LMFDB - Embedded newform 2450.4.a.da.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2450,4,Mod(1,2450)] 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("2450.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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:traces = [8,16,0,32,0,0,0,64,76,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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			1.8
Root			$7.24569$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} +8.65990 q^{3} +4.00000 q^{4} +17.3198 q^{6} +8.00000 q^{8} +47.9939 q^{9} +22.1723 q^{11} +34.6396 q^{12} +66.1544 q^{13} +16.0000 q^{16} +132.325 q^{17} +95.9878 q^{18} -56.5761 q^{19} +44.3445 q^{22} +21.9085 q^{23} +69.2792 q^{24} +132.309 q^{26} +181.805 q^{27} -93.9344 q^{29} -299.832 q^{31} +32.0000 q^{32} +192.010 q^{33} +264.650 q^{34} +191.976 q^{36} -170.797 q^{37} -113.152 q^{38} +572.890 q^{39} +137.875 q^{41} +275.761 q^{43} +88.6891 q^{44} +43.8170 q^{46} +100.717 q^{47} +138.558 q^{48} +1145.92 q^{51} +264.617 q^{52} +36.2240 q^{53} +363.610 q^{54} -489.944 q^{57} -187.869 q^{58} +717.346 q^{59} -722.798 q^{61} -599.663 q^{62} +64.0000 q^{64} +384.019 q^{66} +123.976 q^{67} +529.300 q^{68} +189.726 q^{69} -907.172 q^{71} +383.951 q^{72} -794.370 q^{73} -341.594 q^{74} -226.304 q^{76} +1145.78 q^{78} -1163.27 q^{79} +278.579 q^{81} +275.750 q^{82} +1130.09 q^{83} +551.521 q^{86} -813.463 q^{87} +177.378 q^{88} +897.342 q^{89} +87.6340 q^{92} -2596.51 q^{93} +201.435 q^{94} +277.117 q^{96} +1517.68 q^{97} +1064.13 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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
$3$	8.65990	1.66660	0.833299	−	0.552822i	$-0.186449\pi$
$3$	8.65990	1.66660	0.833299	+	0.552822i	$0.186449\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	17.3198	1.17846
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	47.9939	1.77755
$10$	0	0
$11$	22.1723	0.607745	0.303872	−	0.952713i	$-0.401720\pi$
$11$	22.1723	0.607745	0.303872	+	0.952713i	$0.401720\pi$
$12$	34.6396	0.833299
$13$	66.1544	1.41138	0.705689	−	0.708522i	$-0.250638\pi$
$13$	66.1544	1.41138	0.705689	+	0.708522i	$0.250638\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	132.325	1.88785	0.943927	−	0.330155i	$-0.107101\pi$
$17$	132.325	1.88785	0.943927	+	0.330155i	$0.107101\pi$
$18$	95.9878	1.25692
$19$	−56.5761	−0.683129	−0.341565	−	0.939858i	$-0.610957\pi$
$19$	−56.5761	−0.683129	−0.341565	+	0.939858i	$0.610957\pi$
$20$	0	0
$21$	0	0
$22$	44.3445	0.429740
$23$	21.9085	0.198619	0.0993096	−	0.995057i	$-0.468337\pi$
$23$	21.9085	0.198619	0.0993096	+	0.995057i	$0.468337\pi$
$24$	69.2792	0.589232
$25$	0	0
$26$	132.309	0.997995
$27$	181.805	1.29587
$28$	0	0
$29$	−93.9344	−0.601489	−0.300744	−	0.953705i	$-0.597235\pi$
$29$	−93.9344	−0.601489	−0.300744	+	0.953705i	$0.597235\pi$
$30$	0	0
$31$	−299.832	−1.73714	−0.868570	−	0.495566i	$-0.834961\pi$
$31$	−299.832	−1.73714	−0.868570	+	0.495566i	$0.834961\pi$
$32$	32.0000	0.176777
$33$	192.010	1.01287
$34$	264.650	1.33491
$35$	0	0
$36$	191.976	0.888776
$37$	−170.797	−0.758889	−0.379445	−	0.925214i	$-0.623885\pi$
$37$	−170.797	−0.758889	−0.379445	+	0.925214i	$0.623885\pi$
$38$	−113.152	−0.483045
$39$	572.890	2.35220
$40$	0	0
$41$	137.875	0.525182	0.262591	−	0.964907i	$-0.415423\pi$
$41$	137.875	0.525182	0.262591	+	0.964907i	$0.415423\pi$
$42$	0	0
$43$	275.761	0.977979	0.488990	−	0.872290i	$-0.337366\pi$
$43$	275.761	0.977979	0.488990	+	0.872290i	$0.337366\pi$
$44$	88.6891	0.303872
$45$	0	0
$46$	43.8170	0.140445
$47$	100.717	0.312577	0.156289	−	0.987711i	$-0.450047\pi$
$47$	100.717	0.312577	0.156289	+	0.987711i	$0.450047\pi$
$48$	138.558	0.416650
$49$	0	0
$50$	0	0
$51$	1145.92	3.14629
$52$	264.617	0.705689
$53$	36.2240	0.0938822	0.0469411	−	0.998898i	$-0.485053\pi$
$53$	36.2240	0.0938822	0.0469411	+	0.998898i	$0.485053\pi$
$54$	363.610	0.916316
$55$	0	0
$56$	0	0
$57$	−489.944	−1.13850
$58$	−187.869	−0.425317
$59$	717.346	1.58289	0.791445	−	0.611241i	$-0.209329\pi$
$59$	717.346	1.58289	0.791445	+	0.611241i	$0.209329\pi$
$60$	0	0
$61$	−722.798	−1.51713	−0.758564	−	0.651599i	$-0.774099\pi$
$61$	−722.798	−1.51713	−0.758564	+	0.651599i	$0.774099\pi$
$62$	−599.663	−1.22834
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	384.019	0.716205
$67$	123.976	0.226060	0.113030	−	0.993592i	$-0.463944\pi$
$67$	123.976	0.226060	0.113030	+	0.993592i	$0.463944\pi$
$68$	529.300	0.943927
$69$	189.726	0.331018
$70$	0	0
$71$	−907.172	−1.51636	−0.758180	−	0.652046i	$-0.773911\pi$
$71$	−907.172	−1.51636	−0.758180	+	0.652046i	$0.773911\pi$
$72$	383.951	0.628460
$73$	−794.370	−1.27362	−0.636808	−	0.771023i	$-0.719745\pi$
$73$	−794.370	−1.27362	−0.636808	+	0.771023i	$0.719745\pi$
$74$	−341.594	−0.536616
$75$	0	0
$76$	−226.304	−0.341565
$77$	0	0
$78$	1145.78	1.66326
$79$	−1163.27	−1.65669	−0.828346	−	0.560217i	$-0.810718\pi$
$79$	−1163.27	−1.65669	−0.828346	+	0.560217i	$0.810718\pi$
$80$	0	0
$81$	278.579	0.382139
$82$	275.750	0.371360
$83$	1130.09	1.49450	0.747249	−	0.664544i	$-0.231374\pi$
$83$	1130.09	1.49450	0.747249	+	0.664544i	$0.231374\pi$
$84$	0	0
$85$	0	0
$86$	551.521	0.691536
$87$	−813.463	−1.00244
$88$	177.378	0.214870
$89$	897.342	1.06874	0.534371	−	0.845250i	$-0.320548\pi$
$89$	897.342	1.06874	0.534371	+	0.845250i	$0.320548\pi$
$90$	0	0
$91$	0	0
$92$	87.6340	0.0993096
$93$	−2596.51	−2.89512
$94$	201.435	0.221026
$95$	0	0
$96$	277.117	0.294616
$97$	1517.68	1.58863	0.794315	−	0.607506i	$-0.207830\pi$
$97$	1517.68	1.58863	0.794315	+	0.607506i	$0.207830\pi$
$98$	0	0
$99$	1064.13	1.08030
$100$	0	0
$101$	45.5877	0.0449123	0.0224562	−	0.999748i	$-0.492851\pi$
$101$	45.5877	0.0449123	0.0224562	+	0.999748i	$0.492851\pi$
$102$	2291.84	2.22477
$103$	−1063.01	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$103$	−1063.01	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$104$	529.235	0.498998
$105$	0	0
$106$	72.4481	0.0663847
$107$	−264.881	−0.239318	−0.119659	−	0.992815i	$-0.538180\pi$
$107$	−264.881	−0.239318	−0.119659	+	0.992815i	$0.538180\pi$
$108$	727.221	0.647934
$109$	885.061	0.777738	0.388869	−	0.921293i	$-0.372866\pi$
$109$	885.061	0.777738	0.388869	+	0.921293i	$0.372866\pi$
$110$	0	0
$111$	−1479.09	−1.26476
$112$	0	0
$113$	1605.55	1.33661	0.668305	−	0.743887i	$-0.267020\pi$
$113$	1605.55	1.33661	0.668305	+	0.743887i	$0.267020\pi$
$114$	−979.887	−0.805043
$115$	0	0
$116$	−375.738	−0.300744
$117$	3175.01	2.50880
$118$	1434.69	1.11927
$119$	0	0
$120$	0	0
$121$	−839.390	−0.630646
$122$	−1445.60	−1.07277
$123$	1193.98	0.875268
$124$	−1199.33	−0.868570
$125$	0	0
$126$	0	0
$127$	2468.88	1.72502	0.862511	−	0.506038i	$-0.168890\pi$
$127$	2468.88	1.72502	0.862511	+	0.506038i	$0.168890\pi$
$128$	128.000	0.0883883
$129$	2388.06	1.62990
$130$	0	0
$131$	−2763.11	−1.84285	−0.921427	−	0.388551i	$-0.872976\pi$
$131$	−2763.11	−1.84285	−0.921427	+	0.388551i	$0.872976\pi$
$132$	768.039	0.506433
$133$	0	0
$134$	247.951	0.159849
$135$	0	0
$136$	1058.60	0.667457
$137$	337.270	0.210328	0.105164	−	0.994455i	$-0.466463\pi$
$137$	337.270	0.210328	0.105164	+	0.994455i	$0.466463\pi$
$138$	379.451	0.234065
$139$	−1644.19	−1.00329	−0.501647	−	0.865072i	$-0.667272\pi$
$139$	−1644.19	−1.00329	−0.501647	+	0.865072i	$0.667272\pi$
$140$	0	0
$141$	872.203	0.520941
$142$	−1814.34	−1.07223
$143$	1466.79	0.857758
$144$	767.902	0.444388
$145$	0	0
$146$	−1588.74	−0.900582
$147$	0	0
$148$	−683.189	−0.379445
$149$	−1894.41	−1.04158	−0.520792	−	0.853683i	$-0.674363\pi$
$149$	−1894.41	−1.04158	−0.520792	+	0.853683i	$0.674363\pi$
$150$	0	0
$151$	859.608	0.463271	0.231635	−	0.972803i	$-0.425592\pi$
$151$	859.608	0.463271	0.231635	+	0.972803i	$0.425592\pi$
$152$	−452.609	−0.241523
$153$	6350.79	3.35576
$154$	0	0
$155$	0	0
$156$	2291.56	1.17610
$157$	−1389.94	−0.706554	−0.353277	−	0.935519i	$-0.614933\pi$
$157$	−1389.94	−0.706554	−0.353277	+	0.935519i	$0.614933\pi$
$158$	−2326.55	−1.17146
$159$	313.697	0.156464
$160$	0	0
$161$	0	0
$162$	557.159	0.270213
$163$	−3170.43	−1.52348	−0.761740	−	0.647883i	$-0.775655\pi$
$163$	−3170.43	−1.52348	−0.761740	+	0.647883i	$0.775655\pi$
$164$	551.500	0.262591
$165$	0	0
$166$	2260.18	1.05677
$167$	−497.526	−0.230537	−0.115269	−	0.993334i	$-0.536773\pi$
$167$	−497.526	−0.230537	−0.115269	+	0.993334i	$0.536773\pi$
$168$	0	0
$169$	2179.40	0.991988
$170$	0	0
$171$	−2715.31	−1.21430
$172$	1103.04	0.488990
$173$	4231.03	1.85942	0.929709	−	0.368294i	$-0.120058\pi$
$173$	4231.03	1.85942	0.929709	+	0.368294i	$0.120058\pi$
$174$	−1626.93	−0.708833
$175$	0	0
$176$	354.756	0.151936
$177$	6212.14	2.63804
$178$	1794.68	0.755715
$179$	797.157	0.332862	0.166431	−	0.986053i	$-0.446776\pi$
$179$	797.157	0.332862	0.166431	+	0.986053i	$0.446776\pi$
$180$	0	0
$181$	−1893.79	−0.777703	−0.388851	−	0.921301i	$-0.627128\pi$
$181$	−1893.79	−0.777703	−0.388851	+	0.921301i	$0.627128\pi$
$182$	0	0
$183$	−6259.36	−2.52844
$184$	175.268	0.0702225
$185$	0	0
$186$	−5193.02	−2.04716
$187$	2933.94	1.14733
$188$	402.870	0.156289
$189$	0	0
$190$	0	0
$191$	3988.00	1.51079	0.755397	−	0.655268i	$-0.227444\pi$
$191$	3988.00	1.51079	0.755397	+	0.655268i	$0.227444\pi$
$192$	554.234	0.208325
$193$	4051.43	1.51103	0.755513	−	0.655134i	$-0.227388\pi$
$193$	4051.43	1.51103	0.755513	+	0.655134i	$0.227388\pi$
$194$	3035.36	1.12333
$195$	0	0
$196$	0	0
$197$	1749.83	0.632842	0.316421	−	0.948619i	$-0.397519\pi$
$197$	1749.83	0.632842	0.316421	+	0.948619i	$0.397519\pi$
$198$	2128.27	0.763886
$199$	−2317.90	−0.825685	−0.412843	−	0.910802i	$-0.635464\pi$
$199$	−2317.90	−0.825685	−0.412843	+	0.910802i	$0.635464\pi$
$200$	0	0
$201$	1073.62	0.376752
$202$	91.1753	0.0317578
$203$	0	0
$204$	4583.68	1.57315
$205$	0	0
$206$	−2126.03	−0.719065
$207$	1051.47	0.353056
$208$	1058.47	0.352845
$209$	−1254.42	−0.415168
$210$	0	0
$211$	2196.02	0.716493	0.358246	−	0.933627i	$-0.383375\pi$
$211$	2196.02	0.716493	0.358246	+	0.933627i	$0.383375\pi$
$212$	144.896	0.0469411
$213$	−7856.02	−2.52716
$214$	−529.762	−0.169223
$215$	0	0
$216$	1454.44	0.458158
$217$	0	0
$218$	1770.12	0.549944
$219$	−6879.16	−2.12261
$220$	0	0
$221$	8753.87	2.66447
$222$	−2958.17	−0.894323
$223$	−2111.54	−0.634077	−0.317038	−	0.948413i	$-0.602688\pi$
$223$	−2111.54	−0.634077	−0.317038	+	0.948413i	$0.602688\pi$
$224$	0	0
$225$	0	0
$226$	3211.09	0.945127
$227$	3728.06	1.09004	0.545022	−	0.838422i	$-0.316521\pi$
$227$	3728.06	1.09004	0.545022	+	0.838422i	$0.316521\pi$
$228$	−1959.77	−0.569251
$229$	−700.918	−0.202262	−0.101131	−	0.994873i	$-0.532246\pi$
$229$	−700.918	−0.202262	−0.101131	+	0.994873i	$0.532246\pi$
$230$	0	0
$231$	0	0
$232$	−751.475	−0.212658
$233$	4355.40	1.22460	0.612300	−	0.790625i	$-0.290244\pi$
$233$	4355.40	1.22460	0.612300	+	0.790625i	$0.290244\pi$
$234$	6350.01	1.77399
$235$	0	0
$236$	2869.38	0.791445
$237$	−10073.8	−2.76104
$238$	0	0
$239$	2722.45	0.736823	0.368412	−	0.929663i	$-0.379902\pi$
$239$	2722.45	0.736823	0.368412	+	0.929663i	$0.379902\pi$
$240$	0	0
$241$	−2477.11	−0.662094	−0.331047	−	0.943614i	$-0.607402\pi$
$241$	−2477.11	−0.662094	−0.331047	+	0.943614i	$0.607402\pi$
$242$	−1678.78	−0.445934
$243$	−2496.27	−0.658995
$244$	−2891.19	−0.758564
$245$	0	0
$246$	2387.97	0.618908
$247$	−3742.76	−0.964153
$248$	−2398.65	−0.614172
$249$	9786.45	2.49073
$250$	0	0
$251$	−249.512	−0.0627453	−0.0313727	−	0.999508i	$-0.509988\pi$
$251$	−249.512	−0.0627453	−0.0313727	+	0.999508i	$0.509988\pi$
$252$	0	0
$253$	485.761	0.120710
$254$	4937.77	1.21978
$255$	0	0
$256$	256.000	0.0625000
$257$	960.811	0.233205	0.116603	−	0.993179i	$-0.462800\pi$
$257$	960.811	0.233205	0.116603	+	0.993179i	$0.462800\pi$
$258$	4776.12	1.15251
$259$	0	0
$260$	0	0
$261$	−4508.28	−1.06918
$262$	−5526.22	−1.30309
$263$	−3956.63	−0.927665	−0.463833	−	0.885923i	$-0.653526\pi$
$263$	−3956.63	−0.927665	−0.463833	+	0.885923i	$0.653526\pi$
$264$	1536.08	0.358102
$265$	0	0
$266$	0	0
$267$	7770.90	1.78117
$268$	495.902	0.113030
$269$	−1832.93	−0.415449	−0.207725	−	0.978187i	$-0.566606\pi$
$269$	−1832.93	−0.415449	−0.207725	+	0.978187i	$0.566606\pi$
$270$	0	0
$271$	1907.68	0.427614	0.213807	−	0.976876i	$-0.431414\pi$
$271$	1907.68	0.427614	0.213807	+	0.976876i	$0.431414\pi$
$272$	2117.20	0.471963
$273$	0	0
$274$	674.539	0.148724
$275$	0	0
$276$	758.902	0.165509
$277$	7418.68	1.60919	0.804594	−	0.593825i	$-0.202383\pi$
$277$	7418.68	1.60919	0.804594	+	0.593825i	$0.202383\pi$
$278$	−3288.37	−0.709436
$279$	−14390.1	−3.08786
$280$	0	0
$281$	−7297.51	−1.54923	−0.774614	−	0.632435i	$-0.782056\pi$
$281$	−7297.51	−1.54923	−0.774614	+	0.632435i	$0.782056\pi$
$282$	1744.41	0.368361
$283$	−4034.47	−0.847435	−0.423718	−	0.905794i	$-0.639275\pi$
$283$	−4034.47	−0.847435	−0.423718	+	0.905794i	$0.639275\pi$
$284$	−3628.69	−0.758180
$285$	0	0
$286$	2933.58	0.606526
$287$	0	0
$288$	1535.80	0.314230
$289$	12596.9	2.56399
$290$	0	0
$291$	13143.0	2.64761
$292$	−3177.48	−0.636808
$293$	−5854.31	−1.16728	−0.583639	−	0.812013i	$-0.698372\pi$
$293$	−5854.31	−1.16728	−0.583639	+	0.812013i	$0.698372\pi$
$294$	0	0
$295$	0	0
$296$	−1366.38	−0.268308
$297$	4031.03	0.787556
$298$	−3788.82	−0.736511
$299$	1449.34	0.280327
$300$	0	0
$301$	0	0
$302$	1719.22	0.327582
$303$	394.785	0.0748508
$304$	−905.218	−0.170782
$305$	0	0
$306$	12701.6	2.37288
$307$	2659.84	0.494479	0.247240	−	0.968954i	$-0.420476\pi$
$307$	2659.84	0.494479	0.247240	+	0.968954i	$0.420476\pi$
$308$	0	0
$309$	−9205.60	−1.69478
$310$	0	0
$311$	−4574.29	−0.834032	−0.417016	−	0.908899i	$-0.636924\pi$
$311$	−4574.29	−0.834032	−0.417016	+	0.908899i	$0.636924\pi$
$312$	4583.12	0.831629
$313$	−4987.46	−0.900665	−0.450332	−	0.892861i	$-0.648694\pi$
$313$	−4987.46	−0.900665	−0.450332	+	0.892861i	$0.648694\pi$
$314$	−2779.87	−0.499609
$315$	0	0
$316$	−4653.10	−0.828346
$317$	5692.82	1.00865	0.504323	−	0.863515i	$-0.331742\pi$
$317$	5692.82	1.00865	0.504323	+	0.863515i	$0.331742\pi$
$318$	627.393	0.110637
$319$	−2082.74	−0.365552
$320$	0	0
$321$	−2293.85	−0.398847
$322$	0	0
$323$	−7486.43	−1.28965
$324$	1114.32	0.191069
$325$	0	0
$326$	−6340.86	−1.07726
$327$	7664.54	1.29618
$328$	1103.00	0.185680
$329$	0	0
$330$	0	0
$331$	−6368.23	−1.05749	−0.528746	−	0.848780i	$-0.677338\pi$
$331$	−6368.23	−1.05749	−0.528746	+	0.848780i	$0.677338\pi$
$332$	4520.35	0.747249
$333$	−8197.23	−1.34896
$334$	−995.051	−0.163014
$335$	0	0
$336$	0	0
$337$	−6228.66	−1.00682	−0.503408	−	0.864049i	$-0.667921\pi$
$337$	−6228.66	−1.00682	−0.503408	+	0.864049i	$0.667921\pi$
$338$	4358.80	0.701442
$339$	13903.9	2.22759
$340$	0	0
$341$	−6647.95	−1.05574
$342$	−5430.62	−0.858638
$343$	0	0
$344$	2206.09	0.345768
$345$	0	0
$346$	8462.06	1.31481
$347$	3405.25	0.526812	0.263406	−	0.964685i	$-0.415154\pi$
$347$	3405.25	0.526812	0.263406	+	0.964685i	$0.415154\pi$
$348$	−3253.85	−0.501220
$349$	−11985.4	−1.83829	−0.919143	−	0.393925i	$-0.871117\pi$
$349$	−11985.4	−1.83829	−0.919143	+	0.393925i	$0.871117\pi$
$350$	0	0
$351$	12027.2	1.82896
$352$	709.513	0.107435
$353$	−8400.71	−1.26664	−0.633321	−	0.773889i	$-0.718309\pi$
$353$	−8400.71	−1.26664	−0.633321	+	0.773889i	$0.718309\pi$
$354$	12424.3	1.86538
$355$	0	0
$356$	3589.37	0.534371
$357$	0	0
$358$	1594.31	0.235369
$359$	−716.465	−0.105330	−0.0526652	−	0.998612i	$-0.516772\pi$
$359$	−716.465	−0.105330	−0.0526652	+	0.998612i	$0.516772\pi$
$360$	0	0
$361$	−3658.14	−0.533335
$362$	−3787.58	−0.549919
$363$	−7269.04	−1.05103
$364$	0	0
$365$	0	0
$366$	−12518.7	−1.78788
$367$	2509.69	0.356962	0.178481	−	0.983943i	$-0.442882\pi$
$367$	2509.69	0.356962	0.178481	+	0.983943i	$0.442882\pi$
$368$	350.536	0.0496548
$369$	6617.16	0.933539
$370$	0	0
$371$	0	0
$372$	−10386.0	−1.44756
$373$	802.750	0.111434	0.0557169	−	0.998447i	$-0.482256\pi$
$373$	802.750	0.111434	0.0557169	+	0.998447i	$0.482256\pi$
$374$	5867.89	0.811287
$375$	0	0
$376$	805.739	0.110513
$377$	−6214.17	−0.848928
$378$	0	0
$379$	−5322.77	−0.721404	−0.360702	−	0.932681i	$-0.617463\pi$
$379$	−5322.77	−0.721404	−0.360702	+	0.932681i	$0.617463\pi$
$380$	0	0
$381$	21380.3	2.87492
$382$	7976.00	1.06829
$383$	−13490.7	−1.79985	−0.899923	−	0.436049i	$-0.856377\pi$
$383$	−13490.7	−1.79985	−0.899923	+	0.436049i	$0.856377\pi$
$384$	1108.47	0.147308
$385$	0	0
$386$	8102.85	1.06846
$387$	13234.8	1.73841
$388$	6070.72	0.794315
$389$	5129.21	0.668538	0.334269	−	0.942478i	$-0.391511\pi$
$389$	5129.21	0.668538	0.334269	+	0.942478i	$0.391511\pi$
$390$	0	0
$391$	2899.04	0.374964
$392$	0	0
$393$	−23928.2	−3.07130
$394$	3499.65	0.447487
$395$	0	0
$396$	4256.54	0.540149
$397$	6546.69	0.827630	0.413815	−	0.910361i	$-0.364196\pi$
$397$	6546.69	0.827630	0.413815	+	0.910361i	$0.364196\pi$
$398$	−4635.79	−0.583848
$399$	0	0
$400$	0	0
$401$	26.4671	0.00329602	0.00164801	−	0.999999i	$-0.499475\pi$
$401$	26.4671	0.00329602	0.00164801	+	0.999999i	$0.499475\pi$
$402$	2147.23	0.266404
$403$	−19835.2	−2.45176
$404$	182.351	0.0224562
$405$	0	0
$406$	0	0
$407$	−3786.96	−0.461211
$408$	9167.37	1.11238
$409$	5918.35	0.715510	0.357755	−	0.933816i	$-0.383542\pi$
$409$	5918.35	0.715510	0.357755	+	0.933816i	$0.383542\pi$
$410$	0	0
$411$	2920.72	0.350532
$412$	−4252.06	−0.508456
$413$	0	0
$414$	2102.95	0.249648
$415$	0	0
$416$	2116.94	0.249499
$417$	−14238.5	−1.67209
$418$	−2508.84	−0.293568
$419$	−4228.31	−0.492999	−0.246499	−	0.969143i	$-0.579280\pi$
$419$	−4228.31	−0.492999	−0.246499	+	0.969143i	$0.579280\pi$
$420$	0	0
$421$	4466.68	0.517084	0.258542	−	0.966000i	$-0.416758\pi$
$421$	4466.68	0.517084	0.258542	+	0.966000i	$0.416758\pi$
$422$	4392.03	0.506637
$423$	4833.82	0.555623
$424$	289.792	0.0331924
$425$	0	0
$426$	−15712.0	−1.78697
$427$	0	0
$428$	−1059.52	−0.119659
$429$	12702.3	1.42954
$430$	0	0
$431$	−1962.97	−0.219381	−0.109690	−	0.993966i	$-0.534986\pi$
$431$	−1962.97	−0.219381	−0.109690	+	0.993966i	$0.534986\pi$
$432$	2908.88	0.323967
$433$	−2734.59	−0.303501	−0.151751	−	0.988419i	$-0.548491\pi$
$433$	−2734.59	−0.303501	−0.151751	+	0.988419i	$0.548491\pi$
$434$	0	0
$435$	0	0
$436$	3540.24	0.388869
$437$	−1239.50	−0.135682
$438$	−13758.3	−1.50091
$439$	8595.68	0.934509	0.467255	−	0.884123i	$-0.345243\pi$
$439$	8595.68	0.934509	0.467255	+	0.884123i	$0.345243\pi$
$440$	0	0
$441$	0	0
$442$	17507.7	1.88407
$443$	−1284.76	−0.137790	−0.0688951	−	0.997624i	$-0.521947\pi$
$443$	−1284.76	−0.137790	−0.0688951	+	0.997624i	$0.521947\pi$
$444$	−5916.35	−0.632382
$445$	0	0
$446$	−4223.08	−0.448360
$447$	−16405.4	−1.73590
$448$	0	0
$449$	−2633.49	−0.276798	−0.138399	−	0.990377i	$-0.544196\pi$
$449$	−2633.49	−0.276798	−0.138399	+	0.990377i	$0.544196\pi$
$450$	0	0
$451$	3057.00	0.319177
$452$	6422.18	0.668305
$453$	7444.12	0.772087
$454$	7456.12	0.770778
$455$	0	0
$456$	−3919.55	−0.402521
$457$	−687.957	−0.0704185	−0.0352093	−	0.999380i	$-0.511210\pi$
$457$	−687.957	−0.0704185	−0.0352093	+	0.999380i	$0.511210\pi$
$458$	−1401.84	−0.143021
$459$	24057.3	2.44641
$460$	0	0
$461$	−4355.06	−0.439989	−0.219995	−	0.975501i	$-0.570604\pi$
$461$	−4355.06	−0.439989	−0.219995	+	0.975501i	$0.570604\pi$
$462$	0	0
$463$	−10900.6	−1.09416	−0.547080	−	0.837081i	$-0.684260\pi$
$463$	−10900.6	−1.09416	−0.547080	+	0.837081i	$0.684260\pi$
$464$	−1502.95	−0.150372
$465$	0	0
$466$	8710.81	0.865923
$467$	−15746.8	−1.56033	−0.780166	−	0.625572i	$-0.784866\pi$
$467$	−15746.8	−1.56033	−0.780166	+	0.625572i	$0.784866\pi$
$468$	12700.0	1.25440
$469$	0	0
$470$	0	0
$471$	−12036.7	−1.17754
$472$	5738.77	0.559636
$473$	6114.24	0.594362
$474$	−20147.7	−1.95235
$475$	0	0
$476$	0	0
$477$	1738.53	0.166880
$478$	5444.90	0.521013
$479$	3752.10	0.357907	0.178954	−	0.983857i	$-0.442729\pi$
$479$	3752.10	0.357907	0.178954	+	0.983857i	$0.442729\pi$
$480$	0	0
$481$	−11299.0	−1.07108
$482$	−4954.22	−0.468171
$483$	0	0
$484$	−3357.56	−0.315323
$485$	0	0
$486$	−4992.54	−0.465980
$487$	−7497.49	−0.697625	−0.348813	−	0.937192i	$-0.613415\pi$
$487$	−7497.49	−0.697625	−0.348813	+	0.937192i	$0.613415\pi$
$488$	−5782.38	−0.536386
$489$	−27455.6	−2.53903
$490$	0	0
$491$	2364.21	0.217302	0.108651	−	0.994080i	$-0.465347\pi$
$491$	2364.21	0.217302	0.108651	+	0.994080i	$0.465347\pi$
$492$	4775.94	0.437634
$493$	−12429.9	−1.13552
$494$	−7485.51	−0.681759
$495$	0	0
$496$	−4797.31	−0.434285
$497$	0	0
$498$	19572.9	1.76121
$499$	−2964.11	−0.265916	−0.132958	−	0.991122i	$-0.542448\pi$
$499$	−2964.11	−0.265916	−0.132958	+	0.991122i	$0.542448\pi$
$500$	0	0
$501$	−4308.52	−0.384213
$502$	−499.025	−0.0443677
$503$	2777.43	0.246202	0.123101	−	0.992394i	$-0.460716\pi$
$503$	2777.43	0.246202	0.123101	+	0.992394i	$0.460716\pi$
$504$	0	0
$505$	0	0
$506$	971.523	0.0853547
$507$	18873.4	1.65325
$508$	9875.53	0.862511
$509$	−6479.21	−0.564216	−0.282108	−	0.959383i	$-0.591034\pi$
$509$	−6479.21	−0.564216	−0.282108	+	0.959383i	$0.591034\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	−10285.8	−0.885245
$514$	1921.62	0.164901
$515$	0	0
$516$	9552.24	0.814950
$517$	2233.13	0.189967
$518$	0	0
$519$	36640.3	3.09891
$520$	0	0
$521$	−1303.17	−0.109584	−0.0547919	−	0.998498i	$-0.517450\pi$
$521$	−1303.17	−0.109584	−0.0547919	+	0.998498i	$0.517450\pi$
$522$	−9016.56	−0.756023
$523$	−6088.73	−0.509066	−0.254533	−	0.967064i	$-0.581922\pi$
$523$	−6088.73	−0.509066	−0.254533	+	0.967064i	$0.581922\pi$
$524$	−11052.4	−0.921427
$525$	0	0
$526$	−7913.25	−0.655958
$527$	−39675.2	−3.27947
$528$	3072.16	0.253217
$529$	−11687.0	−0.960550
$530$	0	0
$531$	34428.2	2.81367
$532$	0	0
$533$	9121.04	0.741231
$534$	15541.8	1.25947
$535$	0	0
$536$	991.805	0.0799243
$537$	6903.30	0.554748
$538$	−3665.86	−0.293767
$539$	0	0
$540$	0	0
$541$	18411.9	1.46320	0.731598	−	0.681736i	$-0.238775\pi$
$541$	18411.9	1.46320	0.731598	+	0.681736i	$0.238775\pi$
$542$	3815.36	0.302368
$543$	−16400.0	−1.29612
$544$	4234.40	0.333728
$545$	0	0
$546$	0	0
$547$	−14541.5	−1.13666	−0.568328	−	0.822802i	$-0.692409\pi$
$547$	−14541.5	−1.13666	−0.568328	+	0.822802i	$0.692409\pi$
$548$	1349.08	0.105164
$549$	−34689.9	−2.69677
$550$	0	0
$551$	5314.44	0.410895
$552$	1517.80	0.117033
$553$	0	0
$554$	14837.4	1.13787
$555$	0	0
$556$	−6576.74	−0.501647
$557$	19433.6	1.47833	0.739164	−	0.673525i	$-0.235221\pi$
$557$	19433.6	1.47833	0.739164	+	0.673525i	$0.235221\pi$
$558$	−28780.2	−2.18344
$559$	18242.8	1.38030
$560$	0	0
$561$	25407.7	1.91214
$562$	−14595.0	−1.09547
$563$	−3896.11	−0.291654	−0.145827	−	0.989310i	$-0.546584\pi$
$563$	−3896.11	−0.291654	−0.145827	+	0.989310i	$0.546584\pi$
$564$	3488.81	0.260471
$565$	0	0
$566$	−8068.93	−0.599227
$567$	0	0
$568$	−7257.38	−0.536114
$569$	8980.15	0.661630	0.330815	−	0.943696i	$-0.392676\pi$
$569$	8980.15	0.661630	0.330815	+	0.943696i	$0.392676\pi$
$570$	0	0
$571$	18716.1	1.37170	0.685851	−	0.727742i	$-0.259430\pi$
$571$	18716.1	1.37170	0.685851	+	0.727742i	$0.259430\pi$
$572$	5867.17	0.428879
$573$	34535.7	2.51789
$574$	0	0
$575$	0	0
$576$	3071.61	0.222194
$577$	−9696.89	−0.699631	−0.349815	−	0.936819i	$-0.613756\pi$
$577$	−9696.89	−0.699631	−0.349815	+	0.936819i	$0.613756\pi$
$578$	25193.8	1.81301
$579$	35085.0	2.51827
$580$	0	0
$581$	0	0
$582$	26285.9	1.87214
$583$	803.169	0.0570564
$584$	−6354.96	−0.450291
$585$	0	0
$586$	−11708.6	−0.825390
$587$	−73.3324	−0.00515631	−0.00257815	−	0.999997i	$-0.500821\pi$
$587$	−73.3324	−0.00515631	−0.00257815	+	0.999997i	$0.500821\pi$
$588$	0	0
$589$	16963.3	1.18669
$590$	0	0
$591$	15153.3	1.05469
$592$	−2732.76	−0.189722
$593$	27701.9	1.91835	0.959174	−	0.282816i	$-0.0912685\pi$
$593$	27701.9	1.91835	0.959174	+	0.282816i	$0.0912685\pi$
$594$	8062.07	0.556886
$595$	0	0
$596$	−7577.64	−0.520792
$597$	−20072.8	−1.37609
$598$	2898.69	0.198221
$599$	15982.2	1.09018	0.545088	−	0.838379i	$-0.316496\pi$
$599$	15982.2	1.09018	0.545088	+	0.838379i	$0.316496\pi$
$600$	0	0
$601$	−17877.9	−1.21340	−0.606700	−	0.794931i	$-0.707507\pi$
$601$	−17877.9	−1.21340	−0.606700	+	0.794931i	$0.707507\pi$
$602$	0	0
$603$	5950.07	0.401834
$604$	3438.43	0.231635
$605$	0	0
$606$	789.570	0.0529275
$607$	−4146.93	−0.277296	−0.138648	−	0.990342i	$-0.544276\pi$
$607$	−4146.93	−0.277296	−0.138648	+	0.990342i	$0.544276\pi$
$608$	−1810.44	−0.120761
$609$	0	0
$610$	0	0
$611$	6662.89	0.441165
$612$	25403.2	1.67788
$613$	662.364	0.0436421	0.0218211	−	0.999762i	$-0.493054\pi$
$613$	662.364	0.0436421	0.0218211	+	0.999762i	$0.493054\pi$
$614$	5319.68	0.349650
$615$	0	0
$616$	0	0
$617$	8693.21	0.567221	0.283611	−	0.958940i	$-0.408468\pi$
$617$	8693.21	0.567221	0.283611	+	0.958940i	$0.408468\pi$
$618$	−18411.2	−1.19839
$619$	−2914.63	−0.189255	−0.0946275	−	0.995513i	$-0.530166\pi$
$619$	−2914.63	−0.189255	−0.0946275	+	0.995513i	$0.530166\pi$
$620$	0	0
$621$	3983.08	0.257384
$622$	−9148.58	−0.589750
$623$	0	0
$624$	9166.24	0.588050
$625$	0	0
$626$	−9974.92	−0.636866
$627$	−10863.2	−0.691919
$628$	−5559.75	−0.353277
$629$	−22600.7	−1.43267
$630$	0	0
$631$	−894.747	−0.0564490	−0.0282245	−	0.999602i	$-0.508985\pi$
$631$	−894.747	−0.0564490	−0.0282245	+	0.999602i	$0.508985\pi$
$632$	−9306.20	−0.585729
$633$	19017.3	1.19411
$634$	11385.6	0.713221
$635$	0	0
$636$	1254.79	0.0782320
$637$	0	0
$638$	−4165.48	−0.258484
$639$	−43538.7	−2.69541
$640$	0	0
$641$	−20085.0	−1.23762	−0.618808	−	0.785542i	$-0.712384\pi$
$641$	−20085.0	−1.23762	−0.618808	+	0.785542i	$0.712384\pi$
$642$	−4587.69	−0.282028
$643$	−851.943	−0.0522510	−0.0261255	−	0.999659i	$-0.508317\pi$
$643$	−851.943	−0.0522510	−0.0261255	+	0.999659i	$0.508317\pi$
$644$	0	0
$645$	0	0
$646$	−14972.9	−0.911918
$647$	−20767.9	−1.26193	−0.630966	−	0.775810i	$-0.717341\pi$
$647$	−20767.9	−1.26193	−0.630966	+	0.775810i	$0.717341\pi$
$648$	2228.63	0.135107
$649$	15905.2	0.961993
$650$	0	0
$651$	0	0
$652$	−12681.7	−0.761740
$653$	−14705.7	−0.881282	−0.440641	−	0.897683i	$-0.645249\pi$
$653$	−14705.7	−0.881282	−0.440641	+	0.897683i	$0.645249\pi$
$654$	15329.1	0.916536
$655$	0	0
$656$	2206.00	0.131296
$657$	−38124.9	−2.26392
$658$	0	0
$659$	−27657.2	−1.63486	−0.817428	−	0.576031i	$-0.804601\pi$
$659$	−27657.2	−1.63486	−0.817428	+	0.576031i	$0.804601\pi$
$660$	0	0
$661$	−19683.3	−1.15823	−0.579117	−	0.815244i	$-0.696603\pi$
$661$	−19683.3	−1.15823	−0.579117	+	0.815244i	$0.696603\pi$
$662$	−12736.5	−0.747760
$663$	75807.6	4.44061
$664$	9040.71	0.528385
$665$	0	0
$666$	−16394.5	−0.953862
$667$	−2057.96	−0.119467
$668$	−1990.10	−0.115269
$669$	−18285.7	−1.05675
$670$	0	0
$671$	−16026.1	−0.922026
$672$	0	0
$673$	3443.85	0.197252	0.0986260	−	0.995125i	$-0.468555\pi$
$673$	3443.85	0.197252	0.0986260	+	0.995125i	$0.468555\pi$
$674$	−12457.3	−0.711926
$675$	0	0
$676$	8717.59	0.495994
$677$	−28254.1	−1.60398	−0.801988	−	0.597340i	$-0.796224\pi$
$677$	−28254.1	−1.60398	−0.801988	+	0.597340i	$0.796224\pi$
$678$	27807.7	1.57515
$679$	0	0
$680$	0	0
$681$	32284.6	1.81667
$682$	−13295.9	−0.746519
$683$	23937.4	1.34105	0.670525	−	0.741887i	$-0.266069\pi$
$683$	23937.4	1.34105	0.670525	+	0.741887i	$0.266069\pi$
$684$	−10861.2	−0.607149
$685$	0	0
$686$	0	0
$687$	−6069.88	−0.337089
$688$	4412.17	0.244495
$689$	2396.38	0.132503
$690$	0	0
$691$	−35157.3	−1.93552	−0.967762	−	0.251867i	$-0.918956\pi$
$691$	−35157.3	−1.93552	−0.967762	+	0.251867i	$0.918956\pi$
$692$	16924.1	0.929709
$693$	0	0
$694$	6810.51	0.372512
$695$	0	0
$696$	−6507.70	−0.354416
$697$	18244.3	0.991467
$698$	−23970.7	−1.29986
$699$	37717.4	2.04092
$700$	0	0
$701$	−4703.07	−0.253399	−0.126699	−	0.991941i	$-0.540438\pi$
$701$	−4703.07	−0.253399	−0.126699	+	0.991941i	$0.540438\pi$
$702$	24054.4	1.29327
$703$	9663.04	0.518419
$704$	1419.03	0.0759681
$705$	0	0
$706$	−16801.4	−0.895651
$707$	0	0
$708$	24848.6	1.31902
$709$	13101.7	0.693997	0.346999	−	0.937866i	$-0.387201\pi$
$709$	13101.7	0.693997	0.346999	+	0.937866i	$0.387201\pi$
$710$	0	0
$711$	−55830.1	−2.94485
$712$	7178.74	0.377858
$713$	−6568.86	−0.345029
$714$	0	0
$715$	0	0
$716$	3188.63	0.166431
$717$	23576.2	1.22799
$718$	−1432.93	−0.0744798
$719$	22401.9	1.16196	0.580981	−	0.813917i	$-0.302669\pi$
$719$	22401.9	1.16196	0.580981	+	0.813917i	$0.302669\pi$
$720$	0	0
$721$	0	0
$722$	−7316.29	−0.377125
$723$	−21451.5	−1.10344
$724$	−7575.15	−0.388851
$725$	0	0
$726$	−14538.1	−0.743194
$727$	13870.4	0.707600	0.353800	−	0.935321i	$-0.384889\pi$
$727$	13870.4	0.707600	0.353800	+	0.935321i	$0.384889\pi$
$728$	0	0
$729$	−29139.1	−1.48042
$730$	0	0
$731$	36490.0	1.84628
$732$	−25037.4	−1.26422
$733$	−10216.2	−0.514791	−0.257396	−	0.966306i	$-0.582864\pi$
$733$	−10216.2	−0.514791	−0.257396	+	0.966306i	$0.582864\pi$
$734$	5019.39	0.252410
$735$	0	0
$736$	701.072	0.0351112
$737$	2748.82	0.137387
$738$	13234.3	0.660112
$739$	19171.5	0.954310	0.477155	−	0.878819i	$-0.341668\pi$
$739$	19171.5	0.954310	0.477155	+	0.878819i	$0.341668\pi$
$740$	0	0
$741$	−32411.9	−1.60686
$742$	0	0
$743$	−19779.8	−0.976650	−0.488325	−	0.872662i	$-0.662392\pi$
$743$	−19779.8	−0.976650	−0.488325	+	0.872662i	$0.662392\pi$
$744$	−20772.1	−1.02358
$745$	0	0
$746$	1605.50	0.0787956
$747$	54237.3	2.65655
$748$	11735.8	0.573666
$749$	0	0
$750$	0	0
$751$	20104.8	0.976875	0.488438	−	0.872599i	$-0.337567\pi$
$751$	20104.8	0.976875	0.488438	+	0.872599i	$0.337567\pi$
$752$	1611.48	0.0781444
$753$	−2160.75	−0.104571
$754$	−12428.3	−0.600283
$755$	0	0
$756$	0	0
$757$	−10958.5	−0.526149	−0.263074	−	0.964776i	$-0.584736\pi$
$757$	−10958.5	−0.526149	−0.263074	+	0.964776i	$0.584736\pi$
$758$	−10645.5	−0.510110
$759$	4206.65	0.201175
$760$	0	0
$761$	−16633.4	−0.792327	−0.396164	−	0.918180i	$-0.629659\pi$
$761$	−16633.4	−0.792327	−0.396164	+	0.918180i	$0.629659\pi$
$762$	42760.6	2.03288
$763$	0	0
$764$	15952.0	0.755397
$765$	0	0
$766$	−26981.3	−1.27268
$767$	47455.6	2.23406
$768$	2216.93	0.104162
$769$	24813.6	1.16359	0.581796	−	0.813335i	$-0.302350\pi$
$769$	24813.6	1.16359	0.581796	+	0.813335i	$0.302350\pi$
$770$	0	0
$771$	8320.53	0.388660
$772$	16205.7	0.755513
$773$	35576.5	1.65537	0.827683	−	0.561196i	$-0.189659\pi$
$773$	35576.5	1.65537	0.827683	+	0.561196i	$0.189659\pi$
$774$	26469.7	1.22924
$775$	0	0
$776$	12141.4	0.561666
$777$	0	0
$778$	10258.4	0.472728
$779$	−7800.44	−0.358767
$780$	0	0
$781$	−20114.1	−0.921559
$782$	5798.08	0.265139
$783$	−17077.8	−0.779450
$784$	0	0
$785$	0	0
$786$	−47856.5	−2.17174
$787$	−3112.14	−0.140960	−0.0704802	−	0.997513i	$-0.522453\pi$
$787$	−3112.14	−0.140960	−0.0704802	+	0.997513i	$0.522453\pi$
$788$	6999.30	0.316421
$789$	−34264.0	−1.54605
$790$	0	0
$791$	0	0
$792$	8513.07	0.381943
$793$	−47816.2	−2.14124
$794$	13093.4	0.585223
$795$	0	0
$796$	−9271.59	−0.412843
$797$	4340.97	0.192930	0.0964649	−	0.995336i	$-0.469246\pi$
$797$	4340.97	0.192930	0.0964649	+	0.995336i	$0.469246\pi$
$798$	0	0
$799$	13327.4	0.590100
$800$	0	0
$801$	43067.0	1.89975
$802$	52.9342	0.00233064
$803$	−17613.0	−0.774033
$804$	4294.47	0.188376
$805$	0	0
$806$	−39670.3	−1.73366
$807$	−15873.0	−0.692387
$808$	364.701	0.0158789
$809$	−34714.6	−1.50865	−0.754327	−	0.656499i	$-0.772037\pi$
$809$	−34714.6	−1.50865	−0.754327	+	0.656499i	$0.772037\pi$
$810$	0	0
$811$	−8124.96	−0.351795	−0.175898	−	0.984408i	$-0.556283\pi$
$811$	−8124.96	−0.351795	−0.175898	+	0.984408i	$0.556283\pi$
$812$	0	0
$813$	16520.3	0.712660
$814$	−7573.93	−0.326125
$815$	0	0
$816$	18334.7	0.786574
$817$	−15601.5	−0.668086
$818$	11836.7	0.505942
$819$	0	0
$820$	0	0
$821$	−4762.15	−0.202436	−0.101218	−	0.994864i	$-0.532274\pi$
$821$	−4762.15	−0.202436	−0.101218	+	0.994864i	$0.532274\pi$
$822$	5841.44	0.247863
$823$	−13577.1	−0.575051	−0.287526	−	0.957773i	$-0.592833\pi$
$823$	−13577.1	−0.575051	−0.287526	+	0.957773i	$0.592833\pi$
$824$	−8504.12	−0.359533
$825$	0	0
$826$	0	0
$827$	22576.6	0.949294	0.474647	−	0.880176i	$-0.342576\pi$
$827$	22576.6	0.949294	0.474647	+	0.880176i	$0.342576\pi$
$828$	4205.90	0.176528
$829$	18235.2	0.763976	0.381988	−	0.924167i	$-0.375240\pi$
$829$	18235.2	0.763976	0.381988	+	0.924167i	$0.375240\pi$
$830$	0	0
$831$	64245.0	2.68187
$832$	4233.88	0.176422
$833$	0	0
$834$	−28477.0	−1.18235
$835$	0	0
$836$	−5017.68	−0.207584
$837$	−54510.9	−2.25110
$838$	−8456.62	−0.348603
$839$	34955.5	1.43838	0.719189	−	0.694815i	$-0.244514\pi$
$839$	34955.5	1.43838	0.719189	+	0.694815i	$0.244514\pi$
$840$	0	0
$841$	−15565.3	−0.638211
$842$	8933.35	0.365634
$843$	−63195.7	−2.58194
$844$	8784.06	0.358246
$845$	0	0
$846$	9667.64	0.392885
$847$	0	0
$848$	579.585	0.0234705
$849$	−34938.1	−1.41233
$850$	0	0
$851$	−3741.91	−0.150730
$852$	−31424.1	−1.26358
$853$	8243.14	0.330879	0.165439	−	0.986220i	$-0.447096\pi$
$853$	8243.14	0.330879	0.165439	+	0.986220i	$0.447096\pi$
$854$	0	0
$855$	0	0
$856$	−2119.05	−0.0846117
$857$	6675.90	0.266096	0.133048	−	0.991110i	$-0.457524\pi$
$857$	6675.90	0.266096	0.133048	+	0.991110i	$0.457524\pi$
$858$	25404.6	1.01084
$859$	29038.8	1.15342	0.576712	−	0.816948i	$-0.304336\pi$
$859$	29038.8	1.15342	0.576712	+	0.816948i	$0.304336\pi$
$860$	0	0
$861$	0	0
$862$	−3925.94	−0.155125
$863$	9681.42	0.381876	0.190938	−	0.981602i	$-0.438847\pi$
$863$	9681.42	0.381876	0.190938	+	0.981602i	$0.438847\pi$
$864$	5817.76	0.229079
$865$	0	0
$866$	−5469.18	−0.214608
$867$	109088.	4.27314
$868$	0	0
$869$	−25792.4	−1.00685
$870$	0	0
$871$	8201.53	0.319056
$872$	7080.49	0.274972
$873$	72839.4	2.82387
$874$	−2479.00	−0.0959420
$875$	0	0
$876$	−27516.7	−1.06130
$877$	20310.9	0.782040	0.391020	−	0.920382i	$-0.372122\pi$
$877$	20310.9	0.782040	0.391020	+	0.920382i	$0.372122\pi$
$878$	17191.4	0.660798
$879$	−50697.7	−1.94538
$880$	0	0
$881$	25065.3	0.958539	0.479269	−	0.877668i	$-0.340902\pi$
$881$	25065.3	0.958539	0.479269	+	0.877668i	$0.340902\pi$
$882$	0	0
$883$	13015.5	0.496045	0.248023	−	0.968754i	$-0.420219\pi$
$883$	13015.5	0.496045	0.248023	+	0.968754i	$0.420219\pi$
$884$	35015.5	1.33224
$885$	0	0
$886$	−2569.53	−0.0974323
$887$	31630.5	1.19735	0.598674	−	0.800992i	$-0.295694\pi$
$887$	31630.5	1.19735	0.598674	+	0.800992i	$0.295694\pi$
$888$	−11832.7	−0.447161
$889$	0	0
$890$	0	0
$891$	6176.74	0.232243
$892$	−8446.15	−0.317038
$893$	−5698.20	−0.213531
$894$	−32810.8	−1.22747
$895$	0	0
$896$	0	0
$897$	12551.2	0.467192
$898$	−5266.98	−0.195725
$899$	28164.5	1.04487
$900$	0	0
$901$	4793.34	0.177236
$902$	6114.01	0.225692
$903$	0	0
$904$	12844.4	0.472563
$905$	0	0
$906$	14888.2	0.545948
$907$	48111.3	1.76131	0.880656	−	0.473757i	$-0.157102\pi$
$907$	48111.3	1.76131	0.880656	+	0.473757i	$0.157102\pi$
$908$	14912.2	0.545022
$909$	2187.93	0.0798340
$910$	0	0
$911$	−11465.5	−0.416982	−0.208491	−	0.978024i	$-0.566855\pi$
$911$	−11465.5	−0.416982	−0.208491	+	0.978024i	$0.566855\pi$
$912$	−7839.10	−0.284626
$913$	25056.6	0.908273
$914$	−1375.91	−0.0497934
$915$	0	0
$916$	−2803.67	−0.101131
$917$	0	0
$918$	48114.7	1.72987
$919$	26141.9	0.938347	0.469174	−	0.883106i	$-0.344552\pi$
$919$	26141.9	0.938347	0.469174	+	0.883106i	$0.344552\pi$
$920$	0	0
$921$	23034.0	0.824099
$922$	−8710.11	−0.311120
$923$	−60013.4	−2.14016
$924$	0	0
$925$	0	0
$926$	−21801.3	−0.773687
$927$	−51018.2	−1.80761
$928$	−3005.90	−0.106329
$929$	3155.76	0.111450	0.0557251	−	0.998446i	$-0.482253\pi$
$929$	3155.76	0.111450	0.0557251	+	0.998446i	$0.482253\pi$
$930$	0	0
$931$	0	0
$932$	17421.6	0.612300
$933$	−39612.9	−1.39000
$934$	−31493.6	−1.10332
$935$	0	0
$936$	25400.0	0.886994
$937$	−50554.9	−1.76260	−0.881301	−	0.472556i	$-0.843331\pi$
$937$	−50554.9	−1.76260	−0.881301	+	0.472556i	$0.843331\pi$
$938$	0	0
$939$	−43190.9	−1.50105
$940$	0	0
$941$	−53927.4	−1.86821	−0.934104	−	0.357002i	$-0.883799\pi$
$941$	−53927.4	−1.86821	−0.934104	+	0.357002i	$0.883799\pi$
$942$	−24073.4	−0.832648
$943$	3020.64	0.104311
$944$	11477.5	0.395722
$945$	0	0
$946$	12228.5	0.420277
$947$	−29301.0	−1.00544	−0.502722	−	0.864448i	$-0.667668\pi$
$947$	−29301.0	−1.00544	−0.502722	+	0.864448i	$0.667668\pi$
$948$	−40295.4	−1.38052
$949$	−52551.0	−1.79755
$950$	0	0
$951$	49299.3	1.68101
$952$	0	0
$953$	35739.2	1.21480	0.607401	−	0.794396i	$-0.292212\pi$
$953$	35739.2	1.21480	0.607401	+	0.794396i	$0.292212\pi$
$954$	3477.07	0.118002
$955$	0	0
$956$	10889.8	0.368412
$957$	−18036.3	−0.609228
$958$	7504.19	0.253079
$959$	0	0
$960$	0	0
$961$	60108.0	2.01766
$962$	−22598.0	−0.757368
$963$	−12712.7	−0.425400
$964$	−9908.43	−0.331047
$965$	0	0
$966$	0	0
$967$	−47240.9	−1.57101	−0.785503	−	0.618857i	$-0.787596\pi$
$967$	−47240.9	−1.57101	−0.785503	+	0.618857i	$0.787596\pi$
$968$	−6715.12	−0.222967
$969$	−64831.7	−2.14932
$970$	0	0
$971$	−35861.0	−1.18521	−0.592603	−	0.805494i	$-0.701900\pi$
$971$	−35861.0	−1.18521	−0.592603	+	0.805494i	$0.701900\pi$
$972$	−9985.08	−0.329497
$973$	0	0
$974$	−14995.0	−0.493296
$975$	0	0
$976$	−11564.8	−0.379282
$977$	22986.5	0.752714	0.376357	−	0.926475i	$-0.377177\pi$
$977$	22986.5	0.752714	0.376357	+	0.926475i	$0.377177\pi$
$978$	−54911.2	−1.79537
$979$	19896.1	0.649523
$980$	0	0
$981$	42477.5	1.38247
$982$	4728.41	0.153656
$983$	−45451.8	−1.47476	−0.737379	−	0.675479i	$-0.763937\pi$
$983$	−45451.8	−1.47476	−0.737379	+	0.675479i	$0.763937\pi$
$984$	9551.88	0.309454
$985$	0	0
$986$	−24859.7	−0.802936
$987$	0	0
$988$	−14971.0	−0.482077
$989$	6041.50	0.194245
$990$	0	0
$991$	10088.7	0.323389	0.161694	−	0.986841i	$-0.448304\pi$
$991$	10088.7	0.323389	0.161694	+	0.986841i	$0.448304\pi$
$992$	−9594.61	−0.307086
$993$	−55148.3	−1.76241
$994$	0	0
$995$	0	0
$996$	39145.8	1.24536
$997$	−45887.2	−1.45763	−0.728817	−	0.684709i	$-0.759929\pi$
$997$	−45887.2	−1.45763	−0.728817	+	0.684709i	$0.759929\pi$
$998$	−5928.23	−0.188031
$999$	−31051.8	−0.983419

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.da.1.8	yes	8
5.4	even	2		2450.4.a.cz.1.1	✓	8
7.6	odd	2	inner	2450.4.a.da.1.1	yes	8
35.34	odd	2		2450.4.a.cz.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2450.4.a.cz.1.1	✓	8	5.4	even	2
2450.4.a.cz.1.8	yes	8	35.34	odd	2
2450.4.a.da.1.1	yes	8	7.6	odd	2	inner
2450.4.a.da.1.8	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +8.65990 q^{3} +4.00000 q^{4} +17.3198 q^{6} +8.00000 q^{8} +47.9939 q^{9} +22.1723 q^{11} +34.6396 q^{12} +66.1544 q^{13} +16.0000 q^{16} +132.325 q^{17} +95.9878 q^{18} -56.5761 q^{19} +44.3445 q^{22} +21.9085 q^{23} +69.2792 q^{24} +132.309 q^{26} +181.805 q^{27} -93.9344 q^{29} -299.832 q^{31} +32.0000 q^{32} +192.010 q^{33} +264.650 q^{34} +191.976 q^{36} -170.797 q^{37} -113.152 q^{38} +572.890 q^{39} +137.875 q^{41} +275.761 q^{43} +88.6891 q^{44} +43.8170 q^{46} +100.717 q^{47} +138.558 q^{48} +1145.92 q^{51} +264.617 q^{52} +36.2240 q^{53} +363.610 q^{54} -489.944 q^{57} -187.869 q^{58} +717.346 q^{59} -722.798 q^{61} -599.663 q^{62} +64.0000 q^{64} +384.019 q^{66} +123.976 q^{67} +529.300 q^{68} +189.726 q^{69} -907.172 q^{71} +383.951 q^{72} -794.370 q^{73} -341.594 q^{74} -226.304 q^{76} +1145.78 q^{78} -1163.27 q^{79} +278.579 q^{81} +275.750 q^{82} +1130.09 q^{83} +551.521 q^{86} -813.463 q^{87} +177.378 q^{88} +897.342 q^{89} +87.6340 q^{92} -2596.51 q^{93} +201.435 q^{94} +277.117 q^{96} +1517.68 q^{97} +1064.13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{2} + 32 q^{4} + 64 q^{8} + 76 q^{9} + 36 q^{11} + 128 q^{16} + 152 q^{18} + 72 q^{22} + 164 q^{23} + 392 q^{29} + 256 q^{32} + 304 q^{36} + 32 q^{37} + 832 q^{39} + 752 q^{43} + 144 q^{44} + 328 q^{46}+ \cdots + 2892 q^{99}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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