LMFDB - Embedded newform 2401.4.a.d.1.26

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2401,4,Mod(1,2401)] 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("2401.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2401, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2401 = 7^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2401.a (trivial)

Newform invariants

comment:select newform

sage:traces = [39,1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$141.663585924$
Analytic rank:	$0$
Dimension:	$39$
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.26
Character	$\chi$	$=$	2401.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.90844 q^{2} +0.182265 q^{3} -4.35784 q^{4} +17.8670 q^{5} +0.347842 q^{6} -23.5842 q^{8} -26.9668 q^{9} +34.0981 q^{10} -20.0706 q^{11} -0.794282 q^{12} -44.6450 q^{13} +3.25652 q^{15} -10.1464 q^{16} -10.1793 q^{17} -51.4646 q^{18} -51.5845 q^{19} -77.8615 q^{20} -38.3036 q^{22} +155.516 q^{23} -4.29858 q^{24} +194.229 q^{25} -85.2024 q^{26} -9.83625 q^{27} -261.686 q^{29} +6.21489 q^{30} -211.476 q^{31} +169.310 q^{32} -3.65816 q^{33} -19.4266 q^{34} +117.517 q^{36} +327.672 q^{37} -98.4461 q^{38} -8.13721 q^{39} -421.379 q^{40} +252.419 q^{41} -19.5461 q^{43} +87.4645 q^{44} -481.815 q^{45} +296.793 q^{46} +7.94782 q^{47} -1.84934 q^{48} +370.675 q^{50} -1.85533 q^{51} +194.556 q^{52} +259.621 q^{53} -18.7719 q^{54} -358.601 q^{55} -9.40204 q^{57} -499.413 q^{58} +888.219 q^{59} -14.1914 q^{60} +198.923 q^{61} -403.590 q^{62} +404.290 q^{64} -797.671 q^{65} -6.98139 q^{66} +454.487 q^{67} +44.3598 q^{68} +28.3451 q^{69} +10.6318 q^{71} +635.991 q^{72} +29.9151 q^{73} +625.343 q^{74} +35.4011 q^{75} +224.797 q^{76} -15.5294 q^{78} +906.648 q^{79} -181.286 q^{80} +726.310 q^{81} +481.727 q^{82} +117.480 q^{83} -181.874 q^{85} -37.3026 q^{86} -47.6961 q^{87} +473.349 q^{88} +210.166 q^{89} -919.516 q^{90} -677.714 q^{92} -38.5446 q^{93} +15.1680 q^{94} -921.659 q^{95} +30.8593 q^{96} +1208.12 q^{97} +541.239 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 39 q + q^{2} + q^{3} + 145 q^{4} + 27 q^{5} + 41 q^{6} - 12 q^{8} + 312 q^{9} + 78 q^{10} + q^{11} - 91 q^{12} + 77 q^{13} - 161 q^{15} + 461 q^{16} + 211 q^{17} + 8 q^{18} + 314 q^{19} + 476 q^{20}+ \cdots + 5384 q^{99}+O(q^{100}) $ 39 * q + q^2 + q^3 + 145 * q^4 + 27 * q^5 + 41 * q^6 - 12 * q^8 + 312 * q^9 + 78 * q^10 + q^11 - 91 * q^12 + 77 * q^13 - 161 * q^15 + 461 * q^16 + 211 * q^17 + 8 * q^18 + 314 * q^19 + 476 * q^20 - 61 * q^22 - 69 * q^23 + 330 * q^24 + 606 * q^25 + 504 * q^26 - 50 * q^27 + 57 * q^29 + 42 * q^30 + 638 * q^31 - 1600 * q^32 + 1574 * q^33 + 1343 * q^34 + 782 * q^36 + 71 * q^37 + 1359 * q^38 - 84 * q^39 - 155 * q^40 + 1393 * q^41 - 125 * q^43 + 52 * q^44 + 1129 * q^45 - 1454 * q^46 + 1483 * q^47 - 974 * q^48 + 3074 * q^50 - 2044 * q^51 + 3899 * q^52 + 2213 * q^53 + 1142 * q^54 + 1604 * q^55 + 98 * q^57 + 2403 * q^58 + 2073 * q^59 - 1519 * q^60 + 2575 * q^61 + 1742 * q^62 + 1358 * q^64 + 1876 * q^65 - 48 * q^66 + 176 * q^67 + 3038 * q^68 - 638 * q^69 - 1259 * q^71 - 3799 * q^72 - 307 * q^73 + 3845 * q^74 + 131 * q^75 + 1974 * q^76 - 6041 * q^78 + 22 * q^79 - 804 * q^80 + 795 * q^81 + 8043 * q^82 + 6349 * q^83 + 3094 * q^85 + 1745 * q^86 + 9508 * q^87 + 1299 * q^88 + 2253 * q^89 + 11156 * q^90 + 1284 * q^92 - 3430 * q^93 + 2738 * q^94 - 3290 * q^95 + 3031 * q^96 - 770 * q^97 + 5384 * 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 $

$2$	1.90844	0.674737	0.337368	−	0.941373i	$-0.390463\pi$
$2$	1.90844	0.674737	0.337368	+	0.941373i	$0.390463\pi$
$3$	0.182265	0.0350769	0.0175384	−	0.999846i	$-0.494417\pi$
$3$	0.182265	0.0350769	0.0175384	+	0.999846i	$0.494417\pi$
$4$	−4.35784	−0.544730
$5$	17.8670	1.59807	0.799035	−	0.601284i	$-0.205344\pi$
$5$	17.8670	1.59807	0.799035	+	0.601284i	$0.205344\pi$
$6$	0.347842	0.0236677
$7$	0	0
$7$	0	0
$8$	−23.5842	−1.04229
$9$	−26.9668	−0.998770
$10$	34.0981	1.07828
$11$	−20.0706	−0.550137	−0.275069	−	0.961425i	$-0.588701\pi$
$11$	−20.0706	−0.550137	−0.275069	+	0.961425i	$0.588701\pi$
$12$	−0.794282	−0.0191074
$13$	−44.6450	−0.952484	−0.476242	−	0.879314i	$-0.658001\pi$
$13$	−44.6450	−0.952484	−0.476242	+	0.879314i	$0.658001\pi$
$14$	0	0
$15$	3.25652	0.0560554
$16$	−10.1464	−0.158538
$17$	−10.1793	−0.145226	−0.0726131	−	0.997360i	$-0.523134\pi$
$17$	−10.1793	−0.145226	−0.0726131	+	0.997360i	$0.523134\pi$
$18$	−51.4646	−0.673906
$19$	−51.5845	−0.622858	−0.311429	−	0.950269i	$-0.600808\pi$
$19$	−51.5845	−0.622858	−0.311429	+	0.950269i	$0.600808\pi$
$20$	−77.8615	−0.870518
$21$	0	0
$22$	−38.3036	−0.371198
$23$	155.516	1.40988	0.704942	−	0.709265i	$-0.250973\pi$
$23$	155.516	1.40988	0.704942	+	0.709265i	$0.250973\pi$
$24$	−4.29858	−0.0365602
$25$	194.229	1.55383
$26$	−85.2024	−0.642676
$27$	−9.83625	−0.0701106
$28$	0	0
$29$	−261.686	−1.67565	−0.837825	−	0.545939i	$-0.816173\pi$
$29$	−261.686	−1.67565	−0.837825	+	0.545939i	$0.816173\pi$
$30$	6.21489	0.0378226
$31$	−211.476	−1.22523	−0.612616	−	0.790381i	$-0.709883\pi$
$31$	−211.476	−1.22523	−0.612616	+	0.790381i	$0.709883\pi$
$32$	169.310	0.935315
$33$	−3.65816	−0.0192971
$34$	−19.4266	−0.0979894
$35$	0	0
$36$	117.517	0.544060
$37$	327.672	1.45592	0.727959	−	0.685621i	$-0.240469\pi$
$37$	327.672	1.45592	0.727959	+	0.685621i	$0.240469\pi$
$38$	−98.4461	−0.420265
$39$	−8.13721	−0.0334102
$40$	−421.379	−1.66565
$41$	252.419	0.961493	0.480747	−	0.876860i	$-0.340366\pi$
$41$	252.419	0.961493	0.480747	+	0.876860i	$0.340366\pi$
$42$	0	0
$43$	−19.5461	−0.0693197	−0.0346598	−	0.999399i	$-0.511035\pi$
$43$	−19.5461	−0.0693197	−0.0346598	+	0.999399i	$0.511035\pi$
$44$	87.4645	0.299676
$45$	−481.815	−1.59610
$46$	296.793	0.951300
$47$	7.94782	0.0246662	0.0123331	−	0.999924i	$-0.496074\pi$
$47$	7.94782	0.0246662	0.0123331	+	0.999924i	$0.496074\pi$
$48$	−1.84934	−0.00556103
$49$	0	0
$50$	370.675	1.04843
$51$	−1.85533	−0.00509408
$52$	194.556	0.518847
$53$	259.621	0.672862	0.336431	−	0.941708i	$-0.390780\pi$
$53$	259.621	0.672862	0.336431	+	0.941708i	$0.390780\pi$
$54$	−18.7719	−0.0473062
$55$	−358.601	−0.879158
$56$	0	0
$57$	−9.40204	−0.0218479
$58$	−499.413	−1.13062
$59$	888.219	1.95994	0.979968	−	0.199154i	$-0.0638195\pi$
$59$	888.219	1.95994	0.979968	+	0.199154i	$0.0638195\pi$
$60$	−14.1914	−0.0305351
$61$	198.923	0.417533	0.208767	−	0.977965i	$-0.433055\pi$
$61$	198.923	0.417533	0.208767	+	0.977965i	$0.433055\pi$
$62$	−403.590	−0.826709
$63$	0	0
$64$	404.290	0.789629
$65$	−797.671	−1.52214
$66$	−6.98139	−0.0130205
$67$	454.487	0.828723	0.414362	−	0.910112i	$-0.364005\pi$
$67$	454.487	0.828723	0.414362	+	0.910112i	$0.364005\pi$
$68$	44.3598	0.0791091
$69$	28.3451	0.0494543
$70$	0	0
$71$	10.6318	0.0177712	0.00888561	−	0.999961i	$-0.497172\pi$
$71$	10.6318	0.0177712	0.00888561	+	0.999961i	$0.497172\pi$
$72$	635.991	1.04100
$73$	29.9151	0.0479631	0.0239815	−	0.999712i	$-0.492366\pi$
$73$	29.9151	0.0479631	0.0239815	+	0.999712i	$0.492366\pi$
$74$	625.343	0.982361
$75$	35.4011	0.0545035
$76$	224.797	0.339290
$77$	0	0
$78$	−15.5294	−0.0225431
$79$	906.648	1.29121	0.645607	−	0.763670i	$-0.276604\pi$
$79$	906.648	1.29121	0.645607	+	0.763670i	$0.276604\pi$
$80$	−181.286	−0.253355
$81$	726.310	0.996310
$82$	481.727	0.648755
$83$	117.480	0.155363	0.0776814	−	0.996978i	$-0.475248\pi$
$83$	117.480	0.155363	0.0776814	+	0.996978i	$0.475248\pi$
$84$	0	0
$85$	−181.874	−0.232082
$86$	−37.3026	−0.0467725
$87$	−47.6961	−0.0587766
$88$	473.349	0.573400
$89$	210.166	0.250310	0.125155	−	0.992137i	$-0.460057\pi$
$89$	210.166	0.250310	0.125155	+	0.992137i	$0.460057\pi$
$90$	−919.516	−1.07695
$91$	0	0
$92$	−677.714	−0.768007
$93$	−38.5446	−0.0429773
$94$	15.1680	0.0166432
$95$	−921.659	−0.995371
$96$	30.8593	0.0328079
$97$	1208.12	1.26460	0.632298	−	0.774726i	$-0.282112\pi$
$97$	1208.12	1.26460	0.632298	+	0.774726i	$0.282112\pi$
$98$	0	0
$99$	541.239	0.549460
$100$	−846.419	−0.846419
$101$	1696.00	1.67087	0.835437	−	0.549586i	$-0.185214\pi$
$101$	1696.00	1.67087	0.835437	+	0.549586i	$0.185214\pi$
$102$	−3.54079	−0.00343716
$103$	−1820.90	−1.74192	−0.870962	−	0.491350i	$-0.836504\pi$
$103$	−1820.90	−1.74192	−0.870962	+	0.491350i	$0.836504\pi$
$104$	1052.92	0.992761
$105$	0	0
$106$	495.472	0.454005
$107$	−767.919	−0.693809	−0.346904	−	0.937901i	$-0.612767\pi$
$107$	−767.919	−0.693809	−0.346904	+	0.937901i	$0.612767\pi$
$108$	42.8648	0.0381914
$109$	485.788	0.426882	0.213441	−	0.976956i	$-0.431533\pi$
$109$	485.788	0.426882	0.213441	+	0.976956i	$0.431533\pi$
$110$	−684.369	−0.593200
$111$	59.7231	0.0510690
$112$	0	0
$113$	−1018.84	−0.848179	−0.424090	−	0.905620i	$-0.639406\pi$
$113$	−1018.84	−0.848179	−0.424090	+	0.905620i	$0.639406\pi$
$114$	−17.9433	−0.0147416
$115$	2778.60	2.25309
$116$	1140.39	0.912777
$117$	1203.93	0.951312
$118$	1695.12	1.32244
$119$	0	0
$120$	−76.8026	−0.0584257
$121$	−928.172	−0.697349
$122$	379.634	0.281725
$123$	46.0071	0.0337262
$124$	921.579	0.667421
$125$	1236.91	0.885061
$126$	0	0
$127$	1278.17	0.893063	0.446531	−	0.894768i	$-0.352659\pi$
$127$	1278.17	0.893063	0.446531	+	0.894768i	$0.352659\pi$
$128$	−582.915	−0.402523
$129$	−3.56256	−0.00243152
$130$	−1522.31	−1.02704
$131$	−1496.57	−0.998135	−0.499068	−	0.866563i	$-0.666324\pi$
$131$	−1496.57	−0.998135	−0.499068	+	0.866563i	$0.666324\pi$
$132$	15.9417	0.0105117
$133$	0	0
$134$	867.363	0.559170
$135$	−175.744	−0.112042
$136$	240.071	0.151367
$137$	−1260.82	−0.786273	−0.393137	−	0.919480i	$-0.628610\pi$
$137$	−1260.82	−0.786273	−0.393137	+	0.919480i	$0.628610\pi$
$138$	54.0950	0.0333686
$139$	2070.94	1.26370	0.631851	−	0.775090i	$-0.282295\pi$
$139$	2070.94	1.26370	0.631851	+	0.775090i	$0.282295\pi$
$140$	0	0
$141$	1.44861	0.000865212 0
$142$	20.2901	0.0119909
$143$	896.051	0.523997
$144$	273.617	0.158343
$145$	−4675.53	−2.67781
$146$	57.0914	0.0323624
$147$	0	0
$148$	−1427.94	−0.793082
$149$	1517.48	0.834340	0.417170	−	0.908829i	$-0.363022\pi$
$149$	1517.48	0.834340	0.417170	+	0.908829i	$0.363022\pi$
$150$	67.5610	0.0367755
$151$	−1546.35	−0.833379	−0.416690	−	0.909049i	$-0.636810\pi$
$151$	−1546.35	−0.833379	−0.416690	+	0.909049i	$0.636810\pi$
$152$	1216.58	0.649196
$153$	274.503	0.145048
$154$	0	0
$155$	−3778.43	−1.95801
$156$	35.4607	0.0181995
$157$	−2999.90	−1.52495	−0.762477	−	0.647015i	$-0.776017\pi$
$157$	−2999.90	−1.52495	−0.762477	+	0.647015i	$0.776017\pi$
$158$	1730.29	0.871229
$159$	47.3198	0.0236019
$160$	3025.06	1.49470
$161$	0	0
$162$	1386.12	0.672247
$163$	2322.92	1.11623	0.558114	−	0.829764i	$-0.311525\pi$
$163$	2322.92	1.11623	0.558114	+	0.829764i	$0.311525\pi$
$164$	−1100.00	−0.523755
$165$	−65.3603	−0.0308381
$166$	224.204	0.104829
$167$	−321.875	−0.149146	−0.0745731	−	0.997216i	$-0.523759\pi$
$167$	−321.875	−0.149146	−0.0745731	+	0.997216i	$0.523759\pi$
$168$	0	0
$169$	−203.825	−0.0927742
$170$	−347.095	−0.156594
$171$	1391.07	0.622092
$172$	85.1787	0.0377605
$173$	2374.02	1.04331	0.521657	−	0.853155i	$-0.325314\pi$
$173$	2374.02	1.04331	0.521657	+	0.853155i	$0.325314\pi$
$174$	−91.0253	−0.0396587
$175$	0	0
$176$	203.645	0.0872178
$177$	161.891	0.0687485
$178$	401.091	0.168893
$179$	2078.40	0.867861	0.433931	−	0.900946i	$-0.357126\pi$
$179$	2078.40	0.867861	0.433931	+	0.900946i	$0.357126\pi$
$180$	2099.67	0.869447
$181$	−1212.22	−0.497811	−0.248905	−	0.968528i	$-0.580071\pi$
$181$	−1212.22	−0.497811	−0.248905	+	0.968528i	$0.580071\pi$
$182$	0	0
$183$	36.2567	0.0146458
$184$	−3667.73	−1.46950
$185$	5854.51	2.32666
$186$	−73.5602	−0.0289984
$187$	204.305	0.0798943
$188$	−34.6354	−0.0134364
$189$	0	0
$190$	−1758.93	−0.671613
$191$	1098.98	0.416331	0.208165	−	0.978094i	$-0.433251\pi$
$191$	1098.98	0.416331	0.208165	+	0.978094i	$0.433251\pi$
$192$	73.6879	0.0276977
$193$	−3659.94	−1.36502	−0.682508	−	0.730878i	$-0.739111\pi$
$193$	−3659.94	−1.36502	−0.682508	+	0.730878i	$0.739111\pi$
$194$	2305.62	0.853269
$195$	−145.387	−0.0533918
$196$	0	0
$197$	3701.09	1.33854	0.669269	−	0.743020i	$-0.266607\pi$
$197$	3701.09	1.33854	0.669269	+	0.743020i	$0.266607\pi$
$198$	1032.92	0.370741
$199$	684.085	0.243686	0.121843	−	0.992549i	$-0.461120\pi$
$199$	684.085	0.243686	0.121843	+	0.992549i	$0.461120\pi$
$200$	−4580.74	−1.61954
$201$	82.8370	0.0290690
$202$	3236.72	1.12740
$203$	0	0
$204$	8.08524	0.00277490
$205$	4509.96	1.53653
$206$	−3475.08	−1.17534
$207$	−4193.77	−1.40815
$208$	452.988	0.151005
$209$	1035.33	0.342657
$210$	0	0
$211$	1935.09	0.631359	0.315680	−	0.948866i	$-0.397768\pi$
$211$	1935.09	0.631359	0.315680	+	0.948866i	$0.397768\pi$
$212$	−1131.39	−0.366528
$213$	1.93780	0.000623359 0
$214$	−1465.53	−0.468138
$215$	−349.229	−0.110778
$216$	231.980	0.0730753
$217$	0	0
$218$	927.100	0.288033
$219$	5.45248	0.00168239
$220$	1562.73	0.478904
$221$	454.455	0.138326
$222$	113.978	0.0344582
$223$	428.965	0.128814	0.0644072	−	0.997924i	$-0.479484\pi$
$223$	428.965	0.128814	0.0644072	+	0.997924i	$0.479484\pi$
$224$	0	0
$225$	−5237.73	−1.55192
$226$	−1944.40	−0.572298
$227$	833.226	0.243626	0.121813	−	0.992553i	$-0.461129\pi$
$227$	833.226	0.243626	0.121813	+	0.992553i	$0.461129\pi$
$228$	40.9726	0.0119012
$229$	−1586.46	−0.457800	−0.228900	−	0.973450i	$-0.573513\pi$
$229$	−1586.46	−0.457800	−0.228900	+	0.973450i	$0.573513\pi$
$230$	5302.80	1.52025
$231$	0	0
$232$	6171.66	1.74651
$233$	2281.38	0.641451	0.320725	−	0.947172i	$-0.396073\pi$
$233$	2281.38	0.641451	0.320725	+	0.947172i	$0.396073\pi$
$234$	2297.64	0.641885
$235$	142.004	0.0394183
$236$	−3870.72	−1.06764
$237$	165.250	0.0452918
$238$	0	0
$239$	−3817.19	−1.03311	−0.516555	−	0.856254i	$-0.672786\pi$
$239$	−3817.19	−1.03311	−0.516555	+	0.856254i	$0.672786\pi$
$240$	−33.0421	−0.00888692
$241$	−4519.57	−1.20801	−0.604006	−	0.796980i	$-0.706430\pi$
$241$	−4519.57	−1.20801	−0.604006	+	0.796980i	$0.706430\pi$
$242$	−1771.36	−0.470527
$243$	397.959	0.105058
$244$	−866.877	−0.227443
$245$	0	0
$246$	87.8020	0.0227563
$247$	2302.99	0.593262
$248$	4987.50	1.27704
$249$	21.4125	0.00544965
$250$	2360.57	0.597183
$251$	−3152.41	−0.792743	−0.396371	−	0.918090i	$-0.629731\pi$
$251$	−3152.41	−0.792743	−0.396371	+	0.918090i	$0.629731\pi$
$252$	0	0
$253$	−3121.30	−0.775629
$254$	2439.31	0.602582
$255$	−33.1491	−0.00814071
$256$	−4346.78	−1.06123
$257$	2980.85	0.723502	0.361751	−	0.932275i	$-0.382179\pi$
$257$	2980.85	0.723502	0.361751	+	0.932275i	$0.382179\pi$
$258$	−6.79894	−0.00164063
$259$	0	0
$260$	3476.13	0.829154
$261$	7056.82	1.67359
$262$	−2856.11	−0.673478
$263$	1107.38	0.259635	0.129818	−	0.991538i	$-0.458561\pi$
$263$	1107.38	0.259635	0.129818	+	0.991538i	$0.458561\pi$
$264$	86.2750	0.0201131
$265$	4638.64	1.07528
$266$	0	0
$267$	38.3059	0.00878009
$268$	−1980.58	−0.451431
$269$	−236.514	−0.0536079	−0.0268039	−	0.999641i	$-0.508533\pi$
$269$	−236.514	−0.0536079	−0.0268039	+	0.999641i	$0.508533\pi$
$270$	−335.397	−0.0755987
$271$	−3530.34	−0.791338	−0.395669	−	0.918393i	$-0.629487\pi$
$271$	−3530.34	−0.791338	−0.395669	+	0.918393i	$0.629487\pi$
$272$	103.284	0.0230239
$273$	0	0
$274$	−2406.21	−0.530527
$275$	−3898.29	−0.854820
$276$	−123.524	−0.0269393
$277$	−2805.66	−0.608577	−0.304289	−	0.952580i	$-0.598419\pi$
$277$	−2805.66	−0.608577	−0.304289	+	0.952580i	$0.598419\pi$
$278$	3952.27	0.852666
$279$	5702.82	1.22372
$280$	0	0
$281$	241.742	0.0513207	0.0256604	−	0.999671i	$-0.491831\pi$
$281$	241.742	0.0513207	0.0256604	+	0.999671i	$0.491831\pi$
$282$	2.76459	0.000583790 0
$283$	8390.53	1.76242	0.881211	−	0.472723i	$-0.156729\pi$
$283$	8390.53	1.76242	0.881211	+	0.472723i	$0.156729\pi$
$284$	−46.3315	−0.00968053
$285$	−167.986	−0.0349145
$286$	1710.06	0.353560
$287$	0	0
$288$	−4565.75	−0.934164
$289$	−4809.38	−0.978909
$290$	−8922.99	−1.80681
$291$	220.197	0.0443581
$292$	−130.366	−0.0261269
$293$	4575.19	0.912237	0.456118	−	0.889919i	$-0.349239\pi$
$293$	4575.19	0.912237	0.456118	+	0.889919i	$0.349239\pi$
$294$	0	0
$295$	15869.8	3.13212
$296$	−7727.89	−1.51748
$297$	197.419	0.0385704
$298$	2896.02	0.562960
$299$	−6943.01	−1.34289
$300$	−154.272	−0.0296897
$301$	0	0
$302$	−2951.12	−0.562312
$303$	309.121	0.0586091
$304$	523.400	0.0987468
$305$	3554.16	0.667248
$306$	523.874	0.0978689
$307$	32.4752	0.00603732	0.00301866	−	0.999995i	$-0.499039\pi$
$307$	32.4752	0.00603732	0.00301866	+	0.999995i	$0.499039\pi$
$308$	0	0
$309$	−331.885	−0.0611013
$310$	−7210.93	−1.32114
$311$	−3559.86	−0.649070	−0.324535	−	0.945874i	$-0.605208\pi$
$311$	−3559.86	−0.649070	−0.324535	+	0.945874i	$0.605208\pi$
$312$	191.910	0.0348230
$313$	7720.29	1.39417	0.697087	−	0.716986i	$-0.254479\pi$
$313$	7720.29	1.39417	0.697087	+	0.716986i	$0.254479\pi$
$314$	−5725.13	−1.02894
$315$	0	0
$316$	−3951.03	−0.703363
$317$	1112.95	0.197191	0.0985956	−	0.995128i	$-0.468565\pi$
$317$	1112.95	0.197191	0.0985956	+	0.995128i	$0.468565\pi$
$318$	90.3071	0.0159251
$319$	5252.19	0.921837
$320$	7223.44	1.26188
$321$	−139.965	−0.0243366
$322$	0	0
$323$	525.095	0.0904553
$324$	−3165.15	−0.542721
$325$	−8671.34	−1.48000
$326$	4433.16	0.753160
$327$	88.5422	0.0149737
$328$	−5953.11	−1.00215
$329$	0	0
$330$	−124.736	−0.0208076
$331$	5026.82	0.834740	0.417370	−	0.908737i	$-0.362952\pi$
$331$	5026.82	0.834740	0.417370	+	0.908737i	$0.362952\pi$
$332$	−511.960	−0.0846309
$333$	−8836.26	−1.45413
$334$	−614.280	−0.100634
$335$	8120.31	1.32436
$336$	0	0
$337$	−2717.78	−0.439309	−0.219654	−	0.975578i	$-0.570493\pi$
$337$	−2717.78	−0.439309	−0.219654	+	0.975578i	$0.570493\pi$
$338$	−388.988	−0.0625981
$339$	−185.698	−0.0297515
$340$	792.576	0.126422
$341$	4244.44	0.674045
$342$	2654.77	0.419748
$343$	0	0
$344$	460.979	0.0722510
$345$	506.441	0.0790315
$346$	4530.68	0.703962
$347$	6582.52	1.01835	0.509176	−	0.860662i	$-0.329950\pi$
$347$	6582.52	1.01835	0.509176	+	0.860662i	$0.329950\pi$
$348$	207.852	0.0320174
$349$	9824.88	1.50692	0.753458	−	0.657496i	$-0.228384\pi$
$349$	9824.88	1.50692	0.753458	+	0.657496i	$0.228384\pi$
$350$	0	0
$351$	439.139	0.0667792
$352$	−3398.15	−0.514551
$353$	−1550.49	−0.233780	−0.116890	−	0.993145i	$-0.537292\pi$
$353$	−1550.49	−0.233780	−0.116890	+	0.993145i	$0.537292\pi$
$354$	308.960	0.0463871
$355$	189.957	0.0283997
$356$	−915.872	−0.136351
$357$	0	0
$358$	3966.52	0.585578
$359$	−7338.36	−1.07884	−0.539420	−	0.842037i	$-0.681357\pi$
$359$	−7338.36	−1.07884	−0.539420	+	0.842037i	$0.681357\pi$
$360$	11363.2	1.66360
$361$	−4198.04	−0.612048
$362$	−2313.46	−0.335891
$363$	−169.173	−0.0244608
$364$	0	0
$365$	534.493	0.0766484
$366$	69.1939	0.00988204
$367$	3272.49	0.465457	0.232728	−	0.972542i	$-0.425235\pi$
$367$	3272.49	0.465457	0.232728	+	0.972542i	$0.425235\pi$
$368$	−1577.93	−0.223520
$369$	−6806.93	−0.960310
$370$	11173.0	1.56988
$371$	0	0
$372$	167.971	0.0234111
$373$	11240.2	1.56032	0.780158	−	0.625583i	$-0.215139\pi$
$373$	11240.2	1.56032	0.780158	+	0.625583i	$0.215139\pi$
$374$	389.904	0.0539076
$375$	225.445	0.0310452
$376$	−187.443	−0.0257092
$377$	11683.0	1.59603
$378$	0	0
$379$	7088.85	0.960764	0.480382	−	0.877059i	$-0.340498\pi$
$379$	7088.85	0.960764	0.480382	+	0.877059i	$0.340498\pi$
$380$	4016.45	0.542209
$381$	232.965	0.0313259
$382$	2097.34	0.280914
$383$	12249.2	1.63422	0.817111	−	0.576480i	$-0.195574\pi$
$383$	12249.2	1.63422	0.817111	+	0.576480i	$0.195574\pi$
$384$	−106.245	−0.0141192
$385$	0	0
$386$	−6984.78	−0.921026
$387$	527.094	0.0692344
$388$	−5264.79	−0.688863
$389$	11772.9	1.53448	0.767239	−	0.641361i	$-0.221630\pi$
$389$	11772.9	1.53448	0.767239	+	0.641361i	$0.221630\pi$
$390$	−277.464	−0.0360254
$391$	−1583.05	−0.204752
$392$	0	0
$393$	−272.772	−0.0350115
$394$	7063.33	0.903161
$395$	16199.1	2.06345
$396$	−2358.63	−0.299308
$397$	1634.70	0.206659	0.103329	−	0.994647i	$-0.467050\pi$
$397$	1634.70	0.206659	0.103329	+	0.994647i	$0.467050\pi$
$398$	1305.54	0.164424
$399$	0	0
$400$	−1970.73	−0.246342
$401$	−8098.26	−1.00850	−0.504249	−	0.863558i	$-0.668231\pi$
$401$	−8098.26	−1.00850	−0.504249	+	0.863558i	$0.668231\pi$
$402$	158.090	0.0196139
$403$	9441.34	1.16701
$404$	−7390.91	−0.910176
$405$	12977.0	1.59217
$406$	0	0
$407$	−6576.57	−0.800954
$408$	43.7566	0.00530949
$409$	14477.1	1.75024	0.875120	−	0.483907i	$-0.160783\pi$
$409$	14477.1	1.75024	0.875120	+	0.483907i	$0.160783\pi$
$410$	8607.01	1.03676
$411$	−229.804	−0.0275800
$412$	7935.18	0.948879
$413$	0	0
$414$	−8003.56	−0.950130
$415$	2099.01	0.248281
$416$	−7558.85	−0.890872
$417$	377.459	0.0443267
$418$	1975.87	0.231203
$419$	2376.60	0.277100	0.138550	−	0.990355i	$-0.455756\pi$
$419$	2376.60	0.277100	0.138550	+	0.990355i	$0.455756\pi$
$420$	0	0
$421$	−522.786	−0.0605202	−0.0302601	−	0.999542i	$-0.509634\pi$
$421$	−522.786	−0.0605202	−0.0302601	+	0.999542i	$0.509634\pi$
$422$	3693.00	0.426001
$423$	−214.327	−0.0246358
$424$	−6122.97	−0.701315
$425$	−1977.12	−0.225657
$426$	3.69817	0.000420603 0
$427$	0	0
$428$	3346.47	0.377939
$429$	163.319	0.0183802
$430$	−666.484	−0.0747458
$431$	16091.9	1.79842	0.899211	−	0.437514i	$-0.144141\pi$
$431$	16091.9	1.79842	0.899211	+	0.437514i	$0.144141\pi$
$432$	99.8030	0.0111152
$433$	12073.8	1.34002	0.670009	−	0.742353i	$-0.266290\pi$
$433$	12073.8	1.34002	0.670009	+	0.742353i	$0.266290\pi$
$434$	0	0
$435$	−852.185	−0.0939291
$436$	−2116.99	−0.232535
$437$	−8022.22	−0.878157
$438$	10.4057	0.00113517
$439$	−1041.84	−0.113267	−0.0566337	−	0.998395i	$-0.518037\pi$
$439$	−1041.84	−0.113267	−0.0566337	+	0.998395i	$0.518037\pi$
$440$	8457.32	0.916334
$441$	0	0
$442$	867.302	0.0933334
$443$	−13652.2	−1.46419	−0.732097	−	0.681201i	$-0.761458\pi$
$443$	−13652.2	−1.46419	−0.732097	+	0.681201i	$0.761458\pi$
$444$	−260.264	−0.0278189
$445$	3755.04	0.400013
$446$	818.655	0.0869158
$447$	276.583	0.0292660
$448$	0	0
$449$	−16569.6	−1.74158	−0.870790	−	0.491656i	$-0.836392\pi$
$449$	−16569.6	−1.74158	−0.870790	+	0.491656i	$0.836392\pi$
$450$	−9995.90	−1.04714
$451$	−5066.20	−0.528953
$452$	4439.94	0.462029
$453$	−281.845	−0.0292324
$454$	1590.17	0.164384
$455$	0	0
$456$	221.740	0.0227718
$457$	9887.34	1.01206	0.506029	−	0.862517i	$-0.331113\pi$
$457$	9887.34	1.01206	0.506029	+	0.862517i	$0.331113\pi$
$458$	−3027.67	−0.308894
$459$	100.126	0.0101819
$460$	−12108.7	−1.22733
$461$	16423.8	1.65929	0.829646	−	0.558290i	$-0.188542\pi$
$461$	16423.8	1.65929	0.829646	+	0.558290i	$0.188542\pi$
$462$	0	0
$463$	14439.9	1.44941	0.724705	−	0.689059i	$-0.241976\pi$
$463$	14439.9	1.44941	0.724705	+	0.689059i	$0.241976\pi$
$464$	2655.18	0.265654
$465$	−688.676	−0.0686808
$466$	4353.88	0.432810
$467$	−3098.43	−0.307019	−0.153510	−	0.988147i	$-0.549058\pi$
$467$	−3098.43	−0.307019	−0.153510	+	0.988147i	$0.549058\pi$
$468$	−5246.55	−0.518209
$469$	0	0
$470$	271.006	0.0265969
$471$	−546.776	−0.0534906
$472$	−20948.0	−2.04281
$473$	392.301	0.0381353
$474$	315.370	0.0305600
$475$	−10019.2	−0.967816
$476$	0	0
$477$	−7001.14	−0.672034
$478$	−7284.88	−0.697077
$479$	−14810.4	−1.41275	−0.706374	−	0.707838i	$-0.749670\pi$
$479$	−14810.4	−1.41275	−0.706374	+	0.707838i	$0.749670\pi$
$480$	551.362	0.0524294
$481$	−14628.9	−1.38674
$482$	−8625.34	−0.815090
$483$	0	0
$484$	4044.83	0.379867
$485$	21585.4	2.02091
$486$	759.483	0.0708865
$487$	1886.33	0.175519	0.0877597	−	0.996142i	$-0.472029\pi$
$487$	1886.33	0.175519	0.0877597	+	0.996142i	$0.472029\pi$
$488$	−4691.46	−0.435189
$489$	423.387	0.0391538
$490$	0	0
$491$	−3139.09	−0.288524	−0.144262	−	0.989540i	$-0.546081\pi$
$491$	−3139.09	−0.288524	−0.144262	+	0.989540i	$0.546081\pi$
$492$	−200.492	−0.0183717
$493$	2663.78	0.243348
$494$	4395.13	0.400296
$495$	9670.30	0.878076
$496$	2145.73	0.194246
$497$	0	0
$498$	40.8645	0.00367708
$499$	1675.22	0.150287	0.0751434	−	0.997173i	$-0.476059\pi$
$499$	1675.22	0.150287	0.0751434	+	0.997173i	$0.476059\pi$
$500$	−5390.26	−0.482120
$501$	−58.6664	−0.00523158
$502$	−6016.20	−0.534893
$503$	3419.78	0.303142	0.151571	−	0.988446i	$-0.451567\pi$
$503$	3419.78	0.303142	0.151571	+	0.988446i	$0.451567\pi$
$504$	0	0
$505$	30302.4	2.67018
$506$	−5956.82	−0.523346
$507$	−37.1501	−0.00325423
$508$	−5570.05	−0.486479
$509$	−16653.8	−1.45023	−0.725114	−	0.688629i	$-0.758213\pi$
$509$	−16653.8	−1.45023	−0.725114	+	0.688629i	$0.758213\pi$
$510$	−63.2633	−0.00549283
$511$	0	0
$512$	−3632.27	−0.313525
$513$	507.398	0.0436689
$514$	5688.78	0.488173
$515$	−32533.9	−2.78372
$516$	15.5251	0.00132452
$517$	−159.517	−0.0135698
$518$	0	0
$519$	432.700	0.0365962
$520$	18812.5	1.58650
$521$	−9695.11	−0.815260	−0.407630	−	0.913147i	$-0.633645\pi$
$521$	−9695.11	−0.815260	−0.407630	+	0.913147i	$0.633645\pi$
$522$	13467.5	1.12923
$523$	−14073.4	−1.17665	−0.588323	−	0.808626i	$-0.700212\pi$
$523$	−14073.4	−1.17665	−0.588323	+	0.808626i	$0.700212\pi$
$524$	6521.81	0.543715
$525$	0	0
$526$	2113.37	0.175185
$527$	2152.68	0.177936
$528$	37.1173	0.00305933
$529$	12018.2	0.987772
$530$	8852.59	0.725532
$531$	−23952.4	−1.95752
$532$	0	0
$533$	−11269.2	−0.915807
$534$	73.1047	0.00592425
$535$	−13720.4	−1.10876
$536$	−10718.7	−0.863767
$537$	378.820	0.0304419
$538$	−451.374	−0.0361712
$539$	0	0
$540$	765.865	0.0610325
$541$	−2644.56	−0.210163	−0.105082	−	0.994464i	$-0.533510\pi$
$541$	−2644.56	−0.210163	−0.105082	+	0.994464i	$0.533510\pi$
$542$	−6737.45	−0.533945
$543$	−220.946	−0.0174617
$544$	−1723.46	−0.135832
$545$	8679.57	0.682187
$546$	0	0
$547$	−109.838	−0.00858561	−0.00429281	−	0.999991i	$-0.501366\pi$
$547$	−109.838	−0.00858561	−0.00429281	+	0.999991i	$0.501366\pi$
$548$	5494.47	0.428307
$549$	−5364.32	−0.417020
$550$	−7439.66	−0.576778
$551$	13498.9	1.04369
$552$	−668.498	−0.0515456
$553$	0	0
$554$	−5354.45	−0.410629
$555$	1067.07	0.0816119
$556$	−9024.82	−0.688377
$557$	748.378	0.0569296	0.0284648	−	0.999595i	$-0.490938\pi$
$557$	748.378	0.0569296	0.0284648	+	0.999595i	$0.490938\pi$
$558$	10883.5	0.825692
$559$	872.634	0.0660259
$560$	0	0
$561$	37.2376	0.00280244
$562$	461.351	0.0346280
$563$	−7293.14	−0.545949	−0.272974	−	0.962021i	$-0.588007\pi$
$563$	−7293.14	−0.545949	−0.272974	+	0.962021i	$0.588007\pi$
$564$	−6.31281	−0.000471307 0
$565$	−18203.6	−1.35545
$566$	16012.9	1.18917
$567$	0	0
$568$	−250.742	−0.0185227
$569$	−9917.94	−0.730723	−0.365362	−	0.930866i	$-0.619055\pi$
$569$	−9917.94	−0.730723	−0.365362	+	0.930866i	$0.619055\pi$
$570$	−320.592	−0.0235581
$571$	−24664.3	−1.80765	−0.903824	−	0.427904i	$-0.859252\pi$
$571$	−24664.3	−1.80765	−0.903824	+	0.427904i	$0.859252\pi$
$572$	−3904.85	−0.285437
$573$	200.305	0.0146036
$574$	0	0
$575$	30205.7	2.19072
$576$	−10902.4	−0.788658
$577$	17852.1	1.28803	0.644016	−	0.765012i	$-0.277267\pi$
$577$	17852.1	1.28803	0.644016	+	0.765012i	$0.277267\pi$
$578$	−9178.43	−0.660506
$579$	−667.078	−0.0478805
$580$	20375.2	1.45868
$581$	0	0
$582$	420.234	0.0299300
$583$	−5210.74	−0.370166
$584$	−705.526	−0.0499912
$585$	21510.6	1.52026
$586$	8731.49	0.615520
$587$	14869.5	1.04554	0.522768	−	0.852475i	$-0.324899\pi$
$587$	14869.5	1.04554	0.522768	+	0.852475i	$0.324899\pi$
$588$	0	0
$589$	10908.9	0.763145
$590$	30286.6	2.11335
$591$	674.579	0.0469518
$592$	−3324.71	−0.230819
$593$	21383.3	1.48079	0.740393	−	0.672174i	$-0.234639\pi$
$593$	21383.3	1.48079	0.740393	+	0.672174i	$0.234639\pi$
$594$	376.763	0.0260249
$595$	0	0
$596$	−6612.93	−0.454490
$597$	124.685	0.00854774
$598$	−13250.3	−0.906098
$599$	−3868.64	−0.263887	−0.131944	−	0.991257i	$-0.542122\pi$
$599$	−3868.64	−0.263887	−0.131944	+	0.991257i	$0.542122\pi$
$600$	−834.908	−0.0568083
$601$	−6058.07	−0.411171	−0.205586	−	0.978639i	$-0.565910\pi$
$601$	−6058.07	−0.411171	−0.205586	+	0.978639i	$0.565910\pi$
$602$	0	0
$603$	−12256.1	−0.827703
$604$	6738.76	0.453967
$605$	−16583.6	−1.11441
$606$	589.940	0.0395457
$607$	−4622.27	−0.309081	−0.154540	−	0.987986i	$-0.549390\pi$
$607$	−4622.27	−0.309081	−0.154540	+	0.987986i	$0.549390\pi$
$608$	−8733.78	−0.582568
$609$	0	0
$610$	6782.91	0.450217
$611$	−354.830	−0.0234941
$612$	−1196.24	−0.0790118
$613$	10667.6	0.702874	0.351437	−	0.936212i	$-0.385693\pi$
$613$	10667.6	0.702874	0.351437	+	0.936212i	$0.385693\pi$
$614$	61.9771	0.00407360
$615$	822.008	0.0538968
$616$	0	0
$617$	−3376.49	−0.220312	−0.110156	−	0.993914i	$-0.535135\pi$
$617$	−3376.49	−0.220312	−0.110156	+	0.993914i	$0.535135\pi$
$618$	−633.384	−0.0412273
$619$	15534.0	1.00867	0.504333	−	0.863509i	$-0.331738\pi$
$619$	15534.0	1.00867	0.504333	+	0.863509i	$0.331738\pi$
$620$	16465.8	1.06659
$621$	−1529.69	−0.0988478
$622$	−6793.78	−0.437952
$623$	0	0
$624$	82.5638	0.00529679
$625$	−2178.76	−0.139441
$626$	14733.7	0.940701
$627$	188.704	0.0120193
$628$	13073.1	0.830689
$629$	−3335.47	−0.211437
$630$	0	0
$631$	−6676.00	−0.421185	−0.210592	−	0.977574i	$-0.567539\pi$
$631$	−6676.00	−0.421185	−0.210592	+	0.977574i	$0.567539\pi$
$632$	−21382.6	−1.34581
$633$	352.698	0.0221461
$634$	2124.01	0.133052
$635$	22837.0	1.42718
$636$	−206.212	−0.0128567
$637$	0	0
$638$	10023.5	0.621997
$639$	−286.704	−0.0177494
$640$	−10414.9	−0.643260
$641$	−7119.52	−0.438696	−0.219348	−	0.975647i	$-0.570393\pi$
$641$	−7119.52	−0.438696	−0.219348	+	0.975647i	$0.570393\pi$
$642$	−267.115	−0.0164208
$643$	−15897.5	−0.975020	−0.487510	−	0.873117i	$-0.662095\pi$
$643$	−15897.5	−0.975020	−0.487510	+	0.873117i	$0.662095\pi$
$644$	0	0
$645$	−63.6522	−0.00388574
$646$	1002.11	0.0610335
$647$	−10212.9	−0.620575	−0.310287	−	0.950643i	$-0.600425\pi$
$647$	−10212.9	−0.620575	−0.310287	+	0.950643i	$0.600425\pi$
$648$	−17129.5	−1.03844
$649$	−17827.1	−1.07823
$650$	−16548.8	−0.998610
$651$	0	0
$652$	−10122.9	−0.608044
$653$	−8773.78	−0.525795	−0.262898	−	0.964824i	$-0.584678\pi$
$653$	−8773.78	−0.525795	−0.262898	+	0.964824i	$0.584678\pi$
$654$	168.978	0.0101033
$655$	−26739.1	−1.59509
$656$	−2561.16	−0.152433
$657$	−806.715	−0.0479040
$658$	0	0
$659$	−4611.84	−0.272613	−0.136306	−	0.990667i	$-0.543523\pi$
$659$	−4611.84	−0.272613	−0.136306	+	0.990667i	$0.543523\pi$
$660$	284.830	0.0167985
$661$	13456.9	0.791852	0.395926	−	0.918282i	$-0.370424\pi$
$661$	13456.9	0.791852	0.395926	+	0.918282i	$0.370424\pi$
$662$	9593.40	0.563230
$663$	82.8312	0.00485203
$664$	−2770.68	−0.161933
$665$	0	0
$666$	−16863.5	−0.981152
$667$	−40696.3	−2.36247
$668$	1402.68	0.0812445
$669$	78.1852	0.00451841
$670$	15497.2	0.893593
$671$	−3992.51	−0.229701
$672$	0	0
$673$	−24304.7	−1.39209	−0.696045	−	0.717998i	$-0.745059\pi$
$673$	−24304.7	−1.39209	−0.696045	+	0.717998i	$0.745059\pi$
$674$	−5186.74	−0.296418
$675$	−1910.48	−0.108940
$676$	888.237	0.0505369
$677$	2712.82	0.154006	0.0770032	−	0.997031i	$-0.475465\pi$
$677$	2712.82	0.154006	0.0770032	+	0.997031i	$0.475465\pi$
$678$	−354.395	−0.0200744
$679$	0	0
$680$	4289.35	0.241896
$681$	151.868	0.00854565
$682$	8100.28	0.454803
$683$	−15993.4	−0.896006	−0.448003	−	0.894032i	$-0.647865\pi$
$683$	−15993.4	−0.896006	−0.448003	+	0.894032i	$0.647865\pi$
$684$	−6062.06	−0.338872
$685$	−22527.1	−1.25652
$686$	0	0
$687$	−289.156	−0.0160582
$688$	198.323	0.0109898
$689$	−11590.8	−0.640890
$690$	966.514	0.0533255
$691$	−19302.8	−1.06268	−0.531342	−	0.847158i	$-0.678312\pi$
$691$	−19302.8	−1.06268	−0.531342	+	0.847158i	$0.678312\pi$
$692$	−10345.6	−0.568325
$693$	0	0
$694$	12562.4	0.687119
$695$	37001.4	2.01949
$696$	1124.88	0.0612620
$697$	−2569.45	−0.139634
$698$	18750.2	1.01677
$699$	415.815	0.0225001
$700$	0	0
$701$	−3137.29	−0.169036	−0.0845178	−	0.996422i	$-0.526935\pi$
$701$	−3137.29	−0.169036	−0.0845178	+	0.996422i	$0.526935\pi$
$702$	838.072	0.0450584
$703$	−16902.8	−0.906829
$704$	−8114.34	−0.434404
$705$	25.8823	0.00138267
$706$	−2959.02	−0.157740
$707$	0	0
$708$	−705.496	−0.0374494
$709$	−17632.7	−0.934003	−0.467002	−	0.884256i	$-0.654666\pi$
$709$	−17632.7	−0.934003	−0.467002	+	0.884256i	$0.654666\pi$
$710$	362.523	0.0191623
$711$	−24449.4	−1.28963
$712$	−4956.61	−0.260895
$713$	−32887.9	−1.72743
$714$	0	0
$715$	16009.7	0.837384
$716$	−9057.36	−0.472750
$717$	−695.739	−0.0362383
$718$	−14004.8	−0.727933
$719$	3752.83	0.194655	0.0973276	−	0.995252i	$-0.468971\pi$
$719$	3752.83	0.194655	0.0973276	+	0.995252i	$0.468971\pi$
$720$	4888.71	0.253044
$721$	0	0
$722$	−8011.72	−0.412971
$723$	−823.758	−0.0423733
$724$	5282.68	0.271173
$725$	−50826.9	−2.60368
$726$	−322.857	−0.0165046
$727$	34938.5	1.78239	0.891195	−	0.453620i	$-0.149868\pi$
$727$	34938.5	1.78239	0.891195	+	0.453620i	$0.149868\pi$
$728$	0	0
$729$	−19537.8	−0.992625
$730$	1020.05	0.0517175
$731$	198.965	0.0100670
$732$	−158.001	−0.00797800
$733$	−2702.69	−0.136188	−0.0680941	−	0.997679i	$-0.521692\pi$
$733$	−2702.69	−0.136188	−0.0680941	+	0.997679i	$0.521692\pi$
$734$	6245.36	0.314061
$735$	0	0
$736$	26330.4	1.31868
$737$	−9121.82	−0.455911
$738$	−12990.6	−0.647956
$739$	30983.7	1.54229	0.771147	−	0.636657i	$-0.219683\pi$
$739$	30983.7	1.54229	0.771147	+	0.636657i	$0.219683\pi$
$740$	−25513.0	−1.26740
$741$	419.754	0.0208098
$742$	0	0
$743$	15007.3	0.741004	0.370502	−	0.928832i	$-0.379186\pi$
$743$	15007.3	0.741004	0.370502	+	0.928832i	$0.379186\pi$
$744$	909.046	0.0447947
$745$	27112.7	1.33333
$746$	21451.4	1.05280
$747$	−3168.06	−0.155172
$748$	−890.328	−0.0435209
$749$	0	0
$750$	430.249	0.0209473
$751$	−5059.57	−0.245841	−0.122920	−	0.992417i	$-0.539226\pi$
$751$	−5059.57	−0.245841	−0.122920	+	0.992417i	$0.539226\pi$
$752$	−80.6422	−0.00391053
$753$	−574.574	−0.0278070
$754$	22296.3	1.07690
$755$	−27628.6	−1.33180
$756$	0	0
$757$	−40096.6	−1.92515	−0.962573	−	0.271024i	$-0.912638\pi$
$757$	−40096.6	−1.92515	−0.962573	+	0.271024i	$0.912638\pi$
$758$	13528.7	0.648263
$759$	−568.903	−0.0272067
$760$	21736.6	1.03746
$761$	−7732.11	−0.368317	−0.184158	−	0.982897i	$-0.558956\pi$
$761$	−7732.11	−0.368317	−0.184158	+	0.982897i	$0.558956\pi$
$762$	444.600	0.0211367
$763$	0	0
$764$	−4789.17	−0.226788
$765$	4904.54	0.231796
$766$	23377.0	1.10267
$767$	−39654.5	−1.86681
$768$	−792.266	−0.0372245
$769$	−3864.05	−0.181198	−0.0905990	−	0.995887i	$-0.528878\pi$
$769$	−3864.05	−0.181198	−0.0905990	+	0.995887i	$0.528878\pi$
$770$	0	0
$771$	543.303	0.0253782
$772$	15949.4	0.743566
$773$	1936.91	0.0901240	0.0450620	−	0.998984i	$-0.485651\pi$
$773$	1936.91	0.0901240	0.0450620	+	0.998984i	$0.485651\pi$
$774$	1005.93	0.0467150
$775$	−41074.7	−1.90380
$776$	−28492.5	−1.31807
$777$	0	0
$778$	22468.0	1.03537
$779$	−13020.9	−0.598874
$780$	633.576	0.0290842
$781$	−213.385	−0.00977661
$782$	−3021.15	−0.138154
$783$	2574.01	0.117481
$784$	0	0
$785$	−53599.1	−2.43699
$786$	−520.569	−0.0236235
$787$	14946.9	0.677001	0.338501	−	0.940966i	$-0.390080\pi$
$787$	14946.9	0.677001	0.338501	+	0.940966i	$0.390080\pi$
$788$	−16128.8	−0.729143
$789$	201.837	0.00910719
$790$	30915.0	1.39229
$791$	0	0
$792$	−12764.7	−0.572695
$793$	−8880.93	−0.397694
$794$	3119.74	0.139440
$795$	845.461	0.0377175
$796$	−2981.13	−0.132743
$797$	−4235.37	−0.188237	−0.0941183	−	0.995561i	$-0.530003\pi$
$797$	−4235.37	−0.188237	−0.0941183	+	0.995561i	$0.530003\pi$
$798$	0	0
$799$	−80.9034	−0.00358217
$800$	32884.9	1.45332
$801$	−5667.51	−0.250002
$802$	−15455.1	−0.680471
$803$	−600.414	−0.0263863
$804$	−360.991	−0.0158348
$805$	0	0
$806$	18018.3	0.787427
$807$	−43.1082	−0.00188040
$808$	−39998.9	−1.74153
$809$	−22979.9	−0.998677	−0.499338	−	0.866407i	$-0.666424\pi$
$809$	−22979.9	−0.998677	−0.499338	+	0.866407i	$0.666424\pi$
$810$	24765.8	1.07430
$811$	36597.9	1.58462	0.792310	−	0.610118i	$-0.208878\pi$
$811$	36597.9	1.58462	0.792310	+	0.610118i	$0.208878\pi$
$812$	0	0
$813$	−643.456	−0.0277577
$814$	−12551.0	−0.540433
$815$	41503.6	1.78381
$816$	18.8250	0.000807607 0
$817$	1008.27	0.0431763
$818$	27628.8	1.18095
$819$	0	0
$820$	−19653.7	−0.836997
$821$	−12382.2	−0.526361	−0.263180	−	0.964747i	$-0.584771\pi$
$821$	−12382.2	−0.526361	−0.263180	+	0.964747i	$0.584771\pi$
$822$	−438.568	−0.0186092
$823$	30177.2	1.27814	0.639071	−	0.769148i	$-0.279319\pi$
$823$	30177.2	1.27814	0.639071	+	0.769148i	$0.279319\pi$
$824$	42944.5	1.81558
$825$	−710.520	−0.0299844
$826$	0	0
$827$	31266.7	1.31469	0.657345	−	0.753590i	$-0.271680\pi$
$827$	31266.7	1.31469	0.657345	+	0.753590i	$0.271680\pi$
$828$	18275.8	0.767062
$829$	24670.8	1.03360	0.516799	−	0.856107i	$-0.327124\pi$
$829$	24670.8	1.03360	0.516799	+	0.856107i	$0.327124\pi$
$830$	4005.85	0.167524
$831$	−511.373	−0.0213470
$832$	−18049.5	−0.752109
$833$	0	0
$834$	720.359	0.0299089
$835$	−5750.93	−0.238346
$836$	−4511.81	−0.186656
$837$	2080.13	0.0859017
$838$	4535.61	0.186969
$839$	−41310.7	−1.69988	−0.849942	−	0.526876i	$-0.823363\pi$
$839$	−41310.7	−1.69988	−0.849942	+	0.526876i	$0.823363\pi$
$840$	0	0
$841$	44090.4	1.80780
$842$	−997.707	−0.0408352
$843$	44.0611	0.00180017
$844$	−8432.80	−0.343921
$845$	−3641.73	−0.148260
$846$	−409.031	−0.0166227
$847$	0	0
$848$	−2634.23	−0.106674
$849$	1529.30	0.0618203
$850$	−3773.21	−0.152259
$851$	50958.2	2.05267
$852$	−8.44461	−0.000339563 0
$853$	22106.5	0.887353	0.443676	−	0.896187i	$-0.353674\pi$
$853$	22106.5	0.887353	0.443676	+	0.896187i	$0.353674\pi$
$854$	0	0
$855$	24854.2	0.994146
$856$	18110.8	0.723147
$857$	14242.1	0.567679	0.283840	−	0.958872i	$-0.408392\pi$
$857$	14242.1	0.567679	0.283840	+	0.958872i	$0.408392\pi$
$858$	311.684	0.0124018
$859$	19870.6	0.789260	0.394630	−	0.918840i	$-0.370873\pi$
$859$	19870.6	0.789260	0.394630	+	0.918840i	$0.370873\pi$
$860$	1521.89	0.0603440
$861$	0	0
$862$	30710.5	1.21346
$863$	33504.8	1.32157	0.660787	−	0.750574i	$-0.270223\pi$
$863$	33504.8	1.32157	0.660787	+	0.750574i	$0.270223\pi$
$864$	−1665.38	−0.0655755
$865$	42416.5	1.66729
$866$	23042.1	0.904160
$867$	−876.581	−0.0343371
$868$	0	0
$869$	−18197.0	−0.710345
$870$	−1626.35	−0.0633774
$871$	−20290.6	−0.789346
$872$	−11457.0	−0.444933
$873$	−32579.0	−1.26304
$874$	−15309.9	−0.592525
$875$	0	0
$876$	−23.7611	−0.000916452 0
$877$	−3211.26	−0.123645	−0.0618225	−	0.998087i	$-0.519691\pi$
$877$	−3211.26	−0.123645	−0.0618225	+	0.998087i	$0.519691\pi$
$878$	−1988.29	−0.0764256
$879$	833.896	0.0319984
$880$	3638.52	0.139380
$881$	−18004.0	−0.688501	−0.344250	−	0.938878i	$-0.611867\pi$
$881$	−18004.0	−0.688501	−0.344250	+	0.938878i	$0.611867\pi$
$882$	0	0
$883$	−31882.3	−1.21509	−0.607546	−	0.794285i	$-0.707846\pi$
$883$	−31882.3	−1.21509	−0.607546	+	0.794285i	$0.707846\pi$
$884$	−1980.44	−0.0753502
$885$	2892.50	0.109865
$886$	−26054.5	−0.987945
$887$	20934.7	0.792469	0.396234	−	0.918149i	$-0.370317\pi$
$887$	20934.7	0.792469	0.396234	+	0.918149i	$0.370317\pi$
$888$	−1408.52	−0.0532286
$889$	0	0
$890$	7166.28	0.269904
$891$	−14577.5	−0.548107
$892$	−1869.36	−0.0701691
$893$	−409.985	−0.0153635
$894$	527.843	0.0197469
$895$	37134.8	1.38690
$896$	0	0
$897$	−1265.47	−0.0471045
$898$	−31622.2	−1.17511
$899$	55340.2	2.05306
$900$	22825.2	0.845378
$901$	−2642.76	−0.0977172
$902$	−9668.55	−0.356904
$903$	0	0
$904$	24028.5	0.884046
$905$	−21658.7	−0.795537
$906$	−537.886	−0.0197241
$907$	28156.9	1.03080	0.515400	−	0.856950i	$-0.327643\pi$
$907$	28156.9	1.03080	0.515400	+	0.856950i	$0.327643\pi$
$908$	−3631.07	−0.132711
$909$	−45735.7	−1.66882
$910$	0	0
$911$	−12208.8	−0.444011	−0.222006	−	0.975045i	$-0.571260\pi$
$911$	−12208.8	−0.444011	−0.222006	+	0.975045i	$0.571260\pi$
$912$	95.3973	0.00346373
$913$	−2357.89	−0.0854709
$914$	18869.4	0.682872
$915$	647.798	0.0234050
$916$	6913.54	0.249377
$917$	0	0
$918$	191.085	0.00687010
$919$	21563.2	0.773998	0.386999	−	0.922080i	$-0.373512\pi$
$919$	21563.2	0.773998	0.386999	+	0.922080i	$0.373512\pi$
$920$	−65531.2	−2.34837
$921$	5.91909	0.000211771 0
$922$	31343.9	1.11958
$923$	−474.655	−0.0169268
$924$	0	0
$925$	63643.3	2.26225
$926$	27557.6	0.977970
$927$	49103.7	1.73978
$928$	−44306.0	−1.56726
$929$	−22322.2	−0.788339	−0.394170	−	0.919038i	$-0.628968\pi$
$929$	−22322.2	−0.788339	−0.394170	+	0.919038i	$0.628968\pi$
$930$	−1314.30	−0.0463414
$931$	0	0
$932$	−9941.89	−0.349418
$933$	−648.837	−0.0227674
$934$	−5913.17	−0.207157
$935$	3650.31	0.127677
$936$	−28393.8	−0.991540
$937$	24171.5	0.842741	0.421371	−	0.906889i	$-0.361549\pi$
$937$	24171.5	0.842741	0.421371	+	0.906889i	$0.361549\pi$
$938$	0	0
$939$	1407.14	0.0489033
$940$	−618.829	−0.0214723
$941$	32350.2	1.12071	0.560355	−	0.828253i	$-0.310665\pi$
$941$	32350.2	1.12071	0.560355	+	0.828253i	$0.310665\pi$
$942$	−1043.49	−0.0360921
$943$	39255.2	1.35559
$944$	−9012.26	−0.310725
$945$	0	0
$946$	748.684	0.0257313
$947$	4958.49	0.170147	0.0850735	−	0.996375i	$-0.472887\pi$
$947$	4958.49	0.170147	0.0850735	+	0.996375i	$0.472887\pi$
$948$	−720.134	−0.0246718
$949$	−1335.56	−0.0456840
$950$	−19121.1	−0.653021
$951$	202.852	0.00691685
$952$	0	0
$953$	−21622.5	−0.734963	−0.367482	−	0.930031i	$-0.619780\pi$
$953$	−21622.5	−0.734963	−0.367482	+	0.930031i	$0.619780\pi$
$954$	−13361.3	−0.453446
$955$	19635.4	0.665326
$956$	16634.7	0.562766
$957$	957.289	0.0323352
$958$	−28264.9	−0.953233
$959$	0	0
$960$	1316.58	0.0442629
$961$	14931.0	0.501193
$962$	−27918.4	−0.935683
$963$	20708.3	0.692955
$964$	19695.6	0.658041
$965$	−65392.0	−2.18139
$966$	0	0
$967$	17261.5	0.574035	0.287018	−	0.957925i	$-0.407336\pi$
$967$	17261.5	0.574035	0.287018	+	0.957925i	$0.407336\pi$
$968$	21890.2	0.726837
$969$	95.7063	0.00317289
$970$	41194.5	1.36358
$971$	−33301.3	−1.10061	−0.550304	−	0.834965i	$-0.685488\pi$
$971$	−33301.3	−1.10061	−0.550304	+	0.834965i	$0.685488\pi$
$972$	−1734.25	−0.0572283
$973$	0	0
$974$	3599.96	0.118429
$975$	−1580.48	−0.0519138
$976$	−2018.37	−0.0661950
$977$	27382.3	0.896661	0.448330	−	0.893868i	$-0.352019\pi$
$977$	27382.3	0.896661	0.448330	+	0.893868i	$0.352019\pi$
$978$	808.010	0.0264185
$979$	−4218.16	−0.137705
$980$	0	0
$981$	−13100.1	−0.426356
$982$	−5990.78	−0.194678
$983$	−39159.8	−1.27060	−0.635302	−	0.772264i	$-0.719124\pi$
$983$	−39159.8	−1.27060	−0.635302	+	0.772264i	$0.719124\pi$
$984$	−1085.04	−0.0351523
$985$	66127.4	2.13908
$986$	5083.68	0.164196
$987$	0	0
$988$	−10036.1	−0.323168
$989$	−3039.73	−0.0977327
$990$	18455.2	0.592470
$991$	12906.1	0.413698	0.206849	−	0.978373i	$-0.433679\pi$
$991$	12906.1	0.413698	0.206849	+	0.978373i	$0.433679\pi$
$992$	−35805.0	−1.14598
$993$	916.212	0.0292801
$994$	0	0
$995$	12222.5	0.389427
$996$	−93.3123	−0.00296859
$997$	25772.6	0.818684	0.409342	−	0.912381i	$-0.365758\pi$
$997$	25772.6	0.818684	0.409342	+	0.912381i	$0.365758\pi$
$998$	3197.06	0.101404
$999$	−3223.06	−0.102075

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2401.4.a.d.1.26		39
7.6	odd	2		2401.4.a.c.1.26		39
49.8	even	7		49.4.e.a.15.5	✓	78
49.43	even	7		49.4.e.a.36.5	yes	78

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.15.5	✓	78	49.8	even	7
49.4.e.a.36.5	yes	78	49.43	even	7
2401.4.a.c.1.26		39	7.6	odd	2
2401.4.a.d.1.26		39	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:	\( N \)	\(=\)	\( 2401 = 7^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2401.a (trivial)

\(f(q)\)	\(=\)	\(q+1.90844 q^{2} +0.182265 q^{3} -4.35784 q^{4} +17.8670 q^{5} +0.347842 q^{6} -23.5842 q^{8} -26.9668 q^{9} +34.0981 q^{10} -20.0706 q^{11} -0.794282 q^{12} -44.6450 q^{13} +3.25652 q^{15} -10.1464 q^{16} -10.1793 q^{17} -51.4646 q^{18} -51.5845 q^{19} -77.8615 q^{20} -38.3036 q^{22} +155.516 q^{23} -4.29858 q^{24} +194.229 q^{25} -85.2024 q^{26} -9.83625 q^{27} -261.686 q^{29} +6.21489 q^{30} -211.476 q^{31} +169.310 q^{32} -3.65816 q^{33} -19.4266 q^{34} +117.517 q^{36} +327.672 q^{37} -98.4461 q^{38} -8.13721 q^{39} -421.379 q^{40} +252.419 q^{41} -19.5461 q^{43} +87.4645 q^{44} -481.815 q^{45} +296.793 q^{46} +7.94782 q^{47} -1.84934 q^{48} +370.675 q^{50} -1.85533 q^{51} +194.556 q^{52} +259.621 q^{53} -18.7719 q^{54} -358.601 q^{55} -9.40204 q^{57} -499.413 q^{58} +888.219 q^{59} -14.1914 q^{60} +198.923 q^{61} -403.590 q^{62} +404.290 q^{64} -797.671 q^{65} -6.98139 q^{66} +454.487 q^{67} +44.3598 q^{68} +28.3451 q^{69} +10.6318 q^{71} +635.991 q^{72} +29.9151 q^{73} +625.343 q^{74} +35.4011 q^{75} +224.797 q^{76} -15.5294 q^{78} +906.648 q^{79} -181.286 q^{80} +726.310 q^{81} +481.727 q^{82} +117.480 q^{83} -181.874 q^{85} -37.3026 q^{86} -47.6961 q^{87} +473.349 q^{88} +210.166 q^{89} -919.516 q^{90} -677.714 q^{92} -38.5446 q^{93} +15.1680 q^{94} -921.659 q^{95} +30.8593 q^{96} +1208.12 q^{97} +541.239 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 39 q + q^{2} + q^{3} + 145 q^{4} + 27 q^{5} + 41 q^{6} - 12 q^{8} + 312 q^{9} + 78 q^{10} + q^{11} - 91 q^{12} + 77 q^{13} - 161 q^{15} + 461 q^{16} + 211 q^{17} + 8 q^{18} + 314 q^{19} + 476 q^{20}+ \cdots + 5384 q^{99}+O(q^{100}) \) 39 * q + q^2 + q^3 + 145 * q^4 + 27 * q^5 + 41 * q^6 - 12 * q^8 + 312 * q^9 + 78 * q^10 + q^11 - 91 * q^12 + 77 * q^13 - 161 * q^15 + 461 * q^16 + 211 * q^17 + 8 * q^18 + 314 * q^19 + 476 * q^20 - 61 * q^22 - 69 * q^23 + 330 * q^24 + 606 * q^25 + 504 * q^26 - 50 * q^27 + 57 * q^29 + 42 * q^30 + 638 * q^31 - 1600 * q^32 + 1574 * q^33 + 1343 * q^34 + 782 * q^36 + 71 * q^37 + 1359 * q^38 - 84 * q^39 - 155 * q^40 + 1393 * q^41 - 125 * q^43 + 52 * q^44 + 1129 * q^45 - 1454 * q^46 + 1483 * q^47 - 974 * q^48 + 3074 * q^50 - 2044 * q^51 + 3899 * q^52 + 2213 * q^53 + 1142 * q^54 + 1604 * q^55 + 98 * q^57 + 2403 * q^58 + 2073 * q^59 - 1519 * q^60 + 2575 * q^61 + 1742 * q^62 + 1358 * q^64 + 1876 * q^65 - 48 * q^66 + 176 * q^67 + 3038 * q^68 - 638 * q^69 - 1259 * q^71 - 3799 * q^72 - 307 * q^73 + 3845 * q^74 + 131 * q^75 + 1974 * q^76 - 6041 * q^78 + 22 * q^79 - 804 * q^80 + 795 * q^81 + 8043 * q^82 + 6349 * q^83 + 3094 * q^85 + 1745 * q^86 + 9508 * q^87 + 1299 * q^88 + 2253 * q^89 + 11156 * q^90 + 1284 * q^92 - 3430 * q^93 + 2738 * q^94 - 3290 * q^95 + 3031 * q^96 - 770 * q^97 + 5384 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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