LMFDB - Embedded newform 2107.4.a.h.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2107,4,Mod(1,2107)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2107, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2107.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2107 = 7^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2107.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$124.317024382$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	2107.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93072 q^{2} +10.2626 q^{3} -4.27231 q^{4} -8.49116 q^{5} -19.8142 q^{6} +23.6944 q^{8} +78.3210 q^{9} +O(q^{10})$	\(q-1.93072 q^{2} +10.2626 q^{3} -4.27231 q^{4} -8.49116 q^{5} -19.8142 q^{6} +23.6944 q^{8} +78.3210 q^{9} +16.3941 q^{10} +39.2068 q^{11} -43.8450 q^{12} -4.34216 q^{13} -87.1414 q^{15} -11.5690 q^{16} -62.1818 q^{17} -151.216 q^{18} -126.086 q^{19} +36.2768 q^{20} -75.6976 q^{22} -45.4291 q^{23} +243.166 q^{24} -52.9002 q^{25} +8.38352 q^{26} +526.687 q^{27} +24.0276 q^{29} +168.246 q^{30} -237.931 q^{31} -167.219 q^{32} +402.364 q^{33} +120.056 q^{34} -334.611 q^{36} -114.981 q^{37} +243.438 q^{38} -44.5619 q^{39} -201.193 q^{40} +368.331 q^{41} -43.0000 q^{43} -167.504 q^{44} -665.036 q^{45} +87.7110 q^{46} +404.000 q^{47} -118.728 q^{48} +102.136 q^{50} -638.147 q^{51} +18.5510 q^{52} -386.318 q^{53} -1016.89 q^{54} -332.912 q^{55} -1293.98 q^{57} -46.3907 q^{58} -764.340 q^{59} +372.295 q^{60} -552.606 q^{61} +459.378 q^{62} +415.405 q^{64} +36.8700 q^{65} -776.854 q^{66} -582.026 q^{67} +265.660 q^{68} -466.220 q^{69} -369.762 q^{71} +1855.77 q^{72} +493.859 q^{73} +221.996 q^{74} -542.893 q^{75} +538.680 q^{76} +86.0367 q^{78} -824.338 q^{79} +98.2339 q^{80} +3290.51 q^{81} -711.146 q^{82} +1036.80 q^{83} +527.995 q^{85} +83.0211 q^{86} +246.586 q^{87} +928.984 q^{88} -243.275 q^{89} +1284.00 q^{90} +194.087 q^{92} -2441.79 q^{93} -780.013 q^{94} +1070.62 q^{95} -1716.10 q^{96} +99.1014 q^{97} +3070.72 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9}+O(q^{10}) $ 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9	$ 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9} - 60 q^{11} - 96 q^{15} + 372 q^{16} - 1008 q^{18} - 234 q^{22} - 214 q^{23} + 520 q^{25} - 870 q^{29} - 12 q^{30} - 1548 q^{32} + 1142 q^{36} - 1246 q^{37} - 416 q^{39} - 1290 q^{43} - 446 q^{44} - 660 q^{46} - 278 q^{50} - 3702 q^{51} - 2960 q^{53} + 620 q^{57} - 3634 q^{58} - 898 q^{60} + 2578 q^{64} - 4848 q^{65} + 928 q^{67} - 1708 q^{71} - 7900 q^{72} + 1714 q^{74} - 138 q^{78} - 3562 q^{79} + 2210 q^{81} - 948 q^{85} + 344 q^{86} + 2502 q^{88} - 3848 q^{92} - 11986 q^{93} - 2894 q^{95} - 2804 q^{99}+O(q^{100}) $ 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9 - 60 * q^11 - 96 * q^15 + 372 * q^16 - 1008 * q^18 - 234 * q^22 - 214 * q^23 + 520 * q^25 - 870 * q^29 - 12 * q^30 - 1548 * q^32 + 1142 * q^36 - 1246 * q^37 - 416 * q^39 - 1290 * q^43 - 446 * q^44 - 660 * q^46 - 278 * q^50 - 3702 * q^51 - 2960 * q^53 + 620 * q^57 - 3634 * q^58 - 898 * q^60 + 2578 * q^64 - 4848 * q^65 + 928 * q^67 - 1708 * q^71 - 7900 * q^72 + 1714 * q^74 - 138 * q^78 - 3562 * q^79 + 2210 * q^81 - 948 * q^85 + 344 * q^86 + 2502 * q^88 - 3848 * q^92 - 11986 * q^93 - 2894 * q^95 - 2804 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−1.93072	−0.682614	−0.341307	−	0.939952i	$-0.610870\pi$
$2$	−1.93072	−0.682614	−0.341307	+	0.939952i	$0.610870\pi$
$3$	10.2626	1.97504	0.987519	−	0.157499i	$-0.0503430\pi$
$3$	10.2626	1.97504	0.987519	+	0.157499i	$0.0503430\pi$
$4$	−4.27231	−0.534038
$5$	−8.49116	−0.759473	−0.379736	−	0.925095i	$-0.623985\pi$
$5$	−8.49116	−0.759473	−0.379736	+	0.925095i	$0.623985\pi$
$6$	−19.8142	−1.34819
$7$	0	0
$7$	0	0
$8$	23.6944	1.04716
$9$	78.3210	2.90078
$10$	16.3941	0.518427
$11$	39.2068	1.07466	0.537332	−	0.843371i	$-0.319432\pi$
$11$	39.2068	1.07466	0.537332	+	0.843371i	$0.319432\pi$
$12$	−43.8450	−1.05475
$13$	−4.34216	−0.0926384	−0.0463192	−	0.998927i	$-0.514749\pi$
$13$	−4.34216	−0.0926384	−0.0463192	+	0.998927i	$0.514749\pi$
$14$	0	0
$15$	−87.1414	−1.49999
$16$	−11.5690	−0.180765
$17$	−62.1818	−0.887135	−0.443567	−	0.896241i	$-0.646287\pi$
$17$	−62.1818	−0.887135	−0.443567	+	0.896241i	$0.646287\pi$
$18$	−151.216	−1.98011
$19$	−126.086	−1.52243	−0.761216	−	0.648498i	$-0.775397\pi$
$19$	−126.086	−1.52243	−0.761216	+	0.648498i	$0.775397\pi$
$20$	36.2768	0.405587
$21$	0	0
$22$	−75.6976	−0.733581
$23$	−45.4291	−0.411853	−0.205926	−	0.978567i	$-0.566021\pi$
$23$	−45.4291	−0.411853	−0.205926	+	0.978567i	$0.566021\pi$
$24$	243.166	2.06817
$25$	−52.9002	−0.423201
$26$	8.38352	0.0632363
$27$	526.687	3.75411
$28$	0	0
$29$	24.0276	0.153856	0.0769279	−	0.997037i	$-0.475489\pi$
$29$	24.0276	0.153856	0.0769279	+	0.997037i	$0.475489\pi$
$30$	168.246	1.02391
$31$	−237.931	−1.37850	−0.689252	−	0.724522i	$-0.742061\pi$
$31$	−237.931	−1.37850	−0.689252	+	0.724522i	$0.742061\pi$
$32$	−167.219	−0.923763
$33$	402.364	2.12250
$34$	120.056	0.605571
$35$	0	0
$36$	−334.611	−1.54913
$37$	−114.981	−0.510883	−0.255442	−	0.966824i	$-0.582221\pi$
$37$	−114.981	−0.510883	−0.255442	+	0.966824i	$0.582221\pi$
$38$	243.438	1.03923
$39$	−44.5619	−0.182964
$40$	−201.193	−0.795286
$41$	368.331	1.40302	0.701508	−	0.712661i	$-0.252510\pi$
$41$	368.331	1.40302	0.701508	+	0.712661i	$0.252510\pi$
$42$	0	0
$43$	−43.0000	−0.152499
$43$	−43.0000	−0.152499
$44$	−167.504	−0.573912
$45$	−665.036	−2.20306
$46$	87.7110	0.281136
$47$	404.000	1.25382	0.626909	−	0.779092i	$-0.284320\pi$
$47$	404.000	1.25382	0.626909	+	0.779092i	$0.284320\pi$
$48$	−118.728	−0.357018
$49$	0	0
$50$	102.136	0.288883
$51$	−638.147	−1.75213
$52$	18.5510	0.0494724
$53$	−386.318	−1.00122	−0.500611	−	0.865672i	$-0.666891\pi$
$53$	−386.318	−1.00122	−0.500611	+	0.865672i	$0.666891\pi$
$54$	−1016.89	−2.56261
$55$	−332.912	−0.816178
$56$	0	0
$57$	−1293.98	−3.00686
$58$	−46.3907	−0.105024
$59$	−764.340	−1.68659	−0.843293	−	0.537454i	$-0.819386\pi$
$59$	−764.340	−1.68659	−0.843293	+	0.537454i	$0.819386\pi$
$60$	372.295	0.801051
$61$	−552.606	−1.15990	−0.579951	−	0.814652i	$-0.696928\pi$
$61$	−552.606	−1.15990	−0.579951	+	0.814652i	$0.696928\pi$
$62$	459.378	0.940985
$63$	0	0
$64$	415.405	0.811339
$65$	36.8700	0.0703563
$66$	−776.854	−1.44885
$67$	−582.026	−1.06128	−0.530640	−	0.847597i	$-0.678049\pi$
$67$	−582.026	−1.06128	−0.530640	+	0.847597i	$0.678049\pi$
$68$	265.660	0.473764
$69$	−466.220	−0.813425
$70$	0	0
$71$	−369.762	−0.618066	−0.309033	−	0.951051i	$-0.600005\pi$
$71$	−369.762	−0.618066	−0.309033	+	0.951051i	$0.600005\pi$
$72$	1855.77	3.03757
$73$	493.859	0.791806	0.395903	−	0.918292i	$-0.370432\pi$
$73$	493.859	0.791806	0.395903	+	0.918292i	$0.370432\pi$
$74$	221.996	0.348736
$75$	−542.893	−0.835839
$76$	538.680	0.813037
$77$	0	0
$78$	86.0367	0.124894
$79$	−824.338	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$79$	−824.338	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$80$	98.2339	0.137286
$81$	3290.51	4.51373
$82$	−711.146	−0.957718
$83$	1036.80	1.37112	0.685561	−	0.728015i	$-0.259557\pi$
$83$	1036.80	1.37112	0.685561	+	0.728015i	$0.259557\pi$
$84$	0	0
$85$	527.995	0.673755
$86$	83.0211	0.104098
$87$	246.586	0.303871
$88$	928.984	1.12534
$89$	−243.275	−0.289743	−0.144871	−	0.989450i	$-0.546277\pi$
$89$	−243.275	−0.289743	−0.144871	+	0.989450i	$0.546277\pi$
$90$	1284.00	1.50384
$91$	0	0
$92$	194.087	0.219945
$93$	−2441.79	−2.72260
$94$	−780.013	−0.855874
$95$	1070.62	1.15625
$96$	−1716.10	−1.82447
$97$	99.1014	0.103734	0.0518671	−	0.998654i	$-0.483483\pi$
$97$	99.1014	0.103734	0.0518671	+	0.998654i	$0.483483\pi$
$98$	0	0
$99$	3070.72	3.11736
$100$	226.006	0.226006
$101$	542.591	0.534553	0.267276	−	0.963620i	$-0.413876\pi$
$101$	542.591	0.534553	0.267276	+	0.963620i	$0.413876\pi$
$102$	1232.08	1.19603
$103$	147.878	0.141465	0.0707324	−	0.997495i	$-0.477466\pi$
$103$	147.878	0.141465	0.0707324	+	0.997495i	$0.477466\pi$
$104$	−102.885	−0.0970069
$105$	0	0
$106$	745.873	0.683449
$107$	1470.84	1.32889	0.664446	−	0.747336i	$-0.268667\pi$
$107$	1470.84	1.32889	0.664446	+	0.747336i	$0.268667\pi$
$108$	−2250.17	−2.00484
$109$	−1537.61	−1.35116	−0.675579	−	0.737287i	$-0.736106\pi$
$109$	−1537.61	−1.35116	−0.675579	+	0.737287i	$0.736106\pi$
$110$	642.760	0.557134
$111$	−1180.00	−1.00901
$112$	0	0
$113$	−392.578	−0.326820	−0.163410	−	0.986558i	$-0.552249\pi$
$113$	−392.578	−0.326820	−0.163410	+	0.986558i	$0.552249\pi$
$114$	2498.31	2.05253
$115$	385.745	0.312791
$116$	−102.653	−0.0821648
$117$	−340.082	−0.268723
$118$	1475.73	1.15129
$119$	0	0
$120$	−2064.77	−1.57072
$121$	206.176	0.154903
$122$	1066.93	0.791765
$123$	3780.04	2.77101
$124$	1016.51	0.736173
$125$	1510.58	1.08088
$126$	0	0
$127$	469.175	0.327816	0.163908	−	0.986476i	$-0.447590\pi$
$127$	469.175	0.327816	0.163908	+	0.986476i	$0.447590\pi$
$128$	535.719	0.369932
$129$	−441.292	−0.301191
$130$	−71.1858	−0.0480262
$131$	−1954.93	−1.30384	−0.651919	−	0.758289i	$-0.726036\pi$
$131$	−1954.93	−1.30384	−0.651919	+	0.758289i	$0.726036\pi$
$132$	−1719.02	−1.13350
$133$	0	0
$134$	1123.73	0.724445
$135$	−4472.18	−2.85114
$136$	−1473.36	−0.928969
$137$	1526.52	0.951965	0.475983	−	0.879455i	$-0.342093\pi$
$137$	1526.52	0.951965	0.475983	+	0.879455i	$0.342093\pi$
$138$	900.143	0.555255
$139$	−519.878	−0.317234	−0.158617	−	0.987340i	$-0.550703\pi$
$139$	−519.878	−0.317234	−0.158617	+	0.987340i	$0.550703\pi$
$140$	0	0
$141$	4146.09	2.47634
$142$	713.908	0.421900
$143$	−170.242	−0.0995552
$144$	−906.092	−0.524359
$145$	−204.022	−0.116849
$146$	−953.505	−0.540498
$147$	0	0
$148$	491.232	0.272831
$149$	−2524.82	−1.38819	−0.694097	−	0.719881i	$-0.744196\pi$
$149$	−2524.82	−1.38819	−0.694097	+	0.719881i	$0.744196\pi$
$150$	1048.18	0.570555
$151$	−3417.16	−1.84162	−0.920809	−	0.390014i	$-0.872470\pi$
$151$	−3417.16	−1.84162	−0.920809	+	0.390014i	$0.872470\pi$
$152$	−2987.55	−1.59422
$153$	−4870.14	−2.57338
$154$	0	0
$155$	2020.31	1.04694
$156$	190.382	0.0977100
$157$	3052.93	1.55191	0.775956	−	0.630787i	$-0.217268\pi$
$157$	3052.93	1.55191	0.775956	+	0.630787i	$0.217268\pi$
$158$	1591.57	0.801382
$159$	−3964.62	−1.97745
$160$	1419.88	0.701573
$161$	0	0
$162$	−6353.06	−3.08113
$163$	66.9017	0.0321481	0.0160741	−	0.999871i	$-0.494883\pi$
$163$	66.9017	0.0321481	0.0160741	+	0.999871i	$0.494883\pi$
$164$	−1573.62	−0.749264
$165$	−3416.54	−1.61198
$166$	−2001.77	−0.935947
$167$	2335.93	1.08239	0.541196	−	0.840897i	$-0.317972\pi$
$167$	2335.93	1.08239	0.541196	+	0.840897i	$0.317972\pi$
$168$	0	0
$169$	−2178.15	−0.991418
$170$	−1019.41	−0.459914
$171$	−9875.22	−4.41624
$172$	183.709	0.0814401
$173$	168.429	0.0740198	0.0370099	−	0.999315i	$-0.488217\pi$
$173$	168.429	0.0740198	0.0370099	+	0.999315i	$0.488217\pi$
$174$	−476.089	−0.207427
$175$	0	0
$176$	−453.582	−0.194262
$177$	−7844.12	−3.33107
$178$	469.697	0.197782
$179$	−2378.23	−0.993056	−0.496528	−	0.868021i	$-0.665392\pi$
$179$	−2378.23	−0.993056	−0.496528	+	0.868021i	$0.665392\pi$
$180$	2841.24	1.17652
$181$	−2357.45	−0.968109	−0.484054	−	0.875038i	$-0.660836\pi$
$181$	−2357.45	−0.968109	−0.484054	+	0.875038i	$0.660836\pi$
$182$	0	0
$183$	−5671.18	−2.29085
$184$	−1076.42	−0.431274
$185$	976.318	0.388002
$186$	4714.42	1.85848
$187$	−2437.95	−0.953372
$188$	−1726.01	−0.669587
$189$	0	0
$190$	−2067.07	−0.789270
$191$	3493.71	1.32354	0.661770	−	0.749707i	$-0.269806\pi$
$191$	3493.71	1.32354	0.661770	+	0.749707i	$0.269806\pi$
$192$	4263.14	1.60242
$193$	1011.70	0.377323	0.188662	−	0.982042i	$-0.439585\pi$
$193$	1011.70	0.377323	0.188662	+	0.982042i	$0.439585\pi$
$194$	−191.337	−0.0708105
$195$	378.382	0.138956
$196$	0	0
$197$	−653.917	−0.236496	−0.118248	−	0.992984i	$-0.537728\pi$
$197$	−653.917	−0.236496	−0.118248	+	0.992984i	$0.537728\pi$
$198$	−5928.71	−2.12795
$199$	886.815	0.315903	0.157951	−	0.987447i	$-0.449511\pi$
$199$	886.815	0.315903	0.157951	+	0.987447i	$0.449511\pi$
$200$	−1253.44	−0.443158
$201$	−5973.10	−2.09607
$202$	−1047.59	−0.364893
$203$	0	0
$204$	2726.36	0.935702
$205$	−3127.56	−1.06555
$206$	−285.512	−0.0965659
$207$	−3558.05	−1.19469
$208$	50.2343	0.0167458
$209$	−4943.45	−1.63610
$210$	0	0
$211$	−2418.36	−0.789036	−0.394518	−	0.918888i	$-0.629088\pi$
$211$	−2418.36	−0.789036	−0.394518	+	0.918888i	$0.629088\pi$
$212$	1650.47	0.534691
$213$	−3794.72	−1.22070
$214$	−2839.79	−0.907121
$215$	365.120	0.115818
$216$	12479.5	3.93114
$217$	0	0
$218$	2968.70	0.922320
$219$	5068.28	1.56385
$220$	1422.30	0.435870
$221$	270.003	0.0821828
$222$	2278.25	0.688767
$223$	3431.23	1.03037	0.515183	−	0.857080i	$-0.327724\pi$
$223$	3431.23	1.03037	0.515183	+	0.857080i	$0.327724\pi$
$224$	0	0
$225$	−4143.19	−1.22761
$226$	757.959	0.223092
$227$	−838.154	−0.245067	−0.122534	−	0.992464i	$-0.539102\pi$
$227$	−838.154	−0.245067	−0.122534	+	0.992464i	$0.539102\pi$
$228$	5528.26	1.60578
$229$	5.37035	0.00154971	0.000774854	−	1.00000i	$-0.499753\pi$
$229$	5.37035	0.00154971	0.000774854		1.00000i	$0.499753\pi$
$230$	−744.768	−0.213515
$231$	0	0
$232$	569.321	0.161111
$233$	−5259.94	−1.47893	−0.739464	−	0.673196i	$-0.764921\pi$
$233$	−5259.94	−1.47893	−0.739464	+	0.673196i	$0.764921\pi$
$234$	656.605	0.183434
$235$	−3430.43	−0.952241
$236$	3265.49	0.900702
$237$	−8459.85	−2.31868
$238$	0	0
$239$	−6423.57	−1.73852	−0.869259	−	0.494357i	$-0.835404\pi$
$239$	−6423.57	−1.73852	−0.869259	+	0.494357i	$0.835404\pi$
$240$	1008.14	0.271145
$241$	4286.69	1.14577	0.572883	−	0.819637i	$-0.305825\pi$
$241$	4286.69	1.14577	0.572883	+	0.819637i	$0.305825\pi$
$242$	−398.069	−0.105739
$243$	19548.6	5.16068
$244$	2360.90	0.619432
$245$	0	0
$246$	−7298.21	−1.89153
$247$	547.488	0.141036
$248$	−5637.63	−1.44351
$249$	10640.2	2.70802
$250$	−2916.51	−0.737825
$251$	−2037.38	−0.512343	−0.256171	−	0.966631i	$-0.582461\pi$
$251$	−2037.38	−0.512343	−0.256171	+	0.966631i	$0.582461\pi$
$252$	0	0
$253$	−1781.13	−0.442603
$254$	−905.848	−0.223772
$255$	5418.61	1.33069
$256$	−4357.57	−1.06386
$257$	3052.45	0.740881	0.370440	−	0.928856i	$-0.379207\pi$
$257$	3052.45	0.740881	0.370440	+	0.928856i	$0.379207\pi$
$258$	852.013	0.205597
$259$	0	0
$260$	−157.520	−0.0375730
$261$	1881.87	0.446301
$262$	3774.42	0.890018
$263$	4425.62	1.03763	0.518813	−	0.854888i	$-0.326374\pi$
$263$	4425.62	1.03763	0.518813	+	0.854888i	$0.326374\pi$
$264$	9533.79	2.22259
$265$	3280.29	0.760401
$266$	0	0
$267$	−2496.64	−0.572253
$268$	2486.59	0.566764
$269$	−672.775	−0.152490	−0.0762450	−	0.997089i	$-0.524293\pi$
$269$	−672.775	−0.152490	−0.0762450	+	0.997089i	$0.524293\pi$
$270$	8634.55	1.94623
$271$	−374.852	−0.0840245	−0.0420122	−	0.999117i	$-0.513377\pi$
$271$	−374.852	−0.0840245	−0.0420122	+	0.999117i	$0.513377\pi$
$272$	719.378	0.160363
$273$	0	0
$274$	−2947.28	−0.649825
$275$	−2074.05	−0.454799
$276$	1991.84	0.434400
$277$	−7320.41	−1.58787	−0.793937	−	0.608000i	$-0.791972\pi$
$277$	−7320.41	−1.58787	−0.793937	+	0.608000i	$0.791972\pi$
$278$	1003.74	0.216548
$279$	−18635.0	−3.99873
$280$	0	0
$281$	1788.79	0.379752	0.189876	−	0.981808i	$-0.439191\pi$
$281$	1788.79	0.379752	0.189876	+	0.981808i	$0.439191\pi$
$282$	−8004.96	−1.69038
$283$	−5450.90	−1.14496	−0.572478	−	0.819920i	$-0.694018\pi$
$283$	−5450.90	−1.14496	−0.572478	+	0.819920i	$0.694018\pi$
$284$	1579.74	0.330071
$285$	10987.4	2.28363
$286$	328.691	0.0679578
$287$	0	0
$288$	−13096.8	−2.67963
$289$	−1046.43	−0.212992
$290$	393.911	0.0797629
$291$	1017.04	0.204879
$292$	−2109.92	−0.422855
$293$	−7867.96	−1.56878	−0.784388	−	0.620270i	$-0.787023\pi$
$293$	−7867.96	−1.56878	−0.784388	+	0.620270i	$0.787023\pi$
$294$	0	0
$295$	6490.13	1.28092
$296$	−2724.40	−0.534975
$297$	20649.7	4.03440
$298$	4874.72	0.947601
$299$	197.260	0.0381534
$300$	2319.41	0.446370
$301$	0	0
$302$	6597.59	1.25711
$303$	5568.39	1.05576
$304$	1458.69	0.275203
$305$	4692.27	0.880913
$306$	9402.89	1.75663
$307$	−1695.56	−0.315215	−0.157607	−	0.987502i	$-0.550378\pi$
$307$	−1695.56	−0.315215	−0.157607	+	0.987502i	$0.550378\pi$
$308$	0	0
$309$	1517.62	0.279399
$310$	−3900.65	−0.714653
$311$	−1447.43	−0.263911	−0.131956	−	0.991256i	$-0.542126\pi$
$311$	−1447.43	−0.263911	−0.131956	+	0.991256i	$0.542126\pi$
$312$	−1055.87	−0.191592
$313$	2056.47	0.371368	0.185684	−	0.982609i	$-0.440550\pi$
$313$	2056.47	0.371368	0.185684	+	0.982609i	$0.440550\pi$
$314$	−5894.36	−1.05936
$315$	0	0
$316$	3521.82	0.626956
$317$	−7161.24	−1.26882	−0.634409	−	0.772997i	$-0.718757\pi$
$317$	−7161.24	−1.26882	−0.634409	+	0.772997i	$0.718757\pi$
$318$	7654.59	1.34984
$319$	942.047	0.165343
$320$	−3527.27	−0.616189
$321$	15094.7	2.62461
$322$	0	0
$323$	7840.28	1.35060
$324$	−14058.1	−2.41050
$325$	229.701	0.0392047
$326$	−129.169	−0.0219448
$327$	−15779.9	−2.66859
$328$	8727.40	1.46918
$329$	0	0
$330$	6596.39	1.10036
$331$	1281.27	0.212764	0.106382	−	0.994325i	$-0.466073\pi$
$331$	1281.27	0.212764	0.106382	+	0.994325i	$0.466073\pi$
$332$	−4429.51	−0.732231
$333$	−9005.39	−1.48196
$334$	−4510.03	−0.738856
$335$	4942.08	0.806014
$336$	0	0
$337$	5550.35	0.897171	0.448586	−	0.893740i	$-0.351928\pi$
$337$	5550.35	0.897171	0.448586	+	0.893740i	$0.351928\pi$
$338$	4205.40	0.676756
$339$	−4028.87	−0.645481
$340$	−2255.76	−0.359811
$341$	−9328.51	−1.48143
$342$	19066.3	3.01459
$343$	0	0
$344$	−1018.86	−0.159690
$345$	3958.75	0.617774
$346$	−325.190	−0.0505270
$347$	−923.286	−0.142837	−0.0714187	−	0.997446i	$-0.522753\pi$
$347$	−923.286	−0.142837	−0.0714187	+	0.997446i	$0.522753\pi$
$348$	−1053.49	−0.162279
$349$	−6708.21	−1.02889	−0.514444	−	0.857524i	$-0.672002\pi$
$349$	−6708.21	−1.02889	−0.514444	+	0.857524i	$0.672002\pi$
$350$	0	0
$351$	−2286.96	−0.347774
$352$	−6556.13	−0.992735
$353$	6010.82	0.906299	0.453149	−	0.891435i	$-0.350300\pi$
$353$	6010.82	0.906299	0.453149	+	0.891435i	$0.350300\pi$
$354$	15144.8	2.27384
$355$	3139.71	0.469404
$356$	1039.35	0.154734
$357$	0	0
$358$	4591.70	0.677874
$359$	3491.44	0.513290	0.256645	−	0.966506i	$-0.417383\pi$
$359$	3491.44	0.513290	0.256645	+	0.966506i	$0.417383\pi$
$360$	−15757.7	−2.30695
$361$	9038.80	1.31780
$362$	4551.58	0.660845
$363$	2115.90	0.305940
$364$	0	0
$365$	−4193.44	−0.601355
$366$	10949.5	1.56377
$367$	−3540.86	−0.503628	−0.251814	−	0.967776i	$-0.581027\pi$
$367$	−3540.86	−0.503628	−0.251814	+	0.967776i	$0.581027\pi$
$368$	525.567	0.0744486
$369$	28848.1	4.06984
$370$	−1885.00	−0.264856
$371$	0	0
$372$	10432.1	1.45397
$373$	−7484.88	−1.03901	−0.519507	−	0.854466i	$-0.673884\pi$
$373$	−7484.88	−1.03901	−0.519507	+	0.854466i	$0.673884\pi$
$374$	4707.01	0.650785
$375$	15502.5	2.13478
$376$	9572.55	1.31294
$377$	−104.332	−0.0142530
$378$	0	0
$379$	−13204.5	−1.78963	−0.894814	−	0.446439i	$-0.852692\pi$
$379$	−13204.5	−1.78963	−0.894814	+	0.446439i	$0.852692\pi$
$380$	−4574.02	−0.617479
$381$	4814.96	0.647449
$382$	−6745.39	−0.903467
$383$	298.122	0.0397736	0.0198868	−	0.999802i	$-0.493669\pi$
$383$	298.122	0.0397736	0.0198868	+	0.999802i	$0.493669\pi$
$384$	5497.87	0.730630
$385$	0	0
$386$	−1953.30	−0.257566
$387$	−3367.80	−0.442364
$388$	−423.392	−0.0553981
$389$	−11644.9	−1.51779	−0.758896	−	0.651212i	$-0.774261\pi$
$389$	−11644.9	−1.51779	−0.758896	+	0.651212i	$0.774261\pi$
$390$	−730.551	−0.0948536
$391$	2824.86	0.365369
$392$	0	0
$393$	−20062.6	−2.57513
$394$	1262.53	0.161435
$395$	6999.58	0.891613
$396$	−13119.0	−1.66479
$397$	−6651.72	−0.840907	−0.420453	−	0.907314i	$-0.638129\pi$
$397$	−6651.72	−0.840907	−0.420453	+	0.907314i	$0.638129\pi$
$398$	−1712.19	−0.215640
$399$	0	0
$400$	612.000	0.0765000
$401$	−3353.24	−0.417587	−0.208794	−	0.977960i	$-0.566954\pi$
$401$	−3353.24	−0.417587	−0.208794	+	0.977960i	$0.566954\pi$
$402$	11532.4	1.43081
$403$	1033.13	0.127702
$404$	−2318.11	−0.285471
$405$	−27940.2	−3.42805
$406$	0	0
$407$	−4508.02	−0.549028
$408$	−15120.5	−1.83475
$409$	−11698.5	−1.41431	−0.707153	−	0.707060i	$-0.750021\pi$
$409$	−11698.5	−1.41431	−0.707153	+	0.707060i	$0.750021\pi$
$410$	6038.45	0.727361
$411$	15666.0	1.88017
$412$	−631.781	−0.0755477
$413$	0	0
$414$	6869.61	0.815514
$415$	−8803.60	−1.04133
$416$	726.092	0.0855759
$417$	−5335.30	−0.626549
$418$	9544.44	1.11683
$419$	12034.6	1.40317	0.701587	−	0.712584i	$-0.252475\pi$
$419$	12034.6	1.40317	0.701587	+	0.712584i	$0.252475\pi$
$420$	0	0
$421$	−13799.8	−1.59754	−0.798769	−	0.601638i	$-0.794515\pi$
$421$	−13799.8	−1.59754	−0.798769	+	0.601638i	$0.794515\pi$
$422$	4669.18	0.538607
$423$	31641.7	3.63705
$424$	−9153.58	−1.04844
$425$	3289.43	0.375437
$426$	7326.55	0.833269
$427$	0	0
$428$	−6283.88	−0.709680
$429$	−1747.13	−0.196625
$430$	−704.946	−0.0790593
$431$	11002.2	1.22960	0.614798	−	0.788685i	$-0.289238\pi$
$431$	11002.2	1.22960	0.614798	+	0.788685i	$0.289238\pi$
$432$	−6093.22	−0.678611
$433$	−83.9305	−0.00931511	−0.00465755	−	0.999989i	$-0.501483\pi$
$433$	−83.9305	−0.00931511	−0.00465755	+	0.999989i	$0.501483\pi$
$434$	0	0
$435$	−2093.80	−0.230782
$436$	6569.14	0.721570
$437$	5727.99	0.627018
$438$	−9785.44	−1.06750
$439$	−5986.72	−0.650867	−0.325434	−	0.945565i	$-0.605510\pi$
$439$	−5986.72	−0.650867	−0.325434	+	0.945565i	$0.605510\pi$
$440$	−7888.15	−0.854666
$441$	0	0
$442$	−521.302	−0.0560991
$443$	8745.27	0.937924	0.468962	−	0.883218i	$-0.344628\pi$
$443$	8745.27	0.937924	0.468962	+	0.883218i	$0.344628\pi$
$444$	5041.32	0.538852
$445$	2065.69	0.220052
$446$	−6624.75	−0.703343
$447$	−25911.2	−2.74174
$448$	0	0
$449$	−14454.4	−1.51926	−0.759628	−	0.650357i	$-0.774619\pi$
$449$	−14454.4	−1.51926	−0.759628	+	0.650357i	$0.774619\pi$
$450$	7999.36	0.837986
$451$	14441.1	1.50777
$452$	1677.21	0.174534
$453$	−35068.9	−3.63727
$454$	1618.24	0.167286
$455$	0	0
$456$	−30660.0	−3.14865
$457$	9138.14	0.935370	0.467685	−	0.883895i	$-0.345088\pi$
$457$	9138.14	0.935370	0.467685	+	0.883895i	$0.345088\pi$
$458$	−10.3687	−0.00105785
$459$	−32750.3	−3.33040
$460$	−1648.02	−0.167042
$461$	16974.0	1.71487	0.857436	−	0.514591i	$-0.172056\pi$
$461$	16974.0	1.71487	0.857436	+	0.514591i	$0.172056\pi$
$462$	0	0
$463$	15510.1	1.55683	0.778417	−	0.627748i	$-0.216023\pi$
$463$	15510.1	1.55683	0.778417	+	0.627748i	$0.216023\pi$
$464$	−277.974	−0.0278117
$465$	20733.6	2.06774
$466$	10155.5	1.00954
$467$	−16738.6	−1.65861	−0.829307	−	0.558794i	$-0.811264\pi$
$467$	−16738.6	−1.65861	−0.829307	+	0.558794i	$0.811264\pi$
$468$	1452.94	0.143509
$469$	0	0
$470$	6623.21	0.650013
$471$	31331.0	3.06509
$472$	−18110.6	−1.76612
$473$	−1685.89	−0.163885
$474$	16333.6	1.58276
$475$	6670.00	0.644296
$476$	0	0
$477$	−30256.8	−2.90432
$478$	12402.1	1.18674
$479$	4045.08	0.385855	0.192928	−	0.981213i	$-0.438202\pi$
$479$	4045.08	0.385855	0.192928	+	0.981213i	$0.438202\pi$
$480$	14571.7	1.38563
$481$	499.264	0.0473274
$482$	−8276.41	−0.782116
$483$	0	0
$484$	−880.848	−0.0827242
$485$	−841.486	−0.0787833
$486$	−37743.0	−3.52275
$487$	−15124.6	−1.40731	−0.703657	−	0.710540i	$-0.748451\pi$
$487$	−15124.6	−1.40731	−0.703657	+	0.710540i	$0.748451\pi$
$488$	−13093.7	−1.21460
$489$	686.586	0.0634938
$490$	0	0
$491$	3030.99	0.278588	0.139294	−	0.990251i	$-0.455517\pi$
$491$	3030.99	0.278588	0.139294	+	0.990251i	$0.455517\pi$
$492$	−16149.5	−1.47983
$493$	−1494.08	−0.136491
$494$	−1057.05	−0.0962730
$495$	−26074.0	−2.36755
$496$	2752.61	0.249185
$497$	0	0
$498$	−20543.3	−1.84853
$499$	22233.0	1.99456	0.997280	−	0.0737115i	$-0.0234844\pi$
$499$	22233.0	1.99456	0.997280	+	0.0737115i	$0.0234844\pi$
$500$	−6453.66	−0.577233
$501$	23972.7	2.13776
$502$	3933.61	0.349732
$503$	−15783.0	−1.39907	−0.699533	−	0.714600i	$-0.746609\pi$
$503$	−15783.0	−1.39907	−0.699533	+	0.714600i	$0.746609\pi$
$504$	0	0
$505$	−4607.23	−0.405978
$506$	3438.87	0.302127
$507$	−22353.4	−1.95809
$508$	−2004.46	−0.175066
$509$	−9475.26	−0.825115	−0.412558	−	0.910932i	$-0.635364\pi$
$509$	−9475.26	−0.825115	−0.412558	+	0.910932i	$0.635364\pi$
$510$	−10461.8	−0.908348
$511$	0	0
$512$	4127.51	0.356273
$513$	−66408.1	−5.71538
$514$	−5893.43	−0.505735
$515$	−1255.66	−0.107439
$516$	1885.33	0.160847
$517$	15839.6	1.34743
$518$	0	0
$519$	1728.52	0.146192
$520$	873.614	0.0736740
$521$	2355.06	0.198036	0.0990181	−	0.995086i	$-0.468430\pi$
$521$	2355.06	0.198036	0.0990181	+	0.995086i	$0.468430\pi$
$522$	−3633.36	−0.304651
$523$	4592.63	0.383980	0.191990	−	0.981397i	$-0.438506\pi$
$523$	4592.63	0.383980	0.191990	+	0.981397i	$0.438506\pi$
$524$	8352.04	0.696299
$525$	0	0
$526$	−8544.66	−0.708298
$527$	14794.9	1.22292
$528$	−4654.93	−0.383674
$529$	−10103.2	−0.830377
$530$	−6333.32	−0.519060
$531$	−59863.9	−4.89241
$532$	0	0
$533$	−1599.35	−0.129973
$534$	4820.31	0.390628
$535$	−12489.1	−1.00926
$536$	−13790.8	−1.11133
$537$	−24406.8	−1.96132
$538$	1298.94	0.104092
$539$	0	0
$540$	19106.5	1.52262
$541$	2815.37	0.223738	0.111869	−	0.993723i	$-0.464316\pi$
$541$	2815.37	0.223738	0.111869	+	0.993723i	$0.464316\pi$
$542$	723.736	0.0573563
$543$	−24193.5	−1.91205
$544$	10398.0	0.819503
$545$	13056.1	1.02617
$546$	0	0
$547$	14095.6	1.10180	0.550902	−	0.834570i	$-0.314284\pi$
$547$	14095.6	1.10180	0.550902	+	0.834570i	$0.314284\pi$
$548$	−6521.75	−0.508386
$549$	−43280.6	−3.36461
$550$	4004.42	0.310452
$551$	−3029.56	−0.234235
$552$	−11046.8	−0.851783
$553$	0	0
$554$	14133.7	1.08390
$555$	10019.6	0.766319
$556$	2221.08	0.169415
$557$	18889.1	1.43690	0.718452	−	0.695576i	$-0.244851\pi$
$557$	18889.1	1.43690	0.718452	+	0.695576i	$0.244851\pi$
$558$	35979.0	2.72959
$559$	186.713	0.0141272
$560$	0	0
$561$	−25019.7	−1.88295
$562$	−3453.66	−0.259224
$563$	−626.656	−0.0469101	−0.0234551	−	0.999725i	$-0.507467\pi$
$563$	−626.656	−0.0469101	−0.0234551	+	0.999725i	$0.507467\pi$
$564$	−17713.4	−1.32246
$565$	3333.44	0.248210
$566$	10524.2	0.781563
$567$	0	0
$568$	−8761.30	−0.647211
$569$	21276.4	1.56758	0.783791	−	0.621025i	$-0.213283\pi$
$569$	21276.4	1.56758	0.783791	+	0.621025i	$0.213283\pi$
$570$	−21213.5	−1.55884
$571$	7036.31	0.515692	0.257846	−	0.966186i	$-0.416987\pi$
$571$	7036.31	0.515692	0.257846	+	0.966186i	$0.416987\pi$
$572$	727.328	0.0531663
$573$	35854.6	2.61404
$574$	0	0
$575$	2403.21	0.174297
$576$	32535.0	2.35351
$577$	−4185.59	−0.301991	−0.150995	−	0.988534i	$-0.548248\pi$
$577$	−4185.59	−0.301991	−0.150995	+	0.988534i	$0.548248\pi$
$578$	2020.36	0.145391
$579$	10382.6	0.745228
$580$	871.646	0.0624019
$581$	0	0
$582$	−1963.62	−0.139853
$583$	−15146.3	−1.07598
$584$	11701.7	0.829144
$585$	2887.69	0.204088
$586$	15190.9	1.07087
$587$	7786.33	0.547489	0.273745	−	0.961802i	$-0.411738\pi$
$587$	7786.33	0.547489	0.273745	+	0.961802i	$0.411738\pi$
$588$	0	0
$589$	29999.8	2.09868
$590$	−12530.7	−0.874371
$591$	−6710.89	−0.467088
$592$	1330.21	0.0923498
$593$	−16511.2	−1.14340	−0.571698	−	0.820464i	$-0.693715\pi$
$593$	−16511.2	−1.14340	−0.571698	+	0.820464i	$0.693715\pi$
$594$	−39868.9	−2.75394
$595$	0	0
$596$	10786.8	0.741349
$597$	9101.03	0.623920
$598$	−380.855	−0.0260440
$599$	−1604.00	−0.109412	−0.0547060	−	0.998503i	$-0.517422\pi$
$599$	−1604.00	−0.109412	−0.0547060	+	0.998503i	$0.517422\pi$
$600$	−12863.6	−0.875254
$601$	23798.6	1.61525	0.807625	−	0.589696i	$-0.200753\pi$
$601$	23798.6	1.61525	0.807625	+	0.589696i	$0.200753\pi$
$602$	0	0
$603$	−45584.9	−3.07854
$604$	14599.1	0.983494
$605$	−1750.68	−0.117645
$606$	−10751.0	−0.720678
$607$	6453.70	0.431544	0.215772	−	0.976444i	$-0.430773\pi$
$607$	6453.70	0.431544	0.215772	+	0.976444i	$0.430773\pi$
$608$	21084.1	1.40637
$609$	0	0
$610$	−9059.47	−0.601324
$611$	−1754.23	−0.116152
$612$	20806.7	1.37428
$613$	5727.51	0.377377	0.188688	−	0.982037i	$-0.439576\pi$
$613$	5727.51	0.377377	0.188688	+	0.982037i	$0.439576\pi$
$614$	3273.66	0.215170
$615$	−32096.9	−2.10451
$616$	0	0
$617$	877.626	0.0572640	0.0286320	−	0.999590i	$-0.490885\pi$
$617$	877.626	0.0572640	0.0286320	+	0.999590i	$0.490885\pi$
$618$	−2930.10	−0.190721
$619$	−286.367	−0.0185946	−0.00929730	−	0.999957i	$-0.502959\pi$
$619$	−286.367	−0.0185946	−0.00929730	+	0.999957i	$0.502959\pi$
$620$	−8631.37	−0.559103
$621$	−23926.9	−1.54614
$622$	2794.59	0.180149
$623$	0	0
$624$	515.535	0.0330736
$625$	−6214.05	−0.397699
$626$	−3970.47	−0.253501
$627$	−50732.7	−3.23137
$628$	−13043.0	−0.828780
$629$	7149.69	0.453223
$630$	0	0
$631$	13414.1	0.846284	0.423142	−	0.906063i	$-0.360927\pi$
$631$	13414.1	0.846284	0.423142	+	0.906063i	$0.360927\pi$
$632$	−19532.2	−1.22935
$633$	−24818.6	−1.55838
$634$	13826.4	0.866113
$635$	−3983.84	−0.248967
$636$	16938.1	1.05604
$637$	0	0
$638$	−1818.83	−0.112866
$639$	−28960.1	−1.79287
$640$	−4548.88	−0.280953
$641$	17315.6	1.06696	0.533482	−	0.845811i	$-0.320883\pi$
$641$	17315.6	1.06696	0.533482	+	0.845811i	$0.320883\pi$
$642$	−29143.6	−1.79160
$643$	19391.8	1.18933	0.594663	−	0.803975i	$-0.297286\pi$
$643$	19391.8	1.18933	0.594663	+	0.803975i	$0.297286\pi$
$644$	0	0
$645$	3747.08	0.228746
$646$	−15137.4	−0.921941
$647$	18817.8	1.14344	0.571718	−	0.820450i	$-0.306277\pi$
$647$	18817.8	1.14344	0.571718	+	0.820450i	$0.306277\pi$
$648$	77966.7	4.72658
$649$	−29967.4	−1.81251
$650$	−443.490	−0.0267617
$651$	0	0
$652$	−285.825	−0.0171683
$653$	−30440.2	−1.82422	−0.912112	−	0.409941i	$-0.865549\pi$
$653$	−30440.2	−1.82422	−0.912112	+	0.409941i	$0.865549\pi$
$654$	30466.6	1.82162
$655$	16599.6	0.990229
$656$	−4261.21	−0.253616
$657$	38679.5	2.29685
$658$	0	0
$659$	25337.5	1.49774	0.748870	−	0.662717i	$-0.230597\pi$
$659$	25337.5	1.49774	0.748870	+	0.662717i	$0.230597\pi$
$660$	14596.5	0.860860
$661$	23072.9	1.35769	0.678843	−	0.734283i	$-0.262482\pi$
$661$	23072.9	1.35769	0.678843	+	0.734283i	$0.262482\pi$
$662$	−2473.78	−0.145236
$663$	2770.94	0.162314
$664$	24566.3	1.43578
$665$	0	0
$666$	17386.9	1.01161
$667$	−1091.55	−0.0633659
$668$	−9979.79	−0.578038
$669$	35213.3	2.03501
$670$	−9541.79	−0.550196
$671$	−21665.9	−1.24650
$672$	0	0
$673$	4316.85	0.247254	0.123627	−	0.992329i	$-0.460547\pi$
$673$	4316.85	0.247254	0.123627	+	0.992329i	$0.460547\pi$
$674$	−10716.2	−0.612422
$675$	−27861.8	−1.58874
$676$	9305.70	0.529455
$677$	2197.08	0.124728	0.0623638	−	0.998053i	$-0.480136\pi$
$677$	2197.08	0.124728	0.0623638	+	0.998053i	$0.480136\pi$
$678$	7778.63	0.440614
$679$	0	0
$680$	12510.6	0.705526
$681$	−8601.64	−0.484017
$682$	18010.8	1.01124
$683$	6326.64	0.354440	0.177220	−	0.984171i	$-0.443290\pi$
$683$	6326.64	0.354440	0.177220	+	0.984171i	$0.443290\pi$
$684$	42189.9	2.35844
$685$	−12961.9	−0.722991
$686$	0	0
$687$	55.1138	0.00306073
$688$	497.465	0.0275664
$689$	1677.45	0.0927517
$690$	−7643.26	−0.421701
$691$	−26875.8	−1.47960	−0.739801	−	0.672826i	$-0.765080\pi$
$691$	−26875.8	−1.47960	−0.739801	+	0.672826i	$0.765080\pi$
$692$	−719.581	−0.0395294
$693$	0	0
$694$	1782.61	0.0975028
$695$	4414.37	0.240930
$696$	5842.71	0.318200
$697$	−22903.5	−1.24466
$698$	12951.7	0.702334
$699$	−53980.7	−2.92094
$700$	0	0
$701$	−29410.3	−1.58461	−0.792304	−	0.610127i	$-0.791119\pi$
$701$	−29410.3	−1.58461	−0.792304	+	0.610127i	$0.791119\pi$
$702$	4415.49	0.237396
$703$	14497.5	0.777786
$704$	16286.7	0.871917
$705$	−35205.1	−1.88071
$706$	−11605.2	−0.618652
$707$	0	0
$708$	33512.5	1.77892
$709$	28782.4	1.52461	0.762304	−	0.647219i	$-0.224068\pi$
$709$	28782.4	1.52461	0.762304	+	0.647219i	$0.224068\pi$
$710$	−6061.91	−0.320422
$711$	−64562.9	−3.40548
$712$	−5764.27	−0.303406
$713$	10809.0	0.567740
$714$	0	0
$715$	1445.56	0.0756094
$716$	10160.5	0.530330
$717$	−65922.5	−3.43364
$718$	−6741.00	−0.350379
$719$	−20041.8	−1.03955	−0.519773	−	0.854304i	$-0.673984\pi$
$719$	−20041.8	−1.03955	−0.519773	+	0.854304i	$0.673984\pi$
$720$	7693.77	0.398236
$721$	0	0
$722$	−17451.4	−0.899550
$723$	43992.5	2.26293
$724$	10071.7	0.517007
$725$	−1271.07	−0.0651120
$726$	−4085.23	−0.208839
$727$	17140.7	0.874434	0.437217	−	0.899356i	$-0.355964\pi$
$727$	17140.7	0.874434	0.437217	+	0.899356i	$0.355964\pi$
$728$	0	0
$729$	111776.	5.67881
$730$	8096.37	0.410493
$731$	2673.82	0.135287
$732$	24229.0	1.22340
$733$	27141.0	1.36763	0.683816	−	0.729655i	$-0.260319\pi$
$733$	27141.0	1.36763	0.683816	+	0.729655i	$0.260319\pi$
$734$	6836.43	0.343784
$735$	0	0
$736$	7596.60	0.380454
$737$	−22819.4	−1.14052
$738$	−55697.6	−2.77813
$739$	−20351.1	−1.01303	−0.506513	−	0.862232i	$-0.669066\pi$
$739$	−20351.1	−1.01303	−0.506513	+	0.862232i	$0.669066\pi$
$740$	−4171.13	−0.207208
$741$	5618.65	0.278551
$742$	0	0
$743$	2917.99	0.144079	0.0720394	−	0.997402i	$-0.477049\pi$
$743$	2917.99	0.144079	0.0720394	+	0.997402i	$0.477049\pi$
$744$	−57856.7	−2.85098
$745$	21438.6	1.05430
$746$	14451.2	0.709246
$747$	81202.8	3.97732
$748$	10415.7	0.509137
$749$	0	0
$750$	−29931.0	−1.45723
$751$	−6154.55	−0.299045	−0.149522	−	0.988758i	$-0.547774\pi$
$751$	−6154.55	−0.299045	−0.149522	+	0.988758i	$0.547774\pi$
$752$	−4673.86	−0.226646
$753$	−20908.8	−1.01190
$754$	201.436	0.00972926
$755$	29015.6	1.39866
$756$	0	0
$757$	8896.79	0.427159	0.213580	−	0.976926i	$-0.431488\pi$
$757$	8896.79	0.427159	0.213580	+	0.976926i	$0.431488\pi$
$758$	25494.2	1.22162
$759$	−18279.0	−0.874159
$760$	25367.7	1.21077
$761$	847.470	0.0403689	0.0201845	−	0.999796i	$-0.493575\pi$
$761$	847.470	0.0403689	0.0201845	+	0.999796i	$0.493575\pi$
$762$	−9296.36	−0.441957
$763$	0	0
$764$	−14926.2	−0.706821
$765$	41353.1	1.95441
$766$	−575.590	−0.0271500
$767$	3318.89	0.156243
$768$	−44720.0	−2.10116
$769$	32891.0	1.54237	0.771184	−	0.636612i	$-0.219665\pi$
$769$	32891.0	1.54237	0.771184	+	0.636612i	$0.219665\pi$
$770$	0	0
$771$	31326.0	1.46327
$772$	−4322.27	−0.201505
$773$	36866.6	1.71540	0.857698	−	0.514154i	$-0.171894\pi$
$773$	36866.6	1.71540	0.857698	+	0.514154i	$0.171894\pi$
$774$	6502.29	0.301964
$775$	12586.6	0.583385
$776$	2348.15	0.108626
$777$	0	0
$778$	22483.1	1.03607
$779$	−46441.6	−2.13600
$780$	−1616.56	−0.0742081
$781$	−14497.2	−0.664213
$782$	−5454.02	−0.249406
$783$	12655.0	0.577591
$784$	0	0
$785$	−25922.9	−1.17863
$786$	38735.4	1.75782
$787$	−22669.9	−1.02680	−0.513402	−	0.858148i	$-0.671615\pi$
$787$	−22669.9	−1.02680	−0.513402	+	0.858148i	$0.671615\pi$
$788$	2793.73	0.126298
$789$	45418.4	2.04935
$790$	−13514.3	−0.608628
$791$	0	0
$792$	72758.9	3.26436
$793$	2399.51	0.107451
$794$	12842.6	0.574015
$795$	33664.3	1.50182
$796$	−3788.74	−0.168704
$797$	−28401.2	−1.26226	−0.631130	−	0.775677i	$-0.717408\pi$
$797$	−28401.2	−1.26226	−0.631130	+	0.775677i	$0.717408\pi$
$798$	0	0
$799$	−25121.4	−1.11231
$800$	8845.92	0.390938
$801$	−19053.5	−0.840479
$802$	6474.17	0.285051
$803$	19362.7	0.850925
$804$	25518.9	1.11938
$805$	0	0
$806$	−1994.70	−0.0871714
$807$	−6904.42	−0.301174
$808$	12856.4	0.559760
$809$	−25004.7	−1.08667	−0.543336	−	0.839515i	$-0.682839\pi$
$809$	−25004.7	−1.08667	−0.543336	+	0.839515i	$0.682839\pi$
$810$	53944.9	2.34004
$811$	15508.1	0.671473	0.335736	−	0.941956i	$-0.391015\pi$
$811$	15508.1	0.671473	0.335736	+	0.941956i	$0.391015\pi$
$812$	0	0
$813$	−3846.96	−0.165952
$814$	8703.75	0.374774
$815$	−568.073	−0.0244156
$816$	7382.69	0.316723
$817$	5421.72	0.232169
$818$	22586.5	0.965426
$819$	0	0
$820$	13361.9	0.569046
$821$	15621.2	0.664049	0.332025	−	0.943271i	$-0.392268\pi$
$821$	15621.2	0.664049	0.332025	+	0.943271i	$0.392268\pi$
$822$	−30246.8	−1.28343
$823$	3018.93	0.127866	0.0639328	−	0.997954i	$-0.479636\pi$
$823$	3018.93	0.127866	0.0639328	+	0.997954i	$0.479636\pi$
$824$	3503.89	0.148136
$825$	−21285.1	−0.898246
$826$	0	0
$827$	11832.0	0.497507	0.248754	−	0.968567i	$-0.419979\pi$
$827$	11832.0	0.497507	0.248754	+	0.968567i	$0.419979\pi$
$828$	15201.1	0.638012
$829$	8763.50	0.367152	0.183576	−	0.983006i	$-0.441233\pi$
$829$	8763.50	0.367152	0.183576	+	0.983006i	$0.441233\pi$
$830$	16997.3	0.710826
$831$	−75126.5	−3.13611
$832$	−1803.76	−0.0751611
$833$	0	0
$834$	10301.0	0.427691
$835$	−19834.7	−0.822047
$836$	21119.9	0.873742
$837$	−125315.	−5.17505
$838$	−23235.5	−0.957827
$839$	32356.6	1.33144	0.665718	−	0.746203i	$-0.268125\pi$
$839$	32356.6	1.33144	0.665718	+	0.746203i	$0.268125\pi$
$840$	0	0
$841$	−23811.7	−0.976328
$842$	26643.7	1.09050
$843$	18357.6	0.750025
$844$	10332.0	0.421375
$845$	18495.0	0.752955
$846$	−61091.3	−2.48270
$847$	0	0
$848$	4469.29	0.180986
$849$	−55940.5	−2.26133
$850$	−6350.97	−0.256278
$851$	5223.46	0.210409
$852$	16212.2	0.651902
$853$	6263.80	0.251429	0.125714	−	0.992066i	$-0.459878\pi$
$853$	6263.80	0.251429	0.125714	+	0.992066i	$0.459878\pi$
$854$	0	0
$855$	83852.0	3.35401
$856$	34850.7	1.39156
$857$	12255.0	0.488473	0.244236	−	0.969716i	$-0.421463\pi$
$857$	12255.0	0.488473	0.244236	+	0.969716i	$0.421463\pi$
$858$	3373.23	0.134219
$859$	−34594.4	−1.37409	−0.687047	−	0.726613i	$-0.741093\pi$
$859$	−34594.4	−1.37409	−0.687047	+	0.726613i	$0.741093\pi$
$860$	−1559.90	−0.0618515
$861$	0	0
$862$	−21242.1	−0.839339
$863$	−26051.3	−1.02757	−0.513787	−	0.857918i	$-0.671758\pi$
$863$	−26051.3	−1.02757	−0.513787	+	0.857918i	$0.671758\pi$
$864$	−88072.0	−3.46791
$865$	−1430.16	−0.0562160
$866$	162.047	0.00635862
$867$	−10739.1	−0.420667
$868$	0	0
$869$	−32319.7	−1.26165
$870$	4042.55	0.157535
$871$	2527.25	0.0983154
$872$	−36432.8	−1.41487
$873$	7761.72	0.300910
$874$	−11059.2	−0.428011
$875$	0	0
$876$	−21653.2	−0.835154
$877$	−22928.5	−0.882828	−0.441414	−	0.897304i	$-0.645523\pi$
$877$	−22928.5	−0.882828	−0.441414	+	0.897304i	$0.645523\pi$
$878$	11558.7	0.444291
$879$	−80745.8	−3.09839
$880$	3851.44	0.147536
$881$	16929.0	0.647392	0.323696	−	0.946161i	$-0.395075\pi$
$881$	16929.0	0.647392	0.323696	+	0.946161i	$0.395075\pi$
$882$	0	0
$883$	19754.4	0.752874	0.376437	−	0.926442i	$-0.377149\pi$
$883$	19754.4	0.752874	0.376437	+	0.926442i	$0.377149\pi$
$884$	−1153.54	−0.0438887
$885$	66605.7	2.52986
$886$	−16884.7	−0.640240
$887$	−236.500	−0.00895254	−0.00447627	−	0.999990i	$-0.501425\pi$
$887$	−236.500	−0.00895254	−0.00447627	+	0.999990i	$0.501425\pi$
$888$	−27959.4	−1.05660
$889$	0	0
$890$	−3988.27	−0.150210
$891$	129010.	4.85074
$892$	−14659.2	−0.550255
$893$	−50939.0	−1.90885
$894$	50027.3	1.87155
$895$	20193.9	0.754199
$896$	0	0
$897$	2024.40	0.0753544
$898$	27907.5	1.03707
$899$	−5716.91	−0.212091
$900$	17701.0	0.655592
$901$	24021.9	0.888220
$902$	−27881.8	−1.02923
$903$	0	0
$904$	−9301.91	−0.342231
$905$	20017.5	0.735252
$906$	67708.4	2.48285
$907$	33466.2	1.22517	0.612584	−	0.790406i	$-0.290130\pi$
$907$	33466.2	1.22517	0.612584	+	0.790406i	$0.290130\pi$
$908$	3580.85	0.130875
$909$	42496.2	1.55062
$910$	0	0
$911$	−15840.6	−0.576095	−0.288048	−	0.957616i	$-0.593006\pi$
$911$	−15840.6	−0.576095	−0.288048	+	0.957616i	$0.593006\pi$
$912$	14969.9	0.543536
$913$	40649.5	1.47350
$914$	−17643.2	−0.638496
$915$	48154.9	1.73984
$916$	−22.9438	−0.000827603 0
$917$	0	0
$918$	63231.8	2.27338
$919$	−18784.5	−0.674258	−0.337129	−	0.941458i	$-0.609456\pi$
$919$	−18784.5	−0.674258	−0.337129	+	0.941458i	$0.609456\pi$
$920$	9140.02	0.327541
$921$	−17400.9	−0.622561
$922$	−32772.0	−1.17060
$923$	1605.57	0.0572566
$924$	0	0
$925$	6082.49	0.216207
$926$	−29945.7	−1.06272
$927$	11582.0	0.410358
$928$	−4017.87	−0.142126
$929$	13479.3	0.476041	0.238021	−	0.971260i	$-0.423501\pi$
$929$	13479.3	0.476041	0.238021	+	0.971260i	$0.423501\pi$
$930$	−40030.9	−1.41147
$931$	0	0
$932$	22472.1	0.789804
$933$	−14854.4	−0.521235
$934$	32317.7	1.13219
$935$	20701.0	0.724060
$936$	−8058.06	−0.281395
$937$	−31229.2	−1.08881	−0.544403	−	0.838823i	$-0.683244\pi$
$937$	−31229.2	−1.08881	−0.544403	+	0.838823i	$0.683244\pi$
$938$	0	0
$939$	21104.7	0.733467
$940$	14655.8	0.508533
$941$	4595.15	0.159190	0.0795949	−	0.996827i	$-0.474637\pi$
$941$	4595.15	0.159190	0.0795949	+	0.996827i	$0.474637\pi$
$942$	−60491.5	−2.09227
$943$	−16732.9	−0.577836
$944$	8842.62	0.304876
$945$	0	0
$946$	3255.00	0.111870
$947$	−44968.8	−1.54307	−0.771536	−	0.636186i	$-0.780511\pi$
$947$	−44968.8	−1.54307	−0.771536	+	0.636186i	$0.780511\pi$
$948$	36143.1	1.23826
$949$	−2144.42	−0.0733516
$950$	−12877.9	−0.439805
$951$	−73493.0	−2.50597
$952$	0	0
$953$	6081.61	0.206719	0.103359	−	0.994644i	$-0.467041\pi$
$953$	6081.61	0.206719	0.103359	+	0.994644i	$0.467041\pi$
$954$	58417.5	1.98253
$955$	−29665.7	−1.00519
$956$	27443.4	0.928435
$957$	9667.85	0.326559
$958$	−7809.94	−0.263390
$959$	0	0
$960$	−36199.0	−1.21700
$961$	26820.0	0.900271
$962$	−963.941	−0.0323064
$963$	115198.	3.85482
$964$	−18314.0	−0.611883
$965$	−8590.47	−0.286567
$966$	0	0
$967$	44649.4	1.48483	0.742413	−	0.669942i	$-0.233681\pi$
$967$	44649.4	1.48483	0.742413	+	0.669942i	$0.233681\pi$
$968$	4885.23	0.162208
$969$	80461.7	2.66749
$970$	1624.68	0.0537786
$971$	9668.38	0.319540	0.159770	−	0.987154i	$-0.448925\pi$
$971$	9668.38	0.319540	0.159770	+	0.987154i	$0.448925\pi$
$972$	−83517.7	−2.75600
$973$	0	0
$974$	29201.5	0.960652
$975$	2357.33	0.0774308
$976$	6393.08	0.209669
$977$	20382.9	0.667460	0.333730	−	0.942669i	$-0.391693\pi$
$977$	20382.9	0.667460	0.333730	+	0.942669i	$0.391693\pi$
$978$	−1325.61	−0.0433418
$979$	−9538.05	−0.311376
$980$	0	0
$981$	−120427.	−3.91941
$982$	−5852.01	−0.190168
$983$	−29759.1	−0.965583	−0.482791	−	0.875735i	$-0.660377\pi$
$983$	−29759.1	−0.965583	−0.482791	+	0.875735i	$0.660377\pi$
$984$	89565.8	2.90168
$985$	5552.51	0.179612
$986$	2884.66	0.0931705
$987$	0	0
$988$	−2339.04	−0.0753185
$989$	1953.45	0.0628070
$990$	50341.6	1.61612
$991$	13568.8	0.434943	0.217471	−	0.976067i	$-0.430219\pi$
$991$	13568.8	0.434943	0.217471	+	0.976067i	$0.430219\pi$
$992$	39786.5	1.27341
$993$	13149.2	0.420218
$994$	0	0
$995$	−7530.09	−0.239919
$996$	−45458.3	−1.44618
$997$	26264.3	0.834302	0.417151	−	0.908837i	$-0.363029\pi$
$997$	26264.3	0.834302	0.417151	+	0.908837i	$0.363029\pi$
$998$	−42925.8	−1.36151
$999$	−60558.7	−1.91791

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2107.4.a.h.1.12	yes	30
7.6	odd	2	inner	2107.4.a.h.1.11	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2107.4.a.h.1.11	✓	30	7.6	odd	2	inner
2107.4.a.h.1.12	yes	30	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2107 = 7^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2107.a (trivial)

\(f(q)\)	\(=\)	\(q-1.93072 q^{2} +10.2626 q^{3} -4.27231 q^{4} -8.49116 q^{5} -19.8142 q^{6} +23.6944 q^{8} +78.3210 q^{9} +O(q^{10})\)	\(q-1.93072 q^{2} +10.2626 q^{3} -4.27231 q^{4} -8.49116 q^{5} -19.8142 q^{6} +23.6944 q^{8} +78.3210 q^{9} +16.3941 q^{10} +39.2068 q^{11} -43.8450 q^{12} -4.34216 q^{13} -87.1414 q^{15} -11.5690 q^{16} -62.1818 q^{17} -151.216 q^{18} -126.086 q^{19} +36.2768 q^{20} -75.6976 q^{22} -45.4291 q^{23} +243.166 q^{24} -52.9002 q^{25} +8.38352 q^{26} +526.687 q^{27} +24.0276 q^{29} +168.246 q^{30} -237.931 q^{31} -167.219 q^{32} +402.364 q^{33} +120.056 q^{34} -334.611 q^{36} -114.981 q^{37} +243.438 q^{38} -44.5619 q^{39} -201.193 q^{40} +368.331 q^{41} -43.0000 q^{43} -167.504 q^{44} -665.036 q^{45} +87.7110 q^{46} +404.000 q^{47} -118.728 q^{48} +102.136 q^{50} -638.147 q^{51} +18.5510 q^{52} -386.318 q^{53} -1016.89 q^{54} -332.912 q^{55} -1293.98 q^{57} -46.3907 q^{58} -764.340 q^{59} +372.295 q^{60} -552.606 q^{61} +459.378 q^{62} +415.405 q^{64} +36.8700 q^{65} -776.854 q^{66} -582.026 q^{67} +265.660 q^{68} -466.220 q^{69} -369.762 q^{71} +1855.77 q^{72} +493.859 q^{73} +221.996 q^{74} -542.893 q^{75} +538.680 q^{76} +86.0367 q^{78} -824.338 q^{79} +98.2339 q^{80} +3290.51 q^{81} -711.146 q^{82} +1036.80 q^{83} +527.995 q^{85} +83.0211 q^{86} +246.586 q^{87} +928.984 q^{88} -243.275 q^{89} +1284.00 q^{90} +194.087 q^{92} -2441.79 q^{93} -780.013 q^{94} +1070.62 q^{95} -1716.10 q^{96} +99.1014 q^{97} +3070.72 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9}+O(q^{10}) \) 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9	\( 30 q - 8 q^{2} + 100 q^{4} - 90 q^{8} + 264 q^{9} - 60 q^{11} - 96 q^{15} + 372 q^{16} - 1008 q^{18} - 234 q^{22} - 214 q^{23} + 520 q^{25} - 870 q^{29} - 12 q^{30} - 1548 q^{32} + 1142 q^{36} - 1246 q^{37} - 416 q^{39} - 1290 q^{43} - 446 q^{44} - 660 q^{46} - 278 q^{50} - 3702 q^{51} - 2960 q^{53} + 620 q^{57} - 3634 q^{58} - 898 q^{60} + 2578 q^{64} - 4848 q^{65} + 928 q^{67} - 1708 q^{71} - 7900 q^{72} + 1714 q^{74} - 138 q^{78} - 3562 q^{79} + 2210 q^{81} - 948 q^{85} + 344 q^{86} + 2502 q^{88} - 3848 q^{92} - 11986 q^{93} - 2894 q^{95} - 2804 q^{99}+O(q^{100}) \) 30 * q - 8 * q^2 + 100 * q^4 - 90 * q^8 + 264 * q^9 - 60 * q^11 - 96 * q^15 + 372 * q^16 - 1008 * q^18 - 234 * q^22 - 214 * q^23 + 520 * q^25 - 870 * q^29 - 12 * q^30 - 1548 * q^32 + 1142 * q^36 - 1246 * q^37 - 416 * q^39 - 1290 * q^43 - 446 * q^44 - 660 * q^46 - 278 * q^50 - 3702 * q^51 - 2960 * q^53 + 620 * q^57 - 3634 * q^58 - 898 * q^60 + 2578 * q^64 - 4848 * q^65 + 928 * q^67 - 1708 * q^71 - 7900 * q^72 + 1714 * q^74 - 138 * q^78 - 3562 * q^79 + 2210 * q^81 - 948 * q^85 + 344 * q^86 + 2502 * q^88 - 3848 * q^92 - 11986 * q^93 - 2894 * q^95 - 2804 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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