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Properties

Label 3.31.b.a.2.1
Level $3$
Weight $31$
Character 3.2
Self dual yes
Analytic conductor $17.104$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3,31,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 31, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 31);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 31 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

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: \(17.1042785708\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 2.1
Character \(\chi\) \(=\) 3.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.43489e7 q^{3} +1.07374e9 q^{4} -7.94535e12 q^{7} +2.05891e14 q^{9} +O(q^{10})\) \(q-1.43489e7 q^{3} +1.07374e9 q^{4} -7.94535e12 q^{7} +2.05891e14 q^{9} -1.54070e16 q^{12} +5.58507e16 q^{13} +1.15292e18 q^{16} +2.89405e19 q^{19} +1.14007e20 q^{21} +9.31323e20 q^{25} -2.95431e21 q^{27} -8.53125e21 q^{28} -1.27086e21 q^{31} +2.21074e23 q^{36} +5.21554e23 q^{37} -8.01396e23 q^{39} -4.27866e24 q^{43} -1.65432e25 q^{48} +4.05892e25 q^{49} +5.99692e25 q^{52} -4.15264e26 q^{57} +4.14395e26 q^{61} -1.63588e27 q^{63} +1.23794e27 q^{64} +4.89057e27 q^{67} -1.77373e28 q^{73} -1.33635e28 q^{75} +3.10746e28 q^{76} -5.04025e28 q^{79} +4.23912e28 q^{81} +1.22414e29 q^{84} -4.43753e29 q^{91} +1.82354e28 q^{93} -1.95075e29 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −1.43489e7 −1.00000
\(4\) 1.07374e9 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −7.94535e12 −1.67356 −0.836782 − 0.547536i \(-0.815566\pi\)
−0.836782 + 0.547536i \(0.815566\pi\)
\(8\) 0 0
\(9\) 2.05891e14 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −1.54070e16 −1.00000
\(13\) 5.58507e16 1.09113 0.545567 − 0.838067i \(-0.316314\pi\)
0.545567 + 0.838067i \(0.316314\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.15292e18 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 2.89405e19 1.90635 0.953173 − 0.302424i \(-0.0977959\pi\)
0.953173 + 0.302424i \(0.0977959\pi\)
\(20\) 0 0
\(21\) 1.14007e20 1.67356
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 9.31323e20 1.00000
\(26\) 0 0
\(27\) −2.95431e21 −1.00000
\(28\) −8.53125e21 −1.67356
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.27086e21 −0.0541591 −0.0270795 − 0.999633i \(-0.508621\pi\)
−0.0270795 + 0.999633i \(0.508621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 2.21074e23 1.00000
\(37\) 5.21554e23 1.56413 0.782066 − 0.623196i \(-0.214166\pi\)
0.782066 + 0.623196i \(0.214166\pi\)
\(38\) 0 0
\(39\) −8.01396e23 −1.09113
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −4.27866e24 −1.34673 −0.673365 − 0.739310i \(-0.735152\pi\)
−0.673365 + 0.739310i \(0.735152\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −1.65432e25 −1.00000
\(49\) 4.05892e25 1.80082
\(50\) 0 0
\(51\) 0 0
\(52\) 5.99692e25 1.09113
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.15264e26 −1.90635
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 4.14395e26 0.687808 0.343904 − 0.939005i \(-0.388251\pi\)
0.343904 + 0.939005i \(0.388251\pi\)
\(62\) 0 0
\(63\) −1.63588e27 −1.67356
\(64\) 1.23794e27 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.89057e27 1.98718 0.993591 − 0.113038i \(-0.0360581\pi\)
0.993591 + 0.113038i \(0.0360581\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.77373e28 −1.99088 −0.995439 − 0.0953981i \(-0.969588\pi\)
−0.995439 + 0.0953981i \(0.969588\pi\)
\(74\) 0 0
\(75\) −1.33635e28 −1.00000
\(76\) 3.10746e28 1.90635
\(77\) 0 0
\(78\) 0 0
\(79\) −5.04025e28 −1.73000 −0.864999 − 0.501773i \(-0.832681\pi\)
−0.864999 + 0.501773i \(0.832681\pi\)
\(80\) 0 0
\(81\) 4.23912e28 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 1.22414e29 1.67356
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −4.43753e29 −1.82608
\(92\) 0 0
\(93\) 1.82354e28 0.0541591
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.95075e29 −0.308053 −0.154026 − 0.988067i \(-0.549224\pi\)
−0.154026 + 0.988067i \(0.549224\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e30 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 1.30663e30 0.838679 0.419340 − 0.907829i \(-0.362262\pi\)
0.419340 + 0.907829i \(0.362262\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −3.17217e30 −1.00000
\(109\) 7.03314e30 1.93086 0.965432 − 0.260656i \(-0.0839389\pi\)
0.965432 + 0.260656i \(0.0839389\pi\)
\(110\) 0 0
\(111\) −7.48373e30 −1.56413
\(112\) −9.16036e30 −1.67356
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.14992e31 1.09113
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.74494e31 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −1.36457e30 −0.0541591
\(125\) 0 0
\(126\) 0 0
\(127\) −1.70970e31 −0.474093 −0.237046 − 0.971498i \(-0.576179\pi\)
−0.237046 + 0.971498i \(0.576179\pi\)
\(128\) 0 0
\(129\) 6.13940e31 1.34673
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −2.29942e32 −3.19039
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 2.59224e32 1.85547 0.927733 − 0.373244i \(-0.121755\pi\)
0.927733 + 0.373244i \(0.121755\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 2.37376e32 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) −5.82411e32 −1.80082
\(148\) 5.60014e32 1.56413
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −1.84787e32 −0.381960 −0.190980 − 0.981594i \(-0.561166\pi\)
−0.190980 + 0.981594i \(0.561166\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −8.60492e32 −1.09113
\(157\) −8.38852e32 −0.966475 −0.483237 − 0.875489i \(-0.660539\pi\)
−0.483237 + 0.875489i \(0.660539\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.01894e33 1.98171 0.990856 − 0.134921i \(-0.0430782\pi\)
0.990856 + 0.134921i \(0.0430782\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 4.99300e32 0.190573
\(170\) 0 0
\(171\) 5.95859e33 1.90635
\(172\) −4.59417e33 −1.34673
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −7.39968e33 −1.67356
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 6.07390e32 0.0828493 0.0414247 − 0.999142i \(-0.486810\pi\)
0.0414247 + 0.999142i \(0.486810\pi\)
\(182\) 0 0
\(183\) −5.94612e33 −0.687808
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.34730e34 1.67356
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −1.77631e34 −1.00000
\(193\) 2.13837e34 1.11359 0.556793 − 0.830651i \(-0.312032\pi\)
0.556793 + 0.830651i \(0.312032\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 4.35823e34 1.80082
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −5.17867e34 −1.70381 −0.851907 − 0.523693i \(-0.824554\pi\)
−0.851907 + 0.523693i \(0.824554\pi\)
\(200\) 0 0
\(201\) −7.01743e34 −1.98718
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 6.43914e34 1.09113
\(209\) 0 0
\(210\) 0 0
\(211\) −3.17651e34 −0.434223 −0.217111 − 0.976147i \(-0.569664\pi\)
−0.217111 + 0.976147i \(0.569664\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.00974e34 0.0906387
\(218\) 0 0
\(219\) 2.54511e35 1.99088
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.09881e35 −0.655165 −0.327582 − 0.944823i \(-0.606234\pi\)
−0.327582 + 0.944823i \(0.606234\pi\)
\(224\) 0 0
\(225\) 1.91751e35 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) −4.45887e35 −1.90635
\(229\) −3.75942e35 −1.50518 −0.752590 − 0.658489i \(-0.771196\pi\)
−0.752590 + 0.658489i \(0.771196\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 7.23221e35 1.73000
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.11183e35 −0.206911 −0.103455 − 0.994634i \(-0.532990\pi\)
−0.103455 + 0.994634i \(0.532990\pi\)
\(242\) 0 0
\(243\) −6.08267e35 −1.00000
\(244\) 4.44953e35 0.687808
\(245\) 0 0
\(246\) 0 0
\(247\) 1.61635e36 2.08008
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −1.75651e36 −1.67356
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.32923e36 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −4.14393e36 −2.61767
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 5.25121e36 1.98718
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −3.82663e36 −1.22540 −0.612699 − 0.790316i \(-0.709916\pi\)
−0.612699 + 0.790316i \(0.709916\pi\)
\(272\) 0 0
\(273\) 6.36737e36 1.82608
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.62929e36 0.375666 0.187833 − 0.982201i \(-0.439854\pi\)
0.187833 + 0.982201i \(0.439854\pi\)
\(278\) 0 0
\(279\) −2.61658e35 −0.0541591
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −1.19625e37 −1.99998 −0.999991 − 0.00434644i \(-0.998616\pi\)
−0.999991 + 0.00434644i \(0.998616\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.19347e36 1.00000
\(290\) 0 0
\(291\) 2.79911e36 0.308053
\(292\) −1.90453e37 −1.99088
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −1.43489e37 −1.00000
\(301\) 3.39954e37 2.25384
\(302\) 0 0
\(303\) 0 0
\(304\) 3.33661e37 1.90635
\(305\) 0 0
\(306\) 0 0
\(307\) 3.54015e36 0.174561 0.0872803 − 0.996184i \(-0.472182\pi\)
0.0872803 + 0.996184i \(0.472182\pi\)
\(308\) 0 0
\(309\) −1.87488e37 −0.838679
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −1.55492e37 −0.573514 −0.286757 − 0.958003i \(-0.592577\pi\)
−0.286757 + 0.958003i \(0.592577\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −5.41193e37 −1.73000
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 4.55172e37 1.00000
\(325\) 5.20150e37 1.09113
\(326\) 0 0
\(327\) −1.00918e38 −1.93086
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −8.51597e37 −1.35773 −0.678867 − 0.734262i \(-0.737529\pi\)
−0.678867 + 0.734262i \(0.737529\pi\)
\(332\) 0 0
\(333\) 1.07383e38 1.56413
\(334\) 0 0
\(335\) 0 0
\(336\) 1.31441e38 1.67356
\(337\) 9.46447e37 1.15252 0.576259 − 0.817267i \(-0.304512\pi\)
0.576259 + 0.817267i \(0.304512\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.43412e38 −1.34022
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −1.39534e38 −1.00531 −0.502655 − 0.864487i \(-0.667643\pi\)
−0.502655 + 0.864487i \(0.667643\pi\)
\(350\) 0 0
\(351\) −1.65000e38 −1.09113
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 6.07086e38 2.63416
\(362\) 0 0
\(363\) −2.50380e38 −1.00000
\(364\) −4.76476e38 −1.82608
\(365\) 0 0
\(366\) 0 0
\(367\) 5.90225e38 1.99998 0.999992 − 0.00390072i \(-0.00124164\pi\)
0.999992 + 0.00390072i \(0.00124164\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 1.95801e37 0.0541591
\(373\) 4.83692e38 1.28510 0.642549 − 0.766244i \(-0.277877\pi\)
0.642549 + 0.766244i \(0.277877\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −9.53665e38 −1.99438 −0.997190 − 0.0749137i \(-0.976132\pi\)
−0.997190 + 0.0749137i \(0.976132\pi\)
\(380\) 0 0
\(381\) 2.45323e38 0.474093
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.80937e38 −1.34673
\(388\) −2.09460e38 −0.308053
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.39970e39 −1.45941 −0.729704 − 0.683763i \(-0.760342\pi\)
−0.729704 + 0.683763i \(0.760342\pi\)
\(398\) 0 0
\(399\) 3.29942e39 3.19039
\(400\) 1.07374e39 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −7.09782e37 −0.0590948
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.45271e39 −0.969015 −0.484507 − 0.874787i \(-0.661001\pi\)
−0.484507 + 0.874787i \(0.661001\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.40299e39 0.838679
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3.71958e39 −1.85547
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 1.38578e39 0.599051 0.299525 − 0.954088i \(-0.403172\pi\)
0.299525 + 0.954088i \(0.403172\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −3.29251e39 −1.15109
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −3.40609e39 −1.00000
\(433\) −9.03995e38 −0.256358 −0.128179 − 0.991751i \(-0.540913\pi\)
−0.128179 + 0.991751i \(0.540913\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7.55177e39 1.93086
\(437\) 0 0
\(438\) 0 0
\(439\) 8.66325e39 1.99854 0.999270 − 0.0382008i \(-0.0121626\pi\)
0.999270 + 0.0382008i \(0.0121626\pi\)
\(440\) 0 0
\(441\) 8.35696e39 1.80082
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −8.03559e39 −1.56413
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −9.83586e39 −1.67356
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 2.65149e39 0.381960
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.10463e40 −1.39467 −0.697337 − 0.716744i \(-0.745632\pi\)
−0.697337 + 0.716744i \(0.745632\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 6.84099e39 0.710239 0.355119 − 0.934821i \(-0.384440\pi\)
0.355119 + 0.934821i \(0.384440\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 1.23471e40 1.09113
\(469\) −3.88573e40 −3.32567
\(470\) 0 0
\(471\) 1.20366e40 0.966475
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.69529e40 1.90635
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 2.91291e40 1.70668
\(482\) 0 0
\(483\) 0 0
\(484\) 1.87362e40 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −2.90944e40 −1.41539 −0.707694 − 0.706519i \(-0.750265\pi\)
−0.707694 + 0.706519i \(0.750265\pi\)
\(488\) 0 0
\(489\) −4.33185e40 −1.98171
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.46520e39 −0.0541591
\(497\) 0 0
\(498\) 0 0
\(499\) −2.25636e40 −0.761904 −0.380952 − 0.924595i \(-0.624404\pi\)
−0.380952 + 0.924595i \(0.624404\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −7.16442e39 −0.190573
\(508\) −1.83577e40 −0.474093
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 1.40929e41 3.33186
\(512\) 0 0
\(513\) −8.54993e40 −1.90635
\(514\) 0 0
\(515\) 0 0
\(516\) 6.59214e40 1.34673
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −6.38509e40 −1.06572 −0.532858 − 0.846205i \(-0.678882\pi\)
−0.532858 + 0.846205i \(0.678882\pi\)
\(524\) 0 0
\(525\) 1.06177e41 1.67356
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.10943e40 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −2.46899e41 −3.19039
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −4.49365e40 −0.451482 −0.225741 − 0.974187i \(-0.572480\pi\)
−0.225741 + 0.974187i \(0.572480\pi\)
\(542\) 0 0
\(543\) −8.71538e39 −0.0828493
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.88197e41 −1.60251 −0.801257 − 0.598320i \(-0.795835\pi\)
−0.801257 + 0.598320i \(0.795835\pi\)
\(548\) 0 0
\(549\) 8.53203e40 0.687808
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 4.00465e41 2.89526
\(554\) 0 0
\(555\) 0 0
\(556\) 2.78340e41 1.85547
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −2.38966e41 −1.46946
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −3.36812e41 −1.67356
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 3.06223e41 1.36929 0.684645 − 0.728876i \(-0.259957\pi\)
0.684645 + 0.728876i \(0.259957\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 2.54881e41 1.00000
\(577\) −4.17353e41 −1.59539 −0.797693 − 0.603064i \(-0.793947\pi\)
−0.797693 + 0.603064i \(0.793947\pi\)
\(578\) 0 0
\(579\) −3.06833e41 −1.11359
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) −6.25359e41 −1.80082
\(589\) −3.67792e40 −0.103246
\(590\) 0 0
\(591\) 0 0
\(592\) 6.01311e41 1.56413
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.43083e41 1.70381
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 4.71350e41 0.977748 0.488874 − 0.872354i \(-0.337408\pi\)
0.488874 + 0.872354i \(0.337408\pi\)
\(602\) 0 0
\(603\) 1.00693e42 1.98718
\(604\) −1.98414e41 −0.381960
\(605\) 0 0
\(606\) 0 0
\(607\) −1.11864e42 −1.99922 −0.999611 − 0.0278764i \(-0.991126\pi\)
−0.999611 + 0.0278764i \(0.991126\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.27538e42 −1.96668 −0.983341 − 0.181771i \(-0.941817\pi\)
−0.983341 + 0.181771i \(0.941817\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 7.66077e41 1.02073 0.510367 − 0.859957i \(-0.329510\pi\)
0.510367 + 0.859957i \(0.329510\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −9.23947e41 −1.09113
\(625\) 8.67362e41 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −9.00710e41 −0.966475
\(629\) 0 0
\(630\) 0 0
\(631\) −1.65553e42 −1.65385 −0.826925 − 0.562313i \(-0.809912\pi\)
−0.826925 + 0.562313i \(0.809912\pi\)
\(632\) 0 0
\(633\) 4.55794e41 0.434223
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.26693e42 1.96493
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 1.40696e42 1.05954 0.529772 − 0.848140i \(-0.322278\pi\)
0.529772 + 0.848140i \(0.322278\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −1.44887e41 −0.0906387
\(652\) 3.24156e42 1.98171
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.65195e42 −1.99088
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −3.95015e42 −1.96604 −0.983018 − 0.183512i \(-0.941253\pi\)
−0.983018 + 0.183512i \(0.941253\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 1.57668e42 0.655165
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.33440e42 −0.507059 −0.253530 − 0.967328i \(-0.581592\pi\)
−0.253530 + 0.967328i \(0.581592\pi\)
\(674\) 0 0
\(675\) −2.75142e42 −1.00000
\(676\) 5.36120e41 0.190573
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 1.54994e42 0.515546
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 6.39799e42 1.90635
\(685\) 0 0
\(686\) 0 0
\(687\) 5.39435e42 1.50518
\(688\) −4.93295e42 −1.34673
\(689\) 0 0
\(690\) 0 0
\(691\) 1.01040e42 0.258417 0.129208 − 0.991617i \(-0.458756\pi\)
0.129208 + 0.991617i \(0.458756\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −7.94535e42 −1.67356
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 1.50940e43 2.98178
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.12486e43 −1.95617 −0.978084 − 0.208209i \(-0.933236\pi\)
−0.978084 + 0.208209i \(0.933236\pi\)
\(710\) 0 0
\(711\) −1.03774e43 −1.73000
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) −1.03817e43 −1.40358
\(722\) 0 0
\(723\) 1.59536e42 0.206911
\(724\) 6.52180e41 0.0828493
\(725\) 0 0
\(726\) 0 0
\(727\) 1.57009e43 1.87460 0.937300 − 0.348524i \(-0.113317\pi\)
0.937300 + 0.348524i \(0.113317\pi\)
\(728\) 0 0
\(729\) 8.72796e42 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −6.38459e42 −0.687808
\(733\) 1.38964e43 1.46671 0.733353 − 0.679848i \(-0.237954\pi\)
0.733353 + 0.679848i \(0.237954\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.91540e43 1.78893 0.894463 − 0.447141i \(-0.147558\pi\)
0.894463 + 0.447141i \(0.147558\pi\)
\(740\) 0 0
\(741\) −2.31928e43 −2.08008
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.66029e43 −1.95132 −0.975661 − 0.219282i \(-0.929628\pi\)
−0.975661 + 0.219282i \(0.929628\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 2.52040e43 1.67356
\(757\) 1.80098e43 1.17239 0.586194 − 0.810170i \(-0.300625\pi\)
0.586194 + 0.810170i \(0.300625\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −5.58807e43 −3.23142
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.90730e43 −1.00000
\(769\) −1.82629e43 −0.939017 −0.469509 − 0.882928i \(-0.655569\pi\)
−0.469509 + 0.882928i \(0.655569\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.29606e43 1.11359
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −1.18358e42 −0.0541591
\(776\) 0 0
\(777\) 5.94608e43 2.61767
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 4.67962e43 1.80082
\(785\) 0 0
\(786\) 0 0
\(787\) −1.53477e43 −0.557728 −0.278864 − 0.960331i \(-0.589958\pi\)
−0.278864 + 0.960331i \(0.589958\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.31442e43 0.750490
\(794\) 0 0
\(795\) 0 0
\(796\) −5.56056e43 −1.70381
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −7.53491e43 −1.98718
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 6.14898e43 1.42393 0.711967 − 0.702213i \(-0.247804\pi\)
0.711967 + 0.702213i \(0.247804\pi\)
\(812\) 0 0
\(813\) 5.49080e43 1.22540
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.23826e44 −2.56733
\(818\) 0 0
\(819\) −9.13648e43 −1.82608
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 9.32123e43 1.73171 0.865856 − 0.500293i \(-0.166774\pi\)
0.865856 + 0.500293i \(0.166774\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 1.04520e44 1.74132 0.870662 − 0.491882i \(-0.163691\pi\)
0.870662 + 0.491882i \(0.163691\pi\)
\(830\) 0 0
\(831\) −2.33785e43 −0.375666
\(832\) 6.91398e43 1.09113
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.75451e42 0.0541591
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 7.44629e43 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −3.41075e43 −0.434223
\(845\) 0 0
\(846\) 0 0
\(847\) −1.38642e44 −1.67356
\(848\) 0 0
\(849\) 1.71649e44 1.99998
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.67799e44 −1.82203 −0.911014 − 0.412376i \(-0.864699\pi\)
−0.911014 + 0.412376i \(0.864699\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1.86331e44 −1.82132 −0.910662 − 0.413153i \(-0.864428\pi\)
−0.910662 + 0.413153i \(0.864428\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1.17567e44 −1.00000
\(868\) 1.08420e43 0.0906387
\(869\) 0 0
\(870\) 0 0
\(871\) 2.73142e44 2.16828
\(872\) 0 0
\(873\) −4.01641e43 −0.308053
\(874\) 0 0
\(875\) 0 0
\(876\) 2.73279e44 1.99088
\(877\) −2.44237e44 −1.74911 −0.874554 − 0.484928i \(-0.838846\pi\)
−0.874554 + 0.484928i \(0.838846\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1.90096e44 1.22903 0.614515 − 0.788905i \(-0.289352\pi\)
0.614515 + 0.788905i \(0.289352\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1.35841e44 0.793425
\(890\) 0 0
\(891\) 0 0
\(892\) −1.17984e44 −0.655165
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 2.05891e44 1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) −4.87797e44 −2.25384
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.53041e44 1.95897 0.979485 − 0.201515i \(-0.0645864\pi\)
0.979485 + 0.201515i \(0.0645864\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −4.78767e44 −1.90635
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −4.03664e44 −1.50518
\(917\) 0 0
\(918\) 0 0
\(919\) −4.95759e44 −1.76010 −0.880051 − 0.474879i \(-0.842492\pi\)
−0.880051 + 0.474879i \(0.842492\pi\)
\(920\) 0 0
\(921\) −5.07973e43 −0.174561
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 4.85735e44 1.56413
\(926\) 0 0
\(927\) 2.69024e44 0.838679
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1.17467e45 3.43298
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −6.25222e44 −1.65936 −0.829680 − 0.558239i \(-0.811477\pi\)
−0.829680 + 0.558239i \(0.811477\pi\)
\(938\) 0 0
\(939\) 2.23115e44 0.573514
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 7.76553e44 1.73000
\(949\) −9.90640e44 −2.17231
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −5.49003e44 −0.997067
\(962\) 0 0
\(963\) 0 0
\(964\) −1.19382e44 −0.206911
\(965\) 0 0
\(966\) 0 0
\(967\) 1.11493e45 1.84437 0.922187 − 0.386744i \(-0.126400\pi\)
0.922187 + 0.386744i \(0.126400\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −6.53121e44 −1.00000
\(973\) −2.05962e45 −3.10524
\(974\) 0 0
\(975\) −7.46358e44 −1.09113
\(976\) 4.77765e44 0.687808
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.44806e45 1.93086
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 1.73554e45 2.08008
\(989\) 0 0
\(990\) 0 0
\(991\) 5.95812e44 0.682346 0.341173 − 0.940001i \(-0.389176\pi\)
0.341173 + 0.940001i \(0.389176\pi\)
\(992\) 0 0
\(993\) 1.22195e45 1.35773
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.11283e44 −0.116413 −0.0582067 − 0.998305i \(-0.518538\pi\)
−0.0582067 + 0.998305i \(0.518538\pi\)
\(998\) 0 0
\(999\) −1.54083e45 −1.56413
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 3.31.b.a.2.1 ✓ 1
3.2 odd 2 CM 3.31.b.a.2.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.31.b.a.2.1 ✓ 1 1.1 even 1 trivial
3.31.b.a.2.1 ✓ 1 3.2 odd 2 CM