Properties

Label 2320.2.a.n.1.1
Level $2320$
Weight $2$
Character 2320.1
Self dual yes
Analytic conductor $18.525$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2320,2,Mod(1,2320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2320.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.5252932689\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 2320.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.90321 q^{3} +1.00000 q^{5} +0.903212 q^{7} +5.42864 q^{9} +O(q^{10})\) \(q-2.90321 q^{3} +1.00000 q^{5} +0.903212 q^{7} +5.42864 q^{9} +1.52543 q^{11} -0.622216 q^{13} -2.90321 q^{15} -7.95407 q^{17} +1.09679 q^{19} -2.62222 q^{21} -7.52543 q^{23} +1.00000 q^{25} -7.05086 q^{27} -1.00000 q^{29} +6.90321 q^{31} -4.42864 q^{33} +0.903212 q^{35} +3.95407 q^{37} +1.80642 q^{39} +3.67307 q^{41} +10.5161 q^{43} +5.42864 q^{45} -6.90321 q^{47} -6.18421 q^{49} +23.0923 q^{51} +6.42864 q^{53} +1.52543 q^{55} -3.18421 q^{57} +1.67307 q^{59} -1.86665 q^{61} +4.90321 q^{63} -0.622216 q^{65} -11.5254 q^{67} +21.8479 q^{69} -13.6731 q^{71} +10.1891 q^{73} -2.90321 q^{75} +1.37778 q^{77} -9.13828 q^{79} +4.18421 q^{81} -10.7096 q^{83} -7.95407 q^{85} +2.90321 q^{87} -7.80642 q^{89} -0.561993 q^{91} -20.0415 q^{93} +1.09679 q^{95} -4.08742 q^{97} +8.28100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 2 q^{11} - 2 q^{13} - 2 q^{15} - 4 q^{17} + 10 q^{19} - 8 q^{21} - 16 q^{23} + 3 q^{25} - 8 q^{27} - 3 q^{29} + 14 q^{31} - 4 q^{35} - 8 q^{37} - 8 q^{39} - 2 q^{41} - 2 q^{43} + 3 q^{45} - 14 q^{47} - 5 q^{49} + 16 q^{51} + 6 q^{53} - 2 q^{55} + 4 q^{57} - 8 q^{59} - 6 q^{61} + 8 q^{63} - 2 q^{65} - 28 q^{67} + 12 q^{69} - 28 q^{71} - 16 q^{73} - 2 q^{75} + 4 q^{77} + 6 q^{79} - q^{81} - 12 q^{83} - 4 q^{85} + 2 q^{87} - 10 q^{89} + 12 q^{91} - 20 q^{93} + 10 q^{95} + 8 q^{97} + 18 q^{99}+O(q^{100}) \) 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
\(3\) −2.90321 −1.67617 −0.838085 0.545540i \(-0.816325\pi\)
−0.838085 + 0.545540i \(0.816325\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.903212 0.341382 0.170691 0.985325i \(-0.445400\pi\)
0.170691 + 0.985325i \(0.445400\pi\)
\(8\) 0 0
\(9\) 5.42864 1.80955
\(10\) 0 0
\(11\) 1.52543 0.459934 0.229967 0.973198i \(-0.426138\pi\)
0.229967 + 0.973198i \(0.426138\pi\)
\(12\) 0 0
\(13\) −0.622216 −0.172572 −0.0862858 0.996270i \(-0.527500\pi\)
−0.0862858 + 0.996270i \(0.527500\pi\)
\(14\) 0 0
\(15\) −2.90321 −0.749606
\(16\) 0 0
\(17\) −7.95407 −1.92914 −0.964572 0.263819i \(-0.915018\pi\)
−0.964572 + 0.263819i \(0.915018\pi\)
\(18\) 0 0
\(19\) 1.09679 0.251620 0.125810 0.992054i \(-0.459847\pi\)
0.125810 + 0.992054i \(0.459847\pi\)
\(20\) 0 0
\(21\) −2.62222 −0.572214
\(22\) 0 0
\(23\) −7.52543 −1.56916 −0.784580 0.620028i \(-0.787121\pi\)
−0.784580 + 0.620028i \(0.787121\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −7.05086 −1.35694
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 6.90321 1.23985 0.619927 0.784660i \(-0.287162\pi\)
0.619927 + 0.784660i \(0.287162\pi\)
\(32\) 0 0
\(33\) −4.42864 −0.770927
\(34\) 0 0
\(35\) 0.903212 0.152671
\(36\) 0 0
\(37\) 3.95407 0.650045 0.325022 0.945706i \(-0.394628\pi\)
0.325022 + 0.945706i \(0.394628\pi\)
\(38\) 0 0
\(39\) 1.80642 0.289259
\(40\) 0 0
\(41\) 3.67307 0.573637 0.286819 0.957985i \(-0.407402\pi\)
0.286819 + 0.957985i \(0.407402\pi\)
\(42\) 0 0
\(43\) 10.5161 1.60368 0.801842 0.597536i \(-0.203854\pi\)
0.801842 + 0.597536i \(0.203854\pi\)
\(44\) 0 0
\(45\) 5.42864 0.809254
\(46\) 0 0
\(47\) −6.90321 −1.00694 −0.503468 0.864014i \(-0.667943\pi\)
−0.503468 + 0.864014i \(0.667943\pi\)
\(48\) 0 0
\(49\) −6.18421 −0.883458
\(50\) 0 0
\(51\) 23.0923 3.23357
\(52\) 0 0
\(53\) 6.42864 0.883042 0.441521 0.897251i \(-0.354439\pi\)
0.441521 + 0.897251i \(0.354439\pi\)
\(54\) 0 0
\(55\) 1.52543 0.205689
\(56\) 0 0
\(57\) −3.18421 −0.421759
\(58\) 0 0
\(59\) 1.67307 0.217815 0.108908 0.994052i \(-0.465265\pi\)
0.108908 + 0.994052i \(0.465265\pi\)
\(60\) 0 0
\(61\) −1.86665 −0.239000 −0.119500 0.992834i \(-0.538129\pi\)
−0.119500 + 0.992834i \(0.538129\pi\)
\(62\) 0 0
\(63\) 4.90321 0.617747
\(64\) 0 0
\(65\) −0.622216 −0.0771764
\(66\) 0 0
\(67\) −11.5254 −1.40806 −0.704028 0.710173i \(-0.748617\pi\)
−0.704028 + 0.710173i \(0.748617\pi\)
\(68\) 0 0
\(69\) 21.8479 2.63018
\(70\) 0 0
\(71\) −13.6731 −1.62269 −0.811347 0.584564i \(-0.801266\pi\)
−0.811347 + 0.584564i \(0.801266\pi\)
\(72\) 0 0
\(73\) 10.1891 1.19255 0.596274 0.802781i \(-0.296647\pi\)
0.596274 + 0.802781i \(0.296647\pi\)
\(74\) 0 0
\(75\) −2.90321 −0.335234
\(76\) 0 0
\(77\) 1.37778 0.157013
\(78\) 0 0
\(79\) −9.13828 −1.02814 −0.514068 0.857749i \(-0.671862\pi\)
−0.514068 + 0.857749i \(0.671862\pi\)
\(80\) 0 0
\(81\) 4.18421 0.464912
\(82\) 0 0
\(83\) −10.7096 −1.17554 −0.587768 0.809030i \(-0.699993\pi\)
−0.587768 + 0.809030i \(0.699993\pi\)
\(84\) 0 0
\(85\) −7.95407 −0.862740
\(86\) 0 0
\(87\) 2.90321 0.311257
\(88\) 0 0
\(89\) −7.80642 −0.827479 −0.413740 0.910395i \(-0.635778\pi\)
−0.413740 + 0.910395i \(0.635778\pi\)
\(90\) 0 0
\(91\) −0.561993 −0.0589128
\(92\) 0 0
\(93\) −20.0415 −2.07821
\(94\) 0 0
\(95\) 1.09679 0.112528
\(96\) 0 0
\(97\) −4.08742 −0.415015 −0.207507 0.978233i \(-0.566535\pi\)
−0.207507 + 0.978233i \(0.566535\pi\)
\(98\) 0 0
\(99\) 8.28100 0.832271
\(100\) 0 0
\(101\) 13.9081 1.38391 0.691956 0.721940i \(-0.256749\pi\)
0.691956 + 0.721940i \(0.256749\pi\)
\(102\) 0 0
\(103\) 12.9447 1.27548 0.637740 0.770252i \(-0.279870\pi\)
0.637740 + 0.770252i \(0.279870\pi\)
\(104\) 0 0
\(105\) −2.62222 −0.255902
\(106\) 0 0
\(107\) −11.0049 −1.06389 −0.531943 0.846780i \(-0.678538\pi\)
−0.531943 + 0.846780i \(0.678538\pi\)
\(108\) 0 0
\(109\) −18.0415 −1.72806 −0.864031 0.503439i \(-0.832068\pi\)
−0.864031 + 0.503439i \(0.832068\pi\)
\(110\) 0 0
\(111\) −11.4795 −1.08959
\(112\) 0 0
\(113\) −10.2810 −0.967155 −0.483577 0.875302i \(-0.660663\pi\)
−0.483577 + 0.875302i \(0.660663\pi\)
\(114\) 0 0
\(115\) −7.52543 −0.701750
\(116\) 0 0
\(117\) −3.37778 −0.312276
\(118\) 0 0
\(119\) −7.18421 −0.658575
\(120\) 0 0
\(121\) −8.67307 −0.788461
\(122\) 0 0
\(123\) −10.6637 −0.961514
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −6.22077 −0.552004 −0.276002 0.961157i \(-0.589010\pi\)
−0.276002 + 0.961157i \(0.589010\pi\)
\(128\) 0 0
\(129\) −30.5303 −2.68805
\(130\) 0 0
\(131\) 11.7605 1.02752 0.513759 0.857934i \(-0.328252\pi\)
0.513759 + 0.857934i \(0.328252\pi\)
\(132\) 0 0
\(133\) 0.990632 0.0858987
\(134\) 0 0
\(135\) −7.05086 −0.606841
\(136\) 0 0
\(137\) −3.56691 −0.304742 −0.152371 0.988323i \(-0.548691\pi\)
−0.152371 + 0.988323i \(0.548691\pi\)
\(138\) 0 0
\(139\) 8.56199 0.726219 0.363109 0.931747i \(-0.381715\pi\)
0.363109 + 0.931747i \(0.381715\pi\)
\(140\) 0 0
\(141\) 20.0415 1.68780
\(142\) 0 0
\(143\) −0.949145 −0.0793715
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) 17.9541 1.48083
\(148\) 0 0
\(149\) −5.61285 −0.459822 −0.229911 0.973212i \(-0.573844\pi\)
−0.229911 + 0.973212i \(0.573844\pi\)
\(150\) 0 0
\(151\) −10.7971 −0.878652 −0.439326 0.898328i \(-0.644783\pi\)
−0.439326 + 0.898328i \(0.644783\pi\)
\(152\) 0 0
\(153\) −43.1798 −3.49088
\(154\) 0 0
\(155\) 6.90321 0.554479
\(156\) 0 0
\(157\) 2.28100 0.182043 0.0910217 0.995849i \(-0.470987\pi\)
0.0910217 + 0.995849i \(0.470987\pi\)
\(158\) 0 0
\(159\) −18.6637 −1.48013
\(160\) 0 0
\(161\) −6.79706 −0.535683
\(162\) 0 0
\(163\) −16.3225 −1.27848 −0.639238 0.769009i \(-0.720750\pi\)
−0.639238 + 0.769009i \(0.720750\pi\)
\(164\) 0 0
\(165\) −4.42864 −0.344769
\(166\) 0 0
\(167\) 4.76986 0.369103 0.184551 0.982823i \(-0.440917\pi\)
0.184551 + 0.982823i \(0.440917\pi\)
\(168\) 0 0
\(169\) −12.6128 −0.970219
\(170\) 0 0
\(171\) 5.95407 0.455319
\(172\) 0 0
\(173\) 4.23506 0.321986 0.160993 0.986956i \(-0.448530\pi\)
0.160993 + 0.986956i \(0.448530\pi\)
\(174\) 0 0
\(175\) 0.903212 0.0682764
\(176\) 0 0
\(177\) −4.85728 −0.365095
\(178\) 0 0
\(179\) −9.71456 −0.726100 −0.363050 0.931770i \(-0.618265\pi\)
−0.363050 + 0.931770i \(0.618265\pi\)
\(180\) 0 0
\(181\) 0.326929 0.0243005 0.0121502 0.999926i \(-0.496132\pi\)
0.0121502 + 0.999926i \(0.496132\pi\)
\(182\) 0 0
\(183\) 5.41927 0.400604
\(184\) 0 0
\(185\) 3.95407 0.290709
\(186\) 0 0
\(187\) −12.1334 −0.887279
\(188\) 0 0
\(189\) −6.36842 −0.463234
\(190\) 0 0
\(191\) 14.9447 1.08136 0.540680 0.841228i \(-0.318167\pi\)
0.540680 + 0.841228i \(0.318167\pi\)
\(192\) 0 0
\(193\) −14.1476 −1.01837 −0.509185 0.860657i \(-0.670053\pi\)
−0.509185 + 0.860657i \(0.670053\pi\)
\(194\) 0 0
\(195\) 1.80642 0.129361
\(196\) 0 0
\(197\) −5.70471 −0.406444 −0.203222 0.979133i \(-0.565141\pi\)
−0.203222 + 0.979133i \(0.565141\pi\)
\(198\) 0 0
\(199\) 22.1432 1.56969 0.784845 0.619692i \(-0.212743\pi\)
0.784845 + 0.619692i \(0.212743\pi\)
\(200\) 0 0
\(201\) 33.4608 2.36014
\(202\) 0 0
\(203\) −0.903212 −0.0633930
\(204\) 0 0
\(205\) 3.67307 0.256538
\(206\) 0 0
\(207\) −40.8528 −2.83947
\(208\) 0 0
\(209\) 1.67307 0.115729
\(210\) 0 0
\(211\) 20.8430 1.43489 0.717445 0.696615i \(-0.245311\pi\)
0.717445 + 0.696615i \(0.245311\pi\)
\(212\) 0 0
\(213\) 39.6958 2.71991
\(214\) 0 0
\(215\) 10.5161 0.717189
\(216\) 0 0
\(217\) 6.23506 0.423264
\(218\) 0 0
\(219\) −29.5812 −1.99891
\(220\) 0 0
\(221\) 4.94914 0.332916
\(222\) 0 0
\(223\) −9.03657 −0.605133 −0.302567 0.953128i \(-0.597843\pi\)
−0.302567 + 0.953128i \(0.597843\pi\)
\(224\) 0 0
\(225\) 5.42864 0.361909
\(226\) 0 0
\(227\) −19.4050 −1.28795 −0.643977 0.765045i \(-0.722717\pi\)
−0.643977 + 0.765045i \(0.722717\pi\)
\(228\) 0 0
\(229\) −25.6128 −1.69254 −0.846272 0.532751i \(-0.821158\pi\)
−0.846272 + 0.532751i \(0.821158\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) −3.12399 −0.204659 −0.102330 0.994751i \(-0.532630\pi\)
−0.102330 + 0.994751i \(0.532630\pi\)
\(234\) 0 0
\(235\) −6.90321 −0.450316
\(236\) 0 0
\(237\) 26.5303 1.72333
\(238\) 0 0
\(239\) −13.9398 −0.901689 −0.450845 0.892602i \(-0.648877\pi\)
−0.450845 + 0.892602i \(0.648877\pi\)
\(240\) 0 0
\(241\) −18.4701 −1.18977 −0.594883 0.803813i \(-0.702802\pi\)
−0.594883 + 0.803813i \(0.702802\pi\)
\(242\) 0 0
\(243\) 9.00492 0.577666
\(244\) 0 0
\(245\) −6.18421 −0.395095
\(246\) 0 0
\(247\) −0.682439 −0.0434225
\(248\) 0 0
\(249\) 31.0923 1.97040
\(250\) 0 0
\(251\) 13.7921 0.870552 0.435276 0.900297i \(-0.356651\pi\)
0.435276 + 0.900297i \(0.356651\pi\)
\(252\) 0 0
\(253\) −11.4795 −0.721710
\(254\) 0 0
\(255\) 23.0923 1.44610
\(256\) 0 0
\(257\) −1.47949 −0.0922883 −0.0461442 0.998935i \(-0.514693\pi\)
−0.0461442 + 0.998935i \(0.514693\pi\)
\(258\) 0 0
\(259\) 3.57136 0.221914
\(260\) 0 0
\(261\) −5.42864 −0.336024
\(262\) 0 0
\(263\) 0.442930 0.0273122 0.0136561 0.999907i \(-0.495653\pi\)
0.0136561 + 0.999907i \(0.495653\pi\)
\(264\) 0 0
\(265\) 6.42864 0.394908
\(266\) 0 0
\(267\) 22.6637 1.38700
\(268\) 0 0
\(269\) 3.93978 0.240212 0.120106 0.992761i \(-0.461676\pi\)
0.120106 + 0.992761i \(0.461676\pi\)
\(270\) 0 0
\(271\) −6.20787 −0.377101 −0.188551 0.982063i \(-0.560379\pi\)
−0.188551 + 0.982063i \(0.560379\pi\)
\(272\) 0 0
\(273\) 1.63158 0.0987479
\(274\) 0 0
\(275\) 1.52543 0.0919867
\(276\) 0 0
\(277\) 5.57136 0.334751 0.167375 0.985893i \(-0.446471\pi\)
0.167375 + 0.985893i \(0.446471\pi\)
\(278\) 0 0
\(279\) 37.4750 2.24357
\(280\) 0 0
\(281\) 6.69535 0.399411 0.199705 0.979856i \(-0.436001\pi\)
0.199705 + 0.979856i \(0.436001\pi\)
\(282\) 0 0
\(283\) −25.8020 −1.53377 −0.766884 0.641785i \(-0.778194\pi\)
−0.766884 + 0.641785i \(0.778194\pi\)
\(284\) 0 0
\(285\) −3.18421 −0.188616
\(286\) 0 0
\(287\) 3.31756 0.195829
\(288\) 0 0
\(289\) 46.2672 2.72160
\(290\) 0 0
\(291\) 11.8666 0.695635
\(292\) 0 0
\(293\) −18.8430 −1.10082 −0.550410 0.834895i \(-0.685528\pi\)
−0.550410 + 0.834895i \(0.685528\pi\)
\(294\) 0 0
\(295\) 1.67307 0.0974099
\(296\) 0 0
\(297\) −10.7556 −0.624101
\(298\) 0 0
\(299\) 4.68244 0.270792
\(300\) 0 0
\(301\) 9.49823 0.547469
\(302\) 0 0
\(303\) −40.3783 −2.31967
\(304\) 0 0
\(305\) −1.86665 −0.106884
\(306\) 0 0
\(307\) 1.65878 0.0946716 0.0473358 0.998879i \(-0.484927\pi\)
0.0473358 + 0.998879i \(0.484927\pi\)
\(308\) 0 0
\(309\) −37.5812 −2.13792
\(310\) 0 0
\(311\) −21.3002 −1.20782 −0.603912 0.797051i \(-0.706392\pi\)
−0.603912 + 0.797051i \(0.706392\pi\)
\(312\) 0 0
\(313\) 8.62222 0.487356 0.243678 0.969856i \(-0.421646\pi\)
0.243678 + 0.969856i \(0.421646\pi\)
\(314\) 0 0
\(315\) 4.90321 0.276265
\(316\) 0 0
\(317\) −27.5955 −1.54992 −0.774959 0.632012i \(-0.782229\pi\)
−0.774959 + 0.632012i \(0.782229\pi\)
\(318\) 0 0
\(319\) −1.52543 −0.0854075
\(320\) 0 0
\(321\) 31.9496 1.78325
\(322\) 0 0
\(323\) −8.72393 −0.485412
\(324\) 0 0
\(325\) −0.622216 −0.0345143
\(326\) 0 0
\(327\) 52.3783 2.89652
\(328\) 0 0
\(329\) −6.23506 −0.343750
\(330\) 0 0
\(331\) −16.9131 −0.929626 −0.464813 0.885409i \(-0.653878\pi\)
−0.464813 + 0.885409i \(0.653878\pi\)
\(332\) 0 0
\(333\) 21.4652 1.17629
\(334\) 0 0
\(335\) −11.5254 −0.629701
\(336\) 0 0
\(337\) 11.9956 0.653439 0.326720 0.945121i \(-0.394057\pi\)
0.326720 + 0.945121i \(0.394057\pi\)
\(338\) 0 0
\(339\) 29.8479 1.62112
\(340\) 0 0
\(341\) 10.5303 0.570250
\(342\) 0 0
\(343\) −11.9081 −0.642979
\(344\) 0 0
\(345\) 21.8479 1.17625
\(346\) 0 0
\(347\) −6.14764 −0.330023 −0.165011 0.986292i \(-0.552766\pi\)
−0.165011 + 0.986292i \(0.552766\pi\)
\(348\) 0 0
\(349\) −7.12399 −0.381338 −0.190669 0.981654i \(-0.561066\pi\)
−0.190669 + 0.981654i \(0.561066\pi\)
\(350\) 0 0
\(351\) 4.38715 0.234169
\(352\) 0 0
\(353\) −16.9175 −0.900428 −0.450214 0.892921i \(-0.648652\pi\)
−0.450214 + 0.892921i \(0.648652\pi\)
\(354\) 0 0
\(355\) −13.6731 −0.725691
\(356\) 0 0
\(357\) 20.8573 1.10388
\(358\) 0 0
\(359\) −36.7096 −1.93746 −0.968730 0.248116i \(-0.920188\pi\)
−0.968730 + 0.248116i \(0.920188\pi\)
\(360\) 0 0
\(361\) −17.7971 −0.936687
\(362\) 0 0
\(363\) 25.1798 1.32159
\(364\) 0 0
\(365\) 10.1891 0.533323
\(366\) 0 0
\(367\) −8.41435 −0.439225 −0.219613 0.975587i \(-0.570479\pi\)
−0.219613 + 0.975587i \(0.570479\pi\)
\(368\) 0 0
\(369\) 19.9398 1.03802
\(370\) 0 0
\(371\) 5.80642 0.301455
\(372\) 0 0
\(373\) −8.66370 −0.448590 −0.224295 0.974521i \(-0.572008\pi\)
−0.224295 + 0.974521i \(0.572008\pi\)
\(374\) 0 0
\(375\) −2.90321 −0.149921
\(376\) 0 0
\(377\) 0.622216 0.0320457
\(378\) 0 0
\(379\) 2.76986 0.142278 0.0711390 0.997466i \(-0.477337\pi\)
0.0711390 + 0.997466i \(0.477337\pi\)
\(380\) 0 0
\(381\) 18.0602 0.925253
\(382\) 0 0
\(383\) −1.67752 −0.0857171 −0.0428585 0.999081i \(-0.513646\pi\)
−0.0428585 + 0.999081i \(0.513646\pi\)
\(384\) 0 0
\(385\) 1.37778 0.0702184
\(386\) 0 0
\(387\) 57.0879 2.90194
\(388\) 0 0
\(389\) −5.77478 −0.292793 −0.146397 0.989226i \(-0.546768\pi\)
−0.146397 + 0.989226i \(0.546768\pi\)
\(390\) 0 0
\(391\) 59.8578 3.02714
\(392\) 0 0
\(393\) −34.1432 −1.72230
\(394\) 0 0
\(395\) −9.13828 −0.459797
\(396\) 0 0
\(397\) 29.9081 1.50105 0.750523 0.660844i \(-0.229802\pi\)
0.750523 + 0.660844i \(0.229802\pi\)
\(398\) 0 0
\(399\) −2.87601 −0.143981
\(400\) 0 0
\(401\) 8.53035 0.425985 0.212993 0.977054i \(-0.431679\pi\)
0.212993 + 0.977054i \(0.431679\pi\)
\(402\) 0 0
\(403\) −4.29529 −0.213963
\(404\) 0 0
\(405\) 4.18421 0.207915
\(406\) 0 0
\(407\) 6.03164 0.298977
\(408\) 0 0
\(409\) 5.09234 0.251800 0.125900 0.992043i \(-0.459818\pi\)
0.125900 + 0.992043i \(0.459818\pi\)
\(410\) 0 0
\(411\) 10.3555 0.510800
\(412\) 0 0
\(413\) 1.51114 0.0743582
\(414\) 0 0
\(415\) −10.7096 −0.525715
\(416\) 0 0
\(417\) −24.8573 −1.21727
\(418\) 0 0
\(419\) 24.3368 1.18893 0.594465 0.804122i \(-0.297364\pi\)
0.594465 + 0.804122i \(0.297364\pi\)
\(420\) 0 0
\(421\) 24.5018 1.19414 0.597072 0.802188i \(-0.296331\pi\)
0.597072 + 0.802188i \(0.296331\pi\)
\(422\) 0 0
\(423\) −37.4750 −1.82210
\(424\) 0 0
\(425\) −7.95407 −0.385829
\(426\) 0 0
\(427\) −1.68598 −0.0815902
\(428\) 0 0
\(429\) 2.75557 0.133040
\(430\) 0 0
\(431\) −4.26671 −0.205520 −0.102760 0.994706i \(-0.532767\pi\)
−0.102760 + 0.994706i \(0.532767\pi\)
\(432\) 0 0
\(433\) 27.0049 1.29777 0.648887 0.760885i \(-0.275235\pi\)
0.648887 + 0.760885i \(0.275235\pi\)
\(434\) 0 0
\(435\) 2.90321 0.139198
\(436\) 0 0
\(437\) −8.25380 −0.394833
\(438\) 0 0
\(439\) 2.03164 0.0969650 0.0484825 0.998824i \(-0.484561\pi\)
0.0484825 + 0.998824i \(0.484561\pi\)
\(440\) 0 0
\(441\) −33.5718 −1.59866
\(442\) 0 0
\(443\) 3.46520 0.164637 0.0823184 0.996606i \(-0.473768\pi\)
0.0823184 + 0.996606i \(0.473768\pi\)
\(444\) 0 0
\(445\) −7.80642 −0.370060
\(446\) 0 0
\(447\) 16.2953 0.770741
\(448\) 0 0
\(449\) 37.3590 1.76308 0.881541 0.472107i \(-0.156506\pi\)
0.881541 + 0.472107i \(0.156506\pi\)
\(450\) 0 0
\(451\) 5.60300 0.263835
\(452\) 0 0
\(453\) 31.3461 1.47277
\(454\) 0 0
\(455\) −0.561993 −0.0263466
\(456\) 0 0
\(457\) 13.4509 0.629207 0.314604 0.949223i \(-0.398128\pi\)
0.314604 + 0.949223i \(0.398128\pi\)
\(458\) 0 0
\(459\) 56.0830 2.61773
\(460\) 0 0
\(461\) −16.2766 −0.758075 −0.379037 0.925381i \(-0.623745\pi\)
−0.379037 + 0.925381i \(0.623745\pi\)
\(462\) 0 0
\(463\) 30.3926 1.41246 0.706231 0.707982i \(-0.250394\pi\)
0.706231 + 0.707982i \(0.250394\pi\)
\(464\) 0 0
\(465\) −20.0415 −0.929402
\(466\) 0 0
\(467\) 1.18865 0.0550043 0.0275022 0.999622i \(-0.491245\pi\)
0.0275022 + 0.999622i \(0.491245\pi\)
\(468\) 0 0
\(469\) −10.4099 −0.480685
\(470\) 0 0
\(471\) −6.62222 −0.305136
\(472\) 0 0
\(473\) 16.0415 0.737588
\(474\) 0 0
\(475\) 1.09679 0.0503241
\(476\) 0 0
\(477\) 34.8988 1.59790
\(478\) 0 0
\(479\) −41.0464 −1.87546 −0.937729 0.347367i \(-0.887076\pi\)
−0.937729 + 0.347367i \(0.887076\pi\)
\(480\) 0 0
\(481\) −2.46028 −0.112179
\(482\) 0 0
\(483\) 19.7333 0.897896
\(484\) 0 0
\(485\) −4.08742 −0.185600
\(486\) 0 0
\(487\) 10.1476 0.459834 0.229917 0.973210i \(-0.426155\pi\)
0.229917 + 0.973210i \(0.426155\pi\)
\(488\) 0 0
\(489\) 47.3876 2.14294
\(490\) 0 0
\(491\) 29.2083 1.31815 0.659077 0.752075i \(-0.270947\pi\)
0.659077 + 0.752075i \(0.270947\pi\)
\(492\) 0 0
\(493\) 7.95407 0.358233
\(494\) 0 0
\(495\) 8.28100 0.372203
\(496\) 0 0
\(497\) −12.3497 −0.553959
\(498\) 0 0
\(499\) −21.9813 −0.984017 −0.492008 0.870591i \(-0.663737\pi\)
−0.492008 + 0.870591i \(0.663737\pi\)
\(500\) 0 0
\(501\) −13.8479 −0.618679
\(502\) 0 0
\(503\) −5.77923 −0.257683 −0.128841 0.991665i \(-0.541126\pi\)
−0.128841 + 0.991665i \(0.541126\pi\)
\(504\) 0 0
\(505\) 13.9081 0.618904
\(506\) 0 0
\(507\) 36.6178 1.62625
\(508\) 0 0
\(509\) 13.6543 0.605218 0.302609 0.953115i \(-0.402142\pi\)
0.302609 + 0.953115i \(0.402142\pi\)
\(510\) 0 0
\(511\) 9.20294 0.407114
\(512\) 0 0
\(513\) −7.73329 −0.341433
\(514\) 0 0
\(515\) 12.9447 0.570412
\(516\) 0 0
\(517\) −10.5303 −0.463124
\(518\) 0 0
\(519\) −12.2953 −0.539703
\(520\) 0 0
\(521\) −19.6731 −0.861893 −0.430946 0.902378i \(-0.641820\pi\)
−0.430946 + 0.902378i \(0.641820\pi\)
\(522\) 0 0
\(523\) 15.1383 0.661951 0.330975 0.943639i \(-0.392622\pi\)
0.330975 + 0.943639i \(0.392622\pi\)
\(524\) 0 0
\(525\) −2.62222 −0.114443
\(526\) 0 0
\(527\) −54.9086 −2.39186
\(528\) 0 0
\(529\) 33.6321 1.46226
\(530\) 0 0
\(531\) 9.08250 0.394147
\(532\) 0 0
\(533\) −2.28544 −0.0989935
\(534\) 0 0
\(535\) −11.0049 −0.475784
\(536\) 0 0
\(537\) 28.2034 1.21707
\(538\) 0 0
\(539\) −9.43356 −0.406332
\(540\) 0 0
\(541\) 2.68244 0.115327 0.0576635 0.998336i \(-0.481635\pi\)
0.0576635 + 0.998336i \(0.481635\pi\)
\(542\) 0 0
\(543\) −0.949145 −0.0407317
\(544\) 0 0
\(545\) −18.0415 −0.772812
\(546\) 0 0
\(547\) 15.3635 0.656896 0.328448 0.944522i \(-0.393475\pi\)
0.328448 + 0.944522i \(0.393475\pi\)
\(548\) 0 0
\(549\) −10.1334 −0.432481
\(550\) 0 0
\(551\) −1.09679 −0.0467247
\(552\) 0 0
\(553\) −8.25380 −0.350987
\(554\) 0 0
\(555\) −11.4795 −0.487277
\(556\) 0 0
\(557\) −9.87955 −0.418610 −0.209305 0.977850i \(-0.567120\pi\)
−0.209305 + 0.977850i \(0.567120\pi\)
\(558\) 0 0
\(559\) −6.54326 −0.276750
\(560\) 0 0
\(561\) 35.2257 1.48723
\(562\) 0 0
\(563\) 27.4938 1.15872 0.579362 0.815070i \(-0.303302\pi\)
0.579362 + 0.815070i \(0.303302\pi\)
\(564\) 0 0
\(565\) −10.2810 −0.432525
\(566\) 0 0
\(567\) 3.77923 0.158713
\(568\) 0 0
\(569\) 17.3590 0.727729 0.363865 0.931452i \(-0.381457\pi\)
0.363865 + 0.931452i \(0.381457\pi\)
\(570\) 0 0
\(571\) 25.4479 1.06496 0.532480 0.846443i \(-0.321260\pi\)
0.532480 + 0.846443i \(0.321260\pi\)
\(572\) 0 0
\(573\) −43.3876 −1.81254
\(574\) 0 0
\(575\) −7.52543 −0.313832
\(576\) 0 0
\(577\) −10.6178 −0.442024 −0.221012 0.975271i \(-0.570936\pi\)
−0.221012 + 0.975271i \(0.570936\pi\)
\(578\) 0 0
\(579\) 41.0736 1.70696
\(580\) 0 0
\(581\) −9.67307 −0.401307
\(582\) 0 0
\(583\) 9.80642 0.406141
\(584\) 0 0
\(585\) −3.37778 −0.139654
\(586\) 0 0
\(587\) 8.94470 0.369187 0.184594 0.982815i \(-0.440903\pi\)
0.184594 + 0.982815i \(0.440903\pi\)
\(588\) 0 0
\(589\) 7.57136 0.311972
\(590\) 0 0
\(591\) 16.5620 0.681269
\(592\) 0 0
\(593\) 14.1619 0.581561 0.290780 0.956790i \(-0.406085\pi\)
0.290780 + 0.956790i \(0.406085\pi\)
\(594\) 0 0
\(595\) −7.18421 −0.294524
\(596\) 0 0
\(597\) −64.2864 −2.63107
\(598\) 0 0
\(599\) 22.5575 0.921676 0.460838 0.887484i \(-0.347549\pi\)
0.460838 + 0.887484i \(0.347549\pi\)
\(600\) 0 0
\(601\) −40.6133 −1.65665 −0.828326 0.560246i \(-0.810706\pi\)
−0.828326 + 0.560246i \(0.810706\pi\)
\(602\) 0 0
\(603\) −62.5674 −2.54794
\(604\) 0 0
\(605\) −8.67307 −0.352610
\(606\) 0 0
\(607\) −13.5955 −0.551824 −0.275912 0.961183i \(-0.588980\pi\)
−0.275912 + 0.961183i \(0.588980\pi\)
\(608\) 0 0
\(609\) 2.62222 0.106258
\(610\) 0 0
\(611\) 4.29529 0.173769
\(612\) 0 0
\(613\) −42.0830 −1.69972 −0.849858 0.527012i \(-0.823312\pi\)
−0.849858 + 0.527012i \(0.823312\pi\)
\(614\) 0 0
\(615\) −10.6637 −0.430002
\(616\) 0 0
\(617\) 33.5067 1.34893 0.674464 0.738307i \(-0.264375\pi\)
0.674464 + 0.738307i \(0.264375\pi\)
\(618\) 0 0
\(619\) 14.6780 0.589958 0.294979 0.955504i \(-0.404687\pi\)
0.294979 + 0.955504i \(0.404687\pi\)
\(620\) 0 0
\(621\) 53.0607 2.12925
\(622\) 0 0
\(623\) −7.05086 −0.282487
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −4.85728 −0.193981
\(628\) 0 0
\(629\) −31.4509 −1.25403
\(630\) 0 0
\(631\) −11.3176 −0.450545 −0.225273 0.974296i \(-0.572327\pi\)
−0.225273 + 0.974296i \(0.572327\pi\)
\(632\) 0 0
\(633\) −60.5116 −2.40512
\(634\) 0 0
\(635\) −6.22077 −0.246864
\(636\) 0 0
\(637\) 3.84791 0.152460
\(638\) 0 0
\(639\) −74.2262 −2.93634
\(640\) 0 0
\(641\) 34.8988 1.37842 0.689209 0.724562i \(-0.257958\pi\)
0.689209 + 0.724562i \(0.257958\pi\)
\(642\) 0 0
\(643\) 41.9768 1.65540 0.827702 0.561168i \(-0.189648\pi\)
0.827702 + 0.561168i \(0.189648\pi\)
\(644\) 0 0
\(645\) −30.5303 −1.20213
\(646\) 0 0
\(647\) −5.46520 −0.214859 −0.107430 0.994213i \(-0.534262\pi\)
−0.107430 + 0.994213i \(0.534262\pi\)
\(648\) 0 0
\(649\) 2.55215 0.100181
\(650\) 0 0
\(651\) −18.1017 −0.709462
\(652\) 0 0
\(653\) −8.76986 −0.343191 −0.171596 0.985167i \(-0.554892\pi\)
−0.171596 + 0.985167i \(0.554892\pi\)
\(654\) 0 0
\(655\) 11.7605 0.459520
\(656\) 0 0
\(657\) 55.3131 2.15797
\(658\) 0 0
\(659\) −3.29036 −0.128174 −0.0640872 0.997944i \(-0.520414\pi\)
−0.0640872 + 0.997944i \(0.520414\pi\)
\(660\) 0 0
\(661\) 19.7560 0.768421 0.384211 0.923246i \(-0.374474\pi\)
0.384211 + 0.923246i \(0.374474\pi\)
\(662\) 0 0
\(663\) −14.3684 −0.558023
\(664\) 0 0
\(665\) 0.990632 0.0384151
\(666\) 0 0
\(667\) 7.52543 0.291386
\(668\) 0 0
\(669\) 26.2351 1.01431
\(670\) 0 0
\(671\) −2.84743 −0.109924
\(672\) 0 0
\(673\) 44.3970 1.71138 0.855689 0.517490i \(-0.173134\pi\)
0.855689 + 0.517490i \(0.173134\pi\)
\(674\) 0 0
\(675\) −7.05086 −0.271388
\(676\) 0 0
\(677\) 6.09726 0.234337 0.117168 0.993112i \(-0.462618\pi\)
0.117168 + 0.993112i \(0.462618\pi\)
\(678\) 0 0
\(679\) −3.69181 −0.141679
\(680\) 0 0
\(681\) 56.3368 2.15883
\(682\) 0 0
\(683\) −37.9224 −1.45106 −0.725531 0.688190i \(-0.758406\pi\)
−0.725531 + 0.688190i \(0.758406\pi\)
\(684\) 0 0
\(685\) −3.56691 −0.136285
\(686\) 0 0
\(687\) 74.3595 2.83699
\(688\) 0 0
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −13.3145 −0.506507 −0.253254 0.967400i \(-0.581501\pi\)
−0.253254 + 0.967400i \(0.581501\pi\)
\(692\) 0 0
\(693\) 7.47949 0.284123
\(694\) 0 0
\(695\) 8.56199 0.324775
\(696\) 0 0
\(697\) −29.2159 −1.10663
\(698\) 0 0
\(699\) 9.06959 0.343043
\(700\) 0 0
\(701\) −23.4893 −0.887180 −0.443590 0.896230i \(-0.646295\pi\)
−0.443590 + 0.896230i \(0.646295\pi\)
\(702\) 0 0
\(703\) 4.33677 0.163565
\(704\) 0 0
\(705\) 20.0415 0.754806
\(706\) 0 0
\(707\) 12.5620 0.472442
\(708\) 0 0
\(709\) 11.6731 0.438391 0.219196 0.975681i \(-0.429657\pi\)
0.219196 + 0.975681i \(0.429657\pi\)
\(710\) 0 0
\(711\) −49.6084 −1.86046
\(712\) 0 0
\(713\) −51.9496 −1.94553
\(714\) 0 0
\(715\) −0.949145 −0.0354960
\(716\) 0 0
\(717\) 40.4701 1.51138
\(718\) 0 0
\(719\) −29.5526 −1.10213 −0.551063 0.834463i \(-0.685778\pi\)
−0.551063 + 0.834463i \(0.685778\pi\)
\(720\) 0 0
\(721\) 11.6918 0.435426
\(722\) 0 0
\(723\) 53.6227 1.99425
\(724\) 0 0
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) −3.88094 −0.143936 −0.0719680 0.997407i \(-0.522928\pi\)
−0.0719680 + 0.997407i \(0.522928\pi\)
\(728\) 0 0
\(729\) −38.6958 −1.43318
\(730\) 0 0
\(731\) −83.6454 −3.09374
\(732\) 0 0
\(733\) 14.8845 0.549771 0.274885 0.961477i \(-0.411360\pi\)
0.274885 + 0.961477i \(0.411360\pi\)
\(734\) 0 0
\(735\) 17.9541 0.662246
\(736\) 0 0
\(737\) −17.5812 −0.647612
\(738\) 0 0
\(739\) −2.24935 −0.0827438 −0.0413719 0.999144i \(-0.513173\pi\)
−0.0413719 + 0.999144i \(0.513173\pi\)
\(740\) 0 0
\(741\) 1.98126 0.0727836
\(742\) 0 0
\(743\) 3.46520 0.127126 0.0635630 0.997978i \(-0.479754\pi\)
0.0635630 + 0.997978i \(0.479754\pi\)
\(744\) 0 0
\(745\) −5.61285 −0.205639
\(746\) 0 0
\(747\) −58.1388 −2.12719
\(748\) 0 0
\(749\) −9.93978 −0.363192
\(750\) 0 0
\(751\) −3.16992 −0.115672 −0.0578360 0.998326i \(-0.518420\pi\)
−0.0578360 + 0.998326i \(0.518420\pi\)
\(752\) 0 0
\(753\) −40.0415 −1.45919
\(754\) 0 0
\(755\) −10.7971 −0.392945
\(756\) 0 0
\(757\) −52.0785 −1.89283 −0.946413 0.322958i \(-0.895323\pi\)
−0.946413 + 0.322958i \(0.895323\pi\)
\(758\) 0 0
\(759\) 33.3274 1.20971
\(760\) 0 0
\(761\) −14.9777 −0.542942 −0.271471 0.962447i \(-0.587510\pi\)
−0.271471 + 0.962447i \(0.587510\pi\)
\(762\) 0 0
\(763\) −16.2953 −0.589929
\(764\) 0 0
\(765\) −43.1798 −1.56117
\(766\) 0 0
\(767\) −1.04101 −0.0375887
\(768\) 0 0
\(769\) −1.90813 −0.0688091 −0.0344045 0.999408i \(-0.510953\pi\)
−0.0344045 + 0.999408i \(0.510953\pi\)
\(770\) 0 0
\(771\) 4.29529 0.154691
\(772\) 0 0
\(773\) 21.7891 0.783698 0.391849 0.920029i \(-0.371835\pi\)
0.391849 + 0.920029i \(0.371835\pi\)
\(774\) 0 0
\(775\) 6.90321 0.247971
\(776\) 0 0
\(777\) −10.3684 −0.371965
\(778\) 0 0
\(779\) 4.02858 0.144339
\(780\) 0 0
\(781\) −20.8573 −0.746332
\(782\) 0 0
\(783\) 7.05086 0.251977
\(784\) 0 0
\(785\) 2.28100 0.0814122
\(786\) 0 0
\(787\) 18.1388 0.646577 0.323288 0.946301i \(-0.395212\pi\)
0.323288 + 0.946301i \(0.395212\pi\)
\(788\) 0 0
\(789\) −1.28592 −0.0457799
\(790\) 0 0
\(791\) −9.28592 −0.330169
\(792\) 0 0
\(793\) 1.16146 0.0412445
\(794\) 0 0
\(795\) −18.6637 −0.661933
\(796\) 0 0
\(797\) 2.96343 0.104970 0.0524851 0.998622i \(-0.483286\pi\)
0.0524851 + 0.998622i \(0.483286\pi\)
\(798\) 0 0
\(799\) 54.9086 1.94253
\(800\) 0 0
\(801\) −42.3783 −1.49736
\(802\) 0 0
\(803\) 15.5428 0.548493
\(804\) 0 0
\(805\) −6.79706 −0.239565
\(806\) 0 0
\(807\) −11.4380 −0.402637
\(808\) 0 0
\(809\) −26.2953 −0.924493 −0.462247 0.886751i \(-0.652956\pi\)
−0.462247 + 0.886751i \(0.652956\pi\)
\(810\) 0 0
\(811\) −24.3783 −0.856037 −0.428018 0.903770i \(-0.640788\pi\)
−0.428018 + 0.903770i \(0.640788\pi\)
\(812\) 0 0
\(813\) 18.0228 0.632085
\(814\) 0 0
\(815\) −16.3225 −0.571752
\(816\) 0 0
\(817\) 11.5339 0.403520
\(818\) 0 0
\(819\) −3.05086 −0.106606
\(820\) 0 0
\(821\) 1.52987 0.0533929 0.0266965 0.999644i \(-0.491501\pi\)
0.0266965 + 0.999644i \(0.491501\pi\)
\(822\) 0 0
\(823\) −46.7195 −1.62854 −0.814269 0.580487i \(-0.802862\pi\)
−0.814269 + 0.580487i \(0.802862\pi\)
\(824\) 0 0
\(825\) −4.42864 −0.154185
\(826\) 0 0
\(827\) 29.6499 1.03103 0.515514 0.856881i \(-0.327601\pi\)
0.515514 + 0.856881i \(0.327601\pi\)
\(828\) 0 0
\(829\) −8.79706 −0.305534 −0.152767 0.988262i \(-0.548818\pi\)
−0.152767 + 0.988262i \(0.548818\pi\)
\(830\) 0 0
\(831\) −16.1748 −0.561099
\(832\) 0 0
\(833\) 49.1896 1.70432
\(834\) 0 0
\(835\) 4.76986 0.165068
\(836\) 0 0
\(837\) −48.6735 −1.68240
\(838\) 0 0
\(839\) −11.3319 −0.391219 −0.195609 0.980682i \(-0.562668\pi\)
−0.195609 + 0.980682i \(0.562668\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −19.4380 −0.669481
\(844\) 0 0
\(845\) −12.6128 −0.433895
\(846\) 0 0
\(847\) −7.83362 −0.269166
\(848\) 0 0
\(849\) 74.9086 2.57086
\(850\) 0 0
\(851\) −29.7560 −1.02002
\(852\) 0 0
\(853\) 54.8845 1.87921 0.939604 0.342263i \(-0.111193\pi\)
0.939604 + 0.342263i \(0.111193\pi\)
\(854\) 0 0
\(855\) 5.95407 0.203625
\(856\) 0 0
\(857\) 36.4385 1.24471 0.622357 0.782733i \(-0.286175\pi\)
0.622357 + 0.782733i \(0.286175\pi\)
\(858\) 0 0
\(859\) −1.72885 −0.0589875 −0.0294938 0.999565i \(-0.509390\pi\)
−0.0294938 + 0.999565i \(0.509390\pi\)
\(860\) 0 0
\(861\) −9.63158 −0.328243
\(862\) 0 0
\(863\) 9.40192 0.320045 0.160023 0.987113i \(-0.448843\pi\)
0.160023 + 0.987113i \(0.448843\pi\)
\(864\) 0 0
\(865\) 4.23506 0.143996
\(866\) 0 0
\(867\) −134.323 −4.56186
\(868\) 0 0
\(869\) −13.9398 −0.472875
\(870\) 0 0
\(871\) 7.17130 0.242990
\(872\) 0 0
\(873\) −22.1891 −0.750988
\(874\) 0 0
\(875\) 0.903212 0.0305341
\(876\) 0 0
\(877\) 8.91750 0.301123 0.150561 0.988601i \(-0.451892\pi\)
0.150561 + 0.988601i \(0.451892\pi\)
\(878\) 0 0
\(879\) 54.7052 1.84516
\(880\) 0 0
\(881\) −42.1245 −1.41921 −0.709605 0.704600i \(-0.751126\pi\)
−0.709605 + 0.704600i \(0.751126\pi\)
\(882\) 0 0
\(883\) 38.4340 1.29341 0.646704 0.762741i \(-0.276147\pi\)
0.646704 + 0.762741i \(0.276147\pi\)
\(884\) 0 0
\(885\) −4.85728 −0.163276
\(886\) 0 0
\(887\) 38.6365 1.29729 0.648643 0.761092i \(-0.275337\pi\)
0.648643 + 0.761092i \(0.275337\pi\)
\(888\) 0 0
\(889\) −5.61868 −0.188444
\(890\) 0 0
\(891\) 6.38271 0.213829
\(892\) 0 0
\(893\) −7.57136 −0.253366
\(894\) 0 0
\(895\) −9.71456 −0.324722
\(896\) 0 0
\(897\) −13.5941 −0.453894
\(898\) 0 0
\(899\) −6.90321 −0.230235
\(900\) 0 0
\(901\) −51.1338 −1.70351
\(902\) 0 0
\(903\) −27.5754 −0.917651
\(904\) 0 0
\(905\) 0.326929 0.0108675
\(906\) 0 0
\(907\) −0.534795 −0.0177576 −0.00887880 0.999961i \(-0.502826\pi\)
−0.00887880 + 0.999961i \(0.502826\pi\)
\(908\) 0 0
\(909\) 75.5022 2.50425
\(910\) 0 0
\(911\) 23.6686 0.784177 0.392088 0.919928i \(-0.371753\pi\)
0.392088 + 0.919928i \(0.371753\pi\)
\(912\) 0 0
\(913\) −16.3368 −0.540668
\(914\) 0 0
\(915\) 5.41927 0.179156
\(916\) 0 0
\(917\) 10.6222 0.350776
\(918\) 0 0
\(919\) −35.7748 −1.18010 −0.590051 0.807366i \(-0.700892\pi\)
−0.590051 + 0.807366i \(0.700892\pi\)
\(920\) 0 0
\(921\) −4.81579 −0.158686
\(922\) 0 0
\(923\) 8.50760 0.280031
\(924\) 0 0
\(925\) 3.95407 0.130009
\(926\) 0 0
\(927\) 70.2721 2.30804
\(928\) 0 0
\(929\) −52.7753 −1.73150 −0.865750 0.500477i \(-0.833158\pi\)
−0.865750 + 0.500477i \(0.833158\pi\)
\(930\) 0 0
\(931\) −6.78277 −0.222296
\(932\) 0 0
\(933\) 61.8390 2.02452
\(934\) 0 0
\(935\) −12.1334 −0.396803
\(936\) 0 0
\(937\) 42.1245 1.37615 0.688073 0.725641i \(-0.258457\pi\)
0.688073 + 0.725641i \(0.258457\pi\)
\(938\) 0 0
\(939\) −25.0321 −0.816892
\(940\) 0 0
\(941\) −3.89829 −0.127081 −0.0635403 0.997979i \(-0.520239\pi\)
−0.0635403 + 0.997979i \(0.520239\pi\)
\(942\) 0 0
\(943\) −27.6414 −0.900129
\(944\) 0 0
\(945\) −6.36842 −0.207165
\(946\) 0 0
\(947\) −9.56691 −0.310883 −0.155441 0.987845i \(-0.549680\pi\)
−0.155441 + 0.987845i \(0.549680\pi\)
\(948\) 0 0
\(949\) −6.33984 −0.205800
\(950\) 0 0
\(951\) 80.1156 2.59793
\(952\) 0 0
\(953\) 27.2070 0.881320 0.440660 0.897674i \(-0.354744\pi\)
0.440660 + 0.897674i \(0.354744\pi\)
\(954\) 0 0
\(955\) 14.9447 0.483599
\(956\) 0 0
\(957\) 4.42864 0.143158
\(958\) 0 0
\(959\) −3.22168 −0.104033
\(960\) 0 0
\(961\) 16.6543 0.537237
\(962\) 0 0
\(963\) −59.7418 −1.92515
\(964\) 0 0
\(965\) −14.1476 −0.455429
\(966\) 0 0
\(967\) 16.8015 0.540300 0.270150 0.962818i \(-0.412927\pi\)
0.270150 + 0.962818i \(0.412927\pi\)
\(968\) 0 0
\(969\) 25.3274 0.813633
\(970\) 0 0
\(971\) 17.4465 0.559884 0.279942 0.960017i \(-0.409685\pi\)
0.279942 + 0.960017i \(0.409685\pi\)
\(972\) 0 0
\(973\) 7.73329 0.247918
\(974\) 0 0
\(975\) 1.80642 0.0578519
\(976\) 0 0
\(977\) 32.0513 1.02541 0.512706 0.858564i \(-0.328643\pi\)
0.512706 + 0.858564i \(0.328643\pi\)
\(978\) 0 0
\(979\) −11.9081 −0.380586
\(980\) 0 0
\(981\) −97.9407 −3.12701
\(982\) 0 0
\(983\) 16.5259 0.527094 0.263547 0.964646i \(-0.415108\pi\)
0.263547 + 0.964646i \(0.415108\pi\)
\(984\) 0 0
\(985\) −5.70471 −0.181767
\(986\) 0 0
\(987\) 18.1017 0.576184
\(988\) 0 0
\(989\) −79.1378 −2.51644
\(990\) 0 0
\(991\) 9.34920 0.296987 0.148494 0.988913i \(-0.452558\pi\)
0.148494 + 0.988913i \(0.452558\pi\)
\(992\) 0 0
\(993\) 49.1022 1.55821
\(994\) 0 0
\(995\) 22.1432 0.701987
\(996\) 0 0
\(997\) −15.9956 −0.506584 −0.253292 0.967390i \(-0.581513\pi\)
−0.253292 + 0.967390i \(0.581513\pi\)
\(998\) 0 0
\(999\) −27.8796 −0.882070
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2320.2.a.n.1.1 3
4.3 odd 2 145.2.a.c.1.2 3
8.3 odd 2 9280.2.a.bj.1.1 3
8.5 even 2 9280.2.a.br.1.3 3
12.11 even 2 1305.2.a.p.1.2 3
20.3 even 4 725.2.b.e.349.3 6
20.7 even 4 725.2.b.e.349.4 6
20.19 odd 2 725.2.a.e.1.2 3
28.27 even 2 7105.2.a.o.1.2 3
60.59 even 2 6525.2.a.be.1.2 3
116.115 odd 2 4205.2.a.f.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.2 3 4.3 odd 2
725.2.a.e.1.2 3 20.19 odd 2
725.2.b.e.349.3 6 20.3 even 4
725.2.b.e.349.4 6 20.7 even 4
1305.2.a.p.1.2 3 12.11 even 2
2320.2.a.n.1.1 3 1.1 even 1 trivial
4205.2.a.f.1.2 3 116.115 odd 2
6525.2.a.be.1.2 3 60.59 even 2
7105.2.a.o.1.2 3 28.27 even 2
9280.2.a.bj.1.1 3 8.3 odd 2
9280.2.a.br.1.3 3 8.5 even 2