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Properties

Label 275.3.c.d.76.2
Level $275$
Weight $3$
Character 275.76
Self dual yes
Analytic conductor $7.493$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,3,Mod(76,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.76"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 275.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.49320726991\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 76.2
Root \(3.31662\) of defining polynomial
Character \(\chi\) \(=\) 275.76

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.31662 q^{3} +4.00000 q^{4} +2.00000 q^{9} +11.0000 q^{11} +13.2665 q^{12} +16.0000 q^{16} -29.8496 q^{23} -23.2164 q^{27} +37.0000 q^{31} +36.4829 q^{33} +8.00000 q^{36} -69.6491 q^{37} +44.0000 q^{44} +79.5990 q^{47} +53.0660 q^{48} +49.0000 q^{49} -79.5990 q^{53} -107.000 q^{59} +64.0000 q^{64} +129.348 q^{67} -99.0000 q^{69} -133.000 q^{71} -95.0000 q^{81} -97.0000 q^{89} -119.398 q^{92} +122.715 q^{93} -169.148 q^{97} +22.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{4} + 4 q^{9} + 22 q^{11} + 32 q^{16} + 74 q^{31} + 16 q^{36} + 88 q^{44} + 98 q^{49} - 214 q^{59} + 128 q^{64} - 198 q^{69} - 266 q^{71} - 190 q^{81} - 194 q^{89} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\pi\)
\(3\) 3.31662 1.10554 0.552771 − 0.833333i \(-0.313571\pi\)
0.552771 + 0.833333i \(0.313571\pi\)
\(4\) 4.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 2.00000 0.222222
\(10\) 0 0
\(11\) 11.0000 1.00000
\(12\) 13.2665 1.10554
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −29.8496 −1.29781 −0.648905 − 0.760870i \(-0.724773\pi\)
−0.648905 + 0.760870i \(0.724773\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −23.2164 −0.859866
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 37.0000 1.19355 0.596774 − 0.802409i \(-0.296449\pi\)
0.596774 + 0.802409i \(0.296449\pi\)
\(32\) 0 0
\(33\) 36.4829 1.10554
\(34\) 0 0
\(35\) 0 0
\(36\) 8.00000 0.222222
\(37\) −69.6491 −1.88241 −0.941204 − 0.337838i \(-0.890304\pi\)
−0.941204 + 0.337838i \(0.890304\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 44.0000 1.00000
\(45\) 0 0
\(46\) 0 0
\(47\) 79.5990 1.69360 0.846798 − 0.531915i \(-0.178527\pi\)
0.846798 + 0.531915i \(0.178527\pi\)
\(48\) 53.0660 1.10554
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −79.5990 −1.50187 −0.750934 − 0.660377i \(-0.770396\pi\)
−0.750934 + 0.660377i \(0.770396\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −107.000 −1.81356 −0.906780 − 0.421605i \(-0.861467\pi\)
−0.906780 + 0.421605i \(0.861467\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 129.348 1.93057 0.965286 − 0.261194i \(-0.0841162\pi\)
0.965286 + 0.261194i \(0.0841162\pi\)
\(68\) 0 0
\(69\) −99.0000 −1.43478
\(70\) 0 0
\(71\) −133.000 −1.87324 −0.936620 − 0.350348i \(-0.886063\pi\)
−0.936620 + 0.350348i \(0.886063\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −95.0000 −1.17284
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −97.0000 −1.08989 −0.544944 − 0.838473i \(-0.683449\pi\)
−0.544944 + 0.838473i \(0.683449\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −119.398 −1.29781
\(93\) 122.715 1.31952
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −169.148 −1.74379 −0.871896 − 0.489691i \(-0.837110\pi\)
−0.871896 + 0.489691i \(0.837110\pi\)
\(98\) 0 0
\(99\) 22.0000 0.222222
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −79.5990 −0.772806 −0.386403 − 0.922330i \(-0.626283\pi\)
−0.386403 + 0.922330i \(0.626283\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −92.8655 −0.859866
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −231.000 −2.08108
\(112\) 0 0
\(113\) 69.6491 0.616364 0.308182 − 0.951327i \(-0.400279\pi\)
0.308182 + 0.951327i \(0.400279\pi\)
\(114\) 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
\(124\) 148.000 1.19355
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 145.931 1.10554
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −69.6491 −0.508388 −0.254194 − 0.967153i \(-0.581810\pi\)
−0.254194 + 0.967153i \(0.581810\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 264.000 1.87234
\(142\) 0 0
\(143\) 0 0
\(144\) 32.0000 0.222222
\(145\) 0 0
\(146\) 0 0
\(147\) 162.515 1.10554
\(148\) −278.596 −1.88241
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 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
\(157\) 228.847 1.45762 0.728812 − 0.684713i \(-0.240073\pi\)
0.728812 + 0.684713i \(0.240073\pi\)
\(158\) 0 0
\(159\) −264.000 −1.66038
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 318.396 1.95335 0.976675 − 0.214724i \(-0.0688852\pi\)
0.976675 + 0.214724i \(0.0688852\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 176.000 1.00000
\(177\) −354.879 −2.00497
\(178\) 0 0
\(179\) 83.0000 0.463687 0.231844 − 0.972753i \(-0.425524\pi\)
0.231844 + 0.972753i \(0.425524\pi\)
\(180\) 0 0
\(181\) −263.000 −1.45304 −0.726519 − 0.687146i \(-0.758863\pi\)
−0.726519 + 0.687146i \(0.758863\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 318.396 1.69360
\(189\) 0 0
\(190\) 0 0
\(191\) 157.000 0.821990 0.410995 − 0.911638i \(-0.365181\pi\)
0.410995 + 0.911638i \(0.365181\pi\)
\(192\) 212.264 1.10554
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 196.000 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.0100503 −0.00502513 − 0.999987i \(-0.501600\pi\)
−0.00502513 + 0.999987i \(0.501600\pi\)
\(200\) 0 0
\(201\) 429.000 2.13433
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −59.6992 −0.288402
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −318.396 −1.50187
\(213\) −441.111 −2.07094
\(214\) 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
\(223\) −29.8496 −0.133855 −0.0669274 − 0.997758i \(-0.521320\pi\)
−0.0669274 + 0.997758i \(0.521320\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 433.000 1.89083 0.945415 − 0.325869i \(-0.105657\pi\)
0.945415 + 0.325869i \(0.105657\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −428.000 −1.81356
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −106.132 −0.436757
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 227.000 0.904382 0.452191 − 0.891921i \(-0.350642\pi\)
0.452191 + 0.891921i \(0.350642\pi\)
\(252\) 0 0
\(253\) −328.346 −1.29781
\(254\) 0 0
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) 477.594 1.85834 0.929171 − 0.369650i \(-0.120522\pi\)
0.929171 + 0.369650i \(0.120522\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 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
\(267\) −321.713 −1.20492
\(268\) 517.393 1.93057
\(269\) −362.000 −1.34572 −0.672862 − 0.739768i \(-0.734935\pi\)
−0.672862 + 0.739768i \(0.734935\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −396.000 −1.43478
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 74.0000 0.265233
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −532.000 −1.87324
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) −561.000 −1.92784
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −255.380 −0.859866
\(298\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) −264.000 −0.854369
\(310\) 0 0
\(311\) −478.000 −1.53698 −0.768489 − 0.639863i \(-0.778991\pi\)
−0.768489 + 0.639863i \(0.778991\pi\)
\(312\) 0 0
\(313\) 567.143 1.81196 0.905979 − 0.423323i \(-0.139136\pi\)
0.905979 + 0.423323i \(0.139136\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 626.842 1.97742 0.988710 − 0.149842i \(-0.0478766\pi\)
0.988710 + 0.149842i \(0.0478766\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −380.000 −1.17284
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −563.000 −1.70091 −0.850453 − 0.526051i \(-0.823672\pi\)
−0.850453 + 0.526051i \(0.823672\pi\)
\(332\) 0 0
\(333\) −139.298 −0.418313
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 231.000 0.681416
\(340\) 0 0
\(341\) 407.000 1.19355
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −328.346 −0.930158 −0.465079 − 0.885269i \(-0.653974\pi\)
−0.465079 + 0.885269i \(0.653974\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −388.000 −1.08989
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 401.312 1.10554
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −368.145 −1.00312 −0.501560 − 0.865123i \(-0.667241\pi\)
−0.501560 + 0.865123i \(0.667241\pi\)
\(368\) −477.594 −1.29781
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 490.860 1.31952
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 133.000 0.350923 0.175462 − 0.984486i \(-0.443858\pi\)
0.175462 + 0.984486i \(0.443858\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −129.348 −0.337724 −0.168862 − 0.985640i \(-0.554009\pi\)
−0.168862 + 0.985640i \(0.554009\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −676.591 −1.74379
\(389\) 553.000 1.42159 0.710797 − 0.703397i \(-0.248334\pi\)
0.710797 + 0.703397i \(0.248334\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 88.0000 0.222222
\(397\) 79.5990 0.200501 0.100251 − 0.994962i \(-0.468036\pi\)
0.100251 + 0.994962i \(0.468036\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −98.0000 −0.244389 −0.122195 − 0.992506i \(-0.538993\pi\)
−0.122195 + 0.992506i \(0.538993\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −766.140 −1.88241
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −231.000 −0.562044
\(412\) −318.396 −0.772806
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −262.000 −0.625298 −0.312649 − 0.949869i \(-0.601216\pi\)
−0.312649 + 0.949869i \(0.601216\pi\)
\(420\) 0 0
\(421\) 742.000 1.76247 0.881235 − 0.472678i \(-0.156713\pi\)
0.881235 + 0.472678i \(0.156713\pi\)
\(422\) 0 0
\(423\) 159.198 0.376355
\(424\) 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 \(\pi\)
\(432\) −371.462 −0.859866
\(433\) 865.639 1.99917 0.999583 − 0.0288684i \(-0.00919036\pi\)
0.999583 + 0.0288684i \(0.00919036\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 98.0000 0.222222
\(442\) 0 0
\(443\) 766.140 1.72944 0.864718 − 0.502257i \(-0.167497\pi\)
0.864718 + 0.502257i \(0.167497\pi\)
\(444\) −924.000 −2.08108
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 623.000 1.38753 0.693764 − 0.720202i \(-0.255951\pi\)
0.693764 + 0.720202i \(0.255951\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 278.596 0.616364
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −925.338 −1.99857 −0.999285 − 0.0377970i \(-0.987966\pi\)
−0.999285 + 0.0377970i \(0.987966\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 129.348 0.276977 0.138489 − 0.990364i \(-0.455776\pi\)
0.138489 + 0.990364i \(0.455776\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 759.000 1.61146
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −159.198 −0.333748
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 484.000 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 427.845 0.878531 0.439266 − 0.898357i \(-0.355239\pi\)
0.439266 + 0.898357i \(0.355239\pi\)
\(488\) 0 0
\(489\) 1056.00 2.15951
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 592.000 1.19355
\(497\) 0 0
\(498\) 0 0
\(499\) −602.000 −1.20641 −0.603206 − 0.797585i \(-0.706110\pi\)
−0.603206 + 0.797585i \(0.706110\pi\)
\(500\) 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
\(507\) 560.510 1.10554
\(508\) 0 0
\(509\) −1007.00 −1.97839 −0.989194 − 0.146609i \(-0.953164\pi\)
−0.989194 + 0.146609i \(0.953164\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 875.589 1.69360
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 817.000 1.56814 0.784069 − 0.620674i \(-0.213141\pi\)
0.784069 + 0.620674i \(0.213141\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 583.726 1.10554
\(529\) 362.000 0.684310
\(530\) 0 0
\(531\) −214.000 −0.403013
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 275.280 0.512625
\(538\) 0 0
\(539\) 539.000 1.00000
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −872.272 −1.60639
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −278.596 −0.508388
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 1056.00 1.87234
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 520.710 0.908744
\(574\) 0 0
\(575\) 0 0
\(576\) 128.000 0.222222
\(577\) −467.644 −0.810475 −0.405238 − 0.914211i \(-0.632811\pi\)
−0.405238 + 0.914211i \(0.632811\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −875.589 −1.50187
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −318.396 −0.542412 −0.271206 − 0.962521i \(-0.587423\pi\)
−0.271206 + 0.962521i \(0.587423\pi\)
\(588\) 650.058 1.10554
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1114.39 −1.88241
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6.63325 −0.0111110
\(598\) 0 0
\(599\) 98.0000 0.163606 0.0818030 − 0.996649i \(-0.473932\pi\)
0.0818030 + 0.996649i \(0.473932\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 258.697 0.429016
\(604\) 0 0
\(605\) 0 0
\(606\) 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
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1114.39 −1.80614 −0.903068 − 0.429498i \(-0.858691\pi\)
−0.903068 + 0.429498i \(0.858691\pi\)
\(618\) 0 0
\(619\) −1237.00 −1.99838 −0.999192 − 0.0401853i \(-0.987205\pi\)
−0.999192 + 0.0401853i \(0.987205\pi\)
\(620\) 0 0
\(621\) 693.000 1.11594
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 915.388 1.45762
\(629\) 0 0
\(630\) 0 0
\(631\) −1213.00 −1.92235 −0.961173 − 0.275947i \(-0.911008\pi\)
−0.961173 + 0.275947i \(0.911008\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −1056.00 −1.66038
\(637\) 0 0
\(638\) 0 0
\(639\) −266.000 −0.416275
\(640\) 0 0
\(641\) −743.000 −1.15913 −0.579563 − 0.814927i \(-0.696777\pi\)
−0.579563 + 0.814927i \(0.696777\pi\)
\(642\) 0 0
\(643\) −1223.83 −1.90332 −0.951660 − 0.307154i \(-0.900623\pi\)
−0.951660 + 0.307154i \(0.900623\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1164.14 −1.79928 −0.899641 − 0.436631i \(-0.856172\pi\)
−0.899641 + 0.436631i \(0.856172\pi\)
\(648\) 0 0
\(649\) −1177.00 −1.81356
\(650\) 0 0
\(651\) 0 0
\(652\) 1273.58 1.95335
\(653\) 169.148 0.259032 0.129516 − 0.991577i \(-0.458658\pi\)
0.129516 + 0.991577i \(0.458658\pi\)
\(654\) 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
\(661\) −1153.00 −1.74433 −0.872163 − 0.489215i \(-0.837283\pi\)
−0.872163 + 0.489215i \(0.837283\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −99.0000 −0.147982
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 676.000 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1114.39 1.63160 0.815802 − 0.578331i \(-0.196296\pi\)
0.815802 + 0.578331i \(0.196296\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1436.10 2.09039
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1093.00 −1.58177 −0.790883 − 0.611968i \(-0.790378\pi\)
−0.790883 + 0.611968i \(0.790378\pi\)
\(692\) 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
\(703\) 0 0
\(704\) 704.000 1.00000
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) −1419.52 −2.00497
\(709\) −1057.00 −1.49083 −0.745416 − 0.666599i \(-0.767749\pi\)
−0.745416 + 0.666599i \(0.767749\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1104.44 −1.54900
\(714\) 0 0
\(715\) 0 0
\(716\) 332.000 0.463687
\(717\) 0 0
\(718\) 0 0
\(719\) 1163.00 1.61752 0.808762 − 0.588136i \(-0.200138\pi\)
0.808762 + 0.588136i \(0.200138\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −1052.00 −1.45304
\(725\) 0 0
\(726\) 0 0
\(727\) 527.343 0.725369 0.362685 − 0.931912i \(-0.381860\pi\)
0.362685 + 0.931912i \(0.381860\pi\)
\(728\) 0 0
\(729\) 503.000 0.689986
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1422.83 1.93057
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 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
\(751\) −973.000 −1.29561 −0.647803 − 0.761808i \(-0.724312\pi\)
−0.647803 + 0.761808i \(0.724312\pi\)
\(752\) 1273.58 1.69360
\(753\) 752.874 0.999832
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1512.38 −1.99786 −0.998931 − 0.0462351i \(-0.985278\pi\)
−0.998931 + 0.0462351i \(0.985278\pi\)
\(758\) 0 0
\(759\) −1089.00 −1.43478
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 628.000 0.821990
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 849.056 1.10554
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 1584.00 2.05447
\(772\) 0 0
\(773\) 716.391 0.926767 0.463384 − 0.886158i \(-0.346635\pi\)
0.463384 + 0.886158i \(0.346635\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −1463.00 −1.87324
\(782\) 0 0
\(783\) 0 0
\(784\) 784.000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −8.00000 −0.0100503
\(797\) −169.148 −0.212231 −0.106115 − 0.994354i \(-0.533841\pi\)
−0.106115 + 0.994354i \(0.533841\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −194.000 −0.242197
\(802\) 0 0
\(803\) 0 0
\(804\) 1716.00 2.13433
\(805\) 0 0
\(806\) 0 0
\(807\) −1200.62 −1.48775
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 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
\(823\) 1462.63 1.77720 0.888598 − 0.458688i \(-0.151680\pi\)
0.888598 + 0.458688i \(0.151680\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −238.797 −0.288402
\(829\) −817.000 −0.985525 −0.492762 − 0.870164i \(-0.664013\pi\)
−0.492762 + 0.870164i \(0.664013\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −859.006 −1.02629
\(838\) 0 0
\(839\) −347.000 −0.413588 −0.206794 − 0.978385i \(-0.566303\pi\)
−0.206794 + 0.978385i \(0.566303\pi\)
\(840\) 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
\(848\) −1273.58 −1.50187
\(849\) 0 0
\(850\) 0 0
\(851\) 2079.00 2.44301
\(852\) −1764.44 −2.07094
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −757.000 −0.881257 −0.440629 − 0.897689i \(-0.645244\pi\)
−0.440629 + 0.897689i \(0.645244\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1671.58 −1.93694 −0.968470 − 0.249131i \(-0.919855\pi\)
−0.968470 + 0.249131i \(0.919855\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 958.505 1.10554
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −338.296 −0.387509
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1487.00 1.68785 0.843927 − 0.536457i \(-0.180238\pi\)
0.843927 + 0.536457i \(0.180238\pi\)
\(882\) 0 0
\(883\) 1114.39 1.26205 0.631023 − 0.775764i \(-0.282636\pi\)
0.631023 + 0.775764i \(0.282636\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1045.00 −1.17284
\(892\) −119.398 −0.133855
\(893\) 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
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 477.594 0.526564 0.263282 − 0.964719i \(-0.415195\pi\)
0.263282 + 0.964719i \(0.415195\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1778.00 −1.95170 −0.975851 − 0.218439i \(-0.929904\pi\)
−0.975851 + 0.218439i \(0.929904\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1732.00 1.89083
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −159.198 −0.171735
\(928\) 0 0
\(929\) 958.000 1.03122 0.515608 − 0.856824i \(-0.327566\pi\)
0.515608 + 0.856824i \(0.327566\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −1585.35 −1.69919
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 1881.00 2.00319
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1712.00 −1.81356
\(945\) 0 0
\(946\) 0 0
\(947\) 1323.33 1.39740 0.698698 − 0.715417i \(-0.253763\pi\)
0.698698 + 0.715417i \(0.253763\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 2079.00 2.18612
\(952\) 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
\(961\) 408.000 0.424558
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −83.0000 −0.0854789 −0.0427394 − 0.999086i \(-0.513609\pi\)
−0.0427394 + 0.999086i \(0.513609\pi\)
\(972\) −424.528 −0.436757
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1522.33 1.55817 0.779084 − 0.626919i \(-0.215684\pi\)
0.779084 + 0.626919i \(0.215684\pi\)
\(978\) 0 0
\(979\) −1067.00 −1.08989
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1860.63 1.89280 0.946402 − 0.322991i \(-0.104688\pi\)
0.946402 + 0.322991i \(0.104688\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1582.00 1.59637 0.798184 − 0.602414i \(-0.205794\pi\)
0.798184 + 0.602414i \(0.205794\pi\)
\(992\) 0 0
\(993\) −1867.26 −1.88042
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 1617.00 1.61862
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 275.3.c.d.76.2 2
5.2 odd 4 55.3.d.a.54.1 ✓ 2
5.3 odd 4 55.3.d.a.54.2 yes 2
5.4 even 2 inner 275.3.c.d.76.1 2
11.10 odd 2 CM 275.3.c.d.76.2 2
15.2 even 4 495.3.h.a.109.2 2
15.8 even 4 495.3.h.a.109.1 2
20.3 even 4 880.3.i.b.769.1 2
20.7 even 4 880.3.i.b.769.2 2
55.32 even 4 55.3.d.a.54.1 ✓ 2
55.43 even 4 55.3.d.a.54.2 yes 2
55.54 odd 2 inner 275.3.c.d.76.1 2
165.32 odd 4 495.3.h.a.109.2 2
165.98 odd 4 495.3.h.a.109.1 2
220.43 odd 4 880.3.i.b.769.1 2
220.87 odd 4 880.3.i.b.769.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.3.d.a.54.1 ✓ 2 5.2 odd 4
55.3.d.a.54.1 ✓ 2 55.32 even 4
55.3.d.a.54.2 yes 2 5.3 odd 4
55.3.d.a.54.2 yes 2 55.43 even 4
275.3.c.d.76.1 2 5.4 even 2 inner
275.3.c.d.76.1 2 55.54 odd 2 inner
275.3.c.d.76.2 2 1.1 even 1 trivial
275.3.c.d.76.2 2 11.10 odd 2 CM
495.3.h.a.109.1 2 15.8 even 4
495.3.h.a.109.1 2 165.98 odd 4
495.3.h.a.109.2 2 15.2 even 4
495.3.h.a.109.2 2 165.32 odd 4
880.3.i.b.769.1 2 20.3 even 4
880.3.i.b.769.1 2 220.43 odd 4
880.3.i.b.769.2 2 20.7 even 4
880.3.i.b.769.2 2 220.87 odd 4