Properties

Label 2240.2.k.b.1791.2
Level $2240$
Weight $2$
Character 2240.1791
Analytic conductor $17.886$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(1791,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.1791");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.k (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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1791.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2240.1791
Dual form 2240.2.k.b.1791.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} +1.00000i q^{5} +(1.73205 + 2.00000i) q^{7} +O(q^{10})\) \(q-1.73205 q^{3} +1.00000i q^{5} +(1.73205 + 2.00000i) q^{7} -0.267949i q^{11} +0.464102i q^{13} -1.73205i q^{15} -6.46410i q^{17} +6.00000 q^{19} +(-3.00000 - 3.46410i) q^{21} +1.46410i q^{23} -1.00000 q^{25} +5.19615 q^{27} -7.92820 q^{29} +6.00000 q^{31} +0.464102i q^{33} +(-2.00000 + 1.73205i) q^{35} +9.46410 q^{37} -0.803848i q^{39} -3.46410i q^{41} -2.00000i q^{43} -1.73205 q^{47} +(-1.00000 + 6.92820i) q^{49} +11.1962i q^{51} -2.00000 q^{53} +0.267949 q^{55} -10.3923 q^{57} -3.46410 q^{59} -9.46410i q^{61} -0.464102 q^{65} +3.46410i q^{67} -2.53590i q^{69} +7.46410i q^{71} +12.9282i q^{73} +1.73205 q^{75} +(0.535898 - 0.464102i) q^{77} +14.6603i q^{79} -9.00000 q^{81} +15.4641 q^{83} +6.46410 q^{85} +13.7321 q^{87} +2.53590i q^{89} +(-0.928203 + 0.803848i) q^{91} -10.3923 q^{93} +6.00000i q^{95} -13.3923i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{19} - 12 q^{21} - 4 q^{25} - 4 q^{29} + 24 q^{31} - 8 q^{35} + 24 q^{37} - 4 q^{49} - 8 q^{53} + 8 q^{55} + 12 q^{65} + 16 q^{77} - 36 q^{81} + 48 q^{83} + 12 q^{85} + 48 q^{87} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(3\) −1.73205 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 1.73205 + 2.00000i 0.654654 + 0.755929i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.267949i 0.0807897i −0.999184 0.0403949i \(-0.987138\pi\)
0.999184 0.0403949i \(-0.0128616\pi\)
\(12\) 0 0
\(13\) 0.464102i 0.128719i 0.997927 + 0.0643593i \(0.0205004\pi\)
−0.997927 + 0.0643593i \(0.979500\pi\)
\(14\) 0 0
\(15\) 1.73205i 0.447214i
\(16\) 0 0
\(17\) 6.46410i 1.56777i −0.620903 0.783887i \(-0.713234\pi\)
0.620903 0.783887i \(-0.286766\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) −3.00000 3.46410i −0.654654 0.755929i
\(22\) 0 0
\(23\) 1.46410i 0.305286i 0.988281 + 0.152643i \(0.0487785\pi\)
−0.988281 + 0.152643i \(0.951221\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) −7.92820 −1.47223 −0.736115 0.676856i \(-0.763342\pi\)
−0.736115 + 0.676856i \(0.763342\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0 0
\(33\) 0.464102i 0.0807897i
\(34\) 0 0
\(35\) −2.00000 + 1.73205i −0.338062 + 0.292770i
\(36\) 0 0
\(37\) 9.46410 1.55589 0.777944 0.628333i \(-0.216263\pi\)
0.777944 + 0.628333i \(0.216263\pi\)
\(38\) 0 0
\(39\) 0.803848i 0.128719i
\(40\) 0 0
\(41\) 3.46410i 0.541002i −0.962720 0.270501i \(-0.912811\pi\)
0.962720 0.270501i \(-0.0871893\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i −0.988304 0.152499i \(-0.951268\pi\)
0.988304 0.152499i \(-0.0487319\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.73205 −0.252646 −0.126323 0.991989i \(-0.540318\pi\)
−0.126323 + 0.991989i \(0.540318\pi\)
\(48\) 0 0
\(49\) −1.00000 + 6.92820i −0.142857 + 0.989743i
\(50\) 0 0
\(51\) 11.1962i 1.56777i
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 0.267949 0.0361303
\(56\) 0 0
\(57\) −10.3923 −1.37649
\(58\) 0 0
\(59\) −3.46410 −0.450988 −0.225494 0.974245i \(-0.572400\pi\)
−0.225494 + 0.974245i \(0.572400\pi\)
\(60\) 0 0
\(61\) 9.46410i 1.21175i −0.795558 0.605877i \(-0.792822\pi\)
0.795558 0.605877i \(-0.207178\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.464102 −0.0575647
\(66\) 0 0
\(67\) 3.46410i 0.423207i 0.977356 + 0.211604i \(0.0678686\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) 2.53590i 0.305286i
\(70\) 0 0
\(71\) 7.46410i 0.885826i 0.896565 + 0.442913i \(0.146055\pi\)
−0.896565 + 0.442913i \(0.853945\pi\)
\(72\) 0 0
\(73\) 12.9282i 1.51313i 0.653917 + 0.756566i \(0.273124\pi\)
−0.653917 + 0.756566i \(0.726876\pi\)
\(74\) 0 0
\(75\) 1.73205 0.200000
\(76\) 0 0
\(77\) 0.535898 0.464102i 0.0610713 0.0528893i
\(78\) 0 0
\(79\) 14.6603i 1.64941i 0.565565 + 0.824704i \(0.308658\pi\)
−0.565565 + 0.824704i \(0.691342\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 15.4641 1.69741 0.848703 0.528870i \(-0.177384\pi\)
0.848703 + 0.528870i \(0.177384\pi\)
\(84\) 0 0
\(85\) 6.46410 0.701130
\(86\) 0 0
\(87\) 13.7321 1.47223
\(88\) 0 0
\(89\) 2.53590i 0.268805i 0.990927 + 0.134402i \(0.0429115\pi\)
−0.990927 + 0.134402i \(0.957089\pi\)
\(90\) 0 0
\(91\) −0.928203 + 0.803848i −0.0973021 + 0.0842661i
\(92\) 0 0
\(93\) −10.3923 −1.07763
\(94\) 0 0
\(95\) 6.00000i 0.615587i
\(96\) 0 0
\(97\) 13.3923i 1.35978i −0.733313 0.679891i \(-0.762027\pi\)
0.733313 0.679891i \(-0.237973\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.4641i 1.53874i 0.638806 + 0.769368i \(0.279429\pi\)
−0.638806 + 0.769368i \(0.720571\pi\)
\(102\) 0 0
\(103\) 6.80385 0.670403 0.335202 0.942146i \(-0.391196\pi\)
0.335202 + 0.942146i \(0.391196\pi\)
\(104\) 0 0
\(105\) 3.46410 3.00000i 0.338062 0.292770i
\(106\) 0 0
\(107\) 2.39230i 0.231273i 0.993292 + 0.115636i \(0.0368907\pi\)
−0.993292 + 0.115636i \(0.963109\pi\)
\(108\) 0 0
\(109\) −2.07180 −0.198442 −0.0992211 0.995065i \(-0.531635\pi\)
−0.0992211 + 0.995065i \(0.531635\pi\)
\(110\) 0 0
\(111\) −16.3923 −1.55589
\(112\) 0 0
\(113\) 5.46410 0.514019 0.257010 0.966409i \(-0.417263\pi\)
0.257010 + 0.966409i \(0.417263\pi\)
\(114\) 0 0
\(115\) −1.46410 −0.136528
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.9282 11.1962i 1.18513 1.02635i
\(120\) 0 0
\(121\) 10.9282 0.993473
\(122\) 0 0
\(123\) 6.00000i 0.541002i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 15.4641i 1.37222i −0.727499 0.686109i \(-0.759317\pi\)
0.727499 0.686109i \(-0.240683\pi\)
\(128\) 0 0
\(129\) 3.46410i 0.304997i
\(130\) 0 0
\(131\) 2.53590 0.221562 0.110781 0.993845i \(-0.464665\pi\)
0.110781 + 0.993845i \(0.464665\pi\)
\(132\) 0 0
\(133\) 10.3923 + 12.0000i 0.901127 + 1.04053i
\(134\) 0 0
\(135\) 5.19615i 0.447214i
\(136\) 0 0
\(137\) 20.3923 1.74223 0.871116 0.491077i \(-0.163397\pi\)
0.871116 + 0.491077i \(0.163397\pi\)
\(138\) 0 0
\(139\) 6.92820 0.587643 0.293821 0.955860i \(-0.405073\pi\)
0.293821 + 0.955860i \(0.405073\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 0 0
\(143\) 0.124356 0.0103991
\(144\) 0 0
\(145\) 7.92820i 0.658401i
\(146\) 0 0
\(147\) 1.73205 12.0000i 0.142857 0.989743i
\(148\) 0 0
\(149\) −3.07180 −0.251651 −0.125826 0.992052i \(-0.540158\pi\)
−0.125826 + 0.992052i \(0.540158\pi\)
\(150\) 0 0
\(151\) 15.1962i 1.23665i 0.785924 + 0.618323i \(0.212188\pi\)
−0.785924 + 0.618323i \(0.787812\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.00000i 0.481932i
\(156\) 0 0
\(157\) 7.85641i 0.627009i −0.949587 0.313505i \(-0.898497\pi\)
0.949587 0.313505i \(-0.101503\pi\)
\(158\) 0 0
\(159\) 3.46410 0.274721
\(160\) 0 0
\(161\) −2.92820 + 2.53590i −0.230775 + 0.199857i
\(162\) 0 0
\(163\) 20.7846i 1.62798i 0.580881 + 0.813988i \(0.302708\pi\)
−0.580881 + 0.813988i \(0.697292\pi\)
\(164\) 0 0
\(165\) −0.464102 −0.0361303
\(166\) 0 0
\(167\) 5.19615 0.402090 0.201045 0.979582i \(-0.435566\pi\)
0.201045 + 0.979582i \(0.435566\pi\)
\(168\) 0 0
\(169\) 12.7846 0.983432
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 14.3205i 1.08877i 0.838836 + 0.544384i \(0.183237\pi\)
−0.838836 + 0.544384i \(0.816763\pi\)
\(174\) 0 0
\(175\) −1.73205 2.00000i −0.130931 0.151186i
\(176\) 0 0
\(177\) 6.00000 0.450988
\(178\) 0 0
\(179\) 6.39230i 0.477783i 0.971046 + 0.238892i \(0.0767841\pi\)
−0.971046 + 0.238892i \(0.923216\pi\)
\(180\) 0 0
\(181\) 0.928203i 0.0689928i 0.999405 + 0.0344964i \(0.0109827\pi\)
−0.999405 + 0.0344964i \(0.989017\pi\)
\(182\) 0 0
\(183\) 16.3923i 1.21175i
\(184\) 0 0
\(185\) 9.46410i 0.695815i
\(186\) 0 0
\(187\) −1.73205 −0.126660
\(188\) 0 0
\(189\) 9.00000 + 10.3923i 0.654654 + 0.755929i
\(190\) 0 0
\(191\) 7.19615i 0.520695i 0.965515 + 0.260348i \(0.0838372\pi\)
−0.965515 + 0.260348i \(0.916163\pi\)
\(192\) 0 0
\(193\) −9.46410 −0.681241 −0.340620 0.940201i \(-0.610637\pi\)
−0.340620 + 0.940201i \(0.610637\pi\)
\(194\) 0 0
\(195\) 0.803848 0.0575647
\(196\) 0 0
\(197\) 13.3205 0.949047 0.474523 0.880243i \(-0.342620\pi\)
0.474523 + 0.880243i \(0.342620\pi\)
\(198\) 0 0
\(199\) −3.46410 −0.245564 −0.122782 0.992434i \(-0.539182\pi\)
−0.122782 + 0.992434i \(0.539182\pi\)
\(200\) 0 0
\(201\) 6.00000i 0.423207i
\(202\) 0 0
\(203\) −13.7321 15.8564i −0.963801 1.11290i
\(204\) 0 0
\(205\) 3.46410 0.241943
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.60770i 0.111207i
\(210\) 0 0
\(211\) 3.19615i 0.220032i −0.993930 0.110016i \(-0.964910\pi\)
0.993930 0.110016i \(-0.0350902\pi\)
\(212\) 0 0
\(213\) 12.9282i 0.885826i
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 10.3923 + 12.0000i 0.705476 + 0.814613i
\(218\) 0 0
\(219\) 22.3923i 1.51313i
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) 13.7321 0.919566 0.459783 0.888031i \(-0.347927\pi\)
0.459783 + 0.888031i \(0.347927\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 20.6603 1.37127 0.685635 0.727946i \(-0.259525\pi\)
0.685635 + 0.727946i \(0.259525\pi\)
\(228\) 0 0
\(229\) 8.53590i 0.564068i 0.959404 + 0.282034i \(0.0910091\pi\)
−0.959404 + 0.282034i \(0.908991\pi\)
\(230\) 0 0
\(231\) −0.928203 + 0.803848i −0.0610713 + 0.0528893i
\(232\) 0 0
\(233\) −9.07180 −0.594313 −0.297157 0.954829i \(-0.596038\pi\)
−0.297157 + 0.954829i \(0.596038\pi\)
\(234\) 0 0
\(235\) 1.73205i 0.112987i
\(236\) 0 0
\(237\) 25.3923i 1.64941i
\(238\) 0 0
\(239\) 23.9808i 1.55119i −0.631233 0.775593i \(-0.717451\pi\)
0.631233 0.775593i \(-0.282549\pi\)
\(240\) 0 0
\(241\) 16.3923i 1.05592i 0.849269 + 0.527961i \(0.177043\pi\)
−0.849269 + 0.527961i \(0.822957\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.92820 1.00000i −0.442627 0.0638877i
\(246\) 0 0
\(247\) 2.78461i 0.177180i
\(248\) 0 0
\(249\) −26.7846 −1.69741
\(250\) 0 0
\(251\) −25.8564 −1.63204 −0.816021 0.578022i \(-0.803825\pi\)
−0.816021 + 0.578022i \(0.803825\pi\)
\(252\) 0 0
\(253\) 0.392305 0.0246640
\(254\) 0 0
\(255\) −11.1962 −0.701130
\(256\) 0 0
\(257\) 6.00000i 0.374270i 0.982334 + 0.187135i \(0.0599201\pi\)
−0.982334 + 0.187135i \(0.940080\pi\)
\(258\) 0 0
\(259\) 16.3923 + 18.9282i 1.01857 + 1.17614i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 11.4641i 0.706907i −0.935452 0.353453i \(-0.885007\pi\)
0.935452 0.353453i \(-0.114993\pi\)
\(264\) 0 0
\(265\) 2.00000i 0.122859i
\(266\) 0 0
\(267\) 4.39230i 0.268805i
\(268\) 0 0
\(269\) 12.0000i 0.731653i 0.930683 + 0.365826i \(0.119214\pi\)
−0.930683 + 0.365826i \(0.880786\pi\)
\(270\) 0 0
\(271\) −9.46410 −0.574903 −0.287452 0.957795i \(-0.592808\pi\)
−0.287452 + 0.957795i \(0.592808\pi\)
\(272\) 0 0
\(273\) 1.60770 1.39230i 0.0973021 0.0842661i
\(274\) 0 0
\(275\) 0.267949i 0.0161579i
\(276\) 0 0
\(277\) −16.7846 −1.00849 −0.504245 0.863561i \(-0.668229\pi\)
−0.504245 + 0.863561i \(0.668229\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.92820 −0.472957 −0.236478 0.971637i \(-0.575993\pi\)
−0.236478 + 0.971637i \(0.575993\pi\)
\(282\) 0 0
\(283\) 12.1244 0.720718 0.360359 0.932814i \(-0.382654\pi\)
0.360359 + 0.932814i \(0.382654\pi\)
\(284\) 0 0
\(285\) 10.3923i 0.615587i
\(286\) 0 0
\(287\) 6.92820 6.00000i 0.408959 0.354169i
\(288\) 0 0
\(289\) −24.7846 −1.45792
\(290\) 0 0
\(291\) 23.1962i 1.35978i
\(292\) 0 0
\(293\) 20.3205i 1.18714i −0.804784 0.593568i \(-0.797719\pi\)
0.804784 0.593568i \(-0.202281\pi\)
\(294\) 0 0
\(295\) 3.46410i 0.201688i
\(296\) 0 0
\(297\) 1.39230i 0.0807897i
\(298\) 0 0
\(299\) −0.679492 −0.0392960
\(300\) 0 0
\(301\) 4.00000 3.46410i 0.230556 0.199667i
\(302\) 0 0
\(303\) 26.7846i 1.53874i
\(304\) 0 0
\(305\) 9.46410 0.541913
\(306\) 0 0
\(307\) 1.73205 0.0988534 0.0494267 0.998778i \(-0.484261\pi\)
0.0494267 + 0.998778i \(0.484261\pi\)
\(308\) 0 0
\(309\) −11.7846 −0.670403
\(310\) 0 0
\(311\) −7.85641 −0.445496 −0.222748 0.974876i \(-0.571503\pi\)
−0.222748 + 0.974876i \(0.571503\pi\)
\(312\) 0 0
\(313\) 17.5359i 0.991188i −0.868554 0.495594i \(-0.834950\pi\)
0.868554 0.495594i \(-0.165050\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.07180 0.172529 0.0862646 0.996272i \(-0.472507\pi\)
0.0862646 + 0.996272i \(0.472507\pi\)
\(318\) 0 0
\(319\) 2.12436i 0.118941i
\(320\) 0 0
\(321\) 4.14359i 0.231273i
\(322\) 0 0
\(323\) 38.7846i 2.15803i
\(324\) 0 0
\(325\) 0.464102i 0.0257437i
\(326\) 0 0
\(327\) 3.58846 0.198442
\(328\) 0 0
\(329\) −3.00000 3.46410i −0.165395 0.190982i
\(330\) 0 0
\(331\) 26.3923i 1.45065i 0.688405 + 0.725326i \(0.258311\pi\)
−0.688405 + 0.725326i \(0.741689\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.46410 −0.189264
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) −9.46410 −0.514019
\(340\) 0 0
\(341\) 1.60770i 0.0870616i
\(342\) 0 0
\(343\) −15.5885 + 10.0000i −0.841698 + 0.539949i
\(344\) 0 0
\(345\) 2.53590 0.136528
\(346\) 0 0
\(347\) 34.2487i 1.83857i −0.393596 0.919284i \(-0.628769\pi\)
0.393596 0.919284i \(-0.371231\pi\)
\(348\) 0 0
\(349\) 5.32051i 0.284800i 0.989809 + 0.142400i \(0.0454820\pi\)
−0.989809 + 0.142400i \(0.954518\pi\)
\(350\) 0 0
\(351\) 2.41154i 0.128719i
\(352\) 0 0
\(353\) 7.39230i 0.393453i −0.980458 0.196726i \(-0.936969\pi\)
0.980458 0.196726i \(-0.0630311\pi\)
\(354\) 0 0
\(355\) −7.46410 −0.396153
\(356\) 0 0
\(357\) −22.3923 + 19.3923i −1.18513 + 1.02635i
\(358\) 0 0
\(359\) 25.3205i 1.33637i 0.743997 + 0.668183i \(0.232928\pi\)
−0.743997 + 0.668183i \(0.767072\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) −18.9282 −0.993473
\(364\) 0 0
\(365\) −12.9282 −0.676693
\(366\) 0 0
\(367\) −11.8756 −0.619904 −0.309952 0.950752i \(-0.600313\pi\)
−0.309952 + 0.950752i \(0.600313\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.46410 4.00000i −0.179847 0.207670i
\(372\) 0 0
\(373\) 3.60770 0.186799 0.0933997 0.995629i \(-0.470227\pi\)
0.0933997 + 0.995629i \(0.470227\pi\)
\(374\) 0 0
\(375\) 1.73205i 0.0894427i
\(376\) 0 0
\(377\) 3.67949i 0.189503i
\(378\) 0 0
\(379\) 5.60770i 0.288048i −0.989574 0.144024i \(-0.953996\pi\)
0.989574 0.144024i \(-0.0460042\pi\)
\(380\) 0 0
\(381\) 26.7846i 1.37222i
\(382\) 0 0
\(383\) −27.4641 −1.40335 −0.701675 0.712497i \(-0.747564\pi\)
−0.701675 + 0.712497i \(0.747564\pi\)
\(384\) 0 0
\(385\) 0.464102 + 0.535898i 0.0236528 + 0.0273119i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 20.8564 1.05746 0.528731 0.848790i \(-0.322668\pi\)
0.528731 + 0.848790i \(0.322668\pi\)
\(390\) 0 0
\(391\) 9.46410 0.478620
\(392\) 0 0
\(393\) −4.39230 −0.221562
\(394\) 0 0
\(395\) −14.6603 −0.737637
\(396\) 0 0
\(397\) 12.4641i 0.625555i −0.949826 0.312778i \(-0.898741\pi\)
0.949826 0.312778i \(-0.101259\pi\)
\(398\) 0 0
\(399\) −18.0000 20.7846i −0.901127 1.04053i
\(400\) 0 0
\(401\) −10.0718 −0.502962 −0.251481 0.967862i \(-0.580918\pi\)
−0.251481 + 0.967862i \(0.580918\pi\)
\(402\) 0 0
\(403\) 2.78461i 0.138711i
\(404\) 0 0
\(405\) 9.00000i 0.447214i
\(406\) 0 0
\(407\) 2.53590i 0.125700i
\(408\) 0 0
\(409\) 4.14359i 0.204888i 0.994739 + 0.102444i \(0.0326662\pi\)
−0.994739 + 0.102444i \(0.967334\pi\)
\(410\) 0 0
\(411\) −35.3205 −1.74223
\(412\) 0 0
\(413\) −6.00000 6.92820i −0.295241 0.340915i
\(414\) 0 0
\(415\) 15.4641i 0.759103i
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) −24.2487 −1.18463 −0.592314 0.805708i \(-0.701785\pi\)
−0.592314 + 0.805708i \(0.701785\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.46410i 0.313555i
\(426\) 0 0
\(427\) 18.9282 16.3923i 0.916000 0.793279i
\(428\) 0 0
\(429\) −0.215390 −0.0103991
\(430\) 0 0
\(431\) 13.5885i 0.654533i −0.944932 0.327266i \(-0.893873\pi\)
0.944932 0.327266i \(-0.106127\pi\)
\(432\) 0 0
\(433\) 31.8564i 1.53092i −0.643483 0.765461i \(-0.722511\pi\)
0.643483 0.765461i \(-0.277489\pi\)
\(434\) 0 0
\(435\) 13.7321i 0.658401i
\(436\) 0 0
\(437\) 8.78461i 0.420225i
\(438\) 0 0
\(439\) 39.7128 1.89539 0.947695 0.319179i \(-0.103407\pi\)
0.947695 + 0.319179i \(0.103407\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 26.0000i 1.23530i 0.786454 + 0.617649i \(0.211915\pi\)
−0.786454 + 0.617649i \(0.788085\pi\)
\(444\) 0 0
\(445\) −2.53590 −0.120213
\(446\) 0 0
\(447\) 5.32051 0.251651
\(448\) 0 0
\(449\) 11.9282 0.562927 0.281463 0.959572i \(-0.409180\pi\)
0.281463 + 0.959572i \(0.409180\pi\)
\(450\) 0 0
\(451\) −0.928203 −0.0437074
\(452\) 0 0
\(453\) 26.3205i 1.23665i
\(454\) 0 0
\(455\) −0.803848 0.928203i −0.0376850 0.0435148i
\(456\) 0 0
\(457\) 20.5359 0.960629 0.480314 0.877096i \(-0.340523\pi\)
0.480314 + 0.877096i \(0.340523\pi\)
\(458\) 0 0
\(459\) 33.5885i 1.56777i
\(460\) 0 0
\(461\) 27.7128i 1.29071i −0.763881 0.645357i \(-0.776709\pi\)
0.763881 0.645357i \(-0.223291\pi\)
\(462\) 0 0
\(463\) 16.3923i 0.761815i −0.924613 0.380908i \(-0.875612\pi\)
0.924613 0.380908i \(-0.124388\pi\)
\(464\) 0 0
\(465\) 10.3923i 0.481932i
\(466\) 0 0
\(467\) 22.5167 1.04195 0.520973 0.853573i \(-0.325569\pi\)
0.520973 + 0.853573i \(0.325569\pi\)
\(468\) 0 0
\(469\) −6.92820 + 6.00000i −0.319915 + 0.277054i
\(470\) 0 0
\(471\) 13.6077i 0.627009i
\(472\) 0 0
\(473\) −0.535898 −0.0246406
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −25.1769 −1.15036 −0.575181 0.818026i \(-0.695068\pi\)
−0.575181 + 0.818026i \(0.695068\pi\)
\(480\) 0 0
\(481\) 4.39230i 0.200272i
\(482\) 0 0
\(483\) 5.07180 4.39230i 0.230775 0.199857i
\(484\) 0 0
\(485\) 13.3923 0.608113
\(486\) 0 0
\(487\) 12.7846i 0.579326i −0.957129 0.289663i \(-0.906457\pi\)
0.957129 0.289663i \(-0.0935432\pi\)
\(488\) 0 0
\(489\) 36.0000i 1.62798i
\(490\) 0 0
\(491\) 9.87564i 0.445682i 0.974855 + 0.222841i \(0.0715330\pi\)
−0.974855 + 0.222841i \(0.928467\pi\)
\(492\) 0 0
\(493\) 51.2487i 2.30813i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −14.9282 + 12.9282i −0.669621 + 0.579909i
\(498\) 0 0
\(499\) 25.5885i 1.14550i −0.819731 0.572748i \(-0.805877\pi\)
0.819731 0.572748i \(-0.194123\pi\)
\(500\) 0 0
\(501\) −9.00000 −0.402090
\(502\) 0 0
\(503\) 15.5885 0.695055 0.347527 0.937670i \(-0.387021\pi\)
0.347527 + 0.937670i \(0.387021\pi\)
\(504\) 0 0
\(505\) −15.4641 −0.688143
\(506\) 0 0
\(507\) −22.1436 −0.983432
\(508\) 0 0
\(509\) 25.8564i 1.14607i −0.819533 0.573033i \(-0.805767\pi\)
0.819533 0.573033i \(-0.194233\pi\)
\(510\) 0 0
\(511\) −25.8564 + 22.3923i −1.14382 + 0.990577i
\(512\) 0 0
\(513\) 31.1769 1.37649
\(514\) 0 0
\(515\) 6.80385i 0.299813i
\(516\) 0 0
\(517\) 0.464102i 0.0204112i
\(518\) 0 0
\(519\) 24.8038i 1.08877i
\(520\) 0 0
\(521\) 13.6077i 0.596164i 0.954540 + 0.298082i \(0.0963469\pi\)
−0.954540 + 0.298082i \(0.903653\pi\)
\(522\) 0 0
\(523\) −24.2487 −1.06032 −0.530161 0.847897i \(-0.677869\pi\)
−0.530161 + 0.847897i \(0.677869\pi\)
\(524\) 0 0
\(525\) 3.00000 + 3.46410i 0.130931 + 0.151186i
\(526\) 0 0
\(527\) 38.7846i 1.68948i
\(528\) 0 0
\(529\) 20.8564 0.906800
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.60770 0.0696370
\(534\) 0 0
\(535\) −2.39230 −0.103428
\(536\) 0 0
\(537\) 11.0718i 0.477783i
\(538\) 0 0
\(539\) 1.85641 + 0.267949i 0.0799611 + 0.0115414i
\(540\) 0 0
\(541\) −7.78461 −0.334687 −0.167343 0.985899i \(-0.553519\pi\)
−0.167343 + 0.985899i \(0.553519\pi\)
\(542\) 0 0
\(543\) 1.60770i 0.0689928i
\(544\) 0 0
\(545\) 2.07180i 0.0887460i
\(546\) 0 0
\(547\) 21.4641i 0.917739i −0.888504 0.458869i \(-0.848255\pi\)
0.888504 0.458869i \(-0.151745\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −47.5692 −2.02652
\(552\) 0 0
\(553\) −29.3205 + 25.3923i −1.24683 + 1.07979i
\(554\) 0 0
\(555\) 16.3923i 0.695815i
\(556\) 0 0
\(557\) 21.8564 0.926086 0.463043 0.886336i \(-0.346758\pi\)
0.463043 + 0.886336i \(0.346758\pi\)
\(558\) 0 0
\(559\) 0.928203 0.0392588
\(560\) 0 0
\(561\) 3.00000 0.126660
\(562\) 0 0
\(563\) 19.1769 0.808211 0.404105 0.914712i \(-0.367583\pi\)
0.404105 + 0.914712i \(0.367583\pi\)
\(564\) 0 0
\(565\) 5.46410i 0.229876i
\(566\) 0 0
\(567\) −15.5885 18.0000i −0.654654 0.755929i
\(568\) 0 0
\(569\) 7.07180 0.296465 0.148233 0.988953i \(-0.452642\pi\)
0.148233 + 0.988953i \(0.452642\pi\)
\(570\) 0 0
\(571\) 6.67949i 0.279528i 0.990185 + 0.139764i \(0.0446344\pi\)
−0.990185 + 0.139764i \(0.955366\pi\)
\(572\) 0 0
\(573\) 12.4641i 0.520695i
\(574\) 0 0
\(575\) 1.46410i 0.0610573i
\(576\) 0 0
\(577\) 19.3923i 0.807312i 0.914911 + 0.403656i \(0.132261\pi\)
−0.914911 + 0.403656i \(0.867739\pi\)
\(578\) 0 0
\(579\) 16.3923 0.681241
\(580\) 0 0
\(581\) 26.7846 + 30.9282i 1.11121 + 1.28312i
\(582\) 0 0
\(583\) 0.535898i 0.0221946i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20.5359 −0.847607 −0.423804 0.905754i \(-0.639305\pi\)
−0.423804 + 0.905754i \(0.639305\pi\)
\(588\) 0 0
\(589\) 36.0000 1.48335
\(590\) 0 0
\(591\) −23.0718 −0.949047
\(592\) 0 0
\(593\) 30.4641i 1.25101i 0.780220 + 0.625505i \(0.215107\pi\)
−0.780220 + 0.625505i \(0.784893\pi\)
\(594\) 0 0
\(595\) 11.1962 + 12.9282i 0.458997 + 0.530005i
\(596\) 0 0
\(597\) 6.00000 0.245564
\(598\) 0 0
\(599\) 10.1244i 0.413670i −0.978376 0.206835i \(-0.933684\pi\)
0.978376 0.206835i \(-0.0663163\pi\)
\(600\) 0 0
\(601\) 14.7846i 0.603077i −0.953454 0.301538i \(-0.902500\pi\)
0.953454 0.301538i \(-0.0975001\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.9282i 0.444295i
\(606\) 0 0
\(607\) 29.1962 1.18504 0.592518 0.805557i \(-0.298134\pi\)
0.592518 + 0.805557i \(0.298134\pi\)
\(608\) 0 0
\(609\) 23.7846 + 27.4641i 0.963801 + 1.11290i
\(610\) 0 0
\(611\) 0.803848i 0.0325202i
\(612\) 0 0
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 0 0
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) −22.9282 −0.923055 −0.461527 0.887126i \(-0.652698\pi\)
−0.461527 + 0.887126i \(0.652698\pi\)
\(618\) 0 0
\(619\) 0.679492 0.0273111 0.0136555 0.999907i \(-0.495653\pi\)
0.0136555 + 0.999907i \(0.495653\pi\)
\(620\) 0 0
\(621\) 7.60770i 0.305286i
\(622\) 0 0
\(623\) −5.07180 + 4.39230i −0.203197 + 0.175974i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 2.78461i 0.111207i
\(628\) 0 0
\(629\) 61.1769i 2.43928i
\(630\) 0 0
\(631\) 38.9090i 1.54894i 0.632610 + 0.774471i \(0.281984\pi\)
−0.632610 + 0.774471i \(0.718016\pi\)
\(632\) 0 0
\(633\) 5.53590i 0.220032i
\(634\) 0 0
\(635\) 15.4641 0.613674
\(636\) 0 0
\(637\) −3.21539 0.464102i −0.127398 0.0183884i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 8.92820 0.352643 0.176321 0.984333i \(-0.443580\pi\)
0.176321 + 0.984333i \(0.443580\pi\)
\(642\) 0 0
\(643\) 7.05256 0.278126 0.139063 0.990284i \(-0.455591\pi\)
0.139063 + 0.990284i \(0.455591\pi\)
\(644\) 0 0
\(645\) −3.46410 −0.136399
\(646\) 0 0
\(647\) 10.3923 0.408564 0.204282 0.978912i \(-0.434514\pi\)
0.204282 + 0.978912i \(0.434514\pi\)
\(648\) 0 0
\(649\) 0.928203i 0.0364352i
\(650\) 0 0
\(651\) −18.0000 20.7846i −0.705476 0.814613i
\(652\) 0 0
\(653\) −17.6077 −0.689042 −0.344521 0.938779i \(-0.611959\pi\)
−0.344521 + 0.938779i \(0.611959\pi\)
\(654\) 0 0
\(655\) 2.53590i 0.0990857i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 31.1962i 1.21523i 0.794232 + 0.607615i \(0.207874\pi\)
−0.794232 + 0.607615i \(0.792126\pi\)
\(660\) 0 0
\(661\) 39.7128i 1.54465i −0.635228 0.772325i \(-0.719094\pi\)
0.635228 0.772325i \(-0.280906\pi\)
\(662\) 0 0
\(663\) −5.19615 −0.201802
\(664\) 0 0
\(665\) −12.0000 + 10.3923i −0.465340 + 0.402996i
\(666\) 0 0
\(667\) 11.6077i 0.449452i
\(668\) 0 0
\(669\) −23.7846 −0.919566
\(670\) 0 0
\(671\) −2.53590 −0.0978973
\(672\) 0 0
\(673\) −13.1769 −0.507933 −0.253966 0.967213i \(-0.581735\pi\)
−0.253966 + 0.967213i \(0.581735\pi\)
\(674\) 0 0
\(675\) −5.19615 −0.200000
\(676\) 0 0
\(677\) 25.3923i 0.975906i 0.872870 + 0.487953i \(0.162256\pi\)
−0.872870 + 0.487953i \(0.837744\pi\)
\(678\) 0 0
\(679\) 26.7846 23.1962i 1.02790 0.890187i
\(680\) 0 0
\(681\) −35.7846 −1.37127
\(682\) 0 0
\(683\) 7.32051i 0.280111i −0.990144 0.140056i \(-0.955272\pi\)
0.990144 0.140056i \(-0.0447282\pi\)
\(684\) 0 0
\(685\) 20.3923i 0.779150i
\(686\) 0 0
\(687\) 14.7846i 0.564068i
\(688\) 0 0
\(689\) 0.928203i 0.0353617i
\(690\) 0 0
\(691\) −22.6410 −0.861305 −0.430652 0.902518i \(-0.641717\pi\)
−0.430652 + 0.902518i \(0.641717\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.92820i 0.262802i
\(696\) 0 0
\(697\) −22.3923 −0.848169
\(698\) 0 0
\(699\) 15.7128 0.594313
\(700\) 0 0
\(701\) 37.7846 1.42711 0.713553 0.700602i \(-0.247085\pi\)
0.713553 + 0.700602i \(0.247085\pi\)
\(702\) 0 0
\(703\) 56.7846 2.14167
\(704\) 0 0
\(705\) 3.00000i 0.112987i
\(706\) 0 0
\(707\) −30.9282 + 26.7846i −1.16317 + 1.00734i
\(708\) 0 0
\(709\) −21.0000 −0.788672 −0.394336 0.918966i \(-0.629025\pi\)
−0.394336 + 0.918966i \(0.629025\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.78461i 0.328986i
\(714\) 0 0
\(715\) 0.124356i 0.00465064i
\(716\) 0 0
\(717\) 41.5359i 1.55119i
\(718\) 0 0
\(719\) −38.5359 −1.43715 −0.718573 0.695451i \(-0.755205\pi\)
−0.718573 + 0.695451i \(0.755205\pi\)
\(720\) 0 0
\(721\) 11.7846 + 13.6077i 0.438882 + 0.506777i
\(722\) 0 0
\(723\) 28.3923i 1.05592i
\(724\) 0 0
\(725\) 7.92820 0.294446
\(726\) 0 0
\(727\) 13.6077 0.504681 0.252341 0.967638i \(-0.418800\pi\)
0.252341 + 0.967638i \(0.418800\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −12.9282 −0.478167
\(732\) 0 0
\(733\) 11.5359i 0.426088i −0.977043 0.213044i \(-0.931662\pi\)
0.977043 0.213044i \(-0.0683378\pi\)
\(734\) 0 0
\(735\) 12.0000 + 1.73205i 0.442627 + 0.0638877i
\(736\) 0 0
\(737\) 0.928203 0.0341908
\(738\) 0 0
\(739\) 4.26795i 0.156999i 0.996914 + 0.0784995i \(0.0250129\pi\)
−0.996914 + 0.0784995i \(0.974987\pi\)
\(740\) 0 0
\(741\) 4.82309i 0.177180i
\(742\) 0 0
\(743\) 30.3923i 1.11499i 0.830182 + 0.557493i \(0.188237\pi\)
−0.830182 + 0.557493i \(0.811763\pi\)
\(744\) 0 0
\(745\) 3.07180i 0.112542i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.78461 + 4.14359i −0.174826 + 0.151404i
\(750\) 0 0
\(751\) 25.5885i 0.933736i −0.884327 0.466868i \(-0.845382\pi\)
0.884327 0.466868i \(-0.154618\pi\)
\(752\) 0 0
\(753\) 44.7846 1.63204
\(754\) 0 0
\(755\) −15.1962 −0.553045
\(756\) 0 0
\(757\) −37.8564 −1.37591 −0.687957 0.725751i \(-0.741492\pi\)
−0.687957 + 0.725751i \(0.741492\pi\)
\(758\) 0 0
\(759\) −0.679492 −0.0246640
\(760\) 0 0
\(761\) 42.2487i 1.53151i −0.643130 0.765757i \(-0.722364\pi\)
0.643130 0.765757i \(-0.277636\pi\)
\(762\) 0 0
\(763\) −3.58846 4.14359i −0.129911 0.150008i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.60770i 0.0580505i
\(768\) 0 0
\(769\) 18.0000i 0.649097i 0.945869 + 0.324548i \(0.105212\pi\)
−0.945869 + 0.324548i \(0.894788\pi\)
\(770\) 0 0
\(771\) 10.3923i 0.374270i
\(772\) 0 0
\(773\) 5.53590i 0.199112i 0.995032 + 0.0995562i \(0.0317423\pi\)
−0.995032 + 0.0995562i \(0.968258\pi\)
\(774\) 0 0
\(775\) −6.00000 −0.215526
\(776\) 0 0
\(777\) −28.3923 32.7846i −1.01857 1.17614i
\(778\) 0 0
\(779\) 20.7846i 0.744686i
\(780\) 0 0
\(781\) 2.00000 0.0715656
\(782\) 0 0
\(783\) −41.1962 −1.47223
\(784\) 0 0
\(785\) 7.85641 0.280407
\(786\) 0 0
\(787\) 32.6603 1.16421 0.582106 0.813113i \(-0.302229\pi\)
0.582106 + 0.813113i \(0.302229\pi\)
\(788\) 0 0
\(789\) 19.8564i 0.706907i
\(790\) 0 0
\(791\) 9.46410 + 10.9282i 0.336505 + 0.388562i
\(792\) 0 0
\(793\) 4.39230 0.155975
\(794\) 0 0
\(795\) 3.46410i 0.122859i
\(796\) 0 0
\(797\) 22.1769i 0.785547i 0.919635 + 0.392773i \(0.128484\pi\)
−0.919635 + 0.392773i \(0.871516\pi\)
\(798\) 0 0
\(799\) 11.1962i 0.396091i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.46410 0.122245
\(804\) 0 0
\(805\) −2.53590 2.92820i −0.0893787 0.103206i
\(806\) 0 0
\(807\) 20.7846i 0.731653i
\(808\) 0 0
\(809\) 7.92820 0.278741 0.139370 0.990240i \(-0.455492\pi\)
0.139370 + 0.990240i \(0.455492\pi\)
\(810\) 0 0
\(811\) 29.0718 1.02085 0.510424 0.859923i \(-0.329488\pi\)
0.510424 + 0.859923i \(0.329488\pi\)
\(812\) 0 0
\(813\) 16.3923 0.574903
\(814\) 0 0
\(815\) −20.7846 −0.728053
\(816\) 0 0
\(817\) 12.0000i 0.419827i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −27.7846 −0.969690 −0.484845 0.874600i \(-0.661124\pi\)
−0.484845 + 0.874600i \(0.661124\pi\)
\(822\) 0 0
\(823\) 11.0718i 0.385939i 0.981205 + 0.192969i \(0.0618118\pi\)
−0.981205 + 0.192969i \(0.938188\pi\)
\(824\) 0 0
\(825\) 0.464102i 0.0161579i
\(826\) 0 0
\(827\) 14.1436i 0.491821i −0.969293 0.245910i \(-0.920913\pi\)
0.969293 0.245910i \(-0.0790869\pi\)
\(828\) 0 0
\(829\) 0.248711i 0.00863810i −0.999991 0.00431905i \(-0.998625\pi\)
0.999991 0.00431905i \(-0.00137480\pi\)
\(830\) 0 0
\(831\) 29.0718 1.00849
\(832\) 0 0
\(833\) 44.7846 + 6.46410i 1.55169 + 0.223968i
\(834\) 0 0
\(835\) 5.19615i 0.179820i
\(836\) 0 0
\(837\) 31.1769 1.07763
\(838\) 0 0
\(839\) −4.39230 −0.151639 −0.0758196 0.997122i \(-0.524157\pi\)
−0.0758196 + 0.997122i \(0.524157\pi\)
\(840\) 0 0
\(841\) 33.8564 1.16746
\(842\) 0 0
\(843\) 13.7321 0.472957
\(844\) 0 0
\(845\) 12.7846i 0.439804i
\(846\) 0 0
\(847\) 18.9282 + 21.8564i 0.650381 + 0.750995i
\(848\) 0 0
\(849\) −21.0000 −0.720718
\(850\) 0 0
\(851\) 13.8564i 0.474991i
\(852\) 0 0
\(853\) 2.78461i 0.0953432i 0.998863 + 0.0476716i \(0.0151801\pi\)
−0.998863 + 0.0476716i \(0.984820\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 57.7128i 1.97143i 0.168416 + 0.985716i \(0.446135\pi\)
−0.168416 + 0.985716i \(0.553865\pi\)
\(858\) 0 0
\(859\) 39.7128 1.35498 0.677492 0.735530i \(-0.263067\pi\)
0.677492 + 0.735530i \(0.263067\pi\)
\(860\) 0 0
\(861\) −12.0000 + 10.3923i −0.408959 + 0.354169i
\(862\) 0 0
\(863\) 39.8564i 1.35673i −0.734726 0.678364i \(-0.762689\pi\)
0.734726 0.678364i \(-0.237311\pi\)
\(864\) 0 0
\(865\) −14.3205 −0.486912
\(866\) 0 0
\(867\) 42.9282 1.45792
\(868\) 0 0
\(869\) 3.92820 0.133255
\(870\) 0 0
\(871\) −1.60770 −0.0544747
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.00000 1.73205i 0.0676123 0.0585540i
\(876\) 0 0
\(877\) 18.3923 0.621064 0.310532 0.950563i \(-0.399493\pi\)
0.310532 + 0.950563i \(0.399493\pi\)
\(878\) 0 0
\(879\) 35.1962i 1.18714i
\(880\) 0 0
\(881\) 29.0718i 0.979454i −0.871876 0.489727i \(-0.837096\pi\)
0.871876 0.489727i \(-0.162904\pi\)
\(882\) 0 0
\(883\) 23.6077i 0.794462i 0.917719 + 0.397231i \(0.130029\pi\)
−0.917719 + 0.397231i \(0.869971\pi\)
\(884\) 0 0
\(885\) 6.00000i 0.201688i
\(886\) 0 0
\(887\) 8.53590 0.286607 0.143304 0.989679i \(-0.454227\pi\)
0.143304 + 0.989679i \(0.454227\pi\)
\(888\) 0 0
\(889\) 30.9282 26.7846i 1.03730 0.898327i
\(890\) 0 0
\(891\) 2.41154i 0.0807897i
\(892\) 0 0
\(893\) −10.3923 −0.347765
\(894\) 0 0
\(895\) −6.39230 −0.213671
\(896\) 0 0
\(897\) 1.17691 0.0392960
\(898\) 0 0
\(899\) −47.5692 −1.58652
\(900\) 0 0
\(901\) 12.9282i 0.430701i
\(902\) 0 0
\(903\) −6.92820 + 6.00000i −0.230556 + 0.199667i
\(904\) 0 0
\(905\) −0.928203 −0.0308545
\(906\) 0 0
\(907\) 32.3923i 1.07557i 0.843082 + 0.537784i \(0.180739\pi\)
−0.843082 + 0.537784i \(0.819261\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 20.2487i 0.670870i −0.942063 0.335435i \(-0.891117\pi\)
0.942063 0.335435i \(-0.108883\pi\)
\(912\) 0 0
\(913\) 4.14359i 0.137133i
\(914\) 0 0
\(915\) −16.3923 −0.541913
\(916\) 0 0
\(917\) 4.39230 + 5.07180i 0.145047 + 0.167485i
\(918\) 0 0
\(919\) 6.41154i 0.211497i −0.994393 0.105749i \(-0.966276\pi\)
0.994393 0.105749i \(-0.0337239\pi\)
\(920\) 0 0
\(921\) −3.00000 −0.0988534
\(922\) 0 0
\(923\) −3.46410 −0.114022
\(924\) 0 0
\(925\) −9.46410 −0.311178
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 54.4974i 1.78800i −0.448065 0.894001i \(-0.647887\pi\)
0.448065 0.894001i \(-0.352113\pi\)
\(930\) 0 0
\(931\) −6.00000 + 41.5692i −0.196642 + 1.36238i
\(932\) 0 0
\(933\) 13.6077 0.445496
\(934\) 0 0
\(935\) 1.73205i 0.0566441i
\(936\) 0 0
\(937\) 25.3923i 0.829530i −0.909928 0.414765i \(-0.863864\pi\)
0.909928 0.414765i \(-0.136136\pi\)
\(938\) 0 0
\(939\) 30.3731i 0.991188i
\(940\) 0 0
\(941\) 39.0333i 1.27245i −0.771504 0.636225i \(-0.780495\pi\)
0.771504 0.636225i \(-0.219505\pi\)
\(942\) 0 0
\(943\) 5.07180 0.165160
\(944\) 0 0
\(945\) −10.3923 + 9.00000i −0.338062 + 0.292770i
\(946\) 0 0
\(947\) 22.7846i 0.740400i −0.928952 0.370200i \(-0.879289\pi\)
0.928952 0.370200i \(-0.120711\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) −5.32051 −0.172529
\(952\) 0 0
\(953\) 21.3205 0.690639 0.345320 0.938485i \(-0.387771\pi\)
0.345320 + 0.938485i \(0.387771\pi\)
\(954\) 0 0
\(955\) −7.19615 −0.232862
\(956\) 0 0
\(957\) 3.67949i 0.118941i
\(958\) 0 0
\(959\) 35.3205 + 40.7846i 1.14056 + 1.31700i
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 9.46410i 0.304660i
\(966\) 0 0
\(967\) 31.1769i 1.00258i 0.865279 + 0.501291i \(0.167141\pi\)
−0.865279 + 0.501291i \(0.832859\pi\)
\(968\) 0 0
\(969\) 67.1769i 2.15803i
\(970\) 0 0
\(971\) 14.7846 0.474461 0.237230 0.971453i \(-0.423760\pi\)
0.237230 + 0.971453i \(0.423760\pi\)
\(972\) 0 0
\(973\) 12.0000 + 13.8564i 0.384702 + 0.444216i
\(974\) 0 0
\(975\) 0.803848i 0.0257437i
\(976\) 0 0
\(977\) −4.24871 −0.135928 −0.0679642 0.997688i \(-0.521650\pi\)
−0.0679642 + 0.997688i \(0.521650\pi\)
\(978\) 0 0
\(979\) 0.679492 0.0217167
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 27.5885 0.879935 0.439968 0.898014i \(-0.354990\pi\)
0.439968 + 0.898014i \(0.354990\pi\)
\(984\) 0 0
\(985\) 13.3205i 0.424427i
\(986\) 0 0
\(987\) 5.19615 + 6.00000i 0.165395 + 0.190982i
\(988\) 0 0
\(989\) 2.92820 0.0931114
\(990\) 0 0
\(991\) 47.1769i 1.49862i 0.662217 + 0.749312i \(0.269616\pi\)
−0.662217 + 0.749312i \(0.730384\pi\)
\(992\) 0 0
\(993\) 45.7128i 1.45065i
\(994\) 0 0
\(995\) 3.46410i 0.109819i
\(996\) 0 0
\(997\) 5.53590i 0.175324i −0.996150 0.0876618i \(-0.972061\pi\)
0.996150 0.0876618i \(-0.0279395\pi\)
\(998\) 0 0
\(999\) 49.1769 1.55589
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 2240.2.k.b.1791.2 4
4.3 odd 2 2240.2.k.a.1791.4 4
7.6 odd 2 2240.2.k.a.1791.3 4
8.3 odd 2 140.2.g.a.111.3 4
8.5 even 2 140.2.g.b.111.4 yes 4
24.5 odd 2 1260.2.c.b.811.1 4
24.11 even 2 1260.2.c.a.811.2 4
28.27 even 2 inner 2240.2.k.b.1791.1 4
40.3 even 4 700.2.c.b.699.1 4
40.13 odd 4 700.2.c.f.699.3 4
40.19 odd 2 700.2.g.f.251.2 4
40.27 even 4 700.2.c.e.699.4 4
40.29 even 2 700.2.g.g.251.1 4
40.37 odd 4 700.2.c.c.699.2 4
56.3 even 6 980.2.o.d.411.2 4
56.5 odd 6 980.2.o.c.31.1 4
56.11 odd 6 980.2.o.c.411.2 4
56.13 odd 2 140.2.g.a.111.4 yes 4
56.19 even 6 980.2.o.b.31.1 4
56.27 even 2 140.2.g.b.111.3 yes 4
56.37 even 6 980.2.o.d.31.1 4
56.45 odd 6 980.2.o.a.411.1 4
56.51 odd 6 980.2.o.a.31.1 4
56.53 even 6 980.2.o.b.411.1 4
168.83 odd 2 1260.2.c.b.811.2 4
168.125 even 2 1260.2.c.a.811.1 4
280.13 even 4 700.2.c.e.699.3 4
280.27 odd 4 700.2.c.f.699.4 4
280.69 odd 2 700.2.g.f.251.1 4
280.83 odd 4 700.2.c.c.699.1 4
280.139 even 2 700.2.g.g.251.2 4
280.237 even 4 700.2.c.b.699.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.g.a.111.3 4 8.3 odd 2
140.2.g.a.111.4 yes 4 56.13 odd 2
140.2.g.b.111.3 yes 4 56.27 even 2
140.2.g.b.111.4 yes 4 8.5 even 2
700.2.c.b.699.1 4 40.3 even 4
700.2.c.b.699.2 4 280.237 even 4
700.2.c.c.699.1 4 280.83 odd 4
700.2.c.c.699.2 4 40.37 odd 4
700.2.c.e.699.3 4 280.13 even 4
700.2.c.e.699.4 4 40.27 even 4
700.2.c.f.699.3 4 40.13 odd 4
700.2.c.f.699.4 4 280.27 odd 4
700.2.g.f.251.1 4 280.69 odd 2
700.2.g.f.251.2 4 40.19 odd 2
700.2.g.g.251.1 4 40.29 even 2
700.2.g.g.251.2 4 280.139 even 2
980.2.o.a.31.1 4 56.51 odd 6
980.2.o.a.411.1 4 56.45 odd 6
980.2.o.b.31.1 4 56.19 even 6
980.2.o.b.411.1 4 56.53 even 6
980.2.o.c.31.1 4 56.5 odd 6
980.2.o.c.411.2 4 56.11 odd 6
980.2.o.d.31.1 4 56.37 even 6
980.2.o.d.411.2 4 56.3 even 6
1260.2.c.a.811.1 4 168.125 even 2
1260.2.c.a.811.2 4 24.11 even 2
1260.2.c.b.811.1 4 24.5 odd 2
1260.2.c.b.811.2 4 168.83 odd 2
2240.2.k.a.1791.3 4 7.6 odd 2
2240.2.k.a.1791.4 4 4.3 odd 2
2240.2.k.b.1791.1 4 28.27 even 2 inner
2240.2.k.b.1791.2 4 1.1 even 1 trivial