Properties

Label 700.2.r.c.149.1
Level $700$
Weight $2$
Character 700.149
Analytic conductor $5.590$
Analytic rank $0$
Dimension $4$
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] = [700,2,Mod(149,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.r (of order \(6\), degree \(2\), 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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 149.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 700.149
Dual form 700.2.r.c.249.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.59808 + 1.50000i) q^{3} +(-2.59808 - 0.500000i) q^{7} +(3.00000 - 5.19615i) q^{9} +O(q^{10})\) \(q+(-2.59808 + 1.50000i) q^{3} +(-2.59808 - 0.500000i) q^{7} +(3.00000 - 5.19615i) q^{9} +(1.00000 + 1.73205i) q^{11} -6.00000i q^{13} +(-1.73205 + 1.00000i) q^{17} +(7.50000 - 2.59808i) q^{21} +(7.79423 + 4.50000i) q^{23} +9.00000i q^{27} -3.00000 q^{29} +(-1.00000 - 1.73205i) q^{31} +(-5.19615 - 3.00000i) q^{33} +(6.92820 + 4.00000i) q^{37} +(9.00000 + 15.5885i) q^{39} +5.00000 q^{41} +1.00000i q^{43} +(6.92820 + 4.00000i) q^{47} +(6.50000 + 2.59808i) q^{49} +(3.00000 - 5.19615i) q^{51} +(3.46410 - 2.00000i) q^{53} +(-4.00000 - 6.92820i) q^{59} +(-3.50000 + 6.06218i) q^{61} +(-10.3923 + 12.0000i) q^{63} +(2.59808 - 1.50000i) q^{67} -27.0000 q^{69} +8.00000 q^{71} +(12.1244 - 7.00000i) q^{73} +(-1.73205 - 5.00000i) q^{77} +(2.00000 - 3.46410i) q^{79} +(-4.50000 - 7.79423i) q^{81} -1.00000i q^{83} +(7.79423 - 4.50000i) q^{87} +(6.50000 - 11.2583i) q^{89} +(-3.00000 + 15.5885i) q^{91} +(5.19615 + 3.00000i) q^{93} +10.0000i q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} + 4 q^{11} + 30 q^{21} - 12 q^{29} - 4 q^{31} + 36 q^{39} + 20 q^{41} + 26 q^{49} + 12 q^{51} - 16 q^{59} - 14 q^{61} - 108 q^{69} + 32 q^{71} + 8 q^{79} - 18 q^{81} + 26 q^{89} - 12 q^{91} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\)

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
\(3\) −2.59808 + 1.50000i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.59808 0.500000i −0.981981 0.188982i
\(8\) 0 0
\(9\) 3.00000 5.19615i 1.00000 1.73205i
\(10\) 0 0
\(11\) 1.00000 + 1.73205i 0.301511 + 0.522233i 0.976478 0.215615i \(-0.0691756\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.73205 + 1.00000i −0.420084 + 0.242536i −0.695113 0.718900i \(-0.744646\pi\)
0.275029 + 0.961436i \(0.411312\pi\)
\(18\) 0 0
\(19\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) 0 0
\(21\) 7.50000 2.59808i 1.63663 0.566947i
\(22\) 0 0
\(23\) 7.79423 + 4.50000i 1.62521 + 0.938315i 0.985496 + 0.169701i \(0.0542803\pi\)
0.639713 + 0.768613i \(0.279053\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.00000i 1.73205i
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −1.00000 1.73205i −0.179605 0.311086i 0.762140 0.647412i \(-0.224149\pi\)
−0.941745 + 0.336327i \(0.890815\pi\)
\(32\) 0 0
\(33\) −5.19615 3.00000i −0.904534 0.522233i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.92820 + 4.00000i 1.13899 + 0.657596i 0.946180 0.323640i \(-0.104907\pi\)
0.192809 + 0.981236i \(0.438240\pi\)
\(38\) 0 0
\(39\) 9.00000 + 15.5885i 1.44115 + 2.49615i
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.92820 + 4.00000i 1.01058 + 0.583460i 0.911362 0.411606i \(-0.135032\pi\)
0.0992202 + 0.995066i \(0.468365\pi\)
\(48\) 0 0
\(49\) 6.50000 + 2.59808i 0.928571 + 0.371154i
\(50\) 0 0
\(51\) 3.00000 5.19615i 0.420084 0.727607i
\(52\) 0 0
\(53\) 3.46410 2.00000i 0.475831 0.274721i −0.242846 0.970065i \(-0.578081\pi\)
0.718677 + 0.695344i \(0.244748\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 6.92820i −0.520756 0.901975i −0.999709 0.0241347i \(-0.992317\pi\)
0.478953 0.877841i \(-0.341016\pi\)
\(60\) 0 0
\(61\) −3.50000 + 6.06218i −0.448129 + 0.776182i −0.998264 0.0588933i \(-0.981243\pi\)
0.550135 + 0.835076i \(0.314576\pi\)
\(62\) 0 0
\(63\) −10.3923 + 12.0000i −1.30931 + 1.51186i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.59808 1.50000i 0.317406 0.183254i −0.332830 0.942987i \(-0.608004\pi\)
0.650236 + 0.759733i \(0.274670\pi\)
\(68\) 0 0
\(69\) −27.0000 −3.25042
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 12.1244 7.00000i 1.41905 0.819288i 0.422833 0.906208i \(-0.361036\pi\)
0.996215 + 0.0869195i \(0.0277023\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.73205 5.00000i −0.197386 0.569803i
\(78\) 0 0
\(79\) 2.00000 3.46410i 0.225018 0.389742i −0.731307 0.682048i \(-0.761089\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) 1.00000i 0.109764i −0.998493 0.0548821i \(-0.982522\pi\)
0.998493 0.0548821i \(-0.0174783\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.79423 4.50000i 0.835629 0.482451i
\(88\) 0 0
\(89\) 6.50000 11.2583i 0.688999 1.19338i −0.283164 0.959072i \(-0.591384\pi\)
0.972162 0.234309i \(-0.0752827\pi\)
\(90\) 0 0
\(91\) −3.00000 + 15.5885i −0.314485 + 1.63411i
\(92\) 0 0
\(93\) 5.19615 + 3.00000i 0.538816 + 0.311086i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 12.0000 1.20605
\(100\) 0 0
\(101\) 1.50000 + 2.59808i 0.149256 + 0.258518i 0.930953 0.365140i \(-0.118979\pi\)
−0.781697 + 0.623658i \(0.785646\pi\)
\(102\) 0 0
\(103\) −11.2583 6.50000i −1.10932 0.640464i −0.170664 0.985329i \(-0.554591\pi\)
−0.938652 + 0.344865i \(0.887925\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.9904 7.50000i −1.25583 0.725052i −0.283567 0.958952i \(-0.591518\pi\)
−0.972261 + 0.233900i \(0.924851\pi\)
\(108\) 0 0
\(109\) 4.50000 + 7.79423i 0.431022 + 0.746552i 0.996962 0.0778949i \(-0.0248199\pi\)
−0.565940 + 0.824447i \(0.691487\pi\)
\(110\) 0 0
\(111\) −24.0000 −2.27798
\(112\) 0 0
\(113\) 4.00000i 0.376288i −0.982141 0.188144i \(-0.939753\pi\)
0.982141 0.188144i \(-0.0602472\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −31.1769 18.0000i −2.88231 1.66410i
\(118\) 0 0
\(119\) 5.00000 1.73205i 0.458349 0.158777i
\(120\) 0 0
\(121\) 3.50000 6.06218i 0.318182 0.551107i
\(122\) 0 0
\(123\) −12.9904 + 7.50000i −1.17130 + 0.676252i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000i 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 0 0
\(129\) −1.50000 2.59808i −0.132068 0.228748i
\(130\) 0 0
\(131\) 2.00000 3.46410i 0.174741 0.302660i −0.765331 0.643637i \(-0.777425\pi\)
0.940072 + 0.340977i \(0.110758\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.3923 6.00000i 0.887875 0.512615i 0.0146279 0.999893i \(-0.495344\pi\)
0.873247 + 0.487278i \(0.162010\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) 0 0
\(143\) 10.3923 6.00000i 0.869048 0.501745i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −20.7846 + 3.00000i −1.71429 + 0.247436i
\(148\) 0 0
\(149\) 4.50000 7.79423i 0.368654 0.638528i −0.620701 0.784047i \(-0.713152\pi\)
0.989355 + 0.145519i \(0.0464853\pi\)
\(150\) 0 0
\(151\) 5.00000 + 8.66025i 0.406894 + 0.704761i 0.994540 0.104357i \(-0.0332784\pi\)
−0.587646 + 0.809118i \(0.699945\pi\)
\(152\) 0 0
\(153\) 12.0000i 0.970143i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.73205 1.00000i 0.138233 0.0798087i −0.429289 0.903167i \(-0.641236\pi\)
0.567521 + 0.823359i \(0.307902\pi\)
\(158\) 0 0
\(159\) −6.00000 + 10.3923i −0.475831 + 0.824163i
\(160\) 0 0
\(161\) −18.0000 15.5885i −1.41860 1.22854i
\(162\) 0 0
\(163\) −6.92820 4.00000i −0.542659 0.313304i 0.203497 0.979076i \(-0.434769\pi\)
−0.746156 + 0.665771i \(0.768103\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.00000i 0.696441i 0.937413 + 0.348220i \(0.113214\pi\)
−0.937413 + 0.348220i \(0.886786\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.8564 + 8.00000i 1.05348 + 0.608229i 0.923622 0.383304i \(-0.125214\pi\)
0.129861 + 0.991532i \(0.458547\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 20.7846 + 12.0000i 1.56227 + 0.901975i
\(178\) 0 0
\(179\) −3.00000 5.19615i −0.224231 0.388379i 0.731858 0.681457i \(-0.238654\pi\)
−0.956088 + 0.293079i \(0.905320\pi\)
\(180\) 0 0
\(181\) 1.00000 0.0743294 0.0371647 0.999309i \(-0.488167\pi\)
0.0371647 + 0.999309i \(0.488167\pi\)
\(182\) 0 0
\(183\) 21.0000i 1.55236i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.46410 2.00000i −0.253320 0.146254i
\(188\) 0 0
\(189\) 4.50000 23.3827i 0.327327 1.70084i
\(190\) 0 0
\(191\) −3.00000 + 5.19615i −0.217072 + 0.375980i −0.953912 0.300088i \(-0.902984\pi\)
0.736839 + 0.676068i \(0.236317\pi\)
\(192\) 0 0
\(193\) 1.73205 1.00000i 0.124676 0.0719816i −0.436365 0.899770i \(-0.643734\pi\)
0.561041 + 0.827788i \(0.310401\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.0000i 0.997459i −0.866758 0.498729i \(-0.833800\pi\)
0.866758 0.498729i \(-0.166200\pi\)
\(198\) 0 0
\(199\) 10.0000 + 17.3205i 0.708881 + 1.22782i 0.965272 + 0.261245i \(0.0841331\pi\)
−0.256391 + 0.966573i \(0.582534\pi\)
\(200\) 0 0
\(201\) −4.50000 + 7.79423i −0.317406 + 0.549762i
\(202\) 0 0
\(203\) 7.79423 + 1.50000i 0.547048 + 0.105279i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 46.7654 27.0000i 3.25042 1.87663i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) −20.7846 + 12.0000i −1.42414 + 0.822226i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.73205 + 5.00000i 0.117579 + 0.339422i
\(218\) 0 0
\(219\) −21.0000 + 36.3731i −1.41905 + 2.45786i
\(220\) 0 0
\(221\) 6.00000 + 10.3923i 0.403604 + 0.699062i
\(222\) 0 0
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.46410 2.00000i 0.229920 0.132745i −0.380615 0.924734i \(-0.624288\pi\)
0.610535 + 0.791989i \(0.290954\pi\)
\(228\) 0 0
\(229\) −7.00000 + 12.1244i −0.462573 + 0.801200i −0.999088 0.0426906i \(-0.986407\pi\)
0.536515 + 0.843891i \(0.319740\pi\)
\(230\) 0 0
\(231\) 12.0000 + 10.3923i 0.789542 + 0.683763i
\(232\) 0 0
\(233\) 15.5885 + 9.00000i 1.02123 + 0.589610i 0.914461 0.404674i \(-0.132615\pi\)
0.106773 + 0.994283i \(0.465948\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 12.0000i 0.779484i
\(238\) 0 0
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) −5.00000 8.66025i −0.322078 0.557856i 0.658838 0.752285i \(-0.271048\pi\)
−0.980917 + 0.194429i \(0.937715\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 1.50000 + 2.59808i 0.0950586 + 0.164646i
\(250\) 0 0
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) 0 0
\(253\) 18.0000i 1.13165i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.92820 + 4.00000i 0.432169 + 0.249513i 0.700270 0.713878i \(-0.253063\pi\)
−0.268101 + 0.963391i \(0.586396\pi\)
\(258\) 0 0
\(259\) −16.0000 13.8564i −0.994192 0.860995i
\(260\) 0 0
\(261\) −9.00000 + 15.5885i −0.557086 + 0.964901i
\(262\) 0 0
\(263\) 14.7224 8.50000i 0.907824 0.524132i 0.0280936 0.999605i \(-0.491056\pi\)
0.879730 + 0.475473i \(0.157723\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 39.0000i 2.38676i
\(268\) 0 0
\(269\) 4.50000 + 7.79423i 0.274370 + 0.475223i 0.969976 0.243201i \(-0.0781974\pi\)
−0.695606 + 0.718423i \(0.744864\pi\)
\(270\) 0 0
\(271\) 12.0000 20.7846i 0.728948 1.26258i −0.228380 0.973572i \(-0.573343\pi\)
0.957328 0.289003i \(-0.0933238\pi\)
\(272\) 0 0
\(273\) −15.5885 45.0000i −0.943456 2.72352i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −15.5885 + 9.00000i −0.936620 + 0.540758i −0.888899 0.458103i \(-0.848529\pi\)
−0.0477206 + 0.998861i \(0.515196\pi\)
\(278\) 0 0
\(279\) −12.0000 −0.718421
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) −3.46410 + 2.00000i −0.205919 + 0.118888i −0.599414 0.800439i \(-0.704600\pi\)
0.393494 + 0.919327i \(0.371266\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.9904 2.50000i −0.766798 0.147570i
\(288\) 0 0
\(289\) −6.50000 + 11.2583i −0.382353 + 0.662255i
\(290\) 0 0
\(291\) −15.0000 25.9808i −0.879316 1.52302i
\(292\) 0 0
\(293\) 4.00000i 0.233682i 0.993151 + 0.116841i \(0.0372769\pi\)
−0.993151 + 0.116841i \(0.962723\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −15.5885 + 9.00000i −0.904534 + 0.522233i
\(298\) 0 0
\(299\) 27.0000 46.7654i 1.56145 2.70451i
\(300\) 0 0
\(301\) 0.500000 2.59808i 0.0288195 0.149751i
\(302\) 0 0
\(303\) −7.79423 4.50000i −0.447767 0.258518i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.00000i 0.0570730i −0.999593 0.0285365i \(-0.990915\pi\)
0.999593 0.0285365i \(-0.00908469\pi\)
\(308\) 0 0
\(309\) 39.0000 2.21863
\(310\) 0 0
\(311\) −12.0000 20.7846i −0.680458 1.17859i −0.974841 0.222900i \(-0.928448\pi\)
0.294384 0.955687i \(-0.404886\pi\)
\(312\) 0 0
\(313\) 3.46410 + 2.00000i 0.195803 + 0.113047i 0.594696 0.803951i \(-0.297272\pi\)
−0.398894 + 0.916997i \(0.630606\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.66025 + 5.00000i 0.486408 + 0.280828i 0.723083 0.690761i \(-0.242724\pi\)
−0.236675 + 0.971589i \(0.576058\pi\)
\(318\) 0 0
\(319\) −3.00000 5.19615i −0.167968 0.290929i
\(320\) 0 0
\(321\) 45.0000 2.51166
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −23.3827 13.5000i −1.29307 0.746552i
\(328\) 0 0
\(329\) −16.0000 13.8564i −0.882109 0.763928i
\(330\) 0 0
\(331\) −5.00000 + 8.66025i −0.274825 + 0.476011i −0.970091 0.242742i \(-0.921953\pi\)
0.695266 + 0.718752i \(0.255287\pi\)
\(332\) 0 0
\(333\) 41.5692 24.0000i 2.27798 1.31519i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 28.0000i 1.52526i −0.646837 0.762629i \(-0.723908\pi\)
0.646837 0.762629i \(-0.276092\pi\)
\(338\) 0 0
\(339\) 6.00000 + 10.3923i 0.325875 + 0.564433i
\(340\) 0 0
\(341\) 2.00000 3.46410i 0.108306 0.187592i
\(342\) 0 0
\(343\) −15.5885 10.0000i −0.841698 0.539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −21.6506 + 12.5000i −1.16227 + 0.671035i −0.951846 0.306576i \(-0.900817\pi\)
−0.210421 + 0.977611i \(0.567483\pi\)
\(348\) 0 0
\(349\) 17.0000 0.909989 0.454995 0.890494i \(-0.349641\pi\)
0.454995 + 0.890494i \(0.349641\pi\)
\(350\) 0 0
\(351\) 54.0000 2.88231
\(352\) 0 0
\(353\) −31.1769 + 18.0000i −1.65938 + 0.958043i −0.686378 + 0.727245i \(0.740800\pi\)
−0.973002 + 0.230799i \(0.925866\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −10.3923 + 12.0000i −0.550019 + 0.635107i
\(358\) 0 0
\(359\) −5.00000 + 8.66025i −0.263890 + 0.457071i −0.967272 0.253741i \(-0.918339\pi\)
0.703382 + 0.710812i \(0.251672\pi\)
\(360\) 0 0
\(361\) 9.50000 + 16.4545i 0.500000 + 0.866025i
\(362\) 0 0
\(363\) 21.0000i 1.10221i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.866025 + 0.500000i −0.0452062 + 0.0260998i −0.522433 0.852680i \(-0.674975\pi\)
0.477227 + 0.878780i \(0.341642\pi\)
\(368\) 0 0
\(369\) 15.0000 25.9808i 0.780869 1.35250i
\(370\) 0 0
\(371\) −10.0000 + 3.46410i −0.519174 + 0.179847i
\(372\) 0 0
\(373\) −27.7128 16.0000i −1.43492 0.828449i −0.437425 0.899255i \(-0.644109\pi\)
−0.997490 + 0.0708063i \(0.977443\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.0000i 0.927047i
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 24.0000 + 41.5692i 1.22956 + 2.12966i
\(382\) 0 0
\(383\) 7.79423 + 4.50000i 0.398266 + 0.229939i 0.685736 0.727851i \(-0.259481\pi\)
−0.287469 + 0.957790i \(0.592814\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.19615 + 3.00000i 0.264135 + 0.152499i
\(388\) 0 0
\(389\) −5.00000 8.66025i −0.253510 0.439092i 0.710980 0.703213i \(-0.248252\pi\)
−0.964490 + 0.264120i \(0.914918\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 12.0000i 0.605320i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 8.66025 + 5.00000i 0.434646 + 0.250943i 0.701324 0.712843i \(-0.252593\pi\)
−0.266678 + 0.963786i \(0.585926\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.5000 25.1147i 0.724095 1.25417i −0.235250 0.971935i \(-0.575591\pi\)
0.959345 0.282235i \(-0.0910758\pi\)
\(402\) 0 0
\(403\) −10.3923 + 6.00000i −0.517678 + 0.298881i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16.0000i 0.793091i
\(408\) 0 0
\(409\) 1.50000 + 2.59808i 0.0741702 + 0.128467i 0.900725 0.434389i \(-0.143036\pi\)
−0.826555 + 0.562856i \(0.809703\pi\)
\(410\) 0 0
\(411\) −18.0000 + 31.1769i −0.887875 + 1.53784i
\(412\) 0 0
\(413\) 6.92820 + 20.0000i 0.340915 + 0.984136i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 25.9808 15.0000i 1.27228 0.734553i
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 0 0
\(423\) 41.5692 24.0000i 2.02116 1.16692i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 12.1244 14.0000i 0.586739 0.677507i
\(428\) 0 0
\(429\) −18.0000 + 31.1769i −0.869048 + 1.50524i
\(430\) 0 0
\(431\) −3.00000 5.19615i −0.144505 0.250290i 0.784683 0.619897i \(-0.212826\pi\)
−0.929188 + 0.369607i \(0.879492\pi\)
\(432\) 0 0
\(433\) 4.00000i 0.192228i −0.995370 0.0961139i \(-0.969359\pi\)
0.995370 0.0961139i \(-0.0306413\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 2.00000 3.46410i 0.0954548 0.165333i −0.814344 0.580383i \(-0.802903\pi\)
0.909798 + 0.415051i \(0.136236\pi\)
\(440\) 0 0
\(441\) 33.0000 25.9808i 1.57143 1.23718i
\(442\) 0 0
\(443\) −32.0429 18.5000i −1.52241 0.878962i −0.999649 0.0264796i \(-0.991570\pi\)
−0.522757 0.852482i \(-0.675096\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 27.0000i 1.27706i
\(448\) 0 0
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) 5.00000 + 8.66025i 0.235441 + 0.407795i
\(452\) 0 0
\(453\) −25.9808 15.0000i −1.22068 0.704761i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 24.2487 + 14.0000i 1.13431 + 0.654892i 0.945015 0.327028i \(-0.106047\pi\)
0.189292 + 0.981921i \(0.439381\pi\)
\(458\) 0 0
\(459\) −9.00000 15.5885i −0.420084 0.727607i
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 17.0000i 0.790057i 0.918669 + 0.395029i \(0.129265\pi\)
−0.918669 + 0.395029i \(0.870735\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.33013 + 2.50000i 0.200374 + 0.115686i 0.596830 0.802368i \(-0.296427\pi\)
−0.396456 + 0.918054i \(0.629760\pi\)
\(468\) 0 0
\(469\) −7.50000 + 2.59808i −0.346318 + 0.119968i
\(470\) 0 0
\(471\) −3.00000 + 5.19615i −0.138233 + 0.239426i
\(472\) 0 0
\(473\) −1.73205 + 1.00000i −0.0796398 + 0.0459800i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 24.0000i 1.09888i
\(478\) 0 0
\(479\) −15.0000 25.9808i −0.685367 1.18709i −0.973321 0.229447i \(-0.926308\pi\)
0.287954 0.957644i \(-0.407025\pi\)
\(480\) 0 0
\(481\) 24.0000 41.5692i 1.09431 1.89539i
\(482\) 0 0
\(483\) 70.1481 + 13.5000i 3.19185 + 0.614271i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −27.7128 + 16.0000i −1.25579 + 0.725029i −0.972253 0.233933i \(-0.924840\pi\)
−0.283535 + 0.958962i \(0.591507\pi\)
\(488\) 0 0
\(489\) 24.0000 1.08532
\(490\) 0 0
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 0 0
\(493\) 5.19615 3.00000i 0.234023 0.135113i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −20.7846 4.00000i −0.932317 0.179425i
\(498\) 0 0
\(499\) −3.00000 + 5.19615i −0.134298 + 0.232612i −0.925329 0.379165i \(-0.876211\pi\)
0.791031 + 0.611776i \(0.209545\pi\)
\(500\) 0 0
\(501\) −13.5000 23.3827i −0.603136 1.04466i
\(502\) 0 0
\(503\) 27.0000i 1.20387i 0.798545 + 0.601935i \(0.205603\pi\)
−0.798545 + 0.601935i \(0.794397\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 59.7558 34.5000i 2.65385 1.53220i
\(508\) 0 0
\(509\) −20.5000 + 35.5070i −0.908647 + 1.57382i −0.0927004 + 0.995694i \(0.529550\pi\)
−0.815946 + 0.578128i \(0.803783\pi\)
\(510\) 0 0
\(511\) −35.0000 + 12.1244i −1.54831 + 0.536350i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) −48.0000 −2.10697
\(520\) 0 0
\(521\) −7.00000 12.1244i −0.306676 0.531178i 0.670957 0.741496i \(-0.265883\pi\)
−0.977633 + 0.210318i \(0.932550\pi\)
\(522\) 0 0
\(523\) −3.46410 2.00000i −0.151475 0.0874539i 0.422347 0.906434i \(-0.361206\pi\)
−0.573822 + 0.818980i \(0.694540\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.46410 + 2.00000i 0.150899 + 0.0871214i
\(528\) 0 0
\(529\) 29.0000 + 50.2295i 1.26087 + 2.18389i
\(530\) 0 0
\(531\) −48.0000 −2.08302
\(532\) 0 0
\(533\) 30.0000i 1.29944i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.5885 + 9.00000i 0.672692 + 0.388379i
\(538\) 0 0
\(539\) 2.00000 + 13.8564i 0.0861461 + 0.596838i
\(540\) 0 0
\(541\) −16.5000 + 28.5788i −0.709390 + 1.22870i 0.255693 + 0.966758i \(0.417696\pi\)
−0.965084 + 0.261942i \(0.915637\pi\)
\(542\) 0 0
\(543\) −2.59808 + 1.50000i −0.111494 + 0.0643712i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 15.0000i 0.641354i 0.947189 + 0.320677i \(0.103910\pi\)
−0.947189 + 0.320677i \(0.896090\pi\)
\(548\) 0 0
\(549\) 21.0000 + 36.3731i 0.896258 + 1.55236i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −6.92820 + 8.00000i −0.294617 + 0.340195i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.73205 1.00000i 0.0733893 0.0423714i −0.462856 0.886433i \(-0.653175\pi\)
0.536246 + 0.844062i \(0.319842\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) −11.2583 + 6.50000i −0.474482 + 0.273942i −0.718114 0.695925i \(-0.754994\pi\)
0.243632 + 0.969868i \(0.421661\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 7.79423 + 22.5000i 0.327327 + 0.944911i
\(568\) 0 0
\(569\) 15.0000 25.9808i 0.628833 1.08917i −0.358954 0.933355i \(-0.616866\pi\)
0.987786 0.155815i \(-0.0498003\pi\)
\(570\) 0 0
\(571\) −12.0000 20.7846i −0.502184 0.869809i −0.999997 0.00252413i \(-0.999197\pi\)
0.497812 0.867285i \(-0.334137\pi\)
\(572\) 0 0
\(573\) 18.0000i 0.751961i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −19.0526 + 11.0000i −0.793168 + 0.457936i −0.841077 0.540916i \(-0.818078\pi\)
0.0479084 + 0.998852i \(0.484744\pi\)
\(578\) 0 0
\(579\) −3.00000 + 5.19615i −0.124676 + 0.215945i
\(580\) 0 0
\(581\) −0.500000 + 2.59808i −0.0207435 + 0.107786i
\(582\) 0 0
\(583\) 6.92820 + 4.00000i 0.286937 + 0.165663i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20.0000i 0.825488i −0.910847 0.412744i \(-0.864570\pi\)
0.910847 0.412744i \(-0.135430\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 21.0000 + 36.3731i 0.863825 + 1.49619i
\(592\) 0 0
\(593\) 31.1769 + 18.0000i 1.28028 + 0.739171i 0.976900 0.213697i \(-0.0685507\pi\)
0.303383 + 0.952869i \(0.401884\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −51.9615 30.0000i −2.12664 1.22782i
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) 18.0000i 0.733017i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0.866025 + 0.500000i 0.0351509 + 0.0202944i 0.517472 0.855700i \(-0.326873\pi\)
−0.482322 + 0.875994i \(0.660206\pi\)
\(608\) 0 0
\(609\) −22.5000 + 7.79423i −0.911746 + 0.315838i
\(610\) 0 0
\(611\) 24.0000 41.5692i 0.970936 1.68171i
\(612\) 0 0
\(613\) −5.19615 + 3.00000i −0.209871 + 0.121169i −0.601251 0.799060i \(-0.705331\pi\)
0.391381 + 0.920229i \(0.371998\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 20.0000i 0.805170i −0.915383 0.402585i \(-0.868112\pi\)
0.915383 0.402585i \(-0.131888\pi\)
\(618\) 0 0
\(619\) −5.00000 8.66025i −0.200967 0.348085i 0.747873 0.663842i \(-0.231075\pi\)
−0.948840 + 0.315757i \(0.897742\pi\)
\(620\) 0 0
\(621\) −40.5000 + 70.1481i −1.62521 + 2.81494i
\(622\) 0 0
\(623\) −22.5167 + 26.0000i −0.902111 + 1.04167i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 0 0
\(633\) −10.3923 + 6.00000i −0.413057 + 0.238479i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 15.5885 39.0000i 0.617637 1.54524i
\(638\) 0 0
\(639\) 24.0000 41.5692i 0.949425 1.64445i
\(640\) 0 0
\(641\) 15.5000 + 26.8468i 0.612213 + 1.06038i 0.990867 + 0.134846i \(0.0430539\pi\)
−0.378653 + 0.925539i \(0.623613\pi\)
\(642\) 0 0
\(643\) 20.0000i 0.788723i −0.918955 0.394362i \(-0.870966\pi\)
0.918955 0.394362i \(-0.129034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.866025 + 0.500000i −0.0340470 + 0.0196570i −0.516927 0.856030i \(-0.672924\pi\)
0.482880 + 0.875687i \(0.339591\pi\)
\(648\) 0 0
\(649\) 8.00000 13.8564i 0.314027 0.543912i
\(650\) 0 0
\(651\) −12.0000 10.3923i −0.470317 0.407307i
\(652\) 0 0
\(653\) 29.4449 + 17.0000i 1.15227 + 0.665261i 0.949439 0.313953i \(-0.101653\pi\)
0.202828 + 0.979214i \(0.434987\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 84.0000i 3.27715i
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) −15.5000 26.8468i −0.602880 1.04422i −0.992383 0.123194i \(-0.960686\pi\)
0.389503 0.921025i \(-0.372647\pi\)
\(662\) 0 0
\(663\) −31.1769 18.0000i −1.21081 0.699062i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −23.3827 13.5000i −0.905381 0.522722i
\(668\) 0 0
\(669\) −24.0000 41.5692i −0.927894 1.60716i
\(670\) 0 0
\(671\) −14.0000 −0.540464
\(672\) 0 0
\(673\) 16.0000i 0.616755i 0.951264 + 0.308377i \(0.0997859\pi\)
−0.951264 + 0.308377i \(0.900214\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.19615 + 3.00000i 0.199704 + 0.115299i 0.596518 0.802600i \(-0.296551\pi\)
−0.396813 + 0.917899i \(0.629884\pi\)
\(678\) 0 0
\(679\) 5.00000 25.9808i 0.191882 0.997050i
\(680\) 0 0
\(681\) −6.00000 + 10.3923i −0.229920 + 0.398234i
\(682\) 0 0
\(683\) 25.1147 14.5000i 0.960989 0.554827i 0.0645115 0.997917i \(-0.479451\pi\)
0.896477 + 0.443090i \(0.146118\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 42.0000i 1.60240i
\(688\) 0 0
\(689\) −12.0000 20.7846i −0.457164 0.791831i
\(690\) 0 0
\(691\) 13.0000 22.5167i 0.494543 0.856574i −0.505437 0.862864i \(-0.668669\pi\)
0.999980 + 0.00628943i \(0.00200200\pi\)
\(692\) 0 0
\(693\) −31.1769 6.00000i −1.18431 0.227921i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −8.66025 + 5.00000i −0.328031 + 0.189389i
\(698\) 0 0
\(699\) −54.0000 −2.04247
\(700\) 0 0
\(701\) −29.0000 −1.09531 −0.547657 0.836703i \(-0.684480\pi\)
−0.547657 + 0.836703i \(0.684480\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.59808 7.50000i −0.0977107 0.282067i
\(708\) 0 0
\(709\) 15.5000 26.8468i 0.582115 1.00825i −0.413114 0.910679i \(-0.635559\pi\)
0.995228 0.0975728i \(-0.0311079\pi\)
\(710\) 0 0
\(711\) −12.0000 20.7846i −0.450035 0.779484i
\(712\) 0 0
\(713\) 18.0000i 0.674105i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −67.5500 + 39.0000i −2.52270 + 1.45648i
\(718\) 0 0
\(719\) −3.00000 + 5.19615i −0.111881 + 0.193784i −0.916529 0.399969i \(-0.869021\pi\)
0.804648 + 0.593753i \(0.202354\pi\)
\(720\) 0 0
\(721\) 26.0000 + 22.5167i 0.968291 + 0.838564i
\(722\) 0 0
\(723\) 25.9808 + 15.0000i 0.966235 + 0.557856i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 3.00000i 0.111264i −0.998451 0.0556319i \(-0.982283\pi\)
0.998451 0.0556319i \(-0.0177173\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −1.00000 1.73205i −0.0369863 0.0640622i
\(732\) 0 0
\(733\) −29.4449 17.0000i −1.08757 0.627909i −0.154642 0.987971i \(-0.549422\pi\)
−0.932929 + 0.360061i \(0.882756\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.19615 + 3.00000i 0.191403 + 0.110506i
\(738\) 0 0
\(739\) 5.00000 + 8.66025i 0.183928 + 0.318573i 0.943215 0.332184i \(-0.107785\pi\)
−0.759287 + 0.650756i \(0.774452\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.00000i 0.110059i −0.998485 0.0550297i \(-0.982475\pi\)
0.998485 0.0550297i \(-0.0175253\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −5.19615 3.00000i −0.190117 0.109764i
\(748\) 0 0
\(749\) 30.0000 + 25.9808i 1.09618 + 0.949316i
\(750\) 0 0
\(751\) −10.0000 + 17.3205i −0.364905 + 0.632034i −0.988761 0.149505i \(-0.952232\pi\)
0.623856 + 0.781540i \(0.285565\pi\)
\(752\) 0 0
\(753\) −77.9423 + 45.0000i −2.84037 + 1.63989i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 38.0000i 1.38113i 0.723269 + 0.690567i \(0.242639\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(758\) 0 0
\(759\) −27.0000 46.7654i −0.980038 1.69748i
\(760\) 0 0
\(761\) −15.0000 + 25.9808i −0.543750 + 0.941802i 0.454935 + 0.890525i \(0.349663\pi\)
−0.998684 + 0.0512772i \(0.983671\pi\)
\(762\) 0 0
\(763\) −7.79423 22.5000i −0.282170 0.814555i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −41.5692 + 24.0000i −1.50098 + 0.866590i
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) −24.0000 −0.864339
\(772\) 0 0
\(773\) −15.5885 + 9.00000i −0.560678 + 0.323708i −0.753418 0.657542i \(-0.771596\pi\)
0.192740 + 0.981250i \(0.438263\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 62.3538 + 12.0000i 2.23693 + 0.430498i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 8.00000 + 13.8564i 0.286263 + 0.495821i
\(782\) 0 0
\(783\) 27.0000i 0.964901i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 9.52628 5.50000i 0.339575 0.196054i −0.320509 0.947245i \(-0.603854\pi\)
0.660084 + 0.751192i \(0.270521\pi\)
\(788\) 0 0
\(789\) −25.5000 + 44.1673i −0.907824 + 1.57240i
\(790\) 0 0
\(791\) −2.00000 + 10.3923i −0.0711118 + 0.369508i
\(792\) 0 0
\(793\) 36.3731 + 21.0000i 1.29165 + 0.745732i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 36.0000i 1.27519i −0.770374 0.637593i \(-0.779930\pi\)
0.770374 0.637593i \(-0.220070\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −39.0000 67.5500i −1.37800 2.38676i
\(802\) 0 0
\(803\) 24.2487 + 14.0000i 0.855718 + 0.494049i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −23.3827 13.5000i −0.823110 0.475223i
\(808\) 0 0
\(809\) 22.5000 + 38.9711i 0.791058 + 1.37015i 0.925312 + 0.379206i \(0.123803\pi\)
−0.134255 + 0.990947i \(0.542864\pi\)
\(810\) 0 0
\(811\) 50.0000 1.75574 0.877869 0.478901i \(-0.158965\pi\)
0.877869 + 0.478901i \(0.158965\pi\)
\(812\) 0 0
\(813\) 72.0000i 2.52515i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 72.0000 + 62.3538i 2.51588 + 2.17882i
\(820\) 0 0
\(821\) −9.00000 + 15.5885i −0.314102 + 0.544041i −0.979246 0.202674i \(-0.935037\pi\)
0.665144 + 0.746715i \(0.268370\pi\)
\(822\) 0 0
\(823\) 16.4545 9.50000i 0.573567 0.331149i −0.185006 0.982737i \(-0.559230\pi\)
0.758573 + 0.651588i \(0.225897\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 23.0000i 0.799788i −0.916561 0.399894i \(-0.869047\pi\)
0.916561 0.399894i \(-0.130953\pi\)
\(828\) 0 0
\(829\) −5.00000 8.66025i −0.173657 0.300783i 0.766039 0.642795i \(-0.222225\pi\)
−0.939696 + 0.342012i \(0.888892\pi\)
\(830\) 0 0
\(831\) 27.0000 46.7654i 0.936620 1.62227i
\(832\) 0 0
\(833\) −13.8564 + 2.00000i −0.480096 + 0.0692959i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 15.5885 9.00000i 0.538816 0.311086i
\(838\) 0 0
\(839\) −14.0000 −0.483334 −0.241667 0.970359i \(-0.577694\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 57.1577 33.0000i 1.96861 1.13658i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −12.1244 + 14.0000i −0.416598 + 0.481046i
\(848\) 0 0
\(849\) 6.00000 10.3923i 0.205919 0.356663i
\(850\) 0 0
\(851\) 36.0000 + 62.3538i 1.23406 + 2.13746i
\(852\) 0 0
\(853\) 40.0000i 1.36957i −0.728743 0.684787i \(-0.759895\pi\)
0.728743 0.684787i \(-0.240105\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 46.7654 27.0000i 1.59747 0.922302i 0.605503 0.795843i \(-0.292972\pi\)
0.991972 0.126459i \(-0.0403613\pi\)
\(858\) 0 0
\(859\) −18.0000 + 31.1769i −0.614152 + 1.06374i 0.376381 + 0.926465i \(0.377169\pi\)
−0.990533 + 0.137277i \(0.956165\pi\)
\(860\) 0 0
\(861\) 37.5000 12.9904i 1.27800 0.442711i
\(862\) 0 0
\(863\) −32.0429 18.5000i −1.09075 0.629747i −0.156977 0.987602i \(-0.550175\pi\)
−0.933777 + 0.357855i \(0.883508\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 39.0000i 1.32451i
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) −9.00000 15.5885i −0.304953 0.528195i
\(872\) 0 0
\(873\) 51.9615 + 30.0000i 1.75863 + 1.01535i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 27.7128 + 16.0000i 0.935795 + 0.540282i 0.888640 0.458606i \(-0.151651\pi\)
0.0471555 + 0.998888i \(0.484984\pi\)
\(878\) 0 0
\(879\) −6.00000 10.3923i −0.202375 0.350524i
\(880\) 0 0
\(881\) −31.0000 −1.04442 −0.522208 0.852818i \(-0.674892\pi\)
−0.522208 + 0.852818i \(0.674892\pi\)
\(882\) 0 0
\(883\) 36.0000i 1.21150i −0.795656 0.605748i \(-0.792874\pi\)
0.795656 0.605748i \(-0.207126\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 45.8993 + 26.5000i 1.54115 + 0.889783i 0.998767 + 0.0496513i \(0.0158110\pi\)
0.542383 + 0.840132i \(0.317522\pi\)
\(888\) 0 0
\(889\) −8.00000 + 41.5692i −0.268311 + 1.39419i
\(890\) 0 0
\(891\) 9.00000 15.5885i 0.301511 0.522233i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 162.000i 5.40902i
\(898\) 0 0
\(899\) 3.00000 + 5.19615i 0.100056 + 0.173301i
\(900\) 0 0
\(901\) −4.00000 + 6.92820i −0.133259 + 0.230812i
\(902\) 0 0
\(903\) 2.59808 + 7.50000i 0.0864586 + 0.249584i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 32.0429 18.5000i 1.06397 0.614282i 0.137441 0.990510i \(-0.456112\pi\)
0.926527 + 0.376228i \(0.122779\pi\)
\(908\) 0 0
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 54.0000 1.78910 0.894550 0.446968i \(-0.147496\pi\)
0.894550 + 0.446968i \(0.147496\pi\)
\(912\) 0 0
\(913\) 1.73205 1.00000i 0.0573225 0.0330952i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −6.92820 + 8.00000i −0.228789 + 0.264183i
\(918\) 0 0
\(919\) 16.0000 27.7128i 0.527791 0.914161i −0.471684 0.881768i \(-0.656354\pi\)
0.999475 0.0323936i \(-0.0103130\pi\)
\(920\) 0 0
\(921\) 1.50000 + 2.59808i 0.0494267 + 0.0856095i
\(922\) 0 0
\(923\) 48.0000i 1.57994i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −67.5500 + 39.0000i −2.21863 + 1.28093i
\(928\) 0 0
\(929\) −2.50000 + 4.33013i −0.0820223 + 0.142067i −0.904118 0.427282i \(-0.859471\pi\)
0.822096 + 0.569349i \(0.192805\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 62.3538 + 36.0000i 2.04137 + 1.17859i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 52.0000i 1.69877i 0.527777 + 0.849383i \(0.323026\pi\)
−0.527777 + 0.849383i \(0.676974\pi\)
\(938\) 0 0
\(939\) −12.0000 −0.391605
\(940\) 0 0
\(941\) −19.0000 32.9090i −0.619382 1.07280i −0.989599 0.143856i \(-0.954050\pi\)
0.370216 0.928946i \(-0.379284\pi\)
\(942\) 0 0
\(943\) 38.9711 + 22.5000i 1.26908 + 0.732701i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 25.1147 + 14.5000i 0.816119 + 0.471187i 0.849076 0.528270i \(-0.177159\pi\)
−0.0329571 + 0.999457i \(0.510492\pi\)
\(948\) 0 0
\(949\) −42.0000 72.7461i −1.36338 2.36144i
\(950\) 0 0
\(951\) −30.0000 −0.972817
\(952\) 0 0
\(953\) 60.0000i 1.94359i −0.235826 0.971795i \(-0.575780\pi\)
0.235826 0.971795i \(-0.424220\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 15.5885 + 9.00000i 0.503903 + 0.290929i
\(958\) 0 0
\(959\) −30.0000 + 10.3923i −0.968751 + 0.335585i
\(960\) 0 0
\(961\) 13.5000 23.3827i 0.435484 0.754280i
\(962\) 0 0
\(963\) −77.9423 + 45.0000i −2.51166 + 1.45010i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 37.0000i 1.18984i −0.803785 0.594920i \(-0.797184\pi\)
0.803785 0.594920i \(-0.202816\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 6.00000 10.3923i 0.192549 0.333505i −0.753545 0.657396i \(-0.771658\pi\)
0.946094 + 0.323891i \(0.104991\pi\)
\(972\) 0 0
\(973\) 25.9808 + 5.00000i 0.832905 + 0.160293i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5.19615 3.00000i 0.166240 0.0959785i −0.414572 0.910017i \(-0.636069\pi\)
0.580812 + 0.814038i \(0.302735\pi\)
\(978\) 0 0
\(979\) 26.0000 0.830964
\(980\) 0 0
\(981\) 54.0000 1.72409
\(982\) 0 0
\(983\) −16.4545 + 9.50000i −0.524816 + 0.303003i −0.738903 0.673812i \(-0.764656\pi\)
0.214087 + 0.976815i \(0.431323\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 62.3538 + 12.0000i 1.98474 + 0.381964i
\(988\) 0 0
\(989\) −4.50000 + 7.79423i −0.143092 + 0.247842i
\(990\) 0 0
\(991\) 19.0000 + 32.9090i 0.603555 + 1.04539i 0.992278 + 0.124033i \(0.0395829\pi\)
−0.388723 + 0.921355i \(0.627084\pi\)
\(992\) 0 0
\(993\) 30.0000i 0.952021i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 34.6410 20.0000i 1.09709 0.633406i 0.161636 0.986850i \(-0.448323\pi\)
0.935456 + 0.353444i \(0.114990\pi\)
\(998\) 0 0
\(999\) −36.0000 + 62.3538i −1.13899 + 1.97279i
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 700.2.r.c.149.1 4
5.2 odd 4 140.2.i.b.121.1 yes 2
5.3 odd 4 700.2.i.a.401.1 2
5.4 even 2 inner 700.2.r.c.149.2 4
7.2 even 3 4900.2.e.c.2549.1 2
7.4 even 3 inner 700.2.r.c.249.2 4
7.5 odd 6 4900.2.e.b.2549.2 2
15.2 even 4 1260.2.s.b.541.1 2
20.7 even 4 560.2.q.a.401.1 2
35.2 odd 12 980.2.a.a.1.1 1
35.4 even 6 inner 700.2.r.c.249.1 4
35.9 even 6 4900.2.e.c.2549.2 2
35.12 even 12 980.2.a.i.1.1 1
35.17 even 12 980.2.i.a.361.1 2
35.18 odd 12 700.2.i.a.501.1 2
35.19 odd 6 4900.2.e.b.2549.1 2
35.23 odd 12 4900.2.a.v.1.1 1
35.27 even 4 980.2.i.a.961.1 2
35.32 odd 12 140.2.i.b.81.1 2
35.33 even 12 4900.2.a.a.1.1 1
105.2 even 12 8820.2.a.w.1.1 1
105.32 even 12 1260.2.s.b.361.1 2
105.47 odd 12 8820.2.a.k.1.1 1
140.47 odd 12 3920.2.a.d.1.1 1
140.67 even 12 560.2.q.a.81.1 2
140.107 even 12 3920.2.a.bi.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.i.b.81.1 2 35.32 odd 12
140.2.i.b.121.1 yes 2 5.2 odd 4
560.2.q.a.81.1 2 140.67 even 12
560.2.q.a.401.1 2 20.7 even 4
700.2.i.a.401.1 2 5.3 odd 4
700.2.i.a.501.1 2 35.18 odd 12
700.2.r.c.149.1 4 1.1 even 1 trivial
700.2.r.c.149.2 4 5.4 even 2 inner
700.2.r.c.249.1 4 35.4 even 6 inner
700.2.r.c.249.2 4 7.4 even 3 inner
980.2.a.a.1.1 1 35.2 odd 12
980.2.a.i.1.1 1 35.12 even 12
980.2.i.a.361.1 2 35.17 even 12
980.2.i.a.961.1 2 35.27 even 4
1260.2.s.b.361.1 2 105.32 even 12
1260.2.s.b.541.1 2 15.2 even 4
3920.2.a.d.1.1 1 140.47 odd 12
3920.2.a.bi.1.1 1 140.107 even 12
4900.2.a.a.1.1 1 35.33 even 12
4900.2.a.v.1.1 1 35.23 odd 12
4900.2.e.b.2549.1 2 35.19 odd 6
4900.2.e.b.2549.2 2 7.5 odd 6
4900.2.e.c.2549.1 2 7.2 even 3
4900.2.e.c.2549.2 2 35.9 even 6
8820.2.a.k.1.1 1 105.47 odd 12
8820.2.a.w.1.1 1 105.2 even 12