687$	14202.4	0.788726
$688$	−8345.83	−0.462474
$689$	−28143.4	−1.55614
$690$	−2437.49	−0.134484
$691$	18685.1	1.02867	0.514337	−	0.857588i	$-0.328038\pi$
$691$	18685.1	1.02867	0.514337	+	0.857588i	$0.328038\pi$
$692$	−1750.97	−0.0961877
$693$	151.411	0.00829960
$694$	−6082.06	−0.332668
$695$	−9015.71	−0.492065
$696$	−1713.07	−0.0932955
$697$	−29918.9	−1.62591
$698$	4898.67	0.265641
$699$	−12919.8	−0.699102
$700$	760.586	0.0410678
$701$	11947.4	0.643717	0.321859	−	0.946788i	$-0.395692\pi$
$701$	11947.4	0.643717	0.321859	+	0.946788i	$0.395692\pi$
$702$	−1041.07	−0.0559723
$703$	13616.4	0.730518
$704$	−3447.07	−0.184540
$705$	13260.7	0.708406
$706$	−2822.08	−0.150440
$707$	−300.235	−0.0159710
$708$	4579.02	0.243065
$709$	3781.34	0.200298	0.100149	−	0.994972i	$-0.468068\pi$
$709$	3781.34	0.200298	0.100149	+	0.994972i	$0.468068\pi$
$710$	3156.19	0.166830
$711$	5460.07	0.288001
$712$	6022.72	0.317010
$713$	11501.5	0.604117
$714$	−404.198	−0.0211859
$715$	4372.85	0.228720
$716$	33936.2	1.77130
$717$	−13852.0	−0.721494
$718$	−5294.36	−0.275187
$719$	30006.2	1.55639	0.778193	−	0.628026i	$-0.216137\pi$
$719$	30006.2	1.55639	0.778193	+	0.628026i	$0.216137\pi$
$720$	3513.56	0.181865
$721$	680.709	0.0351608
$722$	4052.50	0.208890
$723$	−8738.84	−0.449517
$724$	20536.5	1.05419
$725$	3331.13	0.170641
$726$	−268.751	−0.0137387
$727$	−9079.19	−0.463176	−0.231588	−	0.972814i	$-0.574392\pi$
$727$	−9079.19	−0.463176	−0.231588	+	0.972814i	$0.574392\pi$
$728$	911.208	0.0463896
$729$	729.000	0.0370370
$730$	−3400.92	−0.172430
$731$	−19416.6	−0.982419
$732$	1363.69	0.0688572
$733$	−11163.7	−0.562540	−0.281270	−	0.959629i	$-0.590756\pi$
$733$	−11163.7	−0.562540	−0.281270	+	0.959629i	$0.590756\pi$
$734$	−2358.95	−0.118624
$735$	−7800.87	−0.391482
$736$	−18602.2	−0.931640
$737$	−5872.88	−0.293528
$738$	1675.41	0.0835675
$739$	−474.923	−0.0236405	−0.0118202	−	0.999930i	$-0.503763\pi$
$739$	−474.923	−0.0236405	−0.0118202	+	0.999930i	$0.503763\pi$
$740$	−20809.2	−1.03373
$741$	5815.22	0.288296
$742$	611.884	0.0302735
$743$	−19450.8	−0.960404	−0.480202	−	0.877158i	$-0.659437\pi$
$743$	−19450.8	−0.960404	−0.480202	+	0.877158i	$0.659437\pi$
$744$	2745.50	0.135289
$745$	12412.9	0.610435
$746$	−2831.45	−0.138963
$747$	3213.82	0.157413
$748$	−9753.63	−0.476776
$749$	−1230.50	−0.0600288
$750$	3250.63	0.158262
$751$	−35040.2	−1.70258	−0.851289	−	0.524697i	$-0.824179\pi$
$751$	−35040.2	−1.70258	−0.851289	+	0.524697i	$0.824179\pi$
$752$	29617.7	1.43623
$753$	−30.6359	−0.00148265
$754$	1924.62	0.0929581
$755$	15666.1	0.755164
$756$	−307.716	−0.0148036
$757$	13092.2	0.628591	0.314295	−	0.949325i	$-0.398232\pi$
$757$	13092.2	0.628591	0.314295	+	0.949325i	$0.398232\pi$
$758$	9802.69	0.469723
$759$	4744.53	0.226898
$760$	3250.10	0.155123
$761$	−6029.20	−0.287199	−0.143600	−	0.989636i	$-0.545868\pi$
$761$	−6029.20	−0.287199	−0.143600	+	0.989636i	$0.545868\pi$
$762$	4361.86	0.207367
$763$	1513.84	0.0718280
$764$	−20979.8	−0.993487
$765$	8174.30	0.386330
$766$	−101.840	−0.00480368
$767$	−10667.4	−0.502187
$768$	4706.58	0.221138
$769$	−35550.8	−1.66709	−0.833545	−	0.552451i	$-0.813693\pi$
$769$	−35550.8	−1.66709	−0.833545	+	0.552451i	$0.813693\pi$
$770$	−95.0729	−0.00444959
$771$	−6419.01	−0.299838
$772$	28514.4	1.32935
$773$	−20175.6	−0.938764	−0.469382	−	0.882995i	$-0.655523\pi$
$773$	−20175.6	−0.938764	−0.469382	+	0.882995i	$0.655523\pi$
$774$	1087.30	0.0504938
$775$	−5338.72	−0.247448
$776$	4059.06	0.187773
$777$	1678.55	0.0775001
$778$	−6608.95	−0.304553
$779$	−9358.57	−0.430431
$780$	−8887.05	−0.407958
$781$	−6143.46	−0.281473
$782$	−12665.8	−0.579190
$783$	−1347.70	−0.0615107
$784$	−17423.2	−0.793695
$785$	21552.6	0.979931
$786$	2553.97	0.115900
$787$	−3957.91	−0.179268	−0.0896342	−	0.995975i	$-0.528570\pi$
$787$	−3957.91	−0.179268	−0.0896342	+	0.995975i	$0.528570\pi$
$788$	−33815.3	−1.52870
$789$	−21726.0	−0.980311
$790$	−3428.45	−0.154404
$791$	434.944	0.0195510
$792$	1132.55	0.0508126
$793$	−3176.89	−0.142263
$794$	11386.3	0.508925
$795$	−12374.4	−0.552045
$796$	−31408.4	−1.39854
$797$	−38747.7	−1.72210	−0.861049	−	0.508521i	$-0.830192\pi$
$797$	−38747.7	−1.72210	−0.861049	+	0.508521i	$0.830192\pi$
$798$	−126.433	−0.00560860
$799$	68905.5	3.05094
$800$	8634.70	0.381603
$801$	4738.18	0.209008
$802$	2497.60	0.109967
$803$	6619.83	0.290920
$804$	11935.6	0.523553
$805$	1678.42	0.0734863
$806$	−3084.54	−0.134799
$807$	13082.8	0.570678
$808$	−2245.76	−0.0977792
$809$	31201.9	1.35599	0.677997	−	0.735065i	$-0.262848\pi$
$809$	31201.9	1.35599	0.677997	+	0.735065i	$0.262848\pi$
$810$	−457.749	−0.0198564
$811$	7526.19	0.325870	0.162935	−	0.986637i	$-0.447904\pi$
$811$	7526.19	0.325870	0.162935	+	0.986637i	$0.447904\pi$
$812$	568.874	0.0245857
$813$	−4147.18	−0.178903
$814$	−2979.38	−0.128289
$815$	5569.49	0.239375
$816$	18257.2	0.783249
$817$	−6073.47	−0.260078
$818$	−2022.18	−0.0864349
$819$	716.863	0.0305851
$820$	14302.1	0.609088
$821$	−9448.42	−0.401647	−0.200823	−	0.979627i	$-0.564362\pi$
$821$	−9448.42	−0.401647	−0.200823	+	0.979627i	$0.564362\pi$
$822$	73.0346	0.00309899
$823$	24069.2	1.01944	0.509720	−	0.860340i	$-0.329749\pi$
$823$	24069.2	1.01944	0.509720	+	0.860340i	$0.329749\pi$
$824$	5091.71	0.215265
$825$	−2202.30	−0.0929383
$826$	231.927	0.00976969
$827$	23845.3	1.00264	0.501320	−	0.865262i	$-0.332848\pi$
$827$	23845.3	1.00264	0.501320	+	0.865262i	$0.332848\pi$
$828$	−9642.44	−0.404708
$829$	25876.0	1.08409	0.542046	−	0.840349i	$-0.317650\pi$
$829$	25876.0	1.08409	0.542046	+	0.840349i	$0.317650\pi$
$830$	−2018.00	−0.0843927
$831$	6683.94	0.279017
$832$	−16320.4	−0.680056
$833$	−40535.1	−1.68602
$834$	2623.40	0.108922
$835$	−11984.9	−0.496711
$836$	−3050.92	−0.126218
$837$	2159.93	0.0891972
$838$	825.878	0.0340447
$839$	17616.6	0.724903	0.362451	−	0.932003i	$-0.381940\pi$
$839$	17616.6	0.724903	0.362451	+	0.932003i	$0.381940\pi$
$840$	400.651	0.0164569
$841$	−21897.5	−0.897844
$842$	8880.40	0.363467
$843$	−5571.48	−0.227630
$844$	−36170.3	−1.47516
$845$	3933.63	0.160143
$846$	−3858.61	−0.156811
$847$	185.058	0.00750727
$848$	−27638.2	−1.11922
$849$	26321.9	1.06404
$850$	5879.13	0.237238
$851$	52598.1	2.11873
$852$	12485.5	0.502051
$853$	22868.6	0.917942	0.458971	−	0.888451i	$-0.348218\pi$
$853$	22868.6	0.917942	0.458971	+	0.888451i	$0.348218\pi$
$854$	69.0708	0.00276763
$855$	2556.91	0.102274
$856$	−9204.17	−0.367514
$857$	−4949.83	−0.197296	−0.0986481	−	0.995122i	$-0.531452\pi$
$857$	−4949.83	−0.197296	−0.0986481	+	0.995122i	$0.531452\pi$
$858$	−1272.42	−0.0506289
$859$	9218.38	0.366155	0.183077	−	0.983099i	$-0.441394\pi$
$859$	9218.38	0.366155	0.183077	+	0.983099i	$0.441394\pi$
$860$	9281.72	0.368028
$861$	−1153.66	−0.0456641
$862$	−3707.12	−0.146479
$863$	−2823.80	−0.111383	−0.0556914	−	0.998448i	$-0.517736\pi$
$863$	−2823.80	−0.111383	−0.0556914	+	0.998448i	$0.517736\pi$
$864$	−3493.41	−0.137556
$865$	1793.55	0.0704999
$866$	−5662.87	−0.222208
$867$	27736.5	1.08648
$868$	−911.722	−0.0356519
$869$	6673.42	0.260507
$870$	846.239	0.0329772
$871$	−27805.5	−1.08169
$872$	11323.6	0.439752
$873$	3193.33	0.123801
$874$	−3961.83	−0.153330
$875$	−2238.34	−0.0864795
$876$	−13453.7	−0.518901
$877$	−20440.6	−0.787037	−0.393518	−	0.919317i	$-0.628742\pi$
$877$	−20440.6	−0.787037	−0.393518	+	0.919317i	$0.628742\pi$
$878$	−7010.33	−0.269461
$879$	−1867.31	−0.0716526
$880$	4294.35	0.164503
$881$	−15206.6	−0.581525	−0.290762	−	0.956795i	$-0.593909\pi$
$881$	−15206.6	−0.581525	−0.290762	+	0.956795i	$0.593909\pi$
$882$	2269.91	0.0866573
$883$	−13801.9	−0.526016	−0.263008	−	0.964794i	$-0.584715\pi$
$883$	−13801.9	−0.526016	−0.263008	+	0.964794i	$0.584715\pi$
$884$	−46179.1	−1.75698
$885$	−4690.37	−0.178153
$886$	−11448.8	−0.434120
$887$	−13127.5	−0.496933	−0.248467	−	0.968640i	$-0.579927\pi$
$887$	−13127.5	−0.496933	−0.248467	+	0.968640i	$0.579927\pi$
$888$	12555.6	0.474479
$889$	−3003.51	−0.113312
$890$	−2975.17	−0.112054
$891$	891.000	0.0335013
$892$	−8314.11	−0.312082
$893$	21553.5	0.807684
$894$	−3611.92	−0.135124
$895$	−34761.4	−1.29826
$896$	1937.89	0.0722549
$897$	22463.3	0.836150
$898$	−5333.89	−0.198212
$899$	−3993.05	−0.148138
$900$	4475.79	0.165770
$901$	−64300.3	−2.37753
$902$	2047.73	0.0755897
$903$	−748.699	−0.0275915
$904$	3253.39	0.119697
$905$	−21035.9	−0.772660
$906$	−4558.55	−0.167161
$907$	−45557.6	−1.66782	−0.833912	−	0.551897i	$-0.813904\pi$
$907$	−45557.6	−1.66782	−0.833912	+	0.551897i	$0.813904\pi$
$908$	−1580.56	−0.0577673
$909$	−1766.78	−0.0644668
$910$	−450.128	−0.0163974
$911$	35825.0	1.30289	0.651446	−	0.758695i	$-0.274163\pi$
$911$	35825.0	1.30289	0.651446	+	0.758695i	$0.274163\pi$
$912$	5710.84	0.207352
$913$	3928.01	0.142386
$914$	−4404.56	−0.159398
$915$	−1396.85	−0.0504683
$916$	−35278.1	−1.27251
$917$	−1758.62	−0.0633314
$918$	−2378.57	−0.0855168
$919$	−24754.1	−0.888533	−0.444267	−	0.895895i	$-0.646536\pi$
$919$	−24754.1	−0.888533	−0.444267	+	0.895895i	$0.646536\pi$
$920$	12554.6	0.449905
$921$	−28959.6	−1.03610
$922$	4541.97	0.162236
$923$	−29086.6	−1.03727
$924$	−376.098	−0.0133904
$925$	−24414.8	−0.867841
$926$	−233.350	−0.00828117
$927$	4005.74	0.141926
$928$	6458.25	0.228451
$929$	−43645.4	−1.54140	−0.770699	−	0.637199i	$-0.780093\pi$
$929$	−43645.4	−1.54140	−0.770699	+	0.637199i	$0.780093\pi$
$930$	−1356.25	−0.0478206
$931$	−12679.3	−0.446345
$932$	32092.3	1.12792
$933$	26862.2	0.942580
$934$	2521.17	0.0883247
$935$	9990.81	0.349448
$936$	5362.15	0.187251
$937$	43969.9	1.53301	0.766507	−	0.642236i	$-0.221993\pi$
$937$	43969.9	1.53301	0.766507	+	0.642236i	$0.221993\pi$
$938$	604.537	0.0210435
$939$	−1457.27	−0.0506457
$940$	−32939.0	−1.14293
$941$	−1932.43	−0.0669454	−0.0334727	−	0.999440i	$-0.510657\pi$
$941$	−1932.43	−0.0669454	−0.0334727	+	0.999440i	$0.510657\pi$
$942$	−6271.40	−0.216914
$943$	−36150.7	−1.24839
$944$	−10475.9	−0.361188
$945$	315.199	0.0108502
$946$	1328.92	0.0456734
$947$	−36563.1	−1.25464	−0.627319	−	0.778763i	$-0.715848\pi$
$947$	−36563.1	−1.25464	−0.627319	+	0.778763i	$0.715848\pi$
$948$	−13562.6	−0.464654
$949$	31342.0	1.07208
$950$	1838.98	0.0628047
$951$	472.795	0.0161214
$952$	2081.87	0.0708759
$953$	−21483.5	−0.730240	−0.365120	−	0.930960i	$-0.618972\pi$
$953$	−21483.5	−0.730240	−0.365120	+	0.930960i	$0.618972\pi$
$954$	3600.73	0.122199
$955$	21490.0	0.728167
$956$	34407.7	1.16404
$957$	−1647.19	−0.0556385
$958$	−12028.0	−0.405646
$959$	−50.2905	−0.00169339
$960$	−7175.93	−0.241252
$961$	−23391.4	−0.785184
$962$	−14106.1	−0.472763
$963$	−7241.08	−0.242306
$964$	21706.9	0.725241
$965$	−29207.8	−0.974333
$966$	−488.388	−0.0162667
$967$	−20215.0	−0.672254	−0.336127	−	0.941817i	$-0.609117\pi$
$967$	−20215.0	−0.672254	−0.336127	+	0.941817i	$0.609117\pi$
$968$	1384.23	0.0459617
$969$	13286.3	0.440471
$970$	−2005.14	−0.0663722
$971$	−1023.55	−0.0338284	−0.0169142	−	0.999857i	$-0.505384\pi$
$971$	−1023.55	−0.0338284	−0.0169142	+	0.999857i	$0.505384\pi$
$972$	−1810.80	−0.0597547
$973$	−1806.43	−0.0595186
$974$	10625.0	0.349536
$975$	−10426.9	−0.342490
$976$	−3119.86	−0.102320
$977$	−24681.0	−0.808203	−0.404102	−	0.914714i	$-0.632416\pi$
$977$	−24681.0	−0.808203	−0.404102	+	0.914714i	$0.632416\pi$
$978$	−1620.62	−0.0529873
$979$	5791.11	0.189055
$980$	19377.0	0.631608
$981$	8908.44	0.289933
$982$	−1095.78	−0.0356086
$983$	36374.9	1.18024	0.590122	−	0.807314i	$-0.299080\pi$
$983$	36374.9	1.18024	0.590122	+	0.807314i	$0.299080\pi$
$984$	−8629.42	−0.279569
$985$	34637.5	1.12045
$986$	4397.25	0.142025
$987$	2656.98	0.0856866
$988$	−14444.8	−0.465130
$989$	−23460.8	−0.754309
$990$	−559.471	−0.0179608
$991$	−36053.4	−1.15568	−0.577838	−	0.816151i	$-0.696104\pi$
$991$	−36053.4	−1.15568	−0.577838	+	0.816151i	$0.696104\pi$
$992$	−10350.5	−0.331279
$993$	16561.7	0.529275
$994$	632.390	0.0201793
$995$	32172.2	1.02505
$996$	−7983.00	−0.253967
$997$	−28547.4	−0.906824	−0.453412	−	0.891301i	$-0.649793\pi$
$997$	−28547.4	−0.906824	−0.453412	+	0.891301i	$0.649793\pi$
$998$	−10349.2	−0.328256
$999$	9877.67	0.312829

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.f.1.16	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.f.1.16	✓	38	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:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2013.a (trivial)

\(f(q)\)	\(=\)	\(q-0.740360 q^{2} +3.00000 q^{3} -7.45187 q^{4} +7.63307 q^{5} -2.22108 q^{6} +1.52940 q^{7} +11.4399 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-0.740360 q^{2} +3.00000 q^{3} -7.45187 q^{4} +7.63307 q^{5} -2.22108 q^{6} +1.52940 q^{7} +11.4399 q^{8} +9.00000 q^{9} -5.65122 q^{10} +11.0000 q^{11} -22.3556 q^{12} +52.0801 q^{13} -1.13231 q^{14} +22.8992 q^{15} +51.1453 q^{16} +118.989 q^{17} -6.66324 q^{18} +37.2197 q^{19} -56.8806 q^{20} +4.58821 q^{21} -8.14396 q^{22} +143.774 q^{23} +34.3198 q^{24} -66.7362 q^{25} -38.5581 q^{26} +27.0000 q^{27} -11.3969 q^{28} -49.9148 q^{29} -16.9537 q^{30} +79.9974 q^{31} -129.385 q^{32} +33.0000 q^{33} -88.0950 q^{34} +11.6740 q^{35} -67.0668 q^{36} +365.840 q^{37} -27.5560 q^{38} +156.240 q^{39} +87.3219 q^{40} -251.441 q^{41} -3.39692 q^{42} -163.179 q^{43} -81.9705 q^{44} +68.6976 q^{45} -106.444 q^{46} +579.089 q^{47} +153.436 q^{48} -340.661 q^{49} +49.4088 q^{50} +356.968 q^{51} -388.094 q^{52} -540.387 q^{53} -19.9897 q^{54} +83.9638 q^{55} +17.4963 q^{56} +111.659 q^{57} +36.9549 q^{58} -204.827 q^{59} -170.642 q^{60} -61.0000 q^{61} -59.2268 q^{62} +13.7646 q^{63} -313.370 q^{64} +397.531 q^{65} -24.4319 q^{66} -533.898 q^{67} -886.694 q^{68} +431.321 q^{69} -8.64299 q^{70} -558.497 q^{71} +102.959 q^{72} +601.803 q^{73} -270.853 q^{74} -200.209 q^{75} -277.356 q^{76} +16.8234 q^{77} -115.674 q^{78} +606.675 q^{79} +390.395 q^{80} +81.0000 q^{81} +186.157 q^{82} +357.092 q^{83} -34.1907 q^{84} +908.255 q^{85} +120.811 q^{86} -149.744 q^{87} +125.839 q^{88} +526.464 q^{89} -50.8610 q^{90} +79.6515 q^{91} -1071.38 q^{92} +239.992 q^{93} -428.735 q^{94} +284.101 q^{95} -388.156 q^{96} +354.815 q^{97} +252.212 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9}+O(q^{10}) \) 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9	\( 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9} + 99 q^{10} + 418 q^{11} + 510 q^{12} + 209 q^{13} + 128 q^{14} + 105 q^{15} + 798 q^{16} + 512 q^{17} + 126 q^{18} + 487 q^{19} + 328 q^{20} + 315 q^{21} + 154 q^{22} + 417 q^{23} + 441 q^{24} + 925 q^{25} + 177 q^{26} + 1026 q^{27} + 902 q^{28} + 626 q^{29} + 297 q^{30} + 300 q^{31} + 1625 q^{32} + 1254 q^{33} - 180 q^{34} + 1086 q^{35} + 1530 q^{36} + 554 q^{37} + 845 q^{38} + 627 q^{39} + 329 q^{40} + 1378 q^{41} + 384 q^{42} + 1979 q^{43} + 1870 q^{44} + 315 q^{45} + 937 q^{46} + 1345 q^{47} + 2394 q^{48} + 2635 q^{49} + 800 q^{50} + 1536 q^{51} + 2006 q^{52} + 1497 q^{53} + 378 q^{54} + 385 q^{55} + 415 q^{56} + 1461 q^{57} + 1241 q^{58} + 2827 q^{59} + 984 q^{60} - 2318 q^{61} + 509 q^{62} + 945 q^{63} + 1003 q^{64} + 2810 q^{65} + 462 q^{66} + 369 q^{67} + 3936 q^{68} + 1251 q^{69} + 922 q^{70} + 965 q^{71} + 1323 q^{72} + 3081 q^{73} + 722 q^{74} + 2775 q^{75} + 2210 q^{76} + 1155 q^{77} + 531 q^{78} + 3795 q^{79} + 3793 q^{80} + 3078 q^{81} - 1678 q^{82} + 3869 q^{83} + 2706 q^{84} + 3553 q^{85} + 3305 q^{86} + 1878 q^{87} + 1617 q^{88} + 2849 q^{89} + 891 q^{90} + 1252 q^{91} + 4519 q^{92} + 900 q^{93} + 340 q^{94} + 1504 q^{95} + 4875 q^{96} + 2562 q^{97} + 6164 q^{98} + 3762 q^{99}+O(q^{100}) \) 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9 + 99 * q^10 + 418 * q^11 + 510 * q^12 + 209 * q^13 + 128 * q^14 + 105 * q^15 + 798 * q^16 + 512 * q^17 + 126 * q^18 + 487 * q^19 + 328 * q^20 + 315 * q^21 + 154 * q^22 + 417 * q^23 + 441 * q^24 + 925 * q^25 + 177 * q^26 + 1026 * q^27 + 902 * q^28 + 626 * q^29 + 297 * q^30 + 300 * q^31 + 1625 * q^32 + 1254 * q^33 - 180 * q^34 + 1086 * q^35 + 1530 * q^36 + 554 * q^37 + 845 * q^38 + 627 * q^39 + 329 * q^40 + 1378 * q^41 + 384 * q^42 + 1979 * q^43 + 1870 * q^44 + 315 * q^45 + 937 * q^46 + 1345 * q^47 + 2394 * q^48 + 2635 * q^49 + 800 * q^50 + 1536 * q^51 + 2006 * q^52 + 1497 * q^53 + 378 * q^54 + 385 * q^55 + 415 * q^56 + 1461 * q^57 + 1241 * q^58 + 2827 * q^59 + 984 * q^60 - 2318 * q^61 + 509 * q^62 + 945 * q^63 + 1003 * q^64 + 2810 * q^65 + 462 * q^66 + 369 * q^67 + 3936 * q^68 + 1251 * q^69 + 922 * q^70 + 965 * q^71 + 1323 * q^72 + 3081 * q^73 + 722 * q^74 + 2775 * q^75 + 2210 * q^76 + 1155 * q^77 + 531 * q^78 + 3795 * q^79 + 3793 * q^80 + 3078 * q^81 - 1678 * q^82 + 3869 * q^83 + 2706 * q^84 + 3553 * q^85 + 3305 * q^86 + 1878 * q^87 + 1617 * q^88 + 2849 * q^89 + 891 * q^90 + 1252 * q^91 + 4519 * q^92 + 900 * q^93 + 340 * q^94 + 1504 * q^95 + 4875 * q^96 + 2562 * q^97 + 6164 * q^98 + 3762 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more