Properties

Label 225.3.g.d.82.1
Level $225$
Weight $3$
Character 225.82
Analytic conductor $6.131$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,3,Mod(82,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.82");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 225.g (of order \(4\), degree \(2\), 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: \(6.13080594811\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 82.1
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 225.82
Dual form 225.3.g.d.118.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000i q^{4} +(-6.12372 - 6.12372i) q^{7} +(-18.3712 + 18.3712i) q^{13} -16.0000 q^{16} -37.0000i q^{19} +(-24.4949 + 24.4949i) q^{28} +13.0000 q^{31} +(-48.9898 - 48.9898i) q^{37} +(42.8661 - 42.8661i) q^{43} +26.0000i q^{49} +(73.4847 + 73.4847i) q^{52} +47.0000 q^{61} +64.0000i q^{64} +(55.1135 + 55.1135i) q^{67} +(97.9796 - 97.9796i) q^{73} -148.000 q^{76} +142.000i q^{79} +225.000 q^{91} +(-67.3610 - 67.3610i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 64 q^{16} + 52 q^{31} + 188 q^{61} - 592 q^{76} + 900 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\)

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 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 4.00000i 1.00000i
\(5\) 0 0
\(6\) 0 0
\(7\) −6.12372 6.12372i −0.874818 0.874818i 0.118175 0.992993i \(-0.462296\pi\)
−0.992993 + 0.118175i \(0.962296\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −18.3712 + 18.3712i −1.41317 + 1.41317i −0.679387 + 0.733780i \(0.737754\pi\)
−0.733780 + 0.679387i \(0.762246\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −16.0000 −1.00000
\(17\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0 0
\(19\) 37.0000i 1.94737i −0.227901 0.973684i \(-0.573186\pi\)
0.227901 0.973684i \(-0.426814\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) −24.4949 + 24.4949i −0.874818 + 0.874818i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 13.0000 0.419355 0.209677 0.977771i \(-0.432759\pi\)
0.209677 + 0.977771i \(0.432759\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −48.9898 48.9898i −1.32405 1.32405i −0.910467 0.413581i \(-0.864278\pi\)
−0.413581 0.910467i \(-0.635722\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 42.8661 42.8661i 0.996885 0.996885i −0.00310980 0.999995i \(-0.500990\pi\)
0.999995 + 0.00310980i \(0.000989883\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0 0
\(49\) 26.0000i 0.530612i
\(50\) 0 0
\(51\) 0 0
\(52\) 73.4847 + 73.4847i 1.41317 + 1.41317i
\(53\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 47.0000 0.770492 0.385246 0.922814i \(-0.374117\pi\)
0.385246 + 0.922814i \(0.374117\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 55.1135 + 55.1135i 0.822590 + 0.822590i 0.986479 0.163889i \(-0.0524039\pi\)
−0.163889 + 0.986479i \(0.552404\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 97.9796 97.9796i 1.34219 1.34219i 0.448306 0.893880i \(-0.352027\pi\)
0.893880 0.448306i \(-0.147973\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −148.000 −1.94737
\(77\) 0 0
\(78\) 0 0
\(79\) 142.000i 1.79747i 0.438494 + 0.898734i \(0.355512\pi\)
−0.438494 + 0.898734i \(0.644488\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 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
\(91\) 225.000 2.47253
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −67.3610 67.3610i −0.694443 0.694443i 0.268763 0.963206i \(-0.413385\pi\)
−0.963206 + 0.268763i \(0.913385\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 48.9898 48.9898i 0.475629 0.475629i −0.428102 0.903731i \(-0.640817\pi\)
0.903731 + 0.428102i \(0.140817\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) 0 0
\(109\) 143.000i 1.31193i −0.754793 0.655963i \(-0.772263\pi\)
0.754793 0.655963i \(-0.227737\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 97.9796 + 97.9796i 0.874818 + 0.874818i
\(113\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(114\) 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
\(124\) 52.0000i 0.419355i
\(125\) 0 0
\(126\) 0 0
\(127\) −146.969 146.969i −1.15724 1.15724i −0.985067 0.172172i \(-0.944921\pi\)
−0.172172 0.985067i \(-0.555079\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −226.578 + 226.578i −1.70359 + 1.70359i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(138\) 0 0
\(139\) 22.0000i 0.158273i −0.996864 0.0791367i \(-0.974784\pi\)
0.996864 0.0791367i \(-0.0252164\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −195.959 + 195.959i −1.32405 + 1.32405i
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −227.000 −1.50331 −0.751656 0.659556i \(-0.770744\pi\)
−0.751656 + 0.659556i \(0.770744\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 30.6186 + 30.6186i 0.195023 + 0.195023i 0.797863 0.602839i \(-0.205964\pi\)
−0.602839 + 0.797863i \(0.705964\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 91.8559 91.8559i 0.563533 0.563533i −0.366776 0.930309i \(-0.619539\pi\)
0.930309 + 0.366776i \(0.119539\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 506.000i 2.99408i
\(170\) 0 0
\(171\) 0 0
\(172\) −171.464 171.464i −0.996885 0.996885i
\(173\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) 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
\(181\) 313.000 1.72928 0.864641 0.502390i \(-0.167546\pi\)
0.864641 + 0.502390i \(0.167546\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −214.330 + 214.330i −1.11052 + 1.11052i −0.117440 + 0.993080i \(0.537469\pi\)
−0.993080 + 0.117440i \(0.962531\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 104.000 0.530612
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 0 0
\(199\) 277.000i 1.39196i 0.718061 + 0.695980i \(0.245030\pi\)
−0.718061 + 0.695980i \(0.754970\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 293.939 293.939i 1.41317 1.41317i
\(209\) 0 0
\(210\) 0 0
\(211\) −253.000 −1.19905 −0.599526 0.800355i \(-0.704644\pi\)
−0.599526 + 0.800355i \(0.704644\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −79.6084 79.6084i −0.366859 0.366859i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 104.103 104.103i 0.466831 0.466831i −0.434055 0.900886i \(-0.642918\pi\)
0.900886 + 0.434055i \(0.142918\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) 383.000i 1.67249i 0.548357 + 0.836245i \(0.315254\pi\)
−0.548357 + 0.836245i \(0.684746\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −193.000 −0.800830 −0.400415 0.916334i \(-0.631134\pi\)
−0.400415 + 0.916334i \(0.631134\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 188.000i 0.770492i
\(245\) 0 0
\(246\) 0 0
\(247\) 679.733 + 679.733i 2.75196 + 2.75196i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(258\) 0 0
\(259\) 600.000i 2.31660i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 220.454 220.454i 0.822590 0.822590i
\(269\) 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
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 116.351 + 116.351i 0.420039 + 0.420039i 0.885217 0.465178i \(-0.154010\pi\)
−0.465178 + 0.885217i \(0.654010\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 398.042 398.042i 1.40651 1.40651i 0.629546 0.776964i \(-0.283241\pi\)
0.776964 0.629546i \(-0.216759\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000i 1.00000i
\(290\) 0 0
\(291\) 0 0
\(292\) −391.918 391.918i −1.34219 1.34219i
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −525.000 −1.74419
\(302\) 0 0
\(303\) 0 0
\(304\) 592.000i 1.94737i
\(305\) 0 0
\(306\) 0 0
\(307\) −42.8661 42.8661i −0.139629 0.139629i 0.633837 0.773466i \(-0.281479\pi\)
−0.773466 + 0.633837i \(0.781479\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −128.598 + 128.598i −0.410857 + 0.410857i −0.882037 0.471180i \(-0.843828\pi\)
0.471180 + 0.882037i \(0.343828\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 568.000 1.79747
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 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
\(327\) 0 0
\(328\) 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
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −128.598 128.598i −0.381597 0.381597i 0.490080 0.871677i \(-0.336967\pi\)
−0.871677 + 0.490080i \(0.836967\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −140.846 + 140.846i −0.410629 + 0.410629i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(348\) 0 0
\(349\) 502.000i 1.43840i −0.694805 0.719198i \(-0.744510\pi\)
0.694805 0.719198i \(-0.255490\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(354\) 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
\(361\) −1008.00 −2.79224
\(362\) 0 0
\(363\) 0 0
\(364\) 900.000i 2.47253i
\(365\) 0 0
\(366\) 0 0
\(367\) 385.795 + 385.795i 1.05121 + 1.05121i 0.998616 + 0.0525956i \(0.0167494\pi\)
0.0525956 + 0.998616i \(0.483251\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −520.517 + 520.517i −1.39549 + 1.39549i −0.583052 + 0.812435i \(0.698142\pi\)
−0.812435 + 0.583052i \(0.801858\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 83.0000i 0.218997i 0.993987 + 0.109499i \(0.0349245\pi\)
−0.993987 + 0.109499i \(0.965075\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −269.444 + 269.444i −0.694443 + 0.694443i
\(389\) 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
\(397\) −471.527 471.527i −1.18772 1.18772i −0.977695 0.210030i \(-0.932644\pi\)
−0.210030 0.977695i \(-0.567356\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −238.825 + 238.825i −0.592618 + 0.592618i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 143.000i 0.349633i 0.984601 + 0.174817i \(0.0559333\pi\)
−0.984601 + 0.174817i \(0.944067\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −195.959 195.959i −0.475629 0.475629i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 358.000 0.850356 0.425178 0.905110i \(-0.360211\pi\)
0.425178 + 0.905110i \(0.360211\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −287.815 287.815i −0.674040 0.674040i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −557.259 + 557.259i −1.28697 + 1.28697i −0.350355 + 0.936617i \(0.613939\pi\)
−0.936617 + 0.350355i \(0.886061\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −572.000 −1.31193
\(437\) 0 0
\(438\) 0 0
\(439\) 803.000i 1.82916i −0.404408 0.914579i \(-0.632522\pi\)
0.404408 0.914579i \(-0.367478\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 391.918 391.918i 0.874818 0.874818i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −293.939 293.939i −0.643192 0.643192i 0.308147 0.951339i \(-0.400291\pi\)
−0.951339 + 0.308147i \(0.900291\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 538.888 538.888i 1.16390 1.16390i 0.180291 0.983613i \(-0.442296\pi\)
0.983613 0.180291i \(-0.0577040\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(468\) 0 0
\(469\) 675.000i 1.43923i
\(470\) 0 0
\(471\) 0 0
\(472\) 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
\(481\) 1800.00 3.74220
\(482\) 0 0
\(483\) 0 0
\(484\) 484.000i 1.00000i
\(485\) 0 0
\(486\) 0 0
\(487\) 642.991 + 642.991i 1.32031 + 1.32031i 0.913523 + 0.406787i \(0.133351\pi\)
0.406787 + 0.913523i \(0.366649\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −208.000 −0.419355
\(497\) 0 0
\(498\) 0 0
\(499\) 877.000i 1.75752i 0.477269 + 0.878758i \(0.341627\pi\)
−0.477269 + 0.878758i \(0.658373\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −587.878 + 587.878i −1.15724 + 1.15724i
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −1200.00 −2.34834
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 728.723 728.723i 1.39335 1.39335i 0.575670 0.817682i \(-0.304741\pi\)
0.817682 0.575670i \(-0.195259\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) 906.311 + 906.311i 1.70359 + 1.70359i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −793.000 −1.46580 −0.732902 0.680334i \(-0.761835\pi\)
−0.732902 + 0.680334i \(0.761835\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −685.857 685.857i −1.25385 1.25385i −0.953979 0.299873i \(-0.903056\pi\)
−0.299873 0.953979i \(-0.596944\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 869.569 869.569i 1.57246 1.57246i
\(554\) 0 0
\(555\) 0 0
\(556\) −88.0000 −0.158273
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) 1575.00i 2.81753i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 1067.00 1.86865 0.934326 0.356420i \(-0.116003\pi\)
0.934326 + 0.356420i \(0.116003\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 814.455 + 814.455i 1.41153 + 1.41153i 0.749272 + 0.662262i \(0.230404\pi\)
0.662262 + 0.749272i \(0.269596\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(588\) 0 0
\(589\) 481.000i 0.816638i
\(590\) 0 0
\(591\) 0 0
\(592\) 783.837 + 783.837i 1.32405 + 1.32405i
\(593\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 673.000 1.11980 0.559900 0.828560i \(-0.310839\pi\)
0.559900 + 0.828560i \(0.310839\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 908.000i 1.50331i
\(605\) 0 0
\(606\) 0 0
\(607\) −636.867 636.867i −1.04920 1.04920i −0.998725 0.0504797i \(-0.983925\pi\)
−0.0504797 0.998725i \(-0.516075\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 342.929 342.929i 0.559427 0.559427i −0.369718 0.929144i \(-0.620546\pi\)
0.929144 + 0.369718i \(0.120546\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(618\) 0 0
\(619\) 1163.00i 1.87884i −0.342773 0.939418i \(-0.611366\pi\)
0.342773 0.939418i \(-0.388634\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 122.474 122.474i 0.195023 0.195023i
\(629\) 0 0
\(630\) 0 0
\(631\) 587.000 0.930269 0.465135 0.885240i \(-0.346006\pi\)
0.465135 + 0.885240i \(0.346006\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −477.650 477.650i −0.749844 0.749844i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 881.816 881.816i 1.37141 1.37141i 0.513052 0.858358i \(-0.328515\pi\)
0.858358 0.513052i \(-0.171485\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −367.423 367.423i −0.563533 0.563533i
\(653\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(654\) 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
\(661\) −122.000 −0.184569 −0.0922844 0.995733i \(-0.529417\pi\)
−0.0922844 + 0.995733i \(0.529417\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −489.898 + 489.898i −0.727932 + 0.727932i −0.970207 0.242276i \(-0.922106\pi\)
0.242276 + 0.970207i \(0.422106\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −2024.00 −2.99408
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) 825.000i 1.21502i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −685.857 + 685.857i −0.996885 + 0.996885i
\(689\) 0 0
\(690\) 0 0
\(691\) 1318.00 1.90738 0.953690 0.300790i \(-0.0972504\pi\)
0.953690 + 0.300790i \(0.0972504\pi\)
\(692\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −1812.62 + 1812.62i −2.57841 + 2.57841i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 457.000i 0.644570i −0.946643 0.322285i \(-0.895549\pi\)
0.946643 0.322285i \(-0.104451\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 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
\(721\) −600.000 −0.832178
\(722\) 0 0
\(723\) 0 0
\(724\) 1252.00i 1.72928i
\(725\) 0 0
\(726\) 0 0
\(727\) −189.835 189.835i −0.261122 0.261122i 0.564388 0.825510i \(-0.309112\pi\)
−0.825510 + 0.564388i \(0.809112\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 734.847 734.847i 1.00252 1.00252i 0.00252286 0.999997i \(-0.499197\pi\)
0.999997 0.00252286i \(-0.000803052\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1222.00i 1.65359i −0.562506 0.826793i \(-0.690163\pi\)
0.562506 0.826793i \(-0.309837\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 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 \(-0.795309\pi\)
−0.800266 + 0.599645i \(0.795309\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 67.3610 + 67.3610i 0.0889841 + 0.0889841i 0.750198 0.661214i \(-0.229958\pi\)
−0.661214 + 0.750198i \(0.729958\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −875.693 + 875.693i −1.14770 + 1.14770i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 863.000i 1.12224i −0.827736 0.561118i \(-0.810371\pi\)
0.827736 0.561118i \(-0.189629\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 857.321 + 857.321i 1.11052 + 1.11052i
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 416.000i 0.530612i
\(785\) 0 0
\(786\) 0 0
\(787\) −887.940 887.940i −1.12826 1.12826i −0.990460 0.137799i \(-0.955997\pi\)
−0.137799 0.990460i \(-0.544003\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −863.445 + 863.445i −1.08883 + 1.08883i
\(794\) 0 0
\(795\) 0 0
\(796\) 1108.00 1.39196
\(797\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 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
\(811\) 253.000 0.311961 0.155980 0.987760i \(-0.450146\pi\)
0.155980 + 0.987760i \(0.450146\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1586.04 1586.04i −1.94130 1.94130i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −202.083 + 202.083i −0.245544 + 0.245544i −0.819139 0.573595i \(-0.805548\pi\)
0.573595 + 0.819139i \(0.305548\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(828\) 0 0
\(829\) 458.000i 0.552473i 0.961090 + 0.276236i \(0.0890873\pi\)
−0.961090 + 0.276236i \(0.910913\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1175.76 1175.76i −1.41317 1.41317i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 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
\(844\) 1012.00i 1.19905i
\(845\) 0 0
\(846\) 0 0
\(847\) 740.971 + 740.971i 0.874818 + 0.874818i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1157.38 1157.38i 1.35684 1.35684i 0.479054 0.877786i \(-0.340980\pi\)
0.877786 0.479054i \(-0.159020\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(858\) 0 0
\(859\) 1418.00i 1.65076i −0.564580 0.825378i \(-0.690962\pi\)
0.564580 0.825378i \(-0.309038\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) −318.434 + 318.434i −0.366859 + 0.366859i
\(869\) 0 0
\(870\) 0 0
\(871\) −2025.00 −2.32491
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 949.177 + 949.177i 1.08230 + 1.08230i 0.996295 + 0.0860055i \(0.0274103\pi\)
0.0860055 + 0.996295i \(0.472590\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 875.693 875.693i 0.991724 0.991724i −0.00824171 0.999966i \(-0.502623\pi\)
0.999966 + 0.00824171i \(0.00262345\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(888\) 0 0
\(889\) 1800.00i 2.02475i
\(890\) 0 0
\(891\) 0 0
\(892\) −416.413 416.413i −0.466831 0.466831i
\(893\) 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
\(907\) 1273.73 + 1273.73i 1.40434 + 1.40434i 0.785587 + 0.618751i \(0.212361\pi\)
0.618751 + 0.785587i \(0.287639\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1532.00 1.67249
\(917\) 0 0
\(918\) 0 0
\(919\) 1837.00i 1.99891i −0.0329825 0.999456i \(-0.510501\pi\)
0.0329825 0.999456i \(-0.489499\pi\)
\(920\) 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
\(931\) 962.000 1.03330
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1243.12 + 1243.12i 1.32670 + 1.32670i 0.908233 + 0.418465i \(0.137432\pi\)
0.418465 + 0.908233i \(0.362568\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(948\) 0 0
\(949\) 3600.00i 3.79347i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −792.000 −0.824142
\(962\) 0 0
\(963\) 0 0
\(964\) 772.000i 0.800830i
\(965\) 0 0
\(966\) 0 0
\(967\) 832.827 + 832.827i 0.861248 + 0.861248i 0.991483 0.130235i \(-0.0415733\pi\)
−0.130235 + 0.991483i \(0.541573\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −134.722 + 134.722i −0.138460 + 0.138460i
\(974\) 0 0
\(975\) 0 0
\(976\) −752.000 −0.770492
\(977\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 2718.93 2718.93i 2.75196 2.75196i
\(989\) 0 0
\(990\) 0 0
\(991\) −1693.00 −1.70838 −0.854188 0.519965i \(-0.825945\pi\)
−0.854188 + 0.519965i \(0.825945\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 440.908 + 440.908i 0.442235 + 0.442235i 0.892762 0.450528i \(-0.148764\pi\)
−0.450528 + 0.892762i \(0.648764\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 225.3.g.d.82.1 4
3.2 odd 2 CM 225.3.g.d.82.1 4
5.2 odd 4 inner 225.3.g.d.118.2 yes 4
5.3 odd 4 inner 225.3.g.d.118.1 yes 4
5.4 even 2 inner 225.3.g.d.82.2 yes 4
15.2 even 4 inner 225.3.g.d.118.2 yes 4
15.8 even 4 inner 225.3.g.d.118.1 yes 4
15.14 odd 2 inner 225.3.g.d.82.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.3.g.d.82.1 4 1.1 even 1 trivial
225.3.g.d.82.1 4 3.2 odd 2 CM
225.3.g.d.82.2 yes 4 5.4 even 2 inner
225.3.g.d.82.2 yes 4 15.14 odd 2 inner
225.3.g.d.118.1 yes 4 5.3 odd 4 inner
225.3.g.d.118.1 yes 4 15.8 even 4 inner
225.3.g.d.118.2 yes 4 5.2 odd 4 inner
225.3.g.d.118.2 yes 4 15.2 even 4 inner