Properties

Label 3330.2.e.c.739.5
Level $3330$
Weight $2$
Character 3330.739
Analytic conductor $26.590$
Analytic rank $0$
Dimension $10$
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] = [3330,2,Mod(739,3330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3330, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3330.739");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.e (of order \(2\), degree \(1\), 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: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 19x^{8} + 103x^{6} + 210x^{4} + 140x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 370)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 739.5
Root \(3.40359i\) of defining polynomial
Character \(\chi\) \(=\) 3330.739
Dual form 3330.2.e.c.739.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +(-1.28269 - 1.83159i) q^{5} -2.06225i q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +(-1.28269 - 1.83159i) q^{5} -2.06225i q^{7} -1.00000 q^{8} +(1.28269 + 1.83159i) q^{10} -3.77719 q^{11} +2.88351 q^{13} +2.06225i q^{14} +1.00000 q^{16} +5.80724 q^{17} +0.157282i q^{19} +(-1.28269 - 1.83159i) q^{20} +3.77719 q^{22} +5.41883 q^{23} +(-1.70942 + 4.69871i) q^{25} -2.88351 q^{26} -2.06225i q^{28} -4.29061i q^{29} -0.425694i q^{31} -1.00000 q^{32} -5.80724 q^{34} +(-3.77719 + 2.64522i) q^{35} +(4.80724 + 3.72698i) q^{37} -0.157282i q^{38} +(1.28269 + 1.83159i) q^{40} -0.923733 q^{41} +10.8263 q^{43} -3.77719 q^{44} -5.41883 q^{46} +0.676445i q^{47} +2.74713 q^{49} +(1.70942 - 4.69871i) q^{50} +2.88351 q^{52} +9.87810i q^{53} +(4.84496 + 6.91824i) q^{55} +2.06225i q^{56} +4.29061i q^{58} -8.47192i q^{59} -1.23904i q^{61} +0.425694i q^{62} +1.00000 q^{64} +(-3.69865 - 5.28140i) q^{65} -6.45516i q^{67} +5.80724 q^{68} +(3.77719 - 2.64522i) q^{70} +3.28329 q^{71} +0.980489i q^{73} +(-4.80724 - 3.72698i) q^{74} +0.157282i q^{76} +7.78950i q^{77} -8.04725i q^{79} +(-1.28269 - 1.83159i) q^{80} +0.923733 q^{82} -11.9496i q^{83} +(-7.44889 - 10.6365i) q^{85} -10.8263 q^{86} +3.77719 q^{88} -7.65857i q^{89} -5.94652i q^{91} +5.41883 q^{92} -0.676445i q^{94} +(0.288075 - 0.201743i) q^{95} -13.9571 q^{97} -2.74713 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} + 10 q^{4} - 3 q^{5} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{2} + 10 q^{4} - 3 q^{5} - 10 q^{8} + 3 q^{10} + 2 q^{13} + 10 q^{16} + 18 q^{17} - 3 q^{20} + 10 q^{23} + 5 q^{25} - 2 q^{26} - 10 q^{32} - 18 q^{34} + 8 q^{37} + 3 q^{40} + 4 q^{41} + 10 q^{43} - 10 q^{46} - 8 q^{49} - 5 q^{50} + 2 q^{52} - 5 q^{55} + 10 q^{64} - 2 q^{65} + 18 q^{68} + 20 q^{71} - 8 q^{74} - 3 q^{80} - 4 q^{82} - 28 q^{85} - 10 q^{86} + 10 q^{92} - 2 q^{95} - 2 q^{97} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.28269 1.83159i −0.573636 0.819110i
\(6\) 0 0
\(7\) 2.06225i 0.779457i −0.920930 0.389728i \(-0.872569\pi\)
0.920930 0.389728i \(-0.127431\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.28269 + 1.83159i 0.405622 + 0.579198i
\(11\) −3.77719 −1.13886 −0.569432 0.822038i \(-0.692837\pi\)
−0.569432 + 0.822038i \(0.692837\pi\)
\(12\) 0 0
\(13\) 2.88351 0.799742 0.399871 0.916571i \(-0.369055\pi\)
0.399871 + 0.916571i \(0.369055\pi\)
\(14\) 2.06225i 0.551159i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.80724 1.40846 0.704232 0.709970i \(-0.251292\pi\)
0.704232 + 0.709970i \(0.251292\pi\)
\(18\) 0 0
\(19\) 0.157282i 0.0360829i 0.999837 + 0.0180414i \(0.00574308\pi\)
−0.999837 + 0.0180414i \(0.994257\pi\)
\(20\) −1.28269 1.83159i −0.286818 0.409555i
\(21\) 0 0
\(22\) 3.77719 0.805299
\(23\) 5.41883 1.12990 0.564952 0.825124i \(-0.308895\pi\)
0.564952 + 0.825124i \(0.308895\pi\)
\(24\) 0 0
\(25\) −1.70942 + 4.69871i −0.341883 + 0.939742i
\(26\) −2.88351 −0.565503
\(27\) 0 0
\(28\) 2.06225i 0.389728i
\(29\) 4.29061i 0.796746i −0.917223 0.398373i \(-0.869575\pi\)
0.917223 0.398373i \(-0.130425\pi\)
\(30\) 0 0
\(31\) 0.425694i 0.0764569i −0.999269 0.0382285i \(-0.987829\pi\)
0.999269 0.0382285i \(-0.0121715\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −5.80724 −0.995934
\(35\) −3.77719 + 2.64522i −0.638461 + 0.447125i
\(36\) 0 0
\(37\) 4.80724 + 3.72698i 0.790306 + 0.612713i
\(38\) 0.157282i 0.0255144i
\(39\) 0 0
\(40\) 1.28269 + 1.83159i 0.202811 + 0.289599i
\(41\) −0.923733 −0.144263 −0.0721314 0.997395i \(-0.522980\pi\)
−0.0721314 + 0.997395i \(0.522980\pi\)
\(42\) 0 0
\(43\) 10.8263 1.65099 0.825497 0.564406i \(-0.190895\pi\)
0.825497 + 0.564406i \(0.190895\pi\)
\(44\) −3.77719 −0.569432
\(45\) 0 0
\(46\) −5.41883 −0.798963
\(47\) 0.676445i 0.0986697i 0.998782 + 0.0493348i \(0.0157101\pi\)
−0.998782 + 0.0493348i \(0.984290\pi\)
\(48\) 0 0
\(49\) 2.74713 0.392447
\(50\) 1.70942 4.69871i 0.241748 0.664498i
\(51\) 0 0
\(52\) 2.88351 0.399871
\(53\) 9.87810i 1.35686i 0.734664 + 0.678431i \(0.237340\pi\)
−0.734664 + 0.678431i \(0.762660\pi\)
\(54\) 0 0
\(55\) 4.84496 + 6.91824i 0.653294 + 0.932856i
\(56\) 2.06225i 0.275580i
\(57\) 0 0
\(58\) 4.29061i 0.563385i
\(59\) 8.47192i 1.10295i −0.834192 0.551475i \(-0.814065\pi\)
0.834192 0.551475i \(-0.185935\pi\)
\(60\) 0 0
\(61\) 1.23904i 0.158643i −0.996849 0.0793215i \(-0.974725\pi\)
0.996849 0.0793215i \(-0.0252754\pi\)
\(62\) 0.425694i 0.0540632i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.69865 5.28140i −0.458761 0.655077i
\(66\) 0 0
\(67\) 6.45516i 0.788623i −0.918977 0.394312i \(-0.870983\pi\)
0.918977 0.394312i \(-0.129017\pi\)
\(68\) 5.80724 0.704232
\(69\) 0 0
\(70\) 3.77719 2.64522i 0.451460 0.316165i
\(71\) 3.28329 0.389655 0.194828 0.980838i \(-0.437585\pi\)
0.194828 + 0.980838i \(0.437585\pi\)
\(72\) 0 0
\(73\) 0.980489i 0.114758i 0.998352 + 0.0573788i \(0.0182743\pi\)
−0.998352 + 0.0573788i \(0.981726\pi\)
\(74\) −4.80724 3.72698i −0.558831 0.433253i
\(75\) 0 0
\(76\) 0.157282i 0.0180414i
\(77\) 7.78950i 0.887696i
\(78\) 0 0
\(79\) 8.04725i 0.905387i −0.891666 0.452693i \(-0.850463\pi\)
0.891666 0.452693i \(-0.149537\pi\)
\(80\) −1.28269 1.83159i −0.143409 0.204778i
\(81\) 0 0
\(82\) 0.923733 0.102009
\(83\) 11.9496i 1.31164i −0.754916 0.655821i \(-0.772323\pi\)
0.754916 0.655821i \(-0.227677\pi\)
\(84\) 0 0
\(85\) −7.44889 10.6365i −0.807945 1.15369i
\(86\) −10.8263 −1.16743
\(87\) 0 0
\(88\) 3.77719 0.402649
\(89\) 7.65857i 0.811807i −0.913916 0.405903i \(-0.866957\pi\)
0.913916 0.405903i \(-0.133043\pi\)
\(90\) 0 0
\(91\) 5.94652i 0.623364i
\(92\) 5.41883 0.564952
\(93\) 0 0
\(94\) 0.676445i 0.0697700i
\(95\) 0.288075 0.201743i 0.0295558 0.0206984i
\(96\) 0 0
\(97\) −13.9571 −1.41712 −0.708562 0.705649i \(-0.750656\pi\)
−0.708562 + 0.705649i \(0.750656\pi\)
\(98\) −2.74713 −0.277502
\(99\) 0 0
\(100\) −1.70942 + 4.69871i −0.170942 + 0.469871i
\(101\) 8.38279 0.834119 0.417059 0.908879i \(-0.363061\pi\)
0.417059 + 0.908879i \(0.363061\pi\)
\(102\) 0 0
\(103\) −13.4727 −1.32751 −0.663753 0.747951i \(-0.731038\pi\)
−0.663753 + 0.747951i \(0.731038\pi\)
\(104\) −2.88351 −0.282751
\(105\) 0 0
\(106\) 9.87810i 0.959446i
\(107\) 1.89614i 0.183306i −0.995791 0.0916532i \(-0.970785\pi\)
0.995791 0.0916532i \(-0.0292151\pi\)
\(108\) 0 0
\(109\) 17.2044i 1.64789i 0.566672 + 0.823943i \(0.308231\pi\)
−0.566672 + 0.823943i \(0.691769\pi\)
\(110\) −4.84496 6.91824i −0.461949 0.659629i
\(111\) 0 0
\(112\) 2.06225i 0.194864i
\(113\) −1.17098 −0.110157 −0.0550783 0.998482i \(-0.517541\pi\)
−0.0550783 + 0.998482i \(0.517541\pi\)
\(114\) 0 0
\(115\) −6.95068 9.92506i −0.648154 0.925516i
\(116\) 4.29061i 0.398373i
\(117\) 0 0
\(118\) 8.47192i 0.779903i
\(119\) 11.9760i 1.09784i
\(120\) 0 0
\(121\) 3.26714 0.297012
\(122\) 1.23904i 0.112178i
\(123\) 0 0
\(124\) 0.425694i 0.0382285i
\(125\) 10.7987 2.89605i 0.965869 0.259030i
\(126\) 0 0
\(127\) 16.1128i 1.42978i −0.699236 0.714891i \(-0.746476\pi\)
0.699236 0.714891i \(-0.253524\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 3.69865 + 5.28140i 0.324393 + 0.463209i
\(131\) 9.64715i 0.842875i 0.906857 + 0.421438i \(0.138474\pi\)
−0.906857 + 0.421438i \(0.861526\pi\)
\(132\) 0 0
\(133\) 0.324354 0.0281250
\(134\) 6.45516i 0.557641i
\(135\) 0 0
\(136\) −5.80724 −0.497967
\(137\) 2.53592i 0.216658i −0.994115 0.108329i \(-0.965450\pi\)
0.994115 0.108329i \(-0.0345501\pi\)
\(138\) 0 0
\(139\) 7.56000 0.641230 0.320615 0.947210i \(-0.396110\pi\)
0.320615 + 0.947210i \(0.396110\pi\)
\(140\) −3.77719 + 2.64522i −0.319231 + 0.223562i
\(141\) 0 0
\(142\) −3.28329 −0.275528
\(143\) −10.8916 −0.910798
\(144\) 0 0
\(145\) −7.85862 + 5.50352i −0.652623 + 0.457042i
\(146\) 0.980489i 0.0811458i
\(147\) 0 0
\(148\) 4.80724 + 3.72698i 0.395153 + 0.306356i
\(149\) −6.72596 −0.551012 −0.275506 0.961299i \(-0.588845\pi\)
−0.275506 + 0.961299i \(0.588845\pi\)
\(150\) 0 0
\(151\) −15.8051 −1.28620 −0.643101 0.765781i \(-0.722353\pi\)
−0.643101 + 0.765781i \(0.722353\pi\)
\(152\) 0.157282i 0.0127572i
\(153\) 0 0
\(154\) 7.78950i 0.627696i
\(155\) −0.779696 + 0.546033i −0.0626267 + 0.0438585i
\(156\) 0 0
\(157\) 8.50755i 0.678976i −0.940610 0.339488i \(-0.889746\pi\)
0.940610 0.339488i \(-0.110254\pi\)
\(158\) 8.04725i 0.640205i
\(159\) 0 0
\(160\) 1.28269 + 1.83159i 0.101405 + 0.144800i
\(161\) 11.1750i 0.880712i
\(162\) 0 0
\(163\) 5.41321 0.423995 0.211998 0.977270i \(-0.432003\pi\)
0.211998 + 0.977270i \(0.432003\pi\)
\(164\) −0.923733 −0.0721314
\(165\) 0 0
\(166\) 11.9496i 0.927471i
\(167\) 2.08165 0.161083 0.0805415 0.996751i \(-0.474335\pi\)
0.0805415 + 0.996751i \(0.474335\pi\)
\(168\) 0 0
\(169\) −4.68537 −0.360413
\(170\) 7.44889 + 10.6365i 0.571304 + 0.815780i
\(171\) 0 0
\(172\) 10.8263 0.825497
\(173\) 12.3562i 0.939423i 0.882820 + 0.469712i \(0.155642\pi\)
−0.882820 + 0.469712i \(0.844358\pi\)
\(174\) 0 0
\(175\) 9.68991 + 3.52524i 0.732489 + 0.266483i
\(176\) −3.77719 −0.284716
\(177\) 0 0
\(178\) 7.65857i 0.574034i
\(179\) 13.4571i 1.00583i 0.864336 + 0.502915i \(0.167739\pi\)
−0.864336 + 0.502915i \(0.832261\pi\)
\(180\) 0 0
\(181\) −3.66608 −0.272498 −0.136249 0.990675i \(-0.543505\pi\)
−0.136249 + 0.990675i \(0.543505\pi\)
\(182\) 5.94652i 0.440785i
\(183\) 0 0
\(184\) −5.41883 −0.399482
\(185\) 0.660094 13.5854i 0.0485311 0.998822i
\(186\) 0 0
\(187\) −21.9350 −1.60405
\(188\) 0.676445i 0.0493348i
\(189\) 0 0
\(190\) −0.288075 + 0.201743i −0.0208991 + 0.0146360i
\(191\) 18.5540i 1.34252i −0.741221 0.671261i \(-0.765753\pi\)
0.741221 0.671261i \(-0.234247\pi\)
\(192\) 0 0
\(193\) −4.89778 −0.352550 −0.176275 0.984341i \(-0.556405\pi\)
−0.176275 + 0.984341i \(0.556405\pi\)
\(194\) 13.9571 1.00206
\(195\) 0 0
\(196\) 2.74713 0.196224
\(197\) 14.7803i 1.05305i −0.850158 0.526527i \(-0.823494\pi\)
0.850158 0.526527i \(-0.176506\pi\)
\(198\) 0 0
\(199\) 12.3298i 0.874039i −0.899452 0.437019i \(-0.856034\pi\)
0.899452 0.437019i \(-0.143966\pi\)
\(200\) 1.70942 4.69871i 0.120874 0.332249i
\(201\) 0 0
\(202\) −8.38279 −0.589811
\(203\) −8.84831 −0.621029
\(204\) 0 0
\(205\) 1.18486 + 1.69190i 0.0827544 + 0.118167i
\(206\) 13.4727 0.938689
\(207\) 0 0
\(208\) 2.88351 0.199935
\(209\) 0.594082i 0.0410935i
\(210\) 0 0
\(211\) 14.1805 0.976224 0.488112 0.872781i \(-0.337686\pi\)
0.488112 + 0.872781i \(0.337686\pi\)
\(212\) 9.87810i 0.678431i
\(213\) 0 0
\(214\) 1.89614i 0.129617i
\(215\) −13.8868 19.8293i −0.947070 1.35235i
\(216\) 0 0
\(217\) −0.877887 −0.0595949
\(218\) 17.2044i 1.16523i
\(219\) 0 0
\(220\) 4.84496 + 6.91824i 0.326647 + 0.466428i
\(221\) 16.7452 1.12641
\(222\) 0 0
\(223\) 7.29071i 0.488222i 0.969747 + 0.244111i \(0.0784962\pi\)
−0.969747 + 0.244111i \(0.921504\pi\)
\(224\) 2.06225i 0.137790i
\(225\) 0 0
\(226\) 1.17098 0.0778925
\(227\) 26.3807 1.75095 0.875473 0.483267i \(-0.160550\pi\)
0.875473 + 0.483267i \(0.160550\pi\)
\(228\) 0 0
\(229\) −19.5925 −1.29471 −0.647354 0.762190i \(-0.724124\pi\)
−0.647354 + 0.762190i \(0.724124\pi\)
\(230\) 6.95068 + 9.92506i 0.458314 + 0.654439i
\(231\) 0 0
\(232\) 4.29061i 0.281692i
\(233\) 3.69983i 0.242384i −0.992629 0.121192i \(-0.961328\pi\)
0.992629 0.121192i \(-0.0386717\pi\)
\(234\) 0 0
\(235\) 1.23897 0.867669i 0.0808213 0.0566005i
\(236\) 8.47192i 0.551475i
\(237\) 0 0
\(238\) 11.9760i 0.776287i
\(239\) 8.68397i 0.561719i −0.959749 0.280860i \(-0.909380\pi\)
0.959749 0.280860i \(-0.0906195\pi\)
\(240\) 0 0
\(241\) 27.5610i 1.77536i −0.460464 0.887678i \(-0.652317\pi\)
0.460464 0.887678i \(-0.347683\pi\)
\(242\) −3.26714 −0.210020
\(243\) 0 0
\(244\) 1.23904i 0.0793215i
\(245\) −3.52371 5.03161i −0.225122 0.321457i
\(246\) 0 0
\(247\) 0.453523i 0.0288570i
\(248\) 0.425694i 0.0270316i
\(249\) 0 0
\(250\) −10.7987 + 2.89605i −0.682973 + 0.183162i
\(251\) 20.4512i 1.29087i 0.763816 + 0.645434i \(0.223324\pi\)
−0.763816 + 0.645434i \(0.776676\pi\)
\(252\) 0 0
\(253\) −20.4679 −1.28681
\(254\) 16.1128i 1.01101i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 22.8951 1.42815 0.714077 0.700067i \(-0.246847\pi\)
0.714077 + 0.700067i \(0.246847\pi\)
\(258\) 0 0
\(259\) 7.68597 9.91373i 0.477583 0.616009i
\(260\) −3.69865 5.28140i −0.229380 0.327538i
\(261\) 0 0
\(262\) 9.64715i 0.596003i
\(263\) 22.1600i 1.36644i −0.730212 0.683221i \(-0.760579\pi\)
0.730212 0.683221i \(-0.239421\pi\)
\(264\) 0 0
\(265\) 18.0926 12.6705i 1.11142 0.778345i
\(266\) −0.324354 −0.0198874
\(267\) 0 0
\(268\) 6.45516i 0.394312i
\(269\) −23.9657 −1.46121 −0.730607 0.682798i \(-0.760763\pi\)
−0.730607 + 0.682798i \(0.760763\pi\)
\(270\) 0 0
\(271\) 9.82713 0.596956 0.298478 0.954417i \(-0.403521\pi\)
0.298478 + 0.954417i \(0.403521\pi\)
\(272\) 5.80724 0.352116
\(273\) 0 0
\(274\) 2.53592i 0.153201i
\(275\) 6.45678 17.7479i 0.389359 1.07024i
\(276\) 0 0
\(277\) 0.807608 0.0485245 0.0242622 0.999706i \(-0.492276\pi\)
0.0242622 + 0.999706i \(0.492276\pi\)
\(278\) −7.56000 −0.453418
\(279\) 0 0
\(280\) 3.77719 2.64522i 0.225730 0.158082i
\(281\) 1.03833i 0.0619414i 0.999520 + 0.0309707i \(0.00985986\pi\)
−0.999520 + 0.0309707i \(0.990140\pi\)
\(282\) 0 0
\(283\) −6.26151 −0.372208 −0.186104 0.982530i \(-0.559586\pi\)
−0.186104 + 0.982530i \(0.559586\pi\)
\(284\) 3.28329 0.194828
\(285\) 0 0
\(286\) 10.8916 0.644031
\(287\) 1.90497i 0.112447i
\(288\) 0 0
\(289\) 16.7241 0.983769
\(290\) 7.85862 5.50352i 0.461474 0.323178i
\(291\) 0 0
\(292\) 0.980489i 0.0573788i
\(293\) 4.84496i 0.283045i 0.989935 + 0.141523i \(0.0451998\pi\)
−0.989935 + 0.141523i \(0.954800\pi\)
\(294\) 0 0
\(295\) −15.5170 + 10.8668i −0.903437 + 0.632692i
\(296\) −4.80724 3.72698i −0.279415 0.216627i
\(297\) 0 0
\(298\) 6.72596 0.389624
\(299\) 15.6253 0.903632
\(300\) 0 0
\(301\) 22.3265i 1.28688i
\(302\) 15.8051 0.909483
\(303\) 0 0
\(304\) 0.157282i 0.00902072i
\(305\) −2.26941 + 1.58931i −0.129946 + 0.0910034i
\(306\) 0 0
\(307\) 2.87515i 0.164093i −0.996628 0.0820467i \(-0.973854\pi\)
0.996628 0.0820467i \(-0.0261457\pi\)
\(308\) 7.78950i 0.443848i
\(309\) 0 0
\(310\) 0.779696 0.546033i 0.0442837 0.0310126i
\(311\) 6.07102i 0.344256i 0.985075 + 0.172128i \(0.0550643\pi\)
−0.985075 + 0.172128i \(0.944936\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 8.50755i 0.480109i
\(315\) 0 0
\(316\) 8.04725i 0.452693i
\(317\) 25.5691i 1.43610i 0.695989 + 0.718052i \(0.254966\pi\)
−0.695989 + 0.718052i \(0.745034\pi\)
\(318\) 0 0
\(319\) 16.2064i 0.907386i
\(320\) −1.28269 1.83159i −0.0717045 0.102389i
\(321\) 0 0
\(322\) 11.1750i 0.622757i
\(323\) 0.913372i 0.0508214i
\(324\) 0 0
\(325\) −4.92912 + 13.5488i −0.273418 + 0.751551i
\(326\) −5.41321 −0.299810
\(327\) 0 0
\(328\) 0.923733 0.0510046
\(329\) 1.39500 0.0769087
\(330\) 0 0
\(331\) 16.8578i 0.926589i −0.886204 0.463295i \(-0.846667\pi\)
0.886204 0.463295i \(-0.153333\pi\)
\(332\) 11.9496i 0.655821i
\(333\) 0 0
\(334\) −2.08165 −0.113903
\(335\) −11.8232 + 8.27996i −0.645969 + 0.452383i
\(336\) 0 0
\(337\) 14.7105i 0.801334i −0.916224 0.400667i \(-0.868779\pi\)
0.916224 0.400667i \(-0.131221\pi\)
\(338\) 4.68537 0.254851
\(339\) 0 0
\(340\) −7.44889 10.6365i −0.403973 0.576843i
\(341\) 1.60793i 0.0870741i
\(342\) 0 0
\(343\) 20.1010i 1.08535i
\(344\) −10.8263 −0.583715
\(345\) 0 0
\(346\) 12.3562i 0.664273i
\(347\) 3.26296 0.175165 0.0875824 0.996157i \(-0.472086\pi\)
0.0875824 + 0.996157i \(0.472086\pi\)
\(348\) 0 0
\(349\) −17.2097 −0.921213 −0.460607 0.887604i \(-0.652368\pi\)
−0.460607 + 0.887604i \(0.652368\pi\)
\(350\) −9.68991 3.52524i −0.517948 0.188432i
\(351\) 0 0
\(352\) 3.77719 0.201325
\(353\) −4.88641 −0.260077 −0.130039 0.991509i \(-0.541510\pi\)
−0.130039 + 0.991509i \(0.541510\pi\)
\(354\) 0 0
\(355\) −4.21144 6.01363i −0.223520 0.319170i
\(356\) 7.65857i 0.405903i
\(357\) 0 0
\(358\) 13.4571i 0.711229i
\(359\) −3.39107 −0.178974 −0.0894870 0.995988i \(-0.528523\pi\)
−0.0894870 + 0.995988i \(0.528523\pi\)
\(360\) 0 0
\(361\) 18.9753 0.998698
\(362\) 3.66608 0.192685
\(363\) 0 0
\(364\) 5.94652i 0.311682i
\(365\) 1.79585 1.25766i 0.0939991 0.0658291i
\(366\) 0 0
\(367\) 9.25169i 0.482934i 0.970409 + 0.241467i \(0.0776286\pi\)
−0.970409 + 0.241467i \(0.922371\pi\)
\(368\) 5.41883 0.282476
\(369\) 0 0
\(370\) −0.660094 + 13.5854i −0.0343167 + 0.706274i
\(371\) 20.3711 1.05761
\(372\) 0 0
\(373\) 11.4101i 0.590792i 0.955375 + 0.295396i \(0.0954515\pi\)
−0.955375 + 0.295396i \(0.904548\pi\)
\(374\) 21.9350 1.13423
\(375\) 0 0
\(376\) 0.676445i 0.0348850i
\(377\) 12.3720i 0.637191i
\(378\) 0 0
\(379\) 36.3643 1.86791 0.933954 0.357394i \(-0.116335\pi\)
0.933954 + 0.357394i \(0.116335\pi\)
\(380\) 0.288075 0.201743i 0.0147779 0.0103492i
\(381\) 0 0
\(382\) 18.5540i 0.949307i
\(383\) −13.6526 −0.697615 −0.348807 0.937194i \(-0.613413\pi\)
−0.348807 + 0.937194i \(0.613413\pi\)
\(384\) 0 0
\(385\) 14.2671 9.99151i 0.727121 0.509214i
\(386\) 4.89778 0.249290
\(387\) 0 0
\(388\) −13.9571 −0.708562
\(389\) 2.36833i 0.120079i 0.998196 + 0.0600396i \(0.0191227\pi\)
−0.998196 + 0.0600396i \(0.980877\pi\)
\(390\) 0 0
\(391\) 31.4685 1.59143
\(392\) −2.74713 −0.138751
\(393\) 0 0
\(394\) 14.7803i 0.744621i
\(395\) −14.7392 + 10.3221i −0.741611 + 0.519362i
\(396\) 0 0
\(397\) 34.4978i 1.73140i −0.500567 0.865698i \(-0.666875\pi\)
0.500567 0.865698i \(-0.333125\pi\)
\(398\) 12.3298i 0.618039i
\(399\) 0 0
\(400\) −1.70942 + 4.69871i −0.0854708 + 0.234936i
\(401\) 29.0777i 1.45207i −0.687658 0.726035i \(-0.741361\pi\)
0.687658 0.726035i \(-0.258639\pi\)
\(402\) 0 0
\(403\) 1.22749i 0.0611458i
\(404\) 8.38279 0.417059
\(405\) 0 0
\(406\) 8.84831 0.439134
\(407\) −18.1579 14.0775i −0.900051 0.697797i
\(408\) 0 0
\(409\) 16.4980i 0.815773i 0.913033 + 0.407887i \(0.133734\pi\)
−0.913033 + 0.407887i \(0.866266\pi\)
\(410\) −1.18486 1.69190i −0.0585162 0.0835568i
\(411\) 0 0
\(412\) −13.4727 −0.663753
\(413\) −17.4712 −0.859702
\(414\) 0 0
\(415\) −21.8868 + 15.3277i −1.07438 + 0.752405i
\(416\) −2.88351 −0.141376
\(417\) 0 0
\(418\) 0.594082i 0.0290575i
\(419\) −22.9409 −1.12074 −0.560368 0.828244i \(-0.689340\pi\)
−0.560368 + 0.828244i \(0.689340\pi\)
\(420\) 0 0
\(421\) 1.72030i 0.0838422i −0.999121 0.0419211i \(-0.986652\pi\)
0.999121 0.0419211i \(-0.0133478\pi\)
\(422\) −14.1805 −0.690294
\(423\) 0 0
\(424\) 9.87810i 0.479723i
\(425\) −9.92699 + 27.2866i −0.481530 + 1.32359i
\(426\) 0 0
\(427\) −2.55521 −0.123655
\(428\) 1.89614i 0.0916532i
\(429\) 0 0
\(430\) 13.8868 + 19.8293i 0.669680 + 0.956253i
\(431\) 2.55447i 0.123045i −0.998106 0.0615223i \(-0.980404\pi\)
0.998106 0.0615223i \(-0.0195955\pi\)
\(432\) 0 0
\(433\) 30.0446i 1.44385i 0.691970 + 0.721926i \(0.256743\pi\)
−0.691970 + 0.721926i \(0.743257\pi\)
\(434\) 0.877887 0.0421399
\(435\) 0 0
\(436\) 17.2044i 0.823943i
\(437\) 0.852283i 0.0407702i
\(438\) 0 0
\(439\) 38.4265i 1.83400i −0.398889 0.916999i \(-0.630604\pi\)
0.398889 0.916999i \(-0.369396\pi\)
\(440\) −4.84496 6.91824i −0.230974 0.329814i
\(441\) 0 0
\(442\) −16.7452 −0.796490
\(443\) 1.00809i 0.0478957i 0.999713 + 0.0239479i \(0.00762357\pi\)
−0.999713 + 0.0239479i \(0.992376\pi\)
\(444\) 0 0
\(445\) −14.0273 + 9.82357i −0.664959 + 0.465682i
\(446\) 7.29071i 0.345225i
\(447\) 0 0
\(448\) 2.06225i 0.0974321i
\(449\) 32.1157i 1.51563i −0.652468 0.757817i \(-0.726266\pi\)
0.652468 0.757817i \(-0.273734\pi\)
\(450\) 0 0
\(451\) 3.48911 0.164296
\(452\) −1.17098 −0.0550783
\(453\) 0 0
\(454\) −26.3807 −1.23811
\(455\) −10.8916 + 7.62753i −0.510604 + 0.357584i
\(456\) 0 0
\(457\) −7.85884 −0.367621 −0.183810 0.982962i \(-0.558843\pi\)
−0.183810 + 0.982962i \(0.558843\pi\)
\(458\) 19.5925 0.915496
\(459\) 0 0
\(460\) −6.95068 9.92506i −0.324077 0.462758i
\(461\) 16.7952i 0.782233i −0.920341 0.391116i \(-0.872089\pi\)
0.920341 0.391116i \(-0.127911\pi\)
\(462\) 0 0
\(463\) 2.46094 0.114370 0.0571848 0.998364i \(-0.481788\pi\)
0.0571848 + 0.998364i \(0.481788\pi\)
\(464\) 4.29061i 0.199187i
\(465\) 0 0
\(466\) 3.69983i 0.171391i
\(467\) −7.57699 −0.350621 −0.175311 0.984513i \(-0.556093\pi\)
−0.175311 + 0.984513i \(0.556093\pi\)
\(468\) 0 0
\(469\) −13.3121 −0.614698
\(470\) −1.23897 + 0.867669i −0.0571493 + 0.0400226i
\(471\) 0 0
\(472\) 8.47192i 0.389952i
\(473\) −40.8929 −1.88026
\(474\) 0 0
\(475\) −0.739021 0.268860i −0.0339086 0.0123361i
\(476\) 11.9760i 0.548918i
\(477\) 0 0
\(478\) 8.68397i 0.397195i
\(479\) 12.6993i 0.580246i 0.956989 + 0.290123i \(0.0936962\pi\)
−0.956989 + 0.290123i \(0.906304\pi\)
\(480\) 0 0
\(481\) 13.8617 + 10.7468i 0.632041 + 0.490012i
\(482\) 27.5610i 1.25537i
\(483\) 0 0
\(484\) 3.26714 0.148506
\(485\) 17.9026 + 25.5635i 0.812913 + 1.16078i
\(486\) 0 0
\(487\) 10.8489 0.491611 0.245806 0.969319i \(-0.420948\pi\)
0.245806 + 0.969319i \(0.420948\pi\)
\(488\) 1.23904i 0.0560888i
\(489\) 0 0
\(490\) 3.52371 + 5.03161i 0.159185 + 0.227305i
\(491\) 17.5266 0.790965 0.395482 0.918474i \(-0.370577\pi\)
0.395482 + 0.918474i \(0.370577\pi\)
\(492\) 0 0
\(493\) 24.9166i 1.12219i
\(494\) 0.453523i 0.0204050i
\(495\) 0 0
\(496\) 0.425694i 0.0191142i
\(497\) 6.77096i 0.303719i
\(498\) 0 0
\(499\) 32.4860i 1.45427i 0.686494 + 0.727135i \(0.259149\pi\)
−0.686494 + 0.727135i \(0.740851\pi\)
\(500\) 10.7987 2.89605i 0.482935 0.129515i
\(501\) 0 0
\(502\) 20.4512i 0.912782i
\(503\) 18.1375 0.808712 0.404356 0.914602i \(-0.367496\pi\)
0.404356 + 0.914602i \(0.367496\pi\)
\(504\) 0 0
\(505\) −10.7525 15.3538i −0.478481 0.683235i
\(506\) 20.4679 0.909911
\(507\) 0 0
\(508\) 16.1128i 0.714891i
\(509\) 15.2070 0.674037 0.337018 0.941498i \(-0.390582\pi\)
0.337018 + 0.941498i \(0.390582\pi\)
\(510\) 0 0
\(511\) 2.02201 0.0894485
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −22.8951 −1.00986
\(515\) 17.2813 + 24.6765i 0.761506 + 1.08737i
\(516\) 0 0
\(517\) 2.55506i 0.112371i
\(518\) −7.68597 + 9.91373i −0.337702 + 0.435584i
\(519\) 0 0
\(520\) 3.69865 + 5.28140i 0.162196 + 0.231605i
\(521\) −21.5850 −0.945657 −0.472829 0.881154i \(-0.656767\pi\)
−0.472829 + 0.881154i \(0.656767\pi\)
\(522\) 0 0
\(523\) −6.64279 −0.290469 −0.145234 0.989397i \(-0.546394\pi\)
−0.145234 + 0.989397i \(0.546394\pi\)
\(524\) 9.64715i 0.421438i
\(525\) 0 0
\(526\) 22.1600i 0.966220i
\(527\) 2.47211i 0.107687i
\(528\) 0 0
\(529\) 6.36374 0.276684
\(530\) −18.0926 + 12.6705i −0.785892 + 0.550373i
\(531\) 0 0
\(532\) 0.324354 0.0140625
\(533\) −2.66359 −0.115373
\(534\) 0 0
\(535\) −3.47294 + 2.43215i −0.150148 + 0.105151i
\(536\) 6.45516i 0.278820i
\(537\) 0 0
\(538\) 23.9657 1.03323
\(539\) −10.3764 −0.446944
\(540\) 0 0
\(541\) 12.3214i 0.529737i 0.964285 + 0.264868i \(0.0853286\pi\)
−0.964285 + 0.264868i \(0.914671\pi\)
\(542\) −9.82713 −0.422112
\(543\) 0 0
\(544\) −5.80724 −0.248983
\(545\) 31.5114 22.0680i 1.34980 0.945287i
\(546\) 0 0
\(547\) 38.2186 1.63411 0.817054 0.576561i \(-0.195606\pi\)
0.817054 + 0.576561i \(0.195606\pi\)
\(548\) 2.53592i 0.108329i
\(549\) 0 0
\(550\) −6.45678 + 17.7479i −0.275318 + 0.756774i
\(551\) 0.674834 0.0287489
\(552\) 0 0
\(553\) −16.5954 −0.705710
\(554\) −0.807608 −0.0343120
\(555\) 0 0
\(556\) 7.56000 0.320615
\(557\) 32.9212 1.39492 0.697458 0.716626i \(-0.254315\pi\)
0.697458 + 0.716626i \(0.254315\pi\)
\(558\) 0 0
\(559\) 31.2177 1.32037
\(560\) −3.77719 + 2.64522i −0.159615 + 0.111781i
\(561\) 0 0
\(562\) 1.03833i 0.0437992i
\(563\) 5.39047 0.227181 0.113591 0.993528i \(-0.463765\pi\)
0.113591 + 0.993528i \(0.463765\pi\)
\(564\) 0 0
\(565\) 1.50200 + 2.14475i 0.0631898 + 0.0902304i
\(566\) 6.26151 0.263191
\(567\) 0 0
\(568\) −3.28329 −0.137764
\(569\) 2.18118i 0.0914399i 0.998954 + 0.0457200i \(0.0145582\pi\)
−0.998954 + 0.0457200i \(0.985442\pi\)
\(570\) 0 0
\(571\) 41.7356 1.74658 0.873291 0.487200i \(-0.161982\pi\)
0.873291 + 0.487200i \(0.161982\pi\)
\(572\) −10.8916 −0.455399
\(573\) 0 0
\(574\) 1.90497i 0.0795118i
\(575\) −9.26304 + 25.4615i −0.386295 + 1.06182i
\(576\) 0 0
\(577\) 27.9657 1.16423 0.582114 0.813107i \(-0.302226\pi\)
0.582114 + 0.813107i \(0.302226\pi\)
\(578\) −16.7241 −0.695630
\(579\) 0 0
\(580\) −7.85862 + 5.50352i −0.326312 + 0.228521i
\(581\) −24.6431 −1.02237
\(582\) 0 0
\(583\) 37.3114i 1.54528i
\(584\) 0.980489i 0.0405729i
\(585\) 0 0
\(586\) 4.84496i 0.200143i
\(587\) −19.4024 −0.800825 −0.400412 0.916335i \(-0.631133\pi\)
−0.400412 + 0.916335i \(0.631133\pi\)
\(588\) 0 0
\(589\) 0.0669539 0.00275879
\(590\) 15.5170 10.8668i 0.638827 0.447381i
\(591\) 0 0
\(592\) 4.80724 + 3.72698i 0.197576 + 0.153178i
\(593\) 12.5182i 0.514061i 0.966403 + 0.257030i \(0.0827440\pi\)
−0.966403 + 0.257030i \(0.917256\pi\)
\(594\) 0 0
\(595\) −21.9350 + 15.3615i −0.899249 + 0.629759i
\(596\) −6.72596 −0.275506
\(597\) 0 0
\(598\) −15.6253 −0.638964
\(599\) −11.7177 −0.478771 −0.239386 0.970925i \(-0.576946\pi\)
−0.239386 + 0.970925i \(0.576946\pi\)
\(600\) 0 0
\(601\) 1.37239 0.0559808 0.0279904 0.999608i \(-0.491089\pi\)
0.0279904 + 0.999608i \(0.491089\pi\)
\(602\) 22.3265i 0.909961i
\(603\) 0 0
\(604\) −15.8051 −0.643101
\(605\) −4.19072 5.98404i −0.170377 0.243286i
\(606\) 0 0
\(607\) −24.6378 −1.00002 −0.500008 0.866021i \(-0.666670\pi\)
−0.500008 + 0.866021i \(0.666670\pi\)
\(608\) 0.157282i 0.00637861i
\(609\) 0 0
\(610\) 2.26941 1.58931i 0.0918858 0.0643491i
\(611\) 1.95054i 0.0789102i
\(612\) 0 0
\(613\) 27.6733i 1.11771i 0.829264 + 0.558857i \(0.188760\pi\)
−0.829264 + 0.558857i \(0.811240\pi\)
\(614\) 2.87515i 0.116032i
\(615\) 0 0
\(616\) 7.78950i 0.313848i
\(617\) 17.9433i 0.722371i 0.932494 + 0.361185i \(0.117628\pi\)
−0.932494 + 0.361185i \(0.882372\pi\)
\(618\) 0 0
\(619\) −25.1044 −1.00903 −0.504515 0.863403i \(-0.668329\pi\)
−0.504515 + 0.863403i \(0.668329\pi\)
\(620\) −0.779696 + 0.546033i −0.0313133 + 0.0219292i
\(621\) 0 0
\(622\) 6.07102i 0.243426i
\(623\) −15.7939 −0.632768
\(624\) 0 0
\(625\) −19.1558 16.0641i −0.766232 0.642564i
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) 8.50755i 0.339488i
\(629\) 27.9168 + 21.6435i 1.11312 + 0.862983i
\(630\) 0 0
\(631\) 47.8987i 1.90682i −0.301679 0.953409i \(-0.597547\pi\)
0.301679 0.953409i \(-0.402453\pi\)
\(632\) 8.04725i 0.320102i
\(633\) 0 0
\(634\) 25.5691i 1.01548i
\(635\) −29.5120 + 20.6677i −1.17115 + 0.820175i
\(636\) 0 0
\(637\) 7.92138 0.313856
\(638\) 16.2064i 0.641619i
\(639\) 0 0
\(640\) 1.28269 + 1.83159i 0.0507027 + 0.0723998i
\(641\) 19.4541 0.768390 0.384195 0.923252i \(-0.374479\pi\)
0.384195 + 0.923252i \(0.374479\pi\)
\(642\) 0 0
\(643\) −26.0553 −1.02752 −0.513760 0.857934i \(-0.671748\pi\)
−0.513760 + 0.857934i \(0.671748\pi\)
\(644\) 11.1750i 0.440356i
\(645\) 0 0
\(646\) 0.913372i 0.0359362i
\(647\) 14.7320 0.579176 0.289588 0.957151i \(-0.406482\pi\)
0.289588 + 0.957151i \(0.406482\pi\)
\(648\) 0 0
\(649\) 32.0000i 1.25611i
\(650\) 4.92912 13.5488i 0.193336 0.531427i
\(651\) 0 0
\(652\) 5.41321 0.211998
\(653\) −30.5877 −1.19699 −0.598494 0.801127i \(-0.704234\pi\)
−0.598494 + 0.801127i \(0.704234\pi\)
\(654\) 0 0
\(655\) 17.6696 12.3743i 0.690408 0.483504i
\(656\) −0.923733 −0.0360657
\(657\) 0 0
\(658\) −1.39500 −0.0543827
\(659\) −29.3533 −1.14344 −0.571720 0.820448i \(-0.693724\pi\)
−0.571720 + 0.820448i \(0.693724\pi\)
\(660\) 0 0
\(661\) 16.7963i 0.653299i 0.945145 + 0.326650i \(0.105920\pi\)
−0.945145 + 0.326650i \(0.894080\pi\)
\(662\) 16.8578i 0.655198i
\(663\) 0 0
\(664\) 11.9496i 0.463735i
\(665\) −0.416045 0.594082i −0.0161335 0.0230375i
\(666\) 0 0
\(667\) 23.2501i 0.900247i
\(668\) 2.08165 0.0805415
\(669\) 0 0
\(670\) 11.8232 8.27996i 0.456769 0.319883i
\(671\) 4.68009i 0.180673i
\(672\) 0 0
\(673\) 21.2015i 0.817259i −0.912700 0.408629i \(-0.866007\pi\)
0.912700 0.408629i \(-0.133993\pi\)
\(674\) 14.7105i 0.566628i
\(675\) 0 0
\(676\) −4.68537 −0.180207
\(677\) 45.9734i 1.76690i −0.468526 0.883450i \(-0.655215\pi\)
0.468526 0.883450i \(-0.344785\pi\)
\(678\) 0 0
\(679\) 28.7829i 1.10459i
\(680\) 7.44889 + 10.6365i 0.285652 + 0.407890i
\(681\) 0 0
\(682\) 1.60793i 0.0615707i
\(683\) 3.06580 0.117309 0.0586547 0.998278i \(-0.481319\pi\)
0.0586547 + 0.998278i \(0.481319\pi\)
\(684\) 0 0
\(685\) −4.64476 + 3.25280i −0.177467 + 0.124283i
\(686\) 20.1010i 0.767460i
\(687\) 0 0
\(688\) 10.8263 0.412749
\(689\) 28.4836i 1.08514i
\(690\) 0 0
\(691\) 8.43426 0.320854 0.160427 0.987048i \(-0.448713\pi\)
0.160427 + 0.987048i \(0.448713\pi\)
\(692\) 12.3562i 0.469712i
\(693\) 0 0
\(694\) −3.26296 −0.123860
\(695\) −9.69713 13.8468i −0.367833 0.525238i
\(696\) 0 0
\(697\) −5.36434 −0.203189
\(698\) 17.2097 0.651396
\(699\) 0 0
\(700\) 9.68991 + 3.52524i 0.366244 + 0.133242i
\(701\) 20.3110i 0.767136i 0.923513 + 0.383568i \(0.125305\pi\)
−0.923513 + 0.383568i \(0.874695\pi\)
\(702\) 0 0
\(703\) −0.586186 + 0.756091i −0.0221084 + 0.0285165i
\(704\) −3.77719 −0.142358
\(705\) 0 0
\(706\) 4.88641 0.183902
\(707\) 17.2874i 0.650159i
\(708\) 0 0
\(709\) 23.8789i 0.896790i 0.893836 + 0.448395i \(0.148004\pi\)
−0.893836 + 0.448395i \(0.851996\pi\)
\(710\) 4.21144 + 6.01363i 0.158053 + 0.225688i
\(711\) 0 0
\(712\) 7.65857i 0.287017i
\(713\) 2.30677i 0.0863890i
\(714\) 0 0
\(715\) 13.9705 + 19.9488i 0.522466 + 0.746044i
\(716\) 13.4571i 0.502915i
\(717\) 0 0
\(718\) 3.39107 0.126554
\(719\) −14.6971 −0.548110 −0.274055 0.961714i \(-0.588365\pi\)
−0.274055 + 0.961714i \(0.588365\pi\)
\(720\) 0 0
\(721\) 27.7841i 1.03473i
\(722\) −18.9753 −0.706186
\(723\) 0 0
\(724\) −3.66608 −0.136249
\(725\) 20.1603 + 7.33444i 0.748736 + 0.272394i
\(726\) 0 0
\(727\) −5.61569 −0.208274 −0.104137 0.994563i \(-0.533208\pi\)
−0.104137 + 0.994563i \(0.533208\pi\)
\(728\) 5.94652i 0.220393i
\(729\) 0 0
\(730\) −1.79585 + 1.25766i −0.0664674 + 0.0465482i
\(731\) 62.8709 2.32536
\(732\) 0 0
\(733\) 16.7565i 0.618917i 0.950913 + 0.309459i \(0.100148\pi\)
−0.950913 + 0.309459i \(0.899852\pi\)
\(734\) 9.25169i 0.341486i
\(735\) 0 0
\(736\) −5.41883 −0.199741
\(737\) 24.3823i 0.898135i
\(738\) 0 0
\(739\) −13.2106 −0.485960 −0.242980 0.970031i \(-0.578125\pi\)
−0.242980 + 0.970031i \(0.578125\pi\)
\(740\) 0.660094 13.5854i 0.0242656 0.499411i
\(741\) 0 0
\(742\) −20.3711 −0.747847
\(743\) 23.0799i 0.846721i 0.905961 + 0.423360i \(0.139150\pi\)
−0.905961 + 0.423360i \(0.860850\pi\)
\(744\) 0 0
\(745\) 8.62731 + 12.3192i 0.316080 + 0.451340i
\(746\) 11.4101i 0.417753i
\(747\) 0 0
\(748\) −21.9350 −0.802024
\(749\) −3.91030 −0.142879
\(750\) 0 0
\(751\) 25.0437 0.913856 0.456928 0.889504i \(-0.348950\pi\)
0.456928 + 0.889504i \(0.348950\pi\)
\(752\) 0.676445i 0.0246674i
\(753\) 0 0
\(754\) 12.3720i 0.450562i
\(755\) 20.2731 + 28.9484i 0.737812 + 1.05354i
\(756\) 0 0
\(757\) 33.5882 1.22078 0.610392 0.792099i \(-0.291012\pi\)
0.610392 + 0.792099i \(0.291012\pi\)
\(758\) −36.3643 −1.32081
\(759\) 0 0
\(760\) −0.288075 + 0.201743i −0.0104496 + 0.00731800i
\(761\) 14.8529 0.538416 0.269208 0.963082i \(-0.413238\pi\)
0.269208 + 0.963082i \(0.413238\pi\)
\(762\) 0 0
\(763\) 35.4798 1.28446
\(764\) 18.5540i 0.671261i
\(765\) 0 0
\(766\) 13.6526 0.493288
\(767\) 24.4289i 0.882075i
\(768\) 0 0
\(769\) 8.80892i 0.317658i −0.987306 0.158829i \(-0.949228\pi\)
0.987306 0.158829i \(-0.0507718\pi\)
\(770\) −14.2671 + 9.99151i −0.514152 + 0.360069i
\(771\) 0 0
\(772\) −4.89778 −0.176275
\(773\) 38.8401i 1.39698i 0.715619 + 0.698491i \(0.246145\pi\)
−0.715619 + 0.698491i \(0.753855\pi\)
\(774\) 0 0
\(775\) 2.00021 + 0.727688i 0.0718498 + 0.0261393i
\(776\) 13.9571 0.501029
\(777\) 0 0
\(778\) 2.36833i 0.0849089i
\(779\) 0.145286i 0.00520542i
\(780\) 0 0
\(781\) −12.4016 −0.443764
\(782\) −31.4685 −1.12531
\(783\) 0 0
\(784\) 2.74713 0.0981118
\(785\) −15.5823 + 10.9125i −0.556156 + 0.389485i
\(786\) 0 0
\(787\) 34.2645i 1.22140i 0.791864 + 0.610698i \(0.209111\pi\)
−0.791864 + 0.610698i \(0.790889\pi\)
\(788\) 14.7803i 0.526527i
\(789\) 0 0
\(790\) 14.7392 10.3221i 0.524398 0.367245i
\(791\) 2.41485i 0.0858623i
\(792\) 0 0
\(793\) 3.57279i 0.126873i
\(794\) 34.4978i 1.22428i
\(795\) 0 0
\(796\) 12.3298i 0.437019i
\(797\) 34.0208 1.20508 0.602539 0.798089i \(-0.294156\pi\)
0.602539 + 0.798089i \(0.294156\pi\)
\(798\) 0 0
\(799\) 3.92828i 0.138973i
\(800\) 1.70942 4.69871i 0.0604370 0.166125i
\(801\) 0 0
\(802\) 29.0777i 1.02677i
\(803\) 3.70349i 0.130693i
\(804\) 0 0
\(805\) −20.4679 + 14.3340i −0.721400 + 0.505208i
\(806\) 1.22749i 0.0432366i
\(807\) 0 0
\(808\) −8.38279 −0.294905
\(809\) 23.0866i 0.811680i 0.913944 + 0.405840i \(0.133021\pi\)
−0.913944 + 0.405840i \(0.866979\pi\)
\(810\) 0 0
\(811\) −14.0477 −0.493280 −0.246640 0.969107i \(-0.579327\pi\)
−0.246640 + 0.969107i \(0.579327\pi\)
\(812\) −8.84831 −0.310515
\(813\) 0 0
\(814\) 18.1579 + 14.0775i 0.636432 + 0.493417i
\(815\) −6.94347 9.91476i −0.243219 0.347299i
\(816\) 0 0
\(817\) 1.70278i 0.0595726i
\(818\) 16.4980i 0.576839i
\(819\) 0 0
\(820\) 1.18486 + 1.69190i 0.0413772 + 0.0590836i
\(821\) 5.87069 0.204888 0.102444 0.994739i \(-0.467334\pi\)
0.102444 + 0.994739i \(0.467334\pi\)
\(822\) 0 0
\(823\) 36.6536i 1.27766i 0.769346 + 0.638832i \(0.220582\pi\)
−0.769346 + 0.638832i \(0.779418\pi\)
\(824\) 13.4727 0.469345
\(825\) 0 0
\(826\) 17.4712 0.607901
\(827\) −31.2291 −1.08594 −0.542971 0.839751i \(-0.682701\pi\)
−0.542971 + 0.839751i \(0.682701\pi\)
\(828\) 0 0
\(829\) 5.76907i 0.200368i 0.994969 + 0.100184i \(0.0319431\pi\)
−0.994969 + 0.100184i \(0.968057\pi\)
\(830\) 21.8868 15.3277i 0.759701 0.532031i
\(831\) 0 0
\(832\) 2.88351 0.0999677
\(833\) 15.9533 0.552747
\(834\) 0 0
\(835\) −2.67011 3.81272i −0.0924030 0.131945i
\(836\) 0.594082i 0.0205468i
\(837\) 0 0
\(838\) 22.9409 0.792480
\(839\) 19.1631 0.661582 0.330791 0.943704i \(-0.392684\pi\)
0.330791 + 0.943704i \(0.392684\pi\)
\(840\) 0 0
\(841\) 10.5907 0.365195
\(842\) 1.72030i 0.0592854i
\(843\) 0 0
\(844\) 14.1805 0.488112
\(845\) 6.00987 + 8.58166i 0.206746 + 0.295218i
\(846\) 0 0
\(847\) 6.73765i 0.231508i
\(848\) 9.87810i 0.339215i
\(849\) 0 0
\(850\) 9.92699 27.2866i 0.340493 0.935921i
\(851\) 26.0496 + 20.1959i 0.892970 + 0.692307i
\(852\) 0 0
\(853\) −30.1823 −1.03342 −0.516711 0.856160i \(-0.672844\pi\)
−0.516711 + 0.856160i \(0.672844\pi\)
\(854\) 2.55521 0.0874376
\(855\) 0 0
\(856\) 1.89614i 0.0648086i
\(857\) −13.4228 −0.458513 −0.229257 0.973366i \(-0.573630\pi\)
−0.229257 + 0.973366i \(0.573630\pi\)
\(858\) 0 0
\(859\) 25.1243i 0.857230i 0.903487 + 0.428615i \(0.140998\pi\)
−0.903487 + 0.428615i \(0.859002\pi\)
\(860\) −13.8868 19.8293i −0.473535 0.676173i
\(861\) 0 0
\(862\) 2.55447i 0.0870057i
\(863\) 6.80983i 0.231809i 0.993260 + 0.115905i \(0.0369767\pi\)
−0.993260 + 0.115905i \(0.963023\pi\)
\(864\) 0 0
\(865\) 22.6314 15.8491i 0.769491 0.538887i
\(866\) 30.0446i 1.02096i
\(867\) 0 0
\(868\) −0.877887 −0.0297974
\(869\) 30.3960i 1.03111i
\(870\) 0 0
\(871\) 18.6135i 0.630695i
\(872\) 17.2044i 0.582616i
\(873\) 0 0
\(874\) 0.852283i 0.0288289i
\(875\) −5.97237 22.2697i −0.201903 0.752853i
\(876\) 0 0
\(877\) 29.4096i 0.993090i 0.868011 + 0.496545i \(0.165398\pi\)
−0.868011 + 0.496545i \(0.834602\pi\)
\(878\) 38.4265i 1.29683i
\(879\) 0 0
\(880\) 4.84496 + 6.91824i 0.163323 + 0.233214i
\(881\) −50.5602 −1.70342 −0.851708 0.524016i \(-0.824433\pi\)
−0.851708 + 0.524016i \(0.824433\pi\)
\(882\) 0 0
\(883\) −40.4904 −1.36261 −0.681305 0.732000i \(-0.738587\pi\)
−0.681305 + 0.732000i \(0.738587\pi\)
\(884\) 16.7452 0.563203
\(885\) 0 0
\(886\) 1.00809i 0.0338674i
\(887\) 8.64379i 0.290230i 0.989415 + 0.145115i \(0.0463552\pi\)
−0.989415 + 0.145115i \(0.953645\pi\)
\(888\) 0 0
\(889\) −33.2287 −1.11445
\(890\) 14.0273 9.82357i 0.470197 0.329287i
\(891\) 0 0
\(892\) 7.29071i 0.244111i
\(893\) −0.106392 −0.00356028
\(894\) 0 0
\(895\) 24.6478 17.2613i 0.823885 0.576980i
\(896\) 2.06225i 0.0688949i
\(897\) 0 0
\(898\) 32.1157i 1.07171i
\(899\) −1.82649 −0.0609168
\(900\) 0 0
\(901\) 57.3645i 1.91109i
\(902\) −3.48911 −0.116175
\(903\) 0 0
\(904\) 1.17098 0.0389462
\(905\) 4.70244 + 6.71474i 0.156314 + 0.223206i
\(906\) 0 0
\(907\) 33.0245 1.09656 0.548281 0.836294i \(-0.315283\pi\)
0.548281 + 0.836294i \(0.315283\pi\)
\(908\) 26.3807 0.875473
\(909\) 0 0
\(910\) 10.8916 7.62753i 0.361052 0.252850i
\(911\) 0.940972i 0.0311758i −0.999879 0.0155879i \(-0.995038\pi\)
0.999879 0.0155879i \(-0.00496198\pi\)
\(912\) 0 0
\(913\) 45.1360i 1.49378i
\(914\) 7.85884 0.259947
\(915\) 0 0
\(916\) −19.5925 −0.647354
\(917\) 19.8948 0.656985
\(918\) 0 0
\(919\) 54.5548i 1.79960i −0.436306 0.899798i \(-0.643713\pi\)
0.436306 0.899798i \(-0.356287\pi\)
\(920\) 6.95068 + 9.92506i 0.229157 + 0.327219i
\(921\) 0 0
\(922\) 16.7952i 0.553122i
\(923\) 9.46740 0.311623
\(924\) 0 0
\(925\) −25.7296 + 16.2169i −0.845984 + 0.533208i
\(926\) −2.46094 −0.0808716
\(927\) 0 0
\(928\) 4.29061i 0.140846i
\(929\) 22.7416 0.746128 0.373064 0.927806i \(-0.378307\pi\)
0.373064 + 0.927806i \(0.378307\pi\)
\(930\) 0 0
\(931\) 0.432073i 0.0141606i
\(932\) 3.69983i 0.121192i
\(933\) 0 0
\(934\) 7.57699 0.247927
\(935\) 28.1358 + 40.1759i 0.920140 + 1.31389i
\(936\) 0 0
\(937\) 43.2122i 1.41168i 0.708370 + 0.705841i \(0.249431\pi\)
−0.708370 + 0.705841i \(0.750569\pi\)
\(938\) 13.3121 0.434657
\(939\) 0 0
\(940\) 1.23897 0.867669i 0.0404107 0.0283002i
\(941\) −32.0847 −1.04593 −0.522966 0.852354i \(-0.675175\pi\)
−0.522966 + 0.852354i \(0.675175\pi\)
\(942\) 0 0
\(943\) −5.00555 −0.163003
\(944\) 8.47192i 0.275737i
\(945\) 0 0
\(946\) 40.8929 1.32954
\(947\) −8.26212 −0.268483 −0.134241 0.990949i \(-0.542860\pi\)
−0.134241 + 0.990949i \(0.542860\pi\)
\(948\) 0 0
\(949\) 2.82725i 0.0917764i
\(950\) 0.739021 + 0.268860i 0.0239770 + 0.00872296i
\(951\) 0 0
\(952\) 11.9760i 0.388144i
\(953\) 51.1883i 1.65815i 0.559136 + 0.829076i \(0.311133\pi\)
−0.559136 + 0.829076i \(0.688867\pi\)
\(954\) 0 0
\(955\) −33.9833 + 23.7991i −1.09967 + 0.770120i
\(956\) 8.68397i 0.280860i
\(957\) 0 0
\(958\) 12.6993i 0.410296i
\(959\) −5.22970 −0.168876
\(960\) 0 0
\(961\) 30.8188 0.994154
\(962\) −13.8617 10.7468i −0.446920 0.346491i
\(963\) 0 0
\(964\) 27.5610i 0.887678i
\(965\) 6.28233 + 8.97070i 0.202235 + 0.288777i
\(966\) 0 0
\(967\) −1.84215 −0.0592395 −0.0296197 0.999561i \(-0.509430\pi\)
−0.0296197 + 0.999561i \(0.509430\pi\)
\(968\) −3.26714 −0.105010
\(969\) 0 0
\(970\) −17.9026 25.5635i −0.574817 0.820796i
\(971\) −52.6384 −1.68925 −0.844624 0.535361i \(-0.820176\pi\)
−0.844624 + 0.535361i \(0.820176\pi\)
\(972\) 0 0
\(973\) 15.5906i 0.499811i
\(974\) −10.8489 −0.347621
\(975\) 0 0
\(976\) 1.23904i 0.0396608i
\(977\) −47.7629 −1.52807 −0.764035 0.645174i \(-0.776785\pi\)
−0.764035 + 0.645174i \(0.776785\pi\)
\(978\) 0 0
\(979\) 28.9278i 0.924538i
\(980\) −3.52371 5.03161i −0.112561 0.160729i
\(981\) 0 0
\(982\) −17.5266 −0.559297
\(983\) 34.6640i 1.10561i 0.833310 + 0.552805i \(0.186443\pi\)
−0.833310 + 0.552805i \(0.813557\pi\)
\(984\) 0 0
\(985\) −27.0714 + 18.9586i −0.862567 + 0.604070i
\(986\) 24.9166i 0.793507i
\(987\) 0 0
\(988\) 0.453523i 0.0144285i
\(989\) 58.6659 1.86547
\(990\) 0 0
\(991\) 45.9061i 1.45826i −0.684378 0.729128i \(-0.739926\pi\)
0.684378 0.729128i \(-0.260074\pi\)
\(992\) 0.425694i 0.0135158i
\(993\) 0 0
\(994\) 6.77096i 0.214762i
\(995\) −22.5831 + 15.8153i −0.715934 + 0.501380i
\(996\) 0 0
\(997\) 24.7232 0.782993 0.391496 0.920180i \(-0.371958\pi\)
0.391496 + 0.920180i \(0.371958\pi\)
\(998\) 32.4860i 1.02832i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3330.2.e.c.739.5 10
3.2 odd 2 370.2.c.b.369.1 yes 10
5.4 even 2 3330.2.e.d.739.5 10
15.2 even 4 1850.2.d.i.1701.11 20
15.8 even 4 1850.2.d.i.1701.10 20
15.14 odd 2 370.2.c.a.369.10 yes 10
37.36 even 2 3330.2.e.d.739.6 10
111.110 odd 2 370.2.c.a.369.1 10
185.184 even 2 inner 3330.2.e.c.739.6 10
555.332 even 4 1850.2.d.i.1701.1 20
555.443 even 4 1850.2.d.i.1701.20 20
555.554 odd 2 370.2.c.b.369.10 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.c.a.369.1 10 111.110 odd 2
370.2.c.a.369.10 yes 10 15.14 odd 2
370.2.c.b.369.1 yes 10 3.2 odd 2
370.2.c.b.369.10 yes 10 555.554 odd 2
1850.2.d.i.1701.1 20 555.332 even 4
1850.2.d.i.1701.10 20 15.8 even 4
1850.2.d.i.1701.11 20 15.2 even 4
1850.2.d.i.1701.20 20 555.443 even 4
3330.2.e.c.739.5 10 1.1 even 1 trivial
3330.2.e.c.739.6 10 185.184 even 2 inner
3330.2.e.d.739.5 10 5.4 even 2
3330.2.e.d.739.6 10 37.36 even 2