Properties

Label 300.3.b.b.149.1
Level $300$
Weight $3$
Character 300.149
Analytic conductor $8.174$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,3,Mod(149,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.149");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 149.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 300.149
Dual form 300.3.b.b.149.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} +13.0000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +13.0000i q^{7} -9.00000 q^{9} +23.0000i q^{13} -11.0000 q^{19} +39.0000 q^{21} +27.0000i q^{27} +59.0000 q^{31} -26.0000i q^{37} +69.0000 q^{39} +83.0000i q^{43} -120.000 q^{49} +33.0000i q^{57} -121.000 q^{61} -117.000i q^{63} +13.0000i q^{67} -46.0000i q^{73} +142.000 q^{79} +81.0000 q^{81} -299.000 q^{91} -177.000i q^{93} -167.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} - 22 q^{19} + 78 q^{21} + 118 q^{31} + 138 q^{39} - 240 q^{49} - 242 q^{61} + 284 q^{79} + 162 q^{81} - 598 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) − 3.00000i − 1.00000i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 13.0000i 1.85714i 0.371154 + 0.928571i \(0.378962\pi\)
−0.371154 + 0.928571i \(0.621038\pi\)
\(8\) 0 0
\(9\) −9.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 23.0000i 1.76923i 0.466321 + 0.884615i \(0.345579\pi\)
−0.466321 + 0.884615i \(0.654421\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −11.0000 −0.578947 −0.289474 0.957186i \(-0.593480\pi\)
−0.289474 + 0.957186i \(0.593480\pi\)
\(20\) 0 0
\(21\) 39.0000 1.85714
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 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
\(31\) 59.0000 1.90323 0.951613 0.307299i \(-0.0994253\pi\)
0.951613 + 0.307299i \(0.0994253\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 26.0000i − 0.702703i −0.936244 0.351351i \(-0.885722\pi\)
0.936244 0.351351i \(-0.114278\pi\)
\(38\) 0 0
\(39\) 69.0000 1.76923
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 83.0000i 1.93023i 0.261822 + 0.965116i \(0.415677\pi\)
−0.261822 + 0.965116i \(0.584323\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −120.000 −2.44898
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 33.0000i 0.578947i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −121.000 −1.98361 −0.991803 0.127774i \(-0.959217\pi\)
−0.991803 + 0.127774i \(0.959217\pi\)
\(62\) 0 0
\(63\) − 117.000i − 1.85714i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.0000i 0.194030i 0.995283 + 0.0970149i \(0.0309295\pi\)
−0.995283 + 0.0970149i \(0.969071\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) − 46.0000i − 0.630137i −0.949069 0.315068i \(-0.897973\pi\)
0.949069 0.315068i \(-0.102027\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 142.000 1.79747 0.898734 0.438494i \(-0.144488\pi\)
0.898734 + 0.438494i \(0.144488\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 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
\(91\) −299.000 −3.28571
\(92\) 0 0
\(93\) − 177.000i − 1.90323i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 167.000i − 1.72165i −0.508902 0.860825i \(-0.669948\pi\)
0.508902 0.860825i \(-0.330052\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 194.000i 1.88350i 0.336321 + 0.941748i \(0.390817\pi\)
−0.336321 + 0.941748i \(0.609183\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −71.0000 −0.651376 −0.325688 0.945477i \(-0.605596\pi\)
−0.325688 + 0.945477i \(0.605596\pi\)
\(110\) 0 0
\(111\) −78.0000 −0.702703
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 207.000i − 1.76923i
\(118\) 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
\(127\) − 146.000i − 1.14961i −0.818292 0.574803i \(-0.805079\pi\)
0.818292 0.574803i \(-0.194921\pi\)
\(128\) 0 0
\(129\) 249.000 1.93023
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) − 143.000i − 1.07519i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 22.0000 0.158273 0.0791367 0.996864i \(-0.474784\pi\)
0.0791367 + 0.996864i \(0.474784\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 360.000i 2.44898i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 59.0000 0.390728 0.195364 0.980731i \(-0.437411\pi\)
0.195364 + 0.980731i \(0.437411\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 193.000i 1.22930i 0.788800 + 0.614650i \(0.210703\pi\)
−0.788800 + 0.614650i \(0.789297\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 37.0000i − 0.226994i −0.993538 0.113497i \(-0.963795\pi\)
0.993538 0.113497i \(-0.0362052\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −360.000 −2.13018
\(170\) 0 0
\(171\) 99.0000 0.578947
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 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
\(181\) −1.00000 −0.00552486 −0.00276243 0.999996i \(-0.500879\pi\)
−0.00276243 + 0.999996i \(0.500879\pi\)
\(182\) 0 0
\(183\) 363.000i 1.98361i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −351.000 −1.85714
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 143.000i 0.740933i 0.928846 + 0.370466i \(0.120802\pi\)
−0.928846 + 0.370466i \(0.879198\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 109.000 0.547739 0.273869 0.961767i \(-0.411696\pi\)
0.273869 + 0.961767i \(0.411696\pi\)
\(200\) 0 0
\(201\) 39.0000 0.194030
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 419.000 1.98578 0.992891 0.119027i \(-0.0379776\pi\)
0.992891 + 0.119027i \(0.0379776\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 767.000i 3.53456i
\(218\) 0 0
\(219\) −138.000 −0.630137
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 83.0000i 0.372197i 0.982531 + 0.186099i \(0.0595844\pi\)
−0.982531 + 0.186099i \(0.940416\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 409.000 1.78603 0.893013 0.450031i \(-0.148587\pi\)
0.893013 + 0.450031i \(0.148587\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 426.000i − 1.79747i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 479.000 1.98755 0.993776 0.111397i \(-0.0355327\pi\)
0.993776 + 0.111397i \(0.0355327\pi\)
\(242\) 0 0
\(243\) − 243.000i − 1.00000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 253.000i − 1.02429i
\(248\) 0 0
\(249\) 0 0
\(250\) 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
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 338.000 1.30502
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 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
\(271\) 242.000 0.892989 0.446494 0.894786i \(-0.352672\pi\)
0.446494 + 0.894786i \(0.352672\pi\)
\(272\) 0 0
\(273\) 897.000i 3.28571i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 407.000i − 1.46931i −0.678439 0.734657i \(-0.737343\pi\)
0.678439 0.734657i \(-0.262657\pi\)
\(278\) 0 0
\(279\) −531.000 −1.90323
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) − 517.000i − 1.82686i −0.407001 0.913428i \(-0.633426\pi\)
0.407001 0.913428i \(-0.366574\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −289.000 −1.00000
\(290\) 0 0
\(291\) −501.000 −1.72165
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −1079.00 −3.58472
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 253.000i 0.824104i 0.911160 + 0.412052i \(0.135188\pi\)
−0.911160 + 0.412052i \(0.864812\pi\)
\(308\) 0 0
\(309\) 582.000 1.88350
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) − 457.000i − 1.46006i −0.683413 0.730032i \(-0.739505\pi\)
0.683413 0.730032i \(-0.260495\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 213.000i 0.651376i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 362.000 1.09366 0.546828 0.837245i \(-0.315835\pi\)
0.546828 + 0.837245i \(0.315835\pi\)
\(332\) 0 0
\(333\) 234.000i 0.702703i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 167.000i − 0.495549i −0.968818 0.247774i \(-0.920301\pi\)
0.968818 0.247774i \(-0.0796992\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 923.000i − 2.69096i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 502.000 1.43840 0.719198 0.694805i \(-0.244510\pi\)
0.719198 + 0.694805i \(0.244510\pi\)
\(350\) 0 0
\(351\) −621.000 −1.76923
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 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
\(361\) −240.000 −0.664820
\(362\) 0 0
\(363\) − 363.000i − 1.00000i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 227.000i − 0.618529i −0.950976 0.309264i \(-0.899917\pi\)
0.950976 0.309264i \(-0.100083\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 577.000i − 1.54692i −0.633847 0.773458i \(-0.718525\pi\)
0.633847 0.773458i \(-0.281475\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −611.000 −1.61214 −0.806069 0.591822i \(-0.798409\pi\)
−0.806069 + 0.591822i \(0.798409\pi\)
\(380\) 0 0
\(381\) −438.000 −1.14961
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 747.000i − 1.93023i
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 793.000i 1.99748i 0.0501728 + 0.998741i \(0.484023\pi\)
−0.0501728 + 0.998741i \(0.515977\pi\)
\(398\) 0 0
\(399\) −429.000 −1.07519
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 1357.00i 3.36725i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 769.000 1.88020 0.940098 0.340905i \(-0.110733\pi\)
0.940098 + 0.340905i \(0.110733\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 66.0000i − 0.158273i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −358.000 −0.850356 −0.425178 0.905110i \(-0.639789\pi\)
−0.425178 + 0.905110i \(0.639789\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 1573.00i − 3.68384i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 503.000i 1.16166i 0.814024 + 0.580831i \(0.197272\pi\)
−0.814024 + 0.580831i \(0.802728\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 709.000 1.61503 0.807517 0.589844i \(-0.200811\pi\)
0.807517 + 0.589844i \(0.200811\pi\)
\(440\) 0 0
\(441\) 1080.00 2.44898
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 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
\(453\) − 177.000i − 0.390728i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 814.000i 1.78118i 0.454805 + 0.890591i \(0.349709\pi\)
−0.454805 + 0.890591i \(0.650291\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) − 526.000i − 1.13607i −0.823005 0.568035i \(-0.807704\pi\)
0.823005 0.568035i \(-0.192296\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −169.000 −0.360341
\(470\) 0 0
\(471\) 579.000 1.22930
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 598.000 1.24324
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 613.000i 1.25873i 0.777111 + 0.629363i \(0.216684\pi\)
−0.777111 + 0.629363i \(0.783316\pi\)
\(488\) 0 0
\(489\) −111.000 −0.226994
\(490\) 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
\(499\) −851.000 −1.70541 −0.852705 0.522392i \(-0.825040\pi\)
−0.852705 + 0.522392i \(0.825040\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1080.00i 2.13018i
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 598.000 1.17025
\(512\) 0 0
\(513\) − 297.000i − 0.578947i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 803.000i 1.53537i 0.640826 + 0.767686i \(0.278592\pi\)
−0.640826 + 0.767686i \(0.721408\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −529.000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 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
\(541\) −241.000 −0.445471 −0.222736 0.974879i \(-0.571499\pi\)
−0.222736 + 0.974879i \(0.571499\pi\)
\(542\) 0 0
\(543\) 3.00000i 0.00552486i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 506.000i − 0.925046i −0.886607 0.462523i \(-0.846944\pi\)
0.886607 0.462523i \(-0.153056\pi\)
\(548\) 0 0
\(549\) 1089.00 1.98361
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1846.00i 3.33816i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −1909.00 −3.41503
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1053.00i 1.85714i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −181.000 −0.316988 −0.158494 0.987360i \(-0.550664\pi\)
−0.158494 + 0.987360i \(0.550664\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1033.00i 1.79029i 0.445770 + 0.895147i \(0.352930\pi\)
−0.445770 + 0.895147i \(0.647070\pi\)
\(578\) 0 0
\(579\) 429.000 0.740933
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −649.000 −1.10187
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 327.000i − 0.547739i
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 1199.00 1.99501 0.997504 0.0706077i \(-0.0224939\pi\)
0.997504 + 0.0706077i \(0.0224939\pi\)
\(602\) 0 0
\(603\) − 117.000i − 0.194030i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 814.000i 1.34102i 0.741900 + 0.670511i \(0.233925\pi\)
−0.741900 + 0.670511i \(0.766075\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 1126.00i − 1.83687i −0.395574 0.918434i \(-0.629454\pi\)
0.395574 0.918434i \(-0.370546\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 949.000 1.53312 0.766559 0.642174i \(-0.221967\pi\)
0.766559 + 0.642174i \(0.221967\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 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
\(631\) −1261.00 −1.99842 −0.999208 0.0398015i \(-0.987327\pi\)
−0.999208 + 0.0398015i \(0.987327\pi\)
\(632\) 0 0
\(633\) − 1257.00i − 1.98578i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 2760.00i − 4.33281i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 314.000i 0.488336i 0.969733 + 0.244168i \(0.0785148\pi\)
−0.969733 + 0.244168i \(0.921485\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 2301.00 3.53456
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 414.000i 0.630137i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 122.000 0.184569 0.0922844 0.995733i \(-0.470583\pi\)
0.0922844 + 0.995733i \(0.470583\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 249.000 0.372197
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1154.00i 1.71471i 0.514725 + 0.857355i \(0.327894\pi\)
−0.514725 + 0.857355i \(0.672106\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 2171.00 3.19735
\(680\) 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
\(687\) − 1227.00i − 1.78603i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1318.00 −1.90738 −0.953690 0.300790i \(-0.902750\pi\)
−0.953690 + 0.300790i \(0.902750\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 286.000i 0.406828i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1391.00 −1.96192 −0.980959 0.194214i \(-0.937784\pi\)
−0.980959 + 0.194214i \(0.937784\pi\)
\(710\) 0 0
\(711\) −1278.00 −1.79747
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) −2522.00 −3.49792
\(722\) 0 0
\(723\) − 1437.00i − 1.98755i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 947.000i − 1.30261i −0.758815 0.651307i \(-0.774221\pi\)
0.758815 0.651307i \(-0.225779\pi\)
\(728\) 0 0
\(729\) −729.000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1034.00i 1.41064i 0.708888 + 0.705321i \(0.249197\pi\)
−0.708888 + 0.705321i \(0.750803\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1222.00 1.65359 0.826793 0.562506i \(-0.190163\pi\)
0.826793 + 0.562506i \(0.190163\pi\)
\(740\) 0 0
\(741\) −759.000 −1.02429
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1202.00 1.60053 0.800266 0.599645i \(-0.204691\pi\)
0.800266 + 0.599645i \(0.204691\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 673.000i 0.889036i 0.895770 + 0.444518i \(0.146625\pi\)
−0.895770 + 0.444518i \(0.853375\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) − 923.000i − 1.20970i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −671.000 −0.872562 −0.436281 0.899811i \(-0.643705\pi\)
−0.436281 + 0.899811i \(0.643705\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 1014.00i − 1.30502i
\(778\) 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
\(787\) 613.000i 0.778907i 0.921046 + 0.389454i \(0.127336\pi\)
−0.921046 + 0.389454i \(0.872664\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) − 2783.00i − 3.50946i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 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
\(811\) −1261.00 −1.55487 −0.777435 0.628963i \(-0.783480\pi\)
−0.777435 + 0.628963i \(0.783480\pi\)
\(812\) 0 0
\(813\) − 726.000i − 0.892989i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 913.000i − 1.11750i
\(818\) 0 0
\(819\) 2691.00 3.28571
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 563.000i 0.684083i 0.939685 + 0.342041i \(0.111118\pi\)
−0.939685 + 0.342041i \(0.888882\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −458.000 −0.552473 −0.276236 0.961090i \(-0.589087\pi\)
−0.276236 + 0.961090i \(0.589087\pi\)
\(830\) 0 0
\(831\) −1221.00 −1.46931
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1593.00i 1.90323i
\(838\) 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
\(847\) 1573.00i 1.85714i
\(848\) 0 0
\(849\) −1551.00 −1.82686
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 1177.00i − 1.37984i −0.723888 0.689918i \(-0.757647\pi\)
0.723888 0.689918i \(-0.242353\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −1418.00 −1.65076 −0.825378 0.564580i \(-0.809038\pi\)
−0.825378 + 0.564580i \(0.809038\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 867.000i 1.00000i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −299.000 −0.343284
\(872\) 0 0
\(873\) 1503.00i 1.72165i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 1727.00i − 1.96921i −0.174785 0.984607i \(-0.555923\pi\)
0.174785 0.984607i \(-0.444077\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 443.000i 0.501699i 0.968026 + 0.250849i \(0.0807099\pi\)
−0.968026 + 0.250849i \(0.919290\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 1898.00 2.13498
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 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
\(903\) 3237.00i 3.58472i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 214.000i 0.235943i 0.993017 + 0.117971i \(0.0376391\pi\)
−0.993017 + 0.117971i \(0.962361\pi\)
\(908\) 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
\(919\) −971.000 −1.05658 −0.528292 0.849063i \(-0.677167\pi\)
−0.528292 + 0.849063i \(0.677167\pi\)
\(920\) 0 0
\(921\) 759.000 0.824104
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 1746.00i − 1.88350i
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1320.00 1.41783
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 1847.00i − 1.97118i −0.169138 0.985592i \(-0.554098\pi\)
0.169138 0.985592i \(-0.445902\pi\)
\(938\) 0 0
\(939\) −1371.00 −1.46006
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 1058.00 1.11486
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2520.00 2.62227
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1534.00i 1.58635i 0.608994 + 0.793175i \(0.291573\pi\)
−0.608994 + 0.793175i \(0.708427\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 286.000i 0.293936i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 639.000 0.651376
\(982\) 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
\(991\) 1739.00 1.75479 0.877397 0.479766i \(-0.159278\pi\)
0.877397 + 0.479766i \(0.159278\pi\)
\(992\) 0 0
\(993\) − 1086.00i − 1.09366i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1894.00i 1.89970i 0.312707 + 0.949850i \(0.398764\pi\)
−0.312707 + 0.949850i \(0.601236\pi\)
\(998\) 0 0
\(999\) 702.000 0.702703
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 300.3.b.b.149.1 2
3.2 odd 2 CM 300.3.b.b.149.1 2
4.3 odd 2 1200.3.c.b.449.2 2
5.2 odd 4 300.3.g.a.101.1 1
5.3 odd 4 300.3.g.c.101.1 yes 1
5.4 even 2 inner 300.3.b.b.149.2 2
12.11 even 2 1200.3.c.b.449.2 2
15.2 even 4 300.3.g.a.101.1 1
15.8 even 4 300.3.g.c.101.1 yes 1
15.14 odd 2 inner 300.3.b.b.149.2 2
20.3 even 4 1200.3.l.a.401.1 1
20.7 even 4 1200.3.l.e.401.1 1
20.19 odd 2 1200.3.c.b.449.1 2
60.23 odd 4 1200.3.l.a.401.1 1
60.47 odd 4 1200.3.l.e.401.1 1
60.59 even 2 1200.3.c.b.449.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.3.b.b.149.1 2 1.1 even 1 trivial
300.3.b.b.149.1 2 3.2 odd 2 CM
300.3.b.b.149.2 2 5.4 even 2 inner
300.3.b.b.149.2 2 15.14 odd 2 inner
300.3.g.a.101.1 1 5.2 odd 4
300.3.g.a.101.1 1 15.2 even 4
300.3.g.c.101.1 yes 1 5.3 odd 4
300.3.g.c.101.1 yes 1 15.8 even 4
1200.3.c.b.449.1 2 20.19 odd 2
1200.3.c.b.449.1 2 60.59 even 2
1200.3.c.b.449.2 2 4.3 odd 2
1200.3.c.b.449.2 2 12.11 even 2
1200.3.l.a.401.1 1 20.3 even 4
1200.3.l.a.401.1 1 60.23 odd 4
1200.3.l.e.401.1 1 20.7 even 4
1200.3.l.e.401.1 1 60.47 odd 4