LMFDB - Embedded newform 25.14.a.b.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [25,14,Mod(1,25)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("25.1"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 25 = 5^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	25.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-142] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.8077322380$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4466x - 18720 $ x^3 - x^2 - 4466*x - 18720
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$69.3208$ of defining polynomial
Character	$\chi$	$=$	25.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+90.6415 q^{2} +1125.79 q^{3} +23.8902 q^{4} +102044. q^{6} -324482. q^{7} -740370. q^{8} -326912. q^{9} -1.64726e6 q^{11} +26895.4 q^{12} -6.26700e6 q^{13} -2.94116e7 q^{14} -6.73040e7 q^{16} -1.66481e8 q^{17} -2.96318e7 q^{18} +3.12929e8 q^{19} -3.65300e8 q^{21} -1.49311e8 q^{22} +6.32351e8 q^{23} -8.33504e8 q^{24} -5.68051e8 q^{26} -2.16291e9 q^{27} -7.75194e6 q^{28} -2.82750e9 q^{29} +7.61629e9 q^{31} -3.54269e7 q^{32} -1.85448e9 q^{33} -1.50901e10 q^{34} -7.80998e6 q^{36} -1.99161e10 q^{37} +2.83644e10 q^{38} -7.05535e9 q^{39} -4.69877e10 q^{41} -3.31114e10 q^{42} +7.85897e9 q^{43} -3.93534e7 q^{44} +5.73173e10 q^{46} -8.31265e10 q^{47} -7.57704e10 q^{48} +8.39984e9 q^{49} -1.87423e11 q^{51} -1.49720e8 q^{52} +1.19285e11 q^{53} -1.96050e11 q^{54} +2.40237e11 q^{56} +3.52294e11 q^{57} -2.56289e11 q^{58} +4.20299e11 q^{59} +4.15504e11 q^{61} +6.90352e11 q^{62} +1.06077e11 q^{63} +5.48143e11 q^{64} -1.68093e11 q^{66} +1.02968e11 q^{67} -3.97727e9 q^{68} +7.11897e11 q^{69} -4.00383e11 q^{71} +2.42036e11 q^{72} -5.55011e11 q^{73} -1.80523e12 q^{74} +7.47593e9 q^{76} +5.34508e11 q^{77} -6.39508e11 q^{78} +1.60313e12 q^{79} -1.91379e12 q^{81} -4.25904e12 q^{82} +2.64201e11 q^{83} -8.72709e9 q^{84} +7.12349e11 q^{86} -3.18318e12 q^{87} +1.21958e12 q^{88} -3.69637e12 q^{89} +2.03353e12 q^{91} +1.51070e10 q^{92} +8.57437e12 q^{93} -7.53472e12 q^{94} -3.98834e10 q^{96} +1.00920e13 q^{97} +7.61375e11 q^{98} +5.38510e11 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 142 q^{2} - 416 q^{3} + 17876 q^{4} + 120376 q^{6} - 448292 q^{7} - 2580360 q^{8} + 1286119 q^{9} - 6604004 q^{11} + 23722448 q^{12} + 33501974 q^{13} - 46562928 q^{14} + 199912208 q^{16} - 83129542 q^{17}+ \cdots + 126787366508 q^{99}+O(q^{100}) $ 3 * q - 142 * q^2 - 416 * q^3 + 17876 * q^4 + 120376 * q^6 - 448292 * q^7 - 2580360 * q^8 + 1286119 * q^9 - 6604004 * q^11 + 23722448 * q^12 + 33501974 * q^13 - 46562928 * q^14 + 199912208 * q^16 - 83129542 * q^17 + 30983914 * q^18 + 97491100 * q^19 + 438200736 * q^21 + 530907256 * q^22 - 316255836 * q^23 - 3890359200 * q^24 - 3746814044 * q^26 - 6518951360 * q^27 + 6227646976 * q^28 + 2236171850 * q^29 + 7482994376 * q^31 - 32169857312 * q^32 - 359182912 * q^33 - 30236073988 * q^34 - 43361833852 * q^36 - 31447174242 * q^37 + 42518132360 * q^38 - 70188571072 * q^39 - 10752884434 * q^41 - 92283853824 * q^42 - 16930554856 * q^43 - 68395825168 * q^44 + 246828204336 * q^46 - 31934201692 * q^47 + 80078828864 * q^48 - 38956926629 * q^49 - 129369882944 * q^51 + 18152597928 * q^52 + 221149123934 * q^53 + 246121937200 * q^54 - 522876451200 * q^56 + 763110695680 * q^57 - 854269976260 * q^58 - 55436423900 * q^59 + 496161392746 * q^61 + 598069940736 * q^62 - 1085454385236 * q^63 + 3297565494336 * q^64 - 22883875168 * q^66 - 459297824792 * q^67 + 2008021178776 * q^68 - 333379292832 * q^69 + 521997878336 * q^71 + 5658040941720 * q^72 - 2505025571086 * q^73 - 2418323586508 * q^74 + 621688050000 * q^76 + 385562457456 * q^77 + 2386950332528 * q^78 + 2990636883200 * q^79 + 320549052763 * q^81 - 5784848206924 * q^82 - 5137135467696 * q^83 - 814234130688 * q^84 + 2343313645096 * q^86 - 6885420178880 * q^87 + 8660607864480 * q^88 - 19423025958450 * q^89 - 6792522370184 * q^91 - 26870022340992 * q^92 + 11095902136128 * q^93 - 7003557875968 * q^94 - 2208331214464 * q^96 + 11088325396458 * q^97 + 10176508990306 * q^98 + 126787366508 * 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 $

$2$	90.6415	1.00146	0.500729	−	0.865604i	$-0.333065\pi$
$2$	90.6415	1.00146	0.500729	+	0.865604i	$0.333065\pi$
$3$	1125.79	0.891601	0.445801	−	0.895132i	$-0.352919\pi$
$3$	1125.79	0.891601	0.445801	+	0.895132i	$0.352919\pi$
$4$	23.8902	0.00291628
$5$	0	0
$5$	0	0
$6$	102044.	0.892900
$7$	−324482.	−1.04245	−0.521223	−	0.853420i	$-0.674524\pi$
$7$	−324482.	−1.04245	−0.521223	+	0.853420i	$0.674524\pi$
$8$	−740370.	−0.998537
$9$	−326912.	−0.205047
$10$	0	0
$11$	−1.64726e6	−0.280356	−0.140178	−	0.990126i	$-0.544768\pi$
$11$	−1.64726e6	−0.280356	−0.140178	+	0.990126i	$0.544768\pi$
$12$	26895.4	0.00260016
$13$	−6.26700e6	−0.360104	−0.180052	−	0.983657i	$-0.557627\pi$
$13$	−6.26700e6	−0.360104	−0.180052	+	0.983657i	$0.557627\pi$
$14$	−2.94116e7	−1.04397
$15$	0	0
$16$	−6.73040e7	−1.00291
$17$	−1.66481e8	−1.67281	−0.836406	−	0.548110i	$-0.815347\pi$
$17$	−1.66481e8	−1.67281	−0.836406	+	0.548110i	$0.815347\pi$
$18$	−2.96318e7	−0.205346
$19$	3.12929e8	1.52598	0.762988	−	0.646413i	$-0.223732\pi$
$19$	3.12929e8	1.52598	0.762988	+	0.646413i	$0.223732\pi$
$20$	0	0
$21$	−3.65300e8	−0.929447
$22$	−1.49311e8	−0.280765
$23$	6.32351e8	0.890692	0.445346	−	0.895359i	$-0.353081\pi$
$23$	6.32351e8	0.890692	0.445346	+	0.895359i	$0.353081\pi$
$24$	−8.33504e8	−0.890296
$25$	0	0
$26$	−5.68051e8	−0.360629
$27$	−2.16291e9	−1.07442
$28$	−7.75194e6	−0.00304007
$29$	−2.82750e9	−0.882704	−0.441352	−	0.897334i	$-0.645501\pi$
$29$	−2.82750e9	−0.882704	−0.441352	+	0.897334i	$0.645501\pi$
$30$	0	0
$31$	7.61629e9	1.54132	0.770659	−	0.637247i	$-0.219927\pi$
$31$	7.61629e9	1.54132	0.770659	+	0.637247i	$0.219927\pi$
$32$	−3.54269e7	−0.00583255
$33$	−1.85448e9	−0.249966
$34$	−1.50901e10	−1.67525
$35$	0	0
$36$	−7.80998e6	−0.000597976 0
$37$	−1.99161e10	−1.27613	−0.638063	−	0.769984i	$-0.720264\pi$
$37$	−1.99161e10	−1.27613	−0.638063	+	0.769984i	$0.720264\pi$
$38$	2.83644e10	1.52820
$39$	−7.05535e9	−0.321069
$40$	0	0
$41$	−4.69877e10	−1.54486	−0.772430	−	0.635099i	$-0.780959\pi$
$41$	−4.69877e10	−1.54486	−0.772430	+	0.635099i	$0.780959\pi$
$42$	−3.31114e10	−0.930801
$43$	7.85897e9	0.189592	0.0947961	−	0.995497i	$-0.469780\pi$
$43$	7.85897e9	0.189592	0.0947961	+	0.995497i	$0.469780\pi$
$44$	−3.93534e7	−0.000817599 0
$45$	0	0
$46$	5.73173e10	0.891990
$47$	−8.31265e10	−1.12487	−0.562437	−	0.826840i	$-0.690136\pi$
$47$	−8.31265e10	−1.12487	−0.562437	+	0.826840i	$0.690136\pi$
$48$	−7.57704e10	−0.894194
$49$	8.39984e9	0.0866955
$50$	0	0
$51$	−1.87423e11	−1.49148
$52$	−1.49720e8	−0.00105017
$53$	1.19285e11	0.739255	0.369628	−	0.929180i	$-0.379485\pi$
$53$	1.19285e11	0.739255	0.369628	+	0.929180i	$0.379485\pi$
$54$	−1.96050e11	−1.07599
$55$	0	0
$56$	2.40237e11	1.04092
$57$	3.52294e11	1.36056
$58$	−2.56289e11	−0.883990
$59$	4.20299e11	1.29724	0.648620	−	0.761112i	$-0.275346\pi$
$59$	4.20299e11	1.29724	0.648620	+	0.761112i	$0.275346\pi$
$60$	0	0
$61$	4.15504e11	1.03260	0.516299	−	0.856409i	$-0.327309\pi$
$61$	4.15504e11	1.03260	0.516299	+	0.856409i	$0.327309\pi$
$62$	6.90352e11	1.54356
$63$	1.06077e11	0.213751
$64$	5.48143e11	0.997067
$65$	0	0
$66$	−1.68093e11	−0.250330
$67$	1.02968e11	0.139065	0.0695323	−	0.997580i	$-0.477849\pi$
$67$	1.02968e11	0.139065	0.0695323	+	0.997580i	$0.477849\pi$
$68$	−3.97727e9	−0.00487839
$69$	7.11897e11	0.794142
$70$	0	0
$71$	−4.00383e11	−0.370933	−0.185467	−	0.982651i	$-0.559380\pi$
$71$	−4.00383e11	−0.370933	−0.185467	+	0.982651i	$0.559380\pi$
$72$	2.42036e11	0.204747
$73$	−5.55011e11	−0.429243	−0.214621	−	0.976697i	$-0.568852\pi$
$73$	−5.55011e11	−0.429243	−0.214621	+	0.976697i	$0.568852\pi$
$74$	−1.80523e12	−1.27798
$75$	0	0
$76$	7.47593e9	0.00445017
$77$	5.34508e11	0.292257
$78$	−6.39508e11	−0.321537
$79$	1.60313e12	0.741980	0.370990	−	0.928637i	$-0.379018\pi$
$79$	1.60313e12	0.741980	0.370990	+	0.928637i	$0.379018\pi$
$80$	0	0
$81$	−1.91379e12	−0.752908
$82$	−4.25904e12	−1.54711
$83$	2.64201e11	0.0887005	0.0443503	−	0.999016i	$-0.485878\pi$
$83$	2.64201e11	0.0887005	0.0443503	+	0.999016i	$0.485878\pi$
$84$	−8.72709e9	−0.00271053
$85$	0	0
$86$	7.12349e11	0.189868
$87$	−3.18318e12	−0.787020
$88$	1.21958e12	0.279946
$89$	−3.69637e12	−0.788388	−0.394194	−	0.919027i	$-0.628976\pi$
$89$	−3.69637e12	−0.788388	−0.394194	+	0.919027i	$0.628976\pi$
$90$	0	0
$91$	2.03353e12	0.375390
$92$	1.51070e10	0.00259751
$93$	8.57437e12	1.37424
$94$	−7.53472e12	−1.12651
$95$	0	0
$96$	−3.98834e10	−0.00520030
$97$	1.00920e13	1.23016	0.615081	−	0.788464i	$-0.289123\pi$
$97$	1.00920e13	1.23016	0.615081	+	0.788464i	$0.289123\pi$
$98$	7.61375e11	0.0868218
$99$	5.38510e11	0.0574863
$100$	0	0
$101$	2.56737e12	0.240658	0.120329	−	0.992734i	$-0.461605\pi$
$101$	2.56737e12	0.240658	0.120329	+	0.992734i	$0.461605\pi$
$102$	−1.69884e13	−1.49365
$103$	−1.21037e13	−0.998791	−0.499395	−	0.866374i	$-0.666444\pi$
$103$	−1.21037e13	−0.998791	−0.499395	+	0.866374i	$0.666444\pi$
$104$	4.63990e12	0.359577
$105$	0	0
$106$	1.08122e13	0.740332
$107$	−9.37488e12	−0.603909	−0.301954	−	0.953322i	$-0.597639\pi$
$107$	−9.37488e12	−0.603909	−0.301954	+	0.953322i	$0.597639\pi$
$108$	−5.16724e10	−0.00313332
$109$	−2.49908e13	−1.42727	−0.713637	−	0.700516i	$-0.752953\pi$
$109$	−2.49908e13	−1.42727	−0.713637	+	0.700516i	$0.752953\pi$
$110$	0	0
$111$	−2.24214e13	−1.13779
$112$	2.18390e13	1.04548
$113$	−1.35600e13	−0.612703	−0.306352	−	0.951918i	$-0.599108\pi$
$113$	−1.35600e13	−0.612703	−0.306352	+	0.951918i	$0.599108\pi$
$114$	3.19324e13	1.36254
$115$	0	0
$116$	−6.75494e10	−0.00257421
$117$	2.04876e12	0.0738384
$118$	3.80966e13	1.29913
$119$	5.40202e13	1.74382
$120$	0	0
$121$	−3.18092e13	−0.921400
$122$	3.76619e13	1.03410
$123$	−5.28985e13	−1.37740
$124$	1.81955e11	0.00449492
$125$	0	0
$126$	9.61499e12	0.214062
$127$	1.16836e13	0.247088	0.123544	−	0.992339i	$-0.460574\pi$
$127$	1.16836e13	0.247088	0.123544	+	0.992339i	$0.460574\pi$
$128$	4.99748e13	1.00435
$129$	8.84758e12	0.169041
$130$	0	0
$131$	5.12734e13	0.886398	0.443199	−	0.896423i	$-0.353843\pi$
$131$	5.12734e13	0.886398	0.443199	+	0.896423i	$0.353843\pi$
$132$	−4.43038e10	−0.000728972 0
$133$	−1.01540e14	−1.59075
$134$	9.33319e12	0.139267
$135$	0	0
$136$	1.23258e14	1.67036
$137$	7.15579e13	0.924642	0.462321	−	0.886713i	$-0.347017\pi$
$137$	7.15579e13	0.924642	0.462321	+	0.886713i	$0.347017\pi$
$138$	6.45275e13	0.795299
$139$	−1.16286e14	−1.36751	−0.683754	−	0.729712i	$-0.739654\pi$
$139$	−1.16286e14	−1.36751	−0.683754	+	0.729712i	$0.739654\pi$
$140$	0	0
$141$	−9.35833e13	−1.00294
$142$	−3.62913e13	−0.371474
$143$	1.03234e13	0.100958
$144$	2.20025e13	0.205644
$145$	0	0
$146$	−5.03070e13	−0.429868
$147$	9.45649e12	0.0772978
$148$	−4.75800e11	−0.00372154
$149$	−1.43657e14	−1.07552	−0.537758	−	0.843099i	$-0.680729\pi$
$149$	−1.43657e14	−1.07552	−0.537758	+	0.843099i	$0.680729\pi$
$150$	0	0
$151$	−1.99719e13	−0.137110	−0.0685551	−	0.997647i	$-0.521839\pi$
$151$	−1.99719e13	−0.137110	−0.0685551	+	0.997647i	$0.521839\pi$
$152$	−2.31683e14	−1.52374
$153$	5.44246e13	0.343006
$154$	4.84486e13	0.292683
$155$	0	0
$156$	−1.68554e11	−0.000936329 0
$157$	−2.46452e14	−1.31336	−0.656682	−	0.754168i	$-0.728041\pi$
$157$	−2.46452e14	−1.31336	−0.656682	+	0.754168i	$0.728041\pi$
$158$	1.45310e14	0.743061
$159$	1.34291e14	0.659121
$160$	0	0
$161$	−2.05187e14	−0.928499
$162$	−1.73469e14	−0.754005
$163$	−1.33661e14	−0.558194	−0.279097	−	0.960263i	$-0.590035\pi$
$163$	−1.33661e14	−0.558194	−0.279097	+	0.960263i	$0.590035\pi$
$164$	−1.12255e12	−0.00450525
$165$	0	0
$166$	2.39476e13	0.0888298
$167$	1.57170e14	0.560676	0.280338	−	0.959901i	$-0.409553\pi$
$167$	1.57170e14	0.560676	0.280338	+	0.959901i	$0.409553\pi$
$168$	2.70457e14	0.928087
$169$	−2.63600e14	−0.870325
$170$	0	0
$171$	−1.02300e14	−0.312897
$172$	1.87752e11	0.000552904 0
$173$	−3.32272e14	−0.942309	−0.471155	−	0.882051i	$-0.656163\pi$
$173$	−3.32272e14	−0.942309	−0.471155	+	0.882051i	$0.656163\pi$
$174$	−2.88528e14	−0.788167
$175$	0	0
$176$	1.10867e14	0.281172
$177$	4.73170e14	1.15662
$178$	−3.35044e14	−0.789537
$179$	4.96701e14	1.12863	0.564313	−	0.825561i	$-0.309141\pi$
$179$	4.96701e14	1.12863	0.564313	+	0.825561i	$0.309141\pi$
$180$	0	0
$181$	5.73895e14	1.21317	0.606585	−	0.795019i	$-0.292539\pi$
$181$	5.73895e14	1.21317	0.606585	+	0.795019i	$0.292539\pi$
$182$	1.84323e14	0.375937
$183$	4.67771e14	0.920665
$184$	−4.68174e14	−0.889388
$185$	0	0
$186$	7.77194e14	1.37624
$187$	2.74238e14	0.468984
$188$	−1.98591e12	−0.00328045
$189$	7.01828e14	1.12003
$190$	0	0
$191$	−9.21109e14	−1.37276	−0.686379	−	0.727244i	$-0.740801\pi$
$191$	−9.21109e14	−1.37276	−0.686379	+	0.727244i	$0.740801\pi$
$192$	6.17096e14	0.888986
$193$	−8.88779e14	−1.23786	−0.618929	−	0.785447i	$-0.712433\pi$
$193$	−8.88779e14	−1.23786	−0.618929	+	0.785447i	$0.712433\pi$
$194$	9.14757e14	1.23195
$195$	0	0
$196$	2.00674e11	0.000252829 0
$197$	4.70943e13	0.0574035	0.0287017	−	0.999588i	$-0.490863\pi$
$197$	4.70943e13	0.0574035	0.0287017	+	0.999588i	$0.490863\pi$
$198$	4.88113e13	0.0575701
$199$	4.94174e14	0.564072	0.282036	−	0.959404i	$-0.408990\pi$
$199$	4.94174e14	0.564072	0.282036	+	0.959404i	$0.408990\pi$
$200$	0	0
$201$	1.15921e14	0.123990
$202$	2.32710e14	0.241008
$203$	9.17473e14	0.920172
$204$	−4.47758e12	−0.00434958
$205$	0	0
$206$	−1.09709e15	−1.00025
$207$	−2.06723e14	−0.182634
$208$	4.21795e14	0.361151
$209$	−5.15477e14	−0.427817
$210$	0	0
$211$	−1.22476e15	−0.955466	−0.477733	−	0.878505i	$-0.658541\pi$
$211$	−1.22476e15	−0.955466	−0.477733	+	0.878505i	$0.658541\pi$
$212$	2.84975e12	0.00215588
$213$	−4.50748e14	−0.330725
$214$	−8.49754e14	−0.604789
$215$	0	0
$216$	1.60136e15	1.07285
$217$	−2.47135e15	−1.60674
$218$	−2.26520e15	−1.42935
$219$	−6.24828e14	−0.382713
$220$	0	0
$221$	1.04334e15	0.602387
$222$	−2.03231e15	−1.13945
$223$	1.23977e15	0.675087	0.337544	−	0.941310i	$-0.390404\pi$
$223$	1.23977e15	0.675087	0.337544	+	0.941310i	$0.390404\pi$
$224$	1.14954e13	0.00608012
$225$	0	0
$226$	−1.22910e15	−0.613596
$227$	3.31953e14	0.161031	0.0805154	−	0.996753i	$-0.474343\pi$
$227$	3.31953e14	0.161031	0.0805154	+	0.996753i	$0.474343\pi$
$228$	8.41636e12	0.00396778
$229$	−1.92799e15	−0.883433	−0.441717	−	0.897155i	$-0.645630\pi$
$229$	−1.92799e15	−0.883433	−0.441717	+	0.897155i	$0.645630\pi$
$230$	0	0
$231$	6.01746e14	0.260576
$232$	2.09339e15	0.881412
$233$	1.12721e15	0.461522	0.230761	−	0.973010i	$-0.425878\pi$
$233$	1.12721e15	0.461522	0.230761	+	0.973010i	$0.425878\pi$
$234$	1.85703e14	0.0739460
$235$	0	0
$236$	1.00410e13	0.00378312
$237$	1.80479e15	0.661550
$238$	4.89648e15	1.74636
$239$	−2.13044e15	−0.739405	−0.369703	−	0.929150i	$-0.620540\pi$
$239$	−2.13044e15	−0.739405	−0.369703	+	0.929150i	$0.620540\pi$
$240$	0	0
$241$	3.70709e15	1.21877	0.609386	−	0.792874i	$-0.291416\pi$
$241$	3.70709e15	1.21877	0.609386	+	0.792874i	$0.291416\pi$
$242$	−2.88324e15	−0.922743
$243$	1.29385e15	0.403128
$244$	9.92646e12	0.00301134
$245$	0	0
$246$	−4.79480e15	−1.37941
$247$	−1.96113e15	−0.549510
$248$	−5.63887e15	−1.53906
$249$	2.97435e14	0.0790855
$250$	0	0
$251$	1.69706e15	0.428368	0.214184	−	0.976793i	$-0.431291\pi$
$251$	1.69706e15	0.428368	0.214184	+	0.976793i	$0.431291\pi$
$252$	2.53420e12	0.000623358 0
$253$	−1.04165e15	−0.249711
$254$	1.05902e15	0.247448
$255$	0	0
$256$	3.94010e13	0.00874877
$257$	3.42768e14	0.0742053	0.0371027	−	0.999311i	$-0.488187\pi$
$257$	3.42768e14	0.0742053	0.0371027	+	0.999311i	$0.488187\pi$
$258$	8.01958e14	0.169287
$259$	6.46243e15	1.33029
$260$	0	0
$261$	9.24342e14	0.180996
$262$	4.64750e15	0.887690
$263$	−4.63399e15	−0.863460	−0.431730	−	0.902003i	$-0.642097\pi$
$263$	−4.63399e15	−0.863460	−0.431730	+	0.902003i	$0.642097\pi$
$264$	1.37300e15	0.249600
$265$	0	0
$266$	−9.20374e15	−1.59307
$267$	−4.16134e15	−0.702927
$268$	2.45993e12	0.000405552 0
$269$	−5.38103e15	−0.865915	−0.432957	−	0.901414i	$-0.642530\pi$
$269$	−5.38103e15	−0.865915	−0.432957	+	0.901414i	$0.642530\pi$
$270$	0	0
$271$	8.17255e15	1.25331	0.626653	−	0.779298i	$-0.284424\pi$
$271$	8.17255e15	1.25331	0.626653	+	0.779298i	$0.284424\pi$
$272$	1.12049e16	1.67768
$273$	2.28934e15	0.334698
$274$	6.48612e15	0.925990
$275$	0	0
$276$	1.70074e13	0.00231594
$277$	7.61841e14	0.101332	0.0506659	−	0.998716i	$-0.483866\pi$
$277$	7.61841e14	0.101332	0.0506659	+	0.998716i	$0.483866\pi$
$278$	−1.05403e16	−1.36950
$279$	−2.48985e15	−0.316043
$280$	0	0
$281$	−9.73851e15	−1.18005	−0.590027	−	0.807384i	$-0.700883\pi$
$281$	−9.73851e15	−1.18005	−0.590027	+	0.807384i	$0.700883\pi$
$282$	−8.48254e15	−1.00440
$283$	1.30296e16	1.50771	0.753857	−	0.657038i	$-0.228191\pi$
$283$	1.30296e16	1.50771	0.753857	+	0.657038i	$0.228191\pi$
$284$	−9.56522e12	−0.00108175
$285$	0	0
$286$	9.35730e14	0.101105
$287$	1.52467e16	1.61044
$288$	1.15815e13	0.00119595
$289$	1.78114e16	1.79830
$290$	0	0
$291$	1.13615e16	1.09681
$292$	−1.32593e13	−0.00125179
$293$	4.27472e15	0.394701	0.197351	−	0.980333i	$-0.436766\pi$
$293$	4.27472e15	0.394701	0.197351	+	0.980333i	$0.436766\pi$
$294$	8.57151e14	0.0774104
$295$	0	0
$296$	1.47453e16	1.27426
$297$	3.56289e15	0.301221
$298$	−1.30213e16	−1.07708
$299$	−3.96295e15	−0.320742
$300$	0	0
$301$	−2.55010e15	−0.197640
$302$	−1.81029e15	−0.137310
$303$	2.89033e15	0.214571
$304$	−2.10614e16	−1.53041
$305$	0	0
$306$	4.93313e15	0.343505
$307$	−1.84450e16	−1.25742	−0.628709	−	0.777640i	$-0.716417\pi$
$307$	−1.84450e16	−1.25742	−0.628709	+	0.777640i	$0.716417\pi$
$308$	1.27695e13	0.000852303 0
$309$	−1.36262e16	−0.890523
$310$	0	0
$311$	−2.21772e16	−1.38984	−0.694918	−	0.719089i	$-0.744559\pi$
$311$	−2.21772e16	−1.38984	−0.694918	+	0.719089i	$0.744559\pi$
$312$	5.22357e15	0.320600
$313$	2.02883e16	1.21957	0.609785	−	0.792567i	$-0.291256\pi$
$313$	2.02883e16	1.21957	0.609785	+	0.792567i	$0.291256\pi$
$314$	−2.23388e16	−1.31528
$315$	0	0
$316$	3.82990e13	0.00216382
$317$	−1.93347e16	−1.07017	−0.535085	−	0.844798i	$-0.679720\pi$
$317$	−1.93347e16	−1.07017	−0.535085	+	0.844798i	$0.679720\pi$
$318$	1.21723e16	0.660081
$319$	4.65763e15	0.247472
$320$	0	0
$321$	−1.05542e16	−0.538446
$322$	−1.85985e16	−0.929852
$323$	−5.20968e16	−2.55267
$324$	−4.57208e13	−0.00219569
$325$	0	0
$326$	−1.21152e16	−0.559008
$327$	−2.81344e16	−1.27256
$328$	3.47883e16	1.54260
$329$	2.69731e16	1.17262
$330$	0	0
$331$	4.61159e15	0.192739	0.0963693	−	0.995346i	$-0.469277\pi$
$331$	4.61159e15	0.192739	0.0963693	+	0.995346i	$0.469277\pi$
$332$	6.31180e12	0.000258676 0
$333$	6.51081e15	0.261666
$334$	1.42461e16	0.561493
$335$	0	0
$336$	2.45862e16	0.932150
$337$	3.91103e16	1.45444	0.727222	−	0.686402i	$-0.240811\pi$
$337$	3.91103e16	1.45444	0.727222	+	0.686402i	$0.240811\pi$
$338$	−2.38931e16	−0.871593
$339$	−1.52658e16	−0.546287
$340$	0	0
$341$	−1.25460e16	−0.432119
$342$	−9.27264e15	−0.313353
$343$	2.87132e16	0.952071
$344$	−5.81854e15	−0.189315
$345$	0	0
$346$	−3.01176e16	−0.943682
$347$	−5.96375e15	−0.183391	−0.0916955	−	0.995787i	$-0.529229\pi$
$347$	−5.96375e15	−0.183391	−0.0916955	+	0.995787i	$0.529229\pi$
$348$	−7.60467e13	−0.00229517
$349$	2.76636e15	0.0819490	0.0409745	−	0.999160i	$-0.486954\pi$
$349$	2.76636e15	0.0819490	0.0409745	+	0.999160i	$0.486954\pi$
$350$	0	0
$351$	1.35550e16	0.386904
$352$	5.83574e13	0.00163519
$353$	−1.98418e16	−0.545816	−0.272908	−	0.962040i	$-0.587985\pi$
$353$	−1.98418e16	−0.545816	−0.272908	+	0.962040i	$0.587985\pi$
$354$	4.28889e16	1.15831
$355$	0	0
$356$	−8.83068e13	−0.00229916
$357$	6.08156e16	1.55479
$358$	4.50217e16	1.13027
$359$	−3.22798e16	−0.795824	−0.397912	−	0.917424i	$-0.630265\pi$
$359$	−3.22798e16	−0.795824	−0.397912	+	0.917424i	$0.630265\pi$
$360$	0	0
$361$	5.58716e16	1.32860
$362$	5.20187e16	1.21494
$363$	−3.58106e16	−0.821522
$364$	4.85815e13	0.00109474
$365$	0	0
$366$	4.23995e16	0.922006
$367$	4.42436e16	0.945194	0.472597	−	0.881279i	$-0.343317\pi$
$367$	4.42436e16	0.945194	0.472597	+	0.881279i	$0.343317\pi$
$368$	−4.25598e16	−0.893282
$369$	1.53608e16	0.316770
$370$	0	0
$371$	−3.87060e16	−0.770634
$372$	2.04843e14	0.00400768
$373$	−9.26267e16	−1.78086	−0.890428	−	0.455125i	$-0.849595\pi$
$373$	−9.26267e16	−1.78086	−0.890428	+	0.455125i	$0.849595\pi$
$374$	2.48574e16	0.469667
$375$	0	0
$376$	6.15444e16	1.12323
$377$	1.77199e16	0.317865
$378$	6.36147e16	1.12166
$379$	−3.71415e16	−0.643732	−0.321866	−	0.946785i	$-0.604310\pi$
$379$	−3.71415e16	−0.643732	−0.321866	+	0.946785i	$0.604310\pi$
$380$	0	0
$381$	1.31533e16	0.220304
$382$	−8.34907e16	−1.37476
$383$	1.22127e16	0.197706	0.0988532	−	0.995102i	$-0.468483\pi$
$383$	1.22127e16	0.197706	0.0988532	+	0.995102i	$0.468483\pi$
$384$	5.62613e16	0.895482
$385$	0	0
$386$	−8.05603e16	−1.23966
$387$	−2.56919e15	−0.0388754
$388$	2.41100e14	0.00358750
$389$	7.53716e16	1.10290	0.551449	−	0.834209i	$-0.314075\pi$
$389$	7.53716e16	1.10290	0.551449	+	0.834209i	$0.314075\pi$
$390$	0	0
$391$	−1.05275e17	−1.48996
$392$	−6.21899e15	−0.0865686
$393$	5.77233e16	0.790314
$394$	4.26870e15	0.0574871
$395$	0	0
$396$	1.28651e13	0.000167646 0
$397$	−8.59809e16	−1.10221	−0.551104	−	0.834437i	$-0.685793\pi$
$397$	−8.59809e16	−1.10221	−0.551104	+	0.834437i	$0.685793\pi$
$398$	4.47927e16	0.564894
$399$	−1.14313e17	−1.41831
$400$	0	0
$401$	−1.13598e17	−1.36437	−0.682186	−	0.731178i	$-0.738971\pi$
$401$	−1.13598e17	−1.36437	−0.682186	+	0.731178i	$0.738971\pi$
$402$	1.05072e16	0.124171
$403$	−4.77313e16	−0.555035
$404$	6.13349e13	0.000701825 0
$405$	0	0
$406$	8.31612e16	0.921513
$407$	3.28071e16	0.357770
$408$	1.38763e17	1.48930
$409$	−3.50572e16	−0.370318	−0.185159	−	0.982709i	$-0.559280\pi$
$409$	−3.50572e16	−0.370318	−0.185159	+	0.982709i	$0.559280\pi$
$410$	0	0
$411$	8.05594e16	0.824412
$412$	−2.89159e14	−0.00291276
$413$	−1.36380e17	−1.35230
$414$	−1.87377e16	−0.182900
$415$	0	0
$416$	2.22020e14	0.00210032
$417$	−1.30914e17	−1.21927
$418$	−4.67236e16	−0.428440
$419$	−2.82535e15	−0.0255083	−0.0127541	−	0.999919i	$-0.504060\pi$
$419$	−2.82535e15	−0.0255083	−0.0127541	+	0.999919i	$0.504060\pi$
$420$	0	0
$421$	8.17054e16	0.715183	0.357591	−	0.933878i	$-0.383598\pi$
$421$	8.17054e16	0.715183	0.357591	+	0.933878i	$0.383598\pi$
$422$	−1.11014e17	−0.956859
$423$	2.71750e16	0.230652
$424$	−8.83154e16	−0.738173
$425$	0	0
$426$	−4.08565e16	−0.331207
$427$	−1.34824e17	−1.07643
$428$	−2.23968e14	−0.00176117
$429$	1.16220e16	0.0900139
$430$	0	0
$431$	4.31353e16	0.324139	0.162069	−	0.986779i	$-0.448183\pi$
$431$	4.31353e16	0.324139	0.162069	+	0.986779i	$0.448183\pi$
$432$	1.45573e17	1.07755
$433$	6.25222e16	0.455893	0.227947	−	0.973674i	$-0.426799\pi$
$433$	6.25222e16	0.455893	0.227947	+	0.973674i	$0.426799\pi$
$434$	−2.24007e17	−1.60908
$435$	0	0
$436$	−5.97034e14	−0.00416233
$437$	1.97881e17	1.35917
$438$	−5.66353e16	−0.383271
$439$	−7.25265e16	−0.483590	−0.241795	−	0.970327i	$-0.577736\pi$
$439$	−7.25265e16	−0.483590	−0.241795	+	0.970327i	$0.577736\pi$
$440$	0	0
$441$	−2.74601e15	−0.0177767
$442$	9.45698e16	0.603265
$443$	2.76459e17	1.73783	0.868914	−	0.494962i	$-0.164818\pi$
$443$	2.76459e17	1.73783	0.868914	+	0.494962i	$0.164818\pi$
$444$	−5.35652e14	−0.00331813
$445$	0	0
$446$	1.12375e17	0.676071
$447$	−1.61728e17	−0.958931
$448$	−1.77863e17	−1.03939
$449$	2.46120e17	1.41757	0.708786	−	0.705424i	$-0.249243\pi$
$449$	2.46120e17	1.41757	0.708786	+	0.705424i	$0.249243\pi$
$450$	0	0
$451$	7.74012e16	0.433112
$452$	−3.23951e14	−0.00178682
$453$	−2.24843e16	−0.122248
$454$	3.00888e16	0.161265
$455$	0	0
$456$	−2.60828e17	−1.35857
$457$	3.45983e17	1.77664	0.888319	−	0.459226i	$-0.151873\pi$
$457$	3.45983e17	1.77664	0.888319	+	0.459226i	$0.151873\pi$
$458$	−1.74756e17	−0.884721
$459$	3.60084e17	1.79731
$460$	0	0
$461$	−1.04114e17	−0.505188	−0.252594	−	0.967572i	$-0.581284\pi$
$461$	−1.04114e17	−0.505188	−0.252594	+	0.967572i	$0.581284\pi$
$462$	5.45432e16	0.260956
$463$	1.70140e17	0.802656	0.401328	−	0.915934i	$-0.368549\pi$
$463$	1.70140e17	0.802656	0.401328	+	0.915934i	$0.368549\pi$
$464$	1.90302e17	0.885271
$465$	0	0
$466$	1.02172e17	0.462195
$467$	−1.84060e17	−0.821106	−0.410553	−	0.911837i	$-0.634664\pi$
$467$	−1.84060e17	−0.821106	−0.410553	+	0.911837i	$0.634664\pi$
$468$	4.89452e13	0.000215334 0
$469$	−3.34114e16	−0.144968
$470$	0	0
$471$	−2.77454e17	−1.17100
$472$	−3.11177e17	−1.29534
$473$	−1.29458e16	−0.0531534
$474$	1.63589e17	0.662514
$475$	0	0
$476$	1.29055e15	0.00508546
$477$	−3.89958e16	−0.151582
$478$	−1.93106e17	−0.740482
$479$	−2.13215e17	−0.806562	−0.403281	−	0.915076i	$-0.632130\pi$
$479$	−2.13215e17	−0.806562	−0.403281	+	0.915076i	$0.632130\pi$
$480$	0	0
$481$	1.24814e17	0.459538
$482$	3.36016e17	1.22055
$483$	−2.30998e17	−0.827851
$484$	−7.59928e14	−0.00268706
$485$	0	0
$486$	1.17276e17	0.403715
$487$	3.74978e17	1.27370	0.636851	−	0.770987i	$-0.280237\pi$
$487$	3.74978e17	1.27370	0.636851	+	0.770987i	$0.280237\pi$
$488$	−3.07626e17	−1.03109
$489$	−1.50475e17	−0.497687
$490$	0	0
$491$	4.11003e17	1.32378	0.661889	−	0.749602i	$-0.269755\pi$
$491$	4.11003e17	1.32378	0.661889	+	0.749602i	$0.269755\pi$
$492$	−1.26375e15	−0.00401689
$493$	4.70725e17	1.47660
$494$	−1.77760e17	−0.550311
$495$	0	0
$496$	−5.12607e17	−1.54580
$497$	1.29917e17	0.386678
$498$	2.69600e16	0.0792007
$499$	1.73839e17	0.504072	0.252036	−	0.967718i	$-0.418900\pi$
$499$	1.73839e17	0.504072	0.252036	+	0.967718i	$0.418900\pi$
$500$	0	0
$501$	1.76941e17	0.499899
$502$	1.53824e17	0.428992
$503$	3.28753e17	0.905062	0.452531	−	0.891749i	$-0.350521\pi$
$503$	3.28753e17	0.905062	0.452531	+	0.891749i	$0.350521\pi$
$504$	−7.85363e16	−0.213438
$505$	0	0
$506$	−9.44167e16	−0.250075
$507$	−2.96759e17	−0.775983
$508$	2.79123e14	0.000720579 0
$509$	6.89978e16	0.175861	0.0879305	−	0.996127i	$-0.471975\pi$
$509$	6.89978e16	0.175861	0.0879305	+	0.996127i	$0.471975\pi$
$510$	0	0
$511$	1.80091e17	0.447463
$512$	−4.05822e17	−0.995591
$513$	−6.76839e17	−1.63954
$514$	3.10690e16	0.0743135
$515$	0	0
$516$	2.11370e14	0.000492970 0
$517$	1.36931e17	0.315365
$518$	5.85765e17	1.33223
$519$	−3.74069e17	−0.840164
$520$	0	0
$521$	8.38335e17	1.83642	0.918211	−	0.396092i	$-0.129634\pi$
$521$	8.38335e17	1.83642	0.918211	+	0.396092i	$0.129634\pi$
$522$	8.37838e16	0.181260
$523$	1.91687e17	0.409573	0.204786	−	0.978807i	$-0.434350\pi$
$523$	1.91687e17	0.409573	0.204786	+	0.978807i	$0.434350\pi$
$524$	1.22493e15	0.00258499
$525$	0	0
$526$	−4.20032e17	−0.864719
$527$	−1.26797e18	−2.57834
$528$	1.24814e17	0.250693
$529$	−1.04168e17	−0.206668
$530$	0	0
$531$	−1.37401e17	−0.265996
$532$	−2.42581e15	−0.00463907
$533$	2.94472e17	0.556311
$534$	−3.77191e17	−0.703952
$535$	0	0
$536$	−7.62345e16	−0.138861
$537$	5.59183e17	1.00628
$538$	−4.87744e17	−0.867177
$539$	−1.38368e16	−0.0243056
$540$	0	0
$541$	−8.65938e17	−1.48492	−0.742462	−	0.669888i	$-0.766342\pi$
$541$	−8.65938e17	−1.48492	−0.742462	+	0.669888i	$0.766342\pi$
$542$	7.40773e17	1.25513
$543$	6.46087e17	1.08166
$544$	5.89791e15	0.00975675
$545$	0	0
$546$	2.07509e17	0.335185
$547$	−1.16237e18	−1.85536	−0.927678	−	0.373382i	$-0.878198\pi$
$547$	−1.16237e18	−1.85536	−0.927678	+	0.373382i	$0.878198\pi$
$548$	1.70953e15	0.00269652
$549$	−1.35833e17	−0.211731
$550$	0	0
$551$	−8.84806e17	−1.34698
$552$	−5.27067e17	−0.792980
$553$	−5.20187e17	−0.773475
$554$	6.90545e16	0.101479
$555$	0	0
$556$	−2.77809e15	−0.00398804
$557$	1.02746e17	0.145783	0.0728917	−	0.997340i	$-0.476777\pi$
$557$	1.02746e17	0.145783	0.0728917	+	0.997340i	$0.476777\pi$
$558$	−2.25684e17	−0.316504
$559$	−4.92522e16	−0.0682730
$560$	0	0
$561$	3.08736e17	0.418146
$562$	−8.82714e17	−1.18177
$563$	−7.17343e17	−0.949341	−0.474671	−	0.880164i	$-0.657433\pi$
$563$	−7.17343e17	−0.949341	−0.474671	+	0.880164i	$0.657433\pi$
$564$	−2.23572e15	−0.00292485
$565$	0	0
$566$	1.18102e18	1.50991
$567$	6.20992e17	0.784867
$568$	2.96431e17	0.370391
$569$	−1.25353e17	−0.154848	−0.0774238	−	0.996998i	$-0.524669\pi$
$569$	−1.25353e17	−0.154848	−0.0774238	+	0.996998i	$0.524669\pi$
$570$	0	0
$571$	−1.00269e17	−0.121069	−0.0605344	−	0.998166i	$-0.519280\pi$
$571$	−1.00269e17	−0.121069	−0.0605344	+	0.998166i	$0.519280\pi$
$572$	2.46628e14	0.000294421 0
$573$	−1.03698e18	−1.22395
$574$	1.38198e18	1.61278
$575$	0	0
$576$	−1.79194e17	−0.204446
$577$	1.36098e18	1.53535	0.767675	−	0.640839i	$-0.221413\pi$
$577$	1.36098e18	1.53535	0.767675	+	0.640839i	$0.221413\pi$
$578$	1.61445e18	1.80092
$579$	−1.00058e18	−1.10368
$580$	0	0
$581$	−8.57285e16	−0.0924656
$582$	1.02983e18	1.09841
$583$	−1.96495e17	−0.207255
$584$	4.10913e17	0.428614
$585$	0	0
$586$	3.87468e17	0.395276
$587$	6.87785e16	0.0693913	0.0346956	−	0.999398i	$-0.488954\pi$
$587$	6.87785e16	0.0693913	0.0346956	+	0.999398i	$0.488954\pi$
$588$	2.25917e14	0.000225422 0
$589$	2.38336e18	2.35201
$590$	0	0
$591$	5.30185e16	0.0511810
$592$	1.34043e18	1.27984
$593$	3.46338e17	0.327073	0.163536	−	0.986537i	$-0.447710\pi$
$593$	3.46338e17	0.327073	0.163536	+	0.986537i	$0.447710\pi$
$594$	3.22946e17	0.301660
$595$	0	0
$596$	−3.43200e15	−0.00313651
$597$	5.56338e17	0.502928
$598$	−3.59208e17	−0.321209
$599$	−1.08982e18	−0.964004	−0.482002	−	0.876170i	$-0.660090\pi$
$599$	−1.08982e18	−0.964004	−0.482002	+	0.876170i	$0.660090\pi$
$600$	0	0
$601$	2.73058e17	0.236358	0.118179	−	0.992992i	$-0.462294\pi$
$601$	2.73058e17	0.236358	0.118179	+	0.992992i	$0.462294\pi$
$602$	−2.31145e17	−0.197928
$603$	−3.36615e16	−0.0285148
$604$	−4.77133e14	−0.000399852 0
$605$	0	0
$606$	2.61984e17	0.214883
$607$	1.19160e18	0.966951	0.483476	−	0.875358i	$-0.339374\pi$
$607$	1.19160e18	0.966951	0.483476	+	0.875358i	$0.339374\pi$
$608$	−1.10861e16	−0.00890032
$609$	1.03289e18	0.820426
$610$	0	0
$611$	5.20954e17	0.405072
$612$	1.30021e15	0.00100030
$613$	−3.63023e17	−0.276338	−0.138169	−	0.990409i	$-0.544122\pi$
$613$	−3.63023e17	−0.276338	−0.138169	+	0.990409i	$0.544122\pi$
$614$	−1.67189e18	−1.25925
$615$	0	0
$616$	−3.95734e17	−0.291829
$617$	−6.06720e17	−0.442726	−0.221363	−	0.975192i	$-0.571051\pi$
$617$	−6.06720e17	−0.442726	−0.221363	+	0.975192i	$0.571051\pi$
$618$	−1.23510e18	−0.891821
$619$	−2.29300e18	−1.63838	−0.819190	−	0.573523i	$-0.805576\pi$
$619$	−2.29300e18	−1.63838	−0.819190	+	0.573523i	$0.805576\pi$
$620$	0	0
$621$	−1.36772e18	−0.956979
$622$	−2.01017e18	−1.39186
$623$	1.19941e18	0.821852
$624$	4.74854e17	0.322003
$625$	0	0
$626$	1.83896e18	1.22135
$627$	−5.80320e17	−0.381442
$628$	−5.88779e15	−0.00383014
$629$	3.31566e18	2.13472
$630$	0	0
$631$	3.33398e17	0.210268	0.105134	−	0.994458i	$-0.466473\pi$
$631$	3.33398e17	0.210268	0.105134	+	0.994458i	$0.466473\pi$
$632$	−1.18691e18	−0.740894
$633$	−1.37883e18	−0.851895
$634$	−1.75253e18	−1.07173
$635$	0	0
$636$	3.20823e15	0.00192218
$637$	−5.26418e16	−0.0312194
$638$	4.22175e17	0.247832
$639$	1.30890e17	0.0760589
$640$	0	0
$641$	−8.91188e17	−0.507449	−0.253724	−	0.967277i	$-0.581656\pi$
$641$	−8.91188e17	−0.507449	−0.253724	+	0.967277i	$0.581656\pi$
$642$	−9.56648e17	−0.539230
$643$	−5.58943e17	−0.311886	−0.155943	−	0.987766i	$-0.549842\pi$
$643$	−5.58943e17	−0.311886	−0.155943	+	0.987766i	$0.549842\pi$
$644$	−4.90195e15	−0.00270776
$645$	0	0
$646$	−4.72213e18	−2.55639
$647$	−1.62698e18	−0.871973	−0.435987	−	0.899953i	$-0.643601\pi$
$647$	−1.62698e18	−0.871973	−0.435987	+	0.899953i	$0.643601\pi$
$648$	1.41691e18	0.751806
$649$	−6.92344e17	−0.363690
$650$	0	0
$651$	−2.78223e18	−1.43257
$652$	−3.19319e15	−0.00162785
$653$	6.71206e17	0.338782	0.169391	−	0.985549i	$-0.445820\pi$
$653$	6.71206e17	0.338782	0.169391	+	0.985549i	$0.445820\pi$
$654$	−2.55015e18	−1.27441
$655$	0	0
$656$	3.16246e18	1.54935
$657$	1.81439e17	0.0880150
$658$	2.44488e18	1.17433
$659$	−1.08505e18	−0.516053	−0.258027	−	0.966138i	$-0.583072\pi$
$659$	−1.08505e18	−0.516053	−0.258027	+	0.966138i	$0.583072\pi$
$660$	0	0
$661$	2.34214e18	1.09220	0.546100	−	0.837720i	$-0.316112\pi$
$661$	2.34214e18	1.09220	0.546100	+	0.837720i	$0.316112\pi$
$662$	4.18002e17	0.193019
$663$	1.17458e18	0.537089
$664$	−1.95606e17	−0.0885707
$665$	0	0
$666$	5.90150e17	0.262047
$667$	−1.78797e18	−0.786217
$668$	3.75481e15	0.00163509
$669$	1.39573e18	0.601909
$670$	0	0
$671$	−6.84444e17	−0.289495
$672$	1.29415e16	0.00542104
$673$	−7.92495e17	−0.328775	−0.164387	−	0.986396i	$-0.552565\pi$
$673$	−7.92495e17	−0.328775	−0.164387	+	0.986396i	$0.552565\pi$
$674$	3.54502e18	1.45656
$675$	0	0
$676$	−6.29745e15	−0.00253811
$677$	2.35105e18	0.938503	0.469251	−	0.883065i	$-0.344524\pi$
$677$	2.35105e18	0.938503	0.469251	+	0.883065i	$0.344524\pi$
$678$	−1.38371e18	−0.547083
$679$	−3.27469e18	−1.28238
$680$	0	0
$681$	3.73711e17	0.143575
$682$	−1.13719e18	−0.432748
$683$	3.22912e17	0.121717	0.0608583	−	0.998146i	$-0.480616\pi$
$683$	3.22912e17	0.121717	0.0608583	+	0.998146i	$0.480616\pi$
$684$	−2.44397e15	−0.000912496 0
$685$	0	0
$686$	2.60261e18	0.953459
$687$	−2.17052e18	−0.787670
$688$	−5.28940e17	−0.190144
$689$	−7.47563e17	−0.266209
$690$	0	0
$691$	3.36976e18	1.17758	0.588792	−	0.808284i	$-0.299604\pi$
$691$	3.36976e18	1.17758	0.588792	+	0.808284i	$0.299604\pi$
$692$	−7.93803e15	−0.00274804
$693$	−1.74737e17	−0.0599265
$694$	−5.40564e17	−0.183658
$695$	0	0
$696$	2.35673e18	0.785868
$697$	7.82257e18	2.58426
$698$	2.50747e17	0.0820684
$699$	1.26901e18	0.411494
$700$	0	0
$701$	−5.38800e18	−1.71499	−0.857493	−	0.514495i	$-0.827979\pi$
$701$	−5.38800e18	−1.71499	−0.857493	+	0.514495i	$0.827979\pi$
$702$	1.22865e18	0.387468
$703$	−6.23233e18	−1.94734
$704$	−9.02936e17	−0.279534
$705$	0	0
$706$	−1.79849e18	−0.546611
$707$	−8.33067e17	−0.250873
$708$	1.13041e16	0.00337303
$709$	−1.40376e18	−0.415044	−0.207522	−	0.978230i	$-0.566540\pi$
$709$	−1.40376e18	−0.415044	−0.207522	+	0.978230i	$0.566540\pi$
$710$	0	0
$711$	−5.24082e17	−0.152141
$712$	2.73668e18	0.787234
$713$	4.81617e18	1.37284
$714$	5.51242e18	1.55706
$715$	0	0
$716$	1.18663e16	0.00329139
$717$	−2.39843e18	−0.659254
$718$	−2.92589e18	−0.796984
$719$	−1.38614e18	−0.374170	−0.187085	−	0.982344i	$-0.559904\pi$
$719$	−1.38614e18	−0.374170	−0.187085	+	0.982344i	$0.559904\pi$
$720$	0	0
$721$	3.92742e18	1.04119
$722$	5.06429e18	1.33054
$723$	4.17342e18	1.08666
$724$	1.37104e16	0.00353794
$725$	0	0
$726$	−3.24593e18	−0.822719
$727$	−4.20824e18	−1.05712	−0.528562	−	0.848894i	$-0.677269\pi$
$727$	−4.20824e18	−1.05712	−0.528562	+	0.848894i	$0.677269\pi$
$728$	−1.50557e18	−0.374840
$729$	4.50781e18	1.11234
$730$	0	0
$731$	−1.30837e18	−0.317152
$732$	1.11751e16	0.00268492
$733$	−7.61043e17	−0.181231	−0.0906156	−	0.995886i	$-0.528883\pi$
$733$	−7.61043e17	−0.181231	−0.0906156	+	0.995886i	$0.528883\pi$
$734$	4.01031e18	0.946571
$735$	0	0
$736$	−2.24022e16	−0.00519500
$737$	−1.69616e17	−0.0389877
$738$	1.39233e18	0.317231
$739$	6.25215e18	1.41202	0.706010	−	0.708202i	$-0.250493\pi$
$739$	6.25215e18	1.41202	0.706010	+	0.708202i	$0.250493\pi$
$740$	0	0
$741$	−2.20783e18	−0.489944
$742$	−3.50838e18	−0.771757
$743$	−5.60504e18	−1.22223	−0.611113	−	0.791543i	$-0.709278\pi$
$743$	−5.60504e18	−1.22223	−0.611113	+	0.791543i	$0.709278\pi$
$744$	−6.34821e18	−1.37223
$745$	0	0
$746$	−8.39583e18	−1.78345
$747$	−8.63703e16	−0.0181878
$748$	6.55160e15	0.00136769
$749$	3.04199e18	0.629543
$750$	0	0
$751$	4.79388e18	0.975053	0.487526	−	0.873108i	$-0.337899\pi$
$751$	4.79388e18	0.975053	0.487526	+	0.873108i	$0.337899\pi$
$752$	5.59475e18	1.12814
$753$	1.91053e18	0.381933
$754$	1.60616e18	0.318329
$755$	0	0
$756$	1.67668e16	0.00326632
$757$	−2.47656e18	−0.478328	−0.239164	−	0.970979i	$-0.576873\pi$
$757$	−2.47656e18	−0.478328	−0.239164	+	0.970979i	$0.576873\pi$
$758$	−3.36656e18	−0.644669
$759$	−1.17268e18	−0.222643
$760$	0	0
$761$	1.02209e19	1.90761	0.953804	−	0.300430i	$-0.0971301\pi$
$761$	1.02209e19	1.90761	0.953804	+	0.300430i	$0.0971301\pi$
$762$	1.19224e18	0.220625
$763$	8.10906e18	1.48786
$764$	−2.20055e16	−0.00400335
$765$	0	0
$766$	1.10698e18	0.197994
$767$	−2.63402e18	−0.467142
$768$	4.43574e16	0.00780042
$769$	−9.71765e18	−1.69449	−0.847247	−	0.531199i	$-0.821742\pi$
$769$	−9.71765e18	−1.69449	−0.847247	+	0.531199i	$0.821742\pi$
$770$	0	0
$771$	3.85886e17	0.0661616
$772$	−2.12331e16	−0.00360995
$773$	−6.33996e18	−1.06886	−0.534429	−	0.845213i	$-0.679473\pi$
$773$	−6.33996e18	−1.06886	−0.534429	+	0.845213i	$0.679473\pi$
$774$	−2.32875e17	−0.0389320
$775$	0	0
$776$	−7.47184e18	−1.22836
$777$	7.27536e18	1.18609
$778$	6.83180e18	1.10450
$779$	−1.47038e19	−2.35742
$780$	0	0
$781$	6.59536e17	0.103994
$782$	−9.54225e18	−1.49213
$783$	6.11563e18	0.948396
$784$	−5.65343e17	−0.0869476
$785$	0	0
$786$	5.23213e18	0.791465
$787$	−6.11756e17	−0.0917789	−0.0458894	−	0.998947i	$-0.514612\pi$
$787$	−6.11756e17	−0.0917789	−0.0458894	+	0.998947i	$0.514612\pi$
$788$	1.12509e15	0.000167405 0
$789$	−5.21691e18	−0.769862
$790$	0	0
$791$	4.39999e18	0.638711
$792$	−3.98696e17	−0.0574022
$793$	−2.60396e18	−0.371843
$794$	−7.79344e18	−1.10381
$795$	0	0
$796$	1.18059e16	0.00164499
$797$	−1.19147e18	−0.164665	−0.0823327	−	0.996605i	$-0.526237\pi$
$797$	−1.19147e18	−0.164665	−0.0823327	+	0.996605i	$0.526237\pi$
$798$	−1.03615e19	−1.42038
$799$	1.38390e19	1.88170
$800$	0	0
$801$	1.20839e18	0.161657
$802$	−1.02967e19	−1.36636
$803$	9.14249e17	0.120341
$804$	2.76937e15	0.000361590 0
$805$	0	0
$806$	−4.32644e18	−0.555844
$807$	−6.05792e18	−0.772051
$808$	−1.90080e18	−0.240305
$809$	7.74613e18	0.971447	0.485724	−	0.874112i	$-0.338556\pi$
$809$	7.74613e18	0.971447	0.485724	+	0.874112i	$0.338556\pi$
$810$	0	0
$811$	−4.31453e18	−0.532474	−0.266237	−	0.963908i	$-0.585780\pi$
$811$	−4.31453e18	−0.532474	−0.266237	+	0.963908i	$0.585780\pi$
$812$	2.19186e16	0.00268348
$813$	9.20061e18	1.11745
$814$	2.97368e18	0.358291
$815$	0	0
$816$	1.26143e19	1.49582
$817$	2.45930e18	0.289313
$818$	−3.17764e18	−0.370858
$819$	−6.64786e17	−0.0769726
$820$	0	0
$821$	−1.32454e19	−1.50950	−0.754752	−	0.656011i	$-0.772243\pi$
$821$	−1.32454e19	−1.50950	−0.754752	+	0.656011i	$0.772243\pi$
$822$	7.30203e18	0.825613
$823$	1.59982e19	1.79461	0.897307	−	0.441407i	$-0.145520\pi$
$823$	1.59982e19	1.79461	0.897307	+	0.441407i	$0.145520\pi$
$824$	8.96118e18	0.997329
$825$	0	0
$826$	−1.23617e19	−1.35427
$827$	7.72194e18	0.839345	0.419672	−	0.907676i	$-0.362145\pi$
$827$	7.72194e18	0.839345	0.419672	+	0.907676i	$0.362145\pi$
$828$	−4.93865e15	−0.000532612 0
$829$	−8.75341e18	−0.936640	−0.468320	−	0.883559i	$-0.655141\pi$
$829$	−8.75341e18	−0.936640	−0.468320	+	0.883559i	$0.655141\pi$
$830$	0	0
$831$	8.57676e17	0.0903476
$832$	−3.43522e18	−0.359048
$833$	−1.39842e18	−0.145025
$834$	−1.18662e19	−1.22105
$835$	0	0
$836$	−1.23148e16	−0.00124764
$837$	−1.64734e19	−1.65603
$838$	−2.56094e17	−0.0255454
$839$	1.14368e19	1.13201	0.566007	−	0.824400i	$-0.308487\pi$
$839$	1.14368e19	1.13201	0.566007	+	0.824400i	$0.308487\pi$
$840$	0	0
$841$	−2.26589e18	−0.220834
$842$	7.40590e18	0.716225
$843$	−1.09636e19	−1.05214
$844$	−2.92598e16	−0.00278641
$845$	0	0
$846$	2.46319e18	0.230988
$847$	1.03215e19	0.960511
$848$	−8.02839e18	−0.741405
$849$	1.46686e19	1.34428
$850$	0	0
$851$	−1.25940e19	−1.13663
$852$	−1.07685e16	−0.000964487 0
$853$	9.31629e17	0.0828084	0.0414042	−	0.999142i	$-0.486817\pi$
$853$	9.31629e17	0.0828084	0.0414042	+	0.999142i	$0.486817\pi$
$854$	−1.22206e19	−1.07800
$855$	0	0
$856$	6.94088e18	0.603025
$857$	−9.56376e18	−0.824619	−0.412309	−	0.911044i	$-0.635278\pi$
$857$	−9.56376e18	−0.824619	−0.412309	+	0.911044i	$0.635278\pi$
$858$	1.05344e18	0.0901450
$859$	7.17114e18	0.609022	0.304511	−	0.952509i	$-0.401507\pi$
$859$	7.17114e18	0.609022	0.304511	+	0.952509i	$0.401507\pi$
$860$	0	0
$861$	1.71646e19	1.43587
$862$	3.90985e18	0.324611
$863$	−1.34010e19	−1.10425	−0.552125	−	0.833761i	$-0.686183\pi$
$863$	−1.34010e19	−1.10425	−0.552125	+	0.833761i	$0.686183\pi$
$864$	7.66253e16	0.00626661
$865$	0	0
$866$	5.66711e18	0.456557
$867$	2.00520e19	1.60337
$868$	−5.90411e16	−0.00468572
$869$	−2.64078e18	−0.208019
$870$	0	0
$871$	−6.45302e17	−0.0500778
$872$	1.85024e19	1.42518
$873$	−3.29920e18	−0.252241
$874$	1.79362e19	1.36115
$875$	0	0
$876$	−1.49272e16	−0.00111610
$877$	8.23622e18	0.611266	0.305633	−	0.952149i	$-0.401132\pi$
$877$	8.23622e18	0.611266	0.305633	+	0.952149i	$0.401132\pi$
$878$	−6.57391e18	−0.484295
$879$	4.81246e18	0.351916
$880$	0	0
$881$	7.58214e18	0.546322	0.273161	−	0.961968i	$-0.411931\pi$
$881$	7.58214e18	0.546322	0.273161	+	0.961968i	$0.411931\pi$
$882$	−2.48902e17	−0.0178026
$883$	1.22388e19	0.868946	0.434473	−	0.900685i	$-0.356935\pi$
$883$	1.22388e19	0.868946	0.434473	+	0.900685i	$0.356935\pi$
$884$	2.49255e16	0.00175673
$885$	0	0
$886$	2.50587e19	1.74036
$887$	−2.48370e18	−0.171236	−0.0856182	−	0.996328i	$-0.527287\pi$
$887$	−2.48370e18	−0.171236	−0.0856182	+	0.996328i	$0.527287\pi$
$888$	1.66002e19	1.13613
$889$	−3.79112e18	−0.257576
$890$	0	0
$891$	3.15252e18	0.211083
$892$	2.96184e16	0.00196875
$893$	−2.60127e19	−1.71653
$894$	−1.46593e19	−0.960329
$895$	0	0
$896$	−1.62159e19	−1.04698
$897$	−4.46146e18	−0.285974
$898$	2.23087e19	1.41964
$899$	−2.15350e19	−1.36053
$900$	0	0
$901$	−1.98588e19	−1.23664
$902$	7.01576e18	0.433743
$903$	−2.87088e18	−0.176216
$904$	1.00394e19	0.611807
$905$	0	0
$906$	−2.03801e18	−0.122426
$907$	6.93837e18	0.413819	0.206909	−	0.978360i	$-0.433659\pi$
$907$	6.93837e18	0.413819	0.206909	+	0.978360i	$0.433659\pi$
$908$	7.93042e15	0.000469611 0
$909$	−8.39303e17	−0.0493462
$910$	0	0
$911$	−4.63486e18	−0.268638	−0.134319	−	0.990938i	$-0.542885\pi$
$911$	−4.63486e18	−0.268638	−0.134319	+	0.990938i	$0.542885\pi$
$912$	−2.37108e19	−1.36452
$913$	−4.35208e17	−0.0248678
$914$	3.13604e19	1.77923
$915$	0	0
$916$	−4.60600e16	−0.00257634
$917$	−1.66373e19	−0.924023
$918$	3.26386e19	1.79992
$919$	−4.92534e18	−0.269703	−0.134851	−	0.990866i	$-0.543056\pi$
$919$	−4.92534e18	−0.269703	−0.134851	+	0.990866i	$0.543056\pi$
$920$	0	0
$921$	−2.07653e19	−1.12112
$922$	−9.43705e18	−0.505924
$923$	2.50920e18	0.133575
$924$	1.43758e16	0.000759914 0
$925$	0	0
$926$	1.54217e19	0.803826
$927$	3.95683e18	0.204799
$928$	1.00169e17	0.00514841
$929$	−1.91203e19	−0.975870	−0.487935	−	0.872880i	$-0.662250\pi$
$929$	−1.91203e19	−0.975870	−0.487935	+	0.872880i	$0.662250\pi$
$930$	0	0
$931$	2.62855e18	0.132295
$932$	2.69293e16	0.00134593
$933$	−2.49669e19	−1.23918
$934$	−1.66835e19	−0.822303
$935$	0	0
$936$	−1.51684e18	−0.0737304
$937$	−4.30002e18	−0.207569	−0.103785	−	0.994600i	$-0.533095\pi$
$937$	−4.30002e18	−0.207569	−0.103785	+	0.994600i	$0.533095\pi$
$938$	−3.02846e18	−0.145179
$939$	2.28404e19	1.08737
$940$	0	0
$941$	5.18107e18	0.243269	0.121635	−	0.992575i	$-0.461186\pi$
$941$	5.18107e18	0.243269	0.121635	+	0.992575i	$0.461186\pi$
$942$	−2.51489e19	−1.17270
$943$	−2.97128e19	−1.37600
$944$	−2.82878e19	−1.30101
$945$	0	0
$946$	−1.17343e18	−0.0532309
$947$	−1.94885e19	−0.878020	−0.439010	−	0.898482i	$-0.644671\pi$
$947$	−1.94885e19	−0.878020	−0.439010	+	0.898482i	$0.644671\pi$
$948$	4.31168e16	0.00192927
$949$	3.47825e18	0.154572
$950$	0	0
$951$	−2.17669e19	−0.954165
$952$	−3.99950e19	−1.74127
$953$	2.96510e19	1.28214	0.641070	−	0.767483i	$-0.278491\pi$
$953$	2.96510e19	1.28214	0.641070	+	0.767483i	$0.278491\pi$
$954$	−3.53464e18	−0.151803
$955$	0	0
$956$	−5.08965e16	−0.00215631
$957$	5.24353e18	0.220646
$958$	−1.93262e19	−0.807737
$959$	−2.32193e19	−0.963890
$960$	0	0
$961$	3.35903e19	1.37566
$962$	1.13134e19	0.460208
$963$	3.06476e18	0.123830
$964$	8.85631e16	0.00355428
$965$	0	0
$966$	−2.09380e19	−0.829057
$967$	−1.25349e18	−0.0493004	−0.0246502	−	0.999696i	$-0.507847\pi$
$967$	−1.25349e18	−0.0493004	−0.0246502	+	0.999696i	$0.507847\pi$
$968$	2.35506e19	0.920052
$969$	−5.86502e19	−2.27596
$970$	0	0
$971$	−2.17263e19	−0.831879	−0.415940	−	0.909392i	$-0.636547\pi$
$971$	−2.17263e19	−0.831879	−0.415940	+	0.909392i	$0.636547\pi$
$972$	3.09103e16	0.00117563
$973$	3.77327e19	1.42555
$974$	3.39886e19	1.27556
$975$	0	0
$976$	−2.79651e19	−1.03560
$977$	2.56555e19	0.943768	0.471884	−	0.881661i	$-0.343574\pi$
$977$	2.56555e19	0.943768	0.471884	+	0.881661i	$0.343574\pi$
$978$	−1.36393e19	−0.498412
$979$	6.08889e18	0.221030
$980$	0	0
$981$	8.16977e18	0.292659
$982$	3.72539e19	1.32571
$983$	−1.02856e19	−0.363608	−0.181804	−	0.983335i	$-0.558194\pi$
$983$	−1.02856e19	−0.363608	−0.181804	+	0.983335i	$0.558194\pi$
$984$	3.91645e19	1.37538
$985$	0	0
$986$	4.26672e19	1.47875
$987$	3.03661e19	1.04551
$988$	−4.68517e16	−0.00160253
$989$	4.96963e18	0.168868
$990$	0	0
$991$	3.91892e19	1.31428	0.657140	−	0.753769i	$-0.271766\pi$
$991$	3.91892e19	1.31428	0.657140	+	0.753769i	$0.271766\pi$
$992$	−2.69821e17	−0.00898981
$993$	5.19170e18	0.171846
$994$	1.17759e19	0.387242
$995$	0	0
$996$	7.10578e15	0.000230636 0
$997$	−1.60895e19	−0.518829	−0.259414	−	0.965766i	$-0.583530\pi$
$997$	−1.60895e19	−0.518829	−0.259414	+	0.965766i	$0.583530\pi$
$998$	1.57570e19	0.504807
$999$	4.30768e19	1.37110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.14.a.b.1.3		3
5.2	odd	4		25.14.b.b.24.5		6
5.3	odd	4		25.14.b.b.24.2		6
5.4	even	2		5.14.a.b.1.1	✓	3
15.14	odd	2		45.14.a.e.1.3		3
20.19	odd	2		80.14.a.g.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.14.a.b.1.1	✓	3	5.4	even	2
25.14.a.b.1.3		3	1.1	even	1	trivial
25.14.b.b.24.2		6	5.3	odd	4
25.14.b.b.24.5		6	5.2	odd	4
45.14.a.e.1.3		3	15.14	odd	2
80.14.a.g.1.3		3	20.19	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	25.a (trivial)

\(f(q)\)	\(=\)	\(q+90.6415 q^{2} +1125.79 q^{3} +23.8902 q^{4} +102044. q^{6} -324482. q^{7} -740370. q^{8} -326912. q^{9} -1.64726e6 q^{11} +26895.4 q^{12} -6.26700e6 q^{13} -2.94116e7 q^{14} -6.73040e7 q^{16} -1.66481e8 q^{17} -2.96318e7 q^{18} +3.12929e8 q^{19} -3.65300e8 q^{21} -1.49311e8 q^{22} +6.32351e8 q^{23} -8.33504e8 q^{24} -5.68051e8 q^{26} -2.16291e9 q^{27} -7.75194e6 q^{28} -2.82750e9 q^{29} +7.61629e9 q^{31} -3.54269e7 q^{32} -1.85448e9 q^{33} -1.50901e10 q^{34} -7.80998e6 q^{36} -1.99161e10 q^{37} +2.83644e10 q^{38} -7.05535e9 q^{39} -4.69877e10 q^{41} -3.31114e10 q^{42} +7.85897e9 q^{43} -3.93534e7 q^{44} +5.73173e10 q^{46} -8.31265e10 q^{47} -7.57704e10 q^{48} +8.39984e9 q^{49} -1.87423e11 q^{51} -1.49720e8 q^{52} +1.19285e11 q^{53} -1.96050e11 q^{54} +2.40237e11 q^{56} +3.52294e11 q^{57} -2.56289e11 q^{58} +4.20299e11 q^{59} +4.15504e11 q^{61} +6.90352e11 q^{62} +1.06077e11 q^{63} +5.48143e11 q^{64} -1.68093e11 q^{66} +1.02968e11 q^{67} -3.97727e9 q^{68} +7.11897e11 q^{69} -4.00383e11 q^{71} +2.42036e11 q^{72} -5.55011e11 q^{73} -1.80523e12 q^{74} +7.47593e9 q^{76} +5.34508e11 q^{77} -6.39508e11 q^{78} +1.60313e12 q^{79} -1.91379e12 q^{81} -4.25904e12 q^{82} +2.64201e11 q^{83} -8.72709e9 q^{84} +7.12349e11 q^{86} -3.18318e12 q^{87} +1.21958e12 q^{88} -3.69637e12 q^{89} +2.03353e12 q^{91} +1.51070e10 q^{92} +8.57437e12 q^{93} -7.53472e12 q^{94} -3.98834e10 q^{96} +1.00920e13 q^{97} +7.61375e11 q^{98} +5.38510e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 142 q^{2} - 416 q^{3} + 17876 q^{4} + 120376 q^{6} - 448292 q^{7} - 2580360 q^{8} + 1286119 q^{9} - 6604004 q^{11} + 23722448 q^{12} + 33501974 q^{13} - 46562928 q^{14} + 199912208 q^{16} - 83129542 q^{17}+ \cdots + 126787366508 q^{99}+O(q^{100}) \) 3 * q - 142 * q^2 - 416 * q^3 + 17876 * q^4 + 120376 * q^6 - 448292 * q^7 - 2580360 * q^8 + 1286119 * q^9 - 6604004 * q^11 + 23722448 * q^12 + 33501974 * q^13 - 46562928 * q^14 + 199912208 * q^16 - 83129542 * q^17 + 30983914 * q^18 + 97491100 * q^19 + 438200736 * q^21 + 530907256 * q^22 - 316255836 * q^23 - 3890359200 * q^24 - 3746814044 * q^26 - 6518951360 * q^27 + 6227646976 * q^28 + 2236171850 * q^29 + 7482994376 * q^31 - 32169857312 * q^32 - 359182912 * q^33 - 30236073988 * q^34 - 43361833852 * q^36 - 31447174242 * q^37 + 42518132360 * q^38 - 70188571072 * q^39 - 10752884434 * q^41 - 92283853824 * q^42 - 16930554856 * q^43 - 68395825168 * q^44 + 246828204336 * q^46 - 31934201692 * q^47 + 80078828864 * q^48 - 38956926629 * q^49 - 129369882944 * q^51 + 18152597928 * q^52 + 221149123934 * q^53 + 246121937200 * q^54 - 522876451200 * q^56 + 763110695680 * q^57 - 854269976260 * q^58 - 55436423900 * q^59 + 496161392746 * q^61 + 598069940736 * q^62 - 1085454385236 * q^63 + 3297565494336 * q^64 - 22883875168 * q^66 - 459297824792 * q^67 + 2008021178776 * q^68 - 333379292832 * q^69 + 521997878336 * q^71 + 5658040941720 * q^72 - 2505025571086 * q^73 - 2418323586508 * q^74 + 621688050000 * q^76 + 385562457456 * q^77 + 2386950332528 * q^78 + 2990636883200 * q^79 + 320549052763 * q^81 - 5784848206924 * q^82 - 5137135467696 * q^83 - 814234130688 * q^84 + 2343313645096 * q^86 - 6885420178880 * q^87 + 8660607864480 * q^88 - 19423025958450 * q^89 - 6792522370184 * q^91 - 26870022340992 * q^92 + 11095902136128 * q^93 - 7003557875968 * q^94 - 2208331214464 * q^96 + 11088325396458 * q^97 + 10176508990306 * q^98 + 126787366508 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more