Properties

Label 1183.2.c.h.337.11
Level $1183$
Weight $2$
Character 1183.337
Analytic conductor $9.446$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 16x^{10} + 96x^{8} + 266x^{6} + 332x^{4} + 141x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.11
Root \(-0.908891i\) of defining polynomial
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.h.337.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.08281i q^{2} -0.0911085 q^{3} -2.33809 q^{4} -2.63777i q^{5} -0.189762i q^{6} -1.00000i q^{7} -0.704173i q^{8} -2.99170 q^{9} +O(q^{10})\) \(q+2.08281i q^{2} -0.0911085 q^{3} -2.33809 q^{4} -2.63777i q^{5} -0.189762i q^{6} -1.00000i q^{7} -0.704173i q^{8} -2.99170 q^{9} +5.49396 q^{10} +3.44676i q^{11} +0.213020 q^{12} +2.08281 q^{14} +0.240323i q^{15} -3.20952 q^{16} +4.83445 q^{17} -6.23113i q^{18} +2.42330i q^{19} +6.16733i q^{20} +0.0911085i q^{21} -7.17893 q^{22} +6.84789 q^{23} +0.0641561i q^{24} -1.95781 q^{25} +0.545895 q^{27} +2.33809i q^{28} +4.64785 q^{29} -0.500546 q^{30} +8.43394i q^{31} -8.09316i q^{32} -0.314029i q^{33} +10.0692i q^{34} -2.63777 q^{35} +6.99486 q^{36} +1.51646i q^{37} -5.04727 q^{38} -1.85744 q^{40} +10.4384i q^{41} -0.189762 q^{42} +0.451785 q^{43} -8.05882i q^{44} +7.89140i q^{45} +14.2628i q^{46} +2.56938i q^{47} +0.292415 q^{48} -1.00000 q^{49} -4.07774i q^{50} -0.440460 q^{51} -10.3402 q^{53} +1.13699i q^{54} +9.09173 q^{55} -0.704173 q^{56} -0.220783i q^{57} +9.68058i q^{58} +12.7994i q^{59} -0.561896i q^{60} -0.984619 q^{61} -17.5663 q^{62} +2.99170i q^{63} +10.4375 q^{64} +0.654061 q^{66} +1.28567i q^{67} -11.3034 q^{68} -0.623901 q^{69} -5.49396i q^{70} -12.0794i q^{71} +2.10667i q^{72} +5.18381i q^{73} -3.15850 q^{74} +0.178373 q^{75} -5.66589i q^{76} +3.44676 q^{77} +9.74733 q^{79} +8.46596i q^{80} +8.92536 q^{81} -21.7412 q^{82} +13.2658i q^{83} -0.213020i q^{84} -12.7522i q^{85} +0.940982i q^{86} -0.423459 q^{87} +2.42711 q^{88} -16.0403i q^{89} -16.4363 q^{90} -16.0110 q^{92} -0.768404i q^{93} -5.35152 q^{94} +6.39210 q^{95} +0.737356i q^{96} -14.9407i q^{97} -2.08281i q^{98} -10.3117i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{3} - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{3} - 16 q^{4} + 28 q^{10} + 46 q^{12} - 4 q^{14} + 46 q^{17} - 8 q^{22} + 36 q^{23} + 20 q^{25} - 20 q^{27} - 30 q^{29} - 28 q^{30} - 4 q^{35} - 44 q^{36} + 22 q^{38} - 28 q^{40} + 16 q^{42} + 36 q^{43} - 22 q^{48} - 12 q^{49} - 28 q^{51} - 50 q^{53} + 6 q^{56} + 32 q^{61} + 18 q^{62} + 14 q^{64} + 32 q^{66} - 68 q^{68} + 2 q^{69} - 28 q^{74} - 30 q^{75} + 16 q^{77} + 4 q^{79} - 12 q^{81} + 20 q^{82} + 26 q^{87} + 96 q^{88} - 64 q^{92} - 28 q^{94} + 14 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.08281i 1.47277i 0.676564 + 0.736384i \(0.263468\pi\)
−0.676564 + 0.736384i \(0.736532\pi\)
\(3\) −0.0911085 −0.0526015 −0.0263008 0.999654i \(-0.508373\pi\)
−0.0263008 + 0.999654i \(0.508373\pi\)
\(4\) −2.33809 −1.16904
\(5\) − 2.63777i − 1.17964i −0.807533 0.589822i \(-0.799198\pi\)
0.807533 0.589822i \(-0.200802\pi\)
\(6\) − 0.189762i − 0.0774698i
\(7\) − 1.00000i − 0.377964i
\(8\) − 0.704173i − 0.248963i
\(9\) −2.99170 −0.997233
\(10\) 5.49396 1.73734
\(11\) 3.44676i 1.03924i 0.854399 + 0.519618i \(0.173926\pi\)
−0.854399 + 0.519618i \(0.826074\pi\)
\(12\) 0.213020 0.0614935
\(13\) 0 0
\(14\) 2.08281 0.556654
\(15\) 0.240323i 0.0620511i
\(16\) −3.20952 −0.802380
\(17\) 4.83445 1.17253 0.586264 0.810120i \(-0.300598\pi\)
0.586264 + 0.810120i \(0.300598\pi\)
\(18\) − 6.23113i − 1.46869i
\(19\) 2.42330i 0.555944i 0.960589 + 0.277972i \(0.0896622\pi\)
−0.960589 + 0.277972i \(0.910338\pi\)
\(20\) 6.16733i 1.37906i
\(21\) 0.0911085i 0.0198815i
\(22\) −7.17893 −1.53055
\(23\) 6.84789 1.42788 0.713942 0.700205i \(-0.246908\pi\)
0.713942 + 0.700205i \(0.246908\pi\)
\(24\) 0.0641561i 0.0130958i
\(25\) −1.95781 −0.391562
\(26\) 0 0
\(27\) 0.545895 0.105057
\(28\) 2.33809i 0.441857i
\(29\) 4.64785 0.863084 0.431542 0.902093i \(-0.357970\pi\)
0.431542 + 0.902093i \(0.357970\pi\)
\(30\) −0.500546 −0.0913869
\(31\) 8.43394i 1.51478i 0.652962 + 0.757390i \(0.273526\pi\)
−0.652962 + 0.757390i \(0.726474\pi\)
\(32\) − 8.09316i − 1.43068i
\(33\) − 0.314029i − 0.0546654i
\(34\) 10.0692i 1.72686i
\(35\) −2.63777 −0.445864
\(36\) 6.99486 1.16581
\(37\) 1.51646i 0.249305i 0.992200 + 0.124652i \(0.0397816\pi\)
−0.992200 + 0.124652i \(0.960218\pi\)
\(38\) −5.04727 −0.818776
\(39\) 0 0
\(40\) −1.85744 −0.293687
\(41\) 10.4384i 1.63020i 0.579317 + 0.815102i \(0.303319\pi\)
−0.579317 + 0.815102i \(0.696681\pi\)
\(42\) −0.189762 −0.0292808
\(43\) 0.451785 0.0688966 0.0344483 0.999406i \(-0.489033\pi\)
0.0344483 + 0.999406i \(0.489033\pi\)
\(44\) − 8.05882i − 1.21491i
\(45\) 7.89140i 1.17638i
\(46\) 14.2628i 2.10294i
\(47\) 2.56938i 0.374782i 0.982285 + 0.187391i \(0.0600032\pi\)
−0.982285 + 0.187391i \(0.939997\pi\)
\(48\) 0.292415 0.0422064
\(49\) −1.00000 −0.142857
\(50\) − 4.07774i − 0.576679i
\(51\) −0.440460 −0.0616767
\(52\) 0 0
\(53\) −10.3402 −1.42033 −0.710166 0.704035i \(-0.751380\pi\)
−0.710166 + 0.704035i \(0.751380\pi\)
\(54\) 1.13699i 0.154725i
\(55\) 9.09173 1.22593
\(56\) −0.704173 −0.0940990
\(57\) − 0.220783i − 0.0292435i
\(58\) 9.68058i 1.27112i
\(59\) 12.7994i 1.66634i 0.553016 + 0.833171i \(0.313477\pi\)
−0.553016 + 0.833171i \(0.686523\pi\)
\(60\) − 0.561896i − 0.0725405i
\(61\) −0.984619 −0.126068 −0.0630338 0.998011i \(-0.520078\pi\)
−0.0630338 + 0.998011i \(0.520078\pi\)
\(62\) −17.5663 −2.23092
\(63\) 2.99170i 0.376919i
\(64\) 10.4375 1.30468
\(65\) 0 0
\(66\) 0.654061 0.0805094
\(67\) 1.28567i 0.157069i 0.996911 + 0.0785346i \(0.0250241\pi\)
−0.996911 + 0.0785346i \(0.974976\pi\)
\(68\) −11.3034 −1.37074
\(69\) −0.623901 −0.0751089
\(70\) − 5.49396i − 0.656654i
\(71\) − 12.0794i − 1.43356i −0.697301 0.716779i \(-0.745616\pi\)
0.697301 0.716779i \(-0.254384\pi\)
\(72\) 2.10667i 0.248274i
\(73\) 5.18381i 0.606719i 0.952876 + 0.303360i \(0.0981083\pi\)
−0.952876 + 0.303360i \(0.901892\pi\)
\(74\) −3.15850 −0.367168
\(75\) 0.178373 0.0205967
\(76\) − 5.66589i − 0.649923i
\(77\) 3.44676 0.392794
\(78\) 0 0
\(79\) 9.74733 1.09666 0.548330 0.836262i \(-0.315264\pi\)
0.548330 + 0.836262i \(0.315264\pi\)
\(80\) 8.46596i 0.946523i
\(81\) 8.92536 0.991707
\(82\) −21.7412 −2.40091
\(83\) 13.2658i 1.45611i 0.685520 + 0.728054i \(0.259575\pi\)
−0.685520 + 0.728054i \(0.740425\pi\)
\(84\) − 0.213020i − 0.0232424i
\(85\) − 12.7522i − 1.38317i
\(86\) 0.940982i 0.101469i
\(87\) −0.423459 −0.0453995
\(88\) 2.42711 0.258731
\(89\) − 16.0403i − 1.70027i −0.526567 0.850134i \(-0.676521\pi\)
0.526567 0.850134i \(-0.323479\pi\)
\(90\) −16.4363 −1.73254
\(91\) 0 0
\(92\) −16.0110 −1.66926
\(93\) − 0.768404i − 0.0796798i
\(94\) −5.35152 −0.551967
\(95\) 6.39210 0.655816
\(96\) 0.737356i 0.0752560i
\(97\) − 14.9407i − 1.51700i −0.651672 0.758501i \(-0.725932\pi\)
0.651672 0.758501i \(-0.274068\pi\)
\(98\) − 2.08281i − 0.210395i
\(99\) − 10.3117i − 1.03636i
\(100\) 4.57753 0.457753
\(101\) 5.58103 0.555334 0.277667 0.960677i \(-0.410439\pi\)
0.277667 + 0.960677i \(0.410439\pi\)
\(102\) − 0.917393i − 0.0908355i
\(103\) 15.0233 1.48029 0.740144 0.672448i \(-0.234757\pi\)
0.740144 + 0.672448i \(0.234757\pi\)
\(104\) 0 0
\(105\) 0.240323 0.0234531
\(106\) − 21.5366i − 2.09182i
\(107\) −12.0550 −1.16540 −0.582701 0.812687i \(-0.698004\pi\)
−0.582701 + 0.812687i \(0.698004\pi\)
\(108\) −1.27635 −0.122817
\(109\) 14.7918i 1.41679i 0.705815 + 0.708397i \(0.250581\pi\)
−0.705815 + 0.708397i \(0.749419\pi\)
\(110\) 18.9363i 1.80551i
\(111\) − 0.138163i − 0.0131138i
\(112\) 3.20952i 0.303271i
\(113\) −10.4878 −0.986608 −0.493304 0.869857i \(-0.664211\pi\)
−0.493304 + 0.869857i \(0.664211\pi\)
\(114\) 0.459850 0.0430689
\(115\) − 18.0631i − 1.68440i
\(116\) −10.8671 −1.00898
\(117\) 0 0
\(118\) −26.6587 −2.45413
\(119\) − 4.83445i − 0.443174i
\(120\) 0.169229 0.0154484
\(121\) −0.880123 −0.0800112
\(122\) − 2.05077i − 0.185668i
\(123\) − 0.951027i − 0.0857512i
\(124\) − 19.7193i − 1.77085i
\(125\) − 8.02459i − 0.717741i
\(126\) −6.23113 −0.555114
\(127\) 5.86506 0.520440 0.260220 0.965549i \(-0.416205\pi\)
0.260220 + 0.965549i \(0.416205\pi\)
\(128\) 5.55289i 0.490811i
\(129\) −0.0411615 −0.00362407
\(130\) 0 0
\(131\) 1.38898 0.121356 0.0606778 0.998157i \(-0.480674\pi\)
0.0606778 + 0.998157i \(0.480674\pi\)
\(132\) 0.734227i 0.0639062i
\(133\) 2.42330 0.210127
\(134\) −2.67780 −0.231326
\(135\) − 1.43994i − 0.123931i
\(136\) − 3.40429i − 0.291915i
\(137\) − 12.4476i − 1.06347i −0.846910 0.531736i \(-0.821540\pi\)
0.846910 0.531736i \(-0.178460\pi\)
\(138\) − 1.29947i − 0.110618i
\(139\) 7.98054 0.676900 0.338450 0.940984i \(-0.390097\pi\)
0.338450 + 0.940984i \(0.390097\pi\)
\(140\) 6.16733 0.521234
\(141\) − 0.234092i − 0.0197141i
\(142\) 25.1590 2.11130
\(143\) 0 0
\(144\) 9.60192 0.800160
\(145\) − 12.2599i − 1.01813i
\(146\) −10.7969 −0.893556
\(147\) 0.0911085 0.00751450
\(148\) − 3.54562i − 0.291448i
\(149\) 9.36477i 0.767192i 0.923501 + 0.383596i \(0.125314\pi\)
−0.923501 + 0.383596i \(0.874686\pi\)
\(150\) 0.371517i 0.0303342i
\(151\) 12.5890i 1.02448i 0.858843 + 0.512239i \(0.171184\pi\)
−0.858843 + 0.512239i \(0.828816\pi\)
\(152\) 1.70642 0.138409
\(153\) −14.4632 −1.16928
\(154\) 7.17893i 0.578495i
\(155\) 22.2468 1.78690
\(156\) 0 0
\(157\) 6.31025 0.503613 0.251807 0.967778i \(-0.418975\pi\)
0.251807 + 0.967778i \(0.418975\pi\)
\(158\) 20.3018i 1.61513i
\(159\) 0.942077 0.0747116
\(160\) −21.3479 −1.68770
\(161\) − 6.84789i − 0.539690i
\(162\) 18.5898i 1.46055i
\(163\) − 11.5904i − 0.907829i −0.891045 0.453915i \(-0.850027\pi\)
0.891045 0.453915i \(-0.149973\pi\)
\(164\) − 24.4059i − 1.90578i
\(165\) −0.828334 −0.0644857
\(166\) −27.6301 −2.14451
\(167\) − 4.38945i − 0.339666i −0.985473 0.169833i \(-0.945677\pi\)
0.985473 0.169833i \(-0.0543228\pi\)
\(168\) 0.0641561 0.00494975
\(169\) 0 0
\(170\) 26.5603 2.03708
\(171\) − 7.24979i − 0.554405i
\(172\) −1.05631 −0.0805432
\(173\) 19.7818 1.50398 0.751990 0.659175i \(-0.229094\pi\)
0.751990 + 0.659175i \(0.229094\pi\)
\(174\) − 0.881983i − 0.0668630i
\(175\) 1.95781i 0.147996i
\(176\) − 11.0624i − 0.833862i
\(177\) − 1.16614i − 0.0876521i
\(178\) 33.4088 2.50410
\(179\) −6.46685 −0.483355 −0.241678 0.970357i \(-0.577698\pi\)
−0.241678 + 0.970357i \(0.577698\pi\)
\(180\) − 18.4508i − 1.37524i
\(181\) −20.9234 −1.55522 −0.777612 0.628745i \(-0.783569\pi\)
−0.777612 + 0.628745i \(0.783569\pi\)
\(182\) 0 0
\(183\) 0.0897072 0.00663135
\(184\) − 4.82210i − 0.355490i
\(185\) 4.00007 0.294091
\(186\) 1.60044 0.117350
\(187\) 16.6632i 1.21853i
\(188\) − 6.00743i − 0.438137i
\(189\) − 0.545895i − 0.0397080i
\(190\) 13.3135i 0.965865i
\(191\) −19.6754 −1.42366 −0.711831 0.702351i \(-0.752134\pi\)
−0.711831 + 0.702351i \(0.752134\pi\)
\(192\) −0.950941 −0.0686282
\(193\) 14.8701i 1.07038i 0.844733 + 0.535188i \(0.179759\pi\)
−0.844733 + 0.535188i \(0.820241\pi\)
\(194\) 31.1187 2.23419
\(195\) 0 0
\(196\) 2.33809 0.167006
\(197\) − 14.9442i − 1.06473i −0.846515 0.532365i \(-0.821303\pi\)
0.846515 0.532365i \(-0.178697\pi\)
\(198\) 21.4772 1.52632
\(199\) −9.08903 −0.644305 −0.322152 0.946688i \(-0.604406\pi\)
−0.322152 + 0.946688i \(0.604406\pi\)
\(200\) 1.37864i 0.0974842i
\(201\) − 0.117135i − 0.00826207i
\(202\) 11.6242i 0.817877i
\(203\) − 4.64785i − 0.326215i
\(204\) 1.02983 0.0721028
\(205\) 27.5341 1.92306
\(206\) 31.2906i 2.18012i
\(207\) −20.4868 −1.42393
\(208\) 0 0
\(209\) −8.35253 −0.577757
\(210\) 0.500546i 0.0345410i
\(211\) −3.39540 −0.233749 −0.116875 0.993147i \(-0.537288\pi\)
−0.116875 + 0.993147i \(0.537288\pi\)
\(212\) 24.1762 1.66043
\(213\) 1.10053i 0.0754073i
\(214\) − 25.1083i − 1.71637i
\(215\) − 1.19170i − 0.0812735i
\(216\) − 0.384404i − 0.0261554i
\(217\) 8.43394 0.572533
\(218\) −30.8084 −2.08661
\(219\) − 0.472289i − 0.0319144i
\(220\) −21.2573 −1.43317
\(221\) 0 0
\(222\) 0.287766 0.0193136
\(223\) 5.64878i 0.378270i 0.981951 + 0.189135i \(0.0605684\pi\)
−0.981951 + 0.189135i \(0.939432\pi\)
\(224\) −8.09316 −0.540747
\(225\) 5.85718 0.390478
\(226\) − 21.8440i − 1.45304i
\(227\) − 13.2468i − 0.879222i −0.898188 0.439611i \(-0.855116\pi\)
0.898188 0.439611i \(-0.144884\pi\)
\(228\) 0.516211i 0.0341869i
\(229\) − 1.66636i − 0.110116i −0.998483 0.0550580i \(-0.982466\pi\)
0.998483 0.0550580i \(-0.0175344\pi\)
\(230\) 37.6220 2.48072
\(231\) −0.314029 −0.0206616
\(232\) − 3.27289i − 0.214876i
\(233\) −12.0833 −0.791600 −0.395800 0.918337i \(-0.629533\pi\)
−0.395800 + 0.918337i \(0.629533\pi\)
\(234\) 0 0
\(235\) 6.77742 0.442110
\(236\) − 29.9262i − 1.94803i
\(237\) −0.888065 −0.0576860
\(238\) 10.0692 0.652692
\(239\) − 14.8457i − 0.960287i −0.877190 0.480143i \(-0.840585\pi\)
0.877190 0.480143i \(-0.159415\pi\)
\(240\) − 0.771321i − 0.0497886i
\(241\) 20.4472i 1.31712i 0.752528 + 0.658560i \(0.228834\pi\)
−0.752528 + 0.658560i \(0.771166\pi\)
\(242\) − 1.83313i − 0.117838i
\(243\) −2.45086 −0.157223
\(244\) 2.30213 0.147379
\(245\) 2.63777i 0.168521i
\(246\) 1.98081 0.126292
\(247\) 0 0
\(248\) 5.93895 0.377124
\(249\) − 1.20862i − 0.0765935i
\(250\) 16.7137 1.05707
\(251\) 6.03670 0.381033 0.190517 0.981684i \(-0.438984\pi\)
0.190517 + 0.981684i \(0.438984\pi\)
\(252\) − 6.99486i − 0.440635i
\(253\) 23.6030i 1.48391i
\(254\) 12.2158i 0.766487i
\(255\) 1.16183i 0.0727566i
\(256\) 9.30930 0.581831
\(257\) 1.99555 0.124479 0.0622396 0.998061i \(-0.480176\pi\)
0.0622396 + 0.998061i \(0.480176\pi\)
\(258\) − 0.0857314i − 0.00533741i
\(259\) 1.51646 0.0942283
\(260\) 0 0
\(261\) −13.9050 −0.860696
\(262\) 2.89298i 0.178729i
\(263\) 12.8872 0.794661 0.397331 0.917676i \(-0.369937\pi\)
0.397331 + 0.917676i \(0.369937\pi\)
\(264\) −0.221130 −0.0136096
\(265\) 27.2749i 1.67549i
\(266\) 5.04727i 0.309468i
\(267\) 1.46141i 0.0894367i
\(268\) − 3.00600i − 0.183621i
\(269\) −24.4694 −1.49193 −0.745963 0.665988i \(-0.768010\pi\)
−0.745963 + 0.665988i \(0.768010\pi\)
\(270\) 2.99912 0.182521
\(271\) 14.2847i 0.867734i 0.900977 + 0.433867i \(0.142851\pi\)
−0.900977 + 0.433867i \(0.857149\pi\)
\(272\) −15.5163 −0.940813
\(273\) 0 0
\(274\) 25.9260 1.56625
\(275\) − 6.74809i − 0.406925i
\(276\) 1.45874 0.0878056
\(277\) 26.5784 1.59694 0.798469 0.602035i \(-0.205643\pi\)
0.798469 + 0.602035i \(0.205643\pi\)
\(278\) 16.6219i 0.996917i
\(279\) − 25.2318i − 1.51059i
\(280\) 1.85744i 0.111003i
\(281\) − 9.68696i − 0.577876i −0.957348 0.288938i \(-0.906698\pi\)
0.957348 0.288938i \(-0.0933021\pi\)
\(282\) 0.487569 0.0290343
\(283\) 9.01853 0.536096 0.268048 0.963406i \(-0.413621\pi\)
0.268048 + 0.963406i \(0.413621\pi\)
\(284\) 28.2426i 1.67589i
\(285\) −0.582375 −0.0344969
\(286\) 0 0
\(287\) 10.4384 0.616159
\(288\) 24.2123i 1.42672i
\(289\) 6.37195 0.374820
\(290\) 25.5351 1.49947
\(291\) 1.36123i 0.0797966i
\(292\) − 12.1202i − 0.709282i
\(293\) 3.36723i 0.196716i 0.995151 + 0.0983579i \(0.0313590\pi\)
−0.995151 + 0.0983579i \(0.968641\pi\)
\(294\) 0.189762i 0.0110671i
\(295\) 33.7619 1.96569
\(296\) 1.06785 0.0620675
\(297\) 1.88157i 0.109180i
\(298\) −19.5050 −1.12989
\(299\) 0 0
\(300\) −0.417052 −0.0240785
\(301\) − 0.451785i − 0.0260405i
\(302\) −26.2204 −1.50882
\(303\) −0.508480 −0.0292114
\(304\) − 7.77764i − 0.446078i
\(305\) 2.59720i 0.148715i
\(306\) − 30.1241i − 1.72208i
\(307\) 5.27364i 0.300983i 0.988611 + 0.150491i \(0.0480856\pi\)
−0.988611 + 0.150491i \(0.951914\pi\)
\(308\) −8.05882 −0.459194
\(309\) −1.36875 −0.0778654
\(310\) 46.3357i 2.63169i
\(311\) −9.31654 −0.528292 −0.264146 0.964483i \(-0.585090\pi\)
−0.264146 + 0.964483i \(0.585090\pi\)
\(312\) 0 0
\(313\) −31.0327 −1.75407 −0.877037 0.480423i \(-0.840483\pi\)
−0.877037 + 0.480423i \(0.840483\pi\)
\(314\) 13.1430i 0.741705i
\(315\) 7.89140 0.444630
\(316\) −22.7901 −1.28204
\(317\) − 9.22067i − 0.517884i −0.965893 0.258942i \(-0.916626\pi\)
0.965893 0.258942i \(-0.0833739\pi\)
\(318\) 1.96217i 0.110033i
\(319\) 16.0200i 0.896948i
\(320\) − 27.5316i − 1.53906i
\(321\) 1.09831 0.0613019
\(322\) 14.2628 0.794837
\(323\) 11.7153i 0.651859i
\(324\) −20.8683 −1.15935
\(325\) 0 0
\(326\) 24.1406 1.33702
\(327\) − 1.34765i − 0.0745255i
\(328\) 7.35043 0.405860
\(329\) 2.56938 0.141654
\(330\) − 1.72526i − 0.0949725i
\(331\) − 15.3888i − 0.845848i −0.906165 0.422924i \(-0.861004\pi\)
0.906165 0.422924i \(-0.138996\pi\)
\(332\) − 31.0166i − 1.70225i
\(333\) − 4.53680i − 0.248615i
\(334\) 9.14238 0.500249
\(335\) 3.39129 0.185286
\(336\) − 0.292415i − 0.0159525i
\(337\) −9.97038 −0.543121 −0.271561 0.962421i \(-0.587540\pi\)
−0.271561 + 0.962421i \(0.587540\pi\)
\(338\) 0 0
\(339\) 0.955527 0.0518971
\(340\) 29.8157i 1.61698i
\(341\) −29.0697 −1.57421
\(342\) 15.0999 0.816510
\(343\) 1.00000i 0.0539949i
\(344\) − 0.318135i − 0.0171527i
\(345\) 1.64571i 0.0886018i
\(346\) 41.2016i 2.21501i
\(347\) 18.3587 0.985547 0.492773 0.870158i \(-0.335983\pi\)
0.492773 + 0.870158i \(0.335983\pi\)
\(348\) 0.990084 0.0530741
\(349\) − 3.91068i − 0.209334i −0.994507 0.104667i \(-0.966622\pi\)
0.994507 0.104667i \(-0.0333776\pi\)
\(350\) −4.07774 −0.217964
\(351\) 0 0
\(352\) 27.8951 1.48682
\(353\) 25.6384i 1.36459i 0.731075 + 0.682297i \(0.239019\pi\)
−0.731075 + 0.682297i \(0.760981\pi\)
\(354\) 2.42884 0.129091
\(355\) −31.8625 −1.69109
\(356\) 37.5036i 1.98769i
\(357\) 0.440460i 0.0233116i
\(358\) − 13.4692i − 0.711870i
\(359\) − 17.5435i − 0.925908i −0.886382 0.462954i \(-0.846790\pi\)
0.886382 0.462954i \(-0.153210\pi\)
\(360\) 5.55691 0.292875
\(361\) 13.1276 0.690927
\(362\) − 43.5794i − 2.29048i
\(363\) 0.0801867 0.00420871
\(364\) 0 0
\(365\) 13.6737 0.715713
\(366\) 0.186843i 0.00976643i
\(367\) 12.0095 0.626890 0.313445 0.949606i \(-0.398517\pi\)
0.313445 + 0.949606i \(0.398517\pi\)
\(368\) −21.9784 −1.14571
\(369\) − 31.2286i − 1.62569i
\(370\) 8.33138i 0.433128i
\(371\) 10.3402i 0.536835i
\(372\) 1.79660i 0.0931492i
\(373\) 6.58590 0.341005 0.170503 0.985357i \(-0.445461\pi\)
0.170503 + 0.985357i \(0.445461\pi\)
\(374\) −34.7062 −1.79461
\(375\) 0.731108i 0.0377543i
\(376\) 1.80929 0.0933068
\(377\) 0 0
\(378\) 1.13699 0.0584807
\(379\) − 11.1382i − 0.572130i −0.958210 0.286065i \(-0.907653\pi\)
0.958210 0.286065i \(-0.0923473\pi\)
\(380\) −14.9453 −0.766678
\(381\) −0.534357 −0.0273759
\(382\) − 40.9801i − 2.09672i
\(383\) 5.16337i 0.263836i 0.991261 + 0.131918i \(0.0421135\pi\)
−0.991261 + 0.131918i \(0.957886\pi\)
\(384\) − 0.505916i − 0.0258174i
\(385\) − 9.09173i − 0.463358i
\(386\) −30.9716 −1.57642
\(387\) −1.35161 −0.0687060
\(388\) 34.9327i 1.77344i
\(389\) 3.90202 0.197840 0.0989200 0.995095i \(-0.468461\pi\)
0.0989200 + 0.995095i \(0.468461\pi\)
\(390\) 0 0
\(391\) 33.1058 1.67423
\(392\) 0.704173i 0.0355661i
\(393\) −0.126548 −0.00638349
\(394\) 31.1259 1.56810
\(395\) − 25.7112i − 1.29367i
\(396\) 24.1096i 1.21155i
\(397\) − 6.09215i − 0.305756i −0.988245 0.152878i \(-0.951146\pi\)
0.988245 0.152878i \(-0.0488542\pi\)
\(398\) − 18.9307i − 0.948911i
\(399\) −0.220783 −0.0110530
\(400\) 6.28363 0.314181
\(401\) 14.5534i 0.726761i 0.931641 + 0.363380i \(0.118377\pi\)
−0.931641 + 0.363380i \(0.881623\pi\)
\(402\) 0.243970 0.0121681
\(403\) 0 0
\(404\) −13.0489 −0.649210
\(405\) − 23.5430i − 1.16986i
\(406\) 9.68058 0.480439
\(407\) −5.22687 −0.259086
\(408\) 0.310160i 0.0153552i
\(409\) − 29.5278i − 1.46006i −0.683417 0.730028i \(-0.739507\pi\)
0.683417 0.730028i \(-0.260493\pi\)
\(410\) 57.3481i 2.83222i
\(411\) 1.13408i 0.0559403i
\(412\) −35.1258 −1.73052
\(413\) 12.7994 0.629818
\(414\) − 42.6701i − 2.09712i
\(415\) 34.9920 1.71769
\(416\) 0 0
\(417\) −0.727095 −0.0356060
\(418\) − 17.3967i − 0.850901i
\(419\) 31.8847 1.55767 0.778835 0.627229i \(-0.215811\pi\)
0.778835 + 0.627229i \(0.215811\pi\)
\(420\) −0.561896 −0.0274177
\(421\) 16.0318i 0.781344i 0.920530 + 0.390672i \(0.127757\pi\)
−0.920530 + 0.390672i \(0.872243\pi\)
\(422\) − 7.07197i − 0.344258i
\(423\) − 7.68681i − 0.373745i
\(424\) 7.28126i 0.353609i
\(425\) −9.46494 −0.459117
\(426\) −2.29220 −0.111057
\(427\) 0.984619i 0.0476491i
\(428\) 28.1857 1.36241
\(429\) 0 0
\(430\) 2.48209 0.119697
\(431\) − 20.3260i − 0.979069i −0.871984 0.489535i \(-0.837167\pi\)
0.871984 0.489535i \(-0.162833\pi\)
\(432\) −1.75206 −0.0842960
\(433\) 13.9077 0.668364 0.334182 0.942509i \(-0.391540\pi\)
0.334182 + 0.942509i \(0.391540\pi\)
\(434\) 17.5663i 0.843208i
\(435\) 1.11698i 0.0535553i
\(436\) − 34.5844i − 1.65629i
\(437\) 16.5945i 0.793823i
\(438\) 0.983688 0.0470024
\(439\) 11.3492 0.541669 0.270835 0.962626i \(-0.412700\pi\)
0.270835 + 0.962626i \(0.412700\pi\)
\(440\) − 6.40215i − 0.305210i
\(441\) 2.99170 0.142462
\(442\) 0 0
\(443\) −21.8184 −1.03662 −0.518311 0.855192i \(-0.673439\pi\)
−0.518311 + 0.855192i \(0.673439\pi\)
\(444\) 0.323036i 0.0153306i
\(445\) −42.3105 −2.00571
\(446\) −11.7653 −0.557104
\(447\) − 0.853210i − 0.0403554i
\(448\) − 10.4375i − 0.493123i
\(449\) − 11.0754i − 0.522680i −0.965247 0.261340i \(-0.915836\pi\)
0.965247 0.261340i \(-0.0841644\pi\)
\(450\) 12.1994i 0.575084i
\(451\) −35.9786 −1.69417
\(452\) 24.5214 1.15339
\(453\) − 1.14696i − 0.0538891i
\(454\) 27.5906 1.29489
\(455\) 0 0
\(456\) −0.155470 −0.00728053
\(457\) − 6.02903i − 0.282026i −0.990008 0.141013i \(-0.954964\pi\)
0.990008 0.141013i \(-0.0450359\pi\)
\(458\) 3.47070 0.162175
\(459\) 2.63910 0.123183
\(460\) 42.2332i 1.96913i
\(461\) 31.2543i 1.45566i 0.685759 + 0.727829i \(0.259471\pi\)
−0.685759 + 0.727829i \(0.740529\pi\)
\(462\) − 0.654061i − 0.0304297i
\(463\) − 16.0896i − 0.747746i −0.927480 0.373873i \(-0.878030\pi\)
0.927480 0.373873i \(-0.121970\pi\)
\(464\) −14.9174 −0.692521
\(465\) −2.02687 −0.0939938
\(466\) − 25.1671i − 1.16584i
\(467\) 5.18604 0.239981 0.119991 0.992775i \(-0.461714\pi\)
0.119991 + 0.992775i \(0.461714\pi\)
\(468\) 0 0
\(469\) 1.28567 0.0593665
\(470\) 14.1161i 0.651125i
\(471\) −0.574918 −0.0264908
\(472\) 9.01299 0.414857
\(473\) 1.55719i 0.0715998i
\(474\) − 1.84967i − 0.0849581i
\(475\) − 4.74436i − 0.217686i
\(476\) 11.3034i 0.518090i
\(477\) 30.9347 1.41640
\(478\) 30.9207 1.41428
\(479\) 4.15275i 0.189744i 0.995489 + 0.0948720i \(0.0302442\pi\)
−0.995489 + 0.0948720i \(0.969756\pi\)
\(480\) 1.94497 0.0887754
\(481\) 0 0
\(482\) −42.5876 −1.93981
\(483\) 0.623901i 0.0283885i
\(484\) 2.05780 0.0935366
\(485\) −39.4101 −1.78952
\(486\) − 5.10467i − 0.231553i
\(487\) 6.52104i 0.295496i 0.989025 + 0.147748i \(0.0472025\pi\)
−0.989025 + 0.147748i \(0.952797\pi\)
\(488\) 0.693342i 0.0313861i
\(489\) 1.05598i 0.0477532i
\(490\) −5.49396 −0.248192
\(491\) −18.9254 −0.854091 −0.427046 0.904230i \(-0.640446\pi\)
−0.427046 + 0.904230i \(0.640446\pi\)
\(492\) 2.22358i 0.100247i
\(493\) 22.4698 1.01199
\(494\) 0 0
\(495\) −27.1997 −1.22254
\(496\) − 27.0689i − 1.21543i
\(497\) −12.0794 −0.541834
\(498\) 2.51733 0.112804
\(499\) − 36.1116i − 1.61658i −0.588785 0.808290i \(-0.700394\pi\)
0.588785 0.808290i \(-0.299606\pi\)
\(500\) 18.7622i 0.839071i
\(501\) 0.399916i 0.0178669i
\(502\) 12.5733i 0.561173i
\(503\) 13.8545 0.617743 0.308872 0.951104i \(-0.400049\pi\)
0.308872 + 0.951104i \(0.400049\pi\)
\(504\) 2.10667 0.0938386
\(505\) − 14.7215i − 0.655096i
\(506\) −49.1605 −2.18545
\(507\) 0 0
\(508\) −13.7130 −0.608417
\(509\) − 10.1964i − 0.451948i −0.974133 0.225974i \(-0.927444\pi\)
0.974133 0.225974i \(-0.0725564\pi\)
\(510\) −2.41987 −0.107154
\(511\) 5.18381 0.229318
\(512\) 30.4953i 1.34771i
\(513\) 1.32287i 0.0584061i
\(514\) 4.15635i 0.183329i
\(515\) − 39.6279i − 1.74621i
\(516\) 0.0962392 0.00423669
\(517\) −8.85602 −0.389487
\(518\) 3.15850i 0.138776i
\(519\) −1.80229 −0.0791116
\(520\) 0 0
\(521\) −40.6356 −1.78028 −0.890139 0.455688i \(-0.849393\pi\)
−0.890139 + 0.455688i \(0.849393\pi\)
\(522\) − 28.9614i − 1.26760i
\(523\) 25.0743 1.09642 0.548210 0.836340i \(-0.315309\pi\)
0.548210 + 0.836340i \(0.315309\pi\)
\(524\) −3.24755 −0.141870
\(525\) − 0.178373i − 0.00778484i
\(526\) 26.8416i 1.17035i
\(527\) 40.7735i 1.77612i
\(528\) 1.00788i 0.0438624i
\(529\) 23.8936 1.03885
\(530\) −56.8085 −2.46760
\(531\) − 38.2920i − 1.66173i
\(532\) −5.66589 −0.245648
\(533\) 0 0
\(534\) −3.04383 −0.131719
\(535\) 31.7983i 1.37476i
\(536\) 0.905331 0.0391043
\(537\) 0.589185 0.0254252
\(538\) − 50.9650i − 2.19726i
\(539\) − 3.44676i − 0.148462i
\(540\) 3.36671i 0.144880i
\(541\) 22.0722i 0.948959i 0.880267 + 0.474479i \(0.157364\pi\)
−0.880267 + 0.474479i \(0.842636\pi\)
\(542\) −29.7523 −1.27797
\(543\) 1.90630 0.0818071
\(544\) − 39.1260i − 1.67751i
\(545\) 39.0172 1.67131
\(546\) 0 0
\(547\) −12.9107 −0.552022 −0.276011 0.961154i \(-0.589013\pi\)
−0.276011 + 0.961154i \(0.589013\pi\)
\(548\) 29.1036i 1.24325i
\(549\) 2.94568 0.125719
\(550\) 14.0550 0.599306
\(551\) 11.2631i 0.479826i
\(552\) 0.439334i 0.0186993i
\(553\) − 9.74733i − 0.414499i
\(554\) 55.3576i 2.35192i
\(555\) −0.364440 −0.0154696
\(556\) −18.6592 −0.791326
\(557\) 23.4061i 0.991746i 0.868395 + 0.495873i \(0.165152\pi\)
−0.868395 + 0.495873i \(0.834848\pi\)
\(558\) 52.5530 2.22475
\(559\) 0 0
\(560\) 8.46596 0.357752
\(561\) − 1.51816i − 0.0640967i
\(562\) 20.1761 0.851077
\(563\) −42.3634 −1.78541 −0.892703 0.450645i \(-0.851194\pi\)
−0.892703 + 0.450645i \(0.851194\pi\)
\(564\) 0.547328i 0.0230467i
\(565\) 27.6643i 1.16385i
\(566\) 18.7839i 0.789545i
\(567\) − 8.92536i − 0.374830i
\(568\) −8.50596 −0.356902
\(569\) −24.3356 −1.02020 −0.510101 0.860115i \(-0.670392\pi\)
−0.510101 + 0.860115i \(0.670392\pi\)
\(570\) − 1.21298i − 0.0508059i
\(571\) −9.71777 −0.406676 −0.203338 0.979109i \(-0.565179\pi\)
−0.203338 + 0.979109i \(0.565179\pi\)
\(572\) 0 0
\(573\) 1.79260 0.0748868
\(574\) 21.7412i 0.907459i
\(575\) −13.4069 −0.559105
\(576\) −31.2257 −1.30107
\(577\) − 4.39680i − 0.183041i −0.995803 0.0915206i \(-0.970827\pi\)
0.995803 0.0915206i \(-0.0291727\pi\)
\(578\) 13.2715i 0.552023i
\(579\) − 1.35480i − 0.0563034i
\(580\) 28.6648i 1.19024i
\(581\) 13.2658 0.550357
\(582\) −2.83518 −0.117522
\(583\) − 35.6400i − 1.47606i
\(584\) 3.65030 0.151050
\(585\) 0 0
\(586\) −7.01330 −0.289717
\(587\) 13.2174i 0.545540i 0.962079 + 0.272770i \(0.0879397\pi\)
−0.962079 + 0.272770i \(0.912060\pi\)
\(588\) −0.213020 −0.00878479
\(589\) −20.4380 −0.842133
\(590\) 70.3194i 2.89501i
\(591\) 1.36154i 0.0560065i
\(592\) − 4.86711i − 0.200037i
\(593\) 5.35235i 0.219795i 0.993943 + 0.109897i \(0.0350522\pi\)
−0.993943 + 0.109897i \(0.964948\pi\)
\(594\) −3.91894 −0.160796
\(595\) −12.7522 −0.522787
\(596\) − 21.8957i − 0.896881i
\(597\) 0.828088 0.0338914
\(598\) 0 0
\(599\) 9.81933 0.401207 0.200604 0.979673i \(-0.435710\pi\)
0.200604 + 0.979673i \(0.435710\pi\)
\(600\) − 0.125605i − 0.00512782i
\(601\) −28.2788 −1.15352 −0.576758 0.816915i \(-0.695682\pi\)
−0.576758 + 0.816915i \(0.695682\pi\)
\(602\) 0.940982 0.0383516
\(603\) − 3.84633i − 0.156635i
\(604\) − 29.4342i − 1.19766i
\(605\) 2.32156i 0.0943848i
\(606\) − 1.05907i − 0.0430216i
\(607\) 8.70338 0.353259 0.176630 0.984277i \(-0.443480\pi\)
0.176630 + 0.984277i \(0.443480\pi\)
\(608\) 19.6122 0.795379
\(609\) 0.423459i 0.0171594i
\(610\) −5.40946 −0.219023
\(611\) 0 0
\(612\) 33.8163 1.36694
\(613\) 28.8798i 1.16644i 0.812313 + 0.583222i \(0.198208\pi\)
−0.812313 + 0.583222i \(0.801792\pi\)
\(614\) −10.9840 −0.443277
\(615\) −2.50859 −0.101156
\(616\) − 2.42711i − 0.0977911i
\(617\) − 24.5573i − 0.988641i −0.869280 0.494320i \(-0.835417\pi\)
0.869280 0.494320i \(-0.164583\pi\)
\(618\) − 2.85084i − 0.114678i
\(619\) − 16.7008i − 0.671262i −0.941994 0.335631i \(-0.891051\pi\)
0.941994 0.335631i \(-0.108949\pi\)
\(620\) −52.0149 −2.08897
\(621\) 3.73823 0.150010
\(622\) − 19.4046i − 0.778052i
\(623\) −16.0403 −0.642641
\(624\) 0 0
\(625\) −30.9560 −1.23824
\(626\) − 64.6352i − 2.58334i
\(627\) 0.760987 0.0303909
\(628\) −14.7539 −0.588746
\(629\) 7.33126i 0.292317i
\(630\) 16.4363i 0.654837i
\(631\) 26.7848i 1.06629i 0.846025 + 0.533143i \(0.178989\pi\)
−0.846025 + 0.533143i \(0.821011\pi\)
\(632\) − 6.86380i − 0.273027i
\(633\) 0.309350 0.0122956
\(634\) 19.2049 0.762723
\(635\) − 15.4707i − 0.613934i
\(636\) −2.20266 −0.0873411
\(637\) 0 0
\(638\) −33.3666 −1.32100
\(639\) 36.1378i 1.42959i
\(640\) 14.6472 0.578982
\(641\) −2.23814 −0.0884012 −0.0442006 0.999023i \(-0.514074\pi\)
−0.0442006 + 0.999023i \(0.514074\pi\)
\(642\) 2.28758i 0.0902834i
\(643\) 34.6442i 1.36623i 0.730309 + 0.683117i \(0.239376\pi\)
−0.730309 + 0.683117i \(0.760624\pi\)
\(644\) 16.0110i 0.630921i
\(645\) 0.108574i 0.00427511i
\(646\) −24.4008 −0.960037
\(647\) 19.0383 0.748473 0.374236 0.927333i \(-0.377905\pi\)
0.374236 + 0.927333i \(0.377905\pi\)
\(648\) − 6.28499i − 0.246898i
\(649\) −44.1164 −1.73172
\(650\) 0 0
\(651\) −0.768404 −0.0301161
\(652\) 27.0994i 1.06129i
\(653\) 0.112802 0.00441429 0.00220715 0.999998i \(-0.499297\pi\)
0.00220715 + 0.999998i \(0.499297\pi\)
\(654\) 2.80691 0.109759
\(655\) − 3.66380i − 0.143157i
\(656\) − 33.5023i − 1.30804i
\(657\) − 15.5084i − 0.605041i
\(658\) 5.35152i 0.208624i
\(659\) 8.44907 0.329129 0.164565 0.986366i \(-0.447378\pi\)
0.164565 + 0.986366i \(0.447378\pi\)
\(660\) 1.93672 0.0753867
\(661\) − 19.0576i − 0.741256i −0.928781 0.370628i \(-0.879142\pi\)
0.928781 0.370628i \(-0.120858\pi\)
\(662\) 32.0520 1.24574
\(663\) 0 0
\(664\) 9.34139 0.362516
\(665\) − 6.39210i − 0.247875i
\(666\) 9.44927 0.366152
\(667\) 31.8280 1.23238
\(668\) 10.2629i 0.397084i
\(669\) − 0.514652i − 0.0198976i
\(670\) 7.06340i 0.272883i
\(671\) − 3.39374i − 0.131014i
\(672\) 0.737356 0.0284441
\(673\) −4.66145 −0.179686 −0.0898428 0.995956i \(-0.528636\pi\)
−0.0898428 + 0.995956i \(0.528636\pi\)
\(674\) − 20.7664i − 0.799891i
\(675\) −1.06876 −0.0411365
\(676\) 0 0
\(677\) −14.2159 −0.546360 −0.273180 0.961963i \(-0.588075\pi\)
−0.273180 + 0.961963i \(0.588075\pi\)
\(678\) 1.99018i 0.0764323i
\(679\) −14.9407 −0.573373
\(680\) −8.97972 −0.344357
\(681\) 1.20690i 0.0462484i
\(682\) − 60.5467i − 2.31845i
\(683\) − 47.4878i − 1.81707i −0.417807 0.908536i \(-0.637201\pi\)
0.417807 0.908536i \(-0.362799\pi\)
\(684\) 16.9507i 0.648124i
\(685\) −32.8339 −1.25452
\(686\) −2.08281 −0.0795220
\(687\) 0.151819i 0.00579227i
\(688\) −1.45001 −0.0552813
\(689\) 0 0
\(690\) −3.42769 −0.130490
\(691\) − 6.11055i − 0.232456i −0.993223 0.116228i \(-0.962920\pi\)
0.993223 0.116228i \(-0.0370804\pi\)
\(692\) −46.2515 −1.75822
\(693\) −10.3117 −0.391707
\(694\) 38.2376i 1.45148i
\(695\) − 21.0508i − 0.798502i
\(696\) 0.298188i 0.0113028i
\(697\) 50.4640i 1.91146i
\(698\) 8.14519 0.308300
\(699\) 1.10089 0.0416394
\(700\) − 4.57753i − 0.173014i
\(701\) −46.9469 −1.77316 −0.886580 0.462575i \(-0.846926\pi\)
−0.886580 + 0.462575i \(0.846926\pi\)
\(702\) 0 0
\(703\) −3.67484 −0.138599
\(704\) 35.9753i 1.35587i
\(705\) −0.617481 −0.0232557
\(706\) −53.3998 −2.00973
\(707\) − 5.58103i − 0.209896i
\(708\) 2.72653i 0.102469i
\(709\) 42.2551i 1.58692i 0.608620 + 0.793462i \(0.291724\pi\)
−0.608620 + 0.793462i \(0.708276\pi\)
\(710\) − 66.3635i − 2.49058i
\(711\) −29.1611 −1.09363
\(712\) −11.2951 −0.423303
\(713\) 57.7547i 2.16293i
\(714\) −0.917393 −0.0343326
\(715\) 0 0
\(716\) 15.1201 0.565063
\(717\) 1.35257i 0.0505125i
\(718\) 36.5396 1.36365
\(719\) −29.1865 −1.08847 −0.544237 0.838932i \(-0.683181\pi\)
−0.544237 + 0.838932i \(0.683181\pi\)
\(720\) − 25.3276i − 0.943904i
\(721\) − 15.0233i − 0.559496i
\(722\) 27.3423i 1.01757i
\(723\) − 1.86291i − 0.0692825i
\(724\) 48.9207 1.81812
\(725\) −9.09960 −0.337951
\(726\) 0.167013i 0.00619845i
\(727\) 21.5693 0.799961 0.399981 0.916524i \(-0.369017\pi\)
0.399981 + 0.916524i \(0.369017\pi\)
\(728\) 0 0
\(729\) −26.5528 −0.983437
\(730\) 28.4796i 1.05408i
\(731\) 2.18413 0.0807831
\(732\) −0.209743 −0.00775234
\(733\) 7.73466i 0.285686i 0.989745 + 0.142843i \(0.0456244\pi\)
−0.989745 + 0.142843i \(0.954376\pi\)
\(734\) 25.0135i 0.923263i
\(735\) − 0.240323i − 0.00886444i
\(736\) − 55.4211i − 2.04285i
\(737\) −4.43138 −0.163232
\(738\) 65.0431 2.39427
\(739\) − 17.0608i − 0.627591i −0.949491 0.313796i \(-0.898399\pi\)
0.949491 0.313796i \(-0.101601\pi\)
\(740\) −9.35251 −0.343805
\(741\) 0 0
\(742\) −21.5366 −0.790633
\(743\) − 42.3773i − 1.55467i −0.629085 0.777337i \(-0.716570\pi\)
0.629085 0.777337i \(-0.283430\pi\)
\(744\) −0.541089 −0.0198373
\(745\) 24.7021 0.905014
\(746\) 13.7172i 0.502221i
\(747\) − 39.6872i − 1.45208i
\(748\) − 38.9600i − 1.42452i
\(749\) 12.0550i 0.440480i
\(750\) −1.52276 −0.0556033
\(751\) −13.6368 −0.497615 −0.248807 0.968553i \(-0.580039\pi\)
−0.248807 + 0.968553i \(0.580039\pi\)
\(752\) − 8.24647i − 0.300718i
\(753\) −0.549995 −0.0200429
\(754\) 0 0
\(755\) 33.2068 1.20852
\(756\) 1.27635i 0.0464204i
\(757\) −3.57825 −0.130054 −0.0650269 0.997884i \(-0.520713\pi\)
−0.0650269 + 0.997884i \(0.520713\pi\)
\(758\) 23.1987 0.842614
\(759\) − 2.15044i − 0.0780558i
\(760\) − 4.50114i − 0.163274i
\(761\) − 10.5161i − 0.381207i −0.981667 0.190603i \(-0.938956\pi\)
0.981667 0.190603i \(-0.0610444\pi\)
\(762\) − 1.11296i − 0.0403184i
\(763\) 14.7918 0.535497
\(764\) 46.0028 1.66432
\(765\) 38.1506i 1.37934i
\(766\) −10.7543 −0.388569
\(767\) 0 0
\(768\) −0.848157 −0.0306052
\(769\) − 19.9912i − 0.720901i −0.932778 0.360451i \(-0.882623\pi\)
0.932778 0.360451i \(-0.117377\pi\)
\(770\) 18.9363 0.682418
\(771\) −0.181812 −0.00654779
\(772\) − 34.7677i − 1.25132i
\(773\) 44.5679i 1.60300i 0.597998 + 0.801498i \(0.295963\pi\)
−0.597998 + 0.801498i \(0.704037\pi\)
\(774\) − 2.81513i − 0.101188i
\(775\) − 16.5120i − 0.593130i
\(776\) −10.5209 −0.377677
\(777\) −0.138163 −0.00495655
\(778\) 8.12715i 0.291372i
\(779\) −25.2954 −0.906302
\(780\) 0 0
\(781\) 41.6346 1.48980
\(782\) 68.9531i 2.46576i
\(783\) 2.53724 0.0906734
\(784\) 3.20952 0.114626
\(785\) − 16.6450i − 0.594085i
\(786\) − 0.263575i − 0.00940140i
\(787\) 37.0091i 1.31923i 0.751602 + 0.659617i \(0.229281\pi\)
−0.751602 + 0.659617i \(0.770719\pi\)
\(788\) 34.9409i 1.24472i
\(789\) −1.17414 −0.0418004
\(790\) 53.5514 1.90527
\(791\) 10.4878i 0.372903i
\(792\) −7.26118 −0.258015
\(793\) 0 0
\(794\) 12.6888 0.450308
\(795\) − 2.48498i − 0.0881331i
\(796\) 21.2510 0.753220
\(797\) 5.86398 0.207713 0.103856 0.994592i \(-0.466882\pi\)
0.103856 + 0.994592i \(0.466882\pi\)
\(798\) − 0.459850i − 0.0162785i
\(799\) 12.4215i 0.439443i
\(800\) 15.8449i 0.560200i
\(801\) 47.9877i 1.69556i
\(802\) −30.3119 −1.07035
\(803\) −17.8673 −0.630524
\(804\) 0.273872i 0.00965873i
\(805\) −18.0631 −0.636642
\(806\) 0 0
\(807\) 2.22937 0.0784775
\(808\) − 3.93001i − 0.138257i
\(809\) 1.17991 0.0414835 0.0207417 0.999785i \(-0.493397\pi\)
0.0207417 + 0.999785i \(0.493397\pi\)
\(810\) 49.0356 1.72293
\(811\) − 24.4596i − 0.858894i −0.903092 0.429447i \(-0.858708\pi\)
0.903092 0.429447i \(-0.141292\pi\)
\(812\) 10.8671i 0.381360i
\(813\) − 1.30146i − 0.0456441i
\(814\) − 10.8866i − 0.381574i
\(815\) −30.5727 −1.07092
\(816\) 1.41366 0.0494882
\(817\) 1.09481i 0.0383026i
\(818\) 61.5008 2.15032
\(819\) 0 0
\(820\) −64.3770 −2.24814
\(821\) − 45.2922i − 1.58071i −0.612651 0.790354i \(-0.709897\pi\)
0.612651 0.790354i \(-0.290103\pi\)
\(822\) −2.36208 −0.0823870
\(823\) 28.5033 0.993563 0.496781 0.867876i \(-0.334515\pi\)
0.496781 + 0.867876i \(0.334515\pi\)
\(824\) − 10.5790i − 0.368536i
\(825\) 0.614808i 0.0214049i
\(826\) 26.6587i 0.927575i
\(827\) − 0.594562i − 0.0206749i −0.999947 0.0103375i \(-0.996709\pi\)
0.999947 0.0103375i \(-0.00329058\pi\)
\(828\) 47.9000 1.66464
\(829\) 40.7651 1.41583 0.707915 0.706298i \(-0.249636\pi\)
0.707915 + 0.706298i \(0.249636\pi\)
\(830\) 72.8816i 2.52976i
\(831\) −2.42151 −0.0840014
\(832\) 0 0
\(833\) −4.83445 −0.167504
\(834\) − 1.51440i − 0.0524393i
\(835\) −11.5783 −0.400685
\(836\) 19.5290 0.675423
\(837\) 4.60404i 0.159139i
\(838\) 66.4097i 2.29408i
\(839\) − 48.1410i − 1.66201i −0.556264 0.831005i \(-0.687766\pi\)
0.556264 0.831005i \(-0.312234\pi\)
\(840\) − 0.169229i − 0.00583895i
\(841\) −7.39749 −0.255086
\(842\) −33.3912 −1.15074
\(843\) 0.882565i 0.0303971i
\(844\) 7.93875 0.273263
\(845\) 0 0
\(846\) 16.0101 0.550440
\(847\) 0.880123i 0.0302414i
\(848\) 33.1870 1.13965
\(849\) −0.821665 −0.0281995
\(850\) − 19.7136i − 0.676172i
\(851\) 10.3846i 0.355978i
\(852\) − 2.57314i − 0.0881544i
\(853\) − 15.8090i − 0.541291i −0.962679 0.270645i \(-0.912763\pi\)
0.962679 0.270645i \(-0.0872371\pi\)
\(854\) −2.05077 −0.0701760
\(855\) −19.1233 −0.654002
\(856\) 8.48880i 0.290141i
\(857\) 43.5809 1.48870 0.744348 0.667792i \(-0.232761\pi\)
0.744348 + 0.667792i \(0.232761\pi\)
\(858\) 0 0
\(859\) 57.5101 1.96222 0.981110 0.193449i \(-0.0619673\pi\)
0.981110 + 0.193449i \(0.0619673\pi\)
\(860\) 2.78631i 0.0950123i
\(861\) −0.951027 −0.0324109
\(862\) 42.3352 1.44194
\(863\) − 16.1116i − 0.548444i −0.961666 0.274222i \(-0.911580\pi\)
0.961666 0.274222i \(-0.0884203\pi\)
\(864\) − 4.41801i − 0.150304i
\(865\) − 52.1797i − 1.77416i
\(866\) 28.9672i 0.984344i
\(867\) −0.580538 −0.0197161
\(868\) −19.7193 −0.669317
\(869\) 33.5967i 1.13969i
\(870\) −2.32646 −0.0788745
\(871\) 0 0
\(872\) 10.4159 0.352728
\(873\) 44.6982i 1.51280i
\(874\) −34.5632 −1.16912
\(875\) −8.02459 −0.271281
\(876\) 1.10425i 0.0373093i
\(877\) 44.0382i 1.48706i 0.668700 + 0.743532i \(0.266851\pi\)
−0.668700 + 0.743532i \(0.733149\pi\)
\(878\) 23.6383i 0.797753i
\(879\) − 0.306784i − 0.0103476i
\(880\) −29.1801 −0.983661
\(881\) 21.2905 0.717295 0.358647 0.933473i \(-0.383238\pi\)
0.358647 + 0.933473i \(0.383238\pi\)
\(882\) 6.23113i 0.209813i
\(883\) −4.28392 −0.144165 −0.0720827 0.997399i \(-0.522965\pi\)
−0.0720827 + 0.997399i \(0.522965\pi\)
\(884\) 0 0
\(885\) −3.07599 −0.103398
\(886\) − 45.4434i − 1.52670i
\(887\) −37.0660 −1.24455 −0.622276 0.782798i \(-0.713792\pi\)
−0.622276 + 0.782798i \(0.713792\pi\)
\(888\) −0.0972902 −0.00326485
\(889\) − 5.86506i − 0.196708i
\(890\) − 88.1247i − 2.95395i
\(891\) 30.7635i 1.03062i
\(892\) − 13.2073i − 0.442215i
\(893\) −6.22638 −0.208358
\(894\) 1.77707 0.0594342
\(895\) 17.0580i 0.570187i
\(896\) 5.55289 0.185509
\(897\) 0 0
\(898\) 23.0679 0.769787
\(899\) 39.1997i 1.30738i
\(900\) −13.6946 −0.456486
\(901\) −49.9891 −1.66538
\(902\) − 74.9365i − 2.49511i
\(903\) 0.0411615i 0.00136977i
\(904\) 7.38521i 0.245629i
\(905\) 55.1910i 1.83461i
\(906\) 2.38891 0.0793661
\(907\) 0.906355 0.0300950 0.0150475 0.999887i \(-0.495210\pi\)
0.0150475 + 0.999887i \(0.495210\pi\)
\(908\) 30.9722i 1.02785i
\(909\) −16.6968 −0.553797
\(910\) 0 0
\(911\) 47.1318 1.56155 0.780773 0.624815i \(-0.214826\pi\)
0.780773 + 0.624815i \(0.214826\pi\)
\(912\) 0.708609i 0.0234644i
\(913\) −45.7239 −1.51324
\(914\) 12.5573 0.415359
\(915\) − 0.236627i − 0.00782263i
\(916\) 3.89609i 0.128730i
\(917\) − 1.38898i − 0.0458681i
\(918\) 5.49674i 0.181420i
\(919\) 52.5207 1.73250 0.866249 0.499613i \(-0.166524\pi\)
0.866249 + 0.499613i \(0.166524\pi\)
\(920\) −12.7196 −0.419352
\(921\) − 0.480474i − 0.0158321i
\(922\) −65.0967 −2.14385
\(923\) 0 0
\(924\) 0.734227 0.0241543
\(925\) − 2.96894i − 0.0976182i
\(926\) 33.5115 1.10126
\(927\) −44.9451 −1.47619
\(928\) − 37.6158i − 1.23480i
\(929\) − 4.76389i − 0.156298i −0.996942 0.0781491i \(-0.975099\pi\)
0.996942 0.0781491i \(-0.0249010\pi\)
\(930\) − 4.22158i − 0.138431i
\(931\) − 2.42330i − 0.0794205i
\(932\) 28.2517 0.925415
\(933\) 0.848816 0.0277890
\(934\) 10.8015i 0.353437i
\(935\) 43.9536 1.43744
\(936\) 0 0
\(937\) 47.8844 1.56432 0.782158 0.623080i \(-0.214119\pi\)
0.782158 + 0.623080i \(0.214119\pi\)
\(938\) 2.67780i 0.0874331i
\(939\) 2.82735 0.0922670
\(940\) −15.8462 −0.516846
\(941\) − 8.54785i − 0.278652i −0.990247 0.139326i \(-0.955506\pi\)
0.990247 0.139326i \(-0.0444936\pi\)
\(942\) − 1.19744i − 0.0390148i
\(943\) 71.4810i 2.32774i
\(944\) − 41.0800i − 1.33704i
\(945\) −1.43994 −0.0468413
\(946\) −3.24333 −0.105450
\(947\) − 11.7381i − 0.381437i −0.981645 0.190718i \(-0.938918\pi\)
0.981645 0.190718i \(-0.0610817\pi\)
\(948\) 2.07637 0.0674375
\(949\) 0 0
\(950\) 9.88160 0.320601
\(951\) 0.840081i 0.0272415i
\(952\) −3.40429 −0.110334
\(953\) 39.2912 1.27277 0.636383 0.771373i \(-0.280430\pi\)
0.636383 + 0.771373i \(0.280430\pi\)
\(954\) 64.4310i 2.08603i
\(955\) 51.8991i 1.67942i
\(956\) 34.7105i 1.12262i
\(957\) − 1.45956i − 0.0471808i
\(958\) −8.64938 −0.279449
\(959\) −12.4476 −0.401955
\(960\) 2.50836i 0.0809569i
\(961\) −40.1314 −1.29456
\(962\) 0 0
\(963\) 36.0650 1.16218
\(964\) − 47.8073i − 1.53977i
\(965\) 39.2240 1.26266
\(966\) −1.29947 −0.0418096
\(967\) − 6.78097i − 0.218061i −0.994038 0.109031i \(-0.965225\pi\)
0.994038 0.109031i \(-0.0347747\pi\)
\(968\) 0.619758i 0.0199198i
\(969\) − 1.06737i − 0.0342888i
\(970\) − 82.0838i − 2.63555i
\(971\) −5.36908 −0.172302 −0.0861511 0.996282i \(-0.527457\pi\)
−0.0861511 + 0.996282i \(0.527457\pi\)
\(972\) 5.73033 0.183800
\(973\) − 7.98054i − 0.255844i
\(974\) −13.5821 −0.435197
\(975\) 0 0
\(976\) 3.16016 0.101154
\(977\) − 22.6247i − 0.723828i −0.932212 0.361914i \(-0.882123\pi\)
0.932212 0.361914i \(-0.117877\pi\)
\(978\) −2.19941 −0.0703294
\(979\) 55.2870 1.76698
\(980\) − 6.16733i − 0.197008i
\(981\) − 44.2525i − 1.41287i
\(982\) − 39.4180i − 1.25788i
\(983\) − 25.8113i − 0.823254i −0.911352 0.411627i \(-0.864961\pi\)
0.911352 0.411627i \(-0.135039\pi\)
\(984\) −0.669687 −0.0213488
\(985\) −39.4193 −1.25600
\(986\) 46.8003i 1.49043i
\(987\) −0.234092 −0.00745124
\(988\) 0 0
\(989\) 3.09378 0.0983764
\(990\) − 56.6518i − 1.80051i
\(991\) 45.2014 1.43587 0.717936 0.696110i \(-0.245087\pi\)
0.717936 + 0.696110i \(0.245087\pi\)
\(992\) 68.2572 2.16717
\(993\) 1.40206i 0.0444929i
\(994\) − 25.1590i − 0.797995i
\(995\) 23.9747i 0.760050i
\(996\) 2.82587i 0.0895412i
\(997\) 12.9965 0.411604 0.205802 0.978594i \(-0.434020\pi\)
0.205802 + 0.978594i \(0.434020\pi\)
\(998\) 75.2136 2.38085
\(999\) 0.827828i 0.0261913i
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.h.337.11 12
13.5 odd 4 1183.2.a.n.1.6 6
13.8 odd 4 1183.2.a.o.1.1 yes 6
13.12 even 2 inner 1183.2.c.h.337.2 12
91.34 even 4 8281.2.a.cg.1.1 6
91.83 even 4 8281.2.a.cb.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1183.2.a.n.1.6 6 13.5 odd 4
1183.2.a.o.1.1 yes 6 13.8 odd 4
1183.2.c.h.337.2 12 13.12 even 2 inner
1183.2.c.h.337.11 12 1.1 even 1 trivial
8281.2.a.cb.1.6 6 91.83 even 4
8281.2.a.cg.1.1 6 91.34 even 4