LMFDB - Embedded newform 5625.2.a.bf.1.17

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:	$24$
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q+1.54077 q^{2} +0.373979 q^{4} -3.37636 q^{7} -2.50533 q^{8} +O(q^{10})$	$q+1.54077 q^{2} +0.373979 q^{4} -3.37636 q^{7} -2.50533 q^{8} -0.275476 q^{11} -5.74270 q^{13} -5.20221 q^{14} -4.60810 q^{16} +6.11846 q^{17} -5.04824 q^{19} -0.424446 q^{22} +2.66559 q^{23} -8.84819 q^{26} -1.26269 q^{28} +6.27782 q^{29} +0.987076 q^{31} -2.08937 q^{32} +9.42716 q^{34} -6.20480 q^{37} -7.77820 q^{38} +0.876571 q^{41} -0.0269678 q^{43} -0.103022 q^{44} +4.10706 q^{46} +11.2757 q^{47} +4.39983 q^{49} -2.14765 q^{52} -2.16918 q^{53} +8.45890 q^{56} +9.67269 q^{58} +13.3441 q^{59} +5.33819 q^{61} +1.52086 q^{62} +5.99695 q^{64} -2.95192 q^{67} +2.28818 q^{68} +5.03195 q^{71} +4.45304 q^{73} -9.56019 q^{74} -1.88794 q^{76} +0.930107 q^{77} +13.6841 q^{79} +1.35060 q^{82} -8.71814 q^{83} -0.0415513 q^{86} +0.690158 q^{88} -14.2494 q^{89} +19.3894 q^{91} +0.996873 q^{92} +17.3733 q^{94} -5.72170 q^{97} +6.77913 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{4}+O(q^{10}) $ 24 * q + 32 * q^4	$ 24 q + 32 q^{4} + 56 q^{16} + 36 q^{19} + 52 q^{31} + 60 q^{34} + 60 q^{46} + 72 q^{49} + 68 q^{61} + 108 q^{64} + 88 q^{76} + 84 q^{79} + 80 q^{91} + 100 q^{94}+O(q^{100}) $ 24 * q + 32 * q^4 + 56 * q^16 + 36 * q^19 + 52 * q^31 + 60 * q^34 + 60 * q^46 + 72 * q^49 + 68 * q^61 + 108 * q^64 + 88 * q^76 + 84 * q^79 + 80 * q^91 + 100 * 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 $

$2$	1.54077	1.08949	0.544745	−	0.838602i	$-0.316626\pi$
$2$	1.54077	1.08949	0.544745	+	0.838602i	$0.316626\pi$
$3$	0	0
$3$	0	0
$4$	0.373979	0.186989
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.37636	−1.27615	−0.638073	−	0.769976i	$-0.720268\pi$
$7$	−3.37636	−1.27615	−0.638073	+	0.769976i	$0.720268\pi$
$8$	−2.50533	−0.885767
$9$	0	0
$10$	0	0
$11$	−0.275476	−0.0830591	−0.0415296	−	0.999137i	$-0.513223\pi$
$11$	−0.275476	−0.0830591	−0.0415296	+	0.999137i	$0.513223\pi$
$12$	0	0
$13$	−5.74270	−1.59274	−0.796369	−	0.604811i	$-0.793248\pi$
$13$	−5.74270	−1.59274	−0.796369	+	0.604811i	$0.793248\pi$
$14$	−5.20221	−1.39035
$15$	0	0
$16$	−4.60810	−1.15202
$17$	6.11846	1.48395	0.741973	−	0.670430i	$-0.233890\pi$
$17$	6.11846	1.48395	0.741973	+	0.670430i	$0.233890\pi$
$18$	0	0
$19$	−5.04824	−1.15815	−0.579073	−	0.815275i	$-0.696586\pi$
$19$	−5.04824	−1.15815	−0.579073	+	0.815275i	$0.696586\pi$
$20$	0	0
$21$	0	0
$22$	−0.424446	−0.0904921
$23$	2.66559	0.555813	0.277907	−	0.960608i	$-0.410359\pi$
$23$	2.66559	0.555813	0.277907	+	0.960608i	$0.410359\pi$
$24$	0	0
$25$	0	0
$26$	−8.84819	−1.73527
$27$	0	0
$28$	−1.26269	−0.238626
$29$	6.27782	1.16576	0.582881	−	0.812558i	$-0.301925\pi$
$29$	6.27782	1.16576	0.582881	+	0.812558i	$0.301925\pi$
$30$	0	0
$31$	0.987076	0.177284	0.0886421	−	0.996064i	$-0.471747\pi$
$31$	0.987076	0.177284	0.0886421	+	0.996064i	$0.471747\pi$
$32$	−2.08937	−0.369352
$33$	0	0
$34$	9.42716	1.61674
$35$	0	0
$36$	0	0
$37$	−6.20480	−1.02006	−0.510032	−	0.860156i	$-0.670366\pi$
$37$	−6.20480	−1.02006	−0.510032	+	0.860156i	$0.670366\pi$
$38$	−7.77820	−1.26179
$39$	0	0
$40$	0	0
$41$	0.876571	0.136897	0.0684487	−	0.997655i	$-0.478195\pi$
$41$	0.876571	0.136897	0.0684487	+	0.997655i	$0.478195\pi$
$42$	0	0
$43$	−0.0269678	−0.00411255	−0.00205628	−	0.999998i	$-0.500655\pi$
$43$	−0.0269678	−0.00411255	−0.00205628	+	0.999998i	$0.500655\pi$
$44$	−0.103022	−0.0155312
$45$	0	0
$46$	4.10706	0.605553
$47$	11.2757	1.64473	0.822365	−	0.568961i	$-0.192654\pi$
$47$	11.2757	1.64473	0.822365	+	0.568961i	$0.192654\pi$
$48$	0	0
$49$	4.39983	0.628547
$50$	0	0
$51$	0	0
$52$	−2.14765	−0.297825
$53$	−2.16918	−0.297960	−0.148980	−	0.988840i	$-0.547599\pi$
$53$	−2.16918	−0.297960	−0.148980	+	0.988840i	$0.547599\pi$
$54$	0	0
$55$	0	0
$56$	8.45890	1.13037
$57$	0	0
$58$	9.67269	1.27009
$59$	13.3441	1.73725	0.868625	−	0.495470i	$-0.165004\pi$
$59$	13.3441	1.73725	0.868625	+	0.495470i	$0.165004\pi$
$60$	0	0
$61$	5.33819	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$61$	5.33819	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$62$	1.52086	0.193149
$63$	0	0
$64$	5.99695	0.749619
$65$	0	0
$66$	0	0
$67$	−2.95192	−0.360634	−0.180317	−	0.983609i	$-0.557712\pi$
$67$	−2.95192	−0.360634	−0.180317	+	0.983609i	$0.557712\pi$
$68$	2.28818	0.277482
$69$	0	0
$70$	0	0
$71$	5.03195	0.597183	0.298591	−	0.954381i	$-0.403483\pi$
$71$	5.03195	0.597183	0.298591	+	0.954381i	$0.403483\pi$
$72$	0	0
$73$	4.45304	0.521189	0.260594	−	0.965448i	$-0.416081\pi$
$73$	4.45304	0.521189	0.260594	+	0.965448i	$0.416081\pi$
$74$	−9.56019	−1.11135
$75$	0	0
$76$	−1.88794	−0.216561
$77$	0.930107	0.105996
$78$	0	0
$79$	13.6841	1.53959	0.769793	−	0.638294i	$-0.220359\pi$
$79$	13.6841	1.53959	0.769793	+	0.638294i	$0.220359\pi$
$80$	0	0
$81$	0	0
$82$	1.35060	0.149148
$83$	−8.71814	−0.956940	−0.478470	−	0.878104i	$-0.658808\pi$
$83$	−8.71814	−0.956940	−0.478470	+	0.878104i	$0.658808\pi$
$84$	0	0
$85$	0	0
$86$	−0.0415513	−0.00448059
$87$	0	0
$88$	0.690158	0.0735711
$89$	−14.2494	−1.51044	−0.755219	−	0.655473i	$-0.772469\pi$
$89$	−14.2494	−1.51044	−0.755219	+	0.655473i	$0.772469\pi$
$90$	0	0
$91$	19.3894	2.03256
$92$	0.996873	0.103931
$93$	0	0
$94$	17.3733	1.79192
$95$	0	0
$96$	0	0
$97$	−5.72170	−0.580951	−0.290475	−	0.956882i	$-0.593813\pi$
$97$	−5.72170	−0.580951	−0.290475	+	0.956882i	$0.593813\pi$
$98$	6.77913	0.684795
$99$	0	0
$100$	0	0
$101$	9.34839	0.930199	0.465100	−	0.885258i	$-0.346019\pi$
$101$	9.34839	0.930199	0.465100	+	0.885258i	$0.346019\pi$
$102$	0	0
$103$	−12.5178	−1.23342	−0.616708	−	0.787192i	$-0.711534\pi$
$103$	−12.5178	−1.23342	−0.616708	+	0.787192i	$0.711534\pi$
$104$	14.3873	1.41079
$105$	0	0
$106$	−3.34221	−0.324624
$107$	3.10152	0.299835	0.149918	−	0.988699i	$-0.452099\pi$
$107$	3.10152	0.299835	0.149918	+	0.988699i	$0.452099\pi$
$108$	0	0
$109$	−1.88328	−0.180386	−0.0901929	−	0.995924i	$-0.528748\pi$
$109$	−1.88328	−0.180386	−0.0901929	+	0.995924i	$0.528748\pi$
$110$	0	0
$111$	0	0
$112$	15.5586	1.47015
$113$	9.54846	0.898244	0.449122	−	0.893470i	$-0.351737\pi$
$113$	9.54846	0.898244	0.449122	+	0.893470i	$0.351737\pi$
$114$	0	0
$115$	0	0
$116$	2.34777	0.217985
$117$	0	0
$118$	20.5602	1.89272
$119$	−20.6582	−1.89373
$120$	0	0
$121$	−10.9241	−0.993101
$122$	8.22494	0.744651
$123$	0	0
$124$	0.369146	0.0331503
$125$	0	0
$126$	0	0
$127$	4.32755	0.384008	0.192004	−	0.981394i	$-0.438501\pi$
$127$	4.32755	0.384008	0.192004	+	0.981394i	$0.438501\pi$
$128$	13.4187	1.18605
$129$	0	0
$130$	0	0
$131$	−3.62288	−0.316532	−0.158266	−	0.987396i	$-0.550590\pi$
$131$	−3.62288	−0.316532	−0.158266	+	0.987396i	$0.550590\pi$
$132$	0	0
$133$	17.0447	1.47796
$134$	−4.54823	−0.392907
$135$	0	0
$136$	−15.3288	−1.31443
$137$	16.2787	1.39079	0.695394	−	0.718629i	$-0.255230\pi$
$137$	16.2787	1.39079	0.695394	+	0.718629i	$0.255230\pi$
$138$	0	0
$139$	8.02585	0.680743	0.340372	−	0.940291i	$-0.389447\pi$
$139$	8.02585	0.680743	0.340372	+	0.940291i	$0.389447\pi$
$140$	0	0
$141$	0	0
$142$	7.75309	0.650625
$143$	1.58198	0.132291
$144$	0	0
$145$	0	0
$146$	6.86112	0.567830
$147$	0	0
$148$	−2.32047	−0.190741
$149$	20.6766	1.69389	0.846947	−	0.531678i	$-0.178438\pi$
$149$	20.6766	1.69389	0.846947	+	0.531678i	$0.178438\pi$
$150$	0	0
$151$	−3.42836	−0.278996	−0.139498	−	0.990222i	$-0.544549\pi$
$151$	−3.42836	−0.278996	−0.139498	+	0.990222i	$0.544549\pi$
$152$	12.6475	1.02585
$153$	0	0
$154$	1.43308	0.115481
$155$	0	0
$156$	0	0
$157$	14.8901	1.18836	0.594180	−	0.804332i	$-0.297477\pi$
$157$	14.8901	1.18836	0.594180	+	0.804332i	$0.297477\pi$
$158$	21.0841	1.67736
$159$	0	0
$160$	0	0
$161$	−8.99999	−0.709299
$162$	0	0
$163$	−23.7139	−1.85742	−0.928708	−	0.370812i	$-0.879080\pi$
$163$	−23.7139	−1.85742	−0.928708	+	0.370812i	$0.879080\pi$
$164$	0.327819	0.0255983
$165$	0	0
$166$	−13.4327	−1.04258
$167$	17.3913	1.34578	0.672888	−	0.739744i	$-0.265054\pi$
$167$	17.3913	1.34578	0.672888	+	0.739744i	$0.265054\pi$
$168$	0	0
$169$	19.9786	1.53681
$170$	0	0
$171$	0	0
$172$	−0.0100854	−0.000769004 0
$173$	−12.2246	−0.929421	−0.464710	−	0.885463i	$-0.653842\pi$
$173$	−12.2246	−0.929421	−0.464710	+	0.885463i	$0.653842\pi$
$174$	0	0
$175$	0	0
$176$	1.26942	0.0956862
$177$	0	0
$178$	−21.9551	−1.64561
$179$	11.2504	0.840896	0.420448	−	0.907317i	$-0.361873\pi$
$179$	11.2504	0.840896	0.420448	+	0.907317i	$0.361873\pi$
$180$	0	0
$181$	4.69070	0.348657	0.174329	−	0.984688i	$-0.444225\pi$
$181$	4.69070	0.348657	0.174329	+	0.984688i	$0.444225\pi$
$182$	29.8747	2.21446
$183$	0	0
$184$	−6.67817	−0.492321
$185$	0	0
$186$	0	0
$187$	−1.68549	−0.123255
$188$	4.21687	0.307547
$189$	0	0
$190$	0	0
$191$	−21.7614	−1.57460	−0.787301	−	0.616569i	$-0.788522\pi$
$191$	−21.7614	−1.57460	−0.787301	+	0.616569i	$0.788522\pi$
$192$	0	0
$193$	−0.450776	−0.0324476	−0.0162238	−	0.999868i	$-0.505164\pi$
$193$	−0.450776	−0.0324476	−0.0162238	+	0.999868i	$0.505164\pi$
$194$	−8.81584	−0.632940
$195$	0	0
$196$	1.64544	0.117532
$197$	−9.45165	−0.673402	−0.336701	−	0.941612i	$-0.609311\pi$
$197$	−9.45165	−0.673402	−0.336701	+	0.941612i	$0.609311\pi$
$198$	0	0
$199$	19.8515	1.40724	0.703618	−	0.710578i	$-0.251567\pi$
$199$	19.8515	1.40724	0.703618	+	0.710578i	$0.251567\pi$
$200$	0	0
$201$	0	0
$202$	14.4037	1.01344
$203$	−21.1962	−1.48768
$204$	0	0
$205$	0	0
$206$	−19.2871	−1.34379
$207$	0	0
$208$	26.4629	1.83487
$209$	1.39067	0.0961947
$210$	0	0
$211$	10.1175	0.696515	0.348257	−	0.937399i	$-0.386773\pi$
$211$	10.1175	0.696515	0.348257	+	0.937399i	$0.386773\pi$
$212$	−0.811227	−0.0557153
$213$	0	0
$214$	4.77873	0.326667
$215$	0	0
$216$	0	0
$217$	−3.33273	−0.226240
$218$	−2.90171	−0.196529
$219$	0	0
$220$	0	0
$221$	−35.1365	−2.36354
$222$	0	0
$223$	−13.0964	−0.876998	−0.438499	−	0.898732i	$-0.644490\pi$
$223$	−13.0964	−0.876998	−0.438499	+	0.898732i	$0.644490\pi$
$224$	7.05448	0.471347
$225$	0	0
$226$	14.7120	0.978628
$227$	12.6754	0.841299	0.420649	−	0.907223i	$-0.361802\pi$
$227$	12.6754	0.841299	0.420649	+	0.907223i	$0.361802\pi$
$228$	0	0
$229$	−11.3525	−0.750197	−0.375099	−	0.926985i	$-0.622391\pi$
$229$	−11.3525	−0.750197	−0.375099	+	0.926985i	$0.622391\pi$
$230$	0	0
$231$	0	0
$232$	−15.7280	−1.03259
$233$	3.02165	0.197955	0.0989774	−	0.995090i	$-0.468443\pi$
$233$	3.02165	0.197955	0.0989774	+	0.995090i	$0.468443\pi$
$234$	0	0
$235$	0	0
$236$	4.99040	0.324847
$237$	0	0
$238$	−31.8295	−2.06320
$239$	2.81485	0.182078	0.0910389	−	0.995847i	$-0.470981\pi$
$239$	2.81485	0.182078	0.0910389	+	0.995847i	$0.470981\pi$
$240$	0	0
$241$	−19.0303	−1.22585	−0.612923	−	0.790142i	$-0.710007\pi$
$241$	−19.0303	−1.22585	−0.612923	+	0.790142i	$0.710007\pi$
$242$	−16.8316	−1.08197
$243$	0	0
$244$	1.99637	0.127805
$245$	0	0
$246$	0	0
$247$	28.9905	1.84462
$248$	−2.47295	−0.157032
$249$	0	0
$250$	0	0
$251$	29.5606	1.86585	0.932924	−	0.360074i	$-0.117249\pi$
$251$	29.5606	1.86585	0.932924	+	0.360074i	$0.117249\pi$
$252$	0	0
$253$	−0.734306	−0.0461654
$254$	6.66776	0.418373
$255$	0	0
$256$	8.68122	0.542576
$257$	−6.05640	−0.377788	−0.188894	−	0.981998i	$-0.560490\pi$
$257$	−6.05640	−0.377788	−0.188894	+	0.981998i	$0.560490\pi$
$258$	0	0
$259$	20.9497	1.30175
$260$	0	0
$261$	0	0
$262$	−5.58203	−0.344859
$263$	−2.79610	−0.172415	−0.0862074	−	0.996277i	$-0.527475\pi$
$263$	−2.79610	−0.172415	−0.0862074	+	0.996277i	$0.527475\pi$
$264$	0	0
$265$	0	0
$266$	26.2620	1.61023
$267$	0	0
$268$	−1.10395	−0.0674347
$269$	−16.4711	−1.00426	−0.502130	−	0.864792i	$-0.667450\pi$
$269$	−16.4711	−1.00426	−0.502130	+	0.864792i	$0.667450\pi$
$270$	0	0
$271$	21.1663	1.28576	0.642880	−	0.765967i	$-0.277739\pi$
$271$	21.1663	1.28576	0.642880	+	0.765967i	$0.277739\pi$
$272$	−28.1945	−1.70954
$273$	0	0
$274$	25.0818	1.51525
$275$	0	0
$276$	0	0
$277$	−24.4330	−1.46803	−0.734017	−	0.679131i	$-0.762357\pi$
$277$	−24.4330	−1.46803	−0.734017	+	0.679131i	$0.762357\pi$
$278$	12.3660	0.741664
$279$	0	0
$280$	0	0
$281$	1.89064	0.112786	0.0563931	−	0.998409i	$-0.482040\pi$
$281$	1.89064	0.112786	0.0563931	+	0.998409i	$0.482040\pi$
$282$	0	0
$283$	−6.75517	−0.401553	−0.200777	−	0.979637i	$-0.564347\pi$
$283$	−6.75517	−0.401553	−0.200777	+	0.979637i	$0.564347\pi$
$284$	1.88184	0.111667
$285$	0	0
$286$	2.43746	0.144130
$287$	−2.95962	−0.174701
$288$	0	0
$289$	20.4356	1.20209
$290$	0	0
$291$	0	0
$292$	1.66534	0.0974568
$293$	−9.98977	−0.583609	−0.291804	−	0.956478i	$-0.594256\pi$
$293$	−9.98977	−0.583609	−0.291804	+	0.956478i	$0.594256\pi$
$294$	0	0
$295$	0	0
$296$	15.5451	0.903539
$297$	0	0
$298$	31.8579	1.84548
$299$	−15.3077	−0.885265
$300$	0	0
$301$	0.0910531	0.00524822
$302$	−5.28233	−0.303964
$303$	0	0
$304$	23.2628	1.33421
$305$	0	0
$306$	0	0
$307$	15.4510	0.881832	0.440916	−	0.897548i	$-0.354654\pi$
$307$	15.4510	0.881832	0.440916	+	0.897548i	$0.354654\pi$
$308$	0.347840	0.0198200
$309$	0	0
$310$	0	0
$311$	24.5330	1.39114	0.695570	−	0.718458i	$-0.255152\pi$
$311$	24.5330	1.39114	0.695570	+	0.718458i	$0.255152\pi$
$312$	0	0
$313$	−9.90435	−0.559827	−0.279914	−	0.960025i	$-0.590306\pi$
$313$	−9.90435	−0.559827	−0.279914	+	0.960025i	$0.590306\pi$
$314$	22.9423	1.29471
$315$	0	0
$316$	5.11758	0.287886
$317$	−11.0669	−0.621576	−0.310788	−	0.950479i	$-0.600593\pi$
$317$	−11.0669	−0.621576	−0.310788	+	0.950479i	$0.600593\pi$
$318$	0	0
$319$	−1.72939	−0.0968272
$320$	0	0
$321$	0	0
$322$	−13.8669	−0.772774
$323$	−30.8875	−1.71863
$324$	0	0
$325$	0	0
$326$	−36.5377	−2.02364
$327$	0	0
$328$	−2.19610	−0.121259
$329$	−38.0708	−2.09891
$330$	0	0
$331$	−13.0271	−0.716034	−0.358017	−	0.933715i	$-0.616547\pi$
$331$	−13.0271	−0.716034	−0.358017	+	0.933715i	$0.616547\pi$
$332$	−3.26040	−0.178938
$333$	0	0
$334$	26.7960	1.46621
$335$	0	0
$336$	0	0
$337$	−16.4458	−0.895858	−0.447929	−	0.894069i	$-0.647838\pi$
$337$	−16.4458	−0.895858	−0.447929	+	0.894069i	$0.647838\pi$
$338$	30.7824	1.67434
$339$	0	0
$340$	0	0
$341$	−0.271916	−0.0147251
$342$	0	0
$343$	8.77913	0.474029
$344$	0.0675632	0.00364277
$345$	0	0
$346$	−18.8354	−1.01260
$347$	18.4439	0.990119	0.495059	−	0.868859i	$-0.335146\pi$
$347$	18.4439	0.990119	0.495059	+	0.868859i	$0.335146\pi$
$348$	0	0
$349$	−10.3683	−0.555005	−0.277502	−	0.960725i	$-0.589507\pi$
$349$	−10.3683	−0.555005	−0.277502	+	0.960725i	$0.589507\pi$
$350$	0	0
$351$	0	0
$352$	0.575572	0.0306781
$353$	−6.65119	−0.354007	−0.177003	−	0.984210i	$-0.556640\pi$
$353$	−6.65119	−0.354007	−0.177003	+	0.984210i	$0.556640\pi$
$354$	0	0
$355$	0	0
$356$	−5.32899	−0.282436
$357$	0	0
$358$	17.3343	0.916149
$359$	−13.4702	−0.710932	−0.355466	−	0.934689i	$-0.615678\pi$
$359$	−13.4702	−0.710932	−0.355466	+	0.934689i	$0.615678\pi$
$360$	0	0
$361$	6.48478	0.341304
$362$	7.22730	0.379859
$363$	0	0
$364$	7.25123	0.380068
$365$	0	0
$366$	0	0
$367$	19.2948	1.00718	0.503591	−	0.863942i	$-0.332012\pi$
$367$	19.2948	1.00718	0.503591	+	0.863942i	$0.332012\pi$
$368$	−12.2833	−0.640311
$369$	0	0
$370$	0	0
$371$	7.32394	0.380240
$372$	0	0
$373$	12.0180	0.622270	0.311135	−	0.950366i	$-0.399291\pi$
$373$	12.0180	0.622270	0.311135	+	0.950366i	$0.399291\pi$
$374$	−2.59696	−0.134285
$375$	0	0
$376$	−28.2493	−1.45685
$377$	−36.0516	−1.85675
$378$	0	0
$379$	18.3248	0.941282	0.470641	−	0.882325i	$-0.344023\pi$
$379$	18.3248	0.941282	0.470641	+	0.882325i	$0.344023\pi$
$380$	0	0
$381$	0	0
$382$	−33.5294	−1.71551
$383$	−4.47653	−0.228740	−0.114370	−	0.993438i	$-0.536485\pi$
$383$	−4.47653	−0.228740	−0.114370	+	0.993438i	$0.536485\pi$
$384$	0	0
$385$	0	0
$386$	−0.694543	−0.0353513
$387$	0	0
$388$	−2.13979	−0.108632
$389$	14.3455	0.727344	0.363672	−	0.931527i	$-0.381523\pi$
$389$	14.3455	0.727344	0.363672	+	0.931527i	$0.381523\pi$
$390$	0	0
$391$	16.3093	0.824797
$392$	−11.0230	−0.556746
$393$	0	0
$394$	−14.5628	−0.733665
$395$	0	0
$396$	0	0
$397$	4.06703	0.204119	0.102059	−	0.994778i	$-0.467457\pi$
$397$	4.06703	0.204119	0.102059	+	0.994778i	$0.467457\pi$
$398$	30.5867	1.53317
$399$	0	0
$400$	0	0
$401$	15.8735	0.792685	0.396342	−	0.918103i	$-0.370279\pi$
$401$	15.8735	0.792685	0.396342	+	0.918103i	$0.370279\pi$
$402$	0	0
$403$	−5.66848	−0.282367
$404$	3.49610	0.173937
$405$	0	0
$406$	−32.6585	−1.62081
$407$	1.70928	0.0847256
$408$	0	0
$409$	11.2178	0.554682	0.277341	−	0.960772i	$-0.410547\pi$
$409$	11.2178	0.554682	0.277341	+	0.960772i	$0.410547\pi$
$410$	0	0
$411$	0	0
$412$	−4.68139	−0.230636
$413$	−45.0544	−2.21698
$414$	0	0
$415$	0	0
$416$	11.9986	0.588281
$417$	0	0
$418$	2.14271	0.104803
$419$	34.0818	1.66500	0.832502	−	0.554022i	$-0.186908\pi$
$419$	34.0818	1.66500	0.832502	+	0.554022i	$0.186908\pi$
$420$	0	0
$421$	1.20959	0.0589518	0.0294759	−	0.999565i	$-0.490616\pi$
$421$	1.20959	0.0589518	0.0294759	+	0.999565i	$0.490616\pi$
$422$	15.5887	0.758846
$423$	0	0
$424$	5.43451	0.263923
$425$	0	0
$426$	0	0
$427$	−18.0237	−0.872227
$428$	1.15990	0.0560660
$429$	0	0
$430$	0	0
$431$	17.8571	0.860144	0.430072	−	0.902795i	$-0.358488\pi$
$431$	17.8571	0.860144	0.430072	+	0.902795i	$0.358488\pi$
$432$	0	0
$433$	31.8056	1.52848	0.764240	−	0.644931i	$-0.223114\pi$
$433$	31.8056	1.52848	0.764240	+	0.644931i	$0.223114\pi$
$434$	−5.13497	−0.246487
$435$	0	0
$436$	−0.704308	−0.0337302
$437$	−13.4565	−0.643714
$438$	0	0
$439$	−8.65773	−0.413211	−0.206605	−	0.978424i	$-0.566242\pi$
$439$	−8.65773	−0.413211	−0.206605	+	0.978424i	$0.566242\pi$
$440$	0	0
$441$	0	0
$442$	−54.1373	−2.57505
$443$	14.2382	0.676477	0.338238	−	0.941061i	$-0.390169\pi$
$443$	14.2382	0.676477	0.338238	+	0.941061i	$0.390169\pi$
$444$	0	0
$445$	0	0
$446$	−20.1785	−0.955481
$447$	0	0
$448$	−20.2479	−0.956622
$449$	13.4056	0.632650	0.316325	−	0.948651i	$-0.397551\pi$
$449$	13.4056	0.632650	0.316325	+	0.948651i	$0.397551\pi$
$450$	0	0
$451$	−0.241474	−0.0113706
$452$	3.57092	0.167962
$453$	0	0
$454$	19.5300	0.916587
$455$	0	0
$456$	0	0
$457$	5.05447	0.236438	0.118219	−	0.992988i	$-0.462281\pi$
$457$	5.05447	0.236438	0.118219	+	0.992988i	$0.462281\pi$
$458$	−17.4917	−0.817333
$459$	0	0
$460$	0	0
$461$	−8.48890	−0.395368	−0.197684	−	0.980266i	$-0.563342\pi$
$461$	−8.48890	−0.395368	−0.197684	+	0.980266i	$0.563342\pi$
$462$	0	0
$463$	21.1118	0.981148	0.490574	−	0.871399i	$-0.336787\pi$
$463$	21.1118	0.981148	0.490574	+	0.871399i	$0.336787\pi$
$464$	−28.9288	−1.34299
$465$	0	0
$466$	4.65567	0.215670
$467$	−10.0250	−0.463901	−0.231951	−	0.972728i	$-0.574511\pi$
$467$	−10.0250	−0.463901	−0.231951	+	0.972728i	$0.574511\pi$
$468$	0	0
$469$	9.96674	0.460221
$470$	0	0
$471$	0	0
$472$	−33.4313	−1.53880
$473$	0.00742899	0.000341585 0
$474$	0	0
$475$	0	0
$476$	−7.72571	−0.354107
$477$	0	0
$478$	4.33705	0.198372
$479$	6.34922	0.290103	0.145052	−	0.989424i	$-0.453665\pi$
$479$	6.34922	0.290103	0.145052	+	0.989424i	$0.453665\pi$
$480$	0	0
$481$	35.6323	1.62469
$482$	−29.3213	−1.33555
$483$	0	0
$484$	−4.08539	−0.185699
$485$	0	0
$486$	0	0
$487$	25.9418	1.17554	0.587768	−	0.809030i	$-0.300007\pi$
$487$	25.9418	1.17554	0.587768	+	0.809030i	$0.300007\pi$
$488$	−13.3739	−0.605409
$489$	0	0
$490$	0	0
$491$	−14.4375	−0.651554	−0.325777	−	0.945447i	$-0.605626\pi$
$491$	−14.4375	−0.651554	−0.325777	+	0.945447i	$0.605626\pi$
$492$	0	0
$493$	38.4106	1.72993
$494$	44.6678	2.00970
$495$	0	0
$496$	−4.54854	−0.204236
$497$	−16.9897	−0.762092
$498$	0	0
$499$	−27.4648	−1.22949	−0.614746	−	0.788725i	$-0.710741\pi$
$499$	−27.4648	−1.22949	−0.614746	+	0.788725i	$0.710741\pi$
$500$	0	0
$501$	0	0
$502$	45.5461	2.03282
$503$	−24.3612	−1.08621	−0.543106	−	0.839664i	$-0.682752\pi$
$503$	−24.3612	−1.08621	−0.543106	+	0.839664i	$0.682752\pi$
$504$	0	0
$505$	0	0
$506$	−1.13140	−0.0502968
$507$	0	0
$508$	1.61841	0.0718054
$509$	5.86899	0.260138	0.130069	−	0.991505i	$-0.458480\pi$
$509$	5.86899	0.260138	0.130069	+	0.991505i	$0.458480\pi$
$510$	0	0
$511$	−15.0351	−0.665113
$512$	−13.4616	−0.594923
$513$	0	0
$514$	−9.33153	−0.411596
$515$	0	0
$516$	0	0
$517$	−3.10618	−0.136610
$518$	32.2787	1.41824
$519$	0	0
$520$	0	0
$521$	29.3976	1.28793	0.643965	−	0.765055i	$-0.277288\pi$
$521$	29.3976	1.28793	0.643965	+	0.765055i	$0.277288\pi$
$522$	0	0
$523$	−12.9982	−0.568373	−0.284187	−	0.958769i	$-0.591724\pi$
$523$	−12.9982	−0.568373	−0.284187	+	0.958769i	$0.591724\pi$
$524$	−1.35488	−0.0591882
$525$	0	0
$526$	−4.30815	−0.187844
$527$	6.03939	0.263080
$528$	0	0
$529$	−15.8946	−0.691071
$530$	0	0
$531$	0	0
$532$	6.37436	0.276363
$533$	−5.03388	−0.218041
$534$	0	0
$535$	0	0
$536$	7.39552	0.319438
$537$	0	0
$538$	−25.3782	−1.09413
$539$	−1.21205	−0.0522065
$540$	0	0
$541$	9.83813	0.422974	0.211487	−	0.977381i	$-0.432169\pi$
$541$	9.83813	0.422974	0.211487	+	0.977381i	$0.432169\pi$
$542$	32.6124	1.40082
$543$	0	0
$544$	−12.7837	−0.548099
$545$	0	0
$546$	0	0
$547$	−20.9058	−0.893868	−0.446934	−	0.894567i	$-0.647484\pi$
$547$	−20.9058	−0.893868	−0.446934	+	0.894567i	$0.647484\pi$
$548$	6.08791	0.260062
$549$	0	0
$550$	0	0
$551$	−31.6920	−1.35012
$552$	0	0
$553$	−46.2026	−1.96474
$554$	−37.6456	−1.59941
$555$	0	0
$556$	3.00150	0.127292
$557$	−20.8969	−0.885431	−0.442716	−	0.896662i	$-0.645985\pi$
$557$	−20.8969	−0.885431	−0.442716	+	0.896662i	$0.645985\pi$
$558$	0	0
$559$	0.154868	0.00655022
$560$	0	0
$561$	0	0
$562$	2.91304	0.122879
$563$	18.5889	0.783428	0.391714	−	0.920087i	$-0.371882\pi$
$563$	18.5889	0.783428	0.391714	+	0.920087i	$0.371882\pi$
$564$	0	0
$565$	0	0
$566$	−10.4082	−0.437488
$567$	0	0
$568$	−12.6067	−0.528965
$569$	30.6786	1.28611	0.643057	−	0.765819i	$-0.277666\pi$
$569$	30.6786	1.28611	0.643057	+	0.765819i	$0.277666\pi$
$570$	0	0
$571$	14.2650	0.596970	0.298485	−	0.954414i	$-0.403519\pi$
$571$	14.2650	0.596970	0.298485	+	0.954414i	$0.403519\pi$
$572$	0.591625	0.0247371
$573$	0	0
$574$	−4.56010	−0.190335
$575$	0	0
$576$	0	0
$577$	−37.1484	−1.54651	−0.773254	−	0.634096i	$-0.781372\pi$
$577$	−37.1484	−1.54651	−0.773254	+	0.634096i	$0.781372\pi$
$578$	31.4866	1.30967
$579$	0	0
$580$	0	0
$581$	29.4356	1.22119
$582$	0	0
$583$	0.597557	0.0247483
$584$	−11.1563	−0.461652
$585$	0	0
$586$	−15.3920	−0.635836
$587$	−30.7079	−1.26745	−0.633726	−	0.773558i	$-0.718475\pi$
$587$	−30.7079	−1.26745	−0.633726	+	0.773558i	$0.718475\pi$
$588$	0	0
$589$	−4.98300	−0.205321
$590$	0	0
$591$	0	0
$592$	28.5923	1.17514
$593$	−30.6705	−1.25949	−0.629744	−	0.776803i	$-0.716840\pi$
$593$	−30.6705	−1.25949	−0.629744	+	0.776803i	$0.716840\pi$
$594$	0	0
$595$	0	0
$596$	7.73261	0.316740
$597$	0	0
$598$	−23.5856	−0.964488
$599$	−32.8329	−1.34152	−0.670759	−	0.741676i	$-0.734031\pi$
$599$	−32.8329	−1.34152	−0.670759	+	0.741676i	$0.734031\pi$
$600$	0	0
$601$	42.8051	1.74606	0.873029	−	0.487669i	$-0.162153\pi$
$601$	42.8051	1.74606	0.873029	+	0.487669i	$0.162153\pi$
$602$	0.140292	0.00571788
$603$	0	0
$604$	−1.28214	−0.0521694
$605$	0	0
$606$	0	0
$607$	41.5366	1.68592	0.842959	−	0.537977i	$-0.180811\pi$
$607$	41.5366	1.68592	0.842959	+	0.537977i	$0.180811\pi$
$608$	10.5477	0.427764
$609$	0	0
$610$	0	0
$611$	−64.7529	−2.61962
$612$	0	0
$613$	15.1045	0.610067	0.305033	−	0.952342i	$-0.401332\pi$
$613$	15.1045	0.610067	0.305033	+	0.952342i	$0.401332\pi$
$614$	23.8064	0.960748
$615$	0	0
$616$	−2.33022	−0.0938874
$617$	16.5874	0.667782	0.333891	−	0.942612i	$-0.391638\pi$
$617$	16.5874	0.667782	0.333891	+	0.942612i	$0.391638\pi$
$618$	0	0
$619$	20.3284	0.817068	0.408534	−	0.912743i	$-0.366040\pi$
$619$	20.3284	0.817068	0.408534	+	0.912743i	$0.366040\pi$
$620$	0	0
$621$	0	0
$622$	37.7998	1.51563
$623$	48.1113	1.92754
$624$	0	0
$625$	0	0
$626$	−15.2603	−0.609926
$627$	0	0
$628$	5.56858	0.222211
$629$	−37.9639	−1.51372
$630$	0	0
$631$	21.0922	0.839668	0.419834	−	0.907601i	$-0.362088\pi$
$631$	21.0922	0.839668	0.419834	+	0.907601i	$0.362088\pi$
$632$	−34.2833	−1.36371
$633$	0	0
$634$	−17.0515	−0.677202
$635$	0	0
$636$	0	0
$637$	−25.2669	−1.00111
$638$	−2.66459	−0.105492
$639$	0	0
$640$	0	0
$641$	−12.6154	−0.498279	−0.249140	−	0.968468i	$-0.580148\pi$
$641$	−12.6154	−0.498279	−0.249140	+	0.968468i	$0.580148\pi$
$642$	0	0
$643$	−18.3450	−0.723457	−0.361728	−	0.932284i	$-0.617813\pi$
$643$	−18.3450	−0.723457	−0.361728	+	0.932284i	$0.617813\pi$
$644$	−3.36581	−0.132631
$645$	0	0
$646$	−47.5906	−1.87243
$647$	−32.1546	−1.26413	−0.632064	−	0.774916i	$-0.717792\pi$
$647$	−32.1546	−1.26413	−0.632064	+	0.774916i	$0.717792\pi$
$648$	0	0
$649$	−3.67597	−0.144295
$650$	0	0
$651$	0	0
$652$	−8.86850	−0.347317
$653$	49.5727	1.93993	0.969964	−	0.243249i	$-0.0782131\pi$
$653$	49.5727	1.93993	0.969964	+	0.243249i	$0.0782131\pi$
$654$	0	0
$655$	0	0
$656$	−4.03932	−0.157709
$657$	0	0
$658$	−58.6585	−2.28675
$659$	−26.8273	−1.04504	−0.522521	−	0.852626i	$-0.675008\pi$
$659$	−26.8273	−1.04504	−0.522521	+	0.852626i	$0.675008\pi$
$660$	0	0
$661$	6.66845	0.259373	0.129686	−	0.991555i	$-0.458603\pi$
$661$	6.66845	0.259373	0.129686	+	0.991555i	$0.458603\pi$
$662$	−20.0718	−0.780112
$663$	0	0
$664$	21.8418	0.847626
$665$	0	0
$666$	0	0
$667$	16.7341	0.647946
$668$	6.50396	0.251646
$669$	0	0
$670$	0	0
$671$	−1.47054	−0.0567697
$672$	0	0
$673$	27.4981	1.05997	0.529987	−	0.848006i	$-0.322197\pi$
$673$	27.4981	1.05997	0.529987	+	0.848006i	$0.322197\pi$
$674$	−25.3392	−0.976029
$675$	0	0
$676$	7.47156	0.287368
$677$	−20.1952	−0.776163	−0.388082	−	0.921625i	$-0.626862\pi$
$677$	−20.1952	−0.776163	−0.388082	+	0.921625i	$0.626862\pi$
$678$	0	0
$679$	19.3185	0.741377
$680$	0	0
$681$	0	0
$682$	−0.418960	−0.0160428
$683$	−44.3125	−1.69557	−0.847786	−	0.530339i	$-0.822065\pi$
$683$	−44.3125	−1.69557	−0.847786	+	0.530339i	$0.822065\pi$
$684$	0	0
$685$	0	0
$686$	13.5266	0.516450
$687$	0	0
$688$	0.124270	0.00473776
$689$	12.4569	0.474572
$690$	0	0
$691$	−27.7685	−1.05636	−0.528182	−	0.849131i	$-0.677126\pi$
$691$	−27.7685	−1.05636	−0.528182	+	0.849131i	$0.677126\pi$
$692$	−4.57175	−0.173792
$693$	0	0
$694$	28.4178	1.07872
$695$	0	0
$696$	0	0
$697$	5.36327	0.203148
$698$	−15.9753	−0.604672
$699$	0	0
$700$	0	0
$701$	34.8116	1.31481	0.657407	−	0.753535i	$-0.271653\pi$
$701$	34.8116	1.31481	0.657407	+	0.753535i	$0.271653\pi$
$702$	0	0
$703$	31.3234	1.18138
$704$	−1.65202	−0.0622627
$705$	0	0
$706$	−10.2480	−0.385687
$707$	−31.5635	−1.18707
$708$	0	0
$709$	−44.0887	−1.65579	−0.827893	−	0.560886i	$-0.810461\pi$
$709$	−44.0887	−1.65579	−0.827893	+	0.560886i	$0.810461\pi$
$710$	0	0
$711$	0	0
$712$	35.6995	1.33790
$713$	2.63114	0.0985369
$714$	0	0
$715$	0	0
$716$	4.20742	0.157239
$717$	0	0
$718$	−20.7546	−0.774554
$719$	38.4272	1.43309	0.716547	−	0.697539i	$-0.245722\pi$
$719$	38.4272	1.43309	0.716547	+	0.697539i	$0.245722\pi$
$720$	0	0
$721$	42.2646	1.57402
$722$	9.99156	0.371848
$723$	0	0
$724$	1.75422	0.0651952
$725$	0	0
$726$	0	0
$727$	28.3236	1.05046	0.525232	−	0.850959i	$-0.323979\pi$
$727$	28.3236	1.05046	0.525232	+	0.850959i	$0.323979\pi$
$728$	−48.5769	−1.80038
$729$	0	0
$730$	0	0
$731$	−0.165002	−0.00610281
$732$	0	0
$733$	−2.46695	−0.0911190	−0.0455595	−	0.998962i	$-0.514507\pi$
$733$	−2.46695	−0.0911190	−0.0455595	+	0.998962i	$0.514507\pi$
$734$	29.7289	1.09732
$735$	0	0
$736$	−5.56940	−0.205291
$737$	0.813182	0.0299540
$738$	0	0
$739$	−20.6368	−0.759136	−0.379568	−	0.925164i	$-0.623927\pi$
$739$	−20.6368	−0.759136	−0.379568	+	0.925164i	$0.623927\pi$
$740$	0	0
$741$	0	0
$742$	11.2845	0.414268
$743$	29.8469	1.09498	0.547488	−	0.836814i	$-0.315584\pi$
$743$	29.8469	1.09498	0.547488	+	0.836814i	$0.315584\pi$
$744$	0	0
$745$	0	0
$746$	18.5170	0.677957
$747$	0	0
$748$	−0.630338	−0.0230474
$749$	−10.4718	−0.382633
$750$	0	0
$751$	27.8172	1.01506	0.507532	−	0.861633i	$-0.330558\pi$
$751$	27.8172	1.01506	0.507532	+	0.861633i	$0.330558\pi$
$752$	−51.9595	−1.89477
$753$	0	0
$754$	−55.5473	−2.02291
$755$	0	0
$756$	0	0
$757$	−31.1959	−1.13383	−0.566916	−	0.823775i	$-0.691864\pi$
$757$	−31.1959	−1.13383	−0.566916	+	0.823775i	$0.691864\pi$
$758$	28.2344	1.02552
$759$	0	0
$760$	0	0
$761$	−9.40270	−0.340848	−0.170424	−	0.985371i	$-0.554514\pi$
$761$	−9.40270	−0.340848	−0.170424	+	0.985371i	$0.554514\pi$
$762$	0	0
$763$	6.35864	0.230198
$764$	−8.13831	−0.294434
$765$	0	0
$766$	−6.89732	−0.249210
$767$	−76.6310	−2.76698
$768$	0	0
$769$	−19.0633	−0.687439	−0.343720	−	0.939072i	$-0.611687\pi$
$769$	−19.0633	−0.687439	−0.343720	+	0.939072i	$0.611687\pi$
$770$	0	0
$771$	0	0
$772$	−0.168581	−0.00606735
$773$	−0.364033	−0.0130934	−0.00654668	−	0.999979i	$-0.502084\pi$
$773$	−0.364033	−0.0130934	−0.00654668	+	0.999979i	$0.502084\pi$
$774$	0	0
$775$	0	0
$776$	14.3347	0.514587
$777$	0	0
$778$	22.1031	0.792434
$779$	−4.42514	−0.158547
$780$	0	0
$781$	−1.38618	−0.0496015
$782$	25.1289	0.898608
$783$	0	0
$784$	−20.2748	−0.724101
$785$	0	0
$786$	0	0
$787$	−23.0565	−0.821876	−0.410938	−	0.911663i	$-0.634799\pi$
$787$	−23.0565	−0.821876	−0.410938	+	0.911663i	$0.634799\pi$
$788$	−3.53472	−0.125919
$789$	0	0
$790$	0	0
$791$	−32.2391	−1.14629
$792$	0	0
$793$	−30.6556	−1.08861
$794$	6.26637	0.222385
$795$	0	0
$796$	7.42405	0.263138
$797$	4.33846	0.153676	0.0768380	−	0.997044i	$-0.475518\pi$
$797$	4.33846	0.153676	0.0768380	+	0.997044i	$0.475518\pi$
$798$	0	0
$799$	68.9899	2.44069
$800$	0	0
$801$	0	0
$802$	24.4574	0.863623
$803$	−1.22671	−0.0432895
$804$	0	0
$805$	0	0
$806$	−8.73384	−0.307636
$807$	0	0
$808$	−23.4208	−0.823940
$809$	11.0758	0.389406	0.194703	−	0.980862i	$-0.437626\pi$
$809$	11.0758	0.389406	0.194703	+	0.980862i	$0.437626\pi$
$810$	0	0
$811$	44.0100	1.54540	0.772701	−	0.634771i	$-0.218905\pi$
$811$	44.0100	1.54540	0.772701	+	0.634771i	$0.218905\pi$
$812$	−7.92693	−0.278181
$813$	0	0
$814$	2.63360	0.0923077
$815$	0	0
$816$	0	0
$817$	0.136140	0.00476294
$818$	17.2840	0.604321
$819$	0	0
$820$	0	0
$821$	1.81681	0.0634072	0.0317036	−	0.999497i	$-0.489907\pi$
$821$	1.81681	0.0634072	0.0317036	+	0.999497i	$0.489907\pi$
$822$	0	0
$823$	10.1162	0.352627	0.176314	−	0.984334i	$-0.443583\pi$
$823$	10.1162	0.352627	0.176314	+	0.984334i	$0.443583\pi$
$824$	31.3612	1.09252
$825$	0	0
$826$	−69.4186	−2.41538
$827$	24.7123	0.859332	0.429666	−	0.902988i	$-0.358631\pi$
$827$	24.7123	0.859332	0.429666	+	0.902988i	$0.358631\pi$
$828$	0	0
$829$	−45.2271	−1.57080	−0.785400	−	0.618988i	$-0.787543\pi$
$829$	−45.2271	−1.57080	−0.785400	+	0.618988i	$0.787543\pi$
$830$	0	0
$831$	0	0
$832$	−34.4387	−1.19395
$833$	26.9202	0.932729
$834$	0	0
$835$	0	0
$836$	0.520081	0.0179874
$837$	0	0
$838$	52.5122	1.81401
$839$	38.1856	1.31831	0.659157	−	0.752005i	$-0.270913\pi$
$839$	38.1856	1.31831	0.659157	+	0.752005i	$0.270913\pi$
$840$	0	0
$841$	10.4110	0.359001
$842$	1.86370	0.0642274
$843$	0	0
$844$	3.78372	0.130241
$845$	0	0
$846$	0	0
$847$	36.8838	1.26734
$848$	9.99579	0.343257
$849$	0	0
$850$	0	0
$851$	−16.5395	−0.566965
$852$	0	0
$853$	45.8622	1.57029	0.785145	−	0.619312i	$-0.212588\pi$
$853$	45.8622	1.57029	0.785145	+	0.619312i	$0.212588\pi$
$854$	−27.7704	−0.950283
$855$	0	0
$856$	−7.77032	−0.265584
$857$	33.5199	1.14502	0.572508	−	0.819899i	$-0.305970\pi$
$857$	33.5199	1.14502	0.572508	+	0.819899i	$0.305970\pi$
$858$	0	0
$859$	45.9508	1.56782	0.783911	−	0.620873i	$-0.213222\pi$
$859$	45.9508	1.56782	0.783911	+	0.620873i	$0.213222\pi$
$860$	0	0
$861$	0	0
$862$	27.5137	0.937119
$863$	−11.3593	−0.386675	−0.193337	−	0.981132i	$-0.561931\pi$
$863$	−11.3593	−0.386675	−0.193337	+	0.981132i	$0.561931\pi$
$864$	0	0
$865$	0	0
$866$	49.0052	1.66527
$867$	0	0
$868$	−1.24637	−0.0423045
$869$	−3.76965	−0.127877
$870$	0	0
$871$	16.9520	0.574395
$872$	4.71824	0.159780
$873$	0	0
$874$	−20.7335	−0.701320
$875$	0	0
$876$	0	0
$877$	9.38169	0.316797	0.158399	−	0.987375i	$-0.449367\pi$
$877$	9.38169	0.316797	0.158399	+	0.987375i	$0.449367\pi$
$878$	−13.3396	−0.450189
$879$	0	0
$880$	0	0
$881$	−6.77079	−0.228114	−0.114057	−	0.993474i	$-0.536385\pi$
$881$	−6.77079	−0.228114	−0.114057	+	0.993474i	$0.536385\pi$
$882$	0	0
$883$	−15.7878	−0.531301	−0.265651	−	0.964069i	$-0.585587\pi$
$883$	−15.7878	−0.531301	−0.265651	+	0.964069i	$0.585587\pi$
$884$	−13.1403	−0.441956
$885$	0	0
$886$	21.9378	0.737015
$887$	9.65613	0.324221	0.162111	−	0.986773i	$-0.448170\pi$
$887$	9.65613	0.324221	0.162111	+	0.986773i	$0.448170\pi$
$888$	0	0
$889$	−14.6114	−0.490050
$890$	0	0
$891$	0	0
$892$	−4.89777	−0.163989
$893$	−56.9225	−1.90484
$894$	0	0
$895$	0	0
$896$	−45.3063	−1.51358
$897$	0	0
$898$	20.6550	0.689266
$899$	6.19669	0.206671
$900$	0	0
$901$	−13.2721	−0.442156
$902$	−0.372057	−0.0123881
$903$	0	0
$904$	−23.9220	−0.795635
$905$	0	0
$906$	0	0
$907$	36.2743	1.20447	0.602233	−	0.798320i	$-0.294278\pi$
$907$	36.2743	1.20447	0.602233	+	0.798320i	$0.294278\pi$
$908$	4.74035	0.157314
$909$	0	0
$910$	0	0
$911$	56.7978	1.88179	0.940897	−	0.338692i	$-0.109984\pi$
$911$	56.7978	1.88179	0.940897	+	0.338692i	$0.109984\pi$
$912$	0	0
$913$	2.40164	0.0794826
$914$	7.78779	0.257597
$915$	0	0
$916$	−4.24561	−0.140279
$917$	12.2322	0.403941
$918$	0	0
$919$	−40.8300	−1.34686	−0.673429	−	0.739252i	$-0.735179\pi$
$919$	−40.8300	−1.34686	−0.673429	+	0.739252i	$0.735179\pi$
$920$	0	0
$921$	0	0
$922$	−13.0795	−0.430749
$923$	−28.8970	−0.951155
$924$	0	0
$925$	0	0
$926$	32.5285	1.06895
$927$	0	0
$928$	−13.1167	−0.430577
$929$	33.5585	1.10102	0.550509	−	0.834829i	$-0.314434\pi$
$929$	33.5585	1.10102	0.550509	+	0.834829i	$0.314434\pi$
$930$	0	0
$931$	−22.2114	−0.727949
$932$	1.13003	0.0370154
$933$	0	0
$934$	−15.4462	−0.505416
$935$	0	0
$936$	0	0
$937$	−13.0417	−0.426053	−0.213026	−	0.977046i	$-0.568332\pi$
$937$	−13.0417	−0.426053	−0.213026	+	0.977046i	$0.568332\pi$
$938$	15.3565	0.501407
$939$	0	0
$940$	0	0
$941$	34.2301	1.11587	0.557935	−	0.829884i	$-0.311594\pi$
$941$	34.2301	1.11587	0.557935	+	0.829884i	$0.311594\pi$
$942$	0	0
$943$	2.33658	0.0760894
$944$	−61.4908	−2.00136
$945$	0	0
$946$	0.0114464	0.000372154 0
$947$	3.22739	0.104876	0.0524381	−	0.998624i	$-0.483301\pi$
$947$	3.22739	0.104876	0.0524381	+	0.998624i	$0.483301\pi$
$948$	0	0
$949$	−25.5725	−0.830117
$950$	0	0
$951$	0	0
$952$	51.7555	1.67740
$953$	6.33220	0.205120	0.102560	−	0.994727i	$-0.467297\pi$
$953$	6.33220	0.205120	0.102560	+	0.994727i	$0.467297\pi$
$954$	0	0
$955$	0	0
$956$	1.05270	0.0340466
$957$	0	0
$958$	9.78270	0.316065
$959$	−54.9630	−1.77485
$960$	0	0
$961$	−30.0257	−0.968570
$962$	54.9013	1.77009
$963$	0	0
$964$	−7.11691	−0.229220
$965$	0	0
$966$	0	0
$967$	−11.1469	−0.358460	−0.179230	−	0.983807i	$-0.557361\pi$
$967$	−11.1469	−0.358460	−0.179230	+	0.983807i	$0.557361\pi$
$968$	27.3685	0.879657
$969$	0	0
$970$	0	0
$971$	30.8854	0.991159	0.495580	−	0.868563i	$-0.334956\pi$
$971$	30.8854	0.991159	0.495580	+	0.868563i	$0.334956\pi$
$972$	0	0
$973$	−27.0982	−0.868728
$974$	39.9704	1.28073
$975$	0	0
$976$	−24.5989	−0.787392
$977$	24.5207	0.784486	0.392243	−	0.919862i	$-0.371699\pi$
$977$	24.5207	0.784486	0.392243	+	0.919862i	$0.371699\pi$
$978$	0	0
$979$	3.92538	0.125456
$980$	0	0
$981$	0	0
$982$	−22.2449	−0.709862
$983$	9.12390	0.291007	0.145504	−	0.989358i	$-0.453520\pi$
$983$	9.12390	0.291007	0.145504	+	0.989358i	$0.453520\pi$
$984$	0	0
$985$	0	0
$986$	59.1820	1.88474
$987$	0	0
$988$	10.8418	0.344925
$989$	−0.0718851	−0.00228581
$990$	0	0
$991$	7.78580	0.247324	0.123662	−	0.992324i	$-0.460536\pi$
$991$	7.78580	0.247324	0.123662	+	0.992324i	$0.460536\pi$
$992$	−2.06237	−0.0654803
$993$	0	0
$994$	−26.1773	−0.830292
$995$	0	0
$996$	0	0
$997$	−17.7566	−0.562357	−0.281178	−	0.959656i	$-0.590725\pi$
$997$	−17.7566	−0.562357	−0.281178	+	0.959656i	$0.590725\pi$
$998$	−42.3169	−1.33952
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.bf.1.17		24
3.2	odd	2	inner	5625.2.a.bf.1.7		24
5.4	even	2	inner	5625.2.a.bf.1.8		24
15.14	odd	2	inner	5625.2.a.bf.1.18		24
25.12	odd	20		225.2.m.c.19.2	✓	24
25.23	odd	20		225.2.m.c.154.2	yes	24
75.23	even	20		225.2.m.c.154.5	yes	24
75.62	even	20		225.2.m.c.19.5	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.m.c.19.2	✓	24	25.12	odd	20
225.2.m.c.19.5	yes	24	75.62	even	20
225.2.m.c.154.2	yes	24	25.23	odd	20
225.2.m.c.154.5	yes	24	75.23	even	20
5625.2.a.bf.1.7		24	3.2	odd	2	inner
5625.2.a.bf.1.8		24	5.4	even	2	inner
5625.2.a.bf.1.17		24	1.1	even	1	trivial
5625.2.a.bf.1.18		24	15.14	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+1.54077 q^{2} +0.373979 q^{4} -3.37636 q^{7} -2.50533 q^{8} +O(q^{10})\)	\(q+1.54077 q^{2} +0.373979 q^{4} -3.37636 q^{7} -2.50533 q^{8} -0.275476 q^{11} -5.74270 q^{13} -5.20221 q^{14} -4.60810 q^{16} +6.11846 q^{17} -5.04824 q^{19} -0.424446 q^{22} +2.66559 q^{23} -8.84819 q^{26} -1.26269 q^{28} +6.27782 q^{29} +0.987076 q^{31} -2.08937 q^{32} +9.42716 q^{34} -6.20480 q^{37} -7.77820 q^{38} +0.876571 q^{41} -0.0269678 q^{43} -0.103022 q^{44} +4.10706 q^{46} +11.2757 q^{47} +4.39983 q^{49} -2.14765 q^{52} -2.16918 q^{53} +8.45890 q^{56} +9.67269 q^{58} +13.3441 q^{59} +5.33819 q^{61} +1.52086 q^{62} +5.99695 q^{64} -2.95192 q^{67} +2.28818 q^{68} +5.03195 q^{71} +4.45304 q^{73} -9.56019 q^{74} -1.88794 q^{76} +0.930107 q^{77} +13.6841 q^{79} +1.35060 q^{82} -8.71814 q^{83} -0.0415513 q^{86} +0.690158 q^{88} -14.2494 q^{89} +19.3894 q^{91} +0.996873 q^{92} +17.3733 q^{94} -5.72170 q^{97} +6.77913 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4}+O(q^{10}) \) 24 * q + 32 * q^4	\( 24 q + 32 q^{4} + 56 q^{16} + 36 q^{19} + 52 q^{31} + 60 q^{34} + 60 q^{46} + 72 q^{49} + 68 q^{61} + 108 q^{64} + 88 q^{76} + 84 q^{79} + 80 q^{91} + 100 q^{94}+O(q^{100}) \) 24 * q + 32 * q^4 + 56 * q^16 + 36 * q^19 + 52 * q^31 + 60 * q^34 + 60 * q^46 + 72 * q^49 + 68 * q^61 + 108 * q^64 + 88 * q^76 + 84 * q^79 + 80 * q^91 + 100 * q^94

Introduction

L-functions

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Database

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