LMFDB - Embedded newform 2025.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(1,2025)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2025.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:	$16.1697064093$
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			1.3
Root			$0.792287$ 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)

$f(q)$	$=$	$q+0.792287 q^{2} -1.37228 q^{4} -3.46410 q^{7} -2.67181 q^{8} +O(q^{10})$	$q+0.792287 q^{2} -1.37228 q^{4} -3.46410 q^{7} -2.67181 q^{8} +4.37228 q^{11} -5.84096 q^{13} -2.74456 q^{14} +0.627719 q^{16} +0.792287 q^{17} -0.372281 q^{19} +3.46410 q^{22} +1.58457 q^{23} -4.62772 q^{26} +4.75372 q^{28} +5.74456 q^{29} +6.37228 q^{31} +5.84096 q^{32} +0.627719 q^{34} +2.37686 q^{37} -0.294954 q^{38} -4.37228 q^{41} +3.46410 q^{43} -6.00000 q^{44} +1.25544 q^{46} +1.87953 q^{47} +5.00000 q^{49} +8.01544 q^{52} +11.9769 q^{53} +9.25544 q^{56} +4.55134 q^{58} -1.62772 q^{59} +9.37228 q^{61} +5.04868 q^{62} +3.37228 q^{64} -11.6819 q^{67} -1.08724 q^{68} -13.1168 q^{71} +2.37686 q^{73} +1.88316 q^{74} +0.510875 q^{76} -15.1460 q^{77} +6.74456 q^{79} -3.46410 q^{82} +11.9769 q^{83} +2.74456 q^{86} -11.6819 q^{88} +3.00000 q^{89} +20.2337 q^{91} -2.17448 q^{92} +1.48913 q^{94} +1.28962 q^{97} +3.96143 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4}+O(q^{10}) $ 4 * q + 6 * q^4	$ 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}+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

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 $

$2$	0.792287	0.560232	0.280116	−	0.959966i	$-0.409627\pi$
$2$	0.792287	0.560232	0.280116	+	0.959966i	$0.409627\pi$
$3$	0	0
$3$	0	0
$4$	−1.37228	−0.686141
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	−2.67181	−0.944629
$9$	0	0
$10$	0	0
$11$	4.37228	1.31829	0.659146	−	0.752015i	$-0.270918\pi$
$11$	4.37228	1.31829	0.659146	+	0.752015i	$0.270918\pi$
$12$	0	0
$13$	−5.84096	−1.61999	−0.809996	−	0.586436i	$-0.800531\pi$
$13$	−5.84096	−1.61999	−0.809996	+	0.586436i	$0.800531\pi$
$14$	−2.74456	−0.733515
$15$	0	0
$16$	0.627719	0.156930
$17$	0.792287	0.192158	0.0960789	−	0.995374i	$-0.469370\pi$
$17$	0.792287	0.192158	0.0960789	+	0.995374i	$0.469370\pi$
$18$	0	0
$19$	−0.372281	−0.0854072	−0.0427036	−	0.999088i	$-0.513597\pi$
$19$	−0.372281	−0.0854072	−0.0427036	+	0.999088i	$0.513597\pi$
$20$	0	0
$21$	0	0
$22$	3.46410	0.738549
$23$	1.58457	0.330407	0.165203	−	0.986260i	$-0.447172\pi$
$23$	1.58457	0.330407	0.165203	+	0.986260i	$0.447172\pi$
$24$	0	0
$25$	0	0
$26$	−4.62772	−0.907570
$27$	0	0
$28$	4.75372	0.898369
$29$	5.74456	1.06674	0.533369	−	0.845883i	$-0.320926\pi$
$29$	5.74456	1.06674	0.533369	+	0.845883i	$0.320926\pi$
$30$	0	0
$31$	6.37228	1.14450	0.572248	−	0.820081i	$-0.306072\pi$
$31$	6.37228	1.14450	0.572248	+	0.820081i	$0.306072\pi$
$32$	5.84096	1.03255
$33$	0	0
$34$	0.627719	0.107653
$35$	0	0
$36$	0	0
$37$	2.37686	0.390754	0.195377	−	0.980728i	$-0.437407\pi$
$37$	2.37686	0.390754	0.195377	+	0.980728i	$0.437407\pi$
$38$	−0.294954	−0.0478478
$39$	0	0
$40$	0	0
$41$	−4.37228	−0.682836	−0.341418	−	0.939912i	$-0.610907\pi$
$41$	−4.37228	−0.682836	−0.341418	+	0.939912i	$0.610907\pi$
$42$	0	0
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	1.25544	0.185104
$47$	1.87953	0.274157	0.137079	−	0.990560i	$-0.456229\pi$
$47$	1.87953	0.274157	0.137079	+	0.990560i	$0.456229\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	8.01544	1.11154
$53$	11.9769	1.64515	0.822575	−	0.568656i	$-0.192536\pi$
$53$	11.9769	1.64515	0.822575	+	0.568656i	$0.192536\pi$
$54$	0	0
$55$	0	0
$56$	9.25544	1.23681
$57$	0	0
$58$	4.55134	0.597621
$59$	−1.62772	−0.211911	−0.105955	−	0.994371i	$-0.533790\pi$
$59$	−1.62772	−0.211911	−0.105955	+	0.994371i	$0.533790\pi$
$60$	0	0
$61$	9.37228	1.20000	0.599999	−	0.800001i	$-0.295168\pi$
$61$	9.37228	1.20000	0.599999	+	0.800001i	$0.295168\pi$
$62$	5.04868	0.641182
$63$	0	0
$64$	3.37228	0.421535
$65$	0	0
$66$	0	0
$67$	−11.6819	−1.42717	−0.713587	−	0.700566i	$-0.752931\pi$
$67$	−11.6819	−1.42717	−0.713587	+	0.700566i	$0.752931\pi$
$68$	−1.08724	−0.131847
$69$	0	0
$70$	0	0
$71$	−13.1168	−1.55668	−0.778341	−	0.627841i	$-0.783939\pi$
$71$	−13.1168	−1.55668	−0.778341	+	0.627841i	$0.783939\pi$
$72$	0	0
$73$	2.37686	0.278191	0.139095	−	0.990279i	$-0.455581\pi$
$73$	2.37686	0.278191	0.139095	+	0.990279i	$0.455581\pi$
$74$	1.88316	0.218912
$75$	0	0
$76$	0.510875	0.0586013
$77$	−15.1460	−1.72605
$78$	0	0
$79$	6.74456	0.758823	0.379411	−	0.925228i	$-0.376127\pi$
$79$	6.74456	0.758823	0.379411	+	0.925228i	$0.376127\pi$
$80$	0	0
$81$	0	0
$82$	−3.46410	−0.382546
$83$	11.9769	1.31463	0.657317	−	0.753615i	$-0.271691\pi$
$83$	11.9769	1.31463	0.657317	+	0.753615i	$0.271691\pi$
$84$	0	0
$85$	0	0
$86$	2.74456	0.295954
$87$	0	0
$88$	−11.6819	−1.24530
$89$	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
$91$	20.2337	2.12107
$92$	−2.17448	−0.226705
$93$	0	0
$94$	1.48913	0.153592
$95$	0	0
$96$	0	0
$97$	1.28962	0.130941	0.0654706	−	0.997855i	$-0.479145\pi$
$97$	1.28962	0.130941	0.0654706	+	0.997855i	$0.479145\pi$
$98$	3.96143	0.400165
$99$	0	0
$100$	0	0
$101$	16.3723	1.62910	0.814551	−	0.580091i	$-0.196983\pi$
$101$	16.3723	1.62910	0.814551	+	0.580091i	$0.196983\pi$
$102$	0	0
$103$	6.92820	0.682656	0.341328	−	0.939944i	$-0.389123\pi$
$103$	6.92820	0.682656	0.341328	+	0.939944i	$0.389123\pi$
$104$	15.6060	1.53029
$105$	0	0
$106$	9.48913	0.921665
$107$	−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
$109$	9.74456	0.933360	0.466680	−	0.884426i	$-0.345450\pi$
$109$	9.74456	0.933360	0.466680	+	0.884426i	$0.345450\pi$
$110$	0	0
$111$	0	0
$112$	−2.17448	−0.205469
$113$	6.13592	0.577218	0.288609	−	0.957447i	$-0.406807\pi$
$113$	6.13592	0.577218	0.288609	+	0.957447i	$0.406807\pi$
$114$	0	0
$115$	0	0
$116$	−7.88316	−0.731933
$117$	0	0
$118$	−1.28962	−0.118719
$119$	−2.74456	−0.251594
$120$	0	0
$121$	8.11684	0.737895
$122$	7.42554	0.672276
$123$	0	0
$124$	−8.74456	−0.785285
$125$	0	0
$126$	0	0
$127$	10.3923	0.922168	0.461084	−	0.887357i	$-0.347461\pi$
$127$	10.3923	0.922168	0.461084	+	0.887357i	$0.347461\pi$
$128$	−9.01011	−0.796389
$129$	0	0
$130$	0	0
$131$	−4.37228	−0.382008	−0.191004	−	0.981589i	$-0.561174\pi$
$131$	−4.37228	−0.382008	−0.191004	+	0.981589i	$0.561174\pi$
$132$	0	0
$133$	1.28962	0.111824
$134$	−9.25544	−0.799548
$135$	0	0
$136$	−2.11684	−0.181518
$137$	−14.3537	−1.22632	−0.613161	−	0.789958i	$-0.710102\pi$
$137$	−14.3537	−1.22632	−0.613161	+	0.789958i	$0.710102\pi$
$138$	0	0
$139$	11.1168	0.942918	0.471459	−	0.881888i	$-0.343727\pi$
$139$	11.1168	0.942918	0.471459	+	0.881888i	$0.343727\pi$
$140$	0	0
$141$	0	0
$142$	−10.3923	−0.872103
$143$	−25.5383	−2.13562
$144$	0	0
$145$	0	0
$146$	1.88316	0.155851
$147$	0	0
$148$	−3.26172	−0.268112
$149$	−10.6277	−0.870657	−0.435328	−	0.900272i	$-0.643368\pi$
$149$	−10.6277	−0.870657	−0.435328	+	0.900272i	$0.643368\pi$
$150$	0	0
$151$	0.883156	0.0718702	0.0359351	−	0.999354i	$-0.488559\pi$
$151$	0.883156	0.0718702	0.0359351	+	0.999354i	$0.488559\pi$
$152$	0.994667	0.0806781
$153$	0	0
$154$	−12.0000	−0.966988
$155$	0	0
$156$	0	0
$157$	11.4795	0.916167	0.458084	−	0.888909i	$-0.348536\pi$
$157$	11.4795	0.916167	0.458084	+	0.888909i	$0.348536\pi$
$158$	5.34363	0.425116
$159$	0	0
$160$	0	0
$161$	−5.48913	−0.432604
$162$	0	0
$163$	−15.1460	−1.18633	−0.593164	−	0.805082i	$-0.702122\pi$
$163$	−15.1460	−1.18633	−0.593164	+	0.805082i	$0.702122\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	9.48913	0.736499
$167$	13.5615	1.04942	0.524708	−	0.851282i	$-0.324174\pi$
$167$	13.5615	1.04942	0.524708	+	0.851282i	$0.324174\pi$
$168$	0	0
$169$	21.1168	1.62437
$170$	0	0
$171$	0	0
$172$	−4.75372	−0.362468
$173$	−11.1846	−0.850349	−0.425174	−	0.905111i	$-0.639787\pi$
$173$	−11.1846	−0.850349	−0.425174	+	0.905111i	$0.639787\pi$
$174$	0	0
$175$	0	0
$176$	2.74456	0.206879
$177$	0	0
$178$	2.37686	0.178153
$179$	4.88316	0.364984	0.182492	−	0.983207i	$-0.441584\pi$
$179$	4.88316	0.364984	0.182492	+	0.983207i	$0.441584\pi$
$180$	0	0
$181$	−13.8614	−1.03031	−0.515155	−	0.857097i	$-0.672266\pi$
$181$	−13.8614	−1.03031	−0.515155	+	0.857097i	$0.672266\pi$
$182$	16.0309	1.18829
$183$	0	0
$184$	−4.23369	−0.312112
$185$	0	0
$186$	0	0
$187$	3.46410	0.253320
$188$	−2.57924	−0.188110
$189$	0	0
$190$	0	0
$191$	−7.62772	−0.551922	−0.275961	−	0.961169i	$-0.588996\pi$
$191$	−7.62772	−0.551922	−0.275961	+	0.961169i	$0.588996\pi$
$192$	0	0
$193$	−11.4795	−0.826316	−0.413158	−	0.910659i	$-0.635574\pi$
$193$	−11.4795	−0.826316	−0.413158	+	0.910659i	$0.635574\pi$
$194$	1.02175	0.0733573
$195$	0	0
$196$	−6.86141	−0.490100
$197$	6.43087	0.458181	0.229090	−	0.973405i	$-0.426425\pi$
$197$	6.43087	0.458181	0.229090	+	0.973405i	$0.426425\pi$
$198$	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
$202$	12.9715	0.912675
$203$	−19.8997	−1.39669
$204$	0	0
$205$	0	0
$206$	5.48913	0.382445
$207$	0	0
$208$	−3.66648	−0.254225
$209$	−1.62772	−0.112592
$210$	0	0
$211$	−1.86141	−0.128145	−0.0640723	−	0.997945i	$-0.520409\pi$
$211$	−1.86141	−0.128145	−0.0640723	+	0.997945i	$0.520409\pi$
$212$	−16.4356	−1.12880
$213$	0	0
$214$	−10.5109	−0.718509
$215$	0	0
$216$	0	0
$217$	−22.0742	−1.49850
$218$	7.72049	0.522898
$219$	0	0
$220$	0	0
$221$	−4.62772	−0.311294
$222$	0	0
$223$	18.6101	1.24623	0.623113	−	0.782132i	$-0.285868\pi$
$223$	18.6101	1.24623	0.623113	+	0.782132i	$0.285868\pi$
$224$	−20.2337	−1.35192
$225$	0	0
$226$	4.86141	0.323376
$227$	13.5615	0.900105	0.450053	−	0.893002i	$-0.351405\pi$
$227$	13.5615	0.900105	0.450053	+	0.893002i	$0.351405\pi$
$228$	0	0
$229$	−17.6060	−1.16344	−0.581718	−	0.813391i	$-0.697619\pi$
$229$	−17.6060	−1.16344	−0.581718	+	0.813391i	$0.697619\pi$
$230$	0	0
$231$	0	0
$232$	−15.3484	−1.00767
$233$	6.13592	0.401977	0.200989	−	0.979594i	$-0.435585\pi$
$233$	6.13592	0.401977	0.200989	+	0.979594i	$0.435585\pi$
$234$	0	0
$235$	0	0
$236$	2.23369	0.145401
$237$	0	0
$238$	−2.17448	−0.140951
$239$	−22.9783	−1.48634	−0.743170	−	0.669103i	$-0.766679\pi$
$239$	−22.9783	−1.48634	−0.743170	+	0.669103i	$0.766679\pi$
$240$	0	0
$241$	−1.51087	−0.0973240	−0.0486620	−	0.998815i	$-0.515496\pi$
$241$	−1.51087	−0.0973240	−0.0486620	+	0.998815i	$0.515496\pi$
$242$	6.43087	0.413392
$243$	0	0
$244$	−12.8614	−0.823367
$245$	0	0
$246$	0	0
$247$	2.17448	0.138359
$248$	−17.0256	−1.08112
$249$	0	0
$250$	0	0
$251$	20.2337	1.27714	0.638570	−	0.769564i	$-0.279526\pi$
$251$	20.2337	1.27714	0.638570	+	0.769564i	$0.279526\pi$
$252$	0	0
$253$	6.92820	0.435572
$254$	8.23369	0.516628
$255$	0	0
$256$	−13.8832	−0.867697
$257$	6.43087	0.401147	0.200573	−	0.979679i	$-0.435720\pi$
$257$	6.43087	0.401147	0.200573	+	0.979679i	$0.435720\pi$
$258$	0	0
$259$	−8.23369	−0.511616
$260$	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
$266$	1.02175	0.0626475
$267$	0	0
$268$	16.0309	0.979242
$269$	29.2337	1.78241	0.891205	−	0.453601i	$-0.149861\pi$
$269$	29.2337	1.78241	0.891205	+	0.453601i	$0.149861\pi$
$270$	0	0
$271$	5.25544	0.319245	0.159623	−	0.987178i	$-0.448972\pi$
$271$	5.25544	0.319245	0.159623	+	0.987178i	$0.448972\pi$
$272$	0.497333	0.0301553
$273$	0	0
$274$	−11.3723	−0.687025
$275$	0	0
$276$	0	0
$277$	22.0742	1.32631	0.663156	−	0.748481i	$-0.269217\pi$
$277$	22.0742	1.32631	0.663156	+	0.748481i	$0.269217\pi$
$278$	8.80773	0.528253
$279$	0	0
$280$	0	0
$281$	1.37228	0.0818634	0.0409317	−	0.999162i	$-0.486967\pi$
$281$	1.37228	0.0818634	0.0409317	+	0.999162i	$0.486967\pi$
$282$	0	0
$283$	27.7128	1.64736	0.823678	−	0.567058i	$-0.191918\pi$
$283$	27.7128	1.64736	0.823678	+	0.567058i	$0.191918\pi$
$284$	18.0000	1.06810
$285$	0	0
$286$	−20.2337	−1.19644
$287$	15.1460	0.894042
$288$	0	0
$289$	−16.3723	−0.963075
$290$	0	0
$291$	0	0
$292$	−3.26172	−0.190878
$293$	−0.792287	−0.0462859	−0.0231430	−	0.999732i	$-0.507367\pi$
$293$	−0.792287	−0.0462859	−0.0231430	+	0.999732i	$0.507367\pi$
$294$	0	0
$295$	0	0
$296$	−6.35053	−0.369117
$297$	0	0
$298$	−8.42020	−0.487769
$299$	−9.25544	−0.535256
$300$	0	0
$301$	−12.0000	−0.691669
$302$	0.699713	0.0402640
$303$	0	0
$304$	−0.233688	−0.0134029
$305$	0	0
$306$	0	0
$307$	15.1460	0.864429	0.432215	−	0.901771i	$-0.357732\pi$
$307$	15.1460	0.864429	0.432215	+	0.901771i	$0.357732\pi$
$308$	20.7846	1.18431
$309$	0	0
$310$	0	0
$311$	15.8614	0.899418	0.449709	−	0.893175i	$-0.351528\pi$
$311$	15.8614	0.899418	0.449709	+	0.893175i	$0.351528\pi$
$312$	0	0
$313$	−24.4511	−1.38206	−0.691029	−	0.722827i	$-0.742842\pi$
$313$	−24.4511	−1.38206	−0.691029	+	0.722827i	$0.742842\pi$
$314$	9.09509	0.513266
$315$	0	0
$316$	−9.25544	−0.520659
$317$	−29.4998	−1.65687	−0.828436	−	0.560084i	$-0.810769\pi$
$317$	−29.4998	−1.65687	−0.828436	+	0.560084i	$0.810769\pi$
$318$	0	0
$319$	25.1168	1.40627
$320$	0	0
$321$	0	0
$322$	−4.34896	−0.242358
$323$	−0.294954	−0.0164117
$324$	0	0
$325$	0	0
$326$	−12.0000	−0.664619
$327$	0	0
$328$	11.6819	0.645026
$329$	−6.51087	−0.358956
$330$	0	0
$331$	9.11684	0.501107	0.250554	−	0.968103i	$-0.419387\pi$
$331$	9.11684	0.501107	0.250554	+	0.968103i	$0.419387\pi$
$332$	−16.4356	−0.902023
$333$	0	0
$334$	10.7446	0.587916
$335$	0	0
$336$	0	0
$337$	−6.92820	−0.377403	−0.188702	−	0.982034i	$-0.560428\pi$
$337$	−6.92820	−0.377403	−0.188702	+	0.982034i	$0.560428\pi$
$338$	16.7306	0.910025
$339$	0	0
$340$	0	0
$341$	27.8614	1.50878
$342$	0	0
$343$	6.92820	0.374088
$344$	−9.25544	−0.499020
$345$	0	0
$346$	−8.86141	−0.476392
$347$	−2.87419	−0.154295	−0.0771474	−	0.997020i	$-0.524581\pi$
$347$	−2.87419	−0.154295	−0.0771474	+	0.997020i	$0.524581\pi$
$348$	0	0
$349$	−26.6060	−1.42418	−0.712092	−	0.702086i	$-0.752252\pi$
$349$	−26.6060	−1.42418	−0.712092	+	0.702086i	$0.752252\pi$
$350$	0	0
$351$	0	0
$352$	25.5383	1.36120
$353$	−1.87953	−0.100037	−0.0500186	−	0.998748i	$-0.515928\pi$
$353$	−1.87953	−0.100037	−0.0500186	+	0.998748i	$0.515928\pi$
$354$	0	0
$355$	0	0
$356$	−4.11684	−0.218192
$357$	0	0
$358$	3.86886	0.204476
$359$	10.8832	0.574391	0.287196	−	0.957872i	$-0.407277\pi$
$359$	10.8832	0.574391	0.287196	+	0.957872i	$0.407277\pi$
$360$	0	0
$361$	−18.8614	−0.992706
$362$	−10.9822	−0.577212
$363$	0	0
$364$	−27.7663	−1.45535
$365$	0	0
$366$	0	0
$367$	16.4356	0.857934	0.428967	−	0.903320i	$-0.358878\pi$
$367$	16.4356	0.857934	0.428967	+	0.903320i	$0.358878\pi$
$368$	0.994667	0.0518506
$369$	0	0
$370$	0	0
$371$	−41.4891	−2.15401
$372$	0	0
$373$	−8.21782	−0.425503	−0.212751	−	0.977106i	$-0.568242\pi$
$373$	−8.21782	−0.425503	−0.212751	+	0.977106i	$0.568242\pi$
$374$	2.74456	0.141918
$375$	0	0
$376$	−5.02175	−0.258977
$377$	−33.5538	−1.72811
$378$	0	0
$379$	−1.48913	−0.0764912	−0.0382456	−	0.999268i	$-0.512177\pi$
$379$	−1.48913	−0.0764912	−0.0382456	+	0.999268i	$0.512177\pi$
$380$	0	0
$381$	0	0
$382$	−6.04334	−0.309204
$383$	11.0920	0.566776	0.283388	−	0.959005i	$-0.408542\pi$
$383$	11.0920	0.566776	0.283388	+	0.959005i	$0.408542\pi$
$384$	0	0
$385$	0	0
$386$	−9.09509	−0.462928
$387$	0	0
$388$	−1.76972	−0.0898440
$389$	0.510875	0.0259024	0.0129512	−	0.999916i	$-0.495877\pi$
$389$	0.510875	0.0259024	0.0129512	+	0.999916i	$0.495877\pi$
$390$	0	0
$391$	1.25544	0.0634902
$392$	−13.3591	−0.674735
$393$	0	0
$394$	5.09509	0.256687
$395$	0	0
$396$	0	0
$397$	−17.5229	−0.879449	−0.439724	−	0.898133i	$-0.644924\pi$
$397$	−17.5229	−0.879449	−0.439724	+	0.898133i	$0.644924\pi$
$398$	12.6766	0.635420
$399$	0	0
$400$	0	0
$401$	−25.3723	−1.26703	−0.633516	−	0.773730i	$-0.718389\pi$
$401$	−25.3723	−1.26703	−0.633516	+	0.773730i	$0.718389\pi$
$402$	0	0
$403$	−37.2203	−1.85407
$404$	−22.4674	−1.11779
$405$	0	0
$406$	−15.7663	−0.782469
$407$	10.3923	0.515127
$408$	0	0
$409$	22.8614	1.13042	0.565212	−	0.824946i	$-0.308794\pi$
$409$	22.8614	1.13042	0.565212	+	0.824946i	$0.308794\pi$
$410$	0	0
$411$	0	0
$412$	−9.50744	−0.468398
$413$	5.63858	0.277457
$414$	0	0
$415$	0	0
$416$	−34.1168	−1.67272
$417$	0	0
$418$	−1.28962	−0.0630774
$419$	17.4891	0.854400	0.427200	−	0.904157i	$-0.359500\pi$
$419$	17.4891	0.854400	0.427200	+	0.904157i	$0.359500\pi$
$420$	0	0
$421$	−21.7446	−1.05977	−0.529883	−	0.848071i	$-0.677764\pi$
$421$	−21.7446	−1.05977	−0.529883	+	0.848071i	$0.677764\pi$
$422$	−1.47477	−0.0717906
$423$	0	0
$424$	−32.0000	−1.55406
$425$	0	0
$426$	0	0
$427$	−32.4665	−1.57117
$428$	18.2054	0.879990
$429$	0	0
$430$	0	0
$431$	33.3505	1.60644	0.803219	−	0.595683i	$-0.203119\pi$
$431$	33.3505	1.60644	0.803219	+	0.595683i	$0.203119\pi$
$432$	0	0
$433$	12.7692	0.613647	0.306823	−	0.951766i	$-0.400734\pi$
$433$	12.7692	0.613647	0.306823	+	0.951766i	$0.400734\pi$
$434$	−17.4891	−0.839505
$435$	0	0
$436$	−13.3723	−0.640416
$437$	−0.589907	−0.0282191
$438$	0	0
$439$	31.8614	1.52066	0.760331	−	0.649536i	$-0.225037\pi$
$439$	31.8614	1.52066	0.760331	+	0.649536i	$0.225037\pi$
$440$	0	0
$441$	0	0
$442$	−3.66648	−0.174397
$443$	21.4843	1.02075	0.510375	−	0.859952i	$-0.329506\pi$
$443$	21.4843	1.02075	0.510375	+	0.859952i	$0.329506\pi$
$444$	0	0
$445$	0	0
$446$	14.7446	0.698175
$447$	0	0
$448$	−11.6819	−0.551919
$449$	10.8832	0.513608	0.256804	−	0.966464i	$-0.417331\pi$
$449$	10.8832	0.513608	0.256804	+	0.966464i	$0.417331\pi$
$450$	0	0
$451$	−19.1168	−0.900177
$452$	−8.42020	−0.396053
$453$	0	0
$454$	10.7446	0.504267
$455$	0	0
$456$	0	0
$457$	−20.9870	−0.981730	−0.490865	−	0.871236i	$-0.663319\pi$
$457$	−20.9870	−0.981730	−0.490865	+	0.871236i	$0.663319\pi$
$458$	−13.9490	−0.651793
$459$	0	0
$460$	0	0
$461$	−24.0951	−1.12222	−0.561110	−	0.827741i	$-0.689626\pi$
$461$	−24.0951	−1.12222	−0.561110	+	0.827741i	$0.689626\pi$
$462$	0	0
$463$	−13.8564	−0.643962	−0.321981	−	0.946746i	$-0.604349\pi$
$463$	−13.8564	−0.643962	−0.321981	+	0.946746i	$0.604349\pi$
$464$	3.60597	0.167403
$465$	0	0
$466$	4.86141	0.225200
$467$	18.3152	0.847525	0.423763	−	0.905773i	$-0.360709\pi$
$467$	18.3152	0.847525	0.423763	+	0.905773i	$0.360709\pi$
$468$	0	0
$469$	40.4674	1.86861
$470$	0	0
$471$	0	0
$472$	4.34896	0.200177
$473$	15.1460	0.696415
$474$	0	0
$475$	0	0
$476$	3.76631	0.172629
$477$	0	0
$478$	−18.2054	−0.832694
$479$	−18.6060	−0.850128	−0.425064	−	0.905163i	$-0.639748\pi$
$479$	−18.6060	−0.850128	−0.425064	+	0.905163i	$0.639748\pi$
$480$	0	0
$481$	−13.8832	−0.633017
$482$	−1.19705	−0.0545240
$483$	0	0
$484$	−11.1386	−0.506300
$485$	0	0
$486$	0	0
$487$	9.50744	0.430823	0.215412	−	0.976523i	$-0.430891\pi$
$487$	9.50744	0.430823	0.215412	+	0.976523i	$0.430891\pi$
$488$	−25.0410	−1.13355
$489$	0	0
$490$	0	0
$491$	31.6277	1.42734	0.713669	−	0.700483i	$-0.247032\pi$
$491$	31.6277	1.42734	0.713669	+	0.700483i	$0.247032\pi$
$492$	0	0
$493$	4.55134	0.204982
$494$	1.72281	0.0775130
$495$	0	0
$496$	4.00000	0.179605
$497$	45.4381	2.03818
$498$	0	0
$499$	−24.3723	−1.09105	−0.545527	−	0.838094i	$-0.683670\pi$
$499$	−24.3723	−1.09105	−0.545527	+	0.838094i	$0.683670\pi$
$500$	0	0
$501$	0	0
$502$	16.0309	0.715494
$503$	−3.16915	−0.141305	−0.0706527	−	0.997501i	$-0.522508\pi$
$503$	−3.16915	−0.141305	−0.0706527	+	0.997501i	$0.522508\pi$
$504$	0	0
$505$	0	0
$506$	5.48913	0.244021
$507$	0	0
$508$	−14.2612	−0.632737
$509$	−0.510875	−0.0226441	−0.0113221	−	0.999936i	$-0.503604\pi$
$509$	−0.510875	−0.0226441	−0.0113221	+	0.999936i	$0.503604\pi$
$510$	0	0
$511$	−8.23369	−0.364237
$512$	7.02078	0.310277
$513$	0	0
$514$	5.09509	0.224735
$515$	0	0
$516$	0	0
$517$	8.21782	0.361419
$518$	−6.52344	−0.286624
$519$	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
$523$	4.75372	0.207866	0.103933	−	0.994584i	$-0.466857\pi$
$523$	4.75372	0.207866	0.103933	+	0.994584i	$0.466857\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	−21.0217	−0.916592
$527$	5.04868	0.219924
$528$	0	0
$529$	−20.4891	−0.890832
$530$	0	0
$531$	0	0
$532$	−1.76972	−0.0767272
$533$	25.5383	1.10619
$534$	0	0
$535$	0	0
$536$	31.2119	1.34815
$537$	0	0
$538$	23.1615	0.998562
$539$	21.8614	0.941637
$540$	0	0
$541$	19.2337	0.826921	0.413460	−	0.910522i	$-0.364320\pi$
$541$	19.2337	0.826921	0.413460	+	0.910522i	$0.364320\pi$
$542$	4.16381	0.178851
$543$	0	0
$544$	4.62772	0.198412
$545$	0	0
$546$	0	0
$547$	6.92820	0.296229	0.148114	−	0.988970i	$-0.452680\pi$
$547$	6.92820	0.296229	0.148114	+	0.988970i	$0.452680\pi$
$548$	19.6974	0.841430
$549$	0	0
$550$	0	0
$551$	−2.13859	−0.0911071
$552$	0	0
$553$	−23.3639	−0.993532
$554$	17.4891	0.743042
$555$	0	0
$556$	−15.2554	−0.646975
$557$	25.0410	1.06102	0.530511	−	0.847678i	$-0.322000\pi$
$557$	25.0410	1.06102	0.530511	+	0.847678i	$0.322000\pi$
$558$	0	0
$559$	−20.2337	−0.855794
$560$	0	0
$561$	0	0
$562$	1.08724	0.0458625
$563$	−32.1716	−1.35587	−0.677935	−	0.735122i	$-0.737125\pi$
$563$	−32.1716	−1.35587	−0.677935	+	0.735122i	$0.737125\pi$
$564$	0	0
$565$	0	0
$566$	21.9565	0.922901
$567$	0	0
$568$	35.0458	1.47049
$569$	−20.4891	−0.858949	−0.429474	−	0.903079i	$-0.641301\pi$
$569$	−20.4891	−0.858949	−0.429474	+	0.903079i	$0.641301\pi$
$570$	0	0
$571$	4.13859	0.173195	0.0865974	−	0.996243i	$-0.472401\pi$
$571$	4.13859	0.173195	0.0865974	+	0.996243i	$0.472401\pi$
$572$	35.0458	1.46534
$573$	0	0
$574$	12.0000	0.500870
$575$	0	0
$576$	0	0
$577$	−18.4077	−0.766325	−0.383162	−	0.923681i	$-0.625165\pi$
$577$	−18.4077	−0.766325	−0.383162	+	0.923681i	$0.625165\pi$
$578$	−12.9715	−0.539545
$579$	0	0
$580$	0	0
$581$	−41.4891	−1.72126
$582$	0	0
$583$	52.3663	2.16879
$584$	−6.35053	−0.262787
$585$	0	0
$586$	−0.627719	−0.0259308
$587$	−16.7306	−0.690546	−0.345273	−	0.938502i	$-0.612214\pi$
$587$	−16.7306	−0.690546	−0.345273	+	0.938502i	$0.612214\pi$
$588$	0	0
$589$	−2.37228	−0.0977481
$590$	0	0
$591$	0	0
$592$	1.49200	0.0613208
$593$	−1.67715	−0.0688722	−0.0344361	−	0.999407i	$-0.510964\pi$
$593$	−1.67715	−0.0688722	−0.0344361	+	0.999407i	$0.510964\pi$
$594$	0	0
$595$	0	0
$596$	14.5842	0.597393
$597$	0	0
$598$	−7.33296	−0.299867
$599$	−1.62772	−0.0665068	−0.0332534	−	0.999447i	$-0.510587\pi$
$599$	−1.62772	−0.0665068	−0.0332534	+	0.999447i	$0.510587\pi$
$600$	0	0
$601$	−17.9783	−0.733348	−0.366674	−	0.930349i	$-0.619504\pi$
$601$	−17.9783	−0.733348	−0.366674	+	0.930349i	$0.619504\pi$
$602$	−9.50744	−0.387494
$603$	0	0
$604$	−1.21194	−0.0493131
$605$	0	0
$606$	0	0
$607$	43.2636	1.75602	0.878008	−	0.478647i	$-0.158872\pi$
$607$	43.2636	1.75602	0.878008	+	0.478647i	$0.158872\pi$
$608$	−2.17448	−0.0881869
$609$	0	0
$610$	0	0
$611$	−10.9783	−0.444132
$612$	0	0
$613$	30.2921	1.22348	0.611742	−	0.791057i	$-0.290469\pi$
$613$	30.2921	1.22348	0.611742	+	0.791057i	$0.290469\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	40.4674	1.63048
$617$	20.6920	0.833030	0.416515	−	0.909129i	$-0.363251\pi$
$617$	20.6920	0.833030	0.416515	+	0.909129i	$0.363251\pi$
$618$	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
$622$	12.5668	0.503882
$623$	−10.3923	−0.416359
$624$	0	0
$625$	0	0
$626$	−19.3723	−0.774272
$627$	0	0
$628$	−15.7532	−0.628620
$629$	1.88316	0.0750863
$630$	0	0
$631$	6.37228	0.253677	0.126838	−	0.991923i	$-0.459517\pi$
$631$	6.37228	0.253677	0.126838	+	0.991923i	$0.459517\pi$
$632$	−18.0202	−0.716806
$633$	0	0
$634$	−23.3723	−0.928232
$635$	0	0
$636$	0	0
$637$	−29.2048	−1.15714
$638$	19.8997	0.787839
$639$	0	0
$640$	0	0
$641$	−7.97825	−0.315122	−0.157561	−	0.987509i	$-0.550363\pi$
$641$	−7.97825	−0.315122	−0.157561	+	0.987509i	$0.550363\pi$
$642$	0	0
$643$	1.28962	0.0508577	0.0254288	−	0.999677i	$-0.491905\pi$
$643$	1.28962	0.0508577	0.0254288	+	0.999677i	$0.491905\pi$
$644$	7.53262	0.296827
$645$	0	0
$646$	−0.233688	−0.00919433
$647$	28.7075	1.12861	0.564304	−	0.825567i	$-0.309145\pi$
$647$	28.7075	1.12861	0.564304	+	0.825567i	$0.309145\pi$
$648$	0	0
$649$	−7.11684	−0.279361
$650$	0	0
$651$	0	0
$652$	20.7846	0.813988
$653$	6.33830	0.248037	0.124018	−	0.992280i	$-0.460422\pi$
$653$	6.33830	0.248037	0.124018	+	0.992280i	$0.460422\pi$
$654$	0	0
$655$	0	0
$656$	−2.74456	−0.107157
$657$	0	0
$658$	−5.15848	−0.201099
$659$	−21.2554	−0.827994	−0.413997	−	0.910278i	$-0.635868\pi$
$659$	−21.2554	−0.827994	−0.413997	+	0.910278i	$0.635868\pi$
$660$	0	0
$661$	−15.2337	−0.592522	−0.296261	−	0.955107i	$-0.595740\pi$
$661$	−15.2337	−0.592522	−0.296261	+	0.955107i	$0.595740\pi$
$662$	7.22316	0.280736
$663$	0	0
$664$	−32.0000	−1.24184
$665$	0	0
$666$	0	0
$667$	9.10268	0.352457
$668$	−18.6101	−0.720047
$669$	0	0
$670$	0	0
$671$	40.9783	1.58195
$672$	0	0
$673$	10.5947	0.408395	0.204198	−	0.978930i	$-0.434542\pi$
$673$	10.5947	0.408395	0.204198	+	0.978930i	$0.434542\pi$
$674$	−5.48913	−0.211433
$675$	0	0
$676$	−28.9783	−1.11455
$677$	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
$679$	−4.46738	−0.171442
$680$	0	0
$681$	0	0
$682$	22.0742	0.845266
$683$	27.1229	1.03783	0.518915	−	0.854826i	$-0.326336\pi$
$683$	27.1229	1.03783	0.518915	+	0.854826i	$0.326336\pi$
$684$	0	0
$685$	0	0
$686$	5.48913	0.209576
$687$	0	0
$688$	2.17448	0.0829013
$689$	−69.9565	−2.66513
$690$	0	0
$691$	45.7228	1.73938	0.869689	−	0.493600i	$-0.164319\pi$
$691$	45.7228	1.73938	0.869689	+	0.493600i	$0.164319\pi$
$692$	15.3484	0.583459
$693$	0	0
$694$	−2.27719	−0.0864408
$695$	0	0
$696$	0	0
$697$	−3.46410	−0.131212
$698$	−21.0796	−0.797873
$699$	0	0
$700$	0	0
$701$	−17.2337	−0.650907	−0.325454	−	0.945558i	$-0.605517\pi$
$701$	−17.2337	−0.650907	−0.325454	+	0.945558i	$0.605517\pi$
$702$	0	0
$703$	−0.884861	−0.0333732
$704$	14.7446	0.555707
$705$	0	0
$706$	−1.48913	−0.0560440
$707$	−56.7152	−2.13300
$708$	0	0
$709$	52.3505	1.96607	0.983033	−	0.183430i	$-0.0587201\pi$
$709$	52.3505	1.96607	0.983033	+	0.183430i	$0.0587201\pi$
$710$	0	0
$711$	0	0
$712$	−8.01544	−0.300391
$713$	10.0974	0.378149
$714$	0	0
$715$	0	0
$716$	−6.70106	−0.250431
$717$	0	0
$718$	8.62258	0.321792
$719$	−10.8832	−0.405873	−0.202937	−	0.979192i	$-0.565049\pi$
$719$	−10.8832	−0.405873	−0.202937	+	0.979192i	$0.565049\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	−14.9436	−0.556145
$723$	0	0
$724$	19.0217	0.706938
$725$	0	0
$726$	0	0
$727$	−48.9022	−1.81368	−0.906841	−	0.421473i	$-0.861513\pi$
$727$	−48.9022	−1.81368	−0.906841	+	0.421473i	$0.861513\pi$
$728$	−54.0607	−2.00362
$729$	0	0
$730$	0	0
$731$	2.74456	0.101511
$732$	0	0
$733$	16.4356	0.607064	0.303532	−	0.952821i	$-0.401834\pi$
$733$	16.4356	0.607064	0.303532	+	0.952821i	$0.401834\pi$
$734$	13.0217	0.480642
$735$	0	0
$736$	9.25544	0.341160
$737$	−51.0767	−1.88143
$738$	0	0
$739$	5.62772	0.207019	0.103509	−	0.994628i	$-0.466993\pi$
$739$	5.62772	0.207019	0.103509	+	0.994628i	$0.466993\pi$
$740$	0	0
$741$	0	0
$742$	−32.8713	−1.20674
$743$	9.39764	0.344766	0.172383	−	0.985030i	$-0.444853\pi$
$743$	9.39764	0.344766	0.172383	+	0.985030i	$0.444853\pi$
$744$	0	0
$745$	0	0
$746$	−6.51087	−0.238380
$747$	0	0
$748$	−4.75372	−0.173813
$749$	45.9565	1.67921
$750$	0	0
$751$	21.7228	0.792677	0.396338	−	0.918105i	$-0.370281\pi$
$751$	21.7228	0.792677	0.396338	+	0.918105i	$0.370281\pi$
$752$	1.17981	0.0430234
$753$	0	0
$754$	−26.5842	−0.968140
$755$	0	0
$756$	0	0
$757$	41.5692	1.51086	0.755429	−	0.655230i	$-0.227428\pi$
$757$	41.5692	1.51086	0.755429	+	0.655230i	$0.227428\pi$
$758$	−1.17981	−0.0428528
$759$	0	0
$760$	0	0
$761$	−32.4891	−1.17773	−0.588865	−	0.808231i	$-0.700425\pi$
$761$	−32.4891	−1.17773	−0.588865	+	0.808231i	$0.700425\pi$
$762$	0	0
$763$	−33.7562	−1.22205
$764$	10.4674	0.378696
$765$	0	0
$766$	8.78806	0.317526
$767$	9.50744	0.343294
$768$	0	0
$769$	−16.4891	−0.594613	−0.297307	−	0.954782i	$-0.596088\pi$
$769$	−16.4891	−0.594613	−0.297307	+	0.954782i	$0.596088\pi$
$770$	0	0
$771$	0	0
$772$	15.7532	0.566969
$773$	−21.5769	−0.776067	−0.388034	−	0.921645i	$-0.626846\pi$
$773$	−21.5769	−0.776067	−0.388034	+	0.921645i	$0.626846\pi$
$774$	0	0
$775$	0	0
$776$	−3.44563	−0.123691
$777$	0	0
$778$	0.404759	0.0145113
$779$	1.62772	0.0583191
$780$	0	0
$781$	−57.3505	−2.05216
$782$	0.994667	0.0355692
$783$	0	0
$784$	3.13859	0.112093
$785$	0	0
$786$	0	0
$787$	−17.3205	−0.617409	−0.308705	−	0.951158i	$-0.599895\pi$
$787$	−17.3205	−0.617409	−0.308705	+	0.951158i	$0.599895\pi$
$788$	−8.82496	−0.314376
$789$	0	0
$790$	0	0
$791$	−21.2554	−0.755756
$792$	0	0
$793$	−54.7431	−1.94399
$794$	−13.8832	−0.492695
$795$	0	0
$796$	−21.9565	−0.778228
$797$	−25.6309	−0.907893	−0.453947	−	0.891029i	$-0.649984\pi$
$797$	−25.6309	−0.907893	−0.453947	+	0.891029i	$0.649984\pi$
$798$	0	0
$799$	1.48913	0.0526815
$800$	0	0
$801$	0	0
$802$	−20.1021	−0.709831
$803$	10.3923	0.366736
$804$	0	0
$805$	0	0
$806$	−29.4891	−1.03871
$807$	0	0
$808$	−43.7437	−1.53890
$809$	−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
$811$	24.8832	0.873766	0.436883	−	0.899518i	$-0.356082\pi$
$811$	24.8832	0.873766	0.436883	+	0.899518i	$0.356082\pi$
$812$	27.3081	0.958325
$813$	0	0
$814$	8.23369	0.288591
$815$	0	0
$816$	0	0
$817$	−1.28962	−0.0451181
$818$	18.1128	0.633299
$819$	0	0
$820$	0	0
$821$	40.7228	1.42124	0.710618	−	0.703578i	$-0.248415\pi$
$821$	40.7228	1.42124	0.710618	+	0.703578i	$0.248415\pi$
$822$	0	0
$823$	−4.34896	−0.151595	−0.0757977	−	0.997123i	$-0.524150\pi$
$823$	−4.34896	−0.151595	−0.0757977	+	0.997123i	$0.524150\pi$
$824$	−18.5109	−0.644857
$825$	0	0
$826$	4.46738	0.155440
$827$	−22.3692	−0.777853	−0.388926	−	0.921269i	$-0.627154\pi$
$827$	−22.3692	−0.777853	−0.388926	+	0.921269i	$0.627154\pi$
$828$	0	0
$829$	−23.3505	−0.810997	−0.405499	−	0.914096i	$-0.632902\pi$
$829$	−23.3505	−0.810997	−0.405499	+	0.914096i	$0.632902\pi$
$830$	0	0
$831$	0	0
$832$	−19.6974	−0.682883
$833$	3.96143	0.137256
$834$	0	0
$835$	0	0
$836$	2.23369	0.0772537
$837$	0	0
$838$	13.8564	0.478662
$839$	45.8614	1.58331	0.791656	−	0.610967i	$-0.209219\pi$
$839$	45.8614	1.58331	0.791656	+	0.610967i	$0.209219\pi$
$840$	0	0
$841$	4.00000	0.137931
$842$	−17.2279	−0.593714
$843$	0	0
$844$	2.55437	0.0879252
$845$	0	0
$846$	0	0
$847$	−28.1176	−0.966131
$848$	7.51811	0.258173
$849$	0	0
$850$	0	0
$851$	3.76631	0.129108
$852$	0	0
$853$	−10.7971	−0.369684	−0.184842	−	0.982768i	$-0.559177\pi$
$853$	−10.7971	−0.369684	−0.184842	+	0.982768i	$0.559177\pi$
$854$	−25.7228	−0.880217
$855$	0	0
$856$	35.4456	1.21151
$857$	−49.3995	−1.68746	−0.843728	−	0.536772i	$-0.819644\pi$
$857$	−49.3995	−1.68746	−0.843728	+	0.536772i	$0.819644\pi$
$858$	0	0
$859$	48.3288	1.64896	0.824478	−	0.565893i	$-0.191469\pi$
$859$	48.3288	1.64896	0.824478	+	0.565893i	$0.191469\pi$
$860$	0	0
$861$	0	0
$862$	26.4232	0.899978
$863$	16.7306	0.569516	0.284758	−	0.958599i	$-0.408087\pi$
$863$	16.7306	0.569516	0.284758	+	0.958599i	$0.408087\pi$
$864$	0	0
$865$	0	0
$866$	10.1168	0.343784
$867$	0	0
$868$	30.2921	1.02818
$869$	29.4891	1.00035
$870$	0	0
$871$	68.2337	2.31201
$872$	−26.0357	−0.881679
$873$	0	0
$874$	−0.467376	−0.0158092
$875$	0	0
$876$	0	0
$877$	1.08724	0.0367135	0.0183568	−	0.999832i	$-0.494157\pi$
$877$	1.08724	0.0367135	0.0183568	+	0.999832i	$0.494157\pi$
$878$	25.2434	0.851923
$879$	0	0
$880$	0	0
$881$	−10.8832	−0.366663	−0.183331	−	0.983051i	$-0.558688\pi$
$881$	−10.8832	−0.366663	−0.183331	+	0.983051i	$0.558688\pi$
$882$	0	0
$883$	−54.9455	−1.84906	−0.924532	−	0.381104i	$-0.875544\pi$
$883$	−54.9455	−1.84906	−0.924532	+	0.381104i	$0.875544\pi$
$884$	6.35053	0.213592
$885$	0	0
$886$	17.0217	0.571857
$887$	43.8535	1.47246	0.736228	−	0.676733i	$-0.236605\pi$
$887$	43.8535	1.47246	0.736228	+	0.676733i	$0.236605\pi$
$888$	0	0
$889$	−36.0000	−1.20740
$890$	0	0
$891$	0	0
$892$	−25.5383	−0.855087
$893$	−0.699713	−0.0234150
$894$	0	0
$895$	0	0
$896$	31.2119	1.04272
$897$	0	0
$898$	8.62258	0.287739
$899$	36.6060	1.22088
$900$	0	0
$901$	9.48913	0.316129
$902$	−15.1460	−0.504308
$903$	0	0
$904$	−16.3940	−0.545257
$905$	0	0
$906$	0	0
$907$	18.2054	0.604499	0.302250	−	0.953229i	$-0.402262\pi$
$907$	18.2054	0.604499	0.302250	+	0.953229i	$0.402262\pi$
$908$	−18.6101	−0.617599
$909$	0	0
$910$	0	0
$911$	18.6060	0.616443	0.308222	−	0.951315i	$-0.400266\pi$
$911$	18.6060	0.616443	0.308222	+	0.951315i	$0.400266\pi$
$912$	0	0
$913$	52.3663	1.73307
$914$	−16.6277	−0.549996
$915$	0	0
$916$	24.1603	0.798280
$917$	15.1460	0.500166
$918$	0	0
$919$	55.3505	1.82585	0.912923	−	0.408132i	$-0.133820\pi$
$919$	55.3505	1.82585	0.912923	+	0.408132i	$0.133820\pi$
$920$	0	0
$921$	0	0
$922$	−19.0902	−0.628703
$923$	76.6150	2.52181
$924$	0	0
$925$	0	0
$926$	−10.9783	−0.360768
$927$	0	0
$928$	33.5538	1.10146
$929$	3.51087	0.115188	0.0575940	−	0.998340i	$-0.481657\pi$
$929$	3.51087	0.115188	0.0575940	+	0.998340i	$0.481657\pi$
$930$	0	0
$931$	−1.86141	−0.0610051
$932$	−8.42020	−0.275813
$933$	0	0
$934$	14.5109	0.474810
$935$	0	0
$936$	0	0
$937$	22.2766	0.727745	0.363873	−	0.931449i	$-0.381454\pi$
$937$	22.2766	0.727745	0.363873	+	0.931449i	$0.381454\pi$
$938$	32.0618	1.04685
$939$	0	0
$940$	0	0
$941$	6.86141	0.223675	0.111838	−	0.993726i	$-0.464326\pi$
$941$	6.86141	0.223675	0.111838	+	0.993726i	$0.464326\pi$
$942$	0	0
$943$	−6.92820	−0.225613
$944$	−1.02175	−0.0332551
$945$	0	0
$946$	12.0000	0.390154
$947$	−43.1538	−1.40231	−0.701155	−	0.713009i	$-0.747332\pi$
$947$	−43.1538	−1.40231	−0.701155	+	0.713009i	$0.747332\pi$
$948$	0	0
$949$	−13.8832	−0.450666
$950$	0	0
$951$	0	0
$952$	7.33296	0.237663
$953$	−25.0410	−0.811158	−0.405579	−	0.914060i	$-0.632930\pi$
$953$	−25.0410	−0.811158	−0.405579	+	0.914060i	$0.632930\pi$
$954$	0	0
$955$	0	0
$956$	31.5326	1.01984
$957$	0	0
$958$	−14.7413	−0.476269
$959$	49.7228	1.60563
$960$	0	0
$961$	9.60597	0.309870
$962$	−10.9994	−0.354636
$963$	0	0
$964$	2.07335	0.0667780
$965$	0	0
$966$	0	0
$967$	43.2636	1.39126	0.695632	−	0.718399i	$-0.255125\pi$
$967$	43.2636	1.39126	0.695632	+	0.718399i	$0.255125\pi$
$968$	−21.6867	−0.697037
$969$	0	0
$970$	0	0
$971$	−41.5842	−1.33450	−0.667251	−	0.744833i	$-0.732529\pi$
$971$	−41.5842	−1.33450	−0.667251	+	0.744833i	$0.732529\pi$
$972$	0	0
$973$	−38.5099	−1.23457
$974$	7.53262	0.241361
$975$	0	0
$976$	5.88316	0.188315
$977$	28.3027	0.905484	0.452742	−	0.891642i	$-0.350446\pi$
$977$	28.3027	0.905484	0.452742	+	0.891642i	$0.350446\pi$
$978$	0	0
$979$	13.1168	0.419216
$980$	0	0
$981$	0	0
$982$	25.0582	0.799640
$983$	−50.3770	−1.60678	−0.803388	−	0.595456i	$-0.796971\pi$
$983$	−50.3770	−1.60678	−0.803388	+	0.595456i	$0.796971\pi$
$984$	0	0
$985$	0	0
$986$	3.60597	0.114837
$987$	0	0
$988$	−2.98400	−0.0949337
$989$	5.48913	0.174544
$990$	0	0
$991$	38.6060	1.22636	0.613180	−	0.789944i	$-0.289890\pi$
$991$	38.6060	1.22636	0.613180	+	0.789944i	$0.289890\pi$
$992$	37.2203	1.18174
$993$	0	0
$994$	36.0000	1.14185
$995$	0	0
$996$	0	0
$997$	60.3817	1.91231	0.956154	−	0.292864i	$-0.0946082\pi$
$997$	60.3817	1.91231	0.956154	+	0.292864i	$0.0946082\pi$
$998$	−19.3098	−0.611242
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.a.x.1.3		4
3.2	odd	2		2025.2.a.w.1.2		4
5.2	odd	4		405.2.b.b.244.3	yes	4
5.3	odd	4		405.2.b.b.244.2	yes	4
5.4	even	2	inner	2025.2.a.x.1.2		4
15.2	even	4		405.2.b.a.244.2	✓	4
15.8	even	4		405.2.b.a.244.3	yes	4
15.14	odd	2		2025.2.a.w.1.3		4
45.2	even	12		405.2.j.e.109.1		4
45.7	odd	12		405.2.j.a.109.2		4
45.13	odd	12		405.2.j.a.379.2		4
45.22	odd	12		405.2.j.d.379.1		4
45.23	even	12		405.2.j.e.379.1		4
45.32	even	12		405.2.j.b.379.2		4
45.38	even	12		405.2.j.b.109.2		4
45.43	odd	12		405.2.j.d.109.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.b.a.244.2	✓	4	15.2	even	4
405.2.b.a.244.3	yes	4	15.8	even	4
405.2.b.b.244.2	yes	4	5.3	odd	4
405.2.b.b.244.3	yes	4	5.2	odd	4
405.2.j.a.109.2		4	45.7	odd	12
405.2.j.a.379.2		4	45.13	odd	12
405.2.j.b.109.2		4	45.38	even	12
405.2.j.b.379.2		4	45.32	even	12
405.2.j.d.109.1		4	45.43	odd	12
405.2.j.d.379.1		4	45.22	odd	12
405.2.j.e.109.1		4	45.2	even	12
405.2.j.e.379.1		4	45.23	even	12
2025.2.a.w.1.2		4	3.2	odd	2
2025.2.a.w.1.3		4	15.14	odd	2
2025.2.a.x.1.2		4	5.4	even	2	inner
2025.2.a.x.1.3		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.792287 q^{2} -1.37228 q^{4} -3.46410 q^{7} -2.67181 q^{8} +O(q^{10})\)	\(q+0.792287 q^{2} -1.37228 q^{4} -3.46410 q^{7} -2.67181 q^{8} +4.37228 q^{11} -5.84096 q^{13} -2.74456 q^{14} +0.627719 q^{16} +0.792287 q^{17} -0.372281 q^{19} +3.46410 q^{22} +1.58457 q^{23} -4.62772 q^{26} +4.75372 q^{28} +5.74456 q^{29} +6.37228 q^{31} +5.84096 q^{32} +0.627719 q^{34} +2.37686 q^{37} -0.294954 q^{38} -4.37228 q^{41} +3.46410 q^{43} -6.00000 q^{44} +1.25544 q^{46} +1.87953 q^{47} +5.00000 q^{49} +8.01544 q^{52} +11.9769 q^{53} +9.25544 q^{56} +4.55134 q^{58} -1.62772 q^{59} +9.37228 q^{61} +5.04868 q^{62} +3.37228 q^{64} -11.6819 q^{67} -1.08724 q^{68} -13.1168 q^{71} +2.37686 q^{73} +1.88316 q^{74} +0.510875 q^{76} -15.1460 q^{77} +6.74456 q^{79} -3.46410 q^{82} +11.9769 q^{83} +2.74456 q^{86} -11.6819 q^{88} +3.00000 q^{89} +20.2337 q^{91} -2.17448 q^{92} +1.48913 q^{94} +1.28962 q^{97} +3.96143 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4}+O(q^{10}) \) 4 * q + 6 * q^4	\( 4 q + 6 q^{4} + 6 q^{11} + 12 q^{14} + 14 q^{16} + 10 q^{19} - 30 q^{26} + 14 q^{31} + 14 q^{34} - 6 q^{41} - 24 q^{44} + 28 q^{46} + 20 q^{49} + 60 q^{56} - 18 q^{59} + 26 q^{61} + 2 q^{64} - 18 q^{71} + 42 q^{74} + 48 q^{76} + 4 q^{79} - 12 q^{86} + 12 q^{89} + 12 q^{91} - 40 q^{94}+O(q^{100}) \) 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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