Properties

Label 1232.3.m.a.1231.2
Level $1232$
Weight $3$
Character 1232.1231
Self dual yes
Analytic conductor $33.570$
Analytic rank $0$
Dimension $4$
CM discriminant -308
Inner twists $4$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1232,3,Mod(1231,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.1231");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1232.m (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: \(33.5695685692\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{7}, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 9x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1231.2
Root \(-0.335437\) of defining polynomial
Character \(\chi\) \(=\) 1232.1231

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.670873 q^{3} -7.00000 q^{7} -8.54993 q^{9} +O(q^{10})\) \(q-0.670873 q^{3} -7.00000 q^{7} -8.54993 q^{9} -11.0000 q^{11} -24.5204 q^{13} +20.4951 q^{17} +4.69611 q^{21} +25.0000 q^{25} +11.7738 q^{27} -51.0534 q^{31} +7.37961 q^{33} +52.6498 q^{37} +16.4501 q^{39} +38.9777 q^{41} +52.6498 q^{43} +92.0437 q^{47} +49.0000 q^{49} -13.7496 q^{51} +52.6498 q^{53} -114.551 q^{59} -59.1039 q^{61} +59.8495 q^{63} -106.501 q^{73} -16.7718 q^{75} +77.0000 q^{77} -157.949 q^{79} +69.0506 q^{81} +171.643 q^{91} +34.2504 q^{93} +94.0492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{7} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 28 q^{7} + 36 q^{9} - 44 q^{11} + 100 q^{25} + 136 q^{39} + 196 q^{49} + 296 q^{51} - 252 q^{63} + 308 q^{77} + 908 q^{81} + 488 q^{93} - 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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
\(3\) −0.670873 −0.223624 −0.111812 0.993729i \(-0.535666\pi\)
−0.111812 + 0.993729i \(0.535666\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −7.00000 −1.00000
\(8\) 0 0
\(9\) −8.54993 −0.949992
\(10\) 0 0
\(11\) −11.0000 −1.00000
\(12\) 0 0
\(13\) −24.5204 −1.88618 −0.943091 0.332533i \(-0.892097\pi\)
−0.943091 + 0.332533i \(0.892097\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 20.4951 1.20560 0.602798 0.797894i \(-0.294052\pi\)
0.602798 + 0.797894i \(0.294052\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 4.69611 0.223624
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 11.7738 0.436066
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −51.0534 −1.64688 −0.823442 0.567401i \(-0.807949\pi\)
−0.823442 + 0.567401i \(0.807949\pi\)
\(32\) 0 0
\(33\) 7.37961 0.223624
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 52.6498 1.42297 0.711484 0.702703i \(-0.248024\pi\)
0.711484 + 0.702703i \(0.248024\pi\)
\(38\) 0 0
\(39\) 16.4501 0.421797
\(40\) 0 0
\(41\) 38.9777 0.950674 0.475337 0.879804i \(-0.342326\pi\)
0.475337 + 0.879804i \(0.342326\pi\)
\(42\) 0 0
\(43\) 52.6498 1.22441 0.612207 0.790698i \(-0.290282\pi\)
0.612207 + 0.790698i \(0.290282\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 92.0437 1.95838 0.979188 0.202956i \(-0.0650550\pi\)
0.979188 + 0.202956i \(0.0650550\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) −13.7496 −0.269601
\(52\) 0 0
\(53\) 52.6498 0.993392 0.496696 0.867925i \(-0.334546\pi\)
0.496696 + 0.867925i \(0.334546\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −114.551 −1.94155 −0.970775 0.239993i \(-0.922855\pi\)
−0.970775 + 0.239993i \(0.922855\pi\)
\(60\) 0 0
\(61\) −59.1039 −0.968916 −0.484458 0.874815i \(-0.660983\pi\)
−0.484458 + 0.874815i \(0.660983\pi\)
\(62\) 0 0
\(63\) 59.8495 0.949992
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −106.501 −1.45892 −0.729458 0.684025i \(-0.760228\pi\)
−0.729458 + 0.684025i \(0.760228\pi\)
\(74\) 0 0
\(75\) −16.7718 −0.223624
\(76\) 0 0
\(77\) 77.0000 1.00000
\(78\) 0 0
\(79\) −157.949 −1.99936 −0.999679 0.0253165i \(-0.991941\pi\)
−0.999679 + 0.0253165i \(0.991941\pi\)
\(80\) 0 0
\(81\) 69.0506 0.852477
\(82\) 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
\(91\) 171.643 1.88618
\(92\) 0 0
\(93\) 34.2504 0.368283
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 94.0492 0.949992
\(100\) 0 0
\(101\) 134.678 1.33344 0.666721 0.745308i \(-0.267697\pi\)
0.666721 + 0.745308i \(0.267697\pi\)
\(102\) 0 0
\(103\) −0.368933 −0.00358187 −0.00179094 0.999998i \(-0.500570\pi\)
−0.00179094 + 0.999998i \(0.500570\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −38.0000 −0.355140 −0.177570 0.984108i \(-0.556824\pi\)
−0.177570 + 0.984108i \(0.556824\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −35.3213 −0.318210
\(112\) 0 0
\(113\) 82.0000 0.725664 0.362832 0.931855i \(-0.381810\pi\)
0.362832 + 0.931855i \(0.381810\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 209.647 1.79186
\(118\) 0 0
\(119\) −143.466 −1.20560
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) −26.1491 −0.212594
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −142.000 −1.11811 −0.559055 0.829130i \(-0.688836\pi\)
−0.559055 + 0.829130i \(0.688836\pi\)
\(128\) 0 0
\(129\) −35.3213 −0.273809
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 263.249 1.92153 0.960763 0.277372i \(-0.0894635\pi\)
0.960763 + 0.277372i \(0.0894635\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) −61.7496 −0.437941
\(142\) 0 0
\(143\) 269.724 1.88618
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −32.8728 −0.223624
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 263.249 1.74337 0.871685 0.490066i \(-0.163027\pi\)
0.871685 + 0.490066i \(0.163027\pi\)
\(152\) 0 0
\(153\) −175.232 −1.14531
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −35.3213 −0.222147
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 432.249 2.55769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −313.096 −1.80980 −0.904902 0.425621i \(-0.860056\pi\)
−0.904902 + 0.425621i \(0.860056\pi\)
\(174\) 0 0
\(175\) −175.000 −1.00000
\(176\) 0 0
\(177\) 76.8495 0.434178
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 39.6512 0.216673
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −225.447 −1.20560
\(188\) 0 0
\(189\) −82.4165 −0.436066
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 396.720 1.99357 0.996784 0.0801294i \(-0.0255333\pi\)
0.996784 + 0.0801294i \(0.0255333\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 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
\(211\) 26.0000 0.123223 0.0616114 0.998100i \(-0.480376\pi\)
0.0616114 + 0.998100i \(0.480376\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 357.374 1.64688
\(218\) 0 0
\(219\) 71.4486 0.326250
\(220\) 0 0
\(221\) −502.549 −2.27398
\(222\) 0 0
\(223\) −445.761 −1.99893 −0.999464 0.0327349i \(-0.989578\pi\)
−0.999464 + 0.0327349i \(0.989578\pi\)
\(224\) 0 0
\(225\) −213.748 −0.949992
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) −51.6573 −0.223624
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 105.964 0.447106
\(238\) 0 0
\(239\) 226.000 0.945607 0.472803 0.881168i \(-0.343242\pi\)
0.472803 + 0.881168i \(0.343242\pi\)
\(240\) 0 0
\(241\) 452.168 1.87621 0.938107 0.346344i \(-0.112577\pi\)
0.938107 + 0.346344i \(0.112577\pi\)
\(242\) 0 0
\(243\) −152.288 −0.626701
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 346.942 1.38224 0.691119 0.722741i \(-0.257118\pi\)
0.691119 + 0.722741i \(0.257118\pi\)
\(252\) 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
\(259\) −368.549 −1.42297
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 482.000 1.83270 0.916350 0.400379i \(-0.131121\pi\)
0.916350 + 0.400379i \(0.131121\pi\)
\(264\) 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
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) −115.150 −0.421797
\(274\) 0 0
\(275\) −275.000 −1.00000
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 436.503 1.56453
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −272.844 −0.950674
\(288\) 0 0
\(289\) 131.051 0.453462
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −504.496 −1.72183 −0.860915 0.508750i \(-0.830108\pi\)
−0.860915 + 0.508750i \(0.830108\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −129.512 −0.436066
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −368.549 −1.22441
\(302\) 0 0
\(303\) −90.3516 −0.298190
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0.247507 0.000800994 0
\(310\) 0 0
\(311\) −604.959 −1.94521 −0.972603 0.232473i \(-0.925318\pi\)
−0.972603 + 0.232473i \(0.925318\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −598.000 −1.88644 −0.943218 0.332175i \(-0.892217\pi\)
−0.943218 + 0.332175i \(0.892217\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 25.4932 0.0794180
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −613.009 −1.88618
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −644.306 −1.95838
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) −450.152 −1.35181
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) −55.0116 −0.162276
\(340\) 0 0
\(341\) 561.587 1.64688
\(342\) 0 0
\(343\) −343.000 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 52.6498 0.151728 0.0758642 0.997118i \(-0.475828\pi\)
0.0758642 + 0.997118i \(0.475828\pi\)
\(348\) 0 0
\(349\) 624.179 1.78848 0.894240 0.447588i \(-0.147717\pi\)
0.894240 + 0.447588i \(0.147717\pi\)
\(350\) 0 0
\(351\) −288.698 −0.822500
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 96.2475 0.269601
\(358\) 0 0
\(359\) 263.249 0.733284 0.366642 0.930362i \(-0.380507\pi\)
0.366642 + 0.930362i \(0.380507\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) −81.1757 −0.223624
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 442.642 1.20611 0.603054 0.797700i \(-0.293950\pi\)
0.603054 + 0.797700i \(0.293950\pi\)
\(368\) 0 0
\(369\) −333.256 −0.903133
\(370\) 0 0
\(371\) −368.549 −0.993392
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 95.2640 0.250037
\(382\) 0 0
\(383\) 634.611 1.65695 0.828474 0.560027i \(-0.189209\pi\)
0.828474 + 0.560027i \(0.189209\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −450.152 −1.16318
\(388\) 0 0
\(389\) −454.000 −1.16710 −0.583548 0.812079i \(-0.698336\pi\)
−0.583548 + 0.812079i \(0.698336\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −494.000 −1.23192 −0.615960 0.787777i \(-0.711232\pi\)
−0.615960 + 0.787777i \(0.711232\pi\)
\(402\) 0 0
\(403\) 1251.85 3.10632
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −579.148 −1.42297
\(408\) 0 0
\(409\) 387.194 0.946685 0.473342 0.880879i \(-0.343047\pi\)
0.473342 + 0.880879i \(0.343047\pi\)
\(410\) 0 0
\(411\) −176.607 −0.429700
\(412\) 0 0
\(413\) 801.860 1.94155
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −142.560 −0.340239 −0.170119 0.985423i \(-0.554415\pi\)
−0.170119 + 0.985423i \(0.554415\pi\)
\(420\) 0 0
\(421\) −789.747 −1.87588 −0.937942 0.346793i \(-0.887271\pi\)
−0.937942 + 0.346793i \(0.887271\pi\)
\(422\) 0 0
\(423\) −786.967 −1.86044
\(424\) 0 0
\(425\) 512.378 1.20560
\(426\) 0 0
\(427\) 413.727 0.968916
\(428\) 0 0
\(429\) −180.951 −0.421797
\(430\) 0 0
\(431\) 684.447 1.58804 0.794022 0.607889i \(-0.207983\pi\)
0.794022 + 0.607889i \(0.207983\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −418.947 −0.949992
\(442\) 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
\(449\) −157.949 −0.351780 −0.175890 0.984410i \(-0.556280\pi\)
−0.175890 + 0.984410i \(0.556280\pi\)
\(450\) 0 0
\(451\) −428.754 −0.950674
\(452\) 0 0
\(453\) −176.607 −0.389860
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 241.305 0.525720
\(460\) 0 0
\(461\) −264.223 −0.573152 −0.286576 0.958057i \(-0.592517\pi\)
−0.286576 + 0.958057i \(0.592517\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 683.820 1.46428 0.732141 0.681153i \(-0.238521\pi\)
0.732141 + 0.681153i \(0.238521\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −579.148 −1.22441
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −450.152 −0.943715
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −1290.99 −2.68398
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 895.046 1.82291 0.911453 0.411405i \(-0.134962\pi\)
0.911453 + 0.411405i \(0.134962\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 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
\(507\) −289.984 −0.571961
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 745.506 1.45892
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1012.48 −1.95838
\(518\) 0 0
\(519\) 210.048 0.404716
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 117.403 0.223624
\(526\) 0 0
\(527\) −1046.35 −1.98548
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 979.406 1.84446
\(532\) 0 0
\(533\) −955.747 −1.79315
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −539.000 −1.00000
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 698.000 1.27605 0.638026 0.770015i \(-0.279751\pi\)
0.638026 + 0.770015i \(0.279751\pi\)
\(548\) 0 0
\(549\) 505.334 0.920462
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1105.65 1.99936
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −1290.99 −2.30947
\(560\) 0 0
\(561\) 151.246 0.269601
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −483.354 −0.852477
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 442.000 0.774081 0.387040 0.922063i \(-0.373497\pi\)
0.387040 + 0.922063i \(0.373497\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −579.148 −0.993392
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1017.24 −1.73295 −0.866476 0.499218i \(-0.833621\pi\)
−0.866476 + 0.499218i \(0.833621\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1088.96 1.83636 0.918179 0.396167i \(-0.129660\pi\)
0.918179 + 0.396167i \(0.129660\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −266.149 −0.445811
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 53.2670 0.0886306 0.0443153 0.999018i \(-0.485889\pi\)
0.0443153 + 0.999018i \(0.485889\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2256.95 −3.69385
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1105.65 1.79197 0.895985 0.444084i \(-0.146471\pi\)
0.895985 + 0.444084i \(0.146471\pi\)
\(618\) 0 0
\(619\) 1207.91 1.95138 0.975691 0.219151i \(-0.0703288\pi\)
0.975691 + 0.219151i \(0.0703288\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1079.06 1.71552
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −17.4427 −0.0275556
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1201.50 −1.88618
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 684.447 1.06778 0.533890 0.845554i \(-0.320730\pi\)
0.533890 + 0.845554i \(0.320730\pi\)
\(642\) 0 0
\(643\) −259.460 −0.403515 −0.201757 0.979436i \(-0.564665\pi\)
−0.201757 + 0.979436i \(0.564665\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 90.2320 0.139462 0.0697311 0.997566i \(-0.477786\pi\)
0.0697311 + 0.997566i \(0.477786\pi\)
\(648\) 0 0
\(649\) 1260.07 1.94155
\(650\) 0 0
\(651\) −239.752 −0.368283
\(652\) 0 0
\(653\) −998.000 −1.52833 −0.764165 0.645020i \(-0.776849\pi\)
−0.764165 + 0.645020i \(0.776849\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 910.575 1.38596
\(658\) 0 0
\(659\) 1316.24 1.99734 0.998668 0.0515933i \(-0.0164300\pi\)
0.998668 + 0.0515933i \(0.0164300\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 337.146 0.508517
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 299.049 0.447009
\(670\) 0 0
\(671\) 650.142 0.968916
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 294.345 0.436066
\(676\) 0 0
\(677\) −1231.89 −1.81963 −0.909814 0.415015i \(-0.863776\pi\)
−0.909814 + 0.415015i \(0.863776\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 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
\(687\) 0 0
\(688\) 0 0
\(689\) −1290.99 −1.87372
\(690\) 0 0
\(691\) −1319.54 −1.90961 −0.954803 0.297238i \(-0.903934\pi\)
−0.954803 + 0.297238i \(0.903934\pi\)
\(692\) 0 0
\(693\) −658.345 −0.949992
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 798.852 1.14613
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −942.743 −1.33344
\(708\) 0 0
\(709\) 1354.00 1.90973 0.954866 0.297037i \(-0.0959984\pi\)
0.954866 + 0.297037i \(0.0959984\pi\)
\(710\) 0 0
\(711\) 1350.46 1.89938
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −151.617 −0.211461
\(718\) 0 0
\(719\) −1114.75 −1.55042 −0.775212 0.631701i \(-0.782357\pi\)
−0.775212 + 0.631701i \(0.782357\pi\)
\(720\) 0 0
\(721\) 2.58253 0.00358187
\(722\) 0 0
\(723\) −303.347 −0.419568
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 805.717 1.10828 0.554138 0.832425i \(-0.313048\pi\)
0.554138 + 0.832425i \(0.313048\pi\)
\(728\) 0 0
\(729\) −519.290 −0.712331
\(730\) 0 0
\(731\) 1079.06 1.47615
\(732\) 0 0
\(733\) 928.286 1.26642 0.633210 0.773980i \(-0.281737\pi\)
0.633210 + 0.773980i \(0.281737\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 106.000 0.143437 0.0717185 0.997425i \(-0.477152\pi\)
0.0717185 + 0.997425i \(0.477152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −782.000 −1.05249 −0.526245 0.850333i \(-0.676401\pi\)
−0.526245 + 0.850333i \(0.676401\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 266.000 0.355140
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −232.754 −0.309102
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1258.00 1.66182 0.830911 0.556405i \(-0.187820\pi\)
0.830911 + 0.556405i \(0.187820\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1505.27 −1.97801 −0.989007 0.147866i \(-0.952760\pi\)
−0.989007 + 0.147866i \(0.952760\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2808.84 3.66212
\(768\) 0 0
\(769\) −1103.99 −1.43561 −0.717807 0.696242i \(-0.754854\pi\)
−0.717807 + 0.696242i \(0.754854\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −1276.33 −1.64688
\(776\) 0 0
\(777\) 247.249 0.318210
\(778\) 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
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) −323.361 −0.409837
\(790\) 0 0
\(791\) −574.000 −0.725664
\(792\) 0 0
\(793\) 1449.25 1.82755
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 1886.45 2.36101
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1171.51 1.45892
\(804\) 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
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −1467.53 −1.79186
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 184.490 0.223624
\(826\) 0 0
\(827\) −1632.14 −1.97357 −0.986786 0.162031i \(-0.948195\pi\)
−0.986786 + 0.162031i \(0.948195\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1004.26 1.20560
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −601.091 −0.718150
\(838\) 0 0
\(839\) 951.196 1.13373 0.566863 0.823812i \(-0.308157\pi\)
0.566863 + 0.823812i \(0.308157\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −847.000 −1.00000
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1597.85 1.87321 0.936606 0.350384i \(-0.113949\pi\)
0.936606 + 0.350384i \(0.113949\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 626.897 0.731502 0.365751 0.930713i \(-0.380812\pi\)
0.365751 + 0.930713i \(0.380812\pi\)
\(858\) 0 0
\(859\) 761.943 0.887012 0.443506 0.896271i \(-0.353734\pi\)
0.443506 + 0.896271i \(0.353734\pi\)
\(860\) 0 0
\(861\) 183.044 0.212594
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −87.9184 −0.101405
\(868\) 0 0
\(869\) 1737.44 1.99936
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 338.453 0.385043
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 994.000 1.11811
\(890\) 0 0
\(891\) −759.557 −0.852477
\(892\) 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
\(901\) 1079.06 1.19763
\(902\) 0 0
\(903\) 247.249 0.273809
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −1151.48 −1.26676
\(910\) 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
\(919\) −1726.00 −1.87813 −0.939064 0.343742i \(-0.888306\pi\)
−0.939064 + 0.343742i \(0.888306\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1316.24 1.42297
\(926\) 0 0
\(927\) 3.15435 0.00340275
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 405.851 0.434996
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −440.428 −0.470041 −0.235020 0.971990i \(-0.575516\pi\)
−0.235020 + 0.971990i \(0.575516\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −395.412 −0.420205 −0.210102 0.977679i \(-0.567380\pi\)
−0.210102 + 0.977679i \(0.567380\pi\)
\(942\) 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
\(949\) 2611.44 2.75178
\(950\) 0 0
\(951\) 401.182 0.421853
\(952\) 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
\(959\) −1842.74 −1.92153
\(960\) 0 0
\(961\) 1645.45 1.71222
\(962\) 0 0
\(963\) 324.897 0.337380
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 263.249 0.272233 0.136116 0.990693i \(-0.456538\pi\)
0.136116 + 0.990693i \(0.456538\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1940.23 −1.99818 −0.999088 0.0426941i \(-0.986406\pi\)
−0.999088 + 0.0426941i \(0.986406\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 411.252 0.421797
\(976\) 0 0
\(977\) −1646.00 −1.68475 −0.842375 0.538892i \(-0.818843\pi\)
−0.842375 + 0.538892i \(0.818843\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1927.42 −1.96075 −0.980374 0.197146i \(-0.936833\pi\)
−0.980374 + 0.197146i \(0.936833\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 432.248 0.437941
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 733.263 0.735469 0.367735 0.929931i \(-0.380133\pi\)
0.367735 + 0.929931i \(0.380133\pi\)
\(998\) 0 0
\(999\) 619.887 0.620508
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 1232.3.m.a.1231.2 4
4.3 odd 2 1232.3.m.b.1231.3 yes 4
7.6 odd 2 inner 1232.3.m.a.1231.3 yes 4
11.10 odd 2 1232.3.m.b.1231.2 yes 4
28.27 even 2 1232.3.m.b.1231.2 yes 4
44.43 even 2 inner 1232.3.m.a.1231.3 yes 4
77.76 even 2 1232.3.m.b.1231.3 yes 4
308.307 odd 2 CM 1232.3.m.a.1231.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1232.3.m.a.1231.2 4 1.1 even 1 trivial
1232.3.m.a.1231.2 4 308.307 odd 2 CM
1232.3.m.a.1231.3 yes 4 7.6 odd 2 inner
1232.3.m.a.1231.3 yes 4 44.43 even 2 inner
1232.3.m.b.1231.2 yes 4 11.10 odd 2
1232.3.m.b.1231.2 yes 4 28.27 even 2
1232.3.m.b.1231.3 yes 4 4.3 odd 2
1232.3.m.b.1231.3 yes 4 77.76 even 2