Properties

Label 225.3.g.c.118.1
Level $225$
Weight $3$
Character 225.118
Analytic conductor $6.131$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $8$

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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: \(\Q(i, \sqrt{30})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 118.1
Root \(-2.73861 + 2.73861i\) of defining polynomial
Character \(\chi\) \(=\) 225.118
Dual form 225.3.g.c.82.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.73861 + 2.73861i) q^{2} -11.0000i q^{4} +(19.1703 + 19.1703i) q^{8} -61.0000 q^{16} +(-21.9089 + 21.9089i) q^{17} +22.0000i q^{19} +(-21.9089 - 21.9089i) q^{23} -2.00000 q^{31} +(90.3742 - 90.3742i) q^{32} -120.000i q^{34} +(-60.2495 - 60.2495i) q^{38} +120.000 q^{46} +(-65.7267 + 65.7267i) q^{47} +49.0000i q^{49} +(-43.8178 - 43.8178i) q^{53} -118.000 q^{61} +(5.47723 - 5.47723i) q^{62} +251.000i q^{64} +(240.998 + 240.998i) q^{68} +242.000 q^{76} +98.0000i q^{79} +(43.8178 + 43.8178i) q^{83} +(-240.998 + 240.998i) q^{92} -360.000i q^{94} +(-134.192 - 134.192i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 244 q^{16} - 8 q^{31} + 480 q^{46} - 472 q^{61} + 968 q^{76}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.73861 + 2.73861i −1.36931 + 1.36931i −0.507877 + 0.861430i \(0.669569\pi\)
−0.861430 + 0.507877i \(0.830431\pi\)
\(3\) 0 0
\(4\) 11.0000i 2.75000i
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 19.1703 + 19.1703i 2.39629 + 2.39629i
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −61.0000 −3.81250
\(17\) −21.9089 + 21.9089i −1.28876 + 1.28876i −0.353218 + 0.935541i \(0.614913\pi\)
−0.935541 + 0.353218i \(0.885087\pi\)
\(18\) 0 0
\(19\) 22.0000i 1.15789i 0.815365 + 0.578947i \(0.196536\pi\)
−0.815365 + 0.578947i \(0.803464\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −21.9089 21.9089i −0.952561 0.952561i 0.0463637 0.998925i \(-0.485237\pi\)
−0.998925 + 0.0463637i \(0.985237\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.0645161 −0.0322581 0.999480i \(-0.510270\pi\)
−0.0322581 + 0.999480i \(0.510270\pi\)
\(32\) 90.3742 90.3742i 2.82419 2.82419i
\(33\) 0 0
\(34\) 120.000i 3.52941i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) −60.2495 60.2495i −1.58551 1.58551i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 120.000 2.60870
\(47\) −65.7267 + 65.7267i −1.39844 + 1.39844i −0.593907 + 0.804534i \(0.702415\pi\)
−0.804534 + 0.593907i \(0.797585\pi\)
\(48\) 0 0
\(49\) 49.0000i 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −43.8178 43.8178i −0.826751 0.826751i 0.160315 0.987066i \(-0.448749\pi\)
−0.987066 + 0.160315i \(0.948749\pi\)
\(54\) 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
\(61\) −118.000 −1.93443 −0.967213 0.253966i \(-0.918265\pi\)
−0.967213 + 0.253966i \(0.918265\pi\)
\(62\) 5.47723 5.47723i 0.0883423 0.0883423i
\(63\) 0 0
\(64\) 251.000i 3.92188i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) 240.998 + 240.998i 3.54409 + 3.54409i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 242.000 3.18421
\(77\) 0 0
\(78\) 0 0
\(79\) 98.0000i 1.24051i 0.784402 + 0.620253i \(0.212970\pi\)
−0.784402 + 0.620253i \(0.787030\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 43.8178 + 43.8178i 0.527925 + 0.527925i 0.919953 0.392028i \(-0.128226\pi\)
−0.392028 + 0.919953i \(0.628226\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −240.998 + 240.998i −2.61954 + 2.61954i
\(93\) 0 0
\(94\) 360.000i 3.82979i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) −134.192 134.192i −1.36931 1.36931i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 240.000 2.26415
\(107\) 131.453 131.453i 1.22854 1.22854i 0.264019 0.964517i \(-0.414952\pi\)
0.964517 0.264019i \(-0.0850482\pi\)
\(108\) 0 0
\(109\) 22.0000i 0.201835i −0.994895 0.100917i \(-0.967822\pi\)
0.994895 0.100917i \(-0.0321778\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 65.7267 + 65.7267i 0.581652 + 0.581652i 0.935357 0.353705i \(-0.115078\pi\)
−0.353705 + 0.935357i \(0.615078\pi\)
\(114\) 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
\(122\) 323.156 323.156i 2.64882 2.64882i
\(123\) 0 0
\(124\) 22.0000i 0.177419i
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(128\) −325.895 325.895i −2.54605 2.54605i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −840.000 −6.17647
\(137\) 109.545 109.545i 0.799595 0.799595i −0.183437 0.983032i \(-0.558722\pi\)
0.983032 + 0.183437i \(0.0587222\pi\)
\(138\) 0 0
\(139\) 262.000i 1.88489i 0.334358 + 0.942446i \(0.391480\pi\)
−0.334358 + 0.942446i \(0.608520\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 238.000 1.57616 0.788079 0.615574i \(-0.211076\pi\)
0.788079 + 0.615574i \(0.211076\pi\)
\(152\) −421.746 + 421.746i −2.77465 + 2.77465i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(158\) −268.384 268.384i −1.69863 1.69863i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −240.000 −1.44578
\(167\) 153.362 153.362i 0.918337 0.918337i −0.0785713 0.996908i \(-0.525036\pi\)
0.996908 + 0.0785713i \(0.0250358\pi\)
\(168\) 0 0
\(169\) 169.000i 1.00000i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 219.089 + 219.089i 1.26641 + 1.26641i 0.947929 + 0.318481i \(0.103173\pi\)
0.318481 + 0.947929i \(0.396827\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −122.000 −0.674033 −0.337017 0.941499i \(-0.609418\pi\)
−0.337017 + 0.941499i \(0.609418\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 840.000i 4.56522i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 722.994 + 722.994i 3.84571 + 3.84571i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 539.000 2.75000
\(197\) 87.6356 87.6356i 0.444851 0.444851i −0.448788 0.893638i \(-0.648144\pi\)
0.893638 + 0.448788i \(0.148144\pi\)
\(198\) 0 0
\(199\) 142.000i 0.713568i −0.934187 0.356784i \(-0.883873\pi\)
0.934187 0.356784i \(-0.116127\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 362.000 1.71564 0.857820 0.513950i \(-0.171818\pi\)
0.857820 + 0.513950i \(0.171818\pi\)
\(212\) −481.996 + 481.996i −2.27357 + 2.27357i
\(213\) 0 0
\(214\) 720.000i 3.36449i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 60.2495 + 60.2495i 0.276374 + 0.276374i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −360.000 −1.59292
\(227\) −306.725 + 306.725i −1.35121 + 1.35121i −0.466899 + 0.884310i \(0.654629\pi\)
−0.884310 + 0.466899i \(0.845371\pi\)
\(228\) 0 0
\(229\) 218.000i 0.951965i −0.879455 0.475983i \(-0.842093\pi\)
0.879455 0.475983i \(-0.157907\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −328.634 328.634i −1.41044 1.41044i −0.756814 0.653631i \(-0.773245\pi\)
−0.653631 0.756814i \(-0.726755\pi\)
\(234\) 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
\(241\) −478.000 −1.98340 −0.991701 0.128564i \(-0.958963\pi\)
−0.991701 + 0.128564i \(0.958963\pi\)
\(242\) 331.372 331.372i 1.36931 1.36931i
\(243\) 0 0
\(244\) 1298.00i 5.31967i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) −38.3406 38.3406i −0.154599 0.154599i
\(249\) 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
\(256\) 781.000 3.05078
\(257\) −153.362 + 153.362i −0.596741 + 0.596741i −0.939444 0.342703i \(-0.888657\pi\)
0.342703 + 0.939444i \(0.388657\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 197.180 + 197.180i 0.749734 + 0.749734i 0.974429 0.224695i \(-0.0721385\pi\)
−0.224695 + 0.974429i \(0.572139\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 482.000 1.77860 0.889299 0.457326i \(-0.151193\pi\)
0.889299 + 0.457326i \(0.151193\pi\)
\(272\) 1336.44 1336.44i 4.91339 4.91339i
\(273\) 0 0
\(274\) 600.000i 2.18978i
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(278\) −717.517 717.517i −2.58099 2.58099i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 671.000i 2.32180i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −306.725 306.725i −1.04684 1.04684i −0.998848 0.0479941i \(-0.984717\pi\)
−0.0479941 0.998848i \(-0.515283\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −651.790 + 651.790i −2.15824 + 2.15824i
\(303\) 0 0
\(304\) 1342.00i 4.41447i
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1078.00 3.41139
\(317\) −438.178 + 438.178i −1.38227 + 1.38227i −0.541681 + 0.840584i \(0.682212\pi\)
−0.840584 + 0.541681i \(0.817788\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −481.996 481.996i −1.49225 1.49225i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 122.000 0.368580 0.184290 0.982872i \(-0.441001\pi\)
0.184290 + 0.982872i \(0.441001\pi\)
\(332\) 481.996 481.996i 1.45179 1.45179i
\(333\) 0 0
\(334\) 840.000i 2.51497i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(338\) 462.826 + 462.826i 1.36931 + 1.36931i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −1200.00 −3.46821
\(347\) −262.907 + 262.907i −0.757657 + 0.757657i −0.975895 0.218239i \(-0.929969\pi\)
0.218239 + 0.975895i \(0.429969\pi\)
\(348\) 0 0
\(349\) 458.000i 1.31232i −0.754621 0.656160i \(-0.772179\pi\)
0.754621 0.656160i \(-0.227821\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 460.087 + 460.087i 1.30336 + 1.30336i 0.926111 + 0.377252i \(0.123131\pi\)
0.377252 + 0.926111i \(0.376869\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −123.000 −0.340720
\(362\) 334.111 334.111i 0.922958 0.922958i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(368\) 1336.44 + 1336.44i 3.63164 + 3.63164i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −2520.00 −6.70213
\(377\) 0 0
\(378\) 0 0
\(379\) 742.000i 1.95778i 0.204379 + 0.978892i \(0.434482\pi\)
−0.204379 + 0.978892i \(0.565518\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −240.998 240.998i −0.629237 0.629237i 0.318639 0.947876i \(-0.396774\pi\)
−0.947876 + 0.318639i \(0.896774\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 960.000 2.45524
\(392\) −939.344 + 939.344i −2.39629 + 2.39629i
\(393\) 0 0
\(394\) 480.000i 1.21827i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(398\) 388.883 + 388.883i 0.977093 + 0.977093i
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 142.000i 0.347188i 0.984817 + 0.173594i \(0.0555381\pi\)
−0.984817 + 0.173594i \(0.944462\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 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
\(421\) −602.000 −1.42993 −0.714964 0.699161i \(-0.753557\pi\)
−0.714964 + 0.699161i \(0.753557\pi\)
\(422\) −991.378 + 991.378i −2.34924 + 2.34924i
\(423\) 0 0
\(424\) 1680.00i 3.96226i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1445.99 1445.99i −3.37848 3.37848i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −242.000 −0.555046
\(437\) 481.996 481.996i 1.10297 1.10297i
\(438\) 0 0
\(439\) 622.000i 1.41686i −0.705783 0.708428i \(-0.749405\pi\)
0.705783 0.708428i \(-0.250595\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 481.996 + 481.996i 1.08803 + 1.08803i 0.995732 + 0.0922950i \(0.0294203\pi\)
0.0922950 + 0.995732i \(0.470580\pi\)
\(444\) 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
\(452\) 722.994 722.994i 1.59954 1.59954i
\(453\) 0 0
\(454\) 1680.00i 3.70044i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(458\) 597.018 + 597.018i 1.30353 + 1.30353i
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 1800.00 3.86266
\(467\) 613.449 613.449i 1.31360 1.31360i 0.394850 0.918745i \(-0.370796\pi\)
0.918745 0.394850i \(-0.129204\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 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
\(481\) 0 0
\(482\) 1309.06 1309.06i 2.71589 2.71589i
\(483\) 0 0
\(484\) 1331.00i 2.75000i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(488\) −2262.09 2262.09i −4.63544 4.63544i
\(489\) 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
\(496\) 122.000 0.245968
\(497\) 0 0
\(498\) 0 0
\(499\) 938.000i 1.87976i 0.341506 + 0.939880i \(0.389063\pi\)
−0.341506 + 0.939880i \(0.610937\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 109.545 + 109.545i 0.217782 + 0.217782i 0.807563 0.589781i \(-0.200786\pi\)
−0.589781 + 0.807563i \(0.700786\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −835.277 + 835.277i −1.63140 + 1.63140i
\(513\) 0 0
\(514\) 840.000i 1.63424i
\(515\) 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
\(523\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −1080.00 −2.05323
\(527\) 43.8178 43.8178i 0.0831457 0.0831457i
\(528\) 0 0
\(529\) 431.000i 0.814745i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1078.00 −1.99261 −0.996303 0.0859072i \(-0.972621\pi\)
−0.996303 + 0.0859072i \(0.972621\pi\)
\(542\) −1320.01 + 1320.01i −2.43545 + 2.43545i
\(543\) 0 0
\(544\) 3960.00i 7.27941i
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(548\) −1204.99 1204.99i −2.19889 2.19889i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 2882.00 5.18345
\(557\) 657.267 657.267i 1.18001 1.18001i 0.200272 0.979740i \(-0.435817\pi\)
0.979740 0.200272i \(-0.0641827\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 788.720 + 788.720i 1.40092 + 1.40092i 0.797171 + 0.603753i \(0.206329\pi\)
0.603753 + 0.797171i \(0.293671\pi\)
\(564\) 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
\(571\) −358.000 −0.626970 −0.313485 0.949593i \(-0.601497\pi\)
−0.313485 + 0.949593i \(0.601497\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(578\) 1837.61 + 1837.61i 3.17925 + 3.17925i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 1680.00 2.86689
\(587\) 569.631 569.631i 0.970411 0.970411i −0.0291633 0.999575i \(-0.509284\pi\)
0.999575 + 0.0291633i \(0.00928429\pi\)
\(588\) 0 0
\(589\) 44.0000i 0.0747029i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −153.362 153.362i −0.258621 0.258621i 0.565872 0.824493i \(-0.308540\pi\)
−0.824493 + 0.565872i \(0.808540\pi\)
\(594\) 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
\(601\) −242.000 −0.402662 −0.201331 0.979523i \(-0.564527\pi\)
−0.201331 + 0.979523i \(0.564527\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 2618.00i 4.33444i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(608\) 1988.23 + 1988.23i 3.27012 + 3.27012i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 240.998 240.998i 0.390596 0.390596i −0.484304 0.874900i \(-0.660927\pi\)
0.874900 + 0.484304i \(0.160927\pi\)
\(618\) 0 0
\(619\) 698.000i 1.12763i 0.825903 + 0.563813i \(0.190666\pi\)
−0.825903 + 0.563813i \(0.809334\pi\)
\(620\) 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
\(631\) −238.000 −0.377179 −0.188590 0.982056i \(-0.560392\pi\)
−0.188590 + 0.982056i \(0.560392\pi\)
\(632\) −1878.69 + 1878.69i −2.97261 + 2.97261i
\(633\) 0 0
\(634\) 2400.00i 3.78549i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 2640.00 4.08669
\(647\) −766.812 + 766.812i −1.18518 + 1.18518i −0.206796 + 0.978384i \(0.566304\pi\)
−0.978384 + 0.206796i \(0.933696\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 481.996 + 481.996i 0.738125 + 0.738125i 0.972215 0.234090i \(-0.0752109\pi\)
−0.234090 + 0.972215i \(0.575211\pi\)
\(654\) 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
\(661\) 838.000 1.26778 0.633888 0.773425i \(-0.281458\pi\)
0.633888 + 0.773425i \(0.281458\pi\)
\(662\) −334.111 + 334.111i −0.504699 + 0.504699i
\(663\) 0 0
\(664\) 1680.00i 2.53012i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −1686.99 1686.99i −2.52543 2.52543i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −1859.00 −2.75000
\(677\) −920.174 + 920.174i −1.35919 + 1.35919i −0.484281 + 0.874913i \(0.660919\pi\)
−0.874913 + 0.484281i \(0.839081\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −963.992 963.992i −1.41141 1.41141i −0.750222 0.661186i \(-0.770053\pi\)
−0.661186 0.750222i \(-0.729947\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1322.00 −1.91317 −0.956585 0.291455i \(-0.905861\pi\)
−0.956585 + 0.291455i \(0.905861\pi\)
\(692\) 2409.98 2409.98i 3.48263 3.48263i
\(693\) 0 0
\(694\) 1440.00i 2.07493i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 1254.28 + 1254.28i 1.79697 + 1.79697i
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −2520.00 −3.56941
\(707\) 0 0
\(708\) 0 0
\(709\) 742.000i 1.04654i 0.852166 + 0.523272i \(0.175289\pi\)
−0.852166 + 0.523272i \(0.824711\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 43.8178 + 43.8178i 0.0614555 + 0.0614555i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 336.849 336.849i 0.466550 0.466550i
\(723\) 0 0
\(724\) 1342.00i 1.85359i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −3960.00 −5.38043
\(737\) 0 0
\(738\) 0 0
\(739\) 1462.00i 1.97835i 0.146744 + 0.989175i \(0.453121\pi\)
−0.146744 + 0.989175i \(0.546879\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −985.901 985.901i −1.32692 1.32692i −0.908044 0.418875i \(-0.862425\pi\)
−0.418875 0.908044i \(-0.637575\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1438.00 1.91478 0.957390 0.288798i \(-0.0932555\pi\)
0.957390 + 0.288798i \(0.0932555\pi\)
\(752\) 4009.33 4009.33i 5.33155 5.33155i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) −2032.05 2032.05i −2.68081 2.68081i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 1320.00 1.72324
\(767\) 0 0
\(768\) 0 0
\(769\) 578.000i 0.751625i 0.926696 + 0.375813i \(0.122636\pi\)
−0.926696 + 0.375813i \(0.877364\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 175.271 + 175.271i 0.226742 + 0.226742i 0.811330 0.584588i \(-0.198744\pi\)
−0.584588 + 0.811330i \(0.698744\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −2629.07 + 2629.07i −3.36198 + 3.36198i
\(783\) 0 0
\(784\) 2989.00i 3.81250i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(788\) −963.992 963.992i −1.22334 1.22334i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −1562.00 −1.96231
\(797\) −963.992 + 963.992i −1.20953 + 1.20953i −0.238345 + 0.971181i \(0.576605\pi\)
−0.971181 + 0.238345i \(0.923395\pi\)
\(798\) 0 0
\(799\) 2880.00i 3.60451i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −1082.00 −1.33416 −0.667078 0.744988i \(-0.732455\pi\)
−0.667078 + 0.744988i \(0.732455\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −388.883 388.883i −0.475407 0.475407i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1139.26 + 1139.26i −1.37759 + 1.37759i −0.528903 + 0.848683i \(0.677396\pi\)
−0.848683 + 0.528903i \(0.822604\pi\)
\(828\) 0 0
\(829\) 502.000i 0.605549i 0.953062 + 0.302774i \(0.0979129\pi\)
−0.953062 + 0.302774i \(0.902087\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1073.54 1073.54i −1.28876 1.28876i
\(834\) 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
\(842\) 1648.64 1648.64i 1.95801 1.95801i
\(843\) 0 0
\(844\) 3982.00i 4.71801i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 2672.89 + 2672.89i 3.15199 + 3.15199i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 5040.00 5.88785
\(857\) −284.816 + 284.816i −0.332340 + 0.332340i −0.853475 0.521134i \(-0.825509\pi\)
0.521134 + 0.853475i \(0.325509\pi\)
\(858\) 0 0
\(859\) 218.000i 0.253783i 0.991917 + 0.126892i \(0.0405001\pi\)
−0.991917 + 0.126892i \(0.959500\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1204.99 + 1204.99i 1.39628 + 1.39628i 0.810392 + 0.585888i \(0.199254\pi\)
0.585888 + 0.810392i \(0.300746\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 421.746 421.746i 0.483654 0.483654i
\(873\) 0 0
\(874\) 2640.00i 3.02059i
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(878\) 1703.42 + 1703.42i 1.94011 + 1.94011i
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −2640.00 −2.97968
\(887\) 372.451 372.451i 0.419900 0.419900i −0.465269 0.885169i \(-0.654042\pi\)
0.885169 + 0.465269i \(0.154042\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1445.99 1445.99i −1.61925 1.61925i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 1920.00 2.13097
\(902\) 0 0
\(903\) 0 0
\(904\) 2520.00i 2.78761i
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(908\) 3373.97 + 3373.97i 3.71583 + 3.71583i
\(909\) 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
\(916\) −2398.00 −2.61790
\(917\) 0 0
\(918\) 0 0
\(919\) 1298.00i 1.41240i −0.708010 0.706202i \(-0.750407\pi\)
0.708010 0.706202i \(-0.249593\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −1078.00 −1.15789
\(932\) −3614.97 + 3614.97i −3.87872 + 3.87872i
\(933\) 0 0
\(934\) 3360.00i 3.59743i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(938\) 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
\(947\) −744.903 + 744.903i −0.786592 + 0.786592i −0.980934 0.194342i \(-0.937743\pi\)
0.194342 + 0.980934i \(0.437743\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 854.447 + 854.447i 0.896587 + 0.896587i 0.995133 0.0985458i \(-0.0314191\pi\)
−0.0985458 + 0.995133i \(0.531419\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −957.000 −0.995838
\(962\) 0 0
\(963\) 0 0
\(964\) 5258.00i 5.45436i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(968\) −2319.61 2319.61i −2.39629 2.39629i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 7198.00 7.37500
\(977\) 197.180 197.180i 0.201822 0.201822i −0.598958 0.800780i \(-0.704418\pi\)
0.800780 + 0.598958i \(0.204418\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −153.362 153.362i −0.156015 0.156015i 0.624783 0.780798i \(-0.285187\pi\)
−0.780798 + 0.624783i \(0.785187\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −958.000 −0.966700 −0.483350 0.875427i \(-0.660580\pi\)
−0.483350 + 0.875427i \(0.660580\pi\)
\(992\) −180.748 + 180.748i −0.182206 + 0.182206i
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(998\) −2568.82 2568.82i −2.57397 2.57397i
\(999\) 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.c.118.1 yes 4
3.2 odd 2 inner 225.3.g.c.118.2 yes 4
5.2 odd 4 inner 225.3.g.c.82.1 4
5.3 odd 4 inner 225.3.g.c.82.2 yes 4
5.4 even 2 inner 225.3.g.c.118.2 yes 4
15.2 even 4 inner 225.3.g.c.82.2 yes 4
15.8 even 4 inner 225.3.g.c.82.1 4
15.14 odd 2 CM 225.3.g.c.118.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.3.g.c.82.1 4 5.2 odd 4 inner
225.3.g.c.82.1 4 15.8 even 4 inner
225.3.g.c.82.2 yes 4 5.3 odd 4 inner
225.3.g.c.82.2 yes 4 15.2 even 4 inner
225.3.g.c.118.1 yes 4 1.1 even 1 trivial
225.3.g.c.118.1 yes 4 15.14 odd 2 CM
225.3.g.c.118.2 yes 4 3.2 odd 2 inner
225.3.g.c.118.2 yes 4 5.4 even 2 inner