LMFDB - Embedded newform 2401.4.a.c.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:	$1$
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	$-0.505583\pi$
$3$	−0.182265	−0.0350769	−0.0175384	+	0.999846i	$0.505583\pi$
$4$	−4.35784	−0.544730
$5$	−17.8670	−1.59807	−0.799035	−	0.601284i	$-0.794656\pi$
$5$	−17.8670	−1.59807	−0.799035	+	0.601284i	$0.794656\pi$
$6$	−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	$-0.341999\pi$
$13$	44.6450	0.952484	0.476242	+	0.879314i	$0.341999\pi$
$14$	0	0
$15$	3.25652	0.0560554
$16$	−10.1464	−0.158538
$17$	10.1793	0.145226	0.0726131	−	0.997360i	$-0.476866\pi$
$17$	10.1793	0.145226	0.0726131	+	0.997360i	$0.476866\pi$
$18$	−51.4646	−0.673906
$19$	51.5845	0.622858	0.311429	−	0.950269i	$-0.399192\pi$
$19$	51.5845	0.622858	0.311429	+	0.950269i	$0.399192\pi$
$20$	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	$-0.290117\pi$
$31$	211.476	1.22523	0.612616	+	0.790381i	$0.290117\pi$
$32$	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	$-0.659634\pi$
$41$	−252.419	−0.961493	−0.480747	+	0.876860i	$0.659634\pi$
$42$	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	$-0.503926\pi$
$47$	−7.94782	−0.0246662	−0.0123331	+	0.999924i	$0.503926\pi$
$48$	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	$-0.936180\pi$
$59$	−888.219	−1.95994	−0.979968	+	0.199154i	$0.936180\pi$
$60$	−14.1914	−0.0305351
$61$	−198.923	−0.417533	−0.208767	−	0.977965i	$-0.566945\pi$
$61$	−198.923	−0.417533	−0.208767	+	0.977965i	$0.566945\pi$
$62$	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	$-0.507634\pi$
$73$	−29.9151	−0.0479631	−0.0239815	+	0.999712i	$0.507634\pi$
$74$	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	$-0.524752\pi$
$83$	−117.480	−0.155363	−0.0776814	+	0.996978i	$0.524752\pi$
$84$	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	$-0.539943\pi$
$89$	−210.166	−0.250310	−0.125155	+	0.992137i	$0.539943\pi$
$90$	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	$-0.717888\pi$
$97$	−1208.12	−1.26460	−0.632298	+	0.774726i	$0.717888\pi$
$98$	0	0
$99$	541.239	0.549460
$100$	−846.419	−0.846419
$101$	−1696.00	−1.67087	−0.835437	−	0.549586i	$-0.814786\pi$
$101$	−1696.00	−1.67087	−0.835437	+	0.549586i	$0.814786\pi$
$102$	−3.54079	−0.00343716
$103$	1820.90	1.74192	0.870962	−	0.491350i	$-0.163496\pi$
$103$	1820.90	1.74192	0.870962	+	0.491350i	$0.163496\pi$
$104$	−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	$-0.333676\pi$
$131$	1496.57	0.998135	0.499068	+	0.866563i	$0.333676\pi$
$132$	−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	$-0.717705\pi$
$139$	−2070.94	−1.26370	−0.631851	+	0.775090i	$0.717705\pi$
$140$	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	$-0.223983\pi$
$157$	2999.90	1.52495	0.762477	+	0.647015i	$0.223983\pi$
$158$	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	$-0.476241\pi$
$167$	321.875	0.149146	0.0745731	+	0.997216i	$0.476241\pi$
$168$	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	$-0.674686\pi$
$173$	−2374.02	−1.04331	−0.521657	+	0.853155i	$0.674686\pi$
$174$	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	$-0.419929\pi$
$181$	1212.22	0.497811	0.248905	+	0.968528i	$0.419929\pi$
$182$	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	$-0.538880\pi$
$199$	−684.085	−0.243686	−0.121843	+	0.992549i	$0.538880\pi$
$200$	−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	$-0.520516\pi$
$223$	−428.965	−0.128814	−0.0644072	+	0.997924i	$0.520516\pi$
$224$	0	0
$225$	−5237.73	−1.55192
$226$	−1944.40	−0.572298
$227$	−833.226	−0.243626	−0.121813	−	0.992553i	$-0.538871\pi$
$227$	−833.226	−0.243626	−0.121813	+	0.992553i	$0.538871\pi$
$228$	40.9726	0.0119012
$229$	1586.46	0.457800	0.228900	−	0.973450i	$-0.426487\pi$
$229$	1586.46	0.457800	0.228900	+	0.973450i	$0.426487\pi$
$230$	−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	$-0.293570\pi$
$241$	4519.57	1.20801	0.604006	+	0.796980i	$0.293570\pi$
$242$	−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	$-0.370269\pi$
$251$	3152.41	0.792743	0.396371	+	0.918090i	$0.370269\pi$
$252$	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	$-0.617821\pi$
$257$	−2980.85	−0.723502	−0.361751	+	0.932275i	$0.617821\pi$
$258$	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	$-0.491467\pi$
$269$	236.514	0.0536079	0.0268039	+	0.999641i	$0.491467\pi$
$270$	−335.397	−0.0755987
$271$	3530.34	0.791338	0.395669	−	0.918393i	$-0.370513\pi$
$271$	3530.34	0.791338	0.395669	+	0.918393i	$0.370513\pi$
$272$	−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	$-0.843271\pi$
$283$	−8390.53	−1.76242	−0.881211	+	0.472723i	$0.843271\pi$
$284$	−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	$-0.650761\pi$
$293$	−4575.19	−0.912237	−0.456118	+	0.889919i	$0.650761\pi$
$294$	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	$-0.500961\pi$
$307$	−32.4752	−0.00603732	−0.00301866	+	0.999995i	$0.500961\pi$
$308$	0	0
$309$	−331.885	−0.0611013
$310$	−7210.93	−1.32114
$311$	3559.86	0.649070	0.324535	−	0.945874i	$-0.394792\pi$
$311$	3559.86	0.649070	0.324535	+	0.945874i	$0.394792\pi$
$312$	191.910	0.0348230
$313$	−7720.29	−1.39417	−0.697087	−	0.716986i	$-0.745521\pi$
$313$	−7720.29	−1.39417	−0.697087	+	0.716986i	$0.745521\pi$
$314$	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	$-0.771616\pi$
$349$	−9824.88	−1.50692	−0.753458	+	0.657496i	$0.771616\pi$
$350$	0	0
$351$	439.139	0.0667792
$352$	−3398.15	−0.514551
$353$	1550.49	0.233780	0.116890	−	0.993145i	$-0.462708\pi$
$353$	1550.49	0.233780	0.116890	+	0.993145i	$0.462708\pi$
$354$	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	$-0.574765\pi$
$367$	−3272.49	−0.465457	−0.232728	+	0.972542i	$0.574765\pi$
$368$	−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	$-0.804426\pi$
$383$	−12249.2	−1.63422	−0.817111	+	0.576480i	$0.804426\pi$
$384$	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	$-0.532950\pi$
$397$	−1634.70	−0.206659	−0.103329	+	0.994647i	$0.532950\pi$
$398$	−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	$-0.839217\pi$
$409$	−14477.1	−1.75024	−0.875120	+	0.483907i	$0.839217\pi$
$410$	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	$-0.544244\pi$
$419$	−2376.60	−0.277100	−0.138550	+	0.990355i	$0.544244\pi$
$420$	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	$-0.733710\pi$
$433$	−12073.8	−1.34002	−0.670009	+	0.742353i	$0.733710\pi$
$434$	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	$-0.481963\pi$
$439$	1041.84	0.113267	0.0566337	+	0.998395i	$0.481963\pi$
$440$	−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	$-0.811458\pi$
$461$	−16423.8	−1.65929	−0.829646	+	0.558290i	$0.811458\pi$
$462$	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	$-0.450942\pi$
$467$	3098.43	0.307019	0.153510	+	0.988147i	$0.450942\pi$
$468$	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	$-0.250330\pi$
$479$	14810.4	1.41275	0.706374	+	0.707838i	$0.250330\pi$
$480$	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	$-0.548433\pi$
$503$	−3419.78	−0.303142	−0.151571	+	0.988446i	$0.548433\pi$
$504$	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	$-0.241787\pi$
$509$	16653.8	1.45023	0.725114	+	0.688629i	$0.241787\pi$
$510$	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	$-0.366355\pi$
$521$	9695.11	0.815260	0.407630	+	0.913147i	$0.366355\pi$
$522$	13467.5	1.12923
$523$	14073.4	1.17665	0.588323	−	0.808626i	$-0.299788\pi$
$523$	14073.4	1.17665	0.588323	+	0.808626i	$0.299788\pi$
$524$	−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	$-0.411993\pi$
$563$	7293.14	0.545949	0.272974	+	0.962021i	$0.411993\pi$
$564$	−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	$-0.722733\pi$
$577$	−17852.1	−1.28803	−0.644016	+	0.765012i	$0.722733\pi$
$578$	−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	$-0.675101\pi$
$587$	−14869.5	−1.04554	−0.522768	+	0.852475i	$0.675101\pi$
$588$	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	$-0.765361\pi$
$593$	−21383.3	−1.48079	−0.740393	+	0.672174i	$0.765361\pi$
$594$	−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	$-0.434090\pi$
$601$	6058.07	0.411171	0.205586	+	0.978639i	$0.434090\pi$
$602$	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	$-0.450610\pi$
$607$	4622.27	0.309081	0.154540	+	0.987986i	$0.450610\pi$
$608$	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	$-0.668262\pi$
$619$	−15534.0	−1.00867	−0.504333	+	0.863509i	$0.668262\pi$
$620$	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	$-0.337905\pi$
$643$	15897.5	0.975020	0.487510	+	0.873117i	$0.337905\pi$
$644$	0	0
$645$	−63.6522	−0.00388574
$646$	1002.11	0.0610335
$647$	10212.9	0.620575	0.310287	−	0.950643i	$-0.399575\pi$
$647$	10212.9	0.620575	0.310287	+	0.950643i	$0.399575\pi$
$648$	−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	$-0.629576\pi$
$661$	−13456.9	−0.791852	−0.395926	+	0.918282i	$0.629576\pi$
$662$	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	$-0.524535\pi$
$677$	−2712.82	−0.154006	−0.0770032	+	0.997031i	$0.524535\pi$
$678$	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	$-0.321688\pi$
$691$	19302.8	1.06268	0.531342	+	0.847158i	$0.321688\pi$
$692$	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	$-0.531029\pi$
$719$	−3752.83	−0.194655	−0.0973276	+	0.995252i	$0.531029\pi$
$720$	−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	$-0.850132\pi$
$727$	−34938.5	−1.78239	−0.891195	+	0.453620i	$0.850132\pi$
$728$	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	$-0.478308\pi$
$733$	2702.69	0.136188	0.0680941	+	0.997679i	$0.478308\pi$
$734$	−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	$-0.441044\pi$
$761$	7732.11	0.368317	0.184158	+	0.982897i	$0.441044\pi$
$762$	−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	$-0.471122\pi$
$769$	3864.05	0.181198	0.0905990	+	0.995887i	$0.471122\pi$
$770$	0	0
$771$	543.303	0.0253782
$772$	15949.4	0.743566
$773$	−1936.91	−0.0901240	−0.0450620	−	0.998984i	$-0.514349\pi$
$773$	−1936.91	−0.0901240	−0.0450620	+	0.998984i	$0.514349\pi$
$774$	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	$-0.609920\pi$
$787$	−14946.9	−0.677001	−0.338501	+	0.940966i	$0.609920\pi$
$788$	−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	$-0.469997\pi$
$797$	4235.37	0.188237	0.0941183	+	0.995561i	$0.469997\pi$
$798$	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	$-0.791122\pi$
$811$	−36597.9	−1.58462	−0.792310	+	0.610118i	$0.791122\pi$
$812$	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	$-0.672876\pi$
$829$	−24670.8	−1.03360	−0.516799	+	0.856107i	$0.672876\pi$
$830$	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	$-0.176637\pi$
$839$	41310.7	1.69988	0.849942	+	0.526876i	$0.176637\pi$
$840$	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	$-0.646326\pi$
$853$	−22106.5	−0.887353	−0.443676	+	0.896187i	$0.646326\pi$
$854$	0	0
$855$	24854.2	0.994146
$856$	18110.8	0.723147
$857$	−14242.1	−0.567679	−0.283840	−	0.958872i	$-0.591608\pi$
$857$	−14242.1	−0.567679	−0.283840	+	0.958872i	$0.591608\pi$
$858$	311.684	0.0124018
$859$	−19870.6	−0.789260	−0.394630	−	0.918840i	$-0.629127\pi$
$859$	−19870.6	−0.789260	−0.394630	+	0.918840i	$0.629127\pi$
$860$	−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	$-0.388133\pi$
$881$	18004.0	0.688501	0.344250	+	0.938878i	$0.388133\pi$
$882$	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	$-0.629683\pi$
$887$	−20934.7	−0.792469	−0.396234	+	0.918149i	$0.629683\pi$
$888$	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	$-0.371032\pi$
$929$	22322.2	0.788339	0.394170	+	0.919038i	$0.371032\pi$
$930$	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	$-0.638451\pi$
$937$	−24171.5	−0.842741	−0.421371	+	0.906889i	$0.638451\pi$
$938$	0	0
$939$	1407.14	0.0489033
$940$	−618.829	−0.0214723
$941$	−32350.2	−1.12071	−0.560355	−	0.828253i	$-0.689335\pi$
$941$	−32350.2	−1.12071	−0.560355	+	0.828253i	$0.689335\pi$
$942$	−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	$-0.314512\pi$
$971$	33301.3	1.10061	0.550304	+	0.834965i	$0.314512\pi$
$972$	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	$-0.280876\pi$
$983$	39159.8	1.27060	0.635302	+	0.772264i	$0.280876\pi$
$984$	−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	$-0.634242\pi$
$997$	−25772.6	−0.818684	−0.409342	+	0.912381i	$0.634242\pi$
$998$	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.c.1.26		39
7.6	odd	2		2401.4.a.d.1.26		39
49.6	odd	14		49.4.e.a.36.5	yes	78
49.41	odd	14		49.4.e.a.15.5	✓	78

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.15.5	✓	78	49.41	odd	14
49.4.e.a.36.5	yes	78	49.6	odd	14
2401.4.a.c.1.26		39	1.1	even	1	trivial
2401.4.a.d.1.26		39	7.6	odd	2

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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