Properties

Label 45.22.a.g.1.4
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 1.4
Root \(1492.88\) 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)
 
\(f(q)\) \(=\) \(q+1716.88 q^{2} +850541. q^{4} -9.76562e6 q^{5} -6.24477e8 q^{7} -2.14029e9 q^{8} +O(q^{10})\) \(q+1716.88 q^{2} +850541. q^{4} -9.76562e6 q^{5} -6.24477e8 q^{7} -2.14029e9 q^{8} -1.67665e10 q^{10} -1.47398e11 q^{11} -5.50754e11 q^{13} -1.07216e12 q^{14} -5.45834e12 q^{16} +4.04946e12 q^{17} +1.22679e12 q^{19} -8.30606e12 q^{20} -2.53065e14 q^{22} +2.50983e14 q^{23} +9.53674e13 q^{25} -9.45581e14 q^{26} -5.31144e14 q^{28} +3.49299e15 q^{29} -2.93209e15 q^{31} -4.88283e15 q^{32} +6.95245e15 q^{34} +6.09841e15 q^{35} +4.06107e16 q^{37} +2.10626e15 q^{38} +2.09012e16 q^{40} -1.45009e17 q^{41} +8.05941e16 q^{43} -1.25368e17 q^{44} +4.30909e17 q^{46} -4.95994e16 q^{47} -1.68574e17 q^{49} +1.63735e17 q^{50} -4.68439e17 q^{52} +1.24656e18 q^{53} +1.43943e18 q^{55} +1.33656e18 q^{56} +5.99706e18 q^{58} -7.20028e18 q^{59} -4.77418e18 q^{61} -5.03405e18 q^{62} +3.06371e18 q^{64} +5.37846e18 q^{65} +7.70841e18 q^{67} +3.44423e18 q^{68} +1.04703e19 q^{70} +1.63265e19 q^{71} -1.29930e19 q^{73} +6.97240e19 q^{74} +1.04344e18 q^{76} +9.20465e19 q^{77} +9.55293e19 q^{79} +5.33041e19 q^{80} -2.48964e20 q^{82} +2.20974e20 q^{83} -3.95455e19 q^{85} +1.38371e20 q^{86} +3.15473e20 q^{88} -3.15065e19 q^{89} +3.43933e20 q^{91} +2.13471e20 q^{92} -8.51565e19 q^{94} -1.19804e19 q^{95} +2.46548e20 q^{97} -2.89422e20 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 897 q^{2} - 1163123 q^{4} - 39062500 q^{5} - 234577504 q^{7} - 76855629 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1716.88 1.18557 0.592784 0.805362i \(-0.298029\pi\)
0.592784 + 0.805362i \(0.298029\pi\)
\(3\) 0 0
\(4\) 850541. 0.405570
\(5\) −9.76562e6 −0.447214
\(6\) 0 0
\(7\) −6.24477e8 −0.835579 −0.417789 0.908544i \(-0.637195\pi\)
−0.417789 + 0.908544i \(0.637195\pi\)
\(8\) −2.14029e9 −0.704737
\(9\) 0 0
\(10\) −1.67665e10 −0.530202
\(11\) −1.47398e11 −1.71343 −0.856717 0.515787i \(-0.827499\pi\)
−0.856717 + 0.515787i \(0.827499\pi\)
\(12\) 0 0
\(13\) −5.50754e11 −1.10803 −0.554016 0.832506i \(-0.686905\pi\)
−0.554016 + 0.832506i \(0.686905\pi\)
\(14\) −1.07216e12 −0.990635
\(15\) 0 0
\(16\) −5.45834e12 −1.24108
\(17\) 4.04946e12 0.487173 0.243587 0.969879i \(-0.421676\pi\)
0.243587 + 0.969879i \(0.421676\pi\)
\(18\) 0 0
\(19\) 1.22679e12 0.0459048 0.0229524 0.999737i \(-0.492693\pi\)
0.0229524 + 0.999737i \(0.492693\pi\)
\(20\) −8.30606e12 −0.181376
\(21\) 0 0
\(22\) −2.53065e14 −2.03139
\(23\) 2.50983e14 1.26329 0.631644 0.775259i \(-0.282380\pi\)
0.631644 + 0.775259i \(0.282380\pi\)
\(24\) 0 0
\(25\) 9.53674e13 0.200000
\(26\) −9.45581e14 −1.31365
\(27\) 0 0
\(28\) −5.31144e14 −0.338885
\(29\) 3.49299e15 1.54177 0.770883 0.636977i \(-0.219815\pi\)
0.770883 + 0.636977i \(0.219815\pi\)
\(30\) 0 0
\(31\) −2.93209e15 −0.642508 −0.321254 0.946993i \(-0.604104\pi\)
−0.321254 + 0.946993i \(0.604104\pi\)
\(32\) −4.88283e15 −0.766650
\(33\) 0 0
\(34\) 6.95245e15 0.577576
\(35\) 6.09841e15 0.373682
\(36\) 0 0
\(37\) 4.06107e16 1.38843 0.694213 0.719770i \(-0.255753\pi\)
0.694213 + 0.719770i \(0.255753\pi\)
\(38\) 2.10626e15 0.0544232
\(39\) 0 0
\(40\) 2.09012e16 0.315168
\(41\) −1.45009e17 −1.68720 −0.843598 0.536976i \(-0.819567\pi\)
−0.843598 + 0.536976i \(0.819567\pi\)
\(42\) 0 0
\(43\) 8.05941e16 0.568702 0.284351 0.958720i \(-0.408222\pi\)
0.284351 + 0.958720i \(0.408222\pi\)
\(44\) −1.25368e17 −0.694916
\(45\) 0 0
\(46\) 4.30909e17 1.49771
\(47\) −4.95994e16 −0.137546 −0.0687732 0.997632i \(-0.521908\pi\)
−0.0687732 + 0.997632i \(0.521908\pi\)
\(48\) 0 0
\(49\) −1.68574e17 −0.301808
\(50\) 1.63735e17 0.237113
\(51\) 0 0
\(52\) −4.68439e17 −0.449384
\(53\) 1.24656e18 0.979079 0.489540 0.871981i \(-0.337165\pi\)
0.489540 + 0.871981i \(0.337165\pi\)
\(54\) 0 0
\(55\) 1.43943e18 0.766271
\(56\) 1.33656e18 0.588863
\(57\) 0 0
\(58\) 5.99706e18 1.82787
\(59\) −7.20028e18 −1.83402 −0.917009 0.398867i \(-0.869404\pi\)
−0.917009 + 0.398867i \(0.869404\pi\)
\(60\) 0 0
\(61\) −4.77418e18 −0.856910 −0.428455 0.903563i \(-0.640942\pi\)
−0.428455 + 0.903563i \(0.640942\pi\)
\(62\) −5.03405e18 −0.761737
\(63\) 0 0
\(64\) 3.06371e18 0.332168
\(65\) 5.37846e18 0.495527
\(66\) 0 0
\(67\) 7.70841e18 0.516630 0.258315 0.966061i \(-0.416833\pi\)
0.258315 + 0.966061i \(0.416833\pi\)
\(68\) 3.44423e18 0.197583
\(69\) 0 0
\(70\) 1.04703e19 0.443025
\(71\) 1.63265e19 0.595223 0.297612 0.954687i \(-0.403810\pi\)
0.297612 + 0.954687i \(0.403810\pi\)
\(72\) 0 0
\(73\) −1.29930e19 −0.353849 −0.176924 0.984224i \(-0.556615\pi\)
−0.176924 + 0.984224i \(0.556615\pi\)
\(74\) 6.97240e19 1.64607
\(75\) 0 0
\(76\) 1.04344e18 0.0186176
\(77\) 9.20465e19 1.43171
\(78\) 0 0
\(79\) 9.55293e19 1.13515 0.567574 0.823323i \(-0.307882\pi\)
0.567574 + 0.823323i \(0.307882\pi\)
\(80\) 5.33041e19 0.555029
\(81\) 0 0
\(82\) −2.48964e20 −2.00028
\(83\) 2.20974e20 1.56323 0.781613 0.623764i \(-0.214397\pi\)
0.781613 + 0.623764i \(0.214397\pi\)
\(84\) 0 0
\(85\) −3.95455e19 −0.217870
\(86\) 1.38371e20 0.674234
\(87\) 0 0
\(88\) 3.15473e20 1.20752
\(89\) −3.15065e19 −0.107104 −0.0535519 0.998565i \(-0.517054\pi\)
−0.0535519 + 0.998565i \(0.517054\pi\)
\(90\) 0 0
\(91\) 3.43933e20 0.925848
\(92\) 2.13471e20 0.512351
\(93\) 0 0
\(94\) −8.51565e19 −0.163070
\(95\) −1.19804e19 −0.0205292
\(96\) 0 0
\(97\) 2.46548e20 0.339467 0.169733 0.985490i \(-0.445709\pi\)
0.169733 + 0.985490i \(0.445709\pi\)
\(98\) −2.89422e20 −0.357814
\(99\) 0 0
\(100\) 8.11139e19 0.0811139
\(101\) −2.15225e21 −1.93873 −0.969367 0.245616i \(-0.921010\pi\)
−0.969367 + 0.245616i \(0.921010\pi\)
\(102\) 0 0
\(103\) −1.53340e21 −1.12426 −0.562128 0.827050i \(-0.690017\pi\)
−0.562128 + 0.827050i \(0.690017\pi\)
\(104\) 1.17877e21 0.780871
\(105\) 0 0
\(106\) 2.14021e21 1.16076
\(107\) −1.54824e21 −0.760865 −0.380433 0.924809i \(-0.624225\pi\)
−0.380433 + 0.924809i \(0.624225\pi\)
\(108\) 0 0
\(109\) 2.85208e21 1.15394 0.576971 0.816764i \(-0.304234\pi\)
0.576971 + 0.816764i \(0.304234\pi\)
\(110\) 2.47133e21 0.908465
\(111\) 0 0
\(112\) 3.40861e21 1.03702
\(113\) −3.30356e21 −0.915499 −0.457750 0.889081i \(-0.651344\pi\)
−0.457750 + 0.889081i \(0.651344\pi\)
\(114\) 0 0
\(115\) −2.45101e21 −0.564959
\(116\) 2.97093e21 0.625293
\(117\) 0 0
\(118\) −1.23621e22 −2.17435
\(119\) −2.52880e21 −0.407071
\(120\) 0 0
\(121\) 1.43258e22 1.93585
\(122\) −8.19671e21 −1.01592
\(123\) 0 0
\(124\) −2.49386e21 −0.260582
\(125\) −9.31323e20 −0.0894427
\(126\) 0 0
\(127\) −8.37131e20 −0.0680541 −0.0340271 0.999421i \(-0.510833\pi\)
−0.0340271 + 0.999421i \(0.510833\pi\)
\(128\) 1.55001e22 1.16046
\(129\) 0 0
\(130\) 9.23419e21 0.587481
\(131\) −1.90600e22 −1.11886 −0.559429 0.828879i \(-0.688979\pi\)
−0.559429 + 0.828879i \(0.688979\pi\)
\(132\) 0 0
\(133\) −7.66104e20 −0.0383571
\(134\) 1.32344e22 0.612499
\(135\) 0 0
\(136\) −8.66701e21 −0.343329
\(137\) 8.63318e20 0.0316669 0.0158334 0.999875i \(-0.494960\pi\)
0.0158334 + 0.999875i \(0.494960\pi\)
\(138\) 0 0
\(139\) 8.91438e21 0.280825 0.140412 0.990093i \(-0.455157\pi\)
0.140412 + 0.990093i \(0.455157\pi\)
\(140\) 5.18695e21 0.151554
\(141\) 0 0
\(142\) 2.80307e22 0.705677
\(143\) 8.11798e22 1.89854
\(144\) 0 0
\(145\) −3.41112e22 −0.689498
\(146\) −2.23074e22 −0.419512
\(147\) 0 0
\(148\) 3.45411e22 0.563103
\(149\) 9.42240e22 1.43122 0.715609 0.698501i \(-0.246149\pi\)
0.715609 + 0.698501i \(0.246149\pi\)
\(150\) 0 0
\(151\) 2.62382e22 0.346479 0.173239 0.984880i \(-0.444577\pi\)
0.173239 + 0.984880i \(0.444577\pi\)
\(152\) −2.62569e21 −0.0323508
\(153\) 0 0
\(154\) 1.58033e23 1.69739
\(155\) 2.86336e22 0.287338
\(156\) 0 0
\(157\) 1.62221e23 1.42286 0.711428 0.702759i \(-0.248049\pi\)
0.711428 + 0.702759i \(0.248049\pi\)
\(158\) 1.64013e23 1.34579
\(159\) 0 0
\(160\) 4.76839e22 0.342856
\(161\) −1.56733e23 −1.05558
\(162\) 0 0
\(163\) 2.32312e23 1.37436 0.687181 0.726486i \(-0.258848\pi\)
0.687181 + 0.726486i \(0.258848\pi\)
\(164\) −1.23336e23 −0.684275
\(165\) 0 0
\(166\) 3.79387e23 1.85331
\(167\) 1.76825e23 0.811002 0.405501 0.914095i \(-0.367097\pi\)
0.405501 + 0.914095i \(0.367097\pi\)
\(168\) 0 0
\(169\) 5.62653e22 0.227735
\(170\) −6.78951e22 −0.258300
\(171\) 0 0
\(172\) 6.85486e22 0.230648
\(173\) −1.88916e23 −0.598113 −0.299057 0.954235i \(-0.596672\pi\)
−0.299057 + 0.954235i \(0.596672\pi\)
\(174\) 0 0
\(175\) −5.95548e22 −0.167116
\(176\) 8.04546e23 2.12651
\(177\) 0 0
\(178\) −5.40930e22 −0.126979
\(179\) 1.14808e23 0.254107 0.127053 0.991896i \(-0.459448\pi\)
0.127053 + 0.991896i \(0.459448\pi\)
\(180\) 0 0
\(181\) 3.15471e23 0.621347 0.310673 0.950517i \(-0.399445\pi\)
0.310673 + 0.950517i \(0.399445\pi\)
\(182\) 5.90494e23 1.09765
\(183\) 0 0
\(184\) −5.37176e23 −0.890286
\(185\) −3.96589e23 −0.620923
\(186\) 0 0
\(187\) −5.96881e23 −0.834739
\(188\) −4.21863e22 −0.0557846
\(189\) 0 0
\(190\) −2.05689e22 −0.0243388
\(191\) 4.11188e23 0.460458 0.230229 0.973137i \(-0.426053\pi\)
0.230229 + 0.973137i \(0.426053\pi\)
\(192\) 0 0
\(193\) −9.08221e22 −0.0911675 −0.0455837 0.998961i \(-0.514515\pi\)
−0.0455837 + 0.998961i \(0.514515\pi\)
\(194\) 4.23294e23 0.402461
\(195\) 0 0
\(196\) −1.43379e23 −0.122404
\(197\) −1.22586e24 −0.992076 −0.496038 0.868301i \(-0.665212\pi\)
−0.496038 + 0.868301i \(0.665212\pi\)
\(198\) 0 0
\(199\) 1.05217e24 0.765824 0.382912 0.923785i \(-0.374921\pi\)
0.382912 + 0.923785i \(0.374921\pi\)
\(200\) −2.04114e23 −0.140947
\(201\) 0 0
\(202\) −3.69517e24 −2.29850
\(203\) −2.18129e24 −1.28827
\(204\) 0 0
\(205\) 1.41611e24 0.754537
\(206\) −2.63267e24 −1.33288
\(207\) 0 0
\(208\) 3.00620e24 1.37516
\(209\) −1.80826e23 −0.0786548
\(210\) 0 0
\(211\) 5.34118e22 0.0210219 0.0105109 0.999945i \(-0.496654\pi\)
0.0105109 + 0.999945i \(0.496654\pi\)
\(212\) 1.06025e24 0.397085
\(213\) 0 0
\(214\) −2.65814e24 −0.902057
\(215\) −7.87052e23 −0.254331
\(216\) 0 0
\(217\) 1.83102e24 0.536866
\(218\) 4.89670e24 1.36808
\(219\) 0 0
\(220\) 1.22429e24 0.310776
\(221\) −2.23026e24 −0.539803
\(222\) 0 0
\(223\) 1.04158e24 0.229346 0.114673 0.993403i \(-0.463418\pi\)
0.114673 + 0.993403i \(0.463418\pi\)
\(224\) 3.04922e24 0.640596
\(225\) 0 0
\(226\) −5.67182e24 −1.08539
\(227\) −4.64270e24 −0.848201 −0.424101 0.905615i \(-0.639410\pi\)
−0.424101 + 0.905615i \(0.639410\pi\)
\(228\) 0 0
\(229\) −3.77779e24 −0.629456 −0.314728 0.949182i \(-0.601913\pi\)
−0.314728 + 0.949182i \(0.601913\pi\)
\(230\) −4.20810e24 −0.669797
\(231\) 0 0
\(232\) −7.47600e24 −1.08654
\(233\) −1.16866e25 −1.62350 −0.811750 0.584005i \(-0.801485\pi\)
−0.811750 + 0.584005i \(0.801485\pi\)
\(234\) 0 0
\(235\) 4.84369e23 0.0615126
\(236\) −6.12414e24 −0.743822
\(237\) 0 0
\(238\) −4.34165e24 −0.482611
\(239\) 1.39609e25 1.48503 0.742514 0.669831i \(-0.233633\pi\)
0.742514 + 0.669831i \(0.233633\pi\)
\(240\) 0 0
\(241\) 1.83127e25 1.78473 0.892366 0.451312i \(-0.149044\pi\)
0.892366 + 0.451312i \(0.149044\pi\)
\(242\) 2.45958e25 2.29508
\(243\) 0 0
\(244\) −4.06063e24 −0.347537
\(245\) 1.64623e24 0.134973
\(246\) 0 0
\(247\) −6.75660e23 −0.0508640
\(248\) 6.27550e24 0.452800
\(249\) 0 0
\(250\) −1.59897e24 −0.106040
\(251\) −2.08800e25 −1.32787 −0.663936 0.747789i \(-0.731115\pi\)
−0.663936 + 0.747789i \(0.731115\pi\)
\(252\) 0 0
\(253\) −3.69943e25 −2.16456
\(254\) −1.43726e24 −0.0806828
\(255\) 0 0
\(256\) 2.01868e25 1.04363
\(257\) 2.97816e24 0.147792 0.0738959 0.997266i \(-0.476457\pi\)
0.0738959 + 0.997266i \(0.476457\pi\)
\(258\) 0 0
\(259\) −2.53605e25 −1.16014
\(260\) 4.57460e24 0.200971
\(261\) 0 0
\(262\) −3.27238e25 −1.32648
\(263\) 4.48688e25 1.74747 0.873734 0.486404i \(-0.161692\pi\)
0.873734 + 0.486404i \(0.161692\pi\)
\(264\) 0 0
\(265\) −1.21735e25 −0.437858
\(266\) −1.31531e24 −0.0454749
\(267\) 0 0
\(268\) 6.55632e24 0.209529
\(269\) −5.40734e25 −1.66182 −0.830912 0.556404i \(-0.812181\pi\)
−0.830912 + 0.556404i \(0.812181\pi\)
\(270\) 0 0
\(271\) 1.64805e25 0.468591 0.234295 0.972165i \(-0.424722\pi\)
0.234295 + 0.972165i \(0.424722\pi\)
\(272\) −2.21033e25 −0.604622
\(273\) 0 0
\(274\) 1.48222e24 0.0375432
\(275\) −1.40569e25 −0.342687
\(276\) 0 0
\(277\) −2.02544e25 −0.457596 −0.228798 0.973474i \(-0.573480\pi\)
−0.228798 + 0.973474i \(0.573480\pi\)
\(278\) 1.53050e25 0.332937
\(279\) 0 0
\(280\) −1.30524e25 −0.263348
\(281\) −9.60686e25 −1.86709 −0.933545 0.358461i \(-0.883301\pi\)
−0.933545 + 0.358461i \(0.883301\pi\)
\(282\) 0 0
\(283\) −9.26489e24 −0.167141 −0.0835704 0.996502i \(-0.526632\pi\)
−0.0835704 + 0.996502i \(0.526632\pi\)
\(284\) 1.38863e25 0.241404
\(285\) 0 0
\(286\) 1.39376e26 2.25085
\(287\) 9.05550e25 1.40978
\(288\) 0 0
\(289\) −5.26938e25 −0.762662
\(290\) −5.85650e25 −0.817447
\(291\) 0 0
\(292\) −1.10510e25 −0.143510
\(293\) −4.08300e25 −0.511528 −0.255764 0.966739i \(-0.582327\pi\)
−0.255764 + 0.966739i \(0.582327\pi\)
\(294\) 0 0
\(295\) 7.03153e25 0.820198
\(296\) −8.69186e25 −0.978476
\(297\) 0 0
\(298\) 1.61772e26 1.69681
\(299\) −1.38230e26 −1.39976
\(300\) 0 0
\(301\) −5.03292e25 −0.475195
\(302\) 4.50480e25 0.410774
\(303\) 0 0
\(304\) −6.69625e24 −0.0569716
\(305\) 4.66228e25 0.383222
\(306\) 0 0
\(307\) 1.09642e26 0.841445 0.420723 0.907189i \(-0.361777\pi\)
0.420723 + 0.907189i \(0.361777\pi\)
\(308\) 7.82893e25 0.580657
\(309\) 0 0
\(310\) 4.91607e25 0.340659
\(311\) 6.99628e25 0.468687 0.234343 0.972154i \(-0.424706\pi\)
0.234343 + 0.972154i \(0.424706\pi\)
\(312\) 0 0
\(313\) 6.95169e25 0.435386 0.217693 0.976017i \(-0.430147\pi\)
0.217693 + 0.976017i \(0.430147\pi\)
\(314\) 2.78515e26 1.68689
\(315\) 0 0
\(316\) 8.12516e25 0.460381
\(317\) 1.47035e26 0.805933 0.402967 0.915215i \(-0.367979\pi\)
0.402967 + 0.915215i \(0.367979\pi\)
\(318\) 0 0
\(319\) −5.14858e26 −2.64171
\(320\) −2.99190e25 −0.148550
\(321\) 0 0
\(322\) −2.69093e26 −1.25146
\(323\) 4.96784e24 0.0223636
\(324\) 0 0
\(325\) −5.25240e25 −0.221606
\(326\) 3.98853e26 1.62940
\(327\) 0 0
\(328\) 3.10362e26 1.18903
\(329\) 3.09737e25 0.114931
\(330\) 0 0
\(331\) 4.36548e26 1.51998 0.759990 0.649935i \(-0.225204\pi\)
0.759990 + 0.649935i \(0.225204\pi\)
\(332\) 1.87948e26 0.633997
\(333\) 0 0
\(334\) 3.03589e26 0.961497
\(335\) −7.52774e25 −0.231044
\(336\) 0 0
\(337\) 4.15229e26 1.19722 0.598610 0.801041i \(-0.295720\pi\)
0.598610 + 0.801041i \(0.295720\pi\)
\(338\) 9.66010e25 0.269995
\(339\) 0 0
\(340\) −3.36351e25 −0.0883616
\(341\) 4.32182e26 1.10090
\(342\) 0 0
\(343\) 4.54070e26 1.08776
\(344\) −1.72495e26 −0.400785
\(345\) 0 0
\(346\) −3.24346e26 −0.709104
\(347\) −2.48113e26 −0.526246 −0.263123 0.964762i \(-0.584753\pi\)
−0.263123 + 0.964762i \(0.584753\pi\)
\(348\) 0 0
\(349\) −8.30405e24 −0.0165815 −0.00829074 0.999966i \(-0.502639\pi\)
−0.00829074 + 0.999966i \(0.502639\pi\)
\(350\) −1.02249e26 −0.198127
\(351\) 0 0
\(352\) 7.19718e26 1.31360
\(353\) −9.55014e25 −0.169190 −0.0845951 0.996415i \(-0.526960\pi\)
−0.0845951 + 0.996415i \(0.526960\pi\)
\(354\) 0 0
\(355\) −1.59438e26 −0.266192
\(356\) −2.67976e25 −0.0434381
\(357\) 0 0
\(358\) 1.97112e26 0.301260
\(359\) −7.06762e25 −0.104901 −0.0524507 0.998624i \(-0.516703\pi\)
−0.0524507 + 0.998624i \(0.516703\pi\)
\(360\) 0 0
\(361\) −7.12704e26 −0.997893
\(362\) 5.41627e26 0.736648
\(363\) 0 0
\(364\) 2.92529e26 0.375496
\(365\) 1.26884e26 0.158246
\(366\) 0 0
\(367\) −1.33814e27 −1.57583 −0.787915 0.615785i \(-0.788839\pi\)
−0.787915 + 0.615785i \(0.788839\pi\)
\(368\) −1.36995e27 −1.56784
\(369\) 0 0
\(370\) −6.80898e26 −0.736146
\(371\) −7.78451e26 −0.818098
\(372\) 0 0
\(373\) −9.62756e24 −0.00956255 −0.00478127 0.999989i \(-0.501522\pi\)
−0.00478127 + 0.999989i \(0.501522\pi\)
\(374\) −1.02478e27 −0.989639
\(375\) 0 0
\(376\) 1.06157e26 0.0969340
\(377\) −1.92378e27 −1.70833
\(378\) 0 0
\(379\) −1.97813e27 −1.66166 −0.830830 0.556526i \(-0.812134\pi\)
−0.830830 + 0.556526i \(0.812134\pi\)
\(380\) −1.01898e25 −0.00832604
\(381\) 0 0
\(382\) 7.05962e26 0.545904
\(383\) 2.14854e27 1.61643 0.808215 0.588888i \(-0.200434\pi\)
0.808215 + 0.588888i \(0.200434\pi\)
\(384\) 0 0
\(385\) −8.98891e26 −0.640279
\(386\) −1.55931e26 −0.108085
\(387\) 0 0
\(388\) 2.09699e26 0.137677
\(389\) −2.40240e27 −1.53524 −0.767618 0.640907i \(-0.778558\pi\)
−0.767618 + 0.640907i \(0.778558\pi\)
\(390\) 0 0
\(391\) 1.01635e27 0.615440
\(392\) 3.60796e26 0.212696
\(393\) 0 0
\(394\) −2.10466e27 −1.17617
\(395\) −9.32903e26 −0.507653
\(396\) 0 0
\(397\) 2.20346e27 1.13711 0.568557 0.822644i \(-0.307502\pi\)
0.568557 + 0.822644i \(0.307502\pi\)
\(398\) 1.80646e27 0.907935
\(399\) 0 0
\(400\) −5.20548e26 −0.248217
\(401\) 2.56251e27 1.19028 0.595141 0.803622i \(-0.297096\pi\)
0.595141 + 0.803622i \(0.297096\pi\)
\(402\) 0 0
\(403\) 1.61486e27 0.711920
\(404\) −1.83058e27 −0.786292
\(405\) 0 0
\(406\) −3.74503e27 −1.52733
\(407\) −5.98593e27 −2.37898
\(408\) 0 0
\(409\) −4.40538e27 −1.66299 −0.831494 0.555534i \(-0.812514\pi\)
−0.831494 + 0.555534i \(0.812514\pi\)
\(410\) 2.43129e27 0.894554
\(411\) 0 0
\(412\) −1.30422e27 −0.455964
\(413\) 4.49641e27 1.53247
\(414\) 0 0
\(415\) −2.15795e27 −0.699096
\(416\) 2.68924e27 0.849473
\(417\) 0 0
\(418\) −3.10458e26 −0.0932505
\(419\) −1.22792e27 −0.359686 −0.179843 0.983695i \(-0.557559\pi\)
−0.179843 + 0.983695i \(0.557559\pi\)
\(420\) 0 0
\(421\) −2.61500e25 −0.00728633 −0.00364317 0.999993i \(-0.501160\pi\)
−0.00364317 + 0.999993i \(0.501160\pi\)
\(422\) 9.17019e25 0.0249229
\(423\) 0 0
\(424\) −2.66800e27 −0.689994
\(425\) 3.86187e26 0.0974346
\(426\) 0 0
\(427\) 2.98137e27 0.716016
\(428\) −1.31684e27 −0.308584
\(429\) 0 0
\(430\) −1.35128e27 −0.301527
\(431\) 3.68868e27 0.803265 0.401633 0.915801i \(-0.368443\pi\)
0.401633 + 0.915801i \(0.368443\pi\)
\(432\) 0 0
\(433\) 4.87388e27 1.01100 0.505501 0.862826i \(-0.331308\pi\)
0.505501 + 0.862826i \(0.331308\pi\)
\(434\) 3.14365e27 0.636491
\(435\) 0 0
\(436\) 2.42582e27 0.468004
\(437\) 3.07904e26 0.0579910
\(438\) 0 0
\(439\) 8.77061e27 1.57453 0.787267 0.616612i \(-0.211495\pi\)
0.787267 + 0.616612i \(0.211495\pi\)
\(440\) −3.08079e27 −0.540019
\(441\) 0 0
\(442\) −3.82909e27 −0.639973
\(443\) −2.39913e26 −0.0391575 −0.0195787 0.999808i \(-0.506233\pi\)
−0.0195787 + 0.999808i \(0.506233\pi\)
\(444\) 0 0
\(445\) 3.07681e26 0.0478983
\(446\) 1.78827e27 0.271905
\(447\) 0 0
\(448\) −1.91322e27 −0.277552
\(449\) 1.18190e27 0.167491 0.0837457 0.996487i \(-0.473312\pi\)
0.0837457 + 0.996487i \(0.473312\pi\)
\(450\) 0 0
\(451\) 2.13740e28 2.89090
\(452\) −2.80981e27 −0.371299
\(453\) 0 0
\(454\) −7.97098e27 −1.00560
\(455\) −3.35872e27 −0.414052
\(456\) 0 0
\(457\) 1.24883e28 1.47023 0.735114 0.677944i \(-0.237129\pi\)
0.735114 + 0.677944i \(0.237129\pi\)
\(458\) −6.48603e27 −0.746262
\(459\) 0 0
\(460\) −2.08468e27 −0.229130
\(461\) 4.81654e27 0.517459 0.258729 0.965950i \(-0.416696\pi\)
0.258729 + 0.965950i \(0.416696\pi\)
\(462\) 0 0
\(463\) 2.77535e27 0.284917 0.142458 0.989801i \(-0.454499\pi\)
0.142458 + 0.989801i \(0.454499\pi\)
\(464\) −1.90659e28 −1.91346
\(465\) 0 0
\(466\) −2.00646e28 −1.92477
\(467\) 1.30821e28 1.22702 0.613508 0.789688i \(-0.289758\pi\)
0.613508 + 0.789688i \(0.289758\pi\)
\(468\) 0 0
\(469\) −4.81373e27 −0.431685
\(470\) 8.31606e26 0.0729273
\(471\) 0 0
\(472\) 1.54107e28 1.29250
\(473\) −1.18794e28 −0.974433
\(474\) 0 0
\(475\) 1.16996e26 0.00918096
\(476\) −2.15084e27 −0.165096
\(477\) 0 0
\(478\) 2.39692e28 1.76060
\(479\) 1.98117e28 1.42363 0.711817 0.702365i \(-0.247873\pi\)
0.711817 + 0.702365i \(0.247873\pi\)
\(480\) 0 0
\(481\) −2.23665e28 −1.53842
\(482\) 3.14408e28 2.11592
\(483\) 0 0
\(484\) 1.21847e28 0.785123
\(485\) −2.40769e27 −0.151814
\(486\) 0 0
\(487\) −2.14424e28 −1.29485 −0.647425 0.762129i \(-0.724154\pi\)
−0.647425 + 0.762129i \(0.724154\pi\)
\(488\) 1.02181e28 0.603896
\(489\) 0 0
\(490\) 2.82639e27 0.160019
\(491\) 1.64770e28 0.913109 0.456554 0.889695i \(-0.349083\pi\)
0.456554 + 0.889695i \(0.349083\pi\)
\(492\) 0 0
\(493\) 1.41447e28 0.751107
\(494\) −1.16003e27 −0.0603027
\(495\) 0 0
\(496\) 1.60043e28 0.797406
\(497\) −1.01955e28 −0.497356
\(498\) 0 0
\(499\) −2.46207e28 −1.15145 −0.575724 0.817644i \(-0.695280\pi\)
−0.575724 + 0.817644i \(0.695280\pi\)
\(500\) −7.92128e26 −0.0362752
\(501\) 0 0
\(502\) −3.58486e28 −1.57428
\(503\) −9.01822e27 −0.387844 −0.193922 0.981017i \(-0.562121\pi\)
−0.193922 + 0.981017i \(0.562121\pi\)
\(504\) 0 0
\(505\) 2.10181e28 0.867028
\(506\) −6.35150e28 −2.56623
\(507\) 0 0
\(508\) −7.12015e26 −0.0276007
\(509\) 1.26550e28 0.480534 0.240267 0.970707i \(-0.422765\pi\)
0.240267 + 0.970707i \(0.422765\pi\)
\(510\) 0 0
\(511\) 8.11381e27 0.295669
\(512\) 2.15236e27 0.0768388
\(513\) 0 0
\(514\) 5.11316e27 0.175217
\(515\) 1.49746e28 0.502783
\(516\) 0 0
\(517\) 7.31084e27 0.235677
\(518\) −4.35410e28 −1.37542
\(519\) 0 0
\(520\) −1.15114e28 −0.349216
\(521\) 3.11290e28 0.925483 0.462742 0.886493i \(-0.346866\pi\)
0.462742 + 0.886493i \(0.346866\pi\)
\(522\) 0 0
\(523\) −2.19095e28 −0.625699 −0.312850 0.949803i \(-0.601284\pi\)
−0.312850 + 0.949803i \(0.601284\pi\)
\(524\) −1.62113e28 −0.453774
\(525\) 0 0
\(526\) 7.70346e28 2.07174
\(527\) −1.18734e28 −0.313013
\(528\) 0 0
\(529\) 2.35210e28 0.595896
\(530\) −2.09004e28 −0.519109
\(531\) 0 0
\(532\) −6.51602e26 −0.0155565
\(533\) 7.98644e28 1.86947
\(534\) 0 0
\(535\) 1.51195e28 0.340269
\(536\) −1.64982e28 −0.364088
\(537\) 0 0
\(538\) −9.28378e28 −1.97020
\(539\) 2.48474e28 0.517129
\(540\) 0 0
\(541\) 8.64723e28 1.73104 0.865518 0.500879i \(-0.166990\pi\)
0.865518 + 0.500879i \(0.166990\pi\)
\(542\) 2.82952e28 0.555546
\(543\) 0 0
\(544\) −1.97728e28 −0.373491
\(545\) −2.78524e28 −0.516059
\(546\) 0 0
\(547\) 6.69174e28 1.19309 0.596544 0.802581i \(-0.296540\pi\)
0.596544 + 0.802581i \(0.296540\pi\)
\(548\) 7.34288e26 0.0128431
\(549\) 0 0
\(550\) −2.41341e28 −0.406278
\(551\) 4.28517e27 0.0707744
\(552\) 0 0
\(553\) −5.96559e28 −0.948505
\(554\) −3.47745e28 −0.542511
\(555\) 0 0
\(556\) 7.58204e27 0.113894
\(557\) 6.14899e28 0.906409 0.453204 0.891407i \(-0.350281\pi\)
0.453204 + 0.891407i \(0.350281\pi\)
\(558\) 0 0
\(559\) −4.43875e28 −0.630140
\(560\) −3.32872e28 −0.463770
\(561\) 0 0
\(562\) −1.64939e29 −2.21356
\(563\) −1.12154e28 −0.147733 −0.0738663 0.997268i \(-0.523534\pi\)
−0.0738663 + 0.997268i \(0.523534\pi\)
\(564\) 0 0
\(565\) 3.22613e28 0.409424
\(566\) −1.59067e28 −0.198157
\(567\) 0 0
\(568\) −3.49434e28 −0.419476
\(569\) 1.24550e29 1.46779 0.733895 0.679263i \(-0.237700\pi\)
0.733895 + 0.679263i \(0.237700\pi\)
\(570\) 0 0
\(571\) −6.70182e28 −0.761226 −0.380613 0.924734i \(-0.624287\pi\)
−0.380613 + 0.924734i \(0.624287\pi\)
\(572\) 6.90467e28 0.769990
\(573\) 0 0
\(574\) 1.55473e29 1.67139
\(575\) 2.39356e28 0.252658
\(576\) 0 0
\(577\) 1.60912e29 1.63773 0.818864 0.573987i \(-0.194604\pi\)
0.818864 + 0.573987i \(0.194604\pi\)
\(578\) −9.04692e28 −0.904187
\(579\) 0 0
\(580\) −2.90130e28 −0.279640
\(581\) −1.37993e29 −1.30620
\(582\) 0 0
\(583\) −1.83740e29 −1.67759
\(584\) 2.78087e28 0.249370
\(585\) 0 0
\(586\) −7.01004e28 −0.606451
\(587\) −1.88288e29 −1.60001 −0.800006 0.599992i \(-0.795170\pi\)
−0.800006 + 0.599992i \(0.795170\pi\)
\(588\) 0 0
\(589\) −3.59706e27 −0.0294942
\(590\) 1.20723e29 0.972399
\(591\) 0 0
\(592\) −2.21667e29 −1.72315
\(593\) −2.04137e29 −1.55901 −0.779503 0.626398i \(-0.784529\pi\)
−0.779503 + 0.626398i \(0.784529\pi\)
\(594\) 0 0
\(595\) 2.46953e28 0.182048
\(596\) 8.01414e28 0.580459
\(597\) 0 0
\(598\) −2.37325e29 −1.65951
\(599\) −9.66401e28 −0.664012 −0.332006 0.943277i \(-0.607725\pi\)
−0.332006 + 0.943277i \(0.607725\pi\)
\(600\) 0 0
\(601\) −1.22230e29 −0.810951 −0.405476 0.914106i \(-0.632894\pi\)
−0.405476 + 0.914106i \(0.632894\pi\)
\(602\) −8.64095e28 −0.563376
\(603\) 0 0
\(604\) 2.23167e28 0.140521
\(605\) −1.39900e29 −0.865740
\(606\) 0 0
\(607\) −1.06271e29 −0.635235 −0.317617 0.948219i \(-0.602883\pi\)
−0.317617 + 0.948219i \(0.602883\pi\)
\(608\) −5.99022e27 −0.0351929
\(609\) 0 0
\(610\) 8.00460e28 0.454335
\(611\) 2.73171e28 0.152406
\(612\) 0 0
\(613\) 3.11548e29 1.67954 0.839768 0.542945i \(-0.182691\pi\)
0.839768 + 0.542945i \(0.182691\pi\)
\(614\) 1.88244e29 0.997590
\(615\) 0 0
\(616\) −1.97006e29 −1.00898
\(617\) 2.29331e29 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(618\) 0 0
\(619\) 1.75961e29 0.856376 0.428188 0.903690i \(-0.359152\pi\)
0.428188 + 0.903690i \(0.359152\pi\)
\(620\) 2.43541e28 0.116536
\(621\) 0 0
\(622\) 1.20118e29 0.555660
\(623\) 1.96751e28 0.0894937
\(624\) 0 0
\(625\) 9.09495e27 0.0400000
\(626\) 1.19352e29 0.516180
\(627\) 0 0
\(628\) 1.37976e29 0.577067
\(629\) 1.64452e29 0.676404
\(630\) 0 0
\(631\) −1.30487e28 −0.0519108 −0.0259554 0.999663i \(-0.508263\pi\)
−0.0259554 + 0.999663i \(0.508263\pi\)
\(632\) −2.04460e29 −0.799981
\(633\) 0 0
\(634\) 2.52442e29 0.955488
\(635\) 8.17511e27 0.0304347
\(636\) 0 0
\(637\) 9.28427e28 0.334413
\(638\) −8.83952e29 −3.13193
\(639\) 0 0
\(640\) −1.51368e29 −0.518972
\(641\) 3.70997e29 1.25130 0.625650 0.780104i \(-0.284834\pi\)
0.625650 + 0.780104i \(0.284834\pi\)
\(642\) 0 0
\(643\) −1.15073e29 −0.375627 −0.187814 0.982205i \(-0.560140\pi\)
−0.187814 + 0.982205i \(0.560140\pi\)
\(644\) −1.33308e29 −0.428110
\(645\) 0 0
\(646\) 8.52921e27 0.0265135
\(647\) −2.45274e29 −0.750165 −0.375082 0.926991i \(-0.622386\pi\)
−0.375082 + 0.926991i \(0.622386\pi\)
\(648\) 0 0
\(649\) 1.06130e30 3.14247
\(650\) −9.01776e28 −0.262729
\(651\) 0 0
\(652\) 1.97591e29 0.557400
\(653\) −6.64107e29 −1.84353 −0.921764 0.387751i \(-0.873252\pi\)
−0.921764 + 0.387751i \(0.873252\pi\)
\(654\) 0 0
\(655\) 1.86133e29 0.500368
\(656\) 7.91510e29 2.09395
\(657\) 0 0
\(658\) 5.31783e28 0.136258
\(659\) 1.71049e29 0.431344 0.215672 0.976466i \(-0.430806\pi\)
0.215672 + 0.976466i \(0.430806\pi\)
\(660\) 0 0
\(661\) −1.36074e29 −0.332399 −0.166200 0.986092i \(-0.553150\pi\)
−0.166200 + 0.986092i \(0.553150\pi\)
\(662\) 7.49502e29 1.80204
\(663\) 0 0
\(664\) −4.72948e29 −1.10166
\(665\) 7.48148e27 0.0171538
\(666\) 0 0
\(667\) 8.76681e29 1.94769
\(668\) 1.50397e29 0.328918
\(669\) 0 0
\(670\) −1.29243e29 −0.273918
\(671\) 7.03702e29 1.46826
\(672\) 0 0
\(673\) −8.46772e29 −1.71241 −0.856206 0.516634i \(-0.827185\pi\)
−0.856206 + 0.516634i \(0.827185\pi\)
\(674\) 7.12900e29 1.41938
\(675\) 0 0
\(676\) 4.78559e28 0.0923625
\(677\) −2.36669e29 −0.449738 −0.224869 0.974389i \(-0.572195\pi\)
−0.224869 + 0.974389i \(0.572195\pi\)
\(678\) 0 0
\(679\) −1.53963e29 −0.283651
\(680\) 8.46387e28 0.153541
\(681\) 0 0
\(682\) 7.42007e29 1.30519
\(683\) 1.02867e30 1.78180 0.890898 0.454203i \(-0.150076\pi\)
0.890898 + 0.454203i \(0.150076\pi\)
\(684\) 0 0
\(685\) −8.43084e27 −0.0141619
\(686\) 7.79586e29 1.28962
\(687\) 0 0
\(688\) −4.39910e29 −0.705806
\(689\) −6.86550e29 −1.08485
\(690\) 0 0
\(691\) 8.33280e29 1.27724 0.638619 0.769523i \(-0.279506\pi\)
0.638619 + 0.769523i \(0.279506\pi\)
\(692\) −1.60680e29 −0.242577
\(693\) 0 0
\(694\) −4.25981e29 −0.623901
\(695\) −8.70545e28 −0.125589
\(696\) 0 0
\(697\) −5.87209e29 −0.821956
\(698\) −1.42571e28 −0.0196584
\(699\) 0 0
\(700\) −5.06538e28 −0.0677770
\(701\) −6.99649e29 −0.922234 −0.461117 0.887339i \(-0.652551\pi\)
−0.461117 + 0.887339i \(0.652551\pi\)
\(702\) 0 0
\(703\) 4.98209e28 0.0637354
\(704\) −4.51583e29 −0.569148
\(705\) 0 0
\(706\) −1.63965e29 −0.200586
\(707\) 1.34403e30 1.61997
\(708\) 0 0
\(709\) 2.89141e29 0.338318 0.169159 0.985589i \(-0.445895\pi\)
0.169159 + 0.985589i \(0.445895\pi\)
\(710\) −2.73737e29 −0.315588
\(711\) 0 0
\(712\) 6.74330e28 0.0754801
\(713\) −7.35904e29 −0.811673
\(714\) 0 0
\(715\) −7.92771e29 −0.849053
\(716\) 9.76491e28 0.103058
\(717\) 0 0
\(718\) −1.21343e29 −0.124368
\(719\) −1.03572e29 −0.104614 −0.0523070 0.998631i \(-0.516657\pi\)
−0.0523070 + 0.998631i \(0.516657\pi\)
\(720\) 0 0
\(721\) 9.57575e29 0.939404
\(722\) −1.22363e30 −1.18307
\(723\) 0 0
\(724\) 2.68321e29 0.251999
\(725\) 3.33117e29 0.308353
\(726\) 0 0
\(727\) 3.99248e29 0.359030 0.179515 0.983755i \(-0.442547\pi\)
0.179515 + 0.983755i \(0.442547\pi\)
\(728\) −7.36116e29 −0.652479
\(729\) 0 0
\(730\) 2.17846e29 0.187611
\(731\) 3.26363e29 0.277056
\(732\) 0 0
\(733\) 1.02595e30 0.846317 0.423158 0.906056i \(-0.360921\pi\)
0.423158 + 0.906056i \(0.360921\pi\)
\(734\) −2.29744e30 −1.86825
\(735\) 0 0
\(736\) −1.22551e30 −0.968500
\(737\) −1.13620e30 −0.885210
\(738\) 0 0
\(739\) −5.23998e28 −0.0396792 −0.0198396 0.999803i \(-0.506316\pi\)
−0.0198396 + 0.999803i \(0.506316\pi\)
\(740\) −3.37315e29 −0.251827
\(741\) 0 0
\(742\) −1.33651e30 −0.969910
\(743\) −1.53956e30 −1.10158 −0.550788 0.834645i \(-0.685673\pi\)
−0.550788 + 0.834645i \(0.685673\pi\)
\(744\) 0 0
\(745\) −9.20156e29 −0.640060
\(746\) −1.65294e28 −0.0113370
\(747\) 0 0
\(748\) −5.07671e29 −0.338545
\(749\) 9.66839e29 0.635763
\(750\) 0 0
\(751\) 1.28343e30 0.820639 0.410320 0.911942i \(-0.365417\pi\)
0.410320 + 0.911942i \(0.365417\pi\)
\(752\) 2.70731e29 0.170706
\(753\) 0 0
\(754\) −3.30290e30 −2.02533
\(755\) −2.56233e29 −0.154950
\(756\) 0 0
\(757\) 7.23137e29 0.425318 0.212659 0.977126i \(-0.431788\pi\)
0.212659 + 0.977126i \(0.431788\pi\)
\(758\) −3.39622e30 −1.97001
\(759\) 0 0
\(760\) 2.56415e28 0.0144677
\(761\) 2.50610e29 0.139463 0.0697316 0.997566i \(-0.477786\pi\)
0.0697316 + 0.997566i \(0.477786\pi\)
\(762\) 0 0
\(763\) −1.78106e30 −0.964210
\(764\) 3.49732e29 0.186748
\(765\) 0 0
\(766\) 3.68880e30 1.91639
\(767\) 3.96558e30 2.03215
\(768\) 0 0
\(769\) −1.40362e30 −0.699880 −0.349940 0.936772i \(-0.613798\pi\)
−0.349940 + 0.936772i \(0.613798\pi\)
\(770\) −1.54329e30 −0.759094
\(771\) 0 0
\(772\) −7.72479e28 −0.0369747
\(773\) 1.31390e30 0.620410 0.310205 0.950670i \(-0.399602\pi\)
0.310205 + 0.950670i \(0.399602\pi\)
\(774\) 0 0
\(775\) −2.79625e29 −0.128502
\(776\) −5.27682e29 −0.239235
\(777\) 0 0
\(778\) −4.12465e30 −1.82013
\(779\) −1.77896e29 −0.0774503
\(780\) 0 0
\(781\) −2.40648e30 −1.01988
\(782\) 1.74495e30 0.729645
\(783\) 0 0
\(784\) 9.20133e29 0.374569
\(785\) −1.58419e30 −0.636321
\(786\) 0 0
\(787\) −3.99554e30 −1.56257 −0.781287 0.624171i \(-0.785437\pi\)
−0.781287 + 0.624171i \(0.785437\pi\)
\(788\) −1.04264e30 −0.402356
\(789\) 0 0
\(790\) −1.60169e30 −0.601857
\(791\) 2.06300e30 0.764972
\(792\) 0 0
\(793\) 2.62940e30 0.949484
\(794\) 3.78308e30 1.34813
\(795\) 0 0
\(796\) 8.94914e29 0.310595
\(797\) 3.18052e30 1.08940 0.544698 0.838633i \(-0.316644\pi\)
0.544698 + 0.838633i \(0.316644\pi\)
\(798\) 0 0
\(799\) −2.00851e29 −0.0670089
\(800\) −4.65663e29 −0.153330
\(801\) 0 0
\(802\) 4.39953e30 1.41116
\(803\) 1.91513e30 0.606296
\(804\) 0 0
\(805\) 1.53060e30 0.472068
\(806\) 2.77252e30 0.844029
\(807\) 0 0
\(808\) 4.60643e30 1.36630
\(809\) −2.04768e30 −0.599519 −0.299759 0.954015i \(-0.596906\pi\)
−0.299759 + 0.954015i \(0.596906\pi\)
\(810\) 0 0
\(811\) 2.02072e30 0.576485 0.288242 0.957557i \(-0.406929\pi\)
0.288242 + 0.957557i \(0.406929\pi\)
\(812\) −1.85528e30 −0.522481
\(813\) 0 0
\(814\) −1.02771e31 −2.82044
\(815\) −2.26867e30 −0.614634
\(816\) 0 0
\(817\) 9.88722e28 0.0261061
\(818\) −7.56354e30 −1.97158
\(819\) 0 0
\(820\) 1.20446e30 0.306017
\(821\) 3.31897e30 0.832531 0.416265 0.909243i \(-0.363339\pi\)
0.416265 + 0.909243i \(0.363339\pi\)
\(822\) 0 0
\(823\) −8.46852e29 −0.207066 −0.103533 0.994626i \(-0.533015\pi\)
−0.103533 + 0.994626i \(0.533015\pi\)
\(824\) 3.28192e30 0.792305
\(825\) 0 0
\(826\) 7.71983e30 1.81684
\(827\) 7.77607e30 1.80697 0.903487 0.428614i \(-0.140998\pi\)
0.903487 + 0.428614i \(0.140998\pi\)
\(828\) 0 0
\(829\) 4.65905e30 1.05554 0.527770 0.849387i \(-0.323028\pi\)
0.527770 + 0.849387i \(0.323028\pi\)
\(830\) −3.70495e30 −0.828825
\(831\) 0 0
\(832\) −1.68735e30 −0.368053
\(833\) −6.82633e29 −0.147033
\(834\) 0 0
\(835\) −1.72681e30 −0.362691
\(836\) −1.53800e29 −0.0319000
\(837\) 0 0
\(838\) −2.10820e30 −0.426431
\(839\) 2.71344e29 0.0542025 0.0271013 0.999633i \(-0.491372\pi\)
0.0271013 + 0.999633i \(0.491372\pi\)
\(840\) 0 0
\(841\) 7.06813e30 1.37704
\(842\) −4.48965e28 −0.00863843
\(843\) 0 0
\(844\) 4.54289e28 0.00852584
\(845\) −5.49466e29 −0.101846
\(846\) 0 0
\(847\) −8.94614e30 −1.61756
\(848\) −6.80417e30 −1.21512
\(849\) 0 0
\(850\) 6.63038e29 0.115515
\(851\) 1.01926e31 1.75398
\(852\) 0 0
\(853\) 9.28049e30 1.55814 0.779070 0.626937i \(-0.215692\pi\)
0.779070 + 0.626937i \(0.215692\pi\)
\(854\) 5.11866e30 0.848885
\(855\) 0 0
\(856\) 3.31367e30 0.536210
\(857\) −4.26068e30 −0.681053 −0.340526 0.940235i \(-0.610605\pi\)
−0.340526 + 0.940235i \(0.610605\pi\)
\(858\) 0 0
\(859\) −1.38058e30 −0.215344 −0.107672 0.994186i \(-0.534340\pi\)
−0.107672 + 0.994186i \(0.534340\pi\)
\(860\) −6.69420e29 −0.103149
\(861\) 0 0
\(862\) 6.33303e30 0.952325
\(863\) −6.77274e30 −1.00612 −0.503061 0.864251i \(-0.667793\pi\)
−0.503061 + 0.864251i \(0.667793\pi\)
\(864\) 0 0
\(865\) 1.84488e30 0.267484
\(866\) 8.36789e30 1.19861
\(867\) 0 0
\(868\) 1.55736e30 0.217737
\(869\) −1.40808e31 −1.94500
\(870\) 0 0
\(871\) −4.24544e30 −0.572442
\(872\) −6.10428e30 −0.813226
\(873\) 0 0
\(874\) 5.28636e29 0.0687522
\(875\) 5.81590e29 0.0747364
\(876\) 0 0
\(877\) 7.75502e30 0.972943 0.486471 0.873697i \(-0.338284\pi\)
0.486471 + 0.873697i \(0.338284\pi\)
\(878\) 1.50581e31 1.86672
\(879\) 0 0
\(880\) −7.85690e30 −0.951005
\(881\) −1.13987e31 −1.36335 −0.681677 0.731653i \(-0.738749\pi\)
−0.681677 + 0.731653i \(0.738749\pi\)
\(882\) 0 0
\(883\) −5.95571e28 −0.00695579 −0.00347789 0.999994i \(-0.501107\pi\)
−0.00347789 + 0.999994i \(0.501107\pi\)
\(884\) −1.89692e30 −0.218928
\(885\) 0 0
\(886\) −4.11903e29 −0.0464238
\(887\) −1.42633e31 −1.58863 −0.794315 0.607506i \(-0.792170\pi\)
−0.794315 + 0.607506i \(0.792170\pi\)
\(888\) 0 0
\(889\) 5.22770e29 0.0568646
\(890\) 5.28252e29 0.0567867
\(891\) 0 0
\(892\) 8.85904e29 0.0930156
\(893\) −6.08481e28 −0.00631404
\(894\) 0 0
\(895\) −1.12117e30 −0.113640
\(896\) −9.67945e30 −0.969653
\(897\) 0 0
\(898\) 2.02918e30 0.198572
\(899\) −1.02417e31 −0.990597
\(900\) 0 0
\(901\) 5.04791e30 0.476981
\(902\) 3.66967e31 3.42735
\(903\) 0 0
\(904\) 7.07056e30 0.645186
\(905\) −3.08077e30 −0.277875
\(906\) 0 0
\(907\) 4.56820e30 0.402595 0.201297 0.979530i \(-0.435484\pi\)
0.201297 + 0.979530i \(0.435484\pi\)
\(908\) −3.94880e30 −0.344005
\(909\) 0 0
\(910\) −5.76654e30 −0.490886
\(911\) 1.83971e31 1.54813 0.774065 0.633107i \(-0.218220\pi\)
0.774065 + 0.633107i \(0.218220\pi\)
\(912\) 0 0
\(913\) −3.25711e31 −2.67848
\(914\) 2.14410e31 1.74305
\(915\) 0 0
\(916\) −3.21317e30 −0.255288
\(917\) 1.19025e31 0.934893
\(918\) 0 0
\(919\) −6.31613e29 −0.0484884 −0.0242442 0.999706i \(-0.507718\pi\)
−0.0242442 + 0.999706i \(0.507718\pi\)
\(920\) 5.24586e30 0.398148
\(921\) 0 0
\(922\) 8.26945e30 0.613482
\(923\) −8.99187e30 −0.659527
\(924\) 0 0
\(925\) 3.87294e30 0.277685
\(926\) 4.76496e30 0.337788
\(927\) 0 0
\(928\) −1.70557e31 −1.18199
\(929\) −1.48189e31 −1.01543 −0.507717 0.861524i \(-0.669511\pi\)
−0.507717 + 0.861524i \(0.669511\pi\)
\(930\) 0 0
\(931\) −2.06805e29 −0.0138544
\(932\) −9.93996e30 −0.658442
\(933\) 0 0
\(934\) 2.24605e31 1.45471
\(935\) 5.82891e30 0.373306
\(936\) 0 0
\(937\) 1.44221e31 0.903154 0.451577 0.892232i \(-0.350862\pi\)
0.451577 + 0.892232i \(0.350862\pi\)
\(938\) −8.26461e30 −0.511791
\(939\) 0 0
\(940\) 4.11976e29 0.0249476
\(941\) 3.60517e30 0.215891 0.107945 0.994157i \(-0.465573\pi\)
0.107945 + 0.994157i \(0.465573\pi\)
\(942\) 0 0
\(943\) −3.63949e31 −2.13141
\(944\) 3.93016e31 2.27617
\(945\) 0 0
\(946\) −2.03955e31 −1.15526
\(947\) −9.57878e30 −0.536581 −0.268291 0.963338i \(-0.586459\pi\)
−0.268291 + 0.963338i \(0.586459\pi\)
\(948\) 0 0
\(949\) 7.15592e30 0.392076
\(950\) 2.00869e29 0.0108846
\(951\) 0 0
\(952\) 5.41235e30 0.286878
\(953\) −1.99764e31 −1.04723 −0.523614 0.851956i \(-0.675417\pi\)
−0.523614 + 0.851956i \(0.675417\pi\)
\(954\) 0 0
\(955\) −4.01551e30 −0.205923
\(956\) 1.18743e31 0.602282
\(957\) 0 0
\(958\) 3.40143e31 1.68781
\(959\) −5.39123e29 −0.0264602
\(960\) 0 0
\(961\) −1.22284e31 −0.587183
\(962\) −3.84007e31 −1.82390
\(963\) 0 0
\(964\) 1.55757e31 0.723833
\(965\) 8.86935e29 0.0407713
\(966\) 0 0
\(967\) −1.46532e30 −0.0659105 −0.0329552 0.999457i \(-0.510492\pi\)
−0.0329552 + 0.999457i \(0.510492\pi\)
\(968\) −3.06613e31 −1.36427
\(969\) 0 0
\(970\) −4.13373e30 −0.179986
\(971\) 2.87520e31 1.23841 0.619207 0.785228i \(-0.287454\pi\)
0.619207 + 0.785228i \(0.287454\pi\)
\(972\) 0 0
\(973\) −5.56683e30 −0.234651
\(974\) −3.68141e31 −1.53513
\(975\) 0 0
\(976\) 2.60591e31 1.06350
\(977\) −1.39361e31 −0.562662 −0.281331 0.959611i \(-0.590776\pi\)
−0.281331 + 0.959611i \(0.590776\pi\)
\(978\) 0 0
\(979\) 4.64398e30 0.183515
\(980\) 1.40019e30 0.0547409
\(981\) 0 0
\(982\) 2.82891e31 1.08255
\(983\) 1.40200e31 0.530805 0.265403 0.964138i \(-0.414495\pi\)
0.265403 + 0.964138i \(0.414495\pi\)
\(984\) 0 0
\(985\) 1.19713e31 0.443670
\(986\) 2.42848e31 0.890487
\(987\) 0 0
\(988\) −5.74677e29 −0.0206289
\(989\) 2.02278e31 0.718434
\(990\) 0 0
\(991\) −5.54876e31 −1.92940 −0.964701 0.263349i \(-0.915173\pi\)
−0.964701 + 0.263349i \(0.915173\pi\)
\(992\) 1.43169e31 0.492579
\(993\) 0 0
\(994\) −1.75045e31 −0.589649
\(995\) −1.02751e31 −0.342487
\(996\) 0 0
\(997\) −5.26811e31 −1.71932 −0.859659 0.510869i \(-0.829324\pi\)
−0.859659 + 0.510869i \(0.829324\pi\)
\(998\) −4.22709e31 −1.36512
\(999\) 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.1.4 4
3.2 odd 2 15.22.a.e.1.1 4
15.2 even 4 75.22.b.h.49.1 8
15.8 even 4 75.22.b.h.49.8 8
15.14 odd 2 75.22.a.h.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.22.a.e.1.1 4 3.2 odd 2
45.22.a.g.1.4 4 1.1 even 1 trivial
75.22.a.h.1.4 4 15.14 odd 2
75.22.b.h.49.1 8 15.2 even 4
75.22.b.h.49.8 8 15.8 even 4