Properties

Label 25.30.b.a.24.3
Level $25$
Weight $30$
Character 25.24
Analytic conductor $133.195$
Analytic rank $0$
Dimension $4$
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] = [25,30,Mod(24,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 30, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.24");
 
S:= CuspForms(chi, 30);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 25.b (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: \(133.195105958\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25675x^{2} + 164788569 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.3
Root \(112.802i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.30.b.a.24.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+17433.9i q^{2} +9.52434e6i q^{3} +2.32930e8 q^{4} -1.66046e11 q^{6} +2.98628e12i q^{7} +1.34206e13i q^{8} -2.20826e13 q^{9} +O(q^{10})\) \(q+17433.9i q^{2} +9.52434e6i q^{3} +2.32930e8 q^{4} -1.66046e11 q^{6} +2.98628e12i q^{7} +1.34206e13i q^{8} -2.20826e13 q^{9} -1.25811e15 q^{11} +2.21850e15i q^{12} +6.09036e15i q^{13} -5.20625e16 q^{14} -1.08921e17 q^{16} -8.28952e16i q^{17} -3.84986e17i q^{18} -3.72257e18 q^{19} -2.84423e19 q^{21} -2.19338e19i q^{22} -1.69513e19i q^{23} -1.27823e20 q^{24} -1.06179e20 q^{26} +4.43337e20i q^{27} +6.95594e20i q^{28} +8.81402e20 q^{29} +3.71093e21 q^{31} +5.30623e21i q^{32} -1.19827e22i q^{33} +1.44519e21 q^{34} -5.14369e21 q^{36} -1.97746e22i q^{37} -6.48990e22i q^{38} -5.80066e22 q^{39} -1.00635e23 q^{41} -4.95861e23i q^{42} -1.61886e23i q^{43} -2.93051e23 q^{44} +2.95527e23 q^{46} +2.61745e24i q^{47} -1.03740e24i q^{48} -5.69796e24 q^{49} +7.89521e23 q^{51} +1.41863e24i q^{52} -1.44856e25i q^{53} -7.72909e24 q^{54} -4.00778e25 q^{56} -3.54550e25i q^{57} +1.53663e25i q^{58} -5.87125e24 q^{59} -6.18692e25 q^{61} +6.46960e25i q^{62} -6.59448e25i q^{63} -1.50985e26 q^{64} +2.08905e26 q^{66} -1.29243e26i q^{67} -1.93088e25i q^{68} +1.61450e26 q^{69} -3.63672e26 q^{71} -2.96362e26i q^{72} +1.45255e27i q^{73} +3.44748e26 q^{74} -8.67098e26 q^{76} -3.75707e27i q^{77} -1.01128e27i q^{78} +3.93709e27 q^{79} -5.73802e27 q^{81} -1.75447e27i q^{82} +6.61171e27i q^{83} -6.62507e27 q^{84} +2.82231e27 q^{86} +8.39477e27i q^{87} -1.68846e28i q^{88} -6.47986e27 q^{89} -1.81875e28 q^{91} -3.94847e27i q^{92} +3.53442e28i q^{93} -4.56324e28 q^{94} -5.05383e28 q^{96} -9.52029e28i q^{97} -9.93377e28i q^{98} +2.77823e28 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 179904512 q^{4} - 1087817513472 q^{6} - 326939139047412 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 179904512 q^{4} - 1087817513472 q^{6} - 326939139047412 q^{9} - 41\!\cdots\!32 q^{11}+ \cdots + 28\!\cdots\!96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 17433.9i 0.752419i 0.926535 + 0.376209i \(0.122773\pi\)
−0.926535 + 0.376209i \(0.877227\pi\)
\(3\) 9.52434e6i 1.14968i 0.818266 + 0.574839i \(0.194935\pi\)
−0.818266 + 0.574839i \(0.805065\pi\)
\(4\) 2.32930e8 0.433866
\(5\) 0 0
\(6\) −1.66046e11 −0.865040
\(7\) 2.98628e12i 1.66421i 0.554616 + 0.832106i \(0.312865\pi\)
−0.554616 + 0.832106i \(0.687135\pi\)
\(8\) 1.34206e13i 1.07887i
\(9\) −2.20826e13 −0.321761
\(10\) 0 0
\(11\) −1.25811e15 −0.998906 −0.499453 0.866341i \(-0.666466\pi\)
−0.499453 + 0.866341i \(0.666466\pi\)
\(12\) 2.21850e15i 0.498806i
\(13\) 6.09036e15i 0.429007i 0.976723 + 0.214503i \(0.0688133\pi\)
−0.976723 + 0.214503i \(0.931187\pi\)
\(14\) −5.20625e16 −1.25219
\(15\) 0 0
\(16\) −1.08921e17 −0.377895
\(17\) − 8.28952e16i − 0.119404i −0.998216 0.0597021i \(-0.980985\pi\)
0.998216 0.0597021i \(-0.0190151\pi\)
\(18\) − 3.84986e17i − 0.242099i
\(19\) −3.72257e18 −1.06885 −0.534424 0.845217i \(-0.679471\pi\)
−0.534424 + 0.845217i \(0.679471\pi\)
\(20\) 0 0
\(21\) −2.84423e19 −1.91331
\(22\) − 2.19338e19i − 0.751596i
\(23\) − 1.69513e19i − 0.304894i −0.988312 0.152447i \(-0.951285\pi\)
0.988312 0.152447i \(-0.0487154\pi\)
\(24\) −1.27823e20 −1.24035
\(25\) 0 0
\(26\) −1.06179e20 −0.322793
\(27\) 4.43337e20i 0.779757i
\(28\) 6.95594e20i 0.722045i
\(29\) 8.81402e20 0.550051 0.275026 0.961437i \(-0.411314\pi\)
0.275026 + 0.961437i \(0.411314\pi\)
\(30\) 0 0
\(31\) 3.71093e21 0.880518 0.440259 0.897871i \(-0.354887\pi\)
0.440259 + 0.897871i \(0.354887\pi\)
\(32\) 5.30623e21i 0.794532i
\(33\) − 1.19827e22i − 1.14842i
\(34\) 1.44519e21 0.0898420
\(35\) 0 0
\(36\) −5.14369e21 −0.139601
\(37\) − 1.97746e22i − 0.360730i −0.983600 0.180365i \(-0.942272\pi\)
0.983600 0.180365i \(-0.0577279\pi\)
\(38\) − 6.48990e22i − 0.804221i
\(39\) −5.80066e22 −0.493220
\(40\) 0 0
\(41\) −1.00635e23 −0.414367 −0.207183 0.978302i \(-0.566430\pi\)
−0.207183 + 0.978302i \(0.566430\pi\)
\(42\) − 4.95861e23i − 1.43961i
\(43\) − 1.61886e23i − 0.334131i −0.985946 0.167066i \(-0.946571\pi\)
0.985946 0.167066i \(-0.0534291\pi\)
\(44\) −2.93051e23 −0.433391
\(45\) 0 0
\(46\) 2.95527e23 0.229408
\(47\) 2.61745e24i 1.48751i 0.668453 + 0.743755i \(0.266957\pi\)
−0.668453 + 0.743755i \(0.733043\pi\)
\(48\) − 1.03740e24i − 0.434458i
\(49\) −5.69796e24 −1.76960
\(50\) 0 0
\(51\) 7.89521e23 0.137277
\(52\) 1.41863e24i 0.186131i
\(53\) − 1.44856e25i − 1.44190i −0.692986 0.720951i \(-0.743705\pi\)
0.692986 0.720951i \(-0.256295\pi\)
\(54\) −7.72909e24 −0.586704
\(55\) 0 0
\(56\) −4.00778e25 −1.79547
\(57\) − 3.54550e25i − 1.22883i
\(58\) 1.53663e25i 0.413869i
\(59\) −5.87125e24 −0.123417 −0.0617087 0.998094i \(-0.519655\pi\)
−0.0617087 + 0.998094i \(0.519655\pi\)
\(60\) 0 0
\(61\) −6.18692e25 −0.802032 −0.401016 0.916071i \(-0.631343\pi\)
−0.401016 + 0.916071i \(0.631343\pi\)
\(62\) 6.46960e25i 0.662519i
\(63\) − 6.59448e25i − 0.535479i
\(64\) −1.50985e26 −0.975716
\(65\) 0 0
\(66\) 2.08905e26 0.864094
\(67\) − 1.29243e26i − 0.429854i −0.976630 0.214927i \(-0.931049\pi\)
0.976630 0.214927i \(-0.0689514\pi\)
\(68\) − 1.93088e25i − 0.0518054i
\(69\) 1.61450e26 0.350531
\(70\) 0 0
\(71\) −3.63672e26 −0.521751 −0.260876 0.965372i \(-0.584011\pi\)
−0.260876 + 0.965372i \(0.584011\pi\)
\(72\) − 2.96362e26i − 0.347138i
\(73\) 1.45255e27i 1.39299i 0.717561 + 0.696496i \(0.245259\pi\)
−0.717561 + 0.696496i \(0.754741\pi\)
\(74\) 3.44748e26 0.271420
\(75\) 0 0
\(76\) −8.67098e26 −0.463736
\(77\) − 3.75707e27i − 1.66239i
\(78\) − 1.01128e27i − 0.371108i
\(79\) 3.93709e27 1.20111 0.600554 0.799584i \(-0.294947\pi\)
0.600554 + 0.799584i \(0.294947\pi\)
\(80\) 0 0
\(81\) −5.73802e27 −1.21823
\(82\) − 1.75447e27i − 0.311778i
\(83\) 6.61171e27i 0.985556i 0.870155 + 0.492778i \(0.164018\pi\)
−0.870155 + 0.492778i \(0.835982\pi\)
\(84\) −6.62507e27 −0.830120
\(85\) 0 0
\(86\) 2.82231e27 0.251407
\(87\) 8.39477e27i 0.632382i
\(88\) − 1.68846e28i − 1.07769i
\(89\) −6.47986e27 −0.351084 −0.175542 0.984472i \(-0.556168\pi\)
−0.175542 + 0.984472i \(0.556168\pi\)
\(90\) 0 0
\(91\) −1.81875e28 −0.713959
\(92\) − 3.94847e27i − 0.132283i
\(93\) 3.53442e28i 1.01231i
\(94\) −4.56324e28 −1.11923
\(95\) 0 0
\(96\) −5.05383e28 −0.913457
\(97\) − 9.52029e28i − 1.48068i −0.672235 0.740338i \(-0.734666\pi\)
0.672235 0.740338i \(-0.265334\pi\)
\(98\) − 9.93377e28i − 1.33148i
\(99\) 2.77823e28 0.321409
\(100\) 0 0
\(101\) 4.65862e28 0.403271 0.201635 0.979461i \(-0.435374\pi\)
0.201635 + 0.979461i \(0.435374\pi\)
\(102\) 1.37644e28i 0.103289i
\(103\) − 1.34288e29i − 0.874775i −0.899273 0.437388i \(-0.855904\pi\)
0.899273 0.437388i \(-0.144096\pi\)
\(104\) −8.17365e28 −0.462842
\(105\) 0 0
\(106\) 2.52540e29 1.08491
\(107\) 4.69036e29i 1.75850i 0.476364 + 0.879248i \(0.341955\pi\)
−0.476364 + 0.879248i \(0.658045\pi\)
\(108\) 1.03266e29i 0.338310i
\(109\) 5.33865e29 1.53020 0.765098 0.643913i \(-0.222690\pi\)
0.765098 + 0.643913i \(0.222690\pi\)
\(110\) 0 0
\(111\) 1.88340e29 0.414723
\(112\) − 3.25268e29i − 0.628897i
\(113\) 7.95436e29i 1.35197i 0.736914 + 0.675987i \(0.236282\pi\)
−0.736914 + 0.675987i \(0.763718\pi\)
\(114\) 6.18120e29 0.924596
\(115\) 0 0
\(116\) 2.05305e29 0.238648
\(117\) − 1.34491e29i − 0.138038i
\(118\) − 1.02359e29i − 0.0928615i
\(119\) 2.47548e29 0.198714
\(120\) 0 0
\(121\) −3.46951e27 −0.00218716
\(122\) − 1.07862e30i − 0.603464i
\(123\) − 9.58486e29i − 0.476389i
\(124\) 8.64387e29 0.382027
\(125\) 0 0
\(126\) 1.14968e30 0.402905
\(127\) − 1.93923e29i − 0.0606003i −0.999541 0.0303002i \(-0.990354\pi\)
0.999541 0.0303002i \(-0.00964632\pi\)
\(128\) 2.16508e29i 0.0603851i
\(129\) 1.54186e30 0.384143
\(130\) 0 0
\(131\) −2.32229e30 −0.462894 −0.231447 0.972847i \(-0.574346\pi\)
−0.231447 + 0.972847i \(0.574346\pi\)
\(132\) − 2.79112e30i − 0.498260i
\(133\) − 1.11166e31i − 1.77879i
\(134\) 2.25320e30 0.323430
\(135\) 0 0
\(136\) 1.11251e30 0.128821
\(137\) − 1.26990e31i − 1.32228i −0.750263 0.661139i \(-0.770073\pi\)
0.750263 0.661139i \(-0.229927\pi\)
\(138\) 2.81470e30i 0.263746i
\(139\) −1.14547e31 −0.966648 −0.483324 0.875442i \(-0.660571\pi\)
−0.483324 + 0.875442i \(0.660571\pi\)
\(140\) 0 0
\(141\) −2.49295e31 −1.71016
\(142\) − 6.34022e30i − 0.392575i
\(143\) − 7.66234e30i − 0.428538i
\(144\) 2.40525e30 0.121592
\(145\) 0 0
\(146\) −2.53236e31 −1.04811
\(147\) − 5.42693e31i − 2.03448i
\(148\) − 4.60609e30i − 0.156508i
\(149\) 5.74280e31 1.76979 0.884897 0.465787i \(-0.154229\pi\)
0.884897 + 0.465787i \(0.154229\pi\)
\(150\) 0 0
\(151\) 5.07974e31 1.29026 0.645128 0.764075i \(-0.276804\pi\)
0.645128 + 0.764075i \(0.276804\pi\)
\(152\) − 4.99593e31i − 1.15315i
\(153\) 1.83054e30i 0.0384197i
\(154\) 6.55003e31 1.25082
\(155\) 0 0
\(156\) −1.35115e31 −0.213991
\(157\) 9.31373e30i 0.134455i 0.997738 + 0.0672277i \(0.0214154\pi\)
−0.997738 + 0.0672277i \(0.978585\pi\)
\(158\) 6.86388e31i 0.903736i
\(159\) 1.37965e32 1.65772
\(160\) 0 0
\(161\) 5.06213e31 0.507409
\(162\) − 1.00036e32i − 0.916620i
\(163\) − 4.45351e31i − 0.373235i −0.982433 0.186617i \(-0.940248\pi\)
0.982433 0.186617i \(-0.0597525\pi\)
\(164\) −2.34410e31 −0.179780
\(165\) 0 0
\(166\) −1.15268e32 −0.741551
\(167\) − 1.74608e32i − 1.02962i −0.857305 0.514808i \(-0.827863\pi\)
0.857305 0.514808i \(-0.172137\pi\)
\(168\) − 3.81714e32i − 2.06421i
\(169\) 1.64446e32 0.815953
\(170\) 0 0
\(171\) 8.22040e31 0.343914
\(172\) − 3.77081e31i − 0.144968i
\(173\) 4.75360e31i 0.168017i 0.996465 + 0.0840086i \(0.0267723\pi\)
−0.996465 + 0.0840086i \(0.973228\pi\)
\(174\) −1.46354e32 −0.475816
\(175\) 0 0
\(176\) 1.37034e32 0.377481
\(177\) − 5.59197e31i − 0.141890i
\(178\) − 1.12969e32i − 0.264162i
\(179\) 6.60198e32 1.42333 0.711664 0.702520i \(-0.247942\pi\)
0.711664 + 0.702520i \(0.247942\pi\)
\(180\) 0 0
\(181\) 5.34310e32 0.980515 0.490257 0.871578i \(-0.336903\pi\)
0.490257 + 0.871578i \(0.336903\pi\)
\(182\) − 3.17079e32i − 0.537196i
\(183\) − 5.89263e32i − 0.922079i
\(184\) 2.27497e32 0.328941
\(185\) 0 0
\(186\) −6.16187e32 −0.761683
\(187\) 1.04291e32i 0.119274i
\(188\) 6.09683e32i 0.645379i
\(189\) −1.32393e33 −1.29768
\(190\) 0 0
\(191\) −9.58754e32 −0.806721 −0.403360 0.915041i \(-0.632158\pi\)
−0.403360 + 0.915041i \(0.632158\pi\)
\(192\) − 1.43803e33i − 1.12176i
\(193\) 7.81683e31i 0.0565522i 0.999600 + 0.0282761i \(0.00900177\pi\)
−0.999600 + 0.0282761i \(0.990998\pi\)
\(194\) 1.65976e33 1.11409
\(195\) 0 0
\(196\) −1.32722e33 −0.767770
\(197\) 5.60393e32i 0.301115i 0.988601 + 0.150557i \(0.0481068\pi\)
−0.988601 + 0.150557i \(0.951893\pi\)
\(198\) 4.84354e32i 0.241834i
\(199\) −8.91542e32 −0.413783 −0.206891 0.978364i \(-0.566335\pi\)
−0.206891 + 0.978364i \(0.566335\pi\)
\(200\) 0 0
\(201\) 1.23095e33 0.494194
\(202\) 8.12179e32i 0.303429i
\(203\) 2.63211e33i 0.915402i
\(204\) 1.83903e32 0.0595596
\(205\) 0 0
\(206\) 2.34116e33 0.658198
\(207\) 3.74329e32i 0.0981032i
\(208\) − 6.63367e32i − 0.162120i
\(209\) 4.68340e33 1.06768
\(210\) 0 0
\(211\) 1.07645e33 0.213746 0.106873 0.994273i \(-0.465916\pi\)
0.106873 + 0.994273i \(0.465916\pi\)
\(212\) − 3.37412e33i − 0.625592i
\(213\) − 3.46373e33i − 0.599846i
\(214\) −8.17713e33 −1.32313
\(215\) 0 0
\(216\) −5.94986e33 −0.841254
\(217\) 1.10819e34i 1.46537i
\(218\) 9.30735e33i 1.15135i
\(219\) −1.38345e34 −1.60149
\(220\) 0 0
\(221\) 5.04861e32 0.0512253
\(222\) 3.28350e33i 0.312046i
\(223\) − 3.94073e33i − 0.350878i −0.984490 0.175439i \(-0.943866\pi\)
0.984490 0.175439i \(-0.0561344\pi\)
\(224\) −1.58459e34 −1.32227
\(225\) 0 0
\(226\) −1.38676e34 −1.01725
\(227\) 2.60531e34i 1.79260i 0.443445 + 0.896302i \(0.353756\pi\)
−0.443445 + 0.896302i \(0.646244\pi\)
\(228\) − 8.25853e33i − 0.533148i
\(229\) −1.02479e34 −0.620898 −0.310449 0.950590i \(-0.600479\pi\)
−0.310449 + 0.950590i \(0.600479\pi\)
\(230\) 0 0
\(231\) 3.57836e34 1.91122
\(232\) 1.18290e34i 0.593432i
\(233\) 1.94574e34i 0.917113i 0.888665 + 0.458557i \(0.151633\pi\)
−0.888665 + 0.458557i \(0.848367\pi\)
\(234\) 2.34470e33 0.103862
\(235\) 0 0
\(236\) −1.36759e33 −0.0535465
\(237\) 3.74981e34i 1.38089i
\(238\) 4.31573e33i 0.149516i
\(239\) 4.56285e34 1.48753 0.743766 0.668440i \(-0.233037\pi\)
0.743766 + 0.668440i \(0.233037\pi\)
\(240\) 0 0
\(241\) 2.90791e34 0.840107 0.420053 0.907499i \(-0.362011\pi\)
0.420053 + 0.907499i \(0.362011\pi\)
\(242\) − 6.04870e31i − 0.00164566i
\(243\) − 2.42245e34i − 0.620817i
\(244\) −1.44112e34 −0.347974
\(245\) 0 0
\(246\) 1.67101e34 0.358444
\(247\) − 2.26718e34i − 0.458543i
\(248\) 4.98030e34i 0.949963i
\(249\) −6.29721e34 −1.13307
\(250\) 0 0
\(251\) −1.72418e34 −0.276256 −0.138128 0.990414i \(-0.544109\pi\)
−0.138128 + 0.990414i \(0.544109\pi\)
\(252\) − 1.53605e34i − 0.232326i
\(253\) 2.13266e34i 0.304561i
\(254\) 3.38083e33 0.0455968
\(255\) 0 0
\(256\) −8.48339e34 −1.02115
\(257\) − 4.22826e33i − 0.0480985i −0.999711 0.0240493i \(-0.992344\pi\)
0.999711 0.0240493i \(-0.00765586\pi\)
\(258\) 2.68806e34i 0.289037i
\(259\) 5.90524e34 0.600331
\(260\) 0 0
\(261\) −1.94636e34 −0.176985
\(262\) − 4.04865e34i − 0.348290i
\(263\) − 3.28330e34i − 0.267271i −0.991031 0.133635i \(-0.957335\pi\)
0.991031 0.133635i \(-0.0426651\pi\)
\(264\) 1.60815e35 1.23899
\(265\) 0 0
\(266\) 1.93806e35 1.33840
\(267\) − 6.17164e34i − 0.403633i
\(268\) − 3.01045e34i − 0.186499i
\(269\) −2.79966e35 −1.64322 −0.821612 0.570048i \(-0.806925\pi\)
−0.821612 + 0.570048i \(0.806925\pi\)
\(270\) 0 0
\(271\) −3.70768e35 −1.95455 −0.977277 0.211965i \(-0.932014\pi\)
−0.977277 + 0.211965i \(0.932014\pi\)
\(272\) 9.02901e33i 0.0451223i
\(273\) − 1.73224e35i − 0.820823i
\(274\) 2.21394e35 0.994907
\(275\) 0 0
\(276\) 3.76065e34 0.152083
\(277\) − 3.33125e35i − 1.27835i −0.769060 0.639176i \(-0.779275\pi\)
0.769060 0.639176i \(-0.220725\pi\)
\(278\) − 1.99700e35i − 0.727324i
\(279\) −8.19470e34 −0.283317
\(280\) 0 0
\(281\) −5.19345e35 −1.61888 −0.809440 0.587203i \(-0.800229\pi\)
−0.809440 + 0.587203i \(0.800229\pi\)
\(282\) − 4.34618e35i − 1.28676i
\(283\) − 1.04587e34i − 0.0294154i −0.999892 0.0147077i \(-0.995318\pi\)
0.999892 0.0147077i \(-0.00468177\pi\)
\(284\) −8.47100e34 −0.226370
\(285\) 0 0
\(286\) 1.33585e35 0.322440
\(287\) − 3.00526e35i − 0.689595i
\(288\) − 1.17175e35i − 0.255650i
\(289\) 4.75097e35 0.985743
\(290\) 0 0
\(291\) 9.06744e35 1.70230
\(292\) 3.38341e35i 0.604371i
\(293\) − 3.74269e35i − 0.636214i −0.948055 0.318107i \(-0.896953\pi\)
0.948055 0.318107i \(-0.103047\pi\)
\(294\) 9.46125e35 1.53078
\(295\) 0 0
\(296\) 2.65387e35 0.389180
\(297\) − 5.57766e35i − 0.778904i
\(298\) 1.00119e36i 1.33163i
\(299\) 1.03240e35 0.130802
\(300\) 0 0
\(301\) 4.83437e35 0.556065
\(302\) 8.85597e35i 0.970813i
\(303\) 4.43702e35i 0.463632i
\(304\) 4.05465e35 0.403912
\(305\) 0 0
\(306\) −3.19135e34 −0.0289077
\(307\) 5.37563e35i 0.464433i 0.972664 + 0.232217i \(0.0745978\pi\)
−0.972664 + 0.232217i \(0.925402\pi\)
\(308\) − 8.75133e35i − 0.721255i
\(309\) 1.27900e36 1.00571
\(310\) 0 0
\(311\) −4.11459e35 −0.294646 −0.147323 0.989088i \(-0.547066\pi\)
−0.147323 + 0.989088i \(0.547066\pi\)
\(312\) − 7.78486e35i − 0.532119i
\(313\) − 1.66582e36i − 1.08702i −0.839404 0.543508i \(-0.817096\pi\)
0.839404 0.543508i \(-0.182904\pi\)
\(314\) −1.62375e35 −0.101167
\(315\) 0 0
\(316\) 9.17065e35 0.521120
\(317\) − 2.49069e36i − 1.35195i −0.736925 0.675975i \(-0.763723\pi\)
0.736925 0.675975i \(-0.236277\pi\)
\(318\) 2.40528e36i 1.24730i
\(319\) −1.10890e36 −0.549449
\(320\) 0 0
\(321\) −4.46726e36 −2.02171
\(322\) 8.82527e35i 0.381784i
\(323\) 3.08583e35i 0.127625i
\(324\) −1.33656e36 −0.528549
\(325\) 0 0
\(326\) 7.76421e35 0.280829
\(327\) 5.08471e36i 1.75923i
\(328\) − 1.35059e36i − 0.447047i
\(329\) −7.81644e36 −2.47553
\(330\) 0 0
\(331\) 1.70323e36 0.494046 0.247023 0.969010i \(-0.420548\pi\)
0.247023 + 0.969010i \(0.420548\pi\)
\(332\) 1.54006e36i 0.427599i
\(333\) 4.36674e35i 0.116069i
\(334\) 3.04410e36 0.774703
\(335\) 0 0
\(336\) 3.09796e36 0.723030
\(337\) − 1.36026e36i − 0.304080i −0.988374 0.152040i \(-0.951416\pi\)
0.988374 0.152040i \(-0.0485843\pi\)
\(338\) 2.86693e36i 0.613939i
\(339\) −7.57600e36 −1.55433
\(340\) 0 0
\(341\) −4.66876e36 −0.879555
\(342\) 1.43314e36i 0.258767i
\(343\) − 7.40016e36i − 1.28078i
\(344\) 2.17261e36 0.360483
\(345\) 0 0
\(346\) −8.28738e35 −0.126419
\(347\) 6.83587e36i 1.00004i 0.866015 + 0.500019i \(0.166674\pi\)
−0.866015 + 0.500019i \(0.833326\pi\)
\(348\) 1.95539e36i 0.274369i
\(349\) −4.60521e36 −0.619842 −0.309921 0.950762i \(-0.600303\pi\)
−0.309921 + 0.950762i \(0.600303\pi\)
\(350\) 0 0
\(351\) −2.70008e36 −0.334521
\(352\) − 6.67582e36i − 0.793663i
\(353\) 1.14489e37i 1.30627i 0.757243 + 0.653134i \(0.226546\pi\)
−0.757243 + 0.653134i \(0.773454\pi\)
\(354\) 9.74899e35 0.106761
\(355\) 0 0
\(356\) −1.50935e36 −0.152323
\(357\) 2.35773e36i 0.228457i
\(358\) 1.15098e37i 1.07094i
\(359\) 1.33540e37 1.19328 0.596638 0.802510i \(-0.296503\pi\)
0.596638 + 0.802510i \(0.296503\pi\)
\(360\) 0 0
\(361\) 1.72772e36 0.142436
\(362\) 9.31511e36i 0.737758i
\(363\) − 3.30447e34i − 0.00251453i
\(364\) −4.23642e36 −0.309762
\(365\) 0 0
\(366\) 1.02732e37 0.693790
\(367\) 1.51022e37i 0.980353i 0.871623 + 0.490177i \(0.163068\pi\)
−0.871623 + 0.490177i \(0.836932\pi\)
\(368\) 1.84635e36i 0.115218i
\(369\) 2.22229e36 0.133327
\(370\) 0 0
\(371\) 4.32580e37 2.39963
\(372\) 8.23271e36i 0.439208i
\(373\) − 1.09297e36i − 0.0560828i −0.999607 0.0280414i \(-0.991073\pi\)
0.999607 0.0280414i \(-0.00892702\pi\)
\(374\) −1.81820e36 −0.0897437
\(375\) 0 0
\(376\) −3.51279e37 −1.60483
\(377\) 5.36806e36i 0.235976i
\(378\) − 2.30812e37i − 0.976400i
\(379\) −1.45750e37 −0.593392 −0.296696 0.954972i \(-0.595885\pi\)
−0.296696 + 0.954972i \(0.595885\pi\)
\(380\) 0 0
\(381\) 1.84699e36 0.0696709
\(382\) − 1.67148e37i − 0.606992i
\(383\) − 5.82118e36i − 0.203530i −0.994808 0.101765i \(-0.967551\pi\)
0.994808 0.101765i \(-0.0324490\pi\)
\(384\) −2.06210e36 −0.0694235
\(385\) 0 0
\(386\) −1.36278e36 −0.0425510
\(387\) 3.57486e36i 0.107510i
\(388\) − 2.21756e37i − 0.642414i
\(389\) 6.42121e36 0.179204 0.0896020 0.995978i \(-0.471440\pi\)
0.0896020 + 0.995978i \(0.471440\pi\)
\(390\) 0 0
\(391\) −1.40518e36 −0.0364057
\(392\) − 7.64702e37i − 1.90917i
\(393\) − 2.21182e37i − 0.532180i
\(394\) −9.76983e36 −0.226564
\(395\) 0 0
\(396\) 6.47133e36 0.139448
\(397\) − 6.76912e37i − 1.40627i −0.711054 0.703137i \(-0.751782\pi\)
0.711054 0.703137i \(-0.248218\pi\)
\(398\) − 1.55431e37i − 0.311338i
\(399\) 1.05879e38 2.04504
\(400\) 0 0
\(401\) −8.28993e37 −1.48922 −0.744608 0.667502i \(-0.767364\pi\)
−0.744608 + 0.667502i \(0.767364\pi\)
\(402\) 2.14603e37i 0.371841i
\(403\) 2.26009e37i 0.377748i
\(404\) 1.08513e37 0.174965
\(405\) 0 0
\(406\) −4.58880e37 −0.688766
\(407\) 2.48786e37i 0.360335i
\(408\) 1.05959e37i 0.148103i
\(409\) −2.19435e37 −0.296018 −0.148009 0.988986i \(-0.547286\pi\)
−0.148009 + 0.988986i \(0.547286\pi\)
\(410\) 0 0
\(411\) 1.20950e38 1.52019
\(412\) − 3.12796e37i − 0.379535i
\(413\) − 1.75332e37i − 0.205393i
\(414\) −6.52601e36 −0.0738147
\(415\) 0 0
\(416\) −3.23169e37 −0.340860
\(417\) − 1.09098e38i − 1.11133i
\(418\) 8.16500e37i 0.803341i
\(419\) 1.94941e37 0.185268 0.0926338 0.995700i \(-0.470471\pi\)
0.0926338 + 0.995700i \(0.470471\pi\)
\(420\) 0 0
\(421\) −3.14435e37 −0.278895 −0.139447 0.990229i \(-0.544533\pi\)
−0.139447 + 0.990229i \(0.544533\pi\)
\(422\) 1.87667e37i 0.160827i
\(423\) − 5.78001e37i − 0.478623i
\(424\) 1.94406e38 1.55562
\(425\) 0 0
\(426\) 6.03863e37 0.451336
\(427\) − 1.84759e38i − 1.33475i
\(428\) 1.09253e38i 0.762951i
\(429\) 7.29787e37 0.492680
\(430\) 0 0
\(431\) −2.71956e38 −1.71624 −0.858120 0.513449i \(-0.828367\pi\)
−0.858120 + 0.513449i \(0.828367\pi\)
\(432\) − 4.82886e37i − 0.294666i
\(433\) 1.20292e38i 0.709842i 0.934896 + 0.354921i \(0.115492\pi\)
−0.934896 + 0.354921i \(0.884508\pi\)
\(434\) −1.93200e38 −1.10257
\(435\) 0 0
\(436\) 1.24353e38 0.663900
\(437\) 6.31025e37i 0.325886i
\(438\) − 2.41190e38i − 1.20499i
\(439\) 6.48441e37 0.313426 0.156713 0.987644i \(-0.449910\pi\)
0.156713 + 0.987644i \(0.449910\pi\)
\(440\) 0 0
\(441\) 1.25826e38 0.569390
\(442\) 8.80171e36i 0.0385429i
\(443\) − 1.25901e38i − 0.533550i −0.963759 0.266775i \(-0.914042\pi\)
0.963759 0.266775i \(-0.0859580\pi\)
\(444\) 4.38699e37 0.179934
\(445\) 0 0
\(446\) 6.87022e37 0.264007
\(447\) 5.46963e38i 2.03469i
\(448\) − 4.50883e38i − 1.62380i
\(449\) −4.32701e38 −1.50874 −0.754372 0.656447i \(-0.772059\pi\)
−0.754372 + 0.656447i \(0.772059\pi\)
\(450\) 0 0
\(451\) 1.26610e38 0.413914
\(452\) 1.85281e38i 0.586575i
\(453\) 4.83812e38i 1.48338i
\(454\) −4.54207e38 −1.34879
\(455\) 0 0
\(456\) 4.75829e38 1.32575
\(457\) 5.34446e38i 1.44251i 0.692670 + 0.721255i \(0.256434\pi\)
−0.692670 + 0.721255i \(0.743566\pi\)
\(458\) − 1.78661e38i − 0.467176i
\(459\) 3.67505e37 0.0931063
\(460\) 0 0
\(461\) −2.70831e38 −0.644220 −0.322110 0.946702i \(-0.604392\pi\)
−0.322110 + 0.946702i \(0.604392\pi\)
\(462\) 6.23847e38i 1.43804i
\(463\) 5.36275e38i 1.19802i 0.800742 + 0.599009i \(0.204439\pi\)
−0.800742 + 0.599009i \(0.795561\pi\)
\(464\) −9.60030e37 −0.207862
\(465\) 0 0
\(466\) −3.39218e38 −0.690054
\(467\) − 9.88390e38i − 1.94910i −0.224177 0.974549i \(-0.571969\pi\)
0.224177 0.974549i \(-0.428031\pi\)
\(468\) − 3.13270e37i − 0.0598899i
\(469\) 3.85954e38 0.715368
\(470\) 0 0
\(471\) −8.87071e37 −0.154581
\(472\) − 7.87958e37i − 0.133151i
\(473\) 2.03670e38i 0.333765i
\(474\) −6.53739e38 −1.03901
\(475\) 0 0
\(476\) 5.76614e37 0.0862152
\(477\) 3.19879e38i 0.463948i
\(478\) 7.95483e38i 1.11925i
\(479\) 1.66145e38 0.226789 0.113395 0.993550i \(-0.463828\pi\)
0.113395 + 0.993550i \(0.463828\pi\)
\(480\) 0 0
\(481\) 1.20434e38 0.154756
\(482\) 5.06963e38i 0.632112i
\(483\) 4.82135e38i 0.583358i
\(484\) −8.08152e35 −0.000948932 0
\(485\) 0 0
\(486\) 4.22328e38 0.467115
\(487\) 5.21209e38i 0.559554i 0.960065 + 0.279777i \(0.0902605\pi\)
−0.960065 + 0.279777i \(0.909740\pi\)
\(488\) − 8.30323e38i − 0.865286i
\(489\) 4.24167e38 0.429100
\(490\) 0 0
\(491\) −6.08070e38 −0.579791 −0.289896 0.957058i \(-0.593621\pi\)
−0.289896 + 0.957058i \(0.593621\pi\)
\(492\) − 2.23260e38i − 0.206689i
\(493\) − 7.30640e37i − 0.0656785i
\(494\) 3.95258e38 0.345017
\(495\) 0 0
\(496\) −4.04197e38 −0.332743
\(497\) − 1.08602e39i − 0.868305i
\(498\) − 1.09785e39i − 0.852545i
\(499\) −1.92541e39 −1.45233 −0.726165 0.687521i \(-0.758699\pi\)
−0.726165 + 0.687521i \(0.758699\pi\)
\(500\) 0 0
\(501\) 1.66303e39 1.18373
\(502\) − 3.00591e38i − 0.207861i
\(503\) 1.47205e39i 0.988981i 0.869183 + 0.494490i \(0.164645\pi\)
−0.869183 + 0.494490i \(0.835355\pi\)
\(504\) 8.85021e38 0.577711
\(505\) 0 0
\(506\) −3.71806e38 −0.229157
\(507\) 1.56624e39i 0.938084i
\(508\) − 4.51704e37i − 0.0262924i
\(509\) −9.86933e37 −0.0558316 −0.0279158 0.999610i \(-0.508887\pi\)
−0.0279158 + 0.999610i \(0.508887\pi\)
\(510\) 0 0
\(511\) −4.33771e39 −2.31823
\(512\) − 1.36275e39i − 0.707948i
\(513\) − 1.65035e39i − 0.833441i
\(514\) 7.37152e37 0.0361903
\(515\) 0 0
\(516\) 3.59145e38 0.166667
\(517\) − 3.29304e39i − 1.48588i
\(518\) 1.02951e39i 0.451701i
\(519\) −4.52749e38 −0.193166
\(520\) 0 0
\(521\) 1.36208e39 0.549613 0.274806 0.961500i \(-0.411386\pi\)
0.274806 + 0.961500i \(0.411386\pi\)
\(522\) − 3.39327e38i − 0.133167i
\(523\) − 3.92619e39i − 1.49864i −0.662208 0.749320i \(-0.730381\pi\)
0.662208 0.749320i \(-0.269619\pi\)
\(524\) −5.40930e38 −0.200834
\(525\) 0 0
\(526\) 5.72407e38 0.201100
\(527\) − 3.07618e38i − 0.105138i
\(528\) 1.30516e39i 0.433982i
\(529\) 2.80371e39 0.907039
\(530\) 0 0
\(531\) 1.29652e38 0.0397109
\(532\) − 2.58940e39i − 0.771756i
\(533\) − 6.12906e38i − 0.177766i
\(534\) 1.07596e39 0.303701
\(535\) 0 0
\(536\) 1.73452e39 0.463756
\(537\) 6.28794e39i 1.63637i
\(538\) − 4.88089e39i − 1.23639i
\(539\) 7.16866e39 1.76767
\(540\) 0 0
\(541\) 3.64331e39 0.851405 0.425702 0.904863i \(-0.360027\pi\)
0.425702 + 0.904863i \(0.360027\pi\)
\(542\) − 6.46394e39i − 1.47064i
\(543\) 5.08895e39i 1.12728i
\(544\) 4.39861e38 0.0948706
\(545\) 0 0
\(546\) 3.01997e39 0.617603
\(547\) 9.64716e38i 0.192125i 0.995375 + 0.0960624i \(0.0306248\pi\)
−0.995375 + 0.0960624i \(0.969375\pi\)
\(548\) − 2.95799e39i − 0.573691i
\(549\) 1.36623e39 0.258063
\(550\) 0 0
\(551\) −3.28108e39 −0.587921
\(552\) 2.16676e39i 0.378176i
\(553\) 1.17572e40i 1.99890i
\(554\) 5.80767e39 0.961857
\(555\) 0 0
\(556\) −2.66814e39 −0.419395
\(557\) 4.34857e39i 0.665956i 0.942935 + 0.332978i \(0.108054\pi\)
−0.942935 + 0.332978i \(0.891946\pi\)
\(558\) − 1.42866e39i − 0.213173i
\(559\) 9.85944e38 0.143345
\(560\) 0 0
\(561\) −9.93305e38 −0.137126
\(562\) − 9.05422e39i − 1.21808i
\(563\) − 5.22292e39i − 0.684765i −0.939561 0.342382i \(-0.888766\pi\)
0.939561 0.342382i \(-0.111234\pi\)
\(564\) −5.80682e39 −0.741979
\(565\) 0 0
\(566\) 1.82336e38 0.0221327
\(567\) − 1.71353e40i − 2.02740i
\(568\) − 4.88070e39i − 0.562900i
\(569\) −8.52031e39 −0.957916 −0.478958 0.877838i \(-0.658985\pi\)
−0.478958 + 0.877838i \(0.658985\pi\)
\(570\) 0 0
\(571\) 8.06636e38 0.0861894 0.0430947 0.999071i \(-0.486278\pi\)
0.0430947 + 0.999071i \(0.486278\pi\)
\(572\) − 1.78479e39i − 0.185928i
\(573\) − 9.13149e39i − 0.927470i
\(574\) 5.23933e39 0.518864
\(575\) 0 0
\(576\) 3.33413e39 0.313948
\(577\) 1.60079e40i 1.46989i 0.678125 + 0.734947i \(0.262793\pi\)
−0.678125 + 0.734947i \(0.737207\pi\)
\(578\) 8.28280e39i 0.741691i
\(579\) −7.44501e38 −0.0650169
\(580\) 0 0
\(581\) −1.97444e40 −1.64017
\(582\) 1.58081e40i 1.28084i
\(583\) 1.82244e40i 1.44032i
\(584\) −1.94941e40 −1.50285
\(585\) 0 0
\(586\) 6.52496e39 0.478699
\(587\) 7.25470e39i 0.519239i 0.965711 + 0.259619i \(0.0835971\pi\)
−0.965711 + 0.259619i \(0.916403\pi\)
\(588\) − 1.26409e40i − 0.882689i
\(589\) −1.38142e40 −0.941140
\(590\) 0 0
\(591\) −5.33737e39 −0.346185
\(592\) 2.15386e39i 0.136318i
\(593\) 4.67095e39i 0.288478i 0.989543 + 0.144239i \(0.0460734\pi\)
−0.989543 + 0.144239i \(0.953927\pi\)
\(594\) 9.72404e39 0.586062
\(595\) 0 0
\(596\) 1.33767e40 0.767853
\(597\) − 8.49135e39i − 0.475717i
\(598\) 1.79987e39i 0.0984178i
\(599\) −3.02320e40 −1.61353 −0.806766 0.590871i \(-0.798784\pi\)
−0.806766 + 0.590871i \(0.798784\pi\)
\(600\) 0 0
\(601\) −1.75696e40 −0.893476 −0.446738 0.894665i \(-0.647414\pi\)
−0.446738 + 0.894665i \(0.647414\pi\)
\(602\) 8.42819e39i 0.418394i
\(603\) 2.85401e39i 0.138310i
\(604\) 1.18322e40 0.559798
\(605\) 0 0
\(606\) −7.73546e39 −0.348846
\(607\) 2.78325e40i 1.22551i 0.790273 + 0.612755i \(0.209939\pi\)
−0.790273 + 0.612755i \(0.790061\pi\)
\(608\) − 1.97528e40i − 0.849234i
\(609\) −2.50691e40 −1.05242
\(610\) 0 0
\(611\) −1.59412e40 −0.638152
\(612\) 4.26387e38i 0.0166690i
\(613\) 2.63728e40i 1.00688i 0.864030 + 0.503441i \(0.167933\pi\)
−0.864030 + 0.503441i \(0.832067\pi\)
\(614\) −9.37183e39 −0.349448
\(615\) 0 0
\(616\) 5.04222e40 1.79350
\(617\) − 9.21532e38i − 0.0320166i −0.999872 0.0160083i \(-0.994904\pi\)
0.999872 0.0160083i \(-0.00509582\pi\)
\(618\) 2.22980e40i 0.756716i
\(619\) 4.42763e40 1.46777 0.733883 0.679276i \(-0.237706\pi\)
0.733883 + 0.679276i \(0.237706\pi\)
\(620\) 0 0
\(621\) 7.51514e39 0.237744
\(622\) − 7.17333e39i − 0.221697i
\(623\) − 1.93507e40i − 0.584278i
\(624\) 6.31813e39 0.186385
\(625\) 0 0
\(626\) 2.90418e40 0.817891
\(627\) 4.46063e40i 1.22749i
\(628\) 2.16945e39i 0.0583356i
\(629\) −1.63922e39 −0.0430727
\(630\) 0 0
\(631\) 4.23643e40 1.06310 0.531551 0.847026i \(-0.321609\pi\)
0.531551 + 0.847026i \(0.321609\pi\)
\(632\) 5.28382e40i 1.29584i
\(633\) 1.02525e40i 0.245739i
\(634\) 4.34225e40 1.01723
\(635\) 0 0
\(636\) 3.21363e40 0.719229
\(637\) − 3.47026e40i − 0.759172i
\(638\) − 1.93325e40i − 0.413416i
\(639\) 8.03081e39 0.167879
\(640\) 0 0
\(641\) −4.13775e40 −0.826652 −0.413326 0.910583i \(-0.635633\pi\)
−0.413326 + 0.910583i \(0.635633\pi\)
\(642\) − 7.78818e40i − 1.52117i
\(643\) 9.84722e40i 1.88042i 0.340598 + 0.940209i \(0.389370\pi\)
−0.340598 + 0.940209i \(0.610630\pi\)
\(644\) 1.17912e40 0.220147
\(645\) 0 0
\(646\) −5.37981e39 −0.0960275
\(647\) 6.78001e39i 0.118336i 0.998248 + 0.0591682i \(0.0188448\pi\)
−0.998248 + 0.0591682i \(0.981155\pi\)
\(648\) − 7.70079e40i − 1.31431i
\(649\) 7.38667e39 0.123282
\(650\) 0 0
\(651\) −1.05548e41 −1.68470
\(652\) − 1.03736e40i − 0.161934i
\(653\) 3.23913e40i 0.494524i 0.968949 + 0.247262i \(0.0795308\pi\)
−0.968949 + 0.247262i \(0.920469\pi\)
\(654\) −8.86464e40 −1.32368
\(655\) 0 0
\(656\) 1.09613e40 0.156587
\(657\) − 3.20760e40i − 0.448211i
\(658\) − 1.36271e41i − 1.86264i
\(659\) 1.40926e40 0.188431 0.0942156 0.995552i \(-0.469966\pi\)
0.0942156 + 0.995552i \(0.469966\pi\)
\(660\) 0 0
\(661\) −9.56737e40 −1.22425 −0.612127 0.790759i \(-0.709686\pi\)
−0.612127 + 0.790759i \(0.709686\pi\)
\(662\) 2.96940e40i 0.371730i
\(663\) 4.80847e39i 0.0588926i
\(664\) −8.87333e40 −1.06328
\(665\) 0 0
\(666\) −7.61293e39 −0.0873324
\(667\) − 1.49409e40i − 0.167708i
\(668\) − 4.06714e40i − 0.446715i
\(669\) 3.75328e40 0.403397
\(670\) 0 0
\(671\) 7.78382e40 0.801154
\(672\) − 1.50922e41i − 1.52019i
\(673\) − 1.43434e40i − 0.141394i −0.997498 0.0706972i \(-0.977478\pi\)
0.997498 0.0706972i \(-0.0225224\pi\)
\(674\) 2.37146e40 0.228796
\(675\) 0 0
\(676\) 3.83043e40 0.354014
\(677\) − 8.89678e40i − 0.804817i −0.915460 0.402408i \(-0.868173\pi\)
0.915460 0.402408i \(-0.131827\pi\)
\(678\) − 1.32079e41i − 1.16951i
\(679\) 2.84302e41 2.46416
\(680\) 0 0
\(681\) −2.48138e41 −2.06092
\(682\) − 8.13947e40i − 0.661794i
\(683\) 1.49228e41i 1.18782i 0.804531 + 0.593910i \(0.202417\pi\)
−0.804531 + 0.593910i \(0.797583\pi\)
\(684\) 1.91478e40 0.149212
\(685\) 0 0
\(686\) 1.29014e41 0.963687
\(687\) − 9.76045e40i − 0.713834i
\(688\) 1.76328e40i 0.126266i
\(689\) 8.82224e40 0.618586
\(690\) 0 0
\(691\) 2.54015e41 1.70776 0.853880 0.520469i \(-0.174243\pi\)
0.853880 + 0.520469i \(0.174243\pi\)
\(692\) 1.10725e40i 0.0728969i
\(693\) 8.29658e40i 0.534893i
\(694\) −1.19176e41 −0.752447
\(695\) 0 0
\(696\) −1.12663e41 −0.682257
\(697\) 8.34219e39i 0.0494772i
\(698\) − 8.02868e40i − 0.466381i
\(699\) −1.85318e41 −1.05439
\(700\) 0 0
\(701\) −1.56415e40 −0.0853824 −0.0426912 0.999088i \(-0.513593\pi\)
−0.0426912 + 0.999088i \(0.513593\pi\)
\(702\) − 4.70729e40i − 0.251700i
\(703\) 7.36123e40i 0.385565i
\(704\) 1.89955e41 0.974649
\(705\) 0 0
\(706\) −1.99600e41 −0.982860
\(707\) 1.39119e41i 0.671129i
\(708\) − 1.30254e40i − 0.0615613i
\(709\) 7.96882e40 0.368998 0.184499 0.982833i \(-0.440934\pi\)
0.184499 + 0.982833i \(0.440934\pi\)
\(710\) 0 0
\(711\) −8.69411e40 −0.386470
\(712\) − 8.69638e40i − 0.378773i
\(713\) − 6.29051e40i − 0.268465i
\(714\) −4.11045e40 −0.171896
\(715\) 0 0
\(716\) 1.53780e41 0.617533
\(717\) 4.34581e41i 1.71018i
\(718\) 2.32812e41i 0.897844i
\(719\) −6.70217e39 −0.0253306 −0.0126653 0.999920i \(-0.504032\pi\)
−0.0126653 + 0.999920i \(0.504032\pi\)
\(720\) 0 0
\(721\) 4.01021e41 1.45581
\(722\) 3.01209e40i 0.107171i
\(723\) 2.76960e41i 0.965853i
\(724\) 1.24457e41 0.425412
\(725\) 0 0
\(726\) 5.76099e38 0.00189198
\(727\) 3.08325e41i 0.992565i 0.868161 + 0.496283i \(0.165302\pi\)
−0.868161 + 0.496283i \(0.834698\pi\)
\(728\) − 2.44088e41i − 0.770267i
\(729\) −1.63080e41 −0.504490
\(730\) 0 0
\(731\) −1.34196e40 −0.0398967
\(732\) − 1.37257e41i − 0.400058i
\(733\) − 2.79480e41i − 0.798626i −0.916815 0.399313i \(-0.869249\pi\)
0.916815 0.399313i \(-0.130751\pi\)
\(734\) −2.63290e41 −0.737636
\(735\) 0 0
\(736\) 8.99476e40 0.242249
\(737\) 1.62601e41i 0.429384i
\(738\) 3.87432e40i 0.100318i
\(739\) 4.36349e41 1.10787 0.553936 0.832560i \(-0.313125\pi\)
0.553936 + 0.832560i \(0.313125\pi\)
\(740\) 0 0
\(741\) 2.15934e41 0.527177
\(742\) 7.54155e41i 1.80553i
\(743\) − 1.06242e41i − 0.249437i −0.992192 0.124718i \(-0.960197\pi\)
0.992192 0.124718i \(-0.0398027\pi\)
\(744\) −4.74341e41 −1.09215
\(745\) 0 0
\(746\) 1.90547e40 0.0421977
\(747\) − 1.46004e41i − 0.317114i
\(748\) 2.42925e40i 0.0517487i
\(749\) −1.40067e42 −2.92651
\(750\) 0 0
\(751\) −1.53315e41 −0.308181 −0.154091 0.988057i \(-0.549245\pi\)
−0.154091 + 0.988057i \(0.549245\pi\)
\(752\) − 2.85095e41i − 0.562122i
\(753\) − 1.64216e41i − 0.317606i
\(754\) −9.35862e40 −0.177553
\(755\) 0 0
\(756\) −3.08382e41 −0.563019
\(757\) 1.29063e41i 0.231160i 0.993298 + 0.115580i \(0.0368727\pi\)
−0.993298 + 0.115580i \(0.963127\pi\)
\(758\) − 2.54100e41i − 0.446479i
\(759\) −2.03122e41 −0.350147
\(760\) 0 0
\(761\) −1.01002e42 −1.67592 −0.837962 0.545728i \(-0.816253\pi\)
−0.837962 + 0.545728i \(0.816253\pi\)
\(762\) 3.22002e40i 0.0524217i
\(763\) 1.59427e42i 2.54657i
\(764\) −2.23322e41 −0.350008
\(765\) 0 0
\(766\) 1.01486e41 0.153140
\(767\) − 3.57580e40i − 0.0529469i
\(768\) − 8.07987e41i − 1.17400i
\(769\) 8.10252e41 1.15528 0.577641 0.816291i \(-0.303973\pi\)
0.577641 + 0.816291i \(0.303973\pi\)
\(770\) 0 0
\(771\) 4.02714e40 0.0552979
\(772\) 1.82077e40i 0.0245361i
\(773\) 3.13830e41i 0.415042i 0.978231 + 0.207521i \(0.0665395\pi\)
−0.978231 + 0.207521i \(0.933461\pi\)
\(774\) −6.23238e40 −0.0808929
\(775\) 0 0
\(776\) 1.27768e42 1.59745
\(777\) 5.62435e41i 0.690188i
\(778\) 1.11947e41i 0.134836i
\(779\) 3.74623e41 0.442895
\(780\) 0 0
\(781\) 4.57539e41 0.521180
\(782\) − 2.44978e40i − 0.0273923i
\(783\) 3.90758e41i 0.428906i
\(784\) 6.20626e41 0.668724
\(785\) 0 0
\(786\) 3.85607e41 0.400422
\(787\) − 8.07245e41i − 0.822947i −0.911422 0.411473i \(-0.865014\pi\)
0.911422 0.411473i \(-0.134986\pi\)
\(788\) 1.30532e41i 0.130643i
\(789\) 3.12712e41 0.307276
\(790\) 0 0
\(791\) −2.37540e42 −2.24997
\(792\) 3.72856e41i 0.346758i
\(793\) − 3.76805e41i − 0.344077i
\(794\) 1.18012e42 1.05811
\(795\) 0 0
\(796\) −2.07667e41 −0.179526
\(797\) − 2.35976e41i − 0.200319i −0.994971 0.100160i \(-0.968065\pi\)
0.994971 0.100160i \(-0.0319353\pi\)
\(798\) 1.84588e42i 1.53872i
\(799\) 2.16974e41 0.177615
\(800\) 0 0
\(801\) 1.43092e41 0.112965
\(802\) − 1.44526e42i − 1.12051i
\(803\) − 1.82746e42i − 1.39147i
\(804\) 2.86725e41 0.214414
\(805\) 0 0
\(806\) −3.94022e41 −0.284225
\(807\) − 2.66649e42i − 1.88918i
\(808\) 6.25216e41i 0.435076i
\(809\) −2.25512e42 −1.54140 −0.770700 0.637198i \(-0.780093\pi\)
−0.770700 + 0.637198i \(0.780093\pi\)
\(810\) 0 0
\(811\) 8.17269e41 0.538967 0.269484 0.963005i \(-0.413147\pi\)
0.269484 + 0.963005i \(0.413147\pi\)
\(812\) 6.13098e41i 0.397162i
\(813\) − 3.53132e42i − 2.24711i
\(814\) −4.33731e41 −0.271123
\(815\) 0 0
\(816\) −8.59953e40 −0.0518761
\(817\) 6.02632e41i 0.357135i
\(818\) − 3.82560e41i − 0.222729i
\(819\) 4.01627e41 0.229724
\(820\) 0 0
\(821\) 8.86919e39 0.00489675 0.00244838 0.999997i \(-0.499221\pi\)
0.00244838 + 0.999997i \(0.499221\pi\)
\(822\) 2.10863e42i 1.14382i
\(823\) 4.50695e41i 0.240207i 0.992761 + 0.120103i \(0.0383226\pi\)
−0.992761 + 0.120103i \(0.961677\pi\)
\(824\) 1.80223e42 0.943767
\(825\) 0 0
\(826\) 3.05672e41 0.154541
\(827\) − 1.64298e42i − 0.816211i −0.912935 0.408106i \(-0.866190\pi\)
0.912935 0.408106i \(-0.133810\pi\)
\(828\) 8.71923e40i 0.0425636i
\(829\) −9.28693e41 −0.445483 −0.222742 0.974878i \(-0.571501\pi\)
−0.222742 + 0.974878i \(0.571501\pi\)
\(830\) 0 0
\(831\) 3.17280e42 1.46970
\(832\) − 9.19552e41i − 0.418589i
\(833\) 4.72333e41i 0.211298i
\(834\) 1.90201e42 0.836189
\(835\) 0 0
\(836\) 1.09090e42 0.463229
\(837\) 1.64519e42i 0.686590i
\(838\) 3.39858e41i 0.139399i
\(839\) 1.59653e42 0.643620 0.321810 0.946804i \(-0.395709\pi\)
0.321810 + 0.946804i \(0.395709\pi\)
\(840\) 0 0
\(841\) −1.79082e42 −0.697444
\(842\) − 5.48183e41i − 0.209846i
\(843\) − 4.94642e42i − 1.86119i
\(844\) 2.50737e41 0.0927371
\(845\) 0 0
\(846\) 1.00768e42 0.360125
\(847\) − 1.03609e40i − 0.00363989i
\(848\) 1.57778e42i 0.544887i
\(849\) 9.96121e40 0.0338182
\(850\) 0 0
\(851\) −3.35205e41 −0.109985
\(852\) − 8.06806e41i − 0.260253i
\(853\) − 2.50802e42i − 0.795373i −0.917521 0.397686i \(-0.869813\pi\)
0.917521 0.397686i \(-0.130187\pi\)
\(854\) 3.22106e42 1.00429
\(855\) 0 0
\(856\) −6.29476e42 −1.89719
\(857\) − 4.74569e42i − 1.40630i −0.711043 0.703149i \(-0.751777\pi\)
0.711043 0.703149i \(-0.248223\pi\)
\(858\) 1.27230e42i 0.370702i
\(859\) 1.19413e42 0.342098 0.171049 0.985263i \(-0.445284\pi\)
0.171049 + 0.985263i \(0.445284\pi\)
\(860\) 0 0
\(861\) 2.86231e42 0.792812
\(862\) − 4.74125e42i − 1.29133i
\(863\) 5.51388e42i 1.47673i 0.674402 + 0.738364i \(0.264401\pi\)
−0.674402 + 0.738364i \(0.735599\pi\)
\(864\) −2.35245e42 −0.619542
\(865\) 0 0
\(866\) −2.09716e42 −0.534099
\(867\) 4.52498e42i 1.13329i
\(868\) 2.58130e42i 0.635774i
\(869\) −4.95329e42 −1.19979
\(870\) 0 0
\(871\) 7.87134e41 0.184410
\(872\) 7.16481e42i 1.65088i
\(873\) 2.10233e42i 0.476424i
\(874\) −1.10012e42 −0.245203
\(875\) 0 0
\(876\) −3.22248e42 −0.694833
\(877\) − 3.25016e42i − 0.689304i −0.938731 0.344652i \(-0.887997\pi\)
0.938731 0.344652i \(-0.112003\pi\)
\(878\) 1.13049e42i 0.235828i
\(879\) 3.56466e42 0.731441
\(880\) 0 0
\(881\) 1.07364e42 0.213162 0.106581 0.994304i \(-0.466010\pi\)
0.106581 + 0.994304i \(0.466010\pi\)
\(882\) 2.19363e42i 0.428420i
\(883\) 4.03526e42i 0.775250i 0.921817 + 0.387625i \(0.126704\pi\)
−0.921817 + 0.387625i \(0.873296\pi\)
\(884\) 1.17597e41 0.0222249
\(885\) 0 0
\(886\) 2.19495e42 0.401453
\(887\) 1.14783e42i 0.206532i 0.994654 + 0.103266i \(0.0329292\pi\)
−0.994654 + 0.103266i \(0.967071\pi\)
\(888\) 2.52764e42i 0.447432i
\(889\) 5.79108e41 0.100852
\(890\) 0 0
\(891\) 7.21906e42 1.21690
\(892\) − 9.17913e41i − 0.152234i
\(893\) − 9.74365e42i − 1.58992i
\(894\) −9.53571e42 −1.53094
\(895\) 0 0
\(896\) −6.46554e41 −0.100494
\(897\) 9.83288e41i 0.150380i
\(898\) − 7.54366e42i − 1.13521i
\(899\) 3.27082e42 0.484330
\(900\) 0 0
\(901\) −1.20078e42 −0.172169
\(902\) 2.20731e42i 0.311436i
\(903\) 4.60442e42i 0.639296i
\(904\) −1.06753e43 −1.45860
\(905\) 0 0
\(906\) −8.43472e42 −1.11612
\(907\) 1.34894e43i 1.75666i 0.478058 + 0.878328i \(0.341341\pi\)
−0.478058 + 0.878328i \(0.658659\pi\)
\(908\) 6.06854e42i 0.777749i
\(909\) −1.02874e42 −0.129757
\(910\) 0 0
\(911\) 3.01049e42 0.367807 0.183904 0.982944i \(-0.441127\pi\)
0.183904 + 0.982944i \(0.441127\pi\)
\(912\) 3.86179e42i 0.464369i
\(913\) − 8.31825e42i − 0.984477i
\(914\) −9.31747e42 −1.08537
\(915\) 0 0
\(916\) −2.38704e42 −0.269386
\(917\) − 6.93499e42i − 0.770355i
\(918\) 6.40704e41i 0.0700549i
\(919\) −6.28761e42 −0.676723 −0.338361 0.941016i \(-0.609873\pi\)
−0.338361 + 0.941016i \(0.609873\pi\)
\(920\) 0 0
\(921\) −5.11993e42 −0.533949
\(922\) − 4.72163e42i − 0.484723i
\(923\) − 2.21489e42i − 0.223835i
\(924\) 8.33506e42 0.829211
\(925\) 0 0
\(926\) −9.34937e42 −0.901412
\(927\) 2.96542e42i 0.281469i
\(928\) 4.67692e42i 0.437034i
\(929\) −1.66684e43 −1.53344 −0.766720 0.641982i \(-0.778112\pi\)
−0.766720 + 0.641982i \(0.778112\pi\)
\(930\) 0 0
\(931\) 2.12111e43 1.89144
\(932\) 4.53220e42i 0.397904i
\(933\) − 3.91887e42i − 0.338748i
\(934\) 1.72315e43 1.46654
\(935\) 0 0
\(936\) 1.80495e42 0.148925
\(937\) 1.87621e43i 1.52426i 0.647425 + 0.762129i \(0.275846\pi\)
−0.647425 + 0.762129i \(0.724154\pi\)
\(938\) 6.72869e42i 0.538257i
\(939\) 1.58659e43 1.24972
\(940\) 0 0
\(941\) 5.14783e42 0.393165 0.196582 0.980487i \(-0.437016\pi\)
0.196582 + 0.980487i \(0.437016\pi\)
\(942\) − 1.54651e42i − 0.116309i
\(943\) 1.70590e42i 0.126338i
\(944\) 6.39501e41 0.0466388
\(945\) 0 0
\(946\) −3.55077e42 −0.251131
\(947\) 2.96240e42i 0.206333i 0.994664 + 0.103167i \(0.0328975\pi\)
−0.994664 + 0.103167i \(0.967103\pi\)
\(948\) 8.73443e42i 0.599120i
\(949\) −8.84653e42 −0.597603
\(950\) 0 0
\(951\) 2.37222e43 1.55431
\(952\) 3.32225e42i 0.214386i
\(953\) 1.30942e43i 0.832208i 0.909317 + 0.416104i \(0.136605\pi\)
−0.909317 + 0.416104i \(0.863395\pi\)
\(954\) −5.57674e42 −0.349083
\(955\) 0 0
\(956\) 1.06282e43 0.645390
\(957\) − 1.05615e43i − 0.631690i
\(958\) 2.89656e42i 0.170640i
\(959\) 3.79229e43 2.20055
\(960\) 0 0
\(961\) −3.99088e42 −0.224688
\(962\) 2.09964e42i 0.116441i
\(963\) − 1.03575e43i − 0.565816i
\(964\) 6.77340e42 0.364493
\(965\) 0 0
\(966\) −8.40549e42 −0.438929
\(967\) − 2.64169e43i − 1.35894i −0.733706 0.679468i \(-0.762211\pi\)
0.733706 0.679468i \(-0.237789\pi\)
\(968\) − 4.65630e40i − 0.00235965i
\(969\) −2.93905e42 −0.146728
\(970\) 0 0
\(971\) −3.41602e43 −1.65517 −0.827583 0.561343i \(-0.810285\pi\)
−0.827583 + 0.561343i \(0.810285\pi\)
\(972\) − 5.64261e42i − 0.269351i
\(973\) − 3.42069e43i − 1.60871i
\(974\) −9.08670e42 −0.421019
\(975\) 0 0
\(976\) 6.73884e42 0.303084
\(977\) − 2.06179e43i − 0.913636i −0.889560 0.456818i \(-0.848989\pi\)
0.889560 0.456818i \(-0.151011\pi\)
\(978\) 7.39489e42i 0.322863i
\(979\) 8.15238e42 0.350700
\(980\) 0 0
\(981\) −1.17891e43 −0.492358
\(982\) − 1.06010e43i − 0.436246i
\(983\) − 1.21841e43i − 0.494047i −0.969009 0.247023i \(-0.920548\pi\)
0.969009 0.247023i \(-0.0794525\pi\)
\(984\) 1.28635e43 0.513961
\(985\) 0 0
\(986\) 1.27379e42 0.0494177
\(987\) − 7.44464e43i − 2.84607i
\(988\) − 5.28094e42i − 0.198946i
\(989\) −2.74418e42 −0.101875
\(990\) 0 0
\(991\) −1.94686e43 −0.701888 −0.350944 0.936397i \(-0.614139\pi\)
−0.350944 + 0.936397i \(0.614139\pi\)
\(992\) 1.96911e43i 0.699600i
\(993\) 1.62221e43i 0.567994i
\(994\) 1.89337e43 0.653329
\(995\) 0 0
\(996\) −1.46681e43 −0.491601
\(997\) − 2.12798e43i − 0.702891i −0.936208 0.351446i \(-0.885690\pi\)
0.936208 0.351446i \(-0.114310\pi\)
\(998\) − 3.35674e43i − 1.09276i
\(999\) 8.76679e42 0.281282
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 25.30.b.a.24.3 4
5.2 odd 4 1.30.a.a.1.1 2
5.3 odd 4 25.30.a.a.1.2 2
5.4 even 2 inner 25.30.b.a.24.2 4
15.2 even 4 9.30.a.a.1.2 2
20.7 even 4 16.30.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.30.a.a.1.1 2 5.2 odd 4
9.30.a.a.1.2 2 15.2 even 4
16.30.a.c.1.1 2 20.7 even 4
25.30.a.a.1.2 2 5.3 odd 4
25.30.b.a.24.2 4 5.4 even 2 inner
25.30.b.a.24.3 4 1.1 even 1 trivial