Properties

Label 35.7.c.a.34.1
Level $35$
Weight $7$
Character 35.34
Self dual yes
Analytic conductor $8.052$
Analytic rank $0$
Dimension $1$
CM discriminant -35
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [35,7,Mod(34,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.34");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 35.c (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: yes
Analytic conductor: \(8.05189292669\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 34.1
Character \(\chi\) \(=\) 35.34

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-26.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +343.000 q^{7} -53.0000 q^{9} +O(q^{10})\) \(q-26.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +343.000 q^{7} -53.0000 q^{9} +2522.00 q^{11} -1664.00 q^{12} +2774.00 q^{13} +3250.00 q^{15} +4096.00 q^{16} -754.000 q^{17} -8000.00 q^{20} -8918.00 q^{21} +15625.0 q^{25} +20332.0 q^{27} +21952.0 q^{28} -45862.0 q^{29} -65572.0 q^{33} -42875.0 q^{35} -3392.00 q^{36} -72124.0 q^{39} +161408. q^{44} +6625.00 q^{45} +175646. q^{47} -106496. q^{48} +117649. q^{49} +19604.0 q^{51} +177536. q^{52} -315250. q^{55} +208000. q^{60} -18179.0 q^{63} +262144. q^{64} -346750. q^{65} -48256.0 q^{68} -30238.0 q^{71} +504254. q^{73} -406250. q^{75} +865046. q^{77} -930382. q^{79} -512000. q^{80} -489995. q^{81} -1.14131e6 q^{83} -570752. q^{84} +94250.0 q^{85} +1.19241e6 q^{87} +951482. q^{91} -897874. q^{97} -133666. q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −26.0000 −0.962963 −0.481481 0.876456i \(-0.659901\pi\)
−0.481481 + 0.876456i \(0.659901\pi\)
\(4\) 64.0000 1.00000
\(5\) −125.000 −1.00000
\(6\) 0 0
\(7\) 343.000 1.00000
\(8\) 0 0
\(9\) −53.0000 −0.0727023
\(10\) 0 0
\(11\) 2522.00 1.89482 0.947408 0.320028i \(-0.103692\pi\)
0.947408 + 0.320028i \(0.103692\pi\)
\(12\) −1664.00 −0.962963
\(13\) 2774.00 1.26263 0.631315 0.775526i \(-0.282515\pi\)
0.631315 + 0.775526i \(0.282515\pi\)
\(14\) 0 0
\(15\) 3250.00 0.962963
\(16\) 4096.00 1.00000
\(17\) −754.000 −0.153470 −0.0767352 0.997052i \(-0.524450\pi\)
−0.0767352 + 0.997052i \(0.524450\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −8000.00 −1.00000
\(21\) −8918.00 −0.962963
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 15625.0 1.00000
\(26\) 0 0
\(27\) 20332.0 1.03297
\(28\) 21952.0 1.00000
\(29\) −45862.0 −1.88044 −0.940219 0.340571i \(-0.889380\pi\)
−0.940219 + 0.340571i \(0.889380\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) −65572.0 −1.82464
\(34\) 0 0
\(35\) −42875.0 −1.00000
\(36\) −3392.00 −0.0727023
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −72124.0 −1.21587
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 161408. 1.89482
\(45\) 6625.00 0.0727023
\(46\) 0 0
\(47\) 175646. 1.69178 0.845892 0.533355i \(-0.179069\pi\)
0.845892 + 0.533355i \(0.179069\pi\)
\(48\) −106496. −0.962963
\(49\) 117649. 1.00000
\(50\) 0 0
\(51\) 19604.0 0.147786
\(52\) 177536. 1.26263
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −315250. −1.89482
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 208000. 0.962963
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −18179.0 −0.0727023
\(64\) 262144. 1.00000
\(65\) −346750. −1.26263
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −48256.0 −0.153470
\(69\) 0 0
\(70\) 0 0
\(71\) −30238.0 −0.0844847 −0.0422423 0.999107i \(-0.513450\pi\)
−0.0422423 + 0.999107i \(0.513450\pi\)
\(72\) 0 0
\(73\) 504254. 1.29623 0.648113 0.761544i \(-0.275558\pi\)
0.648113 + 0.761544i \(0.275558\pi\)
\(74\) 0 0
\(75\) −406250. −0.962963
\(76\) 0 0
\(77\) 865046. 1.89482
\(78\) 0 0
\(79\) −930382. −1.88704 −0.943518 0.331322i \(-0.892505\pi\)
−0.943518 + 0.331322i \(0.892505\pi\)
\(80\) −512000. −1.00000
\(81\) −489995. −0.922012
\(82\) 0 0
\(83\) −1.14131e6 −1.99603 −0.998017 0.0629490i \(-0.979949\pi\)
−0.998017 + 0.0629490i \(0.979949\pi\)
\(84\) −570752. −0.962963
\(85\) 94250.0 0.153470
\(86\) 0 0
\(87\) 1.19241e6 1.81079
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 951482. 1.26263
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −897874. −0.983785 −0.491892 0.870656i \(-0.663695\pi\)
−0.491892 + 0.870656i \(0.663695\pi\)
\(98\) 0 0
\(99\) −133666. −0.137758
\(100\) 1.00000e6 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −1.54703e6 −1.41575 −0.707874 0.706339i \(-0.750346\pi\)
−0.707874 + 0.706339i \(0.750346\pi\)
\(104\) 0 0
\(105\) 1.11475e6 0.962963
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 1.30125e6 1.03297
\(109\) 2.54470e6 1.96497 0.982487 0.186332i \(-0.0596599\pi\)
0.982487 + 0.186332i \(0.0596599\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.40493e6 1.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.93517e6 −1.88044
\(117\) −147022. −0.0917962
\(118\) 0 0
\(119\) −258622. −0.153470
\(120\) 0 0
\(121\) 4.58892e6 2.59033
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.95312e6 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) −4.19661e6 −1.82464
\(133\) 0 0
\(134\) 0 0
\(135\) −2.54150e6 −1.03297
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −2.74400e6 −1.00000
\(141\) −4.56680e6 −1.62912
\(142\) 0 0
\(143\) 6.99603e6 2.39245
\(144\) −217088. −0.0727023
\(145\) 5.73275e6 1.88044
\(146\) 0 0
\(147\) −3.05887e6 −0.962963
\(148\) 0 0
\(149\) −534742. −0.161654 −0.0808268 0.996728i \(-0.525756\pi\)
−0.0808268 + 0.996728i \(0.525756\pi\)
\(150\) 0 0
\(151\) 887042. 0.257640 0.128820 0.991668i \(-0.458881\pi\)
0.128820 + 0.991668i \(0.458881\pi\)
\(152\) 0 0
\(153\) 39962.0 0.0111577
\(154\) 0 0
\(155\) 0 0
\(156\) −4.61594e6 −1.21587
\(157\) −7.50279e6 −1.93876 −0.969380 0.245565i \(-0.921026\pi\)
−0.969380 + 0.245565i \(0.921026\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 8.19650e6 1.82464
\(166\) 0 0
\(167\) 2.77125e6 0.595012 0.297506 0.954720i \(-0.403845\pi\)
0.297506 + 0.954720i \(0.403845\pi\)
\(168\) 0 0
\(169\) 2.86827e6 0.594237
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.04907e6 −1.74769 −0.873847 0.486201i \(-0.838382\pi\)
−0.873847 + 0.486201i \(0.838382\pi\)
\(174\) 0 0
\(175\) 5.35938e6 1.00000
\(176\) 1.03301e7 1.89482
\(177\) 0 0
\(178\) 0 0
\(179\) −1.05946e7 −1.84725 −0.923623 0.383302i \(-0.874787\pi\)
−0.923623 + 0.383302i \(0.874787\pi\)
\(180\) 424000. 0.0727023
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.90159e6 −0.290798
\(188\) 1.12413e7 1.69178
\(189\) 6.97388e6 1.03297
\(190\) 0 0
\(191\) 3.80520e6 0.546107 0.273053 0.961999i \(-0.411966\pi\)
0.273053 + 0.961999i \(0.411966\pi\)
\(192\) −6.81574e6 −0.962963
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 9.01550e6 1.21587
\(196\) 7.52954e6 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.57307e7 −1.88044
\(204\) 1.25466e6 0.147786
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1.13623e7 1.26263
\(209\) 0 0
\(210\) 0 0
\(211\) −1.30662e7 −1.39092 −0.695460 0.718565i \(-0.744799\pi\)
−0.695460 + 0.718565i \(0.744799\pi\)
\(212\) 0 0
\(213\) 786188. 0.0813556
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.31106e7 −1.24822
\(220\) −2.01760e7 −1.89482
\(221\) −2.09160e6 −0.193776
\(222\) 0 0
\(223\) 4.65721e6 0.419964 0.209982 0.977705i \(-0.432660\pi\)
0.209982 + 0.977705i \(0.432660\pi\)
\(224\) 0 0
\(225\) −828125. −0.0727023
\(226\) 0 0
\(227\) 667046. 0.0570267 0.0285133 0.999593i \(-0.490923\pi\)
0.0285133 + 0.999593i \(0.490923\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) −2.24912e7 −1.82464
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −2.19558e7 −1.69178
\(236\) 0 0
\(237\) 2.41899e7 1.81715
\(238\) 0 0
\(239\) 5.46034e6 0.399969 0.199984 0.979799i \(-0.435911\pi\)
0.199984 + 0.979799i \(0.435911\pi\)
\(240\) 1.33120e7 0.962963
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −2.08216e6 −0.145109
\(244\) 0 0
\(245\) −1.47061e7 −1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 2.96740e7 1.92211
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −1.16346e6 −0.0727023
\(253\) 0 0
\(254\) 0 0
\(255\) −2.45050e6 −0.147786
\(256\) 1.67772e7 1.00000
\(257\) 2.26692e7 1.33548 0.667738 0.744396i \(-0.267263\pi\)
0.667738 + 0.744396i \(0.267263\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.21920e7 −1.26263
\(261\) 2.43069e6 0.136712
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −3.08838e6 −0.153470
\(273\) −2.47385e7 −1.21587
\(274\) 0 0
\(275\) 3.94062e7 1.89482
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.15258e7 0.970155 0.485078 0.874471i \(-0.338791\pi\)
0.485078 + 0.874471i \(0.338791\pi\)
\(282\) 0 0
\(283\) −4.03676e7 −1.78104 −0.890520 0.454943i \(-0.849660\pi\)
−0.890520 + 0.454943i \(0.849660\pi\)
\(284\) −1.93523e6 −0.0844847
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.35691e7 −0.976447
\(290\) 0 0
\(291\) 2.33447e7 0.947349
\(292\) 3.22723e7 1.29623
\(293\) 4.88653e6 0.194267 0.0971333 0.995271i \(-0.469033\pi\)
0.0971333 + 0.995271i \(0.469033\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.12773e7 1.95729
\(298\) 0 0
\(299\) 0 0
\(300\) −2.60000e7 −0.962963
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2.33370e7 0.806546 0.403273 0.915080i \(-0.367872\pi\)
0.403273 + 0.915080i \(0.367872\pi\)
\(308\) 5.53629e7 1.89482
\(309\) 4.02227e7 1.36331
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −5.41596e7 −1.76621 −0.883105 0.469175i \(-0.844551\pi\)
−0.883105 + 0.469175i \(0.844551\pi\)
\(314\) 0 0
\(315\) 2.27238e6 0.0727023
\(316\) −5.95444e7 −1.88704
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) −1.15664e8 −3.56308
\(320\) −3.27680e7 −1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −3.13597e7 −0.922012
\(325\) 4.33438e7 1.26263
\(326\) 0 0
\(327\) −6.61621e7 −1.89220
\(328\) 0 0
\(329\) 6.02466e7 1.69178
\(330\) 0 0
\(331\) −1.72948e7 −0.476903 −0.238452 0.971154i \(-0.576640\pi\)
−0.238452 + 0.971154i \(0.576640\pi\)
\(332\) −7.30436e7 −1.99603
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −3.65281e7 −0.962963
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 6.03200e6 0.153470
\(341\) 0 0
\(342\) 0 0
\(343\) 4.03536e7 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 7.63144e7 1.81079
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 5.64010e7 1.30426
\(352\) 0 0
\(353\) 4.92763e7 1.12025 0.560124 0.828409i \(-0.310753\pi\)
0.560124 + 0.828409i \(0.310753\pi\)
\(354\) 0 0
\(355\) 3.77975e6 0.0844847
\(356\) 0 0
\(357\) 6.72417e6 0.147786
\(358\) 0 0
\(359\) −3.03791e7 −0.656586 −0.328293 0.944576i \(-0.606473\pi\)
−0.328293 + 0.944576i \(0.606473\pi\)
\(360\) 0 0
\(361\) 4.70459e7 1.00000
\(362\) 0 0
\(363\) −1.19312e8 −2.49439
\(364\) 6.08948e7 1.26263
\(365\) −6.30317e7 −1.29623
\(366\) 0 0
\(367\) 9.82137e7 1.98689 0.993445 0.114308i \(-0.0364650\pi\)
0.993445 + 0.114308i \(0.0364650\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 5.07812e7 0.962963
\(376\) 0 0
\(377\) −1.27221e8 −2.37430
\(378\) 0 0
\(379\) 8.98173e7 1.64984 0.824921 0.565247i \(-0.191219\pi\)
0.824921 + 0.565247i \(0.191219\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.03977e7 1.60902 0.804509 0.593940i \(-0.202428\pi\)
0.804509 + 0.593940i \(0.202428\pi\)
\(384\) 0 0
\(385\) −1.08131e8 −1.89482
\(386\) 0 0
\(387\) 0 0
\(388\) −5.74639e7 −0.983785
\(389\) 7.28779e7 1.23808 0.619038 0.785361i \(-0.287523\pi\)
0.619038 + 0.785361i \(0.287523\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.16298e8 1.88704
\(396\) −8.55462e6 −0.137758
\(397\) −1.25066e8 −1.99879 −0.999395 0.0347801i \(-0.988927\pi\)
−0.999395 + 0.0347801i \(0.988927\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 6.40000e7 1.00000
\(401\) 3.48264e7 0.540102 0.270051 0.962846i \(-0.412959\pi\)
0.270051 + 0.962846i \(0.412959\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 6.12494e7 0.922012
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −9.90097e7 −1.41575
\(413\) 0 0
\(414\) 0 0
\(415\) 1.42663e8 1.99603
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 7.13440e7 0.962963
\(421\) −1.33855e8 −1.79386 −0.896928 0.442176i \(-0.854207\pi\)
−0.896928 + 0.442176i \(0.854207\pi\)
\(422\) 0 0
\(423\) −9.30924e6 −0.122997
\(424\) 0 0
\(425\) −1.17812e7 −0.153470
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.81897e8 −2.30384
\(430\) 0 0
\(431\) −1.45288e8 −1.81467 −0.907336 0.420406i \(-0.861888\pi\)
−0.907336 + 0.420406i \(0.861888\pi\)
\(432\) 8.32799e7 1.03297
\(433\) −4.56095e6 −0.0561812 −0.0280906 0.999605i \(-0.508943\pi\)
−0.0280906 + 0.999605i \(0.508943\pi\)
\(434\) 0 0
\(435\) −1.49052e8 −1.81079
\(436\) 1.62861e8 1.96497
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −6.23540e6 −0.0727023
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1.39033e7 0.155666
\(448\) 8.99154e7 1.00000
\(449\) −5.12145e7 −0.565788 −0.282894 0.959151i \(-0.591294\pi\)
−0.282894 + 0.959151i \(0.591294\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −2.30631e7 −0.248098
\(454\) 0 0
\(455\) −1.18935e8 −1.26263
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) −1.53303e7 −0.158531
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −1.87851e8 −1.88044
\(465\) 0 0
\(466\) 0 0
\(467\) −9.09098e7 −0.892606 −0.446303 0.894882i \(-0.647260\pi\)
−0.446303 + 0.894882i \(0.647260\pi\)
\(468\) −9.40941e6 −0.0917962
\(469\) 0 0
\(470\) 0 0
\(471\) 1.95073e8 1.86695
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −1.65518e7 −0.153470
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.93691e8 2.59033
\(485\) 1.12234e8 0.983785
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7.61620e7 −0.643419 −0.321709 0.946838i \(-0.604257\pi\)
−0.321709 + 0.946838i \(0.604257\pi\)
\(492\) 0 0
\(493\) 3.45799e7 0.288592
\(494\) 0 0
\(495\) 1.67082e7 0.137758
\(496\) 0 0
\(497\) −1.03716e7 −0.0844847
\(498\) 0 0
\(499\) −1.91591e8 −1.54196 −0.770981 0.636858i \(-0.780234\pi\)
−0.770981 + 0.636858i \(0.780234\pi\)
\(500\) −1.25000e8 −1.00000
\(501\) −7.20524e7 −0.572974
\(502\) 0 0
\(503\) 2.48608e8 1.95349 0.976746 0.214399i \(-0.0687792\pi\)
0.976746 + 0.214399i \(0.0687792\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −7.45749e7 −0.572228
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 1.72959e8 1.29623
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.93378e8 1.41575
\(516\) 0 0
\(517\) 4.42979e8 3.20562
\(518\) 0 0
\(519\) 2.35276e8 1.68296
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −2.32865e8 −1.62780 −0.813898 0.581008i \(-0.802658\pi\)
−0.813898 + 0.581008i \(0.802658\pi\)
\(524\) 0 0
\(525\) −1.39344e8 −0.962963
\(526\) 0 0
\(527\) 0 0
\(528\) −2.68583e8 −1.82464
\(529\) 1.48036e8 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 2.75459e8 1.77883
\(538\) 0 0
\(539\) 2.96711e8 1.89482
\(540\) −1.62656e8 −1.03297
\(541\) −2.22241e8 −1.40357 −0.701783 0.712391i \(-0.747613\pi\)
−0.701783 + 0.712391i \(0.747613\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.18087e8 −1.96497
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −3.19121e8 −1.88704
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −1.75616e8 −1.00000
\(561\) 4.94413e7 0.280028
\(562\) 0 0
\(563\) 1.63465e8 0.916009 0.458005 0.888950i \(-0.348564\pi\)
0.458005 + 0.888950i \(0.348564\pi\)
\(564\) −2.92275e8 −1.62912
\(565\) 0 0
\(566\) 0 0
\(567\) −1.68068e8 −0.922012
\(568\) 0 0
\(569\) −2.67919e8 −1.45434 −0.727172 0.686455i \(-0.759166\pi\)
−0.727172 + 0.686455i \(0.759166\pi\)
\(570\) 0 0
\(571\) 1.13775e8 0.611140 0.305570 0.952170i \(-0.401153\pi\)
0.305570 + 0.952170i \(0.401153\pi\)
\(572\) 4.47746e8 2.39245
\(573\) −9.89353e7 −0.525881
\(574\) 0 0
\(575\) 0 0
\(576\) −1.38936e7 −0.0727023
\(577\) −2.89404e7 −0.150653 −0.0753265 0.997159i \(-0.524000\pi\)
−0.0753265 + 0.997159i \(0.524000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 3.66896e8 1.88044
\(581\) −3.91468e8 −1.99603
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 1.83778e7 0.0917962
\(586\) 0 0
\(587\) 1.78463e8 0.882336 0.441168 0.897424i \(-0.354564\pi\)
0.441168 + 0.897424i \(0.354564\pi\)
\(588\) −1.95768e8 −0.962963
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.15836e8 1.99415 0.997074 0.0764385i \(-0.0243549\pi\)
0.997074 + 0.0764385i \(0.0243549\pi\)
\(594\) 0 0
\(595\) 3.23278e7 0.153470
\(596\) −3.42235e7 −0.161654
\(597\) 0 0
\(598\) 0 0
\(599\) −3.13980e8 −1.46090 −0.730452 0.682965i \(-0.760690\pi\)
−0.730452 + 0.682965i \(0.760690\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 5.67707e7 0.257640
\(605\) −5.73615e8 −2.59033
\(606\) 0 0
\(607\) −3.64751e8 −1.63091 −0.815455 0.578821i \(-0.803513\pi\)
−0.815455 + 0.578821i \(0.803513\pi\)
\(608\) 0 0
\(609\) 4.08997e8 1.81079
\(610\) 0 0
\(611\) 4.87242e8 2.13610
\(612\) 2.55757e6 0.0111577
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −2.95420e8 −1.21587
\(625\) 2.44141e8 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −4.80179e8 −1.93876
\(629\) 0 0
\(630\) 0 0
\(631\) −2.81897e8 −1.12203 −0.561013 0.827807i \(-0.689588\pi\)
−0.561013 + 0.827807i \(0.689588\pi\)
\(632\) 0 0
\(633\) 3.39721e8 1.33940
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.26358e8 1.26263
\(638\) 0 0
\(639\) 1.60261e6 0.00614223
\(640\) 0 0
\(641\) 3.01654e8 1.14534 0.572671 0.819785i \(-0.305907\pi\)
0.572671 + 0.819785i \(0.305907\pi\)
\(642\) 0 0
\(643\) 3.71970e8 1.39918 0.699592 0.714542i \(-0.253365\pi\)
0.699592 + 0.714542i \(0.253365\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.99165e8 −1.10458 −0.552291 0.833651i \(-0.686246\pi\)
−0.552291 + 0.833651i \(0.686246\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −2.67255e7 −0.0942387
\(658\) 0 0
\(659\) 4.40385e8 1.53878 0.769389 0.638781i \(-0.220561\pi\)
0.769389 + 0.638781i \(0.220561\pi\)
\(660\) 5.24576e8 1.82464
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 5.43815e7 0.186600
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 1.77360e8 0.595012
\(669\) −1.21088e8 −0.404409
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 3.17688e8 1.03297
\(676\) 1.83569e8 0.594237
\(677\) −3.76577e8 −1.21363 −0.606817 0.794842i \(-0.707554\pi\)
−0.606817 + 0.794842i \(0.707554\pi\)
\(678\) 0 0
\(679\) −3.07971e8 −0.983785
\(680\) 0 0
\(681\) −1.73432e7 −0.0549146
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) −5.79140e8 −1.74769
\(693\) −4.58474e7 −0.137758
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 3.43000e8 1.00000
\(701\) 4.30761e8 1.25050 0.625248 0.780426i \(-0.284998\pi\)
0.625248 + 0.780426i \(0.284998\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 6.61127e8 1.89482
\(705\) 5.70850e8 1.62912
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −7.08282e8 −1.98732 −0.993659 0.112436i \(-0.964135\pi\)
−0.993659 + 0.112436i \(0.964135\pi\)
\(710\) 0 0
\(711\) 4.93102e7 0.137192
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −8.74504e8 −2.39245
\(716\) −6.78053e8 −1.84725
\(717\) −1.41969e8 −0.385155
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 2.71360e7 0.0727023
\(721\) −5.30630e8 −1.41575
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −7.16594e8 −1.88044
\(726\) 0 0
\(727\) −6.38690e8 −1.66221 −0.831107 0.556113i \(-0.812292\pi\)
−0.831107 + 0.556113i \(0.812292\pi\)
\(728\) 0 0
\(729\) 4.11342e8 1.06175
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −5.15801e8 −1.30970 −0.654848 0.755761i \(-0.727267\pi\)
−0.654848 + 0.755761i \(0.727267\pi\)
\(734\) 0 0
\(735\) 3.82359e8 0.962963
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −2.75588e8 −0.682852 −0.341426 0.939909i \(-0.610910\pi\)
−0.341426 + 0.939909i \(0.610910\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 6.68427e7 0.161654
\(746\) 0 0
\(747\) 6.04892e7 0.145116
\(748\) −1.21702e8 −0.290798
\(749\) 0 0
\(750\) 0 0
\(751\) −1.13153e8 −0.267145 −0.133572 0.991039i \(-0.542645\pi\)
−0.133572 + 0.991039i \(0.542645\pi\)
\(752\) 7.19446e8 1.69178
\(753\) 0 0
\(754\) 0 0
\(755\) −1.10880e8 −0.257640
\(756\) 4.46328e8 1.03297
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 8.72831e8 1.96497
\(764\) 2.43533e8 0.546107
\(765\) −4.99525e6 −0.0111577
\(766\) 0 0
\(767\) 0 0
\(768\) −4.36208e8 −0.962963
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −5.89398e8 −1.28601
\(772\) 0 0
\(773\) 8.97007e8 1.94204 0.971018 0.239006i \(-0.0768217\pi\)
0.971018 + 0.239006i \(0.0768217\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 5.76992e8 1.21587
\(781\) −7.62602e7 −0.160083
\(782\) 0 0
\(783\) −9.32466e8 −1.94244
\(784\) 4.81890e8 1.00000
\(785\) 9.37849e8 1.93876
\(786\) 0 0
\(787\) −7.43771e8 −1.52586 −0.762931 0.646480i \(-0.776240\pi\)
−0.762931 + 0.646480i \(0.776240\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6.71089e8 −1.32558 −0.662789 0.748806i \(-0.730627\pi\)
−0.662789 + 0.748806i \(0.730627\pi\)
\(798\) 0 0
\(799\) −1.32437e8 −0.259639
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.27173e9 2.45611
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.89541e8 0.357980 0.178990 0.983851i \(-0.442717\pi\)
0.178990 + 0.983851i \(0.442717\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −1.00676e9 −1.88044
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 8.02980e7 0.147786
\(817\) 0 0
\(818\) 0 0
\(819\) −5.04285e7 −0.0917962
\(820\) 0 0
\(821\) 9.00444e8 1.62715 0.813574 0.581461i \(-0.197519\pi\)
0.813574 + 0.581461i \(0.197519\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) −1.02456e9 −1.82464
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 7.27187e8 1.26263
\(833\) −8.87073e7 −0.153470
\(834\) 0 0
\(835\) −3.46406e8 −0.595012
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.50850e9 2.53605
\(842\) 0 0
\(843\) −5.59672e8 −0.934223
\(844\) −8.36237e8 −1.39092
\(845\) −3.58533e8 −0.594237
\(846\) 0 0
\(847\) 1.57400e9 2.59033
\(848\) 0 0
\(849\) 1.04956e9 1.71508
\(850\) 0 0
\(851\) 0 0
\(852\) 5.03160e7 0.0813556
\(853\) −1.87087e8 −0.301437 −0.150719 0.988577i \(-0.548159\pi\)
−0.150719 + 0.988577i \(0.548159\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.20367e9 1.91233 0.956167 0.292820i \(-0.0945938\pi\)
0.956167 + 0.292820i \(0.0945938\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 1.13113e9 1.74769
\(866\) 0 0
\(867\) 6.12795e8 0.940282
\(868\) 0 0
\(869\) −2.34642e9 −3.57558
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 4.75873e7 0.0715235
\(874\) 0 0
\(875\) −6.69922e8 −1.00000
\(876\) −8.39079e8 −1.24822
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) −1.27050e8 −0.187072
\(880\) −1.29126e9 −1.89482
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −1.33862e8 −0.193776
\(885\) 0 0
\(886\) 0 0
\(887\) −1.04544e9 −1.49805 −0.749027 0.662540i \(-0.769479\pi\)
−0.749027 + 0.662540i \(0.769479\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.23577e9 −1.74704
\(892\) 2.98062e8 0.419964
\(893\) 0 0
\(894\) 0 0
\(895\) 1.32432e9 1.84725
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −5.30000e7 −0.0727023
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 4.26909e7 0.0570267
\(909\) 0 0
\(910\) 0 0
\(911\) −5.46895e8 −0.723351 −0.361676 0.932304i \(-0.617795\pi\)
−0.361676 + 0.932304i \(0.617795\pi\)
\(912\) 0 0
\(913\) −2.87837e9 −3.78212
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.53505e9 1.97778 0.988889 0.148658i \(-0.0474953\pi\)
0.988889 + 0.148658i \(0.0474953\pi\)
\(920\) 0 0
\(921\) −6.06761e8 −0.776674
\(922\) 0 0
\(923\) −8.38802e7 −0.106673
\(924\) −1.43944e9 −1.82464
\(925\) 0 0
\(926\) 0 0
\(927\) 8.19924e7 0.102928
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.37698e8 0.290798
\(936\) 0 0
\(937\) 1.30103e9 1.58150 0.790749 0.612141i \(-0.209691\pi\)
0.790749 + 0.612141i \(0.209691\pi\)
\(938\) 0 0
\(939\) 1.40815e9 1.70079
\(940\) −1.40517e9 −1.69178
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −8.71734e8 −1.03297
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 1.54816e9 1.81715
\(949\) 1.39880e9 1.63666
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −4.75650e8 −0.546107
\(956\) 3.49462e8 0.399969
\(957\) 3.00726e9 3.43112
\(958\) 0 0
\(959\) 0 0
\(960\) 8.51968e8 0.962963
\(961\) 8.87504e8 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −1.33258e8 −0.145109
\(973\) 0 0
\(974\) 0 0
\(975\) −1.12694e9 −1.21587
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −9.41192e8 −1.00000
\(981\) −1.34869e8 −0.142858
\(982\) 0 0
\(983\) −1.84094e9 −1.93811 −0.969054 0.246849i \(-0.920605\pi\)
−0.969054 + 0.246849i \(0.920605\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −1.56641e9 −1.62912
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.75082e9 −1.79896 −0.899478 0.436966i \(-0.856053\pi\)
−0.899478 + 0.436966i \(0.856053\pi\)
\(992\) 0 0
\(993\) 4.49664e8 0.459240
\(994\) 0 0
\(995\) 0 0
\(996\) 1.89913e9 1.92211
\(997\) 4.23029e8 0.426859 0.213430 0.976958i \(-0.431537\pi\)
0.213430 + 0.976958i \(0.431537\pi\)
\(998\) 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 35.7.c.a.34.1 1
5.2 odd 4 175.7.d.c.76.2 2
5.3 odd 4 175.7.d.c.76.1 2
5.4 even 2 35.7.c.b.34.1 yes 1
7.6 odd 2 35.7.c.b.34.1 yes 1
35.13 even 4 175.7.d.c.76.2 2
35.27 even 4 175.7.d.c.76.1 2
35.34 odd 2 CM 35.7.c.a.34.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.7.c.a.34.1 1 1.1 even 1 trivial
35.7.c.a.34.1 1 35.34 odd 2 CM
35.7.c.b.34.1 yes 1 5.4 even 2
35.7.c.b.34.1 yes 1 7.6 odd 2
175.7.d.c.76.1 2 5.3 odd 4
175.7.d.c.76.1 2 35.27 even 4
175.7.d.c.76.2 2 5.2 odd 4
175.7.d.c.76.2 2 35.13 even 4