Properties

Label 9200.2.a.cq.1.4
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $4$
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] = [9200,2,Mod(1,9200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.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: \(73.4623698596\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5744.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.291367\) of defining polynomial
Character \(\chi\) \(=\) 9200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.14073 q^{3} -1.20647 q^{7} +6.86420 q^{9} +O(q^{10})\) \(q+3.14073 q^{3} -1.20647 q^{7} +6.86420 q^{9} -3.65773 q^{11} -0.859268 q^{13} -6.72347 q^{17} +1.51699 q^{19} -3.78921 q^{21} -1.00000 q^{23} +12.1364 q^{27} +0.548747 q^{29} +5.99568 q^{31} -11.4879 q^{33} -2.04100 q^{37} -2.69873 q^{39} -7.14998 q^{41} -10.0799 q^{43} -9.17040 q^{47} -5.54442 q^{49} -21.1166 q^{51} -5.37896 q^{53} +4.76447 q^{57} +0.582734 q^{59} -8.83244 q^{61} -8.28146 q^{63} -3.20647 q^{67} -3.14073 q^{69} +12.5784 q^{71} -8.62597 q^{73} +4.41294 q^{77} -0.0700619 q^{79} +17.5246 q^{81} +6.74197 q^{83} +1.72347 q^{87} +4.96393 q^{89} +1.03668 q^{91} +18.8308 q^{93} -11.3380 q^{97} -25.1074 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 6 q^{7} + 4 q^{9} - 2 q^{11} - 14 q^{13} - 14 q^{17} + 4 q^{19} - 2 q^{21} - 4 q^{23} + 14 q^{27} + 4 q^{29} - 14 q^{33} - 2 q^{37} + 8 q^{39} - 8 q^{41} + 4 q^{43} + 2 q^{47} - 10 q^{51} - 4 q^{53} - 8 q^{61} - 12 q^{63} - 2 q^{67} - 2 q^{69} + 24 q^{71} - 18 q^{73} - 4 q^{77} - 24 q^{79} + 8 q^{81} - 6 q^{83} - 6 q^{87} - 8 q^{89} - 26 q^{91} - 2 q^{93} - 34 q^{97} - 36 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.14073 1.81330 0.906651 0.421881i \(-0.138630\pi\)
0.906651 + 0.421881i \(0.138630\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.20647 −0.456004 −0.228002 0.973661i \(-0.573219\pi\)
−0.228002 + 0.973661i \(0.573219\pi\)
\(8\) 0 0
\(9\) 6.86420 2.28807
\(10\) 0 0
\(11\) −3.65773 −1.10285 −0.551423 0.834226i \(-0.685915\pi\)
−0.551423 + 0.834226i \(0.685915\pi\)
\(12\) 0 0
\(13\) −0.859268 −0.238318 −0.119159 0.992875i \(-0.538020\pi\)
−0.119159 + 0.992875i \(0.538020\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.72347 −1.63068 −0.815340 0.578982i \(-0.803450\pi\)
−0.815340 + 0.578982i \(0.803450\pi\)
\(18\) 0 0
\(19\) 1.51699 0.348022 0.174011 0.984744i \(-0.444327\pi\)
0.174011 + 0.984744i \(0.444327\pi\)
\(20\) 0 0
\(21\) −3.78921 −0.826873
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 12.1364 2.33565
\(28\) 0 0
\(29\) 0.548747 0.101900 0.0509498 0.998701i \(-0.483775\pi\)
0.0509498 + 0.998701i \(0.483775\pi\)
\(30\) 0 0
\(31\) 5.99568 1.07686 0.538428 0.842672i \(-0.319018\pi\)
0.538428 + 0.842672i \(0.319018\pi\)
\(32\) 0 0
\(33\) −11.4879 −1.99979
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.04100 −0.335539 −0.167770 0.985826i \(-0.553656\pi\)
−0.167770 + 0.985826i \(0.553656\pi\)
\(38\) 0 0
\(39\) −2.69873 −0.432143
\(40\) 0 0
\(41\) −7.14998 −1.11664 −0.558320 0.829626i \(-0.688554\pi\)
−0.558320 + 0.829626i \(0.688554\pi\)
\(42\) 0 0
\(43\) −10.0799 −1.53717 −0.768587 0.639745i \(-0.779040\pi\)
−0.768587 + 0.639745i \(0.779040\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.17040 −1.33764 −0.668820 0.743424i \(-0.733200\pi\)
−0.668820 + 0.743424i \(0.733200\pi\)
\(48\) 0 0
\(49\) −5.54442 −0.792061
\(50\) 0 0
\(51\) −21.1166 −2.95692
\(52\) 0 0
\(53\) −5.37896 −0.738857 −0.369428 0.929259i \(-0.620446\pi\)
−0.369428 + 0.929259i \(0.620446\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.76447 0.631070
\(58\) 0 0
\(59\) 0.582734 0.0758655 0.0379327 0.999280i \(-0.487923\pi\)
0.0379327 + 0.999280i \(0.487923\pi\)
\(60\) 0 0
\(61\) −8.83244 −1.13088 −0.565439 0.824790i \(-0.691293\pi\)
−0.565439 + 0.824790i \(0.691293\pi\)
\(62\) 0 0
\(63\) −8.28146 −1.04337
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.20647 −0.391733 −0.195866 0.980631i \(-0.562752\pi\)
−0.195866 + 0.980631i \(0.562752\pi\)
\(68\) 0 0
\(69\) −3.14073 −0.378100
\(70\) 0 0
\(71\) 12.5784 1.49278 0.746391 0.665507i \(-0.231785\pi\)
0.746391 + 0.665507i \(0.231785\pi\)
\(72\) 0 0
\(73\) −8.62597 −1.00959 −0.504797 0.863238i \(-0.668433\pi\)
−0.504797 + 0.863238i \(0.668433\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.41294 0.502902
\(78\) 0 0
\(79\) −0.0700619 −0.00788258 −0.00394129 0.999992i \(-0.501255\pi\)
−0.00394129 + 0.999992i \(0.501255\pi\)
\(80\) 0 0
\(81\) 17.5246 1.94718
\(82\) 0 0
\(83\) 6.74197 0.740027 0.370014 0.929026i \(-0.379353\pi\)
0.370014 + 0.929026i \(0.379353\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.72347 0.184775
\(88\) 0 0
\(89\) 4.96393 0.526175 0.263088 0.964772i \(-0.415259\pi\)
0.263088 + 0.964772i \(0.415259\pi\)
\(90\) 0 0
\(91\) 1.03668 0.108674
\(92\) 0 0
\(93\) 18.8308 1.95266
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −11.3380 −1.15119 −0.575597 0.817733i \(-0.695230\pi\)
−0.575597 + 0.817733i \(0.695230\pi\)
\(98\) 0 0
\(99\) −25.1074 −2.52338
\(100\) 0 0
\(101\) 3.44693 0.342983 0.171491 0.985186i \(-0.445141\pi\)
0.171491 + 0.985186i \(0.445141\pi\)
\(102\) 0 0
\(103\) 9.13641 0.900237 0.450119 0.892969i \(-0.351382\pi\)
0.450119 + 0.892969i \(0.351382\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.24539 0.893786 0.446893 0.894588i \(-0.352531\pi\)
0.446893 + 0.894588i \(0.352531\pi\)
\(108\) 0 0
\(109\) 1.64847 0.157895 0.0789476 0.996879i \(-0.474844\pi\)
0.0789476 + 0.996879i \(0.474844\pi\)
\(110\) 0 0
\(111\) −6.41025 −0.608434
\(112\) 0 0
\(113\) −1.20647 −0.113495 −0.0567477 0.998389i \(-0.518073\pi\)
−0.0567477 + 0.998389i \(0.518073\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −5.89819 −0.545287
\(118\) 0 0
\(119\) 8.11167 0.743596
\(120\) 0 0
\(121\) 2.37896 0.216269
\(122\) 0 0
\(123\) −22.4562 −2.02481
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 15.3542 1.36247 0.681233 0.732066i \(-0.261444\pi\)
0.681233 + 0.732066i \(0.261444\pi\)
\(128\) 0 0
\(129\) −31.6583 −2.78736
\(130\) 0 0
\(131\) −13.0734 −1.14223 −0.571113 0.820872i \(-0.693488\pi\)
−0.571113 + 0.820872i \(0.693488\pi\)
\(132\) 0 0
\(133\) −1.83021 −0.158699
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.1840 1.12638 0.563191 0.826327i \(-0.309573\pi\)
0.563191 + 0.826327i \(0.309573\pi\)
\(138\) 0 0
\(139\) 14.1160 1.19730 0.598652 0.801010i \(-0.295703\pi\)
0.598652 + 0.801010i \(0.295703\pi\)
\(140\) 0 0
\(141\) −28.8018 −2.42555
\(142\) 0 0
\(143\) 3.14297 0.262828
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −17.4136 −1.43625
\(148\) 0 0
\(149\) −6.54442 −0.536140 −0.268070 0.963399i \(-0.586386\pi\)
−0.268070 + 0.963399i \(0.586386\pi\)
\(150\) 0 0
\(151\) −2.61672 −0.212946 −0.106473 0.994316i \(-0.533956\pi\)
−0.106473 + 0.994316i \(0.533956\pi\)
\(152\) 0 0
\(153\) −46.1512 −3.73110
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −8.81394 −0.703429 −0.351715 0.936107i \(-0.614401\pi\)
−0.351715 + 0.936107i \(0.614401\pi\)
\(158\) 0 0
\(159\) −16.8939 −1.33977
\(160\) 0 0
\(161\) 1.20647 0.0950833
\(162\) 0 0
\(163\) −10.1747 −0.796946 −0.398473 0.917180i \(-0.630460\pi\)
−0.398473 + 0.917180i \(0.630460\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.6647 −0.980027 −0.490014 0.871715i \(-0.663008\pi\)
−0.490014 + 0.871715i \(0.663008\pi\)
\(168\) 0 0
\(169\) −12.2617 −0.943205
\(170\) 0 0
\(171\) 10.4129 0.796298
\(172\) 0 0
\(173\) 15.6901 1.19290 0.596448 0.802652i \(-0.296578\pi\)
0.596448 + 0.802652i \(0.296578\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.83021 0.137567
\(178\) 0 0
\(179\) 2.39876 0.179292 0.0896460 0.995974i \(-0.471426\pi\)
0.0896460 + 0.995974i \(0.471426\pi\)
\(180\) 0 0
\(181\) −9.87122 −0.733722 −0.366861 0.930276i \(-0.619567\pi\)
−0.366861 + 0.930276i \(0.619567\pi\)
\(182\) 0 0
\(183\) −27.7403 −2.05063
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 24.5926 1.79839
\(188\) 0 0
\(189\) −14.6422 −1.06507
\(190\) 0 0
\(191\) 9.60617 0.695078 0.347539 0.937666i \(-0.387017\pi\)
0.347539 + 0.937666i \(0.387017\pi\)
\(192\) 0 0
\(193\) −24.9709 −1.79745 −0.898724 0.438515i \(-0.855505\pi\)
−0.898724 + 0.438515i \(0.855505\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.92292 0.706979 0.353489 0.935439i \(-0.384995\pi\)
0.353489 + 0.935439i \(0.384995\pi\)
\(198\) 0 0
\(199\) −7.01818 −0.497506 −0.248753 0.968567i \(-0.580021\pi\)
−0.248753 + 0.968567i \(0.580021\pi\)
\(200\) 0 0
\(201\) −10.0707 −0.710330
\(202\) 0 0
\(203\) −0.662047 −0.0464666
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −6.86420 −0.477095
\(208\) 0 0
\(209\) −5.54875 −0.383815
\(210\) 0 0
\(211\) −7.44693 −0.512668 −0.256334 0.966588i \(-0.582515\pi\)
−0.256334 + 0.966588i \(0.582515\pi\)
\(212\) 0 0
\(213\) 39.5054 2.70687
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −7.23362 −0.491050
\(218\) 0 0
\(219\) −27.0919 −1.83070
\(220\) 0 0
\(221\) 5.77726 0.388620
\(222\) 0 0
\(223\) −10.2180 −0.684245 −0.342123 0.939655i \(-0.611146\pi\)
−0.342123 + 0.939655i \(0.611146\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.83291 −0.387144 −0.193572 0.981086i \(-0.562007\pi\)
−0.193572 + 0.981086i \(0.562007\pi\)
\(228\) 0 0
\(229\) 5.87061 0.387941 0.193970 0.981007i \(-0.437863\pi\)
0.193970 + 0.981007i \(0.437863\pi\)
\(230\) 0 0
\(231\) 13.8599 0.911913
\(232\) 0 0
\(233\) 0.839462 0.0549950 0.0274975 0.999622i \(-0.491246\pi\)
0.0274975 + 0.999622i \(0.491246\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −0.220046 −0.0142935
\(238\) 0 0
\(239\) 21.3296 1.37970 0.689850 0.723953i \(-0.257677\pi\)
0.689850 + 0.723953i \(0.257677\pi\)
\(240\) 0 0
\(241\) −20.7037 −1.33364 −0.666820 0.745219i \(-0.732345\pi\)
−0.666820 + 0.745219i \(0.732345\pi\)
\(242\) 0 0
\(243\) 18.6309 1.19517
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.30350 −0.0829400
\(248\) 0 0
\(249\) 21.1747 1.34189
\(250\) 0 0
\(251\) −23.7290 −1.49776 −0.748881 0.662705i \(-0.769408\pi\)
−0.748881 + 0.662705i \(0.769408\pi\)
\(252\) 0 0
\(253\) 3.65773 0.229959
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.2481 1.13828 0.569142 0.822239i \(-0.307275\pi\)
0.569142 + 0.822239i \(0.307275\pi\)
\(258\) 0 0
\(259\) 2.46242 0.153007
\(260\) 0 0
\(261\) 3.76670 0.233153
\(262\) 0 0
\(263\) 25.8718 1.59532 0.797662 0.603104i \(-0.206070\pi\)
0.797662 + 0.603104i \(0.206070\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 15.5904 0.954115
\(268\) 0 0
\(269\) 7.34497 0.447831 0.223915 0.974609i \(-0.428116\pi\)
0.223915 + 0.974609i \(0.428116\pi\)
\(270\) 0 0
\(271\) −30.2278 −1.83621 −0.918105 0.396338i \(-0.870281\pi\)
−0.918105 + 0.396338i \(0.870281\pi\)
\(272\) 0 0
\(273\) 3.25594 0.197059
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.50983 −0.150801 −0.0754005 0.997153i \(-0.524024\pi\)
−0.0754005 + 0.997153i \(0.524024\pi\)
\(278\) 0 0
\(279\) 41.1555 2.46392
\(280\) 0 0
\(281\) −10.4836 −0.625400 −0.312700 0.949852i \(-0.601233\pi\)
−0.312700 + 0.949852i \(0.601233\pi\)
\(282\) 0 0
\(283\) 3.83946 0.228232 0.114116 0.993467i \(-0.463596\pi\)
0.114116 + 0.993467i \(0.463596\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.62626 0.509192
\(288\) 0 0
\(289\) 28.2050 1.65912
\(290\) 0 0
\(291\) −35.6095 −2.08746
\(292\) 0 0
\(293\) −0.544425 −0.0318056 −0.0159028 0.999874i \(-0.505062\pi\)
−0.0159028 + 0.999874i \(0.505062\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −44.3917 −2.57587
\(298\) 0 0
\(299\) 0.859268 0.0496927
\(300\) 0 0
\(301\) 12.1611 0.700957
\(302\) 0 0
\(303\) 10.8259 0.621931
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 16.1850 0.923729 0.461865 0.886950i \(-0.347181\pi\)
0.461865 + 0.886950i \(0.347181\pi\)
\(308\) 0 0
\(309\) 28.6950 1.63240
\(310\) 0 0
\(311\) 7.51923 0.426376 0.213188 0.977011i \(-0.431615\pi\)
0.213188 + 0.977011i \(0.431615\pi\)
\(312\) 0 0
\(313\) −25.2475 −1.42707 −0.713536 0.700619i \(-0.752907\pi\)
−0.713536 + 0.700619i \(0.752907\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.4173 0.753589 0.376794 0.926297i \(-0.377026\pi\)
0.376794 + 0.926297i \(0.377026\pi\)
\(318\) 0 0
\(319\) −2.00716 −0.112380
\(320\) 0 0
\(321\) 29.0373 1.62070
\(322\) 0 0
\(323\) −10.1995 −0.567513
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 5.17741 0.286312
\(328\) 0 0
\(329\) 11.0638 0.609969
\(330\) 0 0
\(331\) 24.3748 1.33976 0.669880 0.742470i \(-0.266346\pi\)
0.669880 + 0.742470i \(0.266346\pi\)
\(332\) 0 0
\(333\) −14.0099 −0.767736
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −19.2454 −1.04836 −0.524182 0.851607i \(-0.675629\pi\)
−0.524182 + 0.851607i \(0.675629\pi\)
\(338\) 0 0
\(339\) −3.78921 −0.205801
\(340\) 0 0
\(341\) −21.9305 −1.18761
\(342\) 0 0
\(343\) 15.1345 0.817186
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.51044 −0.403181 −0.201591 0.979470i \(-0.564611\pi\)
−0.201591 + 0.979470i \(0.564611\pi\)
\(348\) 0 0
\(349\) 12.3208 0.659520 0.329760 0.944065i \(-0.393032\pi\)
0.329760 + 0.944065i \(0.393032\pi\)
\(350\) 0 0
\(351\) −10.4284 −0.556628
\(352\) 0 0
\(353\) 11.7333 0.624502 0.312251 0.950000i \(-0.398917\pi\)
0.312251 + 0.950000i \(0.398917\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 25.4766 1.34836
\(358\) 0 0
\(359\) 18.6879 0.986307 0.493154 0.869942i \(-0.335844\pi\)
0.493154 + 0.869942i \(0.335844\pi\)
\(360\) 0 0
\(361\) −16.6987 −0.878881
\(362\) 0 0
\(363\) 7.47167 0.392161
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −23.0103 −1.20113 −0.600564 0.799576i \(-0.705057\pi\)
−0.600564 + 0.799576i \(0.705057\pi\)
\(368\) 0 0
\(369\) −49.0789 −2.55495
\(370\) 0 0
\(371\) 6.48956 0.336921
\(372\) 0 0
\(373\) −35.7402 −1.85056 −0.925278 0.379290i \(-0.876168\pi\)
−0.925278 + 0.379290i \(0.876168\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.471520 −0.0242845
\(378\) 0 0
\(379\) −9.65178 −0.495779 −0.247889 0.968788i \(-0.579737\pi\)
−0.247889 + 0.968788i \(0.579737\pi\)
\(380\) 0 0
\(381\) 48.2235 2.47056
\(382\) 0 0
\(383\) −25.0954 −1.28232 −0.641158 0.767409i \(-0.721546\pi\)
−0.641158 + 0.767409i \(0.721546\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −69.1906 −3.51715
\(388\) 0 0
\(389\) 16.8259 0.853106 0.426553 0.904462i \(-0.359728\pi\)
0.426553 + 0.904462i \(0.359728\pi\)
\(390\) 0 0
\(391\) 6.72347 0.340020
\(392\) 0 0
\(393\) −41.0599 −2.07120
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −27.8080 −1.39564 −0.697822 0.716272i \(-0.745847\pi\)
−0.697822 + 0.716272i \(0.745847\pi\)
\(398\) 0 0
\(399\) −5.74820 −0.287770
\(400\) 0 0
\(401\) 1.46483 0.0731500 0.0365750 0.999331i \(-0.488355\pi\)
0.0365750 + 0.999331i \(0.488355\pi\)
\(402\) 0 0
\(403\) −5.15189 −0.256634
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.46544 0.370048
\(408\) 0 0
\(409\) 0.514906 0.0254605 0.0127302 0.999919i \(-0.495948\pi\)
0.0127302 + 0.999919i \(0.495948\pi\)
\(410\) 0 0
\(411\) 41.4073 2.04247
\(412\) 0 0
\(413\) −0.703052 −0.0345949
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 44.3346 2.17107
\(418\) 0 0
\(419\) 9.65387 0.471622 0.235811 0.971799i \(-0.424225\pi\)
0.235811 + 0.971799i \(0.424225\pi\)
\(420\) 0 0
\(421\) 14.2788 0.695905 0.347952 0.937512i \(-0.386877\pi\)
0.347952 + 0.937512i \(0.386877\pi\)
\(422\) 0 0
\(423\) −62.9474 −3.06061
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 10.6561 0.515685
\(428\) 0 0
\(429\) 9.87122 0.476587
\(430\) 0 0
\(431\) −5.71198 −0.275136 −0.137568 0.990492i \(-0.543929\pi\)
−0.137568 + 0.990492i \(0.543929\pi\)
\(432\) 0 0
\(433\) 30.3302 1.45758 0.728789 0.684738i \(-0.240083\pi\)
0.728789 + 0.684738i \(0.240083\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.51699 −0.0725676
\(438\) 0 0
\(439\) 19.7579 0.942994 0.471497 0.881868i \(-0.343714\pi\)
0.471497 + 0.881868i \(0.343714\pi\)
\(440\) 0 0
\(441\) −38.0580 −1.81229
\(442\) 0 0
\(443\) 16.0643 0.763236 0.381618 0.924320i \(-0.375367\pi\)
0.381618 + 0.924320i \(0.375367\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −20.5543 −0.972184
\(448\) 0 0
\(449\) 28.5881 1.34916 0.674579 0.738203i \(-0.264325\pi\)
0.674579 + 0.738203i \(0.264325\pi\)
\(450\) 0 0
\(451\) 26.1527 1.23148
\(452\) 0 0
\(453\) −8.21842 −0.386135
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10.1209 −0.473437 −0.236718 0.971578i \(-0.576072\pi\)
−0.236718 + 0.971578i \(0.576072\pi\)
\(458\) 0 0
\(459\) −81.5987 −3.80870
\(460\) 0 0
\(461\) 15.6931 0.730901 0.365450 0.930831i \(-0.380915\pi\)
0.365450 + 0.930831i \(0.380915\pi\)
\(462\) 0 0
\(463\) 12.9124 0.600089 0.300044 0.953925i \(-0.402999\pi\)
0.300044 + 0.953925i \(0.402999\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.10196 −0.328640 −0.164320 0.986407i \(-0.552543\pi\)
−0.164320 + 0.986407i \(0.552543\pi\)
\(468\) 0 0
\(469\) 3.86852 0.178632
\(470\) 0 0
\(471\) −27.6822 −1.27553
\(472\) 0 0
\(473\) 36.8696 1.69527
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −36.9222 −1.69055
\(478\) 0 0
\(479\) −16.7758 −0.766506 −0.383253 0.923643i \(-0.625196\pi\)
−0.383253 + 0.923643i \(0.625196\pi\)
\(480\) 0 0
\(481\) 1.75377 0.0799650
\(482\) 0 0
\(483\) 3.78921 0.172415
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 7.02488 0.318328 0.159164 0.987252i \(-0.449120\pi\)
0.159164 + 0.987252i \(0.449120\pi\)
\(488\) 0 0
\(489\) −31.9561 −1.44510
\(490\) 0 0
\(491\) −11.1500 −0.503192 −0.251596 0.967832i \(-0.580955\pi\)
−0.251596 + 0.967832i \(0.580955\pi\)
\(492\) 0 0
\(493\) −3.68948 −0.166166
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −15.1755 −0.680714
\(498\) 0 0
\(499\) −14.8347 −0.664091 −0.332046 0.943263i \(-0.607739\pi\)
−0.332046 + 0.943263i \(0.607739\pi\)
\(500\) 0 0
\(501\) −39.7766 −1.77709
\(502\) 0 0
\(503\) −3.10196 −0.138310 −0.0691548 0.997606i \(-0.522030\pi\)
−0.0691548 + 0.997606i \(0.522030\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −38.5106 −1.71032
\(508\) 0 0
\(509\) −22.3353 −0.989993 −0.494996 0.868895i \(-0.664831\pi\)
−0.494996 + 0.868895i \(0.664831\pi\)
\(510\) 0 0
\(511\) 10.4070 0.460378
\(512\) 0 0
\(513\) 18.4109 0.812859
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 33.5428 1.47521
\(518\) 0 0
\(519\) 49.2784 2.16308
\(520\) 0 0
\(521\) −14.2147 −0.622758 −0.311379 0.950286i \(-0.600791\pi\)
−0.311379 + 0.950286i \(0.600791\pi\)
\(522\) 0 0
\(523\) 4.09494 0.179059 0.0895297 0.995984i \(-0.471464\pi\)
0.0895297 + 0.995984i \(0.471464\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −40.3117 −1.75601
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 6.14375 0.266115
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 7.53387 0.325111
\(538\) 0 0
\(539\) 20.2800 0.873521
\(540\) 0 0
\(541\) 33.6902 1.44846 0.724228 0.689560i \(-0.242196\pi\)
0.724228 + 0.689560i \(0.242196\pi\)
\(542\) 0 0
\(543\) −31.0028 −1.33046
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −32.9358 −1.40823 −0.704117 0.710084i \(-0.748657\pi\)
−0.704117 + 0.710084i \(0.748657\pi\)
\(548\) 0 0
\(549\) −60.6277 −2.58753
\(550\) 0 0
\(551\) 0.832445 0.0354633
\(552\) 0 0
\(553\) 0.0845278 0.00359449
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −20.8337 −0.882751 −0.441375 0.897323i \(-0.645509\pi\)
−0.441375 + 0.897323i \(0.645509\pi\)
\(558\) 0 0
\(559\) 8.66135 0.366336
\(560\) 0 0
\(561\) 77.2387 3.26102
\(562\) 0 0
\(563\) −30.4368 −1.28276 −0.641380 0.767223i \(-0.721638\pi\)
−0.641380 + 0.767223i \(0.721638\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −21.1430 −0.887921
\(568\) 0 0
\(569\) 39.4801 1.65509 0.827545 0.561399i \(-0.189737\pi\)
0.827545 + 0.561399i \(0.189737\pi\)
\(570\) 0 0
\(571\) −27.0301 −1.13118 −0.565588 0.824688i \(-0.691351\pi\)
−0.565588 + 0.824688i \(0.691351\pi\)
\(572\) 0 0
\(573\) 30.1704 1.26039
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 4.68818 0.195171 0.0975857 0.995227i \(-0.468888\pi\)
0.0975857 + 0.995227i \(0.468888\pi\)
\(578\) 0 0
\(579\) −78.4270 −3.25932
\(580\) 0 0
\(581\) −8.13400 −0.337455
\(582\) 0 0
\(583\) 19.6747 0.814845
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −9.61718 −0.396944 −0.198472 0.980107i \(-0.563598\pi\)
−0.198472 + 0.980107i \(0.563598\pi\)
\(588\) 0 0
\(589\) 9.09541 0.374770
\(590\) 0 0
\(591\) 31.1652 1.28197
\(592\) 0 0
\(593\) 8.68902 0.356815 0.178408 0.983957i \(-0.442905\pi\)
0.178408 + 0.983957i \(0.442905\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −22.0422 −0.902128
\(598\) 0 0
\(599\) −11.7581 −0.480421 −0.240211 0.970721i \(-0.577217\pi\)
−0.240211 + 0.970721i \(0.577217\pi\)
\(600\) 0 0
\(601\) 37.9388 1.54756 0.773778 0.633457i \(-0.218365\pi\)
0.773778 + 0.633457i \(0.218365\pi\)
\(602\) 0 0
\(603\) −22.0099 −0.896311
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −9.01418 −0.365874 −0.182937 0.983125i \(-0.558560\pi\)
−0.182937 + 0.983125i \(0.558560\pi\)
\(608\) 0 0
\(609\) −2.07931 −0.0842580
\(610\) 0 0
\(611\) 7.87983 0.318784
\(612\) 0 0
\(613\) 42.5846 1.71997 0.859987 0.510316i \(-0.170472\pi\)
0.859987 + 0.510316i \(0.170472\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.6190 −0.508020 −0.254010 0.967202i \(-0.581750\pi\)
−0.254010 + 0.967202i \(0.581750\pi\)
\(618\) 0 0
\(619\) 26.0638 1.04759 0.523796 0.851844i \(-0.324515\pi\)
0.523796 + 0.851844i \(0.324515\pi\)
\(620\) 0 0
\(621\) −12.1364 −0.487017
\(622\) 0 0
\(623\) −5.98884 −0.239938
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −17.4271 −0.695972
\(628\) 0 0
\(629\) 13.7226 0.547157
\(630\) 0 0
\(631\) 46.1083 1.83554 0.917771 0.397110i \(-0.129987\pi\)
0.917771 + 0.397110i \(0.129987\pi\)
\(632\) 0 0
\(633\) −23.3888 −0.929622
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4.76415 0.188762
\(638\) 0 0
\(639\) 86.3407 3.41559
\(640\) 0 0
\(641\) 19.3319 0.763563 0.381781 0.924253i \(-0.375311\pi\)
0.381781 + 0.924253i \(0.375311\pi\)
\(642\) 0 0
\(643\) 2.27891 0.0898713 0.0449356 0.998990i \(-0.485692\pi\)
0.0449356 + 0.998990i \(0.485692\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 47.5054 1.86763 0.933815 0.357755i \(-0.116458\pi\)
0.933815 + 0.357755i \(0.116458\pi\)
\(648\) 0 0
\(649\) −2.13148 −0.0836679
\(650\) 0 0
\(651\) −22.7189 −0.890422
\(652\) 0 0
\(653\) −2.73226 −0.106921 −0.0534607 0.998570i \(-0.517025\pi\)
−0.0534607 + 0.998570i \(0.517025\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −59.2104 −2.31002
\(658\) 0 0
\(659\) 12.1375 0.472809 0.236405 0.971655i \(-0.424031\pi\)
0.236405 + 0.971655i \(0.424031\pi\)
\(660\) 0 0
\(661\) 41.1992 1.60246 0.801232 0.598354i \(-0.204178\pi\)
0.801232 + 0.598354i \(0.204178\pi\)
\(662\) 0 0
\(663\) 18.1448 0.704686
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −0.548747 −0.0212476
\(668\) 0 0
\(669\) −32.0919 −1.24074
\(670\) 0 0
\(671\) 32.3067 1.24718
\(672\) 0 0
\(673\) −24.2175 −0.933516 −0.466758 0.884385i \(-0.654578\pi\)
−0.466758 + 0.884385i \(0.654578\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −48.1512 −1.85060 −0.925300 0.379235i \(-0.876187\pi\)
−0.925300 + 0.379235i \(0.876187\pi\)
\(678\) 0 0
\(679\) 13.6789 0.524949
\(680\) 0 0
\(681\) −18.3196 −0.702008
\(682\) 0 0
\(683\) 12.1351 0.464337 0.232169 0.972676i \(-0.425418\pi\)
0.232169 + 0.972676i \(0.425418\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 18.4380 0.703454
\(688\) 0 0
\(689\) 4.62197 0.176083
\(690\) 0 0
\(691\) 13.8289 0.526076 0.263038 0.964785i \(-0.415275\pi\)
0.263038 + 0.964785i \(0.415275\pi\)
\(692\) 0 0
\(693\) 30.2913 1.15067
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 48.0727 1.82088
\(698\) 0 0
\(699\) 2.63653 0.0997226
\(700\) 0 0
\(701\) −15.2263 −0.575089 −0.287544 0.957767i \(-0.592839\pi\)
−0.287544 + 0.957767i \(0.592839\pi\)
\(702\) 0 0
\(703\) −3.09619 −0.116775
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.15863 −0.156401
\(708\) 0 0
\(709\) 15.2317 0.572037 0.286019 0.958224i \(-0.407668\pi\)
0.286019 + 0.958224i \(0.407668\pi\)
\(710\) 0 0
\(711\) −0.480919 −0.0180359
\(712\) 0 0
\(713\) −5.99568 −0.224540
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 66.9907 2.50181
\(718\) 0 0
\(719\) −4.58943 −0.171157 −0.0855784 0.996331i \(-0.527274\pi\)
−0.0855784 + 0.996331i \(0.527274\pi\)
\(720\) 0 0
\(721\) −11.0228 −0.410511
\(722\) 0 0
\(723\) −65.0246 −2.41829
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 21.2414 0.787800 0.393900 0.919153i \(-0.371126\pi\)
0.393900 + 0.919153i \(0.371126\pi\)
\(728\) 0 0
\(729\) 5.94082 0.220030
\(730\) 0 0
\(731\) 67.7720 2.50664
\(732\) 0 0
\(733\) −40.5297 −1.49700 −0.748499 0.663136i \(-0.769225\pi\)
−0.748499 + 0.663136i \(0.769225\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11.7284 0.432021
\(738\) 0 0
\(739\) −30.6476 −1.12739 −0.563695 0.825983i \(-0.690621\pi\)
−0.563695 + 0.825983i \(0.690621\pi\)
\(740\) 0 0
\(741\) −4.09396 −0.150395
\(742\) 0 0
\(743\) 1.01357 0.0371844 0.0185922 0.999827i \(-0.494082\pi\)
0.0185922 + 0.999827i \(0.494082\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 46.2782 1.69323
\(748\) 0 0
\(749\) −11.1543 −0.407569
\(750\) 0 0
\(751\) 21.0644 0.768652 0.384326 0.923197i \(-0.374434\pi\)
0.384326 + 0.923197i \(0.374434\pi\)
\(752\) 0 0
\(753\) −74.5264 −2.71589
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 13.2109 0.480160 0.240080 0.970753i \(-0.422826\pi\)
0.240080 + 0.970753i \(0.422826\pi\)
\(758\) 0 0
\(759\) 11.4879 0.416986
\(760\) 0 0
\(761\) −11.9759 −0.434125 −0.217063 0.976158i \(-0.569648\pi\)
−0.217063 + 0.976158i \(0.569648\pi\)
\(762\) 0 0
\(763\) −1.98884 −0.0720008
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −0.500724 −0.0180801
\(768\) 0 0
\(769\) 39.6569 1.43006 0.715031 0.699092i \(-0.246412\pi\)
0.715031 + 0.699092i \(0.246412\pi\)
\(770\) 0 0
\(771\) 57.3123 2.06405
\(772\) 0 0
\(773\) −16.0305 −0.576575 −0.288288 0.957544i \(-0.593086\pi\)
−0.288288 + 0.957544i \(0.593086\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 7.73379 0.277448
\(778\) 0 0
\(779\) −10.8465 −0.388615
\(780\) 0 0
\(781\) −46.0084 −1.64631
\(782\) 0 0
\(783\) 6.65981 0.238002
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −39.0352 −1.39145 −0.695727 0.718306i \(-0.744918\pi\)
−0.695727 + 0.718306i \(0.744918\pi\)
\(788\) 0 0
\(789\) 81.2565 2.89281
\(790\) 0 0
\(791\) 1.45558 0.0517543
\(792\) 0 0
\(793\) 7.58944 0.269509
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.37687 0.119615 0.0598074 0.998210i \(-0.480951\pi\)
0.0598074 + 0.998210i \(0.480951\pi\)
\(798\) 0 0
\(799\) 61.6569 2.18126
\(800\) 0 0
\(801\) 34.0734 1.20392
\(802\) 0 0
\(803\) 31.5514 1.11343
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 23.0686 0.812053
\(808\) 0 0
\(809\) −49.3406 −1.73472 −0.867361 0.497680i \(-0.834186\pi\)
−0.867361 + 0.497680i \(0.834186\pi\)
\(810\) 0 0
\(811\) −18.3582 −0.644645 −0.322322 0.946630i \(-0.604463\pi\)
−0.322322 + 0.946630i \(0.604463\pi\)
\(812\) 0 0
\(813\) −94.9375 −3.32960
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −15.2912 −0.534971
\(818\) 0 0
\(819\) 7.11600 0.248653
\(820\) 0 0
\(821\) 19.4610 0.679192 0.339596 0.940571i \(-0.389710\pi\)
0.339596 + 0.940571i \(0.389710\pi\)
\(822\) 0 0
\(823\) −47.2973 −1.64868 −0.824341 0.566094i \(-0.808454\pi\)
−0.824341 + 0.566094i \(0.808454\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26.2901 0.914197 0.457098 0.889416i \(-0.348889\pi\)
0.457098 + 0.889416i \(0.348889\pi\)
\(828\) 0 0
\(829\) −20.0140 −0.695116 −0.347558 0.937658i \(-0.612989\pi\)
−0.347558 + 0.937658i \(0.612989\pi\)
\(830\) 0 0
\(831\) −7.88270 −0.273448
\(832\) 0 0
\(833\) 37.2778 1.29160
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 72.7660 2.51516
\(838\) 0 0
\(839\) 40.7210 1.40585 0.702923 0.711266i \(-0.251878\pi\)
0.702923 + 0.711266i \(0.251878\pi\)
\(840\) 0 0
\(841\) −28.6989 −0.989616
\(842\) 0 0
\(843\) −32.9262 −1.13404
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −2.87015 −0.0986194
\(848\) 0 0
\(849\) 12.0587 0.413854
\(850\) 0 0
\(851\) 2.04100 0.0699647
\(852\) 0 0
\(853\) −5.34390 −0.182972 −0.0914858 0.995806i \(-0.529162\pi\)
−0.0914858 + 0.995806i \(0.529162\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −48.4603 −1.65537 −0.827686 0.561192i \(-0.810343\pi\)
−0.827686 + 0.561192i \(0.810343\pi\)
\(858\) 0 0
\(859\) 51.3123 1.75075 0.875377 0.483440i \(-0.160613\pi\)
0.875377 + 0.483440i \(0.160613\pi\)
\(860\) 0 0
\(861\) 27.0928 0.923319
\(862\) 0 0
\(863\) 24.5507 0.835714 0.417857 0.908513i \(-0.362781\pi\)
0.417857 + 0.908513i \(0.362781\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 88.5843 3.00848
\(868\) 0 0
\(869\) 0.256267 0.00869327
\(870\) 0 0
\(871\) 2.75522 0.0933570
\(872\) 0 0
\(873\) −77.8260 −2.63401
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −30.1400 −1.01776 −0.508878 0.860838i \(-0.669940\pi\)
−0.508878 + 0.860838i \(0.669940\pi\)
\(878\) 0 0
\(879\) −1.70989 −0.0576732
\(880\) 0 0
\(881\) 36.4729 1.22880 0.614401 0.788994i \(-0.289398\pi\)
0.614401 + 0.788994i \(0.289398\pi\)
\(882\) 0 0
\(883\) −2.17426 −0.0731696 −0.0365848 0.999331i \(-0.511648\pi\)
−0.0365848 + 0.999331i \(0.511648\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16.1463 −0.542139 −0.271069 0.962560i \(-0.587377\pi\)
−0.271069 + 0.962560i \(0.587377\pi\)
\(888\) 0 0
\(889\) −18.5244 −0.621290
\(890\) 0 0
\(891\) −64.1003 −2.14744
\(892\) 0 0
\(893\) −13.9114 −0.465528
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.69873 0.0901080
\(898\) 0 0
\(899\) 3.29011 0.109731
\(900\) 0 0
\(901\) 36.1652 1.20484
\(902\) 0 0
\(903\) 38.1949 1.27105
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −4.74475 −0.157547 −0.0787734 0.996893i \(-0.525100\pi\)
−0.0787734 + 0.996893i \(0.525100\pi\)
\(908\) 0 0
\(909\) 23.6604 0.784767
\(910\) 0 0
\(911\) −38.8650 −1.28765 −0.643827 0.765171i \(-0.722654\pi\)
−0.643827 + 0.765171i \(0.722654\pi\)
\(912\) 0 0
\(913\) −24.6603 −0.816136
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 15.7727 0.520859
\(918\) 0 0
\(919\) −40.5516 −1.33767 −0.668837 0.743409i \(-0.733207\pi\)
−0.668837 + 0.743409i \(0.733207\pi\)
\(920\) 0 0
\(921\) 50.8329 1.67500
\(922\) 0 0
\(923\) −10.8082 −0.355757
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 62.7141 2.05980
\(928\) 0 0
\(929\) 19.4994 0.639755 0.319878 0.947459i \(-0.396358\pi\)
0.319878 + 0.947459i \(0.396358\pi\)
\(930\) 0 0
\(931\) −8.41086 −0.275655
\(932\) 0 0
\(933\) 23.6159 0.773149
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 24.9361 0.814626 0.407313 0.913289i \(-0.366466\pi\)
0.407313 + 0.913289i \(0.366466\pi\)
\(938\) 0 0
\(939\) −79.2956 −2.58771
\(940\) 0 0
\(941\) −41.9160 −1.36642 −0.683211 0.730221i \(-0.739417\pi\)
−0.683211 + 0.730221i \(0.739417\pi\)
\(942\) 0 0
\(943\) 7.14998 0.232836
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −29.3006 −0.952141 −0.476070 0.879407i \(-0.657939\pi\)
−0.476070 + 0.879407i \(0.657939\pi\)
\(948\) 0 0
\(949\) 7.41202 0.240604
\(950\) 0 0
\(951\) 42.1400 1.36648
\(952\) 0 0
\(953\) 24.9759 0.809048 0.404524 0.914527i \(-0.367437\pi\)
0.404524 + 0.914527i \(0.367437\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −6.30397 −0.203778
\(958\) 0 0
\(959\) −15.9061 −0.513635
\(960\) 0 0
\(961\) 4.94816 0.159618
\(962\) 0 0
\(963\) 63.4622 2.04504
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0.405142 0.0130285 0.00651424 0.999979i \(-0.497926\pi\)
0.00651424 + 0.999979i \(0.497926\pi\)
\(968\) 0 0
\(969\) −32.0338 −1.02907
\(970\) 0 0
\(971\) −38.8032 −1.24525 −0.622627 0.782519i \(-0.713935\pi\)
−0.622627 + 0.782519i \(0.713935\pi\)
\(972\) 0 0
\(973\) −17.0306 −0.545975
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 39.9940 1.27952 0.639760 0.768575i \(-0.279034\pi\)
0.639760 + 0.768575i \(0.279034\pi\)
\(978\) 0 0
\(979\) −18.1567 −0.580290
\(980\) 0 0
\(981\) 11.3155 0.361275
\(982\) 0 0
\(983\) 21.3563 0.681160 0.340580 0.940216i \(-0.389377\pi\)
0.340580 + 0.940216i \(0.389377\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 34.7485 1.10606
\(988\) 0 0
\(989\) 10.0799 0.320523
\(990\) 0 0
\(991\) −4.39444 −0.139594 −0.0697970 0.997561i \(-0.522235\pi\)
−0.0697970 + 0.997561i \(0.522235\pi\)
\(992\) 0 0
\(993\) 76.5547 2.42939
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −28.9420 −0.916603 −0.458302 0.888797i \(-0.651542\pi\)
−0.458302 + 0.888797i \(0.651542\pi\)
\(998\) 0 0
\(999\) −24.7705 −0.783703
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 9200.2.a.cq.1.4 4
4.3 odd 2 575.2.a.i.1.2 4
5.2 odd 4 1840.2.e.d.369.1 8
5.3 odd 4 1840.2.e.d.369.8 8
5.4 even 2 9200.2.a.ck.1.1 4
12.11 even 2 5175.2.a.bv.1.3 4
20.3 even 4 115.2.b.b.24.5 yes 8
20.7 even 4 115.2.b.b.24.4 8
20.19 odd 2 575.2.a.j.1.3 4
60.23 odd 4 1035.2.b.e.829.4 8
60.47 odd 4 1035.2.b.e.829.5 8
60.59 even 2 5175.2.a.bw.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.b.b.24.4 8 20.7 even 4
115.2.b.b.24.5 yes 8 20.3 even 4
575.2.a.i.1.2 4 4.3 odd 2
575.2.a.j.1.3 4 20.19 odd 2
1035.2.b.e.829.4 8 60.23 odd 4
1035.2.b.e.829.5 8 60.47 odd 4
1840.2.e.d.369.1 8 5.2 odd 4
1840.2.e.d.369.8 8 5.3 odd 4
5175.2.a.bv.1.3 4 12.11 even 2
5175.2.a.bw.1.2 4 60.59 even 2
9200.2.a.ck.1.1 4 5.4 even 2
9200.2.a.cq.1.4 4 1.1 even 1 trivial