LMFDB - Embedded newform 2475.2.a.bg.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.93185$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.93185 q^{2} +1.73205 q^{4} +2.44949 q^{7} -0.517638 q^{8} +O(q^{10})$	$q+1.93185 q^{2} +1.73205 q^{4} +2.44949 q^{7} -0.517638 q^{8} -1.00000 q^{11} +4.24264 q^{13} +4.73205 q^{14} -4.46410 q^{16} +5.27792 q^{17} -2.00000 q^{19} -1.93185 q^{22} +2.07055 q^{23} +8.19615 q^{26} +4.24264 q^{28} +9.46410 q^{29} -7.46410 q^{31} -7.58871 q^{32} +10.1962 q^{34} +4.89898 q^{37} -3.86370 q^{38} +9.46410 q^{41} -6.03579 q^{43} -1.73205 q^{44} +4.00000 q^{46} +2.82843 q^{47} -1.00000 q^{49} +7.34847 q^{52} +0.757875 q^{53} -1.26795 q^{56} +18.2832 q^{58} -0.928203 q^{59} +8.92820 q^{61} -14.4195 q^{62} -5.73205 q^{64} +13.3843 q^{67} +9.14162 q^{68} +6.00000 q^{71} -9.14162 q^{73} +9.46410 q^{74} -3.46410 q^{76} -2.44949 q^{77} -15.8564 q^{79} +18.2832 q^{82} -7.07107 q^{83} -11.6603 q^{86} +0.517638 q^{88} +8.53590 q^{89} +10.3923 q^{91} +3.58630 q^{92} +5.46410 q^{94} +3.58630 q^{97} -1.93185 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 4 q^{11} + 12 q^{14} - 4 q^{16} - 8 q^{19} + 12 q^{26} + 24 q^{29} - 16 q^{31} + 20 q^{34} + 24 q^{41} + 16 q^{46} - 4 q^{49} - 12 q^{56} + 24 q^{59} + 8 q^{61} - 16 q^{64} + 24 q^{71} + 24 q^{74} - 8 q^{79} - 12 q^{86} + 48 q^{89} + 8 q^{94}+O(q^{100}) $ 4 * q - 4 * q^11 + 12 * q^14 - 4 * q^16 - 8 * q^19 + 12 * q^26 + 24 * q^29 - 16 * q^31 + 20 * q^34 + 24 * q^41 + 16 * q^46 - 4 * q^49 - 12 * q^56 + 24 * q^59 + 8 * q^61 - 16 * q^64 + 24 * q^71 + 24 * q^74 - 8 * q^79 - 12 * q^86 + 48 * q^89 + 8 * 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 $

$2$	1.93185	1.36603	0.683013	−	0.730406i	$-0.260669\pi$
$2$	1.93185	1.36603	0.683013	+	0.730406i	$0.260669\pi$
$3$	0	0
$3$	0	0
$4$	1.73205	0.866025
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.44949	0.925820	0.462910	−	0.886405i	$-0.346805\pi$
$7$	2.44949	0.925820	0.462910	+	0.886405i	$0.346805\pi$
$8$	−0.517638	−0.183013
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	4.73205	1.26469
$15$	0	0
$16$	−4.46410	−1.11603
$17$	5.27792	1.28008	0.640041	−	0.768340i	$-0.278917\pi$
$17$	5.27792	1.28008	0.640041	+	0.768340i	$0.278917\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	−1.93185	−0.411872
$23$	2.07055	0.431740	0.215870	−	0.976422i	$-0.430741\pi$
$23$	2.07055	0.431740	0.215870	+	0.976422i	$0.430741\pi$
$24$	0	0
$25$	0	0
$26$	8.19615	1.60740
$27$	0	0
$28$	4.24264	0.801784
$29$	9.46410	1.75744	0.878720	−	0.477338i	$-0.158398\pi$
$29$	9.46410	1.75744	0.878720	+	0.477338i	$0.158398\pi$
$30$	0	0
$31$	−7.46410	−1.34059	−0.670296	−	0.742094i	$-0.733833\pi$
$31$	−7.46410	−1.34059	−0.670296	+	0.742094i	$0.733833\pi$
$32$	−7.58871	−1.34151
$33$	0	0
$34$	10.1962	1.74863
$35$	0	0
$36$	0	0
$37$	4.89898	0.805387	0.402694	−	0.915335i	$-0.368074\pi$
$37$	4.89898	0.805387	0.402694	+	0.915335i	$0.368074\pi$
$38$	−3.86370	−0.626775
$39$	0	0
$40$	0	0
$41$	9.46410	1.47804	0.739022	−	0.673681i	$-0.235288\pi$
$41$	9.46410	1.47804	0.739022	+	0.673681i	$0.235288\pi$
$42$	0	0
$43$	−6.03579	−0.920450	−0.460225	−	0.887802i	$-0.652231\pi$
$43$	−6.03579	−0.920450	−0.460225	+	0.887802i	$0.652231\pi$
$44$	−1.73205	−0.261116
$45$	0	0
$46$	4.00000	0.589768
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	7.34847	1.01905
$53$	0.757875	0.104102	0.0520511	−	0.998644i	$-0.483424\pi$
$53$	0.757875	0.104102	0.0520511	+	0.998644i	$0.483424\pi$
$54$	0	0
$55$	0	0
$56$	−1.26795	−0.169437
$57$	0	0
$58$	18.2832	2.40071
$59$	−0.928203	−0.120842	−0.0604209	−	0.998173i	$-0.519244\pi$
$59$	−0.928203	−0.120842	−0.0604209	+	0.998173i	$0.519244\pi$
$60$	0	0
$61$	8.92820	1.14314	0.571570	−	0.820554i	$-0.306335\pi$
$61$	8.92820	1.14314	0.571570	+	0.820554i	$0.306335\pi$
$62$	−14.4195	−1.83128
$63$	0	0
$64$	−5.73205	−0.716506
$65$	0	0
$66$	0	0
$67$	13.3843	1.63515	0.817574	−	0.575824i	$-0.195319\pi$
$67$	13.3843	1.63515	0.817574	+	0.575824i	$0.195319\pi$
$68$	9.14162	1.10858
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−9.14162	−1.06995	−0.534973	−	0.844869i	$-0.679678\pi$
$73$	−9.14162	−1.06995	−0.534973	+	0.844869i	$0.679678\pi$
$74$	9.46410	1.10018
$75$	0	0
$76$	−3.46410	−0.397360
$77$	−2.44949	−0.279145
$78$	0	0
$79$	−15.8564	−1.78399	−0.891993	−	0.452050i	$-0.850693\pi$
$79$	−15.8564	−1.78399	−0.891993	+	0.452050i	$0.850693\pi$
$80$	0	0
$81$	0	0
$82$	18.2832	2.01905
$83$	−7.07107	−0.776151	−0.388075	−	0.921628i	$-0.626860\pi$
$83$	−7.07107	−0.776151	−0.388075	+	0.921628i	$0.626860\pi$
$84$	0	0
$85$	0	0
$86$	−11.6603	−1.25736
$87$	0	0
$88$	0.517638	0.0551804
$89$	8.53590	0.904803	0.452402	−	0.891814i	$-0.350567\pi$
$89$	8.53590	0.904803	0.452402	+	0.891814i	$0.350567\pi$
$90$	0	0
$91$	10.3923	1.08941
$92$	3.58630	0.373898
$93$	0	0
$94$	5.46410	0.563579
$95$	0	0
$96$	0	0
$97$	3.58630	0.364134	0.182067	−	0.983286i	$-0.441721\pi$
$97$	3.58630	0.364134	0.182067	+	0.983286i	$0.441721\pi$
$98$	−1.93185	−0.195146
$99$	0	0
$100$	0	0
$101$	−4.39230	−0.437051	−0.218525	−	0.975831i	$-0.570125\pi$
$101$	−4.39230	−0.437051	−0.218525	+	0.975831i	$0.570125\pi$
$102$	0	0
$103$	−8.48528	−0.836080	−0.418040	−	0.908429i	$-0.637283\pi$
$103$	−8.48528	−0.836080	−0.418040	+	0.908429i	$0.637283\pi$
$104$	−2.19615	−0.215350
$105$	0	0
$106$	1.46410	0.142206
$107$	−1.41421	−0.136717	−0.0683586	−	0.997661i	$-0.521776\pi$
$107$	−1.41421	−0.136717	−0.0683586	+	0.997661i	$0.521776\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	−10.9348	−1.03324
$113$	5.65685	0.532152	0.266076	−	0.963952i	$-0.414273\pi$
$113$	5.65685	0.532152	0.266076	+	0.963952i	$0.414273\pi$
$114$	0	0
$115$	0	0
$116$	16.3923	1.52199
$117$	0	0
$118$	−1.79315	−0.165073
$119$	12.9282	1.18513
$120$	0	0
$121$	1.00000	0.0909091
$122$	17.2480	1.56156
$123$	0	0
$124$	−12.9282	−1.16099
$125$	0	0
$126$	0	0
$127$	−17.1464	−1.52150	−0.760750	−	0.649045i	$-0.775169\pi$
$127$	−17.1464	−1.52150	−0.760750	+	0.649045i	$0.775169\pi$
$128$	4.10394	0.362740
$129$	0	0
$130$	0	0
$131$	−13.8564	−1.21064	−0.605320	−	0.795982i	$-0.706955\pi$
$131$	−13.8564	−1.21064	−0.605320	+	0.795982i	$0.706955\pi$
$132$	0	0
$133$	−4.89898	−0.424795
$134$	25.8564	2.23365
$135$	0	0
$136$	−2.73205	−0.234271
$137$	12.6264	1.07874	0.539372	−	0.842067i	$-0.318661\pi$
$137$	12.6264	1.07874	0.539372	+	0.842067i	$0.318661\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	0	0
$142$	11.5911	0.972704
$143$	−4.24264	−0.354787
$144$	0	0
$145$	0	0
$146$	−17.6603	−1.46157
$147$	0	0
$148$	8.48528	0.697486
$149$	−11.3205	−0.927412	−0.463706	−	0.885989i	$-0.653481\pi$
$149$	−11.3205	−0.927412	−0.463706	+	0.885989i	$0.653481\pi$
$150$	0	0
$151$	22.7846	1.85419	0.927093	−	0.374832i	$-0.122300\pi$
$151$	22.7846	1.85419	0.927093	+	0.374832i	$0.122300\pi$
$152$	1.03528	0.0839720
$153$	0	0
$154$	−4.73205	−0.381320
$155$	0	0
$156$	0	0
$157$	1.31268	0.104763	0.0523815	−	0.998627i	$-0.483319\pi$
$157$	1.31268	0.104763	0.0523815	+	0.998627i	$0.483319\pi$
$158$	−30.6322	−2.43697
$159$	0	0
$160$	0	0
$161$	5.07180	0.399714
$162$	0	0
$163$	−18.2832	−1.43205	−0.716027	−	0.698073i	$-0.754041\pi$
$163$	−18.2832	−1.43205	−0.716027	+	0.698073i	$0.754041\pi$
$164$	16.3923	1.28002
$165$	0	0
$166$	−13.6603	−1.06024
$167$	−13.4858	−1.04356	−0.521781	−	0.853079i	$-0.674732\pi$
$167$	−13.4858	−1.04356	−0.521781	+	0.853079i	$0.674732\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	0	0
$172$	−10.4543	−0.797133
$173$	−3.96524	−0.301472	−0.150736	−	0.988574i	$-0.548164\pi$
$173$	−3.96524	−0.301472	−0.150736	+	0.988574i	$0.548164\pi$
$174$	0	0
$175$	0	0
$176$	4.46410	0.336494
$177$	0	0
$178$	16.4901	1.23598
$179$	−6.92820	−0.517838	−0.258919	−	0.965899i	$-0.583366\pi$
$179$	−6.92820	−0.517838	−0.258919	+	0.965899i	$0.583366\pi$
$180$	0	0
$181$	−24.7846	−1.84223	−0.921113	−	0.389296i	$-0.872718\pi$
$181$	−24.7846	−1.84223	−0.921113	+	0.389296i	$0.872718\pi$
$182$	20.0764	1.48816
$183$	0	0
$184$	−1.07180	−0.0790139
$185$	0	0
$186$	0	0
$187$	−5.27792	−0.385960
$188$	4.89898	0.357295
$189$	0	0
$190$	0	0
$191$	13.8564	1.00261	0.501307	−	0.865269i	$-0.332853\pi$
$191$	13.8564	1.00261	0.501307	+	0.865269i	$0.332853\pi$
$192$	0	0
$193$	−0.656339	−0.0472443	−0.0236222	−	0.999721i	$-0.507520\pi$
$193$	−0.656339	−0.0472443	−0.0236222	+	0.999721i	$0.507520\pi$
$194$	6.92820	0.497416
$195$	0	0
$196$	−1.73205	−0.123718
$197$	−27.7023	−1.97370	−0.986852	−	0.161625i	$-0.948326\pi$
$197$	−27.7023	−1.97370	−0.986852	+	0.161625i	$0.948326\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	−8.48528	−0.597022
$203$	23.1822	1.62707
$204$	0	0
$205$	0	0
$206$	−16.3923	−1.14211
$207$	0	0
$208$	−18.9396	−1.31322
$209$	2.00000	0.138343
$210$	0	0
$211$	−11.8564	−0.816229	−0.408114	−	0.912931i	$-0.633814\pi$
$211$	−11.8564	−0.816229	−0.408114	+	0.912931i	$0.633814\pi$
$212$	1.31268	0.0901551
$213$	0	0
$214$	−2.73205	−0.186759
$215$	0	0
$216$	0	0
$217$	−18.2832	−1.24115
$218$	19.3185	1.30842
$219$	0	0
$220$	0	0
$221$	22.3923	1.50627
$222$	0	0
$223$	−14.6969	−0.984180	−0.492090	−	0.870544i	$-0.663767\pi$
$223$	−14.6969	−0.984180	−0.492090	+	0.870544i	$0.663767\pi$
$224$	−18.5885	−1.24199
$225$	0	0
$226$	10.9282	0.726933
$227$	8.38375	0.556449	0.278224	−	0.960516i	$-0.410254\pi$
$227$	8.38375	0.556449	0.278224	+	0.960516i	$0.410254\pi$
$228$	0	0
$229$	−0.143594	−0.00948893	−0.00474446	−	0.999989i	$-0.501510\pi$
$229$	−0.143594	−0.00948893	−0.00474446	+	0.999989i	$0.501510\pi$
$230$	0	0
$231$	0	0
$232$	−4.89898	−0.321634
$233$	6.79367	0.445068	0.222534	−	0.974925i	$-0.428567\pi$
$233$	6.79367	0.445068	0.222534	+	0.974925i	$0.428567\pi$
$234$	0	0
$235$	0	0
$236$	−1.60770	−0.104652
$237$	0	0
$238$	24.9754	1.61891
$239$	−13.8564	−0.896296	−0.448148	−	0.893959i	$-0.647916\pi$
$239$	−13.8564	−0.896296	−0.448148	+	0.893959i	$0.647916\pi$
$240$	0	0
$241$	−4.92820	−0.317453	−0.158727	−	0.987323i	$-0.550739\pi$
$241$	−4.92820	−0.317453	−0.158727	+	0.987323i	$0.550739\pi$
$242$	1.93185	0.124184
$243$	0	0
$244$	15.4641	0.989988
$245$	0	0
$246$	0	0
$247$	−8.48528	−0.539906
$248$	3.86370	0.245345
$249$	0	0
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	−2.07055	−0.130175
$254$	−33.1244	−2.07841
$255$	0	0
$256$	19.3923	1.21202
$257$	−21.6665	−1.35152	−0.675759	−	0.737123i	$-0.736184\pi$
$257$	−21.6665	−1.35152	−0.675759	+	0.737123i	$0.736184\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	0	0
$261$	0	0
$262$	−26.7685	−1.65376
$263$	−4.79744	−0.295823	−0.147912	−	0.989001i	$-0.547255\pi$
$263$	−4.79744	−0.295823	−0.147912	+	0.989001i	$0.547255\pi$
$264$	0	0
$265$	0	0
$266$	−9.46410	−0.580281
$267$	0	0
$268$	23.1822	1.41608
$269$	25.8564	1.57649	0.788246	−	0.615360i	$-0.210989\pi$
$269$	25.8564	1.57649	0.788246	+	0.615360i	$0.210989\pi$
$270$	0	0
$271$	8.92820	0.542350	0.271175	−	0.962530i	$-0.412588\pi$
$271$	8.92820	0.542350	0.271175	+	0.962530i	$0.412588\pi$
$272$	−23.5612	−1.42860
$273$	0	0
$274$	24.3923	1.47359
$275$	0	0
$276$	0	0
$277$	−23.8386	−1.43232	−0.716160	−	0.697936i	$-0.754102\pi$
$277$	−23.8386	−1.43232	−0.716160	+	0.697936i	$0.754102\pi$
$278$	−3.86370	−0.231730
$279$	0	0
$280$	0	0
$281$	11.3205	0.675325	0.337662	−	0.941267i	$-0.390364\pi$
$281$	11.3205	0.675325	0.337662	+	0.941267i	$0.390364\pi$
$282$	0	0
$283$	15.8338	0.941219	0.470609	−	0.882342i	$-0.344034\pi$
$283$	15.8338	0.941219	0.470609	+	0.882342i	$0.344034\pi$
$284$	10.3923	0.616670
$285$	0	0
$286$	−8.19615	−0.484649
$287$	23.1822	1.36840
$288$	0	0
$289$	10.8564	0.638612
$290$	0	0
$291$	0	0
$292$	−15.8338	−0.926600
$293$	3.20736	0.187376	0.0936881	−	0.995602i	$-0.470134\pi$
$293$	3.20736	0.187376	0.0936881	+	0.995602i	$0.470134\pi$
$294$	0	0
$295$	0	0
$296$	−2.53590	−0.147396
$297$	0	0
$298$	−21.8695	−1.26687
$299$	8.78461	0.508027
$300$	0	0
$301$	−14.7846	−0.852171
$302$	44.0165	2.53286
$303$	0	0
$304$	8.92820	0.512068
$305$	0	0
$306$	0	0
$307$	−10.9348	−0.624080	−0.312040	−	0.950069i	$-0.601012\pi$
$307$	−10.9348	−0.624080	−0.312040	+	0.950069i	$0.601012\pi$
$308$	−4.24264	−0.241747
$309$	0	0
$310$	0	0
$311$	24.9282	1.41355	0.706774	−	0.707439i	$-0.250150\pi$
$311$	24.9282	1.41355	0.706774	+	0.707439i	$0.250150\pi$
$312$	0	0
$313$	−12.0716	−0.682326	−0.341163	−	0.940004i	$-0.610821\pi$
$313$	−12.0716	−0.682326	−0.341163	+	0.940004i	$0.610821\pi$
$314$	2.53590	0.143109
$315$	0	0
$316$	−27.4641	−1.54498
$317$	−32.4254	−1.82119	−0.910595	−	0.413299i	$-0.864376\pi$
$317$	−32.4254	−1.82119	−0.910595	+	0.413299i	$0.864376\pi$
$318$	0	0
$319$	−9.46410	−0.529888
$320$	0	0
$321$	0	0
$322$	9.79796	0.546019
$323$	−10.5558	−0.587342
$324$	0	0
$325$	0	0
$326$	−35.3205	−1.95622
$327$	0	0
$328$	−4.89898	−0.270501
$329$	6.92820	0.381964
$330$	0	0
$331$	18.3923	1.01093	0.505466	−	0.862846i	$-0.331321\pi$
$331$	18.3923	1.01093	0.505466	+	0.862846i	$0.331321\pi$
$332$	−12.2474	−0.672166
$333$	0	0
$334$	−26.0526	−1.42553
$335$	0	0
$336$	0	0
$337$	−17.6269	−0.960199	−0.480099	−	0.877214i	$-0.659399\pi$
$337$	−17.6269	−0.960199	−0.480099	+	0.877214i	$0.659399\pi$
$338$	9.65926	0.525394
$339$	0	0
$340$	0	0
$341$	7.46410	0.404204
$342$	0	0
$343$	−19.5959	−1.05808
$344$	3.12436	0.168454
$345$	0	0
$346$	−7.66025	−0.411818
$347$	24.0416	1.29062	0.645311	−	0.763920i	$-0.276728\pi$
$347$	24.0416	1.29062	0.645311	+	0.763920i	$0.276728\pi$
$348$	0	0
$349$	−19.0718	−1.02089	−0.510445	−	0.859910i	$-0.670519\pi$
$349$	−19.0718	−1.02089	−0.510445	+	0.859910i	$0.670519\pi$
$350$	0	0
$351$	0	0
$352$	7.58871	0.404479
$353$	−26.0106	−1.38441	−0.692204	−	0.721702i	$-0.743360\pi$
$353$	−26.0106	−1.38441	−0.692204	+	0.721702i	$0.743360\pi$
$354$	0	0
$355$	0	0
$356$	14.7846	0.783583
$357$	0	0
$358$	−13.3843	−0.707380
$359$	−5.07180	−0.267679	−0.133840	−	0.991003i	$-0.542731\pi$
$359$	−5.07180	−0.267679	−0.133840	+	0.991003i	$0.542731\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−47.8802	−2.51653
$363$	0	0
$364$	18.0000	0.943456
$365$	0	0
$366$	0	0
$367$	12.0716	0.630132	0.315066	−	0.949070i	$-0.397973\pi$
$367$	12.0716	0.630132	0.315066	+	0.949070i	$0.397973\pi$
$368$	−9.24316	−0.481833
$369$	0	0
$370$	0	0
$371$	1.85641	0.0963798
$372$	0	0
$373$	31.0112	1.60570	0.802849	−	0.596183i	$-0.203317\pi$
$373$	31.0112	1.60570	0.802849	+	0.596183i	$0.203317\pi$
$374$	−10.1962	−0.527230
$375$	0	0
$376$	−1.46410	−0.0755053
$377$	40.1528	2.06797
$378$	0	0
$379$	−33.8564	−1.73909	−0.869543	−	0.493857i	$-0.835587\pi$
$379$	−33.8564	−1.73909	−0.869543	+	0.493857i	$0.835587\pi$
$380$	0	0
$381$	0	0
$382$	26.7685	1.36960
$383$	−26.0106	−1.32908	−0.664541	−	0.747252i	$-0.731373\pi$
$383$	−26.0106	−1.32908	−0.664541	+	0.747252i	$0.731373\pi$
$384$	0	0
$385$	0	0
$386$	−1.26795	−0.0645369
$387$	0	0
$388$	6.21166	0.315349
$389$	−6.92820	−0.351274	−0.175637	−	0.984455i	$-0.556198\pi$
$389$	−6.92820	−0.351274	−0.175637	+	0.984455i	$0.556198\pi$
$390$	0	0
$391$	10.9282	0.552663
$392$	0.517638	0.0261447
$393$	0	0
$394$	−53.5167	−2.69613
$395$	0	0
$396$	0	0
$397$	−30.3548	−1.52346	−0.761732	−	0.647892i	$-0.775651\pi$
$397$	−30.3548	−1.52346	−0.761732	+	0.647892i	$0.775651\pi$
$398$	7.72741	0.387340
$399$	0	0
$400$	0	0
$401$	−17.3205	−0.864945	−0.432472	−	0.901647i	$-0.642359\pi$
$401$	−17.3205	−0.864945	−0.432472	+	0.901647i	$0.642359\pi$
$402$	0	0
$403$	−31.6675	−1.57747
$404$	−7.60770	−0.378497
$405$	0	0
$406$	44.7846	2.22262
$407$	−4.89898	−0.242833
$408$	0	0
$409$	28.9282	1.43041	0.715204	−	0.698916i	$-0.246334\pi$
$409$	28.9282	1.43041	0.715204	+	0.698916i	$0.246334\pi$
$410$	0	0
$411$	0	0
$412$	−14.6969	−0.724066
$413$	−2.27362	−0.111878
$414$	0	0
$415$	0	0
$416$	−32.1962	−1.57855
$417$	0	0
$418$	3.86370	0.188980
$419$	6.92820	0.338465	0.169232	−	0.985576i	$-0.445871\pi$
$419$	6.92820	0.338465	0.169232	+	0.985576i	$0.445871\pi$
$420$	0	0
$421$	−22.9282	−1.11745	−0.558726	−	0.829352i	$-0.688710\pi$
$421$	−22.9282	−1.11745	−0.558726	+	0.829352i	$0.688710\pi$
$422$	−22.9048	−1.11499
$423$	0	0
$424$	−0.392305	−0.0190520
$425$	0	0
$426$	0	0
$427$	21.8695	1.05834
$428$	−2.44949	−0.118401
$429$	0	0
$430$	0	0
$431$	−27.7128	−1.33488	−0.667440	−	0.744664i	$-0.732610\pi$
$431$	−27.7128	−1.33488	−0.667440	+	0.744664i	$0.732610\pi$
$432$	0	0
$433$	−16.0096	−0.769373	−0.384687	−	0.923047i	$-0.625690\pi$
$433$	−16.0096	−0.769373	−0.384687	+	0.923047i	$0.625690\pi$
$434$	−35.3205	−1.69544
$435$	0	0
$436$	17.3205	0.829502
$437$	−4.14110	−0.198096
$438$	0	0
$439$	8.14359	0.388673	0.194336	−	0.980935i	$-0.437745\pi$
$439$	8.14359	0.388673	0.194336	+	0.980935i	$0.437745\pi$
$440$	0	0
$441$	0	0
$442$	43.2586	2.05760
$443$	20.3538	0.967038	0.483519	−	0.875334i	$-0.339358\pi$
$443$	20.3538	0.967038	0.483519	+	0.875334i	$0.339358\pi$
$444$	0	0
$445$	0	0
$446$	−28.3923	−1.34441
$447$	0	0
$448$	−14.0406	−0.663356
$449$	−8.53590	−0.402834	−0.201417	−	0.979506i	$-0.564555\pi$
$449$	−8.53590	−0.402834	−0.201417	+	0.979506i	$0.564555\pi$
$450$	0	0
$451$	−9.46410	−0.445647
$452$	9.79796	0.460857
$453$	0	0
$454$	16.1962	0.760123
$455$	0	0
$456$	0	0
$457$	18.9396	0.885956	0.442978	−	0.896532i	$-0.353922\pi$
$457$	18.9396	0.885956	0.442978	+	0.896532i	$0.353922\pi$
$458$	−0.277401	−0.0129621
$459$	0	0
$460$	0	0
$461$	28.3923	1.32236	0.661181	−	0.750227i	$-0.270056\pi$
$461$	28.3923	1.32236	0.661181	+	0.750227i	$0.270056\pi$
$462$	0	0
$463$	41.4655	1.92706	0.963532	−	0.267594i	$-0.0862287\pi$
$463$	41.4655	1.92706	0.963532	+	0.267594i	$0.0862287\pi$
$464$	−42.2487	−1.96135
$465$	0	0
$466$	13.1244	0.607974
$467$	37.1213	1.71777	0.858884	−	0.512170i	$-0.171158\pi$
$467$	37.1213	1.71777	0.858884	+	0.512170i	$0.171158\pi$
$468$	0	0
$469$	32.7846	1.51385
$470$	0	0
$471$	0	0
$472$	0.480473	0.0221156
$473$	6.03579	0.277526
$474$	0	0
$475$	0	0
$476$	22.3923	1.02635
$477$	0	0
$478$	−26.7685	−1.22436
$479$	−30.9282	−1.41315	−0.706573	−	0.707640i	$-0.749760\pi$
$479$	−30.9282	−1.41315	−0.706573	+	0.707640i	$0.749760\pi$
$480$	0	0
$481$	20.7846	0.947697
$482$	−9.52056	−0.433650
$483$	0	0
$484$	1.73205	0.0787296
$485$	0	0
$486$	0	0
$487$	10.7589	0.487533	0.243766	−	0.969834i	$-0.421617\pi$
$487$	10.7589	0.487533	0.243766	+	0.969834i	$0.421617\pi$
$488$	−4.62158	−0.209209
$489$	0	0
$490$	0	0
$491$	13.8564	0.625331	0.312665	−	0.949863i	$-0.398778\pi$
$491$	13.8564	0.625331	0.312665	+	0.949863i	$0.398778\pi$
$492$	0	0
$493$	49.9507	2.24967
$494$	−16.3923	−0.737525
$495$	0	0
$496$	33.3205	1.49613
$497$	14.6969	0.659248
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	0	0
$502$	34.7733	1.55201
$503$	−23.0807	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$503$	−23.0807	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$504$	0	0
$505$	0	0
$506$	−4.00000	−0.177822
$507$	0	0
$508$	−29.6985	−1.31766
$509$	25.8564	1.14607	0.573033	−	0.819533i	$-0.305767\pi$
$509$	25.8564	1.14607	0.573033	+	0.819533i	$0.305767\pi$
$510$	0	0
$511$	−22.3923	−0.990577
$512$	29.2552	1.29291
$513$	0	0
$514$	−41.8564	−1.84621
$515$	0	0
$516$	0	0
$517$	−2.82843	−0.124394
$518$	23.1822	1.01857
$519$	0	0
$520$	0	0
$521$	−17.3205	−0.758825	−0.379413	−	0.925228i	$-0.623874\pi$
$521$	−17.3205	−0.758825	−0.379413	+	0.925228i	$0.623874\pi$
$522$	0	0
$523$	−23.0064	−1.00600	−0.502999	−	0.864287i	$-0.667770\pi$
$523$	−23.0064	−1.00600	−0.502999	+	0.864287i	$0.667770\pi$
$524$	−24.0000	−1.04844
$525$	0	0
$526$	−9.26795	−0.404102
$527$	−39.3949	−1.71607
$528$	0	0
$529$	−18.7128	−0.813601
$530$	0	0
$531$	0	0
$532$	−8.48528	−0.367884
$533$	40.1528	1.73921
$534$	0	0
$535$	0	0
$536$	−6.92820	−0.299253
$537$	0	0
$538$	49.9507	2.15353
$539$	1.00000	0.0430730
$540$	0	0
$541$	−6.78461	−0.291693	−0.145847	−	0.989307i	$-0.546591\pi$
$541$	−6.78461	−0.291693	−0.145847	+	0.989307i	$0.546591\pi$
$542$	17.2480	0.740863
$543$	0	0
$544$	−40.0526	−1.71724
$545$	0	0
$546$	0	0
$547$	3.76217	0.160859	0.0804293	−	0.996760i	$-0.474371\pi$
$547$	3.76217	0.160859	0.0804293	+	0.996760i	$0.474371\pi$
$548$	21.8695	0.934221
$549$	0	0
$550$	0	0
$551$	−18.9282	−0.806369
$552$	0	0
$553$	−38.8401	−1.65165
$554$	−46.0526	−1.95659
$555$	0	0
$556$	−3.46410	−0.146911
$557$	−13.0053	−0.551053	−0.275527	−	0.961293i	$-0.588852\pi$
$557$	−13.0053	−0.551053	−0.275527	+	0.961293i	$0.588852\pi$
$558$	0	0
$559$	−25.6077	−1.08309
$560$	0	0
$561$	0	0
$562$	21.8695	0.922511
$563$	1.41421	0.0596020	0.0298010	−	0.999556i	$-0.490513\pi$
$563$	1.41421	0.0596020	0.0298010	+	0.999556i	$0.490513\pi$
$564$	0	0
$565$	0	0
$566$	30.5885	1.28573
$567$	0	0
$568$	−3.10583	−0.130318
$569$	9.46410	0.396756	0.198378	−	0.980126i	$-0.436433\pi$
$569$	9.46410	0.396756	0.198378	+	0.980126i	$0.436433\pi$
$570$	0	0
$571$	8.92820	0.373634	0.186817	−	0.982395i	$-0.440183\pi$
$571$	8.92820	0.373634	0.186817	+	0.982395i	$0.440183\pi$
$572$	−7.34847	−0.307255
$573$	0	0
$574$	44.7846	1.86927
$575$	0	0
$576$	0	0
$577$	−15.6579	−0.651846	−0.325923	−	0.945396i	$-0.605675\pi$
$577$	−15.6579	−0.651846	−0.325923	+	0.945396i	$0.605675\pi$
$578$	20.9730	0.872360
$579$	0	0
$580$	0	0
$581$	−17.3205	−0.718576
$582$	0	0
$583$	−0.757875	−0.0313880
$584$	4.73205	0.195814
$585$	0	0
$586$	6.19615	0.255961
$587$	−38.6370	−1.59472	−0.797361	−	0.603503i	$-0.793771\pi$
$587$	−38.6370	−1.59472	−0.797361	+	0.603503i	$0.793771\pi$
$588$	0	0
$589$	14.9282	0.615106
$590$	0	0
$591$	0	0
$592$	−21.8695	−0.898833
$593$	−12.4505	−0.511282	−0.255641	−	0.966772i	$-0.582286\pi$
$593$	−12.4505	−0.511282	−0.255641	+	0.966772i	$0.582286\pi$
$594$	0	0
$595$	0	0
$596$	−19.6077	−0.803162
$597$	0	0
$598$	16.9706	0.693978
$599$	−42.9282	−1.75400	−0.876999	−	0.480491i	$-0.840458\pi$
$599$	−42.9282	−1.75400	−0.876999	+	0.480491i	$0.840458\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	−28.5617	−1.16409
$603$	0	0
$604$	39.4641	1.60577
$605$	0	0
$606$	0	0
$607$	48.8139	1.98130	0.990648	−	0.136441i	$-0.0435666\pi$
$607$	48.8139	1.98130	0.990648	+	0.136441i	$0.0435666\pi$
$608$	15.1774	0.615525
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	2.92996	0.118340	0.0591700	−	0.998248i	$-0.481155\pi$
$613$	2.92996	0.118340	0.0591700	+	0.998248i	$0.481155\pi$
$614$	−21.1244	−0.852510
$615$	0	0
$616$	1.26795	0.0510871
$617$	11.3137	0.455473	0.227736	−	0.973723i	$-0.426868\pi$
$617$	11.3137	0.455473	0.227736	+	0.973723i	$0.426868\pi$
$618$	0	0
$619$	2.39230	0.0961549	0.0480774	−	0.998844i	$-0.484691\pi$
$619$	2.39230	0.0961549	0.0480774	+	0.998844i	$0.484691\pi$
$620$	0	0
$621$	0	0
$622$	48.1576	1.93094
$623$	20.9086	0.837685
$624$	0	0
$625$	0	0
$626$	−23.3205	−0.932075
$627$	0	0
$628$	2.27362	0.0907275
$629$	25.8564	1.03096
$630$	0	0
$631$	−17.8564	−0.710852	−0.355426	−	0.934704i	$-0.615664\pi$
$631$	−17.8564	−0.710852	−0.355426	+	0.934704i	$0.615664\pi$
$632$	8.20788	0.326492
$633$	0	0
$634$	−62.6410	−2.48779
$635$	0	0
$636$	0	0
$637$	−4.24264	−0.168100
$638$	−18.2832	−0.723840
$639$	0	0
$640$	0	0
$641$	−8.53590	−0.337148	−0.168574	−	0.985689i	$-0.553916\pi$
$641$	−8.53590	−0.337148	−0.168574	+	0.985689i	$0.553916\pi$
$642$	0	0
$643$	45.0518	1.77667	0.888334	−	0.459198i	$-0.151863\pi$
$643$	45.0518	1.77667	0.888334	+	0.459198i	$0.151863\pi$
$644$	8.78461	0.346162
$645$	0	0
$646$	−20.3923	−0.802325
$647$	33.1833	1.30457	0.652284	−	0.757975i	$-0.273811\pi$
$647$	33.1833	1.30457	0.652284	+	0.757975i	$0.273811\pi$
$648$	0	0
$649$	0.928203	0.0364352
$650$	0	0
$651$	0	0
$652$	−31.6675	−1.24020
$653$	−34.4959	−1.34993	−0.674965	−	0.737850i	$-0.735841\pi$
$653$	−34.4959	−1.34993	−0.674965	+	0.737850i	$0.735841\pi$
$654$	0	0
$655$	0	0
$656$	−42.2487	−1.64954
$657$	0	0
$658$	13.3843	0.521773
$659$	−17.0718	−0.665023	−0.332511	−	0.943099i	$-0.607896\pi$
$659$	−17.0718	−0.665023	−0.332511	+	0.943099i	$0.607896\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	35.5312	1.38096
$663$	0	0
$664$	3.66025	0.142045
$665$	0	0
$666$	0	0
$667$	19.5959	0.758757
$668$	−23.3581	−0.903751
$669$	0	0
$670$	0	0
$671$	−8.92820	−0.344669
$672$	0	0
$673$	−9.14162	−0.352384	−0.176192	−	0.984356i	$-0.556378\pi$
$673$	−9.14162	−0.352384	−0.176192	+	0.984356i	$0.556378\pi$
$674$	−34.0526	−1.31166
$675$	0	0
$676$	8.66025	0.333087
$677$	31.0855	1.19471	0.597356	−	0.801976i	$-0.296218\pi$
$677$	31.0855	1.19471	0.597356	+	0.801976i	$0.296218\pi$
$678$	0	0
$679$	8.78461	0.337122
$680$	0	0
$681$	0	0
$682$	14.4195	0.552153
$683$	43.5360	1.66586	0.832930	−	0.553379i	$-0.186662\pi$
$683$	43.5360	1.66586	0.832930	+	0.553379i	$0.186662\pi$
$684$	0	0
$685$	0	0
$686$	−37.8564	−1.44536
$687$	0	0
$688$	26.9444	1.02725
$689$	3.21539	0.122497
$690$	0	0
$691$	9.85641	0.374955	0.187478	−	0.982269i	$-0.439969\pi$
$691$	9.85641	0.374955	0.187478	+	0.982269i	$0.439969\pi$
$692$	−6.86800	−0.261082
$693$	0	0
$694$	46.4449	1.76302
$695$	0	0
$696$	0	0
$697$	49.9507	1.89202
$698$	−36.8439	−1.39456
$699$	0	0
$700$	0	0
$701$	26.5359	1.00225	0.501124	−	0.865376i	$-0.332920\pi$
$701$	26.5359	1.00225	0.501124	+	0.865376i	$0.332920\pi$
$702$	0	0
$703$	−9.79796	−0.369537
$704$	5.73205	0.216035
$705$	0	0
$706$	−50.2487	−1.89114
$707$	−10.7589	−0.404630
$708$	0	0
$709$	12.7846	0.480136	0.240068	−	0.970756i	$-0.422830\pi$
$709$	12.7846	0.480136	0.240068	+	0.970756i	$0.422830\pi$
$710$	0	0
$711$	0	0
$712$	−4.41851	−0.165591
$713$	−15.4548	−0.578787
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	−9.79796	−0.365657
$719$	41.5692	1.55027	0.775135	−	0.631795i	$-0.217682\pi$
$719$	41.5692	1.55027	0.775135	+	0.631795i	$0.217682\pi$
$720$	0	0
$721$	−20.7846	−0.774059
$722$	−28.9778	−1.07844
$723$	0	0
$724$	−42.9282	−1.59541
$725$	0	0
$726$	0	0
$727$	−26.7685	−0.992790	−0.496395	−	0.868097i	$-0.665343\pi$
$727$	−26.7685	−0.992790	−0.496395	+	0.868097i	$0.665343\pi$
$728$	−5.37945	−0.199376
$729$	0	0
$730$	0	0
$731$	−31.8564	−1.17825
$732$	0	0
$733$	50.6071	1.86922	0.934608	−	0.355681i	$-0.115751\pi$
$733$	50.6071	1.86922	0.934608	+	0.355681i	$0.115751\pi$
$734$	23.3205	0.860776
$735$	0	0
$736$	−15.7128	−0.579182
$737$	−13.3843	−0.493016
$738$	0	0
$739$	8.14359	0.299567	0.149783	−	0.988719i	$-0.452142\pi$
$739$	8.14359	0.299567	0.149783	+	0.988719i	$0.452142\pi$
$740$	0	0
$741$	0	0
$742$	3.58630	0.131657
$743$	−4.79744	−0.176001	−0.0880006	−	0.996120i	$-0.528048\pi$
$743$	−4.79744	−0.176001	−0.0880006	+	0.996120i	$0.528048\pi$
$744$	0	0
$745$	0	0
$746$	59.9090	2.19342
$747$	0	0
$748$	−9.14162	−0.334251
$749$	−3.46410	−0.126576
$750$	0	0
$751$	−41.8564	−1.52736	−0.763681	−	0.645594i	$-0.776610\pi$
$751$	−41.8564	−1.52736	−0.763681	+	0.645594i	$0.776610\pi$
$752$	−12.6264	−0.460437
$753$	0	0
$754$	77.5692	2.82490
$755$	0	0
$756$	0	0
$757$	20.9086	0.759936	0.379968	−	0.925000i	$-0.375935\pi$
$757$	20.9086	0.759936	0.379968	+	0.925000i	$0.375935\pi$
$758$	−65.4056	−2.37564
$759$	0	0
$760$	0	0
$761$	42.2487	1.53151	0.765757	−	0.643130i	$-0.222364\pi$
$761$	42.2487	1.53151	0.765757	+	0.643130i	$0.222364\pi$
$762$	0	0
$763$	24.4949	0.886775
$764$	24.0000	0.868290
$765$	0	0
$766$	−50.2487	−1.81556
$767$	−3.93803	−0.142194
$768$	0	0
$769$	−17.7128	−0.638740	−0.319370	−	0.947630i	$-0.603471\pi$
$769$	−17.7128	−0.638740	−0.319370	+	0.947630i	$0.603471\pi$
$770$	0	0
$771$	0	0
$772$	−1.13681	−0.0409148
$773$	−31.8706	−1.14630	−0.573152	−	0.819449i	$-0.694280\pi$
$773$	−31.8706	−1.14630	−0.573152	+	0.819449i	$0.694280\pi$
$774$	0	0
$775$	0	0
$776$	−1.85641	−0.0666411
$777$	0	0
$778$	−13.3843	−0.479849
$779$	−18.9282	−0.678173
$780$	0	0
$781$	−6.00000	−0.214697
$782$	21.1117	0.754952
$783$	0	0
$784$	4.46410	0.159432
$785$	0	0
$786$	0	0
$787$	13.5601	0.483366	0.241683	−	0.970355i	$-0.422301\pi$
$787$	13.5601	0.483366	0.241683	+	0.970355i	$0.422301\pi$
$788$	−47.9817	−1.70928
$789$	0	0
$790$	0	0
$791$	13.8564	0.492677
$792$	0	0
$793$	37.8792	1.34513
$794$	−58.6410	−2.08109
$795$	0	0
$796$	6.92820	0.245564
$797$	6.76646	0.239680	0.119840	−	0.992793i	$-0.461762\pi$
$797$	6.76646	0.239680	0.119840	+	0.992793i	$0.461762\pi$
$798$	0	0
$799$	14.9282	0.528122
$800$	0	0
$801$	0	0
$802$	−33.4607	−1.18154
$803$	9.14162	0.322601
$804$	0	0
$805$	0	0
$806$	−61.1769	−2.15486
$807$	0	0
$808$	2.27362	0.0799858
$809$	2.53590	0.0891574	0.0445787	−	0.999006i	$-0.485805\pi$
$809$	2.53590	0.0891574	0.0445787	+	0.999006i	$0.485805\pi$
$810$	0	0
$811$	14.0000	0.491606	0.245803	−	0.969320i	$-0.420948\pi$
$811$	14.0000	0.491606	0.245803	+	0.969320i	$0.420948\pi$
$812$	40.1528	1.40909
$813$	0	0
$814$	−9.46410	−0.331717
$815$	0	0
$816$	0	0
$817$	12.0716	0.422331
$818$	55.8850	1.95397
$819$	0	0
$820$	0	0
$821$	25.1769	0.878680	0.439340	−	0.898321i	$-0.355212\pi$
$821$	25.1769	0.878680	0.439340	+	0.898321i	$0.355212\pi$
$822$	0	0
$823$	−15.6579	−0.545800	−0.272900	−	0.962042i	$-0.587983\pi$
$823$	−15.6579	−0.545800	−0.272900	+	0.962042i	$0.587983\pi$
$824$	4.39230	0.153013
$825$	0	0
$826$	−4.39230	−0.152828
$827$	−39.2934	−1.36636	−0.683182	−	0.730248i	$-0.739405\pi$
$827$	−39.2934	−1.36636	−0.683182	+	0.730248i	$0.739405\pi$
$828$	0	0
$829$	24.7846	0.860805	0.430403	−	0.902637i	$-0.358372\pi$
$829$	24.7846	0.860805	0.430403	+	0.902637i	$0.358372\pi$
$830$	0	0
$831$	0	0
$832$	−24.3190	−0.843111
$833$	−5.27792	−0.182869
$834$	0	0
$835$	0	0
$836$	3.46410	0.119808
$837$	0	0
$838$	13.3843	0.462352
$839$	−26.7846	−0.924707	−0.462354	−	0.886696i	$-0.652995\pi$
$839$	−26.7846	−0.924707	−0.462354	+	0.886696i	$0.652995\pi$
$840$	0	0
$841$	60.5692	2.08859
$842$	−44.2939	−1.52647
$843$	0	0
$844$	−20.5359	−0.706875
$845$	0	0
$846$	0	0
$847$	2.44949	0.0841655
$848$	−3.38323	−0.116181
$849$	0	0
$850$	0	0
$851$	10.1436	0.347718
$852$	0	0
$853$	−8.18067	−0.280101	−0.140050	−	0.990144i	$-0.544726\pi$
$853$	−8.18067	−0.280101	−0.140050	+	0.990144i	$0.544726\pi$
$854$	42.2487	1.44572
$855$	0	0
$856$	0.732051	0.0250210
$857$	−9.41902	−0.321748	−0.160874	−	0.986975i	$-0.551431\pi$
$857$	−9.41902	−0.321748	−0.160874	+	0.986975i	$0.551431\pi$
$858$	0	0
$859$	−15.1769	−0.517830	−0.258915	−	0.965900i	$-0.583365\pi$
$859$	−15.1769	−0.517830	−0.258915	+	0.965900i	$0.583365\pi$
$860$	0	0
$861$	0	0
$862$	−53.5370	−1.82348
$863$	54.2949	1.84822	0.924110	−	0.382126i	$-0.124808\pi$
$863$	54.2949	1.84822	0.924110	+	0.382126i	$0.124808\pi$
$864$	0	0
$865$	0	0
$866$	−30.9282	−1.05098
$867$	0	0
$868$	−31.6675	−1.07487
$869$	15.8564	0.537892
$870$	0	0
$871$	56.7846	1.92407
$872$	−5.17638	−0.175294
$873$	0	0
$874$	−8.00000	−0.270604
$875$	0	0
$876$	0	0
$877$	−33.6365	−1.13582	−0.567912	−	0.823089i	$-0.692249\pi$
$877$	−33.6365	−1.13582	−0.567912	+	0.823089i	$0.692249\pi$
$878$	15.7322	0.530937
$879$	0	0
$880$	0	0
$881$	18.9282	0.637707	0.318854	−	0.947804i	$-0.396702\pi$
$881$	18.9282	0.637707	0.318854	+	0.947804i	$0.396702\pi$
$882$	0	0
$883$	14.3452	0.482755	0.241377	−	0.970431i	$-0.422401\pi$
$883$	14.3452	0.482755	0.241377	+	0.970431i	$0.422401\pi$
$884$	38.7846	1.30447
$885$	0	0
$886$	39.3205	1.32100
$887$	9.34469	0.313764	0.156882	−	0.987617i	$-0.449856\pi$
$887$	9.34469	0.313764	0.156882	+	0.987617i	$0.449856\pi$
$888$	0	0
$889$	−42.0000	−1.40863
$890$	0	0
$891$	0	0
$892$	−25.4558	−0.852325
$893$	−5.65685	−0.189299
$894$	0	0
$895$	0	0
$896$	10.0526	0.335832
$897$	0	0
$898$	−16.4901	−0.550281
$899$	−70.6410	−2.35601
$900$	0	0
$901$	4.00000	0.133259
$902$	−18.2832	−0.608765
$903$	0	0
$904$	−2.92820	−0.0973906
$905$	0	0
$906$	0	0
$907$	19.2442	0.638993	0.319496	−	0.947587i	$-0.396486\pi$
$907$	19.2442	0.638993	0.319496	+	0.947587i	$0.396486\pi$
$908$	14.5211	0.481899
$909$	0	0
$910$	0	0
$911$	−10.1436	−0.336072	−0.168036	−	0.985781i	$-0.553743\pi$
$911$	−10.1436	−0.336072	−0.168036	+	0.985781i	$0.553743\pi$
$912$	0	0
$913$	7.07107	0.234018
$914$	36.5885	1.21024
$915$	0	0
$916$	−0.248711	−0.00821765
$917$	−33.9411	−1.12083
$918$	0	0
$919$	13.2154	0.435936	0.217968	−	0.975956i	$-0.430057\pi$
$919$	13.2154	0.435936	0.217968	+	0.975956i	$0.430057\pi$
$920$	0	0
$921$	0	0
$922$	54.8497	1.80638
$923$	25.4558	0.837889
$924$	0	0
$925$	0	0
$926$	80.1051	2.63242
$927$	0	0
$928$	−71.8203	−2.35762
$929$	46.6410	1.53024	0.765121	−	0.643886i	$-0.222679\pi$
$929$	46.6410	1.53024	0.765121	+	0.643886i	$0.222679\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	11.7670	0.385440
$933$	0	0
$934$	71.7128	2.34651
$935$	0	0
$936$	0	0
$937$	−21.5649	−0.704496	−0.352248	−	0.935907i	$-0.614583\pi$
$937$	−21.5649	−0.704496	−0.352248	+	0.935907i	$0.614583\pi$
$938$	63.3350	2.06796
$939$	0	0
$940$	0	0
$941$	40.3923	1.31675	0.658376	−	0.752689i	$-0.271244\pi$
$941$	40.3923	1.31675	0.658376	+	0.752689i	$0.271244\pi$
$942$	0	0
$943$	19.5959	0.638131
$944$	4.14359	0.134862
$945$	0	0
$946$	11.6603	0.379108
$947$	22.4243	0.728693	0.364347	−	0.931263i	$-0.381292\pi$
$947$	22.4243	0.728693	0.364347	+	0.931263i	$0.381292\pi$
$948$	0	0
$949$	−38.7846	−1.25900
$950$	0	0
$951$	0	0
$952$	−6.69213	−0.216893
$953$	21.4906	0.696149	0.348074	−	0.937467i	$-0.386836\pi$
$953$	21.4906	0.696149	0.348074	+	0.937467i	$0.386836\pi$
$954$	0	0
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	−59.7487	−1.93039
$959$	30.9282	0.998724
$960$	0	0
$961$	24.7128	0.797188
$962$	40.1528	1.29458
$963$	0	0
$964$	−8.53590	−0.274923
$965$	0	0
$966$	0	0
$967$	22.0454	0.708933	0.354466	−	0.935069i	$-0.384663\pi$
$967$	22.0454	0.708933	0.354466	+	0.935069i	$0.384663\pi$
$968$	−0.517638	−0.0166375
$969$	0	0
$970$	0	0
$971$	15.7128	0.504248	0.252124	−	0.967695i	$-0.418871\pi$
$971$	15.7128	0.504248	0.252124	+	0.967695i	$0.418871\pi$
$972$	0	0
$973$	−4.89898	−0.157054
$974$	20.7846	0.665982
$975$	0	0
$976$	−39.8564	−1.27577
$977$	−14.1421	−0.452447	−0.226224	−	0.974075i	$-0.572638\pi$
$977$	−14.1421	−0.452447	−0.226224	+	0.974075i	$0.572638\pi$
$978$	0	0
$979$	−8.53590	−0.272808
$980$	0	0
$981$	0	0
$982$	26.7685	0.854218
$983$	−1.86748	−0.0595634	−0.0297817	−	0.999556i	$-0.509481\pi$
$983$	−1.86748	−0.0595634	−0.0297817	+	0.999556i	$0.509481\pi$
$984$	0	0
$985$	0	0
$986$	96.4974	3.07310
$987$	0	0
$988$	−14.6969	−0.467572
$989$	−12.4974	−0.397395
$990$	0	0
$991$	−0.535898	−0.0170234	−0.00851169	−	0.999964i	$-0.502709\pi$
$991$	−0.535898	−0.0170234	−0.00851169	+	0.999964i	$0.502709\pi$
$992$	56.6429	1.79841
$993$	0	0
$994$	28.3923	0.900549
$995$	0	0
$996$	0	0
$997$	19.9005	0.630256	0.315128	−	0.949049i	$-0.397953\pi$
$997$	19.9005	0.630256	0.315128	+	0.949049i	$0.397953\pi$
$998$	30.9096	0.978427
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.bg.1.4		4
3.2	odd	2		2475.2.a.bh.1.1		4
5.2	odd	4		495.2.c.b.199.4	yes	4
5.3	odd	4		495.2.c.b.199.1	✓	4
5.4	even	2	inner	2475.2.a.bg.1.1		4
15.2	even	4		495.2.c.c.199.1	yes	4
15.8	even	4		495.2.c.c.199.4	yes	4
15.14	odd	2		2475.2.a.bh.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.c.b.199.1	✓	4	5.3	odd	4
495.2.c.b.199.4	yes	4	5.2	odd	4
495.2.c.c.199.1	yes	4	15.2	even	4
495.2.c.c.199.4	yes	4	15.8	even	4
2475.2.a.bg.1.1		4	5.4	even	2	inner
2475.2.a.bg.1.4		4	1.1	even	1	trivial
2475.2.a.bh.1.1		4	3.2	odd	2
2475.2.a.bh.1.4		4	15.14	odd	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+1.93185 q^{2} +1.73205 q^{4} +2.44949 q^{7} -0.517638 q^{8} +O(q^{10})\)	\(q+1.93185 q^{2} +1.73205 q^{4} +2.44949 q^{7} -0.517638 q^{8} -1.00000 q^{11} +4.24264 q^{13} +4.73205 q^{14} -4.46410 q^{16} +5.27792 q^{17} -2.00000 q^{19} -1.93185 q^{22} +2.07055 q^{23} +8.19615 q^{26} +4.24264 q^{28} +9.46410 q^{29} -7.46410 q^{31} -7.58871 q^{32} +10.1962 q^{34} +4.89898 q^{37} -3.86370 q^{38} +9.46410 q^{41} -6.03579 q^{43} -1.73205 q^{44} +4.00000 q^{46} +2.82843 q^{47} -1.00000 q^{49} +7.34847 q^{52} +0.757875 q^{53} -1.26795 q^{56} +18.2832 q^{58} -0.928203 q^{59} +8.92820 q^{61} -14.4195 q^{62} -5.73205 q^{64} +13.3843 q^{67} +9.14162 q^{68} +6.00000 q^{71} -9.14162 q^{73} +9.46410 q^{74} -3.46410 q^{76} -2.44949 q^{77} -15.8564 q^{79} +18.2832 q^{82} -7.07107 q^{83} -11.6603 q^{86} +0.517638 q^{88} +8.53590 q^{89} +10.3923 q^{91} +3.58630 q^{92} +5.46410 q^{94} +3.58630 q^{97} -1.93185 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 4 q^{11} + 12 q^{14} - 4 q^{16} - 8 q^{19} + 12 q^{26} + 24 q^{29} - 16 q^{31} + 20 q^{34} + 24 q^{41} + 16 q^{46} - 4 q^{49} - 12 q^{56} + 24 q^{59} + 8 q^{61} - 16 q^{64} + 24 q^{71} + 24 q^{74} - 8 q^{79} - 12 q^{86} + 48 q^{89} + 8 q^{94}+O(q^{100}) \) 4 * q - 4 * q^11 + 12 * q^14 - 4 * q^16 - 8 * q^19 + 12 * q^26 + 24 * q^29 - 16 * q^31 + 20 * q^34 + 24 * q^41 + 16 * q^46 - 4 * q^49 - 12 * q^56 + 24 * q^59 + 8 * q^61 - 16 * q^64 + 24 * q^71 + 24 * q^74 - 8 * q^79 - 12 * q^86 + 48 * q^89 + 8 * q^94

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