Properties

Label 5520.2.be.c.1471.30
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5520,2,Mod(1471,5520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 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("5520.1471");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.be (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.30
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.c.1471.29

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{5} +1.44765 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{5} +1.44765 q^{7} -1.00000 q^{9} -0.540073 q^{11} +4.25689 q^{13} -1.00000 q^{15} -7.14147i q^{17} -0.941351 q^{19} +1.44765i q^{21} +(2.91287 + 3.80988i) q^{23} -1.00000 q^{25} -1.00000i q^{27} +1.26464 q^{29} -3.15253i q^{31} -0.540073i q^{33} +1.44765i q^{35} +1.30413i q^{37} +4.25689i q^{39} +5.97953 q^{41} +2.03228 q^{43} -1.00000i q^{45} -6.61495i q^{47} -4.90431 q^{49} +7.14147 q^{51} -7.54328i q^{53} -0.540073i q^{55} -0.941351i q^{57} -11.1974i q^{59} +2.00097i q^{61} -1.44765 q^{63} +4.25689i q^{65} +7.68722 q^{67} +(-3.80988 + 2.91287i) q^{69} +8.07158i q^{71} +1.91203 q^{73} -1.00000i q^{75} -0.781836 q^{77} -3.99026 q^{79} +1.00000 q^{81} +1.20991 q^{83} +7.14147 q^{85} +1.26464i q^{87} -17.8161i q^{89} +6.16248 q^{91} +3.15253 q^{93} -0.941351i q^{95} -11.5224i q^{97} +0.540073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 8 q^{7} - 32 q^{9} + 8 q^{11} - 8 q^{13} - 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} - 4 q^{51} + 8 q^{63} + 32 q^{67} - 40 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{81} - 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1841\) \(4417\) \(4831\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 1.44765 0.547160 0.273580 0.961849i \(-0.411792\pi\)
0.273580 + 0.961849i \(0.411792\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −0.540073 −0.162838 −0.0814191 0.996680i \(-0.525945\pi\)
−0.0814191 + 0.996680i \(0.525945\pi\)
\(12\) 0 0
\(13\) 4.25689 1.18065 0.590324 0.807166i \(-0.299000\pi\)
0.590324 + 0.807166i \(0.299000\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 7.14147i 1.73206i −0.499992 0.866030i \(-0.666664\pi\)
0.499992 0.866030i \(-0.333336\pi\)
\(18\) 0 0
\(19\) −0.941351 −0.215961 −0.107980 0.994153i \(-0.534438\pi\)
−0.107980 + 0.994153i \(0.534438\pi\)
\(20\) 0 0
\(21\) 1.44765i 0.315903i
\(22\) 0 0
\(23\) 2.91287 + 3.80988i 0.607376 + 0.794415i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 1.26464 0.234838 0.117419 0.993082i \(-0.462538\pi\)
0.117419 + 0.993082i \(0.462538\pi\)
\(30\) 0 0
\(31\) 3.15253i 0.566212i −0.959089 0.283106i \(-0.908635\pi\)
0.959089 0.283106i \(-0.0913648\pi\)
\(32\) 0 0
\(33\) 0.540073i 0.0940147i
\(34\) 0 0
\(35\) 1.44765i 0.244697i
\(36\) 0 0
\(37\) 1.30413i 0.214397i 0.994238 + 0.107199i \(0.0341881\pi\)
−0.994238 + 0.107199i \(0.965812\pi\)
\(38\) 0 0
\(39\) 4.25689i 0.681648i
\(40\) 0 0
\(41\) 5.97953 0.933845 0.466923 0.884298i \(-0.345363\pi\)
0.466923 + 0.884298i \(0.345363\pi\)
\(42\) 0 0
\(43\) 2.03228 0.309919 0.154960 0.987921i \(-0.450475\pi\)
0.154960 + 0.987921i \(0.450475\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 6.61495i 0.964889i −0.875927 0.482444i \(-0.839749\pi\)
0.875927 0.482444i \(-0.160251\pi\)
\(48\) 0 0
\(49\) −4.90431 −0.700616
\(50\) 0 0
\(51\) 7.14147 1.00001
\(52\) 0 0
\(53\) 7.54328i 1.03615i −0.855335 0.518075i \(-0.826649\pi\)
0.855335 0.518075i \(-0.173351\pi\)
\(54\) 0 0
\(55\) 0.540073i 0.0728234i
\(56\) 0 0
\(57\) 0.941351i 0.124685i
\(58\) 0 0
\(59\) 11.1974i 1.45777i −0.684635 0.728886i \(-0.740038\pi\)
0.684635 0.728886i \(-0.259962\pi\)
\(60\) 0 0
\(61\) 2.00097i 0.256197i 0.991761 + 0.128099i \(0.0408874\pi\)
−0.991761 + 0.128099i \(0.959113\pi\)
\(62\) 0 0
\(63\) −1.44765 −0.182387
\(64\) 0 0
\(65\) 4.25689i 0.528002i
\(66\) 0 0
\(67\) 7.68722 0.939143 0.469572 0.882894i \(-0.344408\pi\)
0.469572 + 0.882894i \(0.344408\pi\)
\(68\) 0 0
\(69\) −3.80988 + 2.91287i −0.458656 + 0.350668i
\(70\) 0 0
\(71\) 8.07158i 0.957920i 0.877837 + 0.478960i \(0.158986\pi\)
−0.877837 + 0.478960i \(0.841014\pi\)
\(72\) 0 0
\(73\) 1.91203 0.223787 0.111893 0.993720i \(-0.464309\pi\)
0.111893 + 0.993720i \(0.464309\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) −0.781836 −0.0890985
\(78\) 0 0
\(79\) −3.99026 −0.448939 −0.224470 0.974481i \(-0.572065\pi\)
−0.224470 + 0.974481i \(0.572065\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.20991 0.132805 0.0664024 0.997793i \(-0.478848\pi\)
0.0664024 + 0.997793i \(0.478848\pi\)
\(84\) 0 0
\(85\) 7.14147 0.774601
\(86\) 0 0
\(87\) 1.26464i 0.135584i
\(88\) 0 0
\(89\) 17.8161i 1.88851i −0.329222 0.944253i \(-0.606786\pi\)
0.329222 0.944253i \(-0.393214\pi\)
\(90\) 0 0
\(91\) 6.16248 0.646004
\(92\) 0 0
\(93\) 3.15253 0.326903
\(94\) 0 0
\(95\) 0.941351i 0.0965806i
\(96\) 0 0
\(97\) 11.5224i 1.16992i −0.811062 0.584960i \(-0.801110\pi\)
0.811062 0.584960i \(-0.198890\pi\)
\(98\) 0 0
\(99\) 0.540073 0.0542794
\(100\) 0 0
\(101\) 13.5957 1.35282 0.676412 0.736524i \(-0.263534\pi\)
0.676412 + 0.736524i \(0.263534\pi\)
\(102\) 0 0
\(103\) 3.76574 0.371049 0.185524 0.982640i \(-0.440602\pi\)
0.185524 + 0.982640i \(0.440602\pi\)
\(104\) 0 0
\(105\) −1.44765 −0.141276
\(106\) 0 0
\(107\) 2.93296 0.283540 0.141770 0.989900i \(-0.454721\pi\)
0.141770 + 0.989900i \(0.454721\pi\)
\(108\) 0 0
\(109\) 14.4914i 1.38803i 0.719962 + 0.694013i \(0.244159\pi\)
−0.719962 + 0.694013i \(0.755841\pi\)
\(110\) 0 0
\(111\) −1.30413 −0.123782
\(112\) 0 0
\(113\) 21.0266i 1.97801i −0.147875 0.989006i \(-0.547243\pi\)
0.147875 0.989006i \(-0.452757\pi\)
\(114\) 0 0
\(115\) −3.80988 + 2.91287i −0.355273 + 0.271627i
\(116\) 0 0
\(117\) −4.25689 −0.393550
\(118\) 0 0
\(119\) 10.3383i 0.947714i
\(120\) 0 0
\(121\) −10.7083 −0.973484
\(122\) 0 0
\(123\) 5.97953i 0.539156i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 1.36794i 0.121385i −0.998157 0.0606924i \(-0.980669\pi\)
0.998157 0.0606924i \(-0.0193309\pi\)
\(128\) 0 0
\(129\) 2.03228i 0.178932i
\(130\) 0 0
\(131\) 11.8087i 1.03173i −0.856671 0.515863i \(-0.827471\pi\)
0.856671 0.515863i \(-0.172529\pi\)
\(132\) 0 0
\(133\) −1.36275 −0.118165
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 12.1571i 1.03865i 0.854576 + 0.519326i \(0.173817\pi\)
−0.854576 + 0.519326i \(0.826183\pi\)
\(138\) 0 0
\(139\) 14.5794i 1.23661i 0.785939 + 0.618304i \(0.212180\pi\)
−0.785939 + 0.618304i \(0.787820\pi\)
\(140\) 0 0
\(141\) 6.61495 0.557079
\(142\) 0 0
\(143\) −2.29903 −0.192255
\(144\) 0 0
\(145\) 1.26464i 0.105023i
\(146\) 0 0
\(147\) 4.90431i 0.404501i
\(148\) 0 0
\(149\) 4.61311i 0.377921i 0.981985 + 0.188960i \(0.0605118\pi\)
−0.981985 + 0.188960i \(0.939488\pi\)
\(150\) 0 0
\(151\) 22.2226i 1.80845i 0.427055 + 0.904226i \(0.359551\pi\)
−0.427055 + 0.904226i \(0.640449\pi\)
\(152\) 0 0
\(153\) 7.14147i 0.577354i
\(154\) 0 0
\(155\) 3.15253 0.253218
\(156\) 0 0
\(157\) 23.2200i 1.85316i −0.376099 0.926579i \(-0.622735\pi\)
0.376099 0.926579i \(-0.377265\pi\)
\(158\) 0 0
\(159\) 7.54328 0.598221
\(160\) 0 0
\(161\) 4.21681 + 5.51537i 0.332332 + 0.434672i
\(162\) 0 0
\(163\) 19.5709i 1.53291i 0.642299 + 0.766454i \(0.277981\pi\)
−0.642299 + 0.766454i \(0.722019\pi\)
\(164\) 0 0
\(165\) 0.540073 0.0420446
\(166\) 0 0
\(167\) 16.9663i 1.31289i 0.754374 + 0.656445i \(0.227941\pi\)
−0.754374 + 0.656445i \(0.772059\pi\)
\(168\) 0 0
\(169\) 5.12112 0.393932
\(170\) 0 0
\(171\) 0.941351 0.0719869
\(172\) 0 0
\(173\) 23.6431 1.79755 0.898774 0.438412i \(-0.144459\pi\)
0.898774 + 0.438412i \(0.144459\pi\)
\(174\) 0 0
\(175\) −1.44765 −0.109432
\(176\) 0 0
\(177\) 11.1974 0.841645
\(178\) 0 0
\(179\) 13.7587i 1.02837i 0.857678 + 0.514187i \(0.171906\pi\)
−0.857678 + 0.514187i \(0.828094\pi\)
\(180\) 0 0
\(181\) 23.9965i 1.78364i −0.452387 0.891822i \(-0.649428\pi\)
0.452387 0.891822i \(-0.350572\pi\)
\(182\) 0 0
\(183\) −2.00097 −0.147916
\(184\) 0 0
\(185\) −1.30413 −0.0958814
\(186\) 0 0
\(187\) 3.85691i 0.282046i
\(188\) 0 0
\(189\) 1.44765i 0.105301i
\(190\) 0 0
\(191\) 0.662043 0.0479037 0.0239519 0.999713i \(-0.492375\pi\)
0.0239519 + 0.999713i \(0.492375\pi\)
\(192\) 0 0
\(193\) 8.81504 0.634521 0.317260 0.948338i \(-0.397237\pi\)
0.317260 + 0.948338i \(0.397237\pi\)
\(194\) 0 0
\(195\) −4.25689 −0.304842
\(196\) 0 0
\(197\) −25.8264 −1.84005 −0.920027 0.391855i \(-0.871833\pi\)
−0.920027 + 0.391855i \(0.871833\pi\)
\(198\) 0 0
\(199\) 19.6257 1.39123 0.695615 0.718415i \(-0.255132\pi\)
0.695615 + 0.718415i \(0.255132\pi\)
\(200\) 0 0
\(201\) 7.68722i 0.542215i
\(202\) 0 0
\(203\) 1.83076 0.128494
\(204\) 0 0
\(205\) 5.97953i 0.417628i
\(206\) 0 0
\(207\) −2.91287 3.80988i −0.202459 0.264805i
\(208\) 0 0
\(209\) 0.508398 0.0351666
\(210\) 0 0
\(211\) 16.4801i 1.13454i 0.823532 + 0.567270i \(0.192000\pi\)
−0.823532 + 0.567270i \(0.808000\pi\)
\(212\) 0 0
\(213\) −8.07158 −0.553055
\(214\) 0 0
\(215\) 2.03228i 0.138600i
\(216\) 0 0
\(217\) 4.56376i 0.309808i
\(218\) 0 0
\(219\) 1.91203i 0.129203i
\(220\) 0 0
\(221\) 30.4005i 2.04496i
\(222\) 0 0
\(223\) 0.687718i 0.0460530i −0.999735 0.0230265i \(-0.992670\pi\)
0.999735 0.0230265i \(-0.00733021\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 17.6458 1.17119 0.585597 0.810603i \(-0.300860\pi\)
0.585597 + 0.810603i \(0.300860\pi\)
\(228\) 0 0
\(229\) 6.99625i 0.462325i −0.972915 0.231163i \(-0.925747\pi\)
0.972915 0.231163i \(-0.0742530\pi\)
\(230\) 0 0
\(231\) 0.781836i 0.0514410i
\(232\) 0 0
\(233\) 3.36671 0.220560 0.110280 0.993901i \(-0.464825\pi\)
0.110280 + 0.993901i \(0.464825\pi\)
\(234\) 0 0
\(235\) 6.61495 0.431511
\(236\) 0 0
\(237\) 3.99026i 0.259195i
\(238\) 0 0
\(239\) 15.0526i 0.973674i 0.873493 + 0.486837i \(0.161849\pi\)
−0.873493 + 0.486837i \(0.838151\pi\)
\(240\) 0 0
\(241\) 6.51887i 0.419917i 0.977710 + 0.209959i \(0.0673329\pi\)
−0.977710 + 0.209959i \(0.932667\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 4.90431i 0.313325i
\(246\) 0 0
\(247\) −4.00723 −0.254974
\(248\) 0 0
\(249\) 1.20991i 0.0766749i
\(250\) 0 0
\(251\) 11.0617 0.698208 0.349104 0.937084i \(-0.386486\pi\)
0.349104 + 0.937084i \(0.386486\pi\)
\(252\) 0 0
\(253\) −1.57316 2.05761i −0.0989039 0.129361i
\(254\) 0 0
\(255\) 7.14147i 0.447216i
\(256\) 0 0
\(257\) −4.60346 −0.287156 −0.143578 0.989639i \(-0.545861\pi\)
−0.143578 + 0.989639i \(0.545861\pi\)
\(258\) 0 0
\(259\) 1.88792i 0.117310i
\(260\) 0 0
\(261\) −1.26464 −0.0782793
\(262\) 0 0
\(263\) 0.952711 0.0587467 0.0293733 0.999569i \(-0.490649\pi\)
0.0293733 + 0.999569i \(0.490649\pi\)
\(264\) 0 0
\(265\) 7.54328 0.463380
\(266\) 0 0
\(267\) 17.8161 1.09033
\(268\) 0 0
\(269\) 26.8660 1.63805 0.819025 0.573757i \(-0.194515\pi\)
0.819025 + 0.573757i \(0.194515\pi\)
\(270\) 0 0
\(271\) 25.2660i 1.53480i −0.641167 0.767401i \(-0.721550\pi\)
0.641167 0.767401i \(-0.278450\pi\)
\(272\) 0 0
\(273\) 6.16248i 0.372970i
\(274\) 0 0
\(275\) 0.540073 0.0325676
\(276\) 0 0
\(277\) −18.2298 −1.09532 −0.547660 0.836701i \(-0.684481\pi\)
−0.547660 + 0.836701i \(0.684481\pi\)
\(278\) 0 0
\(279\) 3.15253i 0.188737i
\(280\) 0 0
\(281\) 10.9007i 0.650282i 0.945665 + 0.325141i \(0.105412\pi\)
−0.945665 + 0.325141i \(0.894588\pi\)
\(282\) 0 0
\(283\) −13.1522 −0.781817 −0.390909 0.920430i \(-0.627839\pi\)
−0.390909 + 0.920430i \(0.627839\pi\)
\(284\) 0 0
\(285\) 0.941351 0.0557608
\(286\) 0 0
\(287\) 8.65626 0.510963
\(288\) 0 0
\(289\) −34.0006 −2.00003
\(290\) 0 0
\(291\) 11.5224 0.675454
\(292\) 0 0
\(293\) 12.4528i 0.727499i 0.931497 + 0.363750i \(0.118504\pi\)
−0.931497 + 0.363750i \(0.881496\pi\)
\(294\) 0 0
\(295\) 11.1974 0.651935
\(296\) 0 0
\(297\) 0.540073i 0.0313382i
\(298\) 0 0
\(299\) 12.3998 + 16.2182i 0.717097 + 0.937925i
\(300\) 0 0
\(301\) 2.94202 0.169575
\(302\) 0 0
\(303\) 13.5957i 0.781053i
\(304\) 0 0
\(305\) −2.00097 −0.114575
\(306\) 0 0
\(307\) 21.9864i 1.25483i 0.778684 + 0.627416i \(0.215887\pi\)
−0.778684 + 0.627416i \(0.784113\pi\)
\(308\) 0 0
\(309\) 3.76574i 0.214225i
\(310\) 0 0
\(311\) 22.6678i 1.28537i −0.766128 0.642687i \(-0.777819\pi\)
0.766128 0.642687i \(-0.222181\pi\)
\(312\) 0 0
\(313\) 17.0883i 0.965886i 0.875652 + 0.482943i \(0.160432\pi\)
−0.875652 + 0.482943i \(0.839568\pi\)
\(314\) 0 0
\(315\) 1.44765i 0.0815658i
\(316\) 0 0
\(317\) 29.7843 1.67285 0.836425 0.548081i \(-0.184641\pi\)
0.836425 + 0.548081i \(0.184641\pi\)
\(318\) 0 0
\(319\) −0.682998 −0.0382406
\(320\) 0 0
\(321\) 2.93296i 0.163702i
\(322\) 0 0
\(323\) 6.72263i 0.374057i
\(324\) 0 0
\(325\) −4.25689 −0.236130
\(326\) 0 0
\(327\) −14.4914 −0.801378
\(328\) 0 0
\(329\) 9.57612i 0.527949i
\(330\) 0 0
\(331\) 17.3613i 0.954264i −0.878832 0.477132i \(-0.841676\pi\)
0.878832 0.477132i \(-0.158324\pi\)
\(332\) 0 0
\(333\) 1.30413i 0.0714658i
\(334\) 0 0
\(335\) 7.68722i 0.419998i
\(336\) 0 0
\(337\) 9.95332i 0.542192i 0.962552 + 0.271096i \(0.0873861\pi\)
−0.962552 + 0.271096i \(0.912614\pi\)
\(338\) 0 0
\(339\) 21.0266 1.14201
\(340\) 0 0
\(341\) 1.70260i 0.0922009i
\(342\) 0 0
\(343\) −17.2333 −0.930509
\(344\) 0 0
\(345\) −2.91287 3.80988i −0.156824 0.205117i
\(346\) 0 0
\(347\) 10.6141i 0.569793i −0.958558 0.284896i \(-0.908041\pi\)
0.958558 0.284896i \(-0.0919592\pi\)
\(348\) 0 0
\(349\) 11.4277 0.611713 0.305856 0.952078i \(-0.401057\pi\)
0.305856 + 0.952078i \(0.401057\pi\)
\(350\) 0 0
\(351\) 4.25689i 0.227216i
\(352\) 0 0
\(353\) −2.08495 −0.110971 −0.0554854 0.998459i \(-0.517671\pi\)
−0.0554854 + 0.998459i \(0.517671\pi\)
\(354\) 0 0
\(355\) −8.07158 −0.428395
\(356\) 0 0
\(357\) 10.3383 0.547163
\(358\) 0 0
\(359\) −21.6384 −1.14203 −0.571015 0.820939i \(-0.693450\pi\)
−0.571015 + 0.820939i \(0.693450\pi\)
\(360\) 0 0
\(361\) −18.1139 −0.953361
\(362\) 0 0
\(363\) 10.7083i 0.562041i
\(364\) 0 0
\(365\) 1.91203i 0.100080i
\(366\) 0 0
\(367\) 25.5833 1.33544 0.667720 0.744413i \(-0.267270\pi\)
0.667720 + 0.744413i \(0.267270\pi\)
\(368\) 0 0
\(369\) −5.97953 −0.311282
\(370\) 0 0
\(371\) 10.9200i 0.566939i
\(372\) 0 0
\(373\) 7.23189i 0.374453i 0.982317 + 0.187227i \(0.0599499\pi\)
−0.982317 + 0.187227i \(0.940050\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 5.38344 0.277261
\(378\) 0 0
\(379\) 29.1313 1.49638 0.748188 0.663487i \(-0.230924\pi\)
0.748188 + 0.663487i \(0.230924\pi\)
\(380\) 0 0
\(381\) 1.36794 0.0700815
\(382\) 0 0
\(383\) −23.5528 −1.20349 −0.601746 0.798688i \(-0.705528\pi\)
−0.601746 + 0.798688i \(0.705528\pi\)
\(384\) 0 0
\(385\) 0.781836i 0.0398461i
\(386\) 0 0
\(387\) −2.03228 −0.103306
\(388\) 0 0
\(389\) 31.4964i 1.59693i −0.602039 0.798466i \(-0.705645\pi\)
0.602039 0.798466i \(-0.294355\pi\)
\(390\) 0 0
\(391\) 27.2081 20.8022i 1.37597 1.05201i
\(392\) 0 0
\(393\) 11.8087 0.595668
\(394\) 0 0
\(395\) 3.99026i 0.200772i
\(396\) 0 0
\(397\) 2.30405 0.115637 0.0578184 0.998327i \(-0.481586\pi\)
0.0578184 + 0.998327i \(0.481586\pi\)
\(398\) 0 0
\(399\) 1.36275i 0.0682226i
\(400\) 0 0
\(401\) 6.47560i 0.323376i 0.986842 + 0.161688i \(0.0516939\pi\)
−0.986842 + 0.161688i \(0.948306\pi\)
\(402\) 0 0
\(403\) 13.4200i 0.668498i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 0.704325i 0.0349121i
\(408\) 0 0
\(409\) 19.1498 0.946897 0.473449 0.880821i \(-0.343009\pi\)
0.473449 + 0.880821i \(0.343009\pi\)
\(410\) 0 0
\(411\) −12.1571 −0.599666
\(412\) 0 0
\(413\) 16.2098i 0.797634i
\(414\) 0 0
\(415\) 1.20991i 0.0593921i
\(416\) 0 0
\(417\) −14.5794 −0.713956
\(418\) 0 0
\(419\) −7.39634 −0.361335 −0.180668 0.983544i \(-0.557826\pi\)
−0.180668 + 0.983544i \(0.557826\pi\)
\(420\) 0 0
\(421\) 3.32719i 0.162157i −0.996708 0.0810787i \(-0.974163\pi\)
0.996708 0.0810787i \(-0.0258365\pi\)
\(422\) 0 0
\(423\) 6.61495i 0.321630i
\(424\) 0 0
\(425\) 7.14147i 0.346412i
\(426\) 0 0
\(427\) 2.89670i 0.140181i
\(428\) 0 0
\(429\) 2.29903i 0.110998i
\(430\) 0 0
\(431\) 37.5082 1.80670 0.903352 0.428899i \(-0.141098\pi\)
0.903352 + 0.428899i \(0.141098\pi\)
\(432\) 0 0
\(433\) 25.6190i 1.23117i 0.788070 + 0.615586i \(0.211081\pi\)
−0.788070 + 0.615586i \(0.788919\pi\)
\(434\) 0 0
\(435\) −1.26464 −0.0606349
\(436\) 0 0
\(437\) −2.74203 3.58643i −0.131169 0.171562i
\(438\) 0 0
\(439\) 10.5861i 0.505246i 0.967565 + 0.252623i \(0.0812932\pi\)
−0.967565 + 0.252623i \(0.918707\pi\)
\(440\) 0 0
\(441\) 4.90431 0.233539
\(442\) 0 0
\(443\) 34.7424i 1.65066i 0.564649 + 0.825331i \(0.309011\pi\)
−0.564649 + 0.825331i \(0.690989\pi\)
\(444\) 0 0
\(445\) 17.8161 0.844565
\(446\) 0 0
\(447\) −4.61311 −0.218193
\(448\) 0 0
\(449\) −6.14217 −0.289867 −0.144933 0.989441i \(-0.546297\pi\)
−0.144933 + 0.989441i \(0.546297\pi\)
\(450\) 0 0
\(451\) −3.22938 −0.152066
\(452\) 0 0
\(453\) −22.2226 −1.04411
\(454\) 0 0
\(455\) 6.16248i 0.288902i
\(456\) 0 0
\(457\) 9.20228i 0.430464i 0.976563 + 0.215232i \(0.0690508\pi\)
−0.976563 + 0.215232i \(0.930949\pi\)
\(458\) 0 0
\(459\) −7.14147 −0.333335
\(460\) 0 0
\(461\) 8.39680 0.391078 0.195539 0.980696i \(-0.437354\pi\)
0.195539 + 0.980696i \(0.437354\pi\)
\(462\) 0 0
\(463\) 34.0880i 1.58421i 0.610387 + 0.792103i \(0.291014\pi\)
−0.610387 + 0.792103i \(0.708986\pi\)
\(464\) 0 0
\(465\) 3.15253i 0.146195i
\(466\) 0 0
\(467\) −30.5287 −1.41270 −0.706351 0.707862i \(-0.749660\pi\)
−0.706351 + 0.707862i \(0.749660\pi\)
\(468\) 0 0
\(469\) 11.1284 0.513862
\(470\) 0 0
\(471\) 23.2200 1.06992
\(472\) 0 0
\(473\) −1.09758 −0.0504667
\(474\) 0 0
\(475\) 0.941351 0.0431921
\(476\) 0 0
\(477\) 7.54328i 0.345383i
\(478\) 0 0
\(479\) 8.47938 0.387433 0.193716 0.981058i \(-0.437946\pi\)
0.193716 + 0.981058i \(0.437946\pi\)
\(480\) 0 0
\(481\) 5.55153i 0.253128i
\(482\) 0 0
\(483\) −5.51537 + 4.21681i −0.250958 + 0.191872i
\(484\) 0 0
\(485\) 11.5224 0.523204
\(486\) 0 0
\(487\) 15.8113i 0.716480i −0.933630 0.358240i \(-0.883377\pi\)
0.933630 0.358240i \(-0.116623\pi\)
\(488\) 0 0
\(489\) −19.5709 −0.885025
\(490\) 0 0
\(491\) 0.139651i 0.00630236i −0.999995 0.00315118i \(-0.998997\pi\)
0.999995 0.00315118i \(-0.00100305\pi\)
\(492\) 0 0
\(493\) 9.03139i 0.406753i
\(494\) 0 0
\(495\) 0.540073i 0.0242745i
\(496\) 0 0
\(497\) 11.6848i 0.524135i
\(498\) 0 0
\(499\) 21.9494i 0.982588i −0.870994 0.491294i \(-0.836524\pi\)
0.870994 0.491294i \(-0.163476\pi\)
\(500\) 0 0
\(501\) −16.9663 −0.757998
\(502\) 0 0
\(503\) −40.7383 −1.81643 −0.908216 0.418501i \(-0.862555\pi\)
−0.908216 + 0.418501i \(0.862555\pi\)
\(504\) 0 0
\(505\) 13.5957i 0.605001i
\(506\) 0 0
\(507\) 5.12112i 0.227437i
\(508\) 0 0
\(509\) −11.4424 −0.507177 −0.253589 0.967312i \(-0.581611\pi\)
−0.253589 + 0.967312i \(0.581611\pi\)
\(510\) 0 0
\(511\) 2.76795 0.122447
\(512\) 0 0
\(513\) 0.941351i 0.0415617i
\(514\) 0 0
\(515\) 3.76574i 0.165938i
\(516\) 0 0
\(517\) 3.57255i 0.157121i
\(518\) 0 0
\(519\) 23.6431i 1.03781i
\(520\) 0 0
\(521\) 8.24736i 0.361323i −0.983545 0.180662i \(-0.942176\pi\)
0.983545 0.180662i \(-0.0578239\pi\)
\(522\) 0 0
\(523\) 27.7257 1.21236 0.606179 0.795328i \(-0.292701\pi\)
0.606179 + 0.795328i \(0.292701\pi\)
\(524\) 0 0
\(525\) 1.44765i 0.0631806i
\(526\) 0 0
\(527\) −22.5137 −0.980713
\(528\) 0 0
\(529\) −6.03037 + 22.1954i −0.262190 + 0.965016i
\(530\) 0 0
\(531\) 11.1974i 0.485924i
\(532\) 0 0
\(533\) 25.4542 1.10254
\(534\) 0 0
\(535\) 2.93296i 0.126803i
\(536\) 0 0
\(537\) −13.7587 −0.593732
\(538\) 0 0
\(539\) 2.64869 0.114087
\(540\) 0 0
\(541\) 0.989076 0.0425237 0.0212619 0.999774i \(-0.493232\pi\)
0.0212619 + 0.999774i \(0.493232\pi\)
\(542\) 0 0
\(543\) 23.9965 1.02979
\(544\) 0 0
\(545\) −14.4914 −0.620744
\(546\) 0 0
\(547\) 22.6052i 0.966530i 0.875474 + 0.483265i \(0.160549\pi\)
−0.875474 + 0.483265i \(0.839451\pi\)
\(548\) 0 0
\(549\) 2.00097i 0.0853992i
\(550\) 0 0
\(551\) −1.19047 −0.0507158
\(552\) 0 0
\(553\) −5.77649 −0.245641
\(554\) 0 0
\(555\) 1.30413i 0.0553572i
\(556\) 0 0
\(557\) 21.4269i 0.907885i −0.891031 0.453943i \(-0.850017\pi\)
0.891031 0.453943i \(-0.149983\pi\)
\(558\) 0 0
\(559\) 8.65118 0.365906
\(560\) 0 0
\(561\) −3.85691 −0.162839
\(562\) 0 0
\(563\) 10.5157 0.443184 0.221592 0.975139i \(-0.428875\pi\)
0.221592 + 0.975139i \(0.428875\pi\)
\(564\) 0 0
\(565\) 21.0266 0.884594
\(566\) 0 0
\(567\) 1.44765 0.0607955
\(568\) 0 0
\(569\) 12.1516i 0.509420i −0.967017 0.254710i \(-0.918020\pi\)
0.967017 0.254710i \(-0.0819801\pi\)
\(570\) 0 0
\(571\) 32.1688 1.34622 0.673112 0.739540i \(-0.264957\pi\)
0.673112 + 0.739540i \(0.264957\pi\)
\(572\) 0 0
\(573\) 0.662043i 0.0276572i
\(574\) 0 0
\(575\) −2.91287 3.80988i −0.121475 0.158883i
\(576\) 0 0
\(577\) 2.14210 0.0891767 0.0445884 0.999005i \(-0.485802\pi\)
0.0445884 + 0.999005i \(0.485802\pi\)
\(578\) 0 0
\(579\) 8.81504i 0.366341i
\(580\) 0 0
\(581\) 1.75152 0.0726654
\(582\) 0 0
\(583\) 4.07392i 0.168725i
\(584\) 0 0
\(585\) 4.25689i 0.176001i
\(586\) 0 0
\(587\) 5.47981i 0.226176i −0.993585 0.113088i \(-0.963926\pi\)
0.993585 0.113088i \(-0.0360742\pi\)
\(588\) 0 0
\(589\) 2.96764i 0.122280i
\(590\) 0 0
\(591\) 25.8264i 1.06236i
\(592\) 0 0
\(593\) 9.42307 0.386959 0.193480 0.981104i \(-0.438023\pi\)
0.193480 + 0.981104i \(0.438023\pi\)
\(594\) 0 0
\(595\) 10.3383 0.423831
\(596\) 0 0
\(597\) 19.6257i 0.803227i
\(598\) 0 0
\(599\) 5.88186i 0.240326i −0.992754 0.120163i \(-0.961658\pi\)
0.992754 0.120163i \(-0.0383418\pi\)
\(600\) 0 0
\(601\) −12.9854 −0.529686 −0.264843 0.964292i \(-0.585320\pi\)
−0.264843 + 0.964292i \(0.585320\pi\)
\(602\) 0 0
\(603\) −7.68722 −0.313048
\(604\) 0 0
\(605\) 10.7083i 0.435355i
\(606\) 0 0
\(607\) 16.2816i 0.660851i 0.943832 + 0.330426i \(0.107192\pi\)
−0.943832 + 0.330426i \(0.892808\pi\)
\(608\) 0 0
\(609\) 1.83076i 0.0741860i
\(610\) 0 0
\(611\) 28.1591i 1.13920i
\(612\) 0 0
\(613\) 13.0103i 0.525483i 0.964866 + 0.262741i \(0.0846266\pi\)
−0.964866 + 0.262741i \(0.915373\pi\)
\(614\) 0 0
\(615\) −5.97953 −0.241118
\(616\) 0 0
\(617\) 25.4926i 1.02629i −0.858301 0.513146i \(-0.828480\pi\)
0.858301 0.513146i \(-0.171520\pi\)
\(618\) 0 0
\(619\) −8.65453 −0.347855 −0.173927 0.984758i \(-0.555646\pi\)
−0.173927 + 0.984758i \(0.555646\pi\)
\(620\) 0 0
\(621\) 3.80988 2.91287i 0.152885 0.116889i
\(622\) 0 0
\(623\) 25.7915i 1.03331i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0.508398i 0.0203035i
\(628\) 0 0
\(629\) 9.31339 0.371349
\(630\) 0 0
\(631\) 12.9000 0.513542 0.256771 0.966472i \(-0.417341\pi\)
0.256771 + 0.966472i \(0.417341\pi\)
\(632\) 0 0
\(633\) −16.4801 −0.655027
\(634\) 0 0
\(635\) 1.36794 0.0542849
\(636\) 0 0
\(637\) −20.8771 −0.827182
\(638\) 0 0
\(639\) 8.07158i 0.319307i
\(640\) 0 0
\(641\) 4.96834i 0.196238i −0.995175 0.0981188i \(-0.968717\pi\)
0.995175 0.0981188i \(-0.0312825\pi\)
\(642\) 0 0
\(643\) −31.8422 −1.25573 −0.627866 0.778321i \(-0.716071\pi\)
−0.627866 + 0.778321i \(0.716071\pi\)
\(644\) 0 0
\(645\) −2.03228 −0.0800208
\(646\) 0 0
\(647\) 21.2108i 0.833883i 0.908933 + 0.416942i \(0.136898\pi\)
−0.908933 + 0.416942i \(0.863102\pi\)
\(648\) 0 0
\(649\) 6.04739i 0.237381i
\(650\) 0 0
\(651\) 4.56376 0.178868
\(652\) 0 0
\(653\) 27.7171 1.08466 0.542328 0.840167i \(-0.317543\pi\)
0.542328 + 0.840167i \(0.317543\pi\)
\(654\) 0 0
\(655\) 11.8087 0.461402
\(656\) 0 0
\(657\) −1.91203 −0.0745955
\(658\) 0 0
\(659\) −5.96186 −0.232241 −0.116121 0.993235i \(-0.537046\pi\)
−0.116121 + 0.993235i \(0.537046\pi\)
\(660\) 0 0
\(661\) 23.4766i 0.913133i 0.889689 + 0.456567i \(0.150921\pi\)
−0.889689 + 0.456567i \(0.849079\pi\)
\(662\) 0 0
\(663\) 30.4005 1.18066
\(664\) 0 0
\(665\) 1.36275i 0.0528450i
\(666\) 0 0
\(667\) 3.68374 + 4.81813i 0.142635 + 0.186559i
\(668\) 0 0
\(669\) 0.687718 0.0265887
\(670\) 0 0
\(671\) 1.08067i 0.0417187i
\(672\) 0 0
\(673\) −12.4021 −0.478068 −0.239034 0.971011i \(-0.576831\pi\)
−0.239034 + 0.971011i \(0.576831\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 39.6505i 1.52389i −0.647639 0.761947i \(-0.724244\pi\)
0.647639 0.761947i \(-0.275756\pi\)
\(678\) 0 0
\(679\) 16.6804i 0.640134i
\(680\) 0 0
\(681\) 17.6458i 0.676189i
\(682\) 0 0
\(683\) 38.5043i 1.47333i 0.676260 + 0.736663i \(0.263599\pi\)
−0.676260 + 0.736663i \(0.736401\pi\)
\(684\) 0 0
\(685\) −12.1571 −0.464499
\(686\) 0 0
\(687\) 6.99625 0.266924
\(688\) 0 0
\(689\) 32.1109i 1.22333i
\(690\) 0 0
\(691\) 14.9388i 0.568300i −0.958780 0.284150i \(-0.908289\pi\)
0.958780 0.284150i \(-0.0917114\pi\)
\(692\) 0 0
\(693\) 0.781836 0.0296995
\(694\) 0 0
\(695\) −14.5794 −0.553028
\(696\) 0 0
\(697\) 42.7026i 1.61748i
\(698\) 0 0
\(699\) 3.36671i 0.127341i
\(700\) 0 0
\(701\) 32.4050i 1.22392i −0.790888 0.611961i \(-0.790381\pi\)
0.790888 0.611961i \(-0.209619\pi\)
\(702\) 0 0
\(703\) 1.22764i 0.0463014i
\(704\) 0 0
\(705\) 6.61495i 0.249133i
\(706\) 0 0
\(707\) 19.6818 0.740211
\(708\) 0 0
\(709\) 3.25259i 0.122154i 0.998133 + 0.0610768i \(0.0194535\pi\)
−0.998133 + 0.0610768i \(0.980547\pi\)
\(710\) 0 0
\(711\) 3.99026 0.149646
\(712\) 0 0
\(713\) 12.0108 9.18293i 0.449807 0.343903i
\(714\) 0 0
\(715\) 2.29903i 0.0859789i
\(716\) 0 0
\(717\) −15.0526 −0.562151
\(718\) 0 0
\(719\) 36.9865i 1.37936i −0.724112 0.689682i \(-0.757750\pi\)
0.724112 0.689682i \(-0.242250\pi\)
\(720\) 0 0
\(721\) 5.45146 0.203023
\(722\) 0 0
\(723\) −6.51887 −0.242439
\(724\) 0 0
\(725\) −1.26464 −0.0469676
\(726\) 0 0
\(727\) 15.0714 0.558966 0.279483 0.960151i \(-0.409837\pi\)
0.279483 + 0.960151i \(0.409837\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 14.5134i 0.536799i
\(732\) 0 0
\(733\) 30.7487i 1.13573i −0.823122 0.567864i \(-0.807770\pi\)
0.823122 0.567864i \(-0.192230\pi\)
\(734\) 0 0
\(735\) 4.90431 0.180898
\(736\) 0 0
\(737\) −4.15166 −0.152928
\(738\) 0 0
\(739\) 47.1416i 1.73413i −0.498195 0.867065i \(-0.666004\pi\)
0.498195 0.867065i \(-0.333996\pi\)
\(740\) 0 0
\(741\) 4.00723i 0.147209i
\(742\) 0 0
\(743\) 4.60643 0.168993 0.0844967 0.996424i \(-0.473072\pi\)
0.0844967 + 0.996424i \(0.473072\pi\)
\(744\) 0 0
\(745\) −4.61311 −0.169011
\(746\) 0 0
\(747\) −1.20991 −0.0442683
\(748\) 0 0
\(749\) 4.24589 0.155142
\(750\) 0 0
\(751\) −44.2542 −1.61486 −0.807430 0.589964i \(-0.799142\pi\)
−0.807430 + 0.589964i \(0.799142\pi\)
\(752\) 0 0
\(753\) 11.0617i 0.403110i
\(754\) 0 0
\(755\) −22.2226 −0.808764
\(756\) 0 0
\(757\) 3.88762i 0.141298i 0.997501 + 0.0706489i \(0.0225070\pi\)
−0.997501 + 0.0706489i \(0.977493\pi\)
\(758\) 0 0
\(759\) 2.05761 1.57316i 0.0746866 0.0571022i
\(760\) 0 0
\(761\) −12.0227 −0.435821 −0.217911 0.975969i \(-0.569924\pi\)
−0.217911 + 0.975969i \(0.569924\pi\)
\(762\) 0 0
\(763\) 20.9785i 0.759473i
\(764\) 0 0
\(765\) −7.14147 −0.258200
\(766\) 0 0
\(767\) 47.6659i 1.72112i
\(768\) 0 0
\(769\) 30.0089i 1.08215i 0.840975 + 0.541074i \(0.181982\pi\)
−0.840975 + 0.541074i \(0.818018\pi\)
\(770\) 0 0
\(771\) 4.60346i 0.165789i
\(772\) 0 0
\(773\) 13.0739i 0.470236i 0.971967 + 0.235118i \(0.0755476\pi\)
−0.971967 + 0.235118i \(0.924452\pi\)
\(774\) 0 0
\(775\) 3.15253i 0.113242i
\(776\) 0 0
\(777\) −1.88792 −0.0677288
\(778\) 0 0
\(779\) −5.62883 −0.201674
\(780\) 0 0
\(781\) 4.35924i 0.155986i
\(782\) 0 0
\(783\) 1.26464i 0.0451946i
\(784\) 0 0
\(785\) 23.2200 0.828758
\(786\) 0 0
\(787\) 6.22428 0.221872 0.110936 0.993828i \(-0.464615\pi\)
0.110936 + 0.993828i \(0.464615\pi\)
\(788\) 0 0
\(789\) 0.952711i 0.0339174i
\(790\) 0 0
\(791\) 30.4391i 1.08229i
\(792\) 0 0
\(793\) 8.51789i 0.302479i
\(794\) 0 0
\(795\) 7.54328i 0.267533i
\(796\) 0 0
\(797\) 14.1298i 0.500502i −0.968181 0.250251i \(-0.919487\pi\)
0.968181 0.250251i \(-0.0805132\pi\)
\(798\) 0 0
\(799\) −47.2404 −1.67125
\(800\) 0 0
\(801\) 17.8161i 0.629502i
\(802\) 0 0
\(803\) −1.03264 −0.0364410
\(804\) 0 0
\(805\) −5.51537 + 4.21681i −0.194391 + 0.148623i
\(806\) 0 0
\(807\) 26.8660i 0.945729i
\(808\) 0 0
\(809\) −35.4454 −1.24620 −0.623098 0.782144i \(-0.714126\pi\)
−0.623098 + 0.782144i \(0.714126\pi\)
\(810\) 0 0
\(811\) 3.09154i 0.108559i 0.998526 + 0.0542794i \(0.0172862\pi\)
−0.998526 + 0.0542794i \(0.982714\pi\)
\(812\) 0 0
\(813\) 25.2660 0.886118
\(814\) 0 0
\(815\) −19.5709 −0.685537
\(816\) 0 0
\(817\) −1.91309 −0.0669304
\(818\) 0 0
\(819\) −6.16248 −0.215335
\(820\) 0 0
\(821\) 37.5294 1.30978 0.654892 0.755722i \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(822\) 0 0
\(823\) 28.7913i 1.00360i 0.864983 + 0.501800i \(0.167329\pi\)
−0.864983 + 0.501800i \(0.832671\pi\)
\(824\) 0 0
\(825\) 0.540073i 0.0188029i
\(826\) 0 0
\(827\) −38.9124 −1.35312 −0.676558 0.736390i \(-0.736529\pi\)
−0.676558 + 0.736390i \(0.736529\pi\)
\(828\) 0 0
\(829\) 13.6829 0.475228 0.237614 0.971360i \(-0.423635\pi\)
0.237614 + 0.971360i \(0.423635\pi\)
\(830\) 0 0
\(831\) 18.2298i 0.632384i
\(832\) 0 0
\(833\) 35.0240i 1.21351i
\(834\) 0 0
\(835\) −16.9663 −0.587142
\(836\) 0 0
\(837\) −3.15253 −0.108968
\(838\) 0 0
\(839\) −46.2129 −1.59545 −0.797724 0.603023i \(-0.793963\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(840\) 0 0
\(841\) −27.4007 −0.944851
\(842\) 0 0
\(843\) −10.9007 −0.375441
\(844\) 0 0
\(845\) 5.12112i 0.176172i
\(846\) 0 0
\(847\) −15.5019 −0.532651
\(848\) 0 0
\(849\) 13.1522i 0.451382i
\(850\) 0 0
\(851\) −4.96857 + 3.79876i −0.170320 + 0.130220i
\(852\) 0 0
\(853\) −7.57168 −0.259249 −0.129625 0.991563i \(-0.541377\pi\)
−0.129625 + 0.991563i \(0.541377\pi\)
\(854\) 0 0
\(855\) 0.941351i 0.0321935i
\(856\) 0 0
\(857\) 4.89067 0.167062 0.0835310 0.996505i \(-0.473380\pi\)
0.0835310 + 0.996505i \(0.473380\pi\)
\(858\) 0 0
\(859\) 23.5029i 0.801908i −0.916098 0.400954i \(-0.868679\pi\)
0.916098 0.400954i \(-0.131321\pi\)
\(860\) 0 0
\(861\) 8.65626i 0.295004i
\(862\) 0 0
\(863\) 4.60282i 0.156682i −0.996927 0.0783409i \(-0.975038\pi\)
0.996927 0.0783409i \(-0.0249623\pi\)
\(864\) 0 0
\(865\) 23.6431i 0.803888i
\(866\) 0 0
\(867\) 34.0006i 1.15472i
\(868\) 0 0
\(869\) 2.15503 0.0731044
\(870\) 0 0
\(871\) 32.7237 1.10880
\(872\) 0 0
\(873\) 11.5224i 0.389973i
\(874\) 0 0
\(875\) 1.44765i 0.0489395i
\(876\) 0 0
\(877\) −11.0430 −0.372895 −0.186448 0.982465i \(-0.559697\pi\)
−0.186448 + 0.982465i \(0.559697\pi\)
\(878\) 0 0
\(879\) −12.4528 −0.420022
\(880\) 0 0
\(881\) 26.0903i 0.879003i −0.898242 0.439502i \(-0.855155\pi\)
0.898242 0.439502i \(-0.144845\pi\)
\(882\) 0 0
\(883\) 38.2157i 1.28606i 0.765841 + 0.643030i \(0.222323\pi\)
−0.765841 + 0.643030i \(0.777677\pi\)
\(884\) 0 0
\(885\) 11.1974i 0.376395i
\(886\) 0 0
\(887\) 49.6348i 1.66657i −0.552842 0.833286i \(-0.686457\pi\)
0.552842 0.833286i \(-0.313543\pi\)
\(888\) 0 0
\(889\) 1.98029i 0.0664169i
\(890\) 0 0
\(891\) −0.540073 −0.0180931
\(892\) 0 0
\(893\) 6.22698i 0.208378i
\(894\) 0 0
\(895\) −13.7587 −0.459903
\(896\) 0 0
\(897\) −16.2182 + 12.3998i −0.541511 + 0.414016i
\(898\) 0 0
\(899\) 3.98682i 0.132968i
\(900\) 0 0
\(901\) −53.8701 −1.79467
\(902\) 0 0
\(903\) 2.94202i 0.0979044i
\(904\) 0 0
\(905\) 23.9965 0.797669
\(906\) 0 0
\(907\) −2.35647 −0.0782454 −0.0391227 0.999234i \(-0.512456\pi\)
−0.0391227 + 0.999234i \(0.512456\pi\)
\(908\) 0 0
\(909\) −13.5957 −0.450941
\(910\) 0 0
\(911\) 30.4576 1.00910 0.504552 0.863381i \(-0.331658\pi\)
0.504552 + 0.863381i \(0.331658\pi\)
\(912\) 0 0
\(913\) −0.653439 −0.0216257
\(914\) 0 0
\(915\) 2.00097i 0.0661499i
\(916\) 0 0
\(917\) 17.0948i 0.564520i
\(918\) 0 0
\(919\) −25.2389 −0.832554 −0.416277 0.909238i \(-0.636665\pi\)
−0.416277 + 0.909238i \(0.636665\pi\)
\(920\) 0 0
\(921\) −21.9864 −0.724477
\(922\) 0 0
\(923\) 34.3598i 1.13097i
\(924\) 0 0
\(925\) 1.30413i 0.0428795i
\(926\) 0 0
\(927\) −3.76574 −0.123683
\(928\) 0 0
\(929\) −28.0820 −0.921341 −0.460671 0.887571i \(-0.652391\pi\)
−0.460671 + 0.887571i \(0.652391\pi\)
\(930\) 0 0
\(931\) 4.61668 0.151306
\(932\) 0 0
\(933\) 22.6678 0.742112
\(934\) 0 0
\(935\) −3.85691 −0.126135
\(936\) 0 0
\(937\) 56.1650i 1.83483i 0.397932 + 0.917415i \(0.369728\pi\)
−0.397932 + 0.917415i \(0.630272\pi\)
\(938\) 0 0
\(939\) −17.0883 −0.557655
\(940\) 0 0
\(941\) 13.7747i 0.449044i −0.974469 0.224522i \(-0.927918\pi\)
0.974469 0.224522i \(-0.0720820\pi\)
\(942\) 0 0
\(943\) 17.4176 + 22.7813i 0.567195 + 0.741860i
\(944\) 0 0
\(945\) 1.44765 0.0470920
\(946\) 0 0
\(947\) 20.6023i 0.669486i −0.942309 0.334743i \(-0.891350\pi\)
0.942309 0.334743i \(-0.108650\pi\)
\(948\) 0 0
\(949\) 8.13932 0.264213
\(950\) 0 0
\(951\) 29.7843i 0.965821i
\(952\) 0 0
\(953\) 34.7439i 1.12546i 0.826639 + 0.562732i \(0.190250\pi\)
−0.826639 + 0.562732i \(0.809750\pi\)
\(954\) 0 0
\(955\) 0.662043i 0.0214232i
\(956\) 0 0
\(957\) 0.682998i 0.0220782i
\(958\) 0 0
\(959\) 17.5992i 0.568308i
\(960\) 0 0
\(961\) 21.0615 0.679404
\(962\) 0 0
\(963\) −2.93296 −0.0945132
\(964\) 0 0
\(965\) 8.81504i 0.283766i
\(966\) 0 0
\(967\) 51.1563i 1.64508i −0.568709 0.822539i \(-0.692557\pi\)
0.568709 0.822539i \(-0.307443\pi\)
\(968\) 0 0
\(969\) −6.72263 −0.215962
\(970\) 0 0
\(971\) −33.1319 −1.06325 −0.531627 0.846979i \(-0.678419\pi\)
−0.531627 + 0.846979i \(0.678419\pi\)
\(972\) 0 0
\(973\) 21.1058i 0.676622i
\(974\) 0 0
\(975\) 4.25689i 0.136330i
\(976\) 0 0
\(977\) 23.6188i 0.755634i 0.925880 + 0.377817i \(0.123325\pi\)
−0.925880 + 0.377817i \(0.876675\pi\)
\(978\) 0 0
\(979\) 9.62201i 0.307521i
\(980\) 0 0
\(981\) 14.4914i 0.462676i
\(982\) 0 0
\(983\) −42.2937 −1.34896 −0.674480 0.738293i \(-0.735632\pi\)
−0.674480 + 0.738293i \(0.735632\pi\)
\(984\) 0 0
\(985\) 25.8264i 0.822897i
\(986\) 0 0
\(987\) 9.57612 0.304811
\(988\) 0 0
\(989\) 5.91976 + 7.74273i 0.188237 + 0.246205i
\(990\) 0 0
\(991\) 18.4372i 0.585678i 0.956162 + 0.292839i \(0.0946000\pi\)
−0.956162 + 0.292839i \(0.905400\pi\)
\(992\) 0 0
\(993\) 17.3613 0.550945
\(994\) 0 0
\(995\) 19.6257i 0.622177i
\(996\) 0 0
\(997\) −0.813525 −0.0257646 −0.0128823 0.999917i \(-0.504101\pi\)
−0.0128823 + 0.999917i \(0.504101\pi\)
\(998\) 0 0
\(999\) 1.30413 0.0412608
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 5520.2.be.c.1471.30 yes 32
4.3 odd 2 5520.2.be.d.1471.29 yes 32
23.22 odd 2 5520.2.be.d.1471.30 yes 32
92.91 even 2 inner 5520.2.be.c.1471.29 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5520.2.be.c.1471.29 32 92.91 even 2 inner
5520.2.be.c.1471.30 yes 32 1.1 even 1 trivial
5520.2.be.d.1471.29 yes 32 4.3 odd 2
5520.2.be.d.1471.30 yes 32 23.22 odd 2