Properties

Label 75.3.c.d.26.2
Level $75$
Weight $3$
Character 75.26
Analytic conductor $2.044$
Analytic rank $0$
Dimension $2$
CM discriminant -15
Inner twists $4$

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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: no
Analytic conductor: \(2.04360198270\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 26.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 75.26
Dual form 75.3.c.d.26.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +3.00000i q^{3} +3.00000 q^{4} -3.00000 q^{6} +7.00000i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +3.00000i q^{3} +3.00000 q^{4} -3.00000 q^{6} +7.00000i q^{8} -9.00000 q^{9} +9.00000i q^{12} +5.00000 q^{16} -14.0000i q^{17} -9.00000i q^{18} +22.0000 q^{19} -34.0000i q^{23} -21.0000 q^{24} -27.0000i q^{27} +2.00000 q^{31} +33.0000i q^{32} +14.0000 q^{34} -27.0000 q^{36} +22.0000i q^{38} +34.0000 q^{46} -14.0000i q^{47} +15.0000i q^{48} -49.0000 q^{49} +42.0000 q^{51} +86.0000i q^{53} +27.0000 q^{54} +66.0000i q^{57} -118.000 q^{61} +2.00000i q^{62} -13.0000 q^{64} -42.0000i q^{68} +102.000 q^{69} -63.0000i q^{72} +66.0000 q^{76} -98.0000 q^{79} +81.0000 q^{81} -154.000i q^{83} -102.000i q^{92} +6.00000i q^{93} +14.0000 q^{94} -99.0000 q^{96} -49.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{4} - 6 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{4} - 6 q^{6} - 18 q^{9} + 10 q^{16} + 44 q^{19} - 42 q^{24} + 4 q^{31} + 28 q^{34} - 54 q^{36} + 68 q^{46} - 98 q^{49} + 84 q^{51} + 54 q^{54} - 236 q^{61} - 26 q^{64} + 204 q^{69} + 132 q^{76} - 196 q^{79} + 162 q^{81} + 28 q^{94} - 198 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.500000i 0.968246 + 0.250000i \(0.0804306\pi\)
−0.968246 + 0.250000i \(0.919569\pi\)
\(3\) 3.00000i 1.00000i
\(4\) 3.00000 0.750000
\(5\) 0 0
\(6\) −3.00000 −0.500000
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 7.00000i 0.875000i
\(9\) −9.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 9.00000i 0.750000i
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 5.00000 0.312500
\(17\) − 14.0000i − 0.823529i −0.911290 0.411765i \(-0.864913\pi\)
0.911290 0.411765i \(-0.135087\pi\)
\(18\) − 9.00000i − 0.500000i
\(19\) 22.0000 1.15789 0.578947 0.815365i \(-0.303464\pi\)
0.578947 + 0.815365i \(0.303464\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 34.0000i − 1.47826i −0.673562 0.739130i \(-0.735237\pi\)
0.673562 0.739130i \(-0.264763\pi\)
\(24\) −21.0000 −0.875000
\(25\) 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 \(-0.489730\pi\)
0.0322581 + 0.999480i \(0.489730\pi\)
\(32\) 33.0000i 1.03125i
\(33\) 0 0
\(34\) 14.0000 0.411765
\(35\) 0 0
\(36\) −27.0000 −0.750000
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 22.0000i 0.578947i
\(39\) 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
\(46\) 34.0000 0.739130
\(47\) − 14.0000i − 0.297872i −0.988847 0.148936i \(-0.952415\pi\)
0.988847 0.148936i \(-0.0475849\pi\)
\(48\) 15.0000i 0.312500i
\(49\) −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 \(0.698749\pi\)
\(54\) 27.0000 0.500000
\(55\) 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 \(0.918265\pi\)
\(62\) 2.00000i 0.0322581i
\(63\) 0 0
\(64\) −13.0000 −0.203125
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) − 42.0000i − 0.617647i
\(69\) 102.000 1.47826
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) − 63.0000i − 0.875000i
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 66.0000 0.868421
\(77\) 0 0
\(78\) 0 0
\(79\) −98.0000 −1.24051 −0.620253 0.784402i \(-0.712970\pi\)
−0.620253 + 0.784402i \(0.712970\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) − 154.000i − 1.85542i −0.373300 0.927711i \(-0.621774\pi\)
0.373300 0.927711i \(-0.378226\pi\)
\(84\) 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
\(92\) − 102.000i − 1.10870i
\(93\) 6.00000i 0.0645161i
\(94\) 14.0000 0.148936
\(95\) 0 0
\(96\) −99.0000 −1.03125
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) − 49.0000i − 0.500000i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 42.0000i 0.411765i
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −86.0000 −0.811321
\(107\) 106.000i 0.990654i 0.868707 + 0.495327i \(0.164952\pi\)
−0.868707 + 0.495327i \(0.835048\pi\)
\(108\) − 81.0000i − 0.750000i
\(109\) 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 \(0.634922\pi\)
\(114\) −66.0000 −0.578947
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) − 118.000i − 0.967213i
\(123\) 0 0
\(124\) 6.00000 0.0483871
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 119.000i 0.929688i
\(129\) 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
\(136\) 98.0000 0.720588
\(137\) 226.000i 1.64964i 0.565399 + 0.824818i \(0.308722\pi\)
−0.565399 + 0.824818i \(0.691278\pi\)
\(138\) 102.000i 0.739130i
\(139\) 262.000 1.88489 0.942446 0.334358i \(-0.108520\pi\)
0.942446 + 0.334358i \(0.108520\pi\)
\(140\) 0 0
\(141\) 42.0000 0.297872
\(142\) 0 0
\(143\) 0 0
\(144\) −45.0000 −0.312500
\(145\) 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 \(-0.788924\pi\)
−0.788079 + 0.615574i \(0.788924\pi\)
\(152\) 154.000i 1.01316i
\(153\) 126.000i 0.823529i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) − 98.0000i − 0.620253i
\(159\) −258.000 −1.62264
\(160\) 0 0
\(161\) 0 0
\(162\) 81.0000i 0.500000i
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 154.000 0.927711
\(167\) − 254.000i − 1.52096i −0.649362 0.760479i \(-0.724964\pi\)
0.649362 0.760479i \(-0.275036\pi\)
\(168\) 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
\(184\) 238.000 1.29348
\(185\) 0 0
\(186\) −6.00000 −0.0322581
\(187\) 0 0
\(188\) − 42.0000i − 0.223404i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) − 39.0000i − 0.203125i
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −147.000 −0.750000
\(197\) − 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 \(-0.383873\pi\)
0.356784 + 0.934187i \(0.383873\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 126.000 0.617647
\(205\) 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 \(-0.171818\pi\)
0.857820 + 0.513950i \(0.171818\pi\)
\(212\) 258.000i 1.21698i
\(213\) 0 0
\(214\) −106.000 −0.495327
\(215\) 0 0
\(216\) 189.000 0.875000
\(217\) 0 0
\(218\) 22.0000i 0.100917i
\(219\) 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
\(226\) −206.000 −0.911504
\(227\) − 134.000i − 0.590308i −0.955450 0.295154i \(-0.904629\pi\)
0.955450 0.295154i \(-0.0953710\pi\)
\(228\) 198.000i 0.868421i
\(229\) −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 \(0.958963\pi\)
\(242\) 121.000i 0.500000i
\(243\) 243.000i 1.00000i
\(244\) −354.000 −1.45082
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 14.0000i 0.0564516i
\(249\) 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
\(256\) −171.000 −0.667969
\(257\) 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 \(0.322139\pi\)
−0.530142 + 0.847909i \(0.677861\pi\)
\(264\) 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 \(-0.151193\pi\)
0.889299 + 0.457326i \(0.151193\pi\)
\(272\) − 70.0000i − 0.257353i
\(273\) 0 0
\(274\) −226.000 −0.824818
\(275\) 0 0
\(276\) 306.000 1.10870
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 262.000i 0.942446i
\(279\) −18.0000 −0.0645161
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 42.0000i 0.148936i
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) − 297.000i − 1.03125i
\(289\) 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 \(-0.234717\pi\)
\(294\) 147.000 0.500000
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) − 238.000i − 0.788079i
\(303\) 0 0
\(304\) 110.000 0.361842
\(305\) 0 0
\(306\) −126.000 −0.411765
\(307\) 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
\(316\) −294.000 −0.930380
\(317\) − 134.000i − 0.422713i −0.977409 0.211356i \(-0.932212\pi\)
0.977409 0.211356i \(-0.0677881\pi\)
\(318\) − 258.000i − 0.811321i
\(319\) 0 0
\(320\) 0 0
\(321\) −318.000 −0.990654
\(322\) 0 0
\(323\) − 308.000i − 0.953560i
\(324\) 243.000 0.750000
\(325\) 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 \(-0.441001\pi\)
0.184290 + 0.982872i \(0.441001\pi\)
\(332\) − 462.000i − 1.39157i
\(333\) 0 0
\(334\) 254.000 0.760479
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) − 169.000i − 0.500000i
\(339\) −618.000 −1.82301
\(340\) 0 0
\(341\) 0 0
\(342\) − 198.000i − 0.578947i
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 154.000 0.445087
\(347\) 586.000i 1.68876i 0.535744 + 0.844380i \(0.320031\pi\)
−0.535744 + 0.844380i \(0.679969\pi\)
\(348\) 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
\(362\) 122.000i 0.337017i
\(363\) 363.000i 1.00000i
\(364\) 0 0
\(365\) 0 0
\(366\) 354.000 0.967213
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) − 170.000i − 0.461957i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 18.0000i 0.0483871i
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 98.0000 0.260638
\(377\) 0 0
\(378\) 0 0
\(379\) 742.000 1.95778 0.978892 0.204379i \(-0.0655175\pi\)
0.978892 + 0.204379i \(0.0655175\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 686.000i 1.79112i 0.444938 + 0.895561i \(0.353226\pi\)
−0.444938 + 0.895561i \(0.646774\pi\)
\(384\) −357.000 −0.929688
\(385\) 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
\(392\) − 343.000i − 0.875000i
\(393\) 0 0
\(394\) 374.000 0.949239
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 142.000i 0.356784i
\(399\) 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
\(408\) 294.000i 0.720588i
\(409\) 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
\(414\) −306.000 −0.739130
\(415\) 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 \(0.246443\pi\)
\(422\) 362.000i 0.857820i
\(423\) 126.000i 0.297872i
\(424\) −602.000 −1.41981
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 318.000i 0.742991i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) − 135.000i − 0.312500i
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 66.0000 0.151376
\(437\) − 748.000i − 1.71167i
\(438\) 0 0
\(439\) 622.000 1.41686 0.708428 0.705783i \(-0.249405\pi\)
0.708428 + 0.705783i \(0.249405\pi\)
\(440\) 0 0
\(441\) 441.000 1.00000
\(442\) 0 0
\(443\) 566.000i 1.27765i 0.769351 + 0.638826i \(0.220580\pi\)
−0.769351 + 0.638826i \(0.779420\pi\)
\(444\) 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
\(452\) 618.000i 1.36726i
\(453\) − 714.000i − 1.57616i
\(454\) 134.000 0.295154
\(455\) 0 0
\(456\) −462.000 −1.01316
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) − 218.000i − 0.475983i
\(459\) −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
\(466\) 34.0000 0.0729614
\(467\) 346.000i 0.740899i 0.928853 + 0.370450i \(0.120796\pi\)
−0.928853 + 0.370450i \(0.879204\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 294.000 0.620253
\(475\) 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
\(482\) − 478.000i − 0.991701i
\(483\) 0 0
\(484\) 363.000 0.750000
\(485\) 0 0
\(486\) −243.000 −0.500000
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) − 826.000i − 1.69262i
\(489\) 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
\(496\) 10.0000 0.0201613
\(497\) 0 0
\(498\) 462.000i 0.927711i
\(499\) −938.000 −1.87976 −0.939880 0.341506i \(-0.889063\pi\)
−0.939880 + 0.341506i \(0.889063\pi\)
\(500\) 0 0
\(501\) 762.000 1.52096
\(502\) 0 0
\(503\) − 994.000i − 1.97614i −0.153995 0.988072i \(-0.549214\pi\)
0.153995 0.988072i \(-0.450786\pi\)
\(504\) 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
\(512\) 305.000i 0.595703i
\(513\) − 594.000i − 1.15789i
\(514\) −466.000 −0.906615
\(515\) 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
\(526\) −446.000 −0.847909
\(527\) − 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 \(0.972621\pi\)
\(542\) 482.000i 0.889299i
\(543\) 366.000i 0.674033i
\(544\) 462.000 0.849265
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 678.000i 1.23723i
\(549\) 1062.00 1.93443
\(550\) 0 0
\(551\) 0 0
\(552\) 714.000i 1.29348i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 786.000 1.41367
\(557\) − 614.000i − 1.10233i −0.834395 0.551167i \(-0.814183\pi\)
0.834395 0.551167i \(-0.185817\pi\)
\(558\) − 18.0000i − 0.0322581i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 154.000i − 0.273535i −0.990603 0.136767i \(-0.956329\pi\)
0.990603 0.136767i \(-0.0436713\pi\)
\(564\) 126.000 0.223404
\(565\) 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 \(-0.601497\pi\)
−0.313485 + 0.949593i \(0.601497\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 117.000 0.203125
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 93.0000i 0.160900i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 394.000 0.672355
\(587\) − 854.000i − 1.45486i −0.686184 0.727428i \(-0.740716\pi\)
0.686184 0.727428i \(-0.259284\pi\)
\(588\) − 441.000i − 0.750000i
\(589\) 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
\(604\) −714.000 −1.18212
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 726.000i 1.19408i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 378.000i 0.617647i
\(613\) 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 \(-0.690666\pi\)
−0.563813 + 0.825903i \(0.690666\pi\)
\(620\) 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 \(-0.560392\pi\)
−0.188590 + 0.982056i \(0.560392\pi\)
\(632\) − 686.000i − 1.08544i
\(633\) 1086.00i 1.71564i
\(634\) 134.000 0.211356
\(635\) 0 0
\(636\) −774.000 −1.21698
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) − 318.000i − 0.495327i
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 308.000 0.476780
\(647\) 706.000i 1.09119i 0.838049 + 0.545595i \(0.183696\pi\)
−0.838049 + 0.545595i \(0.816304\pi\)
\(648\) 567.000i 0.875000i
\(649\) 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 \(-0.325211\pi\)
\(654\) −66.0000 −0.100917
\(655\) 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 \(0.718542\pi\)
\(662\) 122.000i 0.184290i
\(663\) 0 0
\(664\) 1078.00 1.62349
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) − 762.000i − 1.14072i
\(669\) 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
\(676\) −507.000 −0.750000
\(677\) − 374.000i − 0.552437i −0.961095 0.276219i \(-0.910919\pi\)
0.961095 0.276219i \(-0.0890814\pi\)
\(678\) − 618.000i − 0.911504i
\(679\) 0 0
\(680\) 0 0
\(681\) 402.000 0.590308
\(682\) 0 0
\(683\) 86.0000i 0.125915i 0.998016 + 0.0629575i \(0.0200533\pi\)
−0.998016 + 0.0629575i \(0.979947\pi\)
\(684\) −594.000 −0.868421
\(685\) 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 \(-0.0941392\pi\)
0.956585 + 0.291455i \(0.0941392\pi\)
\(692\) − 462.000i − 0.667630i
\(693\) 0 0
\(694\) −586.000 −0.844380
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) − 458.000i − 0.656160i
\(699\) 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
\(706\) 274.000 0.388102
\(707\) 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
\(722\) 123.000i 0.170360i
\(723\) − 1434.00i − 1.98340i
\(724\) 366.000 0.505525
\(725\) 0 0
\(726\) −363.000 −0.500000
\(727\) 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
\(732\) − 1062.00i − 1.45082i
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 1122.00 1.52446
\(737\) 0 0
\(738\) 0 0
\(739\) 1462.00 1.97835 0.989175 0.146744i \(-0.0468792\pi\)
0.989175 + 0.146744i \(0.0468792\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 514.000i − 0.691790i −0.938273 0.345895i \(-0.887575\pi\)
0.938273 0.345895i \(-0.112425\pi\)
\(744\) −42.0000 −0.0564516
\(745\) 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 \(-0.906745\pi\)
−0.957390 + 0.288798i \(0.906745\pi\)
\(752\) − 70.0000i − 0.0930851i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 742.000i 0.978892i
\(759\) 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
\(766\) −686.000 −0.895561
\(767\) 0 0
\(768\) − 513.000i − 0.667969i
\(769\) −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
\(782\) − 476.000i − 0.608696i
\(783\) 0 0
\(784\) −245.000 −0.312500
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) − 1122.00i − 1.42386i
\(789\) −1338.00 −1.69582
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 426.000 0.535176
\(797\) 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 \(-0.267545\pi\)
0.667078 + 0.744988i \(0.267545\pi\)
\(812\) 0 0
\(813\) 1446.00i 1.77860i
\(814\) 0 0
\(815\) 0 0
\(816\) 210.000 0.257353
\(817\) 0 0
\(818\) 142.000i 0.173594i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) − 678.000i − 0.824818i
\(823\) 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 \(-0.927396\pi\)
0.974100 0.226119i \(-0.0726037\pi\)
\(828\) 918.000i 1.10870i
\(829\) 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
\(834\) −786.000 −0.942446
\(835\) 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
\(842\) 602.000i 0.714964i
\(843\) 0 0
\(844\) 1086.00 1.28673
\(845\) 0 0
\(846\) −126.000 −0.148936
\(847\) 0 0
\(848\) 430.000i 0.507075i
\(849\) 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
\(856\) −742.000 −0.866822
\(857\) 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 \(-0.540500\pi\)
−0.126892 + 0.991917i \(0.540500\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 274.000i − 0.317497i −0.987319 0.158749i \(-0.949254\pi\)
0.987319 0.158749i \(-0.0507459\pi\)
\(864\) 891.000 1.03125
\(865\) 0 0
\(866\) 0 0
\(867\) 279.000i 0.321799i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 154.000i 0.176606i
\(873\) 0 0
\(874\) 748.000 0.855835
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 622.000i 0.708428i
\(879\) 1182.00 1.34471
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 441.000i 0.500000i
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −566.000 −0.638826
\(887\) − 1694.00i − 1.90981i −0.296914 0.954904i \(-0.595958\pi\)
0.296914 0.954904i \(-0.404042\pi\)
\(888\) 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
\(904\) −1442.00 −1.59513
\(905\) 0 0
\(906\) 714.000 0.788079
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) − 402.000i − 0.442731i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 330.000i 0.361842i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −654.000 −0.713974
\(917\) 0 0
\(918\) − 378.000i − 0.411765i
\(919\) −1298.00 −1.41240 −0.706202 0.708010i \(-0.749593\pi\)
−0.706202 + 0.708010i \(0.749593\pi\)
\(920\) 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
\(932\) − 102.000i − 0.109442i
\(933\) 0 0
\(934\) −346.000 −0.370450
\(935\) 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 \(-0.687743\pi\)
0.556205 0.831045i \(-0.312257\pi\)
\(948\) − 882.000i − 0.930380i
\(949\) 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 \(-0.281419\pi\)
\(954\) 774.000 0.811321
\(955\) 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
\(964\) −1434.00 −1.48755
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 847.000i 0.875000i
\(969\) 924.000 0.953560
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 729.000i 0.750000i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −590.000 −0.604508
\(977\) − 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 \(-0.535187\pi\)
0.110319 0.993896i \(-0.464813\pi\)
\(984\) 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 \(-0.660580\pi\)
−0.483350 + 0.875427i \(0.660580\pi\)
\(992\) 66.0000i 0.0665323i
\(993\) 366.000i 0.368580i
\(994\) 0 0
\(995\) 0 0
\(996\) 1386.00 1.39157
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) − 938.000i − 0.939880i
\(999\) 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 75.3.c.d.26.2 2
3.2 odd 2 inner 75.3.c.d.26.1 2
4.3 odd 2 1200.3.l.l.401.1 2
5.2 odd 4 15.3.d.a.14.1 1
5.3 odd 4 15.3.d.b.14.1 yes 1
5.4 even 2 inner 75.3.c.d.26.1 2
12.11 even 2 1200.3.l.l.401.2 2
15.2 even 4 15.3.d.b.14.1 yes 1
15.8 even 4 15.3.d.a.14.1 1
15.14 odd 2 CM 75.3.c.d.26.2 2
20.3 even 4 240.3.c.b.209.1 1
20.7 even 4 240.3.c.a.209.1 1
20.19 odd 2 1200.3.l.l.401.2 2
40.3 even 4 960.3.c.a.449.1 1
40.13 odd 4 960.3.c.c.449.1 1
40.27 even 4 960.3.c.d.449.1 1
40.37 odd 4 960.3.c.b.449.1 1
45.2 even 12 405.3.h.a.134.1 2
45.7 odd 12 405.3.h.b.134.1 2
45.13 odd 12 405.3.h.a.269.1 2
45.22 odd 12 405.3.h.b.269.1 2
45.23 even 12 405.3.h.b.269.1 2
45.32 even 12 405.3.h.a.269.1 2
45.38 even 12 405.3.h.b.134.1 2
45.43 odd 12 405.3.h.a.134.1 2
60.23 odd 4 240.3.c.a.209.1 1
60.47 odd 4 240.3.c.b.209.1 1
60.59 even 2 1200.3.l.l.401.1 2
120.53 even 4 960.3.c.b.449.1 1
120.77 even 4 960.3.c.c.449.1 1
120.83 odd 4 960.3.c.d.449.1 1
120.107 odd 4 960.3.c.a.449.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.3.d.a.14.1 1 5.2 odd 4
15.3.d.a.14.1 1 15.8 even 4
15.3.d.b.14.1 yes 1 5.3 odd 4
15.3.d.b.14.1 yes 1 15.2 even 4
75.3.c.d.26.1 2 3.2 odd 2 inner
75.3.c.d.26.1 2 5.4 even 2 inner
75.3.c.d.26.2 2 1.1 even 1 trivial
75.3.c.d.26.2 2 15.14 odd 2 CM
240.3.c.a.209.1 1 20.7 even 4
240.3.c.a.209.1 1 60.23 odd 4
240.3.c.b.209.1 1 20.3 even 4
240.3.c.b.209.1 1 60.47 odd 4
405.3.h.a.134.1 2 45.2 even 12
405.3.h.a.134.1 2 45.43 odd 12
405.3.h.a.269.1 2 45.13 odd 12
405.3.h.a.269.1 2 45.32 even 12
405.3.h.b.134.1 2 45.7 odd 12
405.3.h.b.134.1 2 45.38 even 12
405.3.h.b.269.1 2 45.22 odd 12
405.3.h.b.269.1 2 45.23 even 12
960.3.c.a.449.1 1 40.3 even 4
960.3.c.a.449.1 1 120.107 odd 4
960.3.c.b.449.1 1 40.37 odd 4
960.3.c.b.449.1 1 120.53 even 4
960.3.c.c.449.1 1 40.13 odd 4
960.3.c.c.449.1 1 120.77 even 4
960.3.c.d.449.1 1 40.27 even 4
960.3.c.d.449.1 1 120.83 odd 4
1200.3.l.l.401.1 2 4.3 odd 2
1200.3.l.l.401.1 2 60.59 even 2
1200.3.l.l.401.2 2 12.11 even 2
1200.3.l.l.401.2 2 20.19 odd 2