Properties

Label 1200.3.l.l.401.1
Level $1200$
Weight $3$
Character 1200.401
Analytic conductor $32.698$
Analytic rank $0$
Dimension $2$
CM discriminant -15
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 401.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.401
Dual form 1200.3.l.l.401.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} -9.00000 q^{9} -14.0000i q^{17} -22.0000 q^{19} +34.0000i q^{23} +27.0000i q^{27} -2.00000 q^{31} +14.0000i q^{47} -49.0000 q^{49} -42.0000 q^{51} +86.0000i q^{53} +66.0000i q^{57} -118.000 q^{61} +102.000 q^{69} +98.0000 q^{79} +81.0000 q^{81} +154.000i q^{83} +6.00000i q^{93} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} - 44 q^{19} - 4 q^{31} - 98 q^{49} - 84 q^{51} - 236 q^{61} + 204 q^{69} + 196 q^{79} + 162 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) − 3.00000i − 1.00000i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −9.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 14.0000i − 0.823529i −0.911290 0.411765i \(-0.864913\pi\)
0.911290 0.411765i \(-0.135087\pi\)
\(18\) 0 0
\(19\) −22.0000 −1.15789 −0.578947 0.815365i \(-0.696536\pi\)
−0.578947 + 0.815365i \(0.696536\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 34.0000i 1.47826i 0.673562 + 0.739130i \(0.264763\pi\)
−0.673562 + 0.739130i \(0.735237\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000i 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.0645161 −0.0322581 0.999480i \(-0.510270\pi\)
−0.0322581 + 0.999480i \(0.510270\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 14.0000i 0.297872i 0.988847 + 0.148936i \(0.0475849\pi\)
−0.988847 + 0.148936i \(0.952415\pi\)
\(48\) 0 0
\(49\) −49.0000 −1.00000
\(50\) 0 0
\(51\) −42.0000 −0.823529
\(52\) 0 0
\(53\) 86.0000i 1.62264i 0.584601 + 0.811321i \(0.301251\pi\)
−0.584601 + 0.811321i \(0.698749\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 66.0000i 1.15789i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −118.000 −1.93443 −0.967213 0.253966i \(-0.918265\pi\)
−0.967213 + 0.253966i \(0.918265\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 102.000 1.47826
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 98.0000 1.24051 0.620253 0.784402i \(-0.287030\pi\)
0.620253 + 0.784402i \(0.287030\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 154.000i 1.85542i 0.373300 + 0.927711i \(0.378226\pi\)
−0.373300 + 0.927711i \(0.621774\pi\)
\(84\) 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
\(91\) 0 0
\(92\) 0 0
\(93\) 6.00000i 0.0645161i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 106.000i − 0.990654i −0.868707 0.495327i \(-0.835048\pi\)
0.868707 0.495327i \(-0.164952\pi\)
\(108\) 0 0
\(109\) 22.0000 0.201835 0.100917 0.994895i \(-0.467822\pi\)
0.100917 + 0.994895i \(0.467822\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 206.000i 1.82301i 0.411290 + 0.911504i \(0.365078\pi\)
−0.411290 + 0.911504i \(0.634922\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 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
\(135\) 0 0
\(136\) 0 0
\(137\) 226.000i 1.64964i 0.565399 + 0.824818i \(0.308722\pi\)
−0.565399 + 0.824818i \(0.691278\pi\)
\(138\) 0 0
\(139\) −262.000 −1.88489 −0.942446 0.334358i \(-0.891480\pi\)
−0.942446 + 0.334358i \(0.891480\pi\)
\(140\) 0 0
\(141\) 42.0000 0.297872
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 147.000i 1.00000i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 238.000 1.57616 0.788079 0.615574i \(-0.211076\pi\)
0.788079 + 0.615574i \(0.211076\pi\)
\(152\) 0 0
\(153\) 126.000i 0.823529i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 258.000 1.62264
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 254.000i 1.52096i 0.649362 + 0.760479i \(0.275036\pi\)
−0.649362 + 0.760479i \(0.724964\pi\)
\(168\) 0 0
\(169\) −169.000 −1.00000
\(170\) 0 0
\(171\) 198.000 1.15789
\(172\) 0 0
\(173\) − 154.000i − 0.890173i −0.895487 0.445087i \(-0.853173\pi\)
0.895487 0.445087i \(-0.146827\pi\)
\(174\) 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
\(181\) 122.000 0.674033 0.337017 0.941499i \(-0.390582\pi\)
0.337017 + 0.941499i \(0.390582\pi\)
\(182\) 0 0
\(183\) 354.000i 1.93443i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 374.000i − 1.89848i −0.314557 0.949239i \(-0.601856\pi\)
0.314557 0.949239i \(-0.398144\pi\)
\(198\) 0 0
\(199\) −142.000 −0.713568 −0.356784 0.934187i \(-0.616127\pi\)
−0.356784 + 0.934187i \(0.616127\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 306.000i − 1.47826i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −362.000 −1.71564 −0.857820 0.513950i \(-0.828182\pi\)
−0.857820 + 0.513950i \(0.828182\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 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
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 134.000i 0.590308i 0.955450 + 0.295154i \(0.0953710\pi\)
−0.955450 + 0.295154i \(0.904629\pi\)
\(228\) 0 0
\(229\) −218.000 −0.951965 −0.475983 0.879455i \(-0.657907\pi\)
−0.475983 + 0.879455i \(0.657907\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 34.0000i − 0.145923i −0.997335 0.0729614i \(-0.976755\pi\)
0.997335 0.0729614i \(-0.0232450\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 294.000i − 1.24051i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −478.000 −1.98340 −0.991701 0.128564i \(-0.958963\pi\)
−0.991701 + 0.128564i \(0.958963\pi\)
\(242\) 0 0
\(243\) − 243.000i − 1.00000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 462.000 1.85542
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 466.000i 1.81323i 0.421959 + 0.906615i \(0.361343\pi\)
−0.421959 + 0.906615i \(0.638657\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 446.000i − 1.69582i −0.530142 0.847909i \(-0.677861\pi\)
0.530142 0.847909i \(-0.322139\pi\)
\(264\) 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
\(271\) −482.000 −1.77860 −0.889299 0.457326i \(-0.848807\pi\)
−0.889299 + 0.457326i \(0.848807\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 18.0000 0.0645161
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 93.0000 0.321799
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 394.000i − 1.34471i −0.740229 0.672355i \(-0.765283\pi\)
0.740229 0.672355i \(-0.234717\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 134.000i − 0.422713i −0.977409 0.211356i \(-0.932212\pi\)
0.977409 0.211356i \(-0.0677881\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −318.000 −0.990654
\(322\) 0 0
\(323\) 308.000i 0.953560i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 66.0000i − 0.201835i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −122.000 −0.368580 −0.184290 0.982872i \(-0.558999\pi\)
−0.184290 + 0.982872i \(0.558999\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) 618.000 1.82301
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 586.000i − 1.68876i −0.535744 0.844380i \(-0.679969\pi\)
0.535744 0.844380i \(-0.320031\pi\)
\(348\) 0 0
\(349\) −458.000 −1.31232 −0.656160 0.754621i \(-0.727821\pi\)
−0.656160 + 0.754621i \(0.727821\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 274.000i − 0.776204i −0.921616 0.388102i \(-0.873131\pi\)
0.921616 0.388102i \(-0.126869\pi\)
\(354\) 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
\(361\) 123.000 0.340720
\(362\) 0 0
\(363\) − 363.000i − 1.00000i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −742.000 −1.95778 −0.978892 0.204379i \(-0.934482\pi\)
−0.978892 + 0.204379i \(0.934482\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 686.000i − 1.79112i −0.444938 0.895561i \(-0.646774\pi\)
0.444938 0.895561i \(-0.353226\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 476.000 1.21739
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 142.000 0.347188 0.173594 0.984817i \(-0.444462\pi\)
0.173594 + 0.984817i \(0.444462\pi\)
\(410\) 0 0
\(411\) 678.000 1.64964
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 786.000i 1.88489i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 602.000 1.42993 0.714964 0.699161i \(-0.246443\pi\)
0.714964 + 0.699161i \(0.246443\pi\)
\(422\) 0 0
\(423\) − 126.000i − 0.297872i
\(424\) 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
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 748.000i − 1.71167i
\(438\) 0 0
\(439\) −622.000 −1.41686 −0.708428 0.705783i \(-0.750595\pi\)
−0.708428 + 0.705783i \(0.750595\pi\)
\(440\) 0 0
\(441\) 441.000 1.00000
\(442\) 0 0
\(443\) − 566.000i − 1.27765i −0.769351 0.638826i \(-0.779420\pi\)
0.769351 0.638826i \(-0.220580\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) − 714.000i − 1.57616i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 378.000 0.823529
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 346.000i − 0.740899i −0.928853 0.370450i \(-0.879204\pi\)
0.928853 0.370450i \(-0.120796\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 774.000i − 1.62264i
\(478\) 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
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 938.000 1.87976 0.939880 0.341506i \(-0.110937\pi\)
0.939880 + 0.341506i \(0.110937\pi\)
\(500\) 0 0
\(501\) 762.000 1.52096
\(502\) 0 0
\(503\) 994.000i 1.97614i 0.153995 + 0.988072i \(0.450786\pi\)
−0.153995 + 0.988072i \(0.549214\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 507.000i 1.00000i
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) − 594.000i − 1.15789i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −462.000 −0.890173
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 28.0000i 0.0531309i
\(528\) 0 0
\(529\) −627.000 −1.18526
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1078.00 −1.99261 −0.996303 0.0859072i \(-0.972621\pi\)
−0.996303 + 0.0859072i \(0.972621\pi\)
\(542\) 0 0
\(543\) − 366.000i − 0.674033i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 1062.00 1.93443
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 614.000i − 1.10233i −0.834395 0.551167i \(-0.814183\pi\)
0.834395 0.551167i \(-0.185817\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 154.000i 0.273535i 0.990603 + 0.136767i \(0.0436713\pi\)
−0.990603 + 0.136767i \(0.956329\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 358.000 0.626970 0.313485 0.949593i \(-0.398503\pi\)
0.313485 + 0.949593i \(0.398503\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 854.000i 1.45486i 0.686184 + 0.727428i \(0.259284\pi\)
−0.686184 + 0.727428i \(0.740716\pi\)
\(588\) 0 0
\(589\) 44.0000 0.0747029
\(590\) 0 0
\(591\) −1122.00 −1.89848
\(592\) 0 0
\(593\) 1166.00i 1.96627i 0.182873 + 0.983137i \(0.441460\pi\)
−0.182873 + 0.983137i \(0.558540\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 426.000i 0.713568i
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 242.000 0.402662 0.201331 0.979523i \(-0.435473\pi\)
0.201331 + 0.979523i \(0.435473\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1186.00i 1.92220i 0.276193 + 0.961102i \(0.410927\pi\)
−0.276193 + 0.961102i \(0.589073\pi\)
\(618\) 0 0
\(619\) 698.000 1.12763 0.563813 0.825903i \(-0.309334\pi\)
0.563813 + 0.825903i \(0.309334\pi\)
\(620\) 0 0
\(621\) −918.000 −1.47826
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 238.000 0.377179 0.188590 0.982056i \(-0.439608\pi\)
0.188590 + 0.982056i \(0.439608\pi\)
\(632\) 0 0
\(633\) 1086.00i 1.71564i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 706.000i − 1.09119i −0.838049 0.545595i \(-0.816304\pi\)
0.838049 0.545595i \(-0.183696\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 1114.00i − 1.70597i −0.521933 0.852986i \(-0.674789\pi\)
0.521933 0.852986i \(-0.325211\pi\)
\(654\) 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
\(661\) −838.000 −1.26778 −0.633888 0.773425i \(-0.718542\pi\)
−0.633888 + 0.773425i \(0.718542\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 374.000i − 0.552437i −0.961095 0.276219i \(-0.910919\pi\)
0.961095 0.276219i \(-0.0890814\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 402.000 0.590308
\(682\) 0 0
\(683\) − 86.0000i − 0.125915i −0.998016 0.0629575i \(-0.979947\pi\)
0.998016 0.0629575i \(-0.0200533\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 654.000i 0.951965i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1322.00 −1.91317 −0.956585 0.291455i \(-0.905861\pi\)
−0.956585 + 0.291455i \(0.905861\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −102.000 −0.145923
\(700\) 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
\(707\) 0 0
\(708\) 0 0
\(709\) 742.000 1.04654 0.523272 0.852166i \(-0.324711\pi\)
0.523272 + 0.852166i \(0.324711\pi\)
\(710\) 0 0
\(711\) −882.000 −1.24051
\(712\) 0 0
\(713\) − 68.0000i − 0.0953717i
\(714\) 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
\(721\) 0 0
\(722\) 0 0
\(723\) 1434.00i 1.98340i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −729.000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1462.00 −1.97835 −0.989175 0.146744i \(-0.953121\pi\)
−0.989175 + 0.146744i \(0.953121\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 514.000i 0.691790i 0.938273 + 0.345895i \(0.112425\pi\)
−0.938273 + 0.345895i \(0.887575\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 1386.00i − 1.85542i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1438.00 1.91478 0.957390 0.288798i \(-0.0932555\pi\)
0.957390 + 0.288798i \(0.0932555\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 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
\(769\) −578.000 −0.751625 −0.375813 0.926696i \(-0.622636\pi\)
−0.375813 + 0.926696i \(0.622636\pi\)
\(770\) 0 0
\(771\) 1398.00 1.81323
\(772\) 0 0
\(773\) 1526.00i 1.97413i 0.160330 + 0.987063i \(0.448744\pi\)
−0.160330 + 0.987063i \(0.551256\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) −1338.00 −1.69582
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 826.000i 1.03639i 0.855264 + 0.518193i \(0.173395\pi\)
−0.855264 + 0.518193i \(0.826605\pi\)
\(798\) 0 0
\(799\) 196.000 0.245307
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 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
\(811\) −1082.00 −1.33416 −0.667078 0.744988i \(-0.732455\pi\)
−0.667078 + 0.744988i \(0.732455\pi\)
\(812\) 0 0
\(813\) 1446.00i 1.77860i
\(814\) 0 0
\(815\) 0 0
\(816\) 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
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 374.000i 0.452237i 0.974100 + 0.226119i \(0.0726037\pi\)
−0.974100 + 0.226119i \(0.927396\pi\)
\(828\) 0 0
\(829\) 502.000 0.605549 0.302774 0.953062i \(-0.402087\pi\)
0.302774 + 0.953062i \(0.402087\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 686.000i 0.823529i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 54.0000i − 0.0645161i
\(838\) 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
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1666.00i 1.94399i 0.235000 + 0.971995i \(0.424491\pi\)
−0.235000 + 0.971995i \(0.575509\pi\)
\(858\) 0 0
\(859\) 218.000 0.253783 0.126892 0.991917i \(-0.459500\pi\)
0.126892 + 0.991917i \(0.459500\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 274.000i 0.317497i 0.987319 + 0.158749i \(0.0507459\pi\)
−0.987319 + 0.158749i \(0.949254\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 279.000i − 0.321799i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) −1182.00 −1.34471
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1694.00i 1.90981i 0.296914 + 0.954904i \(0.404042\pi\)
−0.296914 + 0.954904i \(0.595958\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 308.000i − 0.344905i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 1204.00 1.33629
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 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
\(919\) 1298.00 1.41240 0.706202 0.708010i \(-0.250407\pi\)
0.706202 + 0.708010i \(0.250407\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1078.00 1.15789
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 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
\(947\) 1574.00i 1.66209i 0.556205 + 0.831045i \(0.312257\pi\)
−0.556205 + 0.831045i \(0.687743\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −402.000 −0.422713
\(952\) 0 0
\(953\) − 1474.00i − 1.54669i −0.633983 0.773347i \(-0.718581\pi\)
0.633983 0.773347i \(-0.281419\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −957.000 −0.995838
\(962\) 0 0
\(963\) 954.000i 0.990654i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 924.000 0.953560
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1934.00i − 1.97953i −0.142710 0.989765i \(-0.545582\pi\)
0.142710 0.989765i \(-0.454418\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −198.000 −0.201835
\(982\) 0 0
\(983\) 1954.00i 1.98779i 0.110319 + 0.993896i \(0.464813\pi\)
−0.110319 + 0.993896i \(0.535187\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 958.000 0.966700 0.483350 0.875427i \(-0.339420\pi\)
0.483350 + 0.875427i \(0.339420\pi\)
\(992\) 0 0
\(993\) 366.000i 0.368580i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 0 0
\(999\) 0 0
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 1200.3.l.l.401.1 2
3.2 odd 2 inner 1200.3.l.l.401.2 2
4.3 odd 2 75.3.c.d.26.2 2
5.2 odd 4 240.3.c.a.209.1 1
5.3 odd 4 240.3.c.b.209.1 1
5.4 even 2 inner 1200.3.l.l.401.2 2
12.11 even 2 75.3.c.d.26.1 2
15.2 even 4 240.3.c.b.209.1 1
15.8 even 4 240.3.c.a.209.1 1
15.14 odd 2 CM 1200.3.l.l.401.1 2
20.3 even 4 15.3.d.b.14.1 yes 1
20.7 even 4 15.3.d.a.14.1 1
20.19 odd 2 75.3.c.d.26.1 2
40.3 even 4 960.3.c.c.449.1 1
40.13 odd 4 960.3.c.a.449.1 1
40.27 even 4 960.3.c.b.449.1 1
40.37 odd 4 960.3.c.d.449.1 1
60.23 odd 4 15.3.d.a.14.1 1
60.47 odd 4 15.3.d.b.14.1 yes 1
60.59 even 2 75.3.c.d.26.2 2
120.53 even 4 960.3.c.d.449.1 1
120.77 even 4 960.3.c.a.449.1 1
120.83 odd 4 960.3.c.b.449.1 1
120.107 odd 4 960.3.c.c.449.1 1
180.7 even 12 405.3.h.b.134.1 2
180.23 odd 12 405.3.h.b.269.1 2
180.43 even 12 405.3.h.a.134.1 2
180.47 odd 12 405.3.h.a.134.1 2
180.67 even 12 405.3.h.b.269.1 2
180.83 odd 12 405.3.h.b.134.1 2
180.103 even 12 405.3.h.a.269.1 2
180.167 odd 12 405.3.h.a.269.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.3.d.a.14.1 1 20.7 even 4
15.3.d.a.14.1 1 60.23 odd 4
15.3.d.b.14.1 yes 1 20.3 even 4
15.3.d.b.14.1 yes 1 60.47 odd 4
75.3.c.d.26.1 2 12.11 even 2
75.3.c.d.26.1 2 20.19 odd 2
75.3.c.d.26.2 2 4.3 odd 2
75.3.c.d.26.2 2 60.59 even 2
240.3.c.a.209.1 1 5.2 odd 4
240.3.c.a.209.1 1 15.8 even 4
240.3.c.b.209.1 1 5.3 odd 4
240.3.c.b.209.1 1 15.2 even 4
405.3.h.a.134.1 2 180.43 even 12
405.3.h.a.134.1 2 180.47 odd 12
405.3.h.a.269.1 2 180.103 even 12
405.3.h.a.269.1 2 180.167 odd 12
405.3.h.b.134.1 2 180.7 even 12
405.3.h.b.134.1 2 180.83 odd 12
405.3.h.b.269.1 2 180.23 odd 12
405.3.h.b.269.1 2 180.67 even 12
960.3.c.a.449.1 1 40.13 odd 4
960.3.c.a.449.1 1 120.77 even 4
960.3.c.b.449.1 1 40.27 even 4
960.3.c.b.449.1 1 120.83 odd 4
960.3.c.c.449.1 1 40.3 even 4
960.3.c.c.449.1 1 120.107 odd 4
960.3.c.d.449.1 1 40.37 odd 4
960.3.c.d.449.1 1 120.53 even 4
1200.3.l.l.401.1 2 1.1 even 1 trivial
1200.3.l.l.401.1 2 15.14 odd 2 CM
1200.3.l.l.401.2 2 3.2 odd 2 inner
1200.3.l.l.401.2 2 5.4 even 2 inner