LMFDB - Embedded newform 2107.4.a.h.1.11

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			1.11
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	$-0.949657\pi$
$3$	−10.2626	−1.97504	−0.987519	+	0.157499i	$0.949657\pi$
$4$	−4.27231	−0.534038
$5$	8.49116	0.759473	0.379736	−	0.925095i	$-0.376015\pi$
$5$	8.49116	0.759473	0.379736	+	0.925095i	$0.376015\pi$
$6$	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	$-0.485251\pi$
$13$	4.34216	0.0926384	0.0463192	+	0.998927i	$0.485251\pi$
$14$	0	0
$15$	−87.1414	−1.49999
$16$	−11.5690	−0.180765
$17$	62.1818	0.887135	0.443567	−	0.896241i	$-0.353713\pi$
$17$	62.1818	0.887135	0.443567	+	0.896241i	$0.353713\pi$
$18$	−151.216	−1.98011
$19$	126.086	1.52243	0.761216	−	0.648498i	$-0.224603\pi$
$19$	126.086	1.52243	0.761216	+	0.648498i	$0.224603\pi$
$20$	−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	$-0.257939\pi$
$31$	237.931	1.37850	0.689252	+	0.724522i	$0.257939\pi$
$32$	−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	$-0.747490\pi$
$41$	−368.331	−1.40302	−0.701508	+	0.712661i	$0.747490\pi$
$42$	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	$-0.715680\pi$
$47$	−404.000	−1.25382	−0.626909	+	0.779092i	$0.715680\pi$
$48$	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	$-0.180614\pi$
$59$	764.340	1.68659	0.843293	+	0.537454i	$0.180614\pi$
$60$	372.295	0.801051
$61$	552.606	1.15990	0.579951	−	0.814652i	$-0.303072\pi$
$61$	552.606	1.15990	0.579951	+	0.814652i	$0.303072\pi$
$62$	−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	$-0.629568\pi$
$73$	−493.859	−0.791806	−0.395903	+	0.918292i	$0.629568\pi$
$74$	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	$-0.740443\pi$
$83$	−1036.80	−1.37112	−0.685561	+	0.728015i	$0.740443\pi$
$84$	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	$-0.453723\pi$
$89$	243.275	0.289743	0.144871	+	0.989450i	$0.453723\pi$
$90$	−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	$-0.516517\pi$
$97$	−99.1014	−0.103734	−0.0518671	+	0.998654i	$0.516517\pi$
$98$	0	0
$99$	3070.72	3.11736
$100$	226.006	0.226006
$101$	−542.591	−0.534553	−0.267276	−	0.963620i	$-0.586124\pi$
$101$	−542.591	−0.534553	−0.267276	+	0.963620i	$0.586124\pi$
$102$	1232.08	1.19603
$103$	−147.878	−0.141465	−0.0707324	−	0.997495i	$-0.522534\pi$
$103$	−147.878	−0.141465	−0.0707324	+	0.997495i	$0.522534\pi$
$104$	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	$-0.273964\pi$
$131$	1954.93	1.30384	0.651919	+	0.758289i	$0.273964\pi$
$132$	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	$-0.449297\pi$
$139$	519.878	0.317234	0.158617	+	0.987340i	$0.449297\pi$
$140$	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	$-0.782732\pi$
$157$	−3052.93	−1.55191	−0.775956	+	0.630787i	$0.782732\pi$
$158$	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	$-0.682028\pi$
$167$	−2335.93	−1.08239	−0.541196	+	0.840897i	$0.682028\pi$
$168$	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	$-0.511783\pi$
$173$	−168.429	−0.0740198	−0.0370099	+	0.999315i	$0.511783\pi$
$174$	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	$-0.339164\pi$
$181$	2357.45	0.968109	0.484054	+	0.875038i	$0.339164\pi$
$182$	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	$-0.550489\pi$
$199$	−886.815	−0.315903	−0.157951	+	0.987447i	$0.550489\pi$
$200$	−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	$-0.672276\pi$
$223$	−3431.23	−1.03037	−0.515183	+	0.857080i	$0.672276\pi$
$224$	0	0
$225$	−4143.19	−1.22761
$226$	757.959	0.223092
$227$	838.154	0.245067	0.122534	−	0.992464i	$-0.460898\pi$
$227$	838.154	0.245067	0.122534	+	0.992464i	$0.460898\pi$
$228$	5528.26	1.60578
$229$	−5.37035	−0.00154971	−0.000774854	−	1.00000i	$-0.500247\pi$
$229$	−5.37035	−0.00154971	−0.000774854		1.00000i	$0.500247\pi$
$230$	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	$-0.694175\pi$
$241$	−4286.69	−1.14577	−0.572883	+	0.819637i	$0.694175\pi$
$242$	−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	$-0.417539\pi$
$251$	2037.38	0.512343	0.256171	+	0.966631i	$0.417539\pi$
$252$	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	$-0.620793\pi$
$257$	−3052.45	−0.740881	−0.370440	+	0.928856i	$0.620793\pi$
$258$	−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	$-0.475707\pi$
$269$	672.775	0.152490	0.0762450	+	0.997089i	$0.475707\pi$
$270$	8634.55	1.94623
$271$	374.852	0.0840245	0.0420122	−	0.999117i	$-0.486623\pi$
$271$	374.852	0.0840245	0.0420122	+	0.999117i	$0.486623\pi$
$272$	−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	$-0.305982\pi$
$283$	5450.90	1.14496	0.572478	+	0.819920i	$0.305982\pi$
$284$	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	$-0.212977\pi$
$293$	7867.96	1.56878	0.784388	+	0.620270i	$0.212977\pi$
$294$	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	$-0.449622\pi$
$307$	1695.56	0.315215	0.157607	+	0.987502i	$0.449622\pi$
$308$	0	0
$309$	1517.62	0.279399
$310$	−3900.65	−0.714653
$311$	1447.43	0.263911	0.131956	−	0.991256i	$-0.457874\pi$
$311$	1447.43	0.263911	0.131956	+	0.991256i	$0.457874\pi$
$312$	−1055.87	−0.191592
$313$	−2056.47	−0.371368	−0.185684	−	0.982609i	$-0.559450\pi$
$313$	−2056.47	−0.371368	−0.185684	+	0.982609i	$0.559450\pi$
$314$	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	$-0.327998\pi$
$349$	6708.21	1.02889	0.514444	+	0.857524i	$0.327998\pi$
$350$	0	0
$351$	−2286.96	−0.347774
$352$	−6556.13	−0.992735
$353$	−6010.82	−0.906299	−0.453149	−	0.891435i	$-0.649700\pi$
$353$	−6010.82	−0.906299	−0.453149	+	0.891435i	$0.649700\pi$
$354$	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	$-0.418973\pi$
$367$	3540.86	0.503628	0.251814	+	0.967776i	$0.418973\pi$
$368$	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	$-0.506331\pi$
$383$	−298.122	−0.0397736	−0.0198868	+	0.999802i	$0.506331\pi$
$384$	−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	$-0.361871\pi$
$397$	6651.72	0.840907	0.420453	+	0.907314i	$0.361871\pi$
$398$	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	$-0.249979\pi$
$409$	11698.5	1.41431	0.707153	+	0.707060i	$0.249979\pi$
$410$	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	$-0.747525\pi$
$419$	−12034.6	−1.40317	−0.701587	+	0.712584i	$0.747525\pi$
$420$	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	$-0.498517\pi$
$433$	83.9305	0.00931511	0.00465755	+	0.999989i	$0.498517\pi$
$434$	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	$-0.394490\pi$
$439$	5986.72	0.650867	0.325434	+	0.945565i	$0.394490\pi$
$440$	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	$-0.827944\pi$
$461$	−16974.0	−1.71487	−0.857436	+	0.514591i	$0.827944\pi$
$462$	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	$-0.188736\pi$
$467$	16738.6	1.65861	0.829307	+	0.558794i	$0.188736\pi$
$468$	−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	$-0.561798\pi$
$479$	−4045.08	−0.385855	−0.192928	+	0.981213i	$0.561798\pi$
$480$	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	$-0.253391\pi$
$503$	15783.0	1.39907	0.699533	+	0.714600i	$0.253391\pi$
$504$	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	$-0.364636\pi$
$509$	9475.26	0.825115	0.412558	+	0.910932i	$0.364636\pi$
$510$	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	$-0.531570\pi$
$521$	−2355.06	−0.198036	−0.0990181	+	0.995086i	$0.531570\pi$
$522$	−3633.36	−0.304651
$523$	−4592.63	−0.383980	−0.191990	−	0.981397i	$-0.561494\pi$
$523$	−4592.63	−0.383980	−0.191990	+	0.981397i	$0.561494\pi$
$524$	−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	$-0.492533\pi$
$563$	626.656	0.0469101	0.0234551	+	0.999725i	$0.492533\pi$
$564$	−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	$-0.451752\pi$
$577$	4185.59	0.301991	0.150995	+	0.988534i	$0.451752\pi$
$578$	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	$-0.588262\pi$
$587$	−7786.33	−0.547489	−0.273745	+	0.961802i	$0.588262\pi$
$588$	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	$-0.306285\pi$
$593$	16511.2	1.14340	0.571698	+	0.820464i	$0.306285\pi$
$594$	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	$-0.799247\pi$
$601$	−23798.6	−1.61525	−0.807625	+	0.589696i	$0.799247\pi$
$602$	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	$-0.569227\pi$
$607$	−6453.70	−0.431544	−0.215772	+	0.976444i	$0.569227\pi$
$608$	−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	$-0.497041\pi$
$619$	286.367	0.0185946	0.00929730	+	0.999957i	$0.497041\pi$
$620$	−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	$-0.702714\pi$
$643$	−19391.8	−1.18933	−0.594663	+	0.803975i	$0.702714\pi$
$644$	0	0
$645$	3747.08	0.228746
$646$	−15137.4	−0.921941
$647$	−18817.8	−1.14344	−0.571718	−	0.820450i	$-0.693723\pi$
$647$	−18817.8	−1.14344	−0.571718	+	0.820450i	$0.693723\pi$
$648$	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	$-0.737518\pi$
$661$	−23072.9	−1.35769	−0.678843	+	0.734283i	$0.737518\pi$
$662$	−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	$-0.519864\pi$
$677$	−2197.08	−0.124728	−0.0623638	+	0.998053i	$0.519864\pi$
$678$	−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	$-0.234920\pi$
$691$	26875.8	1.47960	0.739801	+	0.672826i	$0.234920\pi$
$692$	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	$-0.326016\pi$
$719$	20041.8	1.03955	0.519773	+	0.854304i	$0.326016\pi$
$720$	−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	$-0.644036\pi$
$727$	−17140.7	−0.874434	−0.437217	+	0.899356i	$0.644036\pi$
$728$	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	$-0.739681\pi$
$733$	−27141.0	−1.36763	−0.683816	+	0.729655i	$0.739681\pi$
$734$	−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	$-0.506425\pi$
$761$	−847.470	−0.0403689	−0.0201845	+	0.999796i	$0.506425\pi$
$762$	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	$-0.780335\pi$
$769$	−32891.0	−1.54237	−0.771184	+	0.636612i	$0.780335\pi$
$770$	0	0
$771$	31326.0	1.46327
$772$	−4322.27	−0.201505
$773$	−36866.6	−1.71540	−0.857698	−	0.514154i	$-0.828106\pi$
$773$	−36866.6	−1.71540	−0.857698	+	0.514154i	$0.828106\pi$
$774$	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	$-0.328385\pi$
$787$	22669.9	1.02680	0.513402	+	0.858148i	$0.328385\pi$
$788$	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	$-0.282592\pi$
$797$	28401.2	1.26226	0.631130	+	0.775677i	$0.282592\pi$
$798$	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	$-0.608985\pi$
$811$	−15508.1	−0.671473	−0.335736	+	0.941956i	$0.608985\pi$
$812$	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	$-0.558767\pi$
$829$	−8763.50	−0.367152	−0.183576	+	0.983006i	$0.558767\pi$
$830$	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	$-0.731875\pi$
$839$	−32356.6	−1.33144	−0.665718	+	0.746203i	$0.731875\pi$
$840$	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	$-0.540122\pi$
$853$	−6263.80	−0.251429	−0.125714	+	0.992066i	$0.540122\pi$
$854$	0	0
$855$	83852.0	3.35401
$856$	34850.7	1.39156
$857$	−12255.0	−0.488473	−0.244236	−	0.969716i	$-0.578537\pi$
$857$	−12255.0	−0.488473	−0.244236	+	0.969716i	$0.578537\pi$
$858$	3373.23	0.134219
$859$	34594.4	1.37409	0.687047	−	0.726613i	$-0.258907\pi$
$859$	34594.4	1.37409	0.687047	+	0.726613i	$0.258907\pi$
$860$	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	$-0.604925\pi$
$881$	−16929.0	−0.647392	−0.323696	+	0.946161i	$0.604925\pi$
$882$	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	$-0.498575\pi$
$887$	236.500	0.00895254	0.00447627	+	0.999990i	$0.498575\pi$
$888$	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	$-0.576499\pi$
$929$	−13479.3	−0.476041	−0.238021	+	0.971260i	$0.576499\pi$
$930$	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	$-0.316756\pi$
$937$	31229.2	1.08881	0.544403	+	0.838823i	$0.316756\pi$
$938$	0	0
$939$	21104.7	0.733467
$940$	14655.8	0.508533
$941$	−4595.15	−0.159190	−0.0795949	−	0.996827i	$-0.525363\pi$
$941$	−4595.15	−0.159190	−0.0795949	+	0.996827i	$0.525363\pi$
$942$	−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	$-0.551075\pi$
$971$	−9668.38	−0.319540	−0.159770	+	0.987154i	$0.551075\pi$
$972$	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	$-0.339623\pi$
$983$	29759.1	0.965583	0.482791	+	0.875735i	$0.339623\pi$
$984$	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	$-0.636971\pi$
$997$	−26264.3	−0.834302	−0.417151	+	0.908837i	$0.636971\pi$
$998$	−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.1.11	✓	30
7.6	odd	2	inner	2107.4.a.h.1.12	yes	30

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

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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