Properties

Label 48.22.a.h.1.2
Level $48$
Weight $22$
Character 48.1
Self dual yes
Analytic conductor $134.149$
Analytic rank $1$
Dimension $2$
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] = [48,22,Mod(1,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("48.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 48.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: \(134.149125258\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{537541}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 134385 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 7 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-366.086\) of defining polynomial
Character \(\chi\) \(=\) 48.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-59049.0 q^{3} +2.57551e7 q^{5} -3.59057e8 q^{7} +3.48678e9 q^{9} +O(q^{10})\) \(q-59049.0 q^{3} +2.57551e7 q^{5} -3.59057e8 q^{7} +3.48678e9 q^{9} +1.36294e11 q^{11} -3.92453e11 q^{13} -1.52081e12 q^{15} -1.39882e13 q^{17} +1.50746e13 q^{19} +2.12020e13 q^{21} +8.76097e13 q^{23} +1.86486e14 q^{25} -2.05891e14 q^{27} -2.21409e15 q^{29} -8.94349e14 q^{31} -8.04800e15 q^{33} -9.24753e15 q^{35} -3.48898e16 q^{37} +2.31739e16 q^{39} +1.42605e17 q^{41} -1.89687e17 q^{43} +8.98023e16 q^{45} +3.48311e17 q^{47} -4.29624e17 q^{49} +8.25990e17 q^{51} +1.82306e18 q^{53} +3.51025e18 q^{55} -8.90138e17 q^{57} -2.09411e18 q^{59} -1.46472e18 q^{61} -1.25195e18 q^{63} -1.01076e19 q^{65} -1.86063e18 q^{67} -5.17327e18 q^{69} +4.89991e19 q^{71} -1.41519e19 q^{73} -1.10118e19 q^{75} -4.89372e19 q^{77} -4.80140e18 q^{79} +1.21577e19 q^{81} +1.73723e20 q^{83} -3.60267e20 q^{85} +1.30740e20 q^{87} -1.41275e20 q^{89} +1.40913e20 q^{91} +5.28104e19 q^{93} +3.88246e20 q^{95} -9.17758e20 q^{97} +4.75226e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 118098 q^{3} + 21948620 q^{5} + 659451408 q^{7} + 6973568802 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 118098 q^{3} + 21948620 q^{5} + 659451408 q^{7} + 6973568802 q^{9} + 29910896872 q^{11} + 4468866812 q^{13} - 1296044062380 q^{15} - 17665404721820 q^{17} + 22467979297496 q^{19} - 38939946190992 q^{21} - 120995273049584 q^{23} - 275862509554850 q^{25} - 411782264189298 q^{27} - 33\!\cdots\!84 q^{29}+ \cdots + 10\!\cdots\!72 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −59049.0 −0.577350
\(4\) 0 0
\(5\) 2.57551e7 1.17944 0.589722 0.807606i \(-0.299237\pi\)
0.589722 + 0.807606i \(0.299237\pi\)
\(6\) 0 0
\(7\) −3.59057e8 −0.480434 −0.240217 0.970719i \(-0.577219\pi\)
−0.240217 + 0.970719i \(0.577219\pi\)
\(8\) 0 0
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) 1.36294e11 1.58435 0.792177 0.610292i \(-0.208948\pi\)
0.792177 + 0.610292i \(0.208948\pi\)
\(12\) 0 0
\(13\) −3.92453e11 −0.789555 −0.394777 0.918777i \(-0.629178\pi\)
−0.394777 + 0.918777i \(0.629178\pi\)
\(14\) 0 0
\(15\) −1.52081e12 −0.680952
\(16\) 0 0
\(17\) −1.39882e13 −1.68286 −0.841432 0.540364i \(-0.818287\pi\)
−0.841432 + 0.540364i \(0.818287\pi\)
\(18\) 0 0
\(19\) 1.50746e13 0.564069 0.282034 0.959404i \(-0.408991\pi\)
0.282034 + 0.959404i \(0.408991\pi\)
\(20\) 0 0
\(21\) 2.12020e13 0.277379
\(22\) 0 0
\(23\) 8.76097e13 0.440971 0.220486 0.975390i \(-0.429236\pi\)
0.220486 + 0.975390i \(0.429236\pi\)
\(24\) 0 0
\(25\) 1.86486e14 0.391089
\(26\) 0 0
\(27\) −2.05891e14 −0.192450
\(28\) 0 0
\(29\) −2.21409e15 −0.977272 −0.488636 0.872488i \(-0.662505\pi\)
−0.488636 + 0.872488i \(0.662505\pi\)
\(30\) 0 0
\(31\) −8.94349e14 −0.195979 −0.0979894 0.995187i \(-0.531241\pi\)
−0.0979894 + 0.995187i \(0.531241\pi\)
\(32\) 0 0
\(33\) −8.04800e15 −0.914727
\(34\) 0 0
\(35\) −9.24753e15 −0.566645
\(36\) 0 0
\(37\) −3.48898e16 −1.19284 −0.596418 0.802674i \(-0.703410\pi\)
−0.596418 + 0.802674i \(0.703410\pi\)
\(38\) 0 0
\(39\) 2.31739e16 0.455850
\(40\) 0 0
\(41\) 1.42605e17 1.65923 0.829613 0.558339i \(-0.188561\pi\)
0.829613 + 0.558339i \(0.188561\pi\)
\(42\) 0 0
\(43\) −1.89687e17 −1.33850 −0.669251 0.743037i \(-0.733385\pi\)
−0.669251 + 0.743037i \(0.733385\pi\)
\(44\) 0 0
\(45\) 8.98023e16 0.393148
\(46\) 0 0
\(47\) 3.48311e17 0.965918 0.482959 0.875643i \(-0.339562\pi\)
0.482959 + 0.875643i \(0.339562\pi\)
\(48\) 0 0
\(49\) −4.29624e17 −0.769183
\(50\) 0 0
\(51\) 8.25990e17 0.971601
\(52\) 0 0
\(53\) 1.82306e18 1.43188 0.715938 0.698164i \(-0.245999\pi\)
0.715938 + 0.698164i \(0.245999\pi\)
\(54\) 0 0
\(55\) 3.51025e18 1.86866
\(56\) 0 0
\(57\) −8.90138e17 −0.325665
\(58\) 0 0
\(59\) −2.09411e18 −0.533402 −0.266701 0.963779i \(-0.585934\pi\)
−0.266701 + 0.963779i \(0.585934\pi\)
\(60\) 0 0
\(61\) −1.46472e18 −0.262900 −0.131450 0.991323i \(-0.541963\pi\)
−0.131450 + 0.991323i \(0.541963\pi\)
\(62\) 0 0
\(63\) −1.25195e18 −0.160145
\(64\) 0 0
\(65\) −1.01076e19 −0.931236
\(66\) 0 0
\(67\) −1.86063e18 −0.124702 −0.0623511 0.998054i \(-0.519860\pi\)
−0.0623511 + 0.998054i \(0.519860\pi\)
\(68\) 0 0
\(69\) −5.17327e18 −0.254595
\(70\) 0 0
\(71\) 4.89991e19 1.78639 0.893194 0.449672i \(-0.148459\pi\)
0.893194 + 0.449672i \(0.148459\pi\)
\(72\) 0 0
\(73\) −1.41519e19 −0.385413 −0.192706 0.981256i \(-0.561726\pi\)
−0.192706 + 0.981256i \(0.561726\pi\)
\(74\) 0 0
\(75\) −1.10118e19 −0.225795
\(76\) 0 0
\(77\) −4.89372e19 −0.761178
\(78\) 0 0
\(79\) −4.80140e18 −0.0570537 −0.0285268 0.999593i \(-0.509082\pi\)
−0.0285268 + 0.999593i \(0.509082\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 0 0
\(83\) 1.73723e20 1.22896 0.614480 0.788932i \(-0.289366\pi\)
0.614480 + 0.788932i \(0.289366\pi\)
\(84\) 0 0
\(85\) −3.60267e20 −1.98484
\(86\) 0 0
\(87\) 1.30740e20 0.564229
\(88\) 0 0
\(89\) −1.41275e20 −0.480252 −0.240126 0.970742i \(-0.577189\pi\)
−0.240126 + 0.970742i \(0.577189\pi\)
\(90\) 0 0
\(91\) 1.40913e20 0.379329
\(92\) 0 0
\(93\) 5.28104e19 0.113148
\(94\) 0 0
\(95\) 3.88246e20 0.665288
\(96\) 0 0
\(97\) −9.17758e20 −1.26364 −0.631822 0.775113i \(-0.717693\pi\)
−0.631822 + 0.775113i \(0.717693\pi\)
\(98\) 0 0
\(99\) 4.75226e20 0.528118
\(100\) 0 0
\(101\) 2.47601e19 0.0223037 0.0111519 0.999938i \(-0.496450\pi\)
0.0111519 + 0.999938i \(0.496450\pi\)
\(102\) 0 0
\(103\) −1.69839e21 −1.24522 −0.622612 0.782531i \(-0.713928\pi\)
−0.622612 + 0.782531i \(0.713928\pi\)
\(104\) 0 0
\(105\) 5.46058e20 0.327153
\(106\) 0 0
\(107\) −2.41055e21 −1.18464 −0.592320 0.805703i \(-0.701788\pi\)
−0.592320 + 0.805703i \(0.701788\pi\)
\(108\) 0 0
\(109\) −1.85834e21 −0.751878 −0.375939 0.926644i \(-0.622680\pi\)
−0.375939 + 0.926644i \(0.622680\pi\)
\(110\) 0 0
\(111\) 2.06021e21 0.688684
\(112\) 0 0
\(113\) 2.42812e21 0.672893 0.336446 0.941703i \(-0.390775\pi\)
0.336446 + 0.941703i \(0.390775\pi\)
\(114\) 0 0
\(115\) 2.25639e21 0.520101
\(116\) 0 0
\(117\) −1.36840e21 −0.263185
\(118\) 0 0
\(119\) 5.02257e21 0.808505
\(120\) 0 0
\(121\) 1.11757e22 1.51018
\(122\) 0 0
\(123\) −8.42071e21 −0.957955
\(124\) 0 0
\(125\) −7.47802e21 −0.718177
\(126\) 0 0
\(127\) −2.03194e22 −1.65185 −0.825926 0.563779i \(-0.809347\pi\)
−0.825926 + 0.563779i \(0.809347\pi\)
\(128\) 0 0
\(129\) 1.12008e22 0.772784
\(130\) 0 0
\(131\) −1.93767e22 −1.13745 −0.568725 0.822528i \(-0.692563\pi\)
−0.568725 + 0.822528i \(0.692563\pi\)
\(132\) 0 0
\(133\) −5.41263e21 −0.270998
\(134\) 0 0
\(135\) −5.30274e21 −0.226984
\(136\) 0 0
\(137\) 1.85640e22 0.680935 0.340468 0.940256i \(-0.389415\pi\)
0.340468 + 0.940256i \(0.389415\pi\)
\(138\) 0 0
\(139\) −3.06791e22 −0.966466 −0.483233 0.875492i \(-0.660538\pi\)
−0.483233 + 0.875492i \(0.660538\pi\)
\(140\) 0 0
\(141\) −2.05674e22 −0.557673
\(142\) 0 0
\(143\) −5.34888e22 −1.25093
\(144\) 0 0
\(145\) −5.70239e22 −1.15264
\(146\) 0 0
\(147\) 2.53689e22 0.444088
\(148\) 0 0
\(149\) −1.03106e23 −1.56614 −0.783069 0.621934i \(-0.786347\pi\)
−0.783069 + 0.621934i \(0.786347\pi\)
\(150\) 0 0
\(151\) −5.43298e22 −0.717431 −0.358716 0.933447i \(-0.616785\pi\)
−0.358716 + 0.933447i \(0.616785\pi\)
\(152\) 0 0
\(153\) −4.87739e22 −0.560954
\(154\) 0 0
\(155\) −2.30340e22 −0.231146
\(156\) 0 0
\(157\) 6.24116e22 0.547417 0.273709 0.961813i \(-0.411750\pi\)
0.273709 + 0.961813i \(0.411750\pi\)
\(158\) 0 0
\(159\) −1.07650e23 −0.826694
\(160\) 0 0
\(161\) −3.14569e22 −0.211858
\(162\) 0 0
\(163\) 2.75747e23 1.63132 0.815662 0.578529i \(-0.196373\pi\)
0.815662 + 0.578529i \(0.196373\pi\)
\(164\) 0 0
\(165\) −2.07277e23 −1.07887
\(166\) 0 0
\(167\) 4.31299e22 0.197813 0.0989067 0.995097i \(-0.468465\pi\)
0.0989067 + 0.995097i \(0.468465\pi\)
\(168\) 0 0
\(169\) −9.30453e22 −0.376603
\(170\) 0 0
\(171\) 5.25618e22 0.188023
\(172\) 0 0
\(173\) −4.96523e23 −1.57201 −0.786005 0.618220i \(-0.787854\pi\)
−0.786005 + 0.618220i \(0.787854\pi\)
\(174\) 0 0
\(175\) −6.69590e22 −0.187892
\(176\) 0 0
\(177\) 1.23655e23 0.307960
\(178\) 0 0
\(179\) −1.59980e23 −0.354086 −0.177043 0.984203i \(-0.556653\pi\)
−0.177043 + 0.984203i \(0.556653\pi\)
\(180\) 0 0
\(181\) −1.04239e23 −0.205307 −0.102654 0.994717i \(-0.532733\pi\)
−0.102654 + 0.994717i \(0.532733\pi\)
\(182\) 0 0
\(183\) 8.64901e22 0.151785
\(184\) 0 0
\(185\) −8.98589e23 −1.40688
\(186\) 0 0
\(187\) −1.90650e24 −2.66625
\(188\) 0 0
\(189\) 7.39267e22 0.0924596
\(190\) 0 0
\(191\) 9.53991e21 0.0106830 0.00534151 0.999986i \(-0.498300\pi\)
0.00534151 + 0.999986i \(0.498300\pi\)
\(192\) 0 0
\(193\) −1.63574e24 −1.64196 −0.820982 0.570955i \(-0.806573\pi\)
−0.820982 + 0.570955i \(0.806573\pi\)
\(194\) 0 0
\(195\) 5.96846e23 0.537649
\(196\) 0 0
\(197\) −1.78920e23 −0.144798 −0.0723990 0.997376i \(-0.523065\pi\)
−0.0723990 + 0.997376i \(0.523065\pi\)
\(198\) 0 0
\(199\) −1.27224e24 −0.926000 −0.463000 0.886358i \(-0.653227\pi\)
−0.463000 + 0.886358i \(0.653227\pi\)
\(200\) 0 0
\(201\) 1.09868e23 0.0719969
\(202\) 0 0
\(203\) 7.94983e23 0.469515
\(204\) 0 0
\(205\) 3.67281e24 1.95697
\(206\) 0 0
\(207\) 3.05476e23 0.146990
\(208\) 0 0
\(209\) 2.05457e24 0.893684
\(210\) 0 0
\(211\) −1.28008e24 −0.503814 −0.251907 0.967751i \(-0.581058\pi\)
−0.251907 + 0.967751i \(0.581058\pi\)
\(212\) 0 0
\(213\) −2.89335e24 −1.03137
\(214\) 0 0
\(215\) −4.88540e24 −1.57869
\(216\) 0 0
\(217\) 3.21122e23 0.0941549
\(218\) 0 0
\(219\) 8.35658e23 0.222518
\(220\) 0 0
\(221\) 5.48972e24 1.32871
\(222\) 0 0
\(223\) −8.50398e24 −1.87250 −0.936249 0.351337i \(-0.885727\pi\)
−0.936249 + 0.351337i \(0.885727\pi\)
\(224\) 0 0
\(225\) 6.50235e23 0.130363
\(226\) 0 0
\(227\) 5.87384e24 1.07313 0.536563 0.843860i \(-0.319722\pi\)
0.536563 + 0.843860i \(0.319722\pi\)
\(228\) 0 0
\(229\) 5.22784e24 0.871064 0.435532 0.900173i \(-0.356560\pi\)
0.435532 + 0.900173i \(0.356560\pi\)
\(230\) 0 0
\(231\) 2.88969e24 0.439466
\(232\) 0 0
\(233\) −1.21665e25 −1.69016 −0.845080 0.534639i \(-0.820447\pi\)
−0.845080 + 0.534639i \(0.820447\pi\)
\(234\) 0 0
\(235\) 8.97078e24 1.13925
\(236\) 0 0
\(237\) 2.83518e23 0.0329400
\(238\) 0 0
\(239\) 1.48569e25 1.58034 0.790172 0.612885i \(-0.209991\pi\)
0.790172 + 0.612885i \(0.209991\pi\)
\(240\) 0 0
\(241\) 1.92207e25 1.87322 0.936611 0.350371i \(-0.113944\pi\)
0.936611 + 0.350371i \(0.113944\pi\)
\(242\) 0 0
\(243\) −7.17898e23 −0.0641500
\(244\) 0 0
\(245\) −1.10650e25 −0.907208
\(246\) 0 0
\(247\) −5.91606e24 −0.445363
\(248\) 0 0
\(249\) −1.02582e25 −0.709541
\(250\) 0 0
\(251\) 1.32687e25 0.843827 0.421913 0.906636i \(-0.361359\pi\)
0.421913 + 0.906636i \(0.361359\pi\)
\(252\) 0 0
\(253\) 1.19406e25 0.698654
\(254\) 0 0
\(255\) 2.12734e25 1.14595
\(256\) 0 0
\(257\) 1.26688e25 0.628690 0.314345 0.949309i \(-0.398215\pi\)
0.314345 + 0.949309i \(0.398215\pi\)
\(258\) 0 0
\(259\) 1.25274e25 0.573079
\(260\) 0 0
\(261\) −7.72004e24 −0.325757
\(262\) 0 0
\(263\) 1.53269e25 0.596925 0.298463 0.954421i \(-0.403526\pi\)
0.298463 + 0.954421i \(0.403526\pi\)
\(264\) 0 0
\(265\) 4.69531e25 1.68882
\(266\) 0 0
\(267\) 8.34213e24 0.277274
\(268\) 0 0
\(269\) −1.89875e25 −0.583537 −0.291769 0.956489i \(-0.594244\pi\)
−0.291769 + 0.956489i \(0.594244\pi\)
\(270\) 0 0
\(271\) 7.77343e24 0.221022 0.110511 0.993875i \(-0.464751\pi\)
0.110511 + 0.993875i \(0.464751\pi\)
\(272\) 0 0
\(273\) −8.32077e24 −0.219006
\(274\) 0 0
\(275\) 2.54168e25 0.619623
\(276\) 0 0
\(277\) −4.49025e25 −1.01445 −0.507227 0.861812i \(-0.669330\pi\)
−0.507227 + 0.861812i \(0.669330\pi\)
\(278\) 0 0
\(279\) −3.11840e24 −0.0653263
\(280\) 0 0
\(281\) −4.01753e25 −0.780804 −0.390402 0.920644i \(-0.627664\pi\)
−0.390402 + 0.920644i \(0.627664\pi\)
\(282\) 0 0
\(283\) −3.95046e25 −0.712673 −0.356336 0.934358i \(-0.615974\pi\)
−0.356336 + 0.934358i \(0.615974\pi\)
\(284\) 0 0
\(285\) −2.29256e25 −0.384104
\(286\) 0 0
\(287\) −5.12035e25 −0.797149
\(288\) 0 0
\(289\) 1.26578e26 1.83203
\(290\) 0 0
\(291\) 5.41927e25 0.729565
\(292\) 0 0
\(293\) 1.06017e26 1.32821 0.664105 0.747639i \(-0.268813\pi\)
0.664105 + 0.747639i \(0.268813\pi\)
\(294\) 0 0
\(295\) −5.39340e25 −0.629117
\(296\) 0 0
\(297\) −2.80616e25 −0.304909
\(298\) 0 0
\(299\) −3.43827e25 −0.348171
\(300\) 0 0
\(301\) 6.81085e25 0.643062
\(302\) 0 0
\(303\) −1.46206e24 −0.0128771
\(304\) 0 0
\(305\) −3.77239e25 −0.310076
\(306\) 0 0
\(307\) −1.27333e26 −0.977207 −0.488603 0.872506i \(-0.662493\pi\)
−0.488603 + 0.872506i \(0.662493\pi\)
\(308\) 0 0
\(309\) 1.00288e26 0.718930
\(310\) 0 0
\(311\) 2.10303e26 1.40884 0.704420 0.709784i \(-0.251207\pi\)
0.704420 + 0.709784i \(0.251207\pi\)
\(312\) 0 0
\(313\) −4.27161e25 −0.267532 −0.133766 0.991013i \(-0.542707\pi\)
−0.133766 + 0.991013i \(0.542707\pi\)
\(314\) 0 0
\(315\) −3.22442e25 −0.188882
\(316\) 0 0
\(317\) −1.06128e25 −0.0581713 −0.0290856 0.999577i \(-0.509260\pi\)
−0.0290856 + 0.999577i \(0.509260\pi\)
\(318\) 0 0
\(319\) −3.01766e26 −1.54835
\(320\) 0 0
\(321\) 1.42340e26 0.683952
\(322\) 0 0
\(323\) −2.10866e26 −0.949250
\(324\) 0 0
\(325\) −7.31868e25 −0.308786
\(326\) 0 0
\(327\) 1.09733e26 0.434097
\(328\) 0 0
\(329\) −1.25064e26 −0.464060
\(330\) 0 0
\(331\) 8.50096e24 0.0295988 0.0147994 0.999890i \(-0.495289\pi\)
0.0147994 + 0.999890i \(0.495289\pi\)
\(332\) 0 0
\(333\) −1.21653e26 −0.397612
\(334\) 0 0
\(335\) −4.79206e25 −0.147079
\(336\) 0 0
\(337\) −5.53863e26 −1.59694 −0.798469 0.602035i \(-0.794357\pi\)
−0.798469 + 0.602035i \(0.794357\pi\)
\(338\) 0 0
\(339\) −1.43378e26 −0.388495
\(340\) 0 0
\(341\) −1.21894e26 −0.310500
\(342\) 0 0
\(343\) 3.54809e26 0.849976
\(344\) 0 0
\(345\) −1.33238e26 −0.300280
\(346\) 0 0
\(347\) −7.87571e26 −1.67044 −0.835218 0.549919i \(-0.814659\pi\)
−0.835218 + 0.549919i \(0.814659\pi\)
\(348\) 0 0
\(349\) −3.59188e26 −0.717223 −0.358612 0.933487i \(-0.616750\pi\)
−0.358612 + 0.933487i \(0.616750\pi\)
\(350\) 0 0
\(351\) 8.08026e25 0.151950
\(352\) 0 0
\(353\) −1.07852e27 −1.91071 −0.955357 0.295454i \(-0.904529\pi\)
−0.955357 + 0.295454i \(0.904529\pi\)
\(354\) 0 0
\(355\) 1.26198e27 2.10694
\(356\) 0 0
\(357\) −2.96578e26 −0.466791
\(358\) 0 0
\(359\) −7.47071e26 −1.10884 −0.554421 0.832236i \(-0.687060\pi\)
−0.554421 + 0.832236i \(0.687060\pi\)
\(360\) 0 0
\(361\) −4.86967e26 −0.681827
\(362\) 0 0
\(363\) −6.59913e26 −0.871900
\(364\) 0 0
\(365\) −3.64484e26 −0.454573
\(366\) 0 0
\(367\) −5.12481e25 −0.0603510 −0.0301755 0.999545i \(-0.509607\pi\)
−0.0301755 + 0.999545i \(0.509607\pi\)
\(368\) 0 0
\(369\) 4.97234e26 0.553075
\(370\) 0 0
\(371\) −6.54584e26 −0.687922
\(372\) 0 0
\(373\) 1.85350e27 1.84098 0.920492 0.390761i \(-0.127788\pi\)
0.920492 + 0.390761i \(0.127788\pi\)
\(374\) 0 0
\(375\) 4.41569e26 0.414640
\(376\) 0 0
\(377\) 8.68925e26 0.771610
\(378\) 0 0
\(379\) 1.88527e27 1.58366 0.791832 0.610740i \(-0.209128\pi\)
0.791832 + 0.610740i \(0.209128\pi\)
\(380\) 0 0
\(381\) 1.19984e27 0.953697
\(382\) 0 0
\(383\) 1.86685e27 1.40450 0.702252 0.711928i \(-0.252178\pi\)
0.702252 + 0.711928i \(0.252178\pi\)
\(384\) 0 0
\(385\) −1.26038e27 −0.897767
\(386\) 0 0
\(387\) −6.61398e26 −0.446167
\(388\) 0 0
\(389\) −7.56908e26 −0.483696 −0.241848 0.970314i \(-0.577753\pi\)
−0.241848 + 0.970314i \(0.577753\pi\)
\(390\) 0 0
\(391\) −1.22550e27 −0.742094
\(392\) 0 0
\(393\) 1.14418e27 0.656707
\(394\) 0 0
\(395\) −1.23660e26 −0.0672916
\(396\) 0 0
\(397\) −3.42415e27 −1.76706 −0.883532 0.468371i \(-0.844841\pi\)
−0.883532 + 0.468371i \(0.844841\pi\)
\(398\) 0 0
\(399\) 3.19610e26 0.156461
\(400\) 0 0
\(401\) 4.10952e27 1.90886 0.954431 0.298430i \(-0.0964631\pi\)
0.954431 + 0.298430i \(0.0964631\pi\)
\(402\) 0 0
\(403\) 3.50990e26 0.154736
\(404\) 0 0
\(405\) 3.13121e26 0.131049
\(406\) 0 0
\(407\) −4.75526e27 −1.88987
\(408\) 0 0
\(409\) 2.66757e27 1.00698 0.503490 0.864001i \(-0.332049\pi\)
0.503490 + 0.864001i \(0.332049\pi\)
\(410\) 0 0
\(411\) −1.09619e27 −0.393138
\(412\) 0 0
\(413\) 7.51907e26 0.256264
\(414\) 0 0
\(415\) 4.47425e27 1.44949
\(416\) 0 0
\(417\) 1.81157e27 0.557990
\(418\) 0 0
\(419\) −2.19630e27 −0.643346 −0.321673 0.946851i \(-0.604245\pi\)
−0.321673 + 0.946851i \(0.604245\pi\)
\(420\) 0 0
\(421\) 5.66044e27 1.57721 0.788603 0.614903i \(-0.210805\pi\)
0.788603 + 0.614903i \(0.210805\pi\)
\(422\) 0 0
\(423\) 1.21449e27 0.321973
\(424\) 0 0
\(425\) −2.60860e27 −0.658149
\(426\) 0 0
\(427\) 5.25917e26 0.126306
\(428\) 0 0
\(429\) 3.15846e27 0.722227
\(430\) 0 0
\(431\) 1.11367e27 0.242519 0.121260 0.992621i \(-0.461307\pi\)
0.121260 + 0.992621i \(0.461307\pi\)
\(432\) 0 0
\(433\) −1.02630e27 −0.212888 −0.106444 0.994319i \(-0.533946\pi\)
−0.106444 + 0.994319i \(0.533946\pi\)
\(434\) 0 0
\(435\) 3.36721e27 0.665476
\(436\) 0 0
\(437\) 1.32068e27 0.248738
\(438\) 0 0
\(439\) −8.22966e26 −0.147742 −0.0738710 0.997268i \(-0.523535\pi\)
−0.0738710 + 0.997268i \(0.523535\pi\)
\(440\) 0 0
\(441\) −1.49801e27 −0.256394
\(442\) 0 0
\(443\) −3.65938e27 −0.597267 −0.298633 0.954368i \(-0.596531\pi\)
−0.298633 + 0.954368i \(0.596531\pi\)
\(444\) 0 0
\(445\) −3.63854e27 −0.566431
\(446\) 0 0
\(447\) 6.08834e27 0.904211
\(448\) 0 0
\(449\) −4.66489e27 −0.661081 −0.330541 0.943792i \(-0.607231\pi\)
−0.330541 + 0.943792i \(0.607231\pi\)
\(450\) 0 0
\(451\) 1.94362e28 2.62880
\(452\) 0 0
\(453\) 3.20812e27 0.414209
\(454\) 0 0
\(455\) 3.62922e27 0.447398
\(456\) 0 0
\(457\) 2.25704e27 0.265717 0.132859 0.991135i \(-0.457584\pi\)
0.132859 + 0.991135i \(0.457584\pi\)
\(458\) 0 0
\(459\) 2.88005e27 0.323867
\(460\) 0 0
\(461\) −3.22216e27 −0.346168 −0.173084 0.984907i \(-0.555373\pi\)
−0.173084 + 0.984907i \(0.555373\pi\)
\(462\) 0 0
\(463\) −1.52787e28 −1.56850 −0.784252 0.620442i \(-0.786953\pi\)
−0.784252 + 0.620442i \(0.786953\pi\)
\(464\) 0 0
\(465\) 1.36013e27 0.133452
\(466\) 0 0
\(467\) 5.38717e27 0.505282 0.252641 0.967560i \(-0.418701\pi\)
0.252641 + 0.967560i \(0.418701\pi\)
\(468\) 0 0
\(469\) 6.68072e26 0.0599113
\(470\) 0 0
\(471\) −3.68534e27 −0.316052
\(472\) 0 0
\(473\) −2.58531e28 −2.12066
\(474\) 0 0
\(475\) 2.81119e27 0.220601
\(476\) 0 0
\(477\) 6.35663e27 0.477292
\(478\) 0 0
\(479\) −1.99749e27 −0.143536 −0.0717680 0.997421i \(-0.522864\pi\)
−0.0717680 + 0.997421i \(0.522864\pi\)
\(480\) 0 0
\(481\) 1.36926e28 0.941809
\(482\) 0 0
\(483\) 1.85750e27 0.122316
\(484\) 0 0
\(485\) −2.36369e28 −1.49040
\(486\) 0 0
\(487\) −6.07531e27 −0.366872 −0.183436 0.983032i \(-0.558722\pi\)
−0.183436 + 0.983032i \(0.558722\pi\)
\(488\) 0 0
\(489\) −1.62826e28 −0.941845
\(490\) 0 0
\(491\) −1.63125e28 −0.903990 −0.451995 0.892020i \(-0.649288\pi\)
−0.451995 + 0.892020i \(0.649288\pi\)
\(492\) 0 0
\(493\) 3.09711e28 1.64462
\(494\) 0 0
\(495\) 1.22395e28 0.622886
\(496\) 0 0
\(497\) −1.75935e28 −0.858242
\(498\) 0 0
\(499\) 4.96493e27 0.232197 0.116099 0.993238i \(-0.462961\pi\)
0.116099 + 0.993238i \(0.462961\pi\)
\(500\) 0 0
\(501\) −2.54678e27 −0.114208
\(502\) 0 0
\(503\) −3.73937e28 −1.60818 −0.804090 0.594507i \(-0.797347\pi\)
−0.804090 + 0.594507i \(0.797347\pi\)
\(504\) 0 0
\(505\) 6.37697e26 0.0263060
\(506\) 0 0
\(507\) 5.49423e27 0.217432
\(508\) 0 0
\(509\) 2.97440e28 1.12944 0.564719 0.825283i \(-0.308984\pi\)
0.564719 + 0.825283i \(0.308984\pi\)
\(510\) 0 0
\(511\) 5.08135e27 0.185165
\(512\) 0 0
\(513\) −3.10372e27 −0.108555
\(514\) 0 0
\(515\) −4.37422e28 −1.46867
\(516\) 0 0
\(517\) 4.74726e28 1.53035
\(518\) 0 0
\(519\) 2.93192e28 0.907600
\(520\) 0 0
\(521\) −4.22606e28 −1.25643 −0.628217 0.778038i \(-0.716215\pi\)
−0.628217 + 0.778038i \(0.716215\pi\)
\(522\) 0 0
\(523\) −6.42283e28 −1.83425 −0.917125 0.398600i \(-0.869496\pi\)
−0.917125 + 0.398600i \(0.869496\pi\)
\(524\) 0 0
\(525\) 3.95386e27 0.108480
\(526\) 0 0
\(527\) 1.25103e28 0.329805
\(528\) 0 0
\(529\) −3.17961e28 −0.805545
\(530\) 0 0
\(531\) −7.30173e27 −0.177801
\(532\) 0 0
\(533\) −5.59659e28 −1.31005
\(534\) 0 0
\(535\) −6.20838e28 −1.39722
\(536\) 0 0
\(537\) 9.44665e27 0.204431
\(538\) 0 0
\(539\) −5.85550e28 −1.21866
\(540\) 0 0
\(541\) 1.00894e28 0.201974 0.100987 0.994888i \(-0.467800\pi\)
0.100987 + 0.994888i \(0.467800\pi\)
\(542\) 0 0
\(543\) 6.15520e27 0.118534
\(544\) 0 0
\(545\) −4.78617e28 −0.886798
\(546\) 0 0
\(547\) −9.94342e25 −0.00177284 −0.000886419 1.00000i \(-0.500282\pi\)
−0.000886419 1.00000i \(0.500282\pi\)
\(548\) 0 0
\(549\) −5.10715e27 −0.0876333
\(550\) 0 0
\(551\) −3.33764e28 −0.551249
\(552\) 0 0
\(553\) 1.72398e27 0.0274105
\(554\) 0 0
\(555\) 5.30608e28 0.812264
\(556\) 0 0
\(557\) −1.03071e28 −0.151935 −0.0759676 0.997110i \(-0.524205\pi\)
−0.0759676 + 0.997110i \(0.524205\pi\)
\(558\) 0 0
\(559\) 7.44432e28 1.05682
\(560\) 0 0
\(561\) 1.12577e29 1.53936
\(562\) 0 0
\(563\) 8.09973e28 1.06692 0.533461 0.845825i \(-0.320891\pi\)
0.533461 + 0.845825i \(0.320891\pi\)
\(564\) 0 0
\(565\) 6.25363e28 0.793639
\(566\) 0 0
\(567\) −4.36530e27 −0.0533816
\(568\) 0 0
\(569\) 4.57054e28 0.538628 0.269314 0.963052i \(-0.413203\pi\)
0.269314 + 0.963052i \(0.413203\pi\)
\(570\) 0 0
\(571\) −8.57917e28 −0.974465 −0.487232 0.873272i \(-0.661993\pi\)
−0.487232 + 0.873272i \(0.661993\pi\)
\(572\) 0 0
\(573\) −5.63322e26 −0.00616785
\(574\) 0 0
\(575\) 1.63380e28 0.172459
\(576\) 0 0
\(577\) −7.50127e28 −0.763464 −0.381732 0.924273i \(-0.624672\pi\)
−0.381732 + 0.924273i \(0.624672\pi\)
\(578\) 0 0
\(579\) 9.65890e28 0.947988
\(580\) 0 0
\(581\) −6.23766e28 −0.590435
\(582\) 0 0
\(583\) 2.48472e29 2.26860
\(584\) 0 0
\(585\) −3.52432e28 −0.310412
\(586\) 0 0
\(587\) 1.80794e29 1.53633 0.768163 0.640254i \(-0.221171\pi\)
0.768163 + 0.640254i \(0.221171\pi\)
\(588\) 0 0
\(589\) −1.34819e28 −0.110546
\(590\) 0 0
\(591\) 1.05650e28 0.0835991
\(592\) 0 0
\(593\) −2.00538e29 −1.53152 −0.765759 0.643128i \(-0.777636\pi\)
−0.765759 + 0.643128i \(0.777636\pi\)
\(594\) 0 0
\(595\) 1.29357e29 0.953587
\(596\) 0 0
\(597\) 7.51244e28 0.534626
\(598\) 0 0
\(599\) −2.43168e29 −1.67080 −0.835401 0.549641i \(-0.814765\pi\)
−0.835401 + 0.549641i \(0.814765\pi\)
\(600\) 0 0
\(601\) 7.21456e28 0.478661 0.239331 0.970938i \(-0.423072\pi\)
0.239331 + 0.970938i \(0.423072\pi\)
\(602\) 0 0
\(603\) −6.48761e27 −0.0415674
\(604\) 0 0
\(605\) 2.87830e29 1.78117
\(606\) 0 0
\(607\) 3.08349e29 1.84315 0.921575 0.388199i \(-0.126903\pi\)
0.921575 + 0.388199i \(0.126903\pi\)
\(608\) 0 0
\(609\) −4.69430e28 −0.271075
\(610\) 0 0
\(611\) −1.36696e29 −0.762645
\(612\) 0 0
\(613\) 1.74135e29 0.938754 0.469377 0.882998i \(-0.344478\pi\)
0.469377 + 0.882998i \(0.344478\pi\)
\(614\) 0 0
\(615\) −2.16876e29 −1.12985
\(616\) 0 0
\(617\) −1.09708e29 −0.552386 −0.276193 0.961102i \(-0.589073\pi\)
−0.276193 + 0.961102i \(0.589073\pi\)
\(618\) 0 0
\(619\) −4.18534e28 −0.203695 −0.101847 0.994800i \(-0.532475\pi\)
−0.101847 + 0.994800i \(0.532475\pi\)
\(620\) 0 0
\(621\) −1.80381e28 −0.0848649
\(622\) 0 0
\(623\) 5.07257e28 0.230730
\(624\) 0 0
\(625\) −2.81520e29 −1.23814
\(626\) 0 0
\(627\) −1.21320e29 −0.515969
\(628\) 0 0
\(629\) 4.88046e29 2.00738
\(630\) 0 0
\(631\) 7.19034e28 0.286049 0.143025 0.989719i \(-0.454317\pi\)
0.143025 + 0.989719i \(0.454317\pi\)
\(632\) 0 0
\(633\) 7.55873e28 0.290877
\(634\) 0 0
\(635\) −5.23326e29 −1.94827
\(636\) 0 0
\(637\) 1.68607e29 0.607312
\(638\) 0 0
\(639\) 1.70849e29 0.595462
\(640\) 0 0
\(641\) −7.94295e28 −0.267900 −0.133950 0.990988i \(-0.542766\pi\)
−0.133950 + 0.990988i \(0.542766\pi\)
\(642\) 0 0
\(643\) 2.98801e29 0.975363 0.487682 0.873022i \(-0.337843\pi\)
0.487682 + 0.873022i \(0.337843\pi\)
\(644\) 0 0
\(645\) 2.88478e29 0.911456
\(646\) 0 0
\(647\) 1.99749e29 0.610926 0.305463 0.952204i \(-0.401189\pi\)
0.305463 + 0.952204i \(0.401189\pi\)
\(648\) 0 0
\(649\) −2.85414e29 −0.845097
\(650\) 0 0
\(651\) −1.89619e28 −0.0543604
\(652\) 0 0
\(653\) 7.93075e28 0.220154 0.110077 0.993923i \(-0.464890\pi\)
0.110077 + 0.993923i \(0.464890\pi\)
\(654\) 0 0
\(655\) −4.99049e29 −1.34156
\(656\) 0 0
\(657\) −4.93448e28 −0.128471
\(658\) 0 0
\(659\) 5.45555e29 1.37576 0.687878 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(660\) 0 0
\(661\) 3.69429e28 0.0902435 0.0451218 0.998981i \(-0.485632\pi\)
0.0451218 + 0.998981i \(0.485632\pi\)
\(662\) 0 0
\(663\) −3.24162e29 −0.767133
\(664\) 0 0
\(665\) −1.39403e29 −0.319627
\(666\) 0 0
\(667\) −1.93976e29 −0.430949
\(668\) 0 0
\(669\) 5.02152e29 1.08109
\(670\) 0 0
\(671\) −1.99632e29 −0.416526
\(672\) 0 0
\(673\) 1.85053e29 0.374230 0.187115 0.982338i \(-0.440086\pi\)
0.187115 + 0.982338i \(0.440086\pi\)
\(674\) 0 0
\(675\) −3.83958e28 −0.0752651
\(676\) 0 0
\(677\) 4.29001e29 0.815225 0.407612 0.913155i \(-0.366361\pi\)
0.407612 + 0.913155i \(0.366361\pi\)
\(678\) 0 0
\(679\) 3.29527e29 0.607098
\(680\) 0 0
\(681\) −3.46844e29 −0.619570
\(682\) 0 0
\(683\) −1.06550e29 −0.184559 −0.0922796 0.995733i \(-0.529415\pi\)
−0.0922796 + 0.995733i \(0.529415\pi\)
\(684\) 0 0
\(685\) 4.78117e29 0.803125
\(686\) 0 0
\(687\) −3.08699e29 −0.502909
\(688\) 0 0
\(689\) −7.15467e29 −1.13054
\(690\) 0 0
\(691\) 5.17173e29 0.792714 0.396357 0.918096i \(-0.370274\pi\)
0.396357 + 0.918096i \(0.370274\pi\)
\(692\) 0 0
\(693\) −1.70633e29 −0.253726
\(694\) 0 0
\(695\) −7.90141e29 −1.13989
\(696\) 0 0
\(697\) −1.99480e30 −2.79225
\(698\) 0 0
\(699\) 7.18419e29 0.975815
\(700\) 0 0
\(701\) 5.77924e29 0.761784 0.380892 0.924620i \(-0.375617\pi\)
0.380892 + 0.924620i \(0.375617\pi\)
\(702\) 0 0
\(703\) −5.25949e29 −0.672841
\(704\) 0 0
\(705\) −5.29715e29 −0.657744
\(706\) 0 0
\(707\) −8.89028e27 −0.0107155
\(708\) 0 0
\(709\) −1.60050e30 −1.87270 −0.936352 0.351063i \(-0.885820\pi\)
−0.936352 + 0.351063i \(0.885820\pi\)
\(710\) 0 0
\(711\) −1.67415e28 −0.0190179
\(712\) 0 0
\(713\) −7.83536e28 −0.0864210
\(714\) 0 0
\(715\) −1.37761e30 −1.47541
\(716\) 0 0
\(717\) −8.77288e29 −0.912412
\(718\) 0 0
\(719\) 6.62814e29 0.669482 0.334741 0.942310i \(-0.391351\pi\)
0.334741 + 0.942310i \(0.391351\pi\)
\(720\) 0 0
\(721\) 6.09820e29 0.598248
\(722\) 0 0
\(723\) −1.13496e30 −1.08151
\(724\) 0 0
\(725\) −4.12896e29 −0.382200
\(726\) 0 0
\(727\) 3.75931e29 0.338062 0.169031 0.985611i \(-0.445936\pi\)
0.169031 + 0.985611i \(0.445936\pi\)
\(728\) 0 0
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) 2.65338e30 2.25251
\(732\) 0 0
\(733\) 3.05971e29 0.252400 0.126200 0.992005i \(-0.459722\pi\)
0.126200 + 0.992005i \(0.459722\pi\)
\(734\) 0 0
\(735\) 6.53376e29 0.523777
\(736\) 0 0
\(737\) −2.53592e29 −0.197573
\(738\) 0 0
\(739\) −2.09955e30 −1.58987 −0.794933 0.606698i \(-0.792494\pi\)
−0.794933 + 0.606698i \(0.792494\pi\)
\(740\) 0 0
\(741\) 3.49337e29 0.257131
\(742\) 0 0
\(743\) 1.40094e30 1.00239 0.501196 0.865334i \(-0.332893\pi\)
0.501196 + 0.865334i \(0.332893\pi\)
\(744\) 0 0
\(745\) −2.65551e30 −1.84717
\(746\) 0 0
\(747\) 6.05736e29 0.409654
\(748\) 0 0
\(749\) 8.65524e29 0.569141
\(750\) 0 0
\(751\) −2.12216e30 −1.35694 −0.678468 0.734630i \(-0.737356\pi\)
−0.678468 + 0.734630i \(0.737356\pi\)
\(752\) 0 0
\(753\) −7.83502e29 −0.487184
\(754\) 0 0
\(755\) −1.39927e30 −0.846170
\(756\) 0 0
\(757\) 2.95242e30 1.73649 0.868244 0.496138i \(-0.165249\pi\)
0.868244 + 0.496138i \(0.165249\pi\)
\(758\) 0 0
\(759\) −7.05083e29 −0.403368
\(760\) 0 0
\(761\) 9.69651e29 0.539606 0.269803 0.962916i \(-0.413041\pi\)
0.269803 + 0.962916i \(0.413041\pi\)
\(762\) 0 0
\(763\) 6.67250e29 0.361228
\(764\) 0 0
\(765\) −1.25617e30 −0.661614
\(766\) 0 0
\(767\) 8.21841e29 0.421150
\(768\) 0 0
\(769\) 2.29748e30 1.14558 0.572790 0.819702i \(-0.305861\pi\)
0.572790 + 0.819702i \(0.305861\pi\)
\(770\) 0 0
\(771\) −7.48078e29 −0.362974
\(772\) 0 0
\(773\) −1.39639e30 −0.659361 −0.329681 0.944093i \(-0.606941\pi\)
−0.329681 + 0.944093i \(0.606941\pi\)
\(774\) 0 0
\(775\) −1.66783e29 −0.0766451
\(776\) 0 0
\(777\) −7.39732e29 −0.330867
\(778\) 0 0
\(779\) 2.14972e30 0.935918
\(780\) 0 0
\(781\) 6.67826e30 2.83027
\(782\) 0 0
\(783\) 4.55861e29 0.188076
\(784\) 0 0
\(785\) 1.60741e30 0.645648
\(786\) 0 0
\(787\) 1.65709e30 0.648053 0.324027 0.946048i \(-0.394963\pi\)
0.324027 + 0.946048i \(0.394963\pi\)
\(788\) 0 0
\(789\) −9.05040e29 −0.344635
\(790\) 0 0
\(791\) −8.71832e29 −0.323281
\(792\) 0 0
\(793\) 5.74832e29 0.207574
\(794\) 0 0
\(795\) −2.77253e30 −0.975039
\(796\) 0 0
\(797\) 4.83413e30 1.65579 0.827896 0.560882i \(-0.189538\pi\)
0.827896 + 0.560882i \(0.189538\pi\)
\(798\) 0 0
\(799\) −4.87226e30 −1.62551
\(800\) 0 0
\(801\) −4.92595e29 −0.160084
\(802\) 0 0
\(803\) −1.92882e30 −0.610630
\(804\) 0 0
\(805\) −8.10174e29 −0.249874
\(806\) 0 0
\(807\) 1.12119e30 0.336905
\(808\) 0 0
\(809\) −3.24758e30 −0.950823 −0.475412 0.879763i \(-0.657701\pi\)
−0.475412 + 0.879763i \(0.657701\pi\)
\(810\) 0 0
\(811\) 1.41163e30 0.402719 0.201360 0.979517i \(-0.435464\pi\)
0.201360 + 0.979517i \(0.435464\pi\)
\(812\) 0 0
\(813\) −4.59013e29 −0.127607
\(814\) 0 0
\(815\) 7.10187e30 1.92406
\(816\) 0 0
\(817\) −2.85945e30 −0.755007
\(818\) 0 0
\(819\) 4.91333e29 0.126443
\(820\) 0 0
\(821\) 6.90028e30 1.73087 0.865433 0.501024i \(-0.167043\pi\)
0.865433 + 0.501024i \(0.167043\pi\)
\(822\) 0 0
\(823\) 3.25832e30 0.796702 0.398351 0.917233i \(-0.369583\pi\)
0.398351 + 0.917233i \(0.369583\pi\)
\(824\) 0 0
\(825\) −1.50084e30 −0.357740
\(826\) 0 0
\(827\) 1.21005e30 0.281187 0.140593 0.990067i \(-0.455099\pi\)
0.140593 + 0.990067i \(0.455099\pi\)
\(828\) 0 0
\(829\) −2.20920e30 −0.500510 −0.250255 0.968180i \(-0.580514\pi\)
−0.250255 + 0.968180i \(0.580514\pi\)
\(830\) 0 0
\(831\) 2.65145e30 0.585696
\(832\) 0 0
\(833\) 6.00967e30 1.29443
\(834\) 0 0
\(835\) 1.11081e30 0.233310
\(836\) 0 0
\(837\) 1.84138e29 0.0377161
\(838\) 0 0
\(839\) 9.04315e30 1.80642 0.903211 0.429198i \(-0.141204\pi\)
0.903211 + 0.429198i \(0.141204\pi\)
\(840\) 0 0
\(841\) −2.30662e29 −0.0449385
\(842\) 0 0
\(843\) 2.37231e30 0.450798
\(844\) 0 0
\(845\) −2.39639e30 −0.444183
\(846\) 0 0
\(847\) −4.01271e30 −0.725540
\(848\) 0 0
\(849\) 2.33271e30 0.411462
\(850\) 0 0
\(851\) −3.05669e30 −0.526006
\(852\) 0 0
\(853\) 2.44589e30 0.410650 0.205325 0.978694i \(-0.434175\pi\)
0.205325 + 0.978694i \(0.434175\pi\)
\(854\) 0 0
\(855\) 1.35373e30 0.221763
\(856\) 0 0
\(857\) −3.48470e30 −0.557015 −0.278508 0.960434i \(-0.589840\pi\)
−0.278508 + 0.960434i \(0.589840\pi\)
\(858\) 0 0
\(859\) −3.03387e30 −0.473226 −0.236613 0.971604i \(-0.576037\pi\)
−0.236613 + 0.971604i \(0.576037\pi\)
\(860\) 0 0
\(861\) 3.02351e30 0.460234
\(862\) 0 0
\(863\) −1.80822e30 −0.268620 −0.134310 0.990939i \(-0.542882\pi\)
−0.134310 + 0.990939i \(0.542882\pi\)
\(864\) 0 0
\(865\) −1.27880e31 −1.85410
\(866\) 0 0
\(867\) −7.47433e30 −1.05772
\(868\) 0 0
\(869\) −6.54400e29 −0.0903932
\(870\) 0 0
\(871\) 7.30209e29 0.0984593
\(872\) 0 0
\(873\) −3.20002e30 −0.421215
\(874\) 0 0
\(875\) 2.68503e30 0.345037
\(876\) 0 0
\(877\) −1.01880e31 −1.27818 −0.639089 0.769133i \(-0.720689\pi\)
−0.639089 + 0.769133i \(0.720689\pi\)
\(878\) 0 0
\(879\) −6.26021e30 −0.766842
\(880\) 0 0
\(881\) −8.65056e30 −1.03466 −0.517330 0.855786i \(-0.673074\pi\)
−0.517330 + 0.855786i \(0.673074\pi\)
\(882\) 0 0
\(883\) 1.16043e31 1.35529 0.677643 0.735391i \(-0.263002\pi\)
0.677643 + 0.735391i \(0.263002\pi\)
\(884\) 0 0
\(885\) 3.18475e30 0.363221
\(886\) 0 0
\(887\) 3.27819e30 0.365120 0.182560 0.983195i \(-0.441562\pi\)
0.182560 + 0.983195i \(0.441562\pi\)
\(888\) 0 0
\(889\) 7.29581e30 0.793606
\(890\) 0 0
\(891\) 1.65701e30 0.176039
\(892\) 0 0
\(893\) 5.25064e30 0.544844
\(894\) 0 0
\(895\) −4.12029e30 −0.417624
\(896\) 0 0
\(897\) 2.03026e30 0.201016
\(898\) 0 0
\(899\) 1.98017e30 0.191525
\(900\) 0 0
\(901\) −2.55014e31 −2.40965
\(902\) 0 0
\(903\) −4.02174e30 −0.371272
\(904\) 0 0
\(905\) −2.68468e30 −0.242149
\(906\) 0 0
\(907\) 2.71433e30 0.239213 0.119607 0.992821i \(-0.461837\pi\)
0.119607 + 0.992821i \(0.461837\pi\)
\(908\) 0 0
\(909\) 8.63331e28 0.00743458
\(910\) 0 0
\(911\) −1.50470e31 −1.26621 −0.633107 0.774064i \(-0.718221\pi\)
−0.633107 + 0.774064i \(0.718221\pi\)
\(912\) 0 0
\(913\) 2.36774e31 1.94711
\(914\) 0 0
\(915\) 2.22756e30 0.179022
\(916\) 0 0
\(917\) 6.95735e30 0.546470
\(918\) 0 0
\(919\) 1.22782e31 0.942585 0.471292 0.881977i \(-0.343788\pi\)
0.471292 + 0.881977i \(0.343788\pi\)
\(920\) 0 0
\(921\) 7.51886e30 0.564191
\(922\) 0 0
\(923\) −1.92298e31 −1.41045
\(924\) 0 0
\(925\) −6.50645e30 −0.466505
\(926\) 0 0
\(927\) −5.92193e30 −0.415074
\(928\) 0 0
\(929\) 2.63481e31 1.80545 0.902724 0.430221i \(-0.141564\pi\)
0.902724 + 0.430221i \(0.141564\pi\)
\(930\) 0 0
\(931\) −6.47639e30 −0.433872
\(932\) 0 0
\(933\) −1.24182e31 −0.813394
\(934\) 0 0
\(935\) −4.91021e31 −3.14469
\(936\) 0 0
\(937\) 8.75508e30 0.548270 0.274135 0.961691i \(-0.411608\pi\)
0.274135 + 0.961691i \(0.411608\pi\)
\(938\) 0 0
\(939\) 2.52235e30 0.154460
\(940\) 0 0
\(941\) −2.13855e31 −1.28064 −0.640321 0.768107i \(-0.721199\pi\)
−0.640321 + 0.768107i \(0.721199\pi\)
\(942\) 0 0
\(943\) 1.24936e31 0.731671
\(944\) 0 0
\(945\) 1.90399e30 0.109051
\(946\) 0 0
\(947\) 9.33092e30 0.522697 0.261348 0.965245i \(-0.415833\pi\)
0.261348 + 0.965245i \(0.415833\pi\)
\(948\) 0 0
\(949\) 5.55397e30 0.304304
\(950\) 0 0
\(951\) 6.26677e29 0.0335852
\(952\) 0 0
\(953\) −6.95374e30 −0.364538 −0.182269 0.983249i \(-0.558344\pi\)
−0.182269 + 0.983249i \(0.558344\pi\)
\(954\) 0 0
\(955\) 2.45701e29 0.0126000
\(956\) 0 0
\(957\) 1.78190e31 0.893937
\(958\) 0 0
\(959\) −6.66554e30 −0.327145
\(960\) 0 0
\(961\) −2.00256e31 −0.961592
\(962\) 0 0
\(963\) −8.40506e30 −0.394880
\(964\) 0 0
\(965\) −4.21287e31 −1.93660
\(966\) 0 0
\(967\) −6.93817e30 −0.312081 −0.156040 0.987751i \(-0.549873\pi\)
−0.156040 + 0.987751i \(0.549873\pi\)
\(968\) 0 0
\(969\) 1.24514e31 0.548050
\(970\) 0 0
\(971\) −6.25351e30 −0.269353 −0.134676 0.990890i \(-0.543000\pi\)
−0.134676 + 0.990890i \(0.543000\pi\)
\(972\) 0 0
\(973\) 1.10155e31 0.464324
\(974\) 0 0
\(975\) 4.32161e30 0.178278
\(976\) 0 0
\(977\) −3.80828e30 −0.153757 −0.0768787 0.997040i \(-0.524495\pi\)
−0.0768787 + 0.997040i \(0.524495\pi\)
\(978\) 0 0
\(979\) −1.92548e31 −0.760889
\(980\) 0 0
\(981\) −6.47964e30 −0.250626
\(982\) 0 0
\(983\) 1.71631e31 0.649807 0.324903 0.945747i \(-0.394668\pi\)
0.324903 + 0.945747i \(0.394668\pi\)
\(984\) 0 0
\(985\) −4.60808e30 −0.170781
\(986\) 0 0
\(987\) 7.38488e30 0.267925
\(988\) 0 0
\(989\) −1.66184e31 −0.590240
\(990\) 0 0
\(991\) 1.07510e31 0.373830 0.186915 0.982376i \(-0.440151\pi\)
0.186915 + 0.982376i \(0.440151\pi\)
\(992\) 0 0
\(993\) −5.01973e29 −0.0170889
\(994\) 0 0
\(995\) −3.27666e31 −1.09217
\(996\) 0 0
\(997\) −2.35137e31 −0.767399 −0.383699 0.923458i \(-0.625350\pi\)
−0.383699 + 0.923458i \(0.625350\pi\)
\(998\) 0 0
\(999\) 7.18350e30 0.229561
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 48.22.a.h.1.2 2
4.3 odd 2 24.22.a.a.1.2 2
12.11 even 2 72.22.a.a.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.22.a.a.1.2 2 4.3 odd 2
48.22.a.h.1.2 2 1.1 even 1 trivial
72.22.a.a.1.1 2 12.11 even 2