Properties

Label 338.8.b.a.337.2
Level $338$
Weight $8$
Character 338.337
Analytic conductor $105.586$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,8,Mod(337,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.337");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(105.586138614\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 338.337
Dual form 338.8.b.a.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.00000i q^{2} -87.0000 q^{3} -64.0000 q^{4} +321.000i q^{5} -696.000i q^{6} +181.000i q^{7} -512.000i q^{8} +5382.00 q^{9} +O(q^{10})\) \(q+8.00000i q^{2} -87.0000 q^{3} -64.0000 q^{4} +321.000i q^{5} -696.000i q^{6} +181.000i q^{7} -512.000i q^{8} +5382.00 q^{9} -2568.00 q^{10} -7782.00i q^{11} +5568.00 q^{12} -1448.00 q^{14} -27927.0i q^{15} +4096.00 q^{16} -9069.00 q^{17} +43056.0i q^{18} -37150.0i q^{19} -20544.0i q^{20} -15747.0i q^{21} +62256.0 q^{22} -19008.0 q^{23} +44544.0i q^{24} -24916.0 q^{25} -277965. q^{27} -11584.0i q^{28} +174750. q^{29} +223416. q^{30} +29012.0i q^{31} +32768.0i q^{32} +677034. i q^{33} -72552.0i q^{34} -58101.0 q^{35} -344448. q^{36} -323669. i q^{37} +297200. q^{38} +164352. q^{40} +795312. i q^{41} +125976. q^{42} +314137. q^{43} +498048. i q^{44} +1.72762e6i q^{45} -152064. i q^{46} +447441. i q^{47} -356352. q^{48} +790782. q^{49} -199328. i q^{50} +789003. q^{51} -1.46923e6 q^{53} -2.22372e6i q^{54} +2.49802e6 q^{55} +92672.0 q^{56} +3.23205e6i q^{57} +1.39800e6i q^{58} -1.62777e6i q^{59} +1.78733e6i q^{60} -2.39961e6 q^{61} -232096. q^{62} +974142. i q^{63} -262144. q^{64} -5.41627e6 q^{66} -64066.0i q^{67} +580416. q^{68} +1.65370e6 q^{69} -464808. i q^{70} -322383. i q^{71} -2.75558e6i q^{72} +4.45478e6i q^{73} +2.58935e6 q^{74} +2.16769e6 q^{75} +2.37760e6i q^{76} +1.40854e6 q^{77} +753560. q^{79} +1.31482e6i q^{80} +1.24125e7 q^{81} -6.36250e6 q^{82} -1.21909e6i q^{83} +1.00781e6i q^{84} -2.91115e6i q^{85} +2.51310e6i q^{86} -1.52032e7 q^{87} -3.98438e6 q^{88} -3.39033e6i q^{89} -1.38210e7 q^{90} +1.21651e6 q^{92} -2.52404e6i q^{93} -3.57953e6 q^{94} +1.19252e7 q^{95} -2.85082e6i q^{96} +1.62877e6i q^{97} +6.32626e6i q^{98} -4.18827e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 174 q^{3} - 128 q^{4} + 10764 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 174 q^{3} - 128 q^{4} + 10764 q^{9} - 5136 q^{10} + 11136 q^{12} - 2896 q^{14} + 8192 q^{16} - 18138 q^{17} + 124512 q^{22} - 38016 q^{23} - 49832 q^{25} - 555930 q^{27} + 349500 q^{29} + 446832 q^{30} - 116202 q^{35} - 688896 q^{36} + 594400 q^{38} + 328704 q^{40} + 251952 q^{42} + 628274 q^{43} - 712704 q^{48} + 1581564 q^{49} + 1578006 q^{51} - 2938464 q^{53} + 4996044 q^{55} + 185344 q^{56} - 4799216 q^{61} - 464192 q^{62} - 524288 q^{64} - 10832544 q^{66} + 1160832 q^{68} + 3307392 q^{69} + 5178704 q^{74} + 4335384 q^{75} + 2817084 q^{77} + 1507120 q^{79} + 24825042 q^{81} - 12724992 q^{82} - 30406500 q^{87} - 7968768 q^{88} - 27641952 q^{90} + 2433024 q^{92} - 7159056 q^{94} + 23850300 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000i 0.707107i
\(3\) −87.0000 −1.86035 −0.930175 0.367115i \(-0.880345\pi\)
−0.930175 + 0.367115i \(0.880345\pi\)
\(4\) −64.0000 −0.500000
\(5\) 321.000i 1.14844i 0.818699 + 0.574222i \(0.194695\pi\)
−0.818699 + 0.574222i \(0.805305\pi\)
\(6\) − 696.000i − 1.31547i
\(7\) 181.000i 0.199451i 0.995015 + 0.0997253i \(0.0317964\pi\)
−0.995015 + 0.0997253i \(0.968204\pi\)
\(8\) − 512.000i − 0.353553i
\(9\) 5382.00 2.46091
\(10\) −2568.00 −0.812073
\(11\) − 7782.00i − 1.76286i −0.472318 0.881428i \(-0.656583\pi\)
0.472318 0.881428i \(-0.343417\pi\)
\(12\) 5568.00 0.930175
\(13\) 0 0
\(14\) −1448.00 −0.141033
\(15\) − 27927.0i − 2.13651i
\(16\) 4096.00 0.250000
\(17\) −9069.00 −0.447701 −0.223851 0.974623i \(-0.571863\pi\)
−0.223851 + 0.974623i \(0.571863\pi\)
\(18\) 43056.0i 1.74012i
\(19\) − 37150.0i − 1.24257i −0.783584 0.621286i \(-0.786611\pi\)
0.783584 0.621286i \(-0.213389\pi\)
\(20\) − 20544.0i − 0.574222i
\(21\) − 15747.0i − 0.371048i
\(22\) 62256.0 1.24653
\(23\) −19008.0 −0.325753 −0.162877 0.986646i \(-0.552077\pi\)
−0.162877 + 0.986646i \(0.552077\pi\)
\(24\) 44544.0i 0.657733i
\(25\) −24916.0 −0.318925
\(26\) 0 0
\(27\) −277965. −2.71780
\(28\) − 11584.0i − 0.0997253i
\(29\) 174750. 1.33053 0.665264 0.746608i \(-0.268319\pi\)
0.665264 + 0.746608i \(0.268319\pi\)
\(30\) 223416. 1.51074
\(31\) 29012.0i 0.174909i 0.996169 + 0.0874544i \(0.0278732\pi\)
−0.996169 + 0.0874544i \(0.972127\pi\)
\(32\) 32768.0i 0.176777i
\(33\) 677034.i 3.27953i
\(34\) − 72552.0i − 0.316572i
\(35\) −58101.0 −0.229058
\(36\) −344448. −1.23045
\(37\) − 323669.i − 1.05050i −0.850949 0.525249i \(-0.823972\pi\)
0.850949 0.525249i \(-0.176028\pi\)
\(38\) 297200. 0.878630
\(39\) 0 0
\(40\) 164352. 0.406036
\(41\) 795312.i 1.80216i 0.433650 + 0.901081i \(0.357225\pi\)
−0.433650 + 0.901081i \(0.642775\pi\)
\(42\) 125976. 0.262371
\(43\) 314137. 0.602531 0.301266 0.953540i \(-0.402591\pi\)
0.301266 + 0.953540i \(0.402591\pi\)
\(44\) 498048.i 0.881428i
\(45\) 1.72762e6i 2.82621i
\(46\) − 152064.i − 0.230342i
\(47\) 447441.i 0.628627i 0.949319 + 0.314314i \(0.101774\pi\)
−0.949319 + 0.314314i \(0.898226\pi\)
\(48\) −356352. −0.465088
\(49\) 790782. 0.960219
\(50\) − 199328.i − 0.225514i
\(51\) 789003. 0.832881
\(52\) 0 0
\(53\) −1.46923e6 −1.35558 −0.677790 0.735256i \(-0.737062\pi\)
−0.677790 + 0.735256i \(0.737062\pi\)
\(54\) − 2.22372e6i − 1.92177i
\(55\) 2.49802e6 2.02454
\(56\) 92672.0 0.0705165
\(57\) 3.23205e6i 2.31162i
\(58\) 1.39800e6i 0.940826i
\(59\) − 1.62777e6i − 1.03184i −0.856638 0.515918i \(-0.827451\pi\)
0.856638 0.515918i \(-0.172549\pi\)
\(60\) 1.78733e6i 1.06825i
\(61\) −2.39961e6 −1.35359 −0.676793 0.736173i \(-0.736631\pi\)
−0.676793 + 0.736173i \(0.736631\pi\)
\(62\) −232096. −0.123679
\(63\) 974142.i 0.490829i
\(64\) −262144. −0.125000
\(65\) 0 0
\(66\) −5.41627e6 −2.31898
\(67\) − 64066.0i − 0.0260235i −0.999915 0.0130118i \(-0.995858\pi\)
0.999915 0.0130118i \(-0.00414189\pi\)
\(68\) 580416. 0.223851
\(69\) 1.65370e6 0.606016
\(70\) − 464808.i − 0.161968i
\(71\) − 322383.i − 0.106898i −0.998571 0.0534488i \(-0.982979\pi\)
0.998571 0.0534488i \(-0.0170214\pi\)
\(72\) − 2.75558e6i − 0.870061i
\(73\) 4.45478e6i 1.34028i 0.742233 + 0.670141i \(0.233767\pi\)
−0.742233 + 0.670141i \(0.766233\pi\)
\(74\) 2.58935e6 0.742814
\(75\) 2.16769e6 0.593312
\(76\) 2.37760e6i 0.621286i
\(77\) 1.40854e6 0.351603
\(78\) 0 0
\(79\) 753560. 0.171958 0.0859791 0.996297i \(-0.472598\pi\)
0.0859791 + 0.996297i \(0.472598\pi\)
\(80\) 1.31482e6i 0.287111i
\(81\) 1.24125e7 2.59515
\(82\) −6.36250e6 −1.27432
\(83\) − 1.21909e6i − 0.234025i −0.993130 0.117013i \(-0.962668\pi\)
0.993130 0.117013i \(-0.0373318\pi\)
\(84\) 1.00781e6i 0.185524i
\(85\) − 2.91115e6i − 0.514160i
\(86\) 2.51310e6i 0.426054i
\(87\) −1.52032e7 −2.47525
\(88\) −3.98438e6 −0.623264
\(89\) − 3.39033e6i − 0.509773i −0.966971 0.254887i \(-0.917962\pi\)
0.966971 0.254887i \(-0.0820381\pi\)
\(90\) −1.38210e7 −1.99843
\(91\) 0 0
\(92\) 1.21651e6 0.162877
\(93\) − 2.52404e6i − 0.325392i
\(94\) −3.57953e6 −0.444507
\(95\) 1.19252e7 1.42702
\(96\) − 2.85082e6i − 0.328867i
\(97\) 1.62877e6i 0.181201i 0.995887 + 0.0906003i \(0.0288786\pi\)
−0.995887 + 0.0906003i \(0.971121\pi\)
\(98\) 6.32626e6i 0.678978i
\(99\) − 4.18827e7i − 4.33822i
\(100\) 1.59462e6 0.159462
\(101\) 1.53503e7 1.48249 0.741244 0.671236i \(-0.234236\pi\)
0.741244 + 0.671236i \(0.234236\pi\)
\(102\) 6.31202e6i 0.588936i
\(103\) −6.87643e6 −0.620058 −0.310029 0.950727i \(-0.600339\pi\)
−0.310029 + 0.950727i \(0.600339\pi\)
\(104\) 0 0
\(105\) 5.05479e6 0.426128
\(106\) − 1.17539e7i − 0.958539i
\(107\) −1.52027e7 −1.19971 −0.599857 0.800107i \(-0.704776\pi\)
−0.599857 + 0.800107i \(0.704776\pi\)
\(108\) 1.77898e7 1.35890
\(109\) 6.73260e6i 0.497955i 0.968509 + 0.248978i \(0.0800946\pi\)
−0.968509 + 0.248978i \(0.919905\pi\)
\(110\) 1.99842e7i 1.43157i
\(111\) 2.81592e7i 1.95429i
\(112\) 741376.i 0.0498627i
\(113\) −1.15292e7 −0.751667 −0.375833 0.926687i \(-0.622644\pi\)
−0.375833 + 0.926687i \(0.622644\pi\)
\(114\) −2.58564e7 −1.63456
\(115\) − 6.10157e6i − 0.374110i
\(116\) −1.11840e7 −0.665264
\(117\) 0 0
\(118\) 1.30222e7 0.729619
\(119\) − 1.64149e6i − 0.0892943i
\(120\) −1.42986e7 −0.755370
\(121\) −4.10724e7 −2.10766
\(122\) − 1.91969e7i − 0.957130i
\(123\) − 6.91921e7i − 3.35266i
\(124\) − 1.85677e6i − 0.0874544i
\(125\) 1.70801e7i 0.782177i
\(126\) −7.79314e6 −0.347069
\(127\) −2.06699e7 −0.895418 −0.447709 0.894179i \(-0.647760\pi\)
−0.447709 + 0.894179i \(0.647760\pi\)
\(128\) − 2.09715e6i − 0.0883883i
\(129\) −2.73299e7 −1.12092
\(130\) 0 0
\(131\) −1.90949e7 −0.742107 −0.371054 0.928611i \(-0.621003\pi\)
−0.371054 + 0.928611i \(0.621003\pi\)
\(132\) − 4.33302e7i − 1.63977i
\(133\) 6.72415e6 0.247832
\(134\) 512528. 0.0184014
\(135\) − 8.92268e7i − 3.12124i
\(136\) 4.64333e6i 0.158286i
\(137\) − 2.96901e7i − 0.986482i −0.869893 0.493241i \(-0.835812\pi\)
0.869893 0.493241i \(-0.164188\pi\)
\(138\) 1.32296e7i 0.428518i
\(139\) 1.55652e7 0.491591 0.245795 0.969322i \(-0.420951\pi\)
0.245795 + 0.969322i \(0.420951\pi\)
\(140\) 3.71846e6 0.114529
\(141\) − 3.89274e7i − 1.16947i
\(142\) 2.57906e6 0.0755880
\(143\) 0 0
\(144\) 2.20447e7 0.615226
\(145\) 5.60948e7i 1.52804i
\(146\) −3.56383e7 −0.947723
\(147\) −6.87980e7 −1.78635
\(148\) 2.07148e7i 0.525249i
\(149\) − 2.49675e6i − 0.0618334i −0.999522 0.0309167i \(-0.990157\pi\)
0.999522 0.0309167i \(-0.00984266\pi\)
\(150\) 1.73415e7i 0.419535i
\(151\) − 2.39802e7i − 0.566804i −0.959001 0.283402i \(-0.908537\pi\)
0.959001 0.283402i \(-0.0914630\pi\)
\(152\) −1.90208e7 −0.439315
\(153\) −4.88094e7 −1.10175
\(154\) 1.12683e7i 0.248621i
\(155\) −9.31285e6 −0.200873
\(156\) 0 0
\(157\) 1.70550e7 0.351725 0.175863 0.984415i \(-0.443729\pi\)
0.175863 + 0.984415i \(0.443729\pi\)
\(158\) 6.02848e6i 0.121593i
\(159\) 1.27823e8 2.52185
\(160\) −1.05185e7 −0.203018
\(161\) − 3.44045e6i − 0.0649717i
\(162\) 9.93002e7i 1.83505i
\(163\) 7.34586e7i 1.32857i 0.747477 + 0.664287i \(0.231265\pi\)
−0.747477 + 0.664287i \(0.768735\pi\)
\(164\) − 5.09000e7i − 0.901081i
\(165\) −2.17328e8 −3.76636
\(166\) 9.75274e6 0.165481
\(167\) − 4.66860e7i − 0.775674i −0.921728 0.387837i \(-0.873222\pi\)
0.921728 0.387837i \(-0.126778\pi\)
\(168\) −8.06246e6 −0.131185
\(169\) 0 0
\(170\) 2.32892e7 0.363566
\(171\) − 1.99941e8i − 3.05785i
\(172\) −2.01048e7 −0.301266
\(173\) 7.80931e7 1.14670 0.573352 0.819309i \(-0.305643\pi\)
0.573352 + 0.819309i \(0.305643\pi\)
\(174\) − 1.21626e8i − 1.75027i
\(175\) − 4.50980e6i − 0.0636098i
\(176\) − 3.18751e7i − 0.440714i
\(177\) 1.41616e8i 1.91958i
\(178\) 2.71226e7 0.360464
\(179\) 5.56163e7 0.724797 0.362399 0.932023i \(-0.381958\pi\)
0.362399 + 0.932023i \(0.381958\pi\)
\(180\) − 1.10568e8i − 1.41311i
\(181\) 1.19435e8 1.49711 0.748557 0.663070i \(-0.230747\pi\)
0.748557 + 0.663070i \(0.230747\pi\)
\(182\) 0 0
\(183\) 2.08766e8 2.51815
\(184\) 9.73210e6i 0.115171i
\(185\) 1.03898e8 1.20644
\(186\) 2.01924e7 0.230087
\(187\) 7.05750e7i 0.789233i
\(188\) − 2.86362e7i − 0.314314i
\(189\) − 5.03117e7i − 0.542066i
\(190\) 9.54012e7i 1.00906i
\(191\) 1.05485e8 1.09540 0.547700 0.836675i \(-0.315504\pi\)
0.547700 + 0.836675i \(0.315504\pi\)
\(192\) 2.28065e7 0.232544
\(193\) − 2.12059e7i − 0.212327i −0.994349 0.106164i \(-0.966143\pi\)
0.994349 0.106164i \(-0.0338567\pi\)
\(194\) −1.30302e7 −0.128128
\(195\) 0 0
\(196\) −5.06100e7 −0.480110
\(197\) − 1.66535e8i − 1.55194i −0.630771 0.775969i \(-0.717261\pi\)
0.630771 0.775969i \(-0.282739\pi\)
\(198\) 3.35062e8 3.06759
\(199\) 1.26351e8 1.13656 0.568279 0.822836i \(-0.307609\pi\)
0.568279 + 0.822836i \(0.307609\pi\)
\(200\) 1.27570e7i 0.112757i
\(201\) 5.57374e6i 0.0484129i
\(202\) 1.22802e8i 1.04828i
\(203\) 3.16298e7i 0.265375i
\(204\) −5.04962e7 −0.416441
\(205\) −2.55295e8 −2.06968
\(206\) − 5.50114e7i − 0.438448i
\(207\) −1.02301e8 −0.801648
\(208\) 0 0
\(209\) −2.89101e8 −2.19047
\(210\) 4.04383e7i 0.301318i
\(211\) 1.08571e8 0.795655 0.397828 0.917460i \(-0.369764\pi\)
0.397828 + 0.917460i \(0.369764\pi\)
\(212\) 9.40308e7 0.677790
\(213\) 2.80473e7i 0.198867i
\(214\) − 1.21622e8i − 0.848326i
\(215\) 1.00838e8i 0.691974i
\(216\) 1.42318e8i 0.960886i
\(217\) −5.25117e6 −0.0348857
\(218\) −5.38608e7 −0.352108
\(219\) − 3.87566e8i − 2.49340i
\(220\) −1.59873e8 −1.01227
\(221\) 0 0
\(222\) −2.25274e8 −1.38189
\(223\) − 1.25603e8i − 0.758459i −0.925303 0.379229i \(-0.876189\pi\)
0.925303 0.379229i \(-0.123811\pi\)
\(224\) −5.93101e6 −0.0352582
\(225\) −1.34098e8 −0.784844
\(226\) − 9.22338e7i − 0.531509i
\(227\) 1.90774e8i 1.08250i 0.840861 + 0.541252i \(0.182049\pi\)
−0.840861 + 0.541252i \(0.817951\pi\)
\(228\) − 2.06851e8i − 1.15581i
\(229\) 5.28911e7i 0.291044i 0.989355 + 0.145522i \(0.0464861\pi\)
−0.989355 + 0.145522i \(0.953514\pi\)
\(230\) 4.88125e7 0.264536
\(231\) −1.22543e8 −0.654104
\(232\) − 8.94720e7i − 0.470413i
\(233\) −1.51254e8 −0.783359 −0.391680 0.920102i \(-0.628106\pi\)
−0.391680 + 0.920102i \(0.628106\pi\)
\(234\) 0 0
\(235\) −1.43629e8 −0.721944
\(236\) 1.04177e8i 0.515918i
\(237\) −6.55597e7 −0.319903
\(238\) 1.31319e7 0.0631406
\(239\) 2.61917e8i 1.24100i 0.784208 + 0.620498i \(0.213070\pi\)
−0.784208 + 0.620498i \(0.786930\pi\)
\(240\) − 1.14389e8i − 0.534127i
\(241\) 1.31752e8i 0.606312i 0.952941 + 0.303156i \(0.0980404\pi\)
−0.952941 + 0.303156i \(0.901960\pi\)
\(242\) − 3.28579e8i − 1.49034i
\(243\) −4.71980e8 −2.11009
\(244\) 1.53575e8 0.676793
\(245\) 2.53841e8i 1.10276i
\(246\) 5.53537e8 2.37069
\(247\) 0 0
\(248\) 1.48541e7 0.0618396
\(249\) 1.06061e8i 0.435370i
\(250\) −1.36641e8 −0.553083
\(251\) −2.47061e8 −0.986159 −0.493080 0.869984i \(-0.664129\pi\)
−0.493080 + 0.869984i \(0.664129\pi\)
\(252\) − 6.23451e7i − 0.245415i
\(253\) 1.47920e8i 0.574256i
\(254\) − 1.65359e8i − 0.633156i
\(255\) 2.53270e8i 0.956518i
\(256\) 1.67772e7 0.0625000
\(257\) −2.27286e8 −0.835231 −0.417616 0.908624i \(-0.637134\pi\)
−0.417616 + 0.908624i \(0.637134\pi\)
\(258\) − 2.18639e8i − 0.792610i
\(259\) 5.85841e7 0.209522
\(260\) 0 0
\(261\) 9.40504e8 3.27430
\(262\) − 1.52759e8i − 0.524749i
\(263\) −4.25872e8 −1.44356 −0.721779 0.692124i \(-0.756675\pi\)
−0.721779 + 0.692124i \(0.756675\pi\)
\(264\) 3.46641e8 1.15949
\(265\) − 4.71623e8i − 1.55681i
\(266\) 5.37932e7i 0.175243i
\(267\) 2.94959e8i 0.948357i
\(268\) 4.10022e6i 0.0130118i
\(269\) −5.14154e8 −1.61050 −0.805250 0.592936i \(-0.797969\pi\)
−0.805250 + 0.592936i \(0.797969\pi\)
\(270\) 7.13814e8 2.20705
\(271\) − 4.57096e7i − 0.139513i −0.997564 0.0697565i \(-0.977778\pi\)
0.997564 0.0697565i \(-0.0222222\pi\)
\(272\) −3.71466e7 −0.111925
\(273\) 0 0
\(274\) 2.37521e8 0.697548
\(275\) 1.93896e8i 0.562218i
\(276\) −1.05837e8 −0.303008
\(277\) 2.73964e8 0.774487 0.387244 0.921977i \(-0.373427\pi\)
0.387244 + 0.921977i \(0.373427\pi\)
\(278\) 1.24522e8i 0.347607i
\(279\) 1.56143e8i 0.430434i
\(280\) 2.97477e7i 0.0809842i
\(281\) − 4.21707e8i − 1.13381i −0.823784 0.566903i \(-0.808141\pi\)
0.823784 0.566903i \(-0.191859\pi\)
\(282\) 3.11419e8 0.826938
\(283\) −3.81957e8 −1.00176 −0.500878 0.865518i \(-0.666990\pi\)
−0.500878 + 0.865518i \(0.666990\pi\)
\(284\) 2.06325e7i 0.0534488i
\(285\) −1.03749e9 −2.65477
\(286\) 0 0
\(287\) −1.43951e8 −0.359443
\(288\) 1.76357e8i 0.435031i
\(289\) −3.28092e8 −0.799564
\(290\) −4.48758e8 −1.08049
\(291\) − 1.41703e8i − 0.337097i
\(292\) − 2.85106e8i − 0.670141i
\(293\) 4.04833e8i 0.940240i 0.882602 + 0.470120i \(0.155789\pi\)
−0.882602 + 0.470120i \(0.844211\pi\)
\(294\) − 5.50384e8i − 1.26314i
\(295\) 5.22514e8 1.18501
\(296\) −1.65719e8 −0.371407
\(297\) 2.16312e9i 4.79108i
\(298\) 1.99740e7 0.0437228
\(299\) 0 0
\(300\) −1.38732e8 −0.296656
\(301\) 5.68588e7i 0.120175i
\(302\) 1.91841e8 0.400791
\(303\) −1.33547e9 −2.75795
\(304\) − 1.52166e8i − 0.310643i
\(305\) − 7.70274e8i − 1.55452i
\(306\) − 3.90475e8i − 0.779055i
\(307\) 4.75520e7i 0.0937960i 0.998900 + 0.0468980i \(0.0149336\pi\)
−0.998900 + 0.0468980i \(0.985066\pi\)
\(308\) −9.01467e7 −0.175801
\(309\) 5.98249e8 1.15353
\(310\) − 7.45028e7i − 0.142039i
\(311\) 3.02841e8 0.570892 0.285446 0.958395i \(-0.407858\pi\)
0.285446 + 0.958395i \(0.407858\pi\)
\(312\) 0 0
\(313\) −6.31685e8 −1.16438 −0.582191 0.813052i \(-0.697804\pi\)
−0.582191 + 0.813052i \(0.697804\pi\)
\(314\) 1.36440e8i 0.248707i
\(315\) −3.12700e8 −0.563690
\(316\) −4.82278e7 −0.0859791
\(317\) 7.93332e8i 1.39877i 0.714743 + 0.699387i \(0.246544\pi\)
−0.714743 + 0.699387i \(0.753456\pi\)
\(318\) 1.02259e9i 1.78322i
\(319\) − 1.35990e9i − 2.34553i
\(320\) − 8.41482e7i − 0.143556i
\(321\) 1.32264e9 2.23189
\(322\) 2.75236e7 0.0459420
\(323\) 3.36913e8i 0.556300i
\(324\) −7.94401e8 −1.29757
\(325\) 0 0
\(326\) −5.87669e8 −0.939444
\(327\) − 5.85737e8i − 0.926372i
\(328\) 4.07200e8 0.637161
\(329\) −8.09868e7 −0.125380
\(330\) − 1.73862e9i − 2.66322i
\(331\) − 1.21628e9i − 1.84346i −0.387829 0.921731i \(-0.626775\pi\)
0.387829 0.921731i \(-0.373225\pi\)
\(332\) 7.80219e7i 0.117013i
\(333\) − 1.74199e9i − 2.58518i
\(334\) 3.73488e8 0.548484
\(335\) 2.05652e7 0.0298866
\(336\) − 6.44997e7i − 0.0927620i
\(337\) 1.51221e8 0.215232 0.107616 0.994193i \(-0.465678\pi\)
0.107616 + 0.994193i \(0.465678\pi\)
\(338\) 0 0
\(339\) 1.00304e9 1.39836
\(340\) 1.86314e8i 0.257080i
\(341\) 2.25771e8 0.308339
\(342\) 1.59953e9 2.16223
\(343\) 2.92193e8i 0.390967i
\(344\) − 1.60838e8i − 0.213027i
\(345\) 5.30836e8i 0.695975i
\(346\) 6.24745e8i 0.810842i
\(347\) −5.97234e8 −0.767347 −0.383673 0.923469i \(-0.625341\pi\)
−0.383673 + 0.923469i \(0.625341\pi\)
\(348\) 9.73008e8 1.23762
\(349\) − 1.19600e8i − 0.150606i −0.997161 0.0753029i \(-0.976008\pi\)
0.997161 0.0753029i \(-0.0239924\pi\)
\(350\) 3.60784e7 0.0449789
\(351\) 0 0
\(352\) 2.55001e8 0.311632
\(353\) − 4.66414e8i − 0.564366i −0.959361 0.282183i \(-0.908942\pi\)
0.959361 0.282183i \(-0.0910585\pi\)
\(354\) −1.13293e9 −1.35735
\(355\) 1.03485e8 0.122766
\(356\) 2.16981e8i 0.254887i
\(357\) 1.42810e8i 0.166119i
\(358\) 4.44931e8i 0.512509i
\(359\) 7.70102e8i 0.878451i 0.898377 + 0.439225i \(0.144747\pi\)
−0.898377 + 0.439225i \(0.855253\pi\)
\(360\) 8.84542e8 0.999217
\(361\) −4.86251e8 −0.543983
\(362\) 9.55477e8i 1.05862i
\(363\) 3.57329e9 3.92099
\(364\) 0 0
\(365\) −1.42999e9 −1.53924
\(366\) 1.67013e9i 1.78060i
\(367\) −8.55319e8 −0.903227 −0.451613 0.892214i \(-0.649151\pi\)
−0.451613 + 0.892214i \(0.649151\pi\)
\(368\) −7.78568e7 −0.0814384
\(369\) 4.28037e9i 4.43495i
\(370\) 8.31182e8i 0.853081i
\(371\) − 2.65931e8i − 0.270371i
\(372\) 1.61539e8i 0.162696i
\(373\) −5.29609e8 −0.528414 −0.264207 0.964466i \(-0.585110\pi\)
−0.264207 + 0.964466i \(0.585110\pi\)
\(374\) −5.64600e8 −0.558072
\(375\) − 1.48597e9i − 1.45512i
\(376\) 2.29090e8 0.222253
\(377\) 0 0
\(378\) 4.02493e8 0.383299
\(379\) 1.98358e9i 1.87159i 0.352540 + 0.935797i \(0.385318\pi\)
−0.352540 + 0.935797i \(0.614682\pi\)
\(380\) −7.63210e8 −0.713512
\(381\) 1.79828e9 1.66579
\(382\) 8.43877e8i 0.774564i
\(383\) 8.98756e8i 0.817422i 0.912664 + 0.408711i \(0.134022\pi\)
−0.912664 + 0.408711i \(0.865978\pi\)
\(384\) 1.82452e8i 0.164433i
\(385\) 4.52142e8i 0.403796i
\(386\) 1.69647e8 0.150138
\(387\) 1.69069e9 1.48277
\(388\) − 1.04242e8i − 0.0906003i
\(389\) −1.82475e9 −1.57174 −0.785868 0.618395i \(-0.787783\pi\)
−0.785868 + 0.618395i \(0.787783\pi\)
\(390\) 0 0
\(391\) 1.72384e8 0.145840
\(392\) − 4.04880e8i − 0.339489i
\(393\) 1.66125e9 1.38058
\(394\) 1.33228e9 1.09739
\(395\) 2.41893e8i 0.197485i
\(396\) 2.68049e9i 2.16911i
\(397\) − 4.93083e8i − 0.395506i −0.980252 0.197753i \(-0.936636\pi\)
0.980252 0.197753i \(-0.0633645\pi\)
\(398\) 1.01081e9i 0.803668i
\(399\) −5.85001e8 −0.461054
\(400\) −1.02056e8 −0.0797312
\(401\) 5.68280e8i 0.440105i 0.975488 + 0.220053i \(0.0706229\pi\)
−0.975488 + 0.220053i \(0.929377\pi\)
\(402\) −4.45899e7 −0.0342331
\(403\) 0 0
\(404\) −9.82417e8 −0.741244
\(405\) 3.98442e9i 2.98039i
\(406\) −2.53038e8 −0.187648
\(407\) −2.51879e9 −1.85188
\(408\) − 4.03970e8i − 0.294468i
\(409\) − 1.28472e9i − 0.928489i −0.885707 0.464245i \(-0.846326\pi\)
0.885707 0.464245i \(-0.153674\pi\)
\(410\) − 2.04236e9i − 1.46349i
\(411\) 2.58304e9i 1.83520i
\(412\) 4.40091e8 0.310029
\(413\) 2.94626e8 0.205801
\(414\) − 8.18408e8i − 0.566851i
\(415\) 3.91329e8 0.268765
\(416\) 0 0
\(417\) −1.35418e9 −0.914532
\(418\) − 2.31281e9i − 1.54890i
\(419\) 2.74847e8 0.182533 0.0912667 0.995826i \(-0.470908\pi\)
0.0912667 + 0.995826i \(0.470908\pi\)
\(420\) −3.23506e8 −0.213064
\(421\) 7.51368e8i 0.490756i 0.969428 + 0.245378i \(0.0789120\pi\)
−0.969428 + 0.245378i \(0.921088\pi\)
\(422\) 8.68568e8i 0.562613i
\(423\) 2.40813e9i 1.54699i
\(424\) 7.52247e8i 0.479270i
\(425\) 2.25963e8 0.142783
\(426\) −2.24379e8 −0.140620
\(427\) − 4.34329e8i − 0.269974i
\(428\) 9.72974e8 0.599857
\(429\) 0 0
\(430\) −8.06704e8 −0.489299
\(431\) − 1.30756e8i − 0.0786668i −0.999226 0.0393334i \(-0.987477\pi\)
0.999226 0.0393334i \(-0.0125234\pi\)
\(432\) −1.13854e9 −0.679449
\(433\) −1.66736e9 −0.987010 −0.493505 0.869743i \(-0.664284\pi\)
−0.493505 + 0.869743i \(0.664284\pi\)
\(434\) − 4.20094e7i − 0.0246679i
\(435\) − 4.88024e9i − 2.84269i
\(436\) − 4.30887e8i − 0.248978i
\(437\) 7.06147e8i 0.404772i
\(438\) 3.10053e9 1.76310
\(439\) −2.31478e9 −1.30582 −0.652910 0.757436i \(-0.726452\pi\)
−0.652910 + 0.757436i \(0.726452\pi\)
\(440\) − 1.27899e9i − 0.715784i
\(441\) 4.25599e9 2.36301
\(442\) 0 0
\(443\) −6.90047e8 −0.377108 −0.188554 0.982063i \(-0.560380\pi\)
−0.188554 + 0.982063i \(0.560380\pi\)
\(444\) − 1.80219e9i − 0.977147i
\(445\) 1.08830e9 0.585446
\(446\) 1.00482e9 0.536311
\(447\) 2.17217e8i 0.115032i
\(448\) − 4.74481e7i − 0.0249313i
\(449\) − 2.63806e9i − 1.37538i −0.726004 0.687690i \(-0.758625\pi\)
0.726004 0.687690i \(-0.241375\pi\)
\(450\) − 1.07278e9i − 0.554968i
\(451\) 6.18912e9 3.17695
\(452\) 7.37870e8 0.375833
\(453\) 2.08627e9i 1.05445i
\(454\) −1.52619e9 −0.765445
\(455\) 0 0
\(456\) 1.65481e9 0.817280
\(457\) − 6.16222e8i − 0.302016i −0.988533 0.151008i \(-0.951748\pi\)
0.988533 0.151008i \(-0.0482520\pi\)
\(458\) −4.23129e8 −0.205799
\(459\) 2.52086e9 1.21676
\(460\) 3.90500e8i 0.187055i
\(461\) 1.23621e9i 0.587679i 0.955855 + 0.293839i \(0.0949330\pi\)
−0.955855 + 0.293839i \(0.905067\pi\)
\(462\) − 9.80345e8i − 0.462522i
\(463\) − 6.78469e7i − 0.0317685i −0.999874 0.0158843i \(-0.994944\pi\)
0.999874 0.0158843i \(-0.00505633\pi\)
\(464\) 7.15776e8 0.332632
\(465\) 8.10218e8 0.373694
\(466\) − 1.21003e9i − 0.553919i
\(467\) 1.17502e9 0.533869 0.266934 0.963715i \(-0.413989\pi\)
0.266934 + 0.963715i \(0.413989\pi\)
\(468\) 0 0
\(469\) 1.15959e7 0.00519040
\(470\) − 1.14903e9i − 0.510491i
\(471\) −1.48379e9 −0.654332
\(472\) −8.33418e8 −0.364809
\(473\) − 2.44461e9i − 1.06218i
\(474\) − 5.24478e8i − 0.226205i
\(475\) 9.25629e8i 0.396287i
\(476\) 1.05055e8i 0.0446471i
\(477\) −7.90741e9 −3.33595
\(478\) −2.09533e9 −0.877517
\(479\) − 3.96154e8i − 0.164699i −0.996604 0.0823494i \(-0.973758\pi\)
0.996604 0.0823494i \(-0.0262423\pi\)
\(480\) 9.15112e8 0.377685
\(481\) 0 0
\(482\) −1.05401e9 −0.428728
\(483\) 2.99319e8i 0.120870i
\(484\) 2.62863e9 1.05383
\(485\) −5.22836e8 −0.208099
\(486\) − 3.77584e9i − 1.49206i
\(487\) − 3.03665e9i − 1.19136i −0.803222 0.595680i \(-0.796883\pi\)
0.803222 0.595680i \(-0.203117\pi\)
\(488\) 1.22860e9i 0.478565i
\(489\) − 6.39090e9i − 2.47162i
\(490\) −2.03073e9 −0.779768
\(491\) 2.91974e9 1.11316 0.556582 0.830793i \(-0.312113\pi\)
0.556582 + 0.830793i \(0.312113\pi\)
\(492\) 4.42830e9i 1.67633i
\(493\) −1.58481e9 −0.595679
\(494\) 0 0
\(495\) 1.34444e10 4.98221
\(496\) 1.18833e8i 0.0437272i
\(497\) 5.83513e7 0.0213208
\(498\) −8.48488e8 −0.307853
\(499\) − 1.62343e9i − 0.584898i −0.956281 0.292449i \(-0.905530\pi\)
0.956281 0.292449i \(-0.0944702\pi\)
\(500\) − 1.09313e9i − 0.391089i
\(501\) 4.06168e9i 1.44303i
\(502\) − 1.97649e9i − 0.697320i
\(503\) 4.75888e9 1.66731 0.833655 0.552285i \(-0.186244\pi\)
0.833655 + 0.552285i \(0.186244\pi\)
\(504\) 4.98761e8 0.173534
\(505\) 4.92744e9i 1.70256i
\(506\) −1.18336e9 −0.406061
\(507\) 0 0
\(508\) 1.32288e9 0.447709
\(509\) − 9.19375e8i − 0.309016i −0.987992 0.154508i \(-0.950621\pi\)
0.987992 0.154508i \(-0.0493792\pi\)
\(510\) −2.02616e9 −0.676360
\(511\) −8.06316e8 −0.267320
\(512\) 1.34218e8i 0.0441942i
\(513\) 1.03264e10i 3.37706i
\(514\) − 1.81829e9i − 0.590598i
\(515\) − 2.20733e9i − 0.712103i
\(516\) 1.74911e9 0.560460
\(517\) 3.48199e9 1.10818
\(518\) 4.68673e8i 0.148155i
\(519\) −6.79410e9 −2.13327
\(520\) 0 0
\(521\) −1.46089e9 −0.452569 −0.226284 0.974061i \(-0.572658\pi\)
−0.226284 + 0.974061i \(0.572658\pi\)
\(522\) 7.52404e9i 2.31528i
\(523\) 2.12856e9 0.650624 0.325312 0.945607i \(-0.394531\pi\)
0.325312 + 0.945607i \(0.394531\pi\)
\(524\) 1.22207e9 0.371054
\(525\) 3.92352e8i 0.118336i
\(526\) − 3.40698e9i − 1.02075i
\(527\) − 2.63110e8i − 0.0783069i
\(528\) 2.77313e9i 0.819883i
\(529\) −3.04352e9 −0.893885
\(530\) 3.77299e9 1.10083
\(531\) − 8.76066e9i − 2.53925i
\(532\) −4.30346e8 −0.123916
\(533\) 0 0
\(534\) −2.35967e9 −0.670590
\(535\) − 4.88007e9i − 1.37781i
\(536\) −3.28018e7 −0.00920070
\(537\) −4.83862e9 −1.34838
\(538\) − 4.11323e9i − 1.13879i
\(539\) − 6.15387e9i − 1.69273i
\(540\) 5.71051e9i 1.56062i
\(541\) 1.72479e7i 0.00468324i 0.999997 + 0.00234162i \(0.000745361\pi\)
−0.999997 + 0.00234162i \(0.999255\pi\)
\(542\) 3.65677e8 0.0986507
\(543\) −1.03908e10 −2.78516
\(544\) − 2.97173e8i − 0.0791431i
\(545\) −2.16117e9 −0.571874
\(546\) 0 0
\(547\) 7.51154e8 0.196234 0.0981168 0.995175i \(-0.468718\pi\)
0.0981168 + 0.995175i \(0.468718\pi\)
\(548\) 1.90016e9i 0.493241i
\(549\) −1.29147e10 −3.33105
\(550\) −1.55117e9 −0.397549
\(551\) − 6.49196e9i − 1.65328i
\(552\) − 8.46692e8i − 0.214259i
\(553\) 1.36394e8i 0.0342972i
\(554\) 2.19171e9i 0.547645i
\(555\) −9.03910e9 −2.24440
\(556\) −9.96175e8 −0.245795
\(557\) − 3.00701e9i − 0.737295i −0.929569 0.368647i \(-0.879821\pi\)
0.929569 0.368647i \(-0.120179\pi\)
\(558\) −1.24914e9 −0.304363
\(559\) 0 0
\(560\) −2.37982e8 −0.0572645
\(561\) − 6.14002e9i − 1.46825i
\(562\) 3.37366e9 0.801722
\(563\) −2.82880e9 −0.668070 −0.334035 0.942561i \(-0.608410\pi\)
−0.334035 + 0.942561i \(0.608410\pi\)
\(564\) 2.49135e9i 0.584734i
\(565\) − 3.70088e9i − 0.863248i
\(566\) − 3.05566e9i − 0.708349i
\(567\) 2.24667e9i 0.517604i
\(568\) −1.65060e8 −0.0377940
\(569\) −7.67290e9 −1.74609 −0.873045 0.487639i \(-0.837858\pi\)
−0.873045 + 0.487639i \(0.837858\pi\)
\(570\) − 8.29990e9i − 1.87720i
\(571\) 3.09363e9 0.695411 0.347706 0.937604i \(-0.386961\pi\)
0.347706 + 0.937604i \(0.386961\pi\)
\(572\) 0 0
\(573\) −9.17717e9 −2.03783
\(574\) − 1.15161e9i − 0.254164i
\(575\) 4.73603e8 0.103891
\(576\) −1.41086e9 −0.307613
\(577\) 3.71815e9i 0.805770i 0.915251 + 0.402885i \(0.131993\pi\)
−0.915251 + 0.402885i \(0.868007\pi\)
\(578\) − 2.62474e9i − 0.565377i
\(579\) 1.84491e9i 0.395003i
\(580\) − 3.59006e9i − 0.764019i
\(581\) 2.20656e8 0.0466765
\(582\) 1.13363e9 0.238363
\(583\) 1.14336e10i 2.38969i
\(584\) 2.28085e9 0.473862
\(585\) 0 0
\(586\) −3.23866e9 −0.664850
\(587\) − 2.74853e9i − 0.560876i −0.959872 0.280438i \(-0.909520\pi\)
0.959872 0.280438i \(-0.0904796\pi\)
\(588\) 4.40307e9 0.893173
\(589\) 1.07780e9 0.217337
\(590\) 4.18011e9i 0.837927i
\(591\) 1.44886e10i 2.88715i
\(592\) − 1.32575e9i − 0.262624i
\(593\) 9.11262e9i 1.79453i 0.441488 + 0.897267i \(0.354451\pi\)
−0.441488 + 0.897267i \(0.645549\pi\)
\(594\) −1.73050e10 −3.38781
\(595\) 5.26918e8 0.102550
\(596\) 1.59792e8i 0.0309167i
\(597\) −1.09925e10 −2.11440
\(598\) 0 0
\(599\) −5.52493e9 −1.05035 −0.525174 0.850995i \(-0.676000\pi\)
−0.525174 + 0.850995i \(0.676000\pi\)
\(600\) − 1.10986e9i − 0.209767i
\(601\) −1.78219e9 −0.334883 −0.167441 0.985882i \(-0.553551\pi\)
−0.167441 + 0.985882i \(0.553551\pi\)
\(602\) −4.54870e8 −0.0849767
\(603\) − 3.44803e8i − 0.0640414i
\(604\) 1.53473e9i 0.283402i
\(605\) − 1.31842e10i − 2.42053i
\(606\) − 1.06838e10i − 1.95016i
\(607\) 9.53705e9 1.73083 0.865414 0.501058i \(-0.167056\pi\)
0.865414 + 0.501058i \(0.167056\pi\)
\(608\) 1.21733e9 0.219658
\(609\) − 2.75179e9i − 0.493690i
\(610\) 6.16219e9 1.09921
\(611\) 0 0
\(612\) 3.12380e9 0.550875
\(613\) − 1.18627e9i − 0.208004i −0.994577 0.104002i \(-0.966835\pi\)
0.994577 0.104002i \(-0.0331648\pi\)
\(614\) −3.80416e8 −0.0663238
\(615\) 2.22107e10 3.85034
\(616\) − 7.21174e8i − 0.124310i
\(617\) 1.32256e9i 0.226682i 0.993556 + 0.113341i \(0.0361552\pi\)
−0.993556 + 0.113341i \(0.963845\pi\)
\(618\) 4.78599e9i 0.815666i
\(619\) 3.59450e9i 0.609147i 0.952489 + 0.304573i \(0.0985138\pi\)
−0.952489 + 0.304573i \(0.901486\pi\)
\(620\) 5.96023e8 0.100437
\(621\) 5.28356e9 0.885332
\(622\) 2.42273e9i 0.403681i
\(623\) 6.13650e8 0.101675
\(624\) 0 0
\(625\) −7.42927e9 −1.21721
\(626\) − 5.05348e9i − 0.823343i
\(627\) 2.51518e10 4.07505
\(628\) −1.09152e9 −0.175863
\(629\) 2.93535e9i 0.470309i
\(630\) − 2.50160e9i − 0.398589i
\(631\) − 7.49102e6i − 0.00118697i −1.00000 0.000593483i \(-0.999811\pi\)
1.00000 0.000593483i \(-0.000188911\pi\)
\(632\) − 3.85823e8i − 0.0607964i
\(633\) −9.44567e9 −1.48020
\(634\) −6.34666e9 −0.989083
\(635\) − 6.63505e9i − 1.02834i
\(636\) −8.18068e9 −1.26093
\(637\) 0 0
\(638\) 1.08792e10 1.65854
\(639\) − 1.73507e9i − 0.263065i
\(640\) 6.73186e8 0.101509
\(641\) −4.06396e9 −0.609462 −0.304731 0.952438i \(-0.598567\pi\)
−0.304731 + 0.952438i \(0.598567\pi\)
\(642\) 1.05811e10i 1.57818i
\(643\) 1.56544e6i 0 0.000232219i 1.00000 0.000116109i \(3.69588e-5\pi\)
−1.00000 0.000116109i \(0.999963\pi\)
\(644\) 2.20189e8i 0.0324859i
\(645\) − 8.77290e9i − 1.28731i
\(646\) −2.69531e9 −0.393364
\(647\) −1.31025e10 −1.90191 −0.950956 0.309325i \(-0.899897\pi\)
−0.950956 + 0.309325i \(0.899897\pi\)
\(648\) − 6.35521e9i − 0.917524i
\(649\) −1.26673e10 −1.81898
\(650\) 0 0
\(651\) 4.56852e8 0.0648996
\(652\) − 4.70135e9i − 0.664287i
\(653\) 7.63326e9 1.07279 0.536394 0.843968i \(-0.319786\pi\)
0.536394 + 0.843968i \(0.319786\pi\)
\(654\) 4.68589e9 0.655044
\(655\) − 6.12945e9i − 0.852269i
\(656\) 3.25760e9i 0.450541i
\(657\) 2.39756e10i 3.29831i
\(658\) − 6.47895e8i − 0.0886571i
\(659\) −9.25900e9 −1.26027 −0.630137 0.776484i \(-0.717001\pi\)
−0.630137 + 0.776484i \(0.717001\pi\)
\(660\) 1.39090e10 1.88318
\(661\) − 4.79962e9i − 0.646401i −0.946330 0.323201i \(-0.895241\pi\)
0.946330 0.323201i \(-0.104759\pi\)
\(662\) 9.73021e9 1.30352
\(663\) 0 0
\(664\) −6.24175e8 −0.0827405
\(665\) 2.15845e9i 0.284621i
\(666\) 1.39359e10 1.82799
\(667\) −3.32165e9 −0.433424
\(668\) 2.98791e9i 0.387837i
\(669\) 1.09274e10i 1.41100i
\(670\) 1.64521e8i 0.0211330i
\(671\) 1.86737e10i 2.38618i
\(672\) 5.15998e8 0.0655927
\(673\) 1.08997e10 1.37836 0.689182 0.724589i \(-0.257970\pi\)
0.689182 + 0.724589i \(0.257970\pi\)
\(674\) 1.20977e9i 0.152192i
\(675\) 6.92578e9 0.866773
\(676\) 0 0
\(677\) 3.44099e9 0.426210 0.213105 0.977029i \(-0.431642\pi\)
0.213105 + 0.977029i \(0.431642\pi\)
\(678\) 8.02434e9i 0.988793i
\(679\) −2.94808e8 −0.0361406
\(680\) −1.49051e9 −0.181783
\(681\) − 1.65974e10i − 2.01384i
\(682\) 1.80617e9i 0.218029i
\(683\) − 5.53553e9i − 0.664794i −0.943140 0.332397i \(-0.892143\pi\)
0.943140 0.332397i \(-0.107857\pi\)
\(684\) 1.27962e10i 1.52892i
\(685\) 9.53051e9 1.13292
\(686\) −2.33754e9 −0.276455
\(687\) − 4.60153e9i − 0.541444i
\(688\) 1.28671e9 0.150633
\(689\) 0 0
\(690\) −4.24669e9 −0.492129
\(691\) 4.21595e8i 0.0486097i 0.999705 + 0.0243048i \(0.00773723\pi\)
−0.999705 + 0.0243048i \(0.992263\pi\)
\(692\) −4.99796e9 −0.573352
\(693\) 7.58077e9 0.865261
\(694\) − 4.77788e9i − 0.542596i
\(695\) 4.99644e9i 0.564565i
\(696\) 7.78406e9i 0.875133i
\(697\) − 7.21268e9i − 0.806830i
\(698\) 9.56799e8 0.106494
\(699\) 1.31591e10 1.45732
\(700\) 2.88627e8i 0.0318049i
\(701\) 5.14995e9 0.564663 0.282332 0.959317i \(-0.408892\pi\)
0.282332 + 0.959317i \(0.408892\pi\)
\(702\) 0 0
\(703\) −1.20243e10 −1.30532
\(704\) 2.04000e9i 0.220357i
\(705\) 1.24957e10 1.34307
\(706\) 3.73132e9 0.399067
\(707\) 2.77840e9i 0.295683i
\(708\) − 9.06342e9i − 0.959789i
\(709\) − 1.05683e10i − 1.11363i −0.830635 0.556817i \(-0.812022\pi\)
0.830635 0.556817i \(-0.187978\pi\)
\(710\) 8.27880e8i 0.0868086i
\(711\) 4.05566e9 0.423173
\(712\) −1.73585e9 −0.180232
\(713\) − 5.51460e8i − 0.0569772i
\(714\) −1.14248e9 −0.117464
\(715\) 0 0
\(716\) −3.55944e9 −0.362399
\(717\) − 2.27868e10i − 2.30869i
\(718\) −6.16081e9 −0.621159
\(719\) −1.53690e10 −1.54204 −0.771020 0.636811i \(-0.780253\pi\)
−0.771020 + 0.636811i \(0.780253\pi\)
\(720\) 7.07634e9i 0.706553i
\(721\) − 1.24463e9i − 0.123671i
\(722\) − 3.89001e9i − 0.384654i
\(723\) − 1.14624e10i − 1.12795i
\(724\) −7.64381e9 −0.748557
\(725\) −4.35407e9 −0.424339
\(726\) 2.85864e10i 2.77256i
\(727\) −4.88599e9 −0.471609 −0.235804 0.971801i \(-0.575772\pi\)
−0.235804 + 0.971801i \(0.575772\pi\)
\(728\) 0 0
\(729\) 1.39161e10 1.33036
\(730\) − 1.14399e10i − 1.08841i
\(731\) −2.84891e9 −0.269754
\(732\) −1.33610e10 −1.25907
\(733\) 3.59889e9i 0.337524i 0.985657 + 0.168762i \(0.0539769\pi\)
−0.985657 + 0.168762i \(0.946023\pi\)
\(734\) − 6.84255e9i − 0.638678i
\(735\) − 2.20842e10i − 2.05152i
\(736\) − 6.22854e8i − 0.0575856i
\(737\) −4.98562e8 −0.0458757
\(738\) −3.42430e10 −3.13598
\(739\) − 2.78886e9i − 0.254198i −0.991890 0.127099i \(-0.959433\pi\)
0.991890 0.127099i \(-0.0405666\pi\)
\(740\) −6.64946e9 −0.603219
\(741\) 0 0
\(742\) 2.12745e9 0.191181
\(743\) − 3.08130e9i − 0.275597i −0.990460 0.137798i \(-0.955997\pi\)
0.990460 0.137798i \(-0.0440026\pi\)
\(744\) −1.29231e9 −0.115043
\(745\) 8.01457e8 0.0710122
\(746\) − 4.23687e9i − 0.373645i
\(747\) − 6.56115e9i − 0.575915i
\(748\) − 4.51680e9i − 0.394616i
\(749\) − 2.75169e9i − 0.239284i
\(750\) 1.18877e10 1.02893
\(751\) 6.41281e8 0.0552470 0.0276235 0.999618i \(-0.491206\pi\)
0.0276235 + 0.999618i \(0.491206\pi\)
\(752\) 1.83272e9i 0.157157i
\(753\) 2.14943e10 1.83460
\(754\) 0 0
\(755\) 7.69763e9 0.650943
\(756\) 3.21995e9i 0.271033i
\(757\) −1.60219e10 −1.34239 −0.671195 0.741280i \(-0.734219\pi\)
−0.671195 + 0.741280i \(0.734219\pi\)
\(758\) −1.58686e10 −1.32342
\(759\) − 1.28691e10i − 1.06832i
\(760\) − 6.10568e9i − 0.504529i
\(761\) 5.73623e9i 0.471824i 0.971774 + 0.235912i \(0.0758077\pi\)
−0.971774 + 0.235912i \(0.924192\pi\)
\(762\) 1.43863e10i 1.17789i
\(763\) −1.21860e9 −0.0993175
\(764\) −6.75102e9 −0.547700
\(765\) − 1.56678e10i − 1.26530i
\(766\) −7.19005e9 −0.578005
\(767\) 0 0
\(768\) −1.45962e9 −0.116272
\(769\) 2.45874e10i 1.94971i 0.222832 + 0.974857i \(0.428470\pi\)
−0.222832 + 0.974857i \(0.571530\pi\)
\(770\) −3.61714e9 −0.285527
\(771\) 1.97739e10 1.55382
\(772\) 1.35718e9i 0.106164i
\(773\) − 1.31517e10i − 1.02413i −0.858947 0.512065i \(-0.828881\pi\)
0.858947 0.512065i \(-0.171119\pi\)
\(774\) 1.35255e10i 1.04848i
\(775\) − 7.22863e8i − 0.0557828i
\(776\) 8.33932e8 0.0640641
\(777\) −5.09682e9 −0.389785
\(778\) − 1.45980e10i − 1.11138i
\(779\) 2.95458e10 2.23932
\(780\) 0 0
\(781\) −2.50878e9 −0.188445
\(782\) 1.37907e9i 0.103125i
\(783\) −4.85744e10 −3.61611
\(784\) 3.23904e9 0.240055
\(785\) 5.47466e9i 0.403937i
\(786\) 1.32900e10i 0.976218i
\(787\) 7.38863e9i 0.540322i 0.962815 + 0.270161i \(0.0870769\pi\)
−0.962815 + 0.270161i \(0.912923\pi\)
\(788\) 1.06583e10i 0.775969i
\(789\) 3.70509e10 2.68552
\(790\) −1.93514e9 −0.139643
\(791\) − 2.08679e9i − 0.149920i
\(792\) −2.14440e10 −1.53379
\(793\) 0 0
\(794\) 3.94467e9 0.279665
\(795\) 4.10312e10i 2.89621i
\(796\) −8.08645e9 −0.568279
\(797\) 5.22399e9 0.365509 0.182754 0.983159i \(-0.441499\pi\)
0.182754 + 0.983159i \(0.441499\pi\)
\(798\) − 4.68001e9i − 0.326014i
\(799\) − 4.05784e9i − 0.281437i
\(800\) − 8.16447e8i − 0.0563785i
\(801\) − 1.82468e10i − 1.25450i
\(802\) −4.54624e9 −0.311202
\(803\) 3.46671e10 2.36273
\(804\) − 3.56719e8i − 0.0242064i
\(805\) 1.10438e9 0.0746164
\(806\) 0 0
\(807\) 4.47314e10 2.99609
\(808\) − 7.85934e9i − 0.524139i
\(809\) −7.92102e9 −0.525970 −0.262985 0.964800i \(-0.584707\pi\)
−0.262985 + 0.964800i \(0.584707\pi\)
\(810\) −3.18754e10 −2.10745
\(811\) − 8.16607e9i − 0.537576i −0.963199 0.268788i \(-0.913377\pi\)
0.963199 0.268788i \(-0.0866232\pi\)
\(812\) − 2.02430e9i − 0.132687i
\(813\) 3.97674e9i 0.259543i
\(814\) − 2.01503e10i − 1.30947i
\(815\) −2.35802e10 −1.52579
\(816\) 3.23176e9 0.208220
\(817\) − 1.16702e10i − 0.748688i
\(818\) 1.02778e10 0.656541
\(819\) 0 0
\(820\) 1.63389e10 1.03484
\(821\) 2.63749e10i 1.66338i 0.555244 + 0.831688i \(0.312625\pi\)
−0.555244 + 0.831688i \(0.687375\pi\)
\(822\) −2.06643e10 −1.29768
\(823\) −2.04085e10 −1.27618 −0.638090 0.769962i \(-0.720275\pi\)
−0.638090 + 0.769962i \(0.720275\pi\)
\(824\) 3.52073e9i 0.219224i
\(825\) − 1.68690e10i − 1.04592i
\(826\) 2.35701e9i 0.145523i
\(827\) 2.55307e10i 1.56962i 0.619738 + 0.784809i \(0.287239\pi\)
−0.619738 + 0.784809i \(0.712761\pi\)
\(828\) 6.54727e9 0.400824
\(829\) −8.48208e9 −0.517085 −0.258542 0.966000i \(-0.583242\pi\)
−0.258542 + 0.966000i \(0.583242\pi\)
\(830\) 3.13063e9i 0.190046i
\(831\) −2.38349e10 −1.44082
\(832\) 0 0
\(833\) −7.17160e9 −0.429891
\(834\) − 1.08334e10i − 0.646672i
\(835\) 1.49862e10 0.890819
\(836\) 1.85025e10 1.09524
\(837\) − 8.06432e9i − 0.475367i
\(838\) 2.19878e9i 0.129071i
\(839\) 2.29323e10i 1.34055i 0.742115 + 0.670273i \(0.233823\pi\)
−0.742115 + 0.670273i \(0.766177\pi\)
\(840\) − 2.58805e9i − 0.150659i
\(841\) 1.32877e10 0.770306
\(842\) −6.01094e9 −0.347017
\(843\) 3.66885e10i 2.10928i
\(844\) −6.94854e9 −0.397828
\(845\) 0 0
\(846\) −1.92650e10 −1.09389
\(847\) − 7.43410e9i − 0.420374i
\(848\) −6.01797e9 −0.338895
\(849\) 3.32303e10 1.86362
\(850\) 1.80771e9i 0.100963i
\(851\) 6.15230e9i 0.342203i
\(852\) − 1.79503e9i − 0.0994335i
\(853\) − 2.47175e10i − 1.36358i −0.731546 0.681792i \(-0.761201\pi\)
0.731546 0.681792i \(-0.238799\pi\)
\(854\) 3.47463e9 0.190900
\(855\) 6.41812e10 3.51177
\(856\) 7.78379e9i 0.424163i
\(857\) −1.19081e10 −0.646265 −0.323133 0.946354i \(-0.604736\pi\)
−0.323133 + 0.946354i \(0.604736\pi\)
\(858\) 0 0
\(859\) −4.94214e9 −0.266035 −0.133018 0.991114i \(-0.542467\pi\)
−0.133018 + 0.991114i \(0.542467\pi\)
\(860\) − 6.45363e9i − 0.345987i
\(861\) 1.25238e10 0.668689
\(862\) 1.04605e9 0.0556259
\(863\) − 2.05387e10i − 1.08776i −0.839162 0.543881i \(-0.816954\pi\)
0.839162 0.543881i \(-0.183046\pi\)
\(864\) − 9.10836e9i − 0.480443i
\(865\) 2.50679e10i 1.31693i
\(866\) − 1.33389e10i − 0.697921i
\(867\) 2.85440e10 1.48747
\(868\) 3.36075e8 0.0174428
\(869\) − 5.86420e9i − 0.303138i
\(870\) 3.90419e10 2.01008
\(871\) 0 0
\(872\) 3.44709e9 0.176054
\(873\) 8.76606e9i 0.445918i
\(874\) −5.64918e9 −0.286217
\(875\) −3.09150e9 −0.156006
\(876\) 2.48042e10i 1.24670i
\(877\) − 1.42584e10i − 0.713791i −0.934144 0.356895i \(-0.883835\pi\)
0.934144 0.356895i \(-0.116165\pi\)
\(878\) − 1.85182e10i − 0.923354i
\(879\) − 3.52204e10i − 1.74918i
\(880\) 1.02319e10 0.506136
\(881\) −1.78398e10 −0.878971 −0.439486 0.898250i \(-0.644839\pi\)
−0.439486 + 0.898250i \(0.644839\pi\)
\(882\) 3.40479e10i 1.67090i
\(883\) −3.79954e10 −1.85724 −0.928622 0.371028i \(-0.879005\pi\)
−0.928622 + 0.371028i \(0.879005\pi\)
\(884\) 0 0
\(885\) −4.54587e10 −2.20453
\(886\) − 5.52038e9i − 0.266656i
\(887\) 8.45195e7 0.00406653 0.00203327 0.999998i \(-0.499353\pi\)
0.00203327 + 0.999998i \(0.499353\pi\)
\(888\) 1.44175e10 0.690947
\(889\) − 3.74126e9i − 0.178592i
\(890\) 8.70637e9i 0.413973i
\(891\) − 9.65942e10i − 4.57488i
\(892\) 8.03857e9i 0.379229i
\(893\) 1.66224e10 0.781114
\(894\) −1.73774e9 −0.0813398
\(895\) 1.78528e10i 0.832390i
\(896\) 3.79585e8 0.0176291
\(897\) 0 0
\(898\) 2.11045e10 0.972541
\(899\) 5.06985e9i 0.232721i
\(900\) 8.58227e9 0.392422
\(901\) 1.33245e10 0.606894
\(902\) 4.95129e10i 2.24645i
\(903\) − 4.94672e9i − 0.223568i
\(904\) 5.90296e9i 0.265754i
\(905\) 3.83385e10i 1.71935i
\(906\) −1.66902e10 −0.745611
\(907\) −1.82024e10 −0.810033 −0.405017 0.914309i \(-0.632734\pi\)
−0.405017 + 0.914309i \(0.632734\pi\)
\(908\) − 1.22096e10i − 0.541252i
\(909\) 8.26151e10 3.64826
\(910\) 0 0
\(911\) −3.66963e10 −1.60808 −0.804040 0.594575i \(-0.797320\pi\)
−0.804040 + 0.594575i \(0.797320\pi\)
\(912\) 1.32385e10i 0.577905i
\(913\) −9.48697e9 −0.412553
\(914\) 4.92978e9 0.213558
\(915\) 6.70139e10i 2.89195i
\(916\) − 3.38503e9i − 0.145522i
\(917\) − 3.45617e9i − 0.148014i
\(918\) 2.01669e10i 0.860380i
\(919\) 1.33474e10 0.567275 0.283638 0.958932i \(-0.408459\pi\)
0.283638 + 0.958932i \(0.408459\pi\)
\(920\) −3.12400e9 −0.132268
\(921\) − 4.13702e9i − 0.174493i
\(922\) −9.88970e9 −0.415552
\(923\) 0 0
\(924\) 7.84276e9 0.327052
\(925\) 8.06454e9i 0.335030i
\(926\) 5.42775e8 0.0224637
\(927\) −3.70089e10 −1.52591
\(928\) 5.72621e9i 0.235206i
\(929\) 2.71771e10i 1.11211i 0.831146 + 0.556055i \(0.187686\pi\)
−0.831146 + 0.556055i \(0.812314\pi\)
\(930\) 6.48174e9i 0.264242i
\(931\) − 2.93776e10i − 1.19314i
\(932\) 9.68025e9 0.391680
\(933\) −2.63472e10 −1.06206
\(934\) 9.40012e9i 0.377502i
\(935\) −2.26546e10 −0.906390
\(936\) 0 0
\(937\) 4.04333e10 1.60565 0.802825 0.596214i \(-0.203329\pi\)
0.802825 + 0.596214i \(0.203329\pi\)
\(938\) 9.27676e7i 0.00367017i
\(939\) 5.49566e10 2.16616
\(940\) 9.19223e9 0.360972
\(941\) − 8.49843e9i − 0.332487i −0.986085 0.166244i \(-0.946836\pi\)
0.986085 0.166244i \(-0.0531638\pi\)
\(942\) − 1.18703e10i − 0.462683i
\(943\) − 1.51173e10i − 0.587061i
\(944\) − 6.66735e9i − 0.257959i
\(945\) 1.61500e10 0.622533
\(946\) 1.95569e10 0.751072
\(947\) 4.40082e9i 0.168387i 0.996449 + 0.0841935i \(0.0268314\pi\)
−0.996449 + 0.0841935i \(0.973169\pi\)
\(948\) 4.19582e9 0.159951
\(949\) 0 0
\(950\) −7.40504e9 −0.280217
\(951\) − 6.90199e10i − 2.60221i
\(952\) −8.40442e8 −0.0315703
\(953\) 1.73133e10 0.647970 0.323985 0.946062i \(-0.394977\pi\)
0.323985 + 0.946062i \(0.394977\pi\)
\(954\) − 6.32593e10i − 2.35887i
\(955\) 3.38606e10i 1.25801i
\(956\) − 1.67627e10i − 0.620498i
\(957\) 1.18312e11i 4.36351i
\(958\) 3.16924e9 0.116460
\(959\) 5.37390e9 0.196754
\(960\) 7.32090e9i 0.267064i
\(961\) 2.66709e10 0.969407
\(962\) 0 0
\(963\) −8.18210e10 −2.95238
\(964\) − 8.43211e9i − 0.303156i
\(965\) 6.80708e9 0.243846
\(966\) −2.39455e9 −0.0854681
\(967\) 1.40918e10i 0.501158i 0.968096 + 0.250579i \(0.0806210\pi\)
−0.968096 + 0.250579i \(0.919379\pi\)
\(968\) 2.10290e10i 0.745171i
\(969\) − 2.93115e10i − 1.03491i
\(970\) − 4.18269e9i − 0.147148i
\(971\) −7.27843e9 −0.255135 −0.127568 0.991830i \(-0.540717\pi\)
−0.127568 + 0.991830i \(0.540717\pi\)
\(972\) 3.02067e10 1.05505
\(973\) 2.81731e9i 0.0980481i
\(974\) 2.42932e10 0.842419
\(975\) 0 0
\(976\) −9.82879e9 −0.338397
\(977\) 2.43791e10i 0.836348i 0.908367 + 0.418174i \(0.137330\pi\)
−0.908367 + 0.418174i \(0.862670\pi\)
\(978\) 5.11272e10 1.74770
\(979\) −2.63835e10 −0.898657
\(980\) − 1.62458e10i − 0.551379i
\(981\) 3.62349e10i 1.22542i
\(982\) 2.33579e10i 0.787126i
\(983\) − 4.06556e10i − 1.36516i −0.730811 0.682579i \(-0.760858\pi\)
0.730811 0.682579i \(-0.239142\pi\)
\(984\) −3.54264e10 −1.18534
\(985\) 5.34578e10 1.78231
\(986\) − 1.26785e10i − 0.421209i
\(987\) 7.04585e9 0.233251
\(988\) 0 0
\(989\) −5.97112e9 −0.196277
\(990\) 1.07555e11i 3.52295i
\(991\) −4.86636e10 −1.58835 −0.794175 0.607689i \(-0.792097\pi\)
−0.794175 + 0.607689i \(0.792097\pi\)
\(992\) −9.50665e8 −0.0309198
\(993\) 1.05816e11i 3.42949i
\(994\) 4.66811e8i 0.0150761i
\(995\) 4.05586e10i 1.30527i
\(996\) − 6.78790e9i − 0.217685i
\(997\) −1.76682e10 −0.564622 −0.282311 0.959323i \(-0.591101\pi\)
−0.282311 + 0.959323i \(0.591101\pi\)
\(998\) 1.29874e10 0.413586
\(999\) 8.99687e10i 2.85504i
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 338.8.b.a.337.2 2
13.5 odd 4 26.8.a.b.1.1 1
13.8 odd 4 338.8.a.a.1.1 1
13.12 even 2 inner 338.8.b.a.337.1 2
39.5 even 4 234.8.a.a.1.1 1
52.31 even 4 208.8.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.8.a.b.1.1 1 13.5 odd 4
208.8.a.e.1.1 1 52.31 even 4
234.8.a.a.1.1 1 39.5 even 4
338.8.a.a.1.1 1 13.8 odd 4
338.8.b.a.337.1 2 13.12 even 2 inner
338.8.b.a.337.2 2 1.1 even 1 trivial