Properties

Label 6960.2.a.bw.1.1
Level $6960$
Weight $2$
Character 6960.1
Self dual yes
Analytic conductor $55.576$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6960,2,Mod(1,6960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6960.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: \(55.5758798068\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.79129\) of defining polynomial
Character \(\chi\) \(=\) 6960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} -4.58258 q^{13} -1.00000 q^{15} -3.00000 q^{17} -3.58258 q^{19} +1.00000 q^{21} +4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.00000 q^{29} -4.00000 q^{31} +5.00000 q^{33} -1.00000 q^{35} -4.00000 q^{37} +4.58258 q^{39} -9.16515 q^{41} +9.58258 q^{43} +1.00000 q^{45} -10.5826 q^{47} -6.00000 q^{49} +3.00000 q^{51} +0.417424 q^{53} -5.00000 q^{55} +3.58258 q^{57} +7.58258 q^{59} +12.7477 q^{61} -1.00000 q^{63} -4.58258 q^{65} +4.16515 q^{67} -4.00000 q^{69} +9.58258 q^{71} +4.00000 q^{73} -1.00000 q^{75} +5.00000 q^{77} -7.58258 q^{79} +1.00000 q^{81} +11.5826 q^{83} -3.00000 q^{85} -1.00000 q^{87} +1.41742 q^{89} +4.58258 q^{91} +4.00000 q^{93} -3.58258 q^{95} +11.5826 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 10 q^{11} - 2 q^{15} - 6 q^{17} + 2 q^{19} + 2 q^{21} + 8 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{29} - 8 q^{31} + 10 q^{33} - 2 q^{35} - 8 q^{37} + 10 q^{43} + 2 q^{45} - 12 q^{47} - 12 q^{49} + 6 q^{51} + 10 q^{53} - 10 q^{55} - 2 q^{57} + 6 q^{59} - 2 q^{61} - 2 q^{63} - 10 q^{67} - 8 q^{69} + 10 q^{71} + 8 q^{73} - 2 q^{75} + 10 q^{77} - 6 q^{79} + 2 q^{81} + 14 q^{83} - 6 q^{85} - 2 q^{87} + 12 q^{89} + 8 q^{93} + 2 q^{95} + 14 q^{97} - 10 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) −4.58258 −1.27098 −0.635489 0.772110i \(-0.719201\pi\)
−0.635489 + 0.772110i \(0.719201\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −3.58258 −0.821899 −0.410950 0.911658i \(-0.634803\pi\)
−0.410950 + 0.911658i \(0.634803\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 5.00000 0.870388
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 4.58258 0.733799
\(40\) 0 0
\(41\) −9.16515 −1.43136 −0.715678 0.698430i \(-0.753882\pi\)
−0.715678 + 0.698430i \(0.753882\pi\)
\(42\) 0 0
\(43\) 9.58258 1.46133 0.730665 0.682737i \(-0.239210\pi\)
0.730665 + 0.682737i \(0.239210\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −10.5826 −1.54363 −0.771814 0.635849i \(-0.780650\pi\)
−0.771814 + 0.635849i \(0.780650\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) 0.417424 0.0573376 0.0286688 0.999589i \(-0.490873\pi\)
0.0286688 + 0.999589i \(0.490873\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.674200
\(56\) 0 0
\(57\) 3.58258 0.474524
\(58\) 0 0
\(59\) 7.58258 0.987167 0.493584 0.869698i \(-0.335687\pi\)
0.493584 + 0.869698i \(0.335687\pi\)
\(60\) 0 0
\(61\) 12.7477 1.63218 0.816090 0.577925i \(-0.196138\pi\)
0.816090 + 0.577925i \(0.196138\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −4.58258 −0.568399
\(66\) 0 0
\(67\) 4.16515 0.508854 0.254427 0.967092i \(-0.418113\pi\)
0.254427 + 0.967092i \(0.418113\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 9.58258 1.13724 0.568621 0.822599i \(-0.307477\pi\)
0.568621 + 0.822599i \(0.307477\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 5.00000 0.569803
\(78\) 0 0
\(79\) −7.58258 −0.853106 −0.426553 0.904462i \(-0.640272\pi\)
−0.426553 + 0.904462i \(0.640272\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.5826 1.27135 0.635676 0.771956i \(-0.280721\pi\)
0.635676 + 0.771956i \(0.280721\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 1.41742 0.150247 0.0751233 0.997174i \(-0.476065\pi\)
0.0751233 + 0.997174i \(0.476065\pi\)
\(90\) 0 0
\(91\) 4.58258 0.480384
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) −3.58258 −0.367565
\(96\) 0 0
\(97\) 11.5826 1.17603 0.588016 0.808849i \(-0.299909\pi\)
0.588016 + 0.808849i \(0.299909\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) −0.582576 −0.0579684 −0.0289842 0.999580i \(-0.509227\pi\)
−0.0289842 + 0.999580i \(0.509227\pi\)
\(102\) 0 0
\(103\) −15.1652 −1.49427 −0.747133 0.664674i \(-0.768570\pi\)
−0.747133 + 0.664674i \(0.768570\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) −5.16515 −0.499334 −0.249667 0.968332i \(-0.580321\pi\)
−0.249667 + 0.968332i \(0.580321\pi\)
\(108\) 0 0
\(109\) 14.1652 1.35678 0.678388 0.734704i \(-0.262679\pi\)
0.678388 + 0.734704i \(0.262679\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) −14.1652 −1.33255 −0.666273 0.745708i \(-0.732111\pi\)
−0.666273 + 0.745708i \(0.732111\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) −4.58258 −0.423659
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 9.16515 0.826394
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) −9.58258 −0.843699
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) 3.58258 0.310649
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 16.3303 1.39519 0.697596 0.716491i \(-0.254253\pi\)
0.697596 + 0.716491i \(0.254253\pi\)
\(138\) 0 0
\(139\) 9.41742 0.798776 0.399388 0.916782i \(-0.369223\pi\)
0.399388 + 0.916782i \(0.369223\pi\)
\(140\) 0 0
\(141\) 10.5826 0.891214
\(142\) 0 0
\(143\) 22.9129 1.91607
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) 6.00000 0.494872
\(148\) 0 0
\(149\) 16.7477 1.37203 0.686014 0.727589i \(-0.259359\pi\)
0.686014 + 0.727589i \(0.259359\pi\)
\(150\) 0 0
\(151\) −7.16515 −0.583092 −0.291546 0.956557i \(-0.594170\pi\)
−0.291546 + 0.956557i \(0.594170\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 10.7477 0.857762 0.428881 0.903361i \(-0.358908\pi\)
0.428881 + 0.903361i \(0.358908\pi\)
\(158\) 0 0
\(159\) −0.417424 −0.0331039
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) −7.58258 −0.593913 −0.296957 0.954891i \(-0.595972\pi\)
−0.296957 + 0.954891i \(0.595972\pi\)
\(164\) 0 0
\(165\) 5.00000 0.389249
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 8.00000 0.615385
\(170\) 0 0
\(171\) −3.58258 −0.273966
\(172\) 0 0
\(173\) −3.16515 −0.240642 −0.120321 0.992735i \(-0.538392\pi\)
−0.120321 + 0.992735i \(0.538392\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −7.58258 −0.569941
\(178\) 0 0
\(179\) −4.74773 −0.354862 −0.177431 0.984133i \(-0.556779\pi\)
−0.177431 + 0.984133i \(0.556779\pi\)
\(180\) 0 0
\(181\) 16.1652 1.20155 0.600773 0.799420i \(-0.294860\pi\)
0.600773 + 0.799420i \(0.294860\pi\)
\(182\) 0 0
\(183\) −12.7477 −0.942339
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 15.0000 1.09691
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) −20.3303 −1.46341 −0.731704 0.681623i \(-0.761274\pi\)
−0.731704 + 0.681623i \(0.761274\pi\)
\(194\) 0 0
\(195\) 4.58258 0.328165
\(196\) 0 0
\(197\) −16.3303 −1.16349 −0.581743 0.813373i \(-0.697629\pi\)
−0.581743 + 0.813373i \(0.697629\pi\)
\(198\) 0 0
\(199\) −13.4174 −0.951136 −0.475568 0.879679i \(-0.657757\pi\)
−0.475568 + 0.879679i \(0.657757\pi\)
\(200\) 0 0
\(201\) −4.16515 −0.293787
\(202\) 0 0
\(203\) −1.00000 −0.0701862
\(204\) 0 0
\(205\) −9.16515 −0.640122
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 17.9129 1.23906
\(210\) 0 0
\(211\) −20.3303 −1.39960 −0.699798 0.714341i \(-0.746727\pi\)
−0.699798 + 0.714341i \(0.746727\pi\)
\(212\) 0 0
\(213\) −9.58258 −0.656587
\(214\) 0 0
\(215\) 9.58258 0.653526
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 13.7477 0.924772
\(222\) 0 0
\(223\) −7.00000 −0.468755 −0.234377 0.972146i \(-0.575305\pi\)
−0.234377 + 0.972146i \(0.575305\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 7.58258 0.503273 0.251637 0.967822i \(-0.419031\pi\)
0.251637 + 0.967822i \(0.419031\pi\)
\(228\) 0 0
\(229\) −17.1652 −1.13431 −0.567153 0.823613i \(-0.691955\pi\)
−0.567153 + 0.823613i \(0.691955\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) 0 0
\(233\) −13.1652 −0.862478 −0.431239 0.902238i \(-0.641923\pi\)
−0.431239 + 0.902238i \(0.641923\pi\)
\(234\) 0 0
\(235\) −10.5826 −0.690331
\(236\) 0 0
\(237\) 7.58258 0.492541
\(238\) 0 0
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) 29.3303 1.88933 0.944665 0.328035i \(-0.106387\pi\)
0.944665 + 0.328035i \(0.106387\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 16.4174 1.04462
\(248\) 0 0
\(249\) −11.5826 −0.734016
\(250\) 0 0
\(251\) −16.1652 −1.02034 −0.510168 0.860075i \(-0.670417\pi\)
−0.510168 + 0.860075i \(0.670417\pi\)
\(252\) 0 0
\(253\) −20.0000 −1.25739
\(254\) 0 0
\(255\) 3.00000 0.187867
\(256\) 0 0
\(257\) 12.7477 0.795181 0.397591 0.917563i \(-0.369846\pi\)
0.397591 + 0.917563i \(0.369846\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −30.3303 −1.87025 −0.935123 0.354322i \(-0.884712\pi\)
−0.935123 + 0.354322i \(0.884712\pi\)
\(264\) 0 0
\(265\) 0.417424 0.0256422
\(266\) 0 0
\(267\) −1.41742 −0.0867450
\(268\) 0 0
\(269\) 22.5826 1.37688 0.688442 0.725291i \(-0.258295\pi\)
0.688442 + 0.725291i \(0.258295\pi\)
\(270\) 0 0
\(271\) −1.16515 −0.0707779 −0.0353890 0.999374i \(-0.511267\pi\)
−0.0353890 + 0.999374i \(0.511267\pi\)
\(272\) 0 0
\(273\) −4.58258 −0.277350
\(274\) 0 0
\(275\) −5.00000 −0.301511
\(276\) 0 0
\(277\) −24.9129 −1.49687 −0.748435 0.663208i \(-0.769194\pi\)
−0.748435 + 0.663208i \(0.769194\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −0.417424 −0.0249014 −0.0124507 0.999922i \(-0.503963\pi\)
−0.0124507 + 0.999922i \(0.503963\pi\)
\(282\) 0 0
\(283\) 0.834849 0.0496266 0.0248133 0.999692i \(-0.492101\pi\)
0.0248133 + 0.999692i \(0.492101\pi\)
\(284\) 0 0
\(285\) 3.58258 0.212213
\(286\) 0 0
\(287\) 9.16515 0.541002
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −11.5826 −0.678983
\(292\) 0 0
\(293\) 30.1652 1.76227 0.881133 0.472868i \(-0.156781\pi\)
0.881133 + 0.472868i \(0.156781\pi\)
\(294\) 0 0
\(295\) 7.58258 0.441475
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) −18.3303 −1.06007
\(300\) 0 0
\(301\) −9.58258 −0.552330
\(302\) 0 0
\(303\) 0.582576 0.0334681
\(304\) 0 0
\(305\) 12.7477 0.729933
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 15.1652 0.862715
\(310\) 0 0
\(311\) 3.00000 0.170114 0.0850572 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) 0 0
\(313\) 10.5826 0.598163 0.299081 0.954228i \(-0.403320\pi\)
0.299081 + 0.954228i \(0.403320\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) −25.0000 −1.40414 −0.702070 0.712108i \(-0.747741\pi\)
−0.702070 + 0.712108i \(0.747741\pi\)
\(318\) 0 0
\(319\) −5.00000 −0.279946
\(320\) 0 0
\(321\) 5.16515 0.288291
\(322\) 0 0
\(323\) 10.7477 0.598020
\(324\) 0 0
\(325\) −4.58258 −0.254196
\(326\) 0 0
\(327\) −14.1652 −0.783335
\(328\) 0 0
\(329\) 10.5826 0.583436
\(330\) 0 0
\(331\) 8.33030 0.457875 0.228937 0.973441i \(-0.426475\pi\)
0.228937 + 0.973441i \(0.426475\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 4.16515 0.227567
\(336\) 0 0
\(337\) −21.1652 −1.15294 −0.576470 0.817119i \(-0.695570\pi\)
−0.576470 + 0.817119i \(0.695570\pi\)
\(338\) 0 0
\(339\) 14.1652 0.769345
\(340\) 0 0
\(341\) 20.0000 1.08306
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) −7.16515 −0.384645 −0.192323 0.981332i \(-0.561602\pi\)
−0.192323 + 0.981332i \(0.561602\pi\)
\(348\) 0 0
\(349\) −4.33030 −0.231796 −0.115898 0.993261i \(-0.536975\pi\)
−0.115898 + 0.993261i \(0.536975\pi\)
\(350\) 0 0
\(351\) 4.58258 0.244600
\(352\) 0 0
\(353\) −14.8348 −0.789579 −0.394790 0.918772i \(-0.629183\pi\)
−0.394790 + 0.918772i \(0.629183\pi\)
\(354\) 0 0
\(355\) 9.58258 0.508590
\(356\) 0 0
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) 27.1652 1.43372 0.716861 0.697216i \(-0.245578\pi\)
0.716861 + 0.697216i \(0.245578\pi\)
\(360\) 0 0
\(361\) −6.16515 −0.324482
\(362\) 0 0
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) 37.4955 1.95725 0.978623 0.205661i \(-0.0659343\pi\)
0.978623 + 0.205661i \(0.0659343\pi\)
\(368\) 0 0
\(369\) −9.16515 −0.477119
\(370\) 0 0
\(371\) −0.417424 −0.0216716
\(372\) 0 0
\(373\) 28.3303 1.46689 0.733444 0.679750i \(-0.237912\pi\)
0.733444 + 0.679750i \(0.237912\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −4.58258 −0.236015
\(378\) 0 0
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 0 0
\(383\) −3.58258 −0.183061 −0.0915305 0.995802i \(-0.529176\pi\)
−0.0915305 + 0.995802i \(0.529176\pi\)
\(384\) 0 0
\(385\) 5.00000 0.254824
\(386\) 0 0
\(387\) 9.58258 0.487110
\(388\) 0 0
\(389\) −6.58258 −0.333750 −0.166875 0.985978i \(-0.553368\pi\)
−0.166875 + 0.985978i \(0.553368\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) −15.0000 −0.756650
\(394\) 0 0
\(395\) −7.58258 −0.381521
\(396\) 0 0
\(397\) 10.8348 0.543785 0.271893 0.962328i \(-0.412350\pi\)
0.271893 + 0.962328i \(0.412350\pi\)
\(398\) 0 0
\(399\) −3.58258 −0.179353
\(400\) 0 0
\(401\) 12.4174 0.620097 0.310048 0.950721i \(-0.399655\pi\)
0.310048 + 0.950721i \(0.399655\pi\)
\(402\) 0 0
\(403\) 18.3303 0.913097
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) −2.74773 −0.135866 −0.0679332 0.997690i \(-0.521640\pi\)
−0.0679332 + 0.997690i \(0.521640\pi\)
\(410\) 0 0
\(411\) −16.3303 −0.805514
\(412\) 0 0
\(413\) −7.58258 −0.373114
\(414\) 0 0
\(415\) 11.5826 0.568566
\(416\) 0 0
\(417\) −9.41742 −0.461173
\(418\) 0 0
\(419\) 37.1652 1.81564 0.907818 0.419364i \(-0.137747\pi\)
0.907818 + 0.419364i \(0.137747\pi\)
\(420\) 0 0
\(421\) 4.41742 0.215292 0.107646 0.994189i \(-0.465669\pi\)
0.107646 + 0.994189i \(0.465669\pi\)
\(422\) 0 0
\(423\) −10.5826 −0.514542
\(424\) 0 0
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) −12.7477 −0.616906
\(428\) 0 0
\(429\) −22.9129 −1.10624
\(430\) 0 0
\(431\) 39.1652 1.88652 0.943259 0.332057i \(-0.107742\pi\)
0.943259 + 0.332057i \(0.107742\pi\)
\(432\) 0 0
\(433\) 25.0780 1.20517 0.602587 0.798053i \(-0.294137\pi\)
0.602587 + 0.798053i \(0.294137\pi\)
\(434\) 0 0
\(435\) −1.00000 −0.0479463
\(436\) 0 0
\(437\) −14.3303 −0.685511
\(438\) 0 0
\(439\) 16.9129 0.807208 0.403604 0.914934i \(-0.367757\pi\)
0.403604 + 0.914934i \(0.367757\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) −0.582576 −0.0276790 −0.0138395 0.999904i \(-0.504405\pi\)
−0.0138395 + 0.999904i \(0.504405\pi\)
\(444\) 0 0
\(445\) 1.41742 0.0671924
\(446\) 0 0
\(447\) −16.7477 −0.792140
\(448\) 0 0
\(449\) −30.0780 −1.41947 −0.709735 0.704469i \(-0.751185\pi\)
−0.709735 + 0.704469i \(0.751185\pi\)
\(450\) 0 0
\(451\) 45.8258 2.15785
\(452\) 0 0
\(453\) 7.16515 0.336648
\(454\) 0 0
\(455\) 4.58258 0.214834
\(456\) 0 0
\(457\) −3.74773 −0.175311 −0.0876556 0.996151i \(-0.527938\pi\)
−0.0876556 + 0.996151i \(0.527938\pi\)
\(458\) 0 0
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) −9.16515 −0.426864 −0.213432 0.976958i \(-0.568464\pi\)
−0.213432 + 0.976958i \(0.568464\pi\)
\(462\) 0 0
\(463\) 0.165151 0.00767524 0.00383762 0.999993i \(-0.498778\pi\)
0.00383762 + 0.999993i \(0.498778\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −4.16515 −0.192329
\(470\) 0 0
\(471\) −10.7477 −0.495229
\(472\) 0 0
\(473\) −47.9129 −2.20304
\(474\) 0 0
\(475\) −3.58258 −0.164380
\(476\) 0 0
\(477\) 0.417424 0.0191125
\(478\) 0 0
\(479\) 19.1652 0.875678 0.437839 0.899053i \(-0.355744\pi\)
0.437839 + 0.899053i \(0.355744\pi\)
\(480\) 0 0
\(481\) 18.3303 0.835790
\(482\) 0 0
\(483\) 4.00000 0.182006
\(484\) 0 0
\(485\) 11.5826 0.525938
\(486\) 0 0
\(487\) −2.33030 −0.105596 −0.0527980 0.998605i \(-0.516814\pi\)
−0.0527980 + 0.998605i \(0.516814\pi\)
\(488\) 0 0
\(489\) 7.58258 0.342896
\(490\) 0 0
\(491\) −16.0000 −0.722070 −0.361035 0.932552i \(-0.617576\pi\)
−0.361035 + 0.932552i \(0.617576\pi\)
\(492\) 0 0
\(493\) −3.00000 −0.135113
\(494\) 0 0
\(495\) −5.00000 −0.224733
\(496\) 0 0
\(497\) −9.58258 −0.429837
\(498\) 0 0
\(499\) 13.4174 0.600646 0.300323 0.953837i \(-0.402905\pi\)
0.300323 + 0.953837i \(0.402905\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 22.9129 1.02163 0.510817 0.859689i \(-0.329343\pi\)
0.510817 + 0.859689i \(0.329343\pi\)
\(504\) 0 0
\(505\) −0.582576 −0.0259243
\(506\) 0 0
\(507\) −8.00000 −0.355292
\(508\) 0 0
\(509\) 26.7477 1.18557 0.592786 0.805360i \(-0.298028\pi\)
0.592786 + 0.805360i \(0.298028\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) 0 0
\(513\) 3.58258 0.158175
\(514\) 0 0
\(515\) −15.1652 −0.668256
\(516\) 0 0
\(517\) 52.9129 2.32711
\(518\) 0 0
\(519\) 3.16515 0.138935
\(520\) 0 0
\(521\) 43.0780 1.88728 0.943641 0.330970i \(-0.107376\pi\)
0.943641 + 0.330970i \(0.107376\pi\)
\(522\) 0 0
\(523\) 33.3303 1.45743 0.728716 0.684816i \(-0.240117\pi\)
0.728716 + 0.684816i \(0.240117\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 7.58258 0.329056
\(532\) 0 0
\(533\) 42.0000 1.81922
\(534\) 0 0
\(535\) −5.16515 −0.223309
\(536\) 0 0
\(537\) 4.74773 0.204880
\(538\) 0 0
\(539\) 30.0000 1.29219
\(540\) 0 0
\(541\) −31.4955 −1.35410 −0.677048 0.735939i \(-0.736741\pi\)
−0.677048 + 0.735939i \(0.736741\pi\)
\(542\) 0 0
\(543\) −16.1652 −0.693713
\(544\) 0 0
\(545\) 14.1652 0.606768
\(546\) 0 0
\(547\) −4.16515 −0.178089 −0.0890445 0.996028i \(-0.528381\pi\)
−0.0890445 + 0.996028i \(0.528381\pi\)
\(548\) 0 0
\(549\) 12.7477 0.544060
\(550\) 0 0
\(551\) −3.58258 −0.152623
\(552\) 0 0
\(553\) 7.58258 0.322444
\(554\) 0 0
\(555\) 4.00000 0.169791
\(556\) 0 0
\(557\) −22.7477 −0.963852 −0.481926 0.876212i \(-0.660063\pi\)
−0.481926 + 0.876212i \(0.660063\pi\)
\(558\) 0 0
\(559\) −43.9129 −1.85732
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) 14.5826 0.614582 0.307291 0.951616i \(-0.400577\pi\)
0.307291 + 0.951616i \(0.400577\pi\)
\(564\) 0 0
\(565\) −14.1652 −0.595932
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 28.5826 1.19824 0.599122 0.800658i \(-0.295516\pi\)
0.599122 + 0.800658i \(0.295516\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) −4.00000 −0.167102
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −19.1652 −0.797856 −0.398928 0.916982i \(-0.630618\pi\)
−0.398928 + 0.916982i \(0.630618\pi\)
\(578\) 0 0
\(579\) 20.3303 0.844899
\(580\) 0 0
\(581\) −11.5826 −0.480526
\(582\) 0 0
\(583\) −2.08712 −0.0864397
\(584\) 0 0
\(585\) −4.58258 −0.189466
\(586\) 0 0
\(587\) 11.5826 0.478064 0.239032 0.971012i \(-0.423170\pi\)
0.239032 + 0.971012i \(0.423170\pi\)
\(588\) 0 0
\(589\) 14.3303 0.590470
\(590\) 0 0
\(591\) 16.3303 0.671739
\(592\) 0 0
\(593\) 8.41742 0.345662 0.172831 0.984951i \(-0.444709\pi\)
0.172831 + 0.984951i \(0.444709\pi\)
\(594\) 0 0
\(595\) 3.00000 0.122988
\(596\) 0 0
\(597\) 13.4174 0.549139
\(598\) 0 0
\(599\) −0.165151 −0.00674790 −0.00337395 0.999994i \(-0.501074\pi\)
−0.00337395 + 0.999994i \(0.501074\pi\)
\(600\) 0 0
\(601\) −32.3303 −1.31878 −0.659390 0.751801i \(-0.729185\pi\)
−0.659390 + 0.751801i \(0.729185\pi\)
\(602\) 0 0
\(603\) 4.16515 0.169618
\(604\) 0 0
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) −3.58258 −0.145412 −0.0727061 0.997353i \(-0.523164\pi\)
−0.0727061 + 0.997353i \(0.523164\pi\)
\(608\) 0 0
\(609\) 1.00000 0.0405220
\(610\) 0 0
\(611\) 48.4955 1.96192
\(612\) 0 0
\(613\) −43.7477 −1.76695 −0.883477 0.468474i \(-0.844804\pi\)
−0.883477 + 0.468474i \(0.844804\pi\)
\(614\) 0 0
\(615\) 9.16515 0.369575
\(616\) 0 0
\(617\) 15.4955 0.623823 0.311912 0.950111i \(-0.399031\pi\)
0.311912 + 0.950111i \(0.399031\pi\)
\(618\) 0 0
\(619\) −13.1652 −0.529152 −0.264576 0.964365i \(-0.585232\pi\)
−0.264576 + 0.964365i \(0.585232\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) −1.41742 −0.0567879
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −17.9129 −0.715371
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −10.9129 −0.434435 −0.217217 0.976123i \(-0.569698\pi\)
−0.217217 + 0.976123i \(0.569698\pi\)
\(632\) 0 0
\(633\) 20.3303 0.808057
\(634\) 0 0
\(635\) −2.00000 −0.0793676
\(636\) 0 0
\(637\) 27.4955 1.08941
\(638\) 0 0
\(639\) 9.58258 0.379081
\(640\) 0 0
\(641\) −6.58258 −0.259996 −0.129998 0.991514i \(-0.541497\pi\)
−0.129998 + 0.991514i \(0.541497\pi\)
\(642\) 0 0
\(643\) −43.6606 −1.72181 −0.860903 0.508769i \(-0.830101\pi\)
−0.860903 + 0.508769i \(0.830101\pi\)
\(644\) 0 0
\(645\) −9.58258 −0.377314
\(646\) 0 0
\(647\) −24.7477 −0.972934 −0.486467 0.873699i \(-0.661715\pi\)
−0.486467 + 0.873699i \(0.661715\pi\)
\(648\) 0 0
\(649\) −37.9129 −1.48821
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 0 0
\(653\) 26.1652 1.02392 0.511961 0.859009i \(-0.328919\pi\)
0.511961 + 0.859009i \(0.328919\pi\)
\(654\) 0 0
\(655\) 15.0000 0.586098
\(656\) 0 0
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) −18.1652 −0.707614 −0.353807 0.935318i \(-0.615113\pi\)
−0.353807 + 0.935318i \(0.615113\pi\)
\(660\) 0 0
\(661\) 25.0000 0.972387 0.486194 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) 0 0
\(663\) −13.7477 −0.533917
\(664\) 0 0
\(665\) 3.58258 0.138926
\(666\) 0 0
\(667\) 4.00000 0.154881
\(668\) 0 0
\(669\) 7.00000 0.270636
\(670\) 0 0
\(671\) −63.7386 −2.46060
\(672\) 0 0
\(673\) 1.74773 0.0673699 0.0336850 0.999433i \(-0.489276\pi\)
0.0336850 + 0.999433i \(0.489276\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 44.8258 1.72279 0.861397 0.507932i \(-0.169590\pi\)
0.861397 + 0.507932i \(0.169590\pi\)
\(678\) 0 0
\(679\) −11.5826 −0.444498
\(680\) 0 0
\(681\) −7.58258 −0.290565
\(682\) 0 0
\(683\) −7.16515 −0.274167 −0.137083 0.990560i \(-0.543773\pi\)
−0.137083 + 0.990560i \(0.543773\pi\)
\(684\) 0 0
\(685\) 16.3303 0.623949
\(686\) 0 0
\(687\) 17.1652 0.654891
\(688\) 0 0
\(689\) −1.91288 −0.0728749
\(690\) 0 0
\(691\) −42.9129 −1.63248 −0.816241 0.577711i \(-0.803946\pi\)
−0.816241 + 0.577711i \(0.803946\pi\)
\(692\) 0 0
\(693\) 5.00000 0.189934
\(694\) 0 0
\(695\) 9.41742 0.357223
\(696\) 0 0
\(697\) 27.4955 1.04146
\(698\) 0 0
\(699\) 13.1652 0.497952
\(700\) 0 0
\(701\) −23.0780 −0.871645 −0.435823 0.900033i \(-0.643542\pi\)
−0.435823 + 0.900033i \(0.643542\pi\)
\(702\) 0 0
\(703\) 14.3303 0.540478
\(704\) 0 0
\(705\) 10.5826 0.398563
\(706\) 0 0
\(707\) 0.582576 0.0219100
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) −7.58258 −0.284369
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 22.9129 0.856893
\(716\) 0 0
\(717\) −26.0000 −0.970988
\(718\) 0 0
\(719\) −7.91288 −0.295101 −0.147550 0.989055i \(-0.547139\pi\)
−0.147550 + 0.989055i \(0.547139\pi\)
\(720\) 0 0
\(721\) 15.1652 0.564780
\(722\) 0 0
\(723\) −29.3303 −1.09081
\(724\) 0 0
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) −9.66970 −0.358629 −0.179315 0.983792i \(-0.557388\pi\)
−0.179315 + 0.983792i \(0.557388\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −28.7477 −1.06327
\(732\) 0 0
\(733\) −38.4174 −1.41898 −0.709490 0.704716i \(-0.751075\pi\)
−0.709490 + 0.704716i \(0.751075\pi\)
\(734\) 0 0
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) −20.8258 −0.767127
\(738\) 0 0
\(739\) −19.2523 −0.708206 −0.354103 0.935206i \(-0.615214\pi\)
−0.354103 + 0.935206i \(0.615214\pi\)
\(740\) 0 0
\(741\) −16.4174 −0.603109
\(742\) 0 0
\(743\) −29.7477 −1.09134 −0.545669 0.838001i \(-0.683724\pi\)
−0.545669 + 0.838001i \(0.683724\pi\)
\(744\) 0 0
\(745\) 16.7477 0.613589
\(746\) 0 0
\(747\) 11.5826 0.423784
\(748\) 0 0
\(749\) 5.16515 0.188731
\(750\) 0 0
\(751\) 17.4955 0.638418 0.319209 0.947684i \(-0.396583\pi\)
0.319209 + 0.947684i \(0.396583\pi\)
\(752\) 0 0
\(753\) 16.1652 0.589091
\(754\) 0 0
\(755\) −7.16515 −0.260767
\(756\) 0 0
\(757\) −2.33030 −0.0846963 −0.0423481 0.999103i \(-0.513484\pi\)
−0.0423481 + 0.999103i \(0.513484\pi\)
\(758\) 0 0
\(759\) 20.0000 0.725954
\(760\) 0 0
\(761\) 36.4174 1.32013 0.660065 0.751208i \(-0.270529\pi\)
0.660065 + 0.751208i \(0.270529\pi\)
\(762\) 0 0
\(763\) −14.1652 −0.512813
\(764\) 0 0
\(765\) −3.00000 −0.108465
\(766\) 0 0
\(767\) −34.7477 −1.25467
\(768\) 0 0
\(769\) 25.5826 0.922531 0.461266 0.887262i \(-0.347396\pi\)
0.461266 + 0.887262i \(0.347396\pi\)
\(770\) 0 0
\(771\) −12.7477 −0.459098
\(772\) 0 0
\(773\) −5.16515 −0.185778 −0.0928888 0.995676i \(-0.529610\pi\)
−0.0928888 + 0.995676i \(0.529610\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) 0 0
\(779\) 32.8348 1.17643
\(780\) 0 0
\(781\) −47.9129 −1.71446
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 10.7477 0.383603
\(786\) 0 0
\(787\) 24.0000 0.855508 0.427754 0.903895i \(-0.359305\pi\)
0.427754 + 0.903895i \(0.359305\pi\)
\(788\) 0 0
\(789\) 30.3303 1.07979
\(790\) 0 0
\(791\) 14.1652 0.503655
\(792\) 0 0
\(793\) −58.4174 −2.07446
\(794\) 0 0
\(795\) −0.417424 −0.0148045
\(796\) 0 0
\(797\) 32.3303 1.14520 0.572599 0.819836i \(-0.305935\pi\)
0.572599 + 0.819836i \(0.305935\pi\)
\(798\) 0 0
\(799\) 31.7477 1.12315
\(800\) 0 0
\(801\) 1.41742 0.0500822
\(802\) 0 0
\(803\) −20.0000 −0.705785
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) −22.5826 −0.794944
\(808\) 0 0
\(809\) 44.0780 1.54970 0.774851 0.632145i \(-0.217825\pi\)
0.774851 + 0.632145i \(0.217825\pi\)
\(810\) 0 0
\(811\) −32.5826 −1.14413 −0.572064 0.820209i \(-0.693857\pi\)
−0.572064 + 0.820209i \(0.693857\pi\)
\(812\) 0 0
\(813\) 1.16515 0.0408636
\(814\) 0 0
\(815\) −7.58258 −0.265606
\(816\) 0 0
\(817\) −34.3303 −1.20107
\(818\) 0 0
\(819\) 4.58258 0.160128
\(820\) 0 0
\(821\) −31.4955 −1.09920 −0.549599 0.835428i \(-0.685220\pi\)
−0.549599 + 0.835428i \(0.685220\pi\)
\(822\) 0 0
\(823\) −51.0780 −1.78047 −0.890234 0.455503i \(-0.849459\pi\)
−0.890234 + 0.455503i \(0.849459\pi\)
\(824\) 0 0
\(825\) 5.00000 0.174078
\(826\) 0 0
\(827\) 43.8258 1.52397 0.761985 0.647594i \(-0.224225\pi\)
0.761985 + 0.647594i \(0.224225\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) 24.9129 0.864218
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) 0 0
\(839\) 48.4955 1.67425 0.837125 0.547012i \(-0.184235\pi\)
0.837125 + 0.547012i \(0.184235\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 0.417424 0.0143769
\(844\) 0 0
\(845\) 8.00000 0.275208
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) 0 0
\(849\) −0.834849 −0.0286519
\(850\) 0 0
\(851\) −16.0000 −0.548473
\(852\) 0 0
\(853\) 20.7477 0.710389 0.355194 0.934792i \(-0.384415\pi\)
0.355194 + 0.934792i \(0.384415\pi\)
\(854\) 0 0
\(855\) −3.58258 −0.122522
\(856\) 0 0
\(857\) −46.6606 −1.59390 −0.796948 0.604048i \(-0.793554\pi\)
−0.796948 + 0.604048i \(0.793554\pi\)
\(858\) 0 0
\(859\) 47.5826 1.62350 0.811748 0.584007i \(-0.198516\pi\)
0.811748 + 0.584007i \(0.198516\pi\)
\(860\) 0 0
\(861\) −9.16515 −0.312348
\(862\) 0 0
\(863\) 6.41742 0.218452 0.109226 0.994017i \(-0.465163\pi\)
0.109226 + 0.994017i \(0.465163\pi\)
\(864\) 0 0
\(865\) −3.16515 −0.107618
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 37.9129 1.28611
\(870\) 0 0
\(871\) −19.0871 −0.646742
\(872\) 0 0
\(873\) 11.5826 0.392011
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 19.6697 0.664198 0.332099 0.943244i \(-0.392243\pi\)
0.332099 + 0.943244i \(0.392243\pi\)
\(878\) 0 0
\(879\) −30.1652 −1.01745
\(880\) 0 0
\(881\) −32.0780 −1.08074 −0.540368 0.841429i \(-0.681715\pi\)
−0.540368 + 0.841429i \(0.681715\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) −7.58258 −0.254885
\(886\) 0 0
\(887\) 44.9129 1.50803 0.754013 0.656859i \(-0.228115\pi\)
0.754013 + 0.656859i \(0.228115\pi\)
\(888\) 0 0
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 0 0
\(893\) 37.9129 1.26871
\(894\) 0 0
\(895\) −4.74773 −0.158699
\(896\) 0 0
\(897\) 18.3303 0.612031
\(898\) 0 0
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) −1.25227 −0.0417193
\(902\) 0 0
\(903\) 9.58258 0.318888
\(904\) 0 0
\(905\) 16.1652 0.537348
\(906\) 0 0
\(907\) −23.5826 −0.783047 −0.391523 0.920168i \(-0.628052\pi\)
−0.391523 + 0.920168i \(0.628052\pi\)
\(908\) 0 0
\(909\) −0.582576 −0.0193228
\(910\) 0 0
\(911\) 50.8258 1.68393 0.841966 0.539530i \(-0.181398\pi\)
0.841966 + 0.539530i \(0.181398\pi\)
\(912\) 0 0
\(913\) −57.9129 −1.91664
\(914\) 0 0
\(915\) −12.7477 −0.421427
\(916\) 0 0
\(917\) −15.0000 −0.495344
\(918\) 0 0
\(919\) −18.9129 −0.623878 −0.311939 0.950102i \(-0.600979\pi\)
−0.311939 + 0.950102i \(0.600979\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −43.9129 −1.44541
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) −15.1652 −0.498089
\(928\) 0 0
\(929\) 15.6697 0.514106 0.257053 0.966397i \(-0.417248\pi\)
0.257053 + 0.966397i \(0.417248\pi\)
\(930\) 0 0
\(931\) 21.4955 0.704485
\(932\) 0 0
\(933\) −3.00000 −0.0982156
\(934\) 0 0
\(935\) 15.0000 0.490552
\(936\) 0 0
\(937\) −18.5826 −0.607066 −0.303533 0.952821i \(-0.598166\pi\)
−0.303533 + 0.952821i \(0.598166\pi\)
\(938\) 0 0
\(939\) −10.5826 −0.345349
\(940\) 0 0
\(941\) 19.9129 0.649141 0.324571 0.945861i \(-0.394780\pi\)
0.324571 + 0.945861i \(0.394780\pi\)
\(942\) 0 0
\(943\) −36.6606 −1.19383
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) 54.9129 1.78443 0.892214 0.451612i \(-0.149151\pi\)
0.892214 + 0.451612i \(0.149151\pi\)
\(948\) 0 0
\(949\) −18.3303 −0.595027
\(950\) 0 0
\(951\) 25.0000 0.810681
\(952\) 0 0
\(953\) −37.5826 −1.21742 −0.608710 0.793393i \(-0.708313\pi\)
−0.608710 + 0.793393i \(0.708313\pi\)
\(954\) 0 0
\(955\) 4.00000 0.129437
\(956\) 0 0
\(957\) 5.00000 0.161627
\(958\) 0 0
\(959\) −16.3303 −0.527333
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −5.16515 −0.166445
\(964\) 0 0
\(965\) −20.3303 −0.654456
\(966\) 0 0
\(967\) −9.25227 −0.297533 −0.148767 0.988872i \(-0.547530\pi\)
−0.148767 + 0.988872i \(0.547530\pi\)
\(968\) 0 0
\(969\) −10.7477 −0.345267
\(970\) 0 0
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) −9.41742 −0.301909
\(974\) 0 0
\(975\) 4.58258 0.146760
\(976\) 0 0
\(977\) −2.50455 −0.0801275 −0.0400638 0.999197i \(-0.512756\pi\)
−0.0400638 + 0.999197i \(0.512756\pi\)
\(978\) 0 0
\(979\) −7.08712 −0.226505
\(980\) 0 0
\(981\) 14.1652 0.452258
\(982\) 0 0
\(983\) −55.1652 −1.75950 −0.879748 0.475441i \(-0.842288\pi\)
−0.879748 + 0.475441i \(0.842288\pi\)
\(984\) 0 0
\(985\) −16.3303 −0.520327
\(986\) 0 0
\(987\) −10.5826 −0.336847
\(988\) 0 0
\(989\) 38.3303 1.21883
\(990\) 0 0
\(991\) −16.0780 −0.510735 −0.255368 0.966844i \(-0.582197\pi\)
−0.255368 + 0.966844i \(0.582197\pi\)
\(992\) 0 0
\(993\) −8.33030 −0.264354
\(994\) 0 0
\(995\) −13.4174 −0.425361
\(996\) 0 0
\(997\) −0.834849 −0.0264399 −0.0132200 0.999913i \(-0.504208\pi\)
−0.0132200 + 0.999913i \(0.504208\pi\)
\(998\) 0 0
\(999\) 4.00000 0.126554
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 6960.2.a.bw.1.1 2
4.3 odd 2 435.2.a.f.1.2 2
12.11 even 2 1305.2.a.m.1.1 2
20.3 even 4 2175.2.c.f.349.2 4
20.7 even 4 2175.2.c.f.349.3 4
20.19 odd 2 2175.2.a.r.1.1 2
60.59 even 2 6525.2.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.f.1.2 2 4.3 odd 2
1305.2.a.m.1.1 2 12.11 even 2
2175.2.a.r.1.1 2 20.19 odd 2
2175.2.c.f.349.2 4 20.3 even 4
2175.2.c.f.349.3 4 20.7 even 4
6525.2.a.t.1.2 2 60.59 even 2
6960.2.a.bw.1.1 2 1.1 even 1 trivial