Properties

Label 600.2.b.d.251.1
Level $600$
Weight $2$
Character 600.251
Analytic conductor $4.791$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,2,Mod(251,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.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: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 251.1
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 600.251
Dual form 600.2.b.d.251.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} +(-0.724745 - 1.57313i) q^{3} -2.00000 q^{4} +(-2.22474 + 1.02494i) q^{6} +2.82843i q^{8} +(-1.94949 + 2.28024i) q^{9} +O(q^{10})\) \(q-1.41421i q^{2} +(-0.724745 - 1.57313i) q^{3} -2.00000 q^{4} +(-2.22474 + 1.02494i) q^{6} +2.82843i q^{8} +(-1.94949 + 2.28024i) q^{9} +6.61037i q^{11} +(1.44949 + 3.14626i) q^{12} +4.00000 q^{16} +2.36773i q^{17} +(3.22474 + 2.75699i) q^{18} -8.34847 q^{19} +9.34847 q^{22} +(4.44949 - 2.04989i) q^{24} +(5.00000 + 1.41421i) q^{27} -5.65685i q^{32} +(10.3990 - 4.79083i) q^{33} +3.34847 q^{34} +(3.89898 - 4.56048i) q^{36} +11.8065i q^{38} +0.460702i q^{41} -10.0000 q^{43} -13.2207i q^{44} +(-2.89898 - 6.29253i) q^{48} +7.00000 q^{49} +(3.72474 - 1.71600i) q^{51} +(2.00000 - 7.07107i) q^{54} +(6.05051 + 13.1332i) q^{57} +14.1421i q^{59} -8.00000 q^{64} +(-6.77526 - 14.7064i) q^{66} -14.3485 q^{67} -4.73545i q^{68} +(-6.44949 - 5.51399i) q^{72} +13.6969 q^{73} +16.6969 q^{76} +(-1.39898 - 8.89060i) q^{81} +0.651531 q^{82} -14.1742i q^{83} +14.1421i q^{86} -18.6969 q^{88} +12.7600i q^{89} +(-8.89898 + 4.09978i) q^{96} -10.0000 q^{97} -9.89949i q^{98} +(-15.0732 - 12.8868i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 8 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 8 q^{4} - 4 q^{6} + 2 q^{9} - 4 q^{12} + 16 q^{16} + 8 q^{18} - 4 q^{19} + 8 q^{22} + 8 q^{24} + 20 q^{27} + 22 q^{33} - 16 q^{34} - 4 q^{36} - 40 q^{43} + 8 q^{48} + 28 q^{49} + 10 q^{51} + 8 q^{54} + 34 q^{57} - 32 q^{64} - 32 q^{66} - 28 q^{67} - 16 q^{72} - 4 q^{73} + 8 q^{76} + 14 q^{81} + 32 q^{82} - 16 q^{88} - 16 q^{96} - 40 q^{97} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) −0.724745 1.57313i −0.418432 0.908248i
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) −2.22474 + 1.02494i −0.908248 + 0.418432i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 2.82843i 1.00000i
\(9\) −1.94949 + 2.28024i −0.649830 + 0.760080i
\(10\) 0 0
\(11\) 6.61037i 1.99310i 0.0829925 + 0.996550i \(0.473552\pi\)
−0.0829925 + 0.996550i \(0.526448\pi\)
\(12\) 1.44949 + 3.14626i 0.418432 + 0.908248i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 2.36773i 0.574258i 0.957892 + 0.287129i \(0.0927008\pi\)
−0.957892 + 0.287129i \(0.907299\pi\)
\(18\) 3.22474 + 2.75699i 0.760080 + 0.649830i
\(19\) −8.34847 −1.91527 −0.957635 0.287984i \(-0.907015\pi\)
−0.957635 + 0.287984i \(0.907015\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 9.34847 1.99310
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 4.44949 2.04989i 0.908248 0.418432i
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000 + 1.41421i 0.962250 + 0.272166i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 10.3990 4.79083i 1.81023 0.833976i
\(34\) 3.34847 0.574258
\(35\) 0 0
\(36\) 3.89898 4.56048i 0.649830 0.760080i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 11.8065i 1.91527i
\(39\) 0 0
\(40\) 0 0
\(41\) 0.460702i 0.0719495i 0.999353 + 0.0359748i \(0.0114536\pi\)
−0.999353 + 0.0359748i \(0.988546\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 13.2207i 1.99310i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −2.89898 6.29253i −0.418432 0.908248i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 3.72474 1.71600i 0.521569 0.240288i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 2.00000 7.07107i 0.272166 0.962250i
\(55\) 0 0
\(56\) 0 0
\(57\) 6.05051 + 13.1332i 0.801410 + 1.73954i
\(58\) 0 0
\(59\) 14.1421i 1.84115i 0.390567 + 0.920575i \(0.372279\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) −6.77526 14.7064i −0.833976 1.81023i
\(67\) −14.3485 −1.75294 −0.876472 0.481452i \(-0.840109\pi\)
−0.876472 + 0.481452i \(0.840109\pi\)
\(68\) 4.73545i 0.574258i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −6.44949 5.51399i −0.760080 0.649830i
\(73\) 13.6969 1.60311 0.801553 0.597924i \(-0.204008\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 16.6969 1.91527
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −1.39898 8.89060i −0.155442 0.987845i
\(82\) 0.651531 0.0719495
\(83\) 14.1742i 1.55583i −0.628372 0.777913i \(-0.716279\pi\)
0.628372 0.777913i \(-0.283721\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 14.1421i 1.52499i
\(87\) 0 0
\(88\) −18.6969 −1.99310
\(89\) 12.7600i 1.35256i 0.736644 + 0.676280i \(0.236409\pi\)
−0.736644 + 0.676280i \(0.763591\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −8.89898 + 4.09978i −0.908248 + 0.418432i
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 9.89949i 1.00000i
\(99\) −15.0732 12.8868i −1.51492 1.29518i
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −2.42679 5.26758i −0.240288 0.521569i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.70334i 0.454689i 0.973814 + 0.227345i \(0.0730044\pi\)
−0.973814 + 0.227345i \(0.926996\pi\)
\(108\) −10.0000 2.82843i −0.962250 0.272166i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 21.2453i 1.99859i 0.0375328 + 0.999295i \(0.488050\pi\)
−0.0375328 + 0.999295i \(0.511950\pi\)
\(114\) 18.5732 8.55671i 1.73954 0.801410i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 20.0000 1.84115
\(119\) 0 0
\(120\) 0 0
\(121\) −32.6969 −2.97245
\(122\) 0 0
\(123\) 0.724745 0.333891i 0.0653480 0.0301060i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 11.3137i 1.00000i
\(129\) 7.24745 + 15.7313i 0.638102 + 1.38507i
\(130\) 0 0
\(131\) 14.1421i 1.23560i 0.786334 + 0.617802i \(0.211977\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) −20.7980 + 9.58166i −1.81023 + 0.833976i
\(133\) 0 0
\(134\) 20.2918i 1.75294i
\(135\) 0 0
\(136\) −6.69694 −0.574258
\(137\) 16.5099i 1.41053i −0.708942 0.705266i \(-0.750827\pi\)
0.708942 0.705266i \(-0.249173\pi\)
\(138\) 0 0
\(139\) 18.3485 1.55630 0.778148 0.628080i \(-0.216159\pi\)
0.778148 + 0.628080i \(0.216159\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −7.79796 + 9.12096i −0.649830 + 0.760080i
\(145\) 0 0
\(146\) 19.3704i 1.60311i
\(147\) −5.07321 11.0119i −0.418432 0.908248i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 23.6130i 1.91527i
\(153\) −5.39898 4.61586i −0.436482 0.373170i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −12.5732 + 1.97846i −0.987845 + 0.155442i
\(163\) −23.0454 −1.80506 −0.902528 0.430632i \(-0.858291\pi\)
−0.902528 + 0.430632i \(0.858291\pi\)
\(164\) 0.921404i 0.0719495i
\(165\) 0 0
\(166\) −20.0454 −1.55583
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 16.2753 19.0365i 1.24460 1.45576i
\(172\) 20.0000 1.52499
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 26.4415i 1.99310i
\(177\) 22.2474 10.2494i 1.67222 0.770395i
\(178\) 18.0454 1.35256
\(179\) 5.68896i 0.425213i −0.977138 0.212607i \(-0.931805\pi\)
0.977138 0.212607i \(-0.0681952\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −15.6515 −1.14455
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 5.79796 + 12.5851i 0.418432 + 0.908248i
\(193\) −3.69694 −0.266111 −0.133056 0.991109i \(-0.542479\pi\)
−0.133056 + 0.991109i \(0.542479\pi\)
\(194\) 14.1421i 1.01535i
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) −18.2247 + 21.3167i −1.29518 + 1.51492i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 10.3990 + 22.5720i 0.733487 + 1.59211i
\(202\) 0 0
\(203\) 0 0
\(204\) −7.44949 + 3.43199i −0.521569 + 0.240288i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 55.1864i 3.81733i
\(210\) 0 0
\(211\) 15.0454 1.03577 0.517884 0.855451i \(-0.326720\pi\)
0.517884 + 0.855451i \(0.326720\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 6.65153 0.454689
\(215\) 0 0
\(216\) −4.00000 + 14.1421i −0.272166 + 0.962250i
\(217\) 0 0
\(218\) 0 0
\(219\) −9.92679 21.5471i −0.670790 1.45602i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 30.0454 1.99859
\(227\) 2.82843i 0.187729i −0.995585 0.0938647i \(-0.970078\pi\)
0.995585 0.0938647i \(-0.0299221\pi\)
\(228\) −12.1010 26.2665i −0.801410 1.73954i
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.65685i 0.370593i 0.982683 + 0.185296i \(0.0593245\pi\)
−0.982683 + 0.185296i \(0.940675\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 28.2843i 1.84115i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 1.69694 0.109309 0.0546547 0.998505i \(-0.482594\pi\)
0.0546547 + 0.998505i \(0.482594\pi\)
\(242\) 46.2405i 2.97245i
\(243\) −12.9722 + 8.64420i −0.832167 + 0.554526i
\(244\) 0 0
\(245\) 0 0
\(246\) −0.472194 1.02494i −0.0301060 0.0653480i
\(247\) 0 0
\(248\) 0 0
\(249\) −22.2980 + 10.2727i −1.41308 + 0.651007i
\(250\) 0 0
\(251\) 20.7525i 1.30989i −0.755678 0.654943i \(-0.772693\pi\)
0.755678 0.654943i \(-0.227307\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 11.3137i 0.705730i −0.935674 0.352865i \(-0.885208\pi\)
0.935674 0.352865i \(-0.114792\pi\)
\(258\) 22.2474 10.2494i 1.38507 0.638102i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 20.0000 1.23560
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 13.5505 + 29.4128i 0.833976 + 1.81023i
\(265\) 0 0
\(266\) 0 0
\(267\) 20.0732 9.24777i 1.22846 0.565954i
\(268\) 28.6969 1.75294
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 9.47090i 0.574258i
\(273\) 0 0
\(274\) −23.3485 −1.41053
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 25.9487i 1.55630i
\(279\) 0 0
\(280\) 0 0
\(281\) 28.2843i 1.68730i −0.536895 0.843649i \(-0.680403\pi\)
0.536895 0.843649i \(-0.319597\pi\)
\(282\) 0 0
\(283\) 33.0454 1.96435 0.982173 0.187980i \(-0.0601941\pi\)
0.982173 + 0.187980i \(0.0601941\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 12.8990 + 11.0280i 0.760080 + 0.649830i
\(289\) 11.3939 0.670228
\(290\) 0 0
\(291\) 7.24745 + 15.7313i 0.424853 + 0.922186i
\(292\) −27.3939 −1.60311
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) −15.5732 + 7.17461i −0.908248 + 0.418432i
\(295\) 0 0
\(296\) 0 0
\(297\) −9.34847 + 33.0518i −0.542453 + 1.91786i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −33.3939 −1.91527
\(305\) 0 0
\(306\) −6.52781 + 7.63531i −0.373170 + 0.436482i
\(307\) 24.3485 1.38964 0.694820 0.719183i \(-0.255484\pi\)
0.694820 + 0.719183i \(0.255484\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 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
\(320\) 0 0
\(321\) 7.39898 3.40872i 0.412971 0.190256i
\(322\) 0 0
\(323\) 19.7669i 1.09986i
\(324\) 2.79796 + 17.7812i 0.155442 + 0.987845i
\(325\) 0 0
\(326\) 32.5911i 1.80506i
\(327\) 0 0
\(328\) −1.30306 −0.0719495
\(329\) 0 0
\(330\) 0 0
\(331\) −35.0454 −1.92627 −0.963135 0.269019i \(-0.913301\pi\)
−0.963135 + 0.269019i \(0.913301\pi\)
\(332\) 28.3485i 1.55583i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 22.3939 1.21987 0.609936 0.792451i \(-0.291195\pi\)
0.609936 + 0.792451i \(0.291195\pi\)
\(338\) 18.3848i 1.00000i
\(339\) 33.4217 15.3974i 1.81522 0.836274i
\(340\) 0 0
\(341\) 0 0
\(342\) −26.9217 23.0167i −1.45576 1.24460i
\(343\) 0 0
\(344\) 28.2843i 1.52499i
\(345\) 0 0
\(346\) 0 0
\(347\) 23.5809i 1.26589i 0.774197 + 0.632945i \(0.218154\pi\)
−0.774197 + 0.632945i \(0.781846\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 37.3939 1.99310
\(353\) 22.6274i 1.20434i 0.798369 + 0.602168i \(0.205696\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) −14.4949 31.4626i −0.770395 1.67222i
\(355\) 0 0
\(356\) 25.5201i 1.35256i
\(357\) 0 0
\(358\) −8.04541 −0.425213
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 50.6969 2.66826
\(362\) 0 0
\(363\) 23.6969 + 51.4366i 1.24377 + 2.69972i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −1.05051 0.898133i −0.0546874 0.0467550i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 22.1346i 1.14455i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −11.6515 −0.598499 −0.299249 0.954175i \(-0.596736\pi\)
−0.299249 + 0.954175i \(0.596736\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 17.7980 8.19955i 0.908248 0.418432i
\(385\) 0 0
\(386\) 5.22826i 0.266111i
\(387\) 19.4949 22.8024i 0.990981 1.15911i
\(388\) 20.0000 1.01535
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 19.7990i 1.00000i
\(393\) 22.2474 10.2494i 1.12224 0.517016i
\(394\) 0 0
\(395\) 0 0
\(396\) 30.1464 + 25.7737i 1.51492 + 1.29518i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.6028i 0.729231i −0.931158 0.364615i \(-0.881200\pi\)
0.931158 0.364615i \(-0.118800\pi\)
\(402\) 31.9217 14.7064i 1.59211 0.733487i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 4.85357 + 10.5352i 0.240288 + 0.521569i
\(409\) −18.3939 −0.909519 −0.454759 0.890614i \(-0.650275\pi\)
−0.454759 + 0.890614i \(0.650275\pi\)
\(410\) 0 0
\(411\) −25.9722 + 11.9654i −1.28111 + 0.590211i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −13.2980 28.8646i −0.651204 1.41350i
\(418\) −78.0454 −3.81733
\(419\) 33.9732i 1.65970i 0.557986 + 0.829851i \(0.311574\pi\)
−0.557986 + 0.829851i \(0.688426\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 21.2774i 1.03577i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 9.40669i 0.454689i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 20.0000 + 5.65685i 0.962250 + 0.272166i
\(433\) −33.6969 −1.61937 −0.809686 0.586864i \(-0.800362\pi\)
−0.809686 + 0.586864i \(0.800362\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −30.4722 + 14.0386i −1.45602 + 0.670790i
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −13.6464 + 15.9617i −0.649830 + 0.760080i
\(442\) 0 0
\(443\) 37.7873i 1.79533i 0.440681 + 0.897664i \(0.354737\pi\)
−0.440681 + 0.897664i \(0.645263\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 39.2015i 1.85003i −0.379927 0.925016i \(-0.624051\pi\)
0.379927 0.925016i \(-0.375949\pi\)
\(450\) 0 0
\(451\) −3.04541 −0.143403
\(452\) 42.4906i 1.99859i
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) −37.1464 + 17.1134i −1.73954 + 0.801410i
\(457\) −42.3939 −1.98310 −0.991551 0.129718i \(-0.958593\pi\)
−0.991551 + 0.129718i \(0.958593\pi\)
\(458\) 0 0
\(459\) −3.34847 + 11.8386i −0.156293 + 0.552580i
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 8.00000 0.370593
\(467\) 31.1127i 1.43972i 0.694117 + 0.719862i \(0.255795\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −40.0000 −1.84115
\(473\) 66.1037i 3.03945i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 2.39983i 0.109309i
\(483\) 0 0
\(484\) 65.3939 2.97245
\(485\) 0 0
\(486\) 12.2247 + 18.3455i 0.554526 + 0.832167i
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 16.7020 + 36.2535i 0.755292 + 1.63944i
\(490\) 0 0
\(491\) 14.1421i 0.638226i 0.947717 + 0.319113i \(0.103385\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) −1.44949 + 0.667783i −0.0653480 + 0.0301060i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 14.5278 + 31.5341i 0.651007 + 1.41308i
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −29.3485 −1.30989
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −9.42168 20.4507i −0.418432 0.908248i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) −41.7423 11.8065i −1.84297 0.521271i
\(514\) −16.0000 −0.705730
\(515\) 0 0
\(516\) −14.4949 31.4626i −0.638102 1.38507i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 27.8236i 1.21897i 0.792797 + 0.609486i \(0.208624\pi\)
−0.792797 + 0.609486i \(0.791376\pi\)
\(522\) 0 0
\(523\) 3.04541 0.133166 0.0665832 0.997781i \(-0.478790\pi\)
0.0665832 + 0.997781i \(0.478790\pi\)
\(524\) 28.2843i 1.23560i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 41.5959 19.1633i 1.81023 0.833976i
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −32.2474 27.5699i −1.39942 1.19643i
\(532\) 0 0
\(533\) 0 0
\(534\) −13.0783 28.3878i −0.565954 1.22846i
\(535\) 0 0
\(536\) 40.5836i 1.75294i
\(537\) −8.94949 + 4.12305i −0.386199 + 0.177923i
\(538\) 0 0
\(539\) 46.2726i 1.99310i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 13.3939 0.574258
\(545\) 0 0
\(546\) 0 0
\(547\) 15.6515 0.669211 0.334606 0.942358i \(-0.391397\pi\)
0.334606 + 0.942358i \(0.391397\pi\)
\(548\) 33.0197i 1.41053i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −36.6969 −1.55630
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 11.3434 + 24.6219i 0.478917 + 1.03954i
\(562\) −40.0000 −1.68730
\(563\) 36.7696i 1.54965i −0.632175 0.774826i \(-0.717837\pi\)
0.632175 0.774826i \(-0.282163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 46.7333i 1.96435i
\(567\) 0 0
\(568\) 0 0
\(569\) 25.0594i 1.05054i 0.850935 + 0.525271i \(0.176036\pi\)
−0.850935 + 0.525271i \(0.823964\pi\)
\(570\) 0 0
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 15.5959 18.2419i 0.649830 0.760080i
\(577\) −12.3939 −0.515964 −0.257982 0.966150i \(-0.583058\pi\)
−0.257982 + 0.966150i \(0.583058\pi\)
\(578\) 16.1134i 0.670228i
\(579\) 2.67934 + 5.81577i 0.111349 + 0.241695i
\(580\) 0 0
\(581\) 0 0
\(582\) 22.2474 10.2494i 0.922186 0.424853i
\(583\) 0 0
\(584\) 38.7408i 1.60311i
\(585\) 0 0
\(586\) 0 0
\(587\) 18.8455i 0.777836i −0.921272 0.388918i \(-0.872849\pi\)
0.921272 0.388918i \(-0.127151\pi\)
\(588\) 10.1464 + 22.0239i 0.418432 + 0.908248i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 7.03896i 0.289055i 0.989501 + 0.144528i \(0.0461663\pi\)
−0.989501 + 0.144528i \(0.953834\pi\)
\(594\) 46.7423 + 13.2207i 1.91786 + 0.542453i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 8.30306 0.338689 0.169344 0.985557i \(-0.445835\pi\)
0.169344 + 0.985557i \(0.445835\pi\)
\(602\) 0 0
\(603\) 27.9722 32.7179i 1.13912 1.33238i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 47.2261i 1.91527i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 10.7980 + 9.23171i 0.436482 + 0.373170i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 34.4339i 1.38964i
\(615\) 0 0
\(616\) 0 0
\(617\) 39.5980i 1.59415i 0.603877 + 0.797077i \(0.293622\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 14.1421i 0.565233i
\(627\) −86.8156 + 39.9961i −3.46708 + 1.59729i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −10.9041 23.6684i −0.433398 0.940735i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 28.2843i 1.11716i −0.829450 0.558581i \(-0.811346\pi\)
0.829450 0.558581i \(-0.188654\pi\)
\(642\) −4.82066 10.4637i −0.190256 0.412971i
\(643\) 50.0000 1.97181 0.985904 0.167313i \(-0.0535092\pi\)
0.985904 + 0.167313i \(0.0535092\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −27.9546 −1.09986
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 25.1464 3.95691i 0.987845 0.155442i
\(649\) −93.4847 −3.66960
\(650\) 0 0
\(651\) 0 0
\(652\) 46.0908 1.80506
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.84281i 0.0719495i
\(657\) −26.7020 + 31.2323i −1.04175 + 1.21849i
\(658\) 0 0
\(659\) 8.45317i 0.329289i −0.986353 0.164644i \(-0.947352\pi\)
0.986353 0.164644i \(-0.0526477\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 49.5617i 1.92627i
\(663\) 0 0
\(664\) 40.0908 1.55583
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 31.6697i 1.21987i
\(675\) 0 0
\(676\) −26.0000 −1.00000
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) −21.7753 47.2654i −0.836274 1.81522i
\(679\) 0 0
\(680\) 0 0
\(681\) −4.44949 + 2.04989i −0.170505 + 0.0785519i
\(682\) 0 0
\(683\) 51.9294i 1.98702i −0.113728 0.993512i \(-0.536279\pi\)
0.113728 0.993512i \(-0.463721\pi\)
\(684\) −32.5505 + 38.0730i −1.24460 + 1.45576i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −40.0000 −1.52499
\(689\) 0 0
\(690\) 0 0
\(691\) 45.0454 1.71361 0.856804 0.515642i \(-0.172447\pi\)
0.856804 + 0.515642i \(0.172447\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 33.3485 1.26589
\(695\) 0 0
\(696\) 0 0
\(697\) −1.09082 −0.0413176
\(698\) 0 0
\(699\) 8.89898 4.09978i 0.336590 0.155068i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 52.8829i 1.99310i
\(705\) 0 0
\(706\) 32.0000 1.20434
\(707\) 0 0
\(708\) −44.4949 + 20.4989i −1.67222 + 0.770395i
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −36.0908 −1.35256
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 11.3779i 0.425213i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 71.6963i 2.66826i
\(723\) −1.22985 2.66951i −0.0457385 0.0992801i
\(724\) 0 0
\(725\) 0 0
\(726\) 72.7423 33.5125i 2.69972 1.24377i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 23.0000 + 14.1421i 0.851852 + 0.523783i
\(730\) 0 0
\(731\) 23.6773i 0.875735i
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 94.8486i 3.49379i
\(738\) −1.27015 + 1.48565i −0.0467550 + 0.0546874i
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 32.3207 + 27.6325i 1.18255 + 1.01102i
\(748\) 31.3031 1.14455
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −32.6464 + 15.0403i −1.18970 + 0.548098i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 16.4778i 0.598499i
\(759\) 0 0
\(760\) 0 0
\(761\) 52.4222i 1.90030i 0.311787 + 0.950152i \(0.399073\pi\)
−0.311787 + 0.950152i \(0.600927\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −11.5959 25.1701i −0.418432 0.908248i
\(769\) 55.0908 1.98663 0.993313 0.115454i \(-0.0368323\pi\)
0.993313 + 0.115454i \(0.0368323\pi\)
\(770\) 0 0
\(771\) −17.7980 + 8.19955i −0.640978 + 0.295300i
\(772\) 7.39388 0.266111
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) −32.2474 27.5699i −1.15911 0.990981i
\(775\) 0 0
\(776\) 28.2843i 1.01535i
\(777\) 0 0
\(778\) 0 0
\(779\) 3.84616i 0.137803i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 28.0000 1.00000
\(785\) 0 0
\(786\) −14.4949 31.4626i −0.517016 1.12224i
\(787\) 50.0000 1.78231 0.891154 0.453701i \(-0.149897\pi\)
0.891154 + 0.453701i \(0.149897\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 36.4495 42.6335i 1.29518 1.51492i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −29.0959 24.8755i −1.02805 0.878934i
\(802\) −20.6515 −0.729231
\(803\) 90.5418i 3.19515i
\(804\) −20.7980 45.1441i −0.733487 1.59211i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 56.5685i 1.98884i 0.105474 + 0.994422i \(0.466364\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 14.8990 6.86399i 0.521569 0.240288i
\(817\) 83.4847 2.92076
\(818\) 26.0129i 0.909519i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 16.9217 + 36.7302i 0.590211 + 1.28111i
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 56.6649i 1.97043i 0.171321 + 0.985215i \(0.445196\pi\)
−0.171321 + 0.985215i \(0.554804\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 16.5741i 0.574258i
\(834\) −40.8207 + 18.8062i −1.41350 + 0.651204i
\(835\) 0 0
\(836\) 110.373i 3.81733i
\(837\) 0 0
\(838\) 48.0454 1.65970
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −44.4949 + 20.4989i −1.53249 + 0.706019i
\(844\) −30.0908 −1.03577
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −23.9495 51.9848i −0.821944 1.78411i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −13.3031 −0.454689
\(857\) 35.4517i 1.21101i 0.795843 + 0.605503i \(0.207028\pi\)
−0.795843 + 0.605503i \(0.792972\pi\)
\(858\) 0 0
\(859\) 21.6515 0.738741 0.369370 0.929282i \(-0.379573\pi\)
0.369370 + 0.929282i \(0.379573\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 8.00000 28.2843i 0.272166 0.962250i
\(865\) 0 0
\(866\) 47.6547i 1.61937i
\(867\) −8.25765 17.9241i −0.280445 0.608733i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 19.4949 22.8024i 0.659802 0.771744i
\(874\) 0 0
\(875\) 0 0
\(876\) 19.8536 + 43.0942i 0.670790 + 1.45602i
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 56.5685i 1.90584i 0.303218 + 0.952921i \(0.401939\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 22.5732 + 19.2990i 0.760080 + 0.649830i
\(883\) 50.4393 1.69742 0.848709 0.528861i \(-0.177381\pi\)
0.848709 + 0.528861i \(0.177381\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 53.4393 1.79533
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 58.7702 9.24777i 1.96887 0.309812i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −55.4393 −1.85003
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 4.30686i 0.143403i
\(903\) 0 0
\(904\) −60.0908 −1.99859
\(905\) 0 0
\(906\) 0 0
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) 5.65685i 0.187729i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 24.2020 + 52.5330i 0.801410 + 1.73954i
\(913\) 93.6969 3.10092
\(914\) 59.9540i 1.98310i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 16.7423 + 4.73545i 0.552580 + 0.156293i
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −17.6464 38.3034i −0.581470 1.26214i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 28.2843i 0.927977i −0.885841 0.463988i \(-0.846418\pi\)
0.885841 0.463988i \(-0.153582\pi\)
\(930\) 0 0
\(931\) −58.4393 −1.91527
\(932\) 11.3137i 0.370593i
\(933\) 0 0
\(934\) 44.0000 1.43972
\(935\) 0 0
\(936\) 0 0
\(937\) 61.0908 1.99575 0.997875 0.0651578i \(-0.0207551\pi\)
0.997875 + 0.0651578i \(0.0207551\pi\)
\(938\) 0 0
\(939\) 7.24745 + 15.7313i 0.236512 + 0.513372i
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 56.5685i 1.84115i
\(945\) 0 0
\(946\) −93.4847 −3.03945
\(947\) 53.7401i 1.74632i −0.487435 0.873160i \(-0.662067\pi\)
0.487435 0.873160i \(-0.337933\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 59.0005i 1.91121i 0.294646 + 0.955607i \(0.404798\pi\)
−0.294646 + 0.955607i \(0.595202\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) −10.7247 9.16912i −0.345600 0.295471i
\(964\) −3.39388 −0.109309
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 92.4809i 2.97245i
\(969\) −31.0959 + 14.3259i −0.998945 + 0.460216i
\(970\) 0 0
\(971\) 31.2090i 1.00155i 0.865579 + 0.500773i \(0.166951\pi\)
−0.865579 + 0.500773i \(0.833049\pi\)
\(972\) 25.9444 17.2884i 0.832167 0.554526i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 25.9165i 0.829144i 0.910017 + 0.414572i \(0.136069\pi\)
−0.910017 + 0.414572i \(0.863931\pi\)
\(978\) 51.2702 23.6203i 1.63944 0.755292i
\(979\) −84.3485 −2.69579
\(980\) 0 0
\(981\) 0 0
\(982\) 20.0000 0.638226
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0.944387 + 2.04989i 0.0301060 + 0.0653480i
\(985\) 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 \(\pi\)
\(992\) 0 0
\(993\) 25.3990 + 55.1311i 0.806012 + 1.74953i
\(994\) 0 0
\(995\) 0 0
\(996\) 44.5959 20.5454i 1.41308 0.651007i
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 19.7990i 0.626726i
\(999\) 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 600.2.b.d.251.1 yes 4
3.2 odd 2 inner 600.2.b.d.251.3 yes 4
4.3 odd 2 2400.2.b.b.2351.4 4
5.2 odd 4 600.2.m.b.299.6 8
5.3 odd 4 600.2.m.b.299.3 8
5.4 even 2 600.2.b.b.251.4 yes 4
8.3 odd 2 CM 600.2.b.d.251.1 yes 4
8.5 even 2 2400.2.b.b.2351.4 4
12.11 even 2 2400.2.b.b.2351.3 4
15.2 even 4 600.2.m.b.299.4 8
15.8 even 4 600.2.m.b.299.5 8
15.14 odd 2 600.2.b.b.251.2 4
20.3 even 4 2400.2.m.b.1199.2 8
20.7 even 4 2400.2.m.b.1199.7 8
20.19 odd 2 2400.2.b.d.2351.1 4
24.5 odd 2 2400.2.b.b.2351.3 4
24.11 even 2 inner 600.2.b.d.251.3 yes 4
40.3 even 4 600.2.m.b.299.3 8
40.13 odd 4 2400.2.m.b.1199.2 8
40.19 odd 2 600.2.b.b.251.4 yes 4
40.27 even 4 600.2.m.b.299.6 8
40.29 even 2 2400.2.b.d.2351.1 4
40.37 odd 4 2400.2.m.b.1199.7 8
60.23 odd 4 2400.2.m.b.1199.8 8
60.47 odd 4 2400.2.m.b.1199.1 8
60.59 even 2 2400.2.b.d.2351.2 4
120.29 odd 2 2400.2.b.d.2351.2 4
120.53 even 4 2400.2.m.b.1199.8 8
120.59 even 2 600.2.b.b.251.2 4
120.77 even 4 2400.2.m.b.1199.1 8
120.83 odd 4 600.2.m.b.299.5 8
120.107 odd 4 600.2.m.b.299.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.2.b.b.251.2 4 15.14 odd 2
600.2.b.b.251.2 4 120.59 even 2
600.2.b.b.251.4 yes 4 5.4 even 2
600.2.b.b.251.4 yes 4 40.19 odd 2
600.2.b.d.251.1 yes 4 1.1 even 1 trivial
600.2.b.d.251.1 yes 4 8.3 odd 2 CM
600.2.b.d.251.3 yes 4 3.2 odd 2 inner
600.2.b.d.251.3 yes 4 24.11 even 2 inner
600.2.m.b.299.3 8 5.3 odd 4
600.2.m.b.299.3 8 40.3 even 4
600.2.m.b.299.4 8 15.2 even 4
600.2.m.b.299.4 8 120.107 odd 4
600.2.m.b.299.5 8 15.8 even 4
600.2.m.b.299.5 8 120.83 odd 4
600.2.m.b.299.6 8 5.2 odd 4
600.2.m.b.299.6 8 40.27 even 4
2400.2.b.b.2351.3 4 12.11 even 2
2400.2.b.b.2351.3 4 24.5 odd 2
2400.2.b.b.2351.4 4 4.3 odd 2
2400.2.b.b.2351.4 4 8.5 even 2
2400.2.b.d.2351.1 4 20.19 odd 2
2400.2.b.d.2351.1 4 40.29 even 2
2400.2.b.d.2351.2 4 60.59 even 2
2400.2.b.d.2351.2 4 120.29 odd 2
2400.2.m.b.1199.1 8 60.47 odd 4
2400.2.m.b.1199.1 8 120.77 even 4
2400.2.m.b.1199.2 8 20.3 even 4
2400.2.m.b.1199.2 8 40.13 odd 4
2400.2.m.b.1199.7 8 20.7 even 4
2400.2.m.b.1199.7 8 40.37 odd 4
2400.2.m.b.1199.8 8 60.23 odd 4
2400.2.m.b.1199.8 8 120.53 even 4