Properties

Label 9680.2.a.dc.1.6
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $6$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.25903625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(3.29288\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.29288 q^{3} -1.00000 q^{5} -3.51508 q^{7} +7.84307 q^{9} +O(q^{10})\) \(q+3.29288 q^{3} -1.00000 q^{5} -3.51508 q^{7} +7.84307 q^{9} -4.27118 q^{13} -3.29288 q^{15} +1.58998 q^{17} -5.55094 q^{19} -11.5747 q^{21} +6.39042 q^{23} +1.00000 q^{25} +15.9477 q^{27} -0.381220 q^{29} +0.760465 q^{31} +3.51508 q^{35} -1.45191 q^{37} -14.0645 q^{39} -6.12208 q^{41} +7.19067 q^{43} -7.84307 q^{45} -4.32164 q^{47} +5.35577 q^{49} +5.23561 q^{51} -6.94904 q^{53} -18.2786 q^{57} -12.7110 q^{59} -8.89599 q^{61} -27.5690 q^{63} +4.27118 q^{65} -12.5135 q^{67} +21.0429 q^{69} -7.35890 q^{71} +7.03210 q^{73} +3.29288 q^{75} +6.99381 q^{79} +28.9846 q^{81} +10.7509 q^{83} -1.58998 q^{85} -1.25531 q^{87} +0.451594 q^{89} +15.0135 q^{91} +2.50412 q^{93} +5.55094 q^{95} -2.77956 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} - q^{13} - 3 q^{15} - 6 q^{17} - 7 q^{19} - 14 q^{21} + 9 q^{23} + 6 q^{25} + 21 q^{27} - 10 q^{29} - q^{31} + 7 q^{35} + 3 q^{37} - q^{39} + 6 q^{41} + 18 q^{43} - 5 q^{45} + 3 q^{47} + 17 q^{49} + 15 q^{51} - 23 q^{53} - 9 q^{57} + 2 q^{59} - 6 q^{61} - 49 q^{63} + q^{65} + 22 q^{67} + 2 q^{69} + 13 q^{71} - 10 q^{73} + 3 q^{75} - 22 q^{79} + 10 q^{81} + 10 q^{83} + 6 q^{85} - 3 q^{87} - 25 q^{89} - 12 q^{91} + 19 q^{93} + 7 q^{95} - 33 q^{97}+O(q^{100}) \) 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
\(3\) 3.29288 1.90115 0.950573 0.310501i \(-0.100497\pi\)
0.950573 + 0.310501i \(0.100497\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.51508 −1.32857 −0.664287 0.747477i \(-0.731265\pi\)
−0.664287 + 0.747477i \(0.731265\pi\)
\(8\) 0 0
\(9\) 7.84307 2.61436
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −4.27118 −1.18461 −0.592306 0.805713i \(-0.701782\pi\)
−0.592306 + 0.805713i \(0.701782\pi\)
\(14\) 0 0
\(15\) −3.29288 −0.850218
\(16\) 0 0
\(17\) 1.58998 0.385626 0.192813 0.981236i \(-0.438239\pi\)
0.192813 + 0.981236i \(0.438239\pi\)
\(18\) 0 0
\(19\) −5.55094 −1.27347 −0.636736 0.771082i \(-0.719716\pi\)
−0.636736 + 0.771082i \(0.719716\pi\)
\(20\) 0 0
\(21\) −11.5747 −2.52581
\(22\) 0 0
\(23\) 6.39042 1.33249 0.666247 0.745731i \(-0.267900\pi\)
0.666247 + 0.745731i \(0.267900\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 15.9477 3.06913
\(28\) 0 0
\(29\) −0.381220 −0.0707908 −0.0353954 0.999373i \(-0.511269\pi\)
−0.0353954 + 0.999373i \(0.511269\pi\)
\(30\) 0 0
\(31\) 0.760465 0.136584 0.0682918 0.997665i \(-0.478245\pi\)
0.0682918 + 0.997665i \(0.478245\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.51508 0.594156
\(36\) 0 0
\(37\) −1.45191 −0.238692 −0.119346 0.992853i \(-0.538080\pi\)
−0.119346 + 0.992853i \(0.538080\pi\)
\(38\) 0 0
\(39\) −14.0645 −2.25212
\(40\) 0 0
\(41\) −6.12208 −0.956108 −0.478054 0.878330i \(-0.658658\pi\)
−0.478054 + 0.878330i \(0.658658\pi\)
\(42\) 0 0
\(43\) 7.19067 1.09657 0.548284 0.836293i \(-0.315281\pi\)
0.548284 + 0.836293i \(0.315281\pi\)
\(44\) 0 0
\(45\) −7.84307 −1.16918
\(46\) 0 0
\(47\) −4.32164 −0.630376 −0.315188 0.949029i \(-0.602068\pi\)
−0.315188 + 0.949029i \(0.602068\pi\)
\(48\) 0 0
\(49\) 5.35577 0.765110
\(50\) 0 0
\(51\) 5.23561 0.733132
\(52\) 0 0
\(53\) −6.94904 −0.954523 −0.477262 0.878761i \(-0.658371\pi\)
−0.477262 + 0.878761i \(0.658371\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −18.2786 −2.42106
\(58\) 0 0
\(59\) −12.7110 −1.65484 −0.827419 0.561586i \(-0.810192\pi\)
−0.827419 + 0.561586i \(0.810192\pi\)
\(60\) 0 0
\(61\) −8.89599 −1.13901 −0.569507 0.821986i \(-0.692866\pi\)
−0.569507 + 0.821986i \(0.692866\pi\)
\(62\) 0 0
\(63\) −27.5690 −3.47337
\(64\) 0 0
\(65\) 4.27118 0.529775
\(66\) 0 0
\(67\) −12.5135 −1.52877 −0.764383 0.644763i \(-0.776956\pi\)
−0.764383 + 0.644763i \(0.776956\pi\)
\(68\) 0 0
\(69\) 21.0429 2.53327
\(70\) 0 0
\(71\) −7.35890 −0.873340 −0.436670 0.899622i \(-0.643842\pi\)
−0.436670 + 0.899622i \(0.643842\pi\)
\(72\) 0 0
\(73\) 7.03210 0.823045 0.411523 0.911399i \(-0.364997\pi\)
0.411523 + 0.911399i \(0.364997\pi\)
\(74\) 0 0
\(75\) 3.29288 0.380229
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.99381 0.786865 0.393432 0.919354i \(-0.371288\pi\)
0.393432 + 0.919354i \(0.371288\pi\)
\(80\) 0 0
\(81\) 28.9846 3.22051
\(82\) 0 0
\(83\) 10.7509 1.18007 0.590034 0.807379i \(-0.299114\pi\)
0.590034 + 0.807379i \(0.299114\pi\)
\(84\) 0 0
\(85\) −1.58998 −0.172457
\(86\) 0 0
\(87\) −1.25531 −0.134584
\(88\) 0 0
\(89\) 0.451594 0.0478689 0.0239344 0.999714i \(-0.492381\pi\)
0.0239344 + 0.999714i \(0.492381\pi\)
\(90\) 0 0
\(91\) 15.0135 1.57385
\(92\) 0 0
\(93\) 2.50412 0.259665
\(94\) 0 0
\(95\) 5.55094 0.569514
\(96\) 0 0
\(97\) −2.77956 −0.282222 −0.141111 0.989994i \(-0.545067\pi\)
−0.141111 + 0.989994i \(0.545067\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.33731 0.531082 0.265541 0.964100i \(-0.414449\pi\)
0.265541 + 0.964100i \(0.414449\pi\)
\(102\) 0 0
\(103\) −17.3817 −1.71267 −0.856334 0.516422i \(-0.827263\pi\)
−0.856334 + 0.516422i \(0.827263\pi\)
\(104\) 0 0
\(105\) 11.5747 1.12958
\(106\) 0 0
\(107\) −3.49783 −0.338148 −0.169074 0.985603i \(-0.554078\pi\)
−0.169074 + 0.985603i \(0.554078\pi\)
\(108\) 0 0
\(109\) 4.91209 0.470493 0.235247 0.971936i \(-0.424410\pi\)
0.235247 + 0.971936i \(0.424410\pi\)
\(110\) 0 0
\(111\) −4.78096 −0.453789
\(112\) 0 0
\(113\) −17.1718 −1.61539 −0.807694 0.589602i \(-0.799285\pi\)
−0.807694 + 0.589602i \(0.799285\pi\)
\(114\) 0 0
\(115\) −6.39042 −0.595910
\(116\) 0 0
\(117\) −33.4992 −3.09700
\(118\) 0 0
\(119\) −5.58889 −0.512333
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −20.1593 −1.81770
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −15.4883 −1.37437 −0.687183 0.726484i \(-0.741153\pi\)
−0.687183 + 0.726484i \(0.741153\pi\)
\(128\) 0 0
\(129\) 23.6780 2.08473
\(130\) 0 0
\(131\) 3.64734 0.318670 0.159335 0.987225i \(-0.449065\pi\)
0.159335 + 0.987225i \(0.449065\pi\)
\(132\) 0 0
\(133\) 19.5120 1.69190
\(134\) 0 0
\(135\) −15.9477 −1.37256
\(136\) 0 0
\(137\) −16.2579 −1.38901 −0.694503 0.719490i \(-0.744376\pi\)
−0.694503 + 0.719490i \(0.744376\pi\)
\(138\) 0 0
\(139\) 7.49232 0.635490 0.317745 0.948176i \(-0.397074\pi\)
0.317745 + 0.948176i \(0.397074\pi\)
\(140\) 0 0
\(141\) −14.2306 −1.19844
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0.381220 0.0316586
\(146\) 0 0
\(147\) 17.6359 1.45459
\(148\) 0 0
\(149\) −11.7025 −0.958708 −0.479354 0.877622i \(-0.659129\pi\)
−0.479354 + 0.877622i \(0.659129\pi\)
\(150\) 0 0
\(151\) −12.6007 −1.02543 −0.512715 0.858559i \(-0.671360\pi\)
−0.512715 + 0.858559i \(0.671360\pi\)
\(152\) 0 0
\(153\) 12.4703 1.00816
\(154\) 0 0
\(155\) −0.760465 −0.0610820
\(156\) 0 0
\(157\) −6.47653 −0.516883 −0.258441 0.966027i \(-0.583209\pi\)
−0.258441 + 0.966027i \(0.583209\pi\)
\(158\) 0 0
\(159\) −22.8824 −1.81469
\(160\) 0 0
\(161\) −22.4628 −1.77032
\(162\) 0 0
\(163\) 21.7737 1.70545 0.852724 0.522361i \(-0.174949\pi\)
0.852724 + 0.522361i \(0.174949\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19.0881 −1.47708 −0.738541 0.674208i \(-0.764485\pi\)
−0.738541 + 0.674208i \(0.764485\pi\)
\(168\) 0 0
\(169\) 5.24299 0.403307
\(170\) 0 0
\(171\) −43.5364 −3.32931
\(172\) 0 0
\(173\) −1.44630 −0.109960 −0.0549800 0.998487i \(-0.517509\pi\)
−0.0549800 + 0.998487i \(0.517509\pi\)
\(174\) 0 0
\(175\) −3.51508 −0.265715
\(176\) 0 0
\(177\) −41.8560 −3.14609
\(178\) 0 0
\(179\) 9.02979 0.674918 0.337459 0.941340i \(-0.390433\pi\)
0.337459 + 0.941340i \(0.390433\pi\)
\(180\) 0 0
\(181\) −1.14725 −0.0852744 −0.0426372 0.999091i \(-0.513576\pi\)
−0.0426372 + 0.999091i \(0.513576\pi\)
\(182\) 0 0
\(183\) −29.2934 −2.16543
\(184\) 0 0
\(185\) 1.45191 0.106746
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −56.0573 −4.07757
\(190\) 0 0
\(191\) 14.9868 1.08440 0.542202 0.840248i \(-0.317591\pi\)
0.542202 + 0.840248i \(0.317591\pi\)
\(192\) 0 0
\(193\) −15.1821 −1.09283 −0.546414 0.837515i \(-0.684007\pi\)
−0.546414 + 0.837515i \(0.684007\pi\)
\(194\) 0 0
\(195\) 14.0645 1.00718
\(196\) 0 0
\(197\) −2.95776 −0.210731 −0.105366 0.994434i \(-0.533601\pi\)
−0.105366 + 0.994434i \(0.533601\pi\)
\(198\) 0 0
\(199\) 5.07974 0.360093 0.180046 0.983658i \(-0.442375\pi\)
0.180046 + 0.983658i \(0.442375\pi\)
\(200\) 0 0
\(201\) −41.2054 −2.90641
\(202\) 0 0
\(203\) 1.34002 0.0940509
\(204\) 0 0
\(205\) 6.12208 0.427585
\(206\) 0 0
\(207\) 50.1205 3.48362
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 8.50814 0.585724 0.292862 0.956155i \(-0.405392\pi\)
0.292862 + 0.956155i \(0.405392\pi\)
\(212\) 0 0
\(213\) −24.2320 −1.66035
\(214\) 0 0
\(215\) −7.19067 −0.490400
\(216\) 0 0
\(217\) −2.67309 −0.181461
\(218\) 0 0
\(219\) 23.1559 1.56473
\(220\) 0 0
\(221\) −6.79108 −0.456817
\(222\) 0 0
\(223\) −15.0443 −1.00744 −0.503719 0.863867i \(-0.668035\pi\)
−0.503719 + 0.863867i \(0.668035\pi\)
\(224\) 0 0
\(225\) 7.84307 0.522871
\(226\) 0 0
\(227\) 15.7035 1.04228 0.521138 0.853473i \(-0.325508\pi\)
0.521138 + 0.853473i \(0.325508\pi\)
\(228\) 0 0
\(229\) 21.4760 1.41918 0.709588 0.704617i \(-0.248881\pi\)
0.709588 + 0.704617i \(0.248881\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.07615 −0.463574 −0.231787 0.972767i \(-0.574457\pi\)
−0.231787 + 0.972767i \(0.574457\pi\)
\(234\) 0 0
\(235\) 4.32164 0.281913
\(236\) 0 0
\(237\) 23.0298 1.49595
\(238\) 0 0
\(239\) −12.9846 −0.839905 −0.419952 0.907546i \(-0.637953\pi\)
−0.419952 + 0.907546i \(0.637953\pi\)
\(240\) 0 0
\(241\) 4.91442 0.316565 0.158283 0.987394i \(-0.449404\pi\)
0.158283 + 0.987394i \(0.449404\pi\)
\(242\) 0 0
\(243\) 47.5998 3.05353
\(244\) 0 0
\(245\) −5.35577 −0.342167
\(246\) 0 0
\(247\) 23.7091 1.50857
\(248\) 0 0
\(249\) 35.4015 2.24348
\(250\) 0 0
\(251\) −4.68561 −0.295753 −0.147877 0.989006i \(-0.547244\pi\)
−0.147877 + 0.989006i \(0.547244\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −5.23561 −0.327866
\(256\) 0 0
\(257\) 1.92457 0.120051 0.0600255 0.998197i \(-0.480882\pi\)
0.0600255 + 0.998197i \(0.480882\pi\)
\(258\) 0 0
\(259\) 5.10357 0.317120
\(260\) 0 0
\(261\) −2.98994 −0.185072
\(262\) 0 0
\(263\) 16.6865 1.02894 0.514468 0.857510i \(-0.327990\pi\)
0.514468 + 0.857510i \(0.327990\pi\)
\(264\) 0 0
\(265\) 6.94904 0.426876
\(266\) 0 0
\(267\) 1.48705 0.0910058
\(268\) 0 0
\(269\) −4.58774 −0.279719 −0.139860 0.990171i \(-0.544665\pi\)
−0.139860 + 0.990171i \(0.544665\pi\)
\(270\) 0 0
\(271\) −20.2427 −1.22966 −0.614829 0.788660i \(-0.710775\pi\)
−0.614829 + 0.788660i \(0.710775\pi\)
\(272\) 0 0
\(273\) 49.4378 2.99211
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4.79906 0.288348 0.144174 0.989552i \(-0.453948\pi\)
0.144174 + 0.989552i \(0.453948\pi\)
\(278\) 0 0
\(279\) 5.96438 0.357078
\(280\) 0 0
\(281\) −21.5980 −1.28843 −0.644214 0.764845i \(-0.722815\pi\)
−0.644214 + 0.764845i \(0.722815\pi\)
\(282\) 0 0
\(283\) 12.2540 0.728422 0.364211 0.931316i \(-0.381339\pi\)
0.364211 + 0.931316i \(0.381339\pi\)
\(284\) 0 0
\(285\) 18.2786 1.08273
\(286\) 0 0
\(287\) 21.5196 1.27026
\(288\) 0 0
\(289\) −14.4720 −0.851293
\(290\) 0 0
\(291\) −9.15277 −0.536545
\(292\) 0 0
\(293\) 14.3193 0.836544 0.418272 0.908322i \(-0.362636\pi\)
0.418272 + 0.908322i \(0.362636\pi\)
\(294\) 0 0
\(295\) 12.7110 0.740066
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −27.2946 −1.57849
\(300\) 0 0
\(301\) −25.2758 −1.45687
\(302\) 0 0
\(303\) 17.5751 1.00967
\(304\) 0 0
\(305\) 8.89599 0.509383
\(306\) 0 0
\(307\) −0.568487 −0.0324453 −0.0162226 0.999868i \(-0.505164\pi\)
−0.0162226 + 0.999868i \(0.505164\pi\)
\(308\) 0 0
\(309\) −57.2358 −3.25603
\(310\) 0 0
\(311\) −10.2047 −0.578654 −0.289327 0.957230i \(-0.593432\pi\)
−0.289327 + 0.957230i \(0.593432\pi\)
\(312\) 0 0
\(313\) −9.33625 −0.527716 −0.263858 0.964561i \(-0.584995\pi\)
−0.263858 + 0.964561i \(0.584995\pi\)
\(314\) 0 0
\(315\) 27.5690 1.55334
\(316\) 0 0
\(317\) −28.2884 −1.58884 −0.794418 0.607372i \(-0.792224\pi\)
−0.794418 + 0.607372i \(0.792224\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −11.5179 −0.642869
\(322\) 0 0
\(323\) −8.82586 −0.491084
\(324\) 0 0
\(325\) −4.27118 −0.236922
\(326\) 0 0
\(327\) 16.1749 0.894476
\(328\) 0 0
\(329\) 15.1909 0.837501
\(330\) 0 0
\(331\) 5.31846 0.292329 0.146164 0.989260i \(-0.453307\pi\)
0.146164 + 0.989260i \(0.453307\pi\)
\(332\) 0 0
\(333\) −11.3874 −0.624026
\(334\) 0 0
\(335\) 12.5135 0.683685
\(336\) 0 0
\(337\) −17.6340 −0.960584 −0.480292 0.877109i \(-0.659469\pi\)
−0.480292 + 0.877109i \(0.659469\pi\)
\(338\) 0 0
\(339\) −56.5448 −3.07109
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 5.77961 0.312069
\(344\) 0 0
\(345\) −21.0429 −1.13291
\(346\) 0 0
\(347\) 11.8289 0.635006 0.317503 0.948257i \(-0.397156\pi\)
0.317503 + 0.948257i \(0.397156\pi\)
\(348\) 0 0
\(349\) −9.35132 −0.500565 −0.250282 0.968173i \(-0.580523\pi\)
−0.250282 + 0.968173i \(0.580523\pi\)
\(350\) 0 0
\(351\) −68.1154 −3.63573
\(352\) 0 0
\(353\) −17.7336 −0.943863 −0.471932 0.881635i \(-0.656443\pi\)
−0.471932 + 0.881635i \(0.656443\pi\)
\(354\) 0 0
\(355\) 7.35890 0.390570
\(356\) 0 0
\(357\) −18.4036 −0.974020
\(358\) 0 0
\(359\) 18.8840 0.996658 0.498329 0.866988i \(-0.333947\pi\)
0.498329 + 0.866988i \(0.333947\pi\)
\(360\) 0 0
\(361\) 11.8129 0.621731
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −7.03210 −0.368077
\(366\) 0 0
\(367\) 26.3936 1.37773 0.688867 0.724887i \(-0.258108\pi\)
0.688867 + 0.724887i \(0.258108\pi\)
\(368\) 0 0
\(369\) −48.0159 −2.49961
\(370\) 0 0
\(371\) 24.4264 1.26816
\(372\) 0 0
\(373\) −19.2368 −0.996043 −0.498022 0.867165i \(-0.665940\pi\)
−0.498022 + 0.867165i \(0.665940\pi\)
\(374\) 0 0
\(375\) −3.29288 −0.170044
\(376\) 0 0
\(377\) 1.62826 0.0838597
\(378\) 0 0
\(379\) 34.4665 1.77042 0.885212 0.465189i \(-0.154014\pi\)
0.885212 + 0.465189i \(0.154014\pi\)
\(380\) 0 0
\(381\) −51.0012 −2.61287
\(382\) 0 0
\(383\) 22.3171 1.14035 0.570176 0.821523i \(-0.306875\pi\)
0.570176 + 0.821523i \(0.306875\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 56.3970 2.86682
\(388\) 0 0
\(389\) −16.1067 −0.816644 −0.408322 0.912838i \(-0.633886\pi\)
−0.408322 + 0.912838i \(0.633886\pi\)
\(390\) 0 0
\(391\) 10.1606 0.513845
\(392\) 0 0
\(393\) 12.0103 0.605838
\(394\) 0 0
\(395\) −6.99381 −0.351897
\(396\) 0 0
\(397\) 16.8311 0.844731 0.422365 0.906426i \(-0.361200\pi\)
0.422365 + 0.906426i \(0.361200\pi\)
\(398\) 0 0
\(399\) 64.2506 3.21655
\(400\) 0 0
\(401\) 5.26721 0.263032 0.131516 0.991314i \(-0.458016\pi\)
0.131516 + 0.991314i \(0.458016\pi\)
\(402\) 0 0
\(403\) −3.24808 −0.161799
\(404\) 0 0
\(405\) −28.9846 −1.44025
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 23.3376 1.15397 0.576985 0.816755i \(-0.304229\pi\)
0.576985 + 0.816755i \(0.304229\pi\)
\(410\) 0 0
\(411\) −53.5353 −2.64070
\(412\) 0 0
\(413\) 44.6803 2.19857
\(414\) 0 0
\(415\) −10.7509 −0.527742
\(416\) 0 0
\(417\) 24.6713 1.20816
\(418\) 0 0
\(419\) −9.21755 −0.450307 −0.225153 0.974323i \(-0.572288\pi\)
−0.225153 + 0.974323i \(0.572288\pi\)
\(420\) 0 0
\(421\) 20.9901 1.02299 0.511496 0.859285i \(-0.329091\pi\)
0.511496 + 0.859285i \(0.329091\pi\)
\(422\) 0 0
\(423\) −33.8949 −1.64803
\(424\) 0 0
\(425\) 1.58998 0.0771252
\(426\) 0 0
\(427\) 31.2701 1.51327
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −36.8179 −1.77346 −0.886729 0.462290i \(-0.847028\pi\)
−0.886729 + 0.462290i \(0.847028\pi\)
\(432\) 0 0
\(433\) 11.6339 0.559087 0.279544 0.960133i \(-0.409817\pi\)
0.279544 + 0.960133i \(0.409817\pi\)
\(434\) 0 0
\(435\) 1.25531 0.0601877
\(436\) 0 0
\(437\) −35.4728 −1.69689
\(438\) 0 0
\(439\) 12.4765 0.595471 0.297735 0.954648i \(-0.403769\pi\)
0.297735 + 0.954648i \(0.403769\pi\)
\(440\) 0 0
\(441\) 42.0057 2.00027
\(442\) 0 0
\(443\) −8.22715 −0.390884 −0.195442 0.980715i \(-0.562614\pi\)
−0.195442 + 0.980715i \(0.562614\pi\)
\(444\) 0 0
\(445\) −0.451594 −0.0214076
\(446\) 0 0
\(447\) −38.5350 −1.82264
\(448\) 0 0
\(449\) −34.2395 −1.61586 −0.807930 0.589278i \(-0.799412\pi\)
−0.807930 + 0.589278i \(0.799412\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −41.4926 −1.94949
\(454\) 0 0
\(455\) −15.0135 −0.703845
\(456\) 0 0
\(457\) 24.4317 1.14287 0.571434 0.820648i \(-0.306387\pi\)
0.571434 + 0.820648i \(0.306387\pi\)
\(458\) 0 0
\(459\) 25.3564 1.18354
\(460\) 0 0
\(461\) −15.4406 −0.719141 −0.359571 0.933118i \(-0.617077\pi\)
−0.359571 + 0.933118i \(0.617077\pi\)
\(462\) 0 0
\(463\) 25.6507 1.19209 0.596046 0.802951i \(-0.296738\pi\)
0.596046 + 0.802951i \(0.296738\pi\)
\(464\) 0 0
\(465\) −2.50412 −0.116126
\(466\) 0 0
\(467\) 7.80052 0.360965 0.180482 0.983578i \(-0.442234\pi\)
0.180482 + 0.983578i \(0.442234\pi\)
\(468\) 0 0
\(469\) 43.9859 2.03108
\(470\) 0 0
\(471\) −21.3264 −0.982670
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −5.55094 −0.254694
\(476\) 0 0
\(477\) −54.5018 −2.49547
\(478\) 0 0
\(479\) 38.9353 1.77900 0.889501 0.456934i \(-0.151052\pi\)
0.889501 + 0.456934i \(0.151052\pi\)
\(480\) 0 0
\(481\) 6.20136 0.282758
\(482\) 0 0
\(483\) −73.9674 −3.36563
\(484\) 0 0
\(485\) 2.77956 0.126213
\(486\) 0 0
\(487\) 17.8252 0.807737 0.403869 0.914817i \(-0.367665\pi\)
0.403869 + 0.914817i \(0.367665\pi\)
\(488\) 0 0
\(489\) 71.6983 3.24231
\(490\) 0 0
\(491\) −4.37606 −0.197489 −0.0987443 0.995113i \(-0.531483\pi\)
−0.0987443 + 0.995113i \(0.531483\pi\)
\(492\) 0 0
\(493\) −0.606131 −0.0272988
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 25.8671 1.16030
\(498\) 0 0
\(499\) −40.6175 −1.81829 −0.909145 0.416479i \(-0.863264\pi\)
−0.909145 + 0.416479i \(0.863264\pi\)
\(500\) 0 0
\(501\) −62.8549 −2.80815
\(502\) 0 0
\(503\) −6.32076 −0.281829 −0.140914 0.990022i \(-0.545004\pi\)
−0.140914 + 0.990022i \(0.545004\pi\)
\(504\) 0 0
\(505\) −5.33731 −0.237507
\(506\) 0 0
\(507\) 17.2645 0.766745
\(508\) 0 0
\(509\) −12.9331 −0.573250 −0.286625 0.958043i \(-0.592533\pi\)
−0.286625 + 0.958043i \(0.592533\pi\)
\(510\) 0 0
\(511\) −24.7184 −1.09348
\(512\) 0 0
\(513\) −88.5245 −3.90845
\(514\) 0 0
\(515\) 17.3817 0.765928
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −4.76249 −0.209050
\(520\) 0 0
\(521\) −16.4286 −0.719748 −0.359874 0.933001i \(-0.617180\pi\)
−0.359874 + 0.933001i \(0.617180\pi\)
\(522\) 0 0
\(523\) 21.3601 0.934012 0.467006 0.884254i \(-0.345333\pi\)
0.467006 + 0.884254i \(0.345333\pi\)
\(524\) 0 0
\(525\) −11.5747 −0.505163
\(526\) 0 0
\(527\) 1.20912 0.0526702
\(528\) 0 0
\(529\) 17.8375 0.775542
\(530\) 0 0
\(531\) −99.6937 −4.32634
\(532\) 0 0
\(533\) 26.1485 1.13262
\(534\) 0 0
\(535\) 3.49783 0.151224
\(536\) 0 0
\(537\) 29.7340 1.28312
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 24.1003 1.03615 0.518076 0.855334i \(-0.326648\pi\)
0.518076 + 0.855334i \(0.326648\pi\)
\(542\) 0 0
\(543\) −3.77776 −0.162119
\(544\) 0 0
\(545\) −4.91209 −0.210411
\(546\) 0 0
\(547\) −22.1889 −0.948727 −0.474364 0.880329i \(-0.657322\pi\)
−0.474364 + 0.880329i \(0.657322\pi\)
\(548\) 0 0
\(549\) −69.7719 −2.97779
\(550\) 0 0
\(551\) 2.11613 0.0901501
\(552\) 0 0
\(553\) −24.5838 −1.04541
\(554\) 0 0
\(555\) 4.78096 0.202940
\(556\) 0 0
\(557\) 35.3751 1.49889 0.749445 0.662067i \(-0.230321\pi\)
0.749445 + 0.662067i \(0.230321\pi\)
\(558\) 0 0
\(559\) −30.7127 −1.29901
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.79389 0.286328 0.143164 0.989699i \(-0.454272\pi\)
0.143164 + 0.989699i \(0.454272\pi\)
\(564\) 0 0
\(565\) 17.1718 0.722424
\(566\) 0 0
\(567\) −101.883 −4.27868
\(568\) 0 0
\(569\) 36.5320 1.53150 0.765750 0.643139i \(-0.222368\pi\)
0.765750 + 0.643139i \(0.222368\pi\)
\(570\) 0 0
\(571\) −30.4606 −1.27474 −0.637369 0.770559i \(-0.719977\pi\)
−0.637369 + 0.770559i \(0.719977\pi\)
\(572\) 0 0
\(573\) 49.3496 2.06161
\(574\) 0 0
\(575\) 6.39042 0.266499
\(576\) 0 0
\(577\) 10.5075 0.437434 0.218717 0.975788i \(-0.429813\pi\)
0.218717 + 0.975788i \(0.429813\pi\)
\(578\) 0 0
\(579\) −49.9927 −2.07763
\(580\) 0 0
\(581\) −37.7903 −1.56781
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 33.4992 1.38502
\(586\) 0 0
\(587\) 21.9284 0.905083 0.452542 0.891743i \(-0.350517\pi\)
0.452542 + 0.891743i \(0.350517\pi\)
\(588\) 0 0
\(589\) −4.22129 −0.173935
\(590\) 0 0
\(591\) −9.73955 −0.400631
\(592\) 0 0
\(593\) −21.6482 −0.888987 −0.444493 0.895782i \(-0.646616\pi\)
−0.444493 + 0.895782i \(0.646616\pi\)
\(594\) 0 0
\(595\) 5.58889 0.229122
\(596\) 0 0
\(597\) 16.7270 0.684589
\(598\) 0 0
\(599\) 43.7105 1.78596 0.892982 0.450092i \(-0.148609\pi\)
0.892982 + 0.450092i \(0.148609\pi\)
\(600\) 0 0
\(601\) 4.09877 0.167192 0.0835962 0.996500i \(-0.473359\pi\)
0.0835962 + 0.996500i \(0.473359\pi\)
\(602\) 0 0
\(603\) −98.1442 −3.99674
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −42.1631 −1.71135 −0.855674 0.517515i \(-0.826857\pi\)
−0.855674 + 0.517515i \(0.826857\pi\)
\(608\) 0 0
\(609\) 4.41252 0.178804
\(610\) 0 0
\(611\) 18.4585 0.746751
\(612\) 0 0
\(613\) −10.6572 −0.430441 −0.215220 0.976565i \(-0.569047\pi\)
−0.215220 + 0.976565i \(0.569047\pi\)
\(614\) 0 0
\(615\) 20.1593 0.812901
\(616\) 0 0
\(617\) 6.71898 0.270496 0.135248 0.990812i \(-0.456817\pi\)
0.135248 + 0.990812i \(0.456817\pi\)
\(618\) 0 0
\(619\) 34.3204 1.37945 0.689727 0.724069i \(-0.257730\pi\)
0.689727 + 0.724069i \(0.257730\pi\)
\(620\) 0 0
\(621\) 101.912 4.08960
\(622\) 0 0
\(623\) −1.58739 −0.0635974
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.30850 −0.0920459
\(630\) 0 0
\(631\) −11.0326 −0.439201 −0.219601 0.975590i \(-0.570475\pi\)
−0.219601 + 0.975590i \(0.570475\pi\)
\(632\) 0 0
\(633\) 28.0163 1.11355
\(634\) 0 0
\(635\) 15.4883 0.614636
\(636\) 0 0
\(637\) −22.8755 −0.906358
\(638\) 0 0
\(639\) −57.7163 −2.28322
\(640\) 0 0
\(641\) −25.2836 −0.998644 −0.499322 0.866417i \(-0.666417\pi\)
−0.499322 + 0.866417i \(0.666417\pi\)
\(642\) 0 0
\(643\) 5.33932 0.210562 0.105281 0.994442i \(-0.466426\pi\)
0.105281 + 0.994442i \(0.466426\pi\)
\(644\) 0 0
\(645\) −23.6780 −0.932322
\(646\) 0 0
\(647\) −3.45155 −0.135694 −0.0678471 0.997696i \(-0.521613\pi\)
−0.0678471 + 0.997696i \(0.521613\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −8.80218 −0.344985
\(652\) 0 0
\(653\) 11.3245 0.443161 0.221580 0.975142i \(-0.428878\pi\)
0.221580 + 0.975142i \(0.428878\pi\)
\(654\) 0 0
\(655\) −3.64734 −0.142513
\(656\) 0 0
\(657\) 55.1533 2.15174
\(658\) 0 0
\(659\) −12.0511 −0.469444 −0.234722 0.972063i \(-0.575418\pi\)
−0.234722 + 0.972063i \(0.575418\pi\)
\(660\) 0 0
\(661\) −11.9301 −0.464027 −0.232013 0.972713i \(-0.574531\pi\)
−0.232013 + 0.972713i \(0.574531\pi\)
\(662\) 0 0
\(663\) −22.3622 −0.868477
\(664\) 0 0
\(665\) −19.5120 −0.756642
\(666\) 0 0
\(667\) −2.43616 −0.0943284
\(668\) 0 0
\(669\) −49.5390 −1.91529
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −33.3541 −1.28571 −0.642854 0.765989i \(-0.722250\pi\)
−0.642854 + 0.765989i \(0.722250\pi\)
\(674\) 0 0
\(675\) 15.9477 0.613826
\(676\) 0 0
\(677\) 29.9924 1.15270 0.576351 0.817202i \(-0.304476\pi\)
0.576351 + 0.817202i \(0.304476\pi\)
\(678\) 0 0
\(679\) 9.77038 0.374953
\(680\) 0 0
\(681\) 51.7096 1.98152
\(682\) 0 0
\(683\) −13.1505 −0.503190 −0.251595 0.967833i \(-0.580955\pi\)
−0.251595 + 0.967833i \(0.580955\pi\)
\(684\) 0 0
\(685\) 16.2579 0.621182
\(686\) 0 0
\(687\) 70.7180 2.69806
\(688\) 0 0
\(689\) 29.6806 1.13074
\(690\) 0 0
\(691\) 26.1763 0.995792 0.497896 0.867237i \(-0.334106\pi\)
0.497896 + 0.867237i \(0.334106\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.49232 −0.284200
\(696\) 0 0
\(697\) −9.73397 −0.368700
\(698\) 0 0
\(699\) −23.3009 −0.881322
\(700\) 0 0
\(701\) 1.41481 0.0534368 0.0267184 0.999643i \(-0.491494\pi\)
0.0267184 + 0.999643i \(0.491494\pi\)
\(702\) 0 0
\(703\) 8.05944 0.303968
\(704\) 0 0
\(705\) 14.2306 0.535957
\(706\) 0 0
\(707\) −18.7611 −0.705582
\(708\) 0 0
\(709\) 3.68174 0.138271 0.0691353 0.997607i \(-0.477976\pi\)
0.0691353 + 0.997607i \(0.477976\pi\)
\(710\) 0 0
\(711\) 54.8529 2.05715
\(712\) 0 0
\(713\) 4.85969 0.181997
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −42.7568 −1.59678
\(718\) 0 0
\(719\) 7.95387 0.296629 0.148315 0.988940i \(-0.452615\pi\)
0.148315 + 0.988940i \(0.452615\pi\)
\(720\) 0 0
\(721\) 61.0980 2.27541
\(722\) 0 0
\(723\) 16.1826 0.601837
\(724\) 0 0
\(725\) −0.381220 −0.0141582
\(726\) 0 0
\(727\) 35.6495 1.32217 0.661083 0.750313i \(-0.270097\pi\)
0.661083 + 0.750313i \(0.270097\pi\)
\(728\) 0 0
\(729\) 69.7867 2.58469
\(730\) 0 0
\(731\) 11.4330 0.422865
\(732\) 0 0
\(733\) 9.54263 0.352465 0.176233 0.984349i \(-0.443609\pi\)
0.176233 + 0.984349i \(0.443609\pi\)
\(734\) 0 0
\(735\) −17.6359 −0.650510
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −34.9555 −1.28586 −0.642929 0.765926i \(-0.722281\pi\)
−0.642929 + 0.765926i \(0.722281\pi\)
\(740\) 0 0
\(741\) 78.0711 2.86801
\(742\) 0 0
\(743\) −33.7663 −1.23876 −0.619382 0.785089i \(-0.712617\pi\)
−0.619382 + 0.785089i \(0.712617\pi\)
\(744\) 0 0
\(745\) 11.7025 0.428747
\(746\) 0 0
\(747\) 84.3203 3.08512
\(748\) 0 0
\(749\) 12.2951 0.449255
\(750\) 0 0
\(751\) 13.2772 0.484491 0.242246 0.970215i \(-0.422116\pi\)
0.242246 + 0.970215i \(0.422116\pi\)
\(752\) 0 0
\(753\) −15.4292 −0.562270
\(754\) 0 0
\(755\) 12.6007 0.458586
\(756\) 0 0
\(757\) 16.3056 0.592636 0.296318 0.955089i \(-0.404241\pi\)
0.296318 + 0.955089i \(0.404241\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −21.2704 −0.771053 −0.385526 0.922697i \(-0.625980\pi\)
−0.385526 + 0.922697i \(0.625980\pi\)
\(762\) 0 0
\(763\) −17.2664 −0.625085
\(764\) 0 0
\(765\) −12.4703 −0.450865
\(766\) 0 0
\(767\) 54.2912 1.96034
\(768\) 0 0
\(769\) −5.81494 −0.209692 −0.104846 0.994488i \(-0.533435\pi\)
−0.104846 + 0.994488i \(0.533435\pi\)
\(770\) 0 0
\(771\) 6.33737 0.228235
\(772\) 0 0
\(773\) 18.6709 0.671544 0.335772 0.941943i \(-0.391003\pi\)
0.335772 + 0.941943i \(0.391003\pi\)
\(774\) 0 0
\(775\) 0.760465 0.0273167
\(776\) 0 0
\(777\) 16.8054 0.602892
\(778\) 0 0
\(779\) 33.9833 1.21758
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −6.07957 −0.217266
\(784\) 0 0
\(785\) 6.47653 0.231157
\(786\) 0 0
\(787\) 46.3213 1.65118 0.825588 0.564274i \(-0.190844\pi\)
0.825588 + 0.564274i \(0.190844\pi\)
\(788\) 0 0
\(789\) 54.9468 1.95616
\(790\) 0 0
\(791\) 60.3602 2.14616
\(792\) 0 0
\(793\) 37.9964 1.34929
\(794\) 0 0
\(795\) 22.8824 0.811553
\(796\) 0 0
\(797\) −6.11777 −0.216702 −0.108351 0.994113i \(-0.534557\pi\)
−0.108351 + 0.994113i \(0.534557\pi\)
\(798\) 0 0
\(799\) −6.87131 −0.243089
\(800\) 0 0
\(801\) 3.54189 0.125146
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 22.4628 0.791710
\(806\) 0 0
\(807\) −15.1069 −0.531788
\(808\) 0 0
\(809\) 14.2124 0.499682 0.249841 0.968287i \(-0.419622\pi\)
0.249841 + 0.968287i \(0.419622\pi\)
\(810\) 0 0
\(811\) −30.6405 −1.07593 −0.537967 0.842966i \(-0.680807\pi\)
−0.537967 + 0.842966i \(0.680807\pi\)
\(812\) 0 0
\(813\) −66.6570 −2.33776
\(814\) 0 0
\(815\) −21.7737 −0.762700
\(816\) 0 0
\(817\) −39.9150 −1.39645
\(818\) 0 0
\(819\) 117.752 4.11460
\(820\) 0 0
\(821\) −3.53169 −0.123257 −0.0616285 0.998099i \(-0.519629\pi\)
−0.0616285 + 0.998099i \(0.519629\pi\)
\(822\) 0 0
\(823\) −29.7928 −1.03851 −0.519256 0.854619i \(-0.673791\pi\)
−0.519256 + 0.854619i \(0.673791\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 50.4631 1.75477 0.877386 0.479785i \(-0.159285\pi\)
0.877386 + 0.479785i \(0.159285\pi\)
\(828\) 0 0
\(829\) −19.2980 −0.670246 −0.335123 0.942174i \(-0.608778\pi\)
−0.335123 + 0.942174i \(0.608778\pi\)
\(830\) 0 0
\(831\) 15.8027 0.548191
\(832\) 0 0
\(833\) 8.51555 0.295046
\(834\) 0 0
\(835\) 19.0881 0.660572
\(836\) 0 0
\(837\) 12.1276 0.419193
\(838\) 0 0
\(839\) 16.8989 0.583414 0.291707 0.956508i \(-0.405777\pi\)
0.291707 + 0.956508i \(0.405777\pi\)
\(840\) 0 0
\(841\) −28.8547 −0.994989
\(842\) 0 0
\(843\) −71.1197 −2.44949
\(844\) 0 0
\(845\) −5.24299 −0.180364
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 40.3508 1.38484
\(850\) 0 0
\(851\) −9.27830 −0.318056
\(852\) 0 0
\(853\) −47.5816 −1.62916 −0.814582 0.580048i \(-0.803034\pi\)
−0.814582 + 0.580048i \(0.803034\pi\)
\(854\) 0 0
\(855\) 43.5364 1.48891
\(856\) 0 0
\(857\) 21.8041 0.744813 0.372406 0.928070i \(-0.378533\pi\)
0.372406 + 0.928070i \(0.378533\pi\)
\(858\) 0 0
\(859\) 23.3331 0.796116 0.398058 0.917360i \(-0.369684\pi\)
0.398058 + 0.917360i \(0.369684\pi\)
\(860\) 0 0
\(861\) 70.8615 2.41495
\(862\) 0 0
\(863\) 21.9086 0.745777 0.372889 0.927876i \(-0.378367\pi\)
0.372889 + 0.927876i \(0.378367\pi\)
\(864\) 0 0
\(865\) 1.44630 0.0491756
\(866\) 0 0
\(867\) −47.6545 −1.61843
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 53.4474 1.81099
\(872\) 0 0
\(873\) −21.8003 −0.737829
\(874\) 0 0
\(875\) 3.51508 0.118831
\(876\) 0 0
\(877\) 52.7358 1.78076 0.890381 0.455216i \(-0.150438\pi\)
0.890381 + 0.455216i \(0.150438\pi\)
\(878\) 0 0
\(879\) 47.1518 1.59039
\(880\) 0 0
\(881\) 34.3244 1.15642 0.578210 0.815888i \(-0.303752\pi\)
0.578210 + 0.815888i \(0.303752\pi\)
\(882\) 0 0
\(883\) −49.7184 −1.67316 −0.836579 0.547846i \(-0.815448\pi\)
−0.836579 + 0.547846i \(0.815448\pi\)
\(884\) 0 0
\(885\) 41.8560 1.40697
\(886\) 0 0
\(887\) 11.9815 0.402300 0.201150 0.979560i \(-0.435532\pi\)
0.201150 + 0.979560i \(0.435532\pi\)
\(888\) 0 0
\(889\) 54.4427 1.82595
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 23.9891 0.802766
\(894\) 0 0
\(895\) −9.02979 −0.301833
\(896\) 0 0
\(897\) −89.8780 −3.00094
\(898\) 0 0
\(899\) −0.289905 −0.00966886
\(900\) 0 0
\(901\) −11.0488 −0.368089
\(902\) 0 0
\(903\) −83.2301 −2.76972
\(904\) 0 0
\(905\) 1.14725 0.0381359
\(906\) 0 0
\(907\) −47.2674 −1.56949 −0.784744 0.619820i \(-0.787206\pi\)
−0.784744 + 0.619820i \(0.787206\pi\)
\(908\) 0 0
\(909\) 41.8609 1.38844
\(910\) 0 0
\(911\) 8.85300 0.293313 0.146656 0.989187i \(-0.453149\pi\)
0.146656 + 0.989187i \(0.453149\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 29.2934 0.968411
\(916\) 0 0
\(917\) −12.8207 −0.423376
\(918\) 0 0
\(919\) 16.5141 0.544750 0.272375 0.962191i \(-0.412191\pi\)
0.272375 + 0.962191i \(0.412191\pi\)
\(920\) 0 0
\(921\) −1.87196 −0.0616832
\(922\) 0 0
\(923\) 31.4312 1.03457
\(924\) 0 0
\(925\) −1.45191 −0.0477384
\(926\) 0 0
\(927\) −136.326 −4.47753
\(928\) 0 0
\(929\) −11.2898 −0.370408 −0.185204 0.982700i \(-0.559295\pi\)
−0.185204 + 0.982700i \(0.559295\pi\)
\(930\) 0 0
\(931\) −29.7295 −0.974346
\(932\) 0 0
\(933\) −33.6028 −1.10011
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −21.5727 −0.704748 −0.352374 0.935859i \(-0.614626\pi\)
−0.352374 + 0.935859i \(0.614626\pi\)
\(938\) 0 0
\(939\) −30.7432 −1.00327
\(940\) 0 0
\(941\) 50.3062 1.63994 0.819968 0.572409i \(-0.193991\pi\)
0.819968 + 0.572409i \(0.193991\pi\)
\(942\) 0 0
\(943\) −39.1227 −1.27401
\(944\) 0 0
\(945\) 56.0573 1.82354
\(946\) 0 0
\(947\) 41.7335 1.35615 0.678077 0.734990i \(-0.262813\pi\)
0.678077 + 0.734990i \(0.262813\pi\)
\(948\) 0 0
\(949\) −30.0354 −0.974990
\(950\) 0 0
\(951\) −93.1504 −3.02061
\(952\) 0 0
\(953\) −35.2481 −1.14180 −0.570899 0.821021i \(-0.693405\pi\)
−0.570899 + 0.821021i \(0.693405\pi\)
\(954\) 0 0
\(955\) −14.9868 −0.484960
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 57.1477 1.84540
\(960\) 0 0
\(961\) −30.4217 −0.981345
\(962\) 0 0
\(963\) −27.4337 −0.884039
\(964\) 0 0
\(965\) 15.1821 0.488728
\(966\) 0 0
\(967\) 21.6156 0.695111 0.347556 0.937659i \(-0.387012\pi\)
0.347556 + 0.937659i \(0.387012\pi\)
\(968\) 0 0
\(969\) −29.0625 −0.933623
\(970\) 0 0
\(971\) 11.8335 0.379755 0.189878 0.981808i \(-0.439191\pi\)
0.189878 + 0.981808i \(0.439191\pi\)
\(972\) 0 0
\(973\) −26.3361 −0.844296
\(974\) 0 0
\(975\) −14.0645 −0.450424
\(976\) 0 0
\(977\) −51.6998 −1.65402 −0.827011 0.562185i \(-0.809961\pi\)
−0.827011 + 0.562185i \(0.809961\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 38.5259 1.23004
\(982\) 0 0
\(983\) 51.1243 1.63061 0.815306 0.579030i \(-0.196569\pi\)
0.815306 + 0.579030i \(0.196569\pi\)
\(984\) 0 0
\(985\) 2.95776 0.0942420
\(986\) 0 0
\(987\) 50.0218 1.59221
\(988\) 0 0
\(989\) 45.9514 1.46117
\(990\) 0 0
\(991\) −3.16402 −0.100508 −0.0502542 0.998736i \(-0.516003\pi\)
−0.0502542 + 0.998736i \(0.516003\pi\)
\(992\) 0 0
\(993\) 17.5130 0.555760
\(994\) 0 0
\(995\) −5.07974 −0.161038
\(996\) 0 0
\(997\) −5.68777 −0.180133 −0.0900667 0.995936i \(-0.528708\pi\)
−0.0900667 + 0.995936i \(0.528708\pi\)
\(998\) 0 0
\(999\) −23.1545 −0.732577
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 9680.2.a.dc.1.6 6
4.3 odd 2 4840.2.a.bb.1.1 6
11.5 even 5 880.2.bo.i.641.3 12
11.9 even 5 880.2.bo.i.81.3 12
11.10 odd 2 9680.2.a.dd.1.6 6
44.27 odd 10 440.2.y.c.201.1 yes 12
44.31 odd 10 440.2.y.c.81.1 12
44.43 even 2 4840.2.a.ba.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.81.1 12 44.31 odd 10
440.2.y.c.201.1 yes 12 44.27 odd 10
880.2.bo.i.81.3 12 11.9 even 5
880.2.bo.i.641.3 12 11.5 even 5
4840.2.a.ba.1.1 6 44.43 even 2
4840.2.a.bb.1.1 6 4.3 odd 2
9680.2.a.dc.1.6 6 1.1 even 1 trivial
9680.2.a.dd.1.6 6 11.10 odd 2