Properties

Label 124.5.o.a.13.7
Level $124$
Weight $5$
Character 124.13
Analytic conductor $12.818$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [124,5,Mod(13,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("124.13");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 124.o (of order \(30\), degree \(8\), 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: \(12.8178754224\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(11\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label 13.7
Character \(\chi\) \(=\) 124.13
Dual form 124.5.o.a.105.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.72821 + 3.88162i) q^{3} +(7.33763 + 12.7091i) q^{5} +(-56.6769 - 62.9461i) q^{7} +(42.1193 - 46.7783i) q^{9} +O(q^{10})\) \(q+(1.72821 + 3.88162i) q^{3} +(7.33763 + 12.7091i) q^{5} +(-56.6769 - 62.9461i) q^{7} +(42.1193 - 46.7783i) q^{9} +(-27.3674 - 128.753i) q^{11} +(286.317 + 30.0931i) q^{13} +(-36.6511 + 50.4459i) q^{15} +(-42.0787 + 197.965i) q^{17} +(-4.62881 - 44.0402i) q^{19} +(146.383 - 328.782i) q^{21} +(492.853 - 160.138i) q^{23} +(204.818 - 354.756i) q^{25} +(581.687 + 189.002i) q^{27} +(-285.727 - 393.270i) q^{29} +(-323.087 - 905.061i) q^{31} +(452.475 - 328.742i) q^{33} +(384.117 - 1182.19i) q^{35} +(1336.70 + 771.741i) q^{37} +(378.005 + 1163.38i) q^{39} +(-2144.71 - 954.886i) q^{41} +(1196.64 - 125.771i) q^{43} +(903.567 + 192.059i) q^{45} +(-653.336 - 474.676i) q^{47} +(-498.965 + 4747.34i) q^{49} +(-841.144 + 178.791i) q^{51} +(-3356.62 - 3022.31i) q^{53} +(1435.53 - 1292.56i) q^{55} +(162.948 - 94.0778i) q^{57} +(2633.61 - 1172.56i) q^{59} -1953.56i q^{61} -5331.70 q^{63} +(1718.43 + 3859.65i) q^{65} +(-1871.10 - 3240.84i) q^{67} +(1473.34 + 1636.32i) q^{69} +(-6104.64 + 6779.89i) q^{71} +(1624.55 + 7642.89i) q^{73} +(1731.00 + 181.935i) q^{75} +(-6553.43 + 9020.02i) q^{77} +(-1532.51 + 7209.92i) q^{79} +(-261.311 - 2486.20i) q^{81} +(388.781 - 873.218i) q^{83} +(-2824.72 + 917.808i) q^{85} +(1032.73 - 1788.74i) q^{87} +(13606.0 + 4420.86i) q^{89} +(-14333.3 - 19728.1i) q^{91} +(2954.74 - 2818.23i) q^{93} +(525.749 - 381.979i) q^{95} +(2835.75 - 8727.54i) q^{97} +(-7175.56 - 4142.81i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 88 q - 9 q^{3} + 3 q^{5} - 215 q^{7} - 254 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 88 q - 9 q^{3} + 3 q^{5} - 215 q^{7} - 254 q^{9} - 42 q^{11} + 6 q^{13} + 665 q^{15} - 585 q^{17} - 153 q^{19} - 402 q^{21} - 1365 q^{23} - 5933 q^{25} - 9225 q^{27} - 1140 q^{29} + 117 q^{31} + 5151 q^{33} + 2898 q^{35} + 6594 q^{37} + 3173 q^{39} - 9393 q^{41} - 5322 q^{43} + 2010 q^{45} - 5112 q^{47} - 5210 q^{49} - 1829 q^{51} + 7395 q^{53} + 10585 q^{55} + 40485 q^{57} + 5625 q^{59} - 14954 q^{63} - 17094 q^{65} + 8909 q^{67} - 35370 q^{69} - 11811 q^{71} - 22105 q^{73} + 79377 q^{75} + 71490 q^{77} + 219 q^{79} - 5422 q^{81} + 10545 q^{83} - 53630 q^{85} + 13732 q^{87} - 40305 q^{89} + 42760 q^{91} - 1028 q^{93} + 62319 q^{95} + 35201 q^{97} + 16197 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\)
\(\chi(n)\) \(1\) \(e\left(\frac{11}{30}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.72821 + 3.88162i 0.192023 + 0.431291i 0.983740 0.179597i \(-0.0574794\pi\)
−0.791717 + 0.610888i \(0.790813\pi\)
\(4\) 0 0
\(5\) 7.33763 + 12.7091i 0.293505 + 0.508366i 0.974636 0.223796i \(-0.0718449\pi\)
−0.681131 + 0.732162i \(0.738512\pi\)
\(6\) 0 0
\(7\) −56.6769 62.9461i −1.15667 1.28461i −0.952108 0.305761i \(-0.901089\pi\)
−0.204564 0.978853i \(-0.565578\pi\)
\(8\) 0 0
\(9\) 42.1193 46.7783i 0.519992 0.577509i
\(10\) 0 0
\(11\) −27.3674 128.753i −0.226177 1.06408i −0.933885 0.357573i \(-0.883604\pi\)
0.707708 0.706505i \(-0.249729\pi\)
\(12\) 0 0
\(13\) 286.317 + 30.0931i 1.69418 + 0.178066i 0.901857 0.432034i \(-0.142204\pi\)
0.792325 + 0.610100i \(0.208871\pi\)
\(14\) 0 0
\(15\) −36.6511 + 50.4459i −0.162894 + 0.224204i
\(16\) 0 0
\(17\) −42.0787 + 197.965i −0.145601 + 0.685000i 0.843424 + 0.537249i \(0.180537\pi\)
−0.989025 + 0.147750i \(0.952797\pi\)
\(18\) 0 0
\(19\) −4.62881 44.0402i −0.0128222 0.121995i 0.986238 0.165329i \(-0.0528686\pi\)
−0.999061 + 0.0433342i \(0.986202\pi\)
\(20\) 0 0
\(21\) 146.383 328.782i 0.331935 0.745537i
\(22\) 0 0
\(23\) 492.853 160.138i 0.931669 0.302718i 0.196424 0.980519i \(-0.437067\pi\)
0.735245 + 0.677801i \(0.237067\pi\)
\(24\) 0 0
\(25\) 204.818 354.756i 0.327710 0.567610i
\(26\) 0 0
\(27\) 581.687 + 189.002i 0.797925 + 0.259261i
\(28\) 0 0
\(29\) −285.727 393.270i −0.339747 0.467622i 0.604620 0.796514i \(-0.293325\pi\)
−0.944368 + 0.328892i \(0.893325\pi\)
\(30\) 0 0
\(31\) −323.087 905.061i −0.336198 0.941791i
\(32\) 0 0
\(33\) 452.475 328.742i 0.415496 0.301875i
\(34\) 0 0
\(35\) 384.117 1182.19i 0.313565 0.965053i
\(36\) 0 0
\(37\) 1336.70 + 771.741i 0.976403 + 0.563726i 0.901182 0.433441i \(-0.142701\pi\)
0.0752205 + 0.997167i \(0.476034\pi\)
\(38\) 0 0
\(39\) 378.005 + 1163.38i 0.248524 + 0.764878i
\(40\) 0 0
\(41\) −2144.71 954.886i −1.27585 0.568046i −0.346781 0.937946i \(-0.612725\pi\)
−0.929072 + 0.369900i \(0.879392\pi\)
\(42\) 0 0
\(43\) 1196.64 125.771i 0.647180 0.0680213i 0.224749 0.974417i \(-0.427844\pi\)
0.422430 + 0.906395i \(0.361177\pi\)
\(44\) 0 0
\(45\) 903.567 + 192.059i 0.446206 + 0.0948440i
\(46\) 0 0
\(47\) −653.336 474.676i −0.295761 0.214883i 0.430002 0.902828i \(-0.358513\pi\)
−0.725763 + 0.687945i \(0.758513\pi\)
\(48\) 0 0
\(49\) −498.965 + 4747.34i −0.207816 + 1.97723i
\(50\) 0 0
\(51\) −841.144 + 178.791i −0.323393 + 0.0687392i
\(52\) 0 0
\(53\) −3356.62 3022.31i −1.19495 1.07594i −0.995376 0.0960557i \(-0.969377\pi\)
−0.199575 0.979883i \(-0.563956\pi\)
\(54\) 0 0
\(55\) 1435.53 1292.56i 0.474557 0.427293i
\(56\) 0 0
\(57\) 162.948 94.0778i 0.0501532 0.0289559i
\(58\) 0 0
\(59\) 2633.61 1172.56i 0.756567 0.336845i 0.00805565 0.999968i \(-0.497436\pi\)
0.748511 + 0.663122i \(0.230769\pi\)
\(60\) 0 0
\(61\) 1953.56i 0.525009i −0.964931 0.262505i \(-0.915452\pi\)
0.964931 0.262505i \(-0.0845485\pi\)
\(62\) 0 0
\(63\) −5331.70 −1.34334
\(64\) 0 0
\(65\) 1718.43 + 3859.65i 0.406729 + 0.913527i
\(66\) 0 0
\(67\) −1871.10 3240.84i −0.416818 0.721950i 0.578799 0.815470i \(-0.303522\pi\)
−0.995617 + 0.0935197i \(0.970188\pi\)
\(68\) 0 0
\(69\) 1473.34 + 1636.32i 0.309461 + 0.343692i
\(70\) 0 0
\(71\) −6104.64 + 6779.89i −1.21100 + 1.34495i −0.289203 + 0.957268i \(0.593390\pi\)
−0.921794 + 0.387681i \(0.873276\pi\)
\(72\) 0 0
\(73\) 1624.55 + 7642.89i 0.304850 + 1.43421i 0.817658 + 0.575704i \(0.195272\pi\)
−0.512808 + 0.858503i \(0.671395\pi\)
\(74\) 0 0
\(75\) 1731.00 + 181.935i 0.307733 + 0.0323440i
\(76\) 0 0
\(77\) −6553.43 + 9020.02i −1.10532 + 1.52134i
\(78\) 0 0
\(79\) −1532.51 + 7209.92i −0.245556 + 1.15525i 0.666603 + 0.745413i \(0.267748\pi\)
−0.912159 + 0.409837i \(0.865586\pi\)
\(80\) 0 0
\(81\) −261.311 2486.20i −0.0398279 0.378937i
\(82\) 0 0
\(83\) 388.781 873.218i 0.0564351 0.126755i −0.883127 0.469134i \(-0.844566\pi\)
0.939562 + 0.342379i \(0.111233\pi\)
\(84\) 0 0
\(85\) −2824.72 + 917.808i −0.390965 + 0.127032i
\(86\) 0 0
\(87\) 1032.73 1788.74i 0.136442 0.236324i
\(88\) 0 0
\(89\) 13606.0 + 4420.86i 1.71771 + 0.558118i 0.991587 0.129444i \(-0.0413193\pi\)
0.726125 + 0.687563i \(0.241319\pi\)
\(90\) 0 0
\(91\) −14333.3 19728.1i −1.73087 2.38233i
\(92\) 0 0
\(93\) 2954.74 2818.23i 0.341628 0.325845i
\(94\) 0 0
\(95\) 525.749 381.979i 0.0582547 0.0423245i
\(96\) 0 0
\(97\) 2835.75 8727.54i 0.301387 0.927574i −0.679614 0.733570i \(-0.737853\pi\)
0.981001 0.194004i \(-0.0621474\pi\)
\(98\) 0 0
\(99\) −7175.56 4142.81i −0.732125 0.422693i
\(100\) 0 0
\(101\) 4079.90 + 12556.6i 0.399951 + 1.23092i 0.925038 + 0.379873i \(0.124032\pi\)
−0.525088 + 0.851048i \(0.675968\pi\)
\(102\) 0 0
\(103\) −9290.54 4136.42i −0.875723 0.389897i −0.0808883 0.996723i \(-0.525776\pi\)
−0.794834 + 0.606826i \(0.792442\pi\)
\(104\) 0 0
\(105\) 5252.64 552.075i 0.476430 0.0500748i
\(106\) 0 0
\(107\) 6342.76 + 1348.20i 0.554001 + 0.117757i 0.476401 0.879228i \(-0.341941\pi\)
0.0776000 + 0.996985i \(0.475274\pi\)
\(108\) 0 0
\(109\) −1123.57 816.322i −0.0945687 0.0687082i 0.539496 0.841988i \(-0.318615\pi\)
−0.634065 + 0.773280i \(0.718615\pi\)
\(110\) 0 0
\(111\) −685.518 + 6522.27i −0.0556382 + 0.529362i
\(112\) 0 0
\(113\) −17964.8 + 3818.54i −1.40691 + 0.299048i −0.847919 0.530126i \(-0.822144\pi\)
−0.558990 + 0.829174i \(0.688811\pi\)
\(114\) 0 0
\(115\) 5651.58 + 5088.71i 0.427341 + 0.384779i
\(116\) 0 0
\(117\) 13467.2 12125.9i 0.983795 0.885813i
\(118\) 0 0
\(119\) 14846.0 8571.35i 1.04837 0.605278i
\(120\) 0 0
\(121\) −2453.26 + 1092.26i −0.167561 + 0.0746029i
\(122\) 0 0
\(123\) 9975.18i 0.659341i
\(124\) 0 0
\(125\) 15183.6 0.971748
\(126\) 0 0
\(127\) −4787.78 10753.5i −0.296843 0.666721i 0.702129 0.712050i \(-0.252233\pi\)
−0.998972 + 0.0453292i \(0.985566\pi\)
\(128\) 0 0
\(129\) 2556.23 + 4427.52i 0.153610 + 0.266061i
\(130\) 0 0
\(131\) 12711.2 + 14117.2i 0.740700 + 0.822631i 0.989288 0.145978i \(-0.0466330\pi\)
−0.248587 + 0.968609i \(0.579966\pi\)
\(132\) 0 0
\(133\) −2509.81 + 2787.43i −0.141885 + 0.157580i
\(134\) 0 0
\(135\) 1866.15 + 8779.57i 0.102395 + 0.481732i
\(136\) 0 0
\(137\) 7885.23 + 828.771i 0.420120 + 0.0441564i 0.312230 0.950006i \(-0.398924\pi\)
0.107890 + 0.994163i \(0.465591\pi\)
\(138\) 0 0
\(139\) 11800.8 16242.4i 0.610775 0.840659i −0.385866 0.922555i \(-0.626097\pi\)
0.996641 + 0.0818953i \(0.0260973\pi\)
\(140\) 0 0
\(141\) 713.411 3356.34i 0.0358841 0.168821i
\(142\) 0 0
\(143\) −3961.15 37687.8i −0.193709 1.84302i
\(144\) 0 0
\(145\) 2901.56 6517.02i 0.138005 0.309965i
\(146\) 0 0
\(147\) −19289.7 + 6267.59i −0.892668 + 0.290045i
\(148\) 0 0
\(149\) −9589.56 + 16609.6i −0.431943 + 0.748146i −0.997041 0.0768774i \(-0.975505\pi\)
0.565098 + 0.825024i \(0.308838\pi\)
\(150\) 0 0
\(151\) 31638.2 + 10279.9i 1.38758 + 0.450852i 0.905154 0.425084i \(-0.139755\pi\)
0.482427 + 0.875936i \(0.339755\pi\)
\(152\) 0 0
\(153\) 7488.12 + 10306.5i 0.319882 + 0.440280i
\(154\) 0 0
\(155\) 9131.86 10747.2i 0.380098 0.447332i
\(156\) 0 0
\(157\) −3451.58 + 2507.72i −0.140029 + 0.101737i −0.655595 0.755113i \(-0.727582\pi\)
0.515565 + 0.856850i \(0.327582\pi\)
\(158\) 0 0
\(159\) 5930.52 18252.3i 0.234584 0.721976i
\(160\) 0 0
\(161\) −38013.4 21947.1i −1.46651 0.846691i
\(162\) 0 0
\(163\) −7264.17 22356.8i −0.273408 0.841462i −0.989636 0.143596i \(-0.954133\pi\)
0.716229 0.697866i \(-0.245867\pi\)
\(164\) 0 0
\(165\) 7498.13 + 3338.38i 0.275413 + 0.122622i
\(166\) 0 0
\(167\) −14989.5 + 1575.46i −0.537470 + 0.0564904i −0.369375 0.929280i \(-0.620428\pi\)
−0.168095 + 0.985771i \(0.553762\pi\)
\(168\) 0 0
\(169\) 53134.8 + 11294.2i 1.86040 + 0.395440i
\(170\) 0 0
\(171\) −2255.09 1638.42i −0.0771207 0.0560315i
\(172\) 0 0
\(173\) −2489.09 + 23682.1i −0.0831665 + 0.791276i 0.870855 + 0.491539i \(0.163566\pi\)
−0.954022 + 0.299737i \(0.903101\pi\)
\(174\) 0 0
\(175\) −33939.0 + 7213.95i −1.10821 + 0.235558i
\(176\) 0 0
\(177\) 9102.84 + 8196.23i 0.290556 + 0.261618i
\(178\) 0 0
\(179\) −39033.6 + 35146.0i −1.21824 + 1.09691i −0.225799 + 0.974174i \(0.572499\pi\)
−0.992440 + 0.122733i \(0.960834\pi\)
\(180\) 0 0
\(181\) −145.497 + 84.0030i −0.00444118 + 0.00256411i −0.502219 0.864740i \(-0.667483\pi\)
0.497778 + 0.867305i \(0.334150\pi\)
\(182\) 0 0
\(183\) 7582.97 3376.15i 0.226432 0.100814i
\(184\) 0 0
\(185\) 22651.0i 0.661826i
\(186\) 0 0
\(187\) 26640.2 0.761825
\(188\) 0 0
\(189\) −21071.3 47327.0i −0.589886 1.32491i
\(190\) 0 0
\(191\) 7131.85 + 12352.7i 0.195495 + 0.338607i 0.947063 0.321049i \(-0.104035\pi\)
−0.751568 + 0.659656i \(0.770702\pi\)
\(192\) 0 0
\(193\) −28453.0 31600.3i −0.763859 0.848352i 0.228266 0.973599i \(-0.426694\pi\)
−0.992126 + 0.125247i \(0.960028\pi\)
\(194\) 0 0
\(195\) −12011.9 + 13340.6i −0.315895 + 0.350836i
\(196\) 0 0
\(197\) 5680.15 + 26723.0i 0.146362 + 0.688577i 0.988734 + 0.149682i \(0.0478249\pi\)
−0.842373 + 0.538895i \(0.818842\pi\)
\(198\) 0 0
\(199\) −15954.1 1676.84i −0.402871 0.0423435i −0.0990738 0.995080i \(-0.531588\pi\)
−0.303798 + 0.952737i \(0.598255\pi\)
\(200\) 0 0
\(201\) 9346.04 12863.7i 0.231332 0.318401i
\(202\) 0 0
\(203\) −8560.67 + 40274.8i −0.207738 + 0.977329i
\(204\) 0 0
\(205\) −3601.29 34264.0i −0.0856940 0.815324i
\(206\) 0 0
\(207\) 13267.7 29799.7i 0.309638 0.695458i
\(208\) 0 0
\(209\) −5543.65 + 1801.24i −0.126912 + 0.0412362i
\(210\) 0 0
\(211\) −23342.2 + 40429.9i −0.524297 + 0.908109i 0.475303 + 0.879822i \(0.342339\pi\)
−0.999600 + 0.0282868i \(0.990995\pi\)
\(212\) 0 0
\(213\) −36867.0 11978.8i −0.812603 0.264031i
\(214\) 0 0
\(215\) 10378.9 + 14285.3i 0.224530 + 0.309039i
\(216\) 0 0
\(217\) −38658.5 + 71633.1i −0.820967 + 1.52123i
\(218\) 0 0
\(219\) −26859.2 + 19514.4i −0.560022 + 0.406880i
\(220\) 0 0
\(221\) −18005.2 + 55414.4i −0.368650 + 1.13459i
\(222\) 0 0
\(223\) 71962.4 + 41547.5i 1.44709 + 0.835478i 0.998307 0.0581625i \(-0.0185242\pi\)
0.448783 + 0.893641i \(0.351857\pi\)
\(224\) 0 0
\(225\) −7968.05 24523.1i −0.157394 0.484408i
\(226\) 0 0
\(227\) 31271.6 + 13923.0i 0.606873 + 0.270197i 0.687081 0.726581i \(-0.258892\pi\)
−0.0802080 + 0.996778i \(0.525558\pi\)
\(228\) 0 0
\(229\) −55340.5 + 5816.52i −1.05529 + 0.110916i −0.616249 0.787551i \(-0.711349\pi\)
−0.439041 + 0.898467i \(0.644682\pi\)
\(230\) 0 0
\(231\) −46337.9 9849.43i −0.868386 0.184581i
\(232\) 0 0
\(233\) 55205.1 + 40108.8i 1.01687 + 0.738802i 0.965640 0.259885i \(-0.0836845\pi\)
0.0512342 + 0.998687i \(0.483685\pi\)
\(234\) 0 0
\(235\) 1238.79 11786.3i 0.0224317 0.213424i
\(236\) 0 0
\(237\) −30634.6 + 6511.59i −0.545401 + 0.115929i
\(238\) 0 0
\(239\) 24868.9 + 22392.1i 0.435372 + 0.392011i 0.857466 0.514541i \(-0.172038\pi\)
−0.422093 + 0.906552i \(0.638704\pi\)
\(240\) 0 0
\(241\) 56981.3 51306.2i 0.981066 0.883356i −0.0121164 0.999927i \(-0.503857\pi\)
0.993182 + 0.116571i \(0.0371902\pi\)
\(242\) 0 0
\(243\) 52103.0 30081.7i 0.882369 0.509436i
\(244\) 0 0
\(245\) −63995.8 + 28492.8i −1.06615 + 0.474682i
\(246\) 0 0
\(247\) 12748.7i 0.208965i
\(248\) 0 0
\(249\) 4061.39 0.0655052
\(250\) 0 0
\(251\) 13057.9 + 29328.6i 0.207265 + 0.465525i 0.987028 0.160550i \(-0.0513268\pi\)
−0.779762 + 0.626076i \(0.784660\pi\)
\(252\) 0 0
\(253\) −34106.4 59074.0i −0.532837 0.922901i
\(254\) 0 0
\(255\) −8444.28 9378.33i −0.129862 0.144226i
\(256\) 0 0
\(257\) 34406.4 38212.1i 0.520922 0.578542i −0.424073 0.905628i \(-0.639400\pi\)
0.944995 + 0.327086i \(0.106067\pi\)
\(258\) 0 0
\(259\) −27181.7 127880.i −0.405207 1.90635i
\(260\) 0 0
\(261\) −30431.1 3198.44i −0.446722 0.0469524i
\(262\) 0 0
\(263\) −11211.8 + 15431.7i −0.162093 + 0.223101i −0.882336 0.470620i \(-0.844030\pi\)
0.720243 + 0.693722i \(0.244030\pi\)
\(264\) 0 0
\(265\) 13781.4 64836.3i 0.196246 0.923265i
\(266\) 0 0
\(267\) 6353.91 + 60453.4i 0.0891289 + 0.848005i
\(268\) 0 0
\(269\) −10720.4 + 24078.4i −0.148152 + 0.332754i −0.972343 0.233557i \(-0.924963\pi\)
0.824191 + 0.566311i \(0.191630\pi\)
\(270\) 0 0
\(271\) −34992.7 + 11369.8i −0.476474 + 0.154816i −0.537402 0.843326i \(-0.680594\pi\)
0.0609277 + 0.998142i \(0.480594\pi\)
\(272\) 0 0
\(273\) 51806.0 89730.7i 0.695112 1.20397i
\(274\) 0 0
\(275\) −51281.4 16662.3i −0.678101 0.220328i
\(276\) 0 0
\(277\) 40007.7 + 55065.9i 0.521415 + 0.717667i 0.985792 0.167971i \(-0.0537215\pi\)
−0.464376 + 0.885638i \(0.653722\pi\)
\(278\) 0 0
\(279\) −55945.4 23007.2i −0.718714 0.295566i
\(280\) 0 0
\(281\) 106420. 77318.4i 1.34775 0.979197i 0.348628 0.937261i \(-0.386648\pi\)
0.999120 0.0419356i \(-0.0133524\pi\)
\(282\) 0 0
\(283\) −2982.39 + 9178.86i −0.0372385 + 0.114608i −0.967948 0.251151i \(-0.919191\pi\)
0.930709 + 0.365760i \(0.119191\pi\)
\(284\) 0 0
\(285\) 2391.30 + 1380.62i 0.0294404 + 0.0169974i
\(286\) 0 0
\(287\) 61449.1 + 189121.i 0.746023 + 2.29602i
\(288\) 0 0
\(289\) 38880.8 + 17310.8i 0.465521 + 0.207263i
\(290\) 0 0
\(291\) 38777.7 4075.71i 0.457927 0.0481301i
\(292\) 0 0
\(293\) 58974.3 + 12535.4i 0.686954 + 0.146017i 0.538149 0.842850i \(-0.319124\pi\)
0.148806 + 0.988866i \(0.452457\pi\)
\(294\) 0 0
\(295\) 34226.6 + 24867.1i 0.393297 + 0.285747i
\(296\) 0 0
\(297\) 8415.35 80066.7i 0.0954024 0.907693i
\(298\) 0 0
\(299\) 145931. 31018.6i 1.63232 0.346960i
\(300\) 0 0
\(301\) −75738.4 68195.2i −0.835956 0.752698i
\(302\) 0 0
\(303\) −41689.1 + 37537.0i −0.454085 + 0.408860i
\(304\) 0 0
\(305\) 24828.1 14334.5i 0.266897 0.154093i
\(306\) 0 0
\(307\) 132289. 58898.9i 1.40361 0.624929i 0.441421 0.897300i \(-0.354474\pi\)
0.962192 + 0.272371i \(0.0878078\pi\)
\(308\) 0 0
\(309\) 43210.9i 0.452560i
\(310\) 0 0
\(311\) −132941. −1.37448 −0.687242 0.726428i \(-0.741179\pi\)
−0.687242 + 0.726428i \(0.741179\pi\)
\(312\) 0 0
\(313\) 22496.1 + 50527.1i 0.229625 + 0.515745i 0.991207 0.132323i \(-0.0422437\pi\)
−0.761582 + 0.648069i \(0.775577\pi\)
\(314\) 0 0
\(315\) −39122.0 67761.4i −0.394276 0.682906i
\(316\) 0 0
\(317\) −119220. 132407.i −1.18640 1.31763i −0.937040 0.349221i \(-0.886446\pi\)
−0.249360 0.968411i \(-0.580220\pi\)
\(318\) 0 0
\(319\) −42815.3 + 47551.2i −0.420743 + 0.467283i
\(320\) 0 0
\(321\) 5728.43 + 26950.1i 0.0555937 + 0.261548i
\(322\) 0 0
\(323\) 8913.19 + 936.814i 0.0854334 + 0.00897942i
\(324\) 0 0
\(325\) 69318.7 95409.0i 0.656271 0.903280i
\(326\) 0 0
\(327\) 1226.89 5772.04i 0.0114738 0.0539801i
\(328\) 0 0
\(329\) 7150.04 + 68028.1i 0.0660567 + 0.628487i
\(330\) 0 0
\(331\) −14483.3 + 32529.9i −0.132194 + 0.296911i −0.967499 0.252874i \(-0.918624\pi\)
0.835306 + 0.549786i \(0.185291\pi\)
\(332\) 0 0
\(333\) 92401.4 30023.0i 0.833279 0.270749i
\(334\) 0 0
\(335\) 27458.8 47560.1i 0.244677 0.423792i
\(336\) 0 0
\(337\) −78467.8 25495.7i −0.690926 0.224496i −0.0575534 0.998342i \(-0.518330\pi\)
−0.633373 + 0.773847i \(0.718330\pi\)
\(338\) 0 0
\(339\) −45869.0 63133.3i −0.399135 0.549363i
\(340\) 0 0
\(341\) −107688. + 66367.7i −0.926099 + 0.570753i
\(342\) 0 0
\(343\) 162576. 118118.i 1.38187 1.00399i
\(344\) 0 0
\(345\) −9985.31 + 30731.6i −0.0838925 + 0.258195i
\(346\) 0 0
\(347\) 112129. + 64737.9i 0.931238 + 0.537650i 0.887203 0.461380i \(-0.152645\pi\)
0.0440349 + 0.999030i \(0.485979\pi\)
\(348\) 0 0
\(349\) 7736.70 + 23811.1i 0.0635192 + 0.195492i 0.977780 0.209635i \(-0.0672275\pi\)
−0.914261 + 0.405127i \(0.867227\pi\)
\(350\) 0 0
\(351\) 160859. + 71619.1i 1.30566 + 0.581319i
\(352\) 0 0
\(353\) 12119.2 1273.78i 0.0972575 0.0102222i −0.0557747 0.998443i \(-0.517763\pi\)
0.153032 + 0.988221i \(0.451096\pi\)
\(354\) 0 0
\(355\) −130960. 27836.4i −1.03916 0.220880i
\(356\) 0 0
\(357\) 58927.7 + 42813.5i 0.462363 + 0.335926i
\(358\) 0 0
\(359\) −1551.81 + 14764.5i −0.0120406 + 0.114559i −0.998891 0.0470755i \(-0.985010\pi\)
0.986851 + 0.161634i \(0.0516765\pi\)
\(360\) 0 0
\(361\) 125555. 26687.6i 0.963429 0.204783i
\(362\) 0 0
\(363\) −8479.48 7634.96i −0.0643511 0.0579420i
\(364\) 0 0
\(365\) −85214.3 + 76727.3i −0.639627 + 0.575923i
\(366\) 0 0
\(367\) −50861.1 + 29364.7i −0.377619 + 0.218018i −0.676782 0.736184i \(-0.736626\pi\)
0.299163 + 0.954202i \(0.403293\pi\)
\(368\) 0 0
\(369\) −135002. + 60106.6i −0.991485 + 0.441438i
\(370\) 0 0
\(371\) 382581.i 2.77956i
\(372\) 0 0
\(373\) −179222. −1.28817 −0.644086 0.764953i \(-0.722762\pi\)
−0.644086 + 0.764953i \(0.722762\pi\)
\(374\) 0 0
\(375\) 26240.3 + 58936.8i 0.186598 + 0.419106i
\(376\) 0 0
\(377\) −69973.8 121198.i −0.492326 0.852734i
\(378\) 0 0
\(379\) 48747.4 + 54139.5i 0.339370 + 0.376909i 0.888538 0.458804i \(-0.151722\pi\)
−0.549168 + 0.835712i \(0.685055\pi\)
\(380\) 0 0
\(381\) 33466.8 37168.7i 0.230550 0.256051i
\(382\) 0 0
\(383\) −35531.2 167161.i −0.242221 1.13956i −0.916172 0.400786i \(-0.868737\pi\)
0.673951 0.738776i \(-0.264596\pi\)
\(384\) 0 0
\(385\) −162723. 17102.9i −1.09781 0.115385i
\(386\) 0 0
\(387\) 44518.1 61273.9i 0.297245 0.409123i
\(388\) 0 0
\(389\) −10878.8 + 51180.6i −0.0718921 + 0.338226i −0.999364 0.0356679i \(-0.988644\pi\)
0.927472 + 0.373894i \(0.121977\pi\)
\(390\) 0 0
\(391\) 10963.0 + 104306.i 0.0717094 + 0.682269i
\(392\) 0 0
\(393\) −32829.9 + 73737.2i −0.212562 + 0.477421i
\(394\) 0 0
\(395\) −102877. + 33426.7i −0.659361 + 0.214240i
\(396\) 0 0
\(397\) −108321. + 187617.i −0.687275 + 1.19040i 0.285441 + 0.958396i \(0.407860\pi\)
−0.972716 + 0.231999i \(0.925473\pi\)
\(398\) 0 0
\(399\) −15157.2 4924.87i −0.0952080 0.0309349i
\(400\) 0 0
\(401\) 8151.98 + 11220.2i 0.0506961 + 0.0697772i 0.833612 0.552350i \(-0.186269\pi\)
−0.782916 + 0.622127i \(0.786269\pi\)
\(402\) 0 0
\(403\) −65269.0 268857.i −0.401880 1.65543i
\(404\) 0 0
\(405\) 29680.1 21563.9i 0.180949 0.131467i
\(406\) 0 0
\(407\) 62782.5 193225.i 0.379009 1.16647i
\(408\) 0 0
\(409\) 28054.2 + 16197.1i 0.167707 + 0.0968256i 0.581504 0.813543i \(-0.302464\pi\)
−0.413797 + 0.910369i \(0.635798\pi\)
\(410\) 0 0
\(411\) 10410.3 + 32039.7i 0.0616285 + 0.189673i
\(412\) 0 0
\(413\) −223073. 99318.4i −1.30782 0.582277i
\(414\) 0 0
\(415\) 13950.6 1466.26i 0.0810021 0.00851366i
\(416\) 0 0
\(417\) 83440.9 + 17735.9i 0.479851 + 0.101996i
\(418\) 0 0
\(419\) 49044.5 + 35633.0i 0.279359 + 0.202966i 0.718638 0.695385i \(-0.244766\pi\)
−0.439279 + 0.898351i \(0.644766\pi\)
\(420\) 0 0
\(421\) 18714.5 178056.i 0.105588 1.00460i −0.805558 0.592517i \(-0.798134\pi\)
0.911146 0.412084i \(-0.135199\pi\)
\(422\) 0 0
\(423\) −49722.6 + 10568.9i −0.277890 + 0.0590673i
\(424\) 0 0
\(425\) 61610.7 + 55474.5i 0.341097 + 0.307126i
\(426\) 0 0
\(427\) −122969. + 110722.i −0.674434 + 0.607263i
\(428\) 0 0
\(429\) 139444. 80508.1i 0.757679 0.437446i
\(430\) 0 0
\(431\) −243437. + 108385.i −1.31048 + 0.583465i −0.938662 0.344840i \(-0.887933\pi\)
−0.371821 + 0.928304i \(0.621267\pi\)
\(432\) 0 0
\(433\) 114033.i 0.608211i −0.952638 0.304106i \(-0.901642\pi\)
0.952638 0.304106i \(-0.0983576\pi\)
\(434\) 0 0
\(435\) 30311.1 0.160185
\(436\) 0 0
\(437\) −9333.82 20964.1i −0.0488761 0.109777i
\(438\) 0 0
\(439\) 82062.8 + 142137.i 0.425811 + 0.737527i 0.996496 0.0836424i \(-0.0266553\pi\)
−0.570684 + 0.821170i \(0.693322\pi\)
\(440\) 0 0
\(441\) 201056. + 223295.i 1.03381 + 1.14816i
\(442\) 0 0
\(443\) −201201. + 223456.i −1.02523 + 1.13863i −0.0349733 + 0.999388i \(0.511135\pi\)
−0.990258 + 0.139246i \(0.955532\pi\)
\(444\) 0 0
\(445\) 43650.4 + 205359.i 0.220429 + 1.03704i
\(446\) 0 0
\(447\) −81044.8 8518.15i −0.405611 0.0426315i
\(448\) 0 0
\(449\) 5341.56 7352.03i 0.0264957 0.0364682i −0.795564 0.605870i \(-0.792825\pi\)
0.822059 + 0.569402i \(0.192825\pi\)
\(450\) 0 0
\(451\) −64249.8 + 302271.i −0.315877 + 1.48609i
\(452\) 0 0
\(453\) 14774.8 + 140573.i 0.0719990 + 0.685025i
\(454\) 0 0
\(455\) 145555. 326922.i 0.703079 1.57914i
\(456\) 0 0
\(457\) 67030.8 21779.6i 0.320954 0.104284i −0.144109 0.989562i \(-0.546032\pi\)
0.465063 + 0.885278i \(0.346032\pi\)
\(458\) 0 0
\(459\) −61892.3 + 107201.i −0.293773 + 0.508829i
\(460\) 0 0
\(461\) −91792.7 29825.3i −0.431923 0.140340i 0.0849831 0.996382i \(-0.472916\pi\)
−0.516906 + 0.856042i \(0.672916\pi\)
\(462\) 0 0
\(463\) −155498. 214025.i −0.725378 0.998397i −0.999328 0.0366527i \(-0.988330\pi\)
0.273951 0.961744i \(-0.411670\pi\)
\(464\) 0 0
\(465\) 57498.1 + 16873.1i 0.265918 + 0.0780348i
\(466\) 0 0
\(467\) 62454.3 45375.7i 0.286371 0.208060i −0.435321 0.900275i \(-0.643365\pi\)
0.721691 + 0.692215i \(0.243365\pi\)
\(468\) 0 0
\(469\) −97949.9 + 301459.i −0.445306 + 1.37051i
\(470\) 0 0
\(471\) −15699.1 9063.86i −0.0707672 0.0408574i
\(472\) 0 0
\(473\) −48942.3 150629.i −0.218757 0.673265i
\(474\) 0 0
\(475\) −16571.6 7378.15i −0.0734475 0.0327009i
\(476\) 0 0
\(477\) −282757. + 29718.9i −1.24273 + 0.130616i
\(478\) 0 0
\(479\) 307049. + 65265.3i 1.33825 + 0.284453i 0.820778 0.571248i \(-0.193540\pi\)
0.517470 + 0.855701i \(0.326874\pi\)
\(480\) 0 0
\(481\) 359494. + 261188.i 1.55382 + 1.12892i
\(482\) 0 0
\(483\) 19495.0 185483.i 0.0835659 0.795077i
\(484\) 0 0
\(485\) 131727. 27999.5i 0.560005 0.119033i
\(486\) 0 0
\(487\) −125581. 113073.i −0.529499 0.476763i 0.360479 0.932768i \(-0.382613\pi\)
−0.889977 + 0.456005i \(0.849280\pi\)
\(488\) 0 0
\(489\) 74226.6 66833.9i 0.310414 0.279498i
\(490\) 0 0
\(491\) −82998.1 + 47919.0i −0.344275 + 0.198767i −0.662161 0.749362i \(-0.730360\pi\)
0.317886 + 0.948129i \(0.397027\pi\)
\(492\) 0 0
\(493\) 89876.7 40015.7i 0.369788 0.164640i
\(494\) 0 0
\(495\) 121594.i 0.496250i
\(496\) 0 0
\(497\) 772759. 3.12847
\(498\) 0 0
\(499\) −191080. 429174.i −0.767388 1.72358i −0.686814 0.726833i \(-0.740991\pi\)
−0.0805743 0.996749i \(-0.525675\pi\)
\(500\) 0 0
\(501\) −32020.3 55460.8i −0.127570 0.220958i
\(502\) 0 0
\(503\) −281835. 313009.i −1.11393 1.23715i −0.968829 0.247730i \(-0.920315\pi\)
−0.145103 0.989417i \(-0.546351\pi\)
\(504\) 0 0
\(505\) −129647. + 143988.i −0.508371 + 0.564603i
\(506\) 0 0
\(507\) 47988.4 + 225768.i 0.186690 + 0.878306i
\(508\) 0 0
\(509\) −9981.34 1049.08i −0.0385260 0.00404924i 0.0852459 0.996360i \(-0.472832\pi\)
−0.123772 + 0.992311i \(0.539499\pi\)
\(510\) 0 0
\(511\) 389016. 535434.i 1.48979 2.05052i
\(512\) 0 0
\(513\) 5631.15 26492.5i 0.0213975 0.100667i
\(514\) 0 0
\(515\) −15600.2 148426.i −0.0588189 0.559624i
\(516\) 0 0
\(517\) −43236.1 + 97109.9i −0.161758 + 0.363314i
\(518\) 0 0
\(519\) −96226.5 + 31265.9i −0.357240 + 0.116074i
\(520\) 0 0
\(521\) −3078.02 + 5331.29i −0.0113396 + 0.0196407i −0.871640 0.490147i \(-0.836943\pi\)
0.860300 + 0.509788i \(0.170276\pi\)
\(522\) 0 0
\(523\) −17689.5 5747.66i −0.0646713 0.0210130i 0.276503 0.961013i \(-0.410825\pi\)
−0.341174 + 0.940000i \(0.610825\pi\)
\(524\) 0 0
\(525\) −86655.4 119271.i −0.314396 0.432729i
\(526\) 0 0
\(527\) 192765. 25876.0i 0.694077 0.0931699i
\(528\) 0 0
\(529\) −9136.12 + 6637.78i −0.0326476 + 0.0237198i
\(530\) 0 0
\(531\) 56075.6 172583.i 0.198877 0.612081i
\(532\) 0 0
\(533\) −585330. 337941.i −2.06038 1.18956i
\(534\) 0 0
\(535\) 29406.4 + 90503.6i 0.102739 + 0.316197i
\(536\) 0 0
\(537\) −203881. 90773.9i −0.707016 0.314784i
\(538\) 0 0
\(539\) 624891. 65678.7i 2.15093 0.226072i
\(540\) 0 0
\(541\) 277728. + 59033.0i 0.948911 + 0.201697i 0.656271 0.754525i \(-0.272133\pi\)
0.292640 + 0.956223i \(0.405466\pi\)
\(542\) 0 0
\(543\) −577.517 419.591i −0.00195869 0.00142307i
\(544\) 0 0
\(545\) 2130.41 20269.5i 0.00717249 0.0682417i
\(546\) 0 0
\(547\) −194284. + 41296.3i −0.649325 + 0.138018i −0.520787 0.853686i \(-0.674362\pi\)
−0.128538 + 0.991705i \(0.541028\pi\)
\(548\) 0 0
\(549\) −91384.1 82282.6i −0.303198 0.273000i
\(550\) 0 0
\(551\) −15997.1 + 14403.9i −0.0526912 + 0.0474434i
\(552\) 0 0
\(553\) 540694. 312170.i 1.76808 1.02080i
\(554\) 0 0
\(555\) −87922.5 + 39145.6i −0.285439 + 0.127086i
\(556\) 0 0
\(557\) 455378.i 1.46778i 0.679267 + 0.733891i \(0.262298\pi\)
−0.679267 + 0.733891i \(0.737702\pi\)
\(558\) 0 0
\(559\) 346402. 1.10855
\(560\) 0 0
\(561\) 46039.9 + 103407.i 0.146288 + 0.328568i
\(562\) 0 0
\(563\) 134219. + 232475.i 0.423446 + 0.733430i 0.996274 0.0862455i \(-0.0274870\pi\)
−0.572828 + 0.819676i \(0.694154\pi\)
\(564\) 0 0
\(565\) −180350. 200298.i −0.564961 0.627452i
\(566\) 0 0
\(567\) −141687. + 157359.i −0.440720 + 0.489469i
\(568\) 0 0
\(569\) 82263.1 + 387017.i 0.254086 + 1.19538i 0.901342 + 0.433107i \(0.142583\pi\)
−0.647257 + 0.762272i \(0.724084\pi\)
\(570\) 0 0
\(571\) −82043.3 8623.10i −0.251635 0.0264479i −0.0221289 0.999755i \(-0.507044\pi\)
−0.229506 + 0.973307i \(0.573711\pi\)
\(572\) 0 0
\(573\) −35623.2 + 49031.2i −0.108499 + 0.149336i
\(574\) 0 0
\(575\) 44135.6 207642.i 0.133491 0.628028i
\(576\) 0 0
\(577\) 6155.44 + 58565.1i 0.0184887 + 0.175909i 0.999869 0.0161646i \(-0.00514558\pi\)
−0.981381 + 0.192073i \(0.938479\pi\)
\(578\) 0 0
\(579\) 73487.4 165055.i 0.219208 0.492348i
\(580\) 0 0
\(581\) −77000.6 + 25019.0i −0.228109 + 0.0741170i
\(582\) 0 0
\(583\) −297271. + 514889.i −0.874612 + 1.51487i
\(584\) 0 0
\(585\) 252927. + 82180.9i 0.739066 + 0.240137i
\(586\) 0 0
\(587\) −272099. 374512.i −0.789678 1.08690i −0.994148 0.108026i \(-0.965547\pi\)
0.204470 0.978873i \(-0.434453\pi\)
\(588\) 0 0
\(589\) −38363.6 + 18418.2i −0.110583 + 0.0530903i
\(590\) 0 0
\(591\) −93911.9 + 68231.0i −0.268872 + 0.195347i
\(592\) 0 0
\(593\) 114330. 351871.i 0.325125 1.00063i −0.646259 0.763118i \(-0.723667\pi\)
0.971384 0.237514i \(-0.0763326\pi\)
\(594\) 0 0
\(595\) 217869. + 125787.i 0.615406 + 0.355305i
\(596\) 0 0
\(597\) −21063.1 64825.7i −0.0590982 0.181886i
\(598\) 0 0
\(599\) 223616. + 99560.2i 0.623231 + 0.277480i 0.693958 0.720015i \(-0.255865\pi\)
−0.0707271 + 0.997496i \(0.522532\pi\)
\(600\) 0 0
\(601\) 124049. 13038.1i 0.343434 0.0360964i 0.0687586 0.997633i \(-0.478096\pi\)
0.274676 + 0.961537i \(0.411430\pi\)
\(602\) 0 0
\(603\) −230410. 48975.2i −0.633675 0.134692i
\(604\) 0 0
\(605\) −31882.8 23164.2i −0.0871055 0.0632859i
\(606\) 0 0
\(607\) −66506.3 + 632765.i −0.180503 + 1.71738i 0.411475 + 0.911421i \(0.365014\pi\)
−0.591978 + 0.805954i \(0.701653\pi\)
\(608\) 0 0
\(609\) −171126. + 36373.9i −0.461404 + 0.0980743i
\(610\) 0 0
\(611\) −172776. 155569.i −0.462809 0.416715i
\(612\) 0 0
\(613\) 437817. 394213.i 1.16512 1.04908i 0.167120 0.985937i \(-0.446553\pi\)
0.998004 0.0631457i \(-0.0201133\pi\)
\(614\) 0 0
\(615\) 126776. 73194.1i 0.335187 0.193520i
\(616\) 0 0
\(617\) −128974. + 57422.8i −0.338790 + 0.150839i −0.569078 0.822283i \(-0.692700\pi\)
0.230288 + 0.973123i \(0.426033\pi\)
\(618\) 0 0
\(619\) 675106.i 1.76194i −0.473174 0.880969i \(-0.656892\pi\)
0.473174 0.880969i \(-0.343108\pi\)
\(620\) 0 0
\(621\) 316953. 0.821885
\(622\) 0 0
\(623\) −492870. 1.10700e6i −1.26986 2.85216i
\(624\) 0 0
\(625\) −16600.3 28752.5i −0.0424966 0.0736064i
\(626\) 0 0
\(627\) −16572.3 18405.4i −0.0421549 0.0468177i
\(628\) 0 0
\(629\) −209024. + 232145.i −0.528318 + 0.586756i
\(630\) 0 0
\(631\) 124706. + 586697.i 0.313206 + 1.47352i 0.800000 + 0.600000i \(0.204833\pi\)
−0.486794 + 0.873517i \(0.661834\pi\)
\(632\) 0 0
\(633\) −197274. 20734.3i −0.492336 0.0517466i
\(634\) 0 0
\(635\) 101537. 139754.i 0.251813 0.346591i
\(636\) 0 0
\(637\) −285724. + 1.34423e6i −0.704155 + 3.31279i
\(638\) 0 0
\(639\) 60028.0 + 571128.i 0.147012 + 1.39872i
\(640\) 0 0
\(641\) −269259. + 604765.i −0.655321 + 1.47187i 0.213599 + 0.976921i \(0.431481\pi\)
−0.868920 + 0.494953i \(0.835185\pi\)
\(642\) 0 0
\(643\) −134558. + 43720.6i −0.325453 + 0.105746i −0.467187 0.884159i \(-0.654732\pi\)
0.141734 + 0.989905i \(0.454732\pi\)
\(644\) 0 0
\(645\) −37513.3 + 64975.0i −0.0901708 + 0.156180i
\(646\) 0 0
\(647\) −230663. 74946.8i −0.551021 0.179038i 0.0202559 0.999795i \(-0.493552\pi\)
−0.571277 + 0.820757i \(0.693552\pi\)
\(648\) 0 0
\(649\) −223046. 306996.i −0.529547 0.728859i
\(650\) 0 0
\(651\) −344862. 26260.7i −0.813736 0.0619647i
\(652\) 0 0
\(653\) 600060. 435969.i 1.40724 1.02242i 0.413524 0.910493i \(-0.364298\pi\)
0.993716 0.111927i \(-0.0357022\pi\)
\(654\) 0 0
\(655\) −86147.4 + 265134.i −0.200798 + 0.617993i
\(656\) 0 0
\(657\) 425946. + 245920.i 0.986788 + 0.569722i
\(658\) 0 0
\(659\) 139528. + 429422.i 0.321284 + 0.988810i 0.973090 + 0.230425i \(0.0740116\pi\)
−0.651806 + 0.758386i \(0.725988\pi\)
\(660\) 0 0
\(661\) −173963. 77453.5i −0.398158 0.177271i 0.197883 0.980226i \(-0.436593\pi\)
−0.596040 + 0.802955i \(0.703260\pi\)
\(662\) 0 0
\(663\) −246214. + 25878.1i −0.560126 + 0.0588716i
\(664\) 0 0
\(665\) −53841.9 11444.4i −0.121752 0.0258792i
\(666\) 0 0
\(667\) −203799. 148069.i −0.458089 0.332821i
\(668\) 0 0
\(669\) −36905.6 + 351133.i −0.0824593 + 0.784548i
\(670\) 0 0
\(671\) −251527. + 53463.8i −0.558651 + 0.118745i
\(672\) 0 0
\(673\) 65982.3 + 59410.7i 0.145679 + 0.131170i 0.738748 0.673982i \(-0.235417\pi\)
−0.593069 + 0.805152i \(0.702084\pi\)
\(674\) 0 0
\(675\) 186190. 167646.i 0.408647 0.367947i
\(676\) 0 0
\(677\) −179014. + 103354.i −0.390579 + 0.225501i −0.682411 0.730969i \(-0.739068\pi\)
0.291832 + 0.956470i \(0.405735\pi\)
\(678\) 0 0
\(679\) −710087. + 316151.i −1.54018 + 0.685733i
\(680\) 0 0
\(681\) 145446.i 0.313623i
\(682\) 0 0
\(683\) 610583. 1.30889 0.654445 0.756110i \(-0.272902\pi\)
0.654445 + 0.756110i \(0.272902\pi\)
\(684\) 0 0
\(685\) 47325.9 + 106296.i 0.100860 + 0.226535i
\(686\) 0 0
\(687\) −118217. 204758.i −0.250477 0.433839i
\(688\) 0 0
\(689\) −870105. 966349.i −1.83288 2.03561i
\(690\) 0 0
\(691\) −394330. + 437948.i −0.825855 + 0.917205i −0.997691 0.0679209i \(-0.978363\pi\)
0.171836 + 0.985126i \(0.445030\pi\)
\(692\) 0 0
\(693\) 145915. + 686475.i 0.303832 + 1.42942i
\(694\) 0 0
\(695\) 293016. + 30797.3i 0.606628 + 0.0637592i
\(696\) 0 0
\(697\) 279280. 384397.i 0.574877 0.791250i
\(698\) 0 0
\(699\) −60281.3 + 283601.i −0.123375 + 0.580435i
\(700\) 0 0
\(701\) −83111.8 790756.i −0.169132 1.60919i −0.669121 0.743154i \(-0.733329\pi\)
0.499988 0.866032i \(-0.333338\pi\)
\(702\) 0 0
\(703\) 27800.3 62440.6i 0.0562522 0.126344i
\(704\) 0 0
\(705\) 47890.9 15560.7i 0.0963551 0.0313077i
\(706\) 0 0
\(707\) 559155. 968485.i 1.11865 1.93755i
\(708\) 0 0
\(709\) −267771. 87004.2i −0.532686 0.173080i 0.0303087 0.999541i \(-0.490351\pi\)
−0.562995 + 0.826460i \(0.690351\pi\)
\(710\) 0 0
\(711\) 272719. + 375365.i 0.539481 + 0.742531i
\(712\) 0 0
\(713\) −304169. 394324.i −0.598322 0.775665i
\(714\) 0 0
\(715\) 449915. 326882.i 0.880072 0.639410i
\(716\) 0 0
\(717\) −43938.8 + 135230.i −0.0854692 + 0.263047i
\(718\) 0 0
\(719\) −400512. 231236.i −0.774743 0.447298i 0.0598209 0.998209i \(-0.480947\pi\)
−0.834564 + 0.550911i \(0.814280\pi\)
\(720\) 0 0
\(721\) 266188. + 819243.i 0.512057 + 1.57595i
\(722\) 0 0
\(723\) 297626. + 132512.i 0.569370 + 0.253500i
\(724\) 0 0
\(725\) −198037. + 20814.5i −0.376765 + 0.0395996i
\(726\) 0 0
\(727\) −98889.9 21019.7i −0.187104 0.0397702i 0.113407 0.993549i \(-0.463824\pi\)
−0.300511 + 0.953778i \(0.597157\pi\)
\(728\) 0 0
\(729\) 42991.0 + 31234.8i 0.0808951 + 0.0587737i
\(730\) 0 0
\(731\) −25454.6 + 242184.i −0.0476355 + 0.453222i
\(732\) 0 0
\(733\) 645989. 137309.i 1.20231 0.255559i 0.437145 0.899391i \(-0.355990\pi\)
0.765167 + 0.643832i \(0.222656\pi\)
\(734\) 0 0
\(735\) −221196. 199166.i −0.409451 0.368672i
\(736\) 0 0
\(737\) −366062. + 329603.i −0.673937 + 0.606816i
\(738\) 0 0
\(739\) −242080. + 139765.i −0.443271 + 0.255923i −0.704984 0.709223i \(-0.749046\pi\)
0.261713 + 0.965146i \(0.415713\pi\)
\(740\) 0 0
\(741\) 49485.7 22032.5i 0.0901246 0.0401261i
\(742\) 0 0
\(743\) 809492.i 1.46634i 0.680045 + 0.733170i \(0.261960\pi\)
−0.680045 + 0.733170i \(0.738040\pi\)
\(744\) 0 0
\(745\) −281458. −0.507109
\(746\) 0 0
\(747\) −24472.4 54965.9i −0.0438566 0.0985035i
\(748\) 0 0
\(749\) −274625. 475664.i −0.489526 0.847884i
\(750\) 0 0
\(751\) 341205. + 378947.i 0.604973 + 0.671890i 0.965363 0.260911i \(-0.0840231\pi\)
−0.360390 + 0.932802i \(0.617356\pi\)
\(752\) 0 0
\(753\) −91275.5 + 101372.i −0.160977 + 0.178783i
\(754\) 0 0
\(755\) 101501. + 477525.i 0.178064 + 0.837726i
\(756\) 0 0
\(757\) −844916. 88804.3i −1.47442 0.154968i −0.667051 0.745012i \(-0.732444\pi\)
−0.807371 + 0.590044i \(0.799110\pi\)
\(758\) 0 0
\(759\) 170360. 234480.i 0.295722 0.407026i
\(760\) 0 0
\(761\) −110698. + 520793.i −0.191148 + 0.899281i 0.773101 + 0.634283i \(0.218705\pi\)
−0.964249 + 0.264998i \(0.914629\pi\)
\(762\) 0 0
\(763\) 12296.2 + 116991.i 0.0211214 + 0.200957i
\(764\) 0 0
\(765\) −76042.0 + 170793.i −0.129936 + 0.291842i
\(766\) 0 0
\(767\) 789332. 256470.i 1.34174 0.435958i
\(768\) 0 0
\(769\) 442983. 767269.i 0.749090 1.29746i −0.199169 0.979965i \(-0.563824\pi\)
0.948259 0.317498i \(-0.102843\pi\)
\(770\) 0 0
\(771\) 207786. + 67513.8i 0.349549 + 0.113575i
\(772\) 0 0
\(773\) −615805. 847583.i −1.03059 1.41848i −0.904507 0.426460i \(-0.859761\pi\)
−0.126079 0.992020i \(-0.540239\pi\)
\(774\) 0 0
\(775\) −387250. 70756.4i −0.644745 0.117805i
\(776\) 0 0
\(777\) 449404. 326511.i 0.744381 0.540824i
\(778\) 0 0
\(779\) −32125.9 + 98873.4i −0.0529396 + 0.162931i
\(780\) 0 0
\(781\) 1.04000e6 + 600445.i 1.70503 + 0.984399i
\(782\) 0 0
\(783\) −91875.3 282763.i −0.149856 0.461210i
\(784\) 0 0
\(785\) −57197.4 25465.9i −0.0928190 0.0413257i
\(786\) 0 0
\(787\) −160961. + 16917.7i −0.259879 + 0.0273144i −0.233572 0.972340i \(-0.575041\pi\)
−0.0263071 + 0.999654i \(0.508375\pi\)
\(788\) 0 0
\(789\) −79276.2 16850.7i −0.127347 0.0270684i
\(790\) 0 0
\(791\) 1.25855e6 + 914392.i 2.01149 + 1.46144i
\(792\) 0 0
\(793\) 58788.7 559337.i 0.0934861 0.889461i
\(794\) 0 0
\(795\) 275487. 58556.5i 0.435879 0.0926490i
\(796\) 0 0
\(797\) −170383. 153414.i −0.268232 0.241517i 0.524028 0.851701i \(-0.324429\pi\)
−0.792260 + 0.610184i \(0.791095\pi\)
\(798\) 0 0
\(799\) 121461. 109364.i 0.190258 0.171309i
\(800\) 0 0
\(801\) 779875. 450261.i 1.21551 0.701778i
\(802\) 0 0
\(803\) 939589. 418332.i 1.45716 0.648769i
\(804\) 0 0
\(805\) 644157.i 0.994032i
\(806\) 0 0
\(807\) −111990. −0.171962
\(808\) 0 0
\(809\) −62168.6 139633.i −0.0949892 0.213349i 0.859801 0.510629i \(-0.170587\pi\)
−0.954790 + 0.297280i \(0.903921\pi\)
\(810\) 0 0
\(811\) 178097. + 308474.i 0.270780 + 0.469004i 0.969062 0.246819i \(-0.0793852\pi\)
−0.698282 + 0.715823i \(0.746052\pi\)
\(812\) 0 0
\(813\) −104608. 116179.i −0.158265 0.175771i
\(814\) 0 0
\(815\) 230834. 256367.i 0.347524 0.385964i
\(816\) 0 0
\(817\) −11078.0 52117.9i −0.0165965 0.0780805i
\(818\) 0 0
\(819\) −1.52656e6 160447.i −2.27586 0.239202i
\(820\) 0 0
\(821\) −137382. + 189090.i −0.203819 + 0.280532i −0.898674 0.438617i \(-0.855468\pi\)
0.694855 + 0.719150i \(0.255468\pi\)
\(822\) 0 0
\(823\) 12882.3 60606.2i 0.0190192 0.0894783i −0.967614 0.252433i \(-0.918769\pi\)
0.986634 + 0.162955i \(0.0521025\pi\)
\(824\) 0 0
\(825\) −23948.1 227851.i −0.0351854 0.334767i
\(826\) 0 0
\(827\) −75065.9 + 168601.i −0.109757 + 0.246518i −0.960070 0.279758i \(-0.909746\pi\)
0.850314 + 0.526276i \(0.176412\pi\)
\(828\) 0 0
\(829\) −178024. + 57843.4i −0.259041 + 0.0841676i −0.435659 0.900112i \(-0.643484\pi\)
0.176617 + 0.984280i \(0.443484\pi\)
\(830\) 0 0
\(831\) −144603. + 250460.i −0.209399 + 0.362690i
\(832\) 0 0
\(833\) −918810. 298539.i −1.32415 0.430241i
\(834\) 0 0
\(835\) −130010. 178944.i −0.186468 0.256651i
\(836\) 0 0
\(837\) −16877.3 587526.i −0.0240908 0.838642i
\(838\) 0 0
\(839\) 543230. 394679.i 0.771720 0.560687i −0.130763 0.991414i \(-0.541743\pi\)
0.902482 + 0.430727i \(0.141743\pi\)
\(840\) 0 0
\(841\) 145541. 447928.i 0.205775 0.633310i
\(842\) 0 0
\(843\) 484035. + 279458.i 0.681117 + 0.393243i
\(844\) 0 0
\(845\) 246344. + 758170.i 0.345008 + 1.06183i
\(846\) 0 0
\(847\) 207797. + 92517.1i 0.289649 + 0.128960i
\(848\) 0 0
\(849\) −40783.0 + 4286.47i −0.0565801 + 0.00594681i
\(850\) 0 0
\(851\) 782379. + 166300.i 1.08033 + 0.229632i
\(852\) 0 0
\(853\) −246636. 179191.i −0.338968 0.246274i 0.405258 0.914202i \(-0.367182\pi\)
−0.744226 + 0.667928i \(0.767182\pi\)
\(854\) 0 0
\(855\) 4275.88 40682.3i 0.00584916 0.0556510i
\(856\) 0 0
\(857\) 1.07960e6 229477.i 1.46995 0.312447i 0.597784 0.801658i \(-0.296048\pi\)
0.872166 + 0.489210i \(0.162715\pi\)
\(858\) 0 0
\(859\) 377549. + 339946.i 0.511666 + 0.460706i 0.884063 0.467367i \(-0.154797\pi\)
−0.372397 + 0.928073i \(0.621464\pi\)
\(860\) 0 0
\(861\) −627898. + 565362.i −0.846999 + 0.762642i
\(862\) 0 0
\(863\) 659445. 380731.i 0.885436 0.511207i 0.0129890 0.999916i \(-0.495865\pi\)
0.872447 + 0.488709i \(0.162532\pi\)
\(864\) 0 0
\(865\) −319243. + 142136.i −0.426667 + 0.189965i
\(866\) 0 0
\(867\) 180837.i 0.240574i
\(868\) 0 0
\(869\) 970242. 1.28482
\(870\) 0 0
\(871\) −438200. 984213.i −0.577611 1.29734i
\(872\) 0 0
\(873\) −288819. 500250.i −0.378964 0.656385i
\(874\) 0 0
\(875\) −860557. 955746.i −1.12399 1.24832i
\(876\) 0 0
\(877\) 578849. 642877.i 0.752603 0.835850i −0.238192 0.971218i \(-0.576555\pi\)
0.990795 + 0.135368i \(0.0432215\pi\)
\(878\) 0 0
\(879\) 53262.3 + 250580.i 0.0689354 + 0.324316i
\(880\) 0 0
\(881\) −1.24686e6 131050.i −1.60645 0.168844i −0.741586 0.670857i \(-0.765926\pi\)
−0.864860 + 0.502013i \(0.832593\pi\)
\(882\) 0 0
\(883\) 377249. 519239.i 0.483846 0.665956i −0.495393 0.868669i \(-0.664976\pi\)
0.979238 + 0.202713i \(0.0649758\pi\)
\(884\) 0 0
\(885\) −37373.9 + 175830.i −0.0477179 + 0.224495i
\(886\) 0 0
\(887\) 125471. + 1.19377e6i 0.159476 + 1.51731i 0.722789 + 0.691069i \(0.242860\pi\)
−0.563313 + 0.826243i \(0.690474\pi\)
\(888\) 0 0
\(889\) −405536. + 910850.i −0.513129 + 1.15251i
\(890\) 0 0
\(891\) −312956. + 101686.i −0.394210 + 0.128087i
\(892\) 0 0
\(893\) −17880.7 + 30970.2i −0.0224223 + 0.0388366i
\(894\) 0 0
\(895\) −733090. 238195.i −0.915189 0.297363i
\(896\) 0 0
\(897\) 372602. + 512842.i 0.463084 + 0.637380i
\(898\) 0 0
\(899\) −263619. + 385661.i −0.326180 + 0.477185i
\(900\) 0 0
\(901\) 739554. 537317.i 0.911003 0.661883i
\(902\) 0 0
\(903\) 133816. 411843.i 0.164109 0.505075i
\(904\) 0 0
\(905\) −2135.21 1232.76i −0.00260702 0.00150516i
\(906\) 0 0
\(907\) 274583. + 845079.i 0.333779 + 1.02727i 0.967320 + 0.253557i \(0.0816006\pi\)
−0.633541 + 0.773709i \(0.718399\pi\)
\(908\) 0 0
\(909\) 759220. + 338026.i 0.918840 + 0.409094i
\(910\) 0 0
\(911\) −83140.5 + 8738.42i −0.100179 + 0.0105292i −0.154485 0.987995i \(-0.549372\pi\)
0.0543064 + 0.998524i \(0.482705\pi\)
\(912\) 0 0
\(913\) −123070. 26159.3i −0.147642 0.0313823i
\(914\) 0 0
\(915\) 98549.0 + 71600.1i 0.117709 + 0.0855207i
\(916\) 0 0
\(917\) 168192. 1.60024e6i 0.200016 1.90303i
\(918\) 0 0
\(919\) 96939.9 20605.2i 0.114781 0.0243975i −0.150163 0.988661i \(-0.547980\pi\)
0.264944 + 0.964264i \(0.414646\pi\)
\(920\) 0 0
\(921\) 457246. + 411706.i 0.539052 + 0.485365i
\(922\) 0 0
\(923\) −1.95189e6 + 1.75749e6i −2.29114 + 2.06295i
\(924\) 0 0
\(925\) 547560. 316134.i 0.639953 0.369477i
\(926\) 0 0
\(927\) −584806. + 260372.i −0.680538 + 0.302995i
\(928\) 0 0
\(929\) 483398.i 0.560110i 0.959984 + 0.280055i \(0.0903527\pi\)
−0.959984 + 0.280055i \(0.909647\pi\)
\(930\) 0 0
\(931\) 211383. 0.243877
\(932\) 0 0
\(933\) −229750. 516028.i −0.263933 0.592802i
\(934\) 0 0
\(935\) 195476. + 338575.i 0.223599 + 0.387286i
\(936\) 0 0
\(937\) −11621.2 12906.7i −0.0132365 0.0147006i 0.736490 0.676448i \(-0.236482\pi\)
−0.749727 + 0.661748i \(0.769815\pi\)
\(938\) 0 0
\(939\) −157249. + 174642.i −0.178343 + 0.198070i
\(940\) 0 0
\(941\) −262529. 1.23510e6i −0.296482 1.39484i −0.834084 0.551638i \(-0.814003\pi\)
0.537602 0.843199i \(-0.319330\pi\)
\(942\) 0 0
\(943\) −1.20994e6 127170.i −1.36063 0.143008i
\(944\) 0 0
\(945\) 446872. 615066.i 0.500402 0.688745i
\(946\) 0 0
\(947\) −200340. + 942526.i −0.223392 + 1.05098i 0.713309 + 0.700849i \(0.247195\pi\)
−0.936702 + 0.350129i \(0.886138\pi\)
\(948\) 0 0
\(949\) 235137. + 2.23718e6i 0.261089 + 2.48409i
\(950\) 0 0
\(951\) 307918. 691595.i 0.340466 0.764699i
\(952\) 0 0
\(953\) 269633. 87609.2i 0.296885 0.0964637i −0.156787 0.987632i \(-0.550114\pi\)
0.453672 + 0.891169i \(0.350114\pi\)
\(954\) 0 0
\(955\) −104662. + 181279.i −0.114758 + 0.198766i
\(956\) 0 0
\(957\) −258569. 84014.2i −0.282327 0.0917337i
\(958\) 0 0
\(959\) −394743. 543317.i −0.429217 0.590767i
\(960\) 0 0
\(961\) −714751. + 584826.i −0.773941 + 0.633257i
\(962\) 0 0
\(963\) 330219. 239918.i 0.356082 0.258708i
\(964\) 0 0
\(965\) 192835. 593484.i 0.207076 0.637315i
\(966\) 0 0
\(967\) −558541. 322474.i −0.597313 0.344859i 0.170671 0.985328i \(-0.445406\pi\)
−0.767984 + 0.640469i \(0.778740\pi\)
\(968\) 0 0
\(969\) 11767.5 + 36216.6i 0.0125324 + 0.0385709i
\(970\) 0 0
\(971\) −262626. 116929.i −0.278548 0.124018i 0.262708 0.964875i \(-0.415384\pi\)
−0.541256 + 0.840858i \(0.682051\pi\)
\(972\) 0 0
\(973\) −1.69123e6 + 177755.i −1.78639 + 0.187757i
\(974\) 0 0
\(975\) 490138. + 104182.i 0.515596 + 0.109593i
\(976\) 0 0
\(977\) 727060. + 528240.i 0.761695 + 0.553404i 0.899430 0.437065i \(-0.143982\pi\)
−0.137735 + 0.990469i \(0.543982\pi\)
\(978\) 0 0
\(979\) 196840. 1.87281e6i 0.205375 1.95401i
\(980\) 0 0
\(981\) −85510.1 + 18175.7i −0.0888545 + 0.0188866i
\(982\) 0 0
\(983\) 153998. + 138660.i 0.159370 + 0.143498i 0.744958 0.667111i \(-0.232469\pi\)
−0.585588 + 0.810609i \(0.699136\pi\)
\(984\) 0 0
\(985\) −297948. + 268273.i −0.307091 + 0.276506i
\(986\) 0 0
\(987\) −251702. + 145320.i −0.258376 + 0.149174i
\(988\) 0 0
\(989\) 569624. 253613.i 0.582366 0.259286i
\(990\) 0 0
\(991\) 632201.i 0.643736i 0.946785 + 0.321868i \(0.104311\pi\)
−0.946785 + 0.321868i \(0.895689\pi\)
\(992\) 0 0
\(993\) −151299. −0.153439
\(994\) 0 0
\(995\) −95754.0 215067.i −0.0967188 0.217234i
\(996\) 0 0
\(997\) 902937. + 1.56393e6i 0.908379 + 1.57336i 0.816316 + 0.577605i \(0.196013\pi\)
0.0920626 + 0.995753i \(0.470654\pi\)
\(998\) 0 0
\(999\) 631678. + 701550.i 0.632943 + 0.702955i
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 124.5.o.a.13.7 88
31.12 odd 30 inner 124.5.o.a.105.7 yes 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
124.5.o.a.13.7 88 1.1 even 1 trivial
124.5.o.a.105.7 yes 88 31.12 odd 30 inner