Properties

Label 300.2.i.b.257.2
Level $300$
Weight $2$
Character 300.257
Analytic conductor $2.396$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,2,Mod(257,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.i (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: \(2.39551206064\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 257.2
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 300.257
Dual form 300.2.i.b.293.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.22474 - 1.22474i) q^{3} +(-2.44949 - 2.44949i) q^{7} -3.00000i q^{9} +O(q^{10})\) \(q+(1.22474 - 1.22474i) q^{3} +(-2.44949 - 2.44949i) q^{7} -3.00000i q^{9} +(4.89898 - 4.89898i) q^{13} +8.00000i q^{19} -6.00000 q^{21} +(-3.67423 - 3.67423i) q^{27} +4.00000 q^{31} +(4.89898 + 4.89898i) q^{37} -12.0000i q^{39} +(-7.34847 + 7.34847i) q^{43} +5.00000i q^{49} +(9.79796 + 9.79796i) q^{57} +14.0000 q^{61} +(-7.34847 + 7.34847i) q^{63} +(-2.44949 - 2.44949i) q^{67} +(-9.79796 + 9.79796i) q^{73} +4.00000i q^{79} -9.00000 q^{81} -24.0000 q^{91} +(4.89898 - 4.89898i) q^{93} +(9.79796 + 9.79796i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 24 q^{21} + 16 q^{31} + 56 q^{61} - 36 q^{81} - 96 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) 1.22474 1.22474i 0.707107 0.707107i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.44949 2.44949i −0.925820 0.925820i 0.0716124 0.997433i \(-0.477186\pi\)
−0.997433 + 0.0716124i \(0.977186\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 4.89898 4.89898i 1.35873 1.35873i 0.483250 0.875482i \(-0.339456\pi\)
0.875482 0.483250i \(-0.160544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 0 0
\(19\) 8.00000i 1.83533i 0.397360 + 0.917663i \(0.369927\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) −6.00000 −1.30931
\(22\) 0 0
\(23\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.67423 3.67423i −0.707107 0.707107i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.89898 + 4.89898i 0.805387 + 0.805387i 0.983932 0.178545i \(-0.0571389\pi\)
−0.178545 + 0.983932i \(0.557139\pi\)
\(38\) 0 0
\(39\) 12.0000i 1.92154i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −7.34847 + 7.34847i −1.12063 + 1.12063i −0.128984 + 0.991647i \(0.541172\pi\)
−0.991647 + 0.128984i \(0.958828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) 0 0
\(49\) 5.00000i 0.714286i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.79796 + 9.79796i 1.29777 + 1.29777i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) −7.34847 + 7.34847i −0.925820 + 0.925820i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.44949 2.44949i −0.299253 0.299253i 0.541468 0.840721i \(-0.317869\pi\)
−0.840721 + 0.541468i \(0.817869\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −9.79796 + 9.79796i −1.14676 + 1.14676i −0.159579 + 0.987185i \(0.551014\pi\)
−0.987185 + 0.159579i \(0.948986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000i 0.450035i 0.974355 + 0.225018i \(0.0722440\pi\)
−0.974355 + 0.225018i \(0.927756\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −24.0000 −2.51588
\(92\) 0 0
\(93\) 4.89898 4.89898i 0.508001 0.508001i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.79796 + 9.79796i 0.994832 + 0.994832i 0.999987 0.00515471i \(-0.00164080\pi\)
−0.00515471 + 0.999987i \(0.501641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 2.44949 2.44949i 0.241355 0.241355i −0.576055 0.817411i \(-0.695409\pi\)
0.817411 + 0.576055i \(0.195409\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) 0 0
\(109\) 2.00000i 0.191565i −0.995402 0.0957826i \(-0.969465\pi\)
0.995402 0.0957826i \(-0.0305354\pi\)
\(110\) 0 0
\(111\) 12.0000 1.13899
\(112\) 0 0
\(113\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −14.6969 14.6969i −1.35873 1.35873i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −7.34847 7.34847i −0.652071 0.652071i 0.301420 0.953491i \(-0.402539\pi\)
−0.953491 + 0.301420i \(0.902539\pi\)
\(128\) 0 0
\(129\) 18.0000i 1.58481i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 19.5959 19.5959i 1.69918 1.69918i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(138\) 0 0
\(139\) 16.0000i 1.35710i −0.734553 0.678551i \(-0.762608\pi\)
0.734553 0.678551i \(-0.237392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.12372 + 6.12372i 0.505076 + 0.505076i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.6969 14.6969i −1.17294 1.17294i −0.981505 0.191439i \(-0.938685\pi\)
−0.191439 0.981505i \(-0.561315\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 17.1464 17.1464i 1.34301 1.34301i 0.449966 0.893045i \(-0.351436\pi\)
0.893045 0.449966i \(-0.148564\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(168\) 0 0
\(169\) 35.0000i 2.69231i
\(170\) 0 0
\(171\) 24.0000 1.83533
\(172\) 0 0
\(173\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −26.0000 −1.93256 −0.966282 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 0 0
\(183\) 17.1464 17.1464i 1.26750 1.26750i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 18.0000i 1.30931i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −19.5959 + 19.5959i −1.41055 + 1.41055i −0.654374 + 0.756171i \(0.727068\pi\)
−0.756171 + 0.654374i \(0.772932\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) 28.0000i 1.98487i 0.122782 + 0.992434i \(0.460818\pi\)
−0.122782 + 0.992434i \(0.539182\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) 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
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −9.79796 9.79796i −0.665129 0.665129i
\(218\) 0 0
\(219\) 24.0000i 1.62177i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −7.34847 + 7.34847i −0.492090 + 0.492090i −0.908964 0.416874i \(-0.863126\pi\)
0.416874 + 0.908964i \(0.363126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(228\) 0 0
\(229\) 22.0000i 1.45380i −0.686743 0.726900i \(-0.740960\pi\)
0.686743 0.726900i \(-0.259040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.89898 + 4.89898i 0.318223 + 0.318223i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) −11.0227 + 11.0227i −0.707107 + 0.707107i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 39.1918 + 39.1918i 2.49372 + 2.49372i
\(248\) 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 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(258\) 0 0
\(259\) 24.0000i 1.49129i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 0 0
\(273\) −29.3939 + 29.3939i −1.77900 + 1.77900i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −14.6969 14.6969i −0.883053 0.883053i 0.110790 0.993844i \(-0.464662\pi\)
−0.993844 + 0.110790i \(0.964662\pi\)
\(278\) 0 0
\(279\) 12.0000i 0.718421i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −7.34847 + 7.34847i −0.436821 + 0.436821i −0.890941 0.454120i \(-0.849954\pi\)
0.454120 + 0.890941i \(0.349954\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000i 1.00000i
\(290\) 0 0
\(291\) 24.0000 1.40690
\(292\) 0 0
\(293\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 36.0000 2.07501
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 22.0454 + 22.0454i 1.25820 + 1.25820i 0.951953 + 0.306245i \(0.0990727\pi\)
0.306245 + 0.951953i \(0.400927\pi\)
\(308\) 0 0
\(309\) 6.00000i 0.341328i
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −19.5959 + 19.5959i −1.10763 + 1.10763i −0.114165 + 0.993462i \(0.536419\pi\)
−0.993462 + 0.114165i \(0.963581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(318\) 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
\(327\) −2.44949 2.44949i −0.135457 0.135457i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) 0 0
\(333\) 14.6969 14.6969i 0.805387 0.805387i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9.79796 + 9.79796i 0.533729 + 0.533729i 0.921680 0.387951i \(-0.126817\pi\)
−0.387951 + 0.921680i \(0.626817\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −4.89898 + 4.89898i −0.264520 + 0.264520i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 0 0
\(349\) 14.0000i 0.749403i 0.927146 + 0.374701i \(0.122255\pi\)
−0.927146 + 0.374701i \(0.877745\pi\)
\(350\) 0 0
\(351\) −36.0000 −1.92154
\(352\) 0 0
\(353\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −45.0000 −2.36842
\(362\) 0 0
\(363\) 13.4722 13.4722i 0.707107 0.707107i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −26.9444 26.9444i −1.40649 1.40649i −0.777064 0.629421i \(-0.783292\pi\)
−0.629421 0.777064i \(-0.716708\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4.89898 4.89898i 0.253660 0.253660i −0.568810 0.822469i \(-0.692596\pi\)
0.822469 + 0.568810i \(0.192596\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.00000i 0.410932i 0.978664 + 0.205466i \(0.0658711\pi\)
−0.978664 + 0.205466i \(0.934129\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 22.0454 + 22.0454i 1.12063 + 1.12063i
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −14.6969 14.6969i −0.737618 0.737618i 0.234498 0.972117i \(-0.424655\pi\)
−0.972117 + 0.234498i \(0.924655\pi\)
\(398\) 0 0
\(399\) 48.0000i 2.40301i
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 19.5959 19.5959i 0.976142 0.976142i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 38.0000i 1.87898i 0.342578 + 0.939490i \(0.388700\pi\)
−0.342578 + 0.939490i \(0.611300\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −19.5959 19.5959i −0.959616 0.959616i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −34.2929 34.2929i −1.65955 1.65955i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 29.3939 29.3939i 1.41258 1.41258i 0.672308 0.740271i \(-0.265303\pi\)
0.740271 0.672308i \(-0.234697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 28.0000i 1.33637i 0.743996 + 0.668184i \(0.232928\pi\)
−0.743996 + 0.668184i \(0.767072\pi\)
\(440\) 0 0
\(441\) 15.0000 0.714286
\(442\) 0 0
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 4.89898 4.89898i 0.230174 0.230174i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 29.3939 + 29.3939i 1.37499 + 1.37499i 0.852879 + 0.522108i \(0.174854\pi\)
0.522108 + 0.852879i \(0.325146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 26.9444 26.9444i 1.25221 1.25221i 0.297486 0.954726i \(-0.403852\pi\)
0.954726 0.297486i \(-0.0961480\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(468\) 0 0
\(469\) 12.0000i 0.554109i
\(470\) 0 0
\(471\) −36.0000 −1.65879
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 48.0000 2.18861
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.44949 2.44949i −0.110997 0.110997i 0.649427 0.760424i \(-0.275009\pi\)
−0.760424 + 0.649427i \(0.775009\pi\)
\(488\) 0 0
\(489\) 42.0000i 1.89931i
\(490\) 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
\(499\) 32.0000i 1.43252i −0.697835 0.716258i \(-0.745853\pi\)
0.697835 0.716258i \(-0.254147\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −42.8661 42.8661i −1.90375 1.90375i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 48.0000 2.12339
\(512\) 0 0
\(513\) 29.3939 29.3939i 1.29777 1.29777i
\(514\) 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
\(523\) −31.8434 + 31.8434i −1.39241 + 1.39241i −0.572528 + 0.819885i \(0.694037\pi\)
−0.819885 + 0.572528i \(0.805963\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000i 1.00000i
\(530\) 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
\(541\) −46.0000 −1.97769 −0.988847 0.148933i \(-0.952416\pi\)
−0.988847 + 0.148933i \(0.952416\pi\)
\(542\) 0 0
\(543\) −31.8434 + 31.8434i −1.36653 + 1.36653i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 17.1464 + 17.1464i 0.733128 + 0.733128i 0.971238 0.238110i \(-0.0765278\pi\)
−0.238110 + 0.971238i \(0.576528\pi\)
\(548\) 0 0
\(549\) 42.0000i 1.79252i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 9.79796 9.79796i 0.416652 0.416652i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 0 0
\(559\) 72.0000i 3.04528i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 22.0454 + 22.0454i 0.925820 + 0.925820i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 9.79796 + 9.79796i 0.407894 + 0.407894i 0.881004 0.473109i \(-0.156868\pi\)
−0.473109 + 0.881004i \(0.656868\pi\)
\(578\) 0 0
\(579\) 48.0000i 1.99481i
\(580\) 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 \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 0 0
\(589\) 32.0000i 1.31854i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 34.2929 + 34.2929i 1.40351 + 1.40351i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) −7.34847 + 7.34847i −0.299253 + 0.299253i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −31.8434 31.8434i −1.29248 1.29248i −0.933247 0.359235i \(-0.883038\pi\)
−0.359235 0.933247i \(-0.616962\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −34.2929 + 34.2929i −1.38508 + 1.38508i −0.549739 + 0.835337i \(0.685273\pi\)
−0.835337 + 0.549739i \(0.814727\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(618\) 0 0
\(619\) 32.0000i 1.28619i −0.765787 0.643094i \(-0.777650\pi\)
0.765787 0.643094i \(-0.222350\pi\)
\(620\) 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
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) 0 0
\(633\) −19.5959 + 19.5959i −0.778868 + 0.778868i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 24.4949 + 24.4949i 0.970523 + 0.970523i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −22.0454 + 22.0454i −0.869386 + 0.869386i −0.992404 0.123018i \(-0.960743\pi\)
0.123018 + 0.992404i \(0.460743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −24.0000 −0.940634
\(652\) 0 0
\(653\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 29.3939 + 29.3939i 1.14676 + 1.14676i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 18.0000i 0.695920i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −9.79796 + 9.79796i −0.377684 + 0.377684i −0.870266 0.492582i \(-0.836053\pi\)
0.492582 + 0.870266i \(0.336053\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(678\) 0 0
\(679\) 48.0000i 1.84207i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −26.9444 26.9444i −1.02799 1.02799i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 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
\(703\) −39.1918 + 39.1918i −1.47815 + 1.47815i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 22.0000i 0.826227i −0.910679 0.413114i \(-0.864441\pi\)
0.910679 0.413114i \(-0.135559\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) 0 0
\(723\) 17.1464 17.1464i 0.637683 0.637683i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 22.0454 + 22.0454i 0.817619 + 0.817619i 0.985763 0.168144i \(-0.0537772\pi\)
−0.168144 + 0.985763i \(0.553777\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 14.6969 14.6969i 0.542844 0.542844i −0.381518 0.924362i \(-0.624598\pi\)
0.924362 + 0.381518i \(0.124598\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 16.0000i 0.588570i −0.955718 0.294285i \(-0.904919\pi\)
0.955718 0.294285i \(-0.0950814\pi\)
\(740\) 0 0
\(741\) 96.0000 3.52665
\(742\) 0 0
\(743\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 52.0000 1.89751 0.948753 0.316017i \(-0.102346\pi\)
0.948753 + 0.316017i \(0.102346\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 34.2929 + 34.2929i 1.24640 + 1.24640i 0.957301 + 0.289095i \(0.0933542\pi\)
0.289095 + 0.957301i \(0.406646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −4.89898 + 4.89898i −0.177355 + 0.177355i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.00000i 0.0721218i −0.999350 0.0360609i \(-0.988519\pi\)
0.999350 0.0360609i \(-0.0114810\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −29.3939 29.3939i −1.05450 1.05450i
\(778\) 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
\(787\) −2.44949 2.44949i −0.0873149 0.0873149i 0.662100 0.749415i \(-0.269665\pi\)
−0.749415 + 0.662100i \(0.769665\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 68.5857 68.5857i 2.43555 2.43555i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(798\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −56.0000 −1.96643 −0.983213 0.182462i \(-0.941593\pi\)
−0.983213 + 0.182462i \(0.941593\pi\)
\(812\) 0 0
\(813\) −34.2929 + 34.2929i −1.20270 + 1.20270i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −58.7878 58.7878i −2.05672 2.05672i
\(818\) 0 0
\(819\) 72.0000i 2.51588i
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 17.1464 17.1464i 0.597687 0.597687i −0.342010 0.939696i \(-0.611108\pi\)
0.939696 + 0.342010i \(0.111108\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(828\) 0 0
\(829\) 46.0000i 1.59765i −0.601566 0.798823i \(-0.705456\pi\)
0.601566 0.798823i \(-0.294544\pi\)
\(830\) 0 0
\(831\) −36.0000 −1.24883
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −14.6969 14.6969i −0.508001 0.508001i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −26.9444 26.9444i −0.925820 0.925820i
\(848\) 0 0
\(849\) 18.0000i 0.617758i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 4.89898 4.89898i 0.167738 0.167738i −0.618246 0.785984i \(-0.712157\pi\)
0.785984 + 0.618246i \(0.212157\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 0 0
\(859\) 56.0000i 1.91070i −0.295484 0.955348i \(-0.595481\pi\)
0.295484 0.955348i \(-0.404519\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 20.8207 + 20.8207i 0.707107 + 0.707107i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) 29.3939 29.3939i 0.994832 0.994832i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 34.2929 + 34.2929i 1.15799 + 1.15799i 0.984908 + 0.173080i \(0.0553718\pi\)
0.173080 + 0.984908i \(0.444628\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 41.6413 41.6413i 1.40134 1.40134i 0.605487 0.795855i \(-0.292978\pi\)
0.795855 0.605487i \(-0.207022\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(888\) 0 0
\(889\) 36.0000i 1.20740i
\(890\) 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
\(903\) 44.0908 44.0908i 1.46725 1.46725i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −31.8434 31.8434i −1.05734 1.05734i −0.998253 0.0590889i \(-0.981180\pi\)
−0.0590889 0.998253i \(-0.518820\pi\)
\(908\) 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
\(919\) 52.0000i 1.71532i −0.514216 0.857661i \(-0.671917\pi\)
0.514216 0.857661i \(-0.328083\pi\)
\(920\) 0 0
\(921\) 54.0000 1.77936
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −7.34847 7.34847i −0.241355 0.241355i
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) −40.0000 −1.31095
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −39.1918 39.1918i −1.28034 1.28034i −0.940473 0.339869i \(-0.889617\pi\)
−0.339869 0.940473i \(-0.610383\pi\)
\(938\) 0 0
\(939\) 48.0000i 1.56642i
\(940\) 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 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(948\) 0 0
\(949\) 96.0000i 3.11629i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 41.6413 + 41.6413i 1.33909 + 1.33909i 0.896938 + 0.442157i \(0.145787\pi\)
0.442157 + 0.896938i \(0.354213\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) −39.1918 + 39.1918i −1.25643 + 1.25643i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 0 0
\(983\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) 0 0
\(993\) 39.1918 39.1918i 1.24372 1.24372i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −44.0908 44.0908i −1.39637 1.39637i −0.810157 0.586214i \(-0.800618\pi\)
−0.586214 0.810157i \(-0.699382\pi\)
\(998\) 0 0
\(999\) 36.0000i 1.13899i
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 300.2.i.b.257.2 yes 4
3.2 odd 2 CM 300.2.i.b.257.2 yes 4
4.3 odd 2 1200.2.v.e.257.1 4
5.2 odd 4 inner 300.2.i.b.293.1 yes 4
5.3 odd 4 inner 300.2.i.b.293.2 yes 4
5.4 even 2 inner 300.2.i.b.257.1 4
12.11 even 2 1200.2.v.e.257.1 4
15.2 even 4 inner 300.2.i.b.293.1 yes 4
15.8 even 4 inner 300.2.i.b.293.2 yes 4
15.14 odd 2 inner 300.2.i.b.257.1 4
20.3 even 4 1200.2.v.e.593.1 4
20.7 even 4 1200.2.v.e.593.2 4
20.19 odd 2 1200.2.v.e.257.2 4
60.23 odd 4 1200.2.v.e.593.1 4
60.47 odd 4 1200.2.v.e.593.2 4
60.59 even 2 1200.2.v.e.257.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.2.i.b.257.1 4 5.4 even 2 inner
300.2.i.b.257.1 4 15.14 odd 2 inner
300.2.i.b.257.2 yes 4 1.1 even 1 trivial
300.2.i.b.257.2 yes 4 3.2 odd 2 CM
300.2.i.b.293.1 yes 4 5.2 odd 4 inner
300.2.i.b.293.1 yes 4 15.2 even 4 inner
300.2.i.b.293.2 yes 4 5.3 odd 4 inner
300.2.i.b.293.2 yes 4 15.8 even 4 inner
1200.2.v.e.257.1 4 4.3 odd 2
1200.2.v.e.257.1 4 12.11 even 2
1200.2.v.e.257.2 4 20.19 odd 2
1200.2.v.e.257.2 4 60.59 even 2
1200.2.v.e.593.1 4 20.3 even 4
1200.2.v.e.593.1 4 60.23 odd 4
1200.2.v.e.593.2 4 20.7 even 4
1200.2.v.e.593.2 4 60.47 odd 4