Properties

Label 2320.2.a.m.1.2
Level $2320$
Weight $2$
Character 2320.1
Self dual yes
Analytic conductor $18.525$
Analytic rank $0$
Dimension $3$
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] = [2320,2,Mod(1,2320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2320, 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("2320.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.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: \(18.5252932689\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 580)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.08613\) of defining polynomial
Character \(\chi\) \(=\) 2320.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.35194 q^{3} -1.00000 q^{5} +0.648061 q^{7} -1.17226 q^{9} +O(q^{10})\) \(q-1.35194 q^{3} -1.00000 q^{5} +0.648061 q^{7} -1.17226 q^{9} -3.35194 q^{11} +4.17226 q^{13} +1.35194 q^{15} +4.82032 q^{17} -6.82032 q^{19} -0.876139 q^{21} -5.52420 q^{23} +1.00000 q^{25} +5.64064 q^{27} -1.00000 q^{29} +2.82032 q^{31} +4.53162 q^{33} -0.648061 q^{35} -10.2281 q^{37} -5.64064 q^{39} +8.17226 q^{41} +5.69646 q^{43} +1.17226 q^{45} +2.64806 q^{47} -6.58002 q^{49} -6.51678 q^{51} -2.87614 q^{53} +3.35194 q^{55} +9.22066 q^{57} +13.2207 q^{59} -1.12386 q^{61} -0.759696 q^{63} -4.17226 q^{65} -1.52420 q^{67} +7.46838 q^{69} +8.87614 q^{71} +9.69646 q^{73} -1.35194 q^{75} -2.17226 q^{77} -8.99258 q^{79} -4.10902 q^{81} -1.94418 q^{83} -4.82032 q^{85} +1.35194 q^{87} +17.0484 q^{89} +2.70388 q^{91} -3.81290 q^{93} +6.82032 q^{95} -13.3519 q^{97} +3.92935 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 11 q^{9} - 8 q^{11} - 2 q^{13} + 2 q^{15} + 2 q^{17} - 8 q^{19} + 16 q^{21} + 3 q^{25} - 8 q^{27} - 3 q^{29} - 4 q^{31} + 24 q^{33} - 4 q^{35} - 10 q^{37} + 8 q^{39} + 10 q^{41} - 14 q^{43} - 11 q^{45} + 10 q^{47} + 3 q^{49} + 24 q^{51} + 10 q^{53} + 8 q^{55} - 20 q^{57} - 8 q^{59} - 22 q^{61} + 8 q^{63} + 2 q^{65} + 12 q^{67} + 12 q^{69} + 8 q^{71} - 2 q^{73} - 2 q^{75} + 8 q^{77} + 23 q^{81} - 12 q^{83} - 2 q^{85} + 2 q^{87} + 18 q^{89} + 4 q^{91} + 28 q^{93} + 8 q^{95} - 38 q^{97} - 36 q^{99}+O(q^{100}) \) 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
\(3\) −1.35194 −0.780542 −0.390271 0.920700i \(-0.627619\pi\)
−0.390271 + 0.920700i \(0.627619\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.648061 0.244944 0.122472 0.992472i \(-0.460918\pi\)
0.122472 + 0.992472i \(0.460918\pi\)
\(8\) 0 0
\(9\) −1.17226 −0.390753
\(10\) 0 0
\(11\) −3.35194 −1.01065 −0.505324 0.862930i \(-0.668627\pi\)
−0.505324 + 0.862930i \(0.668627\pi\)
\(12\) 0 0
\(13\) 4.17226 1.15718 0.578588 0.815620i \(-0.303604\pi\)
0.578588 + 0.815620i \(0.303604\pi\)
\(14\) 0 0
\(15\) 1.35194 0.349069
\(16\) 0 0
\(17\) 4.82032 1.16910 0.584550 0.811358i \(-0.301271\pi\)
0.584550 + 0.811358i \(0.301271\pi\)
\(18\) 0 0
\(19\) −6.82032 −1.56469 −0.782344 0.622846i \(-0.785976\pi\)
−0.782344 + 0.622846i \(0.785976\pi\)
\(20\) 0 0
\(21\) −0.876139 −0.191189
\(22\) 0 0
\(23\) −5.52420 −1.15188 −0.575938 0.817494i \(-0.695363\pi\)
−0.575938 + 0.817494i \(0.695363\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.64064 1.08554
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 2.82032 0.506545 0.253272 0.967395i \(-0.418493\pi\)
0.253272 + 0.967395i \(0.418493\pi\)
\(32\) 0 0
\(33\) 4.53162 0.788853
\(34\) 0 0
\(35\) −0.648061 −0.109542
\(36\) 0 0
\(37\) −10.2281 −1.68149 −0.840743 0.541435i \(-0.817881\pi\)
−0.840743 + 0.541435i \(0.817881\pi\)
\(38\) 0 0
\(39\) −5.64064 −0.903226
\(40\) 0 0
\(41\) 8.17226 1.27629 0.638146 0.769915i \(-0.279701\pi\)
0.638146 + 0.769915i \(0.279701\pi\)
\(42\) 0 0
\(43\) 5.69646 0.868702 0.434351 0.900744i \(-0.356978\pi\)
0.434351 + 0.900744i \(0.356978\pi\)
\(44\) 0 0
\(45\) 1.17226 0.174750
\(46\) 0 0
\(47\) 2.64806 0.386259 0.193130 0.981173i \(-0.438136\pi\)
0.193130 + 0.981173i \(0.438136\pi\)
\(48\) 0 0
\(49\) −6.58002 −0.940002
\(50\) 0 0
\(51\) −6.51678 −0.912532
\(52\) 0 0
\(53\) −2.87614 −0.395068 −0.197534 0.980296i \(-0.563293\pi\)
−0.197534 + 0.980296i \(0.563293\pi\)
\(54\) 0 0
\(55\) 3.35194 0.451975
\(56\) 0 0
\(57\) 9.22066 1.22131
\(58\) 0 0
\(59\) 13.2207 1.72118 0.860592 0.509296i \(-0.170094\pi\)
0.860592 + 0.509296i \(0.170094\pi\)
\(60\) 0 0
\(61\) −1.12386 −0.143896 −0.0719478 0.997408i \(-0.522922\pi\)
−0.0719478 + 0.997408i \(0.522922\pi\)
\(62\) 0 0
\(63\) −0.759696 −0.0957127
\(64\) 0 0
\(65\) −4.17226 −0.517505
\(66\) 0 0
\(67\) −1.52420 −0.186211 −0.0931053 0.995656i \(-0.529679\pi\)
−0.0931053 + 0.995656i \(0.529679\pi\)
\(68\) 0 0
\(69\) 7.46838 0.899088
\(70\) 0 0
\(71\) 8.87614 1.05340 0.526702 0.850050i \(-0.323428\pi\)
0.526702 + 0.850050i \(0.323428\pi\)
\(72\) 0 0
\(73\) 9.69646 1.13488 0.567442 0.823413i \(-0.307933\pi\)
0.567442 + 0.823413i \(0.307933\pi\)
\(74\) 0 0
\(75\) −1.35194 −0.156108
\(76\) 0 0
\(77\) −2.17226 −0.247552
\(78\) 0 0
\(79\) −8.99258 −1.01174 −0.505872 0.862608i \(-0.668829\pi\)
−0.505872 + 0.862608i \(0.668829\pi\)
\(80\) 0 0
\(81\) −4.10902 −0.456558
\(82\) 0 0
\(83\) −1.94418 −0.213402 −0.106701 0.994291i \(-0.534029\pi\)
−0.106701 + 0.994291i \(0.534029\pi\)
\(84\) 0 0
\(85\) −4.82032 −0.522837
\(86\) 0 0
\(87\) 1.35194 0.144943
\(88\) 0 0
\(89\) 17.0484 1.80713 0.903563 0.428455i \(-0.140942\pi\)
0.903563 + 0.428455i \(0.140942\pi\)
\(90\) 0 0
\(91\) 2.70388 0.283443
\(92\) 0 0
\(93\) −3.81290 −0.395380
\(94\) 0 0
\(95\) 6.82032 0.699750
\(96\) 0 0
\(97\) −13.3519 −1.35568 −0.677842 0.735208i \(-0.737085\pi\)
−0.677842 + 0.735208i \(0.737085\pi\)
\(98\) 0 0
\(99\) 3.92935 0.394914
\(100\) 0 0
\(101\) 18.4562 1.83646 0.918228 0.396052i \(-0.129620\pi\)
0.918228 + 0.396052i \(0.129620\pi\)
\(102\) 0 0
\(103\) 14.8203 1.46029 0.730145 0.683292i \(-0.239453\pi\)
0.730145 + 0.683292i \(0.239453\pi\)
\(104\) 0 0
\(105\) 0.876139 0.0855024
\(106\) 0 0
\(107\) 3.35194 0.324044 0.162022 0.986787i \(-0.448198\pi\)
0.162022 + 0.986787i \(0.448198\pi\)
\(108\) 0 0
\(109\) 2.76450 0.264791 0.132396 0.991197i \(-0.457733\pi\)
0.132396 + 0.991197i \(0.457733\pi\)
\(110\) 0 0
\(111\) 13.8277 1.31247
\(112\) 0 0
\(113\) 7.94418 0.747326 0.373663 0.927565i \(-0.378102\pi\)
0.373663 + 0.927565i \(0.378102\pi\)
\(114\) 0 0
\(115\) 5.52420 0.515134
\(116\) 0 0
\(117\) −4.89098 −0.452171
\(118\) 0 0
\(119\) 3.12386 0.286364
\(120\) 0 0
\(121\) 0.235496 0.0214088
\(122\) 0 0
\(123\) −11.0484 −0.996201
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 17.6965 1.57031 0.785153 0.619301i \(-0.212584\pi\)
0.785153 + 0.619301i \(0.212584\pi\)
\(128\) 0 0
\(129\) −7.70127 −0.678059
\(130\) 0 0
\(131\) −5.52420 −0.482652 −0.241326 0.970444i \(-0.577582\pi\)
−0.241326 + 0.970444i \(0.577582\pi\)
\(132\) 0 0
\(133\) −4.41998 −0.383261
\(134\) 0 0
\(135\) −5.64064 −0.485469
\(136\) 0 0
\(137\) −3.86872 −0.330527 −0.165264 0.986249i \(-0.552847\pi\)
−0.165264 + 0.986249i \(0.552847\pi\)
\(138\) 0 0
\(139\) −5.64064 −0.478433 −0.239217 0.970966i \(-0.576891\pi\)
−0.239217 + 0.970966i \(0.576891\pi\)
\(140\) 0 0
\(141\) −3.58002 −0.301492
\(142\) 0 0
\(143\) −13.9852 −1.16950
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) 8.89578 0.733712
\(148\) 0 0
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 17.2207 1.40140 0.700699 0.713457i \(-0.252872\pi\)
0.700699 + 0.713457i \(0.252872\pi\)
\(152\) 0 0
\(153\) −5.65067 −0.456830
\(154\) 0 0
\(155\) −2.82032 −0.226534
\(156\) 0 0
\(157\) −5.69646 −0.454627 −0.227314 0.973822i \(-0.572994\pi\)
−0.227314 + 0.973822i \(0.572994\pi\)
\(158\) 0 0
\(159\) 3.88836 0.308367
\(160\) 0 0
\(161\) −3.58002 −0.282145
\(162\) 0 0
\(163\) −16.4003 −1.28457 −0.642287 0.766464i \(-0.722014\pi\)
−0.642287 + 0.766464i \(0.722014\pi\)
\(164\) 0 0
\(165\) −4.53162 −0.352786
\(166\) 0 0
\(167\) 23.2765 1.80119 0.900594 0.434661i \(-0.143132\pi\)
0.900594 + 0.434661i \(0.143132\pi\)
\(168\) 0 0
\(169\) 4.40776 0.339058
\(170\) 0 0
\(171\) 7.99519 0.611408
\(172\) 0 0
\(173\) −13.9245 −1.05866 −0.529332 0.848415i \(-0.677557\pi\)
−0.529332 + 0.848415i \(0.677557\pi\)
\(174\) 0 0
\(175\) 0.648061 0.0489888
\(176\) 0 0
\(177\) −17.8735 −1.34346
\(178\) 0 0
\(179\) 4.34452 0.324725 0.162362 0.986731i \(-0.448089\pi\)
0.162362 + 0.986731i \(0.448089\pi\)
\(180\) 0 0
\(181\) 20.5168 1.52500 0.762500 0.646988i \(-0.223972\pi\)
0.762500 + 0.646988i \(0.223972\pi\)
\(182\) 0 0
\(183\) 1.51939 0.112317
\(184\) 0 0
\(185\) 10.2281 0.751983
\(186\) 0 0
\(187\) −16.1574 −1.18155
\(188\) 0 0
\(189\) 3.65548 0.265897
\(190\) 0 0
\(191\) 9.10422 0.658758 0.329379 0.944198i \(-0.393161\pi\)
0.329379 + 0.944198i \(0.393161\pi\)
\(192\) 0 0
\(193\) −25.1648 −1.81140 −0.905702 0.423914i \(-0.860656\pi\)
−0.905702 + 0.423914i \(0.860656\pi\)
\(194\) 0 0
\(195\) 5.64064 0.403935
\(196\) 0 0
\(197\) −0.703878 −0.0501493 −0.0250746 0.999686i \(-0.507982\pi\)
−0.0250746 + 0.999686i \(0.507982\pi\)
\(198\) 0 0
\(199\) 5.22066 0.370083 0.185041 0.982731i \(-0.440758\pi\)
0.185041 + 0.982731i \(0.440758\pi\)
\(200\) 0 0
\(201\) 2.06063 0.145345
\(202\) 0 0
\(203\) −0.648061 −0.0454850
\(204\) 0 0
\(205\) −8.17226 −0.570775
\(206\) 0 0
\(207\) 6.47580 0.450099
\(208\) 0 0
\(209\) 22.8613 1.58135
\(210\) 0 0
\(211\) 13.1042 0.902131 0.451066 0.892491i \(-0.351044\pi\)
0.451066 + 0.892491i \(0.351044\pi\)
\(212\) 0 0
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) −5.69646 −0.388495
\(216\) 0 0
\(217\) 1.82774 0.124075
\(218\) 0 0
\(219\) −13.1090 −0.885826
\(220\) 0 0
\(221\) 20.1116 1.35285
\(222\) 0 0
\(223\) 20.5726 1.37764 0.688822 0.724931i \(-0.258128\pi\)
0.688822 + 0.724931i \(0.258128\pi\)
\(224\) 0 0
\(225\) −1.17226 −0.0781507
\(226\) 0 0
\(227\) 9.41256 0.624734 0.312367 0.949962i \(-0.398878\pi\)
0.312367 + 0.949962i \(0.398878\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 2.93676 0.193225
\(232\) 0 0
\(233\) 10.3445 0.677692 0.338846 0.940842i \(-0.389963\pi\)
0.338846 + 0.940842i \(0.389963\pi\)
\(234\) 0 0
\(235\) −2.64806 −0.172740
\(236\) 0 0
\(237\) 12.1574 0.789710
\(238\) 0 0
\(239\) −20.8761 −1.35037 −0.675183 0.737651i \(-0.735935\pi\)
−0.675183 + 0.737651i \(0.735935\pi\)
\(240\) 0 0
\(241\) 21.1600 1.36304 0.681519 0.731801i \(-0.261320\pi\)
0.681519 + 0.731801i \(0.261320\pi\)
\(242\) 0 0
\(243\) −11.3668 −0.729179
\(244\) 0 0
\(245\) 6.58002 0.420382
\(246\) 0 0
\(247\) −28.4562 −1.81062
\(248\) 0 0
\(249\) 2.62842 0.166569
\(250\) 0 0
\(251\) 8.64806 0.545861 0.272930 0.962034i \(-0.412007\pi\)
0.272930 + 0.962034i \(0.412007\pi\)
\(252\) 0 0
\(253\) 18.5168 1.16414
\(254\) 0 0
\(255\) 6.51678 0.408097
\(256\) 0 0
\(257\) 27.4535 1.71251 0.856253 0.516557i \(-0.172787\pi\)
0.856253 + 0.516557i \(0.172787\pi\)
\(258\) 0 0
\(259\) −6.62842 −0.411870
\(260\) 0 0
\(261\) 1.17226 0.0725611
\(262\) 0 0
\(263\) 11.3371 0.699076 0.349538 0.936922i \(-0.386339\pi\)
0.349538 + 0.936922i \(0.386339\pi\)
\(264\) 0 0
\(265\) 2.87614 0.176680
\(266\) 0 0
\(267\) −23.0484 −1.41054
\(268\) 0 0
\(269\) −20.1723 −1.22992 −0.614962 0.788557i \(-0.710829\pi\)
−0.614962 + 0.788557i \(0.710829\pi\)
\(270\) 0 0
\(271\) 10.7449 0.652704 0.326352 0.945248i \(-0.394181\pi\)
0.326352 + 0.945248i \(0.394181\pi\)
\(272\) 0 0
\(273\) −3.65548 −0.221240
\(274\) 0 0
\(275\) −3.35194 −0.202130
\(276\) 0 0
\(277\) −12.6284 −0.758768 −0.379384 0.925239i \(-0.623864\pi\)
−0.379384 + 0.925239i \(0.623864\pi\)
\(278\) 0 0
\(279\) −3.30615 −0.197934
\(280\) 0 0
\(281\) 6.76450 0.403536 0.201768 0.979433i \(-0.435331\pi\)
0.201768 + 0.979433i \(0.435331\pi\)
\(282\) 0 0
\(283\) −1.06804 −0.0634886 −0.0317443 0.999496i \(-0.510106\pi\)
−0.0317443 + 0.999496i \(0.510106\pi\)
\(284\) 0 0
\(285\) −9.22066 −0.546185
\(286\) 0 0
\(287\) 5.29612 0.312620
\(288\) 0 0
\(289\) 6.23550 0.366794
\(290\) 0 0
\(291\) 18.0510 1.05817
\(292\) 0 0
\(293\) 5.46357 0.319185 0.159593 0.987183i \(-0.448982\pi\)
0.159593 + 0.987183i \(0.448982\pi\)
\(294\) 0 0
\(295\) −13.2207 −0.769737
\(296\) 0 0
\(297\) −18.9071 −1.09710
\(298\) 0 0
\(299\) −23.0484 −1.33292
\(300\) 0 0
\(301\) 3.69165 0.212783
\(302\) 0 0
\(303\) −24.9516 −1.43343
\(304\) 0 0
\(305\) 1.12386 0.0643521
\(306\) 0 0
\(307\) −6.30354 −0.359762 −0.179881 0.983688i \(-0.557571\pi\)
−0.179881 + 0.983688i \(0.557571\pi\)
\(308\) 0 0
\(309\) −20.0362 −1.13982
\(310\) 0 0
\(311\) −24.6842 −1.39971 −0.699857 0.714283i \(-0.746753\pi\)
−0.699857 + 0.714283i \(0.746753\pi\)
\(312\) 0 0
\(313\) −24.1723 −1.36630 −0.683148 0.730280i \(-0.739390\pi\)
−0.683148 + 0.730280i \(0.739390\pi\)
\(314\) 0 0
\(315\) 0.759696 0.0428040
\(316\) 0 0
\(317\) 1.42740 0.0801708 0.0400854 0.999196i \(-0.487237\pi\)
0.0400854 + 0.999196i \(0.487237\pi\)
\(318\) 0 0
\(319\) 3.35194 0.187673
\(320\) 0 0
\(321\) −4.53162 −0.252930
\(322\) 0 0
\(323\) −32.8761 −1.82928
\(324\) 0 0
\(325\) 4.17226 0.231435
\(326\) 0 0
\(327\) −3.73744 −0.206681
\(328\) 0 0
\(329\) 1.71610 0.0946119
\(330\) 0 0
\(331\) −34.2132 −1.88053 −0.940265 0.340444i \(-0.889423\pi\)
−0.940265 + 0.340444i \(0.889423\pi\)
\(332\) 0 0
\(333\) 11.9900 0.657046
\(334\) 0 0
\(335\) 1.52420 0.0832759
\(336\) 0 0
\(337\) 4.40034 0.239702 0.119851 0.992792i \(-0.461758\pi\)
0.119851 + 0.992792i \(0.461758\pi\)
\(338\) 0 0
\(339\) −10.7401 −0.583320
\(340\) 0 0
\(341\) −9.45355 −0.511938
\(342\) 0 0
\(343\) −8.80068 −0.475192
\(344\) 0 0
\(345\) −7.46838 −0.402084
\(346\) 0 0
\(347\) −12.9926 −0.697478 −0.348739 0.937220i \(-0.613390\pi\)
−0.348739 + 0.937220i \(0.613390\pi\)
\(348\) 0 0
\(349\) −20.0968 −1.07576 −0.537878 0.843022i \(-0.680774\pi\)
−0.537878 + 0.843022i \(0.680774\pi\)
\(350\) 0 0
\(351\) 23.5342 1.25616
\(352\) 0 0
\(353\) −17.8129 −0.948085 −0.474043 0.880502i \(-0.657206\pi\)
−0.474043 + 0.880502i \(0.657206\pi\)
\(354\) 0 0
\(355\) −8.87614 −0.471097
\(356\) 0 0
\(357\) −4.22327 −0.223519
\(358\) 0 0
\(359\) −5.52420 −0.291556 −0.145778 0.989317i \(-0.546569\pi\)
−0.145778 + 0.989317i \(0.546569\pi\)
\(360\) 0 0
\(361\) 27.5168 1.44825
\(362\) 0 0
\(363\) −0.318377 −0.0167104
\(364\) 0 0
\(365\) −9.69646 −0.507536
\(366\) 0 0
\(367\) 20.0558 1.04691 0.523453 0.852055i \(-0.324644\pi\)
0.523453 + 0.852055i \(0.324644\pi\)
\(368\) 0 0
\(369\) −9.58002 −0.498716
\(370\) 0 0
\(371\) −1.86391 −0.0967695
\(372\) 0 0
\(373\) −1.88836 −0.0977758 −0.0488879 0.998804i \(-0.515568\pi\)
−0.0488879 + 0.998804i \(0.515568\pi\)
\(374\) 0 0
\(375\) 1.35194 0.0698138
\(376\) 0 0
\(377\) −4.17226 −0.214882
\(378\) 0 0
\(379\) 15.0894 0.775089 0.387545 0.921851i \(-0.373323\pi\)
0.387545 + 0.921851i \(0.373323\pi\)
\(380\) 0 0
\(381\) −23.9245 −1.22569
\(382\) 0 0
\(383\) −17.3371 −0.885885 −0.442942 0.896550i \(-0.646065\pi\)
−0.442942 + 0.896550i \(0.646065\pi\)
\(384\) 0 0
\(385\) 2.17226 0.110709
\(386\) 0 0
\(387\) −6.67773 −0.339448
\(388\) 0 0
\(389\) −0.890976 −0.0451743 −0.0225871 0.999745i \(-0.507190\pi\)
−0.0225871 + 0.999745i \(0.507190\pi\)
\(390\) 0 0
\(391\) −26.6284 −1.34666
\(392\) 0 0
\(393\) 7.46838 0.376730
\(394\) 0 0
\(395\) 8.99258 0.452466
\(396\) 0 0
\(397\) 5.39292 0.270663 0.135331 0.990800i \(-0.456790\pi\)
0.135331 + 0.990800i \(0.456790\pi\)
\(398\) 0 0
\(399\) 5.97555 0.299152
\(400\) 0 0
\(401\) 5.23550 0.261448 0.130724 0.991419i \(-0.458270\pi\)
0.130724 + 0.991419i \(0.458270\pi\)
\(402\) 0 0
\(403\) 11.7671 0.586162
\(404\) 0 0
\(405\) 4.10902 0.204179
\(406\) 0 0
\(407\) 34.2839 1.69939
\(408\) 0 0
\(409\) −16.0606 −0.794147 −0.397073 0.917787i \(-0.629974\pi\)
−0.397073 + 0.917787i \(0.629974\pi\)
\(410\) 0 0
\(411\) 5.23027 0.257990
\(412\) 0 0
\(413\) 8.56779 0.421593
\(414\) 0 0
\(415\) 1.94418 0.0954362
\(416\) 0 0
\(417\) 7.62581 0.373437
\(418\) 0 0
\(419\) 19.9245 0.973377 0.486689 0.873575i \(-0.338205\pi\)
0.486689 + 0.873575i \(0.338205\pi\)
\(420\) 0 0
\(421\) −15.5652 −0.758600 −0.379300 0.925274i \(-0.623835\pi\)
−0.379300 + 0.925274i \(0.623835\pi\)
\(422\) 0 0
\(423\) −3.10422 −0.150932
\(424\) 0 0
\(425\) 4.82032 0.233820
\(426\) 0 0
\(427\) −0.728330 −0.0352464
\(428\) 0 0
\(429\) 18.9071 0.912843
\(430\) 0 0
\(431\) 13.4078 0.645829 0.322914 0.946428i \(-0.395337\pi\)
0.322914 + 0.946428i \(0.395337\pi\)
\(432\) 0 0
\(433\) −34.6842 −1.66682 −0.833409 0.552657i \(-0.813614\pi\)
−0.833409 + 0.552657i \(0.813614\pi\)
\(434\) 0 0
\(435\) −1.35194 −0.0648205
\(436\) 0 0
\(437\) 37.6768 1.80233
\(438\) 0 0
\(439\) 15.2355 0.727151 0.363575 0.931565i \(-0.381556\pi\)
0.363575 + 0.931565i \(0.381556\pi\)
\(440\) 0 0
\(441\) 7.71349 0.367309
\(442\) 0 0
\(443\) −5.00742 −0.237910 −0.118955 0.992900i \(-0.537954\pi\)
−0.118955 + 0.992900i \(0.537954\pi\)
\(444\) 0 0
\(445\) −17.0484 −0.808172
\(446\) 0 0
\(447\) −24.3349 −1.15100
\(448\) 0 0
\(449\) −17.8129 −0.840643 −0.420321 0.907375i \(-0.638083\pi\)
−0.420321 + 0.907375i \(0.638083\pi\)
\(450\) 0 0
\(451\) −27.3929 −1.28988
\(452\) 0 0
\(453\) −23.2813 −1.09385
\(454\) 0 0
\(455\) −2.70388 −0.126760
\(456\) 0 0
\(457\) 24.4051 1.14162 0.570812 0.821081i \(-0.306628\pi\)
0.570812 + 0.821081i \(0.306628\pi\)
\(458\) 0 0
\(459\) 27.1897 1.26911
\(460\) 0 0
\(461\) −7.98516 −0.371906 −0.185953 0.982559i \(-0.559537\pi\)
−0.185953 + 0.982559i \(0.559537\pi\)
\(462\) 0 0
\(463\) −11.6210 −0.540074 −0.270037 0.962850i \(-0.587036\pi\)
−0.270037 + 0.962850i \(0.587036\pi\)
\(464\) 0 0
\(465\) 3.81290 0.176819
\(466\) 0 0
\(467\) −1.24030 −0.0573944 −0.0286972 0.999588i \(-0.509136\pi\)
−0.0286972 + 0.999588i \(0.509136\pi\)
\(468\) 0 0
\(469\) −0.987774 −0.0456112
\(470\) 0 0
\(471\) 7.70127 0.354856
\(472\) 0 0
\(473\) −19.0942 −0.877952
\(474\) 0 0
\(475\) −6.82032 −0.312938
\(476\) 0 0
\(477\) 3.37158 0.154374
\(478\) 0 0
\(479\) −19.6965 −0.899954 −0.449977 0.893040i \(-0.648568\pi\)
−0.449977 + 0.893040i \(0.648568\pi\)
\(480\) 0 0
\(481\) −42.6742 −1.94578
\(482\) 0 0
\(483\) 4.83997 0.220226
\(484\) 0 0
\(485\) 13.3519 0.606280
\(486\) 0 0
\(487\) 15.0894 0.683765 0.341883 0.939743i \(-0.388936\pi\)
0.341883 + 0.939743i \(0.388936\pi\)
\(488\) 0 0
\(489\) 22.1723 1.00266
\(490\) 0 0
\(491\) 12.2281 0.551845 0.275923 0.961180i \(-0.411017\pi\)
0.275923 + 0.961180i \(0.411017\pi\)
\(492\) 0 0
\(493\) −4.82032 −0.217096
\(494\) 0 0
\(495\) −3.92935 −0.176611
\(496\) 0 0
\(497\) 5.75228 0.258025
\(498\) 0 0
\(499\) 5.52901 0.247512 0.123756 0.992313i \(-0.460506\pi\)
0.123756 + 0.992313i \(0.460506\pi\)
\(500\) 0 0
\(501\) −31.4684 −1.40590
\(502\) 0 0
\(503\) −24.2887 −1.08298 −0.541490 0.840707i \(-0.682140\pi\)
−0.541490 + 0.840707i \(0.682140\pi\)
\(504\) 0 0
\(505\) −18.4562 −0.821288
\(506\) 0 0
\(507\) −5.95902 −0.264649
\(508\) 0 0
\(509\) −9.58002 −0.424627 −0.212313 0.977202i \(-0.568100\pi\)
−0.212313 + 0.977202i \(0.568100\pi\)
\(510\) 0 0
\(511\) 6.28390 0.277983
\(512\) 0 0
\(513\) −38.4710 −1.69854
\(514\) 0 0
\(515\) −14.8203 −0.653061
\(516\) 0 0
\(517\) −8.87614 −0.390372
\(518\) 0 0
\(519\) 18.8251 0.826331
\(520\) 0 0
\(521\) −21.5800 −0.945438 −0.472719 0.881213i \(-0.656727\pi\)
−0.472719 + 0.881213i \(0.656727\pi\)
\(522\) 0 0
\(523\) 42.2132 1.84586 0.922928 0.384972i \(-0.125789\pi\)
0.922928 + 0.384972i \(0.125789\pi\)
\(524\) 0 0
\(525\) −0.876139 −0.0382378
\(526\) 0 0
\(527\) 13.5949 0.592201
\(528\) 0 0
\(529\) 7.51678 0.326817
\(530\) 0 0
\(531\) −15.4981 −0.672558
\(532\) 0 0
\(533\) 34.0968 1.47690
\(534\) 0 0
\(535\) −3.35194 −0.144917
\(536\) 0 0
\(537\) −5.87353 −0.253461
\(538\) 0 0
\(539\) 22.0558 0.950011
\(540\) 0 0
\(541\) 36.5530 1.57153 0.785767 0.618522i \(-0.212268\pi\)
0.785767 + 0.618522i \(0.212268\pi\)
\(542\) 0 0
\(543\) −27.7374 −1.19033
\(544\) 0 0
\(545\) −2.76450 −0.118418
\(546\) 0 0
\(547\) −26.6236 −1.13834 −0.569172 0.822219i \(-0.692736\pi\)
−0.569172 + 0.822219i \(0.692736\pi\)
\(548\) 0 0
\(549\) 1.31746 0.0562277
\(550\) 0 0
\(551\) 6.82032 0.290555
\(552\) 0 0
\(553\) −5.82774 −0.247821
\(554\) 0 0
\(555\) −13.8277 −0.586955
\(556\) 0 0
\(557\) 22.2233 0.941630 0.470815 0.882232i \(-0.343960\pi\)
0.470815 + 0.882232i \(0.343960\pi\)
\(558\) 0 0
\(559\) 23.7671 1.00524
\(560\) 0 0
\(561\) 21.8439 0.922248
\(562\) 0 0
\(563\) −1.11905 −0.0471625 −0.0235812 0.999722i \(-0.507507\pi\)
−0.0235812 + 0.999722i \(0.507507\pi\)
\(564\) 0 0
\(565\) −7.94418 −0.334214
\(566\) 0 0
\(567\) −2.66290 −0.111831
\(568\) 0 0
\(569\) −10.5316 −0.441508 −0.220754 0.975329i \(-0.570852\pi\)
−0.220754 + 0.975329i \(0.570852\pi\)
\(570\) 0 0
\(571\) 1.18449 0.0495692 0.0247846 0.999693i \(-0.492110\pi\)
0.0247846 + 0.999693i \(0.492110\pi\)
\(572\) 0 0
\(573\) −12.3083 −0.514189
\(574\) 0 0
\(575\) −5.52420 −0.230375
\(576\) 0 0
\(577\) 11.8687 0.494101 0.247051 0.969003i \(-0.420539\pi\)
0.247051 + 0.969003i \(0.420539\pi\)
\(578\) 0 0
\(579\) 34.0213 1.41388
\(580\) 0 0
\(581\) −1.25995 −0.0522715
\(582\) 0 0
\(583\) 9.64064 0.399275
\(584\) 0 0
\(585\) 4.89098 0.202217
\(586\) 0 0
\(587\) −32.4610 −1.33981 −0.669904 0.742448i \(-0.733665\pi\)
−0.669904 + 0.742448i \(0.733665\pi\)
\(588\) 0 0
\(589\) −19.2355 −0.792585
\(590\) 0 0
\(591\) 0.951601 0.0391436
\(592\) 0 0
\(593\) 9.12386 0.374672 0.187336 0.982296i \(-0.440015\pi\)
0.187336 + 0.982296i \(0.440015\pi\)
\(594\) 0 0
\(595\) −3.12386 −0.128066
\(596\) 0 0
\(597\) −7.05801 −0.288865
\(598\) 0 0
\(599\) 30.2887 1.23756 0.618781 0.785563i \(-0.287627\pi\)
0.618781 + 0.785563i \(0.287627\pi\)
\(600\) 0 0
\(601\) 20.5168 0.836897 0.418448 0.908241i \(-0.362574\pi\)
0.418448 + 0.908241i \(0.362574\pi\)
\(602\) 0 0
\(603\) 1.78676 0.0727624
\(604\) 0 0
\(605\) −0.235496 −0.00957429
\(606\) 0 0
\(607\) 13.9293 0.565375 0.282687 0.959212i \(-0.408774\pi\)
0.282687 + 0.959212i \(0.408774\pi\)
\(608\) 0 0
\(609\) 0.876139 0.0355029
\(610\) 0 0
\(611\) 11.0484 0.446970
\(612\) 0 0
\(613\) 19.7523 0.797787 0.398893 0.916997i \(-0.369394\pi\)
0.398893 + 0.916997i \(0.369394\pi\)
\(614\) 0 0
\(615\) 11.0484 0.445514
\(616\) 0 0
\(617\) −37.2010 −1.49766 −0.748828 0.662764i \(-0.769383\pi\)
−0.748828 + 0.662764i \(0.769383\pi\)
\(618\) 0 0
\(619\) −7.46357 −0.299986 −0.149993 0.988687i \(-0.547925\pi\)
−0.149993 + 0.988687i \(0.547925\pi\)
\(620\) 0 0
\(621\) −31.1600 −1.25041
\(622\) 0 0
\(623\) 11.0484 0.442645
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −30.9071 −1.23431
\(628\) 0 0
\(629\) −49.3026 −1.96582
\(630\) 0 0
\(631\) −44.4562 −1.76977 −0.884886 0.465808i \(-0.845764\pi\)
−0.884886 + 0.465808i \(0.845764\pi\)
\(632\) 0 0
\(633\) −17.7161 −0.704152
\(634\) 0 0
\(635\) −17.6965 −0.702263
\(636\) 0 0
\(637\) −27.4535 −1.08775
\(638\) 0 0
\(639\) −10.4051 −0.411621
\(640\) 0 0
\(641\) −40.7401 −1.60914 −0.804568 0.593861i \(-0.797603\pi\)
−0.804568 + 0.593861i \(0.797603\pi\)
\(642\) 0 0
\(643\) −20.3493 −0.802499 −0.401250 0.915969i \(-0.631424\pi\)
−0.401250 + 0.915969i \(0.631424\pi\)
\(644\) 0 0
\(645\) 7.70127 0.303237
\(646\) 0 0
\(647\) −41.7933 −1.64306 −0.821531 0.570163i \(-0.806880\pi\)
−0.821531 + 0.570163i \(0.806880\pi\)
\(648\) 0 0
\(649\) −44.3148 −1.73951
\(650\) 0 0
\(651\) −2.47099 −0.0968458
\(652\) 0 0
\(653\) 23.7933 0.931102 0.465551 0.885021i \(-0.345856\pi\)
0.465551 + 0.885021i \(0.345856\pi\)
\(654\) 0 0
\(655\) 5.52420 0.215848
\(656\) 0 0
\(657\) −11.3668 −0.443460
\(658\) 0 0
\(659\) −7.50936 −0.292523 −0.146262 0.989246i \(-0.546724\pi\)
−0.146262 + 0.989246i \(0.546724\pi\)
\(660\) 0 0
\(661\) 45.9245 1.78626 0.893129 0.449801i \(-0.148505\pi\)
0.893129 + 0.449801i \(0.148505\pi\)
\(662\) 0 0
\(663\) −27.1897 −1.05596
\(664\) 0 0
\(665\) 4.41998 0.171400
\(666\) 0 0
\(667\) 5.52420 0.213898
\(668\) 0 0
\(669\) −27.8129 −1.07531
\(670\) 0 0
\(671\) 3.76711 0.145428
\(672\) 0 0
\(673\) 7.29612 0.281245 0.140622 0.990063i \(-0.455090\pi\)
0.140622 + 0.990063i \(0.455090\pi\)
\(674\) 0 0
\(675\) 5.64064 0.217108
\(676\) 0 0
\(677\) 36.2494 1.39318 0.696589 0.717470i \(-0.254700\pi\)
0.696589 + 0.717470i \(0.254700\pi\)
\(678\) 0 0
\(679\) −8.65287 −0.332067
\(680\) 0 0
\(681\) −12.7252 −0.487631
\(682\) 0 0
\(683\) 5.40295 0.206738 0.103369 0.994643i \(-0.467038\pi\)
0.103369 + 0.994643i \(0.467038\pi\)
\(684\) 0 0
\(685\) 3.86872 0.147816
\(686\) 0 0
\(687\) −2.70388 −0.103159
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 36.0458 1.37125 0.685623 0.727957i \(-0.259530\pi\)
0.685623 + 0.727957i \(0.259530\pi\)
\(692\) 0 0
\(693\) 2.54645 0.0967318
\(694\) 0 0
\(695\) 5.64064 0.213962
\(696\) 0 0
\(697\) 39.3929 1.49211
\(698\) 0 0
\(699\) −13.9852 −0.528967
\(700\) 0 0
\(701\) 3.33229 0.125859 0.0629295 0.998018i \(-0.479956\pi\)
0.0629295 + 0.998018i \(0.479956\pi\)
\(702\) 0 0
\(703\) 69.7588 2.63100
\(704\) 0 0
\(705\) 3.58002 0.134831
\(706\) 0 0
\(707\) 11.9607 0.449829
\(708\) 0 0
\(709\) 7.48322 0.281038 0.140519 0.990078i \(-0.455123\pi\)
0.140519 + 0.990078i \(0.455123\pi\)
\(710\) 0 0
\(711\) 10.5416 0.395343
\(712\) 0 0
\(713\) −15.5800 −0.583476
\(714\) 0 0
\(715\) 13.9852 0.523015
\(716\) 0 0
\(717\) 28.2233 1.05402
\(718\) 0 0
\(719\) 39.6620 1.47914 0.739571 0.673078i \(-0.235028\pi\)
0.739571 + 0.673078i \(0.235028\pi\)
\(720\) 0 0
\(721\) 9.60447 0.357689
\(722\) 0 0
\(723\) −28.6071 −1.06391
\(724\) 0 0
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) 8.05582 0.298774 0.149387 0.988779i \(-0.452270\pi\)
0.149387 + 0.988779i \(0.452270\pi\)
\(728\) 0 0
\(729\) 27.6943 1.02571
\(730\) 0 0
\(731\) 27.4588 1.01560
\(732\) 0 0
\(733\) −28.9320 −1.06863 −0.534313 0.845287i \(-0.679430\pi\)
−0.534313 + 0.845287i \(0.679430\pi\)
\(734\) 0 0
\(735\) −8.89578 −0.328126
\(736\) 0 0
\(737\) 5.10902 0.188193
\(738\) 0 0
\(739\) −36.0772 −1.32712 −0.663560 0.748123i \(-0.730955\pi\)
−0.663560 + 0.748123i \(0.730955\pi\)
\(740\) 0 0
\(741\) 38.4710 1.41327
\(742\) 0 0
\(743\) 40.9681 1.50297 0.751487 0.659747i \(-0.229337\pi\)
0.751487 + 0.659747i \(0.229337\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) 2.27909 0.0833875
\(748\) 0 0
\(749\) 2.17226 0.0793727
\(750\) 0 0
\(751\) −23.2861 −0.849722 −0.424861 0.905259i \(-0.639677\pi\)
−0.424861 + 0.905259i \(0.639677\pi\)
\(752\) 0 0
\(753\) −11.6917 −0.426067
\(754\) 0 0
\(755\) −17.2207 −0.626724
\(756\) 0 0
\(757\) 27.3275 0.993234 0.496617 0.867970i \(-0.334575\pi\)
0.496617 + 0.867970i \(0.334575\pi\)
\(758\) 0 0
\(759\) −25.0336 −0.908661
\(760\) 0 0
\(761\) 44.5626 1.61539 0.807696 0.589599i \(-0.200714\pi\)
0.807696 + 0.589599i \(0.200714\pi\)
\(762\) 0 0
\(763\) 1.79157 0.0648591
\(764\) 0 0
\(765\) 5.65067 0.204300
\(766\) 0 0
\(767\) 55.1600 1.99171
\(768\) 0 0
\(769\) 51.6110 1.86114 0.930570 0.366115i \(-0.119312\pi\)
0.930570 + 0.366115i \(0.119312\pi\)
\(770\) 0 0
\(771\) −37.1155 −1.33668
\(772\) 0 0
\(773\) −19.6358 −0.706252 −0.353126 0.935576i \(-0.614881\pi\)
−0.353126 + 0.935576i \(0.614881\pi\)
\(774\) 0 0
\(775\) 2.82032 0.101309
\(776\) 0 0
\(777\) 8.96122 0.321482
\(778\) 0 0
\(779\) −55.7374 −1.99700
\(780\) 0 0
\(781\) −29.7523 −1.06462
\(782\) 0 0
\(783\) −5.64064 −0.201580
\(784\) 0 0
\(785\) 5.69646 0.203315
\(786\) 0 0
\(787\) −37.3371 −1.33092 −0.665462 0.746432i \(-0.731765\pi\)
−0.665462 + 0.746432i \(0.731765\pi\)
\(788\) 0 0
\(789\) −15.3271 −0.545658
\(790\) 0 0
\(791\) 5.14831 0.183053
\(792\) 0 0
\(793\) −4.68904 −0.166513
\(794\) 0 0
\(795\) −3.88836 −0.137906
\(796\) 0 0
\(797\) −11.2159 −0.397286 −0.198643 0.980072i \(-0.563653\pi\)
−0.198643 + 0.980072i \(0.563653\pi\)
\(798\) 0 0
\(799\) 12.7645 0.451576
\(800\) 0 0
\(801\) −19.9852 −0.706141
\(802\) 0 0
\(803\) −32.5019 −1.14697
\(804\) 0 0
\(805\) 3.58002 0.126179
\(806\) 0 0
\(807\) 27.2717 0.960008
\(808\) 0 0
\(809\) 11.1749 0.392888 0.196444 0.980515i \(-0.437061\pi\)
0.196444 + 0.980515i \(0.437061\pi\)
\(810\) 0 0
\(811\) −17.6406 −0.619447 −0.309723 0.950827i \(-0.600236\pi\)
−0.309723 + 0.950827i \(0.600236\pi\)
\(812\) 0 0
\(813\) −14.5264 −0.509463
\(814\) 0 0
\(815\) 16.4003 0.574479
\(816\) 0 0
\(817\) −38.8517 −1.35925
\(818\) 0 0
\(819\) −3.16965 −0.110757
\(820\) 0 0
\(821\) 24.2477 0.846251 0.423126 0.906071i \(-0.360933\pi\)
0.423126 + 0.906071i \(0.360933\pi\)
\(822\) 0 0
\(823\) −16.9777 −0.591807 −0.295903 0.955218i \(-0.595621\pi\)
−0.295903 + 0.955218i \(0.595621\pi\)
\(824\) 0 0
\(825\) 4.53162 0.157771
\(826\) 0 0
\(827\) 5.58482 0.194203 0.0971017 0.995274i \(-0.469043\pi\)
0.0971017 + 0.995274i \(0.469043\pi\)
\(828\) 0 0
\(829\) 18.6136 0.646476 0.323238 0.946318i \(-0.395229\pi\)
0.323238 + 0.946318i \(0.395229\pi\)
\(830\) 0 0
\(831\) 17.0729 0.592251
\(832\) 0 0
\(833\) −31.7178 −1.09896
\(834\) 0 0
\(835\) −23.2765 −0.805516
\(836\) 0 0
\(837\) 15.9084 0.549876
\(838\) 0 0
\(839\) 15.9293 0.549942 0.274971 0.961453i \(-0.411332\pi\)
0.274971 + 0.961453i \(0.411332\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −9.14520 −0.314977
\(844\) 0 0
\(845\) −4.40776 −0.151631
\(846\) 0 0
\(847\) 0.152616 0.00524395
\(848\) 0 0
\(849\) 1.44393 0.0495555
\(850\) 0 0
\(851\) 56.5019 1.93686
\(852\) 0 0
\(853\) −27.0288 −0.925447 −0.462723 0.886503i \(-0.653128\pi\)
−0.462723 + 0.886503i \(0.653128\pi\)
\(854\) 0 0
\(855\) −7.99519 −0.273430
\(856\) 0 0
\(857\) 30.2691 1.03397 0.516986 0.855994i \(-0.327054\pi\)
0.516986 + 0.855994i \(0.327054\pi\)
\(858\) 0 0
\(859\) 41.4487 1.41421 0.707106 0.707107i \(-0.250000\pi\)
0.707106 + 0.707107i \(0.250000\pi\)
\(860\) 0 0
\(861\) −7.16003 −0.244013
\(862\) 0 0
\(863\) −42.2494 −1.43819 −0.719093 0.694913i \(-0.755443\pi\)
−0.719093 + 0.694913i \(0.755443\pi\)
\(864\) 0 0
\(865\) 13.9245 0.473448
\(866\) 0 0
\(867\) −8.43001 −0.286298
\(868\) 0 0
\(869\) 30.1426 1.02252
\(870\) 0 0
\(871\) −6.35936 −0.215479
\(872\) 0 0
\(873\) 15.6519 0.529738
\(874\) 0 0
\(875\) −0.648061 −0.0219085
\(876\) 0 0
\(877\) 37.8129 1.27685 0.638426 0.769684i \(-0.279586\pi\)
0.638426 + 0.769684i \(0.279586\pi\)
\(878\) 0 0
\(879\) −7.38642 −0.249138
\(880\) 0 0
\(881\) −5.35675 −0.180473 −0.0902367 0.995920i \(-0.528762\pi\)
−0.0902367 + 0.995920i \(0.528762\pi\)
\(882\) 0 0
\(883\) −8.53643 −0.287274 −0.143637 0.989630i \(-0.545880\pi\)
−0.143637 + 0.989630i \(0.545880\pi\)
\(884\) 0 0
\(885\) 17.8735 0.600812
\(886\) 0 0
\(887\) 22.7597 0.764196 0.382098 0.924122i \(-0.375202\pi\)
0.382098 + 0.924122i \(0.375202\pi\)
\(888\) 0 0
\(889\) 11.4684 0.384637
\(890\) 0 0
\(891\) 13.7732 0.461420
\(892\) 0 0
\(893\) −18.0606 −0.604376
\(894\) 0 0
\(895\) −4.34452 −0.145221
\(896\) 0 0
\(897\) 31.1600 1.04040
\(898\) 0 0
\(899\) −2.82032 −0.0940630
\(900\) 0 0
\(901\) −13.8639 −0.461874
\(902\) 0 0
\(903\) −4.99089 −0.166086
\(904\) 0 0
\(905\) −20.5168 −0.682001
\(906\) 0 0
\(907\) −3.94418 −0.130964 −0.0654822 0.997854i \(-0.520859\pi\)
−0.0654822 + 0.997854i \(0.520859\pi\)
\(908\) 0 0
\(909\) −21.6354 −0.717602
\(910\) 0 0
\(911\) 40.6546 1.34695 0.673473 0.739212i \(-0.264802\pi\)
0.673473 + 0.739212i \(0.264802\pi\)
\(912\) 0 0
\(913\) 6.51678 0.215674
\(914\) 0 0
\(915\) −1.51939 −0.0502296
\(916\) 0 0
\(917\) −3.58002 −0.118223
\(918\) 0 0
\(919\) −27.3471 −0.902099 −0.451049 0.892499i \(-0.648950\pi\)
−0.451049 + 0.892499i \(0.648950\pi\)
\(920\) 0 0
\(921\) 8.52200 0.280810
\(922\) 0 0
\(923\) 37.0336 1.21897
\(924\) 0 0
\(925\) −10.2281 −0.336297
\(926\) 0 0
\(927\) −17.3733 −0.570613
\(928\) 0 0
\(929\) 20.9368 0.686913 0.343456 0.939169i \(-0.388402\pi\)
0.343456 + 0.939169i \(0.388402\pi\)
\(930\) 0 0
\(931\) 44.8778 1.47081
\(932\) 0 0
\(933\) 33.3716 1.09254
\(934\) 0 0
\(935\) 16.1574 0.528404
\(936\) 0 0
\(937\) 36.4051 1.18930 0.594652 0.803983i \(-0.297290\pi\)
0.594652 + 0.803983i \(0.297290\pi\)
\(938\) 0 0
\(939\) 32.6794 1.06645
\(940\) 0 0
\(941\) 28.0968 0.915929 0.457965 0.888970i \(-0.348579\pi\)
0.457965 + 0.888970i \(0.348579\pi\)
\(942\) 0 0
\(943\) −45.1452 −1.47013
\(944\) 0 0
\(945\) −3.65548 −0.118913
\(946\) 0 0
\(947\) 10.8809 0.353583 0.176792 0.984248i \(-0.443428\pi\)
0.176792 + 0.984248i \(0.443428\pi\)
\(948\) 0 0
\(949\) 40.4562 1.31326
\(950\) 0 0
\(951\) −1.92976 −0.0625767
\(952\) 0 0
\(953\) 38.8103 1.25719 0.628594 0.777733i \(-0.283631\pi\)
0.628594 + 0.777733i \(0.283631\pi\)
\(954\) 0 0
\(955\) −9.10422 −0.294606
\(956\) 0 0
\(957\) −4.53162 −0.146486
\(958\) 0 0
\(959\) −2.50717 −0.0809606
\(960\) 0 0
\(961\) −23.0458 −0.743413
\(962\) 0 0
\(963\) −3.92935 −0.126621
\(964\) 0 0
\(965\) 25.1648 0.810085
\(966\) 0 0
\(967\) −47.1862 −1.51741 −0.758703 0.651437i \(-0.774166\pi\)
−0.758703 + 0.651437i \(0.774166\pi\)
\(968\) 0 0
\(969\) 44.4465 1.42783
\(970\) 0 0
\(971\) −40.0772 −1.28614 −0.643069 0.765809i \(-0.722339\pi\)
−0.643069 + 0.765809i \(0.722339\pi\)
\(972\) 0 0
\(973\) −3.65548 −0.117189
\(974\) 0 0
\(975\) −5.64064 −0.180645
\(976\) 0 0
\(977\) 2.29873 0.0735430 0.0367715 0.999324i \(-0.488293\pi\)
0.0367715 + 0.999324i \(0.488293\pi\)
\(978\) 0 0
\(979\) −57.1452 −1.82637
\(980\) 0 0
\(981\) −3.24072 −0.103468
\(982\) 0 0
\(983\) 8.89578 0.283731 0.141866 0.989886i \(-0.454690\pi\)
0.141866 + 0.989886i \(0.454690\pi\)
\(984\) 0 0
\(985\) 0.703878 0.0224274
\(986\) 0 0
\(987\) −2.32007 −0.0738486
\(988\) 0 0
\(989\) −31.4684 −1.00064
\(990\) 0 0
\(991\) −37.5652 −1.19330 −0.596649 0.802503i \(-0.703501\pi\)
−0.596649 + 0.802503i \(0.703501\pi\)
\(992\) 0 0
\(993\) 46.2542 1.46783
\(994\) 0 0
\(995\) −5.22066 −0.165506
\(996\) 0 0
\(997\) −25.8081 −0.817351 −0.408675 0.912680i \(-0.634009\pi\)
−0.408675 + 0.912680i \(0.634009\pi\)
\(998\) 0 0
\(999\) −57.6929 −1.82532
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 2320.2.a.m.1.2 3
4.3 odd 2 580.2.a.c.1.2 3
8.3 odd 2 9280.2.a.bk.1.2 3
8.5 even 2 9280.2.a.bw.1.2 3
12.11 even 2 5220.2.a.x.1.2 3
20.3 even 4 2900.2.c.f.349.4 6
20.7 even 4 2900.2.c.f.349.3 6
20.19 odd 2 2900.2.a.g.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
580.2.a.c.1.2 3 4.3 odd 2
2320.2.a.m.1.2 3 1.1 even 1 trivial
2900.2.a.g.1.2 3 20.19 odd 2
2900.2.c.f.349.3 6 20.7 even 4
2900.2.c.f.349.4 6 20.3 even 4
5220.2.a.x.1.2 3 12.11 even 2
9280.2.a.bk.1.2 3 8.3 odd 2
9280.2.a.bw.1.2 3 8.5 even 2