Properties

Label 75.22.a.m.1.7
Level $75$
Weight $22$
Character 75.1
Self dual yes
Analytic conductor $209.608$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

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

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(209.608008215\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5 x^{9} - 15486550 x^{8} - 930225020 x^{7} + 80619202368510 x^{6} + \cdots + 47\!\cdots\!18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{14}\cdot 5^{24}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(1063.71\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+998.710 q^{2} +59049.0 q^{3} -1.09973e6 q^{4} +5.89728e7 q^{6} +1.20991e9 q^{7} -3.19276e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\) \(q+998.710 q^{2} +59049.0 q^{3} -1.09973e6 q^{4} +5.89728e7 q^{6} +1.20991e9 q^{7} -3.19276e9 q^{8} +3.48678e9 q^{9} +4.80573e10 q^{11} -6.49380e10 q^{12} -1.71413e11 q^{13} +1.20835e12 q^{14} -8.82334e11 q^{16} -1.14949e13 q^{17} +3.48228e12 q^{18} +7.38155e12 q^{19} +7.14442e13 q^{21} +4.79953e13 q^{22} -2.65776e14 q^{23} -1.88529e14 q^{24} -1.71192e14 q^{26} +2.05891e14 q^{27} -1.33058e15 q^{28} +3.36415e14 q^{29} +2.12517e15 q^{31} +5.81450e15 q^{32} +2.83774e15 q^{33} -1.14801e16 q^{34} -3.83453e15 q^{36} -5.00544e16 q^{37} +7.37203e15 q^{38} -1.01218e16 q^{39} -8.46448e16 q^{41} +7.13520e16 q^{42} -5.45785e16 q^{43} -5.28501e16 q^{44} -2.65433e17 q^{46} +2.16734e17 q^{47} -5.21009e16 q^{48} +9.05346e17 q^{49} -6.78761e17 q^{51} +1.88508e17 q^{52} -7.54978e17 q^{53} +2.05625e17 q^{54} -3.86296e18 q^{56} +4.35873e17 q^{57} +3.35980e17 q^{58} +6.39259e18 q^{59} +5.01150e18 q^{61} +2.12243e18 q^{62} +4.21871e18 q^{63} +7.65739e18 q^{64} +2.83407e18 q^{66} +1.05392e19 q^{67} +1.26413e19 q^{68} -1.56938e19 q^{69} -1.86086e19 q^{71} -1.11325e19 q^{72} -3.26224e18 q^{73} -4.99898e19 q^{74} -8.11772e18 q^{76} +5.81452e19 q^{77} -1.01087e19 q^{78} -1.64810e20 q^{79} +1.21577e19 q^{81} -8.45356e19 q^{82} -3.17359e18 q^{83} -7.85694e19 q^{84} -5.45080e19 q^{86} +1.98649e19 q^{87} -1.53435e20 q^{88} +8.25383e19 q^{89} -2.07395e20 q^{91} +2.92283e20 q^{92} +1.25489e20 q^{93} +2.16455e20 q^{94} +3.43341e20 q^{96} -1.37703e21 q^{97} +9.04178e20 q^{98} +1.67565e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 645 q^{2} + 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} - 115357290 q^{7} - 314142735 q^{8} + 34867844010 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 645 q^{2} + 590490 q^{3} + 10043205 q^{4} - 38086605 q^{6} - 115357290 q^{7} - 314142735 q^{8} + 34867844010 q^{9} + 5976221790 q^{11} + 593041212045 q^{12} - 842570747430 q^{13} + 1605059874570 q^{14} + 6061861547825 q^{16} - 12910340404230 q^{17} - 2248975938645 q^{18} - 76155422176280 q^{19} - 6811732617210 q^{21} - 461780887241010 q^{22} - 82640920915920 q^{23} - 18549814359015 q^{24} + 286555331159670 q^{26} + 20\!\cdots\!90 q^{27}+ \cdots + 20\!\cdots\!90 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 998.710 0.689643 0.344821 0.938668i \(-0.387939\pi\)
0.344821 + 0.938668i \(0.387939\pi\)
\(3\) 59049.0 0.577350
\(4\) −1.09973e6 −0.524393
\(5\) 0 0
\(6\) 5.89728e7 0.398165
\(7\) 1.20991e9 1.61892 0.809459 0.587176i \(-0.199760\pi\)
0.809459 + 0.587176i \(0.199760\pi\)
\(8\) −3.19276e9 −1.05129
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) 4.80573e10 0.558645 0.279323 0.960197i \(-0.409890\pi\)
0.279323 + 0.960197i \(0.409890\pi\)
\(12\) −6.49380e10 −0.302758
\(13\) −1.71413e11 −0.344857 −0.172429 0.985022i \(-0.555161\pi\)
−0.172429 + 0.985022i \(0.555161\pi\)
\(14\) 1.20835e12 1.11648
\(15\) 0 0
\(16\) −8.82334e11 −0.200619
\(17\) −1.14949e13 −1.38290 −0.691450 0.722424i \(-0.743028\pi\)
−0.691450 + 0.722424i \(0.743028\pi\)
\(18\) 3.48228e12 0.229881
\(19\) 7.38155e12 0.276207 0.138104 0.990418i \(-0.455899\pi\)
0.138104 + 0.990418i \(0.455899\pi\)
\(20\) 0 0
\(21\) 7.14442e13 0.934683
\(22\) 4.79953e13 0.385266
\(23\) −2.65776e14 −1.33775 −0.668874 0.743376i \(-0.733223\pi\)
−0.668874 + 0.743376i \(0.733223\pi\)
\(24\) −1.88529e14 −0.606961
\(25\) 0 0
\(26\) −1.71192e14 −0.237828
\(27\) 2.05891e14 0.192450
\(28\) −1.33058e15 −0.848949
\(29\) 3.36415e14 0.148489 0.0742447 0.997240i \(-0.476345\pi\)
0.0742447 + 0.997240i \(0.476345\pi\)
\(30\) 0 0
\(31\) 2.12517e15 0.465689 0.232845 0.972514i \(-0.425197\pi\)
0.232845 + 0.972514i \(0.425197\pi\)
\(32\) 5.81450e15 0.912931
\(33\) 2.83774e15 0.322534
\(34\) −1.14801e16 −0.953707
\(35\) 0 0
\(36\) −3.83453e15 −0.174798
\(37\) −5.00544e16 −1.71129 −0.855646 0.517562i \(-0.826840\pi\)
−0.855646 + 0.517562i \(0.826840\pi\)
\(38\) 7.37203e15 0.190484
\(39\) −1.01218e16 −0.199103
\(40\) 0 0
\(41\) −8.46448e16 −0.984850 −0.492425 0.870355i \(-0.663889\pi\)
−0.492425 + 0.870355i \(0.663889\pi\)
\(42\) 7.13520e16 0.644598
\(43\) −5.45785e16 −0.385126 −0.192563 0.981285i \(-0.561680\pi\)
−0.192563 + 0.981285i \(0.561680\pi\)
\(44\) −5.28501e16 −0.292950
\(45\) 0 0
\(46\) −2.65433e17 −0.922568
\(47\) 2.16734e17 0.601036 0.300518 0.953776i \(-0.402841\pi\)
0.300518 + 0.953776i \(0.402841\pi\)
\(48\) −5.21009e16 −0.115828
\(49\) 9.05346e17 1.62090
\(50\) 0 0
\(51\) −6.78761e17 −0.798418
\(52\) 1.88508e17 0.180841
\(53\) −7.54978e17 −0.592977 −0.296488 0.955036i \(-0.595816\pi\)
−0.296488 + 0.955036i \(0.595816\pi\)
\(54\) 2.05625e17 0.132722
\(55\) 0 0
\(56\) −3.86296e18 −1.70195
\(57\) 4.35873e17 0.159468
\(58\) 3.35980e17 0.102405
\(59\) 6.39259e18 1.62829 0.814143 0.580664i \(-0.197207\pi\)
0.814143 + 0.580664i \(0.197207\pi\)
\(60\) 0 0
\(61\) 5.01150e18 0.899507 0.449754 0.893153i \(-0.351512\pi\)
0.449754 + 0.893153i \(0.351512\pi\)
\(62\) 2.12243e18 0.321159
\(63\) 4.21871e18 0.539640
\(64\) 7.65739e18 0.830216
\(65\) 0 0
\(66\) 2.83407e18 0.222433
\(67\) 1.05392e19 0.706351 0.353175 0.935557i \(-0.385102\pi\)
0.353175 + 0.935557i \(0.385102\pi\)
\(68\) 1.26413e19 0.725183
\(69\) −1.56938e19 −0.772349
\(70\) 0 0
\(71\) −1.86086e19 −0.678425 −0.339212 0.940710i \(-0.610161\pi\)
−0.339212 + 0.940710i \(0.610161\pi\)
\(72\) −1.11325e19 −0.350429
\(73\) −3.26224e18 −0.0888437 −0.0444218 0.999013i \(-0.514145\pi\)
−0.0444218 + 0.999013i \(0.514145\pi\)
\(74\) −4.99898e19 −1.18018
\(75\) 0 0
\(76\) −8.11772e18 −0.144841
\(77\) 5.81452e19 0.904402
\(78\) −1.01087e19 −0.137310
\(79\) −1.64810e20 −1.95839 −0.979194 0.202926i \(-0.934955\pi\)
−0.979194 + 0.202926i \(0.934955\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) −8.45356e19 −0.679195
\(83\) −3.17359e18 −0.0224508 −0.0112254 0.999937i \(-0.503573\pi\)
−0.0112254 + 0.999937i \(0.503573\pi\)
\(84\) −7.85694e19 −0.490141
\(85\) 0 0
\(86\) −5.45080e19 −0.265599
\(87\) 1.98649e19 0.0857305
\(88\) −1.53435e20 −0.587296
\(89\) 8.25383e19 0.280582 0.140291 0.990110i \(-0.455196\pi\)
0.140291 + 0.990110i \(0.455196\pi\)
\(90\) 0 0
\(91\) −2.07395e20 −0.558296
\(92\) 2.92283e20 0.701505
\(93\) 1.25489e20 0.268866
\(94\) 2.16455e20 0.414500
\(95\) 0 0
\(96\) 3.43341e20 0.527081
\(97\) −1.37703e21 −1.89600 −0.948001 0.318268i \(-0.896899\pi\)
−0.948001 + 0.318268i \(0.896899\pi\)
\(98\) 9.04178e20 1.11784
\(99\) 1.67565e20 0.186215
\(100\) 0 0
\(101\) −1.10058e21 −0.991394 −0.495697 0.868495i \(-0.665087\pi\)
−0.495697 + 0.868495i \(0.665087\pi\)
\(102\) −6.77885e20 −0.550623
\(103\) −2.30652e21 −1.69109 −0.845546 0.533903i \(-0.820725\pi\)
−0.845546 + 0.533903i \(0.820725\pi\)
\(104\) 5.47281e20 0.362544
\(105\) 0 0
\(106\) −7.54003e20 −0.408942
\(107\) 3.50991e21 1.72491 0.862455 0.506134i \(-0.168926\pi\)
0.862455 + 0.506134i \(0.168926\pi\)
\(108\) −2.26425e20 −0.100919
\(109\) −1.42742e21 −0.577531 −0.288765 0.957400i \(-0.593245\pi\)
−0.288765 + 0.957400i \(0.593245\pi\)
\(110\) 0 0
\(111\) −2.95566e21 −0.988015
\(112\) −1.06755e21 −0.324787
\(113\) 8.18492e20 0.226825 0.113412 0.993548i \(-0.463822\pi\)
0.113412 + 0.993548i \(0.463822\pi\)
\(114\) 4.35311e20 0.109976
\(115\) 0 0
\(116\) −3.69966e20 −0.0778668
\(117\) −5.97681e20 −0.114952
\(118\) 6.38434e21 1.12294
\(119\) −1.39078e22 −2.23880
\(120\) 0 0
\(121\) −5.09075e21 −0.687915
\(122\) 5.00504e21 0.620339
\(123\) −4.99819e21 −0.568603
\(124\) −2.33712e21 −0.244204
\(125\) 0 0
\(126\) 4.21327e21 0.372159
\(127\) −1.55268e22 −1.26225 −0.631123 0.775683i \(-0.717406\pi\)
−0.631123 + 0.775683i \(0.717406\pi\)
\(128\) −4.54639e21 −0.340378
\(129\) −3.22280e21 −0.222352
\(130\) 0 0
\(131\) 1.80456e22 1.05931 0.529653 0.848214i \(-0.322322\pi\)
0.529653 + 0.848214i \(0.322322\pi\)
\(132\) −3.12075e21 −0.169135
\(133\) 8.93104e21 0.447157
\(134\) 1.05256e22 0.487130
\(135\) 0 0
\(136\) 3.67004e22 1.45382
\(137\) 3.76764e22 1.38199 0.690993 0.722861i \(-0.257173\pi\)
0.690993 + 0.722861i \(0.257173\pi\)
\(138\) −1.56736e22 −0.532645
\(139\) −2.07959e22 −0.655121 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(140\) 0 0
\(141\) 1.27979e22 0.347008
\(142\) −1.85846e22 −0.467871
\(143\) −8.23766e21 −0.192653
\(144\) −3.07651e21 −0.0668732
\(145\) 0 0
\(146\) −3.25803e21 −0.0612704
\(147\) 5.34598e22 0.935826
\(148\) 5.50464e22 0.897389
\(149\) −8.31054e22 −1.26233 −0.631166 0.775648i \(-0.717423\pi\)
−0.631166 + 0.775648i \(0.717423\pi\)
\(150\) 0 0
\(151\) −9.83660e22 −1.29893 −0.649467 0.760390i \(-0.725008\pi\)
−0.649467 + 0.760390i \(0.725008\pi\)
\(152\) −2.35675e22 −0.290373
\(153\) −4.00802e22 −0.460967
\(154\) 5.80702e22 0.623714
\(155\) 0 0
\(156\) 1.11312e22 0.104408
\(157\) −1.08861e23 −0.954827 −0.477414 0.878679i \(-0.658426\pi\)
−0.477414 + 0.878679i \(0.658426\pi\)
\(158\) −1.64597e23 −1.35059
\(159\) −4.45807e22 −0.342355
\(160\) 0 0
\(161\) −3.21567e23 −2.16570
\(162\) 1.21420e22 0.0766270
\(163\) 3.20827e23 1.89802 0.949010 0.315245i \(-0.102087\pi\)
0.949010 + 0.315245i \(0.102087\pi\)
\(164\) 9.30866e22 0.516448
\(165\) 0 0
\(166\) −3.16950e21 −0.0154830
\(167\) −2.16820e23 −0.994437 −0.497219 0.867625i \(-0.665645\pi\)
−0.497219 + 0.867625i \(0.665645\pi\)
\(168\) −2.28104e23 −0.982620
\(169\) −2.17682e23 −0.881074
\(170\) 0 0
\(171\) 2.57379e22 0.0920690
\(172\) 6.00216e22 0.201957
\(173\) −3.92562e23 −1.24287 −0.621433 0.783468i \(-0.713449\pi\)
−0.621433 + 0.783468i \(0.713449\pi\)
\(174\) 1.98393e22 0.0591234
\(175\) 0 0
\(176\) −4.24026e22 −0.112075
\(177\) 3.77476e23 0.940092
\(178\) 8.24318e22 0.193502
\(179\) 2.94913e23 0.652735 0.326368 0.945243i \(-0.394175\pi\)
0.326368 + 0.945243i \(0.394175\pi\)
\(180\) 0 0
\(181\) 5.84557e23 1.15134 0.575668 0.817684i \(-0.304742\pi\)
0.575668 + 0.817684i \(0.304742\pi\)
\(182\) −2.07128e23 −0.385025
\(183\) 2.95924e23 0.519331
\(184\) 8.48560e23 1.40636
\(185\) 0 0
\(186\) 1.25327e23 0.185421
\(187\) −5.52413e23 −0.772551
\(188\) −2.38350e23 −0.315179
\(189\) 2.49111e23 0.311561
\(190\) 0 0
\(191\) 1.13125e24 1.26680 0.633400 0.773824i \(-0.281659\pi\)
0.633400 + 0.773824i \(0.281659\pi\)
\(192\) 4.52161e23 0.479325
\(193\) −1.59566e24 −1.60173 −0.800866 0.598843i \(-0.795627\pi\)
−0.800866 + 0.598843i \(0.795627\pi\)
\(194\) −1.37525e24 −1.30756
\(195\) 0 0
\(196\) −9.95637e23 −0.849987
\(197\) 2.74899e23 0.222473 0.111236 0.993794i \(-0.464519\pi\)
0.111236 + 0.993794i \(0.464519\pi\)
\(198\) 1.67349e23 0.128422
\(199\) −1.12111e24 −0.816000 −0.408000 0.912982i \(-0.633774\pi\)
−0.408000 + 0.912982i \(0.633774\pi\)
\(200\) 0 0
\(201\) 6.22327e23 0.407812
\(202\) −1.09916e24 −0.683708
\(203\) 4.07033e23 0.240392
\(204\) 7.46455e23 0.418685
\(205\) 0 0
\(206\) −2.30355e24 −1.16625
\(207\) −9.26705e23 −0.445916
\(208\) 1.51244e23 0.0691850
\(209\) 3.54737e23 0.154302
\(210\) 0 0
\(211\) 1.53568e24 0.604415 0.302208 0.953242i \(-0.402276\pi\)
0.302208 + 0.953242i \(0.402276\pi\)
\(212\) 8.30273e23 0.310953
\(213\) −1.09882e24 −0.391689
\(214\) 3.50538e24 1.18957
\(215\) 0 0
\(216\) −6.57361e23 −0.202320
\(217\) 2.57127e24 0.753913
\(218\) −1.42558e24 −0.398290
\(219\) −1.92632e23 −0.0512939
\(220\) 0 0
\(221\) 1.97038e24 0.476903
\(222\) −2.95185e24 −0.681377
\(223\) −7.77339e24 −1.71163 −0.855814 0.517284i \(-0.826943\pi\)
−0.855814 + 0.517284i \(0.826943\pi\)
\(224\) 7.03505e24 1.47796
\(225\) 0 0
\(226\) 8.17436e23 0.156428
\(227\) −2.76707e23 −0.0505533 −0.0252766 0.999680i \(-0.508047\pi\)
−0.0252766 + 0.999680i \(0.508047\pi\)
\(228\) −4.79343e23 −0.0836240
\(229\) 5.04294e24 0.840255 0.420128 0.907465i \(-0.361985\pi\)
0.420128 + 0.907465i \(0.361985\pi\)
\(230\) 0 0
\(231\) 3.43342e24 0.522156
\(232\) −1.07409e24 −0.156105
\(233\) −3.34935e24 −0.465290 −0.232645 0.972562i \(-0.574738\pi\)
−0.232645 + 0.972562i \(0.574738\pi\)
\(234\) −5.96910e23 −0.0792761
\(235\) 0 0
\(236\) −7.03013e24 −0.853862
\(237\) −9.73186e24 −1.13068
\(238\) −1.38899e25 −1.54398
\(239\) −3.36899e24 −0.358362 −0.179181 0.983816i \(-0.557345\pi\)
−0.179181 + 0.983816i \(0.557345\pi\)
\(240\) 0 0
\(241\) 1.50937e25 1.47101 0.735505 0.677519i \(-0.236945\pi\)
0.735505 + 0.677519i \(0.236945\pi\)
\(242\) −5.08418e24 −0.474416
\(243\) 7.17898e23 0.0641500
\(244\) −5.51131e24 −0.471695
\(245\) 0 0
\(246\) −4.99174e24 −0.392133
\(247\) −1.26530e24 −0.0952520
\(248\) −6.78516e24 −0.489573
\(249\) −1.87398e23 −0.0129620
\(250\) 0 0
\(251\) −3.55363e24 −0.225994 −0.112997 0.993595i \(-0.536045\pi\)
−0.112997 + 0.993595i \(0.536045\pi\)
\(252\) −4.63945e24 −0.282983
\(253\) −1.27725e25 −0.747326
\(254\) −1.55068e25 −0.870499
\(255\) 0 0
\(256\) −2.05992e25 −1.06496
\(257\) −2.83931e25 −1.40901 −0.704506 0.709698i \(-0.748831\pi\)
−0.704506 + 0.709698i \(0.748831\pi\)
\(258\) −3.21864e24 −0.153344
\(259\) −6.05615e25 −2.77044
\(260\) 0 0
\(261\) 1.17300e24 0.0494965
\(262\) 1.80223e25 0.730543
\(263\) −3.64456e25 −1.41941 −0.709707 0.704497i \(-0.751173\pi\)
−0.709707 + 0.704497i \(0.751173\pi\)
\(264\) −9.06020e24 −0.339076
\(265\) 0 0
\(266\) 8.91952e24 0.308379
\(267\) 4.87380e24 0.161994
\(268\) −1.15902e25 −0.370405
\(269\) −1.65928e25 −0.509943 −0.254972 0.966949i \(-0.582066\pi\)
−0.254972 + 0.966949i \(0.582066\pi\)
\(270\) 0 0
\(271\) −2.72800e25 −0.775652 −0.387826 0.921733i \(-0.626774\pi\)
−0.387826 + 0.921733i \(0.626774\pi\)
\(272\) 1.01423e25 0.277437
\(273\) −1.22465e25 −0.322332
\(274\) 3.76278e25 0.953077
\(275\) 0 0
\(276\) 1.72590e25 0.405014
\(277\) 1.80011e25 0.406688 0.203344 0.979107i \(-0.434819\pi\)
0.203344 + 0.979107i \(0.434819\pi\)
\(278\) −2.07690e25 −0.451800
\(279\) 7.41001e24 0.155230
\(280\) 0 0
\(281\) −2.41233e25 −0.468835 −0.234418 0.972136i \(-0.575318\pi\)
−0.234418 + 0.972136i \(0.575318\pi\)
\(282\) 1.27814e25 0.239312
\(283\) −2.76536e25 −0.498878 −0.249439 0.968390i \(-0.580246\pi\)
−0.249439 + 0.968390i \(0.580246\pi\)
\(284\) 2.04645e25 0.355761
\(285\) 0 0
\(286\) −8.22703e24 −0.132862
\(287\) −1.02413e26 −1.59439
\(288\) 2.02739e25 0.304310
\(289\) 6.30404e25 0.912414
\(290\) 0 0
\(291\) −8.13120e25 −1.09466
\(292\) 3.58759e24 0.0465890
\(293\) −8.83764e25 −1.10720 −0.553600 0.832783i \(-0.686746\pi\)
−0.553600 + 0.832783i \(0.686746\pi\)
\(294\) 5.33908e25 0.645386
\(295\) 0 0
\(296\) 1.59811e26 1.79906
\(297\) 9.89457e24 0.107511
\(298\) −8.29982e25 −0.870558
\(299\) 4.55576e25 0.461332
\(300\) 0 0
\(301\) −6.60352e25 −0.623487
\(302\) −9.82390e25 −0.895800
\(303\) −6.49880e25 −0.572382
\(304\) −6.51299e24 −0.0554125
\(305\) 0 0
\(306\) −4.00285e25 −0.317902
\(307\) −7.20639e25 −0.553050 −0.276525 0.961007i \(-0.589183\pi\)
−0.276525 + 0.961007i \(0.589183\pi\)
\(308\) −6.39441e25 −0.474262
\(309\) −1.36198e26 −0.976352
\(310\) 0 0
\(311\) −1.91089e26 −1.28012 −0.640062 0.768323i \(-0.721091\pi\)
−0.640062 + 0.768323i \(0.721091\pi\)
\(312\) 3.23164e25 0.209315
\(313\) −1.41091e25 −0.0883655 −0.0441827 0.999023i \(-0.514068\pi\)
−0.0441827 + 0.999023i \(0.514068\pi\)
\(314\) −1.08720e26 −0.658490
\(315\) 0 0
\(316\) 1.81247e26 1.02696
\(317\) 2.75275e26 1.50884 0.754421 0.656391i \(-0.227918\pi\)
0.754421 + 0.656391i \(0.227918\pi\)
\(318\) −4.45232e25 −0.236103
\(319\) 1.61672e25 0.0829530
\(320\) 0 0
\(321\) 2.07257e26 0.995877
\(322\) −3.21152e26 −1.49356
\(323\) −8.48501e25 −0.381967
\(324\) −1.33702e25 −0.0582659
\(325\) 0 0
\(326\) 3.20413e26 1.30896
\(327\) −8.42880e25 −0.333437
\(328\) 2.70250e26 1.03536
\(329\) 2.62230e26 0.973028
\(330\) 0 0
\(331\) 2.91314e26 1.01430 0.507151 0.861857i \(-0.330699\pi\)
0.507151 + 0.861857i \(0.330699\pi\)
\(332\) 3.49010e24 0.0117730
\(333\) −1.74529e26 −0.570430
\(334\) −2.16541e26 −0.685806
\(335\) 0 0
\(336\) −6.30376e25 −0.187516
\(337\) 1.20851e26 0.348446 0.174223 0.984706i \(-0.444259\pi\)
0.174223 + 0.984706i \(0.444259\pi\)
\(338\) −2.17401e26 −0.607626
\(339\) 4.83311e25 0.130957
\(340\) 0 0
\(341\) 1.02130e26 0.260155
\(342\) 2.57047e25 0.0634948
\(343\) 4.19599e26 1.00518
\(344\) 1.74256e26 0.404877
\(345\) 0 0
\(346\) −3.92055e26 −0.857133
\(347\) −5.90558e25 −0.125257 −0.0626286 0.998037i \(-0.519948\pi\)
−0.0626286 + 0.998037i \(0.519948\pi\)
\(348\) −2.18461e25 −0.0449564
\(349\) 5.09244e26 1.01686 0.508428 0.861105i \(-0.330227\pi\)
0.508428 + 0.861105i \(0.330227\pi\)
\(350\) 0 0
\(351\) −3.52925e25 −0.0663678
\(352\) 2.79429e26 0.510005
\(353\) 5.10387e26 0.904201 0.452100 0.891967i \(-0.350675\pi\)
0.452100 + 0.891967i \(0.350675\pi\)
\(354\) 3.76989e26 0.648327
\(355\) 0 0
\(356\) −9.07700e25 −0.147135
\(357\) −8.21243e26 −1.29257
\(358\) 2.94533e26 0.450154
\(359\) 4.81726e26 0.715004 0.357502 0.933912i \(-0.383628\pi\)
0.357502 + 0.933912i \(0.383628\pi\)
\(360\) 0 0
\(361\) −6.59722e26 −0.923710
\(362\) 5.83803e26 0.794010
\(363\) −3.00603e26 −0.397168
\(364\) 2.28079e26 0.292766
\(365\) 0 0
\(366\) 2.95542e26 0.358153
\(367\) 1.68764e27 1.98740 0.993701 0.112062i \(-0.0357456\pi\)
0.993701 + 0.112062i \(0.0357456\pi\)
\(368\) 2.34504e26 0.268378
\(369\) −2.95138e26 −0.328283
\(370\) 0 0
\(371\) −9.13458e26 −0.959981
\(372\) −1.38004e26 −0.140991
\(373\) −7.80302e25 −0.0775033 −0.0387516 0.999249i \(-0.512338\pi\)
−0.0387516 + 0.999249i \(0.512338\pi\)
\(374\) −5.51700e26 −0.532784
\(375\) 0 0
\(376\) −6.91980e26 −0.631861
\(377\) −5.76659e25 −0.0512076
\(378\) 2.48789e26 0.214866
\(379\) −1.17814e27 −0.989656 −0.494828 0.868991i \(-0.664769\pi\)
−0.494828 + 0.868991i \(0.664769\pi\)
\(380\) 0 0
\(381\) −9.16844e26 −0.728758
\(382\) 1.12979e27 0.873640
\(383\) 1.32936e27 1.00013 0.500065 0.865988i \(-0.333309\pi\)
0.500065 + 0.865988i \(0.333309\pi\)
\(384\) −2.68460e26 −0.196518
\(385\) 0 0
\(386\) −1.59361e27 −1.10462
\(387\) −1.90303e26 −0.128375
\(388\) 1.51436e27 0.994250
\(389\) −3.12812e27 −1.99900 −0.999501 0.0315999i \(-0.989940\pi\)
−0.999501 + 0.0315999i \(0.989940\pi\)
\(390\) 0 0
\(391\) 3.05507e27 1.84997
\(392\) −2.89055e27 −1.70403
\(393\) 1.06557e27 0.611591
\(394\) 2.74544e26 0.153427
\(395\) 0 0
\(396\) −1.84277e26 −0.0976499
\(397\) 1.58428e27 0.817584 0.408792 0.912628i \(-0.365950\pi\)
0.408792 + 0.912628i \(0.365950\pi\)
\(398\) −1.11966e27 −0.562748
\(399\) 5.27369e26 0.258166
\(400\) 0 0
\(401\) 3.23835e27 1.50421 0.752103 0.659045i \(-0.229039\pi\)
0.752103 + 0.659045i \(0.229039\pi\)
\(402\) 6.21524e26 0.281245
\(403\) −3.64282e26 −0.160596
\(404\) 1.21034e27 0.519880
\(405\) 0 0
\(406\) 4.06507e26 0.165785
\(407\) −2.40548e27 −0.956005
\(408\) 2.16712e27 0.839366
\(409\) −1.32012e27 −0.498333 −0.249166 0.968461i \(-0.580157\pi\)
−0.249166 + 0.968461i \(0.580157\pi\)
\(410\) 0 0
\(411\) 2.22476e27 0.797890
\(412\) 2.53656e27 0.886796
\(413\) 7.73449e27 2.63606
\(414\) −9.25509e26 −0.307523
\(415\) 0 0
\(416\) −9.96683e26 −0.314831
\(417\) −1.22798e27 −0.378234
\(418\) 3.54280e26 0.106413
\(419\) −4.15945e26 −0.121840 −0.0609198 0.998143i \(-0.519403\pi\)
−0.0609198 + 0.998143i \(0.519403\pi\)
\(420\) 0 0
\(421\) −1.74327e27 −0.485740 −0.242870 0.970059i \(-0.578089\pi\)
−0.242870 + 0.970059i \(0.578089\pi\)
\(422\) 1.53370e27 0.416831
\(423\) 7.55706e26 0.200345
\(424\) 2.41046e27 0.623388
\(425\) 0 0
\(426\) −1.09740e27 −0.270125
\(427\) 6.06349e27 1.45623
\(428\) −3.85996e27 −0.904530
\(429\) −4.86425e26 −0.111228
\(430\) 0 0
\(431\) −9.48313e26 −0.206510 −0.103255 0.994655i \(-0.532926\pi\)
−0.103255 + 0.994655i \(0.532926\pi\)
\(432\) −1.81665e26 −0.0386092
\(433\) −1.08053e26 −0.0224137 −0.0112069 0.999937i \(-0.503567\pi\)
−0.0112069 + 0.999937i \(0.503567\pi\)
\(434\) 2.56796e27 0.519931
\(435\) 0 0
\(436\) 1.56978e27 0.302853
\(437\) −1.96184e27 −0.369495
\(438\) −1.92384e26 −0.0353745
\(439\) −5.87233e27 −1.05422 −0.527112 0.849796i \(-0.676725\pi\)
−0.527112 + 0.849796i \(0.676725\pi\)
\(440\) 0 0
\(441\) 3.15675e27 0.540299
\(442\) 1.96783e27 0.328893
\(443\) −1.92716e27 −0.314541 −0.157271 0.987556i \(-0.550270\pi\)
−0.157271 + 0.987556i \(0.550270\pi\)
\(444\) 3.25043e27 0.518108
\(445\) 0 0
\(446\) −7.76335e27 −1.18041
\(447\) −4.90729e27 −0.728808
\(448\) 9.26478e27 1.34405
\(449\) 9.45588e27 1.34003 0.670016 0.742346i \(-0.266287\pi\)
0.670016 + 0.742346i \(0.266287\pi\)
\(450\) 0 0
\(451\) −4.06780e27 −0.550182
\(452\) −9.00121e26 −0.118945
\(453\) −5.80841e27 −0.749940
\(454\) −2.76350e26 −0.0348637
\(455\) 0 0
\(456\) −1.39164e27 −0.167647
\(457\) 2.59303e27 0.305273 0.152636 0.988282i \(-0.451224\pi\)
0.152636 + 0.988282i \(0.451224\pi\)
\(458\) 5.03643e27 0.579476
\(459\) −2.36669e27 −0.266139
\(460\) 0 0
\(461\) 6.56399e27 0.705194 0.352597 0.935775i \(-0.385299\pi\)
0.352597 + 0.935775i \(0.385299\pi\)
\(462\) 3.42899e27 0.360101
\(463\) 9.11873e27 0.936125 0.468063 0.883695i \(-0.344952\pi\)
0.468063 + 0.883695i \(0.344952\pi\)
\(464\) −2.96830e26 −0.0297899
\(465\) 0 0
\(466\) −3.34503e27 −0.320884
\(467\) 1.69430e28 1.58914 0.794572 0.607170i \(-0.207695\pi\)
0.794572 + 0.607170i \(0.207695\pi\)
\(468\) 6.57288e26 0.0602802
\(469\) 1.27515e28 1.14352
\(470\) 0 0
\(471\) −6.42811e27 −0.551270
\(472\) −2.04100e28 −1.71180
\(473\) −2.62289e27 −0.215149
\(474\) −9.71930e27 −0.779763
\(475\) 0 0
\(476\) 1.52949e28 1.17401
\(477\) −2.63244e27 −0.197659
\(478\) −3.36465e27 −0.247142
\(479\) −9.99660e27 −0.718339 −0.359169 0.933272i \(-0.616940\pi\)
−0.359169 + 0.933272i \(0.616940\pi\)
\(480\) 0 0
\(481\) 8.57998e27 0.590151
\(482\) 1.50742e28 1.01447
\(483\) −1.89882e28 −1.25037
\(484\) 5.59845e27 0.360738
\(485\) 0 0
\(486\) 7.16972e26 0.0442406
\(487\) −1.13449e28 −0.685088 −0.342544 0.939502i \(-0.611289\pi\)
−0.342544 + 0.939502i \(0.611289\pi\)
\(488\) −1.60005e28 −0.945640
\(489\) 1.89445e28 1.09582
\(490\) 0 0
\(491\) 7.77588e27 0.430917 0.215459 0.976513i \(-0.430875\pi\)
0.215459 + 0.976513i \(0.430875\pi\)
\(492\) 5.49667e27 0.298171
\(493\) −3.86705e27 −0.205346
\(494\) −1.26366e27 −0.0656898
\(495\) 0 0
\(496\) −1.87511e27 −0.0934263
\(497\) −2.25148e28 −1.09832
\(498\) −1.87156e26 −0.00893913
\(499\) −4.94546e27 −0.231287 −0.115643 0.993291i \(-0.536893\pi\)
−0.115643 + 0.993291i \(0.536893\pi\)
\(500\) 0 0
\(501\) −1.28030e28 −0.574139
\(502\) −3.54904e27 −0.155855
\(503\) 1.49983e27 0.0645028 0.0322514 0.999480i \(-0.489732\pi\)
0.0322514 + 0.999480i \(0.489732\pi\)
\(504\) −1.34693e28 −0.567316
\(505\) 0 0
\(506\) −1.27560e28 −0.515388
\(507\) −1.28539e28 −0.508688
\(508\) 1.70754e28 0.661913
\(509\) 3.17947e28 1.20731 0.603653 0.797247i \(-0.293711\pi\)
0.603653 + 0.797247i \(0.293711\pi\)
\(510\) 0 0
\(511\) −3.94704e27 −0.143831
\(512\) −1.10382e28 −0.394060
\(513\) 1.51980e27 0.0531561
\(514\) −2.83564e28 −0.971715
\(515\) 0 0
\(516\) 3.54422e27 0.116600
\(517\) 1.04157e28 0.335766
\(518\) −6.04833e28 −1.91062
\(519\) −2.31804e28 −0.717568
\(520\) 0 0
\(521\) 3.91614e28 1.16429 0.582147 0.813084i \(-0.302213\pi\)
0.582147 + 0.813084i \(0.302213\pi\)
\(522\) 1.17149e27 0.0341349
\(523\) −5.01202e28 −1.43135 −0.715674 0.698435i \(-0.753880\pi\)
−0.715674 + 0.698435i \(0.753880\pi\)
\(524\) −1.98453e28 −0.555493
\(525\) 0 0
\(526\) −3.63985e28 −0.978889
\(527\) −2.44286e28 −0.644002
\(528\) −2.50383e27 −0.0647066
\(529\) 3.11655e28 0.789568
\(530\) 0 0
\(531\) 2.22896e28 0.542762
\(532\) −9.82175e27 −0.234486
\(533\) 1.45092e28 0.339632
\(534\) 4.86751e27 0.111718
\(535\) 0 0
\(536\) −3.36490e28 −0.742577
\(537\) 1.74143e28 0.376857
\(538\) −1.65714e28 −0.351679
\(539\) 4.35085e28 0.905507
\(540\) 0 0
\(541\) −3.55350e27 −0.0711352 −0.0355676 0.999367i \(-0.511324\pi\)
−0.0355676 + 0.999367i \(0.511324\pi\)
\(542\) −2.72448e28 −0.534923
\(543\) 3.45175e28 0.664724
\(544\) −6.68370e28 −1.26249
\(545\) 0 0
\(546\) −1.22307e28 −0.222294
\(547\) −3.63220e27 −0.0647594 −0.0323797 0.999476i \(-0.510309\pi\)
−0.0323797 + 0.999476i \(0.510309\pi\)
\(548\) −4.14339e28 −0.724704
\(549\) 1.74740e28 0.299836
\(550\) 0 0
\(551\) 2.48326e27 0.0410139
\(552\) 5.01066e28 0.811960
\(553\) −1.99406e29 −3.17047
\(554\) 1.79779e28 0.280470
\(555\) 0 0
\(556\) 2.28699e28 0.343541
\(557\) −7.81368e27 −0.115180 −0.0575899 0.998340i \(-0.518342\pi\)
−0.0575899 + 0.998340i \(0.518342\pi\)
\(558\) 7.40045e27 0.107053
\(559\) 9.35547e27 0.132813
\(560\) 0 0
\(561\) −3.26194e28 −0.446032
\(562\) −2.40922e28 −0.323329
\(563\) 4.73621e28 0.623868 0.311934 0.950104i \(-0.399023\pi\)
0.311934 + 0.950104i \(0.399023\pi\)
\(564\) −1.40743e28 −0.181969
\(565\) 0 0
\(566\) −2.76180e28 −0.344048
\(567\) 1.47097e28 0.179880
\(568\) 5.94129e28 0.713219
\(569\) 3.94549e28 0.464967 0.232484 0.972600i \(-0.425315\pi\)
0.232484 + 0.972600i \(0.425315\pi\)
\(570\) 0 0
\(571\) 9.57091e28 1.08711 0.543556 0.839373i \(-0.317078\pi\)
0.543556 + 0.839373i \(0.317078\pi\)
\(572\) 9.05921e27 0.101026
\(573\) 6.67992e28 0.731387
\(574\) −1.02281e29 −1.09956
\(575\) 0 0
\(576\) 2.66997e28 0.276739
\(577\) 2.32691e28 0.236828 0.118414 0.992964i \(-0.462219\pi\)
0.118414 + 0.992964i \(0.462219\pi\)
\(578\) 6.29591e28 0.629240
\(579\) −9.42224e28 −0.924761
\(580\) 0 0
\(581\) −3.83978e27 −0.0363460
\(582\) −8.12070e28 −0.754922
\(583\) −3.62822e28 −0.331264
\(584\) 1.04156e28 0.0934002
\(585\) 0 0
\(586\) −8.82623e28 −0.763573
\(587\) 4.21414e28 0.358104 0.179052 0.983840i \(-0.442697\pi\)
0.179052 + 0.983840i \(0.442697\pi\)
\(588\) −5.87914e28 −0.490740
\(589\) 1.56871e28 0.128627
\(590\) 0 0
\(591\) 1.62325e28 0.128445
\(592\) 4.41647e28 0.343318
\(593\) −6.60899e28 −0.504732 −0.252366 0.967632i \(-0.581209\pi\)
−0.252366 + 0.967632i \(0.581209\pi\)
\(594\) 9.88180e27 0.0741444
\(595\) 0 0
\(596\) 9.13936e28 0.661958
\(597\) −6.62003e28 −0.471118
\(598\) 4.54988e28 0.318154
\(599\) −1.39484e29 −0.958394 −0.479197 0.877707i \(-0.659072\pi\)
−0.479197 + 0.877707i \(0.659072\pi\)
\(600\) 0 0
\(601\) 1.18282e29 0.784761 0.392380 0.919803i \(-0.371652\pi\)
0.392380 + 0.919803i \(0.371652\pi\)
\(602\) −6.59500e28 −0.429983
\(603\) 3.67478e28 0.235450
\(604\) 1.08176e29 0.681151
\(605\) 0 0
\(606\) −6.49042e28 −0.394739
\(607\) −6.52603e28 −0.390093 −0.195046 0.980794i \(-0.562486\pi\)
−0.195046 + 0.980794i \(0.562486\pi\)
\(608\) 4.29201e28 0.252158
\(609\) 2.40349e28 0.138791
\(610\) 0 0
\(611\) −3.71511e28 −0.207271
\(612\) 4.40774e28 0.241728
\(613\) −5.95584e28 −0.321076 −0.160538 0.987030i \(-0.551323\pi\)
−0.160538 + 0.987030i \(0.551323\pi\)
\(614\) −7.19709e28 −0.381407
\(615\) 0 0
\(616\) −1.85644e29 −0.950785
\(617\) 4.33753e28 0.218398 0.109199 0.994020i \(-0.465171\pi\)
0.109199 + 0.994020i \(0.465171\pi\)
\(618\) −1.36022e29 −0.673334
\(619\) −2.80609e29 −1.36568 −0.682842 0.730566i \(-0.739256\pi\)
−0.682842 + 0.730566i \(0.739256\pi\)
\(620\) 0 0
\(621\) −5.47210e28 −0.257450
\(622\) −1.90843e29 −0.882828
\(623\) 9.98643e28 0.454240
\(624\) 8.93079e27 0.0399440
\(625\) 0 0
\(626\) −1.40909e28 −0.0609406
\(627\) 2.09469e28 0.0890862
\(628\) 1.19718e29 0.500704
\(629\) 5.75369e29 2.36655
\(630\) 0 0
\(631\) −1.91070e29 −0.760124 −0.380062 0.924961i \(-0.624097\pi\)
−0.380062 + 0.924961i \(0.624097\pi\)
\(632\) 5.26198e29 2.05883
\(633\) 9.06805e28 0.348959
\(634\) 2.74919e29 1.04056
\(635\) 0 0
\(636\) 4.90268e28 0.179529
\(637\) −1.55188e29 −0.558978
\(638\) 1.61463e28 0.0572079
\(639\) −6.48843e28 −0.226142
\(640\) 0 0
\(641\) 3.08795e28 0.104150 0.0520752 0.998643i \(-0.483416\pi\)
0.0520752 + 0.998643i \(0.483416\pi\)
\(642\) 2.06989e29 0.686800
\(643\) 3.31619e29 1.08249 0.541245 0.840865i \(-0.317953\pi\)
0.541245 + 0.840865i \(0.317953\pi\)
\(644\) 3.53637e29 1.13568
\(645\) 0 0
\(646\) −8.47406e28 −0.263421
\(647\) 1.60958e29 0.492287 0.246143 0.969233i \(-0.420837\pi\)
0.246143 + 0.969233i \(0.420837\pi\)
\(648\) −3.88165e28 −0.116810
\(649\) 3.07211e29 0.909635
\(650\) 0 0
\(651\) 1.51831e29 0.435272
\(652\) −3.52823e29 −0.995308
\(653\) 4.10395e29 1.13924 0.569618 0.821910i \(-0.307091\pi\)
0.569618 + 0.821910i \(0.307091\pi\)
\(654\) −8.41792e28 −0.229953
\(655\) 0 0
\(656\) 7.46850e28 0.197580
\(657\) −1.13747e28 −0.0296146
\(658\) 2.61891e29 0.671042
\(659\) 4.59733e29 1.15933 0.579667 0.814853i \(-0.303183\pi\)
0.579667 + 0.814853i \(0.303183\pi\)
\(660\) 0 0
\(661\) 3.47451e29 0.848747 0.424374 0.905487i \(-0.360494\pi\)
0.424374 + 0.905487i \(0.360494\pi\)
\(662\) 2.90938e29 0.699506
\(663\) 1.16349e29 0.275340
\(664\) 1.01325e28 0.0236022
\(665\) 0 0
\(666\) −1.74304e29 −0.393393
\(667\) −8.94110e28 −0.198641
\(668\) 2.38444e29 0.521476
\(669\) −4.59011e29 −0.988208
\(670\) 0 0
\(671\) 2.40839e29 0.502506
\(672\) 4.15413e29 0.853301
\(673\) 5.46802e29 1.10579 0.552894 0.833252i \(-0.313524\pi\)
0.552894 + 0.833252i \(0.313524\pi\)
\(674\) 1.20695e29 0.240304
\(675\) 0 0
\(676\) 2.39392e29 0.462029
\(677\) −3.19471e29 −0.607086 −0.303543 0.952818i \(-0.598170\pi\)
−0.303543 + 0.952818i \(0.598170\pi\)
\(678\) 4.82688e28 0.0903139
\(679\) −1.66608e30 −3.06947
\(680\) 0 0
\(681\) −1.63393e28 −0.0291869
\(682\) 1.01998e29 0.179414
\(683\) −2.97244e29 −0.514869 −0.257434 0.966296i \(-0.582877\pi\)
−0.257434 + 0.966296i \(0.582877\pi\)
\(684\) −2.83048e28 −0.0482803
\(685\) 0 0
\(686\) 4.19057e29 0.693218
\(687\) 2.97780e29 0.485121
\(688\) 4.81564e28 0.0772637
\(689\) 1.29413e29 0.204492
\(690\) 0 0
\(691\) 9.77979e28 0.149903 0.0749514 0.997187i \(-0.476120\pi\)
0.0749514 + 0.997187i \(0.476120\pi\)
\(692\) 4.31713e29 0.651749
\(693\) 2.02740e29 0.301467
\(694\) −5.89796e28 −0.0863828
\(695\) 0 0
\(696\) −6.34239e28 −0.0901273
\(697\) 9.72983e29 1.36195
\(698\) 5.08587e29 0.701267
\(699\) −1.97776e29 −0.268635
\(700\) 0 0
\(701\) 7.22259e29 0.952037 0.476019 0.879435i \(-0.342079\pi\)
0.476019 + 0.879435i \(0.342079\pi\)
\(702\) −3.52469e28 −0.0457701
\(703\) −3.69479e29 −0.472671
\(704\) 3.67993e29 0.463796
\(705\) 0 0
\(706\) 5.09728e29 0.623575
\(707\) −1.33160e30 −1.60499
\(708\) −4.15122e29 −0.492977
\(709\) 4.17404e29 0.488395 0.244197 0.969726i \(-0.421475\pi\)
0.244197 + 0.969726i \(0.421475\pi\)
\(710\) 0 0
\(711\) −5.74657e29 −0.652796
\(712\) −2.63525e29 −0.294973
\(713\) −5.64820e29 −0.622975
\(714\) −8.20183e29 −0.891414
\(715\) 0 0
\(716\) −3.24325e29 −0.342290
\(717\) −1.98936e29 −0.206900
\(718\) 4.81105e29 0.493098
\(719\) −7.89811e29 −0.797756 −0.398878 0.917004i \(-0.630600\pi\)
−0.398878 + 0.917004i \(0.630600\pi\)
\(720\) 0 0
\(721\) −2.79070e30 −2.73774
\(722\) −6.58871e29 −0.637030
\(723\) 8.91266e29 0.849288
\(724\) −6.42856e29 −0.603752
\(725\) 0 0
\(726\) −3.00216e29 −0.273904
\(727\) 4.46262e29 0.401309 0.200654 0.979662i \(-0.435693\pi\)
0.200654 + 0.979662i \(0.435693\pi\)
\(728\) 6.62163e29 0.586929
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) 6.27373e29 0.532590
\(732\) −3.25437e29 −0.272333
\(733\) −4.52156e29 −0.372990 −0.186495 0.982456i \(-0.559713\pi\)
−0.186495 + 0.982456i \(0.559713\pi\)
\(734\) 1.68546e30 1.37060
\(735\) 0 0
\(736\) −1.54536e30 −1.22127
\(737\) 5.06484e29 0.394600
\(738\) −2.94757e29 −0.226398
\(739\) −5.68446e29 −0.430450 −0.215225 0.976564i \(-0.569049\pi\)
−0.215225 + 0.976564i \(0.569049\pi\)
\(740\) 0 0
\(741\) −7.47144e28 −0.0549937
\(742\) −9.12279e29 −0.662044
\(743\) −9.68858e29 −0.693231 −0.346615 0.938007i \(-0.612669\pi\)
−0.346615 + 0.938007i \(0.612669\pi\)
\(744\) −4.00657e29 −0.282655
\(745\) 0 0
\(746\) −7.79295e28 −0.0534496
\(747\) −1.10656e28 −0.00748359
\(748\) 6.07506e29 0.405120
\(749\) 4.24669e30 2.79249
\(750\) 0 0
\(751\) 1.04636e30 0.669052 0.334526 0.942387i \(-0.391424\pi\)
0.334526 + 0.942387i \(0.391424\pi\)
\(752\) −1.91232e29 −0.120579
\(753\) −2.09838e29 −0.130478
\(754\) −5.75915e28 −0.0353150
\(755\) 0 0
\(756\) −2.73955e29 −0.163380
\(757\) 2.74898e30 1.61683 0.808415 0.588612i \(-0.200326\pi\)
0.808415 + 0.588612i \(0.200326\pi\)
\(758\) −1.17662e30 −0.682509
\(759\) −7.54203e29 −0.431469
\(760\) 0 0
\(761\) −1.54630e30 −0.860508 −0.430254 0.902708i \(-0.641576\pi\)
−0.430254 + 0.902708i \(0.641576\pi\)
\(762\) −9.15661e29 −0.502583
\(763\) −1.72706e30 −0.934975
\(764\) −1.24407e30 −0.664301
\(765\) 0 0
\(766\) 1.32765e30 0.689733
\(767\) −1.09577e30 −0.561526
\(768\) −1.21636e30 −0.614852
\(769\) −1.67246e30 −0.833928 −0.416964 0.908923i \(-0.636906\pi\)
−0.416964 + 0.908923i \(0.636906\pi\)
\(770\) 0 0
\(771\) −1.67658e30 −0.813493
\(772\) 1.75480e30 0.839937
\(773\) 1.23290e30 0.582159 0.291080 0.956699i \(-0.405986\pi\)
0.291080 + 0.956699i \(0.405986\pi\)
\(774\) −1.90058e29 −0.0885330
\(775\) 0 0
\(776\) 4.39651e30 1.99324
\(777\) −3.57610e30 −1.59952
\(778\) −3.12409e30 −1.37860
\(779\) −6.24810e29 −0.272023
\(780\) 0 0
\(781\) −8.94281e29 −0.378999
\(782\) 3.05113e30 1.27582
\(783\) 6.92648e28 0.0285768
\(784\) −7.98817e29 −0.325184
\(785\) 0 0
\(786\) 1.06420e30 0.421779
\(787\) −2.10384e30 −0.822769 −0.411385 0.911462i \(-0.634955\pi\)
−0.411385 + 0.911462i \(0.634955\pi\)
\(788\) −3.02315e29 −0.116663
\(789\) −2.15207e30 −0.819499
\(790\) 0 0
\(791\) 9.90305e29 0.367211
\(792\) −5.34996e29 −0.195765
\(793\) −8.59038e29 −0.310201
\(794\) 1.58224e30 0.563841
\(795\) 0 0
\(796\) 1.23292e30 0.427904
\(797\) −4.27207e30 −1.46327 −0.731637 0.681694i \(-0.761243\pi\)
−0.731637 + 0.681694i \(0.761243\pi\)
\(798\) 5.26689e29 0.178042
\(799\) −2.49134e30 −0.831172
\(800\) 0 0
\(801\) 2.87793e29 0.0935275
\(802\) 3.23417e30 1.03737
\(803\) −1.56775e29 −0.0496321
\(804\) −6.84392e29 −0.213854
\(805\) 0 0
\(806\) −3.63812e29 −0.110754
\(807\) −9.79791e29 −0.294416
\(808\) 3.51388e30 1.04224
\(809\) 6.44960e30 1.88831 0.944154 0.329503i \(-0.106881\pi\)
0.944154 + 0.329503i \(0.106881\pi\)
\(810\) 0 0
\(811\) −3.43726e28 −0.00980604 −0.00490302 0.999988i \(-0.501561\pi\)
−0.00490302 + 0.999988i \(0.501561\pi\)
\(812\) −4.47627e29 −0.126060
\(813\) −1.61086e30 −0.447823
\(814\) −2.40237e30 −0.659302
\(815\) 0 0
\(816\) 5.98894e29 0.160178
\(817\) −4.02874e29 −0.106374
\(818\) −1.31842e30 −0.343672
\(819\) −7.23143e29 −0.186099
\(820\) 0 0
\(821\) −4.56685e30 −1.14555 −0.572775 0.819713i \(-0.694133\pi\)
−0.572775 + 0.819713i \(0.694133\pi\)
\(822\) 2.22188e30 0.550259
\(823\) −7.18647e30 −1.75718 −0.878592 0.477572i \(-0.841517\pi\)
−0.878592 + 0.477572i \(0.841517\pi\)
\(824\) 7.36417e30 1.77782
\(825\) 0 0
\(826\) 7.72450e30 1.81794
\(827\) 2.77401e30 0.644614 0.322307 0.946635i \(-0.395542\pi\)
0.322307 + 0.946635i \(0.395542\pi\)
\(828\) 1.01913e30 0.233835
\(829\) 5.61496e30 1.27211 0.636054 0.771644i \(-0.280565\pi\)
0.636054 + 0.771644i \(0.280565\pi\)
\(830\) 0 0
\(831\) 1.06295e30 0.234801
\(832\) −1.31258e30 −0.286306
\(833\) −1.04068e31 −2.24154
\(834\) −1.22639e30 −0.260847
\(835\) 0 0
\(836\) −3.90116e29 −0.0809148
\(837\) 4.37554e29 0.0896219
\(838\) −4.15408e29 −0.0840258
\(839\) 3.99858e30 0.798740 0.399370 0.916790i \(-0.369229\pi\)
0.399370 + 0.916790i \(0.369229\pi\)
\(840\) 0 0
\(841\) −5.01967e30 −0.977951
\(842\) −1.74102e30 −0.334987
\(843\) −1.42446e30 −0.270682
\(844\) −1.68884e30 −0.316951
\(845\) 0 0
\(846\) 7.54731e29 0.138167
\(847\) −6.15936e30 −1.11368
\(848\) 6.66142e29 0.118963
\(849\) −1.63292e30 −0.288028
\(850\) 0 0
\(851\) 1.33033e31 2.28928
\(852\) 1.20841e30 0.205399
\(853\) 6.64773e30 1.11611 0.558057 0.829803i \(-0.311547\pi\)
0.558057 + 0.829803i \(0.311547\pi\)
\(854\) 6.05566e30 1.00428
\(855\) 0 0
\(856\) −1.12063e31 −1.81337
\(857\) −8.33064e30 −1.33162 −0.665809 0.746122i \(-0.731914\pi\)
−0.665809 + 0.746122i \(0.731914\pi\)
\(858\) −4.85798e29 −0.0767077
\(859\) 4.73149e30 0.738023 0.369012 0.929425i \(-0.379696\pi\)
0.369012 + 0.929425i \(0.379696\pi\)
\(860\) 0 0
\(861\) −6.04738e30 −0.920523
\(862\) −9.47089e29 −0.142418
\(863\) 5.45168e30 0.809874 0.404937 0.914345i \(-0.367294\pi\)
0.404937 + 0.914345i \(0.367294\pi\)
\(864\) 1.19715e30 0.175694
\(865\) 0 0
\(866\) −1.07914e29 −0.0154575
\(867\) 3.72247e30 0.526782
\(868\) −2.82771e30 −0.395347
\(869\) −7.92032e30 −1.09404
\(870\) 0 0
\(871\) −1.80655e30 −0.243590
\(872\) 4.55742e30 0.607150
\(873\) −4.80139e30 −0.632001
\(874\) −1.95931e30 −0.254820
\(875\) 0 0
\(876\) 2.11844e29 0.0268982
\(877\) 1.97851e30 0.248224 0.124112 0.992268i \(-0.460392\pi\)
0.124112 + 0.992268i \(0.460392\pi\)
\(878\) −5.86475e30 −0.727038
\(879\) −5.21854e30 −0.639242
\(880\) 0 0
\(881\) −4.03712e30 −0.482864 −0.241432 0.970418i \(-0.577617\pi\)
−0.241432 + 0.970418i \(0.577617\pi\)
\(882\) 3.15267e30 0.372614
\(883\) 9.81177e30 1.14593 0.572967 0.819578i \(-0.305792\pi\)
0.572967 + 0.819578i \(0.305792\pi\)
\(884\) −2.16688e30 −0.250084
\(885\) 0 0
\(886\) −1.92467e30 −0.216921
\(887\) −6.61087e30 −0.736310 −0.368155 0.929764i \(-0.620010\pi\)
−0.368155 + 0.929764i \(0.620010\pi\)
\(888\) 9.43671e30 1.03869
\(889\) −1.87861e31 −2.04347
\(890\) 0 0
\(891\) 5.84265e29 0.0620717
\(892\) 8.54864e30 0.897565
\(893\) 1.59984e30 0.166010
\(894\) −4.90096e30 −0.502617
\(895\) 0 0
\(896\) −5.50074e30 −0.551045
\(897\) 2.69013e30 0.266350
\(898\) 9.44368e30 0.924144
\(899\) 7.14938e29 0.0691500
\(900\) 0 0
\(901\) 8.67838e30 0.820028
\(902\) −4.06255e30 −0.379429
\(903\) −3.89931e30 −0.359971
\(904\) −2.61325e30 −0.238458
\(905\) 0 0
\(906\) −5.80092e30 −0.517191
\(907\) −7.36764e30 −0.649310 −0.324655 0.945833i \(-0.605248\pi\)
−0.324655 + 0.945833i \(0.605248\pi\)
\(908\) 3.04304e29 0.0265098
\(909\) −3.83748e30 −0.330465
\(910\) 0 0
\(911\) 3.51177e30 0.295518 0.147759 0.989023i \(-0.452794\pi\)
0.147759 + 0.989023i \(0.452794\pi\)
\(912\) −3.84586e29 −0.0319924
\(913\) −1.52514e29 −0.0125420
\(914\) 2.58969e30 0.210529
\(915\) 0 0
\(916\) −5.54588e30 −0.440624
\(917\) 2.18336e31 1.71493
\(918\) −2.36364e30 −0.183541
\(919\) −1.76504e31 −1.35501 −0.677505 0.735519i \(-0.736939\pi\)
−0.677505 + 0.735519i \(0.736939\pi\)
\(920\) 0 0
\(921\) −4.25530e30 −0.319304
\(922\) 6.55552e30 0.486332
\(923\) 3.18977e30 0.233960
\(924\) −3.77584e30 −0.273815
\(925\) 0 0
\(926\) 9.10696e30 0.645592
\(927\) −8.04235e30 −0.563697
\(928\) 1.95608e30 0.135561
\(929\) 1.81466e31 1.24346 0.621729 0.783232i \(-0.286430\pi\)
0.621729 + 0.783232i \(0.286430\pi\)
\(930\) 0 0
\(931\) 6.68286e30 0.447704
\(932\) 3.68339e30 0.243995
\(933\) −1.12836e31 −0.739080
\(934\) 1.69211e31 1.09594
\(935\) 0 0
\(936\) 1.90825e30 0.120848
\(937\) −1.90377e31 −1.19220 −0.596101 0.802910i \(-0.703284\pi\)
−0.596101 + 0.802910i \(0.703284\pi\)
\(938\) 1.27350e31 0.788624
\(939\) −8.33126e29 −0.0510178
\(940\) 0 0
\(941\) 7.95092e30 0.476131 0.238065 0.971249i \(-0.423487\pi\)
0.238065 + 0.971249i \(0.423487\pi\)
\(942\) −6.41982e30 −0.380179
\(943\) 2.24966e31 1.31748
\(944\) −5.64040e30 −0.326666
\(945\) 0 0
\(946\) −2.61951e30 −0.148376
\(947\) 1.53126e31 0.857779 0.428890 0.903357i \(-0.358905\pi\)
0.428890 + 0.903357i \(0.358905\pi\)
\(948\) 1.07024e31 0.592918
\(949\) 5.59192e29 0.0306384
\(950\) 0 0
\(951\) 1.62547e31 0.871131
\(952\) 4.44043e31 2.35362
\(953\) 2.71409e31 1.42282 0.711409 0.702778i \(-0.248057\pi\)
0.711409 + 0.702778i \(0.248057\pi\)
\(954\) −2.62905e30 −0.136314
\(955\) 0 0
\(956\) 3.70499e30 0.187923
\(957\) 9.54655e29 0.0478929
\(958\) −9.98369e30 −0.495397
\(959\) 4.55852e31 2.23732
\(960\) 0 0
\(961\) −1.63092e31 −0.783134
\(962\) 8.56891e30 0.406993
\(963\) 1.22383e31 0.574970
\(964\) −1.65990e31 −0.771387
\(965\) 0 0
\(966\) −1.89637e31 −0.862309
\(967\) −3.62577e31 −1.63088 −0.815439 0.578842i \(-0.803505\pi\)
−0.815439 + 0.578842i \(0.803505\pi\)
\(968\) 1.62535e31 0.723196
\(969\) −5.01031e30 −0.220529
\(970\) 0 0
\(971\) −4.06711e29 −0.0175180 −0.00875899 0.999962i \(-0.502788\pi\)
−0.00875899 + 0.999962i \(0.502788\pi\)
\(972\) −7.89495e29 −0.0336398
\(973\) −2.51612e31 −1.06059
\(974\) −1.13303e31 −0.472466
\(975\) 0 0
\(976\) −4.42182e30 −0.180459
\(977\) −3.57816e30 −0.144466 −0.0722331 0.997388i \(-0.523013\pi\)
−0.0722331 + 0.997388i \(0.523013\pi\)
\(978\) 1.89201e31 0.755726
\(979\) 3.96657e30 0.156746
\(980\) 0 0
\(981\) −4.97712e30 −0.192510
\(982\) 7.76585e30 0.297179
\(983\) 1.79552e31 0.679796 0.339898 0.940462i \(-0.389607\pi\)
0.339898 + 0.940462i \(0.389607\pi\)
\(984\) 1.59580e31 0.597765
\(985\) 0 0
\(986\) −3.86206e30 −0.141616
\(987\) 1.54844e31 0.561778
\(988\) 1.39149e30 0.0499494
\(989\) 1.45057e31 0.515201
\(990\) 0 0
\(991\) 1.08018e31 0.375599 0.187800 0.982207i \(-0.439864\pi\)
0.187800 + 0.982207i \(0.439864\pi\)
\(992\) 1.23568e31 0.425142
\(993\) 1.72018e31 0.585608
\(994\) −2.24858e31 −0.757445
\(995\) 0 0
\(996\) 2.06087e29 0.00679716
\(997\) −3.78917e31 −1.23664 −0.618322 0.785925i \(-0.712187\pi\)
−0.618322 + 0.785925i \(0.712187\pi\)
\(998\) −4.93907e30 −0.159505
\(999\) −1.03058e31 −0.329338
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 75.22.a.m.1.7 10
5.2 odd 4 15.22.b.a.4.13 yes 20
5.3 odd 4 15.22.b.a.4.8 20
5.4 even 2 75.22.a.n.1.4 10
15.2 even 4 45.22.b.d.19.8 20
15.8 even 4 45.22.b.d.19.13 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.22.b.a.4.8 20 5.3 odd 4
15.22.b.a.4.13 yes 20 5.2 odd 4
45.22.b.d.19.8 20 15.2 even 4
45.22.b.d.19.13 20 15.8 even 4
75.22.a.m.1.7 10 1.1 even 1 trivial
75.22.a.n.1.4 10 5.4 even 2