Properties

Label 75.22.a.f.1.1
Level $75$
Weight $22$
Character 75.1
Self dual yes
Analytic conductor $209.608$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,22,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(209.608008215\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 157936x - 9799664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{4}\cdot 5 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(425.878\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2787.27 q^{2} +59049.0 q^{3} +5.67172e6 q^{4} -1.64586e8 q^{6} +1.37938e9 q^{7} -9.96329e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\) \(q-2787.27 q^{2} +59049.0 q^{3} +5.67172e6 q^{4} -1.64586e8 q^{6} +1.37938e9 q^{7} -9.96329e9 q^{8} +3.48678e9 q^{9} +2.99797e10 q^{11} +3.34910e11 q^{12} +7.63100e11 q^{13} -3.84470e12 q^{14} +1.58759e13 q^{16} +1.14939e13 q^{17} -9.71861e12 q^{18} -5.19898e12 q^{19} +8.14510e13 q^{21} -8.35614e13 q^{22} +8.03048e13 q^{23} -5.88322e14 q^{24} -2.12696e15 q^{26} +2.05891e14 q^{27} +7.82346e15 q^{28} -2.01015e14 q^{29} +1.53535e15 q^{31} -2.33559e16 q^{32} +1.77027e15 q^{33} -3.20366e16 q^{34} +1.97761e16 q^{36} -1.53969e16 q^{37} +1.44910e16 q^{38} +4.50603e16 q^{39} -7.88656e16 q^{41} -2.27026e17 q^{42} +1.61914e17 q^{43} +1.70036e17 q^{44} -2.23831e17 q^{46} +3.76922e17 q^{47} +9.37457e17 q^{48} +1.34414e18 q^{49} +6.78704e17 q^{51} +4.32809e18 q^{52} -3.24880e17 q^{53} -5.73874e17 q^{54} -1.37432e19 q^{56} -3.06994e17 q^{57} +5.60284e17 q^{58} -9.88461e17 q^{59} +1.01280e19 q^{61} -4.27945e18 q^{62} +4.80960e18 q^{63} +3.18051e19 q^{64} -4.93422e18 q^{66} +1.02596e19 q^{67} +6.51902e19 q^{68} +4.74192e18 q^{69} +3.23535e19 q^{71} -3.47399e19 q^{72} +2.89708e19 q^{73} +4.29154e19 q^{74} -2.94872e19 q^{76} +4.13533e19 q^{77} -1.25595e20 q^{78} +3.51920e19 q^{79} +1.21577e19 q^{81} +2.19820e20 q^{82} -2.86096e19 q^{83} +4.61967e20 q^{84} -4.51299e20 q^{86} -1.18698e19 q^{87} -2.98696e20 q^{88} +5.20282e19 q^{89} +1.05260e21 q^{91} +4.55467e20 q^{92} +9.06611e19 q^{93} -1.05058e21 q^{94} -1.37915e21 q^{96} -7.90419e20 q^{97} -3.74649e21 q^{98} +1.04533e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 702 q^{2} + 177147 q^{3} + 5244228 q^{4} - 41452398 q^{6} + 2072418204 q^{7} - 11461067832 q^{8} + 10460353203 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 702 q^{2} + 177147 q^{3} + 5244228 q^{4} - 41452398 q^{6} + 2072418204 q^{7} - 11461067832 q^{8} + 10460353203 q^{9} - 8467600440 q^{11} + 309666419172 q^{12} + 469447548570 q^{13} - 2389270234872 q^{14} + 14993057845776 q^{16} + 6865096035486 q^{17} - 2447722649502 q^{18} + 33215593555044 q^{19} + 122374222527996 q^{21} - 301288671211728 q^{22} - 48760125154728 q^{23} - 676764594411768 q^{24} - 21\!\cdots\!24 q^{26}+ \cdots - 29\!\cdots\!40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2787.27 −1.92470 −0.962352 0.271805i \(-0.912379\pi\)
−0.962352 + 0.271805i \(0.912379\pi\)
\(3\) 59049.0 0.577350
\(4\) 5.67172e6 2.70449
\(5\) 0 0
\(6\) −1.64586e8 −1.11123
\(7\) 1.37938e9 1.84567 0.922836 0.385194i \(-0.125866\pi\)
0.922836 + 0.385194i \(0.125866\pi\)
\(8\) −9.96329e9 −3.28064
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) 2.99797e10 0.348501 0.174250 0.984701i \(-0.444250\pi\)
0.174250 + 0.984701i \(0.444250\pi\)
\(12\) 3.34910e11 1.56144
\(13\) 7.63100e11 1.53524 0.767619 0.640906i \(-0.221441\pi\)
0.767619 + 0.640906i \(0.221441\pi\)
\(14\) −3.84470e12 −3.55237
\(15\) 0 0
\(16\) 1.58759e13 3.60977
\(17\) 1.14939e13 1.38278 0.691391 0.722480i \(-0.256998\pi\)
0.691391 + 0.722480i \(0.256998\pi\)
\(18\) −9.71861e12 −0.641568
\(19\) −5.19898e12 −0.194538 −0.0972692 0.995258i \(-0.531011\pi\)
−0.0972692 + 0.995258i \(0.531011\pi\)
\(20\) 0 0
\(21\) 8.14510e13 1.06560
\(22\) −8.35614e13 −0.670761
\(23\) 8.03048e13 0.404203 0.202101 0.979365i \(-0.435223\pi\)
0.202101 + 0.979365i \(0.435223\pi\)
\(24\) −5.88322e14 −1.89408
\(25\) 0 0
\(26\) −2.12696e15 −2.95488
\(27\) 2.05891e14 0.192450
\(28\) 7.82346e15 4.99159
\(29\) −2.01015e14 −0.0887259 −0.0443629 0.999015i \(-0.514126\pi\)
−0.0443629 + 0.999015i \(0.514126\pi\)
\(30\) 0 0
\(31\) 1.53535e15 0.336442 0.168221 0.985749i \(-0.446198\pi\)
0.168221 + 0.985749i \(0.446198\pi\)
\(32\) −2.33559e16 −3.66710
\(33\) 1.77027e15 0.201207
\(34\) −3.20366e16 −2.66145
\(35\) 0 0
\(36\) 1.97761e16 0.901496
\(37\) −1.53969e16 −0.526401 −0.263200 0.964741i \(-0.584778\pi\)
−0.263200 + 0.964741i \(0.584778\pi\)
\(38\) 1.44910e16 0.374429
\(39\) 4.50603e16 0.886371
\(40\) 0 0
\(41\) −7.88656e16 −0.917608 −0.458804 0.888538i \(-0.651722\pi\)
−0.458804 + 0.888538i \(0.651722\pi\)
\(42\) −2.27026e17 −2.05096
\(43\) 1.61914e17 1.14253 0.571264 0.820766i \(-0.306453\pi\)
0.571264 + 0.820766i \(0.306453\pi\)
\(44\) 1.70036e17 0.942516
\(45\) 0 0
\(46\) −2.23831e17 −0.777971
\(47\) 3.76922e17 1.04526 0.522629 0.852560i \(-0.324951\pi\)
0.522629 + 0.852560i \(0.324951\pi\)
\(48\) 9.37457e17 2.08410
\(49\) 1.34414e18 2.40650
\(50\) 0 0
\(51\) 6.78704e17 0.798350
\(52\) 4.32809e18 4.15204
\(53\) −3.24880e17 −0.255168 −0.127584 0.991828i \(-0.540722\pi\)
−0.127584 + 0.991828i \(0.540722\pi\)
\(54\) −5.73874e17 −0.370410
\(55\) 0 0
\(56\) −1.37432e19 −6.05497
\(57\) −3.06994e17 −0.112317
\(58\) 5.60284e17 0.170771
\(59\) −9.88461e17 −0.251775 −0.125888 0.992044i \(-0.540178\pi\)
−0.125888 + 0.992044i \(0.540178\pi\)
\(60\) 0 0
\(61\) 1.01280e19 1.81786 0.908929 0.416951i \(-0.136901\pi\)
0.908929 + 0.416951i \(0.136901\pi\)
\(62\) −4.27945e18 −0.647552
\(63\) 4.80960e18 0.615224
\(64\) 3.18051e19 3.44832
\(65\) 0 0
\(66\) −4.93422e18 −0.387264
\(67\) 1.02596e19 0.687616 0.343808 0.939040i \(-0.388283\pi\)
0.343808 + 0.939040i \(0.388283\pi\)
\(68\) 6.51902e19 3.73972
\(69\) 4.74192e18 0.233367
\(70\) 0 0
\(71\) 3.23535e19 1.17953 0.589765 0.807575i \(-0.299220\pi\)
0.589765 + 0.807575i \(0.299220\pi\)
\(72\) −3.47399e19 −1.09355
\(73\) 2.89708e19 0.788987 0.394494 0.918899i \(-0.370920\pi\)
0.394494 + 0.918899i \(0.370920\pi\)
\(74\) 4.29154e19 1.01317
\(75\) 0 0
\(76\) −2.94872e19 −0.526126
\(77\) 4.13533e19 0.643218
\(78\) −1.25595e20 −1.70600
\(79\) 3.51920e19 0.418176 0.209088 0.977897i \(-0.432950\pi\)
0.209088 + 0.977897i \(0.432950\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 2.19820e20 1.76612
\(83\) −2.86096e19 −0.202391 −0.101196 0.994867i \(-0.532267\pi\)
−0.101196 + 0.994867i \(0.532267\pi\)
\(84\) 4.61967e20 2.88190
\(85\) 0 0
\(86\) −4.51299e20 −2.19903
\(87\) −1.18698e19 −0.0512259
\(88\) −2.98696e20 −1.14330
\(89\) 5.20282e19 0.176866 0.0884328 0.996082i \(-0.471814\pi\)
0.0884328 + 0.996082i \(0.471814\pi\)
\(90\) 0 0
\(91\) 1.05260e21 2.83355
\(92\) 4.55467e20 1.09316
\(93\) 9.06611e19 0.194245
\(94\) −1.05058e21 −2.01181
\(95\) 0 0
\(96\) −1.37915e21 −2.11720
\(97\) −7.90419e20 −1.08831 −0.544157 0.838983i \(-0.683150\pi\)
−0.544157 + 0.838983i \(0.683150\pi\)
\(98\) −3.74649e21 −4.63180
\(99\) 1.04533e20 0.116167
\(100\) 0 0
\(101\) −1.18585e21 −1.06821 −0.534105 0.845418i \(-0.679351\pi\)
−0.534105 + 0.845418i \(0.679351\pi\)
\(102\) −1.89173e21 −1.53659
\(103\) −1.17111e21 −0.858632 −0.429316 0.903154i \(-0.641245\pi\)
−0.429316 + 0.903154i \(0.641245\pi\)
\(104\) −7.60298e21 −5.03656
\(105\) 0 0
\(106\) 9.05529e20 0.491123
\(107\) −1.02156e21 −0.502035 −0.251018 0.967983i \(-0.580765\pi\)
−0.251018 + 0.967983i \(0.580765\pi\)
\(108\) 1.16776e21 0.520479
\(109\) −8.82895e20 −0.357216 −0.178608 0.983920i \(-0.557159\pi\)
−0.178608 + 0.983920i \(0.557159\pi\)
\(110\) 0 0
\(111\) −9.09174e20 −0.303917
\(112\) 2.18989e22 6.66244
\(113\) −3.34874e21 −0.928022 −0.464011 0.885829i \(-0.653590\pi\)
−0.464011 + 0.885829i \(0.653590\pi\)
\(114\) 8.55676e20 0.216177
\(115\) 0 0
\(116\) −1.14010e21 −0.239958
\(117\) 2.66076e21 0.511746
\(118\) 2.75511e21 0.484593
\(119\) 1.58545e22 2.55216
\(120\) 0 0
\(121\) −6.50147e21 −0.878547
\(122\) −2.82294e22 −3.49884
\(123\) −4.65694e21 −0.529781
\(124\) 8.70810e21 0.909905
\(125\) 0 0
\(126\) −1.34056e22 −1.18412
\(127\) −1.00016e20 −0.00813076 −0.00406538 0.999992i \(-0.501294\pi\)
−0.00406538 + 0.999992i \(0.501294\pi\)
\(128\) −3.96684e22 −2.96989
\(129\) 9.56089e21 0.659639
\(130\) 0 0
\(131\) 1.82022e21 0.106850 0.0534252 0.998572i \(-0.482986\pi\)
0.0534252 + 0.998572i \(0.482986\pi\)
\(132\) 1.00405e22 0.544162
\(133\) −7.17136e21 −0.359054
\(134\) −2.85964e22 −1.32346
\(135\) 0 0
\(136\) −1.14517e23 −4.53641
\(137\) 3.39671e22 1.24593 0.622963 0.782251i \(-0.285929\pi\)
0.622963 + 0.782251i \(0.285929\pi\)
\(138\) −1.32170e22 −0.449162
\(139\) 5.60010e21 0.176417 0.0882085 0.996102i \(-0.471886\pi\)
0.0882085 + 0.996102i \(0.471886\pi\)
\(140\) 0 0
\(141\) 2.22568e22 0.603480
\(142\) −9.01780e22 −2.27025
\(143\) 2.28775e22 0.535032
\(144\) 5.53559e22 1.20326
\(145\) 0 0
\(146\) −8.07494e22 −1.51857
\(147\) 7.93702e22 1.38939
\(148\) −8.73272e22 −1.42364
\(149\) −3.33164e22 −0.506060 −0.253030 0.967458i \(-0.581427\pi\)
−0.253030 + 0.967458i \(0.581427\pi\)
\(150\) 0 0
\(151\) −8.19754e22 −1.08249 −0.541247 0.840863i \(-0.682048\pi\)
−0.541247 + 0.840863i \(0.682048\pi\)
\(152\) 5.17989e22 0.638209
\(153\) 4.00768e22 0.460928
\(154\) −1.15263e23 −1.23800
\(155\) 0 0
\(156\) 2.55569e23 2.39718
\(157\) 1.47933e23 1.29754 0.648769 0.760986i \(-0.275284\pi\)
0.648769 + 0.760986i \(0.275284\pi\)
\(158\) −9.80896e22 −0.804866
\(159\) −1.91839e22 −0.147321
\(160\) 0 0
\(161\) 1.10771e23 0.746025
\(162\) −3.38867e22 −0.213856
\(163\) −5.52958e22 −0.327132 −0.163566 0.986532i \(-0.552300\pi\)
−0.163566 + 0.986532i \(0.552300\pi\)
\(164\) −4.47304e23 −2.48166
\(165\) 0 0
\(166\) 7.97426e22 0.389543
\(167\) 4.62356e22 0.212058 0.106029 0.994363i \(-0.466186\pi\)
0.106029 + 0.994363i \(0.466186\pi\)
\(168\) −8.11520e23 −3.49584
\(169\) 3.35256e23 1.35696
\(170\) 0 0
\(171\) −1.81277e22 −0.0648461
\(172\) 9.18334e23 3.08995
\(173\) 4.87339e23 1.54293 0.771466 0.636270i \(-0.219524\pi\)
0.771466 + 0.636270i \(0.219524\pi\)
\(174\) 3.30842e22 0.0985947
\(175\) 0 0
\(176\) 4.75955e23 1.25801
\(177\) −5.83676e22 −0.145363
\(178\) −1.45017e23 −0.340414
\(179\) −4.33940e23 −0.960446 −0.480223 0.877146i \(-0.659444\pi\)
−0.480223 + 0.877146i \(0.659444\pi\)
\(180\) 0 0
\(181\) −7.08757e23 −1.39596 −0.697979 0.716118i \(-0.745917\pi\)
−0.697979 + 0.716118i \(0.745917\pi\)
\(182\) −2.93389e24 −5.45374
\(183\) 5.98048e23 1.04954
\(184\) −8.00100e23 −1.32604
\(185\) 0 0
\(186\) −2.52697e23 −0.373865
\(187\) 3.44583e23 0.481901
\(188\) 2.13779e24 2.82689
\(189\) 2.84002e23 0.355200
\(190\) 0 0
\(191\) −1.37779e24 −1.54289 −0.771443 0.636299i \(-0.780464\pi\)
−0.771443 + 0.636299i \(0.780464\pi\)
\(192\) 1.87806e24 1.99089
\(193\) −4.06115e23 −0.407659 −0.203829 0.979006i \(-0.565339\pi\)
−0.203829 + 0.979006i \(0.565339\pi\)
\(194\) 2.20311e24 2.09468
\(195\) 0 0
\(196\) 7.62360e24 6.50835
\(197\) −1.19621e24 −0.968083 −0.484041 0.875045i \(-0.660832\pi\)
−0.484041 + 0.875045i \(0.660832\pi\)
\(198\) −2.91361e23 −0.223587
\(199\) 1.89020e24 1.37579 0.687894 0.725811i \(-0.258535\pi\)
0.687894 + 0.725811i \(0.258535\pi\)
\(200\) 0 0
\(201\) 6.05821e23 0.396995
\(202\) 3.30529e24 2.05599
\(203\) −2.77276e23 −0.163759
\(204\) 3.84942e24 2.15913
\(205\) 0 0
\(206\) 3.26420e24 1.65261
\(207\) 2.80006e23 0.134734
\(208\) 1.21149e25 5.54185
\(209\) −1.55864e23 −0.0677967
\(210\) 0 0
\(211\) −3.61070e24 −1.42110 −0.710552 0.703645i \(-0.751555\pi\)
−0.710552 + 0.703645i \(0.751555\pi\)
\(212\) −1.84263e24 −0.690099
\(213\) 1.91044e24 0.681002
\(214\) 2.84736e24 0.966270
\(215\) 0 0
\(216\) −2.05135e24 −0.631359
\(217\) 2.11784e24 0.620962
\(218\) 2.46087e24 0.687535
\(219\) 1.71070e24 0.455522
\(220\) 0 0
\(221\) 8.77099e24 2.12290
\(222\) 2.53411e24 0.584951
\(223\) −2.51046e24 −0.552780 −0.276390 0.961046i \(-0.589138\pi\)
−0.276390 + 0.961046i \(0.589138\pi\)
\(224\) −3.22167e25 −6.76826
\(225\) 0 0
\(226\) 9.33385e24 1.78617
\(227\) −9.57954e24 −1.75014 −0.875071 0.483995i \(-0.839185\pi\)
−0.875071 + 0.483995i \(0.839185\pi\)
\(228\) −1.74119e24 −0.303759
\(229\) −9.71373e24 −1.61850 −0.809251 0.587463i \(-0.800127\pi\)
−0.809251 + 0.587463i \(0.800127\pi\)
\(230\) 0 0
\(231\) 2.44187e24 0.371362
\(232\) 2.00277e24 0.291077
\(233\) 9.68019e24 1.34477 0.672383 0.740204i \(-0.265271\pi\)
0.672383 + 0.740204i \(0.265271\pi\)
\(234\) −7.41627e24 −0.984961
\(235\) 0 0
\(236\) −5.60628e24 −0.680924
\(237\) 2.07805e24 0.241434
\(238\) −4.41907e25 −4.91216
\(239\) −1.36780e25 −1.45494 −0.727471 0.686138i \(-0.759305\pi\)
−0.727471 + 0.686138i \(0.759305\pi\)
\(240\) 0 0
\(241\) −1.69789e25 −1.65474 −0.827372 0.561654i \(-0.810165\pi\)
−0.827372 + 0.561654i \(0.810165\pi\)
\(242\) 1.81214e25 1.69094
\(243\) 7.17898e23 0.0641500
\(244\) 5.74432e25 4.91638
\(245\) 0 0
\(246\) 1.29801e25 1.01967
\(247\) −3.96734e24 −0.298663
\(248\) −1.52972e25 −1.10375
\(249\) −1.68937e24 −0.116851
\(250\) 0 0
\(251\) 1.20909e25 0.768929 0.384465 0.923140i \(-0.374386\pi\)
0.384465 + 0.923140i \(0.374386\pi\)
\(252\) 2.72787e25 1.66386
\(253\) 2.40751e24 0.140865
\(254\) 2.78772e23 0.0156493
\(255\) 0 0
\(256\) 4.38665e25 2.26785
\(257\) 1.47338e25 0.731167 0.365584 0.930779i \(-0.380869\pi\)
0.365584 + 0.930779i \(0.380869\pi\)
\(258\) −2.66488e25 −1.26961
\(259\) −2.12382e25 −0.971562
\(260\) 0 0
\(261\) −7.00897e23 −0.0295753
\(262\) −5.07346e24 −0.205656
\(263\) −3.43475e25 −1.33770 −0.668852 0.743396i \(-0.733214\pi\)
−0.668852 + 0.743396i \(0.733214\pi\)
\(264\) −1.76377e25 −0.660087
\(265\) 0 0
\(266\) 1.99885e25 0.691072
\(267\) 3.07221e24 0.102113
\(268\) 5.81898e25 1.85965
\(269\) 3.49311e25 1.07353 0.536764 0.843733i \(-0.319647\pi\)
0.536764 + 0.843733i \(0.319647\pi\)
\(270\) 0 0
\(271\) −5.04797e25 −1.43529 −0.717644 0.696410i \(-0.754779\pi\)
−0.717644 + 0.696410i \(0.754779\pi\)
\(272\) 1.82476e26 4.99152
\(273\) 6.21552e25 1.63595
\(274\) −9.46755e25 −2.39804
\(275\) 0 0
\(276\) 2.68949e25 0.631137
\(277\) 3.43914e24 0.0776985 0.0388492 0.999245i \(-0.487631\pi\)
0.0388492 + 0.999245i \(0.487631\pi\)
\(278\) −1.56090e25 −0.339551
\(279\) 5.35345e24 0.112147
\(280\) 0 0
\(281\) −8.07634e25 −1.56963 −0.784817 0.619727i \(-0.787243\pi\)
−0.784817 + 0.619727i \(0.787243\pi\)
\(282\) −6.20358e25 −1.16152
\(283\) 2.52971e25 0.456366 0.228183 0.973618i \(-0.426722\pi\)
0.228183 + 0.973618i \(0.426722\pi\)
\(284\) 1.83500e26 3.19002
\(285\) 0 0
\(286\) −6.37657e25 −1.02978
\(287\) −1.08786e26 −1.69360
\(288\) −8.14371e25 −1.22237
\(289\) 6.30180e25 0.912088
\(290\) 0 0
\(291\) −4.66735e25 −0.628338
\(292\) 1.64314e26 2.13381
\(293\) −1.39891e26 −1.75259 −0.876295 0.481775i \(-0.839992\pi\)
−0.876295 + 0.481775i \(0.839992\pi\)
\(294\) −2.21226e26 −2.67417
\(295\) 0 0
\(296\) 1.53404e26 1.72693
\(297\) 6.17255e24 0.0670690
\(298\) 9.28617e25 0.974016
\(299\) 6.12806e25 0.620548
\(300\) 0 0
\(301\) 2.23341e26 2.10873
\(302\) 2.28488e26 2.08348
\(303\) −7.00235e25 −0.616731
\(304\) −8.25386e25 −0.702238
\(305\) 0 0
\(306\) −1.11705e26 −0.887150
\(307\) 1.66063e26 1.27444 0.637222 0.770681i \(-0.280084\pi\)
0.637222 + 0.770681i \(0.280084\pi\)
\(308\) 2.34545e26 1.73957
\(309\) −6.91529e25 −0.495732
\(310\) 0 0
\(311\) 2.85915e26 1.91537 0.957685 0.287819i \(-0.0929302\pi\)
0.957685 + 0.287819i \(0.0929302\pi\)
\(312\) −4.48949e26 −2.90786
\(313\) −6.78268e25 −0.424801 −0.212401 0.977183i \(-0.568128\pi\)
−0.212401 + 0.977183i \(0.568128\pi\)
\(314\) −4.12330e26 −2.49738
\(315\) 0 0
\(316\) 1.99599e26 1.13095
\(317\) 3.05982e25 0.167716 0.0838579 0.996478i \(-0.473276\pi\)
0.0838579 + 0.996478i \(0.473276\pi\)
\(318\) 5.34706e25 0.283550
\(319\) −6.02637e24 −0.0309210
\(320\) 0 0
\(321\) −6.03221e25 −0.289850
\(322\) −3.08748e26 −1.43588
\(323\) −5.97566e25 −0.269004
\(324\) 6.89549e25 0.300499
\(325\) 0 0
\(326\) 1.54124e26 0.629632
\(327\) −5.21341e25 −0.206239
\(328\) 7.85761e26 3.01034
\(329\) 5.19918e26 1.92920
\(330\) 0 0
\(331\) 2.97902e26 1.03724 0.518620 0.855005i \(-0.326446\pi\)
0.518620 + 0.855005i \(0.326446\pi\)
\(332\) −1.62266e26 −0.547364
\(333\) −5.36858e25 −0.175467
\(334\) −1.28871e26 −0.408148
\(335\) 0 0
\(336\) 1.29311e27 3.84656
\(337\) −1.35212e26 −0.389852 −0.194926 0.980818i \(-0.562447\pi\)
−0.194926 + 0.980818i \(0.562447\pi\)
\(338\) −9.34450e26 −2.61174
\(339\) −1.97740e26 −0.535794
\(340\) 0 0
\(341\) 4.60294e25 0.117250
\(342\) 5.05268e25 0.124810
\(343\) 1.08363e27 2.59594
\(344\) −1.61320e27 −3.74822
\(345\) 0 0
\(346\) −1.35834e27 −2.96969
\(347\) −2.89431e26 −0.613884 −0.306942 0.951728i \(-0.599306\pi\)
−0.306942 + 0.951728i \(0.599306\pi\)
\(348\) −6.73220e25 −0.138540
\(349\) 5.89250e26 1.17661 0.588306 0.808639i \(-0.299795\pi\)
0.588306 + 0.808639i \(0.299795\pi\)
\(350\) 0 0
\(351\) 1.57115e26 0.295457
\(352\) −7.00203e26 −1.27799
\(353\) −6.63803e25 −0.117599 −0.0587996 0.998270i \(-0.518727\pi\)
−0.0587996 + 0.998270i \(0.518727\pi\)
\(354\) 1.62686e26 0.279780
\(355\) 0 0
\(356\) 2.95089e26 0.478331
\(357\) 9.36190e26 1.47349
\(358\) 1.20951e27 1.84858
\(359\) 9.35962e26 1.38921 0.694603 0.719393i \(-0.255580\pi\)
0.694603 + 0.719393i \(0.255580\pi\)
\(360\) 0 0
\(361\) −6.87180e26 −0.962155
\(362\) 1.97550e27 2.68681
\(363\) −3.83905e26 −0.507230
\(364\) 5.97008e27 7.66329
\(365\) 0 0
\(366\) −1.66692e27 −2.02006
\(367\) −1.41202e27 −1.66282 −0.831411 0.555658i \(-0.812466\pi\)
−0.831411 + 0.555658i \(0.812466\pi\)
\(368\) 1.27491e27 1.45908
\(369\) −2.74987e26 −0.305869
\(370\) 0 0
\(371\) −4.48133e26 −0.470957
\(372\) 5.14205e26 0.525334
\(373\) 4.21905e26 0.419056 0.209528 0.977803i \(-0.432807\pi\)
0.209528 + 0.977803i \(0.432807\pi\)
\(374\) −9.60447e26 −0.927517
\(375\) 0 0
\(376\) −3.75538e27 −3.42911
\(377\) −1.53395e26 −0.136215
\(378\) −7.91590e26 −0.683654
\(379\) −1.82216e27 −1.53065 −0.765325 0.643644i \(-0.777422\pi\)
−0.765325 + 0.643644i \(0.777422\pi\)
\(380\) 0 0
\(381\) −5.90586e24 −0.00469430
\(382\) 3.84028e27 2.96960
\(383\) 3.47058e26 0.261105 0.130552 0.991441i \(-0.458325\pi\)
0.130552 + 0.991441i \(0.458325\pi\)
\(384\) −2.34238e27 −1.71467
\(385\) 0 0
\(386\) 1.13195e27 0.784623
\(387\) 5.64561e26 0.380843
\(388\) −4.48304e27 −2.94333
\(389\) −1.66189e27 −1.06202 −0.531009 0.847366i \(-0.678187\pi\)
−0.531009 + 0.847366i \(0.678187\pi\)
\(390\) 0 0
\(391\) 9.23016e26 0.558925
\(392\) −1.33921e28 −7.89485
\(393\) 1.07482e26 0.0616902
\(394\) 3.33416e27 1.86327
\(395\) 0 0
\(396\) 5.92880e26 0.314172
\(397\) 3.53949e27 1.82659 0.913293 0.407304i \(-0.133531\pi\)
0.913293 + 0.407304i \(0.133531\pi\)
\(398\) −5.26851e27 −2.64798
\(399\) −4.23462e26 −0.207300
\(400\) 0 0
\(401\) −1.58374e27 −0.735644 −0.367822 0.929896i \(-0.619896\pi\)
−0.367822 + 0.929896i \(0.619896\pi\)
\(402\) −1.68859e27 −0.764099
\(403\) 1.17163e27 0.516520
\(404\) −6.72583e27 −2.88896
\(405\) 0 0
\(406\) 7.72844e26 0.315187
\(407\) −4.61595e26 −0.183451
\(408\) −6.76212e27 −2.61910
\(409\) 2.05726e27 0.776596 0.388298 0.921534i \(-0.373063\pi\)
0.388298 + 0.921534i \(0.373063\pi\)
\(410\) 0 0
\(411\) 2.00572e27 0.719336
\(412\) −6.64222e27 −2.32216
\(413\) −1.36346e27 −0.464695
\(414\) −7.80451e26 −0.259324
\(415\) 0 0
\(416\) −1.78229e28 −5.62987
\(417\) 3.30680e26 0.101854
\(418\) 4.34434e26 0.130489
\(419\) 1.53990e27 0.451071 0.225536 0.974235i \(-0.427587\pi\)
0.225536 + 0.974235i \(0.427587\pi\)
\(420\) 0 0
\(421\) 3.15043e27 0.877824 0.438912 0.898530i \(-0.355364\pi\)
0.438912 + 0.898530i \(0.355364\pi\)
\(422\) 1.00640e28 2.73521
\(423\) 1.31424e27 0.348419
\(424\) 3.23688e27 0.837114
\(425\) 0 0
\(426\) −5.32492e27 −1.31073
\(427\) 1.39703e28 3.35517
\(428\) −5.79400e27 −1.35775
\(429\) 1.35089e27 0.308901
\(430\) 0 0
\(431\) 1.98201e27 0.431614 0.215807 0.976436i \(-0.430762\pi\)
0.215807 + 0.976436i \(0.430762\pi\)
\(432\) 3.26871e27 0.694700
\(433\) −5.59999e27 −1.16162 −0.580810 0.814039i \(-0.697264\pi\)
−0.580810 + 0.814039i \(0.697264\pi\)
\(434\) −5.90298e27 −1.19517
\(435\) 0 0
\(436\) −5.00754e27 −0.966087
\(437\) −4.17503e26 −0.0786329
\(438\) −4.76817e27 −0.876746
\(439\) 1.53894e27 0.276277 0.138138 0.990413i \(-0.455888\pi\)
0.138138 + 0.990413i \(0.455888\pi\)
\(440\) 0 0
\(441\) 4.68673e27 0.802167
\(442\) −2.44471e28 −4.08596
\(443\) 3.27390e27 0.534351 0.267175 0.963648i \(-0.413910\pi\)
0.267175 + 0.963648i \(0.413910\pi\)
\(444\) −5.15658e27 −0.821941
\(445\) 0 0
\(446\) 6.99733e27 1.06394
\(447\) −1.96730e27 −0.292174
\(448\) 4.38713e28 6.36446
\(449\) 3.04397e27 0.431374 0.215687 0.976463i \(-0.430801\pi\)
0.215687 + 0.976463i \(0.430801\pi\)
\(450\) 0 0
\(451\) −2.36436e27 −0.319787
\(452\) −1.89931e28 −2.50982
\(453\) −4.84057e27 −0.624979
\(454\) 2.67008e28 3.36850
\(455\) 0 0
\(456\) 3.05868e27 0.368470
\(457\) 4.76153e26 0.0560566 0.0280283 0.999607i \(-0.491077\pi\)
0.0280283 + 0.999607i \(0.491077\pi\)
\(458\) 2.70748e28 3.11514
\(459\) 2.36649e27 0.266117
\(460\) 0 0
\(461\) −9.99698e27 −1.07401 −0.537006 0.843578i \(-0.680445\pi\)
−0.537006 + 0.843578i \(0.680445\pi\)
\(462\) −6.80616e27 −0.714762
\(463\) −1.40995e28 −1.44745 −0.723723 0.690090i \(-0.757571\pi\)
−0.723723 + 0.690090i \(0.757571\pi\)
\(464\) −3.19130e27 −0.320280
\(465\) 0 0
\(466\) −2.69813e28 −2.58828
\(467\) −3.82873e25 −0.00359110 −0.00179555 0.999998i \(-0.500572\pi\)
−0.00179555 + 0.999998i \(0.500572\pi\)
\(468\) 1.50911e28 1.38401
\(469\) 1.41519e28 1.26911
\(470\) 0 0
\(471\) 8.73532e27 0.749133
\(472\) 9.84832e27 0.825983
\(473\) 4.85414e27 0.398172
\(474\) −5.79209e27 −0.464690
\(475\) 0 0
\(476\) 8.99221e28 6.90229
\(477\) −1.13279e27 −0.0850561
\(478\) 3.81244e28 2.80033
\(479\) 9.30936e27 0.668955 0.334477 0.942404i \(-0.391440\pi\)
0.334477 + 0.942404i \(0.391440\pi\)
\(480\) 0 0
\(481\) −1.17494e28 −0.808151
\(482\) 4.73248e28 3.18490
\(483\) 6.54091e27 0.430718
\(484\) −3.68745e28 −2.37602
\(485\) 0 0
\(486\) −2.00098e27 −0.123470
\(487\) 1.33180e28 0.804239 0.402119 0.915587i \(-0.368274\pi\)
0.402119 + 0.915587i \(0.368274\pi\)
\(488\) −1.00908e29 −5.96373
\(489\) −3.26516e27 −0.188869
\(490\) 0 0
\(491\) −5.75994e27 −0.319199 −0.159600 0.987182i \(-0.551020\pi\)
−0.159600 + 0.987182i \(0.551020\pi\)
\(492\) −2.64128e28 −1.43279
\(493\) −2.31045e27 −0.122689
\(494\) 1.10580e28 0.574838
\(495\) 0 0
\(496\) 2.43752e28 1.21448
\(497\) 4.46278e28 2.17702
\(498\) 4.70872e27 0.224903
\(499\) 1.10496e28 0.516763 0.258382 0.966043i \(-0.416811\pi\)
0.258382 + 0.966043i \(0.416811\pi\)
\(500\) 0 0
\(501\) 2.73017e27 0.122432
\(502\) −3.37007e28 −1.47996
\(503\) −2.38869e28 −1.02730 −0.513649 0.858001i \(-0.671706\pi\)
−0.513649 + 0.858001i \(0.671706\pi\)
\(504\) −4.79194e28 −2.01832
\(505\) 0 0
\(506\) −6.71039e27 −0.271123
\(507\) 1.97966e28 0.783440
\(508\) −5.67264e26 −0.0219896
\(509\) 7.62092e27 0.289381 0.144691 0.989477i \(-0.453781\pi\)
0.144691 + 0.989477i \(0.453781\pi\)
\(510\) 0 0
\(511\) 3.99617e28 1.45621
\(512\) −3.90771e28 −1.39504
\(513\) −1.07042e27 −0.0374389
\(514\) −4.10671e28 −1.40728
\(515\) 0 0
\(516\) 5.42267e28 1.78399
\(517\) 1.13000e28 0.364273
\(518\) 5.91967e28 1.86997
\(519\) 2.87769e28 0.890813
\(520\) 0 0
\(521\) −2.68092e28 −0.797055 −0.398527 0.917156i \(-0.630479\pi\)
−0.398527 + 0.917156i \(0.630479\pi\)
\(522\) 1.95359e27 0.0569237
\(523\) 1.54301e28 0.440657 0.220328 0.975426i \(-0.429287\pi\)
0.220328 + 0.975426i \(0.429287\pi\)
\(524\) 1.03238e28 0.288976
\(525\) 0 0
\(526\) 9.57358e28 2.57469
\(527\) 1.76472e28 0.465227
\(528\) 2.81047e28 0.726310
\(529\) −3.30227e28 −0.836620
\(530\) 0 0
\(531\) −3.44655e27 −0.0839251
\(532\) −4.06740e28 −0.971056
\(533\) −6.01823e28 −1.40875
\(534\) −8.56308e27 −0.196538
\(535\) 0 0
\(536\) −1.02220e29 −2.25582
\(537\) −2.56237e28 −0.554514
\(538\) −9.73623e28 −2.06622
\(539\) 4.02969e28 0.838667
\(540\) 0 0
\(541\) −5.53635e28 −1.10829 −0.554143 0.832421i \(-0.686954\pi\)
−0.554143 + 0.832421i \(0.686954\pi\)
\(542\) 1.40700e29 2.76250
\(543\) −4.18514e28 −0.805957
\(544\) −2.68451e29 −5.07080
\(545\) 0 0
\(546\) −1.73243e29 −3.14872
\(547\) 3.14951e28 0.561535 0.280767 0.959776i \(-0.409411\pi\)
0.280767 + 0.959776i \(0.409411\pi\)
\(548\) 1.92652e29 3.36959
\(549\) 3.53141e28 0.605953
\(550\) 0 0
\(551\) 1.04507e27 0.0172606
\(552\) −4.72451e28 −0.765591
\(553\) 4.85431e28 0.771816
\(554\) −9.58581e27 −0.149547
\(555\) 0 0
\(556\) 3.17622e28 0.477118
\(557\) −4.25159e28 −0.626717 −0.313358 0.949635i \(-0.601454\pi\)
−0.313358 + 0.949635i \(0.601454\pi\)
\(558\) −1.49215e28 −0.215851
\(559\) 1.23557e29 1.75405
\(560\) 0 0
\(561\) 2.03473e28 0.278226
\(562\) 2.25110e29 3.02108
\(563\) 7.19780e28 0.948116 0.474058 0.880494i \(-0.342789\pi\)
0.474058 + 0.880494i \(0.342789\pi\)
\(564\) 1.26235e29 1.63210
\(565\) 0 0
\(566\) −7.05098e28 −0.878369
\(567\) 1.67700e28 0.205075
\(568\) −3.22348e29 −3.86961
\(569\) 1.32568e29 1.56228 0.781142 0.624353i \(-0.214637\pi\)
0.781142 + 0.624353i \(0.214637\pi\)
\(570\) 0 0
\(571\) 5.77857e28 0.656359 0.328179 0.944615i \(-0.393565\pi\)
0.328179 + 0.944615i \(0.393565\pi\)
\(572\) 1.29755e29 1.44699
\(573\) −8.13573e28 −0.890785
\(574\) 3.03215e29 3.25968
\(575\) 0 0
\(576\) 1.10898e29 1.14944
\(577\) 1.02641e29 1.04466 0.522332 0.852742i \(-0.325062\pi\)
0.522332 + 0.852742i \(0.325062\pi\)
\(578\) −1.75648e29 −1.75550
\(579\) −2.39807e28 −0.235362
\(580\) 0 0
\(581\) −3.94635e28 −0.373547
\(582\) 1.30092e29 1.20937
\(583\) −9.73980e27 −0.0889263
\(584\) −2.88644e29 −2.58838
\(585\) 0 0
\(586\) 3.89914e29 3.37322
\(587\) −1.42995e29 −1.21512 −0.607562 0.794272i \(-0.707852\pi\)
−0.607562 + 0.794272i \(0.707852\pi\)
\(588\) 4.50166e29 3.75760
\(589\) −7.98227e27 −0.0654510
\(590\) 0 0
\(591\) −7.06351e28 −0.558923
\(592\) −2.44441e29 −1.90018
\(593\) −1.90982e29 −1.45854 −0.729270 0.684226i \(-0.760140\pi\)
−0.729270 + 0.684226i \(0.760140\pi\)
\(594\) −1.72046e28 −0.129088
\(595\) 0 0
\(596\) −1.88961e29 −1.36863
\(597\) 1.11615e29 0.794311
\(598\) −1.70806e29 −1.19437
\(599\) 1.17313e29 0.806053 0.403027 0.915188i \(-0.367958\pi\)
0.403027 + 0.915188i \(0.367958\pi\)
\(600\) 0 0
\(601\) −2.19161e29 −1.45405 −0.727027 0.686609i \(-0.759099\pi\)
−0.727027 + 0.686609i \(0.759099\pi\)
\(602\) −6.22513e29 −4.05868
\(603\) 3.57731e28 0.229205
\(604\) −4.64942e29 −2.92759
\(605\) 0 0
\(606\) 1.95174e29 1.18703
\(607\) −1.48966e29 −0.890445 −0.445222 0.895420i \(-0.646875\pi\)
−0.445222 + 0.895420i \(0.646875\pi\)
\(608\) 1.21427e29 0.713391
\(609\) −1.63729e28 −0.0945462
\(610\) 0 0
\(611\) 2.87629e29 1.60472
\(612\) 2.27304e29 1.24657
\(613\) −2.26313e29 −1.22004 −0.610022 0.792385i \(-0.708839\pi\)
−0.610022 + 0.792385i \(0.708839\pi\)
\(614\) −4.62863e29 −2.45293
\(615\) 0 0
\(616\) −4.12015e29 −2.11016
\(617\) 3.85893e29 1.94300 0.971500 0.237040i \(-0.0761773\pi\)
0.971500 + 0.237040i \(0.0761773\pi\)
\(618\) 1.92748e29 0.954137
\(619\) −1.80368e29 −0.877823 −0.438912 0.898530i \(-0.644636\pi\)
−0.438912 + 0.898530i \(0.644636\pi\)
\(620\) 0 0
\(621\) 1.65341e28 0.0777889
\(622\) −7.96922e29 −3.68652
\(623\) 7.17666e28 0.326436
\(624\) 7.15373e29 3.19959
\(625\) 0 0
\(626\) 1.89052e29 0.817617
\(627\) −9.20359e27 −0.0391425
\(628\) 8.39037e29 3.50917
\(629\) −1.76971e29 −0.727898
\(630\) 0 0
\(631\) 2.76208e29 1.09882 0.549411 0.835552i \(-0.314852\pi\)
0.549411 + 0.835552i \(0.314852\pi\)
\(632\) −3.50628e29 −1.37188
\(633\) −2.13208e29 −0.820475
\(634\) −8.52855e28 −0.322803
\(635\) 0 0
\(636\) −1.08805e29 −0.398429
\(637\) 1.02571e30 3.69455
\(638\) 1.67971e28 0.0595138
\(639\) 1.12810e29 0.393177
\(640\) 0 0
\(641\) 1.28643e29 0.433889 0.216944 0.976184i \(-0.430391\pi\)
0.216944 + 0.976184i \(0.430391\pi\)
\(642\) 1.68134e29 0.557876
\(643\) 3.99417e29 1.30380 0.651900 0.758305i \(-0.273972\pi\)
0.651900 + 0.758305i \(0.273972\pi\)
\(644\) 6.28261e29 2.01762
\(645\) 0 0
\(646\) 1.66558e29 0.517754
\(647\) 2.80444e29 0.857730 0.428865 0.903368i \(-0.358914\pi\)
0.428865 + 0.903368i \(0.358914\pi\)
\(648\) −1.21130e29 −0.364515
\(649\) −2.96337e28 −0.0877439
\(650\) 0 0
\(651\) 1.25056e29 0.358513
\(652\) −3.13623e29 −0.884723
\(653\) 1.79013e29 0.496931 0.248466 0.968641i \(-0.420074\pi\)
0.248466 + 0.968641i \(0.420074\pi\)
\(654\) 1.45312e29 0.396949
\(655\) 0 0
\(656\) −1.25206e30 −3.31235
\(657\) 1.01015e29 0.262996
\(658\) −1.44915e30 −3.71314
\(659\) 7.33826e29 1.85053 0.925265 0.379320i \(-0.123842\pi\)
0.925265 + 0.379320i \(0.123842\pi\)
\(660\) 0 0
\(661\) −5.96638e29 −1.45746 −0.728728 0.684803i \(-0.759888\pi\)
−0.728728 + 0.684803i \(0.759888\pi\)
\(662\) −8.30333e29 −1.99638
\(663\) 5.17918e29 1.22566
\(664\) 2.85046e29 0.663971
\(665\) 0 0
\(666\) 1.49637e29 0.337722
\(667\) −1.61425e28 −0.0358632
\(668\) 2.62236e29 0.573507
\(669\) −1.48240e29 −0.319148
\(670\) 0 0
\(671\) 3.03634e29 0.633525
\(672\) −1.90236e30 −3.90766
\(673\) 1.89245e28 0.0382706 0.0191353 0.999817i \(-0.493909\pi\)
0.0191353 + 0.999817i \(0.493909\pi\)
\(674\) 3.76871e29 0.750350
\(675\) 0 0
\(676\) 1.90148e30 3.66988
\(677\) 2.47018e28 0.0469405 0.0234703 0.999725i \(-0.492529\pi\)
0.0234703 + 0.999725i \(0.492529\pi\)
\(678\) 5.51155e29 1.03124
\(679\) −1.09029e30 −2.00867
\(680\) 0 0
\(681\) −5.65662e29 −1.01044
\(682\) −1.28296e29 −0.225672
\(683\) 2.88518e29 0.499752 0.249876 0.968278i \(-0.419610\pi\)
0.249876 + 0.968278i \(0.419610\pi\)
\(684\) −1.02815e29 −0.175375
\(685\) 0 0
\(686\) −3.02038e30 −4.99642
\(687\) −5.73586e29 −0.934443
\(688\) 2.57054e30 4.12426
\(689\) −2.47916e29 −0.391744
\(690\) 0 0
\(691\) −3.59152e29 −0.550502 −0.275251 0.961372i \(-0.588761\pi\)
−0.275251 + 0.961372i \(0.588761\pi\)
\(692\) 2.76405e30 4.17284
\(693\) 1.44190e29 0.214406
\(694\) 8.06723e29 1.18154
\(695\) 0 0
\(696\) 1.18262e29 0.168054
\(697\) −9.06474e29 −1.26885
\(698\) −1.64240e30 −2.26463
\(699\) 5.71605e29 0.776401
\(700\) 0 0
\(701\) 1.21736e30 1.60465 0.802327 0.596885i \(-0.203595\pi\)
0.802327 + 0.596885i \(0.203595\pi\)
\(702\) −4.37923e29 −0.568667
\(703\) 8.00483e28 0.102405
\(704\) 9.53506e29 1.20174
\(705\) 0 0
\(706\) 1.85020e29 0.226344
\(707\) −1.63574e30 −1.97156
\(708\) −3.31045e29 −0.393131
\(709\) 1.19822e30 1.40202 0.701008 0.713154i \(-0.252734\pi\)
0.701008 + 0.713154i \(0.252734\pi\)
\(710\) 0 0
\(711\) 1.22707e29 0.139392
\(712\) −5.18372e29 −0.580232
\(713\) 1.23296e29 0.135991
\(714\) −2.60941e30 −2.83604
\(715\) 0 0
\(716\) −2.46119e30 −2.59752
\(717\) −8.07674e29 −0.840011
\(718\) −2.60878e30 −2.67381
\(719\) −2.33520e29 −0.235869 −0.117935 0.993021i \(-0.537627\pi\)
−0.117935 + 0.993021i \(0.537627\pi\)
\(720\) 0 0
\(721\) −1.61541e30 −1.58475
\(722\) 1.91536e30 1.85186
\(723\) −1.00259e30 −0.955367
\(724\) −4.01987e30 −3.77535
\(725\) 0 0
\(726\) 1.07005e30 0.976267
\(727\) 5.30551e29 0.477107 0.238554 0.971129i \(-0.423327\pi\)
0.238554 + 0.971129i \(0.423327\pi\)
\(728\) −1.04874e31 −9.29583
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) 1.86103e30 1.57987
\(732\) 3.39196e30 2.83847
\(733\) −4.20011e28 −0.0346473 −0.0173236 0.999850i \(-0.505515\pi\)
−0.0173236 + 0.999850i \(0.505515\pi\)
\(734\) 3.93567e30 3.20044
\(735\) 0 0
\(736\) −1.87559e30 −1.48225
\(737\) 3.07580e29 0.239635
\(738\) 7.66464e29 0.588708
\(739\) −8.41533e28 −0.0637243 −0.0318621 0.999492i \(-0.510144\pi\)
−0.0318621 + 0.999492i \(0.510144\pi\)
\(740\) 0 0
\(741\) −2.34267e29 −0.172433
\(742\) 1.24907e30 0.906452
\(743\) 1.91121e30 1.36749 0.683746 0.729720i \(-0.260350\pi\)
0.683746 + 0.729720i \(0.260350\pi\)
\(744\) −9.03283e29 −0.637248
\(745\) 0 0
\(746\) −1.17596e30 −0.806559
\(747\) −9.97554e28 −0.0674637
\(748\) 1.95438e30 1.30329
\(749\) −1.40912e30 −0.926592
\(750\) 0 0
\(751\) −1.07482e29 −0.0687250 −0.0343625 0.999409i \(-0.510940\pi\)
−0.0343625 + 0.999409i \(0.510940\pi\)
\(752\) 5.98398e30 3.77314
\(753\) 7.13958e29 0.443941
\(754\) 4.27553e29 0.262174
\(755\) 0 0
\(756\) 1.61078e30 0.960633
\(757\) −3.00991e30 −1.77030 −0.885149 0.465307i \(-0.845944\pi\)
−0.885149 + 0.465307i \(0.845944\pi\)
\(758\) 5.07887e30 2.94605
\(759\) 1.42161e29 0.0813284
\(760\) 0 0
\(761\) −1.13756e29 −0.0633049 −0.0316524 0.999499i \(-0.510077\pi\)
−0.0316524 + 0.999499i \(0.510077\pi\)
\(762\) 1.64612e28 0.00903514
\(763\) −1.21785e30 −0.659303
\(764\) −7.81446e30 −4.17271
\(765\) 0 0
\(766\) −9.67344e29 −0.502549
\(767\) −7.54294e29 −0.386535
\(768\) 2.59027e30 1.30934
\(769\) 2.78880e29 0.139056 0.0695281 0.997580i \(-0.477851\pi\)
0.0695281 + 0.997580i \(0.477851\pi\)
\(770\) 0 0
\(771\) 8.70016e29 0.422140
\(772\) −2.30337e30 −1.10251
\(773\) −1.53967e30 −0.727012 −0.363506 0.931592i \(-0.618420\pi\)
−0.363506 + 0.931592i \(0.618420\pi\)
\(774\) −1.57358e30 −0.733010
\(775\) 0 0
\(776\) 7.87518e30 3.57036
\(777\) −1.25410e30 −0.560932
\(778\) 4.63214e30 2.04407
\(779\) 4.10021e29 0.178510
\(780\) 0 0
\(781\) 9.69948e29 0.411067
\(782\) −2.57270e30 −1.07577
\(783\) −4.13873e28 −0.0170753
\(784\) 2.13395e31 8.68691
\(785\) 0 0
\(786\) −2.99583e29 −0.118735
\(787\) −7.03087e29 −0.274963 −0.137482 0.990504i \(-0.543901\pi\)
−0.137482 + 0.990504i \(0.543901\pi\)
\(788\) −6.78458e30 −2.61817
\(789\) −2.02819e30 −0.772324
\(790\) 0 0
\(791\) −4.61919e30 −1.71282
\(792\) −1.04149e30 −0.381101
\(793\) 7.72867e30 2.79085
\(794\) −9.86551e30 −3.51564
\(795\) 0 0
\(796\) 1.07207e31 3.72080
\(797\) 2.01448e30 0.690001 0.345001 0.938602i \(-0.387879\pi\)
0.345001 + 0.938602i \(0.387879\pi\)
\(798\) 1.18030e30 0.398991
\(799\) 4.33230e30 1.44536
\(800\) 0 0
\(801\) 1.81411e29 0.0589552
\(802\) 4.41431e30 1.41590
\(803\) 8.68534e29 0.274963
\(804\) 3.43605e30 1.07367
\(805\) 0 0
\(806\) −3.26564e30 −0.994148
\(807\) 2.06264e30 0.619801
\(808\) 1.18150e31 3.50441
\(809\) −5.38496e30 −1.57660 −0.788302 0.615288i \(-0.789040\pi\)
−0.788302 + 0.615288i \(0.789040\pi\)
\(810\) 0 0
\(811\) 1.66080e30 0.473804 0.236902 0.971534i \(-0.423868\pi\)
0.236902 + 0.971534i \(0.423868\pi\)
\(812\) −1.57263e30 −0.442884
\(813\) −2.98077e30 −0.828664
\(814\) 1.28659e30 0.353089
\(815\) 0 0
\(816\) 1.07750e31 2.88186
\(817\) −8.41790e29 −0.222265
\(818\) −5.73415e30 −1.49472
\(819\) 3.67020e30 0.944515
\(820\) 0 0
\(821\) −3.71407e30 −0.931638 −0.465819 0.884880i \(-0.654240\pi\)
−0.465819 + 0.884880i \(0.654240\pi\)
\(822\) −5.59049e30 −1.38451
\(823\) −1.83682e30 −0.449126 −0.224563 0.974460i \(-0.572096\pi\)
−0.224563 + 0.974460i \(0.572096\pi\)
\(824\) 1.16681e31 2.81686
\(825\) 0 0
\(826\) 3.80034e30 0.894400
\(827\) −4.85542e30 −1.12828 −0.564142 0.825678i \(-0.690793\pi\)
−0.564142 + 0.825678i \(0.690793\pi\)
\(828\) 1.58811e30 0.364387
\(829\) −6.43767e30 −1.45850 −0.729249 0.684248i \(-0.760130\pi\)
−0.729249 + 0.684248i \(0.760130\pi\)
\(830\) 0 0
\(831\) 2.03078e29 0.0448592
\(832\) 2.42705e31 5.29399
\(833\) 1.54494e31 3.32767
\(834\) −9.21696e29 −0.196040
\(835\) 0 0
\(836\) −8.84015e29 −0.183355
\(837\) 3.16116e29 0.0647484
\(838\) −4.29212e30 −0.868179
\(839\) 3.48676e30 0.696500 0.348250 0.937402i \(-0.386776\pi\)
0.348250 + 0.937402i \(0.386776\pi\)
\(840\) 0 0
\(841\) −5.09244e30 −0.992128
\(842\) −8.78109e30 −1.68955
\(843\) −4.76900e30 −0.906229
\(844\) −2.04789e31 −3.84336
\(845\) 0 0
\(846\) −3.66315e30 −0.670604
\(847\) −8.96799e30 −1.62151
\(848\) −5.15777e30 −0.921098
\(849\) 1.49377e30 0.263483
\(850\) 0 0
\(851\) −1.23645e30 −0.212773
\(852\) 1.08355e31 1.84176
\(853\) 5.20313e30 0.873575 0.436788 0.899565i \(-0.356116\pi\)
0.436788 + 0.899565i \(0.356116\pi\)
\(854\) −3.89391e31 −6.45771
\(855\) 0 0
\(856\) 1.01781e31 1.64699
\(857\) −6.67513e30 −1.06699 −0.533496 0.845803i \(-0.679122\pi\)
−0.533496 + 0.845803i \(0.679122\pi\)
\(858\) −3.76530e30 −0.594543
\(859\) 7.89066e29 0.123079 0.0615397 0.998105i \(-0.480399\pi\)
0.0615397 + 0.998105i \(0.480399\pi\)
\(860\) 0 0
\(861\) −6.42368e30 −0.977802
\(862\) −5.52440e30 −0.830729
\(863\) −1.23936e31 −1.84113 −0.920564 0.390591i \(-0.872271\pi\)
−0.920564 + 0.390591i \(0.872271\pi\)
\(864\) −4.80878e30 −0.705734
\(865\) 0 0
\(866\) 1.56087e31 2.23578
\(867\) 3.72115e30 0.526594
\(868\) 1.20118e31 1.67938
\(869\) 1.05504e30 0.145735
\(870\) 0 0
\(871\) 7.82912e30 1.05566
\(872\) 8.79654e30 1.17190
\(873\) −2.75602e30 −0.362771
\(874\) 1.16369e30 0.151345
\(875\) 0 0
\(876\) 9.70259e30 1.23195
\(877\) 8.03560e30 1.00814 0.504072 0.863662i \(-0.331835\pi\)
0.504072 + 0.863662i \(0.331835\pi\)
\(878\) −4.28944e30 −0.531751
\(879\) −8.26043e30 −1.01186
\(880\) 0 0
\(881\) 6.43033e30 0.769106 0.384553 0.923103i \(-0.374356\pi\)
0.384553 + 0.923103i \(0.374356\pi\)
\(882\) −1.30632e31 −1.54393
\(883\) −1.16447e31 −1.36000 −0.680002 0.733210i \(-0.738021\pi\)
−0.680002 + 0.733210i \(0.738021\pi\)
\(884\) 4.97466e31 5.74136
\(885\) 0 0
\(886\) −9.12525e30 −1.02847
\(887\) 1.24369e30 0.138520 0.0692601 0.997599i \(-0.477936\pi\)
0.0692601 + 0.997599i \(0.477936\pi\)
\(888\) 9.05837e30 0.997043
\(889\) −1.37960e29 −0.0150067
\(890\) 0 0
\(891\) 3.64483e29 0.0387223
\(892\) −1.42386e31 −1.49499
\(893\) −1.95961e30 −0.203343
\(894\) 5.48339e30 0.562348
\(895\) 0 0
\(896\) −5.47178e31 −5.48144
\(897\) 3.61856e30 0.358274
\(898\) −8.48436e30 −0.830267
\(899\) −3.08630e29 −0.0298512
\(900\) 0 0
\(901\) −3.73414e30 −0.352842
\(902\) 6.59012e30 0.615495
\(903\) 1.31881e31 1.21748
\(904\) 3.33645e31 3.04450
\(905\) 0 0
\(906\) 1.34920e31 1.20290
\(907\) 3.37302e30 0.297264 0.148632 0.988893i \(-0.452513\pi\)
0.148632 + 0.988893i \(0.452513\pi\)
\(908\) −5.43325e31 −4.73324
\(909\) −4.13482e30 −0.356070
\(910\) 0 0
\(911\) −1.99391e31 −1.67789 −0.838945 0.544216i \(-0.816827\pi\)
−0.838945 + 0.544216i \(0.816827\pi\)
\(912\) −4.87382e30 −0.405437
\(913\) −8.57705e29 −0.0705334
\(914\) −1.32717e30 −0.107892
\(915\) 0 0
\(916\) −5.50936e31 −4.37722
\(917\) 2.51078e30 0.197211
\(918\) −6.59606e30 −0.512196
\(919\) −3.09702e30 −0.237756 −0.118878 0.992909i \(-0.537930\pi\)
−0.118878 + 0.992909i \(0.537930\pi\)
\(920\) 0 0
\(921\) 9.80587e30 0.735800
\(922\) 2.78643e31 2.06716
\(923\) 2.46890e31 1.81086
\(924\) 1.38496e31 1.00434
\(925\) 0 0
\(926\) 3.92990e31 2.78591
\(927\) −4.08341e30 −0.286211
\(928\) 4.69490e30 0.325367
\(929\) 3.21440e30 0.220260 0.110130 0.993917i \(-0.464873\pi\)
0.110130 + 0.993917i \(0.464873\pi\)
\(930\) 0 0
\(931\) −6.98816e30 −0.468157
\(932\) 5.49033e31 3.63690
\(933\) 1.68830e31 1.10584
\(934\) 1.06717e29 0.00691181
\(935\) 0 0
\(936\) −2.65100e31 −1.67885
\(937\) −4.96558e30 −0.310960 −0.155480 0.987839i \(-0.549692\pi\)
−0.155480 + 0.987839i \(0.549692\pi\)
\(938\) −3.94452e31 −2.44267
\(939\) −4.00511e30 −0.245259
\(940\) 0 0
\(941\) 1.65186e30 0.0989193 0.0494597 0.998776i \(-0.484250\pi\)
0.0494597 + 0.998776i \(0.484250\pi\)
\(942\) −2.43477e31 −1.44186
\(943\) −6.33329e30 −0.370900
\(944\) −1.56927e31 −0.908851
\(945\) 0 0
\(946\) −1.35298e31 −0.766363
\(947\) −1.15884e31 −0.649155 −0.324578 0.945859i \(-0.605222\pi\)
−0.324578 + 0.945859i \(0.605222\pi\)
\(948\) 1.17861e31 0.652956
\(949\) 2.21076e31 1.21128
\(950\) 0 0
\(951\) 1.80679e30 0.0968308
\(952\) −1.57963e32 −8.37271
\(953\) 3.38294e31 1.77345 0.886724 0.462299i \(-0.152975\pi\)
0.886724 + 0.462299i \(0.152975\pi\)
\(954\) 3.15738e30 0.163708
\(955\) 0 0
\(956\) −7.75780e31 −3.93487
\(957\) −3.55851e29 −0.0178523
\(958\) −2.59477e31 −1.28754
\(959\) 4.68535e31 2.29957
\(960\) 0 0
\(961\) −1.84682e31 −0.886806
\(962\) 3.27487e31 1.55545
\(963\) −3.56196e30 −0.167345
\(964\) −9.62997e31 −4.47524
\(965\) 0 0
\(966\) −1.82313e31 −0.829005
\(967\) 1.15852e30 0.0521104 0.0260552 0.999661i \(-0.491705\pi\)
0.0260552 + 0.999661i \(0.491705\pi\)
\(968\) 6.47760e31 2.88219
\(969\) −3.52857e30 −0.155310
\(970\) 0 0
\(971\) 5.52187e30 0.237840 0.118920 0.992904i \(-0.462057\pi\)
0.118920 + 0.992904i \(0.462057\pi\)
\(972\) 4.07172e30 0.173493
\(973\) 7.72467e30 0.325608
\(974\) −3.71209e31 −1.54792
\(975\) 0 0
\(976\) 1.60791e32 6.56204
\(977\) −4.51123e29 −0.0182139 −0.00910693 0.999959i \(-0.502899\pi\)
−0.00910693 + 0.999959i \(0.502899\pi\)
\(978\) 9.10089e30 0.363518
\(979\) 1.55979e30 0.0616378
\(980\) 0 0
\(981\) −3.07847e30 −0.119072
\(982\) 1.60545e31 0.614365
\(983\) −4.25669e31 −1.61161 −0.805805 0.592180i \(-0.798267\pi\)
−0.805805 + 0.592180i \(0.798267\pi\)
\(984\) 4.63984e31 1.73802
\(985\) 0 0
\(986\) 6.43985e30 0.236139
\(987\) 3.07006e31 1.11383
\(988\) −2.25016e31 −0.807730
\(989\) 1.30025e31 0.461813
\(990\) 0 0
\(991\) 3.26274e30 0.113451 0.0567256 0.998390i \(-0.481934\pi\)
0.0567256 + 0.998390i \(0.481934\pi\)
\(992\) −3.58596e31 −1.23377
\(993\) 1.75908e31 0.598851
\(994\) −1.24390e32 −4.19013
\(995\) 0 0
\(996\) −9.58162e30 −0.316021
\(997\) 3.92458e31 1.28084 0.640419 0.768026i \(-0.278761\pi\)
0.640419 + 0.768026i \(0.278761\pi\)
\(998\) −3.07983e31 −0.994617
\(999\) −3.17009e30 −0.101306
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 75.22.a.f.1.1 3
5.2 odd 4 75.22.b.f.49.1 6
5.3 odd 4 75.22.b.f.49.6 6
5.4 even 2 15.22.a.c.1.3 3
15.14 odd 2 45.22.a.c.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.22.a.c.1.3 3 5.4 even 2
45.22.a.c.1.1 3 15.14 odd 2
75.22.a.f.1.1 3 1.1 even 1 trivial
75.22.b.f.49.1 6 5.2 odd 4
75.22.b.f.49.6 6 5.3 odd 4