Properties

Label 45.22.a.g.1.3
Level $45$
Weight $22$
Character 45.1
Self dual yes
Analytic conductor $125.765$
Analytic rank $0$
Dimension $4$
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] = [45,22,Mod(1,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, 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("45.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 45.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: \(125.764804929\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3512166x^{2} + 363520480x + 2321089280000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1069.36\) of defining polynomial
Character \(\chi\) \(=\) 45.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1293.36 q^{2} -424379. q^{4} -9.76562e6 q^{5} +1.27960e9 q^{7} -3.26124e9 q^{8} +O(q^{10})\) \(q+1293.36 q^{2} -424379. q^{4} -9.76562e6 q^{5} +1.27960e9 q^{7} -3.26124e9 q^{8} -1.26304e10 q^{10} +1.21178e11 q^{11} +7.37152e11 q^{13} +1.65498e12 q^{14} -3.32796e12 q^{16} -6.05462e12 q^{17} -3.23277e13 q^{19} +4.14433e12 q^{20} +1.56726e14 q^{22} +3.98109e13 q^{23} +9.53674e13 q^{25} +9.53401e14 q^{26} -5.43035e14 q^{28} -1.40928e15 q^{29} -2.62721e15 q^{31} +2.53508e15 q^{32} -7.83079e15 q^{34} -1.24961e16 q^{35} -3.44728e15 q^{37} -4.18113e16 q^{38} +3.18481e16 q^{40} -7.99269e16 q^{41} +2.07610e17 q^{43} -5.14254e16 q^{44} +5.14897e16 q^{46} +1.95408e17 q^{47} +1.07882e18 q^{49} +1.23344e17 q^{50} -3.12832e17 q^{52} +2.33305e17 q^{53} -1.18338e18 q^{55} -4.17307e18 q^{56} -1.82271e18 q^{58} +3.93099e18 q^{59} +8.99462e18 q^{61} -3.39792e18 q^{62} +1.02580e19 q^{64} -7.19875e18 q^{65} +9.21439e18 q^{67} +2.56946e18 q^{68} -1.61619e19 q^{70} +4.75444e19 q^{71} +1.81566e19 q^{73} -4.45857e18 q^{74} +1.37192e19 q^{76} +1.55059e20 q^{77} -1.11814e20 q^{79} +3.24996e19 q^{80} -1.03374e20 q^{82} +3.39283e19 q^{83} +5.91272e19 q^{85} +2.68514e20 q^{86} -3.95190e20 q^{88} -3.38998e20 q^{89} +9.43257e20 q^{91} -1.68949e19 q^{92} +2.52733e20 q^{94} +3.15700e20 q^{95} +1.12006e20 q^{97} +1.39530e21 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 897 q^{2} - 1163123 q^{4} - 39062500 q^{5} - 234577504 q^{7} - 76855629 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 897 q^{2} - 1163123 q^{4} - 39062500 q^{5} - 234577504 q^{7} - 76855629 q^{8} - 8759765625 q^{10} - 31491830256 q^{11} - 27017977768 q^{13} + 2233308126672 q^{14} - 11324681311775 q^{16} + 2946095028888 q^{17} - 24270353300752 q^{19} + 11358623046875 q^{20} - 56303932793676 q^{22} - 10350924920928 q^{23} + 381469726562500 q^{25} - 474751622871378 q^{26} - 18\!\cdots\!68 q^{28}+ \cdots - 19\!\cdots\!63 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1293.36 0.893107 0.446553 0.894757i \(-0.352651\pi\)
0.446553 + 0.894757i \(0.352651\pi\)
\(3\) 0 0
\(4\) −424379. −0.202360
\(5\) −9.76562e6 −0.447214
\(6\) 0 0
\(7\) 1.27960e9 1.71216 0.856079 0.516846i \(-0.172894\pi\)
0.856079 + 0.516846i \(0.172894\pi\)
\(8\) −3.26124e9 −1.07384
\(9\) 0 0
\(10\) −1.26304e10 −0.399410
\(11\) 1.21178e11 1.40864 0.704320 0.709883i \(-0.251252\pi\)
0.704320 + 0.709883i \(0.251252\pi\)
\(12\) 0 0
\(13\) 7.37152e11 1.48304 0.741518 0.670933i \(-0.234106\pi\)
0.741518 + 0.670933i \(0.234106\pi\)
\(14\) 1.65498e12 1.52914
\(15\) 0 0
\(16\) −3.32796e12 −0.756691
\(17\) −6.05462e12 −0.728406 −0.364203 0.931320i \(-0.618659\pi\)
−0.364203 + 0.931320i \(0.618659\pi\)
\(18\) 0 0
\(19\) −3.23277e13 −1.20966 −0.604828 0.796356i \(-0.706758\pi\)
−0.604828 + 0.796356i \(0.706758\pi\)
\(20\) 4.14433e12 0.0904981
\(21\) 0 0
\(22\) 1.56726e14 1.25807
\(23\) 3.98109e13 0.200382 0.100191 0.994968i \(-0.468055\pi\)
0.100191 + 0.994968i \(0.468055\pi\)
\(24\) 0 0
\(25\) 9.53674e13 0.200000
\(26\) 9.53401e14 1.32451
\(27\) 0 0
\(28\) −5.43035e14 −0.346472
\(29\) −1.40928e15 −0.622041 −0.311021 0.950403i \(-0.600671\pi\)
−0.311021 + 0.950403i \(0.600671\pi\)
\(30\) 0 0
\(31\) −2.62721e15 −0.575701 −0.287850 0.957675i \(-0.592941\pi\)
−0.287850 + 0.957675i \(0.592941\pi\)
\(32\) 2.53508e15 0.398030
\(33\) 0 0
\(34\) −7.83079e15 −0.650544
\(35\) −1.24961e16 −0.765700
\(36\) 0 0
\(37\) −3.44728e15 −0.117858 −0.0589290 0.998262i \(-0.518769\pi\)
−0.0589290 + 0.998262i \(0.518769\pi\)
\(38\) −4.18113e16 −1.08035
\(39\) 0 0
\(40\) 3.18481e16 0.480234
\(41\) −7.99269e16 −0.929956 −0.464978 0.885322i \(-0.653938\pi\)
−0.464978 + 0.885322i \(0.653938\pi\)
\(42\) 0 0
\(43\) 2.07610e17 1.46498 0.732488 0.680780i \(-0.238359\pi\)
0.732488 + 0.680780i \(0.238359\pi\)
\(44\) −5.14254e16 −0.285052
\(45\) 0 0
\(46\) 5.14897e16 0.178963
\(47\) 1.95408e17 0.541895 0.270948 0.962594i \(-0.412663\pi\)
0.270948 + 0.962594i \(0.412663\pi\)
\(48\) 0 0
\(49\) 1.07882e18 1.93148
\(50\) 1.23344e17 0.178621
\(51\) 0 0
\(52\) −3.12832e17 −0.300107
\(53\) 2.33305e17 0.183243 0.0916214 0.995794i \(-0.470795\pi\)
0.0916214 + 0.995794i \(0.470795\pi\)
\(54\) 0 0
\(55\) −1.18338e18 −0.629963
\(56\) −4.17307e18 −1.83858
\(57\) 0 0
\(58\) −1.82271e18 −0.555550
\(59\) 3.93099e18 1.00128 0.500640 0.865656i \(-0.333098\pi\)
0.500640 + 0.865656i \(0.333098\pi\)
\(60\) 0 0
\(61\) 8.99462e18 1.61443 0.807216 0.590256i \(-0.200973\pi\)
0.807216 + 0.590256i \(0.200973\pi\)
\(62\) −3.39792e18 −0.514162
\(63\) 0 0
\(64\) 1.02580e19 1.11217
\(65\) −7.19875e18 −0.663234
\(66\) 0 0
\(67\) 9.21439e18 0.617563 0.308781 0.951133i \(-0.400079\pi\)
0.308781 + 0.951133i \(0.400079\pi\)
\(68\) 2.56946e18 0.147400
\(69\) 0 0
\(70\) −1.61619e19 −0.683852
\(71\) 4.75444e19 1.73335 0.866676 0.498871i \(-0.166252\pi\)
0.866676 + 0.498871i \(0.166252\pi\)
\(72\) 0 0
\(73\) 1.81566e19 0.494474 0.247237 0.968955i \(-0.420477\pi\)
0.247237 + 0.968955i \(0.420477\pi\)
\(74\) −4.45857e18 −0.105260
\(75\) 0 0
\(76\) 1.37192e19 0.244786
\(77\) 1.55059e20 2.41181
\(78\) 0 0
\(79\) −1.11814e20 −1.32865 −0.664327 0.747442i \(-0.731282\pi\)
−0.664327 + 0.747442i \(0.731282\pi\)
\(80\) 3.24996e19 0.338402
\(81\) 0 0
\(82\) −1.03374e20 −0.830550
\(83\) 3.39283e19 0.240017 0.120009 0.992773i \(-0.461708\pi\)
0.120009 + 0.992773i \(0.461708\pi\)
\(84\) 0 0
\(85\) 5.91272e19 0.325753
\(86\) 2.68514e20 1.30838
\(87\) 0 0
\(88\) −3.95190e20 −1.51265
\(89\) −3.38998e20 −1.15240 −0.576199 0.817310i \(-0.695465\pi\)
−0.576199 + 0.817310i \(0.695465\pi\)
\(90\) 0 0
\(91\) 9.43257e20 2.53919
\(92\) −1.68949e19 −0.0405494
\(93\) 0 0
\(94\) 2.52733e20 0.483970
\(95\) 3.15700e20 0.540975
\(96\) 0 0
\(97\) 1.12006e20 0.154219 0.0771097 0.997023i \(-0.475431\pi\)
0.0771097 + 0.997023i \(0.475431\pi\)
\(98\) 1.39530e21 1.72502
\(99\) 0 0
\(100\) −4.04720e19 −0.0404720
\(101\) −2.51635e20 −0.226672 −0.113336 0.993557i \(-0.536154\pi\)
−0.113336 + 0.993557i \(0.536154\pi\)
\(102\) 0 0
\(103\) 2.29653e21 1.68377 0.841883 0.539660i \(-0.181447\pi\)
0.841883 + 0.539660i \(0.181447\pi\)
\(104\) −2.40403e21 −1.59254
\(105\) 0 0
\(106\) 3.01746e20 0.163655
\(107\) −3.75913e21 −1.84739 −0.923694 0.383130i \(-0.874846\pi\)
−0.923694 + 0.383130i \(0.874846\pi\)
\(108\) 0 0
\(109\) −1.44930e21 −0.586380 −0.293190 0.956054i \(-0.594717\pi\)
−0.293190 + 0.956054i \(0.594717\pi\)
\(110\) −1.53053e21 −0.562624
\(111\) 0 0
\(112\) −4.25845e21 −1.29557
\(113\) 5.37147e21 1.48857 0.744285 0.667862i \(-0.232790\pi\)
0.744285 + 0.667862i \(0.232790\pi\)
\(114\) 0 0
\(115\) −3.88778e20 −0.0896137
\(116\) 5.98071e20 0.125876
\(117\) 0 0
\(118\) 5.08417e21 0.894250
\(119\) −7.74747e21 −1.24715
\(120\) 0 0
\(121\) 7.28381e21 0.984266
\(122\) 1.16333e22 1.44186
\(123\) 0 0
\(124\) 1.11493e21 0.116499
\(125\) −9.31323e20 −0.0894427
\(126\) 0 0
\(127\) 1.99248e22 1.61978 0.809888 0.586585i \(-0.199528\pi\)
0.809888 + 0.586585i \(0.199528\pi\)
\(128\) 7.95081e21 0.595260
\(129\) 0 0
\(130\) −9.31056e21 −0.592339
\(131\) −1.95301e22 −1.14645 −0.573225 0.819398i \(-0.694308\pi\)
−0.573225 + 0.819398i \(0.694308\pi\)
\(132\) 0 0
\(133\) −4.13664e22 −2.07112
\(134\) 1.19175e22 0.551549
\(135\) 0 0
\(136\) 1.97456e22 0.782188
\(137\) −2.54688e22 −0.934206 −0.467103 0.884203i \(-0.654702\pi\)
−0.467103 + 0.884203i \(0.654702\pi\)
\(138\) 0 0
\(139\) −3.28445e22 −1.03468 −0.517341 0.855780i \(-0.673078\pi\)
−0.517341 + 0.855780i \(0.673078\pi\)
\(140\) 5.30307e21 0.154947
\(141\) 0 0
\(142\) 6.14919e22 1.54807
\(143\) 8.93265e22 2.08906
\(144\) 0 0
\(145\) 1.37625e22 0.278185
\(146\) 2.34829e22 0.441618
\(147\) 0 0
\(148\) 1.46296e21 0.0238497
\(149\) 4.10815e22 0.624009 0.312004 0.950081i \(-0.399000\pi\)
0.312004 + 0.950081i \(0.399000\pi\)
\(150\) 0 0
\(151\) 8.76777e22 1.15779 0.578897 0.815401i \(-0.303483\pi\)
0.578897 + 0.815401i \(0.303483\pi\)
\(152\) 1.05428e23 1.29897
\(153\) 0 0
\(154\) 2.00546e23 2.15401
\(155\) 2.56563e22 0.257461
\(156\) 0 0
\(157\) 1.06605e23 0.935040 0.467520 0.883982i \(-0.345148\pi\)
0.467520 + 0.883982i \(0.345148\pi\)
\(158\) −1.44615e23 −1.18663
\(159\) 0 0
\(160\) −2.47566e22 −0.178005
\(161\) 5.09419e22 0.343086
\(162\) 0 0
\(163\) 2.93390e23 1.73571 0.867853 0.496822i \(-0.165500\pi\)
0.867853 + 0.496822i \(0.165500\pi\)
\(164\) 3.39193e22 0.188186
\(165\) 0 0
\(166\) 4.38814e22 0.214361
\(167\) −1.40812e23 −0.645830 −0.322915 0.946428i \(-0.604663\pi\)
−0.322915 + 0.946428i \(0.604663\pi\)
\(168\) 0 0
\(169\) 2.96329e23 1.19940
\(170\) 7.64725e22 0.290932
\(171\) 0 0
\(172\) −8.81056e22 −0.296452
\(173\) −3.14336e21 −0.00995198 −0.00497599 0.999988i \(-0.501584\pi\)
−0.00497599 + 0.999988i \(0.501584\pi\)
\(174\) 0 0
\(175\) 1.22032e23 0.342431
\(176\) −4.03275e23 −1.06590
\(177\) 0 0
\(178\) −4.38446e23 −1.02921
\(179\) 7.66967e23 1.69754 0.848770 0.528763i \(-0.177344\pi\)
0.848770 + 0.528763i \(0.177344\pi\)
\(180\) 0 0
\(181\) 1.51630e23 0.298649 0.149325 0.988788i \(-0.452290\pi\)
0.149325 + 0.988788i \(0.452290\pi\)
\(182\) 1.21997e24 2.26777
\(183\) 0 0
\(184\) −1.29833e23 −0.215178
\(185\) 3.36649e22 0.0527077
\(186\) 0 0
\(187\) −7.33686e23 −1.02606
\(188\) −8.29272e22 −0.109658
\(189\) 0 0
\(190\) 4.08313e23 0.483148
\(191\) −6.06790e23 −0.679498 −0.339749 0.940516i \(-0.610342\pi\)
−0.339749 + 0.940516i \(0.610342\pi\)
\(192\) 0 0
\(193\) 1.87596e24 1.88309 0.941544 0.336889i \(-0.109375\pi\)
0.941544 + 0.336889i \(0.109375\pi\)
\(194\) 1.44864e23 0.137734
\(195\) 0 0
\(196\) −4.57830e23 −0.390855
\(197\) −2.26924e23 −0.183647 −0.0918237 0.995775i \(-0.529270\pi\)
−0.0918237 + 0.995775i \(0.529270\pi\)
\(198\) 0 0
\(199\) −6.13967e23 −0.446876 −0.223438 0.974718i \(-0.571728\pi\)
−0.223438 + 0.974718i \(0.571728\pi\)
\(200\) −3.11016e23 −0.214767
\(201\) 0 0
\(202\) −3.25454e23 −0.202442
\(203\) −1.80331e24 −1.06503
\(204\) 0 0
\(205\) 7.80536e23 0.415889
\(206\) 2.97024e24 1.50378
\(207\) 0 0
\(208\) −2.45321e24 −1.12220
\(209\) −3.91740e24 −1.70397
\(210\) 0 0
\(211\) 1.99189e24 0.783970 0.391985 0.919972i \(-0.371788\pi\)
0.391985 + 0.919972i \(0.371788\pi\)
\(212\) −9.90097e22 −0.0370810
\(213\) 0 0
\(214\) −4.86190e24 −1.64992
\(215\) −2.02745e24 −0.655157
\(216\) 0 0
\(217\) −3.36177e24 −0.985690
\(218\) −1.87446e24 −0.523700
\(219\) 0 0
\(220\) 5.02201e23 0.127479
\(221\) −4.46318e24 −1.08025
\(222\) 0 0
\(223\) 3.29187e24 0.724839 0.362420 0.932015i \(-0.381951\pi\)
0.362420 + 0.932015i \(0.381951\pi\)
\(224\) 3.24388e24 0.681491
\(225\) 0 0
\(226\) 6.94722e24 1.32945
\(227\) 1.53623e24 0.280663 0.140332 0.990105i \(-0.455183\pi\)
0.140332 + 0.990105i \(0.455183\pi\)
\(228\) 0 0
\(229\) −3.70932e24 −0.618048 −0.309024 0.951054i \(-0.600002\pi\)
−0.309024 + 0.951054i \(0.600002\pi\)
\(230\) −5.02829e23 −0.0800346
\(231\) 0 0
\(232\) 4.59601e24 0.667971
\(233\) 1.12524e25 1.56317 0.781587 0.623796i \(-0.214411\pi\)
0.781587 + 0.623796i \(0.214411\pi\)
\(234\) 0 0
\(235\) −1.90828e24 −0.242343
\(236\) −1.66823e24 −0.202619
\(237\) 0 0
\(238\) −1.00202e25 −1.11383
\(239\) −1.31053e25 −1.39402 −0.697008 0.717063i \(-0.745486\pi\)
−0.697008 + 0.717063i \(0.745486\pi\)
\(240\) 0 0
\(241\) 1.66470e24 0.162240 0.0811200 0.996704i \(-0.474150\pi\)
0.0811200 + 0.996704i \(0.474150\pi\)
\(242\) 9.42057e24 0.879055
\(243\) 0 0
\(244\) −3.81713e24 −0.326696
\(245\) −1.05354e25 −0.863785
\(246\) 0 0
\(247\) −2.38304e25 −1.79396
\(248\) 8.56796e24 0.618208
\(249\) 0 0
\(250\) −1.20453e24 −0.0798819
\(251\) −2.32403e25 −1.47798 −0.738990 0.673717i \(-0.764697\pi\)
−0.738990 + 0.673717i \(0.764697\pi\)
\(252\) 0 0
\(253\) 4.82420e24 0.282267
\(254\) 2.57699e25 1.44663
\(255\) 0 0
\(256\) −1.12293e25 −0.580543
\(257\) 2.65650e25 1.31829 0.659147 0.752014i \(-0.270918\pi\)
0.659147 + 0.752014i \(0.270918\pi\)
\(258\) 0 0
\(259\) −4.41113e24 −0.201791
\(260\) 3.05500e24 0.134212
\(261\) 0 0
\(262\) −2.52593e25 −1.02390
\(263\) 9.75846e24 0.380055 0.190027 0.981779i \(-0.439142\pi\)
0.190027 + 0.981779i \(0.439142\pi\)
\(264\) 0 0
\(265\) −2.27837e24 −0.0819487
\(266\) −5.35015e25 −1.84973
\(267\) 0 0
\(268\) −3.91040e24 −0.124970
\(269\) 1.15143e25 0.353866 0.176933 0.984223i \(-0.443382\pi\)
0.176933 + 0.984223i \(0.443382\pi\)
\(270\) 0 0
\(271\) 1.57414e25 0.447575 0.223788 0.974638i \(-0.428158\pi\)
0.223788 + 0.974638i \(0.428158\pi\)
\(272\) 2.01495e25 0.551178
\(273\) 0 0
\(274\) −3.29403e25 −0.834346
\(275\) 1.15564e25 0.281728
\(276\) 0 0
\(277\) 1.64872e25 0.372486 0.186243 0.982504i \(-0.440369\pi\)
0.186243 + 0.982504i \(0.440369\pi\)
\(278\) −4.24796e25 −0.924081
\(279\) 0 0
\(280\) 4.07527e25 0.822236
\(281\) −6.63562e25 −1.28963 −0.644815 0.764339i \(-0.723066\pi\)
−0.644815 + 0.764339i \(0.723066\pi\)
\(282\) 0 0
\(283\) −6.72915e25 −1.21395 −0.606977 0.794719i \(-0.707618\pi\)
−0.606977 + 0.794719i \(0.707618\pi\)
\(284\) −2.01769e25 −0.350761
\(285\) 0 0
\(286\) 1.15531e26 1.86576
\(287\) −1.02274e26 −1.59223
\(288\) 0 0
\(289\) −3.24335e25 −0.469425
\(290\) 1.77999e25 0.248449
\(291\) 0 0
\(292\) −7.70527e24 −0.100062
\(293\) 1.51857e25 0.190250 0.0951248 0.995465i \(-0.469675\pi\)
0.0951248 + 0.995465i \(0.469675\pi\)
\(294\) 0 0
\(295\) −3.83885e25 −0.447786
\(296\) 1.12424e25 0.126560
\(297\) 0 0
\(298\) 5.31331e25 0.557307
\(299\) 2.93467e25 0.297174
\(300\) 0 0
\(301\) 2.65658e26 2.50827
\(302\) 1.13399e26 1.03403
\(303\) 0 0
\(304\) 1.07585e26 0.915335
\(305\) −8.78381e25 −0.721996
\(306\) 0 0
\(307\) −2.16909e26 −1.66465 −0.832327 0.554285i \(-0.812992\pi\)
−0.832327 + 0.554285i \(0.812992\pi\)
\(308\) −6.58037e25 −0.488054
\(309\) 0 0
\(310\) 3.31828e25 0.229940
\(311\) 9.86475e25 0.660848 0.330424 0.943833i \(-0.392808\pi\)
0.330424 + 0.943833i \(0.392808\pi\)
\(312\) 0 0
\(313\) 1.28416e26 0.804273 0.402137 0.915580i \(-0.368268\pi\)
0.402137 + 0.915580i \(0.368268\pi\)
\(314\) 1.37878e26 0.835091
\(315\) 0 0
\(316\) 4.74516e25 0.268866
\(317\) 1.45513e26 0.797588 0.398794 0.917041i \(-0.369429\pi\)
0.398794 + 0.917041i \(0.369429\pi\)
\(318\) 0 0
\(319\) −1.70774e26 −0.876232
\(320\) −1.00176e26 −0.497379
\(321\) 0 0
\(322\) 6.58860e25 0.306413
\(323\) 1.95732e26 0.881120
\(324\) 0 0
\(325\) 7.03003e25 0.296607
\(326\) 3.79458e26 1.55017
\(327\) 0 0
\(328\) 2.60661e26 0.998620
\(329\) 2.50044e26 0.927810
\(330\) 0 0
\(331\) −4.40991e26 −1.53545 −0.767725 0.640780i \(-0.778611\pi\)
−0.767725 + 0.640780i \(0.778611\pi\)
\(332\) −1.43985e25 −0.0485699
\(333\) 0 0
\(334\) −1.82121e26 −0.576795
\(335\) −8.99842e25 −0.276182
\(336\) 0 0
\(337\) 4.81798e26 1.38915 0.694577 0.719418i \(-0.255591\pi\)
0.694577 + 0.719418i \(0.255591\pi\)
\(338\) 3.83259e26 1.07119
\(339\) 0 0
\(340\) −2.50924e25 −0.0659193
\(341\) −3.18359e26 −0.810955
\(342\) 0 0
\(343\) 6.65743e26 1.59485
\(344\) −6.77068e26 −1.57314
\(345\) 0 0
\(346\) −4.06548e24 −0.00888819
\(347\) 3.64414e26 0.772922 0.386461 0.922306i \(-0.373697\pi\)
0.386461 + 0.922306i \(0.373697\pi\)
\(348\) 0 0
\(349\) −3.60335e26 −0.719515 −0.359758 0.933046i \(-0.617141\pi\)
−0.359758 + 0.933046i \(0.617141\pi\)
\(350\) 1.57831e26 0.305828
\(351\) 0 0
\(352\) 3.07195e26 0.560681
\(353\) 3.52364e26 0.624247 0.312124 0.950042i \(-0.398960\pi\)
0.312124 + 0.950042i \(0.398960\pi\)
\(354\) 0 0
\(355\) −4.64301e26 −0.775179
\(356\) 1.43864e26 0.233199
\(357\) 0 0
\(358\) 9.91962e26 1.51608
\(359\) −1.17616e27 −1.74571 −0.872857 0.487977i \(-0.837735\pi\)
−0.872857 + 0.487977i \(0.837735\pi\)
\(360\) 0 0
\(361\) 3.30870e26 0.463268
\(362\) 1.96112e26 0.266726
\(363\) 0 0
\(364\) −4.00299e26 −0.513831
\(365\) −1.77310e26 −0.221136
\(366\) 0 0
\(367\) 1.52241e26 0.179282 0.0896411 0.995974i \(-0.471428\pi\)
0.0896411 + 0.995974i \(0.471428\pi\)
\(368\) −1.32489e26 −0.151627
\(369\) 0 0
\(370\) 4.35407e25 0.0470736
\(371\) 2.98536e26 0.313740
\(372\) 0 0
\(373\) −8.84632e26 −0.878658 −0.439329 0.898326i \(-0.644784\pi\)
−0.439329 + 0.898326i \(0.644784\pi\)
\(374\) −9.48918e26 −0.916382
\(375\) 0 0
\(376\) −6.37273e26 −0.581907
\(377\) −1.03886e27 −0.922510
\(378\) 0 0
\(379\) −3.83294e26 −0.321974 −0.160987 0.986957i \(-0.551468\pi\)
−0.160987 + 0.986957i \(0.551468\pi\)
\(380\) −1.33977e26 −0.109472
\(381\) 0 0
\(382\) −7.84796e26 −0.606864
\(383\) −1.86870e27 −1.40589 −0.702947 0.711243i \(-0.748133\pi\)
−0.702947 + 0.711243i \(0.748133\pi\)
\(384\) 0 0
\(385\) −1.51425e27 −1.07860
\(386\) 2.42628e27 1.68180
\(387\) 0 0
\(388\) −4.75332e25 −0.0312078
\(389\) 7.58738e26 0.484865 0.242432 0.970168i \(-0.422055\pi\)
0.242432 + 0.970168i \(0.422055\pi\)
\(390\) 0 0
\(391\) −2.41040e26 −0.145960
\(392\) −3.51830e27 −2.07410
\(393\) 0 0
\(394\) −2.93494e26 −0.164017
\(395\) 1.09193e27 0.594192
\(396\) 0 0
\(397\) 5.16201e26 0.266391 0.133195 0.991090i \(-0.457476\pi\)
0.133195 + 0.991090i \(0.457476\pi\)
\(398\) −7.94078e26 −0.399108
\(399\) 0 0
\(400\) −3.17379e26 −0.151338
\(401\) −2.28825e27 −1.06289 −0.531443 0.847094i \(-0.678350\pi\)
−0.531443 + 0.847094i \(0.678350\pi\)
\(402\) 0 0
\(403\) −1.93665e27 −0.853785
\(404\) 1.06789e26 0.0458692
\(405\) 0 0
\(406\) −2.33233e27 −0.951188
\(407\) −4.17734e26 −0.166019
\(408\) 0 0
\(409\) −1.71019e27 −0.645581 −0.322791 0.946470i \(-0.604621\pi\)
−0.322791 + 0.946470i \(0.604621\pi\)
\(410\) 1.00951e27 0.371433
\(411\) 0 0
\(412\) −9.74601e26 −0.340727
\(413\) 5.03008e27 1.71435
\(414\) 0 0
\(415\) −3.31331e26 −0.107339
\(416\) 1.86874e27 0.590294
\(417\) 0 0
\(418\) −5.06660e27 −1.52183
\(419\) 7.77024e25 0.0227608 0.0113804 0.999935i \(-0.496377\pi\)
0.0113804 + 0.999935i \(0.496377\pi\)
\(420\) 0 0
\(421\) 2.53023e27 0.705015 0.352508 0.935809i \(-0.385329\pi\)
0.352508 + 0.935809i \(0.385329\pi\)
\(422\) 2.57622e27 0.700169
\(423\) 0 0
\(424\) −7.60863e26 −0.196773
\(425\) −5.77414e26 −0.145681
\(426\) 0 0
\(427\) 1.15095e28 2.76416
\(428\) 1.59530e27 0.373837
\(429\) 0 0
\(430\) −2.62221e27 −0.585125
\(431\) 4.79626e26 0.104446 0.0522229 0.998635i \(-0.483369\pi\)
0.0522229 + 0.998635i \(0.483369\pi\)
\(432\) 0 0
\(433\) 8.00695e27 1.66090 0.830451 0.557092i \(-0.188083\pi\)
0.830451 + 0.557092i \(0.188083\pi\)
\(434\) −4.34797e27 −0.880327
\(435\) 0 0
\(436\) 6.15052e26 0.118660
\(437\) −1.28699e27 −0.242394
\(438\) 0 0
\(439\) −9.55381e27 −1.71514 −0.857569 0.514368i \(-0.828026\pi\)
−0.857569 + 0.514368i \(0.828026\pi\)
\(440\) 3.85928e27 0.676477
\(441\) 0 0
\(442\) −5.77248e27 −0.964781
\(443\) −5.53824e27 −0.903925 −0.451962 0.892037i \(-0.649276\pi\)
−0.451962 + 0.892037i \(0.649276\pi\)
\(444\) 0 0
\(445\) 3.31053e27 0.515368
\(446\) 4.25756e27 0.647359
\(447\) 0 0
\(448\) 1.31261e28 1.90422
\(449\) −5.56138e27 −0.788127 −0.394063 0.919083i \(-0.628931\pi\)
−0.394063 + 0.919083i \(0.628931\pi\)
\(450\) 0 0
\(451\) −9.68536e27 −1.30997
\(452\) −2.27954e27 −0.301227
\(453\) 0 0
\(454\) 1.98690e27 0.250662
\(455\) −9.21150e27 −1.13556
\(456\) 0 0
\(457\) 5.87968e27 0.692203 0.346101 0.938197i \(-0.387505\pi\)
0.346101 + 0.938197i \(0.387505\pi\)
\(458\) −4.79748e27 −0.551983
\(459\) 0 0
\(460\) 1.64989e26 0.0181342
\(461\) 1.13415e28 1.21846 0.609231 0.792993i \(-0.291478\pi\)
0.609231 + 0.792993i \(0.291478\pi\)
\(462\) 0 0
\(463\) 1.74305e28 1.78941 0.894705 0.446658i \(-0.147386\pi\)
0.894705 + 0.446658i \(0.147386\pi\)
\(464\) 4.69004e27 0.470693
\(465\) 0 0
\(466\) 1.45533e28 1.39608
\(467\) −1.60704e27 −0.150730 −0.0753651 0.997156i \(-0.524012\pi\)
−0.0753651 + 0.997156i \(0.524012\pi\)
\(468\) 0 0
\(469\) 1.17907e28 1.05736
\(470\) −2.46809e27 −0.216438
\(471\) 0 0
\(472\) −1.28199e28 −1.07521
\(473\) 2.51578e28 2.06362
\(474\) 0 0
\(475\) −3.08301e27 −0.241931
\(476\) 3.28787e27 0.252372
\(477\) 0 0
\(478\) −1.69498e28 −1.24501
\(479\) 5.39349e27 0.387567 0.193783 0.981044i \(-0.437924\pi\)
0.193783 + 0.981044i \(0.437924\pi\)
\(480\) 0 0
\(481\) −2.54117e27 −0.174788
\(482\) 2.15306e27 0.144898
\(483\) 0 0
\(484\) −3.09110e27 −0.199176
\(485\) −1.09381e27 −0.0689690
\(486\) 0 0
\(487\) 1.39994e28 0.845386 0.422693 0.906273i \(-0.361085\pi\)
0.422693 + 0.906273i \(0.361085\pi\)
\(488\) −2.93336e28 −1.73363
\(489\) 0 0
\(490\) −1.36260e28 −0.771453
\(491\) −1.66296e27 −0.0921567 −0.0460783 0.998938i \(-0.514672\pi\)
−0.0460783 + 0.998938i \(0.514672\pi\)
\(492\) 0 0
\(493\) 8.53268e27 0.453099
\(494\) −3.08213e28 −1.60220
\(495\) 0 0
\(496\) 8.74325e27 0.435627
\(497\) 6.08377e28 2.96777
\(498\) 0 0
\(499\) −2.24968e28 −1.05212 −0.526059 0.850448i \(-0.676331\pi\)
−0.526059 + 0.850448i \(0.676331\pi\)
\(500\) 3.95234e26 0.0180996
\(501\) 0 0
\(502\) −3.00581e28 −1.31999
\(503\) 2.16964e28 0.933090 0.466545 0.884497i \(-0.345499\pi\)
0.466545 + 0.884497i \(0.345499\pi\)
\(504\) 0 0
\(505\) 2.45738e27 0.101371
\(506\) 6.23941e27 0.252094
\(507\) 0 0
\(508\) −8.45567e27 −0.327778
\(509\) 1.86305e28 0.707436 0.353718 0.935352i \(-0.384917\pi\)
0.353718 + 0.935352i \(0.384917\pi\)
\(510\) 0 0
\(511\) 2.32331e28 0.846618
\(512\) −3.11976e28 −1.11375
\(513\) 0 0
\(514\) 3.43580e28 1.17738
\(515\) −2.24271e28 −0.753003
\(516\) 0 0
\(517\) 2.36791e28 0.763335
\(518\) −5.70517e27 −0.180221
\(519\) 0 0
\(520\) 2.34769e28 0.712205
\(521\) −6.39183e28 −1.90033 −0.950165 0.311747i \(-0.899086\pi\)
−0.950165 + 0.311747i \(0.899086\pi\)
\(522\) 0 0
\(523\) −9.84531e27 −0.281165 −0.140583 0.990069i \(-0.544898\pi\)
−0.140583 + 0.990069i \(0.544898\pi\)
\(524\) 8.28815e27 0.231995
\(525\) 0 0
\(526\) 1.26212e28 0.339429
\(527\) 1.59068e28 0.419344
\(528\) 0 0
\(529\) −3.78867e28 −0.959847
\(530\) −2.94674e27 −0.0731889
\(531\) 0 0
\(532\) 1.75551e28 0.419112
\(533\) −5.89183e28 −1.37916
\(534\) 0 0
\(535\) 3.67103e28 0.826177
\(536\) −3.00503e28 −0.663161
\(537\) 0 0
\(538\) 1.48921e28 0.316040
\(539\) 1.30729e29 2.72076
\(540\) 0 0
\(541\) 6.36576e28 1.27432 0.637161 0.770731i \(-0.280109\pi\)
0.637161 + 0.770731i \(0.280109\pi\)
\(542\) 2.03593e28 0.399732
\(543\) 0 0
\(544\) −1.53489e28 −0.289928
\(545\) 1.41533e28 0.262237
\(546\) 0 0
\(547\) −7.99629e28 −1.42568 −0.712839 0.701328i \(-0.752591\pi\)
−0.712839 + 0.701328i \(0.752591\pi\)
\(548\) 1.08084e28 0.189046
\(549\) 0 0
\(550\) 1.49466e28 0.251613
\(551\) 4.55589e28 0.752456
\(552\) 0 0
\(553\) −1.43077e29 −2.27486
\(554\) 2.13238e28 0.332669
\(555\) 0 0
\(556\) 1.39385e28 0.209378
\(557\) −6.15885e28 −0.907862 −0.453931 0.891037i \(-0.649979\pi\)
−0.453931 + 0.891037i \(0.649979\pi\)
\(558\) 0 0
\(559\) 1.53040e29 2.17261
\(560\) 4.15864e28 0.579398
\(561\) 0 0
\(562\) −8.58222e28 −1.15178
\(563\) −2.00390e28 −0.263960 −0.131980 0.991252i \(-0.542134\pi\)
−0.131980 + 0.991252i \(0.542134\pi\)
\(564\) 0 0
\(565\) −5.24557e28 −0.665709
\(566\) −8.70319e28 −1.08419
\(567\) 0 0
\(568\) −1.55054e29 −1.86134
\(569\) −6.14459e28 −0.724126 −0.362063 0.932154i \(-0.617927\pi\)
−0.362063 + 0.932154i \(0.617927\pi\)
\(570\) 0 0
\(571\) −4.65211e28 −0.528410 −0.264205 0.964467i \(-0.585110\pi\)
−0.264205 + 0.964467i \(0.585110\pi\)
\(572\) −3.79083e28 −0.422743
\(573\) 0 0
\(574\) −1.32277e29 −1.42203
\(575\) 3.79666e27 0.0400765
\(576\) 0 0
\(577\) 1.04152e28 0.106004 0.0530020 0.998594i \(-0.483121\pi\)
0.0530020 + 0.998594i \(0.483121\pi\)
\(578\) −4.19481e28 −0.419247
\(579\) 0 0
\(580\) −5.84054e27 −0.0562936
\(581\) 4.34146e28 0.410947
\(582\) 0 0
\(583\) 2.82714e28 0.258123
\(584\) −5.92129e28 −0.530984
\(585\) 0 0
\(586\) 1.96405e28 0.169913
\(587\) 4.54423e28 0.386154 0.193077 0.981184i \(-0.438153\pi\)
0.193077 + 0.981184i \(0.438153\pi\)
\(588\) 0 0
\(589\) 8.49316e28 0.696400
\(590\) −4.96501e28 −0.399921
\(591\) 0 0
\(592\) 1.14724e28 0.0891820
\(593\) −3.29422e28 −0.251581 −0.125791 0.992057i \(-0.540147\pi\)
−0.125791 + 0.992057i \(0.540147\pi\)
\(594\) 0 0
\(595\) 7.56589e28 0.557740
\(596\) −1.74342e28 −0.126274
\(597\) 0 0
\(598\) 3.79557e28 0.265409
\(599\) 1.44151e29 0.990458 0.495229 0.868763i \(-0.335084\pi\)
0.495229 + 0.868763i \(0.335084\pi\)
\(600\) 0 0
\(601\) 1.75962e28 0.116745 0.0583723 0.998295i \(-0.481409\pi\)
0.0583723 + 0.998295i \(0.481409\pi\)
\(602\) 3.43590e29 2.24015
\(603\) 0 0
\(604\) −3.72086e28 −0.234291
\(605\) −7.11310e28 −0.440177
\(606\) 0 0
\(607\) −7.40539e28 −0.442656 −0.221328 0.975199i \(-0.571039\pi\)
−0.221328 + 0.975199i \(0.571039\pi\)
\(608\) −8.19532e28 −0.481480
\(609\) 0 0
\(610\) −1.13606e29 −0.644819
\(611\) 1.44046e29 0.803650
\(612\) 0 0
\(613\) −2.04497e29 −1.10243 −0.551216 0.834363i \(-0.685836\pi\)
−0.551216 + 0.834363i \(0.685836\pi\)
\(614\) −2.80540e29 −1.48671
\(615\) 0 0
\(616\) −5.05684e29 −2.58989
\(617\) −3.28969e29 −1.65638 −0.828192 0.560445i \(-0.810630\pi\)
−0.828192 + 0.560445i \(0.810630\pi\)
\(618\) 0 0
\(619\) 1.51436e29 0.737018 0.368509 0.929624i \(-0.379868\pi\)
0.368509 + 0.929624i \(0.379868\pi\)
\(620\) −1.08880e28 −0.0520998
\(621\) 0 0
\(622\) 1.27586e29 0.590208
\(623\) −4.33781e29 −1.97309
\(624\) 0 0
\(625\) 9.09495e27 0.0400000
\(626\) 1.66088e29 0.718302
\(627\) 0 0
\(628\) −4.52409e28 −0.189215
\(629\) 2.08720e28 0.0858484
\(630\) 0 0
\(631\) 1.00288e29 0.398969 0.199485 0.979901i \(-0.436073\pi\)
0.199485 + 0.979901i \(0.436073\pi\)
\(632\) 3.64652e29 1.42676
\(633\) 0 0
\(634\) 1.88200e29 0.712331
\(635\) −1.94578e29 −0.724386
\(636\) 0 0
\(637\) 7.95256e29 2.86446
\(638\) −2.20872e29 −0.782569
\(639\) 0 0
\(640\) −7.76447e28 −0.266208
\(641\) 2.60873e29 0.879871 0.439936 0.898029i \(-0.355001\pi\)
0.439936 + 0.898029i \(0.355001\pi\)
\(642\) 0 0
\(643\) 5.30533e28 0.173180 0.0865898 0.996244i \(-0.472403\pi\)
0.0865898 + 0.996244i \(0.472403\pi\)
\(644\) −2.16187e28 −0.0694269
\(645\) 0 0
\(646\) 2.53151e29 0.786935
\(647\) 7.13885e27 0.0218340 0.0109170 0.999940i \(-0.496525\pi\)
0.0109170 + 0.999940i \(0.496525\pi\)
\(648\) 0 0
\(649\) 4.76348e29 1.41044
\(650\) 9.09234e28 0.264902
\(651\) 0 0
\(652\) −1.24509e29 −0.351237
\(653\) 8.18691e28 0.227264 0.113632 0.993523i \(-0.463751\pi\)
0.113632 + 0.993523i \(0.463751\pi\)
\(654\) 0 0
\(655\) 1.90723e29 0.512708
\(656\) 2.65993e29 0.703689
\(657\) 0 0
\(658\) 3.23396e29 0.828633
\(659\) 3.25306e28 0.0820342 0.0410171 0.999158i \(-0.486940\pi\)
0.0410171 + 0.999158i \(0.486940\pi\)
\(660\) 0 0
\(661\) 8.22623e28 0.200949 0.100474 0.994940i \(-0.467964\pi\)
0.100474 + 0.994940i \(0.467964\pi\)
\(662\) −5.70359e29 −1.37132
\(663\) 0 0
\(664\) −1.10648e29 −0.257739
\(665\) 4.03969e29 0.926234
\(666\) 0 0
\(667\) −5.61048e28 −0.124646
\(668\) 5.97579e28 0.130690
\(669\) 0 0
\(670\) −1.16382e29 −0.246660
\(671\) 1.08995e30 2.27415
\(672\) 0 0
\(673\) 1.16937e29 0.236479 0.118239 0.992985i \(-0.462275\pi\)
0.118239 + 0.992985i \(0.462275\pi\)
\(674\) 6.23136e29 1.24066
\(675\) 0 0
\(676\) −1.25756e29 −0.242710
\(677\) −4.28660e28 −0.0814577 −0.0407289 0.999170i \(-0.512968\pi\)
−0.0407289 + 0.999170i \(0.512968\pi\)
\(678\) 0 0
\(679\) 1.43323e29 0.264048
\(680\) −1.92828e29 −0.349805
\(681\) 0 0
\(682\) −4.11752e29 −0.724270
\(683\) −5.45367e29 −0.944651 −0.472325 0.881424i \(-0.656585\pi\)
−0.472325 + 0.881424i \(0.656585\pi\)
\(684\) 0 0
\(685\) 2.48719e29 0.417790
\(686\) 8.61044e29 1.42437
\(687\) 0 0
\(688\) −6.90919e29 −1.10853
\(689\) 1.71981e29 0.271756
\(690\) 0 0
\(691\) −9.02075e29 −1.38269 −0.691343 0.722527i \(-0.742980\pi\)
−0.691343 + 0.722527i \(0.742980\pi\)
\(692\) 1.33398e27 0.00201388
\(693\) 0 0
\(694\) 4.71318e29 0.690302
\(695\) 3.20747e29 0.462724
\(696\) 0 0
\(697\) 4.83927e29 0.677385
\(698\) −4.66042e29 −0.642604
\(699\) 0 0
\(700\) −5.17878e28 −0.0692944
\(701\) 1.36429e29 0.179833 0.0899164 0.995949i \(-0.471340\pi\)
0.0899164 + 0.995949i \(0.471340\pi\)
\(702\) 0 0
\(703\) 1.11443e29 0.142568
\(704\) 1.24304e30 1.56665
\(705\) 0 0
\(706\) 4.55732e29 0.557520
\(707\) −3.21992e29 −0.388097
\(708\) 0 0
\(709\) −1.96075e29 −0.229423 −0.114712 0.993399i \(-0.536594\pi\)
−0.114712 + 0.993399i \(0.536594\pi\)
\(710\) −6.00507e29 −0.692317
\(711\) 0 0
\(712\) 1.10555e30 1.23749
\(713\) −1.04591e29 −0.115360
\(714\) 0 0
\(715\) −8.72329e29 −0.934258
\(716\) −3.25485e29 −0.343514
\(717\) 0 0
\(718\) −1.52119e30 −1.55911
\(719\) −6.61361e29 −0.668013 −0.334007 0.942571i \(-0.608401\pi\)
−0.334007 + 0.942571i \(0.608401\pi\)
\(720\) 0 0
\(721\) 2.93864e30 2.88287
\(722\) 4.27933e29 0.413748
\(723\) 0 0
\(724\) −6.43488e28 −0.0604346
\(725\) −1.34400e29 −0.124408
\(726\) 0 0
\(727\) −5.62124e29 −0.505500 −0.252750 0.967532i \(-0.581335\pi\)
−0.252750 + 0.967532i \(0.581335\pi\)
\(728\) −3.07619e30 −2.72668
\(729\) 0 0
\(730\) −2.29325e29 −0.197498
\(731\) −1.25700e30 −1.06710
\(732\) 0 0
\(733\) −1.44462e30 −1.19169 −0.595843 0.803101i \(-0.703182\pi\)
−0.595843 + 0.803101i \(0.703182\pi\)
\(734\) 1.96902e29 0.160118
\(735\) 0 0
\(736\) 1.00924e29 0.0797583
\(737\) 1.11658e30 0.869923
\(738\) 0 0
\(739\) 4.75852e29 0.360334 0.180167 0.983636i \(-0.442336\pi\)
0.180167 + 0.983636i \(0.442336\pi\)
\(740\) −1.42867e28 −0.0106659
\(741\) 0 0
\(742\) 3.86114e29 0.280204
\(743\) 1.64704e30 1.17848 0.589239 0.807959i \(-0.299428\pi\)
0.589239 + 0.807959i \(0.299428\pi\)
\(744\) 0 0
\(745\) −4.01187e29 −0.279065
\(746\) −1.14415e30 −0.784736
\(747\) 0 0
\(748\) 3.11361e29 0.207634
\(749\) −4.81018e30 −3.16302
\(750\) 0 0
\(751\) −1.15249e30 −0.736914 −0.368457 0.929645i \(-0.620114\pi\)
−0.368457 + 0.929645i \(0.620114\pi\)
\(752\) −6.50311e29 −0.410047
\(753\) 0 0
\(754\) −1.34361e30 −0.823900
\(755\) −8.56228e29 −0.517781
\(756\) 0 0
\(757\) 6.06438e29 0.356681 0.178340 0.983969i \(-0.442927\pi\)
0.178340 + 0.983969i \(0.442927\pi\)
\(758\) −4.95736e29 −0.287557
\(759\) 0 0
\(760\) −1.02957e30 −0.580918
\(761\) 3.30053e30 1.83673 0.918364 0.395737i \(-0.129511\pi\)
0.918364 + 0.395737i \(0.129511\pi\)
\(762\) 0 0
\(763\) −1.85452e30 −1.00398
\(764\) 2.57509e29 0.137503
\(765\) 0 0
\(766\) −2.41690e30 −1.25561
\(767\) 2.89773e30 1.48493
\(768\) 0 0
\(769\) −2.35995e30 −1.17673 −0.588363 0.808597i \(-0.700228\pi\)
−0.588363 + 0.808597i \(0.700228\pi\)
\(770\) −1.95846e30 −0.963301
\(771\) 0 0
\(772\) −7.96117e29 −0.381062
\(773\) −2.75286e30 −1.29987 −0.649935 0.759990i \(-0.725204\pi\)
−0.649935 + 0.759990i \(0.725204\pi\)
\(774\) 0 0
\(775\) −2.50550e29 −0.115140
\(776\) −3.65279e29 −0.165606
\(777\) 0 0
\(778\) 9.81319e29 0.433036
\(779\) 2.58385e30 1.12493
\(780\) 0 0
\(781\) 5.76133e30 2.44167
\(782\) −3.11751e29 −0.130358
\(783\) 0 0
\(784\) −3.59028e30 −1.46153
\(785\) −1.04106e30 −0.418163
\(786\) 0 0
\(787\) −1.96178e30 −0.767213 −0.383606 0.923497i \(-0.625318\pi\)
−0.383606 + 0.923497i \(0.625318\pi\)
\(788\) 9.63019e28 0.0371629
\(789\) 0 0
\(790\) 1.41226e30 0.530677
\(791\) 6.87331e30 2.54867
\(792\) 0 0
\(793\) 6.63040e30 2.39426
\(794\) 6.67633e29 0.237915
\(795\) 0 0
\(796\) 2.60555e29 0.0904299
\(797\) 3.68558e30 1.26239 0.631195 0.775624i \(-0.282565\pi\)
0.631195 + 0.775624i \(0.282565\pi\)
\(798\) 0 0
\(799\) −1.18312e30 −0.394720
\(800\) 2.41764e29 0.0796061
\(801\) 0 0
\(802\) −2.95952e30 −0.949271
\(803\) 2.20017e30 0.696536
\(804\) 0 0
\(805\) −4.97479e29 −0.153433
\(806\) −2.50478e30 −0.762522
\(807\) 0 0
\(808\) 8.20643e29 0.243408
\(809\) 1.59133e30 0.465909 0.232954 0.972488i \(-0.425161\pi\)
0.232954 + 0.972488i \(0.425161\pi\)
\(810\) 0 0
\(811\) 2.03075e30 0.579345 0.289673 0.957126i \(-0.406454\pi\)
0.289673 + 0.957126i \(0.406454\pi\)
\(812\) 7.65290e29 0.215520
\(813\) 0 0
\(814\) −5.40280e29 −0.148273
\(815\) −2.86514e30 −0.776231
\(816\) 0 0
\(817\) −6.71157e30 −1.77212
\(818\) −2.21189e30 −0.576573
\(819\) 0 0
\(820\) −3.31243e29 −0.0841592
\(821\) 1.16660e30 0.292631 0.146315 0.989238i \(-0.453259\pi\)
0.146315 + 0.989238i \(0.453259\pi\)
\(822\) 0 0
\(823\) −4.40906e30 −1.07807 −0.539036 0.842283i \(-0.681211\pi\)
−0.539036 + 0.842283i \(0.681211\pi\)
\(824\) −7.48954e30 −1.80809
\(825\) 0 0
\(826\) 6.50568e30 1.53110
\(827\) 2.06510e30 0.479881 0.239941 0.970788i \(-0.422872\pi\)
0.239941 + 0.970788i \(0.422872\pi\)
\(828\) 0 0
\(829\) −3.07768e30 −0.697271 −0.348635 0.937258i \(-0.613355\pi\)
−0.348635 + 0.937258i \(0.613355\pi\)
\(830\) −4.28530e29 −0.0958652
\(831\) 0 0
\(832\) 7.56170e30 1.64940
\(833\) −6.53186e30 −1.40690
\(834\) 0 0
\(835\) 1.37512e30 0.288824
\(836\) 1.66246e30 0.344815
\(837\) 0 0
\(838\) 1.00497e29 0.0203278
\(839\) 4.30426e30 0.859800 0.429900 0.902876i \(-0.358549\pi\)
0.429900 + 0.902876i \(0.358549\pi\)
\(840\) 0 0
\(841\) −3.14676e30 −0.613064
\(842\) 3.27249e30 0.629654
\(843\) 0 0
\(844\) −8.45317e29 −0.158644
\(845\) −2.89384e30 −0.536387
\(846\) 0 0
\(847\) 9.32035e30 1.68522
\(848\) −7.76429e29 −0.138658
\(849\) 0 0
\(850\) −7.46802e29 −0.130109
\(851\) −1.37239e29 −0.0236167
\(852\) 0 0
\(853\) −5.29542e30 −0.889069 −0.444534 0.895762i \(-0.646631\pi\)
−0.444534 + 0.895762i \(0.646631\pi\)
\(854\) 1.48859e31 2.46869
\(855\) 0 0
\(856\) 1.22594e31 1.98379
\(857\) −8.07412e29 −0.129062 −0.0645308 0.997916i \(-0.520555\pi\)
−0.0645308 + 0.997916i \(0.520555\pi\)
\(858\) 0 0
\(859\) 4.76822e30 0.743751 0.371876 0.928283i \(-0.378715\pi\)
0.371876 + 0.928283i \(0.378715\pi\)
\(860\) 8.60406e29 0.132578
\(861\) 0 0
\(862\) 6.20328e29 0.0932814
\(863\) −1.13640e31 −1.68817 −0.844085 0.536209i \(-0.819856\pi\)
−0.844085 + 0.536209i \(0.819856\pi\)
\(864\) 0 0
\(865\) 3.06969e28 0.00445066
\(866\) 1.03558e31 1.48336
\(867\) 0 0
\(868\) 1.42667e30 0.199464
\(869\) −1.35494e31 −1.87159
\(870\) 0 0
\(871\) 6.79240e30 0.915868
\(872\) 4.72651e30 0.629676
\(873\) 0 0
\(874\) −1.66454e30 −0.216484
\(875\) −1.19172e30 −0.153140
\(876\) 0 0
\(877\) 6.52492e30 0.818615 0.409307 0.912397i \(-0.365770\pi\)
0.409307 + 0.912397i \(0.365770\pi\)
\(878\) −1.23565e31 −1.53180
\(879\) 0 0
\(880\) 3.93823e30 0.476687
\(881\) −6.24200e30 −0.746581 −0.373290 0.927715i \(-0.621770\pi\)
−0.373290 + 0.927715i \(0.621770\pi\)
\(882\) 0 0
\(883\) −5.35499e30 −0.625419 −0.312709 0.949849i \(-0.601237\pi\)
−0.312709 + 0.949849i \(0.601237\pi\)
\(884\) 1.89408e30 0.218600
\(885\) 0 0
\(886\) −7.16292e30 −0.807302
\(887\) 1.58739e31 1.76802 0.884009 0.467470i \(-0.154834\pi\)
0.884009 + 0.467470i \(0.154834\pi\)
\(888\) 0 0
\(889\) 2.54957e31 2.77331
\(890\) 4.28170e30 0.460279
\(891\) 0 0
\(892\) −1.39700e30 −0.146678
\(893\) −6.31710e30 −0.655507
\(894\) 0 0
\(895\) −7.48991e30 −0.759163
\(896\) 1.01738e31 1.01918
\(897\) 0 0
\(898\) −7.19285e30 −0.703881
\(899\) 3.70248e30 0.358110
\(900\) 0 0
\(901\) −1.41257e30 −0.133475
\(902\) −1.25266e31 −1.16995
\(903\) 0 0
\(904\) −1.75176e31 −1.59848
\(905\) −1.48077e30 −0.133560
\(906\) 0 0
\(907\) −2.03878e31 −1.79677 −0.898386 0.439206i \(-0.855260\pi\)
−0.898386 + 0.439206i \(0.855260\pi\)
\(908\) −6.51945e29 −0.0567950
\(909\) 0 0
\(910\) −1.19138e31 −1.01418
\(911\) −1.16077e31 −0.976792 −0.488396 0.872622i \(-0.662418\pi\)
−0.488396 + 0.872622i \(0.662418\pi\)
\(912\) 0 0
\(913\) 4.11136e30 0.338098
\(914\) 7.60452e30 0.618211
\(915\) 0 0
\(916\) 1.57416e30 0.125068
\(917\) −2.49906e31 −1.96290
\(918\) 0 0
\(919\) 1.11455e31 0.855631 0.427816 0.903866i \(-0.359283\pi\)
0.427816 + 0.903866i \(0.359283\pi\)
\(920\) 1.26790e30 0.0962304
\(921\) 0 0
\(922\) 1.46687e31 1.08822
\(923\) 3.50475e31 2.57062
\(924\) 0 0
\(925\) −3.28759e29 −0.0235716
\(926\) 2.25439e31 1.59813
\(927\) 0 0
\(928\) −3.57264e30 −0.247591
\(929\) 1.58810e30 0.108821 0.0544105 0.998519i \(-0.482672\pi\)
0.0544105 + 0.998519i \(0.482672\pi\)
\(930\) 0 0
\(931\) −3.48758e31 −2.33643
\(932\) −4.77528e30 −0.316324
\(933\) 0 0
\(934\) −2.07848e30 −0.134618
\(935\) 7.16490e30 0.458869
\(936\) 0 0
\(937\) −3.78856e29 −0.0237251 −0.0118626 0.999930i \(-0.503776\pi\)
−0.0118626 + 0.999930i \(0.503776\pi\)
\(938\) 1.52496e31 0.944339
\(939\) 0 0
\(940\) 8.09836e29 0.0490405
\(941\) 2.33225e31 1.39664 0.698320 0.715786i \(-0.253931\pi\)
0.698320 + 0.715786i \(0.253931\pi\)
\(942\) 0 0
\(943\) −3.18196e30 −0.186347
\(944\) −1.30822e31 −0.757659
\(945\) 0 0
\(946\) 3.25380e31 1.84304
\(947\) 1.89428e31 1.06113 0.530566 0.847643i \(-0.321979\pi\)
0.530566 + 0.847643i \(0.321979\pi\)
\(948\) 0 0
\(949\) 1.33841e31 0.733323
\(950\) −3.98743e30 −0.216070
\(951\) 0 0
\(952\) 2.52664e31 1.33923
\(953\) −2.74511e31 −1.43908 −0.719538 0.694453i \(-0.755646\pi\)
−0.719538 + 0.694453i \(0.755646\pi\)
\(954\) 0 0
\(955\) 5.92569e30 0.303881
\(956\) 5.56161e30 0.282093
\(957\) 0 0
\(958\) 6.97570e30 0.346139
\(959\) −3.25898e31 −1.59951
\(960\) 0 0
\(961\) −1.39233e31 −0.668569
\(962\) −3.28664e30 −0.156104
\(963\) 0 0
\(964\) −7.06466e29 −0.0328309
\(965\) −1.83199e31 −0.842143
\(966\) 0 0
\(967\) 3.03551e31 1.36538 0.682690 0.730708i \(-0.260810\pi\)
0.682690 + 0.730708i \(0.260810\pi\)
\(968\) −2.37543e31 −1.05694
\(969\) 0 0
\(970\) −1.41469e30 −0.0615967
\(971\) −3.36946e31 −1.45130 −0.725651 0.688063i \(-0.758461\pi\)
−0.725651 + 0.688063i \(0.758461\pi\)
\(972\) 0 0
\(973\) −4.20277e31 −1.77154
\(974\) 1.81062e31 0.755020
\(975\) 0 0
\(976\) −2.99337e31 −1.22163
\(977\) −1.47911e31 −0.597184 −0.298592 0.954381i \(-0.596517\pi\)
−0.298592 + 0.954381i \(0.596517\pi\)
\(978\) 0 0
\(979\) −4.10791e31 −1.62331
\(980\) 4.47099e30 0.174796
\(981\) 0 0
\(982\) −2.15080e30 −0.0823057
\(983\) −3.37648e31 −1.27836 −0.639180 0.769058i \(-0.720726\pi\)
−0.639180 + 0.769058i \(0.720726\pi\)
\(984\) 0 0
\(985\) 2.21605e30 0.0821297
\(986\) 1.10358e31 0.404665
\(987\) 0 0
\(988\) 1.01131e31 0.363026
\(989\) 8.26515e30 0.293555
\(990\) 0 0
\(991\) −5.48368e31 −1.90677 −0.953387 0.301751i \(-0.902429\pi\)
−0.953387 + 0.301751i \(0.902429\pi\)
\(992\) −6.66017e30 −0.229146
\(993\) 0 0
\(994\) 7.86848e31 2.65054
\(995\) 5.99577e30 0.199849
\(996\) 0 0
\(997\) −2.82839e31 −0.923082 −0.461541 0.887119i \(-0.652703\pi\)
−0.461541 + 0.887119i \(0.652703\pi\)
\(998\) −2.90964e31 −0.939654
\(999\) 0 0
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 45.22.a.g.1.3 4
3.2 odd 2 15.22.a.e.1.2 4
15.2 even 4 75.22.b.h.49.3 8
15.8 even 4 75.22.b.h.49.6 8
15.14 odd 2 75.22.a.h.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.22.a.e.1.2 4 3.2 odd 2
45.22.a.g.1.3 4 1.1 even 1 trivial
75.22.a.h.1.3 4 15.14 odd 2
75.22.b.h.49.3 8 15.2 even 4
75.22.b.h.49.6 8 15.8 even 4