Properties

Label 572.2.b.b.571.15
Level $572$
Weight $2$
Character 572.571
Analytic conductor $4.567$
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] = [572,2,Mod(571,572)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(572, 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("572.571");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.56744299562\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 571.15
Root \(-1.09266 - 0.897823i\) of defining polynomial
Character \(\chi\) \(=\) 572.571
Dual form 572.2.b.b.571.16

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.09266 - 0.897823i) q^{2} -3.43055i q^{3} +(0.387829 - 1.96204i) q^{4} +(-3.08003 - 3.74844i) q^{6} +0.803843i q^{7} +(-1.33779 - 2.49205i) q^{8} -8.76868 q^{9} +O(q^{10})\) \(q+(1.09266 - 0.897823i) q^{2} -3.43055i q^{3} +(0.387829 - 1.96204i) q^{4} +(-3.08003 - 3.74844i) q^{6} +0.803843i q^{7} +(-1.33779 - 2.49205i) q^{8} -8.76868 q^{9} +3.31662i q^{11} +(-6.73087 - 1.33047i) q^{12} +3.60555 q^{13} +(0.721709 + 0.878331i) q^{14} +(-3.69918 - 1.52187i) q^{16} +(-9.58122 + 7.87272i) q^{18} -5.18645i q^{19} +2.75763 q^{21} +(2.97774 + 3.62396i) q^{22} -6.19820i q^{23} +(-8.54910 + 4.58937i) q^{24} +5.00000 q^{25} +(3.93966 - 3.23715i) q^{26} +19.7897i q^{27} +(1.57717 + 0.311754i) q^{28} +(-5.40833 + 1.65831i) q^{32} +11.3779 q^{33} +(-3.40075 + 17.2045i) q^{36} +(-4.65651 - 5.66704i) q^{38} -12.3690i q^{39} +10.3391 q^{41} +(3.01316 - 2.47586i) q^{42} +(6.50734 + 1.28628i) q^{44} +(-5.56488 - 6.77255i) q^{46} +(-5.22085 + 12.6902i) q^{48} +6.35384 q^{49} +(5.46332 - 4.48911i) q^{50} +(1.39834 - 7.07422i) q^{52} -14.3158 q^{53} +(17.7677 + 21.6235i) q^{54} +(2.00322 - 1.07538i) q^{56} -17.7924 q^{57} -7.04865i q^{63} +(-4.42061 + 6.66770i) q^{64} +(12.4322 - 10.2153i) q^{66} -21.2632 q^{69} +(11.7307 + 21.8520i) q^{72} -6.10477 q^{73} -17.1528i q^{75} +(-10.1760 - 2.01145i) q^{76} -2.66605 q^{77} +(-11.1052 - 13.5152i) q^{78} +41.5837 q^{81} +(11.2972 - 9.28267i) q^{82} +17.5249i q^{83} +(1.06949 - 5.41056i) q^{84} +(8.26519 - 4.43696i) q^{88} +2.89830i q^{91} +(-12.1611 - 2.40384i) q^{92} +(5.68893 + 18.5535i) q^{96} +(6.94261 - 5.70462i) q^{98} -29.0824i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 60 q^{9} + 100 q^{25} + 10 q^{36} + 30 q^{38} - 50 q^{42} - 70 q^{48} - 140 q^{49} + 90 q^{56} + 110 q^{66} - 130 q^{78} + 180 q^{81} - 150 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(353\) \(365\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.09266 0.897823i 0.772630 0.634856i
\(3\) 3.43055i 1.98063i −0.138841 0.990315i \(-0.544338\pi\)
0.138841 0.990315i \(-0.455662\pi\)
\(4\) 0.387829 1.96204i 0.193915 0.981018i
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −3.08003 3.74844i −1.25742 1.53029i
\(7\) 0.803843i 0.303824i 0.988394 + 0.151912i \(0.0485431\pi\)
−0.988394 + 0.151912i \(0.951457\pi\)
\(8\) −1.33779 2.49205i −0.472982 0.881072i
\(9\) −8.76868 −2.92289
\(10\) 0 0
\(11\) 3.31662i 1.00000i
\(12\) −6.73087 1.33047i −1.94303 0.384073i
\(13\) 3.60555 1.00000
\(14\) 0.721709 + 0.878331i 0.192885 + 0.234744i
\(15\) 0 0
\(16\) −3.69918 1.52187i −0.924794 0.380467i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −9.58122 + 7.87272i −2.25832 + 1.85562i
\(19\) 5.18645i 1.18985i −0.803780 0.594926i \(-0.797181\pi\)
0.803780 0.594926i \(-0.202819\pi\)
\(20\) 0 0
\(21\) 2.75763 0.601763
\(22\) 2.97774 + 3.62396i 0.634856 + 0.772630i
\(23\) 6.19820i 1.29241i −0.763162 0.646207i \(-0.776354\pi\)
0.763162 0.646207i \(-0.223646\pi\)
\(24\) −8.54910 + 4.58937i −1.74508 + 0.936801i
\(25\) 5.00000 1.00000
\(26\) 3.93966 3.23715i 0.772630 0.634856i
\(27\) 19.7897i 3.80854i
\(28\) 1.57717 + 0.311754i 0.298057 + 0.0589159i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −5.40833 + 1.65831i −0.956066 + 0.293151i
\(33\) 11.3779 1.98063
\(34\) 0 0
\(35\) 0 0
\(36\) −3.40075 + 17.2045i −0.566792 + 2.86741i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −4.65651 5.66704i −0.755385 0.919316i
\(39\) 12.3690i 1.98063i
\(40\) 0 0
\(41\) 10.3391 1.61469 0.807347 0.590076i \(-0.200902\pi\)
0.807347 + 0.590076i \(0.200902\pi\)
\(42\) 3.01316 2.47586i 0.464940 0.382033i
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 6.50734 + 1.28628i 0.981018 + 0.193915i
\(45\) 0 0
\(46\) −5.56488 6.77255i −0.820497 0.998558i
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −5.22085 + 12.6902i −0.753565 + 1.83167i
\(49\) 6.35384 0.907691
\(50\) 5.46332 4.48911i 0.772630 0.634856i
\(51\) 0 0
\(52\) 1.39834 7.07422i 0.193915 0.981018i
\(53\) −14.3158 −1.96642 −0.983210 0.182479i \(-0.941588\pi\)
−0.983210 + 0.182479i \(0.941588\pi\)
\(54\) 17.7677 + 21.6235i 2.41788 + 2.94259i
\(55\) 0 0
\(56\) 2.00322 1.07538i 0.267691 0.143703i
\(57\) −17.7924 −2.35666
\(58\) 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
\(63\) 7.04865i 0.888046i
\(64\) −4.42061 + 6.66770i −0.552577 + 0.833462i
\(65\) 0 0
\(66\) 12.4322 10.2153i 1.53029 1.25742i
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −21.2632 −2.55979
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 11.7307 + 21.8520i 1.38247 + 2.57528i
\(73\) −6.10477 −0.714510 −0.357255 0.934007i \(-0.616287\pi\)
−0.357255 + 0.934007i \(0.616287\pi\)
\(74\) 0 0
\(75\) 17.1528i 1.98063i
\(76\) −10.1760 2.01145i −1.16727 0.230730i
\(77\) −2.66605 −0.303824
\(78\) −11.1052 13.5152i −1.25742 1.53029i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 41.5837 4.62041
\(82\) 11.2972 9.28267i 1.24756 1.02510i
\(83\) 17.5249i 1.92360i 0.273747 + 0.961802i \(0.411737\pi\)
−0.273747 + 0.961802i \(0.588263\pi\)
\(84\) 1.06949 5.41056i 0.116691 0.590341i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 8.26519 4.43696i 0.881072 0.472982i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 2.89830i 0.303824i
\(92\) −12.1611 2.40384i −1.26788 0.250618i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 5.68893 + 18.5535i 0.580623 + 1.89361i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 6.94261 5.70462i 0.701309 0.576253i
\(99\) 29.0824i 2.92289i
\(100\) 1.93915 9.81018i 0.193915 0.981018i
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 12.7980i 1.26102i −0.776179 0.630512i \(-0.782845\pi\)
0.776179 0.630512i \(-0.217155\pi\)
\(104\) −4.82349 8.98521i −0.472982 0.881072i
\(105\) 0 0
\(106\) −15.6423 + 12.8530i −1.51932 + 1.24839i
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 38.8282 + 7.67504i 3.73625 + 0.738531i
\(109\) 20.1192 1.92707 0.963533 0.267589i \(-0.0862270\pi\)
0.963533 + 0.267589i \(0.0862270\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.22335 2.97356i 0.115595 0.280975i
\(113\) 11.4300 1.07524 0.537622 0.843186i \(-0.319323\pi\)
0.537622 + 0.843186i \(0.319323\pi\)
\(114\) −19.4411 + 15.9744i −1.82082 + 1.49614i
\(115\) 0 0
\(116\) 0 0
\(117\) −31.6159 −2.92289
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 35.4688i 3.19811i
\(124\) 0 0
\(125\) 0 0
\(126\) −6.32843 7.70180i −0.563782 0.686131i
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.15616 + 11.2545i 0.102191 + 0.994765i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 4.41266 22.3238i 0.384073 1.94303i
\(133\) 4.16909 0.361506
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) −23.2336 + 19.0906i −1.97777 + 1.62510i
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 11.9583i 1.00000i
\(144\) 32.4369 + 13.3448i 2.70307 + 1.11207i
\(145\) 0 0
\(146\) −6.67047 + 5.48100i −0.552052 + 0.453611i
\(147\) 21.7972i 1.79780i
\(148\) 0 0
\(149\) 21.0414 1.72378 0.861891 0.507093i \(-0.169280\pi\)
0.861891 + 0.507093i \(0.169280\pi\)
\(150\) −15.4001 18.7422i −1.25742 1.53029i
\(151\) 19.8997i 1.61942i 0.586831 + 0.809709i \(0.300375\pi\)
−0.586831 + 0.809709i \(0.699625\pi\)
\(152\) −12.9249 + 6.93840i −1.04835 + 0.562778i
\(153\) 0 0
\(154\) −2.91309 + 2.39364i −0.234744 + 0.192885i
\(155\) 0 0
\(156\) −24.2685 4.79707i −1.94303 0.384073i
\(157\) 14.2034 1.13356 0.566778 0.823871i \(-0.308190\pi\)
0.566778 + 0.823871i \(0.308190\pi\)
\(158\) 0 0
\(159\) 49.1109i 3.89475i
\(160\) 0 0
\(161\) 4.98238 0.392667
\(162\) 45.4370 37.3348i 3.56987 2.93330i
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 4.00980 20.2857i 0.313113 1.58405i
\(165\) 0 0
\(166\) 15.7342 + 19.1488i 1.22121 + 1.48623i
\(167\) 22.7419i 1.75982i 0.475138 + 0.879911i \(0.342398\pi\)
−0.475138 + 0.879911i \(0.657602\pi\)
\(168\) −3.68914 6.87214i −0.284623 0.530197i
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 45.4783i 3.47781i
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 4.01922i 0.303824i
\(176\) 5.04747 12.2688i 0.380467 0.924794i
\(177\) 0 0
\(178\) 0 0
\(179\) 23.9165i 1.78760i −0.448461 0.893802i \(-0.648028\pi\)
0.448461 0.893802i \(-0.351972\pi\)
\(180\) 0 0
\(181\) −10.4681 −0.778087 −0.389044 0.921219i \(-0.627194\pi\)
−0.389044 + 0.921219i \(0.627194\pi\)
\(182\) 2.60216 + 3.16687i 0.192885 + 0.234744i
\(183\) 0 0
\(184\) −15.4462 + 8.29191i −1.13871 + 0.611288i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −15.9079 −1.15713
\(190\) 0 0
\(191\) 4.97779i 0.360181i −0.983650 0.180090i \(-0.942361\pi\)
0.983650 0.180090i \(-0.0576390\pi\)
\(192\) 22.8739 + 15.1651i 1.65078 + 1.09445i
\(193\) −10.0969 −0.726792 −0.363396 0.931635i \(-0.618383\pi\)
−0.363396 + 0.931635i \(0.618383\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2.46420 12.4665i 0.176014 0.890461i
\(197\) −25.3922 −1.80912 −0.904561 0.426344i \(-0.859801\pi\)
−0.904561 + 0.426344i \(0.859801\pi\)
\(198\) −26.1109 31.7773i −1.85562 2.25832i
\(199\) 1.88331i 0.133504i −0.997770 0.0667521i \(-0.978736\pi\)
0.997770 0.0667521i \(-0.0212636\pi\)
\(200\) −6.68897 12.4602i −0.472982 0.881072i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −11.4903 13.9839i −0.800569 0.974305i
\(207\) 54.3500i 3.77759i
\(208\) −13.3376 5.48718i −0.924794 0.380467i
\(209\) 17.2015 1.18985
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −5.55206 + 28.0880i −0.381317 + 1.92909i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 49.3170 26.4746i 3.35560 1.80137i
\(217\) 0 0
\(218\) 21.9835 18.0634i 1.48891 1.22341i
\(219\) 20.9427i 1.41518i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) −1.33302 4.34745i −0.0890664 0.290476i
\(225\) −43.8434 −2.92289
\(226\) 12.4892 10.2621i 0.830766 0.682626i
\(227\) 29.9245i 1.98616i 0.117445 + 0.993079i \(0.462530\pi\)
−0.117445 + 0.993079i \(0.537470\pi\)
\(228\) −6.90040 + 34.9093i −0.456990 + 2.31192i
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 9.14601i 0.601763i
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) −34.5456 + 28.3855i −2.25832 + 1.85562i
\(235\) 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 \(-0.386657\pi\)
0.348600 + 0.937272i \(0.386657\pi\)
\(242\) −12.0193 + 9.87605i −0.772630 + 0.634856i
\(243\) 83.2858i 5.34278i
\(244\) 0 0
\(245\) 0 0
\(246\) −31.8447 38.7555i −2.03034 2.47096i
\(247\) 18.7000i 1.18985i
\(248\) 0 0
\(249\) 60.1199 3.80995
\(250\) 0 0
\(251\) 16.0979i 1.01609i 0.861330 + 0.508045i \(0.169632\pi\)
−0.861330 + 0.508045i \(0.830368\pi\)
\(252\) −13.8297 2.73367i −0.871189 0.172205i
\(253\) 20.5571 1.29241
\(254\) 0 0
\(255\) 0 0
\(256\) 11.3678 + 11.2593i 0.710489 + 0.703708i
\(257\) −11.9739 −0.746914 −0.373457 0.927647i \(-0.621828\pi\)
−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 \(0.5\pi\)
\(264\) −15.2212 28.3542i −0.936801 1.74508i
\(265\) 0 0
\(266\) 4.55542 3.74310i 0.279310 0.229504i
\(267\) 0 0
\(268\) 0 0
\(269\) −31.9460 −1.94778 −0.973892 0.227012i \(-0.927104\pi\)
−0.973892 + 0.227012i \(0.927104\pi\)
\(270\) 0 0
\(271\) 26.7342i 1.62399i −0.583667 0.811993i \(-0.698383\pi\)
0.583667 0.811993i \(-0.301617\pi\)
\(272\) 0 0
\(273\) 9.94276 0.601763
\(274\) 0 0
\(275\) 16.5831i 1.00000i
\(276\) −8.24650 + 41.7192i −0.496381 + 2.51120i
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.831696 0.0496148 0.0248074 0.999692i \(-0.492103\pi\)
0.0248074 + 0.999692i \(0.492103\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 10.7364 + 13.0664i 0.634856 + 0.772630i
\(287\) 8.31101i 0.490583i
\(288\) 47.4239 14.5412i 2.79448 0.856849i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −2.36761 + 11.9778i −0.138554 + 0.700947i
\(293\) −21.6333 −1.26383 −0.631916 0.775037i \(-0.717731\pi\)
−0.631916 + 0.775037i \(0.717731\pi\)
\(294\) −19.5700 23.8170i −1.14134 1.38903i
\(295\) 0 0
\(296\) 0 0
\(297\) −65.6352 −3.80854
\(298\) 22.9912 18.8915i 1.33185 1.09435i
\(299\) 22.3479i 1.29241i
\(300\) −33.6543 6.65234i −1.94303 0.384073i
\(301\) 0 0
\(302\) 17.8664 + 21.7437i 1.02810 + 1.25121i
\(303\) 0 0
\(304\) −7.89310 + 19.1856i −0.452700 + 1.10037i
\(305\) 0 0
\(306\) 0 0
\(307\) 19.8997i 1.13574i −0.823119 0.567869i \(-0.807768\pi\)
0.823119 0.567869i \(-0.192232\pi\)
\(308\) −1.03397 + 5.23088i −0.0589159 + 0.298057i
\(309\) −43.9042 −2.49762
\(310\) 0 0
\(311\) 25.5611i 1.44944i 0.689045 + 0.724719i \(0.258030\pi\)
−0.689045 + 0.724719i \(0.741970\pi\)
\(312\) −30.8242 + 16.5472i −1.74508 + 0.936801i
\(313\) 23.5113 1.32894 0.664469 0.747316i \(-0.268658\pi\)
0.664469 + 0.747316i \(0.268658\pi\)
\(314\) 15.5196 12.7521i 0.875819 0.719645i
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 44.0929 + 53.6617i 2.47261 + 3.00920i
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 5.44407 4.47329i 0.303386 0.249287i
\(323\) 0 0
\(324\) 16.1274 81.5888i 0.895965 4.53271i
\(325\) 18.0278 1.00000
\(326\) 0 0
\(327\) 69.0198i 3.81680i
\(328\) −13.8316 25.7655i −0.763721 1.42266i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 34.3844 + 6.79665i 1.88709 + 0.373015i
\(333\) 0 0
\(334\) 20.4182 + 24.8493i 1.11723 + 1.35969i
\(335\) 0 0
\(336\) −10.2009 4.19675i −0.556507 0.228951i
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 14.2046 11.6717i 0.772630 0.634856i
\(339\) 39.2112i 2.12966i
\(340\) 0 0
\(341\) 0 0
\(342\) 40.8314 + 49.6925i 2.20791 + 2.68706i
\(343\) 10.7344i 0.579603i
\(344\) 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 \(-0.692919\pi\)
−0.569643 + 0.821892i \(0.692919\pi\)
\(350\) 3.60854 + 4.39165i 0.192885 + 0.234744i
\(351\) 71.3530i 3.80854i
\(352\) −5.50000 17.9374i −0.293151 0.956066i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −21.4728 26.1327i −1.13487 1.38116i
\(359\) 37.4612i 1.97713i 0.150798 + 0.988565i \(0.451816\pi\)
−0.150798 + 0.988565i \(0.548184\pi\)
\(360\) 0 0
\(361\) −7.89923 −0.415749
\(362\) −11.4381 + 9.39849i −0.601173 + 0.493974i
\(363\) 37.7361i 1.98063i
\(364\) 5.68657 + 1.12404i 0.298057 + 0.0589159i
\(365\) 0 0
\(366\) 0 0
\(367\) 22.4666i 1.17275i 0.810041 + 0.586374i \(0.199445\pi\)
−0.810041 + 0.586374i \(0.800555\pi\)
\(368\) −9.43285 + 22.9282i −0.491721 + 1.19522i
\(369\) −90.6602 −4.71958
\(370\) 0 0
\(371\) 11.5076i 0.597446i
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) −17.3819 + 14.2824i −0.894031 + 0.734609i
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4.46918 5.43906i −0.228663 0.278286i
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 38.6091 3.96628i 1.97026 0.202403i
\(385\) 0 0
\(386\) −11.0325 + 9.06524i −0.561541 + 0.461409i
\(387\) 0 0
\(388\) 0 0
\(389\) −8.54426 −0.433211 −0.216606 0.976259i \(-0.569499\pi\)
−0.216606 + 0.976259i \(0.569499\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −8.50012 15.8341i −0.429321 0.799741i
\(393\) 0 0
\(394\) −27.7452 + 22.7977i −1.39778 + 1.14853i
\(395\) 0 0
\(396\) −57.0608 11.2790i −2.86741 0.566792i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) −1.69088 2.05782i −0.0847559 0.103149i
\(399\) 14.3023i 0.716009i
\(400\) −18.4959 7.60935i −0.924794 0.380467i
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 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
\(412\) −25.1101 4.96344i −1.23709 0.244531i
\(413\) 0 0
\(414\) 48.7967 + 59.3863i 2.39823 + 2.91868i
\(415\) 0 0
\(416\) −19.5000 + 5.97913i −0.956066 + 0.293151i
\(417\) 0 0
\(418\) 18.7955 15.4439i 0.919316 0.755385i
\(419\) 40.8907i 1.99764i 0.0485577 + 0.998820i \(0.484538\pi\)
−0.0485577 + 0.998820i \(0.515462\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 19.1515 + 35.6755i 0.930080 + 1.73256i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 41.0234 1.98063
\(430\) 0 0
\(431\) 32.6382i 1.57213i −0.618146 0.786063i \(-0.712116\pi\)
0.618146 0.786063i \(-0.287884\pi\)
\(432\) 30.1174 73.2058i 1.44903 3.52212i
\(433\) 19.1253 0.919105 0.459552 0.888151i \(-0.348010\pi\)
0.459552 + 0.888151i \(0.348010\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7.80280 39.4745i 0.373686 1.89049i
\(437\) −32.1466 −1.53778
\(438\) 18.8029 + 22.8834i 0.898435 + 1.09341i
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −55.7148 −2.65308
\(442\) 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
\(447\) 72.1837i 3.41417i
\(448\) −5.35978 3.55348i −0.253226 0.167886i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) −47.9061 + 39.3636i −2.25832 + 1.85562i
\(451\) 34.2909i 1.61469i
\(452\) 4.43289 22.4261i 0.208506 1.05483i
\(453\) 68.2671 3.20747
\(454\) 26.8669 + 32.6974i 1.26093 + 1.53457i
\(455\) 0 0
\(456\) 23.8025 + 44.3394i 1.11466 + 2.07638i
\(457\) 21.9240 1.02556 0.512781 0.858520i \(-0.328615\pi\)
0.512781 + 0.858520i \(0.328615\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −42.8749 −1.99688 −0.998441 0.0558223i \(-0.982222\pi\)
−0.998441 + 0.0558223i \(0.982222\pi\)
\(462\) 8.21149 + 9.99352i 0.382033 + 0.464940i
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.9165i 1.10672i −0.832941 0.553362i \(-0.813345\pi\)
0.832941 0.553362i \(-0.186655\pi\)
\(468\) −12.2616 + 62.0316i −0.566792 + 2.86741i
\(469\) 0 0
\(470\) 0 0
\(471\) 48.7255i 2.24515i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 25.9322i 1.18985i
\(476\) 0 0
\(477\) 125.530 5.74763
\(478\) 7.52084 + 9.15298i 0.343995 + 0.418648i
\(479\) 6.63325i 0.303081i −0.988451 0.151540i \(-0.951577\pi\)
0.988451 0.151540i \(-0.0484234\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 11.8264 9.71753i 0.538677 0.442621i
\(483\) 17.0923i 0.777727i
\(484\) −4.26612 + 21.5824i −0.193915 + 0.981018i
\(485\) 0 0
\(486\) −74.7758 91.0034i −3.39190 4.12800i
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −69.5911 13.7558i −3.13741 0.620161i
\(493\) 0 0
\(494\) −16.7893 20.4328i −0.755385 0.919316i
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 65.6909 53.9770i 2.94368 2.41877i
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 78.0173 3.48556
\(502\) 14.4531 + 17.5896i 0.645071 + 0.785062i
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −17.5656 + 9.42964i −0.782433 + 0.420029i
\(505\) 0 0
\(506\) 22.4620 18.4566i 0.998558 0.820497i
\(507\) 44.5972i 1.98063i
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 4.90728i 0.217085i
\(512\) 22.5301 + 2.09638i 0.995699 + 0.0926477i
\(513\) 102.638 4.53160
\(514\) −13.0835 + 10.7505i −0.577088 + 0.474183i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −22.6381 −0.991795 −0.495897 0.868381i \(-0.665161\pi\)
−0.495897 + 0.868381i \(0.665161\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 13.7881 0.601763
\(526\) 0 0
\(527\) 0 0
\(528\) −42.0887 17.3156i −1.83167 0.753565i
\(529\) −15.4177 −0.670333
\(530\) 0 0
\(531\) 0 0
\(532\) 1.61689 8.17991i 0.0701013 0.354644i
\(533\) 37.2781 1.61469
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −82.0468 −3.54058
\(538\) −34.9063 + 28.6819i −1.50492 + 1.23656i
\(539\) 21.0733i 0.907691i
\(540\) 0 0
\(541\) 41.4775 1.78326 0.891628 0.452770i \(-0.149564\pi\)
0.891628 + 0.452770i \(0.149564\pi\)
\(542\) −24.0026 29.2115i −1.03100 1.25474i
\(543\) 35.9113i 1.54110i
\(544\) 0 0
\(545\) 0 0
\(546\) 10.8641 8.92684i 0.464940 0.382033i
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 14.8887 + 18.1198i 0.634856 + 0.772630i
\(551\) 0 0
\(552\) 28.4458 + 52.9890i 1.21073 + 2.25536i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −20.3149 −0.860771 −0.430385 0.902645i \(-0.641622\pi\)
−0.430385 + 0.902645i \(0.641622\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0.908764 0.746716i 0.0383339 0.0314983i
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 33.4268i 1.40379i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 23.4625 + 4.63776i 0.981018 + 0.193915i
\(573\) −17.0766 −0.713384
\(574\) 7.46181 + 9.08114i 0.311450 + 0.379040i
\(575\) 30.9910i 1.29241i
\(576\) 38.7629 58.4669i 1.61512 2.43612i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 18.5753 15.2630i 0.772630 0.634856i
\(579\) 34.6380i 1.43951i
\(580\) 0 0
\(581\) −14.0872 −0.584437
\(582\) 0 0
\(583\) 47.4800i 1.96642i
\(584\) 8.16693 + 15.2134i 0.337950 + 0.629535i
\(585\) 0 0
\(586\) −23.6379 + 19.4229i −0.976474 + 0.802351i
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −42.7668 8.45357i −1.76367 0.348619i
\(589\) 0 0
\(590\) 0 0
\(591\) 87.1094i 3.58320i
\(592\) 0 0
\(593\) −11.0656 −0.454410 −0.227205 0.973847i \(-0.572959\pi\)
−0.227205 + 0.973847i \(0.572959\pi\)
\(594\) −71.7172 + 58.9287i −2.94259 + 2.41788i
\(595\) 0 0
\(596\) 8.16048 41.2841i 0.334266 1.69106i
\(597\) −6.46078 −0.264422
\(598\) −20.0645 24.4188i −0.820497 0.998558i
\(599\) 36.1888i 1.47863i 0.673357 + 0.739317i \(0.264852\pi\)
−0.673357 + 0.739317i \(0.735148\pi\)
\(600\) −42.7455 + 22.9469i −1.74508 + 0.936801i
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 39.0440 + 7.71770i 1.58868 + 0.314029i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 8.60075 + 28.0500i 0.348806 + 1.13758i
\(609\) 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 \(0.726436\pi\)
\(614\) −17.8664 21.7437i −0.721031 0.877506i
\(615\) 0 0
\(616\) 3.56662 + 6.64392i 0.143703 + 0.267691i
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) −47.9725 + 39.4182i −1.92974 + 1.58563i
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 122.661 4.92221
\(622\) 22.9493 + 27.9297i 0.920184 + 1.11988i
\(623\) 0 0
\(624\) −18.8241 + 45.7552i −0.753565 + 1.83167i
\(625\) 25.0000 1.00000
\(626\) 25.6900 21.1090i 1.02678 0.843685i
\(627\) 59.0106i 2.35666i
\(628\) 5.50849 27.8676i 0.219813 1.11204i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 96.3574 + 19.0466i 3.82082 + 0.755248i
\(637\) 22.9091 0.907691
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 46.5860 1.84004 0.920019 0.391875i \(-0.128173\pi\)
0.920019 + 0.391875i \(0.128173\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 1.93231 9.77561i 0.0761438 0.385213i
\(645\) 0 0
\(646\) 0 0
\(647\) 46.1444i 1.81412i −0.420997 0.907062i \(-0.638320\pi\)
0.420997 0.907062i \(-0.361680\pi\)
\(648\) −55.6304 103.629i −2.18537 4.07092i
\(649\) 0 0
\(650\) 19.6983 16.1857i 0.772630 0.634856i
\(651\) 0 0
\(652\) 0 0
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −61.9675 75.4155i −2.42312 2.94898i
\(655\) 0 0
\(656\) −38.2461 15.7348i −1.49326 0.614339i
\(657\) 53.5308 2.08844
\(658\) 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
\(664\) 43.6728 23.4447i 1.69483 0.909829i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 44.6205 + 8.81998i 1.72642 + 0.341255i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) −14.9141 + 4.57300i −0.575325 + 0.176407i
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 98.9487i 3.80854i
\(676\) 5.04178 25.5065i 0.193915 0.981018i
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) −35.2047 42.8447i −1.35203 1.64544i
\(679\) 0 0
\(680\) 0 0
\(681\) 102.658 3.93384
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 89.2301 + 17.6378i 3.41180 + 0.674398i
\(685\) 0 0
\(686\) 9.63758 + 11.7291i 0.367965 + 0.447819i
\(687\) 0 0
\(688\) 0 0
\(689\) −51.6162 −1.96642
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 23.3777 0.888046
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −23.2558 + 19.1089i −0.880247 + 0.723283i
\(699\) 0 0
\(700\) 7.88585 + 1.55877i 0.298057 + 0.0589159i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 64.0623 + 77.9648i 2.41788 + 2.94259i
\(703\) 0 0
\(704\) −22.1142 14.6615i −0.833462 0.552577i
\(705\) 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
\(716\) −46.9251 9.27552i −1.75367 0.346643i
\(717\) 28.7369 1.07320
\(718\) 33.6336 + 40.9326i 1.25519 + 1.52759i
\(719\) 23.9165i 0.891936i 0.895049 + 0.445968i \(0.147140\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) 10.2876 0.383130
\(722\) −8.63120 + 7.09210i −0.321220 + 0.263941i
\(723\) 37.1304i 1.38089i
\(724\) −4.05983 + 20.5388i −0.150882 + 0.763318i
\(725\) 0 0
\(726\) 33.8803 + 41.2328i 1.25742 + 1.53029i
\(727\) 7.00140i 0.259668i 0.991536 + 0.129834i \(0.0414444\pi\)
−0.991536 + 0.129834i \(0.958556\pi\)
\(728\) 7.22270 3.87733i 0.267691 0.143703i
\(729\) −160.965 −5.96167
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −42.2040 −1.55884 −0.779420 0.626502i \(-0.784486\pi\)
−0.779420 + 0.626502i \(0.784486\pi\)
\(734\) 20.1710 + 24.5485i 0.744526 + 0.906100i
\(735\) 0 0
\(736\) 10.2785 + 33.5219i 0.378872 + 1.23563i
\(737\) 0 0
\(738\) −99.0611 + 81.3968i −3.64649 + 2.99626i
\(739\) 54.1823i 1.99313i 0.0828332 + 0.996563i \(0.473603\pi\)
−0.0828332 + 0.996563i \(0.526397\pi\)
\(740\) 0 0
\(741\) −64.1513 −2.35666
\(742\) −10.3318 12.5740i −0.379292 0.461605i
\(743\) 33.1662i 1.21675i −0.793649 0.608376i \(-0.791821\pi\)
0.793649 0.608376i \(-0.208179\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 153.670i 5.62249i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 43.0499i 1.57091i −0.618916 0.785457i \(-0.712428\pi\)
0.618916 0.785457i \(-0.287572\pi\)
\(752\) 0 0
\(753\) 55.2247 2.01250
\(754\) 0 0
\(755\) 0 0
\(756\) −6.16953 + 31.2118i −0.224384 + 1.13516i
\(757\) −55.0208 −1.99976 −0.999882 0.0153878i \(-0.995102\pi\)
−0.999882 + 0.0153878i \(0.995102\pi\)
\(758\) 0 0
\(759\) 70.5222i 2.55979i
\(760\) 0 0
\(761\) −52.4220 −1.90030 −0.950148 0.311800i \(-0.899068\pi\)
−0.950148 + 0.311800i \(0.899068\pi\)
\(762\) 0 0
\(763\) 16.1727i 0.585489i
\(764\) −9.76662 1.93053i −0.353344 0.0698443i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 38.6257 38.9979i 1.39379 1.40722i
\(769\) −51.9376 −1.87292 −0.936459 0.350776i \(-0.885918\pi\)
−0.936459 + 0.350776i \(0.885918\pi\)
\(770\) 0 0
\(771\) 41.0772i 1.47936i
\(772\) −3.91588 + 19.8105i −0.140936 + 0.712996i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −9.33600 + 7.67123i −0.334712 + 0.275027i
\(779\) 53.6231i 1.92125i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −23.5040 9.66971i −0.839427 0.345347i
\(785\) 0 0
\(786\) 0 0
\(787\) 55.7900i 1.98870i −0.106165 0.994349i \(-0.533857\pi\)
0.106165 0.994349i \(-0.466143\pi\)
\(788\) −9.84785 + 49.8205i −0.350815 + 1.77478i
\(789\) 0 0
\(790\) 0 0
\(791\) 9.18793i 0.326685i
\(792\) −72.4748 + 38.9063i −2.57528 + 1.38247i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −3.69512 0.730401i −0.130970 0.0258884i
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) −12.8409 15.6276i −0.454563 0.553210i
\(799\) 0 0
\(800\) −27.0416 + 8.29156i −0.956066 + 0.293151i
\(801\) 0 0
\(802\) 0 0
\(803\) 20.2472i 0.714510i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 109.592i 3.85784i
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 50.9669i 1.78969i 0.446377 + 0.894845i \(0.352714\pi\)
−0.446377 + 0.894845i \(0.647286\pi\)
\(812\) 0 0
\(813\) −91.7130 −3.21652
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −7.87931 + 6.47429i −0.275494 + 0.226368i
\(819\) 25.4143i 0.888046i
\(820\) 0 0
\(821\) −21.6333 −0.755008 −0.377504 0.926008i \(-0.623217\pi\)
−0.377504 + 0.926008i \(0.623217\pi\)
\(822\) 0 0
\(823\) 49.9872i 1.74244i 0.490890 + 0.871222i \(0.336672\pi\)
−0.490890 + 0.871222i \(0.663328\pi\)
\(824\) −31.8932 + 17.1211i −1.11105 + 0.596441i
\(825\) 56.8893 1.98063
\(826\) 0 0
\(827\) 13.1710i 0.458000i −0.973426 0.229000i \(-0.926454\pi\)
0.973426 0.229000i \(-0.0735456\pi\)
\(828\) 106.637 + 21.0785i 3.70588 + 0.732529i
\(829\) −48.8155 −1.69543 −0.847716 0.530450i \(-0.822023\pi\)
−0.847716 + 0.530450i \(0.822023\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −15.9387 + 24.0407i −0.552577 + 0.833462i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 6.67124 33.7500i 0.230730 1.16727i
\(837\) 0 0
\(838\) 36.7126 + 44.6798i 1.26822 + 1.54344i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 2.85318i 0.0982686i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 8.84228i 0.303824i
\(848\) 52.9565 + 21.7867i 1.81853 + 0.748159i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −4.29993 −0.147227 −0.0736134 0.997287i \(-0.523453\pi\)
−0.0736134 + 0.997287i \(0.523453\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 44.8248 36.8318i 1.53029 1.25742i
\(859\) 27.1083i 0.924925i 0.886639 + 0.462462i \(0.153034\pi\)
−0.886639 + 0.462462i \(0.846966\pi\)
\(860\) 0 0
\(861\) 28.5113 0.971664
\(862\) −29.3033 35.6626i −0.998074 1.21467i
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −32.8176 107.029i −1.11648 3.64122i
\(865\) 0 0
\(866\) 20.8976 17.1712i 0.710128 0.583499i
\(867\) 58.3194i 1.98063i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −26.9153 50.1379i −0.911467 1.69788i
\(873\) 0 0
\(874\) −35.1255 + 28.8620i −1.18814 + 0.976270i
\(875\) 0 0
\(876\) 41.0904 + 8.12220i 1.38832 + 0.274424i
\(877\) −35.9384 −1.21355 −0.606777 0.794872i \(-0.707538\pi\)
−0.606777 + 0.794872i \(0.707538\pi\)
\(878\) 0 0
\(879\) 74.2142i 2.50318i
\(880\) 0 0
\(881\) 5.65851 0.190640 0.0953201 0.995447i \(-0.469613\pi\)
0.0953201 + 0.995447i \(0.469613\pi\)
\(882\) −60.8775 + 50.0220i −2.04985 + 1.68433i
\(883\) 14.0583i 0.473098i −0.971620 0.236549i \(-0.923984\pi\)
0.971620 0.236549i \(-0.0760164\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −3.32329 4.04450i −0.111648 0.135877i
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 137.918i 4.62041i
\(892\) 0 0
\(893\) 0 0
\(894\) −64.8082 78.8726i −2.16751 2.63789i
\(895\) 0 0
\(896\) −9.04684 + 0.929374i −0.302234 + 0.0310482i
\(897\) −76.6657 −2.55979
\(898\) 0 0
\(899\) 0 0
\(900\) −17.0037 + 86.0224i −0.566792 + 2.86741i
\(901\) 0 0
\(902\) 30.7871 + 37.4684i 1.02510 + 1.24756i
\(903\) 0 0
\(904\) −15.2910 28.4841i −0.508571 0.947368i
\(905\) 0 0
\(906\) 74.5930 61.2917i 2.47819 2.03628i
\(907\) 59.8869i 1.98851i −0.107032 0.994256i \(-0.534135\pi\)
0.107032 0.994256i \(-0.465865\pi\)
\(908\) 58.7130 + 11.6056i 1.94846 + 0.385145i
\(909\) 0 0
\(910\) 0 0
\(911\) 50.7904i 1.68276i −0.540444 0.841380i \(-0.681744\pi\)
0.540444 0.841380i \(-0.318256\pi\)
\(912\) 65.8171 + 27.0777i 2.17942 + 0.896631i
\(913\) −58.1234 −1.92360
\(914\) 23.9556 19.6839i 0.792380 0.651084i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −68.2671 −2.24948
\(922\) −46.8478 + 38.4940i −1.54285 + 1.26773i
\(923\) 0 0
\(924\) 17.9448 + 3.54709i 0.590341 + 0.116691i
\(925\) 0 0
\(926\) 0 0
\(927\) 112.222i 3.68584i
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 32.9538i 1.08002i
\(932\) 0 0
\(933\) 87.6887 2.87080
\(934\) −21.4728 26.1327i −0.702611 0.855089i
\(935\) 0 0
\(936\) 42.2956 + 78.7884i 1.38247 + 2.57528i
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 80.6567i 2.63213i
\(940\) 0 0
\(941\) 27.0556 0.881988 0.440994 0.897510i \(-0.354626\pi\)
0.440994 + 0.897510i \(0.354626\pi\)
\(942\) −43.7469 53.2406i −1.42535 1.73467i
\(943\) 64.0837i 2.08685i
\(944\) 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
\(950\) −23.2825 28.3352i −0.755385 0.919316i
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 137.162 112.704i 4.44080 3.64892i
\(955\) 0 0
\(956\) 16.4355 + 3.24875i 0.531562 + 0.105072i
\(957\) 0 0
\(958\) −5.95548 7.24791i −0.192413 0.234169i
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 4.19765 21.2360i 0.135197 0.683965i
\(965\) 0 0
\(966\) −15.3459 18.6762i −0.493745 0.600895i
\(967\) 57.8529i 1.86042i 0.367024 + 0.930211i \(0.380377\pi\)
−0.367024 + 0.930211i \(0.619623\pi\)
\(968\) 14.7157 + 27.4125i 0.472982 + 0.881072i
\(969\) 0 0
\(970\) 0 0
\(971\) 20.9194i 0.671335i 0.941981 + 0.335667i \(0.108962\pi\)
−0.941981 + 0.335667i \(0.891038\pi\)
\(972\) −163.410 32.3006i −5.24137 1.03604i
\(973\) 0 0
\(974\) 0 0
\(975\) 61.8451i 1.98063i
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −176.418 −5.63261
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −88.3899 + 47.4499i −2.81777 + 1.51265i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −36.6901 7.25240i −1.16727 0.230730i
\(989\) 0 0
\(990\) 0 0
\(991\) 59.8666i 1.90173i −0.309613 0.950863i \(-0.600199\pi\)
0.309613 0.950863i \(-0.399801\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 23.3163 117.958i 0.738804 3.73763i
\(997\) 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 572.2.b.b.571.15 yes 20
4.3 odd 2 inner 572.2.b.b.571.16 yes 20
11.10 odd 2 inner 572.2.b.b.571.6 yes 20
13.12 even 2 inner 572.2.b.b.571.6 yes 20
44.43 even 2 inner 572.2.b.b.571.5 20
52.51 odd 2 inner 572.2.b.b.571.5 20
143.142 odd 2 CM 572.2.b.b.571.15 yes 20
572.571 even 2 inner 572.2.b.b.571.16 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
572.2.b.b.571.5 20 44.43 even 2 inner
572.2.b.b.571.5 20 52.51 odd 2 inner
572.2.b.b.571.6 yes 20 11.10 odd 2 inner
572.2.b.b.571.6 yes 20 13.12 even 2 inner
572.2.b.b.571.15 yes 20 1.1 even 1 trivial
572.2.b.b.571.15 yes 20 143.142 odd 2 CM
572.2.b.b.571.16 yes 20 4.3 odd 2 inner
572.2.b.b.571.16 yes 20 572.571 even 2 inner