Properties

Label 507.2.f.d.437.4
Level $507$
Weight $2$
Character 507.437
Analytic conductor $4.048$
Analytic rank $0$
Dimension $8$
CM discriminant -39
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(239,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.f (of order \(4\), degree \(2\), 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: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.56070144.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 437.4
Root \(0.500000 - 1.19293i\) of defining polynomial
Character \(\chi\) \(=\) 507.437
Dual form 507.2.f.d.239.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.69293 - 1.69293i) q^{2} -1.73205 q^{3} -3.73205i q^{4} +(1.23931 - 1.23931i) q^{5} +(-2.93225 + 2.93225i) q^{6} +(-2.93225 - 2.93225i) q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+(1.69293 - 1.69293i) q^{2} -1.73205 q^{3} -3.73205i q^{4} +(1.23931 - 1.23931i) q^{5} +(-2.93225 + 2.93225i) q^{6} +(-2.93225 - 2.93225i) q^{8} +3.00000 q^{9} -4.19615i q^{10} +(-4.62518 - 4.62518i) q^{11} +6.46410i q^{12} +(-2.14655 + 2.14655i) q^{15} -2.46410 q^{16} +(5.07880 - 5.07880i) q^{18} +(-4.62518 - 4.62518i) q^{20} -15.6603 q^{22} +(5.07880 + 5.07880i) q^{24} +1.92820i q^{25} -5.19615 q^{27} +7.26795i q^{30} +(1.69293 - 1.69293i) q^{32} +(8.01105 + 8.01105i) q^{33} -11.1962i q^{36} -7.26795 q^{40} +(5.53242 - 5.53242i) q^{41} -4.00000i q^{43} +(-17.2614 + 17.2614i) q^{44} +(3.71794 - 3.71794i) q^{45} +(7.10381 + 7.10381i) q^{47} +4.26795 q^{48} +7.00000i q^{49} +(3.26432 + 3.26432i) q^{50} +(-8.79674 + 8.79674i) q^{54} -11.4641 q^{55} +(0.332073 + 0.332073i) q^{59} +(8.01105 + 8.01105i) q^{60} +13.8564 q^{61} -10.6603i q^{64} +27.1244 q^{66} +(11.3969 - 11.3969i) q^{71} +(-8.79674 - 8.79674i) q^{72} -3.33975i q^{75} -10.3923 q^{79} +(-3.05379 + 3.05379i) q^{80} +9.00000 q^{81} -18.7321i q^{82} +(8.91829 - 8.91829i) q^{83} +(-6.77174 - 6.77174i) q^{86} +27.1244i q^{88} +(3.05379 + 3.05379i) q^{89} -12.5885i q^{90} +24.0526 q^{94} +(-2.93225 + 2.93225i) q^{96} +(11.8505 + 11.8505i) q^{98} +(-13.8755 - 13.8755i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} + 8 q^{16} - 56 q^{22} - 72 q^{40} + 48 q^{48} - 64 q^{55} + 120 q^{66} + 72 q^{81} + 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.69293 1.69293i 1.19709 1.19709i 0.222050 0.975035i \(-0.428725\pi\)
0.975035 0.222050i \(-0.0712747\pi\)
\(3\) −1.73205 −1.00000
\(4\) 3.73205i 1.86603i
\(5\) 1.23931 1.23931i 0.554238 0.554238i −0.373423 0.927661i \(-0.621816\pi\)
0.927661 + 0.373423i \(0.121816\pi\)
\(6\) −2.93225 + 2.93225i −1.19709 + 1.19709i
\(7\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) −2.93225 2.93225i −1.03671 1.03671i
\(9\) 3.00000 1.00000
\(10\) 4.19615i 1.32694i
\(11\) −4.62518 4.62518i −1.39454 1.39454i −0.814794 0.579751i \(-0.803150\pi\)
−0.579751 0.814794i \(-0.696850\pi\)
\(12\) 6.46410i 1.86603i
\(13\) 0 0
\(14\) 0 0
\(15\) −2.14655 + 2.14655i −0.554238 + 0.554238i
\(16\) −2.46410 −0.616025
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 5.07880 5.07880i 1.19709 1.19709i
\(19\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(20\) −4.62518 4.62518i −1.03422 1.03422i
\(21\) 0 0
\(22\) −15.6603 −3.33878
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 5.07880 + 5.07880i 1.03671 + 1.03671i
\(25\) 1.92820i 0.385641i
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 7.26795i 1.32694i
\(31\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(32\) 1.69293 1.69293i 0.299271 0.299271i
\(33\) 8.01105 + 8.01105i 1.39454 + 1.39454i
\(34\) 0 0
\(35\) 0 0
\(36\) 11.1962i 1.86603i
\(37\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −7.26795 −1.14916
\(41\) 5.53242 5.53242i 0.864019 0.864019i −0.127783 0.991802i \(-0.540786\pi\)
0.991802 + 0.127783i \(0.0407861\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) −17.2614 + 17.2614i −2.60226 + 2.60226i
\(45\) 3.71794 3.71794i 0.554238 0.554238i
\(46\) 0 0
\(47\) 7.10381 + 7.10381i 1.03620 + 1.03620i 0.999320 + 0.0368772i \(0.0117410\pi\)
0.0368772 + 0.999320i \(0.488259\pi\)
\(48\) 4.26795 0.616025
\(49\) 7.00000i 1.00000i
\(50\) 3.26432 + 3.26432i 0.461645 + 0.461645i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −8.79674 + 8.79674i −1.19709 + 1.19709i
\(55\) −11.4641 −1.54582
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.332073 + 0.332073i 0.0432322 + 0.0432322i 0.728392 0.685160i \(-0.240268\pi\)
−0.685160 + 0.728392i \(0.740268\pi\)
\(60\) 8.01105 + 8.01105i 1.03422 + 1.03422i
\(61\) 13.8564 1.77413 0.887066 0.461644i \(-0.152740\pi\)
0.887066 + 0.461644i \(0.152740\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 10.6603i 1.33253i
\(65\) 0 0
\(66\) 27.1244 3.33878
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.3969 11.3969i 1.35257 1.35257i 0.469784 0.882782i \(-0.344332\pi\)
0.882782 0.469784i \(-0.155668\pi\)
\(72\) −8.79674 8.79674i −1.03671 1.03671i
\(73\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) 3.33975i 0.385641i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.3923 −1.16923 −0.584613 0.811312i \(-0.698754\pi\)
−0.584613 + 0.811312i \(0.698754\pi\)
\(80\) −3.05379 + 3.05379i −0.341425 + 0.341425i
\(81\) 9.00000 1.00000
\(82\) 18.7321i 2.06861i
\(83\) 8.91829 8.91829i 0.978910 0.978910i −0.0208726 0.999782i \(-0.506644\pi\)
0.999782 + 0.0208726i \(0.00664445\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.77174 6.77174i −0.730215 0.730215i
\(87\) 0 0
\(88\) 27.1244i 2.89147i
\(89\) 3.05379 + 3.05379i 0.323702 + 0.323702i 0.850185 0.526484i \(-0.176490\pi\)
−0.526484 + 0.850185i \(0.676490\pi\)
\(90\) 12.5885i 1.32694i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 24.0526 2.48083
\(95\) 0 0
\(96\) −2.93225 + 2.93225i −0.299271 + 0.299271i
\(97\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) 11.8505 + 11.8505i 1.19709 + 1.19709i
\(99\) −13.8755 13.8755i −1.39454 1.39454i
\(100\) 7.19615 0.719615
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 19.3923i 1.86603i
\(109\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(110\) −19.4080 + 19.4080i −1.85048 + 1.85048i
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 1.12436 0.103505
\(119\) 0 0
\(120\) 12.5885 1.14916
\(121\) 31.7846i 2.88951i
\(122\) 23.4580 23.4580i 2.12379 2.12379i
\(123\) −9.58244 + 9.58244i −0.864019 + 0.864019i
\(124\) 0 0
\(125\) 8.58622 + 8.58622i 0.767975 + 0.767975i
\(126\) 0 0
\(127\) 17.3205i 1.53695i 0.639882 + 0.768473i \(0.278983\pi\)
−0.639882 + 0.768473i \(0.721017\pi\)
\(128\) −14.6612 14.6612i −1.29588 1.29588i
\(129\) 6.92820i 0.609994i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 29.8976 29.8976i 2.60226 2.60226i
\(133\) 0 0
\(134\) 0 0
\(135\) −6.43966 + 6.43966i −0.554238 + 0.554238i
\(136\) 0 0
\(137\) −12.9683 12.9683i −1.10796 1.10796i −0.993419 0.114538i \(-0.963461\pi\)
−0.114538 0.993419i \(-0.536539\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) −12.3042 12.3042i −1.03620 1.03620i
\(142\) 38.5885i 3.23827i
\(143\) 0 0
\(144\) −7.39230 −0.616025
\(145\) 0 0
\(146\) 0 0
\(147\) 12.1244i 1.00000i
\(148\) 0 0
\(149\) −17.2614 + 17.2614i −1.41411 + 1.41411i −0.698499 + 0.715611i \(0.746148\pi\)
−0.715611 + 0.698499i \(0.753852\pi\)
\(150\) −5.65397 5.65397i −0.461645 0.461645i
\(151\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −17.5935 + 17.5935i −1.39966 + 1.39966i
\(159\) 0 0
\(160\) 4.19615i 0.331735i
\(161\) 0 0
\(162\) 15.2364 15.2364i 1.19709 1.19709i
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) −20.6473 20.6473i −1.61228 1.61228i
\(165\) 19.8564 1.54582
\(166\) 30.1962i 2.34368i
\(167\) 16.3542 + 16.3542i 1.26552 + 1.26552i 0.948376 + 0.317148i \(0.102725\pi\)
0.317148 + 0.948376i \(0.397275\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) −14.9282 −1.13826
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 11.3969 + 11.3969i 0.859075 + 0.859075i
\(177\) −0.575167 0.575167i −0.0432322 0.0432322i
\(178\) 10.3397 0.774997
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −13.8755 13.8755i −1.03422 1.03422i
\(181\) 10.0000i 0.743294i −0.928374 0.371647i \(-0.878793\pi\)
0.928374 0.371647i \(-0.121207\pi\)
\(182\) 0 0
\(183\) −24.0000 −1.77413
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 26.5118 26.5118i 1.93357 1.93357i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 18.4641i 1.33253i
\(193\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 26.1244 1.86603
\(197\) −9.82553 + 9.82553i −0.700040 + 0.700040i −0.964419 0.264379i \(-0.914833\pi\)
0.264379 + 0.964419i \(0.414833\pi\)
\(198\) −46.9808 −3.33878
\(199\) 24.2487i 1.71895i −0.511182 0.859473i \(-0.670792\pi\)
0.511182 0.859473i \(-0.329208\pi\)
\(200\) 5.65397 5.65397i 0.399796 0.399796i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 13.7128i 0.957744i
\(206\) 27.0869 + 27.0869i 1.88724 + 1.88724i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −3.46410 −0.238479 −0.119239 0.992866i \(-0.538046\pi\)
−0.119239 + 0.992866i \(0.538046\pi\)
\(212\) 0 0
\(213\) −19.7400 + 19.7400i −1.35257 + 1.35257i
\(214\) 0 0
\(215\) −4.95725 4.95725i −0.338082 0.338082i
\(216\) 15.2364 + 15.2364i 1.03671 + 1.03671i
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 42.7846i 2.88454i
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(224\) 0 0
\(225\) 5.78461i 0.385641i
\(226\) 0 0
\(227\) −3.96104 + 3.96104i −0.262903 + 0.262903i −0.826232 0.563329i \(-0.809520\pi\)
0.563329 + 0.826232i \(0.309520\pi\)
\(228\) 0 0
\(229\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 17.6077 1.14860
\(236\) 1.23931 1.23931i 0.0806724 0.0806724i
\(237\) 18.0000 1.16923
\(238\) 0 0
\(239\) −12.0611 + 12.0611i −0.780165 + 0.780165i −0.979858 0.199693i \(-0.936005\pi\)
0.199693 + 0.979858i \(0.436005\pi\)
\(240\) 5.28933 5.28933i 0.341425 0.341425i
\(241\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(242\) 53.8092 + 53.8092i 3.45899 + 3.45899i
\(243\) −15.5885 −1.00000
\(244\) 51.7128i 3.31057i
\(245\) 8.67520 + 8.67520i 0.554238 + 0.554238i
\(246\) 32.4449i 2.06861i
\(247\) 0 0
\(248\) 0 0
\(249\) −15.4469 + 15.4469i −0.978910 + 0.978910i
\(250\) 29.0718 1.83866
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 29.3225 + 29.3225i 1.83986 + 1.83986i
\(255\) 0 0
\(256\) −28.3205 −1.77003
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 11.7290 + 11.7290i 0.730215 + 0.730215i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 46.9808i 2.89147i
\(265\) 0 0
\(266\) 0 0
\(267\) −5.28933 5.28933i −0.323702 0.323702i
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 21.8038i 1.32694i
\(271\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −43.9090 −2.65264
\(275\) 8.91829 8.91829i 0.537793 0.537793i
\(276\) 0 0
\(277\) 22.0000i 1.32185i 0.750451 + 0.660926i \(0.229836\pi\)
−0.750451 + 0.660926i \(0.770164\pi\)
\(278\) −33.8587 + 33.8587i −2.03071 + 2.03071i
\(279\) 0 0
\(280\) 0 0
\(281\) −21.5545 21.5545i −1.28583 1.28583i −0.937293 0.348542i \(-0.886677\pi\)
−0.348542 0.937293i \(-0.613323\pi\)
\(282\) −41.6603 −2.48083
\(283\) 17.3205i 1.02960i 0.857311 + 0.514799i \(0.172133\pi\)
−0.857311 + 0.514799i \(0.827867\pi\)
\(284\) −42.5339 42.5339i −2.52392 2.52392i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 5.07880 5.07880i 0.299271 0.299271i
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −7.34690 7.34690i −0.429211 0.429211i 0.459149 0.888359i \(-0.348154\pi\)
−0.888359 + 0.459149i \(0.848154\pi\)
\(294\) −20.5257 20.5257i −1.19709 1.19709i
\(295\) 0.823085 0.0479219
\(296\) 0 0
\(297\) 24.0331 + 24.0331i 1.39454 + 1.39454i
\(298\) 58.4449i 3.38562i
\(299\) 0 0
\(300\) −12.4641 −0.719615
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 17.1724 17.1724i 0.983291 0.983291i
\(306\) 0 0
\(307\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) 0 0
\(309\) 27.7128i 1.57653i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 34.6410 1.95803 0.979013 0.203798i \(-0.0653285\pi\)
0.979013 + 0.203798i \(0.0653285\pi\)
\(314\) 3.38587 3.38587i 0.191076 0.191076i
\(315\) 0 0
\(316\) 38.7846i 2.18180i
\(317\) 22.2187 22.2187i 1.24792 1.24792i 0.291290 0.956635i \(-0.405916\pi\)
0.956635 0.291290i \(-0.0940844\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −13.2114 13.2114i −0.738540 0.738540i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 33.5885i 1.86603i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) −32.4449 −1.79147
\(329\) 0 0
\(330\) 33.6156 33.6156i 1.85048 1.85048i
\(331\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(332\) −33.2835 33.2835i −1.82667 1.82667i
\(333\) 0 0
\(334\) 55.3731 3.02988
\(335\) 0 0
\(336\) 0 0
\(337\) 6.92820i 0.377403i 0.982034 + 0.188702i \(0.0604279\pi\)
−0.982034 + 0.188702i \(0.939572\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −11.7290 + 11.7290i −0.632385 + 0.632385i
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −15.6603 −0.834694
\(353\) 26.5118 26.5118i 1.41108 1.41108i 0.658490 0.752590i \(-0.271196\pi\)
0.752590 0.658490i \(-0.228804\pi\)
\(354\) −1.94744 −0.103505
\(355\) 28.2487i 1.49929i
\(356\) 11.3969 11.3969i 0.604035 0.604035i
\(357\) 0 0
\(358\) 0 0
\(359\) 1.48241 + 1.48241i 0.0782385 + 0.0782385i 0.745143 0.666905i \(-0.232381\pi\)
−0.666905 + 0.745143i \(0.732381\pi\)
\(360\) −21.8038 −1.14916
\(361\) 19.0000i 1.00000i
\(362\) −16.9293 16.9293i −0.889786 0.889786i
\(363\) 55.0526i 2.88951i
\(364\) 0 0
\(365\) 0 0
\(366\) −40.6304 + 40.6304i −2.12379 + 2.12379i
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 16.5973 16.5973i 0.864019 0.864019i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 13.8564 0.717458 0.358729 0.933442i \(-0.383210\pi\)
0.358729 + 0.933442i \(0.383210\pi\)
\(374\) 0 0
\(375\) −14.8718 14.8718i −0.767975 0.767975i
\(376\) 41.6603i 2.14846i
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(380\) 0 0
\(381\) 30.0000i 1.53695i
\(382\) 0 0
\(383\) −24.9404 + 24.9404i −1.27439 + 1.27439i −0.330636 + 0.943758i \(0.607263\pi\)
−0.943758 + 0.330636i \(0.892737\pi\)
\(384\) 25.3940 + 25.3940i 1.29588 + 1.29588i
\(385\) 0 0
\(386\) 0 0
\(387\) 12.0000i 0.609994i
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 20.5257 20.5257i 1.03671 1.03671i
\(393\) 0 0
\(394\) 33.2679i 1.67602i
\(395\) −12.8793 + 12.8793i −0.648029 + 0.648029i
\(396\) −51.7842 + 51.7842i −2.60226 + 2.60226i
\(397\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(398\) −41.0515 41.0515i −2.05772 2.05772i
\(399\) 0 0
\(400\) 4.75129i 0.237564i
\(401\) −17.9256 17.9256i −0.895160 0.895160i 0.0998435 0.995003i \(-0.468166\pi\)
−0.995003 + 0.0998435i \(0.968166\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 11.1538 11.1538i 0.554238 0.554238i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(410\) −23.2149 23.2149i −1.14650 1.14650i
\(411\) 22.4618 + 22.4618i 1.10796 + 1.10796i
\(412\) 59.7128 2.94184
\(413\) 0 0
\(414\) 0 0
\(415\) 22.1051i 1.08510i
\(416\) 0 0
\(417\) 34.6410 1.69638
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(422\) −5.86450 + 5.86450i −0.285479 + 0.285479i
\(423\) 21.3114 + 21.3114i 1.03620 + 1.03620i
\(424\) 0 0
\(425\) 0 0
\(426\) 66.8372i 3.23827i
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) −16.7846 −0.809426
\(431\) −7.76796 + 7.76796i −0.374169 + 0.374169i −0.868993 0.494824i \(-0.835233\pi\)
0.494824 + 0.868993i \(0.335233\pi\)
\(432\) 12.8038 0.616025
\(433\) 20.7846i 0.998845i 0.866359 + 0.499422i \(0.166454\pi\)
−0.866359 + 0.499422i \(0.833546\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 40.0000i 1.90910i 0.298057 + 0.954548i \(0.403661\pi\)
−0.298057 + 0.954548i \(0.596339\pi\)
\(440\) 33.6156 + 33.6156i 1.60256 + 1.60256i
\(441\) 21.0000i 1.00000i
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 7.56922 0.358815
\(446\) 0 0
\(447\) 29.8976 29.8976i 1.41411 1.41411i
\(448\) 0 0
\(449\) 28.9904 + 28.9904i 1.36814 + 1.36814i 0.863088 + 0.505054i \(0.168527\pi\)
0.505054 + 0.863088i \(0.331473\pi\)
\(450\) 9.79296 + 9.79296i 0.461645 + 0.461645i
\(451\) −51.1769 −2.40983
\(452\) 0 0
\(453\) 0 0
\(454\) 13.4115i 0.629435i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 24.6973 24.6973i 1.15027 1.15027i 0.163769 0.986499i \(-0.447635\pi\)
0.986499 0.163769i \(-0.0523652\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 29.8087 29.8087i 1.37497 1.37497i
\(471\) −3.46410 −0.159617
\(472\) 1.94744i 0.0896382i
\(473\) −18.5007 + 18.5007i −0.850664 + 0.850664i
\(474\) 30.4728 30.4728i 1.39966 1.39966i
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 40.8372i 1.86785i
\(479\) −19.9831 19.9831i −0.913053 0.913053i 0.0834585 0.996511i \(-0.473403\pi\)
−0.996511 + 0.0834585i \(0.973403\pi\)
\(480\) 7.26795i 0.331735i
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 118.622 5.39190
\(485\) 0 0
\(486\) −26.3902 + 26.3902i −1.19709 + 1.19709i
\(487\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) −40.6304 40.6304i −1.83925 1.83925i
\(489\) 0 0
\(490\) 29.3731 1.32694
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 35.7621 + 35.7621i 1.61228 + 1.61228i
\(493\) 0 0
\(494\) 0 0
\(495\) −34.3923 −1.54582
\(496\) 0 0
\(497\) 0 0
\(498\) 52.3013i 2.34368i
\(499\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(500\) 32.0442 32.0442i 1.43306 1.43306i
\(501\) −28.3263 28.3263i −1.26552 1.26552i
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 64.6410 2.86798
\(509\) −30.8049 + 30.8049i −1.36540 + 1.36540i −0.498530 + 0.866872i \(0.666127\pi\)
−0.866872 + 0.498530i \(0.833873\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −18.6223 + 18.6223i −0.822996 + 0.822996i
\(513\) 0 0
\(514\) 0 0
\(515\) 19.8290 + 19.8290i 0.873771 + 0.873771i
\(516\) 25.8564 1.13826
\(517\) 65.7128i 2.89005i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 44.0000 1.92399 0.961993 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −19.7400 19.7400i −0.859075 0.859075i
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0.996219 + 0.996219i 0.0432322 + 0.0432322i
\(532\) 0 0
\(533\) 0 0
\(534\) −17.9090 −0.774997
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 32.3763 32.3763i 1.39454 1.39454i
\(540\) 24.0331 + 24.0331i 1.03422 + 1.03422i
\(541\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(542\) 0 0
\(543\) 17.3205i 0.743294i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −48.3984 + 48.3984i −2.06748 + 2.06748i
\(549\) 41.5692 1.77413
\(550\) 30.1962i 1.28757i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 37.2445 + 37.2445i 1.58237 + 1.58237i
\(555\) 0 0
\(556\) 74.6410i 3.16548i
\(557\) −33.2835 33.2835i −1.41027 1.41027i −0.757919 0.652349i \(-0.773784\pi\)
−0.652349 0.757919i \(-0.726216\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −72.9808 −3.07851
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −45.9197 + 45.9197i −1.93357 + 1.93357i
\(565\) 0 0
\(566\) 29.3225 + 29.3225i 1.23252 + 1.23252i
\(567\) 0 0
\(568\) −66.8372 −2.80443
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 38.1051i 1.59465i −0.603550 0.797325i \(-0.706248\pi\)
0.603550 0.797325i \(-0.293752\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 31.9808i 1.33253i
\(577\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(578\) −28.7799 + 28.7799i −1.19709 + 1.19709i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −24.8756 −1.02760
\(587\) −17.5045 + 17.5045i −0.722488 + 0.722488i −0.969111 0.246623i \(-0.920679\pi\)
0.246623 + 0.969111i \(0.420679\pi\)
\(588\) −45.2487 −1.86603
\(589\) 0 0
\(590\) 1.39343 1.39343i 0.0573666 0.0573666i
\(591\) 17.0183 17.0183i 0.700040 0.700040i
\(592\) 0 0
\(593\) 20.4042 + 20.4042i 0.837900 + 0.837900i 0.988582 0.150683i \(-0.0481472\pi\)
−0.150683 + 0.988582i \(0.548147\pi\)
\(594\) 81.3731 3.33878
\(595\) 0 0
\(596\) 64.4205 + 64.4205i 2.63877 + 2.63877i
\(597\) 42.0000i 1.71895i
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) −9.79296 + 9.79296i −0.399796 + 0.399796i
\(601\) −48.4974 −1.97825 −0.989126 0.147074i \(-0.953015\pi\)
−0.989126 + 0.147074i \(0.953015\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 39.3911 + 39.3911i 1.60148 + 1.60148i
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 58.1436i 2.35417i
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(614\) 0 0
\(615\) 23.7513i 0.957744i
\(616\) 0 0
\(617\) −11.6400 + 11.6400i −0.468609 + 0.468609i −0.901464 0.432855i \(-0.857506\pi\)
0.432855 + 0.901464i \(0.357506\pi\)
\(618\) −46.9160 46.9160i −1.88724 1.88724i
\(619\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.6410 0.465641
\(626\) 58.6450 58.6450i 2.34392 2.34392i
\(627\) 0 0
\(628\) 7.46410i 0.297850i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(632\) 30.4728 + 30.4728i 1.21214 + 1.21214i
\(633\) 6.00000 0.238479
\(634\) 75.2295i 2.98774i
\(635\) 21.4655 + 21.4655i 0.851834 + 0.851834i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 34.1908 34.1908i 1.35257 1.35257i
\(640\) −36.3397 −1.43645
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 0 0
\(645\) 8.58622 + 8.58622i 0.338082 + 0.338082i
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −26.3902 26.3902i −1.03671 1.03671i
\(649\) 3.07180i 0.120579i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −13.6325 + 13.6325i −0.532258 + 0.532258i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 74.1051i 2.88454i
\(661\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −52.3013 −2.02968
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 61.0346 61.0346i 2.36150 2.36150i
\(669\) 0 0
\(670\) 0 0
\(671\) −64.0884 64.0884i −2.47411 2.47411i
\(672\) 0 0
\(673\) 14.0000i 0.539660i 0.962908 + 0.269830i \(0.0869676\pi\)
−0.962908 + 0.269830i \(0.913032\pi\)
\(674\) 11.7290 + 11.7290i 0.451784 + 0.451784i
\(675\) 10.0192i 0.385641i
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 6.86071 6.86071i 0.262903 0.262903i
\(682\) 0 0
\(683\) 36.6694 + 36.6694i 1.40311 + 1.40311i 0.789972 + 0.613142i \(0.210095\pi\)
0.613142 + 0.789972i \(0.289905\pi\)
\(684\) 0 0
\(685\) −32.1436 −1.22814
\(686\) 0 0
\(687\) 0 0
\(688\) 9.85641i 0.375772i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −24.7863 + 24.7863i −0.940197 + 0.940197i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −49.3056 + 49.3056i −1.85828 + 1.85828i
\(705\) −30.4974 −1.14860
\(706\) 89.7654i 3.37836i
\(707\) 0 0
\(708\) −2.14655 + 2.14655i −0.0806724 + 0.0806724i
\(709\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(710\) −47.8232 47.8232i −1.79477 1.79477i
\(711\) −31.1769 −1.16923
\(712\) 17.9090i 0.671167i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 20.8904 20.8904i 0.780165 0.780165i
\(718\) 5.01924 0.187316
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −9.16138 + 9.16138i −0.341425 + 0.341425i
\(721\) 0 0
\(722\) −32.1657 32.1657i −1.19709 1.19709i
\(723\) 0 0
\(724\) −37.3205 −1.38701
\(725\) 0 0
\(726\) −93.2003 93.2003i −3.45899 3.45899i
\(727\) 51.9615i 1.92715i 0.267445 + 0.963573i \(0.413821\pi\)
−0.267445 + 0.963573i \(0.586179\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 89.5692i 3.31057i
\(733\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) −13.5435 + 13.5435i −0.499899 + 0.499899i
\(735\) −15.0259 15.0259i −0.554238 0.554238i
\(736\) 0 0
\(737\) 0 0
\(738\) 56.1962i 2.06861i
\(739\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.47485 3.47485i 0.127480 0.127480i −0.640488 0.767968i \(-0.721268\pi\)
0.767968 + 0.640488i \(0.221268\pi\)
\(744\) 0 0
\(745\) 42.7846i 1.56751i
\(746\) 23.4580 23.4580i 0.858858 0.858858i
\(747\) 26.7549 26.7549i 0.978910 0.978910i
\(748\) 0 0
\(749\) 0 0
\(750\) −50.3538 −1.83866
\(751\) 40.0000i 1.45962i 0.683650 + 0.729810i \(0.260392\pi\)
−0.683650 + 0.729810i \(0.739608\pi\)
\(752\) −17.5045 17.5045i −0.638324 0.638324i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 41.5692 1.51086 0.755429 0.655230i \(-0.227428\pi\)
0.755429 + 0.655230i \(0.227428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −33.9477 33.9477i −1.23060 1.23060i −0.963733 0.266869i \(-0.914011\pi\)
−0.266869 0.963733i \(-0.585989\pi\)
\(762\) −50.7880 50.7880i −1.83986 1.83986i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 84.4449i 3.05112i
\(767\) 0 0
\(768\) 49.0526 1.77003
\(769\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −38.2408 + 38.2408i −1.37542 + 1.37542i −0.523238 + 0.852186i \(0.675276\pi\)
−0.852186 + 0.523238i \(0.824724\pi\)
\(774\) −20.3152 20.3152i −0.730215 0.730215i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −105.426 −3.77243
\(782\) 0 0
\(783\) 0 0
\(784\) 17.2487i 0.616025i
\(785\) 2.47863 2.47863i 0.0884660 0.0884660i
\(786\) 0 0
\(787\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(788\) 36.6694 + 36.6694i 1.30629 + 1.30629i
\(789\) 0 0
\(790\) 43.6077i 1.55149i
\(791\) 0 0
\(792\) 81.3731i 2.89147i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −90.4974 −3.20760
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 3.26432 + 3.26432i 0.115411 + 0.115411i
\(801\) 9.16138 + 9.16138i 0.323702 + 0.323702i
\(802\) −60.6936 −2.14316
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 37.7654i 1.32694i
\(811\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) −51.1769 −1.78718
\(821\) −0.0889787 + 0.0889787i −0.00310538 + 0.00310538i −0.708658 0.705552i \(-0.750699\pi\)
0.705552 + 0.708658i \(0.250699\pi\)
\(822\) 76.0526 2.65264
\(823\) 56.0000i 1.95204i −0.217687 0.976019i \(-0.569851\pi\)
0.217687 0.976019i \(-0.430149\pi\)
\(824\) 46.9160 46.9160i 1.63439 1.63439i
\(825\) −15.4469 + 15.4469i −0.537793 + 0.537793i
\(826\) 0 0
\(827\) −19.4969 19.4969i −0.677975 0.677975i 0.281566 0.959542i \(-0.409146\pi\)
−0.959542 + 0.281566i \(0.909146\pi\)
\(828\) 0 0
\(829\) 27.7128i 0.962506i −0.876582 0.481253i \(-0.840182\pi\)
0.876582 0.481253i \(-0.159818\pi\)
\(830\) −37.4225 37.4225i −1.29895 1.29895i
\(831\) 38.1051i 1.32185i
\(832\) 0 0
\(833\) 0 0
\(834\) 58.6450 58.6450i 2.03071 2.03071i
\(835\) 40.5359 1.40280
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −26.2687 26.2687i −0.906896 0.906896i 0.0891249 0.996020i \(-0.471593\pi\)
−0.996020 + 0.0891249i \(0.971593\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 37.3335 + 37.3335i 1.28583 + 1.28583i
\(844\) 12.9282i 0.445007i
\(845\) 0 0
\(846\) 72.1577 2.48083
\(847\) 0 0
\(848\) 0 0
\(849\) 30.0000i 1.02960i
\(850\) 0 0
\(851\) 0 0
\(852\) 73.6708 + 73.6708i 2.52392 + 2.52392i
\(853\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −10.3923 −0.354581 −0.177290 0.984159i \(-0.556733\pi\)
−0.177290 + 0.984159i \(0.556733\pi\)
\(860\) −18.5007 + 18.5007i −0.630870 + 0.630870i
\(861\) 0 0
\(862\) 26.3013i 0.895825i
\(863\) 24.2762 24.2762i 0.826373 0.826373i −0.160640 0.987013i \(-0.551356\pi\)
0.987013 + 0.160640i \(0.0513559\pi\)
\(864\) −8.79674 + 8.79674i −0.299271 + 0.299271i
\(865\) 0 0
\(866\) 35.1870 + 35.1870i 1.19570 + 1.19570i
\(867\) 29.4449 1.00000
\(868\) 0 0
\(869\) 48.0663 + 48.0663i 1.63054 + 1.63054i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(878\) 67.7174 + 67.7174i 2.28535 + 2.28535i
\(879\) 12.7252 + 12.7252i 0.429211 + 0.429211i
\(880\) 28.2487 0.952264
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 35.5516 + 35.5516i 1.19709 + 1.19709i
\(883\) 51.9615i 1.74864i 0.485346 + 0.874322i \(0.338694\pi\)
−0.485346 + 0.874322i \(0.661306\pi\)
\(884\) 0 0
\(885\) −1.42563 −0.0479219
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 12.8142 12.8142i 0.429533 0.429533i
\(891\) −41.6266 41.6266i −1.39454 1.39454i
\(892\) 0 0
\(893\) 0 0
\(894\) 101.229i 3.38562i
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 98.1577 3.27556
\(899\) 0 0
\(900\) 21.5885 0.719615
\(901\) 0 0
\(902\) −86.6391 + 86.6391i −2.88477 + 2.88477i
\(903\) 0 0
\(904\) 0 0
\(905\) −12.3931 12.3931i −0.411962 0.411962i
\(906\) 0 0
\(907\) 17.3205i 0.575118i 0.957763 + 0.287559i \(0.0928437\pi\)
−0.957763 + 0.287559i \(0.907156\pi\)
\(908\) 14.7828 + 14.7828i 0.490584 + 0.490584i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −82.4974 −2.73027
\(914\) 0 0
\(915\) −29.7435 + 29.7435i −0.983291 + 0.983291i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 58.8897 1.94259 0.971296 0.237872i \(-0.0764500\pi\)
0.971296 + 0.237872i \(0.0764500\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 83.6218i 2.75394i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 48.0000i 1.57653i
\(928\) 0 0
\(929\) 9.33934 9.33934i 0.306414 0.306414i −0.537103 0.843517i \(-0.680481\pi\)
0.843517 + 0.537103i \(0.180481\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 34.6410 1.13167 0.565836 0.824518i \(-0.308553\pi\)
0.565836 + 0.824518i \(0.308553\pi\)
\(938\) 0 0
\(939\) −60.0000 −1.95803
\(940\) 65.7128i 2.14332i
\(941\) −40.7194 + 40.7194i −1.32741 + 1.32741i −0.419796 + 0.907619i \(0.637898\pi\)
−0.907619 + 0.419796i \(0.862102\pi\)
\(942\) −5.86450 + 5.86450i −0.191076 + 0.191076i
\(943\) 0 0
\(944\) −0.818262 0.818262i −0.0266322 0.0266322i
\(945\) 0 0
\(946\) 62.6410i 2.03664i
\(947\) −40.9625 40.9625i −1.33110 1.33110i −0.904385 0.426717i \(-0.859670\pi\)
−0.426717 0.904385i \(-0.640330\pi\)
\(948\) 67.1769i 2.18180i
\(949\) 0 0
\(950\) 0 0
\(951\) −38.4839 + 38.4839i −1.24792 + 1.24792i
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 45.0125 + 45.0125i 1.45581 + 1.45581i
\(957\) 0 0
\(958\) −67.6603 −2.18600
\(959\) 0 0
\(960\) 22.8828 + 22.8828i 0.738540 + 0.738540i
\(961\) 31.0000i 1.00000i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(968\) 93.2003 93.2003i 2.99557 2.99557i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 58.1769i 1.86603i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −34.1436 −1.09291
\(977\) −36.4263 + 36.4263i −1.16538 + 1.16538i −0.182100 + 0.983280i \(0.558289\pi\)
−0.983280 + 0.182100i \(0.941711\pi\)
\(978\) 0 0
\(979\) 28.2487i 0.902833i
\(980\) 32.3763 32.3763i 1.03422 1.03422i
\(981\) 0 0
\(982\) 0 0
\(983\) 43.4411 + 43.4411i 1.38556 + 1.38556i 0.834406 + 0.551151i \(0.185811\pi\)
0.551151 + 0.834406i \(0.314189\pi\)
\(984\) 56.1962 1.79147
\(985\) 24.3538i 0.775978i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) −58.2239 + 58.2239i −1.85048 + 1.85048i
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −30.0518 30.0518i −0.952705 0.952705i
\(996\) 57.6487 + 57.6487i 1.82667 + 1.82667i
\(997\) 58.0000 1.83688 0.918439 0.395562i \(-0.129450\pi\)
0.918439 + 0.395562i \(0.129450\pi\)
\(998\) 0 0
\(999\) 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 507.2.f.d.437.4 yes 8
3.2 odd 2 inner 507.2.f.d.437.1 yes 8
13.2 odd 12 507.2.k.h.188.2 8
13.3 even 3 507.2.k.g.488.2 8
13.4 even 6 507.2.k.h.89.2 8
13.5 odd 4 inner 507.2.f.d.239.1 8
13.6 odd 12 507.2.k.g.80.1 8
13.7 odd 12 507.2.k.g.80.2 8
13.8 odd 4 inner 507.2.f.d.239.4 yes 8
13.9 even 3 507.2.k.h.89.1 8
13.10 even 6 507.2.k.g.488.1 8
13.11 odd 12 507.2.k.h.188.1 8
13.12 even 2 inner 507.2.f.d.437.1 yes 8
39.2 even 12 507.2.k.h.188.1 8
39.5 even 4 inner 507.2.f.d.239.4 yes 8
39.8 even 4 inner 507.2.f.d.239.1 8
39.11 even 12 507.2.k.h.188.2 8
39.17 odd 6 507.2.k.h.89.1 8
39.20 even 12 507.2.k.g.80.1 8
39.23 odd 6 507.2.k.g.488.2 8
39.29 odd 6 507.2.k.g.488.1 8
39.32 even 12 507.2.k.g.80.2 8
39.35 odd 6 507.2.k.h.89.2 8
39.38 odd 2 CM 507.2.f.d.437.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.f.d.239.1 8 13.5 odd 4 inner
507.2.f.d.239.1 8 39.8 even 4 inner
507.2.f.d.239.4 yes 8 13.8 odd 4 inner
507.2.f.d.239.4 yes 8 39.5 even 4 inner
507.2.f.d.437.1 yes 8 3.2 odd 2 inner
507.2.f.d.437.1 yes 8 13.12 even 2 inner
507.2.f.d.437.4 yes 8 1.1 even 1 trivial
507.2.f.d.437.4 yes 8 39.38 odd 2 CM
507.2.k.g.80.1 8 13.6 odd 12
507.2.k.g.80.1 8 39.20 even 12
507.2.k.g.80.2 8 13.7 odd 12
507.2.k.g.80.2 8 39.32 even 12
507.2.k.g.488.1 8 13.10 even 6
507.2.k.g.488.1 8 39.29 odd 6
507.2.k.g.488.2 8 13.3 even 3
507.2.k.g.488.2 8 39.23 odd 6
507.2.k.h.89.1 8 13.9 even 3
507.2.k.h.89.1 8 39.17 odd 6
507.2.k.h.89.2 8 13.4 even 6
507.2.k.h.89.2 8 39.35 odd 6
507.2.k.h.188.1 8 13.11 odd 12
507.2.k.h.188.1 8 39.2 even 12
507.2.k.h.188.2 8 13.2 odd 12
507.2.k.h.188.2 8 39.11 even 12