Properties

Label 960.2.h.g.191.8
Level $960$
Weight $2$
Character 960.191
Analytic conductor $7.666$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,2,Mod(191,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.342102016.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 191.8
Root \(1.17915 + 0.780776i\) of defining polynomial
Character \(\chi\) \(=\) 960.191
Dual form 960.2.h.g.191.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.51022 + 0.848071i) q^{3} -1.00000i q^{5} +3.02045i q^{7} +(1.56155 + 2.56155i) q^{9} +O(q^{10})\) \(q+(1.51022 + 0.848071i) q^{3} -1.00000i q^{5} +3.02045i q^{7} +(1.56155 + 2.56155i) q^{9} -1.32431 q^{11} +5.12311 q^{13} +(0.848071 - 1.51022i) q^{15} -2.00000i q^{17} +1.32431i q^{19} +(-2.56155 + 4.56155i) q^{21} +0.371834 q^{23} -1.00000 q^{25} +(0.185917 + 5.19283i) q^{27} +3.12311i q^{29} +4.71659i q^{31} +(-2.00000 - 1.12311i) q^{33} +3.02045 q^{35} -5.12311 q^{37} +(7.73704 + 4.34475i) q^{39} +1.12311i q^{41} +7.73704i q^{43} +(2.56155 - 1.56155i) q^{45} +3.02045 q^{47} -2.12311 q^{49} +(1.69614 - 3.02045i) q^{51} -12.2462i q^{53} +1.32431i q^{55} +(-1.12311 + 2.00000i) q^{57} +14.1498 q^{59} -3.12311 q^{61} +(-7.73704 + 4.71659i) q^{63} -5.12311i q^{65} -4.34475i q^{67} +(0.561553 + 0.315342i) q^{69} +3.39228 q^{71} +8.24621 q^{73} +(-1.51022 - 0.848071i) q^{75} -4.00000i q^{77} -8.10887i q^{79} +(-4.12311 + 8.00000i) q^{81} -15.1022 q^{83} -2.00000 q^{85} +(-2.64861 + 4.71659i) q^{87} -10.2462i q^{89} +15.4741i q^{91} +(-4.00000 + 7.12311i) q^{93} +1.32431 q^{95} -6.00000 q^{97} +(-2.06798 - 3.39228i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{9} + 8 q^{13} - 4 q^{21} - 8 q^{25} - 16 q^{33} - 8 q^{37} + 4 q^{45} + 16 q^{49} + 24 q^{57} + 8 q^{61} - 12 q^{69} - 16 q^{85} - 32 q^{93} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.51022 + 0.848071i 0.871928 + 0.489634i
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 3.02045i 1.14162i 0.821081 + 0.570811i \(0.193371\pi\)
−0.821081 + 0.570811i \(0.806629\pi\)
\(8\) 0 0
\(9\) 1.56155 + 2.56155i 0.520518 + 0.853851i
\(10\) 0 0
\(11\) −1.32431 −0.399294 −0.199647 0.979868i \(-0.563979\pi\)
−0.199647 + 0.979868i \(0.563979\pi\)
\(12\) 0 0
\(13\) 5.12311 1.42089 0.710447 0.703751i \(-0.248493\pi\)
0.710447 + 0.703751i \(0.248493\pi\)
\(14\) 0 0
\(15\) 0.848071 1.51022i 0.218971 0.389938i
\(16\) 0 0
\(17\) 2.00000i 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) 1.32431i 0.303817i 0.988395 + 0.151908i \(0.0485419\pi\)
−0.988395 + 0.151908i \(0.951458\pi\)
\(20\) 0 0
\(21\) −2.56155 + 4.56155i −0.558977 + 0.995412i
\(22\) 0 0
\(23\) 0.371834 0.0775328 0.0387664 0.999248i \(-0.487657\pi\)
0.0387664 + 0.999248i \(0.487657\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0.185917 + 5.19283i 0.0357798 + 0.999360i
\(28\) 0 0
\(29\) 3.12311i 0.579946i 0.957035 + 0.289973i \(0.0936464\pi\)
−0.957035 + 0.289973i \(0.906354\pi\)
\(30\) 0 0
\(31\) 4.71659i 0.847124i 0.905867 + 0.423562i \(0.139220\pi\)
−0.905867 + 0.423562i \(0.860780\pi\)
\(32\) 0 0
\(33\) −2.00000 1.12311i −0.348155 0.195508i
\(34\) 0 0
\(35\) 3.02045 0.510549
\(36\) 0 0
\(37\) −5.12311 −0.842233 −0.421117 0.907006i \(-0.638362\pi\)
−0.421117 + 0.907006i \(0.638362\pi\)
\(38\) 0 0
\(39\) 7.73704 + 4.34475i 1.23892 + 0.695718i
\(40\) 0 0
\(41\) 1.12311i 0.175400i 0.996147 + 0.0876998i \(0.0279516\pi\)
−0.996147 + 0.0876998i \(0.972048\pi\)
\(42\) 0 0
\(43\) 7.73704i 1.17989i 0.807445 + 0.589944i \(0.200850\pi\)
−0.807445 + 0.589944i \(0.799150\pi\)
\(44\) 0 0
\(45\) 2.56155 1.56155i 0.381854 0.232783i
\(46\) 0 0
\(47\) 3.02045 0.440578 0.220289 0.975435i \(-0.429300\pi\)
0.220289 + 0.975435i \(0.429300\pi\)
\(48\) 0 0
\(49\) −2.12311 −0.303301
\(50\) 0 0
\(51\) 1.69614 3.02045i 0.237507 0.422947i
\(52\) 0 0
\(53\) 12.2462i 1.68215i −0.540921 0.841073i \(-0.681924\pi\)
0.540921 0.841073i \(-0.318076\pi\)
\(54\) 0 0
\(55\) 1.32431i 0.178570i
\(56\) 0 0
\(57\) −1.12311 + 2.00000i −0.148759 + 0.264906i
\(58\) 0 0
\(59\) 14.1498 1.84214 0.921071 0.389394i \(-0.127315\pi\)
0.921071 + 0.389394i \(0.127315\pi\)
\(60\) 0 0
\(61\) −3.12311 −0.399873 −0.199936 0.979809i \(-0.564074\pi\)
−0.199936 + 0.979809i \(0.564074\pi\)
\(62\) 0 0
\(63\) −7.73704 + 4.71659i −0.974775 + 0.594234i
\(64\) 0 0
\(65\) 5.12311i 0.635443i
\(66\) 0 0
\(67\) 4.34475i 0.530796i −0.964139 0.265398i \(-0.914497\pi\)
0.964139 0.265398i \(-0.0855034\pi\)
\(68\) 0 0
\(69\) 0.561553 + 0.315342i 0.0676030 + 0.0379627i
\(70\) 0 0
\(71\) 3.39228 0.402590 0.201295 0.979531i \(-0.435485\pi\)
0.201295 + 0.979531i \(0.435485\pi\)
\(72\) 0 0
\(73\) 8.24621 0.965146 0.482573 0.875856i \(-0.339702\pi\)
0.482573 + 0.875856i \(0.339702\pi\)
\(74\) 0 0
\(75\) −1.51022 0.848071i −0.174386 0.0979267i
\(76\) 0 0
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) 8.10887i 0.912319i −0.889898 0.456160i \(-0.849225\pi\)
0.889898 0.456160i \(-0.150775\pi\)
\(80\) 0 0
\(81\) −4.12311 + 8.00000i −0.458123 + 0.888889i
\(82\) 0 0
\(83\) −15.1022 −1.65769 −0.828843 0.559481i \(-0.811000\pi\)
−0.828843 + 0.559481i \(0.811000\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) −2.64861 + 4.71659i −0.283961 + 0.505671i
\(88\) 0 0
\(89\) 10.2462i 1.08610i −0.839702 0.543048i \(-0.817270\pi\)
0.839702 0.543048i \(-0.182730\pi\)
\(90\) 0 0
\(91\) 15.4741i 1.62212i
\(92\) 0 0
\(93\) −4.00000 + 7.12311i −0.414781 + 0.738632i
\(94\) 0 0
\(95\) 1.32431 0.135871
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) −2.06798 3.39228i −0.207839 0.340937i
\(100\) 0 0
\(101\) 0.876894i 0.0872543i −0.999048 0.0436271i \(-0.986109\pi\)
0.999048 0.0436271i \(-0.0138914\pi\)
\(102\) 0 0
\(103\) 9.80501i 0.966117i 0.875588 + 0.483058i \(0.160474\pi\)
−0.875588 + 0.483058i \(0.839526\pi\)
\(104\) 0 0
\(105\) 4.56155 + 2.56155i 0.445162 + 0.249982i
\(106\) 0 0
\(107\) 3.02045 0.291998 0.145999 0.989285i \(-0.453360\pi\)
0.145999 + 0.989285i \(0.453360\pi\)
\(108\) 0 0
\(109\) 0.876894 0.0839912 0.0419956 0.999118i \(-0.486628\pi\)
0.0419956 + 0.999118i \(0.486628\pi\)
\(110\) 0 0
\(111\) −7.73704 4.34475i −0.734367 0.412386i
\(112\) 0 0
\(113\) 14.0000i 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 0 0
\(115\) 0.371834i 0.0346737i
\(116\) 0 0
\(117\) 8.00000 + 13.1231i 0.739600 + 1.21323i
\(118\) 0 0
\(119\) 6.04090 0.553768
\(120\) 0 0
\(121\) −9.24621 −0.840565
\(122\) 0 0
\(123\) −0.952473 + 1.69614i −0.0858816 + 0.152936i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 15.1022i 1.34011i −0.742313 0.670054i \(-0.766271\pi\)
0.742313 0.670054i \(-0.233729\pi\)
\(128\) 0 0
\(129\) −6.56155 + 11.6847i −0.577713 + 1.02878i
\(130\) 0 0
\(131\) −5.46026 −0.477065 −0.238532 0.971135i \(-0.576666\pi\)
−0.238532 + 0.971135i \(0.576666\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 0 0
\(135\) 5.19283 0.185917i 0.446927 0.0160012i
\(136\) 0 0
\(137\) 8.24621i 0.704521i 0.935902 + 0.352261i \(0.114587\pi\)
−0.935902 + 0.352261i \(0.885413\pi\)
\(138\) 0 0
\(139\) 17.5420i 1.48790i −0.668237 0.743949i \(-0.732951\pi\)
0.668237 0.743949i \(-0.267049\pi\)
\(140\) 0 0
\(141\) 4.56155 + 2.56155i 0.384152 + 0.215722i
\(142\) 0 0
\(143\) −6.78456 −0.567354
\(144\) 0 0
\(145\) 3.12311 0.259360
\(146\) 0 0
\(147\) −3.20636 1.80054i −0.264457 0.148506i
\(148\) 0 0
\(149\) 14.0000i 1.14692i −0.819232 0.573462i \(-0.805600\pi\)
0.819232 0.573462i \(-0.194400\pi\)
\(150\) 0 0
\(151\) 7.36520i 0.599372i 0.954038 + 0.299686i \(0.0968819\pi\)
−0.954038 + 0.299686i \(0.903118\pi\)
\(152\) 0 0
\(153\) 5.12311 3.12311i 0.414179 0.252488i
\(154\) 0 0
\(155\) 4.71659 0.378846
\(156\) 0 0
\(157\) 3.36932 0.268901 0.134450 0.990920i \(-0.457073\pi\)
0.134450 + 0.990920i \(0.457073\pi\)
\(158\) 0 0
\(159\) 10.3857 18.4945i 0.823636 1.46671i
\(160\) 0 0
\(161\) 1.12311i 0.0885131i
\(162\) 0 0
\(163\) 15.6829i 1.22838i −0.789159 0.614189i \(-0.789483\pi\)
0.789159 0.614189i \(-0.210517\pi\)
\(164\) 0 0
\(165\) −1.12311 + 2.00000i −0.0874337 + 0.155700i
\(166\) 0 0
\(167\) −9.06134 −0.701188 −0.350594 0.936528i \(-0.614020\pi\)
−0.350594 + 0.936528i \(0.614020\pi\)
\(168\) 0 0
\(169\) 13.2462 1.01894
\(170\) 0 0
\(171\) −3.39228 + 2.06798i −0.259414 + 0.158142i
\(172\) 0 0
\(173\) 2.00000i 0.152057i 0.997106 + 0.0760286i \(0.0242240\pi\)
−0.997106 + 0.0760286i \(0.975776\pi\)
\(174\) 0 0
\(175\) 3.02045i 0.228324i
\(176\) 0 0
\(177\) 21.3693 + 12.0000i 1.60622 + 0.901975i
\(178\) 0 0
\(179\) 10.0138 0.748468 0.374234 0.927334i \(-0.377906\pi\)
0.374234 + 0.927334i \(0.377906\pi\)
\(180\) 0 0
\(181\) 12.2462 0.910254 0.455127 0.890427i \(-0.349594\pi\)
0.455127 + 0.890427i \(0.349594\pi\)
\(182\) 0 0
\(183\) −4.71659 2.64861i −0.348660 0.195791i
\(184\) 0 0
\(185\) 5.12311i 0.376658i
\(186\) 0 0
\(187\) 2.64861i 0.193686i
\(188\) 0 0
\(189\) −15.6847 + 0.561553i −1.14089 + 0.0408470i
\(190\) 0 0
\(191\) −24.9073 −1.80223 −0.901113 0.433585i \(-0.857248\pi\)
−0.901113 + 0.433585i \(0.857248\pi\)
\(192\) 0 0
\(193\) −0.246211 −0.0177227 −0.00886134 0.999961i \(-0.502821\pi\)
−0.00886134 + 0.999961i \(0.502821\pi\)
\(194\) 0 0
\(195\) 4.34475 7.73704i 0.311134 0.554061i
\(196\) 0 0
\(197\) 4.24621i 0.302530i 0.988493 + 0.151265i \(0.0483347\pi\)
−0.988493 + 0.151265i \(0.951665\pi\)
\(198\) 0 0
\(199\) 5.46026i 0.387067i −0.981094 0.193534i \(-0.938005\pi\)
0.981094 0.193534i \(-0.0619949\pi\)
\(200\) 0 0
\(201\) 3.68466 6.56155i 0.259896 0.462816i
\(202\) 0 0
\(203\) −9.43318 −0.662079
\(204\) 0 0
\(205\) 1.12311 0.0784411
\(206\) 0 0
\(207\) 0.580639 + 0.952473i 0.0403572 + 0.0662014i
\(208\) 0 0
\(209\) 1.75379i 0.121312i
\(210\) 0 0
\(211\) 16.7984i 1.15645i −0.815878 0.578224i \(-0.803746\pi\)
0.815878 0.578224i \(-0.196254\pi\)
\(212\) 0 0
\(213\) 5.12311 + 2.87689i 0.351029 + 0.197122i
\(214\) 0 0
\(215\) 7.73704 0.527662
\(216\) 0 0
\(217\) −14.2462 −0.967096
\(218\) 0 0
\(219\) 12.4536 + 6.99337i 0.841538 + 0.472568i
\(220\) 0 0
\(221\) 10.2462i 0.689235i
\(222\) 0 0
\(223\) 8.31768i 0.556993i −0.960437 0.278496i \(-0.910164\pi\)
0.960437 0.278496i \(-0.0898360\pi\)
\(224\) 0 0
\(225\) −1.56155 2.56155i −0.104104 0.170770i
\(226\) 0 0
\(227\) −21.8868 −1.45268 −0.726339 0.687337i \(-0.758780\pi\)
−0.726339 + 0.687337i \(0.758780\pi\)
\(228\) 0 0
\(229\) 16.2462 1.07358 0.536790 0.843716i \(-0.319637\pi\)
0.536790 + 0.843716i \(0.319637\pi\)
\(230\) 0 0
\(231\) 3.39228 6.04090i 0.223196 0.397462i
\(232\) 0 0
\(233\) 10.0000i 0.655122i 0.944830 + 0.327561i \(0.106227\pi\)
−0.944830 + 0.327561i \(0.893773\pi\)
\(234\) 0 0
\(235\) 3.02045i 0.197032i
\(236\) 0 0
\(237\) 6.87689 12.2462i 0.446702 0.795477i
\(238\) 0 0
\(239\) 17.3790 1.12416 0.562078 0.827084i \(-0.310002\pi\)
0.562078 + 0.827084i \(0.310002\pi\)
\(240\) 0 0
\(241\) 13.3693 0.861193 0.430597 0.902544i \(-0.358303\pi\)
0.430597 + 0.902544i \(0.358303\pi\)
\(242\) 0 0
\(243\) −13.0114 + 8.58511i −0.834680 + 0.550735i
\(244\) 0 0
\(245\) 2.12311i 0.135640i
\(246\) 0 0
\(247\) 6.78456i 0.431691i
\(248\) 0 0
\(249\) −22.8078 12.8078i −1.44538 0.811659i
\(250\) 0 0
\(251\) −18.7033 −1.18054 −0.590272 0.807205i \(-0.700979\pi\)
−0.590272 + 0.807205i \(0.700979\pi\)
\(252\) 0 0
\(253\) −0.492423 −0.0309583
\(254\) 0 0
\(255\) −3.02045 1.69614i −0.189148 0.106216i
\(256\) 0 0
\(257\) 30.4924i 1.90207i −0.309091 0.951033i \(-0.600025\pi\)
0.309091 0.951033i \(-0.399975\pi\)
\(258\) 0 0
\(259\) 15.4741i 0.961512i
\(260\) 0 0
\(261\) −8.00000 + 4.87689i −0.495188 + 0.301872i
\(262\) 0 0
\(263\) −23.7917 −1.46706 −0.733531 0.679656i \(-0.762129\pi\)
−0.733531 + 0.679656i \(0.762129\pi\)
\(264\) 0 0
\(265\) −12.2462 −0.752279
\(266\) 0 0
\(267\) 8.68951 15.4741i 0.531789 0.946998i
\(268\) 0 0
\(269\) 14.0000i 0.853595i 0.904347 + 0.426798i \(0.140358\pi\)
−0.904347 + 0.426798i \(0.859642\pi\)
\(270\) 0 0
\(271\) 15.3110i 0.930080i −0.885290 0.465040i \(-0.846040\pi\)
0.885290 0.465040i \(-0.153960\pi\)
\(272\) 0 0
\(273\) −13.1231 + 23.3693i −0.794246 + 1.41438i
\(274\) 0 0
\(275\) 1.32431 0.0798587
\(276\) 0 0
\(277\) −23.3693 −1.40413 −0.702063 0.712115i \(-0.747738\pi\)
−0.702063 + 0.712115i \(0.747738\pi\)
\(278\) 0 0
\(279\) −12.0818 + 7.36520i −0.723318 + 0.440943i
\(280\) 0 0
\(281\) 13.6155i 0.812234i −0.913821 0.406117i \(-0.866882\pi\)
0.913821 0.406117i \(-0.133118\pi\)
\(282\) 0 0
\(283\) 23.2111i 1.37976i 0.723925 + 0.689879i \(0.242336\pi\)
−0.723925 + 0.689879i \(0.757664\pi\)
\(284\) 0 0
\(285\) 2.00000 + 1.12311i 0.118470 + 0.0665270i
\(286\) 0 0
\(287\) −3.39228 −0.200240
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −9.06134 5.08842i −0.531185 0.298289i
\(292\) 0 0
\(293\) 2.49242i 0.145609i 0.997346 + 0.0728044i \(0.0231949\pi\)
−0.997346 + 0.0728044i \(0.976805\pi\)
\(294\) 0 0
\(295\) 14.1498i 0.823831i
\(296\) 0 0
\(297\) −0.246211 6.87689i −0.0142866 0.399038i
\(298\) 0 0
\(299\) 1.90495 0.110166
\(300\) 0 0
\(301\) −23.3693 −1.34699
\(302\) 0 0
\(303\) 0.743668 1.32431i 0.0427226 0.0760794i
\(304\) 0 0
\(305\) 3.12311i 0.178829i
\(306\) 0 0
\(307\) 11.1293i 0.635184i 0.948227 + 0.317592i \(0.102874\pi\)
−0.948227 + 0.317592i \(0.897126\pi\)
\(308\) 0 0
\(309\) −8.31534 + 14.8078i −0.473043 + 0.842384i
\(310\) 0 0
\(311\) 20.7713 1.17783 0.588916 0.808194i \(-0.299555\pi\)
0.588916 + 0.808194i \(0.299555\pi\)
\(312\) 0 0
\(313\) −22.4924 −1.27135 −0.635673 0.771958i \(-0.719278\pi\)
−0.635673 + 0.771958i \(0.719278\pi\)
\(314\) 0 0
\(315\) 4.71659 + 7.73704i 0.265750 + 0.435933i
\(316\) 0 0
\(317\) 16.7386i 0.940135i −0.882630 0.470068i \(-0.844230\pi\)
0.882630 0.470068i \(-0.155770\pi\)
\(318\) 0 0
\(319\) 4.13595i 0.231569i
\(320\) 0 0
\(321\) 4.56155 + 2.56155i 0.254601 + 0.142972i
\(322\) 0 0
\(323\) 2.64861 0.147373
\(324\) 0 0
\(325\) −5.12311 −0.284179
\(326\) 0 0
\(327\) 1.32431 + 0.743668i 0.0732343 + 0.0411249i
\(328\) 0 0
\(329\) 9.12311i 0.502973i
\(330\) 0 0
\(331\) 3.22925i 0.177496i −0.996054 0.0887479i \(-0.971713\pi\)
0.996054 0.0887479i \(-0.0282865\pi\)
\(332\) 0 0
\(333\) −8.00000 13.1231i −0.438397 0.719142i
\(334\) 0 0
\(335\) −4.34475 −0.237379
\(336\) 0 0
\(337\) −1.50758 −0.0821230 −0.0410615 0.999157i \(-0.513074\pi\)
−0.0410615 + 0.999157i \(0.513074\pi\)
\(338\) 0 0
\(339\) 11.8730 21.1431i 0.644852 1.14834i
\(340\) 0 0
\(341\) 6.24621i 0.338251i
\(342\) 0 0
\(343\) 14.7304i 0.795367i
\(344\) 0 0
\(345\) 0.315342 0.561553i 0.0169774 0.0302330i
\(346\) 0 0
\(347\) 22.6305 1.21487 0.607434 0.794370i \(-0.292199\pi\)
0.607434 + 0.794370i \(0.292199\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 0.952473 + 26.6034i 0.0508392 + 1.41998i
\(352\) 0 0
\(353\) 20.2462i 1.07760i 0.842435 + 0.538799i \(0.181122\pi\)
−0.842435 + 0.538799i \(0.818878\pi\)
\(354\) 0 0
\(355\) 3.39228i 0.180044i
\(356\) 0 0
\(357\) 9.12311 + 5.12311i 0.482846 + 0.271144i
\(358\) 0 0
\(359\) 21.5150 1.13552 0.567758 0.823195i \(-0.307811\pi\)
0.567758 + 0.823195i \(0.307811\pi\)
\(360\) 0 0
\(361\) 17.2462 0.907695
\(362\) 0 0
\(363\) −13.9638 7.84144i −0.732912 0.411569i
\(364\) 0 0
\(365\) 8.24621i 0.431626i
\(366\) 0 0
\(367\) 10.9663i 0.572436i 0.958165 + 0.286218i \(0.0923981\pi\)
−0.958165 + 0.286218i \(0.907602\pi\)
\(368\) 0 0
\(369\) −2.87689 + 1.75379i −0.149765 + 0.0912986i
\(370\) 0 0
\(371\) 36.9890 1.92038
\(372\) 0 0
\(373\) 9.12311 0.472377 0.236188 0.971707i \(-0.424102\pi\)
0.236188 + 0.971707i \(0.424102\pi\)
\(374\) 0 0
\(375\) −0.848071 + 1.51022i −0.0437942 + 0.0779876i
\(376\) 0 0
\(377\) 16.0000i 0.824042i
\(378\) 0 0
\(379\) 18.7033i 0.960725i 0.877070 + 0.480363i \(0.159495\pi\)
−0.877070 + 0.480363i \(0.840505\pi\)
\(380\) 0 0
\(381\) 12.8078 22.8078i 0.656162 1.16848i
\(382\) 0 0
\(383\) −15.1022 −0.771688 −0.385844 0.922564i \(-0.626090\pi\)
−0.385844 + 0.922564i \(0.626090\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) 0 0
\(387\) −19.8188 + 12.0818i −1.00745 + 0.614152i
\(388\) 0 0
\(389\) 20.7386i 1.05149i −0.850642 0.525745i \(-0.823787\pi\)
0.850642 0.525745i \(-0.176213\pi\)
\(390\) 0 0
\(391\) 0.743668i 0.0376089i
\(392\) 0 0
\(393\) −8.24621 4.63068i −0.415966 0.233587i
\(394\) 0 0
\(395\) −8.10887 −0.408002
\(396\) 0 0
\(397\) −14.8769 −0.746650 −0.373325 0.927701i \(-0.621782\pi\)
−0.373325 + 0.927701i \(0.621782\pi\)
\(398\) 0 0
\(399\) −6.04090 3.39228i −0.302423 0.169827i
\(400\) 0 0
\(401\) 24.0000i 1.19850i −0.800561 0.599251i \(-0.795465\pi\)
0.800561 0.599251i \(-0.204535\pi\)
\(402\) 0 0
\(403\) 24.1636i 1.20367i
\(404\) 0 0
\(405\) 8.00000 + 4.12311i 0.397523 + 0.204879i
\(406\) 0 0
\(407\) 6.78456 0.336298
\(408\) 0 0
\(409\) 25.3693 1.25443 0.627216 0.778845i \(-0.284194\pi\)
0.627216 + 0.778845i \(0.284194\pi\)
\(410\) 0 0
\(411\) −6.99337 + 12.4536i −0.344957 + 0.614292i
\(412\) 0 0
\(413\) 42.7386i 2.10303i
\(414\) 0 0
\(415\) 15.1022i 0.741340i
\(416\) 0 0
\(417\) 14.8769 26.4924i 0.728525 1.29734i
\(418\) 0 0
\(419\) −7.36520 −0.359814 −0.179907 0.983684i \(-0.557580\pi\)
−0.179907 + 0.983684i \(0.557580\pi\)
\(420\) 0 0
\(421\) 25.3693 1.23642 0.618212 0.786011i \(-0.287857\pi\)
0.618212 + 0.786011i \(0.287857\pi\)
\(422\) 0 0
\(423\) 4.71659 + 7.73704i 0.229328 + 0.376188i
\(424\) 0 0
\(425\) 2.00000i 0.0970143i
\(426\) 0 0
\(427\) 9.43318i 0.456503i
\(428\) 0 0
\(429\) −10.2462 5.75379i −0.494692 0.277796i
\(430\) 0 0
\(431\) −16.6354 −0.801297 −0.400648 0.916232i \(-0.631215\pi\)
−0.400648 + 0.916232i \(0.631215\pi\)
\(432\) 0 0
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 0 0
\(435\) 4.71659 + 2.64861i 0.226143 + 0.126991i
\(436\) 0 0
\(437\) 0.492423i 0.0235558i
\(438\) 0 0
\(439\) 9.27015i 0.442440i −0.975224 0.221220i \(-0.928996\pi\)
0.975224 0.221220i \(-0.0710039\pi\)
\(440\) 0 0
\(441\) −3.31534 5.43845i −0.157873 0.258974i
\(442\) 0 0
\(443\) 16.5896 0.788195 0.394097 0.919069i \(-0.371057\pi\)
0.394097 + 0.919069i \(0.371057\pi\)
\(444\) 0 0
\(445\) −10.2462 −0.485717
\(446\) 0 0
\(447\) 11.8730 21.1431i 0.561573 1.00004i
\(448\) 0 0
\(449\) 27.3693i 1.29164i 0.763491 + 0.645819i \(0.223484\pi\)
−0.763491 + 0.645819i \(0.776516\pi\)
\(450\) 0 0
\(451\) 1.48734i 0.0700359i
\(452\) 0 0
\(453\) −6.24621 + 11.1231i −0.293473 + 0.522609i
\(454\) 0 0
\(455\) 15.4741 0.725436
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) 10.3857 0.371834i 0.484761 0.0173557i
\(460\) 0 0
\(461\) 41.8617i 1.94970i 0.222872 + 0.974848i \(0.428457\pi\)
−0.222872 + 0.974848i \(0.571543\pi\)
\(462\) 0 0
\(463\) 3.02045i 0.140372i −0.997534 0.0701861i \(-0.977641\pi\)
0.997534 0.0701861i \(-0.0223593\pi\)
\(464\) 0 0
\(465\) 7.12311 + 4.00000i 0.330326 + 0.185496i
\(466\) 0 0
\(467\) −2.27678 −0.105357 −0.0526784 0.998612i \(-0.516776\pi\)
−0.0526784 + 0.998612i \(0.516776\pi\)
\(468\) 0 0
\(469\) 13.1231 0.605969
\(470\) 0 0
\(471\) 5.08842 + 2.85742i 0.234462 + 0.131663i
\(472\) 0 0
\(473\) 10.2462i 0.471121i
\(474\) 0 0
\(475\) 1.32431i 0.0607634i
\(476\) 0 0
\(477\) 31.3693 19.1231i 1.43630 0.875587i
\(478\) 0 0
\(479\) 25.6509 1.17202 0.586010 0.810304i \(-0.300698\pi\)
0.586010 + 0.810304i \(0.300698\pi\)
\(480\) 0 0
\(481\) −26.2462 −1.19672
\(482\) 0 0
\(483\) −0.952473 + 1.69614i −0.0433390 + 0.0771771i
\(484\) 0 0
\(485\) 6.00000i 0.272446i
\(486\) 0 0
\(487\) 25.2791i 1.14550i 0.819728 + 0.572752i \(0.194124\pi\)
−0.819728 + 0.572752i \(0.805876\pi\)
\(488\) 0 0
\(489\) 13.3002 23.6847i 0.601455 1.07106i
\(490\) 0 0
\(491\) −26.9752 −1.21737 −0.608687 0.793410i \(-0.708304\pi\)
−0.608687 + 0.793410i \(0.708304\pi\)
\(492\) 0 0
\(493\) 6.24621 0.281315
\(494\) 0 0
\(495\) −3.39228 + 2.06798i −0.152472 + 0.0929486i
\(496\) 0 0
\(497\) 10.2462i 0.459605i
\(498\) 0 0
\(499\) 32.2725i 1.44471i 0.691521 + 0.722357i \(0.256941\pi\)
−0.691521 + 0.722357i \(0.743059\pi\)
\(500\) 0 0
\(501\) −13.6847 7.68466i −0.611385 0.343325i
\(502\) 0 0
\(503\) 14.3586 0.640217 0.320109 0.947381i \(-0.396281\pi\)
0.320109 + 0.947381i \(0.396281\pi\)
\(504\) 0 0
\(505\) −0.876894 −0.0390213
\(506\) 0 0
\(507\) 20.0047 + 11.2337i 0.888442 + 0.498907i
\(508\) 0 0
\(509\) 11.1231i 0.493023i 0.969140 + 0.246511i \(0.0792843\pi\)
−0.969140 + 0.246511i \(0.920716\pi\)
\(510\) 0 0
\(511\) 24.9073i 1.10183i
\(512\) 0 0
\(513\) −6.87689 + 0.246211i −0.303622 + 0.0108705i
\(514\) 0 0
\(515\) 9.80501 0.432060
\(516\) 0 0
\(517\) −4.00000 −0.175920
\(518\) 0 0
\(519\) −1.69614 + 3.02045i −0.0744523 + 0.132583i
\(520\) 0 0
\(521\) 38.2462i 1.67560i 0.545980 + 0.837798i \(0.316158\pi\)
−0.545980 + 0.837798i \(0.683842\pi\)
\(522\) 0 0
\(523\) 35.2929i 1.54325i −0.636077 0.771625i \(-0.719444\pi\)
0.636077 0.771625i \(-0.280556\pi\)
\(524\) 0 0
\(525\) 2.56155 4.56155i 0.111795 0.199082i
\(526\) 0 0
\(527\) 9.43318 0.410916
\(528\) 0 0
\(529\) −22.8617 −0.993989
\(530\) 0 0
\(531\) 22.0956 + 36.2454i 0.958868 + 1.57292i
\(532\) 0 0
\(533\) 5.75379i 0.249224i
\(534\) 0 0
\(535\) 3.02045i 0.130585i
\(536\) 0 0
\(537\) 15.1231 + 8.49242i 0.652610 + 0.366475i
\(538\) 0 0
\(539\) 2.81164 0.121106
\(540\) 0 0
\(541\) 26.9848 1.16017 0.580085 0.814556i \(-0.303020\pi\)
0.580085 + 0.814556i \(0.303020\pi\)
\(542\) 0 0
\(543\) 18.4945 + 10.3857i 0.793676 + 0.445691i
\(544\) 0 0
\(545\) 0.876894i 0.0375620i
\(546\) 0 0
\(547\) 5.83209i 0.249362i 0.992197 + 0.124681i \(0.0397908\pi\)
−0.992197 + 0.124681i \(0.960209\pi\)
\(548\) 0 0
\(549\) −4.87689 8.00000i −0.208141 0.341432i
\(550\) 0 0
\(551\) −4.13595 −0.176197
\(552\) 0 0
\(553\) 24.4924 1.04152
\(554\) 0 0
\(555\) −4.34475 + 7.73704i −0.184425 + 0.328419i
\(556\) 0 0
\(557\) 36.2462i 1.53580i 0.640569 + 0.767901i \(0.278699\pi\)
−0.640569 + 0.767901i \(0.721301\pi\)
\(558\) 0 0
\(559\) 39.6377i 1.67649i
\(560\) 0 0
\(561\) −2.24621 + 4.00000i −0.0948351 + 0.168880i
\(562\) 0 0
\(563\) 7.90007 0.332948 0.166474 0.986046i \(-0.446762\pi\)
0.166474 + 0.986046i \(0.446762\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) 0 0
\(567\) −24.1636 12.4536i −1.01478 0.523003i
\(568\) 0 0
\(569\) 13.1231i 0.550149i 0.961423 + 0.275075i \(0.0887026\pi\)
−0.961423 + 0.275075i \(0.911297\pi\)
\(570\) 0 0
\(571\) 33.0161i 1.38168i 0.723007 + 0.690841i \(0.242759\pi\)
−0.723007 + 0.690841i \(0.757241\pi\)
\(572\) 0 0
\(573\) −37.6155 21.1231i −1.57141 0.882430i
\(574\) 0 0
\(575\) −0.371834 −0.0155066
\(576\) 0 0
\(577\) −32.2462 −1.34243 −0.671214 0.741264i \(-0.734227\pi\)
−0.671214 + 0.741264i \(0.734227\pi\)
\(578\) 0 0
\(579\) −0.371834 0.208805i −0.0154529 0.00867762i
\(580\) 0 0
\(581\) 45.6155i 1.89245i
\(582\) 0 0
\(583\) 16.2177i 0.671670i
\(584\) 0 0
\(585\) 13.1231 8.00000i 0.542574 0.330759i
\(586\) 0 0
\(587\) −1.85917 −0.0767362 −0.0383681 0.999264i \(-0.512216\pi\)
−0.0383681 + 0.999264i \(0.512216\pi\)
\(588\) 0 0
\(589\) −6.24621 −0.257371
\(590\) 0 0
\(591\) −3.60109 + 6.41273i −0.148129 + 0.263784i
\(592\) 0 0
\(593\) 8.24621i 0.338631i −0.985562 0.169316i \(-0.945844\pi\)
0.985562 0.169316i \(-0.0541557\pi\)
\(594\) 0 0
\(595\) 6.04090i 0.247653i
\(596\) 0 0
\(597\) 4.63068 8.24621i 0.189521 0.337495i
\(598\) 0 0
\(599\) −44.1912 −1.80560 −0.902802 0.430056i \(-0.858494\pi\)
−0.902802 + 0.430056i \(0.858494\pi\)
\(600\) 0 0
\(601\) 23.1231 0.943211 0.471606 0.881810i \(-0.343675\pi\)
0.471606 + 0.881810i \(0.343675\pi\)
\(602\) 0 0
\(603\) 11.1293 6.78456i 0.453221 0.276289i
\(604\) 0 0
\(605\) 9.24621i 0.375912i
\(606\) 0 0
\(607\) 4.50778i 0.182965i −0.995807 0.0914827i \(-0.970839\pi\)
0.995807 0.0914827i \(-0.0291606\pi\)
\(608\) 0 0
\(609\) −14.2462 8.00000i −0.577286 0.324176i
\(610\) 0 0
\(611\) 15.4741 0.626014
\(612\) 0 0
\(613\) −9.12311 −0.368479 −0.184239 0.982881i \(-0.558982\pi\)
−0.184239 + 0.982881i \(0.558982\pi\)
\(614\) 0 0
\(615\) 1.69614 + 0.952473i 0.0683950 + 0.0384074i
\(616\) 0 0
\(617\) 14.0000i 0.563619i 0.959470 + 0.281809i \(0.0909346\pi\)
−0.959470 + 0.281809i \(0.909065\pi\)
\(618\) 0 0
\(619\) 28.1365i 1.13090i −0.824782 0.565451i \(-0.808702\pi\)
0.824782 0.565451i \(-0.191298\pi\)
\(620\) 0 0
\(621\) 0.0691303 + 1.93087i 0.00277410 + 0.0774831i
\(622\) 0 0
\(623\) 30.9481 1.23991
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.48734 2.64861i 0.0593985 0.105775i
\(628\) 0 0
\(629\) 10.2462i 0.408543i
\(630\) 0 0
\(631\) 39.8007i 1.58444i −0.610235 0.792220i \(-0.708925\pi\)
0.610235 0.792220i \(-0.291075\pi\)
\(632\) 0 0
\(633\) 14.2462 25.3693i 0.566236 1.00834i
\(634\) 0 0
\(635\) −15.1022 −0.599314
\(636\) 0 0
\(637\) −10.8769 −0.430958
\(638\) 0 0
\(639\) 5.29723 + 8.68951i 0.209555 + 0.343752i
\(640\) 0 0
\(641\) 6.38447i 0.252171i 0.992019 + 0.126086i \(0.0402414\pi\)
−0.992019 + 0.126086i \(0.959759\pi\)
\(642\) 0 0
\(643\) 3.60109i 0.142013i 0.997476 + 0.0710065i \(0.0226211\pi\)
−0.997476 + 0.0710065i \(0.977379\pi\)
\(644\) 0 0
\(645\) 11.6847 + 6.56155i 0.460083 + 0.258361i
\(646\) 0 0
\(647\) −36.6172 −1.43957 −0.719786 0.694197i \(-0.755760\pi\)
−0.719786 + 0.694197i \(0.755760\pi\)
\(648\) 0 0
\(649\) −18.7386 −0.735556
\(650\) 0 0
\(651\) −21.5150 12.0818i −0.843238 0.473523i
\(652\) 0 0
\(653\) 38.9848i 1.52559i −0.646638 0.762797i \(-0.723825\pi\)
0.646638 0.762797i \(-0.276175\pi\)
\(654\) 0 0
\(655\) 5.46026i 0.213350i
\(656\) 0 0
\(657\) 12.8769 + 21.1231i 0.502375 + 0.824091i
\(658\) 0 0
\(659\) −24.7442 −0.963898 −0.481949 0.876199i \(-0.660071\pi\)
−0.481949 + 0.876199i \(0.660071\pi\)
\(660\) 0 0
\(661\) −28.1080 −1.09327 −0.546636 0.837370i \(-0.684092\pi\)
−0.546636 + 0.837370i \(0.684092\pi\)
\(662\) 0 0
\(663\) 8.68951 15.4741i 0.337473 0.600963i
\(664\) 0 0
\(665\) 4.00000i 0.155113i
\(666\) 0 0
\(667\) 1.16128i 0.0449648i
\(668\) 0 0
\(669\) 7.05398 12.5616i 0.272722 0.485658i
\(670\) 0 0
\(671\) 4.13595 0.159667
\(672\) 0 0
\(673\) 22.4924 0.867019 0.433510 0.901149i \(-0.357275\pi\)
0.433510 + 0.901149i \(0.357275\pi\)
\(674\) 0 0
\(675\) −0.185917 5.19283i −0.00715595 0.199872i
\(676\) 0 0
\(677\) 1.50758i 0.0579409i 0.999580 + 0.0289705i \(0.00922287\pi\)
−0.999580 + 0.0289705i \(0.990777\pi\)
\(678\) 0 0
\(679\) 18.1227i 0.695485i
\(680\) 0 0
\(681\) −33.0540 18.5616i −1.26663 0.711280i
\(682\) 0 0
\(683\) −7.90007 −0.302288 −0.151144 0.988512i \(-0.548296\pi\)
−0.151144 + 0.988512i \(0.548296\pi\)
\(684\) 0 0
\(685\) 8.24621 0.315072
\(686\) 0 0
\(687\) 24.5354 + 13.7779i 0.936085 + 0.525661i
\(688\) 0 0
\(689\) 62.7386i 2.39015i
\(690\) 0 0
\(691\) 18.2857i 0.695621i −0.937565 0.347811i \(-0.886925\pi\)
0.937565 0.347811i \(-0.113075\pi\)
\(692\) 0 0
\(693\) 10.2462 6.24621i 0.389221 0.237274i
\(694\) 0 0
\(695\) −17.5420 −0.665408
\(696\) 0 0
\(697\) 2.24621 0.0850813
\(698\) 0 0
\(699\) −8.48071 + 15.1022i −0.320770 + 0.571219i
\(700\) 0 0
\(701\) 17.5076i 0.661252i 0.943762 + 0.330626i \(0.107260\pi\)
−0.943762 + 0.330626i \(0.892740\pi\)
\(702\) 0 0
\(703\) 6.78456i 0.255885i
\(704\) 0 0
\(705\) 2.56155 4.56155i 0.0964737 0.171798i
\(706\) 0 0
\(707\) 2.64861 0.0996114
\(708\) 0 0
\(709\) −6.49242 −0.243828 −0.121914 0.992541i \(-0.538903\pi\)
−0.121914 + 0.992541i \(0.538903\pi\)
\(710\) 0 0
\(711\) 20.7713 12.6624i 0.778985 0.474878i
\(712\) 0 0
\(713\) 1.75379i 0.0656799i
\(714\) 0 0
\(715\) 6.78456i 0.253728i
\(716\) 0 0
\(717\) 26.2462 + 14.7386i 0.980183 + 0.550424i
\(718\) 0 0
\(719\) −30.9481 −1.15417 −0.577086 0.816684i \(-0.695810\pi\)
−0.577086 + 0.816684i \(0.695810\pi\)
\(720\) 0 0
\(721\) −29.6155 −1.10294
\(722\) 0 0
\(723\) 20.1907 + 11.3381i 0.750899 + 0.421669i
\(724\) 0 0
\(725\) 3.12311i 0.115989i
\(726\) 0 0
\(727\) 10.9663i 0.406717i −0.979104 0.203359i \(-0.934814\pi\)
0.979104 0.203359i \(-0.0651857\pi\)
\(728\) 0 0
\(729\) −26.9309 + 1.93087i −0.997440 + 0.0715137i
\(730\) 0 0
\(731\) 15.4741 0.572329
\(732\) 0 0
\(733\) 26.8769 0.992721 0.496360 0.868117i \(-0.334669\pi\)
0.496360 + 0.868117i \(0.334669\pi\)
\(734\) 0 0
\(735\) −1.80054 + 3.20636i −0.0664140 + 0.118269i
\(736\) 0 0
\(737\) 5.75379i 0.211944i
\(738\) 0 0
\(739\) 26.9752i 0.992300i 0.868237 + 0.496150i \(0.165253\pi\)
−0.868237 + 0.496150i \(0.834747\pi\)
\(740\) 0 0
\(741\) −5.75379 + 10.2462i −0.211371 + 0.376404i
\(742\) 0 0
\(743\) −9.80501 −0.359711 −0.179856 0.983693i \(-0.557563\pi\)
−0.179856 + 0.983693i \(0.557563\pi\)
\(744\) 0 0
\(745\) −14.0000 −0.512920
\(746\) 0 0
\(747\) −23.5829 38.6852i −0.862855 1.41542i
\(748\) 0 0
\(749\) 9.12311i 0.333351i
\(750\) 0 0
\(751\) 11.5012i 0.419683i 0.977735 + 0.209842i \(0.0672948\pi\)
−0.977735 + 0.209842i \(0.932705\pi\)
\(752\) 0 0
\(753\) −28.2462 15.8617i −1.02935 0.578034i
\(754\) 0 0
\(755\) 7.36520 0.268047
\(756\) 0 0
\(757\) −10.8769 −0.395327 −0.197664 0.980270i \(-0.563335\pi\)
−0.197664 + 0.980270i \(0.563335\pi\)
\(758\) 0 0
\(759\) −0.743668 0.417609i −0.0269934 0.0151582i
\(760\) 0 0
\(761\) 31.2311i 1.13212i 0.824362 + 0.566062i \(0.191534\pi\)
−0.824362 + 0.566062i \(0.808466\pi\)
\(762\) 0 0
\(763\) 2.64861i 0.0958863i
\(764\) 0 0
\(765\) −3.12311 5.12311i −0.112916 0.185226i
\(766\) 0 0
\(767\) 72.4908 2.61749
\(768\) 0 0
\(769\) 38.9848 1.40583 0.702915 0.711274i \(-0.251882\pi\)
0.702915 + 0.711274i \(0.251882\pi\)
\(770\) 0 0
\(771\) 25.8597 46.0504i 0.931315 1.65846i
\(772\) 0 0
\(773\) 0.246211i 0.00885560i −0.999990 0.00442780i \(-0.998591\pi\)
0.999990 0.00442780i \(-0.00140942\pi\)
\(774\) 0 0
\(775\) 4.71659i 0.169425i
\(776\) 0 0
\(777\) 13.1231 23.3693i 0.470789 0.838370i
\(778\) 0 0
\(779\) −1.48734 −0.0532894
\(780\) 0 0
\(781\) −4.49242 −0.160752
\(782\) 0 0
\(783\) −16.2177 + 0.580639i −0.579575 + 0.0207503i
\(784\) 0 0
\(785\) 3.36932i 0.120256i
\(786\) 0 0
\(787\) 42.0775i 1.49990i 0.661495 + 0.749950i \(0.269922\pi\)
−0.661495 + 0.749950i \(0.730078\pi\)
\(788\) 0 0
\(789\) −35.9309 20.1771i −1.27917 0.718323i
\(790\) 0 0
\(791\) 42.2863 1.50353
\(792\) 0 0
\(793\) −16.0000 −0.568177
\(794\) 0 0
\(795\) −18.4945 10.3857i −0.655933 0.368341i
\(796\) 0 0
\(797\) 12.7386i 0.451226i 0.974217 + 0.225613i \(0.0724384\pi\)
−0.974217 + 0.225613i \(0.927562\pi\)
\(798\) 0 0
\(799\) 6.04090i 0.213712i
\(800\) 0 0
\(801\) 26.2462 16.0000i 0.927364 0.565332i
\(802\) 0 0
\(803\) −10.9205 −0.385377
\(804\) 0 0
\(805\) 1.12311 0.0395843
\(806\) 0 0
\(807\) −11.8730 + 21.1431i −0.417949 + 0.744274i
\(808\) 0 0
\(809\) 29.7538i 1.04609i 0.852306 + 0.523044i \(0.175204\pi\)
−0.852306 + 0.523044i \(0.824796\pi\)
\(810\) 0 0
\(811\) 46.2592i 1.62438i −0.583393 0.812190i \(-0.698275\pi\)
0.583393 0.812190i \(-0.301725\pi\)
\(812\) 0 0
\(813\) 12.9848 23.1231i 0.455398 0.810963i
\(814\) 0 0
\(815\) −15.6829 −0.549347
\(816\) 0 0
\(817\) −10.2462 −0.358470
\(818\) 0 0
\(819\) −39.6377 + 24.1636i −1.38505 + 0.844344i
\(820\) 0 0
\(821\) 53.2311i 1.85778i 0.370360 + 0.928888i \(0.379234\pi\)
−0.370360 + 0.928888i \(0.620766\pi\)
\(822\) 0 0
\(823\) 48.2814i 1.68298i −0.540270 0.841492i \(-0.681678\pi\)
0.540270 0.841492i \(-0.318322\pi\)
\(824\) 0 0
\(825\) 2.00000 + 1.12311i 0.0696311 + 0.0391015i
\(826\) 0 0
\(827\) 17.7509 0.617258 0.308629 0.951183i \(-0.400130\pi\)
0.308629 + 0.951183i \(0.400130\pi\)
\(828\) 0 0
\(829\) 8.87689 0.308307 0.154154 0.988047i \(-0.450735\pi\)
0.154154 + 0.988047i \(0.450735\pi\)
\(830\) 0 0
\(831\) −35.2929 19.8188i −1.22430 0.687508i
\(832\) 0 0
\(833\) 4.24621i 0.147122i
\(834\) 0 0
\(835\) 9.06134i 0.313581i
\(836\) 0 0
\(837\) −24.4924 + 0.876894i −0.846582 + 0.0303099i
\(838\) 0 0
\(839\) −17.7051 −0.611247 −0.305624 0.952152i \(-0.598865\pi\)
−0.305624 + 0.952152i \(0.598865\pi\)
\(840\) 0 0
\(841\) 19.2462 0.663662
\(842\) 0 0
\(843\) 11.5469 20.5625i 0.397697 0.708210i
\(844\) 0 0
\(845\) 13.2462i 0.455684i
\(846\) 0 0
\(847\) 27.9277i 0.959607i
\(848\) 0 0
\(849\) −19.6847 + 35.0540i −0.675576 + 1.20305i
\(850\) 0 0
\(851\) −1.90495 −0.0653007
\(852\) 0 0
\(853\) 7.86174 0.269181 0.134590 0.990901i \(-0.457028\pi\)
0.134590 + 0.990901i \(0.457028\pi\)
\(854\) 0 0
\(855\) 2.06798 + 3.39228i 0.0707233 + 0.116014i
\(856\) 0 0
\(857\) 20.7386i 0.708418i 0.935166 + 0.354209i \(0.115250\pi\)
−0.935166 + 0.354209i \(0.884750\pi\)
\(858\) 0 0
\(859\) 33.4337i 1.14074i −0.821386 0.570372i \(-0.806799\pi\)
0.821386 0.570372i \(-0.193201\pi\)
\(860\) 0 0
\(861\) −5.12311 2.87689i −0.174595 0.0980443i
\(862\) 0 0
\(863\) 10.5487 0.359081 0.179541 0.983751i \(-0.442539\pi\)
0.179541 + 0.983751i \(0.442539\pi\)
\(864\) 0 0
\(865\) 2.00000 0.0680020
\(866\) 0 0
\(867\) 19.6329 + 11.0249i 0.666769 + 0.374426i
\(868\) 0 0
\(869\) 10.7386i 0.364283i
\(870\) 0 0
\(871\) 22.2586i 0.754205i
\(872\) 0 0
\(873\) −9.36932 15.3693i −0.317103 0.520173i
\(874\) 0 0
\(875\) −3.02045 −0.102110
\(876\) 0 0
\(877\) −37.6155 −1.27019 −0.635093 0.772436i \(-0.719038\pi\)
−0.635093 + 0.772436i \(0.719038\pi\)
\(878\) 0 0
\(879\) −2.11375 + 3.76412i −0.0712950 + 0.126960i
\(880\) 0 0
\(881\) 0.630683i 0.0212483i −0.999944 0.0106241i \(-0.996618\pi\)
0.999944 0.0106241i \(-0.00338183\pi\)
\(882\) 0 0
\(883\) 22.4674i 0.756090i −0.925787 0.378045i \(-0.876597\pi\)
0.925787 0.378045i \(-0.123403\pi\)
\(884\) 0 0
\(885\) 12.0000 21.3693i 0.403376 0.718322i
\(886\) 0 0
\(887\) −51.6737 −1.73503 −0.867516 0.497409i \(-0.834285\pi\)
−0.867516 + 0.497409i \(0.834285\pi\)
\(888\) 0 0
\(889\) 45.6155 1.52990
\(890\) 0 0
\(891\) 5.46026 10.5945i 0.182925 0.354928i
\(892\) 0 0
\(893\) 4.00000i 0.133855i
\(894\) 0 0
\(895\) 10.0138i 0.334725i
\(896\) 0 0
\(897\) 2.87689 + 1.61553i 0.0960567 + 0.0539409i
\(898\) 0 0
\(899\) −14.7304 −0.491287
\(900\) 0 0
\(901\) −24.4924 −0.815961
\(902\) 0 0
\(903\) −35.2929 19.8188i −1.17447 0.659529i
\(904\) 0 0
\(905\) 12.2462i 0.407078i
\(906\) 0 0
\(907\) 46.2134i 1.53449i −0.641353 0.767246i \(-0.721627\pi\)
0.641353 0.767246i \(-0.278373\pi\)
\(908\) 0 0
\(909\) 2.24621 1.36932i 0.0745021 0.0454174i
\(910\) 0 0
\(911\) −14.3128 −0.474204 −0.237102 0.971485i \(-0.576198\pi\)
−0.237102 + 0.971485i \(0.576198\pi\)
\(912\) 0 0
\(913\) 20.0000 0.661903
\(914\) 0 0
\(915\) −2.64861 + 4.71659i −0.0875605 + 0.155926i
\(916\) 0 0
\(917\) 16.4924i 0.544628i
\(918\) 0 0
\(919\) 49.9775i 1.64861i 0.566148 + 0.824303i \(0.308433\pi\)
−0.566148 + 0.824303i \(0.691567\pi\)
\(920\) 0 0
\(921\) −9.43845 + 16.8078i −0.311007 + 0.553835i
\(922\) 0 0
\(923\) 17.3790 0.572037
\(924\) 0 0
\(925\) 5.12311 0.168447
\(926\) 0 0
\(927\) −25.1161 + 15.3110i −0.824920 + 0.502881i
\(928\) 0 0
\(929\) 33.1231i 1.08673i 0.839495 + 0.543367i \(0.182851\pi\)
−0.839495 + 0.543367i \(0.817149\pi\)
\(930\) 0 0
\(931\) 2.81164i 0.0921479i
\(932\) 0 0
\(933\) 31.3693 + 17.6155i 1.02699 + 0.576707i
\(934\) 0 0
\(935\) 2.64861 0.0866189
\(936\) 0 0
\(937\) −22.4924 −0.734795 −0.367398 0.930064i \(-0.619751\pi\)
−0.367398 + 0.930064i \(0.619751\pi\)
\(938\) 0 0
\(939\) −33.9686 19.0752i −1.10852 0.622494i
\(940\) 0 0
\(941\) 0.876894i 0.0285859i 0.999898 + 0.0142930i \(0.00454975\pi\)
−0.999898 + 0.0142930i \(0.995450\pi\)
\(942\) 0 0
\(943\) 0.417609i 0.0135992i
\(944\) 0 0
\(945\) 0.561553 + 15.6847i 0.0182673 + 0.510222i
\(946\) 0 0
\(947\) −32.4813 −1.05550 −0.527750 0.849400i \(-0.676964\pi\)
−0.527750 + 0.849400i \(0.676964\pi\)
\(948\) 0 0
\(949\) 42.2462 1.37137
\(950\) 0 0
\(951\) 14.1955 25.2791i 0.460322 0.819731i
\(952\) 0 0
\(953\) 22.4924i 0.728601i 0.931281 + 0.364301i \(0.118692\pi\)
−0.931281 + 0.364301i \(0.881308\pi\)
\(954\) 0 0
\(955\) 24.9073i 0.805980i
\(956\) 0 0
\(957\) 3.50758 6.24621i 0.113384 0.201911i
\(958\) 0 0
\(959\) −24.9073 −0.804297
\(960\) 0 0
\(961\) 8.75379 0.282380
\(962\) 0 0
\(963\) 4.71659 + 7.73704i 0.151990 + 0.249323i
\(964\) 0 0
\(965\) 0.246211i 0.00792582i
\(966\) 0 0
\(967\) 26.4404i 0.850265i −0.905131 0.425132i \(-0.860228\pi\)
0.905131 0.425132i \(-0.139772\pi\)
\(968\) 0 0
\(969\) 4.00000 + 2.24621i 0.128499 + 0.0721587i
\(970\) 0 0
\(971\) −52.6261 −1.68885 −0.844427 0.535671i \(-0.820059\pi\)
−0.844427 + 0.535671i \(0.820059\pi\)
\(972\) 0 0
\(973\) 52.9848 1.69862
\(974\) 0 0
\(975\) −7.73704 4.34475i −0.247783 0.139144i
\(976\) 0 0
\(977\) 31.7538i 1.01589i −0.861388 0.507947i \(-0.830405\pi\)
0.861388 0.507947i \(-0.169595\pi\)
\(978\) 0 0
\(979\) 13.5691i 0.433671i
\(980\) 0 0
\(981\) 1.36932 + 2.24621i 0.0437189 + 0.0717160i
\(982\) 0 0
\(983\) 40.0095 1.27610 0.638052 0.769993i \(-0.279740\pi\)
0.638052 + 0.769993i \(0.279740\pi\)
\(984\) 0 0
\(985\) 4.24621 0.135296
\(986\) 0 0
\(987\) −7.73704 + 13.7779i −0.246273 + 0.438556i
\(988\) 0 0
\(989\) 2.87689i 0.0914799i
\(990\) 0 0
\(991\) 33.0161i 1.04879i 0.851475 + 0.524396i \(0.175709\pi\)
−0.851475 + 0.524396i \(0.824291\pi\)
\(992\) 0 0
\(993\) 2.73863 4.87689i 0.0869079 0.154764i
\(994\) 0 0
\(995\) −5.46026 −0.173102
\(996\) 0 0
\(997\) 33.6155 1.06461 0.532307 0.846551i \(-0.321325\pi\)
0.532307 + 0.846551i \(0.321325\pi\)
\(998\) 0 0
\(999\) −0.952473 26.6034i −0.0301349 0.841694i
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 960.2.h.g.191.8 8
3.2 odd 2 inner 960.2.h.g.191.2 8
4.3 odd 2 inner 960.2.h.g.191.1 8
8.3 odd 2 60.2.e.a.11.1 8
8.5 even 2 60.2.e.a.11.7 yes 8
12.11 even 2 inner 960.2.h.g.191.7 8
24.5 odd 2 60.2.e.a.11.2 yes 8
24.11 even 2 60.2.e.a.11.8 yes 8
40.3 even 4 300.2.h.b.299.3 8
40.13 odd 4 300.2.h.b.299.2 8
40.19 odd 2 300.2.e.c.251.8 8
40.27 even 4 300.2.h.a.299.6 8
40.29 even 2 300.2.e.c.251.2 8
40.37 odd 4 300.2.h.a.299.7 8
120.29 odd 2 300.2.e.c.251.7 8
120.53 even 4 300.2.h.a.299.8 8
120.59 even 2 300.2.e.c.251.1 8
120.77 even 4 300.2.h.b.299.1 8
120.83 odd 4 300.2.h.a.299.5 8
120.107 odd 4 300.2.h.b.299.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.2.e.a.11.1 8 8.3 odd 2
60.2.e.a.11.2 yes 8 24.5 odd 2
60.2.e.a.11.7 yes 8 8.5 even 2
60.2.e.a.11.8 yes 8 24.11 even 2
300.2.e.c.251.1 8 120.59 even 2
300.2.e.c.251.2 8 40.29 even 2
300.2.e.c.251.7 8 120.29 odd 2
300.2.e.c.251.8 8 40.19 odd 2
300.2.h.a.299.5 8 120.83 odd 4
300.2.h.a.299.6 8 40.27 even 4
300.2.h.a.299.7 8 40.37 odd 4
300.2.h.a.299.8 8 120.53 even 4
300.2.h.b.299.1 8 120.77 even 4
300.2.h.b.299.2 8 40.13 odd 4
300.2.h.b.299.3 8 40.3 even 4
300.2.h.b.299.4 8 120.107 odd 4
960.2.h.g.191.1 8 4.3 odd 2 inner
960.2.h.g.191.2 8 3.2 odd 2 inner
960.2.h.g.191.7 8 12.11 even 2 inner
960.2.h.g.191.8 8 1.1 even 1 trivial