Properties

Label 7500.2.d.g.1249.10
Level $7500$
Weight $2$
Character 7500.1249
Analytic conductor $59.888$
Analytic rank $0$
Dimension $24$
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] = [7500,2,Mod(1249,7500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7500, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7500.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7500.d (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: \(59.8878015160\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.10
Character \(\chi\) \(=\) 7500.1249
Dual form 7500.2.d.g.1249.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +3.80992i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +3.80992i q^{7} -1.00000 q^{9} +0.190733 q^{11} -1.67943i q^{13} -4.60324i q^{17} +2.64126 q^{19} +3.80992 q^{21} +6.35567i q^{23} +1.00000i q^{27} -2.52008 q^{29} +3.74937 q^{31} -0.190733i q^{33} +11.8626i q^{37} -1.67943 q^{39} -7.18953 q^{41} -9.22619i q^{43} +4.54848i q^{47} -7.51545 q^{49} -4.60324 q^{51} +9.43006i q^{53} -2.64126i q^{57} -7.14057 q^{59} +9.53012 q^{61} -3.80992i q^{63} -6.05168i q^{67} +6.35567 q^{69} +13.2983 q^{71} -5.21152i q^{73} +0.726676i q^{77} +3.11168 q^{79} +1.00000 q^{81} +4.67837i q^{83} +2.52008i q^{87} -13.9423 q^{89} +6.39850 q^{91} -3.74937i q^{93} -6.69507i q^{97} -0.190733 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 24 q^{9} + 4 q^{11} - 20 q^{19} + 16 q^{21} - 16 q^{29} - 4 q^{31} + 20 q^{41} - 56 q^{49} + 16 q^{51} + 4 q^{59} + 68 q^{61} - 36 q^{69} - 12 q^{79} + 24 q^{81} - 20 q^{89} + 40 q^{91} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times\).

\(n\) \(2501\) \(3751\) \(6877\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.80992i 1.44001i 0.693968 + 0.720006i \(0.255861\pi\)
−0.693968 + 0.720006i \(0.744139\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0.190733 0.0575081 0.0287541 0.999587i \(-0.490846\pi\)
0.0287541 + 0.999587i \(0.490846\pi\)
\(12\) 0 0
\(13\) − 1.67943i − 0.465791i −0.972502 0.232896i \(-0.925180\pi\)
0.972502 0.232896i \(-0.0748200\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.60324i − 1.11645i −0.829690 0.558225i \(-0.811483\pi\)
0.829690 0.558225i \(-0.188517\pi\)
\(18\) 0 0
\(19\) 2.64126 0.605946 0.302973 0.952999i \(-0.402021\pi\)
0.302973 + 0.952999i \(0.402021\pi\)
\(20\) 0 0
\(21\) 3.80992 0.831392
\(22\) 0 0
\(23\) 6.35567i 1.32525i 0.748951 + 0.662625i \(0.230558\pi\)
−0.748951 + 0.662625i \(0.769442\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −2.52008 −0.467966 −0.233983 0.972241i \(-0.575176\pi\)
−0.233983 + 0.972241i \(0.575176\pi\)
\(30\) 0 0
\(31\) 3.74937 0.673407 0.336704 0.941611i \(-0.390688\pi\)
0.336704 + 0.941611i \(0.390688\pi\)
\(32\) 0 0
\(33\) − 0.190733i − 0.0332023i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 11.8626i 1.95019i 0.221778 + 0.975097i \(0.428814\pi\)
−0.221778 + 0.975097i \(0.571186\pi\)
\(38\) 0 0
\(39\) −1.67943 −0.268925
\(40\) 0 0
\(41\) −7.18953 −1.12282 −0.561408 0.827539i \(-0.689740\pi\)
−0.561408 + 0.827539i \(0.689740\pi\)
\(42\) 0 0
\(43\) − 9.22619i − 1.40698i −0.710705 0.703491i \(-0.751624\pi\)
0.710705 0.703491i \(-0.248376\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.54848i 0.663464i 0.943374 + 0.331732i \(0.107633\pi\)
−0.943374 + 0.331732i \(0.892367\pi\)
\(48\) 0 0
\(49\) −7.51545 −1.07364
\(50\) 0 0
\(51\) −4.60324 −0.644583
\(52\) 0 0
\(53\) 9.43006i 1.29532i 0.761930 + 0.647659i \(0.224252\pi\)
−0.761930 + 0.647659i \(0.775748\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.64126i − 0.349843i
\(58\) 0 0
\(59\) −7.14057 −0.929623 −0.464812 0.885410i \(-0.653878\pi\)
−0.464812 + 0.885410i \(0.653878\pi\)
\(60\) 0 0
\(61\) 9.53012 1.22021 0.610103 0.792322i \(-0.291128\pi\)
0.610103 + 0.792322i \(0.291128\pi\)
\(62\) 0 0
\(63\) − 3.80992i − 0.480004i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 6.05168i − 0.739330i −0.929165 0.369665i \(-0.879472\pi\)
0.929165 0.369665i \(-0.120528\pi\)
\(68\) 0 0
\(69\) 6.35567 0.765133
\(70\) 0 0
\(71\) 13.2983 1.57822 0.789110 0.614252i \(-0.210542\pi\)
0.789110 + 0.614252i \(0.210542\pi\)
\(72\) 0 0
\(73\) − 5.21152i − 0.609963i −0.952358 0.304981i \(-0.901350\pi\)
0.952358 0.304981i \(-0.0986502\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.726676i 0.0828124i
\(78\) 0 0
\(79\) 3.11168 0.350092 0.175046 0.984560i \(-0.443993\pi\)
0.175046 + 0.984560i \(0.443993\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.67837i 0.513518i 0.966475 + 0.256759i \(0.0826546\pi\)
−0.966475 + 0.256759i \(0.917345\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.52008i 0.270181i
\(88\) 0 0
\(89\) −13.9423 −1.47788 −0.738939 0.673772i \(-0.764673\pi\)
−0.738939 + 0.673772i \(0.764673\pi\)
\(90\) 0 0
\(91\) 6.39850 0.670745
\(92\) 0 0
\(93\) − 3.74937i − 0.388792i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 6.69507i − 0.679782i −0.940465 0.339891i \(-0.889610\pi\)
0.940465 0.339891i \(-0.110390\pi\)
\(98\) 0 0
\(99\) −0.190733 −0.0191694
\(100\) 0 0
\(101\) 10.5147 1.04625 0.523127 0.852255i \(-0.324765\pi\)
0.523127 + 0.852255i \(0.324765\pi\)
\(102\) 0 0
\(103\) − 12.9648i − 1.27746i −0.769429 0.638732i \(-0.779459\pi\)
0.769429 0.638732i \(-0.220541\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.64606i 0.932520i 0.884648 + 0.466260i \(0.154399\pi\)
−0.884648 + 0.466260i \(0.845601\pi\)
\(108\) 0 0
\(109\) −20.4454 −1.95831 −0.979157 0.203103i \(-0.934898\pi\)
−0.979157 + 0.203103i \(0.934898\pi\)
\(110\) 0 0
\(111\) 11.8626 1.12595
\(112\) 0 0
\(113\) − 4.04417i − 0.380444i −0.981741 0.190222i \(-0.939079\pi\)
0.981741 0.190222i \(-0.0609207\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.67943i 0.155264i
\(118\) 0 0
\(119\) 17.5380 1.60770
\(120\) 0 0
\(121\) −10.9636 −0.996693
\(122\) 0 0
\(123\) 7.18953i 0.648258i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.7537i 1.48665i 0.668928 + 0.743327i \(0.266753\pi\)
−0.668928 + 0.743327i \(0.733247\pi\)
\(128\) 0 0
\(129\) −9.22619 −0.812321
\(130\) 0 0
\(131\) −3.30750 −0.288978 −0.144489 0.989506i \(-0.546154\pi\)
−0.144489 + 0.989506i \(0.546154\pi\)
\(132\) 0 0
\(133\) 10.0630i 0.872570i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.09916i 0.179343i 0.995971 + 0.0896717i \(0.0285818\pi\)
−0.995971 + 0.0896717i \(0.971418\pi\)
\(138\) 0 0
\(139\) 13.2849 1.12681 0.563404 0.826181i \(-0.309491\pi\)
0.563404 + 0.826181i \(0.309491\pi\)
\(140\) 0 0
\(141\) 4.54848 0.383051
\(142\) 0 0
\(143\) − 0.320323i − 0.0267868i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 7.51545i 0.619864i
\(148\) 0 0
\(149\) 2.79913 0.229313 0.114657 0.993405i \(-0.463423\pi\)
0.114657 + 0.993405i \(0.463423\pi\)
\(150\) 0 0
\(151\) −6.71330 −0.546320 −0.273160 0.961969i \(-0.588069\pi\)
−0.273160 + 0.961969i \(0.588069\pi\)
\(152\) 0 0
\(153\) 4.60324i 0.372150i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 13.9495i 1.11329i 0.830750 + 0.556646i \(0.187912\pi\)
−0.830750 + 0.556646i \(0.812088\pi\)
\(158\) 0 0
\(159\) 9.43006 0.747853
\(160\) 0 0
\(161\) −24.2146 −1.90838
\(162\) 0 0
\(163\) 8.87503i 0.695146i 0.937653 + 0.347573i \(0.112994\pi\)
−0.937653 + 0.347573i \(0.887006\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 24.5944i 1.90318i 0.307378 + 0.951588i \(0.400548\pi\)
−0.307378 + 0.951588i \(0.599452\pi\)
\(168\) 0 0
\(169\) 10.1795 0.783039
\(170\) 0 0
\(171\) −2.64126 −0.201982
\(172\) 0 0
\(173\) 16.9350i 1.28755i 0.765216 + 0.643774i \(0.222632\pi\)
−0.765216 + 0.643774i \(0.777368\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 7.14057i 0.536718i
\(178\) 0 0
\(179\) −14.7497 −1.10244 −0.551222 0.834358i \(-0.685838\pi\)
−0.551222 + 0.834358i \(0.685838\pi\)
\(180\) 0 0
\(181\) −10.9888 −0.816791 −0.408396 0.912805i \(-0.633912\pi\)
−0.408396 + 0.912805i \(0.633912\pi\)
\(182\) 0 0
\(183\) − 9.53012i − 0.704486i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 0.877989i − 0.0642049i
\(188\) 0 0
\(189\) −3.80992 −0.277131
\(190\) 0 0
\(191\) −24.0355 −1.73915 −0.869575 0.493801i \(-0.835607\pi\)
−0.869575 + 0.493801i \(0.835607\pi\)
\(192\) 0 0
\(193\) 7.50843i 0.540469i 0.962795 + 0.270234i \(0.0871012\pi\)
−0.962795 + 0.270234i \(0.912899\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.5522i 0.894306i 0.894457 + 0.447153i \(0.147562\pi\)
−0.894457 + 0.447153i \(0.852438\pi\)
\(198\) 0 0
\(199\) 19.7618 1.40088 0.700440 0.713711i \(-0.252987\pi\)
0.700440 + 0.713711i \(0.252987\pi\)
\(200\) 0 0
\(201\) −6.05168 −0.426852
\(202\) 0 0
\(203\) − 9.60128i − 0.673877i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 6.35567i − 0.441750i
\(208\) 0 0
\(209\) 0.503774 0.0348468
\(210\) 0 0
\(211\) −11.2479 −0.774336 −0.387168 0.922009i \(-0.626547\pi\)
−0.387168 + 0.922009i \(0.626547\pi\)
\(212\) 0 0
\(213\) − 13.2983i − 0.911186i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 14.2848i 0.969715i
\(218\) 0 0
\(219\) −5.21152 −0.352162
\(220\) 0 0
\(221\) −7.73084 −0.520033
\(222\) 0 0
\(223\) − 7.27196i − 0.486967i −0.969905 0.243483i \(-0.921710\pi\)
0.969905 0.243483i \(-0.0782901\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 9.46929i − 0.628499i −0.949340 0.314249i \(-0.898247\pi\)
0.949340 0.314249i \(-0.101753\pi\)
\(228\) 0 0
\(229\) −21.2301 −1.40292 −0.701461 0.712708i \(-0.747469\pi\)
−0.701461 + 0.712708i \(0.747469\pi\)
\(230\) 0 0
\(231\) 0.726676 0.0478118
\(232\) 0 0
\(233\) 7.08932i 0.464437i 0.972664 + 0.232218i \(0.0745984\pi\)
−0.972664 + 0.232218i \(0.925402\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 3.11168i − 0.202126i
\(238\) 0 0
\(239\) 20.0117 1.29445 0.647224 0.762300i \(-0.275930\pi\)
0.647224 + 0.762300i \(0.275930\pi\)
\(240\) 0 0
\(241\) 20.8240 1.34139 0.670696 0.741732i \(-0.265995\pi\)
0.670696 + 0.741732i \(0.265995\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 4.43582i − 0.282244i
\(248\) 0 0
\(249\) 4.67837 0.296480
\(250\) 0 0
\(251\) −29.7741 −1.87932 −0.939662 0.342104i \(-0.888861\pi\)
−0.939662 + 0.342104i \(0.888861\pi\)
\(252\) 0 0
\(253\) 1.21224i 0.0762126i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 21.2853i 1.32774i 0.747849 + 0.663869i \(0.231087\pi\)
−0.747849 + 0.663869i \(0.768913\pi\)
\(258\) 0 0
\(259\) −45.1954 −2.80830
\(260\) 0 0
\(261\) 2.52008 0.155989
\(262\) 0 0
\(263\) 21.8653i 1.34827i 0.738608 + 0.674135i \(0.235483\pi\)
−0.738608 + 0.674135i \(0.764517\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 13.9423i 0.853253i
\(268\) 0 0
\(269\) 4.56132 0.278109 0.139054 0.990285i \(-0.455594\pi\)
0.139054 + 0.990285i \(0.455594\pi\)
\(270\) 0 0
\(271\) −12.9193 −0.784791 −0.392395 0.919797i \(-0.628354\pi\)
−0.392395 + 0.919797i \(0.628354\pi\)
\(272\) 0 0
\(273\) − 6.39850i − 0.387255i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 8.83521i − 0.530856i −0.964131 0.265428i \(-0.914487\pi\)
0.964131 0.265428i \(-0.0855133\pi\)
\(278\) 0 0
\(279\) −3.74937 −0.224469
\(280\) 0 0
\(281\) 0.0305495 0.00182243 0.000911215 1.00000i \(-0.499710\pi\)
0.000911215 1.00000i \(0.499710\pi\)
\(282\) 0 0
\(283\) − 4.80719i − 0.285758i −0.989740 0.142879i \(-0.954364\pi\)
0.989740 0.142879i \(-0.0456359\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 27.3915i − 1.61687i
\(288\) 0 0
\(289\) −4.18984 −0.246461
\(290\) 0 0
\(291\) −6.69507 −0.392472
\(292\) 0 0
\(293\) 14.9705i 0.874587i 0.899319 + 0.437294i \(0.144063\pi\)
−0.899319 + 0.437294i \(0.855937\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0.190733i 0.0110674i
\(298\) 0 0
\(299\) 10.6739 0.617290
\(300\) 0 0
\(301\) 35.1510 2.02607
\(302\) 0 0
\(303\) − 10.5147i − 0.604055i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.47622i 0.255472i 0.991808 + 0.127736i \(0.0407710\pi\)
−0.991808 + 0.127736i \(0.959229\pi\)
\(308\) 0 0
\(309\) −12.9648 −0.737544
\(310\) 0 0
\(311\) 0.296114 0.0167911 0.00839554 0.999965i \(-0.497328\pi\)
0.00839554 + 0.999965i \(0.497328\pi\)
\(312\) 0 0
\(313\) 21.1569i 1.19586i 0.801548 + 0.597930i \(0.204010\pi\)
−0.801548 + 0.597930i \(0.795990\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 25.9166i 1.45562i 0.685777 + 0.727812i \(0.259463\pi\)
−0.685777 + 0.727812i \(0.740537\pi\)
\(318\) 0 0
\(319\) −0.480661 −0.0269119
\(320\) 0 0
\(321\) 9.64606 0.538391
\(322\) 0 0
\(323\) − 12.1583i − 0.676509i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 20.4454i 1.13063i
\(328\) 0 0
\(329\) −17.3293 −0.955397
\(330\) 0 0
\(331\) −3.07039 −0.168764 −0.0843819 0.996433i \(-0.526892\pi\)
−0.0843819 + 0.996433i \(0.526892\pi\)
\(332\) 0 0
\(333\) − 11.8626i − 0.650065i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 28.2521i 1.53899i 0.638652 + 0.769495i \(0.279492\pi\)
−0.638652 + 0.769495i \(0.720508\pi\)
\(338\) 0 0
\(339\) −4.04417 −0.219649
\(340\) 0 0
\(341\) 0.715129 0.0387264
\(342\) 0 0
\(343\) − 1.96383i − 0.106037i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 30.6011i 1.64275i 0.570387 + 0.821376i \(0.306793\pi\)
−0.570387 + 0.821376i \(0.693207\pi\)
\(348\) 0 0
\(349\) 16.1178 0.862764 0.431382 0.902169i \(-0.358026\pi\)
0.431382 + 0.902169i \(0.358026\pi\)
\(350\) 0 0
\(351\) 1.67943 0.0896416
\(352\) 0 0
\(353\) 0.431010i 0.0229404i 0.999934 + 0.0114702i \(0.00365115\pi\)
−0.999934 + 0.0114702i \(0.996349\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 17.5380i − 0.928207i
\(358\) 0 0
\(359\) −18.4738 −0.975008 −0.487504 0.873121i \(-0.662093\pi\)
−0.487504 + 0.873121i \(0.662093\pi\)
\(360\) 0 0
\(361\) −12.0238 −0.632829
\(362\) 0 0
\(363\) 10.9636i 0.575441i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 10.2757i 0.536386i 0.963365 + 0.268193i \(0.0864265\pi\)
−0.963365 + 0.268193i \(0.913573\pi\)
\(368\) 0 0
\(369\) 7.18953 0.374272
\(370\) 0 0
\(371\) −35.9277 −1.86528
\(372\) 0 0
\(373\) − 14.6433i − 0.758201i −0.925356 0.379100i \(-0.876233\pi\)
0.925356 0.379100i \(-0.123767\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.23230i 0.217975i
\(378\) 0 0
\(379\) 10.7051 0.549884 0.274942 0.961461i \(-0.411341\pi\)
0.274942 + 0.961461i \(0.411341\pi\)
\(380\) 0 0
\(381\) 16.7537 0.858320
\(382\) 0 0
\(383\) − 26.3174i − 1.34476i −0.740208 0.672378i \(-0.765273\pi\)
0.740208 0.672378i \(-0.234727\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.22619i 0.468994i
\(388\) 0 0
\(389\) −0.291780 −0.0147938 −0.00739692 0.999973i \(-0.502355\pi\)
−0.00739692 + 0.999973i \(0.502355\pi\)
\(390\) 0 0
\(391\) 29.2567 1.47958
\(392\) 0 0
\(393\) 3.30750i 0.166841i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.01611i 0.101185i 0.998719 + 0.0505927i \(0.0161110\pi\)
−0.998719 + 0.0505927i \(0.983889\pi\)
\(398\) 0 0
\(399\) 10.0630 0.503778
\(400\) 0 0
\(401\) 9.88760 0.493763 0.246882 0.969046i \(-0.420594\pi\)
0.246882 + 0.969046i \(0.420594\pi\)
\(402\) 0 0
\(403\) − 6.29683i − 0.313667i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.26258i 0.112152i
\(408\) 0 0
\(409\) 37.6809 1.86320 0.931601 0.363483i \(-0.118413\pi\)
0.931601 + 0.363483i \(0.118413\pi\)
\(410\) 0 0
\(411\) 2.09916 0.103544
\(412\) 0 0
\(413\) − 27.2050i − 1.33867i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 13.2849i − 0.650563i
\(418\) 0 0
\(419\) 15.6537 0.764734 0.382367 0.924011i \(-0.375109\pi\)
0.382367 + 0.924011i \(0.375109\pi\)
\(420\) 0 0
\(421\) 14.5277 0.708036 0.354018 0.935239i \(-0.384815\pi\)
0.354018 + 0.935239i \(0.384815\pi\)
\(422\) 0 0
\(423\) − 4.54848i − 0.221155i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 36.3089i 1.75711i
\(428\) 0 0
\(429\) −0.320323 −0.0154653
\(430\) 0 0
\(431\) −12.6264 −0.608192 −0.304096 0.952641i \(-0.598354\pi\)
−0.304096 + 0.952641i \(0.598354\pi\)
\(432\) 0 0
\(433\) 3.49014i 0.167725i 0.996477 + 0.0838626i \(0.0267257\pi\)
−0.996477 + 0.0838626i \(0.973274\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 16.7870i 0.803030i
\(438\) 0 0
\(439\) 19.5630 0.933689 0.466844 0.884340i \(-0.345391\pi\)
0.466844 + 0.884340i \(0.345391\pi\)
\(440\) 0 0
\(441\) 7.51545 0.357879
\(442\) 0 0
\(443\) 2.77485i 0.131837i 0.997825 + 0.0659185i \(0.0209977\pi\)
−0.997825 + 0.0659185i \(0.979002\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 2.79913i − 0.132394i
\(448\) 0 0
\(449\) 38.4261 1.81344 0.906721 0.421732i \(-0.138578\pi\)
0.906721 + 0.421732i \(0.138578\pi\)
\(450\) 0 0
\(451\) −1.37128 −0.0645710
\(452\) 0 0
\(453\) 6.71330i 0.315418i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 36.0296i − 1.68539i −0.538389 0.842696i \(-0.680967\pi\)
0.538389 0.842696i \(-0.319033\pi\)
\(458\) 0 0
\(459\) 4.60324 0.214861
\(460\) 0 0
\(461\) 0.857592 0.0399421 0.0199710 0.999801i \(-0.493643\pi\)
0.0199710 + 0.999801i \(0.493643\pi\)
\(462\) 0 0
\(463\) − 10.4068i − 0.483643i −0.970321 0.241822i \(-0.922255\pi\)
0.970321 0.241822i \(-0.0777449\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 20.5705i − 0.951889i −0.879475 0.475944i \(-0.842106\pi\)
0.879475 0.475944i \(-0.157894\pi\)
\(468\) 0 0
\(469\) 23.0564 1.06464
\(470\) 0 0
\(471\) 13.9495 0.642759
\(472\) 0 0
\(473\) − 1.75974i − 0.0809128i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 9.43006i − 0.431773i
\(478\) 0 0
\(479\) −31.9380 −1.45929 −0.729643 0.683828i \(-0.760314\pi\)
−0.729643 + 0.683828i \(0.760314\pi\)
\(480\) 0 0
\(481\) 19.9224 0.908383
\(482\) 0 0
\(483\) 24.2146i 1.10180i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 15.6815i 0.710599i 0.934753 + 0.355299i \(0.115621\pi\)
−0.934753 + 0.355299i \(0.884379\pi\)
\(488\) 0 0
\(489\) 8.87503 0.401343
\(490\) 0 0
\(491\) 16.4831 0.743870 0.371935 0.928259i \(-0.378694\pi\)
0.371935 + 0.928259i \(0.378694\pi\)
\(492\) 0 0
\(493\) 11.6005i 0.522461i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 50.6655i 2.27266i
\(498\) 0 0
\(499\) −12.4339 −0.556618 −0.278309 0.960492i \(-0.589774\pi\)
−0.278309 + 0.960492i \(0.589774\pi\)
\(500\) 0 0
\(501\) 24.5944 1.09880
\(502\) 0 0
\(503\) − 4.33327i − 0.193211i −0.995323 0.0966055i \(-0.969201\pi\)
0.995323 0.0966055i \(-0.0307985\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 10.1795i − 0.452088i
\(508\) 0 0
\(509\) −14.0120 −0.621072 −0.310536 0.950562i \(-0.600508\pi\)
−0.310536 + 0.950562i \(0.600508\pi\)
\(510\) 0 0
\(511\) 19.8555 0.878354
\(512\) 0 0
\(513\) 2.64126i 0.116614i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0.867545i 0.0381546i
\(518\) 0 0
\(519\) 16.9350 0.743366
\(520\) 0 0
\(521\) 1.92829 0.0844801 0.0422400 0.999107i \(-0.486551\pi\)
0.0422400 + 0.999107i \(0.486551\pi\)
\(522\) 0 0
\(523\) 1.95894i 0.0856583i 0.999082 + 0.0428291i \(0.0136371\pi\)
−0.999082 + 0.0428291i \(0.986363\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 17.2593i − 0.751826i
\(528\) 0 0
\(529\) −17.3946 −0.756287
\(530\) 0 0
\(531\) 7.14057 0.309874
\(532\) 0 0
\(533\) 12.0743i 0.522998i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 14.7497i 0.636497i
\(538\) 0 0
\(539\) −1.43344 −0.0617428
\(540\) 0 0
\(541\) −34.6132 −1.48814 −0.744069 0.668103i \(-0.767107\pi\)
−0.744069 + 0.668103i \(0.767107\pi\)
\(542\) 0 0
\(543\) 10.9888i 0.471575i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 15.9494i 0.681947i 0.940073 + 0.340973i \(0.110757\pi\)
−0.940073 + 0.340973i \(0.889243\pi\)
\(548\) 0 0
\(549\) −9.53012 −0.406735
\(550\) 0 0
\(551\) −6.65617 −0.283562
\(552\) 0 0
\(553\) 11.8552i 0.504136i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 43.0343i 1.82342i 0.410831 + 0.911711i \(0.365239\pi\)
−0.410831 + 0.911711i \(0.634761\pi\)
\(558\) 0 0
\(559\) −15.4948 −0.655359
\(560\) 0 0
\(561\) −0.877989 −0.0370687
\(562\) 0 0
\(563\) − 18.8560i − 0.794686i −0.917670 0.397343i \(-0.869932\pi\)
0.917670 0.397343i \(-0.130068\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.80992i 0.160001i
\(568\) 0 0
\(569\) 24.3652 1.02144 0.510721 0.859747i \(-0.329379\pi\)
0.510721 + 0.859747i \(0.329379\pi\)
\(570\) 0 0
\(571\) 15.3391 0.641923 0.320962 0.947092i \(-0.395994\pi\)
0.320962 + 0.947092i \(0.395994\pi\)
\(572\) 0 0
\(573\) 24.0355i 1.00410i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 14.3401i 0.596985i 0.954412 + 0.298493i \(0.0964838\pi\)
−0.954412 + 0.298493i \(0.903516\pi\)
\(578\) 0 0
\(579\) 7.50843 0.312040
\(580\) 0 0
\(581\) −17.8242 −0.739472
\(582\) 0 0
\(583\) 1.79862i 0.0744913i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 10.7026i − 0.441744i −0.975303 0.220872i \(-0.929110\pi\)
0.975303 0.220872i \(-0.0708903\pi\)
\(588\) 0 0
\(589\) 9.90306 0.408048
\(590\) 0 0
\(591\) 12.5522 0.516328
\(592\) 0 0
\(593\) − 23.5756i − 0.968135i −0.875031 0.484067i \(-0.839159\pi\)
0.875031 0.484067i \(-0.160841\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 19.7618i − 0.808798i
\(598\) 0 0
\(599\) −13.8055 −0.564078 −0.282039 0.959403i \(-0.591011\pi\)
−0.282039 + 0.959403i \(0.591011\pi\)
\(600\) 0 0
\(601\) 9.61536 0.392219 0.196109 0.980582i \(-0.437169\pi\)
0.196109 + 0.980582i \(0.437169\pi\)
\(602\) 0 0
\(603\) 6.05168i 0.246443i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 23.7884i 0.965539i 0.875747 + 0.482770i \(0.160369\pi\)
−0.875747 + 0.482770i \(0.839631\pi\)
\(608\) 0 0
\(609\) −9.60128 −0.389063
\(610\) 0 0
\(611\) 7.63888 0.309036
\(612\) 0 0
\(613\) 16.0716i 0.649127i 0.945864 + 0.324563i \(0.105217\pi\)
−0.945864 + 0.324563i \(0.894783\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 9.34595i − 0.376254i −0.982145 0.188127i \(-0.939758\pi\)
0.982145 0.188127i \(-0.0602416\pi\)
\(618\) 0 0
\(619\) 18.4768 0.742645 0.371323 0.928504i \(-0.378904\pi\)
0.371323 + 0.928504i \(0.378904\pi\)
\(620\) 0 0
\(621\) −6.35567 −0.255044
\(622\) 0 0
\(623\) − 53.1189i − 2.12816i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 0.503774i − 0.0201188i
\(628\) 0 0
\(629\) 54.6063 2.17729
\(630\) 0 0
\(631\) 38.5690 1.53541 0.767704 0.640804i \(-0.221399\pi\)
0.767704 + 0.640804i \(0.221399\pi\)
\(632\) 0 0
\(633\) 11.2479i 0.447063i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 12.6217i 0.500090i
\(638\) 0 0
\(639\) −13.2983 −0.526073
\(640\) 0 0
\(641\) −12.8461 −0.507389 −0.253695 0.967284i \(-0.581646\pi\)
−0.253695 + 0.967284i \(0.581646\pi\)
\(642\) 0 0
\(643\) 20.1030i 0.792784i 0.918081 + 0.396392i \(0.129738\pi\)
−0.918081 + 0.396392i \(0.870262\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 39.1895i − 1.54070i −0.637622 0.770350i \(-0.720082\pi\)
0.637622 0.770350i \(-0.279918\pi\)
\(648\) 0 0
\(649\) −1.36194 −0.0534609
\(650\) 0 0
\(651\) 14.2848 0.559865
\(652\) 0 0
\(653\) 25.9894i 1.01704i 0.861049 + 0.508522i \(0.169808\pi\)
−0.861049 + 0.508522i \(0.830192\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 5.21152i 0.203321i
\(658\) 0 0
\(659\) 6.42356 0.250226 0.125113 0.992142i \(-0.460071\pi\)
0.125113 + 0.992142i \(0.460071\pi\)
\(660\) 0 0
\(661\) 21.5875 0.839657 0.419829 0.907603i \(-0.362090\pi\)
0.419829 + 0.907603i \(0.362090\pi\)
\(662\) 0 0
\(663\) 7.73084i 0.300241i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 16.0168i − 0.620172i
\(668\) 0 0
\(669\) −7.27196 −0.281150
\(670\) 0 0
\(671\) 1.81771 0.0701718
\(672\) 0 0
\(673\) 5.94040i 0.228986i 0.993424 + 0.114493i \(0.0365243\pi\)
−0.993424 + 0.114493i \(0.963476\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 37.1784i − 1.42888i −0.699696 0.714441i \(-0.746681\pi\)
0.699696 0.714441i \(-0.253319\pi\)
\(678\) 0 0
\(679\) 25.5077 0.978894
\(680\) 0 0
\(681\) −9.46929 −0.362864
\(682\) 0 0
\(683\) − 45.1024i − 1.72580i −0.505377 0.862899i \(-0.668647\pi\)
0.505377 0.862899i \(-0.331353\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 21.2301i 0.809977i
\(688\) 0 0
\(689\) 15.8372 0.603348
\(690\) 0 0
\(691\) −28.5713 −1.08691 −0.543453 0.839440i \(-0.682883\pi\)
−0.543453 + 0.839440i \(0.682883\pi\)
\(692\) 0 0
\(693\) − 0.726676i − 0.0276041i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 33.0952i 1.25357i
\(698\) 0 0
\(699\) 7.08932 0.268143
\(700\) 0 0
\(701\) −44.2636 −1.67181 −0.835907 0.548871i \(-0.815058\pi\)
−0.835907 + 0.548871i \(0.815058\pi\)
\(702\) 0 0
\(703\) 31.3321i 1.18171i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 40.0602i 1.50662i
\(708\) 0 0
\(709\) −17.0124 −0.638913 −0.319457 0.947601i \(-0.603500\pi\)
−0.319457 + 0.947601i \(0.603500\pi\)
\(710\) 0 0
\(711\) −3.11168 −0.116697
\(712\) 0 0
\(713\) 23.8298i 0.892433i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 20.0117i − 0.747350i
\(718\) 0 0
\(719\) −8.29496 −0.309350 −0.154675 0.987965i \(-0.549433\pi\)
−0.154675 + 0.987965i \(0.549433\pi\)
\(720\) 0 0
\(721\) 49.3950 1.83956
\(722\) 0 0
\(723\) − 20.8240i − 0.774454i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 7.46319i 0.276794i 0.990377 + 0.138397i \(0.0441950\pi\)
−0.990377 + 0.138397i \(0.955805\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −42.4704 −1.57082
\(732\) 0 0
\(733\) − 12.0201i − 0.443973i −0.975050 0.221987i \(-0.928746\pi\)
0.975050 0.221987i \(-0.0712542\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 1.15425i − 0.0425175i
\(738\) 0 0
\(739\) −25.3335 −0.931908 −0.465954 0.884809i \(-0.654289\pi\)
−0.465954 + 0.884809i \(0.654289\pi\)
\(740\) 0 0
\(741\) −4.43582 −0.162954
\(742\) 0 0
\(743\) − 21.0959i − 0.773935i −0.922093 0.386968i \(-0.873523\pi\)
0.922093 0.386968i \(-0.126477\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 4.67837i − 0.171173i
\(748\) 0 0
\(749\) −36.7507 −1.34284
\(750\) 0 0
\(751\) −34.3897 −1.25490 −0.627449 0.778658i \(-0.715901\pi\)
−0.627449 + 0.778658i \(0.715901\pi\)
\(752\) 0 0
\(753\) 29.7741i 1.08503i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 33.9762i − 1.23488i −0.786616 0.617442i \(-0.788169\pi\)
0.786616 0.617442i \(-0.211831\pi\)
\(758\) 0 0
\(759\) 1.21224 0.0440014
\(760\) 0 0
\(761\) −13.3872 −0.485284 −0.242642 0.970116i \(-0.578014\pi\)
−0.242642 + 0.970116i \(0.578014\pi\)
\(762\) 0 0
\(763\) − 77.8953i − 2.82000i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11.9921i 0.433010i
\(768\) 0 0
\(769\) 5.94485 0.214377 0.107188 0.994239i \(-0.465815\pi\)
0.107188 + 0.994239i \(0.465815\pi\)
\(770\) 0 0
\(771\) 21.2853 0.766570
\(772\) 0 0
\(773\) 34.0966i 1.22637i 0.789940 + 0.613184i \(0.210112\pi\)
−0.789940 + 0.613184i \(0.789888\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 45.1954i 1.62138i
\(778\) 0 0
\(779\) −18.9894 −0.680366
\(780\) 0 0
\(781\) 2.53643 0.0907604
\(782\) 0 0
\(783\) − 2.52008i − 0.0900602i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1.97301i 0.0703301i 0.999382 + 0.0351651i \(0.0111957\pi\)
−0.999382 + 0.0351651i \(0.988804\pi\)
\(788\) 0 0
\(789\) 21.8653 0.778424
\(790\) 0 0
\(791\) 15.4079 0.547843
\(792\) 0 0
\(793\) − 16.0052i − 0.568361i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 29.5512i − 1.04676i −0.852100 0.523379i \(-0.824671\pi\)
0.852100 0.523379i \(-0.175329\pi\)
\(798\) 0 0
\(799\) 20.9378 0.740725
\(800\) 0 0
\(801\) 13.9423 0.492626
\(802\) 0 0
\(803\) − 0.994009i − 0.0350778i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 4.56132i − 0.160566i
\(808\) 0 0
\(809\) −7.94507 −0.279334 −0.139667 0.990199i \(-0.544603\pi\)
−0.139667 + 0.990199i \(0.544603\pi\)
\(810\) 0 0
\(811\) 18.5021 0.649698 0.324849 0.945766i \(-0.394687\pi\)
0.324849 + 0.945766i \(0.394687\pi\)
\(812\) 0 0
\(813\) 12.9193i 0.453099i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 24.3687i − 0.852555i
\(818\) 0 0
\(819\) −6.39850 −0.223582
\(820\) 0 0
\(821\) −15.4242 −0.538307 −0.269154 0.963097i \(-0.586744\pi\)
−0.269154 + 0.963097i \(0.586744\pi\)
\(822\) 0 0
\(823\) 41.5847i 1.44955i 0.688986 + 0.724775i \(0.258056\pi\)
−0.688986 + 0.724775i \(0.741944\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 34.8857i − 1.21309i −0.795048 0.606547i \(-0.792554\pi\)
0.795048 0.606547i \(-0.207446\pi\)
\(828\) 0 0
\(829\) −46.6095 −1.61882 −0.809408 0.587247i \(-0.800212\pi\)
−0.809408 + 0.587247i \(0.800212\pi\)
\(830\) 0 0
\(831\) −8.83521 −0.306490
\(832\) 0 0
\(833\) 34.5955i 1.19866i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.74937i 0.129597i
\(838\) 0 0
\(839\) −42.3953 −1.46365 −0.731825 0.681493i \(-0.761331\pi\)
−0.731825 + 0.681493i \(0.761331\pi\)
\(840\) 0 0
\(841\) −22.6492 −0.781007
\(842\) 0 0
\(843\) − 0.0305495i − 0.00105218i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 41.7705i − 1.43525i
\(848\) 0 0
\(849\) −4.80719 −0.164982
\(850\) 0 0
\(851\) −75.3946 −2.58449
\(852\) 0 0
\(853\) 3.74484i 0.128221i 0.997943 + 0.0641104i \(0.0204210\pi\)
−0.997943 + 0.0641104i \(0.979579\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 39.9648i 1.36517i 0.730806 + 0.682586i \(0.239145\pi\)
−0.730806 + 0.682586i \(0.760855\pi\)
\(858\) 0 0
\(859\) 56.1078 1.91437 0.957186 0.289472i \(-0.0934798\pi\)
0.957186 + 0.289472i \(0.0934798\pi\)
\(860\) 0 0
\(861\) −27.3915 −0.933500
\(862\) 0 0
\(863\) − 18.5927i − 0.632904i −0.948609 0.316452i \(-0.897508\pi\)
0.948609 0.316452i \(-0.102492\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 4.18984i 0.142294i
\(868\) 0 0
\(869\) 0.593500 0.0201331
\(870\) 0 0
\(871\) −10.1634 −0.344373
\(872\) 0 0
\(873\) 6.69507i 0.226594i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 10.8792i 0.367365i 0.982986 + 0.183682i \(0.0588018\pi\)
−0.982986 + 0.183682i \(0.941198\pi\)
\(878\) 0 0
\(879\) 14.9705 0.504943
\(880\) 0 0
\(881\) 33.9938 1.14528 0.572641 0.819806i \(-0.305919\pi\)
0.572641 + 0.819806i \(0.305919\pi\)
\(882\) 0 0
\(883\) − 15.2549i − 0.513367i −0.966496 0.256683i \(-0.917370\pi\)
0.966496 0.256683i \(-0.0826298\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 28.5046i − 0.957090i −0.878063 0.478545i \(-0.841164\pi\)
0.878063 0.478545i \(-0.158836\pi\)
\(888\) 0 0
\(889\) −63.8303 −2.14080
\(890\) 0 0
\(891\) 0.190733 0.00638979
\(892\) 0 0
\(893\) 12.0137i 0.402024i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 10.6739i − 0.356392i
\(898\) 0 0
\(899\) −9.44871 −0.315132
\(900\) 0 0
\(901\) 43.4089 1.44616
\(902\) 0 0
\(903\) − 35.1510i − 1.16975i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 22.1919i − 0.736868i −0.929654 0.368434i \(-0.879894\pi\)
0.929654 0.368434i \(-0.120106\pi\)
\(908\) 0 0
\(909\) −10.5147 −0.348751
\(910\) 0 0
\(911\) 40.6047 1.34529 0.672647 0.739963i \(-0.265157\pi\)
0.672647 + 0.739963i \(0.265157\pi\)
\(912\) 0 0
\(913\) 0.892319i 0.0295314i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 12.6013i − 0.416132i
\(918\) 0 0
\(919\) −12.5514 −0.414031 −0.207016 0.978338i \(-0.566375\pi\)
−0.207016 + 0.978338i \(0.566375\pi\)
\(920\) 0 0
\(921\) 4.47622 0.147497
\(922\) 0 0
\(923\) − 22.3336i − 0.735121i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 12.9648i 0.425821i
\(928\) 0 0
\(929\) 56.4129 1.85085 0.925424 0.378932i \(-0.123709\pi\)
0.925424 + 0.378932i \(0.123709\pi\)
\(930\) 0 0
\(931\) −19.8502 −0.650565
\(932\) 0 0
\(933\) − 0.296114i − 0.00969434i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 42.1247i − 1.37616i −0.725637 0.688078i \(-0.758455\pi\)
0.725637 0.688078i \(-0.241545\pi\)
\(938\) 0 0
\(939\) 21.1569 0.690430
\(940\) 0 0
\(941\) −30.7834 −1.00351 −0.501755 0.865010i \(-0.667312\pi\)
−0.501755 + 0.865010i \(0.667312\pi\)
\(942\) 0 0
\(943\) − 45.6943i − 1.48801i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0.741722i 0.0241027i 0.999927 + 0.0120514i \(0.00383616\pi\)
−0.999927 + 0.0120514i \(0.996164\pi\)
\(948\) 0 0
\(949\) −8.75241 −0.284115
\(950\) 0 0
\(951\) 25.9166 0.840405
\(952\) 0 0
\(953\) − 39.3063i − 1.27326i −0.771171 0.636628i \(-0.780329\pi\)
0.771171 0.636628i \(-0.219671\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0.480661i 0.0155376i
\(958\) 0 0
\(959\) −7.99763 −0.258257
\(960\) 0 0
\(961\) −16.9422 −0.546522
\(962\) 0 0
\(963\) − 9.64606i − 0.310840i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 48.7793i 1.56864i 0.620358 + 0.784319i \(0.286987\pi\)
−0.620358 + 0.784319i \(0.713013\pi\)
\(968\) 0 0
\(969\) −12.1583 −0.390582
\(970\) 0 0
\(971\) −61.2669 −1.96615 −0.983074 0.183210i \(-0.941351\pi\)
−0.983074 + 0.183210i \(0.941351\pi\)
\(972\) 0 0
\(973\) 50.6143i 1.62262i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 11.1646i 0.357186i 0.983923 + 0.178593i \(0.0571545\pi\)
−0.983923 + 0.178593i \(0.942845\pi\)
\(978\) 0 0
\(979\) −2.65925 −0.0849900
\(980\) 0 0
\(981\) 20.4454 0.652772
\(982\) 0 0
\(983\) 31.0074i 0.988983i 0.869182 + 0.494492i \(0.164646\pi\)
−0.869182 + 0.494492i \(0.835354\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 17.3293i 0.551599i
\(988\) 0 0
\(989\) 58.6387 1.86460
\(990\) 0 0
\(991\) 29.5575 0.938926 0.469463 0.882952i \(-0.344447\pi\)
0.469463 + 0.882952i \(0.344447\pi\)
\(992\) 0 0
\(993\) 3.07039i 0.0974358i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 54.0506i − 1.71180i −0.517142 0.855899i \(-0.673004\pi\)
0.517142 0.855899i \(-0.326996\pi\)
\(998\) 0 0
\(999\) −11.8626 −0.375315
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 7500.2.d.g.1249.10 24
5.2 odd 4 7500.2.a.m.1.3 12
5.3 odd 4 7500.2.a.n.1.10 12
5.4 even 2 inner 7500.2.d.g.1249.15 24
25.2 odd 20 1500.2.m.d.601.2 24
25.9 even 10 300.2.o.a.169.3 24
25.11 even 5 300.2.o.a.229.3 yes 24
25.12 odd 20 1500.2.m.d.901.2 24
25.13 odd 20 1500.2.m.c.901.5 24
25.14 even 10 1500.2.o.c.649.4 24
25.16 even 5 1500.2.o.c.349.4 24
25.23 odd 20 1500.2.m.c.601.5 24
75.11 odd 10 900.2.w.c.829.1 24
75.59 odd 10 900.2.w.c.469.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.2.o.a.169.3 24 25.9 even 10
300.2.o.a.229.3 yes 24 25.11 even 5
900.2.w.c.469.1 24 75.59 odd 10
900.2.w.c.829.1 24 75.11 odd 10
1500.2.m.c.601.5 24 25.23 odd 20
1500.2.m.c.901.5 24 25.13 odd 20
1500.2.m.d.601.2 24 25.2 odd 20
1500.2.m.d.901.2 24 25.12 odd 20
1500.2.o.c.349.4 24 25.16 even 5
1500.2.o.c.649.4 24 25.14 even 10
7500.2.a.m.1.3 12 5.2 odd 4
7500.2.a.n.1.10 12 5.3 odd 4
7500.2.d.g.1249.10 24 1.1 even 1 trivial
7500.2.d.g.1249.15 24 5.4 even 2 inner