LMFDB - Embedded newform 90.18.a.n.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [90,18,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, 18); 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, 18, names="a")

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	90.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,512,0,131072,781250,0,6543844] 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:	$164.899878610$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{36061}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9015 $ x^2 - x - 9015
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$95.4487$ 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)

$f(q)$	$=$	\(q+256.000 q^{2} +65536.0 q^{4} +390625. q^{5} -9.53475e6 q^{7} +1.67772e7 q^{8} +1.00000e8 q^{10} -4.01191e8 q^{11} +8.56166e8 q^{13} -2.44090e9 q^{14} +4.29497e9 q^{16} +3.89127e10 q^{17} -1.13839e11 q^{19} +2.56000e10 q^{20} -1.02705e11 q^{22} -1.64834e10 q^{23} +1.52588e11 q^{25} +2.19179e11 q^{26} -6.24870e11 q^{28} +2.27472e12 q^{29} -1.63788e12 q^{31} +1.09951e12 q^{32} +9.96164e12 q^{34} -3.72451e12 q^{35} -1.75967e13 q^{37} -2.91428e13 q^{38} +6.55360e12 q^{40} +2.95532e13 q^{41} +1.37690e14 q^{43} -2.62925e13 q^{44} -4.21974e12 q^{46} +1.65452e14 q^{47} -1.41719e14 q^{49} +3.90625e13 q^{50} +5.61097e13 q^{52} +7.25259e14 q^{53} -1.56715e14 q^{55} -1.59967e14 q^{56} +5.82328e14 q^{58} -1.62177e15 q^{59} +2.46915e15 q^{61} -4.19296e14 q^{62} +2.81475e14 q^{64} +3.34440e14 q^{65} -2.03244e14 q^{67} +2.55018e15 q^{68} -9.53475e14 q^{70} -9.39117e15 q^{71} +1.54865e15 q^{73} -4.50475e15 q^{74} -7.46056e15 q^{76} +3.82526e15 q^{77} +8.30977e15 q^{79} +1.67772e15 q^{80} +7.56562e15 q^{82} +6.14697e15 q^{83} +1.52003e16 q^{85} +3.52485e16 q^{86} -6.73087e15 q^{88} -4.67428e15 q^{89} -8.16334e15 q^{91} -1.08025e15 q^{92} +4.23558e16 q^{94} -4.44684e16 q^{95} +1.01799e17 q^{97} -3.62801e16 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 512 q^{2} + 131072 q^{4} + 781250 q^{5} + 6543844 q^{7} + 33554432 q^{8} + 200000000 q^{10} - 1189408704 q^{11} - 2017919228 q^{13} + 1675224064 q^{14} + 8589934592 q^{16} + 18755639436 q^{17} - 136704830600 q^{19}+ \cdots - 29\!\cdots\!76 q^{98}+O(q^{100}) $ 2 * q + 512 * q^2 + 131072 * q^4 + 781250 * q^5 + 6543844 * q^7 + 33554432 * q^8 + 200000000 * q^10 - 1189408704 * q^11 - 2017919228 * q^13 + 1675224064 * q^14 + 8589934592 * q^16 + 18755639436 * q^17 - 136704830600 * q^19 + 51200000000 * q^20 - 304488628224 * q^22 + 649234170708 * q^23 + 305175781250 * q^25 - 516587322368 * q^26 + 428857360384 * q^28 + 4696543420020 * q^29 + 7120867378744 * q^31 + 2199023255552 * q^32 + 4801443695616 * q^34 + 2556189062500 * q^35 + 9933114637444 * q^37 - 34996436633600 * q^38 + 13107200000000 * q^40 - 9622711880844 * q^41 + 7708436011852 * q^43 - 77949088825344 * q^44 + 166203947701248 * q^46 + 466072575837996 * q^47 - 115828146887046 * q^49 + 78125000000000 * q^50 - 132246354526208 * q^52 + 623333120284428 * q^53 - 464612775000000 * q^55 + 109787484258304 * q^56 + 1202315115525120 * q^58 - 654616071995160 * q^59 + 1329040385162884 * q^61 + 1822942048958464 * q^62 + 562949953421312 * q^64 - 788249698437500 * q^65 - 4249016411877596 * q^67 + 1229169586077696 * q^68 + 654384400000000 * q^70 - 2035131949347624 * q^71 + 2248593461347972 * q^73 + 2542877347185664 * q^74 - 8959087778201600 * q^76 - 8848168987836288 * q^77 + 19188427678495120 * q^79 + 3355443200000000 * q^80 - 2463414241496064 * q^82 + 18474649086611988 * q^83 + 7326421654687500 * q^85 + 1973359619034112 * q^86 - 19954966739288064 * q^88 + 17458912916395020 * q^89 - 54374608211433016 * q^91 + 42548210611519488 * q^92 + 119314579414526976 * q^94 - 53400324453125000 * q^95 + 195934325822490244 * q^97 - 29652005603083776 * 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 $

$2$	256.000	0.707107
$2$	256.000	0.707107
$3$	0	0
$3$	0	0
$4$	65536.0	0.500000
$5$	390625.	0.447214
$5$	390625.	0.447214
$6$	0	0
$7$	−9.53475e6	−0.625138	−0.312569	−	0.949895i	$-0.601190\pi$
$7$	−9.53475e6	−0.625138	−0.312569	+	0.949895i	$0.601190\pi$
$8$	1.67772e7	0.353553
$9$	0	0
$10$	1.00000e8	0.316228
$11$	−4.01191e8	−0.564305	−0.282152	−	0.959370i	$-0.591048\pi$
$11$	−4.01191e8	−0.564305	−0.282152	+	0.959370i	$0.591048\pi$
$12$	0	0
$13$	8.56166e8	0.291098	0.145549	−	0.989351i	$-0.453505\pi$
$13$	8.56166e8	0.291098	0.145549	+	0.989351i	$0.453505\pi$
$14$	−2.44090e9	−0.442040
$15$	0	0
$16$	4.29497e9	0.250000
$17$	3.89127e10	1.35293	0.676465	−	0.736475i	$-0.263511\pi$
$17$	3.89127e10	1.35293	0.676465	+	0.736475i	$0.263511\pi$
$18$	0	0
$19$	−1.13839e11	−1.53775	−0.768876	−	0.639398i	$-0.779184\pi$
$19$	−1.13839e11	−1.53775	−0.768876	+	0.639398i	$0.779184\pi$
$20$	2.56000e10	0.223607
$21$	0	0
$22$	−1.02705e11	−0.399024
$23$	−1.64834e10	−0.0438894	−0.0219447	−	0.999759i	$-0.506986\pi$
$23$	−1.64834e10	−0.0438894	−0.0219447	+	0.999759i	$0.506986\pi$
$24$	0	0
$25$	1.52588e11	0.200000
$26$	2.19179e11	0.205838
$27$	0	0
$28$	−6.24870e11	−0.312569
$29$	2.27472e12	0.844394	0.422197	−	0.906504i	$-0.361259\pi$
$29$	2.27472e12	0.844394	0.422197	+	0.906504i	$0.361259\pi$
$30$	0	0
$31$	−1.63788e12	−0.344911	−0.172455	−	0.985017i	$-0.555170\pi$
$31$	−1.63788e12	−0.344911	−0.172455	+	0.985017i	$0.555170\pi$
$32$	1.09951e12	0.176777
$33$	0	0
$34$	9.96164e12	0.956665
$35$	−3.72451e12	−0.279570
$36$	0	0
$37$	−1.75967e13	−0.823599	−0.411799	−	0.911275i	$-0.635100\pi$
$37$	−1.75967e13	−0.823599	−0.411799	+	0.911275i	$0.635100\pi$
$38$	−2.91428e13	−1.08735
$39$	0	0
$40$	6.55360e12	0.158114
$41$	2.95532e13	0.578018	0.289009	−	0.957326i	$-0.406674\pi$
$41$	2.95532e13	0.578018	0.289009	+	0.957326i	$0.406674\pi$
$42$	0	0
$43$	1.37690e14	1.79647	0.898233	−	0.439519i	$-0.144851\pi$
$43$	1.37690e14	1.79647	0.898233	+	0.439519i	$0.144851\pi$
$44$	−2.62925e13	−0.282152
$45$	0	0
$46$	−4.21974e12	−0.0310345
$47$	1.65452e14	1.01354	0.506770	−	0.862081i	$-0.330839\pi$
$47$	1.65452e14	1.01354	0.506770	+	0.862081i	$0.330839\pi$
$48$	0	0
$49$	−1.41719e14	−0.609202
$50$	3.90625e13	0.141421
$51$	0	0
$52$	5.61097e13	0.145549
$53$	7.25259e14	1.60010	0.800052	−	0.599931i	$-0.204805\pi$
$53$	7.25259e14	1.60010	0.800052	+	0.599931i	$0.204805\pi$
$54$	0	0
$55$	−1.56715e14	−0.252365
$56$	−1.59967e14	−0.221020
$57$	0	0
$58$	5.82328e14	0.597076
$59$	−1.62177e15	−1.43796	−0.718981	−	0.695030i	$-0.755391\pi$
$59$	−1.62177e15	−1.43796	−0.718981	+	0.695030i	$0.755391\pi$
$60$	0	0
$61$	2.46915e15	1.64909	0.824545	−	0.565796i	$-0.191431\pi$
$61$	2.46915e15	1.64909	0.824545	+	0.565796i	$0.191431\pi$
$62$	−4.19296e14	−0.243889
$63$	0	0
$64$	2.81475e14	0.125000
$65$	3.34440e14	0.130183
$66$	0	0
$67$	−2.03244e14	−0.0611479	−0.0305739	−	0.999533i	$-0.509734\pi$
$67$	−2.03244e14	−0.0611479	−0.0305739	+	0.999533i	$0.509734\pi$
$68$	2.55018e15	0.676465
$69$	0	0
$70$	−9.53475e14	−0.197686
$71$	−9.39117e15	−1.72593	−0.862966	−	0.505262i	$-0.831396\pi$
$71$	−9.39117e15	−1.72593	−0.862966	+	0.505262i	$0.831396\pi$
$72$	0	0
$73$	1.54865e15	0.224754	0.112377	−	0.993666i	$-0.464153\pi$
$73$	1.54865e15	0.224754	0.112377	+	0.993666i	$0.464153\pi$
$74$	−4.50475e15	−0.582372
$75$	0	0
$76$	−7.46056e15	−0.768876
$77$	3.82526e15	0.352769
$78$	0	0
$79$	8.30977e15	0.616252	0.308126	−	0.951345i	$-0.400298\pi$
$79$	8.30977e15	0.616252	0.308126	+	0.951345i	$0.400298\pi$
$80$	1.67772e15	0.111803
$81$	0	0
$82$	7.56562e15	0.408721
$83$	6.14697e15	0.299569	0.149785	−	0.988719i	$-0.452142\pi$
$83$	6.14697e15	0.299569	0.149785	+	0.988719i	$0.452142\pi$
$84$	0	0
$85$	1.52003e16	0.605048
$86$	3.52485e16	1.27029
$87$	0	0
$88$	−6.73087e15	−0.199512
$89$	−4.67428e15	−0.125863	−0.0629317	−	0.998018i	$-0.520045\pi$
$89$	−4.67428e15	−0.125863	−0.0629317	+	0.998018i	$0.520045\pi$
$90$	0	0
$91$	−8.16334e15	−0.181977
$92$	−1.08025e15	−0.0219447
$93$	0	0
$94$	4.23558e16	0.716681
$95$	−4.44684e16	−0.687703
$96$	0	0
$97$	1.01799e17	1.31881	0.659407	−	0.751786i	$-0.270808\pi$
$97$	1.01799e17	1.31881	0.659407	+	0.751786i	$0.270808\pi$
$98$	−3.62801e16	−0.430771
$99$	0	0
$100$	1.00000e16	0.100000
$101$	−3.11898e16	−0.286603	−0.143302	−	0.989679i	$-0.545772\pi$
$101$	−3.11898e16	−0.286603	−0.143302	+	0.989679i	$0.545772\pi$
$102$	0	0
$103$	−1.71633e17	−1.33501	−0.667505	−	0.744606i	$-0.732638\pi$
$103$	−1.71633e17	−1.33501	−0.667505	+	0.744606i	$0.732638\pi$
$104$	1.43641e16	0.102919
$105$	0	0
$106$	1.85666e17	1.13144
$107$	1.64074e17	0.923159	0.461579	−	0.887099i	$-0.347283\pi$
$107$	1.64074e17	0.923159	0.461579	+	0.887099i	$0.347283\pi$
$108$	0	0
$109$	2.80332e17	1.34756	0.673779	−	0.738933i	$-0.264670\pi$
$109$	2.80332e17	1.34756	0.673779	+	0.738933i	$0.264670\pi$
$110$	−4.01191e16	−0.178449
$111$	0	0
$112$	−4.09515e16	−0.156285
$113$	5.30659e17	1.87780	0.938899	−	0.344193i	$-0.111847\pi$
$113$	5.30659e17	1.87780	0.938899	+	0.344193i	$0.111847\pi$
$114$	0	0
$115$	−6.43882e15	−0.0196279
$116$	1.49076e17	0.422197
$117$	0	0
$118$	−4.15173e17	−1.01679
$119$	−3.71023e17	−0.845768
$120$	0	0
$121$	−3.44493e17	−0.681560
$122$	6.32103e17	1.16608
$123$	0	0
$124$	−1.07340e17	−0.172455
$125$	5.96046e16	0.0894427
$126$	0	0
$127$	3.45518e17	0.453043	0.226522	−	0.974006i	$-0.427265\pi$
$127$	3.45518e17	0.453043	0.226522	+	0.974006i	$0.427265\pi$
$128$	7.20576e16	0.0883883
$129$	0	0
$130$	8.56166e16	0.0920534
$131$	5.92983e17	0.597360	0.298680	−	0.954353i	$-0.403454\pi$
$131$	5.92983e17	0.597360	0.298680	+	0.954353i	$0.403454\pi$
$132$	0	0
$133$	1.08543e18	0.961307
$134$	−5.20304e16	−0.0432381
$135$	0	0
$136$	6.52846e17	0.478333
$137$	5.64525e17	0.388650	0.194325	−	0.980937i	$-0.437748\pi$
$137$	5.64525e17	0.388650	0.194325	+	0.980937i	$0.437748\pi$
$138$	0	0
$139$	2.13511e18	1.29956	0.649778	−	0.760124i	$-0.274862\pi$
$139$	2.13511e18	1.29956	0.649778	+	0.760124i	$0.274862\pi$
$140$	−2.44090e17	−0.139785
$141$	0	0
$142$	−2.40414e18	−1.22042
$143$	−3.43487e17	−0.164268
$144$	0	0
$145$	8.88563e17	0.377624
$146$	3.96454e17	0.158925
$147$	0	0
$148$	−1.15321e18	−0.411799
$149$	−2.11167e18	−0.712103	−0.356051	−	0.934466i	$-0.615877\pi$
$149$	−2.11167e18	−0.712103	−0.356051	+	0.934466i	$0.615877\pi$
$150$	0	0
$151$	1.09446e18	0.329532	0.164766	−	0.986333i	$-0.447313\pi$
$151$	1.09446e18	0.329532	0.164766	+	0.986333i	$0.447313\pi$
$152$	−1.90990e18	−0.543677
$153$	0	0
$154$	9.79267e17	0.249445
$155$	−6.39796e17	−0.154249
$156$	0	0
$157$	2.67975e18	0.579359	0.289680	−	0.957124i	$-0.406451\pi$
$157$	2.67975e18	0.579359	0.289680	+	0.957124i	$0.406451\pi$
$158$	2.12730e18	0.435756
$159$	0	0
$160$	4.29497e17	0.0790569
$161$	1.57165e17	0.0274369
$162$	0	0
$163$	−9.47539e18	−1.48937	−0.744685	−	0.667416i	$-0.767400\pi$
$163$	−9.47539e18	−1.48937	−0.744685	+	0.667416i	$0.767400\pi$
$164$	1.93680e18	0.289009
$165$	0	0
$166$	1.57362e18	0.211827
$167$	6.70665e18	0.857859	0.428929	−	0.903338i	$-0.358891\pi$
$167$	6.70665e18	0.857859	0.428929	+	0.903338i	$0.358891\pi$
$168$	0	0
$169$	−7.91739e18	−0.915262
$170$	3.89127e18	0.427834
$171$	0	0
$172$	9.02362e18	0.898233
$173$	−1.05695e19	−1.00152	−0.500762	−	0.865585i	$-0.666947\pi$
$173$	−1.05695e19	−1.00152	−0.500762	+	0.865585i	$0.666947\pi$
$174$	0	0
$175$	−1.45489e18	−0.125028
$176$	−1.72310e18	−0.141076
$177$	0	0
$178$	−1.19662e18	−0.0889989
$179$	1.98718e19	1.40924	0.704621	−	0.709583i	$-0.251117\pi$
$179$	1.98718e19	1.40924	0.704621	+	0.709583i	$0.251117\pi$
$180$	0	0
$181$	2.50176e19	1.61428	0.807138	−	0.590363i	$-0.201015\pi$
$181$	2.50176e19	1.61428	0.807138	+	0.590363i	$0.201015\pi$
$182$	−2.08981e18	−0.128677
$183$	0	0
$184$	−2.76545e17	−0.0155172
$185$	−6.87370e18	−0.368324
$186$	0	0
$187$	−1.56114e19	−0.763464
$188$	1.08431e19	0.506770
$189$	0	0
$190$	−1.13839e19	−0.486280
$191$	2.63385e19	1.07599	0.537995	−	0.842948i	$-0.319182\pi$
$191$	2.63385e19	1.07599	0.537995	+	0.842948i	$0.319182\pi$
$192$	0	0
$193$	3.68856e19	1.37918	0.689589	−	0.724201i	$-0.257791\pi$
$193$	3.68856e19	1.37918	0.689589	+	0.724201i	$0.257791\pi$
$194$	2.60605e19	0.932542
$195$	0	0
$196$	−9.28769e18	−0.304601
$197$	4.04920e19	1.27176	0.635882	−	0.771786i	$-0.280636\pi$
$197$	4.04920e19	1.27176	0.635882	+	0.771786i	$0.280636\pi$
$198$	0	0
$199$	−3.32772e19	−0.959171	−0.479586	−	0.877495i	$-0.659213\pi$
$199$	−3.32772e19	−0.959171	−0.479586	+	0.877495i	$0.659213\pi$
$200$	2.56000e18	0.0707107
$201$	0	0
$202$	−7.98459e18	−0.202659
$203$	−2.16889e19	−0.527863
$204$	0	0
$205$	1.15442e19	0.258498
$206$	−4.39380e19	−0.943994
$207$	0	0
$208$	3.67721e18	0.0727746
$209$	4.56713e19	0.867761
$210$	0	0
$211$	−2.42162e19	−0.424331	−0.212166	−	0.977234i	$-0.568052\pi$
$211$	−2.42162e19	−0.424331	−0.212166	+	0.977234i	$0.568052\pi$
$212$	4.75306e19	0.800052
$213$	0	0
$214$	4.20028e19	0.652772
$215$	5.37850e19	0.803404
$216$	0	0
$217$	1.56168e19	0.215617
$218$	7.17650e19	0.952867
$219$	0	0
$220$	−1.02705e19	−0.126182
$221$	3.33157e19	0.393835
$222$	0	0
$223$	7.42273e19	0.812779	0.406390	−	0.913700i	$-0.366788\pi$
$223$	7.42273e19	0.812779	0.406390	+	0.913700i	$0.366788\pi$
$224$	−1.04836e19	−0.110510
$225$	0	0
$226$	1.35849e20	1.32780
$227$	−9.57531e18	−0.0901432	−0.0450716	−	0.998984i	$-0.514352\pi$
$227$	−9.57531e18	−0.0901432	−0.0450716	+	0.998984i	$0.514352\pi$
$228$	0	0
$229$	1.97753e20	1.72791	0.863955	−	0.503569i	$-0.167980\pi$
$229$	1.97753e20	1.72791	0.863955	+	0.503569i	$0.167980\pi$
$230$	−1.64834e18	−0.0138790
$231$	0	0
$232$	3.81635e19	0.298538
$233$	1.79778e20	1.35585	0.677925	−	0.735131i	$-0.262880\pi$
$233$	1.79778e20	1.35585	0.677925	+	0.735131i	$0.262880\pi$
$234$	0	0
$235$	6.46298e19	0.453269
$236$	−1.06284e20	−0.718981
$237$	0	0
$238$	−9.49818e19	−0.598048
$239$	2.00207e20	1.21646	0.608229	−	0.793761i	$-0.291880\pi$
$239$	2.00207e20	1.21646	0.608229	+	0.793761i	$0.291880\pi$
$240$	0	0
$241$	−1.81616e20	−1.02804	−0.514018	−	0.857779i	$-0.671844\pi$
$241$	−1.81616e20	−1.02804	−0.514018	+	0.857779i	$0.671844\pi$
$242$	−8.81901e19	−0.481936
$243$	0	0
$244$	1.61818e20	0.824545
$245$	−5.53590e19	−0.272443
$246$	0	0
$247$	−9.74653e19	−0.447637
$248$	−2.74790e19	−0.121944
$249$	0	0
$250$	1.52588e19	0.0632456
$251$	3.20192e20	1.28288	0.641438	−	0.767175i	$-0.278338\pi$
$251$	3.20192e20	1.28288	0.641438	+	0.767175i	$0.278338\pi$
$252$	0	0
$253$	6.61299e18	0.0247670
$254$	8.84527e19	0.320350
$255$	0	0
$256$	1.84467e19	0.0625000
$257$	3.34539e20	1.09652	0.548259	−	0.836309i	$-0.315291\pi$
$257$	3.34539e20	1.09652	0.548259	+	0.836309i	$0.315291\pi$
$258$	0	0
$259$	1.67780e20	0.514863
$260$	2.19179e19	0.0650916
$261$	0	0
$262$	1.51804e20	0.422397
$263$	−2.95034e20	−0.794782	−0.397391	−	0.917649i	$-0.630084\pi$
$263$	−2.95034e20	−0.794782	−0.397391	+	0.917649i	$0.630084\pi$
$264$	0	0
$265$	2.83304e20	0.715588
$266$	2.77870e20	0.679747
$267$	0	0
$268$	−1.33198e19	−0.0305739
$269$	−7.56967e20	−1.68338	−0.841690	−	0.539960i	$-0.818439\pi$
$269$	−7.56967e20	−1.68338	−0.841690	+	0.539960i	$0.818439\pi$
$270$	0	0
$271$	−1.18908e20	−0.248298	−0.124149	−	0.992264i	$-0.539620\pi$
$271$	−1.18908e20	−0.248298	−0.124149	+	0.992264i	$0.539620\pi$
$272$	1.67129e20	0.338232
$273$	0	0
$274$	1.44518e20	0.274817
$275$	−6.12169e19	−0.112861
$276$	0	0
$277$	−1.96379e20	−0.340421	−0.170211	−	0.985408i	$-0.554445\pi$
$277$	−1.96379e20	−0.340421	−0.170211	+	0.985408i	$0.554445\pi$
$278$	5.46589e20	0.918925
$279$	0	0
$280$	−6.24870e19	−0.0988431
$281$	3.39035e20	0.520284	0.260142	−	0.965570i	$-0.416231\pi$
$281$	3.39035e20	0.520284	0.260142	+	0.965570i	$0.416231\pi$
$282$	0	0
$283$	4.03445e20	0.582908	0.291454	−	0.956585i	$-0.405861\pi$
$283$	4.03445e20	0.582908	0.291454	+	0.956585i	$0.405861\pi$
$284$	−6.15460e20	−0.862966
$285$	0	0
$286$	−8.79326e19	−0.116155
$287$	−2.81782e20	−0.361341
$288$	0	0
$289$	6.86955e20	0.830418
$290$	2.27472e20	0.267021
$291$	0	0
$292$	1.01492e20	0.112377
$293$	1.05777e20	0.113767	0.0568836	−	0.998381i	$-0.481884\pi$
$293$	1.05777e20	0.113767	0.0568836	+	0.998381i	$0.481884\pi$
$294$	0	0
$295$	−6.33504e20	−0.643076
$296$	−2.95223e20	−0.291186
$297$	0	0
$298$	−5.40587e20	−0.503533
$299$	−1.41125e19	−0.0127761
$300$	0	0
$301$	−1.31284e21	−1.12304
$302$	2.80183e20	0.233014
$303$	0	0
$304$	−4.88936e20	−0.384438
$305$	9.64513e20	0.737495
$306$	0	0
$307$	−1.82911e21	−1.32301	−0.661504	−	0.749941i	$-0.730082\pi$
$307$	−1.82911e21	−1.32301	−0.661504	+	0.749941i	$0.730082\pi$
$308$	2.50692e20	0.176384
$309$	0	0
$310$	−1.63788e20	−0.109070
$311$	−2.00748e21	−1.30073	−0.650366	−	0.759621i	$-0.725384\pi$
$311$	−2.00748e21	−1.30073	−0.650366	+	0.759621i	$0.725384\pi$
$312$	0	0
$313$	−2.84834e21	−1.74769	−0.873846	−	0.486203i	$-0.838381\pi$
$313$	−2.84834e21	−1.74769	−0.873846	+	0.486203i	$0.838381\pi$
$314$	6.86017e20	0.409669
$315$	0	0
$316$	5.44589e20	0.308126
$317$	−2.74347e20	−0.151111	−0.0755554	−	0.997142i	$-0.524073\pi$
$317$	−2.74347e20	−0.151111	−0.0755554	+	0.997142i	$0.524073\pi$
$318$	0	0
$319$	−9.12598e20	−0.476495
$320$	1.09951e20	0.0559017
$321$	0	0
$322$	4.02342e19	0.0194008
$323$	−4.42979e21	−2.08047
$324$	0	0
$325$	1.30641e20	0.0582197
$326$	−2.42570e21	−1.05314
$327$	0	0
$328$	4.95820e20	0.204360
$329$	−1.57755e21	−0.633603
$330$	0	0
$331$	1.83521e20	0.0700082	0.0350041	−	0.999387i	$-0.488856\pi$
$331$	1.83521e20	0.0700082	0.0350041	+	0.999387i	$0.488856\pi$
$332$	4.02848e20	0.149785
$333$	0	0
$334$	1.71690e21	0.606598
$335$	−7.93921e19	−0.0273462
$336$	0	0
$337$	−6.21202e20	−0.203413	−0.101706	−	0.994814i	$-0.532430\pi$
$337$	−6.21202e20	−0.203413	−0.101706	+	0.994814i	$0.532430\pi$
$338$	−2.02685e21	−0.647188
$339$	0	0
$340$	9.96164e20	0.302524
$341$	6.57102e20	0.194635
$342$	0	0
$343$	3.56933e21	1.00597
$344$	2.31005e21	0.635147
$345$	0	0
$346$	−2.70579e21	−0.708185
$347$	−5.08978e21	−1.29987	−0.649933	−	0.759992i	$-0.725203\pi$
$347$	−5.08978e21	−1.29987	−0.649933	+	0.759992i	$0.725203\pi$
$348$	0	0
$349$	−3.81066e20	−0.0926796	−0.0463398	−	0.998926i	$-0.514756\pi$
$349$	−3.81066e20	−0.0926796	−0.0463398	+	0.998926i	$0.514756\pi$
$350$	−3.72451e20	−0.0884079
$351$	0	0
$352$	−4.41115e20	−0.0997559
$353$	−2.34699e21	−0.518115	−0.259058	−	0.965862i	$-0.583412\pi$
$353$	−2.34699e21	−0.518115	−0.259058	+	0.965862i	$0.583412\pi$
$354$	0	0
$355$	−3.66842e21	−0.771860
$356$	−3.06334e20	−0.0629317
$357$	0	0
$358$	5.08717e21	0.996485
$359$	6.19792e19	0.0118561	0.00592807	−	0.999982i	$-0.498113\pi$
$359$	6.19792e19	0.0118561	0.00592807	+	0.999982i	$0.498113\pi$
$360$	0	0
$361$	7.47897e21	1.36468
$362$	6.40451e21	1.14147
$363$	0	0
$364$	−5.34992e20	−0.0909884
$365$	6.04941e20	0.100513
$366$	0	0
$367$	−3.40565e21	−0.540180	−0.270090	−	0.962835i	$-0.587054\pi$
$367$	−3.40565e21	−0.540180	−0.270090	+	0.962835i	$0.587054\pi$
$368$	−7.07955e19	−0.0109723
$369$	0	0
$370$	−1.75967e21	−0.260445
$371$	−6.91517e21	−1.00029
$372$	0	0
$373$	−6.44986e21	−0.891302	−0.445651	−	0.895207i	$-0.647028\pi$
$373$	−6.44986e21	−0.891302	−0.445651	+	0.895207i	$0.647028\pi$
$374$	−3.99652e21	−0.539851
$375$	0	0
$376$	2.77583e21	0.358340
$377$	1.94754e21	0.245802
$378$	0	0
$379$	−5.68235e21	−0.685638	−0.342819	−	0.939401i	$-0.611382\pi$
$379$	−5.68235e21	−0.685638	−0.342819	+	0.939401i	$0.611382\pi$
$380$	−2.91428e21	−0.343852
$381$	0	0
$382$	6.74266e21	0.760839
$383$	1.28548e22	1.41865	0.709327	−	0.704879i	$-0.248999\pi$
$383$	1.28548e22	1.41865	0.709327	+	0.704879i	$0.248999\pi$
$384$	0	0
$385$	1.49424e21	0.157763
$386$	9.44272e21	0.975226
$387$	0	0
$388$	6.67149e21	0.659407
$389$	−1.70189e22	−1.64573	−0.822867	−	0.568234i	$-0.807627\pi$
$389$	−1.70189e22	−1.64573	−0.822867	+	0.568234i	$0.807627\pi$
$390$	0	0
$391$	−6.41412e20	−0.0593792
$392$	−2.37765e21	−0.215385
$393$	0	0
$394$	1.03660e22	0.899273
$395$	3.24600e21	0.275596
$396$	0	0
$397$	−1.81182e22	−1.47365	−0.736827	−	0.676081i	$-0.763677\pi$
$397$	−1.81182e22	−1.47365	−0.736827	+	0.676081i	$0.763677\pi$
$398$	−8.51897e21	−0.678236
$399$	0	0
$400$	6.55360e20	0.0500000
$401$	−1.75008e22	−1.30717	−0.653585	−	0.756853i	$-0.726736\pi$
$401$	−1.75008e22	−1.30717	−0.653585	+	0.756853i	$0.726736\pi$
$402$	0	0
$403$	−1.40230e21	−0.100403
$404$	−2.04406e21	−0.143302
$405$	0	0
$406$	−5.55236e21	−0.373255
$407$	7.05963e21	0.464761
$408$	0	0
$409$	9.06225e21	0.572253	0.286127	−	0.958192i	$-0.407632\pi$
$409$	9.06225e21	0.572253	0.286127	+	0.958192i	$0.407632\pi$
$410$	2.95532e21	0.182785
$411$	0	0
$412$	−1.12481e22	−0.667505
$413$	1.54632e22	0.898925
$414$	0	0
$415$	2.40116e21	0.133971
$416$	9.41365e20	0.0514594
$417$	0	0
$418$	1.16919e22	0.613599
$419$	−7.49706e21	−0.385542	−0.192771	−	0.981244i	$-0.561747\pi$
$419$	−7.49706e21	−0.385542	−0.192771	+	0.981244i	$0.561747\pi$
$420$	0	0
$421$	−2.95580e22	−1.45975	−0.729873	−	0.683583i	$-0.760421\pi$
$421$	−2.95580e22	−1.45975	−0.729873	+	0.683583i	$0.760421\pi$
$422$	−6.19935e21	−0.300048
$423$	0	0
$424$	1.21678e22	0.565722
$425$	5.93760e21	0.270586
$426$	0	0
$427$	−2.35428e22	−1.03091
$428$	1.07527e22	0.461579
$429$	0	0
$430$	1.37690e22	0.568093
$431$	7.37016e21	0.298140	0.149070	−	0.988827i	$-0.452372\pi$
$431$	7.37016e21	0.298140	0.149070	+	0.988827i	$0.452372\pi$
$432$	0	0
$433$	1.77584e21	0.0690650	0.0345325	−	0.999404i	$-0.489006\pi$
$433$	1.77584e21	0.0690650	0.0345325	+	0.999404i	$0.489006\pi$
$434$	3.99789e21	0.152464
$435$	0	0
$436$	1.83718e22	0.673779
$437$	1.87645e21	0.0674909
$438$	0	0
$439$	−2.92011e22	−1.01030	−0.505151	−	0.863031i	$-0.668563\pi$
$439$	−2.92011e22	−1.01030	−0.505151	+	0.863031i	$0.668563\pi$
$440$	−2.62925e21	−0.0892244
$441$	0	0
$442$	8.52882e21	0.278484
$443$	4.06554e22	1.30223	0.651113	−	0.758980i	$-0.274302\pi$
$443$	4.06554e22	1.30223	0.651113	+	0.758980i	$0.274302\pi$
$444$	0	0
$445$	−1.82589e21	−0.0562879
$446$	1.90022e22	0.574722
$447$	0	0
$448$	−2.68379e21	−0.0781423
$449$	−4.62783e21	−0.132216	−0.0661079	−	0.997812i	$-0.521058\pi$
$449$	−4.62783e21	−0.132216	−0.0661079	+	0.997812i	$0.521058\pi$
$450$	0	0
$451$	−1.18565e22	−0.326179
$452$	3.47773e22	0.938899
$453$	0	0
$454$	−2.45128e21	−0.0637409
$455$	−3.18880e21	−0.0813825
$456$	0	0
$457$	6.30841e22	1.55107	0.775536	−	0.631303i	$-0.217480\pi$
$457$	6.30841e22	1.55107	0.775536	+	0.631303i	$0.217480\pi$
$458$	5.06247e22	1.22182
$459$	0	0
$460$	−4.21974e20	−0.00981396
$461$	−5.62982e22	−1.28539	−0.642697	−	0.766120i	$-0.722185\pi$
$461$	−5.62982e22	−1.28539	−0.642697	+	0.766120i	$0.722185\pi$
$462$	0	0
$463$	5.90180e21	0.129881	0.0649405	−	0.997889i	$-0.479314\pi$
$463$	5.90180e21	0.129881	0.0649405	+	0.997889i	$0.479314\pi$
$464$	9.76985e21	0.211098
$465$	0	0
$466$	4.60232e22	0.958730
$467$	−2.51115e22	−0.513663	−0.256832	−	0.966456i	$-0.582679\pi$
$467$	−2.51115e22	−0.513663	−0.256832	+	0.966456i	$0.582679\pi$
$468$	0	0
$469$	1.93788e21	0.0382259
$470$	1.65452e22	0.320509
$471$	0	0
$472$	−2.72088e22	−0.508396
$473$	−5.52399e22	−1.01375
$474$	0	0
$475$	−1.73705e22	−0.307550
$476$	−2.43153e22	−0.422884
$477$	0	0
$478$	5.12530e22	0.860166
$479$	3.54309e22	0.584158	0.292079	−	0.956394i	$-0.405653\pi$
$479$	3.54309e22	0.584158	0.292079	+	0.956394i	$0.405653\pi$
$480$	0	0
$481$	−1.50657e22	−0.239748
$482$	−4.64936e22	−0.726931
$483$	0	0
$484$	−2.25767e22	−0.340780
$485$	3.97652e22	0.589791
$486$	0	0
$487$	−1.51696e22	−0.217259	−0.108629	−	0.994082i	$-0.534646\pi$
$487$	−1.51696e22	−0.217259	−0.108629	+	0.994082i	$0.534646\pi$
$488$	4.14255e22	0.583041
$489$	0	0
$490$	−1.41719e22	−0.192647
$491$	6.96906e22	0.931068	0.465534	−	0.885030i	$-0.345862\pi$
$491$	6.96906e22	0.931068	0.465534	+	0.885030i	$0.345862\pi$
$492$	0	0
$493$	8.85154e22	1.14240
$494$	−2.49511e22	−0.316527
$495$	0	0
$496$	−7.03463e21	−0.0862277
$497$	8.95425e22	1.07895
$498$	0	0
$499$	−5.10974e22	−0.595037	−0.297518	−	0.954716i	$-0.596159\pi$
$499$	−5.10974e22	−0.595037	−0.297518	+	0.954716i	$0.596159\pi$
$500$	3.90625e21	0.0447214
$501$	0	0
$502$	8.19693e22	0.907130
$503$	−6.26474e21	−0.0681671	−0.0340836	−	0.999419i	$-0.510851\pi$
$503$	−6.26474e21	−0.0681671	−0.0340836	+	0.999419i	$0.510851\pi$
$504$	0	0
$505$	−1.21835e22	−0.128173
$506$	1.69292e21	0.0175129
$507$	0	0
$508$	2.26439e22	0.226522
$509$	7.41555e22	0.729528	0.364764	−	0.931100i	$-0.381150\pi$
$509$	7.41555e22	0.729528	0.364764	+	0.931100i	$0.381150\pi$
$510$	0	0
$511$	−1.47660e22	−0.140503
$512$	4.72237e21	0.0441942
$513$	0	0
$514$	8.56421e22	0.775355
$515$	−6.70441e22	−0.597034
$516$	0	0
$517$	−6.63780e22	−0.571945
$518$	4.29516e22	0.364063
$519$	0	0
$520$	5.61097e21	0.0460267
$521$	−9.53386e21	−0.0769393	−0.0384696	−	0.999260i	$-0.512248\pi$
$521$	−9.53386e21	−0.0769393	−0.0384696	+	0.999260i	$0.512248\pi$
$522$	0	0
$523$	1.49123e23	1.16487	0.582437	−	0.812876i	$-0.302099\pi$
$523$	1.49123e23	1.16487	0.582437	+	0.812876i	$0.302099\pi$
$524$	3.88617e22	0.298680
$525$	0	0
$526$	−7.55287e22	−0.561996
$527$	−6.37341e22	−0.466640
$528$	0	0
$529$	−1.40778e23	−0.998074
$530$	7.25259e22	0.505997
$531$	0	0
$532$	7.11347e22	0.480654
$533$	2.53025e22	0.168260
$534$	0	0
$535$	6.40912e22	0.412849
$536$	−3.40986e21	−0.0216190
$537$	0	0
$538$	−1.93783e23	−1.19033
$539$	5.68564e22	0.343776
$540$	0	0
$541$	8.46775e22	0.496125	0.248063	−	0.968744i	$-0.420206\pi$
$541$	8.46775e22	0.496125	0.248063	+	0.968744i	$0.420206\pi$
$542$	−3.04405e22	−0.175573
$543$	0	0
$544$	4.27849e22	0.239166
$545$	1.09505e23	0.602646
$546$	0	0
$547$	4.72359e22	0.251988	0.125994	−	0.992031i	$-0.459788\pi$
$547$	4.72359e22	0.251988	0.125994	+	0.992031i	$0.459788\pi$
$548$	3.69967e22	0.194325
$549$	0	0
$550$	−1.56715e22	−0.0798048
$551$	−2.58952e23	−1.29847
$552$	0	0
$553$	−7.92316e22	−0.385243
$554$	−5.02730e22	−0.240714
$555$	0	0
$556$	1.39927e23	0.649778
$557$	−3.77820e23	−1.72789	−0.863945	−	0.503587i	$-0.832013\pi$
$557$	−3.77820e23	−1.72789	−0.863945	+	0.503587i	$0.832013\pi$
$558$	0	0
$559$	1.17885e23	0.522949
$560$	−1.59967e22	−0.0698926
$561$	0	0
$562$	8.67929e22	0.367896
$563$	4.35959e23	1.82022	0.910112	−	0.414363i	$-0.135995\pi$
$563$	4.35959e23	1.82022	0.910112	+	0.414363i	$0.135995\pi$
$564$	0	0
$565$	2.07289e23	0.839777
$566$	1.03282e23	0.412178
$567$	0	0
$568$	−1.57558e23	−0.610209
$569$	1.25095e23	0.477293	0.238647	−	0.971106i	$-0.423296\pi$
$569$	1.25095e23	0.477293	0.238647	+	0.971106i	$0.423296\pi$
$570$	0	0
$571$	−3.10316e23	−1.14920	−0.574602	−	0.818433i	$-0.694843\pi$
$571$	−3.10316e23	−1.14920	−0.574602	+	0.818433i	$0.694843\pi$
$572$	−2.25107e22	−0.0821341
$573$	0	0
$574$	−7.21363e22	−0.255507
$575$	−2.51516e21	−0.00877787
$576$	0	0
$577$	−2.11578e23	−0.716928	−0.358464	−	0.933544i	$-0.616699\pi$
$577$	−2.11578e23	−0.716928	−0.358464	+	0.933544i	$0.616699\pi$
$578$	1.75860e23	0.587194
$579$	0	0
$580$	5.82328e22	0.188812
$581$	−5.86099e22	−0.187272
$582$	0	0
$583$	−2.90968e23	−0.902947
$584$	2.59820e22	0.0794627
$585$	0	0
$586$	2.70790e22	0.0804456
$587$	−2.15996e23	−0.632443	−0.316222	−	0.948685i	$-0.602414\pi$
$587$	−2.15996e23	−0.632443	−0.316222	+	0.948685i	$0.602414\pi$
$588$	0	0
$589$	1.86455e23	0.530387
$590$	−1.62177e23	−0.454723
$591$	0	0
$592$	−7.55771e22	−0.205900
$593$	−3.56471e23	−0.957326	−0.478663	−	0.877999i	$-0.658878\pi$
$593$	−3.56471e23	−0.957326	−0.478663	+	0.877999i	$0.658878\pi$
$594$	0	0
$595$	−1.44931e23	−0.378239
$596$	−1.38390e23	−0.356051
$597$	0	0
$598$	−3.61280e21	−0.00903408
$599$	−3.17118e23	−0.781796	−0.390898	−	0.920434i	$-0.627835\pi$
$599$	−3.17118e23	−0.781796	−0.390898	+	0.920434i	$0.627835\pi$
$600$	0	0
$601$	−2.47473e23	−0.593055	−0.296528	−	0.955024i	$-0.595829\pi$
$601$	−2.47473e23	−0.593055	−0.296528	+	0.955024i	$0.595829\pi$
$602$	−3.36086e23	−0.794109
$603$	0	0
$604$	7.17268e22	0.164766
$605$	−1.34567e23	−0.304803
$606$	0	0
$607$	1.27464e23	0.280726	0.140363	−	0.990100i	$-0.455173\pi$
$607$	1.27464e23	0.280726	0.140363	+	0.990100i	$0.455173\pi$
$608$	−1.25168e23	−0.271839
$609$	0	0
$610$	2.46915e23	0.521488
$611$	1.41655e23	0.295040
$612$	0	0
$613$	2.08992e23	0.423366	0.211683	−	0.977338i	$-0.432105\pi$
$613$	2.08992e23	0.423366	0.211683	+	0.977338i	$0.432105\pi$
$614$	−4.68251e23	−0.935508
$615$	0	0
$616$	6.41772e22	0.124723
$617$	−2.52173e23	−0.483364	−0.241682	−	0.970356i	$-0.577699\pi$
$617$	−2.52173e23	−0.483364	−0.241682	+	0.970356i	$0.577699\pi$
$618$	0	0
$619$	−8.32171e23	−1.55182	−0.775911	−	0.630842i	$-0.782709\pi$
$619$	−8.32171e23	−1.55182	−0.775911	+	0.630842i	$0.782709\pi$
$620$	−4.19296e22	−0.0771244
$621$	0	0
$622$	−5.13915e23	−0.919756
$623$	4.45681e22	0.0786821
$624$	0	0
$625$	2.32831e22	0.0400000
$626$	−7.29175e23	−1.23580
$627$	0	0
$628$	1.75620e23	0.289680
$629$	−6.84733e23	−1.11427
$630$	0	0
$631$	2.44878e23	0.387883	0.193942	−	0.981013i	$-0.437873\pi$
$631$	2.44878e23	0.387883	0.193942	+	0.981013i	$0.437873\pi$
$632$	1.39415e23	0.217878
$633$	0	0
$634$	−7.02327e22	−0.106852
$635$	1.34968e23	0.202607
$636$	0	0
$637$	−1.21335e23	−0.177338
$638$	−2.33625e23	−0.336933
$639$	0	0
$640$	2.81475e22	0.0395285
$641$	−8.76132e23	−1.21416	−0.607081	−	0.794640i	$-0.707660\pi$
$641$	−8.76132e23	−1.21416	−0.607081	+	0.794640i	$0.707660\pi$
$642$	0	0
$643$	−3.14015e23	−0.423796	−0.211898	−	0.977292i	$-0.567964\pi$
$643$	−3.14015e23	−0.423796	−0.211898	+	0.977292i	$0.567964\pi$
$644$	1.03000e22	0.0137185
$645$	0	0
$646$	−1.13403e24	−1.47111
$647$	7.19139e22	0.0920718	0.0460359	−	0.998940i	$-0.485341\pi$
$647$	7.19139e22	0.0920718	0.0460359	+	0.998940i	$0.485341\pi$
$648$	0	0
$649$	6.50640e23	0.811448
$650$	3.34440e22	0.0411675
$651$	0	0
$652$	−6.20979e23	−0.744685
$653$	7.53337e23	0.891717	0.445858	−	0.895103i	$-0.352898\pi$
$653$	7.53337e23	0.891717	0.445858	+	0.895103i	$0.352898\pi$
$654$	0	0
$655$	2.31634e23	0.267148
$656$	1.26930e23	0.144505
$657$	0	0
$658$	−4.03852e23	−0.448025
$659$	1.78954e24	1.95981	0.979906	−	0.199459i	$-0.0639186\pi$
$659$	1.78954e24	1.95981	0.979906	+	0.199459i	$0.0639186\pi$
$660$	0	0
$661$	3.55263e23	0.379173	0.189586	−	0.981864i	$-0.439285\pi$
$661$	3.55263e23	0.379173	0.189586	+	0.981864i	$0.439285\pi$
$662$	4.69815e22	0.0495033
$663$	0	0
$664$	1.03129e23	0.105914
$665$	4.23996e23	0.429910
$666$	0	0
$667$	−3.74951e22	−0.0370599
$668$	4.39527e23	0.428929
$669$	0	0
$670$	−2.03244e22	−0.0193367
$671$	−9.90603e23	−0.930589
$672$	0	0
$673$	5.15130e23	0.471833	0.235917	−	0.971773i	$-0.424191\pi$
$673$	5.15130e23	0.471833	0.235917	+	0.971773i	$0.424191\pi$
$674$	−1.59028e23	−0.143835
$675$	0	0
$676$	−5.18874e23	−0.457631
$677$	−1.69205e23	−0.147370	−0.0736851	−	0.997282i	$-0.523476\pi$
$677$	−1.69205e23	−0.147370	−0.0736851	+	0.997282i	$0.523476\pi$
$678$	0	0
$679$	−9.70628e23	−0.824441
$680$	2.55018e23	0.213917
$681$	0	0
$682$	1.68218e23	0.137628
$683$	−1.97763e23	−0.159797	−0.0798987	−	0.996803i	$-0.525460\pi$
$683$	−1.97763e23	−0.159797	−0.0798987	+	0.996803i	$0.525460\pi$
$684$	0	0
$685$	2.20518e23	0.173809
$686$	9.13749e23	0.711331
$687$	0	0
$688$	5.91372e23	0.449117
$689$	6.20943e23	0.465788
$690$	0	0
$691$	2.62293e24	1.91966	0.959828	−	0.280589i	$-0.0905298\pi$
$691$	2.62293e24	1.91966	0.959828	+	0.280589i	$0.0905298\pi$
$692$	−6.92681e23	−0.500762
$693$	0	0
$694$	−1.30298e24	−0.919144
$695$	8.34028e23	0.581179
$696$	0	0
$697$	1.14999e24	0.782018
$698$	−9.75529e22	−0.0655344
$699$	0	0
$700$	−9.53475e22	−0.0625138
$701$	4.05441e22	0.0262618	0.0131309	−	0.999914i	$-0.495820\pi$
$701$	4.05441e22	0.0262618	0.0131309	+	0.999914i	$0.495820\pi$
$702$	0	0
$703$	2.00319e24	1.26649
$704$	−1.12925e23	−0.0705381
$705$	0	0
$706$	−6.00830e23	−0.366363
$707$	2.97387e23	0.179167
$708$	0	0
$709$	−7.92278e23	−0.465999	−0.232999	−	0.972477i	$-0.574854\pi$
$709$	−7.92278e23	−0.465999	−0.232999	+	0.972477i	$0.574854\pi$
$710$	−9.39117e23	−0.545788
$711$	0	0
$712$	−7.84214e22	−0.0444995
$713$	2.69977e22	0.0151379
$714$	0	0
$715$	−1.34174e23	−0.0734630
$716$	1.30232e24	0.704621
$717$	0	0
$718$	1.58667e22	0.00838355
$719$	7.47670e23	0.390404	0.195202	−	0.980763i	$-0.437464\pi$
$719$	7.47670e23	0.390404	0.195202	+	0.980763i	$0.437464\pi$
$720$	0	0
$721$	1.63648e24	0.834566
$722$	1.91462e24	0.964974
$723$	0	0
$724$	1.63955e24	0.807138
$725$	3.47095e23	0.168879
$726$	0	0
$727$	5.40980e23	0.257122	0.128561	−	0.991702i	$-0.458964\pi$
$727$	5.40980e23	0.257122	0.128561	+	0.991702i	$0.458964\pi$
$728$	−1.36958e23	−0.0643385
$729$	0	0
$730$	1.54865e23	0.0710736
$731$	5.35787e24	2.43049
$732$	0	0
$733$	2.02014e24	0.895361	0.447680	−	0.894194i	$-0.352250\pi$
$733$	2.02014e24	0.895361	0.447680	+	0.894194i	$0.352250\pi$
$734$	−8.71847e23	−0.381965
$735$	0	0
$736$	−1.81237e22	−0.00775862
$737$	8.15396e22	0.0345060
$738$	0	0
$739$	−2.11021e24	−0.872666	−0.436333	−	0.899785i	$-0.643723\pi$
$739$	−2.11021e24	−0.872666	−0.436333	+	0.899785i	$0.643723\pi$
$740$	−4.50475e23	−0.184162
$741$	0	0
$742$	−1.77028e24	−0.707309
$743$	−1.34376e24	−0.530784	−0.265392	−	0.964141i	$-0.585501\pi$
$743$	−1.34376e24	−0.530784	−0.265392	+	0.964141i	$0.585501\pi$
$744$	0	0
$745$	−8.24871e23	−0.318462
$746$	−1.65116e24	−0.630246
$747$	0	0
$748$	−1.02311e24	−0.381732
$749$	−1.56440e24	−0.577102
$750$	0	0
$751$	2.72368e24	0.982236	0.491118	−	0.871093i	$-0.336588\pi$
$751$	2.72368e24	0.982236	0.491118	+	0.871093i	$0.336588\pi$
$752$	7.10612e23	0.253385
$753$	0	0
$754$	4.98570e23	0.173808
$755$	4.27525e23	0.147371
$756$	0	0
$757$	−3.29400e24	−1.11022	−0.555109	−	0.831777i	$-0.687324\pi$
$757$	−3.29400e24	−1.11022	−0.555109	+	0.831777i	$0.687324\pi$
$758$	−1.45468e24	−0.484819
$759$	0	0
$760$	−7.46056e23	−0.243140
$761$	4.29576e24	1.38443	0.692215	−	0.721692i	$-0.256635\pi$
$761$	4.29576e24	1.38443	0.692215	+	0.721692i	$0.256635\pi$
$762$	0	0
$763$	−2.67290e24	−0.842410
$764$	1.72612e24	0.537995
$765$	0	0
$766$	3.29083e24	1.00314
$767$	−1.38851e24	−0.418588
$768$	0	0
$769$	5.93781e24	1.75086	0.875432	−	0.483341i	$-0.160577\pi$
$769$	5.93781e24	1.75086	0.875432	+	0.483341i	$0.160577\pi$
$770$	3.82526e23	0.111555
$771$	0	0
$772$	2.41734e24	0.689589
$773$	1.98352e24	0.559642	0.279821	−	0.960052i	$-0.409725\pi$
$773$	1.98352e24	0.559642	0.279821	+	0.960052i	$0.409725\pi$
$774$	0	0
$775$	−2.49920e23	−0.0689822
$776$	1.70790e24	0.466271
$777$	0	0
$778$	−4.35684e24	−1.16371
$779$	−3.36431e24	−0.888849
$780$	0	0
$781$	3.76766e24	0.973952
$782$	−1.64201e23	−0.0419874
$783$	0	0
$784$	−6.08678e23	−0.152300
$785$	1.04678e24	0.259097
$786$	0	0
$787$	3.61371e24	0.875322	0.437661	−	0.899140i	$-0.355807\pi$
$787$	3.61371e24	0.875322	0.437661	+	0.899140i	$0.355807\pi$
$788$	2.65369e24	0.635882
$789$	0	0
$790$	8.30977e23	0.194876
$791$	−5.05970e24	−1.17388
$792$	0	0
$793$	2.11401e24	0.480047
$794$	−4.63826e24	−1.04203
$795$	0	0
$796$	−2.18086e24	−0.479586
$797$	4.63064e24	1.00750	0.503751	−	0.863849i	$-0.331953\pi$
$797$	4.63064e24	1.00750	0.503751	+	0.863849i	$0.331953\pi$
$798$	0	0
$799$	6.43819e24	1.37125
$800$	1.67772e23	0.0353553
$801$	0	0
$802$	−4.48022e24	−0.924308
$803$	−6.21304e23	−0.126830
$804$	0	0
$805$	6.13925e22	0.0122702
$806$	−3.58988e23	−0.0709956
$807$	0	0
$808$	−5.23278e23	−0.101330
$809$	1.22642e24	0.235005	0.117502	−	0.993073i	$-0.462511\pi$
$809$	1.22642e24	0.235005	0.117502	+	0.993073i	$0.462511\pi$
$810$	0	0
$811$	−2.42686e24	−0.455373	−0.227686	−	0.973735i	$-0.573116\pi$
$811$	−2.42686e24	−0.455373	−0.227686	+	0.973735i	$0.573116\pi$
$812$	−1.42140e24	−0.263931
$813$	0	0
$814$	1.80726e24	0.328635
$815$	−3.70132e24	−0.666066
$816$	0	0
$817$	−1.56745e25	−2.76252
$818$	2.31994e24	0.404644
$819$	0	0
$820$	7.56562e23	0.129249
$821$	9.87401e24	1.66946	0.834731	−	0.550658i	$-0.185623\pi$
$821$	9.87401e24	1.66946	0.834731	+	0.550658i	$0.185623\pi$
$822$	0	0
$823$	−3.09810e24	−0.513094	−0.256547	−	0.966532i	$-0.582585\pi$
$823$	−3.09810e24	−0.513094	−0.256547	+	0.966532i	$0.582585\pi$
$824$	−2.87952e24	−0.471997
$825$	0	0
$826$	3.95857e24	0.635636
$827$	3.89248e24	0.618628	0.309314	−	0.950960i	$-0.399901\pi$
$827$	3.89248e24	0.618628	0.309314	+	0.950960i	$0.399901\pi$
$828$	0	0
$829$	−1.04208e25	−1.62251	−0.811256	−	0.584691i	$-0.801216\pi$
$829$	−1.04208e25	−1.62251	−0.811256	+	0.584691i	$0.801216\pi$
$830$	6.14697e23	0.0947321
$831$	0	0
$832$	2.40989e23	0.0363873
$833$	−5.51466e24	−0.824207
$834$	0	0
$835$	2.61979e24	0.383646
$836$	2.99311e24	0.433880
$837$	0	0
$838$	−1.91925e24	−0.272619
$839$	6.67595e24	0.938721	0.469360	−	0.883007i	$-0.344485\pi$
$839$	6.67595e24	0.938721	0.469360	+	0.883007i	$0.344485\pi$
$840$	0	0
$841$	−2.08280e24	−0.286999
$842$	−7.56685e24	−1.03220
$843$	0	0
$844$	−1.58703e24	−0.212166
$845$	−3.09273e24	−0.409318
$846$	0	0
$847$	3.28465e24	0.426069
$848$	3.11496e24	0.400026
$849$	0	0
$850$	1.52003e24	0.191333
$851$	2.90052e23	0.0361472
$852$	0	0
$853$	9.79573e24	1.19666	0.598329	−	0.801250i	$-0.295831\pi$
$853$	9.79573e24	1.19666	0.598329	+	0.801250i	$0.295831\pi$
$854$	−6.02695e24	−0.728963
$855$	0	0
$856$	2.75270e24	0.326386
$857$	4.38126e24	0.514354	0.257177	−	0.966364i	$-0.417208\pi$
$857$	4.38126e24	0.514354	0.257177	+	0.966364i	$0.417208\pi$
$858$	0	0
$859$	−1.50626e25	−1.73364	−0.866821	−	0.498620i	$-0.833841\pi$
$859$	−1.50626e25	−1.73364	−0.866821	+	0.498620i	$0.833841\pi$
$860$	3.52485e24	0.401702
$861$	0	0
$862$	1.88676e24	0.210817
$863$	−1.37694e25	−1.52344	−0.761718	−	0.647909i	$-0.775644\pi$
$863$	−1.37694e25	−1.52344	−0.761718	+	0.647909i	$0.775644\pi$
$864$	0	0
$865$	−4.12870e24	−0.447896
$866$	4.54616e23	0.0488363
$867$	0	0
$868$	1.02346e24	0.107808
$869$	−3.33381e24	−0.347754
$870$	0	0
$871$	−1.74010e23	−0.0178000
$872$	4.70319e24	0.476434
$873$	0	0
$874$	4.80372e23	0.0477233
$875$	−5.68316e23	−0.0559141
$876$	0	0
$877$	−7.56082e24	−0.729579	−0.364790	−	0.931090i	$-0.618859\pi$
$877$	−7.56082e24	−0.729579	−0.364790	+	0.931090i	$0.618859\pi$
$878$	−7.47549e24	−0.714391
$879$	0	0
$880$	−6.73087e23	−0.0630912
$881$	−5.11196e24	−0.474561	−0.237281	−	0.971441i	$-0.576256\pi$
$881$	−5.11196e24	−0.474561	−0.237281	+	0.971441i	$0.576256\pi$
$882$	0	0
$883$	−1.37334e25	−1.25058	−0.625289	−	0.780393i	$-0.715019\pi$
$883$	−1.37334e25	−1.25058	−0.625289	+	0.780393i	$0.715019\pi$
$884$	2.18338e24	0.196918
$885$	0	0
$886$	1.04078e25	0.920813
$887$	−1.15450e25	−1.01168	−0.505838	−	0.862628i	$-0.668817\pi$
$887$	−1.15450e25	−1.01168	−0.505838	+	0.862628i	$0.668817\pi$
$888$	0	0
$889$	−3.29443e24	−0.283215
$890$	−4.67428e23	−0.0398015
$891$	0	0
$892$	4.86456e24	0.406390
$893$	−1.88349e25	−1.55857
$894$	0	0
$895$	7.76241e24	0.630233
$896$	−6.87051e23	−0.0552550
$897$	0	0
$898$	−1.18472e24	−0.0934907
$899$	−3.72571e24	−0.291240
$900$	0	0
$901$	2.82218e25	2.16483
$902$	−3.03526e24	−0.230643
$903$	0	0
$904$	8.90298e24	0.663902
$905$	9.77250e24	0.721926
$906$	0	0
$907$	−1.74990e25	−1.26868	−0.634338	−	0.773056i	$-0.718727\pi$
$907$	−1.74990e25	−1.26868	−0.634338	+	0.773056i	$0.718727\pi$
$908$	−6.27527e23	−0.0450716
$909$	0	0
$910$	−8.16334e23	−0.0575461
$911$	1.12560e25	0.786102	0.393051	−	0.919517i	$-0.371420\pi$
$911$	1.12560e25	0.786102	0.393051	+	0.919517i	$0.371420\pi$
$912$	0	0
$913$	−2.46611e24	−0.169048
$914$	1.61495e25	1.09677
$915$	0	0
$916$	1.29599e25	0.863955
$917$	−5.65395e24	−0.373433
$918$	0	0
$919$	−2.33947e24	−0.151682	−0.0758412	−	0.997120i	$-0.524164\pi$
$919$	−2.33947e24	−0.151682	−0.0758412	+	0.997120i	$0.524164\pi$
$920$	−1.08025e23	−0.00693952
$921$	0	0
$922$	−1.44123e25	−0.908911
$923$	−8.04040e24	−0.502416
$924$	0	0
$925$	−2.68504e24	−0.164720
$926$	1.51086e24	0.0918398
$927$	0	0
$928$	2.50108e24	0.149269
$929$	−3.62091e24	−0.214133	−0.107067	−	0.994252i	$-0.534146\pi$
$929$	−3.62091e24	−0.214133	−0.107067	+	0.994252i	$0.534146\pi$
$930$	0	0
$931$	1.61332e25	0.936801
$932$	1.17819e25	0.677925
$933$	0	0
$934$	−6.42853e24	−0.363215
$935$	−6.09821e24	−0.341432
$936$	0	0
$937$	4.50917e24	0.247919	0.123960	−	0.992287i	$-0.460441\pi$
$937$	4.50917e24	0.247919	0.123960	+	0.992287i	$0.460441\pi$
$938$	4.96097e23	0.0270298
$939$	0	0
$940$	4.23558e24	0.226634
$941$	−2.46547e25	−1.30734	−0.653669	−	0.756780i	$-0.726771\pi$
$941$	−2.46547e25	−1.30734	−0.653669	+	0.756780i	$0.726771\pi$
$942$	0	0
$943$	−4.87136e23	−0.0253689
$944$	−6.96545e24	−0.359490
$945$	0	0
$946$	−1.41414e25	−0.716833
$947$	−2.52153e25	−1.26675	−0.633373	−	0.773847i	$-0.718330\pi$
$947$	−2.52153e25	−1.26675	−0.633373	+	0.773847i	$0.718330\pi$
$948$	0	0
$949$	1.32590e24	0.0654257
$950$	−4.44684e24	−0.217471
$951$	0	0
$952$	−6.22473e24	−0.299024
$953$	−2.22635e25	−1.05999	−0.529996	−	0.848000i	$-0.677807\pi$
$953$	−2.22635e25	−1.05999	−0.529996	+	0.848000i	$0.677807\pi$
$954$	0	0
$955$	1.02885e25	0.481197
$956$	1.31208e25	0.608229
$957$	0	0
$958$	9.07031e24	0.413062
$959$	−5.38261e24	−0.242960
$960$	0	0
$961$	−1.98675e25	−0.881037
$962$	−3.85681e24	−0.169528
$963$	0	0
$964$	−1.19024e25	−0.514018
$965$	1.44085e25	0.616787
$966$	0	0
$967$	−4.73776e24	−0.199273	−0.0996363	−	0.995024i	$-0.531768\pi$
$967$	−4.73776e24	−0.199273	−0.0996363	+	0.995024i	$0.531768\pi$
$968$	−5.77963e24	−0.240968
$969$	0	0
$970$	1.01799e25	0.417045
$971$	3.41703e25	1.38767	0.693833	−	0.720136i	$-0.255921\pi$
$971$	3.41703e25	1.38767	0.693833	+	0.720136i	$0.255921\pi$
$972$	0	0
$973$	−2.03578e25	−0.812402
$974$	−3.88341e24	−0.153625
$975$	0	0
$976$	1.06049e25	0.412273
$977$	−6.32382e24	−0.243711	−0.121856	−	0.992548i	$-0.538884\pi$
$977$	−6.32382e24	−0.243711	−0.121856	+	0.992548i	$0.538884\pi$
$978$	0	0
$979$	1.87528e24	0.0710254
$980$	−3.62801e24	−0.136222
$981$	0	0
$982$	1.78408e25	0.658365
$983$	5.13743e25	1.87949	0.939747	−	0.341870i	$-0.111060\pi$
$983$	5.13743e25	1.87949	0.939747	+	0.341870i	$0.111060\pi$
$984$	0	0
$985$	1.58172e25	0.568750
$986$	2.26599e25	0.807802
$987$	0	0
$988$	−6.38749e24	−0.223818
$989$	−2.26959e24	−0.0788458
$990$	0	0
$991$	3.11936e25	1.06522	0.532610	−	0.846361i	$-0.321211\pi$
$991$	3.11936e25	1.06522	0.532610	+	0.846361i	$0.321211\pi$
$992$	−1.80086e24	−0.0609722
$993$	0	0
$994$	2.29229e25	0.762930
$995$	−1.29989e25	−0.428954
$996$	0	0
$997$	−4.66217e25	−1.51244	−0.756222	−	0.654315i	$-0.772957\pi$
$997$	−4.66217e25	−1.51244	−0.756222	+	0.654315i	$0.772957\pi$
$998$	−1.30809e25	−0.420755
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.18.a.n.1.1		2
3.2	odd	2		10.18.a.b.1.2	✓	2
12.11	even	2		80.18.a.e.1.1		2
15.2	even	4		50.18.b.e.49.1		4
15.8	even	4		50.18.b.e.49.4		4
15.14	odd	2		50.18.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.18.a.b.1.2	✓	2	3.2	odd	2
50.18.a.g.1.1		2	15.14	odd	2
50.18.b.e.49.1		4	15.2	even	4
50.18.b.e.49.4		4	15.8	even	4
80.18.a.e.1.1		2	12.11	even	2
90.18.a.n.1.1		2	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)

\(f(q)\)	\(=\)	\(q+256.000 q^{2} +65536.0 q^{4} +390625. q^{5} -9.53475e6 q^{7} +1.67772e7 q^{8} +1.00000e8 q^{10} -4.01191e8 q^{11} +8.56166e8 q^{13} -2.44090e9 q^{14} +4.29497e9 q^{16} +3.89127e10 q^{17} -1.13839e11 q^{19} +2.56000e10 q^{20} -1.02705e11 q^{22} -1.64834e10 q^{23} +1.52588e11 q^{25} +2.19179e11 q^{26} -6.24870e11 q^{28} +2.27472e12 q^{29} -1.63788e12 q^{31} +1.09951e12 q^{32} +9.96164e12 q^{34} -3.72451e12 q^{35} -1.75967e13 q^{37} -2.91428e13 q^{38} +6.55360e12 q^{40} +2.95532e13 q^{41} +1.37690e14 q^{43} -2.62925e13 q^{44} -4.21974e12 q^{46} +1.65452e14 q^{47} -1.41719e14 q^{49} +3.90625e13 q^{50} +5.61097e13 q^{52} +7.25259e14 q^{53} -1.56715e14 q^{55} -1.59967e14 q^{56} +5.82328e14 q^{58} -1.62177e15 q^{59} +2.46915e15 q^{61} -4.19296e14 q^{62} +2.81475e14 q^{64} +3.34440e14 q^{65} -2.03244e14 q^{67} +2.55018e15 q^{68} -9.53475e14 q^{70} -9.39117e15 q^{71} +1.54865e15 q^{73} -4.50475e15 q^{74} -7.46056e15 q^{76} +3.82526e15 q^{77} +8.30977e15 q^{79} +1.67772e15 q^{80} +7.56562e15 q^{82} +6.14697e15 q^{83} +1.52003e16 q^{85} +3.52485e16 q^{86} -6.73087e15 q^{88} -4.67428e15 q^{89} -8.16334e15 q^{91} -1.08025e15 q^{92} +4.23558e16 q^{94} -4.44684e16 q^{95} +1.01799e17 q^{97} -3.62801e16 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 512 q^{2} + 131072 q^{4} + 781250 q^{5} + 6543844 q^{7} + 33554432 q^{8} + 200000000 q^{10} - 1189408704 q^{11} - 2017919228 q^{13} + 1675224064 q^{14} + 8589934592 q^{16} + 18755639436 q^{17} - 136704830600 q^{19}+ \cdots - 29\!\cdots\!76 q^{98}+O(q^{100}) \) 2 * q + 512 * q^2 + 131072 * q^4 + 781250 * q^5 + 6543844 * q^7 + 33554432 * q^8 + 200000000 * q^10 - 1189408704 * q^11 - 2017919228 * q^13 + 1675224064 * q^14 + 8589934592 * q^16 + 18755639436 * q^17 - 136704830600 * q^19 + 51200000000 * q^20 - 304488628224 * q^22 + 649234170708 * q^23 + 305175781250 * q^25 - 516587322368 * q^26 + 428857360384 * q^28 + 4696543420020 * q^29 + 7120867378744 * q^31 + 2199023255552 * q^32 + 4801443695616 * q^34 + 2556189062500 * q^35 + 9933114637444 * q^37 - 34996436633600 * q^38 + 13107200000000 * q^40 - 9622711880844 * q^41 + 7708436011852 * q^43 - 77949088825344 * q^44 + 166203947701248 * q^46 + 466072575837996 * q^47 - 115828146887046 * q^49 + 78125000000000 * q^50 - 132246354526208 * q^52 + 623333120284428 * q^53 - 464612775000000 * q^55 + 109787484258304 * q^56 + 1202315115525120 * q^58 - 654616071995160 * q^59 + 1329040385162884 * q^61 + 1822942048958464 * q^62 + 562949953421312 * q^64 - 788249698437500 * q^65 - 4249016411877596 * q^67 + 1229169586077696 * q^68 + 654384400000000 * q^70 - 2035131949347624 * q^71 + 2248593461347972 * q^73 + 2542877347185664 * q^74 - 8959087778201600 * q^76 - 8848168987836288 * q^77 + 19188427678495120 * q^79 + 3355443200000000 * q^80 - 2463414241496064 * q^82 + 18474649086611988 * q^83 + 7326421654687500 * q^85 + 1973359619034112 * q^86 - 19954966739288064 * q^88 + 17458912916395020 * q^89 - 54374608211433016 * q^91 + 42548210611519488 * q^92 + 119314579414526976 * q^94 - 53400324453125000 * q^95 + 195934325822490244 * q^97 - 29652005603083776 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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