Properties

Label 75.3.c.a.26.1
Level $75$
Weight $3$
Character 75.26
Self dual yes
Analytic conductor $2.044$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(2.04360198270\)
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 26.1
Character \(\chi\) \(=\) 75.26

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} +4.00000 q^{4} +11.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +4.00000 q^{4} +11.0000 q^{7} +9.00000 q^{9} -12.0000 q^{12} -1.00000 q^{13} +16.0000 q^{16} -37.0000 q^{19} -33.0000 q^{21} -27.0000 q^{27} +44.0000 q^{28} -13.0000 q^{31} +36.0000 q^{36} +26.0000 q^{37} +3.00000 q^{39} -61.0000 q^{43} -48.0000 q^{48} +72.0000 q^{49} -4.00000 q^{52} +111.000 q^{57} +47.0000 q^{61} +99.0000 q^{63} +64.0000 q^{64} -109.000 q^{67} -46.0000 q^{73} -148.000 q^{76} -142.000 q^{79} +81.0000 q^{81} -132.000 q^{84} -11.0000 q^{91} +39.0000 q^{93} -169.000 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −3.00000 −1.00000
\(4\) 4.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 11.0000 1.57143 0.785714 0.618590i \(-0.212296\pi\)
0.785714 + 0.618590i \(0.212296\pi\)
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −12.0000 −1.00000
\(13\) −1.00000 −0.0769231 −0.0384615 0.999260i \(-0.512246\pi\)
−0.0384615 + 0.999260i \(0.512246\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −37.0000 −1.94737 −0.973684 0.227901i \(-0.926814\pi\)
−0.973684 + 0.227901i \(0.926814\pi\)
\(20\) 0 0
\(21\) −33.0000 −1.57143
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −1.00000
\(28\) 44.0000 1.57143
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −13.0000 −0.419355 −0.209677 0.977771i \(-0.567241\pi\)
−0.209677 + 0.977771i \(0.567241\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 36.0000 1.00000
\(37\) 26.0000 0.702703 0.351351 0.936244i \(-0.385722\pi\)
0.351351 + 0.936244i \(0.385722\pi\)
\(38\) 0 0
\(39\) 3.00000 0.0769231
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −61.0000 −1.41860 −0.709302 0.704904i \(-0.750990\pi\)
−0.709302 + 0.704904i \(0.750990\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −48.0000 −1.00000
\(49\) 72.0000 1.46939
\(50\) 0 0
\(51\) 0 0
\(52\) −4.00000 −0.0769231
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 111.000 1.94737
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 47.0000 0.770492 0.385246 0.922814i \(-0.374117\pi\)
0.385246 + 0.922814i \(0.374117\pi\)
\(62\) 0 0
\(63\) 99.0000 1.57143
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −109.000 −1.62687 −0.813433 0.581659i \(-0.802404\pi\)
−0.813433 + 0.581659i \(0.802404\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −46.0000 −0.630137 −0.315068 0.949069i \(-0.602027\pi\)
−0.315068 + 0.949069i \(0.602027\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −148.000 −1.94737
\(77\) 0 0
\(78\) 0 0
\(79\) −142.000 −1.79747 −0.898734 0.438494i \(-0.855512\pi\)
−0.898734 + 0.438494i \(0.855512\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −132.000 −1.57143
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −11.0000 −0.120879
\(92\) 0 0
\(93\) 39.0000 0.419355
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −169.000 −1.74227 −0.871134 0.491045i \(-0.836615\pi\)
−0.871134 + 0.491045i \(0.836615\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 194.000 1.88350 0.941748 0.336321i \(-0.109183\pi\)
0.941748 + 0.336321i \(0.109183\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −108.000 −1.00000
\(109\) 143.000 1.31193 0.655963 0.754793i \(-0.272263\pi\)
0.655963 + 0.754793i \(0.272263\pi\)
\(110\) 0 0
\(111\) −78.0000 −0.702703
\(112\) 176.000 1.57143
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −9.00000 −0.0769231
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −52.0000 −0.419355
\(125\) 0 0
\(126\) 0 0
\(127\) 146.000 1.14961 0.574803 0.818292i \(-0.305079\pi\)
0.574803 + 0.818292i \(0.305079\pi\)
\(128\) 0 0
\(129\) 183.000 1.41860
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −407.000 −3.06015
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −22.0000 −0.158273 −0.0791367 0.996864i \(-0.525216\pi\)
−0.0791367 + 0.996864i \(0.525216\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 144.000 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) −216.000 −1.46939
\(148\) 104.000 0.702703
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 227.000 1.50331 0.751656 0.659556i \(-0.229256\pi\)
0.751656 + 0.659556i \(0.229256\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 12.0000 0.0769231
\(157\) 311.000 1.98089 0.990446 0.137902i \(-0.0440359\pi\)
0.990446 + 0.137902i \(0.0440359\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 299.000 1.83436 0.917178 0.398478i \(-0.130461\pi\)
0.917178 + 0.398478i \(0.130461\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −168.000 −0.994083
\(170\) 0 0
\(171\) −333.000 −1.94737
\(172\) −244.000 −1.41860
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −313.000 −1.72928 −0.864641 0.502390i \(-0.832454\pi\)
−0.864641 + 0.502390i \(0.832454\pi\)
\(182\) 0 0
\(183\) −141.000 −0.770492
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −297.000 −1.57143
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −192.000 −1.00000
\(193\) 239.000 1.23834 0.619171 0.785256i \(-0.287469\pi\)
0.619171 + 0.785256i \(0.287469\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 288.000 1.46939
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −277.000 −1.39196 −0.695980 0.718061i \(-0.745030\pi\)
−0.695980 + 0.718061i \(0.745030\pi\)
\(200\) 0 0
\(201\) 327.000 1.62687
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −16.0000 −0.0769231
\(209\) 0 0
\(210\) 0 0
\(211\) −253.000 −1.19905 −0.599526 0.800355i \(-0.704644\pi\)
−0.599526 + 0.800355i \(0.704644\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −143.000 −0.658986
\(218\) 0 0
\(219\) 138.000 0.630137
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −421.000 −1.88789 −0.943946 0.330099i \(-0.892918\pi\)
−0.943946 + 0.330099i \(0.892918\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 444.000 1.94737
\(229\) 383.000 1.67249 0.836245 0.548357i \(-0.184746\pi\)
0.836245 + 0.548357i \(0.184746\pi\)
\(230\) 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
\(237\) 426.000 1.79747
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −193.000 −0.800830 −0.400415 0.916334i \(-0.631134\pi\)
−0.400415 + 0.916334i \(0.631134\pi\)
\(242\) 0 0
\(243\) −243.000 −1.00000
\(244\) 188.000 0.770492
\(245\) 0 0
\(246\) 0 0
\(247\) 37.0000 0.149798
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 396.000 1.57143
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 286.000 1.10425
\(260\) 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
\(268\) −436.000 −1.62687
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 242.000 0.892989 0.446494 0.894786i \(-0.352672\pi\)
0.446494 + 0.894786i \(0.352672\pi\)
\(272\) 0 0
\(273\) 33.0000 0.120879
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −529.000 −1.90975 −0.954874 0.297012i \(-0.904010\pi\)
−0.954874 + 0.297012i \(0.904010\pi\)
\(278\) 0 0
\(279\) −117.000 −0.419355
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 59.0000 0.208481 0.104240 0.994552i \(-0.466759\pi\)
0.104240 + 0.994552i \(0.466759\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 507.000 1.74227
\(292\) −184.000 −0.630137
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −671.000 −2.22924
\(302\) 0 0
\(303\) 0 0
\(304\) −592.000 −1.94737
\(305\) 0 0
\(306\) 0 0
\(307\) 611.000 1.99023 0.995114 0.0987325i \(-0.0314788\pi\)
0.995114 + 0.0987325i \(0.0314788\pi\)
\(308\) 0 0
\(309\) −582.000 −1.88350
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 599.000 1.91374 0.956869 0.290520i \(-0.0938282\pi\)
0.956869 + 0.290520i \(0.0938282\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −568.000 −1.79747
\(317\) 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
\(324\) 324.000 1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) −429.000 −1.31193
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 362.000 1.09366 0.546828 0.837245i \(-0.315835\pi\)
0.546828 + 0.837245i \(0.315835\pi\)
\(332\) 0 0
\(333\) 234.000 0.702703
\(334\) 0 0
\(335\) 0 0
\(336\) −528.000 −1.57143
\(337\) −649.000 −1.92582 −0.962908 0.269830i \(-0.913033\pi\)
−0.962908 + 0.269830i \(0.913033\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 253.000 0.737609
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −502.000 −1.43840 −0.719198 0.694805i \(-0.755490\pi\)
−0.719198 + 0.694805i \(0.755490\pi\)
\(350\) 0 0
\(351\) 27.0000 0.0769231
\(352\) 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
\(361\) 1008.00 2.79224
\(362\) 0 0
\(363\) −363.000 −1.00000
\(364\) −44.0000 −0.120879
\(365\) 0 0
\(366\) 0 0
\(367\) 491.000 1.33787 0.668937 0.743319i \(-0.266749\pi\)
0.668937 + 0.743319i \(0.266749\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 156.000 0.419355
\(373\) −121.000 −0.324397 −0.162198 0.986758i \(-0.551858\pi\)
−0.162198 + 0.986758i \(0.551858\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 83.0000 0.218997 0.109499 0.993987i \(-0.465075\pi\)
0.109499 + 0.993987i \(0.465075\pi\)
\(380\) 0 0
\(381\) −438.000 −1.14961
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −549.000 −1.41860
\(388\) −676.000 −1.74227
\(389\) 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
\(397\) 431.000 1.08564 0.542821 0.839848i \(-0.317356\pi\)
0.542821 + 0.839848i \(0.317356\pi\)
\(398\) 0 0
\(399\) 1221.00 3.06015
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 13.0000 0.0322581
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 143.000 0.349633 0.174817 0.984601i \(-0.444067\pi\)
0.174817 + 0.984601i \(0.444067\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 776.000 1.88350
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 66.0000 0.158273
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −358.000 −0.850356 −0.425178 0.905110i \(-0.639789\pi\)
−0.425178 + 0.905110i \(0.639789\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 517.000 1.21077
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −432.000 −1.00000
\(433\) 359.000 0.829099 0.414550 0.910027i \(-0.363939\pi\)
0.414550 + 0.910027i \(0.363939\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 572.000 1.31193
\(437\) 0 0
\(438\) 0 0
\(439\) 803.000 1.82916 0.914579 0.404408i \(-0.132522\pi\)
0.914579 + 0.404408i \(0.132522\pi\)
\(440\) 0 0
\(441\) 648.000 1.46939
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −312.000 −0.702703
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 704.000 1.57143
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −681.000 −1.50331
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −814.000 −1.78118 −0.890591 0.454805i \(-0.849709\pi\)
−0.890591 + 0.454805i \(0.849709\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −526.000 −1.13607 −0.568035 0.823005i \(-0.692296\pi\)
−0.568035 + 0.823005i \(0.692296\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −36.0000 −0.0769231
\(469\) −1199.00 −2.55650
\(470\) 0 0
\(471\) −933.000 −1.98089
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −26.0000 −0.0540541
\(482\) 0 0
\(483\) 0 0
\(484\) 484.000 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −349.000 −0.716632 −0.358316 0.933600i \(-0.616649\pi\)
−0.358316 + 0.933600i \(0.616649\pi\)
\(488\) 0 0
\(489\) −897.000 −1.83436
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −208.000 −0.419355
\(497\) 0 0
\(498\) 0 0
\(499\) −877.000 −1.75752 −0.878758 0.477269i \(-0.841627\pi\)
−0.878758 + 0.477269i \(0.841627\pi\)
\(500\) 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
\(507\) 504.000 0.994083
\(508\) 584.000 1.14961
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −506.000 −0.990215
\(512\) 0 0
\(513\) 999.000 1.94737
\(514\) 0 0
\(515\) 0 0
\(516\) 732.000 1.41860
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 179.000 0.342256 0.171128 0.985249i \(-0.445259\pi\)
0.171128 + 0.985249i \(0.445259\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −1628.00 −3.06015
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −793.000 −1.46580 −0.732902 0.680334i \(-0.761835\pi\)
−0.732902 + 0.680334i \(0.761835\pi\)
\(542\) 0 0
\(543\) 939.000 1.72928
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 506.000 0.925046 0.462523 0.886607i \(-0.346944\pi\)
0.462523 + 0.886607i \(0.346944\pi\)
\(548\) 0 0
\(549\) 423.000 0.770492
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1562.00 −2.82459
\(554\) 0 0
\(555\) 0 0
\(556\) −88.0000 −0.158273
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 61.0000 0.109123
\(560\) 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
\(567\) 891.000 1.57143
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 1067.00 1.86865 0.934326 0.356420i \(-0.116003\pi\)
0.934326 + 0.356420i \(0.116003\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 576.000 1.00000
\(577\) 71.0000 0.123050 0.0615251 0.998106i \(-0.480404\pi\)
0.0615251 + 0.998106i \(0.480404\pi\)
\(578\) 0 0
\(579\) −717.000 −1.23834
\(580\) 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 \(\pi\)
\(588\) −864.000 −1.46939
\(589\) 481.000 0.816638
\(590\) 0 0
\(591\) 0 0
\(592\) 416.000 0.702703
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 831.000 1.39196
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −673.000 −1.11980 −0.559900 0.828560i \(-0.689161\pi\)
−0.559900 + 0.828560i \(0.689161\pi\)
\(602\) 0 0
\(603\) −981.000 −1.62687
\(604\) 908.000 1.50331
\(605\) 0 0
\(606\) 0 0
\(607\) −814.000 −1.34102 −0.670511 0.741900i \(-0.733925\pi\)
−0.670511 + 0.741900i \(0.733925\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1126.00 −1.83687 −0.918434 0.395574i \(-0.870546\pi\)
−0.918434 + 0.395574i \(0.870546\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 1163.00 1.87884 0.939418 0.342773i \(-0.111366\pi\)
0.939418 + 0.342773i \(0.111366\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 48.0000 0.0769231
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 1244.00 1.98089
\(629\) 0 0
\(630\) 0 0
\(631\) 587.000 0.930269 0.465135 0.885240i \(-0.346006\pi\)
0.465135 + 0.885240i \(0.346006\pi\)
\(632\) 0 0
\(633\) 759.000 1.19905
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −72.0000 −0.113030
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 314.000 0.488336 0.244168 0.969733i \(-0.421485\pi\)
0.244168 + 0.969733i \(0.421485\pi\)
\(644\) 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
\(651\) 429.000 0.658986
\(652\) 1196.00 1.83436
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −414.000 −0.630137
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 122.000 0.184569 0.0922844 0.995733i \(-0.470583\pi\)
0.0922844 + 0.995733i \(0.470583\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 1263.00 1.88789
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1154.00 1.71471 0.857355 0.514725i \(-0.172106\pi\)
0.857355 + 0.514725i \(0.172106\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −672.000 −0.994083
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −1859.00 −2.73785
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −1332.00 −1.94737
\(685\) 0 0
\(686\) 0 0
\(687\) −1149.00 −1.67249
\(688\) −976.000 −1.41860
\(689\) 0 0
\(690\) 0 0
\(691\) −1318.00 −1.90738 −0.953690 0.300790i \(-0.902750\pi\)
−0.953690 + 0.300790i \(0.902750\pi\)
\(692\) 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
\(703\) −962.000 −1.36842
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −457.000 −0.644570 −0.322285 0.946643i \(-0.604451\pi\)
−0.322285 + 0.946643i \(0.604451\pi\)
\(710\) 0 0
\(711\) −1278.00 −1.79747
\(712\) 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
\(721\) 2134.00 2.95978
\(722\) 0 0
\(723\) 579.000 0.800830
\(724\) −1252.00 −1.72928
\(725\) 0 0
\(726\) 0 0
\(727\) −1429.00 −1.96561 −0.982806 0.184641i \(-0.940888\pi\)
−0.982806 + 0.184641i \(0.940888\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −564.000 −0.770492
\(733\) 1034.00 1.41064 0.705321 0.708888i \(-0.250803\pi\)
0.705321 + 0.708888i \(0.250803\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1222.00 −1.65359 −0.826793 0.562506i \(-0.809837\pi\)
−0.826793 + 0.562506i \(0.809837\pi\)
\(740\) 0 0
\(741\) −111.000 −0.149798
\(742\) 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
\(751\) 1202.00 1.60053 0.800266 0.599645i \(-0.204691\pi\)
0.800266 + 0.599645i \(0.204691\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −1188.00 −1.57143
\(757\) 1511.00 1.99604 0.998018 0.0629213i \(-0.0200417\pi\)
0.998018 + 0.0629213i \(0.0200417\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1573.00 2.06160
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −768.000 −1.00000
\(769\) 863.000 1.12224 0.561118 0.827736i \(-0.310371\pi\)
0.561118 + 0.827736i \(0.310371\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 956.000 1.23834
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −858.000 −1.10425
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1152.00 1.46939
\(785\) 0 0
\(786\) 0 0
\(787\) −949.000 −1.20584 −0.602922 0.797800i \(-0.705997\pi\)
−0.602922 + 0.797800i \(0.705997\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −47.0000 −0.0592686
\(794\) 0 0
\(795\) 0 0
\(796\) −1108.00 −1.39196
\(797\) 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
\(804\) 1308.00 1.62687
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −253.000 −0.311961 −0.155980 0.987760i \(-0.549854\pi\)
−0.155980 + 0.987760i \(0.549854\pi\)
\(812\) 0 0
\(813\) −726.000 −0.892989
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2257.00 2.76255
\(818\) 0 0
\(819\) −99.0000 −0.120879
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −1621.00 −1.96962 −0.984812 0.173626i \(-0.944452\pi\)
−0.984812 + 0.173626i \(0.944452\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 458.000 0.552473 0.276236 0.961090i \(-0.410913\pi\)
0.276236 + 0.961090i \(0.410913\pi\)
\(830\) 0 0
\(831\) 1587.00 1.90975
\(832\) −64.0000 −0.0769231
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 351.000 0.419355
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −1012.00 −1.19905
\(845\) 0 0
\(846\) 0 0
\(847\) 1331.00 1.57143
\(848\) 0 0
\(849\) −177.000 −0.208481
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −481.000 −0.563892 −0.281946 0.959430i \(-0.590980\pi\)
−0.281946 + 0.959430i \(0.590980\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 1418.00 1.65076 0.825378 0.564580i \(-0.190962\pi\)
0.825378 + 0.564580i \(0.190962\pi\)
\(860\) 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
\(867\) −867.000 −1.00000
\(868\) −572.000 −0.658986
\(869\) 0 0
\(870\) 0 0
\(871\) 109.000 0.125144
\(872\) 0 0
\(873\) −1521.00 −1.74227
\(874\) 0 0
\(875\) 0 0
\(876\) 552.000 0.630137
\(877\) −1129.00 −1.28734 −0.643672 0.765302i \(-0.722590\pi\)
−0.643672 + 0.765302i \(0.722590\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1259.00 1.42582 0.712911 0.701255i \(-0.247377\pi\)
0.712911 + 0.701255i \(0.247377\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1606.00 1.80652
\(890\) 0 0
\(891\) 0 0
\(892\) −1684.00 −1.88789
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 2013.00 2.22924
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −214.000 −0.235943 −0.117971 0.993017i \(-0.537639\pi\)
−0.117971 + 0.993017i \(0.537639\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 1776.00 1.94737
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1532.00 1.67249
\(917\) 0 0
\(918\) 0 0
\(919\) −1837.00 −1.99891 −0.999456 0.0329825i \(-0.989499\pi\)
−0.999456 + 0.0329825i \(0.989499\pi\)
\(920\) 0 0
\(921\) −1833.00 −1.99023
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1746.00 1.88350
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −2664.00 −2.86144
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −649.000 −0.692636 −0.346318 0.938117i \(-0.612568\pi\)
−0.346318 + 0.938117i \(0.612568\pi\)
\(938\) 0 0
\(939\) −1797.00 −1.91374
\(940\) 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 \(\pi\)
\(948\) 1704.00 1.79747
\(949\) 46.0000 0.0484721
\(950\) 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
\(961\) −792.000 −0.824142
\(962\) 0 0
\(963\) 0 0
\(964\) −772.000 −0.800830
\(965\) 0 0
\(966\) 0 0
\(967\) −1534.00 −1.58635 −0.793175 0.608994i \(-0.791573\pi\)
−0.793175 + 0.608994i \(0.791573\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −972.000 −1.00000
\(973\) −242.000 −0.248715
\(974\) 0 0
\(975\) 0 0
\(976\) 752.000 0.770492
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1287.00 1.31193
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 148.000 0.149798
\(989\) 0 0
\(990\) 0 0
\(991\) −1693.00 −1.70838 −0.854188 0.519965i \(-0.825945\pi\)
−0.854188 + 0.519965i \(0.825945\pi\)
\(992\) 0 0
\(993\) −1086.00 −1.09366
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1894.00 −1.89970 −0.949850 0.312707i \(-0.898764\pi\)
−0.949850 + 0.312707i \(0.898764\pi\)
\(998\) 0 0
\(999\) −702.000 −0.702703
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 75.3.c.a.26.1 1
3.2 odd 2 CM 75.3.c.a.26.1 1
4.3 odd 2 1200.3.l.d.401.1 1
5.2 odd 4 75.3.d.a.74.2 2
5.3 odd 4 75.3.d.a.74.1 2
5.4 even 2 75.3.c.b.26.1 yes 1
12.11 even 2 1200.3.l.d.401.1 1
15.2 even 4 75.3.d.a.74.2 2
15.8 even 4 75.3.d.a.74.1 2
15.14 odd 2 75.3.c.b.26.1 yes 1
20.3 even 4 1200.3.c.a.449.2 2
20.7 even 4 1200.3.c.a.449.1 2
20.19 odd 2 1200.3.l.c.401.1 1
60.23 odd 4 1200.3.c.a.449.2 2
60.47 odd 4 1200.3.c.a.449.1 2
60.59 even 2 1200.3.l.c.401.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.3.c.a.26.1 1 1.1 even 1 trivial
75.3.c.a.26.1 1 3.2 odd 2 CM
75.3.c.b.26.1 yes 1 5.4 even 2
75.3.c.b.26.1 yes 1 15.14 odd 2
75.3.d.a.74.1 2 5.3 odd 4
75.3.d.a.74.1 2 15.8 even 4
75.3.d.a.74.2 2 5.2 odd 4
75.3.d.a.74.2 2 15.2 even 4
1200.3.c.a.449.1 2 20.7 even 4
1200.3.c.a.449.1 2 60.47 odd 4
1200.3.c.a.449.2 2 20.3 even 4
1200.3.c.a.449.2 2 60.23 odd 4
1200.3.l.c.401.1 1 20.19 odd 2
1200.3.l.c.401.1 1 60.59 even 2
1200.3.l.d.401.1 1 4.3 odd 2
1200.3.l.d.401.1 1 12.11 even 2