Properties

Label 1250.2.b.b.1249.4
Level $1250$
Weight $2$
Character 1250.1249
Analytic conductor $9.981$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1250,2,Mod(1249,1250)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1250, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1250.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1250 = 2 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1250.b (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: \(9.98130025266\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.4
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 1250.1249
Dual form 1250.2.b.b.1249.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -0.381966i q^{3} -1.00000 q^{4} +0.381966 q^{6} +3.00000i q^{7} -1.00000i q^{8} +2.85410 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -0.381966i q^{3} -1.00000 q^{4} +0.381966 q^{6} +3.00000i q^{7} -1.00000i q^{8} +2.85410 q^{9} +4.23607 q^{11} +0.381966i q^{12} +1.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} -1.14590i q^{17} +2.85410i q^{18} -5.85410 q^{19} +1.14590 q^{21} +4.23607i q^{22} -1.76393i q^{23} -0.381966 q^{24} -1.00000 q^{26} -2.23607i q^{27} -3.00000i q^{28} +9.47214 q^{29} -0.236068 q^{31} +1.00000i q^{32} -1.61803i q^{33} +1.14590 q^{34} -2.85410 q^{36} +8.32624i q^{37} -5.85410i q^{38} +0.381966 q^{39} +1.47214 q^{41} +1.14590i q^{42} -6.23607i q^{43} -4.23607 q^{44} +1.76393 q^{46} +11.9443i q^{47} -0.381966i q^{48} -2.00000 q^{49} -0.437694 q^{51} -1.00000i q^{52} +10.4721i q^{53} +2.23607 q^{54} +3.00000 q^{56} +2.23607i q^{57} +9.47214i q^{58} +4.47214 q^{59} -8.85410 q^{61} -0.236068i q^{62} +8.56231i q^{63} -1.00000 q^{64} +1.61803 q^{66} +10.2361i q^{67} +1.14590i q^{68} -0.673762 q^{69} -3.00000 q^{71} -2.85410i q^{72} +7.70820i q^{73} -8.32624 q^{74} +5.85410 q^{76} +12.7082i q^{77} +0.381966i q^{78} +7.23607 q^{79} +7.70820 q^{81} +1.47214i q^{82} -4.52786i q^{83} -1.14590 q^{84} +6.23607 q^{86} -3.61803i q^{87} -4.23607i q^{88} -4.47214 q^{89} -3.00000 q^{91} +1.76393i q^{92} +0.0901699i q^{93} -11.9443 q^{94} +0.381966 q^{96} -9.56231i q^{97} -2.00000i q^{98} +12.0902 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 6 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 6 q^{6} - 2 q^{9} + 8 q^{11} - 12 q^{14} + 4 q^{16} - 10 q^{19} + 18 q^{21} - 6 q^{24} - 4 q^{26} + 20 q^{29} + 8 q^{31} + 18 q^{34} + 2 q^{36} + 6 q^{39} - 12 q^{41} - 8 q^{44} + 16 q^{46} - 8 q^{49} - 42 q^{51} + 12 q^{56} - 22 q^{61} - 4 q^{64} + 2 q^{66} - 34 q^{69} - 12 q^{71} - 2 q^{74} + 10 q^{76} + 20 q^{79} + 4 q^{81} - 18 q^{84} + 16 q^{86} - 12 q^{91} - 12 q^{94} + 6 q^{96} + 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1250\mathbb{Z}\right)^\times\).

\(n\) \(627\)
\(\chi(n)\) \(-1\)

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
\(3\) − 0.381966i − 0.220528i −0.993902 0.110264i \(-0.964830\pi\)
0.993902 0.110264i \(-0.0351697\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0.381966 0.155937
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 2.85410 0.951367
\(10\) 0 0
\(11\) 4.23607 1.27722 0.638611 0.769529i \(-0.279509\pi\)
0.638611 + 0.769529i \(0.279509\pi\)
\(12\) 0.381966i 0.110264i
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 1.14590i − 0.277921i −0.990298 0.138961i \(-0.955624\pi\)
0.990298 0.138961i \(-0.0443761\pi\)
\(18\) 2.85410i 0.672718i
\(19\) −5.85410 −1.34302 −0.671512 0.740994i \(-0.734355\pi\)
−0.671512 + 0.740994i \(0.734355\pi\)
\(20\) 0 0
\(21\) 1.14590 0.250055
\(22\) 4.23607i 0.903133i
\(23\) − 1.76393i − 0.367805i −0.982944 0.183903i \(-0.941127\pi\)
0.982944 0.183903i \(-0.0588731\pi\)
\(24\) −0.381966 −0.0779685
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) − 2.23607i − 0.430331i
\(28\) − 3.00000i − 0.566947i
\(29\) 9.47214 1.75893 0.879466 0.475962i \(-0.157900\pi\)
0.879466 + 0.475962i \(0.157900\pi\)
\(30\) 0 0
\(31\) −0.236068 −0.0423991 −0.0211995 0.999775i \(-0.506749\pi\)
−0.0211995 + 0.999775i \(0.506749\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 1.61803i − 0.281664i
\(34\) 1.14590 0.196520
\(35\) 0 0
\(36\) −2.85410 −0.475684
\(37\) 8.32624i 1.36883i 0.729095 + 0.684413i \(0.239941\pi\)
−0.729095 + 0.684413i \(0.760059\pi\)
\(38\) − 5.85410i − 0.949661i
\(39\) 0.381966 0.0611635
\(40\) 0 0
\(41\) 1.47214 0.229909 0.114955 0.993371i \(-0.463328\pi\)
0.114955 + 0.993371i \(0.463328\pi\)
\(42\) 1.14590i 0.176816i
\(43\) − 6.23607i − 0.950991i −0.879718 0.475496i \(-0.842269\pi\)
0.879718 0.475496i \(-0.157731\pi\)
\(44\) −4.23607 −0.638611
\(45\) 0 0
\(46\) 1.76393 0.260078
\(47\) 11.9443i 1.74225i 0.491060 + 0.871126i \(0.336609\pi\)
−0.491060 + 0.871126i \(0.663391\pi\)
\(48\) − 0.381966i − 0.0551320i
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −0.437694 −0.0612894
\(52\) − 1.00000i − 0.138675i
\(53\) 10.4721i 1.43846i 0.694773 + 0.719229i \(0.255505\pi\)
−0.694773 + 0.719229i \(0.744495\pi\)
\(54\) 2.23607 0.304290
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) 2.23607i 0.296174i
\(58\) 9.47214i 1.24375i
\(59\) 4.47214 0.582223 0.291111 0.956689i \(-0.405975\pi\)
0.291111 + 0.956689i \(0.405975\pi\)
\(60\) 0 0
\(61\) −8.85410 −1.13365 −0.566826 0.823838i \(-0.691829\pi\)
−0.566826 + 0.823838i \(0.691829\pi\)
\(62\) − 0.236068i − 0.0299807i
\(63\) 8.56231i 1.07875i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.61803 0.199166
\(67\) 10.2361i 1.25053i 0.780411 + 0.625267i \(0.215010\pi\)
−0.780411 + 0.625267i \(0.784990\pi\)
\(68\) 1.14590i 0.138961i
\(69\) −0.673762 −0.0811114
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) − 2.85410i − 0.336359i
\(73\) 7.70820i 0.902177i 0.892479 + 0.451089i \(0.148964\pi\)
−0.892479 + 0.451089i \(0.851036\pi\)
\(74\) −8.32624 −0.967905
\(75\) 0 0
\(76\) 5.85410 0.671512
\(77\) 12.7082i 1.44823i
\(78\) 0.381966i 0.0432491i
\(79\) 7.23607 0.814121 0.407061 0.913401i \(-0.366554\pi\)
0.407061 + 0.913401i \(0.366554\pi\)
\(80\) 0 0
\(81\) 7.70820 0.856467
\(82\) 1.47214i 0.162570i
\(83\) − 4.52786i − 0.496998i −0.968632 0.248499i \(-0.920063\pi\)
0.968632 0.248499i \(-0.0799372\pi\)
\(84\) −1.14590 −0.125028
\(85\) 0 0
\(86\) 6.23607 0.672453
\(87\) − 3.61803i − 0.387894i
\(88\) − 4.23607i − 0.451566i
\(89\) −4.47214 −0.474045 −0.237023 0.971504i \(-0.576172\pi\)
−0.237023 + 0.971504i \(0.576172\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 1.76393i 0.183903i
\(93\) 0.0901699i 0.00935019i
\(94\) −11.9443 −1.23196
\(95\) 0 0
\(96\) 0.381966 0.0389842
\(97\) − 9.56231i − 0.970905i −0.874263 0.485453i \(-0.838655\pi\)
0.874263 0.485453i \(-0.161345\pi\)
\(98\) − 2.00000i − 0.202031i
\(99\) 12.0902 1.21511
\(100\) 0 0
\(101\) 0.618034 0.0614967 0.0307483 0.999527i \(-0.490211\pi\)
0.0307483 + 0.999527i \(0.490211\pi\)
\(102\) − 0.437694i − 0.0433382i
\(103\) − 13.1459i − 1.29530i −0.761936 0.647652i \(-0.775751\pi\)
0.761936 0.647652i \(-0.224249\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −10.4721 −1.01714
\(107\) 1.09017i 0.105391i 0.998611 + 0.0526954i \(0.0167812\pi\)
−0.998611 + 0.0526954i \(0.983219\pi\)
\(108\) 2.23607i 0.215166i
\(109\) 15.0000 1.43674 0.718370 0.695662i \(-0.244889\pi\)
0.718370 + 0.695662i \(0.244889\pi\)
\(110\) 0 0
\(111\) 3.18034 0.301865
\(112\) 3.00000i 0.283473i
\(113\) 3.76393i 0.354081i 0.984204 + 0.177040i \(0.0566523\pi\)
−0.984204 + 0.177040i \(0.943348\pi\)
\(114\) −2.23607 −0.209427
\(115\) 0 0
\(116\) −9.47214 −0.879466
\(117\) 2.85410i 0.263862i
\(118\) 4.47214i 0.411693i
\(119\) 3.43769 0.315133
\(120\) 0 0
\(121\) 6.94427 0.631297
\(122\) − 8.85410i − 0.801613i
\(123\) − 0.562306i − 0.0507014i
\(124\) 0.236068 0.0211995
\(125\) 0 0
\(126\) −8.56231 −0.762791
\(127\) 3.52786i 0.313047i 0.987674 + 0.156524i \(0.0500287\pi\)
−0.987674 + 0.156524i \(0.949971\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −2.38197 −0.209720
\(130\) 0 0
\(131\) −16.4164 −1.43431 −0.717154 0.696915i \(-0.754556\pi\)
−0.717154 + 0.696915i \(0.754556\pi\)
\(132\) 1.61803i 0.140832i
\(133\) − 17.5623i − 1.52285i
\(134\) −10.2361 −0.884262
\(135\) 0 0
\(136\) −1.14590 −0.0982599
\(137\) − 5.09017i − 0.434883i −0.976073 0.217441i \(-0.930229\pi\)
0.976073 0.217441i \(-0.0697711\pi\)
\(138\) − 0.673762i − 0.0573544i
\(139\) 5.32624 0.451766 0.225883 0.974154i \(-0.427473\pi\)
0.225883 + 0.974154i \(0.427473\pi\)
\(140\) 0 0
\(141\) 4.56231 0.384215
\(142\) − 3.00000i − 0.251754i
\(143\) 4.23607i 0.354238i
\(144\) 2.85410 0.237842
\(145\) 0 0
\(146\) −7.70820 −0.637935
\(147\) 0.763932i 0.0630081i
\(148\) − 8.32624i − 0.684413i
\(149\) −2.23607 −0.183186 −0.0915929 0.995797i \(-0.529196\pi\)
−0.0915929 + 0.995797i \(0.529196\pi\)
\(150\) 0 0
\(151\) −9.70820 −0.790042 −0.395021 0.918672i \(-0.629263\pi\)
−0.395021 + 0.918672i \(0.629263\pi\)
\(152\) 5.85410i 0.474830i
\(153\) − 3.27051i − 0.264405i
\(154\) −12.7082 −1.02406
\(155\) 0 0
\(156\) −0.381966 −0.0305818
\(157\) − 4.56231i − 0.364112i −0.983288 0.182056i \(-0.941725\pi\)
0.983288 0.182056i \(-0.0582752\pi\)
\(158\) 7.23607i 0.575671i
\(159\) 4.00000 0.317221
\(160\) 0 0
\(161\) 5.29180 0.417052
\(162\) 7.70820i 0.605614i
\(163\) 1.85410i 0.145224i 0.997360 + 0.0726122i \(0.0231336\pi\)
−0.997360 + 0.0726122i \(0.976866\pi\)
\(164\) −1.47214 −0.114955
\(165\) 0 0
\(166\) 4.52786 0.351430
\(167\) − 13.3820i − 1.03553i −0.855524 0.517764i \(-0.826765\pi\)
0.855524 0.517764i \(-0.173235\pi\)
\(168\) − 1.14590i − 0.0884080i
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −16.7082 −1.27771
\(172\) 6.23607i 0.475496i
\(173\) − 1.23607i − 0.0939765i −0.998895 0.0469883i \(-0.985038\pi\)
0.998895 0.0469883i \(-0.0149623\pi\)
\(174\) 3.61803 0.274282
\(175\) 0 0
\(176\) 4.23607 0.319306
\(177\) − 1.70820i − 0.128396i
\(178\) − 4.47214i − 0.335201i
\(179\) −11.1803 −0.835658 −0.417829 0.908526i \(-0.637209\pi\)
−0.417829 + 0.908526i \(0.637209\pi\)
\(180\) 0 0
\(181\) 5.41641 0.402598 0.201299 0.979530i \(-0.435484\pi\)
0.201299 + 0.979530i \(0.435484\pi\)
\(182\) − 3.00000i − 0.222375i
\(183\) 3.38197i 0.250002i
\(184\) −1.76393 −0.130039
\(185\) 0 0
\(186\) −0.0901699 −0.00661158
\(187\) − 4.85410i − 0.354967i
\(188\) − 11.9443i − 0.871126i
\(189\) 6.70820 0.487950
\(190\) 0 0
\(191\) −13.6525 −0.987858 −0.493929 0.869502i \(-0.664440\pi\)
−0.493929 + 0.869502i \(0.664440\pi\)
\(192\) 0.381966i 0.0275660i
\(193\) − 14.6525i − 1.05471i −0.849646 0.527354i \(-0.823184\pi\)
0.849646 0.527354i \(-0.176816\pi\)
\(194\) 9.56231 0.686534
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) − 12.0000i − 0.854965i −0.904024 0.427482i \(-0.859401\pi\)
0.904024 0.427482i \(-0.140599\pi\)
\(198\) 12.0902i 0.859211i
\(199\) −2.56231 −0.181637 −0.0908185 0.995867i \(-0.528948\pi\)
−0.0908185 + 0.995867i \(0.528948\pi\)
\(200\) 0 0
\(201\) 3.90983 0.275778
\(202\) 0.618034i 0.0434847i
\(203\) 28.4164i 1.99444i
\(204\) 0.437694 0.0306447
\(205\) 0 0
\(206\) 13.1459 0.915918
\(207\) − 5.03444i − 0.349918i
\(208\) 1.00000i 0.0693375i
\(209\) −24.7984 −1.71534
\(210\) 0 0
\(211\) −22.2705 −1.53317 −0.766583 0.642146i \(-0.778044\pi\)
−0.766583 + 0.642146i \(0.778044\pi\)
\(212\) − 10.4721i − 0.719229i
\(213\) 1.14590i 0.0785156i
\(214\) −1.09017 −0.0745225
\(215\) 0 0
\(216\) −2.23607 −0.152145
\(217\) − 0.708204i − 0.0480760i
\(218\) 15.0000i 1.01593i
\(219\) 2.94427 0.198955
\(220\) 0 0
\(221\) 1.14590 0.0770814
\(222\) 3.18034i 0.213450i
\(223\) − 9.00000i − 0.602685i −0.953516 0.301342i \(-0.902565\pi\)
0.953516 0.301342i \(-0.0974347\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) −3.76393 −0.250373
\(227\) − 29.3607i − 1.94874i −0.224958 0.974368i \(-0.572225\pi\)
0.224958 0.974368i \(-0.427775\pi\)
\(228\) − 2.23607i − 0.148087i
\(229\) 2.56231 0.169322 0.0846610 0.996410i \(-0.473019\pi\)
0.0846610 + 0.996410i \(0.473019\pi\)
\(230\) 0 0
\(231\) 4.85410 0.319376
\(232\) − 9.47214i − 0.621876i
\(233\) 14.2918i 0.936287i 0.883653 + 0.468143i \(0.155077\pi\)
−0.883653 + 0.468143i \(0.844923\pi\)
\(234\) −2.85410 −0.186578
\(235\) 0 0
\(236\) −4.47214 −0.291111
\(237\) − 2.76393i − 0.179537i
\(238\) 3.43769i 0.222833i
\(239\) 6.90983 0.446960 0.223480 0.974709i \(-0.428258\pi\)
0.223480 + 0.974709i \(0.428258\pi\)
\(240\) 0 0
\(241\) 15.6180 1.00605 0.503023 0.864273i \(-0.332221\pi\)
0.503023 + 0.864273i \(0.332221\pi\)
\(242\) 6.94427i 0.446395i
\(243\) − 9.65248i − 0.619207i
\(244\) 8.85410 0.566826
\(245\) 0 0
\(246\) 0.562306 0.0358513
\(247\) − 5.85410i − 0.372488i
\(248\) 0.236068i 0.0149903i
\(249\) −1.72949 −0.109602
\(250\) 0 0
\(251\) 0.819660 0.0517365 0.0258682 0.999665i \(-0.491765\pi\)
0.0258682 + 0.999665i \(0.491765\pi\)
\(252\) − 8.56231i − 0.539375i
\(253\) − 7.47214i − 0.469769i
\(254\) −3.52786 −0.221358
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 25.7426i − 1.60578i −0.596126 0.802891i \(-0.703294\pi\)
0.596126 0.802891i \(-0.296706\pi\)
\(258\) − 2.38197i − 0.148295i
\(259\) −24.9787 −1.55210
\(260\) 0 0
\(261\) 27.0344 1.67339
\(262\) − 16.4164i − 1.01421i
\(263\) 4.61803i 0.284760i 0.989812 + 0.142380i \(0.0454755\pi\)
−0.989812 + 0.142380i \(0.954524\pi\)
\(264\) −1.61803 −0.0995831
\(265\) 0 0
\(266\) 17.5623 1.07681
\(267\) 1.70820i 0.104540i
\(268\) − 10.2361i − 0.625267i
\(269\) −1.90983 −0.116444 −0.0582222 0.998304i \(-0.518543\pi\)
−0.0582222 + 0.998304i \(0.518543\pi\)
\(270\) 0 0
\(271\) 28.5066 1.73165 0.865826 0.500346i \(-0.166794\pi\)
0.865826 + 0.500346i \(0.166794\pi\)
\(272\) − 1.14590i − 0.0694803i
\(273\) 1.14590i 0.0693529i
\(274\) 5.09017 0.307508
\(275\) 0 0
\(276\) 0.673762 0.0405557
\(277\) − 0.291796i − 0.0175323i −0.999962 0.00876616i \(-0.997210\pi\)
0.999962 0.00876616i \(-0.00279039\pi\)
\(278\) 5.32624i 0.319447i
\(279\) −0.673762 −0.0403371
\(280\) 0 0
\(281\) 3.18034 0.189723 0.0948616 0.995490i \(-0.469759\pi\)
0.0948616 + 0.995490i \(0.469759\pi\)
\(282\) 4.56231i 0.271681i
\(283\) − 12.2918i − 0.730671i −0.930876 0.365336i \(-0.880954\pi\)
0.930876 0.365336i \(-0.119046\pi\)
\(284\) 3.00000 0.178017
\(285\) 0 0
\(286\) −4.23607 −0.250484
\(287\) 4.41641i 0.260692i
\(288\) 2.85410i 0.168180i
\(289\) 15.6869 0.922760
\(290\) 0 0
\(291\) −3.65248 −0.214112
\(292\) − 7.70820i − 0.451089i
\(293\) 8.56231i 0.500215i 0.968218 + 0.250108i \(0.0804660\pi\)
−0.968218 + 0.250108i \(0.919534\pi\)
\(294\) −0.763932 −0.0445534
\(295\) 0 0
\(296\) 8.32624 0.483953
\(297\) − 9.47214i − 0.549629i
\(298\) − 2.23607i − 0.129532i
\(299\) 1.76393 0.102011
\(300\) 0 0
\(301\) 18.7082 1.07832
\(302\) − 9.70820i − 0.558644i
\(303\) − 0.236068i − 0.0135618i
\(304\) −5.85410 −0.335756
\(305\) 0 0
\(306\) 3.27051 0.186963
\(307\) 23.1246i 1.31979i 0.751357 + 0.659896i \(0.229400\pi\)
−0.751357 + 0.659896i \(0.770600\pi\)
\(308\) − 12.7082i − 0.724117i
\(309\) −5.02129 −0.285651
\(310\) 0 0
\(311\) −9.90983 −0.561935 −0.280967 0.959717i \(-0.590655\pi\)
−0.280967 + 0.959717i \(0.590655\pi\)
\(312\) − 0.381966i − 0.0216246i
\(313\) − 16.5623i − 0.936157i −0.883687 0.468078i \(-0.844946\pi\)
0.883687 0.468078i \(-0.155054\pi\)
\(314\) 4.56231 0.257466
\(315\) 0 0
\(316\) −7.23607 −0.407061
\(317\) − 8.38197i − 0.470778i −0.971901 0.235389i \(-0.924364\pi\)
0.971901 0.235389i \(-0.0756364\pi\)
\(318\) 4.00000i 0.224309i
\(319\) 40.1246 2.24655
\(320\) 0 0
\(321\) 0.416408 0.0232416
\(322\) 5.29180i 0.294900i
\(323\) 6.70820i 0.373254i
\(324\) −7.70820 −0.428234
\(325\) 0 0
\(326\) −1.85410 −0.102689
\(327\) − 5.72949i − 0.316842i
\(328\) − 1.47214i − 0.0812851i
\(329\) −35.8328 −1.97553
\(330\) 0 0
\(331\) −15.5623 −0.855382 −0.427691 0.903925i \(-0.640673\pi\)
−0.427691 + 0.903925i \(0.640673\pi\)
\(332\) 4.52786i 0.248499i
\(333\) 23.7639i 1.30226i
\(334\) 13.3820 0.732229
\(335\) 0 0
\(336\) 1.14590 0.0625139
\(337\) 4.58359i 0.249684i 0.992177 + 0.124842i \(0.0398424\pi\)
−0.992177 + 0.124842i \(0.960158\pi\)
\(338\) 12.0000i 0.652714i
\(339\) 1.43769 0.0780848
\(340\) 0 0
\(341\) −1.00000 −0.0541530
\(342\) − 16.7082i − 0.903476i
\(343\) 15.0000i 0.809924i
\(344\) −6.23607 −0.336226
\(345\) 0 0
\(346\) 1.23607 0.0664514
\(347\) 21.0902i 1.13218i 0.824344 + 0.566090i \(0.191544\pi\)
−0.824344 + 0.566090i \(0.808456\pi\)
\(348\) 3.61803i 0.193947i
\(349\) 17.7639 0.950881 0.475441 0.879748i \(-0.342289\pi\)
0.475441 + 0.879748i \(0.342289\pi\)
\(350\) 0 0
\(351\) 2.23607 0.119352
\(352\) 4.23607i 0.225783i
\(353\) − 21.2361i − 1.13028i −0.824994 0.565141i \(-0.808822\pi\)
0.824994 0.565141i \(-0.191178\pi\)
\(354\) 1.70820 0.0907900
\(355\) 0 0
\(356\) 4.47214 0.237023
\(357\) − 1.31308i − 0.0694957i
\(358\) − 11.1803i − 0.590899i
\(359\) 8.09017 0.426983 0.213491 0.976945i \(-0.431516\pi\)
0.213491 + 0.976945i \(0.431516\pi\)
\(360\) 0 0
\(361\) 15.2705 0.803711
\(362\) 5.41641i 0.284680i
\(363\) − 2.65248i − 0.139219i
\(364\) 3.00000 0.157243
\(365\) 0 0
\(366\) −3.38197 −0.176778
\(367\) − 35.5410i − 1.85523i −0.373543 0.927613i \(-0.621857\pi\)
0.373543 0.927613i \(-0.378143\pi\)
\(368\) − 1.76393i − 0.0919513i
\(369\) 4.20163 0.218728
\(370\) 0 0
\(371\) −31.4164 −1.63106
\(372\) − 0.0901699i − 0.00467509i
\(373\) − 11.7639i − 0.609113i −0.952494 0.304557i \(-0.901492\pi\)
0.952494 0.304557i \(-0.0985083\pi\)
\(374\) 4.85410 0.251000
\(375\) 0 0
\(376\) 11.9443 0.615979
\(377\) 9.47214i 0.487840i
\(378\) 6.70820i 0.345033i
\(379\) −9.47214 −0.486551 −0.243275 0.969957i \(-0.578222\pi\)
−0.243275 + 0.969957i \(0.578222\pi\)
\(380\) 0 0
\(381\) 1.34752 0.0690358
\(382\) − 13.6525i − 0.698521i
\(383\) 11.3262i 0.578744i 0.957217 + 0.289372i \(0.0934464\pi\)
−0.957217 + 0.289372i \(0.906554\pi\)
\(384\) −0.381966 −0.0194921
\(385\) 0 0
\(386\) 14.6525 0.745791
\(387\) − 17.7984i − 0.904742i
\(388\) 9.56231i 0.485453i
\(389\) −18.2148 −0.923526 −0.461763 0.887003i \(-0.652783\pi\)
−0.461763 + 0.887003i \(0.652783\pi\)
\(390\) 0 0
\(391\) −2.02129 −0.102221
\(392\) 2.00000i 0.101015i
\(393\) 6.27051i 0.316305i
\(394\) 12.0000 0.604551
\(395\) 0 0
\(396\) −12.0902 −0.607554
\(397\) 10.2361i 0.513734i 0.966447 + 0.256867i \(0.0826902\pi\)
−0.966447 + 0.256867i \(0.917310\pi\)
\(398\) − 2.56231i − 0.128437i
\(399\) −6.70820 −0.335830
\(400\) 0 0
\(401\) −14.1803 −0.708132 −0.354066 0.935220i \(-0.615201\pi\)
−0.354066 + 0.935220i \(0.615201\pi\)
\(402\) 3.90983i 0.195005i
\(403\) − 0.236068i − 0.0117594i
\(404\) −0.618034 −0.0307483
\(405\) 0 0
\(406\) −28.4164 −1.41028
\(407\) 35.2705i 1.74829i
\(408\) 0.437694i 0.0216691i
\(409\) 6.70820 0.331699 0.165850 0.986151i \(-0.446963\pi\)
0.165850 + 0.986151i \(0.446963\pi\)
\(410\) 0 0
\(411\) −1.94427 −0.0959039
\(412\) 13.1459i 0.647652i
\(413\) 13.4164i 0.660178i
\(414\) 5.03444 0.247429
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) − 2.03444i − 0.0996270i
\(418\) − 24.7984i − 1.21293i
\(419\) −22.7639 −1.11209 −0.556045 0.831152i \(-0.687682\pi\)
−0.556045 + 0.831152i \(0.687682\pi\)
\(420\) 0 0
\(421\) 21.2705 1.03666 0.518331 0.855180i \(-0.326554\pi\)
0.518331 + 0.855180i \(0.326554\pi\)
\(422\) − 22.2705i − 1.08411i
\(423\) 34.0902i 1.65752i
\(424\) 10.4721 0.508572
\(425\) 0 0
\(426\) −1.14590 −0.0555189
\(427\) − 26.5623i − 1.28544i
\(428\) − 1.09017i − 0.0526954i
\(429\) 1.61803 0.0781194
\(430\) 0 0
\(431\) 26.0689 1.25569 0.627847 0.778337i \(-0.283936\pi\)
0.627847 + 0.778337i \(0.283936\pi\)
\(432\) − 2.23607i − 0.107583i
\(433\) − 21.3607i − 1.02653i −0.858231 0.513264i \(-0.828436\pi\)
0.858231 0.513264i \(-0.171564\pi\)
\(434\) 0.708204 0.0339949
\(435\) 0 0
\(436\) −15.0000 −0.718370
\(437\) 10.3262i 0.493971i
\(438\) 2.94427i 0.140683i
\(439\) −25.6525 −1.22433 −0.612163 0.790732i \(-0.709700\pi\)
−0.612163 + 0.790732i \(0.709700\pi\)
\(440\) 0 0
\(441\) −5.70820 −0.271819
\(442\) 1.14590i 0.0545048i
\(443\) 19.4164i 0.922501i 0.887270 + 0.461251i \(0.152599\pi\)
−0.887270 + 0.461251i \(0.847401\pi\)
\(444\) −3.18034 −0.150932
\(445\) 0 0
\(446\) 9.00000 0.426162
\(447\) 0.854102i 0.0403976i
\(448\) − 3.00000i − 0.141737i
\(449\) 3.94427 0.186142 0.0930709 0.995659i \(-0.470332\pi\)
0.0930709 + 0.995659i \(0.470332\pi\)
\(450\) 0 0
\(451\) 6.23607 0.293645
\(452\) − 3.76393i − 0.177040i
\(453\) 3.70820i 0.174227i
\(454\) 29.3607 1.37796
\(455\) 0 0
\(456\) 2.23607 0.104713
\(457\) − 40.2148i − 1.88117i −0.339561 0.940584i \(-0.610278\pi\)
0.339561 0.940584i \(-0.389722\pi\)
\(458\) 2.56231i 0.119729i
\(459\) −2.56231 −0.119598
\(460\) 0 0
\(461\) 5.81966 0.271049 0.135524 0.990774i \(-0.456728\pi\)
0.135524 + 0.990774i \(0.456728\pi\)
\(462\) 4.85410i 0.225833i
\(463\) − 34.9787i − 1.62560i −0.582544 0.812799i \(-0.697943\pi\)
0.582544 0.812799i \(-0.302057\pi\)
\(464\) 9.47214 0.439733
\(465\) 0 0
\(466\) −14.2918 −0.662055
\(467\) 19.1803i 0.887560i 0.896136 + 0.443780i \(0.146363\pi\)
−0.896136 + 0.443780i \(0.853637\pi\)
\(468\) − 2.85410i − 0.131931i
\(469\) −30.7082 −1.41797
\(470\) 0 0
\(471\) −1.74265 −0.0802969
\(472\) − 4.47214i − 0.205847i
\(473\) − 26.4164i − 1.21463i
\(474\) 2.76393 0.126952
\(475\) 0 0
\(476\) −3.43769 −0.157566
\(477\) 29.8885i 1.36850i
\(478\) 6.90983i 0.316048i
\(479\) 15.2016 0.694580 0.347290 0.937758i \(-0.387102\pi\)
0.347290 + 0.937758i \(0.387102\pi\)
\(480\) 0 0
\(481\) −8.32624 −0.379644
\(482\) 15.6180i 0.711382i
\(483\) − 2.02129i − 0.0919717i
\(484\) −6.94427 −0.315649
\(485\) 0 0
\(486\) 9.65248 0.437845
\(487\) − 22.1246i − 1.00256i −0.865285 0.501281i \(-0.832862\pi\)
0.865285 0.501281i \(-0.167138\pi\)
\(488\) 8.85410i 0.400806i
\(489\) 0.708204 0.0320261
\(490\) 0 0
\(491\) 19.2361 0.868112 0.434056 0.900886i \(-0.357082\pi\)
0.434056 + 0.900886i \(0.357082\pi\)
\(492\) 0.562306i 0.0253507i
\(493\) − 10.8541i − 0.488844i
\(494\) 5.85410 0.263388
\(495\) 0 0
\(496\) −0.236068 −0.0105998
\(497\) − 9.00000i − 0.403705i
\(498\) − 1.72949i − 0.0775003i
\(499\) −34.1459 −1.52858 −0.764290 0.644873i \(-0.776910\pi\)
−0.764290 + 0.644873i \(0.776910\pi\)
\(500\) 0 0
\(501\) −5.11146 −0.228363
\(502\) 0.819660i 0.0365832i
\(503\) − 30.5066i − 1.36022i −0.733110 0.680111i \(-0.761932\pi\)
0.733110 0.680111i \(-0.238068\pi\)
\(504\) 8.56231 0.381395
\(505\) 0 0
\(506\) 7.47214 0.332177
\(507\) − 4.58359i − 0.203564i
\(508\) − 3.52786i − 0.156524i
\(509\) 1.90983 0.0846517 0.0423259 0.999104i \(-0.486523\pi\)
0.0423259 + 0.999104i \(0.486523\pi\)
\(510\) 0 0
\(511\) −23.1246 −1.02297
\(512\) 1.00000i 0.0441942i
\(513\) 13.0902i 0.577945i
\(514\) 25.7426 1.13546
\(515\) 0 0
\(516\) 2.38197 0.104860
\(517\) 50.5967i 2.22524i
\(518\) − 24.9787i − 1.09750i
\(519\) −0.472136 −0.0207245
\(520\) 0 0
\(521\) −23.4508 −1.02740 −0.513700 0.857970i \(-0.671726\pi\)
−0.513700 + 0.857970i \(0.671726\pi\)
\(522\) 27.0344i 1.18327i
\(523\) − 4.65248i − 0.203439i −0.994813 0.101719i \(-0.967566\pi\)
0.994813 0.101719i \(-0.0324343\pi\)
\(524\) 16.4164 0.717154
\(525\) 0 0
\(526\) −4.61803 −0.201356
\(527\) 0.270510i 0.0117836i
\(528\) − 1.61803i − 0.0704159i
\(529\) 19.8885 0.864719
\(530\) 0 0
\(531\) 12.7639 0.553907
\(532\) 17.5623i 0.761423i
\(533\) 1.47214i 0.0637653i
\(534\) −1.70820 −0.0739212
\(535\) 0 0
\(536\) 10.2361 0.442131
\(537\) 4.27051i 0.184286i
\(538\) − 1.90983i − 0.0823386i
\(539\) −8.47214 −0.364921
\(540\) 0 0
\(541\) 15.2918 0.657446 0.328723 0.944426i \(-0.393382\pi\)
0.328723 + 0.944426i \(0.393382\pi\)
\(542\) 28.5066i 1.22446i
\(543\) − 2.06888i − 0.0887843i
\(544\) 1.14590 0.0491300
\(545\) 0 0
\(546\) −1.14590 −0.0490399
\(547\) − 8.38197i − 0.358387i −0.983814 0.179193i \(-0.942651\pi\)
0.983814 0.179193i \(-0.0573488\pi\)
\(548\) 5.09017i 0.217441i
\(549\) −25.2705 −1.07852
\(550\) 0 0
\(551\) −55.4508 −2.36229
\(552\) 0.673762i 0.0286772i
\(553\) 21.7082i 0.923127i
\(554\) 0.291796 0.0123972
\(555\) 0 0
\(556\) −5.32624 −0.225883
\(557\) − 24.8885i − 1.05456i −0.849691 0.527281i \(-0.823212\pi\)
0.849691 0.527281i \(-0.176788\pi\)
\(558\) − 0.673762i − 0.0285226i
\(559\) 6.23607 0.263758
\(560\) 0 0
\(561\) −1.85410 −0.0782802
\(562\) 3.18034i 0.134155i
\(563\) − 28.1459i − 1.18621i −0.805126 0.593104i \(-0.797902\pi\)
0.805126 0.593104i \(-0.202098\pi\)
\(564\) −4.56231 −0.192108
\(565\) 0 0
\(566\) 12.2918 0.516663
\(567\) 23.1246i 0.971142i
\(568\) 3.00000i 0.125877i
\(569\) −17.7639 −0.744703 −0.372351 0.928092i \(-0.621448\pi\)
−0.372351 + 0.928092i \(0.621448\pi\)
\(570\) 0 0
\(571\) 17.3262 0.725080 0.362540 0.931968i \(-0.381910\pi\)
0.362540 + 0.931968i \(0.381910\pi\)
\(572\) − 4.23607i − 0.177119i
\(573\) 5.21478i 0.217851i
\(574\) −4.41641 −0.184337
\(575\) 0 0
\(576\) −2.85410 −0.118921
\(577\) 43.0000i 1.79011i 0.445952 + 0.895057i \(0.352865\pi\)
−0.445952 + 0.895057i \(0.647135\pi\)
\(578\) 15.6869i 0.652490i
\(579\) −5.59675 −0.232593
\(580\) 0 0
\(581\) 13.5836 0.563542
\(582\) − 3.65248i − 0.151400i
\(583\) 44.3607i 1.83723i
\(584\) 7.70820 0.318968
\(585\) 0 0
\(586\) −8.56231 −0.353706
\(587\) 34.1803i 1.41077i 0.708823 + 0.705387i \(0.249227\pi\)
−0.708823 + 0.705387i \(0.750773\pi\)
\(588\) − 0.763932i − 0.0315040i
\(589\) 1.38197 0.0569429
\(590\) 0 0
\(591\) −4.58359 −0.188544
\(592\) 8.32624i 0.342206i
\(593\) 29.0132i 1.19143i 0.803197 + 0.595714i \(0.203131\pi\)
−0.803197 + 0.595714i \(0.796869\pi\)
\(594\) 9.47214 0.388646
\(595\) 0 0
\(596\) 2.23607 0.0915929
\(597\) 0.978714i 0.0400561i
\(598\) 1.76393i 0.0721325i
\(599\) −8.94427 −0.365453 −0.182727 0.983164i \(-0.558492\pi\)
−0.182727 + 0.983164i \(0.558492\pi\)
\(600\) 0 0
\(601\) 38.8328 1.58402 0.792012 0.610506i \(-0.209034\pi\)
0.792012 + 0.610506i \(0.209034\pi\)
\(602\) 18.7082i 0.762489i
\(603\) 29.2148i 1.18972i
\(604\) 9.70820 0.395021
\(605\) 0 0
\(606\) 0.236068 0.00958961
\(607\) 33.8541i 1.37410i 0.726612 + 0.687048i \(0.241094\pi\)
−0.726612 + 0.687048i \(0.758906\pi\)
\(608\) − 5.85410i − 0.237415i
\(609\) 10.8541 0.439830
\(610\) 0 0
\(611\) −11.9443 −0.483214
\(612\) 3.27051i 0.132203i
\(613\) − 21.4377i − 0.865860i −0.901427 0.432930i \(-0.857480\pi\)
0.901427 0.432930i \(-0.142520\pi\)
\(614\) −23.1246 −0.933233
\(615\) 0 0
\(616\) 12.7082 0.512028
\(617\) − 14.3607i − 0.578139i −0.957308 0.289070i \(-0.906654\pi\)
0.957308 0.289070i \(-0.0933459\pi\)
\(618\) − 5.02129i − 0.201986i
\(619\) −46.9574 −1.88738 −0.943689 0.330833i \(-0.892670\pi\)
−0.943689 + 0.330833i \(0.892670\pi\)
\(620\) 0 0
\(621\) −3.94427 −0.158278
\(622\) − 9.90983i − 0.397348i
\(623\) − 13.4164i − 0.537517i
\(624\) 0.381966 0.0152909
\(625\) 0 0
\(626\) 16.5623 0.661963
\(627\) 9.47214i 0.378281i
\(628\) 4.56231i 0.182056i
\(629\) 9.54102 0.380425
\(630\) 0 0
\(631\) −44.8328 −1.78477 −0.892383 0.451279i \(-0.850968\pi\)
−0.892383 + 0.451279i \(0.850968\pi\)
\(632\) − 7.23607i − 0.287835i
\(633\) 8.50658i 0.338106i
\(634\) 8.38197 0.332890
\(635\) 0 0
\(636\) −4.00000 −0.158610
\(637\) − 2.00000i − 0.0792429i
\(638\) 40.1246i 1.58855i
\(639\) −8.56231 −0.338720
\(640\) 0 0
\(641\) −30.3607 −1.19917 −0.599587 0.800309i \(-0.704669\pi\)
−0.599587 + 0.800309i \(0.704669\pi\)
\(642\) 0.416408i 0.0164343i
\(643\) − 31.7639i − 1.25265i −0.779563 0.626324i \(-0.784559\pi\)
0.779563 0.626324i \(-0.215441\pi\)
\(644\) −5.29180 −0.208526
\(645\) 0 0
\(646\) −6.70820 −0.263931
\(647\) 23.6525i 0.929875i 0.885344 + 0.464937i \(0.153923\pi\)
−0.885344 + 0.464937i \(0.846077\pi\)
\(648\) − 7.70820i − 0.302807i
\(649\) 18.9443 0.743628
\(650\) 0 0
\(651\) −0.270510 −0.0106021
\(652\) − 1.85410i − 0.0726122i
\(653\) − 8.59675i − 0.336417i −0.985751 0.168208i \(-0.946202\pi\)
0.985751 0.168208i \(-0.0537981\pi\)
\(654\) 5.72949 0.224041
\(655\) 0 0
\(656\) 1.47214 0.0574773
\(657\) 22.0000i 0.858302i
\(658\) − 35.8328i − 1.39691i
\(659\) 25.6525 0.999279 0.499639 0.866234i \(-0.333466\pi\)
0.499639 + 0.866234i \(0.333466\pi\)
\(660\) 0 0
\(661\) −27.2705 −1.06070 −0.530350 0.847779i \(-0.677939\pi\)
−0.530350 + 0.847779i \(0.677939\pi\)
\(662\) − 15.5623i − 0.604846i
\(663\) − 0.437694i − 0.0169986i
\(664\) −4.52786 −0.175715
\(665\) 0 0
\(666\) −23.7639 −0.920834
\(667\) − 16.7082i − 0.646944i
\(668\) 13.3820i 0.517764i
\(669\) −3.43769 −0.132909
\(670\) 0 0
\(671\) −37.5066 −1.44793
\(672\) 1.14590i 0.0442040i
\(673\) 3.23607i 0.124741i 0.998053 + 0.0623706i \(0.0198661\pi\)
−0.998053 + 0.0623706i \(0.980134\pi\)
\(674\) −4.58359 −0.176553
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) − 8.38197i − 0.322145i −0.986943 0.161073i \(-0.948505\pi\)
0.986943 0.161073i \(-0.0514953\pi\)
\(678\) 1.43769i 0.0552143i
\(679\) 28.6869 1.10090
\(680\) 0 0
\(681\) −11.2148 −0.429751
\(682\) − 1.00000i − 0.0382920i
\(683\) − 11.3607i − 0.434704i −0.976093 0.217352i \(-0.930258\pi\)
0.976093 0.217352i \(-0.0697420\pi\)
\(684\) 16.7082 0.638854
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) − 0.978714i − 0.0373403i
\(688\) − 6.23607i − 0.237748i
\(689\) −10.4721 −0.398957
\(690\) 0 0
\(691\) −32.2705 −1.22763 −0.613814 0.789451i \(-0.710366\pi\)
−0.613814 + 0.789451i \(0.710366\pi\)
\(692\) 1.23607i 0.0469883i
\(693\) 36.2705i 1.37780i
\(694\) −21.0902 −0.800572
\(695\) 0 0
\(696\) −3.61803 −0.137141
\(697\) − 1.68692i − 0.0638966i
\(698\) 17.7639i 0.672375i
\(699\) 5.45898 0.206478
\(700\) 0 0
\(701\) 18.1803 0.686662 0.343331 0.939214i \(-0.388445\pi\)
0.343331 + 0.939214i \(0.388445\pi\)
\(702\) 2.23607i 0.0843949i
\(703\) − 48.7426i − 1.83836i
\(704\) −4.23607 −0.159653
\(705\) 0 0
\(706\) 21.2361 0.799230
\(707\) 1.85410i 0.0697307i
\(708\) 1.70820i 0.0641982i
\(709\) −12.2361 −0.459535 −0.229768 0.973246i \(-0.573797\pi\)
−0.229768 + 0.973246i \(0.573797\pi\)
\(710\) 0 0
\(711\) 20.6525 0.774528
\(712\) 4.47214i 0.167600i
\(713\) 0.416408i 0.0155946i
\(714\) 1.31308 0.0491409
\(715\) 0 0
\(716\) 11.1803 0.417829
\(717\) − 2.63932i − 0.0985672i
\(718\) 8.09017i 0.301922i
\(719\) −11.5066 −0.429123 −0.214561 0.976710i \(-0.568832\pi\)
−0.214561 + 0.976710i \(0.568832\pi\)
\(720\) 0 0
\(721\) 39.4377 1.46874
\(722\) 15.2705i 0.568310i
\(723\) − 5.96556i − 0.221861i
\(724\) −5.41641 −0.201299
\(725\) 0 0
\(726\) 2.65248 0.0984426
\(727\) − 2.65248i − 0.0983749i −0.998790 0.0491874i \(-0.984337\pi\)
0.998790 0.0491874i \(-0.0156632\pi\)
\(728\) 3.00000i 0.111187i
\(729\) 19.4377 0.719915
\(730\) 0 0
\(731\) −7.14590 −0.264301
\(732\) − 3.38197i − 0.125001i
\(733\) 31.7771i 1.17371i 0.809691 + 0.586857i \(0.199635\pi\)
−0.809691 + 0.586857i \(0.800365\pi\)
\(734\) 35.5410 1.31184
\(735\) 0 0
\(736\) 1.76393 0.0650194
\(737\) 43.3607i 1.59721i
\(738\) 4.20163i 0.154664i
\(739\) −30.0000 −1.10357 −0.551784 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) 0 0
\(741\) −2.23607 −0.0821440
\(742\) − 31.4164i − 1.15333i
\(743\) 15.2705i 0.560221i 0.959968 + 0.280110i \(0.0903711\pi\)
−0.959968 + 0.280110i \(0.909629\pi\)
\(744\) 0.0901699 0.00330579
\(745\) 0 0
\(746\) 11.7639 0.430708
\(747\) − 12.9230i − 0.472827i
\(748\) 4.85410i 0.177484i
\(749\) −3.27051 −0.119502
\(750\) 0 0
\(751\) 7.85410 0.286600 0.143300 0.989679i \(-0.454229\pi\)
0.143300 + 0.989679i \(0.454229\pi\)
\(752\) 11.9443i 0.435563i
\(753\) − 0.313082i − 0.0114094i
\(754\) −9.47214 −0.344955
\(755\) 0 0
\(756\) −6.70820 −0.243975
\(757\) 16.4164i 0.596664i 0.954462 + 0.298332i \(0.0964303\pi\)
−0.954462 + 0.298332i \(0.903570\pi\)
\(758\) − 9.47214i − 0.344043i
\(759\) −2.85410 −0.103597
\(760\) 0 0
\(761\) 26.9230 0.975957 0.487979 0.872856i \(-0.337734\pi\)
0.487979 + 0.872856i \(0.337734\pi\)
\(762\) 1.34752i 0.0488156i
\(763\) 45.0000i 1.62911i
\(764\) 13.6525 0.493929
\(765\) 0 0
\(766\) −11.3262 −0.409234
\(767\) 4.47214i 0.161479i
\(768\) − 0.381966i − 0.0137830i
\(769\) −3.41641 −0.123199 −0.0615994 0.998101i \(-0.519620\pi\)
−0.0615994 + 0.998101i \(0.519620\pi\)
\(770\) 0 0
\(771\) −9.83282 −0.354120
\(772\) 14.6525i 0.527354i
\(773\) 30.7984i 1.10774i 0.832603 + 0.553870i \(0.186850\pi\)
−0.832603 + 0.553870i \(0.813150\pi\)
\(774\) 17.7984 0.639749
\(775\) 0 0
\(776\) −9.56231 −0.343267
\(777\) 9.54102i 0.342282i
\(778\) − 18.2148i − 0.653032i
\(779\) −8.61803 −0.308773
\(780\) 0 0
\(781\) −12.7082 −0.454735
\(782\) − 2.02129i − 0.0722810i
\(783\) − 21.1803i − 0.756924i
\(784\) −2.00000 −0.0714286
\(785\) 0 0
\(786\) −6.27051 −0.223662
\(787\) 11.4164i 0.406951i 0.979080 + 0.203475i \(0.0652237\pi\)
−0.979080 + 0.203475i \(0.934776\pi\)
\(788\) 12.0000i 0.427482i
\(789\) 1.76393 0.0627976
\(790\) 0 0
\(791\) −11.2918 −0.401490
\(792\) − 12.0902i − 0.429605i
\(793\) − 8.85410i − 0.314418i
\(794\) −10.2361 −0.363264
\(795\) 0 0
\(796\) 2.56231 0.0908185
\(797\) − 5.61803i − 0.199001i −0.995038 0.0995005i \(-0.968276\pi\)
0.995038 0.0995005i \(-0.0317245\pi\)
\(798\) − 6.70820i − 0.237468i
\(799\) 13.6869 0.484208
\(800\) 0 0
\(801\) −12.7639 −0.450991
\(802\) − 14.1803i − 0.500725i
\(803\) 32.6525i 1.15228i
\(804\) −3.90983 −0.137889
\(805\) 0 0
\(806\) 0.236068 0.00831514
\(807\) 0.729490i 0.0256793i
\(808\) − 0.618034i − 0.0217424i
\(809\) 2.23607 0.0786160 0.0393080 0.999227i \(-0.487485\pi\)
0.0393080 + 0.999227i \(0.487485\pi\)
\(810\) 0 0
\(811\) 40.9443 1.43775 0.718874 0.695140i \(-0.244658\pi\)
0.718874 + 0.695140i \(0.244658\pi\)
\(812\) − 28.4164i − 0.997220i
\(813\) − 10.8885i − 0.381878i
\(814\) −35.2705 −1.23623
\(815\) 0 0
\(816\) −0.437694 −0.0153224
\(817\) 36.5066i 1.27720i
\(818\) 6.70820i 0.234547i
\(819\) −8.56231 −0.299191
\(820\) 0 0
\(821\) 39.1591 1.36666 0.683330 0.730109i \(-0.260531\pi\)
0.683330 + 0.730109i \(0.260531\pi\)
\(822\) − 1.94427i − 0.0678143i
\(823\) − 20.8328i − 0.726186i −0.931753 0.363093i \(-0.881721\pi\)
0.931753 0.363093i \(-0.118279\pi\)
\(824\) −13.1459 −0.457959
\(825\) 0 0
\(826\) −13.4164 −0.466817
\(827\) 5.03444i 0.175065i 0.996162 + 0.0875323i \(0.0278981\pi\)
−0.996162 + 0.0875323i \(0.972102\pi\)
\(828\) 5.03444i 0.174959i
\(829\) −53.4164 −1.85523 −0.927614 0.373540i \(-0.878144\pi\)
−0.927614 + 0.373540i \(0.878144\pi\)
\(830\) 0 0
\(831\) −0.111456 −0.00386637
\(832\) − 1.00000i − 0.0346688i
\(833\) 2.29180i 0.0794060i
\(834\) 2.03444 0.0704470
\(835\) 0 0
\(836\) 24.7984 0.857670
\(837\) 0.527864i 0.0182457i
\(838\) − 22.7639i − 0.786367i
\(839\) 55.2492 1.90742 0.953708 0.300736i \(-0.0972322\pi\)
0.953708 + 0.300736i \(0.0972322\pi\)
\(840\) 0 0
\(841\) 60.7214 2.09384
\(842\) 21.2705i 0.733030i
\(843\) − 1.21478i − 0.0418393i
\(844\) 22.2705 0.766583
\(845\) 0 0
\(846\) −34.0902 −1.17204
\(847\) 20.8328i 0.715824i
\(848\) 10.4721i 0.359615i
\(849\) −4.69505 −0.161134
\(850\) 0 0
\(851\) 14.6869 0.503461
\(852\) − 1.14590i − 0.0392578i
\(853\) − 18.1459i − 0.621304i −0.950524 0.310652i \(-0.899453\pi\)
0.950524 0.310652i \(-0.100547\pi\)
\(854\) 26.5623 0.908943
\(855\) 0 0
\(856\) 1.09017 0.0372612
\(857\) 56.3394i 1.92452i 0.272137 + 0.962259i \(0.412270\pi\)
−0.272137 + 0.962259i \(0.587730\pi\)
\(858\) 1.61803i 0.0552388i
\(859\) −34.3951 −1.17355 −0.586773 0.809751i \(-0.699602\pi\)
−0.586773 + 0.809751i \(0.699602\pi\)
\(860\) 0 0
\(861\) 1.68692 0.0574900
\(862\) 26.0689i 0.887910i
\(863\) − 11.2361i − 0.382480i −0.981543 0.191240i \(-0.938749\pi\)
0.981543 0.191240i \(-0.0612509\pi\)
\(864\) 2.23607 0.0760726
\(865\) 0 0
\(866\) 21.3607 0.725865
\(867\) − 5.99187i − 0.203495i
\(868\) 0.708204i 0.0240380i
\(869\) 30.6525 1.03981
\(870\) 0 0
\(871\) −10.2361 −0.346836
\(872\) − 15.0000i − 0.507964i
\(873\) − 27.2918i − 0.923687i
\(874\) −10.3262 −0.349290
\(875\) 0 0
\(876\) −2.94427 −0.0994777
\(877\) − 4.43769i − 0.149850i −0.997189 0.0749251i \(-0.976128\pi\)
0.997189 0.0749251i \(-0.0238718\pi\)
\(878\) − 25.6525i − 0.865729i
\(879\) 3.27051 0.110312
\(880\) 0 0
\(881\) 38.1803 1.28633 0.643164 0.765728i \(-0.277621\pi\)
0.643164 + 0.765728i \(0.277621\pi\)
\(882\) − 5.70820i − 0.192205i
\(883\) − 41.6869i − 1.40288i −0.712730 0.701438i \(-0.752542\pi\)
0.712730 0.701438i \(-0.247458\pi\)
\(884\) −1.14590 −0.0385407
\(885\) 0 0
\(886\) −19.4164 −0.652307
\(887\) − 33.1803i − 1.11409i −0.830483 0.557043i \(-0.811936\pi\)
0.830483 0.557043i \(-0.188064\pi\)
\(888\) − 3.18034i − 0.106725i
\(889\) −10.5836 −0.354962
\(890\) 0 0
\(891\) 32.6525 1.09390
\(892\) 9.00000i 0.301342i
\(893\) − 69.9230i − 2.33988i
\(894\) −0.854102 −0.0285654
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) − 0.673762i − 0.0224963i
\(898\) 3.94427i 0.131622i
\(899\) −2.23607 −0.0745770
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 6.23607i 0.207638i
\(903\) − 7.14590i − 0.237801i
\(904\) 3.76393 0.125187
\(905\) 0 0
\(906\) −3.70820 −0.123197
\(907\) − 25.6180i − 0.850633i −0.905045 0.425316i \(-0.860163\pi\)
0.905045 0.425316i \(-0.139837\pi\)
\(908\) 29.3607i 0.974368i
\(909\) 1.76393 0.0585059
\(910\) 0 0
\(911\) −16.4164 −0.543900 −0.271950 0.962311i \(-0.587669\pi\)
−0.271950 + 0.962311i \(0.587669\pi\)
\(912\) 2.23607i 0.0740436i
\(913\) − 19.1803i − 0.634777i
\(914\) 40.2148 1.33019
\(915\) 0 0
\(916\) −2.56231 −0.0846610
\(917\) − 49.2492i − 1.62635i
\(918\) − 2.56231i − 0.0845687i
\(919\) 5.65248 0.186458 0.0932290 0.995645i \(-0.470281\pi\)
0.0932290 + 0.995645i \(0.470281\pi\)
\(920\) 0 0
\(921\) 8.83282 0.291051
\(922\) 5.81966i 0.191660i
\(923\) − 3.00000i − 0.0987462i
\(924\) −4.85410 −0.159688
\(925\) 0 0
\(926\) 34.9787 1.14947
\(927\) − 37.5197i − 1.23231i
\(928\) 9.47214i 0.310938i
\(929\) 52.8885 1.73522 0.867608 0.497248i \(-0.165656\pi\)
0.867608 + 0.497248i \(0.165656\pi\)
\(930\) 0 0
\(931\) 11.7082 0.383721
\(932\) − 14.2918i − 0.468143i
\(933\) 3.78522i 0.123922i
\(934\) −19.1803 −0.627600
\(935\) 0 0
\(936\) 2.85410 0.0932892
\(937\) − 41.9230i − 1.36956i −0.728748 0.684782i \(-0.759897\pi\)
0.728748 0.684782i \(-0.240103\pi\)
\(938\) − 30.7082i − 1.00266i
\(939\) −6.32624 −0.206449
\(940\) 0 0
\(941\) −9.38197 −0.305843 −0.152922 0.988238i \(-0.548868\pi\)
−0.152922 + 0.988238i \(0.548868\pi\)
\(942\) − 1.74265i − 0.0567785i
\(943\) − 2.59675i − 0.0845617i
\(944\) 4.47214 0.145556
\(945\) 0 0
\(946\) 26.4164 0.858872
\(947\) 39.3820i 1.27974i 0.768482 + 0.639871i \(0.221012\pi\)
−0.768482 + 0.639871i \(0.778988\pi\)
\(948\) 2.76393i 0.0897683i
\(949\) −7.70820 −0.250219
\(950\) 0 0
\(951\) −3.20163 −0.103820
\(952\) − 3.43769i − 0.111416i
\(953\) 44.6180i 1.44532i 0.691204 + 0.722660i \(0.257081\pi\)
−0.691204 + 0.722660i \(0.742919\pi\)
\(954\) −29.8885 −0.967677
\(955\) 0 0
\(956\) −6.90983 −0.223480
\(957\) − 15.3262i − 0.495427i
\(958\) 15.2016i 0.491142i
\(959\) 15.2705 0.493110
\(960\) 0 0
\(961\) −30.9443 −0.998202
\(962\) − 8.32624i − 0.268449i
\(963\) 3.11146i 0.100265i
\(964\) −15.6180 −0.503023
\(965\) 0 0
\(966\) 2.02129 0.0650338
\(967\) 13.1246i 0.422059i 0.977480 + 0.211030i \(0.0676816\pi\)
−0.977480 + 0.211030i \(0.932318\pi\)
\(968\) − 6.94427i − 0.223197i
\(969\) 2.56231 0.0823131
\(970\) 0 0
\(971\) 6.06888 0.194760 0.0973799 0.995247i \(-0.468954\pi\)
0.0973799 + 0.995247i \(0.468954\pi\)
\(972\) 9.65248i 0.309603i
\(973\) 15.9787i 0.512254i
\(974\) 22.1246 0.708918
\(975\) 0 0
\(976\) −8.85410 −0.283413
\(977\) − 50.0132i − 1.60006i −0.599958 0.800031i \(-0.704816\pi\)
0.599958 0.800031i \(-0.295184\pi\)
\(978\) 0.708204i 0.0226459i
\(979\) −18.9443 −0.605462
\(980\) 0 0
\(981\) 42.8115 1.36687
\(982\) 19.2361i 0.613848i
\(983\) 38.4853i 1.22749i 0.789504 + 0.613745i \(0.210338\pi\)
−0.789504 + 0.613745i \(0.789662\pi\)
\(984\) −0.562306 −0.0179257
\(985\) 0 0
\(986\) 10.8541 0.345665
\(987\) 13.6869i 0.435659i
\(988\) 5.85410i 0.186244i
\(989\) −11.0000 −0.349780
\(990\) 0 0
\(991\) −3.45085 −0.109620 −0.0548099 0.998497i \(-0.517455\pi\)
−0.0548099 + 0.998497i \(0.517455\pi\)
\(992\) − 0.236068i − 0.00749517i
\(993\) 5.94427i 0.188636i
\(994\) 9.00000 0.285463
\(995\) 0 0
\(996\) 1.72949 0.0548010
\(997\) − 27.2492i − 0.862992i −0.902115 0.431496i \(-0.857986\pi\)
0.902115 0.431496i \(-0.142014\pi\)
\(998\) − 34.1459i − 1.08087i
\(999\) 18.6180 0.589049
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 1250.2.b.b.1249.4 4
5.2 odd 4 1250.2.a.a.1.2 2
5.3 odd 4 1250.2.a.d.1.1 2
5.4 even 2 inner 1250.2.b.b.1249.1 4
20.3 even 4 10000.2.a.a.1.2 2
20.7 even 4 10000.2.a.n.1.1 2
25.2 odd 20 50.2.d.a.21.1 4
25.9 even 10 250.2.e.b.99.2 8
25.11 even 5 250.2.e.b.149.2 8
25.12 odd 20 50.2.d.a.31.1 yes 4
25.13 odd 20 250.2.d.a.151.1 4
25.14 even 10 250.2.e.b.149.1 8
25.16 even 5 250.2.e.b.99.1 8
25.23 odd 20 250.2.d.a.101.1 4
75.2 even 20 450.2.h.a.271.1 4
75.62 even 20 450.2.h.a.181.1 4
100.27 even 20 400.2.u.c.321.1 4
100.87 even 20 400.2.u.c.81.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.2.d.a.21.1 4 25.2 odd 20
50.2.d.a.31.1 yes 4 25.12 odd 20
250.2.d.a.101.1 4 25.23 odd 20
250.2.d.a.151.1 4 25.13 odd 20
250.2.e.b.99.1 8 25.16 even 5
250.2.e.b.99.2 8 25.9 even 10
250.2.e.b.149.1 8 25.14 even 10
250.2.e.b.149.2 8 25.11 even 5
400.2.u.c.81.1 4 100.87 even 20
400.2.u.c.321.1 4 100.27 even 20
450.2.h.a.181.1 4 75.62 even 20
450.2.h.a.271.1 4 75.2 even 20
1250.2.a.a.1.2 2 5.2 odd 4
1250.2.a.d.1.1 2 5.3 odd 4
1250.2.b.b.1249.1 4 5.4 even 2 inner
1250.2.b.b.1249.4 4 1.1 even 1 trivial
10000.2.a.a.1.2 2 20.3 even 4
10000.2.a.n.1.1 2 20.7 even 4