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Properties

Label 12.17.c.a.5.1
Level $12$
Weight $17$
Character 12.5
Self dual yes
Analytic conductor $19.479$
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] = [12,17,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, 17); 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, 17, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 17 \)
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: \(19.4789452628\)
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)
 
\(f(q)\) \(=\) \(q+6561.00 q^{3} +4.74355e6 q^{7} +4.30467e7 q^{9} -3.49392e8 q^{13} +3.28686e10 q^{19} +3.11225e10 q^{21} +1.52588e11 q^{25} +2.82430e11 q^{27} +1.56817e12 q^{31} -6.77142e12 q^{37} -2.29236e12 q^{39} -1.11810e13 q^{43} -1.07316e13 q^{49} +2.15651e14 q^{57} +1.84289e14 q^{61} +2.04194e14 q^{63} -7.82744e14 q^{67} -1.35147e15 q^{73} +1.00113e15 q^{75} -2.67750e15 q^{79} +1.85302e15 q^{81} -1.65736e15 q^{91} +1.02888e16 q^{93} +1.56216e16 q^{97} +O(q^{100})\)

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
\(3\) 6561.00 1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 4.74355e6 0.822848 0.411424 − 0.911444i \(-0.365032\pi\)
0.411424 + 0.911444i \(0.365032\pi\)
\(8\) 0 0
\(9\) 4.30467e7 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −3.49392e8 −0.428318 −0.214159 − 0.976799i \(-0.568701\pi\)
−0.214159 + 0.976799i \(0.568701\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 3.28686e10 1.93532 0.967658 − 0.252265i \(-0.0811755\pi\)
0.967658 + 0.252265i \(0.0811755\pi\)
\(20\) 0 0
\(21\) 3.11225e10 0.822848
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 1.52588e11 1.00000
\(26\) 0 0
\(27\) 2.82430e11 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.56817e12 1.83865 0.919325 − 0.393499i \(-0.128736\pi\)
0.919325 + 0.393499i \(0.128736\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.77142e12 −1.92782 −0.963909 − 0.266230i \(-0.914222\pi\)
−0.963909 + 0.266230i \(0.914222\pi\)
\(38\) 0 0
\(39\) −2.29236e12 −0.428318
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −1.11810e13 −0.956604 −0.478302 − 0.878195i \(-0.658748\pi\)
−0.478302 + 0.878195i \(0.658748\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −1.07316e13 −0.322921
\(50\) 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
\(57\) 2.15651e14 1.93532
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 1.84289e14 0.961302 0.480651 − 0.876912i \(-0.340400\pi\)
0.480651 + 0.876912i \(0.340400\pi\)
\(62\) 0 0
\(63\) 2.04194e14 0.822848
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.82744e14 −1.92762 −0.963809 − 0.266592i \(-0.914102\pi\)
−0.963809 + 0.266592i \(0.914102\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.35147e15 −1.67581 −0.837905 − 0.545816i \(-0.816220\pi\)
−0.837905 + 0.545816i \(0.816220\pi\)
\(74\) 0 0
\(75\) 1.00113e15 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.67750e15 −1.76487 −0.882434 − 0.470436i \(-0.844097\pi\)
−0.882434 + 0.470436i \(0.844097\pi\)
\(80\) 0 0
\(81\) 1.85302e15 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
\(91\) −1.65736e15 −0.352440
\(92\) 0 0
\(93\) 1.02888e16 1.83865
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.56216e16 1.99320 0.996601 − 0.0823822i \(-0.0262528\pi\)
0.996601 + 0.0823822i \(0.0262528\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −2.33597e16 −1.84404 −0.922018 − 0.387147i \(-0.873460\pi\)
−0.922018 + 0.387147i \(0.873460\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 1.43120e15 0.0718270 0.0359135 − 0.999355i \(-0.488566\pi\)
0.0359135 + 0.999355i \(0.488566\pi\)
\(110\) 0 0
\(111\) −4.44273e16 −1.92782
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.50402e16 −0.428318
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 4.59497e16 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.50994e16 0.370880 0.185440 − 0.982656i \(-0.440629\pi\)
0.185440 + 0.982656i \(0.440629\pi\)
\(128\) 0 0
\(129\) −7.33584e16 −0.956604
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 1.55914e17 1.59247
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 2.24585e17 1.61162 0.805809 − 0.592176i \(-0.201731\pi\)
0.805809 + 0.592176i \(0.201731\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −7.04102e16 −0.322921
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −4.67535e17 −1.72981 −0.864906 − 0.501934i \(-0.832622\pi\)
−0.864906 + 0.501934i \(0.832622\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −7.36982e17 −1.99646 −0.998228 − 0.0595113i \(-0.981046\pi\)
−0.998228 + 0.0595113i \(0.981046\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.75526e17 0.753597 0.376798 − 0.926295i \(-0.377025\pi\)
0.376798 + 0.926295i \(0.377025\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −5.43342e17 −0.816544
\(170\) 0 0
\(171\) 1.41488e18 1.93532
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 7.23809e17 0.822848
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.19574e18 −1.03803 −0.519015 − 0.854765i \(-0.673701\pi\)
−0.519015 + 0.854765i \(0.673701\pi\)
\(182\) 0 0
\(183\) 1.20912e18 0.961302
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.33972e18 0.822848
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 1.56328e18 0.812039 0.406020 − 0.913864i \(-0.366916\pi\)
0.406020 + 0.913864i \(0.366916\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.90948e18 −0.776409 −0.388205 − 0.921573i \(-0.626905\pi\)
−0.388205 + 0.921573i \(0.626905\pi\)
\(200\) 0 0
\(201\) −5.13558e18 −1.92762
\(202\) 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
\(211\) −7.82385e18 −1.99141 −0.995705 − 0.0925844i \(-0.970487\pi\)
−0.995705 + 0.0925844i \(0.970487\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.43869e18 1.51293
\(218\) 0 0
\(219\) −8.86702e18 −1.67581
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.01164e19 1.65419 0.827096 − 0.562061i \(-0.189991\pi\)
0.827096 + 0.562061i \(0.189991\pi\)
\(224\) 0 0
\(225\) 6.56841e18 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 1.35908e19 1.79706 0.898528 − 0.438917i \(-0.144638\pi\)
0.898528 + 0.438917i \(0.144638\pi\)
\(230\) 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
\(237\) −1.75671e19 −1.76487
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 8.21825e18 0.722176 0.361088 − 0.932532i \(-0.382405\pi\)
0.361088 + 0.932532i \(0.382405\pi\)
\(242\) 0 0
\(243\) 1.21577e19 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.14840e19 −0.828930
\(248\) 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
\(259\) −3.21206e19 −1.58630
\(260\) 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
\(271\) −4.92587e19 −1.69328 −0.846640 − 0.532166i \(-0.821378\pi\)
−0.846640 + 0.532166i \(0.821378\pi\)
\(272\) 0 0
\(273\) −1.08739e19 −0.352440
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.41452e19 −0.408105 −0.204053 − 0.978960i \(-0.565411\pi\)
−0.204053 + 0.978960i \(0.565411\pi\)
\(278\) 0 0
\(279\) 6.75045e19 1.83865
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 2.52460e19 0.613623 0.306812 − 0.951770i \(-0.400738\pi\)
0.306812 + 0.951770i \(0.400738\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 4.86612e19 1.00000
\(290\) 0 0
\(291\) 1.02493e20 1.99320
\(292\) 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
\(301\) −5.30376e19 −0.787140
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.17204e19 0.528740 0.264370 − 0.964421i \(-0.414836\pi\)
0.264370 + 0.964421i \(0.414836\pi\)
\(308\) 0 0
\(309\) −1.53263e20 −1.84404
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −4.73351e19 −0.513841 −0.256921 − 0.966433i \(-0.582708\pi\)
−0.256921 + 0.966433i \(0.582708\pi\)
\(314\) 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
\(325\) −5.33130e19 −0.428318
\(326\) 0 0
\(327\) 9.39009e18 0.0718270
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.41275e19 −0.167451 −0.0837255 − 0.996489i \(-0.526682\pi\)
−0.0837255 + 0.996489i \(0.526682\pi\)
\(332\) 0 0
\(333\) −2.91488e20 −1.92782
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.31326e20 1.99166 0.995832 − 0.0912087i \(-0.0290730\pi\)
0.995832 + 0.0912087i \(0.0290730\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −2.08548e20 −1.08856
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 4.35999e20 1.98099 0.990494 − 0.137553i \(-0.0439236\pi\)
0.990494 + 0.137553i \(0.0439236\pi\)
\(350\) 0 0
\(351\) −9.86786e19 −0.428318
\(352\) 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
\(361\) 7.91901e20 2.74545
\(362\) 0 0
\(363\) 3.01476e20 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −6.74388e19 −0.204918 −0.102459 − 0.994737i \(-0.532671\pi\)
−0.102459 + 0.994737i \(0.532671\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −7.24928e20 −1.93475 −0.967373 − 0.253356i \(-0.918466\pi\)
−0.967373 + 0.253356i \(0.918466\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −8.39226e20 −1.97136 −0.985678 − 0.168638i \(-0.946063\pi\)
−0.985678 + 0.168638i \(0.946063\pi\)
\(380\) 0 0
\(381\) 1.64677e20 0.370880
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.81305e20 −0.956604
\(388\) 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
\(397\) −9.85697e20 −1.59742 −0.798711 − 0.601715i \(-0.794484\pi\)
−0.798711 + 0.601715i \(0.794484\pi\)
\(398\) 0 0
\(399\) 1.02295e21 1.59247
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −5.47905e20 −0.787526
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.20920e21 1.54422 0.772112 − 0.635487i \(-0.219201\pi\)
0.772112 + 0.635487i \(0.219201\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.47350e21 1.61162
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −1.83896e21 −1.86344 −0.931720 − 0.363178i \(-0.881692\pi\)
−0.931720 + 0.363178i \(0.881692\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.74183e20 0.791006
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 1.77557e21 1.43692 0.718462 − 0.695566i \(-0.244846\pi\)
0.718462 + 0.695566i \(0.244846\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 1.80394e21 1.30770 0.653848 − 0.756626i \(-0.273153\pi\)
0.653848 + 0.756626i \(0.273153\pi\)
\(440\) 0 0
\(441\) −4.61961e20 −0.322921
\(442\) 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
\(453\) −3.06750e21 −1.72981
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.06189e21 −1.60939 −0.804693 − 0.593691i \(-0.797670\pi\)
−0.804693 + 0.593691i \(0.797670\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 5.07863e20 0.240491 0.120245 − 0.992744i \(-0.461632\pi\)
0.120245 + 0.992744i \(0.461632\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −3.71299e21 −1.58614
\(470\) 0 0
\(471\) −4.83534e21 −1.99646
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 5.01535e21 1.93532
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 2.36588e21 0.825719
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.95277e21 0.617191 0.308595 − 0.951193i \(-0.400141\pi\)
0.308595 + 0.951193i \(0.400141\pi\)
\(488\) 0 0
\(489\) 2.46382e21 0.753597
\(490\) 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
\(499\) 7.52196e21 1.95671 0.978355 − 0.206934i \(-0.0663486\pi\)
0.978355 + 0.206934i \(0.0663486\pi\)
\(500\) 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
\(507\) −3.56487e21 −0.816544
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −6.41079e21 −1.37894
\(512\) 0 0
\(513\) 9.28305e21 1.93532
\(514\) 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
\(523\) −1.05965e22 −1.89299 −0.946495 − 0.322717i \(-0.895404\pi\)
−0.946495 + 0.322717i \(0.895404\pi\)
\(524\) 0 0
\(525\) 4.74891e21 0.822848
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 6.13261e21 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
\(541\) −1.07407e22 −1.46370 −0.731852 − 0.681463i \(-0.761344\pi\)
−0.731852 + 0.681463i \(0.761344\pi\)
\(542\) 0 0
\(543\) −7.84528e21 −1.03803
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.22073e22 −1.52307 −0.761536 − 0.648122i \(-0.775555\pi\)
−0.761536 + 0.648122i \(0.775555\pi\)
\(548\) 0 0
\(549\) 7.93302e21 0.961302
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.27009e22 −1.45222
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 3.90654e21 0.409730
\(560\) 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
\(567\) 8.78990e21 0.822848
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 1.54008e22 1.36287 0.681435 − 0.731879i \(-0.261356\pi\)
0.681435 + 0.731879i \(0.261356\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −7.63732e20 −0.0621633 −0.0310817 − 0.999517i \(-0.509895\pi\)
−0.0310817 + 0.999517i \(0.509895\pi\)
\(578\) 0 0
\(579\) 1.02566e22 0.812039
\(580\) 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
\(589\) 5.15434e22 3.55837
\(590\) 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
\(597\) −1.25281e22 −0.776409
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −3.01664e22 −1.77226 −0.886132 − 0.463433i \(-0.846618\pi\)
−0.886132 + 0.463433i \(0.846618\pi\)
\(602\) 0 0
\(603\) −3.36946e22 −1.92762
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.38912e22 1.83898 0.919492 − 0.393108i \(-0.128600\pi\)
0.919492 + 0.393108i \(0.128600\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −3.96266e22 −1.98748 −0.993740 − 0.111719i \(-0.964364\pi\)
−0.993740 + 0.111719i \(0.964364\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 7.75741e21 0.359908 0.179954 − 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.32831e22 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.02373e22 −0.407336 −0.203668 − 0.979040i \(-0.565286\pi\)
−0.203668 + 0.979040i \(0.565286\pi\)
\(632\) 0 0
\(633\) −5.13322e22 −1.99141
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.74954e21 0.138313
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −2.28923e22 −0.783433 −0.391716 − 0.920086i \(-0.628119\pi\)
−0.391716 + 0.920086i \(0.628119\pi\)
\(644\) 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
\(651\) 4.88052e22 1.51293
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −5.81765e22 −1.67581
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 5.38568e22 1.47784 0.738922 − 0.673791i \(-0.235335\pi\)
0.738922 + 0.673791i \(0.235335\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 6.63735e22 1.65419
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −3.17578e22 −0.754622 −0.377311 − 0.926087i \(-0.623151\pi\)
−0.377311 + 0.926087i \(0.623151\pi\)
\(674\) 0 0
\(675\) 4.30953e22 1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 7.41018e22 1.64010
\(680\) 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
\(687\) 8.91693e22 1.79706
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −7.96831e22 −1.53300 −0.766500 − 0.642244i \(-0.778003\pi\)
−0.766500 + 0.642244i \(0.778003\pi\)
\(692\) 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
\(703\) −2.22567e23 −3.73094
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.10135e23 1.72486 0.862431 − 0.506174i \(-0.168941\pi\)
0.862431 + 0.506174i \(0.168941\pi\)
\(710\) 0 0
\(711\) −1.15257e23 −1.76487
\(712\) 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
\(721\) −1.10808e23 −1.51736
\(722\) 0 0
\(723\) 5.39200e22 0.722176
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.41303e23 −1.81082 −0.905409 − 0.424540i \(-0.860436\pi\)
−0.905409 + 0.424540i \(0.860436\pi\)
\(728\) 0 0
\(729\) 7.97664e22 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.66638e23 1.99959 0.999796 − 0.0201815i \(-0.00642441\pi\)
0.999796 + 0.0201815i \(0.00642441\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.18971e22 0.133748 0.0668739 − 0.997761i \(-0.478697\pi\)
0.0668739 + 0.997761i \(0.478697\pi\)
\(740\) 0 0
\(741\) −7.53466e22 −0.828930
\(742\) 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
\(751\) 8.47430e22 0.837499 0.418750 − 0.908102i \(-0.362468\pi\)
0.418750 + 0.908102i \(0.362468\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −4.29074e21 −0.0397892 −0.0198946 − 0.999802i \(-0.506333\pi\)
−0.0198946 + 0.999802i \(0.506333\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 6.78897e21 0.0591027
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.04979e23 1.67610 0.838048 − 0.545596i \(-0.183697\pi\)
0.838048 + 0.545596i \(0.183697\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 2.39283e23 1.83865
\(776\) 0 0
\(777\) −2.10743e23 −1.58630
\(778\) 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
\(787\) 1.61866e23 1.09991 0.549957 − 0.835193i \(-0.314644\pi\)
0.549957 + 0.835193i \(0.314644\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −6.43890e22 −0.411743
\(794\) 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
\(811\) −3.66385e23 −1.95781 −0.978906 − 0.204312i \(-0.934504\pi\)
−0.978906 + 0.204312i \(0.934504\pi\)
\(812\) 0 0
\(813\) −3.23186e23 −1.69328
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −3.67503e23 −1.85133
\(818\) 0 0
\(819\) −7.13439e22 −0.352440
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 3.22416e23 1.53186 0.765929 − 0.642926i \(-0.222280\pi\)
0.765929 + 0.642926i \(0.222280\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −2.76418e23 −1.23917 −0.619583 − 0.784931i \(-0.712698\pi\)
−0.619583 + 0.784931i \(0.712698\pi\)
\(830\) 0 0
\(831\) −9.28069e22 −0.408105
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 4.42897e23 1.83865
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 2.50246e23 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.17965e23 0.822848
\(848\) 0 0
\(849\) 1.65639e23 0.613623
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.82398e23 −0.650770 −0.325385 − 0.945582i \(-0.605494\pi\)
−0.325385 + 0.945582i \(0.605494\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 5.15198e22 0.173792 0.0868958 − 0.996217i \(-0.472305\pi\)
0.0868958 + 0.996217i \(0.472305\pi\)
\(860\) 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
\(867\) 3.19266e23 1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 2.73484e23 0.825633
\(872\) 0 0
\(873\) 6.72458e23 1.99320
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6.55438e23 −1.87299 −0.936494 − 0.350685i \(-0.885949\pi\)
−0.936494 + 0.350685i \(0.885949\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −4.10931e23 −1.11195 −0.555973 − 0.831200i \(-0.687654\pi\)
−0.555973 + 0.831200i \(0.687654\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1.19060e23 0.305178
\(890\) 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
\(903\) −3.47980e23 −0.787140
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.35809e23 1.16991 0.584953 − 0.811067i \(-0.301113\pi\)
0.584953 + 0.811067i \(0.301113\pi\)
\(908\) 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
\(919\) −7.21029e23 −1.41719 −0.708596 − 0.705614i \(-0.750671\pi\)
−0.708596 + 0.705614i \(0.750671\pi\)
\(920\) 0 0
\(921\) 2.73728e23 0.528740
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.03324e24 −1.92782
\(926\) 0 0
\(927\) −1.00556e24 −1.84404
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −3.52733e23 −0.624955
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 8.81757e23 1.48399 0.741997 − 0.670403i \(-0.233879\pi\)
0.741997 + 0.670403i \(0.233879\pi\)
\(938\) 0 0
\(939\) −3.10566e23 −0.513841
\(940\) 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
\(949\) 4.72194e23 0.717779
\(950\) 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
\(961\) 1.73173e24 2.38063
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 7.67665e23 1.00406 0.502031 − 0.864850i \(-0.332586\pi\)
0.502031 + 0.864850i \(0.332586\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 1.06533e24 1.32612
\(974\) 0 0
\(975\) −3.49786e23 −0.428318
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 6.16084e22 0.0718270
\(982\) 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
\(991\) 1.82847e24 1.96562 0.982810 − 0.184622i \(-0.0591062\pi\)
0.982810 + 0.184622i \(0.0591062\pi\)
\(992\) 0 0
\(993\) −1.58300e23 −0.167451
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.61449e24 −1.65376 −0.826881 − 0.562377i \(-0.809887\pi\)
−0.826881 + 0.562377i \(0.809887\pi\)
\(998\) 0 0
\(999\) −1.91245e24 −1.92782
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 12.17.c.a.5.1 ✓ 1
3.2 odd 2 CM 12.17.c.a.5.1 ✓ 1
4.3 odd 2 48.17.e.a.17.1 1
12.11 even 2 48.17.e.a.17.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.17.c.a.5.1 ✓ 1 1.1 even 1 trivial
12.17.c.a.5.1 ✓ 1 3.2 odd 2 CM
48.17.e.a.17.1 1 4.3 odd 2
48.17.e.a.17.1 1 12.11 even 2