LMFDB - Embedded newform 2025.2.a.y.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:	4.4.11661.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 3 $ x^4 - 2x^3 - 4x^2 + 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.473255$ 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)

$f(q)$	$=$	$q+1.47325 q^{2} +0.170479 q^{4} +3.86583 q^{7} -2.69535 q^{8} +O(q^{10})$	$q+1.47325 q^{2} +0.170479 q^{4} +3.86583 q^{7} -2.69535 q^{8} +0.260278 q^{11} -4.07881 q^{13} +5.69535 q^{14} -4.31189 q^{16} +3.26028 q^{17} +4.24928 q^{19} +0.383456 q^{22} +8.69535 q^{23} -6.00912 q^{26} +0.659042 q^{28} +4.22210 q^{29} +2.65285 q^{31} -0.961818 q^{32} +4.80322 q^{34} -2.27559 q^{37} +6.26028 q^{38} +5.64186 q^{41} -9.07262 q^{43} +0.0443719 q^{44} +12.8105 q^{46} -1.42888 q^{47} +7.94463 q^{49} -0.695350 q^{52} +11.3816 q^{53} -10.4198 q^{56} +6.22022 q^{58} -7.12423 q^{59} +2.52487 q^{61} +3.90833 q^{62} +7.20679 q^{64} +11.2856 q^{67} +0.555809 q^{68} -8.38158 q^{71} +0.403568 q^{73} -3.35252 q^{74} +0.724413 q^{76} +1.00619 q^{77} -3.04250 q^{79} +8.31189 q^{82} +4.58024 q^{83} -13.3663 q^{86} -0.701540 q^{88} +7.17772 q^{89} -15.7680 q^{91} +1.48237 q^{92} -2.10511 q^{94} +3.11511 q^{97} +11.7045 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{4} - q^{7} + 9 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 4 * q^4 - q^7 + 9 * q^8	$ 4 q + 2 q^{2} + 4 q^{4} - q^{7} + 9 q^{8} - q^{11} + 2 q^{13} + 3 q^{14} + 4 q^{16} + 11 q^{17} + 2 q^{19} + 3 q^{22} + 15 q^{23} - 10 q^{26} - 4 q^{28} + q^{29} - 4 q^{31} + 10 q^{32} + 9 q^{34} - q^{37} + 23 q^{38} - 5 q^{41} - 10 q^{43} + 22 q^{44} + 20 q^{47} - 3 q^{49} + 17 q^{52} + 20 q^{53} - 30 q^{56} - 18 q^{58} + 17 q^{59} - 13 q^{61} - 6 q^{62} + 19 q^{64} + 17 q^{67} + 34 q^{68} - 8 q^{71} + 2 q^{73} + 40 q^{74} + 11 q^{76} + 12 q^{77} - 7 q^{79} + 12 q^{82} + 30 q^{83} - 34 q^{86} + 9 q^{88} - 9 q^{89} - 17 q^{91} - 12 q^{92} + 3 q^{94} - 19 q^{97} + 13 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^4 - q^7 + 9 * q^8 - q^11 + 2 * q^13 + 3 * q^14 + 4 * q^16 + 11 * q^17 + 2 * q^19 + 3 * q^22 + 15 * q^23 - 10 * q^26 - 4 * q^28 + q^29 - 4 * q^31 + 10 * q^32 + 9 * q^34 - q^37 + 23 * q^38 - 5 * q^41 - 10 * q^43 + 22 * q^44 + 20 * q^47 - 3 * q^49 + 17 * q^52 + 20 * q^53 - 30 * q^56 - 18 * q^58 + 17 * q^59 - 13 * q^61 - 6 * q^62 + 19 * q^64 + 17 * q^67 + 34 * q^68 - 8 * q^71 + 2 * q^73 + 40 * q^74 + 11 * q^76 + 12 * q^77 - 7 * q^79 + 12 * q^82 + 30 * q^83 - 34 * q^86 + 9 * q^88 - 9 * q^89 - 17 * q^91 - 12 * q^92 + 3 * q^94 - 19 * q^97 + 13 * q^98

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 $

$2$	1.47325	1.04175	0.520874	−	0.853633i	$-0.325606\pi$
$2$	1.47325	1.04175	0.520874	+	0.853633i	$0.325606\pi$
$3$	0	0
$3$	0	0
$4$	0.170479	0.0852394
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.86583	1.46115	0.730573	−	0.682835i	$-0.239253\pi$
$7$	3.86583	1.46115	0.730573	+	0.682835i	$0.239253\pi$
$8$	−2.69535	−0.952950
$9$	0	0
$10$	0	0
$11$	0.260278	0.0784767	0.0392384	−	0.999230i	$-0.487507\pi$
$11$	0.260278	0.0784767	0.0392384	+	0.999230i	$0.487507\pi$
$12$	0	0
$13$	−4.07881	−1.13126	−0.565629	−	0.824660i	$-0.691366\pi$
$13$	−4.07881	−1.13126	−0.565629	+	0.824660i	$0.691366\pi$
$14$	5.69535	1.52215
$15$	0	0
$16$	−4.31189	−1.07797
$17$	3.26028	0.790734	0.395367	−	0.918523i	$-0.370617\pi$
$17$	3.26028	0.790734	0.395367	+	0.918523i	$0.370617\pi$
$18$	0	0
$19$	4.24928	0.974853	0.487426	−	0.873164i	$-0.337936\pi$
$19$	4.24928	0.974853	0.487426	+	0.873164i	$0.337936\pi$
$20$	0	0
$21$	0	0
$22$	0.383456	0.0817530
$23$	8.69535	1.81311	0.906553	−	0.422092i	$-0.138704\pi$
$23$	8.69535	1.81311	0.906553	+	0.422092i	$0.138704\pi$
$24$	0	0
$25$	0	0
$26$	−6.00912	−1.17849
$27$	0	0
$28$	0.659042	0.124547
$29$	4.22210	0.784023	0.392012	−	0.919960i	$-0.371779\pi$
$29$	4.22210	0.784023	0.392012	+	0.919960i	$0.371779\pi$
$30$	0	0
$31$	2.65285	0.476466	0.238233	−	0.971208i	$-0.423432\pi$
$31$	2.65285	0.476466	0.238233	+	0.971208i	$0.423432\pi$
$32$	−0.961818	−0.170027
$33$	0	0
$34$	4.80322	0.823745
$35$	0	0
$36$	0	0
$37$	−2.27559	−0.374104	−0.187052	−	0.982350i	$-0.559893\pi$
$37$	−2.27559	−0.374104	−0.187052	+	0.982350i	$0.559893\pi$
$38$	6.26028	1.01555
$39$	0	0
$40$	0	0
$41$	5.64186	0.881110	0.440555	−	0.897726i	$-0.354782\pi$
$41$	5.64186	0.881110	0.440555	+	0.897726i	$0.354782\pi$
$42$	0	0
$43$	−9.07262	−1.38356	−0.691780	−	0.722108i	$-0.743173\pi$
$43$	−9.07262	−1.38356	−0.691780	+	0.722108i	$0.743173\pi$
$44$	0.0443719	0.00668931
$45$	0	0
$46$	12.8105	1.88880
$47$	−1.42888	−0.208424	−0.104212	−	0.994555i	$-0.533232\pi$
$47$	−1.42888	−0.208424	−0.104212	+	0.994555i	$0.533232\pi$
$48$	0	0
$49$	7.94463	1.13495
$50$	0	0
$51$	0	0
$52$	−0.695350	−0.0964277
$53$	11.3816	1.56338	0.781690	−	0.623667i	$-0.214358\pi$
$53$	11.3816	1.56338	0.781690	+	0.623667i	$0.214358\pi$
$54$	0	0
$55$	0	0
$56$	−10.4198	−1.39240
$57$	0	0
$58$	6.22022	0.816755
$59$	−7.12423	−0.927496	−0.463748	−	0.885967i	$-0.653496\pi$
$59$	−7.12423	−0.927496	−0.463748	+	0.885967i	$0.653496\pi$
$60$	0	0
$61$	2.52487	0.323277	0.161638	−	0.986850i	$-0.448322\pi$
$61$	2.52487	0.323277	0.161638	+	0.986850i	$0.448322\pi$
$62$	3.90833	0.496358
$63$	0	0
$64$	7.20679	0.900848
$65$	0	0
$66$	0	0
$67$	11.2856	1.37875	0.689377	−	0.724402i	$-0.257884\pi$
$67$	11.2856	1.37875	0.689377	+	0.724402i	$0.257884\pi$
$68$	0.555809	0.0674017
$69$	0	0
$70$	0	0
$71$	−8.38158	−0.994711	−0.497355	−	0.867547i	$-0.665695\pi$
$71$	−8.38158	−0.994711	−0.497355	+	0.867547i	$0.665695\pi$
$72$	0	0
$73$	0.403568	0.0472340	0.0236170	−	0.999721i	$-0.492482\pi$
$73$	0.403568	0.0472340	0.0236170	+	0.999721i	$0.492482\pi$
$74$	−3.35252	−0.389722
$75$	0	0
$76$	0.724413	0.0830959
$77$	1.00619	0.114666
$78$	0	0
$79$	−3.04250	−0.342308	−0.171154	−	0.985244i	$-0.554750\pi$
$79$	−3.04250	−0.342308	−0.171154	+	0.985244i	$0.554750\pi$
$80$	0	0
$81$	0	0
$82$	8.31189	0.917895
$83$	4.58024	0.502746	0.251373	−	0.967890i	$-0.419118\pi$
$83$	4.58024	0.502746	0.251373	+	0.967890i	$0.419118\pi$
$84$	0	0
$85$	0	0
$86$	−13.3663	−1.44132
$87$	0	0
$88$	−0.701540	−0.0747844
$89$	7.17772	0.760837	0.380419	−	0.924814i	$-0.375780\pi$
$89$	7.17772	0.760837	0.380419	+	0.924814i	$0.375780\pi$
$90$	0	0
$91$	−15.7680	−1.65293
$92$	1.48237	0.154548
$93$	0	0
$94$	−2.10511	−0.217125
$95$	0	0
$96$	0	0
$97$	3.11511	0.316292	0.158146	−	0.987416i	$-0.449448\pi$
$97$	3.11511	0.316292	0.158146	+	0.987416i	$0.449448\pi$
$98$	11.7045	1.18233
$99$	0	0
$100$	0	0
$101$	3.84572	0.382663	0.191332	−	0.981525i	$-0.438719\pi$
$101$	3.84572	0.382663	0.191332	+	0.981525i	$0.438719\pi$
$102$	0	0
$103$	−4.20679	−0.414507	−0.207254	−	0.978287i	$-0.566452\pi$
$103$	−4.20679	−0.414507	−0.207254	+	0.978287i	$0.566452\pi$
$104$	10.9938	1.07803
$105$	0	0
$106$	16.7680	1.62865
$107$	−1.62655	−0.157245	−0.0786223	−	0.996904i	$-0.525052\pi$
$107$	−1.62655	−0.157245	−0.0786223	+	0.996904i	$0.525052\pi$
$108$	0	0
$109$	−12.9021	−1.23580	−0.617900	−	0.786256i	$-0.712016\pi$
$109$	−12.9021	−1.23580	−0.617900	+	0.786256i	$0.712016\pi$
$110$	0	0
$111$	0	0
$112$	−16.6690	−1.57508
$113$	1.32908	0.125029	0.0625146	−	0.998044i	$-0.480088\pi$
$113$	1.32908	0.125029	0.0625146	+	0.998044i	$0.480088\pi$
$114$	0	0
$115$	0	0
$116$	0.719778	0.0668297
$117$	0	0
$118$	−10.4958	−0.966217
$119$	12.6037	1.15538
$120$	0	0
$121$	−10.9323	−0.993841
$122$	3.71978	0.336773
$123$	0	0
$124$	0.452255	0.0406137
$125$	0	0
$126$	0	0
$127$	−1.65285	−0.146667	−0.0733335	−	0.997307i	$-0.523364\pi$
$127$	−1.65285	−0.146667	−0.0733335	+	0.997307i	$0.523364\pi$
$128$	12.5411	1.10848
$129$	0	0
$130$	0	0
$131$	−13.1777	−1.15134	−0.575672	−	0.817681i	$-0.695259\pi$
$131$	−13.1777	−1.15134	−0.575672	+	0.817681i	$0.695259\pi$
$132$	0	0
$133$	16.4270	1.42440
$134$	16.6266	1.43632
$135$	0	0
$136$	−8.78759	−0.753530
$137$	20.2928	1.73373	0.866867	−	0.498539i	$-0.166130\pi$
$137$	20.2928	1.73373	0.866867	+	0.498539i	$0.166130\pi$
$138$	0	0
$139$	3.06880	0.260292	0.130146	−	0.991495i	$-0.458455\pi$
$139$	3.06880	0.260292	0.130146	+	0.991495i	$0.458455\pi$
$140$	0	0
$141$	0	0
$142$	−12.3482	−1.03624
$143$	−1.06162	−0.0887774
$144$	0	0
$145$	0	0
$146$	0.594558	0.0492060
$147$	0	0
$148$	−0.387939	−0.0318884
$149$	4.06162	0.332741	0.166371	−	0.986063i	$-0.446795\pi$
$149$	4.06162	0.332741	0.166371	+	0.986063i	$0.446795\pi$
$150$	0	0
$151$	−13.6199	−1.10837	−0.554185	−	0.832394i	$-0.686970\pi$
$151$	−13.6199	−1.10837	−0.554185	+	0.832394i	$0.686970\pi$
$152$	−11.4533	−0.928986
$153$	0	0
$154$	1.48237	0.119453
$155$	0	0
$156$	0	0
$157$	2.06261	0.164614	0.0823071	−	0.996607i	$-0.473771\pi$
$157$	2.06261	0.164614	0.0823071	+	0.996607i	$0.473771\pi$
$158$	−4.48237	−0.356598
$159$	0	0
$160$	0	0
$161$	33.6147	2.64921
$162$	0	0
$163$	−3.50525	−0.274552	−0.137276	−	0.990533i	$-0.543835\pi$
$163$	−3.50525	−0.274552	−0.137276	+	0.990533i	$0.543835\pi$
$164$	0.961818	0.0751054
$165$	0	0
$166$	6.74785	0.523735
$167$	20.5349	1.58904	0.794518	−	0.607240i	$-0.207723\pi$
$167$	20.5349	1.58904	0.794518	+	0.607240i	$0.207723\pi$
$168$	0	0
$169$	3.63666	0.279743
$170$	0	0
$171$	0	0
$172$	−1.54669	−0.117934
$173$	−7.75177	−0.589356	−0.294678	−	0.955597i	$-0.595212\pi$
$173$	−7.75177	−0.589356	−0.294678	+	0.955597i	$0.595212\pi$
$174$	0	0
$175$	0	0
$176$	−1.12229	−0.0845958
$177$	0	0
$178$	10.5746	0.792601
$179$	−10.7632	−0.804477	−0.402238	−	0.915535i	$-0.631768\pi$
$179$	−10.7632	−0.804477	−0.402238	+	0.915535i	$0.631768\pi$
$180$	0	0
$181$	−7.84572	−0.583168	−0.291584	−	0.956545i	$-0.594182\pi$
$181$	−7.84572	−0.583168	−0.291584	+	0.956545i	$0.594182\pi$
$182$	−23.2302	−1.72194
$183$	0	0
$184$	−23.4370	−1.72780
$185$	0	0
$186$	0	0
$187$	0.848578	0.0620542
$188$	−0.243594	−0.0177659
$189$	0	0
$190$	0	0
$191$	−5.73255	−0.414792	−0.207396	−	0.978257i	$-0.566499\pi$
$191$	−5.73255	−0.414792	−0.207396	+	0.978257i	$0.566499\pi$
$192$	0	0
$193$	−8.48237	−0.610575	−0.305287	−	0.952260i	$-0.598752\pi$
$193$	−8.48237	−0.610575	−0.305287	+	0.952260i	$0.598752\pi$
$194$	4.58936	0.329497
$195$	0	0
$196$	1.35439	0.0967423
$197$	10.6266	0.757110	0.378555	−	0.925579i	$-0.376421\pi$
$197$	10.6266	0.757110	0.378555	+	0.925579i	$0.376421\pi$
$198$	0	0
$199$	−18.5784	−1.31699	−0.658495	−	0.752585i	$-0.728807\pi$
$199$	−18.5784	−1.31699	−0.658495	+	0.752585i	$0.728807\pi$
$200$	0	0
$201$	0	0
$202$	5.66572	0.398639
$203$	16.3219	1.14557
$204$	0	0
$205$	0	0
$206$	−6.19767	−0.431812
$207$	0	0
$208$	17.5874	1.21947
$209$	1.10599	0.0765033
$210$	0	0
$211$	10.4533	0.719636	0.359818	−	0.933023i	$-0.382839\pi$
$211$	10.4533	0.719636	0.359818	+	0.933023i	$0.382839\pi$
$212$	1.94032	0.133262
$213$	0	0
$214$	−2.39632	−0.163809
$215$	0	0
$216$	0	0
$217$	10.2555	0.696187
$218$	−19.0081	−1.28739
$219$	0	0
$220$	0	0
$221$	−13.2980	−0.894523
$222$	0	0
$223$	3.93120	0.263253	0.131626	−	0.991299i	$-0.457980\pi$
$223$	3.93120	0.263253	0.131626	+	0.991299i	$0.457980\pi$
$224$	−3.71822	−0.248434
$225$	0	0
$226$	1.95807	0.130249
$227$	4.83140	0.320671	0.160335	−	0.987063i	$-0.448742\pi$
$227$	4.83140	0.320671	0.160335	+	0.987063i	$0.448742\pi$
$228$	0	0
$229$	−18.8530	−1.24584	−0.622919	−	0.782286i	$-0.714054\pi$
$229$	−18.8530	−1.24584	−0.622919	+	0.782286i	$0.714054\pi$
$230$	0	0
$231$	0	0
$232$	−11.3800	−0.747135
$233$	−11.9021	−0.779735	−0.389867	−	0.920871i	$-0.627479\pi$
$233$	−11.9021	−0.779735	−0.389867	+	0.920871i	$0.627479\pi$
$234$	0	0
$235$	0	0
$236$	−1.21453	−0.0790593
$237$	0	0
$238$	18.5684	1.20361
$239$	−21.6295	−1.39909	−0.699547	−	0.714586i	$-0.746615\pi$
$239$	−21.6295	−1.39909	−0.699547	+	0.714586i	$0.746615\pi$
$240$	0	0
$241$	3.89832	0.251113	0.125556	−	0.992086i	$-0.459928\pi$
$241$	3.89832	0.251113	0.125556	+	0.992086i	$0.459928\pi$
$242$	−16.1060	−1.03533
$243$	0	0
$244$	0.430437	0.0275559
$245$	0	0
$246$	0	0
$247$	−17.3320	−1.10281
$248$	−7.15037	−0.454049
$249$	0	0
$250$	0	0
$251$	−30.1033	−1.90010	−0.950052	−	0.312092i	$-0.898970\pi$
$251$	−30.1033	−1.90010	−0.950052	+	0.312092i	$0.898970\pi$
$252$	0	0
$253$	2.26321	0.142287
$254$	−2.43507	−0.152790
$255$	0	0
$256$	4.06261	0.253913
$257$	−16.4141	−1.02389	−0.511943	−	0.859019i	$-0.671074\pi$
$257$	−16.4141	−1.02389	−0.511943	+	0.859019i	$0.671074\pi$
$258$	0	0
$259$	−8.79703	−0.546621
$260$	0	0
$261$	0	0
$262$	−19.4141	−1.19941
$263$	25.8072	1.59134	0.795670	−	0.605730i	$-0.207119\pi$
$263$	25.8072	1.59134	0.795670	+	0.605730i	$0.207119\pi$
$264$	0	0
$265$	0	0
$266$	24.2012	1.48387
$267$	0	0
$268$	1.92396	0.117524
$269$	−12.5206	−0.763392	−0.381696	−	0.924288i	$-0.624660\pi$
$269$	−12.5206	−0.763392	−0.381696	+	0.924288i	$0.624660\pi$
$270$	0	0
$271$	19.6462	1.19342	0.596710	−	0.802457i	$-0.296474\pi$
$271$	19.6462	1.19342	0.596710	+	0.802457i	$0.296474\pi$
$272$	−14.0580	−0.852390
$273$	0	0
$274$	29.8965	1.80611
$275$	0	0
$276$	0	0
$277$	−20.8301	−1.25156	−0.625779	−	0.780000i	$-0.715219\pi$
$277$	−20.8301	−1.25156	−0.625779	+	0.780000i	$0.715219\pi$
$278$	4.52112	0.271159
$279$	0	0
$280$	0	0
$281$	4.72441	0.281835	0.140917	−	0.990021i	$-0.454995\pi$
$281$	4.72441	0.281835	0.140917	+	0.990021i	$0.454995\pi$
$282$	0	0
$283$	23.1525	1.37627	0.688136	−	0.725582i	$-0.258429\pi$
$283$	23.1525	1.37627	0.688136	+	0.725582i	$0.258429\pi$
$284$	−1.42888	−0.0847886
$285$	0	0
$286$	−1.56404	−0.0924837
$287$	21.8105	1.28743
$288$	0	0
$289$	−6.37059	−0.374740
$290$	0	0
$291$	0	0
$292$	0.0687998	0.00402620
$293$	−16.8793	−0.986097	−0.493049	−	0.870002i	$-0.664118\pi$
$293$	−16.8793	−0.986097	−0.493049	+	0.870002i	$0.664118\pi$
$294$	0	0
$295$	0	0
$296$	6.13350	0.356503
$297$	0	0
$298$	5.98380	0.346632
$299$	−35.4666	−2.05109
$300$	0	0
$301$	−35.0732	−2.02158
$302$	−20.0655	−1.15464
$303$	0	0
$304$	−18.3225	−1.05087
$305$	0	0
$306$	0	0
$307$	−22.7177	−1.29657	−0.648285	−	0.761398i	$-0.724513\pi$
$307$	−22.7177	−1.29657	−0.648285	+	0.761398i	$0.724513\pi$
$308$	0.171534	0.00977406
$309$	0	0
$310$	0	0
$311$	31.5936	1.79151	0.895754	−	0.444551i	$-0.146637\pi$
$311$	31.5936	1.79151	0.895754	+	0.444551i	$0.146637\pi$
$312$	0	0
$313$	−30.4996	−1.72394	−0.861970	−	0.506959i	$-0.830770\pi$
$313$	−30.4996	−1.72394	−0.861970	+	0.506959i	$0.830770\pi$
$314$	3.03875	0.171487
$315$	0	0
$316$	−0.518682	−0.0291781
$317$	22.0890	1.24064	0.620320	−	0.784349i	$-0.287003\pi$
$317$	22.0890	1.24064	0.620320	+	0.784349i	$0.287003\pi$
$318$	0	0
$319$	1.09892	0.0615276
$320$	0	0
$321$	0	0
$322$	49.5231	2.75981
$323$	13.8538	0.770849
$324$	0	0
$325$	0	0
$326$	−5.16412	−0.286014
$327$	0	0
$328$	−15.2068	−0.839654
$329$	−5.52382	−0.304538
$330$	0	0
$331$	−29.6047	−1.62722	−0.813612	−	0.581409i	$-0.802502\pi$
$331$	−29.6047	−1.62722	−0.813612	+	0.581409i	$0.802502\pi$
$332$	0.780834	0.0428538
$333$	0	0
$334$	30.2531	1.65538
$335$	0	0
$336$	0	0
$337$	12.5311	0.682610	0.341305	−	0.939953i	$-0.389131\pi$
$337$	12.5311	0.682610	0.341305	+	0.939953i	$0.389131\pi$
$338$	5.35772	0.291422
$339$	0	0
$340$	0	0
$341$	0.690479	0.0373915
$342$	0	0
$343$	3.65180	0.197179
$344$	24.4539	1.31846
$345$	0	0
$346$	−11.4203	−0.613961
$347$	−17.0974	−0.917839	−0.458919	−	0.888478i	$-0.651763\pi$
$347$	−17.0974	−0.917839	−0.458919	+	0.888478i	$0.651763\pi$
$348$	0	0
$349$	−18.4046	−0.985177	−0.492588	−	0.870262i	$-0.663949\pi$
$349$	−18.4046	−0.985177	−0.492588	+	0.870262i	$0.663949\pi$
$350$	0	0
$351$	0	0
$352$	−0.250340	−0.0133432
$353$	−31.7188	−1.68822	−0.844110	−	0.536169i	$-0.819871\pi$
$353$	−31.7188	−1.68822	−0.844110	+	0.536169i	$0.819871\pi$
$354$	0	0
$355$	0	0
$356$	1.22365	0.0648533
$357$	0	0
$358$	−15.8569	−0.838062
$359$	−11.4533	−0.604483	−0.302241	−	0.953231i	$-0.597735\pi$
$359$	−11.4533	−0.604483	−0.302241	+	0.953231i	$0.597735\pi$
$360$	0	0
$361$	−0.943580	−0.0496621
$362$	−11.5587	−0.607514
$363$	0	0
$364$	−2.68811	−0.140895
$365$	0	0
$366$	0	0
$367$	2.49238	0.130101	0.0650506	−	0.997882i	$-0.479279\pi$
$367$	2.49238	0.130101	0.0650506	+	0.997882i	$0.479279\pi$
$368$	−37.4934	−1.95448
$369$	0	0
$370$	0	0
$371$	43.9992	2.28433
$372$	0	0
$373$	−15.0374	−0.778605	−0.389303	−	0.921110i	$-0.627284\pi$
$373$	−15.0374	−0.778605	−0.389303	+	0.921110i	$0.627284\pi$
$374$	1.25017	0.0646448
$375$	0	0
$376$	3.85134	0.198618
$377$	−17.2211	−0.886932
$378$	0	0
$379$	6.27273	0.322208	0.161104	−	0.986937i	$-0.448495\pi$
$379$	6.27273	0.322208	0.161104	+	0.986937i	$0.448495\pi$
$380$	0	0
$381$	0	0
$382$	−8.44550	−0.432109
$383$	−22.1888	−1.13379	−0.566897	−	0.823789i	$-0.691856\pi$
$383$	−22.1888	−1.13379	−0.566897	+	0.823789i	$0.691856\pi$
$384$	0	0
$385$	0	0
$386$	−12.4967	−0.636065
$387$	0	0
$388$	0.531061	0.0269605
$389$	−30.0922	−1.52574	−0.762869	−	0.646554i	$-0.776210\pi$
$389$	−30.0922	−1.52574	−0.762869	+	0.646554i	$0.776210\pi$
$390$	0	0
$391$	28.3493	1.43368
$392$	−21.4136	−1.08155
$393$	0	0
$394$	15.6556	0.788718
$395$	0	0
$396$	0	0
$397$	29.2313	1.46708	0.733538	−	0.679648i	$-0.237868\pi$
$397$	29.2313	1.46708	0.733538	+	0.679648i	$0.237868\pi$
$398$	−27.3708	−1.37197
$399$	0	0
$400$	0	0
$401$	24.2341	1.21020	0.605098	−	0.796151i	$-0.293134\pi$
$401$	24.2341	1.21020	0.605098	+	0.796151i	$0.293134\pi$
$402$	0	0
$403$	−10.8205	−0.539006
$404$	0.655614	0.0326180
$405$	0	0
$406$	24.0463	1.19340
$407$	−0.592285	−0.0293585
$408$	0	0
$409$	2.33990	0.115701	0.0578504	−	0.998325i	$-0.481575\pi$
$409$	2.33990	0.115701	0.0578504	+	0.998325i	$0.481575\pi$
$410$	0	0
$411$	0	0
$412$	−0.717168	−0.0353323
$413$	−27.5411	−1.35521
$414$	0	0
$415$	0	0
$416$	3.92307	0.192344
$417$	0	0
$418$	1.62941	0.0796971
$419$	22.8425	1.11593	0.557964	−	0.829865i	$-0.311583\pi$
$419$	22.8425	1.11593	0.557964	+	0.829865i	$0.311583\pi$
$420$	0	0
$421$	11.8758	0.578793	0.289396	−	0.957209i	$-0.406545\pi$
$421$	11.8758	0.578793	0.289396	+	0.957209i	$0.406545\pi$
$422$	15.4004	0.749679
$423$	0	0
$424$	−30.6773	−1.48982
$425$	0	0
$426$	0	0
$427$	9.76072	0.472354
$428$	−0.277293	−0.0134034
$429$	0	0
$430$	0	0
$431$	8.86916	0.427212	0.213606	−	0.976920i	$-0.431479\pi$
$431$	8.86916	0.427212	0.213606	+	0.976920i	$0.431479\pi$
$432$	0	0
$433$	9.37059	0.450322	0.225161	−	0.974322i	$-0.427709\pi$
$433$	9.37059	0.450322	0.225161	+	0.974322i	$0.427709\pi$
$434$	15.1089	0.725252
$435$	0	0
$436$	−2.19954	−0.105339
$437$	36.9490	1.76751
$438$	0	0
$439$	19.4231	0.927014	0.463507	−	0.886093i	$-0.346591\pi$
$439$	19.4231	0.927014	0.463507	+	0.886093i	$0.346591\pi$
$440$	0	0
$441$	0	0
$442$	−19.5914	−0.931868
$443$	10.8793	0.516889	0.258445	−	0.966026i	$-0.416790\pi$
$443$	10.8793	0.516889	0.258445	+	0.966026i	$0.416790\pi$
$444$	0	0
$445$	0	0
$446$	5.79166	0.274243
$447$	0	0
$448$	27.8602	1.31627
$449$	1.34014	0.0632451	0.0316225	−	0.999500i	$-0.489933\pi$
$449$	1.34014	0.0632451	0.0316225	+	0.999500i	$0.489933\pi$
$450$	0	0
$451$	1.46845	0.0691467
$452$	0.226580	0.0106574
$453$	0	0
$454$	7.11787	0.334058
$455$	0	0
$456$	0	0
$457$	20.1113	0.940767	0.470383	−	0.882462i	$-0.344116\pi$
$457$	20.1113	0.940767	0.470383	+	0.882462i	$0.344116\pi$
$458$	−27.7752	−1.29785
$459$	0	0
$460$	0	0
$461$	−33.7533	−1.57205	−0.786024	−	0.618196i	$-0.787864\pi$
$461$	−33.7533	−1.57205	−0.786024	+	0.618196i	$0.787864\pi$
$462$	0	0
$463$	−5.24537	−0.243773	−0.121886	−	0.992544i	$-0.538894\pi$
$463$	−5.24537	−0.243773	−0.121886	+	0.992544i	$0.538894\pi$
$464$	−18.2052	−0.845157
$465$	0	0
$466$	−17.5349	−0.812288
$467$	14.2120	0.657652	0.328826	−	0.944390i	$-0.393347\pi$
$467$	14.2120	0.657652	0.328826	+	0.944390i	$0.393347\pi$
$468$	0	0
$469$	43.6282	2.01456
$470$	0	0
$471$	0	0
$472$	19.2023	0.883858
$473$	−2.36140	−0.108577
$474$	0	0
$475$	0	0
$476$	2.14866	0.0984837
$477$	0	0
$478$	−31.8657	−1.45750
$479$	20.4833	0.935907	0.467954	−	0.883753i	$-0.344991\pi$
$479$	20.4833	0.935907	0.467954	+	0.883753i	$0.344991\pi$
$480$	0	0
$481$	9.28168	0.423208
$482$	5.74322	0.261596
$483$	0	0
$484$	−1.86372	−0.0847145
$485$	0	0
$486$	0	0
$487$	31.3554	1.42085	0.710425	−	0.703772i	$-0.248503\pi$
$487$	31.3554	1.42085	0.710425	+	0.703772i	$0.248503\pi$
$488$	−6.80541	−0.308067
$489$	0	0
$490$	0	0
$491$	−10.3921	−0.468988	−0.234494	−	0.972118i	$-0.575343\pi$
$491$	−10.3921	−0.468988	−0.234494	+	0.972118i	$0.575343\pi$
$492$	0	0
$493$	13.7652	0.619954
$494$	−25.5345	−1.14885
$495$	0	0
$496$	−11.4388	−0.513618
$497$	−32.4018	−1.45342
$498$	0	0
$499$	−3.82570	−0.171262	−0.0856310	−	0.996327i	$-0.527291\pi$
$499$	−3.82570	−0.171262	−0.0856310	+	0.996327i	$0.527291\pi$
$500$	0	0
$501$	0	0
$502$	−44.3498	−1.97943
$503$	1.00236	0.0446931	0.0223466	−	0.999750i	$-0.492886\pi$
$503$	1.00236	0.0446931	0.0223466	+	0.999750i	$0.492886\pi$
$504$	0	0
$505$	0	0
$506$	3.33428	0.148227
$507$	0	0
$508$	−0.281776	−0.0125018
$509$	4.56322	0.202261	0.101131	−	0.994873i	$-0.467754\pi$
$509$	4.56322	0.202261	0.101131	+	0.994873i	$0.467754\pi$
$510$	0	0
$511$	1.56012	0.0690158
$512$	−19.0969	−0.843971
$513$	0	0
$514$	−24.1822	−1.06663
$515$	0	0
$516$	0	0
$517$	−0.371907	−0.0163564
$518$	−12.9603	−0.569441
$519$	0	0
$520$	0	0
$521$	39.3708	1.72486	0.862432	−	0.506173i	$-0.168940\pi$
$521$	39.3708	1.72486	0.862432	+	0.506173i	$0.168940\pi$
$522$	0	0
$523$	−10.3998	−0.454749	−0.227375	−	0.973807i	$-0.573014\pi$
$523$	−10.3998	−0.454749	−0.227375	+	0.973807i	$0.573014\pi$
$524$	−2.24652	−0.0981398
$525$	0	0
$526$	38.0206	1.65778
$527$	8.64904	0.376758
$528$	0	0
$529$	52.6091	2.28735
$530$	0	0
$531$	0	0
$532$	2.80046	0.121415
$533$	−23.0120	−0.996762
$534$	0	0
$535$	0	0
$536$	−30.4186	−1.31388
$537$	0	0
$538$	−18.4460	−0.795262
$539$	2.06781	0.0890670
$540$	0	0
$541$	13.7093	0.589408	0.294704	−	0.955589i	$-0.404779\pi$
$541$	13.7093	0.589408	0.294704	+	0.955589i	$0.404779\pi$
$542$	28.9438	1.24324
$543$	0	0
$544$	−3.13579	−0.134446
$545$	0	0
$546$	0	0
$547$	22.7847	0.974205	0.487102	−	0.873345i	$-0.338054\pi$
$547$	22.7847	0.974205	0.487102	+	0.873345i	$0.338054\pi$
$548$	3.45950	0.147783
$549$	0	0
$550$	0	0
$551$	17.9409	0.764307
$552$	0	0
$553$	−11.7618	−0.500162
$554$	−30.6880	−1.30381
$555$	0	0
$556$	0.523166	0.0221872
$557$	−18.2341	−0.772605	−0.386303	−	0.922372i	$-0.626248\pi$
$557$	−18.2341	−0.772605	−0.386303	+	0.922372i	$0.626248\pi$
$558$	0	0
$559$	37.0054	1.56516
$560$	0	0
$561$	0	0
$562$	6.96026	0.293601
$563$	24.1112	1.01617	0.508083	−	0.861308i	$-0.330354\pi$
$563$	24.1112	1.01617	0.508083	+	0.861308i	$0.330354\pi$
$564$	0	0
$565$	0	0
$566$	34.1095	1.43373
$567$	0	0
$568$	22.5913	0.947910
$569$	−32.0049	−1.34171	−0.670857	−	0.741587i	$-0.734074\pi$
$569$	−32.0049	−1.34171	−0.670857	+	0.741587i	$0.734074\pi$
$570$	0	0
$571$	−19.7808	−0.827802	−0.413901	−	0.910322i	$-0.635834\pi$
$571$	−19.7808	−0.827802	−0.413901	+	0.910322i	$0.635834\pi$
$572$	−0.180984	−0.00756733
$573$	0	0
$574$	32.1324	1.34118
$575$	0	0
$576$	0	0
$577$	−35.4119	−1.47422	−0.737108	−	0.675775i	$-0.763809\pi$
$577$	−35.4119	−1.47422	−0.737108	+	0.675775i	$0.763809\pi$
$578$	−9.38550	−0.390385
$579$	0	0
$580$	0	0
$581$	17.7064	0.734586
$582$	0	0
$583$	2.96237	0.122689
$584$	−1.08776	−0.0450117
$585$	0	0
$586$	−24.8675	−1.02726
$587$	−32.3851	−1.33668	−0.668338	−	0.743858i	$-0.732994\pi$
$587$	−32.3851	−1.33668	−0.668338	+	0.743858i	$0.732994\pi$
$588$	0	0
$589$	11.2727	0.464485
$590$	0	0
$591$	0	0
$592$	9.81209	0.403274
$593$	29.2504	1.20117	0.600585	−	0.799561i	$-0.294934\pi$
$593$	29.2504	1.20117	0.600585	+	0.799561i	$0.294934\pi$
$594$	0	0
$595$	0	0
$596$	0.692421	0.0283627
$597$	0	0
$598$	−52.2514	−2.13672
$599$	4.06162	0.165953	0.0829767	−	0.996551i	$-0.473557\pi$
$599$	4.06162	0.165953	0.0829767	+	0.996551i	$0.473557\pi$
$600$	0	0
$601$	46.9076	1.91340	0.956700	−	0.291076i	$-0.0940132\pi$
$601$	46.9076	1.91340	0.956700	+	0.291076i	$0.0940132\pi$
$602$	−51.6717	−2.10598
$603$	0	0
$604$	−2.32190	−0.0944768
$605$	0	0
$606$	0	0
$607$	−40.9466	−1.66197	−0.830987	−	0.556293i	$-0.812223\pi$
$607$	−40.9466	−1.66197	−0.830987	+	0.556293i	$0.812223\pi$
$608$	−4.08704	−0.165751
$609$	0	0
$610$	0	0
$611$	5.82813	0.235781
$612$	0	0
$613$	−33.3827	−1.34831	−0.674157	−	0.738588i	$-0.735493\pi$
$613$	−33.3827	−1.34831	−0.674157	+	0.738588i	$0.735493\pi$
$614$	−33.4690	−1.35070
$615$	0	0
$616$	−2.71203	−0.109271
$617$	2.24880	0.0905332	0.0452666	−	0.998975i	$-0.485586\pi$
$617$	2.24880	0.0905332	0.0452666	+	0.998975i	$0.485586\pi$
$618$	0	0
$619$	−34.2934	−1.37837	−0.689184	−	0.724586i	$-0.742031\pi$
$619$	−34.2934	−1.37837	−0.689184	+	0.724586i	$0.742031\pi$
$620$	0	0
$621$	0	0
$622$	46.5454	1.86630
$623$	27.7479	1.11169
$624$	0	0
$625$	0	0
$626$	−44.9337	−1.79591
$627$	0	0
$628$	0.351631	0.0140316
$629$	−7.41904	−0.295817
$630$	0	0
$631$	−18.7552	−0.746633	−0.373316	−	0.927704i	$-0.621779\pi$
$631$	−18.7552	−0.746633	−0.373316	+	0.927704i	$0.621779\pi$
$632$	8.20060	0.326202
$633$	0	0
$634$	32.5427	1.29244
$635$	0	0
$636$	0	0
$637$	−32.4046	−1.28392
$638$	1.61899	0.0640963
$639$	0	0
$640$	0	0
$641$	20.6350	0.815034	0.407517	−	0.913198i	$-0.366395\pi$
$641$	20.6350	0.815034	0.407517	+	0.913198i	$0.366395\pi$
$642$	0	0
$643$	27.1939	1.07242	0.536212	−	0.844084i	$-0.319855\pi$
$643$	27.1939	1.07242	0.536212	+	0.844084i	$0.319855\pi$
$644$	5.73060	0.225817
$645$	0	0
$646$	20.4102	0.803030
$647$	16.7316	0.657787	0.328893	−	0.944367i	$-0.393324\pi$
$647$	16.7316	0.657787	0.328893	+	0.944367i	$0.393324\pi$
$648$	0	0
$649$	−1.85428	−0.0727869
$650$	0	0
$651$	0	0
$652$	−0.597571	−0.0234027
$653$	−45.7331	−1.78967	−0.894837	−	0.446392i	$-0.852709\pi$
$653$	−45.7331	−1.78967	−0.894837	+	0.446392i	$0.852709\pi$
$654$	0	0
$655$	0	0
$656$	−24.3271	−0.949814
$657$	0	0
$658$	−8.13799	−0.317252
$659$	18.6109	0.724976	0.362488	−	0.931988i	$-0.381927\pi$
$659$	18.6109	0.724976	0.362488	+	0.931988i	$0.381927\pi$
$660$	0	0
$661$	16.7960	0.653288	0.326644	−	0.945148i	$-0.394082\pi$
$661$	16.7960	0.653288	0.326644	+	0.945148i	$0.394082\pi$
$662$	−43.6153	−1.69516
$663$	0	0
$664$	−12.3453	−0.479092
$665$	0	0
$666$	0	0
$667$	36.7126	1.42152
$668$	3.50076	0.135449
$669$	0	0
$670$	0	0
$671$	0.657168	0.0253697
$672$	0	0
$673$	49.9480	1.92535	0.962676	−	0.270656i	$-0.0872405\pi$
$673$	49.9480	1.92535	0.962676	+	0.270656i	$0.0872405\pi$
$674$	18.4614	0.711108
$675$	0	0
$676$	0.619973	0.0238451
$677$	10.8311	0.416272	0.208136	−	0.978100i	$-0.433260\pi$
$677$	10.8311	0.416272	0.208136	+	0.978100i	$0.433260\pi$
$678$	0	0
$679$	12.0425	0.462149
$680$	0	0
$681$	0	0
$682$	1.01725	0.0389526
$683$	0.429870	0.0164485	0.00822426	−	0.999966i	$-0.497382\pi$
$683$	0.429870	0.0164485	0.00822426	+	0.999966i	$0.497382\pi$
$684$	0	0
$685$	0	0
$686$	5.38003	0.205410
$687$	0	0
$688$	39.1202	1.49144
$689$	−46.4233	−1.76859
$690$	0	0
$691$	34.7036	1.32019	0.660093	−	0.751184i	$-0.270517\pi$
$691$	34.7036	1.32019	0.660093	+	0.751184i	$0.270517\pi$
$692$	−1.32151	−0.0502364
$693$	0	0
$694$	−25.1889	−0.956157
$695$	0	0
$696$	0	0
$697$	18.3940	0.696724
$698$	−27.1147	−1.02631
$699$	0	0
$700$	0	0
$701$	1.84808	0.0698010	0.0349005	−	0.999391i	$-0.488889\pi$
$701$	1.84808	0.0698010	0.0349005	+	0.999391i	$0.488889\pi$
$702$	0	0
$703$	−9.66962	−0.364696
$704$	1.87577	0.0706956
$705$	0	0
$706$	−46.7299	−1.75870
$707$	14.8669	0.559127
$708$	0	0
$709$	−6.30676	−0.236855	−0.118428	−	0.992963i	$-0.537785\pi$
$709$	−6.30676	−0.236855	−0.118428	+	0.992963i	$0.537785\pi$
$710$	0	0
$711$	0	0
$712$	−19.3465	−0.725040
$713$	23.0675	0.863884
$714$	0	0
$715$	0	0
$716$	−1.83489	−0.0685731
$717$	0	0
$718$	−16.8736	−0.629719
$719$	18.0129	0.671770	0.335885	−	0.941903i	$-0.390965\pi$
$719$	18.0129	0.671770	0.335885	+	0.941903i	$0.390965\pi$
$720$	0	0
$721$	−16.2627	−0.605655
$722$	−1.39013	−0.0517354
$723$	0	0
$724$	−1.33753	−0.0497089
$725$	0	0
$726$	0	0
$727$	26.2823	0.974758	0.487379	−	0.873190i	$-0.337953\pi$
$727$	26.2823	0.974758	0.487379	+	0.873190i	$0.337953\pi$
$728$	42.5002	1.57516
$729$	0	0
$730$	0	0
$731$	−29.5792	−1.09403
$732$	0	0
$733$	47.6633	1.76049	0.880243	−	0.474524i	$-0.157380\pi$
$733$	47.6633	1.76049	0.880243	+	0.474524i	$0.157380\pi$
$734$	3.67191	0.135533
$735$	0	0
$736$	−8.36334	−0.308277
$737$	2.93739	0.108200
$738$	0	0
$739$	−10.0273	−0.368859	−0.184429	−	0.982846i	$-0.559044\pi$
$739$	−10.0273	−0.368859	−0.184429	+	0.982846i	$0.559044\pi$
$740$	0	0
$741$	0	0
$742$	64.8221	2.37969
$743$	8.27266	0.303494	0.151747	−	0.988419i	$-0.451510\pi$
$743$	8.27266	0.303494	0.151747	+	0.988419i	$0.451510\pi$
$744$	0	0
$745$	0	0
$746$	−22.1539	−0.811111
$747$	0	0
$748$	0.144665	0.00528946
$749$	−6.28797	−0.229757
$750$	0	0
$751$	−5.79760	−0.211557	−0.105779	−	0.994390i	$-0.533734\pi$
$751$	−5.79760	−0.211557	−0.105779	+	0.994390i	$0.533734\pi$
$752$	6.16119	0.224676
$753$	0	0
$754$	−25.3711	−0.923960
$755$	0	0
$756$	0	0
$757$	−25.2804	−0.918830	−0.459415	−	0.888222i	$-0.651941\pi$
$757$	−25.2804	−0.918830	−0.459415	+	0.888222i	$0.651941\pi$
$758$	9.24132	0.335660
$759$	0	0
$760$	0	0
$761$	19.4638	0.705562	0.352781	−	0.935706i	$-0.385236\pi$
$761$	19.4638	0.705562	0.352781	+	0.935706i	$0.385236\pi$
$762$	0	0
$763$	−49.8775	−1.80569
$764$	−0.977278	−0.0353567
$765$	0	0
$766$	−32.6897	−1.18113
$767$	29.0584	1.04924
$768$	0	0
$769$	−49.3431	−1.77936	−0.889678	−	0.456588i	$-0.849071\pi$
$769$	−49.3431	−1.77936	−0.889678	+	0.456588i	$0.849071\pi$
$770$	0	0
$771$	0	0
$772$	−1.44607	−0.0520450
$773$	20.8502	0.749930	0.374965	−	0.927039i	$-0.377655\pi$
$773$	20.8502	0.749930	0.374965	+	0.927039i	$0.377655\pi$
$774$	0	0
$775$	0	0
$776$	−8.39632	−0.301410
$777$	0	0
$778$	−44.3335	−1.58943
$779$	23.9739	0.858953
$780$	0	0
$781$	−2.18154	−0.0780616
$782$	41.7657	1.49354
$783$	0	0
$784$	−34.2564	−1.22344
$785$	0	0
$786$	0	0
$787$	−44.1883	−1.57514	−0.787571	−	0.616223i	$-0.788662\pi$
$787$	−44.1883	−1.57514	−0.787571	+	0.616223i	$0.788662\pi$
$788$	1.81160	0.0645357
$789$	0	0
$790$	0	0
$791$	5.13799	0.182686
$792$	0	0
$793$	−10.2985	−0.365709
$794$	43.0651	1.52832
$795$	0	0
$796$	−3.16723	−0.112259
$797$	31.0374	1.09940	0.549701	−	0.835361i	$-0.314742\pi$
$797$	31.0374	1.09940	0.549701	+	0.835361i	$0.314742\pi$
$798$	0	0
$799$	−4.65855	−0.164808
$800$	0	0
$801$	0	0
$802$	35.7031	1.26072
$803$	0.105040	0.00370677
$804$	0	0
$805$	0	0
$806$	−15.9413	−0.561509
$807$	0	0
$808$	−10.3656	−0.364659
$809$	14.6229	0.514114	0.257057	−	0.966396i	$-0.417247\pi$
$809$	14.6229	0.514114	0.257057	+	0.966396i	$0.417247\pi$
$810$	0	0
$811$	26.7177	0.938187	0.469093	−	0.883149i	$-0.344581\pi$
$811$	26.7177	0.938187	0.469093	+	0.883149i	$0.344581\pi$
$812$	2.78254	0.0976480
$813$	0	0
$814$	−0.872586	−0.0305841
$815$	0	0
$816$	0	0
$817$	−38.5521	−1.34877
$818$	3.44727	0.120531
$819$	0	0
$820$	0	0
$821$	−18.5981	−0.649077	−0.324538	−	0.945873i	$-0.605209\pi$
$821$	−18.5981	−0.649077	−0.324538	+	0.945873i	$0.605209\pi$
$822$	0	0
$823$	−3.06204	−0.106736	−0.0533680	−	0.998575i	$-0.516996\pi$
$823$	−3.06204	−0.106736	−0.0533680	+	0.998575i	$0.516996\pi$
$824$	11.3388	0.395005
$825$	0	0
$826$	−40.5750	−1.41178
$827$	7.27526	0.252985	0.126493	−	0.991968i	$-0.459628\pi$
$827$	7.27526	0.252985	0.126493	+	0.991968i	$0.459628\pi$
$828$	0	0
$829$	10.5211	0.365411	0.182706	−	0.983168i	$-0.441514\pi$
$829$	10.5211	0.365411	0.182706	+	0.983168i	$0.441514\pi$
$830$	0	0
$831$	0	0
$832$	−29.3951	−1.01909
$833$	25.9017	0.897441
$834$	0	0
$835$	0	0
$836$	0.188549	0.00652109
$837$	0	0
$838$	33.6528	1.16252
$839$	15.1807	0.524094	0.262047	−	0.965055i	$-0.415602\pi$
$839$	15.1807	0.524094	0.262047	+	0.965055i	$0.415602\pi$
$840$	0	0
$841$	−11.1739	−0.385307
$842$	17.4961	0.602956
$843$	0	0
$844$	1.78207	0.0613413
$845$	0	0
$846$	0	0
$847$	−42.2622	−1.45215
$848$	−49.0762	−1.68528
$849$	0	0
$850$	0	0
$851$	−19.7870	−0.678290
$852$	0	0
$853$	10.4862	0.359040	0.179520	−	0.983754i	$-0.442545\pi$
$853$	10.4862	0.359040	0.179520	+	0.983754i	$0.442545\pi$
$854$	14.3800	0.492074
$855$	0	0
$856$	4.38412	0.149846
$857$	−8.84075	−0.301994	−0.150997	−	0.988534i	$-0.548248\pi$
$857$	−8.84075	−0.301994	−0.150997	+	0.988534i	$0.548248\pi$
$858$	0	0
$859$	−2.06831	−0.0705699	−0.0352849	−	0.999377i	$-0.511234\pi$
$859$	−2.06831	−0.0705699	−0.0352849	+	0.999377i	$0.511234\pi$
$860$	0	0
$861$	0	0
$862$	13.0665	0.445048
$863$	−22.4434	−0.763984	−0.381992	−	0.924166i	$-0.624762\pi$
$863$	−22.4434	−0.763984	−0.381992	+	0.924166i	$0.624762\pi$
$864$	0	0
$865$	0	0
$866$	13.8053	0.469122
$867$	0	0
$868$	1.74834	0.0593426
$869$	−0.791895	−0.0268632
$870$	0	0
$871$	−46.0317	−1.55973
$872$	34.7758	1.17766
$873$	0	0
$874$	54.4353	1.84130
$875$	0	0
$876$	0	0
$877$	27.8932	0.941886	0.470943	−	0.882164i	$-0.343914\pi$
$877$	27.8932	0.941886	0.470943	+	0.882164i	$0.343914\pi$
$878$	28.6152	0.965715
$879$	0	0
$880$	0	0
$881$	−9.22153	−0.310681	−0.155341	−	0.987861i	$-0.549647\pi$
$881$	−9.22153	−0.310681	−0.155341	+	0.987861i	$0.549647\pi$
$882$	0	0
$883$	49.2436	1.65718	0.828589	−	0.559858i	$-0.189144\pi$
$883$	49.2436	1.65718	0.828589	+	0.559858i	$0.189144\pi$
$884$	−2.26704	−0.0762486
$885$	0	0
$886$	16.0279	0.538469
$887$	10.7681	0.361556	0.180778	−	0.983524i	$-0.442138\pi$
$887$	10.7681	0.361556	0.180778	+	0.983524i	$0.442138\pi$
$888$	0	0
$889$	−6.38965	−0.214302
$890$	0	0
$891$	0	0
$892$	0.670187	0.0224395
$893$	−6.07173	−0.203183
$894$	0	0
$895$	0	0
$896$	48.4816	1.61966
$897$	0	0
$898$	1.97437	0.0658854
$899$	11.2006	0.373561
$900$	0	0
$901$	37.1071	1.23622
$902$	2.16340	0.0720334
$903$	0	0
$904$	−3.58233	−0.119147
$905$	0	0
$906$	0	0
$907$	−31.9703	−1.06156	−0.530779	−	0.847510i	$-0.678101\pi$
$907$	−31.9703	−1.06156	−0.530779	+	0.847510i	$0.678101\pi$
$908$	0.823651	0.0273338
$909$	0	0
$910$	0	0
$911$	−10.0802	−0.333972	−0.166986	−	0.985959i	$-0.553403\pi$
$911$	−10.0802	−0.333972	−0.166986	+	0.985959i	$0.553403\pi$
$912$	0	0
$913$	1.19213	0.0394539
$914$	29.6291	0.980042
$915$	0	0
$916$	−3.21403	−0.106195
$917$	−50.9428	−1.68228
$918$	0	0
$919$	−29.7976	−0.982932	−0.491466	−	0.870897i	$-0.663539\pi$
$919$	−29.7976	−0.982932	−0.491466	+	0.870897i	$0.663539\pi$
$920$	0	0
$921$	0	0
$922$	−49.7272	−1.63768
$923$	34.1868	1.12527
$924$	0	0
$925$	0	0
$926$	−7.72776	−0.253950
$927$	0	0
$928$	−4.06089	−0.133305
$929$	−12.3855	−0.406355	−0.203178	−	0.979142i	$-0.565127\pi$
$929$	−12.3855	−0.406355	−0.203178	+	0.979142i	$0.565127\pi$
$930$	0	0
$931$	33.7590	1.10641
$932$	−2.02906	−0.0664642
$933$	0	0
$934$	20.9379	0.685108
$935$	0	0
$936$	0	0
$937$	−44.4280	−1.45140	−0.725699	−	0.688012i	$-0.758484\pi$
$937$	−44.4280	−1.45140	−0.725699	+	0.688012i	$0.758484\pi$
$938$	64.2754	2.09867
$939$	0	0
$940$	0	0
$941$	15.3323	0.499820	0.249910	−	0.968269i	$-0.419599\pi$
$941$	15.3323	0.499820	0.249910	+	0.968269i	$0.419599\pi$
$942$	0	0
$943$	49.0579	1.59755
$944$	30.7189	0.999816
$945$	0	0
$946$	−3.47894	−0.113110
$947$	−21.6997	−0.705144	−0.352572	−	0.935785i	$-0.614693\pi$
$947$	−21.6997	−0.705144	−0.352572	+	0.935785i	$0.614693\pi$
$948$	0	0
$949$	−1.64607	−0.0534338
$950$	0	0
$951$	0	0
$952$	−33.9713	−1.10102
$953$	−36.9099	−1.19563	−0.597815	−	0.801634i	$-0.703964\pi$
$953$	−36.9099	−1.19563	−0.597815	+	0.801634i	$0.703964\pi$
$954$	0	0
$955$	0	0
$956$	−3.68737	−0.119258
$957$	0	0
$958$	30.1772	0.974980
$959$	78.4486	2.53324
$960$	0	0
$961$	−23.9624	−0.772980
$962$	13.6743	0.440876
$963$	0	0
$964$	0.664581	0.0214047
$965$	0	0
$966$	0	0
$967$	21.3382	0.686190	0.343095	−	0.939301i	$-0.388525\pi$
$967$	21.3382	0.686190	0.343095	+	0.939301i	$0.388525\pi$
$968$	29.4663	0.947081
$969$	0	0
$970$	0	0
$971$	−42.5851	−1.36662	−0.683311	−	0.730128i	$-0.739460\pi$
$971$	−42.5851	−1.36662	−0.683311	+	0.730128i	$0.739460\pi$
$972$	0	0
$973$	11.8635	0.380325
$974$	46.1946	1.48017
$975$	0	0
$976$	−10.8870	−0.348484
$977$	−48.9392	−1.56570	−0.782852	−	0.622209i	$-0.786236\pi$
$977$	−48.9392	−1.56570	−0.782852	+	0.622209i	$0.786236\pi$
$978$	0	0
$979$	1.86820	0.0597080
$980$	0	0
$981$	0	0
$982$	−15.3102	−0.488567
$983$	36.1089	1.15170	0.575848	−	0.817557i	$-0.304672\pi$
$983$	36.1089	1.15170	0.575848	+	0.817557i	$0.304672\pi$
$984$	0	0
$985$	0	0
$986$	20.2797	0.645836
$987$	0	0
$988$	−2.95474	−0.0940028
$989$	−78.8896	−2.50854
$990$	0	0
$991$	32.0054	1.01669	0.508343	−	0.861155i	$-0.330258\pi$
$991$	32.0054	1.01669	0.508343	+	0.861155i	$0.330258\pi$
$992$	−2.55156	−0.0810121
$993$	0	0
$994$	−47.7360	−1.51410
$995$	0	0
$996$	0	0
$997$	51.7680	1.63951	0.819754	−	0.572716i	$-0.194110\pi$
$997$	51.7680	1.63951	0.819754	+	0.572716i	$0.194110\pi$
$998$	−5.63624	−0.178412
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.a.y.1.3		4
3.2	odd	2		2025.2.a.p.1.2		4
5.2	odd	4		2025.2.b.n.649.6		8
5.3	odd	4		2025.2.b.n.649.3		8
5.4	even	2		2025.2.a.q.1.2		4
9.2	odd	6		675.2.e.e.226.3		8
9.4	even	3		225.2.e.c.151.2	yes	8
9.5	odd	6		675.2.e.e.451.3		8
9.7	even	3		225.2.e.c.76.2	✓	8
15.2	even	4		2025.2.b.o.649.3		8
15.8	even	4		2025.2.b.o.649.6		8
15.14	odd	2		2025.2.a.z.1.3		4
45.2	even	12		675.2.k.c.199.3		16
45.4	even	6		225.2.e.e.151.3	yes	8
45.7	odd	12		225.2.k.c.49.6		16
45.13	odd	12		225.2.k.c.124.6		16
45.14	odd	6		675.2.e.c.451.2		8
45.22	odd	12		225.2.k.c.124.3		16
45.23	even	12		675.2.k.c.424.3		16
45.29	odd	6		675.2.e.c.226.2		8
45.32	even	12		675.2.k.c.424.6		16
45.34	even	6		225.2.e.e.76.3	yes	8
45.38	even	12		675.2.k.c.199.6		16
45.43	odd	12		225.2.k.c.49.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.e.c.76.2	✓	8	9.7	even	3
225.2.e.c.151.2	yes	8	9.4	even	3
225.2.e.e.76.3	yes	8	45.34	even	6
225.2.e.e.151.3	yes	8	45.4	even	6
225.2.k.c.49.3		16	45.43	odd	12
225.2.k.c.49.6		16	45.7	odd	12
225.2.k.c.124.3		16	45.22	odd	12
225.2.k.c.124.6		16	45.13	odd	12
675.2.e.c.226.2		8	45.29	odd	6
675.2.e.c.451.2		8	45.14	odd	6
675.2.e.e.226.3		8	9.2	odd	6
675.2.e.e.451.3		8	9.5	odd	6
675.2.k.c.199.3		16	45.2	even	12
675.2.k.c.199.6		16	45.38	even	12
675.2.k.c.424.3		16	45.23	even	12
675.2.k.c.424.6		16	45.32	even	12
2025.2.a.p.1.2		4	3.2	odd	2
2025.2.a.q.1.2		4	5.4	even	2
2025.2.a.y.1.3		4	1.1	even	1	trivial
2025.2.a.z.1.3		4	15.14	odd	2
2025.2.b.n.649.3		8	5.3	odd	4
2025.2.b.n.649.6		8	5.2	odd	4
2025.2.b.o.649.3		8	15.2	even	4
2025.2.b.o.649.6		8	15.8	even	4

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)

\(f(q)\)	\(=\)	\(q+1.47325 q^{2} +0.170479 q^{4} +3.86583 q^{7} -2.69535 q^{8} +O(q^{10})\)	\(q+1.47325 q^{2} +0.170479 q^{4} +3.86583 q^{7} -2.69535 q^{8} +0.260278 q^{11} -4.07881 q^{13} +5.69535 q^{14} -4.31189 q^{16} +3.26028 q^{17} +4.24928 q^{19} +0.383456 q^{22} +8.69535 q^{23} -6.00912 q^{26} +0.659042 q^{28} +4.22210 q^{29} +2.65285 q^{31} -0.961818 q^{32} +4.80322 q^{34} -2.27559 q^{37} +6.26028 q^{38} +5.64186 q^{41} -9.07262 q^{43} +0.0443719 q^{44} +12.8105 q^{46} -1.42888 q^{47} +7.94463 q^{49} -0.695350 q^{52} +11.3816 q^{53} -10.4198 q^{56} +6.22022 q^{58} -7.12423 q^{59} +2.52487 q^{61} +3.90833 q^{62} +7.20679 q^{64} +11.2856 q^{67} +0.555809 q^{68} -8.38158 q^{71} +0.403568 q^{73} -3.35252 q^{74} +0.724413 q^{76} +1.00619 q^{77} -3.04250 q^{79} +8.31189 q^{82} +4.58024 q^{83} -13.3663 q^{86} -0.701540 q^{88} +7.17772 q^{89} -15.7680 q^{91} +1.48237 q^{92} -2.10511 q^{94} +3.11511 q^{97} +11.7045 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{4} - q^{7} + 9 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 4 * q^4 - q^7 + 9 * q^8	\( 4 q + 2 q^{2} + 4 q^{4} - q^{7} + 9 q^{8} - q^{11} + 2 q^{13} + 3 q^{14} + 4 q^{16} + 11 q^{17} + 2 q^{19} + 3 q^{22} + 15 q^{23} - 10 q^{26} - 4 q^{28} + q^{29} - 4 q^{31} + 10 q^{32} + 9 q^{34} - q^{37} + 23 q^{38} - 5 q^{41} - 10 q^{43} + 22 q^{44} + 20 q^{47} - 3 q^{49} + 17 q^{52} + 20 q^{53} - 30 q^{56} - 18 q^{58} + 17 q^{59} - 13 q^{61} - 6 q^{62} + 19 q^{64} + 17 q^{67} + 34 q^{68} - 8 q^{71} + 2 q^{73} + 40 q^{74} + 11 q^{76} + 12 q^{77} - 7 q^{79} + 12 q^{82} + 30 q^{83} - 34 q^{86} + 9 q^{88} - 9 q^{89} - 17 q^{91} - 12 q^{92} + 3 q^{94} - 19 q^{97} + 13 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^4 - q^7 + 9 * q^8 - q^11 + 2 * q^13 + 3 * q^14 + 4 * q^16 + 11 * q^17 + 2 * q^19 + 3 * q^22 + 15 * q^23 - 10 * q^26 - 4 * q^28 + q^29 - 4 * q^31 + 10 * q^32 + 9 * q^34 - q^37 + 23 * q^38 - 5 * q^41 - 10 * q^43 + 22 * q^44 + 20 * q^47 - 3 * q^49 + 17 * q^52 + 20 * q^53 - 30 * q^56 - 18 * q^58 + 17 * q^59 - 13 * q^61 - 6 * q^62 + 19 * q^64 + 17 * q^67 + 34 * q^68 - 8 * q^71 + 2 * q^73 + 40 * q^74 + 11 * q^76 + 12 * q^77 - 7 * q^79 + 12 * q^82 + 30 * q^83 - 34 * q^86 + 9 * q^88 - 9 * q^89 - 17 * q^91 - 12 * q^92 + 3 * q^94 - 19 * q^97 + 13 * q^98

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