LMFDB - Embedded newform 5625.2.a.z.1.2

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:	$1$
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			1.2
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	$-0.817581\pi$
$2$	−2.37653	−1.68046	−0.840231	+	0.542228i	$0.817581\pi$
$3$	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	$-0.798472\pi$
$7$	−4.26594	−1.61237	−0.806187	+	0.591661i	$0.798472\pi$
$8$	−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	$-0.590830\pi$
$13$	−2.02987	−0.562985	−0.281493	+	0.959563i	$0.590830\pi$
$14$	10.1381	2.70953
$15$	0	0
$16$	2.01141	0.502853
$17$	7.24447	1.75704	0.878521	−	0.477704i	$-0.158531\pi$
$17$	7.24447	1.75704	0.878521	+	0.477704i	$0.158531\pi$
$18$	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	$-0.721592\pi$
$23$	−6.15085	−1.28254	−0.641270	+	0.767315i	$0.721592\pi$
$24$	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	$-0.792304\pi$
$37$	−9.66637	−1.58914	−0.794571	+	0.607172i	$0.792304\pi$
$38$	−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	$-0.707323\pi$
$43$	−7.95077	−1.21248	−0.606241	+	0.795281i	$0.707323\pi$
$44$	−9.08840	−1.37013
$45$	0	0
$46$	14.6177	2.15526
$47$	−2.93756	−0.428487	−0.214243	−	0.976780i	$-0.568729\pi$
$47$	−2.93756	−0.428487	−0.214243	+	0.976780i	$0.568729\pi$
$48$	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	$-0.173780\pi$
$53$	12.4437	1.70927	0.854636	+	0.519228i	$0.173780\pi$
$54$	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	$-0.259235\pi$
$67$	11.2352	1.37260	0.686298	+	0.727321i	$0.259235\pi$
$68$	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	$-0.660232\pi$
$73$	−8.24312	−0.964784	−0.482392	+	0.875955i	$0.660232\pi$
$74$	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	$-0.308257\pi$
$83$	10.3240	1.13321	0.566604	+	0.823990i	$0.308257\pi$
$84$	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	$-0.413045\pi$
$97$	5.31428	0.539583	0.269791	+	0.962919i	$0.413045\pi$
$98$	−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	$-0.710032\pi$
$103$	−12.4423	−1.22597	−0.612986	+	0.790094i	$0.710032\pi$
$104$	7.94960	0.779522
$105$	0	0
$106$	−29.5728	−2.87237
$107$	−0.0593251	−0.00573518	−0.00286759	−	0.999996i	$-0.500913\pi$
$107$	−0.0593251	−0.00573518	−0.00286759	+	0.999996i	$0.500913\pi$
$108$	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	$-0.487752\pi$
$113$	0.817881	0.0769398	0.0384699	+	0.999260i	$0.487752\pi$
$114$	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	$-0.466078\pi$
$127$	2.39739	0.212734	0.106367	+	0.994327i	$0.466078\pi$
$128$	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	$-0.453013\pi$
$137$	3.44303	0.294158	0.147079	+	0.989125i	$0.453013\pi$
$138$	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	$-0.278667\pi$
$157$	16.0545	1.28129	0.640644	+	0.767838i	$0.278667\pi$
$158$	−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	$-0.427223\pi$
$163$	5.78730	0.453297	0.226648	+	0.973977i	$0.427223\pi$
$164$	44.3875	3.46608
$165$	0	0
$166$	−24.5353	−1.90431
$167$	12.1313	0.938747	0.469373	−	0.883000i	$-0.344480\pi$
$167$	12.1313	0.938747	0.469373	+	0.883000i	$0.344480\pi$
$168$	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	$-0.675127\pi$
$173$	−13.7537	−1.04568	−0.522838	+	0.852432i	$0.675127\pi$
$174$	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	$-0.313449\pi$
$193$	15.3675	1.10618	0.553089	+	0.833122i	$0.313449\pi$
$194$	−12.6295	−0.906749
$195$	0	0
$196$	40.8502	2.91787
$197$	−6.50925	−0.463765	−0.231882	−	0.972744i	$-0.574488\pi$
$197$	−6.50925	−0.463765	−0.231882	+	0.972744i	$0.574488\pi$
$198$	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	$-0.335465\pi$
$223$	14.7597	0.988380	0.494190	+	0.869354i	$0.335465\pi$
$224$	−13.0215	−0.870033
$225$	0	0
$226$	−1.94372	−0.129294
$227$	−7.24447	−0.480832	−0.240416	−	0.970670i	$-0.577284\pi$
$227$	−7.24447	−0.480832	−0.240416	+	0.970670i	$0.577284\pi$
$228$	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	$-0.752711\pi$
$233$	−21.7701	−1.42620	−0.713102	+	0.701060i	$0.752711\pi$
$234$	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	$-0.486858\pi$
$257$	1.32336	0.0825489	0.0412744	+	0.999148i	$0.486858\pi$
$258$	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	$-0.763135\pi$
$263$	−23.8613	−1.47135	−0.735676	+	0.677334i	$0.763135\pi$
$264$	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	$-0.411356\pi$
$277$	9.15047	0.549798	0.274899	+	0.961473i	$0.411356\pi$
$278$	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	$-0.265853\pi$
$283$	22.5768	1.34205	0.671027	+	0.741433i	$0.265853\pi$
$284$	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	$-0.285625\pi$
$293$	21.3523	1.24742	0.623709	+	0.781657i	$0.285625\pi$
$294$	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	$-0.422633\pi$
$307$	8.43377	0.481341	0.240670	+	0.970607i	$0.422633\pi$
$308$	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	$-0.701899\pi$
$313$	−20.9683	−1.18520	−0.592600	+	0.805497i	$0.701899\pi$
$314$	−38.1540	−2.15316
$315$	0	0
$316$	27.9246	1.57088
$317$	12.7561	0.716453	0.358227	−	0.933635i	$-0.383382\pi$
$317$	12.7561	0.716453	0.358227	+	0.933635i	$0.383382\pi$
$318$	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	$-0.409830\pi$
$337$	10.2620	0.559008	0.279504	+	0.960145i	$0.409830\pi$
$338$	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	$-0.863888\pi$
$347$	−33.9014	−1.81992	−0.909960	+	0.414696i	$0.863888\pi$
$348$	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	$-0.685467\pi$
$353$	−20.6764	−1.10050	−0.550248	+	0.835001i	$0.685467\pi$
$354$	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	$-0.788633\pi$
$367$	−30.1733	−1.57503	−0.787516	+	0.616294i	$0.788633\pi$
$368$	−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	$-0.691302\pi$
$373$	−21.8418	−1.13093	−0.565463	+	0.824774i	$0.691302\pi$
$374$	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	$-0.468551\pi$
$383$	3.86076	0.197276	0.0986378	+	0.995123i	$0.468551\pi$
$384$	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	$-0.555317\pi$
$397$	−6.89048	−0.345823	−0.172912	+	0.984937i	$0.555317\pi$
$398$	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	$-0.620109\pi$
$433$	−15.3336	−0.736887	−0.368444	+	0.929650i	$0.620109\pi$
$434$	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	$-0.618163\pi$
$443$	−15.2702	−0.725507	−0.362753	+	0.931885i	$0.618163\pi$
$444$	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	$-0.565175\pi$
$457$	−8.69321	−0.406651	−0.203326	+	0.979111i	$0.565175\pi$
$458$	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	$-0.355713\pi$
$463$	18.8462	0.875855	0.437928	+	0.899010i	$0.355713\pi$
$464$	1.91412	0.0888608
$465$	0	0
$466$	51.7373	2.39668
$467$	18.3055	0.847076	0.423538	−	0.905878i	$-0.360788\pi$
$467$	18.3055	0.847076	0.423538	+	0.905878i	$0.360788\pi$
$468$	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	$-0.589609\pi$
$487$	−12.2616	−0.555625	−0.277813	+	0.960635i	$0.589609\pi$
$488$	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	$-0.491978\pi$
$503$	1.13028	0.0503969	0.0251984	+	0.999682i	$0.491978\pi$
$504$	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	$-0.565978\pi$
$523$	−9.41273	−0.411590	−0.205795	+	0.978595i	$0.565978\pi$
$524$	−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	$-0.408631\pi$
$547$	13.2431	0.566235	0.283117	+	0.959085i	$0.408631\pi$
$548$	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	$-0.493272\pi$
$557$	0.997628	0.0422709	0.0211354	+	0.999777i	$0.493272\pi$
$558$	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	$-0.463347\pi$
$563$	5.45230	0.229787	0.114893	+	0.993378i	$0.463347\pi$
$564$	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	$-0.302370\pi$
$577$	27.9479	1.16349	0.581744	+	0.813372i	$0.302370\pi$
$578$	−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	$-0.142734\pi$
$587$	43.6656	1.80227	0.901137	+	0.433534i	$0.142734\pi$
$588$	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	$-0.693527\pi$
$593$	−27.8198	−1.14242	−0.571212	+	0.820803i	$0.693527\pi$
$594$	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	$-0.748110\pi$
$607$	−34.6351	−1.40579	−0.702897	+	0.711292i	$0.748110\pi$
$608$	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	$-0.742788\pi$
$613$	−34.2121	−1.38181	−0.690907	+	0.722944i	$0.742788\pi$
$614$	−20.0431	−0.808875
$615$	0	0
$616$	−41.6232	−1.67705
$617$	12.6968	0.511152	0.255576	−	0.966789i	$-0.417735\pi$
$617$	12.6968	0.511152	0.255576	+	0.966789i	$0.417735\pi$
$618$	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	$-0.701780\pi$
$643$	−30.0385	−1.18460	−0.592301	+	0.805717i	$0.701780\pi$
$644$	95.7180	3.77182
$645$	0	0
$646$	−56.2288	−2.21229
$647$	2.12791	0.0836569	0.0418284	−	0.999125i	$-0.486682\pi$
$647$	2.12791	0.0836569	0.0418284	+	0.999125i	$0.486682\pi$
$648$	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	$-0.598119\pi$
$653$	−15.5057	−0.606783	−0.303392	+	0.952866i	$0.598119\pi$
$654$	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	$-0.568701\pi$
$673$	−11.1115	−0.428319	−0.214159	+	0.976799i	$0.568701\pi$
$674$	−24.3880	−0.939391
$675$	0	0
$676$	−32.3920	−1.24585
$677$	28.8492	1.10877	0.554383	−	0.832262i	$-0.312954\pi$
$677$	28.8492	1.10877	0.554383	+	0.832262i	$0.312954\pi$
$678$	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	$-0.382187\pi$
$683$	18.9069	0.723454	0.361727	+	0.932284i	$0.382187\pi$
$684$	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	$-0.416650\pi$
$727$	13.9598	0.517741	0.258871	+	0.965912i	$0.416650\pi$
$728$	−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	$-0.538407\pi$
$733$	−6.51756	−0.240731	−0.120366	+	0.992730i	$0.538407\pi$
$734$	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	$-0.757539\pi$
$743$	−39.4507	−1.44731	−0.723653	+	0.690163i	$0.757539\pi$
$744$	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	$-0.658903\pi$
$757$	−26.3433	−0.957462	−0.478731	+	0.877962i	$0.658903\pi$
$758$	−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	$-0.463347\pi$
$773$	6.38883	0.229790	0.114895	+	0.993378i	$0.463347\pi$
$774$	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	$-0.639698\pi$
$787$	−23.8410	−0.849839	−0.424920	+	0.905231i	$0.639698\pi$
$788$	−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	$-0.376192\pi$
$797$	21.4117	0.758440	0.379220	+	0.925306i	$0.376192\pi$
$798$	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	$-0.697222\pi$
$823$	−33.3183	−1.16140	−0.580701	+	0.814117i	$0.697222\pi$
$824$	48.7277	1.69751
$825$	0	0
$826$	−82.4918	−2.87026
$827$	47.6602	1.65731	0.828653	−	0.559763i	$-0.189108\pi$
$827$	47.6602	1.65731	0.828653	+	0.559763i	$0.189108\pi$
$828$	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	$-0.242262\pi$
$853$	42.2955	1.44817	0.724085	+	0.689711i	$0.242262\pi$
$854$	−33.7968	−1.15650
$855$	0	0
$856$	0.232335	0.00794106
$857$	39.7714	1.35856	0.679282	−	0.733877i	$-0.262291\pi$
$857$	39.7714	1.35856	0.679282	+	0.733877i	$0.262291\pi$
$858$	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	$-0.380548\pi$
$863$	21.5346	0.733045	0.366522	+	0.930409i	$0.380548\pi$
$864$	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	$-0.309186\pi$
$877$	33.4164	1.12839	0.564196	+	0.825641i	$0.309186\pi$
$878$	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	$-0.454482\pi$
$883$	8.46965	0.285027	0.142513	+	0.989793i	$0.454482\pi$
$884$	−53.6437	−1.80423
$885$	0	0
$886$	36.2900	1.21919
$887$	0.671898	0.0225601	0.0112801	−	0.999936i	$-0.496409\pi$
$887$	0.671898	0.0225601	0.0112801	+	0.999936i	$0.496409\pi$
$888$	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	$-0.723251\pi$
$907$	−38.8659	−1.29052	−0.645260	+	0.763963i	$0.723251\pi$
$908$	−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	$-0.394385\pi$
$937$	19.9423	0.651488	0.325744	+	0.945458i	$0.394385\pi$
$938$	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	$-0.592071\pi$
$947$	−17.5552	−0.570466	−0.285233	+	0.958458i	$0.592071\pi$
$948$	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	$-0.554876\pi$
$953$	−10.5915	−0.343093	−0.171546	+	0.985176i	$0.554876\pi$
$954$	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	$-0.770015\pi$
$967$	−46.6537	−1.50028	−0.750141	+	0.661278i	$0.770015\pi$
$968$	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	$-0.548820\pi$
$977$	−9.55036	−0.305543	−0.152771	+	0.988262i	$0.548820\pi$
$978$	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	$-0.680655\pi$
$983$	−33.7083	−1.07513	−0.537564	+	0.843223i	$0.680655\pi$
$984$	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	$-0.616605\pi$
$997$	−22.6197	−0.716374	−0.358187	+	0.933650i	$0.616605\pi$
$998$	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.z.1.2	✓	8
3.2	odd	2	inner	5625.2.a.z.1.7	yes	8
5.4	even	2		5625.2.a.bb.1.7	yes	8
15.14	odd	2		5625.2.a.bb.1.2	yes	8

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

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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