LMFDB - Embedded newform 90.14.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [90,14,Mod(1,90)] 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("90.1"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	90.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,128,0,8192,31250,0,-236912] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$96.5078360567$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5778852 $ x^2 - x - 5778852
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2403.42$ of defining polynomial
Character	$\chi$	$=$	90.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+64.0000 q^{2} +4096.00 q^{4} +15625.0 q^{5} +170015. q^{7} +262144. q^{8} +1.00000e6 q^{10} +6.06791e6 q^{11} +3.29464e7 q^{13} +1.08810e7 q^{14} +1.67772e7 q^{16} -5.09064e7 q^{17} -3.38083e8 q^{19} +6.40000e7 q^{20} +3.88346e8 q^{22} +9.47034e8 q^{23} +2.44141e8 q^{25} +2.10857e9 q^{26} +6.96381e8 q^{28} -5.81785e9 q^{29} +3.79039e9 q^{31} +1.07374e9 q^{32} -3.25801e9 q^{34} +2.65648e9 q^{35} +2.84389e10 q^{37} -2.16373e10 q^{38} +4.09600e9 q^{40} -2.20260e10 q^{41} +3.49140e10 q^{43} +2.48541e10 q^{44} +6.06102e10 q^{46} -5.50002e10 q^{47} -6.79839e10 q^{49} +1.56250e10 q^{50} +1.34949e11 q^{52} +3.46323e9 q^{53} +9.48110e10 q^{55} +4.45684e10 q^{56} -3.72342e11 q^{58} -2.80284e11 q^{59} +5.28916e11 q^{61} +2.42585e11 q^{62} +6.87195e10 q^{64} +5.14788e11 q^{65} -7.51430e9 q^{67} -2.08513e11 q^{68} +1.70015e11 q^{70} -1.83209e11 q^{71} +2.21482e12 q^{73} +1.82009e12 q^{74} -1.38479e12 q^{76} +1.03163e12 q^{77} -6.58669e11 q^{79} +2.62144e11 q^{80} -1.40966e12 q^{82} +4.01969e12 q^{83} -7.95413e11 q^{85} +2.23450e12 q^{86} +1.59067e12 q^{88} +2.09375e12 q^{89} +5.60138e12 q^{91} +3.87905e12 q^{92} -3.52001e12 q^{94} -5.28254e12 q^{95} -8.64949e12 q^{97} -4.35097e12 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 128 q^{2} + 8192 q^{4} + 31250 q^{5} - 236912 q^{7} + 524288 q^{8} + 2000000 q^{10} - 4018560 q^{11} + 13391116 q^{13} - 15162368 q^{14} + 33554432 q^{16} + 47615052 q^{17} - 90569648 q^{19} + 128000000 q^{20}+ \cdots + 45860598912 q^{98}+O(q^{100}) $ 2 * q + 128 * q^2 + 8192 * q^4 + 31250 * q^5 - 236912 * q^7 + 524288 * q^8 + 2000000 * q^10 - 4018560 * q^11 + 13391116 * q^13 - 15162368 * q^14 + 33554432 * q^16 + 47615052 * q^17 - 90569648 * q^19 + 128000000 * q^20 - 257187840 * q^22 + 1025770200 * q^23 + 488281250 * q^25 + 857031424 * q^26 - 970391552 * q^28 - 2072310228 * q^29 + 11728406776 * q^31 + 2147483648 * q^32 + 3047363328 * q^34 - 3701750000 * q^35 + 47778848716 * q^37 - 5796457472 * q^38 + 8192000000 * q^40 - 67871568228 * q^41 + 77614412728 * q^43 - 16460021760 * q^44 + 65649292800 * q^46 - 106902703320 * q^47 + 716571858 * q^49 + 31250000000 * q^50 + 54850011136 * q^52 - 4551230532 * q^53 - 62790000000 * q^55 - 62105059328 * q^56 - 132627854592 * q^58 + 4334346240 * q^59 + 887411684860 * q^61 + 750618033664 * q^62 + 137438953472 * q^64 + 209236187500 * q^65 + 1467845781688 * q^67 + 195031252992 * q^68 - 236912000000 * q^70 - 936151332480 * q^71 + 813936935956 * q^73 + 3057846317824 * q^74 - 370973278208 * q^76 + 5136088997760 * q^77 - 1248629414744 * q^79 + 524288000000 * q^80 - 4343780366592 * q^82 - 603864311256 * q^83 + 743985187500 * q^85 + 4967322414592 * q^86 - 1053441392640 * q^88 + 8016259169484 * q^89 + 13558957939904 * q^91 + 4201554739200 * q^92 - 6841773012480 * q^94 - 1415150750000 * q^95 - 1682542380284 * q^97 + 45860598912 * q^98

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 $

$2$	64.0000	0.707107
$2$	64.0000	0.707107
$3$	0	0
$3$	0	0
$4$	4096.00	0.500000
$5$	15625.0	0.447214
$5$	15625.0	0.447214
$6$	0	0
$7$	170015.	0.546198	0.273099	−	0.961986i	$-0.411951\pi$
$7$	170015.	0.546198	0.273099	+	0.961986i	$0.411951\pi$
$8$	262144.	0.353553
$9$	0	0
$10$	1.00000e6	0.316228
$11$	6.06791e6	1.03273	0.516365	−	0.856369i	$-0.327285\pi$
$11$	6.06791e6	1.03273	0.516365	+	0.856369i	$0.327285\pi$
$12$	0	0
$13$	3.29464e7	1.89311	0.946556	−	0.322539i	$-0.104536\pi$
$13$	3.29464e7	1.89311	0.946556	+	0.322539i	$0.104536\pi$
$14$	1.08810e7	0.386220
$15$	0	0
$16$	1.67772e7	0.250000
$17$	−5.09064e7	−0.511511	−0.255755	−	0.966742i	$-0.582324\pi$
$17$	−5.09064e7	−0.511511	−0.255755	+	0.966742i	$0.582324\pi$
$18$	0	0
$19$	−3.38083e8	−1.64864	−0.824318	−	0.566127i	$-0.808441\pi$
$19$	−3.38083e8	−1.64864	−0.824318	+	0.566127i	$0.808441\pi$
$20$	6.40000e7	0.223607
$21$	0	0
$22$	3.88346e8	0.730250
$23$	9.47034e8	1.33393	0.666967	−	0.745087i	$-0.267592\pi$
$23$	9.47034e8	1.33393	0.666967	+	0.745087i	$0.267592\pi$
$24$	0	0
$25$	2.44141e8	0.200000
$26$	2.10857e9	1.33863
$27$	0	0
$28$	6.96381e8	0.273099
$29$	−5.81785e9	−1.81625	−0.908125	−	0.418700i	$-0.862486\pi$
$29$	−5.81785e9	−1.81625	−0.908125	+	0.418700i	$0.862486\pi$
$30$	0	0
$31$	3.79039e9	0.767065	0.383533	−	0.923527i	$-0.374707\pi$
$31$	3.79039e9	0.767065	0.383533	+	0.923527i	$0.374707\pi$
$32$	1.07374e9	0.176777
$33$	0	0
$34$	−3.25801e9	−0.361693
$35$	2.65648e9	0.244267
$36$	0	0
$37$	2.84389e10	1.82222	0.911112	−	0.412160i	$-0.135225\pi$
$37$	2.84389e10	1.82222	0.911112	+	0.412160i	$0.135225\pi$
$38$	−2.16373e10	−1.16576
$39$	0	0
$40$	4.09600e9	0.158114
$41$	−2.20260e10	−0.724169	−0.362085	−	0.932145i	$-0.617935\pi$
$41$	−2.20260e10	−0.724169	−0.362085	+	0.932145i	$0.617935\pi$
$42$	0	0
$43$	3.49140e10	0.842276	0.421138	−	0.906996i	$-0.361631\pi$
$43$	3.49140e10	0.842276	0.421138	+	0.906996i	$0.361631\pi$
$44$	2.48541e10	0.516365
$45$	0	0
$46$	6.06102e10	0.943234
$47$	−5.50002e10	−0.744265	−0.372133	−	0.928180i	$-0.621373\pi$
$47$	−5.50002e10	−0.744265	−0.372133	+	0.928180i	$0.621373\pi$
$48$	0	0
$49$	−6.79839e10	−0.701668
$50$	1.56250e10	0.141421
$51$	0	0
$52$	1.34949e11	0.946556
$53$	3.46323e9	0.0214629	0.0107314	−	0.999942i	$-0.496584\pi$
$53$	3.46323e9	0.0214629	0.0107314	+	0.999942i	$0.496584\pi$
$54$	0	0
$55$	9.48110e10	0.461850
$56$	4.45684e10	0.193110
$57$	0	0
$58$	−3.72342e11	−1.28428
$59$	−2.80284e11	−0.865088	−0.432544	−	0.901613i	$-0.642384\pi$
$59$	−2.80284e11	−0.865088	−0.432544	+	0.901613i	$0.642384\pi$
$60$	0	0
$61$	5.28916e11	1.31445	0.657223	−	0.753696i	$-0.271731\pi$
$61$	5.28916e11	1.31445	0.657223	+	0.753696i	$0.271731\pi$
$62$	2.42585e11	0.542397
$63$	0	0
$64$	6.87195e10	0.125000
$65$	5.14788e11	0.846625
$66$	0	0
$67$	−7.51430e9	−0.0101485	−0.00507426	−	0.999987i	$-0.501615\pi$
$67$	−7.51430e9	−0.0101485	−0.00507426	+	0.999987i	$0.501615\pi$
$68$	−2.08513e11	−0.255755
$69$	0	0
$70$	1.70015e11	0.172723
$71$	−1.83209e11	−0.169734	−0.0848670	−	0.996392i	$-0.527047\pi$
$71$	−1.83209e11	−0.169734	−0.0848670	+	0.996392i	$0.527047\pi$
$72$	0	0
$73$	2.21482e12	1.71293	0.856467	−	0.516202i	$-0.172654\pi$
$73$	2.21482e12	1.71293	0.856467	+	0.516202i	$0.172654\pi$
$74$	1.82009e12	1.28851
$75$	0	0
$76$	−1.38479e12	−0.824318
$77$	1.03163e12	0.564074
$78$	0	0
$79$	−6.58669e11	−0.304853	−0.152427	−	0.988315i	$-0.548709\pi$
$79$	−6.58669e11	−0.304853	−0.152427	+	0.988315i	$0.548709\pi$
$80$	2.62144e11	0.111803
$81$	0	0
$82$	−1.40966e12	−0.512065
$83$	4.01969e12	1.34954	0.674768	−	0.738030i	$-0.264244\pi$
$83$	4.01969e12	1.34954	0.674768	+	0.738030i	$0.264244\pi$
$84$	0	0
$85$	−7.95413e11	−0.228755
$86$	2.23450e12	0.595579
$87$	0	0
$88$	1.59067e12	0.365125
$89$	2.09375e12	0.446571	0.223285	−	0.974753i	$-0.428322\pi$
$89$	2.09375e12	0.446571	0.223285	+	0.974753i	$0.428322\pi$
$90$	0	0
$91$	5.60138e12	1.03401
$92$	3.87905e12	0.666967
$93$	0	0
$94$	−3.52001e12	−0.526275
$95$	−5.28254e12	−0.737292
$96$	0	0
$97$	−8.64949e12	−1.05432	−0.527162	−	0.849765i	$-0.676744\pi$
$97$	−8.64949e12	−1.05432	−0.527162	+	0.849765i	$0.676744\pi$
$98$	−4.35097e12	−0.496154
$99$	0	0
$100$	1.00000e12	0.100000
$101$	−9.92703e12	−0.930530	−0.465265	−	0.885172i	$-0.654041\pi$
$101$	−9.92703e12	−0.930530	−0.465265	+	0.885172i	$0.654041\pi$
$102$	0	0
$103$	9.95146e12	0.821192	0.410596	−	0.911817i	$-0.365321\pi$
$103$	9.95146e12	0.821192	0.410596	+	0.911817i	$0.365321\pi$
$104$	8.63670e12	0.669316
$105$	0	0
$106$	2.21646e11	0.0151765
$107$	1.03046e13	0.663801	0.331900	−	0.943314i	$-0.392310\pi$
$107$	1.03046e13	0.663801	0.331900	+	0.943314i	$0.392310\pi$
$108$	0	0
$109$	2.36639e13	1.35149	0.675747	−	0.737134i	$-0.263821\pi$
$109$	2.36639e13	1.35149	0.675747	+	0.737134i	$0.263821\pi$
$110$	6.06791e12	0.326578
$111$	0	0
$112$	2.85238e12	0.136549
$113$	2.70737e12	0.122331	0.0611656	−	0.998128i	$-0.480518\pi$
$113$	2.70737e12	0.122331	0.0611656	+	0.998128i	$0.480518\pi$
$114$	0	0
$115$	1.47974e13	0.596554
$116$	−2.38299e13	−0.908125
$117$	0	0
$118$	−1.79382e13	−0.611709
$119$	−8.65485e12	−0.279386
$120$	0	0
$121$	2.29677e12	0.0665292
$122$	3.38506e13	0.929454
$123$	0	0
$124$	1.55254e13	0.383533
$125$	3.81470e12	0.0894427
$126$	0	0
$127$	6.46305e12	0.136683	0.0683413	−	0.997662i	$-0.478229\pi$
$127$	6.46305e12	0.136683	0.0683413	+	0.997662i	$0.478229\pi$
$128$	4.39805e12	0.0883883
$129$	0	0
$130$	3.29464e13	0.598655
$131$	8.36401e12	0.144594	0.0722972	−	0.997383i	$-0.476967\pi$
$131$	8.36401e12	0.144594	0.0722972	+	0.997383i	$0.476967\pi$
$132$	0	0
$133$	−5.74791e13	−0.900481
$134$	−4.80915e11	−0.00717608
$135$	0	0
$136$	−1.33448e13	−0.180846
$137$	1.23769e14	1.59930	0.799649	−	0.600468i	$-0.205019\pi$
$137$	1.23769e14	1.59930	0.799649	+	0.600468i	$0.205019\pi$
$138$	0	0
$139$	−1.98109e13	−0.232975	−0.116487	−	0.993192i	$-0.537163\pi$
$139$	−1.98109e13	−0.232975	−0.116487	+	0.993192i	$0.537163\pi$
$140$	1.08810e13	0.122133
$141$	0	0
$142$	−1.17254e13	−0.120020
$143$	1.99916e14	1.95507
$144$	0	0
$145$	−9.09039e13	−0.812251
$146$	1.41749e14	1.21123
$147$	0	0
$148$	1.16486e14	0.911112
$149$	1.36275e14	1.02024	0.510122	−	0.860102i	$-0.329600\pi$
$149$	1.36275e14	1.02024	0.510122	+	0.860102i	$0.329600\pi$
$150$	0	0
$151$	1.46787e14	1.00771	0.503856	−	0.863788i	$-0.331914\pi$
$151$	1.46787e14	1.00771	0.503856	+	0.863788i	$0.331914\pi$
$152$	−8.86264e13	−0.582881
$153$	0	0
$154$	6.60246e13	0.398861
$155$	5.92248e13	0.343042
$156$	0	0
$157$	−2.46869e14	−1.31559	−0.657793	−	0.753199i	$-0.728510\pi$
$157$	−2.46869e14	−1.31559	−0.657793	+	0.753199i	$0.728510\pi$
$158$	−4.21548e13	−0.215564
$159$	0	0
$160$	1.67772e13	0.0790569
$161$	1.61010e14	0.728592
$162$	0	0
$163$	−9.21834e13	−0.384976	−0.192488	−	0.981299i	$-0.561656\pi$
$163$	−9.21834e13	−0.384976	−0.192488	+	0.981299i	$0.561656\pi$
$164$	−9.02184e13	−0.362085
$165$	0	0
$166$	2.57260e14	0.954266
$167$	7.83286e13	0.279424	0.139712	−	0.990192i	$-0.455382\pi$
$167$	7.83286e13	0.279424	0.139712	+	0.990192i	$0.455382\pi$
$168$	0	0
$169$	7.82591e14	2.58387
$170$	−5.09064e13	−0.161754
$171$	0	0
$172$	1.43008e14	0.421138
$173$	−4.49950e14	−1.27604	−0.638020	−	0.770020i	$-0.720246\pi$
$173$	−4.49950e14	−1.27604	−0.638020	+	0.770020i	$0.720246\pi$
$174$	0	0
$175$	4.15075e13	0.109240
$176$	1.01803e14	0.258182
$177$	0	0
$178$	1.34000e14	0.315773
$179$	1.39179e14	0.316249	0.158124	−	0.987419i	$-0.449455\pi$
$179$	1.39179e14	0.316249	0.158124	+	0.987419i	$0.449455\pi$
$180$	0	0
$181$	−5.20706e14	−1.10073	−0.550367	−	0.834923i	$-0.685512\pi$
$181$	−5.20706e14	−1.10073	−0.550367	+	0.834923i	$0.685512\pi$
$182$	3.58488e14	0.731158
$183$	0	0
$184$	2.48259e14	0.471617
$185$	4.44358e14	0.814923
$186$	0	0
$187$	−3.08895e14	−0.528252
$188$	−2.25281e14	−0.372133
$189$	0	0
$190$	−3.38083e14	−0.521344
$191$	−1.09463e15	−1.63136	−0.815682	−	0.578501i	$-0.803638\pi$
$191$	−1.09463e15	−1.63136	−0.815682	+	0.578501i	$0.803638\pi$
$192$	0	0
$193$	6.26624e14	0.872739	0.436370	−	0.899767i	$-0.356264\pi$
$193$	6.26624e14	0.872739	0.436370	+	0.899767i	$0.356264\pi$
$194$	−5.53567e14	−0.745520
$195$	0	0
$196$	−2.78462e14	−0.350834
$197$	3.32275e14	0.405012	0.202506	−	0.979281i	$-0.435091\pi$
$197$	3.32275e14	0.405012	0.202506	+	0.979281i	$0.435091\pi$
$198$	0	0
$199$	5.22579e13	0.0596495	0.0298248	−	0.999555i	$-0.490505\pi$
$199$	5.22579e13	0.0596495	0.0298248	+	0.999555i	$0.490505\pi$
$200$	6.40000e13	0.0707107
$201$	0	0
$202$	−6.35330e14	−0.657984
$203$	−9.89121e14	−0.992031
$204$	0	0
$205$	−3.44156e14	−0.323858
$206$	6.36893e14	0.580670
$207$	0	0
$208$	5.52749e14	0.473278
$209$	−2.05145e15	−1.70259
$210$	0	0
$211$	−1.36362e15	−1.06379	−0.531896	−	0.846810i	$-0.678520\pi$
$211$	−1.36362e15	−1.06379	−0.531896	+	0.846810i	$0.678520\pi$
$212$	1.41854e13	0.0107314
$213$	0	0
$214$	6.59496e14	0.469378
$215$	5.45531e14	0.376677
$216$	0	0
$217$	6.44422e14	0.418969
$218$	1.51449e15	0.955650
$219$	0	0
$220$	3.88346e14	0.230925
$221$	−1.67718e15	−0.968347
$222$	0	0
$223$	−2.47259e15	−1.34639	−0.673196	−	0.739464i	$-0.735079\pi$
$223$	−2.47259e15	−1.34639	−0.673196	+	0.739464i	$0.735079\pi$
$224$	1.82552e14	0.0965550
$225$	0	0
$226$	1.73271e14	0.0865012
$227$	1.16478e15	0.565034	0.282517	−	0.959262i	$-0.408831\pi$
$227$	1.16478e15	0.565034	0.282517	+	0.959262i	$0.408831\pi$
$228$	0	0
$229$	−2.63317e15	−1.20656	−0.603279	−	0.797530i	$-0.706140\pi$
$229$	−2.63317e15	−1.20656	−0.603279	+	0.797530i	$0.706140\pi$
$230$	9.47034e14	0.421827
$231$	0	0
$232$	−1.52511e15	−0.642141
$233$	−1.81315e15	−0.742369	−0.371185	−	0.928559i	$-0.621048\pi$
$233$	−1.81315e15	−0.742369	−0.371185	+	0.928559i	$0.621048\pi$
$234$	0	0
$235$	−8.59377e14	−0.332846
$236$	−1.14804e15	−0.432544
$237$	0	0
$238$	−5.53911e14	−0.197556
$239$	−5.57074e15	−1.93342	−0.966710	−	0.255874i	$-0.917637\pi$
$239$	−5.57074e15	−1.93342	−0.966710	+	0.255874i	$0.917637\pi$
$240$	0	0
$241$	−6.85386e14	−0.225333	−0.112666	−	0.993633i	$-0.535939\pi$
$241$	−6.85386e14	−0.225333	−0.112666	+	0.993633i	$0.535939\pi$
$242$	1.46993e14	0.0470433
$243$	0	0
$244$	2.16644e15	0.657223
$245$	−1.06225e15	−0.313796
$246$	0	0
$247$	−1.11386e16	−3.12105
$248$	9.93627e14	0.271199
$249$	0	0
$250$	2.44141e14	0.0632456
$251$	−7.27308e14	−0.183586	−0.0917929	−	0.995778i	$-0.529260\pi$
$251$	−7.27308e14	−0.183586	−0.0917929	+	0.995778i	$0.529260\pi$
$252$	0	0
$253$	5.74651e15	1.37759
$254$	4.13635e14	0.0966492
$255$	0	0
$256$	2.81475e14	0.0625000
$257$	2.83845e14	0.0614491	0.0307245	−	0.999528i	$-0.490219\pi$
$257$	2.83845e14	0.0614491	0.0307245	+	0.999528i	$0.490219\pi$
$258$	0	0
$259$	4.83504e15	0.995294
$260$	2.10857e15	0.423313
$261$	0	0
$262$	5.35297e14	0.102244
$263$	3.35049e15	0.624303	0.312152	−	0.950032i	$-0.398950\pi$
$263$	3.35049e15	0.624303	0.312152	+	0.950032i	$0.398950\pi$
$264$	0	0
$265$	5.41129e13	0.00959848
$266$	−3.67866e15	−0.636736
$267$	0	0
$268$	−3.07786e13	−0.00507426
$269$	−7.65805e15	−1.23233	−0.616167	−	0.787615i	$-0.711315\pi$
$269$	−7.65805e15	−1.23233	−0.616167	+	0.787615i	$0.711315\pi$
$270$	0	0
$271$	4.60099e14	0.0705588	0.0352794	−	0.999377i	$-0.488768\pi$
$271$	4.60099e14	0.0705588	0.0352794	+	0.999377i	$0.488768\pi$
$272$	−8.54068e14	−0.127878
$273$	0	0
$274$	7.92124e15	1.13087
$275$	1.48142e15	0.206546
$276$	0	0
$277$	−7.55123e15	−1.00438	−0.502191	−	0.864757i	$-0.667473\pi$
$277$	−7.55123e15	−1.00438	−0.502191	+	0.864757i	$0.667473\pi$
$278$	−1.26790e15	−0.164738
$279$	0	0
$280$	6.96381e14	0.0863614
$281$	−5.96982e15	−0.723386	−0.361693	−	0.932297i	$-0.617801\pi$
$281$	−5.96982e15	−0.723386	−0.361693	+	0.932297i	$0.617801\pi$
$282$	0	0
$283$	1.32479e16	1.53297	0.766486	−	0.642260i	$-0.222003\pi$
$283$	1.32479e16	1.53297	0.766486	+	0.642260i	$0.222003\pi$
$284$	−7.50426e14	−0.0848670
$285$	0	0
$286$	1.27946e16	1.38244
$287$	−3.74474e15	−0.395539
$288$	0	0
$289$	−7.31311e15	−0.738357
$290$	−5.81785e15	−0.574348
$291$	0	0
$292$	9.07192e15	0.856467
$293$	4.74056e15	0.437714	0.218857	−	0.975757i	$-0.429767\pi$
$293$	4.74056e15	0.437714	0.218857	+	0.975757i	$0.429767\pi$
$294$	0	0
$295$	−4.37944e15	−0.386879
$296$	7.45509e15	0.644253
$297$	0	0
$298$	8.72157e15	0.721421
$299$	3.12014e16	2.52529
$300$	0	0
$301$	5.93590e15	0.460049
$302$	9.39435e15	0.712560
$303$	0	0
$304$	−5.67209e15	−0.412159
$305$	8.26431e15	0.587838
$306$	0	0
$307$	−3.78865e15	−0.258276	−0.129138	−	0.991627i	$-0.541221\pi$
$307$	−3.78865e15	−0.258276	−0.129138	+	0.991627i	$0.541221\pi$
$308$	4.22558e15	0.282037
$309$	0	0
$310$	3.79039e15	0.242567
$311$	−1.46650e16	−0.919050	−0.459525	−	0.888165i	$-0.651980\pi$
$311$	−1.46650e16	−0.919050	−0.459525	+	0.888165i	$0.651980\pi$
$312$	0	0
$313$	−5.58982e15	−0.336016	−0.168008	−	0.985786i	$-0.553733\pi$
$313$	−5.58982e15	−0.336016	−0.168008	+	0.985786i	$0.553733\pi$
$314$	−1.57996e16	−0.930260
$315$	0	0
$316$	−2.69791e15	−0.152427
$317$	1.75354e16	0.970578	0.485289	−	0.874354i	$-0.338714\pi$
$317$	1.75354e16	0.970578	0.485289	+	0.874354i	$0.338714\pi$
$318$	0	0
$319$	−3.53022e16	−1.87569
$320$	1.07374e15	0.0559017
$321$	0	0
$322$	1.03046e16	0.515192
$323$	1.72106e16	0.843295
$324$	0	0
$325$	8.04356e15	0.378622
$326$	−5.89974e15	−0.272219
$327$	0	0
$328$	−5.77398e15	−0.256032
$329$	−9.35085e15	−0.406516
$330$	0	0
$331$	−2.66488e16	−1.11377	−0.556886	−	0.830589i	$-0.688004\pi$
$331$	−2.66488e16	−1.11377	−0.556886	+	0.830589i	$0.688004\pi$
$332$	1.64646e16	0.674768
$333$	0	0
$334$	5.01303e15	0.197582
$335$	−1.17411e14	−0.00453855
$336$	0	0
$337$	2.82770e16	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$337$	2.82770e16	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$338$	5.00858e16	1.82707
$339$	0	0
$340$	−3.25801e15	−0.114377
$341$	2.29997e16	0.792171
$342$	0	0
$343$	−2.80309e16	−0.929447
$344$	9.15250e15	0.297790
$345$	0	0
$346$	−2.87968e16	−0.902297
$347$	−4.37490e16	−1.34532	−0.672662	−	0.739950i	$-0.734849\pi$
$347$	−4.37490e16	−1.34532	−0.672662	+	0.739950i	$0.734849\pi$
$348$	0	0
$349$	−9.77276e15	−0.289502	−0.144751	−	0.989468i	$-0.546238\pi$
$349$	−9.77276e15	−0.289502	−0.144751	+	0.989468i	$0.546238\pi$
$350$	2.65648e15	0.0772440
$351$	0	0
$352$	6.51536e15	0.182562
$353$	−3.36370e16	−0.925297	−0.462649	−	0.886542i	$-0.653101\pi$
$353$	−3.36370e16	−0.925297	−0.462649	+	0.886542i	$0.653101\pi$
$354$	0	0
$355$	−2.86265e15	−0.0759073
$356$	8.57601e15	0.223285
$357$	0	0
$358$	8.90746e15	0.223622
$359$	5.73427e16	1.41372	0.706861	−	0.707353i	$-0.250111\pi$
$359$	5.73427e16	1.41372	0.706861	+	0.707353i	$0.250111\pi$
$360$	0	0
$361$	7.22470e16	1.71800
$362$	−3.33252e16	−0.778336
$363$	0	0
$364$	2.29433e16	0.517007
$365$	3.46066e16	0.766048
$366$	0	0
$367$	1.30762e15	0.0279352	0.0139676	−	0.999902i	$-0.495554\pi$
$367$	1.30762e15	0.0279352	0.0139676	+	0.999902i	$0.495554\pi$
$368$	1.58886e16	0.333484
$369$	0	0
$370$	2.84389e16	0.576238
$371$	5.88800e14	0.0117230
$372$	0	0
$373$	5.05383e16	0.971658	0.485829	−	0.874054i	$-0.338518\pi$
$373$	5.05383e16	0.971658	0.485829	+	0.874054i	$0.338518\pi$
$374$	−1.97693e16	−0.373531
$375$	0	0
$376$	−1.44180e16	−0.263138
$377$	−1.91677e17	−3.43836
$378$	0	0
$379$	3.58137e16	0.620719	0.310359	−	0.950619i	$-0.399551\pi$
$379$	3.58137e16	0.620719	0.310359	+	0.950619i	$0.399551\pi$
$380$	−2.16373e16	−0.368646
$381$	0	0
$382$	−7.00564e16	−1.15355
$383$	1.16220e17	1.88143	0.940713	−	0.339203i	$-0.110157\pi$
$383$	1.16220e17	1.88143	0.940713	+	0.339203i	$0.110157\pi$
$384$	0	0
$385$	1.61193e16	0.252262
$386$	4.01039e16	0.617120
$387$	0	0
$388$	−3.54283e16	−0.527162
$389$	−2.44732e16	−0.358112	−0.179056	−	0.983839i	$-0.557304\pi$
$389$	−2.44732e16	−0.358112	−0.179056	+	0.983839i	$0.557304\pi$
$390$	0	0
$391$	−4.82101e16	−0.682322
$392$	−1.78216e16	−0.248077
$393$	0	0
$394$	2.12656e16	0.286386
$395$	−1.02917e16	−0.136335
$396$	0	0
$397$	7.19607e16	0.922481	0.461240	−	0.887275i	$-0.347405\pi$
$397$	7.19607e16	0.922481	0.461240	+	0.887275i	$0.347405\pi$
$398$	3.34451e15	0.0421786
$399$	0	0
$400$	4.09600e15	0.0500000
$401$	−2.77695e16	−0.333526	−0.166763	−	0.985997i	$-0.553331\pi$
$401$	−2.77695e16	−0.333526	−0.166763	+	0.985997i	$0.553331\pi$
$402$	0	0
$403$	1.24880e17	1.45214
$404$	−4.06611e16	−0.465265
$405$	0	0
$406$	−6.33038e16	−0.701472
$407$	1.72565e17	1.88186
$408$	0	0
$409$	−1.50306e17	−1.58772	−0.793859	−	0.608102i	$-0.791931\pi$
$409$	−1.50306e17	−1.58772	−0.793859	+	0.608102i	$0.791931\pi$
$410$	−2.20260e16	−0.229002
$411$	0	0
$412$	4.07612e16	0.410596
$413$	−4.76525e16	−0.472509
$414$	0	0
$415$	6.28076e16	0.603531
$416$	3.53759e16	0.334658
$417$	0	0
$418$	−1.31293e17	−1.20392
$419$	9.26732e16	0.836687	0.418343	−	0.908289i	$-0.362611\pi$
$419$	9.26732e16	0.836687	0.418343	+	0.908289i	$0.362611\pi$
$420$	0	0
$421$	4.68568e16	0.410146	0.205073	−	0.978747i	$-0.434257\pi$
$421$	4.68568e16	0.410146	0.205073	+	0.978747i	$0.434257\pi$
$422$	−8.72715e16	−0.752214
$423$	0	0
$424$	9.07864e14	0.00758827
$425$	−1.24283e16	−0.102302
$426$	0	0
$427$	8.99236e16	0.717947
$428$	4.22078e16	0.331900
$429$	0	0
$430$	3.49140e16	0.266351
$431$	−2.33764e17	−1.75661	−0.878304	−	0.478103i	$-0.841324\pi$
$431$	−2.33764e17	−1.75661	−0.878304	+	0.478103i	$0.841324\pi$
$432$	0	0
$433$	−1.57411e17	−1.14779	−0.573897	−	0.818927i	$-0.694569\pi$
$433$	−1.57411e17	−1.14779	−0.573897	+	0.818927i	$0.694569\pi$
$434$	4.12430e16	0.296256
$435$	0	0
$436$	9.69273e16	0.675747
$437$	−3.20176e17	−2.19917
$438$	0	0
$439$	−2.21873e17	−1.47940	−0.739698	−	0.672939i	$-0.765032\pi$
$439$	−2.21873e17	−1.47940	−0.739698	+	0.672939i	$0.765032\pi$
$440$	2.48541e16	0.163289
$441$	0	0
$442$	−1.07340e17	−0.684725
$443$	−1.42772e17	−0.897470	−0.448735	−	0.893665i	$-0.648125\pi$
$443$	−1.42772e17	−0.897470	−0.448735	+	0.893665i	$0.648125\pi$
$444$	0	0
$445$	3.27149e16	0.199712
$446$	−1.58246e17	−0.952042
$447$	0	0
$448$	1.16833e16	0.0682747
$449$	1.94696e17	1.12139	0.560693	−	0.828024i	$-0.310535\pi$
$449$	1.94696e17	1.12139	0.560693	+	0.828024i	$0.310535\pi$
$450$	0	0
$451$	−1.33652e17	−0.747871
$452$	1.10894e16	0.0611656
$453$	0	0
$454$	7.45457e16	0.399539
$455$	8.75216e16	0.462425
$456$	0	0
$457$	−2.06061e17	−1.05813	−0.529067	−	0.848580i	$-0.677458\pi$
$457$	−2.06061e17	−1.05813	−0.529067	+	0.848580i	$0.677458\pi$
$458$	−1.68523e17	−0.853166
$459$	0	0
$460$	6.06102e16	0.298277
$461$	3.50887e17	1.70260	0.851298	−	0.524682i	$-0.175816\pi$
$461$	3.50887e17	1.70260	0.851298	+	0.524682i	$0.175816\pi$
$462$	0	0
$463$	3.11707e16	0.147052	0.0735259	−	0.997293i	$-0.476575\pi$
$463$	3.11707e16	0.147052	0.0735259	+	0.997293i	$0.476575\pi$
$464$	−9.76073e16	−0.454062
$465$	0	0
$466$	−1.16042e17	−0.524934
$467$	3.16096e17	1.41013	0.705066	−	0.709141i	$-0.250917\pi$
$467$	3.16096e17	1.41013	0.705066	+	0.709141i	$0.250917\pi$
$468$	0	0
$469$	−1.27754e15	−0.00554309
$470$	−5.50002e16	−0.235357
$471$	0	0
$472$	−7.34748e16	−0.305855
$473$	2.11855e17	0.869843
$474$	0	0
$475$	−8.25397e16	−0.329727
$476$	−3.54503e16	−0.139693
$477$	0	0
$478$	−3.56527e17	−1.36713
$479$	−7.59224e16	−0.287203	−0.143601	−	0.989636i	$-0.545868\pi$
$479$	−7.59224e16	−0.287203	−0.143601	+	0.989636i	$0.545868\pi$
$480$	0	0
$481$	9.36960e17	3.44967
$482$	−4.38647e16	−0.159334
$483$	0	0
$484$	9.40756e15	0.0332646
$485$	−1.35148e17	−0.471508
$486$	0	0
$487$	−3.20670e17	−1.08923	−0.544615	−	0.838686i	$-0.683324\pi$
$487$	−3.20670e17	−1.08923	−0.544615	+	0.838686i	$0.683324\pi$
$488$	1.38652e17	0.464727
$489$	0	0
$490$	−6.79839e16	−0.221887
$491$	−4.08390e17	−1.31536	−0.657681	−	0.753296i	$-0.728463\pi$
$491$	−4.08390e17	−1.31536	−0.657681	+	0.753296i	$0.728463\pi$
$492$	0	0
$493$	2.96166e17	0.929031
$494$	−7.12871e17	−2.20692
$495$	0	0
$496$	6.35921e16	0.191766
$497$	−3.11483e16	−0.0927083
$498$	0	0
$499$	−2.29462e17	−0.665361	−0.332680	−	0.943040i	$-0.607953\pi$
$499$	−2.29462e17	−0.665361	−0.332680	+	0.943040i	$0.607953\pi$
$500$	1.56250e16	0.0447214
$501$	0	0
$502$	−4.65477e16	−0.129815
$503$	−6.88434e17	−1.89527	−0.947633	−	0.319361i	$-0.896532\pi$
$503$	−6.88434e17	−1.89527	−0.947633	+	0.319361i	$0.896532\pi$
$504$	0	0
$505$	−1.55110e17	−0.416145
$506$	3.67777e17	0.974105
$507$	0	0
$508$	2.64726e16	0.0683413
$509$	2.88019e17	0.734101	0.367051	−	0.930201i	$-0.380368\pi$
$509$	2.88019e17	0.734101	0.367051	+	0.930201i	$0.380368\pi$
$510$	0	0
$511$	3.76553e17	0.935601
$512$	1.80144e16	0.0441942
$513$	0	0
$514$	1.81661e16	0.0434511
$515$	1.55492e17	0.367248
$516$	0	0
$517$	−3.33736e17	−0.768624
$518$	3.09442e17	0.703779
$519$	0	0
$520$	1.34949e17	0.299327
$521$	3.07194e17	0.672926	0.336463	−	0.941697i	$-0.390769\pi$
$521$	3.07194e17	0.672926	0.336463	+	0.941697i	$0.390769\pi$
$522$	0	0
$523$	5.25192e17	1.12217	0.561083	−	0.827760i	$-0.310385\pi$
$523$	5.25192e17	1.12217	0.561083	+	0.827760i	$0.310385\pi$
$524$	3.42590e16	0.0722972
$525$	0	0
$526$	2.14431e17	0.441449
$527$	−1.92955e17	−0.392362
$528$	0	0
$529$	3.92837e17	0.779382
$530$	3.46323e15	0.00678715
$531$	0	0
$532$	−2.35434e17	−0.450240
$533$	−7.25677e17	−1.37093
$534$	0	0
$535$	1.61010e17	0.296861
$536$	−1.96983e15	−0.00358804
$537$	0	0
$538$	−4.90115e17	−0.871392
$539$	−4.12520e17	−0.724633
$540$	0	0
$541$	−6.85158e16	−0.117492	−0.0587460	−	0.998273i	$-0.518710\pi$
$541$	−6.85158e16	−0.117492	−0.0587460	+	0.998273i	$0.518710\pi$
$542$	2.94463e16	0.0498926
$543$	0	0
$544$	−5.46604e16	−0.0904232
$545$	3.69748e17	0.604406
$546$	0	0
$547$	−7.30717e17	−1.16636	−0.583179	−	0.812344i	$-0.698191\pi$
$547$	−7.30717e17	−1.16636	−0.583179	+	0.812344i	$0.698191\pi$
$548$	5.06959e17	0.799649
$549$	0	0
$550$	9.48110e16	0.146050
$551$	1.96691e18	2.99433
$552$	0	0
$553$	−1.11984e17	−0.166510
$554$	−4.83279e17	−0.710206
$555$	0	0
$556$	−8.11456e16	−0.116487
$557$	1.23097e18	1.74658	0.873291	−	0.487199i	$-0.161981\pi$
$557$	1.23097e18	1.74658	0.873291	+	0.487199i	$0.161981\pi$
$558$	0	0
$559$	1.15029e18	1.59452
$560$	4.45684e16	0.0610667
$561$	0	0
$562$	−3.82069e17	−0.511511
$563$	−4.06336e16	−0.0537751	−0.0268876	−	0.999638i	$-0.508560\pi$
$563$	−4.06336e16	−0.0537751	−0.0268876	+	0.999638i	$0.508560\pi$
$564$	0	0
$565$	4.23026e16	0.0547082
$566$	8.47864e17	1.08398
$567$	0	0
$568$	−4.80273e16	−0.0600100
$569$	2.95112e17	0.364550	0.182275	−	0.983248i	$-0.441654\pi$
$569$	2.95112e17	0.364550	0.182275	+	0.983248i	$0.441654\pi$
$570$	0	0
$571$	1.05610e18	1.27518	0.637590	−	0.770376i	$-0.279932\pi$
$571$	1.05610e18	1.27518	0.637590	+	0.770376i	$0.279932\pi$
$572$	8.18855e17	0.977536
$573$	0	0
$574$	−2.39664e17	−0.279689
$575$	2.31209e17	0.266787
$576$	0	0
$577$	5.39539e17	0.608667	0.304334	−	0.952566i	$-0.401566\pi$
$577$	5.39539e17	0.608667	0.304334	+	0.952566i	$0.401566\pi$
$578$	−4.68039e17	−0.522097
$579$	0	0
$580$	−3.72342e17	−0.406126
$581$	6.83406e17	0.737113
$582$	0	0
$583$	2.10145e16	0.0221653
$584$	5.80603e17	0.605614
$585$	0	0
$586$	3.03396e17	0.309510
$587$	8.49866e16	0.0857438	0.0428719	−	0.999081i	$-0.486349\pi$
$587$	8.49866e16	0.0857438	0.0428719	+	0.999081i	$0.486349\pi$
$588$	0	0
$589$	−1.28146e18	−1.26461
$590$	−2.80284e17	−0.273565
$591$	0	0
$592$	4.77126e17	0.455556
$593$	4.33934e17	0.409796	0.204898	−	0.978783i	$-0.434314\pi$
$593$	4.33934e17	0.409796	0.204898	+	0.978783i	$0.434314\pi$
$594$	0	0
$595$	−1.35232e17	−0.124945
$596$	5.58180e17	0.510122
$597$	0	0
$598$	1.99689e18	1.78565
$599$	−1.08180e18	−0.956909	−0.478455	−	0.878112i	$-0.658803\pi$
$599$	−1.08180e18	−0.956909	−0.478455	+	0.878112i	$0.658803\pi$
$600$	0	0
$601$	8.22862e17	0.712267	0.356134	−	0.934435i	$-0.384095\pi$
$601$	8.22862e17	0.712267	0.356134	+	0.934435i	$0.384095\pi$
$602$	3.79898e17	0.325304
$603$	0	0
$604$	6.01238e17	0.503856
$605$	3.58870e16	0.0297528
$606$	0	0
$607$	−6.79810e17	−0.551647	−0.275823	−	0.961208i	$-0.588951\pi$
$607$	−6.79810e17	−0.551647	−0.275823	+	0.961208i	$0.588951\pi$
$608$	−3.63014e17	−0.291440
$609$	0	0
$610$	5.28916e17	0.415664
$611$	−1.81206e18	−1.40898
$612$	0	0
$613$	−9.66682e17	−0.735852	−0.367926	−	0.929855i	$-0.619932\pi$
$613$	−9.66682e17	−0.735852	−0.367926	+	0.929855i	$0.619932\pi$
$614$	−2.42473e17	−0.182629
$615$	0	0
$616$	2.70437e17	0.199430
$617$	−7.79322e17	−0.568674	−0.284337	−	0.958724i	$-0.591773\pi$
$617$	−7.79322e17	−0.568674	−0.284337	+	0.958724i	$0.591773\pi$
$618$	0	0
$619$	−1.30527e18	−0.932636	−0.466318	−	0.884617i	$-0.654420\pi$
$619$	−1.30527e18	−0.932636	−0.466318	+	0.884617i	$0.654420\pi$
$620$	2.42585e17	0.171521
$621$	0	0
$622$	−9.38559e17	−0.649866
$623$	3.55969e17	0.243916
$624$	0	0
$625$	5.96046e16	0.0400000
$626$	−3.57749e17	−0.237599
$627$	0	0
$628$	−1.01118e18	−0.657793
$629$	−1.44772e18	−0.932087
$630$	0	0
$631$	2.48848e18	1.56944	0.784718	−	0.619853i	$-0.212808\pi$
$631$	2.48848e18	1.56944	0.784718	+	0.619853i	$0.212808\pi$
$632$	−1.72666e17	−0.107782
$633$	0	0
$634$	1.12227e18	0.686302
$635$	1.00985e17	0.0611263
$636$	0	0
$637$	−2.23983e18	−1.32834
$638$	−2.25934e18	−1.32632
$639$	0	0
$640$	6.87195e16	0.0395285
$641$	1.11455e18	0.634634	0.317317	−	0.948320i	$-0.397218\pi$
$641$	1.11455e18	0.634634	0.317317	+	0.948320i	$0.397218\pi$
$642$	0	0
$643$	1.21634e18	0.678710	0.339355	−	0.940658i	$-0.389791\pi$
$643$	1.21634e18	0.678710	0.339355	+	0.940658i	$0.389791\pi$
$644$	6.59496e17	0.364296
$645$	0	0
$646$	1.10148e18	0.596299
$647$	−3.56728e18	−1.91188	−0.955939	−	0.293567i	$-0.905158\pi$
$647$	−3.56728e18	−1.91188	−0.955939	+	0.293567i	$0.905158\pi$
$648$	0	0
$649$	−1.70074e18	−0.893401
$650$	5.14788e17	0.267726
$651$	0	0
$652$	−3.77583e17	−0.192488
$653$	−1.05284e18	−0.531408	−0.265704	−	0.964055i	$-0.585604\pi$
$653$	−1.05284e18	−0.531408	−0.265704	+	0.964055i	$0.585604\pi$
$654$	0	0
$655$	1.30688e17	0.0646645
$656$	−3.69535e17	−0.181042
$657$	0	0
$658$	−5.98454e17	−0.287450
$659$	1.06619e18	0.507085	0.253542	−	0.967324i	$-0.418404\pi$
$659$	1.06619e18	0.507085	0.253542	+	0.967324i	$0.418404\pi$
$660$	0	0
$661$	−3.83025e17	−0.178615	−0.0893075	−	0.996004i	$-0.528465\pi$
$661$	−3.83025e17	−0.178615	−0.0893075	+	0.996004i	$0.528465\pi$
$662$	−1.70553e18	−0.787556
$663$	0	0
$664$	1.05374e18	0.477133
$665$	−8.98111e17	−0.402707
$666$	0	0
$667$	−5.50970e18	−2.42276
$668$	3.20834e17	0.139712
$669$	0	0
$670$	−7.51430e15	−0.00320924
$671$	3.20941e18	1.35747
$672$	0	0
$673$	−3.12295e18	−1.29559	−0.647793	−	0.761816i	$-0.724308\pi$
$673$	−3.12295e18	−1.29559	−0.647793	+	0.761816i	$0.724308\pi$
$674$	1.80973e18	0.743573
$675$	0	0
$676$	3.20549e18	1.29194
$677$	−1.03569e18	−0.413431	−0.206716	−	0.978401i	$-0.566277\pi$
$677$	−1.03569e18	−0.413431	−0.206716	+	0.978401i	$0.566277\pi$
$678$	0	0
$679$	−1.47054e18	−0.575869
$680$	−2.08513e17	−0.0808769
$681$	0	0
$682$	1.47198e18	0.560149
$683$	−2.86963e18	−1.08166	−0.540831	−	0.841131i	$-0.681890\pi$
$683$	−2.86963e18	−1.08166	−0.540831	+	0.841131i	$0.681890\pi$
$684$	0	0
$685$	1.93390e18	0.715228
$686$	−1.79398e18	−0.657218
$687$	0	0
$688$	5.85760e17	0.210569
$689$	1.14101e17	0.0406316
$690$	0	0
$691$	1.70272e18	0.595026	0.297513	−	0.954718i	$-0.403843\pi$
$691$	1.70272e18	0.595026	0.297513	+	0.954718i	$0.403843\pi$
$692$	−1.84299e18	−0.638020
$693$	0	0
$694$	−2.79994e18	−0.951288
$695$	−3.09546e17	−0.104190
$696$	0	0
$697$	1.12126e18	0.370420
$698$	−6.25457e17	−0.204709
$699$	0	0
$700$	1.70015e17	0.0546198
$701$	1.89524e18	0.603249	0.301624	−	0.953427i	$-0.402471\pi$
$701$	1.89524e18	0.603249	0.301624	+	0.953427i	$0.402471\pi$
$702$	0	0
$703$	−9.61470e18	−3.00418
$704$	4.16983e17	0.129091
$705$	0	0
$706$	−2.15277e18	−0.654284
$707$	−1.68774e18	−0.508253
$708$	0	0
$709$	−1.81610e18	−0.536957	−0.268479	−	0.963286i	$-0.586521\pi$
$709$	−1.81610e18	−0.536957	−0.268479	+	0.963286i	$0.586521\pi$
$710$	−1.83209e17	−0.0536746
$711$	0	0
$712$	5.48864e17	0.157887
$713$	3.58962e18	1.02322
$714$	0	0
$715$	3.12368e18	0.874335
$716$	5.70077e17	0.158124
$717$	0	0
$718$	3.66993e18	0.999652
$719$	−4.07912e18	−1.10110	−0.550552	−	0.834801i	$-0.685583\pi$
$719$	−4.07912e18	−1.10110	−0.550552	+	0.834801i	$0.685583\pi$
$720$	0	0
$721$	1.69190e18	0.448533
$722$	4.62381e18	1.21481
$723$	0	0
$724$	−2.13281e18	−0.550367
$725$	−1.42037e18	−0.363250
$726$	0	0
$727$	−3.30463e18	−0.830136	−0.415068	−	0.909790i	$-0.636242\pi$
$727$	−3.30463e18	−0.830136	−0.415068	+	0.909790i	$0.636242\pi$
$728$	1.46837e18	0.365579
$729$	0	0
$730$	2.21482e18	0.541677
$731$	−1.77735e18	−0.430833
$732$	0	0
$733$	−2.10951e18	−0.502350	−0.251175	−	0.967942i	$-0.580817\pi$
$733$	−2.10951e18	−0.502350	−0.251175	+	0.967942i	$0.580817\pi$
$734$	8.36876e16	0.0197532
$735$	0	0
$736$	1.01687e18	0.235809
$737$	−4.55961e16	−0.0104807
$738$	0	0
$739$	3.64707e18	0.823673	0.411837	−	0.911258i	$-0.364887\pi$
$739$	3.64707e18	0.823673	0.411837	+	0.911258i	$0.364887\pi$
$740$	1.82009e18	0.407461
$741$	0	0
$742$	3.76832e16	0.00828939
$743$	−4.43149e18	−0.966324	−0.483162	−	0.875531i	$-0.660512\pi$
$743$	−4.43149e18	−0.966324	−0.483162	+	0.875531i	$0.660512\pi$
$744$	0	0
$745$	2.12929e18	0.456267
$746$	3.23445e18	0.687066
$747$	0	0
$748$	−1.26524e18	−0.264126
$749$	1.75194e18	0.362566
$750$	0	0
$751$	4.80237e18	0.976779	0.488389	−	0.872626i	$-0.337585\pi$
$751$	4.80237e18	0.976779	0.488389	+	0.872626i	$0.337585\pi$
$752$	−9.22749e17	−0.186066
$753$	0	0
$754$	−1.22673e19	−2.43129
$755$	2.29354e18	0.450663
$756$	0	0
$757$	5.04669e18	0.974729	0.487364	−	0.873199i	$-0.337958\pi$
$757$	5.04669e18	0.974729	0.487364	+	0.873199i	$0.337958\pi$
$758$	2.29208e18	0.438914
$759$	0	0
$760$	−1.38479e18	−0.260672
$761$	−1.68626e18	−0.314720	−0.157360	−	0.987541i	$-0.550298\pi$
$761$	−1.68626e18	−0.314720	−0.157360	+	0.987541i	$0.550298\pi$
$762$	0	0
$763$	4.02322e18	0.738183
$764$	−4.48361e18	−0.815682
$765$	0	0
$766$	7.43805e18	1.33037
$767$	−9.23435e18	−1.63771
$768$	0	0
$769$	1.11315e18	0.194104	0.0970519	−	0.995279i	$-0.469059\pi$
$769$	1.11315e18	0.194104	0.0970519	+	0.995279i	$0.469059\pi$
$770$	1.03163e18	0.178376
$771$	0	0
$772$	2.56665e18	0.436370
$773$	9.65963e18	1.62852	0.814261	−	0.580499i	$-0.197142\pi$
$773$	9.65963e18	1.62852	0.814261	+	0.580499i	$0.197142\pi$
$774$	0	0
$775$	9.25387e17	0.153413
$776$	−2.26741e18	−0.372760
$777$	0	0
$778$	−1.56629e18	−0.253223
$779$	7.44660e18	1.19389
$780$	0	0
$781$	−1.11170e18	−0.175289
$782$	−3.08545e18	−0.482474
$783$	0	0
$784$	−1.14058e18	−0.175417
$785$	−3.85733e18	−0.588348
$786$	0	0
$787$	1.40037e18	0.210091	0.105045	−	0.994467i	$-0.466501\pi$
$787$	1.40037e18	0.210091	0.105045	+	0.994467i	$0.466501\pi$
$788$	1.36100e18	0.202506
$789$	0	0
$790$	−6.58669e17	−0.0964031
$791$	4.60293e17	0.0668170
$792$	0	0
$793$	1.74259e19	2.48839
$794$	4.60549e18	0.652292
$795$	0	0
$796$	2.14048e17	0.0298248
$797$	−9.30010e18	−1.28531	−0.642656	−	0.766155i	$-0.722168\pi$
$797$	−9.30010e18	−1.28531	−0.642656	+	0.766155i	$0.722168\pi$
$798$	0	0
$799$	2.79986e18	0.380700
$800$	2.62144e17	0.0353553
$801$	0	0
$802$	−1.77725e18	−0.235838
$803$	1.34393e19	1.76900
$804$	0	0
$805$	2.51578e18	0.325836
$806$	7.99230e18	1.02682
$807$	0	0
$808$	−2.60231e18	−0.328992
$809$	−3.89357e18	−0.488296	−0.244148	−	0.969738i	$-0.578508\pi$
$809$	−3.89357e18	−0.488296	−0.244148	+	0.969738i	$0.578508\pi$
$810$	0	0
$811$	−1.53890e19	−1.89922	−0.949610	−	0.313434i	$-0.898521\pi$
$811$	−1.53890e19	−1.89922	−0.949610	+	0.313434i	$0.898521\pi$
$812$	−4.05144e18	−0.496016
$813$	0	0
$814$	1.10441e19	1.33068
$815$	−1.44037e18	−0.172166
$816$	0	0
$817$	−1.18038e19	−1.38861
$818$	−9.61955e18	−1.12269
$819$	0	0
$820$	−1.40966e18	−0.161929
$821$	−3.58141e18	−0.408153	−0.204077	−	0.978955i	$-0.565419\pi$
$821$	−3.58141e18	−0.408153	−0.204077	+	0.978955i	$0.565419\pi$
$822$	0	0
$823$	−8.98746e18	−1.00818	−0.504090	−	0.863651i	$-0.668172\pi$
$823$	−8.98746e18	−1.00818	−0.504090	+	0.863651i	$0.668172\pi$
$824$	2.60871e18	0.290335
$825$	0	0
$826$	−3.04976e18	−0.334114
$827$	−1.73077e18	−0.188129	−0.0940643	−	0.995566i	$-0.529986\pi$
$827$	−1.73077e18	−0.188129	−0.0940643	+	0.995566i	$0.529986\pi$
$828$	0	0
$829$	7.78822e18	0.833362	0.416681	−	0.909053i	$-0.363193\pi$
$829$	7.78822e18	0.833362	0.416681	+	0.909053i	$0.363193\pi$
$830$	4.01969e18	0.426761
$831$	0	0
$832$	2.26406e18	0.236639
$833$	3.46082e18	0.358911
$834$	0	0
$835$	1.22388e18	0.124962
$836$	−8.40276e18	−0.851297
$837$	0	0
$838$	5.93108e18	0.591627
$839$	3.26310e18	0.322981	0.161491	−	0.986874i	$-0.448370\pi$
$839$	3.26310e18	0.322981	0.161491	+	0.986874i	$0.448370\pi$
$840$	0	0
$841$	2.35867e19	2.29876
$842$	2.99883e18	0.290017
$843$	0	0
$844$	−5.58538e18	−0.531896
$845$	1.22280e19	1.15554
$846$	0	0
$847$	3.90485e17	0.0363381
$848$	5.81033e16	0.00536571
$849$	0	0
$850$	−7.95413e17	−0.0723385
$851$	2.69326e19	2.43073
$852$	0	0
$853$	5.82090e18	0.517394	0.258697	−	0.965958i	$-0.416707\pi$
$853$	5.82090e18	0.517394	0.258697	+	0.965958i	$0.416707\pi$
$854$	5.75511e18	0.507666
$855$	0	0
$856$	2.70130e18	0.234689
$857$	9.30539e18	0.802342	0.401171	−	0.916003i	$-0.368603\pi$
$857$	9.30539e18	0.802342	0.401171	+	0.916003i	$0.368603\pi$
$858$	0	0
$859$	−1.20727e19	−1.02530	−0.512649	−	0.858598i	$-0.671336\pi$
$859$	−1.20727e19	−1.02530	−0.512649	+	0.858598i	$0.671336\pi$
$860$	2.23450e18	0.188339
$861$	0	0
$862$	−1.49609e19	−1.24211
$863$	6.75985e18	0.557015	0.278508	−	0.960434i	$-0.410160\pi$
$863$	6.75985e18	0.557015	0.278508	+	0.960434i	$0.410160\pi$
$864$	0	0
$865$	−7.03047e18	−0.570663
$866$	−1.00743e19	−0.811613
$867$	0	0
$868$	2.63955e18	0.209485
$869$	−3.99674e18	−0.314831
$870$	0	0
$871$	−2.47569e17	−0.0192123
$872$	6.20335e18	0.477825
$873$	0	0
$874$	−2.04913e19	−1.55505
$875$	6.48555e17	0.0488534
$876$	0	0
$877$	−6.38272e18	−0.473706	−0.236853	−	0.971546i	$-0.576116\pi$
$877$	−6.38272e18	−0.473706	−0.236853	+	0.971546i	$0.576116\pi$
$878$	−1.41998e19	−1.04609
$879$	0	0
$880$	1.59067e18	0.115463
$881$	1.38731e19	0.999609	0.499804	−	0.866138i	$-0.333405\pi$
$881$	1.38731e19	0.999609	0.499804	+	0.866138i	$0.333405\pi$
$882$	0	0
$883$	−6.03335e18	−0.428365	−0.214183	−	0.976794i	$-0.568709\pi$
$883$	−6.03335e18	−0.428365	−0.214183	+	0.976794i	$0.568709\pi$
$884$	−6.86975e18	−0.484174
$885$	0	0
$886$	−9.13744e18	−0.634607
$887$	1.56332e19	1.07781	0.538907	−	0.842365i	$-0.318838\pi$
$887$	1.56332e19	1.07781	0.538907	+	0.842365i	$0.318838\pi$
$888$	0	0
$889$	1.09881e18	0.0746557
$890$	2.09375e18	0.141218
$891$	0	0
$892$	−1.01277e19	−0.673196
$893$	1.85946e19	1.22702
$894$	0	0
$895$	2.17467e18	0.141431
$896$	7.47734e17	0.0482775
$897$	0	0
$898$	1.24605e19	0.792940
$899$	−2.20519e19	−1.39318
$900$	0	0
$901$	−1.76300e17	−0.0109785
$902$	−8.55370e18	−0.528824
$903$	0	0
$904$	7.09720e17	0.0432506
$905$	−8.13604e18	−0.492263
$906$	0	0
$907$	−1.67708e19	−1.00025	−0.500123	−	0.865954i	$-0.666712\pi$
$907$	−1.67708e19	−1.00025	−0.500123	+	0.865954i	$0.666712\pi$
$908$	4.77092e18	0.282517
$909$	0	0
$910$	5.60138e18	0.326984
$911$	−1.90416e19	−1.10366	−0.551828	−	0.833958i	$-0.686070\pi$
$911$	−1.90416e19	−1.10366	−0.551828	+	0.833958i	$0.686070\pi$
$912$	0	0
$913$	2.43911e19	1.39370
$914$	−1.31879e19	−0.748213
$915$	0	0
$916$	−1.07855e19	−0.603279
$917$	1.42201e18	0.0789771
$918$	0	0
$919$	−1.55602e19	−0.852050	−0.426025	−	0.904711i	$-0.640086\pi$
$919$	−1.55602e19	−0.852050	−0.426025	+	0.904711i	$0.640086\pi$
$920$	3.87905e18	0.210914
$921$	0	0
$922$	2.24568e19	1.20392
$923$	−6.03609e18	−0.321325
$924$	0	0
$925$	6.94309e18	0.364445
$926$	1.99493e18	0.103981
$927$	0	0
$928$	−6.24687e18	−0.321071
$929$	−2.03127e19	−1.03673	−0.518364	−	0.855160i	$-0.673459\pi$
$929$	−2.03127e19	−1.03673	−0.518364	+	0.855160i	$0.673459\pi$
$930$	0	0
$931$	2.29842e19	1.15680
$932$	−7.42666e18	−0.371185
$933$	0	0
$934$	2.02302e19	0.997114
$935$	−4.82649e18	−0.236241
$936$	0	0
$937$	3.27979e19	1.58321	0.791605	−	0.611033i	$-0.209246\pi$
$937$	3.27979e19	1.58321	0.791605	+	0.611033i	$0.209246\pi$
$938$	−8.17628e16	−0.00391956
$939$	0	0
$940$	−3.52001e18	−0.166423
$941$	−2.39618e19	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$941$	−2.39618e19	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$942$	0	0
$943$	−2.08593e19	−0.965994
$944$	−4.70238e18	−0.216272
$945$	0	0
$946$	1.35587e19	0.615072
$947$	1.88259e19	0.848166	0.424083	−	0.905623i	$-0.360597\pi$
$947$	1.88259e19	0.848166	0.424083	+	0.905623i	$0.360597\pi$
$948$	0	0
$949$	7.29705e19	3.24278
$950$	−5.28254e18	−0.233152
$951$	0	0
$952$	−2.26882e18	−0.0987778
$953$	−1.74009e19	−0.752431	−0.376215	−	0.926532i	$-0.622775\pi$
$953$	−1.74009e19	−0.752431	−0.376215	+	0.926532i	$0.622775\pi$
$954$	0	0
$955$	−1.71036e19	−0.729568
$956$	−2.28177e19	−0.966710
$957$	0	0
$958$	−4.85903e18	−0.203083
$959$	2.10426e19	0.873532
$960$	0	0
$961$	−1.00505e19	−0.411611
$962$	5.99654e19	2.43929
$963$	0	0
$964$	−2.80734e18	−0.112666
$965$	9.79100e18	0.390301
$966$	0	0
$967$	−3.44802e19	−1.35612	−0.678059	−	0.735007i	$-0.737179\pi$
$967$	−3.44802e19	−1.35612	−0.678059	+	0.735007i	$0.737179\pi$
$968$	6.02084e17	0.0235216
$969$	0	0
$970$	−8.64949e18	−0.333406
$971$	−4.55517e19	−1.74413	−0.872067	−	0.489387i	$-0.837220\pi$
$971$	−4.55517e19	−1.74413	−0.872067	+	0.489387i	$0.837220\pi$
$972$	0	0
$973$	−3.36816e18	−0.127250
$974$	−2.05229e19	−0.770202
$975$	0	0
$976$	8.87374e18	0.328612
$977$	−3.55127e19	−1.30638	−0.653189	−	0.757194i	$-0.726569\pi$
$977$	−3.55127e19	−1.30638	−0.653189	+	0.757194i	$0.726569\pi$
$978$	0	0
$979$	1.27047e19	0.461186
$980$	−4.35097e18	−0.156898
$981$	0	0
$982$	−2.61370e19	−0.930102
$983$	−5.22615e19	−1.84750	−0.923749	−	0.382998i	$-0.874892\pi$
$983$	−5.22615e19	−1.84750	−0.923749	+	0.382998i	$0.874892\pi$
$984$	0	0
$985$	5.19180e18	0.181127
$986$	1.89546e19	0.656924
$987$	0	0
$988$	−4.56238e19	−1.56053
$989$	3.30647e19	1.12354
$990$	0	0
$991$	−2.90393e18	−0.0973885	−0.0486943	−	0.998814i	$-0.515506\pi$
$991$	−2.90393e18	−0.0973885	−0.0486943	+	0.998814i	$0.515506\pi$
$992$	4.06990e18	0.135599
$993$	0	0
$994$	−1.99349e18	−0.0655546
$995$	8.16530e17	0.0266761
$996$	0	0
$997$	−1.88720e19	−0.608555	−0.304278	−	0.952583i	$-0.598415\pi$
$997$	−1.88720e19	−0.608555	−0.304278	+	0.952583i	$0.598415\pi$
$998$	−1.46856e19	−0.470481
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.14.a.m.1.2		2
3.2	odd	2		30.14.a.g.1.2	✓	2
15.2	even	4		150.14.c.j.49.2		4
15.8	even	4		150.14.c.j.49.3		4
15.14	odd	2		150.14.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.14.a.g.1.2	✓	2	3.2	odd	2
90.14.a.m.1.2		2	1.1	even	1	trivial
150.14.a.n.1.1		2	15.14	odd	2
150.14.c.j.49.2		4	15.2	even	4
150.14.c.j.49.3		4	15.8	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)

\(f(q)\)	\(=\)	\(q+64.0000 q^{2} +4096.00 q^{4} +15625.0 q^{5} +170015. q^{7} +262144. q^{8} +1.00000e6 q^{10} +6.06791e6 q^{11} +3.29464e7 q^{13} +1.08810e7 q^{14} +1.67772e7 q^{16} -5.09064e7 q^{17} -3.38083e8 q^{19} +6.40000e7 q^{20} +3.88346e8 q^{22} +9.47034e8 q^{23} +2.44141e8 q^{25} +2.10857e9 q^{26} +6.96381e8 q^{28} -5.81785e9 q^{29} +3.79039e9 q^{31} +1.07374e9 q^{32} -3.25801e9 q^{34} +2.65648e9 q^{35} +2.84389e10 q^{37} -2.16373e10 q^{38} +4.09600e9 q^{40} -2.20260e10 q^{41} +3.49140e10 q^{43} +2.48541e10 q^{44} +6.06102e10 q^{46} -5.50002e10 q^{47} -6.79839e10 q^{49} +1.56250e10 q^{50} +1.34949e11 q^{52} +3.46323e9 q^{53} +9.48110e10 q^{55} +4.45684e10 q^{56} -3.72342e11 q^{58} -2.80284e11 q^{59} +5.28916e11 q^{61} +2.42585e11 q^{62} +6.87195e10 q^{64} +5.14788e11 q^{65} -7.51430e9 q^{67} -2.08513e11 q^{68} +1.70015e11 q^{70} -1.83209e11 q^{71} +2.21482e12 q^{73} +1.82009e12 q^{74} -1.38479e12 q^{76} +1.03163e12 q^{77} -6.58669e11 q^{79} +2.62144e11 q^{80} -1.40966e12 q^{82} +4.01969e12 q^{83} -7.95413e11 q^{85} +2.23450e12 q^{86} +1.59067e12 q^{88} +2.09375e12 q^{89} +5.60138e12 q^{91} +3.87905e12 q^{92} -3.52001e12 q^{94} -5.28254e12 q^{95} -8.64949e12 q^{97} -4.35097e12 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 128 q^{2} + 8192 q^{4} + 31250 q^{5} - 236912 q^{7} + 524288 q^{8} + 2000000 q^{10} - 4018560 q^{11} + 13391116 q^{13} - 15162368 q^{14} + 33554432 q^{16} + 47615052 q^{17} - 90569648 q^{19} + 128000000 q^{20}+ \cdots + 45860598912 q^{98}+O(q^{100}) \) 2 * q + 128 * q^2 + 8192 * q^4 + 31250 * q^5 - 236912 * q^7 + 524288 * q^8 + 2000000 * q^10 - 4018560 * q^11 + 13391116 * q^13 - 15162368 * q^14 + 33554432 * q^16 + 47615052 * q^17 - 90569648 * q^19 + 128000000 * q^20 - 257187840 * q^22 + 1025770200 * q^23 + 488281250 * q^25 + 857031424 * q^26 - 970391552 * q^28 - 2072310228 * q^29 + 11728406776 * q^31 + 2147483648 * q^32 + 3047363328 * q^34 - 3701750000 * q^35 + 47778848716 * q^37 - 5796457472 * q^38 + 8192000000 * q^40 - 67871568228 * q^41 + 77614412728 * q^43 - 16460021760 * q^44 + 65649292800 * q^46 - 106902703320 * q^47 + 716571858 * q^49 + 31250000000 * q^50 + 54850011136 * q^52 - 4551230532 * q^53 - 62790000000 * q^55 - 62105059328 * q^56 - 132627854592 * q^58 + 4334346240 * q^59 + 887411684860 * q^61 + 750618033664 * q^62 + 137438953472 * q^64 + 209236187500 * q^65 + 1467845781688 * q^67 + 195031252992 * q^68 - 236912000000 * q^70 - 936151332480 * q^71 + 813936935956 * q^73 + 3057846317824 * q^74 - 370973278208 * q^76 + 5136088997760 * q^77 - 1248629414744 * q^79 + 524288000000 * q^80 - 4343780366592 * q^82 - 603864311256 * q^83 + 743985187500 * q^85 + 4967322414592 * q^86 - 1053441392640 * q^88 + 8016259169484 * q^89 + 13558957939904 * q^91 + 4201554739200 * q^92 - 6841773012480 * q^94 - 1415150750000 * q^95 - 1682542380284 * q^97 + 45860598912 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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