LMFDB - Embedded newform 2025.2.a.p.1.2

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:	$1$
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			1.2
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	$-0.674394\pi$
$2$	−1.47325	−1.04175	−0.520874	+	0.853633i	$0.674394\pi$
$3$	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	$-0.512493\pi$
$11$	−0.260278	−0.0784767	−0.0392384	+	0.999230i	$0.512493\pi$
$12$	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	$-0.629383\pi$
$17$	−3.26028	−0.790734	−0.395367	+	0.918523i	$0.629383\pi$
$18$	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	$-0.861296\pi$
$23$	−8.69535	−1.81311	−0.906553	+	0.422092i	$0.861296\pi$
$24$	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	$-0.628221\pi$
$29$	−4.22210	−0.784023	−0.392012	+	0.919960i	$0.628221\pi$
$30$	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	$-0.645218\pi$
$41$	−5.64186	−0.881110	−0.440555	+	0.897726i	$0.645218\pi$
$42$	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	$-0.466768\pi$
$47$	1.42888	0.208424	0.104212	+	0.994555i	$0.466768\pi$
$48$	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	$-0.785642\pi$
$53$	−11.3816	−1.56338	−0.781690	+	0.623667i	$0.785642\pi$
$54$	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	$-0.346504\pi$
$59$	7.12423	0.927496	0.463748	+	0.885967i	$0.346504\pi$
$60$	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	$-0.334305\pi$
$71$	8.38158	0.994711	0.497355	+	0.867547i	$0.334305\pi$
$72$	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	$-0.580882\pi$
$83$	−4.58024	−0.502746	−0.251373	+	0.967890i	$0.580882\pi$
$84$	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	$-0.624220\pi$
$89$	−7.17772	−0.760837	−0.380419	+	0.924814i	$0.624220\pi$
$90$	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	$-0.561281\pi$
$101$	−3.84572	−0.382663	−0.191332	+	0.981525i	$0.561281\pi$
$102$	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	$-0.474948\pi$
$107$	1.62655	0.157245	0.0786223	+	0.996904i	$0.474948\pi$
$108$	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	$-0.519912\pi$
$113$	−1.32908	−0.125029	−0.0625146	+	0.998044i	$0.519912\pi$
$114$	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	$-0.304741\pi$
$131$	13.1777	1.15134	0.575672	+	0.817681i	$0.304741\pi$
$132$	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	$-0.833870\pi$
$137$	−20.2928	−1.73373	−0.866867	+	0.498539i	$0.833870\pi$
$138$	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	$-0.553205\pi$
$149$	−4.06162	−0.332741	−0.166371	+	0.986063i	$0.553205\pi$
$150$	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	$-0.792277\pi$
$167$	−20.5349	−1.58904	−0.794518	+	0.607240i	$0.792277\pi$
$168$	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	$-0.404788\pi$
$173$	7.75177	0.589356	0.294678	+	0.955597i	$0.404788\pi$
$174$	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	$-0.368232\pi$
$179$	10.7632	0.804477	0.402238	+	0.915535i	$0.368232\pi$
$180$	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	$-0.433501\pi$
$191$	5.73255	0.414792	0.207396	+	0.978257i	$0.433501\pi$
$192$	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	$-0.623579\pi$
$197$	−10.6266	−0.757110	−0.378555	+	0.925579i	$0.623579\pi$
$198$	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	$-0.551258\pi$
$227$	−4.83140	−0.320671	−0.160335	+	0.987063i	$0.551258\pi$
$228$	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	$-0.372521\pi$
$233$	11.9021	0.779735	0.389867	+	0.920871i	$0.372521\pi$
$234$	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	$-0.253385\pi$
$239$	21.6295	1.39909	0.699547	+	0.714586i	$0.253385\pi$
$240$	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	$-0.101030\pi$
$251$	30.1033	1.90010	0.950052	+	0.312092i	$0.101030\pi$
$252$	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	$-0.328926\pi$
$257$	16.4141	1.02389	0.511943	+	0.859019i	$0.328926\pi$
$258$	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	$-0.792881\pi$
$263$	−25.8072	−1.59134	−0.795670	+	0.605730i	$0.792881\pi$
$264$	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	$-0.375340\pi$
$269$	12.5206	0.763392	0.381696	+	0.924288i	$0.375340\pi$
$270$	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	$-0.545005\pi$
$281$	−4.72441	−0.281835	−0.140917	+	0.990021i	$0.545005\pi$
$282$	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	$-0.335882\pi$
$293$	16.8793	0.986097	0.493049	+	0.870002i	$0.335882\pi$
$294$	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	$-0.853363\pi$
$311$	−31.5936	−1.79151	−0.895754	+	0.444551i	$0.853363\pi$
$312$	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	$-0.712997\pi$
$317$	−22.0890	−1.24064	−0.620320	+	0.784349i	$0.712997\pi$
$318$	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	$-0.348237\pi$
$347$	17.0974	0.917839	0.458919	+	0.888478i	$0.348237\pi$
$348$	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	$-0.180129\pi$
$353$	31.7188	1.68822	0.844110	+	0.536169i	$0.180129\pi$
$354$	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	$-0.402265\pi$
$359$	11.4533	0.604483	0.302241	+	0.953231i	$0.402265\pi$
$360$	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	$-0.308144\pi$
$383$	22.1888	1.13379	0.566897	+	0.823789i	$0.308144\pi$
$384$	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	$-0.223790\pi$
$389$	30.0922	1.52574	0.762869	+	0.646554i	$0.223790\pi$
$390$	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	$-0.706866\pi$
$401$	−24.2341	−1.21020	−0.605098	+	0.796151i	$0.706866\pi$
$402$	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	$-0.688417\pi$
$419$	−22.8425	−1.11593	−0.557964	+	0.829865i	$0.688417\pi$
$420$	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	$-0.568521\pi$
$431$	−8.86916	−0.427212	−0.213606	+	0.976920i	$0.568521\pi$
$432$	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	$-0.583210\pi$
$443$	−10.8793	−0.516889	−0.258445	+	0.966026i	$0.583210\pi$
$444$	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	$-0.510067\pi$
$449$	−1.34014	−0.0632451	−0.0316225	+	0.999500i	$0.510067\pi$
$450$	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	$-0.212136\pi$
$461$	33.7533	1.57205	0.786024	+	0.618196i	$0.212136\pi$
$462$	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	$-0.606653\pi$
$467$	−14.2120	−0.657652	−0.328826	+	0.944390i	$0.606653\pi$
$468$	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	$-0.655009\pi$
$479$	−20.4833	−0.935907	−0.467954	+	0.883753i	$0.655009\pi$
$480$	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	$-0.424657\pi$
$491$	10.3921	0.468988	0.234494	+	0.972118i	$0.424657\pi$
$492$	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	$-0.507114\pi$
$503$	−1.00236	−0.0446931	−0.0223466	+	0.999750i	$0.507114\pi$
$504$	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	$-0.532246\pi$
$509$	−4.56322	−0.202261	−0.101131	+	0.994873i	$0.532246\pi$
$510$	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	$-0.831060\pi$
$521$	−39.3708	−1.72486	−0.862432	+	0.506173i	$0.831060\pi$
$522$	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	$-0.373752\pi$
$557$	18.2341	0.772605	0.386303	+	0.922372i	$0.373752\pi$
$558$	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	$-0.669646\pi$
$563$	−24.1112	−1.01617	−0.508083	+	0.861308i	$0.669646\pi$
$564$	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	$-0.265926\pi$
$569$	32.0049	1.34171	0.670857	+	0.741587i	$0.265926\pi$
$570$	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	$-0.267006\pi$
$587$	32.3851	1.33668	0.668338	+	0.743858i	$0.267006\pi$
$588$	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	$-0.705066\pi$
$593$	−29.2504	−1.20117	−0.600585	+	0.799561i	$0.705066\pi$
$594$	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	$-0.526443\pi$
$599$	−4.06162	−0.165953	−0.0829767	+	0.996551i	$0.526443\pi$
$600$	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	$-0.514414\pi$
$617$	−2.24880	−0.0905332	−0.0452666	+	0.998975i	$0.514414\pi$
$618$	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	$-0.633605\pi$
$641$	−20.6350	−0.815034	−0.407517	+	0.913198i	$0.633605\pi$
$642$	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	$-0.606676\pi$
$647$	−16.7316	−0.657787	−0.328893	+	0.944367i	$0.606676\pi$
$648$	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	$-0.147291\pi$
$653$	45.7331	1.78967	0.894837	+	0.446392i	$0.147291\pi$
$654$	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	$-0.618073\pi$
$659$	−18.6109	−0.724976	−0.362488	+	0.931988i	$0.618073\pi$
$660$	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	$-0.566740\pi$
$677$	−10.8311	−0.416272	−0.208136	+	0.978100i	$0.566740\pi$
$678$	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	$-0.502618\pi$
$683$	−0.429870	−0.0164485	−0.00822426	+	0.999966i	$0.502618\pi$
$684$	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	$-0.511111\pi$
$701$	−1.84808	−0.0698010	−0.0349005	+	0.999391i	$0.511111\pi$
$702$	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	$-0.609035\pi$
$719$	−18.0129	−0.671770	−0.335885	+	0.941903i	$0.609035\pi$
$720$	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	$-0.548490\pi$
$743$	−8.27266	−0.303494	−0.151747	+	0.988419i	$0.548490\pi$
$744$	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	$-0.614764\pi$
$761$	−19.4638	−0.705562	−0.352781	+	0.935706i	$0.614764\pi$
$762$	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	$-0.622345\pi$
$773$	−20.8502	−0.749930	−0.374965	+	0.927039i	$0.622345\pi$
$774$	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	$-0.685258\pi$
$797$	−31.0374	−1.09940	−0.549701	+	0.835361i	$0.685258\pi$
$798$	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	$-0.582753\pi$
$809$	−14.6229	−0.514114	−0.257057	+	0.966396i	$0.582753\pi$
$810$	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	$-0.394791\pi$
$821$	18.5981	0.649077	0.324538	+	0.945873i	$0.394791\pi$
$822$	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	$-0.540372\pi$
$827$	−7.27526	−0.252985	−0.126493	+	0.991968i	$0.540372\pi$
$828$	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	$-0.584398\pi$
$839$	−15.1807	−0.524094	−0.262047	+	0.965055i	$0.584398\pi$
$840$	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	$-0.451752\pi$
$857$	8.84075	0.301994	0.150997	+	0.988534i	$0.451752\pi$
$858$	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	$-0.375238\pi$
$863$	22.4434	0.763984	0.381992	+	0.924166i	$0.375238\pi$
$864$	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	$-0.450353\pi$
$881$	9.22153	0.310681	0.155341	+	0.987861i	$0.450353\pi$
$882$	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	$-0.557862\pi$
$887$	−10.7681	−0.361556	−0.180778	+	0.983524i	$0.557862\pi$
$888$	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	$-0.446597\pi$
$911$	10.0802	0.333972	0.166986	+	0.985959i	$0.446597\pi$
$912$	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	$-0.434873\pi$
$929$	12.3855	0.406355	0.203178	+	0.979142i	$0.434873\pi$
$930$	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	$-0.580401\pi$
$941$	−15.3323	−0.499820	−0.249910	+	0.968269i	$0.580401\pi$
$942$	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	$-0.385307\pi$
$947$	21.6997	0.705144	0.352572	+	0.935785i	$0.385307\pi$
$948$	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	$-0.296036\pi$
$953$	36.9099	1.19563	0.597815	+	0.801634i	$0.296036\pi$
$954$	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	$-0.260540\pi$
$971$	42.5851	1.36662	0.683311	+	0.730128i	$0.260540\pi$
$972$	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	$-0.213764\pi$
$977$	48.9392	1.56570	0.782852	+	0.622209i	$0.213764\pi$
$978$	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	$-0.695328\pi$
$983$	−36.1089	−1.15170	−0.575848	+	0.817557i	$0.695328\pi$
$984$	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.p.1.2		4
3.2	odd	2		2025.2.a.y.1.3		4
5.2	odd	4		2025.2.b.o.649.3		8
5.3	odd	4		2025.2.b.o.649.6		8
5.4	even	2		2025.2.a.z.1.3		4
9.2	odd	6		225.2.e.c.76.2	✓	8
9.4	even	3		675.2.e.e.451.3		8
9.5	odd	6		225.2.e.c.151.2	yes	8
9.7	even	3		675.2.e.e.226.3		8
15.2	even	4		2025.2.b.n.649.6		8
15.8	even	4		2025.2.b.n.649.3		8
15.14	odd	2		2025.2.a.q.1.2		4
45.2	even	12		225.2.k.c.49.6		16
45.4	even	6		675.2.e.c.451.2		8
45.7	odd	12		675.2.k.c.199.3		16
45.13	odd	12		675.2.k.c.424.3		16
45.14	odd	6		225.2.e.e.151.3	yes	8
45.22	odd	12		675.2.k.c.424.6		16
45.23	even	12		225.2.k.c.124.6		16
45.29	odd	6		225.2.e.e.76.3	yes	8
45.32	even	12		225.2.k.c.124.3		16
45.34	even	6		675.2.e.c.226.2		8
45.38	even	12		225.2.k.c.49.3		16
45.43	odd	12		675.2.k.c.199.6		16

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