Properties

Label 1305.2.a.i.1.1
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.a (trivial)

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: yes
Analytic conductor: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{5} -3.12311 q^{7} -6.56155 q^{8} +O(q^{10})\) \(q-2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{5} -3.12311 q^{7} -6.56155 q^{8} +2.56155 q^{10} +5.56155 q^{11} -2.00000 q^{13} +8.00000 q^{14} +7.68466 q^{16} -1.12311 q^{17} -3.12311 q^{19} -4.56155 q^{20} -14.2462 q^{22} -2.43845 q^{23} +1.00000 q^{25} +5.12311 q^{26} -14.2462 q^{28} -1.00000 q^{29} +4.00000 q^{31} -6.56155 q^{32} +2.87689 q^{34} +3.12311 q^{35} +10.6847 q^{37} +8.00000 q^{38} +6.56155 q^{40} -10.6847 q^{41} -4.68466 q^{43} +25.3693 q^{44} +6.24621 q^{46} +4.87689 q^{47} +2.75379 q^{49} -2.56155 q^{50} -9.12311 q^{52} +7.56155 q^{53} -5.56155 q^{55} +20.4924 q^{56} +2.56155 q^{58} -12.0000 q^{59} +9.12311 q^{61} -10.2462 q^{62} +1.43845 q^{64} +2.00000 q^{65} +13.3693 q^{67} -5.12311 q^{68} -8.00000 q^{70} +11.1231 q^{71} -10.6847 q^{73} -27.3693 q^{74} -14.2462 q^{76} -17.3693 q^{77} -12.0000 q^{79} -7.68466 q^{80} +27.3693 q^{82} -1.56155 q^{83} +1.12311 q^{85} +12.0000 q^{86} -36.4924 q^{88} +4.24621 q^{89} +6.24621 q^{91} -11.1231 q^{92} -12.4924 q^{94} +3.12311 q^{95} +6.68466 q^{97} -7.05398 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 5 q^{4} - 2 q^{5} + 2 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 5 q^{4} - 2 q^{5} + 2 q^{7} - 9 q^{8} + q^{10} + 7 q^{11} - 4 q^{13} + 16 q^{14} + 3 q^{16} + 6 q^{17} + 2 q^{19} - 5 q^{20} - 12 q^{22} - 9 q^{23} + 2 q^{25} + 2 q^{26} - 12 q^{28} - 2 q^{29} + 8 q^{31} - 9 q^{32} + 14 q^{34} - 2 q^{35} + 9 q^{37} + 16 q^{38} + 9 q^{40} - 9 q^{41} + 3 q^{43} + 26 q^{44} - 4 q^{46} + 18 q^{47} + 22 q^{49} - q^{50} - 10 q^{52} + 11 q^{53} - 7 q^{55} + 8 q^{56} + q^{58} - 24 q^{59} + 10 q^{61} - 4 q^{62} + 7 q^{64} + 4 q^{65} + 2 q^{67} - 2 q^{68} - 16 q^{70} + 14 q^{71} - 9 q^{73} - 30 q^{74} - 12 q^{76} - 10 q^{77} - 24 q^{79} - 3 q^{80} + 30 q^{82} + q^{83} - 6 q^{85} + 24 q^{86} - 40 q^{88} - 8 q^{89} - 4 q^{91} - 14 q^{92} + 8 q^{94} - 2 q^{95} + q^{97} + 23 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.56155 −1.81129 −0.905646 0.424035i \(-0.860613\pi\)
−0.905646 + 0.424035i \(0.860613\pi\)
\(3\) 0 0
\(4\) 4.56155 2.28078
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.12311 −1.18042 −0.590211 0.807249i \(-0.700956\pi\)
−0.590211 + 0.807249i \(0.700956\pi\)
\(8\) −6.56155 −2.31986
\(9\) 0 0
\(10\) 2.56155 0.810034
\(11\) 5.56155 1.67687 0.838436 0.545001i \(-0.183471\pi\)
0.838436 + 0.545001i \(0.183471\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 8.00000 2.13809
\(15\) 0 0
\(16\) 7.68466 1.92116
\(17\) −1.12311 −0.272393 −0.136197 0.990682i \(-0.543488\pi\)
−0.136197 + 0.990682i \(0.543488\pi\)
\(18\) 0 0
\(19\) −3.12311 −0.716490 −0.358245 0.933628i \(-0.616625\pi\)
−0.358245 + 0.933628i \(0.616625\pi\)
\(20\) −4.56155 −1.01999
\(21\) 0 0
\(22\) −14.2462 −3.03730
\(23\) −2.43845 −0.508451 −0.254226 0.967145i \(-0.581821\pi\)
−0.254226 + 0.967145i \(0.581821\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 5.12311 1.00472
\(27\) 0 0
\(28\) −14.2462 −2.69228
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −6.56155 −1.15993
\(33\) 0 0
\(34\) 2.87689 0.493383
\(35\) 3.12311 0.527901
\(36\) 0 0
\(37\) 10.6847 1.75655 0.878274 0.478159i \(-0.158696\pi\)
0.878274 + 0.478159i \(0.158696\pi\)
\(38\) 8.00000 1.29777
\(39\) 0 0
\(40\) 6.56155 1.03747
\(41\) −10.6847 −1.66866 −0.834332 0.551263i \(-0.814146\pi\)
−0.834332 + 0.551263i \(0.814146\pi\)
\(42\) 0 0
\(43\) −4.68466 −0.714404 −0.357202 0.934027i \(-0.616269\pi\)
−0.357202 + 0.934027i \(0.616269\pi\)
\(44\) 25.3693 3.82457
\(45\) 0 0
\(46\) 6.24621 0.920954
\(47\) 4.87689 0.711368 0.355684 0.934606i \(-0.384248\pi\)
0.355684 + 0.934606i \(0.384248\pi\)
\(48\) 0 0
\(49\) 2.75379 0.393398
\(50\) −2.56155 −0.362258
\(51\) 0 0
\(52\) −9.12311 −1.26515
\(53\) 7.56155 1.03866 0.519330 0.854574i \(-0.326182\pi\)
0.519330 + 0.854574i \(0.326182\pi\)
\(54\) 0 0
\(55\) −5.56155 −0.749920
\(56\) 20.4924 2.73842
\(57\) 0 0
\(58\) 2.56155 0.336348
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 9.12311 1.16809 0.584047 0.811720i \(-0.301468\pi\)
0.584047 + 0.811720i \(0.301468\pi\)
\(62\) −10.2462 −1.30127
\(63\) 0 0
\(64\) 1.43845 0.179806
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 13.3693 1.63332 0.816661 0.577118i \(-0.195823\pi\)
0.816661 + 0.577118i \(0.195823\pi\)
\(68\) −5.12311 −0.621268
\(69\) 0 0
\(70\) −8.00000 −0.956183
\(71\) 11.1231 1.32007 0.660035 0.751235i \(-0.270541\pi\)
0.660035 + 0.751235i \(0.270541\pi\)
\(72\) 0 0
\(73\) −10.6847 −1.25054 −0.625272 0.780407i \(-0.715012\pi\)
−0.625272 + 0.780407i \(0.715012\pi\)
\(74\) −27.3693 −3.18162
\(75\) 0 0
\(76\) −14.2462 −1.63415
\(77\) −17.3693 −1.97942
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −7.68466 −0.859171
\(81\) 0 0
\(82\) 27.3693 3.02244
\(83\) −1.56155 −0.171403 −0.0857013 0.996321i \(-0.527313\pi\)
−0.0857013 + 0.996321i \(0.527313\pi\)
\(84\) 0 0
\(85\) 1.12311 0.121818
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) −36.4924 −3.89011
\(89\) 4.24621 0.450097 0.225049 0.974348i \(-0.427746\pi\)
0.225049 + 0.974348i \(0.427746\pi\)
\(90\) 0 0
\(91\) 6.24621 0.654781
\(92\) −11.1231 −1.15966
\(93\) 0 0
\(94\) −12.4924 −1.28849
\(95\) 3.12311 0.320424
\(96\) 0 0
\(97\) 6.68466 0.678724 0.339362 0.940656i \(-0.389789\pi\)
0.339362 + 0.940656i \(0.389789\pi\)
\(98\) −7.05398 −0.712559
\(99\) 0 0
\(100\) 4.56155 0.456155
\(101\) 4.43845 0.441642 0.220821 0.975314i \(-0.429126\pi\)
0.220821 + 0.975314i \(0.429126\pi\)
\(102\) 0 0
\(103\) −11.1231 −1.09599 −0.547996 0.836481i \(-0.684609\pi\)
−0.547996 + 0.836481i \(0.684609\pi\)
\(104\) 13.1231 1.28683
\(105\) 0 0
\(106\) −19.3693 −1.88131
\(107\) 16.4924 1.59438 0.797191 0.603727i \(-0.206318\pi\)
0.797191 + 0.603727i \(0.206318\pi\)
\(108\) 0 0
\(109\) 6.68466 0.640274 0.320137 0.947371i \(-0.396271\pi\)
0.320137 + 0.947371i \(0.396271\pi\)
\(110\) 14.2462 1.35832
\(111\) 0 0
\(112\) −24.0000 −2.26779
\(113\) 13.1231 1.23452 0.617259 0.786760i \(-0.288243\pi\)
0.617259 + 0.786760i \(0.288243\pi\)
\(114\) 0 0
\(115\) 2.43845 0.227386
\(116\) −4.56155 −0.423530
\(117\) 0 0
\(118\) 30.7386 2.82972
\(119\) 3.50758 0.321539
\(120\) 0 0
\(121\) 19.9309 1.81190
\(122\) −23.3693 −2.11576
\(123\) 0 0
\(124\) 18.2462 1.63856
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.6847 1.48052 0.740262 0.672318i \(-0.234701\pi\)
0.740262 + 0.672318i \(0.234701\pi\)
\(128\) 9.43845 0.834249
\(129\) 0 0
\(130\) −5.12311 −0.449326
\(131\) 6.24621 0.545734 0.272867 0.962052i \(-0.412028\pi\)
0.272867 + 0.962052i \(0.412028\pi\)
\(132\) 0 0
\(133\) 9.75379 0.845761
\(134\) −34.2462 −2.95842
\(135\) 0 0
\(136\) 7.36932 0.631914
\(137\) 6.87689 0.587533 0.293766 0.955877i \(-0.405091\pi\)
0.293766 + 0.955877i \(0.405091\pi\)
\(138\) 0 0
\(139\) 10.9309 0.927144 0.463572 0.886059i \(-0.346567\pi\)
0.463572 + 0.886059i \(0.346567\pi\)
\(140\) 14.2462 1.20402
\(141\) 0 0
\(142\) −28.4924 −2.39103
\(143\) −11.1231 −0.930161
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 27.3693 2.26510
\(147\) 0 0
\(148\) 48.7386 4.00629
\(149\) 16.2462 1.33094 0.665471 0.746424i \(-0.268231\pi\)
0.665471 + 0.746424i \(0.268231\pi\)
\(150\) 0 0
\(151\) 16.6847 1.35778 0.678889 0.734241i \(-0.262462\pi\)
0.678889 + 0.734241i \(0.262462\pi\)
\(152\) 20.4924 1.66215
\(153\) 0 0
\(154\) 44.4924 3.58530
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 30.7386 2.44543
\(159\) 0 0
\(160\) 6.56155 0.518736
\(161\) 7.61553 0.600188
\(162\) 0 0
\(163\) 10.9309 0.856172 0.428086 0.903738i \(-0.359188\pi\)
0.428086 + 0.903738i \(0.359188\pi\)
\(164\) −48.7386 −3.80585
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.87689 −0.220648
\(171\) 0 0
\(172\) −21.3693 −1.62940
\(173\) −8.43845 −0.641563 −0.320782 0.947153i \(-0.603946\pi\)
−0.320782 + 0.947153i \(0.603946\pi\)
\(174\) 0 0
\(175\) −3.12311 −0.236085
\(176\) 42.7386 3.22155
\(177\) 0 0
\(178\) −10.8769 −0.815258
\(179\) 7.12311 0.532406 0.266203 0.963917i \(-0.414231\pi\)
0.266203 + 0.963917i \(0.414231\pi\)
\(180\) 0 0
\(181\) 13.3153 0.989722 0.494861 0.868972i \(-0.335219\pi\)
0.494861 + 0.868972i \(0.335219\pi\)
\(182\) −16.0000 −1.18600
\(183\) 0 0
\(184\) 16.0000 1.17954
\(185\) −10.6847 −0.785552
\(186\) 0 0
\(187\) −6.24621 −0.456768
\(188\) 22.2462 1.62247
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) −18.9309 −1.36979 −0.684895 0.728642i \(-0.740152\pi\)
−0.684895 + 0.728642i \(0.740152\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −17.1231 −1.22937
\(195\) 0 0
\(196\) 12.5616 0.897254
\(197\) −0.438447 −0.0312381 −0.0156190 0.999878i \(-0.504972\pi\)
−0.0156190 + 0.999878i \(0.504972\pi\)
\(198\) 0 0
\(199\) 26.0540 1.84692 0.923459 0.383698i \(-0.125350\pi\)
0.923459 + 0.383698i \(0.125350\pi\)
\(200\) −6.56155 −0.463972
\(201\) 0 0
\(202\) −11.3693 −0.799942
\(203\) 3.12311 0.219199
\(204\) 0 0
\(205\) 10.6847 0.746249
\(206\) 28.4924 1.98516
\(207\) 0 0
\(208\) −15.3693 −1.06567
\(209\) −17.3693 −1.20146
\(210\) 0 0
\(211\) −11.1231 −0.765746 −0.382873 0.923801i \(-0.625065\pi\)
−0.382873 + 0.923801i \(0.625065\pi\)
\(212\) 34.4924 2.36895
\(213\) 0 0
\(214\) −42.2462 −2.88789
\(215\) 4.68466 0.319491
\(216\) 0 0
\(217\) −12.4924 −0.848041
\(218\) −17.1231 −1.15972
\(219\) 0 0
\(220\) −25.3693 −1.71040
\(221\) 2.24621 0.151097
\(222\) 0 0
\(223\) −1.75379 −0.117442 −0.0587212 0.998274i \(-0.518702\pi\)
−0.0587212 + 0.998274i \(0.518702\pi\)
\(224\) 20.4924 1.36921
\(225\) 0 0
\(226\) −33.6155 −2.23607
\(227\) 23.8078 1.58018 0.790088 0.612993i \(-0.210035\pi\)
0.790088 + 0.612993i \(0.210035\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) −6.24621 −0.411863
\(231\) 0 0
\(232\) 6.56155 0.430787
\(233\) 22.6847 1.48612 0.743061 0.669224i \(-0.233373\pi\)
0.743061 + 0.669224i \(0.233373\pi\)
\(234\) 0 0
\(235\) −4.87689 −0.318134
\(236\) −54.7386 −3.56318
\(237\) 0 0
\(238\) −8.98485 −0.582401
\(239\) 4.87689 0.315460 0.157730 0.987482i \(-0.449582\pi\)
0.157730 + 0.987482i \(0.449582\pi\)
\(240\) 0 0
\(241\) 12.0540 0.776465 0.388232 0.921562i \(-0.373086\pi\)
0.388232 + 0.921562i \(0.373086\pi\)
\(242\) −51.0540 −3.28187
\(243\) 0 0
\(244\) 41.6155 2.66416
\(245\) −2.75379 −0.175933
\(246\) 0 0
\(247\) 6.24621 0.397437
\(248\) −26.2462 −1.66664
\(249\) 0 0
\(250\) 2.56155 0.162007
\(251\) 14.2462 0.899213 0.449606 0.893227i \(-0.351564\pi\)
0.449606 + 0.893227i \(0.351564\pi\)
\(252\) 0 0
\(253\) −13.5616 −0.852608
\(254\) −42.7386 −2.68166
\(255\) 0 0
\(256\) −27.0540 −1.69087
\(257\) −15.5616 −0.970703 −0.485351 0.874319i \(-0.661308\pi\)
−0.485351 + 0.874319i \(0.661308\pi\)
\(258\) 0 0
\(259\) −33.3693 −2.07347
\(260\) 9.12311 0.565791
\(261\) 0 0
\(262\) −16.0000 −0.988483
\(263\) 22.2462 1.37176 0.685880 0.727715i \(-0.259417\pi\)
0.685880 + 0.727715i \(0.259417\pi\)
\(264\) 0 0
\(265\) −7.56155 −0.464502
\(266\) −24.9848 −1.53192
\(267\) 0 0
\(268\) 60.9848 3.72524
\(269\) −28.2462 −1.72220 −0.861101 0.508434i \(-0.830225\pi\)
−0.861101 + 0.508434i \(0.830225\pi\)
\(270\) 0 0
\(271\) 8.87689 0.539233 0.269616 0.962968i \(-0.413103\pi\)
0.269616 + 0.962968i \(0.413103\pi\)
\(272\) −8.63068 −0.523312
\(273\) 0 0
\(274\) −17.6155 −1.06419
\(275\) 5.56155 0.335374
\(276\) 0 0
\(277\) 9.12311 0.548154 0.274077 0.961708i \(-0.411628\pi\)
0.274077 + 0.961708i \(0.411628\pi\)
\(278\) −28.0000 −1.67933
\(279\) 0 0
\(280\) −20.4924 −1.22466
\(281\) −17.6155 −1.05085 −0.525427 0.850839i \(-0.676094\pi\)
−0.525427 + 0.850839i \(0.676094\pi\)
\(282\) 0 0
\(283\) 15.1231 0.898975 0.449488 0.893287i \(-0.351607\pi\)
0.449488 + 0.893287i \(0.351607\pi\)
\(284\) 50.7386 3.01078
\(285\) 0 0
\(286\) 28.4924 1.68479
\(287\) 33.3693 1.96973
\(288\) 0 0
\(289\) −15.7386 −0.925802
\(290\) −2.56155 −0.150420
\(291\) 0 0
\(292\) −48.7386 −2.85221
\(293\) −3.36932 −0.196838 −0.0984188 0.995145i \(-0.531378\pi\)
−0.0984188 + 0.995145i \(0.531378\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) −70.1080 −4.07494
\(297\) 0 0
\(298\) −41.6155 −2.41072
\(299\) 4.87689 0.282038
\(300\) 0 0
\(301\) 14.6307 0.843299
\(302\) −42.7386 −2.45933
\(303\) 0 0
\(304\) −24.0000 −1.37649
\(305\) −9.12311 −0.522388
\(306\) 0 0
\(307\) −19.3153 −1.10238 −0.551192 0.834378i \(-0.685827\pi\)
−0.551192 + 0.834378i \(0.685827\pi\)
\(308\) −79.2311 −4.51461
\(309\) 0 0
\(310\) 10.2462 0.581946
\(311\) −28.6847 −1.62656 −0.813279 0.581874i \(-0.802320\pi\)
−0.813279 + 0.581874i \(0.802320\pi\)
\(312\) 0 0
\(313\) 3.36932 0.190445 0.0952225 0.995456i \(-0.469644\pi\)
0.0952225 + 0.995456i \(0.469644\pi\)
\(314\) 35.8617 2.02380
\(315\) 0 0
\(316\) −54.7386 −3.07929
\(317\) −13.1231 −0.737067 −0.368534 0.929614i \(-0.620140\pi\)
−0.368534 + 0.929614i \(0.620140\pi\)
\(318\) 0 0
\(319\) −5.56155 −0.311387
\(320\) −1.43845 −0.0804116
\(321\) 0 0
\(322\) −19.5076 −1.08711
\(323\) 3.50758 0.195167
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) −28.0000 −1.55078
\(327\) 0 0
\(328\) 70.1080 3.87106
\(329\) −15.2311 −0.839715
\(330\) 0 0
\(331\) −6.24621 −0.343323 −0.171661 0.985156i \(-0.554914\pi\)
−0.171661 + 0.985156i \(0.554914\pi\)
\(332\) −7.12311 −0.390931
\(333\) 0 0
\(334\) 0 0
\(335\) −13.3693 −0.730444
\(336\) 0 0
\(337\) −24.2462 −1.32078 −0.660388 0.750925i \(-0.729608\pi\)
−0.660388 + 0.750925i \(0.729608\pi\)
\(338\) 23.0540 1.25397
\(339\) 0 0
\(340\) 5.12311 0.277839
\(341\) 22.2462 1.20470
\(342\) 0 0
\(343\) 13.2614 0.716046
\(344\) 30.7386 1.65732
\(345\) 0 0
\(346\) 21.6155 1.16206
\(347\) −33.1771 −1.78104 −0.890520 0.454945i \(-0.849659\pi\)
−0.890520 + 0.454945i \(0.849659\pi\)
\(348\) 0 0
\(349\) 28.9309 1.54863 0.774317 0.632798i \(-0.218094\pi\)
0.774317 + 0.632798i \(0.218094\pi\)
\(350\) 8.00000 0.427618
\(351\) 0 0
\(352\) −36.4924 −1.94505
\(353\) 2.49242 0.132658 0.0663291 0.997798i \(-0.478871\pi\)
0.0663291 + 0.997798i \(0.478871\pi\)
\(354\) 0 0
\(355\) −11.1231 −0.590353
\(356\) 19.3693 1.02657
\(357\) 0 0
\(358\) −18.2462 −0.964342
\(359\) 23.8078 1.25653 0.628263 0.778001i \(-0.283766\pi\)
0.628263 + 0.778001i \(0.283766\pi\)
\(360\) 0 0
\(361\) −9.24621 −0.486643
\(362\) −34.1080 −1.79267
\(363\) 0 0
\(364\) 28.4924 1.49341
\(365\) 10.6847 0.559261
\(366\) 0 0
\(367\) −7.31534 −0.381858 −0.190929 0.981604i \(-0.561150\pi\)
−0.190929 + 0.981604i \(0.561150\pi\)
\(368\) −18.7386 −0.976819
\(369\) 0 0
\(370\) 27.3693 1.42286
\(371\) −23.6155 −1.22606
\(372\) 0 0
\(373\) −11.3693 −0.588681 −0.294340 0.955701i \(-0.595100\pi\)
−0.294340 + 0.955701i \(0.595100\pi\)
\(374\) 16.0000 0.827340
\(375\) 0 0
\(376\) −32.0000 −1.65027
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 14.2462 0.730815
\(381\) 0 0
\(382\) 48.4924 2.48109
\(383\) 26.0540 1.33130 0.665648 0.746266i \(-0.268155\pi\)
0.665648 + 0.746266i \(0.268155\pi\)
\(384\) 0 0
\(385\) 17.3693 0.885222
\(386\) 25.6155 1.30380
\(387\) 0 0
\(388\) 30.4924 1.54802
\(389\) 5.80776 0.294465 0.147233 0.989102i \(-0.452963\pi\)
0.147233 + 0.989102i \(0.452963\pi\)
\(390\) 0 0
\(391\) 2.73863 0.138499
\(392\) −18.0691 −0.912629
\(393\) 0 0
\(394\) 1.12311 0.0565812
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) 7.75379 0.389152 0.194576 0.980887i \(-0.437667\pi\)
0.194576 + 0.980887i \(0.437667\pi\)
\(398\) −66.7386 −3.34531
\(399\) 0 0
\(400\) 7.68466 0.384233
\(401\) 5.61553 0.280426 0.140213 0.990121i \(-0.455221\pi\)
0.140213 + 0.990121i \(0.455221\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 20.2462 1.00729
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) 59.4233 2.94550
\(408\) 0 0
\(409\) −7.36932 −0.364389 −0.182195 0.983262i \(-0.558320\pi\)
−0.182195 + 0.983262i \(0.558320\pi\)
\(410\) −27.3693 −1.35167
\(411\) 0 0
\(412\) −50.7386 −2.49971
\(413\) 37.4773 1.84414
\(414\) 0 0
\(415\) 1.56155 0.0766536
\(416\) 13.1231 0.643413
\(417\) 0 0
\(418\) 44.4924 2.17620
\(419\) 2.63068 0.128517 0.0642586 0.997933i \(-0.479532\pi\)
0.0642586 + 0.997933i \(0.479532\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 28.4924 1.38699
\(423\) 0 0
\(424\) −49.6155 −2.40954
\(425\) −1.12311 −0.0544786
\(426\) 0 0
\(427\) −28.4924 −1.37884
\(428\) 75.2311 3.63643
\(429\) 0 0
\(430\) −12.0000 −0.578691
\(431\) 12.8769 0.620258 0.310129 0.950694i \(-0.399628\pi\)
0.310129 + 0.950694i \(0.399628\pi\)
\(432\) 0 0
\(433\) −32.9309 −1.58256 −0.791278 0.611456i \(-0.790584\pi\)
−0.791278 + 0.611456i \(0.790584\pi\)
\(434\) 32.0000 1.53605
\(435\) 0 0
\(436\) 30.4924 1.46032
\(437\) 7.61553 0.364300
\(438\) 0 0
\(439\) −20.4924 −0.978050 −0.489025 0.872270i \(-0.662647\pi\)
−0.489025 + 0.872270i \(0.662647\pi\)
\(440\) 36.4924 1.73971
\(441\) 0 0
\(442\) −5.75379 −0.273680
\(443\) 10.2462 0.486812 0.243406 0.969924i \(-0.421735\pi\)
0.243406 + 0.969924i \(0.421735\pi\)
\(444\) 0 0
\(445\) −4.24621 −0.201290
\(446\) 4.49242 0.212722
\(447\) 0 0
\(448\) −4.49242 −0.212247
\(449\) −15.5616 −0.734395 −0.367198 0.930143i \(-0.619683\pi\)
−0.367198 + 0.930143i \(0.619683\pi\)
\(450\) 0 0
\(451\) −59.4233 −2.79813
\(452\) 59.8617 2.81566
\(453\) 0 0
\(454\) −60.9848 −2.86216
\(455\) −6.24621 −0.292827
\(456\) 0 0
\(457\) −17.1231 −0.800985 −0.400493 0.916300i \(-0.631161\pi\)
−0.400493 + 0.916300i \(0.631161\pi\)
\(458\) −56.3542 −2.63326
\(459\) 0 0
\(460\) 11.1231 0.518617
\(461\) −27.1771 −1.26576 −0.632881 0.774249i \(-0.718128\pi\)
−0.632881 + 0.774249i \(0.718128\pi\)
\(462\) 0 0
\(463\) −15.6155 −0.725715 −0.362858 0.931845i \(-0.618199\pi\)
−0.362858 + 0.931845i \(0.618199\pi\)
\(464\) −7.68466 −0.356751
\(465\) 0 0
\(466\) −58.1080 −2.69180
\(467\) −21.3693 −0.988854 −0.494427 0.869219i \(-0.664622\pi\)
−0.494427 + 0.869219i \(0.664622\pi\)
\(468\) 0 0
\(469\) −41.7538 −1.92801
\(470\) 12.4924 0.576232
\(471\) 0 0
\(472\) 78.7386 3.62424
\(473\) −26.0540 −1.19796
\(474\) 0 0
\(475\) −3.12311 −0.143298
\(476\) 16.0000 0.733359
\(477\) 0 0
\(478\) −12.4924 −0.571390
\(479\) 5.75379 0.262897 0.131449 0.991323i \(-0.458037\pi\)
0.131449 + 0.991323i \(0.458037\pi\)
\(480\) 0 0
\(481\) −21.3693 −0.974357
\(482\) −30.8769 −1.40640
\(483\) 0 0
\(484\) 90.9157 4.13253
\(485\) −6.68466 −0.303535
\(486\) 0 0
\(487\) 17.7538 0.804501 0.402250 0.915530i \(-0.368228\pi\)
0.402250 + 0.915530i \(0.368228\pi\)
\(488\) −59.8617 −2.70981
\(489\) 0 0
\(490\) 7.05398 0.318666
\(491\) 42.7386 1.92877 0.964384 0.264507i \(-0.0852092\pi\)
0.964384 + 0.264507i \(0.0852092\pi\)
\(492\) 0 0
\(493\) 1.12311 0.0505821
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) 30.7386 1.38021
\(497\) −34.7386 −1.55824
\(498\) 0 0
\(499\) −6.73863 −0.301663 −0.150831 0.988560i \(-0.548195\pi\)
−0.150831 + 0.988560i \(0.548195\pi\)
\(500\) −4.56155 −0.203999
\(501\) 0 0
\(502\) −36.4924 −1.62874
\(503\) 14.6307 0.652350 0.326175 0.945309i \(-0.394240\pi\)
0.326175 + 0.945309i \(0.394240\pi\)
\(504\) 0 0
\(505\) −4.43845 −0.197508
\(506\) 34.7386 1.54432
\(507\) 0 0
\(508\) 76.1080 3.37674
\(509\) 14.4924 0.642365 0.321183 0.947017i \(-0.395920\pi\)
0.321183 + 0.947017i \(0.395920\pi\)
\(510\) 0 0
\(511\) 33.3693 1.47617
\(512\) 50.4233 2.22842
\(513\) 0 0
\(514\) 39.8617 1.75823
\(515\) 11.1231 0.490143
\(516\) 0 0
\(517\) 27.1231 1.19287
\(518\) 85.4773 3.75566
\(519\) 0 0
\(520\) −13.1231 −0.575486
\(521\) −16.2462 −0.711759 −0.355880 0.934532i \(-0.615819\pi\)
−0.355880 + 0.934532i \(0.615819\pi\)
\(522\) 0 0
\(523\) −30.7386 −1.34411 −0.672053 0.740503i \(-0.734587\pi\)
−0.672053 + 0.740503i \(0.734587\pi\)
\(524\) 28.4924 1.24470
\(525\) 0 0
\(526\) −56.9848 −2.48466
\(527\) −4.49242 −0.195693
\(528\) 0 0
\(529\) −17.0540 −0.741477
\(530\) 19.3693 0.841349
\(531\) 0 0
\(532\) 44.4924 1.92899
\(533\) 21.3693 0.925608
\(534\) 0 0
\(535\) −16.4924 −0.713030
\(536\) −87.7235 −3.78908
\(537\) 0 0
\(538\) 72.3542 3.11941
\(539\) 15.3153 0.659678
\(540\) 0 0
\(541\) 8.73863 0.375703 0.187852 0.982197i \(-0.439848\pi\)
0.187852 + 0.982197i \(0.439848\pi\)
\(542\) −22.7386 −0.976708
\(543\) 0 0
\(544\) 7.36932 0.315957
\(545\) −6.68466 −0.286339
\(546\) 0 0
\(547\) 6.73863 0.288123 0.144062 0.989569i \(-0.453984\pi\)
0.144062 + 0.989569i \(0.453984\pi\)
\(548\) 31.3693 1.34003
\(549\) 0 0
\(550\) −14.2462 −0.607460
\(551\) 3.12311 0.133049
\(552\) 0 0
\(553\) 37.4773 1.59370
\(554\) −23.3693 −0.992867
\(555\) 0 0
\(556\) 49.8617 2.11461
\(557\) −41.4233 −1.75516 −0.877581 0.479429i \(-0.840844\pi\)
−0.877581 + 0.479429i \(0.840844\pi\)
\(558\) 0 0
\(559\) 9.36932 0.396280
\(560\) 24.0000 1.01419
\(561\) 0 0
\(562\) 45.1231 1.90340
\(563\) −0.876894 −0.0369567 −0.0184783 0.999829i \(-0.505882\pi\)
−0.0184783 + 0.999829i \(0.505882\pi\)
\(564\) 0 0
\(565\) −13.1231 −0.552093
\(566\) −38.7386 −1.62831
\(567\) 0 0
\(568\) −72.9848 −3.06238
\(569\) −0.246211 −0.0103217 −0.00516086 0.999987i \(-0.501643\pi\)
−0.00516086 + 0.999987i \(0.501643\pi\)
\(570\) 0 0
\(571\) 1.94602 0.0814386 0.0407193 0.999171i \(-0.487035\pi\)
0.0407193 + 0.999171i \(0.487035\pi\)
\(572\) −50.7386 −2.12149
\(573\) 0 0
\(574\) −85.4773 −3.56775
\(575\) −2.43845 −0.101690
\(576\) 0 0
\(577\) 2.49242 0.103761 0.0518805 0.998653i \(-0.483479\pi\)
0.0518805 + 0.998653i \(0.483479\pi\)
\(578\) 40.3153 1.67690
\(579\) 0 0
\(580\) 4.56155 0.189408
\(581\) 4.87689 0.202328
\(582\) 0 0
\(583\) 42.0540 1.74170
\(584\) 70.1080 2.90109
\(585\) 0 0
\(586\) 8.63068 0.356530
\(587\) 8.49242 0.350520 0.175260 0.984522i \(-0.443923\pi\)
0.175260 + 0.984522i \(0.443923\pi\)
\(588\) 0 0
\(589\) −12.4924 −0.514741
\(590\) −30.7386 −1.26549
\(591\) 0 0
\(592\) 82.1080 3.37462
\(593\) −19.7538 −0.811191 −0.405595 0.914053i \(-0.632936\pi\)
−0.405595 + 0.914053i \(0.632936\pi\)
\(594\) 0 0
\(595\) −3.50758 −0.143797
\(596\) 74.1080 3.03558
\(597\) 0 0
\(598\) −12.4924 −0.510853
\(599\) −44.9848 −1.83803 −0.919015 0.394221i \(-0.871014\pi\)
−0.919015 + 0.394221i \(0.871014\pi\)
\(600\) 0 0
\(601\) 16.6307 0.678380 0.339190 0.940718i \(-0.389847\pi\)
0.339190 + 0.940718i \(0.389847\pi\)
\(602\) −37.4773 −1.52746
\(603\) 0 0
\(604\) 76.1080 3.09679
\(605\) −19.9309 −0.810305
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 20.4924 0.831077
\(609\) 0 0
\(610\) 23.3693 0.946196
\(611\) −9.75379 −0.394596
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 49.4773 1.99674
\(615\) 0 0
\(616\) 113.970 4.59197
\(617\) 8.24621 0.331980 0.165990 0.986127i \(-0.446918\pi\)
0.165990 + 0.986127i \(0.446918\pi\)
\(618\) 0 0
\(619\) 20.8769 0.839113 0.419557 0.907729i \(-0.362186\pi\)
0.419557 + 0.907729i \(0.362186\pi\)
\(620\) −18.2462 −0.732785
\(621\) 0 0
\(622\) 73.4773 2.94617
\(623\) −13.2614 −0.531305
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −8.63068 −0.344951
\(627\) 0 0
\(628\) −63.8617 −2.54836
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 78.7386 3.13205
\(633\) 0 0
\(634\) 33.6155 1.33504
\(635\) −16.6847 −0.662110
\(636\) 0 0
\(637\) −5.50758 −0.218218
\(638\) 14.2462 0.564013
\(639\) 0 0
\(640\) −9.43845 −0.373087
\(641\) −23.5616 −0.930625 −0.465313 0.885146i \(-0.654058\pi\)
−0.465313 + 0.885146i \(0.654058\pi\)
\(642\) 0 0
\(643\) 30.7386 1.21221 0.606107 0.795383i \(-0.292730\pi\)
0.606107 + 0.795383i \(0.292730\pi\)
\(644\) 34.7386 1.36889
\(645\) 0 0
\(646\) −8.98485 −0.353504
\(647\) 22.9309 0.901506 0.450753 0.892649i \(-0.351156\pi\)
0.450753 + 0.892649i \(0.351156\pi\)
\(648\) 0 0
\(649\) −66.7386 −2.61972
\(650\) 5.12311 0.200945
\(651\) 0 0
\(652\) 49.8617 1.95274
\(653\) 4.24621 0.166167 0.0830835 0.996543i \(-0.473523\pi\)
0.0830835 + 0.996543i \(0.473523\pi\)
\(654\) 0 0
\(655\) −6.24621 −0.244060
\(656\) −82.1080 −3.20578
\(657\) 0 0
\(658\) 39.0152 1.52097
\(659\) −2.43845 −0.0949884 −0.0474942 0.998872i \(-0.515124\pi\)
−0.0474942 + 0.998872i \(0.515124\pi\)
\(660\) 0 0
\(661\) −36.0540 −1.40234 −0.701169 0.712996i \(-0.747338\pi\)
−0.701169 + 0.712996i \(0.747338\pi\)
\(662\) 16.0000 0.621858
\(663\) 0 0
\(664\) 10.2462 0.397630
\(665\) −9.75379 −0.378236
\(666\) 0 0
\(667\) 2.43845 0.0944171
\(668\) 0 0
\(669\) 0 0
\(670\) 34.2462 1.32305
\(671\) 50.7386 1.95874
\(672\) 0 0
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) 62.1080 2.39231
\(675\) 0 0
\(676\) −41.0540 −1.57900
\(677\) 43.8617 1.68574 0.842872 0.538114i \(-0.180863\pi\)
0.842872 + 0.538114i \(0.180863\pi\)
\(678\) 0 0
\(679\) −20.8769 −0.801182
\(680\) −7.36932 −0.282600
\(681\) 0 0
\(682\) −56.9848 −2.18206
\(683\) 25.5616 0.978086 0.489043 0.872260i \(-0.337346\pi\)
0.489043 + 0.872260i \(0.337346\pi\)
\(684\) 0 0
\(685\) −6.87689 −0.262753
\(686\) −33.9697 −1.29697
\(687\) 0 0
\(688\) −36.0000 −1.37249
\(689\) −15.1231 −0.576144
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) −38.4924 −1.46326
\(693\) 0 0
\(694\) 84.9848 3.22598
\(695\) −10.9309 −0.414632
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) −74.1080 −2.80503
\(699\) 0 0
\(700\) −14.2462 −0.538456
\(701\) −20.2462 −0.764689 −0.382344 0.924020i \(-0.624883\pi\)
−0.382344 + 0.924020i \(0.624883\pi\)
\(702\) 0 0
\(703\) −33.3693 −1.25855
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) −6.38447 −0.240283
\(707\) −13.8617 −0.521324
\(708\) 0 0
\(709\) 49.4233 1.85613 0.928065 0.372417i \(-0.121471\pi\)
0.928065 + 0.372417i \(0.121471\pi\)
\(710\) 28.4924 1.06930
\(711\) 0 0
\(712\) −27.8617 −1.04416
\(713\) −9.75379 −0.365282
\(714\) 0 0
\(715\) 11.1231 0.415981
\(716\) 32.4924 1.21430
\(717\) 0 0
\(718\) −60.9848 −2.27593
\(719\) −28.4924 −1.06259 −0.531294 0.847187i \(-0.678294\pi\)
−0.531294 + 0.847187i \(0.678294\pi\)
\(720\) 0 0
\(721\) 34.7386 1.29373
\(722\) 23.6847 0.881452
\(723\) 0 0
\(724\) 60.7386 2.25733
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) 10.7386 0.398274 0.199137 0.979972i \(-0.436186\pi\)
0.199137 + 0.979972i \(0.436186\pi\)
\(728\) −40.9848 −1.51900
\(729\) 0 0
\(730\) −27.3693 −1.01298
\(731\) 5.26137 0.194599
\(732\) 0 0
\(733\) 42.9848 1.58768 0.793841 0.608126i \(-0.208078\pi\)
0.793841 + 0.608126i \(0.208078\pi\)
\(734\) 18.7386 0.691656
\(735\) 0 0
\(736\) 16.0000 0.589768
\(737\) 74.3542 2.73887
\(738\) 0 0
\(739\) −24.0000 −0.882854 −0.441427 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) −48.7386 −1.79167
\(741\) 0 0
\(742\) 60.4924 2.22075
\(743\) −43.1231 −1.58203 −0.791017 0.611795i \(-0.790448\pi\)
−0.791017 + 0.611795i \(0.790448\pi\)
\(744\) 0 0
\(745\) −16.2462 −0.595215
\(746\) 29.1231 1.06627
\(747\) 0 0
\(748\) −28.4924 −1.04179
\(749\) −51.5076 −1.88205
\(750\) 0 0
\(751\) 15.5076 0.565880 0.282940 0.959138i \(-0.408690\pi\)
0.282940 + 0.959138i \(0.408690\pi\)
\(752\) 37.4773 1.36666
\(753\) 0 0
\(754\) −5.12311 −0.186573
\(755\) −16.6847 −0.607217
\(756\) 0 0
\(757\) −6.68466 −0.242958 −0.121479 0.992594i \(-0.538764\pi\)
−0.121479 + 0.992594i \(0.538764\pi\)
\(758\) −20.4924 −0.744318
\(759\) 0 0
\(760\) −20.4924 −0.743338
\(761\) 33.1231 1.20071 0.600356 0.799733i \(-0.295026\pi\)
0.600356 + 0.799733i \(0.295026\pi\)
\(762\) 0 0
\(763\) −20.8769 −0.755794
\(764\) −86.3542 −3.12418
\(765\) 0 0
\(766\) −66.7386 −2.41136
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) 2.38447 0.0859863 0.0429931 0.999075i \(-0.486311\pi\)
0.0429931 + 0.999075i \(0.486311\pi\)
\(770\) −44.4924 −1.60340
\(771\) 0 0
\(772\) −45.6155 −1.64174
\(773\) 24.7386 0.889787 0.444893 0.895584i \(-0.353242\pi\)
0.444893 + 0.895584i \(0.353242\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) −43.8617 −1.57454
\(777\) 0 0
\(778\) −14.8769 −0.533363
\(779\) 33.3693 1.19558
\(780\) 0 0
\(781\) 61.8617 2.21359
\(782\) −7.01515 −0.250861
\(783\) 0 0
\(784\) 21.1619 0.755783
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) 51.2311 1.82619 0.913095 0.407747i \(-0.133685\pi\)
0.913095 + 0.407747i \(0.133685\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) −30.7386 −1.09363
\(791\) −40.9848 −1.45725
\(792\) 0 0
\(793\) −18.2462 −0.647942
\(794\) −19.8617 −0.704867
\(795\) 0 0
\(796\) 118.847 4.21241
\(797\) −5.50758 −0.195088 −0.0975442 0.995231i \(-0.531099\pi\)
−0.0975442 + 0.995231i \(0.531099\pi\)
\(798\) 0 0
\(799\) −5.47727 −0.193772
\(800\) −6.56155 −0.231986
\(801\) 0 0
\(802\) −14.3845 −0.507933
\(803\) −59.4233 −2.09700
\(804\) 0 0
\(805\) −7.61553 −0.268412
\(806\) 20.4924 0.721815
\(807\) 0 0
\(808\) −29.1231 −1.02455
\(809\) −23.1771 −0.814863 −0.407431 0.913236i \(-0.633575\pi\)
−0.407431 + 0.913236i \(0.633575\pi\)
\(810\) 0 0
\(811\) −1.17708 −0.0413329 −0.0206665 0.999786i \(-0.506579\pi\)
−0.0206665 + 0.999786i \(0.506579\pi\)
\(812\) 14.2462 0.499944
\(813\) 0 0
\(814\) −152.216 −5.33516
\(815\) −10.9309 −0.382892
\(816\) 0 0
\(817\) 14.6307 0.511863
\(818\) 18.8769 0.660015
\(819\) 0 0
\(820\) 48.7386 1.70203
\(821\) −14.3845 −0.502022 −0.251011 0.967984i \(-0.580763\pi\)
−0.251011 + 0.967984i \(0.580763\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 72.9848 2.54255
\(825\) 0 0
\(826\) −96.0000 −3.34027
\(827\) −12.3845 −0.430650 −0.215325 0.976542i \(-0.569081\pi\)
−0.215325 + 0.976542i \(0.569081\pi\)
\(828\) 0 0
\(829\) 12.2462 0.425328 0.212664 0.977125i \(-0.431786\pi\)
0.212664 + 0.977125i \(0.431786\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) −2.87689 −0.0997384
\(833\) −3.09280 −0.107159
\(834\) 0 0
\(835\) 0 0
\(836\) −79.2311 −2.74026
\(837\) 0 0
\(838\) −6.73863 −0.232782
\(839\) 6.73863 0.232643 0.116322 0.993212i \(-0.462890\pi\)
0.116322 + 0.993212i \(0.462890\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 25.6155 0.882769
\(843\) 0 0
\(844\) −50.7386 −1.74650
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −62.2462 −2.13881
\(848\) 58.1080 1.99544
\(849\) 0 0
\(850\) 2.87689 0.0986767
\(851\) −26.0540 −0.893119
\(852\) 0 0
\(853\) 23.5616 0.806732 0.403366 0.915039i \(-0.367840\pi\)
0.403366 + 0.915039i \(0.367840\pi\)
\(854\) 72.9848 2.49749
\(855\) 0 0
\(856\) −108.216 −3.69874
\(857\) 19.5616 0.668210 0.334105 0.942536i \(-0.391566\pi\)
0.334105 + 0.942536i \(0.391566\pi\)
\(858\) 0 0
\(859\) 1.36932 0.0467205 0.0233602 0.999727i \(-0.492564\pi\)
0.0233602 + 0.999727i \(0.492564\pi\)
\(860\) 21.3693 0.728688
\(861\) 0 0
\(862\) −32.9848 −1.12347
\(863\) 24.9848 0.850494 0.425247 0.905077i \(-0.360187\pi\)
0.425247 + 0.905077i \(0.360187\pi\)
\(864\) 0 0
\(865\) 8.43845 0.286916
\(866\) 84.3542 2.86647
\(867\) 0 0
\(868\) −56.9848 −1.93419
\(869\) −66.7386 −2.26395
\(870\) 0 0
\(871\) −26.7386 −0.906004
\(872\) −43.8617 −1.48535
\(873\) 0 0
\(874\) −19.5076 −0.659854
\(875\) 3.12311 0.105580
\(876\) 0 0
\(877\) 17.1231 0.578206 0.289103 0.957298i \(-0.406643\pi\)
0.289103 + 0.957298i \(0.406643\pi\)
\(878\) 52.4924 1.77153
\(879\) 0 0
\(880\) −42.7386 −1.44072
\(881\) 36.9309 1.24423 0.622116 0.782925i \(-0.286273\pi\)
0.622116 + 0.782925i \(0.286273\pi\)
\(882\) 0 0
\(883\) 6.73863 0.226773 0.113387 0.993551i \(-0.463830\pi\)
0.113387 + 0.993551i \(0.463830\pi\)
\(884\) 10.2462 0.344617
\(885\) 0 0
\(886\) −26.2462 −0.881759
\(887\) −13.8617 −0.465432 −0.232716 0.972545i \(-0.574761\pi\)
−0.232716 + 0.972545i \(0.574761\pi\)
\(888\) 0 0
\(889\) −52.1080 −1.74764
\(890\) 10.8769 0.364594
\(891\) 0 0
\(892\) −8.00000 −0.267860
\(893\) −15.2311 −0.509688
\(894\) 0 0
\(895\) −7.12311 −0.238099
\(896\) −29.4773 −0.984766
\(897\) 0 0
\(898\) 39.8617 1.33020
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) −8.49242 −0.282924
\(902\) 152.216 5.06824
\(903\) 0 0
\(904\) −86.1080 −2.86391
\(905\) −13.3153 −0.442617
\(906\) 0 0
\(907\) −6.05398 −0.201019 −0.100509 0.994936i \(-0.532047\pi\)
−0.100509 + 0.994936i \(0.532047\pi\)
\(908\) 108.600 3.60403
\(909\) 0 0
\(910\) 16.0000 0.530395
\(911\) 14.4384 0.478367 0.239184 0.970974i \(-0.423120\pi\)
0.239184 + 0.970974i \(0.423120\pi\)
\(912\) 0 0
\(913\) −8.68466 −0.287420
\(914\) 43.8617 1.45082
\(915\) 0 0
\(916\) 100.354 3.31579
\(917\) −19.5076 −0.644197
\(918\) 0 0
\(919\) −18.7386 −0.618130 −0.309065 0.951041i \(-0.600016\pi\)
−0.309065 + 0.951041i \(0.600016\pi\)
\(920\) −16.0000 −0.527504
\(921\) 0 0
\(922\) 69.6155 2.29267
\(923\) −22.2462 −0.732243
\(924\) 0 0
\(925\) 10.6847 0.351309
\(926\) 40.0000 1.31448
\(927\) 0 0
\(928\) 6.56155 0.215394
\(929\) −13.1231 −0.430555 −0.215278 0.976553i \(-0.569066\pi\)
−0.215278 + 0.976553i \(0.569066\pi\)
\(930\) 0 0
\(931\) −8.60037 −0.281866
\(932\) 103.477 3.38951
\(933\) 0 0
\(934\) 54.7386 1.79110
\(935\) 6.24621 0.204273
\(936\) 0 0
\(937\) 30.8769 1.00870 0.504352 0.863498i \(-0.331731\pi\)
0.504352 + 0.863498i \(0.331731\pi\)
\(938\) 106.955 3.49219
\(939\) 0 0
\(940\) −22.2462 −0.725591
\(941\) −48.7386 −1.58883 −0.794417 0.607373i \(-0.792223\pi\)
−0.794417 + 0.607373i \(0.792223\pi\)
\(942\) 0 0
\(943\) 26.0540 0.848434
\(944\) −92.2159 −3.00137
\(945\) 0 0
\(946\) 66.7386 2.16986
\(947\) −21.3693 −0.694409 −0.347205 0.937789i \(-0.612869\pi\)
−0.347205 + 0.937789i \(0.612869\pi\)
\(948\) 0 0
\(949\) 21.3693 0.693677
\(950\) 8.00000 0.259554
\(951\) 0 0
\(952\) −23.0152 −0.745925
\(953\) 1.50758 0.0488352 0.0244176 0.999702i \(-0.492227\pi\)
0.0244176 + 0.999702i \(0.492227\pi\)
\(954\) 0 0
\(955\) 18.9309 0.612589
\(956\) 22.2462 0.719494
\(957\) 0 0
\(958\) −14.7386 −0.476184
\(959\) −21.4773 −0.693537
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 54.7386 1.76484
\(963\) 0 0
\(964\) 54.9848 1.77094
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) −10.0540 −0.323314 −0.161657 0.986847i \(-0.551684\pi\)
−0.161657 + 0.986847i \(0.551684\pi\)
\(968\) −130.777 −4.20335
\(969\) 0 0
\(970\) 17.1231 0.549790
\(971\) −33.6695 −1.08051 −0.540253 0.841503i \(-0.681671\pi\)
−0.540253 + 0.841503i \(0.681671\pi\)
\(972\) 0 0
\(973\) −34.1383 −1.09442
\(974\) −45.4773 −1.45719
\(975\) 0 0
\(976\) 70.1080 2.24410
\(977\) 11.5616 0.369887 0.184943 0.982749i \(-0.440790\pi\)
0.184943 + 0.982749i \(0.440790\pi\)
\(978\) 0 0
\(979\) 23.6155 0.754756
\(980\) −12.5616 −0.401264
\(981\) 0 0
\(982\) −109.477 −3.49356
\(983\) 29.8617 0.952442 0.476221 0.879326i \(-0.342006\pi\)
0.476221 + 0.879326i \(0.342006\pi\)
\(984\) 0 0
\(985\) 0.438447 0.0139701
\(986\) −2.87689 −0.0916190
\(987\) 0 0
\(988\) 28.4924 0.906465
\(989\) 11.4233 0.363240
\(990\) 0 0
\(991\) 42.0540 1.33589 0.667944 0.744211i \(-0.267174\pi\)
0.667944 + 0.744211i \(0.267174\pi\)
\(992\) −26.2462 −0.833318
\(993\) 0 0
\(994\) 88.9848 2.82243
\(995\) −26.0540 −0.825967
\(996\) 0 0
\(997\) −49.0388 −1.55308 −0.776538 0.630071i \(-0.783026\pi\)
−0.776538 + 0.630071i \(0.783026\pi\)
\(998\) 17.2614 0.546399
\(999\) 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 1305.2.a.i.1.1 2
3.2 odd 2 435.2.a.h.1.2 2
5.4 even 2 6525.2.a.bc.1.2 2
12.11 even 2 6960.2.a.bx.1.2 2
15.2 even 4 2175.2.c.h.349.4 4
15.8 even 4 2175.2.c.h.349.1 4
15.14 odd 2 2175.2.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.h.1.2 2 3.2 odd 2
1305.2.a.i.1.1 2 1.1 even 1 trivial
2175.2.a.m.1.1 2 15.14 odd 2
2175.2.c.h.349.1 4 15.8 even 4
2175.2.c.h.349.4 4 15.2 even 4
6525.2.a.bc.1.2 2 5.4 even 2
6960.2.a.bx.1.2 2 12.11 even 2