Properties

Label 252.2.b.b.55.3
Level $252$
Weight $2$
Character 252.55
Analytic conductor $2.012$
Analytic rank $0$
Dimension $4$
CM discriminant -84
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,2,Mod(55,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.55");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.b (of order \(2\), degree \(1\), 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: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 55.3
Root \(1.16372i\) of defining polynomial
Character \(\chi\) \(=\) 252.55
Dual form 252.2.b.b.55.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} -3.74166i q^{5} +2.64575 q^{7} -2.82843i q^{8} +O(q^{10})\) \(q+1.41421i q^{2} -2.00000 q^{4} -3.74166i q^{5} +2.64575 q^{7} -2.82843i q^{8} +5.29150 q^{10} -1.41421i q^{11} +3.74166i q^{14} +4.00000 q^{16} -3.74166i q^{17} +5.29150 q^{19} +7.48331i q^{20} +2.00000 q^{22} +7.07107i q^{23} -9.00000 q^{25} -5.29150 q^{28} -10.5830 q^{31} +5.65685i q^{32} +5.29150 q^{34} -9.89949i q^{35} +8.00000 q^{37} +7.48331i q^{38} -10.5830 q^{40} -3.74166i q^{41} +2.82843i q^{44} -10.0000 q^{46} +7.00000 q^{49} -12.7279i q^{50} -5.29150 q^{55} -7.48331i q^{56} -14.9666i q^{62} -8.00000 q^{64} +7.48331i q^{68} +14.0000 q^{70} +15.5563i q^{71} +11.3137i q^{74} -10.5830 q^{76} -3.74166i q^{77} -14.9666i q^{80} +5.29150 q^{82} -14.0000 q^{85} -4.00000 q^{88} +18.7083i q^{89} -14.1421i q^{92} -19.7990i q^{95} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 16 q^{16} + 8 q^{22} - 36 q^{25} + 32 q^{37} - 40 q^{46} + 28 q^{49} - 32 q^{64} + 56 q^{70} - 56 q^{85} - 16 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) − 3.74166i − 1.67332i −0.547723 0.836660i \(-0.684505\pi\)
0.547723 0.836660i \(-0.315495\pi\)
\(6\) 0 0
\(7\) 2.64575 1.00000
\(8\) − 2.82843i − 1.00000i
\(9\) 0 0
\(10\) 5.29150 1.67332
\(11\) − 1.41421i − 0.426401i −0.977008 0.213201i \(-0.931611\pi\)
0.977008 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 3.74166i 1.00000i
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) − 3.74166i − 0.907485i −0.891133 0.453743i \(-0.850089\pi\)
0.891133 0.453743i \(-0.149911\pi\)
\(18\) 0 0
\(19\) 5.29150 1.21395 0.606977 0.794719i \(-0.292382\pi\)
0.606977 + 0.794719i \(0.292382\pi\)
\(20\) 7.48331i 1.67332i
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 7.07107i 1.47442i 0.675664 + 0.737210i \(0.263857\pi\)
−0.675664 + 0.737210i \(0.736143\pi\)
\(24\) 0 0
\(25\) −9.00000 −1.80000
\(26\) 0 0
\(27\) 0 0
\(28\) −5.29150 −1.00000
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −10.5830 −1.90076 −0.950382 0.311086i \(-0.899307\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 0 0
\(34\) 5.29150 0.907485
\(35\) − 9.89949i − 1.67332i
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 7.48331i 1.21395i
\(39\) 0 0
\(40\) −10.5830 −1.67332
\(41\) − 3.74166i − 0.584349i −0.956365 0.292174i \(-0.905621\pi\)
0.956365 0.292174i \(-0.0943788\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 2.82843i 0.426401i
\(45\) 0 0
\(46\) −10.0000 −1.47442
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) − 12.7279i − 1.80000i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −5.29150 −0.713506
\(56\) − 7.48331i − 1.00000i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) − 14.9666i − 1.90076i
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 7.48331i 0.907485i
\(69\) 0 0
\(70\) 14.0000 1.67332
\(71\) 15.5563i 1.84620i 0.384561 + 0.923099i \(0.374353\pi\)
−0.384561 + 0.923099i \(0.625647\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 11.3137i 1.31519i
\(75\) 0 0
\(76\) −10.5830 −1.21395
\(77\) − 3.74166i − 0.426401i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) − 14.9666i − 1.67332i
\(81\) 0 0
\(82\) 5.29150 0.584349
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −14.0000 −1.51851
\(86\) 0 0
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) 18.7083i 1.98307i 0.129823 + 0.991537i \(0.458559\pi\)
−0.129823 + 0.991537i \(0.541441\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 14.1421i − 1.47442i
\(93\) 0 0
\(94\) 0 0
\(95\) − 19.7990i − 2.03133i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 9.89949i 1.00000i
\(99\) 0 0
\(100\) 18.0000 1.80000
\(101\) 18.7083i 1.86154i 0.365600 + 0.930772i \(0.380864\pi\)
−0.365600 + 0.930772i \(0.619136\pi\)
\(102\) 0 0
\(103\) −10.5830 −1.04277 −0.521387 0.853320i \(-0.674585\pi\)
−0.521387 + 0.853320i \(0.674585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.07107i 0.683586i 0.939775 + 0.341793i \(0.111034\pi\)
−0.939775 + 0.341793i \(0.888966\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) − 7.48331i − 0.713506i
\(111\) 0 0
\(112\) 10.5830 1.00000
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 26.4575 2.46718
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 9.89949i − 0.907485i
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 21.1660 1.90076
\(125\) 14.9666i 1.33866i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) − 11.3137i − 1.00000i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 14.0000 1.21395
\(134\) 0 0
\(135\) 0 0
\(136\) −10.5830 −0.907485
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 21.1660 1.79528 0.897639 0.440732i \(-0.145281\pi\)
0.897639 + 0.440732i \(0.145281\pi\)
\(140\) 19.7990i 1.67332i
\(141\) 0 0
\(142\) −22.0000 −1.84620
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −16.0000 −1.31519
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) − 14.9666i − 1.21395i
\(153\) 0 0
\(154\) 5.29150 0.426401
\(155\) 39.5980i 3.18059i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 21.1660 1.67332
\(161\) 18.7083i 1.47442i
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 7.48331i 0.584349i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) − 19.7990i − 1.51851i
\(171\) 0 0
\(172\) 0 0
\(173\) − 26.1916i − 1.99131i −0.0931156 0.995655i \(-0.529683\pi\)
0.0931156 0.995655i \(-0.470317\pi\)
\(174\) 0 0
\(175\) −23.8118 −1.80000
\(176\) − 5.65685i − 0.426401i
\(177\) 0 0
\(178\) −26.4575 −1.98307
\(179\) − 18.3848i − 1.37414i −0.726590 0.687071i \(-0.758896\pi\)
0.726590 0.687071i \(-0.241104\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 20.0000 1.47442
\(185\) − 29.9333i − 2.20074i
\(186\) 0 0
\(187\) −5.29150 −0.386953
\(188\) 0 0
\(189\) 0 0
\(190\) 28.0000 2.03133
\(191\) − 26.8701i − 1.94425i −0.234465 0.972125i \(-0.575334\pi\)
0.234465 0.972125i \(-0.424666\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 5.29150 0.375105 0.187552 0.982255i \(-0.439945\pi\)
0.187552 + 0.982255i \(0.439945\pi\)
\(200\) 25.4558i 1.80000i
\(201\) 0 0
\(202\) −26.4575 −1.86154
\(203\) 0 0
\(204\) 0 0
\(205\) −14.0000 −0.977802
\(206\) − 14.9666i − 1.04277i
\(207\) 0 0
\(208\) 0 0
\(209\) − 7.48331i − 0.517632i
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −10.0000 −0.683586
\(215\) 0 0
\(216\) 0 0
\(217\) −28.0000 −1.90076
\(218\) − 14.1421i − 0.957826i
\(219\) 0 0
\(220\) 10.5830 0.713506
\(221\) 0 0
\(222\) 0 0
\(223\) −26.4575 −1.77173 −0.885863 0.463947i \(-0.846433\pi\)
−0.885863 + 0.463947i \(0.846433\pi\)
\(224\) 14.9666i 1.00000i
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 37.4166i 2.46718i
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 14.0000 0.907485
\(239\) 24.0416i 1.55512i 0.628806 + 0.777562i \(0.283544\pi\)
−0.628806 + 0.777562i \(0.716456\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 12.7279i 0.818182i
\(243\) 0 0
\(244\) 0 0
\(245\) − 26.1916i − 1.67332i
\(246\) 0 0
\(247\) 0 0
\(248\) 29.9333i 1.90076i
\(249\) 0 0
\(250\) −21.1660 −1.33866
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 10.0000 0.628695
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) − 3.74166i − 0.233398i −0.993167 0.116699i \(-0.962769\pi\)
0.993167 0.116699i \(-0.0372313\pi\)
\(258\) 0 0
\(259\) 21.1660 1.31519
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 1.41421i − 0.0872041i −0.999049 0.0436021i \(-0.986117\pi\)
0.999049 0.0436021i \(-0.0138834\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 19.7990i 1.21395i
\(267\) 0 0
\(268\) 0 0
\(269\) 18.7083i 1.14066i 0.821414 + 0.570332i \(0.193186\pi\)
−0.821414 + 0.570332i \(0.806814\pi\)
\(270\) 0 0
\(271\) −10.5830 −0.642872 −0.321436 0.946931i \(-0.604165\pi\)
−0.321436 + 0.946931i \(0.604165\pi\)
\(272\) − 14.9666i − 0.907485i
\(273\) 0 0
\(274\) 0 0
\(275\) 12.7279i 0.767523i
\(276\) 0 0
\(277\) 32.0000 1.92269 0.961347 0.275340i \(-0.0887905\pi\)
0.961347 + 0.275340i \(0.0887905\pi\)
\(278\) 29.9333i 1.79528i
\(279\) 0 0
\(280\) −28.0000 −1.67332
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −26.4575 −1.57274 −0.786368 0.617758i \(-0.788041\pi\)
−0.786368 + 0.617758i \(0.788041\pi\)
\(284\) − 31.1127i − 1.84620i
\(285\) 0 0
\(286\) 0 0
\(287\) − 9.89949i − 0.584349i
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 26.1916i − 1.53013i −0.643953 0.765065i \(-0.722707\pi\)
0.643953 0.765065i \(-0.277293\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) − 22.6274i − 1.31519i
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 21.1660 1.21395
\(305\) 0 0
\(306\) 0 0
\(307\) 5.29150 0.302002 0.151001 0.988534i \(-0.451750\pi\)
0.151001 + 0.988534i \(0.451750\pi\)
\(308\) 7.48331i 0.426401i
\(309\) 0 0
\(310\) −56.0000 −3.18059
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 29.9333i 1.67332i
\(321\) 0 0
\(322\) −26.4575 −1.47442
\(323\) − 19.7990i − 1.10165i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) −10.5830 −0.584349
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 18.3848i 1.00000i
\(339\) 0 0
\(340\) 28.0000 1.51851
\(341\) 14.9666i 0.810488i
\(342\) 0 0
\(343\) 18.5203 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 37.0405 1.99131
\(347\) − 18.3848i − 0.986947i −0.869761 0.493473i \(-0.835727\pi\)
0.869761 0.493473i \(-0.164273\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) − 33.6749i − 1.80000i
\(351\) 0 0
\(352\) 8.00000 0.426401
\(353\) − 26.1916i − 1.39404i −0.717053 0.697019i \(-0.754509\pi\)
0.717053 0.697019i \(-0.245491\pi\)
\(354\) 0 0
\(355\) 58.2065 3.08928
\(356\) − 37.4166i − 1.98307i
\(357\) 0 0
\(358\) 26.0000 1.37414
\(359\) 32.5269i 1.71670i 0.513061 + 0.858352i \(0.328512\pi\)
−0.513061 + 0.858352i \(0.671488\pi\)
\(360\) 0 0
\(361\) 9.00000 0.473684
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −26.4575 −1.38107 −0.690535 0.723299i \(-0.742625\pi\)
−0.690535 + 0.723299i \(0.742625\pi\)
\(368\) 28.2843i 1.47442i
\(369\) 0 0
\(370\) 42.3320 2.20074
\(371\) 0 0
\(372\) 0 0
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) − 7.48331i − 0.386953i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 39.5980i 2.03133i
\(381\) 0 0
\(382\) 38.0000 1.94425
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −14.0000 −0.713506
\(386\) − 5.65685i − 0.287926i
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 26.4575 1.33801
\(392\) − 19.7990i − 1.00000i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 7.48331i 0.375105i
\(399\) 0 0
\(400\) −36.0000 −1.80000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) − 37.4166i − 1.86154i
\(405\) 0 0
\(406\) 0 0
\(407\) − 11.3137i − 0.560800i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) − 19.7990i − 0.977802i
\(411\) 0 0
\(412\) 21.1660 1.04277
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 10.5830 0.517632
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −40.0000 −1.94948 −0.974740 0.223341i \(-0.928304\pi\)
−0.974740 + 0.223341i \(0.928304\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 33.6749i 1.63347i
\(426\) 0 0
\(427\) 0 0
\(428\) − 14.1421i − 0.683586i
\(429\) 0 0
\(430\) 0 0
\(431\) 41.0122i 1.97549i 0.156083 + 0.987744i \(0.450113\pi\)
−0.156083 + 0.987744i \(0.549887\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) − 39.5980i − 1.90076i
\(435\) 0 0
\(436\) 20.0000 0.957826
\(437\) 37.4166i 1.78988i
\(438\) 0 0
\(439\) 5.29150 0.252550 0.126275 0.991995i \(-0.459698\pi\)
0.126275 + 0.991995i \(0.459698\pi\)
\(440\) 14.9666i 0.713506i
\(441\) 0 0
\(442\) 0 0
\(443\) − 26.8701i − 1.27663i −0.769773 0.638317i \(-0.779631\pi\)
0.769773 0.638317i \(-0.220369\pi\)
\(444\) 0 0
\(445\) 70.0000 3.31832
\(446\) − 37.4166i − 1.77173i
\(447\) 0 0
\(448\) −21.1660 −1.00000
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) −5.29150 −0.249167
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −52.9150 −2.46718
\(461\) 41.1582i 1.91693i 0.285210 + 0.958465i \(0.407937\pi\)
−0.285210 + 0.958465i \(0.592063\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −47.6235 −2.18512
\(476\) 19.7990i 0.907485i
\(477\) 0 0
\(478\) −34.0000 −1.55512
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −18.0000 −0.818182
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 37.0405 1.67332
\(491\) − 43.8406i − 1.97850i −0.146236 0.989250i \(-0.546716\pi\)
0.146236 0.989250i \(-0.453284\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −42.3320 −1.90076
\(497\) 41.1582i 1.84620i
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) − 29.9333i − 1.33866i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 70.0000 3.11496
\(506\) 14.1421i 0.628695i
\(507\) 0 0
\(508\) 0 0
\(509\) − 26.1916i − 1.16092i −0.814288 0.580461i \(-0.802872\pi\)
0.814288 0.580461i \(-0.197128\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) 0 0
\(514\) 5.29150 0.233398
\(515\) 39.5980i 1.74490i
\(516\) 0 0
\(517\) 0 0
\(518\) 29.9333i 1.31519i
\(519\) 0 0
\(520\) 0 0
\(521\) 18.7083i 0.819625i 0.912170 + 0.409812i \(0.134406\pi\)
−0.912170 + 0.409812i \(0.865594\pi\)
\(522\) 0 0
\(523\) −42.3320 −1.85105 −0.925525 0.378686i \(-0.876376\pi\)
−0.925525 + 0.378686i \(0.876376\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 2.00000 0.0872041
\(527\) 39.5980i 1.72492i
\(528\) 0 0
\(529\) −27.0000 −1.17391
\(530\) 0 0
\(531\) 0 0
\(532\) −28.0000 −1.21395
\(533\) 0 0
\(534\) 0 0
\(535\) 26.4575 1.14386
\(536\) 0 0
\(537\) 0 0
\(538\) −26.4575 −1.14066
\(539\) − 9.89949i − 0.426401i
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) − 14.9666i − 0.642872i
\(543\) 0 0
\(544\) 21.1660 0.907485
\(545\) 37.4166i 1.60275i
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −18.0000 −0.767523
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 45.2548i 1.92269i
\(555\) 0 0
\(556\) −42.3320 −1.79528
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) − 39.5980i − 1.67332i
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) − 37.4166i − 1.57274i
\(567\) 0 0
\(568\) 44.0000 1.84620
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 14.0000 0.584349
\(575\) − 63.6396i − 2.65396i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 4.24264i 0.176471i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 37.0405 1.53013
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −56.0000 −2.30744
\(590\) 0 0
\(591\) 0 0
\(592\) 32.0000 1.31519
\(593\) − 48.6415i − 1.99747i −0.0502942 0.998734i \(-0.516016\pi\)
0.0502942 0.998734i \(-0.483984\pi\)
\(594\) 0 0
\(595\) −37.0405 −1.51851
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 18.3848i − 0.751182i −0.926786 0.375591i \(-0.877440\pi\)
0.926786 0.375591i \(-0.122560\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 33.6749i − 1.36908i
\(606\) 0 0
\(607\) −26.4575 −1.07388 −0.536939 0.843621i \(-0.680419\pi\)
−0.536939 + 0.843621i \(0.680419\pi\)
\(608\) 29.9333i 1.21395i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −46.0000 −1.85792 −0.928961 0.370177i \(-0.879297\pi\)
−0.928961 + 0.370177i \(0.879297\pi\)
\(614\) 7.48331i 0.302002i
\(615\) 0 0
\(616\) −10.5830 −0.426401
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 21.1660 0.850734 0.425367 0.905021i \(-0.360145\pi\)
0.425367 + 0.905021i \(0.360145\pi\)
\(620\) − 79.1960i − 3.18059i
\(621\) 0 0
\(622\) 0 0
\(623\) 49.4975i 1.98307i
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 29.9333i − 1.19352i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −42.3320 −1.67332
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 37.0405 1.46074 0.730368 0.683054i \(-0.239349\pi\)
0.730368 + 0.683054i \(0.239349\pi\)
\(644\) − 37.4166i − 1.47442i
\(645\) 0 0
\(646\) 28.0000 1.10165
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) − 14.9666i − 0.584349i
\(657\) 0 0
\(658\) 0 0
\(659\) 24.0416i 0.936529i 0.883588 + 0.468264i \(0.155121\pi\)
−0.883588 + 0.468264i \(0.844879\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 52.3832i − 2.03133i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 44.0000 1.69608 0.848038 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) 2.82843i 0.108947i
\(675\) 0 0
\(676\) −26.0000 −1.00000
\(677\) 41.1582i 1.58184i 0.611920 + 0.790920i \(0.290397\pi\)
−0.611920 + 0.790920i \(0.709603\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 39.5980i 1.51851i
\(681\) 0 0
\(682\) −21.1660 −0.810488
\(683\) 41.0122i 1.56929i 0.619947 + 0.784644i \(0.287154\pi\)
−0.619947 + 0.784644i \(0.712846\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 26.1916i 1.00000i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −42.3320 −1.61039 −0.805193 0.593013i \(-0.797938\pi\)
−0.805193 + 0.593013i \(0.797938\pi\)
\(692\) 52.3832i 1.99131i
\(693\) 0 0
\(694\) 26.0000 0.986947
\(695\) − 79.1960i − 3.00407i
\(696\) 0 0
\(697\) −14.0000 −0.530288
\(698\) 0 0
\(699\) 0 0
\(700\) 47.6235 1.80000
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 42.3320 1.59658
\(704\) 11.3137i 0.426401i
\(705\) 0 0
\(706\) 37.0405 1.39404
\(707\) 49.4975i 1.86154i
\(708\) 0 0
\(709\) 50.0000 1.87779 0.938895 0.344204i \(-0.111851\pi\)
0.938895 + 0.344204i \(0.111851\pi\)
\(710\) 82.3165i 3.08928i
\(711\) 0 0
\(712\) 52.9150 1.98307
\(713\) − 74.8331i − 2.80252i
\(714\) 0 0
\(715\) 0 0
\(716\) 36.7696i 1.37414i
\(717\) 0 0
\(718\) −46.0000 −1.71670
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −28.0000 −1.04277
\(722\) 12.7279i 0.473684i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 52.9150 1.96251 0.981255 0.192715i \(-0.0617292\pi\)
0.981255 + 0.192715i \(0.0617292\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) − 37.4166i − 1.38107i
\(735\) 0 0
\(736\) −40.0000 −1.47442
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 59.8665i 2.20074i
\(741\) 0 0
\(742\) 0 0
\(743\) − 43.8406i − 1.60836i −0.594388 0.804178i \(-0.702606\pi\)
0.594388 0.804178i \(-0.297394\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) − 48.0833i − 1.76045i
\(747\) 0 0
\(748\) 10.5830 0.386953
\(749\) 18.7083i 0.683586i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) −56.0000 −2.03133
\(761\) 41.1582i 1.49198i 0.665955 + 0.745992i \(0.268024\pi\)
−0.665955 + 0.745992i \(0.731976\pi\)
\(762\) 0 0
\(763\) −26.4575 −0.957826
\(764\) 53.7401i 1.94425i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) − 19.7990i − 0.713506i
\(771\) 0 0
\(772\) 8.00000 0.287926
\(773\) − 48.6415i − 1.74951i −0.484561 0.874757i \(-0.661021\pi\)
0.484561 0.874757i \(-0.338979\pi\)
\(774\) 0 0
\(775\) 95.2470 3.42137
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 19.7990i − 0.709372i
\(780\) 0 0
\(781\) 22.0000 0.787222
\(782\) 37.4166i 1.33801i
\(783\) 0 0
\(784\) 28.0000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 21.1660 0.754487 0.377243 0.926114i \(-0.376872\pi\)
0.377243 + 0.926114i \(0.376872\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −10.5830 −0.375105
\(797\) − 3.74166i − 0.132536i −0.997802 0.0662682i \(-0.978891\pi\)
0.997802 0.0662682i \(-0.0211093\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) − 50.9117i − 1.80000i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 70.0000 2.46718
\(806\) 0 0
\(807\) 0 0
\(808\) 52.9150 1.86154
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −42.3320 −1.48648 −0.743239 0.669026i \(-0.766712\pi\)
−0.743239 + 0.669026i \(0.766712\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 16.0000 0.560800
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 28.0000 0.977802
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 29.9333i 1.04277i
\(825\) 0 0
\(826\) 0 0
\(827\) − 35.3553i − 1.22943i −0.788751 0.614713i \(-0.789272\pi\)
0.788751 0.614713i \(-0.210728\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 26.1916i − 0.907485i
\(834\) 0 0
\(835\) 0 0
\(836\) 14.9666i 0.517632i
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) − 56.5685i − 1.94948i
\(843\) 0 0
\(844\) 0 0
\(845\) − 48.6415i − 1.67332i
\(846\) 0 0
\(847\) 23.8118 0.818182
\(848\) 0 0
\(849\) 0 0
\(850\) −47.6235 −1.63347
\(851\) 56.5685i 1.93914i
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 20.0000 0.683586
\(857\) 41.1582i 1.40594i 0.711220 + 0.702969i \(0.248143\pi\)
−0.711220 + 0.702969i \(0.751857\pi\)
\(858\) 0 0
\(859\) −58.2065 −1.98598 −0.992991 0.118194i \(-0.962290\pi\)
−0.992991 + 0.118194i \(0.962290\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −58.0000 −1.97549
\(863\) 7.07107i 0.240702i 0.992731 + 0.120351i \(0.0384020\pi\)
−0.992731 + 0.120351i \(0.961598\pi\)
\(864\) 0 0
\(865\) −98.0000 −3.33210
\(866\) 0 0
\(867\) 0 0
\(868\) 56.0000 1.90076
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 28.2843i 0.957826i
\(873\) 0 0
\(874\) −52.9150 −1.78988
\(875\) 39.5980i 1.33866i
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 7.48331i 0.252550i
\(879\) 0 0
\(880\) −21.1660 −0.713506
\(881\) 18.7083i 0.630298i 0.949042 + 0.315149i \(0.102055\pi\)
−0.949042 + 0.315149i \(0.897945\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 38.0000 1.27663
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 98.9949i 3.31832i
\(891\) 0 0
\(892\) 52.9150 1.77173
\(893\) 0 0
\(894\) 0 0
\(895\) −68.7895 −2.29938
\(896\) − 29.9333i − 1.00000i
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) − 7.48331i − 0.249167i
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 15.5563i 0.515405i 0.966224 + 0.257702i \(0.0829654\pi\)
−0.966224 + 0.257702i \(0.917035\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) − 31.1127i − 1.02912i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) − 74.8331i − 2.46718i
\(921\) 0 0
\(922\) −58.2065 −1.91693
\(923\) 0 0
\(924\) 0 0
\(925\) −72.0000 −2.36735
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 48.6415i − 1.59588i −0.602739 0.797939i \(-0.705924\pi\)
0.602739 0.797939i \(-0.294076\pi\)
\(930\) 0 0
\(931\) 37.0405 1.21395
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 19.7990i 0.647496i
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 3.74166i − 0.121975i −0.998139 0.0609873i \(-0.980575\pi\)
0.998139 0.0609873i \(-0.0194249\pi\)
\(942\) 0 0
\(943\) 26.4575 0.861575
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 52.3259i − 1.70036i −0.526489 0.850182i \(-0.676492\pi\)
0.526489 0.850182i \(-0.323508\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) − 67.3498i − 2.18512i
\(951\) 0 0
\(952\) −28.0000 −0.907485
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) −100.539 −3.25335
\(956\) − 48.0833i − 1.55512i
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 81.0000 2.61290
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 14.9666i 0.481793i
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) − 25.4558i − 0.818182i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 56.0000 1.79528
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 26.4575 0.845586
\(980\) 52.3832i 1.67332i
\(981\) 0 0
\(982\) 62.0000 1.97850
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) − 59.8665i − 1.90076i
\(993\) 0 0
\(994\) −58.2065 −1.84620
\(995\) − 19.7990i − 0.627670i
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 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 252.2.b.b.55.3 yes 4
3.2 odd 2 inner 252.2.b.b.55.2 yes 4
4.3 odd 2 inner 252.2.b.b.55.1 4
7.6 odd 2 inner 252.2.b.b.55.4 yes 4
8.3 odd 2 4032.2.b.k.3583.3 4
8.5 even 2 4032.2.b.k.3583.4 4
12.11 even 2 inner 252.2.b.b.55.4 yes 4
21.20 even 2 inner 252.2.b.b.55.1 4
24.5 odd 2 4032.2.b.k.3583.2 4
24.11 even 2 4032.2.b.k.3583.1 4
28.27 even 2 inner 252.2.b.b.55.2 yes 4
56.13 odd 2 4032.2.b.k.3583.1 4
56.27 even 2 4032.2.b.k.3583.2 4
84.83 odd 2 CM 252.2.b.b.55.3 yes 4
168.83 odd 2 4032.2.b.k.3583.4 4
168.125 even 2 4032.2.b.k.3583.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.b.b.55.1 4 4.3 odd 2 inner
252.2.b.b.55.1 4 21.20 even 2 inner
252.2.b.b.55.2 yes 4 3.2 odd 2 inner
252.2.b.b.55.2 yes 4 28.27 even 2 inner
252.2.b.b.55.3 yes 4 1.1 even 1 trivial
252.2.b.b.55.3 yes 4 84.83 odd 2 CM
252.2.b.b.55.4 yes 4 7.6 odd 2 inner
252.2.b.b.55.4 yes 4 12.11 even 2 inner
4032.2.b.k.3583.1 4 24.11 even 2
4032.2.b.k.3583.1 4 56.13 odd 2
4032.2.b.k.3583.2 4 24.5 odd 2
4032.2.b.k.3583.2 4 56.27 even 2
4032.2.b.k.3583.3 4 8.3 odd 2
4032.2.b.k.3583.3 4 168.125 even 2
4032.2.b.k.3583.4 4 8.5 even 2
4032.2.b.k.3583.4 4 168.83 odd 2