Properties

Label 75.9.d.a.74.2
Level $75$
Weight $9$
Character 75.74
Analytic conductor $30.553$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 75.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.5533957546\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 74.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 75.74
Dual form 75.9.d.a.74.1

$q$-expansion

\(f(q)\) \(=\) \(q+81.0000i q^{3} -256.000 q^{4} +4273.00i q^{7} -6561.00 q^{9} +O(q^{10})\) \(q+81.0000i q^{3} -256.000 q^{4} +4273.00i q^{7} -6561.00 q^{9} -20736.0i q^{12} +56447.0i q^{13} +65536.0 q^{16} -157967. q^{19} -346113. q^{21} -531441. i q^{27} -1.09389e6i q^{28} +1.22597e6 q^{31} +1.67962e6 q^{36} -503522. i q^{37} -4.57221e6 q^{39} -6.83707e6i q^{43} +5.30842e6i q^{48} -1.24937e7 q^{49} -1.44504e7i q^{52} -1.27953e7i q^{57} -307393. q^{61} -2.80352e7i q^{63} -1.67772e7 q^{64} +3.18748e7i q^{67} +1.61693e7i q^{73} +4.04396e7 q^{76} +1.88870e7 q^{79} +4.30467e7 q^{81} +8.86049e7 q^{84} -2.41198e8 q^{91} +9.93033e7i q^{93} +8.21325e7i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 512 q^{4} - 13122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 512 q^{4} - 13122 q^{9} + 131072 q^{16} - 315934 q^{19} - 692226 q^{21} + 2451934 q^{31} + 3359232 q^{36} - 9144414 q^{39} - 24987456 q^{49} - 614786 q^{61} - 33554432 q^{64} + 80879104 q^{76} + 37774076 q^{79} + 86093442 q^{81} + 177209856 q^{84} - 482396062 q^{91}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 81.0000i 1.00000i
\(4\) −256.000 −1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 4273.00i 1.77968i 0.456277 + 0.889838i \(0.349182\pi\)
−0.456277 + 0.889838i \(0.650818\pi\)
\(8\) 0 0
\(9\) −6561.00 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) − 20736.0i − 1.00000i
\(13\) 56447.0i 1.97637i 0.153277 + 0.988183i \(0.451017\pi\)
−0.153277 + 0.988183i \(0.548983\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 65536.0 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −157967. −1.21214 −0.606069 0.795412i \(-0.707254\pi\)
−0.606069 + 0.795412i \(0.707254\pi\)
\(20\) 0 0
\(21\) −346113. −1.77968
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 531441.i − 1.00000i
\(28\) − 1.09389e6i − 1.77968i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.22597e6 1.32749 0.663746 0.747958i \(-0.268966\pi\)
0.663746 + 0.747958i \(0.268966\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.67962e6 1.00000
\(37\) − 503522.i − 0.268665i −0.990936 0.134333i \(-0.957111\pi\)
0.990936 0.134333i \(-0.0428891\pi\)
\(38\) 0 0
\(39\) −4.57221e6 −1.97637
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) − 6.83707e6i − 1.99985i −0.0124389 0.999923i \(-0.503960\pi\)
0.0124389 0.999923i \(-0.496040\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 5.30842e6i 1.00000i
\(49\) −1.24937e7 −2.16724
\(50\) 0 0
\(51\) 0 0
\(52\) − 1.44504e7i − 1.97637i
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.27953e7i − 1.21214i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −307393. −0.0222011 −0.0111006 0.999938i \(-0.503533\pi\)
−0.0111006 + 0.999938i \(0.503533\pi\)
\(62\) 0 0
\(63\) − 2.80352e7i − 1.77968i
\(64\) −1.67772e7 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.18748e7i 1.58179i 0.611952 + 0.790895i \(0.290384\pi\)
−0.611952 + 0.790895i \(0.709616\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.61693e7i 0.569376i 0.958620 + 0.284688i \(0.0918900\pi\)
−0.958620 + 0.284688i \(0.908110\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 4.04396e7 1.21214
\(77\) 0 0
\(78\) 0 0
\(79\) 1.88870e7 0.484904 0.242452 0.970163i \(-0.422048\pi\)
0.242452 + 0.970163i \(0.422048\pi\)
\(80\) 0 0
\(81\) 4.30467e7 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 8.86049e7 1.77968
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −2.41198e8 −3.51729
\(92\) 0 0
\(93\) 9.93033e7i 1.32749i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.21325e7i 0.927744i 0.885902 + 0.463872i \(0.153540\pi\)
−0.885902 + 0.463872i \(0.846460\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 4.44490e7i 0.394923i 0.980311 + 0.197462i \(0.0632698\pi\)
−0.980311 + 0.197462i \(0.936730\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.36049e8i 1.00000i
\(109\) 2.71340e8 1.92224 0.961122 0.276125i \(-0.0890504\pi\)
0.961122 + 0.276125i \(0.0890504\pi\)
\(110\) 0 0
\(111\) 4.07853e7 0.268665
\(112\) 2.80035e8i 1.77968i
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 3.70349e8i − 1.97637i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.14359e8 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −3.13848e8 −1.32749
\(125\) 0 0
\(126\) 0 0
\(127\) 4.00562e8i 1.53977i 0.638185 + 0.769883i \(0.279686\pi\)
−0.638185 + 0.769883i \(0.720314\pi\)
\(128\) 0 0
\(129\) 5.53803e8 1.99985
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) − 6.74993e8i − 2.15721i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −7.09431e8 −1.90043 −0.950213 0.311602i \(-0.899135\pi\)
−0.950213 + 0.311602i \(0.899135\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −4.29982e8 −1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) − 1.01199e9i − 2.16724i
\(148\) 1.28902e8i 0.268665i
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −1.00464e9 −1.93243 −0.966214 0.257740i \(-0.917022\pi\)
−0.966214 + 0.257740i \(0.917022\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.17048e9 1.97637
\(157\) − 1.03379e9i − 1.70151i −0.525559 0.850757i \(-0.676144\pi\)
0.525559 0.850757i \(-0.323856\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 9.68184e7i − 0.137154i −0.997646 0.0685768i \(-0.978154\pi\)
0.997646 0.0685768i \(-0.0218458\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −2.37053e9 −2.90602
\(170\) 0 0
\(171\) 1.03642e9 1.21214
\(172\) 1.75029e9i 1.99985i
\(173\) 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
\(181\) −1.09376e9 −1.01908 −0.509539 0.860448i \(-0.670184\pi\)
−0.509539 + 0.860448i \(0.670184\pi\)
\(182\) 0 0
\(183\) − 2.48988e7i − 0.0222011i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.27085e9 1.77968
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) − 1.35895e9i − 1.00000i
\(193\) − 2.47302e9i − 1.78237i −0.453641 0.891185i \(-0.649875\pi\)
0.453641 0.891185i \(-0.350125\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 3.19839e9 2.16724
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 3.13036e9 1.99610 0.998050 0.0624175i \(-0.0198810\pi\)
0.998050 + 0.0624175i \(0.0198810\pi\)
\(200\) 0 0
\(201\) −2.58186e9 −1.58179
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 3.69931e9i 1.97637i
\(209\) 0 0
\(210\) 0 0
\(211\) −3.33759e9 −1.68385 −0.841924 0.539596i \(-0.818577\pi\)
−0.841924 + 0.539596i \(0.818577\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.23856e9i 2.36251i
\(218\) 0 0
\(219\) −1.30971e9 −0.569376
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.10425e9i 0.446527i 0.974758 + 0.223263i \(0.0716710\pi\)
−0.974758 + 0.223263i \(0.928329\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 3.27560e9i 1.21214i
\(229\) 3.75229e9 1.36444 0.682220 0.731147i \(-0.261014\pi\)
0.682220 + 0.731147i \(0.261014\pi\)
\(230\) 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
\(237\) 1.52985e9i 0.484904i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −5.19544e8 −0.154012 −0.0770059 0.997031i \(-0.524536\pi\)
−0.0770059 + 0.997031i \(0.524536\pi\)
\(242\) 0 0
\(243\) 3.48678e9i 1.00000i
\(244\) 7.86926e7 0.0222011
\(245\) 0 0
\(246\) 0 0
\(247\) − 8.91676e9i − 2.39563i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 7.17700e9i 1.77968i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4.29497e9 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 2.15155e9 0.478137
\(260\) 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
\(268\) − 8.15996e9i − 1.58179i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −2.98709e9 −0.553824 −0.276912 0.960895i \(-0.589311\pi\)
−0.276912 + 0.960895i \(0.589311\pi\)
\(272\) 0 0
\(273\) − 1.95370e10i − 3.51729i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 4.19798e9i − 0.713053i −0.934285 0.356526i \(-0.883961\pi\)
0.934285 0.356526i \(-0.116039\pi\)
\(278\) 0 0
\(279\) −8.04357e9 −1.32749
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 1.17255e10i 1.82803i 0.405678 + 0.914016i \(0.367036\pi\)
−0.405678 + 0.914016i \(0.632964\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −6.97576e9 −1.00000
\(290\) 0 0
\(291\) −6.65273e9 −0.927744
\(292\) − 4.13934e9i − 0.569376i
\(293\) 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
\(301\) 2.92148e10 3.55907
\(302\) 0 0
\(303\) 0 0
\(304\) −1.03525e10 −1.21214
\(305\) 0 0
\(306\) 0 0
\(307\) − 1.63938e10i − 1.84555i −0.385338 0.922775i \(-0.625915\pi\)
0.385338 0.922775i \(-0.374085\pi\)
\(308\) 0 0
\(309\) −3.60037e9 −0.394923
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 7.32851e9i 0.763551i 0.924255 + 0.381776i \(0.124687\pi\)
−0.924255 + 0.381776i \(0.875313\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −4.83508e9 −0.484904
\(317\) 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
\(324\) −1.10200e10 −1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 2.19786e10i 1.92224i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.62495e10 −1.35372 −0.676858 0.736113i \(-0.736659\pi\)
−0.676858 + 0.736113i \(0.736659\pi\)
\(332\) 0 0
\(333\) 3.30361e9i 0.268665i
\(334\) 0 0
\(335\) 0 0
\(336\) −2.26829e10 −1.77968
\(337\) − 1.18646e10i − 0.919886i −0.887948 0.459943i \(-0.847870\pi\)
0.887948 0.459943i \(-0.152130\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 2.87527e10i − 2.07731i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 2.96004e10 1.99524 0.997621 0.0689403i \(-0.0219618\pi\)
0.997621 + 0.0689403i \(0.0219618\pi\)
\(350\) 0 0
\(351\) 2.99983e10 1.97637
\(352\) 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
\(361\) 7.97001e9 0.469278
\(362\) 0 0
\(363\) 1.73631e10i 1.00000i
\(364\) 6.17467e10 3.51729
\(365\) 0 0
\(366\) 0 0
\(367\) 3.54815e10i 1.95586i 0.208929 + 0.977931i \(0.433002\pi\)
−0.208929 + 0.977931i \(0.566998\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) − 2.54217e10i − 1.32749i
\(373\) 3.07802e10i 1.59014i 0.606517 + 0.795070i \(0.292566\pi\)
−0.606517 + 0.795070i \(0.707434\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −3.73548e10 −1.81046 −0.905230 0.424921i \(-0.860302\pi\)
−0.905230 + 0.424921i \(0.860302\pi\)
\(380\) 0 0
\(381\) −3.24455e10 −1.53977
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.48580e10i 1.99985i
\(388\) − 2.10259e10i − 0.927744i
\(389\) 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
\(397\) 3.29221e10i 1.32533i 0.748914 + 0.662667i \(0.230575\pi\)
−0.748914 + 0.662667i \(0.769425\pi\)
\(398\) 0 0
\(399\) 5.46744e10 2.15721
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 6.92022e10i 2.62361i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −4.27011e10 −1.52597 −0.762985 0.646416i \(-0.776267\pi\)
−0.762985 + 0.646416i \(0.776267\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 1.13789e10i − 0.394923i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 5.74639e10i − 1.90043i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −1.16089e10 −0.369541 −0.184771 0.982782i \(-0.559154\pi\)
−0.184771 + 0.982782i \(0.559154\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 1.31349e9i − 0.0395108i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) − 3.48285e10i − 1.00000i
\(433\) − 9.74052e9i − 0.277096i −0.990356 0.138548i \(-0.955756\pi\)
0.990356 0.138548i \(-0.0442435\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.94631e10 −1.92224
\(437\) 0 0
\(438\) 0 0
\(439\) 7.01153e9 0.188779 0.0943897 0.995535i \(-0.469910\pi\)
0.0943897 + 0.995535i \(0.469910\pi\)
\(440\) 0 0
\(441\) 8.19713e10 2.16724
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) −1.04410e10 −0.268665
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) − 7.16890e10i − 1.77968i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) − 8.13760e10i − 1.93243i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.72608e10i 0.624991i 0.949919 + 0.312495i \(0.101165\pi\)
−0.949919 + 0.312495i \(0.898835\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) − 6.87853e10i − 1.49683i −0.663232 0.748413i \(-0.730816\pi\)
0.663232 0.748413i \(-0.269184\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 9.48093e10i 1.97637i
\(469\) −1.36201e11 −2.81507
\(470\) 0 0
\(471\) 8.37374e10 1.70151
\(472\) 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
\(481\) 2.84223e10 0.530981
\(482\) 0 0
\(483\) 0 0
\(484\) −5.48759e10 −1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) − 1.17841e10i − 0.209498i −0.994499 0.104749i \(-0.966596\pi\)
0.994499 0.104749i \(-0.0334039\pi\)
\(488\) 0 0
\(489\) 7.84229e9 0.137154
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 8.03450e10 1.32749
\(497\) 0 0
\(498\) 0 0
\(499\) 5.04931e10 0.814386 0.407193 0.913342i \(-0.366508\pi\)
0.407193 + 0.913342i \(0.366508\pi\)
\(500\) 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
\(507\) − 1.92013e11i − 2.90602i
\(508\) − 1.02544e11i − 1.53977i
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −6.90913e10 −1.01330
\(512\) 0 0
\(513\) 8.39501e10i 1.21214i
\(514\) 0 0
\(515\) 0 0
\(516\) −1.41774e11 −1.99985
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 1.15606e11i 1.54516i 0.634915 + 0.772582i \(0.281035\pi\)
−0.634915 + 0.772582i \(0.718965\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.83110e10 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 1.72798e11i 2.15721i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1.69433e11 −1.97792 −0.988962 0.148171i \(-0.952661\pi\)
−0.988962 + 0.148171i \(0.952661\pi\)
\(542\) 0 0
\(543\) − 8.85944e10i − 1.01908i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 6.18266e10i 0.690599i 0.938492 + 0.345300i \(0.112223\pi\)
−0.938492 + 0.345300i \(0.887777\pi\)
\(548\) 0 0
\(549\) 2.01681e9 0.0222011
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.07043e10i 0.862971i
\(554\) 0 0
\(555\) 0 0
\(556\) 1.81614e11 1.90043
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 3.85932e11 3.95243
\(560\) 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
\(567\) 1.83939e11i 1.77968i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 2.39863e10 0.225641 0.112821 0.993615i \(-0.464011\pi\)
0.112821 + 0.993615i \(0.464011\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.10075e11 1.00000
\(577\) − 2.14996e11i − 1.93966i −0.243775 0.969832i \(-0.578386\pi\)
0.243775 0.969832i \(-0.421614\pi\)
\(578\) 0 0
\(579\) 2.00314e11 1.78237
\(580\) 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 \(0.5\pi\)
\(588\) 2.59070e11i 2.16724i
\(589\) −1.93662e11 −1.60910
\(590\) 0 0
\(591\) 0 0
\(592\) − 3.29988e10i − 0.268665i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2.53559e11i 1.99610i
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1.88317e11 −1.44341 −0.721707 0.692199i \(-0.756642\pi\)
−0.721707 + 0.692199i \(0.756642\pi\)
\(602\) 0 0
\(603\) − 2.09131e11i − 1.58179i
\(604\) 2.57188e11 1.93243
\(605\) 0 0
\(606\) 0 0
\(607\) 2.65989e11i 1.95933i 0.200634 + 0.979666i \(0.435700\pi\)
−0.200634 + 0.979666i \(0.564300\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 1.57998e10i − 0.111894i −0.998434 0.0559472i \(-0.982182\pi\)
0.998434 0.0559472i \(-0.0178179\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −5.00608e10 −0.340985 −0.170493 0.985359i \(-0.554536\pi\)
−0.170493 + 0.985359i \(0.554536\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −2.99644e11 −1.97637
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 2.64651e11i 1.70151i
\(629\) 0 0
\(630\) 0 0
\(631\) −1.12984e11 −0.712686 −0.356343 0.934355i \(-0.615976\pi\)
−0.356343 + 0.934355i \(0.615976\pi\)
\(632\) 0 0
\(633\) − 2.70345e11i − 1.68385i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 7.05233e11i − 4.28327i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 1.88544e11i 1.10298i 0.834181 + 0.551490i \(0.185941\pi\)
−0.834181 + 0.551490i \(0.814059\pi\)
\(644\) 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
\(651\) −4.24323e11 −2.36251
\(652\) 2.47855e10i 0.137154i
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 1.06087e11i − 0.569376i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 3.56009e11 1.86490 0.932449 0.361302i \(-0.117668\pi\)
0.932449 + 0.361302i \(0.117668\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −8.94441e10 −0.446527
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) − 2.28934e11i − 1.11597i −0.829853 0.557983i \(-0.811576\pi\)
0.829853 0.557983i \(-0.188424\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 6.06856e11 2.90602
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) −3.50952e11 −1.65108
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) −2.65324e11 −1.21214
\(685\) 0 0
\(686\) 0 0
\(687\) 3.03936e11i 1.36444i
\(688\) − 4.48074e11i − 1.99985i
\(689\) 0 0
\(690\) 0 0
\(691\) 1.55801e11 0.683374 0.341687 0.939814i \(-0.389002\pi\)
0.341687 + 0.939814i \(0.389002\pi\)
\(692\) 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
\(703\) 7.95399e10i 0.325659i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.29057e11 −0.510735 −0.255367 0.966844i \(-0.582196\pi\)
−0.255367 + 0.966844i \(0.582196\pi\)
\(710\) 0 0
\(711\) −1.23918e11 −0.484904
\(712\) 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
\(721\) −1.89930e11 −0.702835
\(722\) 0 0
\(723\) − 4.20830e10i − 0.154012i
\(724\) 2.80002e11 1.01908
\(725\) 0 0
\(726\) 0 0
\(727\) − 4.11506e11i − 1.47312i −0.676372 0.736560i \(-0.736449\pi\)
0.676372 0.736560i \(-0.263551\pi\)
\(728\) 0 0
\(729\) −2.82430e11 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 6.37410e9i 0.0222011i
\(733\) − 5.77330e11i − 1.99990i −0.0100913 0.999949i \(-0.503212\pi\)
0.0100913 0.999949i \(-0.496788\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 4.35662e11 1.46074 0.730368 0.683054i \(-0.239349\pi\)
0.730368 + 0.683054i \(0.239349\pi\)
\(740\) 0 0
\(741\) 7.22258e11 2.39563
\(742\) 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
\(751\) −5.35831e11 −1.68449 −0.842244 0.539097i \(-0.818766\pi\)
−0.842244 + 0.539097i \(0.818766\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −5.81337e11 −1.77968
\(757\) − 6.36051e11i − 1.93691i −0.249197 0.968453i \(-0.580167\pi\)
0.249197 0.968453i \(-0.419833\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.15944e12i 3.42097i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 3.47892e11i 1.00000i
\(769\) 5.07613e11 1.45153 0.725767 0.687941i \(-0.241485\pi\)
0.725767 + 0.687941i \(0.241485\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 6.33092e11i 1.78237i
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.74276e11i 0.478137i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −8.18789e11 −2.16724
\(785\) 0 0
\(786\) 0 0
\(787\) 6.52899e11i 1.70195i 0.525205 + 0.850976i \(0.323989\pi\)
−0.525205 + 0.850976i \(0.676011\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) − 1.73514e10i − 0.0438775i
\(794\) 0 0
\(795\) 0 0
\(796\) −8.01373e11 −1.99610
\(797\) 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
\(804\) 6.60957e11 1.58179
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 7.00891e11 1.62019 0.810097 0.586296i \(-0.199415\pi\)
0.810097 + 0.586296i \(0.199415\pi\)
\(812\) 0 0
\(813\) − 2.41954e11i − 0.553824i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.08003e12i 2.42409i
\(818\) 0 0
\(819\) 1.58250e12 3.51729
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 7.02937e11i 1.53220i 0.642719 + 0.766102i \(0.277806\pi\)
−0.642719 + 0.766102i \(0.722194\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −4.11968e11 −0.872258 −0.436129 0.899884i \(-0.643651\pi\)
−0.436129 + 0.899884i \(0.643651\pi\)
\(830\) 0 0
\(831\) 3.40037e11 0.713053
\(832\) − 9.47024e11i − 1.97637i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 6.51529e11i − 1.32749i
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.00246e11 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 8.54422e11 1.68385
\(845\) 0 0
\(846\) 0 0
\(847\) 9.15955e11i 1.77968i
\(848\) 0 0
\(849\) −9.49762e11 −1.82803
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 4.38996e11i 0.829210i 0.910001 + 0.414605i \(0.136080\pi\)
−0.910001 + 0.414605i \(0.863920\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 8.02752e11 1.47438 0.737189 0.675686i \(-0.236153\pi\)
0.737189 + 0.675686i \(0.236153\pi\)
\(860\) 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
\(867\) − 5.65036e11i − 1.00000i
\(868\) − 1.34107e12i − 2.36251i
\(869\) 0 0
\(870\) 0 0
\(871\) −1.79924e12 −3.12620
\(872\) 0 0
\(873\) − 5.38871e11i − 0.927744i
\(874\) 0 0
\(875\) 0 0
\(876\) 3.35286e11 0.569376
\(877\) 1.11362e12i 1.88252i 0.337677 + 0.941262i \(0.390359\pi\)
−0.337677 + 0.941262i \(0.609641\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) − 1.21517e12i − 1.99891i −0.0329612 0.999457i \(-0.510494\pi\)
0.0329612 0.999457i \(-0.489506\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −1.71160e12 −2.74028
\(890\) 0 0
\(891\) 0 0
\(892\) − 2.82688e11i − 0.446527i
\(893\) 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
\(903\) 2.36640e12i 3.55907i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 1.20490e12i − 1.78042i −0.455547 0.890212i \(-0.650556\pi\)
0.455547 0.890212i \(-0.349444\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) − 8.38555e11i − 1.21214i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −9.60587e11 −1.36444
\(917\) 0 0
\(918\) 0 0
\(919\) −1.41417e12 −1.98261 −0.991307 0.131571i \(-0.957998\pi\)
−0.991307 + 0.131571i \(0.957998\pi\)
\(920\) 0 0
\(921\) 1.32790e12 1.84555
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 2.91630e11i − 0.394923i
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1.97360e12 2.62700
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 2.39864e11i − 0.311176i −0.987822 0.155588i \(-0.950273\pi\)
0.987822 0.155588i \(-0.0497272\pi\)
\(938\) 0 0
\(939\) −5.93609e11 −0.763551
\(940\) 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 \(0.5\pi\)
\(948\) − 3.91642e11i − 0.484904i
\(949\) −9.12707e11 −1.12530
\(950\) 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
\(961\) 6.50104e11 0.762236
\(962\) 0 0
\(963\) 0 0
\(964\) 1.33003e11 0.154012
\(965\) 0 0
\(966\) 0 0
\(967\) 1.51552e12i 1.73322i 0.498984 + 0.866611i \(0.333707\pi\)
−0.498984 + 0.866611i \(0.666293\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) − 8.92617e11i − 1.00000i
\(973\) − 3.03140e12i − 3.38214i
\(974\) 0 0
\(975\) 0 0
\(976\) −2.01453e10 −0.0222011
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.78026e12 −1.92224
\(982\) 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
\(988\) 2.28269e12i 2.39563i
\(989\) 0 0
\(990\) 0 0
\(991\) −1.11521e12 −1.15627 −0.578136 0.815940i \(-0.696220\pi\)
−0.578136 + 0.815940i \(0.696220\pi\)
\(992\) 0 0
\(993\) − 1.31621e12i − 1.35372i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 5.81390e11i − 0.588420i −0.955741 0.294210i \(-0.904944\pi\)
0.955741 0.294210i \(-0.0950565\pi\)
\(998\) 0 0
\(999\) −2.67592e11 −0.268665
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 75.9.d.a.74.2 2
3.2 odd 2 CM 75.9.d.a.74.2 2
5.2 odd 4 75.9.c.b.26.1 yes 1
5.3 odd 4 75.9.c.a.26.1 1
5.4 even 2 inner 75.9.d.a.74.1 2
15.2 even 4 75.9.c.b.26.1 yes 1
15.8 even 4 75.9.c.a.26.1 1
15.14 odd 2 inner 75.9.d.a.74.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.9.c.a.26.1 1 5.3 odd 4
75.9.c.a.26.1 1 15.8 even 4
75.9.c.b.26.1 yes 1 5.2 odd 4
75.9.c.b.26.1 yes 1 15.2 even 4
75.9.d.a.74.1 2 5.4 even 2 inner
75.9.d.a.74.1 2 15.14 odd 2 inner
75.9.d.a.74.2 2 1.1 even 1 trivial
75.9.d.a.74.2 2 3.2 odd 2 CM