Properties

Label 1232.3.m.a.1231.4
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 1231.4
Root \(2.98119\) 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)
 
\(f(q)\) \(=\) \(q+5.96238 q^{3} -7.00000 q^{7} +26.5499 q^{9} +O(q^{10})\) \(q+5.96238 q^{3} -7.00000 q^{7} +26.5499 q^{9} -11.0000 q^{11} +8.64587 q^{13} +27.1284 q^{17} -41.7366 q^{21} +25.0000 q^{25} +104.639 q^{27} +35.1789 q^{31} -65.5861 q^{33} -52.6498 q^{37} +51.5499 q^{39} +72.1439 q^{41} -52.6498 q^{43} +19.0779 q^{47} +49.0000 q^{49} +161.750 q^{51} -52.6498 q^{53} -28.3192 q^{59} +106.727 q^{61} -185.850 q^{63} -99.8677 q^{73} +149.059 q^{75} +77.0000 q^{77} +157.949 q^{79} +384.949 q^{81} -60.5211 q^{91} +209.750 q^{93} -292.049 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{7} + 36 q^{9}+O(q^{10}) \) 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
\(3\) 5.96238 1.98746 0.993729 0.111812i \(-0.0356655\pi\)
0.993729 + 0.111812i \(0.0356655\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −7.00000 −1.00000
\(8\) 0 0
\(9\) 26.5499 2.94999
\(10\) 0 0
\(11\) −11.0000 −1.00000
\(12\) 0 0
\(13\) 8.64587 0.665067 0.332533 0.943091i \(-0.392097\pi\)
0.332533 + 0.943091i \(0.392097\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 27.1284 1.59579 0.797894 0.602798i \(-0.205948\pi\)
0.797894 + 0.602798i \(0.205948\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −41.7366 −1.98746
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 104.639 3.87553
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 35.1789 1.13480 0.567401 0.823442i \(-0.307949\pi\)
0.567401 + 0.823442i \(0.307949\pi\)
\(32\) 0 0
\(33\) −65.5861 −1.98746
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −52.6498 −1.42297 −0.711484 0.702703i \(-0.751976\pi\)
−0.711484 + 0.702703i \(0.751976\pi\)
\(38\) 0 0
\(39\) 51.5499 1.32179
\(40\) 0 0
\(41\) 72.1439 1.75961 0.879804 0.475337i \(-0.157674\pi\)
0.879804 + 0.475337i \(0.157674\pi\)
\(42\) 0 0
\(43\) −52.6498 −1.22441 −0.612207 0.790698i \(-0.709718\pi\)
−0.612207 + 0.790698i \(0.709718\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 19.0779 0.405913 0.202956 0.979188i \(-0.434945\pi\)
0.202956 + 0.979188i \(0.434945\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 161.750 3.17156
\(52\) 0 0
\(53\) −52.6498 −0.993392 −0.496696 0.867925i \(-0.665454\pi\)
−0.496696 + 0.867925i \(0.665454\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −28.3192 −0.479986 −0.239993 0.970775i \(-0.577145\pi\)
−0.239993 + 0.970775i \(0.577145\pi\)
\(60\) 0 0
\(61\) 106.727 1.74963 0.874815 0.484458i \(-0.160983\pi\)
0.874815 + 0.484458i \(0.160983\pi\)
\(62\) 0 0
\(63\) −185.850 −2.94999
\(64\) 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
\(73\) −99.8677 −1.36805 −0.684025 0.729458i \(-0.739772\pi\)
−0.684025 + 0.729458i \(0.739772\pi\)
\(74\) 0 0
\(75\) 149.059 1.98746
\(76\) 0 0
\(77\) 77.0000 1.00000
\(78\) 0 0
\(79\) 157.949 1.99936 0.999679 0.0253165i \(-0.00805934\pi\)
0.999679 + 0.0253165i \(0.00805934\pi\)
\(80\) 0 0
\(81\) 384.949 4.75246
\(82\) 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
\(91\) −60.5211 −0.665067
\(92\) 0 0
\(93\) 209.750 2.25537
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) −292.049 −2.94999
\(100\) 0 0
\(101\) −150.552 −1.49062 −0.745308 0.666721i \(-0.767697\pi\)
−0.745308 + 0.666721i \(0.767697\pi\)
\(102\) 0 0
\(103\) −206.000 −2.00000 −0.999998 0.00179094i \(-0.999430\pi\)
−0.999998 + 0.00179094i \(0.999430\pi\)
\(104\) 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
\(111\) −313.918 −2.82809
\(112\) 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
\(117\) 229.547 1.96194
\(118\) 0 0
\(119\) −189.899 −1.59579
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 430.149 3.49715
\(124\) 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
\(129\) −313.918 −2.43347
\(130\) 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 \(-0.910536\pi\)
−0.960763 + 0.277372i \(0.910536\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 113.750 0.806735
\(142\) 0 0
\(143\) −95.1046 −0.665067
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 292.156 1.98746
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −263.249 −1.74337 −0.871685 0.490066i \(-0.836973\pi\)
−0.871685 + 0.490066i \(0.836973\pi\)
\(152\) 0 0
\(153\) 720.257 4.70756
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −313.918 −1.97433
\(160\) 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
\(169\) −94.2489 −0.557686
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −147.265 −0.851241 −0.425621 0.904902i \(-0.639944\pi\)
−0.425621 + 0.904902i \(0.639944\pi\)
\(174\) 0 0
\(175\) −175.000 −1.00000
\(176\) 0 0
\(177\) −168.850 −0.953952
\(178\) 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
\(183\) 636.349 3.47732
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −298.412 −1.59579
\(188\) 0 0
\(189\) −732.475 −3.87553
\(190\) 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
\(199\) 31.8915 0.160259 0.0801294 0.996784i \(-0.474467\pi\)
0.0801294 + 0.996784i \(0.474467\pi\)
\(200\) 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
\(217\) −246.252 −1.13480
\(218\) 0 0
\(219\) −595.449 −2.71894
\(220\) 0 0
\(221\) 234.549 1.06131
\(222\) 0 0
\(223\) −14.5998 −0.0654697 −0.0327349 0.999464i \(-0.510422\pi\)
−0.0327349 + 0.999464i \(0.510422\pi\)
\(224\) 0 0
\(225\) 663.748 2.94999
\(226\) 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
\(231\) 459.103 1.98746
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 941.753 3.97364
\(238\) 0 0
\(239\) 226.000 0.945607 0.472803 0.881168i \(-0.343242\pi\)
0.472803 + 0.881168i \(0.343242\pi\)
\(240\) 0 0
\(241\) 166.938 0.692689 0.346344 0.938107i \(-0.387423\pi\)
0.346344 + 0.938107i \(0.387423\pi\)
\(242\) 0 0
\(243\) 1353.46 5.56979
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −362.816 −1.44548 −0.722741 0.691119i \(-0.757118\pi\)
−0.722741 + 0.691119i \(0.757118\pi\)
\(252\) 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
\(273\) −360.850 −1.32179
\(274\) 0 0
\(275\) −275.000 −1.00000
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 933.996 3.34766
\(280\) 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
\(287\) −505.007 −1.75961
\(288\) 0 0
\(289\) 446.949 1.54654
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 298.127 1.01750 0.508750 0.860915i \(-0.330108\pi\)
0.508750 + 0.860915i \(0.330108\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1151.03 −3.87553
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 368.549 1.22441
\(302\) 0 0
\(303\) −897.648 −2.96254
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) −1228.25 −3.97491
\(310\) 0 0
\(311\) 144.598 0.464946 0.232473 0.972603i \(-0.425318\pi\)
0.232473 + 0.972603i \(0.425318\pi\)
\(312\) 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
\(321\) −226.570 −0.705826
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 216.147 0.665067
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −133.545 −0.405913
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) −1397.85 −4.19774
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 488.915 1.44223
\(340\) 0 0
\(341\) −386.968 −1.13480
\(342\) 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 \(-0.524172\pi\)
−0.0758642 + 0.997118i \(0.524172\pi\)
\(348\) 0 0
\(349\) 312.417 0.895177 0.447588 0.894240i \(-0.352283\pi\)
0.447588 + 0.894240i \(0.352283\pi\)
\(350\) 0 0
\(351\) 904.698 2.57749
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1132.25 −3.17156
\(358\) 0 0
\(359\) −263.249 −0.733284 −0.366642 0.930362i \(-0.619493\pi\)
−0.366642 + 0.930362i \(0.619493\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 721.448 1.98746
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −585.512 −1.59540 −0.797700 0.603054i \(-0.793950\pi\)
−0.797700 + 0.603054i \(0.793950\pi\)
\(368\) 0 0
\(369\) 1915.42 5.19083
\(370\) 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
\(381\) −846.657 −2.22220
\(382\) 0 0
\(383\) 428.981 1.12005 0.560027 0.828474i \(-0.310791\pi\)
0.560027 + 0.828474i \(0.310791\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1397.85 −3.61201
\(388\) 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
\(403\) 304.152 0.754719
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 579.148 1.42297
\(408\) 0 0
\(409\) −720.559 −1.76176 −0.880879 0.473342i \(-0.843047\pi\)
−0.880879 + 0.473342i \(0.843047\pi\)
\(410\) 0 0
\(411\) −1569.59 −3.81895
\(412\) 0 0
\(413\) 198.234 0.479986
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −825.785 −1.97085 −0.985423 0.170119i \(-0.945585\pi\)
−0.985423 + 0.170119i \(0.945585\pi\)
\(420\) 0 0
\(421\) 789.747 1.87588 0.937942 0.346793i \(-0.112729\pi\)
0.937942 + 0.346793i \(0.112729\pi\)
\(422\) 0 0
\(423\) 506.517 1.19744
\(424\) 0 0
\(425\) 678.210 1.59579
\(426\) 0 0
\(427\) −747.092 −1.74963
\(428\) 0 0
\(429\) −567.049 −1.32179
\(430\) 0 0
\(431\) −684.447 −1.58804 −0.794022 0.607889i \(-0.792017\pi\)
−0.794022 + 0.607889i \(0.792017\pi\)
\(432\) 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
\(441\) 1300.95 2.94999
\(442\) 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 \(-0.443720\pi\)
0.175890 + 0.984410i \(0.443720\pi\)
\(450\) 0 0
\(451\) −793.583 −1.75961
\(452\) 0 0
\(453\) −1569.59 −3.46488
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 2838.69 6.18452
\(460\) 0 0
\(461\) 883.329 1.91611 0.958057 0.286576i \(-0.0925171\pi\)
0.958057 + 0.286576i \(0.0925171\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −636.197 −1.36231 −0.681153 0.732141i \(-0.738521\pi\)
−0.681153 + 0.732141i \(0.738521\pi\)
\(468\) 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
\(477\) −1397.85 −2.93050
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −455.203 −0.946368
\(482\) 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 \(-0.865038\pi\)
−0.911453 + 0.411405i \(0.865038\pi\)
\(492\) 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
\(507\) −561.948 −1.10838
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 699.074 1.36805
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −209.857 −0.405913
\(518\) 0 0
\(519\) −878.048 −1.69181
\(520\) 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
\(525\) −1043.42 −1.98746
\(526\) 0 0
\(527\) 954.346 1.81090
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) −751.872 −1.41595
\(532\) 0 0
\(533\) 623.747 1.17026
\(534\) 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
\(549\) 2833.60 5.16139
\(550\) 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
\(559\) −455.203 −0.814317
\(560\) 0 0
\(561\) −1779.25 −3.17156
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2694.65 −4.75246
\(568\) 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
\(587\) −586.082 −0.998436 −0.499218 0.866476i \(-0.666379\pi\)
−0.499218 + 0.866476i \(0.666379\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −469.854 −0.792334 −0.396167 0.918179i \(-0.629660\pi\)
−0.396167 + 0.918179i \(0.629660\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 190.149 0.318508
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 1200.82 1.99804 0.999018 0.0443153i \(-0.0141106\pi\)
0.999018 + 0.0443153i \(0.0141106\pi\)
\(602\) 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
\(611\) 164.945 0.269959
\(612\) 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 \(-0.853529\pi\)
−0.895985 + 0.444084i \(0.853529\pi\)
\(618\) 0 0
\(619\) −271.309 −0.438303 −0.219151 0.975691i \(-0.570329\pi\)
−0.219151 + 0.975691i \(0.570329\pi\)
\(620\) 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
\(629\) −1428.30 −2.27075
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 155.022 0.244900
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 423.648 0.665067
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −684.447 −1.06778 −0.533890 0.845554i \(-0.679270\pi\)
−0.533890 + 0.845554i \(0.679270\pi\)
\(642\) 0 0
\(643\) 1259.55 1.95887 0.979436 0.201757i \(-0.0646653\pi\)
0.979436 + 0.201757i \(0.0646653\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1290.85 1.99513 0.997566 0.0697311i \(-0.0222141\pi\)
0.997566 + 0.0697311i \(0.0222141\pi\)
\(648\) 0 0
\(649\) 311.511 0.479986
\(650\) 0 0
\(651\) −1468.25 −2.25537
\(652\) 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
\(657\) −2651.48 −4.03574
\(658\) 0 0
\(659\) −1316.24 −1.99734 −0.998668 0.0515933i \(-0.983570\pi\)
−0.998668 + 0.0515933i \(0.983570\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 1398.47 2.10930
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −87.0492 −0.130118
\(670\) 0 0
\(671\) −1174.00 −1.74963
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 2615.98 3.87553
\(676\) 0 0
\(677\) −561.931 −0.830030 −0.415015 0.909814i \(-0.636224\pi\)
−0.415015 + 0.909814i \(0.636224\pi\)
\(678\) 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
\(689\) −455.203 −0.660672
\(690\) 0 0
\(691\) −410.783 −0.594476 −0.297238 0.954803i \(-0.596066\pi\)
−0.297238 + 0.954803i \(0.596066\pi\)
\(692\) 0 0
\(693\) 2044.34 2.94999
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1957.15 2.80796
\(698\) 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
\(707\) 1053.86 1.49062
\(708\) 0 0
\(709\) 1354.00 1.90973 0.954866 0.297037i \(-0.0959984\pi\)
0.954866 + 0.297037i \(0.0959984\pi\)
\(710\) 0 0
\(711\) 4193.54 5.89809
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1347.50 1.87935
\(718\) 0 0
\(719\) 908.386 1.26340 0.631701 0.775212i \(-0.282357\pi\)
0.631701 + 0.775212i \(0.282357\pi\)
\(720\) 0 0
\(721\) 1442.00 2.00000
\(722\) 0 0
\(723\) 995.347 1.37669
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1210.35 1.66485 0.832425 0.554138i \(-0.186952\pi\)
0.832425 + 0.554138i \(0.186952\pi\)
\(728\) 0 0
\(729\) 4605.29 6.31727
\(730\) 0 0
\(731\) −1428.30 −1.95390
\(732\) 0 0
\(733\) −1134.65 −1.54796 −0.773980 0.633210i \(-0.781737\pi\)
−0.773980 + 0.633210i \(0.781737\pi\)
\(734\) 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
\(753\) −2163.25 −2.87284
\(754\) 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
\(761\) −225.052 −0.295732 −0.147866 0.989007i \(-0.547240\pi\)
−0.147866 + 0.989007i \(0.547240\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −244.844 −0.319223
\(768\) 0 0
\(769\) −1070.82 −1.39248 −0.696242 0.717807i \(-0.745146\pi\)
−0.696242 + 0.717807i \(0.745146\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 879.472 1.13480
\(776\) 0 0
\(777\) 2197.42 2.82809
\(778\) 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
\(789\) 2873.87 3.64241
\(790\) 0 0
\(791\) −574.000 −0.725664
\(792\) 0 0
\(793\) 922.751 1.16362
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 517.553 0.647751
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1098.54 1.36805
\(804\) 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
\(819\) −1606.83 −1.96194
\(820\) 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
\(825\) −1639.65 −1.98746
\(826\) 0 0
\(827\) 1632.14 1.97357 0.986786 0.162031i \(-0.0518046\pi\)
0.986786 + 0.162031i \(0.0518046\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1329.29 1.59579
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3681.09 4.39796
\(838\) 0 0
\(839\) 1382.36 1.64762 0.823812 0.566863i \(-0.191843\pi\)
0.823812 + 0.566863i \(0.191843\pi\)
\(840\) 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
\(853\) −597.756 −0.700769 −0.350384 0.936606i \(-0.613949\pi\)
−0.350384 + 0.936606i \(0.613949\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1595.24 −1.86143 −0.930713 0.365751i \(-0.880812\pi\)
−0.930713 + 0.365751i \(0.880812\pi\)
\(858\) 0 0
\(859\) −1539.79 −1.79254 −0.896271 0.443506i \(-0.853734\pi\)
−0.896271 + 0.443506i \(0.853734\pi\)
\(860\) 0 0
\(861\) −3011.04 −3.49715
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2664.88 3.07368
\(868\) 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
\(879\) 1777.55 2.02224
\(880\) 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
\(891\) −4234.44 −4.75246
\(892\) 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
\(901\) −1428.30 −1.58524
\(902\) 0 0
\(903\) 2197.42 2.43347
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −3997.15 −4.39730
\(910\) 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
\(927\) −5469.28 −5.89997
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 862.149 0.924061
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1821.51 1.94398 0.971990 0.235020i \(-0.0755156\pi\)
0.971990 + 0.235020i \(0.0755156\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1839.99 1.95536 0.977679 0.210102i \(-0.0673797\pi\)
0.977679 + 0.210102i \(0.0673797\pi\)
\(942\) 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
\(949\) −863.443 −0.909845
\(950\) 0 0
\(951\) −3565.50 −3.74921
\(952\) 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
\(961\) 276.553 0.287776
\(962\) 0 0
\(963\) −1008.90 −1.04766
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −263.249 −0.272233 −0.136116 0.990693i \(-0.543462\pi\)
−0.136116 + 0.990693i \(0.543462\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 82.9119 0.0853881 0.0426941 0.999088i \(-0.486406\pi\)
0.0426941 + 0.999088i \(0.486406\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 1288.75 1.32179
\(976\) 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
\(983\) 387.588 0.394291 0.197146 0.980374i \(-0.436833\pi\)
0.197146 + 0.980374i \(0.436833\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −796.248 −0.806735
\(988\) 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
\(997\) 1854.28 1.85986 0.929931 0.367735i \(-0.119867\pi\)
0.929931 + 0.367735i \(0.119867\pi\)
\(998\) 0 0
\(999\) −5509.24 −5.51475
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.1231.4 yes 4
4.3 odd 2 1232.3.m.b.1231.1 yes 4
7.6 odd 2 inner 1232.3.m.a.1231.1 4
11.10 odd 2 1232.3.m.b.1231.4 yes 4
28.27 even 2 1232.3.m.b.1231.4 yes 4
44.43 even 2 inner 1232.3.m.a.1231.1 4
77.76 even 2 1232.3.m.b.1231.1 yes 4
308.307 odd 2 CM 1232.3.m.a.1231.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1232.3.m.a.1231.1 4 7.6 odd 2 inner
1232.3.m.a.1231.1 4 44.43 even 2 inner
1232.3.m.a.1231.4 yes 4 1.1 even 1 trivial
1232.3.m.a.1231.4 yes 4 308.307 odd 2 CM
1232.3.m.b.1231.1 yes 4 4.3 odd 2
1232.3.m.b.1231.1 yes 4 77.76 even 2
1232.3.m.b.1231.4 yes 4 11.10 odd 2
1232.3.m.b.1231.4 yes 4 28.27 even 2