LMFDB - Embedded newform 75.22.a.m.1.5

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:	$1$
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			1.5
Root			$-294.921$ 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)

$f(q)$	$=$	$q-359.921 q^{2} +59049.0 q^{3} -1.96761e6 q^{4} -2.12530e7 q^{6} +2.25801e8 q^{7} +1.46299e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q-359.921 q^{2} +59049.0 q^{3} -1.96761e6 q^{4} -2.12530e7 q^{6} +2.25801e8 q^{7} +1.46299e9 q^{8} +3.48678e9 q^{9} -3.39385e10 q^{11} -1.16185e11 q^{12} -7.87206e11 q^{13} -8.12704e10 q^{14} +3.59981e12 q^{16} +1.90567e12 q^{17} -1.25497e12 q^{18} +3.53051e13 q^{19} +1.33333e13 q^{21} +1.22152e13 q^{22} +1.05097e14 q^{23} +8.63883e13 q^{24} +2.83332e14 q^{26} +2.05891e14 q^{27} -4.44287e14 q^{28} -1.97643e15 q^{29} -6.94447e15 q^{31} -4.36377e15 q^{32} -2.00403e15 q^{33} -6.85892e14 q^{34} -6.86063e15 q^{36} +3.71887e16 q^{37} -1.27070e16 q^{38} -4.64837e16 q^{39} +1.35296e17 q^{41} -4.79894e15 q^{42} -4.58115e16 q^{43} +6.67777e16 q^{44} -3.78265e16 q^{46} +4.29974e17 q^{47} +2.12565e17 q^{48} -5.07560e17 q^{49} +1.12528e17 q^{51} +1.54891e18 q^{52} -2.26519e18 q^{53} -7.41046e16 q^{54} +3.30345e17 q^{56} +2.08473e18 q^{57} +7.11359e17 q^{58} +2.27777e17 q^{59} +3.52948e18 q^{61} +2.49946e18 q^{62} +7.87318e17 q^{63} -5.97874e18 q^{64} +7.21294e17 q^{66} +1.45192e19 q^{67} -3.74962e18 q^{68} +6.20586e18 q^{69} -2.96266e19 q^{71} +5.10114e18 q^{72} +3.47560e19 q^{73} -1.33850e19 q^{74} -6.94666e19 q^{76} -7.66333e18 q^{77} +1.67305e19 q^{78} +3.47111e19 q^{79} +1.21577e19 q^{81} -4.86958e19 q^{82} +1.56621e20 q^{83} -2.62347e19 q^{84} +1.64885e19 q^{86} -1.16706e20 q^{87} -4.96518e19 q^{88} +1.89658e20 q^{89} -1.77752e20 q^{91} -2.06789e20 q^{92} -4.10064e20 q^{93} -1.54757e20 q^{94} -2.57676e20 q^{96} -1.11510e20 q^{97} +1.82682e20 q^{98} -1.18336e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−359.921	−0.248538	−0.124269	−	0.992249i	$-0.539659\pi$
$2$	−359.921	−0.248538	−0.124269	+	0.992249i	$0.539659\pi$
$3$	59049.0	0.577350
$3$	59049.0	0.577350
$4$	−1.96761e6	−0.938229
$5$	0	0
$5$	0	0
$6$	−2.12530e7	−0.143493
$7$	2.25801e8	0.302131	0.151066	−	0.988524i	$-0.451729\pi$
$7$	2.25801e8	0.302131	0.151066	+	0.988524i	$0.451729\pi$
$8$	1.46299e9	0.481723
$9$	3.48678e9	0.333333
$10$	0	0
$11$	−3.39385e10	−0.394520	−0.197260	−	0.980351i	$-0.563204\pi$
$11$	−3.39385e10	−0.394520	−0.197260	+	0.980351i	$0.563204\pi$
$12$	−1.16185e11	−0.541687
$13$	−7.87206e11	−1.58374	−0.791869	−	0.610691i	$-0.790892\pi$
$13$	−7.87206e11	−1.58374	−0.791869	+	0.610691i	$0.790892\pi$
$14$	−8.12704e10	−0.0750911
$15$	0	0
$16$	3.59981e12	0.818503
$17$	1.90567e12	0.229263	0.114632	−	0.993408i	$-0.463431\pi$
$17$	1.90567e12	0.229263	0.114632	+	0.993408i	$0.463431\pi$
$18$	−1.25497e12	−0.0828459
$19$	3.53051e13	1.32107	0.660533	−	0.750797i	$-0.270330\pi$
$19$	3.53051e13	1.32107	0.660533	+	0.750797i	$0.270330\pi$
$20$	0	0
$21$	1.33333e13	0.174436
$22$	1.22152e13	0.0980532
$23$	1.05097e14	0.528989	0.264495	−	0.964387i	$-0.414795\pi$
$23$	1.05097e14	0.528989	0.264495	+	0.964387i	$0.414795\pi$
$24$	8.63883e13	0.278123
$25$	0	0
$26$	2.83332e14	0.393619
$27$	2.05891e14	0.192450
$28$	−4.44287e14	−0.283468
$29$	−1.97643e15	−0.872373	−0.436187	−	0.899856i	$-0.643671\pi$
$29$	−1.97643e15	−0.872373	−0.436187	+	0.899856i	$0.643671\pi$
$30$	0	0
$31$	−6.94447e15	−1.52174	−0.760872	−	0.648902i	$-0.775228\pi$
$31$	−6.94447e15	−1.52174	−0.760872	+	0.648902i	$0.775228\pi$
$32$	−4.36377e15	−0.685152
$33$	−2.00403e15	−0.227776
$34$	−6.85892e14	−0.0569806
$35$	0	0
$36$	−6.86063e15	−0.312743
$37$	3.71887e16	1.27143	0.635715	−	0.771924i	$-0.280705\pi$
$37$	3.71887e16	1.27143	0.635715	+	0.771924i	$0.280705\pi$
$38$	−1.27070e16	−0.328335
$39$	−4.64837e16	−0.914371
$40$	0	0
$41$	1.35296e17	1.57418	0.787089	−	0.616839i	$-0.211587\pi$
$41$	1.35296e17	1.57418	0.787089	+	0.616839i	$0.211587\pi$
$42$	−4.79894e15	−0.0433538
$43$	−4.58115e16	−0.323263	−0.161631	−	0.986851i	$-0.551676\pi$
$43$	−4.58115e16	−0.323263	−0.161631	+	0.986851i	$0.551676\pi$
$44$	6.67777e16	0.370150
$45$	0	0
$46$	−3.78265e16	−0.131474
$47$	4.29974e17	1.19238	0.596190	−	0.802843i	$-0.296680\pi$
$47$	4.29974e17	1.19238	0.596190	+	0.802843i	$0.296680\pi$
$48$	2.12565e17	0.472563
$49$	−5.07560e17	−0.908717
$50$	0	0
$51$	1.12528e17	0.132365
$52$	1.54891e18	1.48591
$53$	−2.26519e18	−1.77913	−0.889565	−	0.456808i	$-0.848993\pi$
$53$	−2.26519e18	−1.77913	−0.889565	+	0.456808i	$0.848993\pi$
$54$	−7.41046e16	−0.0478311
$55$	0	0
$56$	3.30345e17	0.145544
$57$	2.08473e18	0.762718
$58$	7.11359e17	0.216818
$59$	2.27777e17	0.0580180	0.0290090	−	0.999579i	$-0.490765\pi$
$59$	2.27777e17	0.0580180	0.0290090	+	0.999579i	$0.490765\pi$
$60$	0	0
$61$	3.52948e18	0.633502	0.316751	−	0.948509i	$-0.397408\pi$
$61$	3.52948e18	0.633502	0.316751	+	0.948509i	$0.397408\pi$
$62$	2.49946e18	0.378211
$63$	7.87318e17	0.100710
$64$	−5.97874e18	−0.648217
$65$	0	0
$66$	7.21294e17	0.0566110
$67$	1.45192e19	0.973100	0.486550	−	0.873653i	$-0.338255\pi$
$67$	1.45192e19	0.973100	0.486550	+	0.873653i	$0.338255\pi$
$68$	−3.74962e18	−0.215101
$69$	6.20586e18	0.305412
$70$	0	0
$71$	−2.96266e19	−1.08011	−0.540057	−	0.841629i	$-0.681597\pi$
$71$	−2.96266e19	−1.08011	−0.540057	+	0.841629i	$0.681597\pi$
$72$	5.10114e18	0.160574
$73$	3.47560e19	0.946540	0.473270	−	0.880917i	$-0.343073\pi$
$73$	3.47560e19	0.946540	0.473270	+	0.880917i	$0.343073\pi$
$74$	−1.33850e19	−0.315998
$75$	0	0
$76$	−6.94666e19	−1.23946
$77$	−7.66333e18	−0.119197
$78$	1.67305e19	0.227256
$79$	3.47111e19	0.412462	0.206231	−	0.978503i	$-0.433880\pi$
$79$	3.47111e19	0.412462	0.206231	+	0.978503i	$0.433880\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	−4.86958e19	−0.391243
$83$	1.56621e20	1.10798	0.553988	−	0.832525i	$-0.313105\pi$
$83$	1.56621e20	1.10798	0.553988	+	0.832525i	$0.313105\pi$
$84$	−2.62347e19	−0.163661
$85$	0	0
$86$	1.64885e19	0.0803430
$87$	−1.16706e20	−0.503665
$88$	−4.96518e19	−0.190049
$89$	1.89658e20	0.644727	0.322364	−	0.946616i	$-0.395523\pi$
$89$	1.89658e20	0.644727	0.322364	+	0.946616i	$0.395523\pi$
$90$	0	0
$91$	−1.77752e20	−0.478497
$92$	−2.06789e20	−0.496313
$93$	−4.10064e20	−0.878579
$94$	−1.54757e20	−0.296352
$95$	0	0
$96$	−2.57676e20	−0.395573
$97$	−1.11510e20	−0.153536	−0.0767682	−	0.997049i	$-0.524460\pi$
$97$	−1.11510e20	−0.153536	−0.0767682	+	0.997049i	$0.524460\pi$
$98$	1.82682e20	0.225850
$99$	−1.18336e20	−0.131507
$100$	0	0
$101$	−7.95776e20	−0.716830	−0.358415	−	0.933562i	$-0.616683\pi$
$101$	−7.95776e20	−0.716830	−0.358415	+	0.933562i	$0.616683\pi$
$102$	−4.05012e19	−0.0328978
$103$	−5.47853e20	−0.401673	−0.200837	−	0.979625i	$-0.564366\pi$
$103$	−5.47853e20	−0.401673	−0.200837	+	0.979625i	$0.564366\pi$
$104$	−1.15168e21	−0.762923
$105$	0	0
$106$	8.15289e20	0.442181
$107$	1.16424e21	0.572154	0.286077	−	0.958207i	$-0.407649\pi$
$107$	1.16424e21	0.572154	0.286077	+	0.958207i	$0.407649\pi$
$108$	−4.05113e20	−0.180562
$109$	8.74266e20	0.353725	0.176862	−	0.984236i	$-0.443405\pi$
$109$	8.74266e20	0.353725	0.176862	+	0.984236i	$0.443405\pi$
$110$	0	0
$111$	2.19595e21	0.734060
$112$	8.12840e20	0.247295
$113$	−5.05935e21	−1.40208	−0.701038	−	0.713124i	$-0.747279\pi$
$113$	−5.05935e21	−1.40208	−0.701038	+	0.713124i	$0.747279\pi$
$114$	−7.50338e20	−0.189564
$115$	0	0
$116$	3.88884e21	0.818486
$117$	−2.74482e21	−0.527913
$118$	−8.19816e19	−0.0144197
$119$	4.30302e20	0.0692676
$120$	0	0
$121$	−6.24843e21	−0.844354
$122$	−1.27033e21	−0.157449
$123$	7.98908e21	0.908852
$124$	1.36640e22	1.42774
$125$	0	0
$126$	−2.83372e20	−0.0250304
$127$	5.06817e21	0.412014	0.206007	−	0.978551i	$-0.433953\pi$
$127$	5.06817e21	0.412014	0.206007	+	0.978551i	$0.433953\pi$
$128$	1.13034e22	0.846258
$129$	−2.70512e21	−0.186636
$130$	0	0
$131$	7.92504e21	0.465214	0.232607	−	0.972571i	$-0.425274\pi$
$131$	7.92504e21	0.465214	0.232607	+	0.972571i	$0.425274\pi$
$132$	3.94315e21	0.213706
$133$	7.97191e21	0.399135
$134$	−5.22577e21	−0.241852
$135$	0	0
$136$	2.78799e21	0.110441
$137$	−7.84646e21	−0.287811	−0.143906	−	0.989591i	$-0.545966\pi$
$137$	−7.84646e21	−0.287811	−0.143906	+	0.989591i	$0.545966\pi$
$138$	−2.23362e21	−0.0759064
$139$	3.12526e22	0.984534	0.492267	−	0.870444i	$-0.336168\pi$
$139$	3.12526e22	0.984534	0.492267	+	0.870444i	$0.336168\pi$
$140$	0	0
$141$	2.53895e22	0.688421
$142$	1.06632e22	0.268449
$143$	2.67166e22	0.624817
$144$	1.25518e22	0.272834
$145$	0	0
$146$	−1.25094e22	−0.235251
$147$	−2.99709e22	−0.524648
$148$	−7.31727e22	−1.19289
$149$	3.26426e22	0.495826	0.247913	−	0.968782i	$-0.420255\pi$
$149$	3.26426e22	0.495826	0.247913	+	0.968782i	$0.420255\pi$
$150$	0	0
$151$	−3.84186e22	−0.507322	−0.253661	−	0.967293i	$-0.581635\pi$
$151$	−3.84186e22	−0.507322	−0.253661	+	0.967293i	$0.581635\pi$
$152$	5.16511e22	0.636388
$153$	6.64467e21	0.0764211
$154$	2.75820e21	0.0296249
$155$	0	0
$156$	9.14618e22	0.857890
$157$	3.94963e22	0.346426	0.173213	−	0.984884i	$-0.444585\pi$
$157$	3.94963e22	0.346426	0.173213	+	0.984884i	$0.444585\pi$
$158$	−1.24933e22	−0.102512
$159$	−1.33757e23	−1.02718
$160$	0	0
$161$	2.37309e22	0.159824
$162$	−4.37580e21	−0.0276153
$163$	−3.02701e23	−1.79079	−0.895395	−	0.445273i	$-0.853107\pi$
$163$	−3.02701e23	−1.79079	−0.895395	+	0.445273i	$0.853107\pi$
$164$	−2.66209e23	−1.47694
$165$	0	0
$166$	−5.63712e22	−0.275374
$167$	−3.58711e23	−1.64521	−0.822607	−	0.568611i	$-0.807481\pi$
$167$	−3.58711e23	−1.64521	−0.822607	+	0.568611i	$0.807481\pi$
$168$	1.95065e22	0.0840297
$169$	3.72629e23	1.50823
$170$	0	0
$171$	1.23101e23	0.440355
$172$	9.01391e22	0.303295
$173$	2.25208e22	0.0713017	0.0356508	−	0.999364i	$-0.488650\pi$
$173$	2.25208e22	0.0713017	0.0356508	+	0.999364i	$0.488650\pi$
$174$	4.20050e22	0.125180
$175$	0	0
$176$	−1.22172e23	−0.322916
$177$	1.34500e22	0.0334967
$178$	−6.82619e22	−0.160239
$179$	−2.97381e23	−0.658197	−0.329099	−	0.944296i	$-0.606745\pi$
$179$	−2.97381e23	−0.658197	−0.329099	+	0.944296i	$0.606745\pi$
$180$	0	0
$181$	−6.34042e23	−1.24880	−0.624400	−	0.781105i	$-0.714656\pi$
$181$	−6.34042e23	−1.24880	−0.624400	+	0.781105i	$0.714656\pi$
$182$	6.39766e22	0.118925
$183$	2.08412e23	0.365752
$184$	1.53756e23	0.254826
$185$	0	0
$186$	1.47591e23	0.218360
$187$	−6.46756e22	−0.0904490
$188$	−8.46021e23	−1.11873
$189$	4.64904e22	0.0581452
$190$	0	0
$191$	−1.59653e24	−1.78783	−0.893914	−	0.448238i	$-0.852052\pi$
$191$	−1.59653e24	−1.78783	−0.893914	+	0.448238i	$0.852052\pi$
$192$	−3.53039e23	−0.374248
$193$	1.28260e24	1.28748	0.643739	−	0.765245i	$-0.277382\pi$
$193$	1.28260e24	1.28748	0.643739	+	0.765245i	$0.277382\pi$
$194$	4.01349e22	0.0381596
$195$	0	0
$196$	9.98679e23	0.852584
$197$	−1.15374e24	−0.933709	−0.466854	−	0.884334i	$-0.654613\pi$
$197$	−1.15374e24	−0.933709	−0.466854	+	0.884334i	$0.654613\pi$
$198$	4.25917e22	0.0326844
$199$	−8.81552e22	−0.0641639	−0.0320819	−	0.999485i	$-0.510214\pi$
$199$	−8.81552e22	−0.0641639	−0.0320819	+	0.999485i	$0.510214\pi$
$200$	0	0
$201$	8.57344e23	0.561819
$202$	2.86416e23	0.178159
$203$	−4.46279e23	−0.263571
$204$	−2.21411e23	−0.124189
$205$	0	0
$206$	1.97184e23	0.0998310
$207$	3.66450e23	0.176330
$208$	−2.83379e24	−1.29629
$209$	−1.19820e24	−0.521187
$210$	0	0
$211$	1.77022e24	0.696724	0.348362	−	0.937360i	$-0.386738\pi$
$211$	1.77022e24	0.696724	0.348362	+	0.937360i	$0.386738\pi$
$212$	4.45701e24	1.66923
$213$	−1.74942e24	−0.623604
$214$	−4.19034e23	−0.142202
$215$	0	0
$216$	3.01217e23	0.0927076
$217$	−1.56807e24	−0.459767
$218$	−3.14667e23	−0.0879139
$219$	2.05230e24	0.546485
$220$	0	0
$221$	−1.50016e24	−0.363093
$222$	−7.90370e23	−0.182442
$223$	−6.37800e24	−1.40438	−0.702188	−	0.711991i	$-0.747794\pi$
$223$	−6.37800e24	−1.40438	−0.702188	+	0.711991i	$0.747794\pi$
$224$	−9.85342e23	−0.207006
$225$	0	0
$226$	1.82097e24	0.348469
$227$	−5.77167e24	−1.05446	−0.527230	−	0.849723i	$-0.676769\pi$
$227$	−5.77167e24	−1.05446	−0.527230	+	0.849723i	$0.676769\pi$
$228$	−4.10193e24	−0.715604
$229$	−4.49646e24	−0.749201	−0.374600	−	0.927186i	$-0.622220\pi$
$229$	−4.49646e24	−0.749201	−0.374600	+	0.927186i	$0.622220\pi$
$230$	0	0
$231$	−4.52512e23	−0.0688184
$232$	−2.89150e24	−0.420242
$233$	−1.04080e25	−1.44587	−0.722937	−	0.690914i	$-0.757208\pi$
$233$	−1.04080e25	−1.44587	−0.722937	+	0.690914i	$0.757208\pi$
$234$	9.87918e23	0.131206
$235$	0	0
$236$	−4.48175e23	−0.0544342
$237$	2.04966e24	0.238135
$238$	−1.54875e23	−0.0172156
$239$	9.90599e24	1.05371	0.526854	−	0.849956i	$-0.323372\pi$
$239$	9.90599e24	1.05371	0.526854	+	0.849956i	$0.323372\pi$
$240$	0	0
$241$	1.33934e25	1.30531	0.652654	−	0.757656i	$-0.273656\pi$
$241$	1.33934e25	1.30531	0.652654	+	0.757656i	$0.273656\pi$
$242$	2.24894e24	0.209854
$243$	7.17898e23	0.0641500
$244$	−6.94464e24	−0.594369
$245$	0	0
$246$	−2.87544e24	−0.225884
$247$	−2.77924e25	−2.09222
$248$	−1.01597e25	−0.733059
$249$	9.24832e24	0.639690
$250$	0	0
$251$	−2.23731e25	−1.42283	−0.711413	−	0.702774i	$-0.751944\pi$
$251$	−2.23731e25	−1.42283	−0.711413	+	0.702774i	$0.751944\pi$
$252$	−1.54913e24	−0.0944895
$253$	−3.56682e24	−0.208697
$254$	−1.82414e24	−0.102401
$255$	0	0
$256$	8.47001e24	0.437889
$257$	−2.43723e25	−1.20948	−0.604740	−	0.796423i	$-0.706723\pi$
$257$	−2.43723e25	−1.20948	−0.604740	+	0.796423i	$0.706723\pi$
$258$	9.73631e23	0.0463861
$259$	8.39723e24	0.384139
$260$	0	0
$261$	−6.89138e24	−0.290791
$262$	−2.85239e24	−0.115623
$263$	−2.84968e25	−1.10984	−0.554921	−	0.831903i	$-0.687252\pi$
$263$	−2.84968e25	−1.10984	−0.554921	+	0.831903i	$0.687252\pi$
$264$	−2.93189e24	−0.109725
$265$	0	0
$266$	−2.86926e24	−0.0992002
$267$	1.11991e25	0.372233
$268$	−2.85681e25	−0.912990
$269$	−3.91028e24	−0.120174	−0.0600868	−	0.998193i	$-0.519138\pi$
$269$	−3.91028e24	−0.120174	−0.0600868	+	0.998193i	$0.519138\pi$
$270$	0	0
$271$	−4.15108e25	−1.18028	−0.590138	−	0.807302i	$-0.700927\pi$
$271$	−4.15108e25	−1.18028	−0.590138	+	0.807302i	$0.700927\pi$
$272$	6.86006e24	0.187653
$273$	−1.04961e25	−0.276260
$274$	2.82411e24	0.0715319
$275$	0	0
$276$	−1.22107e25	−0.286546
$277$	4.68304e25	1.05801	0.529006	−	0.848618i	$-0.322565\pi$
$277$	4.68304e25	1.05801	0.529006	+	0.848618i	$0.322565\pi$
$278$	−1.12485e25	−0.244694
$279$	−2.42139e25	−0.507248
$280$	0	0
$281$	8.18476e25	1.59070	0.795352	−	0.606148i	$-0.207286\pi$
$281$	8.18476e25	1.59070	0.795352	+	0.606148i	$0.207286\pi$
$282$	−9.13824e24	−0.171099
$283$	−6.88309e25	−1.24173	−0.620863	−	0.783919i	$-0.713218\pi$
$283$	−6.88309e25	−1.24173	−0.620863	+	0.783919i	$0.713218\pi$
$284$	5.82936e25	1.01339
$285$	0	0
$286$	−9.61586e24	−0.155290
$287$	3.05499e25	0.475609
$288$	−1.52155e25	−0.228384
$289$	−6.54603e25	−0.947438
$290$	0	0
$291$	−6.58457e24	−0.0886443
$292$	−6.83861e25	−0.888072
$293$	−1.04180e26	−1.30519	−0.652594	−	0.757708i	$-0.726319\pi$
$293$	−1.04180e26	−1.30519	−0.652594	+	0.757708i	$0.726319\pi$
$294$	1.07872e25	0.130395
$295$	0	0
$296$	5.44068e25	0.612477
$297$	−6.98763e24	−0.0759254
$298$	−1.17488e25	−0.123231
$299$	−8.27328e25	−0.837780
$300$	0	0
$301$	−1.03443e25	−0.0976679
$302$	1.38277e25	0.126089
$303$	−4.69898e25	−0.413862
$304$	1.27092e26	1.08130
$305$	0	0
$306$	−2.39156e24	−0.0189935
$307$	−1.21443e25	−0.0932005	−0.0466003	−	0.998914i	$-0.514839\pi$
$307$	−1.21443e25	−0.0932005	−0.0466003	+	0.998914i	$0.514839\pi$
$308$	1.50784e25	0.111834
$309$	−3.23501e25	−0.231906
$310$	0	0
$311$	5.12643e25	0.343424	0.171712	−	0.985147i	$-0.445070\pi$
$311$	5.12643e25	0.343424	0.171712	+	0.985147i	$0.445070\pi$
$312$	−6.80054e25	−0.440474
$313$	−2.40814e26	−1.50822	−0.754111	−	0.656747i	$-0.771932\pi$
$313$	−2.40814e26	−1.50822	−0.754111	+	0.656747i	$0.771932\pi$
$314$	−1.42156e25	−0.0860999
$315$	0	0
$316$	−6.82979e25	−0.386984
$317$	−2.81853e26	−1.54490	−0.772451	−	0.635074i	$-0.780969\pi$
$317$	−2.81853e26	−1.54490	−0.772451	+	0.635074i	$0.780969\pi$
$318$	4.81420e25	0.255293
$319$	6.70770e25	0.344169
$320$	0	0
$321$	6.87472e25	0.330333
$322$	−8.54126e24	−0.0397224
$323$	6.72799e25	0.302872
$324$	−2.39215e25	−0.104248
$325$	0	0
$326$	1.08949e26	0.445079
$327$	5.16245e25	0.204223
$328$	1.97937e26	0.758318
$329$	9.70885e25	0.360256
$330$	0	0
$331$	−3.97534e26	−1.38414	−0.692070	−	0.721830i	$-0.743301\pi$
$331$	−3.97534e26	−1.38414	−0.692070	+	0.721830i	$0.743301\pi$
$332$	−3.08169e26	−1.03954
$333$	1.29669e26	0.423810
$334$	1.29108e26	0.408898
$335$	0	0
$336$	4.79974e25	0.142776
$337$	−9.29580e25	−0.268023	−0.134012	−	0.990980i	$-0.542786\pi$
$337$	−9.29580e25	−0.268023	−0.134012	+	0.990980i	$0.542786\pi$
$338$	−1.34117e26	−0.374851
$339$	−2.98750e26	−0.809488
$340$	0	0
$341$	2.35685e26	0.600359
$342$	−4.43067e25	−0.109445
$343$	−2.40727e26	−0.576683
$344$	−6.70219e25	−0.155723
$345$	0	0
$346$	−8.10571e24	−0.0177212
$347$	8.71900e26	1.84930	0.924650	−	0.380818i	$-0.124358\pi$
$347$	8.71900e26	1.84930	0.924650	+	0.380818i	$0.124358\pi$
$348$	2.29632e26	0.472553
$349$	−2.14177e26	−0.427667	−0.213834	−	0.976870i	$-0.568595\pi$
$349$	−2.14177e26	−0.427667	−0.213834	+	0.976870i	$0.568595\pi$
$350$	0	0
$351$	−1.62079e26	−0.304790
$352$	1.48100e26	0.270306
$353$	2.63256e26	0.466384	0.233192	−	0.972431i	$-0.425083\pi$
$353$	2.63256e26	0.466384	0.233192	+	0.972431i	$0.425083\pi$
$354$	−4.84093e24	−0.00832520
$355$	0	0
$356$	−3.73173e26	−0.604902
$357$	2.54089e25	0.0399917
$358$	1.07034e26	0.163587
$359$	−2.16196e26	−0.320890	−0.160445	−	0.987045i	$-0.551293\pi$
$359$	−2.16196e26	−0.320890	−0.160445	+	0.987045i	$0.551293\pi$
$360$	0	0
$361$	5.32239e26	0.745214
$362$	2.28205e26	0.310374
$363$	−3.68964e26	−0.487488
$364$	3.49746e26	0.448940
$365$	0	0
$366$	−7.50120e25	−0.0909032
$367$	7.58083e25	0.0892736	0.0446368	−	0.999003i	$-0.485787\pi$
$367$	7.58083e25	0.0892736	0.0446368	+	0.999003i	$0.485787\pi$
$368$	3.78328e26	0.432979
$369$	4.71747e26	0.524726
$370$	0	0
$371$	−5.11481e26	−0.537531
$372$	8.06846e26	0.824309
$373$	4.98462e26	0.495095	0.247548	−	0.968876i	$-0.420375\pi$
$373$	4.98462e26	0.495095	0.247548	+	0.968876i	$0.420375\pi$
$374$	2.32781e25	0.0224800
$375$	0	0
$376$	6.29049e26	0.574397
$377$	1.55586e27	1.38161
$378$	−1.67329e25	−0.0144513
$379$	−1.76506e27	−1.48268	−0.741340	−	0.671130i	$-0.765809\pi$
$379$	−1.76506e27	−1.48268	−0.741340	+	0.671130i	$0.765809\pi$
$380$	0	0
$381$	2.99270e26	0.237876
$382$	5.74624e26	0.444343
$383$	2.00746e27	1.51029	0.755143	−	0.655560i	$-0.227568\pi$
$383$	2.00746e27	1.51029	0.755143	+	0.655560i	$0.227568\pi$
$384$	6.67452e26	0.488587
$385$	0	0
$386$	−4.61635e26	−0.319987
$387$	−1.59735e26	−0.107754
$388$	2.19409e26	0.144052
$389$	−5.35448e26	−0.342174	−0.171087	−	0.985256i	$-0.554728\pi$
$389$	−5.35448e26	−0.342174	−0.171087	+	0.985256i	$0.554728\pi$
$390$	0	0
$391$	2.00280e26	0.121278
$392$	−7.42557e26	−0.437750
$393$	4.67966e26	0.268591
$394$	4.15254e26	0.232062
$395$	0	0
$396$	2.32839e26	0.123383
$397$	−3.53227e27	−1.82286	−0.911431	−	0.411452i	$-0.865022\pi$
$397$	−3.53227e27	−1.82286	−0.911431	+	0.411452i	$0.865022\pi$
$398$	3.17289e25	0.0159471
$399$	4.70733e26	0.230441
$400$	0	0
$401$	−4.12842e27	−1.91765	−0.958823	−	0.284006i	$-0.908336\pi$
$401$	−4.12842e27	−1.91765	−0.958823	+	0.284006i	$0.908336\pi$
$402$	−3.08576e26	−0.139633
$403$	5.46673e27	2.41004
$404$	1.56578e27	0.672551
$405$	0	0
$406$	1.60625e26	0.0655074
$407$	−1.26213e27	−0.501605
$408$	1.64628e26	0.0637634
$409$	1.84613e27	0.696894	0.348447	−	0.937328i	$-0.386709\pi$
$409$	1.84613e27	0.696894	0.348447	+	0.937328i	$0.386709\pi$
$410$	0	0
$411$	−4.63325e26	−0.166168
$412$	1.07796e27	0.376861
$413$	5.14321e25	0.0175291
$414$	−1.31893e26	−0.0438246
$415$	0	0
$416$	3.43518e27	1.08510
$417$	1.84543e27	0.568421
$418$	4.31258e26	0.129535
$419$	3.00419e27	0.879996	0.439998	−	0.897999i	$-0.354979\pi$
$419$	3.00419e27	0.879996	0.439998	+	0.897999i	$0.354979\pi$
$420$	0	0
$421$	6.22566e27	1.73470	0.867348	−	0.497702i	$-0.165823\pi$
$421$	6.22566e27	1.73470	0.867348	+	0.497702i	$0.165823\pi$
$422$	−6.37139e26	−0.173162
$423$	1.49923e27	0.397460
$424$	−3.31396e27	−0.857048
$425$	0	0
$426$	6.29654e26	0.154989
$427$	7.96959e26	0.191401
$428$	−2.29077e27	−0.536811
$429$	1.57759e27	0.360738
$430$	0	0
$431$	5.84115e25	0.0127200	0.00636000	−	0.999980i	$-0.497976\pi$
$431$	5.84115e25	0.0127200	0.00636000	+	0.999980i	$0.497976\pi$
$432$	7.41169e26	0.157521
$433$	2.80239e27	0.581307	0.290653	−	0.956828i	$-0.406127\pi$
$433$	2.80239e27	0.581307	0.290653	+	0.956828i	$0.406127\pi$
$434$	5.64380e26	0.114269
$435$	0	0
$436$	−1.72021e27	−0.331875
$437$	3.71045e27	0.698830
$438$	−7.38668e26	−0.135822
$439$	−6.01312e26	−0.107950	−0.0539749	−	0.998542i	$-0.517189\pi$
$439$	−6.01312e26	−0.107950	−0.0539749	+	0.998542i	$0.517189\pi$
$440$	0	0
$441$	−1.76975e27	−0.302906
$442$	5.39938e26	0.0902423
$443$	8.83755e27	1.44242	0.721212	−	0.692715i	$-0.243586\pi$
$443$	8.83755e27	1.44242	0.721212	+	0.692715i	$0.243586\pi$
$444$	−4.32078e27	−0.688717
$445$	0	0
$446$	2.29558e27	0.349041
$447$	1.92751e27	0.286265
$448$	−1.35000e27	−0.195847
$449$	−5.63925e27	−0.799162	−0.399581	−	0.916698i	$-0.630844\pi$
$449$	−5.63925e27	−0.799162	−0.399581	+	0.916698i	$0.630844\pi$
$450$	0	0
$451$	−4.59173e27	−0.621045
$452$	9.95483e27	1.31547
$453$	−2.26858e27	−0.292902
$454$	2.07735e27	0.262073
$455$	0	0
$456$	3.04995e27	0.367419
$457$	7.44037e27	0.875940	0.437970	−	0.898990i	$-0.355698\pi$
$457$	7.44037e27	0.875940	0.437970	+	0.898990i	$0.355698\pi$
$458$	1.61837e27	0.186205
$459$	3.92361e26	0.0441217
$460$	0	0
$461$	−3.45643e27	−0.371337	−0.185668	−	0.982612i	$-0.559445\pi$
$461$	−3.45643e27	−0.371337	−0.185668	+	0.982612i	$0.559445\pi$
$462$	1.62869e26	0.0171040
$463$	−1.22618e28	−1.25879	−0.629397	−	0.777084i	$-0.716698\pi$
$463$	−1.22618e28	−1.25879	−0.629397	+	0.777084i	$0.716698\pi$
$464$	−7.11478e27	−0.714040
$465$	0	0
$466$	3.74606e27	0.359354
$467$	−4.64105e27	−0.435300	−0.217650	−	0.976027i	$-0.569839\pi$
$467$	−4.64105e27	−0.435300	−0.217650	+	0.976027i	$0.569839\pi$
$468$	5.40073e27	0.495303
$469$	3.27845e27	0.294004
$470$	0	0
$471$	2.33222e27	0.200009
$472$	3.33236e26	0.0279486
$473$	1.55477e27	0.127534
$474$	−7.37715e26	−0.0591856
$475$	0	0
$476$	−8.46666e26	−0.0649889
$477$	−7.89823e27	−0.593044
$478$	−3.56538e27	−0.261886
$479$	4.75956e27	0.342014	0.171007	−	0.985270i	$-0.445298\pi$
$479$	4.75956e27	0.342014	0.171007	+	0.985270i	$0.445298\pi$
$480$	0	0
$481$	−2.92751e28	−2.01361
$482$	−4.82058e27	−0.324418
$483$	1.40129e27	0.0922746
$484$	1.22945e28	0.792197
$485$	0	0
$486$	−2.58387e26	−0.0159437
$487$	−2.21192e28	−1.33572	−0.667860	−	0.744287i	$-0.732790\pi$
$487$	−2.21192e28	−1.33572	−0.667860	+	0.744287i	$0.732790\pi$
$488$	5.16361e27	0.305172
$489$	−1.78742e28	−1.03391
$490$	0	0
$491$	3.24006e28	1.79555	0.897774	−	0.440456i	$-0.145183\pi$
$491$	3.24006e28	1.79555	0.897774	+	0.440456i	$0.145183\pi$
$492$	−1.57194e28	−0.852711
$493$	−3.76643e27	−0.200003
$494$	1.00031e28	0.519996
$495$	0	0
$496$	−2.49988e28	−1.24555
$497$	−6.68971e27	−0.326336
$498$	−3.32867e27	−0.158987
$499$	−2.01743e28	−0.943502	−0.471751	−	0.881732i	$-0.656378\pi$
$499$	−2.01743e28	−0.943502	−0.471751	+	0.881732i	$0.656378\pi$
$500$	0	0
$501$	−2.11816e28	−0.949865
$502$	8.05255e27	0.353626
$503$	−2.90806e28	−1.25066	−0.625330	−	0.780360i	$-0.715036\pi$
$503$	−2.90806e28	−1.25066	−0.625330	+	0.780360i	$0.715036\pi$
$504$	1.15184e27	0.0485146
$505$	0	0
$506$	1.28377e27	0.0518691
$507$	2.20034e28	0.870774
$508$	−9.97217e27	−0.386563
$509$	2.66984e28	1.01379	0.506895	−	0.862008i	$-0.330793\pi$
$509$	2.66984e28	1.01379	0.506895	+	0.862008i	$0.330793\pi$
$510$	0	0
$511$	7.84792e27	0.285980
$512$	−2.67534e28	−0.955090
$513$	7.26900e27	0.254239
$514$	8.77210e27	0.300601
$515$	0	0
$516$	5.32263e27	0.175107
$517$	−1.45927e28	−0.470418
$518$	−3.02234e27	−0.0954730
$519$	1.32983e27	0.0411660
$520$	0	0
$521$	1.59165e28	0.473208	0.236604	−	0.971606i	$-0.423966\pi$
$521$	1.59165e28	0.473208	0.236604	+	0.971606i	$0.423966\pi$
$522$	2.48035e27	0.0722726
$523$	−3.68283e28	−1.05175	−0.525877	−	0.850561i	$-0.676263\pi$
$523$	−3.68283e28	−1.05175	−0.525877	+	0.850561i	$0.676263\pi$
$524$	−1.55934e28	−0.436477
$525$	0	0
$526$	1.02566e28	0.275838
$527$	−1.32339e28	−0.348880
$528$	−7.21414e27	−0.186436
$529$	−2.84263e28	−0.720170
$530$	0	0
$531$	7.94208e26	0.0193393
$532$	−1.56856e28	−0.374480
$533$	−1.06506e29	−2.49309
$534$	−4.03080e27	−0.0925140
$535$	0	0
$536$	2.12415e28	0.468765
$537$	−1.75600e28	−0.380010
$538$	1.40739e27	0.0298677
$539$	1.72258e28	0.358507
$540$	0	0
$541$	8.57342e28	1.71626	0.858130	−	0.513433i	$-0.171627\pi$
$541$	8.57342e28	1.71626	0.858130	+	0.513433i	$0.171627\pi$
$542$	1.49406e28	0.293343
$543$	−3.74395e28	−0.720995
$544$	−8.31591e27	−0.157080
$545$	0	0
$546$	3.77775e27	0.0686611
$547$	9.45029e28	1.68492	0.842458	−	0.538762i	$-0.181108\pi$
$547$	9.45029e28	1.68492	0.842458	+	0.538762i	$0.181108\pi$
$548$	1.54388e28	0.270033
$549$	1.23065e28	0.211167
$550$	0	0
$551$	−6.97780e28	−1.15246
$552$	9.07913e27	0.147124
$553$	7.83780e27	0.124618
$554$	−1.68553e28	−0.262956
$555$	0	0
$556$	−6.14929e28	−0.923718
$557$	5.79016e28	0.853514	0.426757	−	0.904366i	$-0.359656\pi$
$557$	5.79016e28	0.853514	0.426757	+	0.904366i	$0.359656\pi$
$558$	8.71509e27	0.126070
$559$	3.60631e28	0.511964
$560$	0	0
$561$	−3.81903e27	−0.0522207
$562$	−2.94587e28	−0.395350
$563$	3.38680e28	0.446119	0.223060	−	0.974805i	$-0.428396\pi$
$563$	3.38680e28	0.446119	0.223060	+	0.974805i	$0.428396\pi$
$564$	−4.99567e28	−0.645897
$565$	0	0
$566$	2.47737e28	0.308616
$567$	2.74521e27	0.0335702
$568$	−4.33435e28	−0.520316
$569$	1.61113e29	1.89869	0.949343	−	0.314242i	$-0.101750\pi$
$569$	1.61113e29	1.89869	0.949343	+	0.314242i	$0.101750\pi$
$570$	0	0
$571$	7.94989e28	0.902988	0.451494	−	0.892274i	$-0.350891\pi$
$571$	7.94989e28	0.902988	0.451494	+	0.892274i	$0.350891\pi$
$572$	−5.25678e28	−0.586221
$573$	−9.42733e28	−1.03220
$574$	−1.09956e28	−0.118207
$575$	0	0
$576$	−2.08466e28	−0.216072
$577$	−1.61242e29	−1.64109	−0.820544	−	0.571583i	$-0.806330\pi$
$577$	−1.61242e29	−1.64109	−0.820544	+	0.571583i	$0.806330\pi$
$578$	2.35606e28	0.235474
$579$	7.57363e28	0.743326
$580$	0	0
$581$	3.53652e28	0.334754
$582$	2.36993e27	0.0220315
$583$	7.68771e28	0.701903
$584$	5.08477e28	0.455970
$585$	0	0
$586$	3.74965e28	0.324389
$587$	−7.40062e28	−0.628880	−0.314440	−	0.949277i	$-0.601817\pi$
$587$	−7.40062e28	−0.628880	−0.314440	+	0.949277i	$0.601817\pi$
$588$	5.89710e28	0.492240
$589$	−2.45175e29	−2.01032
$590$	0	0
$591$	−6.81270e28	−0.539077
$592$	1.33872e29	1.04067
$593$	2.27346e28	0.173625	0.0868126	−	0.996225i	$-0.472332\pi$
$593$	2.27346e28	0.173625	0.0868126	+	0.996225i	$0.472332\pi$
$594$	2.51500e27	0.0188703
$595$	0	0
$596$	−6.42279e28	−0.465198
$597$	−5.20548e27	−0.0370450
$598$	2.97773e28	0.208220
$599$	−1.57021e28	−0.107889	−0.0539444	−	0.998544i	$-0.517179\pi$
$599$	−1.57021e28	−0.107889	−0.0539444	+	0.998544i	$0.517179\pi$
$600$	0	0
$601$	−6.01910e28	−0.399346	−0.199673	−	0.979863i	$-0.563988\pi$
$601$	−6.01910e28	−0.399346	−0.199673	+	0.979863i	$0.563988\pi$
$602$	3.72312e27	0.0242742
$603$	5.06253e28	0.324367
$604$	7.55928e28	0.475984
$605$	0	0
$606$	1.69126e28	0.102860
$607$	−1.63356e29	−0.976458	−0.488229	−	0.872715i	$-0.662357\pi$
$607$	−1.63356e29	−0.976458	−0.488229	+	0.872715i	$0.662357\pi$
$608$	−1.54063e29	−0.905130
$609$	−2.63523e28	−0.152173
$610$	0	0
$611$	−3.38478e29	−1.88842
$612$	−1.30741e28	−0.0717005
$613$	6.75283e28	0.364042	0.182021	−	0.983295i	$-0.441736\pi$
$613$	6.75283e28	0.364042	0.182021	+	0.983295i	$0.441736\pi$
$614$	4.37098e27	0.0231638
$615$	0	0
$616$	−1.12114e28	−0.0574199
$617$	1.20545e29	0.606955	0.303477	−	0.952839i	$-0.401852\pi$
$617$	1.20545e29	0.606955	0.303477	+	0.952839i	$0.401852\pi$
$618$	1.16435e28	0.0576374
$619$	−1.41237e29	−0.687381	−0.343691	−	0.939083i	$-0.611677\pi$
$619$	−1.41237e29	−0.687381	−0.343691	+	0.939083i	$0.611677\pi$
$620$	0	0
$621$	2.16385e28	0.101804
$622$	−1.84511e28	−0.0853539
$623$	4.28249e28	0.194792
$624$	−1.67333e29	−0.748415
$625$	0	0
$626$	8.66739e28	0.374850
$627$	−7.07526e28	−0.300907
$628$	−7.77133e28	−0.325027
$629$	7.08694e28	0.291492
$630$	0	0
$631$	−8.36528e28	−0.332792	−0.166396	−	0.986059i	$-0.553213\pi$
$631$	−8.36528e28	−0.332792	−0.166396	+	0.986059i	$0.553213\pi$
$632$	5.07821e28	0.198693
$633$	1.04530e29	0.402254
$634$	1.01445e29	0.383966
$635$	0	0
$636$	2.63182e29	0.963731
$637$	3.99554e29	1.43917
$638$	−2.41424e28	−0.0855390
$639$	−1.03302e29	−0.360038
$640$	0	0
$641$	3.35898e29	1.13292	0.566458	−	0.824091i	$-0.308313\pi$
$641$	3.35898e29	1.13292	0.566458	+	0.824091i	$0.308313\pi$
$642$	−2.47436e28	−0.0821002
$643$	1.61665e29	0.527716	0.263858	−	0.964561i	$-0.415005\pi$
$643$	1.61665e29	0.527716	0.263858	+	0.964561i	$0.415005\pi$
$644$	−4.66932e28	−0.149952
$645$	0	0
$646$	−2.42155e28	−0.0752751
$647$	−1.20758e29	−0.369336	−0.184668	−	0.982801i	$-0.559121\pi$
$647$	−1.20758e29	−0.369336	−0.184668	+	0.982801i	$0.559121\pi$
$648$	1.77866e28	0.0535248
$649$	−7.73039e27	−0.0228893
$650$	0	0
$651$	−9.25928e28	−0.265446
$652$	5.95598e29	1.68017
$653$	4.54361e29	1.26128	0.630641	−	0.776074i	$-0.282792\pi$
$653$	4.54361e29	1.26128	0.630641	+	0.776074i	$0.282792\pi$
$654$	−1.85808e28	−0.0507571
$655$	0	0
$656$	4.87040e29	1.28847
$657$	1.21187e29	0.315513
$658$	−3.49442e28	−0.0895371
$659$	1.57194e29	0.396406	0.198203	−	0.980161i	$-0.436489\pi$
$659$	1.57194e29	0.396406	0.198203	+	0.980161i	$0.436489\pi$
$660$	0	0
$661$	−5.03408e27	−0.0122972	−0.00614858	−	0.999981i	$-0.501957\pi$
$661$	−5.03408e27	−0.0122972	−0.00614858	+	0.999981i	$0.501957\pi$
$662$	1.43081e29	0.344011
$663$	−8.85828e28	−0.209632
$664$	2.29136e29	0.533738
$665$	0	0
$666$	−4.66706e28	−0.105333
$667$	−2.07716e29	−0.461476
$668$	7.05804e29	1.54359
$669$	−3.76615e29	−0.810817
$670$	0	0
$671$	−1.19785e29	−0.249929
$672$	−5.81834e28	−0.119515
$673$	4.72877e29	0.956290	0.478145	−	0.878281i	$-0.341309\pi$
$673$	4.72877e29	0.956290	0.478145	+	0.878281i	$0.341309\pi$
$674$	3.34575e28	0.0666139
$675$	0	0
$676$	−7.33188e29	−1.41506
$677$	1.26340e29	0.240082	0.120041	−	0.992769i	$-0.461697\pi$
$677$	1.26340e29	0.240082	0.120041	+	0.992769i	$0.461697\pi$
$678$	1.07526e29	0.201188
$679$	−2.51791e28	−0.0463882
$680$	0	0
$681$	−3.40811e29	−0.608792
$682$	−8.48280e28	−0.149212
$683$	4.69895e29	0.813923	0.406961	−	0.913445i	$-0.366588\pi$
$683$	4.69895e29	0.813923	0.406961	+	0.913445i	$0.366588\pi$
$684$	−2.42215e29	−0.413154
$685$	0	0
$686$	8.66429e28	0.143328
$687$	−2.65511e29	−0.432551
$688$	−1.64913e29	−0.264592
$689$	1.78317e30	2.81768
$690$	0	0
$691$	−1.34128e28	−0.0205589	−0.0102795	−	0.999947i	$-0.503272\pi$
$691$	−1.34128e28	−0.0205589	−0.0102795	+	0.999947i	$0.503272\pi$
$692$	−4.43121e28	−0.0668973
$693$	−2.67204e28	−0.0397323
$694$	−3.13815e29	−0.459621
$695$	0	0
$696$	−1.70740e29	−0.242627
$697$	2.57829e29	0.360901
$698$	7.70868e28	0.106291
$699$	−6.14582e29	−0.834776
$700$	0	0
$701$	1.13909e30	1.50148	0.750742	−	0.660596i	$-0.229696\pi$
$701$	1.13909e30	1.50148	0.750742	+	0.660596i	$0.229696\pi$
$702$	5.83356e28	0.0757519
$703$	1.31295e30	1.67964
$704$	2.02909e29	0.255734
$705$	0	0
$706$	−9.47513e28	−0.115914
$707$	−1.79687e29	−0.216577
$708$	−2.64643e28	−0.0314276
$709$	−1.14479e30	−1.33949	−0.669747	−	0.742589i	$-0.733598\pi$
$709$	−1.14479e30	−1.33949	−0.669747	+	0.742589i	$0.733598\pi$
$710$	0	0
$711$	1.21030e29	0.137487
$712$	2.77468e29	0.310580
$713$	−7.29841e29	−0.804986
$714$	−9.14520e27	−0.00993944
$715$	0	0
$716$	5.85129e29	0.617540
$717$	5.84939e29	0.608358
$718$	7.78135e28	0.0797532
$719$	9.42126e29	0.951603	0.475801	−	0.879553i	$-0.342158\pi$
$719$	9.42126e29	0.951603	0.475801	+	0.879553i	$0.342158\pi$
$720$	0	0
$721$	−1.23706e29	−0.121358
$722$	−1.91564e29	−0.185214
$723$	7.90868e29	0.753620
$724$	1.24755e30	1.17166
$725$	0	0
$726$	1.32798e29	0.121159
$727$	−9.77996e29	−0.879479	−0.439740	−	0.898125i	$-0.644929\pi$
$727$	−9.77996e29	−0.879479	−0.439740	+	0.898125i	$0.644929\pi$
$728$	−2.60050e29	−0.230503
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−8.73017e28	−0.0741123
$732$	−4.10074e29	−0.343159
$733$	−2.16239e30	−1.78378	−0.891892	−	0.452249i	$-0.850622\pi$
$733$	−2.16239e30	−1.78378	−0.891892	+	0.452249i	$0.850622\pi$
$734$	−2.72850e28	−0.0221878
$735$	0	0
$736$	−4.58618e29	−0.362438
$737$	−4.92760e29	−0.383907
$738$	−1.69792e29	−0.130414
$739$	−2.19604e30	−1.66293	−0.831463	−	0.555580i	$-0.812496\pi$
$739$	−2.19604e30	−1.66293	−0.831463	+	0.555580i	$0.812496\pi$
$740$	0	0
$741$	−1.64111e30	−1.20794
$742$	1.84093e29	0.133597
$743$	−2.05357e30	−1.46936	−0.734679	−	0.678414i	$-0.762667\pi$
$743$	−2.05357e30	−1.46936	−0.734679	+	0.678414i	$0.762667\pi$
$744$	−5.99921e29	−0.423232
$745$	0	0
$746$	−1.79407e29	−0.123050
$747$	5.46104e29	0.369325
$748$	1.27256e29	0.0848619
$749$	2.62886e29	0.172866
$750$	0	0
$751$	9.76058e29	0.624103	0.312052	−	0.950065i	$-0.398984\pi$
$751$	9.76058e29	0.624103	0.312052	+	0.950065i	$0.398984\pi$
$752$	1.54783e30	0.975967
$753$	−1.32111e30	−0.821469
$754$	−5.59986e29	−0.343382
$755$	0	0
$756$	−9.14748e28	−0.0545535
$757$	−1.52530e29	−0.0897113	−0.0448557	−	0.998993i	$-0.514283\pi$
$757$	−1.52530e29	−0.0897113	−0.0448557	+	0.998993i	$0.514283\pi$
$758$	6.35282e29	0.368502
$759$	−2.10617e29	−0.120491
$760$	0	0
$761$	1.43376e30	0.797878	0.398939	−	0.916977i	$-0.369378\pi$
$761$	1.43376e30	0.797878	0.398939	+	0.916977i	$0.369378\pi$
$762$	−1.07714e29	−0.0591212
$763$	1.97410e29	0.106871
$764$	3.14134e30	1.67739
$765$	0	0
$766$	−7.22526e29	−0.375363
$767$	−1.79307e29	−0.0918853
$768$	5.00146e29	0.252816
$769$	−3.06925e30	−1.53040	−0.765202	−	0.643790i	$-0.777361\pi$
$769$	−3.06925e30	−1.53040	−0.765202	+	0.643790i	$0.777361\pi$
$770$	0	0
$771$	−1.43916e30	−0.698293
$772$	−2.52366e30	−1.20795
$773$	−1.11134e30	−0.524760	−0.262380	−	0.964965i	$-0.584507\pi$
$773$	−1.11134e30	−0.524760	−0.262380	+	0.964965i	$0.584507\pi$
$774$	5.74920e28	0.0267810
$775$	0	0
$776$	−1.63139e29	−0.0739621
$777$	4.95848e29	0.221783
$778$	1.92719e29	0.0850431
$779$	4.77663e30	2.07959
$780$	0	0
$781$	1.00548e30	0.426126
$782$	−7.20850e28	−0.0301421
$783$	−4.06929e29	−0.167888
$784$	−1.82712e30	−0.743787
$785$	0	0
$786$	−1.68431e29	−0.0667551
$787$	−2.74489e30	−1.07347	−0.536735	−	0.843751i	$-0.680343\pi$
$787$	−2.74489e30	−1.07347	−0.536735	+	0.843751i	$0.680343\pi$
$788$	2.27010e30	0.876032
$789$	−1.68271e30	−0.640768
$790$	0	0
$791$	−1.14241e30	−0.423611
$792$	−1.73125e29	−0.0633498
$793$	−2.77843e30	−1.00330
$794$	1.27134e30	0.453050
$795$	0	0
$796$	1.73455e29	0.0602004
$797$	1.59912e30	0.547731	0.273866	−	0.961768i	$-0.411698\pi$
$797$	1.59912e30	0.547731	0.273866	+	0.961768i	$0.411698\pi$
$798$	−1.69427e29	−0.0572733
$799$	8.19390e29	0.273369
$800$	0	0
$801$	6.61296e29	0.214909
$802$	1.48591e30	0.476607
$803$	−1.17956e30	−0.373429
$804$	−1.68692e30	−0.527115
$805$	0	0
$806$	−1.96759e30	−0.598987
$807$	−2.30898e29	−0.0693823
$808$	−1.16421e30	−0.345314
$809$	−1.45623e30	−0.426354	−0.213177	−	0.977014i	$-0.568381\pi$
$809$	−1.45623e30	−0.426354	−0.213177	+	0.977014i	$0.568381\pi$
$810$	0	0
$811$	4.66640e29	0.133126	0.0665631	−	0.997782i	$-0.478797\pi$
$811$	4.66640e29	0.133126	0.0665631	+	0.997782i	$0.478797\pi$
$812$	8.78103e29	0.247290
$813$	−2.45117e30	−0.681433
$814$	4.54266e29	0.124668
$815$	0	0
$816$	4.05080e29	0.108341
$817$	−1.61738e30	−0.427051
$818$	−6.64460e29	−0.173205
$819$	−6.19782e29	−0.159499
$820$	0	0
$821$	1.92587e30	0.483084	0.241542	−	0.970390i	$-0.422347\pi$
$821$	1.92587e30	0.483084	0.241542	+	0.970390i	$0.422347\pi$
$822$	1.66761e29	0.0412990
$823$	3.60800e29	0.0882202	0.0441101	−	0.999027i	$-0.485955\pi$
$823$	3.60800e29	0.0882202	0.0441101	+	0.999027i	$0.485955\pi$
$824$	−8.01505e29	−0.193495
$825$	0	0
$826$	−1.85115e28	−0.00435663
$827$	−6.95742e30	−1.61674	−0.808371	−	0.588674i	$-0.799650\pi$
$827$	−6.95742e30	−1.61674	−0.808371	+	0.588674i	$0.799650\pi$
$828$	−7.21029e29	−0.165438
$829$	1.71309e30	0.388112	0.194056	−	0.980990i	$-0.437836\pi$
$829$	1.71309e30	0.388112	0.194056	+	0.980990i	$0.437836\pi$
$830$	0	0
$831$	2.76529e30	0.610844
$832$	4.70650e30	1.02661
$833$	−9.67243e29	−0.208335
$834$	−6.64211e29	−0.141274
$835$	0	0
$836$	2.35759e30	0.488993
$837$	−1.42981e30	−0.292860
$838$	−1.08127e30	−0.218712
$839$	8.66052e30	1.72999	0.864995	−	0.501781i	$-0.167322\pi$
$839$	8.66052e30	1.72999	0.864995	+	0.501781i	$0.167322\pi$
$840$	0	0
$841$	−1.22657e30	−0.238965
$842$	−2.24075e30	−0.431138
$843$	4.83302e30	0.918393
$844$	−3.48310e30	−0.653687
$845$	0	0
$846$	−5.39604e29	−0.0987839
$847$	−1.41090e30	−0.255106
$848$	−8.15426e30	−1.45622
$849$	−4.06440e30	−0.716911
$850$	0	0
$851$	3.90841e30	0.672573
$852$	3.44218e30	0.585083
$853$	−1.10259e31	−1.85118	−0.925588	−	0.378532i	$-0.876429\pi$
$853$	−1.10259e31	−1.85118	−0.925588	+	0.378532i	$0.876429\pi$
$854$	−2.86843e29	−0.0475703
$855$	0	0
$856$	1.70327e30	0.275620
$857$	4.05713e30	0.648516	0.324258	−	0.945969i	$-0.394885\pi$
$857$	4.05713e30	0.648516	0.324258	+	0.945969i	$0.394885\pi$
$858$	−5.67807e29	−0.0896570
$859$	−1.70873e30	−0.266530	−0.133265	−	0.991080i	$-0.542546\pi$
$859$	−1.70873e30	−0.266530	−0.133265	+	0.991080i	$0.542546\pi$
$860$	0	0
$861$	1.80394e30	0.274593
$862$	−2.10235e28	−0.00316140
$863$	−6.61629e30	−0.982882	−0.491441	−	0.870911i	$-0.663530\pi$
$863$	−6.61629e30	−0.982882	−0.491441	+	0.870911i	$0.663530\pi$
$864$	−8.98461e29	−0.131858
$865$	0	0
$866$	−1.00864e30	−0.144477
$867$	−3.86537e30	−0.547004
$868$	3.08534e30	0.431366
$869$	−1.17804e30	−0.162725
$870$	0	0
$871$	−1.14296e31	−1.54113
$872$	1.27904e30	0.170397
$873$	−3.88812e29	−0.0511788
$874$	−1.33547e30	−0.173686
$875$	0	0
$876$	−4.03813e30	−0.512728
$877$	−5.96651e30	−0.748556	−0.374278	−	0.927317i	$-0.622109\pi$
$877$	−5.96651e30	−0.748556	−0.374278	+	0.927317i	$0.622109\pi$
$878$	2.16425e29	0.0268296
$879$	−6.15171e30	−0.753551
$880$	0	0
$881$	4.83453e29	0.0578239	0.0289119	−	0.999582i	$-0.490796\pi$
$881$	4.83453e29	0.0578239	0.0289119	+	0.999582i	$0.490796\pi$
$882$	6.36971e29	0.0752835
$883$	9.20409e30	1.07496	0.537482	−	0.843276i	$-0.319376\pi$
$883$	9.20409e30	1.07496	0.537482	+	0.843276i	$0.319376\pi$
$884$	2.95172e30	0.340664
$885$	0	0
$886$	−3.18082e30	−0.358497
$887$	−6.74053e30	−0.750752	−0.375376	−	0.926873i	$-0.622486\pi$
$887$	−6.74053e30	−0.750752	−0.375376	+	0.926873i	$0.622486\pi$
$888$	3.21267e30	0.353614
$889$	1.14440e30	0.124482
$890$	0	0
$891$	−4.12613e29	−0.0438356
$892$	1.25494e31	1.31763
$893$	1.51803e31	1.57521
$894$	−6.93753e29	−0.0711477
$895$	0	0
$896$	2.55231e30	0.255681
$897$	−4.88529e30	−0.483693
$898$	2.02969e30	0.198622
$899$	1.37253e31	1.32753
$900$	0	0
$901$	−4.31671e30	−0.407889
$902$	1.65266e30	0.154353
$903$	−6.10819e29	−0.0563886
$904$	−7.40180e30	−0.675412
$905$	0	0
$906$	8.16510e29	0.0727973
$907$	−3.81758e30	−0.336443	−0.168221	−	0.985749i	$-0.553802\pi$
$907$	−3.81758e30	−0.336443	−0.168221	+	0.985749i	$0.553802\pi$
$908$	1.13564e31	0.989324
$909$	−2.77470e30	−0.238943
$910$	0	0
$911$	3.19309e30	0.268701	0.134350	−	0.990934i	$-0.457105\pi$
$911$	3.19309e30	0.268701	0.134350	+	0.990934i	$0.457105\pi$
$912$	7.50464e30	0.624286
$913$	−5.31548e30	−0.437119
$914$	−2.67795e30	−0.217704
$915$	0	0
$916$	8.84727e30	0.702922
$917$	1.78948e30	0.140556
$918$	−1.41219e29	−0.0109659
$919$	1.69470e31	1.30101	0.650503	−	0.759504i	$-0.274558\pi$
$919$	1.69470e31	1.30101	0.650503	+	0.759504i	$0.274558\pi$
$920$	0	0
$921$	−7.17107e29	−0.0538093
$922$	1.24404e30	0.0922912
$923$	2.33223e31	1.71062
$924$	8.90367e29	0.0645674
$925$	0	0
$926$	4.41329e30	0.312858
$927$	−1.91024e30	−0.133891
$928$	8.62468e30	0.597708
$929$	2.76272e31	1.89309	0.946546	−	0.322569i	$-0.104546\pi$
$929$	2.76272e31	1.89309	0.946546	+	0.322569i	$0.104546\pi$
$930$	0	0
$931$	−1.79194e31	−1.20047
$932$	2.04789e31	1.35656
$933$	3.02711e30	0.198276
$934$	1.67041e30	0.108189
$935$	0	0
$936$	−4.01565e30	−0.254308
$937$	8.51699e30	0.533360	0.266680	−	0.963785i	$-0.414073\pi$
$937$	8.51699e30	0.533360	0.266680	+	0.963785i	$0.414073\pi$
$938$	−1.17998e30	−0.0730711
$939$	−1.42198e31	−0.870772
$940$	0	0
$941$	1.60192e31	0.959288	0.479644	−	0.877463i	$-0.340766\pi$
$941$	1.60192e31	0.959288	0.479644	+	0.877463i	$0.340766\pi$
$942$	−8.39415e29	−0.0497098
$943$	1.42191e31	0.832723
$944$	8.19953e29	0.0474879
$945$	0	0
$946$	−5.59596e29	−0.0316969
$947$	−9.01245e30	−0.504856	−0.252428	−	0.967616i	$-0.581229\pi$
$947$	−9.01245e30	−0.504856	−0.252428	+	0.967616i	$0.581229\pi$
$948$	−4.03292e30	−0.223425
$949$	−2.73601e31	−1.49907
$950$	0	0
$951$	−1.66432e31	−0.891950
$952$	6.29529e29	0.0333678
$953$	1.14770e30	0.0601664	0.0300832	−	0.999547i	$-0.490423\pi$
$953$	1.14770e30	0.0601664	0.0300832	+	0.999547i	$0.490423\pi$
$954$	2.84274e30	0.147394
$955$	0	0
$956$	−1.94911e31	−0.988619
$957$	3.96083e30	0.198706
$958$	−1.71307e30	−0.0850034
$959$	−1.77174e30	−0.0869568
$960$	0	0
$961$	2.74002e31	1.31570
$962$	1.05367e31	0.500459
$963$	4.05945e30	0.190718
$964$	−2.63530e31	−1.22468
$965$	0	0
$966$	−5.04353e29	−0.0229337
$967$	3.66156e30	0.164698	0.0823489	−	0.996604i	$-0.473758\pi$
$967$	3.66156e30	0.164698	0.0823489	+	0.996604i	$0.473758\pi$
$968$	−9.14141e30	−0.406745
$969$	3.97281e30	0.174863
$970$	0	0
$971$	3.28650e31	1.41557	0.707785	−	0.706428i	$-0.249694\pi$
$971$	3.28650e31	1.41557	0.707785	+	0.706428i	$0.249694\pi$
$972$	−1.41254e30	−0.0601874
$973$	7.05686e30	0.297459
$974$	7.96117e30	0.331977
$975$	0	0
$976$	1.27055e31	0.518523
$977$	3.53328e30	0.142655	0.0713273	−	0.997453i	$-0.477277\pi$
$977$	3.53328e30	0.142655	0.0713273	+	0.997453i	$0.477277\pi$
$978$	6.43331e30	0.256966
$979$	−6.43670e30	−0.254358
$980$	0	0
$981$	3.04838e30	0.117908
$982$	−1.16617e31	−0.446262
$983$	−3.20399e31	−1.21305	−0.606526	−	0.795064i	$-0.707437\pi$
$983$	−3.20399e31	−1.21305	−0.606526	+	0.795064i	$0.707437\pi$
$984$	1.16880e31	0.437815
$985$	0	0
$986$	1.35562e30	0.0497083
$987$	5.73298e30	0.207994
$988$	5.46845e31	1.96298
$989$	−4.81464e30	−0.171003
$990$	0	0
$991$	8.12209e30	0.282419	0.141210	−	0.989980i	$-0.454901\pi$
$991$	8.12209e30	0.282419	0.141210	+	0.989980i	$0.454901\pi$
$992$	3.03041e31	1.04263
$993$	−2.34740e31	−0.799134
$994$	2.40777e30	0.0811069
$995$	0	0
$996$	−1.81971e31	−0.600176
$997$	1.48762e31	0.485504	0.242752	−	0.970088i	$-0.421950\pi$
$997$	1.48762e31	0.485504	0.242752	+	0.970088i	$0.421950\pi$
$998$	7.26116e30	0.234496
$999$	7.65682e30	0.244687

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.m.1.5		10
5.2	odd	4		15.22.b.a.4.9	✓	20
5.3	odd	4		15.22.b.a.4.12	yes	20
5.4	even	2		75.22.a.n.1.6		10
15.2	even	4		45.22.b.d.19.12		20
15.8	even	4		45.22.b.d.19.9		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.b.a.4.9	✓	20	5.2	odd	4
15.22.b.a.4.12	yes	20	5.3	odd	4
45.22.b.d.19.9		20	15.8	even	4
45.22.b.d.19.12		20	15.2	even	4
75.22.a.m.1.5		10	1.1	even	1	trivial
75.22.a.n.1.6		10	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q-359.921 q^{2} +59049.0 q^{3} -1.96761e6 q^{4} -2.12530e7 q^{6} +2.25801e8 q^{7} +1.46299e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q-359.921 q^{2} +59049.0 q^{3} -1.96761e6 q^{4} -2.12530e7 q^{6} +2.25801e8 q^{7} +1.46299e9 q^{8} +3.48678e9 q^{9} -3.39385e10 q^{11} -1.16185e11 q^{12} -7.87206e11 q^{13} -8.12704e10 q^{14} +3.59981e12 q^{16} +1.90567e12 q^{17} -1.25497e12 q^{18} +3.53051e13 q^{19} +1.33333e13 q^{21} +1.22152e13 q^{22} +1.05097e14 q^{23} +8.63883e13 q^{24} +2.83332e14 q^{26} +2.05891e14 q^{27} -4.44287e14 q^{28} -1.97643e15 q^{29} -6.94447e15 q^{31} -4.36377e15 q^{32} -2.00403e15 q^{33} -6.85892e14 q^{34} -6.86063e15 q^{36} +3.71887e16 q^{37} -1.27070e16 q^{38} -4.64837e16 q^{39} +1.35296e17 q^{41} -4.79894e15 q^{42} -4.58115e16 q^{43} +6.67777e16 q^{44} -3.78265e16 q^{46} +4.29974e17 q^{47} +2.12565e17 q^{48} -5.07560e17 q^{49} +1.12528e17 q^{51} +1.54891e18 q^{52} -2.26519e18 q^{53} -7.41046e16 q^{54} +3.30345e17 q^{56} +2.08473e18 q^{57} +7.11359e17 q^{58} +2.27777e17 q^{59} +3.52948e18 q^{61} +2.49946e18 q^{62} +7.87318e17 q^{63} -5.97874e18 q^{64} +7.21294e17 q^{66} +1.45192e19 q^{67} -3.74962e18 q^{68} +6.20586e18 q^{69} -2.96266e19 q^{71} +5.10114e18 q^{72} +3.47560e19 q^{73} -1.33850e19 q^{74} -6.94666e19 q^{76} -7.66333e18 q^{77} +1.67305e19 q^{78} +3.47111e19 q^{79} +1.21577e19 q^{81} -4.86958e19 q^{82} +1.56621e20 q^{83} -2.62347e19 q^{84} +1.64885e19 q^{86} -1.16706e20 q^{87} -4.96518e19 q^{88} +1.89658e20 q^{89} -1.77752e20 q^{91} -2.06789e20 q^{92} -4.10064e20 q^{93} -1.54757e20 q^{94} -2.57676e20 q^{96} -1.11510e20 q^{97} +1.82682e20 q^{98} -1.18336e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 645 q^{2} + 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} - 115357290 q^{7} - 314142735 q^{8} + 34867844010 q^{9}+O(q^{10}) \) 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

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