Properties

Label 585.2.h.e.64.1
Level $585$
Weight $2$
Character 585.64
Analytic conductor $4.671$
Analytic rank $0$
Dimension $4$
CM discriminant -195
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-5}, \sqrt{13})\)
Defining polynomial: \( x^{4} - 2x^{3} + 5x^{2} - 4x + 69 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 64.1
Root \(-1.30278 + 2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 585.64
Dual form 585.2.h.e.64.3

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{4} -2.23607i q^{5} -3.60555 q^{7} +O(q^{10})\) \(q-2.00000 q^{4} -2.23607i q^{5} -3.60555 q^{7} +2.23607i q^{11} +3.60555 q^{13} +4.00000 q^{16} +8.06226i q^{17} +4.47214i q^{20} +8.06226i q^{23} -5.00000 q^{25} +7.21110 q^{28} +8.06226i q^{35} -3.60555 q^{37} -11.1803i q^{41} -4.47214i q^{44} +6.00000 q^{49} -7.21110 q^{52} +8.06226i q^{53} +5.00000 q^{55} +8.94427i q^{59} -7.00000 q^{61} -8.00000 q^{64} -8.06226i q^{65} -14.4222 q^{67} -16.1245i q^{68} +15.6525i q^{71} +7.21110 q^{73} -8.06226i q^{77} +11.0000 q^{79} -8.94427i q^{80} +18.0278 q^{85} +2.23607i q^{89} -13.0000 q^{91} -16.1245i q^{92} -3.60555 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 24 q^{49} + 20 q^{55} - 28 q^{61} - 32 q^{64} + 44 q^{79} - 52 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) − 2.23607i − 1.00000i
\(6\) 0 0
\(7\) −3.60555 −1.36277 −0.681385 0.731925i \(-0.738622\pi\)
−0.681385 + 0.731925i \(0.738622\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.23607i 0.674200i 0.941469 + 0.337100i \(0.109446\pi\)
−0.941469 + 0.337100i \(0.890554\pi\)
\(12\) 0 0
\(13\) 3.60555 1.00000
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 8.06226i 1.95538i 0.210042 + 0.977692i \(0.432640\pi\)
−0.210042 + 0.977692i \(0.567360\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 4.47214i 1.00000i
\(21\) 0 0
\(22\) 0 0
\(23\) 8.06226i 1.68110i 0.541736 + 0.840548i \(0.317767\pi\)
−0.541736 + 0.840548i \(0.682233\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 7.21110 1.36277
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8.06226i 1.36277i
\(36\) 0 0
\(37\) −3.60555 −0.592749 −0.296374 0.955072i \(-0.595778\pi\)
−0.296374 + 0.955072i \(0.595778\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 11.1803i − 1.74608i −0.487652 0.873038i \(-0.662147\pi\)
0.487652 0.873038i \(-0.337853\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) − 4.47214i − 0.674200i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) −7.21110 −1.00000
\(53\) 8.06226i 1.10744i 0.832704 + 0.553718i \(0.186791\pi\)
−0.832704 + 0.553718i \(0.813209\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.94427i 1.16445i 0.813029 + 0.582223i \(0.197817\pi\)
−0.813029 + 0.582223i \(0.802183\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) − 8.06226i − 1.00000i
\(66\) 0 0
\(67\) −14.4222 −1.76195 −0.880976 0.473160i \(-0.843113\pi\)
−0.880976 + 0.473160i \(0.843113\pi\)
\(68\) − 16.1245i − 1.95538i
\(69\) 0 0
\(70\) 0 0
\(71\) 15.6525i 1.85761i 0.370572 + 0.928804i \(0.379162\pi\)
−0.370572 + 0.928804i \(0.620838\pi\)
\(72\) 0 0
\(73\) 7.21110 0.843996 0.421998 0.906597i \(-0.361329\pi\)
0.421998 + 0.906597i \(0.361329\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 8.06226i − 0.918780i
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) − 8.94427i − 1.00000i
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 18.0278 1.95538
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.23607i 0.237023i 0.992953 + 0.118511i \(0.0378122\pi\)
−0.992953 + 0.118511i \(0.962188\pi\)
\(90\) 0 0
\(91\) −13.0000 −1.36277
\(92\) − 16.1245i − 1.68110i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.60555 −0.366088 −0.183044 0.983105i \(-0.558595\pi\)
−0.183044 + 0.983105i \(0.558595\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 10.0000 1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.06226i 0.779408i 0.920940 + 0.389704i \(0.127423\pi\)
−0.920940 + 0.389704i \(0.872577\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −14.4222 −1.36277
\(113\) − 16.1245i − 1.51687i −0.651751 0.758433i \(-0.725965\pi\)
0.651751 0.758433i \(-0.274035\pi\)
\(114\) 0 0
\(115\) 18.0278 1.68110
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 29.0689i − 2.66474i
\(120\) 0 0
\(121\) 6.00000 0.545455
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803i 1.00000i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −19.0000 −1.61156 −0.805779 0.592216i \(-0.798253\pi\)
−0.805779 + 0.592216i \(0.798253\pi\)
\(140\) − 16.1245i − 1.36277i
\(141\) 0 0
\(142\) 0 0
\(143\) 8.06226i 0.674200i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 7.21110 0.592749
\(149\) 15.6525i 1.28230i 0.767415 + 0.641150i \(0.221543\pi\)
−0.767415 + 0.641150i \(0.778457\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 29.0689i − 2.29095i
\(162\) 0 0
\(163\) −25.2389 −1.97686 −0.988430 0.151678i \(-0.951532\pi\)
−0.988430 + 0.151678i \(0.951532\pi\)
\(164\) 22.3607i 1.74608i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 16.1245i − 1.22592i −0.790112 0.612962i \(-0.789978\pi\)
0.790112 0.612962i \(-0.210022\pi\)
\(174\) 0 0
\(175\) 18.0278 1.36277
\(176\) 8.94427i 0.674200i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 23.0000 1.70958 0.854788 0.518977i \(-0.173687\pi\)
0.854788 + 0.518977i \(0.173687\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.06226i 0.592749i
\(186\) 0 0
\(187\) −18.0278 −1.31832
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −25.2389 −1.81673 −0.908366 0.418175i \(-0.862670\pi\)
−0.908366 + 0.418175i \(0.862670\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −12.0000 −0.857143
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −25.0000 −1.74608
\(206\) 0 0
\(207\) 0 0
\(208\) 14.4222 1.00000
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) − 16.1245i − 1.10744i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −10.0000 −0.674200
\(221\) 29.0689i 1.95538i
\(222\) 0 0
\(223\) 28.8444 1.93156 0.965782 0.259354i \(-0.0835097\pi\)
0.965782 + 0.259354i \(0.0835097\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.06226i 0.528176i 0.964499 + 0.264088i \(0.0850709\pi\)
−0.964499 + 0.264088i \(0.914929\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) − 17.8885i − 1.16445i
\(237\) 0 0
\(238\) 0 0
\(239\) − 24.5967i − 1.59103i −0.605933 0.795516i \(-0.707200\pi\)
0.605933 0.795516i \(-0.292800\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 14.0000 0.896258
\(245\) − 13.4164i − 0.857143i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −18.0278 −1.13340
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) − 16.1245i − 1.00582i −0.864339 0.502910i \(-0.832263\pi\)
0.864339 0.502910i \(-0.167737\pi\)
\(258\) 0 0
\(259\) 13.0000 0.807781
\(260\) 16.1245i 1.00000i
\(261\) 0 0
\(262\) 0 0
\(263\) 32.2490i 1.98856i 0.106803 + 0.994280i \(0.465939\pi\)
−0.106803 + 0.994280i \(0.534061\pi\)
\(264\) 0 0
\(265\) 18.0278 1.10744
\(266\) 0 0
\(267\) 0 0
\(268\) 28.8444 1.76195
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 32.2490i 1.95538i
\(273\) 0 0
\(274\) 0 0
\(275\) − 11.1803i − 0.674200i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.3607i 1.33393i 0.745091 + 0.666963i \(0.232406\pi\)
−0.745091 + 0.666963i \(0.767594\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) − 31.3050i − 1.85761i
\(285\) 0 0
\(286\) 0 0
\(287\) 40.3113i 2.37950i
\(288\) 0 0
\(289\) −48.0000 −2.82353
\(290\) 0 0
\(291\) 0 0
\(292\) −14.4222 −0.843996
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 20.0000 1.16445
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 29.0689i 1.68110i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 15.6525i 0.896258i
\(306\) 0 0
\(307\) −3.60555 −0.205780 −0.102890 0.994693i \(-0.532809\pi\)
−0.102890 + 0.994693i \(0.532809\pi\)
\(308\) 16.1245i 0.918780i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −22.0000 −1.23760
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 17.8885i 1.00000i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −18.0278 −1.00000
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 32.2490i 1.76195i
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −36.0555 −1.95538
\(341\) 0 0
\(342\) 0 0
\(343\) 3.60555 0.194681
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.06226i 0.432805i 0.976304 + 0.216402i \(0.0694323\pi\)
−0.976304 + 0.216402i \(0.930568\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 35.0000 1.85761
\(356\) − 4.47214i − 0.237023i
\(357\) 0 0
\(358\) 0 0
\(359\) 35.7771i 1.88824i 0.329598 + 0.944121i \(0.393087\pi\)
−0.329598 + 0.944121i \(0.606913\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 26.0000 1.36277
\(365\) − 16.1245i − 0.843996i
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 32.2490i 1.68110i
\(369\) 0 0
\(370\) 0 0
\(371\) − 29.0689i − 1.50918i
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −18.0278 −0.918780
\(386\) 0 0
\(387\) 0 0
\(388\) 7.21110 0.366088
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −65.0000 −3.28719
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 24.5967i − 1.23760i
\(396\) 0 0
\(397\) 39.6611 1.99053 0.995266 0.0971897i \(-0.0309854\pi\)
0.995266 + 0.0971897i \(0.0309854\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −20.0000 −1.00000
\(401\) − 31.3050i − 1.56329i −0.623721 0.781647i \(-0.714380\pi\)
0.623721 0.781647i \(-0.285620\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 8.06226i − 0.399631i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 32.2490i − 1.58687i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 40.3113i − 1.95538i
\(426\) 0 0
\(427\) 25.2389 1.22139
\(428\) − 16.1245i − 0.779408i
\(429\) 0 0
\(430\) 0 0
\(431\) − 17.8885i − 0.861661i −0.902433 0.430830i \(-0.858221\pi\)
0.902433 0.430830i \(-0.141779\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −1.00000 −0.0477274 −0.0238637 0.999715i \(-0.507597\pi\)
−0.0238637 + 0.999715i \(0.507597\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 40.3113i − 1.91525i −0.288023 0.957624i \(-0.592998\pi\)
0.288023 0.957624i \(-0.407002\pi\)
\(444\) 0 0
\(445\) 5.00000 0.237023
\(446\) 0 0
\(447\) 0 0
\(448\) 28.8444 1.36277
\(449\) − 38.0132i − 1.79395i −0.442080 0.896976i \(-0.645759\pi\)
0.442080 0.896976i \(-0.354241\pi\)
\(450\) 0 0
\(451\) 25.0000 1.17720
\(452\) 32.2490i 1.51687i
\(453\) 0 0
\(454\) 0 0
\(455\) 29.0689i 1.36277i
\(456\) 0 0
\(457\) −3.60555 −0.168661 −0.0843303 0.996438i \(-0.526875\pi\)
−0.0843303 + 0.996438i \(0.526875\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −36.0555 −1.68110
\(461\) 42.4853i 1.97874i 0.145429 + 0.989369i \(0.453544\pi\)
−0.145429 + 0.989369i \(0.546456\pi\)
\(462\) 0 0
\(463\) −25.2389 −1.17295 −0.586475 0.809968i \(-0.699485\pi\)
−0.586475 + 0.809968i \(0.699485\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 40.3113i − 1.86538i −0.360674 0.932692i \(-0.617453\pi\)
0.360674 0.932692i \(-0.382547\pi\)
\(468\) 0 0
\(469\) 52.0000 2.40114
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 58.1378i 2.66474i
\(477\) 0 0
\(478\) 0 0
\(479\) 2.23607i 0.102169i 0.998694 + 0.0510843i \(0.0162677\pi\)
−0.998694 + 0.0510843i \(0.983732\pi\)
\(480\) 0 0
\(481\) −13.0000 −0.592749
\(482\) 0 0
\(483\) 0 0
\(484\) −12.0000 −0.545455
\(485\) 8.06226i 0.366088i
\(486\) 0 0
\(487\) 39.6611 1.79721 0.898607 0.438754i \(-0.144580\pi\)
0.898607 + 0.438754i \(0.144580\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 56.4358i − 2.53149i
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) − 22.3607i − 1.00000i
\(501\) 0 0
\(502\) 0 0
\(503\) 32.2490i 1.43791i 0.695055 + 0.718957i \(0.255380\pi\)
−0.695055 + 0.718957i \(0.744620\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 11.1803i − 0.495560i −0.968816 0.247780i \(-0.920299\pi\)
0.968816 0.247780i \(-0.0797010\pi\)
\(510\) 0 0
\(511\) −26.0000 −1.15017
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 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
\(529\) −42.0000 −1.82609
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 40.3113i − 1.74608i
\(534\) 0 0
\(535\) 18.0278 0.779408
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 13.4164i 0.577886i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −39.6611 −1.68656
\(554\) 0 0
\(555\) 0 0
\(556\) 38.0000 1.61156
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 32.2490i 1.36277i
\(561\) 0 0
\(562\) 0 0
\(563\) 8.06226i 0.339784i 0.985463 + 0.169892i \(0.0543418\pi\)
−0.985463 + 0.169892i \(0.945658\pi\)
\(564\) 0 0
\(565\) −36.0555 −1.51687
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 23.0000 0.962520 0.481260 0.876578i \(-0.340179\pi\)
0.481260 + 0.876578i \(0.340179\pi\)
\(572\) − 16.1245i − 0.674200i
\(573\) 0 0
\(574\) 0 0
\(575\) − 40.3113i − 1.68110i
\(576\) 0 0
\(577\) 39.6611 1.65111 0.825556 0.564320i \(-0.190862\pi\)
0.825556 + 0.564320i \(0.190862\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −18.0278 −0.746633
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −14.4222 −0.592749
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) −65.0000 −2.66474
\(596\) − 31.3050i − 1.28230i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 47.0000 1.91717 0.958585 0.284807i \(-0.0919294\pi\)
0.958585 + 0.284807i \(0.0919294\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 13.4164i − 0.545455i
\(606\) 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
\(613\) −25.2389 −1.01939 −0.509694 0.860356i \(-0.670241\pi\)
−0.509694 + 0.860356i \(0.670241\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 8.06226i − 0.323008i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 29.0689i − 1.15905i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 21.6333 0.857143
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −46.8722 −1.84846 −0.924229 0.381839i \(-0.875291\pi\)
−0.924229 + 0.381839i \(0.875291\pi\)
\(644\) 58.1378i 2.29095i
\(645\) 0 0
\(646\) 0 0
\(647\) 8.06226i 0.316960i 0.987362 + 0.158480i \(0.0506593\pi\)
−0.987362 + 0.158480i \(0.949341\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 50.4777 1.97686
\(653\) − 16.1245i − 0.631001i −0.948925 0.315501i \(-0.897828\pi\)
0.948925 0.315501i \(-0.102172\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) − 44.7214i − 1.74608i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 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
\(671\) − 15.6525i − 0.604257i
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −26.0000 −1.00000
\(677\) − 40.3113i − 1.54929i −0.632397 0.774644i \(-0.717929\pi\)
0.632397 0.774644i \(-0.282071\pi\)
\(678\) 0 0
\(679\) 13.0000 0.498894
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 29.0689i 1.10744i
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 32.2490i 1.22592i
\(693\) 0 0
\(694\) 0 0
\(695\) 42.4853i 1.61156i
\(696\) 0 0
\(697\) 90.1388 3.41425
\(698\) 0 0
\(699\) 0 0
\(700\) −36.0555 −1.36277
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) − 17.8885i − 0.674200i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 18.0278 0.674200
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −46.0000 −1.70958
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −46.8722 −1.73126 −0.865631 0.500682i \(-0.833083\pi\)
−0.865631 + 0.500682i \(0.833083\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 32.2490i − 1.18791i
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) − 16.1245i − 0.592749i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 35.0000 1.28230
\(746\) 0 0
\(747\) 0 0
\(748\) 36.0555 1.31832
\(749\) − 29.0689i − 1.06215i
\(750\) 0 0
\(751\) 53.0000 1.93400 0.966999 0.254781i \(-0.0820034\pi\)
0.966999 + 0.254781i \(0.0820034\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 49.1935i 1.78326i 0.452762 + 0.891631i \(0.350439\pi\)
−0.452762 + 0.891631i \(0.649561\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 32.2490i 1.16445i
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 50.4777 1.81673
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −35.0000 −1.25240
\(782\) 0 0
\(783\) 0 0
\(784\) 24.0000 0.857143
\(785\) 0 0
\(786\) 0 0
\(787\) −14.4222 −0.514096 −0.257048 0.966399i \(-0.582750\pi\)
−0.257048 + 0.966399i \(0.582750\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 58.1378i 2.06714i
\(792\) 0 0
\(793\) −25.2389 −0.896258
\(794\) 0 0
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 56.4358i 1.99906i 0.0306762 + 0.999529i \(0.490234\pi\)
−0.0306762 + 0.999529i \(0.509766\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 16.1245i 0.569022i
\(804\) 0 0
\(805\) −65.0000 −2.29095
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 56.4358i 1.97686i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 50.0000 1.74608
\(821\) − 11.1803i − 0.390197i −0.980784 0.195098i \(-0.937497\pi\)
0.980784 0.195098i \(-0.0625026\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −28.8444 −1.00000
\(833\) 48.3735i 1.67604i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 38.0132i − 1.31236i −0.754604 0.656180i \(-0.772171\pi\)
0.754604 0.656180i \(-0.227829\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −16.0000 −0.550743
\(845\) − 29.0689i − 1.00000i
\(846\) 0 0
\(847\) −21.6333 −0.743329
\(848\) 32.2490i 1.10744i
\(849\) 0 0
\(850\) 0 0
\(851\) − 29.0689i − 0.996468i
\(852\) 0 0
\(853\) −46.8722 −1.60487 −0.802436 0.596738i \(-0.796463\pi\)
−0.802436 + 0.596738i \(0.796463\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 56.4358i 1.92781i 0.266246 + 0.963905i \(0.414217\pi\)
−0.266246 + 0.963905i \(0.585783\pi\)
\(858\) 0 0
\(859\) 41.0000 1.39890 0.699451 0.714681i \(-0.253428\pi\)
0.699451 + 0.714681i \(0.253428\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −36.0555 −1.22592
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 24.5967i 0.834388i
\(870\) 0 0
\(871\) −52.0000 −1.76195
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 40.3113i − 1.36277i
\(876\) 0 0
\(877\) 50.4777 1.70451 0.852256 0.523125i \(-0.175234\pi\)
0.852256 + 0.523125i \(0.175234\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 20.0000 0.674200
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) − 58.1378i − 1.95538i
\(885\) 0 0
\(886\) 0 0
\(887\) 56.4358i 1.89493i 0.319861 + 0.947464i \(0.396364\pi\)
−0.319861 + 0.947464i \(0.603636\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −57.6888 −1.93156
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −65.0000 −2.16546
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 51.4296i − 1.70958i
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 59.0000 1.94623 0.973115 0.230319i \(-0.0739769\pi\)
0.973115 + 0.230319i \(0.0739769\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 56.4358i 1.85761i
\(924\) 0 0
\(925\) 18.0278 0.592749
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 42.4853i 1.39390i 0.717121 + 0.696949i \(0.245459\pi\)
−0.717121 + 0.696949i \(0.754541\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 16.1245i − 0.528176i
\(933\) 0 0
\(934\) 0 0
\(935\) 40.3113i 1.31832i
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 24.5967i − 0.801831i −0.916115 0.400916i \(-0.868692\pi\)
0.916115 0.400916i \(-0.131308\pi\)
\(942\) 0 0
\(943\) 90.1388 2.93532
\(944\) 35.7771i 1.16445i
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 26.0000 0.843996
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 40.3113i − 1.30581i −0.757439 0.652905i \(-0.773550\pi\)
0.757439 0.652905i \(-0.226450\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 49.1935i 1.59103i
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 56.4358i 1.81673i
\(966\) 0 0
\(967\) −57.6888 −1.85515 −0.927574 0.373640i \(-0.878109\pi\)
−0.927574 + 0.373640i \(0.878109\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 68.5055 2.19618
\(974\) 0 0
\(975\) 0 0
\(976\) −28.0000 −0.896258
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) −5.00000 −0.159801
\(980\) 26.8328i 0.857143i
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 47.0000 1.49300 0.746502 0.665383i \(-0.231732\pi\)
0.746502 + 0.665383i \(0.231732\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.94427i 0.283552i
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 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 585.2.h.e.64.1 4
3.2 odd 2 inner 585.2.h.e.64.3 yes 4
5.4 even 2 inner 585.2.h.e.64.2 yes 4
13.12 even 2 inner 585.2.h.e.64.4 yes 4
15.14 odd 2 inner 585.2.h.e.64.4 yes 4
39.38 odd 2 inner 585.2.h.e.64.2 yes 4
65.64 even 2 inner 585.2.h.e.64.3 yes 4
195.194 odd 2 CM 585.2.h.e.64.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.h.e.64.1 4 1.1 even 1 trivial
585.2.h.e.64.1 4 195.194 odd 2 CM
585.2.h.e.64.2 yes 4 5.4 even 2 inner
585.2.h.e.64.2 yes 4 39.38 odd 2 inner
585.2.h.e.64.3 yes 4 3.2 odd 2 inner
585.2.h.e.64.3 yes 4 65.64 even 2 inner
585.2.h.e.64.4 yes 4 13.12 even 2 inner
585.2.h.e.64.4 yes 4 15.14 odd 2 inner