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Properties

Label 12.57.c.a.5.1
Level $12$
Weight $57$
Character 12.5
Self dual yes
Analytic conductor $238.333$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [12,57,Mod(5,12)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12.5"); S:= CuspForms(chi, 57); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 57, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 57 \)
Character orbit: \([\chi]\) \(=\) 12.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(238.332749918\)
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 5.1
Character \(\chi\) \(=\) 12.5

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.28768e13 q^{3} -5.91717e23 q^{7} +5.23348e26 q^{9} -3.09919e31 q^{13} -8.01094e35 q^{19} -1.35366e37 q^{21} +1.38778e39 q^{25} +1.19725e40 q^{27} -1.77219e41 q^{31} -1.31479e44 q^{37} -7.08996e44 q^{39} +6.26064e45 q^{43} +1.38542e47 q^{49} -1.83265e49 q^{57} +1.60882e50 q^{61} -3.09674e50 q^{63} +2.18634e51 q^{67} -2.68247e52 q^{73} +3.17479e52 q^{75} +2.69218e53 q^{79} +2.73893e53 q^{81} +1.83385e55 q^{91} -4.05421e54 q^{93} +8.17123e55 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) 2.28768e13 1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −5.91717e23 −1.28638 −0.643190 − 0.765707i \(-0.722389\pi\)
−0.643190 + 0.765707i \(0.722389\pi\)
\(8\) 0 0
\(9\) 5.23348e26 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −3.09919e31 −1.99910 −0.999550 − 0.0299867i \(-0.990453\pi\)
−0.999550 + 0.0299867i \(0.990453\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −8.01094e35 −1.25482 −0.627412 − 0.778688i \(-0.715886\pi\)
−0.627412 + 0.778688i \(0.715886\pi\)
\(20\) 0 0
\(21\) −1.35366e37 −1.28638
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 1.38778e39 1.00000
\(26\) 0 0
\(27\) 1.19725e40 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.77219e41 −0.309302 −0.154651 − 0.987969i \(-0.549425\pi\)
−0.154651 + 0.987969i \(0.549425\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.31479e44 −1.61886 −0.809429 − 0.587217i \(-0.800223\pi\)
−0.809429 + 0.587217i \(0.800223\pi\)
\(38\) 0 0
\(39\) −7.08996e44 −1.99910
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 6.26064e45 1.14684 0.573419 − 0.819262i \(-0.305617\pi\)
0.573419 + 0.819262i \(0.305617\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.38542e47 0.654773
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.83265e49 −1.25482
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 1.60882e50 1.64920 0.824598 − 0.565718i \(-0.191401\pi\)
0.824598 + 0.565718i \(0.191401\pi\)
\(62\) 0 0
\(63\) −3.09674e50 −1.28638
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.18634e51 1.62040 0.810201 − 0.586153i \(-0.199358\pi\)
0.810201 + 0.586153i \(0.199358\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −2.68247e52 −1.80092 −0.900460 − 0.434939i \(-0.856770\pi\)
−0.900460 + 0.434939i \(0.856770\pi\)
\(74\) 0 0
\(75\) 3.17479e52 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.69218e53 1.97946 0.989729 − 0.142958i \(-0.0456615\pi\)
0.989729 + 0.142958i \(0.0456615\pi\)
\(80\) 0 0
\(81\) 2.73893e53 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 1.83385e55 2.57160
\(92\) 0 0
\(93\) −4.05421e54 −0.309302
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.17123e55 1.91725 0.958625 − 0.284673i \(-0.0918849\pi\)
0.958625 + 0.284673i \(0.0918849\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −4.50240e56 −1.96789 −0.983947 − 0.178460i \(-0.942889\pi\)
−0.983947 + 0.178460i \(0.942889\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 1.36878e57 1.22572 0.612859 − 0.790192i \(-0.290019\pi\)
0.612859 + 0.790192i \(0.290019\pi\)
\(110\) 0 0
\(111\) −3.00782e57 −1.61886
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.62195e58 −1.99910
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.07965e58 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2.13176e58 −0.264383 −0.132192 − 0.991224i \(-0.542201\pi\)
−0.132192 + 0.991224i \(0.542201\pi\)
\(128\) 0 0
\(129\) 1.43223e59 1.14684
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 4.74021e59 1.61418
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −1.21847e60 −1.20615 −0.603075 − 0.797685i \(-0.706058\pi\)
−0.603075 + 0.797685i \(0.706058\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.16939e60 0.654773
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −1.97881e61 −1.92774 −0.963872 − 0.266365i \(-0.914177\pi\)
−0.963872 + 0.266365i \(0.914177\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.26472e61 −0.413814 −0.206907 − 0.978361i \(-0.566340\pi\)
−0.206907 + 0.978361i \(0.566340\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 9.37979e61 1.07384 0.536921 − 0.843633i \(-0.319587\pi\)
0.536921 + 0.843633i \(0.319587\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 7.20158e62 2.99640
\(170\) 0 0
\(171\) −4.19251e62 −1.25482
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −8.21173e62 −1.28638
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.41611e63 −0.863173 −0.431586 − 0.902072i \(-0.642046\pi\)
−0.431586 + 0.902072i \(0.642046\pi\)
\(182\) 0 0
\(183\) 3.68047e63 1.64920
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −7.08435e63 −1.28638
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −1.24174e64 −1.25437 −0.627183 − 0.778872i \(-0.715792\pi\)
−0.627183 + 0.778872i \(0.715792\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 3.82425e64 1.63931 0.819653 − 0.572860i \(-0.194166\pi\)
0.819653 + 0.572860i \(0.194166\pi\)
\(200\) 0 0
\(201\) 5.00164e64 1.62040
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 7.66510e64 0.637689 0.318844 − 0.947807i \(-0.396705\pi\)
0.318844 + 0.947807i \(0.396705\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.04864e65 0.397880
\(218\) 0 0
\(219\) −6.13662e65 −1.80092
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 5.60405e65 0.990753 0.495377 − 0.868678i \(-0.335030\pi\)
0.495377 + 0.868678i \(0.335030\pi\)
\(224\) 0 0
\(225\) 7.26291e65 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −4.65769e64 −0.0391541 −0.0195771 − 0.999808i \(-0.506232\pi\)
−0.0195771 + 0.999808i \(0.506232\pi\)
\(230\) 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
\(237\) 6.15885e66 1.97946
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 4.83319e66 0.972202 0.486101 − 0.873903i \(-0.338419\pi\)
0.486101 + 0.873903i \(0.338419\pi\)
\(242\) 0 0
\(243\) 6.26579e66 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.48274e67 2.50852
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 7.77986e67 2.08247
\(260\) 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
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 2.45293e68 1.84733 0.923667 − 0.383197i \(-0.125177\pi\)
0.923667 + 0.383197i \(0.125177\pi\)
\(272\) 0 0
\(273\) 4.19525e68 2.57160
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4.89229e68 1.99562 0.997812 − 0.0661144i \(-0.0210602\pi\)
0.997812 + 0.0661144i \(0.0210602\pi\)
\(278\) 0 0
\(279\) −9.27472e67 −0.309302
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 2.69176e68 0.602581 0.301291 − 0.953532i \(-0.402583\pi\)
0.301291 + 0.953532i \(0.402583\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.03784e68 1.00000
\(290\) 0 0
\(291\) 1.86931e69 1.91725
\(292\) 0 0
\(293\) 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
\(300\) 0 0
\(301\) −3.70453e69 −1.47527
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.31314e69 0.300910 0.150455 − 0.988617i \(-0.451926\pi\)
0.150455 + 0.988617i \(0.451926\pi\)
\(308\) 0 0
\(309\) −1.03000e70 −1.96789
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 1.48910e70 1.98464 0.992319 − 0.123705i \(-0.0394776\pi\)
0.992319 + 0.123705i \(0.0394776\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 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
\(324\) 0 0
\(325\) −4.30099e70 −1.99910
\(326\) 0 0
\(327\) 3.13132e70 1.22572
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −6.32963e70 −1.76277 −0.881383 − 0.472403i \(-0.843387\pi\)
−0.881383 + 0.472403i \(0.843387\pi\)
\(332\) 0 0
\(333\) −6.88094e70 −1.61886
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.12743e71 −1.89868 −0.949338 − 0.314258i \(-0.898244\pi\)
−0.949338 + 0.314258i \(0.898244\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 4.32224e70 0.444093
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −2.80150e71 −1.77124 −0.885622 − 0.464407i \(-0.846267\pi\)
−0.885622 + 0.464407i \(0.846267\pi\)
\(350\) 0 0
\(351\) −3.71051e71 −1.99910
\(352\) 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
\(361\) 2.34182e71 0.574583
\(362\) 0 0
\(363\) 4.75757e71 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.17757e72 1.82112 0.910560 − 0.413378i \(-0.135651\pi\)
0.910560 + 0.413378i \(0.135651\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.59080e72 −1.56231 −0.781157 − 0.624335i \(-0.785370\pi\)
−0.781157 + 0.624335i \(0.785370\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.77938e72 1.11782 0.558908 − 0.829230i \(-0.311221\pi\)
0.558908 + 0.829230i \(0.311221\pi\)
\(380\) 0 0
\(381\) −4.87678e71 −0.264383
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.27649e72 1.14684
\(388\) 0 0
\(389\) 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
\(397\) 9.01004e72 1.54381 0.771903 − 0.635741i \(-0.219305\pi\)
0.771903 + 0.635741i \(0.219305\pi\)
\(398\) 0 0
\(399\) 1.08441e73 1.61418
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 5.49236e72 0.618327
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.99992e73 1.48854 0.744269 − 0.667880i \(-0.232798\pi\)
0.744269 + 0.667880i \(0.232798\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −2.78747e73 −1.20615
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 5.81984e73 1.92758 0.963790 − 0.266661i \(-0.0859204\pi\)
0.963790 + 0.266661i \(0.0859204\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −9.51968e73 −2.12149
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −1.19476e74 −1.80144 −0.900722 − 0.434396i \(-0.856962\pi\)
−0.900722 + 0.434396i \(0.856962\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −1.93144e74 −1.98095 −0.990477 − 0.137675i \(-0.956037\pi\)
−0.990477 + 0.137675i \(0.956037\pi\)
\(440\) 0 0
\(441\) 7.25056e73 0.654773
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −4.52688e74 −1.92774
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.76802e74 1.58739 0.793696 − 0.608314i \(-0.208154\pi\)
0.793696 + 0.608314i \(0.208154\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −3.07763e74 −0.711130 −0.355565 − 0.934652i \(-0.615712\pi\)
−0.355565 + 0.934652i \(0.615712\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −1.29369e75 −2.08445
\(470\) 0 0
\(471\) −2.89328e74 −0.413814
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −1.11174e75 −1.25482
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 4.07480e75 3.23626
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.09583e75 −0.615082 −0.307541 − 0.951535i \(-0.599506\pi\)
−0.307541 + 0.951535i \(0.599506\pi\)
\(488\) 0 0
\(489\) 2.14579e75 1.07384
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 5.25148e75 1.49096 0.745480 − 0.666528i \(-0.232220\pi\)
0.745480 + 0.666528i \(0.232220\pi\)
\(500\) 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
\(507\) 1.64749e76 2.99640
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 1.58726e76 2.31667
\(512\) 0 0
\(513\) −9.59111e75 −1.25482
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −2.39582e76 −1.82561 −0.912803 − 0.408400i \(-0.866087\pi\)
−0.912803 + 0.408400i \(0.866087\pi\)
\(524\) 0 0
\(525\) −1.87858e76 −1.28638
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.80617e76 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −3.34850e76 −0.989299 −0.494649 − 0.869093i \(-0.664703\pi\)
−0.494649 + 0.869093i \(0.664703\pi\)
\(542\) 0 0
\(543\) −3.23961e76 −0.863173
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 5.75135e76 1.24774 0.623871 − 0.781527i \(-0.285559\pi\)
0.623871 + 0.781527i \(0.285559\pi\)
\(548\) 0 0
\(549\) 8.41973e76 1.64920
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.59301e77 −2.54633
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −1.94029e77 −2.29265
\(560\) 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
\(567\) −1.62067e77 −1.28638
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 2.95852e77 1.92869 0.964347 − 0.264641i \(-0.0852534\pi\)
0.964347 + 0.264641i \(0.0852534\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −3.20542e77 −1.55942 −0.779709 − 0.626142i \(-0.784633\pi\)
−0.779709 + 0.626142i \(0.784633\pi\)
\(578\) 0 0
\(579\) −2.84069e77 −1.25437
\(580\) 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 \(\pi\)
\(588\) 0 0
\(589\) 1.41969e77 0.388120
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.74866e77 1.63931
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 1.27820e78 1.98662 0.993310 − 0.115474i \(-0.0368387\pi\)
0.993310 + 0.115474i \(0.0368387\pi\)
\(602\) 0 0
\(603\) 1.14421e78 1.62040
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −2.65323e77 −0.312242 −0.156121 − 0.987738i \(-0.549899\pi\)
−0.156121 + 0.987738i \(0.549899\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 8.54783e77 0.763770 0.381885 − 0.924210i \(-0.375275\pi\)
0.381885 + 0.924210i \(0.375275\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 4.45501e77 0.303047 0.151524 − 0.988454i \(-0.451582\pi\)
0.151524 + 0.988454i \(0.451582\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.92593e78 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −5.02182e78 −1.99544 −0.997720 − 0.0674871i \(-0.978502\pi\)
−0.997720 + 0.0674871i \(0.978502\pi\)
\(632\) 0 0
\(633\) 1.75353e78 0.637689
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.29368e78 −1.30896
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 6.92561e78 1.62387 0.811934 − 0.583750i \(-0.198415\pi\)
0.811934 + 0.583750i \(0.198415\pi\)
\(644\) 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
\(651\) 2.39894e78 0.397880
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.40386e79 −1.80092
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.57153e79 −1.70091 −0.850455 − 0.526048i \(-0.823673\pi\)
−0.850455 + 0.526048i \(0.823673\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 1.28203e79 0.990753
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −2.57673e79 −1.68518 −0.842591 − 0.538554i \(-0.818971\pi\)
−0.842591 + 0.538554i \(0.818971\pi\)
\(674\) 0 0
\(675\) 1.66152e79 1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −4.83506e79 −2.46631
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.06553e78 −0.0391541
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −4.12823e79 −1.28937 −0.644686 − 0.764447i \(-0.723012\pi\)
−0.644686 + 0.764447i \(0.723012\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 1.05327e80 2.03138
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.72166e79 0.565773 0.282887 − 0.959153i \(-0.408708\pi\)
0.282887 + 0.959153i \(0.408708\pi\)
\(710\) 0 0
\(711\) 1.40895e80 1.97946
\(712\) 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
\(721\) 2.66415e80 2.53146
\(722\) 0 0
\(723\) 1.10568e80 0.972202
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −2.65283e80 −1.99869 −0.999345 − 0.0361986i \(-0.988475\pi\)
−0.999345 + 0.0361986i \(0.988475\pi\)
\(728\) 0 0
\(729\) 1.43341e80 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −3.33316e80 −1.99501 −0.997506 − 0.0705814i \(-0.977515\pi\)
−0.997506 + 0.0705814i \(0.977515\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −2.19857e80 −1.04736 −0.523680 − 0.851915i \(-0.675441\pi\)
−0.523680 + 0.851915i \(0.675441\pi\)
\(740\) 0 0
\(741\) 5.67972e80 2.50852
\(742\) 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
\(751\) 4.37976e80 1.32902 0.664511 − 0.747278i \(-0.268640\pi\)
0.664511 + 0.747278i \(0.268640\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −6.21480e80 −1.50919 −0.754593 − 0.656193i \(-0.772166\pi\)
−0.754593 + 0.656193i \(0.772166\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −8.09929e80 −1.57674
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −5.55354e80 −0.868226 −0.434113 − 0.900859i \(-0.642938\pi\)
−0.434113 + 0.900859i \(0.642938\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −2.45941e80 −0.309302
\(776\) 0 0
\(777\) 1.77978e81 2.08247
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −2.32257e81 −1.89966 −0.949831 − 0.312763i \(-0.898745\pi\)
−0.949831 + 0.312763i \(0.898745\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4.98605e81 −3.29691
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 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
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −3.73967e81 −1.31901 −0.659507 − 0.751698i \(-0.729235\pi\)
−0.659507 + 0.751698i \(0.729235\pi\)
\(812\) 0 0
\(813\) 5.61152e81 1.84733
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −5.01537e81 −1.43908
\(818\) 0 0
\(819\) 9.59739e81 2.57160
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 6.55706e81 1.53290 0.766450 − 0.642304i \(-0.222021\pi\)
0.766450 + 0.642304i \(0.222021\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 1.83001e80 0.0349082 0.0174541 − 0.999848i \(-0.494444\pi\)
0.0174541 + 0.999848i \(0.494444\pi\)
\(830\) 0 0
\(831\) 1.11920e82 1.99562
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2.12176e81 −0.309302
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 7.83949e81 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.23057e82 −1.28638
\(848\) 0 0
\(849\) 6.15788e81 0.602581
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 2.16968e82 1.86133 0.930666 − 0.365871i \(-0.119229\pi\)
0.930666 + 0.365871i \(0.119229\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1.31222e82 −0.925117 −0.462558 − 0.886589i \(-0.653068\pi\)
−0.462558 + 0.886589i \(0.653068\pi\)
\(860\) 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
\(867\) 1.83880e82 1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −6.77588e82 −3.23934
\(872\) 0 0
\(873\) 4.27639e82 1.91725
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4.81792e82 −1.90051 −0.950256 − 0.311469i \(-0.899179\pi\)
−0.950256 + 0.311469i \(0.899179\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1.77141e82 0.577323 0.288661 − 0.957431i \(-0.406790\pi\)
0.288661 + 0.957431i \(0.406790\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1.26140e82 0.340098
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −8.47478e82 −1.47527
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.28168e83 −1.97140 −0.985700 − 0.168507i \(-0.946105\pi\)
−0.985700 + 0.168507i \(0.946105\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 7.31747e82 0.778980 0.389490 − 0.921031i \(-0.372651\pi\)
0.389490 + 0.921031i \(0.372651\pi\)
\(920\) 0 0
\(921\) 3.00404e82 0.300910
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.82464e83 −1.61886
\(926\) 0 0
\(927\) −2.35632e83 −1.96789
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −1.10985e83 −0.821625
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.72277e83 1.68384 0.841922 − 0.539599i \(-0.181424\pi\)
0.841922 + 0.539599i \(0.181424\pi\)
\(938\) 0 0
\(939\) 3.40657e83 1.98464
\(940\) 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 \(\pi\)
\(948\) 0 0
\(949\) 8.31348e83 3.60022
\(950\) 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
\(961\) −2.96881e83 −0.904332
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 6.80051e83 1.74020 0.870102 − 0.492872i \(-0.164053\pi\)
0.870102 + 0.492872i \(0.164053\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 7.20991e83 1.55157
\(974\) 0 0
\(975\) −9.83929e83 −1.99910
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 7.16346e83 1.22572
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.23618e84 1.59228 0.796140 − 0.605112i \(-0.206872\pi\)
0.796140 + 0.605112i \(0.206872\pi\)
\(992\) 0 0
\(993\) −1.44802e84 −1.76277
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.59596e84 −1.73603 −0.868016 − 0.496537i \(-0.834605\pi\)
−0.868016 + 0.496537i \(0.834605\pi\)
\(998\) 0 0
\(999\) −1.57414e84 −1.61886
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 12.57.c.a.5.1 ✓ 1
3.2 odd 2 CM 12.57.c.a.5.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.57.c.a.5.1 ✓ 1 1.1 even 1 trivial
12.57.c.a.5.1 ✓ 1 3.2 odd 2 CM