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Properties

Label 6930.2.a.u.1.1
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6930,2,Mod(1,6930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6930, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6930.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.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: \(55.3363286007\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6930.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} +4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{22} -4.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{28} +2.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} +1.00000 q^{35} -10.0000 q^{37} +4.00000 q^{38} -1.00000 q^{40} +6.00000 q^{41} +12.0000 q^{43} +1.00000 q^{44} -4.00000 q^{46} -12.0000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{55} -1.00000 q^{56} +2.00000 q^{58} +12.0000 q^{59} +6.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} +8.00000 q^{67} -6.00000 q^{68} +1.00000 q^{70} +8.00000 q^{71} +14.0000 q^{73} -10.0000 q^{74} +4.00000 q^{76} -1.00000 q^{77} -1.00000 q^{80} +6.00000 q^{82} -4.00000 q^{83} +6.00000 q^{85} +12.0000 q^{86} +1.00000 q^{88} +6.00000 q^{89} -2.00000 q^{91} -4.00000 q^{92} -12.0000 q^{94} -4.00000 q^{95} -14.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 − 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 − 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 − 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 14.0000 1.19610 0.598050 − 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) 18.0000 1.47462 0.737309 − 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 − 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 24.0000 1.87983 0.939913 − 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 16.0000 1.23812 0.619059 − 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) −26.0000 −1.97674 −0.988372 − 0.152057i \(-0.951410\pi\)
−0.988372 + 0.152057i \(0.951410\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 10.0000 0.735215
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 0 0
\(193\) 22.0000 1.58359 0.791797 − 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 14.0000 0.997459 0.498729 − 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) −12.0000 −0.836080
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 − 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −12.0000 −0.818393
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 6.00000 0.406371
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −12.0000 −0.803579 −0.401790 − 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) 10.0000 0.655122 0.327561 − 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 6.00000 0.388922
\(239\) −16.0000 −1.03495 −0.517477 − 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 − 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 14.0000 0.845771
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) 30.0000 1.78965 0.894825 − 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) −2.00000 −0.117444
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) 22.0000 1.28525 0.642627 − 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 − 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 18.0000 1.01742 0.508710 − 0.860938i \(-0.330123\pi\)
0.508710 + 0.860938i \(0.330123\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 0 0
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 24.0000 1.32924
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) 16.0000 0.875481
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 − 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 6.00000 0.325396
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) −26.0000 −1.39777
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) 12.0000 0.626395 0.313197 − 0.949688i \(-0.398600\pi\)
0.313197 + 0.949688i \(0.398600\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 10.0000 0.519875
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) −30.0000 −1.55334 −0.776671 − 0.629907i \(-0.783093\pi\)
−0.776671 + 0.629907i \(0.783093\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 − 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) 24.0000 1.22795
\(383\) 20.0000 1.02195 0.510976 − 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 22.0000 1.11977
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) 10.0000 0.507020 0.253510 − 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 14.0000 0.705310
\(395\) 0 0
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −2.00000 −0.0998752 −0.0499376 − 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 − 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) −12.0000 −0.591198
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 4.00000 0.195646
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 − 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) −28.0000 −1.36302
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) −12.0000 −0.578691
\(431\) −16.0000 −0.770693 −0.385346 − 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 − 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 − 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) 16.0000 0.760183 0.380091 − 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) −12.0000 −0.568216
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −2.00000 −0.0943858 −0.0471929 − 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 − 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) −30.0000 −1.39724 −0.698620 − 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 28.0000 1.30127 0.650635 − 0.759390i \(-0.274503\pi\)
0.650635 + 0.759390i \(0.274503\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 12.0000 0.553519
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 6.00000 0.275010
\(477\) 0 0
\(478\) −16.0000 −0.731823
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 − 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) −4.00000 −0.180517 −0.0902587 − 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 − 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 18.0000 0.797836 0.398918 − 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 12.0000 0.528783
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 10.0000 0.439375
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) −10.0000 −0.438108 −0.219054 − 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) −44.0000 −1.92399 −0.961993 − 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 0 0
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 18.0000 0.776035
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −26.0000 −1.11783 −0.558914 − 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −6.00000 −0.257248
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 − 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 14.0000 0.598050
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 6.00000 0.254228 0.127114 − 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) 4.00000 0.168580 0.0842900 − 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −26.0000 −1.08998 −0.544988 − 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 − 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −46.0000 −1.91501 −0.957503 − 0.288425i \(-0.906868\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) −2.00000 −0.0830455
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) −6.00000 −0.248495
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) 22.0000 0.908812
\(587\) −8.00000 −0.330195 −0.165098 − 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) −12.0000 −0.494032
\(591\) 0 0
\(592\) −10.0000 −0.410997
\(593\) 42.0000 1.72473 0.862367 − 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) −6.00000 −0.245976
\(596\) 18.0000 0.737309
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 − 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) −12.0000 −0.489083
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 − 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) −14.0000 −0.565455 −0.282727 − 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) −18.0000 −0.724653 −0.362326 − 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 − 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 18.0000 0.719425
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) 60.0000 2.39236
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 2.00000 0.0791808
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −34.0000 −1.34292 −0.671460 − 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 − 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 4.00000 0.157256 0.0786281 − 0.996904i \(-0.474946\pi\)
0.0786281 + 0.996904i \(0.474946\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 24.0000 0.939913
\(653\) −30.0000 −1.17399 −0.586995 − 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) −4.00000 −0.156293
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 12.0000 0.467809
\(659\) 28.0000 1.09073 0.545363 − 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 4.00000 0.155113
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) 16.0000 0.619059
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) 6.00000 0.231627
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 − 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −26.0000 −0.999261 −0.499631 − 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) 8.00000 0.306336
\(683\) 24.0000 0.918334 0.459167 − 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) −14.0000 −0.534913
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 12.0000 0.457496
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 − 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) −26.0000 −0.988372
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) −10.0000 −0.378506
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −30.0000 −1.13308 −0.566542 − 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) −40.0000 −1.50863
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 − 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) −2.00000 −0.0747958
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) 16.0000 0.596699 0.298350 − 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) −20.0000 −0.741759 −0.370879 − 0.928681i \(-0.620944\pi\)
−0.370879 + 0.928681i \(0.620944\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 0 0
\(730\) −14.0000 −0.518163
\(731\) −72.0000 −2.66302
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 − 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 12.0000 0.442928
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) −12.0000 −0.441427 −0.220714 − 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 10.0000 0.367607
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) −16.0000 −0.586983 −0.293492 − 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) −30.0000 −1.09838
\(747\) 0 0
\(748\) −6.00000 −0.219382
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 − 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) 46.0000 1.67190 0.835949 − 0.548807i \(-0.184918\pi\)
0.835949 + 0.548807i \(0.184918\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) −2.00000 −0.0724999 −0.0362500 − 0.999343i \(-0.511541\pi\)
−0.0362500 + 0.999343i \(0.511541\pi\)
\(762\) 0 0
\(763\) −6.00000 −0.217215
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) 20.0000 0.722629
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 − 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) 22.0000 0.791797
\(773\) 18.0000 0.647415 0.323708 − 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) −14.0000 −0.502571
\(777\) 0 0
\(778\) 10.0000 0.358517
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 24.0000 0.858238
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 − 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 14.0000 0.498729
\(789\) 0 0
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −14.0000 −0.495905 −0.247953 − 0.968772i \(-0.579758\pi\)
−0.247953 + 0.968772i \(0.579758\pi\)
\(798\) 0 0
\(799\) 72.0000 2.54718
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −2.00000 −0.0706225
\(803\) 14.0000 0.494049
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 16.0000 0.563576
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 30.0000 1.05474 0.527372 − 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 − 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) −10.0000 −0.350500
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) 48.0000 1.67931
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 42.0000 1.46581 0.732905 − 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 − 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −12.0000 −0.418040
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) −20.0000 −0.695468 −0.347734 − 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 − 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 38.0000 1.30957
\(843\) 0 0
\(844\) −28.0000 −0.963800
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −6.00000 −0.205798
\(851\) 40.0000 1.37118
\(852\) 0 0
\(853\) 2.00000 0.0684787 0.0342393 − 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −52.0000 −1.77422 −0.887109 − 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) −12.0000 −0.409197
\(861\) 0 0
\(862\) −16.0000 −0.544962
\(863\) −36.0000 −1.22545 −0.612727 − 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) 26.0000 0.884027
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) 0 0
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) −16.0000 −0.541208
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −6.00000 −0.202606 −0.101303 − 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) 8.00000 0.269987
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) −50.0000 −1.68454 −0.842271 − 0.539054i \(-0.818782\pi\)
−0.842271 + 0.539054i \(0.818782\pi\)
\(882\) 0 0
\(883\) −24.0000 −0.807664 −0.403832 − 0.914833i \(-0.632322\pi\)
−0.403832 + 0.914833i \(0.632322\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) −16.0000 −0.537227 −0.268614 − 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) −12.0000 −0.401790
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −2.00000 −0.0667409
\(899\) 16.0000 0.533630
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 6.00000 0.199778
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −32.0000 −1.06254 −0.531271 − 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) 2.00000 0.0662994
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) −4.00000 −0.132381
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 − 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 4.00000 0.131876
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 28.0000 0.920137
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) 30.0000 0.984268 0.492134 − 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) 0 0
\(935\) 6.00000 0.196221
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 − 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) 12.0000 0.391397
\(941\) −46.0000 −1.49956 −0.749779 − 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 24.0000 0.779895 0.389948 − 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 0 0
\(949\) 28.0000 0.908918
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 6.00000 0.194461
\(953\) −38.0000 −1.23094 −0.615470 − 0.788160i \(-0.711034\pi\)
−0.615470 + 0.788160i \(0.711034\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) −14.0000 −0.452084
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −20.0000 −0.644826
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) −22.0000 −0.708205
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 − 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 14.0000 0.449513
\(971\) −36.0000 −1.15529 −0.577647 − 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) 12.0000 0.384505
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) −42.0000 −1.34370 −0.671850 − 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) −4.00000 −0.127645
\(983\) 12.0000 0.382741 0.191370 − 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) −14.0000 −0.446077
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) −24.0000 −0.762385 −0.381193 − 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) 18.0000 0.570066 0.285033 − 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) −36.0000 −1.13956
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6930.2.a.u.1.1 1
3.2 odd 2 770.2.a.c.1.1 ✓ 1
12.11 even 2 6160.2.a.i.1.1 1
15.2 even 4 3850.2.c.h.1849.1 2
15.8 even 4 3850.2.c.h.1849.2 2
15.14 odd 2 3850.2.a.t.1.1 1
21.20 even 2 5390.2.a.i.1.1 1
33.32 even 2 8470.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.c.1.1 ✓ 1 3.2 odd 2
3850.2.a.t.1.1 1 15.14 odd 2
3850.2.c.h.1849.1 2 15.2 even 4
3850.2.c.h.1849.2 2 15.8 even 4
5390.2.a.i.1.1 1 21.20 even 2
6160.2.a.i.1.1 1 12.11 even 2
6930.2.a.u.1.1 1 1.1 even 1 trivial
8470.2.a.ba.1.1 1 33.32 even 2