LMFDB - Embedded newform 2025.4.a.bf.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2025,4,Mod(1,2025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [12,-4,0,48,0,0,6,-45,0,0,29,0,24,-69] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	yes
Analytic conductor:	$119.478867762$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 64 x^{10} + 241 x^{9} + 1452 x^{8} - 5130 x^{7} - 13834 x^{6} + 45247 x^{5} + \cdots + 73008 $ x^12 - 4x^11 - 64x^10 + 241x^9 + 1452x^8 - 5130x^7 - 13834x^6 + 45247x^5 + 52669x^4 - 144610x^3 - 96816x^2 + 143136*x + 73008
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.63978$ 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)

$f(q)$	$=$	$q-3.63978 q^{2} +5.24803 q^{4} -3.72054 q^{7} +10.0166 q^{8} -28.0133 q^{11} -43.6420 q^{13} +13.5420 q^{14} -78.4424 q^{16} -87.3128 q^{17} +38.5062 q^{19} +101.962 q^{22} +115.947 q^{23} +158.848 q^{26} -19.5255 q^{28} +196.049 q^{29} +173.535 q^{31} +205.381 q^{32} +317.800 q^{34} +181.738 q^{37} -140.154 q^{38} -249.785 q^{41} +460.708 q^{43} -147.015 q^{44} -422.021 q^{46} -328.392 q^{47} -329.158 q^{49} -229.035 q^{52} -667.821 q^{53} -37.2671 q^{56} -713.577 q^{58} +111.117 q^{59} +437.946 q^{61} -631.630 q^{62} -120.003 q^{64} -154.845 q^{67} -458.221 q^{68} +912.989 q^{71} -975.779 q^{73} -661.485 q^{74} +202.082 q^{76} +104.225 q^{77} +856.040 q^{79} +909.164 q^{82} -68.4209 q^{83} -1676.88 q^{86} -280.597 q^{88} +665.452 q^{89} +162.372 q^{91} +608.492 q^{92} +1195.27 q^{94} +1403.67 q^{97} +1198.06 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{2} + 48 q^{4} + 6 q^{7} - 45 q^{8} + 29 q^{11} + 24 q^{13} - 69 q^{14} + 192 q^{16} - 79 q^{17} - 75 q^{19} - 18 q^{22} - 318 q^{23} - 154 q^{26} + 96 q^{28} + 106 q^{29} + 60 q^{31} - 914 q^{32}+ \cdots - 7511 q^{98}+O(q^{100}) $ 12 * q - 4 * q^2 + 48 * q^4 + 6 * q^7 - 45 * q^8 + 29 * q^11 + 24 * q^13 - 69 * q^14 + 192 * q^16 - 79 * q^17 - 75 * q^19 - 18 * q^22 - 318 * q^23 - 154 * q^26 + 96 * q^28 + 106 * q^29 + 60 * q^31 - 914 * q^32 - 108 * q^34 - 84 * q^37 - 640 * q^38 - 353 * q^41 - 426 * q^43 + 571 * q^44 + 270 * q^46 - 1210 * q^47 + 666 * q^49 - 75 * q^52 - 448 * q^53 - 570 * q^56 + 594 * q^58 + 482 * q^59 + 402 * q^61 - 2544 * q^62 + 975 * q^64 - 201 * q^67 - 3437 * q^68 - 944 * q^71 - 453 * q^73 + 10 * q^74 - 462 * q^76 - 2652 * q^77 + 258 * q^79 + 873 * q^82 - 3012 * q^83 + 1952 * q^86 - 216 * q^88 - 738 * q^89 - 618 * q^91 - 5232 * q^92 + 63 * q^94 - 318 * q^97 - 7511 * q^98

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 $

$2$	−3.63978	−1.28686	−0.643429	−	0.765506i	$-0.722489\pi$
$2$	−3.63978	−1.28686	−0.643429	+	0.765506i	$0.722489\pi$
$3$	0	0
$3$	0	0
$4$	5.24803	0.656004
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.72054	−0.200890	−0.100445	−	0.994943i	$-0.532027\pi$
$7$	−3.72054	−0.200890	−0.100445	+	0.994943i	$0.532027\pi$
$8$	10.0166	0.442674
$9$	0	0
$10$	0	0
$11$	−28.0133	−0.767847	−0.383923	−	0.923365i	$-0.625427\pi$
$11$	−28.0133	−0.767847	−0.383923	+	0.923365i	$0.625427\pi$
$12$	0	0
$13$	−43.6420	−0.931086	−0.465543	−	0.885025i	$-0.654141\pi$
$13$	−43.6420	−0.931086	−0.465543	+	0.885025i	$0.654141\pi$
$14$	13.5420	0.258517
$15$	0	0
$16$	−78.4424	−1.22566
$17$	−87.3128	−1.24567	−0.622837	−	0.782351i	$-0.714020\pi$
$17$	−87.3128	−1.24567	−0.622837	+	0.782351i	$0.714020\pi$
$18$	0	0
$19$	38.5062	0.464944	0.232472	−	0.972603i	$-0.425319\pi$
$19$	38.5062	0.464944	0.232472	+	0.972603i	$0.425319\pi$
$20$	0	0
$21$	0	0
$22$	101.962	0.988110
$23$	115.947	1.05115	0.525577	−	0.850746i	$-0.323849\pi$
$23$	115.947	1.05115	0.525577	+	0.850746i	$0.323849\pi$
$24$	0	0
$25$	0	0
$26$	158.848	1.19818
$27$	0	0
$28$	−19.5255	−0.131785
$29$	196.049	1.25536	0.627679	−	0.778472i	$-0.284005\pi$
$29$	196.049	1.25536	0.627679	+	0.778472i	$0.284005\pi$
$30$	0	0
$31$	173.535	1.00541	0.502707	−	0.864457i	$-0.332338\pi$
$31$	173.535	1.00541	0.502707	+	0.864457i	$0.332338\pi$
$32$	205.381	1.13458
$33$	0	0
$34$	317.800	1.60301
$35$	0	0
$36$	0	0
$37$	181.738	0.807499	0.403749	−	0.914870i	$-0.367707\pi$
$37$	181.738	0.807499	0.403749	+	0.914870i	$0.367707\pi$
$38$	−140.154	−0.598317
$39$	0	0
$40$	0	0
$41$	−249.785	−0.951460	−0.475730	−	0.879591i	$-0.657816\pi$
$41$	−249.785	−0.951460	−0.475730	+	0.879591i	$0.657816\pi$
$42$	0	0
$43$	460.708	1.63389	0.816945	−	0.576715i	$-0.195666\pi$
$43$	460.708	1.63389	0.816945	+	0.576715i	$0.195666\pi$
$44$	−147.015	−0.503711
$45$	0	0
$46$	−422.021	−1.35269
$47$	−328.392	−1.01917	−0.509583	−	0.860421i	$-0.670200\pi$
$47$	−328.392	−1.01917	−0.509583	+	0.860421i	$0.670200\pi$
$48$	0	0
$49$	−329.158	−0.959643
$50$	0	0
$51$	0	0
$52$	−229.035	−0.610797
$53$	−667.821	−1.73080	−0.865399	−	0.501083i	$-0.832935\pi$
$53$	−667.821	−1.73080	−0.865399	+	0.501083i	$0.832935\pi$
$54$	0	0
$55$	0	0
$56$	−37.2671	−0.0889289
$57$	0	0
$58$	−713.577	−1.61547
$59$	111.117	0.245191	0.122595	−	0.992457i	$-0.460878\pi$
$59$	111.117	0.245191	0.122595	+	0.992457i	$0.460878\pi$
$60$	0	0
$61$	437.946	0.919233	0.459617	−	0.888117i	$-0.347987\pi$
$61$	437.946	0.919233	0.459617	+	0.888117i	$0.347987\pi$
$62$	−631.630	−1.29382
$63$	0	0
$64$	−120.003	−0.234381
$65$	0	0
$66$	0	0
$67$	−154.845	−0.282348	−0.141174	−	0.989985i	$-0.545088\pi$
$67$	−154.845	−0.282348	−0.141174	+	0.989985i	$0.545088\pi$
$68$	−458.221	−0.817168
$69$	0	0
$70$	0	0
$71$	912.989	1.52608	0.763041	−	0.646350i	$-0.223705\pi$
$71$	912.989	1.52608	0.763041	+	0.646350i	$0.223705\pi$
$72$	0	0
$73$	−975.779	−1.56447	−0.782234	−	0.622984i	$-0.785920\pi$
$73$	−975.779	−1.56447	−0.782234	+	0.622984i	$0.785920\pi$
$74$	−661.485	−1.03914
$75$	0	0
$76$	202.082	0.305005
$77$	104.225	0.154253
$78$	0	0
$79$	856.040	1.21914	0.609570	−	0.792733i	$-0.291342\pi$
$79$	856.040	1.21914	0.609570	+	0.792733i	$0.291342\pi$
$80$	0	0
$81$	0	0
$82$	909.164	1.22439
$83$	−68.4209	−0.0904840	−0.0452420	−	0.998976i	$-0.514406\pi$
$83$	−68.4209	−0.0904840	−0.0452420	+	0.998976i	$0.514406\pi$
$84$	0	0
$85$	0	0
$86$	−1676.88	−2.10259
$87$	0	0
$88$	−280.597	−0.339906
$89$	665.452	0.792559	0.396280	−	0.918130i	$-0.370301\pi$
$89$	665.452	0.792559	0.396280	+	0.918130i	$0.370301\pi$
$90$	0	0
$91$	162.372	0.187046
$92$	608.492	0.689562
$93$	0	0
$94$	1195.27	1.31152
$95$	0	0
$96$	0	0
$97$	1403.67	1.46929	0.734646	−	0.678450i	$-0.237348\pi$
$97$	1403.67	1.46929	0.734646	+	0.678450i	$0.237348\pi$
$98$	1198.06	1.23492
$99$	0	0
$100$	0	0
$101$	−655.178	−0.645472	−0.322736	−	0.946489i	$-0.604603\pi$
$101$	−655.178	−0.645472	−0.322736	+	0.946489i	$0.604603\pi$
$102$	0	0
$103$	−253.523	−0.242528	−0.121264	−	0.992620i	$-0.538695\pi$
$103$	−253.523	−0.242528	−0.121264	+	0.992620i	$0.538695\pi$
$104$	−437.143	−0.412168
$105$	0	0
$106$	2430.73	2.22729
$107$	−1132.73	−1.02341	−0.511706	−	0.859161i	$-0.670986\pi$
$107$	−1132.73	−1.02341	−0.511706	+	0.859161i	$0.670986\pi$
$108$	0	0
$109$	1514.68	1.33101	0.665506	−	0.746392i	$-0.268216\pi$
$109$	1514.68	1.33101	0.665506	+	0.746392i	$0.268216\pi$
$110$	0	0
$111$	0	0
$112$	291.848	0.246224
$113$	−1672.03	−1.39196	−0.695980	−	0.718061i	$-0.745030\pi$
$113$	−1672.03	−1.39196	−0.695980	+	0.718061i	$0.745030\pi$
$114$	0	0
$115$	0	0
$116$	1028.87	0.823521
$117$	0	0
$118$	−404.443	−0.315526
$119$	324.851	0.250244
$120$	0	0
$121$	−546.258	−0.410411
$122$	−1594.03	−1.18292
$123$	0	0
$124$	910.718	0.659555
$125$	0	0
$126$	0	0
$127$	547.016	0.382203	0.191102	−	0.981570i	$-0.438794\pi$
$127$	547.016	0.382203	0.191102	+	0.981570i	$0.438794\pi$
$128$	−1206.26	−0.832965
$129$	0	0
$130$	0	0
$131$	1267.97	0.845673	0.422836	−	0.906206i	$-0.361035\pi$
$131$	1267.97	0.845673	0.422836	+	0.906206i	$0.361035\pi$
$132$	0	0
$133$	−143.264	−0.0934028
$134$	563.602	0.363342
$135$	0	0
$136$	−874.575	−0.551428
$137$	−930.444	−0.580242	−0.290121	−	0.956990i	$-0.593696\pi$
$137$	−930.444	−0.580242	−0.290121	+	0.956990i	$0.593696\pi$
$138$	0	0
$139$	3130.27	1.91011	0.955057	−	0.296423i	$-0.0957939\pi$
$139$	3130.27	1.91011	0.955057	+	0.296423i	$0.0957939\pi$
$140$	0	0
$141$	0	0
$142$	−3323.08	−1.96385
$143$	1222.56	0.714932
$144$	0	0
$145$	0	0
$146$	3551.62	2.01325
$147$	0	0
$148$	953.765	0.529723
$149$	−1631.36	−0.896953	−0.448476	−	0.893795i	$-0.648033\pi$
$149$	−1631.36	−0.896953	−0.448476	+	0.893795i	$0.648033\pi$
$150$	0	0
$151$	1821.64	0.981739	0.490870	−	0.871233i	$-0.336679\pi$
$151$	1821.64	0.981739	0.490870	+	0.871233i	$0.336679\pi$
$152$	385.700	0.205819
$153$	0	0
$154$	−379.355	−0.198502
$155$	0	0
$156$	0	0
$157$	2835.18	1.44122	0.720612	−	0.693338i	$-0.243861\pi$
$157$	2835.18	1.44122	0.720612	+	0.693338i	$0.243861\pi$
$158$	−3115.80	−1.56886
$159$	0	0
$160$	0	0
$161$	−431.385	−0.211167
$162$	0	0
$163$	−3048.62	−1.46495	−0.732473	−	0.680796i	$-0.761634\pi$
$163$	−3048.62	−1.46495	−0.732473	+	0.680796i	$0.761634\pi$
$164$	−1310.88	−0.624162
$165$	0	0
$166$	249.037	0.116440
$167$	2837.92	1.31500	0.657500	−	0.753454i	$-0.271614\pi$
$167$	2837.92	1.31500	0.657500	+	0.753454i	$0.271614\pi$
$168$	0	0
$169$	−292.373	−0.133078
$170$	0	0
$171$	0	0
$172$	2417.81	1.07184
$173$	−510.534	−0.224365	−0.112183	−	0.993688i	$-0.535784\pi$
$173$	−510.534	−0.224365	−0.112183	+	0.993688i	$0.535784\pi$
$174$	0	0
$175$	0	0
$176$	2197.43	0.941121
$177$	0	0
$178$	−2422.10	−1.01991
$179$	−2274.36	−0.949687	−0.474843	−	0.880070i	$-0.657495\pi$
$179$	−2274.36	−0.949687	−0.474843	+	0.880070i	$0.657495\pi$
$180$	0	0
$181$	−975.489	−0.400594	−0.200297	−	0.979735i	$-0.564191\pi$
$181$	−975.489	−0.400594	−0.200297	+	0.979735i	$0.564191\pi$
$182$	−590.999	−0.240702
$183$	0	0
$184$	1161.39	0.465319
$185$	0	0
$186$	0	0
$187$	2445.92	0.956487
$188$	−1723.41	−0.668578
$189$	0	0
$190$	0	0
$191$	−1880.03	−0.712219	−0.356110	−	0.934444i	$-0.615897\pi$
$191$	−1880.03	−0.712219	−0.356110	+	0.934444i	$0.615897\pi$
$192$	0	0
$193$	613.562	0.228835	0.114418	−	0.993433i	$-0.463500\pi$
$193$	613.562	0.228835	0.114418	+	0.993433i	$0.463500\pi$
$194$	−5109.07	−1.89077
$195$	0	0
$196$	−1727.43	−0.629530
$197$	68.3454	0.0247178	0.0123589	−	0.999924i	$-0.496066\pi$
$197$	68.3454	0.0247178	0.0123589	+	0.999924i	$0.496066\pi$
$198$	0	0
$199$	2174.87	0.774736	0.387368	−	0.921925i	$-0.373384\pi$
$199$	2174.87	0.774736	0.387368	+	0.921925i	$0.373384\pi$
$200$	0	0
$201$	0	0
$202$	2384.71	0.830630
$203$	−729.409	−0.252190
$204$	0	0
$205$	0	0
$206$	922.769	0.312099
$207$	0	0
$208$	3423.39	1.14120
$209$	−1078.69	−0.357006
$210$	0	0
$211$	−475.041	−0.154991	−0.0774957	−	0.996993i	$-0.524692\pi$
$211$	−475.041	−0.154991	−0.0774957	+	0.996993i	$0.524692\pi$
$212$	−3504.75	−1.13541
$213$	0	0
$214$	4122.89	1.31699
$215$	0	0
$216$	0	0
$217$	−645.644	−0.201978
$218$	−5513.12	−1.71282
$219$	0	0
$220$	0	0
$221$	3810.51	1.15983
$222$	0	0
$223$	−2966.35	−0.890768	−0.445384	−	0.895340i	$-0.646933\pi$
$223$	−2966.35	−0.890768	−0.445384	+	0.895340i	$0.646933\pi$
$224$	−764.129	−0.227926
$225$	0	0
$226$	6085.84	1.79126
$227$	−2832.23	−0.828113	−0.414056	−	0.910251i	$-0.635888\pi$
$227$	−2832.23	−0.828113	−0.414056	+	0.910251i	$0.635888\pi$
$228$	0	0
$229$	−4292.45	−1.23866	−0.619330	−	0.785131i	$-0.712596\pi$
$229$	−4292.45	−1.23866	−0.619330	+	0.785131i	$0.712596\pi$
$230$	0	0
$231$	0	0
$232$	1963.74	0.555715
$233$	−4267.52	−1.19989	−0.599945	−	0.800041i	$-0.704811\pi$
$233$	−4267.52	−1.19989	−0.599945	+	0.800041i	$0.704811\pi$
$234$	0	0
$235$	0	0
$236$	583.148	0.160846
$237$	0	0
$238$	−1182.39	−0.322029
$239$	2585.03	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$239$	2585.03	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$240$	0	0
$241$	−3904.99	−1.04374	−0.521872	−	0.853024i	$-0.674766\pi$
$241$	−3904.99	−1.04374	−0.521872	+	0.853024i	$0.674766\pi$
$242$	1988.26	0.528141
$243$	0	0
$244$	2298.36	0.603021
$245$	0	0
$246$	0	0
$247$	−1680.49	−0.432903
$248$	1738.22	0.445070
$249$	0	0
$250$	0	0
$251$	1466.63	0.368815	0.184408	−	0.982850i	$-0.440963\pi$
$251$	1466.63	0.368815	0.184408	+	0.982850i	$0.440963\pi$
$252$	0	0
$253$	−3248.04	−0.807126
$254$	−1991.02	−0.491842
$255$	0	0
$256$	5350.56	1.30629
$257$	4447.35	1.07945	0.539723	−	0.841842i	$-0.318529\pi$
$257$	4447.35	1.07945	0.539723	+	0.841842i	$0.318529\pi$
$258$	0	0
$259$	−676.162	−0.162219
$260$	0	0
$261$	0	0
$262$	−4615.14	−1.08826
$263$	1755.14	0.411509	0.205754	−	0.978604i	$-0.434035\pi$
$263$	1755.14	0.411509	0.205754	+	0.978604i	$0.434035\pi$
$264$	0	0
$265$	0	0
$266$	521.451	0.120196
$267$	0	0
$268$	−812.632	−0.185222
$269$	−3412.93	−0.773569	−0.386785	−	0.922170i	$-0.626414\pi$
$269$	−3412.93	−0.773569	−0.386785	+	0.922170i	$0.626414\pi$
$270$	0	0
$271$	206.271	0.0462365	0.0231183	−	0.999733i	$-0.492641\pi$
$271$	206.271	0.0462365	0.0231183	+	0.999733i	$0.492641\pi$
$272$	6849.03	1.52678
$273$	0	0
$274$	3386.62	0.746689
$275$	0	0
$276$	0	0
$277$	5303.65	1.15042	0.575208	−	0.818007i	$-0.304921\pi$
$277$	5303.65	1.15042	0.575208	+	0.818007i	$0.304921\pi$
$278$	−11393.5	−2.45805
$279$	0	0
$280$	0	0
$281$	2849.15	0.604862	0.302431	−	0.953171i	$-0.402202\pi$
$281$	2849.15	0.604862	0.302431	+	0.953171i	$0.402202\pi$
$282$	0	0
$283$	−8075.05	−1.69615	−0.848077	−	0.529873i	$-0.822240\pi$
$283$	−8075.05	−1.69615	−0.848077	+	0.529873i	$0.822240\pi$
$284$	4791.40	1.00112
$285$	0	0
$286$	−4449.84	−0.920016
$287$	929.336	0.191139
$288$	0	0
$289$	2710.53	0.551705
$290$	0	0
$291$	0	0
$292$	−5120.92	−1.02630
$293$	2622.85	0.522964	0.261482	−	0.965208i	$-0.415789\pi$
$293$	2622.85	0.522964	0.261482	+	0.965208i	$0.415789\pi$
$294$	0	0
$295$	0	0
$296$	1820.39	0.357459
$297$	0	0
$298$	5937.79	1.15425
$299$	−5060.15	−0.978716
$300$	0	0
$301$	−1714.08	−0.328233
$302$	−6630.36	−1.26336
$303$	0	0
$304$	−3020.52	−0.569865
$305$	0	0
$306$	0	0
$307$	1699.16	0.315883	0.157941	−	0.987448i	$-0.449514\pi$
$307$	1699.16	0.315883	0.157941	+	0.987448i	$0.449514\pi$
$308$	546.974	0.101191
$309$	0	0
$310$	0	0
$311$	4818.93	0.878639	0.439319	−	0.898331i	$-0.355220\pi$
$311$	4818.93	0.878639	0.439319	+	0.898331i	$0.355220\pi$
$312$	0	0
$313$	6016.46	1.08649	0.543243	−	0.839575i	$-0.317196\pi$
$313$	6016.46	1.08649	0.543243	+	0.839575i	$0.317196\pi$
$314$	−10319.5	−1.85465
$315$	0	0
$316$	4492.52	0.799760
$317$	−7276.72	−1.28928	−0.644639	−	0.764487i	$-0.722992\pi$
$317$	−7276.72	−1.28928	−0.644639	+	0.764487i	$0.722992\pi$
$318$	0	0
$319$	−5491.97	−0.963923
$320$	0	0
$321$	0	0
$322$	1570.15	0.271742
$323$	−3362.09	−0.579169
$324$	0	0
$325$	0	0
$326$	11096.3	1.88518
$327$	0	0
$328$	−2501.99	−0.421187
$329$	1221.80	0.204741
$330$	0	0
$331$	−1594.18	−0.264725	−0.132362	−	0.991201i	$-0.542256\pi$
$331$	−1594.18	−0.264725	−0.132362	+	0.991201i	$0.542256\pi$
$332$	−359.075	−0.0593579
$333$	0	0
$334$	−10329.4	−1.69222
$335$	0	0
$336$	0	0
$337$	−8396.93	−1.35730	−0.678650	−	0.734462i	$-0.737435\pi$
$337$	−8396.93	−1.35730	−0.678650	+	0.734462i	$0.737435\pi$
$338$	1064.17	0.171253
$339$	0	0
$340$	0	0
$341$	−4861.28	−0.772003
$342$	0	0
$343$	2500.79	0.393673
$344$	4614.71	0.723281
$345$	0	0
$346$	1858.23	0.288726
$347$	−10420.9	−1.61217	−0.806083	−	0.591802i	$-0.798417\pi$
$347$	−10420.9	−1.61217	−0.806083	+	0.591802i	$0.798417\pi$
$348$	0	0
$349$	−7907.42	−1.21282	−0.606411	−	0.795152i	$-0.707391\pi$
$349$	−7907.42	−1.21282	−0.606411	+	0.795152i	$0.707391\pi$
$350$	0	0
$351$	0	0
$352$	−5753.39	−0.871184
$353$	−5003.97	−0.754488	−0.377244	−	0.926114i	$-0.623128\pi$
$353$	−5003.97	−0.754488	−0.377244	+	0.926114i	$0.623128\pi$
$354$	0	0
$355$	0	0
$356$	3492.31	0.519922
$357$	0	0
$358$	8278.19	1.22211
$359$	−2385.12	−0.350646	−0.175323	−	0.984511i	$-0.556097\pi$
$359$	−2385.12	−0.350646	−0.175323	+	0.984511i	$0.556097\pi$
$360$	0	0
$361$	−5376.27	−0.783827
$362$	3550.57	0.515508
$363$	0	0
$364$	852.134	0.122703
$365$	0	0
$366$	0	0
$367$	−1808.01	−0.257159	−0.128580	−	0.991699i	$-0.541042\pi$
$367$	−1808.01	−0.257159	−0.128580	+	0.991699i	$0.541042\pi$
$368$	−9095.14	−1.28836
$369$	0	0
$370$	0	0
$371$	2484.66	0.347701
$372$	0	0
$373$	12533.7	1.73987	0.869933	−	0.493170i	$-0.164162\pi$
$373$	12533.7	1.73987	0.869933	+	0.493170i	$0.164162\pi$
$374$	−8902.61	−1.23086
$375$	0	0
$376$	−3289.36	−0.451159
$377$	−8555.98	−1.16885
$378$	0	0
$379$	5221.98	0.707745	0.353872	−	0.935294i	$-0.384865\pi$
$379$	5221.98	0.707745	0.353872	+	0.935294i	$0.384865\pi$
$380$	0	0
$381$	0	0
$382$	6842.89	0.916525
$383$	−8557.69	−1.14172	−0.570859	−	0.821048i	$-0.693390\pi$
$383$	−8557.69	−1.14172	−0.570859	+	0.821048i	$0.693390\pi$
$384$	0	0
$385$	0	0
$386$	−2233.23	−0.294478
$387$	0	0
$388$	7366.52	0.963862
$389$	−5461.33	−0.711826	−0.355913	−	0.934519i	$-0.615830\pi$
$389$	−5461.33	−0.711826	−0.355913	+	0.934519i	$0.615830\pi$
$390$	0	0
$391$	−10123.6	−1.30940
$392$	−3297.03	−0.424809
$393$	0	0
$394$	−248.763	−0.0318083
$395$	0	0
$396$	0	0
$397$	−1998.55	−0.252656	−0.126328	−	0.991989i	$-0.540319\pi$
$397$	−1998.55	−0.252656	−0.126328	+	0.991989i	$0.540319\pi$
$398$	−7916.06	−0.996975
$399$	0	0
$400$	0	0
$401$	−492.166	−0.0612907	−0.0306454	−	0.999530i	$-0.509756\pi$
$401$	−492.166	−0.0612907	−0.0306454	+	0.999530i	$0.509756\pi$
$402$	0	0
$403$	−7573.42	−0.936126
$404$	−3438.40	−0.423432
$405$	0	0
$406$	2654.89	0.324532
$407$	−5091.06	−0.620035
$408$	0	0
$409$	−8000.45	−0.967230	−0.483615	−	0.875281i	$-0.660676\pi$
$409$	−8000.45	−0.967230	−0.483615	+	0.875281i	$0.660676\pi$
$410$	0	0
$411$	0	0
$412$	−1330.50	−0.159099
$413$	−413.417	−0.0492565
$414$	0	0
$415$	0	0
$416$	−8963.24	−1.05639
$417$	0	0
$418$	3926.18	0.459416
$419$	5504.93	0.641846	0.320923	−	0.947105i	$-0.396007\pi$
$419$	5504.93	0.641846	0.320923	+	0.947105i	$0.396007\pi$
$420$	0	0
$421$	−10404.6	−1.20448	−0.602241	−	0.798314i	$-0.705726\pi$
$421$	−10404.6	−1.20448	−0.602241	+	0.798314i	$0.705726\pi$
$422$	1729.05	0.199452
$423$	0	0
$424$	−6689.28	−0.766179
$425$	0	0
$426$	0	0
$427$	−1629.40	−0.184665
$428$	−5944.60	−0.671362
$429$	0	0
$430$	0	0
$431$	6446.05	0.720406	0.360203	−	0.932874i	$-0.382707\pi$
$431$	6446.05	0.720406	0.360203	+	0.932874i	$0.382707\pi$
$432$	0	0
$433$	6307.26	0.700018	0.350009	−	0.936746i	$-0.386179\pi$
$433$	6307.26	0.700018	0.350009	+	0.936746i	$0.386179\pi$
$434$	2350.01	0.259917
$435$	0	0
$436$	7949.11	0.873150
$437$	4464.67	0.488728
$438$	0	0
$439$	−7894.27	−0.858253	−0.429127	−	0.903244i	$-0.641179\pi$
$439$	−7894.27	−0.858253	−0.429127	+	0.903244i	$0.641179\pi$
$440$	0	0
$441$	0	0
$442$	−13869.4	−1.49254
$443$	−8779.17	−0.941560	−0.470780	−	0.882251i	$-0.656027\pi$
$443$	−8779.17	−0.941560	−0.470780	+	0.882251i	$0.656027\pi$
$444$	0	0
$445$	0	0
$446$	10796.9	1.14629
$447$	0	0
$448$	446.477	0.0470850
$449$	16514.2	1.73575	0.867874	−	0.496784i	$-0.165486\pi$
$449$	16514.2	1.73575	0.867874	+	0.496784i	$0.165486\pi$
$450$	0	0
$451$	6997.29	0.730576
$452$	−8774.88	−0.913132
$453$	0	0
$454$	10308.7	1.06566
$455$	0	0
$456$	0	0
$457$	−15496.9	−1.58624	−0.793121	−	0.609064i	$-0.791545\pi$
$457$	−15496.9	−1.58624	−0.793121	+	0.609064i	$0.791545\pi$
$458$	15623.6	1.59398
$459$	0	0
$460$	0	0
$461$	−15282.9	−1.54402	−0.772012	−	0.635608i	$-0.780750\pi$
$461$	−15282.9	−1.54402	−0.772012	+	0.635608i	$0.780750\pi$
$462$	0	0
$463$	13649.7	1.37010	0.685051	−	0.728495i	$-0.259780\pi$
$463$	13649.7	1.37010	0.685051	+	0.728495i	$0.259780\pi$
$464$	−15378.6	−1.53865
$465$	0	0
$466$	15532.8	1.54409
$467$	6841.24	0.677891	0.338945	−	0.940806i	$-0.389930\pi$
$467$	6841.24	0.677891	0.338945	+	0.940806i	$0.389930\pi$
$468$	0	0
$469$	576.107	0.0567210
$470$	0	0
$471$	0	0
$472$	1113.01	0.108540
$473$	−12905.9	−1.25458
$474$	0	0
$475$	0	0
$476$	1704.83	0.164161
$477$	0	0
$478$	−9408.96	−0.900326
$479$	−5777.26	−0.551085	−0.275542	−	0.961289i	$-0.588857\pi$
$479$	−5777.26	−0.551085	−0.275542	+	0.961289i	$0.588857\pi$
$480$	0	0
$481$	−7931.39	−0.751851
$482$	14213.3	1.34315
$483$	0	0
$484$	−2866.78	−0.269232
$485$	0	0
$486$	0	0
$487$	17699.8	1.64693	0.823463	−	0.567370i	$-0.192039\pi$
$487$	17699.8	1.64693	0.823463	+	0.567370i	$0.192039\pi$
$488$	4386.72	0.406921
$489$	0	0
$490$	0	0
$491$	−12091.0	−1.11132	−0.555661	−	0.831409i	$-0.687535\pi$
$491$	−12091.0	−1.11132	−0.555661	+	0.831409i	$0.687535\pi$
$492$	0	0
$493$	−17117.6	−1.56377
$494$	6116.63	0.557085
$495$	0	0
$496$	−13612.5	−1.23230
$497$	−3396.81	−0.306575
$498$	0	0
$499$	−16307.4	−1.46296	−0.731481	−	0.681862i	$-0.761171\pi$
$499$	−16307.4	−1.46296	−0.731481	+	0.681862i	$0.761171\pi$
$500$	0	0
$501$	0	0
$502$	−5338.20	−0.474613
$503$	8574.88	0.760109	0.380054	−	0.924964i	$-0.375905\pi$
$503$	8574.88	0.760109	0.380054	+	0.924964i	$0.375905\pi$
$504$	0	0
$505$	0	0
$506$	11822.2	1.03866
$507$	0	0
$508$	2870.76	0.250727
$509$	4054.42	0.353063	0.176532	−	0.984295i	$-0.443512\pi$
$509$	4054.42	0.353063	0.176532	+	0.984295i	$0.443512\pi$
$510$	0	0
$511$	3630.43	0.314287
$512$	−9824.79	−0.848044
$513$	0	0
$514$	−16187.4	−1.38909
$515$	0	0
$516$	0	0
$517$	9199.32	0.782564
$518$	2461.08	0.208753
$519$	0	0
$520$	0	0
$521$	−22757.9	−1.91371	−0.956854	−	0.290570i	$-0.906155\pi$
$521$	−22757.9	−1.91371	−0.956854	+	0.290570i	$0.906155\pi$
$522$	0	0
$523$	−12468.0	−1.04242	−0.521211	−	0.853428i	$-0.674520\pi$
$523$	−12468.0	−1.04242	−0.521211	+	0.853428i	$0.674520\pi$
$524$	6654.35	0.554765
$525$	0	0
$526$	−6388.35	−0.529554
$527$	−15151.8	−1.25242
$528$	0	0
$529$	1276.64	0.104926
$530$	0	0
$531$	0	0
$532$	−751.855	−0.0612726
$533$	10901.1	0.885892
$534$	0	0
$535$	0	0
$536$	−1551.01	−0.124988
$537$	0	0
$538$	12422.3	0.995474
$539$	9220.77	0.736859
$540$	0	0
$541$	−22965.1	−1.82504	−0.912519	−	0.409034i	$-0.865866\pi$
$541$	−22965.1	−1.82504	−0.912519	+	0.409034i	$0.865866\pi$
$542$	−750.783	−0.0594998
$543$	0	0
$544$	−17932.4	−1.41332
$545$	0	0
$546$	0	0
$547$	−10183.0	−0.795966	−0.397983	−	0.917393i	$-0.630290\pi$
$547$	−10183.0	−0.795966	−0.397983	+	0.917393i	$0.630290\pi$
$548$	−4883.00	−0.380641
$549$	0	0
$550$	0	0
$551$	7549.12	0.583672
$552$	0	0
$553$	−3184.93	−0.244913
$554$	−19304.1	−1.48042
$555$	0	0
$556$	16427.7	1.25304
$557$	−4498.86	−0.342231	−0.171116	−	0.985251i	$-0.554737\pi$
$557$	−4498.86	−0.342231	−0.171116	+	0.985251i	$0.554737\pi$
$558$	0	0
$559$	−20106.2	−1.52129
$560$	0	0
$561$	0	0
$562$	−10370.3	−0.778371
$563$	−12042.2	−0.901452	−0.450726	−	0.892662i	$-0.648835\pi$
$563$	−12042.2	−0.901452	−0.450726	+	0.892662i	$0.648835\pi$
$564$	0	0
$565$	0	0
$566$	29391.4	2.18271
$567$	0	0
$568$	9145.01	0.675557
$569$	3491.33	0.257231	0.128615	−	0.991695i	$-0.458947\pi$
$569$	3491.33	0.257231	0.128615	+	0.991695i	$0.458947\pi$
$570$	0	0
$571$	−16100.2	−1.17999	−0.589993	−	0.807408i	$-0.700870\pi$
$571$	−16100.2	−1.17999	−0.589993	+	0.807408i	$0.700870\pi$
$572$	6416.01	0.468998
$573$	0	0
$574$	−3382.58	−0.245969
$575$	0	0
$576$	0	0
$577$	5196.41	0.374921	0.187460	−	0.982272i	$-0.439974\pi$
$577$	5196.41	0.374921	0.187460	+	0.982272i	$0.439974\pi$
$578$	−9865.74	−0.709966
$579$	0	0
$580$	0	0
$581$	254.563	0.0181774
$582$	0	0
$583$	18707.8	1.32899
$584$	−9773.95	−0.692550
$585$	0	0
$586$	−9546.61	−0.672981
$587$	3897.19	0.274028	0.137014	−	0.990569i	$-0.456249\pi$
$587$	3897.19	0.274028	0.137014	+	0.990569i	$0.456249\pi$
$588$	0	0
$589$	6682.18	0.467461
$590$	0	0
$591$	0	0
$592$	−14255.9	−0.989721
$593$	3023.70	0.209390	0.104695	−	0.994504i	$-0.466613\pi$
$593$	3023.70	0.209390	0.104695	+	0.994504i	$0.466613\pi$
$594$	0	0
$595$	0	0
$596$	−8561.41	−0.588405
$597$	0	0
$598$	18417.9	1.25947
$599$	16629.5	1.13433	0.567164	−	0.823605i	$-0.308041\pi$
$599$	16629.5	1.13433	0.567164	+	0.823605i	$0.308041\pi$
$600$	0	0
$601$	−2958.68	−0.200810	−0.100405	−	0.994947i	$-0.532014\pi$
$601$	−2958.68	−0.200810	−0.100405	+	0.994947i	$0.532014\pi$
$602$	6238.89	0.422389
$603$	0	0
$604$	9560.01	0.644025
$605$	0	0
$606$	0	0
$607$	−7012.36	−0.468901	−0.234451	−	0.972128i	$-0.575329\pi$
$607$	−7012.36	−0.468901	−0.234451	+	0.972128i	$0.575329\pi$
$608$	7908.45	0.527516
$609$	0	0
$610$	0	0
$611$	14331.7	0.948932
$612$	0	0
$613$	−8462.75	−0.557597	−0.278799	−	0.960350i	$-0.589936\pi$
$613$	−8462.75	−0.557597	−0.278799	+	0.960350i	$0.589936\pi$
$614$	−6184.57	−0.406496
$615$	0	0
$616$	1043.97	0.0682838
$617$	8413.43	0.548966	0.274483	−	0.961592i	$-0.411493\pi$
$617$	8413.43	0.548966	0.274483	+	0.961592i	$0.411493\pi$
$618$	0	0
$619$	−1120.47	−0.0727549	−0.0363775	−	0.999338i	$-0.511582\pi$
$619$	−1120.47	−0.0727549	−0.0363775	+	0.999338i	$0.511582\pi$
$620$	0	0
$621$	0	0
$622$	−17539.9	−1.13068
$623$	−2475.84	−0.159218
$624$	0	0
$625$	0	0
$626$	−21898.6	−1.39815
$627$	0	0
$628$	14879.1	0.945450
$629$	−15868.0	−1.00588
$630$	0	0
$631$	7709.69	0.486399	0.243200	−	0.969976i	$-0.421803\pi$
$631$	7709.69	0.486399	0.243200	+	0.969976i	$0.421803\pi$
$632$	8574.58	0.539681
$633$	0	0
$634$	26485.7	1.65912
$635$	0	0
$636$	0	0
$637$	14365.1	0.893511
$638$	19989.6	1.24043
$639$	0	0
$640$	0	0
$641$	20299.6	1.25084	0.625419	−	0.780289i	$-0.284928\pi$
$641$	20299.6	1.25084	0.625419	+	0.780289i	$0.284928\pi$
$642$	0	0
$643$	−31865.2	−1.95434	−0.977169	−	0.212462i	$-0.931852\pi$
$643$	−31865.2	−1.95434	−0.977169	+	0.212462i	$0.931852\pi$
$644$	−2263.92	−0.138526
$645$	0	0
$646$	12237.3	0.745309
$647$	−22400.4	−1.36113	−0.680563	−	0.732689i	$-0.738265\pi$
$647$	−22400.4	−1.36113	−0.680563	+	0.732689i	$0.738265\pi$
$648$	0	0
$649$	−3112.76	−0.188269
$650$	0	0
$651$	0	0
$652$	−15999.3	−0.961011
$653$	−10342.2	−0.619791	−0.309895	−	0.950771i	$-0.600294\pi$
$653$	−10342.2	−0.619791	−0.309895	+	0.950771i	$0.600294\pi$
$654$	0	0
$655$	0	0
$656$	19593.7	1.16617
$657$	0	0
$658$	−4447.07	−0.263472
$659$	237.410	0.0140336	0.00701682	−	0.999975i	$-0.497766\pi$
$659$	237.410	0.0140336	0.00701682	+	0.999975i	$0.497766\pi$
$660$	0	0
$661$	28823.7	1.69608	0.848042	−	0.529930i	$-0.177782\pi$
$661$	28823.7	1.69608	0.848042	+	0.529930i	$0.177782\pi$
$662$	5802.46	0.340663
$663$	0	0
$664$	−685.342	−0.0400549
$665$	0	0
$666$	0	0
$667$	22731.2	1.31958
$668$	14893.5	0.862646
$669$	0	0
$670$	0	0
$671$	−12268.3	−0.705830
$672$	0	0
$673$	7760.84	0.444515	0.222257	−	0.974988i	$-0.428658\pi$
$673$	7760.84	0.444515	0.222257	+	0.974988i	$0.428658\pi$
$674$	30563.0	1.74665
$675$	0	0
$676$	−1534.38	−0.0872998
$677$	23593.3	1.33939	0.669693	−	0.742638i	$-0.266426\pi$
$677$	23593.3	1.33939	0.669693	+	0.742638i	$0.266426\pi$
$678$	0	0
$679$	−5222.43	−0.295167
$680$	0	0
$681$	0	0
$682$	17694.0	0.993459
$683$	−4383.39	−0.245572	−0.122786	−	0.992433i	$-0.539183\pi$
$683$	−4383.39	−0.245572	−0.122786	+	0.992433i	$0.539183\pi$
$684$	0	0
$685$	0	0
$686$	−9102.34	−0.506602
$687$	0	0
$688$	−36139.0	−2.00260
$689$	29145.1	1.61152
$690$	0	0
$691$	−17752.5	−0.977333	−0.488667	−	0.872471i	$-0.662517\pi$
$691$	−17752.5	−0.977333	−0.488667	+	0.872471i	$0.662517\pi$
$692$	−2679.30	−0.147185
$693$	0	0
$694$	37929.7	2.07463
$695$	0	0
$696$	0	0
$697$	21809.4	1.18521
$698$	28781.3	1.56073
$699$	0	0
$700$	0	0
$701$	15650.2	0.843223	0.421611	−	0.906777i	$-0.361465\pi$
$701$	15650.2	0.843223	0.421611	+	0.906777i	$0.361465\pi$
$702$	0	0
$703$	6998.03	0.375442
$704$	3361.68	0.179969
$705$	0	0
$706$	18213.4	0.970920
$707$	2437.62	0.129669
$708$	0	0
$709$	13217.6	0.700137	0.350069	−	0.936724i	$-0.386158\pi$
$709$	13217.6	0.700137	0.350069	+	0.936724i	$0.386158\pi$
$710$	0	0
$711$	0	0
$712$	6665.54	0.350845
$713$	20120.8	1.05684
$714$	0	0
$715$	0	0
$716$	−11935.9	−0.622998
$717$	0	0
$718$	8681.34	0.451232
$719$	−14824.5	−0.768932	−0.384466	−	0.923139i	$-0.625614\pi$
$719$	−14824.5	−0.768932	−0.384466	+	0.923139i	$0.625614\pi$
$720$	0	0
$721$	943.243	0.0487215
$722$	19568.5	1.00867
$723$	0	0
$724$	−5119.40	−0.262791
$725$	0	0
$726$	0	0
$727$	7220.09	0.368333	0.184167	−	0.982895i	$-0.441041\pi$
$727$	7220.09	0.368333	0.184167	+	0.982895i	$0.441041\pi$
$728$	1626.41	0.0828005
$729$	0	0
$730$	0	0
$731$	−40225.7	−2.03530
$732$	0	0
$733$	1861.96	0.0938242	0.0469121	−	0.998899i	$-0.485062\pi$
$733$	1861.96	0.0938242	0.0469121	+	0.998899i	$0.485062\pi$
$734$	6580.77	0.330927
$735$	0	0
$736$	23813.2	1.19262
$737$	4337.71	0.216800
$738$	0	0
$739$	−29412.7	−1.46409	−0.732046	−	0.681255i	$-0.761434\pi$
$739$	−29412.7	−1.46409	−0.732046	+	0.681255i	$0.761434\pi$
$740$	0	0
$741$	0	0
$742$	−9043.62	−0.447442
$743$	34422.1	1.69963	0.849816	−	0.527080i	$-0.176713\pi$
$743$	34422.1	1.69963	0.849816	+	0.527080i	$0.176713\pi$
$744$	0	0
$745$	0	0
$746$	−45619.9	−2.23896
$747$	0	0
$748$	12836.3	0.627460
$749$	4214.37	0.205594
$750$	0	0
$751$	−29376.4	−1.42738	−0.713689	−	0.700462i	$-0.752977\pi$
$751$	−29376.4	−1.42738	−0.713689	+	0.700462i	$0.752977\pi$
$752$	25759.8	1.24915
$753$	0	0
$754$	31141.9	1.50414
$755$	0	0
$756$	0	0
$757$	6235.27	0.299372	0.149686	−	0.988734i	$-0.452174\pi$
$757$	6235.27	0.299372	0.149686	+	0.988734i	$0.452174\pi$
$758$	−19006.9	−0.910767
$759$	0	0
$760$	0	0
$761$	−27970.2	−1.33235	−0.666176	−	0.745795i	$-0.732070\pi$
$761$	−27970.2	−1.33235	−0.666176	+	0.745795i	$0.732070\pi$
$762$	0	0
$763$	−5635.44	−0.267388
$764$	−9866.44	−0.467219
$765$	0	0
$766$	31148.2	1.46923
$767$	−4849.39	−0.228294
$768$	0	0
$769$	−9068.10	−0.425233	−0.212616	−	0.977136i	$-0.568198\pi$
$769$	−9068.10	−0.425233	−0.212616	+	0.977136i	$0.568198\pi$
$770$	0	0
$771$	0	0
$772$	3219.99	0.150117
$773$	−31339.8	−1.45823	−0.729117	−	0.684389i	$-0.760069\pi$
$773$	−31339.8	−1.45823	−0.729117	+	0.684389i	$0.760069\pi$
$774$	0	0
$775$	0	0
$776$	14060.0	0.650418
$777$	0	0
$778$	19878.1	0.916019
$779$	−9618.28	−0.442376
$780$	0	0
$781$	−25575.8	−1.17180
$782$	36847.8	1.68501
$783$	0	0
$784$	25819.9	1.17620
$785$	0	0
$786$	0	0
$787$	10079.4	0.456533	0.228267	−	0.973599i	$-0.426694\pi$
$787$	10079.4	0.456533	0.228267	+	0.973599i	$0.426694\pi$
$788$	358.679	0.0162150
$789$	0	0
$790$	0	0
$791$	6220.87	0.279632
$792$	0	0
$793$	−19112.9	−0.855886
$794$	7274.29	0.325132
$795$	0	0
$796$	11413.8	0.508230
$797$	−7630.80	−0.339143	−0.169571	−	0.985518i	$-0.554238\pi$
$797$	−7630.80	−0.339143	−0.169571	+	0.985518i	$0.554238\pi$
$798$	0	0
$799$	28672.8	1.26955
$800$	0	0
$801$	0	0
$802$	1791.38	0.0788725
$803$	27334.7	1.20127
$804$	0	0
$805$	0	0
$806$	27565.6	1.20466
$807$	0	0
$808$	−6562.63	−0.285733
$809$	−2066.39	−0.0898028	−0.0449014	−	0.998991i	$-0.514297\pi$
$809$	−2066.39	−0.0898028	−0.0449014	+	0.998991i	$0.514297\pi$
$810$	0	0
$811$	−27700.4	−1.19937	−0.599687	−	0.800235i	$-0.704708\pi$
$811$	−27700.4	−1.19937	−0.599687	+	0.800235i	$0.704708\pi$
$812$	−3827.96	−0.165437
$813$	0	0
$814$	18530.4	0.797898
$815$	0	0
$816$	0	0
$817$	17740.1	0.759668
$818$	29119.9	1.24469
$819$	0	0
$820$	0	0
$821$	17103.1	0.727042	0.363521	−	0.931586i	$-0.381574\pi$
$821$	17103.1	0.727042	0.363521	+	0.931586i	$0.381574\pi$
$822$	0	0
$823$	−9190.15	−0.389245	−0.194623	−	0.980878i	$-0.562348\pi$
$823$	−9190.15	−0.389245	−0.194623	+	0.980878i	$0.562348\pi$
$824$	−2539.43	−0.107361
$825$	0	0
$826$	1504.75	0.0633861
$827$	9902.35	0.416371	0.208185	−	0.978089i	$-0.433244\pi$
$827$	9902.35	0.416371	0.208185	+	0.978089i	$0.433244\pi$
$828$	0	0
$829$	26858.5	1.12525	0.562626	−	0.826711i	$-0.309791\pi$
$829$	26858.5	1.12525	0.562626	+	0.826711i	$0.309791\pi$
$830$	0	0
$831$	0	0
$832$	5237.19	0.218229
$833$	28739.7	1.19540
$834$	0	0
$835$	0	0
$836$	−5660.98	−0.234197
$837$	0	0
$838$	−20036.8	−0.825965
$839$	−9300.55	−0.382706	−0.191353	−	0.981521i	$-0.561288\pi$
$839$	−9300.55	−0.382706	−0.191353	+	0.981521i	$0.561288\pi$
$840$	0	0
$841$	14046.3	0.575926
$842$	37870.4	1.55000
$843$	0	0
$844$	−2493.03	−0.101675
$845$	0	0
$846$	0	0
$847$	2032.37	0.0824477
$848$	52385.5	2.12137
$849$	0	0
$850$	0	0
$851$	21071.9	0.848806
$852$	0	0
$853$	−22214.6	−0.891691	−0.445845	−	0.895110i	$-0.647097\pi$
$853$	−22214.6	−0.891691	−0.445845	+	0.895110i	$0.647097\pi$
$854$	5930.65	0.237638
$855$	0	0
$856$	−11346.1	−0.453037
$857$	−34347.9	−1.36908	−0.684541	−	0.728974i	$-0.739997\pi$
$857$	−34347.9	−1.36908	−0.684541	+	0.728974i	$0.739997\pi$
$858$	0	0
$859$	11178.2	0.444001	0.222001	−	0.975047i	$-0.428741\pi$
$859$	11178.2	0.444001	0.222001	+	0.975047i	$0.428741\pi$
$860$	0	0
$861$	0	0
$862$	−23462.2	−0.927061
$863$	33684.0	1.32864	0.664321	−	0.747448i	$-0.268721\pi$
$863$	33684.0	1.32864	0.664321	+	0.747448i	$0.268721\pi$
$864$	0	0
$865$	0	0
$866$	−22957.1	−0.900824
$867$	0	0
$868$	−3388.36	−0.132498
$869$	−23980.5	−0.936112
$870$	0	0
$871$	6757.75	0.262890
$872$	15171.9	0.589204
$873$	0	0
$874$	−16250.4	−0.628924
$875$	0	0
$876$	0	0
$877$	−5409.31	−0.208278	−0.104139	−	0.994563i	$-0.533209\pi$
$877$	−5409.31	−0.208278	−0.104139	+	0.994563i	$0.533209\pi$
$878$	28733.5	1.10445
$879$	0	0
$880$	0	0
$881$	−41501.3	−1.58708	−0.793538	−	0.608520i	$-0.791763\pi$
$881$	−41501.3	−1.58708	−0.793538	+	0.608520i	$0.791763\pi$
$882$	0	0
$883$	−10532.8	−0.401423	−0.200711	−	0.979650i	$-0.564325\pi$
$883$	−10532.8	−0.401423	−0.200711	+	0.979650i	$0.564325\pi$
$884$	19997.7	0.760854
$885$	0	0
$886$	31954.3	1.21165
$887$	−30002.5	−1.13572	−0.567861	−	0.823125i	$-0.692229\pi$
$887$	−30002.5	−1.13572	−0.567861	+	0.823125i	$0.692229\pi$
$888$	0	0
$889$	−2035.20	−0.0767810
$890$	0	0
$891$	0	0
$892$	−15567.5	−0.584347
$893$	−12645.1	−0.473856
$894$	0	0
$895$	0	0
$896$	4487.95	0.167335
$897$	0	0
$898$	−60108.0	−2.23366
$899$	34021.4	1.26215
$900$	0	0
$901$	58309.4	2.15601
$902$	−25468.6	−0.940147
$903$	0	0
$904$	−16748.0	−0.616185
$905$	0	0
$906$	0	0
$907$	−1098.35	−0.0402096	−0.0201048	−	0.999798i	$-0.506400\pi$
$907$	−1098.35	−0.0402096	−0.0201048	+	0.999798i	$0.506400\pi$
$908$	−14863.6	−0.543245
$909$	0	0
$910$	0	0
$911$	−36841.5	−1.33986	−0.669931	−	0.742423i	$-0.733676\pi$
$911$	−36841.5	−1.33986	−0.669931	+	0.742423i	$0.733676\pi$
$912$	0	0
$913$	1916.69	0.0694778
$914$	56405.3	2.04127
$915$	0	0
$916$	−22526.9	−0.812566
$917$	−4717.54	−0.169888
$918$	0	0
$919$	46004.8	1.65131	0.825657	−	0.564172i	$-0.190804\pi$
$919$	46004.8	1.65131	0.825657	+	0.564172i	$0.190804\pi$
$920$	0	0
$921$	0	0
$922$	55626.4	1.98694
$923$	−39844.7	−1.42091
$924$	0	0
$925$	0	0
$926$	−49682.1	−1.76313
$927$	0	0
$928$	40264.8	1.42431
$929$	50902.9	1.79771	0.898853	−	0.438251i	$-0.144402\pi$
$929$	50902.9	1.79771	0.898853	+	0.438251i	$0.144402\pi$
$930$	0	0
$931$	−12674.6	−0.446180
$932$	−22396.1	−0.787133
$933$	0	0
$934$	−24900.7	−0.872349
$935$	0	0
$936$	0	0
$937$	−26913.6	−0.938343	−0.469172	−	0.883107i	$-0.655447\pi$
$937$	−26913.6	−0.938343	−0.469172	+	0.883107i	$0.655447\pi$
$938$	−2096.91	−0.0729919
$939$	0	0
$940$	0	0
$941$	21560.9	0.746935	0.373468	−	0.927643i	$-0.378169\pi$
$941$	21560.9	0.746935	0.373468	+	0.927643i	$0.378169\pi$
$942$	0	0
$943$	−28961.8	−1.00013
$944$	−8716.32	−0.300521
$945$	0	0
$946$	46974.8	1.61446
$947$	28569.0	0.980326	0.490163	−	0.871631i	$-0.336937\pi$
$947$	28569.0	0.980326	0.490163	+	0.871631i	$0.336937\pi$
$948$	0	0
$949$	42585.0	1.45666
$950$	0	0
$951$	0	0
$952$	3253.89	0.110777
$953$	−3807.33	−0.129414	−0.0647069	−	0.997904i	$-0.520611\pi$
$953$	−3807.33	−0.129414	−0.0647069	+	0.997904i	$0.520611\pi$
$954$	0	0
$955$	0	0
$956$	13566.3	0.458961
$957$	0	0
$958$	21028.0	0.709168
$959$	3461.76	0.116565
$960$	0	0
$961$	323.400	0.0108556
$962$	28868.6	0.967526
$963$	0	0
$964$	−20493.5	−0.684701
$965$	0	0
$966$	0	0
$967$	17905.3	0.595447	0.297723	−	0.954652i	$-0.403773\pi$
$967$	17905.3	0.595447	0.297723	+	0.954652i	$0.403773\pi$
$968$	−5471.62	−0.181678
$969$	0	0
$970$	0	0
$971$	−8567.95	−0.283170	−0.141585	−	0.989926i	$-0.545220\pi$
$971$	−8567.95	−0.283170	−0.141585	+	0.989926i	$0.545220\pi$
$972$	0	0
$973$	−11646.3	−0.383723
$974$	−64423.3	−2.11936
$975$	0	0
$976$	−34353.5	−1.12667
$977$	−19620.3	−0.642488	−0.321244	−	0.946997i	$-0.604101\pi$
$977$	−19620.3	−0.642488	−0.321244	+	0.946997i	$0.604101\pi$
$978$	0	0
$979$	−18641.5	−0.608564
$980$	0	0
$981$	0	0
$982$	44008.6	1.43011
$983$	−47336.3	−1.53590	−0.767951	−	0.640508i	$-0.778724\pi$
$983$	−47336.3	−1.53590	−0.767951	+	0.640508i	$0.778724\pi$
$984$	0	0
$985$	0	0
$986$	62304.4	2.01235
$987$	0	0
$988$	−8819.27	−0.283986
$989$	53417.6	1.71747
$990$	0	0
$991$	13735.8	0.440294	0.220147	−	0.975467i	$-0.429346\pi$
$991$	13735.8	0.440294	0.220147	+	0.975467i	$0.429346\pi$
$992$	35640.8	1.14072
$993$	0	0
$994$	12363.7	0.394519
$995$	0	0
$996$	0	0
$997$	35872.6	1.13952	0.569758	−	0.821813i	$-0.307037\pi$
$997$	35872.6	1.13952	0.569758	+	0.821813i	$0.307037\pi$
$998$	59355.3	1.88262
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bf.1.3		12
3.2	odd	2		2025.4.a.bj.1.10		12
5.4	even	2		2025.4.a.bi.1.10		12
9.4	even	3		225.4.e.f.151.10	yes	24
9.7	even	3		225.4.e.f.76.10	yes	24
15.14	odd	2		2025.4.a.be.1.3		12
45.4	even	6		225.4.e.e.151.3	yes	24
45.7	odd	12		225.4.k.e.49.6		48
45.13	odd	12		225.4.k.e.124.6		48
45.22	odd	12		225.4.k.e.124.19		48
45.34	even	6		225.4.e.e.76.3	✓	24
45.43	odd	12		225.4.k.e.49.19		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.e.e.76.3	✓	24	45.34	even	6
225.4.e.e.151.3	yes	24	45.4	even	6
225.4.e.f.76.10	yes	24	9.7	even	3
225.4.e.f.151.10	yes	24	9.4	even	3
225.4.k.e.49.6		48	45.7	odd	12
225.4.k.e.49.19		48	45.43	odd	12
225.4.k.e.124.6		48	45.13	odd	12
225.4.k.e.124.19		48	45.22	odd	12
2025.4.a.be.1.3		12	15.14	odd	2
2025.4.a.bf.1.3		12	1.1	even	1	trivial
2025.4.a.bi.1.10		12	5.4	even	2
2025.4.a.bj.1.10		12	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.63978 q^{2} +5.24803 q^{4} -3.72054 q^{7} +10.0166 q^{8} -28.0133 q^{11} -43.6420 q^{13} +13.5420 q^{14} -78.4424 q^{16} -87.3128 q^{17} +38.5062 q^{19} +101.962 q^{22} +115.947 q^{23} +158.848 q^{26} -19.5255 q^{28} +196.049 q^{29} +173.535 q^{31} +205.381 q^{32} +317.800 q^{34} +181.738 q^{37} -140.154 q^{38} -249.785 q^{41} +460.708 q^{43} -147.015 q^{44} -422.021 q^{46} -328.392 q^{47} -329.158 q^{49} -229.035 q^{52} -667.821 q^{53} -37.2671 q^{56} -713.577 q^{58} +111.117 q^{59} +437.946 q^{61} -631.630 q^{62} -120.003 q^{64} -154.845 q^{67} -458.221 q^{68} +912.989 q^{71} -975.779 q^{73} -661.485 q^{74} +202.082 q^{76} +104.225 q^{77} +856.040 q^{79} +909.164 q^{82} -68.4209 q^{83} -1676.88 q^{86} -280.597 q^{88} +665.452 q^{89} +162.372 q^{91} +608.492 q^{92} +1195.27 q^{94} +1403.67 q^{97} +1198.06 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} + 48 q^{4} + 6 q^{7} - 45 q^{8} + 29 q^{11} + 24 q^{13} - 69 q^{14} + 192 q^{16} - 79 q^{17} - 75 q^{19} - 18 q^{22} - 318 q^{23} - 154 q^{26} + 96 q^{28} + 106 q^{29} + 60 q^{31} - 914 q^{32}+ \cdots - 7511 q^{98}+O(q^{100}) \) 12 * q - 4 * q^2 + 48 * q^4 + 6 * q^7 - 45 * q^8 + 29 * q^11 + 24 * q^13 - 69 * q^14 + 192 * q^16 - 79 * q^17 - 75 * q^19 - 18 * q^22 - 318 * q^23 - 154 * q^26 + 96 * q^28 + 106 * q^29 + 60 * q^31 - 914 * q^32 - 108 * q^34 - 84 * q^37 - 640 * q^38 - 353 * q^41 - 426 * q^43 + 571 * q^44 + 270 * q^46 - 1210 * q^47 + 666 * q^49 - 75 * q^52 - 448 * q^53 - 570 * q^56 + 594 * q^58 + 482 * q^59 + 402 * q^61 - 2544 * q^62 + 975 * q^64 - 201 * q^67 - 3437 * q^68 - 944 * q^71 - 453 * q^73 + 10 * q^74 - 462 * q^76 - 2652 * q^77 + 258 * q^79 + 873 * q^82 - 3012 * q^83 + 1952 * q^86 - 216 * q^88 - 738 * q^89 - 618 * q^91 - 5232 * q^92 + 63 * q^94 - 318 * q^97 - 7511 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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