Properties

Label 2520.2.t.j.1009.3
Level $2520$
Weight $2$
Character 2520.1009
Analytic conductor $20.122$
Analytic rank $0$
Dimension $6$
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] = [2520,2,Mod(1009,2520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2520.1009");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2520.t (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: \(20.1223013094\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.3
Root \(1.45161 + 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 2520.1009
Dual form 2520.2.t.j.1009.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.311108 - 2.21432i) q^{5} -1.00000i q^{7} +O(q^{10})\) \(q+(0.311108 - 2.21432i) q^{5} -1.00000i q^{7} +3.80642 q^{11} +0.622216i q^{13} -4.42864i q^{17} -0.622216 q^{19} -2.62222i q^{23} +(-4.80642 - 1.37778i) q^{25} +9.61285 q^{29} -0.622216 q^{31} +(-2.21432 - 0.311108i) q^{35} +1.24443i q^{37} -4.62222 q^{41} -4.85728i q^{43} +11.6128i q^{47} -1.00000 q^{49} -13.4795i q^{53} +(1.18421 - 8.42864i) q^{55} +11.6128 q^{59} -8.10171 q^{61} +(1.37778 + 0.193576i) q^{65} -2.56199 q^{71} +10.9906i q^{73} -3.80642i q^{77} +6.75557 q^{79} -11.6128i q^{83} +(-9.80642 - 1.37778i) q^{85} -8.23506 q^{89} +0.622216 q^{91} +(-0.193576 + 1.37778i) q^{95} -4.23506i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{5} - 4 q^{11} - 4 q^{19} - 2 q^{25} + 4 q^{29} - 4 q^{31} - 28 q^{41} - 6 q^{49} - 20 q^{55} + 16 q^{59} + 4 q^{61} + 8 q^{65} + 12 q^{71} + 40 q^{79} - 32 q^{85} + 4 q^{89} + 4 q^{91} - 28 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.311108 2.21432i 0.139132 0.990274i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.80642 1.14768 0.573840 0.818967i \(-0.305453\pi\)
0.573840 + 0.818967i \(0.305453\pi\)
\(12\) 0 0
\(13\) 0.622216i 0.172572i 0.996270 + 0.0862858i \(0.0274998\pi\)
−0.996270 + 0.0862858i \(0.972500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.42864i 1.07410i −0.843550 0.537051i \(-0.819538\pi\)
0.843550 0.537051i \(-0.180462\pi\)
\(18\) 0 0
\(19\) −0.622216 −0.142746 −0.0713730 0.997450i \(-0.522738\pi\)
−0.0713730 + 0.997450i \(0.522738\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.62222i 0.546770i −0.961905 0.273385i \(-0.911857\pi\)
0.961905 0.273385i \(-0.0881433\pi\)
\(24\) 0 0
\(25\) −4.80642 1.37778i −0.961285 0.275557i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.61285 1.78506 0.892531 0.450987i \(-0.148928\pi\)
0.892531 + 0.450987i \(0.148928\pi\)
\(30\) 0 0
\(31\) −0.622216 −0.111753 −0.0558766 0.998438i \(-0.517795\pi\)
−0.0558766 + 0.998438i \(0.517795\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.21432 0.311108i −0.374288 0.0525868i
\(36\) 0 0
\(37\) 1.24443i 0.204583i 0.994754 + 0.102292i \(0.0326175\pi\)
−0.994754 + 0.102292i \(0.967383\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.62222 −0.721869 −0.360934 0.932591i \(-0.617542\pi\)
−0.360934 + 0.932591i \(0.617542\pi\)
\(42\) 0 0
\(43\) 4.85728i 0.740728i −0.928887 0.370364i \(-0.879233\pi\)
0.928887 0.370364i \(-0.120767\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.6128i 1.69391i 0.531666 + 0.846954i \(0.321566\pi\)
−0.531666 + 0.846954i \(0.678434\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.4795i 1.85155i −0.378074 0.925775i \(-0.623413\pi\)
0.378074 0.925775i \(-0.376587\pi\)
\(54\) 0 0
\(55\) 1.18421 8.42864i 0.159679 1.13652i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.6128 1.51186 0.755932 0.654650i \(-0.227184\pi\)
0.755932 + 0.654650i \(0.227184\pi\)
\(60\) 0 0
\(61\) −8.10171 −1.03732 −0.518659 0.854981i \(-0.673569\pi\)
−0.518659 + 0.854981i \(0.673569\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.37778 + 0.193576i 0.170893 + 0.0240102i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.56199 −0.304053 −0.152026 0.988376i \(-0.548580\pi\)
−0.152026 + 0.988376i \(0.548580\pi\)
\(72\) 0 0
\(73\) 10.9906i 1.28636i 0.765717 + 0.643178i \(0.222385\pi\)
−0.765717 + 0.643178i \(0.777615\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.80642i 0.433782i
\(78\) 0 0
\(79\) 6.75557 0.760061 0.380030 0.924974i \(-0.375914\pi\)
0.380030 + 0.924974i \(0.375914\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 11.6128i 1.27468i −0.770585 0.637338i \(-0.780036\pi\)
0.770585 0.637338i \(-0.219964\pi\)
\(84\) 0 0
\(85\) −9.80642 1.37778i −1.06366 0.149442i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.23506 −0.872915 −0.436457 0.899725i \(-0.643767\pi\)
−0.436457 + 0.899725i \(0.643767\pi\)
\(90\) 0 0
\(91\) 0.622216 0.0652259
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.193576 + 1.37778i −0.0198605 + 0.141358i
\(96\) 0 0
\(97\) 4.23506i 0.430006i −0.976613 0.215003i \(-0.931024\pi\)
0.976613 0.215003i \(-0.0689761\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −18.7239 −1.86310 −0.931550 0.363613i \(-0.881543\pi\)
−0.931550 + 0.363613i \(0.881543\pi\)
\(102\) 0 0
\(103\) 0.857279i 0.0844702i −0.999108 0.0422351i \(-0.986552\pi\)
0.999108 0.0422351i \(-0.0134479\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.0923i 1.07234i −0.844111 0.536169i \(-0.819871\pi\)
0.844111 0.536169i \(-0.180129\pi\)
\(108\) 0 0
\(109\) −5.61285 −0.537613 −0.268807 0.963194i \(-0.586629\pi\)
−0.268807 + 0.963194i \(0.586629\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.2351i 1.52727i −0.645650 0.763633i \(-0.723414\pi\)
0.645650 0.763633i \(-0.276586\pi\)
\(114\) 0 0
\(115\) −5.80642 0.815792i −0.541452 0.0760730i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4.42864 −0.405973
\(120\) 0 0
\(121\) 3.48886 0.317169
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4.54617 + 10.2143i −0.406622 + 0.913597i
\(126\) 0 0
\(127\) 15.3461i 1.36175i −0.732400 0.680875i \(-0.761600\pi\)
0.732400 0.680875i \(-0.238400\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 0.622216i 0.0539529i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.0923i 1.46030i −0.683288 0.730149i \(-0.739451\pi\)
0.683288 0.730149i \(-0.260549\pi\)
\(138\) 0 0
\(139\) 13.4795 1.14332 0.571658 0.820492i \(-0.306300\pi\)
0.571658 + 0.820492i \(0.306300\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.36842i 0.198057i
\(144\) 0 0
\(145\) 2.99063 21.2859i 0.248358 1.76770i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.34614 −0.765666 −0.382833 0.923818i \(-0.625051\pi\)
−0.382833 + 0.923818i \(0.625051\pi\)
\(150\) 0 0
\(151\) −7.14272 −0.581266 −0.290633 0.956835i \(-0.593866\pi\)
−0.290633 + 0.956835i \(0.593866\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.193576 + 1.37778i −0.0155484 + 0.110666i
\(156\) 0 0
\(157\) 6.99063i 0.557913i 0.960304 + 0.278957i \(0.0899886\pi\)
−0.960304 + 0.278957i \(0.910011\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.62222 −0.206660
\(162\) 0 0
\(163\) 15.6128i 1.22289i 0.791286 + 0.611446i \(0.209412\pi\)
−0.791286 + 0.611446i \(0.790588\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.51114i 0.116935i −0.998289 0.0584677i \(-0.981379\pi\)
0.998289 0.0584677i \(-0.0186215\pi\)
\(168\) 0 0
\(169\) 12.6128 0.970219
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.53035i 0.496493i 0.968697 + 0.248247i \(0.0798543\pi\)
−0.968697 + 0.248247i \(0.920146\pi\)
\(174\) 0 0
\(175\) −1.37778 + 4.80642i −0.104151 + 0.363331i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.29529 −0.470532 −0.235266 0.971931i \(-0.575596\pi\)
−0.235266 + 0.971931i \(0.575596\pi\)
\(180\) 0 0
\(181\) −6.85728 −0.509698 −0.254849 0.966981i \(-0.582026\pi\)
−0.254849 + 0.966981i \(0.582026\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.75557 + 0.387152i 0.202593 + 0.0284640i
\(186\) 0 0
\(187\) 16.8573i 1.23273i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.5620 0.764239 0.382119 0.924113i \(-0.375194\pi\)
0.382119 + 0.924113i \(0.375194\pi\)
\(192\) 0 0
\(193\) 5.24443i 0.377502i 0.982025 + 0.188751i \(0.0604440\pi\)
−0.982025 + 0.188751i \(0.939556\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.7462i 1.26436i 0.774820 + 0.632182i \(0.217841\pi\)
−0.774820 + 0.632182i \(0.782159\pi\)
\(198\) 0 0
\(199\) 20.2351 1.43443 0.717213 0.696854i \(-0.245418\pi\)
0.717213 + 0.696854i \(0.245418\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.61285i 0.674690i
\(204\) 0 0
\(205\) −1.43801 + 10.2351i −0.100435 + 0.714848i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.36842 −0.163827
\(210\) 0 0
\(211\) 21.3274 1.46824 0.734120 0.679020i \(-0.237595\pi\)
0.734120 + 0.679020i \(0.237595\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −10.7556 1.51114i −0.733524 0.103059i
\(216\) 0 0
\(217\) 0.622216i 0.0422387i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.75557 0.185360
\(222\) 0 0
\(223\) 9.71456i 0.650535i −0.945622 0.325267i \(-0.894546\pi\)
0.945622 0.325267i \(-0.105454\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.3461i 0.753070i −0.926402 0.376535i \(-0.877115\pi\)
0.926402 0.376535i \(-0.122885\pi\)
\(228\) 0 0
\(229\) −1.34614 −0.0889555 −0.0444778 0.999010i \(-0.514162\pi\)
−0.0444778 + 0.999010i \(0.514162\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.3778i 1.00743i −0.863869 0.503716i \(-0.831966\pi\)
0.863869 0.503716i \(-0.168034\pi\)
\(234\) 0 0
\(235\) 25.7146 + 3.61285i 1.67743 + 0.235676i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.53972 0.487704 0.243852 0.969812i \(-0.421589\pi\)
0.243852 + 0.969812i \(0.421589\pi\)
\(240\) 0 0
\(241\) 23.9813 1.54477 0.772385 0.635155i \(-0.219064\pi\)
0.772385 + 0.635155i \(0.219064\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.311108 + 2.21432i −0.0198759 + 0.141468i
\(246\) 0 0
\(247\) 0.387152i 0.0246339i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −14.1017 −0.890092 −0.445046 0.895508i \(-0.646813\pi\)
−0.445046 + 0.895508i \(0.646813\pi\)
\(252\) 0 0
\(253\) 9.98126i 0.627517i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 17.0192i 1.06163i 0.847488 + 0.530815i \(0.178114\pi\)
−0.847488 + 0.530815i \(0.821886\pi\)
\(258\) 0 0
\(259\) 1.24443 0.0773252
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.6035i 0.777164i −0.921414 0.388582i \(-0.872965\pi\)
0.921414 0.388582i \(-0.127035\pi\)
\(264\) 0 0
\(265\) −29.8479 4.19358i −1.83354 0.257609i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.76494 0.229552 0.114776 0.993391i \(-0.463385\pi\)
0.114776 + 0.993391i \(0.463385\pi\)
\(270\) 0 0
\(271\) −17.8666 −1.08532 −0.542661 0.839952i \(-0.682583\pi\)
−0.542661 + 0.839952i \(0.682583\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −18.2953 5.24443i −1.10325 0.316251i
\(276\) 0 0
\(277\) 1.24443i 0.0747706i 0.999301 + 0.0373853i \(0.0119029\pi\)
−0.999301 + 0.0373853i \(0.988097\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.95899 0.534448 0.267224 0.963634i \(-0.413894\pi\)
0.267224 + 0.963634i \(0.413894\pi\)
\(282\) 0 0
\(283\) 30.5718i 1.81731i 0.417551 + 0.908654i \(0.362889\pi\)
−0.417551 + 0.908654i \(0.637111\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.62222i 0.272841i
\(288\) 0 0
\(289\) −2.61285 −0.153697
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.67307i 0.331424i 0.986174 + 0.165712i \(0.0529923\pi\)
−0.986174 + 0.165712i \(0.947008\pi\)
\(294\) 0 0
\(295\) 3.61285 25.7146i 0.210348 1.49716i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.63158 0.0943569
\(300\) 0 0
\(301\) −4.85728 −0.279969
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.52051 + 17.9398i −0.144324 + 1.02723i
\(306\) 0 0
\(307\) 4.85728i 0.277220i 0.990347 + 0.138610i \(0.0442634\pi\)
−0.990347 + 0.138610i \(0.955737\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 34.5718 1.96039 0.980195 0.198037i \(-0.0634567\pi\)
0.980195 + 0.198037i \(0.0634567\pi\)
\(312\) 0 0
\(313\) 6.33677i 0.358176i −0.983833 0.179088i \(-0.942685\pi\)
0.983833 0.179088i \(-0.0573146\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.9684i 0.896872i 0.893815 + 0.448436i \(0.148019\pi\)
−0.893815 + 0.448436i \(0.851981\pi\)
\(318\) 0 0
\(319\) 36.5906 2.04868
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.75557i 0.153324i
\(324\) 0 0
\(325\) 0.857279 2.99063i 0.0475533 0.165890i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.6128 0.640237
\(330\) 0 0
\(331\) −27.6128 −1.51774 −0.758870 0.651243i \(-0.774248\pi\)
−0.758870 + 0.651243i \(0.774248\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.0000i 0.871576i −0.900049 0.435788i \(-0.856470\pi\)
0.900049 0.435788i \(-0.143530\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.36842 −0.128257
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.8666i 0.637035i −0.947917 0.318517i \(-0.896815\pi\)
0.947917 0.318517i \(-0.103185\pi\)
\(348\) 0 0
\(349\) −21.8163 −1.16780 −0.583899 0.811826i \(-0.698474\pi\)
−0.583899 + 0.811826i \(0.698474\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.79706i 0.148872i 0.997226 + 0.0744361i \(0.0237157\pi\)
−0.997226 + 0.0744361i \(0.976284\pi\)
\(354\) 0 0
\(355\) −0.797056 + 5.67307i −0.0423033 + 0.301095i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −13.0509 −0.688798 −0.344399 0.938823i \(-0.611917\pi\)
−0.344399 + 0.938823i \(0.611917\pi\)
\(360\) 0 0
\(361\) −18.6128 −0.979624
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 24.3368 + 3.41927i 1.27384 + 0.178973i
\(366\) 0 0
\(367\) 10.4889i 0.547514i 0.961799 + 0.273757i \(0.0882664\pi\)
−0.961799 + 0.273757i \(0.911734\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −13.4795 −0.699820
\(372\) 0 0
\(373\) 30.1847i 1.56290i 0.623966 + 0.781452i \(0.285521\pi\)
−0.623966 + 0.781452i \(0.714479\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.98126i 0.308051i
\(378\) 0 0
\(379\) 12.8573 0.660434 0.330217 0.943905i \(-0.392878\pi\)
0.330217 + 0.943905i \(0.392878\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 24.4701i 1.25037i 0.780479 + 0.625183i \(0.214975\pi\)
−0.780479 + 0.625183i \(0.785025\pi\)
\(384\) 0 0
\(385\) −8.42864 1.18421i −0.429563 0.0603528i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.61285 −0.0817746 −0.0408873 0.999164i \(-0.513018\pi\)
−0.0408873 + 0.999164i \(0.513018\pi\)
\(390\) 0 0
\(391\) −11.6128 −0.587287
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.10171 14.9590i 0.105749 0.752668i
\(396\) 0 0
\(397\) 22.2163i 1.11501i −0.830175 0.557503i \(-0.811760\pi\)
0.830175 0.557503i \(-0.188240\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −19.9813 −0.997817 −0.498908 0.866655i \(-0.666266\pi\)
−0.498908 + 0.866655i \(0.666266\pi\)
\(402\) 0 0
\(403\) 0.387152i 0.0192854i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.73683i 0.234796i
\(408\) 0 0
\(409\) 5.73329 0.283493 0.141747 0.989903i \(-0.454728\pi\)
0.141747 + 0.989903i \(0.454728\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.6128i 0.571431i
\(414\) 0 0
\(415\) −25.7146 3.61285i −1.26228 0.177348i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 26.3684 1.28818 0.644091 0.764949i \(-0.277236\pi\)
0.644091 + 0.764949i \(0.277236\pi\)
\(420\) 0 0
\(421\) −19.3274 −0.941960 −0.470980 0.882144i \(-0.656100\pi\)
−0.470980 + 0.882144i \(0.656100\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.10171 + 21.2859i −0.295976 + 1.03252i
\(426\) 0 0
\(427\) 8.10171i 0.392069i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 26.9491 1.29809 0.649047 0.760748i \(-0.275168\pi\)
0.649047 + 0.760748i \(0.275168\pi\)
\(432\) 0 0
\(433\) 2.13335i 0.102522i −0.998685 0.0512612i \(-0.983676\pi\)
0.998685 0.0512612i \(-0.0163241\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.63158i 0.0780492i
\(438\) 0 0
\(439\) 10.5205 0.502116 0.251058 0.967972i \(-0.419221\pi\)
0.251058 + 0.967972i \(0.419221\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.88892i 0.327303i 0.986518 + 0.163651i \(0.0523272\pi\)
−0.986518 + 0.163651i \(0.947673\pi\)
\(444\) 0 0
\(445\) −2.56199 + 18.2351i −0.121450 + 0.864425i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 39.9180 1.88385 0.941923 0.335829i \(-0.109016\pi\)
0.941923 + 0.335829i \(0.109016\pi\)
\(450\) 0 0
\(451\) −17.5941 −0.828474
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.193576 1.37778i 0.00907499 0.0645915i
\(456\) 0 0
\(457\) 8.47013i 0.396216i 0.980180 + 0.198108i \(0.0634796\pi\)
−0.980180 + 0.198108i \(0.936520\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −40.1146 −1.86832 −0.934162 0.356849i \(-0.883851\pi\)
−0.934162 + 0.356849i \(0.883851\pi\)
\(462\) 0 0
\(463\) 33.5941i 1.56125i 0.624999 + 0.780625i \(0.285099\pi\)
−0.624999 + 0.780625i \(0.714901\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.3461i 0.525037i 0.964927 + 0.262518i \(0.0845530\pi\)
−0.964927 + 0.262518i \(0.915447\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 18.4889i 0.850119i
\(474\) 0 0
\(475\) 2.99063 + 0.857279i 0.137220 + 0.0393347i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 36.2864 1.65797 0.828984 0.559273i \(-0.188919\pi\)
0.828984 + 0.559273i \(0.188919\pi\)
\(480\) 0 0
\(481\) −0.774305 −0.0353053
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −9.37778 1.31756i −0.425823 0.0598274i
\(486\) 0 0
\(487\) 38.8385i 1.75994i 0.475027 + 0.879971i \(0.342438\pi\)
−0.475027 + 0.879971i \(0.657562\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 28.7467 1.29732 0.648660 0.761079i \(-0.275330\pi\)
0.648660 + 0.761079i \(0.275330\pi\)
\(492\) 0 0
\(493\) 42.5718i 1.91734i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.56199i 0.114921i
\(498\) 0 0
\(499\) 5.63158 0.252104 0.126052 0.992024i \(-0.459769\pi\)
0.126052 + 0.992024i \(0.459769\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 34.9590i 1.55874i 0.626561 + 0.779372i \(0.284462\pi\)
−0.626561 + 0.779372i \(0.715538\pi\)
\(504\) 0 0
\(505\) −5.82516 + 41.4608i −0.259216 + 1.84498i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −10.9906 −0.487151 −0.243576 0.969882i \(-0.578320\pi\)
−0.243576 + 0.969882i \(0.578320\pi\)
\(510\) 0 0
\(511\) 10.9906 0.486197
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.89829 0.266706i −0.0836486 0.0117525i
\(516\) 0 0
\(517\) 44.2034i 1.94406i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.90766 −0.302630 −0.151315 0.988486i \(-0.548351\pi\)
−0.151315 + 0.988486i \(0.548351\pi\)
\(522\) 0 0
\(523\) 37.7146i 1.64914i 0.565758 + 0.824571i \(0.308584\pi\)
−0.565758 + 0.824571i \(0.691416\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.75557i 0.120034i
\(528\) 0 0
\(529\) 16.1240 0.701043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.87601i 0.124574i
\(534\) 0 0
\(535\) −24.5620 3.45091i −1.06191 0.149196i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.80642 −0.163954
\(540\) 0 0
\(541\) −3.12399 −0.134311 −0.0671553 0.997743i \(-0.521392\pi\)
−0.0671553 + 0.997743i \(0.521392\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.74620 + 12.4286i −0.0747990 + 0.532384i
\(546\) 0 0
\(547\) 5.51114i 0.235639i 0.993035 + 0.117820i \(0.0375905\pi\)
−0.993035 + 0.117820i \(0.962410\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5.98126 −0.254810
\(552\) 0 0
\(553\) 6.75557i 0.287276i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 36.7052i 1.55525i 0.628729 + 0.777624i \(0.283575\pi\)
−0.628729 + 0.777624i \(0.716425\pi\)
\(558\) 0 0
\(559\) 3.02227 0.127829
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 27.4924i 1.15867i 0.815091 + 0.579333i \(0.196687\pi\)
−0.815091 + 0.579333i \(0.803313\pi\)
\(564\) 0 0
\(565\) −35.9496 5.05086i −1.51241 0.212491i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23.2444 0.974457 0.487229 0.873274i \(-0.338008\pi\)
0.487229 + 0.873274i \(0.338008\pi\)
\(570\) 0 0
\(571\) −25.5111 −1.06761 −0.533804 0.845608i \(-0.679238\pi\)
−0.533804 + 0.845608i \(0.679238\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.61285 + 12.6035i −0.150666 + 0.525601i
\(576\) 0 0
\(577\) 26.0701i 1.08531i −0.839955 0.542656i \(-0.817419\pi\)
0.839955 0.542656i \(-0.182581\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −11.6128 −0.481782
\(582\) 0 0
\(583\) 51.3087i 2.12499i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.2667i 0.506301i −0.967427 0.253151i \(-0.918533\pi\)
0.967427 0.253151i \(-0.0814668\pi\)
\(588\) 0 0
\(589\) 0.387152 0.0159523
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 14.9175i 0.612588i 0.951937 + 0.306294i \(0.0990891\pi\)
−0.951937 + 0.306294i \(0.900911\pi\)
\(594\) 0 0
\(595\) −1.37778 + 9.80642i −0.0564837 + 0.402024i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −26.5620 −1.08529 −0.542647 0.839961i \(-0.682578\pi\)
−0.542647 + 0.839961i \(0.682578\pi\)
\(600\) 0 0
\(601\) 39.7146 1.61999 0.809995 0.586436i \(-0.199470\pi\)
0.809995 + 0.586436i \(0.199470\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.08541 7.72546i 0.0441283 0.314084i
\(606\) 0 0
\(607\) 6.28544i 0.255118i −0.991831 0.127559i \(-0.959286\pi\)
0.991831 0.127559i \(-0.0407143\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7.22570 −0.292320
\(612\) 0 0
\(613\) 45.7146i 1.84639i −0.384328 0.923197i \(-0.625567\pi\)
0.384328 0.923197i \(-0.374433\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.88892i 0.196821i 0.995146 + 0.0984103i \(0.0313758\pi\)
−0.995146 + 0.0984103i \(0.968624\pi\)
\(618\) 0 0
\(619\) −12.2351 −0.491769 −0.245884 0.969299i \(-0.579078\pi\)
−0.245884 + 0.969299i \(0.579078\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.23506i 0.329931i
\(624\) 0 0
\(625\) 21.2034 + 13.2444i 0.848137 + 0.529777i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.51114 0.219743
\(630\) 0 0
\(631\) 1.24443 0.0495400 0.0247700 0.999693i \(-0.492115\pi\)
0.0247700 + 0.999693i \(0.492115\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −33.9813 4.77430i −1.34851 0.189462i
\(636\) 0 0
\(637\) 0.622216i 0.0246531i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 48.1847 1.90318 0.951590 0.307369i \(-0.0994487\pi\)
0.951590 + 0.307369i \(0.0994487\pi\)
\(642\) 0 0
\(643\) 4.85728i 0.191552i 0.995403 + 0.0957762i \(0.0305333\pi\)
−0.995403 + 0.0957762i \(0.969467\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.203420i 0.00799728i 0.999992 + 0.00399864i \(0.00127281\pi\)
−0.999992 + 0.00399864i \(0.998727\pi\)
\(648\) 0 0
\(649\) 44.2034 1.73514
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27.3145i 1.06890i 0.845200 + 0.534449i \(0.179481\pi\)
−0.845200 + 0.534449i \(0.820519\pi\)
\(654\) 0 0
\(655\) 1.24443 8.85728i 0.0486240 0.346083i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 33.3176 1.29787 0.648934 0.760845i \(-0.275215\pi\)
0.648934 + 0.760845i \(0.275215\pi\)
\(660\) 0 0
\(661\) 14.5906 0.567508 0.283754 0.958897i \(-0.408420\pi\)
0.283754 + 0.958897i \(0.408420\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.37778 + 0.193576i 0.0534282 + 0.00750656i
\(666\) 0 0
\(667\) 25.2070i 0.976017i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −30.8385 −1.19051
\(672\) 0 0
\(673\) 4.53341i 0.174750i 0.996175 + 0.0873751i \(0.0278479\pi\)
−0.996175 + 0.0873751i \(0.972152\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 27.2672i 1.04796i −0.851730 0.523981i \(-0.824446\pi\)
0.851730 0.523981i \(-0.175554\pi\)
\(678\) 0 0
\(679\) −4.23506 −0.162527
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 29.5812i 1.13189i 0.824442 + 0.565947i \(0.191489\pi\)
−0.824442 + 0.565947i \(0.808511\pi\)
\(684\) 0 0
\(685\) −37.8479 5.31756i −1.44609 0.203174i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 8.38715 0.319525
\(690\) 0 0
\(691\) 2.99063 0.113769 0.0568845 0.998381i \(-0.481883\pi\)
0.0568845 + 0.998381i \(0.481883\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.19358 29.8479i 0.159071 1.13220i
\(696\) 0 0
\(697\) 20.4701i 0.775361i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 41.0420 1.55013 0.775067 0.631879i \(-0.217716\pi\)
0.775067 + 0.631879i \(0.217716\pi\)
\(702\) 0 0
\(703\) 0.774305i 0.0292035i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.7239i 0.704186i
\(708\) 0 0
\(709\) 41.4291 1.55590 0.777952 0.628324i \(-0.216259\pi\)
0.777952 + 0.628324i \(0.216259\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.63158i 0.0611033i
\(714\) 0 0
\(715\) 5.24443 + 0.736833i 0.196131 + 0.0275560i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −18.9590 −0.707051 −0.353525 0.935425i \(-0.615017\pi\)
−0.353525 + 0.935425i \(0.615017\pi\)
\(720\) 0 0
\(721\) −0.857279 −0.0319267
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −46.2034 13.2444i −1.71595 0.491886i
\(726\) 0 0
\(727\) 41.7975i 1.55018i −0.631848 0.775092i \(-0.717703\pi\)
0.631848 0.775092i \(-0.282297\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −21.5111 −0.795618
\(732\) 0 0
\(733\) 15.3145i 0.565654i 0.959171 + 0.282827i \(0.0912722\pi\)
−0.959171 + 0.282827i \(0.908728\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 32.7368 1.20424 0.602122 0.798404i \(-0.294322\pi\)
0.602122 + 0.798404i \(0.294322\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 37.3778i 1.37126i 0.727951 + 0.685629i \(0.240473\pi\)
−0.727951 + 0.685629i \(0.759527\pi\)
\(744\) 0 0
\(745\) −2.90766 + 20.6953i −0.106528 + 0.758219i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −11.0923 −0.405305
\(750\) 0 0
\(751\) 20.3497 0.742570 0.371285 0.928519i \(-0.378917\pi\)
0.371285 + 0.928519i \(0.378917\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.22216 + 15.8163i −0.0808725 + 0.575613i
\(756\) 0 0
\(757\) 6.95899i 0.252929i 0.991971 + 0.126464i \(0.0403629\pi\)
−0.991971 + 0.126464i \(0.959637\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −48.6419 −1.76327 −0.881634 0.471934i \(-0.843556\pi\)
−0.881634 + 0.471934i \(0.843556\pi\)
\(762\) 0 0
\(763\) 5.61285i 0.203199i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.22570i 0.260905i
\(768\) 0 0
\(769\) 24.6923 0.890426 0.445213 0.895425i \(-0.353128\pi\)
0.445213 + 0.895425i \(0.353128\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 36.0415i 1.29632i −0.761503 0.648161i \(-0.775538\pi\)
0.761503 0.648161i \(-0.224462\pi\)
\(774\) 0 0
\(775\) 2.99063 + 0.857279i 0.107427 + 0.0307944i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.87601 0.103044
\(780\) 0 0
\(781\) −9.75203 −0.348955
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 15.4795 + 2.17484i 0.552487 + 0.0776234i
\(786\) 0 0
\(787\) 32.2034i 1.14793i 0.818881 + 0.573964i \(0.194595\pi\)
−0.818881 + 0.573964i \(0.805405\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −16.2351 −0.577252
\(792\) 0 0
\(793\) 5.04101i 0.179012i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17.5526i 0.621746i −0.950451 0.310873i \(-0.899379\pi\)
0.950451 0.310873i \(-0.100621\pi\)
\(798\) 0 0
\(799\) 51.4291 1.81943
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 41.8350i 1.47633i
\(804\) 0 0
\(805\) −0.815792 + 5.80642i −0.0287529 + 0.204650i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 46.1659 1.62311 0.811554 0.584277i \(-0.198622\pi\)
0.811554 + 0.584277i \(0.198622\pi\)
\(810\) 0 0
\(811\) 18.5205 0.650343 0.325171 0.945655i \(-0.394578\pi\)
0.325171 + 0.945655i \(0.394578\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 34.5718 + 4.85728i 1.21100 + 0.170143i
\(816\) 0 0
\(817\) 3.02227i 0.105736i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −43.0607 −1.50283 −0.751414 0.659831i \(-0.770628\pi\)
−0.751414 + 0.659831i \(0.770628\pi\)
\(822\) 0 0
\(823\) 14.5718i 0.507942i 0.967212 + 0.253971i \(0.0817368\pi\)
−0.967212 + 0.253971i \(0.918263\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 55.8292i 1.94137i −0.240354 0.970685i \(-0.577264\pi\)
0.240354 0.970685i \(-0.422736\pi\)
\(828\) 0 0
\(829\) 37.3087 1.29578 0.647892 0.761732i \(-0.275651\pi\)
0.647892 + 0.761732i \(0.275651\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.42864i 0.153443i
\(834\) 0 0
\(835\) −3.34614 0.470127i −0.115798 0.0162694i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −51.0420 −1.76216 −0.881082 0.472963i \(-0.843184\pi\)
−0.881082 + 0.472963i \(0.843184\pi\)
\(840\) 0 0
\(841\) 63.4068 2.18644
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.92396 27.9289i 0.134988 0.960783i
\(846\) 0 0
\(847\) 3.48886i 0.119879i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.26317 0.111860
\(852\) 0 0
\(853\) 26.4197i 0.904595i 0.891867 + 0.452297i \(0.149395\pi\)
−0.891867 + 0.452297i \(0.850605\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.161933i 0.00553154i −0.999996 0.00276577i \(-0.999120\pi\)
0.999996 0.00276577i \(-0.000880372\pi\)
\(858\) 0 0
\(859\) −51.3403 −1.75171 −0.875854 0.482575i \(-0.839701\pi\)
−0.875854 + 0.482575i \(0.839701\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 12.8702i 0.438106i −0.975713 0.219053i \(-0.929703\pi\)
0.975713 0.219053i \(-0.0702968\pi\)
\(864\) 0 0
\(865\) 14.4603 + 2.03164i 0.491664 + 0.0690779i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 25.7146 0.872307
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 10.2143 + 4.54617i 0.345307 + 0.153689i
\(876\) 0 0
\(877\) 43.2257i 1.45963i 0.683646 + 0.729814i \(0.260393\pi\)
−0.683646 + 0.729814i \(0.739607\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 16.5018 0.555959 0.277979 0.960587i \(-0.410335\pi\)
0.277979 + 0.960587i \(0.410335\pi\)
\(882\) 0 0
\(883\) 46.1847i 1.55424i −0.629353 0.777119i \(-0.716680\pi\)
0.629353 0.777119i \(-0.283320\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 37.3274i 1.25333i −0.779288 0.626666i \(-0.784419\pi\)
0.779288 0.626666i \(-0.215581\pi\)
\(888\) 0 0
\(889\) −15.3461 −0.514693
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7.22570i 0.241799i
\(894\) 0 0
\(895\) −1.95851 + 13.9398i −0.0654659 + 0.465955i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −5.98126 −0.199486
\(900\) 0 0
\(901\) −59.6958 −1.98876
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.13335 + 15.1842i −0.0709151 + 0.504740i
\(906\) 0 0
\(907\) 24.6735i 0.819272i −0.912249 0.409636i \(-0.865656\pi\)
0.912249 0.409636i \(-0.134344\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −29.4380 −0.975325 −0.487662 0.873032i \(-0.662150\pi\)
−0.487662 + 0.873032i \(0.662150\pi\)
\(912\) 0 0
\(913\) 44.2034i 1.46292i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.00000i 0.132092i
\(918\) 0 0
\(919\) −40.0197 −1.32013 −0.660064 0.751209i \(-0.729471\pi\)
−0.660064 + 0.751209i \(0.729471\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.59411i 0.0524708i
\(924\) 0 0
\(925\) 1.71456 5.98126i 0.0563743 0.196663i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −22.4572 −0.736797 −0.368399 0.929668i \(-0.620094\pi\)
−0.368399 + 0.929668i \(0.620094\pi\)
\(930\) 0 0
\(931\) 0.622216 0.0203923
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −37.3274 5.24443i −1.22074 0.171511i
\(936\) 0 0
\(937\) 33.8292i 1.10515i 0.833463 + 0.552575i \(0.186355\pi\)
−0.833463 + 0.552575i \(0.813645\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −47.8479 −1.55980 −0.779899 0.625906i \(-0.784729\pi\)
−0.779899 + 0.625906i \(0.784729\pi\)
\(942\) 0 0
\(943\) 12.1204i 0.394696i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 44.8069i 1.45603i 0.685562 + 0.728014i \(0.259557\pi\)
−0.685562 + 0.728014i \(0.740443\pi\)
\(948\) 0 0
\(949\) −6.83854 −0.221989
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 15.7017i 0.508626i −0.967122 0.254313i \(-0.918151\pi\)
0.967122 0.254313i \(-0.0818494\pi\)
\(954\) 0 0
\(955\) 3.28592 23.3876i 0.106330 0.756806i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −17.0923 −0.551941
\(960\) 0 0
\(961\) −30.6128 −0.987511
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 11.6128 + 1.63158i 0.373831 + 0.0525225i
\(966\) 0 0
\(967\) 46.9590i 1.51010i 0.655668 + 0.755050i \(0.272387\pi\)
−0.655668 + 0.755050i \(0.727613\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 18.7556 0.601895 0.300947 0.953641i \(-0.402697\pi\)
0.300947 + 0.953641i \(0.402697\pi\)
\(972\) 0 0
\(973\) 13.4795i 0.432133i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 37.8292i 1.21026i 0.796126 + 0.605131i \(0.206879\pi\)
−0.796126 + 0.605131i \(0.793121\pi\)
\(978\) 0 0
\(979\) −31.3461 −1.00183
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 17.6316i 0.562360i 0.959655 + 0.281180i \(0.0907258\pi\)
−0.959655 + 0.281180i \(0.909274\pi\)
\(984\) 0 0
\(985\) 39.2958 + 5.52098i 1.25207 + 0.175913i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12.7368 −0.405008
\(990\) 0 0
\(991\) −37.1240 −1.17928 −0.589641 0.807665i \(-0.700731\pi\)
−0.589641 + 0.807665i \(0.700731\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 6.29529 44.8069i 0.199574 1.42047i
\(996\) 0 0
\(997\) 42.4197i 1.34345i −0.740802 0.671723i \(-0.765554\pi\)
0.740802 0.671723i \(-0.234446\pi\)
\(998\) 0 0
\(999\) 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 2520.2.t.j.1009.3 6
3.2 odd 2 840.2.t.e.169.5 yes 6
4.3 odd 2 5040.2.t.ba.1009.3 6
5.4 even 2 inner 2520.2.t.j.1009.4 6
12.11 even 2 1680.2.t.i.1009.2 6
15.2 even 4 4200.2.a.bq.1.1 3
15.8 even 4 4200.2.a.bo.1.1 3
15.14 odd 2 840.2.t.e.169.2 6
20.19 odd 2 5040.2.t.ba.1009.4 6
60.23 odd 4 8400.2.a.dk.1.3 3
60.47 odd 4 8400.2.a.dh.1.3 3
60.59 even 2 1680.2.t.i.1009.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.e.169.2 6 15.14 odd 2
840.2.t.e.169.5 yes 6 3.2 odd 2
1680.2.t.i.1009.2 6 12.11 even 2
1680.2.t.i.1009.5 6 60.59 even 2
2520.2.t.j.1009.3 6 1.1 even 1 trivial
2520.2.t.j.1009.4 6 5.4 even 2 inner
4200.2.a.bo.1.1 3 15.8 even 4
4200.2.a.bq.1.1 3 15.2 even 4
5040.2.t.ba.1009.3 6 4.3 odd 2
5040.2.t.ba.1009.4 6 20.19 odd 2
8400.2.a.dh.1.3 3 60.47 odd 4
8400.2.a.dk.1.3 3 60.23 odd 4