LMFDB - Embedded newform 2401.4.a.d.1.39

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			1.39
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)

$f(q)$	$=$	\(q+5.24450 q^{2} -1.98985 q^{3} +19.5048 q^{4} +1.36203 q^{5} -10.4358 q^{6} +60.3369 q^{8} -23.0405 q^{9} +7.14318 q^{10} -18.4649 q^{11} -38.8116 q^{12} +21.2521 q^{13} -2.71024 q^{15} +160.399 q^{16} +126.077 q^{17} -120.836 q^{18} -50.2875 q^{19} +26.5662 q^{20} -96.8392 q^{22} -82.1963 q^{23} -120.061 q^{24} -123.145 q^{25} +111.457 q^{26} +99.5731 q^{27} +58.4210 q^{29} -14.2139 q^{30} +319.654 q^{31} +358.516 q^{32} +36.7424 q^{33} +661.211 q^{34} -449.400 q^{36} +266.470 q^{37} -263.733 q^{38} -42.2886 q^{39} +82.1808 q^{40} +246.066 q^{41} +360.939 q^{43} -360.154 q^{44} -31.3819 q^{45} -431.079 q^{46} +139.613 q^{47} -319.170 q^{48} -645.833 q^{50} -250.874 q^{51} +414.519 q^{52} -306.421 q^{53} +522.211 q^{54} -25.1498 q^{55} +100.065 q^{57} +306.389 q^{58} -562.646 q^{59} -52.8627 q^{60} +623.766 q^{61} +1676.43 q^{62} +597.049 q^{64} +28.9461 q^{65} +192.696 q^{66} +923.511 q^{67} +2459.11 q^{68} +163.558 q^{69} +196.020 q^{71} -1390.19 q^{72} +62.3114 q^{73} +1397.50 q^{74} +245.040 q^{75} -980.847 q^{76} -221.783 q^{78} -432.393 q^{79} +218.468 q^{80} +423.958 q^{81} +1290.49 q^{82} +747.945 q^{83} +171.721 q^{85} +1892.94 q^{86} -116.249 q^{87} -1114.12 q^{88} -327.402 q^{89} -164.582 q^{90} -1603.22 q^{92} -636.064 q^{93} +732.202 q^{94} -68.4932 q^{95} -713.394 q^{96} -1538.64 q^{97} +425.441 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	5.24450	1.85421	0.927106	−	0.374800i	$-0.122289\pi$
$2$	5.24450	1.85421	0.927106	+	0.374800i	$0.122289\pi$
$3$	−1.98985	−0.382947	−0.191473	−	0.981498i	$-0.561327\pi$
$3$	−1.98985	−0.382947	−0.191473	+	0.981498i	$0.561327\pi$
$4$	19.5048	2.43810
$5$	1.36203	0.121824	0.0609119	−	0.998143i	$-0.480599\pi$
$5$	1.36203	0.121824	0.0609119	+	0.998143i	$0.480599\pi$
$6$	−10.4358	−0.710064
$7$	0	0
$7$	0	0
$8$	60.3369	2.66654
$9$	−23.0405	−0.853352
$10$	7.14318	0.225887
$11$	−18.4649	−0.506125	−0.253063	−	0.967450i	$-0.581438\pi$
$11$	−18.4649	−0.506125	−0.253063	+	0.967450i	$0.581438\pi$
$12$	−38.8116	−0.933663
$13$	21.2521	0.453406	0.226703	−	0.973964i	$-0.427205\pi$
$13$	21.2521	0.453406	0.226703	+	0.973964i	$0.427205\pi$
$14$	0	0
$15$	−2.71024	−0.0466520
$16$	160.399	2.50623
$17$	126.077	1.79872	0.899358	−	0.437214i	$-0.144035\pi$
$17$	126.077	1.79872	0.899358	+	0.437214i	$0.144035\pi$
$18$	−120.836	−1.58229
$19$	−50.2875	−0.607197	−0.303598	−	0.952800i	$-0.598188\pi$
$19$	−50.2875	−0.607197	−0.303598	+	0.952800i	$0.598188\pi$
$20$	26.5662	0.297019
$21$	0	0
$22$	−96.8392	−0.938464
$23$	−82.1963	−0.745179	−0.372590	−	0.927996i	$-0.621530\pi$
$23$	−82.1963	−0.745179	−0.372590	+	0.927996i	$0.621530\pi$
$24$	−120.061	−1.02114
$25$	−123.145	−0.985159
$26$	111.457	0.840711
$27$	99.5731	0.709735
$28$	0	0
$29$	58.4210	0.374086	0.187043	−	0.982352i	$-0.440110\pi$
$29$	58.4210	0.374086	0.187043	+	0.982352i	$0.440110\pi$
$30$	−14.2139	−0.0865028
$31$	319.654	1.85199	0.925993	−	0.377541i	$-0.123230\pi$
$31$	319.654	1.85199	0.925993	+	0.377541i	$0.123230\pi$
$32$	358.516	1.98054
$33$	36.7424	0.193819
$34$	661.211	3.33520
$35$	0	0
$36$	−449.400	−2.08056
$37$	266.470	1.18398	0.591992	−	0.805944i	$-0.298342\pi$
$37$	266.470	1.18398	0.591992	+	0.805944i	$0.298342\pi$
$38$	−263.733	−1.12587
$39$	−42.2886	−0.173631
$40$	82.1808	0.324848
$41$	246.066	0.937292	0.468646	−	0.883386i	$-0.344742\pi$
$41$	246.066	0.937292	0.468646	+	0.883386i	$0.344742\pi$
$42$	0	0
$43$	360.939	1.28006	0.640030	−	0.768350i	$-0.278922\pi$
$43$	360.939	1.28006	0.640030	+	0.768350i	$0.278922\pi$
$44$	−360.154	−1.23398
$45$	−31.3819	−0.103959
$46$	−431.079	−1.38172
$47$	139.613	0.433291	0.216646	−	0.976250i	$-0.430488\pi$
$47$	139.613	0.433291	0.216646	+	0.976250i	$0.430488\pi$
$48$	−319.170	−0.959753
$49$	0	0
$50$	−645.833	−1.82669
$51$	−250.874	−0.688812
$52$	414.519	1.10545
$53$	−306.421	−0.794155	−0.397078	−	0.917785i	$-0.629976\pi$
$53$	−306.421	−0.794155	−0.397078	+	0.917785i	$0.629976\pi$
$54$	522.211	1.31600
$55$	−25.1498	−0.0616581
$56$	0	0
$57$	100.065	0.232524
$58$	306.389	0.693635
$59$	−562.646	−1.24153	−0.620765	−	0.783997i	$-0.713178\pi$
$59$	−562.646	−1.24153	−0.620765	+	0.783997i	$0.713178\pi$
$60$	−52.8627	−0.113742
$61$	623.766	1.30926	0.654632	−	0.755948i	$-0.272824\pi$
$61$	623.766	1.30926	0.654632	+	0.755948i	$0.272824\pi$
$62$	1676.43	3.43397
$63$	0	0
$64$	597.049	1.16611
$65$	28.9461	0.0552357
$66$	192.696	0.359382
$67$	923.511	1.68395	0.841976	−	0.539515i	$-0.181392\pi$
$67$	923.511	1.68395	0.841976	+	0.539515i	$0.181392\pi$
$68$	2459.11	4.38545
$69$	163.558	0.285364
$70$	0	0
$71$	196.020	0.327653	0.163826	−	0.986489i	$-0.447616\pi$
$71$	196.020	0.327653	0.163826	+	0.986489i	$0.447616\pi$
$72$	−1390.19	−2.27550
$73$	62.3114	0.0999041	0.0499520	−	0.998752i	$-0.484093\pi$
$73$	62.3114	0.0999041	0.0499520	+	0.998752i	$0.484093\pi$
$74$	1397.50	2.19536
$75$	245.040	0.377264
$76$	−980.847	−1.48041
$77$	0	0
$78$	−221.783	−0.321948
$79$	−432.393	−0.615798	−0.307899	−	0.951419i	$-0.599626\pi$
$79$	−432.393	−0.615798	−0.307899	+	0.951419i	$0.599626\pi$
$80$	218.468	0.305319
$81$	423.958	0.581561
$82$	1290.49	1.73794
$83$	747.945	0.989128	0.494564	−	0.869141i	$-0.335328\pi$
$83$	747.945	0.989128	0.494564	+	0.869141i	$0.335328\pi$
$84$	0	0
$85$	171.721	0.219126
$86$	1892.94	2.37350
$87$	−116.249	−0.143255
$88$	−1114.12	−1.34960
$89$	−327.402	−0.389938	−0.194969	−	0.980809i	$-0.562461\pi$
$89$	−327.402	−0.389938	−0.194969	+	0.980809i	$0.562461\pi$
$90$	−164.582	−0.192761
$91$	0	0
$92$	−1603.22	−1.81682
$93$	−636.064	−0.709212
$94$	732.202	0.803414
$95$	−68.4932	−0.0739711
$96$	−713.394	−0.758442
$97$	−1538.64	−1.61057	−0.805285	−	0.592889i	$-0.797987\pi$
$97$	−1538.64	−1.61057	−0.805285	+	0.592889i	$0.797987\pi$
$98$	0	0
$99$	425.441	0.431903
$100$	−2401.92	−2.40192
$101$	1496.85	1.47467	0.737337	−	0.675525i	$-0.236083\pi$
$101$	1496.85	1.47467	0.737337	+	0.675525i	$0.236083\pi$
$102$	−1315.71	−1.27720
$103$	−1227.47	−1.17423	−0.587117	−	0.809502i	$-0.699737\pi$
$103$	−1227.47	−1.17423	−0.587117	+	0.809502i	$0.699737\pi$
$104$	1282.29	1.20903
$105$	0	0
$106$	−1607.03	−1.47253
$107$	1457.02	1.31641	0.658204	−	0.752840i	$-0.271316\pi$
$107$	1457.02	1.31641	0.658204	+	0.752840i	$0.271316\pi$
$108$	1942.15	1.73041
$109$	608.725	0.534911	0.267456	−	0.963570i	$-0.413817\pi$
$109$	608.725	0.534911	0.267456	+	0.963570i	$0.413817\pi$
$110$	−131.898	−0.114327
$111$	−530.236	−0.453403
$112$	0	0
$113$	−777.486	−0.647254	−0.323627	−	0.946185i	$-0.604902\pi$
$113$	−777.486	−0.647254	−0.323627	+	0.946185i	$0.604902\pi$
$114$	524.789	0.431149
$115$	−111.954	−0.0907806
$116$	1139.49	0.912060
$117$	−489.660	−0.386915
$118$	−2950.80	−2.30206
$119$	0	0
$120$	−163.528	−0.124400
$121$	−990.047	−0.743837
$122$	3271.34	2.42765
$123$	−489.634	−0.358933
$124$	6234.79	4.51533
$125$	−337.981	−0.241840
$126$	0	0
$127$	−1057.64	−0.738979	−0.369490	−	0.929235i	$-0.620467\pi$
$127$	−1057.64	−0.738979	−0.369490	+	0.929235i	$0.620467\pi$
$128$	263.093	0.181675
$129$	−718.214	−0.490195
$130$	151.808	0.102419
$131$	122.057	0.0814056	0.0407028	−	0.999171i	$-0.487040\pi$
$131$	122.057	0.0814056	0.0407028	+	0.999171i	$0.487040\pi$
$132$	716.653	0.472550
$133$	0	0
$134$	4843.35	3.12240
$135$	135.622	0.0864627
$136$	7607.10	4.79635
$137$	1383.08	0.862516	0.431258	−	0.902229i	$-0.358070\pi$
$137$	1383.08	0.862516	0.431258	+	0.902229i	$0.358070\pi$
$138$	857.782	0.529125
$139$	583.524	0.356071	0.178036	−	0.984024i	$-0.443026\pi$
$139$	583.524	0.356071	0.178036	+	0.984024i	$0.443026\pi$
$140$	0	0
$141$	−277.810	−0.165928
$142$	1028.03	0.607537
$143$	−392.419	−0.229480
$144$	−3695.67	−2.13870
$145$	79.5712	0.0455726
$146$	326.792	0.185243
$147$	0	0
$148$	5197.45	2.88667
$149$	459.659	0.252730	0.126365	−	0.991984i	$-0.459669\pi$
$149$	459.659	0.252730	0.126365	+	0.991984i	$0.459669\pi$
$150$	1285.11	0.699526
$151$	−146.978	−0.0792112	−0.0396056	−	0.999215i	$-0.512610\pi$
$151$	−146.978	−0.0792112	−0.0396056	+	0.999215i	$0.512610\pi$
$152$	−3034.19	−1.61912
$153$	−2904.88	−1.53494
$154$	0	0
$155$	435.379	0.225616
$156$	−824.830	−0.423329
$157$	3320.20	1.68777	0.843887	−	0.536520i	$-0.180261\pi$
$157$	3320.20	1.68777	0.843887	+	0.536520i	$0.180261\pi$
$158$	−2267.69	−1.14182
$159$	609.733	0.304119
$160$	488.310	0.241277
$161$	0	0
$162$	2223.45	1.07834
$163$	1650.53	0.793124	0.396562	−	0.918008i	$-0.370203\pi$
$163$	1650.53	0.793124	0.396562	+	0.918008i	$0.370203\pi$
$164$	4799.46	2.28521
$165$	50.0443	0.0236118
$166$	3922.60	1.83405
$167$	1712.90	0.793701	0.396850	−	0.917883i	$-0.370103\pi$
$167$	1712.90	0.793701	0.396850	+	0.917883i	$0.370103\pi$
$168$	0	0
$169$	−1745.35	−0.794423
$170$	900.590	0.406307
$171$	1158.65	0.518153
$172$	7040.03	3.12092
$173$	786.523	0.345655	0.172827	−	0.984952i	$-0.444710\pi$
$173$	786.523	0.345655	0.172827	+	0.984952i	$0.444710\pi$
$174$	−609.668	−0.265625
$175$	0	0
$176$	−2961.75	−1.26847
$177$	1119.58	0.475440
$178$	−1717.06	−0.723028
$179$	501.066	0.209226	0.104613	−	0.994513i	$-0.466640\pi$
$179$	501.066	0.209226	0.104613	+	0.994513i	$0.466640\pi$
$180$	−612.097	−0.253461
$181$	−1311.98	−0.538776	−0.269388	−	0.963032i	$-0.586821\pi$
$181$	−1311.98	−0.538776	−0.269388	+	0.963032i	$0.586821\pi$
$182$	0	0
$183$	−1241.20	−0.501378
$184$	−4959.48	−1.98705
$185$	362.941	0.144238
$186$	−3335.84	−1.31503
$187$	−2328.00	−0.910376
$188$	2723.13	1.05641
$189$	0	0
$190$	−359.212	−0.137158
$191$	−672.508	−0.254770	−0.127385	−	0.991853i	$-0.540658\pi$
$191$	−672.508	−0.254770	−0.127385	+	0.991853i	$0.540658\pi$
$192$	−1188.04	−0.446558
$193$	−2254.08	−0.840686	−0.420343	−	0.907365i	$-0.638090\pi$
$193$	−2254.08	−0.840686	−0.420343	+	0.907365i	$0.638090\pi$
$194$	−8069.40	−2.98634
$195$	−57.5984	−0.0211523
$196$	0	0
$197$	−979.957	−0.354411	−0.177206	−	0.984174i	$-0.556706\pi$
$197$	−979.957	−0.354411	−0.177206	+	0.984174i	$0.556706\pi$
$198$	2231.22	0.800839
$199$	−3378.08	−1.20334	−0.601672	−	0.798743i	$-0.705499\pi$
$199$	−3378.08	−1.20334	−0.601672	+	0.798743i	$0.705499\pi$
$200$	−7430.18	−2.62697
$201$	−1837.65	−0.644864
$202$	7850.23	2.73436
$203$	0	0
$204$	−4893.25	−1.67939
$205$	335.149	0.114185
$206$	−6437.46	−2.17728
$207$	1893.84	0.635900
$208$	3408.82	1.13634
$209$	928.554	0.307318
$210$	0	0
$211$	1293.93	0.422171	0.211085	−	0.977468i	$-0.432300\pi$
$211$	1293.93	0.422171	0.211085	+	0.977468i	$0.432300\pi$
$212$	−5976.69	−1.93623
$213$	−390.051	−0.125474
$214$	7641.35	2.44090
$215$	491.610	0.155942
$216$	6007.94	1.89254
$217$	0	0
$218$	3192.46	0.991838
$219$	−123.990	−0.0382580
$220$	−490.542	−0.150329
$221$	2679.41	0.815549
$222$	−2780.82	−0.840705
$223$	1871.27	0.561925	0.280962	−	0.959719i	$-0.409346\pi$
$223$	1871.27	0.561925	0.280962	+	0.959719i	$0.409346\pi$
$224$	0	0
$225$	2837.32	0.840687
$226$	−4077.53	−1.20015
$227$	5085.76	1.48702	0.743510	−	0.668724i	$-0.233159\pi$
$227$	5085.76	1.48702	0.743510	+	0.668724i	$0.233159\pi$
$228$	1951.74	0.566917
$229$	−2811.81	−0.811396	−0.405698	−	0.914007i	$-0.632972\pi$
$229$	−2811.81	−0.811396	−0.405698	+	0.914007i	$0.632972\pi$
$230$	−587.143	−0.168326
$231$	0	0
$232$	3524.94	0.997516
$233$	−1172.95	−0.329795	−0.164897	−	0.986311i	$-0.552729\pi$
$233$	−1172.95	−0.329795	−0.164897	+	0.986311i	$0.552729\pi$
$234$	−2568.02	−0.717422
$235$	190.158	0.0527852
$236$	−10974.3	−3.02698
$237$	860.398	0.235818
$238$	0	0
$239$	770.341	0.208490	0.104245	−	0.994552i	$-0.466757\pi$
$239$	770.341	0.208490	0.104245	+	0.994552i	$0.466757\pi$
$240$	−434.719	−0.116921
$241$	−1290.80	−0.345012	−0.172506	−	0.985008i	$-0.555186\pi$
$241$	−1290.80	−0.345012	−0.172506	+	0.985008i	$0.555186\pi$
$242$	−5192.30	−1.37923
$243$	−3532.09	−0.932442
$244$	12166.4	3.19211
$245$	0	0
$246$	−2567.88	−0.665538
$247$	−1068.72	−0.275307
$248$	19286.9	4.93840
$249$	−1488.30	−0.378783
$250$	−1772.54	−0.448422
$251$	−3476.31	−0.874195	−0.437098	−	0.899414i	$-0.643994\pi$
$251$	−3476.31	−0.874195	−0.437098	+	0.899414i	$0.643994\pi$
$252$	0	0
$253$	1517.75	0.377154
$254$	−5546.79	−1.37022
$255$	−341.699	−0.0839137
$256$	−3396.60	−0.829247
$257$	−3272.15	−0.794208	−0.397104	−	0.917774i	$-0.629985\pi$
$257$	−3272.15	−0.794208	−0.397104	+	0.917774i	$0.629985\pi$
$258$	−3766.67	−0.908926
$259$	0	0
$260$	564.588	0.134670
$261$	−1346.05	−0.319227
$262$	640.126	0.150943
$263$	−3494.16	−0.819236	−0.409618	−	0.912257i	$-0.634338\pi$
$263$	−3494.16	−0.819236	−0.409618	+	0.912257i	$0.634338\pi$
$264$	2216.92	0.516827
$265$	−417.356	−0.0967470
$266$	0	0
$267$	651.480	0.149326
$268$	18012.9	4.10564
$269$	−2679.42	−0.607312	−0.303656	−	0.952782i	$-0.598207\pi$
$269$	−2679.42	−0.607312	−0.303656	+	0.952782i	$0.598207\pi$
$270$	711.268	0.160320
$271$	−1413.98	−0.316950	−0.158475	−	0.987363i	$-0.550658\pi$
$271$	−1413.98	−0.316950	−0.158475	+	0.987363i	$0.550658\pi$
$272$	20222.6	4.50800
$273$	0	0
$274$	7253.58	1.59929
$275$	2273.86	0.498614
$276$	3190.17	0.695746
$277$	729.313	0.158195	0.0790977	−	0.996867i	$-0.474796\pi$
$277$	729.313	0.158195	0.0790977	+	0.996867i	$0.474796\pi$
$278$	3060.29	0.660231
$279$	−7364.99	−1.58040
$280$	0	0
$281$	−8055.15	−1.71007	−0.855036	−	0.518569i	$-0.826465\pi$
$281$	−8055.15	−1.71007	−0.855036	+	0.518569i	$0.826465\pi$
$282$	−1456.97	−0.307665
$283$	66.5270	0.0139739	0.00698696	−	0.999976i	$-0.497776\pi$
$283$	66.5270	0.0139739	0.00698696	+	0.999976i	$0.497776\pi$
$284$	3823.34	0.798850
$285$	136.291	0.0283270
$286$	−2058.04	−0.425505
$287$	0	0
$288$	−8260.39	−1.69010
$289$	10982.4	2.23538
$290$	417.311	0.0845013
$291$	3061.66	0.616762
$292$	1215.37	0.243576
$293$	5784.42	1.15334	0.576672	−	0.816976i	$-0.304351\pi$
$293$	5784.42	1.15334	0.576672	+	0.816976i	$0.304351\pi$
$294$	0	0
$295$	−766.342	−0.151248
$296$	16078.0	3.15714
$297$	−1838.61	−0.359215
$298$	2410.68	0.468614
$299$	−1746.85	−0.337869
$300$	4779.45	0.919806
$301$	0	0
$302$	−770.826	−0.146874
$303$	−2978.50	−0.564722
$304$	−8066.05	−1.52178
$305$	849.589	0.159499
$306$	−15234.6	−2.84610
$307$	2066.36	0.384149	0.192074	−	0.981380i	$-0.438479\pi$
$307$	2066.36	0.384149	0.192074	+	0.981380i	$0.438479\pi$
$308$	0	0
$309$	2442.48	0.449669
$310$	2283.35	0.418340
$311$	8619.70	1.57163	0.785817	−	0.618459i	$-0.212243\pi$
$311$	8619.70	1.57163	0.785817	+	0.618459i	$0.212243\pi$
$312$	−2551.56	−0.462993
$313$	−4123.53	−0.744651	−0.372326	−	0.928102i	$-0.621440\pi$
$313$	−4123.53	−0.744651	−0.372326	+	0.928102i	$0.621440\pi$
$314$	17412.8	3.12949
$315$	0	0
$316$	−8433.74	−1.50138
$317$	−9294.74	−1.64683	−0.823414	−	0.567440i	$-0.807934\pi$
$317$	−9294.74	−1.64683	−0.823414	+	0.567440i	$0.807934\pi$
$318$	3197.74	0.563901
$319$	−1078.74	−0.189335
$320$	813.199	0.142060
$321$	−2899.26	−0.504114
$322$	0	0
$323$	−6340.10	−1.09217
$324$	8269.21	1.41790
$325$	−2617.09	−0.446677
$326$	8656.19	1.47062
$327$	−1211.27	−0.204842
$328$	14846.8	2.49933
$329$	0	0
$330$	262.458	0.0437812
$331$	−961.951	−0.159739	−0.0798695	−	0.996805i	$-0.525450\pi$
$331$	−961.951	−0.159739	−0.0798695	+	0.996805i	$0.525450\pi$
$332$	14588.5	2.41159
$333$	−6139.61	−1.01036
$334$	8983.30	1.47169
$335$	1257.85	0.205145
$336$	0	0
$337$	−4253.95	−0.687618	−0.343809	−	0.939040i	$-0.611717\pi$
$337$	−4253.95	−0.687618	−0.343809	+	0.939040i	$0.611717\pi$
$338$	−9153.47	−1.47303
$339$	1547.08	0.247864
$340$	3349.38	0.534252
$341$	−5902.38	−0.937337
$342$	6076.54	0.960764
$343$	0	0
$344$	21777.9	3.41333
$345$	222.772	0.0347641
$346$	4124.92	0.640917
$347$	5860.62	0.906671	0.453335	−	0.891340i	$-0.350234\pi$
$347$	5860.62	0.906671	0.453335	+	0.891340i	$0.350234\pi$
$348$	−2267.41	−0.349270
$349$	813.842	0.124825	0.0624126	−	0.998050i	$-0.480121\pi$
$349$	813.842	0.124825	0.0624126	+	0.998050i	$0.480121\pi$
$350$	0	0
$351$	2116.14	0.321798
$352$	−6619.97	−1.00240
$353$	9786.95	1.47566	0.737828	−	0.674989i	$-0.235851\pi$
$353$	9786.95	1.47566	0.737828	+	0.674989i	$0.235851\pi$
$354$	5871.65	0.881567
$355$	266.986	0.0399159
$356$	−6385.90	−0.950709
$357$	0	0
$358$	2627.84	0.387949
$359$	6992.48	1.02799	0.513996	−	0.857793i	$-0.328165\pi$
$359$	6992.48	1.02799	0.513996	+	0.857793i	$0.328165\pi$
$360$	−1893.49	−0.277210
$361$	−4330.17	−0.631312
$362$	−6880.66	−0.999004
$363$	1970.05	0.284850
$364$	0	0
$365$	84.8701	0.0121707
$366$	−6509.48	−0.929661
$367$	926.124	0.131725	0.0658627	−	0.997829i	$-0.479020\pi$
$367$	926.124	0.131725	0.0658627	+	0.997829i	$0.479020\pi$
$368$	−13184.2	−1.86759
$369$	−5669.47	−0.799840
$370$	1903.44	0.267447
$371$	0	0
$372$	−12406.3	−1.72913
$373$	−8784.82	−1.21947	−0.609733	−	0.792607i	$-0.708723\pi$
$373$	−8784.82	−1.21947	−0.609733	+	0.792607i	$0.708723\pi$
$374$	−12209.2	−1.68803
$375$	672.532	0.0926117
$376$	8423.84	1.15539
$377$	1241.57	0.169613
$378$	0	0
$379$	4146.96	0.562045	0.281023	−	0.959701i	$-0.409326\pi$
$379$	4146.96	0.562045	0.281023	+	0.959701i	$0.409326\pi$
$380$	−1335.95	−0.180349
$381$	2104.54	0.282990
$382$	−3526.97	−0.472397
$383$	−6301.72	−0.840738	−0.420369	−	0.907353i	$-0.638099\pi$
$383$	−6301.72	−0.840738	−0.420369	+	0.907353i	$0.638099\pi$
$384$	−523.516	−0.0695718
$385$	0	0
$386$	−11821.5	−1.55881
$387$	−8316.20	−1.09234
$388$	−30010.9	−3.92673
$389$	−1229.28	−0.160223	−0.0801115	−	0.996786i	$-0.525528\pi$
$389$	−1229.28	−0.160223	−0.0801115	+	0.996786i	$0.525528\pi$
$390$	−302.075	−0.0392209
$391$	−10363.1	−1.34037
$392$	0	0
$393$	−242.874	−0.0311740
$394$	−5139.39	−0.657154
$395$	−588.933	−0.0750189
$396$	8298.13	1.05302
$397$	5683.55	0.718512	0.359256	−	0.933239i	$-0.383031\pi$
$397$	5683.55	0.718512	0.359256	+	0.933239i	$0.383031\pi$
$398$	−17716.3	−2.23125
$399$	0	0
$400$	−19752.3	−2.46904
$401$	7640.35	0.951474	0.475737	−	0.879588i	$-0.342181\pi$
$401$	7640.35	0.951474	0.475737	+	0.879588i	$0.342181\pi$
$402$	−9637.55	−1.19571
$403$	6793.33	0.839702
$404$	29195.7	3.59540
$405$	577.444	0.0708480
$406$	0	0
$407$	−4920.35	−0.599245
$408$	−15137.0	−1.83675
$409$	−8815.09	−1.06572	−0.532858	−	0.846204i	$-0.678882\pi$
$409$	−8815.09	−1.06572	−0.532858	+	0.846204i	$0.678882\pi$
$410$	1757.69	0.211722
$411$	−2752.13	−0.330298
$412$	−23941.5	−2.86290
$413$	0	0
$414$	9932.27	1.17909
$415$	1018.72	0.120499
$416$	7619.24	0.897990
$417$	−1161.13	−0.136356
$418$	4869.80	0.569832
$419$	−13793.6	−1.60826	−0.804129	−	0.594454i	$-0.797368\pi$
$419$	−13793.6	−1.60826	−0.804129	+	0.594454i	$0.797368\pi$
$420$	0	0
$421$	−745.966	−0.0863567	−0.0431783	−	0.999067i	$-0.513748\pi$
$421$	−745.966	−0.0863567	−0.0431783	+	0.999067i	$0.513748\pi$
$422$	6786.03	0.782793
$423$	−3216.76	−0.369750
$424$	−18488.5	−2.11765
$425$	−15525.7	−1.77202
$426$	−2045.62	−0.232655
$427$	0	0
$428$	28418.9	3.20953
$429$	780.855	0.0878788
$430$	2578.25	0.289149
$431$	7841.86	0.876401	0.438201	−	0.898877i	$-0.355616\pi$
$431$	7841.86	0.876401	0.438201	+	0.898877i	$0.355616\pi$
$432$	15971.4	1.77876
$433$	8305.38	0.921781	0.460890	−	0.887457i	$-0.347530\pi$
$433$	8305.38	0.921781	0.460890	+	0.887457i	$0.347530\pi$
$434$	0	0
$435$	−158.335	−0.0174519
$436$	11873.1	1.30417
$437$	4133.45	0.452470
$438$	−650.268	−0.0709383
$439$	10086.7	1.09661	0.548306	−	0.836278i	$-0.315273\pi$
$439$	10086.7	1.09661	0.548306	+	0.836278i	$0.315273\pi$
$440$	−1517.46	−0.164414
$441$	0	0
$442$	14052.1	1.51220
$443$	−12664.8	−1.35829	−0.679146	−	0.734004i	$-0.737649\pi$
$443$	−12664.8	−1.35829	−0.679146	+	0.734004i	$0.737649\pi$
$444$	−10342.1	−1.10544
$445$	−445.932	−0.0475038
$446$	9813.86	1.04193
$447$	−914.653	−0.0967821
$448$	0	0
$449$	−9220.30	−0.969116	−0.484558	−	0.874759i	$-0.661020\pi$
$449$	−9220.30	−0.969116	−0.484558	+	0.874759i	$0.661020\pi$
$450$	14880.3	1.55881
$451$	−4543.58	−0.474387
$452$	−15164.7	−1.57807
$453$	292.464	0.0303337
$454$	26672.3	2.75725
$455$	0	0
$456$	6037.59	0.620035
$457$	15054.6	1.54097	0.770486	−	0.637457i	$-0.220014\pi$
$457$	15054.6	1.54097	0.770486	+	0.637457i	$0.220014\pi$
$458$	−14746.6	−1.50450
$459$	12553.9	1.27661
$460$	−2183.64	−0.221332
$461$	2093.39	0.211495	0.105747	−	0.994393i	$-0.466277\pi$
$461$	2093.39	0.211495	0.105747	+	0.994393i	$0.466277\pi$
$462$	0	0
$463$	7411.65	0.743949	0.371975	−	0.928243i	$-0.378681\pi$
$463$	7411.65	0.743949	0.371975	+	0.928243i	$0.378681\pi$
$464$	9370.65	0.937547
$465$	−866.339	−0.0863989
$466$	−6151.51	−0.611509
$467$	−9250.57	−0.916628	−0.458314	−	0.888790i	$-0.651546\pi$
$467$	−9250.57	−0.916628	−0.458314	+	0.888790i	$0.651546\pi$
$468$	−9550.72	−0.943338
$469$	0	0
$470$	997.283	0.0978749
$471$	−6606.70	−0.646328
$472$	−33948.3	−3.31059
$473$	−6664.70	−0.647871
$474$	4512.36	0.437256
$475$	6192.65	0.598186
$476$	0	0
$477$	7060.10	0.677694
$478$	4040.05	0.386585
$479$	−8154.77	−0.777873	−0.388936	−	0.921265i	$-0.627157\pi$
$479$	−8154.77	−0.777873	−0.388936	+	0.921265i	$0.627157\pi$
$480$	−971.665	−0.0923963
$481$	5663.06	0.536826
$482$	−6769.62	−0.639725
$483$	0	0
$484$	−19310.7	−1.81355
$485$	−2095.68	−0.196206
$486$	−18524.0	−1.72894
$487$	12949.9	1.20496	0.602479	−	0.798135i	$-0.294180\pi$
$487$	12949.9	1.20496	0.602479	+	0.798135i	$0.294180\pi$
$488$	37636.1	3.49120
$489$	−3284.30	−0.303725
$490$	0	0
$491$	1928.10	0.177218	0.0886091	−	0.996066i	$-0.471758\pi$
$491$	1928.10	0.177218	0.0886091	+	0.996066i	$0.471758\pi$
$492$	−9550.20	−0.875115
$493$	7365.54	0.672875
$494$	−5604.89	−0.510477
$495$	579.464	0.0526161
$496$	51272.1	4.64150
$497$	0	0
$498$	−7805.38	−0.702345
$499$	−2068.51	−0.185570	−0.0927850	−	0.995686i	$-0.529577\pi$
$499$	−2068.51	−0.185570	−0.0927850	+	0.995686i	$0.529577\pi$
$500$	−6592.25	−0.589629
$501$	−3408.41	−0.303945
$502$	−18231.5	−1.62094
$503$	4151.98	0.368047	0.184023	−	0.982922i	$-0.441088\pi$
$503$	4151.98	0.368047	0.184023	+	0.982922i	$0.441088\pi$
$504$	0	0
$505$	2038.76	0.179650
$506$	7959.83	0.699323
$507$	3472.98	0.304222
$508$	−20629.1	−1.80170
$509$	9669.43	0.842024	0.421012	−	0.907055i	$-0.361675\pi$
$509$	9669.43	0.842024	0.421012	+	0.907055i	$0.361675\pi$
$510$	−1792.04	−0.155594
$511$	0	0
$512$	−19918.2	−1.71927
$513$	−5007.28	−0.430949
$514$	−17160.8	−1.47263
$515$	−1671.85	−0.143050
$516$	−14008.6	−1.19514
$517$	−2577.95	−0.219300
$518$	0	0
$519$	−1565.06	−0.132367
$520$	1746.52	0.147288
$521$	3992.34	0.335715	0.167857	−	0.985811i	$-0.446315\pi$
$521$	3992.34	0.335715	0.167857	+	0.985811i	$0.446315\pi$
$522$	−7059.35	−0.591915
$523$	14741.1	1.23247	0.616237	−	0.787561i	$-0.288657\pi$
$523$	14741.1	1.23247	0.616237	+	0.787561i	$0.288657\pi$
$524$	2380.69	0.198475
$525$	0	0
$526$	−18325.1	−1.51904
$527$	40301.0	3.33120
$528$	5893.44	0.485756
$529$	−5410.76	−0.444708
$530$	−2188.82	−0.179389
$531$	12963.6	1.05946
$532$	0	0
$533$	5229.42	0.424974
$534$	3416.69	0.276881
$535$	1984.51	0.160370
$536$	55721.8	4.49033
$537$	−997.047	−0.0801224
$538$	−14052.2	−1.12609
$539$	0	0
$540$	2645.27	0.210805
$541$	−17751.2	−1.41069	−0.705346	−	0.708864i	$-0.749208\pi$
$541$	−17751.2	−1.41069	−0.705346	+	0.708864i	$0.749208\pi$
$542$	−7415.64	−0.587692
$543$	2610.64	0.206322
$544$	45200.6	3.56243
$545$	829.103	0.0651649
$546$	0	0
$547$	12013.5	0.939049	0.469525	−	0.882919i	$-0.344425\pi$
$547$	12013.5	0.939049	0.469525	+	0.882919i	$0.344425\pi$
$548$	26976.7	2.10290
$549$	−14371.9	−1.11726
$550$	11925.3	0.924536
$551$	−2937.84	−0.227144
$552$	9868.61	0.760935
$553$	0	0
$554$	3824.88	0.293328
$555$	−722.198	−0.0552353
$556$	11381.5	0.868137
$557$	−18320.4	−1.39365	−0.696824	−	0.717242i	$-0.745404\pi$
$557$	−18320.4	−1.39365	−0.696824	+	0.717242i	$0.745404\pi$
$558$	−38625.7	−2.93039
$559$	7670.72	0.580388
$560$	0	0
$561$	4632.37	0.348625
$562$	−42245.3	−3.17083
$563$	16476.5	1.23339	0.616697	−	0.787200i	$-0.288470\pi$
$563$	16476.5	1.23339	0.616697	+	0.787200i	$0.288470\pi$
$564$	−5418.62	−0.404548
$565$	−1058.96	−0.0788510
$566$	348.901	0.0259106
$567$	0	0
$568$	11827.3	0.873700
$569$	−9794.41	−0.721622	−0.360811	−	0.932639i	$-0.617500\pi$
$569$	−9794.41	−0.721622	−0.360811	+	0.932639i	$0.617500\pi$
$570$	714.779	0.0525242
$571$	−13658.5	−1.00104	−0.500518	−	0.865726i	$-0.666857\pi$
$571$	−13658.5	−1.00104	−0.500518	+	0.865726i	$0.666857\pi$
$572$	−7654.05	−0.559496
$573$	1338.19	0.0975632
$574$	0	0
$575$	10122.1	0.734120
$576$	−13756.3	−0.995103
$577$	1695.84	0.122355	0.0611773	−	0.998127i	$-0.480515\pi$
$577$	1695.84	0.122355	0.0611773	+	0.998127i	$0.480515\pi$
$578$	57597.2	4.14486
$579$	4485.29	0.321938
$580$	1552.02	0.111111
$581$	0	0
$582$	16056.9	1.14361
$583$	5658.04	0.401942
$584$	3759.68	0.266398
$585$	−666.932	−0.0471355
$586$	30336.4	2.13854
$587$	−11850.5	−0.833260	−0.416630	−	0.909076i	$-0.636789\pi$
$587$	−11850.5	−0.833260	−0.416630	+	0.909076i	$0.636789\pi$
$588$	0	0
$589$	−16074.6	−1.12452
$590$	−4019.08	−0.280446
$591$	1949.97	0.135721
$592$	42741.5	2.96734
$593$	−2706.06	−0.187394	−0.0936968	−	0.995601i	$-0.529868\pi$
$593$	−2706.06	−0.187394	−0.0936968	+	0.995601i	$0.529868\pi$
$594$	−9642.58	−0.666061
$595$	0	0
$596$	8965.56	0.616180
$597$	6721.86	0.460817
$598$	−9161.35	−0.626481
$599$	−19463.9	−1.32767	−0.663833	−	0.747881i	$-0.731072\pi$
$599$	−19463.9	−1.32767	−0.663833	+	0.747881i	$0.731072\pi$
$600$	14785.0	1.00599
$601$	19356.7	1.31377	0.656884	−	0.753991i	$-0.271874\pi$
$601$	19356.7	1.31377	0.656884	+	0.753991i	$0.271874\pi$
$602$	0	0
$603$	−21278.1	−1.43700
$604$	−2866.78	−0.193125
$605$	−1348.48	−0.0906171
$606$	−15620.8	−1.04711
$607$	−16016.8	−1.07101	−0.535504	−	0.844532i	$-0.679878\pi$
$607$	−16016.8	−1.07101	−0.535504	+	0.844532i	$0.679878\pi$
$608$	−18028.9	−1.20258
$609$	0	0
$610$	4455.67	0.295746
$611$	2967.08	0.196457
$612$	−56659.0	−3.74233
$613$	−28153.7	−1.85501	−0.927503	−	0.373816i	$-0.878049\pi$
$613$	−28153.7	−1.85501	−0.927503	+	0.373816i	$0.878049\pi$
$614$	10837.0	0.712293
$615$	−666.896	−0.0437266
$616$	0	0
$617$	22850.4	1.49096	0.745481	−	0.666526i	$-0.232220\pi$
$617$	22850.4	1.49096	0.745481	+	0.666526i	$0.232220\pi$
$618$	12809.6	0.833781
$619$	5432.81	0.352768	0.176384	−	0.984321i	$-0.443560\pi$
$619$	5432.81	0.352768	0.176384	+	0.984321i	$0.443560\pi$
$620$	8491.98	0.550074
$621$	−8184.54	−0.528880
$622$	45206.0	2.91414
$623$	0	0
$624$	−6783.04	−0.435158
$625$	14932.8	0.955697
$626$	−21625.9	−1.38074
$627$	−1847.68	−0.117686
$628$	64759.8	4.11496
$629$	33595.8	2.12965
$630$	0	0
$631$	21058.9	1.32859	0.664296	−	0.747470i	$-0.268731\pi$
$631$	21058.9	1.32859	0.664296	+	0.747470i	$0.268731\pi$
$632$	−26089.3	−1.64205
$633$	−2574.73	−0.161669
$634$	−48746.3	−3.05357
$635$	−1440.54	−0.0900253
$636$	11892.7	0.741473
$637$	0	0
$638$	−5657.44	−0.351066
$639$	−4516.41	−0.279603
$640$	358.341	0.0221323
$641$	2484.39	0.153085	0.0765424	−	0.997066i	$-0.475612\pi$
$641$	2484.39	0.153085	0.0765424	+	0.997066i	$0.475612\pi$
$642$	−15205.2	−0.934734
$643$	−27818.0	−1.70612	−0.853059	−	0.521814i	$-0.825256\pi$
$643$	−27818.0	−1.70612	−0.853059	+	0.521814i	$0.825256\pi$
$644$	0	0
$645$	−978.230	−0.0597175
$646$	−33250.6	−2.02512
$647$	−7293.46	−0.443177	−0.221589	−	0.975140i	$-0.571124\pi$
$647$	−7293.46	−0.443177	−0.221589	+	0.975140i	$0.571124\pi$
$648$	25580.3	1.55076
$649$	10389.2	0.628370
$650$	−13725.3	−0.828234
$651$	0	0
$652$	32193.2	1.93372
$653$	−13485.2	−0.808142	−0.404071	−	0.914728i	$-0.632405\pi$
$653$	−13485.2	−0.808142	−0.404071	+	0.914728i	$0.632405\pi$
$654$	−6352.52	−0.379821
$655$	166.245	0.00991714
$656$	39468.6	2.34907
$657$	−1435.69	−0.0852533
$658$	0	0
$659$	−24793.2	−1.46556	−0.732782	−	0.680464i	$-0.761778\pi$
$659$	−24793.2	−1.46556	−0.732782	+	0.680464i	$0.761778\pi$
$660$	976.104	0.0575679
$661$	−8289.36	−0.487774	−0.243887	−	0.969804i	$-0.578423\pi$
$661$	−8289.36	−0.487774	−0.243887	+	0.969804i	$0.578423\pi$
$662$	−5044.95	−0.296190
$663$	−5331.62	−0.312312
$664$	45128.7	2.63755
$665$	0	0
$666$	−32199.2	−1.87341
$667$	−4801.99	−0.278761
$668$	33409.7	1.93512
$669$	−3723.54	−0.215187
$670$	6596.80	0.380383
$671$	−11517.8	−0.662651
$672$	0	0
$673$	−15723.5	−0.900591	−0.450296	−	0.892880i	$-0.648681\pi$
$673$	−15723.5	−0.900591	−0.450296	+	0.892880i	$0.648681\pi$
$674$	−22309.8	−1.27499
$675$	−12261.9	−0.699202
$676$	−34042.6	−1.93688
$677$	−33557.3	−1.90504	−0.952518	−	0.304481i	$-0.901517\pi$
$677$	−33557.3	−1.90504	−0.952518	+	0.304481i	$0.901517\pi$
$678$	8113.67	0.459592
$679$	0	0
$680$	10361.1	0.584309
$681$	−10119.9	−0.569450
$682$	−30955.1	−1.73802
$683$	29679.0	1.66272	0.831359	−	0.555736i	$-0.187564\pi$
$683$	29679.0	1.66272	0.831359	+	0.555736i	$0.187564\pi$
$684$	22599.2	1.26331
$685$	1883.80	0.105075
$686$	0	0
$687$	5595.08	0.310722
$688$	57894.1	3.20813
$689$	−6512.11	−0.360075
$690$	1168.33	0.0644601
$691$	2310.28	0.127188	0.0635941	−	0.997976i	$-0.479744\pi$
$691$	2310.28	0.127188	0.0635941	+	0.997976i	$0.479744\pi$
$692$	15341.0	0.842740
$693$	0	0
$694$	30736.1	1.68116
$695$	794.779	0.0433779
$696$	−7014.11	−0.381996
$697$	31023.2	1.68592
$698$	4268.20	0.231452
$699$	2333.99	0.126294
$700$	0	0
$701$	739.790	0.0398594	0.0199297	−	0.999801i	$-0.493656\pi$
$701$	739.790	0.0398594	0.0199297	+	0.999801i	$0.493656\pi$
$702$	11098.1	0.596682
$703$	−13400.1	−0.718912
$704$	−11024.5	−0.590198
$705$	−378.385	−0.0202139
$706$	51327.7	2.73618
$707$	0	0
$708$	21837.2	1.15917
$709$	−4249.22	−0.225082	−0.112541	−	0.993647i	$-0.535899\pi$
$709$	−4249.22	−0.225082	−0.112541	+	0.993647i	$0.535899\pi$
$710$	1400.21	0.0740125
$711$	9962.56	0.525492
$712$	−19754.4	−1.03979
$713$	−26274.4	−1.38006
$714$	0	0
$715$	−534.487	−0.0279562
$716$	9773.20	0.510114
$717$	−1532.86	−0.0798407
$718$	36672.1	1.90611
$719$	−30944.9	−1.60508	−0.802539	−	0.596600i	$-0.796518\pi$
$719$	−30944.9	−1.60508	−0.802539	+	0.596600i	$0.796518\pi$
$720$	−5033.62	−0.260544
$721$	0	0
$722$	−22709.6	−1.17059
$723$	2568.50	0.132121
$724$	−25589.8	−1.31359
$725$	−7194.24	−0.368534
$726$	10331.9	0.528172
$727$	−10047.8	−0.512588	−0.256294	−	0.966599i	$-0.582502\pi$
$727$	−10047.8	−0.512588	−0.256294	+	0.966599i	$0.582502\pi$
$728$	0	0
$729$	−4418.54	−0.224485
$730$	445.101	0.0225670
$731$	45506.0	2.30246
$732$	−24209.4	−1.22241
$733$	−16434.6	−0.828137	−0.414069	−	0.910246i	$-0.635893\pi$
$733$	−16434.6	−0.828137	−0.414069	+	0.910246i	$0.635893\pi$
$734$	4857.06	0.244247
$735$	0	0
$736$	−29468.7	−1.47586
$737$	−17052.5	−0.852291
$738$	−29733.6	−1.48307
$739$	−7973.72	−0.396912	−0.198456	−	0.980110i	$-0.563593\pi$
$739$	−7973.72	−0.396912	−0.198456	+	0.980110i	$0.563593\pi$
$740$	7079.09	0.351666
$741$	2126.59	0.105428
$742$	0	0
$743$	−29507.6	−1.45697	−0.728486	−	0.685061i	$-0.759776\pi$
$743$	−29507.6	−1.45697	−0.728486	+	0.685061i	$0.759776\pi$
$744$	−38378.1	−1.89114
$745$	626.070	0.0307885
$746$	−46072.0	−2.26115
$747$	−17233.0	−0.844074
$748$	−45407.2	−2.21959
$749$	0	0
$750$	3527.09	0.171722
$751$	34617.6	1.68204	0.841022	−	0.541001i	$-0.181954\pi$
$751$	34617.6	1.68204	0.841022	+	0.541001i	$0.181954\pi$
$752$	22393.8	1.08593
$753$	6917.35	0.334770
$754$	6511.42	0.314499
$755$	−200.189	−0.00964981
$756$	0	0
$757$	−8402.94	−0.403448	−0.201724	−	0.979442i	$-0.564654\pi$
$757$	−8402.94	−0.403448	−0.201724	+	0.979442i	$0.564654\pi$
$758$	21748.8	1.04215
$759$	−3020.09	−0.144430
$760$	−4132.67	−0.197247
$761$	−20792.2	−0.990431	−0.495216	−	0.868770i	$-0.664911\pi$
$761$	−20792.2	−0.990431	−0.495216	+	0.868770i	$0.664911\pi$
$762$	11037.3	0.524723
$763$	0	0
$764$	−13117.1	−0.621154
$765$	−3956.53	−0.186992
$766$	−33049.4	−1.55891
$767$	−11957.4	−0.562918
$768$	6758.72	0.317558
$769$	−18646.1	−0.874375	−0.437188	−	0.899370i	$-0.644025\pi$
$769$	−18646.1	−0.874375	−0.437188	+	0.899370i	$0.644025\pi$
$770$	0	0
$771$	6511.10	0.304139
$772$	−43965.4	−2.04968
$773$	27222.3	1.26665	0.633323	−	0.773888i	$-0.281691\pi$
$773$	27222.3	1.26665	0.633323	+	0.773888i	$0.281691\pi$
$774$	−43614.3	−2.02543
$775$	−39363.8	−1.82450
$776$	−92836.8	−4.29465
$777$	0	0
$778$	−6446.94	−0.297087
$779$	−12374.0	−0.569121
$780$	−1123.44	−0.0515715
$781$	−3619.50	−0.165833
$782$	−54349.1	−2.48532
$783$	5817.16	0.265502
$784$	0	0
$785$	4522.22	0.205611
$786$	−1273.75	−0.0578032
$787$	24604.9	1.11445	0.557223	−	0.830363i	$-0.311867\pi$
$787$	24604.9	1.11445	0.557223	+	0.830363i	$0.311867\pi$
$788$	−19113.9	−0.864090
$789$	6952.85	0.313724
$790$	−3088.66	−0.139101
$791$	0	0
$792$	25669.8	1.15169
$793$	13256.4	0.593628
$794$	29807.4	1.33227
$795$	830.475	0.0370490
$796$	−65888.7	−2.93387
$797$	−5763.61	−0.256157	−0.128079	−	0.991764i	$-0.540881\pi$
$797$	−5763.61	−0.256157	−0.128079	+	0.991764i	$0.540881\pi$
$798$	0	0
$799$	17602.0	0.779368
$800$	−44149.4	−1.95115
$801$	7543.50	0.332755
$802$	40069.9	1.76423
$803$	−1150.57	−0.0505640
$804$	−35843.0	−1.57224
$805$	0	0
$806$	35627.6	1.55699
$807$	5331.64	0.232568
$808$	90315.3	3.93228
$809$	15136.3	0.657805	0.328902	−	0.944364i	$-0.393321\pi$
$809$	15136.3	0.657805	0.328902	+	0.944364i	$0.393321\pi$
$810$	3028.41	0.131367
$811$	4231.60	0.183220	0.0916101	−	0.995795i	$-0.470799\pi$
$811$	4231.60	0.183220	0.0916101	+	0.995795i	$0.470799\pi$
$812$	0	0
$813$	2813.62	0.121375
$814$	−25804.8	−1.11113
$815$	2248.07	0.0966215
$816$	−40239.9	−1.72632
$817$	−18150.7	−0.777249
$818$	−46230.8	−1.97606
$819$	0	0
$820$	6537.02	0.278393
$821$	−9874.29	−0.419751	−0.209875	−	0.977728i	$-0.567306\pi$
$821$	−9874.29	−0.419751	−0.209875	+	0.977728i	$0.567306\pi$
$822$	−14433.5	−0.612442
$823$	20097.2	0.851207	0.425604	−	0.904910i	$-0.360062\pi$
$823$	20097.2	0.851207	0.425604	+	0.904910i	$0.360062\pi$
$824$	−74061.7	−3.13114
$825$	−4524.64	−0.190943
$826$	0	0
$827$	−29525.9	−1.24149	−0.620747	−	0.784011i	$-0.713171\pi$
$827$	−29525.9	−1.24149	−0.620747	+	0.784011i	$0.713171\pi$
$828$	36939.1	1.55039
$829$	−34453.4	−1.44344	−0.721722	−	0.692183i	$-0.756649\pi$
$829$	−34453.4	−1.44344	−0.721722	+	0.692183i	$0.756649\pi$
$830$	5342.70	0.223431
$831$	−1451.22	−0.0605805
$832$	12688.6	0.528722
$833$	0	0
$834$	−6089.53	−0.252833
$835$	2333.02	0.0966917
$836$	18111.3	0.749271
$837$	31828.9	1.31442
$838$	−72340.4	−2.98205
$839$	−39604.6	−1.62968	−0.814840	−	0.579686i	$-0.803175\pi$
$839$	−39604.6	−1.62968	−0.814840	+	0.579686i	$0.803175\pi$
$840$	0	0
$841$	−20976.0	−0.860059
$842$	−3912.22	−0.160124
$843$	16028.5	0.654866
$844$	25237.9	1.02929
$845$	−2377.22	−0.0967796
$846$	−16870.3	−0.685595
$847$	0	0
$848$	−49149.6	−1.99034
$849$	−132.379	−0.00535127
$850$	−81424.7	−3.28570
$851$	−21902.9	−0.882281
$852$	−7607.87	−0.305917
$853$	18796.9	0.754507	0.377254	−	0.926110i	$-0.376869\pi$
$853$	18796.9	0.754507	0.377254	+	0.926110i	$0.376869\pi$
$854$	0	0
$855$	1578.12	0.0631233
$856$	87912.2	3.51026
$857$	20826.4	0.830123	0.415062	−	0.909793i	$-0.363760\pi$
$857$	20826.4	0.830123	0.415062	+	0.909793i	$0.363760\pi$
$858$	4095.19	0.162946
$859$	−32789.1	−1.30239	−0.651194	−	0.758911i	$-0.725731\pi$
$859$	−32789.1	−1.30239	−0.651194	+	0.758911i	$0.725731\pi$
$860$	9588.75	0.380202
$861$	0	0
$862$	41126.6	1.62503
$863$	−27348.7	−1.07875	−0.539374	−	0.842066i	$-0.681339\pi$
$863$	−27348.7	−1.07875	−0.539374	+	0.842066i	$0.681339\pi$
$864$	35698.6	1.40566
$865$	1071.27	0.0421090
$866$	43557.6	1.70918
$867$	−21853.3	−0.856030
$868$	0	0
$869$	7984.10	0.311671
$870$	−830.387	−0.0323595
$871$	19626.6	0.763515
$872$	36728.6	1.42636
$873$	35451.0	1.37438
$874$	21677.9	0.838976
$875$	0	0
$876$	−2418.41	−0.0932767
$877$	20258.9	0.780038	0.390019	−	0.920807i	$-0.372468\pi$
$877$	20258.9	0.780038	0.390019	+	0.920807i	$0.372468\pi$
$878$	52899.8	2.03335
$879$	−11510.1	−0.441669
$880$	−4034.00	−0.154530
$881$	37308.5	1.42674	0.713368	−	0.700790i	$-0.247169\pi$
$881$	37308.5	1.42674	0.713368	+	0.700790i	$0.247169\pi$
$882$	0	0
$883$	8230.79	0.313690	0.156845	−	0.987623i	$-0.449868\pi$
$883$	8230.79	0.313690	0.156845	+	0.987623i	$0.449868\pi$
$884$	52261.3	1.98839
$885$	1524.91	0.0579199
$886$	−66420.6	−2.51856
$887$	−7578.32	−0.286871	−0.143436	−	0.989660i	$-0.545815\pi$
$887$	−7578.32	−0.286871	−0.143436	+	0.989660i	$0.545815\pi$
$888$	−31992.8	−1.20902
$889$	0	0
$890$	−2338.69	−0.0880820
$891$	−7828.34	−0.294343
$892$	36498.7	1.37003
$893$	−7020.80	−0.263093
$894$	−4796.90	−0.179454
$895$	682.468	0.0254887
$896$	0	0
$897$	3475.97	0.129386
$898$	−48355.9	−1.79695
$899$	18674.5	0.692802
$900$	55341.3	2.04968
$901$	−38632.7	−1.42846
$902$	−23828.8	−0.879615
$903$	0	0
$904$	−46911.1	−1.72593
$905$	−1786.95	−0.0656357
$906$	1533.83	0.0562451
$907$	−6966.98	−0.255055	−0.127527	−	0.991835i	$-0.540704\pi$
$907$	−6966.98	−0.255055	−0.127527	+	0.991835i	$0.540704\pi$
$908$	99196.7	3.62551
$909$	−34488.1	−1.25842
$910$	0	0
$911$	−28212.4	−1.02604	−0.513018	−	0.858378i	$-0.671473\pi$
$911$	−28212.4	−1.02604	−0.513018	+	0.858378i	$0.671473\pi$
$912$	16050.2	0.582759
$913$	−13810.7	−0.500623
$914$	78953.8	2.85729
$915$	−1690.55	−0.0610798
$916$	−54843.8	−1.97827
$917$	0	0
$918$	65838.8	2.36711
$919$	2934.95	0.105348	0.0526742	−	0.998612i	$-0.483226\pi$
$919$	2934.95	0.105348	0.0526742	+	0.998612i	$0.483226\pi$
$920$	−6754.96	−0.242070
$921$	−4111.75	−0.147108
$922$	10978.8	0.392156
$923$	4165.85	0.148560
$924$	0	0
$925$	−32814.4	−1.16641
$926$	38870.4	1.37944
$927$	28281.5	1.00203
$928$	20944.9	0.740893
$929$	−41336.3	−1.45985	−0.729924	−	0.683528i	$-0.760445\pi$
$929$	−41336.3	−1.45985	−0.729924	+	0.683528i	$0.760445\pi$
$930$	−4543.52	−0.160202
$931$	0	0
$932$	−22878.1	−0.804073
$933$	−17151.9	−0.601853
$934$	−48514.6	−1.69962
$935$	−3170.81	−0.110905
$936$	−29544.6	−1.03173
$937$	−11917.7	−0.415512	−0.207756	−	0.978181i	$-0.566616\pi$
$937$	−11917.7	−0.415512	−0.207756	+	0.978181i	$0.566616\pi$
$938$	0	0
$939$	8205.21	0.285162
$940$	3708.99	0.128696
$941$	41888.1	1.45113	0.725565	−	0.688154i	$-0.241578\pi$
$941$	41888.1	1.45113	0.725565	+	0.688154i	$0.241578\pi$
$942$	−34648.8	−1.19843
$943$	−20225.7	−0.698451
$944$	−90247.8	−3.11156
$945$	0	0
$946$	−34953.0	−1.20129
$947$	9275.36	0.318277	0.159139	−	0.987256i	$-0.449128\pi$
$947$	9275.36	0.318277	0.159139	+	0.987256i	$0.449128\pi$
$948$	16781.9	0.574948
$949$	1324.25	0.0452972
$950$	32477.3	1.10916
$951$	18495.1	0.630648
$952$	0	0
$953$	29603.5	1.00625	0.503123	−	0.864215i	$-0.332184\pi$
$953$	29603.5	1.00625	0.503123	+	0.864215i	$0.332184\pi$
$954$	37026.7	1.25659
$955$	−915.977	−0.0310370
$956$	15025.3	0.508320
$957$	2146.53	0.0725051
$958$	−42767.7	−1.44234
$959$	0	0
$960$	−1618.14	−0.0544015
$961$	72387.7	2.42985
$962$	29699.9	0.995389
$963$	−33570.5	−1.12336
$964$	−25176.8	−0.841174
$965$	−3070.13	−0.102416
$966$	0	0
$967$	−40626.4	−1.35104	−0.675520	−	0.737342i	$-0.736081\pi$
$967$	−40626.4	−1.35104	−0.675520	+	0.737342i	$0.736081\pi$
$968$	−59736.4	−1.98347
$969$	12615.8	0.418245
$970$	−10990.8	−0.363807
$971$	−21392.7	−0.707029	−0.353515	−	0.935429i	$-0.615014\pi$
$971$	−21392.7	−0.707029	−0.353515	+	0.935429i	$0.615014\pi$
$972$	−68892.6	−2.27339
$973$	0	0
$974$	67915.6	2.23425
$975$	5207.62	0.171054
$976$	100051.	3.28132
$977$	−3351.12	−0.109736	−0.0548679	−	0.998494i	$-0.517474\pi$
$977$	−3351.12	−0.109736	−0.0548679	+	0.998494i	$0.517474\pi$
$978$	−17224.5	−0.563169
$979$	6045.44	0.197358
$980$	0	0
$981$	−14025.3	−0.456467
$982$	10111.9	0.328600
$983$	13663.4	0.443332	0.221666	−	0.975123i	$-0.428851\pi$
$983$	13663.4	0.443332	0.221666	+	0.975123i	$0.428851\pi$
$984$	−29543.0	−0.957110
$985$	−1334.73	−0.0431757
$986$	38628.6	1.24765
$987$	0	0
$988$	−20845.1	−0.671226
$989$	−29667.8	−0.953875
$990$	3039.00	0.0975613
$991$	−2033.13	−0.0651712	−0.0325856	−	0.999469i	$-0.510374\pi$
$991$	−2033.13	−0.0651712	−0.0325856	+	0.999469i	$0.510374\pi$
$992$	114601.	3.66793
$993$	1914.14	0.0611715
$994$	0	0
$995$	−4601.05	−0.146596
$996$	−29029.0	−0.923512
$997$	18352.5	0.582980	0.291490	−	0.956574i	$-0.405849\pi$
$997$	18352.5	0.582980	0.291490	+	0.956574i	$0.405849\pi$
$998$	−10848.3	−0.344086
$999$	26533.3	0.840316

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2401.4.a.d.1.39		39
7.6	odd	2		2401.4.a.c.1.39		39
49.8	even	7		49.4.e.a.15.1	✓	78
49.43	even	7		49.4.e.a.36.1	yes	78

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.15.1	✓	78	49.8	even	7
49.4.e.a.36.1	yes	78	49.43	even	7
2401.4.a.c.1.39		39	7.6	odd	2
2401.4.a.d.1.39		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)

\(f(q)\)	\(=\)	\(q+5.24450 q^{2} -1.98985 q^{3} +19.5048 q^{4} +1.36203 q^{5} -10.4358 q^{6} +60.3369 q^{8} -23.0405 q^{9} +7.14318 q^{10} -18.4649 q^{11} -38.8116 q^{12} +21.2521 q^{13} -2.71024 q^{15} +160.399 q^{16} +126.077 q^{17} -120.836 q^{18} -50.2875 q^{19} +26.5662 q^{20} -96.8392 q^{22} -82.1963 q^{23} -120.061 q^{24} -123.145 q^{25} +111.457 q^{26} +99.5731 q^{27} +58.4210 q^{29} -14.2139 q^{30} +319.654 q^{31} +358.516 q^{32} +36.7424 q^{33} +661.211 q^{34} -449.400 q^{36} +266.470 q^{37} -263.733 q^{38} -42.2886 q^{39} +82.1808 q^{40} +246.066 q^{41} +360.939 q^{43} -360.154 q^{44} -31.3819 q^{45} -431.079 q^{46} +139.613 q^{47} -319.170 q^{48} -645.833 q^{50} -250.874 q^{51} +414.519 q^{52} -306.421 q^{53} +522.211 q^{54} -25.1498 q^{55} +100.065 q^{57} +306.389 q^{58} -562.646 q^{59} -52.8627 q^{60} +623.766 q^{61} +1676.43 q^{62} +597.049 q^{64} +28.9461 q^{65} +192.696 q^{66} +923.511 q^{67} +2459.11 q^{68} +163.558 q^{69} +196.020 q^{71} -1390.19 q^{72} +62.3114 q^{73} +1397.50 q^{74} +245.040 q^{75} -980.847 q^{76} -221.783 q^{78} -432.393 q^{79} +218.468 q^{80} +423.958 q^{81} +1290.49 q^{82} +747.945 q^{83} +171.721 q^{85} +1892.94 q^{86} -116.249 q^{87} -1114.12 q^{88} -327.402 q^{89} -164.582 q^{90} -1603.22 q^{92} -636.064 q^{93} +732.202 q^{94} -68.4932 q^{95} -713.394 q^{96} -1538.64 q^{97} +425.441 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 39 q + q^{2} + q^{3} + 145 q^{4} + 27 q^{5} + 41 q^{6} - 12 q^{8} + 312 q^{9} + 78 q^{10} + q^{11} - 91 q^{12} + 77 q^{13} - 161 q^{15} + 461 q^{16} + 211 q^{17} + 8 q^{18} + 314 q^{19} + 476 q^{20}+ \cdots + 5384 q^{99}+O(q^{100}) \) 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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