LMFDB - Embedded newform 75.22.a.m.1.9

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			1.9
Root			$2403.38$ 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)

$f(q)$	$=$	$q+2338.38 q^{2} +59049.0 q^{3} +3.37088e6 q^{4} +1.38079e8 q^{6} -9.66595e8 q^{7} +2.97845e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$	\(q+2338.38 q^{2} +59049.0 q^{3} +3.37088e6 q^{4} +1.38079e8 q^{6} -9.66595e8 q^{7} +2.97845e9 q^{8} +3.48678e9 q^{9} +5.19141e10 q^{11} +1.99047e11 q^{12} -2.70939e11 q^{13} -2.26027e12 q^{14} -1.04480e11 q^{16} +6.96562e12 q^{17} +8.15343e12 q^{18} +1.69301e13 q^{19} -5.70764e13 q^{21} +1.21395e14 q^{22} -2.26725e14 q^{23} +1.75875e14 q^{24} -6.33559e14 q^{26} +2.05891e14 q^{27} -3.25827e15 q^{28} +1.87885e15 q^{29} -7.51747e15 q^{31} -6.49058e15 q^{32} +3.06547e15 q^{33} +1.62883e16 q^{34} +1.17535e16 q^{36} +4.96590e15 q^{37} +3.95890e16 q^{38} -1.59987e16 q^{39} +9.84771e16 q^{41} -1.33466e17 q^{42} -1.09759e17 q^{43} +1.74996e17 q^{44} -5.30170e17 q^{46} -6.58224e17 q^{47} -6.16943e15 q^{48} +3.75759e17 q^{49} +4.11313e17 q^{51} -9.13302e17 q^{52} -1.50241e18 q^{53} +4.81452e17 q^{54} -2.87896e18 q^{56} +9.99704e17 q^{57} +4.39347e18 q^{58} +3.23685e18 q^{59} -5.86659e18 q^{61} -1.75787e19 q^{62} -3.37031e18 q^{63} -1.49583e19 q^{64} +7.16825e18 q^{66} -2.13011e19 q^{67} +2.34802e19 q^{68} -1.33879e19 q^{69} -2.84783e19 q^{71} +1.03852e19 q^{72} -5.58032e19 q^{73} +1.16122e19 q^{74} +5.70692e19 q^{76} -5.01798e19 q^{77} -3.74110e19 q^{78} +8.86230e19 q^{79} +1.21577e19 q^{81} +2.30277e20 q^{82} -1.63666e20 q^{83} -1.92398e20 q^{84} -2.56658e20 q^{86} +1.10944e20 q^{87} +1.54624e20 q^{88} -1.12975e20 q^{89} +2.61888e20 q^{91} -7.64263e20 q^{92} -4.43899e20 q^{93} -1.53918e21 q^{94} -3.83262e20 q^{96} +6.72145e20 q^{97} +8.78668e20 q^{98} +1.81013e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2338.38	1.61473	0.807366	−	0.590051i	$-0.200892\pi$
$2$	2338.38	1.61473	0.807366	+	0.590051i	$0.200892\pi$
$3$	59049.0	0.577350
$3$	59049.0	0.577350
$4$	3.37088e6	1.60736
$5$	0	0
$5$	0	0
$6$	1.38079e8	0.932266
$7$	−9.66595e8	−1.29335	−0.646673	−	0.762767i	$-0.723840\pi$
$7$	−9.66595e8	−1.29335	−0.646673	+	0.762767i	$0.723840\pi$
$8$	2.97845e9	0.980722
$9$	3.48678e9	0.333333
$10$	0	0
$11$	5.19141e10	0.603478	0.301739	−	0.953390i	$-0.402433\pi$
$11$	5.19141e10	0.603478	0.301739	+	0.953390i	$0.402433\pi$
$12$	1.99047e11	0.928009
$13$	−2.70939e11	−0.545088	−0.272544	−	0.962143i	$-0.587865\pi$
$13$	−2.70939e11	−0.545088	−0.272544	+	0.962143i	$0.587865\pi$
$14$	−2.26027e12	−2.08841
$15$	0	0
$16$	−1.04480e11	−0.0237560
$17$	6.96562e12	0.838004	0.419002	−	0.907985i	$-0.362380\pi$
$17$	6.96562e12	0.838004	0.419002	+	0.907985i	$0.362380\pi$
$18$	8.15343e12	0.538244
$19$	1.69301e13	0.633499	0.316750	−	0.948509i	$-0.397409\pi$
$19$	1.69301e13	0.633499	0.316750	+	0.948509i	$0.397409\pi$
$20$	0	0
$21$	−5.70764e13	−0.746714
$22$	1.21395e14	0.974456
$23$	−2.26725e14	−1.14119	−0.570595	−	0.821232i	$-0.693287\pi$
$23$	−2.26725e14	−1.14119	−0.570595	+	0.821232i	$0.693287\pi$
$24$	1.75875e14	0.566220
$25$	0	0
$26$	−6.33559e14	−0.880171
$27$	2.05891e14	0.192450
$28$	−3.25827e15	−2.07887
$29$	1.87885e15	0.829304	0.414652	−	0.909980i	$-0.363903\pi$
$29$	1.87885e15	0.829304	0.414652	+	0.909980i	$0.363903\pi$
$30$	0	0
$31$	−7.51747e15	−1.64730	−0.823652	−	0.567095i	$-0.808067\pi$
$31$	−7.51747e15	−1.64730	−0.823652	+	0.567095i	$0.808067\pi$
$32$	−6.49058e15	−1.01908
$33$	3.06547e15	0.348418
$34$	1.62883e16	1.35315
$35$	0	0
$36$	1.17535e16	0.535786
$37$	4.96590e15	0.169777	0.0848887	−	0.996390i	$-0.472947\pi$
$37$	4.96590e15	0.169777	0.0848887	+	0.996390i	$0.472947\pi$
$38$	3.95890e16	1.02293
$39$	−1.59987e16	−0.314707
$40$	0	0
$41$	9.84771e16	1.14579	0.572895	−	0.819629i	$-0.305820\pi$
$41$	9.84771e16	1.14579	0.572895	+	0.819629i	$0.305820\pi$
$42$	−1.33466e17	−1.20574
$43$	−1.09759e17	−0.774498	−0.387249	−	0.921975i	$-0.626575\pi$
$43$	−1.09759e17	−0.774498	−0.387249	+	0.921975i	$0.626575\pi$
$44$	1.74996e17	0.970007
$45$	0	0
$46$	−5.30170e17	−1.84272
$47$	−6.58224e17	−1.82535	−0.912676	−	0.408685i	$-0.865988\pi$
$47$	−6.58224e17	−1.82535	−0.912676	+	0.408685i	$0.865988\pi$
$48$	−6.16943e15	−0.0137155
$49$	3.75759e17	0.672745
$50$	0	0
$51$	4.11313e17	0.483822
$52$	−9.13302e17	−0.876152
$53$	−1.50241e18	−1.18003	−0.590013	−	0.807393i	$-0.700878\pi$
$53$	−1.50241e18	−1.18003	−0.590013	+	0.807393i	$0.700878\pi$
$54$	4.81452e17	0.310755
$55$	0	0
$56$	−2.87896e18	−1.26841
$57$	9.99704e17	0.365751
$58$	4.39347e18	1.33910
$59$	3.23685e18	0.824473	0.412236	−	0.911077i	$-0.364748\pi$
$59$	3.23685e18	0.824473	0.412236	+	0.911077i	$0.364748\pi$
$60$	0	0
$61$	−5.86659e18	−1.05299	−0.526493	−	0.850180i	$-0.676493\pi$
$61$	−5.86659e18	−1.05299	−0.526493	+	0.850180i	$0.676493\pi$
$62$	−1.75787e19	−2.65996
$63$	−3.37031e18	−0.431116
$64$	−1.49583e19	−1.62179
$65$	0	0
$66$	7.16825e18	0.562602
$67$	−2.13011e19	−1.42763	−0.713815	−	0.700334i	$-0.753034\pi$
$67$	−2.13011e19	−1.42763	−0.713815	+	0.700334i	$0.753034\pi$
$68$	2.34802e19	1.34697
$69$	−1.33879e19	−0.658866
$70$	0	0
$71$	−2.84783e19	−1.03825	−0.519124	−	0.854699i	$-0.673742\pi$
$71$	−2.84783e19	−1.03825	−0.519124	+	0.854699i	$0.673742\pi$
$72$	1.03852e19	0.326907
$73$	−5.58032e19	−1.51974	−0.759869	−	0.650076i	$-0.774737\pi$
$73$	−5.58032e19	−1.51974	−0.759869	+	0.650076i	$0.774737\pi$
$74$	1.16122e19	0.274145
$75$	0	0
$76$	5.70692e19	1.01826
$77$	−5.01798e19	−0.780507
$78$	−3.74110e19	−0.508167
$79$	8.86230e19	1.05308	0.526541	−	0.850150i	$-0.323489\pi$
$79$	8.86230e19	1.05308	0.526541	+	0.850150i	$0.323489\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	2.30277e20	1.85014
$83$	−1.63666e20	−1.15782	−0.578908	−	0.815393i	$-0.696521\pi$
$83$	−1.63666e20	−1.15782	−0.578908	+	0.815393i	$0.696521\pi$
$84$	−1.92398e20	−1.20024
$85$	0	0
$86$	−2.56658e20	−1.25061
$87$	1.10944e20	0.478799
$88$	1.54624e20	0.591845
$89$	−1.12975e20	−0.384049	−0.192024	−	0.981390i	$-0.561505\pi$
$89$	−1.12975e20	−0.384049	−0.192024	+	0.981390i	$0.561505\pi$
$90$	0	0
$91$	2.61888e20	0.704988
$92$	−7.64263e20	−1.83430
$93$	−4.43899e20	−0.951072
$94$	−1.53918e21	−2.94745
$95$	0	0
$96$	−3.83262e20	−0.588367
$97$	6.72145e20	0.925464	0.462732	−	0.886498i	$-0.346869\pi$
$97$	6.72145e20	0.925464	0.462732	+	0.886498i	$0.346869\pi$
$98$	8.78668e20	1.08630
$99$	1.81013e20	0.201159
$100$	0	0
$101$	−1.70396e21	−1.53492	−0.767461	−	0.641096i	$-0.778480\pi$
$101$	−1.70396e21	−1.53492	−0.767461	+	0.641096i	$0.778480\pi$
$102$	9.61807e20	0.781243
$103$	2.45991e21	1.80355	0.901774	−	0.432207i	$-0.142265\pi$
$103$	2.45991e21	1.80355	0.901774	+	0.432207i	$0.142265\pi$
$104$	−8.06980e20	−0.534580
$105$	0	0
$106$	−3.51321e21	−1.90543
$107$	2.84925e21	1.40024	0.700118	−	0.714027i	$-0.253131\pi$
$107$	2.84925e21	1.40024	0.700118	+	0.714027i	$0.253131\pi$
$108$	6.94034e20	0.309336
$109$	1.04517e21	0.422872	0.211436	−	0.977392i	$-0.432186\pi$
$109$	1.04517e21	0.422872	0.211436	+	0.977392i	$0.432186\pi$
$110$	0	0
$111$	2.93231e20	0.0980210
$112$	1.00990e20	0.0307247
$113$	−5.85624e21	−1.62291	−0.811456	−	0.584413i	$-0.801325\pi$
$113$	−5.85624e21	−1.62291	−0.811456	+	0.584413i	$0.801325\pi$
$114$	2.33769e21	0.590590
$115$	0	0
$116$	6.33338e21	1.33299
$117$	−9.44706e20	−0.181696
$118$	7.56899e21	1.33130
$119$	−6.73293e21	−1.08383
$120$	0	0
$121$	−4.70518e21	−0.635814
$122$	−1.37183e22	−1.70029
$123$	5.81498e21	0.661522
$124$	−2.53405e22	−2.64781
$125$	0	0
$126$	−7.88106e21	−0.696136
$127$	−5.67060e21	−0.460988	−0.230494	−	0.973074i	$-0.574034\pi$
$127$	−5.67060e21	−0.460988	−0.230494	+	0.973074i	$0.574034\pi$
$128$	−2.13666e22	−1.59967
$129$	−6.48115e21	−0.447157
$130$	0	0
$131$	1.75652e22	1.03111	0.515553	−	0.856858i	$-0.327587\pi$
$131$	1.75652e22	1.03111	0.515553	+	0.856858i	$0.327587\pi$
$132$	1.03333e22	0.560034
$133$	−1.63645e22	−0.819334
$134$	−4.98100e22	−2.30524
$135$	0	0
$136$	2.07468e22	0.821849
$137$	5.76804e21	0.211574	0.105787	−	0.994389i	$-0.466264\pi$
$137$	5.76804e21	0.211574	0.105787	+	0.994389i	$0.466264\pi$
$138$	−3.13060e22	−1.06389
$139$	−4.14279e22	−1.30508	−0.652541	−	0.757753i	$-0.726297\pi$
$139$	−4.14279e22	−1.30508	−0.652541	+	0.757753i	$0.726297\pi$
$140$	0	0
$141$	−3.88675e22	−1.05387
$142$	−6.65931e22	−1.67649
$143$	−1.40656e22	−0.328949
$144$	−3.64299e20	−0.00791865
$145$	0	0
$146$	−1.30489e23	−2.45397
$147$	2.21882e22	0.388410
$148$	1.67394e22	0.272893
$149$	3.92046e22	0.595500	0.297750	−	0.954644i	$-0.403764\pi$
$149$	3.92046e22	0.595500	0.297750	+	0.954644i	$0.403764\pi$
$150$	0	0
$151$	1.08828e23	1.43708	0.718542	−	0.695484i	$-0.244810\pi$
$151$	1.08828e23	1.43708	0.718542	+	0.695484i	$0.244810\pi$
$152$	5.04254e22	0.621287
$153$	2.42876e22	0.279335
$154$	−1.17340e23	−1.26031
$155$	0	0
$156$	−5.39296e22	−0.505847
$157$	3.63587e22	0.318906	0.159453	−	0.987206i	$-0.449027\pi$
$157$	3.63587e22	0.318906	0.159453	+	0.987206i	$0.449027\pi$
$158$	2.07234e23	1.70044
$159$	−8.87158e22	−0.681289
$160$	0	0
$161$	2.19151e23	1.47595
$162$	2.84293e22	0.179415
$163$	9.55955e22	0.565546	0.282773	−	0.959187i	$-0.408746\pi$
$163$	9.55955e22	0.565546	0.282773	+	0.959187i	$0.408746\pi$
$164$	3.31954e23	1.84170
$165$	0	0
$166$	−3.82715e23	−1.86956
$167$	1.03344e23	0.473984	0.236992	−	0.971512i	$-0.423838\pi$
$167$	1.03344e23	0.473984	0.236992	+	0.971512i	$0.423838\pi$
$168$	−1.70000e23	−0.732319
$169$	−1.73657e23	−0.702879
$170$	0	0
$171$	5.90315e22	0.211166
$172$	−3.69983e23	−1.24490
$173$	1.53983e23	0.487517	0.243759	−	0.969836i	$-0.421620\pi$
$173$	1.53983e23	0.487517	0.243759	+	0.969836i	$0.421620\pi$
$174$	2.59430e23	0.773131
$175$	0	0
$176$	−5.42397e21	−0.0143362
$177$	1.91133e23	0.476010
$178$	−2.64178e23	−0.620135
$179$	−2.16487e22	−0.0479153	−0.0239577	−	0.999713i	$-0.507627\pi$
$179$	−2.16487e22	−0.0479153	−0.0239577	+	0.999713i	$0.507627\pi$
$180$	0	0
$181$	−2.32732e23	−0.458386	−0.229193	−	0.973381i	$-0.573609\pi$
$181$	−2.32732e23	−0.458386	−0.229193	+	0.973381i	$0.573609\pi$
$182$	6.12395e23	1.13837
$183$	−3.46416e23	−0.607941
$184$	−6.75291e23	−1.11919
$185$	0	0
$186$	−1.03801e24	−1.53573
$187$	3.61614e23	0.505717
$188$	−2.21879e24	−2.93400
$189$	−1.99013e23	−0.248905
$190$	0	0
$191$	1.70987e24	1.91475	0.957376	−	0.288844i	$-0.0932707\pi$
$191$	1.70987e24	1.91475	0.957376	+	0.288844i	$0.0932707\pi$
$192$	−8.83276e23	−0.936339
$193$	6.58890e23	0.661395	0.330698	−	0.943737i	$-0.392716\pi$
$193$	6.58890e23	0.661395	0.330698	+	0.943737i	$0.392716\pi$
$194$	1.57173e24	1.49438
$195$	0	0
$196$	1.26664e24	1.08134
$197$	1.08836e23	0.0880799	0.0440399	−	0.999030i	$-0.485977\pi$
$197$	1.08836e23	0.0880799	0.0440399	+	0.999030i	$0.485977\pi$
$198$	4.23278e23	0.324819
$199$	2.08493e24	1.51752	0.758760	−	0.651370i	$-0.225805\pi$
$199$	2.08493e24	1.51752	0.758760	+	0.651370i	$0.225805\pi$
$200$	0	0
$201$	−1.25781e24	−0.824242
$202$	−3.98452e24	−2.47849
$203$	−1.81609e24	−1.07258
$204$	1.38649e24	0.777676
$205$	0	0
$206$	5.75220e24	2.91225
$207$	−7.90542e23	−0.380397
$208$	2.83077e22	0.0129491
$209$	8.78909e23	0.382303
$210$	0	0
$211$	4.35815e23	0.171529	0.0857643	−	0.996315i	$-0.472667\pi$
$211$	4.35815e23	0.171529	0.0857643	+	0.996315i	$0.472667\pi$
$212$	−5.06444e24	−1.89673
$213$	−1.68161e24	−0.599433
$214$	6.66264e24	2.26100
$215$	0	0
$216$	6.13237e23	0.188740
$217$	7.26635e24	2.13054
$218$	2.44401e24	0.682824
$219$	−3.29512e24	−0.877421
$220$	0	0
$221$	−1.88726e24	−0.456786
$222$	6.85687e23	0.158278
$223$	−6.07649e24	−1.33799	−0.668993	−	0.743268i	$-0.733274\pi$
$223$	−6.07649e24	−1.33799	−0.668993	+	0.743268i	$0.733274\pi$
$224$	6.27376e24	1.31803
$225$	0	0
$226$	−1.36941e25	−2.62057
$227$	−2.80610e24	−0.512662	−0.256331	−	0.966589i	$-0.582514\pi$
$227$	−2.80610e24	−0.512662	−0.256331	+	0.966589i	$0.582514\pi$
$228$	3.36988e24	0.587893
$229$	−1.43777e24	−0.239561	−0.119780	−	0.992800i	$-0.538219\pi$
$229$	−1.43777e24	−0.239561	−0.119780	+	0.992800i	$0.538219\pi$
$230$	0	0
$231$	−2.96307e24	−0.450626
$232$	5.59607e24	0.813316
$233$	−1.45034e24	−0.201480	−0.100740	−	0.994913i	$-0.532121\pi$
$233$	−1.45034e24	−0.201480	−0.100740	+	0.994913i	$0.532121\pi$
$234$	−2.20908e24	−0.293390
$235$	0	0
$236$	1.09110e25	1.32522
$237$	5.23310e24	0.607997
$238$	−1.57442e25	−1.75009
$239$	−2.77647e24	−0.295336	−0.147668	−	0.989037i	$-0.547177\pi$
$239$	−2.77647e24	−0.295336	−0.147668	+	0.989037i	$0.547177\pi$
$240$	0	0
$241$	9.00139e24	0.877265	0.438632	−	0.898667i	$-0.355463\pi$
$241$	9.00139e24	0.877265	0.438632	+	0.898667i	$0.355463\pi$
$242$	−1.10025e25	−1.02667
$243$	7.17898e23	0.0641500
$244$	−1.97755e25	−1.69253
$245$	0	0
$246$	1.35976e25	1.06818
$247$	−4.58702e24	−0.345313
$248$	−2.23904e25	−1.61555
$249$	−9.66434e24	−0.668465
$250$	0	0
$251$	−4.84540e24	−0.308145	−0.154073	−	0.988060i	$-0.549239\pi$
$251$	−4.84540e24	−0.308145	−0.154073	+	0.988060i	$0.549239\pi$
$252$	−1.13609e25	−0.692957
$253$	−1.17702e25	−0.688683
$254$	−1.32600e25	−0.744372
$255$	0	0
$256$	−1.85933e25	−0.961251
$257$	−7.26766e24	−0.360659	−0.180330	−	0.983606i	$-0.557716\pi$
$257$	−7.26766e24	−0.360659	−0.180330	+	0.983606i	$0.557716\pi$
$258$	−1.51554e25	−0.722038
$259$	−4.80001e24	−0.219581
$260$	0	0
$261$	6.55115e24	0.276435
$262$	4.10740e25	1.66496
$263$	1.74397e25	0.679211	0.339606	−	0.940568i	$-0.389706\pi$
$263$	1.74397e25	0.679211	0.339606	+	0.940568i	$0.389706\pi$
$264$	9.13037e24	0.341702
$265$	0	0
$266$	−3.82665e25	−1.32300
$267$	−6.67104e24	−0.221731
$268$	−7.18032e25	−2.29471
$269$	−4.81212e25	−1.47890	−0.739448	−	0.673213i	$-0.764914\pi$
$269$	−4.81212e25	−1.47890	−0.739448	+	0.673213i	$0.764914\pi$
$270$	0	0
$271$	4.28400e24	0.121807	0.0609035	−	0.998144i	$-0.480602\pi$
$271$	4.28400e24	0.121807	0.0609035	+	0.998144i	$0.480602\pi$
$272$	−7.27767e23	−0.0199076
$273$	1.54642e25	0.407025
$274$	1.34879e25	0.341635
$275$	0	0
$276$	−4.51290e25	−1.05903
$277$	−3.95364e25	−0.893222	−0.446611	−	0.894728i	$-0.647369\pi$
$277$	−3.95364e25	−0.893222	−0.446611	+	0.894728i	$0.647369\pi$
$278$	−9.68743e25	−2.10736
$279$	−2.62118e25	−0.549102
$280$	0	0
$281$	−4.85860e25	−0.944266	−0.472133	−	0.881527i	$-0.656516\pi$
$281$	−4.85860e25	−0.944266	−0.472133	+	0.881527i	$0.656516\pi$
$282$	−9.08870e25	−1.70171
$283$	3.98185e25	0.718335	0.359168	−	0.933273i	$-0.383061\pi$
$283$	3.98185e25	0.718335	0.359168	+	0.933273i	$0.383061\pi$
$284$	−9.59967e25	−1.66884
$285$	0	0
$286$	−3.28906e25	−0.531164
$287$	−9.51875e25	−1.48190
$288$	−2.26313e25	−0.339694
$289$	−2.05721e25	−0.297749
$290$	0	0
$291$	3.96895e25	0.534317
$292$	−1.88106e26	−2.44276
$293$	−5.02711e25	−0.629808	−0.314904	−	0.949123i	$-0.601972\pi$
$293$	−5.02711e25	−0.629808	−0.314904	+	0.949123i	$0.601972\pi$
$294$	5.18845e25	0.627178
$295$	0	0
$296$	1.47907e25	0.166504
$297$	1.06886e25	0.116139
$298$	9.16754e25	0.961573
$299$	6.14288e25	0.622049
$300$	0	0
$301$	1.06092e26	1.00169
$302$	2.54481e26	2.32050
$303$	−1.00617e26	−0.886187
$304$	−1.76885e24	−0.0150494
$305$	0	0
$306$	5.67937e25	0.451051
$307$	1.98117e26	1.52044	0.760218	−	0.649668i	$-0.225092\pi$
$307$	1.98117e26	1.52044	0.760218	+	0.649668i	$0.225092\pi$
$308$	−1.69150e26	−1.25455
$309$	1.45255e26	1.04128
$310$	0	0
$311$	−2.53610e26	−1.69896	−0.849480	−	0.527621i	$-0.823084\pi$
$311$	−2.53610e26	−1.69896	−0.849480	+	0.527621i	$0.823084\pi$
$312$	−4.76513e25	−0.308640
$313$	−7.37569e25	−0.461942	−0.230971	−	0.972961i	$-0.574190\pi$
$313$	−7.37569e25	−0.461942	−0.230971	+	0.972961i	$0.574190\pi$
$314$	8.50205e25	0.514947
$315$	0	0
$316$	2.98737e26	1.69268
$317$	−1.82634e26	−1.00106	−0.500528	−	0.865720i	$-0.666861\pi$
$317$	−1.82634e26	−1.00106	−0.500528	+	0.865720i	$0.666861\pi$
$318$	−2.07451e26	−1.10010
$319$	9.75388e25	0.500467
$320$	0	0
$321$	1.68245e26	0.808426
$322$	5.12460e26	2.38327
$323$	1.17928e26	0.530875
$324$	4.09820e25	0.178595
$325$	0	0
$326$	2.23539e26	0.913205
$327$	6.17162e25	0.244145
$328$	2.93310e26	1.12370
$329$	6.36236e26	2.36081
$330$	0	0
$331$	2.23519e26	0.778254	0.389127	−	0.921184i	$-0.372777\pi$
$331$	2.23519e26	0.778254	0.389127	+	0.921184i	$0.372777\pi$
$332$	−5.51699e26	−1.86103
$333$	1.73150e25	0.0565925
$334$	2.41658e26	0.765357
$335$	0	0
$336$	5.96334e24	0.0177389
$337$	5.48760e26	1.58222	0.791112	−	0.611671i	$-0.209502\pi$
$337$	5.48760e26	1.58222	0.791112	+	0.611671i	$0.209502\pi$
$338$	−4.06075e26	−1.13496
$339$	−3.45805e26	−0.936989
$340$	0	0
$341$	−3.90262e26	−0.994113
$342$	1.38038e26	0.340977
$343$	1.76681e26	0.423254
$344$	−3.26911e26	−0.759568
$345$	0	0
$346$	3.60072e26	0.787209
$347$	−3.29409e26	−0.698676	−0.349338	−	0.936997i	$-0.613593\pi$
$347$	−3.29409e26	−0.698676	−0.349338	+	0.936997i	$0.613593\pi$
$348$	3.73980e26	0.769601
$349$	−4.76133e26	−0.950740	−0.475370	−	0.879786i	$-0.657686\pi$
$349$	−4.76133e26	−0.950740	−0.475370	+	0.879786i	$0.657686\pi$
$350$	0	0
$351$	−5.57840e25	−0.104902
$352$	−3.36952e26	−0.614994
$353$	6.43990e26	1.14089	0.570446	−	0.821335i	$-0.306770\pi$
$353$	6.43990e26	1.14089	0.570446	+	0.821335i	$0.306770\pi$
$354$	4.46941e26	0.768628
$355$	0	0
$356$	−3.80824e26	−0.617304
$357$	−3.97573e26	−0.625749
$358$	−5.06229e25	−0.0773704
$359$	1.03268e26	0.153276	0.0766381	−	0.997059i	$-0.475581\pi$
$359$	1.03268e26	0.153276	0.0766381	+	0.997059i	$0.475581\pi$
$360$	0	0
$361$	−4.27582e26	−0.598679
$362$	−5.44217e26	−0.740171
$363$	−2.77836e26	−0.367087
$364$	8.82793e26	1.13317
$365$	0	0
$366$	−8.10053e26	−0.981662
$367$	−6.62148e25	−0.0779760	−0.0389880	−	0.999240i	$-0.512413\pi$
$367$	−6.62148e25	−0.0779760	−0.0389880	+	0.999240i	$0.512413\pi$
$368$	2.36882e25	0.0271101
$369$	3.43369e26	0.381930
$370$	0	0
$371$	1.45222e27	1.52618
$372$	−1.49633e27	−1.52871
$373$	6.16652e26	0.612488	0.306244	−	0.951953i	$-0.400928\pi$
$373$	6.16652e26	0.612488	0.306244	+	0.951953i	$0.400928\pi$
$374$	8.45591e26	0.816598
$375$	0	0
$376$	−1.96049e27	−1.79016
$377$	−5.09054e26	−0.452043
$378$	−4.65369e26	−0.401914
$379$	−8.59079e26	−0.721641	−0.360821	−	0.932635i	$-0.617503\pi$
$379$	−8.59079e26	−0.721641	−0.360821	+	0.932635i	$0.617503\pi$
$380$	0	0
$381$	−3.34843e26	−0.266152
$382$	3.99833e27	3.09181
$383$	−1.31838e27	−0.991867	−0.495933	−	0.868361i	$-0.665174\pi$
$383$	−1.31838e27	−0.991867	−0.495933	+	0.868361i	$0.665174\pi$
$384$	−1.26168e27	−0.923570
$385$	0	0
$386$	1.54074e27	1.06798
$387$	−3.82705e26	−0.258166
$388$	2.26572e27	1.48755
$389$	−1.98680e27	−1.26965	−0.634825	−	0.772656i	$-0.718928\pi$
$389$	−1.98680e27	−1.26965	−0.634825	+	0.772656i	$0.718928\pi$
$390$	0	0
$391$	−1.57928e27	−0.956322
$392$	1.11918e27	0.659776
$393$	1.03720e27	0.595309
$394$	2.54500e26	0.142225
$395$	0	0
$396$	6.10173e26	0.323336
$397$	2.98271e27	1.53926	0.769628	−	0.638492i	$-0.220442\pi$
$397$	2.98271e27	1.53926	0.769628	+	0.638492i	$0.220442\pi$
$398$	4.87537e27	2.45039
$399$	−9.66308e26	−0.473043
$400$	0	0
$401$	2.13996e27	0.994009	0.497004	−	0.867748i	$-0.334433\pi$
$401$	2.13996e27	0.994009	0.497004	+	0.867748i	$0.334433\pi$
$402$	−2.94123e27	−1.33093
$403$	2.03678e27	0.897926
$404$	−5.74385e27	−2.46717
$405$	0	0
$406$	−4.24671e27	−1.73192
$407$	2.57800e26	0.102457
$408$	1.22508e27	0.474495
$409$	−2.30885e27	−0.871569	−0.435785	−	0.900051i	$-0.643529\pi$
$409$	−2.30885e27	−0.871569	−0.435785	+	0.900051i	$0.643529\pi$
$410$	0	0
$411$	3.40597e26	0.122152
$412$	8.29204e27	2.89895
$413$	−3.12872e27	−1.06633
$414$	−1.84859e27	−0.614238
$415$	0	0
$416$	1.75855e27	0.555489
$417$	−2.44628e27	−0.753489
$418$	2.05522e27	0.617317
$419$	9.75473e26	0.285738	0.142869	−	0.989742i	$-0.454367\pi$
$419$	9.75473e26	0.285738	0.142869	+	0.989742i	$0.454367\pi$
$420$	0	0
$421$	3.92709e27	1.09423	0.547116	−	0.837057i	$-0.315726\pi$
$421$	3.92709e27	1.09423	0.547116	+	0.837057i	$0.315726\pi$
$422$	1.01910e27	0.276973
$423$	−2.29509e27	−0.608450
$424$	−4.47486e27	−1.15728
$425$	0	0
$426$	−3.93225e27	−0.967923
$427$	5.67061e27	1.36187
$428$	9.60447e27	2.25068
$429$	−8.30557e26	−0.189919
$430$	0	0
$431$	3.24576e26	0.0706815	0.0353407	−	0.999375i	$-0.488748\pi$
$431$	3.24576e26	0.0706815	0.0353407	+	0.999375i	$0.488748\pi$
$432$	−2.15115e25	−0.00457184
$433$	5.07014e26	0.105171	0.0525856	−	0.998616i	$-0.483254\pi$
$433$	5.07014e26	0.105171	0.0525856	+	0.998616i	$0.483254\pi$
$434$	1.69915e28	3.44024
$435$	0	0
$436$	3.52314e27	0.679707
$437$	−3.83848e27	−0.722943
$438$	−7.70525e27	−1.41680
$439$	7.50424e27	1.34719	0.673596	−	0.739100i	$-0.264749\pi$
$439$	7.50424e27	1.34719	0.673596	+	0.739100i	$0.264749\pi$
$440$	0	0
$441$	1.31019e27	0.224248
$442$	−4.41313e27	−0.737587
$443$	−4.85179e27	−0.791887	−0.395943	−	0.918275i	$-0.629582\pi$
$443$	−4.85179e27	−0.791887	−0.395943	+	0.918275i	$0.629582\pi$
$444$	9.88447e26	0.157555
$445$	0	0
$446$	−1.42092e28	−2.16049
$447$	2.31499e27	0.343812
$448$	1.44587e28	2.09753
$449$	−5.09745e27	−0.722381	−0.361190	−	0.932492i	$-0.617630\pi$
$449$	−5.09745e27	−0.722381	−0.361190	+	0.932492i	$0.617630\pi$
$450$	0	0
$451$	5.11235e27	0.691459
$452$	−1.97407e28	−2.60860
$453$	6.42617e27	0.829700
$454$	−6.56172e27	−0.827811
$455$	0	0
$456$	2.97757e27	0.358700
$457$	−1.49018e27	−0.175436	−0.0877180	−	0.996145i	$-0.527957\pi$
$457$	−1.49018e27	−0.175436	−0.0877180	+	0.996145i	$0.527957\pi$
$458$	−3.36205e27	−0.386827
$459$	1.43416e27	0.161274
$460$	0	0
$461$	1.56660e28	1.68306	0.841529	−	0.540212i	$-0.181656\pi$
$461$	1.56660e28	1.68306	0.841529	+	0.540212i	$0.181656\pi$
$462$	−6.92879e27	−0.727640
$463$	−8.00082e27	−0.821361	−0.410680	−	0.911779i	$-0.634709\pi$
$463$	−8.00082e27	−0.821361	−0.410680	+	0.911779i	$0.634709\pi$
$464$	−1.96302e26	−0.0197009
$465$	0	0
$466$	−3.39145e27	−0.325337
$467$	−8.80606e27	−0.825951	−0.412976	−	0.910742i	$-0.635511\pi$
$467$	−8.80606e27	−0.825951	−0.412976	+	0.910742i	$0.635511\pi$
$468$	−3.18449e27	−0.292051
$469$	2.05895e28	1.84642
$470$	0	0
$471$	2.14694e27	0.184120
$472$	9.64080e27	0.808579
$473$	−5.69802e27	−0.467393
$474$	1.22370e28	0.981752
$475$	0	0
$476$	−2.26959e28	−1.74210
$477$	−5.23858e27	−0.393342
$478$	−6.49246e27	−0.476888
$479$	1.77353e28	1.27443	0.637214	−	0.770687i	$-0.280087\pi$
$479$	1.77353e28	1.27443	0.637214	+	0.770687i	$0.280087\pi$
$480$	0	0
$481$	−1.34546e27	−0.0925436
$482$	2.10487e28	1.41655
$483$	1.29407e28	0.852142
$484$	−1.58606e28	−1.02198
$485$	0	0
$486$	1.67872e27	0.103585
$487$	1.29243e28	0.780467	0.390234	−	0.920716i	$-0.372394\pi$
$487$	1.29243e28	0.780467	0.390234	+	0.920716i	$0.372394\pi$
$488$	−1.74734e28	−1.03269
$489$	5.64482e27	0.326518
$490$	0	0
$491$	2.91624e28	1.61610	0.808048	−	0.589117i	$-0.200524\pi$
$491$	2.91624e28	1.61610	0.808048	+	0.589117i	$0.200524\pi$
$492$	1.96016e28	1.06330
$493$	1.30874e28	0.694960
$494$	−1.07262e28	−0.557588
$495$	0	0
$496$	7.85424e26	0.0391333
$497$	2.75269e28	1.34281
$498$	−2.25989e28	−1.07939
$499$	1.37844e28	0.644662	0.322331	−	0.946627i	$-0.395534\pi$
$499$	1.37844e28	0.644662	0.322331	+	0.946627i	$0.395534\pi$
$500$	0	0
$501$	6.10238e27	0.273655
$502$	−1.13304e28	−0.497572
$503$	−1.31132e28	−0.563956	−0.281978	−	0.959421i	$-0.590991\pi$
$503$	−1.31132e28	−0.563956	−0.281978	+	0.959421i	$0.590991\pi$
$504$	−1.00383e28	−0.422804
$505$	0	0
$506$	−2.75233e28	−1.11204
$507$	−1.02542e28	−0.405807
$508$	−1.91149e28	−0.740974
$509$	−3.93813e28	−1.49539	−0.747694	−	0.664044i	$-0.768839\pi$
$509$	−3.93813e28	−1.49539	−0.747694	+	0.664044i	$0.768839\pi$
$510$	0	0
$511$	5.39390e28	1.96555
$512$	1.33074e27	0.0475073
$513$	3.48575e27	0.121917
$514$	−1.69946e28	−0.582368
$515$	0	0
$516$	−2.18471e28	−0.718742
$517$	−3.41711e28	−1.10156
$518$	−1.12243e28	−0.354564
$519$	9.09257e27	0.281468
$520$	0	0
$521$	−3.26234e28	−0.969913	−0.484957	−	0.874538i	$-0.661165\pi$
$521$	−3.26234e28	−0.969913	−0.484957	+	0.874538i	$0.661165\pi$
$522$	1.53191e28	0.446368
$523$	4.41267e28	1.26018	0.630092	−	0.776520i	$-0.283017\pi$
$523$	4.41267e28	1.26018	0.630092	+	0.776520i	$0.283017\pi$
$524$	5.92100e28	1.65736
$525$	0	0
$526$	4.07808e28	1.09674
$527$	−5.23638e28	−1.38045
$528$	−3.20280e26	−0.00827702
$529$	1.19328e28	0.302314
$530$	0	0
$531$	1.12862e28	0.274824
$532$	−5.51628e28	−1.31696
$533$	−2.66813e28	−0.624556
$534$	−1.55994e28	−0.358035
$535$	0	0
$536$	−6.34442e28	−1.40011
$537$	−1.27833e27	−0.0276639
$538$	−1.12526e29	−2.38802
$539$	1.95072e28	0.405987
$540$	0	0
$541$	−6.85055e28	−1.37137	−0.685684	−	0.727899i	$-0.740497\pi$
$541$	−6.85055e28	−1.37137	−0.685684	+	0.727899i	$0.740497\pi$
$542$	1.00176e28	0.196686
$543$	−1.37426e28	−0.264649
$544$	−4.52109e28	−0.853995
$545$	0	0
$546$	3.61613e28	0.657236
$547$	6.59127e28	1.17517	0.587587	−	0.809161i	$-0.300078\pi$
$547$	6.59127e28	1.17517	0.587587	+	0.809161i	$0.300078\pi$
$548$	1.94434e28	0.340076
$549$	−2.04555e28	−0.350995
$550$	0	0
$551$	3.18091e28	0.525363
$552$	−3.98753e28	−0.646165
$553$	−8.56625e28	−1.36200
$554$	−9.24512e28	−1.44231
$555$	0	0
$556$	−1.39648e29	−2.09774
$557$	−3.49493e27	−0.0515179	−0.0257590	−	0.999668i	$-0.508200\pi$
$557$	−3.49493e27	−0.0515179	−0.0257590	+	0.999668i	$0.508200\pi$
$558$	−6.12932e28	−0.886652
$559$	2.97380e28	0.422170
$560$	0	0
$561$	2.13529e28	0.291976
$562$	−1.13613e29	−1.52474
$563$	9.55924e28	1.25917	0.629586	−	0.776931i	$-0.283224\pi$
$563$	9.55924e28	1.25917	0.629586	+	0.776931i	$0.283224\pi$
$564$	−1.31018e29	−1.69394
$565$	0	0
$566$	9.31108e28	1.15992
$567$	−1.17515e28	−0.143705
$568$	−8.48212e28	−1.01823
$569$	1.31470e29	1.54934	0.774669	−	0.632367i	$-0.217916\pi$
$569$	1.31470e29	1.54934	0.774669	+	0.632367i	$0.217916\pi$
$570$	0	0
$571$	4.02367e28	0.457028	0.228514	−	0.973541i	$-0.426613\pi$
$571$	4.02367e28	0.457028	0.228514	+	0.973541i	$0.426613\pi$
$572$	−4.74132e28	−0.528739
$573$	1.00966e29	1.10548
$574$	−2.22585e29	−2.39288
$575$	0	0
$576$	−5.21565e28	−0.540596
$577$	−1.58150e29	−1.60962	−0.804810	−	0.593533i	$-0.797733\pi$
$577$	−1.58150e29	−1.60962	−0.804810	+	0.593533i	$0.797733\pi$
$578$	−4.81053e28	−0.480785
$579$	3.89068e28	0.381857
$580$	0	0
$581$	1.58199e29	1.49746
$582$	9.28091e28	0.862778
$583$	−7.79962e28	−0.712121
$584$	−1.66207e29	−1.49044
$585$	0	0
$586$	−1.17553e29	−1.01697
$587$	1.32870e29	1.12908	0.564541	−	0.825405i	$-0.309053\pi$
$587$	1.32870e29	1.12908	0.564541	+	0.825405i	$0.309053\pi$
$588$	7.47937e28	0.624314
$589$	−1.27271e29	−1.04357
$590$	0	0
$591$	6.42665e27	0.0508529
$592$	−5.18836e26	−0.00403323
$593$	−9.24977e27	−0.0706410	−0.0353205	−	0.999376i	$-0.511245\pi$
$593$	−9.24977e27	−0.0706410	−0.0353205	+	0.999376i	$0.511245\pi$
$594$	2.49941e28	0.187534
$595$	0	0
$596$	1.32154e29	0.957182
$597$	1.23113e29	0.876141
$598$	1.43644e29	1.00444
$599$	4.78210e28	0.328577	0.164289	−	0.986412i	$-0.447467\pi$
$599$	4.78210e28	0.328577	0.164289	+	0.986412i	$0.447467\pi$
$600$	0	0
$601$	−4.17777e28	−0.277180	−0.138590	−	0.990350i	$-0.544257\pi$
$601$	−4.17777e28	−0.277180	−0.138590	+	0.990350i	$0.544257\pi$
$602$	2.48084e29	1.61747
$603$	−7.42722e28	−0.475877
$604$	3.66845e29	2.30991
$605$	0	0
$606$	−2.35282e29	−1.43095
$607$	1.58223e29	0.945775	0.472887	−	0.881123i	$-0.343212\pi$
$607$	1.58223e29	0.945775	0.472887	+	0.881123i	$0.343212\pi$
$608$	−1.09886e29	−0.645587
$609$	−1.07238e29	−0.619253
$610$	0	0
$611$	1.78339e29	0.994977
$612$	8.18706e28	0.448991
$613$	−1.43461e29	−0.773390	−0.386695	−	0.922208i	$-0.626383\pi$
$613$	−1.43461e29	−0.773390	−0.386695	+	0.922208i	$0.626383\pi$
$614$	4.63273e29	2.45510
$615$	0	0
$616$	−1.49458e29	−0.765460
$617$	3.28289e28	0.165296	0.0826480	−	0.996579i	$-0.473662\pi$
$617$	3.28289e28	0.165296	0.0826480	+	0.996579i	$0.473662\pi$
$618$	3.39662e29	1.68139
$619$	3.54829e29	1.72690	0.863449	−	0.504436i	$-0.168299\pi$
$619$	3.54829e29	1.72690	0.863449	+	0.504436i	$0.168299\pi$
$620$	0	0
$621$	−4.66807e28	−0.219622
$622$	−5.93038e29	−2.74336
$623$	1.09201e29	0.496708
$624$	1.67154e27	0.00747616
$625$	0	0
$626$	−1.72472e29	−0.745912
$627$	5.18987e28	0.220723
$628$	1.22561e29	0.512596
$629$	3.45906e28	0.142274
$630$	0	0
$631$	9.74111e28	0.387525	0.193763	−	0.981048i	$-0.437931\pi$
$631$	9.74111e28	0.387525	0.193763	+	0.981048i	$0.437931\pi$
$632$	2.63960e29	1.03278
$633$	2.57345e28	0.0990321
$634$	−4.27067e29	−1.61644
$635$	0	0
$636$	−2.99050e29	−1.09508
$637$	−1.01808e29	−0.366705
$638$	2.28083e29	0.808120
$639$	−9.92976e28	−0.346083
$640$	0	0
$641$	3.41830e29	1.15292	0.576462	−	0.817124i	$-0.304433\pi$
$641$	3.41830e29	1.15292	0.576462	+	0.817124i	$0.304433\pi$
$642$	3.93422e29	1.30539
$643$	−3.63901e29	−1.18787	−0.593934	−	0.804514i	$-0.702426\pi$
$643$	−3.63901e29	−1.18787	−0.593934	+	0.804514i	$0.702426\pi$
$644$	7.38733e29	2.37239
$645$	0	0
$646$	2.75762e29	0.857221
$647$	−6.30531e29	−1.92846	−0.964232	−	0.265058i	$-0.914609\pi$
$647$	−6.30531e29	−1.92846	−0.964232	+	0.265058i	$0.914609\pi$
$648$	3.62110e28	0.108969
$649$	1.68038e29	0.497552
$650$	0	0
$651$	4.29070e29	1.23007
$652$	3.22241e29	0.909035
$653$	5.65207e29	1.56899	0.784493	−	0.620138i	$-0.212923\pi$
$653$	5.65207e29	1.56899	0.784493	+	0.620138i	$0.212923\pi$
$654$	1.44316e29	0.394229
$655$	0	0
$656$	−1.02889e28	−0.0272193
$657$	−1.94574e29	−0.506579
$658$	1.48776e30	3.81208
$659$	−1.80341e29	−0.454776	−0.227388	−	0.973804i	$-0.573019\pi$
$659$	−1.80341e29	−0.454776	−0.227388	+	0.973804i	$0.573019\pi$
$660$	0	0
$661$	5.31421e29	1.29815	0.649073	−	0.760726i	$-0.275157\pi$
$661$	5.31421e29	1.29815	0.649073	+	0.760726i	$0.275157\pi$
$662$	5.22674e29	1.25667
$663$	−1.11441e29	−0.263725
$664$	−4.87473e29	−1.13550
$665$	0	0
$666$	4.04891e28	0.0913817
$667$	−4.25983e29	−0.946393
$668$	3.48361e29	0.761862
$669$	−3.58811e29	−0.772487
$670$	0	0
$671$	−3.04558e29	−0.635454
$672$	3.70459e29	0.760962
$673$	−3.62908e29	−0.733902	−0.366951	−	0.930240i	$-0.619598\pi$
$673$	−3.62908e29	−0.733902	−0.366951	+	0.930240i	$0.619598\pi$
$674$	1.28321e30	2.55487
$675$	0	0
$676$	−5.85375e29	−1.12978
$677$	−1.36360e29	−0.259124	−0.129562	−	0.991571i	$-0.541357\pi$
$677$	−1.36360e29	−0.259124	−0.129562	+	0.991571i	$0.541357\pi$
$678$	−8.08624e29	−1.51299
$679$	−6.49691e29	−1.19695
$680$	0	0
$681$	−1.65697e29	−0.295985
$682$	−9.12582e29	−1.60523
$683$	−3.14438e29	−0.544650	−0.272325	−	0.962205i	$-0.587793\pi$
$683$	−3.14438e29	−0.544650	−0.272325	+	0.962205i	$0.587793\pi$
$684$	1.98988e29	0.339420
$685$	0	0
$686$	4.13147e29	0.683441
$687$	−8.48987e28	−0.138311
$688$	1.14676e28	0.0183990
$689$	4.07062e29	0.643218
$690$	0	0
$691$	9.44313e28	0.144743	0.0723713	−	0.997378i	$-0.476943\pi$
$691$	9.44313e28	0.144743	0.0723713	+	0.997378i	$0.476943\pi$
$692$	5.19059e29	0.783615
$693$	−1.74966e29	−0.260169
$694$	−7.70283e29	−1.12817
$695$	0	0
$696$	3.30442e29	0.469568
$697$	6.85954e29	0.960177
$698$	−1.11338e30	−1.53519
$699$	−8.56412e28	−0.116325
$700$	0	0
$701$	−1.15385e30	−1.52094	−0.760469	−	0.649374i	$-0.775031\pi$
$701$	−1.15385e30	−1.52094	−0.760469	+	0.649374i	$0.775031\pi$
$702$	−1.30444e29	−0.169389
$703$	8.40731e28	0.107554
$704$	−7.76549e29	−0.978714
$705$	0	0
$706$	1.50589e30	1.84224
$707$	1.64704e30	1.98519
$708$	6.44285e29	0.765118
$709$	−5.50935e28	−0.0644637	−0.0322318	−	0.999480i	$-0.510261\pi$
$709$	−5.50935e28	−0.0644637	−0.0322318	+	0.999480i	$0.510261\pi$
$710$	0	0
$711$	3.09009e29	0.351027
$712$	−3.36490e29	−0.376645
$713$	1.70440e30	1.87989
$714$	−9.29677e29	−1.01042
$715$	0	0
$716$	−7.29750e28	−0.0770172
$717$	−1.63948e29	−0.170512
$718$	2.41480e29	0.247500
$719$	−9.04260e29	−0.913356	−0.456678	−	0.889632i	$-0.650961\pi$
$719$	−9.04260e29	−0.913356	−0.456678	+	0.889632i	$0.650961\pi$
$720$	0	0
$721$	−2.37773e30	−2.33261
$722$	−9.99850e29	−0.966706
$723$	5.31523e29	0.506489
$724$	−7.84511e29	−0.736791
$725$	0	0
$726$	−6.49687e29	−0.592747
$727$	1.25053e30	1.12456	0.562280	−	0.826947i	$-0.309924\pi$
$727$	1.25053e30	1.12456	0.562280	+	0.826947i	$0.309924\pi$
$728$	7.80022e29	0.691397
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−7.64538e29	−0.649033
$732$	−1.16773e30	−0.977180
$733$	3.34408e28	0.0275858	0.0137929	−	0.999905i	$-0.495609\pi$
$733$	3.34408e28	0.0275858	0.0137929	+	0.999905i	$0.495609\pi$
$734$	−1.54835e29	−0.125910
$735$	0	0
$736$	1.47158e30	1.16297
$737$	−1.10582e30	−0.861544
$738$	8.02927e29	0.616714
$739$	1.86392e30	1.41144	0.705719	−	0.708492i	$-0.250624\pi$
$739$	1.86392e30	1.41144	0.705719	+	0.708492i	$0.250624\pi$
$740$	0	0
$741$	−2.70859e29	−0.199366
$742$	3.39585e30	2.46438
$743$	−1.44222e30	−1.03193	−0.515964	−	0.856611i	$-0.672566\pi$
$743$	−1.44222e30	−1.03193	−0.515964	+	0.856611i	$0.672566\pi$
$744$	−1.32213e30	−0.932737
$745$	0	0
$746$	1.44197e30	0.989004
$747$	−5.70669e29	−0.385939
$748$	1.21895e30	0.812870
$749$	−2.75407e30	−1.81099
$750$	0	0
$751$	−2.05877e30	−1.31640	−0.658200	−	0.752843i	$-0.728682\pi$
$751$	−2.05877e30	−1.31640	−0.658200	+	0.752843i	$0.728682\pi$
$752$	6.87712e28	0.0433630
$753$	−2.86116e29	−0.177908
$754$	−1.19036e30	−0.729929
$755$	0	0
$756$	−6.70849e29	−0.400079
$757$	−1.21804e30	−0.716396	−0.358198	−	0.933646i	$-0.616609\pi$
$757$	−1.21804e30	−0.716396	−0.358198	+	0.933646i	$0.616609\pi$
$758$	−2.00885e30	−1.16526
$759$	−6.95021e29	−0.397612
$760$	0	0
$761$	−2.01750e30	−1.12273	−0.561365	−	0.827568i	$-0.689724\pi$
$761$	−2.01750e30	−1.12273	−0.561365	+	0.827568i	$0.689724\pi$
$762$	−7.82991e29	−0.429764
$763$	−1.01026e30	−0.546920
$764$	5.76376e30	3.07769
$765$	0	0
$766$	−3.08287e30	−1.60160
$767$	−8.76989e29	−0.449410
$768$	−1.09792e30	−0.554979
$769$	−7.88254e29	−0.393043	−0.196521	−	0.980500i	$-0.562964\pi$
$769$	−7.88254e29	−0.393043	−0.196521	+	0.980500i	$0.562964\pi$
$770$	0	0
$771$	−4.29148e29	−0.208227
$772$	2.22104e30	1.06310
$773$	−1.88570e29	−0.0890404	−0.0445202	−	0.999008i	$-0.514176\pi$
$773$	−1.88570e29	−0.0890404	−0.0445202	+	0.999008i	$0.514176\pi$
$774$	−8.94911e29	−0.416869
$775$	0	0
$776$	2.00195e30	0.907623
$777$	−2.83436e29	−0.126775
$778$	−4.64591e30	−2.05015
$779$	1.66723e30	0.725857
$780$	0	0
$781$	−1.47842e30	−0.626560
$782$	−3.69297e30	−1.54420
$783$	3.86839e29	0.159600
$784$	−3.92593e28	−0.0159817
$785$	0	0
$786$	2.42538e30	0.961265
$787$	4.17685e29	0.163348	0.0816740	−	0.996659i	$-0.473973\pi$
$787$	4.17685e29	0.163348	0.0816740	+	0.996659i	$0.473973\pi$
$788$	3.66872e29	0.141576
$789$	1.02980e30	0.392143
$790$	0	0
$791$	5.66061e30	2.09899
$792$	5.39139e29	0.197282
$793$	1.58949e30	0.573970
$794$	6.97472e30	2.48549
$795$	0	0
$796$	7.02805e30	2.43920
$797$	−4.90677e30	−1.68067	−0.840336	−	0.542065i	$-0.817642\pi$
$797$	−4.90677e30	−1.68067	−0.840336	+	0.542065i	$0.817642\pi$
$798$	−2.25960e30	−0.763837
$799$	−4.58494e30	−1.52965
$800$	0	0
$801$	−3.93918e29	−0.128016
$802$	5.00405e30	1.60506
$803$	−2.89697e30	−0.917129
$804$	−4.23991e30	−1.32485
$805$	0	0
$806$	4.76276e30	1.44991
$807$	−2.84151e30	−0.853841
$808$	−5.07518e30	−1.50533
$809$	4.13041e30	1.20930	0.604649	−	0.796492i	$-0.293313\pi$
$809$	4.13041e30	1.20930	0.604649	+	0.796492i	$0.293313\pi$
$810$	0	0
$811$	−5.67739e29	−0.161968	−0.0809841	−	0.996715i	$-0.525806\pi$
$811$	−5.67739e29	−0.161968	−0.0809841	+	0.996715i	$0.525806\pi$
$812$	−6.12181e30	−1.72402
$813$	2.52966e29	0.0703253
$814$	6.02835e29	0.165441
$815$	0	0
$816$	−4.29739e28	−0.0114937
$817$	−1.85822e30	−0.490644
$818$	−5.39898e30	−1.40735
$819$	9.13148e29	0.234996
$820$	0	0
$821$	1.24561e30	0.312448	0.156224	−	0.987722i	$-0.450068\pi$
$821$	1.24561e30	0.312448	0.156224	+	0.987722i	$0.450068\pi$
$822$	7.96446e29	0.197243
$823$	1.40410e30	0.343320	0.171660	−	0.985156i	$-0.445087\pi$
$823$	1.40410e30	0.343320	0.171660	+	0.985156i	$0.445087\pi$
$824$	7.32672e30	1.76878
$825$	0	0
$826$	−7.31614e30	−1.72184
$827$	−2.62381e30	−0.609712	−0.304856	−	0.952398i	$-0.598608\pi$
$827$	−2.62381e30	−0.609712	−0.304856	+	0.952398i	$0.598608\pi$
$828$	−2.66482e30	−0.611434
$829$	6.45710e30	1.46290	0.731451	−	0.681894i	$-0.238844\pi$
$829$	6.45710e30	1.46290	0.731451	+	0.681894i	$0.238844\pi$
$830$	0	0
$831$	−2.33458e30	−0.515702
$832$	4.05280e30	0.884017
$833$	2.61740e30	0.563763
$834$	−5.72033e30	−1.21668
$835$	0	0
$836$	2.96269e30	0.614498
$837$	−1.54778e30	−0.317024
$838$	2.28103e30	0.461390
$839$	−1.49564e30	−0.298763	−0.149381	−	0.988780i	$-0.547728\pi$
$839$	−1.49564e30	−0.298763	−0.149381	+	0.988780i	$0.547728\pi$
$840$	0	0
$841$	−1.60276e30	−0.312256
$842$	9.18304e30	1.76689
$843$	−2.86895e30	−0.545172
$844$	1.46908e30	0.275708
$845$	0	0
$846$	−5.36679e30	−0.982484
$847$	4.54800e30	0.822327
$848$	1.56972e29	0.0280327
$849$	2.35124e30	0.414731
$850$	0	0
$851$	−1.12590e30	−0.193748
$852$	−5.66851e30	−0.963503
$853$	4.49367e30	0.754460	0.377230	−	0.926120i	$-0.376877\pi$
$853$	4.49367e30	0.754460	0.377230	+	0.926120i	$0.376877\pi$
$854$	1.32601e31	2.19906
$855$	0	0
$856$	8.48636e30	1.37324
$857$	7.48368e30	1.19624	0.598118	−	0.801408i	$-0.295915\pi$
$857$	7.48368e30	1.19624	0.598118	+	0.801408i	$0.295915\pi$
$858$	−1.94216e30	−0.306668
$859$	−1.21175e31	−1.89010	−0.945048	−	0.326932i	$-0.893985\pi$
$859$	−1.21175e31	−1.89010	−0.945048	+	0.326932i	$0.893985\pi$
$860$	0	0
$861$	−5.62072e30	−0.855577
$862$	7.58983e29	0.114132
$863$	−1.14819e31	−1.70569	−0.852845	−	0.522164i	$-0.825125\pi$
$863$	−1.14819e31	−1.70569	−0.852845	+	0.522164i	$0.825125\pi$
$864$	−1.33635e30	−0.196122
$865$	0	0
$866$	1.18559e30	0.169823
$867$	−1.21476e30	−0.171906
$868$	2.44940e31	3.42454
$869$	4.60078e30	0.635512
$870$	0	0
$871$	5.77129e30	0.778184
$872$	3.11299e30	0.414720
$873$	2.34362e30	0.308488
$874$	−8.97582e30	−1.16736
$875$	0	0
$876$	−1.11074e31	−1.41033
$877$	−7.42378e30	−0.931385	−0.465693	−	0.884947i	$-0.654195\pi$
$877$	−7.42378e30	−0.931385	−0.465693	+	0.884947i	$0.654195\pi$
$878$	1.75478e31	2.17535
$879$	−2.96846e30	−0.363620
$880$	0	0
$881$	−1.35138e30	−0.161633	−0.0808165	−	0.996729i	$-0.525753\pi$
$881$	−1.35138e30	−0.161633	−0.0808165	+	0.996729i	$0.525753\pi$
$882$	3.06373e30	0.362101
$883$	−5.01586e30	−0.585812	−0.292906	−	0.956141i	$-0.594622\pi$
$883$	−5.01586e30	−0.585812	−0.292906	+	0.956141i	$0.594622\pi$
$884$	−6.36172e30	−0.734219
$885$	0	0
$886$	−1.13453e31	−1.27868
$887$	3.97104e30	0.442289	0.221145	−	0.975241i	$-0.429021\pi$
$887$	3.97104e30	0.442289	0.221145	+	0.975241i	$0.429021\pi$
$888$	8.73376e29	0.0961314
$889$	5.48117e30	0.596218
$890$	0	0
$891$	6.31154e29	0.0670532
$892$	−2.04831e31	−2.15062
$893$	−1.11438e31	−1.15636
$894$	5.41334e30	0.555164
$895$	0	0
$896$	2.06528e31	2.06893
$897$	3.62731e30	0.359140
$898$	−1.19198e31	−1.16645
$899$	−1.41242e31	−1.36612
$900$	0	0
$901$	−1.04652e31	−0.988867
$902$	1.19546e31	1.11652
$903$	6.26464e30	0.578329
$904$	−1.74425e31	−1.59163
$905$	0	0
$906$	1.50268e31	1.33974
$907$	−2.69779e30	−0.237757	−0.118878	−	0.992909i	$-0.537930\pi$
$907$	−2.69779e30	−0.237757	−0.118878	+	0.992909i	$0.537930\pi$
$908$	−9.45900e30	−0.824032
$909$	−5.94136e30	−0.511640
$910$	0	0
$911$	−1.19326e31	−1.00414	−0.502069	−	0.864827i	$-0.667428\pi$
$911$	−1.19326e31	−1.00414	−0.502069	+	0.864827i	$0.667428\pi$
$912$	−1.04449e29	−0.00868877
$913$	−8.49659e30	−0.698717
$914$	−3.48461e30	−0.283282
$915$	0	0
$916$	−4.84653e30	−0.385060
$917$	−1.69784e31	−1.33358
$918$	3.35361e30	0.260414
$919$	−1.42331e31	−1.09266	−0.546330	−	0.837570i	$-0.683976\pi$
$919$	−1.42331e31	−1.09266	−0.546330	+	0.837570i	$0.683976\pi$
$920$	0	0
$921$	1.16986e31	0.877824
$922$	3.66331e31	2.71769
$923$	7.71588e30	0.565936
$924$	−9.98814e30	−0.724318
$925$	0	0
$926$	−1.87090e31	−1.32628
$927$	8.57717e30	0.601183
$928$	−1.21948e31	−0.845128
$929$	1.46344e31	1.00279	0.501394	−	0.865219i	$-0.332821\pi$
$929$	1.46344e31	1.00279	0.501394	+	0.865219i	$0.332821\pi$
$930$	0	0
$931$	6.36163e30	0.426184
$932$	−4.88892e30	−0.323851
$933$	−1.49754e31	−0.980895
$934$	−2.05919e31	−1.33369
$935$	0	0
$936$	−2.81376e30	−0.178193
$937$	−2.59339e31	−1.62406	−0.812031	−	0.583615i	$-0.801638\pi$
$937$	−2.59339e31	−1.62406	−0.812031	+	0.583615i	$0.801638\pi$
$938$	4.81461e31	2.98147
$939$	−4.35527e30	−0.266702
$940$	0	0
$941$	−1.80407e31	−1.08035	−0.540173	−	0.841554i	$-0.681641\pi$
$941$	−1.80407e31	−1.08035	−0.540173	+	0.841554i	$0.681641\pi$
$942$	5.02038e30	0.297305
$943$	−2.23273e31	−1.30756
$944$	−3.38185e29	−0.0195861
$945$	0	0
$946$	−1.33242e31	−0.754715
$947$	−1.33146e30	−0.0745854	−0.0372927	−	0.999304i	$-0.511873\pi$
$947$	−1.33146e30	−0.0745854	−0.0372927	+	0.999304i	$0.511873\pi$
$948$	1.76401e31	0.977269
$949$	1.51193e31	0.828391
$950$	0	0
$951$	−1.07843e31	−0.577960
$952$	−2.00537e31	−1.06294
$953$	1.89970e31	0.995884	0.497942	−	0.867210i	$-0.334089\pi$
$953$	1.89970e31	0.995884	0.497942	+	0.867210i	$0.334089\pi$
$954$	−1.22498e31	−0.635142
$955$	0	0
$956$	−9.35915e30	−0.474710
$957$	5.75957e30	0.288945
$958$	4.14719e31	2.05786
$959$	−5.57536e30	−0.273639
$960$	0	0
$961$	3.56869e31	1.71361
$962$	−3.14619e30	−0.149433
$963$	9.93472e30	0.466745
$964$	3.03426e31	1.41008
$965$	0	0
$966$	3.02602e31	1.37598
$967$	−1.78447e31	−0.802661	−0.401330	−	0.915933i	$-0.631452\pi$
$967$	−1.78447e31	−0.802661	−0.401330	+	0.915933i	$0.631452\pi$
$968$	−1.40142e31	−0.623556
$969$	6.96356e30	0.306501
$970$	0	0
$971$	−1.22548e31	−0.527842	−0.263921	−	0.964544i	$-0.585016\pi$
$971$	−1.22548e31	−0.527842	−0.263921	+	0.964544i	$0.585016\pi$
$972$	2.41995e30	0.103112
$973$	4.00440e31	1.68792
$974$	3.02221e31	1.26025
$975$	0	0
$976$	6.12940e29	0.0250147
$977$	−9.23163e30	−0.372722	−0.186361	−	0.982481i	$-0.559669\pi$
$977$	−9.23163e30	−0.372722	−0.186361	+	0.982481i	$0.559669\pi$
$978$	1.31997e31	0.527239
$979$	−5.86498e30	−0.231765
$980$	0	0
$981$	3.64428e30	0.140957
$982$	6.81928e31	2.60956
$983$	9.55687e30	0.361829	0.180915	−	0.983499i	$-0.442094\pi$
$983$	9.55687e30	0.361829	0.180915	+	0.983499i	$0.442094\pi$
$984$	1.73196e31	0.648769
$985$	0	0
$986$	3.06033e31	1.12217
$987$	3.75691e31	1.36302
$988$	−1.54623e31	−0.555042
$989$	2.48851e31	0.883850
$990$	0	0
$991$	3.98073e31	1.38417	0.692085	−	0.721816i	$-0.256692\pi$
$991$	3.98073e31	1.38417	0.692085	+	0.721816i	$0.256692\pi$
$992$	4.87928e31	1.67874
$993$	1.31986e31	0.449325
$994$	6.43685e31	2.16828
$995$	0	0
$996$	−3.25773e31	−1.07446
$997$	−1.05683e31	−0.344911	−0.172456	−	0.985017i	$-0.555170\pi$
$997$	−1.05683e31	−0.344911	−0.172456	+	0.985017i	$0.555170\pi$
$998$	3.22332e31	1.04096
$999$	1.02243e30	0.0326737

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.m.1.9		10
5.2	odd	4		15.22.b.a.4.18	yes	20
5.3	odd	4		15.22.b.a.4.3	✓	20
5.4	even	2		75.22.a.n.1.2		10
15.2	even	4		45.22.b.d.19.3		20
15.8	even	4		45.22.b.d.19.18		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.b.a.4.3	✓	20	5.3	odd	4
15.22.b.a.4.18	yes	20	5.2	odd	4
45.22.b.d.19.3		20	15.2	even	4
45.22.b.d.19.18		20	15.8	even	4
75.22.a.m.1.9		10	1.1	even	1	trivial
75.22.a.n.1.2		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)

\(f(q)\)	\(=\)	\(q+2338.38 q^{2} +59049.0 q^{3} +3.37088e6 q^{4} +1.38079e8 q^{6} -9.66595e8 q^{7} +2.97845e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\)	\(q+2338.38 q^{2} +59049.0 q^{3} +3.37088e6 q^{4} +1.38079e8 q^{6} -9.66595e8 q^{7} +2.97845e9 q^{8} +3.48678e9 q^{9} +5.19141e10 q^{11} +1.99047e11 q^{12} -2.70939e11 q^{13} -2.26027e12 q^{14} -1.04480e11 q^{16} +6.96562e12 q^{17} +8.15343e12 q^{18} +1.69301e13 q^{19} -5.70764e13 q^{21} +1.21395e14 q^{22} -2.26725e14 q^{23} +1.75875e14 q^{24} -6.33559e14 q^{26} +2.05891e14 q^{27} -3.25827e15 q^{28} +1.87885e15 q^{29} -7.51747e15 q^{31} -6.49058e15 q^{32} +3.06547e15 q^{33} +1.62883e16 q^{34} +1.17535e16 q^{36} +4.96590e15 q^{37} +3.95890e16 q^{38} -1.59987e16 q^{39} +9.84771e16 q^{41} -1.33466e17 q^{42} -1.09759e17 q^{43} +1.74996e17 q^{44} -5.30170e17 q^{46} -6.58224e17 q^{47} -6.16943e15 q^{48} +3.75759e17 q^{49} +4.11313e17 q^{51} -9.13302e17 q^{52} -1.50241e18 q^{53} +4.81452e17 q^{54} -2.87896e18 q^{56} +9.99704e17 q^{57} +4.39347e18 q^{58} +3.23685e18 q^{59} -5.86659e18 q^{61} -1.75787e19 q^{62} -3.37031e18 q^{63} -1.49583e19 q^{64} +7.16825e18 q^{66} -2.13011e19 q^{67} +2.34802e19 q^{68} -1.33879e19 q^{69} -2.84783e19 q^{71} +1.03852e19 q^{72} -5.58032e19 q^{73} +1.16122e19 q^{74} +5.70692e19 q^{76} -5.01798e19 q^{77} -3.74110e19 q^{78} +8.86230e19 q^{79} +1.21577e19 q^{81} +2.30277e20 q^{82} -1.63666e20 q^{83} -1.92398e20 q^{84} -2.56658e20 q^{86} +1.10944e20 q^{87} +1.54624e20 q^{88} -1.12975e20 q^{89} +2.61888e20 q^{91} -7.64263e20 q^{92} -4.43899e20 q^{93} -1.53918e21 q^{94} -3.83262e20 q^{96} +6.72145e20 q^{97} +8.78668e20 q^{98} +1.81013e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 645 q^{2} + 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} - 115357290 q^{7} - 314142735 q^{8} + 34867844010 q^{9}+O(q^{10}) \) 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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