Properties

Label 1050.2.g.g.799.1
Level $1050$
Weight $2$
Character 1050.799
Analytic conductor $8.384$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,2,Mod(799,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.g (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: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1050.799
Dual form 1050.2.g.g.799.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -1.00000 q^{21} +4.00000i q^{22} -8.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} -1.00000i q^{28} +2.00000 q^{29} -1.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} +4.00000i q^{38} +2.00000 q^{39} -6.00000 q^{41} +1.00000i q^{42} -4.00000i q^{43} +4.00000 q^{44} -8.00000 q^{46} +1.00000i q^{48} -1.00000 q^{49} +2.00000 q^{51} +2.00000i q^{52} -10.0000i q^{53} -1.00000 q^{54} -1.00000 q^{56} -4.00000i q^{57} -2.00000i q^{58} -12.0000 q^{59} +14.0000 q^{61} -1.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +12.0000i q^{67} +2.00000i q^{68} +8.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} +10.0000i q^{73} -6.00000 q^{74} +4.00000 q^{76} -4.00000i q^{77} -2.00000i q^{78} -16.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -12.0000i q^{83} +1.00000 q^{84} -4.00000 q^{86} +2.00000i q^{87} -4.00000i q^{88} -10.0000 q^{89} +2.00000 q^{91} +8.00000i q^{92} +1.00000 q^{96} -2.00000i q^{97} +1.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{14} + 2 q^{16} - 8 q^{19} - 2 q^{21} - 2 q^{24} - 4 q^{26} + 4 q^{29} - 4 q^{34} + 2 q^{36} + 4 q^{39} - 12 q^{41} + 8 q^{44} - 16 q^{46} - 2 q^{49} + 4 q^{51} - 2 q^{54} - 2 q^{56} - 24 q^{59} + 28 q^{61} - 2 q^{64} - 8 q^{66} + 16 q^{69} - 16 q^{71} - 12 q^{74} + 8 q^{76} - 32 q^{79} + 2 q^{81} + 2 q^{84} - 8 q^{86} - 20 q^{89} + 4 q^{91} + 2 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 1.00000i 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 4.00000i 0.852803i
\(23\) − 8.00000i − 1.66812i −0.551677 0.834058i \(-0.686012\pi\)
0.551677 0.834058i \(-0.313988\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) − 1.00000i − 0.192450i
\(28\) − 1.00000i − 0.188982i
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 4.00000i − 0.696311i
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000i 0.154303i
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 2.00000i 0.277350i
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) − 4.00000i − 0.529813i
\(58\) − 2.00000i − 0.262613i
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) − 1.00000i − 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) − 4.00000i − 0.455842i
\(78\) − 2.00000i − 0.226455i
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000i 0.662589i
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 2.00000i 0.214423i
\(88\) − 4.00000i − 0.426401i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 8.00000i 0.834058i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) − 2.00000i − 0.198030i
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 1.00000i 0.0944911i
\(113\) 18.0000i 1.69330i 0.532152 + 0.846649i \(0.321383\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 2.00000i 0.184900i
\(118\) 12.0000i 1.10469i
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) − 14.0000i − 1.26750i
\(123\) − 6.00000i − 0.541002i
\(124\) 0 0
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 4.00000i 0.348155i
\(133\) − 4.00000i − 0.346844i
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) − 10.0000i − 0.854358i −0.904167 0.427179i \(-0.859507\pi\)
0.904167 0.427179i \(-0.140493\pi\)
\(138\) − 8.00000i − 0.681005i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000i 0.671345i
\(143\) 8.00000i 0.668994i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) − 1.00000i − 0.0824786i
\(148\) 6.00000i 0.493197i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) 2.00000i 0.161690i
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) 16.0000i 1.27289i
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) − 1.00000i − 0.0785674i
\(163\) 20.0000i 1.56652i 0.621694 + 0.783260i \(0.286445\pi\)
−0.621694 + 0.783260i \(0.713555\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) − 1.00000i − 0.0771517i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 4.00000i 0.304997i
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) − 12.0000i − 0.901975i
\(178\) 10.0000i 0.749532i
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) − 2.00000i − 0.148250i
\(183\) 14.0000i 1.03491i
\(184\) 8.00000 0.589768
\(185\) 0 0
\(186\) 0 0
\(187\) 8.00000i 0.585018i
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) − 6.00000i − 0.422159i
\(203\) 2.00000i 0.140372i
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 8.00000i 0.556038i
\(208\) − 2.00000i − 0.138675i
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 10.0000i 0.686803i
\(213\) − 8.00000i − 0.548151i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 14.0000i 0.948200i
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) − 6.00000i − 0.402694i
\(223\) − 16.0000i − 1.07144i −0.844396 0.535720i \(-0.820040\pi\)
0.844396 0.535720i \(-0.179960\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 2.00000i 0.131306i
\(233\) − 22.0000i − 1.44127i −0.693316 0.720634i \(-0.743851\pi\)
0.693316 0.720634i \(-0.256149\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) − 16.0000i − 1.03931i
\(238\) − 2.00000i − 0.129641i
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 1.00000i 0.0641500i
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 8.00000i 0.509028i
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 1.00000i 0.0629941i
\(253\) 32.0000i 2.01182i
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000i 0.873296i 0.899632 + 0.436648i \(0.143834\pi\)
−0.899632 + 0.436648i \(0.856166\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) − 20.0000i − 1.23560i
\(263\) 8.00000i 0.493301i 0.969104 + 0.246651i \(0.0793300\pi\)
−0.969104 + 0.246651i \(0.920670\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) − 10.0000i − 0.611990i
\(268\) − 12.0000i − 0.733017i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 2.00000i 0.121046i
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 28.0000i 1.66443i 0.554455 + 0.832214i \(0.312927\pi\)
−0.554455 + 0.832214i \(0.687073\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) − 6.00000i − 0.354169i
\(288\) 1.00000i 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) − 10.0000i − 0.585206i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 4.00000i 0.232104i
\(298\) − 10.0000i − 0.579284i
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 8.00000i 0.460348i
\(303\) 6.00000i 0.344691i
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) − 20.0000i − 1.14146i −0.821138 0.570730i \(-0.806660\pi\)
0.821138 0.570730i \(-0.193340\pi\)
\(308\) 4.00000i 0.227921i
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 2.00000i 0.113228i
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) − 10.0000i − 0.560772i
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) − 8.00000i − 0.445823i
\(323\) 8.00000i 0.445132i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) − 14.0000i − 0.774202i
\(328\) − 6.00000i − 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 12.0000i 0.658586i
\(333\) 6.00000i 0.328798i
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) − 18.0000i − 0.980522i −0.871576 0.490261i \(-0.836901\pi\)
0.871576 0.490261i \(-0.163099\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) − 4.00000i − 0.216295i
\(343\) − 1.00000i − 0.0539949i
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 4.00000i 0.214731i 0.994220 + 0.107366i \(0.0342415\pi\)
−0.994220 + 0.107366i \(0.965758\pi\)
\(348\) − 2.00000i − 0.107211i
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 4.00000i 0.213201i
\(353\) − 14.0000i − 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 2.00000i 0.105851i
\(358\) 4.00000i 0.211407i
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 6.00000i − 0.315353i
\(363\) 5.00000i 0.262432i
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 14.0000 0.731792
\(367\) 32.0000i 1.67039i 0.549957 + 0.835193i \(0.314644\pi\)
−0.549957 + 0.835193i \(0.685356\pi\)
\(368\) − 8.00000i − 0.417029i
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 10.0000 0.519174
\(372\) 0 0
\(373\) − 10.0000i − 0.517780i −0.965907 0.258890i \(-0.916643\pi\)
0.965907 0.258890i \(-0.0833568\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) − 4.00000i − 0.206010i
\(378\) − 1.00000i − 0.0514344i
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 16.0000i 0.818631i
\(383\) 16.0000i 0.817562i 0.912633 + 0.408781i \(0.134046\pi\)
−0.912633 + 0.408781i \(0.865954\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 4.00000i 0.203331i
\(388\) 2.00000i 0.101535i
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) − 1.00000i − 0.0505076i
\(393\) 20.0000i 1.00887i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) − 30.0000i − 1.50566i −0.658217 0.752828i \(-0.728689\pi\)
0.658217 0.752828i \(-0.271311\pi\)
\(398\) − 24.0000i − 1.20301i
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 12.0000i 0.598506i
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) 24.0000i 1.18964i
\(408\) 2.00000i 0.0990148i
\(409\) 38.0000 1.87898 0.939490 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) 8.00000i 0.394132i
\(413\) − 12.0000i − 0.590481i
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 4.00000i 0.195881i
\(418\) − 16.0000i − 0.782586i
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 0 0
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 14.0000i 0.677507i
\(428\) 12.0000i 0.580042i
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 14.0000i − 0.672797i −0.941720 0.336399i \(-0.890791\pi\)
0.941720 0.336399i \(-0.109209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 32.0000i 1.53077i
\(438\) 10.0000i 0.477818i
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 4.00000i 0.190261i
\(443\) − 36.0000i − 1.71041i −0.518289 0.855206i \(-0.673431\pi\)
0.518289 0.855206i \(-0.326569\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 10.0000i 0.472984i
\(448\) − 1.00000i − 0.0472456i
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) − 18.0000i − 0.846649i
\(453\) − 8.00000i − 0.375873i
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) − 4.00000i − 0.186097i
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) − 12.0000i − 0.552345i
\(473\) 16.0000i 0.735681i
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) 10.0000i 0.457869i
\(478\) 0 0
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 14.0000i 0.637683i
\(483\) 8.00000i 0.364013i
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 8.00000i 0.362515i 0.983436 + 0.181257i \(0.0580167\pi\)
−0.983436 + 0.181257i \(0.941983\pi\)
\(488\) 14.0000i 0.633750i
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 6.00000i 0.270501i
\(493\) − 4.00000i − 0.180151i
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) − 8.00000i − 0.358849i
\(498\) − 12.0000i − 0.537733i
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) − 12.0000i − 0.535586i
\(503\) 40.0000i 1.78351i 0.452517 + 0.891756i \(0.350526\pi\)
−0.452517 + 0.891756i \(0.649474\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 32.0000 1.42257
\(507\) 9.00000i 0.399704i
\(508\) − 16.0000i − 0.709885i
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) − 1.00000i − 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) 14.0000 0.617514
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) − 6.00000i − 0.263625i
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) −20.0000 −0.873704
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) 0 0
\(528\) − 4.00000i − 0.174078i
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 4.00000i 0.173422i
\(533\) 12.0000i 0.519778i
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) − 4.00000i − 0.172613i
\(538\) − 18.0000i − 0.776035i
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) − 16.0000i − 0.687259i
\(543\) 6.00000i 0.257485i
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) 12.0000i 0.513083i 0.966533 + 0.256541i \(0.0825830\pi\)
−0.966533 + 0.256541i \(0.917417\pi\)
\(548\) 10.0000i 0.427179i
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 8.00000i 0.340503i
\(553\) − 16.0000i − 0.680389i
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 6.00000i 0.253095i
\(563\) 20.0000i 0.842900i 0.906852 + 0.421450i \(0.138479\pi\)
−0.906852 + 0.421450i \(0.861521\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 28.0000 1.17693
\(567\) 1.00000i 0.0419961i
\(568\) − 8.00000i − 0.335673i
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) − 8.00000i − 0.334497i
\(573\) − 16.0000i − 0.668410i
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 34.0000i − 1.41544i −0.706494 0.707719i \(-0.749724\pi\)
0.706494 0.707719i \(-0.250276\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) − 2.00000i − 0.0829027i
\(583\) 40.0000i 1.65663i
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 1.00000i 0.0412393i
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) − 6.00000i − 0.246598i
\(593\) − 30.0000i − 1.23195i −0.787765 0.615976i \(-0.788762\pi\)
0.787765 0.615976i \(-0.211238\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 24.0000i 0.982255i
\(598\) 16.0000i 0.654289i
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) − 4.00000i − 0.163028i
\(603\) − 12.0000i − 0.488678i
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) − 16.0000i − 0.649420i −0.945814 0.324710i \(-0.894733\pi\)
0.945814 0.324710i \(-0.105267\pi\)
\(608\) 4.00000i 0.162221i
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) 0 0
\(612\) − 2.00000i − 0.0808452i
\(613\) 6.00000i 0.242338i 0.992632 + 0.121169i \(0.0386643\pi\)
−0.992632 + 0.121169i \(0.961336\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) 22.0000i 0.885687i 0.896599 + 0.442843i \(0.146030\pi\)
−0.896599 + 0.442843i \(0.853970\pi\)
\(618\) − 8.00000i − 0.321807i
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 8.00000i 0.320771i
\(623\) − 10.0000i − 0.400642i
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) 16.0000i 0.638978i
\(628\) − 18.0000i − 0.718278i
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) − 16.0000i − 0.636446i
\(633\) − 12.0000i − 0.476957i
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) 2.00000i 0.0792429i
\(638\) 8.00000i 0.316723i
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 34.0000 1.34292 0.671460 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) − 12.0000i − 0.473602i
\(643\) − 28.0000i − 1.10421i −0.833774 0.552106i \(-0.813824\pi\)
0.833774 0.552106i \(-0.186176\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) − 20.0000i − 0.783260i
\(653\) 14.0000i 0.547862i 0.961749 + 0.273931i \(0.0883240\pi\)
−0.961749 + 0.273931i \(0.911676\pi\)
\(654\) −14.0000 −0.547443
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) − 10.0000i − 0.390137i
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) − 12.0000i − 0.466393i
\(663\) − 4.00000i − 0.155347i
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) − 16.0000i − 0.619522i
\(668\) − 8.00000i − 0.309529i
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −56.0000 −2.16186
\(672\) 1.00000i 0.0385758i
\(673\) − 30.0000i − 1.15642i −0.815890 0.578208i \(-0.803752\pi\)
0.815890 0.578208i \(-0.196248\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 26.0000i 0.999261i 0.866239 + 0.499631i \(0.166531\pi\)
−0.866239 + 0.499631i \(0.833469\pi\)
\(678\) 18.0000i 0.691286i
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 10.0000i 0.381524i
\(688\) − 4.00000i − 0.152499i
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) 36.0000 1.36950 0.684752 0.728776i \(-0.259910\pi\)
0.684752 + 0.728776i \(0.259910\pi\)
\(692\) 18.0000i 0.684257i
\(693\) 4.00000i 0.151947i
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) 12.0000i 0.454532i
\(698\) 14.0000i 0.529908i
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 2.00000i 0.0754851i
\(703\) 24.0000i 0.905177i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 6.00000i 0.225653i
\(708\) 12.0000i 0.450988i
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) − 10.0000i − 0.374766i
\(713\) 0 0
\(714\) 2.00000 0.0748481
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 24.0000i 0.895672i
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 3.00000i 0.111648i
\(723\) − 14.0000i − 0.520666i
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) − 8.00000i − 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) 2.00000i 0.0741249i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) − 14.0000i − 0.517455i
\(733\) 46.0000i 1.69905i 0.527549 + 0.849524i \(0.323111\pi\)
−0.527549 + 0.849524i \(0.676889\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) − 48.0000i − 1.76810i
\(738\) − 6.00000i − 0.220863i
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) − 10.0000i − 0.367112i
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −10.0000 −0.366126
\(747\) 12.0000i 0.439057i
\(748\) − 8.00000i − 0.292509i
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) 12.0000i 0.437304i
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) − 22.0000i − 0.799604i −0.916602 0.399802i \(-0.869079\pi\)
0.916602 0.399802i \(-0.130921\pi\)
\(758\) 28.0000i 1.01701i
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 16.0000i 0.579619i
\(763\) − 14.0000i − 0.506834i
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 24.0000i 0.866590i
\(768\) 1.00000i 0.0360844i
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) − 2.00000i − 0.0719816i
\(773\) − 26.0000i − 0.935155i −0.883952 0.467578i \(-0.845127\pi\)
0.883952 0.467578i \(-0.154873\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 6.00000i 0.215249i
\(778\) − 26.0000i − 0.932145i
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 16.0000i 0.572159i
\(783\) − 2.00000i − 0.0714742i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 20.0000 0.713376
\(787\) − 20.0000i − 0.712923i −0.934310 0.356462i \(-0.883983\pi\)
0.934310 0.356462i \(-0.116017\pi\)
\(788\) 6.00000i 0.213741i
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 4.00000i 0.142134i
\(793\) − 28.0000i − 0.994309i
\(794\) −30.0000 −1.06466
\(795\) 0 0
\(796\) −24.0000 −0.850657
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) − 4.00000i − 0.141598i
\(799\) 0 0
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 14.0000i 0.494357i
\(803\) − 40.0000i − 1.41157i
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 0 0
\(807\) 18.0000i 0.633630i
\(808\) 6.00000i 0.211079i
\(809\) 22.0000 0.773479 0.386739 0.922189i \(-0.373601\pi\)
0.386739 + 0.922189i \(0.373601\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) − 2.00000i − 0.0701862i
\(813\) 16.0000i 0.561144i
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) 16.0000i 0.559769i
\(818\) − 38.0000i − 1.32864i
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) − 10.0000i − 0.348790i
\(823\) − 24.0000i − 0.836587i −0.908312 0.418294i \(-0.862628\pi\)
0.908312 0.418294i \(-0.137372\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) − 28.0000i − 0.973655i −0.873498 0.486828i \(-0.838154\pi\)
0.873498 0.486828i \(-0.161846\pi\)
\(828\) − 8.00000i − 0.278019i
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 2.00000i 0.0693375i
\(833\) 2.00000i 0.0692959i
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) 20.0000i 0.690889i
\(839\) −8.00000 −0.276191 −0.138095 0.990419i \(-0.544098\pi\)
−0.138095 + 0.990419i \(0.544098\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 26.0000i 0.896019i
\(843\) − 6.00000i − 0.206651i
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) 5.00000i 0.171802i
\(848\) − 10.0000i − 0.343401i
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) −48.0000 −1.64542
\(852\) 8.00000i 0.274075i
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) 14.0000 0.479070
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 22.0000i 0.751506i 0.926720 + 0.375753i \(0.122616\pi\)
−0.926720 + 0.375753i \(0.877384\pi\)
\(858\) 8.00000i 0.273115i
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 0 0
\(863\) 32.0000i 1.08929i 0.838666 + 0.544646i \(0.183336\pi\)
−0.838666 + 0.544646i \(0.816664\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) 64.0000 2.17105
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) − 14.0000i − 0.474100i
\(873\) 2.00000i 0.0676897i
\(874\) 32.0000 1.08242
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) − 14.0000i − 0.472746i −0.971662 0.236373i \(-0.924041\pi\)
0.971662 0.236373i \(-0.0759588\pi\)
\(878\) − 8.00000i − 0.269987i
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) − 1.00000i − 0.0336718i
\(883\) − 28.0000i − 0.942275i −0.882060 0.471138i \(-0.843844\pi\)
0.882060 0.471138i \(-0.156156\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) − 8.00000i − 0.268614i −0.990940 0.134307i \(-0.957119\pi\)
0.990940 0.134307i \(-0.0428808\pi\)
\(888\) 6.00000i 0.201347i
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 16.0000i 0.535720i
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) − 16.0000i − 0.534224i
\(898\) 2.00000i 0.0667409i
\(899\) 0 0
\(900\) 0 0
\(901\) −20.0000 −0.666297
\(902\) − 24.0000i − 0.799113i
\(903\) 4.00000i 0.133112i
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) − 28.0000i − 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) 4.00000i 0.132745i
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) 48.0000i 1.58857i
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 20.0000i 0.660458i
\(918\) 2.00000i 0.0660098i
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) − 14.0000i − 0.461065i
\(923\) 16.0000i 0.526646i
\(924\) −4.00000 −0.131590
\(925\) 0 0
\(926\) 0 0
\(927\) 8.00000i 0.262754i
\(928\) − 2.00000i − 0.0656532i
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 22.0000i 0.720634i
\(933\) − 8.00000i − 0.261908i
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) − 10.0000i − 0.326686i −0.986569 0.163343i \(-0.947772\pi\)
0.986569 0.163343i \(-0.0522277\pi\)
\(938\) 12.0000i 0.391814i
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 18.0000i 0.586472i
\(943\) 48.0000i 1.56310i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) − 52.0000i − 1.68977i −0.534946 0.844886i \(-0.679668\pi\)
0.534946 0.844886i \(-0.320332\pi\)
\(948\) 16.0000i 0.519656i
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 2.00000i 0.0648204i
\(953\) 26.0000i 0.842223i 0.907009 + 0.421111i \(0.138360\pi\)
−0.907009 + 0.421111i \(0.861640\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) 0 0
\(957\) − 8.00000i − 0.258603i
\(958\) 32.0000i 1.03387i
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 12.0000i 0.386896i
\(963\) 12.0000i 0.386695i
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) 8.00000i 0.257263i 0.991692 + 0.128631i \(0.0410584\pi\)
−0.991692 + 0.128631i \(0.958942\pi\)
\(968\) 5.00000i 0.160706i
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 4.00000i 0.128234i
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) 20.0000i 0.639529i
\(979\) 40.0000 1.27841
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 36.0000i 1.14881i
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) − 8.00000i − 0.254514i
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 12.0000i 0.380808i
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) − 22.0000i − 0.696747i −0.937356 0.348373i \(-0.886734\pi\)
0.937356 0.348373i \(-0.113266\pi\)
\(998\) − 12.0000i − 0.379853i
\(999\) −6.00000 −0.189832
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 1050.2.g.g.799.1 2
3.2 odd 2 3150.2.g.q.2899.2 2
5.2 odd 4 210.2.a.e.1.1 1
5.3 odd 4 1050.2.a.c.1.1 1
5.4 even 2 inner 1050.2.g.g.799.2 2
15.2 even 4 630.2.a.a.1.1 1
15.8 even 4 3150.2.a.bp.1.1 1
15.14 odd 2 3150.2.g.q.2899.1 2
20.3 even 4 8400.2.a.ce.1.1 1
20.7 even 4 1680.2.a.j.1.1 1
35.2 odd 12 1470.2.i.a.361.1 2
35.12 even 12 1470.2.i.j.361.1 2
35.13 even 4 7350.2.a.w.1.1 1
35.17 even 12 1470.2.i.j.961.1 2
35.27 even 4 1470.2.a.j.1.1 1
35.32 odd 12 1470.2.i.a.961.1 2
40.27 even 4 6720.2.a.bq.1.1 1
40.37 odd 4 6720.2.a.j.1.1 1
60.47 odd 4 5040.2.a.k.1.1 1
105.62 odd 4 4410.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.e.1.1 1 5.2 odd 4
630.2.a.a.1.1 1 15.2 even 4
1050.2.a.c.1.1 1 5.3 odd 4
1050.2.g.g.799.1 2 1.1 even 1 trivial
1050.2.g.g.799.2 2 5.4 even 2 inner
1470.2.a.j.1.1 1 35.27 even 4
1470.2.i.a.361.1 2 35.2 odd 12
1470.2.i.a.961.1 2 35.32 odd 12
1470.2.i.j.361.1 2 35.12 even 12
1470.2.i.j.961.1 2 35.17 even 12
1680.2.a.j.1.1 1 20.7 even 4
3150.2.a.bp.1.1 1 15.8 even 4
3150.2.g.q.2899.1 2 15.14 odd 2
3150.2.g.q.2899.2 2 3.2 odd 2
4410.2.a.t.1.1 1 105.62 odd 4
5040.2.a.k.1.1 1 60.47 odd 4
6720.2.a.j.1.1 1 40.37 odd 4
6720.2.a.bq.1.1 1 40.27 even 4
7350.2.a.w.1.1 1 35.13 even 4
8400.2.a.ce.1.1 1 20.3 even 4