Properties

Label 1183.2.c.f.337.1
Level $1183$
Weight $2$
Character 1183.337
Analytic conductor $9.446$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1183,2,Mod(337,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1183.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(-0.671462 - 1.24464i\) of defining polynomial
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.f.337.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.34292i q^{2} -1.14637 q^{3} -3.48929 q^{4} +1.34292i q^{5} +2.68585i q^{6} -1.00000i q^{7} +3.48929i q^{8} -1.68585 q^{9} +O(q^{10})\) \(q-2.34292i q^{2} -1.14637 q^{3} -3.48929 q^{4} +1.34292i q^{5} +2.68585i q^{6} -1.00000i q^{7} +3.48929i q^{8} -1.68585 q^{9} +3.14637 q^{10} +1.14637i q^{11} +4.00000 q^{12} -2.34292 q^{14} -1.53948i q^{15} +1.19656 q^{16} -5.83221 q^{17} +3.94981i q^{18} +3.34292i q^{19} -4.68585i q^{20} +1.14637i q^{21} +2.68585 q^{22} +3.17513 q^{23} -4.00000i q^{24} +3.19656 q^{25} +5.37169 q^{27} +3.48929i q^{28} +10.4893 q^{29} -3.60688 q^{30} -1.63565i q^{31} +4.17513i q^{32} -1.31415i q^{33} +13.6644i q^{34} +1.34292 q^{35} +5.88240 q^{36} +8.51806i q^{37} +7.83221 q^{38} -4.68585 q^{40} +0.292731i q^{41} +2.68585 q^{42} +8.15371 q^{43} -4.00000i q^{44} -2.26396i q^{45} -7.43910i q^{46} -10.6142i q^{47} -1.37169 q^{48} -1.00000 q^{49} -7.48929i q^{50} +6.68585 q^{51} -0.782020 q^{53} -12.5855i q^{54} -1.53948 q^{55} +3.48929 q^{56} -3.83221i q^{57} -24.5756i q^{58} +12.6430i q^{59} +5.37169i q^{60} -2.00000 q^{61} -3.83221 q^{62} +1.68585i q^{63} +12.1751 q^{64} -3.07896 q^{66} +6.10038i q^{67} +20.3503 q^{68} -3.63986 q^{69} -3.14637i q^{70} -1.53948i q^{71} -5.88240i q^{72} -15.3001i q^{73} +19.9572 q^{74} -3.66442 q^{75} -11.6644i q^{76} +1.14637 q^{77} +0.882404 q^{79} +1.60688i q^{80} -1.10038 q^{81} +0.685846 q^{82} +12.1292i q^{83} -4.00000i q^{84} -7.83221i q^{85} -19.1035i q^{86} -12.0246 q^{87} -4.00000 q^{88} +5.73604i q^{89} -5.30429 q^{90} -11.0790 q^{92} +1.87506i q^{93} -24.8683 q^{94} -4.48929 q^{95} -4.78623i q^{96} +5.34292i q^{97} +2.34292i q^{98} -1.93260i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} - 6 q^{4} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{3} - 6 q^{4} + 14 q^{9} + 16 q^{10} + 24 q^{12} - 2 q^{14} - 2 q^{16} - 8 q^{17} - 8 q^{22} - 20 q^{23} + 10 q^{25} - 16 q^{27} + 48 q^{29} - 40 q^{30} - 4 q^{35} + 2 q^{36} + 20 q^{38} - 4 q^{40} - 8 q^{42} - 20 q^{43} + 40 q^{48} - 6 q^{49} + 16 q^{51} + 16 q^{53} + 12 q^{55} + 6 q^{56} - 12 q^{61} + 4 q^{62} + 34 q^{64} + 24 q^{66} + 44 q^{68} + 12 q^{69} + 60 q^{74} + 32 q^{75} + 4 q^{77} - 28 q^{79} + 6 q^{81} - 20 q^{82} - 52 q^{87} - 24 q^{88} + 56 q^{90} - 24 q^{92} - 20 q^{94} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.34292i − 1.65670i −0.560213 0.828348i \(-0.689281\pi\)
0.560213 0.828348i \(-0.310719\pi\)
\(3\) −1.14637 −0.661854 −0.330927 0.943656i \(-0.607361\pi\)
−0.330927 + 0.943656i \(0.607361\pi\)
\(4\) −3.48929 −1.74464
\(5\) 1.34292i 0.600573i 0.953849 + 0.300287i \(0.0970824\pi\)
−0.953849 + 0.300287i \(0.902918\pi\)
\(6\) 2.68585i 1.09649i
\(7\) − 1.00000i − 0.377964i
\(8\) 3.48929i 1.23365i
\(9\) −1.68585 −0.561949
\(10\) 3.14637 0.994968
\(11\) 1.14637i 0.345642i 0.984953 + 0.172821i \(0.0552882\pi\)
−0.984953 + 0.172821i \(0.944712\pi\)
\(12\) 4.00000 1.15470
\(13\) 0 0
\(14\) −2.34292 −0.626173
\(15\) − 1.53948i − 0.397492i
\(16\) 1.19656 0.299139
\(17\) −5.83221 −1.41452 −0.707260 0.706954i \(-0.750069\pi\)
−0.707260 + 0.706954i \(0.750069\pi\)
\(18\) 3.94981i 0.930979i
\(19\) 3.34292i 0.766919i 0.923558 + 0.383460i \(0.125267\pi\)
−0.923558 + 0.383460i \(0.874733\pi\)
\(20\) − 4.68585i − 1.04779i
\(21\) 1.14637i 0.250157i
\(22\) 2.68585 0.572624
\(23\) 3.17513 0.662061 0.331031 0.943620i \(-0.392604\pi\)
0.331031 + 0.943620i \(0.392604\pi\)
\(24\) − 4.00000i − 0.816497i
\(25\) 3.19656 0.639312
\(26\) 0 0
\(27\) 5.37169 1.03378
\(28\) 3.48929i 0.659414i
\(29\) 10.4893 1.94781 0.973906 0.226952i \(-0.0728760\pi\)
0.973906 + 0.226952i \(0.0728760\pi\)
\(30\) −3.60688 −0.658524
\(31\) − 1.63565i − 0.293772i −0.989153 0.146886i \(-0.953075\pi\)
0.989153 0.146886i \(-0.0469251\pi\)
\(32\) 4.17513i 0.738067i
\(33\) − 1.31415i − 0.228765i
\(34\) 13.6644i 2.34343i
\(35\) 1.34292 0.226995
\(36\) 5.88240 0.980401
\(37\) 8.51806i 1.40036i 0.713966 + 0.700180i \(0.246897\pi\)
−0.713966 + 0.700180i \(0.753103\pi\)
\(38\) 7.83221 1.27055
\(39\) 0 0
\(40\) −4.68585 −0.740897
\(41\) 0.292731i 0.0457169i 0.999739 + 0.0228584i \(0.00727670\pi\)
−0.999739 + 0.0228584i \(0.992723\pi\)
\(42\) 2.68585 0.414435
\(43\) 8.15371 1.24343 0.621715 0.783244i \(-0.286436\pi\)
0.621715 + 0.783244i \(0.286436\pi\)
\(44\) − 4.00000i − 0.603023i
\(45\) − 2.26396i − 0.337491i
\(46\) − 7.43910i − 1.09683i
\(47\) − 10.6142i − 1.54824i −0.633036 0.774122i \(-0.718191\pi\)
0.633036 0.774122i \(-0.281809\pi\)
\(48\) −1.37169 −0.197987
\(49\) −1.00000 −0.142857
\(50\) − 7.48929i − 1.05915i
\(51\) 6.68585 0.936206
\(52\) 0 0
\(53\) −0.782020 −0.107419 −0.0537093 0.998557i \(-0.517104\pi\)
−0.0537093 + 0.998557i \(0.517104\pi\)
\(54\) − 12.5855i − 1.71266i
\(55\) −1.53948 −0.207584
\(56\) 3.48929 0.466276
\(57\) − 3.83221i − 0.507589i
\(58\) − 24.5756i − 3.22693i
\(59\) 12.6430i 1.64598i 0.568057 + 0.822989i \(0.307695\pi\)
−0.568057 + 0.822989i \(0.692305\pi\)
\(60\) 5.37169i 0.693482i
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −3.83221 −0.486691
\(63\) 1.68585i 0.212397i
\(64\) 12.1751 1.52189
\(65\) 0 0
\(66\) −3.07896 −0.378994
\(67\) 6.10038i 0.745281i 0.927976 + 0.372640i \(0.121547\pi\)
−0.927976 + 0.372640i \(0.878453\pi\)
\(68\) 20.3503 2.46783
\(69\) −3.63986 −0.438188
\(70\) − 3.14637i − 0.376063i
\(71\) − 1.53948i − 0.182703i −0.995819 0.0913514i \(-0.970881\pi\)
0.995819 0.0913514i \(-0.0291186\pi\)
\(72\) − 5.88240i − 0.693248i
\(73\) − 15.3001i − 1.79074i −0.445324 0.895369i \(-0.646912\pi\)
0.445324 0.895369i \(-0.353088\pi\)
\(74\) 19.9572 2.31997
\(75\) −3.66442 −0.423131
\(76\) − 11.6644i − 1.33800i
\(77\) 1.14637 0.130640
\(78\) 0 0
\(79\) 0.882404 0.0992782 0.0496391 0.998767i \(-0.484193\pi\)
0.0496391 + 0.998767i \(0.484193\pi\)
\(80\) 1.60688i 0.179655i
\(81\) −1.10038 −0.122265
\(82\) 0.685846 0.0757390
\(83\) 12.1292i 1.33135i 0.746243 + 0.665674i \(0.231856\pi\)
−0.746243 + 0.665674i \(0.768144\pi\)
\(84\) − 4.00000i − 0.436436i
\(85\) − 7.83221i − 0.849523i
\(86\) − 19.1035i − 2.05999i
\(87\) −12.0246 −1.28917
\(88\) −4.00000 −0.426401
\(89\) 5.73604i 0.608019i 0.952669 + 0.304009i \(0.0983254\pi\)
−0.952669 + 0.304009i \(0.901675\pi\)
\(90\) −5.30429 −0.559121
\(91\) 0 0
\(92\) −11.0790 −1.15506
\(93\) 1.87506i 0.194434i
\(94\) −24.8683 −2.56497
\(95\) −4.48929 −0.460591
\(96\) − 4.78623i − 0.488493i
\(97\) 5.34292i 0.542492i 0.962510 + 0.271246i \(0.0874356\pi\)
−0.962510 + 0.271246i \(0.912564\pi\)
\(98\) 2.34292i 0.236671i
\(99\) − 1.93260i − 0.194233i
\(100\) −11.1537 −1.11537
\(101\) −11.1464 −1.10910 −0.554552 0.832149i \(-0.687111\pi\)
−0.554552 + 0.832149i \(0.687111\pi\)
\(102\) − 15.6644i − 1.55101i
\(103\) 3.41454 0.336444 0.168222 0.985749i \(-0.446197\pi\)
0.168222 + 0.985749i \(0.446197\pi\)
\(104\) 0 0
\(105\) −1.53948 −0.150238
\(106\) 1.83221i 0.177960i
\(107\) 4.97858 0.481297 0.240649 0.970612i \(-0.422640\pi\)
0.240649 + 0.970612i \(0.422640\pi\)
\(108\) −18.7434 −1.80358
\(109\) 13.4966i 1.29274i 0.763023 + 0.646372i \(0.223714\pi\)
−0.763023 + 0.646372i \(0.776286\pi\)
\(110\) 3.60688i 0.343903i
\(111\) − 9.76481i − 0.926835i
\(112\) − 1.19656i − 0.113064i
\(113\) 16.4464 1.54715 0.773576 0.633704i \(-0.218466\pi\)
0.773576 + 0.633704i \(0.218466\pi\)
\(114\) −8.97858 −0.840921
\(115\) 4.26396i 0.397616i
\(116\) −36.6002 −3.39824
\(117\) 0 0
\(118\) 29.6216 2.72689
\(119\) 5.83221i 0.534638i
\(120\) 5.37169 0.490366
\(121\) 9.68585 0.880531
\(122\) 4.68585i 0.424237i
\(123\) − 0.335577i − 0.0302579i
\(124\) 5.70727i 0.512528i
\(125\) 11.0073i 0.984527i
\(126\) 3.94981 0.351877
\(127\) −12.0575 −1.06993 −0.534967 0.844873i \(-0.679676\pi\)
−0.534967 + 0.844873i \(0.679676\pi\)
\(128\) − 20.1751i − 1.78325i
\(129\) −9.34713 −0.822969
\(130\) 0 0
\(131\) −3.66442 −0.320162 −0.160081 0.987104i \(-0.551176\pi\)
−0.160081 + 0.987104i \(0.551176\pi\)
\(132\) 4.58546i 0.399113i
\(133\) 3.34292 0.289868
\(134\) 14.2927 1.23470
\(135\) 7.21377i 0.620862i
\(136\) − 20.3503i − 1.74502i
\(137\) − 13.1035i − 1.11951i −0.828658 0.559755i \(-0.810895\pi\)
0.828658 0.559755i \(-0.189105\pi\)
\(138\) 8.52792i 0.725945i
\(139\) 7.49663 0.635856 0.317928 0.948115i \(-0.397013\pi\)
0.317928 + 0.948115i \(0.397013\pi\)
\(140\) −4.68585 −0.396026
\(141\) 12.1678i 1.02471i
\(142\) −3.60688 −0.302683
\(143\) 0 0
\(144\) −2.01721 −0.168101
\(145\) 14.0863i 1.16980i
\(146\) −35.8469 −2.96671
\(147\) 1.14637 0.0945506
\(148\) − 29.7220i − 2.44313i
\(149\) − 2.16779i − 0.177592i −0.996050 0.0887961i \(-0.971698\pi\)
0.996050 0.0887961i \(-0.0283019\pi\)
\(150\) 8.58546i 0.701000i
\(151\) 14.9112i 1.21345i 0.794910 + 0.606727i \(0.207518\pi\)
−0.794910 + 0.606727i \(0.792482\pi\)
\(152\) −11.6644 −0.946110
\(153\) 9.83221 0.794887
\(154\) − 2.68585i − 0.216432i
\(155\) 2.19656 0.176432
\(156\) 0 0
\(157\) 22.8683 1.82509 0.912546 0.408975i \(-0.134114\pi\)
0.912546 + 0.408975i \(0.134114\pi\)
\(158\) − 2.06740i − 0.164474i
\(159\) 0.896480 0.0710955
\(160\) −5.60688 −0.443263
\(161\) − 3.17513i − 0.250236i
\(162\) 2.57812i 0.202556i
\(163\) 7.07896i 0.554467i 0.960803 + 0.277234i \(0.0894176\pi\)
−0.960803 + 0.277234i \(0.910582\pi\)
\(164\) − 1.02142i − 0.0797597i
\(165\) 1.76481 0.137390
\(166\) 28.4177 2.20564
\(167\) 2.61423i 0.202295i 0.994871 + 0.101148i \(0.0322514\pi\)
−0.994871 + 0.101148i \(0.967749\pi\)
\(168\) −4.00000 −0.308607
\(169\) 0 0
\(170\) −18.3503 −1.40740
\(171\) − 5.63565i − 0.430969i
\(172\) −28.4507 −2.16934
\(173\) −11.0031 −0.836553 −0.418276 0.908320i \(-0.637366\pi\)
−0.418276 + 0.908320i \(0.637366\pi\)
\(174\) 28.1726i 2.13576i
\(175\) − 3.19656i − 0.241637i
\(176\) 1.37169i 0.103395i
\(177\) − 14.4935i − 1.08940i
\(178\) 13.4391 1.00730
\(179\) −23.9614 −1.79096 −0.895478 0.445105i \(-0.853166\pi\)
−0.895478 + 0.445105i \(0.853166\pi\)
\(180\) 7.89962i 0.588803i
\(181\) −6.56090 −0.487668 −0.243834 0.969817i \(-0.578405\pi\)
−0.243834 + 0.969817i \(0.578405\pi\)
\(182\) 0 0
\(183\) 2.29273 0.169484
\(184\) 11.0790i 0.816752i
\(185\) −11.4391 −0.841019
\(186\) 4.39312 0.322119
\(187\) − 6.68585i − 0.488917i
\(188\) 37.0361i 2.70114i
\(189\) − 5.37169i − 0.390733i
\(190\) 10.5181i 0.763060i
\(191\) −4.39312 −0.317875 −0.158937 0.987289i \(-0.550807\pi\)
−0.158937 + 0.987289i \(0.550807\pi\)
\(192\) −13.9572 −1.00727
\(193\) − 8.29273i − 0.596924i −0.954422 0.298462i \(-0.903526\pi\)
0.954422 0.298462i \(-0.0964736\pi\)
\(194\) 12.5181 0.898744
\(195\) 0 0
\(196\) 3.48929 0.249235
\(197\) 3.17092i 0.225919i 0.993600 + 0.112959i \(0.0360330\pi\)
−0.993600 + 0.112959i \(0.963967\pi\)
\(198\) −4.52792 −0.321786
\(199\) 13.5970 0.963867 0.481934 0.876208i \(-0.339935\pi\)
0.481934 + 0.876208i \(0.339935\pi\)
\(200\) 11.1537i 0.788687i
\(201\) − 6.99327i − 0.493267i
\(202\) 26.1151i 1.83745i
\(203\) − 10.4893i − 0.736204i
\(204\) −23.3288 −1.63335
\(205\) −0.393115 −0.0274564
\(206\) − 8.00000i − 0.557386i
\(207\) −5.35279 −0.372045
\(208\) 0 0
\(209\) −3.83221 −0.265080
\(210\) 3.60688i 0.248899i
\(211\) 9.27552 0.638553 0.319277 0.947662i \(-0.396560\pi\)
0.319277 + 0.947662i \(0.396560\pi\)
\(212\) 2.72869 0.187407
\(213\) 1.76481i 0.120923i
\(214\) − 11.6644i − 0.797364i
\(215\) 10.9498i 0.746771i
\(216\) 18.7434i 1.27533i
\(217\) −1.63565 −0.111035
\(218\) 31.6216 2.14168
\(219\) 17.5395i 1.18521i
\(220\) 5.37169 0.362159
\(221\) 0 0
\(222\) −22.8782 −1.53548
\(223\) 19.5928i 1.31203i 0.754747 + 0.656016i \(0.227760\pi\)
−0.754747 + 0.656016i \(0.772240\pi\)
\(224\) 4.17513 0.278963
\(225\) −5.38890 −0.359260
\(226\) − 38.5328i − 2.56316i
\(227\) 19.6644i 1.30517i 0.757714 + 0.652587i \(0.226316\pi\)
−0.757714 + 0.652587i \(0.773684\pi\)
\(228\) 13.3717i 0.885562i
\(229\) 7.76481i 0.513113i 0.966529 + 0.256556i \(0.0825880\pi\)
−0.966529 + 0.256556i \(0.917412\pi\)
\(230\) 9.99013 0.658730
\(231\) −1.31415 −0.0864650
\(232\) 36.6002i 2.40292i
\(233\) −12.1966 −0.799023 −0.399512 0.916728i \(-0.630820\pi\)
−0.399512 + 0.916728i \(0.630820\pi\)
\(234\) 0 0
\(235\) 14.2541 0.929835
\(236\) − 44.1151i − 2.87165i
\(237\) −1.01156 −0.0657077
\(238\) 13.6644 0.885733
\(239\) 10.2927i 0.665781i 0.942965 + 0.332891i \(0.108024\pi\)
−0.942965 + 0.332891i \(0.891976\pi\)
\(240\) − 1.84208i − 0.118906i
\(241\) − 4.02877i − 0.259516i −0.991546 0.129758i \(-0.958580\pi\)
0.991546 0.129758i \(-0.0414200\pi\)
\(242\) − 22.6932i − 1.45877i
\(243\) −14.8536 −0.952861
\(244\) 6.97858 0.446758
\(245\) − 1.34292i − 0.0857962i
\(246\) −0.786230 −0.0501282
\(247\) 0 0
\(248\) 5.70727 0.362412
\(249\) − 13.9044i − 0.881158i
\(250\) 25.7894 1.63106
\(251\) −2.91117 −0.183752 −0.0918758 0.995770i \(-0.529286\pi\)
−0.0918758 + 0.995770i \(0.529286\pi\)
\(252\) − 5.88240i − 0.370557i
\(253\) 3.63986i 0.228836i
\(254\) 28.2499i 1.77256i
\(255\) 8.97858i 0.562260i
\(256\) −22.9185 −1.43241
\(257\) 19.5970 1.22243 0.611214 0.791465i \(-0.290681\pi\)
0.611214 + 0.791465i \(0.290681\pi\)
\(258\) 21.8996i 1.36341i
\(259\) 8.51806 0.529286
\(260\) 0 0
\(261\) −17.6833 −1.09457
\(262\) 8.58546i 0.530412i
\(263\) 7.56825 0.466678 0.233339 0.972395i \(-0.425035\pi\)
0.233339 + 0.972395i \(0.425035\pi\)
\(264\) 4.58546 0.282216
\(265\) − 1.05019i − 0.0645128i
\(266\) − 7.83221i − 0.480224i
\(267\) − 6.57560i − 0.402420i
\(268\) − 21.2860i − 1.30025i
\(269\) −9.47208 −0.577523 −0.288761 0.957401i \(-0.593243\pi\)
−0.288761 + 0.957401i \(0.593243\pi\)
\(270\) 16.9013 1.02858
\(271\) 29.3717i 1.78420i 0.451835 + 0.892102i \(0.350770\pi\)
−0.451835 + 0.892102i \(0.649230\pi\)
\(272\) −6.97858 −0.423138
\(273\) 0 0
\(274\) −30.7005 −1.85469
\(275\) 3.66442i 0.220973i
\(276\) 12.7005 0.764483
\(277\) 1.90383 0.114390 0.0571949 0.998363i \(-0.481784\pi\)
0.0571949 + 0.998363i \(0.481784\pi\)
\(278\) − 17.5640i − 1.05342i
\(279\) 2.75746i 0.165085i
\(280\) 4.68585i 0.280033i
\(281\) − 20.5756i − 1.22744i −0.789525 0.613719i \(-0.789673\pi\)
0.789525 0.613719i \(-0.210327\pi\)
\(282\) 28.5082 1.69764
\(283\) 26.9933 1.60458 0.802292 0.596932i \(-0.203614\pi\)
0.802292 + 0.596932i \(0.203614\pi\)
\(284\) 5.37169i 0.318751i
\(285\) 5.14637 0.304844
\(286\) 0 0
\(287\) 0.292731 0.0172794
\(288\) − 7.03863i − 0.414756i
\(289\) 17.0147 1.00086
\(290\) 33.0031 1.93801
\(291\) − 6.12494i − 0.359050i
\(292\) 53.3864i 3.12420i
\(293\) − 14.9070i − 0.870874i −0.900219 0.435437i \(-0.856594\pi\)
0.900219 0.435437i \(-0.143406\pi\)
\(294\) − 2.68585i − 0.156642i
\(295\) −16.9786 −0.988531
\(296\) −29.7220 −1.72755
\(297\) 6.15792i 0.357319i
\(298\) −5.07896 −0.294216
\(299\) 0 0
\(300\) 12.7862 0.738213
\(301\) − 8.15371i − 0.469972i
\(302\) 34.9357 2.01033
\(303\) 12.7778 0.734066
\(304\) 4.00000i 0.229416i
\(305\) − 2.68585i − 0.153791i
\(306\) − 23.0361i − 1.31689i
\(307\) 26.0288i 1.48554i 0.669546 + 0.742770i \(0.266488\pi\)
−0.669546 + 0.742770i \(0.733512\pi\)
\(308\) −4.00000 −0.227921
\(309\) −3.91431 −0.222677
\(310\) − 5.14637i − 0.292294i
\(311\) −19.4966 −1.10555 −0.552776 0.833330i \(-0.686432\pi\)
−0.552776 + 0.833330i \(0.686432\pi\)
\(312\) 0 0
\(313\) −3.48194 −0.196811 −0.0984055 0.995146i \(-0.531374\pi\)
−0.0984055 + 0.995146i \(0.531374\pi\)
\(314\) − 53.5787i − 3.02362i
\(315\) −2.26396 −0.127560
\(316\) −3.07896 −0.173205
\(317\) − 5.02142i − 0.282031i −0.990007 0.141016i \(-0.954963\pi\)
0.990007 0.141016i \(-0.0450368\pi\)
\(318\) − 2.10038i − 0.117784i
\(319\) 12.0246i 0.673246i
\(320\) 16.3503i 0.914008i
\(321\) −5.70727 −0.318549
\(322\) −7.43910 −0.414565
\(323\) − 19.4966i − 1.08482i
\(324\) 3.83956 0.213309
\(325\) 0 0
\(326\) 16.5855 0.918584
\(327\) − 15.4721i − 0.855608i
\(328\) −1.02142 −0.0563986
\(329\) −10.6142 −0.585182
\(330\) − 4.13481i − 0.227614i
\(331\) 6.14950i 0.338007i 0.985615 + 0.169004i \(0.0540549\pi\)
−0.985615 + 0.169004i \(0.945945\pi\)
\(332\) − 42.3221i − 2.32273i
\(333\) − 14.3601i − 0.786931i
\(334\) 6.12494 0.335142
\(335\) −8.19235 −0.447596
\(336\) 1.37169i 0.0748320i
\(337\) 25.6258 1.39593 0.697963 0.716134i \(-0.254090\pi\)
0.697963 + 0.716134i \(0.254090\pi\)
\(338\) 0 0
\(339\) −18.8536 −1.02399
\(340\) 27.3288i 1.48211i
\(341\) 1.87506 0.101540
\(342\) −13.2039 −0.713985
\(343\) 1.00000i 0.0539949i
\(344\) 28.4507i 1.53396i
\(345\) − 4.88806i − 0.263164i
\(346\) 25.7795i 1.38591i
\(347\) −16.7005 −0.896532 −0.448266 0.893900i \(-0.647958\pi\)
−0.448266 + 0.893900i \(0.647958\pi\)
\(348\) 41.9572 2.24914
\(349\) − 23.5500i − 1.26060i −0.776351 0.630300i \(-0.782932\pi\)
0.776351 0.630300i \(-0.217068\pi\)
\(350\) −7.48929 −0.400319
\(351\) 0 0
\(352\) −4.78623 −0.255107
\(353\) 7.64973i 0.407154i 0.979059 + 0.203577i \(0.0652567\pi\)
−0.979059 + 0.203577i \(0.934743\pi\)
\(354\) −33.9572 −1.80480
\(355\) 2.06740 0.109726
\(356\) − 20.0147i − 1.06078i
\(357\) − 6.68585i − 0.353853i
\(358\) 56.1396i 2.96707i
\(359\) 18.3748i 0.969786i 0.874573 + 0.484893i \(0.161142\pi\)
−0.874573 + 0.484893i \(0.838858\pi\)
\(360\) 7.89962 0.416346
\(361\) 7.82487 0.411835
\(362\) 15.3717i 0.807918i
\(363\) −11.1035 −0.582784
\(364\) 0 0
\(365\) 20.5468 1.07547
\(366\) − 5.37169i − 0.280783i
\(367\) 5.33871 0.278679 0.139339 0.990245i \(-0.455502\pi\)
0.139339 + 0.990245i \(0.455502\pi\)
\(368\) 3.79923 0.198049
\(369\) − 0.493499i − 0.0256906i
\(370\) 26.8009i 1.39331i
\(371\) 0.782020i 0.0406004i
\(372\) − 6.54262i − 0.339219i
\(373\) 21.5212 1.11433 0.557163 0.830403i \(-0.311890\pi\)
0.557163 + 0.830403i \(0.311890\pi\)
\(374\) −15.6644 −0.809988
\(375\) − 12.6184i − 0.651614i
\(376\) 37.0361 1.90999
\(377\) 0 0
\(378\) −12.5855 −0.647326
\(379\) − 4.61002i − 0.236801i −0.992966 0.118400i \(-0.962223\pi\)
0.992966 0.118400i \(-0.0377766\pi\)
\(380\) 15.6644 0.803568
\(381\) 13.8223 0.708140
\(382\) 10.2927i 0.526622i
\(383\) 8.33558i 0.425928i 0.977060 + 0.212964i \(0.0683117\pi\)
−0.977060 + 0.212964i \(0.931688\pi\)
\(384\) 23.1281i 1.18025i
\(385\) 1.53948i 0.0784592i
\(386\) −19.4292 −0.988922
\(387\) −13.7459 −0.698744
\(388\) − 18.6430i − 0.946455i
\(389\) 6.44223 0.326634 0.163317 0.986574i \(-0.447781\pi\)
0.163317 + 0.986574i \(0.447781\pi\)
\(390\) 0 0
\(391\) −18.5181 −0.936498
\(392\) − 3.48929i − 0.176236i
\(393\) 4.20077 0.211901
\(394\) 7.42923 0.374279
\(395\) 1.18500i 0.0596238i
\(396\) 6.74338i 0.338868i
\(397\) 1.40046i 0.0702872i 0.999382 + 0.0351436i \(0.0111889\pi\)
−0.999382 + 0.0351436i \(0.988811\pi\)
\(398\) − 31.8568i − 1.59684i
\(399\) −3.83221 −0.191851
\(400\) 3.82487 0.191243
\(401\) − 6.97858i − 0.348494i −0.984702 0.174247i \(-0.944251\pi\)
0.984702 0.174247i \(-0.0557490\pi\)
\(402\) −16.3847 −0.817194
\(403\) 0 0
\(404\) 38.8929 1.93499
\(405\) − 1.47773i − 0.0734291i
\(406\) −24.5756 −1.21967
\(407\) −9.76481 −0.484024
\(408\) 23.3288i 1.15495i
\(409\) 18.3790i 0.908785i 0.890802 + 0.454392i \(0.150144\pi\)
−0.890802 + 0.454392i \(0.849856\pi\)
\(410\) 0.921039i 0.0454869i
\(411\) 15.0214i 0.740952i
\(412\) −11.9143 −0.586976
\(413\) 12.6430 0.622121
\(414\) 12.5412i 0.616365i
\(415\) −16.2885 −0.799572
\(416\) 0 0
\(417\) −8.59388 −0.420844
\(418\) 8.97858i 0.439157i
\(419\) −30.0393 −1.46751 −0.733757 0.679412i \(-0.762235\pi\)
−0.733757 + 0.679412i \(0.762235\pi\)
\(420\) 5.37169 0.262112
\(421\) 8.31729i 0.405360i 0.979245 + 0.202680i \(0.0649651\pi\)
−0.979245 + 0.202680i \(0.935035\pi\)
\(422\) − 21.7318i − 1.05789i
\(423\) 17.8940i 0.870034i
\(424\) − 2.72869i − 0.132517i
\(425\) −18.6430 −0.904318
\(426\) 4.13481 0.200332
\(427\) 2.00000i 0.0967868i
\(428\) −17.3717 −0.839692
\(429\) 0 0
\(430\) 25.6546 1.23717
\(431\) − 9.64973i − 0.464811i −0.972619 0.232406i \(-0.925340\pi\)
0.972619 0.232406i \(-0.0746597\pi\)
\(432\) 6.42754 0.309245
\(433\) −26.3074 −1.26425 −0.632127 0.774865i \(-0.717818\pi\)
−0.632127 + 0.774865i \(0.717818\pi\)
\(434\) 3.83221i 0.183952i
\(435\) − 16.1481i − 0.774240i
\(436\) − 47.0937i − 2.25538i
\(437\) 10.6142i 0.507748i
\(438\) 41.0937 1.96353
\(439\) −33.8139 −1.61385 −0.806925 0.590654i \(-0.798870\pi\)
−0.806925 + 0.590654i \(0.798870\pi\)
\(440\) − 5.37169i − 0.256085i
\(441\) 1.68585 0.0802784
\(442\) 0 0
\(443\) 26.4464 1.25651 0.628254 0.778008i \(-0.283770\pi\)
0.628254 + 0.778008i \(0.283770\pi\)
\(444\) 34.0722i 1.61700i
\(445\) −7.70306 −0.365160
\(446\) 45.9044 2.17364
\(447\) 2.48508i 0.117540i
\(448\) − 12.1751i − 0.575221i
\(449\) 2.64300i 0.124731i 0.998053 + 0.0623655i \(0.0198644\pi\)
−0.998053 + 0.0623655i \(0.980136\pi\)
\(450\) 12.6258i 0.595185i
\(451\) −0.335577 −0.0158017
\(452\) −57.3864 −2.69923
\(453\) − 17.0937i − 0.803130i
\(454\) 46.0722 2.16228
\(455\) 0 0
\(456\) 13.3717 0.626187
\(457\) 33.6890i 1.57590i 0.615737 + 0.787952i \(0.288859\pi\)
−0.615737 + 0.787952i \(0.711141\pi\)
\(458\) 18.1923 0.850073
\(459\) −31.3288 −1.46231
\(460\) − 14.8782i − 0.693699i
\(461\) − 33.0790i − 1.54064i −0.637657 0.770320i \(-0.720096\pi\)
0.637657 0.770320i \(-0.279904\pi\)
\(462\) 3.07896i 0.143246i
\(463\) − 2.51806i − 0.117024i −0.998287 0.0585120i \(-0.981364\pi\)
0.998287 0.0585120i \(-0.0186356\pi\)
\(464\) 12.5510 0.582667
\(465\) −2.51806 −0.116772
\(466\) 28.5756i 1.32374i
\(467\) −2.57560 −0.119184 −0.0595922 0.998223i \(-0.518980\pi\)
−0.0595922 + 0.998223i \(0.518980\pi\)
\(468\) 0 0
\(469\) 6.10038 0.281690
\(470\) − 33.3963i − 1.54045i
\(471\) −26.2155 −1.20794
\(472\) −44.1151 −2.03056
\(473\) 9.34713i 0.429782i
\(474\) 2.37000i 0.108858i
\(475\) 10.6858i 0.490300i
\(476\) − 20.3503i − 0.932753i
\(477\) 1.31836 0.0603638
\(478\) 24.1151 1.10300
\(479\) 0.513847i 0.0234783i 0.999931 + 0.0117391i \(0.00373677\pi\)
−0.999931 + 0.0117391i \(0.996263\pi\)
\(480\) 6.42754 0.293376
\(481\) 0 0
\(482\) −9.43910 −0.429939
\(483\) 3.63986i 0.165620i
\(484\) −33.7967 −1.53621
\(485\) −7.17513 −0.325806
\(486\) 34.8009i 1.57860i
\(487\) − 36.0575i − 1.63392i −0.576692 0.816962i \(-0.695657\pi\)
0.576692 0.816962i \(-0.304343\pi\)
\(488\) − 6.97858i − 0.315905i
\(489\) − 8.11508i − 0.366976i
\(490\) −3.14637 −0.142138
\(491\) 9.22846 0.416475 0.208237 0.978078i \(-0.433227\pi\)
0.208237 + 0.978078i \(0.433227\pi\)
\(492\) 1.17092i 0.0527893i
\(493\) −61.1758 −2.75522
\(494\) 0 0
\(495\) 2.59533 0.116651
\(496\) − 1.95715i − 0.0878788i
\(497\) −1.53948 −0.0690551
\(498\) −32.5770 −1.45981
\(499\) − 1.00314i − 0.0449065i −0.999748 0.0224533i \(-0.992852\pi\)
0.999748 0.0224533i \(-0.00714769\pi\)
\(500\) − 38.4078i − 1.71765i
\(501\) − 2.99686i − 0.133890i
\(502\) 6.82065i 0.304421i
\(503\) −30.3503 −1.35325 −0.676626 0.736327i \(-0.736559\pi\)
−0.676626 + 0.736327i \(0.736559\pi\)
\(504\) −5.88240 −0.262023
\(505\) − 14.9687i − 0.666099i
\(506\) 8.52792 0.379112
\(507\) 0 0
\(508\) 42.0722 1.86665
\(509\) − 10.5995i − 0.469816i −0.972018 0.234908i \(-0.924521\pi\)
0.972018 0.234908i \(-0.0754789\pi\)
\(510\) 21.0361 0.931495
\(511\) −15.3001 −0.676836
\(512\) 13.3461i 0.589818i
\(513\) 17.9572i 0.792828i
\(514\) − 45.9143i − 2.02519i
\(515\) 4.58546i 0.202060i
\(516\) 32.6148 1.43579
\(517\) 12.1678 0.535139
\(518\) − 19.9572i − 0.876867i
\(519\) 12.6136 0.553676
\(520\) 0 0
\(521\) −16.2646 −0.712564 −0.356282 0.934378i \(-0.615956\pi\)
−0.356282 + 0.934378i \(0.615956\pi\)
\(522\) 41.4307i 1.81337i
\(523\) 7.22219 0.315804 0.157902 0.987455i \(-0.449527\pi\)
0.157902 + 0.987455i \(0.449527\pi\)
\(524\) 12.7862 0.558569
\(525\) 3.66442i 0.159929i
\(526\) − 17.7318i − 0.773144i
\(527\) 9.53948i 0.415546i
\(528\) − 1.57246i − 0.0684326i
\(529\) −12.9185 −0.561675
\(530\) −2.46052 −0.106878
\(531\) − 21.3142i − 0.924955i
\(532\) −11.6644 −0.505717
\(533\) 0 0
\(534\) −15.4061 −0.666688
\(535\) 6.68585i 0.289054i
\(536\) −21.2860 −0.919415
\(537\) 27.4685 1.18535
\(538\) 22.1923i 0.956780i
\(539\) − 1.14637i − 0.0493775i
\(540\) − 25.1709i − 1.08318i
\(541\) 17.3534i 0.746081i 0.927815 + 0.373041i \(0.121685\pi\)
−0.927815 + 0.373041i \(0.878315\pi\)
\(542\) 68.8156 2.95588
\(543\) 7.52119 0.322765
\(544\) − 24.3503i − 1.04401i
\(545\) −18.1249 −0.776387
\(546\) 0 0
\(547\) −34.1109 −1.45848 −0.729238 0.684261i \(-0.760125\pi\)
−0.729238 + 0.684261i \(0.760125\pi\)
\(548\) 45.7220i 1.95315i
\(549\) 3.37169 0.143900
\(550\) 8.58546 0.366085
\(551\) 35.0649i 1.49381i
\(552\) − 12.7005i − 0.540571i
\(553\) − 0.882404i − 0.0375236i
\(554\) − 4.46052i − 0.189509i
\(555\) 13.1134 0.556632
\(556\) −26.1579 −1.10934
\(557\) − 13.2222i − 0.560242i −0.959965 0.280121i \(-0.909625\pi\)
0.959965 0.280121i \(-0.0903746\pi\)
\(558\) 6.46052 0.273496
\(559\) 0 0
\(560\) 1.60688 0.0679033
\(561\) 7.66442i 0.323592i
\(562\) −48.2070 −2.03349
\(563\) 41.3717 1.74361 0.871804 0.489854i \(-0.162950\pi\)
0.871804 + 0.489854i \(0.162950\pi\)
\(564\) − 42.4569i − 1.78776i
\(565\) 22.0863i 0.929178i
\(566\) − 63.2432i − 2.65831i
\(567\) 1.10038i 0.0462118i
\(568\) 5.37169 0.225391
\(569\) 2.68164 0.112420 0.0562100 0.998419i \(-0.482098\pi\)
0.0562100 + 0.998419i \(0.482098\pi\)
\(570\) − 12.0575i − 0.505035i
\(571\) 28.9315 1.21075 0.605373 0.795942i \(-0.293024\pi\)
0.605373 + 0.795942i \(0.293024\pi\)
\(572\) 0 0
\(573\) 5.03612 0.210387
\(574\) − 0.685846i − 0.0286267i
\(575\) 10.1495 0.423263
\(576\) −20.5254 −0.855225
\(577\) − 37.5296i − 1.56238i −0.624294 0.781189i \(-0.714613\pi\)
0.624294 0.781189i \(-0.285387\pi\)
\(578\) − 39.8641i − 1.65813i
\(579\) 9.50650i 0.395077i
\(580\) − 49.1512i − 2.04089i
\(581\) 12.1292 0.503202
\(582\) −14.3503 −0.594838
\(583\) − 0.896480i − 0.0371284i
\(584\) 53.3864 2.20914
\(585\) 0 0
\(586\) −34.9259 −1.44277
\(587\) − 23.0649i − 0.951990i −0.879448 0.475995i \(-0.842088\pi\)
0.879448 0.475995i \(-0.157912\pi\)
\(588\) −4.00000 −0.164957
\(589\) 5.46787 0.225299
\(590\) 39.7795i 1.63770i
\(591\) − 3.63504i − 0.149525i
\(592\) 10.1923i 0.418903i
\(593\) − 11.0502i − 0.453777i −0.973921 0.226889i \(-0.927145\pi\)
0.973921 0.226889i \(-0.0728553\pi\)
\(594\) 14.4275 0.591969
\(595\) −7.83221 −0.321089
\(596\) 7.56404i 0.309835i
\(597\) −15.5872 −0.637940
\(598\) 0 0
\(599\) −44.6044 −1.82248 −0.911242 0.411870i \(-0.864876\pi\)
−0.911242 + 0.411870i \(0.864876\pi\)
\(600\) − 12.7862i − 0.521996i
\(601\) 47.8715 1.95272 0.976359 0.216156i \(-0.0693520\pi\)
0.976359 + 0.216156i \(0.0693520\pi\)
\(602\) −19.1035 −0.778601
\(603\) − 10.2843i − 0.418809i
\(604\) − 52.0294i − 2.11705i
\(605\) 13.0073i 0.528824i
\(606\) − 29.9374i − 1.21612i
\(607\) 45.0691 1.82930 0.914649 0.404249i \(-0.132467\pi\)
0.914649 + 0.404249i \(0.132467\pi\)
\(608\) −13.9572 −0.566037
\(609\) 12.0246i 0.487260i
\(610\) −6.29273 −0.254785
\(611\) 0 0
\(612\) −34.3074 −1.38680
\(613\) 25.5212i 1.03079i 0.856952 + 0.515396i \(0.172355\pi\)
−0.856952 + 0.515396i \(0.827645\pi\)
\(614\) 60.9834 2.46109
\(615\) 0.450654 0.0181721
\(616\) 4.00000i 0.161165i
\(617\) − 29.2432i − 1.17729i −0.808393 0.588643i \(-0.799663\pi\)
0.808393 0.588643i \(-0.200337\pi\)
\(618\) 9.17092i 0.368909i
\(619\) 4.78623i 0.192375i 0.995363 + 0.0961874i \(0.0306648\pi\)
−0.995363 + 0.0961874i \(0.969335\pi\)
\(620\) −7.66442 −0.307811
\(621\) 17.0558 0.684428
\(622\) 45.6791i 1.83157i
\(623\) 5.73604 0.229810
\(624\) 0 0
\(625\) 1.20077 0.0480307
\(626\) 8.15792i 0.326056i
\(627\) 4.39312 0.175444
\(628\) −79.7942 −3.18413
\(629\) − 49.6791i − 1.98084i
\(630\) 5.30429i 0.211328i
\(631\) − 28.3931i − 1.13031i −0.824984 0.565156i \(-0.808816\pi\)
0.824984 0.565156i \(-0.191184\pi\)
\(632\) 3.07896i 0.122475i
\(633\) −10.6331 −0.422629
\(634\) −11.7648 −0.467240
\(635\) − 16.1923i − 0.642574i
\(636\) −3.12808 −0.124036
\(637\) 0 0
\(638\) 28.1726 1.11536
\(639\) 2.59533i 0.102670i
\(640\) 27.0937 1.07097
\(641\) −5.96137 −0.235460 −0.117730 0.993046i \(-0.537562\pi\)
−0.117730 + 0.993046i \(0.537562\pi\)
\(642\) 13.3717i 0.527739i
\(643\) − 31.1940i − 1.23017i −0.788460 0.615086i \(-0.789121\pi\)
0.788460 0.615086i \(-0.210879\pi\)
\(644\) 11.0790i 0.436572i
\(645\) − 12.5525i − 0.494253i
\(646\) −45.6791 −1.79722
\(647\) −14.9112 −0.586219 −0.293109 0.956079i \(-0.594690\pi\)
−0.293109 + 0.956079i \(0.594690\pi\)
\(648\) − 3.83956i − 0.150832i
\(649\) −14.4935 −0.568920
\(650\) 0 0
\(651\) 1.87506 0.0734893
\(652\) − 24.7005i − 0.967348i
\(653\) −3.57246 −0.139801 −0.0699006 0.997554i \(-0.522268\pi\)
−0.0699006 + 0.997554i \(0.522268\pi\)
\(654\) −36.2499 −1.41748
\(655\) − 4.92104i − 0.192281i
\(656\) 0.350269i 0.0136757i
\(657\) 25.7936i 1.00630i
\(658\) 24.8683i 0.969468i
\(659\) −3.90383 −0.152071 −0.0760357 0.997105i \(-0.524226\pi\)
−0.0760357 + 0.997105i \(0.524226\pi\)
\(660\) −6.15792 −0.239697
\(661\) 13.7936i 0.536508i 0.963348 + 0.268254i \(0.0864466\pi\)
−0.963348 + 0.268254i \(0.913553\pi\)
\(662\) 14.4078 0.559975
\(663\) 0 0
\(664\) −42.3221 −1.64242
\(665\) 4.48929i 0.174087i
\(666\) −33.6447 −1.30371
\(667\) 33.3049 1.28957
\(668\) − 9.12181i − 0.352933i
\(669\) − 22.4605i − 0.868374i
\(670\) 19.1940i 0.741530i
\(671\) − 2.29273i − 0.0885099i
\(672\) −4.78623 −0.184633
\(673\) −5.70306 −0.219837 −0.109918 0.993941i \(-0.535059\pi\)
−0.109918 + 0.993941i \(0.535059\pi\)
\(674\) − 60.0393i − 2.31263i
\(675\) 17.1709 0.660909
\(676\) 0 0
\(677\) 35.2614 1.35521 0.677604 0.735427i \(-0.263019\pi\)
0.677604 + 0.735427i \(0.263019\pi\)
\(678\) 44.1726i 1.69644i
\(679\) 5.34292 0.205043
\(680\) 27.3288 1.04801
\(681\) − 22.5426i − 0.863835i
\(682\) − 4.39312i − 0.168221i
\(683\) 1.03612i 0.0396459i 0.999804 + 0.0198229i \(0.00631025\pi\)
−0.999804 + 0.0198229i \(0.993690\pi\)
\(684\) 19.6644i 0.751888i
\(685\) 17.5970 0.672348
\(686\) 2.34292 0.0894532
\(687\) − 8.90131i − 0.339606i
\(688\) 9.75639 0.371959
\(689\) 0 0
\(690\) −11.4523 −0.435983
\(691\) − 3.67850i − 0.139937i −0.997549 0.0699684i \(-0.977710\pi\)
0.997549 0.0699684i \(-0.0222898\pi\)
\(692\) 38.3931 1.45949
\(693\) −1.93260 −0.0734132
\(694\) 39.1281i 1.48528i
\(695\) 10.0674i 0.381878i
\(696\) − 41.9572i − 1.59038i
\(697\) − 1.70727i − 0.0646674i
\(698\) −55.1758 −2.08843
\(699\) 13.9817 0.528837
\(700\) 11.1537i 0.421571i
\(701\) 0.0617493 0.00233224 0.00116612 0.999999i \(-0.499629\pi\)
0.00116612 + 0.999999i \(0.499629\pi\)
\(702\) 0 0
\(703\) −28.4752 −1.07396
\(704\) 13.9572i 0.526030i
\(705\) −16.3404 −0.615415
\(706\) 17.9227 0.674531
\(707\) 11.1464i 0.419202i
\(708\) 50.5720i 1.90061i
\(709\) − 42.9834i − 1.61428i −0.590363 0.807138i \(-0.701015\pi\)
0.590363 0.807138i \(-0.298985\pi\)
\(710\) − 4.84377i − 0.181783i
\(711\) −1.48760 −0.0557892
\(712\) −20.0147 −0.750082
\(713\) − 5.19342i − 0.194495i
\(714\) −15.6644 −0.586226
\(715\) 0 0
\(716\) 83.6081 3.12458
\(717\) − 11.7992i − 0.440650i
\(718\) 43.0508 1.60664
\(719\) −17.6546 −0.658404 −0.329202 0.944260i \(-0.606780\pi\)
−0.329202 + 0.944260i \(0.606780\pi\)
\(720\) − 2.70896i − 0.100957i
\(721\) − 3.41454i − 0.127164i
\(722\) − 18.3331i − 0.682286i
\(723\) 4.61844i 0.171762i
\(724\) 22.8929 0.850807
\(725\) 33.5296 1.24526
\(726\) 26.0147i 0.965496i
\(727\) 23.8077 0.882977 0.441488 0.897267i \(-0.354451\pi\)
0.441488 + 0.897267i \(0.354451\pi\)
\(728\) 0 0
\(729\) 20.3288 0.752920
\(730\) − 48.1396i − 1.78173i
\(731\) −47.5542 −1.75885
\(732\) −8.00000 −0.295689
\(733\) − 31.3492i − 1.15791i −0.815360 0.578954i \(-0.803461\pi\)
0.815360 0.578954i \(-0.196539\pi\)
\(734\) − 12.5082i − 0.461686i
\(735\) 1.53948i 0.0567846i
\(736\) 13.2566i 0.488645i
\(737\) −6.99327 −0.257600
\(738\) −1.15623 −0.0425615
\(739\) − 24.8108i − 0.912680i −0.889806 0.456340i \(-0.849160\pi\)
0.889806 0.456340i \(-0.150840\pi\)
\(740\) 39.9143 1.46728
\(741\) 0 0
\(742\) 1.83221 0.0672626
\(743\) − 21.3717i − 0.784051i −0.919954 0.392026i \(-0.871774\pi\)
0.919954 0.392026i \(-0.128226\pi\)
\(744\) −6.54262 −0.239864
\(745\) 2.91117 0.106657
\(746\) − 50.4225i − 1.84610i
\(747\) − 20.4479i − 0.748149i
\(748\) 23.3288i 0.852987i
\(749\) − 4.97858i − 0.181913i
\(750\) −29.5640 −1.07953
\(751\) −19.2243 −0.701503 −0.350751 0.936469i \(-0.614074\pi\)
−0.350751 + 0.936469i \(0.614074\pi\)
\(752\) − 12.7005i − 0.463141i
\(753\) 3.33727 0.121617
\(754\) 0 0
\(755\) −20.0246 −0.728768
\(756\) 18.7434i 0.681690i
\(757\) −19.8610 −0.721860 −0.360930 0.932593i \(-0.617541\pi\)
−0.360930 + 0.932593i \(0.617541\pi\)
\(758\) −10.8009 −0.392307
\(759\) − 4.17262i − 0.151456i
\(760\) − 15.6644i − 0.568208i
\(761\) 6.12073i 0.221876i 0.993827 + 0.110938i \(0.0353856\pi\)
−0.993827 + 0.110938i \(0.964614\pi\)
\(762\) − 32.3847i − 1.17317i
\(763\) 13.4966 0.488611
\(764\) 15.3288 0.554578
\(765\) 13.2039i 0.477388i
\(766\) 19.5296 0.705634
\(767\) 0 0
\(768\) 26.2730 0.948045
\(769\) − 3.82800i − 0.138041i −0.997615 0.0690206i \(-0.978013\pi\)
0.997615 0.0690206i \(-0.0219874\pi\)
\(770\) 3.60688 0.129983
\(771\) −22.4653 −0.809070
\(772\) 28.9357i 1.04142i
\(773\) 6.53635i 0.235096i 0.993067 + 0.117548i \(0.0375034\pi\)
−0.993067 + 0.117548i \(0.962497\pi\)
\(774\) 32.2056i 1.15761i
\(775\) − 5.22846i − 0.187812i
\(776\) −18.6430 −0.669245
\(777\) −9.76481 −0.350311
\(778\) − 15.0937i − 0.541134i
\(779\) −0.978577 −0.0350612
\(780\) 0 0
\(781\) 1.76481 0.0631498
\(782\) 43.3864i 1.55149i
\(783\) 56.3452 2.01361
\(784\) −1.19656 −0.0427342
\(785\) 30.7104i 1.09610i
\(786\) − 9.84208i − 0.351055i
\(787\) 30.8066i 1.09814i 0.835778 + 0.549068i \(0.185017\pi\)
−0.835778 + 0.549068i \(0.814983\pi\)
\(788\) − 11.0643i − 0.394148i
\(789\) −8.67598 −0.308873
\(790\) 2.77636 0.0987786
\(791\) − 16.4464i − 0.584768i
\(792\) 6.74338 0.239616
\(793\) 0 0
\(794\) 3.28117 0.116444
\(795\) 1.20390i 0.0426981i
\(796\) −47.4439 −1.68161
\(797\) 38.8156 1.37492 0.687460 0.726222i \(-0.258726\pi\)
0.687460 + 0.726222i \(0.258726\pi\)
\(798\) 8.97858i 0.317838i
\(799\) 61.9044i 2.19002i
\(800\) 13.3461i 0.471854i
\(801\) − 9.67008i − 0.341675i
\(802\) −16.3503 −0.577348
\(803\) 17.5395 0.618955
\(804\) 24.4015i 0.860576i
\(805\) 4.26396 0.150285
\(806\) 0 0
\(807\) 10.8585 0.382236
\(808\) − 38.8929i − 1.36825i
\(809\) 1.04033 0.0365759 0.0182880 0.999833i \(-0.494178\pi\)
0.0182880 + 0.999833i \(0.494178\pi\)
\(810\) −3.46221 −0.121650
\(811\) − 12.6712i − 0.444944i −0.974939 0.222472i \(-0.928587\pi\)
0.974939 0.222472i \(-0.0714127\pi\)
\(812\) 36.6002i 1.28441i
\(813\) − 33.6707i − 1.18088i
\(814\) 22.8782i 0.801880i
\(815\) −9.50650 −0.332998
\(816\) 8.00000 0.280056
\(817\) 27.2572i 0.953610i
\(818\) 43.0607 1.50558
\(819\) 0 0
\(820\) 1.37169 0.0479016
\(821\) − 27.0361i − 0.943567i −0.881714 0.471783i \(-0.843610\pi\)
0.881714 0.471783i \(-0.156390\pi\)
\(822\) 35.1940 1.22753
\(823\) 31.6363 1.10277 0.551386 0.834251i \(-0.314099\pi\)
0.551386 + 0.834251i \(0.314099\pi\)
\(824\) 11.9143i 0.415055i
\(825\) − 4.20077i − 0.146252i
\(826\) − 29.6216i − 1.03067i
\(827\) 56.4800i 1.96400i 0.188872 + 0.982002i \(0.439517\pi\)
−0.188872 + 0.982002i \(0.560483\pi\)
\(828\) 18.6774 0.649085
\(829\) 42.6760 1.48220 0.741099 0.671396i \(-0.234305\pi\)
0.741099 + 0.671396i \(0.234305\pi\)
\(830\) 38.1627i 1.32465i
\(831\) −2.18248 −0.0757094
\(832\) 0 0
\(833\) 5.83221 0.202074
\(834\) 20.1348i 0.697211i
\(835\) −3.51071 −0.121493
\(836\) 13.3717 0.462470
\(837\) − 8.78623i − 0.303697i
\(838\) 70.3797i 2.43122i
\(839\) − 40.1642i − 1.38662i −0.720639 0.693311i \(-0.756151\pi\)
0.720639 0.693311i \(-0.243849\pi\)
\(840\) − 5.37169i − 0.185341i
\(841\) 81.0252 2.79397
\(842\) 19.4868 0.671558
\(843\) 23.5872i 0.812385i
\(844\) −32.3650 −1.11405
\(845\) 0 0
\(846\) 41.9242 1.44138
\(847\) − 9.68585i − 0.332810i
\(848\) −0.935731 −0.0321331
\(849\) −30.9442 −1.06200
\(850\) 43.6791i 1.49818i
\(851\) 27.0460i 0.927124i
\(852\) − 6.15792i − 0.210967i
\(853\) 19.6932i 0.674282i 0.941454 + 0.337141i \(0.109460\pi\)
−0.941454 + 0.337141i \(0.890540\pi\)
\(854\) 4.68585 0.160346
\(855\) 7.56825 0.258829
\(856\) 17.3717i 0.593752i
\(857\) −1.66442 −0.0568556 −0.0284278 0.999596i \(-0.509050\pi\)
−0.0284278 + 0.999596i \(0.509050\pi\)
\(858\) 0 0
\(859\) −41.2944 −1.40895 −0.704474 0.709730i \(-0.748817\pi\)
−0.704474 + 0.709730i \(0.748817\pi\)
\(860\) − 38.2070i − 1.30285i
\(861\) −0.335577 −0.0114364
\(862\) −22.6086 −0.770051
\(863\) 31.3288i 1.06645i 0.845975 + 0.533223i \(0.179019\pi\)
−0.845975 + 0.533223i \(0.820981\pi\)
\(864\) 22.4275i 0.763000i
\(865\) − 14.7764i − 0.502411i
\(866\) 61.6363i 2.09449i
\(867\) −19.5051 −0.662426
\(868\) 5.70727 0.193717
\(869\) 1.01156i 0.0343147i
\(870\) −37.8337 −1.28268
\(871\) 0 0
\(872\) −47.0937 −1.59479
\(873\) − 9.00735i − 0.304852i
\(874\) 24.8683 0.841184
\(875\) 11.0073 0.372116
\(876\) − 61.2003i − 2.06777i
\(877\) 53.8041i 1.81683i 0.418065 + 0.908417i \(0.362708\pi\)
−0.418065 + 0.908417i \(0.637292\pi\)
\(878\) 79.2234i 2.67366i
\(879\) 17.0888i 0.576392i
\(880\) −1.84208 −0.0620964
\(881\) −8.09196 −0.272625 −0.136313 0.990666i \(-0.543525\pi\)
−0.136313 + 0.990666i \(0.543525\pi\)
\(882\) − 3.94981i − 0.132997i
\(883\) 27.0705 0.910996 0.455498 0.890237i \(-0.349461\pi\)
0.455498 + 0.890237i \(0.349461\pi\)
\(884\) 0 0
\(885\) 19.4637 0.654264
\(886\) − 61.9620i − 2.08165i
\(887\) −38.4935 −1.29249 −0.646243 0.763132i \(-0.723661\pi\)
−0.646243 + 0.763132i \(0.723661\pi\)
\(888\) 34.0722 1.14339
\(889\) 12.0575i 0.404397i
\(890\) 18.0477i 0.604959i
\(891\) − 1.26144i − 0.0422599i
\(892\) − 68.3650i − 2.28903i
\(893\) 35.4826 1.18738
\(894\) 5.82235 0.194728
\(895\) − 32.1783i − 1.07560i
\(896\) −20.1751 −0.674004
\(897\) 0 0
\(898\) 6.19235 0.206641
\(899\) − 17.1568i − 0.572213i
\(900\) 18.8034 0.626781
\(901\) 4.56090 0.151946
\(902\) 0.786230i 0.0261786i
\(903\) 9.34713i 0.311053i
\(904\) 57.3864i 1.90864i
\(905\) − 8.81079i − 0.292881i
\(906\) −40.0491 −1.33054
\(907\) −15.7031 −0.521411 −0.260706 0.965418i \(-0.583955\pi\)
−0.260706 + 0.965418i \(0.583955\pi\)
\(908\) − 68.6148i − 2.27706i
\(909\) 18.7911 0.623260
\(910\) 0 0
\(911\) 44.9399 1.48893 0.744463 0.667663i \(-0.232705\pi\)
0.744463 + 0.667663i \(0.232705\pi\)
\(912\) − 4.58546i − 0.151840i
\(913\) −13.9044 −0.460170
\(914\) 78.9307 2.61080
\(915\) 3.07896i 0.101787i
\(916\) − 27.0937i − 0.895200i
\(917\) 3.66442i 0.121010i
\(918\) 73.4011i 2.42260i
\(919\) −27.2432 −0.898669 −0.449334 0.893364i \(-0.648339\pi\)
−0.449334 + 0.893364i \(0.648339\pi\)
\(920\) −14.8782 −0.490519
\(921\) − 29.8385i − 0.983211i
\(922\) −77.5015 −2.55237
\(923\) 0 0
\(924\) 4.58546 0.150851
\(925\) 27.2285i 0.895266i
\(926\) −5.89962 −0.193873
\(927\) −5.75639 −0.189065
\(928\) 43.7942i 1.43761i
\(929\) 17.1422i 0.562416i 0.959647 + 0.281208i \(0.0907351\pi\)
−0.959647 + 0.281208i \(0.909265\pi\)
\(930\) 5.89962i 0.193456i
\(931\) − 3.34292i − 0.109560i
\(932\) 42.5573 1.39401
\(933\) 22.3503 0.731715
\(934\) 6.03442i 0.197452i
\(935\) 8.97858 0.293631
\(936\) 0 0
\(937\) −51.5197 −1.68308 −0.841538 0.540197i \(-0.818350\pi\)
−0.841538 + 0.540197i \(0.818350\pi\)
\(938\) − 14.2927i − 0.466674i
\(939\) 3.99158 0.130260
\(940\) −49.7367 −1.62223
\(941\) − 32.7575i − 1.06786i −0.845528 0.533931i \(-0.820714\pi\)
0.845528 0.533931i \(-0.179286\pi\)
\(942\) 61.4208i 2.00120i
\(943\) 0.929460i 0.0302674i
\(944\) 15.1281i 0.492377i
\(945\) 7.21377 0.234664
\(946\) 21.8996 0.712018
\(947\) 20.9295i 0.680116i 0.940404 + 0.340058i \(0.110447\pi\)
−0.940404 + 0.340058i \(0.889553\pi\)
\(948\) 3.52962 0.114637
\(949\) 0 0
\(950\) 25.0361 0.812279
\(951\) 5.75639i 0.186664i
\(952\) −20.3503 −0.659556
\(953\) −46.4120 −1.50343 −0.751716 0.659487i \(-0.770774\pi\)
−0.751716 + 0.659487i \(0.770774\pi\)
\(954\) − 3.08883i − 0.100004i
\(955\) − 5.89962i − 0.190907i
\(956\) − 35.9143i − 1.16155i
\(957\) − 13.7845i − 0.445591i
\(958\) 1.20390 0.0388964
\(959\) −13.1035 −0.423135
\(960\) − 18.7434i − 0.604940i
\(961\) 28.3246 0.913698
\(962\) 0 0
\(963\) −8.39312 −0.270464
\(964\) 14.0575i 0.452763i
\(965\) 11.1365 0.358497
\(966\) 8.52792 0.274381
\(967\) 23.2186i 0.746660i 0.927699 + 0.373330i \(0.121784\pi\)
−0.927699 + 0.373330i \(0.878216\pi\)
\(968\) 33.7967i 1.08627i
\(969\) 22.3503i 0.717994i
\(970\) 16.8108i 0.539762i
\(971\) −14.1004 −0.452503 −0.226251 0.974069i \(-0.572647\pi\)
−0.226251 + 0.974069i \(0.572647\pi\)
\(972\) 51.8286 1.66240
\(973\) − 7.49663i − 0.240331i
\(974\) −84.4800 −2.70692
\(975\) 0 0
\(976\) −2.39312 −0.0766018
\(977\) 2.95402i 0.0945074i 0.998883 + 0.0472537i \(0.0150469\pi\)
−0.998883 + 0.0472537i \(0.984953\pi\)
\(978\) −19.0130 −0.607969
\(979\) −6.57560 −0.210157
\(980\) 4.68585i 0.149684i
\(981\) − 22.7533i − 0.726455i
\(982\) − 21.6216i − 0.689972i
\(983\) − 35.0367i − 1.11750i −0.829337 0.558749i \(-0.811281\pi\)
0.829337 0.558749i \(-0.188719\pi\)
\(984\) 1.17092 0.0373277
\(985\) −4.25831 −0.135681
\(986\) 143.330i 4.56456i
\(987\) 12.1678 0.387305
\(988\) 0 0
\(989\) 25.8891 0.823227
\(990\) − 6.08065i − 0.193256i
\(991\) 47.9718 1.52388 0.761938 0.647650i \(-0.224248\pi\)
0.761938 + 0.647650i \(0.224248\pi\)
\(992\) 6.82908 0.216823
\(993\) − 7.04958i − 0.223712i
\(994\) 3.60688i 0.114403i
\(995\) 18.2598i 0.578873i
\(996\) 48.5166i 1.53731i
\(997\) −38.4422 −1.21748 −0.608739 0.793371i \(-0.708324\pi\)
−0.608739 + 0.793371i \(0.708324\pi\)
\(998\) −2.35027 −0.0743965
\(999\) 45.7564i 1.44767i
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 1183.2.c.f.337.1 6
13.5 odd 4 1183.2.a.i.1.1 3
13.8 odd 4 91.2.a.d.1.3 3
13.12 even 2 inner 1183.2.c.f.337.6 6
39.8 even 4 819.2.a.i.1.1 3
52.47 even 4 1456.2.a.t.1.2 3
65.34 odd 4 2275.2.a.m.1.1 3
91.34 even 4 637.2.a.j.1.3 3
91.47 even 12 637.2.e.i.508.1 6
91.60 odd 12 637.2.e.j.79.1 6
91.73 even 12 637.2.e.i.79.1 6
91.83 even 4 8281.2.a.bg.1.1 3
91.86 odd 12 637.2.e.j.508.1 6
104.21 odd 4 5824.2.a.by.1.2 3
104.99 even 4 5824.2.a.bs.1.2 3
273.125 odd 4 5733.2.a.x.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.3 3 13.8 odd 4
637.2.a.j.1.3 3 91.34 even 4
637.2.e.i.79.1 6 91.73 even 12
637.2.e.i.508.1 6 91.47 even 12
637.2.e.j.79.1 6 91.60 odd 12
637.2.e.j.508.1 6 91.86 odd 12
819.2.a.i.1.1 3 39.8 even 4
1183.2.a.i.1.1 3 13.5 odd 4
1183.2.c.f.337.1 6 1.1 even 1 trivial
1183.2.c.f.337.6 6 13.12 even 2 inner
1456.2.a.t.1.2 3 52.47 even 4
2275.2.a.m.1.1 3 65.34 odd 4
5733.2.a.x.1.1 3 273.125 odd 4
5824.2.a.bs.1.2 3 104.99 even 4
5824.2.a.by.1.2 3 104.21 odd 4
8281.2.a.bg.1.1 3 91.83 even 4