Properties

Label 1470.3.f.d.391.2
Level $1470$
Weight $3$
Character 1470.391
Analytic conductor $40.055$
Analytic rank $0$
Dimension $16$
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] = [1470,3,Mod(391,1470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1470, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1470.391");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1470.f (of order \(2\), degree \(1\), 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: \(40.0545988610\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 92 x^{14} - 112 x^{13} + 5846 x^{12} - 7728 x^{11} + 197216 x^{10} - 298200 x^{9} + 4836403 x^{8} - 6808704 x^{7} + 64376800 x^{6} - 91953512 x^{5} + \cdots + 101626561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 391.2
Root \(0.848921 + 1.47037i\) of defining polynomial
Character \(\chi\) \(=\) 1470.391
Dual form 1470.3.f.d.391.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949i q^{6} -2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949i q^{6} -2.82843 q^{8} -3.00000 q^{9} +3.16228i q^{10} +19.9750 q^{11} -3.46410i q^{12} -3.49788i q^{13} -3.87298 q^{15} +4.00000 q^{16} +18.2422i q^{17} +4.24264 q^{18} +24.6116i q^{19} -4.47214i q^{20} -28.2489 q^{22} +25.1904 q^{23} +4.89898i q^{24} -5.00000 q^{25} +4.94675i q^{26} +5.19615i q^{27} -53.1223 q^{29} +5.47723 q^{30} +30.1312i q^{31} -5.65685 q^{32} -34.5977i q^{33} -25.7984i q^{34} -6.00000 q^{36} -46.7693 q^{37} -34.8060i q^{38} -6.05851 q^{39} +6.32456i q^{40} +31.5250i q^{41} +64.4116 q^{43} +39.9499 q^{44} +6.70820i q^{45} -35.6246 q^{46} +28.0626i q^{47} -6.92820i q^{48} +7.07107 q^{50} +31.5965 q^{51} -6.99576i q^{52} -64.8748 q^{53} -7.34847i q^{54} -44.6654i q^{55} +42.6285 q^{57} +75.1263 q^{58} +100.192i q^{59} -7.74597 q^{60} -8.02397i q^{61} -42.6119i q^{62} +8.00000 q^{64} -7.82150 q^{65} +48.9285i q^{66} +16.2633 q^{67} +36.4845i q^{68} -43.6310i q^{69} -107.725 q^{71} +8.48528 q^{72} +51.6607i q^{73} +66.1417 q^{74} +8.66025i q^{75} +49.2232i q^{76} +8.56803 q^{78} -21.9755 q^{79} -8.94427i q^{80} +9.00000 q^{81} -44.5831i q^{82} -0.417479i q^{83} +40.7909 q^{85} -91.0918 q^{86} +92.0105i q^{87} -56.4978 q^{88} +111.210i q^{89} -9.48683i q^{90} +50.3807 q^{92} +52.1887 q^{93} -39.6865i q^{94} +55.0332 q^{95} +9.79796i q^{96} -74.2244i q^{97} -59.9249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} - 48 q^{9} + 8 q^{11} + 64 q^{16} - 48 q^{22} + 24 q^{23} - 80 q^{25} + 72 q^{29} - 96 q^{36} - 88 q^{37} - 72 q^{39} - 56 q^{43} + 16 q^{44} - 16 q^{46} + 24 q^{51} - 64 q^{53} + 144 q^{57} + 176 q^{58} + 128 q^{64} - 40 q^{65} + 328 q^{67} - 136 q^{71} + 224 q^{74} - 560 q^{79} + 144 q^{81} + 120 q^{85} + 176 q^{86} - 96 q^{88} + 48 q^{92} + 240 q^{93} - 400 q^{95} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −0.707107
\(3\) − 1.73205i − 0.577350i
\(4\) 2.00000 0.500000
\(5\) − 2.23607i − 0.447214i
\(6\) 2.44949i 0.408248i
\(7\) 0 0
\(8\) −2.82843 −0.353553
\(9\) −3.00000 −0.333333
\(10\) 3.16228i 0.316228i
\(11\) 19.9750 1.81591 0.907953 0.419071i \(-0.137644\pi\)
0.907953 + 0.419071i \(0.137644\pi\)
\(12\) − 3.46410i − 0.288675i
\(13\) − 3.49788i − 0.269068i −0.990909 0.134534i \(-0.957046\pi\)
0.990909 0.134534i \(-0.0429537\pi\)
\(14\) 0 0
\(15\) −3.87298 −0.258199
\(16\) 4.00000 0.250000
\(17\) 18.2422i 1.07307i 0.843877 + 0.536536i \(0.180268\pi\)
−0.843877 + 0.536536i \(0.819732\pi\)
\(18\) 4.24264 0.235702
\(19\) 24.6116i 1.29535i 0.761918 + 0.647673i \(0.224258\pi\)
−0.761918 + 0.647673i \(0.775742\pi\)
\(20\) − 4.47214i − 0.223607i
\(21\) 0 0
\(22\) −28.2489 −1.28404
\(23\) 25.1904 1.09523 0.547617 0.836729i \(-0.315535\pi\)
0.547617 + 0.836729i \(0.315535\pi\)
\(24\) 4.89898i 0.204124i
\(25\) −5.00000 −0.200000
\(26\) 4.94675i 0.190260i
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) −53.1223 −1.83180 −0.915902 0.401402i \(-0.868523\pi\)
−0.915902 + 0.401402i \(0.868523\pi\)
\(30\) 5.47723 0.182574
\(31\) 30.1312i 0.971974i 0.873966 + 0.485987i \(0.161540\pi\)
−0.873966 + 0.485987i \(0.838460\pi\)
\(32\) −5.65685 −0.176777
\(33\) − 34.5977i − 1.04841i
\(34\) − 25.7984i − 0.758777i
\(35\) 0 0
\(36\) −6.00000 −0.166667
\(37\) −46.7693 −1.26403 −0.632017 0.774954i \(-0.717773\pi\)
−0.632017 + 0.774954i \(0.717773\pi\)
\(38\) − 34.8060i − 0.915948i
\(39\) −6.05851 −0.155346
\(40\) 6.32456i 0.158114i
\(41\) 31.5250i 0.768903i 0.923145 + 0.384452i \(0.125609\pi\)
−0.923145 + 0.384452i \(0.874391\pi\)
\(42\) 0 0
\(43\) 64.4116 1.49794 0.748972 0.662602i \(-0.230548\pi\)
0.748972 + 0.662602i \(0.230548\pi\)
\(44\) 39.9499 0.907953
\(45\) 6.70820i 0.149071i
\(46\) −35.6246 −0.774447
\(47\) 28.0626i 0.597076i 0.954398 + 0.298538i \(0.0964990\pi\)
−0.954398 + 0.298538i \(0.903501\pi\)
\(48\) − 6.92820i − 0.144338i
\(49\) 0 0
\(50\) 7.07107 0.141421
\(51\) 31.5965 0.619539
\(52\) − 6.99576i − 0.134534i
\(53\) −64.8748 −1.22405 −0.612027 0.790837i \(-0.709646\pi\)
−0.612027 + 0.790837i \(0.709646\pi\)
\(54\) − 7.34847i − 0.136083i
\(55\) − 44.6654i − 0.812098i
\(56\) 0 0
\(57\) 42.6285 0.747869
\(58\) 75.1263 1.29528
\(59\) 100.192i 1.69816i 0.528263 + 0.849081i \(0.322844\pi\)
−0.528263 + 0.849081i \(0.677156\pi\)
\(60\) −7.74597 −0.129099
\(61\) − 8.02397i − 0.131540i −0.997835 0.0657702i \(-0.979050\pi\)
0.997835 0.0657702i \(-0.0209504\pi\)
\(62\) − 42.6119i − 0.687289i
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) −7.82150 −0.120331
\(66\) 48.9285i 0.741341i
\(67\) 16.2633 0.242736 0.121368 0.992608i \(-0.461272\pi\)
0.121368 + 0.992608i \(0.461272\pi\)
\(68\) 36.4845i 0.536536i
\(69\) − 43.6310i − 0.632333i
\(70\) 0 0
\(71\) −107.725 −1.51725 −0.758625 0.651528i \(-0.774128\pi\)
−0.758625 + 0.651528i \(0.774128\pi\)
\(72\) 8.48528 0.117851
\(73\) 51.6607i 0.707681i 0.935306 + 0.353840i \(0.115124\pi\)
−0.935306 + 0.353840i \(0.884876\pi\)
\(74\) 66.1417 0.893807
\(75\) 8.66025i 0.115470i
\(76\) 49.2232i 0.647673i
\(77\) 0 0
\(78\) 8.56803 0.109846
\(79\) −21.9755 −0.278170 −0.139085 0.990280i \(-0.544416\pi\)
−0.139085 + 0.990280i \(0.544416\pi\)
\(80\) − 8.94427i − 0.111803i
\(81\) 9.00000 0.111111
\(82\) − 44.5831i − 0.543697i
\(83\) − 0.417479i − 0.00502987i −0.999997 0.00251494i \(-0.999199\pi\)
0.999997 0.00251494i \(-0.000800530\pi\)
\(84\) 0 0
\(85\) 40.7909 0.479893
\(86\) −91.0918 −1.05921
\(87\) 92.0105i 1.05759i
\(88\) −56.4978 −0.642020
\(89\) 111.210i 1.24956i 0.780803 + 0.624778i \(0.214810\pi\)
−0.780803 + 0.624778i \(0.785190\pi\)
\(90\) − 9.48683i − 0.105409i
\(91\) 0 0
\(92\) 50.3807 0.547617
\(93\) 52.1887 0.561169
\(94\) − 39.6865i − 0.422197i
\(95\) 55.0332 0.579297
\(96\) 9.79796i 0.102062i
\(97\) − 74.2244i − 0.765200i −0.923914 0.382600i \(-0.875029\pi\)
0.923914 0.382600i \(-0.124971\pi\)
\(98\) 0 0
\(99\) −59.9249 −0.605302
\(100\) −10.0000 −0.100000
\(101\) 87.5941i 0.867268i 0.901089 + 0.433634i \(0.142769\pi\)
−0.901089 + 0.433634i \(0.857231\pi\)
\(102\) −44.6842 −0.438080
\(103\) − 139.041i − 1.34991i −0.737857 0.674957i \(-0.764162\pi\)
0.737857 0.674957i \(-0.235838\pi\)
\(104\) 9.89350i 0.0951299i
\(105\) 0 0
\(106\) 91.7469 0.865537
\(107\) 152.190 1.42234 0.711168 0.703023i \(-0.248167\pi\)
0.711168 + 0.703023i \(0.248167\pi\)
\(108\) 10.3923i 0.0962250i
\(109\) 64.7554 0.594086 0.297043 0.954864i \(-0.404000\pi\)
0.297043 + 0.954864i \(0.404000\pi\)
\(110\) 63.1664i 0.574240i
\(111\) 81.0068i 0.729791i
\(112\) 0 0
\(113\) 3.25860 0.0288372 0.0144186 0.999896i \(-0.495410\pi\)
0.0144186 + 0.999896i \(0.495410\pi\)
\(114\) −60.2858 −0.528823
\(115\) − 56.3274i − 0.489803i
\(116\) −106.245 −0.915902
\(117\) 10.4936i 0.0896893i
\(118\) − 141.692i − 1.20078i
\(119\) 0 0
\(120\) 10.9545 0.0912871
\(121\) 278.000 2.29752
\(122\) 11.3476i 0.0930131i
\(123\) 54.6029 0.443926
\(124\) 60.2624i 0.485987i
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −88.5772 −0.697458 −0.348729 0.937224i \(-0.613387\pi\)
−0.348729 + 0.937224i \(0.613387\pi\)
\(128\) −11.3137 −0.0883883
\(129\) − 111.564i − 0.864839i
\(130\) 11.0613 0.0850867
\(131\) 125.124i 0.955146i 0.878592 + 0.477573i \(0.158483\pi\)
−0.878592 + 0.477573i \(0.841517\pi\)
\(132\) − 69.1953i − 0.524207i
\(133\) 0 0
\(134\) −22.9998 −0.171640
\(135\) 11.6190 0.0860663
\(136\) − 51.5968i − 0.379388i
\(137\) −38.9538 −0.284334 −0.142167 0.989843i \(-0.545407\pi\)
−0.142167 + 0.989843i \(0.545407\pi\)
\(138\) 61.7035i 0.447127i
\(139\) − 98.9454i − 0.711837i −0.934517 0.355919i \(-0.884168\pi\)
0.934517 0.355919i \(-0.115832\pi\)
\(140\) 0 0
\(141\) 48.6058 0.344722
\(142\) 152.346 1.07286
\(143\) − 69.8701i − 0.488602i
\(144\) −12.0000 −0.0833333
\(145\) 118.785i 0.819208i
\(146\) − 73.0593i − 0.500406i
\(147\) 0 0
\(148\) −93.5385 −0.632017
\(149\) −186.465 −1.25144 −0.625721 0.780047i \(-0.715195\pi\)
−0.625721 + 0.780047i \(0.715195\pi\)
\(150\) − 12.2474i − 0.0816497i
\(151\) 154.840 1.02543 0.512716 0.858558i \(-0.328639\pi\)
0.512716 + 0.858558i \(0.328639\pi\)
\(152\) − 69.6121i − 0.457974i
\(153\) − 54.7267i − 0.357691i
\(154\) 0 0
\(155\) 67.3754 0.434680
\(156\) −12.1170 −0.0776732
\(157\) − 50.2037i − 0.319769i −0.987136 0.159885i \(-0.948888\pi\)
0.987136 0.159885i \(-0.0511122\pi\)
\(158\) 31.0780 0.196696
\(159\) 112.367i 0.706708i
\(160\) 12.6491i 0.0790569i
\(161\) 0 0
\(162\) −12.7279 −0.0785674
\(163\) −115.811 −0.710496 −0.355248 0.934772i \(-0.615604\pi\)
−0.355248 + 0.934772i \(0.615604\pi\)
\(164\) 63.0500i 0.384452i
\(165\) −77.3627 −0.468865
\(166\) 0.590405i 0.00355666i
\(167\) 61.3210i 0.367191i 0.983002 + 0.183596i \(0.0587737\pi\)
−0.983002 + 0.183596i \(0.941226\pi\)
\(168\) 0 0
\(169\) 156.765 0.927602
\(170\) −57.6870 −0.339335
\(171\) − 73.8347i − 0.431782i
\(172\) 128.823 0.748972
\(173\) − 31.7016i − 0.183246i −0.995794 0.0916232i \(-0.970794\pi\)
0.995794 0.0916232i \(-0.0292055\pi\)
\(174\) − 130.123i − 0.747831i
\(175\) 0 0
\(176\) 79.8999 0.453977
\(177\) 173.537 0.980434
\(178\) − 157.275i − 0.883569i
\(179\) 131.894 0.736840 0.368420 0.929659i \(-0.379899\pi\)
0.368420 + 0.929659i \(0.379899\pi\)
\(180\) 13.4164i 0.0745356i
\(181\) − 55.1431i − 0.304658i −0.988330 0.152329i \(-0.951323\pi\)
0.988330 0.152329i \(-0.0486773\pi\)
\(182\) 0 0
\(183\) −13.8979 −0.0759449
\(184\) −71.2491 −0.387223
\(185\) 104.579i 0.565293i
\(186\) −73.8060 −0.396807
\(187\) 364.388i 1.94860i
\(188\) 56.1252i 0.298538i
\(189\) 0 0
\(190\) −77.8287 −0.409625
\(191\) 195.164 1.02180 0.510901 0.859639i \(-0.329312\pi\)
0.510901 + 0.859639i \(0.329312\pi\)
\(192\) − 13.8564i − 0.0721688i
\(193\) 348.760 1.80705 0.903523 0.428540i \(-0.140972\pi\)
0.903523 + 0.428540i \(0.140972\pi\)
\(194\) 104.969i 0.541078i
\(195\) 13.5472i 0.0694730i
\(196\) 0 0
\(197\) −56.3808 −0.286197 −0.143098 0.989708i \(-0.545707\pi\)
−0.143098 + 0.989708i \(0.545707\pi\)
\(198\) 84.7466 0.428013
\(199\) 171.308i 0.860845i 0.902628 + 0.430422i \(0.141635\pi\)
−0.902628 + 0.430422i \(0.858365\pi\)
\(200\) 14.1421 0.0707107
\(201\) − 28.1689i − 0.140144i
\(202\) − 123.877i − 0.613251i
\(203\) 0 0
\(204\) 63.1930 0.309769
\(205\) 70.4921 0.343864
\(206\) 196.634i 0.954533i
\(207\) −75.5711 −0.365078
\(208\) − 13.9915i − 0.0672670i
\(209\) 491.616i 2.35223i
\(210\) 0 0
\(211\) 162.038 0.767954 0.383977 0.923343i \(-0.374554\pi\)
0.383977 + 0.923343i \(0.374554\pi\)
\(212\) −129.750 −0.612027
\(213\) 186.585i 0.875984i
\(214\) −215.229 −1.00574
\(215\) − 144.029i − 0.669901i
\(216\) − 14.6969i − 0.0680414i
\(217\) 0 0
\(218\) −91.5779 −0.420082
\(219\) 89.4789 0.408580
\(220\) − 89.3308i − 0.406049i
\(221\) 63.8092 0.288729
\(222\) − 114.561i − 0.516040i
\(223\) 365.329i 1.63825i 0.573618 + 0.819123i \(0.305539\pi\)
−0.573618 + 0.819123i \(0.694461\pi\)
\(224\) 0 0
\(225\) 15.0000 0.0666667
\(226\) −4.60836 −0.0203910
\(227\) 257.874i 1.13601i 0.823026 + 0.568004i \(0.192284\pi\)
−0.823026 + 0.568004i \(0.807716\pi\)
\(228\) 85.2570 0.373934
\(229\) 15.4308i 0.0673832i 0.999432 + 0.0336916i \(0.0107264\pi\)
−0.999432 + 0.0336916i \(0.989274\pi\)
\(230\) 79.6589i 0.346343i
\(231\) 0 0
\(232\) 150.253 0.647640
\(233\) −125.347 −0.537969 −0.268984 0.963145i \(-0.586688\pi\)
−0.268984 + 0.963145i \(0.586688\pi\)
\(234\) − 14.8403i − 0.0634199i
\(235\) 62.7499 0.267021
\(236\) 200.383i 0.849081i
\(237\) 38.0626i 0.160602i
\(238\) 0 0
\(239\) −3.62565 −0.0151701 −0.00758503 0.999971i \(-0.502414\pi\)
−0.00758503 + 0.999971i \(0.502414\pi\)
\(240\) −15.4919 −0.0645497
\(241\) 96.6387i 0.400990i 0.979695 + 0.200495i \(0.0642551\pi\)
−0.979695 + 0.200495i \(0.935745\pi\)
\(242\) −393.151 −1.62459
\(243\) − 15.5885i − 0.0641500i
\(244\) − 16.0479i − 0.0657702i
\(245\) 0 0
\(246\) −77.2202 −0.313903
\(247\) 86.0884 0.348536
\(248\) − 85.2239i − 0.343645i
\(249\) −0.723095 −0.00290400
\(250\) − 15.8114i − 0.0632456i
\(251\) 29.7311i 0.118450i 0.998245 + 0.0592252i \(0.0188630\pi\)
−0.998245 + 0.0592252i \(0.981137\pi\)
\(252\) 0 0
\(253\) 503.177 1.98884
\(254\) 125.267 0.493178
\(255\) − 70.6519i − 0.277066i
\(256\) 16.0000 0.0625000
\(257\) − 441.623i − 1.71838i −0.511659 0.859189i \(-0.670969\pi\)
0.511659 0.859189i \(-0.329031\pi\)
\(258\) 157.776i 0.611533i
\(259\) 0 0
\(260\) −15.6430 −0.0601654
\(261\) 159.367 0.610601
\(262\) − 176.952i − 0.675390i
\(263\) 343.047 1.30436 0.652180 0.758064i \(-0.273855\pi\)
0.652180 + 0.758064i \(0.273855\pi\)
\(264\) 97.8570i 0.370670i
\(265\) 145.065i 0.547413i
\(266\) 0 0
\(267\) 192.622 0.721431
\(268\) 32.5266 0.121368
\(269\) 463.351i 1.72249i 0.508186 + 0.861247i \(0.330316\pi\)
−0.508186 + 0.861247i \(0.669684\pi\)
\(270\) −16.4317 −0.0608581
\(271\) − 244.581i − 0.902515i −0.892394 0.451257i \(-0.850976\pi\)
0.892394 0.451257i \(-0.149024\pi\)
\(272\) 72.9689i 0.268268i
\(273\) 0 0
\(274\) 55.0889 0.201054
\(275\) −99.8749 −0.363181
\(276\) − 87.2620i − 0.316167i
\(277\) 141.018 0.509092 0.254546 0.967061i \(-0.418074\pi\)
0.254546 + 0.967061i \(0.418074\pi\)
\(278\) 139.930i 0.503345i
\(279\) − 90.3935i − 0.323991i
\(280\) 0 0
\(281\) 84.9953 0.302475 0.151237 0.988497i \(-0.451674\pi\)
0.151237 + 0.988497i \(0.451674\pi\)
\(282\) −68.7390 −0.243755
\(283\) − 116.678i − 0.412291i −0.978521 0.206146i \(-0.933908\pi\)
0.978521 0.206146i \(-0.0660921\pi\)
\(284\) −215.449 −0.758625
\(285\) − 95.3202i − 0.334457i
\(286\) 98.8112i 0.345494i
\(287\) 0 0
\(288\) 16.9706 0.0589256
\(289\) −43.7791 −0.151485
\(290\) − 167.987i − 0.579267i
\(291\) −128.560 −0.441789
\(292\) 103.321i 0.353840i
\(293\) − 131.882i − 0.450110i −0.974346 0.225055i \(-0.927744\pi\)
0.974346 0.225055i \(-0.0722562\pi\)
\(294\) 0 0
\(295\) 224.035 0.759441
\(296\) 132.283 0.446904
\(297\) 103.793i 0.349471i
\(298\) 263.701 0.884903
\(299\) − 88.1129i − 0.294692i
\(300\) 17.3205i 0.0577350i
\(301\) 0 0
\(302\) −218.977 −0.725090
\(303\) 151.717 0.500717
\(304\) 98.4463i 0.323837i
\(305\) −17.9421 −0.0588267
\(306\) 77.3952i 0.252926i
\(307\) − 429.871i − 1.40023i −0.714030 0.700115i \(-0.753132\pi\)
0.714030 0.700115i \(-0.246868\pi\)
\(308\) 0 0
\(309\) −240.826 −0.779373
\(310\) −95.2832 −0.307365
\(311\) − 251.478i − 0.808610i −0.914624 0.404305i \(-0.867513\pi\)
0.914624 0.404305i \(-0.132487\pi\)
\(312\) 17.1361 0.0549232
\(313\) − 13.7033i − 0.0437806i −0.999760 0.0218903i \(-0.993032\pi\)
0.999760 0.0218903i \(-0.00696846\pi\)
\(314\) 70.9988i 0.226111i
\(315\) 0 0
\(316\) −43.9509 −0.139085
\(317\) −32.9301 −0.103881 −0.0519403 0.998650i \(-0.516541\pi\)
−0.0519403 + 0.998650i \(0.516541\pi\)
\(318\) − 158.910i − 0.499718i
\(319\) −1061.12 −3.32638
\(320\) − 17.8885i − 0.0559017i
\(321\) − 263.601i − 0.821186i
\(322\) 0 0
\(323\) −448.970 −1.39000
\(324\) 18.0000 0.0555556
\(325\) 17.4894i 0.0538136i
\(326\) 163.781 0.502397
\(327\) − 112.160i − 0.342996i
\(328\) − 89.1662i − 0.271848i
\(329\) 0 0
\(330\) 109.407 0.331538
\(331\) −417.881 −1.26248 −0.631240 0.775588i \(-0.717454\pi\)
−0.631240 + 0.775588i \(0.717454\pi\)
\(332\) − 0.834959i − 0.00251494i
\(333\) 140.308 0.421345
\(334\) − 86.7209i − 0.259643i
\(335\) − 36.3658i − 0.108555i
\(336\) 0 0
\(337\) −286.688 −0.850705 −0.425353 0.905028i \(-0.639850\pi\)
−0.425353 + 0.905028i \(0.639850\pi\)
\(338\) −221.699 −0.655914
\(339\) − 5.64407i − 0.0166492i
\(340\) 81.5818 0.239946
\(341\) 601.870i 1.76501i
\(342\) 104.418i 0.305316i
\(343\) 0 0
\(344\) −182.184 −0.529603
\(345\) −97.5619 −0.282788
\(346\) 44.8329i 0.129575i
\(347\) 306.796 0.884137 0.442069 0.896981i \(-0.354245\pi\)
0.442069 + 0.896981i \(0.354245\pi\)
\(348\) 184.021i 0.528796i
\(349\) 340.162i 0.974676i 0.873214 + 0.487338i \(0.162032\pi\)
−0.873214 + 0.487338i \(0.837968\pi\)
\(350\) 0 0
\(351\) 18.1755 0.0517821
\(352\) −112.996 −0.321010
\(353\) − 375.002i − 1.06233i −0.847269 0.531165i \(-0.821755\pi\)
0.847269 0.531165i \(-0.178245\pi\)
\(354\) −245.418 −0.693272
\(355\) 240.880i 0.678534i
\(356\) 222.421i 0.624778i
\(357\) 0 0
\(358\) −186.527 −0.521025
\(359\) −329.501 −0.917829 −0.458915 0.888480i \(-0.651762\pi\)
−0.458915 + 0.888480i \(0.651762\pi\)
\(360\) − 18.9737i − 0.0527046i
\(361\) −244.730 −0.677922
\(362\) 77.9841i 0.215426i
\(363\) − 481.509i − 1.32647i
\(364\) 0 0
\(365\) 115.517 0.316484
\(366\) 19.6546 0.0537012
\(367\) 27.1543i 0.0739900i 0.999315 + 0.0369950i \(0.0117786\pi\)
−0.999315 + 0.0369950i \(0.988221\pi\)
\(368\) 100.761 0.273808
\(369\) − 94.5751i − 0.256301i
\(370\) − 147.897i − 0.399723i
\(371\) 0 0
\(372\) 104.377 0.280585
\(373\) 127.498 0.341817 0.170908 0.985287i \(-0.445330\pi\)
0.170908 + 0.985287i \(0.445330\pi\)
\(374\) − 515.323i − 1.37787i
\(375\) 19.3649 0.0516398
\(376\) − 79.3730i − 0.211098i
\(377\) 185.816i 0.492880i
\(378\) 0 0
\(379\) 319.795 0.843785 0.421893 0.906646i \(-0.361366\pi\)
0.421893 + 0.906646i \(0.361366\pi\)
\(380\) 110.066 0.289648
\(381\) 153.420i 0.402678i
\(382\) −276.004 −0.722524
\(383\) − 408.942i − 1.06773i −0.845568 0.533867i \(-0.820738\pi\)
0.845568 0.533867i \(-0.179262\pi\)
\(384\) 19.5959i 0.0510310i
\(385\) 0 0
\(386\) −493.221 −1.27777
\(387\) −193.235 −0.499315
\(388\) − 148.449i − 0.382600i
\(389\) 387.153 0.995252 0.497626 0.867392i \(-0.334205\pi\)
0.497626 + 0.867392i \(0.334205\pi\)
\(390\) − 19.1587i − 0.0491248i
\(391\) 459.529i 1.17526i
\(392\) 0 0
\(393\) 216.721 0.551454
\(394\) 79.7345 0.202372
\(395\) 49.1386i 0.124402i
\(396\) −119.850 −0.302651
\(397\) − 30.1112i − 0.0758468i −0.999281 0.0379234i \(-0.987926\pi\)
0.999281 0.0379234i \(-0.0120743\pi\)
\(398\) − 242.266i − 0.608709i
\(399\) 0 0
\(400\) −20.0000 −0.0500000
\(401\) 136.770 0.341073 0.170537 0.985351i \(-0.445450\pi\)
0.170537 + 0.985351i \(0.445450\pi\)
\(402\) 39.8368i 0.0990965i
\(403\) 105.395 0.261527
\(404\) 175.188i 0.433634i
\(405\) − 20.1246i − 0.0496904i
\(406\) 0 0
\(407\) −934.215 −2.29537
\(408\) −89.3683 −0.219040
\(409\) − 492.870i − 1.20506i −0.798096 0.602531i \(-0.794159\pi\)
0.798096 0.602531i \(-0.205841\pi\)
\(410\) −99.6909 −0.243148
\(411\) 67.4699i 0.164160i
\(412\) − 278.082i − 0.674957i
\(413\) 0 0
\(414\) 106.874 0.258149
\(415\) −0.933512 −0.00224943
\(416\) 19.7870i 0.0475649i
\(417\) −171.378 −0.410979
\(418\) − 695.250i − 1.66328i
\(419\) 311.640i 0.743771i 0.928279 + 0.371885i \(0.121289\pi\)
−0.928279 + 0.371885i \(0.878711\pi\)
\(420\) 0 0
\(421\) −539.935 −1.28250 −0.641252 0.767330i \(-0.721585\pi\)
−0.641252 + 0.767330i \(0.721585\pi\)
\(422\) −229.157 −0.543025
\(423\) − 84.1878i − 0.199025i
\(424\) 183.494 0.432768
\(425\) − 91.2112i − 0.214615i
\(426\) − 263.871i − 0.619414i
\(427\) 0 0
\(428\) 304.380 0.711168
\(429\) −121.019 −0.282095
\(430\) 203.687i 0.473692i
\(431\) −628.041 −1.45717 −0.728586 0.684954i \(-0.759822\pi\)
−0.728586 + 0.684954i \(0.759822\pi\)
\(432\) 20.7846i 0.0481125i
\(433\) − 706.789i − 1.63231i −0.577836 0.816153i \(-0.696103\pi\)
0.577836 0.816153i \(-0.303897\pi\)
\(434\) 0 0
\(435\) 205.742 0.472970
\(436\) 129.511 0.297043
\(437\) 619.975i 1.41871i
\(438\) −126.542 −0.288909
\(439\) − 651.773i − 1.48468i −0.670025 0.742338i \(-0.733717\pi\)
0.670025 0.742338i \(-0.266283\pi\)
\(440\) 126.333i 0.287120i
\(441\) 0 0
\(442\) −90.2398 −0.204162
\(443\) −23.6258 −0.0533314 −0.0266657 0.999644i \(-0.508489\pi\)
−0.0266657 + 0.999644i \(0.508489\pi\)
\(444\) 162.014i 0.364895i
\(445\) 248.674 0.558818
\(446\) − 516.653i − 1.15841i
\(447\) 322.967i 0.722520i
\(448\) 0 0
\(449\) −55.1499 −0.122828 −0.0614141 0.998112i \(-0.519561\pi\)
−0.0614141 + 0.998112i \(0.519561\pi\)
\(450\) −21.2132 −0.0471405
\(451\) 629.712i 1.39626i
\(452\) 6.51721 0.0144186
\(453\) − 268.191i − 0.592034i
\(454\) − 364.689i − 0.803279i
\(455\) 0 0
\(456\) −120.572 −0.264411
\(457\) 350.773 0.767556 0.383778 0.923425i \(-0.374623\pi\)
0.383778 + 0.923425i \(0.374623\pi\)
\(458\) − 21.8224i − 0.0476471i
\(459\) −94.7894 −0.206513
\(460\) − 112.655i − 0.244902i
\(461\) 471.598i 1.02299i 0.859287 + 0.511494i \(0.170908\pi\)
−0.859287 + 0.511494i \(0.829092\pi\)
\(462\) 0 0
\(463\) 387.112 0.836094 0.418047 0.908425i \(-0.362715\pi\)
0.418047 + 0.908425i \(0.362715\pi\)
\(464\) −212.489 −0.457951
\(465\) − 116.698i − 0.250963i
\(466\) 177.267 0.380402
\(467\) 293.346i 0.628150i 0.949398 + 0.314075i \(0.101694\pi\)
−0.949398 + 0.314075i \(0.898306\pi\)
\(468\) 20.9873i 0.0448446i
\(469\) 0 0
\(470\) −88.7417 −0.188812
\(471\) −86.9554 −0.184619
\(472\) − 283.384i − 0.600391i
\(473\) 1286.62 2.72013
\(474\) − 53.8287i − 0.113563i
\(475\) − 123.058i − 0.259069i
\(476\) 0 0
\(477\) 194.625 0.408018
\(478\) 5.12744 0.0107269
\(479\) − 763.032i − 1.59297i −0.604660 0.796484i \(-0.706691\pi\)
0.604660 0.796484i \(-0.293309\pi\)
\(480\) 21.9089 0.0456435
\(481\) 163.593i 0.340111i
\(482\) − 136.668i − 0.283543i
\(483\) 0 0
\(484\) 555.999 1.14876
\(485\) −165.971 −0.342208
\(486\) 22.0454i 0.0453609i
\(487\) −223.889 −0.459730 −0.229865 0.973223i \(-0.573828\pi\)
−0.229865 + 0.973223i \(0.573828\pi\)
\(488\) 22.6952i 0.0465066i
\(489\) 200.590i 0.410205i
\(490\) 0 0
\(491\) −837.694 −1.70610 −0.853049 0.521830i \(-0.825249\pi\)
−0.853049 + 0.521830i \(0.825249\pi\)
\(492\) 109.206 0.221963
\(493\) − 969.070i − 1.96566i
\(494\) −121.747 −0.246452
\(495\) 133.996i 0.270699i
\(496\) 120.525i 0.242993i
\(497\) 0 0
\(498\) 1.02261 0.00205344
\(499\) 174.647 0.349993 0.174997 0.984569i \(-0.444009\pi\)
0.174997 + 0.984569i \(0.444009\pi\)
\(500\) 22.3607i 0.0447214i
\(501\) 106.211 0.211998
\(502\) − 42.0461i − 0.0837571i
\(503\) 747.962i 1.48700i 0.668734 + 0.743501i \(0.266836\pi\)
−0.668734 + 0.743501i \(0.733164\pi\)
\(504\) 0 0
\(505\) 195.866 0.387854
\(506\) −711.600 −1.40632
\(507\) − 271.525i − 0.535552i
\(508\) −177.154 −0.348729
\(509\) 234.429i 0.460567i 0.973124 + 0.230284i \(0.0739654\pi\)
−0.973124 + 0.230284i \(0.926035\pi\)
\(510\) 99.9168i 0.195915i
\(511\) 0 0
\(512\) −22.6274 −0.0441942
\(513\) −127.886 −0.249290
\(514\) 624.549i 1.21508i
\(515\) −310.905 −0.603700
\(516\) − 223.128i − 0.432419i
\(517\) 560.549i 1.08423i
\(518\) 0 0
\(519\) −54.9088 −0.105797
\(520\) 22.1225 0.0425434
\(521\) 214.853i 0.412386i 0.978511 + 0.206193i \(0.0661075\pi\)
−0.978511 + 0.206193i \(0.933893\pi\)
\(522\) −225.379 −0.431760
\(523\) − 925.040i − 1.76872i −0.466806 0.884360i \(-0.654595\pi\)
0.466806 0.884360i \(-0.345405\pi\)
\(524\) 250.248i 0.477573i
\(525\) 0 0
\(526\) −485.141 −0.922321
\(527\) −549.660 −1.04300
\(528\) − 138.391i − 0.262104i
\(529\) 105.555 0.199536
\(530\) − 205.152i − 0.387080i
\(531\) − 300.575i − 0.566054i
\(532\) 0 0
\(533\) 110.271 0.206887
\(534\) −272.409 −0.510129
\(535\) − 340.307i − 0.636088i
\(536\) −45.9996 −0.0858201
\(537\) − 228.448i − 0.425415i
\(538\) − 655.277i − 1.21799i
\(539\) 0 0
\(540\) 23.2379 0.0430331
\(541\) −114.312 −0.211298 −0.105649 0.994404i \(-0.533692\pi\)
−0.105649 + 0.994404i \(0.533692\pi\)
\(542\) 345.890i 0.638174i
\(543\) −95.5106 −0.175894
\(544\) − 103.194i − 0.189694i
\(545\) − 144.797i − 0.265683i
\(546\) 0 0
\(547\) −57.7698 −0.105612 −0.0528060 0.998605i \(-0.516817\pi\)
−0.0528060 + 0.998605i \(0.516817\pi\)
\(548\) −77.9075 −0.142167
\(549\) 24.0719i 0.0438468i
\(550\) 141.244 0.256808
\(551\) − 1307.42i − 2.37282i
\(552\) 123.407i 0.223564i
\(553\) 0 0
\(554\) −199.430 −0.359982
\(555\) 181.137 0.326372
\(556\) − 197.891i − 0.355919i
\(557\) −406.152 −0.729178 −0.364589 0.931169i \(-0.618790\pi\)
−0.364589 + 0.931169i \(0.618790\pi\)
\(558\) 127.836i 0.229096i
\(559\) − 225.304i − 0.403049i
\(560\) 0 0
\(561\) 631.139 1.12502
\(562\) −120.202 −0.213882
\(563\) − 428.292i − 0.760731i −0.924836 0.380366i \(-0.875798\pi\)
0.924836 0.380366i \(-0.124202\pi\)
\(564\) 97.2117 0.172361
\(565\) − 7.28646i − 0.0128964i
\(566\) 165.008i 0.291534i
\(567\) 0 0
\(568\) 304.691 0.536429
\(569\) −915.793 −1.60948 −0.804739 0.593628i \(-0.797695\pi\)
−0.804739 + 0.593628i \(0.797695\pi\)
\(570\) 134.803i 0.236497i
\(571\) −713.893 −1.25025 −0.625125 0.780524i \(-0.714952\pi\)
−0.625125 + 0.780524i \(0.714952\pi\)
\(572\) − 139.740i − 0.244301i
\(573\) − 338.035i − 0.589938i
\(574\) 0 0
\(575\) −125.952 −0.219047
\(576\) −24.0000 −0.0416667
\(577\) − 87.5946i − 0.151810i −0.997115 0.0759052i \(-0.975815\pi\)
0.997115 0.0759052i \(-0.0241846\pi\)
\(578\) 61.9130 0.107116
\(579\) − 604.070i − 1.04330i
\(580\) 237.570i 0.409604i
\(581\) 0 0
\(582\) 181.812 0.312392
\(583\) −1295.87 −2.22277
\(584\) − 146.119i − 0.250203i
\(585\) 23.4645 0.0401103
\(586\) 186.510i 0.318276i
\(587\) 169.908i 0.289452i 0.989472 + 0.144726i \(0.0462300\pi\)
−0.989472 + 0.144726i \(0.953770\pi\)
\(588\) 0 0
\(589\) −741.576 −1.25904
\(590\) −316.833 −0.537006
\(591\) 97.6544i 0.165236i
\(592\) −187.077 −0.316009
\(593\) − 200.253i − 0.337694i −0.985642 0.168847i \(-0.945996\pi\)
0.985642 0.168847i \(-0.0540044\pi\)
\(594\) − 146.785i − 0.247114i
\(595\) 0 0
\(596\) −372.930 −0.625721
\(597\) 296.714 0.497009
\(598\) 124.611i 0.208379i
\(599\) −700.402 −1.16929 −0.584643 0.811291i \(-0.698765\pi\)
−0.584643 + 0.811291i \(0.698765\pi\)
\(600\) − 24.4949i − 0.0408248i
\(601\) − 1039.21i − 1.72914i −0.502515 0.864569i \(-0.667592\pi\)
0.502515 0.864569i \(-0.332408\pi\)
\(602\) 0 0
\(603\) −48.7899 −0.0809119
\(604\) 309.681 0.512716
\(605\) − 621.626i − 1.02748i
\(606\) −214.561 −0.354061
\(607\) − 61.1592i − 0.100757i −0.998730 0.0503783i \(-0.983957\pi\)
0.998730 0.0503783i \(-0.0160427\pi\)
\(608\) − 139.224i − 0.228987i
\(609\) 0 0
\(610\) 25.3740 0.0415967
\(611\) 98.1596 0.160654
\(612\) − 109.453i − 0.178845i
\(613\) −522.719 −0.852722 −0.426361 0.904553i \(-0.640205\pi\)
−0.426361 + 0.904553i \(0.640205\pi\)
\(614\) 607.929i 0.990112i
\(615\) − 122.096i − 0.198530i
\(616\) 0 0
\(617\) −608.200 −0.985738 −0.492869 0.870104i \(-0.664052\pi\)
−0.492869 + 0.870104i \(0.664052\pi\)
\(618\) 340.580 0.551100
\(619\) 1042.31i 1.68387i 0.539580 + 0.841934i \(0.318583\pi\)
−0.539580 + 0.841934i \(0.681417\pi\)
\(620\) 134.751 0.217340
\(621\) 130.893i 0.210778i
\(622\) 355.643i 0.571774i
\(623\) 0 0
\(624\) −24.2340 −0.0388366
\(625\) 25.0000 0.0400000
\(626\) 19.3794i 0.0309576i
\(627\) 851.503 1.35806
\(628\) − 100.407i − 0.159885i
\(629\) − 853.176i − 1.35640i
\(630\) 0 0
\(631\) −235.274 −0.372859 −0.186430 0.982468i \(-0.559692\pi\)
−0.186430 + 0.982468i \(0.559692\pi\)
\(632\) 62.1560 0.0983481
\(633\) − 280.658i − 0.443378i
\(634\) 46.5703 0.0734547
\(635\) 198.065i 0.311913i
\(636\) 224.733i 0.353354i
\(637\) 0 0
\(638\) 1500.65 2.35211
\(639\) 323.174 0.505750
\(640\) 25.2982i 0.0395285i
\(641\) −116.980 −0.182496 −0.0912481 0.995828i \(-0.529086\pi\)
−0.0912481 + 0.995828i \(0.529086\pi\)
\(642\) 372.788i 0.580666i
\(643\) 874.209i 1.35958i 0.733408 + 0.679789i \(0.237929\pi\)
−0.733408 + 0.679789i \(0.762071\pi\)
\(644\) 0 0
\(645\) −249.465 −0.386768
\(646\) 634.940 0.982879
\(647\) − 11.6838i − 0.0180584i −0.999959 0.00902920i \(-0.997126\pi\)
0.999959 0.00902920i \(-0.00287412\pi\)
\(648\) −25.4558 −0.0392837
\(649\) 2001.32i 3.08370i
\(650\) − 24.7338i − 0.0380519i
\(651\) 0 0
\(652\) −231.622 −0.355248
\(653\) 638.135 0.977236 0.488618 0.872498i \(-0.337501\pi\)
0.488618 + 0.872498i \(0.337501\pi\)
\(654\) 158.618i 0.242535i
\(655\) 279.786 0.427154
\(656\) 126.100i 0.192226i
\(657\) − 154.982i − 0.235894i
\(658\) 0 0
\(659\) 870.363 1.32073 0.660367 0.750943i \(-0.270401\pi\)
0.660367 + 0.750943i \(0.270401\pi\)
\(660\) −154.725 −0.234433
\(661\) 482.170i 0.729455i 0.931114 + 0.364728i \(0.118838\pi\)
−0.931114 + 0.364728i \(0.881162\pi\)
\(662\) 590.973 0.892708
\(663\) − 110.521i − 0.166698i
\(664\) 1.18081i 0.00177833i
\(665\) 0 0
\(666\) −198.425 −0.297936
\(667\) −1338.17 −2.00625
\(668\) 122.642i 0.183596i
\(669\) 632.768 0.945842
\(670\) 51.4291i 0.0767598i
\(671\) − 160.279i − 0.238865i
\(672\) 0 0
\(673\) −399.323 −0.593347 −0.296674 0.954979i \(-0.595877\pi\)
−0.296674 + 0.954979i \(0.595877\pi\)
\(674\) 405.438 0.601539
\(675\) − 25.9808i − 0.0384900i
\(676\) 313.530 0.463801
\(677\) − 141.341i − 0.208776i −0.994537 0.104388i \(-0.966712\pi\)
0.994537 0.104388i \(-0.0332884\pi\)
\(678\) 7.98191i 0.0117727i
\(679\) 0 0
\(680\) −115.374 −0.169668
\(681\) 446.650 0.655874
\(682\) − 851.172i − 1.24805i
\(683\) 988.175 1.44682 0.723408 0.690421i \(-0.242575\pi\)
0.723408 + 0.690421i \(0.242575\pi\)
\(684\) − 147.669i − 0.215891i
\(685\) 87.1032i 0.127158i
\(686\) 0 0
\(687\) 26.7268 0.0389037
\(688\) 257.646 0.374486
\(689\) 226.925i 0.329353i
\(690\) 137.973 0.199961
\(691\) − 350.831i − 0.507715i −0.967242 0.253857i \(-0.918301\pi\)
0.967242 0.253857i \(-0.0816994\pi\)
\(692\) − 63.4033i − 0.0916232i
\(693\) 0 0
\(694\) −433.875 −0.625180
\(695\) −221.249 −0.318343
\(696\) − 260.245i − 0.373915i
\(697\) −575.087 −0.825089
\(698\) − 481.061i − 0.689200i
\(699\) 217.107i 0.310597i
\(700\) 0 0
\(701\) 307.500 0.438659 0.219330 0.975651i \(-0.429613\pi\)
0.219330 + 0.975651i \(0.429613\pi\)
\(702\) −25.7041 −0.0366155
\(703\) − 1151.07i − 1.63736i
\(704\) 159.800 0.226988
\(705\) − 108.686i − 0.154164i
\(706\) 530.333i 0.751180i
\(707\) 0 0
\(708\) 347.074 0.490217
\(709\) −98.1423 −0.138424 −0.0692118 0.997602i \(-0.522048\pi\)
−0.0692118 + 0.997602i \(0.522048\pi\)
\(710\) − 340.655i − 0.479796i
\(711\) 65.9264 0.0927235
\(712\) − 314.551i − 0.441784i
\(713\) 759.016i 1.06454i
\(714\) 0 0
\(715\) −156.234 −0.218510
\(716\) 263.789 0.368420
\(717\) 6.27980i 0.00875844i
\(718\) 465.984 0.649003
\(719\) − 707.069i − 0.983406i −0.870763 0.491703i \(-0.836375\pi\)
0.870763 0.491703i \(-0.163625\pi\)
\(720\) 26.8328i 0.0372678i
\(721\) 0 0
\(722\) 346.100 0.479363
\(723\) 167.383 0.231512
\(724\) − 110.286i − 0.152329i
\(725\) 265.612 0.366361
\(726\) 680.957i 0.937957i
\(727\) 1353.85i 1.86225i 0.364705 + 0.931123i \(0.381170\pi\)
−0.364705 + 0.931123i \(0.618830\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) −163.365 −0.223788
\(731\) 1175.01i 1.60740i
\(732\) −27.7958 −0.0379725
\(733\) 553.033i 0.754479i 0.926116 + 0.377240i \(0.123127\pi\)
−0.926116 + 0.377240i \(0.876873\pi\)
\(734\) − 38.4020i − 0.0523188i
\(735\) 0 0
\(736\) −142.498 −0.193612
\(737\) 324.859 0.440786
\(738\) 133.749i 0.181232i
\(739\) 933.479 1.26316 0.631582 0.775309i \(-0.282406\pi\)
0.631582 + 0.775309i \(0.282406\pi\)
\(740\) 209.159i 0.282647i
\(741\) − 149.110i − 0.201227i
\(742\) 0 0
\(743\) 554.921 0.746865 0.373432 0.927657i \(-0.378181\pi\)
0.373432 + 0.927657i \(0.378181\pi\)
\(744\) −147.612 −0.198403
\(745\) 416.948i 0.559662i
\(746\) −180.309 −0.241701
\(747\) 1.25244i 0.00167662i
\(748\) 728.776i 0.974300i
\(749\) 0 0
\(750\) −27.3861 −0.0365148
\(751\) 727.948 0.969305 0.484652 0.874707i \(-0.338946\pi\)
0.484652 + 0.874707i \(0.338946\pi\)
\(752\) 112.250i 0.149269i
\(753\) 51.4957 0.0683874
\(754\) − 262.783i − 0.348518i
\(755\) − 346.233i − 0.458587i
\(756\) 0 0
\(757\) 667.167 0.881330 0.440665 0.897672i \(-0.354743\pi\)
0.440665 + 0.897672i \(0.354743\pi\)
\(758\) −452.258 −0.596646
\(759\) − 871.528i − 1.14826i
\(760\) −155.657 −0.204812
\(761\) 727.532i 0.956021i 0.878354 + 0.478010i \(0.158642\pi\)
−0.878354 + 0.478010i \(0.841358\pi\)
\(762\) − 216.969i − 0.284736i
\(763\) 0 0
\(764\) 390.329 0.510901
\(765\) −122.373 −0.159964
\(766\) 578.332i 0.755003i
\(767\) 350.458 0.456921
\(768\) − 27.7128i − 0.0360844i
\(769\) − 1374.28i − 1.78709i −0.448969 0.893547i \(-0.648209\pi\)
0.448969 0.893547i \(-0.351791\pi\)
\(770\) 0 0
\(771\) −764.913 −0.992106
\(772\) 697.520 0.903523
\(773\) 92.6712i 0.119885i 0.998202 + 0.0599425i \(0.0190917\pi\)
−0.998202 + 0.0599425i \(0.980908\pi\)
\(774\) 273.275 0.353069
\(775\) − 150.656i − 0.194395i
\(776\) 209.938i 0.270539i
\(777\) 0 0
\(778\) −547.517 −0.703749
\(779\) −775.881 −0.995996
\(780\) 27.0945i 0.0347365i
\(781\) −2151.80 −2.75518
\(782\) − 649.872i − 0.831038i
\(783\) − 276.032i − 0.352531i
\(784\) 0 0
\(785\) −112.259 −0.143005
\(786\) −306.490 −0.389937
\(787\) 836.226i 1.06255i 0.847200 + 0.531275i \(0.178287\pi\)
−0.847200 + 0.531275i \(0.821713\pi\)
\(788\) −112.762 −0.143098
\(789\) − 594.174i − 0.753072i
\(790\) − 69.4925i − 0.0879652i
\(791\) 0 0
\(792\) 169.493 0.214007
\(793\) −28.0669 −0.0353933
\(794\) 42.5836i 0.0536318i
\(795\) 251.259 0.316049
\(796\) 342.616i 0.430422i
\(797\) 130.279i 0.163462i 0.996654 + 0.0817310i \(0.0260448\pi\)
−0.996654 + 0.0817310i \(0.973955\pi\)
\(798\) 0 0
\(799\) −511.924 −0.640706
\(800\) 28.2843 0.0353553
\(801\) − 333.631i − 0.416518i
\(802\) −193.423 −0.241175
\(803\) 1031.92i 1.28508i
\(804\) − 56.3377i − 0.0700718i
\(805\) 0 0
\(806\) −149.052 −0.184927
\(807\) 802.548 0.994483
\(808\) − 247.753i − 0.306626i
\(809\) 621.564 0.768311 0.384156 0.923268i \(-0.374493\pi\)
0.384156 + 0.923268i \(0.374493\pi\)
\(810\) 28.4605i 0.0351364i
\(811\) − 445.846i − 0.549748i −0.961480 0.274874i \(-0.911364\pi\)
0.961480 0.274874i \(-0.0886361\pi\)
\(812\) 0 0
\(813\) −423.627 −0.521067
\(814\) 1321.18 1.62307
\(815\) 258.961i 0.317743i
\(816\) 126.386 0.154885
\(817\) 1585.27i 1.94036i
\(818\) 697.023i 0.852107i
\(819\) 0 0
\(820\) 140.984 0.171932
\(821\) 1257.24 1.53136 0.765678 0.643224i \(-0.222403\pi\)
0.765678 + 0.643224i \(0.222403\pi\)
\(822\) − 95.4168i − 0.116079i
\(823\) 120.342 0.146223 0.0731117 0.997324i \(-0.476707\pi\)
0.0731117 + 0.997324i \(0.476707\pi\)
\(824\) 393.268i 0.477267i
\(825\) 172.988i 0.209683i
\(826\) 0 0
\(827\) 151.053 0.182652 0.0913258 0.995821i \(-0.470890\pi\)
0.0913258 + 0.995821i \(0.470890\pi\)
\(828\) −151.142 −0.182539
\(829\) 251.308i 0.303147i 0.988446 + 0.151573i \(0.0484340\pi\)
−0.988446 + 0.151573i \(0.951566\pi\)
\(830\) 1.32019 0.00159058
\(831\) − 244.251i − 0.293924i
\(832\) − 27.9831i − 0.0336335i
\(833\) 0 0
\(834\) 242.366 0.290606
\(835\) 137.118 0.164213
\(836\) 983.231i 1.17611i
\(837\) −156.566 −0.187056
\(838\) − 440.725i − 0.525925i
\(839\) 499.592i 0.595461i 0.954650 + 0.297730i \(0.0962297\pi\)
−0.954650 + 0.297730i \(0.903770\pi\)
\(840\) 0 0
\(841\) 1980.98 2.35550
\(842\) 763.583 0.906868
\(843\) − 147.216i − 0.174634i
\(844\) 324.076 0.383977
\(845\) − 350.537i − 0.414836i
\(846\) 119.059i 0.140732i
\(847\) 0 0
\(848\) −259.499 −0.306013
\(849\) −202.093 −0.238037
\(850\) 128.992i 0.151755i
\(851\) −1178.14 −1.38441
\(852\) 373.169i 0.437992i
\(853\) − 1265.99i − 1.48416i −0.670310 0.742082i \(-0.733839\pi\)
0.670310 0.742082i \(-0.266161\pi\)
\(854\) 0 0
\(855\) −165.100 −0.193099
\(856\) −430.458 −0.502871
\(857\) 369.620i 0.431295i 0.976471 + 0.215648i \(0.0691862\pi\)
−0.976471 + 0.215648i \(0.930814\pi\)
\(858\) 171.146 0.199471
\(859\) 1258.03i 1.46453i 0.681021 + 0.732264i \(0.261536\pi\)
−0.681021 + 0.732264i \(0.738464\pi\)
\(860\) − 288.057i − 0.334951i
\(861\) 0 0
\(862\) 888.184 1.03038
\(863\) −236.666 −0.274236 −0.137118 0.990555i \(-0.543784\pi\)
−0.137118 + 0.990555i \(0.543784\pi\)
\(864\) − 29.3939i − 0.0340207i
\(865\) −70.8870 −0.0819503
\(866\) 999.550i 1.15421i
\(867\) 75.8277i 0.0874598i
\(868\) 0 0
\(869\) −438.959 −0.505132
\(870\) −290.963 −0.334440
\(871\) − 56.8871i − 0.0653124i
\(872\) −183.156 −0.210041
\(873\) 222.673i 0.255067i
\(874\) − 876.777i − 1.00318i
\(875\) 0 0
\(876\) 178.958 0.204290
\(877\) 805.531 0.918508 0.459254 0.888305i \(-0.348117\pi\)
0.459254 + 0.888305i \(0.348117\pi\)
\(878\) 921.746i 1.04982i
\(879\) −228.427 −0.259871
\(880\) − 178.662i − 0.203025i
\(881\) − 763.260i − 0.866356i −0.901308 0.433178i \(-0.857392\pi\)
0.901308 0.433178i \(-0.142608\pi\)
\(882\) 0 0
\(883\) 574.986 0.651173 0.325587 0.945512i \(-0.394438\pi\)
0.325587 + 0.945512i \(0.394438\pi\)
\(884\) 127.618 0.144365
\(885\) − 388.040i − 0.438464i
\(886\) 33.4119 0.0377110
\(887\) 383.502i 0.432358i 0.976354 + 0.216179i \(0.0693595\pi\)
−0.976354 + 0.216179i \(0.930641\pi\)
\(888\) − 229.122i − 0.258020i
\(889\) 0 0
\(890\) −351.678 −0.395144
\(891\) 179.775 0.201767
\(892\) 730.658i 0.819123i
\(893\) −690.665 −0.773421
\(894\) − 456.744i − 0.510899i
\(895\) − 294.925i − 0.329525i
\(896\) 0 0
\(897\) −152.616 −0.170141
\(898\) 77.9937 0.0868526
\(899\) − 1600.64i − 1.78046i
\(900\) 30.0000 0.0333333
\(901\) − 1183.46i − 1.31350i
\(902\) − 890.547i − 0.987302i
\(903\) 0 0
\(904\) −9.21672 −0.0101955
\(905\) −123.304 −0.136247
\(906\) 379.280i 0.418631i
\(907\) −1162.44 −1.28163 −0.640816 0.767695i \(-0.721404\pi\)
−0.640816 + 0.767695i \(0.721404\pi\)
\(908\) 515.747i 0.568004i
\(909\) − 262.782i − 0.289089i
\(910\) 0 0
\(911\) 1070.03 1.17457 0.587284 0.809381i \(-0.300197\pi\)
0.587284 + 0.809381i \(0.300197\pi\)
\(912\) 170.514 0.186967
\(913\) − 8.33914i − 0.00913378i
\(914\) −496.068 −0.542744
\(915\) 31.0767i 0.0339636i
\(916\) 30.8615i 0.0336916i
\(917\) 0 0
\(918\) 134.053 0.146027
\(919\) 1166.72 1.26955 0.634777 0.772695i \(-0.281092\pi\)
0.634777 + 0.772695i \(0.281092\pi\)
\(920\) 159.318i 0.173172i
\(921\) −744.558 −0.808423
\(922\) − 666.940i − 0.723362i
\(923\) 376.808i 0.408243i
\(924\) 0 0
\(925\) 233.846 0.252807
\(926\) −547.458 −0.591208
\(927\) 417.123i 0.449971i
\(928\) 300.505 0.323820
\(929\) − 224.387i − 0.241536i −0.992681 0.120768i \(-0.961464\pi\)
0.992681 0.120768i \(-0.0385358\pi\)
\(930\) 165.035i 0.177457i
\(931\) 0 0
\(932\) −250.694 −0.268984
\(933\) −435.572 −0.466851
\(934\) − 414.854i − 0.444169i
\(935\) 814.797 0.871440
\(936\) − 29.6805i − 0.0317100i
\(937\) 644.185i 0.687497i 0.939062 + 0.343749i \(0.111697\pi\)
−0.939062 + 0.343749i \(0.888303\pi\)
\(938\) 0 0
\(939\) −23.7349 −0.0252767
\(940\) 125.500 0.133510
\(941\) − 307.262i − 0.326527i −0.986582 0.163264i \(-0.947798\pi\)
0.986582 0.163264i \(-0.0522021\pi\)
\(942\) 122.974 0.130545
\(943\) 794.127i 0.842128i
\(944\) 400.766i 0.424540i
\(945\) 0 0
\(946\) −1819.56 −1.92342
\(947\) −981.511 −1.03644 −0.518221 0.855247i \(-0.673406\pi\)
−0.518221 + 0.855247i \(0.673406\pi\)
\(948\) 76.1252i 0.0803009i
\(949\) 180.703 0.190414
\(950\) 174.030i 0.183190i
\(951\) 57.0367i 0.0599755i
\(952\) 0 0
\(953\) 1258.67 1.32075 0.660375 0.750936i \(-0.270397\pi\)
0.660375 + 0.750936i \(0.270397\pi\)
\(954\) −275.241 −0.288512
\(955\) − 436.401i − 0.456964i
\(956\) −7.25129 −0.00758503
\(957\) 1837.91i 1.92049i
\(958\) 1079.09i 1.12640i
\(959\) 0 0
\(960\) −30.9839 −0.0322749
\(961\) 53.1118 0.0552673
\(962\) − 231.356i − 0.240495i
\(963\) −456.570 −0.474112
\(964\) 193.277i 0.200495i
\(965\) − 779.851i − 0.808135i
\(966\) 0 0
\(967\) −992.744 −1.02662 −0.513311 0.858202i \(-0.671581\pi\)
−0.513311 + 0.858202i \(0.671581\pi\)
\(968\) −786.301 −0.812295
\(969\) 777.639i 0.802517i
\(970\) 234.718 0.241978
\(971\) 357.573i 0.368253i 0.982903 + 0.184126i \(0.0589455\pi\)
−0.982903 + 0.184126i \(0.941054\pi\)
\(972\) − 31.1769i − 0.0320750i
\(973\) 0 0
\(974\) 316.626 0.325078
\(975\) 30.2925 0.0310693
\(976\) − 32.0959i − 0.0328851i
\(977\) −877.346 −0.898000 −0.449000 0.893532i \(-0.648220\pi\)
−0.449000 + 0.893532i \(0.648220\pi\)
\(978\) − 283.677i − 0.290059i
\(979\) 2221.43i 2.26908i
\(980\) 0 0
\(981\) −194.266 −0.198029
\(982\) 1184.68 1.20639
\(983\) − 1440.21i − 1.46511i −0.680705 0.732557i \(-0.738327\pi\)
0.680705 0.732557i \(-0.261673\pi\)
\(984\) −154.440 −0.156952
\(985\) 126.071i 0.127991i
\(986\) 1370.47i 1.38993i
\(987\) 0 0
\(988\) 172.177 0.174268
\(989\) 1622.55 1.64060
\(990\) − 189.499i − 0.191413i
\(991\) 1154.36 1.16485 0.582423 0.812886i \(-0.302105\pi\)
0.582423 + 0.812886i \(0.302105\pi\)
\(992\) − 170.448i − 0.171822i
\(993\) 723.791i 0.728893i
\(994\) 0 0
\(995\) 383.057 0.384981
\(996\) −1.44619 −0.00145200
\(997\) 1624.90i 1.62979i 0.579608 + 0.814896i \(0.303206\pi\)
−0.579608 + 0.814896i \(0.696794\pi\)
\(998\) −246.988 −0.247483
\(999\) − 243.020i − 0.243264i
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 1470.3.f.d.391.2 16
7.4 even 3 210.3.o.b.61.7 yes 16
7.5 odd 6 210.3.o.b.31.7 16
7.6 odd 2 inner 1470.3.f.d.391.8 16
21.5 even 6 630.3.v.c.451.1 16
21.11 odd 6 630.3.v.c.271.1 16
35.4 even 6 1050.3.p.i.901.4 16
35.12 even 12 1050.3.q.e.199.1 32
35.18 odd 12 1050.3.q.e.649.1 32
35.19 odd 6 1050.3.p.i.451.4 16
35.32 odd 12 1050.3.q.e.649.10 32
35.33 even 12 1050.3.q.e.199.10 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.3.o.b.31.7 16 7.5 odd 6
210.3.o.b.61.7 yes 16 7.4 even 3
630.3.v.c.271.1 16 21.11 odd 6
630.3.v.c.451.1 16 21.5 even 6
1050.3.p.i.451.4 16 35.19 odd 6
1050.3.p.i.901.4 16 35.4 even 6
1050.3.q.e.199.1 32 35.12 even 12
1050.3.q.e.199.10 32 35.33 even 12
1050.3.q.e.649.1 32 35.18 odd 12
1050.3.q.e.649.10 32 35.32 odd 12
1470.3.f.d.391.2 16 1.1 even 1 trivial
1470.3.f.d.391.8 16 7.6 odd 2 inner