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Properties

Label 176.11.h.a.65.1
Level $176$
Weight $11$
Character 176.65
Self dual yes
Analytic conductor $111.823$
Analytic rank $0$
Dimension $1$
CM discriminant -11
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [176,11,Mod(65,176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(176, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("176.65");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 176.h (of order \(2\), degree \(1\), not 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: \(111.822876471\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 65.1
Character \(\chi\) \(=\) 176.65

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-475.000 q^{3} -3001.00 q^{5} +166576. q^{9} +O(q^{10})\) \(q-475.000 q^{3} -3001.00 q^{5} +166576. q^{9} +161051. q^{11} +1.42548e6 q^{15} +1.19103e7 q^{23} -759624. q^{25} -5.10753e7 q^{27} -3.19232e6 q^{31} -7.64992e7 q^{33} -1.37083e8 q^{37} -4.99895e8 q^{45} -1.51795e8 q^{47} +2.82475e8 q^{49} +3.75067e8 q^{53} -4.83314e8 q^{55} +8.13568e8 q^{59} -2.61664e9 q^{67} -5.65740e9 q^{69} -7.83652e8 q^{71} +3.60821e8 q^{75} +1.44246e10 q^{81} -2.87091e9 q^{89} +1.51635e9 q^{93} +9.45401e9 q^{97} +2.68272e10 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/176\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(133\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) −475.000 −1.95473 −0.977366 − 0.211554i \(-0.932147\pi\)
−0.977366 + 0.211554i \(0.932147\pi\)
\(4\) 0 0
\(5\) −3001.00 −0.960320 −0.480160 − 0.877181i \(-0.659421\pi\)
−0.480160 + 0.877181i \(0.659421\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 166576. 2.82098
\(10\) 0 0
\(11\) 161051. 1.00000
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 1.42548e6 1.87717
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.19103e7 1.85048 0.925240 − 0.379382i \(-0.123863\pi\)
0.925240 + 0.379382i \(0.123863\pi\)
\(24\) 0 0
\(25\) −759624. −0.0777855
\(26\) 0 0
\(27\) −5.10753e7 −3.55953
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −3.19232e6 −0.111506 −0.0557530 − 0.998445i \(-0.517756\pi\)
−0.0557530 + 0.998445i \(0.517756\pi\)
\(32\) 0 0
\(33\) −7.64992e7 −1.95473
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.37083e8 −1.97685 −0.988425 − 0.151709i \(-0.951522\pi\)
−0.988425 + 0.151709i \(0.951522\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −4.99895e8 −2.70904
\(46\) 0 0
\(47\) −1.51795e8 −0.661864 −0.330932 − 0.943655i \(-0.607363\pi\)
−0.330932 + 0.943655i \(0.607363\pi\)
\(48\) 0 0
\(49\) 2.82475e8 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.75067e8 0.896869 0.448435 − 0.893816i \(-0.351982\pi\)
0.448435 + 0.893816i \(0.351982\pi\)
\(54\) 0 0
\(55\) −4.83314e8 −0.960320
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.13568e8 1.13798 0.568989 − 0.822345i \(-0.307335\pi\)
0.568989 + 0.822345i \(0.307335\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.61664e9 −1.93807 −0.969036 − 0.246921i \(-0.920581\pi\)
−0.969036 + 0.246921i \(0.920581\pi\)
\(68\) 0 0
\(69\) −5.65740e9 −3.61719
\(70\) 0 0
\(71\) −7.83652e8 −0.434342 −0.217171 − 0.976134i \(-0.569683\pi\)
−0.217171 + 0.976134i \(0.569683\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 3.60821e8 0.152050
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.44246e10 4.13694
\(82\) 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
\(89\) −2.87091e9 −0.514127 −0.257063 − 0.966395i \(-0.582755\pi\)
−0.257063 + 0.966395i \(0.582755\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.51635e9 0.217964
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.45401e9 1.10092 0.550462 − 0.834860i \(-0.314452\pi\)
0.550462 + 0.834860i \(0.314452\pi\)
\(98\) 0 0
\(99\) 2.68272e10 2.82098
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −9.30242e9 −0.802435 −0.401217 − 0.915983i \(-0.631413\pi\)
−0.401217 + 0.915983i \(0.631413\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 6.51142e10 3.86421
\(112\) 0 0
\(113\) 1.61746e8 0.00877890 0.00438945 − 0.999990i \(-0.498603\pi\)
0.00438945 + 0.999990i \(0.498603\pi\)
\(114\) 0 0
\(115\) −3.57429e10 −1.77705
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.59374e10 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.15863e10 1.03502
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.53277e11 3.41829
\(136\) 0 0
\(137\) −2.72055e10 −0.563708 −0.281854 − 0.959457i \(-0.590949\pi\)
−0.281854 + 0.959457i \(0.590949\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 7.21027e10 1.29377
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.34176e11 −1.95473
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.58016e9 0.107081
\(156\) 0 0
\(157\) −1.12312e11 −1.17742 −0.588708 − 0.808346i \(-0.700363\pi\)
−0.588708 + 0.808346i \(0.700363\pi\)
\(158\) 0 0
\(159\) −1.78157e11 −1.75314
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.03184e11 1.76584 0.882920 − 0.469523i \(-0.155574\pi\)
0.882920 + 0.469523i \(0.155574\pi\)
\(164\) 0 0
\(165\) 2.29574e11 1.87717
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.37858e11 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −3.86445e11 −2.22444
\(178\) 0 0
\(179\) −3.38386e11 −1.84139 −0.920697 − 0.390278i \(-0.872379\pi\)
−0.920697 + 0.390278i \(0.872379\pi\)
\(180\) 0 0
\(181\) −3.10198e11 −1.59679 −0.798393 − 0.602137i \(-0.794316\pi\)
−0.798393 + 0.602137i \(0.794316\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.11385e11 1.89841
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.34229e11 1.70825 0.854126 − 0.520066i \(-0.174093\pi\)
0.854126 + 0.520066i \(0.174093\pi\)
\(192\) 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
\(199\) −1.56808e10 −0.0502462 −0.0251231 − 0.999684i \(-0.507998\pi\)
−0.0251231 + 0.999684i \(0.507998\pi\)
\(200\) 0 0
\(201\) 1.24290e12 3.78841
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.98397e12 5.22017
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 3.72235e11 0.849022
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.04167e12 1.88889 0.944447 − 0.328663i \(-0.106598\pi\)
0.944447 + 0.328663i \(0.106598\pi\)
\(224\) 0 0
\(225\) −1.26535e11 −0.219431
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 1.11773e11 0.177484 0.0887419 − 0.996055i \(-0.471715\pi\)
0.0887419 + 0.996055i \(0.471715\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 4.55538e11 0.635601
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −3.83575e12 −4.52709
\(244\) 0 0
\(245\) −8.47708e11 −0.960320
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.42306e12 −1.42841 −0.714207 − 0.699935i \(-0.753212\pi\)
−0.714207 + 0.699935i \(0.753212\pi\)
\(252\) 0 0
\(253\) 1.91817e12 1.85048
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.12681e12 −1.89698 −0.948489 − 0.316810i \(-0.897388\pi\)
−0.948489 + 0.316810i \(0.897388\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −1.12558e12 −0.861281
\(266\) 0 0
\(267\) 1.36368e12 1.00498
\(268\) 0 0
\(269\) −1.46946e12 −1.04327 −0.521633 − 0.853170i \(-0.674677\pi\)
−0.521633 + 0.853170i \(0.674677\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.22338e11 −0.0777855
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −5.31764e11 −0.314556
\(280\) 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
\(287\) 0 0
\(288\) 0 0
\(289\) 2.01599e12 1.00000
\(290\) 0 0
\(291\) −4.49066e12 −2.15201
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) −2.44152e12 −1.09282
\(296\) 0 0
\(297\) −8.22573e12 −3.55953
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 4.41865e12 1.56855
\(310\) 0 0
\(311\) −5.50479e12 −1.89208 −0.946039 − 0.324052i \(-0.894955\pi\)
−0.946039 + 0.324052i \(0.894955\pi\)
\(312\) 0 0
\(313\) −4.90828e12 −1.63383 −0.816916 − 0.576756i \(-0.804318\pi\)
−0.816916 + 0.576756i \(0.804318\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.37352e12 1.36627 0.683133 − 0.730294i \(-0.260617\pi\)
0.683133 + 0.730294i \(0.260617\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7.40333e12 1.86332 0.931660 − 0.363332i \(-0.118361\pi\)
0.931660 + 0.363332i \(0.118361\pi\)
\(332\) 0 0
\(333\) −2.28347e13 −5.57665
\(334\) 0 0
\(335\) 7.85253e12 1.86117
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) −7.68291e10 −0.0171604
\(340\) 0 0
\(341\) −5.14127e11 −0.111506
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1.69779e13 3.47366
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.22003e12 1.49968 0.749842 − 0.661617i \(-0.230129\pi\)
0.749842 + 0.661617i \(0.230129\pi\)
\(354\) 0 0
\(355\) 2.35174e12 0.417107
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 6.13107e12 1.00000
\(362\) 0 0
\(363\) −1.23203e13 −1.95473
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.15912e13 1.74099 0.870497 − 0.492174i \(-0.163797\pi\)
0.870497 + 0.492174i \(0.163797\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −1.50035e13 −2.02319
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.20727e13 1.54386 0.771931 − 0.635706i \(-0.219291\pi\)
0.771931 + 0.635706i \(0.219291\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.08983e13 −1.32241 −0.661203 − 0.750207i \(-0.729954\pi\)
−0.661203 + 0.750207i \(0.729954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.29221e13 1.45072 0.725362 − 0.688368i \(-0.241672\pi\)
0.725362 + 0.688368i \(0.241672\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.72891e13 −1.75315 −0.876575 − 0.481264i \(-0.840178\pi\)
−0.876575 + 0.481264i \(0.840178\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.19222e13 1.14984 0.574918 − 0.818211i \(-0.305034\pi\)
0.574918 + 0.818211i \(0.305034\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −4.32883e13 −3.97279
\(406\) 0 0
\(407\) −2.20773e13 −1.97685
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 1.29226e13 1.10190
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.58238e13 1.99963 0.999817 − 0.0191062i \(-0.00608206\pi\)
0.999817 + 0.0191062i \(0.00608206\pi\)
\(420\) 0 0
\(421\) 2.05682e13 1.55520 0.777600 − 0.628759i \(-0.216437\pi\)
0.777600 + 0.628759i \(0.216437\pi\)
\(422\) 0 0
\(423\) −2.52854e13 −1.86711
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −4.37938e12 −0.287722 −0.143861 − 0.989598i \(-0.545952\pi\)
−0.143861 + 0.989598i \(0.545952\pi\)
\(434\) 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
\(441\) 4.70536e13 2.82098
\(442\) 0 0
\(443\) 1.66747e13 0.977325 0.488663 − 0.872473i \(-0.337485\pi\)
0.488663 + 0.872473i \(0.337485\pi\)
\(444\) 0 0
\(445\) 8.61561e12 0.493726
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.32859e13 −1.27603 −0.638015 − 0.770024i \(-0.720244\pi\)
−0.638015 + 0.770024i \(0.720244\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −7.99606e12 −0.375812 −0.187906 − 0.982187i \(-0.560170\pi\)
−0.187906 + 0.982187i \(0.560170\pi\)
\(464\) 0 0
\(465\) −4.55058e12 −0.209316
\(466\) 0 0
\(467\) 3.41290e13 1.53652 0.768261 − 0.640136i \(-0.221122\pi\)
0.768261 + 0.640136i \(0.221122\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 5.33484e13 2.30153
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.24771e13 2.53005
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.83715e13 −1.05724
\(486\) 0 0
\(487\) 3.54246e13 1.29318 0.646592 − 0.762836i \(-0.276194\pi\)
0.646592 + 0.762836i \(0.276194\pi\)
\(488\) 0 0
\(489\) −9.65124e13 −3.45175
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −8.05085e13 −2.70904
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 5.93009e12 0.191672 0.0958359 − 0.995397i \(-0.469448\pi\)
0.0958359 + 0.995397i \(0.469448\pi\)
\(500\) 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
\(507\) −6.54828e13 −1.95473
\(508\) 0 0
\(509\) 5.06582e13 1.48273 0.741363 − 0.671104i \(-0.234180\pi\)
0.741363 + 0.671104i \(0.234180\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.79166e13 0.770594
\(516\) 0 0
\(517\) −2.44468e13 −0.661864
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.51450e13 1.95754 0.978772 − 0.204952i \(-0.0657038\pi\)
0.978772 + 0.204952i \(0.0657038\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00429e14 2.42428
\(530\) 0 0
\(531\) 1.35521e14 3.21021
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.60733e14 3.59943
\(538\) 0 0
\(539\) 4.54929e13 1.00000
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 1.47344e14 3.12129
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1.95408e14 −3.71088
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) −4.85398e11 −0.00843055
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) −2.06259e14 −3.33918
\(574\) 0 0
\(575\) −9.04737e12 −0.143941
\(576\) 0 0
\(577\) −6.30450e13 −0.985760 −0.492880 − 0.870097i \(-0.664056\pi\)
−0.492880 + 0.870097i \(0.664056\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 6.04049e13 0.896869
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.73593e13 −0.392568 −0.196284 − 0.980547i \(-0.562887\pi\)
−0.196284 + 0.980547i \(0.562887\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 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
\(597\) 7.44838e12 0.0982178
\(598\) 0 0
\(599\) −6.14022e13 −0.796251 −0.398126 − 0.917331i \(-0.630339\pi\)
−0.398126 + 0.917331i \(0.630339\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −4.35869e14 −5.46726
\(604\) 0 0
\(605\) −7.78382e13 −0.960320
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.42489e14 1.59352 0.796758 − 0.604299i \(-0.206547\pi\)
0.796758 + 0.604299i \(0.206547\pi\)
\(618\) 0 0
\(619\) 1.78095e14 1.95974 0.979871 − 0.199630i \(-0.0639741\pi\)
0.979871 + 0.199630i \(0.0639741\pi\)
\(620\) 0 0
\(621\) −6.08324e14 −6.58683
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −8.73722e13 −0.916164
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 3.44236e13 0.344119 0.172060 − 0.985087i \(-0.444958\pi\)
0.172060 + 0.985087i \(0.444958\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −1.30538e14 −1.22527
\(640\) 0 0
\(641\) 1.09532e13 0.101216 0.0506082 − 0.998719i \(-0.483884\pi\)
0.0506082 + 0.998719i \(0.483884\pi\)
\(642\) 0 0
\(643\) −2.19818e14 −1.99990 −0.999952 − 0.00979126i \(-0.996883\pi\)
−0.999952 + 0.00979126i \(0.996883\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.75105e14 1.54446 0.772229 − 0.635344i \(-0.219142\pi\)
0.772229 + 0.635344i \(0.219142\pi\)
\(648\) 0 0
\(649\) 1.31026e14 1.13798
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.89126e14 1.59289 0.796444 − 0.604712i \(-0.206712\pi\)
0.796444 + 0.604712i \(0.206712\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 2.10313e14 1.66671 0.833355 − 0.552739i \(-0.186417\pi\)
0.833355 + 0.552739i \(0.186417\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −4.94795e14 −3.69228
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 3.87980e13 0.276880
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.72866e13 0.116307 0.0581537 − 0.998308i \(-0.481479\pi\)
0.0581537 + 0.998308i \(0.481479\pi\)
\(684\) 0 0
\(685\) 8.16437e13 0.541340
\(686\) 0 0
\(687\) −5.30921e13 −0.346933
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −3.11488e14 −1.97720 −0.988600 − 0.150563i \(-0.951891\pi\)
−0.988600 + 0.150563i \(0.951891\pi\)
\(692\) 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
\(703\) 0 0
\(704\) 0 0
\(705\) −2.16380e14 −1.24243
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.13308e14 1.74880 0.874402 − 0.485203i \(-0.161254\pi\)
0.874402 + 0.485203i \(0.161254\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3.80216e13 −0.206340
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 3.84303e14 2.00000 0.999998 − 0.00216702i \(-0.000689785\pi\)
0.999998 + 0.00216702i \(0.000689785\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.14172e14 0.562197 0.281098 − 0.959679i \(-0.409301\pi\)
0.281098 + 0.959679i \(0.409301\pi\)
\(728\) 0 0
\(729\) 9.70223e14 4.71231
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 4.02661e14 1.87717
\(736\) 0 0
\(737\) −4.21412e14 −1.93807
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 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
\(751\) −1.78058e14 −0.745353 −0.372677 − 0.927961i \(-0.621560\pi\)
−0.372677 + 0.927961i \(0.621560\pi\)
\(752\) 0 0
\(753\) 6.75952e14 2.79217
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.13954e14 −0.458405 −0.229202 − 0.973379i \(-0.573612\pi\)
−0.229202 + 0.973379i \(0.573612\pi\)
\(758\) 0 0
\(759\) −9.11131e14 −3.61719
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 1.01023e15 3.70808
\(772\) 0 0
\(773\) −4.10387e14 −1.48695 −0.743475 − 0.668764i \(-0.766824\pi\)
−0.743475 + 0.668764i \(0.766824\pi\)
\(774\) 0 0
\(775\) 2.42497e12 0.00867355
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −1.26208e14 −0.434342
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.37050e14 1.13070
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 5.34648e14 1.68357
\(796\) 0 0
\(797\) −5.54413e14 −1.72402 −0.862008 − 0.506894i \(-0.830794\pi\)
−0.862008 + 0.506894i \(0.830794\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −4.78225e14 −1.45034
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.97992e14 2.03931
\(808\) 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
\(815\) −6.09755e14 −1.69577
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −5.19688e14 −1.37640 −0.688199 − 0.725522i \(-0.741598\pi\)
−0.688199 + 0.725522i \(0.741598\pi\)
\(824\) 0 0
\(825\) 5.81106e13 0.152050
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −4.19459e14 −1.07131 −0.535657 − 0.844436i \(-0.679936\pi\)
−0.535657 + 0.844436i \(0.679936\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.63049e14 0.396909
\(838\) 0 0
\(839\) −7.17676e14 −1.72631 −0.863154 − 0.504940i \(-0.831515\pi\)
−0.863154 + 0.504940i \(0.831515\pi\)
\(840\) 0 0
\(841\) 4.20707e14 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −4.13713e14 −0.960320
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1.63270e15 −3.65812
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 7.08943e14 1.51581 0.757907 − 0.652363i \(-0.226222\pi\)
0.757907 + 0.652363i \(0.226222\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9.11193e14 1.90352 0.951758 − 0.306851i \(-0.0992752\pi\)
0.951758 + 0.306851i \(0.0992752\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −9.57597e14 −1.95473
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.57481e15 3.10568
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.01074e15 −1.90441 −0.952204 − 0.305462i \(-0.901189\pi\)
−0.952204 + 0.305462i \(0.901189\pi\)
\(882\) 0 0
\(883\) 1.03389e15 1.92606 0.963031 − 0.269391i \(-0.0868225\pi\)
0.963031 + 0.269391i \(0.0868225\pi\)
\(884\) 0 0
\(885\) 1.15972e15 2.13618
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.32310e15 4.13694
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 1.01550e15 1.76833
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 9.30906e14 1.53343
\(906\) 0 0
\(907\) 2.90238e14 0.472844 0.236422 − 0.971650i \(-0.424025\pi\)
0.236422 + 0.971650i \(0.424025\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.68037e14 −0.905284 −0.452642 − 0.891692i \(-0.649518\pi\)
−0.452642 + 0.891692i \(0.649518\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.04131e14 0.153770
\(926\) 0 0
\(927\) −1.54956e15 −2.26365
\(928\) 0 0
\(929\) −5.80692e14 −0.839203 −0.419601 − 0.907708i \(-0.637830\pi\)
−0.419601 + 0.907708i \(0.637830\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 2.61478e15 3.69851
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 2.33143e15 3.19371
\(940\) 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
\(947\) 1.13888e15 1.49529 0.747647 − 0.664096i \(-0.231184\pi\)
0.747647 + 0.664096i \(0.231184\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −2.07742e15 −2.67068
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −1.30312e15 −1.64047
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −8.09437e14 −0.987566
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.66222e14 −0.424276 −0.212138 − 0.977240i \(-0.568043\pi\)
−0.212138 + 0.977240i \(0.568043\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.34198e14 0.487769 0.243885 − 0.969804i \(-0.421578\pi\)
0.243885 + 0.969804i \(0.421578\pi\)
\(978\) 0 0
\(979\) −4.62363e14 −0.514127
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.83071e15 −1.99458 −0.997290 − 0.0735772i \(-0.976558\pi\)
−0.997290 + 0.0735772i \(0.976558\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.90369e15 −1.99172 −0.995861 − 0.0908931i \(-0.971028\pi\)
−0.995861 + 0.0908931i \(0.971028\pi\)
\(992\) 0 0
\(993\) −3.51658e15 −3.64229
\(994\) 0 0
\(995\) 4.70581e13 0.0482524
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 7.00154e15 7.03665
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 176.11.h.a.65.1 1
4.3 odd 2 11.11.b.a.10.1 ✓ 1
11.10 odd 2 CM 176.11.h.a.65.1 1
12.11 even 2 99.11.c.a.10.1 1
44.43 even 2 11.11.b.a.10.1 ✓ 1
132.131 odd 2 99.11.c.a.10.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.11.b.a.10.1 ✓ 1 4.3 odd 2
11.11.b.a.10.1 ✓ 1 44.43 even 2
99.11.c.a.10.1 1 12.11 even 2
99.11.c.a.10.1 1 132.131 odd 2
176.11.h.a.65.1 1 1.1 even 1 trivial
176.11.h.a.65.1 1 11.10 odd 2 CM