LMFDB - Embedded newform 2025.4.a.bd.1.7

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)

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);

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")

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

Newform invariants

comment:select newform

sage:traces = [8,0,0,24,0,0,0,0,0,0,0,0,0,42] 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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 44x^{6} + 567x^{4} - 2024x^{2} + 1900 $ x^8 - 44x^6 + 567x^4 - 2024*x^2 + 1900
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 405)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.99806$ 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)

$f(q)$	$=$	$q+3.99806 q^{2} +7.98447 q^{4} +7.79840 q^{7} -0.0620886 q^{8} +32.7050 q^{11} -65.9775 q^{13} +31.1784 q^{14} -64.1240 q^{16} +15.5853 q^{17} +51.6661 q^{19} +130.756 q^{22} -49.6796 q^{23} -263.782 q^{26} +62.2661 q^{28} -282.020 q^{29} -147.663 q^{31} -255.875 q^{32} +62.3111 q^{34} +122.498 q^{37} +206.564 q^{38} -109.032 q^{41} +191.493 q^{43} +261.132 q^{44} -198.622 q^{46} -414.420 q^{47} -282.185 q^{49} -526.796 q^{52} +236.381 q^{53} -0.484192 q^{56} -1127.53 q^{58} +784.024 q^{59} -535.138 q^{61} -590.363 q^{62} -510.010 q^{64} +121.069 q^{67} +124.441 q^{68} +590.432 q^{71} +986.078 q^{73} +489.753 q^{74} +412.526 q^{76} +255.046 q^{77} -607.025 q^{79} -435.915 q^{82} -1292.46 q^{83} +765.598 q^{86} -2.03061 q^{88} -1506.78 q^{89} -514.519 q^{91} -396.665 q^{92} -1656.87 q^{94} -610.887 q^{97} -1128.19 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{4} + 42 q^{14} + 4 q^{16} - 118 q^{19} - 36 q^{26} - 318 q^{29} - 416 q^{31} - 638 q^{34} + 486 q^{41} - 852 q^{44} - 598 q^{46} - 350 q^{49} + 1530 q^{56} + 1146 q^{59} - 398 q^{61} - 1640 q^{64}+ \cdots - 1238 q^{94}+O(q^{100}) $ 8 * q + 24 * q^4 + 42 * q^14 + 4 * q^16 - 118 * q^19 - 36 * q^26 - 318 * q^29 - 416 * q^31 - 638 * q^34 + 486 * q^41 - 852 * q^44 - 598 * q^46 - 350 * q^49 + 1530 * q^56 + 1146 * q^59 - 398 * q^61 - 1640 * q^64 - 1728 * q^71 - 1218 * q^74 - 3498 * q^76 - 2596 * q^79 + 480 * q^86 - 1086 * q^89 - 2574 * q^91 - 1238 * q^94

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 $

$2$	3.99806	1.41353	0.706764	−	0.707450i	$-0.250154\pi$
$2$	3.99806	1.41353	0.706764	+	0.707450i	$0.250154\pi$
$3$	0	0
$3$	0	0
$4$	7.98447	0.998059
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.79840	0.421074	0.210537	−	0.977586i	$-0.432479\pi$
$7$	7.79840	0.421074	0.210537	+	0.977586i	$0.432479\pi$
$8$	−0.0620886	−0.00274396
$9$	0	0
$10$	0	0
$11$	32.7050	0.896448	0.448224	−	0.893921i	$-0.352057\pi$
$11$	32.7050	0.896448	0.448224	+	0.893921i	$0.352057\pi$
$12$	0	0
$13$	−65.9775	−1.40761	−0.703803	−	0.710395i	$-0.748516\pi$
$13$	−65.9775	−1.40761	−0.703803	+	0.710395i	$0.748516\pi$
$14$	31.1784	0.595199
$15$	0	0
$16$	−64.1240	−1.00194
$17$	15.5853	0.222353	0.111176	−	0.993801i	$-0.464538\pi$
$17$	15.5853	0.222353	0.111176	+	0.993801i	$0.464538\pi$
$18$	0	0
$19$	51.6661	0.623843	0.311921	−	0.950108i	$-0.399027\pi$
$19$	51.6661	0.623843	0.311921	+	0.950108i	$0.399027\pi$
$20$	0	0
$21$	0	0
$22$	130.756	1.26715
$23$	−49.6796	−0.450387	−0.225194	−	0.974314i	$-0.572301\pi$
$23$	−49.6796	−0.450387	−0.225194	+	0.974314i	$0.572301\pi$
$24$	0	0
$25$	0	0
$26$	−263.782	−1.98969
$27$	0	0
$28$	62.2661	0.420256
$29$	−282.020	−1.80586	−0.902929	−	0.429790i	$-0.858587\pi$
$29$	−282.020	−1.80586	−0.902929	+	0.429790i	$0.858587\pi$
$30$	0	0
$31$	−147.663	−0.855515	−0.427758	−	0.903893i	$-0.640696\pi$
$31$	−147.663	−0.855515	−0.427758	+	0.903893i	$0.640696\pi$
$32$	−255.875	−1.41352
$33$	0	0
$34$	62.3111	0.314302
$35$	0	0
$36$	0	0
$37$	122.498	0.544284	0.272142	−	0.962257i	$-0.412268\pi$
$37$	122.498	0.544284	0.272142	+	0.962257i	$0.412268\pi$
$38$	206.564	0.881818
$39$	0	0
$40$	0	0
$41$	−109.032	−0.415314	−0.207657	−	0.978202i	$-0.566584\pi$
$41$	−109.032	−0.415314	−0.207657	+	0.978202i	$0.566584\pi$
$42$	0	0
$43$	191.493	0.679124	0.339562	−	0.940584i	$-0.389721\pi$
$43$	191.493	0.679124	0.339562	+	0.940584i	$0.389721\pi$
$44$	261.132	0.894707
$45$	0	0
$46$	−198.622	−0.636634
$47$	−414.420	−1.28616	−0.643078	−	0.765800i	$-0.722343\pi$
$47$	−414.420	−1.28616	−0.643078	+	0.765800i	$0.722343\pi$
$48$	0	0
$49$	−282.185	−0.822697
$50$	0	0
$51$	0	0
$52$	−526.796	−1.40487
$53$	236.381	0.612631	0.306315	−	0.951930i	$-0.400904\pi$
$53$	236.381	0.612631	0.306315	+	0.951930i	$0.400904\pi$
$54$	0	0
$55$	0	0
$56$	−0.484192	−0.00115541
$57$	0	0
$58$	−1127.53	−2.55263
$59$	784.024	1.73002	0.865011	−	0.501753i	$-0.167312\pi$
$59$	784.024	1.73002	0.865011	+	0.501753i	$0.167312\pi$
$60$	0	0
$61$	−535.138	−1.12324	−0.561618	−	0.827397i	$-0.689821\pi$
$61$	−535.138	−1.12324	−0.561618	+	0.827397i	$0.689821\pi$
$62$	−590.363	−1.20929
$63$	0	0
$64$	−510.010	−0.996114
$65$	0	0
$66$	0	0
$67$	121.069	0.220761	0.110380	−	0.993889i	$-0.464793\pi$
$67$	121.069	0.220761	0.110380	+	0.993889i	$0.464793\pi$
$68$	124.441	0.221921
$69$	0	0
$70$	0	0
$71$	590.432	0.986921	0.493460	−	0.869768i	$-0.335732\pi$
$71$	590.432	0.986921	0.493460	+	0.869768i	$0.335732\pi$
$72$	0	0
$73$	986.078	1.58098	0.790491	−	0.612473i	$-0.209825\pi$
$73$	986.078	1.58098	0.790491	+	0.612473i	$0.209825\pi$
$74$	489.753	0.769360
$75$	0	0
$76$	412.526	0.622632
$77$	255.046	0.377471
$78$	0	0
$79$	−607.025	−0.864503	−0.432251	−	0.901753i	$-0.642281\pi$
$79$	−607.025	−0.864503	−0.432251	+	0.901753i	$0.642281\pi$
$80$	0	0
$81$	0	0
$82$	−435.915	−0.587058
$83$	−1292.46	−1.70923	−0.854616	−	0.519260i	$-0.826208\pi$
$83$	−1292.46	−1.70923	−0.854616	+	0.519260i	$0.826208\pi$
$84$	0	0
$85$	0	0
$86$	765.598	0.959961
$87$	0	0
$88$	−2.03061	−0.00245981
$89$	−1506.78	−1.79459	−0.897296	−	0.441428i	$-0.854472\pi$
$89$	−1506.78	−1.79459	−0.897296	+	0.441428i	$0.854472\pi$
$90$	0	0
$91$	−514.519	−0.592706
$92$	−396.665	−0.449513
$93$	0	0
$94$	−1656.87	−1.81802
$95$	0	0
$96$	0	0
$97$	−610.887	−0.639445	−0.319723	−	0.947511i	$-0.603590\pi$
$97$	−610.887	−0.639445	−0.319723	+	0.947511i	$0.603590\pi$
$98$	−1128.19	−1.16290
$99$	0	0
$100$	0	0
$101$	−584.247	−0.575592	−0.287796	−	0.957692i	$-0.592923\pi$
$101$	−584.247	−0.575592	−0.287796	+	0.957692i	$0.592923\pi$
$102$	0	0
$103$	−1452.09	−1.38912	−0.694558	−	0.719437i	$-0.744400\pi$
$103$	−1452.09	−1.38912	−0.694558	+	0.719437i	$0.744400\pi$
$104$	4.09645	0.00386241
$105$	0	0
$106$	945.065	0.865970
$107$	−361.964	−0.327032	−0.163516	−	0.986541i	$-0.552284\pi$
$107$	−361.964	−0.327032	−0.163516	+	0.986541i	$0.552284\pi$
$108$	0	0
$109$	−1073.79	−0.943582	−0.471791	−	0.881710i	$-0.656392\pi$
$109$	−1073.79	−0.943582	−0.471791	+	0.881710i	$0.656392\pi$
$110$	0	0
$111$	0	0
$112$	−500.064	−0.421890
$113$	−1416.62	−1.17933	−0.589667	−	0.807646i	$-0.700741\pi$
$113$	−1416.62	−1.17933	−0.589667	+	0.807646i	$0.700741\pi$
$114$	0	0
$115$	0	0
$116$	−2251.78	−1.80235
$117$	0	0
$118$	3134.57	2.44543
$119$	121.541	0.0936270
$120$	0	0
$121$	−261.384	−0.196382
$122$	−2139.51	−1.58772
$123$	0	0
$124$	−1179.01	−0.853854
$125$	0	0
$126$	0	0
$127$	2709.85	1.89339	0.946694	−	0.322134i	$-0.104400\pi$
$127$	2709.85	1.89339	0.946694	+	0.322134i	$0.104400\pi$
$128$	7.94733	0.00548790
$129$	0	0
$130$	0	0
$131$	−621.192	−0.414304	−0.207152	−	0.978309i	$-0.566419\pi$
$131$	−621.192	−0.414304	−0.207152	+	0.978309i	$0.566419\pi$
$132$	0	0
$133$	402.912	0.262684
$134$	484.042	0.312051
$135$	0	0
$136$	−0.967672	−0.000610127 0
$137$	1328.54	0.828502	0.414251	−	0.910163i	$-0.364044\pi$
$137$	1328.54	0.828502	0.414251	+	0.910163i	$0.364044\pi$
$138$	0	0
$139$	2536.79	1.54797	0.773986	−	0.633203i	$-0.218260\pi$
$139$	2536.79	1.54797	0.773986	+	0.633203i	$0.218260\pi$
$140$	0	0
$141$	0	0
$142$	2360.58	1.39504
$143$	−2157.79	−1.26184
$144$	0	0
$145$	0	0
$146$	3942.40	2.23476
$147$	0	0
$148$	978.079	0.543227
$149$	−321.699	−0.176877	−0.0884384	−	0.996082i	$-0.528188\pi$
$149$	−321.699	−0.176877	−0.0884384	+	0.996082i	$0.528188\pi$
$150$	0	0
$151$	−3196.36	−1.72262	−0.861311	−	0.508078i	$-0.830356\pi$
$151$	−3196.36	−1.72262	−0.861311	+	0.508078i	$0.830356\pi$
$152$	−3.20787	−0.00171180
$153$	0	0
$154$	1019.69	0.533565
$155$	0	0
$156$	0	0
$157$	−2607.69	−1.32558	−0.662792	−	0.748804i	$-0.730628\pi$
$157$	−2607.69	−1.32558	−0.662792	+	0.748804i	$0.730628\pi$
$158$	−2426.92	−1.22200
$159$	0	0
$160$	0	0
$161$	−387.421	−0.189646
$162$	0	0
$163$	642.287	0.308637	0.154318	−	0.988021i	$-0.450682\pi$
$163$	642.287	0.308637	0.154318	+	0.988021i	$0.450682\pi$
$164$	−870.560	−0.414508
$165$	0	0
$166$	−5167.34	−2.41605
$167$	1999.64	0.926569	0.463285	−	0.886209i	$-0.346671\pi$
$167$	1999.64	0.926569	0.463285	+	0.886209i	$0.346671\pi$
$168$	0	0
$169$	2156.03	0.981353
$170$	0	0
$171$	0	0
$172$	1528.97	0.677806
$173$	−3070.57	−1.34943	−0.674714	−	0.738079i	$-0.735733\pi$
$173$	−3070.57	−1.34943	−0.674714	+	0.738079i	$0.735733\pi$
$174$	0	0
$175$	0	0
$176$	−2097.17	−0.898184
$177$	0	0
$178$	−6024.21	−2.53671
$179$	656.735	0.274227	0.137114	−	0.990555i	$-0.456217\pi$
$179$	656.735	0.274227	0.137114	+	0.990555i	$0.456217\pi$
$180$	0	0
$181$	−1504.70	−0.617919	−0.308960	−	0.951075i	$-0.599981\pi$
$181$	−1504.70	−0.617919	−0.308960	+	0.951075i	$0.599981\pi$
$182$	−2057.08	−0.837806
$183$	0	0
$184$	3.08454	0.00123584
$185$	0	0
$186$	0	0
$187$	509.718	0.199328
$188$	−3308.92	−1.28366
$189$	0	0
$190$	0	0
$191$	4819.27	1.82571	0.912853	−	0.408288i	$-0.133874\pi$
$191$	4819.27	1.82571	0.912853	+	0.408288i	$0.133874\pi$
$192$	0	0
$193$	−4027.39	−1.50206	−0.751031	−	0.660267i	$-0.770443\pi$
$193$	−4027.39	−1.50206	−0.751031	+	0.660267i	$0.770443\pi$
$194$	−2442.36	−0.903873
$195$	0	0
$196$	−2253.10	−0.821100
$197$	2132.04	0.771075	0.385537	−	0.922692i	$-0.374016\pi$
$197$	2132.04	0.771075	0.385537	+	0.922692i	$0.374016\pi$
$198$	0	0
$199$	3024.96	1.07756	0.538779	−	0.842447i	$-0.318886\pi$
$199$	3024.96	1.07756	0.538779	+	0.842447i	$0.318886\pi$
$200$	0	0
$201$	0	0
$202$	−2335.86	−0.813615
$203$	−2199.31	−0.760399
$204$	0	0
$205$	0	0
$206$	−5805.55	−1.96355
$207$	0	0
$208$	4230.74	1.41033
$209$	1689.74	0.559242
$210$	0	0
$211$	−2191.00	−0.714856	−0.357428	−	0.933941i	$-0.616346\pi$
$211$	−2191.00	−0.714856	−0.357428	+	0.933941i	$0.616346\pi$
$212$	1887.38	0.611442
$213$	0	0
$214$	−1447.15	−0.462268
$215$	0	0
$216$	0	0
$217$	−1151.53	−0.360235
$218$	−4293.08	−1.33378
$219$	0	0
$220$	0	0
$221$	−1028.28	−0.312985
$222$	0	0
$223$	1159.78	0.348271	0.174136	−	0.984722i	$-0.444287\pi$
$223$	1159.78	0.348271	0.174136	+	0.984722i	$0.444287\pi$
$224$	−1995.41	−0.595197
$225$	0	0
$226$	−5663.75	−1.66702
$227$	408.227	0.119361	0.0596805	−	0.998218i	$-0.480992\pi$
$227$	408.227	0.119361	0.0596805	+	0.998218i	$0.480992\pi$
$228$	0	0
$229$	−1858.12	−0.536193	−0.268097	−	0.963392i	$-0.586395\pi$
$229$	−1858.12	−0.536193	−0.268097	+	0.963392i	$0.586395\pi$
$230$	0	0
$231$	0	0
$232$	17.5103	0.00495519
$233$	−2813.15	−0.790968	−0.395484	−	0.918473i	$-0.629423\pi$
$233$	−2813.15	−0.790968	−0.395484	+	0.918473i	$0.629423\pi$
$234$	0	0
$235$	0	0
$236$	6260.02	1.72666
$237$	0	0
$238$	485.927	0.132344
$239$	−5756.48	−1.55797	−0.778987	−	0.627040i	$-0.784266\pi$
$239$	−5756.48	−1.55797	−0.778987	+	0.627040i	$0.784266\pi$
$240$	0	0
$241$	5740.04	1.53423	0.767114	−	0.641511i	$-0.221692\pi$
$241$	5740.04	1.53423	0.767114	+	0.641511i	$0.221692\pi$
$242$	−1045.03	−0.277591
$243$	0	0
$244$	−4272.79	−1.12105
$245$	0	0
$246$	0	0
$247$	−3408.80	−0.878124
$248$	9.16816	0.00234750
$249$	0	0
$250$	0	0
$251$	−1120.86	−0.281866	−0.140933	−	0.990019i	$-0.545010\pi$
$251$	−1120.86	−0.281866	−0.140933	+	0.990019i	$0.545010\pi$
$252$	0	0
$253$	−1624.77	−0.403749
$254$	10834.1	2.67636
$255$	0	0
$256$	4111.86	1.00387
$257$	982.609	0.238496	0.119248	−	0.992864i	$-0.461952\pi$
$257$	982.609	0.238496	0.119248	+	0.992864i	$0.461952\pi$
$258$	0	0
$259$	955.286	0.229184
$260$	0	0
$261$	0	0
$262$	−2483.56	−0.585630
$263$	3419.85	0.801813	0.400907	−	0.916119i	$-0.368695\pi$
$263$	3419.85	0.801813	0.400907	+	0.916119i	$0.368695\pi$
$264$	0	0
$265$	0	0
$266$	1610.87	0.371311
$267$	0	0
$268$	966.674	0.220332
$269$	−1425.68	−0.323142	−0.161571	−	0.986861i	$-0.551656\pi$
$269$	−1425.68	−0.323142	−0.161571	+	0.986861i	$0.551656\pi$
$270$	0	0
$271$	4234.80	0.949247	0.474624	−	0.880189i	$-0.342584\pi$
$271$	4234.80	0.949247	0.474624	+	0.880189i	$0.342584\pi$
$272$	−999.394	−0.222784
$273$	0	0
$274$	5311.58	1.17111
$275$	0	0
$276$	0	0
$277$	−3258.65	−0.706835	−0.353418	−	0.935466i	$-0.614981\pi$
$277$	−3258.65	−0.706835	−0.353418	+	0.935466i	$0.614981\pi$
$278$	10142.2	2.18810
$279$	0	0
$280$	0	0
$281$	936.334	0.198779	0.0993897	−	0.995049i	$-0.468311\pi$
$281$	936.334	0.198779	0.0993897	+	0.995049i	$0.468311\pi$
$282$	0	0
$283$	1368.62	0.287477	0.143739	−	0.989616i	$-0.454088\pi$
$283$	1368.62	0.287477	0.143739	+	0.989616i	$0.454088\pi$
$284$	4714.29	0.985005
$285$	0	0
$286$	−8626.98	−1.78365
$287$	−850.272	−0.174878
$288$	0	0
$289$	−4670.10	−0.950559
$290$	0	0
$291$	0	0
$292$	7873.31	1.57791
$293$	−1667.96	−0.332571	−0.166285	−	0.986078i	$-0.553177\pi$
$293$	−1667.96	−0.332571	−0.166285	+	0.986078i	$0.553177\pi$
$294$	0	0
$295$	0	0
$296$	−7.60571	−0.00149349
$297$	0	0
$298$	−1286.17	−0.250020
$299$	3277.73	0.633967
$300$	0	0
$301$	1493.34	0.285961
$302$	−12779.2	−2.43497
$303$	0	0
$304$	−3313.03	−0.625051
$305$	0	0
$306$	0	0
$307$	6894.86	1.28179	0.640897	−	0.767627i	$-0.278563\pi$
$307$	6894.86	1.28179	0.640897	+	0.767627i	$0.278563\pi$
$308$	2036.41	0.376738
$309$	0	0
$310$	0	0
$311$	697.317	0.127142	0.0635711	−	0.997977i	$-0.479751\pi$
$311$	697.317	0.127142	0.0635711	+	0.997977i	$0.479751\pi$
$312$	0	0
$313$	4889.18	0.882917	0.441459	−	0.897282i	$-0.354461\pi$
$313$	4889.18	0.882917	0.441459	+	0.897282i	$0.354461\pi$
$314$	−10425.7	−1.87375
$315$	0	0
$316$	−4846.78	−0.862824
$317$	5468.56	0.968911	0.484455	−	0.874816i	$-0.339018\pi$
$317$	5468.56	0.968911	0.484455	+	0.874816i	$0.339018\pi$
$318$	0	0
$319$	−9223.47	−1.61886
$320$	0	0
$321$	0	0
$322$	−1548.93	−0.268070
$323$	805.233	0.138713
$324$	0	0
$325$	0	0
$326$	2567.90	0.436266
$327$	0	0
$328$	6.76962	0.00113960
$329$	−3231.81	−0.541567
$330$	0	0
$331$	−5172.64	−0.858955	−0.429478	−	0.903078i	$-0.641302\pi$
$331$	−5172.64	−0.858955	−0.429478	+	0.903078i	$0.641302\pi$
$332$	−10319.6	−1.70591
$333$	0	0
$334$	7994.69	1.30973
$335$	0	0
$336$	0	0
$337$	409.004	0.0661124	0.0330562	−	0.999453i	$-0.489476\pi$
$337$	409.004	0.0661124	0.0330562	+	0.999453i	$0.489476\pi$
$338$	8619.94	1.38717
$339$	0	0
$340$	0	0
$341$	−4829.30	−0.766925
$342$	0	0
$343$	−4875.44	−0.767490
$344$	−11.8895	−0.00186349
$345$	0	0
$346$	−12276.3	−1.90745
$347$	8027.94	1.24197	0.620983	−	0.783824i	$-0.286734\pi$
$347$	8027.94	1.24197	0.620983	+	0.783824i	$0.286734\pi$
$348$	0	0
$349$	−4988.97	−0.765196	−0.382598	−	0.923915i	$-0.624971\pi$
$349$	−4988.97	−0.765196	−0.382598	+	0.923915i	$0.624971\pi$
$350$	0	0
$351$	0	0
$352$	−8368.38	−1.26715
$353$	4238.90	0.639133	0.319567	−	0.947564i	$-0.396463\pi$
$353$	4238.90	0.639133	0.319567	+	0.947564i	$0.396463\pi$
$354$	0	0
$355$	0	0
$356$	−12030.9	−1.79111
$357$	0	0
$358$	2625.66	0.387628
$359$	9453.60	1.38981	0.694905	−	0.719101i	$-0.255446\pi$
$359$	9453.60	1.38981	0.694905	+	0.719101i	$0.255446\pi$
$360$	0	0
$361$	−4189.62	−0.610820
$362$	−6015.88	−0.873446
$363$	0	0
$364$	−4108.16	−0.591555
$365$	0	0
$366$	0	0
$367$	2750.63	0.391231	0.195616	−	0.980681i	$-0.437330\pi$
$367$	2750.63	0.391231	0.195616	+	0.980681i	$0.437330\pi$
$368$	3185.65	0.451260
$369$	0	0
$370$	0	0
$371$	1843.39	0.257963
$372$	0	0
$373$	−3417.55	−0.474407	−0.237204	−	0.971460i	$-0.576231\pi$
$373$	−3417.55	−0.474407	−0.237204	+	0.971460i	$0.576231\pi$
$374$	2037.88	0.281755
$375$	0	0
$376$	25.7308	0.00352916
$377$	18607.0	2.54193
$378$	0	0
$379$	−9983.20	−1.35304	−0.676520	−	0.736424i	$-0.736513\pi$
$379$	−9983.20	−1.35304	−0.676520	+	0.736424i	$0.736513\pi$
$380$	0	0
$381$	0	0
$382$	19267.7	2.58069
$383$	13456.7	1.79531	0.897654	−	0.440701i	$-0.145270\pi$
$383$	13456.7	1.79531	0.897654	+	0.440701i	$0.145270\pi$
$384$	0	0
$385$	0	0
$386$	−16101.7	−2.12321
$387$	0	0
$388$	−4877.61	−0.638204
$389$	−8101.33	−1.05592	−0.527961	−	0.849269i	$-0.677043\pi$
$389$	−8101.33	−1.05592	−0.527961	+	0.849269i	$0.677043\pi$
$390$	0	0
$391$	−774.273	−0.100145
$392$	17.5205	0.00225744
$393$	0	0
$394$	8524.03	1.08994
$395$	0	0
$396$	0	0
$397$	−7043.15	−0.890392	−0.445196	−	0.895433i	$-0.646866\pi$
$397$	−7043.15	−0.890392	−0.445196	+	0.895433i	$0.646866\pi$
$398$	12094.0	1.52316
$399$	0	0
$400$	0	0
$401$	2716.60	0.338305	0.169153	−	0.985590i	$-0.445897\pi$
$401$	2716.60	0.338305	0.169153	+	0.985590i	$0.445897\pi$
$402$	0	0
$403$	9742.41	1.20423
$404$	−4664.91	−0.574475
$405$	0	0
$406$	−8792.96	−1.07485
$407$	4006.28	0.487922
$408$	0	0
$409$	−6062.59	−0.732948	−0.366474	−	0.930428i	$-0.619435\pi$
$409$	−6062.59	−0.732948	−0.366474	+	0.930428i	$0.619435\pi$
$410$	0	0
$411$	0	0
$412$	−11594.2	−1.38642
$413$	6114.13	0.728467
$414$	0	0
$415$	0	0
$416$	16882.0	1.98968
$417$	0	0
$418$	6755.67	0.790504
$419$	7008.83	0.817193	0.408596	−	0.912715i	$-0.366018\pi$
$419$	7008.83	0.817193	0.408596	+	0.912715i	$0.366018\pi$
$420$	0	0
$421$	−10812.3	−1.25168	−0.625842	−	0.779950i	$-0.715245\pi$
$421$	−10812.3	−1.25168	−0.625842	+	0.779950i	$0.715245\pi$
$422$	−8759.74	−1.01047
$423$	0	0
$424$	−14.6766	−0.00168103
$425$	0	0
$426$	0	0
$427$	−4173.22	−0.472965
$428$	−2890.09	−0.326397
$429$	0	0
$430$	0	0
$431$	−8875.68	−0.991941	−0.495970	−	0.868339i	$-0.665188\pi$
$431$	−8875.68	−0.991941	−0.495970	+	0.868339i	$0.665188\pi$
$432$	0	0
$433$	−7039.08	−0.781239	−0.390619	−	0.920552i	$-0.627739\pi$
$433$	−7039.08	−0.781239	−0.390619	+	0.920552i	$0.627739\pi$
$434$	−4603.89	−0.509202
$435$	0	0
$436$	−8573.65	−0.941751
$437$	−2566.75	−0.280971
$438$	0	0
$439$	−5139.05	−0.558710	−0.279355	−	0.960188i	$-0.590121\pi$
$439$	−5139.05	−0.558710	−0.279355	+	0.960188i	$0.590121\pi$
$440$	0	0
$441$	0	0
$442$	−4111.13	−0.442413
$443$	5074.43	0.544229	0.272115	−	0.962265i	$-0.412277\pi$
$443$	5074.43	0.544229	0.272115	+	0.962265i	$0.412277\pi$
$444$	0	0
$445$	0	0
$446$	4636.86	0.492291
$447$	0	0
$448$	−3977.26	−0.419437
$449$	11783.0	1.23848	0.619238	−	0.785203i	$-0.287442\pi$
$449$	11783.0	1.23848	0.619238	+	0.785203i	$0.287442\pi$
$450$	0	0
$451$	−3565.88	−0.372307
$452$	−11311.0	−1.17705
$453$	0	0
$454$	1632.11	0.168720
$455$	0	0
$456$	0	0
$457$	11230.4	1.14953	0.574764	−	0.818320i	$-0.305094\pi$
$457$	11230.4	1.14953	0.574764	+	0.818320i	$0.305094\pi$
$458$	−7428.88	−0.757923
$459$	0	0
$460$	0	0
$461$	7987.75	0.806999	0.403499	−	0.914980i	$-0.367794\pi$
$461$	7987.75	0.806999	0.403499	+	0.914980i	$0.367794\pi$
$462$	0	0
$463$	16415.3	1.64770	0.823848	−	0.566810i	$-0.191823\pi$
$463$	16415.3	1.64770	0.823848	+	0.566810i	$0.191823\pi$
$464$	18084.3	1.80936
$465$	0	0
$466$	−11247.1	−1.11805
$467$	−6002.60	−0.594790	−0.297395	−	0.954755i	$-0.596118\pi$
$467$	−6002.60	−0.594790	−0.297395	+	0.954755i	$0.596118\pi$
$468$	0	0
$469$	944.146	0.0929565
$470$	0	0
$471$	0	0
$472$	−48.6790	−0.00474710
$473$	6262.76	0.608799
$474$	0	0
$475$	0	0
$476$	970.438	0.0934453
$477$	0	0
$478$	−23014.7	−2.20224
$479$	−2516.64	−0.240059	−0.120029	−	0.992770i	$-0.538299\pi$
$479$	−2516.64	−0.240059	−0.120029	+	0.992770i	$0.538299\pi$
$480$	0	0
$481$	−8082.09	−0.766137
$482$	22949.0	2.16867
$483$	0	0
$484$	−2087.01	−0.196000
$485$	0	0
$486$	0	0
$487$	3893.26	0.362260	0.181130	−	0.983459i	$-0.442025\pi$
$487$	3893.26	0.362260	0.181130	+	0.983459i	$0.442025\pi$
$488$	33.2260	0.00308211
$489$	0	0
$490$	0	0
$491$	8911.50	0.819084	0.409542	−	0.912291i	$-0.365688\pi$
$491$	8911.50	0.819084	0.409542	+	0.912291i	$0.365688\pi$
$492$	0	0
$493$	−4395.38	−0.401538
$494$	−13628.6	−1.24125
$495$	0	0
$496$	9468.71	0.857173
$497$	4604.42	0.415567
$498$	0	0
$499$	8639.01	0.775021	0.387510	−	0.921865i	$-0.373335\pi$
$499$	8639.01	0.775021	0.387510	+	0.921865i	$0.373335\pi$
$500$	0	0
$501$	0	0
$502$	−4481.28	−0.398425
$503$	12847.4	1.13884	0.569422	−	0.822046i	$-0.307167\pi$
$503$	12847.4	1.13884	0.569422	+	0.822046i	$0.307167\pi$
$504$	0	0
$505$	0	0
$506$	−6495.92	−0.570709
$507$	0	0
$508$	21636.7	1.88971
$509$	−3956.77	−0.344560	−0.172280	−	0.985048i	$-0.555113\pi$
$509$	−3956.77	−0.344560	−0.172280	+	0.985048i	$0.555113\pi$
$510$	0	0
$511$	7689.83	0.665710
$512$	16375.9	1.41351
$513$	0	0
$514$	3928.53	0.337121
$515$	0	0
$516$	0	0
$517$	−13553.6	−1.15297
$518$	3819.29	0.323957
$519$	0	0
$520$	0	0
$521$	6392.00	0.537502	0.268751	−	0.963210i	$-0.413389\pi$
$521$	6392.00	0.537502	0.268751	+	0.963210i	$0.413389\pi$
$522$	0	0
$523$	−13029.6	−1.08938	−0.544690	−	0.838638i	$-0.683353\pi$
$523$	−13029.6	−1.08938	−0.544690	+	0.838638i	$0.683353\pi$
$524$	−4959.89	−0.413500
$525$	0	0
$526$	13672.8	1.13338
$527$	−2301.37	−0.190226
$528$	0	0
$529$	−9698.94	−0.797151
$530$	0	0
$531$	0	0
$532$	3217.04	0.262174
$533$	7193.64	0.584598
$534$	0	0
$535$	0	0
$536$	−7.51702	−0.000605757 0
$537$	0	0
$538$	−5699.95	−0.456770
$539$	−9228.86	−0.737505
$540$	0	0
$541$	−9481.72	−0.753514	−0.376757	−	0.926312i	$-0.622961\pi$
$541$	−9481.72	−0.753514	−0.376757	+	0.926312i	$0.622961\pi$
$542$	16931.0	1.34179
$543$	0	0
$544$	−3987.90	−0.314301
$545$	0	0
$546$	0	0
$547$	22132.1	1.72998	0.864991	−	0.501788i	$-0.167324\pi$
$547$	22132.1	1.72998	0.864991	+	0.501788i	$0.167324\pi$
$548$	10607.7	0.826893
$549$	0	0
$550$	0	0
$551$	−14570.9	−1.12657
$552$	0	0
$553$	−4733.83	−0.364019
$554$	−13028.3	−0.999131
$555$	0	0
$556$	20255.0	1.54497
$557$	4426.25	0.336708	0.168354	−	0.985727i	$-0.446155\pi$
$557$	4426.25	0.336708	0.168354	+	0.985727i	$0.446155\pi$
$558$	0	0
$559$	−12634.2	−0.955939
$560$	0	0
$561$	0	0
$562$	3743.52	0.280980
$563$	3173.26	0.237543	0.118772	−	0.992922i	$-0.462104\pi$
$563$	3173.26	0.237543	0.118772	+	0.992922i	$0.462104\pi$
$564$	0	0
$565$	0	0
$566$	5471.83	0.406357
$567$	0	0
$568$	−36.6591	−0.00270807
$569$	3665.79	0.270084	0.135042	−	0.990840i	$-0.456883\pi$
$569$	3665.79	0.270084	0.135042	+	0.990840i	$0.456883\pi$
$570$	0	0
$571$	9022.74	0.661279	0.330639	−	0.943757i	$-0.392736\pi$
$571$	9022.74	0.661279	0.330639	+	0.943757i	$0.392736\pi$
$572$	−17228.8	−1.25940
$573$	0	0
$574$	−3399.44	−0.247195
$575$	0	0
$576$	0	0
$577$	−16600.1	−1.19769	−0.598847	−	0.800863i	$-0.704374\pi$
$577$	−16600.1	−1.19769	−0.598847	+	0.800863i	$0.704374\pi$
$578$	−18671.3	−1.34364
$579$	0	0
$580$	0	0
$581$	−10079.1	−0.719713
$582$	0	0
$583$	7730.84	0.549191
$584$	−61.2242	−0.00433815
$585$	0	0
$586$	−6668.60	−0.470098
$587$	18193.9	1.27929	0.639645	−	0.768671i	$-0.279082\pi$
$587$	18193.9	1.27929	0.639645	+	0.768671i	$0.279082\pi$
$588$	0	0
$589$	−7629.14	−0.533707
$590$	0	0
$591$	0	0
$592$	−7855.04	−0.545338
$593$	−448.648	−0.0310688	−0.0155344	−	0.999879i	$-0.504945\pi$
$593$	−448.648	−0.0310688	−0.0155344	+	0.999879i	$0.504945\pi$
$594$	0	0
$595$	0	0
$596$	−2568.60	−0.176533
$597$	0	0
$598$	13104.6	0.896130
$599$	−8787.84	−0.599435	−0.299717	−	0.954028i	$-0.596892\pi$
$599$	−8787.84	−0.599435	−0.299717	+	0.954028i	$0.596892\pi$
$600$	0	0
$601$	9787.87	0.664319	0.332159	−	0.943223i	$-0.392223\pi$
$601$	9787.87	0.664319	0.332159	+	0.943223i	$0.392223\pi$
$602$	5970.44	0.404214
$603$	0	0
$604$	−25521.2	−1.71928
$605$	0	0
$606$	0	0
$607$	−12769.7	−0.853883	−0.426942	−	0.904279i	$-0.640409\pi$
$607$	−12769.7	−0.853883	−0.426942	+	0.904279i	$0.640409\pi$
$608$	−13220.0	−0.881815
$609$	0	0
$610$	0	0
$611$	27342.4	1.81040
$612$	0	0
$613$	22910.6	1.50954	0.754771	−	0.655988i	$-0.227748\pi$
$613$	22910.6	1.50954	0.754771	+	0.655988i	$0.227748\pi$
$614$	27566.1	1.81185
$615$	0	0
$616$	−15.8355	−0.00103576
$617$	22841.4	1.49038	0.745188	−	0.666854i	$-0.232360\pi$
$617$	22841.4	1.49038	0.745188	+	0.666854i	$0.232360\pi$
$618$	0	0
$619$	−495.421	−0.0321691	−0.0160845	−	0.999871i	$-0.505120\pi$
$619$	−495.421	−0.0321691	−0.0160845	+	0.999871i	$0.505120\pi$
$620$	0	0
$621$	0	0
$622$	2787.92	0.179719
$623$	−11750.5	−0.755656
$624$	0	0
$625$	0	0
$626$	19547.2	1.24803
$627$	0	0
$628$	−20821.0	−1.32301
$629$	1909.17	0.121023
$630$	0	0
$631$	25481.5	1.60761	0.803807	−	0.594890i	$-0.202805\pi$
$631$	25481.5	1.60761	0.803807	+	0.594890i	$0.202805\pi$
$632$	37.6894	0.00237216
$633$	0	0
$634$	21863.6	1.36958
$635$	0	0
$636$	0	0
$637$	18617.9	1.15803
$638$	−36876.0	−2.28830
$639$	0	0
$640$	0	0
$641$	−2913.25	−0.179511	−0.0897555	−	0.995964i	$-0.528609\pi$
$641$	−2913.25	−0.179511	−0.0897555	+	0.995964i	$0.528609\pi$
$642$	0	0
$643$	15121.2	0.927404	0.463702	−	0.885991i	$-0.346521\pi$
$643$	15121.2	0.927404	0.463702	+	0.885991i	$0.346521\pi$
$644$	−3093.35	−0.189278
$645$	0	0
$646$	3219.37	0.196075
$647$	8436.06	0.512605	0.256303	−	0.966597i	$-0.417496\pi$
$647$	8436.06	0.512605	0.256303	+	0.966597i	$0.417496\pi$
$648$	0	0
$649$	25641.5	1.55087
$650$	0	0
$651$	0	0
$652$	5128.32	0.308037
$653$	−17146.6	−1.02756	−0.513780	−	0.857922i	$-0.671755\pi$
$653$	−17146.6	−1.02756	−0.513780	+	0.857922i	$0.671755\pi$
$654$	0	0
$655$	0	0
$656$	6991.54	0.416119
$657$	0	0
$658$	−12921.0	−0.765519
$659$	1774.60	0.104900	0.0524498	−	0.998624i	$-0.483297\pi$
$659$	1774.60	0.104900	0.0524498	+	0.998624i	$0.483297\pi$
$660$	0	0
$661$	−7159.71	−0.421302	−0.210651	−	0.977561i	$-0.567558\pi$
$661$	−7159.71	−0.421302	−0.210651	+	0.977561i	$0.567558\pi$
$662$	−20680.5	−1.21416
$663$	0	0
$664$	80.2473	0.00469006
$665$	0	0
$666$	0	0
$667$	14010.6	0.813335
$668$	15966.1	0.924771
$669$	0	0
$670$	0	0
$671$	−17501.7	−1.00692
$672$	0	0
$673$	−15556.7	−0.891034	−0.445517	−	0.895273i	$-0.646980\pi$
$673$	−15556.7	−0.891034	−0.445517	+	0.895273i	$0.646980\pi$
$674$	1635.22	0.0934516
$675$	0	0
$676$	17214.8	0.979448
$677$	4461.98	0.253306	0.126653	−	0.991947i	$-0.459577\pi$
$677$	4461.98	0.253306	0.126653	+	0.991947i	$0.459577\pi$
$678$	0	0
$679$	−4763.94	−0.269254
$680$	0	0
$681$	0	0
$682$	−19307.8	−1.08407
$683$	33470.5	1.87513	0.937566	−	0.347808i	$-0.113074\pi$
$683$	33470.5	1.87513	0.937566	+	0.347808i	$0.113074\pi$
$684$	0	0
$685$	0	0
$686$	−19492.3	−1.08487
$687$	0	0
$688$	−12279.3	−0.680440
$689$	−15595.8	−0.862342
$690$	0	0
$691$	−20280.9	−1.11653	−0.558264	−	0.829664i	$-0.688532\pi$
$691$	−20280.9	−1.11653	−0.558264	+	0.829664i	$0.688532\pi$
$692$	−24516.9	−1.34681
$693$	0	0
$694$	32096.2	1.75555
$695$	0	0
$696$	0	0
$697$	−1699.29	−0.0923463
$698$	−19946.2	−1.08163
$699$	0	0
$700$	0	0
$701$	3762.36	0.202714	0.101357	−	0.994850i	$-0.467682\pi$
$701$	3762.36	0.202714	0.101357	+	0.994850i	$0.467682\pi$
$702$	0	0
$703$	6328.97	0.339547
$704$	−16679.9	−0.892964
$705$	0	0
$706$	16947.4	0.903432
$707$	−4556.19	−0.242367
$708$	0	0
$709$	1643.51	0.0870570	0.0435285	−	0.999052i	$-0.486140\pi$
$709$	1643.51	0.0870570	0.0435285	+	0.999052i	$0.486140\pi$
$710$	0	0
$711$	0	0
$712$	93.5541	0.00492428
$713$	7335.81	0.385313
$714$	0	0
$715$	0	0
$716$	5243.68	0.273695
$717$	0	0
$718$	37796.0	1.96453
$719$	3748.68	0.194440	0.0972200	−	0.995263i	$-0.469005\pi$
$719$	3748.68	0.194440	0.0972200	+	0.995263i	$0.469005\pi$
$720$	0	0
$721$	−11324.0	−0.584921
$722$	−16750.3	−0.863411
$723$	0	0
$724$	−12014.2	−0.616720
$725$	0	0
$726$	0	0
$727$	12221.3	0.623470	0.311735	−	0.950169i	$-0.399090\pi$
$727$	12221.3	0.623470	0.311735	+	0.950169i	$0.399090\pi$
$728$	31.9458	0.00162636
$729$	0	0
$730$	0	0
$731$	2984.48	0.151005
$732$	0	0
$733$	−31862.7	−1.60556	−0.802781	−	0.596274i	$-0.796647\pi$
$733$	−31862.7	−1.60556	−0.802781	+	0.596274i	$0.796647\pi$
$734$	10997.2	0.553016
$735$	0	0
$736$	12711.7	0.636632
$737$	3959.57	0.197900
$738$	0	0
$739$	9905.64	0.493078	0.246539	−	0.969133i	$-0.420707\pi$
$739$	9905.64	0.493078	0.246539	+	0.969133i	$0.420707\pi$
$740$	0	0
$741$	0	0
$742$	7369.99	0.364637
$743$	29282.9	1.44588	0.722938	−	0.690913i	$-0.242791\pi$
$743$	29282.9	1.44588	0.722938	+	0.690913i	$0.242791\pi$
$744$	0	0
$745$	0	0
$746$	−13663.6	−0.670587
$747$	0	0
$748$	4069.83	0.198941
$749$	−2822.74	−0.137705
$750$	0	0
$751$	−26361.7	−1.28090	−0.640448	−	0.768001i	$-0.721251\pi$
$751$	−26361.7	−1.28090	−0.640448	+	0.768001i	$0.721251\pi$
$752$	26574.3	1.28865
$753$	0	0
$754$	74391.9	3.59309
$755$	0	0
$756$	0	0
$757$	5667.66	0.272120	0.136060	−	0.990701i	$-0.456556\pi$
$757$	5667.66	0.272120	0.136060	+	0.990701i	$0.456556\pi$
$758$	−39913.4	−1.91256
$759$	0	0
$760$	0	0
$761$	−36532.3	−1.74020	−0.870101	−	0.492874i	$-0.835946\pi$
$761$	−36532.3	−1.74020	−0.870101	+	0.492874i	$0.835946\pi$
$762$	0	0
$763$	−8373.85	−0.397318
$764$	38479.3	1.82216
$765$	0	0
$766$	53800.5	2.53772
$767$	−51728.0	−2.43519
$768$	0	0
$769$	40900.2	1.91795	0.958973	−	0.283498i	$-0.0914949\pi$
$769$	40900.2	1.91795	0.958973	+	0.283498i	$0.0914949\pi$
$770$	0	0
$771$	0	0
$772$	−32156.6	−1.49915
$773$	−28081.8	−1.30664	−0.653319	−	0.757083i	$-0.726624\pi$
$773$	−28081.8	−1.30664	−0.653319	+	0.757083i	$0.726624\pi$
$774$	0	0
$775$	0	0
$776$	37.9292	0.00175461
$777$	0	0
$778$	−32389.6	−1.49257
$779$	−5633.23	−0.259091
$780$	0	0
$781$	19310.1	0.884723
$782$	−3095.59	−0.141558
$783$	0	0
$784$	18094.8	0.824291
$785$	0	0
$786$	0	0
$787$	−41047.8	−1.85921	−0.929605	−	0.368558i	$-0.879852\pi$
$787$	−41047.8	−1.85921	−0.929605	+	0.368558i	$0.879852\pi$
$788$	17023.2	0.769578
$789$	0	0
$790$	0	0
$791$	−11047.4	−0.496587
$792$	0	0
$793$	35307.0	1.58107
$794$	−28158.9	−1.25859
$795$	0	0
$796$	24152.7	1.07547
$797$	3768.31	0.167478	0.0837392	−	0.996488i	$-0.473314\pi$
$797$	3768.31	0.167478	0.0837392	+	0.996488i	$0.473314\pi$
$798$	0	0
$799$	−6458.88	−0.285981
$800$	0	0
$801$	0	0
$802$	10861.1	0.478203
$803$	32249.7	1.41727
$804$	0	0
$805$	0	0
$806$	38950.7	1.70221
$807$	0	0
$808$	36.2751	0.00157940
$809$	24285.5	1.05542	0.527708	−	0.849426i	$-0.323051\pi$
$809$	24285.5	1.05542	0.527708	+	0.849426i	$0.323051\pi$
$810$	0	0
$811$	−42698.1	−1.84875	−0.924374	−	0.381487i	$-0.875412\pi$
$811$	−42698.1	−1.84875	−0.924374	+	0.381487i	$0.875412\pi$
$812$	−17560.3	−0.758923
$813$	0	0
$814$	16017.4	0.689691
$815$	0	0
$816$	0	0
$817$	9893.67	0.423667
$818$	−24238.6	−1.03604
$819$	0	0
$820$	0	0
$821$	−9160.55	−0.389410	−0.194705	−	0.980862i	$-0.562375\pi$
$821$	−9160.55	−0.389410	−0.194705	+	0.980862i	$0.562375\pi$
$822$	0	0
$823$	−37097.7	−1.57126	−0.785629	−	0.618697i	$-0.787661\pi$
$823$	−37097.7	−1.57126	−0.785629	+	0.618697i	$0.787661\pi$
$824$	90.1585	0.00381167
$825$	0	0
$826$	24444.7	1.02971
$827$	−24878.9	−1.04610	−0.523050	−	0.852302i	$-0.675206\pi$
$827$	−24878.9	−1.04610	−0.523050	+	0.852302i	$0.675206\pi$
$828$	0	0
$829$	−8788.86	−0.368214	−0.184107	−	0.982906i	$-0.558939\pi$
$829$	−8788.86	−0.368214	−0.184107	+	0.982906i	$0.558939\pi$
$830$	0	0
$831$	0	0
$832$	33649.2	1.40214
$833$	−4397.95	−0.182929
$834$	0	0
$835$	0	0
$836$	13491.7	0.558157
$837$	0	0
$838$	28021.7	1.15512
$839$	6041.34	0.248594	0.124297	−	0.992245i	$-0.460332\pi$
$839$	6041.34	0.248594	0.124297	+	0.992245i	$0.460332\pi$
$840$	0	0
$841$	55146.5	2.26112
$842$	−43228.2	−1.76929
$843$	0	0
$844$	−17494.0	−0.713468
$845$	0	0
$846$	0	0
$847$	−2038.38	−0.0826912
$848$	−15157.7	−0.613818
$849$	0	0
$850$	0	0
$851$	−6085.63	−0.245138
$852$	0	0
$853$	−19541.8	−0.784405	−0.392203	−	0.919879i	$-0.628287\pi$
$853$	−19541.8	−0.784405	−0.392203	+	0.919879i	$0.628287\pi$
$854$	−16684.8	−0.668549
$855$	0	0
$856$	22.4739	0.000897361 0
$857$	34985.7	1.39450	0.697252	−	0.716826i	$-0.254406\pi$
$857$	34985.7	1.39450	0.697252	+	0.716826i	$0.254406\pi$
$858$	0	0
$859$	−3118.04	−0.123849	−0.0619244	−	0.998081i	$-0.519724\pi$
$859$	−3118.04	−0.123849	−0.0619244	+	0.998081i	$0.519724\pi$
$860$	0	0
$861$	0	0
$862$	−35485.5	−1.40214
$863$	−8308.05	−0.327705	−0.163852	−	0.986485i	$-0.552392\pi$
$863$	−8308.05	−0.327705	−0.163852	+	0.986485i	$0.552392\pi$
$864$	0	0
$865$	0	0
$866$	−28142.6	−1.10430
$867$	0	0
$868$	−9194.36	−0.359536
$869$	−19852.8	−0.774981
$870$	0	0
$871$	−7987.85	−0.310744
$872$	66.6702	0.00258915
$873$	0	0
$874$	−10262.0	−0.397160
$875$	0	0
$876$	0	0
$877$	31065.2	1.19612	0.598059	−	0.801452i	$-0.295939\pi$
$877$	31065.2	1.19612	0.598059	+	0.801452i	$0.295939\pi$
$878$	−20546.2	−0.789751
$879$	0	0
$880$	0	0
$881$	27176.4	1.03927	0.519634	−	0.854389i	$-0.326068\pi$
$881$	27176.4	1.03927	0.519634	+	0.854389i	$0.326068\pi$
$882$	0	0
$883$	17811.1	0.678813	0.339406	−	0.940640i	$-0.389774\pi$
$883$	17811.1	0.678813	0.339406	+	0.940640i	$0.389774\pi$
$884$	−8210.29	−0.312378
$885$	0	0
$886$	20287.9	0.769283
$887$	−9708.93	−0.367524	−0.183762	−	0.982971i	$-0.558828\pi$
$887$	−9708.93	−0.367524	−0.183762	+	0.982971i	$0.558828\pi$
$888$	0	0
$889$	21132.5	0.797256
$890$	0	0
$891$	0	0
$892$	9260.22	0.347595
$893$	−21411.4	−0.802359
$894$	0	0
$895$	0	0
$896$	61.9764	0.00231081
$897$	0	0
$898$	47109.2	1.75062
$899$	41643.8	1.54494
$900$	0	0
$901$	3684.08	0.136220
$902$	−14256.6	−0.526266
$903$	0	0
$904$	87.9562	0.00323604
$905$	0	0
$906$	0	0
$907$	−1793.35	−0.0656530	−0.0328265	−	0.999461i	$-0.510451\pi$
$907$	−1793.35	−0.0656530	−0.0328265	+	0.999461i	$0.510451\pi$
$908$	3259.47	0.119129
$909$	0	0
$910$	0	0
$911$	−40163.5	−1.46068	−0.730338	−	0.683086i	$-0.760638\pi$
$911$	−40163.5	−1.46068	−0.730338	+	0.683086i	$0.760638\pi$
$912$	0	0
$913$	−42270.0	−1.53224
$914$	44899.6	1.62489
$915$	0	0
$916$	−14836.1	−0.535152
$917$	−4844.30	−0.174453
$918$	0	0
$919$	−19871.3	−0.713268	−0.356634	−	0.934244i	$-0.616076\pi$
$919$	−19871.3	−0.713268	−0.356634	+	0.934244i	$0.616076\pi$
$920$	0	0
$921$	0	0
$922$	31935.5	1.14071
$923$	−38955.2	−1.38920
$924$	0	0
$925$	0	0
$926$	65629.3	2.32906
$927$	0	0
$928$	72161.9	2.55262
$929$	3690.27	0.130327	0.0651636	−	0.997875i	$-0.479243\pi$
$929$	3690.27	0.130327	0.0651636	+	0.997875i	$0.479243\pi$
$930$	0	0
$931$	−14579.4	−0.513233
$932$	−22461.5	−0.789432
$933$	0	0
$934$	−23998.7	−0.840752
$935$	0	0
$936$	0	0
$937$	−15191.5	−0.529653	−0.264826	−	0.964296i	$-0.585315\pi$
$937$	−15191.5	−0.529653	−0.264826	+	0.964296i	$0.585315\pi$
$938$	3774.75	0.131397
$939$	0	0
$940$	0	0
$941$	29396.9	1.01840	0.509198	−	0.860650i	$-0.329942\pi$
$941$	29396.9	1.01840	0.509198	+	0.860650i	$0.329942\pi$
$942$	0	0
$943$	5416.64	0.187052
$944$	−50274.8	−1.73337
$945$	0	0
$946$	25038.9	0.860554
$947$	1602.80	0.0549991	0.0274995	−	0.999622i	$-0.491246\pi$
$947$	1602.80	0.0549991	0.0274995	+	0.999622i	$0.491246\pi$
$948$	0	0
$949$	−65059.0	−2.22540
$950$	0	0
$951$	0	0
$952$	−7.54629	−0.000256908 0
$953$	−14868.1	−0.505377	−0.252689	−	0.967548i	$-0.581315\pi$
$953$	−14868.1	−0.505377	−0.252689	+	0.967548i	$0.581315\pi$
$954$	0	0
$955$	0	0
$956$	−45962.4	−1.55495
$957$	0	0
$958$	−10061.7	−0.339329
$959$	10360.5	0.348860
$960$	0	0
$961$	−7986.78	−0.268094
$962$	−32312.7	−1.08295
$963$	0	0
$964$	45831.2	1.53125
$965$	0	0
$966$	0	0
$967$	−44013.7	−1.46369	−0.731843	−	0.681474i	$-0.761339\pi$
$967$	−44013.7	−1.46369	−0.731843	+	0.681474i	$0.761339\pi$
$968$	16.2290	0.000538862 0
$969$	0	0
$970$	0	0
$971$	50954.6	1.68405	0.842024	−	0.539440i	$-0.181364\pi$
$971$	50954.6	1.68405	0.842024	+	0.539440i	$0.181364\pi$
$972$	0	0
$973$	19782.9	0.651810
$974$	15565.5	0.512064
$975$	0	0
$976$	34315.2	1.12541
$977$	−18626.3	−0.609935	−0.304968	−	0.952363i	$-0.598646\pi$
$977$	−18626.3	−0.609935	−0.304968	+	0.952363i	$0.598646\pi$
$978$	0	0
$979$	−49279.3	−1.60876
$980$	0	0
$981$	0	0
$982$	35628.7	1.15780
$983$	−11506.9	−0.373361	−0.186681	−	0.982421i	$-0.559773\pi$
$983$	−11506.9	−0.373361	−0.186681	+	0.982421i	$0.559773\pi$
$984$	0	0
$985$	0	0
$986$	−17573.0	−0.567584
$987$	0	0
$988$	−27217.5	−0.876420
$989$	−9513.27	−0.305869
$990$	0	0
$991$	31332.2	1.00434	0.502170	−	0.864769i	$-0.332535\pi$
$991$	31332.2	1.00434	0.502170	+	0.864769i	$0.332535\pi$
$992$	37783.1	1.20929
$993$	0	0
$994$	18408.8	0.587415
$995$	0	0
$996$	0	0
$997$	29027.6	0.922080	0.461040	−	0.887379i	$-0.347476\pi$
$997$	29027.6	0.922080	0.461040	+	0.887379i	$0.347476\pi$
$998$	34539.3	1.09551
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bd.1.7		8
3.2	odd	2		2025.4.a.bc.1.2		8
5.2	odd	4		405.4.b.c.244.7	yes	8
5.3	odd	4		405.4.b.c.244.2	yes	8
5.4	even	2	inner	2025.4.a.bd.1.2		8
15.2	even	4		405.4.b.b.244.2	✓	8
15.8	even	4		405.4.b.b.244.7	yes	8
15.14	odd	2		2025.4.a.bc.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.b.b.244.2	✓	8	15.2	even	4
405.4.b.b.244.7	yes	8	15.8	even	4
405.4.b.c.244.2	yes	8	5.3	odd	4
405.4.b.c.244.7	yes	8	5.2	odd	4
2025.4.a.bc.1.2		8	3.2	odd	2
2025.4.a.bc.1.7		8	15.14	odd	2
2025.4.a.bd.1.2		8	5.4	even	2	inner
2025.4.a.bd.1.7		8	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+3.99806 q^{2} +7.98447 q^{4} +7.79840 q^{7} -0.0620886 q^{8} +32.7050 q^{11} -65.9775 q^{13} +31.1784 q^{14} -64.1240 q^{16} +15.5853 q^{17} +51.6661 q^{19} +130.756 q^{22} -49.6796 q^{23} -263.782 q^{26} +62.2661 q^{28} -282.020 q^{29} -147.663 q^{31} -255.875 q^{32} +62.3111 q^{34} +122.498 q^{37} +206.564 q^{38} -109.032 q^{41} +191.493 q^{43} +261.132 q^{44} -198.622 q^{46} -414.420 q^{47} -282.185 q^{49} -526.796 q^{52} +236.381 q^{53} -0.484192 q^{56} -1127.53 q^{58} +784.024 q^{59} -535.138 q^{61} -590.363 q^{62} -510.010 q^{64} +121.069 q^{67} +124.441 q^{68} +590.432 q^{71} +986.078 q^{73} +489.753 q^{74} +412.526 q^{76} +255.046 q^{77} -607.025 q^{79} -435.915 q^{82} -1292.46 q^{83} +765.598 q^{86} -2.03061 q^{88} -1506.78 q^{89} -514.519 q^{91} -396.665 q^{92} -1656.87 q^{94} -610.887 q^{97} -1128.19 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{4} + 42 q^{14} + 4 q^{16} - 118 q^{19} - 36 q^{26} - 318 q^{29} - 416 q^{31} - 638 q^{34} + 486 q^{41} - 852 q^{44} - 598 q^{46} - 350 q^{49} + 1530 q^{56} + 1146 q^{59} - 398 q^{61} - 1640 q^{64}+ \cdots - 1238 q^{94}+O(q^{100}) \) 8 * q + 24 * q^4 + 42 * q^14 + 4 * q^16 - 118 * q^19 - 36 * q^26 - 318 * q^29 - 416 * q^31 - 638 * q^34 + 486 * q^41 - 852 * q^44 - 598 * q^46 - 350 * q^49 + 1530 * q^56 + 1146 * q^59 - 398 * q^61 - 1640 * q^64 - 1728 * q^71 - 1218 * q^74 - 3498 * q^76 - 2596 * q^79 + 480 * q^86 - 1086 * q^89 - 2574 * q^91 - 1238 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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