LMFDB - Embedded newform 90.18.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,18,Mod(1,90)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$164.899878610$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{83281}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 20820 $ x^2 - x - 20820
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$144.792$ 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)

$f(q)$	$=$	\(q+256.000 q^{2} +65536.0 q^{4} -390625. q^{5} +2.54694e7 q^{7} +1.67772e7 q^{8} -1.00000e8 q^{10} +2.81089e8 q^{11} -1.52610e9 q^{13} +6.52016e9 q^{14} +4.29497e9 q^{16} -5.46901e10 q^{17} +6.88947e8 q^{19} -2.56000e10 q^{20} +7.19587e10 q^{22} -3.91035e11 q^{23} +1.52588e11 q^{25} -3.90681e11 q^{26} +1.66916e12 q^{28} -5.12259e12 q^{29} -7.31204e10 q^{31} +1.09951e12 q^{32} -1.40007e13 q^{34} -9.94897e12 q^{35} -6.81637e12 q^{37} +1.76370e11 q^{38} -6.55360e12 q^{40} +5.76386e13 q^{41} +7.57081e13 q^{43} +1.84214e13 q^{44} -1.00105e14 q^{46} +4.60351e13 q^{47} +4.16059e14 q^{49} +3.90625e13 q^{50} -1.00014e14 q^{52} -6.58261e14 q^{53} -1.09800e14 q^{55} +4.27305e14 q^{56} -1.31138e15 q^{58} +2.98287e14 q^{59} +8.50623e14 q^{61} -1.87188e13 q^{62} +2.81475e14 q^{64} +5.96131e14 q^{65} -6.12967e15 q^{67} -3.58417e15 q^{68} -2.54694e15 q^{70} -5.41472e14 q^{71} -7.16849e15 q^{73} -1.74499e15 q^{74} +4.51508e13 q^{76} +7.15916e15 q^{77} +5.45373e15 q^{79} -1.67772e15 q^{80} +1.47555e16 q^{82} +3.64723e15 q^{83} +2.13633e16 q^{85} +1.93813e16 q^{86} +4.71589e15 q^{88} -6.81792e14 q^{89} -3.88687e16 q^{91} -2.56269e16 q^{92} +1.17850e16 q^{94} -2.69120e14 q^{95} -1.20373e17 q^{97} +1.06511e17 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 512 q^{2} + 131072 q^{4} - 781250 q^{5} + 603844 q^{7} + 33554432 q^{8} - 200000000 q^{10} + 471481296 q^{11} - 1541834228 q^{13} + 154584064 q^{14} + 8589934592 q^{16} - 32139900564 q^{17} + 128672529400 q^{19}+ \cdots + 20\!\cdots\!24 q^{98}+O(q^{100}) $ 2 * q + 512 * q^2 + 131072 * q^4 - 781250 * q^5 + 603844 * q^7 + 33554432 * q^8 - 200000000 * q^10 + 471481296 * q^11 - 1541834228 * q^13 + 154584064 * q^14 + 8589934592 * q^16 - 32139900564 * q^17 + 128672529400 * q^19 - 51200000000 * q^20 + 120699211776 * q^22 - 650359859292 * q^23 + 305175781250 * q^25 - 394709562368 * q^26 + 39573520384 * q^28 - 2543054749980 * q^29 + 7839407998744 * q^31 + 2199023255552 * q^32 - 8227814544384 * q^34 - 235876562500 * q^35 - 27805209097556 * q^37 + 32940167526400 * q^38 - 13107200000000 * q^40 + 37826364264156 * q^41 + 29630453926852 * q^43 + 30898998214656 * q^44 - 166492123978752 * q^46 - 220791583022004 * q^47 + 801722584677954 * q^49 + 78125000000000 * q^50 - 101045647966208 * q^52 - 1058306017600572 * q^53 - 184172381250000 * q^55 + 10130821218304 * q^56 - 651022015994880 * q^58 - 1527820717235160 * q^59 - 631738213122116 * q^61 + 2006888447678464 * q^62 + 562949953421312 * q^64 + 602278995312500 * q^65 - 8843763493872596 * q^67 - 2106320523362304 * q^68 - 60384400000000 * q^70 + 3331901660912376 * q^71 - 10026350574292028 * q^73 - 7118133528974336 * q^74 + 8432682886758400 * q^76 + 2424946105073712 * q^77 + 8263606842055120 * q^79 - 3355443200000000 * q^80 + 9683549251623936 * q^82 + 10689526892336988 * q^83 + 12554648657812500 * q^85 + 7585396205274112 * q^86 + 7910143542951936 * q^88 - 55878412521904980 * q^89 - 38477396610603016 * q^91 - 42621983738560512 * q^92 - 56522645253633024 * q^94 - 50262706796875000 * q^95 - 56106580363859756 * q^97 + 205240981677556224 * 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
$7$	2.54694e7	1.66988	0.834939	−	0.550342i	$-0.185503\pi$
$7$	2.54694e7	1.66988	0.834939	+	0.550342i	$0.185503\pi$
$8$	1.67772e7	0.353553
$9$	0	0
$10$	−1.00000e8	−0.316228
$11$	2.81089e8	0.395372	0.197686	−	0.980265i	$-0.436657\pi$
$11$	2.81089e8	0.395372	0.197686	+	0.980265i	$0.436657\pi$
$12$	0	0
$13$	−1.52610e9	−0.518876	−0.259438	−	0.965760i	$-0.583537\pi$
$13$	−1.52610e9	−0.518876	−0.259438	+	0.965760i	$0.583537\pi$
$14$	6.52016e9	1.18078
$15$	0	0
$16$	4.29497e9	0.250000
$17$	−5.46901e10	−1.90148	−0.950742	−	0.309984i	$-0.899676\pi$
$17$	−5.46901e10	−1.90148	−0.950742	+	0.309984i	$0.899676\pi$
$18$	0	0
$19$	6.88947e8	0.00930637	0.00465318	−	0.999989i	$-0.498519\pi$
$19$	6.88947e8	0.00930637	0.00465318	+	0.999989i	$0.498519\pi$
$20$	−2.56000e10	−0.223607
$21$	0	0
$22$	7.19587e10	0.279570
$23$	−3.91035e11	−1.04119	−0.520594	−	0.853804i	$-0.674290\pi$
$23$	−3.91035e11	−1.04119	−0.520594	+	0.853804i	$0.674290\pi$
$24$	0	0
$25$	1.52588e11	0.200000
$26$	−3.90681e11	−0.366901
$27$	0	0
$28$	1.66916e12	0.834939
$29$	−5.12259e12	−1.90154	−0.950772	−	0.309892i	$-0.899707\pi$
$29$	−5.12259e12	−1.90154	−0.950772	+	0.309892i	$0.899707\pi$
$30$	0	0
$31$	−7.31204e10	−0.0153980	−0.00769900	−	0.999970i	$-0.502451\pi$
$31$	−7.31204e10	−0.0153980	−0.00769900	+	0.999970i	$0.502451\pi$
$32$	1.09951e12	0.176777
$33$	0	0
$34$	−1.40007e13	−1.34455
$35$	−9.94897e12	−0.746792
$36$	0	0
$37$	−6.81637e12	−0.319035	−0.159517	−	0.987195i	$-0.550994\pi$
$37$	−6.81637e12	−0.319035	−0.159517	+	0.987195i	$0.550994\pi$
$38$	1.76370e11	0.00658059
$39$	0	0
$40$	−6.55360e12	−0.158114
$41$	5.76386e13	1.12733	0.563665	−	0.826004i	$-0.309391\pi$
$41$	5.76386e13	1.12733	0.563665	+	0.826004i	$0.309391\pi$
$42$	0	0
$43$	7.57081e13	0.987781	0.493890	−	0.869524i	$-0.335574\pi$
$43$	7.57081e13	0.987781	0.493890	+	0.869524i	$0.335574\pi$
$44$	1.84214e13	0.197686
$45$	0	0
$46$	−1.00105e14	−0.736231
$47$	4.60351e13	0.282006	0.141003	−	0.990009i	$-0.454967\pi$
$47$	4.60351e13	0.282006	0.141003	+	0.990009i	$0.454967\pi$
$48$	0	0
$49$	4.16059e14	1.78849
$50$	3.90625e13	0.141421
$51$	0	0
$52$	−1.00014e14	−0.259438
$53$	−6.58261e14	−1.45229	−0.726145	−	0.687541i	$-0.758690\pi$
$53$	−6.58261e14	−1.45229	−0.726145	+	0.687541i	$0.758690\pi$
$54$	0	0
$55$	−1.09800e14	−0.176816
$56$	4.27305e14	0.590391
$57$	0	0
$58$	−1.31138e15	−1.34459
$59$	2.98287e14	0.264479	0.132240	−	0.991218i	$-0.457783\pi$
$59$	2.98287e14	0.264479	0.132240	+	0.991218i	$0.457783\pi$
$60$	0	0
$61$	8.50623e14	0.568111	0.284056	−	0.958808i	$-0.408320\pi$
$61$	8.50623e14	0.568111	0.284056	+	0.958808i	$0.408320\pi$
$62$	−1.87188e13	−0.0108880
$63$	0	0
$64$	2.81475e14	0.125000
$65$	5.96131e14	0.232048
$66$	0	0
$67$	−6.12967e15	−1.84417	−0.922085	−	0.386987i	$-0.873516\pi$
$67$	−6.12967e15	−1.84417	−0.922085	+	0.386987i	$0.873516\pi$
$68$	−3.58417e15	−0.950742
$69$	0	0
$70$	−2.54694e15	−0.528062
$71$	−5.41472e14	−0.0995131	−0.0497565	−	0.998761i	$-0.515845\pi$
$71$	−5.41472e14	−0.0995131	−0.0497565	+	0.998761i	$0.515845\pi$
$72$	0	0
$73$	−7.16849e15	−1.04036	−0.520180	−	0.854057i	$-0.674135\pi$
$73$	−7.16849e15	−1.04036	−0.520180	+	0.854057i	$0.674135\pi$
$74$	−1.74499e15	−0.225592
$75$	0	0
$76$	4.51508e13	0.00465318
$77$	7.15916e15	0.660223
$78$	0	0
$79$	5.45373e15	0.404448	0.202224	−	0.979339i	$-0.435183\pi$
$79$	5.45373e15	0.404448	0.202224	+	0.979339i	$0.435183\pi$
$80$	−1.67772e15	−0.111803
$81$	0	0
$82$	1.47555e16	0.797142
$83$	3.64723e15	0.177746	0.0888728	−	0.996043i	$-0.471674\pi$
$83$	3.64723e15	0.177746	0.0888728	+	0.996043i	$0.471674\pi$
$84$	0	0
$85$	2.13633e16	0.850369
$86$	1.93813e16	0.698467
$87$	0	0
$88$	4.71589e15	0.139785
$89$	−6.81792e14	−0.0183585	−0.00917925	−	0.999958i	$-0.502922\pi$
$89$	−6.81792e14	−0.0183585	−0.00917925	+	0.999958i	$0.502922\pi$
$90$	0	0
$91$	−3.88687e16	−0.866460
$92$	−2.56269e16	−0.520594
$93$	0	0
$94$	1.17850e16	0.199408
$95$	−2.69120e14	−0.00416193
$96$	0	0
$97$	−1.20373e17	−1.55944	−0.779720	−	0.626129i	$-0.784638\pi$
$97$	−1.20373e17	−1.55944	−0.779720	+	0.626129i	$0.784638\pi$
$98$	1.06511e17	1.26466
$99$	0	0
$100$	1.00000e16	0.100000
$101$	−7.74092e16	−0.711314	−0.355657	−	0.934617i	$-0.615743\pi$
$101$	−7.74092e16	−0.711314	−0.355657	+	0.934617i	$0.615743\pi$
$102$	0	0
$103$	−1.39309e17	−1.08359	−0.541793	−	0.840512i	$-0.682254\pi$
$103$	−1.39309e17	−1.08359	−0.541793	+	0.840512i	$0.682254\pi$
$104$	−2.56037e16	−0.183450
$105$	0	0
$106$	−1.68515e17	−1.02692
$107$	−2.75662e17	−1.55101	−0.775505	−	0.631341i	$-0.782505\pi$
$107$	−2.75662e17	−1.55101	−0.775505	+	0.631341i	$0.782505\pi$
$108$	0	0
$109$	1.37550e17	0.661202	0.330601	−	0.943771i	$-0.392749\pi$
$109$	1.37550e17	0.661202	0.330601	+	0.943771i	$0.392749\pi$
$110$	−2.81089e16	−0.125028
$111$	0	0
$112$	1.09390e17	0.417470
$113$	3.90258e17	1.38097	0.690486	−	0.723345i	$-0.257396\pi$
$113$	3.90258e17	1.38097	0.690486	+	0.723345i	$0.257396\pi$
$114$	0	0
$115$	1.52748e17	0.465634
$116$	−3.35714e17	−0.950772
$117$	0	0
$118$	7.63614e16	0.187015
$119$	−1.39292e18	−3.17525
$120$	0	0
$121$	−4.26436e17	−0.843681
$122$	2.17759e17	0.401715
$123$	0	0
$124$	−4.79202e15	−0.00769900
$125$	−5.96046e16	−0.0894427
$126$	0	0
$127$	−7.37824e17	−0.967433	−0.483717	−	0.875225i	$-0.660713\pi$
$127$	−7.37824e17	−0.967433	−0.483717	+	0.875225i	$0.660713\pi$
$128$	7.20576e16	0.0883883
$129$	0	0
$130$	1.52610e17	0.164083
$131$	1.45408e18	1.46482	0.732409	−	0.680865i	$-0.238396\pi$
$131$	1.45408e18	1.46482	0.732409	+	0.680865i	$0.238396\pi$
$132$	0	0
$133$	1.75470e16	0.0155405
$134$	−1.56919e18	−1.30403
$135$	0	0
$136$	−9.17547e17	−0.672276
$137$	2.27534e18	1.56647	0.783236	−	0.621725i	$-0.213568\pi$
$137$	2.27534e18	1.56647	0.783236	+	0.621725i	$0.213568\pi$
$138$	0	0
$139$	1.41598e17	0.0861849	0.0430924	−	0.999071i	$-0.486279\pi$
$139$	1.41598e17	0.0861849	0.0430924	+	0.999071i	$0.486279\pi$
$140$	−6.52016e17	−0.373396
$141$	0	0
$142$	−1.38617e17	−0.0703664
$143$	−4.28969e17	−0.205149
$144$	0	0
$145$	2.00101e18	0.850396
$146$	−1.83513e18	−0.735646
$147$	0	0
$148$	−4.46717e17	−0.159517
$149$	1.31089e18	0.442063	0.221032	−	0.975267i	$-0.429058\pi$
$149$	1.31089e18	0.442063	0.221032	+	0.975267i	$0.429058\pi$
$150$	0	0
$151$	−2.76548e18	−0.832658	−0.416329	−	0.909214i	$-0.636684\pi$
$151$	−2.76548e18	−0.832658	−0.416329	+	0.909214i	$0.636684\pi$
$152$	1.15586e16	0.00329030
$153$	0	0
$154$	1.83274e18	0.466848
$155$	2.85627e16	0.00688619
$156$	0	0
$157$	−3.81235e18	−0.824225	−0.412112	−	0.911133i	$-0.635209\pi$
$157$	−3.81235e18	−0.824225	−0.412112	+	0.911133i	$0.635209\pi$
$158$	1.39615e18	0.285988
$159$	0	0
$160$	−4.29497e17	−0.0790569
$161$	−9.95942e18	−1.73866
$162$	0	0
$163$	1.03773e19	1.63114	0.815570	−	0.578658i	$-0.196423\pi$
$163$	1.03773e19	1.63114	0.815570	+	0.578658i	$0.196423\pi$
$164$	3.77740e18	0.563665
$165$	0	0
$166$	9.33690e17	0.125685
$167$	−1.21920e19	−1.55949	−0.779747	−	0.626095i	$-0.784652\pi$
$167$	−1.21920e19	−1.55949	−0.779747	+	0.626095i	$0.784652\pi$
$168$	0	0
$169$	−6.32145e18	−0.730768
$170$	5.46901e18	0.601302
$171$	0	0
$172$	4.96161e18	0.493890
$173$	3.88210e18	0.367853	0.183927	−	0.982940i	$-0.441119\pi$
$173$	3.88210e18	0.367853	0.183927	+	0.982940i	$0.441119\pi$
$174$	0	0
$175$	3.88632e18	0.333976
$176$	1.20727e18	0.0988430
$177$	0	0
$178$	−1.74539e17	−0.0129814
$179$	1.62354e19	1.15136	0.575680	−	0.817675i	$-0.304737\pi$
$179$	1.62354e19	1.15136	0.575680	+	0.817675i	$0.304737\pi$
$180$	0	0
$181$	−2.32195e19	−1.49825	−0.749127	−	0.662427i	$-0.769527\pi$
$181$	−2.32195e19	−1.49825	−0.749127	+	0.662427i	$0.769527\pi$
$182$	−9.95039e18	−0.612680
$183$	0	0
$184$	−6.56048e18	−0.368116
$185$	2.66264e18	0.142677
$186$	0	0
$187$	−1.53728e19	−0.751793
$188$	3.01696e18	0.141003
$189$	0	0
$190$	−6.88947e16	−0.00294293
$191$	−4.47928e19	−1.82989	−0.914944	−	0.403580i	$-0.867766\pi$
$191$	−4.47928e19	−1.82989	−0.914944	+	0.403580i	$0.867766\pi$
$192$	0	0
$193$	3.61154e19	1.35038	0.675189	−	0.737645i	$-0.264062\pi$
$193$	3.61154e19	1.35038	0.675189	+	0.737645i	$0.264062\pi$
$194$	−3.08154e19	−1.10269
$195$	0	0
$196$	2.72668e19	0.894247
$197$	1.80503e19	0.566920	0.283460	−	0.958984i	$-0.408518\pi$
$197$	1.80503e19	0.566920	0.283460	+	0.958984i	$0.408518\pi$
$198$	0	0
$199$	−6.39636e18	−0.184366	−0.0921832	−	0.995742i	$-0.529385\pi$
$199$	−6.39636e18	−0.184366	−0.0921832	+	0.995742i	$0.529385\pi$
$200$	2.56000e18	0.0707107
$201$	0	0
$202$	−1.98168e19	−0.502975
$203$	−1.30469e20	−3.17535
$204$	0	0
$205$	−2.25151e19	−0.504157
$206$	−3.56631e19	−0.766211
$207$	0	0
$208$	−6.55453e18	−0.129719
$209$	1.93655e17	0.00367948
$210$	0	0
$211$	7.72101e18	0.135292	0.0676462	−	0.997709i	$-0.478451\pi$
$211$	7.72101e18	0.135292	0.0676462	+	0.997709i	$0.478451\pi$
$212$	−4.31398e19	−0.726145
$213$	0	0
$214$	−7.05695e19	−1.09673
$215$	−2.95735e19	−0.441749
$216$	0	0
$217$	−1.86233e18	−0.0257128
$218$	3.52127e19	0.467541
$219$	0	0
$220$	−7.19587e18	−0.0884078
$221$	8.34623e19	0.986634
$222$	0	0
$223$	−1.08588e20	−1.18902	−0.594510	−	0.804088i	$-0.702654\pi$
$223$	−1.08588e20	−1.18902	−0.594510	+	0.804088i	$0.702654\pi$
$224$	2.80039e19	0.295196
$225$	0	0
$226$	9.99061e19	0.976495
$227$	−1.95539e19	−0.184083	−0.0920415	−	0.995755i	$-0.529339\pi$
$227$	−1.95539e19	−0.184083	−0.0920415	+	0.995755i	$0.529339\pi$
$228$	0	0
$229$	1.20601e20	1.05378	0.526890	−	0.849933i	$-0.323358\pi$
$229$	1.20601e20	1.05378	0.526890	+	0.849933i	$0.323358\pi$
$230$	3.91035e19	0.329253
$231$	0	0
$232$	−8.59427e19	−0.672297
$233$	−7.20388e19	−0.543302	−0.271651	−	0.962396i	$-0.587570\pi$
$233$	−7.20388e19	−0.543302	−0.271651	+	0.962396i	$0.587570\pi$
$234$	0	0
$235$	−1.79825e19	−0.126117
$236$	1.95485e19	0.132240
$237$	0	0
$238$	−3.56588e20	−2.24524
$239$	−1.70425e20	−1.03550	−0.517750	−	0.855532i	$-0.673230\pi$
$239$	−1.70425e20	−1.03550	−0.517750	+	0.855532i	$0.673230\pi$
$240$	0	0
$241$	−1.71129e20	−0.968679	−0.484340	−	0.874880i	$-0.660940\pi$
$241$	−1.71129e20	−0.968679	−0.484340	+	0.874880i	$0.660940\pi$
$242$	−1.09168e20	−0.596573
$243$	0	0
$244$	5.57464e19	0.284056
$245$	−1.62523e20	−0.799839
$246$	0	0
$247$	−1.05140e18	−0.00482885
$248$	−1.22676e18	−0.00544401
$249$	0	0
$250$	−1.52588e19	−0.0632456
$251$	−1.85831e20	−0.744546	−0.372273	−	0.928123i	$-0.621422\pi$
$251$	−1.85831e20	−0.744546	−0.372273	+	0.928123i	$0.621422\pi$
$252$	0	0
$253$	−1.09916e20	−0.411657
$254$	−1.88883e20	−0.684078
$255$	0	0
$256$	1.84467e19	0.0625000
$257$	−1.68273e19	−0.0551547	−0.0275773	−	0.999620i	$-0.508779\pi$
$257$	−1.68273e19	−0.0551547	−0.0275773	+	0.999620i	$0.508779\pi$
$258$	0	0
$259$	−1.73609e20	−0.532750
$260$	3.90681e19	0.116024
$261$	0	0
$262$	3.72245e20	1.03578
$263$	−2.77268e20	−0.746924	−0.373462	−	0.927645i	$-0.621829\pi$
$263$	−2.77268e20	−0.746924	−0.373462	+	0.927645i	$0.621829\pi$
$264$	0	0
$265$	2.57133e20	0.649484
$266$	4.49204e18	0.0109888
$267$	0	0
$268$	−4.01714e20	−0.922085
$269$	−9.40799e19	−0.209220	−0.104610	−	0.994513i	$-0.533359\pi$
$269$	−9.40799e19	−0.209220	−0.104610	+	0.994513i	$0.533359\pi$
$270$	0	0
$271$	−6.17628e20	−1.28970	−0.644849	−	0.764310i	$-0.723080\pi$
$271$	−6.17628e20	−1.28970	−0.644849	+	0.764310i	$0.723080\pi$
$272$	−2.34892e20	−0.475371
$273$	0	0
$274$	5.82488e20	1.10766
$275$	4.28907e19	0.0790744
$276$	0	0
$277$	−6.67865e20	−1.15774	−0.578869	−	0.815420i	$-0.696506\pi$
$277$	−6.67865e20	−1.15774	−0.578869	+	0.815420i	$0.696506\pi$
$278$	3.62491e19	0.0609419
$279$	0	0
$280$	−1.66916e20	−0.264031
$281$	1.13640e20	0.174393	0.0871963	−	0.996191i	$-0.472209\pi$
$281$	1.13640e20	0.174393	0.0871963	+	0.996191i	$0.472209\pi$
$282$	0	0
$283$	−3.76957e20	−0.544637	−0.272318	−	0.962207i	$-0.587790\pi$
$283$	−3.76957e20	−0.544637	−0.272318	+	0.962207i	$0.587790\pi$
$284$	−3.54859e19	−0.0497565
$285$	0	0
$286$	−1.09816e20	−0.145062
$287$	1.46802e21	1.88250
$288$	0	0
$289$	2.16376e21	2.61564
$290$	5.12259e20	0.601321
$291$	0	0
$292$	−4.69794e20	−0.520180
$293$	3.57350e20	0.384344	0.192172	−	0.981361i	$-0.438447\pi$
$293$	3.57350e20	0.384344	0.192172	+	0.981361i	$0.438447\pi$
$294$	0	0
$295$	−1.16518e20	−0.118279
$296$	−1.14360e20	−0.112796
$297$	0	0
$298$	3.35589e20	0.312586
$299$	5.96757e20	0.540247
$300$	0	0
$301$	1.92824e21	1.64947
$302$	−7.07963e20	−0.588778
$303$	0	0
$304$	2.95900e18	0.00232659
$305$	−3.32275e20	−0.254067
$306$	0	0
$307$	2.25658e21	1.63220	0.816102	−	0.577908i	$-0.196131\pi$
$307$	2.25658e21	1.63220	0.816102	+	0.577908i	$0.196131\pi$
$308$	4.69182e20	0.330112
$309$	0	0
$310$	7.31204e18	0.00486927
$311$	−5.76844e20	−0.373762	−0.186881	−	0.982383i	$-0.559838\pi$
$311$	−5.76844e20	−0.373762	−0.186881	+	0.982383i	$0.559838\pi$
$312$	0	0
$313$	2.63218e21	1.61506	0.807530	−	0.589827i	$-0.200804\pi$
$313$	2.63218e21	1.61506	0.807530	+	0.589827i	$0.200804\pi$
$314$	−9.75961e20	−0.582815
$315$	0	0
$316$	3.57415e20	0.202224
$317$	9.95998e20	0.548599	0.274299	−	0.961644i	$-0.411554\pi$
$317$	9.95998e20	0.548599	0.274299	+	0.961644i	$0.411554\pi$
$318$	0	0
$319$	−1.43990e21	−0.751817
$320$	−1.09951e20	−0.0559017
$321$	0	0
$322$	−2.54961e21	−1.22942
$323$	−3.76786e19	−0.0176959
$324$	0	0
$325$	−2.32864e20	−0.103775
$326$	2.65660e21	1.15339
$327$	0	0
$328$	9.67016e20	0.398571
$329$	1.17249e21	0.470915
$330$	0	0
$331$	−3.56520e21	−1.36002	−0.680011	−	0.733202i	$-0.738025\pi$
$331$	−3.56520e21	−1.36002	−0.680011	+	0.733202i	$0.738025\pi$
$332$	2.39025e20	0.0888728
$333$	0	0
$334$	−3.12114e21	−1.10273
$335$	2.39440e21	0.824738
$336$	0	0
$337$	3.65283e20	0.119612	0.0598060	−	0.998210i	$-0.480952\pi$
$337$	3.65283e20	0.119612	0.0598060	+	0.998210i	$0.480952\pi$
$338$	−1.61829e21	−0.516731
$339$	0	0
$340$	1.40007e21	0.425185
$341$	−2.05533e19	−0.00608793
$342$	0	0
$343$	4.67180e21	1.31669
$344$	1.27017e21	0.349233
$345$	0	0
$346$	9.93818e20	0.260112
$347$	2.06934e21	0.528483	0.264242	−	0.964457i	$-0.414878\pi$
$347$	2.06934e21	0.528483	0.264242	+	0.964457i	$0.414878\pi$
$348$	0	0
$349$	3.13803e20	0.0763204	0.0381602	−	0.999272i	$-0.487850\pi$
$349$	3.13803e20	0.0763204	0.0381602	+	0.999272i	$0.487850\pi$
$350$	9.94897e20	0.236157
$351$	0	0
$352$	3.09060e20	0.0698925
$353$	6.87901e21	1.51859	0.759296	−	0.650746i	$-0.225544\pi$
$353$	6.87901e21	1.51859	0.759296	+	0.650746i	$0.225544\pi$
$354$	0	0
$355$	2.11513e20	0.0445036
$356$	−4.46820e19	−0.00917925
$357$	0	0
$358$	4.15625e21	0.814134
$359$	9.50907e21	1.81901	0.909505	−	0.415693i	$-0.136461\pi$
$359$	9.50907e21	1.81901	0.909505	+	0.415693i	$0.136461\pi$
$360$	0	0
$361$	−5.47991e21	−0.999913
$362$	−5.94420e21	−1.05943
$363$	0	0
$364$	−2.54730e21	−0.433230
$365$	2.80019e21	0.465263
$366$	0	0
$367$	1.88008e20	0.0298205	0.0149103	−	0.999889i	$-0.495254\pi$
$367$	1.88008e20	0.0298205	0.0149103	+	0.999889i	$0.495254\pi$
$368$	−1.67948e21	−0.260297
$369$	0	0
$370$	6.81637e20	0.100888
$371$	−1.67655e22	−2.42515
$372$	0	0
$373$	9.66745e20	0.133594	0.0667970	−	0.997767i	$-0.478722\pi$
$373$	9.66745e20	0.133594	0.0667970	+	0.997767i	$0.478722\pi$
$374$	−3.93543e21	−0.531598
$375$	0	0
$376$	7.72342e20	0.0997040
$377$	7.81756e21	0.986665
$378$	0	0
$379$	1.09478e22	1.32097	0.660484	−	0.750840i	$-0.270351\pi$
$379$	1.09478e22	1.32097	0.660484	+	0.750840i	$0.270351\pi$
$380$	−1.76370e19	−0.00208097
$381$	0	0
$382$	−1.14670e22	−1.29393
$383$	9.48135e21	1.04636	0.523179	−	0.852223i	$-0.324746\pi$
$383$	9.48135e21	1.04636	0.523179	+	0.852223i	$0.324746\pi$
$384$	0	0
$385$	−2.79655e21	−0.295261
$386$	9.24554e21	0.954861
$387$	0	0
$388$	−7.88875e21	−0.779720
$389$	−8.68048e21	−0.839406	−0.419703	−	0.907661i	$-0.637866\pi$
$389$	−8.68048e21	−0.839406	−0.419703	+	0.907661i	$0.637866\pi$
$390$	0	0
$391$	2.13857e22	1.97980
$392$	6.98030e21	0.632328
$393$	0	0
$394$	4.62088e21	0.400873
$395$	−2.13036e21	−0.180875
$396$	0	0
$397$	−5.61938e21	−0.457055	−0.228528	−	0.973537i	$-0.573391\pi$
$397$	−5.61938e21	−0.457055	−0.228528	+	0.973537i	$0.573391\pi$
$398$	−1.63747e21	−0.130367
$399$	0	0
$400$	6.55360e20	0.0500000
$401$	4.97857e21	0.371858	0.185929	−	0.982563i	$-0.440470\pi$
$401$	4.97857e21	0.371858	0.185929	+	0.982563i	$0.440470\pi$
$402$	0	0
$403$	1.11589e20	0.00798965
$404$	−5.07309e21	−0.355657
$405$	0	0
$406$	−3.34001e22	−2.24531
$407$	−1.91600e21	−0.126137
$408$	0	0
$409$	2.65693e20	0.0167777	0.00838883	−	0.999965i	$-0.497330\pi$
$409$	2.65693e20	0.0167777	0.00838883	+	0.999965i	$0.497330\pi$
$410$	−5.76386e21	−0.356493
$411$	0	0
$412$	−9.12977e21	−0.541793
$413$	7.59717e21	0.441648
$414$	0	0
$415$	−1.42470e21	−0.0794903
$416$	−1.67796e21	−0.0917252
$417$	0	0
$418$	4.95757e19	0.00260178
$419$	−1.71242e22	−0.880626	−0.440313	−	0.897844i	$-0.645133\pi$
$419$	−1.71242e22	−0.880626	−0.440313	+	0.897844i	$0.645133\pi$
$420$	0	0
$421$	2.71839e22	1.34250	0.671249	−	0.741231i	$-0.265758\pi$
$421$	2.71839e22	1.34250	0.671249	+	0.741231i	$0.265758\pi$
$422$	1.97658e21	0.0956662
$423$	0	0
$424$	−1.10438e22	−0.513462
$425$	−8.34504e21	−0.380297
$426$	0	0
$427$	2.16648e22	0.948677
$428$	−1.80658e22	−0.775505
$429$	0	0
$430$	−7.57081e21	−0.312364
$431$	2.34759e22	0.949656	0.474828	−	0.880079i	$-0.342510\pi$
$431$	2.34759e22	0.949656	0.474828	+	0.880079i	$0.342510\pi$
$432$	0	0
$433$	−1.88732e22	−0.734003	−0.367002	−	0.930220i	$-0.619616\pi$
$433$	−1.88732e22	−0.734003	−0.367002	+	0.930220i	$0.619616\pi$
$434$	−4.76757e20	−0.0181817
$435$	0	0
$436$	9.01446e21	0.330601
$437$	−2.69402e20	−0.00968968
$438$	0	0
$439$	5.51895e22	1.90945	0.954724	−	0.297494i	$-0.0961509\pi$
$439$	5.51895e22	1.90945	0.954724	+	0.297494i	$0.0961509\pi$
$440$	−1.84214e21	−0.0625138
$441$	0	0
$442$	2.13664e22	0.697656
$443$	1.20735e22	0.386723	0.193362	−	0.981128i	$-0.438061\pi$
$443$	1.20735e22	0.386723	0.193362	+	0.981128i	$0.438061\pi$
$444$	0	0
$445$	2.66325e20	0.00821017
$446$	−2.77984e22	−0.840764
$447$	0	0
$448$	7.16899e21	0.208735
$449$	−2.57466e22	−0.735573	−0.367787	−	0.929910i	$-0.619884\pi$
$449$	−2.57466e22	−0.735573	−0.367787	+	0.929910i	$0.619884\pi$
$450$	0	0
$451$	1.62016e22	0.445714
$452$	2.55760e22	0.690486
$453$	0	0
$454$	−5.00580e21	−0.130166
$455$	1.51831e22	0.387493
$456$	0	0
$457$	−1.01069e22	−0.248503	−0.124252	−	0.992251i	$-0.539653\pi$
$457$	−1.01069e22	−0.248503	−0.124252	+	0.992251i	$0.539653\pi$
$458$	3.08739e22	0.745136
$459$	0	0
$460$	1.00105e22	0.232817
$461$	−1.68599e21	−0.0384942	−0.0192471	−	0.999815i	$-0.506127\pi$
$461$	−1.68599e21	−0.0384942	−0.0192471	+	0.999815i	$0.506127\pi$
$462$	0	0
$463$	−2.77945e22	−0.611674	−0.305837	−	0.952084i	$-0.598936\pi$
$463$	−2.77945e22	−0.611674	−0.305837	+	0.952084i	$0.598936\pi$
$464$	−2.20013e22	−0.475386
$465$	0	0
$466$	−1.84419e22	−0.384172
$467$	−8.26643e22	−1.69093	−0.845463	−	0.534034i	$-0.820676\pi$
$467$	−8.26643e22	−1.69093	−0.845463	+	0.534034i	$0.820676\pi$
$468$	0	0
$469$	−1.56119e23	−3.07954
$470$	−4.60351e21	−0.0891780
$471$	0	0
$472$	5.00442e21	0.0935075
$473$	2.12807e22	0.390541
$474$	0	0
$475$	1.05125e20	0.00186127
$476$	−9.12865e22	−1.58762
$477$	0	0
$478$	−4.36287e22	−0.732209
$479$	−2.59191e22	−0.427335	−0.213667	−	0.976906i	$-0.568541\pi$
$479$	−2.59191e22	−0.427335	−0.213667	+	0.976906i	$0.568541\pi$
$480$	0	0
$481$	1.04024e22	0.165540
$482$	−4.38091e22	−0.684960
$483$	0	0
$484$	−2.79469e22	−0.421841
$485$	4.70206e22	0.697403
$486$	0	0
$487$	−4.14828e22	−0.594116	−0.297058	−	0.954859i	$-0.596006\pi$
$487$	−4.14828e22	−0.594116	−0.297058	+	0.954859i	$0.596006\pi$
$488$	1.42711e22	0.200858
$489$	0	0
$490$	−4.16059e22	−0.565572
$491$	−3.08644e22	−0.412349	−0.206174	−	0.978515i	$-0.566101\pi$
$491$	−3.08644e22	−0.412349	−0.206174	+	0.978515i	$0.566101\pi$
$492$	0	0
$493$	2.80155e23	3.61575
$494$	−2.69158e20	−0.00341451
$495$	0	0
$496$	−3.14050e20	−0.00384950
$497$	−1.37910e22	−0.166175
$498$	0	0
$499$	3.24182e22	0.377516	0.188758	−	0.982024i	$-0.439554\pi$
$499$	3.24182e22	0.377516	0.188758	+	0.982024i	$0.439554\pi$
$500$	−3.90625e21	−0.0447214
$501$	0	0
$502$	−4.75728e22	−0.526474
$503$	−6.39324e22	−0.695654	−0.347827	−	0.937559i	$-0.613080\pi$
$503$	−6.39324e22	−0.695654	−0.347827	+	0.937559i	$0.613080\pi$
$504$	0	0
$505$	3.02380e22	0.318109
$506$	−2.81384e22	−0.291085
$507$	0	0
$508$	−4.83540e22	−0.483717
$509$	9.33355e22	0.918218	0.459109	−	0.888380i	$-0.348169\pi$
$509$	9.33355e22	0.918218	0.459109	+	0.888380i	$0.348169\pi$
$510$	0	0
$511$	−1.82577e23	−1.73728
$512$	4.72237e21	0.0441942
$513$	0	0
$514$	−4.30778e21	−0.0390003
$515$	5.44176e22	0.484594
$516$	0	0
$517$	1.29400e22	0.111497
$518$	−4.44438e22	−0.376711
$519$	0	0
$520$	1.00014e22	0.0820415
$521$	−3.30201e22	−0.266476	−0.133238	−	0.991084i	$-0.542537\pi$
$521$	−3.30201e22	−0.266476	−0.133238	+	0.991084i	$0.542537\pi$
$522$	0	0
$523$	4.06002e22	0.317149	0.158575	−	0.987347i	$-0.449310\pi$
$523$	4.06002e22	0.317149	0.158575	+	0.987347i	$0.449310\pi$
$524$	9.52948e22	0.732409
$525$	0	0
$526$	−7.09807e22	−0.528155
$527$	3.99896e21	0.0292790
$528$	0	0
$529$	1.18585e22	0.0840729
$530$	6.58261e22	0.459255
$531$	0	0
$532$	1.14996e21	0.00777025
$533$	−8.79621e22	−0.584944
$534$	0	0
$535$	1.07680e23	0.693633
$536$	−1.02839e23	−0.652013
$537$	0	0
$538$	−2.40844e22	−0.147941
$539$	1.16949e23	0.707121
$540$	0	0
$541$	−4.20460e22	−0.246347	−0.123174	−	0.992385i	$-0.539307\pi$
$541$	−4.20460e22	−0.246347	−0.123174	+	0.992385i	$0.539307\pi$
$542$	−1.58113e23	−0.911954
$543$	0	0
$544$	−6.01324e22	−0.336138
$545$	−5.37303e22	−0.295699
$546$	0	0
$547$	−2.01110e23	−1.07285	−0.536427	−	0.843947i	$-0.680226\pi$
$547$	−2.01110e23	−1.07285	−0.536427	+	0.843947i	$0.680226\pi$
$548$	1.49117e23	0.783236
$549$	0	0
$550$	1.09800e22	0.0559140
$551$	−3.52919e21	−0.0176965
$552$	0	0
$553$	1.38903e23	0.675380
$554$	−1.70973e23	−0.818645
$555$	0	0
$556$	9.27976e21	0.0430924
$557$	−1.46727e23	−0.671030	−0.335515	−	0.942035i	$-0.608910\pi$
$557$	−1.46727e23	−0.671030	−0.335515	+	0.942035i	$0.608910\pi$
$558$	0	0
$559$	−1.15538e23	−0.512536
$560$	−4.27305e22	−0.186698
$561$	0	0
$562$	2.90919e22	0.123314
$563$	3.39706e23	1.41834	0.709172	−	0.705036i	$-0.249069\pi$
$563$	3.39706e23	1.41834	0.709172	+	0.705036i	$0.249069\pi$
$564$	0	0
$565$	−1.52445e23	−0.617590
$566$	−9.65010e22	−0.385116
$567$	0	0
$568$	−9.08439e21	−0.0351832
$569$	−2.72602e23	−1.04010	−0.520050	−	0.854136i	$-0.674087\pi$
$569$	−2.72602e23	−1.04010	−0.520050	+	0.854136i	$0.674087\pi$
$570$	0	0
$571$	−2.79882e22	−0.103650	−0.0518249	−	0.998656i	$-0.516504\pi$
$571$	−2.79882e22	−0.103650	−0.0518249	+	0.998656i	$0.516504\pi$
$572$	−2.81129e22	−0.102574
$573$	0	0
$574$	3.75813e23	1.33113
$575$	−5.96672e22	−0.208238
$576$	0	0
$577$	3.86678e23	1.31025	0.655127	−	0.755519i	$-0.272615\pi$
$577$	3.86678e23	1.31025	0.655127	+	0.755519i	$0.272615\pi$
$578$	5.53923e23	1.84954
$579$	0	0
$580$	1.31138e23	0.425198
$581$	9.28926e22	0.296814
$582$	0	0
$583$	−1.85030e23	−0.574195
$584$	−1.20267e23	−0.367823
$585$	0	0
$586$	9.14817e22	0.271772
$587$	−3.64152e23	−1.06625	−0.533125	−	0.846036i	$-0.678983\pi$
$587$	−3.64152e23	−1.06625	−0.533125	+	0.846036i	$0.678983\pi$
$588$	0	0
$589$	−5.03761e19	−0.000143299 0
$590$	−2.98287e22	−0.0836357
$591$	0	0
$592$	−2.92761e22	−0.0797587
$593$	6.86689e23	1.84415	0.922073	−	0.387017i	$-0.126494\pi$
$593$	6.86689e23	1.84415	0.922073	+	0.387017i	$0.126494\pi$
$594$	0	0
$595$	5.44110e23	1.42001
$596$	8.59108e22	0.221032
$597$	0	0
$598$	1.52770e23	0.382013
$599$	3.82850e23	0.943846	0.471923	−	0.881640i	$-0.343560\pi$
$599$	3.82850e23	0.943846	0.471923	+	0.881640i	$0.343560\pi$
$600$	0	0
$601$	−2.66402e23	−0.638417	−0.319209	−	0.947684i	$-0.603417\pi$
$601$	−2.66402e23	−0.638417	−0.319209	+	0.947684i	$0.603417\pi$
$602$	4.93629e23	1.16635
$603$	0	0
$604$	−1.81239e23	−0.416329
$605$	1.66577e23	0.377306
$606$	0	0
$607$	−1.11277e23	−0.245076	−0.122538	−	0.992464i	$-0.539103\pi$
$607$	−1.11277e23	−0.245076	−0.122538	+	0.992464i	$0.539103\pi$
$608$	7.57505e20	0.00164515
$609$	0	0
$610$	−8.50623e22	−0.179653
$611$	−7.02541e22	−0.146326
$612$	0	0
$613$	−3.91229e23	−0.792532	−0.396266	−	0.918136i	$-0.629694\pi$
$613$	−3.91229e23	−0.792532	−0.396266	+	0.918136i	$0.629694\pi$
$614$	5.77684e23	1.15414
$615$	0	0
$616$	1.20111e23	0.233424
$617$	5.72626e23	1.09761	0.548804	−	0.835951i	$-0.315083\pi$
$617$	5.72626e23	1.09761	0.548804	+	0.835951i	$0.315083\pi$
$618$	0	0
$619$	7.86155e23	1.46601	0.733007	−	0.680222i	$-0.238116\pi$
$619$	7.86155e23	1.46601	0.733007	+	0.680222i	$0.238116\pi$
$620$	1.87188e21	0.00344310
$621$	0	0
$622$	−1.47672e23	−0.264290
$623$	−1.73648e22	−0.0306565
$624$	0	0
$625$	2.32831e22	0.0400000
$626$	6.73838e23	1.14202
$627$	0	0
$628$	−2.49846e23	−0.412112
$629$	3.72788e23	0.606640
$630$	0	0
$631$	6.52712e23	1.03389	0.516943	−	0.856020i	$-0.327070\pi$
$631$	6.52712e23	1.03389	0.516943	+	0.856020i	$0.327070\pi$
$632$	9.14983e22	0.142994
$633$	0	0
$634$	2.54975e23	0.387918
$635$	2.88212e23	0.432649
$636$	0	0
$637$	−6.34945e23	−0.928007
$638$	−3.68615e23	−0.531615
$639$	0	0
$640$	−2.81475e22	−0.0395285
$641$	−7.10734e23	−0.984950	−0.492475	−	0.870327i	$-0.663908\pi$
$641$	−7.10734e23	−0.984950	−0.492475	+	0.870327i	$0.663908\pi$
$642$	0	0
$643$	2.34278e23	0.316183	0.158092	−	0.987424i	$-0.449466\pi$
$643$	2.34278e23	0.316183	0.158092	+	0.987424i	$0.449466\pi$
$644$	−6.52701e23	−0.869329
$645$	0	0
$646$	−9.64571e21	−0.0125129
$647$	−1.50806e23	−0.193078	−0.0965391	−	0.995329i	$-0.530777\pi$
$647$	−1.50806e23	−0.193078	−0.0965391	+	0.995329i	$0.530777\pi$
$648$	0	0
$649$	8.38450e22	0.104568
$650$	−5.96131e22	−0.0733801
$651$	0	0
$652$	6.80089e23	0.815570
$653$	−6.74375e22	−0.0798251	−0.0399126	−	0.999203i	$-0.512708\pi$
$653$	−6.74375e22	−0.0798251	−0.0399126	+	0.999203i	$0.512708\pi$
$654$	0	0
$655$	−5.68001e23	−0.655086
$656$	2.47556e23	0.281832
$657$	0	0
$658$	3.00157e23	0.332987
$659$	−7.11301e22	−0.0778981	−0.0389491	−	0.999241i	$-0.512401\pi$
$659$	−7.11301e22	−0.0778981	−0.0389491	+	0.999241i	$0.512401\pi$
$660$	0	0
$661$	1.69805e24	1.81233	0.906164	−	0.422926i	$-0.138997\pi$
$661$	1.69805e24	1.81233	0.906164	+	0.422926i	$0.138997\pi$
$662$	−9.12691e23	−0.961680
$663$	0	0
$664$	6.11903e22	0.0628426
$665$	−6.85431e21	−0.00694992
$666$	0	0
$667$	2.00311e24	1.97986
$668$	−7.99012e23	−0.779747
$669$	0	0
$670$	6.12967e23	0.583178
$671$	2.39101e23	0.224615
$672$	0	0
$673$	−1.39848e24	−1.28094	−0.640469	−	0.767984i	$-0.721260\pi$
$673$	−1.39848e24	−1.28094	−0.640469	+	0.767984i	$0.721260\pi$
$674$	9.35124e22	0.0845785
$675$	0	0
$676$	−4.14282e23	−0.365384
$677$	9.38639e23	0.817514	0.408757	−	0.912643i	$-0.365962\pi$
$677$	9.38639e23	0.817514	0.408757	+	0.912643i	$0.365962\pi$
$678$	0	0
$679$	−3.06582e24	−2.60408
$680$	3.58417e23	0.300651
$681$	0	0
$682$	−5.26165e21	−0.00430482
$683$	−1.23254e24	−0.995922	−0.497961	−	0.867199i	$-0.665918\pi$
$683$	−1.23254e24	−0.995922	−0.497961	+	0.867199i	$0.665918\pi$
$684$	0	0
$685$	−8.88807e23	−0.700547
$686$	1.19598e24	0.931041
$687$	0	0
$688$	3.25164e23	0.246945
$689$	1.00457e24	0.753559
$690$	0	0
$691$	3.83055e23	0.280348	0.140174	−	0.990127i	$-0.455234\pi$
$691$	3.83055e23	0.280348	0.140174	+	0.990127i	$0.455234\pi$
$692$	2.54417e23	0.183927
$693$	0	0
$694$	5.29751e23	0.373694
$695$	−5.53117e22	−0.0385431
$696$	0	0
$697$	−3.15226e24	−2.14360
$698$	8.03335e22	0.0539667
$699$	0	0
$700$	2.54694e23	0.166988
$701$	−1.63235e24	−1.05733	−0.528665	−	0.848831i	$-0.677307\pi$
$701$	−1.63235e24	−1.05733	−0.528665	+	0.848831i	$0.677307\pi$
$702$	0	0
$703$	−4.69611e21	−0.00296906
$704$	7.91195e22	0.0494215
$705$	0	0
$706$	1.76103e24	1.07381
$707$	−1.97156e24	−1.18781
$708$	0	0
$709$	−6.09431e23	−0.358453	−0.179226	−	0.983808i	$-0.557359\pi$
$709$	−6.09431e23	−0.358453	−0.179226	+	0.983808i	$0.557359\pi$
$710$	5.41472e22	0.0314688
$711$	0	0
$712$	−1.14386e22	−0.00649071
$713$	2.85926e22	0.0160322
$714$	0	0
$715$	1.67566e23	0.0917454
$716$	1.06400e24	0.575680
$717$	0	0
$718$	2.43432e24	1.28623
$719$	4.49148e23	0.234527	0.117264	−	0.993101i	$-0.462588\pi$
$719$	4.49148e23	0.234527	0.117264	+	0.993101i	$0.462588\pi$
$720$	0	0
$721$	−3.54812e24	−1.80946
$722$	−1.40286e24	−0.707046
$723$	0	0
$724$	−1.52171e24	−0.749127
$725$	−7.81644e23	−0.380309
$726$	0	0
$727$	−5.00713e23	−0.237983	−0.118992	−	0.992895i	$-0.537966\pi$
$727$	−5.00713e23	−0.237983	−0.118992	+	0.992895i	$0.537966\pi$
$728$	−6.52109e23	−0.306340
$729$	0	0
$730$	7.16849e23	0.328991
$731$	−4.14048e24	−1.87825
$732$	0	0
$733$	1.20329e24	0.533320	0.266660	−	0.963791i	$-0.414080\pi$
$733$	1.20329e24	0.533320	0.266660	+	0.963791i	$0.414080\pi$
$734$	4.81301e22	0.0210863
$735$	0	0
$736$	−4.29948e23	−0.184058
$737$	−1.72298e24	−0.729133
$738$	0	0
$739$	−2.19426e24	−0.907426	−0.453713	−	0.891148i	$-0.649901\pi$
$739$	−2.19426e24	−0.907426	−0.453713	+	0.891148i	$0.649901\pi$
$740$	1.74499e23	0.0713384
$741$	0	0
$742$	−4.29197e24	−1.71484
$743$	9.89633e22	0.0390903	0.0195452	−	0.999809i	$-0.493778\pi$
$743$	9.89633e22	0.0390903	0.0195452	+	0.999809i	$0.493778\pi$
$744$	0	0
$745$	−5.12068e23	−0.197697
$746$	2.47487e23	0.0944652
$747$	0	0
$748$	−1.00747e24	−0.375897
$749$	−7.02094e24	−2.59000
$750$	0	0
$751$	−1.01856e24	−0.367322	−0.183661	−	0.982990i	$-0.558795\pi$
$751$	−1.01856e24	−0.367322	−0.183661	+	0.982990i	$0.558795\pi$
$752$	1.97719e23	0.0705014
$753$	0	0
$754$	2.00130e24	0.697677
$755$	1.08027e24	0.372376
$756$	0	0
$757$	−1.43989e24	−0.485305	−0.242653	−	0.970113i	$-0.578018\pi$
$757$	−1.43989e24	−0.485305	−0.242653	+	0.970113i	$0.578018\pi$
$758$	2.80263e24	0.934066
$759$	0	0
$760$	−4.51508e21	−0.00147147
$761$	7.15424e23	0.230565	0.115283	−	0.993333i	$-0.463223\pi$
$761$	7.15424e23	0.230565	0.115283	+	0.993333i	$0.463223\pi$
$762$	0	0
$763$	3.50330e24	1.10413
$764$	−2.93554e24	−0.914944
$765$	0	0
$766$	2.42722e24	0.739887
$767$	−4.55214e23	−0.137232
$768$	0	0
$769$	1.87177e24	0.551922	0.275961	−	0.961169i	$-0.411004\pi$
$769$	1.87177e24	0.551922	0.275961	+	0.961169i	$0.411004\pi$
$770$	−7.15916e23	−0.208781
$771$	0	0
$772$	2.36686e24	0.675189
$773$	1.86999e24	0.527611	0.263805	−	0.964576i	$-0.415022\pi$
$773$	1.86999e24	0.527611	0.263805	+	0.964576i	$0.415022\pi$
$774$	0	0
$775$	−1.11573e22	−0.00307960
$776$	−2.01952e24	−0.551345
$777$	0	0
$778$	−2.22220e24	−0.593550
$779$	3.97099e22	0.0104913
$780$	0	0
$781$	−1.52202e23	−0.0393447
$782$	5.47475e24	1.39993
$783$	0	0
$784$	1.78696e24	0.447124
$785$	1.48920e24	0.368604
$786$	0	0
$787$	−3.29304e24	−0.797648	−0.398824	−	0.917027i	$-0.630582\pi$
$787$	−3.29304e24	−0.797648	−0.398824	+	0.917027i	$0.630582\pi$
$788$	1.18295e24	0.283460
$789$	0	0
$790$	−5.45373e23	−0.127898
$791$	9.93963e24	2.30606
$792$	0	0
$793$	−1.29813e24	−0.294779
$794$	−1.43856e24	−0.323187
$795$	0	0
$796$	−4.19192e23	−0.0921832
$797$	3.80089e24	0.826969	0.413485	−	0.910511i	$-0.364312\pi$
$797$	3.80089e24	0.826969	0.413485	+	0.910511i	$0.364312\pi$
$798$	0	0
$799$	−2.51767e24	−0.536229
$800$	1.67772e23	0.0353553
$801$	0	0
$802$	1.27451e24	0.262944
$803$	−2.01498e24	−0.411329
$804$	0	0
$805$	3.89040e24	0.777552
$806$	2.85667e22	0.00564953
$807$	0	0
$808$	−1.29871e24	−0.251487
$809$	−1.38713e24	−0.265799	−0.132900	−	0.991129i	$-0.542429\pi$
$809$	−1.38713e24	−0.265799	−0.132900	+	0.991129i	$0.542429\pi$
$810$	0	0
$811$	2.43418e24	0.456747	0.228374	−	0.973574i	$-0.426659\pi$
$811$	2.43418e24	0.456747	0.228374	+	0.973574i	$0.426659\pi$
$812$	−8.55042e24	−1.58767
$813$	0	0
$814$	−4.90497e23	−0.0891926
$815$	−4.05365e24	−0.729468
$816$	0	0
$817$	5.21589e22	0.00919265
$818$	6.80174e22	0.0118636
$819$	0	0
$820$	−1.47555e24	−0.252078
$821$	7.07675e24	1.19651	0.598256	−	0.801305i	$-0.295860\pi$
$821$	7.07675e24	1.19651	0.598256	+	0.801305i	$0.295860\pi$
$822$	0	0
$823$	−5.31425e24	−0.880124	−0.440062	−	0.897968i	$-0.645043\pi$
$823$	−5.31425e24	−0.880124	−0.440062	+	0.897968i	$0.645043\pi$
$824$	−2.33722e24	−0.383105
$825$	0	0
$826$	1.94488e24	0.312292
$827$	−5.13487e24	−0.816080	−0.408040	−	0.912964i	$-0.633788\pi$
$827$	−5.13487e24	−0.816080	−0.408040	+	0.912964i	$0.633788\pi$
$828$	0	0
$829$	1.06738e25	1.66191	0.830954	−	0.556341i	$-0.187795\pi$
$829$	1.06738e25	1.66191	0.830954	+	0.556341i	$0.187795\pi$
$830$	−3.64723e23	−0.0562081
$831$	0	0
$832$	−4.29558e23	−0.0648595
$833$	−2.27543e25	−3.40079
$834$	0	0
$835$	4.76248e24	0.697427
$836$	1.26914e22	0.00183974
$837$	0	0
$838$	−4.38380e24	−0.622697
$839$	6.47302e24	0.910186	0.455093	−	0.890444i	$-0.349606\pi$
$839$	6.47302e24	0.910186	0.455093	+	0.890444i	$0.349606\pi$
$840$	0	0
$841$	1.89837e25	2.61587
$842$	6.95908e24	0.949290
$843$	0	0
$844$	5.06004e23	0.0676462
$845$	2.46931e24	0.326809
$846$	0	0
$847$	−1.08611e25	−1.40885
$848$	−2.82721e24	−0.363073
$849$	0	0
$850$	−2.13633e24	−0.268910
$851$	2.66544e24	0.332175
$852$	0	0
$853$	2.69306e24	0.328988	0.164494	−	0.986378i	$-0.447401\pi$
$853$	2.69306e24	0.328988	0.164494	+	0.986378i	$0.447401\pi$
$854$	5.54620e24	0.670816
$855$	0	0
$856$	−4.62484e24	−0.548365
$857$	1.86785e24	0.219283	0.109641	−	0.993971i	$-0.465030\pi$
$857$	1.86785e24	0.219283	0.109641	+	0.993971i	$0.465030\pi$
$858$	0	0
$859$	−1.13124e25	−1.30200	−0.651000	−	0.759077i	$-0.725650\pi$
$859$	−1.13124e25	−1.30200	−0.651000	+	0.759077i	$0.725650\pi$
$860$	−1.93813e24	−0.220875
$861$	0	0
$862$	6.00984e24	0.671508
$863$	2.37061e24	0.262282	0.131141	−	0.991364i	$-0.458136\pi$
$863$	2.37061e24	0.262282	0.131141	+	0.991364i	$0.458136\pi$
$864$	0	0
$865$	−1.51645e24	−0.164509
$866$	−4.83153e24	−0.519019
$867$	0	0
$868$	−1.22050e23	−0.0128564
$869$	1.53298e24	0.159907
$870$	0	0
$871$	9.35446e24	0.956895
$872$	2.30770e24	0.233770
$873$	0	0
$874$	−6.89670e22	−0.00685164
$875$	−1.51809e24	−0.149358
$876$	0	0
$877$	−6.79534e24	−0.655714	−0.327857	−	0.944727i	$-0.606326\pi$
$877$	−6.79534e24	−0.655714	−0.327857	+	0.944727i	$0.606326\pi$
$878$	1.41285e25	1.35018
$879$	0	0
$880$	−4.71589e23	−0.0442039
$881$	1.88402e25	1.74900	0.874501	−	0.485024i	$-0.161189\pi$
$881$	1.88402e25	1.74900	0.874501	+	0.485024i	$0.161189\pi$
$882$	0	0
$883$	3.85491e24	0.351033	0.175517	−	0.984476i	$-0.443840\pi$
$883$	3.85491e24	0.351033	0.175517	+	0.984476i	$0.443840\pi$
$884$	5.46979e24	0.493317
$885$	0	0
$886$	3.09081e24	0.273455
$887$	−8.32727e24	−0.729712	−0.364856	−	0.931064i	$-0.618882\pi$
$887$	−8.32727e24	−0.729712	−0.364856	+	0.931064i	$0.618882\pi$
$888$	0	0
$889$	−1.87919e25	−1.61550
$890$	6.81792e22	0.00580547
$891$	0	0
$892$	−7.11640e24	−0.594510
$893$	3.17158e22	0.00262445
$894$	0	0
$895$	−6.34194e24	−0.514904
$896$	1.83526e24	0.147598
$897$	0	0
$898$	−6.59113e24	−0.520129
$899$	3.74565e23	0.0292799
$900$	0	0
$901$	3.60004e25	2.76151
$902$	4.14760e24	0.315168
$903$	0	0
$904$	6.54744e24	0.488248
$905$	9.07012e24	0.670039
$906$	0	0
$907$	1.92435e25	1.39515	0.697576	−	0.716511i	$-0.254262\pi$
$907$	1.92435e25	1.39515	0.697576	+	0.716511i	$0.254262\pi$
$908$	−1.28148e24	−0.0920415
$909$	0	0
$910$	3.88687e24	0.273999
$911$	1.57682e25	1.10123	0.550613	−	0.834761i	$-0.314394\pi$
$911$	1.57682e25	1.10123	0.550613	+	0.834761i	$0.314394\pi$
$912$	0	0
$913$	1.02519e24	0.0702756
$914$	−2.58738e24	−0.175718
$915$	0	0
$916$	7.90373e24	0.526890
$917$	3.70346e25	2.44607
$918$	0	0
$919$	1.03058e25	0.668189	0.334094	−	0.942540i	$-0.391570\pi$
$919$	1.03058e25	0.668189	0.334094	+	0.942540i	$0.391570\pi$
$920$	2.56269e24	0.164626
$921$	0	0
$922$	−4.31612e23	−0.0272195
$923$	8.26339e23	0.0516349
$924$	0	0
$925$	−1.04010e24	−0.0638070
$926$	−7.11539e24	−0.432519
$927$	0	0
$928$	−5.63234e24	−0.336149
$929$	−9.73688e24	−0.575819	−0.287910	−	0.957658i	$-0.592960\pi$
$929$	−9.73688e24	−0.575819	−0.287910	+	0.957658i	$0.592960\pi$
$930$	0	0
$931$	2.86642e23	0.0166444
$932$	−4.72113e24	−0.271651
$933$	0	0
$934$	−2.11621e25	−1.19567
$935$	6.00499e24	0.336212
$936$	0	0
$937$	−3.22145e25	−1.77119	−0.885594	−	0.464459i	$-0.846249\pi$
$937$	−3.22145e25	−1.77119	−0.885594	+	0.464459i	$0.846249\pi$
$938$	−3.99664e25	−2.17756
$939$	0	0
$940$	−1.17850e24	−0.0630584
$941$	7.90374e24	0.419103	0.209552	−	0.977798i	$-0.432800\pi$
$941$	7.90374e24	0.419103	0.209552	+	0.977798i	$0.432800\pi$
$942$	0	0
$943$	−2.25387e25	−1.17376
$944$	1.28113e24	0.0661198
$945$	0	0
$946$	5.44786e24	0.276154
$947$	−2.36682e25	−1.18902	−0.594512	−	0.804086i	$-0.702655\pi$
$947$	−2.36682e25	−1.18902	−0.594512	+	0.804086i	$0.702655\pi$
$948$	0	0
$949$	1.09398e25	0.539818
$950$	2.69120e22	0.00131612
$951$	0	0
$952$	−2.33693e25	−1.12262
$953$	−9.09477e24	−0.433014	−0.216507	−	0.976281i	$-0.569466\pi$
$953$	−9.09477e24	−0.433014	−0.216507	+	0.976281i	$0.569466\pi$
$954$	0	0
$955$	1.74972e25	0.818351
$956$	−1.11690e25	−0.517750
$957$	0	0
$958$	−6.63529e24	−0.302171
$959$	5.79516e25	2.61582
$960$	0	0
$961$	−2.25448e25	−0.999763
$962$	2.66302e24	0.117054
$963$	0	0
$964$	−1.12151e25	−0.484340
$965$	−1.41076e25	−0.603907
$966$	0	0
$967$	3.62970e25	1.52667	0.763337	−	0.646001i	$-0.223560\pi$
$967$	3.62970e25	1.52667	0.763337	+	0.646001i	$0.223560\pi$
$968$	−7.15441e24	−0.298286
$969$	0	0
$970$	1.20373e25	0.493138
$971$	9.70396e24	0.394081	0.197040	−	0.980395i	$-0.436867\pi$
$971$	9.70396e24	0.394081	0.197040	+	0.980395i	$0.436867\pi$
$972$	0	0
$973$	3.60641e24	0.143918
$974$	−1.06196e25	−0.420104
$975$	0	0
$976$	3.65340e24	0.142028
$977$	−3.70141e25	−1.42647	−0.713236	−	0.700924i	$-0.752771\pi$
$977$	−3.70141e25	−1.42647	−0.713236	+	0.700924i	$0.752771\pi$
$978$	0	0
$979$	−1.91644e23	−0.00725843
$980$	−1.06511e25	−0.399920
$981$	0	0
$982$	−7.90127e24	−0.291574
$983$	6.43184e24	0.235305	0.117652	−	0.993055i	$-0.462463\pi$
$983$	6.43184e24	0.235305	0.117652	+	0.993055i	$0.462463\pi$
$984$	0	0
$985$	−7.05090e24	−0.253534
$986$	7.17196e25	2.55672
$987$	0	0
$988$	−6.89045e22	−0.00241442
$989$	−2.96045e25	−1.02847
$990$	0	0
$991$	5.17070e25	1.76573	0.882863	−	0.469631i	$-0.155613\pi$
$991$	5.17070e25	1.76573	0.882863	+	0.469631i	$0.155613\pi$
$992$	−8.03967e22	−0.00272201
$993$	0	0
$994$	−3.53048e24	−0.117503
$995$	2.49858e24	0.0824512
$996$	0	0
$997$	−2.77457e25	−0.900093	−0.450047	−	0.893005i	$-0.648593\pi$
$997$	−2.77457e25	−0.900093	−0.450047	+	0.893005i	$0.648593\pi$
$998$	8.29907e24	0.266944
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.18.a.c.1.2	✓	2	3.2	odd	2
50.18.a.f.1.1		2	15.14	odd	2
50.18.b.d.49.1		4	15.2	even	4
50.18.b.d.49.4		4	15.8	even	4
80.18.a.d.1.1		2	12.11	even	2
90.18.a.k.1.2		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+256.000 q^{2} +65536.0 q^{4} -390625. q^{5} +2.54694e7 q^{7} +1.67772e7 q^{8} -1.00000e8 q^{10} +2.81089e8 q^{11} -1.52610e9 q^{13} +6.52016e9 q^{14} +4.29497e9 q^{16} -5.46901e10 q^{17} +6.88947e8 q^{19} -2.56000e10 q^{20} +7.19587e10 q^{22} -3.91035e11 q^{23} +1.52588e11 q^{25} -3.90681e11 q^{26} +1.66916e12 q^{28} -5.12259e12 q^{29} -7.31204e10 q^{31} +1.09951e12 q^{32} -1.40007e13 q^{34} -9.94897e12 q^{35} -6.81637e12 q^{37} +1.76370e11 q^{38} -6.55360e12 q^{40} +5.76386e13 q^{41} +7.57081e13 q^{43} +1.84214e13 q^{44} -1.00105e14 q^{46} +4.60351e13 q^{47} +4.16059e14 q^{49} +3.90625e13 q^{50} -1.00014e14 q^{52} -6.58261e14 q^{53} -1.09800e14 q^{55} +4.27305e14 q^{56} -1.31138e15 q^{58} +2.98287e14 q^{59} +8.50623e14 q^{61} -1.87188e13 q^{62} +2.81475e14 q^{64} +5.96131e14 q^{65} -6.12967e15 q^{67} -3.58417e15 q^{68} -2.54694e15 q^{70} -5.41472e14 q^{71} -7.16849e15 q^{73} -1.74499e15 q^{74} +4.51508e13 q^{76} +7.15916e15 q^{77} +5.45373e15 q^{79} -1.67772e15 q^{80} +1.47555e16 q^{82} +3.64723e15 q^{83} +2.13633e16 q^{85} +1.93813e16 q^{86} +4.71589e15 q^{88} -6.81792e14 q^{89} -3.88687e16 q^{91} -2.56269e16 q^{92} +1.17850e16 q^{94} -2.69120e14 q^{95} -1.20373e17 q^{97} +1.06511e17 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 512 q^{2} + 131072 q^{4} - 781250 q^{5} + 603844 q^{7} + 33554432 q^{8} - 200000000 q^{10} + 471481296 q^{11} - 1541834228 q^{13} + 154584064 q^{14} + 8589934592 q^{16} - 32139900564 q^{17} + 128672529400 q^{19}+ \cdots + 20\!\cdots\!24 q^{98}+O(q^{100}) \) 2 * q + 512 * q^2 + 131072 * q^4 - 781250 * q^5 + 603844 * q^7 + 33554432 * q^8 - 200000000 * q^10 + 471481296 * q^11 - 1541834228 * q^13 + 154584064 * q^14 + 8589934592 * q^16 - 32139900564 * q^17 + 128672529400 * q^19 - 51200000000 * q^20 + 120699211776 * q^22 - 650359859292 * q^23 + 305175781250 * q^25 - 394709562368 * q^26 + 39573520384 * q^28 - 2543054749980 * q^29 + 7839407998744 * q^31 + 2199023255552 * q^32 - 8227814544384 * q^34 - 235876562500 * q^35 - 27805209097556 * q^37 + 32940167526400 * q^38 - 13107200000000 * q^40 + 37826364264156 * q^41 + 29630453926852 * q^43 + 30898998214656 * q^44 - 166492123978752 * q^46 - 220791583022004 * q^47 + 801722584677954 * q^49 + 78125000000000 * q^50 - 101045647966208 * q^52 - 1058306017600572 * q^53 - 184172381250000 * q^55 + 10130821218304 * q^56 - 651022015994880 * q^58 - 1527820717235160 * q^59 - 631738213122116 * q^61 + 2006888447678464 * q^62 + 562949953421312 * q^64 + 602278995312500 * q^65 - 8843763493872596 * q^67 - 2106320523362304 * q^68 - 60384400000000 * q^70 + 3331901660912376 * q^71 - 10026350574292028 * q^73 - 7118133528974336 * q^74 + 8432682886758400 * q^76 + 2424946105073712 * q^77 + 8263606842055120 * q^79 - 3355443200000000 * q^80 + 9683549251623936 * q^82 + 10689526892336988 * q^83 + 12554648657812500 * q^85 + 7585396205274112 * q^86 + 7910143542951936 * q^88 - 55878412521904980 * q^89 - 38477396610603016 * q^91 - 42621983738560512 * q^92 - 56522645253633024 * q^94 - 50262706796875000 * q^95 - 56106580363859756 * q^97 + 205240981677556224 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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