LMFDB - Embedded newform 2450.4.a.da.1.6

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			1.6
Root			$0.424582$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} +1.83880 q^{3} +4.00000 q^{4} +3.67759 q^{6} +8.00000 q^{8} -23.6188 q^{9} -69.0013 q^{11} +7.35518 q^{12} +8.88477 q^{13} +16.0000 q^{16} -87.9891 q^{17} -47.2377 q^{18} +150.474 q^{19} -138.003 q^{22} +56.9429 q^{23} +14.7104 q^{24} +17.7695 q^{26} -93.0777 q^{27} -160.791 q^{29} +121.614 q^{31} +32.0000 q^{32} -126.879 q^{33} -175.978 q^{34} -94.4753 q^{36} +96.0929 q^{37} +300.949 q^{38} +16.3373 q^{39} -165.669 q^{41} +311.670 q^{43} -276.005 q^{44} +113.886 q^{46} +641.297 q^{47} +29.4207 q^{48} -161.794 q^{51} +35.5391 q^{52} +385.445 q^{53} -186.155 q^{54} +276.692 q^{57} -321.583 q^{58} +249.674 q^{59} -453.791 q^{61} +243.228 q^{62} +64.0000 q^{64} -253.759 q^{66} +570.001 q^{67} -351.956 q^{68} +104.706 q^{69} +274.128 q^{71} -188.951 q^{72} -750.031 q^{73} +192.186 q^{74} +601.898 q^{76} +32.6746 q^{78} +427.803 q^{79} +466.558 q^{81} -331.337 q^{82} +676.015 q^{83} +623.341 q^{86} -295.662 q^{87} -552.010 q^{88} -658.636 q^{89} +227.772 q^{92} +223.624 q^{93} +1282.59 q^{94} +58.8415 q^{96} +169.978 q^{97} +1629.73 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.83880	0.353876	0.176938	−	0.984222i	$-0.443381\pi$
$3$	1.83880	0.353876	0.176938	+	0.984222i	$0.443381\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	3.67759	0.250228
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−23.6188	−0.874771
$10$	0	0
$11$	−69.0013	−1.89133	−0.945667	−	0.325138i	$-0.894589\pi$
$11$	−69.0013	−1.89133	−0.945667	+	0.325138i	$0.894589\pi$
$12$	7.35518	0.176938
$13$	8.88477	0.189553	0.0947766	−	0.995499i	$-0.469786\pi$
$13$	8.88477	0.189553	0.0947766	+	0.995499i	$0.469786\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−87.9891	−1.25532	−0.627661	−	0.778487i	$-0.715988\pi$
$17$	−87.9891	−1.25532	−0.627661	+	0.778487i	$0.715988\pi$
$18$	−47.2377	−0.618557
$19$	150.474	1.81690	0.908452	−	0.417988i	$-0.137265\pi$
$19$	150.474	1.81690	0.908452	+	0.417988i	$0.137265\pi$
$20$	0	0
$21$	0	0
$22$	−138.003	−1.33737
$23$	56.9429	0.516236	0.258118	−	0.966113i	$-0.416898\pi$
$23$	56.9429	0.516236	0.258118	+	0.966113i	$0.416898\pi$
$24$	14.7104	0.125114
$25$	0	0
$26$	17.7695	0.134034
$27$	−93.0777	−0.663437
$28$	0	0
$29$	−160.791	−1.02959	−0.514796	−	0.857312i	$-0.672132\pi$
$29$	−160.791	−1.02959	−0.514796	+	0.857312i	$0.672132\pi$
$30$	0	0
$31$	121.614	0.704598	0.352299	−	0.935887i	$-0.385400\pi$
$31$	121.614	0.704598	0.352299	+	0.935887i	$0.385400\pi$
$32$	32.0000	0.176777
$33$	−126.879	−0.669298
$34$	−175.978	−0.887647
$35$	0	0
$36$	−94.4753	−0.437386
$37$	96.0929	0.426961	0.213481	−	0.976947i	$-0.431520\pi$
$37$	96.0929	0.426961	0.213481	+	0.976947i	$0.431520\pi$
$38$	300.949	1.28475
$39$	16.3373	0.0670784
$40$	0	0
$41$	−165.669	−0.631051	−0.315525	−	0.948917i	$-0.602181\pi$
$41$	−165.669	−0.631051	−0.315525	+	0.948917i	$0.602181\pi$
$42$	0	0
$43$	311.670	1.10533	0.552666	−	0.833403i	$-0.313610\pi$
$43$	311.670	1.10533	0.552666	+	0.833403i	$0.313610\pi$
$44$	−276.005	−0.945667
$45$	0	0
$46$	113.886	0.365034
$47$	641.297	1.99027	0.995136	−	0.0985153i	$-0.0314093\pi$
$47$	641.297	1.99027	0.995136	+	0.0985153i	$0.0314093\pi$
$48$	29.4207	0.0884691
$49$	0	0
$50$	0	0
$51$	−161.794	−0.444229
$52$	35.5391	0.0947766
$53$	385.445	0.998960	0.499480	−	0.866325i	$-0.333524\pi$
$53$	385.445	0.998960	0.499480	+	0.866325i	$0.333524\pi$
$54$	−186.155	−0.469121
$55$	0	0
$56$	0	0
$57$	276.692	0.642960
$58$	−321.583	−0.728032
$59$	249.674	0.550928	0.275464	−	0.961311i	$-0.411169\pi$
$59$	249.674	0.550928	0.275464	+	0.961311i	$0.411169\pi$
$60$	0	0
$61$	−453.791	−0.952491	−0.476245	−	0.879312i	$-0.658003\pi$
$61$	−453.791	−0.952491	−0.476245	+	0.879312i	$0.658003\pi$
$62$	243.228	0.498226
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	−253.759	−0.473265
$67$	570.001	1.03935	0.519677	−	0.854363i	$-0.326052\pi$
$67$	570.001	1.03935	0.519677	+	0.854363i	$0.326052\pi$
$68$	−351.956	−0.627661
$69$	104.706	0.182684
$70$	0	0
$71$	274.128	0.458211	0.229106	−	0.973402i	$-0.426420\pi$
$71$	274.128	0.458211	0.229106	+	0.973402i	$0.426420\pi$
$72$	−188.951	−0.309278
$73$	−750.031	−1.20253	−0.601263	−	0.799051i	$-0.705336\pi$
$73$	−750.031	−1.20253	−0.601263	+	0.799051i	$0.705336\pi$
$74$	192.186	0.301907
$75$	0	0
$76$	601.898	0.908452
$77$	0	0
$78$	32.6746	0.0474316
$79$	427.803	0.609261	0.304631	−	0.952471i	$-0.401467\pi$
$79$	427.803	0.609261	0.304631	+	0.952471i	$0.401467\pi$
$80$	0	0
$81$	466.558	0.639997
$82$	−331.337	−0.446220
$83$	676.015	0.894003	0.447002	−	0.894533i	$-0.352492\pi$
$83$	676.015	0.894003	0.447002	+	0.894533i	$0.352492\pi$
$84$	0	0
$85$	0	0
$86$	623.341	0.781588
$87$	−295.662	−0.364349
$88$	−552.010	−0.668687
$89$	−658.636	−0.784442	−0.392221	−	0.919871i	$-0.628293\pi$
$89$	−658.636	−0.784442	−0.392221	+	0.919871i	$0.628293\pi$
$90$	0	0
$91$	0	0
$92$	227.772	0.258118
$93$	223.624	0.249341
$94$	1282.59	1.40733
$95$	0	0
$96$	58.8415	0.0625571
$97$	169.978	0.177924	0.0889619	−	0.996035i	$-0.471645\pi$
$97$	169.978	0.177924	0.0889619	+	0.996035i	$0.471645\pi$
$98$	0	0
$99$	1629.73	1.65448
$100$	0	0
$101$	386.870	0.381139	0.190570	−	0.981674i	$-0.438967\pi$
$101$	386.870	0.381139	0.190570	+	0.981674i	$0.438967\pi$
$102$	−323.588	−0.314117
$103$	1690.37	1.61706	0.808531	−	0.588453i	$-0.200263\pi$
$103$	1690.37	1.61706	0.808531	+	0.588453i	$0.200263\pi$
$104$	71.0782	0.0670172
$105$	0	0
$106$	770.889	0.706372
$107$	984.275	0.889285	0.444642	−	0.895708i	$-0.353331\pi$
$107$	984.275	0.889285	0.444642	+	0.895708i	$0.353331\pi$
$108$	−372.311	−0.331719
$109$	887.811	0.780155	0.390077	−	0.920782i	$-0.372448\pi$
$109$	887.811	0.780155	0.390077	+	0.920782i	$0.372448\pi$
$110$	0	0
$111$	176.695	0.151092
$112$	0	0
$113$	−631.384	−0.525625	−0.262812	−	0.964847i	$-0.584650\pi$
$113$	−631.384	−0.525625	−0.262812	+	0.964847i	$0.584650\pi$
$114$	553.383	0.454641
$115$	0	0
$116$	−643.165	−0.514796
$117$	−209.848	−0.165816
$118$	499.347	0.389565
$119$	0	0
$120$	0	0
$121$	3430.18	2.57714
$122$	−907.581	−0.673512
$123$	−304.631	−0.223314
$124$	486.457	0.352299
$125$	0	0
$126$	0	0
$127$	−520.877	−0.363940	−0.181970	−	0.983304i	$-0.558247\pi$
$127$	−520.877	−0.363940	−0.181970	+	0.983304i	$0.558247\pi$
$128$	128.000	0.0883883
$129$	573.098	0.391151
$130$	0	0
$131$	174.337	0.116274	0.0581370	−	0.998309i	$-0.481484\pi$
$131$	174.337	0.116274	0.0581370	+	0.998309i	$0.481484\pi$
$132$	−507.517	−0.334649
$133$	0	0
$134$	1140.00	0.734934
$135$	0	0
$136$	−703.913	−0.443824
$137$	2185.52	1.36293	0.681465	−	0.731850i	$-0.261343\pi$
$137$	2185.52	1.36293	0.681465	+	0.731850i	$0.261343\pi$
$138$	209.413	0.129177
$139$	−761.596	−0.464732	−0.232366	−	0.972628i	$-0.574647\pi$
$139$	−761.596	−0.464732	−0.232366	+	0.972628i	$0.574647\pi$
$140$	0	0
$141$	1179.21	0.704310
$142$	548.256	0.324004
$143$	−613.060	−0.358508
$144$	−377.901	−0.218693
$145$	0	0
$146$	−1500.06	−0.850315
$147$	0	0
$148$	384.372	0.213481
$149$	−3228.60	−1.77515	−0.887574	−	0.460665i	$-0.847611\pi$
$149$	−3228.60	−1.77515	−0.887574	+	0.460665i	$0.847611\pi$
$150$	0	0
$151$	3205.47	1.72753	0.863767	−	0.503891i	$-0.168099\pi$
$151$	3205.47	1.72753	0.863767	+	0.503891i	$0.168099\pi$
$152$	1203.80	0.642373
$153$	2078.20	1.09812
$154$	0	0
$155$	0	0
$156$	65.3491	0.0335392
$157$	237.052	0.120502	0.0602510	−	0.998183i	$-0.480810\pi$
$157$	237.052	0.120502	0.0602510	+	0.998183i	$0.480810\pi$
$158$	855.606	0.430813
$159$	708.754	0.353508
$160$	0	0
$161$	0	0
$162$	933.115	0.452546
$163$	−741.303	−0.356217	−0.178108	−	0.984011i	$-0.556998\pi$
$163$	−741.303	−0.356217	−0.178108	+	0.984011i	$0.556998\pi$
$164$	−662.674	−0.315525
$165$	0	0
$166$	1352.03	0.632156
$167$	−1723.47	−0.798601	−0.399300	−	0.916820i	$-0.630747\pi$
$167$	−1723.47	−0.798601	−0.399300	+	0.916820i	$0.630747\pi$
$168$	0	0
$169$	−2118.06	−0.964070
$170$	0	0
$171$	−3554.03	−1.58938
$172$	1246.68	0.552666
$173$	−4172.45	−1.83367	−0.916837	−	0.399261i	$-0.869267\pi$
$173$	−4172.45	−1.83367	−0.916837	+	0.399261i	$0.869267\pi$
$174$	−591.325	−0.257633
$175$	0	0
$176$	−1104.02	−0.472833
$177$	459.099	0.194960
$178$	−1317.27	−0.554684
$179$	1387.62	0.579417	0.289709	−	0.957115i	$-0.406442\pi$
$179$	1387.62	0.579417	0.289709	+	0.957115i	$0.406442\pi$
$180$	0	0
$181$	−401.725	−0.164972	−0.0824862	−	0.996592i	$-0.526286\pi$
$181$	−401.725	−0.164972	−0.0824862	+	0.996592i	$0.526286\pi$
$182$	0	0
$183$	−834.428	−0.337064
$184$	455.543	0.182517
$185$	0	0
$186$	447.247	0.176311
$187$	6071.36	2.37423
$188$	2565.19	0.995136
$189$	0	0
$190$	0	0
$191$	−3284.66	−1.24434	−0.622172	−	0.782880i	$-0.713750\pi$
$191$	−3284.66	−1.24434	−0.622172	+	0.782880i	$0.713750\pi$
$192$	117.683	0.0442346
$193$	−1553.77	−0.579496	−0.289748	−	0.957103i	$-0.593572\pi$
$193$	−1553.77	−0.579496	−0.289748	+	0.957103i	$0.593572\pi$
$194$	339.955	0.125811
$195$	0	0
$196$	0	0
$197$	3956.43	1.43088	0.715441	−	0.698673i	$-0.246226\pi$
$197$	3956.43	1.43088	0.715441	+	0.698673i	$0.246226\pi$
$198$	3259.46	1.16990
$199$	3902.76	1.39025	0.695124	−	0.718890i	$-0.255349\pi$
$199$	3902.76	1.39025	0.695124	+	0.718890i	$0.255349\pi$
$200$	0	0
$201$	1048.12	0.367803
$202$	773.741	0.269506
$203$	0	0
$204$	−647.176	−0.222115
$205$	0	0
$206$	3380.75	1.14344
$207$	−1344.92	−0.451588
$208$	142.156	0.0473883
$209$	−10382.9	−3.43637
$210$	0	0
$211$	−1140.04	−0.371961	−0.185980	−	0.982553i	$-0.559546\pi$
$211$	−1140.04	−0.371961	−0.185980	+	0.982553i	$0.559546\pi$
$212$	1541.78	0.499480
$213$	504.065	0.162150
$214$	1968.55	0.628819
$215$	0	0
$216$	−744.622	−0.234561
$217$	0	0
$218$	1775.62	0.551653
$219$	−1379.15	−0.425546
$220$	0	0
$221$	−781.763	−0.237950
$222$	353.390	0.106838
$223$	2267.03	0.680768	0.340384	−	0.940287i	$-0.389443\pi$
$223$	2267.03	0.680768	0.340384	+	0.940287i	$0.389443\pi$
$224$	0	0
$225$	0	0
$226$	−1262.77	−0.371673
$227$	−5404.26	−1.58015	−0.790074	−	0.613012i	$-0.789958\pi$
$227$	−5404.26	−1.58015	−0.790074	+	0.613012i	$0.789958\pi$
$228$	1106.77	0.321480
$229$	4865.51	1.40402	0.702012	−	0.712165i	$-0.252285\pi$
$229$	4865.51	1.40402	0.702012	+	0.712165i	$0.252285\pi$
$230$	0	0
$231$	0	0
$232$	−1286.33	−0.364016
$233$	−4980.44	−1.40034	−0.700170	−	0.713976i	$-0.746892\pi$
$233$	−4980.44	−1.40034	−0.700170	+	0.713976i	$0.746892\pi$
$234$	−419.696	−0.117249
$235$	0	0
$236$	998.695	0.275464
$237$	786.643	0.215603
$238$	0	0
$239$	3528.95	0.955098	0.477549	−	0.878605i	$-0.341525\pi$
$239$	3528.95	0.955098	0.477549	+	0.878605i	$0.341525\pi$
$240$	0	0
$241$	−860.934	−0.230115	−0.115057	−	0.993359i	$-0.536705\pi$
$241$	−860.934	−0.230115	−0.115057	+	0.993359i	$0.536705\pi$
$242$	6860.35	1.82231
$243$	3371.00	0.889917
$244$	−1815.16	−0.476245
$245$	0	0
$246$	−609.261	−0.157907
$247$	1336.93	0.344400
$248$	972.913	0.249113
$249$	1243.05	0.316367
$250$	0	0
$251$	−2359.10	−0.593246	−0.296623	−	0.954995i	$-0.595861\pi$
$251$	−2359.10	−0.593246	−0.296623	+	0.954995i	$0.595861\pi$
$252$	0	0
$253$	−3929.13	−0.976373
$254$	−1041.75	−0.257344
$255$	0	0
$256$	256.000	0.0625000
$257$	−2388.61	−0.579757	−0.289878	−	0.957064i	$-0.593615\pi$
$257$	−2388.61	−0.579757	−0.289878	+	0.957064i	$0.593615\pi$
$258$	1146.20	0.276586
$259$	0	0
$260$	0	0
$261$	3797.70	0.900658
$262$	348.674	0.0822181
$263$	−4813.17	−1.12849	−0.564244	−	0.825608i	$-0.690832\pi$
$263$	−4813.17	−1.12849	−0.564244	+	0.825608i	$0.690832\pi$
$264$	−1015.03	−0.236633
$265$	0	0
$266$	0	0
$267$	−1211.10	−0.277595
$268$	2280.00	0.519677
$269$	7915.05	1.79401	0.897006	−	0.442018i	$-0.145737\pi$
$269$	7915.05	1.79401	0.897006	+	0.442018i	$0.145737\pi$
$270$	0	0
$271$	3063.01	0.686586	0.343293	−	0.939228i	$-0.388458\pi$
$271$	3063.01	0.686586	0.343293	+	0.939228i	$0.388458\pi$
$272$	−1407.83	−0.313831
$273$	0	0
$274$	4371.04	0.963738
$275$	0	0
$276$	418.826	0.0913418
$277$	5054.99	1.09648	0.548240	−	0.836321i	$-0.315298\pi$
$277$	5054.99	1.09648	0.548240	+	0.836321i	$0.315298\pi$
$278$	−1523.19	−0.328615
$279$	−2872.38	−0.616362
$280$	0	0
$281$	3634.47	0.771581	0.385790	−	0.922586i	$-0.373929\pi$
$281$	3634.47	0.771581	0.385790	+	0.922586i	$0.373929\pi$
$282$	2358.43	0.498022
$283$	−8273.74	−1.73789	−0.868945	−	0.494908i	$-0.835202\pi$
$283$	−8273.74	−1.73789	−0.868945	+	0.494908i	$0.835202\pi$
$284$	1096.51	0.229106
$285$	0	0
$286$	−1226.12	−0.253504
$287$	0	0
$288$	−755.803	−0.154639
$289$	2829.08	0.575835
$290$	0	0
$291$	312.554	0.0629631
$292$	−3000.12	−0.601263
$293$	−5476.76	−1.09200	−0.545999	−	0.837786i	$-0.683850\pi$
$293$	−5476.76	−1.09200	−0.545999	+	0.837786i	$0.683850\pi$
$294$	0	0
$295$	0	0
$296$	768.743	0.150954
$297$	6422.48	1.25478
$298$	−6457.20	−1.25522
$299$	505.925	0.0978541
$300$	0	0
$301$	0	0
$302$	6410.95	1.22155
$303$	711.376	0.134876
$304$	2407.59	0.454226
$305$	0	0
$306$	4156.40	0.776488
$307$	−6893.24	−1.28149	−0.640746	−	0.767753i	$-0.721375\pi$
$307$	−6893.24	−1.28149	−0.640746	+	0.767753i	$0.721375\pi$
$308$	0	0
$309$	3108.25	0.572240
$310$	0	0
$311$	7000.66	1.27643	0.638217	−	0.769856i	$-0.279672\pi$
$311$	7000.66	1.27643	0.638217	+	0.769856i	$0.279672\pi$
$312$	130.698	0.0237158
$313$	1537.68	0.277684	0.138842	−	0.990315i	$-0.455662\pi$
$313$	1537.68	0.277684	0.138842	+	0.990315i	$0.455662\pi$
$314$	474.104	0.0852078
$315$	0	0
$316$	1711.21	0.304631
$317$	9820.96	1.74006	0.870032	−	0.492995i	$-0.164098\pi$
$317$	9820.96	1.74006	0.870032	+	0.492995i	$0.164098\pi$
$318$	1417.51	0.249968
$319$	11094.8	1.94730
$320$	0	0
$321$	1809.88	0.314697
$322$	0	0
$323$	−13240.1	−2.28080
$324$	1866.23	0.319998
$325$	0	0
$326$	−1482.61	−0.251883
$327$	1632.50	0.276078
$328$	−1325.35	−0.223110
$329$	0	0
$330$	0	0
$331$	1173.01	0.194787	0.0973933	−	0.995246i	$-0.468950\pi$
$331$	1173.01	0.194787	0.0973933	+	0.995246i	$0.468950\pi$
$332$	2704.06	0.447002
$333$	−2269.60	−0.373494
$334$	−3446.95	−0.564696
$335$	0	0
$336$	0	0
$337$	7801.45	1.26104	0.630522	−	0.776171i	$-0.282841\pi$
$337$	7801.45	1.26104	0.630522	+	0.776171i	$0.282841\pi$
$338$	−4236.12	−0.681700
$339$	−1160.99	−0.186006
$340$	0	0
$341$	−8391.53	−1.33263
$342$	−7108.06	−1.12386
$343$	0	0
$344$	2493.36	0.390794
$345$	0	0
$346$	−8344.91	−1.29660
$347$	6388.29	0.988303	0.494152	−	0.869376i	$-0.335479\pi$
$347$	6388.29	0.988303	0.494152	+	0.869376i	$0.335479\pi$
$348$	−1182.65	−0.182174
$349$	−6800.96	−1.04312	−0.521558	−	0.853216i	$-0.674649\pi$
$349$	−6800.96	−1.04312	−0.521558	+	0.853216i	$0.674649\pi$
$350$	0	0
$351$	−826.974	−0.125757
$352$	−2208.04	−0.334344
$353$	−5435.58	−0.819567	−0.409783	−	0.912183i	$-0.634396\pi$
$353$	−5435.58	−0.819567	−0.409783	+	0.912183i	$0.634396\pi$
$354$	918.198	0.137858
$355$	0	0
$356$	−2634.55	−0.392221
$357$	0	0
$358$	2775.24	0.409710
$359$	5914.56	0.869522	0.434761	−	0.900546i	$-0.356833\pi$
$359$	5914.56	0.869522	0.434761	+	0.900546i	$0.356833\pi$
$360$	0	0
$361$	15783.5	2.30114
$362$	−803.450	−0.116653
$363$	6307.39	0.911990
$364$	0	0
$365$	0	0
$366$	−1668.86	−0.238340
$367$	6135.60	0.872686	0.436343	−	0.899780i	$-0.356273\pi$
$367$	6135.60	0.872686	0.436343	+	0.899780i	$0.356273\pi$
$368$	911.087	0.129059
$369$	3912.90	0.552025
$370$	0	0
$371$	0	0
$372$	894.494	0.124670
$373$	2819.17	0.391343	0.195671	−	0.980670i	$-0.437311\pi$
$373$	2819.17	0.391343	0.195671	+	0.980670i	$0.437311\pi$
$374$	12142.7	1.67884
$375$	0	0
$376$	5130.37	0.703667
$377$	−1428.59	−0.195163
$378$	0	0
$379$	5166.94	0.700285	0.350142	−	0.936697i	$-0.386133\pi$
$379$	5166.94	0.700285	0.350142	+	0.936697i	$0.386133\pi$
$380$	0	0
$381$	−957.786	−0.128790
$382$	−6569.33	−0.879885
$383$	1066.26	0.142254	0.0711270	−	0.997467i	$-0.477340\pi$
$383$	1066.26	0.142254	0.0711270	+	0.997467i	$0.477340\pi$
$384$	235.366	0.0312786
$385$	0	0
$386$	−3107.54	−0.409766
$387$	−7361.29	−0.966913
$388$	679.911	0.0889619
$389$	9097.26	1.18573	0.592865	−	0.805302i	$-0.297997\pi$
$389$	9097.26	1.18573	0.592865	+	0.805302i	$0.297997\pi$
$390$	0	0
$391$	−5010.35	−0.648042
$392$	0	0
$393$	320.570	0.0411466
$394$	7912.86	1.01179
$395$	0	0
$396$	6518.92	0.827242
$397$	4137.44	0.523053	0.261527	−	0.965196i	$-0.415774\pi$
$397$	4137.44	0.523053	0.261527	+	0.965196i	$0.415774\pi$
$398$	7805.52	0.983054
$399$	0	0
$400$	0	0
$401$	−7172.02	−0.893151	−0.446575	−	0.894746i	$-0.647357\pi$
$401$	−7172.02	−0.893151	−0.446575	+	0.894746i	$0.647357\pi$
$402$	2096.23	0.260076
$403$	1080.51	0.133559
$404$	1547.48	0.190570
$405$	0	0
$406$	0	0
$407$	−6630.53	−0.807526
$408$	−1294.35	−0.157059
$409$	−11416.2	−1.38018	−0.690092	−	0.723721i	$-0.742430\pi$
$409$	−11416.2	−1.38018	−0.690092	+	0.723721i	$0.742430\pi$
$410$	0	0
$411$	4018.72	0.482309
$412$	6761.50	0.808531
$413$	0	0
$414$	−2689.85	−0.319321
$415$	0	0
$416$	284.313	0.0335086
$417$	−1400.42	−0.164458
$418$	−20765.8	−2.42988
$419$	−16264.2	−1.89632	−0.948161	−	0.317790i	$-0.897059\pi$
$419$	−16264.2	−1.89632	−0.948161	+	0.317790i	$0.897059\pi$
$420$	0	0
$421$	7877.69	0.911960	0.455980	−	0.889990i	$-0.349289\pi$
$421$	7877.69	0.911960	0.455980	+	0.889990i	$0.349289\pi$
$422$	−2280.08	−0.263016
$423$	−15146.7	−1.74103
$424$	3083.56	0.353186
$425$	0	0
$426$	1008.13	0.114657
$427$	0	0
$428$	3937.10	0.444642
$429$	−1127.29	−0.126868
$430$	0	0
$431$	−4770.51	−0.533150	−0.266575	−	0.963814i	$-0.585892\pi$
$431$	−4770.51	−0.533150	−0.266575	+	0.963814i	$0.585892\pi$
$432$	−1489.24	−0.165859
$433$	13725.8	1.52337	0.761686	−	0.647947i	$-0.224372\pi$
$433$	13725.8	1.52337	0.761686	+	0.647947i	$0.224372\pi$
$434$	0	0
$435$	0	0
$436$	3551.24	0.390077
$437$	8568.45	0.937951
$438$	−2758.31	−0.300906
$439$	11509.6	1.25131	0.625655	−	0.780100i	$-0.284832\pi$
$439$	11509.6	1.25131	0.625655	+	0.780100i	$0.284832\pi$
$440$	0	0
$441$	0	0
$442$	−1563.53	−0.168256
$443$	6746.38	0.723544	0.361772	−	0.932267i	$-0.382172\pi$
$443$	6746.38	0.723544	0.361772	+	0.932267i	$0.382172\pi$
$444$	706.781	0.0755458
$445$	0	0
$446$	4534.05	0.481376
$447$	−5936.73	−0.628183
$448$	0	0
$449$	−9537.33	−1.00244	−0.501219	−	0.865320i	$-0.667115\pi$
$449$	−9537.33	−1.00244	−0.501219	+	0.865320i	$0.667115\pi$
$450$	0	0
$451$	11431.3	1.19353
$452$	−2525.54	−0.262812
$453$	5894.21	0.611334
$454$	−10808.5	−1.11733
$455$	0	0
$456$	2213.53	0.227321
$457$	−10645.8	−1.08969	−0.544846	−	0.838536i	$-0.683412\pi$
$457$	−10645.8	−1.08969	−0.544846	+	0.838536i	$0.683412\pi$
$458$	9731.01	0.992796
$459$	8189.82	0.832828
$460$	0	0
$461$	−767.927	−0.0775834	−0.0387917	−	0.999247i	$-0.512351\pi$
$461$	−767.927	−0.0775834	−0.0387917	+	0.999247i	$0.512351\pi$
$462$	0	0
$463$	−5752.63	−0.577424	−0.288712	−	0.957416i	$-0.593227\pi$
$463$	−5752.63	−0.577424	−0.288712	+	0.957416i	$0.593227\pi$
$464$	−2572.66	−0.257398
$465$	0	0
$466$	−9960.87	−0.990190
$467$	10233.1	1.01399	0.506993	−	0.861950i	$-0.330757\pi$
$467$	10233.1	1.01399	0.506993	+	0.861950i	$0.330757\pi$
$468$	−839.392	−0.0829079
$469$	0	0
$470$	0	0
$471$	435.890	0.0426428
$472$	1997.39	0.194782
$473$	−21505.7	−2.09055
$474$	1573.29	0.152454
$475$	0	0
$476$	0	0
$477$	−9103.75	−0.873862
$478$	7057.89	0.675357
$479$	561.498	0.0535606	0.0267803	−	0.999641i	$-0.491475\pi$
$479$	561.498	0.0535606	0.0267803	+	0.999641i	$0.491475\pi$
$480$	0	0
$481$	853.763	0.0809319
$482$	−1721.87	−0.162716
$483$	0	0
$484$	13720.7	1.28857
$485$	0	0
$486$	6742.00	0.629266
$487$	10026.4	0.932933	0.466466	−	0.884539i	$-0.345527\pi$
$487$	10026.4	0.932933	0.466466	+	0.884539i	$0.345527\pi$
$488$	−3630.32	−0.336756
$489$	−1363.11	−0.126057
$490$	0	0
$491$	7092.54	0.651897	0.325949	−	0.945387i	$-0.394316\pi$
$491$	7092.54	0.651897	0.325949	+	0.945387i	$0.394316\pi$
$492$	−1218.52	−0.111657
$493$	14147.9	1.29247
$494$	2673.86	0.243528
$495$	0	0
$496$	1945.83	0.176150
$497$	0	0
$498$	2486.11	0.223705
$499$	17.0592	0.00153041	0.000765207	−	1.00000i	$-0.499756\pi$
$499$	17.0592	0.00153041	0.000765207		1.00000i	$0.499756\pi$
$500$	0	0
$501$	−3169.12	−0.282606
$502$	−4718.19	−0.419488
$503$	−10954.5	−0.971044	−0.485522	−	0.874224i	$-0.661371\pi$
$503$	−10954.5	−0.971044	−0.485522	+	0.874224i	$0.661371\pi$
$504$	0	0
$505$	0	0
$506$	−7858.27	−0.690400
$507$	−3894.68	−0.341161
$508$	−2083.51	−0.181970
$509$	20554.0	1.78986	0.894930	−	0.446207i	$-0.147226\pi$
$509$	20554.0	1.78986	0.894930	+	0.446207i	$0.147226\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	−14005.8	−1.20540
$514$	−4777.22	−0.409950
$515$	0	0
$516$	2292.39	0.195575
$517$	−44250.3	−3.76427
$518$	0	0
$519$	−7672.29	−0.648894
$520$	0	0
$521$	12869.6	1.08220	0.541102	−	0.840957i	$-0.318007\pi$
$521$	12869.6	1.08220	0.541102	+	0.840957i	$0.318007\pi$
$522$	7595.40	0.636862
$523$	−11156.7	−0.932786	−0.466393	−	0.884578i	$-0.654447\pi$
$523$	−11156.7	−0.932786	−0.466393	+	0.884578i	$0.654447\pi$
$524$	697.348	0.0581370
$525$	0	0
$526$	−9626.33	−0.797962
$527$	−10700.7	−0.884498
$528$	−2030.07	−0.167325
$529$	−8924.51	−0.733501
$530$	0	0
$531$	−5897.00	−0.481936
$532$	0	0
$533$	−1471.93	−0.119618
$534$	−2422.20	−0.196290
$535$	0	0
$536$	4560.01	0.367467
$537$	2551.55	0.205042
$538$	15830.1	1.26856
$539$	0	0
$540$	0	0
$541$	8229.32	0.653985	0.326993	−	0.945027i	$-0.393965\pi$
$541$	8229.32	0.653985	0.326993	+	0.945027i	$0.393965\pi$
$542$	6126.03	0.485490
$543$	−738.691	−0.0583798
$544$	−2815.65	−0.221912
$545$	0	0
$546$	0	0
$547$	11756.3	0.918946	0.459473	−	0.888192i	$-0.348038\pi$
$547$	11756.3	0.918946	0.459473	+	0.888192i	$0.348038\pi$
$548$	8742.08	0.681465
$549$	10718.0	0.833212
$550$	0	0
$551$	−24195.0	−1.87067
$552$	837.651	0.0645884
$553$	0	0
$554$	10110.0	0.775328
$555$	0	0
$556$	−3046.38	−0.232366
$557$	−8931.05	−0.679391	−0.339695	−	0.940536i	$-0.610324\pi$
$557$	−8931.05	−0.679391	−0.339695	+	0.940536i	$0.610324\pi$
$558$	−5744.77	−0.435834
$559$	2769.12	0.209519
$560$	0	0
$561$	11164.0	0.840185
$562$	7268.94	0.545590
$563$	3444.77	0.257868	0.128934	−	0.991653i	$-0.458844\pi$
$563$	3444.77	0.257868	0.128934	+	0.991653i	$0.458844\pi$
$564$	4716.85	0.352155
$565$	0	0
$566$	−16547.5	−1.22887
$567$	0	0
$568$	2193.02	0.162002
$569$	3753.62	0.276555	0.138278	−	0.990393i	$-0.455843\pi$
$569$	3753.62	0.276555	0.138278	+	0.990393i	$0.455843\pi$
$570$	0	0
$571$	−2168.51	−0.158931	−0.0794653	−	0.996838i	$-0.525321\pi$
$571$	−2168.51	−0.158931	−0.0794653	+	0.996838i	$0.525321\pi$
$572$	−2452.24	−0.179254
$573$	−6039.82	−0.440344
$574$	0	0
$575$	0	0
$576$	−1511.61	−0.109346
$577$	−5135.98	−0.370561	−0.185280	−	0.982686i	$-0.559319\pi$
$577$	−5135.98	−0.370561	−0.185280	+	0.982686i	$0.559319\pi$
$578$	5658.15	0.407177
$579$	−2857.06	−0.205070
$580$	0	0
$581$	0	0
$582$	625.108	0.0445216
$583$	−26596.2	−1.88937
$584$	−6000.24	−0.425157
$585$	0	0
$586$	−10953.5	−0.772160
$587$	5619.63	0.395139	0.197570	−	0.980289i	$-0.436695\pi$
$587$	5619.63	0.395139	0.197570	+	0.980289i	$0.436695\pi$
$588$	0	0
$589$	18299.8	1.28019
$590$	0	0
$591$	7275.06	0.506356
$592$	1537.49	0.106740
$593$	13827.3	0.957536	0.478768	−	0.877941i	$-0.341083\pi$
$593$	13827.3	0.957536	0.478768	+	0.877941i	$0.341083\pi$
$594$	12845.0	0.887264
$595$	0	0
$596$	−12914.4	−0.887574
$597$	7176.38	0.491976
$598$	1011.85	0.0691933
$599$	10933.2	0.745771	0.372885	−	0.927877i	$-0.378368\pi$
$599$	10933.2	0.745771	0.372885	+	0.927877i	$0.378368\pi$
$600$	0	0
$601$	−7308.53	−0.496042	−0.248021	−	0.968755i	$-0.579780\pi$
$601$	−7308.53	−0.496042	−0.248021	+	0.968755i	$0.579780\pi$
$602$	0	0
$603$	−13462.8	−0.909197
$604$	12821.9	0.863767
$605$	0	0
$606$	1422.75	0.0953718
$607$	−7584.43	−0.507154	−0.253577	−	0.967315i	$-0.581607\pi$
$607$	−7584.43	−0.507154	−0.253577	+	0.967315i	$0.581607\pi$
$608$	4815.18	0.321186
$609$	0	0
$610$	0	0
$611$	5697.77	0.377262
$612$	8312.80	0.549060
$613$	3199.18	0.210789	0.105395	−	0.994430i	$-0.466389\pi$
$613$	3199.18	0.210789	0.105395	+	0.994430i	$0.466389\pi$
$614$	−13786.5	−0.906152
$615$	0	0
$616$	0	0
$617$	−740.073	−0.0482888	−0.0241444	−	0.999708i	$-0.507686\pi$
$617$	−740.073	−0.0482888	−0.0241444	+	0.999708i	$0.507686\pi$
$618$	6216.50	0.404635
$619$	−2574.39	−0.167162	−0.0835811	−	0.996501i	$-0.526636\pi$
$619$	−2574.39	−0.167162	−0.0835811	+	0.996501i	$0.526636\pi$
$620$	0	0
$621$	−5300.11	−0.342490
$622$	14001.3	0.902575
$623$	0	0
$624$	261.396	0.0167696
$625$	0	0
$626$	3075.37	0.196352
$627$	−19092.1	−1.21605
$628$	948.208	0.0602510
$629$	−8455.12	−0.535974
$630$	0	0
$631$	19747.1	1.24583	0.622916	−	0.782289i	$-0.285948\pi$
$631$	19747.1	1.24583	0.622916	+	0.782289i	$0.285948\pi$
$632$	3422.43	0.215406
$633$	−2096.30	−0.131628
$634$	19641.9	1.23041
$635$	0	0
$636$	2835.02	0.176754
$637$	0	0
$638$	22189.6	1.37695
$639$	−6474.58	−0.400830
$640$	0	0
$641$	7302.18	0.449951	0.224976	−	0.974364i	$-0.427770\pi$
$641$	7302.18	0.449951	0.224976	+	0.974364i	$0.427770\pi$
$642$	3619.76	0.222524
$643$	−18721.9	−1.14824	−0.574121	−	0.818771i	$-0.694656\pi$
$643$	−18721.9	−1.14824	−0.574121	+	0.818771i	$0.694656\pi$
$644$	0	0
$645$	0	0
$646$	−26480.2	−1.61277
$647$	28433.5	1.72772	0.863861	−	0.503730i	$-0.168039\pi$
$647$	28433.5	1.72772	0.863861	+	0.503730i	$0.168039\pi$
$648$	3732.46	0.226273
$649$	−17227.8	−1.04199
$650$	0	0
$651$	0	0
$652$	−2965.21	−0.178108
$653$	−16152.5	−0.967986	−0.483993	−	0.875072i	$-0.660814\pi$
$653$	−16152.5	−0.967986	−0.483993	+	0.875072i	$0.660814\pi$
$654$	3265.01	0.195217
$655$	0	0
$656$	−2650.70	−0.157763
$657$	17714.8	1.05194
$658$	0	0
$659$	15915.4	0.940785	0.470392	−	0.882457i	$-0.344112\pi$
$659$	15915.4	0.940785	0.470392	+	0.882457i	$0.344112\pi$
$660$	0	0
$661$	7589.60	0.446598	0.223299	−	0.974750i	$-0.428317\pi$
$661$	7589.60	0.446598	0.223299	+	0.974750i	$0.428317\pi$
$662$	2346.02	0.137735
$663$	−1437.50	−0.0842050
$664$	5408.12	0.316078
$665$	0	0
$666$	−4539.20	−0.264100
$667$	−9155.92	−0.531512
$668$	−6893.89	−0.399300
$669$	4168.60	0.240908
$670$	0	0
$671$	31312.1	1.80148
$672$	0	0
$673$	−30271.8	−1.73387	−0.866933	−	0.498425i	$-0.833912\pi$
$673$	−30271.8	−1.73387	−0.866933	+	0.498425i	$0.833912\pi$
$674$	15602.9	0.891693
$675$	0	0
$676$	−8472.24	−0.482035
$677$	−5637.68	−0.320050	−0.160025	−	0.987113i	$-0.551157\pi$
$677$	−5637.68	−0.320050	−0.160025	+	0.987113i	$0.551157\pi$
$678$	−2321.97	−0.131526
$679$	0	0
$680$	0	0
$681$	−9937.33	−0.559177
$682$	−16783.1	−0.942312
$683$	956.383	0.0535798	0.0267899	−	0.999641i	$-0.491471\pi$
$683$	956.383	0.0535798	0.0267899	+	0.999641i	$0.491471\pi$
$684$	−14216.1	−0.794688
$685$	0	0
$686$	0	0
$687$	8946.67	0.496851
$688$	4986.73	0.276333
$689$	3424.59	0.189356
$690$	0	0
$691$	−17744.6	−0.976898	−0.488449	−	0.872592i	$-0.662437\pi$
$691$	−17744.6	−0.976898	−0.488449	+	0.872592i	$0.662437\pi$
$692$	−16689.8	−0.916837
$693$	0	0
$694$	12776.6	0.698836
$695$	0	0
$696$	−2365.30	−0.128817
$697$	14577.0	0.792172
$698$	−13601.9	−0.737594
$699$	−9158.01	−0.495547
$700$	0	0
$701$	−16656.8	−0.897459	−0.448729	−	0.893668i	$-0.648123\pi$
$701$	−16656.8	−0.897459	−0.448729	+	0.893668i	$0.648123\pi$
$702$	−1653.95	−0.0889234
$703$	14459.5	0.775748
$704$	−4416.08	−0.236417
$705$	0	0
$706$	−10871.2	−0.579521
$707$	0	0
$708$	1836.40	0.0974802
$709$	7293.75	0.386350	0.193175	−	0.981164i	$-0.438121\pi$
$709$	7293.75	0.386350	0.193175	+	0.981164i	$0.438121\pi$
$710$	0	0
$711$	−10104.2	−0.532964
$712$	−5269.09	−0.277342
$713$	6925.06	0.363739
$714$	0	0
$715$	0	0
$716$	5550.48	0.289709
$717$	6489.01	0.337987
$718$	11829.1	0.614845
$719$	35830.5	1.85849	0.929244	−	0.369467i	$-0.120460\pi$
$719$	35830.5	1.85849	0.929244	+	0.369467i	$0.120460\pi$
$720$	0	0
$721$	0	0
$722$	31567.1	1.62715
$723$	−1583.08	−0.0814322
$724$	−1606.90	−0.0824862
$725$	0	0
$726$	12614.8	0.644874
$727$	31303.6	1.59696	0.798478	−	0.602024i	$-0.205639\pi$
$727$	31303.6	1.59696	0.798478	+	0.602024i	$0.205639\pi$
$728$	0	0
$729$	−6398.47	−0.325076
$730$	0	0
$731$	−27423.6	−1.38755
$732$	−3337.71	−0.168532
$733$	30654.4	1.54467	0.772337	−	0.635213i	$-0.219088\pi$
$733$	30654.4	1.54467	0.772337	+	0.635213i	$0.219088\pi$
$734$	12271.2	0.617082
$735$	0	0
$736$	1822.17	0.0912584
$737$	−39330.8	−1.96576
$738$	7825.80	0.390341
$739$	−16502.7	−0.821461	−0.410731	−	0.911757i	$-0.634726\pi$
$739$	−16502.7	−0.821461	−0.410731	+	0.911757i	$0.634726\pi$
$740$	0	0
$741$	2458.34	0.121875
$742$	0	0
$743$	17047.5	0.841738	0.420869	−	0.907121i	$-0.361725\pi$
$743$	17047.5	0.841738	0.420869	+	0.907121i	$0.361725\pi$
$744$	1788.99	0.0881553
$745$	0	0
$746$	5638.33	0.276721
$747$	−15966.7	−0.782049
$748$	24285.4	1.18712
$749$	0	0
$750$	0	0
$751$	32522.8	1.58026	0.790129	−	0.612941i	$-0.210014\pi$
$751$	32522.8	1.58026	0.790129	+	0.612941i	$0.210014\pi$
$752$	10260.7	0.497568
$753$	−4337.90	−0.209936
$754$	−2857.19	−0.138001
$755$	0	0
$756$	0	0
$757$	−39925.5	−1.91693	−0.958465	−	0.285211i	$-0.907936\pi$
$757$	−39925.5	−1.91693	−0.958465	+	0.285211i	$0.907936\pi$
$758$	10333.9	0.495176
$759$	−7224.87	−0.345516
$760$	0	0
$761$	27457.5	1.30793	0.653963	−	0.756526i	$-0.273105\pi$
$761$	27457.5	1.30793	0.653963	+	0.756526i	$0.273105\pi$
$762$	−1915.57	−0.0910680
$763$	0	0
$764$	−13138.7	−0.622172
$765$	0	0
$766$	2132.52	0.100589
$767$	2218.29	0.104430
$768$	470.732	0.0221173
$769$	27170.3	1.27411	0.637053	−	0.770820i	$-0.280153\pi$
$769$	27170.3	1.27411	0.637053	+	0.770820i	$0.280153\pi$
$770$	0	0
$771$	−4392.17	−0.205162
$772$	−6215.08	−0.289748
$773$	−17904.3	−0.833083	−0.416542	−	0.909117i	$-0.636758\pi$
$773$	−17904.3	−0.833083	−0.416542	+	0.909117i	$0.636758\pi$
$774$	−14722.6	−0.683711
$775$	0	0
$776$	1359.82	0.0629056
$777$	0	0
$778$	18194.5	0.838438
$779$	−24928.9	−1.14656
$780$	0	0
$781$	−18915.2	−0.866630
$782$	−10020.7	−0.458235
$783$	14966.1	0.683070
$784$	0	0
$785$	0	0
$786$	641.140	0.0290951
$787$	−14190.9	−0.642760	−0.321380	−	0.946950i	$-0.604147\pi$
$787$	−14190.9	−0.642760	−0.321380	+	0.946950i	$0.604147\pi$
$788$	15825.7	0.715441
$789$	−8850.43	−0.399346
$790$	0	0
$791$	0	0
$792$	13037.8	0.584949
$793$	−4031.82	−0.180548
$794$	8274.88	0.369854
$795$	0	0
$796$	15611.0	0.695124
$797$	−29777.3	−1.32342	−0.661711	−	0.749759i	$-0.730169\pi$
$797$	−29777.3	−1.32342	−0.661711	+	0.749759i	$0.730169\pi$
$798$	0	0
$799$	−56427.1	−2.49843
$800$	0	0
$801$	15556.2	0.686207
$802$	−14344.0	−0.631553
$803$	51753.1	2.27438
$804$	4192.46	0.183901
$805$	0	0
$806$	2161.03	0.0944404
$807$	14554.2	0.634859
$808$	3094.96	0.134753
$809$	−15905.3	−0.691225	−0.345612	−	0.938377i	$-0.612329\pi$
$809$	−15905.3	−0.691225	−0.345612	+	0.938377i	$0.612329\pi$
$810$	0	0
$811$	10201.6	0.441712	0.220856	−	0.975306i	$-0.429115\pi$
$811$	10201.6	0.441712	0.220856	+	0.975306i	$0.429115\pi$
$812$	0	0
$813$	5632.26	0.242967
$814$	−13261.1	−0.571007
$815$	0	0
$816$	−2588.70	−0.111057
$817$	46898.4	2.00828
$818$	−22832.4	−0.975938
$819$	0	0
$820$	0	0
$821$	21684.6	0.921800	0.460900	−	0.887452i	$-0.347527\pi$
$821$	21684.6	0.921800	0.460900	+	0.887452i	$0.347527\pi$
$822$	8037.45	0.341044
$823$	−40608.3	−1.71995	−0.859974	−	0.510337i	$-0.829521\pi$
$823$	−40608.3	−1.71995	−0.859974	+	0.510337i	$0.829521\pi$
$824$	13523.0	0.571718
$825$	0	0
$826$	0	0
$827$	−30936.3	−1.30080	−0.650400	−	0.759592i	$-0.725399\pi$
$827$	−30936.3	−1.30080	−0.650400	+	0.759592i	$0.725399\pi$
$828$	−5379.70	−0.225794
$829$	−38463.5	−1.61145	−0.805726	−	0.592288i	$-0.798225\pi$
$829$	−38463.5	−1.61145	−0.805726	+	0.592288i	$0.798225\pi$
$830$	0	0
$831$	9295.09	0.388018
$832$	568.625	0.0236942
$833$	0	0
$834$	−2800.84	−0.116289
$835$	0	0
$836$	−41531.7	−1.71819
$837$	−11319.6	−0.467457
$838$	−32528.4	−1.34090
$839$	1623.27	0.0667955	0.0333977	−	0.999442i	$-0.489367\pi$
$839$	1623.27	0.0667955	0.0333977	+	0.999442i	$0.489367\pi$
$840$	0	0
$841$	1464.84	0.0600614
$842$	15755.4	0.644853
$843$	6683.05	0.273044
$844$	−4560.17	−0.185980
$845$	0	0
$846$	−30293.4	−1.23110
$847$	0	0
$848$	6167.11	0.249740
$849$	−15213.7	−0.614998
$850$	0	0
$851$	5471.81	0.220413
$852$	2016.26	0.0810751
$853$	−46305.1	−1.85868	−0.929341	−	0.369222i	$-0.879624\pi$
$853$	−46305.1	−1.85868	−0.929341	+	0.369222i	$0.879624\pi$
$854$	0	0
$855$	0	0
$856$	7874.20	0.314410
$857$	−30937.1	−1.23313	−0.616564	−	0.787305i	$-0.711476\pi$
$857$	−30937.1	−1.23313	−0.616564	+	0.787305i	$0.711476\pi$
$858$	−2254.59	−0.0897090
$859$	36721.9	1.45860	0.729299	−	0.684195i	$-0.239846\pi$
$859$	36721.9	1.45860	0.729299	+	0.684195i	$0.239846\pi$
$860$	0	0
$861$	0	0
$862$	−9541.03	−0.376994
$863$	27945.5	1.10229	0.551145	−	0.834409i	$-0.314191\pi$
$863$	27945.5	1.10229	0.551145	+	0.834409i	$0.314191\pi$
$864$	−2978.49	−0.117280
$865$	0	0
$866$	27451.6	1.07719
$867$	5202.09	0.203774
$868$	0	0
$869$	−29519.0	−1.15232
$870$	0	0
$871$	5064.33	0.197013
$872$	7102.49	0.275826
$873$	−4014.67	−0.155643
$874$	17136.9	0.663231
$875$	0	0
$876$	−5516.61	−0.212773
$877$	18326.9	0.705650	0.352825	−	0.935689i	$-0.385221\pi$
$877$	18326.9	0.705650	0.352825	+	0.935689i	$0.385221\pi$
$878$	23019.3	0.884810
$879$	−10070.6	−0.386433
$880$	0	0
$881$	−19774.4	−0.756205	−0.378102	−	0.925764i	$-0.623423\pi$
$881$	−19774.4	−0.756205	−0.378102	+	0.925764i	$0.623423\pi$
$882$	0	0
$883$	−2629.82	−0.100227	−0.0501136	−	0.998744i	$-0.515958\pi$
$883$	−2629.82	−0.100227	−0.0501136	+	0.998744i	$0.515958\pi$
$884$	−3127.05	−0.118975
$885$	0	0
$886$	13492.8	0.511623
$887$	−41599.9	−1.57473	−0.787366	−	0.616485i	$-0.788556\pi$
$887$	−41599.9	−1.57473	−0.787366	+	0.616485i	$0.788556\pi$
$888$	1413.56	0.0534189
$889$	0	0
$890$	0	0
$891$	−32193.1	−1.21045
$892$	9068.10	0.340384
$893$	96498.7	3.61613
$894$	−11873.5	−0.444192
$895$	0	0
$896$	0	0
$897$	930.292	0.0346283
$898$	−19074.7	−0.708831
$899$	−19554.5	−0.725449
$900$	0	0
$901$	−33914.9	−1.25402
$902$	22862.7	0.843951
$903$	0	0
$904$	−5051.07	−0.185836
$905$	0	0
$906$	11788.4	0.432278
$907$	−21394.7	−0.783240	−0.391620	−	0.920127i	$-0.628085\pi$
$907$	−21394.7	−0.783240	−0.391620	+	0.920127i	$0.628085\pi$
$908$	−21617.0	−0.790074
$909$	−9137.43	−0.333410
$910$	0	0
$911$	−18422.5	−0.669995	−0.334998	−	0.942219i	$-0.608735\pi$
$911$	−18422.5	−0.669995	−0.334998	+	0.942219i	$0.608735\pi$
$912$	4427.07	0.160740
$913$	−46645.9	−1.69086
$914$	−21291.6	−0.770528
$915$	0	0
$916$	19462.0	0.702012
$917$	0	0
$918$	16379.6	0.588898
$919$	−21452.5	−0.770026	−0.385013	−	0.922911i	$-0.625803\pi$
$919$	−21452.5	−0.770026	−0.385013	+	0.922911i	$0.625803\pi$
$920$	0	0
$921$	−12675.3	−0.453490
$922$	−1535.85	−0.0548597
$923$	2435.56	0.0868554
$924$	0	0
$925$	0	0
$926$	−11505.3	−0.408300
$927$	−39924.7	−1.41456
$928$	−5145.32	−0.182008
$929$	−19985.4	−0.705812	−0.352906	−	0.935659i	$-0.614807\pi$
$929$	−19985.4	−0.705812	−0.352906	+	0.935659i	$0.614807\pi$
$930$	0	0
$931$	0	0
$932$	−19921.7	−0.700170
$933$	12872.8	0.451700
$934$	20466.2	0.716996
$935$	0	0
$936$	−1678.78	−0.0586247
$937$	32111.1	1.11956	0.559778	−	0.828643i	$-0.310887\pi$
$937$	32111.1	1.11956	0.559778	+	0.828643i	$0.310887\pi$
$938$	0	0
$939$	2827.49	0.0982658
$940$	0	0
$941$	−47875.3	−1.65855	−0.829273	−	0.558844i	$-0.811245\pi$
$941$	−47875.3	−1.65855	−0.829273	+	0.558844i	$0.811245\pi$
$942$	871.780	0.0301530
$943$	−9433.65	−0.325771
$944$	3994.78	0.137732
$945$	0	0
$946$	−43011.3	−1.47824
$947$	6035.60	0.207107	0.103554	−	0.994624i	$-0.466979\pi$
$947$	6035.60	0.207107	0.103554	+	0.994624i	$0.466979\pi$
$948$	3146.57	0.107802
$949$	−6663.85	−0.227943
$950$	0	0
$951$	18058.7	0.615768
$952$	0	0
$953$	−14260.3	−0.484719	−0.242360	−	0.970186i	$-0.577921\pi$
$953$	−14260.3	−0.484719	−0.242360	+	0.970186i	$0.577921\pi$
$954$	−18207.5	−0.617914
$955$	0	0
$956$	14115.8	0.477549
$957$	20401.1	0.689105
$958$	1123.00	0.0378730
$959$	0	0
$960$	0	0
$961$	−15001.0	−0.503541
$962$	1707.53	0.0572275
$963$	−23247.4	−0.777921
$964$	−3443.74	−0.115057
$965$	0	0
$966$	0	0
$967$	9534.48	0.317072	0.158536	−	0.987353i	$-0.449323\pi$
$967$	9534.48	0.317072	0.158536	+	0.987353i	$0.449323\pi$
$968$	27441.4	0.911157
$969$	−24345.8	−0.807122
$970$	0	0
$971$	−10365.4	−0.342576	−0.171288	−	0.985221i	$-0.554793\pi$
$971$	−10365.4	−0.342576	−0.171288	+	0.985221i	$0.554793\pi$
$972$	13484.0	0.444959
$973$	0	0
$974$	20052.7	0.659683
$975$	0	0
$976$	−7260.65	−0.238123
$977$	−1885.14	−0.0617309	−0.0308655	−	0.999524i	$-0.509826\pi$
$977$	−1885.14	−0.0617309	−0.0308655	+	0.999524i	$0.509826\pi$
$978$	−2726.21	−0.0891356
$979$	45446.7	1.48364
$980$	0	0
$981$	−20969.1	−0.682457
$982$	14185.1	0.460961
$983$	−9556.93	−0.310090	−0.155045	−	0.987907i	$-0.549552\pi$
$983$	−9556.93	−0.310090	−0.155045	+	0.987907i	$0.549552\pi$
$984$	−2437.05	−0.0789534
$985$	0	0
$986$	28295.8	0.913915
$987$	0	0
$988$	5347.72	0.172200
$989$	17747.4	0.570612
$990$	0	0
$991$	30384.8	0.973970	0.486985	−	0.873410i	$-0.338097\pi$
$991$	30384.8	0.973970	0.486985	+	0.873410i	$0.338097\pi$
$992$	3891.65	0.124557
$993$	2156.92	0.0689304
$994$	0	0
$995$	0	0
$996$	4972.21	0.158183
$997$	−43843.4	−1.39271	−0.696357	−	0.717696i	$-0.745197\pi$
$997$	−43843.4	−1.39271	−0.696357	+	0.717696i	$0.745197\pi$
$998$	34.1185	0.00108217
$999$	−8944.10	−0.283262

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.da.1.6	yes	8
5.4	even	2		2450.4.a.cz.1.3	✓	8
7.6	odd	2	inner	2450.4.a.da.1.3	yes	8
35.34	odd	2		2450.4.a.cz.1.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2450.4.a.cz.1.3	✓	8	5.4	even	2
2450.4.a.cz.1.6	yes	8	35.34	odd	2
2450.4.a.da.1.3	yes	8	7.6	odd	2	inner
2450.4.a.da.1.6	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +1.83880 q^{3} +4.00000 q^{4} +3.67759 q^{6} +8.00000 q^{8} -23.6188 q^{9} -69.0013 q^{11} +7.35518 q^{12} +8.88477 q^{13} +16.0000 q^{16} -87.9891 q^{17} -47.2377 q^{18} +150.474 q^{19} -138.003 q^{22} +56.9429 q^{23} +14.7104 q^{24} +17.7695 q^{26} -93.0777 q^{27} -160.791 q^{29} +121.614 q^{31} +32.0000 q^{32} -126.879 q^{33} -175.978 q^{34} -94.4753 q^{36} +96.0929 q^{37} +300.949 q^{38} +16.3373 q^{39} -165.669 q^{41} +311.670 q^{43} -276.005 q^{44} +113.886 q^{46} +641.297 q^{47} +29.4207 q^{48} -161.794 q^{51} +35.5391 q^{52} +385.445 q^{53} -186.155 q^{54} +276.692 q^{57} -321.583 q^{58} +249.674 q^{59} -453.791 q^{61} +243.228 q^{62} +64.0000 q^{64} -253.759 q^{66} +570.001 q^{67} -351.956 q^{68} +104.706 q^{69} +274.128 q^{71} -188.951 q^{72} -750.031 q^{73} +192.186 q^{74} +601.898 q^{76} +32.6746 q^{78} +427.803 q^{79} +466.558 q^{81} -331.337 q^{82} +676.015 q^{83} +623.341 q^{86} -295.662 q^{87} -552.010 q^{88} -658.636 q^{89} +227.772 q^{92} +223.624 q^{93} +1282.59 q^{94} +58.8415 q^{96} +169.978 q^{97} +1629.73 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{2} + 32 q^{4} + 64 q^{8} + 76 q^{9} + 36 q^{11} + 128 q^{16} + 152 q^{18} + 72 q^{22} + 164 q^{23} + 392 q^{29} + 256 q^{32} + 304 q^{36} + 32 q^{37} + 832 q^{39} + 752 q^{43} + 144 q^{44} + 328 q^{46}+ \cdots + 2892 q^{99}+O(q^{100}) \) 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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