LMFDB - Embedded newform 2025.4.a.r.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,4,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, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2025.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$119.478867762$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.7032.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x + 18 $ x^3 - x^2 - 14*x + 18
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 405)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.52348$ 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)

$f(q)$	$=$	$q+3.52348 q^{2} +4.41489 q^{4} -25.4325 q^{7} -12.6321 q^{8} +O(q^{10})$	$q+3.52348 q^{2} +4.41489 q^{4} -25.4325 q^{7} -12.6321 q^{8} -71.3561 q^{11} +51.3935 q^{13} -89.6107 q^{14} -79.8279 q^{16} +33.3191 q^{17} +113.372 q^{19} -251.422 q^{22} -81.9767 q^{23} +181.084 q^{26} -112.281 q^{28} -246.827 q^{29} +222.679 q^{31} -180.215 q^{32} +117.399 q^{34} -22.3910 q^{37} +399.463 q^{38} -434.225 q^{41} +236.850 q^{43} -315.029 q^{44} -288.843 q^{46} +107.984 q^{47} +303.810 q^{49} +226.896 q^{52} +123.961 q^{53} +321.265 q^{56} -869.690 q^{58} +171.091 q^{59} -79.4391 q^{61} +784.604 q^{62} +3.63945 q^{64} +611.506 q^{67} +147.100 q^{68} +511.102 q^{71} +410.012 q^{73} -78.8941 q^{74} +500.524 q^{76} +1814.76 q^{77} -793.597 q^{79} -1529.98 q^{82} -270.081 q^{83} +834.534 q^{86} +901.376 q^{88} +177.400 q^{89} -1307.06 q^{91} -361.918 q^{92} +380.478 q^{94} +881.860 q^{97} +1070.47 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} + 25 q^{7} - 27 q^{8}+O(q^{10}) $ 3 * q + q^2 + 5 * q^4 + 25 * q^7 - 27 * q^8	$ 3 q + q^{2} + 5 q^{4} + 25 q^{7} - 27 q^{8} - 58 q^{11} + 47 q^{13} - 159 q^{14} - 127 q^{16} + 34 q^{17} - 5 q^{19} - 260 q^{22} - 51 q^{23} - 253 q^{26} - 83 q^{28} - 350 q^{29} + 638 q^{31} + 245 q^{32} - 154 q^{34} + 414 q^{37} + 397 q^{38} - 179 q^{41} + 836 q^{43} - 332 q^{44} + 261 q^{46} + 235 q^{47} + 892 q^{49} + 1335 q^{52} + 505 q^{53} - 15 q^{56} - 1876 q^{58} - 535 q^{59} - 104 q^{61} + 348 q^{62} - 303 q^{64} + 40 q^{67} + 830 q^{68} - 452 q^{71} + 710 q^{73} - 1394 q^{74} + 849 q^{76} + 2148 q^{77} - 634 q^{79} - 613 q^{82} + 1734 q^{83} - 460 q^{86} + 768 q^{88} + 852 q^{89} - 1229 q^{91} - 1839 q^{92} + 1751 q^{94} + 38 q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 + 25 * q^7 - 27 * q^8 - 58 * q^11 + 47 * q^13 - 159 * q^14 - 127 * q^16 + 34 * q^17 - 5 * q^19 - 260 * q^22 - 51 * q^23 - 253 * q^26 - 83 * q^28 - 350 * q^29 + 638 * q^31 + 245 * q^32 - 154 * q^34 + 414 * q^37 + 397 * q^38 - 179 * q^41 + 836 * q^43 - 332 * q^44 + 261 * q^46 + 235 * q^47 + 892 * q^49 + 1335 * q^52 + 505 * q^53 - 15 * q^56 - 1876 * q^58 - 535 * q^59 - 104 * q^61 + 348 * q^62 - 303 * q^64 + 40 * q^67 + 830 * q^68 - 452 * q^71 + 710 * q^73 - 1394 * q^74 + 849 * q^76 + 2148 * q^77 - 634 * q^79 - 613 * q^82 + 1734 * q^83 - 460 * q^86 + 768 * q^88 + 852 * q^89 - 1229 * q^91 - 1839 * q^92 + 1751 * q^94 + 38 * 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 $

$2$	3.52348	1.24574	0.622868	−	0.782327i	$-0.285967\pi$
$2$	3.52348	1.24574	0.622868	+	0.782327i	$0.285967\pi$
$3$	0	0
$3$	0	0
$4$	4.41489	0.551861
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−25.4325	−1.37322	−0.686612	−	0.727024i	$-0.740903\pi$
$7$	−25.4325	−1.37322	−0.686612	+	0.727024i	$0.740903\pi$
$8$	−12.6321	−0.558264
$9$	0	0
$10$	0	0
$11$	−71.3561	−1.95588	−0.977940	−	0.208885i	$-0.933017\pi$
$11$	−71.3561	−1.95588	−0.977940	+	0.208885i	$0.933017\pi$
$12$	0	0
$13$	51.3935	1.09646	0.548231	−	0.836327i	$-0.315302\pi$
$13$	51.3935	1.09646	0.548231	+	0.836327i	$0.315302\pi$
$14$	−89.6107	−1.71068
$15$	0	0
$16$	−79.8279	−1.24731
$17$	33.3191	0.475357	0.237678	−	0.971344i	$-0.423614\pi$
$17$	33.3191	0.475357	0.237678	+	0.971344i	$0.423614\pi$
$18$	0	0
$19$	113.372	1.36891	0.684455	−	0.729055i	$-0.260040\pi$
$19$	113.372	1.36891	0.684455	+	0.729055i	$0.260040\pi$
$20$	0	0
$21$	0	0
$22$	−251.422	−2.43651
$23$	−81.9767	−0.743188	−0.371594	−	0.928395i	$-0.621189\pi$
$23$	−81.9767	−0.743188	−0.371594	+	0.928395i	$0.621189\pi$
$24$	0	0
$25$	0	0
$26$	181.084	1.36590
$27$	0	0
$28$	−112.281	−0.757828
$29$	−246.827	−1.58051	−0.790253	−	0.612781i	$-0.790051\pi$
$29$	−246.827	−1.58051	−0.790253	+	0.612781i	$0.790051\pi$
$30$	0	0
$31$	222.679	1.29014	0.645070	−	0.764124i	$-0.276828\pi$
$31$	222.679	1.29014	0.645070	+	0.764124i	$0.276828\pi$
$32$	−180.215	−0.995557
$33$	0	0
$34$	117.399	0.592169
$35$	0	0
$36$	0	0
$37$	−22.3910	−0.0994880	−0.0497440	−	0.998762i	$-0.515841\pi$
$37$	−22.3910	−0.0994880	−0.0497440	+	0.998762i	$0.515841\pi$
$38$	399.463	1.70530
$39$	0	0
$40$	0	0
$41$	−434.225	−1.65401	−0.827007	−	0.562191i	$-0.809959\pi$
$41$	−434.225	−1.65401	−0.827007	+	0.562191i	$0.809959\pi$
$42$	0	0
$43$	236.850	0.839982	0.419991	−	0.907528i	$-0.362033\pi$
$43$	236.850	0.839982	0.419991	+	0.907528i	$0.362033\pi$
$44$	−315.029	−1.07937
$45$	0	0
$46$	−288.843	−0.925817
$47$	107.984	0.335129	0.167564	−	0.985861i	$-0.446410\pi$
$47$	107.984	0.335129	0.167564	+	0.985861i	$0.446410\pi$
$48$	0	0
$49$	303.810	0.885744
$50$	0	0
$51$	0	0
$52$	226.896	0.605094
$53$	123.961	0.321272	0.160636	−	0.987014i	$-0.448646\pi$
$53$	123.961	0.321272	0.160636	+	0.987014i	$0.448646\pi$
$54$	0	0
$55$	0	0
$56$	321.265	0.766621
$57$	0	0
$58$	−869.690	−1.96889
$59$	171.091	0.377528	0.188764	−	0.982023i	$-0.439552\pi$
$59$	171.091	0.377528	0.188764	+	0.982023i	$0.439552\pi$
$60$	0	0
$61$	−79.4391	−0.166740	−0.0833700	−	0.996519i	$-0.526568\pi$
$61$	−79.4391	−0.166740	−0.0833700	+	0.996519i	$0.526568\pi$
$62$	784.604	1.60717
$63$	0	0
$64$	3.63945	0.00710830
$65$	0	0
$66$	0	0
$67$	611.506	1.11503	0.557517	−	0.830165i	$-0.311754\pi$
$67$	611.506	1.11503	0.557517	+	0.830165i	$0.311754\pi$
$68$	147.100	0.262331
$69$	0	0
$70$	0	0
$71$	511.102	0.854318	0.427159	−	0.904176i	$-0.359514\pi$
$71$	511.102	0.854318	0.427159	+	0.904176i	$0.359514\pi$
$72$	0	0
$73$	410.012	0.657373	0.328686	−	0.944439i	$-0.393394\pi$
$73$	410.012	0.657373	0.328686	+	0.944439i	$0.393394\pi$
$74$	−78.8941	−0.123936
$75$	0	0
$76$	500.524	0.755447
$77$	1814.76	2.68586
$78$	0	0
$79$	−793.597	−1.13021	−0.565105	−	0.825019i	$-0.691164\pi$
$79$	−793.597	−1.13021	−0.565105	+	0.825019i	$0.691164\pi$
$80$	0	0
$81$	0	0
$82$	−1529.98	−2.06047
$83$	−270.081	−0.357171	−0.178586	−	0.983924i	$-0.557152\pi$
$83$	−270.081	−0.357171	−0.178586	+	0.983924i	$0.557152\pi$
$84$	0	0
$85$	0	0
$86$	834.534	1.04640
$87$	0	0
$88$	901.376	1.09190
$89$	177.400	0.211284	0.105642	−	0.994404i	$-0.466310\pi$
$89$	177.400	0.211284	0.105642	+	0.994404i	$0.466310\pi$
$90$	0	0
$91$	−1307.06	−1.50569
$92$	−361.918	−0.410136
$93$	0	0
$94$	380.478	0.417482
$95$	0	0
$96$	0	0
$97$	881.860	0.923086	0.461543	−	0.887118i	$-0.347296\pi$
$97$	881.860	0.923086	0.461543	+	0.887118i	$0.347296\pi$
$98$	1070.47	1.10340
$99$	0	0
$100$	0	0
$101$	1026.03	1.01083	0.505417	−	0.862875i	$-0.331339\pi$
$101$	1026.03	1.01083	0.505417	+	0.862875i	$0.331339\pi$
$102$	0	0
$103$	1790.28	1.71263	0.856317	−	0.516451i	$-0.172747\pi$
$103$	1790.28	1.71263	0.856317	+	0.516451i	$0.172747\pi$
$104$	−649.206	−0.612115
$105$	0	0
$106$	436.775	0.400220
$107$	2007.30	1.81358	0.906788	−	0.421587i	$-0.138527\pi$
$107$	2007.30	1.81358	0.906788	+	0.421587i	$0.138527\pi$
$108$	0	0
$109$	−211.654	−0.185989	−0.0929945	−	0.995667i	$-0.529644\pi$
$109$	−211.654	−0.185989	−0.0929945	+	0.995667i	$0.529644\pi$
$110$	0	0
$111$	0	0
$112$	2030.22	1.71284
$113$	−1.94491	−0.00161913	−0.000809567	−	1.00000i	$-0.500258\pi$
$113$	−1.94491	−0.00161913	−0.000809567		1.00000i	$0.500258\pi$
$114$	0	0
$115$	0	0
$116$	−1089.71	−0.872219
$117$	0	0
$118$	602.835	0.470300
$119$	−847.386	−0.652771
$120$	0	0
$121$	3760.70	2.82547
$122$	−279.902	−0.207714
$123$	0	0
$124$	983.102	0.711977
$125$	0	0
$126$	0	0
$127$	1178.02	0.823089	0.411544	−	0.911390i	$-0.364989\pi$
$127$	1178.02	0.823089	0.411544	+	0.911390i	$0.364989\pi$
$128$	1454.54	1.00441
$129$	0	0
$130$	0	0
$131$	521.218	0.347626	0.173813	−	0.984779i	$-0.444391\pi$
$131$	521.218	0.347626	0.173813	+	0.984779i	$0.444391\pi$
$132$	0	0
$133$	−2883.32	−1.87982
$134$	2154.63	1.38904
$135$	0	0
$136$	−420.889	−0.265374
$137$	449.532	0.280336	0.140168	−	0.990128i	$-0.455236\pi$
$137$	449.532	0.280336	0.140168	+	0.990128i	$0.455236\pi$
$138$	0	0
$139$	−2345.45	−1.43121	−0.715605	−	0.698505i	$-0.753849\pi$
$139$	−2345.45	−1.43121	−0.715605	+	0.698505i	$0.753849\pi$
$140$	0	0
$141$	0	0
$142$	1800.85	1.06426
$143$	−3667.24	−2.14455
$144$	0	0
$145$	0	0
$146$	1444.67	0.818914
$147$	0	0
$148$	−98.8537	−0.0549035
$149$	−1269.38	−0.697932	−0.348966	−	0.937135i	$-0.613467\pi$
$149$	−1269.38	−0.697932	−0.348966	+	0.937135i	$0.613467\pi$
$150$	0	0
$151$	2158.32	1.16319	0.581594	−	0.813480i	$-0.302429\pi$
$151$	2158.32	1.16319	0.581594	+	0.813480i	$0.302429\pi$
$152$	−1432.12	−0.764213
$153$	0	0
$154$	6394.27	3.34588
$155$	0	0
$156$	0	0
$157$	2343.31	1.19119	0.595593	−	0.803286i	$-0.296917\pi$
$157$	2343.31	1.19119	0.595593	+	0.803286i	$0.296917\pi$
$158$	−2796.22	−1.40794
$159$	0	0
$160$	0	0
$161$	2084.87	1.02056
$162$	0	0
$163$	−3399.94	−1.63376	−0.816882	−	0.576804i	$-0.804300\pi$
$163$	−3399.94	−1.63376	−0.816882	+	0.576804i	$0.804300\pi$
$164$	−1917.06	−0.912786
$165$	0	0
$166$	−951.623	−0.444942
$167$	1809.22	0.838332	0.419166	−	0.907910i	$-0.362323\pi$
$167$	1809.22	0.838332	0.419166	+	0.907910i	$0.362323\pi$
$168$	0	0
$169$	444.292	0.202227
$170$	0	0
$171$	0	0
$172$	1045.66	0.463553
$173$	1327.41	0.583358	0.291679	−	0.956516i	$-0.405786\pi$
$173$	1327.41	0.583358	0.291679	+	0.956516i	$0.405786\pi$
$174$	0	0
$175$	0	0
$176$	5696.21	2.43959
$177$	0	0
$178$	625.063	0.263205
$179$	−448.734	−0.187374	−0.0936871	−	0.995602i	$-0.529865\pi$
$179$	−448.734	−0.187374	−0.0936871	+	0.995602i	$0.529865\pi$
$180$	0	0
$181$	−3450.55	−1.41700	−0.708502	−	0.705709i	$-0.750629\pi$
$181$	−3450.55	−1.41700	−0.708502	+	0.705709i	$0.750629\pi$
$182$	−4605.41	−1.87569
$183$	0	0
$184$	1035.54	0.414895
$185$	0	0
$186$	0	0
$187$	−2377.52	−0.929741
$188$	476.736	0.184944
$189$	0	0
$190$	0	0
$191$	−1592.09	−0.603141	−0.301570	−	0.953444i	$-0.597511\pi$
$191$	−1592.09	−0.603141	−0.301570	+	0.953444i	$0.597511\pi$
$192$	0	0
$193$	−443.299	−0.165334	−0.0826668	−	0.996577i	$-0.526344\pi$
$193$	−443.299	−0.165334	−0.0826668	+	0.996577i	$0.526344\pi$
$194$	3107.21	1.14992
$195$	0	0
$196$	1341.29	0.488807
$197$	208.992	0.0755839	0.0377919	−	0.999286i	$-0.487968\pi$
$197$	208.992	0.0755839	0.0377919	+	0.999286i	$0.487968\pi$
$198$	0	0
$199$	−479.916	−0.170957	−0.0854783	−	0.996340i	$-0.527242\pi$
$199$	−479.916	−0.170957	−0.0854783	+	0.996340i	$0.527242\pi$
$200$	0	0
$201$	0	0
$202$	3615.21	1.25923
$203$	6277.43	2.17039
$204$	0	0
$205$	0	0
$206$	6308.00	2.13349
$207$	0	0
$208$	−4102.63	−1.36763
$209$	−8089.78	−2.67742
$210$	0	0
$211$	−2632.14	−0.858786	−0.429393	−	0.903118i	$-0.641273\pi$
$211$	−2632.14	−0.858786	−0.429393	+	0.903118i	$0.641273\pi$
$212$	547.275	0.177297
$213$	0	0
$214$	7072.66	2.25924
$215$	0	0
$216$	0	0
$217$	−5663.28	−1.77165
$218$	−745.759	−0.231693
$219$	0	0
$220$	0	0
$221$	1712.38	0.521210
$222$	0	0
$223$	−3353.22	−1.00694	−0.503471	−	0.864012i	$-0.667944\pi$
$223$	−3353.22	−1.00694	−0.503471	+	0.864012i	$0.667944\pi$
$224$	4583.31	1.36712
$225$	0	0
$226$	−6.85286	−0.00201701
$227$	−690.764	−0.201972	−0.100986	−	0.994888i	$-0.532200\pi$
$227$	−690.764	−0.201972	−0.100986	+	0.994888i	$0.532200\pi$
$228$	0	0
$229$	−668.131	−0.192801	−0.0964003	−	0.995343i	$-0.530733\pi$
$229$	−668.131	−0.192801	−0.0964003	+	0.995343i	$0.530733\pi$
$230$	0	0
$231$	0	0
$232$	3117.94	0.882339
$233$	940.537	0.264449	0.132225	−	0.991220i	$-0.457788\pi$
$233$	940.537	0.264449	0.132225	+	0.991220i	$0.457788\pi$
$234$	0	0
$235$	0	0
$236$	755.347	0.208343
$237$	0	0
$238$	−2985.75	−0.813181
$239$	1107.40	0.299715	0.149858	−	0.988708i	$-0.452118\pi$
$239$	1107.40	0.299715	0.149858	+	0.988708i	$0.452118\pi$
$240$	0	0
$241$	1228.86	0.328456	0.164228	−	0.986422i	$-0.447487\pi$
$241$	1228.86	0.328456	0.164228	+	0.986422i	$0.447487\pi$
$242$	13250.7	3.51979
$243$	0	0
$244$	−350.715	−0.0920172
$245$	0	0
$246$	0	0
$247$	5826.58	1.50096
$248$	−2812.90	−0.720238
$249$	0	0
$250$	0	0
$251$	738.480	0.185707	0.0928534	−	0.995680i	$-0.470401\pi$
$251$	738.480	0.185707	0.0928534	+	0.995680i	$0.470401\pi$
$252$	0	0
$253$	5849.54	1.45359
$254$	4150.72	1.02535
$255$	0	0
$256$	5095.94	1.24412
$257$	−6312.57	−1.53217	−0.766084	−	0.642740i	$-0.777797\pi$
$257$	−6312.57	−1.53217	−0.766084	+	0.642740i	$0.777797\pi$
$258$	0	0
$259$	569.458	0.136619
$260$	0	0
$261$	0	0
$262$	1836.50	0.433051
$263$	−5993.62	−1.40526	−0.702628	−	0.711558i	$-0.747990\pi$
$263$	−5993.62	−1.40526	−0.702628	+	0.711558i	$0.747990\pi$
$264$	0	0
$265$	0	0
$266$	−10159.3	−2.34176
$267$	0	0
$268$	2699.73	0.615344
$269$	−1749.70	−0.396584	−0.198292	−	0.980143i	$-0.563539\pi$
$269$	−1749.70	−0.396584	−0.198292	+	0.980143i	$0.563539\pi$
$270$	0	0
$271$	−1931.68	−0.432994	−0.216497	−	0.976283i	$-0.569463\pi$
$271$	−1931.68	−0.432994	−0.216497	+	0.976283i	$0.569463\pi$
$272$	−2659.79	−0.592917
$273$	0	0
$274$	1583.91	0.349225
$275$	0	0
$276$	0	0
$277$	3791.55	0.822427	0.411213	−	0.911539i	$-0.365105\pi$
$277$	3791.55	0.822427	0.411213	+	0.911539i	$0.365105\pi$
$278$	−8264.13	−1.78291
$279$	0	0
$280$	0	0
$281$	4489.36	0.953071	0.476536	−	0.879155i	$-0.341892\pi$
$281$	4489.36	0.953071	0.476536	+	0.879155i	$0.341892\pi$
$282$	0	0
$283$	−3133.86	−0.658264	−0.329132	−	0.944284i	$-0.606756\pi$
$283$	−3133.86	−0.658264	−0.329132	+	0.944284i	$0.606756\pi$
$284$	2256.45	0.471465
$285$	0	0
$286$	−12921.4	−2.67154
$287$	11043.4	2.27133
$288$	0	0
$289$	−3802.84	−0.774036
$290$	0	0
$291$	0	0
$292$	1810.15	0.362778
$293$	−2797.50	−0.557787	−0.278894	−	0.960322i	$-0.589968\pi$
$293$	−2797.50	−0.557787	−0.278894	+	0.960322i	$0.589968\pi$
$294$	0	0
$295$	0	0
$296$	282.845	0.0555406
$297$	0	0
$298$	−4472.64	−0.869439
$299$	−4213.07	−0.814877
$300$	0	0
$301$	−6023.67	−1.15348
$302$	7604.77	1.44903
$303$	0	0
$304$	−9050.23	−1.70746
$305$	0	0
$306$	0	0
$307$	6839.91	1.27158	0.635789	−	0.771863i	$-0.280675\pi$
$307$	6839.91	1.27158	0.635789	+	0.771863i	$0.280675\pi$
$308$	8011.97	1.48222
$309$	0	0
$310$	0	0
$311$	2419.41	0.441132	0.220566	−	0.975372i	$-0.429210\pi$
$311$	2419.41	0.441132	0.220566	+	0.975372i	$0.429210\pi$
$312$	0	0
$313$	−3903.18	−0.704858	−0.352429	−	0.935838i	$-0.614644\pi$
$313$	−3903.18	−0.704858	−0.352429	+	0.935838i	$0.614644\pi$
$314$	8256.59	1.48391
$315$	0	0
$316$	−3503.64	−0.623718
$317$	−9894.73	−1.75313	−0.876567	−	0.481280i	$-0.840172\pi$
$317$	−9894.73	−1.75313	−0.876567	+	0.481280i	$0.840172\pi$
$318$	0	0
$319$	17612.6	3.09128
$320$	0	0
$321$	0	0
$322$	7345.99	1.27135
$323$	3777.44	0.650720
$324$	0	0
$325$	0	0
$326$	−11979.6	−2.03524
$327$	0	0
$328$	5485.16	0.923377
$329$	−2746.29	−0.460207
$330$	0	0
$331$	4163.04	0.691303	0.345651	−	0.938363i	$-0.387658\pi$
$331$	4163.04	0.691303	0.345651	+	0.938363i	$0.387658\pi$
$332$	−1192.38	−0.197109
$333$	0	0
$334$	6374.73	1.04434
$335$	0	0
$336$	0	0
$337$	−9704.93	−1.56873	−0.784364	−	0.620301i	$-0.787011\pi$
$337$	−9704.93	−1.56873	−0.784364	+	0.620301i	$0.787011\pi$
$338$	1565.45	0.251921
$339$	0	0
$340$	0	0
$341$	−15889.5	−2.52336
$342$	0	0
$343$	996.691	0.156899
$344$	−2991.90	−0.468931
$345$	0	0
$346$	4677.09	0.726711
$347$	5432.38	0.840420	0.420210	−	0.907427i	$-0.361956\pi$
$347$	5432.38	0.840420	0.420210	+	0.907427i	$0.361956\pi$
$348$	0	0
$349$	10571.9	1.62150	0.810749	−	0.585394i	$-0.199060\pi$
$349$	10571.9	1.62150	0.810749	+	0.585394i	$0.199060\pi$
$350$	0	0
$351$	0	0
$352$	12859.5	1.94719
$353$	9642.98	1.45395	0.726975	−	0.686664i	$-0.240926\pi$
$353$	9642.98	1.45395	0.726975	+	0.686664i	$0.240926\pi$
$354$	0	0
$355$	0	0
$356$	783.198	0.116600
$357$	0	0
$358$	−1581.10	−0.233419
$359$	−3002.18	−0.441362	−0.220681	−	0.975346i	$-0.570828\pi$
$359$	−3002.18	−0.441362	−0.220681	+	0.975346i	$0.570828\pi$
$360$	0	0
$361$	5994.17	0.873913
$362$	−12157.9	−1.76521
$363$	0	0
$364$	−5770.54	−0.830929
$365$	0	0
$366$	0	0
$367$	1149.28	0.163465	0.0817326	−	0.996654i	$-0.473955\pi$
$367$	1149.28	0.163465	0.0817326	+	0.996654i	$0.473955\pi$
$368$	6544.03	0.926986
$369$	0	0
$370$	0	0
$371$	−3152.64	−0.441178
$372$	0	0
$373$	513.214	0.0712418	0.0356209	−	0.999365i	$-0.488659\pi$
$373$	513.214	0.0712418	0.0356209	+	0.999365i	$0.488659\pi$
$374$	−8377.14	−1.15821
$375$	0	0
$376$	−1364.06	−0.187090
$377$	−12685.3	−1.73296
$378$	0	0
$379$	12560.0	1.70228	0.851138	−	0.524942i	$-0.175913\pi$
$379$	12560.0	1.70228	0.851138	+	0.524942i	$0.175913\pi$
$380$	0	0
$381$	0	0
$382$	−5609.70	−0.751355
$383$	273.990	0.0365542	0.0182771	−	0.999833i	$-0.494182\pi$
$383$	273.990	0.0365542	0.0182771	+	0.999833i	$0.494182\pi$
$384$	0	0
$385$	0	0
$386$	−1561.95	−0.205962
$387$	0	0
$388$	3893.31	0.509415
$389$	−6846.43	−0.892359	−0.446179	−	0.894944i	$-0.647216\pi$
$389$	−6846.43	−0.892359	−0.446179	+	0.894944i	$0.647216\pi$
$390$	0	0
$391$	−2731.39	−0.353279
$392$	−3837.75	−0.494479
$393$	0	0
$394$	736.377	0.0941577
$395$	0	0
$396$	0	0
$397$	12117.5	1.53189	0.765943	−	0.642909i	$-0.222273\pi$
$397$	12117.5	1.53189	0.765943	+	0.642909i	$0.222273\pi$
$398$	−1690.97	−0.212967
$399$	0	0
$400$	0	0
$401$	−3016.80	−0.375691	−0.187845	−	0.982199i	$-0.560150\pi$
$401$	−3016.80	−0.375691	−0.187845	+	0.982199i	$0.560150\pi$
$402$	0	0
$403$	11444.3	1.41459
$404$	4529.82	0.557839
$405$	0	0
$406$	22118.4	2.70373
$407$	1597.74	0.194587
$408$	0	0
$409$	6535.13	0.790076	0.395038	−	0.918665i	$-0.370731\pi$
$409$	6535.13	0.790076	0.395038	+	0.918665i	$0.370731\pi$
$410$	0	0
$411$	0	0
$412$	7903.87	0.945135
$413$	−4351.26	−0.518430
$414$	0	0
$415$	0	0
$416$	−9261.88	−1.09159
$417$	0	0
$418$	−28504.1	−3.33537
$419$	7148.45	0.833471	0.416736	−	0.909028i	$-0.363174\pi$
$419$	7148.45	0.833471	0.416736	+	0.909028i	$0.363174\pi$
$420$	0	0
$421$	14801.5	1.71350	0.856749	−	0.515733i	$-0.172480\pi$
$421$	14801.5	1.71350	0.856749	+	0.515733i	$0.172480\pi$
$422$	−9274.28	−1.06982
$423$	0	0
$424$	−1565.89	−0.179354
$425$	0	0
$426$	0	0
$427$	2020.33	0.228971
$428$	8861.98	1.00084
$429$	0	0
$430$	0	0
$431$	2284.19	0.255280	0.127640	−	0.991821i	$-0.459260\pi$
$431$	2284.19	0.255280	0.127640	+	0.991821i	$0.459260\pi$
$432$	0	0
$433$	−5529.26	−0.613670	−0.306835	−	0.951763i	$-0.599270\pi$
$433$	−5529.26	−0.613670	−0.306835	+	0.951763i	$0.599270\pi$
$434$	−19954.4	−2.20701
$435$	0	0
$436$	−934.429	−0.102640
$437$	−9293.85	−1.01736
$438$	0	0
$439$	−11861.6	−1.28958	−0.644788	−	0.764361i	$-0.723054\pi$
$439$	−11861.6	−1.28958	−0.644788	+	0.764361i	$0.723054\pi$
$440$	0	0
$441$	0	0
$442$	6033.55	0.649291
$443$	15293.6	1.64023	0.820115	−	0.572199i	$-0.193910\pi$
$443$	15293.6	1.64023	0.820115	+	0.572199i	$0.193910\pi$
$444$	0	0
$445$	0	0
$446$	−11815.0	−1.25439
$447$	0	0
$448$	−92.5602	−0.00976129
$449$	12998.8	1.36626	0.683129	−	0.730297i	$-0.260619\pi$
$449$	12998.8	1.36626	0.683129	+	0.730297i	$0.260619\pi$
$450$	0	0
$451$	30984.6	3.23506
$452$	−8.58657	−0.000893536 0
$453$	0	0
$454$	−2433.89	−0.251604
$455$	0	0
$456$	0	0
$457$	−14311.9	−1.46495	−0.732477	−	0.680792i	$-0.761636\pi$
$457$	−14311.9	−1.46495	−0.732477	+	0.680792i	$0.761636\pi$
$458$	−2354.14	−0.240179
$459$	0	0
$460$	0	0
$461$	4290.34	0.433451	0.216725	−	0.976233i	$-0.430462\pi$
$461$	4290.34	0.433451	0.216725	+	0.976233i	$0.430462\pi$
$462$	0	0
$463$	14510.3	1.45648	0.728240	−	0.685322i	$-0.240339\pi$
$463$	14510.3	1.45648	0.728240	+	0.685322i	$0.240339\pi$
$464$	19703.7	1.97138
$465$	0	0
$466$	3313.96	0.329434
$467$	7393.05	0.732569	0.366285	−	0.930503i	$-0.380630\pi$
$467$	7393.05	0.732569	0.366285	+	0.930503i	$0.380630\pi$
$468$	0	0
$469$	−15552.1	−1.53119
$470$	0	0
$471$	0	0
$472$	−2161.23	−0.210760
$473$	−16900.7	−1.64290
$474$	0	0
$475$	0	0
$476$	−3741.11	−0.360239
$477$	0	0
$478$	3901.91	0.373366
$479$	13509.9	1.28869	0.644344	−	0.764736i	$-0.277131\pi$
$479$	13509.9	1.28869	0.644344	+	0.764736i	$0.277131\pi$
$480$	0	0
$481$	−1150.75	−0.109085
$482$	4329.87	0.409170
$483$	0	0
$484$	16603.1	1.55926
$485$	0	0
$486$	0	0
$487$	11140.0	1.03655	0.518277	−	0.855213i	$-0.326573\pi$
$487$	11140.0	1.03655	0.518277	+	0.855213i	$0.326573\pi$
$488$	1003.48	0.0930849
$489$	0	0
$490$	0	0
$491$	2012.74	0.184998	0.0924988	−	0.995713i	$-0.470515\pi$
$491$	2012.74	0.184998	0.0924988	+	0.995713i	$0.470515\pi$
$492$	0	0
$493$	−8224.06	−0.751304
$494$	20529.8	1.86980
$495$	0	0
$496$	−17776.0	−1.60920
$497$	−12998.6	−1.17317
$498$	0	0
$499$	7352.98	0.659648	0.329824	−	0.944042i	$-0.393011\pi$
$499$	7352.98	0.659648	0.329824	+	0.944042i	$0.393011\pi$
$500$	0	0
$501$	0	0
$502$	2602.02	0.231342
$503$	−16898.3	−1.49793	−0.748965	−	0.662609i	$-0.769449\pi$
$503$	−16898.3	−1.49793	−0.748965	+	0.662609i	$0.769449\pi$
$504$	0	0
$505$	0	0
$506$	20610.7	1.81079
$507$	0	0
$508$	5200.82	0.454230
$509$	−19358.6	−1.68577	−0.842884	−	0.538095i	$-0.819144\pi$
$509$	−19358.6	−1.68577	−0.842884	+	0.538095i	$0.819144\pi$
$510$	0	0
$511$	−10427.6	−0.902720
$512$	6319.06	0.545440
$513$	0	0
$514$	−22242.2	−1.90868
$515$	0	0
$516$	0	0
$517$	−7705.30	−0.655471
$518$	2006.47	0.170192
$519$	0	0
$520$	0	0
$521$	−146.772	−0.0123420	−0.00617100	−	0.999981i	$-0.501964\pi$
$521$	−146.772	−0.0123420	−0.00617100	+	0.999981i	$0.501964\pi$
$522$	0	0
$523$	2872.03	0.240125	0.120062	−	0.992766i	$-0.461691\pi$
$523$	2872.03	0.240125	0.120062	+	0.992766i	$0.461691\pi$
$524$	2301.12	0.191841
$525$	0	0
$526$	−21118.4	−1.75058
$527$	7419.46	0.613277
$528$	0	0
$529$	−5446.82	−0.447671
$530$	0	0
$531$	0	0
$532$	−12729.5	−1.03740
$533$	−22316.4	−1.81356
$534$	0	0
$535$	0	0
$536$	−7724.58	−0.622484
$537$	0	0
$538$	−6165.03	−0.494039
$539$	−21678.7	−1.73241
$540$	0	0
$541$	−3810.28	−0.302804	−0.151402	−	0.988472i	$-0.548379\pi$
$541$	−3810.28	−0.302804	−0.151402	+	0.988472i	$0.548379\pi$
$542$	−6806.24	−0.539397
$543$	0	0
$544$	−6004.60	−0.473245
$545$	0	0
$546$	0	0
$547$	−5945.59	−0.464744	−0.232372	−	0.972627i	$-0.574649\pi$
$547$	−5945.59	−0.464744	−0.232372	+	0.972627i	$0.574649\pi$
$548$	1984.63	0.154707
$549$	0	0
$550$	0	0
$551$	−27983.3	−2.16357
$552$	0	0
$553$	20183.1	1.55203
$554$	13359.4	1.02453
$555$	0	0
$556$	−10354.9	−0.789828
$557$	10872.2	0.827059	0.413530	−	0.910491i	$-0.364296\pi$
$557$	10872.2	0.827059	0.413530	+	0.910491i	$0.364296\pi$
$558$	0	0
$559$	12172.5	0.921007
$560$	0	0
$561$	0	0
$562$	15818.2	1.18728
$563$	−13575.1	−1.01620	−0.508100	−	0.861298i	$-0.669652\pi$
$563$	−13575.1	−1.01620	−0.508100	+	0.861298i	$0.669652\pi$
$564$	0	0
$565$	0	0
$566$	−11042.1	−0.820024
$567$	0	0
$568$	−6456.27	−0.476935
$569$	20642.6	1.52088	0.760442	−	0.649406i	$-0.224982\pi$
$569$	20642.6	1.52088	0.760442	+	0.649406i	$0.224982\pi$
$570$	0	0
$571$	−2730.77	−0.200139	−0.100069	−	0.994980i	$-0.531906\pi$
$571$	−2730.77	−0.200139	−0.100069	+	0.994980i	$0.531906\pi$
$572$	−16190.5	−1.18349
$573$	0	0
$574$	38911.2	2.82948
$575$	0	0
$576$	0	0
$577$	−21953.4	−1.58394	−0.791970	−	0.610560i	$-0.790945\pi$
$577$	−21953.4	−1.58394	−0.791970	+	0.610560i	$0.790945\pi$
$578$	−13399.2	−0.964245
$579$	0	0
$580$	0	0
$581$	6868.82	0.490476
$582$	0	0
$583$	−8845.40	−0.628369
$584$	−5179.29	−0.366988
$585$	0	0
$586$	−9856.92	−0.694856
$587$	−11055.3	−0.777348	−0.388674	−	0.921375i	$-0.627067\pi$
$587$	−11055.3	−0.777348	−0.388674	+	0.921375i	$0.627067\pi$
$588$	0	0
$589$	25245.5	1.76608
$590$	0	0
$591$	0	0
$592$	1787.43	0.124092
$593$	−16538.0	−1.14525	−0.572625	−	0.819817i	$-0.694075\pi$
$593$	−16538.0	−1.14525	−0.572625	+	0.819817i	$0.694075\pi$
$594$	0	0
$595$	0	0
$596$	−5604.18	−0.385161
$597$	0	0
$598$	−14844.7	−1.01512
$599$	−6403.59	−0.436800	−0.218400	−	0.975859i	$-0.570084\pi$
$599$	−6403.59	−0.436800	−0.218400	+	0.975859i	$0.570084\pi$
$600$	0	0
$601$	−14210.4	−0.964480	−0.482240	−	0.876039i	$-0.660177\pi$
$601$	−14210.4	−0.964480	−0.482240	+	0.876039i	$0.660177\pi$
$602$	−21224.3	−1.43694
$603$	0	0
$604$	9528.72	0.641917
$605$	0	0
$606$	0	0
$607$	8915.72	0.596174	0.298087	−	0.954539i	$-0.403651\pi$
$607$	8915.72	0.596174	0.298087	+	0.954539i	$0.403651\pi$
$608$	−20431.3	−1.36283
$609$	0	0
$610$	0	0
$611$	5549.66	0.367455
$612$	0	0
$613$	15372.5	1.01287	0.506436	−	0.862277i	$-0.330963\pi$
$613$	15372.5	1.01287	0.506436	+	0.862277i	$0.330963\pi$
$614$	24100.3	1.58405
$615$	0	0
$616$	−22924.2	−1.49942
$617$	−17327.5	−1.13060	−0.565298	−	0.824887i	$-0.691239\pi$
$617$	−17327.5	−1.13060	−0.565298	+	0.824887i	$0.691239\pi$
$618$	0	0
$619$	−28787.3	−1.86924	−0.934621	−	0.355645i	$-0.884261\pi$
$619$	−28787.3	−1.86924	−0.934621	+	0.355645i	$0.884261\pi$
$620$	0	0
$621$	0	0
$622$	8524.73	0.549535
$623$	−4511.71	−0.290141
$624$	0	0
$625$	0	0
$626$	−13752.8	−0.878068
$627$	0	0
$628$	10345.4	0.657369
$629$	−746.047	−0.0472923
$630$	0	0
$631$	18091.0	1.14135	0.570675	−	0.821176i	$-0.306682\pi$
$631$	18091.0	1.14135	0.570675	+	0.821176i	$0.306682\pi$
$632$	10024.8	0.630955
$633$	0	0
$634$	−34863.8	−2.18394
$635$	0	0
$636$	0	0
$637$	15613.9	0.971184
$638$	62057.7	3.85092
$639$	0	0
$640$	0	0
$641$	19577.5	1.20634	0.603171	−	0.797612i	$-0.293904\pi$
$641$	19577.5	1.20634	0.603171	+	0.797612i	$0.293904\pi$
$642$	0	0
$643$	29971.7	1.83821	0.919106	−	0.394011i	$-0.128913\pi$
$643$	29971.7	1.83821	0.919106	+	0.394011i	$0.128913\pi$
$644$	9204.46	0.563209
$645$	0	0
$646$	13309.7	0.810626
$647$	3635.64	0.220915	0.110457	−	0.993881i	$-0.464768\pi$
$647$	3635.64	0.220915	0.110457	+	0.993881i	$0.464768\pi$
$648$	0	0
$649$	−12208.4	−0.738399
$650$	0	0
$651$	0	0
$652$	−15010.3	−0.901610
$653$	19407.3	1.16304	0.581520	−	0.813532i	$-0.302458\pi$
$653$	19407.3	1.16304	0.581520	+	0.813532i	$0.302458\pi$
$654$	0	0
$655$	0	0
$656$	34663.3	2.06307
$657$	0	0
$658$	−9676.49	−0.573296
$659$	−5627.86	−0.332671	−0.166335	−	0.986069i	$-0.553193\pi$
$659$	−5627.86	−0.332671	−0.166335	+	0.986069i	$0.553193\pi$
$660$	0	0
$661$	−16663.0	−0.980507	−0.490253	−	0.871580i	$-0.663096\pi$
$661$	−16663.0	−0.980507	−0.490253	+	0.871580i	$0.663096\pi$
$662$	14668.4	0.861182
$663$	0	0
$664$	3411.68	0.199396
$665$	0	0
$666$	0	0
$667$	20234.1	1.17461
$668$	7987.48	0.462642
$669$	0	0
$670$	0	0
$671$	5668.47	0.326124
$672$	0	0
$673$	12638.8	0.723906	0.361953	−	0.932196i	$-0.382110\pi$
$673$	12638.8	0.723906	0.361953	+	0.932196i	$0.382110\pi$
$674$	−34195.1	−1.95422
$675$	0	0
$676$	1961.50	0.111601
$677$	25316.0	1.43718	0.718591	−	0.695433i	$-0.244787\pi$
$677$	25316.0	1.43718	0.718591	+	0.695433i	$0.244787\pi$
$678$	0	0
$679$	−22427.9	−1.26760
$680$	0	0
$681$	0	0
$682$	−55986.3	−3.14344
$683$	−24818.4	−1.39041	−0.695204	−	0.718812i	$-0.744686\pi$
$683$	−24818.4	−1.39041	−0.695204	+	0.718812i	$0.744686\pi$
$684$	0	0
$685$	0	0
$686$	3511.82	0.195455
$687$	0	0
$688$	−18907.2	−1.04772
$689$	6370.81	0.352262
$690$	0	0
$691$	19958.0	1.09875	0.549375	−	0.835576i	$-0.314866\pi$
$691$	19958.0	1.09875	0.549375	+	0.835576i	$0.314866\pi$
$692$	5860.35	0.321932
$693$	0	0
$694$	19140.9	1.04694
$695$	0	0
$696$	0	0
$697$	−14468.0	−0.786247
$698$	37250.0	2.01996
$699$	0	0
$700$	0	0
$701$	−5624.21	−0.303029	−0.151515	−	0.988455i	$-0.548415\pi$
$701$	−5624.21	−0.303029	−0.151515	+	0.988455i	$0.548415\pi$
$702$	0	0
$703$	−2538.51	−0.136190
$704$	−259.697	−0.0139030
$705$	0	0
$706$	33976.8	1.81124
$707$	−26094.6	−1.38810
$708$	0	0
$709$	−7715.43	−0.408687	−0.204344	−	0.978899i	$-0.565506\pi$
$709$	−7715.43	−0.408687	−0.204344	+	0.978899i	$0.565506\pi$
$710$	0	0
$711$	0	0
$712$	−2240.92	−0.117952
$713$	−18254.5	−0.958817
$714$	0	0
$715$	0	0
$716$	−1981.11	−0.103404
$717$	0	0
$718$	−10578.1	−0.549821
$719$	21647.8	1.12285	0.561424	−	0.827528i	$-0.310254\pi$
$719$	21647.8	1.12285	0.561424	+	0.827528i	$0.310254\pi$
$720$	0	0
$721$	−45531.2	−2.35183
$722$	21120.3	1.08867
$723$	0	0
$724$	−15233.8	−0.781989
$725$	0	0
$726$	0	0
$727$	3071.30	0.156682	0.0783412	−	0.996927i	$-0.475038\pi$
$727$	3071.30	0.156682	0.0783412	+	0.996927i	$0.475038\pi$
$728$	16510.9	0.840570
$729$	0	0
$730$	0	0
$731$	7891.61	0.399291
$732$	0	0
$733$	−14652.0	−0.738314	−0.369157	−	0.929367i	$-0.620354\pi$
$733$	−14652.0	−0.738314	−0.369157	+	0.929367i	$0.620354\pi$
$734$	4049.45	0.203635
$735$	0	0
$736$	14773.4	0.739886
$737$	−43634.7	−2.18087
$738$	0	0
$739$	17610.9	0.876629	0.438314	−	0.898822i	$-0.355576\pi$
$739$	17610.9	0.876629	0.438314	+	0.898822i	$0.355576\pi$
$740$	0	0
$741$	0	0
$742$	−11108.3	−0.549592
$743$	16788.2	0.828935	0.414468	−	0.910064i	$-0.363968\pi$
$743$	16788.2	0.828935	0.414468	+	0.910064i	$0.363968\pi$
$744$	0	0
$745$	0	0
$746$	1808.30	0.0887485
$747$	0	0
$748$	−10496.5	−0.513087
$749$	−51050.5	−2.49045
$750$	0	0
$751$	23514.2	1.14254	0.571269	−	0.820763i	$-0.306451\pi$
$751$	23514.2	1.14254	0.571269	+	0.820763i	$0.306451\pi$
$752$	−8620.11	−0.418009
$753$	0	0
$754$	−44696.4	−2.15882
$755$	0	0
$756$	0	0
$757$	−24817.4	−1.19155	−0.595775	−	0.803151i	$-0.703155\pi$
$757$	−24817.4	−1.19155	−0.595775	+	0.803151i	$0.703155\pi$
$758$	44254.8	2.12059
$759$	0	0
$760$	0	0
$761$	32517.7	1.54897	0.774485	−	0.632592i	$-0.218009\pi$
$761$	32517.7	1.54897	0.774485	+	0.632592i	$0.218009\pi$
$762$	0	0
$763$	5382.89	0.255404
$764$	−7028.91	−0.332850
$765$	0	0
$766$	965.398	0.0455369
$767$	8792.96	0.413944
$768$	0	0
$769$	18057.8	0.846791	0.423395	−	0.905945i	$-0.360838\pi$
$769$	18057.8	0.846791	0.423395	+	0.905945i	$0.360838\pi$
$770$	0	0
$771$	0	0
$772$	−1957.11	−0.0912411
$773$	−38147.2	−1.77498	−0.887491	−	0.460825i	$-0.847553\pi$
$773$	−38147.2	−1.77498	−0.887491	+	0.460825i	$0.847553\pi$
$774$	0	0
$775$	0	0
$776$	−11139.7	−0.515326
$777$	0	0
$778$	−24123.2	−1.11164
$779$	−49228.9	−2.26420
$780$	0	0
$781$	−36470.2	−1.67094
$782$	−9623.98	−0.440093
$783$	0	0
$784$	−24252.5	−1.10480
$785$	0	0
$786$	0	0
$787$	4801.76	0.217489	0.108745	−	0.994070i	$-0.465317\pi$
$787$	4801.76	0.217489	0.108745	+	0.994070i	$0.465317\pi$
$788$	922.673	0.0417118
$789$	0	0
$790$	0	0
$791$	49.4640	0.00222343
$792$	0	0
$793$	−4082.66	−0.182824
$794$	42695.6	1.90833
$795$	0	0
$796$	−2118.77	−0.0943442
$797$	15471.0	0.687592	0.343796	−	0.939044i	$-0.388287\pi$
$797$	15471.0	0.687592	0.343796	+	0.939044i	$0.388287\pi$
$798$	0	0
$799$	3597.92	0.159306
$800$	0	0
$801$	0	0
$802$	−10629.6	−0.468012
$803$	−29256.8	−1.28574
$804$	0	0
$805$	0	0
$806$	40323.6	1.76220
$807$	0	0
$808$	−12960.9	−0.564312
$809$	−17007.6	−0.739129	−0.369564	−	0.929205i	$-0.620493\pi$
$809$	−17007.6	−0.739129	−0.369564	+	0.929205i	$0.620493\pi$
$810$	0	0
$811$	−7552.52	−0.327010	−0.163505	−	0.986543i	$-0.552280\pi$
$811$	−7552.52	−0.327010	−0.163505	+	0.986543i	$0.552280\pi$
$812$	27714.1	1.19775
$813$	0	0
$814$	5629.58	0.242404
$815$	0	0
$816$	0	0
$817$	26852.1	1.14986
$818$	23026.4	0.984227
$819$	0	0
$820$	0	0
$821$	3745.20	0.159206	0.0796031	−	0.996827i	$-0.474635\pi$
$821$	3745.20	0.159206	0.0796031	+	0.996827i	$0.474635\pi$
$822$	0	0
$823$	30021.0	1.27153	0.635764	−	0.771884i	$-0.280685\pi$
$823$	30021.0	1.27153	0.635764	+	0.771884i	$0.280685\pi$
$824$	−22614.9	−0.956101
$825$	0	0
$826$	−15331.6	−0.645828
$827$	7918.68	0.332962	0.166481	−	0.986045i	$-0.446760\pi$
$827$	7918.68	0.332962	0.166481	+	0.986045i	$0.446760\pi$
$828$	0	0
$829$	−6976.17	−0.292271	−0.146135	−	0.989265i	$-0.546683\pi$
$829$	−6976.17	−0.292271	−0.146135	+	0.989265i	$0.546683\pi$
$830$	0	0
$831$	0	0
$832$	187.044	0.00779398
$833$	10122.7	0.421044
$834$	0	0
$835$	0	0
$836$	−35715.4	−1.47756
$837$	0	0
$838$	25187.4	1.03829
$839$	8657.91	0.356263	0.178131	−	0.984007i	$-0.442995\pi$
$839$	8657.91	0.356263	0.178131	+	0.984007i	$0.442995\pi$
$840$	0	0
$841$	36534.7	1.49800
$842$	52152.9	2.13457
$843$	0	0
$844$	−11620.6	−0.473930
$845$	0	0
$846$	0	0
$847$	−95643.8	−3.88000
$848$	−9895.57	−0.400726
$849$	0	0
$850$	0	0
$851$	1835.54	0.0739383
$852$	0	0
$853$	−18984.8	−0.762047	−0.381024	−	0.924565i	$-0.624428\pi$
$853$	−18984.8	−0.762047	−0.381024	+	0.924565i	$0.624428\pi$
$854$	7118.60	0.285238
$855$	0	0
$856$	−25356.3	−1.01245
$857$	36434.4	1.45225	0.726124	−	0.687564i	$-0.241320\pi$
$857$	36434.4	1.45225	0.726124	+	0.687564i	$0.241320\pi$
$858$	0	0
$859$	−17485.0	−0.694507	−0.347253	−	0.937771i	$-0.612886\pi$
$859$	−17485.0	−0.694507	−0.347253	+	0.937771i	$0.612886\pi$
$860$	0	0
$861$	0	0
$862$	8048.29	0.318011
$863$	10423.6	0.411152	0.205576	−	0.978641i	$-0.434093\pi$
$863$	10423.6	0.411152	0.205576	+	0.978641i	$0.434093\pi$
$864$	0	0
$865$	0	0
$866$	−19482.2	−0.764472
$867$	0	0
$868$	−25002.7	−0.977704
$869$	56628.0	2.21056
$870$	0	0
$871$	31427.4	1.22259
$872$	2673.63	0.103831
$873$	0	0
$874$	−32746.7	−1.26736
$875$	0	0
$876$	0	0
$877$	−7873.24	−0.303148	−0.151574	−	0.988446i	$-0.548434\pi$
$877$	−7873.24	−0.303148	−0.151574	+	0.988446i	$0.548434\pi$
$878$	−41794.1	−1.60647
$879$	0	0
$880$	0	0
$881$	−35135.0	−1.34362	−0.671809	−	0.740725i	$-0.734482\pi$
$881$	−35135.0	−1.34362	−0.671809	+	0.740725i	$0.734482\pi$
$882$	0	0
$883$	−33069.4	−1.26033	−0.630167	−	0.776460i	$-0.717013\pi$
$883$	−33069.4	−1.26033	−0.630167	+	0.776460i	$0.717013\pi$
$884$	7559.98	0.287635
$885$	0	0
$886$	53886.7	2.04329
$887$	51479.1	1.94870	0.974351	−	0.225031i	$-0.0722485\pi$
$887$	51479.1	1.94870	0.974351	+	0.225031i	$0.0722485\pi$
$888$	0	0
$889$	−29959.9	−1.13029
$890$	0	0
$891$	0	0
$892$	−14804.1	−0.555692
$893$	12242.3	0.458761
$894$	0	0
$895$	0	0
$896$	−36992.6	−1.37928
$897$	0	0
$898$	45800.9	1.70200
$899$	−54963.3	−2.03907
$900$	0	0
$901$	4130.28	0.152719
$902$	109174.	4.03003
$903$	0	0
$904$	24.5683	0.000903904 0
$905$	0	0
$906$	0	0
$907$	10809.8	0.395737	0.197868	−	0.980229i	$-0.436598\pi$
$907$	10809.8	0.395737	0.197868	+	0.980229i	$0.436598\pi$
$908$	−3049.64	−0.111460
$909$	0	0
$910$	0	0
$911$	31539.8	1.14705	0.573523	−	0.819190i	$-0.305576\pi$
$911$	31539.8	1.14705	0.573523	+	0.819190i	$0.305576\pi$
$912$	0	0
$913$	19271.9	0.698585
$914$	−50427.8	−1.82495
$915$	0	0
$916$	−2949.72	−0.106399
$917$	−13255.9	−0.477368
$918$	0	0
$919$	3592.53	0.128952	0.0644759	−	0.997919i	$-0.479462\pi$
$919$	3592.53	0.128952	0.0644759	+	0.997919i	$0.479462\pi$
$920$	0	0
$921$	0	0
$922$	15116.9	0.539966
$923$	26267.3	0.936727
$924$	0	0
$925$	0	0
$926$	51126.7	1.81439
$927$	0	0
$928$	44482.0	1.57348
$929$	31614.7	1.11652	0.558259	−	0.829667i	$-0.311470\pi$
$929$	31614.7	1.11652	0.558259	+	0.829667i	$0.311470\pi$
$930$	0	0
$931$	34443.5	1.21250
$932$	4152.36	0.145939
$933$	0	0
$934$	26049.2	0.912588
$935$	0	0
$936$	0	0
$937$	20891.5	0.728382	0.364191	−	0.931324i	$-0.381346\pi$
$937$	20891.5	0.728382	0.364191	+	0.931324i	$0.381346\pi$
$938$	−54797.5	−1.90746
$939$	0	0
$940$	0	0
$941$	37769.2	1.30844	0.654219	−	0.756305i	$-0.272998\pi$
$941$	37769.2	1.30844	0.654219	+	0.756305i	$0.272998\pi$
$942$	0	0
$943$	35596.4	1.22924
$944$	−13657.8	−0.470894
$945$	0	0
$946$	−59549.1	−2.04663
$947$	10430.8	0.357927	0.178963	−	0.983856i	$-0.442726\pi$
$947$	10430.8	0.357927	0.178963	+	0.983856i	$0.442726\pi$
$948$	0	0
$949$	21071.9	0.720784
$950$	0	0
$951$	0	0
$952$	10704.2	0.364419
$953$	33865.3	1.15111	0.575553	−	0.817764i	$-0.304787\pi$
$953$	33865.3	1.15111	0.575553	+	0.817764i	$0.304787\pi$
$954$	0	0
$955$	0	0
$956$	4889.06	0.165401
$957$	0	0
$958$	47601.7	1.60537
$959$	−11432.7	−0.384965
$960$	0	0
$961$	19794.9	0.664461
$962$	−4054.65	−0.135891
$963$	0	0
$964$	5425.28	0.181262
$965$	0	0
$966$	0	0
$967$	46581.8	1.54909	0.774544	−	0.632520i	$-0.217979\pi$
$967$	46581.8	1.54909	0.774544	+	0.632520i	$0.217979\pi$
$968$	−47505.4	−1.57736
$969$	0	0
$970$	0	0
$971$	13829.5	0.457065	0.228532	−	0.973536i	$-0.426607\pi$
$971$	13829.5	0.457065	0.228532	+	0.973536i	$0.426607\pi$
$972$	0	0
$973$	59650.5	1.96537
$974$	39251.6	1.29127
$975$	0	0
$976$	6341.46	0.207977
$977$	−15943.6	−0.522089	−0.261045	−	0.965327i	$-0.584067\pi$
$977$	−15943.6	−0.522089	−0.261045	+	0.965327i	$0.584067\pi$
$978$	0	0
$979$	−12658.5	−0.413247
$980$	0	0
$981$	0	0
$982$	7091.85	0.230458
$983$	29111.6	0.944572	0.472286	−	0.881445i	$-0.343429\pi$
$983$	29111.6	0.944572	0.472286	+	0.881445i	$0.343429\pi$
$984$	0	0
$985$	0	0
$986$	−28977.3	−0.935927
$987$	0	0
$988$	25723.7	0.828318
$989$	−19416.1	−0.624265
$990$	0	0
$991$	8745.85	0.280344	0.140172	−	0.990127i	$-0.455234\pi$
$991$	8745.85	0.280344	0.140172	+	0.990127i	$0.455234\pi$
$992$	−40130.1	−1.28441
$993$	0	0
$994$	−45800.2	−1.46146
$995$	0	0
$996$	0	0
$997$	−45026.9	−1.43031	−0.715153	−	0.698968i	$-0.753643\pi$
$997$	−45026.9	−1.43031	−0.715153	+	0.698968i	$0.753643\pi$
$998$	25908.0	0.821748
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.r.1.3		3
3.2	odd	2		2025.4.a.p.1.1		3
5.4	even	2		405.4.a.g.1.1	✓	3
15.14	odd	2		405.4.a.i.1.3	yes	3
45.4	even	6		405.4.e.u.136.3		6
45.14	odd	6		405.4.e.s.136.1		6
45.29	odd	6		405.4.e.s.271.1		6
45.34	even	6		405.4.e.u.271.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.g.1.1	✓	3	5.4	even	2
405.4.a.i.1.3	yes	3	15.14	odd	2
405.4.e.s.136.1		6	45.14	odd	6
405.4.e.s.271.1		6	45.29	odd	6
405.4.e.u.136.3		6	45.4	even	6
405.4.e.u.271.3		6	45.34	even	6
2025.4.a.p.1.1		3	3.2	odd	2
2025.4.a.r.1.3		3	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q+3.52348 q^{2} +4.41489 q^{4} -25.4325 q^{7} -12.6321 q^{8} +O(q^{10})\)	\(q+3.52348 q^{2} +4.41489 q^{4} -25.4325 q^{7} -12.6321 q^{8} -71.3561 q^{11} +51.3935 q^{13} -89.6107 q^{14} -79.8279 q^{16} +33.3191 q^{17} +113.372 q^{19} -251.422 q^{22} -81.9767 q^{23} +181.084 q^{26} -112.281 q^{28} -246.827 q^{29} +222.679 q^{31} -180.215 q^{32} +117.399 q^{34} -22.3910 q^{37} +399.463 q^{38} -434.225 q^{41} +236.850 q^{43} -315.029 q^{44} -288.843 q^{46} +107.984 q^{47} +303.810 q^{49} +226.896 q^{52} +123.961 q^{53} +321.265 q^{56} -869.690 q^{58} +171.091 q^{59} -79.4391 q^{61} +784.604 q^{62} +3.63945 q^{64} +611.506 q^{67} +147.100 q^{68} +511.102 q^{71} +410.012 q^{73} -78.8941 q^{74} +500.524 q^{76} +1814.76 q^{77} -793.597 q^{79} -1529.98 q^{82} -270.081 q^{83} +834.534 q^{86} +901.376 q^{88} +177.400 q^{89} -1307.06 q^{91} -361.918 q^{92} +380.478 q^{94} +881.860 q^{97} +1070.47 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} + 25 q^{7} - 27 q^{8}+O(q^{10}) \) 3 * q + q^2 + 5 * q^4 + 25 * q^7 - 27 * q^8	\( 3 q + q^{2} + 5 q^{4} + 25 q^{7} - 27 q^{8} - 58 q^{11} + 47 q^{13} - 159 q^{14} - 127 q^{16} + 34 q^{17} - 5 q^{19} - 260 q^{22} - 51 q^{23} - 253 q^{26} - 83 q^{28} - 350 q^{29} + 638 q^{31} + 245 q^{32} - 154 q^{34} + 414 q^{37} + 397 q^{38} - 179 q^{41} + 836 q^{43} - 332 q^{44} + 261 q^{46} + 235 q^{47} + 892 q^{49} + 1335 q^{52} + 505 q^{53} - 15 q^{56} - 1876 q^{58} - 535 q^{59} - 104 q^{61} + 348 q^{62} - 303 q^{64} + 40 q^{67} + 830 q^{68} - 452 q^{71} + 710 q^{73} - 1394 q^{74} + 849 q^{76} + 2148 q^{77} - 634 q^{79} - 613 q^{82} + 1734 q^{83} - 460 q^{86} + 768 q^{88} + 852 q^{89} - 1229 q^{91} - 1839 q^{92} + 1751 q^{94} + 38 q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 + 25 * q^7 - 27 * q^8 - 58 * q^11 + 47 * q^13 - 159 * q^14 - 127 * q^16 + 34 * q^17 - 5 * q^19 - 260 * q^22 - 51 * q^23 - 253 * q^26 - 83 * q^28 - 350 * q^29 + 638 * q^31 + 245 * q^32 - 154 * q^34 + 414 * q^37 + 397 * q^38 - 179 * q^41 + 836 * q^43 - 332 * q^44 + 261 * q^46 + 235 * q^47 + 892 * q^49 + 1335 * q^52 + 505 * q^53 - 15 * q^56 - 1876 * q^58 - 535 * q^59 - 104 * q^61 + 348 * q^62 - 303 * q^64 + 40 * q^67 + 830 * q^68 - 452 * q^71 + 710 * q^73 - 1394 * q^74 + 849 * q^76 + 2148 * q^77 - 634 * q^79 - 613 * q^82 + 1734 * q^83 - 460 * q^86 + 768 * q^88 + 852 * q^89 - 1229 * q^91 - 1839 * q^92 + 1751 * q^94 + 38 * q^98

Introduction

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