Properties

Label 2240.2.k.e.1791.2
Level $2240$
Weight $2$
Character 2240.1791
Analytic conductor $17.886$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(1791,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.1791");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.k (of order \(2\), degree \(1\), not 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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.342102016.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1791.2
Root \(1.17915 + 0.780776i\) of defining polynomial
Character \(\chi\) \(=\) 2240.1791
Dual form 2240.2.k.e.1791.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.02045 q^{3} +1.00000i q^{5} +(-2.17238 - 1.51022i) q^{7} +6.12311 q^{9} +O(q^{10})\) \(q-3.02045 q^{3} +1.00000i q^{5} +(-2.17238 - 1.51022i) q^{7} +6.12311 q^{9} -4.71659i q^{11} +2.00000i q^{13} -3.02045i q^{15} -1.12311i q^{17} +4.71659 q^{19} +(6.56155 + 4.56155i) q^{21} +6.41273i q^{23} -1.00000 q^{25} -9.43318 q^{27} +2.00000 q^{29} -3.39228 q^{31} +14.2462i q^{33} +(1.51022 - 2.17238i) q^{35} -2.00000 q^{37} -6.04090i q^{39} +1.12311i q^{41} +0.371834i q^{43} +6.12311i q^{45} +5.08842 q^{47} +(2.43845 + 6.56155i) q^{49} +3.39228i q^{51} -2.00000 q^{53} +4.71659 q^{55} -14.2462 q^{57} +2.06798 q^{59} +2.00000i q^{61} +(-13.3017 - 9.24726i) q^{63} -2.00000 q^{65} +3.76412i q^{67} -19.3693i q^{69} -7.36520i q^{71} -15.3693i q^{73} +3.02045 q^{75} +(-7.12311 + 10.2462i) q^{77} +1.32431i q^{79} +10.1231 q^{81} -3.02045 q^{83} +1.12311 q^{85} -6.04090 q^{87} -12.0000i q^{89} +(3.02045 - 4.34475i) q^{91} +10.2462 q^{93} +4.71659i q^{95} +1.12311i q^{97} -28.8802i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{9} + 36 q^{21} - 8 q^{25} + 16 q^{29} - 16 q^{37} + 36 q^{49} - 16 q^{53} - 48 q^{57} - 16 q^{65} - 24 q^{77} + 48 q^{81} - 24 q^{85} + 16 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.02045 −1.74386 −0.871928 0.489634i \(-0.837130\pi\)
−0.871928 + 0.489634i \(0.837130\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) −2.17238 1.51022i −0.821081 0.570811i
\(8\) 0 0
\(9\) 6.12311 2.04104
\(10\) 0 0
\(11\) 4.71659i 1.42211i −0.703139 0.711053i \(-0.748219\pi\)
0.703139 0.711053i \(-0.251781\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 3.02045i 0.779876i
\(16\) 0 0
\(17\) 1.12311i 0.272393i −0.990682 0.136197i \(-0.956512\pi\)
0.990682 0.136197i \(-0.0434879\pi\)
\(18\) 0 0
\(19\) 4.71659 1.08206 0.541030 0.841003i \(-0.318035\pi\)
0.541030 + 0.841003i \(0.318035\pi\)
\(20\) 0 0
\(21\) 6.56155 + 4.56155i 1.43185 + 0.995412i
\(22\) 0 0
\(23\) 6.41273i 1.33715i 0.743646 + 0.668573i \(0.233095\pi\)
−0.743646 + 0.668573i \(0.766905\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −9.43318 −1.81542
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −3.39228 −0.609272 −0.304636 0.952469i \(-0.598535\pi\)
−0.304636 + 0.952469i \(0.598535\pi\)
\(32\) 0 0
\(33\) 14.2462i 2.47995i
\(34\) 0 0
\(35\) 1.51022 2.17238i 0.255274 0.367199i
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 6.04090i 0.967317i
\(40\) 0 0
\(41\) 1.12311i 0.175400i 0.996147 + 0.0876998i \(0.0279516\pi\)
−0.996147 + 0.0876998i \(0.972048\pi\)
\(42\) 0 0
\(43\) 0.371834i 0.0567042i 0.999598 + 0.0283521i \(0.00902596\pi\)
−0.999598 + 0.0283521i \(0.990974\pi\)
\(44\) 0 0
\(45\) 6.12311i 0.912779i
\(46\) 0 0
\(47\) 5.08842 0.742223 0.371111 0.928588i \(-0.378977\pi\)
0.371111 + 0.928588i \(0.378977\pi\)
\(48\) 0 0
\(49\) 2.43845 + 6.56155i 0.348350 + 0.937365i
\(50\) 0 0
\(51\) 3.39228i 0.475014i
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 4.71659 0.635985
\(56\) 0 0
\(57\) −14.2462 −1.88696
\(58\) 0 0
\(59\) 2.06798 0.269227 0.134614 0.990898i \(-0.457021\pi\)
0.134614 + 0.990898i \(0.457021\pi\)
\(60\) 0 0
\(61\) 2.00000i 0.256074i 0.991769 + 0.128037i \(0.0408676\pi\)
−0.991769 + 0.128037i \(0.959132\pi\)
\(62\) 0 0
\(63\) −13.3017 9.24726i −1.67586 1.16505i
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 3.76412i 0.459860i 0.973207 + 0.229930i \(0.0738497\pi\)
−0.973207 + 0.229930i \(0.926150\pi\)
\(68\) 0 0
\(69\) 19.3693i 2.33179i
\(70\) 0 0
\(71\) 7.36520i 0.874089i −0.899440 0.437044i \(-0.856025\pi\)
0.899440 0.437044i \(-0.143975\pi\)
\(72\) 0 0
\(73\) 15.3693i 1.79884i −0.437083 0.899421i \(-0.643988\pi\)
0.437083 0.899421i \(-0.356012\pi\)
\(74\) 0 0
\(75\) 3.02045 0.348771
\(76\) 0 0
\(77\) −7.12311 + 10.2462i −0.811753 + 1.16766i
\(78\) 0 0
\(79\) 1.32431i 0.148996i 0.997221 + 0.0744981i \(0.0237355\pi\)
−0.997221 + 0.0744981i \(0.976265\pi\)
\(80\) 0 0
\(81\) 10.1231 1.12479
\(82\) 0 0
\(83\) −3.02045 −0.331537 −0.165769 0.986165i \(-0.553010\pi\)
−0.165769 + 0.986165i \(0.553010\pi\)
\(84\) 0 0
\(85\) 1.12311 0.121818
\(86\) 0 0
\(87\) −6.04090 −0.647652
\(88\) 0 0
\(89\) 12.0000i 1.27200i −0.771690 0.635999i \(-0.780588\pi\)
0.771690 0.635999i \(-0.219412\pi\)
\(90\) 0 0
\(91\) 3.02045 4.34475i 0.316629 0.455454i
\(92\) 0 0
\(93\) 10.2462 1.06248
\(94\) 0 0
\(95\) 4.71659i 0.483912i
\(96\) 0 0
\(97\) 1.12311i 0.114034i 0.998373 + 0.0570170i \(0.0181589\pi\)
−0.998373 + 0.0570170i \(0.981841\pi\)
\(98\) 0 0
\(99\) 28.8802i 2.90257i
\(100\) 0 0
\(101\) 15.1231i 1.50481i −0.658703 0.752403i \(-0.728895\pi\)
0.658703 0.752403i \(-0.271105\pi\)
\(102\) 0 0
\(103\) −7.73704 −0.762353 −0.381176 0.924502i \(-0.624481\pi\)
−0.381176 + 0.924502i \(0.624481\pi\)
\(104\) 0 0
\(105\) −4.56155 + 6.56155i −0.445162 + 0.640342i
\(106\) 0 0
\(107\) 5.66906i 0.548049i 0.961723 + 0.274024i \(0.0883549\pi\)
−0.961723 + 0.274024i \(0.911645\pi\)
\(108\) 0 0
\(109\) 7.12311 0.682270 0.341135 0.940014i \(-0.389189\pi\)
0.341135 + 0.940014i \(0.389189\pi\)
\(110\) 0 0
\(111\) 6.04090 0.573376
\(112\) 0 0
\(113\) 18.4924 1.73962 0.869810 0.493386i \(-0.164241\pi\)
0.869810 + 0.493386i \(0.164241\pi\)
\(114\) 0 0
\(115\) −6.41273 −0.597990
\(116\) 0 0
\(117\) 12.2462i 1.13216i
\(118\) 0 0
\(119\) −1.69614 + 2.43981i −0.155485 + 0.223657i
\(120\) 0 0
\(121\) −11.2462 −1.02238
\(122\) 0 0
\(123\) 3.39228i 0.305872i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 17.7509i 1.57513i −0.616229 0.787567i \(-0.711341\pi\)
0.616229 0.787567i \(-0.288659\pi\)
\(128\) 0 0
\(129\) 1.12311i 0.0988839i
\(130\) 0 0
\(131\) −17.5420 −1.53266 −0.766328 0.642450i \(-0.777918\pi\)
−0.766328 + 0.642450i \(0.777918\pi\)
\(132\) 0 0
\(133\) −10.2462 7.12311i −0.888459 0.617652i
\(134\) 0 0
\(135\) 9.43318i 0.811879i
\(136\) 0 0
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) −16.7984 −1.42482 −0.712410 0.701763i \(-0.752396\pi\)
−0.712410 + 0.701763i \(0.752396\pi\)
\(140\) 0 0
\(141\) −15.3693 −1.29433
\(142\) 0 0
\(143\) 9.43318 0.788842
\(144\) 0 0
\(145\) 2.00000i 0.166091i
\(146\) 0 0
\(147\) −7.36520 19.8188i −0.607472 1.63463i
\(148\) 0 0
\(149\) −5.36932 −0.439872 −0.219936 0.975514i \(-0.570585\pi\)
−0.219936 + 0.975514i \(0.570585\pi\)
\(150\) 0 0
\(151\) 10.0138i 0.814913i −0.913225 0.407456i \(-0.866416\pi\)
0.913225 0.407456i \(-0.133584\pi\)
\(152\) 0 0
\(153\) 6.87689i 0.555964i
\(154\) 0 0
\(155\) 3.39228i 0.272475i
\(156\) 0 0
\(157\) 0.246211i 0.0196498i −0.999952 0.00982490i \(-0.996873\pi\)
0.999952 0.00982490i \(-0.00312741\pi\)
\(158\) 0 0
\(159\) 6.04090 0.479074
\(160\) 0 0
\(161\) 9.68466 13.9309i 0.763258 1.09791i
\(162\) 0 0
\(163\) 3.02045i 0.236580i −0.992979 0.118290i \(-0.962259\pi\)
0.992979 0.118290i \(-0.0377412\pi\)
\(164\) 0 0
\(165\) −14.2462 −1.10907
\(166\) 0 0
\(167\) −19.8188 −1.53363 −0.766813 0.641870i \(-0.778159\pi\)
−0.766813 + 0.641870i \(0.778159\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 28.8802 2.20852
\(172\) 0 0
\(173\) 16.2462i 1.23518i −0.786502 0.617588i \(-0.788110\pi\)
0.786502 0.617588i \(-0.211890\pi\)
\(174\) 0 0
\(175\) 2.17238 + 1.51022i 0.164216 + 0.114162i
\(176\) 0 0
\(177\) −6.24621 −0.469494
\(178\) 0 0
\(179\) 10.7575i 0.804052i 0.915628 + 0.402026i \(0.131694\pi\)
−0.915628 + 0.402026i \(0.868306\pi\)
\(180\) 0 0
\(181\) 15.1231i 1.12409i −0.827106 0.562046i \(-0.810014\pi\)
0.827106 0.562046i \(-0.189986\pi\)
\(182\) 0 0
\(183\) 6.04090i 0.446556i
\(184\) 0 0
\(185\) 2.00000i 0.147043i
\(186\) 0 0
\(187\) −5.29723 −0.387372
\(188\) 0 0
\(189\) 20.4924 + 14.2462i 1.49060 + 1.03626i
\(190\) 0 0
\(191\) 16.7984i 1.21549i 0.794133 + 0.607744i \(0.207925\pi\)
−0.794133 + 0.607744i \(0.792075\pi\)
\(192\) 0 0
\(193\) −22.4924 −1.61904 −0.809520 0.587092i \(-0.800273\pi\)
−0.809520 + 0.587092i \(0.800273\pi\)
\(194\) 0 0
\(195\) 6.04090 0.432598
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −26.8122 −1.90067 −0.950333 0.311235i \(-0.899258\pi\)
−0.950333 + 0.311235i \(0.899258\pi\)
\(200\) 0 0
\(201\) 11.3693i 0.801930i
\(202\) 0 0
\(203\) −4.34475 3.02045i −0.304942 0.211994i
\(204\) 0 0
\(205\) −1.12311 −0.0784411
\(206\) 0 0
\(207\) 39.2658i 2.72916i
\(208\) 0 0
\(209\) 22.2462i 1.53880i
\(210\) 0 0
\(211\) 22.0956i 1.52112i −0.649265 0.760562i \(-0.724923\pi\)
0.649265 0.760562i \(-0.275077\pi\)
\(212\) 0 0
\(213\) 22.2462i 1.52429i
\(214\) 0 0
\(215\) −0.371834 −0.0253589
\(216\) 0 0
\(217\) 7.36932 + 5.12311i 0.500262 + 0.347779i
\(218\) 0 0
\(219\) 46.4222i 3.13692i
\(220\) 0 0
\(221\) 2.24621 0.151097
\(222\) 0 0
\(223\) 20.5625 1.37697 0.688483 0.725252i \(-0.258277\pi\)
0.688483 + 0.725252i \(0.258277\pi\)
\(224\) 0 0
\(225\) −6.12311 −0.408207
\(226\) 0 0
\(227\) 3.76412 0.249833 0.124917 0.992167i \(-0.460134\pi\)
0.124917 + 0.992167i \(0.460134\pi\)
\(228\) 0 0
\(229\) 12.8769i 0.850929i −0.904975 0.425465i \(-0.860111\pi\)
0.904975 0.425465i \(-0.139889\pi\)
\(230\) 0 0
\(231\) 21.5150 30.9481i 1.41558 2.03624i
\(232\) 0 0
\(233\) 7.75379 0.507968 0.253984 0.967208i \(-0.418259\pi\)
0.253984 + 0.967208i \(0.418259\pi\)
\(234\) 0 0
\(235\) 5.08842i 0.331932i
\(236\) 0 0
\(237\) 4.00000i 0.259828i
\(238\) 0 0
\(239\) 3.97292i 0.256987i 0.991710 + 0.128493i \(0.0410141\pi\)
−0.991710 + 0.128493i \(0.958986\pi\)
\(240\) 0 0
\(241\) 25.6155i 1.65004i 0.565103 + 0.825021i \(0.308837\pi\)
−0.565103 + 0.825021i \(0.691163\pi\)
\(242\) 0 0
\(243\) −2.27678 −0.146055
\(244\) 0 0
\(245\) −6.56155 + 2.43845i −0.419202 + 0.155787i
\(246\) 0 0
\(247\) 9.43318i 0.600219i
\(248\) 0 0
\(249\) 9.12311 0.578153
\(250\) 0 0
\(251\) 16.0547 1.01336 0.506682 0.862133i \(-0.330872\pi\)
0.506682 + 0.862133i \(0.330872\pi\)
\(252\) 0 0
\(253\) 30.2462 1.90156
\(254\) 0 0
\(255\) −3.39228 −0.212433
\(256\) 0 0
\(257\) 11.3693i 0.709199i 0.935018 + 0.354599i \(0.115383\pi\)
−0.935018 + 0.354599i \(0.884617\pi\)
\(258\) 0 0
\(259\) 4.34475 + 3.02045i 0.269970 + 0.187682i
\(260\) 0 0
\(261\) 12.2462 0.758021
\(262\) 0 0
\(263\) 18.4945i 1.14042i −0.821499 0.570211i \(-0.806862\pi\)
0.821499 0.570211i \(-0.193138\pi\)
\(264\) 0 0
\(265\) 2.00000i 0.122859i
\(266\) 0 0
\(267\) 36.2454i 2.21818i
\(268\) 0 0
\(269\) 0.246211i 0.0150118i −0.999972 0.00750588i \(-0.997611\pi\)
0.999972 0.00750588i \(-0.00238922\pi\)
\(270\) 0 0
\(271\) 1.90495 0.115717 0.0578586 0.998325i \(-0.481573\pi\)
0.0578586 + 0.998325i \(0.481573\pi\)
\(272\) 0 0
\(273\) −9.12311 + 13.1231i −0.552155 + 0.794246i
\(274\) 0 0
\(275\) 4.71659i 0.284421i
\(276\) 0 0
\(277\) −12.2462 −0.735804 −0.367902 0.929865i \(-0.619924\pi\)
−0.367902 + 0.929865i \(0.619924\pi\)
\(278\) 0 0
\(279\) −20.7713 −1.24355
\(280\) 0 0
\(281\) −29.3693 −1.75203 −0.876013 0.482287i \(-0.839806\pi\)
−0.876013 + 0.482287i \(0.839806\pi\)
\(282\) 0 0
\(283\) 4.92539 0.292784 0.146392 0.989227i \(-0.453234\pi\)
0.146392 + 0.989227i \(0.453234\pi\)
\(284\) 0 0
\(285\) 14.2462i 0.843873i
\(286\) 0 0
\(287\) 1.69614 2.43981i 0.100120 0.144017i
\(288\) 0 0
\(289\) 15.7386 0.925802
\(290\) 0 0
\(291\) 3.39228i 0.198859i
\(292\) 0 0
\(293\) 15.7538i 0.920346i −0.887829 0.460173i \(-0.847787\pi\)
0.887829 0.460173i \(-0.152213\pi\)
\(294\) 0 0
\(295\) 2.06798i 0.120402i
\(296\) 0 0
\(297\) 44.4924i 2.58171i
\(298\) 0 0
\(299\) −12.8255 −0.741715
\(300\) 0 0
\(301\) 0.561553 0.807764i 0.0323674 0.0465587i
\(302\) 0 0
\(303\) 45.6786i 2.62416i
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −1.53311 −0.0874993 −0.0437496 0.999043i \(-0.513930\pi\)
−0.0437496 + 0.999043i \(0.513930\pi\)
\(308\) 0 0
\(309\) 23.3693 1.32943
\(310\) 0 0
\(311\) 30.2045 1.71274 0.856369 0.516364i \(-0.172715\pi\)
0.856369 + 0.516364i \(0.172715\pi\)
\(312\) 0 0
\(313\) 33.6155i 1.90006i −0.312156 0.950031i \(-0.601051\pi\)
0.312156 0.950031i \(-0.398949\pi\)
\(314\) 0 0
\(315\) 9.24726 13.3017i 0.521024 0.749466i
\(316\) 0 0
\(317\) −12.2462 −0.687816 −0.343908 0.939003i \(-0.611751\pi\)
−0.343908 + 0.939003i \(0.611751\pi\)
\(318\) 0 0
\(319\) 9.43318i 0.528157i
\(320\) 0 0
\(321\) 17.1231i 0.955719i
\(322\) 0 0
\(323\) 5.29723i 0.294746i
\(324\) 0 0
\(325\) 2.00000i 0.110940i
\(326\) 0 0
\(327\) −21.5150 −1.18978
\(328\) 0 0
\(329\) −11.0540 7.68466i −0.609425 0.423669i
\(330\) 0 0
\(331\) 18.2857i 1.00507i −0.864556 0.502537i \(-0.832400\pi\)
0.864556 0.502537i \(-0.167600\pi\)
\(332\) 0 0
\(333\) −12.2462 −0.671088
\(334\) 0 0
\(335\) −3.76412 −0.205656
\(336\) 0 0
\(337\) −12.2462 −0.667094 −0.333547 0.942734i \(-0.608246\pi\)
−0.333547 + 0.942734i \(0.608246\pi\)
\(338\) 0 0
\(339\) −55.8554 −3.03365
\(340\) 0 0
\(341\) 16.0000i 0.866449i
\(342\) 0 0
\(343\) 4.61219 17.9368i 0.249035 0.968495i
\(344\) 0 0
\(345\) 19.3693 1.04281
\(346\) 0 0
\(347\) 5.66906i 0.304331i 0.988355 + 0.152166i \(0.0486247\pi\)
−0.988355 + 0.152166i \(0.951375\pi\)
\(348\) 0 0
\(349\) 29.8617i 1.59846i 0.601024 + 0.799231i \(0.294760\pi\)
−0.601024 + 0.799231i \(0.705240\pi\)
\(350\) 0 0
\(351\) 18.8664i 1.00701i
\(352\) 0 0
\(353\) 19.3693i 1.03092i 0.856912 + 0.515462i \(0.172380\pi\)
−0.856912 + 0.515462i \(0.827620\pi\)
\(354\) 0 0
\(355\) 7.36520 0.390904
\(356\) 0 0
\(357\) 5.12311 7.36932i 0.271144 0.390026i
\(358\) 0 0
\(359\) 16.0547i 0.847335i 0.905818 + 0.423668i \(0.139258\pi\)
−0.905818 + 0.423668i \(0.860742\pi\)
\(360\) 0 0
\(361\) 3.24621 0.170853
\(362\) 0 0
\(363\) 33.9686 1.78289
\(364\) 0 0
\(365\) 15.3693 0.804467
\(366\) 0 0
\(367\) −10.3857 −0.542127 −0.271063 0.962562i \(-0.587375\pi\)
−0.271063 + 0.962562i \(0.587375\pi\)
\(368\) 0 0
\(369\) 6.87689i 0.357997i
\(370\) 0 0
\(371\) 4.34475 + 3.02045i 0.225568 + 0.156814i
\(372\) 0 0
\(373\) 10.4924 0.543277 0.271639 0.962399i \(-0.412434\pi\)
0.271639 + 0.962399i \(0.412434\pi\)
\(374\) 0 0
\(375\) 3.02045i 0.155975i
\(376\) 0 0
\(377\) 4.00000i 0.206010i
\(378\) 0 0
\(379\) 14.8934i 0.765024i −0.923950 0.382512i \(-0.875059\pi\)
0.923950 0.382512i \(-0.124941\pi\)
\(380\) 0 0
\(381\) 53.6155i 2.74681i
\(382\) 0 0
\(383\) −19.0752 −0.974695 −0.487348 0.873208i \(-0.662036\pi\)
−0.487348 + 0.873208i \(0.662036\pi\)
\(384\) 0 0
\(385\) −10.2462 7.12311i −0.522195 0.363027i
\(386\) 0 0
\(387\) 2.27678i 0.115735i
\(388\) 0 0
\(389\) 7.12311 0.361156 0.180578 0.983561i \(-0.442203\pi\)
0.180578 + 0.983561i \(0.442203\pi\)
\(390\) 0 0
\(391\) 7.20217 0.364230
\(392\) 0 0
\(393\) 52.9848 2.67273
\(394\) 0 0
\(395\) −1.32431 −0.0666331
\(396\) 0 0
\(397\) 14.0000i 0.702640i −0.936255 0.351320i \(-0.885733\pi\)
0.936255 0.351320i \(-0.114267\pi\)
\(398\) 0 0
\(399\) 30.9481 + 21.5150i 1.54935 + 1.07710i
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 6.78456i 0.337963i
\(404\) 0 0
\(405\) 10.1231i 0.503021i
\(406\) 0 0
\(407\) 9.43318i 0.467585i
\(408\) 0 0
\(409\) 17.1231i 0.846683i 0.905970 + 0.423342i \(0.139143\pi\)
−0.905970 + 0.423342i \(0.860857\pi\)
\(410\) 0 0
\(411\) 42.2863 2.08583
\(412\) 0 0
\(413\) −4.49242 3.12311i −0.221058 0.153678i
\(414\) 0 0
\(415\) 3.02045i 0.148268i
\(416\) 0 0
\(417\) 50.7386 2.48468
\(418\) 0 0
\(419\) 23.5829 1.15210 0.576051 0.817414i \(-0.304593\pi\)
0.576051 + 0.817414i \(0.304593\pi\)
\(420\) 0 0
\(421\) −23.6155 −1.15095 −0.575475 0.817819i \(-0.695183\pi\)
−0.575475 + 0.817819i \(0.695183\pi\)
\(422\) 0 0
\(423\) 31.1570 1.51490
\(424\) 0 0
\(425\) 1.12311i 0.0544786i
\(426\) 0 0
\(427\) 3.02045 4.34475i 0.146170 0.210257i
\(428\) 0 0
\(429\) −28.4924 −1.37563
\(430\) 0 0
\(431\) 5.87787i 0.283127i 0.989929 + 0.141563i \(0.0452129\pi\)
−0.989929 + 0.141563i \(0.954787\pi\)
\(432\) 0 0
\(433\) 29.6155i 1.42323i −0.702569 0.711616i \(-0.747964\pi\)
0.702569 0.711616i \(-0.252036\pi\)
\(434\) 0 0
\(435\) 6.04090i 0.289639i
\(436\) 0 0
\(437\) 30.2462i 1.44687i
\(438\) 0 0
\(439\) 16.2177 0.774031 0.387015 0.922073i \(-0.373506\pi\)
0.387015 + 0.922073i \(0.373506\pi\)
\(440\) 0 0
\(441\) 14.9309 + 40.1771i 0.710994 + 1.91319i
\(442\) 0 0
\(443\) 23.0481i 1.09505i 0.836790 + 0.547524i \(0.184429\pi\)
−0.836790 + 0.547524i \(0.815571\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 0 0
\(447\) 16.2177 0.767073
\(448\) 0 0
\(449\) −15.6155 −0.736942 −0.368471 0.929639i \(-0.620119\pi\)
−0.368471 + 0.929639i \(0.620119\pi\)
\(450\) 0 0
\(451\) 5.29723 0.249437
\(452\) 0 0
\(453\) 30.2462i 1.42109i
\(454\) 0 0
\(455\) 4.34475 + 3.02045i 0.203685 + 0.141601i
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) 0 0
\(459\) 10.5945i 0.494507i
\(460\) 0 0
\(461\) 3.61553i 0.168392i 0.996449 + 0.0841960i \(0.0268322\pi\)
−0.996449 + 0.0841960i \(0.973168\pi\)
\(462\) 0 0
\(463\) 28.6714i 1.33247i −0.745741 0.666236i \(-0.767904\pi\)
0.745741 0.666236i \(-0.232096\pi\)
\(464\) 0 0
\(465\) 10.2462i 0.475157i
\(466\) 0 0
\(467\) −18.4945 −0.855824 −0.427912 0.903820i \(-0.640751\pi\)
−0.427912 + 0.903820i \(0.640751\pi\)
\(468\) 0 0
\(469\) 5.68466 8.17708i 0.262493 0.377583i
\(470\) 0 0
\(471\) 0.743668i 0.0342664i
\(472\) 0 0
\(473\) 1.75379 0.0806393
\(474\) 0 0
\(475\) −4.71659 −0.216412
\(476\) 0 0
\(477\) −12.2462 −0.560715
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 4.00000i 0.182384i
\(482\) 0 0
\(483\) −29.2520 + 42.0775i −1.33101 + 1.91459i
\(484\) 0 0
\(485\) −1.12311 −0.0509976
\(486\) 0 0
\(487\) 13.1973i 0.598026i 0.954249 + 0.299013i \(0.0966575\pi\)
−0.954249 + 0.299013i \(0.903343\pi\)
\(488\) 0 0
\(489\) 9.12311i 0.412561i
\(490\) 0 0
\(491\) 31.5288i 1.42287i 0.702750 + 0.711437i \(0.251955\pi\)
−0.702750 + 0.711437i \(0.748045\pi\)
\(492\) 0 0
\(493\) 2.24621i 0.101164i
\(494\) 0 0
\(495\) 28.8802 1.29807
\(496\) 0 0
\(497\) −11.1231 + 16.0000i −0.498939 + 0.717698i
\(498\) 0 0
\(499\) 24.3266i 1.08901i 0.838758 + 0.544504i \(0.183282\pi\)
−0.838758 + 0.544504i \(0.816718\pi\)
\(500\) 0 0
\(501\) 59.8617 2.67443
\(502\) 0 0
\(503\) −17.9139 −0.798741 −0.399370 0.916790i \(-0.630771\pi\)
−0.399370 + 0.916790i \(0.630771\pi\)
\(504\) 0 0
\(505\) 15.1231 0.672969
\(506\) 0 0
\(507\) −27.1840 −1.20729
\(508\) 0 0
\(509\) 23.6155i 1.04674i −0.852106 0.523370i \(-0.824675\pi\)
0.852106 0.523370i \(-0.175325\pi\)
\(510\) 0 0
\(511\) −23.2111 + 33.3880i −1.02680 + 1.47700i
\(512\) 0 0
\(513\) −44.4924 −1.96439
\(514\) 0 0
\(515\) 7.73704i 0.340935i
\(516\) 0 0
\(517\) 24.0000i 1.05552i
\(518\) 0 0
\(519\) 49.0708i 2.15397i
\(520\) 0 0
\(521\) 9.75379i 0.427321i 0.976908 + 0.213661i \(0.0685387\pi\)
−0.976908 + 0.213661i \(0.931461\pi\)
\(522\) 0 0
\(523\) −19.2382 −0.841227 −0.420614 0.907240i \(-0.638185\pi\)
−0.420614 + 0.907240i \(0.638185\pi\)
\(524\) 0 0
\(525\) −6.56155 4.56155i −0.286370 0.199082i
\(526\) 0 0
\(527\) 3.80989i 0.165961i
\(528\) 0 0
\(529\) −18.1231 −0.787961
\(530\) 0 0
\(531\) 12.6624 0.549503
\(532\) 0 0
\(533\) −2.24621 −0.0972942
\(534\) 0 0
\(535\) −5.66906 −0.245095
\(536\) 0 0
\(537\) 32.4924i 1.40215i
\(538\) 0 0
\(539\) 30.9481 11.5012i 1.33303 0.495390i
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 0 0
\(543\) 45.6786i 1.96025i
\(544\) 0 0
\(545\) 7.12311i 0.305120i
\(546\) 0 0
\(547\) 30.5763i 1.30735i −0.756776 0.653674i \(-0.773227\pi\)
0.756776 0.653674i \(-0.226773\pi\)
\(548\) 0 0
\(549\) 12.2462i 0.522656i
\(550\) 0 0
\(551\) 9.43318 0.401867
\(552\) 0 0
\(553\) 2.00000 2.87689i 0.0850487 0.122338i
\(554\) 0 0
\(555\) 6.04090i 0.256422i
\(556\) 0 0
\(557\) 26.4924 1.12252 0.561260 0.827640i \(-0.310317\pi\)
0.561260 + 0.827640i \(0.310317\pi\)
\(558\) 0 0
\(559\) −0.743668 −0.0314538
\(560\) 0 0
\(561\) 16.0000 0.675521
\(562\) 0 0
\(563\) 38.1045 1.60592 0.802958 0.596036i \(-0.203259\pi\)
0.802958 + 0.596036i \(0.203259\pi\)
\(564\) 0 0
\(565\) 18.4924i 0.777982i
\(566\) 0 0
\(567\) −21.9912 15.2882i −0.923544 0.642042i
\(568\) 0 0
\(569\) −28.8769 −1.21058 −0.605291 0.796004i \(-0.706943\pi\)
−0.605291 + 0.796004i \(0.706943\pi\)
\(570\) 0 0
\(571\) 3.22925i 0.135140i −0.997715 0.0675700i \(-0.978475\pi\)
0.997715 0.0675700i \(-0.0215246\pi\)
\(572\) 0 0
\(573\) 50.7386i 2.11964i
\(574\) 0 0
\(575\) 6.41273i 0.267429i
\(576\) 0 0
\(577\) 45.6155i 1.89900i −0.313768 0.949500i \(-0.601591\pi\)
0.313768 0.949500i \(-0.398409\pi\)
\(578\) 0 0
\(579\) 67.9372 2.82337
\(580\) 0 0
\(581\) 6.56155 + 4.56155i 0.272219 + 0.189245i
\(582\) 0 0
\(583\) 9.43318i 0.390682i
\(584\) 0 0
\(585\) −12.2462 −0.506319
\(586\) 0 0
\(587\) 21.8868 0.903365 0.451683 0.892179i \(-0.350824\pi\)
0.451683 + 0.892179i \(0.350824\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 54.3681 2.23640
\(592\) 0 0
\(593\) 11.3693i 0.466882i 0.972371 + 0.233441i \(0.0749986\pi\)
−0.972371 + 0.233441i \(0.925001\pi\)
\(594\) 0 0
\(595\) −2.43981 1.69614i −0.100022 0.0695350i
\(596\) 0 0
\(597\) 80.9848 3.31449
\(598\) 0 0
\(599\) 29.6238i 1.21040i −0.796074 0.605199i \(-0.793094\pi\)
0.796074 0.605199i \(-0.206906\pi\)
\(600\) 0 0
\(601\) 17.1231i 0.698466i −0.937036 0.349233i \(-0.886442\pi\)
0.937036 0.349233i \(-0.113558\pi\)
\(602\) 0 0
\(603\) 23.0481i 0.938590i
\(604\) 0 0
\(605\) 11.2462i 0.457224i
\(606\) 0 0
\(607\) −37.9415 −1.54000 −0.769999 0.638045i \(-0.779743\pi\)
−0.769999 + 0.638045i \(0.779743\pi\)
\(608\) 0 0
\(609\) 13.1231 + 9.12311i 0.531775 + 0.369687i
\(610\) 0 0
\(611\) 10.1768i 0.411711i
\(612\) 0 0
\(613\) −28.2462 −1.14085 −0.570427 0.821348i \(-0.693222\pi\)
−0.570427 + 0.821348i \(0.693222\pi\)
\(614\) 0 0
\(615\) 3.39228 0.136790
\(616\) 0 0
\(617\) −26.0000 −1.04672 −0.523360 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 0 0
\(619\) −20.6083 −0.828316 −0.414158 0.910205i \(-0.635924\pi\)
−0.414158 + 0.910205i \(0.635924\pi\)
\(620\) 0 0
\(621\) 60.4924i 2.42748i
\(622\) 0 0
\(623\) −18.1227 + 26.0685i −0.726070 + 1.04441i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 67.1935i 2.68345i
\(628\) 0 0
\(629\) 2.24621i 0.0895623i
\(630\) 0 0
\(631\) 14.1498i 0.563293i −0.959518 0.281647i \(-0.909119\pi\)
0.959518 0.281647i \(-0.0908806\pi\)
\(632\) 0 0
\(633\) 66.7386i 2.65262i
\(634\) 0 0
\(635\) 17.7509 0.704421
\(636\) 0 0
\(637\) −13.1231 + 4.87689i −0.519956 + 0.193230i
\(638\) 0 0
\(639\) 45.0979i 1.78405i
\(640\) 0 0
\(641\) −7.12311 −0.281346 −0.140673 0.990056i \(-0.544927\pi\)
−0.140673 + 0.990056i \(0.544927\pi\)
\(642\) 0 0
\(643\) −33.9686 −1.33959 −0.669795 0.742546i \(-0.733618\pi\)
−0.669795 + 0.742546i \(0.733618\pi\)
\(644\) 0 0
\(645\) 1.12311 0.0442222
\(646\) 0 0
\(647\) 40.1725 1.57934 0.789672 0.613529i \(-0.210251\pi\)
0.789672 + 0.613529i \(0.210251\pi\)
\(648\) 0 0
\(649\) 9.75379i 0.382870i
\(650\) 0 0
\(651\) −22.2586 15.4741i −0.872385 0.606477i
\(652\) 0 0
\(653\) 22.9848 0.899466 0.449733 0.893163i \(-0.351519\pi\)
0.449733 + 0.893163i \(0.351519\pi\)
\(654\) 0 0
\(655\) 17.5420i 0.685425i
\(656\) 0 0
\(657\) 94.1080i 3.67150i
\(658\) 0 0
\(659\) 1.32431i 0.0515877i −0.999667 0.0257938i \(-0.991789\pi\)
0.999667 0.0257938i \(-0.00821134\pi\)
\(660\) 0 0
\(661\) 12.2462i 0.476322i −0.971226 0.238161i \(-0.923455\pi\)
0.971226 0.238161i \(-0.0765447\pi\)
\(662\) 0 0
\(663\) −6.78456 −0.263491
\(664\) 0 0
\(665\) 7.12311 10.2462i 0.276222 0.397331i
\(666\) 0 0
\(667\) 12.8255i 0.496604i
\(668\) 0 0
\(669\) −62.1080 −2.40123
\(670\) 0 0
\(671\) 9.43318 0.364164
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 9.43318 0.363083
\(676\) 0 0
\(677\) 32.7386i 1.25825i −0.777305 0.629124i \(-0.783414\pi\)
0.777305 0.629124i \(-0.216586\pi\)
\(678\) 0 0
\(679\) 1.69614 2.43981i 0.0650919 0.0936313i
\(680\) 0 0
\(681\) −11.3693 −0.435673
\(682\) 0 0
\(683\) 0.371834i 0.0142278i 0.999975 + 0.00711392i \(0.00226445\pi\)
−0.999975 + 0.00711392i \(0.997736\pi\)
\(684\) 0 0
\(685\) 14.0000i 0.534913i
\(686\) 0 0
\(687\) 38.8940i 1.48390i
\(688\) 0 0
\(689\) 4.00000i 0.152388i
\(690\) 0 0
\(691\) 14.8934 0.566573 0.283286 0.959035i \(-0.408575\pi\)
0.283286 + 0.959035i \(0.408575\pi\)
\(692\) 0 0
\(693\) −43.6155 + 62.7386i −1.65682 + 2.38324i
\(694\) 0 0
\(695\) 16.7984i 0.637199i
\(696\) 0 0
\(697\) 1.26137 0.0477777
\(698\) 0 0
\(699\) −23.4199 −0.885823
\(700\) 0 0
\(701\) −11.1231 −0.420114 −0.210057 0.977689i \(-0.567365\pi\)
−0.210057 + 0.977689i \(0.567365\pi\)
\(702\) 0 0
\(703\) −9.43318 −0.355779
\(704\) 0 0
\(705\) 15.3693i 0.578842i
\(706\) 0 0
\(707\) −22.8393 + 32.8531i −0.858959 + 1.23557i
\(708\) 0 0
\(709\) −44.7386 −1.68019 −0.840097 0.542436i \(-0.817502\pi\)
−0.840097 + 0.542436i \(0.817502\pi\)
\(710\) 0 0
\(711\) 8.10887i 0.304106i
\(712\) 0 0
\(713\) 21.7538i 0.814686i
\(714\) 0 0
\(715\) 9.43318i 0.352781i
\(716\) 0 0
\(717\) 12.0000i 0.448148i
\(718\) 0 0
\(719\) 22.6762 0.845681 0.422841 0.906204i \(-0.361033\pi\)
0.422841 + 0.906204i \(0.361033\pi\)
\(720\) 0 0
\(721\) 16.8078 + 11.6847i 0.625954 + 0.435159i
\(722\) 0 0
\(723\) 77.3704i 2.87743i
\(724\) 0 0
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −9.64198 −0.357601 −0.178801 0.983885i \(-0.557222\pi\)
−0.178801 + 0.983885i \(0.557222\pi\)
\(728\) 0 0
\(729\) −23.4924 −0.870090
\(730\) 0 0
\(731\) 0.417609 0.0154458
\(732\) 0 0
\(733\) 0.738634i 0.0272821i 0.999907 + 0.0136410i \(0.00434221\pi\)
−0.999907 + 0.0136410i \(0.995658\pi\)
\(734\) 0 0
\(735\) 19.8188 7.36520i 0.731029 0.271670i
\(736\) 0 0
\(737\) 17.7538 0.653969
\(738\) 0 0
\(739\) 37.5697i 1.38202i −0.722844 0.691012i \(-0.757165\pi\)
0.722844 0.691012i \(-0.242835\pi\)
\(740\) 0 0
\(741\) 28.4924i 1.04670i
\(742\) 0 0
\(743\) 23.7917i 0.872835i 0.899744 + 0.436417i \(0.143753\pi\)
−0.899744 + 0.436417i \(0.856247\pi\)
\(744\) 0 0
\(745\) 5.36932i 0.196717i
\(746\) 0 0
\(747\) −18.4945 −0.676679
\(748\) 0 0
\(749\) 8.56155 12.3153i 0.312832 0.449993i
\(750\) 0 0
\(751\) 28.8802i 1.05385i −0.849911 0.526926i \(-0.823344\pi\)
0.849911 0.526926i \(-0.176656\pi\)
\(752\) 0 0
\(753\) −48.4924 −1.76716
\(754\) 0 0
\(755\) 10.0138 0.364440
\(756\) 0 0
\(757\) −4.24621 −0.154331 −0.0771656 0.997018i \(-0.524587\pi\)
−0.0771656 + 0.997018i \(0.524587\pi\)
\(758\) 0 0
\(759\) −91.3571 −3.31605
\(760\) 0 0
\(761\) 47.2311i 1.71212i 0.516873 + 0.856062i \(0.327096\pi\)
−0.516873 + 0.856062i \(0.672904\pi\)
\(762\) 0 0
\(763\) −15.4741 10.7575i −0.560199 0.389447i
\(764\) 0 0
\(765\) 6.87689 0.248635
\(766\) 0 0
\(767\) 4.13595i 0.149341i
\(768\) 0 0
\(769\) 27.2311i 0.981977i 0.871166 + 0.490989i \(0.163364\pi\)
−0.871166 + 0.490989i \(0.836636\pi\)
\(770\) 0 0
\(771\) 34.3404i 1.23674i
\(772\) 0 0
\(773\) 32.7386i 1.17753i −0.808305 0.588763i \(-0.799615\pi\)
0.808305 0.588763i \(-0.200385\pi\)
\(774\) 0 0
\(775\) 3.39228 0.121854
\(776\) 0 0
\(777\) −13.1231 9.12311i −0.470789 0.327290i
\(778\) 0 0
\(779\) 5.29723i 0.189793i
\(780\) 0 0
\(781\) −34.7386 −1.24305
\(782\) 0 0
\(783\) −18.8664 −0.674229
\(784\) 0 0
\(785\) 0.246211 0.00878766
\(786\) 0 0
\(787\) −3.02045 −0.107667 −0.0538337 0.998550i \(-0.517144\pi\)
−0.0538337 + 0.998550i \(0.517144\pi\)
\(788\) 0 0
\(789\) 55.8617i 1.98873i
\(790\) 0 0
\(791\) −40.1725 27.9277i −1.42837 0.992995i
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 0 0
\(795\) 6.04090i 0.214248i
\(796\) 0 0
\(797\) 3.75379i 0.132966i −0.997788 0.0664830i \(-0.978822\pi\)
0.997788 0.0664830i \(-0.0211778\pi\)
\(798\) 0 0
\(799\) 5.71484i 0.202176i
\(800\) 0 0
\(801\) 73.4773i 2.59619i
\(802\) 0 0
\(803\) −72.4908 −2.55814
\(804\) 0 0
\(805\) 13.9309 + 9.68466i 0.490999 + 0.341339i
\(806\) 0 0
\(807\) 0.743668i 0.0261784i
\(808\) 0 0
\(809\) 46.9848 1.65190 0.825950 0.563744i \(-0.190640\pi\)
0.825950 + 0.563744i \(0.190640\pi\)
\(810\) 0 0
\(811\) −8.10887 −0.284741 −0.142370 0.989813i \(-0.545472\pi\)
−0.142370 + 0.989813i \(0.545472\pi\)
\(812\) 0 0
\(813\) −5.75379 −0.201794
\(814\) 0 0
\(815\) 3.02045 0.105802
\(816\) 0 0
\(817\) 1.75379i 0.0613573i
\(818\) 0 0
\(819\) 18.4945 26.6034i 0.646251 0.929598i
\(820\) 0 0
\(821\) −15.6155 −0.544986 −0.272493 0.962158i \(-0.587848\pi\)
−0.272493 + 0.962158i \(0.587848\pi\)
\(822\) 0 0
\(823\) 29.0890i 1.01398i −0.861953 0.506989i \(-0.830758\pi\)
0.861953 0.506989i \(-0.169242\pi\)
\(824\) 0 0
\(825\) 14.2462i 0.495989i
\(826\) 0 0
\(827\) 24.5354i 0.853180i 0.904445 + 0.426590i \(0.140285\pi\)
−0.904445 + 0.426590i \(0.859715\pi\)
\(828\) 0 0
\(829\) 15.7538i 0.547152i 0.961850 + 0.273576i \(0.0882065\pi\)
−0.961850 + 0.273576i \(0.911794\pi\)
\(830\) 0 0
\(831\) 36.9890 1.28314
\(832\) 0 0
\(833\) 7.36932 2.73863i 0.255332 0.0948880i
\(834\) 0 0
\(835\) 19.8188i 0.685859i
\(836\) 0 0
\(837\) 32.0000 1.10608
\(838\) 0 0
\(839\) 10.9205 0.377018 0.188509 0.982071i \(-0.439635\pi\)
0.188509 + 0.982071i \(0.439635\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 88.7085 3.05528
\(844\) 0 0
\(845\) 9.00000i 0.309609i
\(846\) 0 0
\(847\) 24.4310 + 16.9843i 0.839460 + 0.583587i
\(848\) 0 0
\(849\) −14.8769 −0.510574
\(850\) 0 0
\(851\) 12.8255i 0.439651i
\(852\) 0 0
\(853\) 24.2462i 0.830174i 0.909782 + 0.415087i \(0.136249\pi\)
−0.909782 + 0.415087i \(0.863751\pi\)
\(854\) 0 0
\(855\) 28.8802i 0.987681i
\(856\) 0 0
\(857\) 25.6155i 0.875010i −0.899216 0.437505i \(-0.855862\pi\)
0.899216 0.437505i \(-0.144138\pi\)
\(858\) 0 0
\(859\) 3.55531 0.121306 0.0606528 0.998159i \(-0.480682\pi\)
0.0606528 + 0.998159i \(0.480682\pi\)
\(860\) 0 0
\(861\) −5.12311 + 7.36932i −0.174595 + 0.251146i
\(862\) 0 0
\(863\) 10.2226i 0.347982i −0.984747 0.173991i \(-0.944334\pi\)
0.984747 0.173991i \(-0.0556664\pi\)
\(864\) 0 0
\(865\) 16.2462 0.552388
\(866\) 0 0
\(867\) −47.5377 −1.61447
\(868\) 0 0
\(869\) 6.24621 0.211888
\(870\) 0 0
\(871\) −7.52823 −0.255084
\(872\) 0 0
\(873\) 6.87689i 0.232748i
\(874\) 0 0
\(875\) −1.51022 + 2.17238i −0.0510549 + 0.0734398i
\(876\) 0 0
\(877\) −0.738634 −0.0249419 −0.0124709 0.999922i \(-0.503970\pi\)
−0.0124709 + 0.999922i \(0.503970\pi\)
\(878\) 0 0
\(879\) 47.5835i 1.60495i
\(880\) 0 0
\(881\) 31.3693i 1.05686i −0.848977 0.528430i \(-0.822781\pi\)
0.848977 0.528430i \(-0.177219\pi\)
\(882\) 0 0
\(883\) 31.3200i 1.05400i 0.849865 + 0.527001i \(0.176683\pi\)
−0.849865 + 0.527001i \(0.823317\pi\)
\(884\) 0 0
\(885\) 6.24621i 0.209964i
\(886\) 0 0
\(887\) 29.9957 1.00716 0.503578 0.863950i \(-0.332017\pi\)
0.503578 + 0.863950i \(0.332017\pi\)
\(888\) 0 0
\(889\) −26.8078 + 38.5616i −0.899104 + 1.29331i
\(890\) 0 0
\(891\) 47.7465i 1.59957i
\(892\) 0 0
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) −10.7575 −0.359583
\(896\) 0 0
\(897\) 38.7386 1.29345
\(898\) 0 0
\(899\) −6.78456 −0.226278
\(900\) 0 0
\(901\) 2.24621i 0.0748321i
\(902\) 0 0
\(903\) −1.69614 + 2.43981i −0.0564440 + 0.0811918i
\(904\) 0 0
\(905\) 15.1231 0.502709
\(906\) 0 0
\(907\) 52.8350i 1.75436i −0.480166 0.877178i \(-0.659423\pi\)
0.480166 0.877178i \(-0.340577\pi\)
\(908\) 0 0
\(909\) 92.6004i 3.07136i
\(910\) 0 0
\(911\) 2.06798i 0.0685151i 0.999413 + 0.0342575i \(0.0109066\pi\)
−0.999413 + 0.0342575i \(0.989093\pi\)
\(912\) 0 0
\(913\) 14.2462i 0.471481i
\(914\) 0 0
\(915\) 6.04090 0.199706
\(916\) 0 0
\(917\) 38.1080 + 26.4924i 1.25844 + 0.874857i
\(918\) 0 0
\(919\) 9.27015i 0.305794i −0.988242 0.152897i \(-0.951140\pi\)
0.988242 0.152897i \(-0.0488603\pi\)
\(920\) 0 0
\(921\) 4.63068 0.152586
\(922\) 0 0
\(923\) 14.7304 0.484857
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) −47.3747 −1.55599
\(928\) 0 0
\(929\) 21.1231i 0.693027i 0.938045 + 0.346513i \(0.112634\pi\)
−0.938045 + 0.346513i \(0.887366\pi\)
\(930\) 0 0
\(931\) 11.5012 + 30.9481i 0.376935 + 1.01428i
\(932\) 0 0
\(933\) −91.2311 −2.98677
\(934\) 0 0
\(935\) 5.29723i 0.173238i
\(936\) 0 0
\(937\) 38.1080i 1.24493i 0.782647 + 0.622466i \(0.213869\pi\)
−0.782647 + 0.622466i \(0.786131\pi\)
\(938\) 0 0
\(939\) 101.534i 3.31344i
\(940\) 0 0
\(941\) 26.6307i 0.868135i 0.900880 + 0.434068i \(0.142922\pi\)
−0.900880 + 0.434068i \(0.857078\pi\)
\(942\) 0 0
\(943\) −7.20217 −0.234535
\(944\) 0 0
\(945\) −14.2462 + 20.4924i −0.463429 + 0.666619i
\(946\) 0 0
\(947\) 35.8735i 1.16573i −0.812568 0.582867i \(-0.801931\pi\)
0.812568 0.582867i \(-0.198069\pi\)
\(948\) 0 0
\(949\) 30.7386 0.997818
\(950\) 0 0
\(951\) 36.9890 1.19945
\(952\) 0 0
\(953\) 28.2462 0.914985 0.457492 0.889214i \(-0.348748\pi\)
0.457492 + 0.889214i \(0.348748\pi\)
\(954\) 0 0
\(955\) −16.7984 −0.543583
\(956\) 0 0
\(957\) 28.4924i 0.921029i
\(958\) 0 0
\(959\) 30.4133 + 21.1431i 0.982096 + 0.682747i
\(960\) 0 0
\(961\) −19.4924 −0.628788
\(962\) 0 0
\(963\) 34.7123i 1.11859i
\(964\) 0 0
\(965\) 22.4924i 0.724057i
\(966\) 0 0
\(967\) 3.43806i 0.110560i 0.998471 + 0.0552802i \(0.0176052\pi\)
−0.998471 + 0.0552802i \(0.982395\pi\)
\(968\) 0 0
\(969\) 16.0000i 0.513994i
\(970\) 0 0
\(971\) 10.7575 0.345224 0.172612 0.984990i \(-0.444779\pi\)
0.172612 + 0.984990i \(0.444779\pi\)
\(972\) 0 0
\(973\) 36.4924 + 25.3693i 1.16989 + 0.813303i
\(974\) 0 0
\(975\) 6.04090i 0.193463i
\(976\) 0 0
\(977\) 22.4924 0.719596 0.359798 0.933030i \(-0.382846\pi\)
0.359798 + 0.933030i \(0.382846\pi\)
\(978\) 0 0
\(979\) −56.5991 −1.80891
\(980\) 0 0
\(981\) 43.6155 1.39254
\(982\) 0 0
\(983\) 28.5083 0.909275 0.454637 0.890677i \(-0.349769\pi\)
0.454637 + 0.890677i \(0.349769\pi\)
\(984\) 0 0
\(985\) 18.0000i 0.573528i
\(986\) 0 0
\(987\) 33.3880 + 23.2111i 1.06275 + 0.738818i
\(988\) 0 0
\(989\) −2.38447 −0.0758218
\(990\) 0 0
\(991\) 48.9078i 1.55361i −0.629743 0.776804i \(-0.716840\pi\)
0.629743 0.776804i \(-0.283160\pi\)
\(992\) 0 0
\(993\) 55.2311i 1.75270i
\(994\) 0 0
\(995\) 26.8122i 0.850004i
\(996\) 0 0
\(997\) 5.50758i 0.174427i −0.996190 0.0872134i \(-0.972204\pi\)
0.996190 0.0872134i \(-0.0277962\pi\)
\(998\) 0 0
\(999\) 18.8664 0.596905
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 2240.2.k.e.1791.2 8
4.3 odd 2 inner 2240.2.k.e.1791.8 8
7.6 odd 2 inner 2240.2.k.e.1791.7 8
8.3 odd 2 140.2.g.c.111.3 yes 8
8.5 even 2 140.2.g.c.111.2 yes 8
24.5 odd 2 1260.2.c.c.811.8 8
24.11 even 2 1260.2.c.c.811.6 8
28.27 even 2 inner 2240.2.k.e.1791.1 8
40.3 even 4 700.2.c.j.699.8 8
40.13 odd 4 700.2.c.j.699.2 8
40.19 odd 2 700.2.g.j.251.6 8
40.27 even 4 700.2.c.i.699.1 8
40.29 even 2 700.2.g.j.251.7 8
40.37 odd 4 700.2.c.i.699.7 8
56.3 even 6 980.2.o.e.411.3 16
56.5 odd 6 980.2.o.e.31.4 16
56.11 odd 6 980.2.o.e.411.4 16
56.13 odd 2 140.2.g.c.111.1 8
56.19 even 6 980.2.o.e.31.7 16
56.27 even 2 140.2.g.c.111.4 yes 8
56.37 even 6 980.2.o.e.31.3 16
56.45 odd 6 980.2.o.e.411.8 16
56.51 odd 6 980.2.o.e.31.8 16
56.53 even 6 980.2.o.e.411.7 16
168.83 odd 2 1260.2.c.c.811.5 8
168.125 even 2 1260.2.c.c.811.7 8
280.13 even 4 700.2.c.i.699.2 8
280.27 odd 4 700.2.c.j.699.1 8
280.69 odd 2 700.2.g.j.251.8 8
280.83 odd 4 700.2.c.i.699.8 8
280.139 even 2 700.2.g.j.251.5 8
280.237 even 4 700.2.c.j.699.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.g.c.111.1 8 56.13 odd 2
140.2.g.c.111.2 yes 8 8.5 even 2
140.2.g.c.111.3 yes 8 8.3 odd 2
140.2.g.c.111.4 yes 8 56.27 even 2
700.2.c.i.699.1 8 40.27 even 4
700.2.c.i.699.2 8 280.13 even 4
700.2.c.i.699.7 8 40.37 odd 4
700.2.c.i.699.8 8 280.83 odd 4
700.2.c.j.699.1 8 280.27 odd 4
700.2.c.j.699.2 8 40.13 odd 4
700.2.c.j.699.7 8 280.237 even 4
700.2.c.j.699.8 8 40.3 even 4
700.2.g.j.251.5 8 280.139 even 2
700.2.g.j.251.6 8 40.19 odd 2
700.2.g.j.251.7 8 40.29 even 2
700.2.g.j.251.8 8 280.69 odd 2
980.2.o.e.31.3 16 56.37 even 6
980.2.o.e.31.4 16 56.5 odd 6
980.2.o.e.31.7 16 56.19 even 6
980.2.o.e.31.8 16 56.51 odd 6
980.2.o.e.411.3 16 56.3 even 6
980.2.o.e.411.4 16 56.11 odd 6
980.2.o.e.411.7 16 56.53 even 6
980.2.o.e.411.8 16 56.45 odd 6
1260.2.c.c.811.5 8 168.83 odd 2
1260.2.c.c.811.6 8 24.11 even 2
1260.2.c.c.811.7 8 168.125 even 2
1260.2.c.c.811.8 8 24.5 odd 2
2240.2.k.e.1791.1 8 28.27 even 2 inner
2240.2.k.e.1791.2 8 1.1 even 1 trivial
2240.2.k.e.1791.7 8 7.6 odd 2 inner
2240.2.k.e.1791.8 8 4.3 odd 2 inner