LMFDB - Embedded newform 2025.2.a.x.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2025,2,Mod(1,2025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2025.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2025.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,6,0,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$16.1697064093$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 405)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.52434$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.52434 q^{2} +4.37228 q^{4} +3.46410 q^{7} +5.98844 q^{8} -1.37228 q^{11} -4.10891 q^{13} +8.74456 q^{14} +6.37228 q^{16} +2.52434 q^{17} +5.37228 q^{19} -3.46410 q^{22} +5.04868 q^{23} -10.3723 q^{26} +15.1460 q^{28} -5.74456 q^{29} +0.627719 q^{31} +4.10891 q^{32} +6.37228 q^{34} +7.57301 q^{37} +13.5615 q^{38} +1.37228 q^{41} -3.46410 q^{43} -6.00000 q^{44} +12.7446 q^{46} -8.51278 q^{47} +5.00000 q^{49} -17.9653 q^{52} -5.34363 q^{53} +20.7446 q^{56} -14.5012 q^{58} -7.37228 q^{59} +3.62772 q^{61} +1.58457 q^{62} -2.37228 q^{64} -8.21782 q^{67} +11.0371 q^{68} +4.11684 q^{71} +7.57301 q^{73} +19.1168 q^{74} +23.4891 q^{76} -4.75372 q^{77} -4.74456 q^{79} +3.46410 q^{82} -5.34363 q^{83} -8.74456 q^{86} -8.21782 q^{88} +3.00000 q^{89} -14.2337 q^{91} +22.0742 q^{92} -21.4891 q^{94} +18.6101 q^{97} +12.6217 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4} + 6 q^{11} + 12 q^{14} + 14 q^{16} + 10 q^{19} - 30 q^{26} + 14 q^{31} + 14 q^{34} - 6 q^{41} - 24 q^{44} + 28 q^{46} + 20 q^{49} + 60 q^{56} - 18 q^{59} + 26 q^{61} + 2 q^{64} - 18 q^{71}+ \cdots - 40 q^{94}+O(q^{100}) $ 4 * q + 6 * q^4 + 6 * q^11 + 12 * q^14 + 14 * q^16 + 10 * q^19 - 30 * q^26 + 14 * q^31 + 14 * q^34 - 6 * q^41 - 24 * q^44 + 28 * q^46 + 20 * q^49 + 60 * q^56 - 18 * q^59 + 26 * q^61 + 2 * q^64 - 18 * q^71 + 42 * q^74 + 48 * q^76 + 4 * q^79 - 12 * q^86 + 12 * q^89 + 12 * q^91 - 40 * q^94

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 $

$2$	2.52434	1.78498	0.892488	−	0.451071i	$-0.148958\pi$
$2$	2.52434	1.78498	0.892488	+	0.451071i	$0.148958\pi$
$3$	0	0
$3$	0	0
$4$	4.37228	2.18614
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	5.98844	2.11723
$9$	0	0
$10$	0	0
$11$	−1.37228	−0.413758	−0.206879	−	0.978366i	$-0.566331\pi$
$11$	−1.37228	−0.413758	−0.206879	+	0.978366i	$0.566331\pi$
$12$	0	0
$13$	−4.10891	−1.13961	−0.569804	−	0.821781i	$-0.692981\pi$
$13$	−4.10891	−1.13961	−0.569804	+	0.821781i	$0.692981\pi$
$14$	8.74456	2.33708
$15$	0	0
$16$	6.37228	1.59307
$17$	2.52434	0.612242	0.306121	−	0.951993i	$-0.400969\pi$
$17$	2.52434	0.612242	0.306121	+	0.951993i	$0.400969\pi$
$18$	0	0
$19$	5.37228	1.23249	0.616243	−	0.787556i	$-0.288654\pi$
$19$	5.37228	1.23249	0.616243	+	0.787556i	$0.288654\pi$
$20$	0	0
$21$	0	0
$22$	−3.46410	−0.738549
$23$	5.04868	1.05272	0.526361	−	0.850261i	$-0.323556\pi$
$23$	5.04868	1.05272	0.526361	+	0.850261i	$0.323556\pi$
$24$	0	0
$25$	0	0
$26$	−10.3723	−2.03417
$27$	0	0
$28$	15.1460	2.86233
$29$	−5.74456	−1.06674	−0.533369	−	0.845883i	$-0.679074\pi$
$29$	−5.74456	−1.06674	−0.533369	+	0.845883i	$0.679074\pi$
$30$	0	0
$31$	0.627719	0.112742	0.0563708	−	0.998410i	$-0.482047\pi$
$31$	0.627719	0.112742	0.0563708	+	0.998410i	$0.482047\pi$
$32$	4.10891	0.726360
$33$	0	0
$34$	6.37228	1.09284
$35$	0	0
$36$	0	0
$37$	7.57301	1.24500	0.622498	−	0.782621i	$-0.286118\pi$
$37$	7.57301	1.24500	0.622498	+	0.782621i	$0.286118\pi$
$38$	13.5615	2.19996
$39$	0	0
$40$	0	0
$41$	1.37228	0.214314	0.107157	−	0.994242i	$-0.465825\pi$
$41$	1.37228	0.214314	0.107157	+	0.994242i	$0.465825\pi$
$42$	0	0
$43$	−3.46410	−0.528271	−0.264135	−	0.964486i	$-0.585087\pi$
$43$	−3.46410	−0.528271	−0.264135	+	0.964486i	$0.585087\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	12.7446	1.87908
$47$	−8.51278	−1.24172	−0.620858	−	0.783923i	$-0.713216\pi$
$47$	−8.51278	−1.24172	−0.620858	+	0.783923i	$0.713216\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	−17.9653	−2.49134
$53$	−5.34363	−0.734004	−0.367002	−	0.930220i	$-0.619616\pi$
$53$	−5.34363	−0.734004	−0.367002	+	0.930220i	$0.619616\pi$
$54$	0	0
$55$	0	0
$56$	20.7446	2.77211
$57$	0	0
$58$	−14.5012	−1.90410
$59$	−7.37228	−0.959789	−0.479895	−	0.877326i	$-0.659325\pi$
$59$	−7.37228	−0.959789	−0.479895	+	0.877326i	$0.659325\pi$
$60$	0	0
$61$	3.62772	0.464482	0.232241	−	0.972658i	$-0.425394\pi$
$61$	3.62772	0.464482	0.232241	+	0.972658i	$0.425394\pi$
$62$	1.58457	0.201241
$63$	0	0
$64$	−2.37228	−0.296535
$65$	0	0
$66$	0	0
$67$	−8.21782	−1.00397	−0.501983	−	0.864877i	$-0.667396\pi$
$67$	−8.21782	−1.00397	−0.501983	+	0.864877i	$0.667396\pi$
$68$	11.0371	1.33845
$69$	0	0
$70$	0	0
$71$	4.11684	0.488579	0.244290	−	0.969702i	$-0.421445\pi$
$71$	4.11684	0.488579	0.244290	+	0.969702i	$0.421445\pi$
$72$	0	0
$73$	7.57301	0.886354	0.443177	−	0.896434i	$-0.353851\pi$
$73$	7.57301	0.886354	0.443177	+	0.896434i	$0.353851\pi$
$74$	19.1168	2.22229
$75$	0	0
$76$	23.4891	2.69439
$77$	−4.75372	−0.541737
$78$	0	0
$79$	−4.74456	−0.533805	−0.266903	−	0.963724i	$-0.586000\pi$
$79$	−4.74456	−0.533805	−0.266903	+	0.963724i	$0.586000\pi$
$80$	0	0
$81$	0	0
$82$	3.46410	0.382546
$83$	−5.34363	−0.586540	−0.293270	−	0.956030i	$-0.594743\pi$
$83$	−5.34363	−0.586540	−0.293270	+	0.956030i	$0.594743\pi$
$84$	0	0
$85$	0	0
$86$	−8.74456	−0.942950
$87$	0	0
$88$	−8.21782	−0.876023
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	−14.2337	−1.49210
$92$	22.0742	2.30140
$93$	0	0
$94$	−21.4891	−2.21643
$95$	0	0
$96$	0	0
$97$	18.6101	1.88957	0.944786	−	0.327688i	$-0.106269\pi$
$97$	18.6101	1.88957	0.944786	+	0.327688i	$0.106269\pi$
$98$	12.6217	1.27498
$99$	0	0
$100$	0	0
$101$	10.6277	1.05750	0.528749	−	0.848778i	$-0.322661\pi$
$101$	10.6277	1.05750	0.528749	+	0.848778i	$0.322661\pi$
$102$	0	0
$103$	−6.92820	−0.682656	−0.341328	−	0.939944i	$-0.610877\pi$
$103$	−6.92820	−0.682656	−0.341328	+	0.939944i	$0.610877\pi$
$104$	−24.6060	−2.41281
$105$	0	0
$106$	−13.4891	−1.31018
$107$	−13.2665	−1.28252	−0.641260	−	0.767323i	$-0.721588\pi$
$107$	−13.2665	−1.28252	−0.641260	+	0.767323i	$0.721588\pi$
$108$	0	0
$109$	−1.74456	−0.167099	−0.0835494	−	0.996504i	$-0.526626\pi$
$109$	−1.74456	−0.167099	−0.0835494	+	0.996504i	$0.526626\pi$
$110$	0	0
$111$	0	0
$112$	22.0742	2.08582
$113$	−9.45254	−0.889220	−0.444610	−	0.895724i	$-0.646658\pi$
$113$	−9.45254	−0.889220	−0.444610	+	0.895724i	$0.646658\pi$
$114$	0	0
$115$	0	0
$116$	−25.1168	−2.33204
$117$	0	0
$118$	−18.6101	−1.71320
$119$	8.74456	0.801613
$120$	0	0
$121$	−9.11684	−0.828804
$122$	9.15759	0.829089
$123$	0	0
$124$	2.74456	0.246469
$125$	0	0
$126$	0	0
$127$	−10.3923	−0.922168	−0.461084	−	0.887357i	$-0.652539\pi$
$127$	−10.3923	−0.922168	−0.461084	+	0.887357i	$0.652539\pi$
$128$	−14.2063	−1.25567
$129$	0	0
$130$	0	0
$131$	1.37228	0.119897	0.0599484	−	0.998201i	$-0.480906\pi$
$131$	1.37228	0.119897	0.0599484	+	0.998201i	$0.480906\pi$
$132$	0	0
$133$	18.6101	1.61370
$134$	−20.7446	−1.79206
$135$	0	0
$136$	15.1168	1.29626
$137$	−2.22938	−0.190469	−0.0952346	−	0.995455i	$-0.530360\pi$
$137$	−2.22938	−0.190469	−0.0952346	+	0.995455i	$0.530360\pi$
$138$	0	0
$139$	−6.11684	−0.518824	−0.259412	−	0.965767i	$-0.583529\pi$
$139$	−6.11684	−0.518824	−0.259412	+	0.965767i	$0.583529\pi$
$140$	0	0
$141$	0	0
$142$	10.3923	0.872103
$143$	5.63858	0.471522
$144$	0	0
$145$	0	0
$146$	19.1168	1.58212
$147$	0	0
$148$	33.1113	2.72174
$149$	−16.3723	−1.34127	−0.670635	−	0.741788i	$-0.733978\pi$
$149$	−16.3723	−1.34127	−0.670635	+	0.741788i	$0.733978\pi$
$150$	0	0
$151$	18.1168	1.47433	0.737164	−	0.675714i	$-0.236165\pi$
$151$	18.1168	1.47433	0.737164	+	0.675714i	$0.236165\pi$
$152$	32.1716	2.60946
$153$	0	0
$154$	−12.0000	−0.966988
$155$	0	0
$156$	0	0
$157$	−21.4294	−1.71025	−0.855127	−	0.518419i	$-0.826521\pi$
$157$	−21.4294	−1.71025	−0.855127	+	0.518419i	$0.826521\pi$
$158$	−11.9769	−0.952829
$159$	0	0
$160$	0	0
$161$	17.4891	1.37834
$162$	0	0
$163$	−4.75372	−0.372340	−0.186170	−	0.982518i	$-0.559608\pi$
$163$	−4.75372	−0.372340	−0.186170	+	0.982518i	$0.559608\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−13.4891	−1.04696
$167$	−0.294954	−0.0228242	−0.0114121	−	0.999935i	$-0.503633\pi$
$167$	−0.294954	−0.0228242	−0.0114121	+	0.999935i	$0.503633\pi$
$168$	0	0
$169$	3.88316	0.298704
$170$	0	0
$171$	0	0
$172$	−15.1460	−1.15487
$173$	7.86797	0.598190	0.299095	−	0.954223i	$-0.403315\pi$
$173$	7.86797	0.598190	0.299095	+	0.954223i	$0.403315\pi$
$174$	0	0
$175$	0	0
$176$	−8.74456	−0.659146
$177$	0	0
$178$	7.57301	0.567621
$179$	22.1168	1.65309	0.826545	−	0.562870i	$-0.190303\pi$
$179$	22.1168	1.65309	0.826545	+	0.562870i	$0.190303\pi$
$180$	0	0
$181$	14.8614	1.10464	0.552320	−	0.833632i	$-0.313743\pi$
$181$	14.8614	1.10464	0.552320	+	0.833632i	$0.313743\pi$
$182$	−35.9306	−2.66336
$183$	0	0
$184$	30.2337	2.22886
$185$	0	0
$186$	0	0
$187$	−3.46410	−0.253320
$188$	−37.2203	−2.71457
$189$	0	0
$190$	0	0
$191$	−13.3723	−0.967584	−0.483792	−	0.875183i	$-0.660741\pi$
$191$	−13.3723	−0.967584	−0.483792	+	0.875183i	$0.660741\pi$
$192$	0	0
$193$	21.4294	1.54252	0.771262	−	0.636518i	$-0.219626\pi$
$193$	21.4294	1.54252	0.771262	+	0.636518i	$0.219626\pi$
$194$	46.9783	3.37284
$195$	0	0
$196$	21.8614	1.56153
$197$	−23.0140	−1.63968	−0.819840	−	0.572593i	$-0.805937\pi$
$197$	−23.0140	−1.63968	−0.819840	+	0.572593i	$0.805937\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	26.8280	1.88761
$203$	−19.8997	−1.39669
$204$	0	0
$205$	0	0
$206$	−17.4891	−1.21853
$207$	0	0
$208$	−26.1831	−1.81547
$209$	−7.37228	−0.509951
$210$	0	0
$211$	26.8614	1.84922	0.924608	−	0.380921i	$-0.124393\pi$
$211$	26.8614	1.84922	0.924608	+	0.380921i	$0.124393\pi$
$212$	−23.3639	−1.60464
$213$	0	0
$214$	−33.4891	−2.28927
$215$	0	0
$216$	0	0
$217$	2.17448	0.147613
$218$	−4.40387	−0.298267
$219$	0	0
$220$	0	0
$221$	−10.3723	−0.697715
$222$	0	0
$223$	1.28962	0.0863594	0.0431797	−	0.999067i	$-0.486251\pi$
$223$	1.28962	0.0863594	0.0431797	+	0.999067i	$0.486251\pi$
$224$	14.2337	0.951028
$225$	0	0
$226$	−23.8614	−1.58724
$227$	−0.294954	−0.0195768	−0.00978838	−	0.999952i	$-0.503116\pi$
$227$	−0.294954	−0.0195768	−0.00978838	+	0.999952i	$0.503116\pi$
$228$	0	0
$229$	22.6060	1.49384	0.746922	−	0.664911i	$-0.231531\pi$
$229$	22.6060	1.49384	0.746922	+	0.664911i	$0.231531\pi$
$230$	0	0
$231$	0	0
$232$	−34.4010	−2.25853
$233$	−9.45254	−0.619257	−0.309628	−	0.950858i	$-0.600205\pi$
$233$	−9.45254	−0.619257	−0.309628	+	0.950858i	$0.600205\pi$
$234$	0	0
$235$	0	0
$236$	−32.2337	−2.09823
$237$	0	0
$238$	22.0742	1.43086
$239$	22.9783	1.48634	0.743170	−	0.669103i	$-0.233321\pi$
$239$	22.9783	1.48634	0.743170	+	0.669103i	$0.233321\pi$
$240$	0	0
$241$	−24.4891	−1.57748	−0.788742	−	0.614725i	$-0.789267\pi$
$241$	−24.4891	−1.57748	−0.788742	+	0.614725i	$0.789267\pi$
$242$	−23.0140	−1.47940
$243$	0	0
$244$	15.8614	1.01542
$245$	0	0
$246$	0	0
$247$	−22.0742	−1.40455
$248$	3.75906	0.238700
$249$	0	0
$250$	0	0
$251$	−14.2337	−0.898422	−0.449211	−	0.893426i	$-0.648295\pi$
$251$	−14.2337	−0.898422	−0.449211	+	0.893426i	$0.648295\pi$
$252$	0	0
$253$	−6.92820	−0.435572
$254$	−26.2337	−1.64605
$255$	0	0
$256$	−31.1168	−1.94480
$257$	−23.0140	−1.43557	−0.717787	−	0.696263i	$-0.754845\pi$
$257$	−23.0140	−1.43557	−0.717787	+	0.696263i	$0.754845\pi$
$258$	0	0
$259$	26.2337	1.63008
$260$	0	0
$261$	0	0
$262$	3.46410	0.214013
$263$	−26.5330	−1.63609	−0.818047	−	0.575151i	$-0.804943\pi$
$263$	−26.5330	−1.63609	−0.818047	+	0.575151i	$0.804943\pi$
$264$	0	0
$265$	0	0
$266$	46.9783	2.88042
$267$	0	0
$268$	−35.9306	−2.19481
$269$	−5.23369	−0.319104	−0.159552	−	0.987190i	$-0.551005\pi$
$269$	−5.23369	−0.319104	−0.159552	+	0.987190i	$0.551005\pi$
$270$	0	0
$271$	16.7446	1.01716	0.508580	−	0.861015i	$-0.330171\pi$
$271$	16.7446	1.01716	0.508580	+	0.861015i	$0.330171\pi$
$272$	16.0858	0.975344
$273$	0	0
$274$	−5.62772	−0.339983
$275$	0	0
$276$	0	0
$277$	−2.17448	−0.130652	−0.0653260	−	0.997864i	$-0.520809\pi$
$277$	−2.17448	−0.130652	−0.0653260	+	0.997864i	$0.520809\pi$
$278$	−15.4410	−0.926088
$279$	0	0
$280$	0	0
$281$	−4.37228	−0.260828	−0.130414	−	0.991460i	$-0.541631\pi$
$281$	−4.37228	−0.260828	−0.130414	+	0.991460i	$0.541631\pi$
$282$	0	0
$283$	−27.7128	−1.64736	−0.823678	−	0.567058i	$-0.808082\pi$
$283$	−27.7128	−1.64736	−0.823678	+	0.567058i	$0.808082\pi$
$284$	18.0000	1.06810
$285$	0	0
$286$	14.2337	0.841656
$287$	4.75372	0.280603
$288$	0	0
$289$	−10.6277	−0.625160
$290$	0	0
$291$	0	0
$292$	33.1113	1.93769
$293$	−2.52434	−0.147473	−0.0737367	−	0.997278i	$-0.523492\pi$
$293$	−2.52434	−0.147473	−0.0737367	+	0.997278i	$0.523492\pi$
$294$	0	0
$295$	0	0
$296$	45.3505	2.63595
$297$	0	0
$298$	−41.3292	−2.39413
$299$	−20.7446	−1.19969
$300$	0	0
$301$	−12.0000	−0.691669
$302$	45.7330	2.63164
$303$	0	0
$304$	34.2337	1.96344
$305$	0	0
$306$	0	0
$307$	4.75372	0.271309	0.135655	−	0.990756i	$-0.456686\pi$
$307$	4.75372	0.271309	0.135655	+	0.990756i	$0.456686\pi$
$308$	−20.7846	−1.18431
$309$	0	0
$310$	0	0
$311$	−12.8614	−0.729303	−0.364652	−	0.931144i	$-0.618812\pi$
$311$	−12.8614	−0.729303	−0.364652	+	0.931144i	$0.618812\pi$
$312$	0	0
$313$	−5.39853	−0.305143	−0.152572	−	0.988292i	$-0.548755\pi$
$313$	−5.39853	−0.305143	−0.152572	+	0.988292i	$0.548755\pi$
$314$	−54.0951	−3.05276
$315$	0	0
$316$	−20.7446	−1.16697
$317$	−6.98311	−0.392210	−0.196105	−	0.980583i	$-0.562829\pi$
$317$	−6.98311	−0.392210	−0.196105	+	0.980583i	$0.562829\pi$
$318$	0	0
$319$	7.88316	0.441372
$320$	0	0
$321$	0	0
$322$	44.1485	2.46030
$323$	13.5615	0.754579
$324$	0	0
$325$	0	0
$326$	−12.0000	−0.664619
$327$	0	0
$328$	8.21782	0.453753
$329$	−29.4891	−1.62579
$330$	0	0
$331$	−8.11684	−0.446142	−0.223071	−	0.974802i	$-0.571608\pi$
$331$	−8.11684	−0.446142	−0.223071	+	0.974802i	$0.571608\pi$
$332$	−23.3639	−1.28226
$333$	0	0
$334$	−0.744563	−0.0407407
$335$	0	0
$336$	0	0
$337$	6.92820	0.377403	0.188702	−	0.982034i	$-0.439572\pi$
$337$	6.92820	0.377403	0.188702	+	0.982034i	$0.439572\pi$
$338$	9.80240	0.533180
$339$	0	0
$340$	0	0
$341$	−0.861407	−0.0466478
$342$	0	0
$343$	−6.92820	−0.374088
$344$	−20.7446	−1.11847
$345$	0	0
$346$	19.8614	1.06776
$347$	−23.6588	−1.27007	−0.635036	−	0.772483i	$-0.719015\pi$
$347$	−23.6588	−1.27007	−0.635036	+	0.772483i	$0.719015\pi$
$348$	0	0
$349$	13.6060	0.728311	0.364155	−	0.931338i	$-0.381358\pi$
$349$	13.6060	0.728311	0.364155	+	0.931338i	$0.381358\pi$
$350$	0	0
$351$	0	0
$352$	−5.63858	−0.300537
$353$	8.51278	0.453089	0.226545	−	0.974001i	$-0.427257\pi$
$353$	8.51278	0.453089	0.226545	+	0.974001i	$0.427257\pi$
$354$	0	0
$355$	0	0
$356$	13.1168	0.695191
$357$	0	0
$358$	55.8304	2.95073
$359$	28.1168	1.48395	0.741975	−	0.670427i	$-0.233889\pi$
$359$	28.1168	1.48395	0.741975	+	0.670427i	$0.233889\pi$
$360$	0	0
$361$	9.86141	0.519021
$362$	37.5152	1.97176
$363$	0	0
$364$	−62.2337	−3.26193
$365$	0	0
$366$	0	0
$367$	23.3639	1.21958	0.609792	−	0.792562i	$-0.291253\pi$
$367$	23.3639	1.21958	0.609792	+	0.792562i	$0.291253\pi$
$368$	32.1716	1.67706
$369$	0	0
$370$	0	0
$371$	−18.5109	−0.961037
$372$	0	0
$373$	−11.6819	−0.604867	−0.302434	−	0.953170i	$-0.597799\pi$
$373$	−11.6819	−0.604867	−0.302434	+	0.953170i	$0.597799\pi$
$374$	−8.74456	−0.452171
$375$	0	0
$376$	−50.9783	−2.62900
$377$	23.6039	1.21566
$378$	0	0
$379$	21.4891	1.10382	0.551911	−	0.833903i	$-0.313899\pi$
$379$	21.4891	1.10382	0.551911	+	0.833903i	$0.313899\pi$
$380$	0	0
$381$	0	0
$382$	−33.7562	−1.72712
$383$	35.3407	1.80583	0.902913	−	0.429822i	$-0.141424\pi$
$383$	35.3407	1.80583	0.902913	+	0.429822i	$0.141424\pi$
$384$	0	0
$385$	0	0
$386$	54.0951	2.75337
$387$	0	0
$388$	81.3687	4.13087
$389$	23.4891	1.19095	0.595473	−	0.803375i	$-0.296965\pi$
$389$	23.4891	1.19095	0.595473	+	0.803375i	$0.296965\pi$
$390$	0	0
$391$	12.7446	0.644520
$392$	29.9422	1.51231
$393$	0	0
$394$	−58.0951	−2.92679
$395$	0	0
$396$	0	0
$397$	−12.3267	−0.618661	−0.309331	−	0.950955i	$-0.600105\pi$
$397$	−12.3267	−0.618661	−0.309331	+	0.950955i	$0.600105\pi$
$398$	40.3894	2.02454
$399$	0	0
$400$	0	0
$401$	−19.6277	−0.980161	−0.490081	−	0.871677i	$-0.663033\pi$
$401$	−19.6277	−0.980161	−0.490081	+	0.871677i	$0.663033\pi$
$402$	0	0
$403$	−2.57924	−0.128481
$404$	46.4674	2.31184
$405$	0	0
$406$	−50.2337	−2.49306
$407$	−10.3923	−0.515127
$408$	0	0
$409$	−5.86141	−0.289828	−0.144914	−	0.989444i	$-0.546291\pi$
$409$	−5.86141	−0.289828	−0.144914	+	0.989444i	$0.546291\pi$
$410$	0	0
$411$	0	0
$412$	−30.2921	−1.49238
$413$	−25.5383	−1.25666
$414$	0	0
$415$	0	0
$416$	−16.8832	−0.827765
$417$	0	0
$418$	−18.6101	−0.910251
$419$	−5.48913	−0.268161	−0.134081	−	0.990970i	$-0.542808\pi$
$419$	−5.48913	−0.268161	−0.134081	+	0.990970i	$0.542808\pi$
$420$	0	0
$421$	−10.2554	−0.499819	−0.249910	−	0.968269i	$-0.580401\pi$
$421$	−10.2554	−0.499819	−0.249910	+	0.968269i	$0.580401\pi$
$422$	67.8073	3.30081
$423$	0	0
$424$	−32.0000	−1.55406
$425$	0	0
$426$	0	0
$427$	12.5668	0.608149
$428$	−58.0049	−2.80377
$429$	0	0
$430$	0	0
$431$	−18.3505	−0.883914	−0.441957	−	0.897036i	$-0.645716\pi$
$431$	−18.3505	−0.883914	−0.441957	+	0.897036i	$0.645716\pi$
$432$	0	0
$433$	−2.81929	−0.135487	−0.0677433	−	0.997703i	$-0.521580\pi$
$433$	−2.81929	−0.135487	−0.0677433	+	0.997703i	$0.521580\pi$
$434$	5.48913	0.263486
$435$	0	0
$436$	−7.62772	−0.365301
$437$	27.1229	1.29746
$438$	0	0
$439$	3.13859	0.149797	0.0748984	−	0.997191i	$-0.476137\pi$
$439$	3.13859	0.149797	0.0748984	+	0.997191i	$0.476137\pi$
$440$	0	0
$441$	0	0
$442$	−26.1831	−1.24541
$443$	24.9484	1.18534	0.592668	−	0.805447i	$-0.298075\pi$
$443$	24.9484	1.18534	0.592668	+	0.805447i	$0.298075\pi$
$444$	0	0
$445$	0	0
$446$	3.25544	0.154149
$447$	0	0
$448$	−8.21782	−0.388256
$449$	28.1168	1.32692	0.663458	−	0.748214i	$-0.269088\pi$
$449$	28.1168	1.32692	0.663458	+	0.748214i	$0.269088\pi$
$450$	0	0
$451$	−1.88316	−0.0886744
$452$	−41.3292	−1.94396
$453$	0	0
$454$	−0.744563	−0.0349441
$455$	0	0
$456$	0	0
$457$	−8.86263	−0.414577	−0.207288	−	0.978280i	$-0.566464\pi$
$457$	−8.86263	−0.414577	−0.207288	+	0.978280i	$0.566464\pi$
$458$	57.0651	2.66648
$459$	0	0
$460$	0	0
$461$	39.0951	1.82084	0.910420	−	0.413685i	$-0.135759\pi$
$461$	39.0951	1.82084	0.910420	+	0.413685i	$0.135759\pi$
$462$	0	0
$463$	13.8564	0.643962	0.321981	−	0.946746i	$-0.395651\pi$
$463$	13.8564	0.643962	0.321981	+	0.946746i	$0.395651\pi$
$464$	−36.6060	−1.69939
$465$	0	0
$466$	−23.8614	−1.10536
$467$	14.8511	0.687226	0.343613	−	0.939111i	$-0.388349\pi$
$467$	14.8511	0.687226	0.343613	+	0.939111i	$0.388349\pi$
$468$	0	0
$469$	−28.4674	−1.31450
$470$	0	0
$471$	0	0
$472$	−44.1485	−2.03210
$473$	4.75372	0.218576
$474$	0	0
$475$	0	0
$476$	38.2337	1.75244
$477$	0	0
$478$	58.0049	2.65308
$479$	21.6060	0.987202	0.493601	−	0.869688i	$-0.335680\pi$
$479$	21.6060	0.987202	0.493601	+	0.869688i	$0.335680\pi$
$480$	0	0
$481$	−31.1168	−1.41881
$482$	−61.8188	−2.81577
$483$	0	0
$484$	−39.8614	−1.81188
$485$	0	0
$486$	0	0
$487$	30.2921	1.37266	0.686332	−	0.727288i	$-0.259220\pi$
$487$	30.2921	1.37266	0.686332	+	0.727288i	$0.259220\pi$
$488$	21.7244	0.983416
$489$	0	0
$490$	0	0
$491$	37.3723	1.68659	0.843294	−	0.537453i	$-0.180613\pi$
$491$	37.3723	1.68659	0.843294	+	0.537453i	$0.180613\pi$
$492$	0	0
$493$	−14.5012	−0.653102
$494$	−55.7228	−2.50709
$495$	0	0
$496$	4.00000	0.179605
$497$	14.2612	0.639701
$498$	0	0
$499$	−18.6277	−0.833891	−0.416946	−	0.908931i	$-0.636899\pi$
$499$	−18.6277	−0.833891	−0.416946	+	0.908931i	$0.636899\pi$
$500$	0	0
$501$	0	0
$502$	−35.9306	−1.60366
$503$	−10.0974	−0.450219	−0.225109	−	0.974334i	$-0.572274\pi$
$503$	−10.0974	−0.450219	−0.225109	+	0.974334i	$0.572274\pi$
$504$	0	0
$505$	0	0
$506$	−17.4891	−0.777486
$507$	0	0
$508$	−45.4381	−2.01599
$509$	−23.4891	−1.04114	−0.520569	−	0.853820i	$-0.674280\pi$
$509$	−23.4891	−1.04114	−0.520569	+	0.853820i	$0.674280\pi$
$510$	0	0
$511$	26.2337	1.16051
$512$	−50.1369	−2.21576
$513$	0	0
$514$	−58.0951	−2.56246
$515$	0	0
$516$	0	0
$517$	11.6819	0.513770
$518$	66.2227	2.90966
$519$	0	0
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	15.1460	0.662290	0.331145	−	0.943580i	$-0.392565\pi$
$523$	15.1460	0.662290	0.331145	+	0.943580i	$0.392565\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	−66.9783	−2.92039
$527$	1.58457	0.0690251
$528$	0	0
$529$	2.48913	0.108223
$530$	0	0
$531$	0	0
$532$	81.3687	3.52778
$533$	−5.63858	−0.244234
$534$	0	0
$535$	0	0
$536$	−49.2119	−2.12563
$537$	0	0
$538$	−13.2116	−0.569592
$539$	−6.86141	−0.295542
$540$	0	0
$541$	−15.2337	−0.654947	−0.327474	−	0.944860i	$-0.606197\pi$
$541$	−15.2337	−0.654947	−0.327474	+	0.944860i	$0.606197\pi$
$542$	42.2689	1.81561
$543$	0	0
$544$	10.3723	0.444708
$545$	0	0
$546$	0	0
$547$	−6.92820	−0.296229	−0.148114	−	0.988970i	$-0.547320\pi$
$547$	−6.92820	−0.296229	−0.148114	+	0.988970i	$0.547320\pi$
$548$	−9.74749	−0.416392
$549$	0	0
$550$	0	0
$551$	−30.8614	−1.31474
$552$	0	0
$553$	−16.4356	−0.698915
$554$	−5.48913	−0.233211
$555$	0	0
$556$	−26.7446	−1.13422
$557$	−21.7244	−0.920491	−0.460246	−	0.887792i	$-0.652239\pi$
$557$	−21.7244	−0.920491	−0.460246	+	0.887792i	$0.652239\pi$
$558$	0	0
$559$	14.2337	0.602021
$560$	0	0
$561$	0	0
$562$	−11.0371	−0.465573
$563$	−0.994667	−0.0419202	−0.0209601	−	0.999780i	$-0.506672\pi$
$563$	−0.994667	−0.0419202	−0.0209601	+	0.999780i	$0.506672\pi$
$564$	0	0
$565$	0	0
$566$	−69.9565	−2.94049
$567$	0	0
$568$	24.6535	1.03444
$569$	2.48913	0.104350	0.0521748	−	0.998638i	$-0.483385\pi$
$569$	2.48913	0.104350	0.0521748	+	0.998638i	$0.483385\pi$
$570$	0	0
$571$	32.8614	1.37521	0.687604	−	0.726086i	$-0.258663\pi$
$571$	32.8614	1.37521	0.687604	+	0.726086i	$0.258663\pi$
$572$	24.6535	1.03081
$573$	0	0
$574$	12.0000	0.500870
$575$	0	0
$576$	0	0
$577$	28.3576	1.18054	0.590272	−	0.807205i	$-0.299021\pi$
$577$	28.3576	1.18054	0.590272	+	0.807205i	$0.299021\pi$
$578$	−26.8280	−1.11590
$579$	0	0
$580$	0	0
$581$	−18.5109	−0.767960
$582$	0	0
$583$	7.33296	0.303700
$584$	45.3505	1.87662
$585$	0	0
$586$	−6.37228	−0.263237
$587$	−9.80240	−0.404588	−0.202294	−	0.979325i	$-0.564840\pi$
$587$	−9.80240	−0.404588	−0.202294	+	0.979325i	$0.564840\pi$
$588$	0	0
$589$	3.37228	0.138952
$590$	0	0
$591$	0	0
$592$	48.2574	1.98337
$593$	38.1600	1.56704	0.783522	−	0.621364i	$-0.213421\pi$
$593$	38.1600	1.56704	0.783522	+	0.621364i	$0.213421\pi$
$594$	0	0
$595$	0	0
$596$	−71.5842	−2.93220
$597$	0	0
$598$	−52.3663	−2.14142
$599$	−7.37228	−0.301223	−0.150612	−	0.988593i	$-0.548124\pi$
$599$	−7.37228	−0.301223	−0.150612	+	0.988593i	$0.548124\pi$
$600$	0	0
$601$	27.9783	1.14126	0.570628	−	0.821208i	$-0.306700\pi$
$601$	27.9783	1.14126	0.570628	+	0.821208i	$0.306700\pi$
$602$	−30.2921	−1.23461
$603$	0	0
$604$	79.2119	3.22309
$605$	0	0
$606$	0	0
$607$	36.3354	1.47481	0.737404	−	0.675452i	$-0.236051\pi$
$607$	36.3354	1.47481	0.737404	+	0.675452i	$0.236051\pi$
$608$	22.0742	0.895228
$609$	0	0
$610$	0	0
$611$	34.9783	1.41507
$612$	0	0
$613$	9.50744	0.384002	0.192001	−	0.981395i	$-0.438502\pi$
$613$	9.50744	0.384002	0.192001	+	0.981395i	$0.438502\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	−28.4674	−1.14698
$617$	22.4241	0.902760	0.451380	−	0.892332i	$-0.350932\pi$
$617$	22.4241	0.902760	0.451380	+	0.892332i	$0.350932\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	−32.4665	−1.30179
$623$	10.3923	0.416359
$624$	0	0
$625$	0	0
$626$	−13.6277	−0.544673
$627$	0	0
$628$	−93.6955	−3.73886
$629$	19.1168	0.762238
$630$	0	0
$631$	0.627719	0.0249891	0.0124945	−	0.999922i	$-0.496023\pi$
$631$	0.627719	0.0249891	0.0124945	+	0.999922i	$0.496023\pi$
$632$	−28.4125	−1.13019
$633$	0	0
$634$	−17.6277	−0.700086
$635$	0	0
$636$	0	0
$637$	−20.5446	−0.814005
$638$	19.8997	0.787839
$639$	0	0
$640$	0	0
$641$	37.9783	1.50005	0.750025	−	0.661409i	$-0.230041\pi$
$641$	37.9783	1.50005	0.750025	+	0.661409i	$0.230041\pi$
$642$	0	0
$643$	18.6101	0.733912	0.366956	−	0.930238i	$-0.380400\pi$
$643$	18.6101	0.733912	0.366956	+	0.930238i	$0.380400\pi$
$644$	76.4674	3.01324
$645$	0	0
$646$	34.2337	1.34691
$647$	4.45877	0.175292	0.0876461	−	0.996152i	$-0.472066\pi$
$647$	4.45877	0.175292	0.0876461	+	0.996152i	$0.472066\pi$
$648$	0	0
$649$	10.1168	0.397121
$650$	0	0
$651$	0	0
$652$	−20.7846	−0.813988
$653$	20.1947	0.790280	0.395140	−	0.918621i	$-0.370696\pi$
$653$	20.1947	0.790280	0.395140	+	0.918621i	$0.370696\pi$
$654$	0	0
$655$	0	0
$656$	8.74456	0.341418
$657$	0	0
$658$	−74.4405	−2.90199
$659$	−32.7446	−1.27555	−0.637774	−	0.770224i	$-0.720144\pi$
$659$	−32.7446	−1.27555	−0.637774	+	0.770224i	$0.720144\pi$
$660$	0	0
$661$	19.2337	0.748104	0.374052	−	0.927408i	$-0.377968\pi$
$661$	19.2337	0.748104	0.374052	+	0.927408i	$0.377968\pi$
$662$	−20.4897	−0.796353
$663$	0	0
$664$	−32.0000	−1.24184
$665$	0	0
$666$	0	0
$667$	−29.0024	−1.12298
$668$	−1.28962	−0.0498969
$669$	0	0
$670$	0	0
$671$	−4.97825	−0.192183
$672$	0	0
$673$	19.2549	0.742223	0.371112	−	0.928588i	$-0.378977\pi$
$673$	19.2549	0.742223	0.371112	+	0.928588i	$0.378977\pi$
$674$	17.4891	0.673656
$675$	0	0
$676$	16.9783	0.653010
$677$	26.5330	1.01975	0.509873	−	0.860250i	$-0.329692\pi$
$677$	26.5330	1.01975	0.509873	+	0.860250i	$0.329692\pi$
$678$	0	0
$679$	64.4674	2.47403
$680$	0	0
$681$	0	0
$682$	−2.17448	−0.0832652
$683$	−0.589907	−0.0225722	−0.0112861	−	0.999936i	$-0.503593\pi$
$683$	−0.589907	−0.0225722	−0.0112861	+	0.999936i	$0.503593\pi$
$684$	0	0
$685$	0	0
$686$	−17.4891	−0.667738
$687$	0	0
$688$	−22.0742	−0.841572
$689$	21.9565	0.836476
$690$	0	0
$691$	−11.7228	−0.445957	−0.222978	−	0.974823i	$-0.571578\pi$
$691$	−11.7228	−0.445957	−0.222978	+	0.974823i	$0.571578\pi$
$692$	34.4010	1.30773
$693$	0	0
$694$	−59.7228	−2.26705
$695$	0	0
$696$	0	0
$697$	3.46410	0.131212
$698$	34.3461	1.30002
$699$	0	0
$700$	0	0
$701$	17.2337	0.650907	0.325454	−	0.945558i	$-0.394483\pi$
$701$	17.2337	0.650907	0.325454	+	0.945558i	$0.394483\pi$
$702$	0	0
$703$	40.6844	1.53444
$704$	3.25544	0.122694
$705$	0	0
$706$	21.4891	0.808754
$707$	36.8155	1.38459
$708$	0	0
$709$	0.649468	0.0243913	0.0121956	−	0.999926i	$-0.496118\pi$
$709$	0.649468	0.0243913	0.0121956	+	0.999926i	$0.496118\pi$
$710$	0	0
$711$	0	0
$712$	17.9653	0.673279
$713$	3.16915	0.118686
$714$	0	0
$715$	0	0
$716$	96.7011	3.61389
$717$	0	0
$718$	70.9764	2.64882
$719$	−28.1168	−1.04858	−0.524291	−	0.851539i	$-0.675669\pi$
$719$	−28.1168	−1.04858	−0.524291	+	0.851539i	$0.675669\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	24.8935	0.926441
$723$	0	0
$724$	64.9783	2.41490
$725$	0	0
$726$	0	0
$727$	−10.7971	−0.400441	−0.200220	−	0.979751i	$-0.564166\pi$
$727$	−10.7971	−0.400441	−0.200220	+	0.979751i	$0.564166\pi$
$728$	−85.2376	−3.15911
$729$	0	0
$730$	0	0
$731$	−8.74456	−0.323429
$732$	0	0
$733$	23.3639	0.862964	0.431482	−	0.902122i	$-0.357991\pi$
$733$	23.3639	0.862964	0.431482	+	0.902122i	$0.357991\pi$
$734$	58.9783	2.17693
$735$	0	0
$736$	20.7446	0.764655
$737$	11.2772	0.415400
$738$	0	0
$739$	11.3723	0.418336	0.209168	−	0.977880i	$-0.432924\pi$
$739$	11.3723	0.418336	0.209168	+	0.977880i	$0.432924\pi$
$740$	0	0
$741$	0	0
$742$	−46.7277	−1.71543
$743$	−42.5639	−1.56152	−0.780759	−	0.624833i	$-0.785167\pi$
$743$	−42.5639	−1.56152	−0.780759	+	0.624833i	$0.785167\pi$
$744$	0	0
$745$	0	0
$746$	−29.4891	−1.07967
$747$	0	0
$748$	−15.1460	−0.553794
$749$	−45.9565	−1.67921
$750$	0	0
$751$	−35.7228	−1.30354	−0.651772	−	0.758415i	$-0.725974\pi$
$751$	−35.7228	−1.30354	−0.651772	+	0.758415i	$0.725974\pi$
$752$	−54.2458	−1.97814
$753$	0	0
$754$	59.5842	2.16993
$755$	0	0
$756$	0	0
$757$	−41.5692	−1.51086	−0.755429	−	0.655230i	$-0.772572\pi$
$757$	−41.5692	−1.51086	−0.755429	+	0.655230i	$0.772572\pi$
$758$	54.2458	1.97030
$759$	0	0
$760$	0	0
$761$	−9.51087	−0.344769	−0.172384	−	0.985030i	$-0.555147\pi$
$761$	−9.51087	−0.344769	−0.172384	+	0.985030i	$0.555147\pi$
$762$	0	0
$763$	−6.04334	−0.218784
$764$	−58.4674	−2.11528
$765$	0	0
$766$	89.2119	3.22336
$767$	30.2921	1.09378
$768$	0	0
$769$	6.48913	0.234004	0.117002	−	0.993132i	$-0.462672\pi$
$769$	6.48913	0.234004	0.117002	+	0.993132i	$0.462672\pi$
$770$	0	0
$771$	0	0
$772$	93.6955	3.37217
$773$	18.2603	0.656776	0.328388	−	0.944543i	$-0.393495\pi$
$773$	18.2603	0.656776	0.328388	+	0.944543i	$0.393495\pi$
$774$	0	0
$775$	0	0
$776$	111.446	4.00066
$777$	0	0
$778$	59.2945	2.12581
$779$	7.37228	0.264139
$780$	0	0
$781$	−5.64947	−0.202154
$782$	32.1716	1.15045
$783$	0	0
$784$	31.8614	1.13791
$785$	0	0
$786$	0	0
$787$	17.3205	0.617409	0.308705	−	0.951158i	$-0.400105\pi$
$787$	17.3205	0.617409	0.308705	+	0.951158i	$0.400105\pi$
$788$	−100.624	−3.58457
$789$	0	0
$790$	0	0
$791$	−32.7446	−1.16426
$792$	0	0
$793$	−14.9060	−0.529327
$794$	−31.1168	−1.10430
$795$	0	0
$796$	69.9565	2.47954
$797$	48.8473	1.73026	0.865130	−	0.501548i	$-0.167236\pi$
$797$	48.8473	1.73026	0.865130	+	0.501548i	$0.167236\pi$
$798$	0	0
$799$	−21.4891	−0.760231
$800$	0	0
$801$	0	0
$802$	−49.5470	−1.74957
$803$	−10.3923	−0.366736
$804$	0	0
$805$	0	0
$806$	−6.51087	−0.229336
$807$	0	0
$808$	63.6434	2.23897
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	0	0
$811$	42.1168	1.47892	0.739461	−	0.673199i	$-0.235080\pi$
$811$	42.1168	1.47892	0.739461	+	0.673199i	$0.235080\pi$
$812$	−87.0073	−3.05336
$813$	0	0
$814$	−26.2337	−0.919490
$815$	0	0
$816$	0	0
$817$	−18.6101	−0.651086
$818$	−14.7962	−0.517336
$819$	0	0
$820$	0	0
$821$	−16.7228	−0.583630	−0.291815	−	0.956475i	$-0.594259\pi$
$821$	−16.7228	−0.583630	−0.291815	+	0.956475i	$0.594259\pi$
$822$	0	0
$823$	44.1485	1.53892	0.769459	−	0.638696i	$-0.220526\pi$
$823$	44.1485	1.53892	0.769459	+	0.638696i	$0.220526\pi$
$824$	−41.4891	−1.44534
$825$	0	0
$826$	−64.4674	−2.24311
$827$	15.7359	0.547192	0.273596	−	0.961845i	$-0.411787\pi$
$827$	15.7359	0.547192	0.273596	+	0.961845i	$0.411787\pi$
$828$	0	0
$829$	28.3505	0.984655	0.492327	−	0.870410i	$-0.336146\pi$
$829$	28.3505	0.984655	0.492327	+	0.870410i	$0.336146\pi$
$830$	0	0
$831$	0	0
$832$	9.74749	0.337934
$833$	12.6217	0.437316
$834$	0	0
$835$	0	0
$836$	−32.2337	−1.11483
$837$	0	0
$838$	−13.8564	−0.478662
$839$	17.1386	0.591690	0.295845	−	0.955236i	$-0.404399\pi$
$839$	17.1386	0.591690	0.295845	+	0.955236i	$0.404399\pi$
$840$	0	0
$841$	4.00000	0.137931
$842$	−25.8882	−0.892166
$843$	0	0
$844$	117.446	4.04265
$845$	0	0
$846$	0	0
$847$	−31.5817	−1.08516
$848$	−34.0511	−1.16932
$849$	0	0
$850$	0	0
$851$	38.2337	1.31063
$852$	0	0
$853$	−48.9022	−1.67438	−0.837189	−	0.546913i	$-0.815803\pi$
$853$	−48.9022	−1.67438	−0.837189	+	0.546913i	$0.815803\pi$
$854$	31.7228	1.08553
$855$	0	0
$856$	−79.4456	−2.71540
$857$	−26.8829	−0.918301	−0.459150	−	0.888359i	$-0.651846\pi$
$857$	−26.8829	−0.918301	−0.459150	+	0.888359i	$0.651846\pi$
$858$	0	0
$859$	−49.3288	−1.68308	−0.841538	−	0.540198i	$-0.818349\pi$
$859$	−49.3288	−1.68308	−0.841538	+	0.540198i	$0.818349\pi$
$860$	0	0
$861$	0	0
$862$	−46.3229	−1.57777
$863$	9.80240	0.333677	0.166839	−	0.985984i	$-0.446644\pi$
$863$	9.80240	0.333677	0.166839	+	0.985984i	$0.446644\pi$
$864$	0	0
$865$	0	0
$866$	−7.11684	−0.241840
$867$	0	0
$868$	9.50744	0.322704
$869$	6.51087	0.220866
$870$	0	0
$871$	33.7663	1.14413
$872$	−10.4472	−0.353787
$873$	0	0
$874$	68.4674	2.31594
$875$	0	0
$876$	0	0
$877$	−11.0371	−0.372697	−0.186348	−	0.982484i	$-0.559665\pi$
$877$	−11.0371	−0.372697	−0.186348	+	0.982484i	$0.559665\pi$
$878$	7.92287	0.267384
$879$	0	0
$880$	0	0
$881$	−28.1168	−0.947281	−0.473640	−	0.880718i	$-0.657060\pi$
$881$	−28.1168	−0.947281	−0.473640	+	0.880718i	$0.657060\pi$
$882$	0	0
$883$	−44.5532	−1.49934	−0.749668	−	0.661815i	$-0.769787\pi$
$883$	−44.5532	−1.49934	−0.749668	+	0.661815i	$0.769787\pi$
$884$	−45.3505	−1.52530
$885$	0	0
$886$	62.9783	2.11580
$887$	9.21249	0.309325	0.154663	−	0.987967i	$-0.450571\pi$
$887$	9.21249	0.309325	0.154663	+	0.987967i	$0.450571\pi$
$888$	0	0
$889$	−36.0000	−1.20740
$890$	0	0
$891$	0	0
$892$	5.63858	0.188794
$893$	−45.7330	−1.53040
$894$	0	0
$895$	0	0
$896$	−49.2119	−1.64406
$897$	0	0
$898$	70.9764	2.36851
$899$	−3.60597	−0.120266
$900$	0	0
$901$	−13.4891	−0.449388
$902$	−4.75372	−0.158282
$903$	0	0
$904$	−56.6060	−1.88269
$905$	0	0
$906$	0	0
$907$	−58.0049	−1.92602	−0.963010	−	0.269466i	$-0.913153\pi$
$907$	−58.0049	−1.92602	−0.963010	+	0.269466i	$0.913153\pi$
$908$	−1.28962	−0.0427976
$909$	0	0
$910$	0	0
$911$	−21.6060	−0.715838	−0.357919	−	0.933753i	$-0.616514\pi$
$911$	−21.6060	−0.715838	−0.357919	+	0.933753i	$0.616514\pi$
$912$	0	0
$913$	7.33296	0.242686
$914$	−22.3723	−0.740009
$915$	0	0
$916$	98.8397	3.26575
$917$	4.75372	0.156982
$918$	0	0
$919$	3.64947	0.120385	0.0601924	−	0.998187i	$-0.480829\pi$
$919$	3.64947	0.120385	0.0601924	+	0.998187i	$0.480829\pi$
$920$	0	0
$921$	0	0
$922$	98.6892	3.25016
$923$	−16.9157	−0.556789
$924$	0	0
$925$	0	0
$926$	34.9783	1.14946
$927$	0	0
$928$	−23.6039	−0.774836
$929$	26.4891	0.869080	0.434540	−	0.900653i	$-0.356911\pi$
$929$	26.4891	0.869080	0.434540	+	0.900653i	$0.356911\pi$
$930$	0	0
$931$	26.8614	0.880347
$932$	−41.3292	−1.35378
$933$	0	0
$934$	37.4891	1.22668
$935$	0	0
$936$	0	0
$937$	27.4728	0.897496	0.448748	−	0.893658i	$-0.351870\pi$
$937$	27.4728	0.897496	0.448748	+	0.893658i	$0.351870\pi$
$938$	−71.8613	−2.34635
$939$	0	0
$940$	0	0
$941$	−21.8614	−0.712661	−0.356331	−	0.934360i	$-0.615972\pi$
$941$	−21.8614	−0.712661	−0.356331	+	0.934360i	$0.615972\pi$
$942$	0	0
$943$	6.92820	0.225613
$944$	−46.9783	−1.52901
$945$	0	0
$946$	12.0000	0.390154
$947$	36.5205	1.18676	0.593379	−	0.804923i	$-0.297794\pi$
$947$	36.5205	1.18676	0.593379	+	0.804923i	$0.297794\pi$
$948$	0	0
$949$	−31.1168	−1.01010
$950$	0	0
$951$	0	0
$952$	52.3663	1.69720
$953$	21.7244	0.703721	0.351861	−	0.936052i	$-0.385549\pi$
$953$	21.7244	0.703721	0.351861	+	0.936052i	$0.385549\pi$
$954$	0	0
$955$	0	0
$956$	100.467	3.24935
$957$	0	0
$958$	54.5408	1.76213
$959$	−7.72281	−0.249383
$960$	0	0
$961$	−30.6060	−0.987289
$962$	−78.5494	−2.53254
$963$	0	0
$964$	−107.073	−3.44860
$965$	0	0
$966$	0	0
$967$	36.3354	1.16847	0.584234	−	0.811585i	$-0.301395\pi$
$967$	36.3354	1.16847	0.584234	+	0.811585i	$0.301395\pi$
$968$	−54.5957	−1.75477
$969$	0	0
$970$	0	0
$971$	44.5842	1.43078	0.715388	−	0.698728i	$-0.246250\pi$
$971$	44.5842	1.43078	0.715388	+	0.698728i	$0.246250\pi$
$972$	0	0
$973$	−21.1894	−0.679300
$974$	76.4674	2.45017
$975$	0	0
$976$	23.1168	0.739952
$977$	−54.8357	−1.75435	−0.877175	−	0.480171i	$-0.840575\pi$
$977$	−54.8357	−1.75435	−0.877175	+	0.480171i	$0.840575\pi$
$978$	0	0
$979$	−4.11684	−0.131575
$980$	0	0
$981$	0	0
$982$	94.3403	3.01052
$983$	57.0102	1.81834	0.909171	−	0.416422i	$-0.136716\pi$
$983$	57.0102	1.81834	0.909171	+	0.416422i	$0.136716\pi$
$984$	0	0
$985$	0	0
$986$	−36.6060	−1.16577
$987$	0	0
$988$	−96.5147	−3.07054
$989$	−17.4891	−0.556122
$990$	0	0
$991$	−1.60597	−0.0510153	−0.0255076	−	0.999675i	$-0.508120\pi$
$991$	−1.60597	−0.0510153	−0.0255076	+	0.999675i	$0.508120\pi$
$992$	2.57924	0.0818910
$993$	0	0
$994$	36.0000	1.14185
$995$	0	0
$996$	0	0
$997$	−10.6324	−0.336730	−0.168365	−	0.985725i	$-0.553849\pi$
$997$	−10.6324	−0.336730	−0.168365	+	0.985725i	$0.553849\pi$
$998$	−47.0227	−1.48848
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.a.x.1.4		4
3.2	odd	2		2025.2.a.w.1.1		4
5.2	odd	4		405.2.b.b.244.4	yes	4
5.3	odd	4		405.2.b.b.244.1	yes	4
5.4	even	2	inner	2025.2.a.x.1.1		4
15.2	even	4		405.2.b.a.244.1	✓	4
15.8	even	4		405.2.b.a.244.4	yes	4
15.14	odd	2		2025.2.a.w.1.4		4
45.2	even	12		405.2.j.b.109.1		4
45.7	odd	12		405.2.j.d.109.2		4
45.13	odd	12		405.2.j.d.379.2		4
45.22	odd	12		405.2.j.a.379.1		4
45.23	even	12		405.2.j.b.379.1		4
45.32	even	12		405.2.j.e.379.2		4
45.38	even	12		405.2.j.e.109.2		4
45.43	odd	12		405.2.j.a.109.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.b.a.244.1	✓	4	15.2	even	4
405.2.b.a.244.4	yes	4	15.8	even	4
405.2.b.b.244.1	yes	4	5.3	odd	4
405.2.b.b.244.4	yes	4	5.2	odd	4
405.2.j.a.109.1		4	45.43	odd	12
405.2.j.a.379.1		4	45.22	odd	12
405.2.j.b.109.1		4	45.2	even	12
405.2.j.b.379.1		4	45.23	even	12
405.2.j.d.109.2		4	45.7	odd	12
405.2.j.d.379.2		4	45.13	odd	12
405.2.j.e.109.2		4	45.38	even	12
405.2.j.e.379.2		4	45.32	even	12
2025.2.a.w.1.1		4	3.2	odd	2
2025.2.a.w.1.4		4	15.14	odd	2
2025.2.a.x.1.1		4	5.4	even	2	inner
2025.2.a.x.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.52434 q^{2} +4.37228 q^{4} +3.46410 q^{7} +5.98844 q^{8} -1.37228 q^{11} -4.10891 q^{13} +8.74456 q^{14} +6.37228 q^{16} +2.52434 q^{17} +5.37228 q^{19} -3.46410 q^{22} +5.04868 q^{23} -10.3723 q^{26} +15.1460 q^{28} -5.74456 q^{29} +0.627719 q^{31} +4.10891 q^{32} +6.37228 q^{34} +7.57301 q^{37} +13.5615 q^{38} +1.37228 q^{41} -3.46410 q^{43} -6.00000 q^{44} +12.7446 q^{46} -8.51278 q^{47} +5.00000 q^{49} -17.9653 q^{52} -5.34363 q^{53} +20.7446 q^{56} -14.5012 q^{58} -7.37228 q^{59} +3.62772 q^{61} +1.58457 q^{62} -2.37228 q^{64} -8.21782 q^{67} +11.0371 q^{68} +4.11684 q^{71} +7.57301 q^{73} +19.1168 q^{74} +23.4891 q^{76} -4.75372 q^{77} -4.74456 q^{79} +3.46410 q^{82} -5.34363 q^{83} -8.74456 q^{86} -8.21782 q^{88} +3.00000 q^{89} -14.2337 q^{91} +22.0742 q^{92} -21.4891 q^{94} +18.6101 q^{97} +12.6217 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} + 6 q^{11} + 12 q^{14} + 14 q^{16} + 10 q^{19} - 30 q^{26} + 14 q^{31} + 14 q^{34} - 6 q^{41} - 24 q^{44} + 28 q^{46} + 20 q^{49} + 60 q^{56} - 18 q^{59} + 26 q^{61} + 2 q^{64} - 18 q^{71}+ \cdots - 40 q^{94}+O(q^{100}) \) 4 * q + 6 * q^4 + 6 * q^11 + 12 * q^14 + 14 * q^16 + 10 * q^19 - 30 * q^26 + 14 * q^31 + 14 * q^34 - 6 * q^41 - 24 * q^44 + 28 * q^46 + 20 * q^49 + 60 * q^56 - 18 * q^59 + 26 * q^61 + 2 * q^64 - 18 * q^71 + 42 * q^74 + 48 * q^76 + 4 * q^79 - 12 * q^86 + 12 * q^89 + 12 * q^91 - 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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