Properties

Label 429.2.e.d.428.5
Level $429$
Weight $2$
Character 429.428
Analytic conductor $3.426$
Analytic rank $0$
Dimension $20$
CM discriminant -143
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [429,2,Mod(428,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.428");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.e (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 53x^{10} + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 428.5
Root \(1.09266 + 0.897823i\) of defining polynomial
Character \(\chi\) \(=\) 429.428
Dual form 429.2.e.d.428.16

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.79565i q^{2} +(-0.240479 - 1.71528i) q^{3} -1.22434 q^{4} +(-3.08003 + 0.431815i) q^{6} +5.23009 q^{7} -1.39281i q^{8} +(-2.88434 + 0.824975i) q^{9} +O(q^{10})\) \(q-1.79565i q^{2} +(-0.240479 - 1.71528i) q^{3} -1.22434 q^{4} +(-3.08003 + 0.431815i) q^{6} +5.23009 q^{7} -1.39281i q^{8} +(-2.88434 + 0.824975i) q^{9} -3.31662i q^{11} +(0.294428 + 2.10008i) q^{12} -3.60555 q^{13} -9.39139i q^{14} -4.94967 q^{16} +(1.48136 + 5.17925i) q^{18} +7.00719 q^{19} +(-1.25773 - 8.97104i) q^{21} -5.95548 q^{22} +6.19820i q^{23} +(-2.38905 + 0.334941i) q^{24} -5.00000 q^{25} +6.47429i q^{26} +(2.10868 + 4.74905i) q^{27} -6.40342 q^{28} +6.10224i q^{32} +(-5.68893 + 0.797578i) q^{33} +(3.53142 - 1.01005i) q^{36} -12.5824i q^{38} +(0.867059 + 6.18451i) q^{39} +7.55666i q^{41} +(-16.1088 + 2.25843i) q^{42} +4.06068i q^{44} +11.1298 q^{46} +(1.19029 + 8.49005i) q^{48} +20.3538 q^{49} +8.97823i q^{50} +4.41443 q^{52} -2.65693i q^{53} +(8.52761 - 3.78644i) q^{54} -7.28451i q^{56} +(-1.68508 - 12.0193i) q^{57} +(-15.0854 + 4.31469i) q^{63} +1.05811 q^{64} +(1.43217 + 10.2153i) q^{66} +(10.6316 - 1.49054i) q^{69} +(1.14903 + 4.01733i) q^{72} -6.10477 q^{73} +(1.20239 + 8.57638i) q^{75} -8.57920 q^{76} -17.3462i q^{77} +(11.1052 - 1.55693i) q^{78} +(7.63883 - 4.75902i) q^{81} +13.5691 q^{82} -17.5249i q^{83} +(1.53989 + 10.9836i) q^{84} -4.61942 q^{88} -18.8574 q^{91} -7.58871i q^{92} +(10.4670 - 1.46746i) q^{96} -36.5483i q^{98} +(2.73613 + 9.56627i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 40 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 40 q^{4} + 80 q^{16} - 100 q^{25} + 10 q^{36} - 50 q^{42} + 70 q^{48} + 140 q^{49} - 160 q^{64} - 110 q^{66} + 130 q^{78}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(79\) \(287\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.79565i 1.26971i −0.772630 0.634856i \(-0.781059\pi\)
0.772630 0.634856i \(-0.218941\pi\)
\(3\) −0.240479 1.71528i −0.138841 0.990315i
\(4\) −1.22434 −0.612171
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −3.08003 + 0.431815i −1.25742 + 0.176288i
\(7\) 5.23009 1.97679 0.988394 0.151912i \(-0.0485431\pi\)
0.988394 + 0.151912i \(0.0485431\pi\)
\(8\) 1.39281i 0.492432i
\(9\) −2.88434 + 0.824975i −0.961447 + 0.274992i
\(10\) 0 0
\(11\) 3.31662i 1.00000i
\(12\) 0.294428 + 2.10008i 0.0849941 + 0.606242i
\(13\) −3.60555 −1.00000
\(14\) 9.39139i 2.50995i
\(15\) 0 0
\(16\) −4.94967 −1.23742
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.48136 + 5.17925i 0.349160 + 1.22076i
\(19\) 7.00719 1.60756 0.803780 0.594926i \(-0.202819\pi\)
0.803780 + 0.594926i \(0.202819\pi\)
\(20\) 0 0
\(21\) −1.25773 8.97104i −0.274458 1.95764i
\(22\) −5.95548 −1.26971
\(23\) 6.19820i 1.29241i 0.763162 + 0.646207i \(0.223646\pi\)
−0.763162 + 0.646207i \(0.776354\pi\)
\(24\) −2.38905 + 0.334941i −0.487662 + 0.0683695i
\(25\) −5.00000 −1.00000
\(26\) 6.47429i 1.26971i
\(27\) 2.10868 + 4.74905i 0.405816 + 0.913955i
\(28\) −6.40342 −1.21013
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 6.10224i 1.07873i
\(33\) −5.68893 + 0.797578i −0.990315 + 0.138841i
\(34\) 0 0
\(35\) 0 0
\(36\) 3.53142 1.01005i 0.588570 0.168342i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 12.5824i 2.04114i
\(39\) 0.867059 + 6.18451i 0.138841 + 0.990315i
\(40\) 0 0
\(41\) 7.55666i 1.18015i 0.807347 + 0.590076i \(0.200902\pi\)
−0.807347 + 0.590076i \(0.799098\pi\)
\(42\) −16.1088 + 2.25843i −2.48564 + 0.348483i
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 4.06068i 0.612171i
\(45\) 0 0
\(46\) 11.1298 1.64099
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.19029 + 8.49005i 0.171804 + 1.22543i
\(49\) 20.3538 2.90769
\(50\) 8.97823i 1.26971i
\(51\) 0 0
\(52\) 4.41443 0.612171
\(53\) 2.65693i 0.364957i −0.983210 0.182479i \(-0.941588\pi\)
0.983210 0.182479i \(-0.0584120\pi\)
\(54\) 8.52761 3.78644i 1.16046 0.515270i
\(55\) 0 0
\(56\) 7.28451i 0.973433i
\(57\) −1.68508 12.0193i −0.223195 1.59199i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −15.0854 + 4.31469i −1.90058 + 0.543600i
\(64\) 1.05811 0.132264
\(65\) 0 0
\(66\) 1.43217 + 10.2153i 0.176288 + 1.25742i
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 10.6316 1.49054i 1.27990 0.179439i
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.14903 + 4.01733i 0.135415 + 0.473447i
\(73\) −6.10477 −0.714510 −0.357255 0.934007i \(-0.616287\pi\)
−0.357255 + 0.934007i \(0.616287\pi\)
\(74\) 0 0
\(75\) 1.20239 + 8.57638i 0.138841 + 0.990315i
\(76\) −8.57920 −0.984102
\(77\) 17.3462i 1.97679i
\(78\) 11.1052 1.55693i 1.25742 0.176288i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 7.63883 4.75902i 0.848759 0.528779i
\(82\) 13.5691 1.49846
\(83\) 17.5249i 1.92360i −0.273747 0.961802i \(-0.588263\pi\)
0.273747 0.961802i \(-0.411737\pi\)
\(84\) 1.53989 + 10.9836i 0.168015 + 1.19841i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −4.61942 −0.492432
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −18.8574 −1.97679
\(92\) 7.58871i 0.791178i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 10.4670 1.46746i 1.06829 0.149772i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 36.5483i 3.69193i
\(99\) 2.73613 + 9.56627i 0.274992 + 0.961447i
\(100\) 6.12171 0.612171
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 15.7547 1.55236 0.776179 0.630512i \(-0.217155\pi\)
0.776179 + 0.630512i \(0.217155\pi\)
\(104\) 5.02184i 0.492432i
\(105\) 0 0
\(106\) −4.77090 −0.463391
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −2.58175 5.81446i −0.248429 0.559497i
\(109\) 20.1192 1.92707 0.963533 0.267589i \(-0.0862270\pi\)
0.963533 + 0.267589i \(0.0862270\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −25.8872 −2.44611
\(113\) 17.9264i 1.68637i 0.537622 + 0.843186i \(0.319323\pi\)
−0.537622 + 0.843186i \(0.680677\pi\)
\(114\) −21.5823 + 3.02581i −2.02137 + 0.283393i
\(115\) 0 0
\(116\) 0 0
\(117\) 10.3996 2.97449i 0.961447 0.274992i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 12.9618 1.81722i 1.16872 0.163853i
\(124\) 0 0
\(125\) 0 0
\(126\) 7.74766 + 27.0879i 0.690216 + 2.41319i
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 10.3045i 0.910796i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 6.96519 0.976508i 0.606242 0.0849941i
\(133\) 36.6483 3.17781
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) −2.67647 19.0906i −0.227836 1.62510i
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 11.9583i 1.00000i
\(144\) 14.2765 4.08335i 1.18971 0.340279i
\(145\) 0 0
\(146\) 10.9620i 0.907222i
\(147\) −4.89467 34.9124i −0.403705 2.87953i
\(148\) 0 0
\(149\) 12.3797i 1.01419i 0.861891 + 0.507093i \(0.169280\pi\)
−0.861891 + 0.507093i \(0.830720\pi\)
\(150\) 15.4001 2.15907i 1.25742 0.176288i
\(151\) 14.4222 1.17366 0.586831 0.809709i \(-0.300375\pi\)
0.586831 + 0.809709i \(0.300375\pi\)
\(152\) 9.75967i 0.791614i
\(153\) 0 0
\(154\) −31.1477 −2.50995
\(155\) 0 0
\(156\) −1.06158 7.57196i −0.0849941 0.606242i
\(157\) −14.2034 −1.13356 −0.566778 0.823871i \(-0.691810\pi\)
−0.566778 + 0.823871i \(0.691810\pi\)
\(158\) 0 0
\(159\) −4.55737 + 0.638935i −0.361423 + 0.0506709i
\(160\) 0 0
\(161\) 32.4171i 2.55483i
\(162\) −8.54550 13.7166i −0.671398 1.07768i
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 9.25194i 0.722455i
\(165\) 0 0
\(166\) −31.4684 −2.44242
\(167\) 22.7419i 1.75982i −0.475138 0.879911i \(-0.657602\pi\)
0.475138 0.879911i \(-0.342398\pi\)
\(168\) −12.4949 + 1.75177i −0.964005 + 0.135152i
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) −20.2111 + 5.78076i −1.54558 + 0.442066i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −26.1504 −1.97679
\(176\) 16.4162i 1.23742i
\(177\) 0 0
\(178\) 0 0
\(179\) 23.9165i 1.78760i 0.448461 + 0.893802i \(0.351972\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 10.4681 0.778087 0.389044 0.921219i \(-0.372806\pi\)
0.389044 + 0.921219i \(0.372806\pi\)
\(182\) 33.8611i 2.50995i
\(183\) 0 0
\(184\) 8.63289 0.636425
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 11.0286 + 24.8379i 0.802212 + 1.80669i
\(190\) 0 0
\(191\) 4.97779i 0.360181i −0.983650 0.180090i \(-0.942361\pi\)
0.983650 0.180090i \(-0.0576390\pi\)
\(192\) −0.254454 1.81496i −0.0183636 0.130983i
\(193\) −10.0969 −0.726792 −0.363396 0.931635i \(-0.618383\pi\)
−0.363396 + 0.931635i \(0.618383\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −24.9201 −1.78000
\(197\) 11.9680i 0.852688i −0.904561 0.426344i \(-0.859801\pi\)
0.904561 0.426344i \(-0.140199\pi\)
\(198\) 17.1776 4.91312i 1.22076 0.349160i
\(199\) −28.1505 −1.99554 −0.997770 0.0667521i \(-0.978736\pi\)
−0.997770 + 0.0667521i \(0.978736\pi\)
\(200\) 6.96403i 0.492432i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 28.2899i 1.97105i
\(207\) −5.11336 17.8777i −0.355403 1.24259i
\(208\) 17.8463 1.23742
\(209\) 23.2402i 1.60756i
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 3.25299i 0.223416i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 6.61451 2.93699i 0.450060 0.199837i
\(217\) 0 0
\(218\) 36.1269i 2.44682i
\(219\) 1.46807 + 10.4714i 0.0992029 + 0.707589i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 31.9153i 2.13243i
\(225\) 14.4217 4.12487i 0.961447 0.274992i
\(226\) 32.1894 2.14121
\(227\) 29.9245i 1.98616i 0.117445 + 0.993079i \(0.462530\pi\)
−0.117445 + 0.993079i \(0.537470\pi\)
\(228\) 2.06312 + 14.7157i 0.136633 + 0.974571i
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) −29.7536 + 4.17140i −1.95764 + 0.274458i
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) −5.34113 18.6741i −0.349160 1.22076i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.37676i 0.541847i 0.962601 + 0.270924i \(0.0873291\pi\)
−0.962601 + 0.270924i \(0.912671\pi\)
\(240\) 0 0
\(241\) −10.8234 −0.697199 −0.348600 0.937272i \(-0.613343\pi\)
−0.348600 + 0.937272i \(0.613343\pi\)
\(242\) 19.7521i 1.26971i
\(243\) −10.0000 11.9583i −0.641500 0.767123i
\(244\) 0 0
\(245\) 0 0
\(246\) −3.26308 23.2747i −0.208046 1.48394i
\(247\) −25.2648 −1.60756
\(248\) 0 0
\(249\) −30.0600 + 4.21436i −1.90497 + 0.267074i
\(250\) 0 0
\(251\) 16.0979i 1.01609i 0.861330 + 0.508045i \(0.169632\pi\)
−0.861330 + 0.508045i \(0.830368\pi\)
\(252\) 18.4696 5.28266i 1.16348 0.332776i
\(253\) 20.5571 1.29241
\(254\) 0 0
\(255\) 0 0
\(256\) 20.6194 1.28871
\(257\) 29.7426i 1.85529i 0.373457 + 0.927647i \(0.378172\pi\)
−0.373457 + 0.927647i \(0.621828\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 1.11087 + 7.92357i 0.0683695 + 0.487662i
\(265\) 0 0
\(266\) 65.8073i 4.03490i
\(267\) 0 0
\(268\) 0 0
\(269\) 7.44655i 0.454024i −0.973892 0.227012i \(-0.927104\pi\)
0.973892 0.227012i \(-0.0728957\pi\)
\(270\) 0 0
\(271\) −19.2167 −1.16733 −0.583667 0.811993i \(-0.698383\pi\)
−0.583667 + 0.811993i \(0.698383\pi\)
\(272\) 0 0
\(273\) 4.53479 + 32.3456i 0.274458 + 1.95764i
\(274\) 0 0
\(275\) 16.5831i 1.00000i
\(276\) −13.0167 + 1.82492i −0.783515 + 0.109848i
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 33.5158i 1.99938i −0.0248074 0.999692i \(-0.507897\pi\)
0.0248074 0.999692i \(-0.492103\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 21.4728 1.26971
\(287\) 39.5220i 2.33291i
\(288\) −5.03419 17.6009i −0.296643 1.03714i
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 7.47433 0.437402
\(293\) 26.5330i 1.55007i −0.631916 0.775037i \(-0.717731\pi\)
0.631916 0.775037i \(-0.282269\pi\)
\(294\) −62.6903 + 8.78908i −3.65618 + 0.512590i
\(295\) 0 0
\(296\) 0 0
\(297\) 15.7508 6.99371i 0.913955 0.405816i
\(298\) 22.2296 1.28773
\(299\) 22.3479i 1.29241i
\(300\) −1.47214 10.5004i −0.0849941 0.606242i
\(301\) 0 0
\(302\) 25.8972i 1.49021i
\(303\) 0 0
\(304\) −34.6833 −1.98922
\(305\) 0 0
\(306\) 0 0
\(307\) −28.8444 −1.64624 −0.823119 0.567869i \(-0.807768\pi\)
−0.823119 + 0.567869i \(0.807768\pi\)
\(308\) 21.2377i 1.21013i
\(309\) −3.78868 27.0237i −0.215530 1.53732i
\(310\) 0 0
\(311\) 25.5611i 1.44944i −0.689045 0.724719i \(-0.741970\pi\)
0.689045 0.724719i \(-0.258030\pi\)
\(312\) 8.61383 1.20765i 0.487662 0.0683695i
\(313\) −23.5113 −1.32894 −0.664469 0.747316i \(-0.731342\pi\)
−0.664469 + 0.747316i \(0.731342\pi\)
\(314\) 25.5043i 1.43929i
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 1.14730 + 8.18341i 0.0643374 + 0.458903i
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 58.2097 3.24390
\(323\) 0 0
\(324\) −9.35254 + 5.82666i −0.519586 + 0.323703i
\(325\) 18.0278 1.00000
\(326\) 0 0
\(327\) −4.83823 34.5099i −0.267555 1.90840i
\(328\) 10.5250 0.581145
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 21.4564i 1.17757i
\(333\) 0 0
\(334\) −40.8364 −2.23447
\(335\) 0 0
\(336\) 6.22533 + 44.4037i 0.339619 + 2.42242i
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 23.3434i 1.26971i
\(339\) 30.7487 4.31091i 1.67004 0.234137i
\(340\) 0 0
\(341\) 0 0
\(342\) 10.3802 + 36.2920i 0.561297 + 1.96245i
\(343\) 69.8418 3.77110
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 21.2836 1.13929 0.569643 0.821892i \(-0.307081\pi\)
0.569643 + 0.821892i \(0.307081\pi\)
\(350\) 46.9569i 2.50995i
\(351\) −7.60296 17.1229i −0.405816 0.913955i
\(352\) 20.2388 1.07873
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 42.9456 2.26974
\(359\) 37.4612i 1.97713i 0.150798 + 0.988565i \(0.451816\pi\)
−0.150798 + 0.988565i \(0.548184\pi\)
\(360\) 0 0
\(361\) 30.1008 1.58425
\(362\) 18.7970i 0.987947i
\(363\) 2.64527 + 18.8680i 0.138841 + 0.990315i
\(364\) 23.0878 1.21013
\(365\) 0 0
\(366\) 0 0
\(367\) −31.0363 −1.62008 −0.810041 0.586374i \(-0.800555\pi\)
−0.810041 + 0.586374i \(0.800555\pi\)
\(368\) 30.6790i 1.59926i
\(369\) −6.23406 21.7960i −0.324532 1.13465i
\(370\) 0 0
\(371\) 13.8960i 0.721443i
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 44.6001 19.8034i 2.29398 1.01858i
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8.93835 −0.457326
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 17.6750 2.47801i 0.901975 0.126455i
\(385\) 0 0
\(386\) 18.1305i 0.922817i
\(387\) 0 0
\(388\) 0 0
\(389\) 38.5097i 1.95252i 0.216606 + 0.976259i \(0.430501\pi\)
−0.216606 + 0.976259i \(0.569499\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 28.3490i 1.43184i
\(393\) 0 0
\(394\) −21.4904 −1.08267
\(395\) 0 0
\(396\) −3.34996 11.7124i −0.168342 0.588570i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 50.5484i 2.53376i
\(399\) −8.81313 62.8619i −0.441208 3.14703i
\(400\) 24.7484 1.23742
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −7.21110 −0.356566 −0.178283 0.983979i \(-0.557054\pi\)
−0.178283 + 0.983979i \(0.557054\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −19.2892 −0.950309
\(413\) 0 0
\(414\) −32.1020 + 9.18178i −1.57773 + 0.451260i
\(415\) 0 0
\(416\) 22.0019i 1.07873i
\(417\) 0 0
\(418\) −41.7312 −2.04114
\(419\) 40.8907i 1.99764i −0.0485577 0.998820i \(-0.515462\pi\)
0.0485577 0.998820i \(-0.484538\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −3.70059 −0.179717
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 20.5117 2.87571i 0.990315 0.138841i
\(430\) 0 0
\(431\) 32.6382i 1.57213i 0.618146 + 0.786063i \(0.287884\pi\)
−0.618146 + 0.786063i \(0.712116\pi\)
\(432\) −10.4373 23.5062i −0.502164 1.13094i
\(433\) 19.1253 0.919105 0.459552 0.888151i \(-0.348010\pi\)
0.459552 + 0.888151i \(0.348010\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −24.6327 −1.17969
\(437\) 43.4320i 2.07763i
\(438\) 18.8029 2.63613i 0.898435 0.125959i
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −58.7074 + 16.7914i −2.79559 + 0.799591i
\(442\) 0 0
\(443\) 3.70150i 0.175864i −0.996126 0.0879318i \(-0.971974\pi\)
0.996126 0.0879318i \(-0.0280258\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 21.2346 2.97706i 1.00436 0.140810i
\(448\) 5.53403 0.261458
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) −7.40681 25.8963i −0.349160 1.22076i
\(451\) 25.0626 1.18015
\(452\) 21.9480i 1.03235i
\(453\) −3.46823 24.7381i −0.162952 1.16230i
\(454\) 53.7338 2.52185
\(455\) 0 0
\(456\) −16.7405 + 2.34699i −0.783947 + 0.109908i
\(457\) −21.9240 −1.02556 −0.512781 0.858520i \(-0.671385\pi\)
−0.512781 + 0.858520i \(0.671385\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.39711i 0.111645i −0.998441 0.0558223i \(-0.982222\pi\)
0.998441 0.0558223i \(-0.0177780\pi\)
\(462\) 7.49036 + 53.4269i 0.348483 + 2.48564i
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.9165i 1.10672i 0.832941 + 0.553362i \(0.186655\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) −12.7327 + 3.64179i −0.588570 + 0.168342i
\(469\) 0 0
\(470\) 0 0
\(471\) 3.41562 + 24.3628i 0.157383 + 1.12258i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −35.0360 −1.60756
\(476\) 0 0
\(477\) 2.19190 + 7.66349i 0.100360 + 0.350887i
\(478\) 15.0417 0.687991
\(479\) 6.63325i 0.303081i −0.988451 0.151540i \(-0.951577\pi\)
0.988451 0.151540i \(-0.0484234\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 19.4351i 0.885243i
\(483\) 55.6043 7.79563i 2.53008 0.354714i
\(484\) 13.4678 0.612171
\(485\) 0 0
\(486\) −21.4728 + 17.9565i −0.974026 + 0.814521i
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −15.8696 + 2.22490i −0.715458 + 0.100306i
\(493\) 0 0
\(494\) 45.3666i 2.04114i
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 7.56749 + 53.9770i 0.339107 + 2.41877i
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) −39.0087 + 5.46895i −1.74278 + 0.244335i
\(502\) 28.9061 1.29014
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 6.00953 + 21.0110i 0.267686 + 0.935904i
\(505\) 0 0
\(506\) 36.9133i 1.64099i
\(507\) −3.12622 22.2986i −0.138841 0.990315i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −31.9285 −1.41243
\(512\) 16.4162i 0.725500i
\(513\) 14.7759 + 33.2775i 0.652374 + 1.46924i
\(514\) 53.4072 2.35569
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 39.6423i 1.73676i 0.495897 + 0.868381i \(0.334839\pi\)
−0.495897 + 0.868381i \(0.665161\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 6.28863 + 44.8552i 0.274458 + 1.95764i
\(526\) 0 0
\(527\) 0 0
\(528\) 28.1583 3.94775i 1.22543 0.171804i
\(529\) −15.4177 −0.670333
\(530\) 0 0
\(531\) 0 0
\(532\) −44.8700 −1.94536
\(533\) 27.2459i 1.18015i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 41.0234 5.75142i 1.77029 0.248192i
\(538\) −13.3714 −0.576480
\(539\) 67.5060i 2.90769i
\(540\) 0 0
\(541\) −41.4775 −1.78326 −0.891628 0.452770i \(-0.850436\pi\)
−0.891628 + 0.452770i \(0.850436\pi\)
\(542\) 34.5064i 1.48218i
\(543\) −2.51735 17.9557i −0.108030 0.770551i
\(544\) 0 0
\(545\) 0 0
\(546\) 58.0812 8.14288i 2.48564 0.348483i
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 29.7774 1.26971
\(551\) 0 0
\(552\) −2.07603 14.8078i −0.0883616 0.630261i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 42.6064i 1.80529i −0.430385 0.902645i \(-0.641622\pi\)
0.430385 0.902645i \(-0.358378\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −60.1825 −2.53864
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 39.9518 24.8901i 1.67782 1.04528i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 14.6410i 0.612171i
\(573\) −8.53829 + 1.19705i −0.356692 + 0.0500077i
\(574\) 70.9675 2.96213
\(575\) 30.9910i 1.29241i
\(576\) −3.05196 + 0.872918i −0.127165 + 0.0363716i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 30.5260i 1.26971i
\(579\) 2.42809 + 17.3190i 0.100908 + 0.719753i
\(580\) 0 0
\(581\) 91.6566i 3.80256i
\(582\) 0 0
\(583\) −8.81204 −0.364957
\(584\) 8.50277i 0.351847i
\(585\) 0 0
\(586\) −47.6439 −1.96815
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 5.99274 + 42.7448i 0.247137 + 1.76276i
\(589\) 0 0
\(590\) 0 0
\(591\) −20.5285 + 2.87806i −0.844430 + 0.118388i
\(592\) 0 0
\(593\) 47.4294i 1.94769i −0.227205 0.973847i \(-0.572959\pi\)
0.227205 0.973847i \(-0.427041\pi\)
\(594\) −12.5582 28.2829i −0.515270 1.16046i
\(595\) 0 0
\(596\) 15.1570i 0.620856i
\(597\) 6.76961 + 48.2859i 0.277062 + 1.97621i
\(598\) −40.1289 −1.64099
\(599\) 36.1888i 1.47863i 0.673357 + 0.739317i \(0.264852\pi\)
−0.673357 + 0.739317i \(0.735148\pi\)
\(600\) 11.9452 1.67470i 0.487662 0.0683695i
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −17.6577 −0.718482
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 42.7596i 1.73413i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −32.3287 −1.30574 −0.652872 0.757468i \(-0.726436\pi\)
−0.652872 + 0.757468i \(0.726436\pi\)
\(614\) 51.7943i 2.09025i
\(615\) 0 0
\(616\) −24.1600 −0.973433
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) −48.5250 + 6.80312i −1.95196 + 0.273662i
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) −29.4355 + 13.0700i −1.18121 + 0.524482i
\(622\) −45.8987 −1.84037
\(623\) 0 0
\(624\) −4.29165 30.6113i −0.171804 1.22543i
\(625\) 25.0000 1.00000
\(626\) 42.2180i 1.68737i
\(627\) −39.8634 + 5.58878i −1.59199 + 0.223195i
\(628\) 17.3898 0.693930
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 5.57977 0.782275i 0.221252 0.0310192i
\(637\) −73.3868 −2.90769
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 19.8429i 0.783749i 0.920019 + 0.391875i \(0.128173\pi\)
−0.920019 + 0.391875i \(0.871827\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 39.6896i 1.56399i
\(645\) 0 0
\(646\) 0 0
\(647\) 46.1444i 1.81412i −0.420997 0.907062i \(-0.638320\pi\)
0.420997 0.907062i \(-0.361680\pi\)
\(648\) −6.62839 10.6394i −0.260388 0.417956i
\(649\) 0 0
\(650\) 32.3715i 1.26971i
\(651\) 0 0
\(652\) 0 0
\(653\) 47.8330i 1.87185i −0.352197 0.935926i \(-0.614565\pi\)
0.352197 0.935926i \(-0.385435\pi\)
\(654\) −61.9675 + 8.68775i −2.42312 + 0.339718i
\(655\) 0 0
\(656\) 37.4030i 1.46034i
\(657\) 17.6082 5.03628i 0.686963 0.196484i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −24.4087 −0.947243
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 27.8439i 1.07731i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 54.7435 7.67494i 2.11177 0.296067i
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −10.5434 23.7452i −0.405816 0.913955i
\(676\) −15.9164 −0.612171
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) −7.74087 55.2137i −0.297286 2.12047i
\(679\) 0 0
\(680\) 0 0
\(681\) 51.3288 7.19621i 1.96692 0.275759i
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 24.7453 7.07762i 0.946161 0.270620i
\(685\) 0 0
\(686\) 125.411i 4.78821i
\(687\) 0 0
\(688\) 0 0
\(689\) 9.57969i 0.364957i
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 14.3102 + 50.0325i 0.543600 + 1.90058i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 38.2178i 1.44657i
\(699\) 0 0
\(700\) 32.0171 1.21013
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) −30.7467 + 13.6522i −1.16046 + 0.515270i
\(703\) 0 0
\(704\) 3.50937i 0.132264i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 29.2820i 1.09432i
\(717\) 14.3684 2.01443i 0.536600 0.0752304i
\(718\) 67.2671 2.51039
\(719\) 23.9165i 0.891936i 0.895049 + 0.445968i \(0.147140\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) 82.3986 3.06868
\(722\) 54.0503i 2.01154i
\(723\) 2.60281 + 18.5652i 0.0967995 + 0.690447i
\(724\) −12.8165 −0.476322
\(725\) 0 0
\(726\) 33.8803 4.74996i 1.25742 0.176288i
\(727\) −53.4694 −1.98307 −0.991536 0.129834i \(-0.958556\pi\)
−0.991536 + 0.129834i \(0.958556\pi\)
\(728\) 26.2647i 0.973433i
\(729\) −18.1069 + 20.0285i −0.670627 + 0.741795i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −42.2040 −1.55884 −0.779420 0.626502i \(-0.784486\pi\)
−0.779420 + 0.626502i \(0.784486\pi\)
\(734\) 55.7302i 2.05704i
\(735\) 0 0
\(736\) −37.8229 −1.39417
\(737\) 0 0
\(738\) −39.1379 + 11.1942i −1.44068 + 0.412063i
\(739\) 4.50357 0.165666 0.0828332 0.996563i \(-0.473603\pi\)
0.0828332 + 0.996563i \(0.473603\pi\)
\(740\) 0 0
\(741\) 6.07565 + 43.3361i 0.223195 + 1.59199i
\(742\) −24.9522 −0.916026
\(743\) 33.1662i 1.21675i 0.793649 + 0.608376i \(0.208179\pi\)
−0.793649 + 0.608376i \(0.791821\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 14.4576 + 50.5477i 0.528975 + 1.84944i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −33.9220 −1.23783 −0.618916 0.785457i \(-0.712428\pi\)
−0.618916 + 0.785457i \(0.712428\pi\)
\(752\) 0 0
\(753\) 27.6123 3.87120i 1.00625 0.141074i
\(754\) 0 0
\(755\) 0 0
\(756\) −13.5028 30.4101i −0.491091 1.10601i
\(757\) 55.0208 1.99976 0.999882 0.0153878i \(-0.00489828\pi\)
0.999882 + 0.0153878i \(0.00489828\pi\)
\(758\) 0 0
\(759\) −4.94355 35.2611i −0.179439 1.27990i
\(760\) 0 0
\(761\) 17.2028i 0.623600i 0.950148 + 0.311800i \(0.100932\pi\)
−0.950148 + 0.311800i \(0.899068\pi\)
\(762\) 0 0
\(763\) 105.225 3.80940
\(764\) 6.09452i 0.220492i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −4.95853 35.3680i −0.178926 1.27623i
\(769\) 51.9376 1.87292 0.936459 0.350776i \(-0.114082\pi\)
0.936459 + 0.350776i \(0.114082\pi\)
\(770\) 0 0
\(771\) 51.0168 7.15247i 1.83733 0.257590i
\(772\) 12.3621 0.444921
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 69.1497 2.47914
\(779\) 52.9510i 1.89717i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −100.745 −3.59803
\(785\) 0 0
\(786\) 0 0
\(787\) 5.95661 0.212330 0.106165 0.994349i \(-0.466143\pi\)
0.106165 + 0.994349i \(0.466143\pi\)
\(788\) 14.6530i 0.521991i
\(789\) 0 0
\(790\) 0 0
\(791\) 93.7566i 3.33360i
\(792\) 13.3240 3.81090i 0.473447 0.135415i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 34.4659 1.22161
\(797\) 47.8330i 1.69433i −0.531327 0.847167i \(-0.678307\pi\)
0.531327 0.847167i \(-0.321693\pi\)
\(798\) −112.878 + 15.8253i −3.99582 + 0.560208i
\(799\) 0 0
\(800\) 30.5112i 1.07873i
\(801\) 0 0
\(802\) 0 0
\(803\) 20.2472i 0.714510i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −12.7729 + 1.79074i −0.449627 + 0.0630369i
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −25.4239 −0.892755 −0.446377 0.894845i \(-0.647286\pi\)
−0.446377 + 0.894845i \(0.647286\pi\)
\(812\) 0 0
\(813\) 4.62122 + 32.9620i 0.162073 + 1.15603i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 12.9486i 0.452737i
\(819\) 54.3910 15.5568i 1.90058 0.543600i
\(820\) 0 0
\(821\) 53.0660i 1.85202i 0.377504 + 0.926008i \(0.376783\pi\)
−0.377504 + 0.926008i \(0.623217\pi\)
\(822\) 0 0
\(823\) −28.1653 −0.981779 −0.490890 0.871222i \(-0.663328\pi\)
−0.490890 + 0.871222i \(0.663328\pi\)
\(824\) 21.9433i 0.764431i
\(825\) 28.4446 3.98789i 0.990315 0.138841i
\(826\) 0 0
\(827\) 13.1710i 0.458000i −0.973426 0.229000i \(-0.926454\pi\)
0.973426 0.229000i \(-0.0735456\pi\)
\(828\) 6.26050 + 21.8884i 0.217567 + 0.760675i
\(829\) −48.8155 −1.69543 −0.847716 0.530450i \(-0.822023\pi\)
−0.847716 + 0.530450i \(0.822023\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −3.81509 −0.132264
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 28.4540i 0.984102i
\(837\) 0 0
\(838\) −73.4252 −2.53643
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −57.4888 + 8.05984i −1.98002 + 0.277596i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −57.5310 −1.97679
\(848\) 13.1509i 0.451605i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 4.29993 0.147227 0.0736134 0.997287i \(-0.476547\pi\)
0.0736134 + 0.997287i \(0.476547\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) −5.16375 36.8318i −0.176288 1.25742i
\(859\) 51.9725 1.77328 0.886639 0.462462i \(-0.153034\pi\)
0.886639 + 0.462462i \(0.153034\pi\)
\(860\) 0 0
\(861\) 67.7912 9.50421i 2.31032 0.323903i
\(862\) 58.6066 1.99615
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −28.9798 + 12.8677i −0.985914 + 0.437767i
\(865\) 0 0
\(866\) 34.3423i 1.16700i
\(867\) 4.08814 + 29.1597i 0.138841 + 0.990315i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 28.0221i 0.948948i
\(873\) 0 0
\(874\) 77.9884 2.63800
\(875\) 0 0
\(876\) −1.79742 12.8205i −0.0607291 0.433166i
\(877\) 35.9384 1.21355 0.606777 0.794872i \(-0.292462\pi\)
0.606777 + 0.794872i \(0.292462\pi\)
\(878\) 0 0
\(879\) −45.5114 + 6.38062i −1.53506 + 0.215213i
\(880\) 0 0
\(881\) 59.0930i 1.99089i 0.0953201 + 0.995447i \(0.469613\pi\)
−0.0953201 + 0.995447i \(0.530387\pi\)
\(882\) 30.1514 + 105.418i 1.01525 + 3.54960i
\(883\) 57.7440 1.94324 0.971620 0.236549i \(-0.0760164\pi\)
0.971620 + 0.236549i \(0.0760164\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −6.64658 −0.223296
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −15.7839 25.3351i −0.528779 0.848759i
\(892\) 0 0
\(893\) 0 0
\(894\) −5.34575 38.1299i −0.178789 1.27525i
\(895\) 0 0
\(896\) 53.8933i 1.80045i
\(897\) −38.3328 + 5.37420i −1.27990 + 0.179439i
\(898\) 0 0
\(899\) 0 0
\(900\) −17.6571 + 5.05026i −0.588570 + 0.168342i
\(901\) 0 0
\(902\) 45.0036i 1.49846i
\(903\) 0 0
\(904\) 24.9680 0.830423
\(905\) 0 0
\(906\) −44.4208 + 6.22772i −1.47578 + 0.206902i
\(907\) 6.44682 0.214063 0.107032 0.994256i \(-0.465865\pi\)
0.107032 + 0.994256i \(0.465865\pi\)
\(908\) 36.6378i 1.21587i
\(909\) 0 0
\(910\) 0 0
\(911\) 50.7904i 1.68276i −0.540444 0.841380i \(-0.681744\pi\)
0.540444 0.841380i \(-0.318256\pi\)
\(912\) 8.34060 + 59.4914i 0.276185 + 1.96996i
\(913\) −58.1234 −1.92360
\(914\) 39.3677i 1.30217i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 6.93647 + 49.4761i 0.228564 + 1.63029i
\(922\) −4.30436 −0.141757
\(923\) 0 0
\(924\) 36.4286 5.10722i 1.19841 0.168015i
\(925\) 0 0
\(926\) 0 0
\(927\) −45.4420 + 12.9972i −1.49251 + 0.426886i
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 142.623 4.67429
\(932\) 0 0
\(933\) −43.8443 + 6.14690i −1.43540 + 0.201241i
\(934\) 42.9456 1.40522
\(935\) 0 0
\(936\) −4.14289 14.4847i −0.135415 0.473447i
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 5.65397 + 40.3284i 0.184510 + 1.31607i
\(940\) 0 0
\(941\) 55.0635i 1.79502i −0.440994 0.897510i \(-0.645374\pi\)
0.440994 0.897510i \(-0.354626\pi\)
\(942\) 43.7469 6.13324i 1.42535 0.199832i
\(943\) −46.8377 −1.52525
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 22.0111 0.714510
\(950\) 62.9122i 2.04114i
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 13.7609 3.93587i 0.445526 0.127429i
\(955\) 0 0
\(956\) 10.2560i 0.331703i
\(957\) 0 0
\(958\) −11.9110 −0.384826
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 13.2516 0.426805
\(965\) 0 0
\(966\) −13.9982 99.8456i −0.450384 3.21248i
\(967\) 22.8264 0.734048 0.367024 0.930211i \(-0.380377\pi\)
0.367024 + 0.930211i \(0.380377\pi\)
\(968\) 15.3209i 0.492432i
\(969\) 0 0
\(970\) 0 0
\(971\) 20.9194i 0.671335i −0.941981 0.335667i \(-0.891038\pi\)
0.941981 0.335667i \(-0.108962\pi\)
\(972\) 12.2434 + 14.6410i 0.392708 + 0.469610i
\(973\) 0 0
\(974\) 0 0
\(975\) −4.33529 30.9226i −0.138841 0.990315i
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −58.0305 + 16.5978i −1.85277 + 0.529927i
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −2.53103 18.0532i −0.0806864 0.575516i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 30.9327 0.984102
\(989\) 0 0
\(990\) 0 0
\(991\) −19.4933 −0.619225 −0.309613 0.950863i \(-0.600199\pi\)
−0.309613 + 0.950863i \(0.600199\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 36.8037 5.15981i 1.16617 0.163495i
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 429.2.e.d.428.5 20
3.2 odd 2 inner 429.2.e.d.428.16 yes 20
11.10 odd 2 inner 429.2.e.d.428.15 yes 20
13.12 even 2 inner 429.2.e.d.428.15 yes 20
33.32 even 2 inner 429.2.e.d.428.6 yes 20
39.38 odd 2 inner 429.2.e.d.428.6 yes 20
143.142 odd 2 CM 429.2.e.d.428.5 20
429.428 even 2 inner 429.2.e.d.428.16 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
429.2.e.d.428.5 20 1.1 even 1 trivial
429.2.e.d.428.5 20 143.142 odd 2 CM
429.2.e.d.428.6 yes 20 33.32 even 2 inner
429.2.e.d.428.6 yes 20 39.38 odd 2 inner
429.2.e.d.428.15 yes 20 11.10 odd 2 inner
429.2.e.d.428.15 yes 20 13.12 even 2 inner
429.2.e.d.428.16 yes 20 3.2 odd 2 inner
429.2.e.d.428.16 yes 20 429.428 even 2 inner