Properties

Label 3040.2.a.x.1.2
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $1$
Dimension $5$
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] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, 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("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.387268.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.81079\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.531822 q^{3} -1.00000 q^{5} +0.0955795 q^{7} -2.71717 q^{9} +O(q^{10})\) \(q-0.531822 q^{3} -1.00000 q^{5} +0.0955795 q^{7} -2.71717 q^{9} -0.337570 q^{11} +4.49098 q^{13} +0.531822 q^{15} -1.68405 q^{17} -1.00000 q^{19} -0.0508313 q^{21} +2.51569 q^{23} +1.00000 q^{25} +3.04051 q^{27} -6.61268 q^{29} +6.79080 q^{31} +0.179527 q^{33} -0.0955795 q^{35} +6.28059 q^{37} -2.38840 q^{39} +1.06364 q^{41} -0.325938 q^{43} +2.71717 q^{45} -8.94884 q^{47} -6.99086 q^{49} +0.895614 q^{51} -14.4587 q^{53} +0.337570 q^{55} +0.531822 q^{57} +1.20007 q^{59} -1.21006 q^{61} -0.259705 q^{63} -4.49098 q^{65} +0.111712 q^{67} -1.33790 q^{69} -14.9003 q^{71} +2.47485 q^{73} -0.531822 q^{75} -0.0322648 q^{77} +8.33842 q^{79} +6.53448 q^{81} +17.9308 q^{83} +1.68405 q^{85} +3.51677 q^{87} -13.6749 q^{89} +0.429246 q^{91} -3.61150 q^{93} +1.00000 q^{95} -10.7213 q^{97} +0.917233 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 4 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9} + 2 q^{11} - 4 q^{13} - 4 q^{15} - 12 q^{17} - 5 q^{19} - 10 q^{21} - 8 q^{23} + 5 q^{25} + 16 q^{27} - 6 q^{29} - 10 q^{31} - 18 q^{33} + 4 q^{35} - 6 q^{37} - 18 q^{39} - 8 q^{41} + 12 q^{43} - 7 q^{45} - 16 q^{47} + 7 q^{49} - 14 q^{51} - 18 q^{53} - 2 q^{55} - 4 q^{57} + 8 q^{59} + 2 q^{61} - 36 q^{63} + 4 q^{65} + 10 q^{67} - 22 q^{69} + 18 q^{71} - 28 q^{73} + 4 q^{75} - 28 q^{77} - 14 q^{79} + 25 q^{81} + 8 q^{83} + 12 q^{85} - 24 q^{87} - 30 q^{89} + 28 q^{91} - 24 q^{93} + 5 q^{95} - 18 q^{97} - 14 q^{99}+O(q^{100}) \) 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
\(3\) −0.531822 −0.307048 −0.153524 0.988145i \(-0.549062\pi\)
−0.153524 + 0.988145i \(0.549062\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.0955795 0.0361257 0.0180628 0.999837i \(-0.494250\pi\)
0.0180628 + 0.999837i \(0.494250\pi\)
\(8\) 0 0
\(9\) −2.71717 −0.905722
\(10\) 0 0
\(11\) −0.337570 −0.101781 −0.0508906 0.998704i \(-0.516206\pi\)
−0.0508906 + 0.998704i \(0.516206\pi\)
\(12\) 0 0
\(13\) 4.49098 1.24557 0.622787 0.782392i \(-0.286000\pi\)
0.622787 + 0.782392i \(0.286000\pi\)
\(14\) 0 0
\(15\) 0.531822 0.137316
\(16\) 0 0
\(17\) −1.68405 −0.408442 −0.204221 0.978925i \(-0.565466\pi\)
−0.204221 + 0.978925i \(0.565466\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −0.0508313 −0.0110923
\(22\) 0 0
\(23\) 2.51569 0.524558 0.262279 0.964992i \(-0.415526\pi\)
0.262279 + 0.964992i \(0.415526\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 3.04051 0.585147
\(28\) 0 0
\(29\) −6.61268 −1.22794 −0.613972 0.789328i \(-0.710429\pi\)
−0.613972 + 0.789328i \(0.710429\pi\)
\(30\) 0 0
\(31\) 6.79080 1.21966 0.609832 0.792531i \(-0.291237\pi\)
0.609832 + 0.792531i \(0.291237\pi\)
\(32\) 0 0
\(33\) 0.179527 0.0312517
\(34\) 0 0
\(35\) −0.0955795 −0.0161559
\(36\) 0 0
\(37\) 6.28059 1.03252 0.516262 0.856431i \(-0.327323\pi\)
0.516262 + 0.856431i \(0.327323\pi\)
\(38\) 0 0
\(39\) −2.38840 −0.382450
\(40\) 0 0
\(41\) 1.06364 0.166113 0.0830567 0.996545i \(-0.473532\pi\)
0.0830567 + 0.996545i \(0.473532\pi\)
\(42\) 0 0
\(43\) −0.325938 −0.0497051 −0.0248525 0.999691i \(-0.507912\pi\)
−0.0248525 + 0.999691i \(0.507912\pi\)
\(44\) 0 0
\(45\) 2.71717 0.405051
\(46\) 0 0
\(47\) −8.94884 −1.30532 −0.652661 0.757650i \(-0.726347\pi\)
−0.652661 + 0.757650i \(0.726347\pi\)
\(48\) 0 0
\(49\) −6.99086 −0.998695
\(50\) 0 0
\(51\) 0.895614 0.125411
\(52\) 0 0
\(53\) −14.4587 −1.98606 −0.993028 0.117875i \(-0.962392\pi\)
−0.993028 + 0.117875i \(0.962392\pi\)
\(54\) 0 0
\(55\) 0.337570 0.0455179
\(56\) 0 0
\(57\) 0.531822 0.0704416
\(58\) 0 0
\(59\) 1.20007 0.156236 0.0781178 0.996944i \(-0.475109\pi\)
0.0781178 + 0.996944i \(0.475109\pi\)
\(60\) 0 0
\(61\) −1.21006 −0.154932 −0.0774658 0.996995i \(-0.524683\pi\)
−0.0774658 + 0.996995i \(0.524683\pi\)
\(62\) 0 0
\(63\) −0.259705 −0.0327198
\(64\) 0 0
\(65\) −4.49098 −0.557037
\(66\) 0 0
\(67\) 0.111712 0.0136478 0.00682389 0.999977i \(-0.497828\pi\)
0.00682389 + 0.999977i \(0.497828\pi\)
\(68\) 0 0
\(69\) −1.33790 −0.161064
\(70\) 0 0
\(71\) −14.9003 −1.76834 −0.884168 0.467169i \(-0.845274\pi\)
−0.884168 + 0.467169i \(0.845274\pi\)
\(72\) 0 0
\(73\) 2.47485 0.289659 0.144829 0.989457i \(-0.453737\pi\)
0.144829 + 0.989457i \(0.453737\pi\)
\(74\) 0 0
\(75\) −0.531822 −0.0614095
\(76\) 0 0
\(77\) −0.0322648 −0.00367691
\(78\) 0 0
\(79\) 8.33842 0.938146 0.469073 0.883159i \(-0.344588\pi\)
0.469073 + 0.883159i \(0.344588\pi\)
\(80\) 0 0
\(81\) 6.53448 0.726054
\(82\) 0 0
\(83\) 17.9308 1.96816 0.984080 0.177725i \(-0.0568737\pi\)
0.984080 + 0.177725i \(0.0568737\pi\)
\(84\) 0 0
\(85\) 1.68405 0.182661
\(86\) 0 0
\(87\) 3.51677 0.377037
\(88\) 0 0
\(89\) −13.6749 −1.44954 −0.724769 0.688992i \(-0.758054\pi\)
−0.724769 + 0.688992i \(0.758054\pi\)
\(90\) 0 0
\(91\) 0.429246 0.0449972
\(92\) 0 0
\(93\) −3.61150 −0.374495
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −10.7213 −1.08859 −0.544293 0.838895i \(-0.683202\pi\)
−0.544293 + 0.838895i \(0.683202\pi\)
\(98\) 0 0
\(99\) 0.917233 0.0921854
\(100\) 0 0
\(101\) −3.66243 −0.364425 −0.182213 0.983259i \(-0.558326\pi\)
−0.182213 + 0.983259i \(0.558326\pi\)
\(102\) 0 0
\(103\) −18.5427 −1.82706 −0.913531 0.406768i \(-0.866656\pi\)
−0.913531 + 0.406768i \(0.866656\pi\)
\(104\) 0 0
\(105\) 0.0508313 0.00496063
\(106\) 0 0
\(107\) 14.5024 1.40200 0.700999 0.713163i \(-0.252738\pi\)
0.700999 + 0.713163i \(0.252738\pi\)
\(108\) 0 0
\(109\) −10.8354 −1.03785 −0.518924 0.854821i \(-0.673667\pi\)
−0.518924 + 0.854821i \(0.673667\pi\)
\(110\) 0 0
\(111\) −3.34016 −0.317034
\(112\) 0 0
\(113\) −17.2193 −1.61986 −0.809928 0.586529i \(-0.800494\pi\)
−0.809928 + 0.586529i \(0.800494\pi\)
\(114\) 0 0
\(115\) −2.51569 −0.234589
\(116\) 0 0
\(117\) −12.2027 −1.12814
\(118\) 0 0
\(119\) −0.160961 −0.0147552
\(120\) 0 0
\(121\) −10.8860 −0.989641
\(122\) 0 0
\(123\) −0.565670 −0.0510047
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −6.78390 −0.601974 −0.300987 0.953628i \(-0.597316\pi\)
−0.300987 + 0.953628i \(0.597316\pi\)
\(128\) 0 0
\(129\) 0.173341 0.0152618
\(130\) 0 0
\(131\) 16.7091 1.45988 0.729941 0.683510i \(-0.239548\pi\)
0.729941 + 0.683510i \(0.239548\pi\)
\(132\) 0 0
\(133\) −0.0955795 −0.00828780
\(134\) 0 0
\(135\) −3.04051 −0.261686
\(136\) 0 0
\(137\) −9.00891 −0.769683 −0.384842 0.922983i \(-0.625744\pi\)
−0.384842 + 0.922983i \(0.625744\pi\)
\(138\) 0 0
\(139\) 7.43282 0.630444 0.315222 0.949018i \(-0.397921\pi\)
0.315222 + 0.949018i \(0.397921\pi\)
\(140\) 0 0
\(141\) 4.75919 0.400796
\(142\) 0 0
\(143\) −1.51602 −0.126776
\(144\) 0 0
\(145\) 6.61268 0.549153
\(146\) 0 0
\(147\) 3.71790 0.306647
\(148\) 0 0
\(149\) 4.98045 0.408014 0.204007 0.978969i \(-0.434603\pi\)
0.204007 + 0.978969i \(0.434603\pi\)
\(150\) 0 0
\(151\) −19.8661 −1.61668 −0.808339 0.588717i \(-0.799633\pi\)
−0.808339 + 0.588717i \(0.799633\pi\)
\(152\) 0 0
\(153\) 4.57584 0.369935
\(154\) 0 0
\(155\) −6.79080 −0.545450
\(156\) 0 0
\(157\) −13.9183 −1.11080 −0.555401 0.831583i \(-0.687435\pi\)
−0.555401 + 0.831583i \(0.687435\pi\)
\(158\) 0 0
\(159\) 7.68946 0.609814
\(160\) 0 0
\(161\) 0.240448 0.0189500
\(162\) 0 0
\(163\) −7.12837 −0.558337 −0.279168 0.960242i \(-0.590059\pi\)
−0.279168 + 0.960242i \(0.590059\pi\)
\(164\) 0 0
\(165\) −0.179527 −0.0139762
\(166\) 0 0
\(167\) −23.0523 −1.78384 −0.891919 0.452195i \(-0.850641\pi\)
−0.891919 + 0.452195i \(0.850641\pi\)
\(168\) 0 0
\(169\) 7.16888 0.551452
\(170\) 0 0
\(171\) 2.71717 0.207787
\(172\) 0 0
\(173\) 10.4279 0.792815 0.396408 0.918075i \(-0.370257\pi\)
0.396408 + 0.918075i \(0.370257\pi\)
\(174\) 0 0
\(175\) 0.0955795 0.00722513
\(176\) 0 0
\(177\) −0.638223 −0.0479718
\(178\) 0 0
\(179\) −13.2047 −0.986967 −0.493484 0.869755i \(-0.664277\pi\)
−0.493484 + 0.869755i \(0.664277\pi\)
\(180\) 0 0
\(181\) 12.3384 0.917108 0.458554 0.888667i \(-0.348368\pi\)
0.458554 + 0.888667i \(0.348368\pi\)
\(182\) 0 0
\(183\) 0.643534 0.0475714
\(184\) 0 0
\(185\) −6.28059 −0.461758
\(186\) 0 0
\(187\) 0.568484 0.0415717
\(188\) 0 0
\(189\) 0.290611 0.0211388
\(190\) 0 0
\(191\) 18.3515 1.32787 0.663933 0.747792i \(-0.268886\pi\)
0.663933 + 0.747792i \(0.268886\pi\)
\(192\) 0 0
\(193\) 5.74437 0.413489 0.206745 0.978395i \(-0.433713\pi\)
0.206745 + 0.978395i \(0.433713\pi\)
\(194\) 0 0
\(195\) 2.38840 0.171037
\(196\) 0 0
\(197\) −22.8975 −1.63138 −0.815688 0.578492i \(-0.803641\pi\)
−0.815688 + 0.578492i \(0.803641\pi\)
\(198\) 0 0
\(199\) 4.38925 0.311146 0.155573 0.987824i \(-0.450278\pi\)
0.155573 + 0.987824i \(0.450278\pi\)
\(200\) 0 0
\(201\) −0.0594109 −0.00419052
\(202\) 0 0
\(203\) −0.632037 −0.0443603
\(204\) 0 0
\(205\) −1.06364 −0.0742881
\(206\) 0 0
\(207\) −6.83554 −0.475103
\(208\) 0 0
\(209\) 0.337570 0.0233502
\(210\) 0 0
\(211\) 20.3932 1.40392 0.701961 0.712215i \(-0.252308\pi\)
0.701961 + 0.712215i \(0.252308\pi\)
\(212\) 0 0
\(213\) 7.92429 0.542963
\(214\) 0 0
\(215\) 0.325938 0.0222288
\(216\) 0 0
\(217\) 0.649061 0.0440611
\(218\) 0 0
\(219\) −1.31618 −0.0889390
\(220\) 0 0
\(221\) −7.56303 −0.508744
\(222\) 0 0
\(223\) 27.1245 1.81639 0.908195 0.418548i \(-0.137461\pi\)
0.908195 + 0.418548i \(0.137461\pi\)
\(224\) 0 0
\(225\) −2.71717 −0.181144
\(226\) 0 0
\(227\) −15.8638 −1.05292 −0.526460 0.850200i \(-0.676481\pi\)
−0.526460 + 0.850200i \(0.676481\pi\)
\(228\) 0 0
\(229\) −11.1600 −0.737472 −0.368736 0.929534i \(-0.620209\pi\)
−0.368736 + 0.929534i \(0.620209\pi\)
\(230\) 0 0
\(231\) 0.0171591 0.00112899
\(232\) 0 0
\(233\) −6.40848 −0.419833 −0.209917 0.977719i \(-0.567319\pi\)
−0.209917 + 0.977719i \(0.567319\pi\)
\(234\) 0 0
\(235\) 8.94884 0.583758
\(236\) 0 0
\(237\) −4.43456 −0.288055
\(238\) 0 0
\(239\) −2.59204 −0.167665 −0.0838327 0.996480i \(-0.526716\pi\)
−0.0838327 + 0.996480i \(0.526716\pi\)
\(240\) 0 0
\(241\) 3.86108 0.248714 0.124357 0.992238i \(-0.460313\pi\)
0.124357 + 0.992238i \(0.460313\pi\)
\(242\) 0 0
\(243\) −12.5967 −0.808080
\(244\) 0 0
\(245\) 6.99086 0.446630
\(246\) 0 0
\(247\) −4.49098 −0.285754
\(248\) 0 0
\(249\) −9.53599 −0.604319
\(250\) 0 0
\(251\) −13.7699 −0.869151 −0.434575 0.900635i \(-0.643102\pi\)
−0.434575 + 0.900635i \(0.643102\pi\)
\(252\) 0 0
\(253\) −0.849221 −0.0533901
\(254\) 0 0
\(255\) −0.895614 −0.0560856
\(256\) 0 0
\(257\) −10.1649 −0.634071 −0.317036 0.948414i \(-0.602687\pi\)
−0.317036 + 0.948414i \(0.602687\pi\)
\(258\) 0 0
\(259\) 0.600296 0.0373006
\(260\) 0 0
\(261\) 17.9677 1.11217
\(262\) 0 0
\(263\) 9.49624 0.585563 0.292782 0.956179i \(-0.405419\pi\)
0.292782 + 0.956179i \(0.405419\pi\)
\(264\) 0 0
\(265\) 14.4587 0.888192
\(266\) 0 0
\(267\) 7.27262 0.445077
\(268\) 0 0
\(269\) 7.83381 0.477636 0.238818 0.971064i \(-0.423240\pi\)
0.238818 + 0.971064i \(0.423240\pi\)
\(270\) 0 0
\(271\) −24.7086 −1.50094 −0.750469 0.660906i \(-0.770172\pi\)
−0.750469 + 0.660906i \(0.770172\pi\)
\(272\) 0 0
\(273\) −0.228282 −0.0138163
\(274\) 0 0
\(275\) −0.337570 −0.0203562
\(276\) 0 0
\(277\) −6.06342 −0.364315 −0.182158 0.983269i \(-0.558308\pi\)
−0.182158 + 0.983269i \(0.558308\pi\)
\(278\) 0 0
\(279\) −18.4517 −1.10468
\(280\) 0 0
\(281\) 24.7958 1.47919 0.739596 0.673051i \(-0.235016\pi\)
0.739596 + 0.673051i \(0.235016\pi\)
\(282\) 0 0
\(283\) −16.1626 −0.960765 −0.480382 0.877059i \(-0.659502\pi\)
−0.480382 + 0.877059i \(0.659502\pi\)
\(284\) 0 0
\(285\) −0.531822 −0.0315024
\(286\) 0 0
\(287\) 0.101663 0.00600096
\(288\) 0 0
\(289\) −14.1640 −0.833175
\(290\) 0 0
\(291\) 5.70184 0.334248
\(292\) 0 0
\(293\) −8.52258 −0.497895 −0.248947 0.968517i \(-0.580085\pi\)
−0.248947 + 0.968517i \(0.580085\pi\)
\(294\) 0 0
\(295\) −1.20007 −0.0698707
\(296\) 0 0
\(297\) −1.02639 −0.0595570
\(298\) 0 0
\(299\) 11.2979 0.653375
\(300\) 0 0
\(301\) −0.0311530 −0.00179563
\(302\) 0 0
\(303\) 1.94776 0.111896
\(304\) 0 0
\(305\) 1.21006 0.0692876
\(306\) 0 0
\(307\) 24.4232 1.39390 0.696952 0.717118i \(-0.254539\pi\)
0.696952 + 0.717118i \(0.254539\pi\)
\(308\) 0 0
\(309\) 9.86140 0.560995
\(310\) 0 0
\(311\) −7.67265 −0.435076 −0.217538 0.976052i \(-0.569803\pi\)
−0.217538 + 0.976052i \(0.569803\pi\)
\(312\) 0 0
\(313\) 2.02932 0.114704 0.0573518 0.998354i \(-0.481734\pi\)
0.0573518 + 0.998354i \(0.481734\pi\)
\(314\) 0 0
\(315\) 0.259705 0.0146327
\(316\) 0 0
\(317\) −5.44515 −0.305830 −0.152915 0.988239i \(-0.548866\pi\)
−0.152915 + 0.988239i \(0.548866\pi\)
\(318\) 0 0
\(319\) 2.23224 0.124982
\(320\) 0 0
\(321\) −7.71268 −0.430480
\(322\) 0 0
\(323\) 1.68405 0.0937030
\(324\) 0 0
\(325\) 4.49098 0.249115
\(326\) 0 0
\(327\) 5.76253 0.318668
\(328\) 0 0
\(329\) −0.855326 −0.0471556
\(330\) 0 0
\(331\) −6.87780 −0.378038 −0.189019 0.981973i \(-0.560531\pi\)
−0.189019 + 0.981973i \(0.560531\pi\)
\(332\) 0 0
\(333\) −17.0654 −0.935179
\(334\) 0 0
\(335\) −0.111712 −0.00610348
\(336\) 0 0
\(337\) −4.21120 −0.229398 −0.114699 0.993400i \(-0.536590\pi\)
−0.114699 + 0.993400i \(0.536590\pi\)
\(338\) 0 0
\(339\) 9.15761 0.497373
\(340\) 0 0
\(341\) −2.29237 −0.124139
\(342\) 0 0
\(343\) −1.33724 −0.0722042
\(344\) 0 0
\(345\) 1.33790 0.0720301
\(346\) 0 0
\(347\) −3.64180 −0.195502 −0.0977510 0.995211i \(-0.531165\pi\)
−0.0977510 + 0.995211i \(0.531165\pi\)
\(348\) 0 0
\(349\) 28.5835 1.53004 0.765020 0.644006i \(-0.222729\pi\)
0.765020 + 0.644006i \(0.222729\pi\)
\(350\) 0 0
\(351\) 13.6549 0.728844
\(352\) 0 0
\(353\) 7.26564 0.386711 0.193356 0.981129i \(-0.438063\pi\)
0.193356 + 0.981129i \(0.438063\pi\)
\(354\) 0 0
\(355\) 14.9003 0.790824
\(356\) 0 0
\(357\) 0.0856024 0.00453056
\(358\) 0 0
\(359\) −4.80851 −0.253784 −0.126892 0.991917i \(-0.540500\pi\)
−0.126892 + 0.991917i \(0.540500\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 5.78944 0.303867
\(364\) 0 0
\(365\) −2.47485 −0.129539
\(366\) 0 0
\(367\) −27.9947 −1.46131 −0.730655 0.682747i \(-0.760785\pi\)
−0.730655 + 0.682747i \(0.760785\pi\)
\(368\) 0 0
\(369\) −2.89010 −0.150452
\(370\) 0 0
\(371\) −1.38196 −0.0717476
\(372\) 0 0
\(373\) −30.9677 −1.60344 −0.801722 0.597697i \(-0.796083\pi\)
−0.801722 + 0.597697i \(0.796083\pi\)
\(374\) 0 0
\(375\) 0.531822 0.0274632
\(376\) 0 0
\(377\) −29.6974 −1.52949
\(378\) 0 0
\(379\) −20.3238 −1.04396 −0.521981 0.852957i \(-0.674807\pi\)
−0.521981 + 0.852957i \(0.674807\pi\)
\(380\) 0 0
\(381\) 3.60783 0.184835
\(382\) 0 0
\(383\) −19.3567 −0.989082 −0.494541 0.869154i \(-0.664664\pi\)
−0.494541 + 0.869154i \(0.664664\pi\)
\(384\) 0 0
\(385\) 0.0322648 0.00164437
\(386\) 0 0
\(387\) 0.885627 0.0450190
\(388\) 0 0
\(389\) −24.5206 −1.24324 −0.621621 0.783318i \(-0.713526\pi\)
−0.621621 + 0.783318i \(0.713526\pi\)
\(390\) 0 0
\(391\) −4.23654 −0.214251
\(392\) 0 0
\(393\) −8.88627 −0.448253
\(394\) 0 0
\(395\) −8.33842 −0.419552
\(396\) 0 0
\(397\) −9.23699 −0.463591 −0.231796 0.972765i \(-0.574460\pi\)
−0.231796 + 0.972765i \(0.574460\pi\)
\(398\) 0 0
\(399\) 0.0508313 0.00254475
\(400\) 0 0
\(401\) 21.0570 1.05154 0.525768 0.850628i \(-0.323778\pi\)
0.525768 + 0.850628i \(0.323778\pi\)
\(402\) 0 0
\(403\) 30.4973 1.51918
\(404\) 0 0
\(405\) −6.53448 −0.324701
\(406\) 0 0
\(407\) −2.12014 −0.105091
\(408\) 0 0
\(409\) 29.4567 1.45654 0.728269 0.685291i \(-0.240325\pi\)
0.728269 + 0.685291i \(0.240325\pi\)
\(410\) 0 0
\(411\) 4.79114 0.236329
\(412\) 0 0
\(413\) 0.114702 0.00564411
\(414\) 0 0
\(415\) −17.9308 −0.880188
\(416\) 0 0
\(417\) −3.95294 −0.193576
\(418\) 0 0
\(419\) 26.4973 1.29448 0.647239 0.762287i \(-0.275923\pi\)
0.647239 + 0.762287i \(0.275923\pi\)
\(420\) 0 0
\(421\) 20.0442 0.976894 0.488447 0.872594i \(-0.337564\pi\)
0.488447 + 0.872594i \(0.337564\pi\)
\(422\) 0 0
\(423\) 24.3155 1.18226
\(424\) 0 0
\(425\) −1.68405 −0.0816884
\(426\) 0 0
\(427\) −0.115657 −0.00559701
\(428\) 0 0
\(429\) 0.806253 0.0389262
\(430\) 0 0
\(431\) 17.6186 0.848660 0.424330 0.905508i \(-0.360510\pi\)
0.424330 + 0.905508i \(0.360510\pi\)
\(432\) 0 0
\(433\) 8.99700 0.432368 0.216184 0.976353i \(-0.430639\pi\)
0.216184 + 0.976353i \(0.430639\pi\)
\(434\) 0 0
\(435\) −3.51677 −0.168616
\(436\) 0 0
\(437\) −2.51569 −0.120342
\(438\) 0 0
\(439\) 16.0456 0.765815 0.382907 0.923787i \(-0.374923\pi\)
0.382907 + 0.923787i \(0.374923\pi\)
\(440\) 0 0
\(441\) 18.9953 0.904540
\(442\) 0 0
\(443\) −17.7087 −0.841365 −0.420683 0.907208i \(-0.638209\pi\)
−0.420683 + 0.907208i \(0.638209\pi\)
\(444\) 0 0
\(445\) 13.6749 0.648253
\(446\) 0 0
\(447\) −2.64871 −0.125280
\(448\) 0 0
\(449\) 29.2271 1.37931 0.689655 0.724138i \(-0.257762\pi\)
0.689655 + 0.724138i \(0.257762\pi\)
\(450\) 0 0
\(451\) −0.359054 −0.0169072
\(452\) 0 0
\(453\) 10.5652 0.496397
\(454\) 0 0
\(455\) −0.429246 −0.0201233
\(456\) 0 0
\(457\) −6.29269 −0.294359 −0.147180 0.989110i \(-0.547020\pi\)
−0.147180 + 0.989110i \(0.547020\pi\)
\(458\) 0 0
\(459\) −5.12038 −0.238999
\(460\) 0 0
\(461\) 8.76201 0.408087 0.204044 0.978962i \(-0.434592\pi\)
0.204044 + 0.978962i \(0.434592\pi\)
\(462\) 0 0
\(463\) 3.21624 0.149471 0.0747357 0.997203i \(-0.476189\pi\)
0.0747357 + 0.997203i \(0.476189\pi\)
\(464\) 0 0
\(465\) 3.61150 0.167479
\(466\) 0 0
\(467\) 31.4584 1.45572 0.727862 0.685724i \(-0.240514\pi\)
0.727862 + 0.685724i \(0.240514\pi\)
\(468\) 0 0
\(469\) 0.0106774 0.000493035 0
\(470\) 0 0
\(471\) 7.40207 0.341069
\(472\) 0 0
\(473\) 0.110027 0.00505904
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 39.2867 1.79881
\(478\) 0 0
\(479\) 13.7313 0.627399 0.313699 0.949522i \(-0.398432\pi\)
0.313699 + 0.949522i \(0.398432\pi\)
\(480\) 0 0
\(481\) 28.2060 1.28608
\(482\) 0 0
\(483\) −0.127876 −0.00581855
\(484\) 0 0
\(485\) 10.7213 0.486831
\(486\) 0 0
\(487\) 6.70766 0.303953 0.151977 0.988384i \(-0.451436\pi\)
0.151977 + 0.988384i \(0.451436\pi\)
\(488\) 0 0
\(489\) 3.79102 0.171436
\(490\) 0 0
\(491\) −16.7613 −0.756429 −0.378214 0.925718i \(-0.623462\pi\)
−0.378214 + 0.925718i \(0.623462\pi\)
\(492\) 0 0
\(493\) 11.1361 0.501543
\(494\) 0 0
\(495\) −0.917233 −0.0412266
\(496\) 0 0
\(497\) −1.42416 −0.0638823
\(498\) 0 0
\(499\) −17.3105 −0.774926 −0.387463 0.921885i \(-0.626648\pi\)
−0.387463 + 0.921885i \(0.626648\pi\)
\(500\) 0 0
\(501\) 12.2597 0.547723
\(502\) 0 0
\(503\) −38.7748 −1.72888 −0.864441 0.502734i \(-0.832327\pi\)
−0.864441 + 0.502734i \(0.832327\pi\)
\(504\) 0 0
\(505\) 3.66243 0.162976
\(506\) 0 0
\(507\) −3.81257 −0.169322
\(508\) 0 0
\(509\) −3.39492 −0.150477 −0.0752385 0.997166i \(-0.523972\pi\)
−0.0752385 + 0.997166i \(0.523972\pi\)
\(510\) 0 0
\(511\) 0.236545 0.0104641
\(512\) 0 0
\(513\) −3.04051 −0.134242
\(514\) 0 0
\(515\) 18.5427 0.817087
\(516\) 0 0
\(517\) 3.02086 0.132857
\(518\) 0 0
\(519\) −5.54576 −0.243432
\(520\) 0 0
\(521\) 5.32985 0.233505 0.116753 0.993161i \(-0.462752\pi\)
0.116753 + 0.993161i \(0.462752\pi\)
\(522\) 0 0
\(523\) −7.55465 −0.330342 −0.165171 0.986265i \(-0.552818\pi\)
−0.165171 + 0.986265i \(0.552818\pi\)
\(524\) 0 0
\(525\) −0.0508313 −0.00221846
\(526\) 0 0
\(527\) −11.4360 −0.498161
\(528\) 0 0
\(529\) −16.6713 −0.724839
\(530\) 0 0
\(531\) −3.26078 −0.141506
\(532\) 0 0
\(533\) 4.77680 0.206906
\(534\) 0 0
\(535\) −14.5024 −0.626992
\(536\) 0 0
\(537\) 7.02256 0.303046
\(538\) 0 0
\(539\) 2.35991 0.101648
\(540\) 0 0
\(541\) 28.8850 1.24186 0.620931 0.783865i \(-0.286755\pi\)
0.620931 + 0.783865i \(0.286755\pi\)
\(542\) 0 0
\(543\) −6.56185 −0.281596
\(544\) 0 0
\(545\) 10.8354 0.464139
\(546\) 0 0
\(547\) 28.7106 1.22757 0.613787 0.789471i \(-0.289645\pi\)
0.613787 + 0.789471i \(0.289645\pi\)
\(548\) 0 0
\(549\) 3.28792 0.140325
\(550\) 0 0
\(551\) 6.61268 0.281709
\(552\) 0 0
\(553\) 0.796983 0.0338911
\(554\) 0 0
\(555\) 3.34016 0.141782
\(556\) 0 0
\(557\) −5.93658 −0.251541 −0.125771 0.992059i \(-0.540140\pi\)
−0.125771 + 0.992059i \(0.540140\pi\)
\(558\) 0 0
\(559\) −1.46378 −0.0619113
\(560\) 0 0
\(561\) −0.302333 −0.0127645
\(562\) 0 0
\(563\) 15.9031 0.670235 0.335117 0.942176i \(-0.391224\pi\)
0.335117 + 0.942176i \(0.391224\pi\)
\(564\) 0 0
\(565\) 17.2193 0.724422
\(566\) 0 0
\(567\) 0.624563 0.0262292
\(568\) 0 0
\(569\) −18.4451 −0.773258 −0.386629 0.922235i \(-0.626361\pi\)
−0.386629 + 0.922235i \(0.626361\pi\)
\(570\) 0 0
\(571\) 34.2148 1.43184 0.715922 0.698180i \(-0.246006\pi\)
0.715922 + 0.698180i \(0.246006\pi\)
\(572\) 0 0
\(573\) −9.75971 −0.407718
\(574\) 0 0
\(575\) 2.51569 0.104912
\(576\) 0 0
\(577\) 7.51233 0.312742 0.156371 0.987698i \(-0.450020\pi\)
0.156371 + 0.987698i \(0.450020\pi\)
\(578\) 0 0
\(579\) −3.05498 −0.126961
\(580\) 0 0
\(581\) 1.71382 0.0711011
\(582\) 0 0
\(583\) 4.88083 0.202143
\(584\) 0 0
\(585\) 12.2027 0.504521
\(586\) 0 0
\(587\) 18.0757 0.746064 0.373032 0.927819i \(-0.378318\pi\)
0.373032 + 0.927819i \(0.378318\pi\)
\(588\) 0 0
\(589\) −6.79080 −0.279810
\(590\) 0 0
\(591\) 12.1774 0.500910
\(592\) 0 0
\(593\) −2.52243 −0.103584 −0.0517919 0.998658i \(-0.516493\pi\)
−0.0517919 + 0.998658i \(0.516493\pi\)
\(594\) 0 0
\(595\) 0.160961 0.00659874
\(596\) 0 0
\(597\) −2.33430 −0.0955366
\(598\) 0 0
\(599\) −34.6184 −1.41447 −0.707235 0.706978i \(-0.750058\pi\)
−0.707235 + 0.706978i \(0.750058\pi\)
\(600\) 0 0
\(601\) 38.0768 1.55318 0.776592 0.630004i \(-0.216947\pi\)
0.776592 + 0.630004i \(0.216947\pi\)
\(602\) 0 0
\(603\) −0.303540 −0.0123611
\(604\) 0 0
\(605\) 10.8860 0.442581
\(606\) 0 0
\(607\) −24.4616 −0.992867 −0.496433 0.868075i \(-0.665357\pi\)
−0.496433 + 0.868075i \(0.665357\pi\)
\(608\) 0 0
\(609\) 0.336131 0.0136207
\(610\) 0 0
\(611\) −40.1890 −1.62587
\(612\) 0 0
\(613\) 16.6663 0.673146 0.336573 0.941657i \(-0.390732\pi\)
0.336573 + 0.941657i \(0.390732\pi\)
\(614\) 0 0
\(615\) 0.565670 0.0228100
\(616\) 0 0
\(617\) 40.2955 1.62224 0.811118 0.584883i \(-0.198860\pi\)
0.811118 + 0.584883i \(0.198860\pi\)
\(618\) 0 0
\(619\) 20.4210 0.820788 0.410394 0.911908i \(-0.365391\pi\)
0.410394 + 0.911908i \(0.365391\pi\)
\(620\) 0 0
\(621\) 7.64899 0.306943
\(622\) 0 0
\(623\) −1.30704 −0.0523655
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −0.179527 −0.00716962
\(628\) 0 0
\(629\) −10.5768 −0.421726
\(630\) 0 0
\(631\) 3.64134 0.144960 0.0724798 0.997370i \(-0.476909\pi\)
0.0724798 + 0.997370i \(0.476909\pi\)
\(632\) 0 0
\(633\) −10.8455 −0.431071
\(634\) 0 0
\(635\) 6.78390 0.269211
\(636\) 0 0
\(637\) −31.3958 −1.24395
\(638\) 0 0
\(639\) 40.4865 1.60162
\(640\) 0 0
\(641\) −30.6730 −1.21151 −0.605756 0.795651i \(-0.707129\pi\)
−0.605756 + 0.795651i \(0.707129\pi\)
\(642\) 0 0
\(643\) −10.9021 −0.429938 −0.214969 0.976621i \(-0.568965\pi\)
−0.214969 + 0.976621i \(0.568965\pi\)
\(644\) 0 0
\(645\) −0.173341 −0.00682530
\(646\) 0 0
\(647\) 5.79113 0.227673 0.113836 0.993500i \(-0.463686\pi\)
0.113836 + 0.993500i \(0.463686\pi\)
\(648\) 0 0
\(649\) −0.405107 −0.0159018
\(650\) 0 0
\(651\) −0.345185 −0.0135289
\(652\) 0 0
\(653\) −27.5926 −1.07978 −0.539890 0.841735i \(-0.681534\pi\)
−0.539890 + 0.841735i \(0.681534\pi\)
\(654\) 0 0
\(655\) −16.7091 −0.652879
\(656\) 0 0
\(657\) −6.72456 −0.262350
\(658\) 0 0
\(659\) −23.8541 −0.929224 −0.464612 0.885514i \(-0.653806\pi\)
−0.464612 + 0.885514i \(0.653806\pi\)
\(660\) 0 0
\(661\) −33.2311 −1.29254 −0.646270 0.763109i \(-0.723672\pi\)
−0.646270 + 0.763109i \(0.723672\pi\)
\(662\) 0 0
\(663\) 4.02218 0.156209
\(664\) 0 0
\(665\) 0.0955795 0.00370642
\(666\) 0 0
\(667\) −16.6354 −0.644127
\(668\) 0 0
\(669\) −14.4254 −0.557718
\(670\) 0 0
\(671\) 0.408478 0.0157691
\(672\) 0 0
\(673\) 25.6622 0.989206 0.494603 0.869119i \(-0.335313\pi\)
0.494603 + 0.869119i \(0.335313\pi\)
\(674\) 0 0
\(675\) 3.04051 0.117029
\(676\) 0 0
\(677\) −9.59071 −0.368601 −0.184300 0.982870i \(-0.559002\pi\)
−0.184300 + 0.982870i \(0.559002\pi\)
\(678\) 0 0
\(679\) −1.02474 −0.0393259
\(680\) 0 0
\(681\) 8.43674 0.323296
\(682\) 0 0
\(683\) 36.4798 1.39586 0.697929 0.716166i \(-0.254105\pi\)
0.697929 + 0.716166i \(0.254105\pi\)
\(684\) 0 0
\(685\) 9.00891 0.344213
\(686\) 0 0
\(687\) 5.93512 0.226439
\(688\) 0 0
\(689\) −64.9338 −2.47378
\(690\) 0 0
\(691\) −20.3671 −0.774799 −0.387400 0.921912i \(-0.626627\pi\)
−0.387400 + 0.921912i \(0.626627\pi\)
\(692\) 0 0
\(693\) 0.0876687 0.00333026
\(694\) 0 0
\(695\) −7.43282 −0.281943
\(696\) 0 0
\(697\) −1.79123 −0.0678476
\(698\) 0 0
\(699\) 3.40817 0.128909
\(700\) 0 0
\(701\) 37.7648 1.42636 0.713178 0.700983i \(-0.247255\pi\)
0.713178 + 0.700983i \(0.247255\pi\)
\(702\) 0 0
\(703\) −6.28059 −0.236877
\(704\) 0 0
\(705\) −4.75919 −0.179241
\(706\) 0 0
\(707\) −0.350053 −0.0131651
\(708\) 0 0
\(709\) 27.8421 1.04563 0.522817 0.852445i \(-0.324881\pi\)
0.522817 + 0.852445i \(0.324881\pi\)
\(710\) 0 0
\(711\) −22.6569 −0.849699
\(712\) 0 0
\(713\) 17.0835 0.639783
\(714\) 0 0
\(715\) 1.51602 0.0566959
\(716\) 0 0
\(717\) 1.37851 0.0514813
\(718\) 0 0
\(719\) 30.1635 1.12491 0.562455 0.826828i \(-0.309857\pi\)
0.562455 + 0.826828i \(0.309857\pi\)
\(720\) 0 0
\(721\) −1.77230 −0.0660039
\(722\) 0 0
\(723\) −2.05341 −0.0763670
\(724\) 0 0
\(725\) −6.61268 −0.245589
\(726\) 0 0
\(727\) −17.8911 −0.663544 −0.331772 0.943360i \(-0.607646\pi\)
−0.331772 + 0.943360i \(0.607646\pi\)
\(728\) 0 0
\(729\) −12.9042 −0.477934
\(730\) 0 0
\(731\) 0.548896 0.0203016
\(732\) 0 0
\(733\) 12.7778 0.471958 0.235979 0.971758i \(-0.424170\pi\)
0.235979 + 0.971758i \(0.424170\pi\)
\(734\) 0 0
\(735\) −3.71790 −0.137137
\(736\) 0 0
\(737\) −0.0377106 −0.00138909
\(738\) 0 0
\(739\) −19.5006 −0.717340 −0.358670 0.933464i \(-0.616770\pi\)
−0.358670 + 0.933464i \(0.616770\pi\)
\(740\) 0 0
\(741\) 2.38840 0.0877401
\(742\) 0 0
\(743\) 41.0316 1.50530 0.752652 0.658418i \(-0.228774\pi\)
0.752652 + 0.658418i \(0.228774\pi\)
\(744\) 0 0
\(745\) −4.98045 −0.182469
\(746\) 0 0
\(747\) −48.7209 −1.78261
\(748\) 0 0
\(749\) 1.38613 0.0506481
\(750\) 0 0
\(751\) 23.6623 0.863449 0.431724 0.902006i \(-0.357905\pi\)
0.431724 + 0.902006i \(0.357905\pi\)
\(752\) 0 0
\(753\) 7.32316 0.266871
\(754\) 0 0
\(755\) 19.8661 0.723000
\(756\) 0 0
\(757\) 7.29305 0.265070 0.132535 0.991178i \(-0.457688\pi\)
0.132535 + 0.991178i \(0.457688\pi\)
\(758\) 0 0
\(759\) 0.451635 0.0163933
\(760\) 0 0
\(761\) 34.8131 1.26197 0.630987 0.775793i \(-0.282650\pi\)
0.630987 + 0.775793i \(0.282650\pi\)
\(762\) 0 0
\(763\) −1.03565 −0.0374929
\(764\) 0 0
\(765\) −4.57584 −0.165440
\(766\) 0 0
\(767\) 5.38948 0.194603
\(768\) 0 0
\(769\) 12.5028 0.450862 0.225431 0.974259i \(-0.427621\pi\)
0.225431 + 0.974259i \(0.427621\pi\)
\(770\) 0 0
\(771\) 5.40594 0.194690
\(772\) 0 0
\(773\) 0.0576140 0.00207223 0.00103612 0.999999i \(-0.499670\pi\)
0.00103612 + 0.999999i \(0.499670\pi\)
\(774\) 0 0
\(775\) 6.79080 0.243933
\(776\) 0 0
\(777\) −0.319251 −0.0114531
\(778\) 0 0
\(779\) −1.06364 −0.0381090
\(780\) 0 0
\(781\) 5.02988 0.179983
\(782\) 0 0
\(783\) −20.1059 −0.718528
\(784\) 0 0
\(785\) 13.9183 0.496766
\(786\) 0 0
\(787\) 6.52348 0.232537 0.116268 0.993218i \(-0.462907\pi\)
0.116268 + 0.993218i \(0.462907\pi\)
\(788\) 0 0
\(789\) −5.05031 −0.179796
\(790\) 0 0
\(791\) −1.64581 −0.0585184
\(792\) 0 0
\(793\) −5.43433 −0.192979
\(794\) 0 0
\(795\) −7.68946 −0.272717
\(796\) 0 0
\(797\) −17.9775 −0.636797 −0.318399 0.947957i \(-0.603145\pi\)
−0.318399 + 0.947957i \(0.603145\pi\)
\(798\) 0 0
\(799\) 15.0703 0.533148
\(800\) 0 0
\(801\) 37.1570 1.31288
\(802\) 0 0
\(803\) −0.835433 −0.0294818
\(804\) 0 0
\(805\) −0.240448 −0.00847470
\(806\) 0 0
\(807\) −4.16619 −0.146657
\(808\) 0 0
\(809\) −9.96787 −0.350452 −0.175226 0.984528i \(-0.556066\pi\)
−0.175226 + 0.984528i \(0.556066\pi\)
\(810\) 0 0
\(811\) −7.56773 −0.265739 −0.132870 0.991134i \(-0.542419\pi\)
−0.132870 + 0.991134i \(0.542419\pi\)
\(812\) 0 0
\(813\) 13.1406 0.460859
\(814\) 0 0
\(815\) 7.12837 0.249696
\(816\) 0 0
\(817\) 0.325938 0.0114031
\(818\) 0 0
\(819\) −1.16633 −0.0407549
\(820\) 0 0
\(821\) 38.1434 1.33122 0.665608 0.746302i \(-0.268172\pi\)
0.665608 + 0.746302i \(0.268172\pi\)
\(822\) 0 0
\(823\) −12.9509 −0.451440 −0.225720 0.974192i \(-0.572473\pi\)
−0.225720 + 0.974192i \(0.572473\pi\)
\(824\) 0 0
\(825\) 0.179527 0.00625033
\(826\) 0 0
\(827\) 13.3538 0.464356 0.232178 0.972673i \(-0.425415\pi\)
0.232178 + 0.972673i \(0.425415\pi\)
\(828\) 0 0
\(829\) −15.5768 −0.541005 −0.270503 0.962719i \(-0.587190\pi\)
−0.270503 + 0.962719i \(0.587190\pi\)
\(830\) 0 0
\(831\) 3.22466 0.111862
\(832\) 0 0
\(833\) 11.7730 0.407909
\(834\) 0 0
\(835\) 23.0523 0.797757
\(836\) 0 0
\(837\) 20.6475 0.713683
\(838\) 0 0
\(839\) −28.7351 −0.992045 −0.496023 0.868310i \(-0.665207\pi\)
−0.496023 + 0.868310i \(0.665207\pi\)
\(840\) 0 0
\(841\) 14.7275 0.507845
\(842\) 0 0
\(843\) −13.1869 −0.454183
\(844\) 0 0
\(845\) −7.16888 −0.246617
\(846\) 0 0
\(847\) −1.04048 −0.0357514
\(848\) 0 0
\(849\) 8.59561 0.295000
\(850\) 0 0
\(851\) 15.8000 0.541618
\(852\) 0 0
\(853\) 52.3067 1.79095 0.895473 0.445116i \(-0.146837\pi\)
0.895473 + 0.445116i \(0.146837\pi\)
\(854\) 0 0
\(855\) −2.71717 −0.0929251
\(856\) 0 0
\(857\) −53.7133 −1.83481 −0.917405 0.397955i \(-0.869720\pi\)
−0.917405 + 0.397955i \(0.869720\pi\)
\(858\) 0 0
\(859\) −28.2653 −0.964399 −0.482199 0.876061i \(-0.660162\pi\)
−0.482199 + 0.876061i \(0.660162\pi\)
\(860\) 0 0
\(861\) −0.0540664 −0.00184258
\(862\) 0 0
\(863\) 21.4628 0.730603 0.365301 0.930889i \(-0.380966\pi\)
0.365301 + 0.930889i \(0.380966\pi\)
\(864\) 0 0
\(865\) −10.4279 −0.354558
\(866\) 0 0
\(867\) 7.53272 0.255824
\(868\) 0 0
\(869\) −2.81480 −0.0954856
\(870\) 0 0
\(871\) 0.501696 0.0169993
\(872\) 0 0
\(873\) 29.1316 0.985957
\(874\) 0 0
\(875\) −0.0955795 −0.00323118
\(876\) 0 0
\(877\) 7.69073 0.259697 0.129849 0.991534i \(-0.458551\pi\)
0.129849 + 0.991534i \(0.458551\pi\)
\(878\) 0 0
\(879\) 4.53250 0.152877
\(880\) 0 0
\(881\) 41.6429 1.40299 0.701493 0.712676i \(-0.252517\pi\)
0.701493 + 0.712676i \(0.252517\pi\)
\(882\) 0 0
\(883\) −9.41566 −0.316862 −0.158431 0.987370i \(-0.550644\pi\)
−0.158431 + 0.987370i \(0.550644\pi\)
\(884\) 0 0
\(885\) 0.638223 0.0214536
\(886\) 0 0
\(887\) −24.2210 −0.813263 −0.406632 0.913592i \(-0.633297\pi\)
−0.406632 + 0.913592i \(0.633297\pi\)
\(888\) 0 0
\(889\) −0.648402 −0.0217467
\(890\) 0 0
\(891\) −2.20585 −0.0738986
\(892\) 0 0
\(893\) 8.94884 0.299461
\(894\) 0 0
\(895\) 13.2047 0.441385
\(896\) 0 0
\(897\) −6.00848 −0.200617
\(898\) 0 0
\(899\) −44.9053 −1.49768
\(900\) 0 0
\(901\) 24.3492 0.811189
\(902\) 0 0
\(903\) 0.0165679 0.000551344 0
\(904\) 0 0
\(905\) −12.3384 −0.410143
\(906\) 0 0
\(907\) 35.2614 1.17083 0.585417 0.810732i \(-0.300931\pi\)
0.585417 + 0.810732i \(0.300931\pi\)
\(908\) 0 0
\(909\) 9.95143 0.330068
\(910\) 0 0
\(911\) 58.6348 1.94266 0.971328 0.237742i \(-0.0764072\pi\)
0.971328 + 0.237742i \(0.0764072\pi\)
\(912\) 0 0
\(913\) −6.05290 −0.200322
\(914\) 0 0
\(915\) −0.643534 −0.0212746
\(916\) 0 0
\(917\) 1.59705 0.0527392
\(918\) 0 0
\(919\) 20.0824 0.662458 0.331229 0.943550i \(-0.392537\pi\)
0.331229 + 0.943550i \(0.392537\pi\)
\(920\) 0 0
\(921\) −12.9888 −0.427995
\(922\) 0 0
\(923\) −66.9168 −2.20259
\(924\) 0 0
\(925\) 6.28059 0.206505
\(926\) 0 0
\(927\) 50.3835 1.65481
\(928\) 0 0
\(929\) −17.7731 −0.583116 −0.291558 0.956553i \(-0.594174\pi\)
−0.291558 + 0.956553i \(0.594174\pi\)
\(930\) 0 0
\(931\) 6.99086 0.229116
\(932\) 0 0
\(933\) 4.08049 0.133589
\(934\) 0 0
\(935\) −0.568484 −0.0185914
\(936\) 0 0
\(937\) −28.1246 −0.918792 −0.459396 0.888232i \(-0.651934\pi\)
−0.459396 + 0.888232i \(0.651934\pi\)
\(938\) 0 0
\(939\) −1.07923 −0.0352195
\(940\) 0 0
\(941\) −22.0914 −0.720160 −0.360080 0.932921i \(-0.617251\pi\)
−0.360080 + 0.932921i \(0.617251\pi\)
\(942\) 0 0
\(943\) 2.67580 0.0871360
\(944\) 0 0
\(945\) −0.290611 −0.00945358
\(946\) 0 0
\(947\) 38.3266 1.24545 0.622724 0.782442i \(-0.286026\pi\)
0.622724 + 0.782442i \(0.286026\pi\)
\(948\) 0 0
\(949\) 11.1145 0.360791
\(950\) 0 0
\(951\) 2.89585 0.0939044
\(952\) 0 0
\(953\) 14.8386 0.480670 0.240335 0.970690i \(-0.422743\pi\)
0.240335 + 0.970690i \(0.422743\pi\)
\(954\) 0 0
\(955\) −18.3515 −0.593839
\(956\) 0 0
\(957\) −1.18716 −0.0383753
\(958\) 0 0
\(959\) −0.861067 −0.0278053
\(960\) 0 0
\(961\) 15.1149 0.487578
\(962\) 0 0
\(963\) −39.4053 −1.26982
\(964\) 0 0
\(965\) −5.74437 −0.184918
\(966\) 0 0
\(967\) −28.2875 −0.909664 −0.454832 0.890577i \(-0.650301\pi\)
−0.454832 + 0.890577i \(0.650301\pi\)
\(968\) 0 0
\(969\) −0.895614 −0.0287713
\(970\) 0 0
\(971\) −41.8695 −1.34366 −0.671828 0.740707i \(-0.734490\pi\)
−0.671828 + 0.740707i \(0.734490\pi\)
\(972\) 0 0
\(973\) 0.710426 0.0227752
\(974\) 0 0
\(975\) −2.38840 −0.0764901
\(976\) 0 0
\(977\) 7.36966 0.235776 0.117888 0.993027i \(-0.462388\pi\)
0.117888 + 0.993027i \(0.462388\pi\)
\(978\) 0 0
\(979\) 4.61624 0.147536
\(980\) 0 0
\(981\) 29.4417 0.940001
\(982\) 0 0
\(983\) 18.5640 0.592098 0.296049 0.955173i \(-0.404331\pi\)
0.296049 + 0.955173i \(0.404331\pi\)
\(984\) 0 0
\(985\) 22.8975 0.729573
\(986\) 0 0
\(987\) 0.454881 0.0144790
\(988\) 0 0
\(989\) −0.819959 −0.0260732
\(990\) 0 0
\(991\) −42.5301 −1.35101 −0.675506 0.737354i \(-0.736075\pi\)
−0.675506 + 0.737354i \(0.736075\pi\)
\(992\) 0 0
\(993\) 3.65776 0.116076
\(994\) 0 0
\(995\) −4.38925 −0.139149
\(996\) 0 0
\(997\) −52.5775 −1.66515 −0.832574 0.553914i \(-0.813134\pi\)
−0.832574 + 0.553914i \(0.813134\pi\)
\(998\) 0 0
\(999\) 19.0962 0.604178
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 3040.2.a.x.1.2 yes 5
4.3 odd 2 3040.2.a.u.1.4 5
8.3 odd 2 6080.2.a.cl.1.2 5
8.5 even 2 6080.2.a.ci.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.u.1.4 5 4.3 odd 2
3040.2.a.x.1.2 yes 5 1.1 even 1 trivial
6080.2.a.ci.1.4 5 8.5 even 2
6080.2.a.cl.1.2 5 8.3 odd 2