Properties

Label 9.5.b.a.8.2
Level $9$
Weight $5$
Character 9.8
Analytic conductor $0.930$
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] = [9,5,Mod(8,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.8");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 9.b (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: \(0.930329667755\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 8.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 9.8
Dual form 9.5.b.a.8.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.24264i q^{2} -2.00000 q^{4} -29.6985i q^{5} -28.0000 q^{7} +59.3970i q^{8} +O(q^{10})\) \(q+4.24264i q^{2} -2.00000 q^{4} -29.6985i q^{5} -28.0000 q^{7} +59.3970i q^{8} +126.000 q^{10} +16.9706i q^{11} -112.000 q^{13} -118.794i q^{14} -284.000 q^{16} -89.0955i q^{17} +560.000 q^{19} +59.3970i q^{20} -72.0000 q^{22} +797.616i q^{23} -257.000 q^{25} -475.176i q^{26} +56.0000 q^{28} -988.535i q^{29} -364.000 q^{31} -254.558i q^{32} +378.000 q^{34} +831.558i q^{35} -826.000 q^{37} +2375.88i q^{38} +1764.00 q^{40} -1811.61i q^{41} +1736.00 q^{43} -33.9411i q^{44} -3384.00 q^{46} -1306.73i q^{47} -1617.00 q^{49} -1090.36i q^{50} +224.000 q^{52} +1794.64i q^{53} +504.000 q^{55} -1663.12i q^{56} +4194.00 q^{58} +4514.17i q^{59} +2618.00 q^{61} -1544.32i q^{62} -3464.00 q^{64} +3326.23i q^{65} -3784.00 q^{67} +178.191i q^{68} -3528.00 q^{70} -8604.08i q^{71} +6608.00 q^{73} -3504.42i q^{74} -1120.00 q^{76} -475.176i q^{77} -4276.00 q^{79} +8434.37i q^{80} +7686.00 q^{82} +118.794i q^{83} -2646.00 q^{85} +7365.22i q^{86} -1008.00 q^{88} +4365.68i q^{89} +3136.00 q^{91} -1595.23i q^{92} +5544.00 q^{94} -16631.2i q^{95} -5824.00 q^{97} -6860.35i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 56 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} - 56 q^{7} + 252 q^{10} - 224 q^{13} - 568 q^{16} + 1120 q^{19} - 144 q^{22} - 514 q^{25} + 112 q^{28} - 728 q^{31} + 756 q^{34} - 1652 q^{37} + 3528 q^{40} + 3472 q^{43} - 6768 q^{46} - 3234 q^{49} + 448 q^{52} + 1008 q^{55} + 8388 q^{58} + 5236 q^{61} - 6928 q^{64} - 7568 q^{67} - 7056 q^{70} + 13216 q^{73} - 2240 q^{76} - 8552 q^{79} + 15372 q^{82} - 5292 q^{85} - 2016 q^{88} + 6272 q^{91} + 11088 q^{94} - 11648 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.24264i 1.06066i 0.847791 + 0.530330i \(0.177932\pi\)
−0.847791 + 0.530330i \(0.822068\pi\)
\(3\) 0 0
\(4\) −2.00000 −0.125000
\(5\) − 29.6985i − 1.18794i −0.804487 0.593970i \(-0.797560\pi\)
0.804487 0.593970i \(-0.202440\pi\)
\(6\) 0 0
\(7\) −28.0000 −0.571429 −0.285714 0.958315i \(-0.592231\pi\)
−0.285714 + 0.958315i \(0.592231\pi\)
\(8\) 59.3970i 0.928078i
\(9\) 0 0
\(10\) 126.000 1.26000
\(11\) 16.9706i 0.140253i 0.997538 + 0.0701263i \(0.0223402\pi\)
−0.997538 + 0.0701263i \(0.977660\pi\)
\(12\) 0 0
\(13\) −112.000 −0.662722 −0.331361 0.943504i \(-0.607508\pi\)
−0.331361 + 0.943504i \(0.607508\pi\)
\(14\) − 118.794i − 0.606092i
\(15\) 0 0
\(16\) −284.000 −1.10938
\(17\) − 89.0955i − 0.308289i −0.988048 0.154144i \(-0.950738\pi\)
0.988048 0.154144i \(-0.0492621\pi\)
\(18\) 0 0
\(19\) 560.000 1.55125 0.775623 0.631196i \(-0.217436\pi\)
0.775623 + 0.631196i \(0.217436\pi\)
\(20\) 59.3970i 0.148492i
\(21\) 0 0
\(22\) −72.0000 −0.148760
\(23\) 797.616i 1.50778i 0.657000 + 0.753891i \(0.271825\pi\)
−0.657000 + 0.753891i \(0.728175\pi\)
\(24\) 0 0
\(25\) −257.000 −0.411200
\(26\) − 475.176i − 0.702923i
\(27\) 0 0
\(28\) 56.0000 0.0714286
\(29\) − 988.535i − 1.17543i −0.809069 0.587714i \(-0.800028\pi\)
0.809069 0.587714i \(-0.199972\pi\)
\(30\) 0 0
\(31\) −364.000 −0.378772 −0.189386 0.981903i \(-0.560650\pi\)
−0.189386 + 0.981903i \(0.560650\pi\)
\(32\) − 254.558i − 0.248592i
\(33\) 0 0
\(34\) 378.000 0.326990
\(35\) 831.558i 0.678823i
\(36\) 0 0
\(37\) −826.000 −0.603360 −0.301680 0.953409i \(-0.597547\pi\)
−0.301680 + 0.953409i \(0.597547\pi\)
\(38\) 2375.88i 1.64535i
\(39\) 0 0
\(40\) 1764.00 1.10250
\(41\) − 1811.61i − 1.07770i −0.842403 0.538848i \(-0.818860\pi\)
0.842403 0.538848i \(-0.181140\pi\)
\(42\) 0 0
\(43\) 1736.00 0.938886 0.469443 0.882963i \(-0.344455\pi\)
0.469443 + 0.882963i \(0.344455\pi\)
\(44\) − 33.9411i − 0.0175316i
\(45\) 0 0
\(46\) −3384.00 −1.59924
\(47\) − 1306.73i − 0.591550i −0.955258 0.295775i \(-0.904422\pi\)
0.955258 0.295775i \(-0.0955778\pi\)
\(48\) 0 0
\(49\) −1617.00 −0.673469
\(50\) − 1090.36i − 0.436143i
\(51\) 0 0
\(52\) 224.000 0.0828402
\(53\) 1794.64i 0.638888i 0.947605 + 0.319444i \(0.103496\pi\)
−0.947605 + 0.319444i \(0.896504\pi\)
\(54\) 0 0
\(55\) 504.000 0.166612
\(56\) − 1663.12i − 0.530330i
\(57\) 0 0
\(58\) 4194.00 1.24673
\(59\) 4514.17i 1.29680i 0.761299 + 0.648401i \(0.224562\pi\)
−0.761299 + 0.648401i \(0.775438\pi\)
\(60\) 0 0
\(61\) 2618.00 0.703574 0.351787 0.936080i \(-0.385574\pi\)
0.351787 + 0.936080i \(0.385574\pi\)
\(62\) − 1544.32i − 0.401748i
\(63\) 0 0
\(64\) −3464.00 −0.845703
\(65\) 3326.23i 0.787273i
\(66\) 0 0
\(67\) −3784.00 −0.842949 −0.421475 0.906840i \(-0.638487\pi\)
−0.421475 + 0.906840i \(0.638487\pi\)
\(68\) 178.191i 0.0385361i
\(69\) 0 0
\(70\) −3528.00 −0.720000
\(71\) − 8604.08i − 1.70682i −0.521241 0.853410i \(-0.674531\pi\)
0.521241 0.853410i \(-0.325469\pi\)
\(72\) 0 0
\(73\) 6608.00 1.24001 0.620004 0.784599i \(-0.287131\pi\)
0.620004 + 0.784599i \(0.287131\pi\)
\(74\) − 3504.42i − 0.639960i
\(75\) 0 0
\(76\) −1120.00 −0.193906
\(77\) − 475.176i − 0.0801443i
\(78\) 0 0
\(79\) −4276.00 −0.685147 −0.342573 0.939491i \(-0.611299\pi\)
−0.342573 + 0.939491i \(0.611299\pi\)
\(80\) 8434.37i 1.31787i
\(81\) 0 0
\(82\) 7686.00 1.14307
\(83\) 118.794i 0.0172440i 0.999963 + 0.00862200i \(0.00274450\pi\)
−0.999963 + 0.00862200i \(0.997255\pi\)
\(84\) 0 0
\(85\) −2646.00 −0.366228
\(86\) 7365.22i 0.995839i
\(87\) 0 0
\(88\) −1008.00 −0.130165
\(89\) 4365.68i 0.551152i 0.961279 + 0.275576i \(0.0888686\pi\)
−0.961279 + 0.275576i \(0.911131\pi\)
\(90\) 0 0
\(91\) 3136.00 0.378698
\(92\) − 1595.23i − 0.188473i
\(93\) 0 0
\(94\) 5544.00 0.627433
\(95\) − 16631.2i − 1.84279i
\(96\) 0 0
\(97\) −5824.00 −0.618982 −0.309491 0.950902i \(-0.600159\pi\)
−0.309491 + 0.950902i \(0.600159\pi\)
\(98\) − 6860.35i − 0.714322i
\(99\) 0 0
\(100\) 514.000 0.0514000
\(101\) 8048.29i 0.788971i 0.918902 + 0.394485i \(0.129077\pi\)
−0.918902 + 0.394485i \(0.870923\pi\)
\(102\) 0 0
\(103\) −14980.0 −1.41201 −0.706004 0.708208i \(-0.749504\pi\)
−0.706004 + 0.708208i \(0.749504\pi\)
\(104\) − 6652.46i − 0.615057i
\(105\) 0 0
\(106\) −7614.00 −0.677643
\(107\) 9775.04i 0.853790i 0.904301 + 0.426895i \(0.140393\pi\)
−0.904301 + 0.426895i \(0.859607\pi\)
\(108\) 0 0
\(109\) 10640.0 0.895548 0.447774 0.894147i \(-0.352217\pi\)
0.447774 + 0.894147i \(0.352217\pi\)
\(110\) 2138.29i 0.176718i
\(111\) 0 0
\(112\) 7952.00 0.633929
\(113\) 2872.27i 0.224941i 0.993655 + 0.112470i \(0.0358763\pi\)
−0.993655 + 0.112470i \(0.964124\pi\)
\(114\) 0 0
\(115\) 23688.0 1.79115
\(116\) 1977.07i 0.146929i
\(117\) 0 0
\(118\) −19152.0 −1.37547
\(119\) 2494.67i 0.176165i
\(120\) 0 0
\(121\) 14353.0 0.980329
\(122\) 11107.2i 0.746253i
\(123\) 0 0
\(124\) 728.000 0.0473465
\(125\) − 10929.0i − 0.699459i
\(126\) 0 0
\(127\) −19420.0 −1.20404 −0.602021 0.798480i \(-0.705638\pi\)
−0.602021 + 0.798480i \(0.705638\pi\)
\(128\) − 18769.4i − 1.14560i
\(129\) 0 0
\(130\) −14112.0 −0.835030
\(131\) 20195.0i 1.17679i 0.808572 + 0.588397i \(0.200241\pi\)
−0.808572 + 0.588397i \(0.799759\pi\)
\(132\) 0 0
\(133\) −15680.0 −0.886427
\(134\) − 16054.2i − 0.894083i
\(135\) 0 0
\(136\) 5292.00 0.286116
\(137\) − 17331.2i − 0.923394i −0.887038 0.461697i \(-0.847241\pi\)
0.887038 0.461697i \(-0.152759\pi\)
\(138\) 0 0
\(139\) 18536.0 0.959371 0.479685 0.877441i \(-0.340751\pi\)
0.479685 + 0.877441i \(0.340751\pi\)
\(140\) − 1663.12i − 0.0848528i
\(141\) 0 0
\(142\) 36504.0 1.81036
\(143\) − 1900.70i − 0.0929485i
\(144\) 0 0
\(145\) −29358.0 −1.39634
\(146\) 28035.4i 1.31523i
\(147\) 0 0
\(148\) 1652.00 0.0754200
\(149\) 3152.28i 0.141988i 0.997477 + 0.0709941i \(0.0226172\pi\)
−0.997477 + 0.0709941i \(0.977383\pi\)
\(150\) 0 0
\(151\) 7028.00 0.308232 0.154116 0.988053i \(-0.450747\pi\)
0.154116 + 0.988053i \(0.450747\pi\)
\(152\) 33262.3i 1.43968i
\(153\) 0 0
\(154\) 2016.00 0.0850059
\(155\) 10810.2i 0.449958i
\(156\) 0 0
\(157\) −5530.00 −0.224350 −0.112175 0.993688i \(-0.535782\pi\)
−0.112175 + 0.993688i \(0.535782\pi\)
\(158\) − 18141.5i − 0.726708i
\(159\) 0 0
\(160\) −7560.00 −0.295312
\(161\) − 22333.3i − 0.861589i
\(162\) 0 0
\(163\) 10856.0 0.408596 0.204298 0.978909i \(-0.434509\pi\)
0.204298 + 0.978909i \(0.434509\pi\)
\(164\) 3623.22i 0.134712i
\(165\) 0 0
\(166\) −504.000 −0.0182900
\(167\) − 29698.5i − 1.06488i −0.846467 0.532441i \(-0.821275\pi\)
0.846467 0.532441i \(-0.178725\pi\)
\(168\) 0 0
\(169\) −16017.0 −0.560800
\(170\) − 11226.0i − 0.388444i
\(171\) 0 0
\(172\) −3472.00 −0.117361
\(173\) − 27589.9i − 0.921845i −0.887440 0.460922i \(-0.847519\pi\)
0.887440 0.460922i \(-0.152481\pi\)
\(174\) 0 0
\(175\) 7196.00 0.234971
\(176\) − 4819.64i − 0.155593i
\(177\) 0 0
\(178\) −18522.0 −0.584585
\(179\) 54781.0i 1.70971i 0.518863 + 0.854857i \(0.326356\pi\)
−0.518863 + 0.854857i \(0.673644\pi\)
\(180\) 0 0
\(181\) 18704.0 0.570923 0.285461 0.958390i \(-0.407853\pi\)
0.285461 + 0.958390i \(0.407853\pi\)
\(182\) 13304.9i 0.401670i
\(183\) 0 0
\(184\) −47376.0 −1.39934
\(185\) 24530.9i 0.716755i
\(186\) 0 0
\(187\) 1512.00 0.0432383
\(188\) 2613.47i 0.0739437i
\(189\) 0 0
\(190\) 70560.0 1.95457
\(191\) 8519.22i 0.233525i 0.993160 + 0.116762i \(0.0372516\pi\)
−0.993160 + 0.116762i \(0.962748\pi\)
\(192\) 0 0
\(193\) 48398.0 1.29931 0.649655 0.760229i \(-0.274913\pi\)
0.649655 + 0.760229i \(0.274913\pi\)
\(194\) − 24709.1i − 0.656529i
\(195\) 0 0
\(196\) 3234.00 0.0841837
\(197\) 10016.9i 0.258107i 0.991638 + 0.129054i \(0.0411939\pi\)
−0.991638 + 0.129054i \(0.958806\pi\)
\(198\) 0 0
\(199\) −13300.0 −0.335850 −0.167925 0.985800i \(-0.553707\pi\)
−0.167925 + 0.985800i \(0.553707\pi\)
\(200\) − 15265.0i − 0.381626i
\(201\) 0 0
\(202\) −34146.0 −0.836830
\(203\) 27679.0i 0.671673i
\(204\) 0 0
\(205\) −53802.0 −1.28024
\(206\) − 63554.8i − 1.49766i
\(207\) 0 0
\(208\) 31808.0 0.735207
\(209\) 9503.52i 0.217566i
\(210\) 0 0
\(211\) −36904.0 −0.828912 −0.414456 0.910069i \(-0.636028\pi\)
−0.414456 + 0.910069i \(0.636028\pi\)
\(212\) − 3589.27i − 0.0798610i
\(213\) 0 0
\(214\) −41472.0 −0.905581
\(215\) − 51556.6i − 1.11534i
\(216\) 0 0
\(217\) 10192.0 0.216441
\(218\) 45141.7i 0.949872i
\(219\) 0 0
\(220\) −1008.00 −0.0208264
\(221\) 9978.69i 0.204310i
\(222\) 0 0
\(223\) 51380.0 1.03320 0.516600 0.856227i \(-0.327197\pi\)
0.516600 + 0.856227i \(0.327197\pi\)
\(224\) 7127.64i 0.142053i
\(225\) 0 0
\(226\) −12186.0 −0.238586
\(227\) − 90402.2i − 1.75439i −0.480130 0.877197i \(-0.659411\pi\)
0.480130 0.877197i \(-0.340589\pi\)
\(228\) 0 0
\(229\) 78512.0 1.49715 0.748575 0.663051i \(-0.230739\pi\)
0.748575 + 0.663051i \(0.230739\pi\)
\(230\) 100500.i 1.89980i
\(231\) 0 0
\(232\) 58716.0 1.09089
\(233\) 52451.8i 0.966158i 0.875577 + 0.483079i \(0.160482\pi\)
−0.875577 + 0.483079i \(0.839518\pi\)
\(234\) 0 0
\(235\) −38808.0 −0.702725
\(236\) − 9028.34i − 0.162100i
\(237\) 0 0
\(238\) −10584.0 −0.186851
\(239\) − 22265.4i − 0.389793i −0.980824 0.194897i \(-0.937563\pi\)
0.980824 0.194897i \(-0.0624371\pi\)
\(240\) 0 0
\(241\) −53200.0 −0.915962 −0.457981 0.888962i \(-0.651427\pi\)
−0.457981 + 0.888962i \(0.651427\pi\)
\(242\) 60894.6i 1.03980i
\(243\) 0 0
\(244\) −5236.00 −0.0879468
\(245\) 48022.4i 0.800041i
\(246\) 0 0
\(247\) −62720.0 −1.02805
\(248\) − 21620.5i − 0.351530i
\(249\) 0 0
\(250\) 46368.0 0.741888
\(251\) − 83037.0i − 1.31803i −0.752131 0.659013i \(-0.770974\pi\)
0.752131 0.659013i \(-0.229026\pi\)
\(252\) 0 0
\(253\) −13536.0 −0.211470
\(254\) − 82392.1i − 1.27708i
\(255\) 0 0
\(256\) 24208.0 0.369385
\(257\) − 84670.4i − 1.28193i −0.767569 0.640966i \(-0.778534\pi\)
0.767569 0.640966i \(-0.221466\pi\)
\(258\) 0 0
\(259\) 23128.0 0.344777
\(260\) − 6652.46i − 0.0984092i
\(261\) 0 0
\(262\) −85680.0 −1.24818
\(263\) 110343.i 1.59526i 0.603146 + 0.797630i \(0.293913\pi\)
−0.603146 + 0.797630i \(0.706087\pi\)
\(264\) 0 0
\(265\) 53298.0 0.758960
\(266\) − 66524.6i − 0.940197i
\(267\) 0 0
\(268\) 7568.00 0.105369
\(269\) 27530.5i 0.380460i 0.981740 + 0.190230i \(0.0609234\pi\)
−0.981740 + 0.190230i \(0.939077\pi\)
\(270\) 0 0
\(271\) −73276.0 −0.997753 −0.498877 0.866673i \(-0.666254\pi\)
−0.498877 + 0.866673i \(0.666254\pi\)
\(272\) 25303.1i 0.342008i
\(273\) 0 0
\(274\) 73530.0 0.979408
\(275\) − 4361.43i − 0.0576719i
\(276\) 0 0
\(277\) 46736.0 0.609105 0.304552 0.952496i \(-0.401493\pi\)
0.304552 + 0.952496i \(0.401493\pi\)
\(278\) 78641.6i 1.01757i
\(279\) 0 0
\(280\) −49392.0 −0.630000
\(281\) 112774.i 1.42822i 0.700034 + 0.714110i \(0.253168\pi\)
−0.700034 + 0.714110i \(0.746832\pi\)
\(282\) 0 0
\(283\) −24304.0 −0.303462 −0.151731 0.988422i \(-0.548485\pi\)
−0.151731 + 0.988422i \(0.548485\pi\)
\(284\) 17208.2i 0.213352i
\(285\) 0 0
\(286\) 8064.00 0.0985867
\(287\) 50725.0i 0.615826i
\(288\) 0 0
\(289\) 75583.0 0.904958
\(290\) − 124555.i − 1.48104i
\(291\) 0 0
\(292\) −13216.0 −0.155001
\(293\) − 9473.82i − 0.110354i −0.998477 0.0551772i \(-0.982428\pi\)
0.998477 0.0551772i \(-0.0175724\pi\)
\(294\) 0 0
\(295\) 134064. 1.54052
\(296\) − 49061.9i − 0.559965i
\(297\) 0 0
\(298\) −13374.0 −0.150601
\(299\) − 89333.0i − 0.999240i
\(300\) 0 0
\(301\) −48608.0 −0.536506
\(302\) 29817.3i 0.326930i
\(303\) 0 0
\(304\) −159040. −1.72091
\(305\) − 77750.6i − 0.835804i
\(306\) 0 0
\(307\) 19208.0 0.203801 0.101900 0.994795i \(-0.467508\pi\)
0.101900 + 0.994795i \(0.467508\pi\)
\(308\) 950.352i 0.0100180i
\(309\) 0 0
\(310\) −45864.0 −0.477253
\(311\) 78641.6i 0.813077i 0.913634 + 0.406538i \(0.133264\pi\)
−0.913634 + 0.406538i \(0.866736\pi\)
\(312\) 0 0
\(313\) −127918. −1.30570 −0.652849 0.757488i \(-0.726427\pi\)
−0.652849 + 0.757488i \(0.726427\pi\)
\(314\) − 23461.8i − 0.237959i
\(315\) 0 0
\(316\) 8552.00 0.0856433
\(317\) − 65875.5i − 0.655549i −0.944756 0.327775i \(-0.893701\pi\)
0.944756 0.327775i \(-0.106299\pi\)
\(318\) 0 0
\(319\) 16776.0 0.164857
\(320\) 102876.i 1.00464i
\(321\) 0 0
\(322\) 94752.0 0.913854
\(323\) − 49893.5i − 0.478232i
\(324\) 0 0
\(325\) 28784.0 0.272511
\(326\) 46058.1i 0.433382i
\(327\) 0 0
\(328\) 107604. 1.00019
\(329\) 36588.5i 0.338028i
\(330\) 0 0
\(331\) 15848.0 0.144650 0.0723250 0.997381i \(-0.476958\pi\)
0.0723250 + 0.997381i \(0.476958\pi\)
\(332\) − 237.588i − 0.00215550i
\(333\) 0 0
\(334\) 126000. 1.12948
\(335\) 112379.i 1.00137i
\(336\) 0 0
\(337\) −127120. −1.11932 −0.559660 0.828722i \(-0.689068\pi\)
−0.559660 + 0.828722i \(0.689068\pi\)
\(338\) − 67954.4i − 0.594818i
\(339\) 0 0
\(340\) 5292.00 0.0457785
\(341\) − 6177.28i − 0.0531238i
\(342\) 0 0
\(343\) 112504. 0.956268
\(344\) 103113.i 0.871359i
\(345\) 0 0
\(346\) 117054. 0.977764
\(347\) 21875.1i 0.181673i 0.995866 + 0.0908365i \(0.0289541\pi\)
−0.995866 + 0.0908365i \(0.971046\pi\)
\(348\) 0 0
\(349\) −200998. −1.65022 −0.825108 0.564975i \(-0.808886\pi\)
−0.825108 + 0.564975i \(0.808886\pi\)
\(350\) 30530.0i 0.249225i
\(351\) 0 0
\(352\) 4320.00 0.0348657
\(353\) − 241775.i − 1.94027i −0.242561 0.970136i \(-0.577988\pi\)
0.242561 0.970136i \(-0.422012\pi\)
\(354\) 0 0
\(355\) −255528. −2.02760
\(356\) − 8731.35i − 0.0688940i
\(357\) 0 0
\(358\) −232416. −1.81343
\(359\) 108187.i 0.839436i 0.907655 + 0.419718i \(0.137871\pi\)
−0.907655 + 0.419718i \(0.862129\pi\)
\(360\) 0 0
\(361\) 183279. 1.40637
\(362\) 79354.4i 0.605555i
\(363\) 0 0
\(364\) −6272.00 −0.0473373
\(365\) − 196248.i − 1.47305i
\(366\) 0 0
\(367\) 221732. 1.64625 0.823126 0.567859i \(-0.192228\pi\)
0.823126 + 0.567859i \(0.192228\pi\)
\(368\) − 226523.i − 1.67270i
\(369\) 0 0
\(370\) −104076. −0.760234
\(371\) − 50249.8i − 0.365079i
\(372\) 0 0
\(373\) 87674.0 0.630163 0.315082 0.949065i \(-0.397968\pi\)
0.315082 + 0.949065i \(0.397968\pi\)
\(374\) 6414.87i 0.0458611i
\(375\) 0 0
\(376\) 77616.0 0.549004
\(377\) 110716.i 0.778982i
\(378\) 0 0
\(379\) −40768.0 −0.283819 −0.141909 0.989880i \(-0.545324\pi\)
−0.141909 + 0.989880i \(0.545324\pi\)
\(380\) 33262.3i 0.230348i
\(381\) 0 0
\(382\) −36144.0 −0.247691
\(383\) 75077.8i 0.511816i 0.966701 + 0.255908i \(0.0823744\pi\)
−0.966701 + 0.255908i \(0.917626\pi\)
\(384\) 0 0
\(385\) −14112.0 −0.0952066
\(386\) 205335.i 1.37813i
\(387\) 0 0
\(388\) 11648.0 0.0773727
\(389\) 123236.i 0.814401i 0.913339 + 0.407201i \(0.133495\pi\)
−0.913339 + 0.407201i \(0.866505\pi\)
\(390\) 0 0
\(391\) 71064.0 0.464832
\(392\) − 96044.9i − 0.625032i
\(393\) 0 0
\(394\) −42498.0 −0.273764
\(395\) 126991.i 0.813913i
\(396\) 0 0
\(397\) −8134.00 −0.0516087 −0.0258044 0.999667i \(-0.508215\pi\)
−0.0258044 + 0.999667i \(0.508215\pi\)
\(398\) − 56427.1i − 0.356223i
\(399\) 0 0
\(400\) 72988.0 0.456175
\(401\) 133639.i 0.831083i 0.909574 + 0.415541i \(0.136408\pi\)
−0.909574 + 0.415541i \(0.863592\pi\)
\(402\) 0 0
\(403\) 40768.0 0.251021
\(404\) − 16096.6i − 0.0986213i
\(405\) 0 0
\(406\) −117432. −0.712417
\(407\) − 14017.7i − 0.0846228i
\(408\) 0 0
\(409\) 35168.0 0.210233 0.105117 0.994460i \(-0.466478\pi\)
0.105117 + 0.994460i \(0.466478\pi\)
\(410\) − 228263.i − 1.35790i
\(411\) 0 0
\(412\) 29960.0 0.176501
\(413\) − 126397.i − 0.741030i
\(414\) 0 0
\(415\) 3528.00 0.0204848
\(416\) 28510.5i 0.164748i
\(417\) 0 0
\(418\) −40320.0 −0.230764
\(419\) − 157640.i − 0.897919i −0.893552 0.448959i \(-0.851795\pi\)
0.893552 0.448959i \(-0.148205\pi\)
\(420\) 0 0
\(421\) 18800.0 0.106070 0.0530351 0.998593i \(-0.483110\pi\)
0.0530351 + 0.998593i \(0.483110\pi\)
\(422\) − 156570.i − 0.879194i
\(423\) 0 0
\(424\) −106596. −0.592938
\(425\) 22897.5i 0.126768i
\(426\) 0 0
\(427\) −73304.0 −0.402042
\(428\) − 19550.1i − 0.106724i
\(429\) 0 0
\(430\) 218736. 1.18300
\(431\) − 72243.7i − 0.388907i −0.980912 0.194453i \(-0.937707\pi\)
0.980912 0.194453i \(-0.0622933\pi\)
\(432\) 0 0
\(433\) −351106. −1.87268 −0.936338 0.351101i \(-0.885807\pi\)
−0.936338 + 0.351101i \(0.885807\pi\)
\(434\) 43241.0i 0.229571i
\(435\) 0 0
\(436\) −21280.0 −0.111943
\(437\) 446665.i 2.33894i
\(438\) 0 0
\(439\) −60508.0 −0.313967 −0.156983 0.987601i \(-0.550177\pi\)
−0.156983 + 0.987601i \(0.550177\pi\)
\(440\) 29936.1i 0.154628i
\(441\) 0 0
\(442\) −42336.0 −0.216703
\(443\) − 57105.9i − 0.290987i −0.989359 0.145494i \(-0.953523\pi\)
0.989359 0.145494i \(-0.0464770\pi\)
\(444\) 0 0
\(445\) 129654. 0.654736
\(446\) 217987.i 1.09587i
\(447\) 0 0
\(448\) 96992.0 0.483259
\(449\) − 210482.i − 1.04405i −0.852930 0.522025i \(-0.825177\pi\)
0.852930 0.522025i \(-0.174823\pi\)
\(450\) 0 0
\(451\) 30744.0 0.151150
\(452\) − 5744.54i − 0.0281176i
\(453\) 0 0
\(454\) 383544. 1.86082
\(455\) − 93134.4i − 0.449871i
\(456\) 0 0
\(457\) 324800. 1.55519 0.777595 0.628765i \(-0.216439\pi\)
0.777595 + 0.628765i \(0.216439\pi\)
\(458\) 333098.i 1.58797i
\(459\) 0 0
\(460\) −47376.0 −0.223894
\(461\) − 165331.i − 0.777954i −0.921248 0.388977i \(-0.872829\pi\)
0.921248 0.388977i \(-0.127171\pi\)
\(462\) 0 0
\(463\) −117220. −0.546814 −0.273407 0.961898i \(-0.588151\pi\)
−0.273407 + 0.961898i \(0.588151\pi\)
\(464\) 280744.i 1.30399i
\(465\) 0 0
\(466\) −222534. −1.02477
\(467\) − 170707.i − 0.782740i −0.920233 0.391370i \(-0.872001\pi\)
0.920233 0.391370i \(-0.127999\pi\)
\(468\) 0 0
\(469\) 105952. 0.481685
\(470\) − 164648.i − 0.745353i
\(471\) 0 0
\(472\) −268128. −1.20353
\(473\) 29460.9i 0.131681i
\(474\) 0 0
\(475\) −143920. −0.637873
\(476\) − 4989.35i − 0.0220206i
\(477\) 0 0
\(478\) 94464.0 0.413438
\(479\) − 99668.1i − 0.434395i −0.976128 0.217198i \(-0.930308\pi\)
0.976128 0.217198i \(-0.0696916\pi\)
\(480\) 0 0
\(481\) 92512.0 0.399860
\(482\) − 225708.i − 0.971525i
\(483\) 0 0
\(484\) −28706.0 −0.122541
\(485\) 172964.i 0.735313i
\(486\) 0 0
\(487\) −416500. −1.75613 −0.878066 0.478540i \(-0.841166\pi\)
−0.878066 + 0.478540i \(0.841166\pi\)
\(488\) 155501.i 0.652972i
\(489\) 0 0
\(490\) −203742. −0.848571
\(491\) − 355024.i − 1.47263i −0.676637 0.736317i \(-0.736563\pi\)
0.676637 0.736317i \(-0.263437\pi\)
\(492\) 0 0
\(493\) −88074.0 −0.362371
\(494\) − 266098.i − 1.09041i
\(495\) 0 0
\(496\) 103376. 0.420200
\(497\) 240914.i 0.975325i
\(498\) 0 0
\(499\) 135968. 0.546054 0.273027 0.962006i \(-0.411975\pi\)
0.273027 + 0.962006i \(0.411975\pi\)
\(500\) 21858.1i 0.0874323i
\(501\) 0 0
\(502\) 352296. 1.39798
\(503\) 170707.i 0.674707i 0.941378 + 0.337353i \(0.109532\pi\)
−0.941378 + 0.337353i \(0.890468\pi\)
\(504\) 0 0
\(505\) 239022. 0.937249
\(506\) − 57428.4i − 0.224298i
\(507\) 0 0
\(508\) 38840.0 0.150505
\(509\) − 193367.i − 0.746357i −0.927760 0.373178i \(-0.878268\pi\)
0.927760 0.373178i \(-0.121732\pi\)
\(510\) 0 0
\(511\) −185024. −0.708576
\(512\) − 197605.i − 0.753804i
\(513\) 0 0
\(514\) 359226. 1.35970
\(515\) 444883.i 1.67738i
\(516\) 0 0
\(517\) 22176.0 0.0829664
\(518\) 98123.8i 0.365691i
\(519\) 0 0
\(520\) −197568. −0.730651
\(521\) 391931.i 1.44389i 0.691951 + 0.721945i \(0.256752\pi\)
−0.691951 + 0.721945i \(0.743248\pi\)
\(522\) 0 0
\(523\) 410816. 1.50191 0.750955 0.660353i \(-0.229593\pi\)
0.750955 + 0.660353i \(0.229593\pi\)
\(524\) − 40389.9i − 0.147099i
\(525\) 0 0
\(526\) −468144. −1.69203
\(527\) 32430.7i 0.116771i
\(528\) 0 0
\(529\) −356351. −1.27341
\(530\) 226124.i 0.804999i
\(531\) 0 0
\(532\) 31360.0 0.110803
\(533\) 202900.i 0.714213i
\(534\) 0 0
\(535\) 290304. 1.01425
\(536\) − 224758.i − 0.782323i
\(537\) 0 0
\(538\) −116802. −0.403539
\(539\) − 27441.4i − 0.0944558i
\(540\) 0 0
\(541\) −57616.0 −0.196856 −0.0984280 0.995144i \(-0.531381\pi\)
−0.0984280 + 0.995144i \(0.531381\pi\)
\(542\) − 310884.i − 1.05828i
\(543\) 0 0
\(544\) −22680.0 −0.0766382
\(545\) − 315992.i − 1.06386i
\(546\) 0 0
\(547\) 78920.0 0.263762 0.131881 0.991266i \(-0.457898\pi\)
0.131881 + 0.991266i \(0.457898\pi\)
\(548\) 34662.4i 0.115424i
\(549\) 0 0
\(550\) 18504.0 0.0611702
\(551\) − 553580.i − 1.82338i
\(552\) 0 0
\(553\) 119728. 0.391512
\(554\) 198284.i 0.646053i
\(555\) 0 0
\(556\) −37072.0 −0.119921
\(557\) − 178025.i − 0.573815i −0.957958 0.286907i \(-0.907373\pi\)
0.957958 0.286907i \(-0.0926272\pi\)
\(558\) 0 0
\(559\) −194432. −0.622220
\(560\) − 236162.i − 0.753069i
\(561\) 0 0
\(562\) −478458. −1.51486
\(563\) 431816.i 1.36233i 0.732131 + 0.681164i \(0.238526\pi\)
−0.732131 + 0.681164i \(0.761474\pi\)
\(564\) 0 0
\(565\) 85302.0 0.267216
\(566\) − 103113.i − 0.321870i
\(567\) 0 0
\(568\) 511056. 1.58406
\(569\) 65756.7i 0.203103i 0.994830 + 0.101551i \(0.0323806\pi\)
−0.994830 + 0.101551i \(0.967619\pi\)
\(570\) 0 0
\(571\) 111632. 0.342386 0.171193 0.985237i \(-0.445238\pi\)
0.171193 + 0.985237i \(0.445238\pi\)
\(572\) 3801.41i 0.0116186i
\(573\) 0 0
\(574\) −215208. −0.653183
\(575\) − 204987.i − 0.620000i
\(576\) 0 0
\(577\) −164878. −0.495235 −0.247617 0.968858i \(-0.579648\pi\)
−0.247617 + 0.968858i \(0.579648\pi\)
\(578\) 320672.i 0.959853i
\(579\) 0 0
\(580\) 58716.0 0.174542
\(581\) − 3326.23i − 0.00985372i
\(582\) 0 0
\(583\) −30456.0 −0.0896057
\(584\) 392495.i 1.15082i
\(585\) 0 0
\(586\) 40194.0 0.117049
\(587\) 568548.i 1.65003i 0.565114 + 0.825013i \(0.308832\pi\)
−0.565114 + 0.825013i \(0.691168\pi\)
\(588\) 0 0
\(589\) −203840. −0.587569
\(590\) 568785.i 1.63397i
\(591\) 0 0
\(592\) 234584. 0.669353
\(593\) − 580279.i − 1.65016i −0.565013 0.825082i \(-0.691129\pi\)
0.565013 0.825082i \(-0.308871\pi\)
\(594\) 0 0
\(595\) 74088.0 0.209273
\(596\) − 6304.56i − 0.0177485i
\(597\) 0 0
\(598\) 379008. 1.05985
\(599\) − 5719.08i − 0.0159394i −0.999968 0.00796971i \(-0.997463\pi\)
0.999968 0.00796971i \(-0.00253686\pi\)
\(600\) 0 0
\(601\) 233198. 0.645618 0.322809 0.946464i \(-0.395373\pi\)
0.322809 + 0.946464i \(0.395373\pi\)
\(602\) − 206226.i − 0.569051i
\(603\) 0 0
\(604\) −14056.0 −0.0385290
\(605\) − 426262.i − 1.16457i
\(606\) 0 0
\(607\) 154868. 0.420324 0.210162 0.977667i \(-0.432601\pi\)
0.210162 + 0.977667i \(0.432601\pi\)
\(608\) − 142553.i − 0.385628i
\(609\) 0 0
\(610\) 329868. 0.886504
\(611\) 146354.i 0.392033i
\(612\) 0 0
\(613\) −333862. −0.888477 −0.444238 0.895909i \(-0.646526\pi\)
−0.444238 + 0.895909i \(0.646526\pi\)
\(614\) 81492.6i 0.216163i
\(615\) 0 0
\(616\) 28224.0 0.0743802
\(617\) − 19325.2i − 0.0507638i −0.999678 0.0253819i \(-0.991920\pi\)
0.999678 0.0253819i \(-0.00808018\pi\)
\(618\) 0 0
\(619\) −473536. −1.23587 −0.617933 0.786230i \(-0.712030\pi\)
−0.617933 + 0.786230i \(0.712030\pi\)
\(620\) − 21620.5i − 0.0562448i
\(621\) 0 0
\(622\) −333648. −0.862398
\(623\) − 122239.i − 0.314944i
\(624\) 0 0
\(625\) −485201. −1.24211
\(626\) − 542710.i − 1.38490i
\(627\) 0 0
\(628\) 11060.0 0.0280437
\(629\) 73592.8i 0.186009i
\(630\) 0 0
\(631\) −384820. −0.966493 −0.483247 0.875484i \(-0.660543\pi\)
−0.483247 + 0.875484i \(0.660543\pi\)
\(632\) − 253981.i − 0.635869i
\(633\) 0 0
\(634\) 279486. 0.695315
\(635\) 576745.i 1.43033i
\(636\) 0 0
\(637\) 181104. 0.446323
\(638\) 71174.5i 0.174857i
\(639\) 0 0
\(640\) −557424. −1.36090
\(641\) − 145832.i − 0.354926i −0.984127 0.177463i \(-0.943211\pi\)
0.984127 0.177463i \(-0.0567890\pi\)
\(642\) 0 0
\(643\) 528584. 1.27847 0.639237 0.769010i \(-0.279250\pi\)
0.639237 + 0.769010i \(0.279250\pi\)
\(644\) 44666.5i 0.107699i
\(645\) 0 0
\(646\) 211680. 0.507242
\(647\) − 176409.i − 0.421417i −0.977549 0.210709i \(-0.932423\pi\)
0.977549 0.210709i \(-0.0675771\pi\)
\(648\) 0 0
\(649\) −76608.0 −0.181880
\(650\) 122120.i 0.289042i
\(651\) 0 0
\(652\) −21712.0 −0.0510746
\(653\) − 403250.i − 0.945689i −0.881146 0.472844i \(-0.843227\pi\)
0.881146 0.472844i \(-0.156773\pi\)
\(654\) 0 0
\(655\) 599760. 1.39796
\(656\) 514497.i 1.19557i
\(657\) 0 0
\(658\) −155232. −0.358533
\(659\) 137020.i 0.315511i 0.987478 + 0.157755i \(0.0504257\pi\)
−0.987478 + 0.157755i \(0.949574\pi\)
\(660\) 0 0
\(661\) 286790. 0.656389 0.328194 0.944610i \(-0.393560\pi\)
0.328194 + 0.944610i \(0.393560\pi\)
\(662\) 67237.4i 0.153425i
\(663\) 0 0
\(664\) −7056.00 −0.0160038
\(665\) 465672.i 1.05302i
\(666\) 0 0
\(667\) 788472. 1.77229
\(668\) 59397.0i 0.133110i
\(669\) 0 0
\(670\) −476784. −1.06212
\(671\) 44428.9i 0.0986781i
\(672\) 0 0
\(673\) 121058. 0.267278 0.133639 0.991030i \(-0.457334\pi\)
0.133639 + 0.991030i \(0.457334\pi\)
\(674\) − 539324.i − 1.18722i
\(675\) 0 0
\(676\) 32034.0 0.0701000
\(677\) 377260.i 0.823120i 0.911383 + 0.411560i \(0.135016\pi\)
−0.911383 + 0.411560i \(0.864984\pi\)
\(678\) 0 0
\(679\) 163072. 0.353704
\(680\) − 157164.i − 0.339888i
\(681\) 0 0
\(682\) 26208.0 0.0563463
\(683\) 335457.i 0.719110i 0.933124 + 0.359555i \(0.117072\pi\)
−0.933124 + 0.359555i \(0.882928\pi\)
\(684\) 0 0
\(685\) −514710. −1.09694
\(686\) 477314.i 1.01428i
\(687\) 0 0
\(688\) −493024. −1.04158
\(689\) − 200999.i − 0.423405i
\(690\) 0 0
\(691\) −578872. −1.21235 −0.606173 0.795333i \(-0.707296\pi\)
−0.606173 + 0.795333i \(0.707296\pi\)
\(692\) 55179.8i 0.115231i
\(693\) 0 0
\(694\) −92808.0 −0.192693
\(695\) − 550491.i − 1.13967i
\(696\) 0 0
\(697\) −161406. −0.332242
\(698\) − 852762.i − 1.75032i
\(699\) 0 0
\(700\) −14392.0 −0.0293714
\(701\) 654279.i 1.33146i 0.746194 + 0.665728i \(0.231879\pi\)
−0.746194 + 0.665728i \(0.768121\pi\)
\(702\) 0 0
\(703\) −462560. −0.935960
\(704\) − 58786.0i − 0.118612i
\(705\) 0 0
\(706\) 1.02577e6 2.05797
\(707\) − 225352.i − 0.450840i
\(708\) 0 0
\(709\) 779408. 1.55050 0.775251 0.631653i \(-0.217623\pi\)
0.775251 + 0.631653i \(0.217623\pi\)
\(710\) − 1.08411e6i − 2.15059i
\(711\) 0 0
\(712\) −259308. −0.511512
\(713\) − 290332.i − 0.571106i
\(714\) 0 0
\(715\) −56448.0 −0.110417
\(716\) − 109562.i − 0.213714i
\(717\) 0 0
\(718\) −459000. −0.890356
\(719\) − 147898.i − 0.286092i −0.989716 0.143046i \(-0.954310\pi\)
0.989716 0.143046i \(-0.0456897\pi\)
\(720\) 0 0
\(721\) 419440. 0.806862
\(722\) 777587.i 1.49168i
\(723\) 0 0
\(724\) −37408.0 −0.0713653
\(725\) 254054.i 0.483336i
\(726\) 0 0
\(727\) −713188. −1.34938 −0.674691 0.738100i \(-0.735723\pi\)
−0.674691 + 0.738100i \(0.735723\pi\)
\(728\) 186269.i 0.351461i
\(729\) 0 0
\(730\) 832608. 1.56241
\(731\) − 154670.i − 0.289448i
\(732\) 0 0
\(733\) −826672. −1.53860 −0.769299 0.638889i \(-0.779394\pi\)
−0.769299 + 0.638889i \(0.779394\pi\)
\(734\) 940729.i 1.74611i
\(735\) 0 0
\(736\) 203040. 0.374823
\(737\) − 64216.6i − 0.118226i
\(738\) 0 0
\(739\) 703280. 1.28777 0.643887 0.765121i \(-0.277321\pi\)
0.643887 + 0.765121i \(0.277321\pi\)
\(740\) − 49061.9i − 0.0895944i
\(741\) 0 0
\(742\) 213192. 0.387225
\(743\) 469643.i 0.850728i 0.905022 + 0.425364i \(0.139854\pi\)
−0.905022 + 0.425364i \(0.860146\pi\)
\(744\) 0 0
\(745\) 93618.0 0.168673
\(746\) 371969.i 0.668389i
\(747\) 0 0
\(748\) −3024.00 −0.00540479
\(749\) − 273701.i − 0.487880i
\(750\) 0 0
\(751\) −963004. −1.70745 −0.853725 0.520723i \(-0.825662\pi\)
−0.853725 + 0.520723i \(0.825662\pi\)
\(752\) 371112.i 0.656250i
\(753\) 0 0
\(754\) −469728. −0.826235
\(755\) − 208721.i − 0.366161i
\(756\) 0 0
\(757\) 899696. 1.57002 0.785008 0.619486i \(-0.212659\pi\)
0.785008 + 0.619486i \(0.212659\pi\)
\(758\) − 172964.i − 0.301035i
\(759\) 0 0
\(760\) 987840. 1.71025
\(761\) 886292.i 1.53041i 0.643787 + 0.765204i \(0.277362\pi\)
−0.643787 + 0.765204i \(0.722638\pi\)
\(762\) 0 0
\(763\) −297920. −0.511741
\(764\) − 17038.4i − 0.0291906i
\(765\) 0 0
\(766\) −318528. −0.542863
\(767\) − 505587.i − 0.859419i
\(768\) 0 0
\(769\) 183218. 0.309824 0.154912 0.987928i \(-0.450491\pi\)
0.154912 + 0.987928i \(0.450491\pi\)
\(770\) − 59872.1i − 0.100982i
\(771\) 0 0
\(772\) −96796.0 −0.162414
\(773\) − 1.11521e6i − 1.86637i −0.359402 0.933183i \(-0.617019\pi\)
0.359402 0.933183i \(-0.382981\pi\)
\(774\) 0 0
\(775\) 93548.0 0.155751
\(776\) − 345928.i − 0.574463i
\(777\) 0 0
\(778\) −522846. −0.863803
\(779\) − 1.01450e6i − 1.67177i
\(780\) 0 0
\(781\) 146016. 0.239386
\(782\) 301499.i 0.493029i
\(783\) 0 0
\(784\) 459228. 0.747130
\(785\) 164233.i 0.266514i
\(786\) 0 0
\(787\) −276304. −0.446106 −0.223053 0.974806i \(-0.571602\pi\)
−0.223053 + 0.974806i \(0.571602\pi\)
\(788\) − 20033.7i − 0.0322634i
\(789\) 0 0
\(790\) −538776. −0.863285
\(791\) − 80423.5i − 0.128538i
\(792\) 0 0
\(793\) −293216. −0.466274
\(794\) − 34509.6i − 0.0547393i
\(795\) 0 0
\(796\) 26600.0 0.0419813
\(797\) 249438.i 0.392686i 0.980535 + 0.196343i \(0.0629066\pi\)
−0.980535 + 0.196343i \(0.937093\pi\)
\(798\) 0 0
\(799\) −116424. −0.182368
\(800\) 65421.5i 0.102221i
\(801\) 0 0
\(802\) −566982. −0.881496
\(803\) 112141.i 0.173914i
\(804\) 0 0
\(805\) −663264. −1.02352
\(806\) 172964.i 0.266248i
\(807\) 0 0
\(808\) −478044. −0.732226
\(809\) 657868.i 1.00518i 0.864526 + 0.502588i \(0.167619\pi\)
−0.864526 + 0.502588i \(0.832381\pi\)
\(810\) 0 0
\(811\) 660800. 1.00468 0.502341 0.864670i \(-0.332472\pi\)
0.502341 + 0.864670i \(0.332472\pi\)
\(812\) − 55358.0i − 0.0839592i
\(813\) 0 0
\(814\) 59472.0 0.0897561
\(815\) − 322407.i − 0.485388i
\(816\) 0 0
\(817\) 972160. 1.45644
\(818\) 149205.i 0.222986i
\(819\) 0 0
\(820\) 107604. 0.160030
\(821\) 1.23522e6i 1.83256i 0.400533 + 0.916282i \(0.368825\pi\)
−0.400533 + 0.916282i \(0.631175\pi\)
\(822\) 0 0
\(823\) 1.07330e6 1.58461 0.792303 0.610127i \(-0.208882\pi\)
0.792303 + 0.610127i \(0.208882\pi\)
\(824\) − 889767.i − 1.31045i
\(825\) 0 0
\(826\) 536256. 0.785981
\(827\) − 356687.i − 0.521527i −0.965403 0.260763i \(-0.916026\pi\)
0.965403 0.260763i \(-0.0839742\pi\)
\(828\) 0 0
\(829\) 91280.0 0.132821 0.0664105 0.997792i \(-0.478845\pi\)
0.0664105 + 0.997792i \(0.478845\pi\)
\(830\) 14968.0i 0.0217274i
\(831\) 0 0
\(832\) 387968. 0.560466
\(833\) 144067.i 0.207623i
\(834\) 0 0
\(835\) −882000. −1.26501
\(836\) − 19007.0i − 0.0271958i
\(837\) 0 0
\(838\) 668808. 0.952387
\(839\) 1.18473e6i 1.68305i 0.540221 + 0.841523i \(0.318341\pi\)
−0.540221 + 0.841523i \(0.681659\pi\)
\(840\) 0 0
\(841\) −269921. −0.381632
\(842\) 79761.6i 0.112505i
\(843\) 0 0
\(844\) 73808.0 0.103614
\(845\) 475681.i 0.666196i
\(846\) 0 0
\(847\) −401884. −0.560188
\(848\) − 509677.i − 0.708767i
\(849\) 0 0
\(850\) −97146.0 −0.134458
\(851\) − 658831.i − 0.909735i
\(852\) 0 0
\(853\) 429254. 0.589951 0.294976 0.955505i \(-0.404688\pi\)
0.294976 + 0.955505i \(0.404688\pi\)
\(854\) − 311003.i − 0.426430i
\(855\) 0 0
\(856\) −580608. −0.792384
\(857\) − 337286.i − 0.459236i −0.973281 0.229618i \(-0.926252\pi\)
0.973281 0.229618i \(-0.0737477\pi\)
\(858\) 0 0
\(859\) −535864. −0.726220 −0.363110 0.931746i \(-0.618285\pi\)
−0.363110 + 0.931746i \(0.618285\pi\)
\(860\) 103113.i 0.139417i
\(861\) 0 0
\(862\) 306504. 0.412498
\(863\) 894722.i 1.20134i 0.799496 + 0.600671i \(0.205100\pi\)
−0.799496 + 0.600671i \(0.794900\pi\)
\(864\) 0 0
\(865\) −819378. −1.09510
\(866\) − 1.48962e6i − 1.98627i
\(867\) 0 0
\(868\) −20384.0 −0.0270552
\(869\) − 72566.1i − 0.0960936i
\(870\) 0 0
\(871\) 423808. 0.558641
\(872\) 631984.i 0.831138i
\(873\) 0 0
\(874\) −1.89504e6 −2.48082
\(875\) 306013.i 0.399691i
\(876\) 0 0
\(877\) −179878. −0.233872 −0.116936 0.993139i \(-0.537307\pi\)
−0.116936 + 0.993139i \(0.537307\pi\)
\(878\) − 256714.i − 0.333012i
\(879\) 0 0
\(880\) −143136. −0.184835
\(881\) − 1.27451e6i − 1.64207i −0.570878 0.821035i \(-0.693397\pi\)
0.570878 0.821035i \(-0.306603\pi\)
\(882\) 0 0
\(883\) 383096. 0.491345 0.245672 0.969353i \(-0.420991\pi\)
0.245672 + 0.969353i \(0.420991\pi\)
\(884\) − 19957.4i − 0.0255387i
\(885\) 0 0
\(886\) 242280. 0.308639
\(887\) − 450229.i − 0.572251i −0.958192 0.286125i \(-0.907633\pi\)
0.958192 0.286125i \(-0.0923674\pi\)
\(888\) 0 0
\(889\) 543760. 0.688024
\(890\) 550075.i 0.694452i
\(891\) 0 0
\(892\) −102760. −0.129150
\(893\) − 731771.i − 0.917639i
\(894\) 0 0
\(895\) 1.62691e6 2.03104
\(896\) 525544.i 0.654626i
\(897\) 0 0
\(898\) 892998. 1.10738
\(899\) 359827.i 0.445219i
\(900\) 0 0
\(901\) 159894. 0.196962
\(902\) 130436.i 0.160318i
\(903\) 0 0
\(904\) −170604. −0.208762
\(905\) − 555480.i − 0.678222i
\(906\) 0 0
\(907\) −1.09900e6 −1.33593 −0.667964 0.744193i \(-0.732834\pi\)
−0.667964 + 0.744193i \(0.732834\pi\)
\(908\) 180804.i 0.219299i
\(909\) 0 0
\(910\) 395136. 0.477160
\(911\) 284461.i 0.342756i 0.985205 + 0.171378i \(0.0548220\pi\)
−0.985205 + 0.171378i \(0.945178\pi\)
\(912\) 0 0
\(913\) −2016.00 −0.00241852
\(914\) 1.37801e6i 1.64953i
\(915\) 0 0
\(916\) −157024. −0.187144
\(917\) − 565459.i − 0.672454i
\(918\) 0 0
\(919\) −490588. −0.580879 −0.290440 0.956893i \(-0.593802\pi\)
−0.290440 + 0.956893i \(0.593802\pi\)
\(920\) 1.40700e6i 1.66233i
\(921\) 0 0
\(922\) 701442. 0.825144
\(923\) 963656.i 1.13115i
\(924\) 0 0
\(925\) 212282. 0.248102
\(926\) − 497322.i − 0.579984i
\(927\) 0 0
\(928\) −251640. −0.292202
\(929\) 651852.i 0.755297i 0.925949 + 0.377648i \(0.123267\pi\)
−0.925949 + 0.377648i \(0.876733\pi\)
\(930\) 0 0
\(931\) −905520. −1.04472
\(932\) − 104904.i − 0.120770i
\(933\) 0 0
\(934\) 724248. 0.830221
\(935\) − 44904.1i − 0.0513645i
\(936\) 0 0
\(937\) −947842. −1.07958 −0.539792 0.841798i \(-0.681497\pi\)
−0.539792 + 0.841798i \(0.681497\pi\)
\(938\) 449516.i 0.510905i
\(939\) 0 0
\(940\) 77616.0 0.0878407
\(941\) − 214750.i − 0.242523i −0.992621 0.121262i \(-0.961306\pi\)
0.992621 0.121262i \(-0.0386940\pi\)
\(942\) 0 0
\(943\) 1.44497e6 1.62493
\(944\) − 1.28202e6i − 1.43864i
\(945\) 0 0
\(946\) −124992. −0.139669
\(947\) − 1.18220e6i − 1.31823i −0.752041 0.659117i \(-0.770930\pi\)
0.752041 0.659117i \(-0.229070\pi\)
\(948\) 0 0
\(949\) −740096. −0.821780
\(950\) − 610601.i − 0.676566i
\(951\) 0 0
\(952\) −148176. −0.163495
\(953\) − 426729.i − 0.469858i −0.972013 0.234929i \(-0.924514\pi\)
0.972013 0.234929i \(-0.0754857\pi\)
\(954\) 0 0
\(955\) 253008. 0.277413
\(956\) 44530.8i 0.0487242i
\(957\) 0 0
\(958\) 422856. 0.460746
\(959\) 485273.i 0.527654i
\(960\) 0 0
\(961\) −791025. −0.856532
\(962\) 392495.i 0.424116i
\(963\) 0 0
\(964\) 106400. 0.114495
\(965\) − 1.43735e6i − 1.54350i
\(966\) 0 0
\(967\) 392828. 0.420097 0.210048 0.977691i \(-0.432638\pi\)
0.210048 + 0.977691i \(0.432638\pi\)
\(968\) 852525.i 0.909822i
\(969\) 0 0
\(970\) −733824. −0.779917
\(971\) − 567716.i − 0.602134i −0.953603 0.301067i \(-0.902657\pi\)
0.953603 0.301067i \(-0.0973427\pi\)
\(972\) 0 0
\(973\) −519008. −0.548212
\(974\) − 1.76706e6i − 1.86266i
\(975\) 0 0
\(976\) −743512. −0.780528
\(977\) − 122370.i − 0.128200i −0.997943 0.0640999i \(-0.979582\pi\)
0.997943 0.0640999i \(-0.0204176\pi\)
\(978\) 0 0
\(979\) −74088.0 −0.0773005
\(980\) − 96044.9i − 0.100005i
\(981\) 0 0
\(982\) 1.50624e6 1.56196
\(983\) 1.58495e6i 1.64024i 0.572189 + 0.820121i \(0.306094\pi\)
−0.572189 + 0.820121i \(0.693906\pi\)
\(984\) 0 0
\(985\) 297486. 0.306615
\(986\) − 373666.i − 0.384353i
\(987\) 0 0
\(988\) 125440. 0.128506
\(989\) 1.38466e6i 1.41563i
\(990\) 0 0
\(991\) 7364.00 0.00749836 0.00374918 0.999993i \(-0.498807\pi\)
0.00374918 + 0.999993i \(0.498807\pi\)
\(992\) 92659.3i 0.0941598i
\(993\) 0 0
\(994\) −1.02211e6 −1.03449
\(995\) 394990.i 0.398970i
\(996\) 0 0
\(997\) −1.43479e6 −1.44344 −0.721719 0.692186i \(-0.756648\pi\)
−0.721719 + 0.692186i \(0.756648\pi\)
\(998\) 576863.i 0.579178i
\(999\) 0 0
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 9.5.b.a.8.2 yes 2
3.2 odd 2 inner 9.5.b.a.8.1 2
4.3 odd 2 144.5.e.c.17.1 2
5.2 odd 4 225.5.d.a.224.1 4
5.3 odd 4 225.5.d.a.224.4 4
5.4 even 2 225.5.c.a.26.1 2
7.6 odd 2 441.5.b.a.197.2 2
8.3 odd 2 576.5.e.g.449.2 2
8.5 even 2 576.5.e.d.449.2 2
9.2 odd 6 81.5.d.c.53.1 4
9.4 even 3 81.5.d.c.26.1 4
9.5 odd 6 81.5.d.c.26.2 4
9.7 even 3 81.5.d.c.53.2 4
12.11 even 2 144.5.e.c.17.2 2
15.2 even 4 225.5.d.a.224.3 4
15.8 even 4 225.5.d.a.224.2 4
15.14 odd 2 225.5.c.a.26.2 2
21.20 even 2 441.5.b.a.197.1 2
24.5 odd 2 576.5.e.d.449.1 2
24.11 even 2 576.5.e.g.449.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.5.b.a.8.1 2 3.2 odd 2 inner
9.5.b.a.8.2 yes 2 1.1 even 1 trivial
81.5.d.c.26.1 4 9.4 even 3
81.5.d.c.26.2 4 9.5 odd 6
81.5.d.c.53.1 4 9.2 odd 6
81.5.d.c.53.2 4 9.7 even 3
144.5.e.c.17.1 2 4.3 odd 2
144.5.e.c.17.2 2 12.11 even 2
225.5.c.a.26.1 2 5.4 even 2
225.5.c.a.26.2 2 15.14 odd 2
225.5.d.a.224.1 4 5.2 odd 4
225.5.d.a.224.2 4 15.8 even 4
225.5.d.a.224.3 4 15.2 even 4
225.5.d.a.224.4 4 5.3 odd 4
441.5.b.a.197.1 2 21.20 even 2
441.5.b.a.197.2 2 7.6 odd 2
576.5.e.d.449.1 2 24.5 odd 2
576.5.e.d.449.2 2 8.5 even 2
576.5.e.g.449.1 2 24.11 even 2
576.5.e.g.449.2 2 8.3 odd 2