LMFDB - Embedded newform 2450.4.a.da.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, 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("2450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.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:	$144.554679514$
Analytic rank:	$0$
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} - 146x^{6} + 4997x^{4} - 4646x^{2} + 676 $ x^8 - 146x^6 + 4997x^4 - 4646*x^2 + 676
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2\cdot 7^{2}\cdot 11^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$9.62018$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +8.20597 q^{3} +4.00000 q^{4} +16.4119 q^{6} +8.00000 q^{8} +40.3380 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +8.20597 q^{3} +4.00000 q^{4} +16.4119 q^{6} +8.00000 q^{8} +40.3380 q^{9} +7.21713 q^{11} +32.8239 q^{12} -15.3045 q^{13} +16.0000 q^{16} +24.0041 q^{17} +80.6759 q^{18} -8.88027 q^{19} +14.4343 q^{22} +195.815 q^{23} +65.6478 q^{24} -30.6089 q^{26} +109.451 q^{27} +202.783 q^{29} +306.868 q^{31} +32.0000 q^{32} +59.2236 q^{33} +48.0082 q^{34} +161.352 q^{36} -147.473 q^{37} -17.7605 q^{38} -125.588 q^{39} -461.836 q^{41} -354.171 q^{43} +28.8685 q^{44} +391.629 q^{46} +174.401 q^{47} +131.296 q^{48} +196.977 q^{51} -61.2179 q^{52} -70.5623 q^{53} +218.902 q^{54} -72.8712 q^{57} +405.567 q^{58} +174.034 q^{59} -207.337 q^{61} +613.737 q^{62} +64.0000 q^{64} +118.447 q^{66} -467.319 q^{67} +96.0164 q^{68} +1606.85 q^{69} +1120.43 q^{71} +322.704 q^{72} +162.041 q^{73} -294.945 q^{74} -35.5211 q^{76} -251.176 q^{78} +496.605 q^{79} -190.974 q^{81} -923.673 q^{82} +595.261 q^{83} -708.342 q^{86} +1664.03 q^{87} +57.7370 q^{88} +684.810 q^{89} +783.258 q^{92} +2518.15 q^{93} +348.802 q^{94} +262.591 q^{96} +32.2295 q^{97} +291.124 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{2} + 32 q^{4} + 64 q^{8} + 76 q^{9}+O(q^{10}) $ 8 * q + 16 * q^2 + 32 * q^4 + 64 * q^8 + 76 * q^9	$ 8 q + 16 q^{2} + 32 q^{4} + 64 q^{8} + 76 q^{9} + 36 q^{11} + 128 q^{16} + 152 q^{18} + 72 q^{22} + 164 q^{23} + 392 q^{29} + 256 q^{32} + 304 q^{36} + 32 q^{37} + 832 q^{39} + 752 q^{43} + 144 q^{44} + 328 q^{46} + 2348 q^{51} + 700 q^{53} - 696 q^{57} + 784 q^{58} + 512 q^{64} + 1552 q^{67} + 2648 q^{71} + 608 q^{72} + 64 q^{74} + 1664 q^{78} - 1916 q^{79} + 2520 q^{81} + 1504 q^{86} + 288 q^{88} + 656 q^{92} + 536 q^{93} + 2892 q^{99}+O(q^{100}) $ 8 * q + 16 * q^2 + 32 * q^4 + 64 * q^8 + 76 * q^9 + 36 * q^11 + 128 * q^16 + 152 * q^18 + 72 * q^22 + 164 * q^23 + 392 * q^29 + 256 * q^32 + 304 * q^36 + 32 * q^37 + 832 * q^39 + 752 * q^43 + 144 * q^44 + 328 * q^46 + 2348 * q^51 + 700 * q^53 - 696 * q^57 + 784 * q^58 + 512 * q^64 + 1552 * q^67 + 2648 * q^71 + 608 * q^72 + 64 * q^74 + 1664 * q^78 - 1916 * q^79 + 2520 * q^81 + 1504 * q^86 + 288 * q^88 + 656 * q^92 + 536 * q^93 + 2892 * q^99

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	8.20597	1.57924	0.789620	−	0.613596i	$-0.210278\pi$
$3$	8.20597	1.57924	0.789620	+	0.613596i	$0.210278\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	16.4119	1.11669
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	40.3380	1.49400
$10$	0	0
$11$	7.21713	0.197822	0.0989112	−	0.995096i	$-0.468464\pi$
$11$	7.21713	0.197822	0.0989112	+	0.995096i	$0.468464\pi$
$12$	32.8239	0.789620
$13$	−15.3045	−0.326515	−0.163258	−	0.986583i	$-0.552200\pi$
$13$	−15.3045	−0.326515	−0.163258	+	0.986583i	$0.552200\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	24.0041	0.342462	0.171231	−	0.985231i	$-0.445226\pi$
$17$	24.0041	0.342462	0.171231	+	0.985231i	$0.445226\pi$
$18$	80.6759	1.05642
$19$	−8.88027	−0.107225	−0.0536125	−	0.998562i	$-0.517074\pi$
$19$	−8.88027	−0.107225	−0.0536125	+	0.998562i	$0.517074\pi$
$20$	0	0
$21$	0	0
$22$	14.4343	0.139882
$23$	195.815	1.77522	0.887612	−	0.460592i	$-0.152363\pi$
$23$	195.815	1.77522	0.887612	+	0.460592i	$0.152363\pi$
$24$	65.6478	0.558346
$25$	0	0
$26$	−30.6089	−0.230881
$27$	109.451	0.780142
$28$	0	0
$29$	202.783	1.29848	0.649240	−	0.760584i	$-0.275087\pi$
$29$	202.783	1.29848	0.649240	+	0.760584i	$0.275087\pi$
$30$	0	0
$31$	306.868	1.77791	0.888955	−	0.457995i	$-0.151432\pi$
$31$	306.868	1.77791	0.888955	+	0.457995i	$0.151432\pi$
$32$	32.0000	0.176777
$33$	59.2236	0.312409
$34$	48.0082	0.242157
$35$	0	0
$36$	161.352	0.746999
$37$	−147.473	−0.655253	−0.327626	−	0.944807i	$-0.606249\pi$
$37$	−147.473	−0.655253	−0.327626	+	0.944807i	$0.606249\pi$
$38$	−17.7605	−0.0758195
$39$	−125.588	−0.515646
$40$	0	0
$41$	−461.836	−1.75919	−0.879594	−	0.475725i	$-0.842186\pi$
$41$	−461.836	−1.75919	−0.879594	+	0.475725i	$0.842186\pi$
$42$	0	0
$43$	−354.171	−1.25606	−0.628030	−	0.778189i	$-0.716138\pi$
$43$	−354.171	−1.25606	−0.628030	+	0.778189i	$0.716138\pi$
$44$	28.8685	0.0989112
$45$	0	0
$46$	391.629	1.25527
$47$	174.401	0.541256	0.270628	−	0.962684i	$-0.412769\pi$
$47$	174.401	0.541256	0.270628	+	0.962684i	$0.412769\pi$
$48$	131.296	0.394810
$49$	0	0
$50$	0	0
$51$	196.977	0.540829
$52$	−61.2179	−0.163258
$53$	−70.5623	−0.182877	−0.0914384	−	0.995811i	$-0.529146\pi$
$53$	−70.5623	−0.182877	−0.0914384	+	0.995811i	$0.529146\pi$
$54$	218.902	0.551644
$55$	0	0
$56$	0	0
$57$	−72.8712	−0.169334
$58$	405.567	0.918164
$59$	174.034	0.384021	0.192011	−	0.981393i	$-0.438499\pi$
$59$	174.034	0.384021	0.192011	+	0.981393i	$0.438499\pi$
$60$	0	0
$61$	−207.337	−0.435193	−0.217596	−	0.976039i	$-0.569822\pi$
$61$	−207.337	−0.435193	−0.217596	+	0.976039i	$0.569822\pi$
$62$	613.737	1.25717
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	118.447	0.220907
$67$	−467.319	−0.852122	−0.426061	−	0.904695i	$-0.640099\pi$
$67$	−467.319	−0.852122	−0.426061	+	0.904695i	$0.640099\pi$
$68$	96.0164	0.171231
$69$	1606.85	2.80350
$70$	0	0
$71$	1120.43	1.87282	0.936412	−	0.350902i	$-0.114125\pi$
$71$	1120.43	1.87282	0.936412	+	0.350902i	$0.114125\pi$
$72$	322.704	0.528208
$73$	162.041	0.259800	0.129900	−	0.991527i	$-0.458534\pi$
$73$	162.041	0.259800	0.129900	+	0.991527i	$0.458534\pi$
$74$	−294.945	−0.463334
$75$	0	0
$76$	−35.5211	−0.0536125
$77$	0	0
$78$	−251.176	−0.364617
$79$	496.605	0.707247	0.353623	−	0.935388i	$-0.384949\pi$
$79$	496.605	0.707247	0.353623	+	0.935388i	$0.384949\pi$
$80$	0	0
$81$	−190.974	−0.261967
$82$	−923.673	−1.24393
$83$	595.261	0.787209	0.393604	−	0.919280i	$-0.371228\pi$
$83$	595.261	0.787209	0.393604	+	0.919280i	$0.371228\pi$
$84$	0	0
$85$	0	0
$86$	−708.342	−0.888168
$87$	1664.03	2.05061
$88$	57.7370	0.0699408
$89$	684.810	0.815614	0.407807	−	0.913068i	$-0.366294\pi$
$89$	684.810	0.815614	0.407807	+	0.913068i	$0.366294\pi$
$90$	0	0
$91$	0	0
$92$	783.258	0.887612
$93$	2518.15	2.80774
$94$	348.802	0.382726
$95$	0	0
$96$	262.591	0.279173
$97$	32.2295	0.0337362	0.0168681	−	0.999858i	$-0.494630\pi$
$97$	32.2295	0.0337362	0.0168681	+	0.999858i	$0.494630\pi$
$98$	0	0
$99$	291.124	0.295546
$100$	0	0
$101$	−1481.21	−1.45926	−0.729632	−	0.683840i	$-0.760308\pi$
$101$	−1481.21	−1.45926	−0.729632	+	0.683840i	$0.760308\pi$
$102$	393.954	0.382424
$103$	1536.92	1.47027	0.735133	−	0.677923i	$-0.237120\pi$
$103$	1536.92	1.47027	0.735133	+	0.677923i	$0.237120\pi$
$104$	−122.436	−0.115441
$105$	0	0
$106$	−141.125	−0.129313
$107$	1987.01	1.79525	0.897623	−	0.440765i	$-0.145293\pi$
$107$	1987.01	1.79525	0.897623	+	0.440765i	$0.145293\pi$
$108$	437.803	0.390071
$109$	−201.610	−0.177162	−0.0885812	−	0.996069i	$-0.528233\pi$
$109$	−201.610	−0.177162	−0.0885812	+	0.996069i	$0.528233\pi$
$110$	0	0
$111$	−1210.16	−1.03480
$112$	0	0
$113$	−1050.60	−0.874621	−0.437311	−	0.899311i	$-0.644069\pi$
$113$	−1050.60	−0.874621	−0.437311	+	0.899311i	$0.644069\pi$
$114$	−145.742	−0.119737
$115$	0	0
$116$	811.133	0.649240
$117$	−617.351	−0.487813
$118$	348.068	0.271544
$119$	0	0
$120$	0	0
$121$	−1278.91	−0.960866
$122$	−414.674	−0.307728
$123$	−3789.82	−2.77818
$124$	1227.47	0.888955
$125$	0	0
$126$	0	0
$127$	−1422.42	−0.993850	−0.496925	−	0.867794i	$-0.665538\pi$
$127$	−1422.42	−0.993850	−0.496925	+	0.867794i	$0.665538\pi$
$128$	128.000	0.0883883
$129$	−2906.32	−1.98362
$130$	0	0
$131$	−1929.92	−1.28716	−0.643579	−	0.765380i	$-0.722551\pi$
$131$	−1929.92	−1.28716	−0.643579	+	0.765380i	$0.722551\pi$
$132$	236.894	0.156205
$133$	0	0
$134$	−934.639	−0.602541
$135$	0	0
$136$	192.033	0.121078
$137$	2679.25	1.67083	0.835414	−	0.549621i	$-0.185228\pi$
$137$	2679.25	1.67083	0.835414	+	0.549621i	$0.185228\pi$
$138$	3213.70	1.98238
$139$	2569.25	1.56778	0.783888	−	0.620902i	$-0.213233\pi$
$139$	2569.25	1.56778	0.783888	+	0.620902i	$0.213233\pi$
$140$	0	0
$141$	1431.13	0.854772
$142$	2240.86	1.32429
$143$	−110.454	−0.0645920
$144$	645.407	0.373500
$145$	0	0
$146$	324.081	0.183707
$147$	0	0
$148$	−589.891	−0.327626
$149$	1336.72	0.734956	0.367478	−	0.930032i	$-0.380221\pi$
$149$	1336.72	0.734956	0.367478	+	0.930032i	$0.380221\pi$
$150$	0	0
$151$	879.445	0.473962	0.236981	−	0.971514i	$-0.423842\pi$
$151$	879.445	0.473962	0.236981	+	0.971514i	$0.423842\pi$
$152$	−71.0422	−0.0379097
$153$	968.276	0.511637
$154$	0	0
$155$	0	0
$156$	−502.352	−0.257823
$157$	1733.98	0.881444	0.440722	−	0.897644i	$-0.354723\pi$
$157$	1733.98	0.881444	0.440722	+	0.897644i	$0.354723\pi$
$158$	993.211	0.500099
$159$	−579.032	−0.288806
$160$	0	0
$161$	0	0
$162$	−381.949	−0.185239
$163$	530.732	0.255032	0.127516	−	0.991837i	$-0.459300\pi$
$163$	530.732	0.255032	0.127516	+	0.991837i	$0.459300\pi$
$164$	−1847.35	−0.879594
$165$	0	0
$166$	1190.52	0.556641
$167$	2044.19	0.947212	0.473606	−	0.880737i	$-0.342952\pi$
$167$	2044.19	0.947212	0.473606	+	0.880737i	$0.342952\pi$
$168$	0	0
$169$	−1962.77	−0.893388
$170$	0	0
$171$	−358.212	−0.160194
$172$	−1416.68	−0.628030
$173$	−1414.18	−0.621494	−0.310747	−	0.950493i	$-0.600579\pi$
$173$	−1414.18	−0.621494	−0.310747	+	0.950493i	$0.600579\pi$
$174$	3328.07	1.45000
$175$	0	0
$176$	115.474	0.0494556
$177$	1428.12	0.606462
$178$	1369.62	0.576726
$179$	−176.305	−0.0736183	−0.0368092	−	0.999322i	$-0.511719\pi$
$179$	−176.305	−0.0736183	−0.0368092	+	0.999322i	$0.511719\pi$
$180$	0	0
$181$	−1494.26	−0.613631	−0.306816	−	0.951769i	$-0.599264\pi$
$181$	−1494.26	−0.613631	−0.306816	+	0.951769i	$0.599264\pi$
$182$	0	0
$183$	−1701.40	−0.687274
$184$	1566.52	0.627637
$185$	0	0
$186$	5036.31	1.98538
$187$	173.241	0.0677466
$188$	697.604	0.270628
$189$	0	0
$190$	0	0
$191$	3434.54	1.30112	0.650561	−	0.759454i	$-0.274534\pi$
$191$	3434.54	1.30112	0.650561	+	0.759454i	$0.274534\pi$
$192$	525.182	0.197405
$193$	−2086.79	−0.778294	−0.389147	−	0.921176i	$-0.627230\pi$
$193$	−2086.79	−0.778294	−0.389147	+	0.921176i	$0.627230\pi$
$194$	64.4591	0.0238551
$195$	0	0
$196$	0	0
$197$	270.202	0.0977212	0.0488606	−	0.998806i	$-0.484441\pi$
$197$	270.202	0.0977212	0.0488606	+	0.998806i	$0.484441\pi$
$198$	582.249	0.208983
$199$	−3875.51	−1.38054	−0.690271	−	0.723551i	$-0.742509\pi$
$199$	−3875.51	−1.38054	−0.690271	+	0.723551i	$0.742509\pi$
$200$	0	0
$201$	−3834.81	−1.34570
$202$	−2962.41	−1.03185
$203$	0	0
$204$	787.908	0.270415
$205$	0	0
$206$	3073.84	1.03964
$207$	7898.76	2.65218
$208$	−244.872	−0.0816288
$209$	−64.0901	−0.0212115
$210$	0	0
$211$	2915.91	0.951373	0.475687	−	0.879615i	$-0.342200\pi$
$211$	2915.91	0.951373	0.475687	+	0.879615i	$0.342200\pi$
$212$	−282.249	−0.0914384
$213$	9194.21	2.95764
$214$	3974.01	1.26943
$215$	0	0
$216$	875.607	0.275822
$217$	0	0
$218$	−403.219	−0.125273
$219$	1329.70	0.410287
$220$	0	0
$221$	−367.370	−0.111819
$222$	−2420.31	−0.731715
$223$	−2286.00	−0.686467	−0.343233	−	0.939250i	$-0.611522\pi$
$223$	−2286.00	−0.686467	−0.343233	+	0.939250i	$0.611522\pi$
$224$	0	0
$225$	0	0
$226$	−2101.20	−0.618450
$227$	−3646.70	−1.06625	−0.533127	−	0.846035i	$-0.678983\pi$
$227$	−3646.70	−1.06625	−0.533127	+	0.846035i	$0.678983\pi$
$228$	−291.485	−0.0846669
$229$	−5512.14	−1.59062	−0.795310	−	0.606202i	$-0.792692\pi$
$229$	−5512.14	−1.59062	−0.795310	+	0.606202i	$0.792692\pi$
$230$	0	0
$231$	0	0
$232$	1622.27	0.459082
$233$	3680.32	1.03479	0.517395	−	0.855747i	$-0.326902\pi$
$233$	3680.32	1.03479	0.517395	+	0.855747i	$0.326902\pi$
$234$	−1234.70	−0.344936
$235$	0	0
$236$	696.135	0.192011
$237$	4075.13	1.11691
$238$	0	0
$239$	327.373	0.0886026	0.0443013	−	0.999018i	$-0.485894\pi$
$239$	327.373	0.0886026	0.0443013	+	0.999018i	$0.485894\pi$
$240$	0	0
$241$	−1261.34	−0.337136	−0.168568	−	0.985690i	$-0.553914\pi$
$241$	−1261.34	−0.337136	−0.168568	+	0.985690i	$0.553914\pi$
$242$	−2557.83	−0.679435
$243$	−4522.30	−1.19385
$244$	−829.347	−0.217596
$245$	0	0
$246$	−7579.63	−1.96447
$247$	135.908	0.0350106
$248$	2454.95	0.628586
$249$	4884.69	1.24319
$250$	0	0
$251$	561.483	0.141197	0.0705986	−	0.997505i	$-0.477509\pi$
$251$	561.483	0.141197	0.0705986	+	0.997505i	$0.477509\pi$
$252$	0	0
$253$	1413.22	0.351179
$254$	−2844.83	−0.702758
$255$	0	0
$256$	256.000	0.0625000
$257$	−1728.68	−0.419581	−0.209791	−	0.977746i	$-0.567278\pi$
$257$	−1728.68	−0.419581	−0.209791	+	0.977746i	$0.567278\pi$
$258$	−5812.63	−1.40263
$259$	0	0
$260$	0	0
$261$	8179.87	1.93993
$262$	−3859.83	−0.910158
$263$	−170.860	−0.0400595	−0.0200298	−	0.999799i	$-0.506376\pi$
$263$	−170.860	−0.0400595	−0.0200298	+	0.999799i	$0.506376\pi$
$264$	473.789	0.110453
$265$	0	0
$266$	0	0
$267$	5619.53	1.28805
$268$	−1869.28	−0.426061
$269$	479.851	0.108762	0.0543811	−	0.998520i	$-0.482681\pi$
$269$	479.851	0.108762	0.0543811	+	0.998520i	$0.482681\pi$
$270$	0	0
$271$	−8398.27	−1.88250	−0.941252	−	0.337705i	$-0.890349\pi$
$271$	−8398.27	−1.88250	−0.941252	+	0.337705i	$0.890349\pi$
$272$	384.066	0.0856154
$273$	0	0
$274$	5358.49	1.18145
$275$	0	0
$276$	6427.39	1.40175
$277$	5085.80	1.10316	0.551581	−	0.834121i	$-0.314025\pi$
$277$	5085.80	1.10316	0.551581	+	0.834121i	$0.314025\pi$
$278$	5138.50	1.10859
$279$	12378.4	2.65619
$280$	0	0
$281$	2606.78	0.553408	0.276704	−	0.960955i	$-0.410758\pi$
$281$	2606.78	0.553408	0.276704	+	0.960955i	$0.410758\pi$
$282$	2862.26	0.604415
$283$	−7456.45	−1.56622	−0.783110	−	0.621884i	$-0.786368\pi$
$283$	−7456.45	−1.56622	−0.783110	+	0.621884i	$0.786368\pi$
$284$	4481.72	0.936412
$285$	0	0
$286$	−220.909	−0.0456735
$287$	0	0
$288$	1290.81	0.264104
$289$	−4336.80	−0.882720
$290$	0	0
$291$	264.475	0.0532776
$292$	648.162	0.129900
$293$	−6916.73	−1.37911	−0.689556	−	0.724232i	$-0.742194\pi$
$293$	−6916.73	−1.37911	−0.689556	+	0.724232i	$0.742194\pi$
$294$	0	0
$295$	0	0
$296$	−1179.78	−0.231667
$297$	789.921	0.154330
$298$	2673.44	0.519693
$299$	−2996.84	−0.579638
$300$	0	0
$301$	0	0
$302$	1758.89	0.335142
$303$	−12154.7	−2.30453
$304$	−142.084	−0.0268062
$305$	0	0
$306$	1936.55	0.361782
$307$	4607.54	0.856568	0.428284	−	0.903644i	$-0.359118\pi$
$307$	4607.54	0.856568	0.428284	+	0.903644i	$0.359118\pi$
$308$	0	0
$309$	12611.9	2.32190
$310$	0	0
$311$	304.624	0.0555423	0.0277711	−	0.999614i	$-0.491159\pi$
$311$	304.624	0.0555423	0.0277711	+	0.999614i	$0.491159\pi$
$312$	−1004.70	−0.182308
$313$	−8910.57	−1.60912	−0.804561	−	0.593870i	$-0.797599\pi$
$313$	−8910.57	−1.60912	−0.804561	+	0.593870i	$0.797599\pi$
$314$	3467.96	0.623275
$315$	0	0
$316$	1986.42	0.353623
$317$	−707.221	−0.125304	−0.0626522	−	0.998035i	$-0.519956\pi$
$317$	−707.221	−0.125304	−0.0626522	+	0.998035i	$0.519956\pi$
$318$	−1158.06	−0.204217
$319$	1463.51	0.256869
$320$	0	0
$321$	16305.3	2.83512
$322$	0	0
$323$	−213.163	−0.0367204
$324$	−763.897	−0.130984
$325$	0	0
$326$	1061.46	0.180335
$327$	−1654.40	−0.279782
$328$	−3694.69	−0.621967
$329$	0	0
$330$	0	0
$331$	6667.11	1.10712	0.553561	−	0.832808i	$-0.313268\pi$
$331$	6667.11	1.10712	0.553561	+	0.832808i	$0.313268\pi$
$332$	2381.04	0.393604
$333$	−5948.74	−0.978946
$334$	4088.39	0.669780
$335$	0	0
$336$	0	0
$337$	5093.71	0.823358	0.411679	−	0.911329i	$-0.364942\pi$
$337$	5093.71	0.823358	0.411679	+	0.911329i	$0.364942\pi$
$338$	−3925.55	−0.631721
$339$	−8621.20	−1.38124
$340$	0	0
$341$	2214.71	0.351710
$342$	−716.424	−0.113274
$343$	0	0
$344$	−2833.37	−0.444084
$345$	0	0
$346$	−2828.37	−0.439463
$347$	−1257.47	−0.194537	−0.0972684	−	0.995258i	$-0.531011\pi$
$347$	−1257.47	−0.194537	−0.0972684	+	0.995258i	$0.531011\pi$
$348$	6656.14	1.02531
$349$	−2982.73	−0.457484	−0.228742	−	0.973487i	$-0.573461\pi$
$349$	−2982.73	−0.457484	−0.228742	+	0.973487i	$0.573461\pi$
$350$	0	0
$351$	−1675.09	−0.254728
$352$	230.948	0.0349704
$353$	−2020.10	−0.304586	−0.152293	−	0.988335i	$-0.548666\pi$
$353$	−2020.10	−0.304586	−0.152293	+	0.988335i	$0.548666\pi$
$354$	2856.23	0.428833
$355$	0	0
$356$	2739.24	0.407807
$357$	0	0
$358$	−352.611	−0.0520560
$359$	−10420.5	−1.53196	−0.765979	−	0.642865i	$-0.777746\pi$
$359$	−10420.5	−1.53196	−0.765979	+	0.642865i	$0.777746\pi$
$360$	0	0
$361$	−6780.14	−0.988503
$362$	−2988.52	−0.433903
$363$	−10494.7	−1.51744
$364$	0	0
$365$	0	0
$366$	−3402.80	−0.485976
$367$	3832.92	0.545169	0.272584	−	0.962132i	$-0.412122\pi$
$367$	3832.92	0.545169	0.272584	+	0.962132i	$0.412122\pi$
$368$	3133.03	0.443806
$369$	−18629.5	−2.62822
$370$	0	0
$371$	0	0
$372$	10072.6	1.40387
$373$	−8585.59	−1.19181	−0.595904	−	0.803055i	$-0.703206\pi$
$373$	−8585.59	−1.19181	−0.595904	+	0.803055i	$0.703206\pi$
$374$	346.481	0.0479041
$375$	0	0
$376$	1395.21	0.191363
$377$	−3103.49	−0.423973
$378$	0	0
$379$	4956.19	0.671721	0.335860	−	0.941912i	$-0.390973\pi$
$379$	4956.19	0.671721	0.335860	+	0.941912i	$0.390973\pi$
$380$	0	0
$381$	−11672.3	−1.56953
$382$	6869.08	0.920033
$383$	−7935.21	−1.05867	−0.529335	−	0.848413i	$-0.677558\pi$
$383$	−7935.21	−1.05867	−0.529335	+	0.848413i	$0.677558\pi$
$384$	1050.36	0.139586
$385$	0	0
$386$	−4173.59	−0.550337
$387$	−14286.5	−1.87655
$388$	128.918	0.0168681
$389$	12035.1	1.56865	0.784327	−	0.620348i	$-0.213009\pi$
$389$	12035.1	1.56865	0.784327	+	0.620348i	$0.213009\pi$
$390$	0	0
$391$	4700.35	0.607946
$392$	0	0
$393$	−15836.8	−2.03273
$394$	540.404	0.0690993
$395$	0	0
$396$	1164.50	0.147773
$397$	−12452.9	−1.57429	−0.787146	−	0.616766i	$-0.788442\pi$
$397$	−12452.9	−1.57429	−0.787146	+	0.616766i	$0.788442\pi$
$398$	−7751.03	−0.976190
$399$	0	0
$400$	0	0
$401$	−9718.15	−1.21023	−0.605114	−	0.796139i	$-0.706872\pi$
$401$	−9718.15	−1.21023	−0.605114	+	0.796139i	$0.706872\pi$
$402$	−7669.62	−0.951557
$403$	−4696.46	−0.580514
$404$	−5924.83	−0.729632
$405$	0	0
$406$	0	0
$407$	−1064.33	−0.129624
$408$	1575.82	0.191212
$409$	7630.43	0.922495	0.461247	−	0.887272i	$-0.347402\pi$
$409$	7630.43	0.922495	0.461247	+	0.887272i	$0.347402\pi$
$410$	0	0
$411$	21985.8	2.63864
$412$	6147.69	0.735133
$413$	0	0
$414$	15797.5	1.87538
$415$	0	0
$416$	−489.743	−0.0577203
$417$	21083.2	2.47590
$418$	−128.180	−0.0149988
$419$	15228.2	1.77552	0.887762	−	0.460303i	$-0.152259\pi$
$419$	15228.2	1.77552	0.887762	+	0.460303i	$0.152259\pi$
$420$	0	0
$421$	6072.84	0.703022	0.351511	−	0.936184i	$-0.385668\pi$
$421$	6072.84	0.703022	0.351511	+	0.936184i	$0.385668\pi$
$422$	5831.83	0.672723
$423$	7034.98	0.808635
$424$	−564.498	−0.0646567
$425$	0	0
$426$	18388.4	2.09137
$427$	0	0
$428$	7948.03	0.897623
$429$	−906.385	−0.102006
$430$	0	0
$431$	−9321.23	−1.04174	−0.520868	−	0.853637i	$-0.674391\pi$
$431$	−9321.23	−1.04174	−0.520868	+	0.853637i	$0.674391\pi$
$432$	1751.21	0.195035
$433$	−12769.3	−1.41722	−0.708609	−	0.705601i	$-0.750677\pi$
$433$	−12769.3	−1.41722	−0.708609	+	0.705601i	$0.750677\pi$
$434$	0	0
$435$	0	0
$436$	−806.438	−0.0885812
$437$	−1738.89	−0.190348
$438$	2659.40	0.290117
$439$	1663.57	0.180861	0.0904304	−	0.995903i	$-0.471176\pi$
$439$	1663.57	0.180861	0.0904304	+	0.995903i	$0.471176\pi$
$440$	0	0
$441$	0	0
$442$	−734.740	−0.0790679
$443$	−17979.1	−1.92824	−0.964120	−	0.265466i	$-0.914474\pi$
$443$	−17979.1	−1.92824	−0.964120	+	0.265466i	$0.914474\pi$
$444$	−4840.62	−0.517401
$445$	0	0
$446$	−4572.01	−0.485405
$447$	10969.1	1.16067
$448$	0	0
$449$	8885.50	0.933926	0.466963	−	0.884277i	$-0.345348\pi$
$449$	8885.50	0.933926	0.466963	+	0.884277i	$0.345348\pi$
$450$	0	0
$451$	−3333.13	−0.348007
$452$	−4202.40	−0.437311
$453$	7216.70	0.748500
$454$	−7293.39	−0.753955
$455$	0	0
$456$	−582.970	−0.0598686
$457$	−3429.79	−0.351070	−0.175535	−	0.984473i	$-0.556166\pi$
$457$	−3429.79	−0.351070	−0.175535	+	0.984473i	$0.556166\pi$
$458$	−11024.3	−1.12474
$459$	2627.27	0.267169
$460$	0	0
$461$	−12061.7	−1.21859	−0.609294	−	0.792944i	$-0.708547\pi$
$461$	−12061.7	−1.21859	−0.609294	+	0.792944i	$0.708547\pi$
$462$	0	0
$463$	−2894.46	−0.290533	−0.145267	−	0.989393i	$-0.546404\pi$
$463$	−2894.46	−0.290533	−0.145267	+	0.989393i	$0.546404\pi$
$464$	3244.53	0.324620
$465$	0	0
$466$	7360.65	0.731707
$467$	13794.1	1.36684	0.683422	−	0.730023i	$-0.260491\pi$
$467$	13794.1	1.36684	0.683422	+	0.730023i	$0.260491\pi$
$468$	−2469.40	−0.243907
$469$	0	0
$470$	0	0
$471$	14229.0	1.39201
$472$	1392.27	0.135772
$473$	−2556.10	−0.248477
$474$	8150.26	0.789776
$475$	0	0
$476$	0	0
$477$	−2846.34	−0.273218
$478$	654.747	0.0626515
$479$	−8622.93	−0.822530	−0.411265	−	0.911516i	$-0.634913\pi$
$479$	−8622.93	−0.822530	−0.411265	+	0.911516i	$0.634913\pi$
$480$	0	0
$481$	2256.99	0.213950
$482$	−2522.67	−0.238391
$483$	0	0
$484$	−5115.65	−0.480433
$485$	0	0
$486$	−9044.60	−0.844180
$487$	−18785.7	−1.74797	−0.873985	−	0.485953i	$-0.838473\pi$
$487$	−18785.7	−1.74797	−0.873985	+	0.485953i	$0.838473\pi$
$488$	−1658.69	−0.153864
$489$	4355.18	0.402756
$490$	0	0
$491$	3284.74	0.301911	0.150955	−	0.988541i	$-0.451765\pi$
$491$	3284.74	0.301911	0.150955	+	0.988541i	$0.451765\pi$
$492$	−15159.3	−1.38909
$493$	4867.63	0.444680
$494$	271.816	0.0247562
$495$	0	0
$496$	4909.89	0.444477
$497$	0	0
$498$	9769.38	0.879069
$499$	12384.6	1.11104	0.555522	−	0.831502i	$-0.312518\pi$
$499$	12384.6	1.11104	0.555522	+	0.831502i	$0.312518\pi$
$500$	0	0
$501$	16774.6	1.49587
$502$	1122.97	0.0998415
$503$	−6438.84	−0.570763	−0.285381	−	0.958414i	$-0.592120\pi$
$503$	−6438.84	−0.570763	−0.285381	+	0.958414i	$0.592120\pi$
$504$	0	0
$505$	0	0
$506$	2826.44	0.248321
$507$	−16106.5	−1.41087
$508$	−5689.66	−0.496925
$509$	−9066.26	−0.789498	−0.394749	−	0.918789i	$-0.629169\pi$
$509$	−9066.26	−0.789498	−0.394749	+	0.918789i	$0.629169\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	−971.953	−0.0836506
$514$	−3457.37	−0.296689
$515$	0	0
$516$	−11625.3	−0.991809
$517$	1258.68	0.107073
$518$	0	0
$519$	−11604.8	−0.981488
$520$	0	0
$521$	−63.8301	−0.00536746	−0.00268373	−	0.999996i	$-0.500854\pi$
$521$	−63.8301	−0.00536746	−0.00268373	+	0.999996i	$0.500854\pi$
$522$	16359.7	1.37174
$523$	7595.57	0.635050	0.317525	−	0.948250i	$-0.397148\pi$
$523$	7595.57	0.635050	0.317525	+	0.948250i	$0.397148\pi$
$524$	−7719.67	−0.643579
$525$	0	0
$526$	−341.719	−0.0283264
$527$	7366.10	0.608866
$528$	947.577	0.0781023
$529$	26176.3	2.15142
$530$	0	0
$531$	7020.17	0.573727
$532$	0	0
$533$	7068.16	0.574402
$534$	11239.1	0.910789
$535$	0	0
$536$	−3738.56	−0.301270
$537$	−1446.76	−0.116261
$538$	959.702	0.0769065
$539$	0	0
$540$	0	0
$541$	−1108.12	−0.0880623	−0.0440312	−	0.999030i	$-0.514020\pi$
$541$	−1108.12	−0.0880623	−0.0440312	+	0.999030i	$0.514020\pi$
$542$	−16796.5	−1.33113
$543$	−12261.8	−0.969071
$544$	768.131	0.0605392
$545$	0	0
$546$	0	0
$547$	−15567.7	−1.21686	−0.608432	−	0.793606i	$-0.708201\pi$
$547$	−15567.7	−1.21686	−0.608432	+	0.793606i	$0.708201\pi$
$548$	10717.0	0.835414
$549$	−8363.54	−0.650177
$550$	0	0
$551$	−1800.77	−0.139229
$552$	12854.8	0.991189
$553$	0	0
$554$	10171.6	0.780054
$555$	0	0
$556$	10277.0	0.783888
$557$	−21223.6	−1.61449	−0.807247	−	0.590214i	$-0.799043\pi$
$557$	−21223.6	−1.61449	−0.807247	+	0.590214i	$0.799043\pi$
$558$	24756.9	1.87821
$559$	5420.40	0.410122
$560$	0	0
$561$	1421.61	0.106988
$562$	5213.56	0.391318
$563$	9595.03	0.718263	0.359132	−	0.933287i	$-0.383073\pi$
$563$	9595.03	0.718263	0.359132	+	0.933287i	$0.383073\pi$
$564$	5724.52	0.427386
$565$	0	0
$566$	−14912.9	−1.10748
$567$	0	0
$568$	8963.44	0.662143
$569$	−3101.05	−0.228476	−0.114238	−	0.993453i	$-0.536443\pi$
$569$	−3101.05	−0.228476	−0.114238	+	0.993453i	$0.536443\pi$
$570$	0	0
$571$	13147.6	0.963592	0.481796	−	0.876283i	$-0.339985\pi$
$571$	13147.6	0.963592	0.481796	+	0.876283i	$0.339985\pi$
$572$	−441.817	−0.0322960
$573$	28183.7	2.05479
$574$	0	0
$575$	0	0
$576$	2581.63	0.186750
$577$	−19908.8	−1.43642	−0.718209	−	0.695827i	$-0.755038\pi$
$577$	−19908.8	−1.43642	−0.718209	+	0.695827i	$0.755038\pi$
$578$	−8673.61	−0.624177
$579$	−17124.2	−1.22911
$580$	0	0
$581$	0	0
$582$	528.949	0.0376729
$583$	−509.257	−0.0361771
$584$	1296.32	0.0918533
$585$	0	0
$586$	−13833.5	−0.975180
$587$	12997.2	0.913887	0.456944	−	0.889496i	$-0.348944\pi$
$587$	12997.2	0.913887	0.456944	+	0.889496i	$0.348944\pi$
$588$	0	0
$589$	−2725.07	−0.190636
$590$	0	0
$591$	2217.27	0.154325
$592$	−2359.56	−0.163813
$593$	−13343.4	−0.924024	−0.462012	−	0.886874i	$-0.652872\pi$
$593$	−13343.4	−0.924024	−0.462012	+	0.886874i	$0.652872\pi$
$594$	1579.84	0.109127
$595$	0	0
$596$	5346.89	0.367478
$597$	−31802.3	−2.18021
$598$	−5993.68	−0.409866
$599$	14900.5	1.01639	0.508195	−	0.861242i	$-0.330313\pi$
$599$	14900.5	1.01639	0.508195	+	0.861242i	$0.330313\pi$
$600$	0	0
$601$	−22423.8	−1.52194	−0.760969	−	0.648789i	$-0.775276\pi$
$601$	−22423.8	−1.52194	−0.760969	+	0.648789i	$0.775276\pi$
$602$	0	0
$603$	−18850.7	−1.27307
$604$	3517.78	0.236981
$605$	0	0
$606$	−24309.5	−1.62955
$607$	−19671.7	−1.31540	−0.657700	−	0.753280i	$-0.728471\pi$
$607$	−19671.7	−1.31540	−0.657700	+	0.753280i	$0.728471\pi$
$608$	−284.169	−0.0189549
$609$	0	0
$610$	0	0
$611$	−2669.12	−0.176728
$612$	3873.10	0.255819
$613$	728.557	0.0480035	0.0240018	−	0.999712i	$-0.492359\pi$
$613$	728.557	0.0480035	0.0240018	+	0.999712i	$0.492359\pi$
$614$	9215.08	0.605685
$615$	0	0
$616$	0	0
$617$	11258.1	0.734577	0.367288	−	0.930107i	$-0.380286\pi$
$617$	11258.1	0.734577	0.367288	+	0.930107i	$0.380286\pi$
$618$	25223.9	1.64183
$619$	27589.0	1.79143	0.895716	−	0.444627i	$-0.146664\pi$
$619$	27589.0	1.79143	0.895716	+	0.444627i	$0.146664\pi$
$620$	0	0
$621$	21432.1	1.38493
$622$	609.248	0.0392743
$623$	0	0
$624$	−2009.41	−0.128911
$625$	0	0
$626$	−17821.1	−1.13782
$627$	−525.921	−0.0334980
$628$	6935.92	0.440722
$629$	−3539.95	−0.224399
$630$	0	0
$631$	6934.14	0.437470	0.218735	−	0.975784i	$-0.429807\pi$
$631$	6934.14	0.437470	0.218735	+	0.975784i	$0.429807\pi$
$632$	3972.84	0.250049
$633$	23927.9	1.50245
$634$	−1414.44	−0.0886036
$635$	0	0
$636$	−2316.13	−0.144403
$637$	0	0
$638$	2927.03	0.181633
$639$	45195.8	2.79800
$640$	0	0
$641$	15357.6	0.946313	0.473157	−	0.880978i	$-0.343114\pi$
$641$	15357.6	0.946313	0.473157	+	0.880978i	$0.343114\pi$
$642$	32610.6	2.00473
$643$	11256.3	0.690364	0.345182	−	0.938536i	$-0.387817\pi$
$643$	11256.3	0.690364	0.345182	+	0.938536i	$0.387817\pi$
$644$	0	0
$645$	0	0
$646$	−426.326	−0.0259653
$647$	−8422.14	−0.511760	−0.255880	−	0.966709i	$-0.582365\pi$
$647$	−8422.14	−0.511760	−0.255880	+	0.966709i	$0.582365\pi$
$648$	−1527.79	−0.0926195
$649$	1256.02	0.0759681
$650$	0	0
$651$	0	0
$652$	2122.93	0.127516
$653$	27972.1	1.67631	0.838155	−	0.545431i	$-0.183634\pi$
$653$	27972.1	1.67631	0.838155	+	0.545431i	$0.183634\pi$
$654$	−3308.80	−0.197836
$655$	0	0
$656$	−7389.38	−0.439797
$657$	6536.39	0.388141
$658$	0	0
$659$	−24042.4	−1.42118	−0.710591	−	0.703605i	$-0.751573\pi$
$659$	−24042.4	−1.42118	−0.710591	+	0.703605i	$0.751573\pi$
$660$	0	0
$661$	−17890.4	−1.05273	−0.526366	−	0.850258i	$-0.676446\pi$
$661$	−17890.4	−1.05273	−0.526366	+	0.850258i	$0.676446\pi$
$662$	13334.2	0.782854
$663$	−3014.63	−0.176589
$664$	4762.08	0.278320
$665$	0	0
$666$	−11897.5	−0.692220
$667$	39707.9	2.30509
$668$	8176.77	0.473606
$669$	−18758.9	−1.08410
$670$	0	0
$671$	−1496.38	−0.0860909
$672$	0	0
$673$	−22758.7	−1.30354	−0.651771	−	0.758415i	$-0.725974\pi$
$673$	−22758.7	−1.30354	−0.651771	+	0.758415i	$0.725974\pi$
$674$	10187.4	0.582202
$675$	0	0
$676$	−7851.09	−0.446694
$677$	−20740.9	−1.17745	−0.588727	−	0.808332i	$-0.700371\pi$
$677$	−20740.9	−1.17745	−0.588727	+	0.808332i	$0.700371\pi$
$678$	−17242.4	−0.976682
$679$	0	0
$680$	0	0
$681$	−29924.7	−1.68387
$682$	4429.42	0.248697
$683$	8482.21	0.475202	0.237601	−	0.971363i	$-0.423639\pi$
$683$	8482.21	0.475202	0.237601	+	0.971363i	$0.423639\pi$
$684$	−1432.85	−0.0800969
$685$	0	0
$686$	0	0
$687$	−45232.4	−2.51197
$688$	−5666.73	−0.314015
$689$	1079.92	0.0597121
$690$	0	0
$691$	15227.3	0.838314	0.419157	−	0.907914i	$-0.362326\pi$
$691$	15227.3	0.838314	0.419157	+	0.907914i	$0.362326\pi$
$692$	−5656.74	−0.310747
$693$	0	0
$694$	−2514.93	−0.137558
$695$	0	0
$696$	13312.3	0.725001
$697$	−11086.0	−0.602455
$698$	−5965.46	−0.323490
$699$	30200.6	1.63418
$700$	0	0
$701$	19408.1	1.04570	0.522848	−	0.852426i	$-0.324870\pi$
$701$	19408.1	1.04570	0.522848	+	0.852426i	$0.324870\pi$
$702$	−3350.17	−0.180120
$703$	1309.60	0.0702594
$704$	461.896	0.0247278
$705$	0	0
$706$	−4040.20	−0.215375
$707$	0	0
$708$	5712.46	0.303231
$709$	−30059.8	−1.59227	−0.796134	−	0.605120i	$-0.793125\pi$
$709$	−30059.8	−1.59227	−0.796134	+	0.605120i	$0.793125\pi$
$710$	0	0
$711$	20032.0	1.05663
$712$	5478.48	0.288363
$713$	60089.3	3.15619
$714$	0	0
$715$	0	0
$716$	−705.221	−0.0368092
$717$	2686.42	0.139925
$718$	−20841.0	−1.08326
$719$	27345.5	1.41838	0.709190	−	0.705018i	$-0.249061\pi$
$719$	27345.5	1.41838	0.709190	+	0.705018i	$0.249061\pi$
$720$	0	0
$721$	0	0
$722$	−13560.3	−0.698977
$723$	−10350.5	−0.532418
$724$	−5977.03	−0.306816
$725$	0	0
$726$	−20989.4	−1.07299
$727$	8169.65	0.416775	0.208388	−	0.978046i	$-0.433178\pi$
$727$	8169.65	0.416775	0.208388	+	0.978046i	$0.433178\pi$
$728$	0	0
$729$	−31953.6	−1.62341
$730$	0	0
$731$	−8501.55	−0.430152
$732$	−6805.60	−0.343637
$733$	28125.3	1.41723	0.708615	−	0.705595i	$-0.249320\pi$
$733$	28125.3	1.41723	0.708615	+	0.705595i	$0.249320\pi$
$734$	7665.85	0.385493
$735$	0	0
$736$	6266.07	0.313818
$737$	−3372.71	−0.168569
$738$	−37259.1	−1.85844
$739$	−11923.8	−0.593539	−0.296769	−	0.954949i	$-0.595909\pi$
$739$	−11923.8	−0.593539	−0.296769	+	0.954949i	$0.595909\pi$
$740$	0	0
$741$	1115.26	0.0552901
$742$	0	0
$743$	−13965.9	−0.689582	−0.344791	−	0.938679i	$-0.612050\pi$
$743$	−13965.9	−0.689582	−0.344791	+	0.938679i	$0.612050\pi$
$744$	20145.2	0.992688
$745$	0	0
$746$	−17171.2	−0.842736
$747$	24011.6	1.17609
$748$	692.963	0.0338733
$749$	0	0
$750$	0	0
$751$	−28536.8	−1.38658	−0.693290	−	0.720659i	$-0.743839\pi$
$751$	−28536.8	−1.38658	−0.693290	+	0.720659i	$0.743839\pi$
$752$	2790.42	0.135314
$753$	4607.51	0.222984
$754$	−6206.98	−0.299794
$755$	0	0
$756$	0	0
$757$	29163.0	1.40019	0.700096	−	0.714048i	$-0.253140\pi$
$757$	29163.0	1.40019	0.700096	+	0.714048i	$0.253140\pi$
$758$	9912.37	0.474978
$759$	11596.8	0.554596
$760$	0	0
$761$	−9551.79	−0.454996	−0.227498	−	0.973779i	$-0.573055\pi$
$761$	−9551.79	−0.454996	−0.227498	+	0.973779i	$0.573055\pi$
$762$	−23344.6	−1.10982
$763$	0	0
$764$	13738.2	0.650561
$765$	0	0
$766$	−15870.4	−0.748592
$767$	−2663.49	−0.125389
$768$	2100.73	0.0987025
$769$	−15133.4	−0.709653	−0.354826	−	0.934932i	$-0.615460\pi$
$769$	−15133.4	−0.709653	−0.354826	+	0.934932i	$0.615460\pi$
$770$	0	0
$771$	−14185.5	−0.662620
$772$	−8347.18	−0.389147
$773$	−28538.7	−1.32790	−0.663950	−	0.747777i	$-0.731121\pi$
$773$	−28538.7	−1.32790	−0.663950	+	0.747777i	$0.731121\pi$
$774$	−28573.1	−1.32692
$775$	0	0
$776$	257.836	0.0119276
$777$	0	0
$778$	24070.3	1.10921
$779$	4101.23	0.188629
$780$	0	0
$781$	8086.29	0.370487
$782$	9400.70	0.429883
$783$	22194.8	1.01300
$784$	0	0
$785$	0	0
$786$	−31673.7	−1.43736
$787$	33253.1	1.50615	0.753077	−	0.657932i	$-0.228569\pi$
$787$	33253.1	1.50615	0.753077	+	0.657932i	$0.228569\pi$
$788$	1080.81	0.0488606
$789$	−1402.07	−0.0632636
$790$	0	0
$791$	0	0
$792$	2328.99	0.104491
$793$	3173.18	0.142097
$794$	−24905.8	−1.11319
$795$	0	0
$796$	−15502.1	−0.690271
$797$	24568.2	1.09191	0.545954	−	0.837815i	$-0.316167\pi$
$797$	24568.2	1.09191	0.545954	+	0.837815i	$0.316167\pi$
$798$	0	0
$799$	4186.34	0.185359
$800$	0	0
$801$	27623.8	1.21853
$802$	−19436.3	−0.855760
$803$	1169.47	0.0513943
$804$	−15339.2	−0.672852
$805$	0	0
$806$	−9392.92	−0.410486
$807$	3937.64	0.171762
$808$	−11849.7	−0.515927
$809$	−30406.5	−1.32143	−0.660713	−	0.750638i	$-0.729746\pi$
$809$	−30406.5	−1.32143	−0.660713	+	0.750638i	$0.729746\pi$
$810$	0	0
$811$	−12997.8	−0.562780	−0.281390	−	0.959593i	$-0.590795\pi$
$811$	−12997.8	−0.562780	−0.281390	+	0.959593i	$0.590795\pi$
$812$	0	0
$813$	−68915.9	−2.97292
$814$	−2128.66	−0.0916578
$815$	0	0
$816$	3151.63	0.135207
$817$	3145.13	0.134681
$818$	15260.9	0.652302
$819$	0	0
$820$	0	0
$821$	−19989.6	−0.849748	−0.424874	−	0.905253i	$-0.639682\pi$
$821$	−19989.6	−0.849748	−0.424874	+	0.905253i	$0.639682\pi$
$822$	43971.6	1.86580
$823$	−2162.33	−0.0915845	−0.0457923	−	0.998951i	$-0.514581\pi$
$823$	−2162.33	−0.0915845	−0.0457923	+	0.998951i	$0.514581\pi$
$824$	12295.4	0.519818
$825$	0	0
$826$	0	0
$827$	46187.9	1.94209	0.971046	−	0.238891i	$-0.0767839\pi$
$827$	46187.9	1.94209	0.971046	+	0.238891i	$0.0767839\pi$
$828$	31595.0	1.32609
$829$	15225.9	0.637898	0.318949	−	0.947772i	$-0.396670\pi$
$829$	15225.9	0.637898	0.318949	+	0.947772i	$0.396670\pi$
$830$	0	0
$831$	41733.9	1.74216
$832$	−979.486	−0.0408144
$833$	0	0
$834$	42166.4	1.75072
$835$	0	0
$836$	−256.360	−0.0106057
$837$	33587.0	1.38702
$838$	30456.3	1.25548
$839$	25038.7	1.03031	0.515155	−	0.857097i	$-0.327734\pi$
$839$	25038.7	1.03031	0.515155	+	0.857097i	$0.327734\pi$
$840$	0	0
$841$	16732.1	0.686051
$842$	12145.7	0.497111
$843$	21391.2	0.873963
$844$	11663.7	0.475687
$845$	0	0
$846$	14070.0	0.571791
$847$	0	0
$848$	−1129.00	−0.0457192
$849$	−61187.4	−2.47344
$850$	0	0
$851$	−28877.3	−1.16322
$852$	36776.8	1.47882
$853$	582.907	0.0233978	0.0116989	−	0.999932i	$-0.496276\pi$
$853$	582.907	0.0233978	0.0116989	+	0.999932i	$0.496276\pi$
$854$	0	0
$855$	0	0
$856$	15896.1	0.634715
$857$	−23113.6	−0.921290	−0.460645	−	0.887585i	$-0.652382\pi$
$857$	−23113.6	−0.921290	−0.460645	+	0.887585i	$0.652382\pi$
$858$	−1812.77	−0.0721293
$859$	28459.7	1.13042	0.565212	−	0.824946i	$-0.308794\pi$
$859$	28459.7	1.13042	0.565212	+	0.824946i	$0.308794\pi$
$860$	0	0
$861$	0	0
$862$	−18642.5	−0.736618
$863$	−14986.6	−0.591134	−0.295567	−	0.955322i	$-0.595509\pi$
$863$	−14986.6	−0.591134	−0.295567	+	0.955322i	$0.595509\pi$
$864$	3502.43	0.137911
$865$	0	0
$866$	−25538.7	−1.00212
$867$	−35587.7	−1.39403
$868$	0	0
$869$	3584.07	0.139909
$870$	0	0
$871$	7152.08	0.278231
$872$	−1612.88	−0.0626363
$873$	1300.07	0.0504018
$874$	−3477.77	−0.134597
$875$	0	0
$876$	5318.80	0.205143
$877$	19343.3	0.744785	0.372393	−	0.928075i	$-0.378537\pi$
$877$	19343.3	0.744785	0.372393	+	0.928075i	$0.378537\pi$
$878$	3327.14	0.127888
$879$	−56758.5	−2.17795
$880$	0	0
$881$	9417.92	0.360156	0.180078	−	0.983652i	$-0.442365\pi$
$881$	9417.92	0.360156	0.180078	+	0.983652i	$0.442365\pi$
$882$	0	0
$883$	17989.3	0.685604	0.342802	−	0.939408i	$-0.388624\pi$
$883$	17989.3	0.685604	0.342802	+	0.939408i	$0.388624\pi$
$884$	−1469.48	−0.0559095
$885$	0	0
$886$	−35958.1	−1.36347
$887$	−30327.1	−1.14801	−0.574005	−	0.818852i	$-0.694611\pi$
$887$	−30327.1	−1.14801	−0.574005	+	0.818852i	$0.694611\pi$
$888$	−9681.25	−0.365857
$889$	0	0
$890$	0	0
$891$	−1378.29	−0.0518230
$892$	−9144.01	−0.343233
$893$	−1548.73	−0.0580361
$894$	21938.2	0.820719
$895$	0	0
$896$	0	0
$897$	−24592.0	−0.915387
$898$	17771.0	0.660386
$899$	62227.8	2.30858
$900$	0	0
$901$	−1693.78	−0.0626283
$902$	−6666.27	−0.246078
$903$	0	0
$904$	−8404.80	−0.309225
$905$	0	0
$906$	14433.4	0.529269
$907$	−49783.2	−1.82252	−0.911258	−	0.411835i	$-0.864888\pi$
$907$	−49783.2	−1.82252	−0.911258	+	0.411835i	$0.864888\pi$
$908$	−14586.8	−0.533127
$909$	−59748.8	−2.18014
$910$	0	0
$911$	−38108.7	−1.38595	−0.692973	−	0.720964i	$-0.743699\pi$
$911$	−38108.7	−1.38595	−0.692973	+	0.720964i	$0.743699\pi$
$912$	−1165.94	−0.0423335
$913$	4296.07	0.155728
$914$	−6859.59	−0.248244
$915$	0	0
$916$	−22048.5	−0.795310
$917$	0	0
$918$	5254.54	0.188917
$919$	49555.9	1.77878	0.889390	−	0.457149i	$-0.151129\pi$
$919$	49555.9	1.77878	0.889390	+	0.457149i	$0.151129\pi$
$920$	0	0
$921$	37809.3	1.35273
$922$	−24123.4	−0.861672
$923$	−17147.6	−0.611505
$924$	0	0
$925$	0	0
$926$	−5788.92	−0.205438
$927$	61996.3	2.19658
$928$	6489.07	0.229541
$929$	−1110.46	−0.0392175	−0.0196088	−	0.999808i	$-0.506242\pi$
$929$	−1110.46	−0.0392175	−0.0196088	+	0.999808i	$0.506242\pi$
$930$	0	0
$931$	0	0
$932$	14721.3	0.517395
$933$	2499.74	0.0877146
$934$	27588.3	0.966505
$935$	0	0
$936$	−4938.81	−0.172468
$937$	−27081.7	−0.944204	−0.472102	−	0.881544i	$-0.656505\pi$
$937$	−27081.7	−0.944204	−0.472102	+	0.881544i	$0.656505\pi$
$938$	0	0
$939$	−73119.9	−2.54119
$940$	0	0
$941$	6340.68	0.219660	0.109830	−	0.993950i	$-0.464969\pi$
$941$	6340.68	0.219660	0.109830	+	0.993950i	$0.464969\pi$
$942$	28458.0	0.984300
$943$	−90434.3	−3.12295
$944$	2784.54	0.0960054
$945$	0	0
$946$	−5112.19	−0.175700
$947$	28522.5	0.978729	0.489365	−	0.872079i	$-0.337229\pi$
$947$	28522.5	0.978729	0.489365	+	0.872079i	$0.337229\pi$
$948$	16300.5	0.558456
$949$	−2479.95	−0.0848287
$950$	0	0
$951$	−5803.43	−0.197886
$952$	0	0
$953$	19390.5	0.659098	0.329549	−	0.944139i	$-0.393103\pi$
$953$	19390.5	0.659098	0.329549	+	0.944139i	$0.393103\pi$
$954$	−5692.67	−0.193194
$955$	0	0
$956$	1309.49	0.0443013
$957$	12009.6	0.405657
$958$	−17245.9	−0.581616
$959$	0	0
$960$	0	0
$961$	64377.2	2.16096
$962$	4513.98	0.151285
$963$	80151.8	2.68209
$964$	−5045.34	−0.168568
$965$	0	0
$966$	0	0
$967$	−23280.1	−0.774187	−0.387093	−	0.922040i	$-0.626521\pi$
$967$	−23280.1	−0.774187	−0.387093	+	0.922040i	$0.626521\pi$
$968$	−10231.3	−0.339718
$969$	−1749.21	−0.0579904
$970$	0	0
$971$	1975.22	0.0652809	0.0326405	−	0.999467i	$-0.489608\pi$
$971$	1975.22	0.0652809	0.0326405	+	0.999467i	$0.489608\pi$
$972$	−18089.2	−0.596926
$973$	0	0
$974$	−37571.4	−1.23600
$975$	0	0
$976$	−3317.39	−0.108798
$977$	−42917.1	−1.40536	−0.702681	−	0.711505i	$-0.748014\pi$
$977$	−42917.1	−1.40536	−0.702681	+	0.711505i	$0.748014\pi$
$978$	8710.35	0.284792
$979$	4942.36	0.161347
$980$	0	0
$981$	−8132.52	−0.264680
$982$	6569.47	0.213483
$983$	983.085	0.0318978	0.0159489	−	0.999873i	$-0.494923\pi$
$983$	983.085	0.0318978	0.0159489	+	0.999873i	$0.494923\pi$
$984$	−30318.5	−0.982235
$985$	0	0
$986$	9735.26	0.314436
$987$	0	0
$988$	543.631	0.0175053
$989$	−69351.8	−2.22979
$990$	0	0
$991$	−8138.73	−0.260883	−0.130442	−	0.991456i	$-0.541639\pi$
$991$	−8138.73	−0.260883	−0.130442	+	0.991456i	$0.541639\pi$
$992$	9819.79	0.314293
$993$	54710.1	1.74841
$994$	0	0
$995$	0	0
$996$	19538.8	0.621596
$997$	4663.64	0.148143	0.0740717	−	0.997253i	$-0.476401\pi$
$997$	4663.64	0.148143	0.0740717	+	0.997253i	$0.476401\pi$
$998$	24769.2	0.785627
$999$	−16141.0	−0.511190

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.da.1.7	yes	8
5.4	even	2		2450.4.a.cz.1.2	✓	8
7.6	odd	2	inner	2450.4.a.da.1.2	yes	8
35.34	odd	2		2450.4.a.cz.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2450.4.a.cz.1.2	✓	8	5.4	even	2
2450.4.a.cz.1.7	yes	8	35.34	odd	2
2450.4.a.da.1.2	yes	8	7.6	odd	2	inner
2450.4.a.da.1.7	yes	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +8.20597 q^{3} +4.00000 q^{4} +16.4119 q^{6} +8.00000 q^{8} +40.3380 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +8.20597 q^{3} +4.00000 q^{4} +16.4119 q^{6} +8.00000 q^{8} +40.3380 q^{9} +7.21713 q^{11} +32.8239 q^{12} -15.3045 q^{13} +16.0000 q^{16} +24.0041 q^{17} +80.6759 q^{18} -8.88027 q^{19} +14.4343 q^{22} +195.815 q^{23} +65.6478 q^{24} -30.6089 q^{26} +109.451 q^{27} +202.783 q^{29} +306.868 q^{31} +32.0000 q^{32} +59.2236 q^{33} +48.0082 q^{34} +161.352 q^{36} -147.473 q^{37} -17.7605 q^{38} -125.588 q^{39} -461.836 q^{41} -354.171 q^{43} +28.8685 q^{44} +391.629 q^{46} +174.401 q^{47} +131.296 q^{48} +196.977 q^{51} -61.2179 q^{52} -70.5623 q^{53} +218.902 q^{54} -72.8712 q^{57} +405.567 q^{58} +174.034 q^{59} -207.337 q^{61} +613.737 q^{62} +64.0000 q^{64} +118.447 q^{66} -467.319 q^{67} +96.0164 q^{68} +1606.85 q^{69} +1120.43 q^{71} +322.704 q^{72} +162.041 q^{73} -294.945 q^{74} -35.5211 q^{76} -251.176 q^{78} +496.605 q^{79} -190.974 q^{81} -923.673 q^{82} +595.261 q^{83} -708.342 q^{86} +1664.03 q^{87} +57.7370 q^{88} +684.810 q^{89} +783.258 q^{92} +2518.15 q^{93} +348.802 q^{94} +262.591 q^{96} +32.2295 q^{97} +291.124 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{2} + 32 q^{4} + 64 q^{8} + 76 q^{9}+O(q^{10}) \) 8 * q + 16 * q^2 + 32 * q^4 + 64 * q^8 + 76 * q^9	\( 8 q + 16 q^{2} + 32 q^{4} + 64 q^{8} + 76 q^{9} + 36 q^{11} + 128 q^{16} + 152 q^{18} + 72 q^{22} + 164 q^{23} + 392 q^{29} + 256 q^{32} + 304 q^{36} + 32 q^{37} + 832 q^{39} + 752 q^{43} + 144 q^{44} + 328 q^{46} + 2348 q^{51} + 700 q^{53} - 696 q^{57} + 784 q^{58} + 512 q^{64} + 1552 q^{67} + 2648 q^{71} + 608 q^{72} + 64 q^{74} + 1664 q^{78} - 1916 q^{79} + 2520 q^{81} + 1504 q^{86} + 288 q^{88} + 656 q^{92} + 536 q^{93} + 2892 q^{99}+O(q^{100}) \) 8 * q + 16 * q^2 + 32 * q^4 + 64 * q^8 + 76 * q^9 + 36 * q^11 + 128 * q^16 + 152 * q^18 + 72 * q^22 + 164 * q^23 + 392 * q^29 + 256 * q^32 + 304 * q^36 + 32 * q^37 + 832 * q^39 + 752 * q^43 + 144 * q^44 + 328 * q^46 + 2348 * q^51 + 700 * q^53 - 696 * q^57 + 784 * q^58 + 512 * q^64 + 1552 * q^67 + 2648 * q^71 + 608 * q^72 + 64 * q^74 + 1664 * q^78 - 1916 * q^79 + 2520 * q^81 + 1504 * q^86 + 288 * q^88 + 656 * q^92 + 536 * q^93 + 2892 * q^99

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