LMFDB - Embedded newform 5625.2.a.bb.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, 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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45 $ x^8 - 15x^6 + 70x^4 - 105*x^2 + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.37653$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37653 q^{2} +3.64791 q^{4} +4.26594 q^{7} +3.91630 q^{8} +O(q^{10})$	$q+2.37653 q^{2} +3.64791 q^{4} +4.26594 q^{7} +3.91630 q^{8} -2.49140 q^{11} +2.02987 q^{13} +10.1381 q^{14} +2.01141 q^{16} -7.24447 q^{17} +3.26594 q^{19} -5.92090 q^{22} +6.15085 q^{23} +4.82406 q^{26} +15.5618 q^{28} +0.951631 q^{29} +2.66637 q^{31} -3.05243 q^{32} -17.2167 q^{34} +9.66637 q^{37} +7.76161 q^{38} +12.1679 q^{41} +7.95077 q^{43} -9.08840 q^{44} +14.6177 q^{46} +2.93756 q^{47} +11.1983 q^{49} +7.40479 q^{52} -12.4437 q^{53} +16.7067 q^{56} +2.26158 q^{58} -8.13677 q^{59} -3.33363 q^{61} +6.33671 q^{62} -11.2770 q^{64} -11.2352 q^{67} -26.4271 q^{68} +9.67655 q^{71} +8.24312 q^{73} +22.9724 q^{74} +11.9138 q^{76} -10.6282 q^{77} +7.65496 q^{79} +28.9175 q^{82} -10.3240 q^{83} +18.8953 q^{86} -9.75709 q^{88} +0.997628 q^{89} +8.65932 q^{91} +22.4377 q^{92} +6.98120 q^{94} -5.31428 q^{97} +26.6130 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{4} + 10 q^{7}+O(q^{10}) $ 8 * q + 14 * q^4 + 10 * q^7	$ 8 q + 14 q^{4} + 10 q^{7} + 10 q^{13} + 22 q^{16} + 2 q^{19} + 10 q^{22} + 70 q^{28} - 6 q^{31} - 50 q^{34} + 50 q^{37} + 14 q^{49} + 80 q^{52} - 30 q^{58} - 54 q^{61} + 36 q^{64} + 10 q^{67} + 30 q^{73} + 56 q^{76} + 28 q^{79} + 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100}) $ 8 * q + 14 * q^4 + 10 * q^7 + 10 * q^13 + 22 * q^16 + 2 * q^19 + 10 * q^22 + 70 * q^28 - 6 * q^31 - 50 * q^34 + 50 * q^37 + 14 * q^49 + 80 * q^52 - 30 * q^58 - 54 * q^61 + 36 * q^64 + 10 * q^67 + 30 * q^73 + 56 * q^76 + 28 * q^79 + 20 * q^88 + 60 * q^91 - 40 * q^94

Display 100 coefficients 10 coefficients

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 $

$2$	2.37653	1.68046	0.840231	−	0.542228i	$-0.182419\pi$
$2$	2.37653	1.68046	0.840231	+	0.542228i	$0.182419\pi$
$3$	0	0
$3$	0	0
$4$	3.64791	1.82395
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.26594	1.61237	0.806187	−	0.591661i	$-0.201528\pi$
$7$	4.26594	1.61237	0.806187	+	0.591661i	$0.201528\pi$
$8$	3.91630	1.38462
$9$	0	0
$10$	0	0
$11$	−2.49140	−0.751186	−0.375593	−	0.926785i	$-0.622561\pi$
$11$	−2.49140	−0.751186	−0.375593	+	0.926785i	$0.622561\pi$
$12$	0	0
$13$	2.02987	0.562985	0.281493	−	0.959563i	$-0.409170\pi$
$13$	2.02987	0.562985	0.281493	+	0.959563i	$0.409170\pi$
$14$	10.1381	2.70953
$15$	0	0
$16$	2.01141	0.502853
$17$	−7.24447	−1.75704	−0.878521	−	0.477704i	$-0.841469\pi$
$17$	−7.24447	−1.75704	−0.878521	+	0.477704i	$0.841469\pi$
$18$	0	0
$19$	3.26594	0.749258	0.374629	−	0.927175i	$-0.377770\pi$
$19$	3.26594	0.749258	0.374629	+	0.927175i	$0.377770\pi$
$20$	0	0
$21$	0	0
$22$	−5.92090	−1.26234
$23$	6.15085	1.28254	0.641270	−	0.767315i	$-0.278408\pi$
$23$	6.15085	1.28254	0.641270	+	0.767315i	$0.278408\pi$
$24$	0	0
$25$	0	0
$26$	4.82406	0.946076
$27$	0	0
$28$	15.5618	2.94090
$29$	0.951631	0.176713	0.0883567	−	0.996089i	$-0.471838\pi$
$29$	0.951631	0.176713	0.0883567	+	0.996089i	$0.471838\pi$
$30$	0	0
$31$	2.66637	0.478894	0.239447	−	0.970909i	$-0.423034\pi$
$31$	2.66637	0.478894	0.239447	+	0.970909i	$0.423034\pi$
$32$	−3.05243	−0.539598
$33$	0	0
$34$	−17.2167	−2.95264
$35$	0	0
$36$	0	0
$37$	9.66637	1.58914	0.794571	−	0.607172i	$-0.207696\pi$
$37$	9.66637	1.58914	0.794571	+	0.607172i	$0.207696\pi$
$38$	7.76161	1.25910
$39$	0	0
$40$	0	0
$41$	12.1679	1.90031	0.950157	−	0.311771i	$-0.100922\pi$
$41$	12.1679	1.90031	0.950157	+	0.311771i	$0.100922\pi$
$42$	0	0
$43$	7.95077	1.21248	0.606241	−	0.795281i	$-0.292677\pi$
$43$	7.95077	1.21248	0.606241	+	0.795281i	$0.292677\pi$
$44$	−9.08840	−1.37013
$45$	0	0
$46$	14.6177	2.15526
$47$	2.93756	0.428487	0.214243	−	0.976780i	$-0.431271\pi$
$47$	2.93756	0.428487	0.214243	+	0.976780i	$0.431271\pi$
$48$	0	0
$49$	11.1983	1.59975
$50$	0	0
$51$	0	0
$52$	7.40479	1.02686
$53$	−12.4437	−1.70927	−0.854636	−	0.519228i	$-0.826220\pi$
$53$	−12.4437	−1.70927	−0.854636	+	0.519228i	$0.826220\pi$
$54$	0	0
$55$	0	0
$56$	16.7067	2.23253
$57$	0	0
$58$	2.26158	0.296960
$59$	−8.13677	−1.05932	−0.529659	−	0.848211i	$-0.677680\pi$
$59$	−8.13677	−1.05932	−0.529659	+	0.848211i	$0.677680\pi$
$60$	0	0
$61$	−3.33363	−0.426828	−0.213414	−	0.976962i	$-0.568458\pi$
$61$	−3.33363	−0.426828	−0.213414	+	0.976962i	$0.568458\pi$
$62$	6.33671	0.804763
$63$	0	0
$64$	−11.2770	−1.40963
$65$	0	0
$66$	0	0
$67$	−11.2352	−1.37260	−0.686298	−	0.727321i	$-0.740765\pi$
$67$	−11.2352	−1.37260	−0.686298	+	0.727321i	$0.740765\pi$
$68$	−26.4271	−3.20476
$69$	0	0
$70$	0	0
$71$	9.67655	1.14839	0.574197	−	0.818717i	$-0.305314\pi$
$71$	9.67655	1.14839	0.574197	+	0.818717i	$0.305314\pi$
$72$	0	0
$73$	8.24312	0.964784	0.482392	−	0.875955i	$-0.339768\pi$
$73$	8.24312	0.964784	0.482392	+	0.875955i	$0.339768\pi$
$74$	22.9724	2.67049
$75$	0	0
$76$	11.9138	1.36661
$77$	−10.6282	−1.21119
$78$	0	0
$79$	7.65496	0.861250	0.430625	−	0.902531i	$-0.358293\pi$
$79$	7.65496	0.861250	0.430625	+	0.902531i	$0.358293\pi$
$80$	0	0
$81$	0	0
$82$	28.9175	3.19341
$83$	−10.3240	−1.13321	−0.566604	−	0.823990i	$-0.691743\pi$
$83$	−10.3240	−1.13321	−0.566604	+	0.823990i	$0.691743\pi$
$84$	0	0
$85$	0	0
$86$	18.8953	2.03753
$87$	0	0
$88$	−9.75709	−1.04011
$89$	0.997628	0.105748	0.0528742	−	0.998601i	$-0.483162\pi$
$89$	0.997628	0.105748	0.0528742	+	0.998601i	$0.483162\pi$
$90$	0	0
$91$	8.65932	0.907743
$92$	22.4377	2.33929
$93$	0	0
$94$	6.98120	0.720055
$95$	0	0
$96$	0	0
$97$	−5.31428	−0.539583	−0.269791	−	0.962919i	$-0.586955\pi$
$97$	−5.31428	−0.539583	−0.269791	+	0.962919i	$0.586955\pi$
$98$	26.6130	2.68832
$99$	0	0
$100$	0	0
$101$	12.5774	1.25150	0.625751	−	0.780023i	$-0.284793\pi$
$101$	12.5774	1.25150	0.625751	+	0.780023i	$0.284793\pi$
$102$	0	0
$103$	12.4423	1.22597	0.612986	−	0.790094i	$-0.289968\pi$
$103$	12.4423	1.22597	0.612986	+	0.790094i	$0.289968\pi$
$104$	7.94960	0.779522
$105$	0	0
$106$	−29.5728	−2.87237
$107$	0.0593251	0.00573518	0.00286759	−	0.999996i	$-0.499087\pi$
$107$	0.0593251	0.00573518	0.00286759	+	0.999996i	$0.499087\pi$
$108$	0	0
$109$	1.05325	0.100883	0.0504413	−	0.998727i	$-0.483937\pi$
$109$	1.05325	0.100883	0.0504413	+	0.998727i	$0.483937\pi$
$110$	0	0
$111$	0	0
$112$	8.58056	0.810786
$113$	−0.817881	−0.0769398	−0.0384699	−	0.999260i	$-0.512248\pi$
$113$	−0.817881	−0.0769398	−0.0384699	+	0.999260i	$0.512248\pi$
$114$	0	0
$115$	0	0
$116$	3.47146	0.322317
$117$	0	0
$118$	−19.3373	−1.78014
$119$	−30.9045	−2.83301
$120$	0	0
$121$	−4.79291	−0.435719
$122$	−7.92248	−0.717268
$123$	0	0
$124$	9.72667	0.873480
$125$	0	0
$126$	0	0
$127$	−2.39739	−0.212734	−0.106367	−	0.994327i	$-0.533922\pi$
$127$	−2.39739	−0.212734	−0.106367	+	0.994327i	$0.533922\pi$
$128$	−20.6953	−1.82923
$129$	0	0
$130$	0	0
$131$	−13.7077	−1.19765	−0.598825	−	0.800880i	$-0.704365\pi$
$131$	−13.7077	−1.19765	−0.598825	+	0.800880i	$0.704365\pi$
$132$	0	0
$133$	13.9323	1.20808
$134$	−26.7008	−2.30659
$135$	0	0
$136$	−28.3715	−2.43284
$137$	−3.44303	−0.294158	−0.147079	−	0.989125i	$-0.546987\pi$
$137$	−3.44303	−0.294158	−0.147079	+	0.989125i	$0.546987\pi$
$138$	0	0
$139$	−1.97414	−0.167445	−0.0837224	−	0.996489i	$-0.526681\pi$
$139$	−1.97414	−0.167445	−0.0837224	+	0.996489i	$0.526681\pi$
$140$	0	0
$141$	0	0
$142$	22.9966	1.92983
$143$	−5.05723	−0.422907
$144$	0	0
$145$	0	0
$146$	19.5900	1.62128
$147$	0	0
$148$	35.2620	2.89852
$149$	−6.52258	−0.534350	−0.267175	−	0.963648i	$-0.586090\pi$
$149$	−6.52258	−0.534350	−0.267175	+	0.963648i	$0.586090\pi$
$150$	0	0
$151$	−8.11513	−0.660400	−0.330200	−	0.943911i	$-0.607116\pi$
$151$	−8.11513	−0.660400	−0.330200	+	0.943911i	$0.607116\pi$
$152$	12.7904	1.03744
$153$	0	0
$154$	−25.2582	−2.03536
$155$	0	0
$156$	0	0
$157$	−16.0545	−1.28129	−0.640644	−	0.767838i	$-0.721333\pi$
$157$	−16.0545	−1.28129	−0.640644	+	0.767838i	$0.721333\pi$
$158$	18.1923	1.44730
$159$	0	0
$160$	0	0
$161$	26.2392	2.06794
$162$	0	0
$163$	−5.78730	−0.453297	−0.226648	−	0.973977i	$-0.572777\pi$
$163$	−5.78730	−0.453297	−0.226648	+	0.973977i	$0.572777\pi$
$164$	44.3875	3.46608
$165$	0	0
$166$	−24.5353	−1.90431
$167$	−12.1313	−0.938747	−0.469373	−	0.883000i	$-0.655520\pi$
$167$	−12.1313	−0.938747	−0.469373	+	0.883000i	$0.655520\pi$
$168$	0	0
$169$	−8.87962	−0.683047
$170$	0	0
$171$	0	0
$172$	29.0037	2.21151
$173$	13.7537	1.04568	0.522838	−	0.852432i	$-0.324873\pi$
$173$	13.7537	1.04568	0.522838	+	0.852432i	$0.324873\pi$
$174$	0	0
$175$	0	0
$176$	−5.01123	−0.377736
$177$	0	0
$178$	2.37090	0.177706
$179$	−16.6086	−1.24139	−0.620693	−	0.784054i	$-0.713149\pi$
$179$	−16.6086	−1.24139	−0.620693	+	0.784054i	$0.713149\pi$
$180$	0	0
$181$	5.78586	0.430060	0.215030	−	0.976607i	$-0.431015\pi$
$181$	5.78586	0.430060	0.215030	+	0.976607i	$0.431015\pi$
$182$	20.5791	1.52543
$183$	0	0
$184$	24.0886	1.77583
$185$	0	0
$186$	0	0
$187$	18.0489	1.31987
$188$	10.7159	0.781539
$189$	0	0
$190$	0	0
$191$	−15.5826	−1.12751	−0.563757	−	0.825941i	$-0.690645\pi$
$191$	−15.5826	−1.12751	−0.563757	+	0.825941i	$0.690645\pi$
$192$	0	0
$193$	−15.3675	−1.10618	−0.553089	−	0.833122i	$-0.686551\pi$
$193$	−15.3675	−1.10618	−0.553089	+	0.833122i	$0.686551\pi$
$194$	−12.6295	−0.906749
$195$	0	0
$196$	40.8502	2.91787
$197$	6.50925	0.463765	0.231882	−	0.972744i	$-0.425512\pi$
$197$	6.50925	0.463765	0.231882	+	0.972744i	$0.425512\pi$
$198$	0	0
$199$	−15.9970	−1.13399	−0.566997	−	0.823720i	$-0.691895\pi$
$199$	−15.9970	−1.13399	−0.566997	+	0.823720i	$0.691895\pi$
$200$	0	0
$201$	0	0
$202$	29.8907	2.10310
$203$	4.05960	0.284928
$204$	0	0
$205$	0	0
$206$	29.5694	2.06020
$207$	0	0
$208$	4.08291	0.283099
$209$	−8.13677	−0.562832
$210$	0	0
$211$	−15.7153	−1.08188	−0.540941	−	0.841060i	$-0.681932\pi$
$211$	−15.7153	−1.08188	−0.540941	+	0.841060i	$0.681932\pi$
$212$	−45.3934	−3.11763
$213$	0	0
$214$	0.140988	0.00963775
$215$	0	0
$216$	0	0
$217$	11.3746	0.772156
$218$	2.50307	0.169529
$219$	0	0
$220$	0	0
$221$	−14.7053	−0.989189
$222$	0	0
$223$	−14.7597	−0.988380	−0.494190	−	0.869354i	$-0.664535\pi$
$223$	−14.7597	−0.988380	−0.494190	+	0.869354i	$0.664535\pi$
$224$	−13.0215	−0.870033
$225$	0	0
$226$	−1.94372	−0.129294
$227$	7.24447	0.480832	0.240416	−	0.970670i	$-0.422716\pi$
$227$	7.24447	0.480832	0.240416	+	0.970670i	$0.422716\pi$
$228$	0	0
$229$	−3.39739	−0.224506	−0.112253	−	0.993680i	$-0.535807\pi$
$229$	−3.39739	−0.224506	−0.112253	+	0.993680i	$0.535807\pi$
$230$	0	0
$231$	0	0
$232$	3.72688	0.244681
$233$	21.7701	1.42620	0.713102	−	0.701060i	$-0.247289\pi$
$233$	21.7701	1.42620	0.713102	+	0.701060i	$0.247289\pi$
$234$	0	0
$235$	0	0
$236$	−29.6822	−1.93215
$237$	0	0
$238$	−73.4455	−4.76076
$239$	8.10835	0.524485	0.262243	−	0.965002i	$-0.415538\pi$
$239$	8.10835	0.524485	0.262243	+	0.965002i	$0.415538\pi$
$240$	0	0
$241$	20.0963	1.29452	0.647259	−	0.762270i	$-0.275915\pi$
$241$	20.0963	1.29452	0.647259	+	0.762270i	$0.275915\pi$
$242$	−11.3905	−0.732210
$243$	0	0
$244$	−12.1608	−0.778514
$245$	0	0
$246$	0	0
$247$	6.62944	0.421821
$248$	10.4423	0.663088
$249$	0	0
$250$	0	0
$251$	20.2303	1.27693	0.638463	−	0.769653i	$-0.279571\pi$
$251$	20.2303	1.27693	0.638463	+	0.769653i	$0.279571\pi$
$252$	0	0
$253$	−15.3242	−0.963427
$254$	−5.69748	−0.357492
$255$	0	0
$256$	−26.6291	−1.66432
$257$	−1.32336	−0.0825489	−0.0412744	−	0.999148i	$-0.513142\pi$
$257$	−1.32336	−0.0825489	−0.0412744	+	0.999148i	$0.513142\pi$
$258$	0	0
$259$	41.2362	2.56229
$260$	0	0
$261$	0	0
$262$	−32.5768	−2.01260
$263$	23.8613	1.47135	0.735676	−	0.677334i	$-0.236865\pi$
$263$	23.8613	1.47135	0.735676	+	0.677334i	$0.236865\pi$
$264$	0	0
$265$	0	0
$266$	33.1106	2.03014
$267$	0	0
$268$	−40.9849	−2.50355
$269$	6.49415	0.395955	0.197978	−	0.980207i	$-0.436563\pi$
$269$	6.49415	0.395955	0.197978	+	0.980207i	$0.436563\pi$
$270$	0	0
$271$	−9.50817	−0.577580	−0.288790	−	0.957392i	$-0.593253\pi$
$271$	−9.50817	−0.577580	−0.288790	+	0.957392i	$0.593253\pi$
$272$	−14.5716	−0.883533
$273$	0	0
$274$	−8.18248	−0.494322
$275$	0	0
$276$	0	0
$277$	−9.15047	−0.549798	−0.274899	−	0.961473i	$-0.588644\pi$
$277$	−9.15047	−0.549798	−0.274899	+	0.961473i	$0.588644\pi$
$278$	−4.69162	−0.281385
$279$	0	0
$280$	0	0
$281$	−17.1968	−1.02587	−0.512936	−	0.858427i	$-0.671442\pi$
$281$	−17.1968	−1.02587	−0.512936	+	0.858427i	$0.671442\pi$
$282$	0	0
$283$	−22.5768	−1.34205	−0.671027	−	0.741433i	$-0.734147\pi$
$283$	−22.5768	−1.34205	−0.671027	+	0.741433i	$0.734147\pi$
$284$	35.2991	2.09462
$285$	0	0
$286$	−12.0187	−0.710679
$287$	51.9077	3.06402
$288$	0	0
$289$	35.4823	2.08719
$290$	0	0
$291$	0	0
$292$	30.0701	1.75972
$293$	−21.3523	−1.24742	−0.623709	−	0.781657i	$-0.714375\pi$
$293$	−21.3523	−1.24742	−0.623709	+	0.781657i	$0.714375\pi$
$294$	0	0
$295$	0	0
$296$	37.8564	2.20036
$297$	0	0
$298$	−15.5011	−0.897956
$299$	12.4854	0.722052
$300$	0	0
$301$	33.9175	1.95497
$302$	−19.2859	−1.10978
$303$	0	0
$304$	6.56915	0.376766
$305$	0	0
$306$	0	0
$307$	−8.43377	−0.481341	−0.240670	−	0.970607i	$-0.577367\pi$
$307$	−8.43377	−0.481341	−0.240670	+	0.970607i	$0.577367\pi$
$308$	−38.7706	−2.20916
$309$	0	0
$310$	0	0
$311$	9.13440	0.517964	0.258982	−	0.965882i	$-0.416613\pi$
$311$	9.13440	0.517964	0.258982	+	0.965882i	$0.416613\pi$
$312$	0	0
$313$	20.9683	1.18520	0.592600	−	0.805497i	$-0.298101\pi$
$313$	20.9683	1.18520	0.592600	+	0.805497i	$0.298101\pi$
$314$	−38.1540	−2.15316
$315$	0	0
$316$	27.9246	1.57088
$317$	−12.7561	−0.716453	−0.358227	−	0.933635i	$-0.616618\pi$
$317$	−12.7561	−0.716453	−0.358227	+	0.933635i	$0.616618\pi$
$318$	0	0
$319$	−2.37090	−0.132745
$320$	0	0
$321$	0	0
$322$	62.3582	3.47509
$323$	−23.6600	−1.31648
$324$	0	0
$325$	0	0
$326$	−13.7537	−0.761748
$327$	0	0
$328$	47.6534	2.63122
$329$	12.5314	0.690881
$330$	0	0
$331$	−15.8024	−0.868578	−0.434289	−	0.900774i	$-0.643000\pi$
$331$	−15.8024	−0.868578	−0.434289	+	0.900774i	$0.643000\pi$
$332$	−37.6610	−2.06692
$333$	0	0
$334$	−28.8304	−1.57753
$335$	0	0
$336$	0	0
$337$	−10.2620	−0.559008	−0.279504	−	0.960145i	$-0.590170\pi$
$337$	−10.2620	−0.559008	−0.279504	+	0.960145i	$0.590170\pi$
$338$	−21.1027	−1.14784
$339$	0	0
$340$	0	0
$341$	−6.64300	−0.359739
$342$	0	0
$343$	17.9095	0.967022
$344$	31.1376	1.67883
$345$	0	0
$346$	32.6862	1.75722
$347$	33.9014	1.81992	0.909960	−	0.414696i	$-0.136112\pi$
$347$	33.9014	1.81992	0.909960	+	0.414696i	$0.136112\pi$
$348$	0	0
$349$	−35.3728	−1.89346	−0.946731	−	0.322026i	$-0.895636\pi$
$349$	−35.3728	−1.89346	−0.946731	+	0.322026i	$0.895636\pi$
$350$	0	0
$351$	0	0
$352$	7.60482	0.405338
$353$	20.6764	1.10050	0.550248	−	0.835001i	$-0.314533\pi$
$353$	20.6764	1.10050	0.550248	+	0.835001i	$0.314533\pi$
$354$	0	0
$355$	0	0
$356$	3.63926	0.192880
$357$	0	0
$358$	−39.4709	−2.08610
$359$	−9.05998	−0.478167	−0.239084	−	0.970999i	$-0.576847\pi$
$359$	−9.05998	−0.478167	−0.239084	+	0.970999i	$0.576847\pi$
$360$	0	0
$361$	−8.33363	−0.438612
$362$	13.7503	0.722699
$363$	0	0
$364$	31.5884	1.65568
$365$	0	0
$366$	0	0
$367$	30.1733	1.57503	0.787516	−	0.616294i	$-0.211367\pi$
$367$	30.1733	1.57503	0.787516	+	0.616294i	$0.211367\pi$
$368$	12.3719	0.644929
$369$	0	0
$370$	0	0
$371$	−53.0840	−2.75599
$372$	0	0
$373$	21.8418	1.13093	0.565463	−	0.824774i	$-0.308698\pi$
$373$	21.8418	1.13093	0.565463	+	0.824774i	$0.308698\pi$
$374$	42.8938	2.21798
$375$	0	0
$376$	11.5044	0.593292
$377$	1.93169	0.0994871
$378$	0	0
$379$	13.9320	0.715637	0.357819	−	0.933791i	$-0.383521\pi$
$379$	13.9320	0.715637	0.357819	+	0.933791i	$0.383521\pi$
$380$	0	0
$381$	0	0
$382$	−37.0324	−1.89474
$383$	−3.86076	−0.197276	−0.0986378	−	0.995123i	$-0.531449\pi$
$383$	−3.86076	−0.197276	−0.0986378	+	0.995123i	$0.531449\pi$
$384$	0	0
$385$	0	0
$386$	−36.5214	−1.85889
$387$	0	0
$388$	−19.3860	−0.984174
$389$	−30.3908	−1.54087	−0.770436	−	0.637517i	$-0.779962\pi$
$389$	−30.3908	−1.54087	−0.770436	+	0.637517i	$0.779962\pi$
$390$	0	0
$391$	−44.5596	−2.25348
$392$	43.8558	2.21505
$393$	0	0
$394$	15.4694	0.779339
$395$	0	0
$396$	0	0
$397$	6.89048	0.345823	0.172912	−	0.984937i	$-0.444683\pi$
$397$	6.89048	0.345823	0.172912	+	0.984937i	$0.444683\pi$
$398$	−38.0173	−1.90564
$399$	0	0
$400$	0	0
$401$	32.5187	1.62390	0.811952	−	0.583724i	$-0.198405\pi$
$401$	32.5187	1.62390	0.811952	+	0.583724i	$0.198405\pi$
$402$	0	0
$403$	5.41239	0.269610
$404$	45.8813	2.28268
$405$	0	0
$406$	9.64778	0.478811
$407$	−24.0828	−1.19374
$408$	0	0
$409$	−4.71415	−0.233100	−0.116550	−	0.993185i	$-0.537184\pi$
$409$	−4.71415	−0.233100	−0.116550	+	0.993185i	$0.537184\pi$
$410$	0	0
$411$	0	0
$412$	45.3882	2.23612
$413$	−34.7110	−1.70802
$414$	0	0
$415$	0	0
$416$	−6.19604	−0.303786
$417$	0	0
$418$	−19.3373	−0.945819
$419$	23.9724	1.17113	0.585564	−	0.810626i	$-0.300873\pi$
$419$	23.9724	1.17113	0.585564	+	0.810626i	$0.300873\pi$
$420$	0	0
$421$	27.6965	1.34984	0.674921	−	0.737890i	$-0.264178\pi$
$421$	27.6965	1.34984	0.674921	+	0.737890i	$0.264178\pi$
$422$	−37.3478	−1.81806
$423$	0	0
$424$	−48.7333	−2.36670
$425$	0	0
$426$	0	0
$427$	−14.2211	−0.688206
$428$	0.216413	0.0104607
$429$	0	0
$430$	0	0
$431$	6.14152	0.295826	0.147913	−	0.989000i	$-0.452744\pi$
$431$	6.14152	0.295826	0.147913	+	0.989000i	$0.452744\pi$
$432$	0	0
$433$	15.3336	0.736887	0.368444	−	0.929650i	$-0.379891\pi$
$433$	15.3336	0.736887	0.368444	+	0.929650i	$0.379891\pi$
$434$	27.0320	1.29758
$435$	0	0
$436$	3.84214	0.184005
$437$	20.0883	0.960954
$438$	0	0
$439$	−26.1170	−1.24650	−0.623248	−	0.782024i	$-0.714187\pi$
$439$	−26.1170	−1.24650	−0.623248	+	0.782024i	$0.714187\pi$
$440$	0	0
$441$	0	0
$442$	−34.9477	−1.66229
$443$	15.2702	0.725507	0.362753	−	0.931885i	$-0.381837\pi$
$443$	15.2702	0.725507	0.362753	+	0.931885i	$0.381837\pi$
$444$	0	0
$445$	0	0
$446$	−35.0768	−1.66094
$447$	0	0
$448$	−48.1071	−2.27284
$449$	−34.4395	−1.62530	−0.812650	−	0.582752i	$-0.801976\pi$
$449$	−34.4395	−1.62530	−0.812650	+	0.582752i	$0.801976\pi$
$450$	0	0
$451$	−30.3153	−1.42749
$452$	−2.98355	−0.140335
$453$	0	0
$454$	17.2167	0.808020
$455$	0	0
$456$	0	0
$457$	8.69321	0.406651	0.203326	−	0.979111i	$-0.434825\pi$
$457$	8.69321	0.406651	0.203326	+	0.979111i	$0.434825\pi$
$458$	−8.07402	−0.377274
$459$	0	0
$460$	0	0
$461$	11.7300	0.546322	0.273161	−	0.961968i	$-0.411931\pi$
$461$	11.7300	0.546322	0.273161	+	0.961968i	$0.411931\pi$
$462$	0	0
$463$	−18.8462	−0.875855	−0.437928	−	0.899010i	$-0.644287\pi$
$463$	−18.8462	−0.875855	−0.437928	+	0.899010i	$0.644287\pi$
$464$	1.91412	0.0888608
$465$	0	0
$466$	51.7373	2.39668
$467$	−18.3055	−0.847076	−0.423538	−	0.905878i	$-0.639212\pi$
$467$	−18.3055	−0.847076	−0.423538	+	0.905878i	$0.639212\pi$
$468$	0	0
$469$	−47.9286	−2.21314
$470$	0	0
$471$	0	0
$472$	−31.8661	−1.46676
$473$	−19.8086	−0.910799
$474$	0	0
$475$	0	0
$476$	−112.737	−5.16727
$477$	0	0
$478$	19.2697	0.881378
$479$	−21.5630	−0.985238	−0.492619	−	0.870245i	$-0.663960\pi$
$479$	−21.5630	−0.985238	−0.492619	+	0.870245i	$0.663960\pi$
$480$	0	0
$481$	19.6215	0.894663
$482$	47.7596	2.17539
$483$	0	0
$484$	−17.4841	−0.794732
$485$	0	0
$486$	0	0
$487$	12.2616	0.555625	0.277813	−	0.960635i	$-0.410391\pi$
$487$	12.2616	0.555625	0.277813	+	0.960635i	$0.410391\pi$
$488$	−13.0555	−0.590995
$489$	0	0
$490$	0	0
$491$	−8.67892	−0.391674	−0.195837	−	0.980636i	$-0.562742\pi$
$491$	−8.67892	−0.391674	−0.195837	+	0.980636i	$0.562742\pi$
$492$	0	0
$493$	−6.89406	−0.310493
$494$	15.7551	0.708855
$495$	0	0
$496$	5.36316	0.240813
$497$	41.2796	1.85164
$498$	0	0
$499$	−6.77589	−0.303331	−0.151665	−	0.988432i	$-0.548464\pi$
$499$	−6.77589	−0.303331	−0.151665	+	0.988432i	$0.548464\pi$
$500$	0	0
$501$	0	0
$502$	48.0780	2.14582
$503$	−1.13028	−0.0503969	−0.0251984	−	0.999682i	$-0.508022\pi$
$503$	−1.13028	−0.0503969	−0.0251984	+	0.999682i	$0.508022\pi$
$504$	0	0
$505$	0	0
$506$	−36.4186	−1.61900
$507$	0	0
$508$	−8.74547	−0.388017
$509$	−5.88844	−0.261000	−0.130500	−	0.991448i	$-0.541658\pi$
$509$	−5.88844	−0.261000	−0.130500	+	0.991448i	$0.541658\pi$
$510$	0	0
$511$	35.1647	1.55559
$512$	−21.8943	−0.967599
$513$	0	0
$514$	−3.14501	−0.138720
$515$	0	0
$516$	0	0
$517$	−7.31863	−0.321873
$518$	97.9991	4.30583
$519$	0	0
$520$	0	0
$521$	8.50026	0.372403	0.186202	−	0.982512i	$-0.440382\pi$
$521$	8.50026	0.372403	0.186202	+	0.982512i	$0.440382\pi$
$522$	0	0
$523$	9.41273	0.411590	0.205795	−	0.978595i	$-0.434022\pi$
$523$	9.41273	0.411590	0.205795	+	0.978595i	$0.434022\pi$
$524$	−50.0045	−2.18446
$525$	0	0
$526$	56.7072	2.47255
$527$	−19.3164	−0.841437
$528$	0	0
$529$	14.8329	0.644911
$530$	0	0
$531$	0	0
$532$	50.8238	2.20349
$533$	24.6994	1.06985
$534$	0	0
$535$	0	0
$536$	−44.0004	−1.90053
$537$	0	0
$538$	15.4336	0.665388
$539$	−27.8994	−1.20171
$540$	0	0
$541$	4.40257	0.189281	0.0946406	−	0.995512i	$-0.469830\pi$
$541$	4.40257	0.189281	0.0946406	+	0.995512i	$0.469830\pi$
$542$	−22.5965	−0.970602
$543$	0	0
$544$	22.1132	0.948096
$545$	0	0
$546$	0	0
$547$	−13.2431	−0.566235	−0.283117	−	0.959085i	$-0.591369\pi$
$547$	−13.2431	−0.566235	−0.283117	+	0.959085i	$0.591369\pi$
$548$	−12.5599	−0.536531
$549$	0	0
$550$	0	0
$551$	3.10797	0.132404
$552$	0	0
$553$	32.6556	1.38866
$554$	−21.7464	−0.923915
$555$	0	0
$556$	−7.20150	−0.305411
$557$	−0.997628	−0.0422709	−0.0211354	−	0.999777i	$-0.506728\pi$
$557$	−0.997628	−0.0422709	−0.0211354	+	0.999777i	$0.506728\pi$
$558$	0	0
$559$	16.1391	0.682609
$560$	0	0
$561$	0	0
$562$	−40.8686	−1.72394
$563$	−5.45230	−0.229787	−0.114893	−	0.993378i	$-0.536653\pi$
$563$	−5.45230	−0.229787	−0.114893	+	0.993378i	$0.536653\pi$
$564$	0	0
$565$	0	0
$566$	−53.6546	−2.25527
$567$	0	0
$568$	37.8963	1.59009
$569$	−8.03392	−0.336799	−0.168400	−	0.985719i	$-0.553860\pi$
$569$	−8.03392	−0.336799	−0.168400	+	0.985719i	$0.553860\pi$
$570$	0	0
$571$	−21.0647	−0.881528	−0.440764	−	0.897623i	$-0.645293\pi$
$571$	−21.0647	−0.881528	−0.440764	+	0.897623i	$0.645293\pi$
$572$	−18.4483	−0.771362
$573$	0	0
$574$	123.360	5.14897
$575$	0	0
$576$	0	0
$577$	−27.9479	−1.16349	−0.581744	−	0.813372i	$-0.697630\pi$
$577$	−27.9479	−1.16349	−0.581744	+	0.813372i	$0.697630\pi$
$578$	84.3249	3.50745
$579$	0	0
$580$	0	0
$581$	−44.0416	−1.82715
$582$	0	0
$583$	31.0022	1.28398
$584$	32.2826	1.33586
$585$	0	0
$586$	−50.7445	−2.09624
$587$	−43.6656	−1.80227	−0.901137	−	0.433534i	$-0.857266\pi$
$587$	−43.6656	−1.80227	−0.901137	+	0.433534i	$0.857266\pi$
$588$	0	0
$589$	8.70820	0.358815
$590$	0	0
$591$	0	0
$592$	19.4430	0.799104
$593$	27.8198	1.14242	0.571212	−	0.820803i	$-0.306473\pi$
$593$	27.8198	1.14242	0.571212	+	0.820803i	$0.306473\pi$
$594$	0	0
$595$	0	0
$596$	−23.7938	−0.974630
$597$	0	0
$598$	29.6721	1.21338
$599$	−46.3084	−1.89211	−0.946054	−	0.324008i	$-0.894970\pi$
$599$	−46.3084	−1.89211	−0.946054	+	0.324008i	$0.894970\pi$
$600$	0	0
$601$	−18.1955	−0.742209	−0.371105	−	0.928591i	$-0.621021\pi$
$601$	−18.1955	−0.742209	−0.371105	+	0.928591i	$0.621021\pi$
$602$	80.6061	3.28526
$603$	0	0
$604$	−29.6032	−1.20454
$605$	0	0
$606$	0	0
$607$	34.6351	1.40579	0.702897	−	0.711292i	$-0.251890\pi$
$607$	34.6351	1.40579	0.702897	+	0.711292i	$0.251890\pi$
$608$	−9.96904	−0.404298
$609$	0	0
$610$	0	0
$611$	5.96286	0.241232
$612$	0	0
$613$	34.2121	1.38181	0.690907	−	0.722944i	$-0.257212\pi$
$613$	34.2121	1.38181	0.690907	+	0.722944i	$0.257212\pi$
$614$	−20.0431	−0.808875
$615$	0	0
$616$	−41.6232	−1.67705
$617$	−12.6968	−0.511152	−0.255576	−	0.966789i	$-0.582265\pi$
$617$	−12.6968	−0.511152	−0.255576	+	0.966789i	$0.582265\pi$
$618$	0	0
$619$	26.8216	1.07805	0.539025	−	0.842290i	$-0.318793\pi$
$619$	26.8216	1.07805	0.539025	+	0.842290i	$0.318793\pi$
$620$	0	0
$621$	0	0
$622$	21.7082	0.870420
$623$	4.25582	0.170506
$624$	0	0
$625$	0	0
$626$	49.8319	1.99169
$627$	0	0
$628$	−58.5653	−2.33701
$629$	−70.0277	−2.79219
$630$	0	0
$631$	−29.6972	−1.18223	−0.591114	−	0.806588i	$-0.701312\pi$
$631$	−29.6972	−1.18223	−0.591114	+	0.806588i	$0.701312\pi$
$632$	29.9791	1.19251
$633$	0	0
$634$	−30.3153	−1.20397
$635$	0	0
$636$	0	0
$637$	22.7310	0.900636
$638$	−5.63451	−0.223072
$639$	0	0
$640$	0	0
$641$	2.85489	0.112762	0.0563808	−	0.998409i	$-0.482044\pi$
$641$	2.85489	0.112762	0.0563808	+	0.998409i	$0.482044\pi$
$642$	0	0
$643$	30.0385	1.18460	0.592301	−	0.805717i	$-0.298220\pi$
$643$	30.0385	1.18460	0.592301	+	0.805717i	$0.298220\pi$
$644$	95.7180	3.77182
$645$	0	0
$646$	−56.2288	−2.21229
$647$	−2.12791	−0.0836569	−0.0418284	−	0.999125i	$-0.513318\pi$
$647$	−2.12791	−0.0836569	−0.0418284	+	0.999125i	$0.513318\pi$
$648$	0	0
$649$	20.2720	0.795745
$650$	0	0
$651$	0	0
$652$	−21.1115	−0.826792
$653$	15.5057	0.606783	0.303392	−	0.952866i	$-0.401881\pi$
$653$	15.5057	0.606783	0.303392	+	0.952866i	$0.401881\pi$
$654$	0	0
$655$	0	0
$656$	24.4747	0.955578
$657$	0	0
$658$	29.7814	1.16100
$659$	8.54626	0.332915	0.166458	−	0.986049i	$-0.446767\pi$
$659$	8.54626	0.332915	0.166458	+	0.986049i	$0.446767\pi$
$660$	0	0
$661$	27.9246	1.08614	0.543070	−	0.839687i	$-0.317262\pi$
$661$	27.9246	1.08614	0.543070	+	0.839687i	$0.317262\pi$
$662$	−37.5549	−1.45961
$663$	0	0
$664$	−40.4320	−1.56906
$665$	0	0
$666$	0	0
$667$	5.85334	0.226642
$668$	−44.2538	−1.71223
$669$	0	0
$670$	0	0
$671$	8.30542	0.320627
$672$	0	0
$673$	11.1115	0.428319	0.214159	−	0.976799i	$-0.431299\pi$
$673$	11.1115	0.428319	0.214159	+	0.976799i	$0.431299\pi$
$674$	−24.3880	−0.939391
$675$	0	0
$676$	−32.3920	−1.24585
$677$	−28.8492	−1.10877	−0.554383	−	0.832262i	$-0.687046\pi$
$677$	−28.8492	−1.10877	−0.554383	+	0.832262i	$0.687046\pi$
$678$	0	0
$679$	−22.6704	−0.870010
$680$	0	0
$681$	0	0
$682$	−15.7873	−0.604527
$683$	−18.9069	−0.723454	−0.361727	−	0.932284i	$-0.617813\pi$
$683$	−18.9069	−0.723454	−0.361727	+	0.932284i	$0.617813\pi$
$684$	0	0
$685$	0	0
$686$	42.5625	1.62504
$687$	0	0
$688$	15.9923	0.609699
$689$	−25.2591	−0.962295
$690$	0	0
$691$	5.03727	0.191627	0.0958133	−	0.995399i	$-0.469455\pi$
$691$	5.03727	0.191627	0.0958133	+	0.995399i	$0.469455\pi$
$692$	50.1723	1.90726
$693$	0	0
$694$	80.5677	3.05831
$695$	0	0
$696$	0	0
$697$	−88.1503	−3.33893
$698$	−84.0646	−3.18189
$699$	0	0
$700$	0	0
$701$	−41.3716	−1.56258	−0.781291	−	0.624167i	$-0.785439\pi$
$701$	−41.3716	−1.56258	−0.781291	+	0.624167i	$0.785439\pi$
$702$	0	0
$703$	31.5698	1.19068
$704$	28.0956	1.05889
$705$	0	0
$706$	49.1383	1.84934
$707$	53.6546	2.01789
$708$	0	0
$709$	−36.1114	−1.35619	−0.678096	−	0.734973i	$-0.737195\pi$
$709$	−36.1114	−1.35619	−0.678096	+	0.734973i	$0.737195\pi$
$710$	0	0
$711$	0	0
$712$	3.90702	0.146422
$713$	16.4004	0.614201
$714$	0	0
$715$	0	0
$716$	−60.5867	−2.26423
$717$	0	0
$718$	−21.5313	−0.803542
$719$	−13.2982	−0.495940	−0.247970	−	0.968768i	$-0.579764\pi$
$719$	−13.2982	−0.495940	−0.247970	+	0.968768i	$0.579764\pi$
$720$	0	0
$721$	53.0780	1.97673
$722$	−19.8051	−0.737071
$723$	0	0
$724$	21.1063	0.784409
$725$	0	0
$726$	0	0
$727$	−13.9598	−0.517741	−0.258871	−	0.965912i	$-0.583350\pi$
$727$	−13.9598	−0.517741	−0.258871	+	0.965912i	$0.583350\pi$
$728$	33.9125	1.25688
$729$	0	0
$730$	0	0
$731$	−57.5991	−2.13038
$732$	0	0
$733$	6.51756	0.240731	0.120366	−	0.992730i	$-0.461593\pi$
$733$	6.51756	0.240731	0.120366	+	0.992730i	$0.461593\pi$
$734$	71.7078	2.64678
$735$	0	0
$736$	−18.7750	−0.692056
$737$	27.9913	1.03107
$738$	0	0
$739$	−46.2414	−1.70102	−0.850509	−	0.525960i	$-0.823706\pi$
$739$	−46.2414	−1.70102	−0.850509	+	0.525960i	$0.823706\pi$
$740$	0	0
$741$	0	0
$742$	−126.156	−4.63133
$743$	39.4507	1.44731	0.723653	−	0.690163i	$-0.242461\pi$
$743$	39.4507	1.44731	0.723653	+	0.690163i	$0.242461\pi$
$744$	0	0
$745$	0	0
$746$	51.9077	1.90048
$747$	0	0
$748$	65.8407	2.40737
$749$	0.253078	0.00924725
$750$	0	0
$751$	−26.9003	−0.981606	−0.490803	−	0.871271i	$-0.663296\pi$
$751$	−26.9003	−0.981606	−0.490803	+	0.871271i	$0.663296\pi$
$752$	5.90863	0.215466
$753$	0	0
$754$	4.59072	0.167184
$755$	0	0
$756$	0	0
$757$	26.3433	0.957462	0.478731	−	0.877962i	$-0.341097\pi$
$757$	26.3433	0.957462	0.478731	+	0.877962i	$0.341097\pi$
$758$	33.1098	1.20260
$759$	0	0
$760$	0	0
$761$	−22.3582	−0.810484	−0.405242	−	0.914209i	$-0.632813\pi$
$761$	−22.3582	−0.810484	−0.405242	+	0.914209i	$0.632813\pi$
$762$	0	0
$763$	4.49308	0.162660
$764$	−56.8437	−2.05653
$765$	0	0
$766$	−9.17522	−0.331514
$767$	−16.5166	−0.596380
$768$	0	0
$769$	−47.9185	−1.72799	−0.863993	−	0.503504i	$-0.832044\pi$
$769$	−47.9185	−1.72799	−0.863993	+	0.503504i	$0.832044\pi$
$770$	0	0
$771$	0	0
$772$	−56.0593	−2.01762
$773$	−6.38883	−0.229790	−0.114895	−	0.993378i	$-0.536653\pi$
$773$	−6.38883	−0.229790	−0.114895	+	0.993378i	$0.536653\pi$
$774$	0	0
$775$	0	0
$776$	−20.8123	−0.747119
$777$	0	0
$778$	−72.2246	−2.58938
$779$	39.7398	1.42383
$780$	0	0
$781$	−24.1082	−0.862658
$782$	−105.897	−3.78688
$783$	0	0
$784$	22.5243	0.804439
$785$	0	0
$786$	0	0
$787$	23.8410	0.849839	0.424920	−	0.905231i	$-0.360302\pi$
$787$	23.8410	0.849839	0.424920	+	0.905231i	$0.360302\pi$
$788$	23.7451	0.845885
$789$	0	0
$790$	0	0
$791$	−3.48903	−0.124056
$792$	0	0
$793$	−6.76685	−0.240298
$794$	16.3754	0.581143
$795$	0	0
$796$	−58.3554	−2.06835
$797$	−21.4117	−0.758440	−0.379220	−	0.925306i	$-0.623808\pi$
$797$	−21.4117	−0.758440	−0.379220	+	0.925306i	$0.623808\pi$
$798$	0	0
$799$	−21.2810	−0.752869
$800$	0	0
$801$	0	0
$802$	77.2817	2.72891
$803$	−20.5369	−0.724733
$804$	0	0
$805$	0	0
$806$	12.8627	0.453070
$807$	0	0
$808$	49.2571	1.73286
$809$	−21.7118	−0.763348	−0.381674	−	0.924297i	$-0.624652\pi$
$809$	−21.7118	−0.763348	−0.381674	+	0.924297i	$0.624652\pi$
$810$	0	0
$811$	13.5518	0.475868	0.237934	−	0.971281i	$-0.423530\pi$
$811$	13.5518	0.475868	0.237934	+	0.971281i	$0.423530\pi$
$812$	14.8091	0.519696
$813$	0	0
$814$	−57.2336	−2.00604
$815$	0	0
$816$	0	0
$817$	25.9668	0.908462
$818$	−11.2033	−0.391716
$819$	0	0
$820$	0	0
$821$	−11.9433	−0.416824	−0.208412	−	0.978041i	$-0.566829\pi$
$821$	−11.9433	−0.416824	−0.208412	+	0.978041i	$0.566829\pi$
$822$	0	0
$823$	33.3183	1.16140	0.580701	−	0.814117i	$-0.302778\pi$
$823$	33.3183	1.16140	0.580701	+	0.814117i	$0.302778\pi$
$824$	48.7277	1.69751
$825$	0	0
$826$	−82.4918	−2.87026
$827$	−47.6602	−1.65731	−0.828653	−	0.559763i	$-0.810892\pi$
$827$	−47.6602	−1.65731	−0.828653	+	0.559763i	$0.810892\pi$
$828$	0	0
$829$	3.26594	0.113431	0.0567154	−	0.998390i	$-0.481937\pi$
$829$	3.26594	0.113431	0.0567154	+	0.998390i	$0.481937\pi$
$830$	0	0
$831$	0	0
$832$	−22.8909	−0.793599
$833$	−81.1254	−2.81083
$834$	0	0
$835$	0	0
$836$	−29.6822	−1.02658
$837$	0	0
$838$	56.9712	1.96804
$839$	−54.3532	−1.87648	−0.938240	−	0.345986i	$-0.887544\pi$
$839$	−54.3532	−1.87648	−0.938240	+	0.345986i	$0.887544\pi$
$840$	0	0
$841$	−28.0944	−0.968772
$842$	65.8215	2.26836
$843$	0	0
$844$	−57.3278	−1.97330
$845$	0	0
$846$	0	0
$847$	−20.4463	−0.702543
$848$	−25.0294	−0.859512
$849$	0	0
$850$	0	0
$851$	59.4564	2.03814
$852$	0	0
$853$	−42.2955	−1.44817	−0.724085	−	0.689711i	$-0.757738\pi$
$853$	−42.2955	−1.44817	−0.724085	+	0.689711i	$0.757738\pi$
$854$	−33.7968	−1.15650
$855$	0	0
$856$	0.232335	0.00794106
$857$	−39.7714	−1.35856	−0.679282	−	0.733877i	$-0.737709\pi$
$857$	−39.7714	−1.35856	−0.679282	+	0.733877i	$0.737709\pi$
$858$	0	0
$859$	20.9842	0.715973	0.357986	−	0.933727i	$-0.383463\pi$
$859$	20.9842	0.715973	0.357986	+	0.933727i	$0.383463\pi$
$860$	0	0
$861$	0	0
$862$	14.5955	0.497125
$863$	−21.5346	−0.733045	−0.366522	−	0.930409i	$-0.619452\pi$
$863$	−21.5346	−0.733045	−0.366522	+	0.930409i	$0.619452\pi$
$864$	0	0
$865$	0	0
$866$	36.4409	1.23831
$867$	0	0
$868$	41.4934	1.40838
$869$	−19.0716	−0.646959
$870$	0	0
$871$	−22.8060	−0.772751
$872$	4.12483	0.139684
$873$	0	0
$874$	47.7405	1.61485
$875$	0	0
$876$	0	0
$877$	−33.4164	−1.12839	−0.564196	−	0.825641i	$-0.690814\pi$
$877$	−33.4164	−1.12839	−0.564196	+	0.825641i	$0.690814\pi$
$878$	−62.0679	−2.09469
$879$	0	0
$880$	0	0
$881$	3.37862	0.113829	0.0569143	−	0.998379i	$-0.481874\pi$
$881$	3.37862	0.113829	0.0569143	+	0.998379i	$0.481874\pi$
$882$	0	0
$883$	−8.46965	−0.285027	−0.142513	−	0.989793i	$-0.545518\pi$
$883$	−8.46965	−0.285027	−0.142513	+	0.989793i	$0.545518\pi$
$884$	−53.6437	−1.80423
$885$	0	0
$886$	36.2900	1.21919
$887$	−0.671898	−0.0225601	−0.0112801	−	0.999936i	$-0.503591\pi$
$887$	−0.671898	−0.0225601	−0.0112801	+	0.999936i	$0.503591\pi$
$888$	0	0
$889$	−10.2271	−0.343007
$890$	0	0
$891$	0	0
$892$	−53.8419	−1.80276
$893$	9.59388	0.321047
$894$	0	0
$895$	0	0
$896$	−88.2851	−2.94940
$897$	0	0
$898$	−81.8466	−2.73126
$899$	2.53740	0.0846270
$900$	0	0
$901$	90.1479	3.00326
$902$	−72.0452	−2.39884
$903$	0	0
$904$	−3.20307	−0.106533
$905$	0	0
$906$	0	0
$907$	38.8659	1.29052	0.645260	−	0.763963i	$-0.276749\pi$
$907$	38.8659	1.29052	0.645260	+	0.763963i	$0.276749\pi$
$908$	26.4271	0.877016
$909$	0	0
$910$	0	0
$911$	−10.6457	−0.352709	−0.176355	−	0.984327i	$-0.556431\pi$
$911$	−10.6457	−0.352709	−0.176355	+	0.984327i	$0.556431\pi$
$912$	0	0
$913$	25.7213	0.851250
$914$	20.6597	0.683362
$915$	0	0
$916$	−12.3934	−0.409489
$917$	−58.4763	−1.93106
$918$	0	0
$919$	−0.508850	−0.0167854	−0.00839271	−	0.999965i	$-0.502672\pi$
$919$	−0.508850	−0.0167854	−0.00839271	+	0.999965i	$0.502672\pi$
$920$	0	0
$921$	0	0
$922$	27.8768	0.918074
$923$	19.6422	0.646529
$924$	0	0
$925$	0	0
$926$	−44.7885	−1.47184
$927$	0	0
$928$	−2.90478	−0.0953542
$929$	36.7185	1.20469	0.602347	−	0.798234i	$-0.294232\pi$
$929$	36.7185	1.20469	0.602347	+	0.798234i	$0.294232\pi$
$930$	0	0
$931$	36.5728	1.19863
$932$	79.4152	2.60133
$933$	0	0
$934$	−43.5035	−1.42348
$935$	0	0
$936$	0	0
$937$	−19.9423	−0.651488	−0.325744	−	0.945458i	$-0.605615\pi$
$937$	−19.9423	−0.651488	−0.325744	+	0.945458i	$0.605615\pi$
$938$	−113.904	−3.71909
$939$	0	0
$940$	0	0
$941$	14.0888	0.459281	0.229641	−	0.973276i	$-0.426245\pi$
$941$	14.0888	0.459281	0.229641	+	0.973276i	$0.426245\pi$
$942$	0	0
$943$	74.8432	2.43723
$944$	−16.3664	−0.532681
$945$	0	0
$946$	−47.0757	−1.53056
$947$	17.5552	0.570466	0.285233	−	0.958458i	$-0.407929\pi$
$947$	17.5552	0.570466	0.285233	+	0.958458i	$0.407929\pi$
$948$	0	0
$949$	16.7325	0.543159
$950$	0	0
$951$	0	0
$952$	−121.031	−3.92265
$953$	10.5915	0.343093	0.171546	−	0.985176i	$-0.445124\pi$
$953$	10.5915	0.343093	0.171546	+	0.985176i	$0.445124\pi$
$954$	0	0
$955$	0	0
$956$	29.5785	0.956637
$957$	0	0
$958$	−51.2451	−1.65566
$959$	−14.6878	−0.474293
$960$	0	0
$961$	−23.8905	−0.770660
$962$	46.6311	1.50345
$963$	0	0
$964$	73.3095	2.36114
$965$	0	0
$966$	0	0
$967$	46.6537	1.50028	0.750141	−	0.661278i	$-0.229985\pi$
$967$	46.6537	1.50028	0.750141	+	0.661278i	$0.229985\pi$
$968$	−18.7705	−0.603307
$969$	0	0
$970$	0	0
$971$	14.7798	0.474305	0.237153	−	0.971472i	$-0.423786\pi$
$971$	14.7798	0.474305	0.237153	+	0.971472i	$0.423786\pi$
$972$	0	0
$973$	−8.42158	−0.269984
$974$	29.1400	0.933707
$975$	0	0
$976$	−6.70530	−0.214631
$977$	9.55036	0.305543	0.152771	−	0.988262i	$-0.451180\pi$
$977$	9.55036	0.305543	0.152771	+	0.988262i	$0.451180\pi$
$978$	0	0
$979$	−2.48549	−0.0794367
$980$	0	0
$981$	0	0
$982$	−20.6257	−0.658193
$983$	33.7083	1.07513	0.537564	−	0.843223i	$-0.319345\pi$
$983$	33.7083	1.07513	0.537564	+	0.843223i	$0.319345\pi$
$984$	0	0
$985$	0	0
$986$	−16.3840	−0.521772
$987$	0	0
$988$	24.1836	0.769383
$989$	48.9040	1.55506
$990$	0	0
$991$	14.5129	0.461016	0.230508	−	0.973070i	$-0.425961\pi$
$991$	14.5129	0.461016	0.230508	+	0.973070i	$0.425961\pi$
$992$	−8.13889	−0.258410
$993$	0	0
$994$	98.1022	3.11161
$995$	0	0
$996$	0	0
$997$	22.6197	0.716374	0.358187	−	0.933650i	$-0.383395\pi$
$997$	22.6197	0.716374	0.358187	+	0.933650i	$0.383395\pi$
$998$	−16.1031	−0.509736
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.bb.1.7	yes	8
3.2	odd	2	inner	5625.2.a.bb.1.2	yes	8
5.4	even	2		5625.2.a.z.1.2	✓	8
15.14	odd	2		5625.2.a.z.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5625.2.a.z.1.2	✓	8	5.4	even	2
5625.2.a.z.1.7	yes	8	15.14	odd	2
5625.2.a.bb.1.2	yes	8	3.2	odd	2	inner
5625.2.a.bb.1.7	yes	8	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37653 q^{2} +3.64791 q^{4} +4.26594 q^{7} +3.91630 q^{8} +O(q^{10})\)	\(q+2.37653 q^{2} +3.64791 q^{4} +4.26594 q^{7} +3.91630 q^{8} -2.49140 q^{11} +2.02987 q^{13} +10.1381 q^{14} +2.01141 q^{16} -7.24447 q^{17} +3.26594 q^{19} -5.92090 q^{22} +6.15085 q^{23} +4.82406 q^{26} +15.5618 q^{28} +0.951631 q^{29} +2.66637 q^{31} -3.05243 q^{32} -17.2167 q^{34} +9.66637 q^{37} +7.76161 q^{38} +12.1679 q^{41} +7.95077 q^{43} -9.08840 q^{44} +14.6177 q^{46} +2.93756 q^{47} +11.1983 q^{49} +7.40479 q^{52} -12.4437 q^{53} +16.7067 q^{56} +2.26158 q^{58} -8.13677 q^{59} -3.33363 q^{61} +6.33671 q^{62} -11.2770 q^{64} -11.2352 q^{67} -26.4271 q^{68} +9.67655 q^{71} +8.24312 q^{73} +22.9724 q^{74} +11.9138 q^{76} -10.6282 q^{77} +7.65496 q^{79} +28.9175 q^{82} -10.3240 q^{83} +18.8953 q^{86} -9.75709 q^{88} +0.997628 q^{89} +8.65932 q^{91} +22.4377 q^{92} +6.98120 q^{94} -5.31428 q^{97} +26.6130 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{4} + 10 q^{7}+O(q^{10}) \) 8 * q + 14 * q^4 + 10 * q^7	\( 8 q + 14 q^{4} + 10 q^{7} + 10 q^{13} + 22 q^{16} + 2 q^{19} + 10 q^{22} + 70 q^{28} - 6 q^{31} - 50 q^{34} + 50 q^{37} + 14 q^{49} + 80 q^{52} - 30 q^{58} - 54 q^{61} + 36 q^{64} + 10 q^{67} + 30 q^{73} + 56 q^{76} + 28 q^{79} + 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100}) \) 8 * q + 14 * q^4 + 10 * q^7 + 10 * q^13 + 22 * q^16 + 2 * q^19 + 10 * q^22 + 70 * q^28 - 6 * q^31 - 50 * q^34 + 50 * q^37 + 14 * q^49 + 80 * q^52 - 30 * q^58 - 54 * q^61 + 36 * q^64 + 10 * q^67 + 30 * q^73 + 56 * q^76 + 28 * q^79 + 20 * q^88 + 60 * q^91 - 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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