Properties

Label 3.145.b.a.2.1
Level $3$
Weight $145$
Character 3.2
Self dual yes
Analytic conductor $393.943$
Analytic rank $0$
Dimension $1$
CM discriminant -3
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] = [3,145,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 145, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 145);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 145 \)
Character orbit: \([\chi]\) \(=\) 3.b (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: \(393.943094413\)
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 2.1
Character \(\chi\) \(=\) 3.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.25284e34 q^{3} +2.23007e43 q^{4} -8.78320e60 q^{7} +5.07529e68 q^{9} +O(q^{10})\) \(q+2.25284e34 q^{3} +2.23007e43 q^{4} -8.78320e60 q^{7} +5.07529e68 q^{9} +5.02400e77 q^{12} -2.98014e80 q^{13} +4.97323e86 q^{16} -1.55808e92 q^{19} -1.97871e95 q^{21} +4.48416e100 q^{25} +1.14338e103 q^{27} -1.95872e104 q^{28} -4.19525e107 q^{31} +1.13183e112 q^{36} +1.22765e113 q^{37} -6.71377e114 q^{39} +7.93926e117 q^{43} +1.12039e121 q^{48} +2.77002e121 q^{49} -6.64593e123 q^{52} -3.51011e126 q^{57} -6.85567e128 q^{61} -4.45773e129 q^{63} +1.10907e130 q^{64} +4.54160e131 q^{67} -1.34253e134 q^{73} +1.01021e135 q^{75} -3.47464e135 q^{76} +2.55151e136 q^{79} +2.57585e137 q^{81} -4.41268e138 q^{84} +2.61752e141 q^{91} -9.45122e141 q^{93} +1.64445e143 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 2.25284e34 1.00000
\(4\) 2.23007e43 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −8.78320e60 −1.24909 −0.624545 0.780988i \(-0.714716\pi\)
−0.624545 + 0.780988i \(0.714716\pi\)
\(8\) 0 0
\(9\) 5.07529e68 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 5.02400e77 1.00000
\(13\) −2.98014e80 −1.86344 −0.931719 0.363180i \(-0.881691\pi\)
−0.931719 + 0.363180i \(0.881691\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.97323e86 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −1.55808e92 −1.32535 −0.662677 0.748906i \(-0.730580\pi\)
−0.662677 + 0.748906i \(0.730580\pi\)
\(20\) 0 0
\(21\) −1.97871e95 −1.24909
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 4.48416e100 1.00000
\(26\) 0 0
\(27\) 1.14338e103 1.00000
\(28\) −1.95872e104 −1.24909
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −4.19525e107 −1.75677 −0.878386 0.477951i \(-0.841380\pi\)
−0.878386 + 0.477951i \(0.841380\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.13183e112 1.00000
\(37\) 1.22765e113 1.50852 0.754258 0.656578i \(-0.227997\pi\)
0.754258 + 0.656578i \(0.227997\pi\)
\(38\) 0 0
\(39\) −6.71377e114 −1.86344
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 7.93926e117 1.95008 0.975038 0.222040i \(-0.0712716\pi\)
0.975038 + 0.222040i \(0.0712716\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.12039e121 1.00000
\(49\) 2.77002e121 0.560228
\(50\) 0 0
\(51\) 0 0
\(52\) −6.64593e123 −1.86344
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.51011e126 −1.32535
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −6.85567e128 −1.96021 −0.980103 0.198487i \(-0.936397\pi\)
−0.980103 + 0.198487i \(0.936397\pi\)
\(62\) 0 0
\(63\) −4.45773e129 −1.24909
\(64\) 1.10907e130 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.54160e131 1.51295 0.756473 0.654025i \(-0.226921\pi\)
0.756473 + 0.654025i \(0.226921\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.34253e134 −0.930416 −0.465208 0.885201i \(-0.654021\pi\)
−0.465208 + 0.885201i \(0.654021\pi\)
\(74\) 0 0
\(75\) 1.01021e135 1.00000
\(76\) −3.47464e135 −1.32535
\(77\) 0 0
\(78\) 0 0
\(79\) 2.55151e136 0.599303 0.299652 0.954049i \(-0.403130\pi\)
0.299652 + 0.954049i \(0.403130\pi\)
\(80\) 0 0
\(81\) 2.57585e137 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −4.41268e138 −1.24909
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 2.61752e141 2.32760
\(92\) 0 0
\(93\) −9.45122e141 −1.75677
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.64445e143 1.47386 0.736928 0.675971i \(-0.236275\pi\)
0.736928 + 0.675971i \(0.236275\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e144 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 1.52267e145 1.81270 0.906351 0.422525i \(-0.138856\pi\)
0.906351 + 0.422525i \(0.138856\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 2.54983e146 1.00000
\(109\) 3.14586e146 0.635378 0.317689 0.948195i \(-0.397093\pi\)
0.317689 + 0.948195i \(0.397093\pi\)
\(110\) 0 0
\(111\) 2.76570e147 1.50852
\(112\) −4.36809e147 −1.24909
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.51251e149 −1.86344
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 9.13160e149 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −9.35571e150 −1.75677
\(125\) 0 0
\(126\) 0 0
\(127\) 5.92060e151 1.98837 0.994187 0.107670i \(-0.0343391\pi\)
0.994187 + 0.107670i \(0.0343391\pi\)
\(128\) 0 0
\(129\) 1.78859e152 1.95008
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 1.36850e153 1.65549
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 3.31679e154 1.67361 0.836803 0.547505i \(-0.184422\pi\)
0.836803 + 0.547505i \(0.184422\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 2.52406e155 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 6.24041e155 0.560228
\(148\) 2.73775e156 1.50852
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −3.09549e155 −0.0402151 −0.0201076 0.999798i \(-0.506401\pi\)
−0.0201076 + 0.999798i \(0.506401\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −1.49722e158 −1.86344
\(157\) −2.18878e158 −1.71960 −0.859800 0.510630i \(-0.829412\pi\)
−0.859800 + 0.510630i \(0.829412\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.24733e159 −0.658377 −0.329189 0.944264i \(-0.606775\pi\)
−0.329189 + 0.944264i \(0.606775\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 6.32356e160 2.47240
\(170\) 0 0
\(171\) −7.90772e160 −1.32535
\(172\) 1.77051e161 1.95008
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −3.93852e161 −1.24909
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 7.00229e162 1.96057 0.980285 0.197587i \(-0.0633103\pi\)
0.980285 + 0.197587i \(0.0633103\pi\)
\(182\) 0 0
\(183\) −1.54447e163 −1.96021
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.00425e164 −1.24909
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 2.49855e164 1.00000
\(193\) −4.22787e164 −1.16411 −0.582056 0.813148i \(-0.697752\pi\)
−0.582056 + 0.813148i \(0.697752\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 6.17734e164 0.560228
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 2.84310e165 0.863715 0.431857 0.901942i \(-0.357858\pi\)
0.431857 + 0.901942i \(0.357858\pi\)
\(200\) 0 0
\(201\) 1.02315e166 1.51295
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.48209e167 −1.86344
\(209\) 0 0
\(210\) 0 0
\(211\) −2.99548e167 −1.34316 −0.671582 0.740930i \(-0.734385\pi\)
−0.671582 + 0.740930i \(0.734385\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3.68477e168 2.19437
\(218\) 0 0
\(219\) −3.02450e168 −0.930416
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.46635e169 1.22543 0.612715 0.790304i \(-0.290077\pi\)
0.612715 + 0.790304i \(0.290077\pi\)
\(224\) 0 0
\(225\) 2.27584e169 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) −7.82782e169 −1.32535
\(229\) −9.44318e169 −1.16671 −0.583356 0.812217i \(-0.698261\pi\)
−0.583356 + 0.812217i \(0.698261\pi\)
\(230\) 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
\(237\) 5.74814e170 0.599303
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.16688e171 −0.364584 −0.182292 0.983244i \(-0.558352\pi\)
−0.182292 + 0.983244i \(0.558352\pi\)
\(242\) 0 0
\(243\) 5.80299e171 1.00000
\(244\) −1.52887e172 −1.96021
\(245\) 0 0
\(246\) 0 0
\(247\) 4.64330e172 2.46971
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −9.94107e172 −1.24909
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 2.47330e173 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −1.07827e174 −1.88427
\(260\) 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
\(268\) 1.01281e175 1.51295
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −9.89621e174 −0.663254 −0.331627 0.943411i \(-0.607598\pi\)
−0.331627 + 0.943411i \(0.607598\pi\)
\(272\) 0 0
\(273\) 5.89684e175 2.32760
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.38831e176 −1.92285 −0.961423 0.275074i \(-0.911298\pi\)
−0.961423 + 0.275074i \(0.911298\pi\)
\(278\) 0 0
\(279\) −2.12921e176 −1.75677
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 2.22112e176 0.657580 0.328790 0.944403i \(-0.393359\pi\)
0.328790 + 0.944403i \(0.393359\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.52984e177 1.00000
\(290\) 0 0
\(291\) 3.70468e177 1.47386
\(292\) −2.99394e177 −0.930416
\(293\) 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
\(300\) 2.25284e178 1.00000
\(301\) −6.97321e178 −2.43582
\(302\) 0 0
\(303\) 0 0
\(304\) −7.74871e178 −1.32535
\(305\) 0 0
\(306\) 0 0
\(307\) 1.58779e179 1.33915 0.669577 0.742743i \(-0.266476\pi\)
0.669577 + 0.742743i \(0.266476\pi\)
\(308\) 0 0
\(309\) 3.43034e179 1.81270
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −6.87469e179 −1.43900 −0.719500 0.694492i \(-0.755629\pi\)
−0.719500 + 0.694492i \(0.755629\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 5.69006e179 0.599303
\(317\) 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
\(324\) 5.74435e180 1.00000
\(325\) −1.33634e181 −1.86344
\(326\) 0 0
\(327\) 7.08713e180 0.635378
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3.66647e181 −1.36972 −0.684861 0.728674i \(-0.740137\pi\)
−0.684861 + 0.728674i \(0.740137\pi\)
\(332\) 0 0
\(333\) 6.23067e181 1.50852
\(334\) 0 0
\(335\) 0 0
\(336\) −9.84061e181 −1.24909
\(337\) 1.32851e182 1.36148 0.680742 0.732523i \(-0.261657\pi\)
0.680742 + 0.732523i \(0.261657\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.90985e182 0.549315
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 7.82794e182 0.645977 0.322988 0.946403i \(-0.395313\pi\)
0.322988 + 0.946403i \(0.395313\pi\)
\(350\) 0 0
\(351\) −3.40743e183 −1.86344
\(352\) 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
\(361\) 1.04559e184 0.756561
\(362\) 0 0
\(363\) 2.05720e184 1.00000
\(364\) 5.83725e184 2.32760
\(365\) 0 0
\(366\) 0 0
\(367\) −7.22635e184 −1.59574 −0.797870 0.602829i \(-0.794040\pi\)
−0.797870 + 0.602829i \(0.794040\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −2.10769e185 −1.75677
\(373\) 1.95117e185 1.34048 0.670240 0.742145i \(-0.266191\pi\)
0.670240 + 0.742145i \(0.266191\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −4.20167e184 −0.0914953 −0.0457476 0.998953i \(-0.514567\pi\)
−0.0457476 + 0.998953i \(0.514567\pi\)
\(380\) 0 0
\(381\) 1.33382e186 1.98837
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.02940e186 1.95008
\(388\) 3.66724e186 1.47386
\(389\) 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
\(397\) −2.30982e187 −1.78099 −0.890496 0.454992i \(-0.849642\pi\)
−0.890496 + 0.454992i \(0.849642\pi\)
\(398\) 0 0
\(399\) 3.08300e187 1.65549
\(400\) 2.23007e187 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 1.25024e188 3.27364
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.20557e188 1.99271 0.996353 0.0853312i \(-0.0271948\pi\)
0.996353 + 0.0853312i \(0.0271948\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 3.39567e188 1.81270
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 7.47221e188 1.67361
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 1.73894e189 1.95873 0.979365 0.202098i \(-0.0647760\pi\)
0.979365 + 0.202098i \(0.0647760\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 6.02148e189 2.44848
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 5.68630e189 1.00000
\(433\) 1.07768e190 1.60457 0.802287 0.596938i \(-0.203616\pi\)
0.802287 + 0.596938i \(0.203616\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7.01551e189 0.635378
\(437\) 0 0
\(438\) 0 0
\(439\) 4.71718e189 0.260758 0.130379 0.991464i \(-0.458381\pi\)
0.130379 + 0.991464i \(0.458381\pi\)
\(440\) 0 0
\(441\) 1.40586e190 0.560228
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 6.16771e190 1.50852
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −9.74117e190 −1.24909
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −6.97365e189 −0.0402151
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −5.52513e191 −1.69190 −0.845952 0.533259i \(-0.820967\pi\)
−0.845952 + 0.533259i \(0.820967\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 1.47715e192 1.76845 0.884226 0.467060i \(-0.154687\pi\)
0.884226 + 0.467060i \(0.154687\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −3.37300e192 −1.86344
\(469\) −3.98898e192 −1.88981
\(470\) 0 0
\(471\) −4.93098e192 −1.71960
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −6.98669e192 −1.32535
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −3.65856e193 −2.81103
\(482\) 0 0
\(483\) 0 0
\(484\) 2.03641e193 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 1.98790e193 0.625617 0.312809 0.949816i \(-0.398730\pi\)
0.312809 + 0.949816i \(0.398730\pi\)
\(488\) 0 0
\(489\) −2.81004e193 −0.658377
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −2.08639e194 −1.75677
\(497\) 0 0
\(498\) 0 0
\(499\) −1.10166e194 −0.600906 −0.300453 0.953797i \(-0.597138\pi\)
−0.300453 + 0.953797i \(0.597138\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.42460e195 2.47240
\(508\) 1.32034e195 1.98837
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 1.17917e195 1.16217
\(512\) 0 0
\(513\) −1.78148e195 −1.32535
\(514\) 0 0
\(515\) 0 0
\(516\) 3.98869e195 1.95008
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 1.06106e196 1.96617 0.983083 0.183163i \(-0.0586336\pi\)
0.983083 + 0.183163i \(0.0586336\pi\)
\(524\) 0 0
\(525\) −8.87286e195 −1.24909
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.22690e196 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 3.05185e196 1.65549
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1.10296e197 −1.78795 −0.893975 0.448117i \(-0.852095\pi\)
−0.893975 + 0.448117i \(0.852095\pi\)
\(542\) 0 0
\(543\) 1.57750e197 1.96057
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2.72432e197 −1.99605 −0.998025 0.0628190i \(-0.979991\pi\)
−0.998025 + 0.0628190i \(0.979991\pi\)
\(548\) 0 0
\(549\) −3.47945e197 −1.96021
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −2.24104e197 −0.748584
\(554\) 0 0
\(555\) 0 0
\(556\) 7.39670e197 1.67361
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −2.36601e198 −3.63384
\(560\) 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
\(567\) −2.26243e198 −1.24909
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 2.69104e198 0.895611 0.447806 0.894131i \(-0.352206\pi\)
0.447806 + 0.894131i \(0.352206\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 5.62884e198 1.00000
\(577\) −3.52229e198 −0.552289 −0.276144 0.961116i \(-0.589057\pi\)
−0.276144 + 0.961116i \(0.589057\pi\)
\(578\) 0 0
\(579\) −9.52471e198 −1.16411
\(580\) 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 \(\pi\)
\(588\) 1.39166e199 0.560228
\(589\) 6.53655e199 2.32834
\(590\) 0 0
\(591\) 0 0
\(592\) 6.10538e199 1.50852
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.40504e199 0.863715
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 8.79952e199 0.733664 0.366832 0.930287i \(-0.380442\pi\)
0.366832 + 0.930287i \(0.380442\pi\)
\(602\) 0 0
\(603\) 2.30499e200 1.51295
\(604\) −6.90318e198 −0.0402151
\(605\) 0 0
\(606\) 0 0
\(607\) −4.31719e200 −1.76042 −0.880209 0.474587i \(-0.842598\pi\)
−0.880209 + 0.474587i \(0.842598\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −5.31684e200 −1.06782 −0.533912 0.845540i \(-0.679279\pi\)
−0.533912 + 0.845540i \(0.679279\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 2.00463e201 1.99668 0.998338 0.0576284i \(-0.0183539\pi\)
0.998338 + 0.0576284i \(0.0183539\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −3.33892e201 −1.86344
\(625\) 2.01076e201 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −4.88115e201 −1.71960
\(629\) 0 0
\(630\) 0 0
\(631\) −7.70008e201 −1.92478 −0.962390 0.271671i \(-0.912424\pi\)
−0.962390 + 0.271671i \(0.912424\pi\)
\(632\) 0 0
\(633\) −6.74833e201 −1.34316
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −8.25503e201 −1.04395
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 1.43676e202 0.925110 0.462555 0.886590i \(-0.346933\pi\)
0.462555 + 0.886590i \(0.346933\pi\)
\(644\) 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
\(651\) 8.30120e202 2.19437
\(652\) −2.78165e202 −0.658377
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −6.81371e202 −0.930416
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 2.11337e203 1.86410 0.932052 0.362324i \(-0.118017\pi\)
0.932052 + 0.362324i \(0.118017\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 3.30346e203 1.22543
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 2.76512e203 0.667774 0.333887 0.942613i \(-0.391640\pi\)
0.333887 + 0.942613i \(0.391640\pi\)
\(674\) 0 0
\(675\) 5.12710e203 1.00000
\(676\) 1.41020e204 2.47240
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −1.44435e204 −1.84098
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −1.76348e204 −1.32535
\(685\) 0 0
\(686\) 0 0
\(687\) −2.12740e204 −1.16671
\(688\) 3.94838e204 1.95008
\(689\) 0 0
\(690\) 0 0
\(691\) −5.53912e204 −1.99996 −0.999980 0.00637929i \(-0.997969\pi\)
−0.999980 + 0.00637929i \(0.997969\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −8.78320e204 −1.24909
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −1.91278e205 −1.99932
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.26686e204 0.128498 0.0642489 0.997934i \(-0.479535\pi\)
0.0642489 + 0.997934i \(0.479535\pi\)
\(710\) 0 0
\(711\) 1.29496e205 0.599303
\(712\) 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
\(721\) −1.33739e206 −2.26423
\(722\) 0 0
\(723\) −2.62879e205 −0.364584
\(724\) 1.56156e206 1.96057
\(725\) 0 0
\(726\) 0 0
\(727\) 1.48778e206 1.38695 0.693475 0.720481i \(-0.256079\pi\)
0.693475 + 0.720481i \(0.256079\pi\)
\(728\) 0 0
\(729\) 1.30732e206 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −3.44429e206 −1.96021
\(733\) 3.81339e206 1.96710 0.983548 0.180649i \(-0.0578197\pi\)
0.983548 + 0.180649i \(0.0578197\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 3.95065e206 1.13310 0.566551 0.824027i \(-0.308277\pi\)
0.566551 + 0.824027i \(0.308277\pi\)
\(740\) 0 0
\(741\) 1.04606e207 2.46971
\(742\) 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
\(751\) −1.51101e207 −1.35891 −0.679455 0.733717i \(-0.737784\pi\)
−0.679455 + 0.733717i \(0.737784\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −2.23956e207 −1.24909
\(757\) −7.02452e206 −0.356216 −0.178108 0.984011i \(-0.556998\pi\)
−0.178108 + 0.984011i \(0.556998\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −2.76308e207 −0.793645
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 5.57196e207 1.00000
\(769\) 5.66785e207 0.926236 0.463118 0.886297i \(-0.346731\pi\)
0.463118 + 0.886297i \(0.346731\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −9.42846e207 −1.16411
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −1.88121e208 −1.75677
\(776\) 0 0
\(777\) −2.42917e208 −1.88427
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.37759e208 0.560228
\(785\) 0 0
\(786\) 0 0
\(787\) −5.59145e208 −1.72721 −0.863603 0.504173i \(-0.831797\pi\)
−0.863603 + 0.504173i \(0.831797\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.04308e209 3.65272
\(794\) 0 0
\(795\) 0 0
\(796\) 6.34032e208 0.863715
\(797\) 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
\(804\) 2.28170e209 1.51295
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 1.56165e209 0.554731 0.277365 0.960764i \(-0.410539\pi\)
0.277365 + 0.960764i \(0.410539\pi\)
\(812\) 0 0
\(813\) −2.22946e209 −0.663254
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.23700e210 −2.58454
\(818\) 0 0
\(819\) 1.32846e210 2.32760
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1.62113e210 2.00000 0.999999 0.00162199i \(-0.000516295\pi\)
0.999999 + 0.00162199i \(0.000516295\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 7.27620e209 0.532082 0.266041 0.963962i \(-0.414284\pi\)
0.266041 + 0.963962i \(0.414284\pi\)
\(830\) 0 0
\(831\) −3.12765e210 −1.92285
\(832\) −3.30517e210 −1.86344
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.79677e210 −1.75677
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 3.84868e210 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −6.68013e210 −1.34316
\(845\) 0 0
\(846\) 0 0
\(847\) −8.02047e210 −1.24909
\(848\) 0 0
\(849\) 5.00383e210 0.657580
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −3.37633e210 −0.316311 −0.158155 0.987414i \(-0.550555\pi\)
−0.158155 + 0.987414i \(0.550555\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 2.49460e211 1.41089 0.705444 0.708765i \(-0.250747\pi\)
0.705444 + 0.708765i \(0.250747\pi\)
\(860\) 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
\(867\) 3.44647e211 1.00000
\(868\) 8.21731e211 2.19437
\(869\) 0 0
\(870\) 0 0
\(871\) −1.35346e212 −2.81928
\(872\) 0 0
\(873\) 8.34605e211 1.47386
\(874\) 0 0
\(875\) 0 0
\(876\) −6.74486e211 −0.930416
\(877\) 1.56855e212 1.99309 0.996547 0.0830310i \(-0.0264600\pi\)
0.996547 + 0.0830310i \(0.0264600\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 2.13194e212 1.65808 0.829039 0.559191i \(-0.188888\pi\)
0.829039 + 0.559191i \(0.188888\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −5.20018e212 −2.48366
\(890\) 0 0
\(891\) 0 0
\(892\) 3.27008e212 1.22543
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 5.07529e212 1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) −1.57095e213 −2.43582
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.08686e213 −1.22589 −0.612944 0.790127i \(-0.710015\pi\)
−0.612944 + 0.790127i \(0.710015\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −1.74566e213 −1.32535
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −2.10590e213 −1.16671
\(917\) 0 0
\(918\) 0 0
\(919\) −3.29092e213 −1.44079 −0.720395 0.693564i \(-0.756040\pi\)
−0.720395 + 0.693564i \(0.756040\pi\)
\(920\) 0 0
\(921\) 3.57704e213 1.33915
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 5.50497e213 1.50852
\(926\) 0 0
\(927\) 7.72800e213 1.81270
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −4.31592e213 −0.742500
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.74684e214 1.89235 0.946177 0.323649i \(-0.104910\pi\)
0.946177 + 0.323649i \(0.104910\pi\)
\(938\) 0 0
\(939\) −1.54876e214 −1.43900
\(940\) 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 \(\pi\)
\(948\) 1.28188e214 0.599303
\(949\) 4.00092e214 1.73377
\(950\) 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
\(961\) 1.18974e215 2.08625
\(962\) 0 0
\(963\) 0 0
\(964\) −2.60222e214 −0.364584
\(965\) 0 0
\(966\) 0 0
\(967\) −1.78500e215 −1.99955 −0.999777 0.0211147i \(-0.993278\pi\)
−0.999777 + 0.0211147i \(0.993278\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 1.29411e215 1.00000
\(973\) −2.91321e215 −2.09048
\(974\) 0 0
\(975\) −3.01056e215 −1.86344
\(976\) −3.40949e215 −1.96021
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.59662e215 0.635378
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 1.03549e216 2.46971
\(989\) 0 0
\(990\) 0 0
\(991\) −1.04546e215 −0.200449 −0.100225 0.994965i \(-0.531956\pi\)
−0.100225 + 0.994965i \(0.531956\pi\)
\(992\) 0 0
\(993\) −8.25997e215 −1.36972
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −9.91421e215 −1.23085 −0.615427 0.788194i \(-0.711016\pi\)
−0.615427 + 0.788194i \(0.711016\pi\)
\(998\) 0 0
\(999\) 1.40367e216 1.50852
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 3.145.b.a.2.1 1
3.2 odd 2 CM 3.145.b.a.2.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.145.b.a.2.1 1 1.1 even 1 trivial
3.145.b.a.2.1 1 3.2 odd 2 CM