Properties

Label 5148.2.a.s.1.3
Level $5148$
Weight $2$
Character 5148.1
Self dual yes
Analytic conductor $41.107$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5148,2,Mod(1,5148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5148, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5148.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5148.a (trivial)

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: yes
Analytic conductor: \(41.1069869606\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.815952.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 9x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.02664\) of defining polynomial
Character \(\chi\) \(=\) 5148.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.577317 q^{5} +2.32465 q^{7} +O(q^{10})\) \(q-0.577317 q^{5} +2.32465 q^{7} +1.00000 q^{11} -1.00000 q^{13} -4.24402 q^{17} +1.26475 q^{19} +3.28521 q^{23} -4.66671 q^{25} -10.0151 q^{29} +6.24910 q^{31} -1.34206 q^{35} -9.82439 q^{37} -3.76603 q^{41} -1.64801 q^{43} -7.17509 q^{47} -1.59602 q^{49} -2.94011 q^{53} -0.577317 q^{55} -0.161250 q^{59} +4.92574 q^{61} +0.577317 q^{65} +13.7438 q^{67} +4.97902 q^{71} +6.63391 q^{73} +2.32465 q^{77} -5.30391 q^{79} -0.0291080 q^{83} +2.45015 q^{85} -2.17537 q^{89} -2.32465 q^{91} -0.730164 q^{95} -7.70460 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{5} - 2 q^{7} + 5 q^{11} - 5 q^{13} + 2 q^{17} - 2 q^{19} - 12 q^{23} - q^{25} - 4 q^{29} - 2 q^{31} + 2 q^{35} - 6 q^{41} + 14 q^{43} - 14 q^{47} - 7 q^{49} - 20 q^{53} - 2 q^{55} - 20 q^{59} + 2 q^{65} + 10 q^{67} - 26 q^{71} + 16 q^{73} - 2 q^{77} + 2 q^{79} - 16 q^{83} - 18 q^{85} - 24 q^{89} + 2 q^{91} - 22 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0
\(4\) 0 0
\(5\) −0.577317 −0.258184 −0.129092 0.991633i \(-0.541206\pi\)
−0.129092 + 0.991633i \(0.541206\pi\)
\(6\) 0 0
\(7\) 2.32465 0.878634 0.439317 0.898332i \(-0.355221\pi\)
0.439317 + 0.898332i \(0.355221\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.24402 −1.02933 −0.514663 0.857392i \(-0.672083\pi\)
−0.514663 + 0.857392i \(0.672083\pi\)
\(18\) 0 0
\(19\) 1.26475 0.290155 0.145077 0.989420i \(-0.453657\pi\)
0.145077 + 0.989420i \(0.453657\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.28521 0.685015 0.342507 0.939515i \(-0.388724\pi\)
0.342507 + 0.939515i \(0.388724\pi\)
\(24\) 0 0
\(25\) −4.66671 −0.933341
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −10.0151 −1.85976 −0.929882 0.367859i \(-0.880091\pi\)
−0.929882 + 0.367859i \(0.880091\pi\)
\(30\) 0 0
\(31\) 6.24910 1.12237 0.561186 0.827690i \(-0.310345\pi\)
0.561186 + 0.827690i \(0.310345\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.34206 −0.226849
\(36\) 0 0
\(37\) −9.82439 −1.61512 −0.807560 0.589786i \(-0.799212\pi\)
−0.807560 + 0.589786i \(0.799212\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.76603 −0.588155 −0.294078 0.955782i \(-0.595012\pi\)
−0.294078 + 0.955782i \(0.595012\pi\)
\(42\) 0 0
\(43\) −1.64801 −0.251318 −0.125659 0.992073i \(-0.540105\pi\)
−0.125659 + 0.992073i \(0.540105\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.17509 −1.04660 −0.523298 0.852150i \(-0.675298\pi\)
−0.523298 + 0.852150i \(0.675298\pi\)
\(48\) 0 0
\(49\) −1.59602 −0.228002
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.94011 −0.403855 −0.201927 0.979400i \(-0.564721\pi\)
−0.201927 + 0.979400i \(0.564721\pi\)
\(54\) 0 0
\(55\) −0.577317 −0.0778454
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.161250 −0.0209929 −0.0104965 0.999945i \(-0.503341\pi\)
−0.0104965 + 0.999945i \(0.503341\pi\)
\(60\) 0 0
\(61\) 4.92574 0.630677 0.315339 0.948979i \(-0.397882\pi\)
0.315339 + 0.948979i \(0.397882\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.577317 0.0716074
\(66\) 0 0
\(67\) 13.7438 1.67907 0.839534 0.543307i \(-0.182828\pi\)
0.839534 + 0.543307i \(0.182828\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.97902 0.590901 0.295451 0.955358i \(-0.404530\pi\)
0.295451 + 0.955358i \(0.404530\pi\)
\(72\) 0 0
\(73\) 6.63391 0.776441 0.388221 0.921567i \(-0.373090\pi\)
0.388221 + 0.921567i \(0.373090\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.32465 0.264918
\(78\) 0 0
\(79\) −5.30391 −0.596737 −0.298368 0.954451i \(-0.596442\pi\)
−0.298368 + 0.954451i \(0.596442\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.0291080 −0.00319502 −0.00159751 0.999999i \(-0.500509\pi\)
−0.00159751 + 0.999999i \(0.500509\pi\)
\(84\) 0 0
\(85\) 2.45015 0.265756
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.17537 −0.230588 −0.115294 0.993331i \(-0.536781\pi\)
−0.115294 + 0.993331i \(0.536781\pi\)
\(90\) 0 0
\(91\) −2.32465 −0.243689
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.730164 −0.0749133
\(96\) 0 0
\(97\) −7.70460 −0.782284 −0.391142 0.920330i \(-0.627920\pi\)
−0.391142 + 0.920330i \(0.627920\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.39866 −0.537186 −0.268593 0.963254i \(-0.586559\pi\)
−0.268593 + 0.963254i \(0.586559\pi\)
\(102\) 0 0
\(103\) −4.89864 −0.482678 −0.241339 0.970441i \(-0.577587\pi\)
−0.241339 + 0.970441i \(0.577587\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.8797 −1.63182 −0.815911 0.578178i \(-0.803764\pi\)
−0.815911 + 0.578178i \(0.803764\pi\)
\(108\) 0 0
\(109\) −6.09117 −0.583429 −0.291714 0.956505i \(-0.594226\pi\)
−0.291714 + 0.956505i \(0.594226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −16.4691 −1.54928 −0.774641 0.632401i \(-0.782070\pi\)
−0.774641 + 0.632401i \(0.782070\pi\)
\(114\) 0 0
\(115\) −1.89661 −0.176860
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −9.86585 −0.904401
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.58075 0.499158
\(126\) 0 0
\(127\) 13.0618 1.15905 0.579523 0.814956i \(-0.303239\pi\)
0.579523 + 0.814956i \(0.303239\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.72558 −0.412876 −0.206438 0.978460i \(-0.566187\pi\)
−0.206438 + 0.978460i \(0.566187\pi\)
\(132\) 0 0
\(133\) 2.94011 0.254940
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 22.8175 1.94943 0.974716 0.223446i \(-0.0717305\pi\)
0.974716 + 0.223446i \(0.0717305\pi\)
\(138\) 0 0
\(139\) −7.41912 −0.629281 −0.314641 0.949211i \(-0.601884\pi\)
−0.314641 + 0.949211i \(0.601884\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) 5.78191 0.480161
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.17663 −0.342163 −0.171081 0.985257i \(-0.554726\pi\)
−0.171081 + 0.985257i \(0.554726\pi\)
\(150\) 0 0
\(151\) 2.80088 0.227932 0.113966 0.993485i \(-0.463644\pi\)
0.113966 + 0.993485i \(0.463644\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.60771 −0.289779
\(156\) 0 0
\(157\) 6.89966 0.550653 0.275326 0.961351i \(-0.411214\pi\)
0.275326 + 0.961351i \(0.411214\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.63696 0.601877
\(162\) 0 0
\(163\) 0.925991 0.0725292 0.0362646 0.999342i \(-0.488454\pi\)
0.0362646 + 0.999342i \(0.488454\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.9868 −1.08233 −0.541164 0.840917i \(-0.682016\pi\)
−0.541164 + 0.840917i \(0.682016\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.86915 0.218137 0.109069 0.994034i \(-0.465213\pi\)
0.109069 + 0.994034i \(0.465213\pi\)
\(174\) 0 0
\(175\) −10.8484 −0.820065
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.61294 0.718505 0.359252 0.933240i \(-0.383032\pi\)
0.359252 + 0.933240i \(0.383032\pi\)
\(180\) 0 0
\(181\) −0.518713 −0.0385557 −0.0192778 0.999814i \(-0.506137\pi\)
−0.0192778 + 0.999814i \(0.506137\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.67179 0.416998
\(186\) 0 0
\(187\) −4.24402 −0.310354
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.2862 0.816643 0.408321 0.912838i \(-0.366114\pi\)
0.408321 + 0.912838i \(0.366114\pi\)
\(192\) 0 0
\(193\) −6.97801 −0.502288 −0.251144 0.967950i \(-0.580807\pi\)
−0.251144 + 0.967950i \(0.580807\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −20.0338 −1.42735 −0.713676 0.700476i \(-0.752971\pi\)
−0.713676 + 0.700476i \(0.752971\pi\)
\(198\) 0 0
\(199\) −8.63035 −0.611789 −0.305895 0.952065i \(-0.598956\pi\)
−0.305895 + 0.952065i \(0.598956\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −23.2816 −1.63405
\(204\) 0 0
\(205\) 2.17419 0.151852
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.26475 0.0874849
\(210\) 0 0
\(211\) −5.46516 −0.376237 −0.188119 0.982146i \(-0.560239\pi\)
−0.188119 + 0.982146i \(0.560239\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.951421 0.0648864
\(216\) 0 0
\(217\) 14.5270 0.986154
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.24402 0.285484
\(222\) 0 0
\(223\) −1.17740 −0.0788445 −0.0394222 0.999223i \(-0.512552\pi\)
−0.0394222 + 0.999223i \(0.512552\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −21.2766 −1.41218 −0.706088 0.708124i \(-0.749542\pi\)
−0.706088 + 0.708124i \(0.749542\pi\)
\(228\) 0 0
\(229\) −2.32973 −0.153953 −0.0769764 0.997033i \(-0.524527\pi\)
−0.0769764 + 0.997033i \(0.524527\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.79489 −0.641685 −0.320842 0.947133i \(-0.603966\pi\)
−0.320842 + 0.947133i \(0.603966\pi\)
\(234\) 0 0
\(235\) 4.14230 0.270214
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.65808 0.171937 0.0859686 0.996298i \(-0.472602\pi\)
0.0859686 + 0.996298i \(0.472602\pi\)
\(240\) 0 0
\(241\) 18.2422 1.17508 0.587542 0.809194i \(-0.300096\pi\)
0.587542 + 0.809194i \(0.300096\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.921408 0.0588666
\(246\) 0 0
\(247\) −1.26475 −0.0804744
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −16.6677 −1.05206 −0.526029 0.850466i \(-0.676320\pi\)
−0.526029 + 0.850466i \(0.676320\pi\)
\(252\) 0 0
\(253\) 3.28521 0.206540
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −16.8608 −1.05174 −0.525872 0.850563i \(-0.676261\pi\)
−0.525872 + 0.850563i \(0.676261\pi\)
\(258\) 0 0
\(259\) −22.8382 −1.41910
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.1127 1.17854 0.589269 0.807937i \(-0.299416\pi\)
0.589269 + 0.807937i \(0.299416\pi\)
\(264\) 0 0
\(265\) 1.69737 0.104269
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.3876 −0.633342 −0.316671 0.948535i \(-0.602565\pi\)
−0.316671 + 0.948535i \(0.602565\pi\)
\(270\) 0 0
\(271\) 8.82592 0.536137 0.268068 0.963400i \(-0.413615\pi\)
0.268068 + 0.963400i \(0.413615\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.66671 −0.281413
\(276\) 0 0
\(277\) −22.3810 −1.34474 −0.672372 0.740213i \(-0.734724\pi\)
−0.672372 + 0.740213i \(0.734724\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.44188 0.443945 0.221973 0.975053i \(-0.428750\pi\)
0.221973 + 0.975053i \(0.428750\pi\)
\(282\) 0 0
\(283\) −5.64342 −0.335467 −0.167733 0.985832i \(-0.553645\pi\)
−0.167733 + 0.985832i \(0.553645\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.75469 −0.516773
\(288\) 0 0
\(289\) 1.01172 0.0595131
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.64977 −0.0963803 −0.0481902 0.998838i \(-0.515345\pi\)
−0.0481902 + 0.998838i \(0.515345\pi\)
\(294\) 0 0
\(295\) 0.0930921 0.00542003
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.28521 −0.189989
\(300\) 0 0
\(301\) −3.83103 −0.220817
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.84372 −0.162831
\(306\) 0 0
\(307\) 15.4603 0.882368 0.441184 0.897417i \(-0.354559\pi\)
0.441184 + 0.897417i \(0.354559\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −13.8895 −0.787603 −0.393801 0.919196i \(-0.628840\pi\)
−0.393801 + 0.919196i \(0.628840\pi\)
\(312\) 0 0
\(313\) −31.0903 −1.75733 −0.878664 0.477441i \(-0.841564\pi\)
−0.878664 + 0.477441i \(0.841564\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.78317 0.212484 0.106242 0.994340i \(-0.466118\pi\)
0.106242 + 0.994340i \(0.466118\pi\)
\(318\) 0 0
\(319\) −10.0151 −0.560740
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.36765 −0.298664
\(324\) 0 0
\(325\) 4.66671 0.258862
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −16.6796 −0.919574
\(330\) 0 0
\(331\) 30.3787 1.66976 0.834882 0.550429i \(-0.185536\pi\)
0.834882 + 0.550429i \(0.185536\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7.93451 −0.433508
\(336\) 0 0
\(337\) 8.95192 0.487642 0.243821 0.969820i \(-0.421599\pi\)
0.243821 + 0.969820i \(0.421599\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.24910 0.338408
\(342\) 0 0
\(343\) −19.9827 −1.07896
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −0.774206 −0.0415616 −0.0207808 0.999784i \(-0.506615\pi\)
−0.0207808 + 0.999784i \(0.506615\pi\)
\(348\) 0 0
\(349\) 19.9289 1.06677 0.533386 0.845872i \(-0.320919\pi\)
0.533386 + 0.845872i \(0.320919\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.4402 −0.555678 −0.277839 0.960628i \(-0.589618\pi\)
−0.277839 + 0.960628i \(0.589618\pi\)
\(354\) 0 0
\(355\) −2.87447 −0.152561
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.7707 1.09624 0.548119 0.836401i \(-0.315344\pi\)
0.548119 + 0.836401i \(0.315344\pi\)
\(360\) 0 0
\(361\) −17.4004 −0.915810
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.82987 −0.200465
\(366\) 0 0
\(367\) 17.0528 0.890147 0.445073 0.895494i \(-0.353178\pi\)
0.445073 + 0.895494i \(0.353178\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.83471 −0.354841
\(372\) 0 0
\(373\) −9.25712 −0.479316 −0.239658 0.970857i \(-0.577035\pi\)
−0.239658 + 0.970857i \(0.577035\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 10.0151 0.515806
\(378\) 0 0
\(379\) −1.77493 −0.0911720 −0.0455860 0.998960i \(-0.514516\pi\)
−0.0455860 + 0.998960i \(0.514516\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −31.0651 −1.58735 −0.793677 0.608340i \(-0.791836\pi\)
−0.793677 + 0.608340i \(0.791836\pi\)
\(384\) 0 0
\(385\) −1.34206 −0.0683976
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −15.3228 −0.776895 −0.388447 0.921471i \(-0.626988\pi\)
−0.388447 + 0.921471i \(0.626988\pi\)
\(390\) 0 0
\(391\) −13.9425 −0.705104
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.06204 0.154068
\(396\) 0 0
\(397\) −15.6759 −0.786750 −0.393375 0.919378i \(-0.628693\pi\)
−0.393375 + 0.919378i \(0.628693\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5.08266 0.253816 0.126908 0.991915i \(-0.459495\pi\)
0.126908 + 0.991915i \(0.459495\pi\)
\(402\) 0 0
\(403\) −6.24910 −0.311290
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −9.82439 −0.486977
\(408\) 0 0
\(409\) −32.5378 −1.60889 −0.804444 0.594028i \(-0.797537\pi\)
−0.804444 + 0.594028i \(0.797537\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −0.374848 −0.0184451
\(414\) 0 0
\(415\) 0.0168046 0.000824903 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −27.2864 −1.33303 −0.666513 0.745493i \(-0.732214\pi\)
−0.666513 + 0.745493i \(0.732214\pi\)
\(420\) 0 0
\(421\) 13.7042 0.667904 0.333952 0.942590i \(-0.391618\pi\)
0.333952 + 0.942590i \(0.391618\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 19.8056 0.960713
\(426\) 0 0
\(427\) 11.4506 0.554134
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −27.7144 −1.33496 −0.667478 0.744629i \(-0.732626\pi\)
−0.667478 + 0.744629i \(0.732626\pi\)
\(432\) 0 0
\(433\) 30.3197 1.45707 0.728535 0.685009i \(-0.240202\pi\)
0.728535 + 0.685009i \(0.240202\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.15499 0.198760
\(438\) 0 0
\(439\) 34.5272 1.64789 0.823947 0.566667i \(-0.191767\pi\)
0.823947 + 0.566667i \(0.191767\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.3329 0.490931 0.245465 0.969405i \(-0.421059\pi\)
0.245465 + 0.969405i \(0.421059\pi\)
\(444\) 0 0
\(445\) 1.25588 0.0595342
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7.34288 0.346532 0.173266 0.984875i \(-0.444568\pi\)
0.173266 + 0.984875i \(0.444568\pi\)
\(450\) 0 0
\(451\) −3.76603 −0.177335
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.34206 0.0629167
\(456\) 0 0
\(457\) −18.7210 −0.875730 −0.437865 0.899041i \(-0.644265\pi\)
−0.437865 + 0.899041i \(0.644265\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −14.3466 −0.668189 −0.334094 0.942540i \(-0.608430\pi\)
−0.334094 + 0.942540i \(0.608430\pi\)
\(462\) 0 0
\(463\) 29.1867 1.35642 0.678210 0.734869i \(-0.262756\pi\)
0.678210 + 0.734869i \(0.262756\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −33.2639 −1.53927 −0.769634 0.638486i \(-0.779561\pi\)
−0.769634 + 0.638486i \(0.779561\pi\)
\(468\) 0 0
\(469\) 31.9494 1.47529
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.64801 −0.0757754
\(474\) 0 0
\(475\) −5.90224 −0.270813
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.13113 0.234447 0.117224 0.993106i \(-0.462601\pi\)
0.117224 + 0.993106i \(0.462601\pi\)
\(480\) 0 0
\(481\) 9.82439 0.447954
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.44800 0.201973
\(486\) 0 0
\(487\) −1.97407 −0.0894537 −0.0447269 0.998999i \(-0.514242\pi\)
−0.0447269 + 0.998999i \(0.514242\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −2.20016 −0.0992920 −0.0496460 0.998767i \(-0.515809\pi\)
−0.0496460 + 0.998767i \(0.515809\pi\)
\(492\) 0 0
\(493\) 42.5044 1.91430
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 11.5745 0.519186
\(498\) 0 0
\(499\) −30.3234 −1.35746 −0.678731 0.734387i \(-0.737470\pi\)
−0.678731 + 0.734387i \(0.737470\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 26.0069 1.15959 0.579795 0.814763i \(-0.303133\pi\)
0.579795 + 0.814763i \(0.303133\pi\)
\(504\) 0 0
\(505\) 3.11674 0.138693
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9.82414 0.435447 0.217724 0.976010i \(-0.430137\pi\)
0.217724 + 0.976010i \(0.430137\pi\)
\(510\) 0 0
\(511\) 15.4215 0.682207
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.82807 0.124620
\(516\) 0 0
\(517\) −7.17509 −0.315560
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −17.0343 −0.746286 −0.373143 0.927774i \(-0.621720\pi\)
−0.373143 + 0.927774i \(0.621720\pi\)
\(522\) 0 0
\(523\) 19.0598 0.833426 0.416713 0.909038i \(-0.363182\pi\)
0.416713 + 0.909038i \(0.363182\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −26.5213 −1.15529
\(528\) 0 0
\(529\) −12.2074 −0.530755
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.76603 0.163125
\(534\) 0 0
\(535\) 9.74494 0.421310
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.59602 −0.0687453
\(540\) 0 0
\(541\) −24.0923 −1.03581 −0.517905 0.855438i \(-0.673288\pi\)
−0.517905 + 0.855438i \(0.673288\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.51654 0.150632
\(546\) 0 0
\(547\) 1.56763 0.0670269 0.0335134 0.999438i \(-0.489330\pi\)
0.0335134 + 0.999438i \(0.489330\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.6667 −0.539619
\(552\) 0 0
\(553\) −12.3297 −0.524313
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.65147 0.324203 0.162102 0.986774i \(-0.448173\pi\)
0.162102 + 0.986774i \(0.448173\pi\)
\(558\) 0 0
\(559\) 1.64801 0.0697032
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 15.6704 0.660427 0.330214 0.943906i \(-0.392879\pi\)
0.330214 + 0.943906i \(0.392879\pi\)
\(564\) 0 0
\(565\) 9.50789 0.400000
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 24.2374 1.01608 0.508042 0.861332i \(-0.330369\pi\)
0.508042 + 0.861332i \(0.330369\pi\)
\(570\) 0 0
\(571\) 21.6748 0.907060 0.453530 0.891241i \(-0.350164\pi\)
0.453530 + 0.891241i \(0.350164\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −15.3311 −0.639352
\(576\) 0 0
\(577\) 12.3568 0.514422 0.257211 0.966355i \(-0.417197\pi\)
0.257211 + 0.966355i \(0.417197\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −0.0676659 −0.00280725
\(582\) 0 0
\(583\) −2.94011 −0.121767
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19.8474 −0.819191 −0.409595 0.912267i \(-0.634330\pi\)
−0.409595 + 0.912267i \(0.634330\pi\)
\(588\) 0 0
\(589\) 7.90358 0.325661
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 35.8721 1.47309 0.736545 0.676388i \(-0.236456\pi\)
0.736545 + 0.676388i \(0.236456\pi\)
\(594\) 0 0
\(595\) 5.69572 0.233502
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.6061 −0.515072 −0.257536 0.966269i \(-0.582911\pi\)
−0.257536 + 0.966269i \(0.582911\pi\)
\(600\) 0 0
\(601\) −1.79471 −0.0732077 −0.0366038 0.999330i \(-0.511654\pi\)
−0.0366038 + 0.999330i \(0.511654\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.577317 −0.0234713
\(606\) 0 0
\(607\) −20.1856 −0.819309 −0.409654 0.912241i \(-0.634351\pi\)
−0.409654 + 0.912241i \(0.634351\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.17509 0.290273
\(612\) 0 0
\(613\) −46.6311 −1.88341 −0.941706 0.336438i \(-0.890778\pi\)
−0.941706 + 0.336438i \(0.890778\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.5397 1.22948 0.614740 0.788730i \(-0.289261\pi\)
0.614740 + 0.788730i \(0.289261\pi\)
\(618\) 0 0
\(619\) −0.372313 −0.0149645 −0.00748226 0.999972i \(-0.502382\pi\)
−0.00748226 + 0.999972i \(0.502382\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −5.05696 −0.202603
\(624\) 0 0
\(625\) 20.1117 0.804466
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 41.6949 1.66249
\(630\) 0 0
\(631\) −36.9298 −1.47015 −0.735076 0.677985i \(-0.762854\pi\)
−0.735076 + 0.677985i \(0.762854\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −7.54080 −0.299247
\(636\) 0 0
\(637\) 1.59602 0.0632365
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 39.3642 1.55479 0.777396 0.629012i \(-0.216540\pi\)
0.777396 + 0.629012i \(0.216540\pi\)
\(642\) 0 0
\(643\) −4.22345 −0.166556 −0.0832782 0.996526i \(-0.526539\pi\)
−0.0832782 + 0.996526i \(0.526539\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.51054 0.373898 0.186949 0.982370i \(-0.440140\pi\)
0.186949 + 0.982370i \(0.440140\pi\)
\(648\) 0 0
\(649\) −0.161250 −0.00632960
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.2209 1.65223 0.826115 0.563502i \(-0.190546\pi\)
0.826115 + 0.563502i \(0.190546\pi\)
\(654\) 0 0
\(655\) 2.72816 0.106598
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 42.7083 1.66368 0.831840 0.555016i \(-0.187288\pi\)
0.831840 + 0.555016i \(0.187288\pi\)
\(660\) 0 0
\(661\) 2.89699 0.112680 0.0563400 0.998412i \(-0.482057\pi\)
0.0563400 + 0.998412i \(0.482057\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.69737 −0.0658213
\(666\) 0 0
\(667\) −32.9019 −1.27397
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.92574 0.190156
\(672\) 0 0
\(673\) −7.71339 −0.297329 −0.148665 0.988888i \(-0.547497\pi\)
−0.148665 + 0.988888i \(0.547497\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −9.15703 −0.351933 −0.175967 0.984396i \(-0.556305\pi\)
−0.175967 + 0.984396i \(0.556305\pi\)
\(678\) 0 0
\(679\) −17.9105 −0.687341
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 30.8023 1.17862 0.589308 0.807908i \(-0.299400\pi\)
0.589308 + 0.807908i \(0.299400\pi\)
\(684\) 0 0
\(685\) −13.1729 −0.503312
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.94011 0.112009
\(690\) 0 0
\(691\) −15.0928 −0.574159 −0.287079 0.957907i \(-0.592684\pi\)
−0.287079 + 0.957907i \(0.592684\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.28318 0.162470
\(696\) 0 0
\(697\) 15.9831 0.605404
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.62906 0.363685 0.181842 0.983328i \(-0.441794\pi\)
0.181842 + 0.983328i \(0.441794\pi\)
\(702\) 0 0
\(703\) −12.4254 −0.468634
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.5500 −0.471990
\(708\) 0 0
\(709\) −1.58482 −0.0595191 −0.0297596 0.999557i \(-0.509474\pi\)
−0.0297596 + 0.999557i \(0.509474\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 20.5296 0.768841
\(714\) 0 0
\(715\) 0.577317 0.0215904
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −41.7830 −1.55824 −0.779122 0.626873i \(-0.784335\pi\)
−0.779122 + 0.626873i \(0.784335\pi\)
\(720\) 0 0
\(721\) −11.3876 −0.424097
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 46.7377 1.73579
\(726\) 0 0
\(727\) −14.0272 −0.520240 −0.260120 0.965576i \(-0.583762\pi\)
−0.260120 + 0.965576i \(0.583762\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 6.99417 0.258689
\(732\) 0 0
\(733\) −1.14183 −0.0421745 −0.0210873 0.999778i \(-0.506713\pi\)
−0.0210873 + 0.999778i \(0.506713\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13.7438 0.506258
\(738\) 0 0
\(739\) 21.2412 0.781370 0.390685 0.920524i \(-0.372238\pi\)
0.390685 + 0.920524i \(0.372238\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −6.10797 −0.224080 −0.112040 0.993704i \(-0.535738\pi\)
−0.112040 + 0.993704i \(0.535738\pi\)
\(744\) 0 0
\(745\) 2.41124 0.0883409
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −39.2393 −1.43377
\(750\) 0 0
\(751\) −37.5690 −1.37091 −0.685457 0.728113i \(-0.740397\pi\)
−0.685457 + 0.728113i \(0.740397\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.61700 −0.0588485
\(756\) 0 0
\(757\) 21.2962 0.774022 0.387011 0.922075i \(-0.373508\pi\)
0.387011 + 0.922075i \(0.373508\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 36.6289 1.32780 0.663899 0.747823i \(-0.268901\pi\)
0.663899 + 0.747823i \(0.268901\pi\)
\(762\) 0 0
\(763\) −14.1598 −0.512620
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.161250 0.00582238
\(768\) 0 0
\(769\) −13.2465 −0.477680 −0.238840 0.971059i \(-0.576767\pi\)
−0.238840 + 0.971059i \(0.576767\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −24.8912 −0.895276 −0.447638 0.894215i \(-0.647735\pi\)
−0.447638 + 0.894215i \(0.647735\pi\)
\(774\) 0 0
\(775\) −29.1627 −1.04756
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.76310 −0.170656
\(780\) 0 0
\(781\) 4.97902 0.178163
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.98329 −0.142170
\(786\) 0 0
\(787\) 34.6156 1.23391 0.616956 0.786998i \(-0.288366\pi\)
0.616956 + 0.786998i \(0.288366\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −38.2848 −1.36125
\(792\) 0 0
\(793\) −4.92574 −0.174918
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 47.4262 1.67992 0.839961 0.542647i \(-0.182578\pi\)
0.839961 + 0.542647i \(0.182578\pi\)
\(798\) 0 0
\(799\) 30.4513 1.07729
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.63391 0.234106
\(804\) 0 0
\(805\) −4.40895 −0.155395
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −45.5375 −1.60101 −0.800507 0.599324i \(-0.795436\pi\)
−0.800507 + 0.599324i \(0.795436\pi\)
\(810\) 0 0
\(811\) 47.6530 1.67332 0.836662 0.547720i \(-0.184504\pi\)
0.836662 + 0.547720i \(0.184504\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −0.534590 −0.0187259
\(816\) 0 0
\(817\) −2.08432 −0.0729212
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −20.4373 −0.713267 −0.356633 0.934244i \(-0.616075\pi\)
−0.356633 + 0.934244i \(0.616075\pi\)
\(822\) 0 0
\(823\) 16.7858 0.585117 0.292559 0.956248i \(-0.405493\pi\)
0.292559 + 0.956248i \(0.405493\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −40.2453 −1.39947 −0.699733 0.714405i \(-0.746698\pi\)
−0.699733 + 0.714405i \(0.746698\pi\)
\(828\) 0 0
\(829\) −19.5946 −0.680549 −0.340275 0.940326i \(-0.610520\pi\)
−0.340275 + 0.940326i \(0.610520\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 6.77353 0.234689
\(834\) 0 0
\(835\) 8.07480 0.279440
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 44.2515 1.52773 0.763866 0.645375i \(-0.223299\pi\)
0.763866 + 0.645375i \(0.223299\pi\)
\(840\) 0 0
\(841\) 71.3029 2.45872
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −0.577317 −0.0198603
\(846\) 0 0
\(847\) 2.32465 0.0798758
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −32.2752 −1.10638
\(852\) 0 0
\(853\) 1.94453 0.0665794 0.0332897 0.999446i \(-0.489402\pi\)
0.0332897 + 0.999446i \(0.489402\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 20.2112 0.690402 0.345201 0.938529i \(-0.387811\pi\)
0.345201 + 0.938529i \(0.387811\pi\)
\(858\) 0 0
\(859\) 39.9473 1.36299 0.681493 0.731825i \(-0.261331\pi\)
0.681493 + 0.731825i \(0.261331\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 38.0748 1.29608 0.648040 0.761606i \(-0.275589\pi\)
0.648040 + 0.761606i \(0.275589\pi\)
\(864\) 0 0
\(865\) −1.65641 −0.0563195
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −5.30391 −0.179923
\(870\) 0 0
\(871\) −13.7438 −0.465690
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12.9733 0.438577
\(876\) 0 0
\(877\) −23.9780 −0.809679 −0.404840 0.914388i \(-0.632673\pi\)
−0.404840 + 0.914388i \(0.632673\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 9.70216 0.326874 0.163437 0.986554i \(-0.447742\pi\)
0.163437 + 0.986554i \(0.447742\pi\)
\(882\) 0 0
\(883\) −25.1937 −0.847835 −0.423917 0.905701i \(-0.639345\pi\)
−0.423917 + 0.905701i \(0.639345\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 42.3766 1.42287 0.711434 0.702753i \(-0.248046\pi\)
0.711434 + 0.702753i \(0.248046\pi\)
\(888\) 0 0
\(889\) 30.3641 1.01838
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −9.07473 −0.303674
\(894\) 0 0
\(895\) −5.54971 −0.185506
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −62.5856 −2.08735
\(900\) 0 0
\(901\) 12.4779 0.415699
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0.299462 0.00995445
\(906\) 0 0
\(907\) −8.41088 −0.279279 −0.139639 0.990202i \(-0.544594\pi\)
−0.139639 + 0.990202i \(0.544594\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −0.389007 −0.0128884 −0.00644419 0.999979i \(-0.502051\pi\)
−0.00644419 + 0.999979i \(0.502051\pi\)
\(912\) 0 0
\(913\) −0.0291080 −0.000963335 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −10.9853 −0.362767
\(918\) 0 0
\(919\) 36.6999 1.21062 0.605309 0.795990i \(-0.293049\pi\)
0.605309 + 0.795990i \(0.293049\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −4.97902 −0.163886
\(924\) 0 0
\(925\) 45.8475 1.50746
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 48.3490 1.58628 0.793140 0.609039i \(-0.208445\pi\)
0.793140 + 0.609039i \(0.208445\pi\)
\(930\) 0 0
\(931\) −2.01857 −0.0661559
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.45015 0.0801283
\(936\) 0 0
\(937\) −37.0605 −1.21071 −0.605357 0.795954i \(-0.706970\pi\)
−0.605357 + 0.795954i \(0.706970\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −12.9386 −0.421788 −0.210894 0.977509i \(-0.567637\pi\)
−0.210894 + 0.977509i \(0.567637\pi\)
\(942\) 0 0
\(943\) −12.3722 −0.402895
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −5.23237 −0.170029 −0.0850145 0.996380i \(-0.527094\pi\)
−0.0850145 + 0.996380i \(0.527094\pi\)
\(948\) 0 0
\(949\) −6.63391 −0.215346
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 31.9464 1.03485 0.517423 0.855730i \(-0.326891\pi\)
0.517423 + 0.855730i \(0.326891\pi\)
\(954\) 0 0
\(955\) −6.51573 −0.210844
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 53.0427 1.71284
\(960\) 0 0
\(961\) 8.05129 0.259719
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.02852 0.129683
\(966\) 0 0
\(967\) −13.4290 −0.431849 −0.215924 0.976410i \(-0.569276\pi\)
−0.215924 + 0.976410i \(0.569276\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 32.7906 1.05230 0.526150 0.850392i \(-0.323635\pi\)
0.526150 + 0.850392i \(0.323635\pi\)
\(972\) 0 0
\(973\) −17.2468 −0.552908
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 23.5258 0.752657 0.376329 0.926486i \(-0.377186\pi\)
0.376329 + 0.926486i \(0.377186\pi\)
\(978\) 0 0
\(979\) −2.17537 −0.0695250
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 48.6643 1.55215 0.776075 0.630641i \(-0.217208\pi\)
0.776075 + 0.630641i \(0.217208\pi\)
\(984\) 0 0
\(985\) 11.5659 0.368519
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −5.41405 −0.172157
\(990\) 0 0
\(991\) 11.8776 0.377306 0.188653 0.982044i \(-0.439588\pi\)
0.188653 + 0.982044i \(0.439588\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.98245 0.157954
\(996\) 0 0
\(997\) −12.6861 −0.401774 −0.200887 0.979614i \(-0.564382\pi\)
−0.200887 + 0.979614i \(0.564382\pi\)
\(998\) 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 5148.2.a.s.1.3 5
3.2 odd 2 5148.2.a.t.1.3 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5148.2.a.s.1.3 5 1.1 even 1 trivial
5148.2.a.t.1.3 yes 5 3.2 odd 2