LMFDB - Embedded newform 75.22.a.n.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	75.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:	$209.608008215$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 5 x^{9} - 15486550 x^{8} - 930225020 x^{7} + 80619202368510 x^{6} + \cdots + 47\!\cdots\!18 $ x^10 - 5x^9 - 15486550x^8 - 930225020x^7 + 80619202368510x^6 + 14206919536731394x^5 - 159833300365521427920x^4 - 53427455410418539963580x^3 + 93639331506281672292832825x^2 + 44269185428005803484908321595*x + 4729332062567824907768533377518
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{14}\cdot 5^{24}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1403.87$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1468.87 q^{2} -59049.0 q^{3} +60422.4 q^{4} -8.67352e7 q^{6} +1.30137e9 q^{7} -2.99169e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q+1468.87 q^{2} -59049.0 q^{3} +60422.4 q^{4} -8.67352e7 q^{6} +1.30137e9 q^{7} -2.99169e9 q^{8} +3.48678e9 q^{9} +1.42507e11 q^{11} -3.56788e9 q^{12} -3.79323e11 q^{13} +1.91154e12 q^{14} -4.52111e12 q^{16} +1.18990e13 q^{17} +5.12163e12 q^{18} -3.41779e12 q^{19} -7.68447e13 q^{21} +2.09323e14 q^{22} +2.05302e14 q^{23} +1.76656e14 q^{24} -5.57175e14 q^{26} -2.05891e14 q^{27} +7.86320e13 q^{28} +1.97632e15 q^{29} -1.01848e15 q^{31} -3.66892e14 q^{32} -8.41487e15 q^{33} +1.74781e16 q^{34} +2.10680e14 q^{36} -6.16147e15 q^{37} -5.02029e15 q^{38} +2.23986e16 q^{39} -4.68002e16 q^{41} -1.12875e17 q^{42} +1.10768e17 q^{43} +8.61058e15 q^{44} +3.01561e17 q^{46} -4.10059e17 q^{47} +2.66967e17 q^{48} +1.13502e18 q^{49} -7.02627e17 q^{51} -2.29196e16 q^{52} -6.01672e17 q^{53} -3.02427e17 q^{54} -3.89330e18 q^{56} +2.01817e17 q^{57} +2.90296e18 q^{58} -6.49215e18 q^{59} +1.32049e18 q^{61} -1.49601e18 q^{62} +4.53760e18 q^{63} +8.94254e18 q^{64} -1.23603e19 q^{66} -1.32066e19 q^{67} +7.18969e17 q^{68} -1.21228e19 q^{69} +1.69431e19 q^{71} -1.04314e19 q^{72} -7.07922e18 q^{73} -9.05038e18 q^{74} -2.06511e17 q^{76} +1.85454e20 q^{77} +3.29006e19 q^{78} +1.25182e20 q^{79} +1.21577e19 q^{81} -6.87433e19 q^{82} -1.70384e20 q^{83} -4.64314e18 q^{84} +1.62703e20 q^{86} -1.16700e20 q^{87} -4.26335e20 q^{88} -3.11012e20 q^{89} -4.93640e20 q^{91} +1.24048e19 q^{92} +6.01400e19 q^{93} -6.02323e20 q^{94} +2.16646e19 q^{96} -1.04258e20 q^{97} +1.66720e21 q^{98} +4.96889e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 645 q^{2} - 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} + 115357290 q^{7} + 314142735 q^{8} + 34867844010 q^{9}+O(q^{10}) $ 10 * q + 645 * q^2 - 590490 * q^3 + 10043205 * q^4 - 38086605 * q^6 + 115357290 * q^7 + 314142735 * q^8 + 34867844010 * q^9	$ 10 q + 645 q^{2} - 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} + 115357290 q^{7} + 314142735 q^{8} + 34867844010 q^{9} + 5976221790 q^{11} - 593041212045 q^{12} + 842570747430 q^{13} + 1605059874570 q^{14} + 6061861547825 q^{16} + 12910340404230 q^{17} + 2248975938645 q^{18} - 76155422176280 q^{19} - 6811732617210 q^{21} + 461780887241010 q^{22} + 82640920915920 q^{23} - 18549814359015 q^{24} + 286555331159670 q^{26} - 20\!\cdots\!90 q^{27}+ \cdots + 20\!\cdots\!90 q^{99}+O(q^{100}) $ 10 * q + 645 * q^2 - 590490 * q^3 + 10043205 * q^4 - 38086605 * q^6 + 115357290 * q^7 + 314142735 * q^8 + 34867844010 * q^9 + 5976221790 * q^11 - 593041212045 * q^12 + 842570747430 * q^13 + 1605059874570 * q^14 + 6061861547825 * q^16 + 12910340404230 * q^17 + 2248975938645 * q^18 - 76155422176280 * q^19 - 6811732617210 * q^21 + 461780887241010 * q^22 + 82640920915920 * q^23 - 18549814359015 * q^24 + 286555331159670 * q^26 - 2058911320946490 * q^27 + 1520428218531930 * q^28 - 2553653160731010 * q^29 - 5579671364724940 * q^31 + 29255856594217815 * q^32 - 352889920477710 * q^33 + 36285968426431010 * q^34 + 35018490530045205 * q^36 + 1797908431222890 * q^37 + 10684960744876320 * q^38 - 49752960064994070 * q^39 - 157319729506021440 * q^41 - 94777180533483930 * q^42 - 148924853700743400 * q^43 + 49118820773767290 * q^44 - 238453286038259540 * q^46 - 43171138083158100 * q^47 - 357946862537518425 * q^48 + 453692886452454950 * q^49 - 762342690529377270 * q^51 + 5077473324432504030 * q^52 + 1835165786666252970 * q^53 - 132799780201048605 * q^54 - 5146515624013530450 * q^56 + 4496901524087157720 * q^57 + 3684042125074285440 * q^58 + 4936135121536323630 * q^59 - 4443080871007408300 * q^61 + 39366851097921260280 * q^62 + 402225999313633290 * q^63 - 14877283608188093455 * q^64 - 27267699610694399490 * q^66 + 33664559574358232580 * q^67 + 44791395195934111290 * q^68 - 4879863739164160080 * q^69 - 74662724971386630180 * q^71 + 1095347988085476735 * q^72 + 16062862692682477620 * q^73 - 130249023546610518030 * q^74 - 258736483662380220720 * q^76 + 56216747719240740180 * q^77 - 16920805749647353830 * q^78 - 49130708010799643300 * q^79 + 121576654590569288010 * q^81 + 117185080045656261930 * q^82 + 118408383608453695560 * q^83 - 89779765876091934570 * q^84 - 1007271272794707382260 * q^86 + 150790665488005409490 * q^87 + 41803519849182967710 * q^88 - 572103976713991581780 * q^89 - 1669810835257196575140 * q^91 - 1206545757146297440260 * q^92 + 329474014415642982060 * q^93 - 1671319720124696594920 * q^94 - 1727529076031967757935 * q^96 - 1299275526640742173920 * q^97 - 2683959884079093505455 * q^98 + 20837796914288297790 * 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 $

$2$	1468.87	1.01430	0.507152	−	0.861857i	$-0.330698\pi$
$2$	1468.87	1.01430	0.507152	+	0.861857i	$0.330698\pi$
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	60422.4	0.0288116
$5$	0	0
$5$	0	0
$6$	−8.67352e7	−0.585608
$7$	1.30137e9	1.74129	0.870647	−	0.491909i	$-0.163701\pi$
$7$	1.30137e9	1.74129	0.870647	+	0.491909i	$0.163701\pi$
$8$	−2.99169e9	−0.985080
$9$	3.48678e9	0.333333
$10$	0	0
$11$	1.42507e11	1.65658	0.828288	−	0.560302i	$-0.189315\pi$
$11$	1.42507e11	1.65658	0.828288	+	0.560302i	$0.189315\pi$
$12$	−3.56788e9	−0.0166344
$13$	−3.79323e11	−0.763139	−0.381569	−	0.924340i	$-0.624616\pi$
$13$	−3.79323e11	−0.763139	−0.381569	+	0.924340i	$0.624616\pi$
$14$	1.91154e12	1.76620
$15$	0	0
$16$	−4.52111e12	−1.02798
$17$	1.18990e13	1.43152	0.715762	−	0.698345i	$-0.246080\pi$
$17$	1.18990e13	1.43152	0.715762	+	0.698345i	$0.246080\pi$
$18$	5.12163e12	0.338101
$19$	−3.41779e12	−0.127889	−0.0639445	−	0.997953i	$-0.520368\pi$
$19$	−3.41779e12	−0.127889	−0.0639445	+	0.997953i	$0.520368\pi$
$20$	0	0
$21$	−7.68447e13	−1.00534
$22$	2.09323e14	1.68027
$23$	2.05302e14	1.03336	0.516678	−	0.856180i	$-0.327168\pi$
$23$	2.05302e14	1.03336	0.516678	+	0.856180i	$0.327168\pi$
$24$	1.76656e14	0.568736
$25$	0	0
$26$	−5.57175e14	−0.774054
$27$	−2.05891e14	−0.192450
$28$	7.86320e13	0.0501695
$29$	1.97632e15	0.872326	0.436163	−	0.899868i	$-0.356337\pi$
$29$	1.97632e15	0.872326	0.436163	+	0.899868i	$0.356337\pi$
$30$	0	0
$31$	−1.01848e15	−0.223179	−0.111589	−	0.993754i	$-0.535594\pi$
$31$	−1.01848e15	−0.223179	−0.111589	+	0.993754i	$0.535594\pi$
$32$	−3.66892e14	−0.0576055
$33$	−8.41487e15	−0.956425
$34$	1.74781e16	1.45200
$35$	0	0
$36$	2.10680e14	0.00960388
$37$	−6.16147e15	−0.210652	−0.105326	−	0.994438i	$-0.533589\pi$
$37$	−6.16147e15	−0.210652	−0.105326	+	0.994438i	$0.533589\pi$
$38$	−5.02029e15	−0.129718
$39$	2.23986e16	0.440598
$40$	0	0
$41$	−4.68002e16	−0.544524	−0.272262	−	0.962223i	$-0.587772\pi$
$41$	−4.68002e16	−0.544524	−0.272262	+	0.962223i	$0.587772\pi$
$42$	−1.12875e17	−1.01972
$43$	1.10768e17	0.781617	0.390808	−	0.920472i	$-0.372196\pi$
$43$	1.10768e17	0.781617	0.390808	+	0.920472i	$0.372196\pi$
$44$	8.61058e15	0.0477287
$45$	0	0
$46$	3.01561e17	1.04814
$47$	−4.10059e17	−1.13715	−0.568576	−	0.822631i	$-0.692506\pi$
$47$	−4.10059e17	−1.13715	−0.568576	+	0.822631i	$0.692506\pi$
$48$	2.66967e17	0.593505
$49$	1.13502e18	2.03210
$50$	0	0
$51$	−7.02627e17	−0.826490
$52$	−2.29196e16	−0.0219873
$53$	−6.01672e17	−0.472567	−0.236284	−	0.971684i	$-0.575929\pi$
$53$	−6.01672e17	−0.472567	−0.236284	+	0.971684i	$0.575929\pi$
$54$	−3.02427e17	−0.195203
$55$	0	0
$56$	−3.89330e18	−1.71531
$57$	2.01817e17	0.0738367
$58$	2.90296e18	0.884803
$59$	−6.49215e18	−1.65365	−0.826823	−	0.562462i	$-0.809854\pi$
$59$	−6.49215e18	−1.65365	−0.826823	+	0.562462i	$0.809854\pi$
$60$	0	0
$61$	1.32049e18	0.237012	0.118506	−	0.992953i	$-0.462189\pi$
$61$	1.32049e18	0.237012	0.118506	+	0.992953i	$0.462189\pi$
$62$	−1.49601e18	−0.226371
$63$	4.53760e18	0.580431
$64$	8.94254e18	0.969552
$65$	0	0
$66$	−1.23603e19	−0.970105
$67$	−1.32066e19	−0.885130	−0.442565	−	0.896736i	$-0.645931\pi$
$67$	−1.32066e19	−0.885130	−0.442565	+	0.896736i	$0.645931\pi$
$68$	7.18969e17	0.0412445
$69$	−1.21228e19	−0.596608
$70$	0	0
$71$	1.69431e19	0.617705	0.308852	−	0.951110i	$-0.400055\pi$
$71$	1.69431e19	0.617705	0.308852	+	0.951110i	$0.400055\pi$
$72$	−1.04314e19	−0.328360
$73$	−7.07922e18	−0.192795	−0.0963975	−	0.995343i	$-0.530732\pi$
$73$	−7.07922e18	−0.192795	−0.0963975	+	0.995343i	$0.530732\pi$
$74$	−9.05038e18	−0.213665
$75$	0	0
$76$	−2.06511e17	−0.00368469
$77$	1.85454e20	2.88458
$78$	3.29006e19	0.446900
$79$	1.25182e20	1.48751	0.743753	−	0.668455i	$-0.233044\pi$
$79$	1.25182e20	1.48751	0.743753	+	0.668455i	$0.233044\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	−6.87433e19	−0.552313
$83$	−1.70384e20	−1.20534	−0.602668	−	0.797992i	$-0.705896\pi$
$83$	−1.70384e20	−1.20534	−0.602668	+	0.797992i	$0.705896\pi$
$84$	−4.64314e18	−0.0289654
$85$	0	0
$86$	1.62703e20	0.792797
$87$	−1.16700e20	−0.503638
$88$	−4.26335e20	−1.63186
$89$	−3.11012e20	−1.05726	−0.528631	−	0.848852i	$-0.677294\pi$
$89$	−3.11012e20	−1.05726	−0.528631	+	0.848852i	$0.677294\pi$
$90$	0	0
$91$	−4.93640e20	−1.32885
$92$	1.24048e19	0.0297727
$93$	6.01400e19	0.128852
$94$	−6.02323e20	−1.15342
$95$	0	0
$96$	2.16646e19	0.0332586
$97$	−1.04258e20	−0.143552	−0.0717758	−	0.997421i	$-0.522867\pi$
$97$	−1.04258e20	−0.143552	−0.0717758	+	0.997421i	$0.522867\pi$
$98$	1.66720e21	2.06117
$99$	4.96889e20	0.552192
$100$	0	0
$101$	−7.73285e20	−0.696571	−0.348285	−	0.937389i	$-0.613236\pi$
$101$	−7.73285e20	−0.696571	−0.348285	+	0.937389i	$0.613236\pi$
$102$	−1.03207e21	−0.838312
$103$	1.79873e20	0.131879	0.0659393	−	0.997824i	$-0.478996\pi$
$103$	1.79873e20	0.131879	0.0659393	+	0.997824i	$0.478996\pi$
$104$	1.13481e21	0.751753
$105$	0	0
$106$	−8.83777e20	−0.479326
$107$	3.38585e21	1.66394	0.831970	−	0.554821i	$-0.187213\pi$
$107$	3.38585e21	1.66394	0.831970	+	0.554821i	$0.187213\pi$
$108$	−1.24404e19	−0.00554480
$109$	4.66356e21	1.88686	0.943430	−	0.331571i	$-0.107579\pi$
$109$	4.66356e21	1.88686	0.943430	+	0.331571i	$0.107579\pi$
$110$	0	0
$111$	3.63828e20	0.121620
$112$	−5.88364e21	−1.79002
$113$	2.66349e21	0.738122	0.369061	−	0.929405i	$-0.379679\pi$
$113$	2.66349e21	0.738122	0.369061	+	0.929405i	$0.379679\pi$
$114$	2.96443e20	0.0748929
$115$	0	0
$116$	1.19414e20	0.0251331
$117$	−1.32262e21	−0.254380
$118$	−9.53612e21	−1.67730
$119$	1.54851e22	2.49270
$120$	0	0
$121$	1.29079e22	1.74425
$122$	1.93962e21	0.240402
$123$	2.76350e21	0.314381
$124$	−6.15388e19	−0.00643015
$125$	0	0
$126$	6.66514e21	0.588733
$127$	1.68717e22	1.37158	0.685788	−	0.727801i	$-0.259458\pi$
$127$	1.68717e22	1.37158	0.685788	+	0.727801i	$0.259458\pi$
$128$	1.39048e22	1.04103
$129$	−6.54071e21	−0.451267
$130$	0	0
$131$	2.95475e22	1.73449	0.867245	−	0.497882i	$-0.165889\pi$
$131$	2.95475e22	1.73449	0.867245	+	0.497882i	$0.165889\pi$
$132$	−5.08446e20	−0.0275562
$133$	−4.44782e21	−0.222692
$134$	−1.93988e22	−0.897791
$135$	0	0
$136$	−3.55982e22	−1.41016
$137$	−4.83093e21	−0.177201	−0.0886003	−	0.996067i	$-0.528239\pi$
$137$	−4.83093e21	−0.177201	−0.0886003	+	0.996067i	$0.528239\pi$
$138$	−1.78069e22	−0.605142
$139$	5.28122e22	1.66371	0.831857	−	0.554990i	$-0.187278\pi$
$139$	5.28122e22	1.66371	0.831857	+	0.554990i	$0.187278\pi$
$140$	0	0
$141$	2.42136e22	0.656535
$142$	2.48872e22	0.626540
$143$	−5.40559e22	−1.26420
$144$	−1.57641e22	−0.342661
$145$	0	0
$146$	−1.03984e22	−0.195553
$147$	−6.70219e22	−1.17323
$148$	−3.72291e20	−0.00606924
$149$	1.75027e22	0.265857	0.132929	−	0.991126i	$-0.457562\pi$
$149$	1.75027e22	0.265857	0.132929	+	0.991126i	$0.457562\pi$
$150$	0	0
$151$	2.38626e22	0.315109	0.157554	−	0.987510i	$-0.449639\pi$
$151$	2.38626e22	0.315109	0.157554	+	0.987510i	$0.449639\pi$
$152$	1.02250e22	0.125981
$153$	4.14894e22	0.477175
$154$	2.72407e23	2.92584
$155$	0	0
$156$	1.35338e21	0.0126944
$157$	7.75106e22	0.679852	0.339926	−	0.940452i	$-0.389598\pi$
$157$	7.75106e22	0.679852	0.339926	+	0.940452i	$0.389598\pi$
$158$	1.83876e23	1.50878
$159$	3.55281e22	0.272837
$160$	0	0
$161$	2.67174e23	1.79938
$162$	1.78580e22	0.112700
$163$	1.97835e23	1.17039	0.585197	−	0.810891i	$-0.301017\pi$
$163$	1.97835e23	1.17039	0.585197	+	0.810891i	$0.301017\pi$
$164$	−2.82778e21	−0.0156886
$165$	0	0
$166$	−2.50271e23	−1.22258
$167$	−2.75129e23	−1.26187	−0.630934	−	0.775837i	$-0.717328\pi$
$167$	−2.75129e23	−1.26187	−0.630934	+	0.775837i	$0.717328\pi$
$168$	2.29895e23	0.990336
$169$	−1.03179e23	−0.417619
$170$	0	0
$171$	−1.19171e22	−0.0426297
$172$	6.69284e21	0.0225197
$173$	−2.54176e23	−0.804729	−0.402365	−	0.915480i	$-0.631812\pi$
$173$	−2.54176e23	−0.804729	−0.402365	+	0.915480i	$0.631812\pi$
$174$	−1.71417e23	−0.510841
$175$	0	0
$176$	−6.44288e23	−1.70293
$177$	3.83355e23	0.954733
$178$	−4.56836e23	−1.07238
$179$	−4.64915e23	−1.02900	−0.514502	−	0.857489i	$-0.672023\pi$
$179$	−4.64915e23	−1.02900	−0.514502	+	0.857489i	$0.672023\pi$
$180$	0	0
$181$	−4.73205e23	−0.932018	−0.466009	−	0.884780i	$-0.654309\pi$
$181$	−4.73205e23	−0.932018	−0.466009	+	0.884780i	$0.654309\pi$
$182$	−7.25092e23	−1.34786
$183$	−7.79734e22	−0.136839
$184$	−6.14198e23	−1.01794
$185$	0	0
$186$	8.83378e22	0.130695
$187$	1.69569e24	2.37143
$188$	−2.47767e22	−0.0327632
$189$	−2.67941e23	−0.335112
$190$	0	0
$191$	6.19013e23	0.693185	0.346593	−	0.938016i	$-0.387339\pi$
$191$	6.19013e23	0.693185	0.346593	+	0.938016i	$0.387339\pi$
$192$	−5.28048e23	−0.559771
$193$	−1.29119e23	−0.129610	−0.0648049	−	0.997898i	$-0.520643\pi$
$193$	−1.29119e23	−0.129610	−0.0648049	+	0.997898i	$0.520643\pi$
$194$	−1.53142e23	−0.145605
$195$	0	0
$196$	6.85807e22	0.0585482
$197$	1.29355e24	1.04686	0.523428	−	0.852070i	$-0.324653\pi$
$197$	1.29355e24	1.04686	0.523428	+	0.852070i	$0.324653\pi$
$198$	7.29865e23	0.560090
$199$	−1.97928e24	−1.44062	−0.720312	−	0.693650i	$-0.756001\pi$
$199$	−1.97928e24	−1.44062	−0.720312	+	0.693650i	$0.756001\pi$
$200$	0	0
$201$	7.79839e23	0.511030
$202$	−1.13585e24	−0.706534
$203$	2.57193e24	1.51897
$204$	−4.24544e22	−0.0238125
$205$	0	0
$206$	2.64209e23	0.133765
$207$	7.15842e23	0.344452
$208$	1.71496e24	0.784493
$209$	−4.87058e23	−0.211858
$210$	0	0
$211$	−4.14575e23	−0.163169	−0.0815845	−	0.996666i	$-0.525998\pi$
$211$	−4.14575e23	−0.163169	−0.0815845	+	0.996666i	$0.525998\pi$
$212$	−3.63545e22	−0.0136154
$213$	−1.00048e24	−0.356632
$214$	4.97336e24	1.68774
$215$	0	0
$216$	6.15962e23	0.189579
$217$	−1.32542e24	−0.388620
$218$	6.85016e24	1.91385
$219$	4.18021e23	0.111310
$220$	0	0
$221$	−4.51358e24	−1.09245
$222$	5.34416e23	0.123360
$223$	5.92745e23	0.130517	0.0652585	−	0.997868i	$-0.479213\pi$
$223$	5.92745e23	0.130517	0.0652585	+	0.997868i	$0.479213\pi$
$224$	−4.77463e23	−0.100308
$225$	0	0
$226$	3.91232e24	0.748680
$227$	6.77143e24	1.23711	0.618556	−	0.785741i	$-0.287718\pi$
$227$	6.77143e24	1.23711	0.618556	+	0.785741i	$0.287718\pi$
$228$	1.21943e22	0.00212736
$229$	−7.62208e24	−1.26999	−0.634996	−	0.772516i	$-0.718998\pi$
$229$	−7.62208e24	−1.26999	−0.634996	+	0.772516i	$0.718998\pi$
$230$	0	0
$231$	−1.09509e25	−1.66542
$232$	−5.91254e24	−0.859311
$233$	−4.21367e24	−0.585360	−0.292680	−	0.956210i	$-0.594547\pi$
$233$	−4.21367e24	−0.585360	−0.292680	+	0.956210i	$0.594547\pi$
$234$	−1.94275e24	−0.258018
$235$	0	0
$236$	−3.92271e23	−0.0476443
$237$	−7.39189e24	−0.858812
$238$	2.27455e25	2.52836
$239$	−2.98655e24	−0.317681	−0.158841	−	0.987304i	$-0.550776\pi$
$239$	−2.98655e24	−0.317681	−0.158841	+	0.987304i	$0.550776\pi$
$240$	0	0
$241$	1.15029e25	1.12106	0.560531	−	0.828133i	$-0.310597\pi$
$241$	1.15029e25	1.12106	0.560531	+	0.828133i	$0.310597\pi$
$242$	1.89599e25	1.76919
$243$	−7.17898e23	−0.0641500
$244$	7.97870e22	0.00682871
$245$	0	0
$246$	4.05922e24	0.318878
$247$	1.29645e24	0.0975970
$248$	3.04696e24	0.219849
$249$	1.00610e25	0.695901
$250$	0	0
$251$	1.74488e25	1.10966	0.554832	−	0.831962i	$-0.312783\pi$
$251$	1.74488e25	1.10966	0.554832	+	0.831962i	$0.312783\pi$
$252$	2.74173e23	0.0167232
$253$	2.92568e25	1.71183
$254$	2.47823e25	1.39119
$255$	0	0
$256$	1.67052e24	0.0863639
$257$	5.15268e24	0.255703	0.127851	−	0.991793i	$-0.459192\pi$
$257$	5.15268e24	0.255703	0.127851	+	0.991793i	$0.459192\pi$
$258$	−9.60745e24	−0.457721
$259$	−8.01836e24	−0.366807
$260$	0	0
$261$	6.89101e24	0.290775
$262$	4.34014e25	1.75930
$263$	−4.87036e24	−0.189682	−0.0948410	−	0.995492i	$-0.530234\pi$
$263$	−4.87036e24	−0.189682	−0.0948410	+	0.995492i	$0.530234\pi$
$264$	2.51747e25	0.942155
$265$	0	0
$266$	−6.53326e24	−0.225877
$267$	1.83650e25	0.610410
$268$	−7.97977e23	−0.0255021
$269$	2.43436e25	0.748147	0.374073	−	0.927399i	$-0.377961\pi$
$269$	2.43436e25	0.748147	0.374073	+	0.927399i	$0.377961\pi$
$270$	0	0
$271$	−4.31389e24	−0.122657	−0.0613284	−	0.998118i	$-0.519534\pi$
$271$	−4.31389e24	−0.122657	−0.0613284	+	0.998118i	$0.519534\pi$
$272$	−5.37969e25	−1.47158
$273$	2.91489e25	0.767211
$274$	−7.09601e24	−0.179735
$275$	0	0
$276$	−7.32492e23	−0.0171893
$277$	3.03114e25	0.684808	0.342404	−	0.939553i	$-0.388759\pi$
$277$	3.03114e25	0.684808	0.342404	+	0.939553i	$0.388759\pi$
$278$	7.75741e25	1.68751
$279$	−3.55121e24	−0.0743930
$280$	0	0
$281$	−5.00890e25	−0.973477	−0.486739	−	0.873548i	$-0.661814\pi$
$281$	−5.00890e25	−0.973477	−0.486739	+	0.873548i	$0.661814\pi$
$282$	3.55665e25	0.665926
$283$	9.52620e25	1.71855	0.859275	−	0.511514i	$-0.170915\pi$
$283$	9.52620e25	1.71855	0.859275	+	0.511514i	$0.170915\pi$
$284$	1.02374e24	0.0177971
$285$	0	0
$286$	−7.94011e25	−1.28228
$287$	−6.09044e25	−0.948176
$288$	−1.27928e24	−0.0192018
$289$	7.24954e25	1.04926
$290$	0	0
$291$	6.15636e24	0.0828795
$292$	−4.27744e23	−0.00555474
$293$	1.10268e26	1.38146	0.690731	−	0.723112i	$-0.257289\pi$
$293$	1.10268e26	1.38146	0.690731	+	0.723112i	$0.257289\pi$
$294$	−9.84464e25	−1.19002
$295$	0	0
$296$	1.84332e25	0.207509
$297$	−2.93408e25	−0.318808
$298$	2.57091e25	0.269660
$299$	−7.78755e25	−0.788594
$300$	0	0
$301$	1.44150e26	1.36102
$302$	3.50511e25	0.319616
$303$	4.56617e25	0.402165
$304$	1.54522e25	0.131468
$305$	0	0
$306$	6.09425e25	0.484000
$307$	1.43610e26	1.10213	0.551064	−	0.834463i	$-0.314222\pi$
$307$	1.43610e26	1.10213	0.551064	+	0.834463i	$0.314222\pi$
$308$	1.12056e25	0.0831096
$309$	−1.06213e25	−0.0761401
$310$	0	0
$311$	2.75110e25	0.184299	0.0921494	−	0.995745i	$-0.470626\pi$
$311$	2.75110e25	0.184299	0.0921494	+	0.995745i	$0.470626\pi$
$312$	−6.70097e25	−0.434025
$313$	−2.51960e26	−1.57803	−0.789017	−	0.614372i	$-0.789410\pi$
$313$	−2.51960e26	−1.57803	−0.789017	+	0.614372i	$0.789410\pi$
$314$	1.13853e26	0.689577
$315$	0	0
$316$	7.56382e24	0.0428575
$317$	−2.73071e25	−0.149676	−0.0748382	−	0.997196i	$-0.523844\pi$
$317$	−2.73071e25	−0.149676	−0.0748382	+	0.997196i	$0.523844\pi$
$318$	5.21862e25	0.276739
$319$	2.81639e26	1.44507
$320$	0	0
$321$	−1.99931e26	−0.960676
$322$	3.92443e26	1.82511
$323$	−4.06685e25	−0.183076
$324$	7.34595e23	0.00320129
$325$	0	0
$326$	2.90593e26	1.18714
$327$	−2.75379e26	−1.08938
$328$	1.40012e26	0.536400
$329$	−5.33639e26	−1.98012
$330$	0	0
$331$	−3.16277e26	−1.10122	−0.550610	−	0.834763i	$-0.685605\pi$
$331$	−3.16277e26	−1.10122	−0.550610	+	0.834763i	$0.685605\pi$
$332$	−1.02950e25	−0.0347277
$333$	−2.14837e25	−0.0702174
$334$	−4.04129e26	−1.27992
$335$	0	0
$336$	3.47423e26	1.03347
$337$	7.09882e25	0.204679	0.102339	−	0.994750i	$-0.467367\pi$
$337$	7.09882e25	0.204679	0.102339	+	0.994750i	$0.467367\pi$
$338$	−1.51556e26	−0.423593
$339$	−1.57277e26	−0.426155
$340$	0	0
$341$	−1.45139e26	−0.369713
$342$	−1.75047e25	−0.0432394
$343$	7.50209e26	1.79719
$344$	−3.31382e26	−0.769955
$345$	0	0
$346$	−3.73350e26	−0.816240
$347$	5.53175e26	1.17328	0.586642	−	0.809846i	$-0.300450\pi$
$347$	5.53175e26	1.17328	0.586642	+	0.809846i	$0.300450\pi$
$348$	−7.05128e24	−0.0145106
$349$	−6.34228e26	−1.26642	−0.633211	−	0.773979i	$-0.718263\pi$
$349$	−6.34228e26	−1.26642	−0.633211	+	0.773979i	$0.718263\pi$
$350$	0	0
$351$	7.80992e25	0.146866
$352$	−5.22846e25	−0.0954279
$353$	4.36368e25	0.0773070	0.0386535	−	0.999253i	$-0.487693\pi$
$353$	4.36368e25	0.0773070	0.0386535	+	0.999253i	$0.487693\pi$
$354$	5.63098e26	0.968389
$355$	0	0
$356$	−1.87921e25	−0.0304614
$357$	−9.14378e26	−1.43916
$358$	−6.82900e26	−1.04372
$359$	1.81956e26	0.270070	0.135035	−	0.990841i	$-0.456885\pi$
$359$	1.81956e26	0.270070	0.135035	+	0.990841i	$0.456885\pi$
$360$	0	0
$361$	−7.02528e26	−0.983644
$362$	−6.95076e26	−0.945349
$363$	−7.62196e26	−1.00704
$364$	−2.98269e25	−0.0382863
$365$	0	0
$366$	−1.14533e26	−0.138796
$367$	−1.20245e26	−0.141603	−0.0708017	−	0.997490i	$-0.522556\pi$
$367$	−1.20245e26	−0.141603	−0.0708017	+	0.997490i	$0.522556\pi$
$368$	−9.28191e26	−1.06227
$369$	−1.63182e26	−0.181508
$370$	0	0
$371$	−7.82999e26	−0.822878
$372$	3.63380e24	0.00371245
$373$	−1.48474e26	−0.147471	−0.0737356	−	0.997278i	$-0.523492\pi$
$373$	−1.48474e26	−0.147471	−0.0737356	+	0.997278i	$0.523492\pi$
$374$	2.49075e27	2.40535
$375$	0	0
$376$	1.22677e27	1.12019
$377$	−7.49664e26	−0.665706
$378$	−3.93570e26	−0.339905
$379$	7.78963e26	0.654342	0.327171	−	0.944965i	$-0.393905\pi$
$379$	7.78963e26	0.654342	0.327171	+	0.944965i	$0.393905\pi$
$380$	0	0
$381$	−9.96257e26	−0.791880
$382$	9.09248e26	0.703100
$383$	9.27138e26	0.697521	0.348760	−	0.937212i	$-0.386603\pi$
$383$	9.27138e26	0.697521	0.348760	+	0.937212i	$0.386603\pi$
$384$	−8.21067e26	−0.601036
$385$	0	0
$386$	−1.89659e26	−0.131464
$387$	3.86223e26	0.260539
$388$	−6.29954e24	−0.00413596
$389$	−8.02029e26	−0.512530	−0.256265	−	0.966607i	$-0.582492\pi$
$389$	−8.02029e26	−0.512530	−0.256265	+	0.966607i	$0.582492\pi$
$390$	0	0
$391$	2.44289e27	1.47927
$392$	−3.39563e27	−2.00178
$393$	−1.74475e27	−1.00141
$394$	1.90005e27	1.06183
$395$	0	0
$396$	3.00232e25	0.0159096
$397$	3.06028e27	1.57929	0.789644	−	0.613566i	$-0.210265\pi$
$397$	3.06028e27	1.57929	0.789644	+	0.613566i	$0.210265\pi$
$398$	−2.90731e27	−1.46123
$399$	2.62639e26	0.128571
$400$	0	0
$401$	−1.88300e26	−0.0874649	−0.0437324	−	0.999043i	$-0.513925\pi$
$401$	−1.88300e26	−0.0874649	−0.0437324	+	0.999043i	$0.513925\pi$
$402$	1.14548e27	0.518340
$403$	3.86331e26	0.170316
$404$	−4.67237e25	−0.0200693
$405$	0	0
$406$	3.77783e27	1.54070
$407$	−8.78049e26	−0.348962
$408$	2.10204e27	0.814159
$409$	−3.08787e27	−1.16564	−0.582820	−	0.812602i	$-0.698051\pi$
$409$	−3.08787e27	−1.16564	−0.582820	+	0.812602i	$0.698051\pi$
$410$	0	0
$411$	2.85262e26	0.102307
$412$	1.08683e25	0.00379964
$413$	−8.44870e27	−2.87948
$414$	1.05148e27	0.349379
$415$	0	0
$416$	1.39171e26	0.0439610
$417$	−3.11851e27	−0.960545
$418$	−7.15424e26	−0.214888
$419$	1.55172e27	0.454534	0.227267	−	0.973833i	$-0.427021\pi$
$419$	1.55172e27	0.454534	0.227267	+	0.973833i	$0.427021\pi$
$420$	0	0
$421$	−4.87778e27	−1.35913	−0.679564	−	0.733617i	$-0.737831\pi$
$421$	−4.87778e27	−1.35913	−0.679564	+	0.733617i	$0.737831\pi$
$422$	−6.08956e26	−0.165503
$423$	−1.42979e27	−0.379051
$424$	1.80002e27	0.465516
$425$	0	0
$426$	−1.46957e27	−0.361733
$427$	1.71844e27	0.412708
$428$	2.04581e26	0.0479408
$429$	3.19195e27	0.729885
$430$	0	0
$431$	6.28554e27	1.36877	0.684386	−	0.729120i	$-0.260070\pi$
$431$	6.28554e27	1.36877	0.684386	+	0.729120i	$0.260070\pi$
$432$	9.30857e26	0.197835
$433$	−5.80864e27	−1.20490	−0.602450	−	0.798156i	$-0.705809\pi$
$433$	−5.80864e27	−1.20490	−0.602450	+	0.798156i	$0.705809\pi$
$434$	−1.94686e27	−0.394178
$435$	0	0
$436$	2.81784e26	0.0543635
$437$	−7.01678e26	−0.132155
$438$	6.14018e26	0.112902
$439$	−5.20866e27	−0.935079	−0.467540	−	0.883972i	$-0.654859\pi$
$439$	−5.20866e27	−0.935079	−0.467540	+	0.883972i	$0.654859\pi$
$440$	0	0
$441$	3.95758e27	0.677367
$442$	−6.62985e27	−1.10808
$443$	7.28076e27	1.18833	0.594166	−	0.804343i	$-0.297482\pi$
$443$	7.28076e27	1.18833	0.594166	+	0.804343i	$0.297482\pi$
$444$	2.19834e25	0.00350408
$445$	0	0
$446$	8.70664e26	0.132384
$447$	−1.03352e27	−0.153493
$448$	1.16376e28	1.68827
$449$	1.11455e28	1.57948	0.789740	−	0.613441i	$-0.210215\pi$
$449$	1.11455e28	1.57948	0.789740	+	0.613441i	$0.210215\pi$
$450$	0	0
$451$	−6.66933e27	−0.902046
$452$	1.60935e26	0.0212665
$453$	−1.40907e27	−0.181928
$454$	9.94635e27	1.25481
$455$	0	0
$456$	−6.03774e26	−0.0727351
$457$	6.97414e27	0.821052	0.410526	−	0.911849i	$-0.365345\pi$
$457$	6.97414e27	0.821052	0.410526	+	0.911849i	$0.365345\pi$
$458$	−1.11958e28	−1.28816
$459$	−2.44991e27	−0.275497
$460$	0	0
$461$	−6.34234e27	−0.681381	−0.340690	−	0.940176i	$-0.610661\pi$
$461$	−6.34234e27	−0.681381	−0.340690	+	0.940176i	$0.610661\pi$
$462$	−1.60854e28	−1.68924
$463$	−8.99687e27	−0.923615	−0.461808	−	0.886980i	$-0.652799\pi$
$463$	−8.99687e27	−0.923615	−0.461808	+	0.886980i	$0.652799\pi$
$464$	−8.93517e27	−0.896735
$465$	0	0
$466$	−6.18932e27	−0.593733
$467$	−1.23844e28	−1.16158	−0.580790	−	0.814054i	$-0.697256\pi$
$467$	−1.23844e28	−1.16158	−0.580790	+	0.814054i	$0.697256\pi$
$468$	−7.99156e25	−0.00732909
$469$	−1.71868e28	−1.54127
$470$	0	0
$471$	−4.57692e27	−0.392513
$472$	1.94225e28	1.62897
$473$	1.57851e28	1.29481
$474$	−1.08577e28	−0.871096
$475$	0	0
$476$	9.35646e26	0.0718188
$477$	−2.09790e27	−0.157522
$478$	−4.38684e27	−0.322225
$479$	−1.97865e28	−1.42183	−0.710913	−	0.703280i	$-0.751718\pi$
$479$	−1.97865e28	−1.42183	−0.710913	+	0.703280i	$0.751718\pi$
$480$	0	0
$481$	2.33718e27	0.160757
$482$	1.68963e28	1.13710
$483$	−1.57763e28	−1.03887
$484$	7.79923e26	0.0502546
$485$	0	0
$486$	−1.05450e27	−0.0650676
$487$	8.40989e27	0.507851	0.253926	−	0.967224i	$-0.418278\pi$
$487$	8.40989e27	0.507851	0.253926	+	0.967224i	$0.418278\pi$
$488$	−3.95048e27	−0.233476
$489$	−1.16819e28	−0.675728
$490$	0	0
$491$	−2.88130e28	−1.59674	−0.798368	−	0.602169i	$-0.794303\pi$
$491$	−2.88130e28	−1.59674	−0.798368	+	0.602169i	$0.794303\pi$
$492$	1.66978e26	0.00905783
$493$	2.35163e28	1.24875
$494$	1.90431e27	0.0989930
$495$	0	0
$496$	4.60464e27	0.229424
$497$	2.20493e28	1.07561
$498$	1.47783e28	0.705855
$499$	1.45488e28	0.680411	0.340206	−	0.940351i	$-0.389503\pi$
$499$	1.45488e28	0.680411	0.340206	+	0.940351i	$0.389503\pi$
$500$	0	0
$501$	1.62461e28	0.728540
$502$	2.56300e28	1.12554
$503$	4.52996e28	1.94819	0.974093	−	0.226150i	$-0.0726138\pi$
$503$	4.52996e28	1.94819	0.974093	+	0.226150i	$0.0726138\pi$
$504$	−1.35751e28	−0.571771
$505$	0	0
$506$	4.29744e28	1.73632
$507$	6.09261e27	0.241113
$508$	1.01943e27	0.0395174
$509$	−1.43922e28	−0.546499	−0.273250	−	0.961943i	$-0.588099\pi$
$509$	−1.43922e28	−0.546499	−0.273250	+	0.961943i	$0.588099\pi$
$510$	0	0
$511$	−9.21270e27	−0.335712
$512$	−2.67068e28	−0.953426
$513$	7.03694e26	0.0246122
$514$	7.56861e27	0.259360
$515$	0	0
$516$	−3.95206e26	−0.0130017
$517$	−5.84361e28	−1.88378
$518$	−1.17779e28	−0.372054
$519$	1.50088e28	0.464611
$520$	0	0
$521$	−1.43400e28	−0.426337	−0.213168	−	0.977015i	$-0.568378\pi$
$521$	−1.43400e28	−0.426337	−0.213168	+	0.977015i	$0.568378\pi$
$522$	1.01220e28	0.294934
$523$	−2.57571e28	−0.735578	−0.367789	−	0.929909i	$-0.619885\pi$
$523$	−2.57571e28	−0.735578	−0.367789	+	0.929909i	$0.619885\pi$
$524$	1.78533e27	0.0499735
$525$	0	0
$526$	−7.15392e27	−0.192395
$527$	−1.21189e28	−0.319486
$528$	3.80445e28	0.983187
$529$	2.67713e27	0.0678243
$530$	0	0
$531$	−2.26367e28	−0.551215
$532$	−2.68748e26	−0.00641613
$533$	1.77524e28	0.415547
$534$	2.69757e28	0.619141
$535$	0	0
$536$	3.95102e28	0.871924
$537$	2.74528e28	0.594096
$538$	3.57576e28	0.758848
$539$	1.61748e29	3.36633
$540$	0	0
$541$	−2.07031e28	−0.414443	−0.207221	−	0.978294i	$-0.566442\pi$
$541$	−2.07031e28	−0.414443	−0.207221	+	0.978294i	$0.566442\pi$
$542$	−6.33654e27	−0.124411
$543$	2.79423e28	0.538101
$544$	−4.36567e27	−0.0824636
$545$	0	0
$546$	4.28159e28	0.778185
$547$	4.74477e28	0.845957	0.422978	−	0.906140i	$-0.360985\pi$
$547$	4.74477e28	0.845957	0.422978	+	0.906140i	$0.360985\pi$
$548$	−2.91897e26	−0.00510544
$549$	4.60425e27	0.0790041
$550$	0	0
$551$	−6.75466e27	−0.111561
$552$	3.62678e28	0.587707
$553$	1.62909e29	2.59018
$554$	4.45235e28	0.694603
$555$	0	0
$556$	3.19104e27	0.0479343
$557$	8.78329e28	1.29473	0.647363	−	0.762182i	$-0.275872\pi$
$557$	8.78329e28	1.29473	0.647363	+	0.762182i	$0.275872\pi$
$558$	−5.21626e27	−0.0754570
$559$	−4.20166e28	−0.596482
$560$	0	0
$561$	−1.00129e29	−1.36914
$562$	−7.35741e28	−0.987401
$563$	−1.74646e28	−0.230050	−0.115025	−	0.993363i	$-0.536695\pi$
$563$	−1.74646e28	−0.230050	−0.115025	+	0.993363i	$0.536695\pi$
$564$	1.46304e27	0.0189159
$565$	0	0
$566$	1.39927e29	1.74313
$567$	1.58216e28	0.193477
$568$	−5.06886e28	−0.608489
$569$	−7.29669e28	−0.859899	−0.429949	−	0.902853i	$-0.641469\pi$
$569$	−7.29669e28	−0.859899	−0.429949	+	0.902853i	$0.641469\pi$
$570$	0	0
$571$	1.34969e29	1.53304	0.766520	−	0.642220i	$-0.221986\pi$
$571$	1.34969e29	1.53304	0.766520	+	0.642220i	$0.221986\pi$
$572$	−3.26619e27	−0.0364236
$573$	−3.65521e28	−0.400211
$574$	−8.94606e28	−0.961738
$575$	0	0
$576$	3.11807e28	0.323184
$577$	−1.08935e29	−1.10872	−0.554361	−	0.832276i	$-0.687037\pi$
$577$	−1.08935e29	−1.10872	−0.554361	+	0.832276i	$0.687037\pi$
$578$	1.06486e29	1.06427
$579$	7.62434e27	0.0748303
$580$	0	0
$581$	−2.21732e29	−2.09884
$582$	9.04288e27	0.0840650
$583$	−8.57422e28	−0.782843
$584$	2.11788e28	0.189918
$585$	0	0
$586$	1.61969e29	1.40122
$587$	4.12838e28	0.350816	0.175408	−	0.984496i	$-0.443875\pi$
$587$	4.12838e28	0.350816	0.175408	+	0.984496i	$0.443875\pi$
$588$	−4.04962e27	−0.0338028
$589$	3.48094e27	0.0285421
$590$	0	0
$591$	−7.63827e28	−0.604403
$592$	2.78567e28	0.216547
$593$	1.78264e28	0.136141	0.0680707	−	0.997680i	$-0.478316\pi$
$593$	1.78264e28	0.136141	0.0680707	+	0.997680i	$0.478316\pi$
$594$	−4.30978e28	−0.323368
$595$	0	0
$596$	1.05755e27	0.00765979
$597$	1.16875e29	0.831745
$598$	−1.14389e29	−0.799874
$599$	7.09563e28	0.487539	0.243770	−	0.969833i	$-0.421616\pi$
$599$	7.09563e28	0.487539	0.243770	+	0.969833i	$0.421616\pi$
$600$	0	0
$601$	−1.79606e28	−0.119162	−0.0595810	−	0.998223i	$-0.518976\pi$
$601$	−1.79606e28	−0.119162	−0.0595810	+	0.998223i	$0.518976\pi$
$602$	2.11737e29	1.38049
$603$	−4.60487e28	−0.295043
$604$	1.44184e27	0.00907881
$605$	0	0
$606$	6.70710e28	0.407918
$607$	−2.99794e29	−1.79201	−0.896006	−	0.444041i	$-0.853544\pi$
$607$	−2.99794e29	−1.79201	−0.896006	+	0.444041i	$0.853544\pi$
$608$	1.25396e27	0.00736711
$609$	−1.51870e29	−0.876981
$610$	0	0
$611$	1.55545e29	0.867805
$612$	2.50689e27	0.0137482
$613$	5.01641e28	0.270432	0.135216	−	0.990816i	$-0.456827\pi$
$613$	5.01641e28	0.270432	0.135216	+	0.990816i	$0.456827\pi$
$614$	2.10944e29	1.11789
$615$	0	0
$616$	−5.54820e29	−2.84155
$617$	4.92149e28	0.247801	0.123900	−	0.992295i	$-0.460460\pi$
$617$	4.92149e28	0.247801	0.123900	+	0.992295i	$0.460460\pi$
$618$	−1.56013e28	−0.0772292
$619$	3.71234e28	0.180674	0.0903372	−	0.995911i	$-0.471206\pi$
$619$	3.71234e28	0.180674	0.0903372	+	0.995911i	$0.471206\pi$
$620$	0	0
$621$	−4.22698e28	−0.198869
$622$	4.04101e28	0.186935
$623$	−4.04742e29	−1.84100
$624$	−1.01267e29	−0.452927
$625$	0	0
$626$	−3.70096e29	−1.60061
$627$	2.87603e28	0.122316
$628$	4.68337e27	0.0195877
$629$	−7.33156e28	−0.301554
$630$	0	0
$631$	8.77350e28	0.349031	0.174516	−	0.984654i	$-0.444164\pi$
$631$	8.77350e28	0.349031	0.174516	+	0.984654i	$0.444164\pi$
$632$	−3.74507e29	−1.46531
$633$	2.44802e28	0.0942056
$634$	−4.01105e28	−0.151817
$635$	0	0
$636$	2.14670e27	0.00786087
$637$	−4.30539e29	−1.55078
$638$	4.13690e29	1.46574
$639$	5.90770e28	0.205902
$640$	0	0
$641$	4.13996e29	1.39633	0.698163	−	0.715939i	$-0.254001\pi$
$641$	4.13996e29	1.39633	0.698163	+	0.715939i	$0.254001\pi$
$642$	−2.93672e29	−0.974417
$643$	−1.13794e29	−0.371455	−0.185727	−	0.982601i	$-0.559464\pi$
$643$	−1.13794e29	−0.371455	−0.185727	+	0.982601i	$0.559464\pi$
$644$	1.61433e28	0.0518430
$645$	0	0
$646$	−5.97367e28	−0.185695
$647$	4.45067e29	1.36123	0.680613	−	0.732643i	$-0.261714\pi$
$647$	4.45067e29	1.36123	0.680613	+	0.732643i	$0.261714\pi$
$648$	−3.63719e28	−0.109453
$649$	−9.25174e29	−2.73939
$650$	0	0
$651$	7.82645e28	0.224370
$652$	1.19536e28	0.0337210
$653$	−1.53373e29	−0.425757	−0.212878	−	0.977079i	$-0.568284\pi$
$653$	−1.53373e29	−0.425757	−0.212878	+	0.977079i	$0.568284\pi$
$654$	−4.04495e29	−1.10496
$655$	0	0
$656$	2.11589e29	0.559761
$657$	−2.46837e28	−0.0642650
$658$	−7.83845e29	−2.00844
$659$	9.50910e28	0.239796	0.119898	−	0.992786i	$-0.461743\pi$
$659$	9.50910e28	0.239796	0.119898	+	0.992786i	$0.461743\pi$
$660$	0	0
$661$	−3.92155e29	−0.957949	−0.478974	−	0.877829i	$-0.658991\pi$
$661$	−3.92155e29	−0.957949	−0.478974	+	0.877829i	$0.658991\pi$
$662$	−4.64570e29	−1.11697
$663$	2.66522e29	0.630727
$664$	5.09735e29	1.18735
$665$	0	0
$666$	−3.15567e28	−0.0712218
$667$	4.05742e29	0.901423
$668$	−1.66240e28	−0.0363565
$669$	−3.50010e28	−0.0753540
$670$	0	0
$671$	1.88178e29	0.392629
$672$	2.81937e28	0.0579129
$673$	−4.90767e29	−0.992469	−0.496235	−	0.868189i	$-0.665284\pi$
$673$	−4.90767e29	−0.992469	−0.496235	+	0.868189i	$0.665284\pi$
$674$	1.04272e29	0.207606
$675$	0	0
$676$	−6.23432e27	−0.0120323
$677$	4.44170e29	0.844051	0.422025	−	0.906584i	$-0.361319\pi$
$677$	4.44170e29	0.844051	0.422025	+	0.906584i	$0.361319\pi$
$678$	−2.31019e29	−0.432251
$679$	−1.35679e29	−0.249965
$680$	0	0
$681$	−3.99846e29	−0.714247
$682$	−2.13191e29	−0.375001
$683$	−5.25616e29	−0.910439	−0.455220	−	0.890379i	$-0.650439\pi$
$683$	−5.25616e29	−0.910439	−0.455220	+	0.890379i	$0.650439\pi$
$684$	−7.20060e26	−0.00122823
$685$	0	0
$686$	1.10196e30	1.82290
$687$	4.50076e29	0.733230
$688$	−5.00792e29	−0.803488
$689$	2.28228e29	0.360634
$690$	0	0
$691$	3.73748e29	0.572875	0.286437	−	0.958099i	$-0.407529\pi$
$691$	3.73748e29	0.572875	0.286437	+	0.958099i	$0.407529\pi$
$692$	−1.53579e28	−0.0231856
$693$	6.46638e29	0.961528
$694$	8.12541e29	1.19007
$695$	0	0
$696$	3.49129e29	0.496123
$697$	−5.56878e29	−0.779499
$698$	−9.31597e29	−1.28454
$699$	2.48813e29	0.337958
$700$	0	0
$701$	−8.40890e29	−1.10841	−0.554204	−	0.832381i	$-0.686977\pi$
$701$	−8.40890e29	−1.10841	−0.554204	+	0.832381i	$0.686977\pi$
$702$	1.14717e29	0.148967
$703$	2.10586e28	0.0269401
$704$	1.27437e30	1.60614
$705$	0	0
$706$	6.40967e28	0.0784127
$707$	−1.00633e30	−1.21293
$708$	2.31632e28	0.0275074
$709$	−4.13351e29	−0.483653	−0.241827	−	0.970320i	$-0.577746\pi$
$709$	−4.13351e29	−0.483653	−0.241827	+	0.970320i	$0.577746\pi$
$710$	0	0
$711$	4.36484e29	0.495835
$712$	9.30451e29	1.04149
$713$	−2.09095e29	−0.230623
$714$	−1.34310e30	−1.45975
$715$	0	0
$716$	−2.80913e28	−0.0296473
$717$	1.76353e29	0.183413
$718$	2.67270e29	0.273933
$719$	1.44007e30	1.45455	0.727275	−	0.686346i	$-0.240786\pi$
$719$	1.44007e30	1.45455	0.727275	+	0.686346i	$0.240786\pi$
$720$	0	0
$721$	2.34081e29	0.229639
$722$	−1.03192e30	−0.997714
$723$	−6.79237e29	−0.647246
$724$	−2.85922e28	−0.0268530
$725$	0	0
$726$	−1.11957e30	−1.02144
$727$	9.76091e29	0.877767	0.438883	−	0.898544i	$-0.355374\pi$
$727$	9.76091e29	0.877767	0.438883	+	0.898544i	$0.355374\pi$
$728$	1.47682e30	1.30902
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	1.31803e30	1.11890
$732$	−4.71134e27	−0.00394256
$733$	9.82057e29	0.810112	0.405056	−	0.914292i	$-0.367252\pi$
$733$	9.82057e29	0.810112	0.405056	+	0.914292i	$0.367252\pi$
$734$	−1.76624e29	−0.143629
$735$	0	0
$736$	−7.53236e28	−0.0595270
$737$	−1.88203e30	−1.46629
$738$	−2.39693e29	−0.184104
$739$	−1.39945e30	−1.05972	−0.529859	−	0.848086i	$-0.677755\pi$
$739$	−1.39945e30	−1.05972	−0.529859	+	0.848086i	$0.677755\pi$
$740$	0	0
$741$	−7.65539e28	−0.0563477
$742$	−1.15012e30	−0.834648
$743$	−8.59602e29	−0.615056	−0.307528	−	0.951539i	$-0.599502\pi$
$743$	−8.59602e29	−0.615056	−0.307528	+	0.951539i	$0.599502\pi$
$744$	−1.79920e29	−0.126930
$745$	0	0
$746$	−2.18089e29	−0.149581
$747$	−5.94091e29	−0.401779
$748$	1.02458e29	0.0683247
$749$	4.40624e30	2.89741
$750$	0	0
$751$	−2.48243e30	−1.58730	−0.793648	−	0.608377i	$-0.791821\pi$
$751$	−2.48243e30	−1.58730	−0.793648	+	0.608377i	$0.791821\pi$
$752$	1.85392e30	1.16897
$753$	−1.03033e30	−0.640665
$754$	−1.10116e30	−0.675228
$755$	0	0
$756$	−1.61896e28	−0.00965513
$757$	1.25613e30	0.738800	0.369400	−	0.929270i	$-0.379563\pi$
$757$	1.25613e30	0.738800	0.369400	+	0.929270i	$0.379563\pi$
$758$	1.14419e30	0.663702
$759$	−1.72758e30	−0.988327
$760$	0	0
$761$	−2.22680e30	−1.23920	−0.619602	−	0.784916i	$-0.712706\pi$
$761$	−2.22680e30	−1.23920	−0.619602	+	0.784916i	$0.712706\pi$
$762$	−1.46337e30	−0.803206
$763$	6.06903e30	3.28558
$764$	3.74022e28	0.0199718
$765$	0	0
$766$	1.36184e30	0.707498
$767$	2.46262e30	1.26196
$768$	−9.86425e28	−0.0498622
$769$	1.77035e30	0.882739	0.441369	−	0.897325i	$-0.354493\pi$
$769$	1.77035e30	0.882739	0.441369	+	0.897325i	$0.354493\pi$
$770$	0	0
$771$	−3.04261e29	−0.147630
$772$	−7.80167e27	−0.00373427
$773$	1.36464e30	0.644369	0.322185	−	0.946677i	$-0.395583\pi$
$773$	1.36464e30	0.644369	0.322185	+	0.946677i	$0.395583\pi$
$774$	5.67310e29	0.264266
$775$	0	0
$776$	3.11909e29	0.141410
$777$	4.73476e29	0.211776
$778$	−1.17808e30	−0.519861
$779$	1.59953e29	0.0696386
$780$	0	0
$781$	2.41451e30	1.02328
$782$	3.58829e30	1.50043
$783$	−4.06907e29	−0.167879
$784$	−5.13156e30	−2.08896
$785$	0	0
$786$	−2.56281e30	−1.01573
$787$	3.17264e30	1.24076	0.620379	−	0.784303i	$-0.286979\pi$
$787$	3.17264e30	1.24076	0.620379	+	0.784303i	$0.286979\pi$
$788$	7.81592e28	0.0301616
$789$	2.87590e29	0.109513
$790$	0	0
$791$	3.46620e30	1.28529
$792$	−1.48654e30	−0.543953
$793$	−5.00890e29	−0.180873
$794$	4.49515e30	1.60188
$795$	0	0
$796$	−1.19593e29	−0.0415068
$797$	−2.34197e30	−0.802175	−0.401087	−	0.916040i	$-0.631368\pi$
$797$	−2.34197e30	−0.802175	−0.401087	+	0.916040i	$0.631368\pi$
$798$	3.85783e29	0.130410
$799$	−4.87931e30	−1.62786
$800$	0	0
$801$	−1.08443e30	−0.352420
$802$	−2.76587e29	−0.0887159
$803$	−1.00884e30	−0.319380
$804$	4.71198e28	0.0147236
$805$	0	0
$806$	5.67469e29	0.172753
$807$	−1.43747e30	−0.431943
$808$	2.31343e30	0.686178
$809$	−2.93728e30	−0.859975	−0.429988	−	0.902835i	$-0.641482\pi$
$809$	−2.93728e30	−0.859975	−0.429988	+	0.902835i	$0.641482\pi$
$810$	0	0
$811$	5.12068e30	1.46086	0.730430	−	0.682988i	$-0.239320\pi$
$811$	5.12068e30	1.46086	0.730430	+	0.682988i	$0.239320\pi$
$812$	1.55402e29	0.0437642
$813$	2.54731e29	0.0708159
$814$	−1.28974e30	−0.353953
$815$	0	0
$816$	3.17665e30	0.849617
$817$	−3.78581e29	−0.0999602
$818$	−4.53567e30	−1.18231
$819$	−1.72121e30	−0.442949
$820$	0	0
$821$	5.69136e30	1.42762	0.713810	−	0.700339i	$-0.246968\pi$
$821$	5.69136e30	1.42762	0.713810	+	0.700339i	$0.246968\pi$
$822$	4.19012e29	0.103770
$823$	7.67686e29	0.187709	0.0938545	−	0.995586i	$-0.470081\pi$
$823$	7.67686e29	0.187709	0.0938545	+	0.995586i	$0.470081\pi$
$824$	−5.38123e29	−0.129911
$825$	0	0
$826$	−1.24100e31	−2.92067
$827$	4.06624e30	0.944899	0.472450	−	0.881358i	$-0.343370\pi$
$827$	4.06624e30	0.944899	0.472450	+	0.881358i	$0.343370\pi$
$828$	4.32529e28	0.00992423
$829$	1.57538e30	0.356914	0.178457	−	0.983948i	$-0.442889\pi$
$829$	1.57538e30	0.356914	0.178457	+	0.983948i	$0.442889\pi$
$830$	0	0
$831$	−1.78986e30	−0.395374
$832$	−3.39211e30	−0.739903
$833$	1.35057e31	2.90900
$834$	−4.58067e30	−0.974285
$835$	0	0
$836$	−2.94292e28	−0.00610397
$837$	2.09695e29	0.0429508
$838$	2.27927e30	0.461035
$839$	4.06398e30	0.811804	0.405902	−	0.913917i	$-0.366957\pi$
$839$	4.06398e30	0.811804	0.405902	+	0.913917i	$0.366957\pi$
$840$	0	0
$841$	−1.22699e30	−0.239048
$842$	−7.16482e30	−1.37857
$843$	2.95770e30	0.562037
$844$	−2.50496e28	−0.00470116
$845$	0	0
$846$	−2.10017e30	−0.384473
$847$	1.67979e31	3.03724
$848$	2.72023e30	0.485790
$849$	−5.62513e30	−0.992205
$850$	0	0
$851$	−1.26496e30	−0.217679
$852$	−6.04511e28	−0.0102752
$853$	4.94621e29	0.0830439	0.0415220	−	0.999138i	$-0.486779\pi$
$853$	4.94621e29	0.0830439	0.0415220	+	0.999138i	$0.486779\pi$
$854$	2.52417e30	0.418611
$855$	0	0
$856$	−1.01294e31	−1.63911
$857$	−3.34073e30	−0.534002	−0.267001	−	0.963696i	$-0.586033\pi$
$857$	−3.34073e30	−0.534002	−0.267001	+	0.963696i	$0.586033\pi$
$858$	4.68855e30	0.740325
$859$	−2.25370e30	−0.351535	−0.175767	−	0.984432i	$-0.556241\pi$
$859$	−2.25370e30	−0.351535	−0.175767	+	0.984432i	$0.556241\pi$
$860$	0	0
$861$	3.59635e30	0.547430
$862$	9.23263e30	1.38835
$863$	−4.01963e30	−0.597135	−0.298568	−	0.954388i	$-0.596509\pi$
$863$	−4.01963e30	−0.597135	−0.298568	+	0.954388i	$0.596509\pi$
$864$	7.55399e28	0.0110862
$865$	0	0
$866$	−8.53212e30	−1.22213
$867$	−4.28078e30	−0.605790
$868$	−8.00848e28	−0.0111968
$869$	1.78393e31	2.46417
$870$	0	0
$871$	5.00958e30	0.675477
$872$	−1.39519e31	−1.85871
$873$	−3.63527e29	−0.0478505
$874$	−1.03067e30	−0.134045
$875$	0	0
$876$	2.52578e28	0.00320703
$877$	−1.01190e31	−1.26952	−0.634760	−	0.772709i	$-0.718901\pi$
$877$	−1.01190e31	−1.26952	−0.634760	+	0.772709i	$0.718901\pi$
$878$	−7.65083e30	−0.948454
$879$	−6.51121e30	−0.797587
$880$	0	0
$881$	9.44764e30	1.12999	0.564997	−	0.825093i	$-0.308877\pi$
$881$	9.44764e30	1.12999	0.564997	+	0.825093i	$0.308877\pi$
$882$	5.81316e30	0.687056
$883$	−1.33501e31	−1.55918	−0.779591	−	0.626290i	$-0.784573\pi$
$883$	−1.33501e31	−1.55918	−0.779591	+	0.626290i	$0.784573\pi$
$884$	−2.72721e29	−0.0314753
$885$	0	0
$886$	1.06945e31	1.20533
$887$	−5.72606e30	−0.637761	−0.318881	−	0.947795i	$-0.603307\pi$
$887$	−5.72606e30	−0.637761	−0.318881	+	0.947795i	$0.603307\pi$
$888$	−1.08846e30	−0.119806
$889$	2.19564e31	2.38832
$890$	0	0
$891$	1.73255e30	0.184064
$892$	3.58151e28	0.00376041
$893$	1.40150e30	0.145429
$894$	−1.51810e30	−0.155688
$895$	0	0
$896$	1.80954e31	1.81273
$897$	4.59847e30	0.455295
$898$	1.63713e31	1.60207
$899$	−2.01284e30	−0.194685
$900$	0	0
$901$	−7.15933e30	−0.676491
$902$	−9.79637e30	−0.914948
$903$	−8.51190e30	−0.785788
$904$	−7.96835e30	−0.727109
$905$	0	0
$906$	−2.06973e30	−0.184530
$907$	9.62032e30	0.847839	0.423919	−	0.905700i	$-0.360654\pi$
$907$	9.62032e30	0.847839	0.423919	+	0.905700i	$0.360654\pi$
$908$	4.09146e29	0.0356432
$909$	−2.69628e30	−0.232190
$910$	0	0
$911$	−7.24114e30	−0.609346	−0.304673	−	0.952457i	$-0.598547\pi$
$911$	−7.24114e30	−0.609346	−0.304673	+	0.952457i	$0.598547\pi$
$912$	−9.12439e29	−0.0759028
$913$	−2.42808e31	−1.99673
$914$	1.02441e31	0.832796
$915$	0	0
$916$	−4.60544e29	−0.0365905
$917$	3.84522e31	3.02026
$918$	−3.59859e30	−0.279437
$919$	−9.66995e30	−0.742354	−0.371177	−	0.928562i	$-0.621046\pi$
$919$	−9.66995e30	−0.742354	−0.371177	+	0.928562i	$0.621046\pi$
$920$	0	0
$921$	−8.48003e30	−0.636314
$922$	−9.31606e30	−0.691127
$923$	−6.42691e30	−0.471395
$924$	−6.61678e29	−0.0479834
$925$	0	0
$926$	−1.32152e31	−0.936826
$927$	6.27177e29	0.0439595
$928$	−7.25098e29	−0.0502508
$929$	−4.05924e30	−0.278150	−0.139075	−	0.990282i	$-0.544413\pi$
$929$	−4.05924e30	−0.278150	−0.139075	+	0.990282i	$0.544413\pi$
$930$	0	0
$931$	−3.87927e30	−0.259883
$932$	−2.54600e29	−0.0168652
$933$	−1.62450e30	−0.106405
$934$	−1.81911e31	−1.17819
$935$	0	0
$936$	3.95685e30	0.250584
$937$	−1.20088e31	−0.752029	−0.376015	−	0.926614i	$-0.622706\pi$
$937$	−1.20088e31	−0.752029	−0.376015	+	0.926614i	$0.622706\pi$
$938$	−2.52451e31	−1.56332
$939$	1.48780e31	0.911078
$940$	0	0
$941$	1.10298e31	0.660505	0.330252	−	0.943893i	$-0.392866\pi$
$941$	1.10298e31	0.660505	0.330252	+	0.943893i	$0.392866\pi$
$942$	−6.72290e30	−0.398127
$943$	−9.60815e30	−0.562687
$944$	2.93517e31	1.69992
$945$	0	0
$946$	2.31862e31	1.31333
$947$	1.55574e31	0.871489	0.435745	−	0.900070i	$-0.356485\pi$
$947$	1.55574e31	0.871489	0.435745	+	0.900070i	$0.356485\pi$
$948$	−4.46636e29	−0.0247438
$949$	2.68531e30	0.147129
$950$	0	0
$951$	1.61246e30	0.0864157
$952$	−4.63265e31	−2.45551
$953$	2.67324e31	1.40140	0.700702	−	0.713454i	$-0.252870\pi$
$953$	2.67324e31	1.40140	0.700702	+	0.713454i	$0.252870\pi$
$954$	−3.08154e30	−0.159775
$955$	0	0
$956$	−1.80454e29	−0.00915291
$957$	−1.66305e31	−0.834314
$958$	−2.90638e31	−1.44216
$959$	−6.28684e30	−0.308558
$960$	0	0
$961$	−1.97882e31	−0.950191
$962$	3.43302e30	0.163056
$963$	1.18057e31	0.554646
$964$	6.95035e29	0.0322996
$965$	0	0
$966$	−2.31734e31	−1.05373
$967$	−1.49208e31	−0.671140	−0.335570	−	0.942015i	$-0.608929\pi$
$967$	−1.49208e31	−0.671140	−0.335570	+	0.942015i	$0.608929\pi$
$968$	−3.86163e31	−1.71822
$969$	2.40143e30	0.105699
$970$	0	0
$971$	2.35234e31	1.01321	0.506603	−	0.862180i	$-0.330901\pi$
$971$	2.35234e31	1.01321	0.506603	+	0.862180i	$0.330901\pi$
$972$	−4.33771e28	−0.00184827
$973$	6.87282e31	2.89701
$974$	1.23530e31	0.515115
$975$	0	0
$976$	−5.97007e30	−0.243644
$977$	2.92377e30	0.118046	0.0590228	−	0.998257i	$-0.481202\pi$
$977$	2.92377e30	0.118046	0.0590228	+	0.998257i	$0.481202\pi$
$978$	−1.71592e31	−0.685393
$979$	−4.43212e31	−1.75143
$980$	0	0
$981$	1.62608e31	0.628953
$982$	−4.23226e31	−1.61958
$983$	−6.45114e30	−0.244244	−0.122122	−	0.992515i	$-0.538970\pi$
$983$	−6.45114e30	−0.244244	−0.122122	+	0.992515i	$0.538970\pi$
$984$	−8.26754e30	−0.309690
$985$	0	0
$986$	3.45424e31	1.26662
$987$	3.15108e31	1.14322
$988$	7.83344e28	0.00281193
$989$	2.27408e31	0.807688
$990$	0	0
$991$	−2.59391e31	−0.901947	−0.450973	−	0.892537i	$-0.648923\pi$
$991$	−2.59391e31	−0.901947	−0.450973	+	0.892537i	$0.648923\pi$
$992$	3.73671e29	0.0128563
$993$	1.86759e31	0.635790
$994$	3.23875e31	1.09099
$995$	0	0
$996$	6.07909e29	0.0200500
$997$	3.12296e31	1.01922	0.509610	−	0.860406i	$-0.329790\pi$
$997$	3.12296e31	1.01922	0.509610	+	0.860406i	$0.329790\pi$
$998$	2.13703e31	0.690144
$999$	1.26859e30	0.0405400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.n.1.8		10
5.2	odd	4		15.22.b.a.4.15	yes	20
5.3	odd	4		15.22.b.a.4.6	✓	20
5.4	even	2		75.22.a.m.1.3		10
15.2	even	4		45.22.b.d.19.6		20
15.8	even	4		45.22.b.d.19.15		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.b.a.4.6	✓	20	5.3	odd	4
15.22.b.a.4.15	yes	20	5.2	odd	4
45.22.b.d.19.6		20	15.2	even	4
45.22.b.d.19.15		20	15.8	even	4
75.22.a.m.1.3		10	5.4	even	2
75.22.a.n.1.8		10	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q+1468.87 q^{2} -59049.0 q^{3} +60422.4 q^{4} -8.67352e7 q^{6} +1.30137e9 q^{7} -2.99169e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q+1468.87 q^{2} -59049.0 q^{3} +60422.4 q^{4} -8.67352e7 q^{6} +1.30137e9 q^{7} -2.99169e9 q^{8} +3.48678e9 q^{9} +1.42507e11 q^{11} -3.56788e9 q^{12} -3.79323e11 q^{13} +1.91154e12 q^{14} -4.52111e12 q^{16} +1.18990e13 q^{17} +5.12163e12 q^{18} -3.41779e12 q^{19} -7.68447e13 q^{21} +2.09323e14 q^{22} +2.05302e14 q^{23} +1.76656e14 q^{24} -5.57175e14 q^{26} -2.05891e14 q^{27} +7.86320e13 q^{28} +1.97632e15 q^{29} -1.01848e15 q^{31} -3.66892e14 q^{32} -8.41487e15 q^{33} +1.74781e16 q^{34} +2.10680e14 q^{36} -6.16147e15 q^{37} -5.02029e15 q^{38} +2.23986e16 q^{39} -4.68002e16 q^{41} -1.12875e17 q^{42} +1.10768e17 q^{43} +8.61058e15 q^{44} +3.01561e17 q^{46} -4.10059e17 q^{47} +2.66967e17 q^{48} +1.13502e18 q^{49} -7.02627e17 q^{51} -2.29196e16 q^{52} -6.01672e17 q^{53} -3.02427e17 q^{54} -3.89330e18 q^{56} +2.01817e17 q^{57} +2.90296e18 q^{58} -6.49215e18 q^{59} +1.32049e18 q^{61} -1.49601e18 q^{62} +4.53760e18 q^{63} +8.94254e18 q^{64} -1.23603e19 q^{66} -1.32066e19 q^{67} +7.18969e17 q^{68} -1.21228e19 q^{69} +1.69431e19 q^{71} -1.04314e19 q^{72} -7.07922e18 q^{73} -9.05038e18 q^{74} -2.06511e17 q^{76} +1.85454e20 q^{77} +3.29006e19 q^{78} +1.25182e20 q^{79} +1.21577e19 q^{81} -6.87433e19 q^{82} -1.70384e20 q^{83} -4.64314e18 q^{84} +1.62703e20 q^{86} -1.16700e20 q^{87} -4.26335e20 q^{88} -3.11012e20 q^{89} -4.93640e20 q^{91} +1.24048e19 q^{92} +6.01400e19 q^{93} -6.02323e20 q^{94} +2.16646e19 q^{96} -1.04258e20 q^{97} +1.66720e21 q^{98} +4.96889e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 645 q^{2} - 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} + 115357290 q^{7} + 314142735 q^{8} + 34867844010 q^{9}+O(q^{10}) \) 10 * q + 645 * q^2 - 590490 * q^3 + 10043205 * q^4 - 38086605 * q^6 + 115357290 * q^7 + 314142735 * q^8 + 34867844010 * q^9	\( 10 q + 645 q^{2} - 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} + 115357290 q^{7} + 314142735 q^{8} + 34867844010 q^{9} + 5976221790 q^{11} - 593041212045 q^{12} + 842570747430 q^{13} + 1605059874570 q^{14} + 6061861547825 q^{16} + 12910340404230 q^{17} + 2248975938645 q^{18} - 76155422176280 q^{19} - 6811732617210 q^{21} + 461780887241010 q^{22} + 82640920915920 q^{23} - 18549814359015 q^{24} + 286555331159670 q^{26} - 20\!\cdots\!90 q^{27}+ \cdots + 20\!\cdots\!90 q^{99}+O(q^{100}) \) 10 * q + 645 * q^2 - 590490 * q^3 + 10043205 * q^4 - 38086605 * q^6 + 115357290 * q^7 + 314142735 * q^8 + 34867844010 * q^9 + 5976221790 * q^11 - 593041212045 * q^12 + 842570747430 * q^13 + 1605059874570 * q^14 + 6061861547825 * q^16 + 12910340404230 * q^17 + 2248975938645 * q^18 - 76155422176280 * q^19 - 6811732617210 * q^21 + 461780887241010 * q^22 + 82640920915920 * q^23 - 18549814359015 * q^24 + 286555331159670 * q^26 - 2058911320946490 * q^27 + 1520428218531930 * q^28 - 2553653160731010 * q^29 - 5579671364724940 * q^31 + 29255856594217815 * q^32 - 352889920477710 * q^33 + 36285968426431010 * q^34 + 35018490530045205 * q^36 + 1797908431222890 * q^37 + 10684960744876320 * q^38 - 49752960064994070 * q^39 - 157319729506021440 * q^41 - 94777180533483930 * q^42 - 148924853700743400 * q^43 + 49118820773767290 * q^44 - 238453286038259540 * q^46 - 43171138083158100 * q^47 - 357946862537518425 * q^48 + 453692886452454950 * q^49 - 762342690529377270 * q^51 + 5077473324432504030 * q^52 + 1835165786666252970 * q^53 - 132799780201048605 * q^54 - 5146515624013530450 * q^56 + 4496901524087157720 * q^57 + 3684042125074285440 * q^58 + 4936135121536323630 * q^59 - 4443080871007408300 * q^61 + 39366851097921260280 * q^62 + 402225999313633290 * q^63 - 14877283608188093455 * q^64 - 27267699610694399490 * q^66 + 33664559574358232580 * q^67 + 44791395195934111290 * q^68 - 4879863739164160080 * q^69 - 74662724971386630180 * q^71 + 1095347988085476735 * q^72 + 16062862692682477620 * q^73 - 130249023546610518030 * q^74 - 258736483662380220720 * q^76 + 56216747719240740180 * q^77 - 16920805749647353830 * q^78 - 49130708010799643300 * q^79 + 121576654590569288010 * q^81 + 117185080045656261930 * q^82 + 118408383608453695560 * q^83 - 89779765876091934570 * q^84 - 1007271272794707382260 * q^86 + 150790665488005409490 * q^87 + 41803519849182967710 * q^88 - 572103976713991581780 * q^89 - 1669810835257196575140 * q^91 - 1206545757146297440260 * q^92 + 329474014415642982060 * q^93 - 1671319720124696594920 * q^94 - 1727529076031967757935 * q^96 - 1299275526640742173920 * q^97 - 2683959884079093505455 * q^98 + 20837796914288297790 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more