Properties

Label 507.2.x.a.176.1
Level $507$
Weight $2$
Character 507.176
Analytic conductor $4.048$
Analytic rank $0$
Dimension $48$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(2,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(156))
 
chi = DirichletCharacter(H, H._module([78, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.x (of order \(156\), degree \(48\), 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: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{156}]$

Embedding invariants

Embedding label 176.1
Character \(\chi\) \(=\) 507.176
Dual form 507.2.x.a.314.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.70962 + 0.277840i) q^{3} +(-0.783933 - 1.83996i) q^{4} +(1.45291 + 4.07679i) q^{7} +(2.84561 + 0.950004i) q^{9} +O(q^{10})\) \(q+(1.70962 + 0.277840i) q^{3} +(-0.783933 - 1.83996i) q^{4} +(1.45291 + 4.07679i) q^{7} +(2.84561 + 0.950004i) q^{9} +(-0.829014 - 3.36344i) q^{12} +(2.59808 + 2.50000i) q^{13} +(-2.77090 + 2.88481i) q^{16} +(2.01264 - 7.51128i) q^{19} +(1.35124 + 7.37345i) q^{21} +(2.32362 + 4.42728i) q^{25} +(4.60096 + 2.41477i) q^{27} +(6.36215 - 5.86924i) q^{28} +(-3.22827 - 10.3599i) q^{31} +(-0.482799 - 5.98054i) q^{36} +(1.88677 - 8.37752i) q^{37} +(3.74712 + 4.99590i) q^{39} +(-6.24401 + 9.87411i) q^{43} +(-5.53870 + 4.16206i) q^{48} +(-9.08706 + 7.41935i) q^{49} +(2.56318 - 6.74019i) q^{52} +(5.52779 - 12.2822i) q^{57} +(-13.9039 - 1.12244i) q^{61} +(0.261457 + 12.9812i) q^{63} +(7.48013 + 2.83684i) q^{64} +(5.26774 + 13.0887i) q^{67} +(-13.6273 + 0.824301i) q^{73} +(2.74243 + 8.21457i) q^{75} +(-15.3982 + 2.18516i) q^{76} +(-0.946919 - 7.79858i) q^{79} +(7.19498 + 5.40668i) q^{81} +(12.5076 - 8.26651i) q^{84} +(-6.41720 + 14.2241i) q^{91} +(-2.64073 - 18.6084i) q^{93} +(4.79404 - 8.70370i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 10 q^{7} + 6 q^{9} - 8 q^{16} - 14 q^{19} - 18 q^{21} + 20 q^{28} + 14 q^{31} + 2 q^{37} + 24 q^{39} + 6 q^{43} - 18 q^{49} - 28 q^{52} - 12 q^{57} - 24 q^{63} - 32 q^{67} + 34 q^{73} + 30 q^{75} + 28 q^{76} + 18 q^{81} + 12 q^{84} - 2 q^{91} - 6 q^{93} + 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(e\left(\frac{107}{156}\right)\)

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 0.551377 0.834256i \(-0.314103\pi\)
−0.551377 + 0.834256i \(0.685897\pi\)
\(3\) 1.70962 + 0.277840i 0.987050 + 0.160411i
\(4\) −0.783933 1.83996i −0.391967 0.919979i
\(5\) 0 0 −0.855781 0.517338i \(-0.826923\pi\)
0.855781 + 0.517338i \(0.173077\pi\)
\(6\) 0 0
\(7\) 1.45291 + 4.07679i 0.549150 + 1.54088i 0.816536 + 0.577294i \(0.195891\pi\)
−0.267386 + 0.963589i \(0.586160\pi\)
\(8\) 0 0
\(9\) 2.84561 + 0.950004i 0.948536 + 0.316668i
\(10\) 0 0
\(11\) 0 0 0.446798 0.894635i \(-0.352564\pi\)
−0.446798 + 0.894635i \(0.647436\pi\)
\(12\) −0.829014 3.36344i −0.239316 0.970942i
\(13\) 2.59808 + 2.50000i 0.720577 + 0.693375i
\(14\) 0 0
\(15\) 0 0
\(16\) −2.77090 + 2.88481i −0.692724 + 0.721202i
\(17\) 0 0 −0.903450 0.428693i \(-0.858974\pi\)
0.903450 + 0.428693i \(0.141026\pi\)
\(18\) 0 0
\(19\) 2.01264 7.51128i 0.461731 1.72321i −0.205773 0.978600i \(-0.565971\pi\)
0.667505 0.744606i \(-0.267362\pi\)
\(20\) 0 0
\(21\) 1.35124 + 7.37345i 0.294864 + 1.60902i
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 2.32362 + 4.42728i 0.464723 + 0.885456i
\(26\) 0 0
\(27\) 4.60096 + 2.41477i 0.885456 + 0.464723i
\(28\) 6.36215 5.86924i 1.20233 1.10918i
\(29\) 0 0 −0.200026 0.979791i \(-0.564103\pi\)
0.200026 + 0.979791i \(0.435897\pi\)
\(30\) 0 0
\(31\) −3.22827 10.3599i −0.579815 1.86069i −0.504264 0.863550i \(-0.668236\pi\)
−0.0755512 0.997142i \(-0.524072\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.482799 5.98054i −0.0804666 0.996757i
\(37\) 1.88677 8.37752i 0.310184 1.37726i −0.535935 0.844260i \(-0.680041\pi\)
0.846118 0.532995i \(-0.178934\pi\)
\(38\) 0 0
\(39\) 3.74712 + 4.99590i 0.600020 + 0.799985i
\(40\) 0 0
\(41\) 0 0 −0.584522 0.811378i \(-0.698718\pi\)
0.584522 + 0.811378i \(0.301282\pi\)
\(42\) 0 0
\(43\) −6.24401 + 9.87411i −0.952202 + 1.50579i −0.0920473 + 0.995755i \(0.529341\pi\)
−0.860155 + 0.510033i \(0.829633\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.787183 0.616719i \(-0.788462\pi\)
0.787183 + 0.616719i \(0.211538\pi\)
\(48\) −5.53870 + 4.16206i −0.799443 + 0.600742i
\(49\) −9.08706 + 7.41935i −1.29815 + 1.05991i
\(50\) 0 0
\(51\) 0 0
\(52\) 2.56318 6.74019i 0.355449 0.934696i
\(53\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.52779 12.2822i 0.732174 1.62682i
\(58\) 0 0
\(59\) 0 0 −0.999797 0.0201371i \(-0.993590\pi\)
0.999797 + 0.0201371i \(0.00641026\pi\)
\(60\) 0 0
\(61\) −13.9039 1.12244i −1.78021 0.143713i −0.853889 0.520455i \(-0.825762\pi\)
−0.926318 + 0.376742i \(0.877044\pi\)
\(62\) 0 0
\(63\) 0.261457 + 12.9812i 0.0329405 + 1.63548i
\(64\) 7.48013 + 2.83684i 0.935016 + 0.354605i
\(65\) 0 0
\(66\) 0 0
\(67\) 5.26774 + 13.0887i 0.643557 + 1.59904i 0.792279 + 0.610158i \(0.208894\pi\)
−0.148722 + 0.988879i \(0.547516\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.994935 0.100522i \(-0.0320513\pi\)
−0.994935 + 0.100522i \(0.967949\pi\)
\(72\) 0 0
\(73\) −13.6273 + 0.824301i −1.59496 + 0.0964771i −0.833791 0.552080i \(-0.813834\pi\)
−0.761166 + 0.648557i \(0.775373\pi\)
\(74\) 0 0
\(75\) 2.74243 + 8.21457i 0.316668 + 0.948536i
\(76\) −15.3982 + 2.18516i −1.76630 + 0.250655i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.946919 7.79858i −0.106537 0.877409i −0.943779 0.330577i \(-0.892757\pi\)
0.837242 0.546832i \(-0.184166\pi\)
\(80\) 0 0
\(81\) 7.19498 + 5.40668i 0.799443 + 0.600742i
\(82\) 0 0
\(83\) 0 0 0.911900 0.410413i \(-0.134615\pi\)
−0.911900 + 0.410413i \(0.865385\pi\)
\(84\) 12.5076 8.26651i 1.36469 0.901951i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(90\) 0 0
\(91\) −6.41720 + 14.2241i −0.672706 + 1.49109i
\(92\) 0 0
\(93\) −2.64073 18.6084i −0.273831 1.92960i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.79404 8.70370i 0.486761 0.883727i −0.512920 0.858437i \(-0.671436\pi\)
0.999680 0.0252904i \(-0.00805106\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 6.32445 7.74605i 0.632445 0.774605i
\(101\) 0 0 −0.999189 0.0402659i \(-0.987179\pi\)
0.999189 + 0.0402659i \(0.0128205\pi\)
\(102\) 0 0
\(103\) 11.2430 4.26390i 1.10780 0.420135i 0.268217 0.963359i \(-0.413566\pi\)
0.839588 + 0.543224i \(0.182796\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.278217 0.960518i \(-0.410256\pi\)
−0.278217 + 0.960518i \(0.589744\pi\)
\(108\) 0.836233 10.3586i 0.0804666 0.996757i
\(109\) −16.4601 5.12916i −1.57659 0.491285i −0.620461 0.784238i \(-0.713054\pi\)
−0.956127 + 0.292953i \(0.905362\pi\)
\(110\) 0 0
\(111\) 5.55328 13.7982i 0.527094 1.30966i
\(112\) −15.7867 7.10500i −1.49170 0.671359i
\(113\) 0 0 0.987050 0.160411i \(-0.0512821\pi\)
−0.987050 + 0.160411i \(0.948718\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 5.01810 + 9.58221i 0.463924 + 0.885875i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −6.60816 8.79387i −0.600742 0.799443i
\(122\) 0 0
\(123\) 0 0
\(124\) −16.5310 + 14.0614i −1.48453 + 1.26275i
\(125\) 0 0
\(126\) 0 0
\(127\) −1.73157 + 4.06415i −0.153652 + 0.360635i −0.978961 0.204047i \(-0.934591\pi\)
0.825309 + 0.564681i \(0.191001\pi\)
\(128\) 0 0
\(129\) −13.4183 + 15.1461i −1.18142 + 1.33354i
\(130\) 0 0
\(131\) 0 0 0.748511 0.663123i \(-0.230769\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(132\) 0 0
\(133\) 33.5461 2.70812i 2.90882 0.234824i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.975564 0.219715i \(-0.0705128\pi\)
−0.975564 + 0.219715i \(0.929487\pi\)
\(138\) 0 0
\(139\) 1.18683 + 4.09740i 0.100665 + 0.347537i 0.995002 0.0998583i \(-0.0318389\pi\)
−0.894336 + 0.447395i \(0.852352\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −10.6255 + 5.57668i −0.885456 + 0.464723i
\(145\) 0 0
\(146\) 0 0
\(147\) −17.5968 + 10.1595i −1.45136 + 0.837944i
\(148\) −16.8934 + 3.09583i −1.38863 + 0.254476i
\(149\) 0 0 −0.990080 0.140502i \(-0.955128\pi\)
0.990080 + 0.140502i \(0.0448718\pi\)
\(150\) 0 0
\(151\) −13.5436 17.2871i −1.10216 1.40681i −0.904467 0.426543i \(-0.859731\pi\)
−0.197696 0.980263i \(-0.563346\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 6.25476 10.8110i 0.500782 0.865574i
\(157\) −20.4156 + 5.03199i −1.62934 + 0.401597i −0.944871 0.327444i \(-0.893813\pi\)
−0.684471 + 0.729040i \(0.739967\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.82039 + 1.67935i 0.142584 + 0.131537i 0.746156 0.665771i \(-0.231897\pi\)
−0.603572 + 0.797309i \(0.706256\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.834256 0.551377i \(-0.814103\pi\)
0.834256 + 0.551377i \(0.185897\pi\)
\(168\) 0 0
\(169\) 0.500000 + 12.9904i 0.0384615 + 0.999260i
\(170\) 0 0
\(171\) 12.8629 19.4621i 0.983653 1.48831i
\(172\) 23.0628 + 3.74808i 1.75852 + 0.285788i
\(173\) 0 0 −0.391967 0.919979i \(-0.628205\pi\)
0.391967 + 0.919979i \(0.371795\pi\)
\(174\) 0 0
\(175\) −14.6731 + 15.9054i −1.10918 + 1.20233i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.903450 0.428693i \(-0.141026\pi\)
−0.903450 + 0.428693i \(0.858974\pi\)
\(180\) 0 0
\(181\) −4.35971 17.6881i −0.324055 1.31474i −0.876584 0.481249i \(-0.840183\pi\)
0.552529 0.833494i \(-0.313663\pi\)
\(182\) 0 0
\(183\) −23.4585 5.78200i −1.73410 0.427417i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −3.15972 + 22.2656i −0.229836 + 1.61959i
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 12.0000 + 6.92820i 0.866025 + 0.500000i
\(193\) 18.3486 + 6.53918i 1.32076 + 0.470701i 0.899770 0.436365i \(-0.143734\pi\)
0.420989 + 0.907066i \(0.361683\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 20.7750 + 10.9035i 1.48393 + 0.778824i
\(197\) 0 0 0.735006 0.678061i \(-0.237179\pi\)
−0.735006 + 0.678061i \(0.762821\pi\)
\(198\) 0 0
\(199\) 26.5641 7.69438i 1.88308 0.545440i 0.886505 0.462719i \(-0.153126\pi\)
0.996573 0.0827210i \(-0.0263610\pi\)
\(200\) 0 0
\(201\) 5.36928 + 23.8403i 0.378720 + 1.68156i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −14.4110 + 0.567712i −0.999225 + 0.0393638i
\(209\) 0 0
\(210\) 0 0
\(211\) −26.7219 11.3851i −1.83961 0.783786i −0.927402 0.374067i \(-0.877963\pi\)
−0.912212 0.409719i \(-0.865627\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 37.5448 28.2130i 2.54870 1.91523i
\(218\) 0 0
\(219\) −23.5266 2.37698i −1.58978 0.160621i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 23.7790 13.0975i 1.59236 0.877077i 0.595053 0.803686i \(-0.297131\pi\)
0.997303 0.0733905i \(-0.0233819\pi\)
\(224\) 0 0
\(225\) 2.40617 + 14.8058i 0.160411 + 0.987050i
\(226\) 0 0
\(227\) 0 0 −0.927686 0.373361i \(-0.878205\pi\)
0.927686 + 0.373361i \(0.121795\pi\)
\(228\) −26.9322 0.542446i −1.78363 0.0359244i
\(229\) 1.90570 6.11561i 0.125932 0.404131i −0.870126 0.492829i \(-0.835963\pi\)
0.996059 + 0.0886981i \(0.0282706\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.935016 0.354605i \(-0.884615\pi\)
0.935016 + 0.354605i \(0.115385\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.547888 13.5957i 0.0355892 0.883136i
\(238\) 0 0
\(239\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(240\) 0 0
\(241\) 11.5455 + 6.35930i 0.743710 + 0.409638i 0.808680 0.588249i \(-0.200182\pi\)
−0.0649701 + 0.997887i \(0.520695\pi\)
\(242\) 0 0
\(243\) 10.7985 + 11.2424i 0.692724 + 0.721202i
\(244\) 8.83446 + 26.4624i 0.565569 + 1.69408i
\(245\) 0 0
\(246\) 0 0
\(247\) 24.0072 14.4833i 1.52754 0.921548i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.999189 0.0402659i \(-0.0128205\pi\)
−0.999189 + 0.0402659i \(0.987179\pi\)
\(252\) 23.6800 10.6575i 1.49170 0.671359i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.644255 15.9870i −0.0402659 0.999189i
\(257\) 0 0 0.600742 0.799443i \(-0.294872\pi\)
−0.600742 + 0.799443i \(0.705128\pi\)
\(258\) 0 0
\(259\) 36.8947 4.47983i 2.29253 0.278363i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.948536 0.316668i \(-0.102564\pi\)
−0.948536 + 0.316668i \(0.897436\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 19.9531 19.9531i 1.21883 1.21883i
\(269\) 0 0 0.632445 0.774605i \(-0.282051\pi\)
−0.632445 + 0.774605i \(0.717949\pi\)
\(270\) 0 0
\(271\) 5.20791 2.09600i 0.316358 0.127323i −0.209715 0.977763i \(-0.567254\pi\)
0.526073 + 0.850439i \(0.323664\pi\)
\(272\) 0 0
\(273\) −14.9230 + 22.5349i −0.903182 + 1.36387i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.54924 + 31.5780i −0.153169 + 1.89734i 0.229277 + 0.973361i \(0.426364\pi\)
−0.382446 + 0.923978i \(0.624918\pi\)
\(278\) 0 0
\(279\) 0.655535 32.5471i 0.0392459 1.94854i
\(280\) 0 0
\(281\) 0 0 −0.911900 0.410413i \(-0.865385\pi\)
0.911900 + 0.410413i \(0.134615\pi\)
\(282\) 0 0
\(283\) 15.8534 + 25.0701i 0.942387 + 1.49027i 0.870096 + 0.492882i \(0.164056\pi\)
0.0722903 + 0.997384i \(0.476969\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 10.7516 + 13.1683i 0.632445 + 0.774605i
\(290\) 0 0
\(291\) 10.6142 13.5481i 0.622217 0.794201i
\(292\) 12.1996 + 24.4275i 0.713927 + 1.42951i
\(293\) 0 0 0.761712 0.647915i \(-0.224359\pi\)
−0.761712 + 0.647915i \(0.775641\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 12.9646 11.4856i 0.748511 0.663123i
\(301\) −49.3267 11.1093i −2.84315 0.640329i
\(302\) 0 0
\(303\) 0 0
\(304\) 16.0918 + 26.6191i 0.922927 + 1.52671i
\(305\) 0 0
\(306\) 0 0
\(307\) −4.39673 + 1.37007i −0.250934 + 0.0781943i −0.420390 0.907344i \(-0.638107\pi\)
0.169456 + 0.985538i \(0.445799\pi\)
\(308\) 0 0
\(309\) 20.4059 4.16590i 1.16085 0.236990i
\(310\) 0 0
\(311\) 0 0 0.464723 0.885456i \(-0.346154\pi\)
−0.464723 + 0.885456i \(0.653846\pi\)
\(312\) 0 0
\(313\) −21.0959 + 11.0720i −1.19241 + 0.625825i −0.939794 0.341741i \(-0.888983\pi\)
−0.252616 + 0.967567i \(0.581291\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −13.6067 + 7.85586i −0.765439 + 0.441926i
\(317\) 0 0 0.983620 0.180255i \(-0.0576923\pi\)
−0.983620 + 0.180255i \(0.942308\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 4.30768 17.4770i 0.239316 0.970942i
\(325\) −5.03127 + 17.3115i −0.279085 + 0.960267i
\(326\) 0 0
\(327\) −26.7154 13.3422i −1.47736 0.737825i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 22.5395 8.03279i 1.23889 0.441522i 0.366535 0.930404i \(-0.380544\pi\)
0.872350 + 0.488882i \(0.162595\pi\)
\(332\) 0 0
\(333\) 13.3277 22.0467i 0.730353 1.20815i
\(334\) 0 0
\(335\) 0 0
\(336\) −25.0151 16.5330i −1.36469 0.901951i
\(337\) 35.0047i 1.90683i −0.301663 0.953414i \(-0.597542\pi\)
0.301663 0.953414i \(-0.402458\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −17.5234 10.5933i −0.946176 0.571983i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.948536 0.316668i \(-0.897436\pi\)
0.948536 + 0.316668i \(0.102564\pi\)
\(348\) 0 0
\(349\) −16.5691 + 33.1767i −0.886925 + 1.77591i −0.334343 + 0.942451i \(0.608514\pi\)
−0.552582 + 0.833459i \(0.686357\pi\)
\(350\) 0 0
\(351\) 5.91673 + 17.7762i 0.315811 + 0.948822i
\(352\) 0 0
\(353\) 0 0 −0.761712 0.647915i \(-0.775641\pi\)
0.761712 + 0.647915i \(0.224359\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.180255 0.983620i \(-0.557692\pi\)
0.180255 + 0.983620i \(0.442308\pi\)
\(360\) 0 0
\(361\) −35.9141 20.7350i −1.89022 1.09132i
\(362\) 0 0
\(363\) −8.85417 16.8702i −0.464723 0.885456i
\(364\) 31.2024 + 0.656637i 1.63545 + 0.0344171i
\(365\) 0 0
\(366\) 0 0
\(367\) 6.41006 + 31.3986i 0.334603 + 1.63899i 0.704345 + 0.709857i \(0.251241\pi\)
−0.369743 + 0.929134i \(0.620554\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −32.1686 + 19.4466i −1.66786 + 1.00826i
\(373\) −1.70440 + 8.34872i −0.0882507 + 0.432281i 0.911528 + 0.411238i \(0.134904\pi\)
−0.999779 + 0.0210423i \(0.993302\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.37985 + 11.6321i 0.430444 + 0.597501i 0.969597 0.244706i \(-0.0786916\pi\)
−0.539153 + 0.842208i \(0.681256\pi\)
\(380\) 0 0
\(381\) −4.08951 + 6.46705i −0.209512 + 0.331317i
\(382\) 0 0
\(383\) 0 0 −0.647915 0.761712i \(-0.724359\pi\)
0.647915 + 0.761712i \(0.275641\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −27.1484 + 22.1660i −1.38003 + 1.12676i
\(388\) −19.7727 1.99771i −1.00380 0.101418i
\(389\) 0 0 −0.822984 0.568065i \(-0.807692\pi\)
0.822984 + 0.568065i \(0.192308\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −31.9694 0.643900i −1.60450 0.0323164i −0.790232 0.612808i \(-0.790040\pi\)
−0.814266 + 0.580492i \(0.802860\pi\)
\(398\) 0 0
\(399\) 58.1036 + 4.69061i 2.90882 + 0.234824i
\(400\) −19.2104 5.56435i −0.960518 0.278217i
\(401\) 0 0 −0.0201371 0.999797i \(-0.506410\pi\)
0.0201371 + 0.999797i \(0.493590\pi\)
\(402\) 0 0
\(403\) 17.5124 34.9865i 0.872356 1.74280i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 27.6643 2.79502i 1.36791 0.138205i 0.610951 0.791668i \(-0.290787\pi\)
0.756958 + 0.653463i \(0.226685\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −16.6592 17.3440i −0.820738 0.854479i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.890600 + 7.33475i 0.0436129 + 0.359184i
\(418\) 0 0
\(419\) 0 0 −0.799443 0.600742i \(-0.794872\pi\)
0.799443 + 0.600742i \(0.205128\pi\)
\(420\) 0 0
\(421\) −2.33202 + 1.04956i −0.113656 + 0.0511522i −0.466443 0.884551i \(-0.654465\pi\)
0.352787 + 0.935704i \(0.385234\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −15.6252 58.3140i −0.756156 2.82201i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.140502 0.990080i \(-0.544872\pi\)
0.140502 + 0.990080i \(0.455128\pi\)
\(432\) −19.7150 + 6.58182i −0.948536 + 0.316668i
\(433\) −8.12548 + 7.80463i −0.390485 + 0.375066i −0.862100 0.506739i \(-0.830851\pi\)
0.471614 + 0.881805i \(0.343671\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.46614 + 34.3067i 0.165998 + 1.64300i
\(437\) 0 0
\(438\) 0 0
\(439\) 38.4080 + 1.54779i 1.83311 + 0.0738719i 0.932590 0.360938i \(-0.117543\pi\)
0.900522 + 0.434810i \(0.143184\pi\)
\(440\) 0 0
\(441\) −32.9066 + 12.4798i −1.56698 + 0.594278i
\(442\) 0 0
\(443\) 0 0 0.354605 0.935016i \(-0.384615\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(444\) −29.7414 + 0.599026i −1.41147 + 0.0284285i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.697219 + 34.6166i −0.0329405 + 1.63548i
\(449\) 0 0 0.373361 0.927686i \(-0.378205\pi\)
−0.373361 + 0.927686i \(0.621795\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −18.3514 33.3174i −0.862222 1.56539i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.08916 + 10.7802i −0.0509488 + 0.504275i 0.936982 + 0.349378i \(0.113607\pi\)
−0.987931 + 0.154897i \(0.950495\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.446798 0.894635i \(-0.647436\pi\)
0.446798 + 0.894635i \(0.352564\pi\)
\(462\) 0 0
\(463\) −0.0523714 + 0.865802i −0.00243390 + 0.0402372i −0.999190 0.0402476i \(-0.987185\pi\)
0.996756 + 0.0804849i \(0.0256469\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.663123 0.748511i \(-0.269231\pi\)
−0.663123 + 0.748511i \(0.730769\pi\)
\(468\) 13.6970 16.7449i 0.633145 0.774034i
\(469\) −45.7063 + 40.4923i −2.11052 + 1.86976i
\(470\) 0 0
\(471\) −36.3010 + 2.93052i −1.67266 + 0.135031i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 37.9311 8.54280i 1.74040 0.391971i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.678061 0.735006i \(-0.737179\pi\)
0.678061 + 0.735006i \(0.262821\pi\)
\(480\) 0 0
\(481\) 25.8458 17.0485i 1.17847 0.777344i
\(482\) 0 0
\(483\) 0 0
\(484\) −11.0000 + 19.0526i −0.500000 + 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) −24.1666 3.42949i −1.09509 0.155405i −0.430486 0.902597i \(-0.641658\pi\)
−0.664608 + 0.747192i \(0.731401\pi\)
\(488\) 0 0
\(489\) 2.64558 + 3.37684i 0.119637 + 0.152706i
\(490\) 0 0
\(491\) 0 0 −0.721202 0.692724i \(-0.756410\pi\)
0.721202 + 0.692724i \(0.243590\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 38.8315 + 19.3932i 1.74359 + 0.870782i
\(497\) 0 0
\(498\) 0 0
\(499\) 7.84362 42.8012i 0.351129 1.91605i −0.0473700 0.998877i \(-0.515084\pi\)
0.398499 0.917169i \(-0.369531\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.919979 0.391967i \(-0.128205\pi\)
−0.919979 + 0.391967i \(0.871795\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −2.75444 + 22.3476i −0.122329 + 0.992490i
\(508\) 8.83530 0.392003
\(509\) 0 0 0.551377 0.834256i \(-0.314103\pi\)
−0.551377 + 0.834256i \(0.685897\pi\)
\(510\) 0 0
\(511\) −23.1598 54.3581i −1.02453 2.40466i
\(512\) 0 0
\(513\) 27.3981 29.6991i 1.20966 1.31125i
\(514\) 0 0
\(515\) 0 0
\(516\) 38.3874 + 12.8156i 1.68991 + 0.564174i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.970942 0.239316i \(-0.923077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(522\) 0 0
\(523\) 24.8347 25.8556i 1.08594 1.13059i 0.0951145 0.995466i \(-0.469678\pi\)
0.990829 0.135121i \(-0.0431422\pi\)
\(524\) 0 0
\(525\) −29.5046 + 23.1154i −1.28769 + 1.00884i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) −31.2808 59.6005i −1.35619 2.58401i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 39.7067 24.0035i 1.70712 1.03199i 0.816329 0.577587i \(-0.196006\pi\)
0.890795 0.454405i \(-0.150148\pi\)
\(542\) 0 0
\(543\) −2.53900 31.4512i −0.108959 1.34970i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −20.4302 18.0996i −0.873533 0.773883i 0.101757 0.994809i \(-0.467553\pi\)
−0.975290 + 0.220926i \(0.929092\pi\)
\(548\) 0 0
\(549\) −38.4986 16.4027i −1.64308 0.700052i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 30.4174 15.1911i 1.29348 0.645990i
\(554\) 0 0
\(555\) 0 0
\(556\) 6.60865 5.39580i 0.280269 0.228833i
\(557\) 0 0 −0.994935 0.100522i \(-0.967949\pi\)
0.994935 + 0.100522i \(0.0320513\pi\)
\(558\) 0 0
\(559\) −40.9077 + 10.0437i −1.73021 + 0.424802i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.160411 0.987050i \(-0.551282\pi\)
0.160411 + 0.987050i \(0.448718\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −11.5882 + 37.1879i −0.486660 + 1.56175i
\(568\) 0 0
\(569\) 0 0 −0.960518 0.278217i \(-0.910256\pi\)
0.960518 + 0.278217i \(0.0897436\pi\)
\(570\) 0 0
\(571\) 34.0656 + 12.9194i 1.42560 + 0.540660i 0.942293 0.334790i \(-0.108665\pi\)
0.483310 + 0.875449i \(0.339434\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 18.5905 + 15.1787i 0.774605 + 0.632445i
\(577\) −19.0312 19.0312i −0.792278 0.792278i 0.189586 0.981864i \(-0.439285\pi\)
−0.981864 + 0.189586i \(0.939285\pi\)
\(578\) 0 0
\(579\) 29.5522 + 16.2775i 1.22815 + 0.676470i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(588\) 32.4879 + 24.4130i 1.33978 + 1.00678i
\(589\) −84.3134 + 3.39771i −3.47407 + 0.140000i
\(590\) 0 0
\(591\) 0 0
\(592\) 18.9395 + 28.6562i 0.778408 + 1.17776i
\(593\) 0 0 −0.410413 0.911900i \(-0.634615\pi\)
0.410413 + 0.911900i \(0.365385\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 47.5523 5.77390i 1.94619 0.236310i
\(598\) 0 0
\(599\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(600\) 0 0
\(601\) −2.10236 + 0.701872i −0.0857571 + 0.0286299i −0.359223 0.933252i \(-0.616958\pi\)
0.273466 + 0.961882i \(0.411830\pi\)
\(602\) 0 0
\(603\) 2.55563 + 42.2497i 0.104073 + 1.72054i
\(604\) −21.1903 + 38.4716i −0.862222 + 1.56539i
\(605\) 0 0
\(606\) 0 0
\(607\) −16.2541 + 19.9077i −0.659734 + 0.808027i −0.990765 0.135593i \(-0.956706\pi\)
0.331031 + 0.943620i \(0.392604\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −42.3174 + 0.852320i −1.70918 + 0.0344249i −0.865043 0.501699i \(-0.832709\pi\)
−0.844140 + 0.536123i \(0.819888\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.0201371 0.999797i \(-0.493590\pi\)
−0.0201371 + 0.999797i \(0.506410\pi\)
\(618\) 0 0
\(619\) 22.9354 + 10.3224i 0.921852 + 0.414892i 0.815051 0.579389i \(-0.196709\pi\)
0.106800 + 0.994280i \(0.465939\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −24.7951 3.03339i −0.992600 0.121433i
\(625\) −14.2016 + 20.5746i −0.568065 + 0.822984i
\(626\) 0 0
\(627\) 0 0
\(628\) 25.2631 + 33.6191i 1.00811 + 1.34155i
\(629\) 0 0
\(630\) 0 0
\(631\) −37.4964 + 31.8946i −1.49271 + 1.26970i −0.616902 + 0.787040i \(0.711612\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) 0 0
\(633\) −42.5211 26.8887i −1.69006 1.06873i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −42.1573 3.44160i −1.67033 0.136361i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.979791 0.200026i \(-0.935897\pi\)
0.979791 + 0.200026i \(0.0641026\pi\)
\(642\) 0 0
\(643\) −8.33162 6.00216i −0.328567 0.236702i 0.408131 0.912923i \(-0.366180\pi\)
−0.736699 + 0.676221i \(0.763616\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.979791 0.200026i \(-0.0641026\pi\)
−0.979791 + 0.200026i \(0.935897\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 72.0260 37.8022i 2.82292 1.48158i
\(652\) 1.66288 4.66594i 0.0651234 0.182732i
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −39.5611 10.6004i −1.54343 0.413560i
\(658\) 0 0
\(659\) 0 0 0.428693 0.903450i \(-0.358974\pi\)
−0.428693 + 0.903450i \(0.641026\pi\)
\(660\) 0 0
\(661\) 8.16469 9.59870i 0.317570 0.373346i −0.579699 0.814830i \(-0.696830\pi\)
0.897269 + 0.441484i \(0.145548\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 44.2920 15.7851i 1.71243 0.610287i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −4.64819 + 28.6015i −0.179175 + 1.10251i 0.729185 + 0.684316i \(0.239899\pi\)
−0.908360 + 0.418189i \(0.862665\pi\)
\(674\) 0 0
\(675\) 25.9808i 1.00000i
\(676\) 23.5098 11.1036i 0.904223 0.427060i
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 42.4485 + 6.89856i 1.62903 + 0.264742i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.335705 0.941967i \(-0.608974\pi\)
0.335705 + 0.941967i \(0.391026\pi\)
\(684\) −45.8932 8.41025i −1.75477 0.321574i
\(685\) 0 0
\(686\) 0 0
\(687\) 4.95719 9.92589i 0.189129 0.378696i
\(688\) −11.1834 45.3729i −0.426364 1.72983i
\(689\) 0 0
\(690\) 0 0
\(691\) −9.12451 7.76134i −0.347113 0.295255i 0.457316 0.889304i \(-0.348811\pi\)
−0.804429 + 0.594049i \(0.797529\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 40.7679 + 14.5291i 1.54088 + 0.549150i
\(701\) 0 0 −0.464723 0.885456i \(-0.653846\pi\)
0.464723 + 0.885456i \(0.346154\pi\)
\(702\) 0 0
\(703\) −59.1285 31.0330i −2.23007 1.17043i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 30.4805 42.3102i 1.14472 1.58899i 0.407708 0.913112i \(-0.366328\pi\)
0.737012 0.675880i \(-0.236236\pi\)
\(710\) 0 0
\(711\) 4.71412 23.0913i 0.176793 0.865991i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.534466 0.845190i \(-0.320513\pi\)
−0.534466 + 0.845190i \(0.679487\pi\)
\(720\) 0 0
\(721\) 33.7182 + 39.6403i 1.25573 + 1.47628i
\(722\) 0 0
\(723\) 17.9715 + 14.0798i 0.668368 + 0.523633i
\(724\) −29.1276 + 21.8880i −1.08252 + 0.813459i
\(725\) 0 0
\(726\) 0 0
\(727\) 7.65731 + 5.28546i 0.283994 + 0.196027i 0.701542 0.712628i \(-0.252495\pi\)
−0.417548 + 0.908655i \(0.637111\pi\)
\(728\) 0 0
\(729\) 15.3377 + 22.2206i 0.568065 + 0.822984i
\(730\) 0 0
\(731\) 0 0
\(732\) 7.75125 + 47.6953i 0.286494 + 1.76287i
\(733\) −6.79803 + 15.1046i −0.251091 + 0.557901i −0.993732 0.111789i \(-0.964342\pi\)
0.742641 + 0.669690i \(0.233573\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.02237 + 50.7605i 0.0376086 + 1.86725i 0.376485 + 0.926423i \(0.377133\pi\)
−0.338876 + 0.940831i \(0.610047\pi\)
\(740\) 0 0
\(741\) 45.0672 18.0907i 1.65559 0.664580i
\(742\) 0 0
\(743\) 0 0 −0.373361 0.927686i \(-0.621795\pi\)
0.373361 + 0.927686i \(0.378205\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.38472 4.14775i −0.0505293 0.151354i 0.920248 0.391335i \(-0.127987\pi\)
−0.970777 + 0.239982i \(0.922859\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 43.4449 11.6410i 1.58008 0.423380i
\(757\) 39.1305 + 29.4047i 1.42222 + 1.06873i 0.985601 + 0.169086i \(0.0540817\pi\)
0.436622 + 0.899645i \(0.356175\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.551377 0.834256i \(-0.685897\pi\)
0.551377 + 0.834256i \(0.314103\pi\)
\(762\) 0 0
\(763\) −3.00452 74.5565i −0.108771 2.69913i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 3.34041 27.5108i 0.120537 0.992709i
\(769\) 2.58507 + 18.2162i 0.0932199 + 0.656894i 0.980624 + 0.195901i \(0.0627631\pi\)
−0.887404 + 0.460993i \(0.847493\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.35222 38.8869i −0.0846584 1.39957i
\(773\) 0 0 0.482459 0.875918i \(-0.339744\pi\)
−0.482459 + 0.875918i \(0.660256\pi\)
\(774\) 0 0
\(775\) 38.3649 38.3649i 1.37811 1.37811i
\(776\) 0 0
\(777\) 64.3207 + 2.59204i 2.30749 + 0.0929888i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.77588 46.7727i 0.134853 1.67045i
\(785\) 0 0
\(786\) 0 0
\(787\) 11.2203 27.8788i 0.399959 0.993772i −0.583414 0.812175i \(-0.698283\pi\)
0.983373 0.181598i \(-0.0581268\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −33.3172 37.6758i −1.18313 1.33791i
\(794\) 0 0
\(795\) 0 0
\(796\) −34.9818 42.8449i −1.23990 1.51860i
\(797\) 0 0 −0.600742 0.799443i \(-0.705128\pi\)
0.600742 + 0.799443i \(0.294872\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 39.6560 28.5684i 1.39856 1.00753i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.996757 0.0804666i \(-0.0256410\pi\)
−0.996757 + 0.0804666i \(0.974359\pi\)
\(810\) 0 0
\(811\) 28.3205 + 46.8479i 0.994468 + 1.64505i 0.733482 + 0.679708i \(0.237894\pi\)
0.260986 + 0.965343i \(0.415952\pi\)
\(812\) 0 0
\(813\) 9.48591 2.13640i 0.332685 0.0749269i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 61.6002 + 66.7735i 2.15512 + 2.33611i
\(818\) 0 0
\(819\) −31.7738 + 34.3799i −1.11027 + 1.20133i
\(820\) 0 0
\(821\) 0 0 0.335705 0.941967i \(-0.391026\pi\)
−0.335705 + 0.941967i \(0.608974\pi\)
\(822\) 0 0
\(823\) 45.5813 26.3164i 1.58887 0.917332i 0.595371 0.803451i \(-0.297005\pi\)
0.993494 0.113881i \(-0.0363284\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.616719 0.787183i \(-0.711538\pi\)
0.616719 + 0.787183i \(0.288462\pi\)
\(828\) 0 0
\(829\) 33.2268 + 31.9147i 1.15401 + 1.10844i 0.992584 + 0.121560i \(0.0387897\pi\)
0.161429 + 0.986884i \(0.448390\pi\)
\(830\) 0 0
\(831\) −13.1319 + 53.2782i −0.455540 + 1.84820i
\(832\) 12.3418 + 26.0706i 0.427877 + 0.903837i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 10.1636 55.4610i 0.351306 1.91701i
\(838\) 0 0
\(839\) 0 0 −0.735006 0.678061i \(-0.762821\pi\)
0.735006 + 0.678061i \(0.237179\pi\)
\(840\) 0 0
\(841\) −26.6794 + 11.3670i −0.919979 + 0.391967i
\(842\) 0 0
\(843\) 0 0
\(844\) 58.0924i 1.99962i
\(845\) 0 0
\(846\) 0 0
\(847\) 26.2497 39.7169i 0.901951 1.36469i
\(848\) 0 0
\(849\) 20.1378 + 47.2652i 0.691128 + 1.62214i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −57.1756 10.4778i −1.95766 0.358754i −0.996599 0.0824022i \(-0.973741\pi\)
−0.961057 0.276352i \(-0.910875\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.239316 0.970942i \(-0.576923\pi\)
0.239316 + 0.970942i \(0.423077\pi\)
\(858\) 0 0
\(859\) −7.87743 1.94161i −0.268774 0.0662469i 0.102625 0.994720i \(-0.467276\pi\)
−0.371399 + 0.928473i \(0.621122\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.787183 0.616719i \(-0.211538\pi\)
−0.787183 + 0.616719i \(0.788462\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 14.7224 + 25.5000i 0.500000 + 0.866025i
\(868\) −81.3434 46.9637i −2.76098 1.59405i
\(869\) 0 0
\(870\) 0 0
\(871\) −19.0357 + 47.1748i −0.645001 + 1.59846i
\(872\) 0 0
\(873\) 21.9105 20.2130i 0.741558 0.684106i
\(874\) 0 0
\(875\) 0 0
\(876\) 14.0697 + 45.1513i 0.475372 + 1.52552i
\(877\) −11.8276 52.5163i −0.399391 1.77335i −0.606554 0.795042i \(-0.707449\pi\)
0.207163 0.978306i \(-0.433577\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.0804666 0.996757i \(-0.525641\pi\)
0.0804666 + 0.996757i \(0.474359\pi\)
\(882\) 0 0
\(883\) 33.0065 + 37.2567i 1.11076 + 1.25379i 0.964369 + 0.264560i \(0.0852267\pi\)
0.146389 + 0.989227i \(0.453235\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.919979 0.391967i \(-0.871795\pi\)
0.919979 + 0.391967i \(0.128205\pi\)
\(888\) 0 0
\(889\) −19.0845 1.15440i −0.640074 0.0387173i
\(890\) 0 0
\(891\) 0 0
\(892\) −42.7401 33.4847i −1.43104 1.12115i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 25.3557 16.0340i 0.845190 0.534466i
\(901\) 0 0
\(902\) 0 0
\(903\) −81.2434 32.6977i −2.70361 1.08811i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −13.9181 4.03143i −0.462143 0.133861i 0.0389751 0.999240i \(-0.487591\pi\)
−0.501118 + 0.865379i \(0.667078\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(912\) 20.1150 + 49.9795i 0.666074 + 1.65499i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −12.7464 + 1.28782i −0.421153 + 0.0425507i
\(917\) 0 0
\(918\) 0 0
\(919\) 2.38310 + 2.48107i 0.0786111 + 0.0818429i 0.759346 0.650687i \(-0.225519\pi\)
−0.680735 + 0.732530i \(0.738339\pi\)
\(920\) 0 0
\(921\) −7.89740 + 1.12072i −0.260228 + 0.0369290i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 41.4737 11.1129i 1.36365 0.365388i
\(926\) 0 0
\(927\) 36.0439 1.45252i 1.18384 0.0477070i
\(928\) 0 0
\(929\) 0 0 0.834256 0.551377i \(-0.185897\pi\)
−0.834256 + 0.551377i \(0.814103\pi\)
\(930\) 0 0
\(931\) 37.4398 + 83.1879i 1.22704 + 2.72637i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.92247 + 24.0687i −0.0954730 + 0.786291i 0.863805 + 0.503826i \(0.168075\pi\)
−0.959278 + 0.282464i \(0.908848\pi\)
\(938\) 0 0
\(939\) −39.1422 + 13.0676i −1.27736 + 0.426445i
\(940\) 0 0
\(941\) 0 0 −0.0603785 0.998176i \(-0.519231\pi\)
0.0603785 + 0.998176i \(0.480769\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.927686 0.373361i \(-0.121795\pi\)
−0.927686 + 0.373361i \(0.878205\pi\)
\(948\) −25.4450 + 9.65003i −0.826417 + 0.313419i
\(949\) −37.4656 31.9267i −1.21618 1.03638i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.0804666 0.996757i \(-0.474359\pi\)
−0.0804666 + 0.996757i \(0.525641\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −71.3931 + 49.2791i −2.30300 + 1.58965i
\(962\) 0 0
\(963\) 0 0
\(964\) 2.64996 26.2285i 0.0853495 0.844762i
\(965\) 0 0
\(966\) 0 0
\(967\) −12.6888 + 16.1961i −0.408045 + 0.520831i −0.946883 0.321578i \(-0.895787\pi\)
0.538838 + 0.842410i \(0.318864\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.845190 0.534466i \(-0.820513\pi\)
0.845190 + 0.534466i \(0.179487\pi\)
\(972\) 12.2203 28.6821i 0.391967 0.919979i
\(973\) −14.9799 + 10.7916i −0.480233 + 0.345963i
\(974\) 0 0
\(975\) −13.4114 + 28.1981i −0.429508 + 0.903063i
\(976\) 41.7642 36.9998i 1.33684 1.18434i
\(977\) 0 0 −0.975564 0.219715i \(-0.929487\pi\)
0.975564 + 0.219715i \(0.0705128\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −41.9662 30.2327i −1.33988 0.965256i
\(982\) 0 0
\(983\) 0 0 0.954721 0.297503i \(-0.0961538\pi\)
−0.954721 + 0.297503i \(0.903846\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −45.4687 32.8183i −1.44655 1.04409i
\(989\) 0 0
\(990\) 0 0
\(991\) 1.53593 2.66031i 0.0487904 0.0845074i −0.840599 0.541658i \(-0.817797\pi\)
0.889389 + 0.457151i \(0.151130\pi\)
\(992\) 0 0
\(993\) 40.7659 7.47063i 1.29367 0.237073i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −17.9099 + 37.7443i −0.567211 + 1.19537i 0.393885 + 0.919160i \(0.371131\pi\)
−0.961096 + 0.276213i \(0.910921\pi\)
\(998\) 0 0
\(999\) 28.9108 33.9885i 0.914696 1.07535i
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.x.a.176.1 48
3.2 odd 2 CM 507.2.x.a.176.1 48
169.145 odd 156 inner 507.2.x.a.314.1 yes 48
507.314 even 156 inner 507.2.x.a.314.1 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.x.a.176.1 48 1.1 even 1 trivial
507.2.x.a.176.1 48 3.2 odd 2 CM
507.2.x.a.314.1 yes 48 169.145 odd 156 inner
507.2.x.a.314.1 yes 48 507.314 even 156 inner