LMFDB - Embedded newform 75.22.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$209.608008215$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3512166x^{2} + 363520480x + 2321089280000 $ x^4 - x^3 - 3512166x^2 + 363520480x + 2321089280000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1492.88$ 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)

$f(q)$	$=$	\(q+1716.88 q^{2} +59049.0 q^{3} +850541. q^{4} +1.01380e8 q^{6} +6.24477e8 q^{7} -2.14029e9 q^{8} +3.48678e9 q^{9} +1.47398e11 q^{11} +5.02236e10 q^{12} +5.50754e11 q^{13} +1.07216e12 q^{14} -5.45834e12 q^{16} +4.04946e12 q^{17} +5.98641e12 q^{18} +1.22679e12 q^{19} +3.68748e13 q^{21} +2.53065e14 q^{22} +2.50983e14 q^{23} -1.26382e14 q^{24} +9.45581e14 q^{26} +2.05891e14 q^{27} +5.31144e14 q^{28} -3.49299e15 q^{29} -2.93209e15 q^{31} -4.88283e15 q^{32} +8.70368e15 q^{33} +6.95245e15 q^{34} +2.96565e15 q^{36} -4.06107e16 q^{37} +2.10626e15 q^{38} +3.25215e16 q^{39} +1.45009e17 q^{41} +6.33097e16 q^{42} -8.05941e16 q^{43} +1.25368e17 q^{44} +4.30909e17 q^{46} -4.95994e16 q^{47} -3.22310e17 q^{48} -1.68574e17 q^{49} +2.39117e17 q^{51} +4.68439e17 q^{52} +1.24656e18 q^{53} +3.53491e17 q^{54} -1.33656e18 q^{56} +7.24408e16 q^{57} -5.99706e18 q^{58} +7.20028e18 q^{59} -4.77418e18 q^{61} -5.03405e18 q^{62} +2.17742e18 q^{63} +3.06371e18 q^{64} +1.49432e19 q^{66} -7.70841e18 q^{67} +3.44423e18 q^{68} +1.48203e19 q^{69} -1.63265e19 q^{71} -7.46272e18 q^{72} +1.29930e19 q^{73} -6.97240e19 q^{74} +1.04344e18 q^{76} +9.20465e19 q^{77} +5.58356e19 q^{78} +9.55293e19 q^{79} +1.21577e19 q^{81} +2.48964e20 q^{82} +2.20974e20 q^{83} +3.13635e19 q^{84} -1.38371e20 q^{86} -2.06258e20 q^{87} -3.15473e20 q^{88} +3.15065e19 q^{89} +3.43933e20 q^{91} +2.13471e20 q^{92} -1.73137e20 q^{93} -8.51565e19 q^{94} -2.88326e20 q^{96} -2.46548e20 q^{97} -2.89422e20 q^{98} +5.13944e20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9} + 31491830256 q^{11} - 68681250027 q^{12} + 27017977768 q^{13} - 2233308126672 q^{14}+ \cdots + 10\!\cdots\!56 q^{99}+O(q^{100}) $ 4 * q + 897 * q^2 + 236196 * q^3 - 1163123 * q^4 + 52966953 * q^6 + 234577504 * q^7 - 76855629 * q^8 + 13947137604 * q^9 + 31491830256 * q^11 - 68681250027 * q^12 + 27017977768 * q^13 - 2233308126672 * q^14 - 11324681311775 * q^16 + 2946095028888 * q^17 + 3127645607697 * q^18 - 24270353300752 * q^19 + 13851567033696 * q^21 + 56303932793676 * q^22 - 10350924920928 * q^23 - 4538248036821 * q^24 + 474751622871378 * q^26 + 823564528378596 * q^27 + 1868468795909968 * q^28 + 4728924677079096 * q^29 - 1094923910405536 * q^31 - 7933327650814221 * q^32 + 1859561084786544 * q^33 + 3862795623387294 * q^34 - 4055559132844323 * q^36 - 4813435696247096 * q^37 - 21307487199674508 * q^38 + 1595384569222632 * q^39 + 289731591445930344 * q^41 - 131874611571854928 * q^42 - 451091912658458000 * q^43 + 245434444420437276 * q^44 + 645145648102459824 * q^46 - 813883435638492480 * q^47 - 668711106779001975 * q^48 + 1545180753004755300 * q^49 + 173963965360807512 * q^51 - 884645329958480822 * q^52 - 697278335404085208 * q^53 + 184684345488900153 * q^54 + 5673275878853478000 * q^56 - 1433140092056104848 * q^57 - 12277858736996428626 * q^58 + 6622888614569598192 * q^59 + 7390887218011683320 * q^61 - 11649094150323079392 * q^62 + 817921181772715104 * q^63 + 21525702776885072473 * q^64 + 3324690927533774124 * q^66 + 24188188449376788688 * q^67 - 18995582935459304346 * q^68 - 611211765655877472 * q^69 - 37390337803999713312 * q^71 - 267979008326243229 * q^72 + 37253672904265201432 * q^73 - 77875663243460100102 * q^74 - 44795472028972724668 * q^76 + 315245139112859776128 * q^77 + 28033608578931999522 * q^78 - 109751291042259570400 * q^79 + 48630661836227715204 * q^81 + 210801413757052759338 * q^82 + 149023592088638482896 * q^83 + 110331213929687700432 * q^84 - 346200583098972939204 * q^86 + 279238273256843539704 * q^87 + 116461091469581805156 * q^88 + 558502237151273959848 * q^89 + 30360075980445717184 * q^91 + 764380916144075558544 * q^92 - 64654161985536495264 * q^93 + 1096939878232058257512 * q^94 - 468455064452928935829 * q^96 + 236244923748673777528 * q^97 - 197058383612642903463 * q^98 + 109805222495560636656 * q^99

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 $

$2$	1716.88	1.18557	0.592784	−	0.805362i	$-0.298029\pi$
$2$	1716.88	1.18557	0.592784	+	0.805362i	$0.298029\pi$
$3$	59049.0	0.577350
$3$	59049.0	0.577350
$4$	850541.	0.405570
$5$	0	0
$5$	0	0
$6$	1.01380e8	0.684488
$7$	6.24477e8	0.835579	0.417789	−	0.908544i	$-0.362805\pi$
$7$	6.24477e8	0.835579	0.417789	+	0.908544i	$0.362805\pi$
$8$	−2.14029e9	−0.704737
$9$	3.48678e9	0.333333
$10$	0	0
$11$	1.47398e11	1.71343	0.856717	−	0.515787i	$-0.172501\pi$
$11$	1.47398e11	1.71343	0.856717	+	0.515787i	$0.172501\pi$
$12$	5.02236e10	0.234156
$13$	5.50754e11	1.10803	0.554016	−	0.832506i	$-0.313095\pi$
$13$	5.50754e11	1.10803	0.554016	+	0.832506i	$0.313095\pi$
$14$	1.07216e12	0.990635
$15$	0	0
$16$	−5.45834e12	−1.24108
$17$	4.04946e12	0.487173	0.243587	−	0.969879i	$-0.421676\pi$
$17$	4.04946e12	0.487173	0.243587	+	0.969879i	$0.421676\pi$
$18$	5.98641e12	0.395189
$19$	1.22679e12	0.0459048	0.0229524	−	0.999737i	$-0.492693\pi$
$19$	1.22679e12	0.0459048	0.0229524	+	0.999737i	$0.492693\pi$
$20$	0	0
$21$	3.68748e13	0.482422
$22$	2.53065e14	2.03139
$23$	2.50983e14	1.26329	0.631644	−	0.775259i	$-0.282380\pi$
$23$	2.50983e14	1.26329	0.631644	+	0.775259i	$0.282380\pi$
$24$	−1.26382e14	−0.406880
$25$	0	0
$26$	9.45581e14	1.31365
$27$	2.05891e14	0.192450
$28$	5.31144e14	0.338885
$29$	−3.49299e15	−1.54177	−0.770883	−	0.636977i	$-0.780185\pi$
$29$	−3.49299e15	−1.54177	−0.770883	+	0.636977i	$0.780185\pi$
$30$	0	0
$31$	−2.93209e15	−0.642508	−0.321254	−	0.946993i	$-0.604104\pi$
$31$	−2.93209e15	−0.642508	−0.321254	+	0.946993i	$0.604104\pi$
$32$	−4.88283e15	−0.766650
$33$	8.70368e15	0.989251
$34$	6.95245e15	0.577576
$35$	0	0
$36$	2.96565e15	0.135190
$37$	−4.06107e16	−1.38843	−0.694213	−	0.719770i	$-0.744247\pi$
$37$	−4.06107e16	−1.38843	−0.694213	+	0.719770i	$0.744247\pi$
$38$	2.10626e15	0.0544232
$39$	3.25215e16	0.639723
$40$	0	0
$41$	1.45009e17	1.68720	0.843598	−	0.536976i	$-0.180433\pi$
$41$	1.45009e17	1.68720	0.843598	+	0.536976i	$0.180433\pi$
$42$	6.33097e16	0.571943
$43$	−8.05941e16	−0.568702	−0.284351	−	0.958720i	$-0.591778\pi$
$43$	−8.05941e16	−0.568702	−0.284351	+	0.958720i	$0.591778\pi$
$44$	1.25368e17	0.694916
$45$	0	0
$46$	4.30909e17	1.49771
$47$	−4.95994e16	−0.137546	−0.0687732	−	0.997632i	$-0.521908\pi$
$47$	−4.95994e16	−0.137546	−0.0687732	+	0.997632i	$0.521908\pi$
$48$	−3.22310e17	−0.716540
$49$	−1.68574e17	−0.301808
$50$	0	0
$51$	2.39117e17	0.281270
$52$	4.68439e17	0.449384
$53$	1.24656e18	0.979079	0.489540	−	0.871981i	$-0.337165\pi$
$53$	1.24656e18	0.979079	0.489540	+	0.871981i	$0.337165\pi$
$54$	3.53491e17	0.228163
$55$	0	0
$56$	−1.33656e18	−0.588863
$57$	7.24408e16	0.0265031
$58$	−5.99706e18	−1.82787
$59$	7.20028e18	1.83402	0.917009	−	0.398867i	$-0.130596\pi$
$59$	7.20028e18	1.83402	0.917009	+	0.398867i	$0.130596\pi$
$60$	0	0
$61$	−4.77418e18	−0.856910	−0.428455	−	0.903563i	$-0.640942\pi$
$61$	−4.77418e18	−0.856910	−0.428455	+	0.903563i	$0.640942\pi$
$62$	−5.03405e18	−0.761737
$63$	2.17742e18	0.278526
$64$	3.06371e18	0.332168
$65$	0	0
$66$	1.49432e19	1.17282
$67$	−7.70841e18	−0.516630	−0.258315	−	0.966061i	$-0.583167\pi$
$67$	−7.70841e18	−0.516630	−0.258315	+	0.966061i	$0.583167\pi$
$68$	3.44423e18	0.197583
$69$	1.48203e19	0.729360
$70$	0	0
$71$	−1.63265e19	−0.595223	−0.297612	−	0.954687i	$-0.596190\pi$
$71$	−1.63265e19	−0.595223	−0.297612	+	0.954687i	$0.596190\pi$
$72$	−7.46272e18	−0.234912
$73$	1.29930e19	0.353849	0.176924	−	0.984224i	$-0.443385\pi$
$73$	1.29930e19	0.353849	0.176924	+	0.984224i	$0.443385\pi$
$74$	−6.97240e19	−1.64607
$75$	0	0
$76$	1.04344e18	0.0186176
$77$	9.20465e19	1.43171
$78$	5.58356e19	0.758434
$79$	9.55293e19	1.13515	0.567574	−	0.823323i	$-0.307882\pi$
$79$	9.55293e19	1.13515	0.567574	+	0.823323i	$0.307882\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	2.48964e20	2.00028
$83$	2.20974e20	1.56323	0.781613	−	0.623764i	$-0.214397\pi$
$83$	2.20974e20	1.56323	0.781613	+	0.623764i	$0.214397\pi$
$84$	3.13635e19	0.195655
$85$	0	0
$86$	−1.38371e20	−0.674234
$87$	−2.06258e20	−0.890139
$88$	−3.15473e20	−1.20752
$89$	3.15065e19	0.107104	0.0535519	−	0.998565i	$-0.482946\pi$
$89$	3.15065e19	0.107104	0.0535519	+	0.998565i	$0.482946\pi$
$90$	0	0
$91$	3.43933e20	0.925848
$92$	2.13471e20	0.512351
$93$	−1.73137e20	−0.370952
$94$	−8.51565e19	−0.163070
$95$	0	0
$96$	−2.88326e20	−0.442626
$97$	−2.46548e20	−0.339467	−0.169733	−	0.985490i	$-0.554291\pi$
$97$	−2.46548e20	−0.339467	−0.169733	+	0.985490i	$0.554291\pi$
$98$	−2.89422e20	−0.357814
$99$	5.13944e20	0.571144
$100$	0	0
$101$	2.15225e21	1.93873	0.969367	−	0.245616i	$-0.0789902\pi$
$101$	2.15225e21	1.93873	0.969367	+	0.245616i	$0.0789902\pi$
$102$	4.10535e20	0.333464
$103$	1.53340e21	1.12426	0.562128	−	0.827050i	$-0.309983\pi$
$103$	1.53340e21	1.12426	0.562128	+	0.827050i	$0.309983\pi$
$104$	−1.17877e21	−0.780871
$105$	0	0
$106$	2.14021e21	1.16076
$107$	−1.54824e21	−0.760865	−0.380433	−	0.924809i	$-0.624225\pi$
$107$	−1.54824e21	−0.760865	−0.380433	+	0.924809i	$0.624225\pi$
$108$	1.75119e20	0.0780519
$109$	2.85208e21	1.15394	0.576971	−	0.816764i	$-0.304234\pi$
$109$	2.85208e21	1.15394	0.576971	+	0.816764i	$0.304234\pi$
$110$	0	0
$111$	−2.39802e21	−0.801608
$112$	−3.40861e21	−1.03702
$113$	−3.30356e21	−0.915499	−0.457750	−	0.889081i	$-0.651344\pi$
$113$	−3.30356e21	−0.915499	−0.457750	+	0.889081i	$0.651344\pi$
$114$	1.24373e20	0.0314213
$115$	0	0
$116$	−2.97093e21	−0.625293
$117$	1.92036e21	0.369344
$118$	1.23621e22	2.17435
$119$	2.52880e21	0.407071
$120$	0	0
$121$	1.43258e22	1.93585
$122$	−8.19671e21	−1.01592
$123$	8.56265e21	0.974102
$124$	−2.49386e21	−0.260582
$125$	0	0
$126$	3.73838e21	0.330212
$127$	8.37131e20	0.0680541	0.0340271	−	0.999421i	$-0.489167\pi$
$127$	8.37131e20	0.0680541	0.0340271	+	0.999421i	$0.489167\pi$
$128$	1.55001e22	1.16046
$129$	−4.75900e21	−0.328340
$130$	0	0
$131$	1.90600e22	1.11886	0.559429	−	0.828879i	$-0.311021\pi$
$131$	1.90600e22	1.11886	0.559429	+	0.828879i	$0.311021\pi$
$132$	7.40284e21	0.401210
$133$	7.66104e20	0.0383571
$134$	−1.32344e22	−0.612499
$135$	0	0
$136$	−8.66701e21	−0.343329
$137$	8.63318e20	0.0316669	0.0158334	−	0.999875i	$-0.494960\pi$
$137$	8.63318e20	0.0316669	0.0158334	+	0.999875i	$0.494960\pi$
$138$	2.54448e22	0.864705
$139$	8.91438e21	0.280825	0.140412	−	0.990093i	$-0.455157\pi$
$139$	8.91438e21	0.280825	0.140412	+	0.990093i	$0.455157\pi$
$140$	0	0
$141$	−2.92880e21	−0.0794124
$142$	−2.80307e22	−0.705677
$143$	8.11798e22	1.89854
$144$	−1.90321e22	−0.413694
$145$	0	0
$146$	2.23074e22	0.419512
$147$	−9.95412e21	−0.174249
$148$	−3.45411e22	−0.563103
$149$	−9.42240e22	−1.43122	−0.715609	−	0.698501i	$-0.753851\pi$
$149$	−9.42240e22	−1.43122	−0.715609	+	0.698501i	$0.753851\pi$
$150$	0	0
$151$	2.62382e22	0.346479	0.173239	−	0.984880i	$-0.444577\pi$
$151$	2.62382e22	0.346479	0.173239	+	0.984880i	$0.444577\pi$
$152$	−2.62569e21	−0.0323508
$153$	1.41196e22	0.162391
$154$	1.58033e23	1.69739
$155$	0	0
$156$	2.76608e22	0.259452
$157$	−1.62221e23	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$157$	−1.62221e23	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$158$	1.64013e23	1.34579
$159$	7.36083e22	0.565272
$160$	0	0
$161$	1.56733e23	1.05558
$162$	2.08733e22	0.131730
$163$	−2.32312e23	−1.37436	−0.687181	−	0.726486i	$-0.741152\pi$
$163$	−2.32312e23	−1.37436	−0.687181	+	0.726486i	$0.741152\pi$
$164$	1.23336e23	0.684275
$165$	0	0
$166$	3.79387e23	1.85331
$167$	1.76825e23	0.811002	0.405501	−	0.914095i	$-0.367097\pi$
$167$	1.76825e23	0.811002	0.405501	+	0.914095i	$0.367097\pi$
$168$	−7.89226e22	−0.339980
$169$	5.62653e22	0.227735
$170$	0	0
$171$	4.27756e21	0.0153016
$172$	−6.85486e22	−0.230648
$173$	−1.88916e23	−0.598113	−0.299057	−	0.954235i	$-0.596672\pi$
$173$	−1.88916e23	−0.598113	−0.299057	+	0.954235i	$0.596672\pi$
$174$	−3.54120e23	−1.05532
$175$	0	0
$176$	−8.04546e23	−2.12651
$177$	4.25170e23	1.05887
$178$	5.40930e22	0.126979
$179$	−1.14808e23	−0.254107	−0.127053	−	0.991896i	$-0.540552\pi$
$179$	−1.14808e23	−0.254107	−0.127053	+	0.991896i	$0.540552\pi$
$180$	0	0
$181$	3.15471e23	0.621347	0.310673	−	0.950517i	$-0.399445\pi$
$181$	3.15471e23	0.621347	0.310673	+	0.950517i	$0.399445\pi$
$182$	5.90494e23	1.09765
$183$	−2.81910e23	−0.494737
$184$	−5.37176e23	−0.890286
$185$	0	0
$186$	−2.97256e23	−0.439789
$187$	5.96881e23	0.834739
$188$	−4.21863e22	−0.0557846
$189$	1.28574e23	0.160807
$190$	0	0
$191$	−4.11188e23	−0.460458	−0.230229	−	0.973137i	$-0.573947\pi$
$191$	−4.11188e23	−0.460458	−0.230229	+	0.973137i	$0.573947\pi$
$192$	1.80909e23	0.191777
$193$	9.08221e22	0.0911675	0.0455837	−	0.998961i	$-0.485485\pi$
$193$	9.08221e22	0.0911675	0.0455837	+	0.998961i	$0.485485\pi$
$194$	−4.23294e23	−0.402461
$195$	0	0
$196$	−1.43379e23	−0.122404
$197$	−1.22586e24	−0.992076	−0.496038	−	0.868301i	$-0.665212\pi$
$197$	−1.22586e24	−0.992076	−0.496038	+	0.868301i	$0.665212\pi$
$198$	8.82382e23	0.677130
$199$	1.05217e24	0.765824	0.382912	−	0.923785i	$-0.374921\pi$
$199$	1.05217e24	0.765824	0.382912	+	0.923785i	$0.374921\pi$
$200$	0	0
$201$	−4.55174e23	−0.298276
$202$	3.69517e24	2.29850
$203$	−2.18129e24	−1.28827
$204$	2.03378e23	0.114074
$205$	0	0
$206$	2.63267e24	1.33288
$207$	8.75124e23	0.421096
$208$	−3.00620e24	−1.37516
$209$	1.80826e23	0.0786548
$210$	0	0
$211$	5.34118e22	0.0210219	0.0105109	−	0.999945i	$-0.496654\pi$
$211$	5.34118e22	0.0210219	0.0105109	+	0.999945i	$0.496654\pi$
$212$	1.06025e24	0.397085
$213$	−9.64062e23	−0.343652
$214$	−2.65814e24	−0.902057
$215$	0	0
$216$	−4.40666e23	−0.135627
$217$	−1.83102e24	−0.536866
$218$	4.89670e24	1.36808
$219$	7.67221e23	0.204295
$220$	0	0
$221$	2.23026e24	0.539803
$222$	−4.11713e24	−0.950360
$223$	−1.04158e24	−0.229346	−0.114673	−	0.993403i	$-0.536582\pi$
$223$	−1.04158e24	−0.229346	−0.114673	+	0.993403i	$0.536582\pi$
$224$	−3.04922e24	−0.640596
$225$	0	0
$226$	−5.67182e24	−1.08539
$227$	−4.64270e24	−0.848201	−0.424101	−	0.905615i	$-0.639410\pi$
$227$	−4.64270e24	−0.848201	−0.424101	+	0.905615i	$0.639410\pi$
$228$	6.16139e22	0.0107489
$229$	−3.77779e24	−0.629456	−0.314728	−	0.949182i	$-0.601913\pi$
$229$	−3.77779e24	−0.629456	−0.314728	+	0.949182i	$0.601913\pi$
$230$	0	0
$231$	5.43525e24	0.826597
$232$	7.47600e24	1.08654
$233$	−1.16866e25	−1.62350	−0.811750	−	0.584005i	$-0.801485\pi$
$233$	−1.16866e25	−1.62350	−0.811750	+	0.584005i	$0.801485\pi$
$234$	3.29704e24	0.437882
$235$	0	0
$236$	6.12414e24	0.743822
$237$	5.64091e24	0.655378
$238$	4.34165e24	0.482611
$239$	−1.39609e25	−1.48503	−0.742514	−	0.669831i	$-0.766367\pi$
$239$	−1.39609e25	−1.48503	−0.742514	+	0.669831i	$0.766367\pi$
$240$	0	0
$241$	1.83127e25	1.78473	0.892366	−	0.451312i	$-0.149044\pi$
$241$	1.83127e25	1.78473	0.892366	+	0.451312i	$0.149044\pi$
$242$	2.45958e25	2.29508
$243$	7.17898e23	0.0641500
$244$	−4.06063e24	−0.347537
$245$	0	0
$246$	1.47011e25	1.15486
$247$	6.75660e23	0.0508640
$248$	6.27550e24	0.452800
$249$	1.30483e25	0.902529
$250$	0	0
$251$	2.08800e25	1.32787	0.663936	−	0.747789i	$-0.268885\pi$
$251$	2.08800e25	1.32787	0.663936	+	0.747789i	$0.268885\pi$
$252$	1.85198e24	0.112962
$253$	3.69943e25	2.16456
$254$	1.43726e24	0.0806828
$255$	0	0
$256$	2.01868e25	1.04363
$257$	2.97816e24	0.147792	0.0738959	−	0.997266i	$-0.476457\pi$
$257$	2.97816e24	0.147792	0.0738959	+	0.997266i	$0.476457\pi$
$258$	−8.17066e24	−0.389269
$259$	−2.53605e25	−1.16014
$260$	0	0
$261$	−1.21793e25	−0.513922
$262$	3.27238e25	1.32648
$263$	4.48688e25	1.74747	0.873734	−	0.486404i	$-0.161692\pi$
$263$	4.48688e25	1.74747	0.873734	+	0.486404i	$0.161692\pi$
$264$	−1.86284e25	−0.697162
$265$	0	0
$266$	1.31531e24	0.0454749
$267$	1.86043e24	0.0618365
$268$	−6.55632e24	−0.209529
$269$	5.40734e25	1.66182	0.830912	−	0.556404i	$-0.187819\pi$
$269$	5.40734e25	1.66182	0.830912	+	0.556404i	$0.187819\pi$
$270$	0	0
$271$	1.64805e25	0.468591	0.234295	−	0.972165i	$-0.424722\pi$
$271$	1.64805e25	0.468591	0.234295	+	0.972165i	$0.424722\pi$
$272$	−2.21033e25	−0.604622
$273$	2.03089e25	0.534539
$274$	1.48222e24	0.0375432
$275$	0	0
$276$	1.26053e25	0.295806
$277$	2.02544e25	0.457596	0.228798	−	0.973474i	$-0.426520\pi$
$277$	2.02544e25	0.457596	0.228798	+	0.973474i	$0.426520\pi$
$278$	1.53050e25	0.332937
$279$	−1.02235e25	−0.214169
$280$	0	0
$281$	9.60686e25	1.86709	0.933545	−	0.358461i	$-0.116699\pi$
$281$	9.60686e25	1.86709	0.933545	+	0.358461i	$0.116699\pi$
$282$	−5.02841e24	−0.0941488
$283$	9.26489e24	0.167141	0.0835704	−	0.996502i	$-0.473368\pi$
$283$	9.26489e24	0.167141	0.0835704	+	0.996502i	$0.473368\pi$
$284$	−1.38863e25	−0.241404
$285$	0	0
$286$	1.39376e26	2.25085
$287$	9.05550e25	1.40978
$288$	−1.70254e25	−0.255550
$289$	−5.26938e25	−0.762662
$290$	0	0
$291$	−1.45584e25	−0.195991
$292$	1.10510e25	0.143510
$293$	−4.08300e25	−0.511528	−0.255764	−	0.966739i	$-0.582327\pi$
$293$	−4.08300e25	−0.511528	−0.255764	+	0.966739i	$0.582327\pi$
$294$	−1.70901e25	−0.206584
$295$	0	0
$296$	8.69186e25	0.978476
$297$	3.03479e25	0.329750
$298$	−1.61772e26	−1.69681
$299$	1.38230e26	1.39976
$300$	0	0
$301$	−5.03292e25	−0.475195
$302$	4.50480e25	0.410774
$303$	1.27088e26	1.11933
$304$	−6.69625e24	−0.0569716
$305$	0	0
$306$	2.42417e25	0.192525
$307$	−1.09642e26	−0.841445	−0.420723	−	0.907189i	$-0.638223\pi$
$307$	−1.09642e26	−0.841445	−0.420723	+	0.907189i	$0.638223\pi$
$308$	7.82893e25	0.580657
$309$	9.05459e25	0.649089
$310$	0	0
$311$	−6.99628e25	−0.468687	−0.234343	−	0.972154i	$-0.575294\pi$
$311$	−6.99628e25	−0.468687	−0.234343	+	0.972154i	$0.575294\pi$
$312$	−6.96053e25	−0.450836
$313$	−6.95169e25	−0.435386	−0.217693	−	0.976017i	$-0.569853\pi$
$313$	−6.95169e25	−0.435386	−0.217693	+	0.976017i	$0.569853\pi$
$314$	−2.78515e26	−1.68689
$315$	0	0
$316$	8.12516e25	0.460381
$317$	1.47035e26	0.805933	0.402967	−	0.915215i	$-0.367979\pi$
$317$	1.47035e26	0.805933	0.402967	+	0.915215i	$0.367979\pi$
$318$	1.26377e26	0.670167
$319$	−5.14858e26	−2.64171
$320$	0	0
$321$	−9.14218e25	−0.439286
$322$	2.69093e26	1.25146
$323$	4.96784e24	0.0223636
$324$	1.03406e25	0.0450633
$325$	0	0
$326$	−3.98853e26	−1.62940
$327$	1.68413e26	0.666229
$328$	−3.10362e26	−1.18903
$329$	−3.09737e25	−0.114931
$330$	0	0
$331$	4.36548e26	1.51998	0.759990	−	0.649935i	$-0.225204\pi$
$331$	4.36548e26	1.51998	0.759990	+	0.649935i	$0.225204\pi$
$332$	1.87948e26	0.633997
$333$	−1.41601e26	−0.462809
$334$	3.03589e26	0.961497
$335$	0	0
$336$	−2.01275e26	−0.598725
$337$	−4.15229e26	−1.19722	−0.598610	−	0.801041i	$-0.704280\pi$
$337$	−4.15229e26	−1.19722	−0.598610	+	0.801041i	$0.704280\pi$
$338$	9.66010e25	0.269995
$339$	−1.95072e26	−0.528564
$340$	0	0
$341$	−4.32182e26	−1.10090
$342$	7.34407e24	0.0181411
$343$	−4.54070e26	−1.08776
$344$	1.72495e26	0.400785
$345$	0	0
$346$	−3.24346e26	−0.709104
$347$	−2.48113e26	−0.526246	−0.263123	−	0.964762i	$-0.584753\pi$
$347$	−2.48113e26	−0.526246	−0.263123	+	0.964762i	$0.584753\pi$
$348$	−1.75430e26	−0.361013
$349$	−8.30405e24	−0.0165815	−0.00829074	−	0.999966i	$-0.502639\pi$
$349$	−8.30405e24	−0.0165815	−0.00829074	+	0.999966i	$0.502639\pi$
$350$	0	0
$351$	1.13395e26	0.213241
$352$	−7.19718e26	−1.31360
$353$	−9.55014e25	−0.169190	−0.0845951	−	0.996415i	$-0.526960\pi$
$353$	−9.55014e25	−0.169190	−0.0845951	+	0.996415i	$0.526960\pi$
$354$	7.29967e26	1.25536
$355$	0	0
$356$	2.67976e25	0.0434381
$357$	1.49323e26	0.235023
$358$	−1.97112e26	−0.301260
$359$	7.06762e25	0.104901	0.0524507	−	0.998624i	$-0.483297\pi$
$359$	7.06762e25	0.104901	0.0524507	+	0.998624i	$0.483297\pi$
$360$	0	0
$361$	−7.12704e26	−0.997893
$362$	5.41627e26	0.736648
$363$	8.45924e26	1.11767
$364$	2.92529e26	0.375496
$365$	0	0
$366$	−4.84008e26	−0.586544
$367$	1.33814e27	1.57583	0.787915	−	0.615785i	$-0.211161\pi$
$367$	1.33814e27	1.57583	0.787915	+	0.615785i	$0.211161\pi$
$368$	−1.36995e27	−1.56784
$369$	5.05616e26	0.562398
$370$	0	0
$371$	7.78451e26	0.818098
$372$	−1.47260e26	−0.150447
$373$	9.62756e24	0.00956255	0.00478127	−	0.999989i	$-0.498478\pi$
$373$	9.62756e24	0.00956255	0.00478127	+	0.999989i	$0.498478\pi$
$374$	1.02478e27	0.989639
$375$	0	0
$376$	1.06157e26	0.0969340
$377$	−1.92378e27	−1.70833
$378$	2.20747e26	0.190648
$379$	−1.97813e27	−1.66166	−0.830830	−	0.556526i	$-0.812134\pi$
$379$	−1.97813e27	−1.66166	−0.830830	+	0.556526i	$0.812134\pi$
$380$	0	0
$381$	4.94318e25	0.0392911
$382$	−7.05962e26	−0.545904
$383$	2.14854e27	1.61643	0.808215	−	0.588888i	$-0.200434\pi$
$383$	2.14854e27	1.61643	0.808215	+	0.588888i	$0.200434\pi$
$384$	9.15264e26	0.669990
$385$	0	0
$386$	1.55931e26	0.108085
$387$	−2.81014e26	−0.189567
$388$	−2.09699e26	−0.137677
$389$	2.40240e27	1.53524	0.767618	−	0.640907i	$-0.221442\pi$
$389$	2.40240e27	1.53524	0.767618	+	0.640907i	$0.221442\pi$
$390$	0	0
$391$	1.01635e27	0.615440
$392$	3.60796e26	0.212696
$393$	1.12547e27	0.645972
$394$	−2.10466e27	−1.17617
$395$	0	0
$396$	4.37130e26	0.231639
$397$	−2.20346e27	−1.13711	−0.568557	−	0.822644i	$-0.692498\pi$
$397$	−2.20346e27	−1.13711	−0.568557	+	0.822644i	$0.692498\pi$
$398$	1.80646e27	0.907935
$399$	4.52376e25	0.0221455
$400$	0	0
$401$	−2.56251e27	−1.19028	−0.595141	−	0.803622i	$-0.702904\pi$
$401$	−2.56251e27	−1.19028	−0.595141	+	0.803622i	$0.702904\pi$
$402$	−7.81481e26	−0.353627
$403$	−1.61486e27	−0.711920
$404$	1.83058e27	0.786292
$405$	0	0
$406$	−3.74503e27	−1.52733
$407$	−5.98593e27	−2.37898
$408$	−5.11778e26	−0.198221
$409$	−4.40538e27	−1.66299	−0.831494	−	0.555534i	$-0.812514\pi$
$409$	−4.40538e27	−1.66299	−0.831494	+	0.555534i	$0.812514\pi$
$410$	0	0
$411$	5.09781e25	0.0182829
$412$	1.30422e27	0.455964
$413$	4.49641e27	1.53247
$414$	1.50249e27	0.499238
$415$	0	0
$416$	−2.68924e27	−0.849473
$417$	5.26385e26	0.162134
$418$	3.10458e26	0.0932505
$419$	1.22792e27	0.359686	0.179843	−	0.983695i	$-0.442441\pi$
$419$	1.22792e27	0.359686	0.179843	+	0.983695i	$0.442441\pi$
$420$	0	0
$421$	−2.61500e25	−0.00728633	−0.00364317	−	0.999993i	$-0.501160\pi$
$421$	−2.61500e25	−0.00728633	−0.00364317	+	0.999993i	$0.501160\pi$
$422$	9.17019e25	0.0249229
$423$	−1.72942e26	−0.0458488
$424$	−2.66800e27	−0.689994
$425$	0	0
$426$	−1.65518e27	−0.407423
$427$	−2.98137e27	−0.716016
$428$	−1.31684e27	−0.308584
$429$	4.79359e27	1.09612
$430$	0	0
$431$	−3.68868e27	−0.803265	−0.401633	−	0.915801i	$-0.631557\pi$
$431$	−3.68868e27	−0.803265	−0.401633	+	0.915801i	$0.631557\pi$
$432$	−1.12382e27	−0.238847
$433$	−4.87388e27	−1.01100	−0.505501	−	0.862826i	$-0.668692\pi$
$433$	−4.87388e27	−1.01100	−0.505501	+	0.862826i	$0.668692\pi$
$434$	−3.14365e27	−0.636491
$435$	0	0
$436$	2.42582e27	0.468004
$437$	3.07904e26	0.0579910
$438$	1.31723e27	0.242205
$439$	8.77061e27	1.57453	0.787267	−	0.616612i	$-0.211495\pi$
$439$	8.77061e27	1.57453	0.787267	+	0.616612i	$0.211495\pi$
$440$	0	0
$441$	−5.87781e26	−0.100603
$442$	3.82909e27	0.639973
$443$	−2.39913e26	−0.0391575	−0.0195787	−	0.999808i	$-0.506233\pi$
$443$	−2.39913e26	−0.0391575	−0.0195787	+	0.999808i	$0.506233\pi$
$444$	−2.03962e27	−0.325108
$445$	0	0
$446$	−1.78827e27	−0.271905
$447$	−5.56383e27	−0.826314
$448$	1.91322e27	0.277552
$449$	−1.18190e27	−0.167491	−0.0837457	−	0.996487i	$-0.526688\pi$
$449$	−1.18190e27	−0.167491	−0.0837457	+	0.996487i	$0.526688\pi$
$450$	0	0
$451$	2.13740e28	2.89090
$452$	−2.80981e27	−0.371299
$453$	1.54934e27	0.200040
$454$	−7.97098e27	−1.00560
$455$	0	0
$456$	−1.55044e26	−0.0186777
$457$	−1.24883e28	−1.47023	−0.735114	−	0.677944i	$-0.762871\pi$
$457$	−1.24883e28	−1.47023	−0.735114	+	0.677944i	$0.762871\pi$
$458$	−6.48603e27	−0.746262
$459$	8.33748e26	0.0937565
$460$	0	0
$461$	−4.81654e27	−0.517459	−0.258729	−	0.965950i	$-0.583304\pi$
$461$	−4.81654e27	−0.517459	−0.258729	+	0.965950i	$0.583304\pi$
$462$	9.33170e27	0.979986
$463$	−2.77535e27	−0.284917	−0.142458	−	0.989801i	$-0.545501\pi$
$463$	−2.77535e27	−0.284917	−0.142458	+	0.989801i	$0.545501\pi$
$464$	1.90659e28	1.91346
$465$	0	0
$466$	−2.00646e28	−1.92477
$467$	1.30821e28	1.22702	0.613508	−	0.789688i	$-0.289758\pi$
$467$	1.30821e28	1.22702	0.613508	+	0.789688i	$0.289758\pi$
$468$	1.63334e27	0.149795
$469$	−4.81373e27	−0.431685
$470$	0	0
$471$	−9.57900e27	−0.821486
$472$	−1.54107e28	−1.29250
$473$	−1.18794e28	−0.974433
$474$	9.68479e27	0.776994
$475$	0	0
$476$	2.15084e27	0.165096
$477$	4.34650e27	0.326360
$478$	−2.39692e28	−1.76060
$479$	−1.98117e28	−1.42363	−0.711817	−	0.702365i	$-0.752127\pi$
$479$	−1.98117e28	−1.42363	−0.711817	+	0.702365i	$0.752127\pi$
$480$	0	0
$481$	−2.23665e28	−1.53842
$482$	3.14408e28	2.11592
$483$	9.25495e27	0.609437
$484$	1.21847e28	0.785123
$485$	0	0
$486$	1.23255e27	0.0760542
$487$	2.14424e28	1.29485	0.647425	−	0.762129i	$-0.275846\pi$
$487$	2.14424e28	1.29485	0.647425	+	0.762129i	$0.275846\pi$
$488$	1.02181e28	0.603896
$489$	−1.37178e28	−0.793489
$490$	0	0
$491$	−1.64770e28	−0.913109	−0.456554	−	0.889695i	$-0.650917\pi$
$491$	−1.64770e28	−0.913109	−0.456554	+	0.889695i	$0.650917\pi$
$492$	7.28289e27	0.395066
$493$	−1.41447e28	−0.751107
$494$	1.16003e27	0.0603027
$495$	0	0
$496$	1.60043e28	0.797406
$497$	−1.01955e28	−0.497356
$498$	2.24024e28	1.07001
$499$	−2.46207e28	−1.15145	−0.575724	−	0.817644i	$-0.695280\pi$
$499$	−2.46207e28	−1.15145	−0.575724	+	0.817644i	$0.695280\pi$
$500$	0	0
$501$	1.04414e28	0.468232
$502$	3.58486e28	1.57428
$503$	−9.01822e27	−0.387844	−0.193922	−	0.981017i	$-0.562121\pi$
$503$	−9.01822e27	−0.387844	−0.193922	+	0.981017i	$0.562121\pi$
$504$	−4.66030e27	−0.196288
$505$	0	0
$506$	6.35150e28	2.56623
$507$	3.32241e27	0.131483
$508$	7.12015e26	0.0276007
$509$	−1.26550e28	−0.480534	−0.240267	−	0.970707i	$-0.577235\pi$
$509$	−1.26550e28	−0.480534	−0.240267	+	0.970707i	$0.577235\pi$
$510$	0	0
$511$	8.11381e27	0.295669
$512$	2.15236e27	0.0768388
$513$	2.52585e26	0.00883438
$514$	5.11316e27	0.175217
$515$	0	0
$516$	−4.04773e27	−0.133165
$517$	−7.31084e27	−0.235677
$518$	−4.35410e28	−1.37542
$519$	−1.11553e28	−0.345321
$520$	0	0
$521$	−3.11290e28	−0.925483	−0.462742	−	0.886493i	$-0.653134\pi$
$521$	−3.11290e28	−0.925483	−0.462742	+	0.886493i	$0.653134\pi$
$522$	−2.09105e28	−0.609289
$523$	2.19095e28	0.625699	0.312850	−	0.949803i	$-0.398716\pi$
$523$	2.19095e28	0.625699	0.312850	+	0.949803i	$0.398716\pi$
$524$	1.62113e28	0.453774
$525$	0	0
$526$	7.70346e28	2.07174
$527$	−1.18734e28	−0.313013
$528$	−4.75077e28	−1.22774
$529$	2.35210e28	0.595896
$530$	0	0
$531$	2.51058e28	0.611339
$532$	6.51602e26	0.0155565
$533$	7.98644e28	1.86947
$534$	3.19414e27	0.0733113
$535$	0	0
$536$	1.64982e28	0.364088
$537$	−6.77931e27	−0.146709
$538$	9.28378e28	1.97020
$539$	−2.48474e28	−0.517129
$540$	0	0
$541$	8.64723e28	1.73104	0.865518	−	0.500879i	$-0.166990\pi$
$541$	8.64723e28	1.73104	0.865518	+	0.500879i	$0.166990\pi$
$542$	2.82952e28	0.555546
$543$	1.86282e28	0.358735
$544$	−1.97728e28	−0.373491
$545$	0	0
$546$	3.48681e28	0.633731
$547$	−6.69174e28	−1.19309	−0.596544	−	0.802581i	$-0.703460\pi$
$547$	−6.69174e28	−1.19309	−0.596544	+	0.802581i	$0.703460\pi$
$548$	7.34288e26	0.0128431
$549$	−1.66465e28	−0.285637
$550$	0	0
$551$	−4.28517e27	−0.0707744
$552$	−3.17197e28	−0.514007
$553$	5.96559e28	0.948505
$554$	3.47745e28	0.542511
$555$	0	0
$556$	7.58204e27	0.113894
$557$	6.14899e28	0.906409	0.453204	−	0.891407i	$-0.350281\pi$
$557$	6.14899e28	0.906409	0.453204	+	0.891407i	$0.350281\pi$
$558$	−1.75527e28	−0.253912
$559$	−4.43875e28	−0.630140
$560$	0	0
$561$	3.52452e28	0.481937
$562$	1.64939e29	2.21356
$563$	−1.12154e28	−0.147733	−0.0738663	−	0.997268i	$-0.523534\pi$
$563$	−1.12154e28	−0.147733	−0.0738663	+	0.997268i	$0.523534\pi$
$564$	−2.49106e27	−0.0322073
$565$	0	0
$566$	1.59067e28	0.198157
$567$	7.59219e27	0.0928421
$568$	3.49434e28	0.419476
$569$	−1.24550e29	−1.46779	−0.733895	−	0.679263i	$-0.762300\pi$
$569$	−1.24550e29	−1.46779	−0.733895	+	0.679263i	$0.762300\pi$
$570$	0	0
$571$	−6.70182e28	−0.761226	−0.380613	−	0.924734i	$-0.624287\pi$
$571$	−6.70182e28	−0.761226	−0.380613	+	0.924734i	$0.624287\pi$
$572$	6.90467e28	0.769990
$573$	−2.42802e28	−0.265845
$574$	1.55473e29	1.67139
$575$	0	0
$576$	1.06825e28	0.110723
$577$	−1.60912e29	−1.63773	−0.818864	−	0.573987i	$-0.805396\pi$
$577$	−1.60912e29	−1.63773	−0.818864	+	0.573987i	$0.805396\pi$
$578$	−9.04692e28	−0.904187
$579$	5.36296e27	0.0526356
$580$	0	0
$581$	1.37993e29	1.30620
$582$	−2.49951e28	−0.232361
$583$	1.83740e29	1.67759
$584$	−2.78087e28	−0.249370
$585$	0	0
$586$	−7.01004e28	−0.606451
$587$	−1.88288e29	−1.60001	−0.800006	−	0.599992i	$-0.795170\pi$
$587$	−1.88288e29	−1.60001	−0.800006	+	0.599992i	$0.795170\pi$
$588$	−8.46638e27	−0.0706702
$589$	−3.59706e27	−0.0294942
$590$	0	0
$591$	−7.23857e28	−0.572775
$592$	2.21667e29	1.72315
$593$	−2.04137e29	−1.55901	−0.779503	−	0.626398i	$-0.784529\pi$
$593$	−2.04137e29	−1.55901	−0.779503	+	0.626398i	$0.784529\pi$
$594$	5.21038e28	0.390941
$595$	0	0
$596$	−8.01414e28	−0.580459
$597$	6.21296e28	0.442149
$598$	2.37325e29	1.65951
$599$	9.66401e28	0.664012	0.332006	−	0.943277i	$-0.392275\pi$
$599$	9.66401e28	0.664012	0.332006	+	0.943277i	$0.392275\pi$
$600$	0	0
$601$	−1.22230e29	−0.810951	−0.405476	−	0.914106i	$-0.632894\pi$
$601$	−1.22230e29	−0.810951	−0.405476	+	0.914106i	$0.632894\pi$
$602$	−8.64095e28	−0.563376
$603$	−2.68776e28	−0.172210
$604$	2.23167e28	0.140521
$605$	0	0
$606$	2.18196e29	1.32704
$607$	1.06271e29	0.635235	0.317617	−	0.948219i	$-0.397117\pi$
$607$	1.06271e29	0.635235	0.317617	+	0.948219i	$0.397117\pi$
$608$	−5.99022e27	−0.0351929
$609$	−1.28803e29	−0.743781
$610$	0	0
$611$	−2.73171e28	−0.152406
$612$	1.20093e28	0.0658609
$613$	−3.11548e29	−1.67954	−0.839768	−	0.542945i	$-0.817309\pi$
$613$	−3.11548e29	−1.67954	−0.839768	+	0.542945i	$0.817309\pi$
$614$	−1.88244e29	−0.997590
$615$	0	0
$616$	−1.97006e29	−1.00898
$617$	2.29331e29	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$617$	2.29331e29	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$618$	1.55457e29	0.769539
$619$	1.75961e29	0.856376	0.428188	−	0.903690i	$-0.359152\pi$
$619$	1.75961e29	0.856376	0.428188	+	0.903690i	$0.359152\pi$
$620$	0	0
$621$	5.16752e28	0.243120
$622$	−1.20118e29	−0.555660
$623$	1.96751e28	0.0894937
$624$	−1.77513e29	−0.793949
$625$	0	0
$626$	−1.19352e29	−0.516180
$627$	1.06776e28	0.0454114
$628$	−1.37976e29	−0.577067
$629$	−1.64452e29	−0.676404
$630$	0	0
$631$	−1.30487e28	−0.0519108	−0.0259554	−	0.999663i	$-0.508263\pi$
$631$	−1.30487e28	−0.0519108	−0.0259554	+	0.999663i	$0.508263\pi$
$632$	−2.04460e29	−0.799981
$633$	3.15391e27	0.0121370
$634$	2.52442e29	0.955488
$635$	0	0
$636$	6.26069e28	0.229257
$637$	−9.28427e28	−0.334413
$638$	−8.83952e29	−3.13193
$639$	−5.69269e28	−0.198408
$640$	0	0
$641$	−3.70997e29	−1.25130	−0.625650	−	0.780104i	$-0.715166\pi$
$641$	−3.70997e29	−1.25130	−0.625650	+	0.780104i	$0.715166\pi$
$642$	−1.56961e29	−0.520803
$643$	1.15073e29	0.375627	0.187814	−	0.982205i	$-0.439860\pi$
$643$	1.15073e29	0.375627	0.187814	+	0.982205i	$0.439860\pi$
$644$	1.33308e29	0.428110
$645$	0	0
$646$	8.52921e27	0.0265135
$647$	−2.45274e29	−0.750165	−0.375082	−	0.926991i	$-0.622386\pi$
$647$	−2.45274e29	−0.750165	−0.375082	+	0.926991i	$0.622386\pi$
$648$	−2.60209e28	−0.0783041
$649$	1.06130e30	3.14247
$650$	0	0
$651$	−1.08120e29	−0.309960
$652$	−1.97591e29	−0.557400
$653$	−6.64107e29	−1.84353	−0.921764	−	0.387751i	$-0.873252\pi$
$653$	−6.64107e29	−1.84353	−0.921764	+	0.387751i	$0.873252\pi$
$654$	2.89145e29	0.789859
$655$	0	0
$656$	−7.91510e29	−2.09395
$657$	4.53036e28	0.117950
$658$	−5.31783e28	−0.136258
$659$	−1.71049e29	−0.431344	−0.215672	−	0.976466i	$-0.569194\pi$
$659$	−1.71049e29	−0.431344	−0.215672	+	0.976466i	$0.569194\pi$
$660$	0	0
$661$	−1.36074e29	−0.332399	−0.166200	−	0.986092i	$-0.553150\pi$
$661$	−1.36074e29	−0.332399	−0.166200	+	0.986092i	$0.553150\pi$
$662$	7.49502e29	1.80204
$663$	1.31694e29	0.311656
$664$	−4.72948e29	−1.10166
$665$	0	0
$666$	−2.43112e29	−0.548691
$667$	−8.76681e29	−1.94769
$668$	1.50397e29	0.328918
$669$	−6.15041e28	−0.132413
$670$	0	0
$671$	−7.03702e29	−1.46826
$672$	−1.80053e29	−0.369848
$673$	8.46772e29	1.71241	0.856206	−	0.516634i	$-0.172815\pi$
$673$	8.46772e29	1.71241	0.856206	+	0.516634i	$0.172815\pi$
$674$	−7.12900e29	−1.41938
$675$	0	0
$676$	4.78559e28	0.0923625
$677$	−2.36669e29	−0.449738	−0.224869	−	0.974389i	$-0.572195\pi$
$677$	−2.36669e29	−0.449738	−0.224869	+	0.974389i	$0.572195\pi$
$678$	−3.34916e29	−0.626648
$679$	−1.53963e29	−0.283651
$680$	0	0
$681$	−2.74147e29	−0.489709
$682$	−7.42007e29	−1.30519
$683$	1.02867e30	1.78180	0.890898	−	0.454203i	$-0.150076\pi$
$683$	1.02867e30	1.78180	0.890898	+	0.454203i	$0.150076\pi$
$684$	3.63824e27	0.00620586
$685$	0	0
$686$	−7.79586e29	−1.28962
$687$	−2.23075e29	−0.363417
$688$	4.39910e29	0.705806
$689$	6.86550e29	1.08485
$690$	0	0
$691$	8.33280e29	1.27724	0.638619	−	0.769523i	$-0.279506\pi$
$691$	8.33280e29	1.27724	0.638619	+	0.769523i	$0.279506\pi$
$692$	−1.60680e29	−0.242577
$693$	3.20946e29	0.477236
$694$	−4.25981e29	−0.623901
$695$	0	0
$696$	4.41450e29	0.627314
$697$	5.87209e29	0.821956
$698$	−1.42571e28	−0.0196584
$699$	−6.90084e29	−0.937328
$700$	0	0
$701$	6.99649e29	0.922234	0.461117	−	0.887339i	$-0.347449\pi$
$701$	6.99649e29	0.922234	0.461117	+	0.887339i	$0.347449\pi$
$702$	1.94687e29	0.252811
$703$	−4.98209e28	−0.0637354
$704$	4.51583e29	0.569148
$705$	0	0
$706$	−1.63965e29	−0.200586
$707$	1.34403e30	1.61997
$708$	3.61624e29	0.429446
$709$	2.89141e29	0.338318	0.169159	−	0.985589i	$-0.445895\pi$
$709$	2.89141e29	0.338318	0.169159	+	0.985589i	$0.445895\pi$
$710$	0	0
$711$	3.33090e29	0.378382
$712$	−6.74330e28	−0.0754801
$713$	−7.35904e29	−0.811673
$714$	2.56370e29	0.278635
$715$	0	0
$716$	−9.76491e28	−0.103058
$717$	−8.24375e29	−0.857381
$718$	1.21343e29	0.124368
$719$	1.03572e29	0.104614	0.0523070	−	0.998631i	$-0.483343\pi$
$719$	1.03572e29	0.104614	0.0523070	+	0.998631i	$0.483343\pi$
$720$	0	0
$721$	9.57575e29	0.939404
$722$	−1.22363e30	−1.18307
$723$	1.08135e30	1.03042
$724$	2.68321e29	0.251999
$725$	0	0
$726$	1.45235e30	1.32507
$727$	−3.99248e29	−0.359030	−0.179515	−	0.983755i	$-0.557453\pi$
$727$	−3.99248e29	−0.359030	−0.179515	+	0.983755i	$0.557453\pi$
$728$	−7.36116e29	−0.652479
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−3.26363e29	−0.277056
$732$	−2.39776e29	−0.200650
$733$	−1.02595e30	−0.846317	−0.423158	−	0.906056i	$-0.639079\pi$
$733$	−1.02595e30	−0.846317	−0.423158	+	0.906056i	$0.639079\pi$
$734$	2.29744e30	1.86825
$735$	0	0
$736$	−1.22551e30	−0.968500
$737$	−1.13620e30	−0.885210
$738$	8.68084e29	0.666761
$739$	−5.23998e28	−0.0396792	−0.0198396	−	0.999803i	$-0.506316\pi$
$739$	−5.23998e28	−0.0396792	−0.0198396	+	0.999803i	$0.506316\pi$
$740$	0	0
$741$	3.98971e28	0.0293663
$742$	1.33651e30	0.969910
$743$	−1.53956e30	−1.10158	−0.550788	−	0.834645i	$-0.685673\pi$
$743$	−1.53956e30	−1.10158	−0.550788	+	0.834645i	$0.685673\pi$
$744$	3.70562e29	0.261424
$745$	0	0
$746$	1.65294e28	0.0113370
$747$	7.70490e29	0.521075
$748$	5.07671e29	0.338545
$749$	−9.66839e29	−0.635763
$750$	0	0
$751$	1.28343e30	0.820639	0.410320	−	0.911942i	$-0.365417\pi$
$751$	1.28343e30	0.820639	0.410320	+	0.911942i	$0.365417\pi$
$752$	2.70731e29	0.170706
$753$	1.23294e30	0.766648
$754$	−3.30290e30	−2.02533
$755$	0	0
$756$	1.09358e29	0.0652185
$757$	−7.23137e29	−0.425318	−0.212659	−	0.977126i	$-0.568212\pi$
$757$	−7.23137e29	−0.425318	−0.212659	+	0.977126i	$0.568212\pi$
$758$	−3.39622e30	−1.97001
$759$	2.18448e30	1.24971
$760$	0	0
$761$	−2.50610e29	−0.139463	−0.0697316	−	0.997566i	$-0.522214\pi$
$761$	−2.50610e29	−0.139463	−0.0697316	+	0.997566i	$0.522214\pi$
$762$	8.48686e28	0.0465822
$763$	1.78106e30	0.964210
$764$	−3.49732e29	−0.186748
$765$	0	0
$766$	3.68880e30	1.91639
$767$	3.96558e30	2.03215
$768$	1.19201e30	0.602541
$769$	−1.40362e30	−0.699880	−0.349940	−	0.936772i	$-0.613798\pi$
$769$	−1.40362e30	−0.699880	−0.349940	+	0.936772i	$0.613798\pi$
$770$	0	0
$771$	1.75857e29	0.0853276
$772$	7.72479e28	0.0369747
$773$	1.31390e30	0.620410	0.310205	−	0.950670i	$-0.399602\pi$
$773$	1.31390e30	0.620410	0.310205	+	0.950670i	$0.399602\pi$
$774$	−4.82469e29	−0.224745
$775$	0	0
$776$	5.27682e29	0.239235
$777$	−1.49751e30	−0.669807
$778$	4.12465e30	1.82013
$779$	1.77896e29	0.0774503
$780$	0	0
$781$	−2.40648e30	−1.01988
$782$	1.74495e30	0.729645
$783$	−7.19175e29	−0.296713
$784$	9.20133e29	0.374569
$785$	0	0
$786$	1.93231e30	0.765844
$787$	3.99554e30	1.56257	0.781287	−	0.624171i	$-0.214563\pi$
$787$	3.99554e30	1.56257	0.781287	+	0.624171i	$0.214563\pi$
$788$	−1.04264e30	−0.402356
$789$	2.64946e30	1.00890
$790$	0	0
$791$	−2.06300e30	−0.764972
$792$	−1.09999e30	−0.402507
$793$	−2.62940e30	−0.949484
$794$	−3.78308e30	−1.34813
$795$	0	0
$796$	8.94914e29	0.310595
$797$	3.18052e30	1.08940	0.544698	−	0.838633i	$-0.316644\pi$
$797$	3.18052e30	1.08940	0.544698	+	0.838633i	$0.316644\pi$
$798$	7.76678e28	0.0262549
$799$	−2.00851e29	−0.0670089
$800$	0	0
$801$	1.09856e29	0.0357013
$802$	−4.39953e30	−1.41116
$803$	1.91513e30	0.606296
$804$	−3.87144e29	−0.120972
$805$	0	0
$806$	−2.77252e30	−0.844029
$807$	3.19298e30	0.959454
$808$	−4.60643e30	−1.36630
$809$	2.04768e30	0.599519	0.299759	−	0.954015i	$-0.403094\pi$
$809$	2.04768e30	0.599519	0.299759	+	0.954015i	$0.403094\pi$
$810$	0	0
$811$	2.02072e30	0.576485	0.288242	−	0.957557i	$-0.406929\pi$
$811$	2.02072e30	0.576485	0.288242	+	0.957557i	$0.406929\pi$
$812$	−1.85528e30	−0.522481
$813$	9.73159e29	0.270541
$814$	−1.02771e31	−2.82044
$815$	0	0
$816$	−1.30518e30	−0.349079
$817$	−9.88722e28	−0.0261061
$818$	−7.56354e30	−1.97158
$819$	1.19922e30	0.308616
$820$	0	0
$821$	−3.31897e30	−0.832531	−0.416265	−	0.909243i	$-0.636661\pi$
$821$	−3.31897e30	−0.832531	−0.416265	+	0.909243i	$0.636661\pi$
$822$	8.75235e28	0.0216756
$823$	8.46852e29	0.207066	0.103533	−	0.994626i	$-0.466985\pi$
$823$	8.46852e29	0.207066	0.103533	+	0.994626i	$0.466985\pi$
$824$	−3.28192e30	−0.792305
$825$	0	0
$826$	7.71983e30	1.81684
$827$	7.77607e30	1.80697	0.903487	−	0.428614i	$-0.140998\pi$
$827$	7.77607e30	1.80697	0.903487	+	0.428614i	$0.140998\pi$
$828$	7.44329e29	0.170784
$829$	4.65905e30	1.05554	0.527770	−	0.849387i	$-0.323028\pi$
$829$	4.65905e30	1.05554	0.527770	+	0.849387i	$0.323028\pi$
$830$	0	0
$831$	1.19600e30	0.264193
$832$	1.68735e30	0.368053
$833$	−6.82633e29	−0.147033
$834$	9.03742e29	0.192221
$835$	0	0
$836$	1.53800e29	0.0319000
$837$	−6.03690e29	−0.123651
$838$	2.10820e30	0.426431
$839$	−2.71344e29	−0.0542025	−0.0271013	−	0.999633i	$-0.508628\pi$
$839$	−2.71344e29	−0.0542025	−0.0271013	+	0.999633i	$0.508628\pi$
$840$	0	0
$841$	7.06813e30	1.37704
$842$	−4.48965e28	−0.00863843
$843$	5.67275e30	1.07796
$844$	4.54289e28	0.00852584
$845$	0	0
$846$	−2.96922e29	−0.0543568
$847$	8.94614e30	1.61756
$848$	−6.80417e30	−1.21512
$849$	5.47082e29	0.0964988
$850$	0	0
$851$	−1.01926e31	−1.75398
$852$	−8.19975e29	−0.139375
$853$	−9.28049e30	−1.55814	−0.779070	−	0.626937i	$-0.784308\pi$
$853$	−9.28049e30	−1.55814	−0.779070	+	0.626937i	$0.784308\pi$
$854$	−5.11866e30	−0.848885
$855$	0	0
$856$	3.31367e30	0.536210
$857$	−4.26068e30	−0.681053	−0.340526	−	0.940235i	$-0.610605\pi$
$857$	−4.26068e30	−0.681053	−0.340526	+	0.940235i	$0.610605\pi$
$858$	8.23003e30	1.29953
$859$	−1.38058e30	−0.215344	−0.107672	−	0.994186i	$-0.534340\pi$
$859$	−1.38058e30	−0.215344	−0.107672	+	0.994186i	$0.534340\pi$
$860$	0	0
$861$	5.34718e30	0.813939
$862$	−6.33303e30	−0.952325
$863$	−6.77274e30	−1.00612	−0.503061	−	0.864251i	$-0.667793\pi$
$863$	−6.77274e30	−1.00612	−0.503061	+	0.864251i	$0.667793\pi$
$864$	−1.00533e30	−0.147542
$865$	0	0
$866$	−8.36789e30	−1.19861
$867$	−3.11152e30	−0.440323
$868$	−1.55736e30	−0.217737
$869$	1.40808e31	1.94500
$870$	0	0
$871$	−4.24544e30	−0.572442
$872$	−6.10428e30	−0.813226
$873$	−8.59658e29	−0.113156
$874$	5.28636e29	0.0687522
$875$	0	0
$876$	6.52553e29	0.0828557
$877$	−7.75502e30	−0.972943	−0.486471	−	0.873697i	$-0.661716\pi$
$877$	−7.75502e30	−0.972943	−0.486471	+	0.873697i	$0.661716\pi$
$878$	1.50581e31	1.86672
$879$	−2.41097e30	−0.295331
$880$	0	0
$881$	1.13987e31	1.36335	0.681677	−	0.731653i	$-0.261251\pi$
$881$	1.13987e31	1.36335	0.681677	+	0.731653i	$0.261251\pi$
$882$	−1.00915e30	−0.119271
$883$	5.95571e28	0.00695579	0.00347789	−	0.999994i	$-0.498893\pi$
$883$	5.95571e28	0.00695579	0.00347789	+	0.999994i	$0.498893\pi$
$884$	1.89692e30	0.218928
$885$	0	0
$886$	−4.11903e29	−0.0464238
$887$	−1.42633e31	−1.58863	−0.794315	−	0.607506i	$-0.792170\pi$
$887$	−1.42633e31	−1.58863	−0.794315	+	0.607506i	$0.792170\pi$
$888$	5.13246e30	0.564923
$889$	5.22770e29	0.0568646
$890$	0	0
$891$	1.79201e30	0.190381
$892$	−8.85904e29	−0.0930156
$893$	−6.08481e28	−0.00631404
$894$	−9.55246e30	−0.979651
$895$	0	0
$896$	9.67945e30	0.969653
$897$	8.16234e30	0.808154
$898$	−2.02918e30	−0.198572
$899$	1.02417e31	0.990597
$900$	0	0
$901$	5.04791e30	0.476981
$902$	3.66967e31	3.42735
$903$	−2.97189e30	−0.274354
$904$	7.07056e30	0.645186
$905$	0	0
$906$	2.66004e30	0.237160
$907$	−4.56820e30	−0.402595	−0.201297	−	0.979530i	$-0.564516\pi$
$907$	−4.56820e30	−0.402595	−0.201297	+	0.979530i	$0.564516\pi$
$908$	−3.94880e30	−0.344005
$909$	7.50443e30	0.646245
$910$	0	0
$911$	−1.83971e31	−1.54813	−0.774065	−	0.633107i	$-0.781780\pi$
$911$	−1.83971e31	−1.54813	−0.774065	+	0.633107i	$0.781780\pi$
$912$	−3.95407e29	−0.0328926
$913$	3.25711e31	2.67848
$914$	−2.14410e31	−1.74305
$915$	0	0
$916$	−3.21317e30	−0.255288
$917$	1.19025e31	0.934893
$918$	1.43145e30	0.111155
$919$	−6.31613e29	−0.0484884	−0.0242442	−	0.999706i	$-0.507718\pi$
$919$	−6.31613e29	−0.0484884	−0.0242442	+	0.999706i	$0.507718\pi$
$920$	0	0
$921$	−6.47428e30	−0.485809
$922$	−8.26945e30	−0.613482
$923$	−8.99187e30	−0.659527
$924$	4.62290e30	0.335243
$925$	0	0
$926$	−4.76496e30	−0.337788
$927$	5.34664e30	0.374752
$928$	1.70557e31	1.18199
$929$	1.48189e31	1.01543	0.507717	−	0.861524i	$-0.330489\pi$
$929$	1.48189e31	1.01543	0.507717	+	0.861524i	$0.330489\pi$
$930$	0	0
$931$	−2.06805e29	−0.0138544
$932$	−9.93996e30	−0.658442
$933$	−4.13123e30	−0.270596
$934$	2.24605e31	1.45471
$935$	0	0
$936$	−4.11012e30	−0.260290
$937$	−1.44221e31	−0.903154	−0.451577	−	0.892232i	$-0.649138\pi$
$937$	−1.44221e31	−0.903154	−0.451577	+	0.892232i	$0.649138\pi$
$938$	−8.26461e30	−0.511791
$939$	−4.10490e30	−0.251370
$940$	0	0
$941$	−3.60517e30	−0.215891	−0.107945	−	0.994157i	$-0.534427\pi$
$941$	−3.60517e30	−0.215891	−0.107945	+	0.994157i	$0.534427\pi$
$942$	−1.64460e31	−0.973927
$943$	3.63949e31	2.13141
$944$	−3.93016e31	−2.27617
$945$	0	0
$946$	−2.03955e31	−1.15526
$947$	−9.57878e30	−0.536581	−0.268291	−	0.963338i	$-0.586459\pi$
$947$	−9.57878e30	−0.536581	−0.268291	+	0.963338i	$0.586459\pi$
$948$	4.79783e30	0.265801
$949$	7.15592e30	0.392076
$950$	0	0
$951$	8.68228e30	0.465306
$952$	−5.41235e30	−0.286878
$953$	−1.99764e31	−1.04723	−0.523614	−	0.851956i	$-0.675417\pi$
$953$	−1.99764e31	−1.04723	−0.523614	+	0.851956i	$0.675417\pi$
$954$	7.46244e30	0.386921
$955$	0	0
$956$	−1.18743e31	−0.602282
$957$	−3.04019e31	−1.52519
$958$	−3.40143e31	−1.68781
$959$	5.39123e29	0.0264602
$960$	0	0
$961$	−1.22284e31	−0.587183
$962$	−3.84007e31	−1.82390
$963$	−5.39837e30	−0.253622
$964$	1.55757e31	0.723833
$965$	0	0
$966$	1.58897e31	0.722529
$967$	1.46532e30	0.0659105	0.0329552	−	0.999457i	$-0.489508\pi$
$967$	1.46532e30	0.0659105	0.0329552	+	0.999457i	$0.489508\pi$
$968$	−3.06613e31	−1.36427
$969$	2.93346e29	0.0129116
$970$	0	0
$971$	−2.87520e31	−1.23841	−0.619207	−	0.785228i	$-0.712546\pi$
$971$	−2.87520e31	−1.23841	−0.619207	+	0.785228i	$0.712546\pi$
$972$	6.10602e29	0.0260173
$973$	5.56683e30	0.234651
$974$	3.68141e31	1.53513
$975$	0	0
$976$	2.60591e31	1.06350
$977$	−1.39361e31	−0.562662	−0.281331	−	0.959611i	$-0.590776\pi$
$977$	−1.39361e31	−0.562662	−0.281331	+	0.959611i	$0.590776\pi$
$978$	−2.35518e31	−0.940734
$979$	4.64398e30	0.183515
$980$	0	0
$981$	9.94460e30	0.384648
$982$	−2.82891e31	−1.08255
$983$	1.40200e31	0.530805	0.265403	−	0.964138i	$-0.414495\pi$
$983$	1.40200e31	0.530805	0.265403	+	0.964138i	$0.414495\pi$
$984$	−1.83265e31	−0.686486
$985$	0	0
$986$	−2.42848e31	−0.890487
$987$	−1.82897e30	−0.0663553
$988$	5.74677e29	0.0206289
$989$	−2.02278e31	−0.718434
$990$	0	0
$991$	−5.54876e31	−1.92940	−0.964701	−	0.263349i	$-0.915173\pi$
$991$	−5.54876e31	−1.92940	−0.964701	+	0.263349i	$0.915173\pi$
$992$	1.43169e31	0.492579
$993$	2.57777e31	0.877561
$994$	−1.75045e31	−0.589649
$995$	0	0
$996$	1.10981e31	0.366038
$997$	5.26811e31	1.71932	0.859659	−	0.510869i	$-0.170676\pi$
$997$	5.26811e31	1.71932	0.859659	+	0.510869i	$0.170676\pi$
$998$	−4.22709e31	−1.36512
$999$	−8.36139e30	−0.267203

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.h.1.4		4
5.2	odd	4		75.22.b.h.49.8		8
5.3	odd	4		75.22.b.h.49.1		8
5.4	even	2		15.22.a.e.1.1	✓	4
15.14	odd	2		45.22.a.g.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.a.e.1.1	✓	4	5.4	even	2
45.22.a.g.1.4		4	15.14	odd	2
75.22.a.h.1.4		4	1.1	even	1	trivial
75.22.b.h.49.1		8	5.3	odd	4
75.22.b.h.49.8		8	5.2	odd	4

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)

\(f(q)\)	\(=\)	\(q+1716.88 q^{2} +59049.0 q^{3} +850541. q^{4} +1.01380e8 q^{6} +6.24477e8 q^{7} -2.14029e9 q^{8} +3.48678e9 q^{9} +1.47398e11 q^{11} +5.02236e10 q^{12} +5.50754e11 q^{13} +1.07216e12 q^{14} -5.45834e12 q^{16} +4.04946e12 q^{17} +5.98641e12 q^{18} +1.22679e12 q^{19} +3.68748e13 q^{21} +2.53065e14 q^{22} +2.50983e14 q^{23} -1.26382e14 q^{24} +9.45581e14 q^{26} +2.05891e14 q^{27} +5.31144e14 q^{28} -3.49299e15 q^{29} -2.93209e15 q^{31} -4.88283e15 q^{32} +8.70368e15 q^{33} +6.95245e15 q^{34} +2.96565e15 q^{36} -4.06107e16 q^{37} +2.10626e15 q^{38} +3.25215e16 q^{39} +1.45009e17 q^{41} +6.33097e16 q^{42} -8.05941e16 q^{43} +1.25368e17 q^{44} +4.30909e17 q^{46} -4.95994e16 q^{47} -3.22310e17 q^{48} -1.68574e17 q^{49} +2.39117e17 q^{51} +4.68439e17 q^{52} +1.24656e18 q^{53} +3.53491e17 q^{54} -1.33656e18 q^{56} +7.24408e16 q^{57} -5.99706e18 q^{58} +7.20028e18 q^{59} -4.77418e18 q^{61} -5.03405e18 q^{62} +2.17742e18 q^{63} +3.06371e18 q^{64} +1.49432e19 q^{66} -7.70841e18 q^{67} +3.44423e18 q^{68} +1.48203e19 q^{69} -1.63265e19 q^{71} -7.46272e18 q^{72} +1.29930e19 q^{73} -6.97240e19 q^{74} +1.04344e18 q^{76} +9.20465e19 q^{77} +5.58356e19 q^{78} +9.55293e19 q^{79} +1.21577e19 q^{81} +2.48964e20 q^{82} +2.20974e20 q^{83} +3.13635e19 q^{84} -1.38371e20 q^{86} -2.06258e20 q^{87} -3.15473e20 q^{88} +3.15065e19 q^{89} +3.43933e20 q^{91} +2.13471e20 q^{92} -1.73137e20 q^{93} -8.51565e19 q^{94} -2.88326e20 q^{96} -2.46548e20 q^{97} -2.89422e20 q^{98} +5.13944e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 897 q^{2} + 236196 q^{3} - 1163123 q^{4} + 52966953 q^{6} + 234577504 q^{7} - 76855629 q^{8} + 13947137604 q^{9} + 31491830256 q^{11} - 68681250027 q^{12} + 27017977768 q^{13} - 2233308126672 q^{14}+ \cdots + 10\!\cdots\!56 q^{99}+O(q^{100}) \) 4 * q + 897 * q^2 + 236196 * q^3 - 1163123 * q^4 + 52966953 * q^6 + 234577504 * q^7 - 76855629 * q^8 + 13947137604 * q^9 + 31491830256 * q^11 - 68681250027 * q^12 + 27017977768 * q^13 - 2233308126672 * q^14 - 11324681311775 * q^16 + 2946095028888 * q^17 + 3127645607697 * q^18 - 24270353300752 * q^19 + 13851567033696 * q^21 + 56303932793676 * q^22 - 10350924920928 * q^23 - 4538248036821 * q^24 + 474751622871378 * q^26 + 823564528378596 * q^27 + 1868468795909968 * q^28 + 4728924677079096 * q^29 - 1094923910405536 * q^31 - 7933327650814221 * q^32 + 1859561084786544 * q^33 + 3862795623387294 * q^34 - 4055559132844323 * q^36 - 4813435696247096 * q^37 - 21307487199674508 * q^38 + 1595384569222632 * q^39 + 289731591445930344 * q^41 - 131874611571854928 * q^42 - 451091912658458000 * q^43 + 245434444420437276 * q^44 + 645145648102459824 * q^46 - 813883435638492480 * q^47 - 668711106779001975 * q^48 + 1545180753004755300 * q^49 + 173963965360807512 * q^51 - 884645329958480822 * q^52 - 697278335404085208 * q^53 + 184684345488900153 * q^54 + 5673275878853478000 * q^56 - 1433140092056104848 * q^57 - 12277858736996428626 * q^58 + 6622888614569598192 * q^59 + 7390887218011683320 * q^61 - 11649094150323079392 * q^62 + 817921181772715104 * q^63 + 21525702776885072473 * q^64 + 3324690927533774124 * q^66 + 24188188449376788688 * q^67 - 18995582935459304346 * q^68 - 611211765655877472 * q^69 - 37390337803999713312 * q^71 - 267979008326243229 * q^72 + 37253672904265201432 * q^73 - 77875663243460100102 * q^74 - 44795472028972724668 * q^76 + 315245139112859776128 * q^77 + 28033608578931999522 * q^78 - 109751291042259570400 * q^79 + 48630661836227715204 * q^81 + 210801413757052759338 * q^82 + 149023592088638482896 * q^83 + 110331213929687700432 * q^84 - 346200583098972939204 * q^86 + 279238273256843539704 * q^87 + 116461091469581805156 * q^88 + 558502237151273959848 * q^89 + 30360075980445717184 * q^91 + 764380916144075558544 * q^92 - 64654161985536495264 * q^93 + 1096939878232058257512 * q^94 - 468455064452928935829 * q^96 + 236244923748673777528 * q^97 - 197058383612642903463 * q^98 + 109805222495560636656 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more