LMFDB - Embedded newform 9801.2.a.cp.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9801,2,Mod(1,9801)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9801, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9801.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9801 = 3^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9801.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:	$78.2613790211$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 2 x^{17} - 22 x^{16} + 42 x^{15} + 198 x^{14} - 357 x^{13} - 944 x^{12} + 1579 x^{11} + \cdots - 55 $ x^18 - 2x^17 - 22x^16 + 42x^15 + 198x^14 - 357x^13 - 944x^12 + 1579x^11 + 2594x^10 - 3897x^9 - 4219x^8 + 5363x^7 + 4035x^6 - 3886x^5 - 2173x^4 + 1303x^3 + 576x^2 - 155*x - 55
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$0.664320$ of defining polynomial
Character	$\chi$	$=$	9801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.664320 q^{2} -1.55868 q^{4} -1.11662 q^{5} -3.90454 q^{7} -2.36410 q^{8} -0.741796 q^{10} +2.67800 q^{13} -2.59387 q^{14} +1.54683 q^{16} -5.54657 q^{17} -5.13234 q^{19} +1.74046 q^{20} -2.11438 q^{23} -3.75315 q^{25} +1.77905 q^{26} +6.08592 q^{28} -1.08304 q^{29} -0.430163 q^{31} +5.75580 q^{32} -3.68470 q^{34} +4.35990 q^{35} -8.70580 q^{37} -3.40952 q^{38} +2.63981 q^{40} +2.91908 q^{41} -2.23257 q^{43} -1.40462 q^{46} -12.8355 q^{47} +8.24545 q^{49} -2.49330 q^{50} -4.17415 q^{52} -5.64118 q^{53} +9.23074 q^{56} -0.719486 q^{58} +2.37537 q^{59} -4.08445 q^{61} -0.285766 q^{62} +0.730027 q^{64} -2.99032 q^{65} -9.75169 q^{67} +8.64532 q^{68} +2.89637 q^{70} -6.31707 q^{71} -11.6727 q^{73} -5.78344 q^{74} +7.99967 q^{76} +4.51471 q^{79} -1.72723 q^{80} +1.93920 q^{82} -0.482004 q^{83} +6.19343 q^{85} -1.48314 q^{86} -16.0830 q^{89} -10.4564 q^{91} +3.29563 q^{92} -8.52687 q^{94} +5.73089 q^{95} +8.01565 q^{97} +5.47762 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 2 q^{2} + 12 q^{4} + q^{5} + q^{7} + 6 q^{8} - 2 q^{10} + 3 q^{13} - 8 q^{16} + 20 q^{17} - 3 q^{19} + 5 q^{20} + 10 q^{23} + 7 q^{25} - 2 q^{26} + 19 q^{28} + 21 q^{29} + 6 q^{31} + 9 q^{32} - 4 q^{34}+ \cdots + 82 q^{98}+O(q^{100}) $ 18 * q + 2 * q^2 + 12 * q^4 + q^5 + q^7 + 6 * q^8 - 2 * q^10 + 3 * q^13 - 8 * q^16 + 20 * q^17 - 3 * q^19 + 5 * q^20 + 10 * q^23 + 7 * q^25 - 2 * q^26 + 19 * q^28 + 21 * q^29 + 6 * q^31 + 9 * q^32 - 4 * q^34 + 38 * q^35 + 7 * q^37 - 13 * q^38 + 20 * q^41 + 4 * q^43 + 8 * q^46 - 7 * q^47 + 7 * q^49 + 25 * q^50 - 19 * q^52 - 31 * q^53 + 57 * q^56 + 12 * q^58 + 12 * q^59 - 16 * q^61 + 19 * q^62 - 16 * q^64 + 42 * q^65 - 5 * q^67 + 51 * q^68 + 8 * q^70 - 13 * q^71 + 9 * q^74 - 8 * q^76 - 2 * q^79 - 46 * q^80 - 34 * q^82 + 36 * q^83 + 25 * q^85 - 26 * q^86 + 14 * q^89 - 15 * q^91 + 15 * q^92 + 4 * q^94 + 64 * q^95 - 16 * q^97 + 82 * 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 $

$2$	0.664320	0.469746	0.234873	−	0.972026i	$-0.424533\pi$
$2$	0.664320	0.469746	0.234873	+	0.972026i	$0.424533\pi$
$3$	0	0
$3$	0	0
$4$	−1.55868	−0.779339
$5$	−1.11662	−0.499369	−0.249685	−	0.968327i	$-0.580327\pi$
$5$	−1.11662	−0.499369	−0.249685	+	0.968327i	$0.580327\pi$
$6$	0	0
$7$	−3.90454	−1.47578	−0.737889	−	0.674922i	$-0.764177\pi$
$7$	−3.90454	−1.47578	−0.737889	+	0.674922i	$0.764177\pi$
$8$	−2.36410	−0.835837
$9$	0	0
$10$	−0.741796	−0.234576
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.67800	0.742745	0.371372	−	0.928484i	$-0.378887\pi$
$13$	2.67800	0.742745	0.371372	+	0.928484i	$0.378887\pi$
$14$	−2.59387	−0.693240
$15$	0	0
$16$	1.54683	0.386709
$17$	−5.54657	−1.34524	−0.672620	−	0.739988i	$-0.734831\pi$
$17$	−5.54657	−1.34524	−0.672620	+	0.739988i	$0.734831\pi$
$18$	0	0
$19$	−5.13234	−1.17744	−0.588720	−	0.808337i	$-0.700368\pi$
$19$	−5.13234	−1.17744	−0.588720	+	0.808337i	$0.700368\pi$
$20$	1.74046	0.389178
$21$	0	0
$22$	0	0
$23$	−2.11438	−0.440878	−0.220439	−	0.975401i	$-0.570749\pi$
$23$	−2.11438	−0.440878	−0.220439	+	0.975401i	$0.570749\pi$
$24$	0	0
$25$	−3.75315	−0.750630
$26$	1.77905	0.348901
$27$	0	0
$28$	6.08592	1.15013
$29$	−1.08304	−0.201116	−0.100558	−	0.994931i	$-0.532063\pi$
$29$	−1.08304	−0.201116	−0.100558	+	0.994931i	$0.532063\pi$
$30$	0	0
$31$	−0.430163	−0.0772595	−0.0386297	−	0.999254i	$-0.512299\pi$
$31$	−0.430163	−0.0772595	−0.0386297	+	0.999254i	$0.512299\pi$
$32$	5.75580	1.01749
$33$	0	0
$34$	−3.68470	−0.631921
$35$	4.35990	0.736958
$36$	0	0
$37$	−8.70580	−1.43122	−0.715612	−	0.698498i	$-0.753852\pi$
$37$	−8.70580	−1.43122	−0.715612	+	0.698498i	$0.753852\pi$
$38$	−3.40952	−0.553097
$39$	0	0
$40$	2.63981	0.417391
$41$	2.91908	0.455883	0.227942	−	0.973675i	$-0.426800\pi$
$41$	2.91908	0.455883	0.227942	+	0.973675i	$0.426800\pi$
$42$	0	0
$43$	−2.23257	−0.340463	−0.170232	−	0.985404i	$-0.554452\pi$
$43$	−2.23257	−0.340463	−0.170232	+	0.985404i	$0.554452\pi$
$44$	0	0
$45$	0	0
$46$	−1.40462	−0.207101
$47$	−12.8355	−1.87225	−0.936123	−	0.351673i	$-0.885613\pi$
$47$	−12.8355	−1.87225	−0.936123	+	0.351673i	$0.885613\pi$
$48$	0	0
$49$	8.24545	1.17792
$50$	−2.49330	−0.352605
$51$	0	0
$52$	−4.17415	−0.578850
$53$	−5.64118	−0.774876	−0.387438	−	0.921896i	$-0.626640\pi$
$53$	−5.64118	−0.774876	−0.387438	+	0.921896i	$0.626640\pi$
$54$	0	0
$55$	0	0
$56$	9.23074	1.23351
$57$	0	0
$58$	−0.719486	−0.0944732
$59$	2.37537	0.309247	0.154623	−	0.987973i	$-0.450584\pi$
$59$	2.37537	0.309247	0.154623	+	0.987973i	$0.450584\pi$
$60$	0	0
$61$	−4.08445	−0.522961	−0.261480	−	0.965209i	$-0.584211\pi$
$61$	−4.08445	−0.522961	−0.261480	+	0.965209i	$0.584211\pi$
$62$	−0.285766	−0.0362923
$63$	0	0
$64$	0.730027	0.0912533
$65$	−2.99032	−0.370904
$66$	0	0
$67$	−9.75169	−1.19136	−0.595679	−	0.803223i	$-0.703117\pi$
$67$	−9.75169	−1.19136	−0.595679	+	0.803223i	$0.703117\pi$
$68$	8.64532	1.04840
$69$	0	0
$70$	2.89637	0.346183
$71$	−6.31707	−0.749699	−0.374849	−	0.927086i	$-0.622306\pi$
$71$	−6.31707	−0.749699	−0.374849	+	0.927086i	$0.622306\pi$
$72$	0	0
$73$	−11.6727	−1.36618	−0.683091	−	0.730333i	$-0.739365\pi$
$73$	−11.6727	−1.36618	−0.683091	+	0.730333i	$0.739365\pi$
$74$	−5.78344	−0.672311
$75$	0	0
$76$	7.99967	0.917625
$77$	0	0
$78$	0	0
$79$	4.51471	0.507944	0.253972	−	0.967212i	$-0.418263\pi$
$79$	4.51471	0.507944	0.253972	+	0.967212i	$0.418263\pi$
$80$	−1.72723	−0.193110
$81$	0	0
$82$	1.93920	0.214149
$83$	−0.482004	−0.0529068	−0.0264534	−	0.999650i	$-0.508421\pi$
$83$	−0.482004	−0.0529068	−0.0264534	+	0.999650i	$0.508421\pi$
$84$	0	0
$85$	6.19343	0.671772
$86$	−1.48314	−0.159931
$87$	0	0
$88$	0	0
$89$	−16.0830	−1.70480	−0.852399	−	0.522891i	$-0.824853\pi$
$89$	−16.0830	−1.70480	−0.852399	+	0.522891i	$0.824853\pi$
$90$	0	0
$91$	−10.4564	−1.09613
$92$	3.29563	0.343594
$93$	0	0
$94$	−8.52687	−0.879479
$95$	5.73089	0.587977
$96$	0	0
$97$	8.01565	0.813866	0.406933	−	0.913458i	$-0.366598\pi$
$97$	8.01565	0.813866	0.406933	+	0.913458i	$0.366598\pi$
$98$	5.47762	0.553323
$99$	0	0
$100$	5.84996	0.584996
$101$	−1.76131	−0.175257	−0.0876287	−	0.996153i	$-0.527929\pi$
$101$	−1.76131	−0.175257	−0.0876287	+	0.996153i	$0.527929\pi$
$102$	0	0
$103$	2.68767	0.264824	0.132412	−	0.991195i	$-0.457728\pi$
$103$	2.68767	0.264824	0.132412	+	0.991195i	$0.457728\pi$
$104$	−6.33108	−0.620813
$105$	0	0
$106$	−3.74755	−0.363995
$107$	10.2764	0.993456	0.496728	−	0.867906i	$-0.334535\pi$
$107$	10.2764	0.993456	0.496728	+	0.867906i	$0.334535\pi$
$108$	0	0
$109$	2.36284	0.226319	0.113160	−	0.993577i	$-0.463903\pi$
$109$	2.36284	0.226319	0.113160	+	0.993577i	$0.463903\pi$
$110$	0	0
$111$	0	0
$112$	−6.03968	−0.570696
$113$	−1.53243	−0.144159	−0.0720796	−	0.997399i	$-0.522964\pi$
$113$	−1.53243	−0.144159	−0.0720796	+	0.997399i	$0.522964\pi$
$114$	0	0
$115$	2.36096	0.220161
$116$	1.68811	0.156737
$117$	0	0
$118$	1.57801	0.145267
$119$	21.6568	1.98528
$120$	0	0
$121$	0	0
$122$	−2.71339	−0.245658
$123$	0	0
$124$	0.670485	0.0602113
$125$	9.77397	0.874211
$126$	0	0
$127$	−7.65133	−0.678946	−0.339473	−	0.940616i	$-0.610249\pi$
$127$	−7.65133	−0.678946	−0.339473	+	0.940616i	$0.610249\pi$
$128$	−11.0266	−0.974625
$129$	0	0
$130$	−1.98653	−0.174230
$131$	5.35912	0.468229	0.234114	−	0.972209i	$-0.424781\pi$
$131$	5.35912	0.468229	0.234114	+	0.972209i	$0.424781\pi$
$132$	0	0
$133$	20.0394	1.73764
$134$	−6.47825	−0.559635
$135$	0	0
$136$	13.1127	1.12440
$137$	−16.7312	−1.42944	−0.714721	−	0.699409i	$-0.753446\pi$
$137$	−16.7312	−1.42944	−0.714721	+	0.699409i	$0.753446\pi$
$138$	0	0
$139$	13.7401	1.16542	0.582711	−	0.812680i	$-0.301992\pi$
$139$	13.7401	1.16542	0.582711	+	0.812680i	$0.301992\pi$
$140$	−6.79569	−0.574340
$141$	0	0
$142$	−4.19656	−0.352168
$143$	0	0
$144$	0	0
$145$	1.20935	0.100431
$146$	−7.75440	−0.641758
$147$	0	0
$148$	13.5695	1.11541
$149$	−5.55966	−0.455465	−0.227732	−	0.973724i	$-0.573131\pi$
$149$	−5.55966	−0.455465	−0.227732	+	0.973724i	$0.573131\pi$
$150$	0	0
$151$	−0.163908	−0.0133386	−0.00666930	−	0.999978i	$-0.502123\pi$
$151$	−0.163908	−0.0133386	−0.00666930	+	0.999978i	$0.502123\pi$
$152$	12.1334	0.984147
$153$	0	0
$154$	0	0
$155$	0.480329	0.0385810
$156$	0	0
$157$	23.8530	1.90368	0.951839	−	0.306598i	$-0.0991909\pi$
$157$	23.8530	1.90368	0.951839	+	0.306598i	$0.0991909\pi$
$158$	2.99921	0.238605
$159$	0	0
$160$	−6.42706	−0.508104
$161$	8.25568	0.650638
$162$	0	0
$163$	−4.19602	−0.328658	−0.164329	−	0.986406i	$-0.552546\pi$
$163$	−4.19602	−0.328658	−0.164329	+	0.986406i	$0.552546\pi$
$164$	−4.54990	−0.355288
$165$	0	0
$166$	−0.320205	−0.0248527
$167$	−15.8404	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$167$	−15.8404	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$168$	0	0
$169$	−5.82829	−0.448330
$170$	4.11442	0.315562
$171$	0	0
$172$	3.47985	0.265336
$173$	2.60289	0.197894	0.0989470	−	0.995093i	$-0.468453\pi$
$173$	2.60289	0.197894	0.0989470	+	0.995093i	$0.468453\pi$
$174$	0	0
$175$	14.6543	1.10776
$176$	0	0
$177$	0	0
$178$	−10.6843	−0.800822
$179$	−7.58802	−0.567155	−0.283578	−	0.958949i	$-0.591521\pi$
$179$	−7.58802	−0.567155	−0.283578	+	0.958949i	$0.591521\pi$
$180$	0	0
$181$	−2.40019	−0.178405	−0.0892023	−	0.996014i	$-0.528432\pi$
$181$	−2.40019	−0.178405	−0.0892023	+	0.996014i	$0.528432\pi$
$182$	−6.94639	−0.514901
$183$	0	0
$184$	4.99861	0.368502
$185$	9.72109	0.714709
$186$	0	0
$187$	0	0
$188$	20.0064	1.45911
$189$	0	0
$190$	3.80715	0.276200
$191$	16.5182	1.19522	0.597609	−	0.801788i	$-0.296118\pi$
$191$	16.5182	1.19522	0.597609	+	0.801788i	$0.296118\pi$
$192$	0	0
$193$	−8.13496	−0.585567	−0.292784	−	0.956179i	$-0.594582\pi$
$193$	−8.13496	−0.585567	−0.292784	+	0.956179i	$0.594582\pi$
$194$	5.32496	0.382310
$195$	0	0
$196$	−12.8520	−0.918000
$197$	22.9072	1.63207	0.816035	−	0.578003i	$-0.196168\pi$
$197$	22.9072	1.63207	0.816035	+	0.578003i	$0.196168\pi$
$198$	0	0
$199$	9.27177	0.657259	0.328629	−	0.944459i	$-0.393413\pi$
$199$	9.27177	0.657259	0.328629	+	0.944459i	$0.393413\pi$
$200$	8.87284	0.627404
$201$	0	0
$202$	−1.17008	−0.0823264
$203$	4.22878	0.296802
$204$	0	0
$205$	−3.25951	−0.227654
$206$	1.78547	0.124400
$207$	0	0
$208$	4.14243	0.287226
$209$	0	0
$210$	0	0
$211$	4.21058	0.289868	0.144934	−	0.989441i	$-0.453703\pi$
$211$	4.21058	0.289868	0.144934	+	0.989441i	$0.453703\pi$
$212$	8.79279	0.603891
$213$	0	0
$214$	6.82681	0.466671
$215$	2.49294	0.170017
$216$	0	0
$217$	1.67959	0.114018
$218$	1.56968	0.106312
$219$	0	0
$220$	0	0
$221$	−14.8537	−0.999170
$222$	0	0
$223$	−6.85906	−0.459317	−0.229658	−	0.973271i	$-0.573761\pi$
$223$	−6.85906	−0.459317	−0.229658	+	0.973271i	$0.573761\pi$
$224$	−22.4738	−1.50159
$225$	0	0
$226$	−1.01803	−0.0677181
$227$	−28.1299	−1.86705	−0.933524	−	0.358515i	$-0.883283\pi$
$227$	−28.1299	−1.86705	−0.933524	+	0.358515i	$0.883283\pi$
$228$	0	0
$229$	−14.0347	−0.927439	−0.463719	−	0.885982i	$-0.653485\pi$
$229$	−14.0347	−0.927439	−0.463719	+	0.885982i	$0.653485\pi$
$230$	1.56844	0.103420
$231$	0	0
$232$	2.56042	0.168100
$233$	14.5096	0.950557	0.475278	−	0.879835i	$-0.342347\pi$
$233$	14.5096	0.950557	0.475278	+	0.879835i	$0.342347\pi$
$234$	0	0
$235$	14.3324	0.934942
$236$	−3.70244	−0.241008
$237$	0	0
$238$	14.3871	0.932575
$239$	25.1481	1.62670	0.813349	−	0.581777i	$-0.197642\pi$
$239$	25.1481	1.62670	0.813349	+	0.581777i	$0.197642\pi$
$240$	0	0
$241$	17.7422	1.14288	0.571438	−	0.820645i	$-0.306386\pi$
$241$	17.7422	1.14288	0.571438	+	0.820645i	$0.306386\pi$
$242$	0	0
$243$	0	0
$244$	6.36635	0.407564
$245$	−9.20706	−0.588217
$246$	0	0
$247$	−13.7444	−0.874537
$248$	1.01695	0.0645763
$249$	0	0
$250$	6.49305	0.410657
$251$	21.8775	1.38090	0.690448	−	0.723382i	$-0.257414\pi$
$251$	21.8775	1.38090	0.690448	+	0.723382i	$0.257414\pi$
$252$	0	0
$253$	0	0
$254$	−5.08294	−0.318932
$255$	0	0
$256$	−8.78527	−0.549079
$257$	−5.16372	−0.322104	−0.161052	−	0.986946i	$-0.551489\pi$
$257$	−5.16372	−0.322104	−0.161052	+	0.986946i	$0.551489\pi$
$258$	0	0
$259$	33.9921	2.11217
$260$	4.66095	0.289060
$261$	0	0
$262$	3.56017	0.219948
$263$	27.8981	1.72027	0.860135	−	0.510066i	$-0.170379\pi$
$263$	27.8981	1.72027	0.860135	+	0.510066i	$0.170379\pi$
$264$	0	0
$265$	6.29908	0.386949
$266$	13.3126	0.816249
$267$	0	0
$268$	15.1997	0.928472
$269$	−3.01959	−0.184108	−0.0920538	−	0.995754i	$-0.529343\pi$
$269$	−3.01959	−0.184108	−0.0920538	+	0.995754i	$0.529343\pi$
$270$	0	0
$271$	−25.3748	−1.54141	−0.770705	−	0.637192i	$-0.780096\pi$
$271$	−25.3748	−1.54141	−0.770705	+	0.637192i	$0.780096\pi$
$272$	−8.57962	−0.520216
$273$	0	0
$274$	−11.1149	−0.671474
$275$	0	0
$276$	0	0
$277$	−10.5644	−0.634752	−0.317376	−	0.948300i	$-0.602802\pi$
$277$	−10.5644	−0.634752	−0.317376	+	0.948300i	$0.602802\pi$
$278$	9.12784	0.547451
$279$	0	0
$280$	−10.3073	−0.615976
$281$	22.5705	1.34644	0.673222	−	0.739440i	$-0.264910\pi$
$281$	22.5705	1.34644	0.673222	+	0.739440i	$0.264910\pi$
$282$	0	0
$283$	−5.87723	−0.349365	−0.174683	−	0.984625i	$-0.555890\pi$
$283$	−5.87723	−0.349365	−0.174683	+	0.984625i	$0.555890\pi$
$284$	9.84628	0.584269
$285$	0	0
$286$	0	0
$287$	−11.3977	−0.672782
$288$	0	0
$289$	13.7644	0.809672
$290$	0.803395	0.0471770
$291$	0	0
$292$	18.1939	1.06472
$293$	−7.09411	−0.414442	−0.207221	−	0.978294i	$-0.566442\pi$
$293$	−7.09411	−0.414442	−0.207221	+	0.978294i	$0.566442\pi$
$294$	0	0
$295$	−2.65239	−0.154428
$296$	20.5814	1.19627
$297$	0	0
$298$	−3.69339	−0.213953
$299$	−5.66231	−0.327460
$300$	0	0
$301$	8.71715	0.502448
$302$	−0.108887	−0.00626575
$303$	0	0
$304$	−7.93888	−0.455326
$305$	4.56080	0.261150
$306$	0	0
$307$	−18.5370	−1.05796	−0.528981	−	0.848633i	$-0.677426\pi$
$307$	−18.5370	−1.05796	−0.528981	+	0.848633i	$0.677426\pi$
$308$	0	0
$309$	0	0
$310$	0.319093	0.0181232
$311$	−6.87624	−0.389916	−0.194958	−	0.980812i	$-0.562457\pi$
$311$	−6.87624	−0.389916	−0.194958	+	0.980812i	$0.562457\pi$
$312$	0	0
$313$	−25.3258	−1.43150	−0.715749	−	0.698357i	$-0.753915\pi$
$313$	−25.3258	−1.43150	−0.715749	+	0.698357i	$0.753915\pi$
$314$	15.8460	0.894244
$315$	0	0
$316$	−7.03698	−0.395861
$317$	−10.9793	−0.616657	−0.308328	−	0.951280i	$-0.599770\pi$
$317$	−10.9793	−0.616657	−0.308328	+	0.951280i	$0.599770\pi$
$318$	0	0
$319$	0	0
$320$	−0.815165	−0.0455691
$321$	0	0
$322$	5.48442	0.305635
$323$	28.4669	1.58394
$324$	0	0
$325$	−10.0510	−0.557527
$326$	−2.78751	−0.154386
$327$	0	0
$328$	−6.90100	−0.381044
$329$	50.1166	2.76302
$330$	0	0
$331$	−3.71025	−0.203934	−0.101967	−	0.994788i	$-0.532514\pi$
$331$	−3.71025	−0.203934	−0.101967	+	0.994788i	$0.532514\pi$
$332$	0.751289	0.0412323
$333$	0	0
$334$	−10.5231	−0.575798
$335$	10.8890	0.594928
$336$	0	0
$337$	−32.1952	−1.75378	−0.876891	−	0.480689i	$-0.840386\pi$
$337$	−32.1952	−1.75378	−0.876891	+	0.480689i	$0.840386\pi$
$338$	−3.87185	−0.210601
$339$	0	0
$340$	−9.65356	−0.523538
$341$	0	0
$342$	0	0
$343$	−4.86291	−0.262572
$344$	5.27802	0.284572
$345$	0	0
$346$	1.72915	0.0929598
$347$	21.5225	1.15539	0.577694	−	0.816254i	$-0.303953\pi$
$347$	21.5225	1.15539	0.577694	+	0.816254i	$0.303953\pi$
$348$	0	0
$349$	−16.8200	−0.900352	−0.450176	−	0.892940i	$-0.648639\pi$
$349$	−16.8200	−0.900352	−0.450176	+	0.892940i	$0.648639\pi$
$350$	9.73518	0.520367
$351$	0	0
$352$	0	0
$353$	−3.13085	−0.166639	−0.0833193	−	0.996523i	$-0.526552\pi$
$353$	−3.13085	−0.166639	−0.0833193	+	0.996523i	$0.526552\pi$
$354$	0	0
$355$	7.05379	0.374376
$356$	25.0683	1.32862
$357$	0	0
$358$	−5.04088	−0.266419
$359$	−8.48869	−0.448016	−0.224008	−	0.974587i	$-0.571914\pi$
$359$	−8.48869	−0.448016	−0.224008	+	0.974587i	$0.571914\pi$
$360$	0	0
$361$	7.34092	0.386364
$362$	−1.59449	−0.0838048
$363$	0	0
$364$	16.2981	0.854254
$365$	13.0340	0.682230
$366$	0	0
$367$	−11.0110	−0.574768	−0.287384	−	0.957815i	$-0.592786\pi$
$367$	−11.0110	−0.574768	−0.287384	+	0.957815i	$0.592786\pi$
$368$	−3.27059	−0.170491
$369$	0	0
$370$	6.45792	0.335731
$371$	22.0262	1.14355
$372$	0	0
$373$	11.0068	0.569912	0.284956	−	0.958541i	$-0.408021\pi$
$373$	11.0068	0.569912	0.284956	+	0.958541i	$0.408021\pi$
$374$	0	0
$375$	0	0
$376$	30.3444	1.56489
$377$	−2.90039	−0.149378
$378$	0	0
$379$	25.3868	1.30403	0.652017	−	0.758204i	$-0.273923\pi$
$379$	25.3868	1.30403	0.652017	+	0.758204i	$0.273923\pi$
$380$	−8.93262	−0.458234
$381$	0	0
$382$	10.9734	0.561448
$383$	12.1147	0.619034	0.309517	−	0.950894i	$-0.399833\pi$
$383$	12.1147	0.619034	0.309517	+	0.950894i	$0.399833\pi$
$384$	0	0
$385$	0	0
$386$	−5.40422	−0.275068
$387$	0	0
$388$	−12.4938	−0.634278
$389$	−14.3511	−0.727629	−0.363814	−	0.931471i	$-0.618526\pi$
$389$	−14.3511	−0.727629	−0.363814	+	0.931471i	$0.618526\pi$
$390$	0	0
$391$	11.7275	0.593087
$392$	−19.4931	−0.984550
$393$	0	0
$394$	15.2177	0.766657
$395$	−5.04123	−0.253652
$396$	0	0
$397$	−5.03882	−0.252891	−0.126446	−	0.991974i	$-0.540357\pi$
$397$	−5.03882	−0.252891	−0.126446	+	0.991974i	$0.540357\pi$
$398$	6.15943	0.308744
$399$	0	0
$400$	−5.80551	−0.290275
$401$	18.2002	0.908874	0.454437	−	0.890779i	$-0.349841\pi$
$401$	18.2002	0.908874	0.454437	+	0.890779i	$0.349841\pi$
$402$	0	0
$403$	−1.15198	−0.0573841
$404$	2.74532	0.136585
$405$	0	0
$406$	2.80926	0.139421
$407$	0	0
$408$	0	0
$409$	16.4305	0.812434	0.406217	−	0.913777i	$-0.366848\pi$
$409$	16.4305	0.812434	0.406217	+	0.913777i	$0.366848\pi$
$410$	−2.16536	−0.106939
$411$	0	0
$412$	−4.18921	−0.206387
$413$	−9.27473	−0.456380
$414$	0	0
$415$	0.538216	0.0264200
$416$	15.4141	0.755736
$417$	0	0
$418$	0	0
$419$	17.4065	0.850362	0.425181	−	0.905108i	$-0.360210\pi$
$419$	17.4065	0.850362	0.425181	+	0.905108i	$0.360210\pi$
$420$	0	0
$421$	4.54865	0.221687	0.110844	−	0.993838i	$-0.464645\pi$
$421$	4.54865	0.221687	0.110844	+	0.993838i	$0.464645\pi$
$422$	2.79717	0.136164
$423$	0	0
$424$	13.3363	0.647670
$425$	20.8171	1.00978
$426$	0	0
$427$	15.9479	0.771774
$428$	−16.0176	−0.774239
$429$	0	0
$430$	1.65611	0.0798647
$431$	−12.4848	−0.601372	−0.300686	−	0.953723i	$-0.597216\pi$
$431$	−12.4848	−0.601372	−0.300686	+	0.953723i	$0.597216\pi$
$432$	0	0
$433$	28.6862	1.37857	0.689284	−	0.724491i	$-0.257925\pi$
$433$	28.6862	1.37857	0.689284	+	0.724491i	$0.257925\pi$
$434$	1.11578	0.0535594
$435$	0	0
$436$	−3.68291	−0.176379
$437$	10.8517	0.519108
$438$	0	0
$439$	−4.10807	−0.196068	−0.0980338	−	0.995183i	$-0.531255\pi$
$439$	−4.10807	−0.196068	−0.0980338	+	0.995183i	$0.531255\pi$
$440$	0	0
$441$	0	0
$442$	−9.86764	−0.469356
$443$	21.2297	1.00865	0.504327	−	0.863513i	$-0.331741\pi$
$443$	21.2297	1.00865	0.504327	+	0.863513i	$0.331741\pi$
$444$	0	0
$445$	17.9587	0.851324
$446$	−4.55661	−0.215762
$447$	0	0
$448$	−2.85042	−0.134670
$449$	−25.8949	−1.22205	−0.611027	−	0.791610i	$-0.709243\pi$
$449$	−25.8949	−1.22205	−0.611027	+	0.791610i	$0.709243\pi$
$450$	0	0
$451$	0	0
$452$	2.38857	0.112349
$453$	0	0
$454$	−18.6873	−0.877037
$455$	11.6758	0.547372
$456$	0	0
$457$	−10.3039	−0.481994	−0.240997	−	0.970526i	$-0.577474\pi$
$457$	−10.3039	−0.481994	−0.240997	+	0.970526i	$0.577474\pi$
$458$	−9.32354	−0.435660
$459$	0	0
$460$	−3.67998	−0.171580
$461$	25.4532	1.18548	0.592738	−	0.805395i	$-0.298047\pi$
$461$	25.4532	1.18548	0.592738	+	0.805395i	$0.298047\pi$
$462$	0	0
$463$	−39.8554	−1.85224	−0.926119	−	0.377231i	$-0.876876\pi$
$463$	−39.8554	−1.85224	−0.926119	+	0.377231i	$0.876876\pi$
$464$	−1.67528	−0.0777732
$465$	0	0
$466$	9.63904	0.446520
$467$	17.3769	0.804108	0.402054	−	0.915616i	$-0.368296\pi$
$467$	17.3769	0.804108	0.402054	+	0.915616i	$0.368296\pi$
$468$	0	0
$469$	38.0759	1.75818
$470$	9.52130	0.439185
$471$	0	0
$472$	−5.61562	−0.258480
$473$	0	0
$474$	0	0
$475$	19.2625	0.883822
$476$	−33.7560	−1.54720
$477$	0	0
$478$	16.7064	0.764134
$479$	21.8960	1.00045	0.500226	−	0.865895i	$-0.333250\pi$
$479$	21.8960	1.00045	0.500226	+	0.865895i	$0.333250\pi$
$480$	0	0
$481$	−23.3142	−1.06303
$482$	11.7865	0.536861
$483$	0	0
$484$	0	0
$485$	−8.95046	−0.406420
$486$	0	0
$487$	3.27116	0.148230	0.0741152	−	0.997250i	$-0.476387\pi$
$487$	3.27116	0.148230	0.0741152	+	0.997250i	$0.476387\pi$
$488$	9.65607	0.437110
$489$	0	0
$490$	−6.11644	−0.276313
$491$	7.22390	0.326010	0.163005	−	0.986625i	$-0.447881\pi$
$491$	7.22390	0.326010	0.163005	+	0.986625i	$0.447881\pi$
$492$	0	0
$493$	6.00716	0.270549
$494$	−9.13071	−0.410810
$495$	0	0
$496$	−0.665390	−0.0298769
$497$	24.6653	1.10639
$498$	0	0
$499$	14.8345	0.664084	0.332042	−	0.943265i	$-0.392262\pi$
$499$	14.8345	0.664084	0.332042	+	0.943265i	$0.392262\pi$
$500$	−15.2345	−0.681307
$501$	0	0
$502$	14.5337	0.648669
$503$	32.9813	1.47057	0.735283	−	0.677760i	$-0.237049\pi$
$503$	32.9813	1.47057	0.735283	+	0.677760i	$0.237049\pi$
$504$	0	0
$505$	1.96673	0.0875181
$506$	0	0
$507$	0	0
$508$	11.9260	0.529129
$509$	−38.6694	−1.71399	−0.856995	−	0.515325i	$-0.827671\pi$
$509$	−38.6694	−1.71399	−0.856995	+	0.515325i	$0.827671\pi$
$510$	0	0
$511$	45.5764	2.01618
$512$	16.2170	0.716698
$513$	0	0
$514$	−3.43036	−0.151307
$515$	−3.00111	−0.132245
$516$	0	0
$517$	0	0
$518$	22.5817	0.992182
$519$	0	0
$520$	7.06943	0.310015
$521$	17.4844	0.766005	0.383002	−	0.923747i	$-0.374890\pi$
$521$	17.4844	0.766005	0.383002	+	0.923747i	$0.374890\pi$
$522$	0	0
$523$	14.4881	0.633522	0.316761	−	0.948505i	$-0.397405\pi$
$523$	14.4881	0.633522	0.316761	+	0.948505i	$0.397405\pi$
$524$	−8.35314	−0.364909
$525$	0	0
$526$	18.5333	0.808089
$527$	2.38593	0.103933
$528$	0	0
$529$	−18.5294	−0.805626
$530$	4.18461	0.181768
$531$	0	0
$532$	−31.2350	−1.35421
$533$	7.81730	0.338605
$534$	0	0
$535$	−11.4749	−0.496101
$536$	23.0540	0.995781
$537$	0	0
$538$	−2.00597	−0.0864837
$539$	0	0
$540$	0	0
$541$	33.0983	1.42301	0.711503	−	0.702683i	$-0.248015\pi$
$541$	33.0983	1.42301	0.711503	+	0.702683i	$0.248015\pi$
$542$	−16.8570	−0.724071
$543$	0	0
$544$	−31.9249	−1.36877
$545$	−2.63840	−0.113017
$546$	0	0
$547$	9.21414	0.393968	0.196984	−	0.980407i	$-0.436885\pi$
$547$	9.21414	0.393968	0.196984	+	0.980407i	$0.436885\pi$
$548$	26.0786	1.11402
$549$	0	0
$550$	0	0
$551$	5.55853	0.236802
$552$	0	0
$553$	−17.6279	−0.749613
$554$	−7.01814	−0.298172
$555$	0	0
$556$	−21.4164	−0.908258
$557$	−37.0518	−1.56994	−0.784968	−	0.619536i	$-0.787321\pi$
$557$	−37.0518	−1.56994	−0.784968	+	0.619536i	$0.787321\pi$
$558$	0	0
$559$	−5.97883	−0.252877
$560$	6.74405	0.284988
$561$	0	0
$562$	14.9941	0.632486
$563$	26.9941	1.13766	0.568832	−	0.822454i	$-0.307395\pi$
$563$	26.9941	1.13766	0.568832	+	0.822454i	$0.307395\pi$
$564$	0	0
$565$	1.71115	0.0719886
$566$	−3.90436	−0.164113
$567$	0	0
$568$	14.9342	0.626625
$569$	27.4757	1.15184	0.575921	−	0.817505i	$-0.304644\pi$
$569$	27.4757	1.15184	0.575921	+	0.817505i	$0.304644\pi$
$570$	0	0
$571$	−39.0103	−1.63253	−0.816266	−	0.577676i	$-0.803960\pi$
$571$	−39.0103	−1.63253	−0.816266	+	0.577676i	$0.803960\pi$
$572$	0	0
$573$	0	0
$574$	−7.57170	−0.316036
$575$	7.93558	0.330937
$576$	0	0
$577$	−29.1659	−1.21419	−0.607096	−	0.794629i	$-0.707666\pi$
$577$	−29.1659	−1.21419	−0.607096	+	0.794629i	$0.707666\pi$
$578$	9.14399	0.380340
$579$	0	0
$580$	−1.88499	−0.0782698
$581$	1.88200	0.0780787
$582$	0	0
$583$	0	0
$584$	27.5954	1.14191
$585$	0	0
$586$	−4.71276	−0.194682
$587$	5.08793	0.210001	0.105001	−	0.994472i	$-0.466516\pi$
$587$	5.08793	0.210001	0.105001	+	0.994472i	$0.466516\pi$
$588$	0	0
$589$	2.20774	0.0909684
$590$	−1.76204	−0.0725420
$591$	0	0
$592$	−13.4664	−0.553467
$593$	−3.46422	−0.142258	−0.0711292	−	0.997467i	$-0.522660\pi$
$593$	−3.46422	−0.142258	−0.0711292	+	0.997467i	$0.522660\pi$
$594$	0	0
$595$	−24.1825	−0.991386
$596$	8.66572	0.354962
$597$	0	0
$598$	−3.76159	−0.153823
$599$	−24.8748	−1.01636	−0.508179	−	0.861251i	$-0.669681\pi$
$599$	−24.8748	−1.01636	−0.508179	+	0.861251i	$0.669681\pi$
$600$	0	0
$601$	11.0681	0.451478	0.225739	−	0.974188i	$-0.427520\pi$
$601$	11.0681	0.451478	0.225739	+	0.974188i	$0.427520\pi$
$602$	5.79098	0.236023
$603$	0	0
$604$	0.255479	0.0103953
$605$	0	0
$606$	0	0
$607$	−27.9610	−1.13490	−0.567451	−	0.823407i	$-0.692070\pi$
$607$	−27.9610	−1.13490	−0.567451	+	0.823407i	$0.692070\pi$
$608$	−29.5407	−1.19803
$609$	0	0
$610$	3.02983	0.122674
$611$	−34.3734	−1.39060
$612$	0	0
$613$	10.9923	0.443974	0.221987	−	0.975050i	$-0.428746\pi$
$613$	10.9923	0.443974	0.221987	+	0.975050i	$0.428746\pi$
$614$	−12.3145	−0.496973
$615$	0	0
$616$	0	0
$617$	−18.6262	−0.749864	−0.374932	−	0.927052i	$-0.622334\pi$
$617$	−18.6262	−0.749864	−0.374932	+	0.927052i	$0.622334\pi$
$618$	0	0
$619$	2.68848	0.108059	0.0540296	−	0.998539i	$-0.482793\pi$
$619$	2.68848	0.108059	0.0540296	+	0.998539i	$0.482793\pi$
$620$	−0.748679	−0.0300677
$621$	0	0
$622$	−4.56803	−0.183161
$623$	62.7969	2.51591
$624$	0	0
$625$	7.85192	0.314077
$626$	−16.8244	−0.672440
$627$	0	0
$628$	−37.1792	−1.48361
$629$	48.2873	1.92534
$630$	0	0
$631$	−39.2289	−1.56168	−0.780839	−	0.624732i	$-0.785208\pi$
$631$	−39.2289	−1.56168	−0.780839	+	0.624732i	$0.785208\pi$
$632$	−10.6732	−0.424558
$633$	0	0
$634$	−7.29375	−0.289672
$635$	8.54365	0.339045
$636$	0	0
$637$	22.0813	0.874895
$638$	0	0
$639$	0	0
$640$	12.3126	0.486698
$641$	−25.7490	−1.01702	−0.508511	−	0.861055i	$-0.669804\pi$
$641$	−25.7490	−1.01702	−0.508511	+	0.861055i	$0.669804\pi$
$642$	0	0
$643$	−13.3321	−0.525765	−0.262883	−	0.964828i	$-0.584673\pi$
$643$	−13.3321	−0.525765	−0.262883	+	0.964828i	$0.584673\pi$
$644$	−12.8679	−0.507068
$645$	0	0
$646$	18.9111	0.744049
$647$	−23.8506	−0.937663	−0.468832	−	0.883288i	$-0.655325\pi$
$647$	−23.8506	−0.937663	−0.468832	+	0.883288i	$0.655325\pi$
$648$	0	0
$649$	0	0
$650$	−6.67706	−0.261896
$651$	0	0
$652$	6.54025	0.256136
$653$	−12.7249	−0.497965	−0.248983	−	0.968508i	$-0.580096\pi$
$653$	−12.7249	−0.497965	−0.248983	+	0.968508i	$0.580096\pi$
$654$	0	0
$655$	−5.98412	−0.233819
$656$	4.51533	0.176294
$657$	0	0
$658$	33.2935	1.29792
$659$	27.2527	1.06161	0.530806	−	0.847493i	$-0.321889\pi$
$659$	27.2527	1.06161	0.530806	+	0.847493i	$0.321889\pi$
$660$	0	0
$661$	43.6440	1.69755	0.848777	−	0.528751i	$-0.177339\pi$
$661$	43.6440	1.69755	0.848777	+	0.528751i	$0.177339\pi$
$662$	−2.46480	−0.0957971
$663$	0	0
$664$	1.13951	0.0442214
$665$	−22.3765	−0.867724
$666$	0	0
$667$	2.28996	0.0886675
$668$	24.6901	0.955288
$669$	0	0
$670$	7.23376	0.279465
$671$	0	0
$672$	0	0
$673$	−26.6867	−1.02870	−0.514348	−	0.857582i	$-0.671966\pi$
$673$	−26.6867	−1.02870	−0.514348	+	0.857582i	$0.671966\pi$
$674$	−21.3879	−0.823831
$675$	0	0
$676$	9.08443	0.349401
$677$	−27.0831	−1.04089	−0.520443	−	0.853896i	$-0.674233\pi$
$677$	−27.0831	−1.04089	−0.520443	+	0.853896i	$0.674233\pi$
$678$	0	0
$679$	−31.2975	−1.20109
$680$	−14.6419	−0.561491
$681$	0	0
$682$	0	0
$683$	16.7343	0.640322	0.320161	−	0.947363i	$-0.396263\pi$
$683$	16.7343	0.640322	0.320161	+	0.947363i	$0.396263\pi$
$684$	0	0
$685$	18.6824	0.713820
$686$	−3.23053	−0.123342
$687$	0	0
$688$	−3.45341	−0.131660
$689$	−15.1071	−0.575535
$690$	0	0
$691$	−37.5196	−1.42731	−0.713656	−	0.700497i	$-0.752962\pi$
$691$	−37.5196	−1.42731	−0.713656	+	0.700497i	$0.752962\pi$
$692$	−4.05707	−0.154226
$693$	0	0
$694$	14.2978	0.542738
$695$	−15.3425	−0.581975
$696$	0	0
$697$	−16.1909	−0.613272
$698$	−11.1738	−0.422936
$699$	0	0
$700$	−22.8414	−0.863324
$701$	−10.2007	−0.385275	−0.192637	−	0.981270i	$-0.561704\pi$
$701$	−10.2007	−0.385275	−0.192637	+	0.981270i	$0.561704\pi$
$702$	0	0
$703$	44.6811	1.68518
$704$	0	0
$705$	0	0
$706$	−2.07989	−0.0782777
$707$	6.87713	0.258641
$708$	0	0
$709$	−14.4977	−0.544472	−0.272236	−	0.962231i	$-0.587763\pi$
$709$	−14.4977	−0.544472	−0.272236	+	0.962231i	$0.587763\pi$
$710$	4.68598	0.175862
$711$	0	0
$712$	38.0220	1.42493
$713$	0.909526	0.0340620
$714$	0	0
$715$	0	0
$716$	11.8273	0.442006
$717$	0	0
$718$	−5.63921	−0.210454
$719$	−19.2266	−0.717032	−0.358516	−	0.933524i	$-0.616717\pi$
$719$	−19.2266	−0.717032	−0.358516	+	0.933524i	$0.616717\pi$
$720$	0	0
$721$	−10.4941	−0.390821
$722$	4.87672	0.181493
$723$	0	0
$724$	3.74112	0.139038
$725$	4.06482	0.150964
$726$	0	0
$727$	0.726827	0.0269565	0.0134783	−	0.999909i	$-0.495710\pi$
$727$	0.726827	0.0269565	0.0134783	+	0.999909i	$0.495710\pi$
$728$	24.7200	0.916183
$729$	0	0
$730$	8.65874	0.320474
$731$	12.3831	0.458005
$732$	0	0
$733$	−1.87589	−0.0692875	−0.0346438	−	0.999400i	$-0.511030\pi$
$733$	−1.87589	−0.0692875	−0.0346438	+	0.999400i	$0.511030\pi$
$734$	−7.31481	−0.269995
$735$	0	0
$736$	−12.1699	−0.448590
$737$	0	0
$738$	0	0
$739$	−52.5826	−1.93428	−0.967140	−	0.254243i	$-0.918174\pi$
$739$	−52.5826	−1.93428	−0.967140	+	0.254243i	$0.918174\pi$
$740$	−15.1521	−0.557001
$741$	0	0
$742$	14.6325	0.537175
$743$	33.3328	1.22286	0.611431	−	0.791298i	$-0.290594\pi$
$743$	33.3328	1.22286	0.611431	+	0.791298i	$0.290594\pi$
$744$	0	0
$745$	6.20804	0.227445
$746$	7.31206	0.267714
$747$	0	0
$748$	0	0
$749$	−40.1246	−1.46612
$750$	0	0
$751$	−23.3641	−0.852567	−0.426284	−	0.904590i	$-0.640177\pi$
$751$	−23.3641	−0.852567	−0.426284	+	0.904590i	$0.640177\pi$
$752$	−19.8543	−0.724014
$753$	0	0
$754$	−1.92679	−0.0701694
$755$	0.183023	0.00666089
$756$	0	0
$757$	−33.8030	−1.22859	−0.614296	−	0.789076i	$-0.710560\pi$
$757$	−33.8030	−1.22859	−0.614296	+	0.789076i	$0.710560\pi$
$758$	16.8650	0.612564
$759$	0	0
$760$	−13.5484	−0.491453
$761$	−20.1119	−0.729056	−0.364528	−	0.931192i	$-0.618770\pi$
$761$	−20.1119	−0.729056	−0.364528	+	0.931192i	$0.618770\pi$
$762$	0	0
$763$	−9.22582	−0.333997
$764$	−25.7466	−0.931480
$765$	0	0
$766$	8.04807	0.290788
$767$	6.36125	0.229691
$768$	0	0
$769$	0.626999	0.0226102	0.0113051	−	0.999936i	$-0.496401\pi$
$769$	0.626999	0.0226102	0.0113051	+	0.999936i	$0.496401\pi$
$770$	0	0
$771$	0	0
$772$	12.6798	0.456356
$773$	−28.7378	−1.03363	−0.516813	−	0.856098i	$-0.672882\pi$
$773$	−28.7378	−1.03363	−0.516813	+	0.856098i	$0.672882\pi$
$774$	0	0
$775$	1.61447	0.0579933
$776$	−18.9498	−0.680259
$777$	0	0
$778$	−9.53372	−0.341800
$779$	−14.9817	−0.536775
$780$	0	0
$781$	0	0
$782$	7.79085	0.278600
$783$	0	0
$784$	12.7543	0.455512
$785$	−26.6348	−0.950638
$786$	0	0
$787$	−14.1029	−0.502716	−0.251358	−	0.967894i	$-0.580877\pi$
$787$	−14.1029	−0.502716	−0.251358	+	0.967894i	$0.580877\pi$
$788$	−35.7049	−1.27194
$789$	0	0
$790$	−3.34899	−0.119152
$791$	5.98345	0.212747
$792$	0	0
$793$	−10.9382	−0.388426
$794$	−3.34739	−0.118795
$795$	0	0
$796$	−14.4517	−0.512227
$797$	−26.1547	−0.926447	−0.463223	−	0.886242i	$-0.653307\pi$
$797$	−26.1547	−0.926447	−0.463223	+	0.886242i	$0.653307\pi$
$798$	0	0
$799$	71.1928	2.51862
$800$	−21.6024	−0.763760
$801$	0	0
$802$	12.0908	0.426939
$803$	0	0
$804$	0	0
$805$	−9.21848	−0.324909
$806$	−0.765282	−0.0269559
$807$	0	0
$808$	4.16393	0.146487
$809$	−15.8533	−0.557374	−0.278687	−	0.960382i	$-0.589899\pi$
$809$	−15.8533	−0.557374	−0.278687	+	0.960382i	$0.589899\pi$
$810$	0	0
$811$	−36.3264	−1.27559	−0.637796	−	0.770206i	$-0.720154\pi$
$811$	−36.3264	−1.27559	−0.637796	+	0.770206i	$0.720154\pi$
$812$	−6.59131	−0.231309
$813$	0	0
$814$	0	0
$815$	4.68538	0.164122
$816$	0	0
$817$	11.4583	0.400875
$818$	10.9151	0.381637
$819$	0	0
$820$	5.08053	0.177420
$821$	16.3567	0.570854	0.285427	−	0.958400i	$-0.407865\pi$
$821$	16.3567	0.570854	0.285427	+	0.958400i	$0.407865\pi$
$822$	0	0
$823$	−26.1745	−0.912386	−0.456193	−	0.889881i	$-0.650787\pi$
$823$	−26.1745	−0.912386	−0.456193	+	0.889881i	$0.650787\pi$
$824$	−6.35392	−0.221349
$825$	0	0
$826$	−6.16139	−0.214382
$827$	−5.53930	−0.192620	−0.0963101	−	0.995351i	$-0.530704\pi$
$827$	−5.53930	−0.192620	−0.0963101	+	0.995351i	$0.530704\pi$
$828$	0	0
$829$	18.5033	0.642647	0.321324	−	0.946969i	$-0.395872\pi$
$829$	18.5033	0.642647	0.321324	+	0.946969i	$0.395872\pi$
$830$	0.357548	0.0124107
$831$	0	0
$832$	1.95501	0.0677779
$833$	−45.7339	−1.58459
$834$	0	0
$835$	17.6878	0.612110
$836$	0	0
$837$	0	0
$838$	11.5635	0.399454
$839$	17.9653	0.620231	0.310115	−	0.950699i	$-0.399632\pi$
$839$	17.9653	0.620231	0.310115	+	0.950699i	$0.399632\pi$
$840$	0	0
$841$	−27.8270	−0.959553
$842$	3.02176	0.104137
$843$	0	0
$844$	−6.56293	−0.225906
$845$	6.50801	0.223882
$846$	0	0
$847$	0	0
$848$	−8.72598	−0.299651
$849$	0	0
$850$	13.8292	0.474339
$851$	18.4073	0.630996
$852$	0	0
$853$	27.9131	0.955726	0.477863	−	0.878434i	$-0.341412\pi$
$853$	27.9131	0.955726	0.477863	+	0.878434i	$0.341412\pi$
$854$	10.5945	0.362537
$855$	0	0
$856$	−24.2944	−0.830367
$857$	8.49967	0.290343	0.145172	−	0.989406i	$-0.453627\pi$
$857$	8.49967	0.290343	0.145172	+	0.989406i	$0.453627\pi$
$858$	0	0
$859$	27.8517	0.950289	0.475145	−	0.879908i	$-0.342396\pi$
$859$	27.8517	0.950289	0.475145	+	0.879908i	$0.342396\pi$
$860$	−3.88569	−0.132501
$861$	0	0
$862$	−8.29391	−0.282492
$863$	−13.5940	−0.462745	−0.231372	−	0.972865i	$-0.574322\pi$
$863$	−13.5940	−0.462745	−0.231372	+	0.972865i	$0.574322\pi$
$864$	0	0
$865$	−2.90645	−0.0988221
$866$	19.0568	0.647576
$867$	0	0
$868$	−2.61794	−0.0888586
$869$	0	0
$870$	0	0
$871$	−26.1151	−0.884875
$872$	−5.58600	−0.189166
$873$	0	0
$874$	7.20901	0.243848
$875$	−38.1629	−1.29014
$876$	0	0
$877$	36.1780	1.22164	0.610822	−	0.791768i	$-0.290839\pi$
$877$	36.1780	1.22164	0.610822	+	0.791768i	$0.290839\pi$
$878$	−2.72908	−0.0921019
$879$	0	0
$880$	0	0
$881$	−26.6423	−0.897601	−0.448800	−	0.893632i	$-0.648149\pi$
$881$	−26.6423	−0.897601	−0.448800	+	0.893632i	$0.648149\pi$
$882$	0	0
$883$	−20.4764	−0.689087	−0.344544	−	0.938770i	$-0.611966\pi$
$883$	−20.4764	−0.689087	−0.344544	+	0.938770i	$0.611966\pi$
$884$	23.1522	0.778693
$885$	0	0
$886$	14.1033	0.473810
$887$	32.0012	1.07449	0.537247	−	0.843425i	$-0.319464\pi$
$887$	32.0012	1.07449	0.537247	+	0.843425i	$0.319464\pi$
$888$	0	0
$889$	29.8749	1.00197
$890$	11.9303	0.399906
$891$	0	0
$892$	10.6911	0.357963
$893$	65.8760	2.20446
$894$	0	0
$895$	8.47296	0.283220
$896$	43.0539	1.43833
$897$	0	0
$898$	−17.2025	−0.574055
$899$	0.465884	0.0155381
$900$	0	0
$901$	31.2892	1.04239
$902$	0	0
$903$	0	0
$904$	3.62283	0.120493
$905$	2.68011	0.0890897
$906$	0	0
$907$	−49.0503	−1.62869	−0.814344	−	0.580383i	$-0.802903\pi$
$907$	−49.0503	−1.62869	−0.814344	+	0.580383i	$0.802903\pi$
$908$	43.8455	1.45506
$909$	0	0
$910$	7.75650	0.257125
$911$	23.3813	0.774656	0.387328	−	0.921942i	$-0.373398\pi$
$911$	23.3813	0.774656	0.387328	+	0.921942i	$0.373398\pi$
$912$	0	0
$913$	0	0
$914$	−6.84506	−0.226415
$915$	0	0
$916$	21.8756	0.722789
$917$	−20.9249	−0.691001
$918$	0	0
$919$	−7.59931	−0.250678	−0.125339	−	0.992114i	$-0.540002\pi$
$919$	−7.59931	−0.250678	−0.125339	+	0.992114i	$0.540002\pi$
$920$	−5.58156	−0.184019
$921$	0	0
$922$	16.9091	0.556872
$923$	−16.9171	−0.556835
$924$	0	0
$925$	32.6742	1.07432
$926$	−26.4768	−0.870081
$927$	0	0
$928$	−6.23377	−0.204633
$929$	14.8767	0.488090	0.244045	−	0.969764i	$-0.421525\pi$
$929$	14.8767	0.488090	0.244045	+	0.969764i	$0.421525\pi$
$930$	0	0
$931$	−42.3185	−1.38693
$932$	−22.6158	−0.740806
$933$	0	0
$934$	11.5438	0.377726
$935$	0	0
$936$	0	0
$937$	17.8231	0.582257	0.291128	−	0.956684i	$-0.405969\pi$
$937$	17.8231	0.582257	0.291128	+	0.956684i	$0.405969\pi$
$938$	25.2946	0.825897
$939$	0	0
$940$	−22.3396	−0.728637
$941$	−42.6591	−1.39065	−0.695324	−	0.718697i	$-0.744739\pi$
$941$	−42.6591	−1.39065	−0.695324	+	0.718697i	$0.744739\pi$
$942$	0	0
$943$	−6.17203	−0.200989
$944$	3.67430	0.119588
$945$	0	0
$946$	0	0
$947$	−6.89340	−0.224006	−0.112003	−	0.993708i	$-0.535727\pi$
$947$	−6.89340	−0.224006	−0.112003	+	0.993708i	$0.535727\pi$
$948$	0	0
$949$	−31.2595	−1.01473
$950$	12.7964	0.415171
$951$	0	0
$952$	−51.1989	−1.65937
$953$	23.6805	0.767085	0.383543	−	0.923523i	$-0.374704\pi$
$953$	23.6805	0.767085	0.383543	+	0.923523i	$0.374704\pi$
$954$	0	0
$955$	−18.4446	−0.596855
$956$	−39.1978	−1.26775
$957$	0	0
$958$	14.5459	0.469958
$959$	65.3277	2.10954
$960$	0	0
$961$	−30.8150	−0.994031
$962$	−15.4881	−0.499356
$963$	0	0
$964$	−27.6544	−0.890688
$965$	9.08369	0.292414
$966$	0	0
$967$	−31.1399	−1.00139	−0.500696	−	0.865623i	$-0.666923\pi$
$967$	−31.1399	−1.00139	−0.500696	+	0.865623i	$0.666923\pi$
$968$	0	0
$969$	0	0
$970$	−5.94598	−0.190914
$971$	−13.3548	−0.428577	−0.214288	−	0.976770i	$-0.568743\pi$
$971$	−13.3548	−0.428577	−0.214288	+	0.976770i	$0.568743\pi$
$972$	0	0
$973$	−53.6488	−1.71990
$974$	2.17310	0.0696306
$975$	0	0
$976$	−6.31797	−0.202233
$977$	8.91593	0.285246	0.142623	−	0.989777i	$-0.454446\pi$
$977$	8.91593	0.285246	0.142623	+	0.989777i	$0.454446\pi$
$978$	0	0
$979$	0	0
$980$	14.3508	0.458421
$981$	0	0
$982$	4.79899	0.153142
$983$	16.8585	0.537704	0.268852	−	0.963181i	$-0.413356\pi$
$983$	16.8585	0.537704	0.268852	+	0.963181i	$0.413356\pi$
$984$	0	0
$985$	−25.5787	−0.815005
$986$	3.99068	0.127089
$987$	0	0
$988$	21.4231	0.681561
$989$	4.72049	0.150103
$990$	0	0
$991$	−33.5356	−1.06529	−0.532647	−	0.846338i	$-0.678803\pi$
$991$	−33.5356	−1.06529	−0.532647	+	0.846338i	$0.678803\pi$
$992$	−2.47593	−0.0786108
$993$	0	0
$994$	16.3856	0.519721
$995$	−10.3531	−0.328215
$996$	0	0
$997$	−3.24367	−0.102728	−0.0513641	−	0.998680i	$-0.516357\pi$
$997$	−3.24367	−0.102728	−0.0513641	+	0.998680i	$0.516357\pi$
$998$	9.85487	0.311950
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9801.2.a.cp.1.11		18
3.2	odd	2		9801.2.a.cm.1.8		18
9.2	odd	6		1089.2.e.p.364.11		36
9.5	odd	6		1089.2.e.p.727.11		36
11.5	even	5		891.2.f.e.487.4		36
11.9	even	5		891.2.f.e.730.4		36
11.10	odd	2		9801.2.a.cn.1.8		18
33.5	odd	10		891.2.f.f.487.6		36
33.20	odd	10		891.2.f.f.730.6		36
33.32	even	2		9801.2.a.co.1.11		18
99.5	odd	30		99.2.m.b.25.6	yes	72
99.16	even	15		297.2.n.b.91.6		72
99.20	odd	30		99.2.m.b.4.6	✓	72
99.31	even	15		297.2.n.b.235.6		72
99.32	even	6		1089.2.e.o.727.8		36
99.38	odd	30		99.2.m.b.58.4	yes	72
99.49	even	15		297.2.n.b.289.4		72
99.65	even	6		1089.2.e.o.364.8		36
99.86	odd	30		99.2.m.b.70.4	yes	72
99.97	even	15		297.2.n.b.37.4		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.m.b.4.6	✓	72	99.20	odd	30
99.2.m.b.25.6	yes	72	99.5	odd	30
99.2.m.b.58.4	yes	72	99.38	odd	30
99.2.m.b.70.4	yes	72	99.86	odd	30
297.2.n.b.37.4		72	99.97	even	15
297.2.n.b.91.6		72	99.16	even	15
297.2.n.b.235.6		72	99.31	even	15
297.2.n.b.289.4		72	99.49	even	15
891.2.f.e.487.4		36	11.5	even	5
891.2.f.e.730.4		36	11.9	even	5
891.2.f.f.487.6		36	33.5	odd	10
891.2.f.f.730.6		36	33.20	odd	10
1089.2.e.o.364.8		36	99.65	even	6
1089.2.e.o.727.8		36	99.32	even	6
1089.2.e.p.364.11		36	9.2	odd	6
1089.2.e.p.727.11		36	9.5	odd	6
9801.2.a.cm.1.8		18	3.2	odd	2
9801.2.a.cn.1.8		18	11.10	odd	2
9801.2.a.co.1.11		18	33.32	even	2
9801.2.a.cp.1.11		18	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:	\( N \)	\(=\)	\( 9801 = 3^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9801.a (trivial)

\(f(q)\)	\(=\)	\(q+0.664320 q^{2} -1.55868 q^{4} -1.11662 q^{5} -3.90454 q^{7} -2.36410 q^{8} -0.741796 q^{10} +2.67800 q^{13} -2.59387 q^{14} +1.54683 q^{16} -5.54657 q^{17} -5.13234 q^{19} +1.74046 q^{20} -2.11438 q^{23} -3.75315 q^{25} +1.77905 q^{26} +6.08592 q^{28} -1.08304 q^{29} -0.430163 q^{31} +5.75580 q^{32} -3.68470 q^{34} +4.35990 q^{35} -8.70580 q^{37} -3.40952 q^{38} +2.63981 q^{40} +2.91908 q^{41} -2.23257 q^{43} -1.40462 q^{46} -12.8355 q^{47} +8.24545 q^{49} -2.49330 q^{50} -4.17415 q^{52} -5.64118 q^{53} +9.23074 q^{56} -0.719486 q^{58} +2.37537 q^{59} -4.08445 q^{61} -0.285766 q^{62} +0.730027 q^{64} -2.99032 q^{65} -9.75169 q^{67} +8.64532 q^{68} +2.89637 q^{70} -6.31707 q^{71} -11.6727 q^{73} -5.78344 q^{74} +7.99967 q^{76} +4.51471 q^{79} -1.72723 q^{80} +1.93920 q^{82} -0.482004 q^{83} +6.19343 q^{85} -1.48314 q^{86} -16.0830 q^{89} -10.4564 q^{91} +3.29563 q^{92} -8.52687 q^{94} +5.73089 q^{95} +8.01565 q^{97} +5.47762 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 2 q^{2} + 12 q^{4} + q^{5} + q^{7} + 6 q^{8} - 2 q^{10} + 3 q^{13} - 8 q^{16} + 20 q^{17} - 3 q^{19} + 5 q^{20} + 10 q^{23} + 7 q^{25} - 2 q^{26} + 19 q^{28} + 21 q^{29} + 6 q^{31} + 9 q^{32} - 4 q^{34}+ \cdots + 82 q^{98}+O(q^{100}) \) 18 * q + 2 * q^2 + 12 * q^4 + q^5 + q^7 + 6 * q^8 - 2 * q^10 + 3 * q^13 - 8 * q^16 + 20 * q^17 - 3 * q^19 + 5 * q^20 + 10 * q^23 + 7 * q^25 - 2 * q^26 + 19 * q^28 + 21 * q^29 + 6 * q^31 + 9 * q^32 - 4 * q^34 + 38 * q^35 + 7 * q^37 - 13 * q^38 + 20 * q^41 + 4 * q^43 + 8 * q^46 - 7 * q^47 + 7 * q^49 + 25 * q^50 - 19 * q^52 - 31 * q^53 + 57 * q^56 + 12 * q^58 + 12 * q^59 - 16 * q^61 + 19 * q^62 - 16 * q^64 + 42 * q^65 - 5 * q^67 + 51 * q^68 + 8 * q^70 - 13 * q^71 + 9 * q^74 - 8 * q^76 - 2 * q^79 - 46 * q^80 - 34 * q^82 + 36 * q^83 + 25 * q^85 - 26 * q^86 + 14 * q^89 - 15 * q^91 + 15 * q^92 + 4 * q^94 + 64 * q^95 - 16 * q^97 + 82 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more