Properties

Label 7.18.a.a.1.2
Level $7$
Weight $18$
Character 7.1
Self dual yes
Analytic conductor $12.826$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7,18,Mod(1,7)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 7.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: \(12.8255461141\)
Analytic rank: \(1\)
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: \( x^{4} - 2x^{3} - 2290x^{2} - 4009x + 1104138 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(23.8876\) of defining polynomial
Character \(\chi\) \(=\) 7.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-181.857 q^{2} +16535.0 q^{3} -98000.2 q^{4} -359290. q^{5} -3.00701e6 q^{6} -5.76480e6 q^{7} +4.16583e7 q^{8} +1.44267e8 q^{9} +O(q^{10})\) \(q-181.857 q^{2} +16535.0 q^{3} -98000.2 q^{4} -359290. q^{5} -3.00701e6 q^{6} -5.76480e6 q^{7} +4.16583e7 q^{8} +1.44267e8 q^{9} +6.53393e7 q^{10} -5.17615e8 q^{11} -1.62044e9 q^{12} -4.13704e9 q^{13} +1.04837e9 q^{14} -5.94088e9 q^{15} +5.26925e9 q^{16} -2.18253e10 q^{17} -2.62360e10 q^{18} -9.10855e10 q^{19} +3.52105e10 q^{20} -9.53212e10 q^{21} +9.41317e10 q^{22} +3.15846e10 q^{23} +6.88821e11 q^{24} -6.33850e11 q^{25} +7.52348e11 q^{26} +2.50131e11 q^{27} +5.64952e11 q^{28} +1.03888e12 q^{29} +1.08039e12 q^{30} +7.26717e12 q^{31} -6.41848e12 q^{32} -8.55879e12 q^{33} +3.96908e12 q^{34} +2.07124e12 q^{35} -1.41382e13 q^{36} -2.70095e13 q^{37} +1.65645e13 q^{38} -6.84062e13 q^{39} -1.49674e13 q^{40} +9.47502e13 q^{41} +1.73348e13 q^{42} +9.10093e13 q^{43} +5.07264e13 q^{44} -5.18339e13 q^{45} -5.74387e12 q^{46} -1.35606e14 q^{47} +8.71273e13 q^{48} +3.32329e13 q^{49} +1.15270e14 q^{50} -3.60883e14 q^{51} +4.05431e14 q^{52} +5.33679e14 q^{53} -4.54879e13 q^{54} +1.85974e14 q^{55} -2.40152e14 q^{56} -1.50610e15 q^{57} -1.88928e14 q^{58} -1.36156e15 q^{59} +5.82208e14 q^{60} +3.58851e14 q^{61} -1.32158e15 q^{62} -8.31673e14 q^{63} +4.76592e14 q^{64} +1.48640e15 q^{65} +1.55647e15 q^{66} +3.62448e15 q^{67} +2.13889e15 q^{68} +5.22253e14 q^{69} -3.76668e14 q^{70} -8.21132e15 q^{71} +6.00993e15 q^{72} +1.69268e15 q^{73} +4.91185e15 q^{74} -1.04807e16 q^{75} +8.92639e15 q^{76} +2.98395e15 q^{77} +1.24401e16 q^{78} -1.40597e16 q^{79} -1.89319e15 q^{80} -1.44948e16 q^{81} -1.72309e16 q^{82} -2.12860e16 q^{83} +9.34150e15 q^{84} +7.84163e15 q^{85} -1.65506e16 q^{86} +1.71780e16 q^{87} -2.15630e16 q^{88} -9.39599e15 q^{89} +9.42634e15 q^{90} +2.38492e16 q^{91} -3.09530e15 q^{92} +1.20163e17 q^{93} +2.46609e16 q^{94} +3.27261e16 q^{95} -1.06130e17 q^{96} +1.65321e16 q^{97} -6.04363e15 q^{98} -7.46750e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 186 q^{2} - 2786 q^{3} - 16300 q^{4} + 274722 q^{5} + 1469804 q^{6} - 23059204 q^{7} + 8649336 q^{8} - 2050976 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 186 q^{2} - 2786 q^{3} - 16300 q^{4} + 274722 q^{5} + 1469804 q^{6} - 23059204 q^{7} + 8649336 q^{8} - 2050976 q^{9} - 893891656 q^{10} + 610110180 q^{11} - 1826039320 q^{12} - 8514921674 q^{13} - 1072252986 q^{14} - 30645264896 q^{15} - 47269015792 q^{16} - 47762899716 q^{17} - 148424524342 q^{18} - 142813479494 q^{19} - 88080723360 q^{20} + 16060735586 q^{21} - 25116572128 q^{22} + 161322432240 q^{23} + 387147758256 q^{24} + 1921891698992 q^{25} + 2984730379008 q^{26} + 2041714521028 q^{27} + 93966256300 q^{28} + 2470023989364 q^{29} + 6457134393152 q^{30} + 3069063677988 q^{31} - 7036366816032 q^{32} - 14819614563824 q^{33} - 9992374959252 q^{34} - 1583717660322 q^{35} - 18927631502956 q^{36} - 53477713304508 q^{37} - 51421850028780 q^{38} - 4140246547640 q^{39} + 22110911913216 q^{40} - 84856086719628 q^{41} - 8473127569004 q^{42} + 14664094189676 q^{43} + 237550257793824 q^{44} + 160924162333018 q^{45} + 187722899918496 q^{46} + 110590112906028 q^{47} + 428386513367456 q^{48} + 132931722278404 q^{49} + 539831164264974 q^{50} - 229270804715244 q^{51} + 68940623118416 q^{52} - 517697020820328 q^{53} - 32330860930648 q^{54} - 17\!\cdots\!44 q^{55}+ \cdots - 57\!\cdots\!28 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −181.857 −0.502312 −0.251156 0.967947i \(-0.580811\pi\)
−0.251156 + 0.967947i \(0.580811\pi\)
\(3\) 16535.0 1.45504 0.727520 0.686087i \(-0.240673\pi\)
0.727520 + 0.686087i \(0.240673\pi\)
\(4\) −98000.2 −0.747682
\(5\) −359290. −0.411340 −0.205670 0.978621i \(-0.565937\pi\)
−0.205670 + 0.978621i \(0.565937\pi\)
\(6\) −3.00701e6 −0.730884
\(7\) −5.76480e6 −0.377964
\(8\) 4.16583e7 0.877883
\(9\) 1.44267e8 1.11714
\(10\) 6.53393e7 0.206621
\(11\) −5.17615e8 −0.728063 −0.364032 0.931387i \(-0.618600\pi\)
−0.364032 + 0.931387i \(0.618600\pi\)
\(12\) −1.62044e9 −1.08791
\(13\) −4.13704e9 −1.40660 −0.703301 0.710892i \(-0.748292\pi\)
−0.703301 + 0.710892i \(0.748292\pi\)
\(14\) 1.04837e9 0.189856
\(15\) −5.94088e9 −0.598515
\(16\) 5.26925e9 0.306711
\(17\) −2.18253e10 −0.758831 −0.379415 0.925226i \(-0.623875\pi\)
−0.379415 + 0.925226i \(0.623875\pi\)
\(18\) −2.62360e10 −0.561153
\(19\) −9.10855e10 −1.23039 −0.615196 0.788374i \(-0.710923\pi\)
−0.615196 + 0.788374i \(0.710923\pi\)
\(20\) 3.52105e10 0.307551
\(21\) −9.53212e10 −0.549953
\(22\) 9.41317e10 0.365715
\(23\) 3.15846e10 0.0840986 0.0420493 0.999116i \(-0.486611\pi\)
0.0420493 + 0.999116i \(0.486611\pi\)
\(24\) 6.88821e11 1.27735
\(25\) −6.33850e11 −0.830800
\(26\) 7.52348e11 0.706554
\(27\) 2.50131e11 0.170441
\(28\) 5.64952e11 0.282597
\(29\) 1.03888e12 0.385641 0.192821 0.981234i \(-0.438236\pi\)
0.192821 + 0.981234i \(0.438236\pi\)
\(30\) 1.08039e12 0.300642
\(31\) 7.26717e12 1.53035 0.765175 0.643822i \(-0.222652\pi\)
0.765175 + 0.643822i \(0.222652\pi\)
\(32\) −6.41848e12 −1.03195
\(33\) −8.55879e12 −1.05936
\(34\) 3.96908e12 0.381170
\(35\) 2.07124e12 0.155472
\(36\) −1.41382e13 −0.835265
\(37\) −2.70095e13 −1.26416 −0.632080 0.774903i \(-0.717798\pi\)
−0.632080 + 0.774903i \(0.717798\pi\)
\(38\) 1.65645e13 0.618041
\(39\) −6.84062e13 −2.04666
\(40\) −1.49674e13 −0.361108
\(41\) 9.47502e13 1.85318 0.926589 0.376075i \(-0.122727\pi\)
0.926589 + 0.376075i \(0.122727\pi\)
\(42\) 1.73348e13 0.276248
\(43\) 9.10093e13 1.18742 0.593709 0.804680i \(-0.297663\pi\)
0.593709 + 0.804680i \(0.297663\pi\)
\(44\) 5.07264e13 0.544360
\(45\) −5.18339e13 −0.459524
\(46\) −5.74387e12 −0.0422438
\(47\) −1.35606e14 −0.830707 −0.415353 0.909660i \(-0.636342\pi\)
−0.415353 + 0.909660i \(0.636342\pi\)
\(48\) 8.71273e13 0.446276
\(49\) 3.32329e13 0.142857
\(50\) 1.15270e14 0.417321
\(51\) −3.60883e14 −1.10413
\(52\) 4.05431e14 1.05169
\(53\) 5.33679e14 1.17743 0.588715 0.808341i \(-0.299634\pi\)
0.588715 + 0.808341i \(0.299634\pi\)
\(54\) −4.54879e13 −0.0856148
\(55\) 1.85974e14 0.299481
\(56\) −2.40152e14 −0.331808
\(57\) −1.50610e15 −1.79027
\(58\) −1.88928e14 −0.193713
\(59\) −1.36156e15 −1.20724 −0.603619 0.797273i \(-0.706275\pi\)
−0.603619 + 0.797273i \(0.706275\pi\)
\(60\) 5.82208e14 0.447499
\(61\) 3.58851e14 0.239668 0.119834 0.992794i \(-0.461764\pi\)
0.119834 + 0.992794i \(0.461764\pi\)
\(62\) −1.32158e15 −0.768714
\(63\) −8.31673e14 −0.422239
\(64\) 4.76592e14 0.211649
\(65\) 1.48640e15 0.578592
\(66\) 1.55647e15 0.532130
\(67\) 3.62448e15 1.09046 0.545231 0.838286i \(-0.316442\pi\)
0.545231 + 0.838286i \(0.316442\pi\)
\(68\) 2.13889e15 0.567364
\(69\) 5.22253e14 0.122367
\(70\) −3.76668e14 −0.0780954
\(71\) −8.21132e15 −1.50910 −0.754548 0.656245i \(-0.772144\pi\)
−0.754548 + 0.656245i \(0.772144\pi\)
\(72\) 6.00993e15 0.980717
\(73\) 1.69268e15 0.245658 0.122829 0.992428i \(-0.460803\pi\)
0.122829 + 0.992428i \(0.460803\pi\)
\(74\) 4.91185e15 0.635003
\(75\) −1.04807e16 −1.20885
\(76\) 8.92639e15 0.919942
\(77\) 2.98395e15 0.275182
\(78\) 1.24401e16 1.02806
\(79\) −1.40597e16 −1.04267 −0.521335 0.853352i \(-0.674566\pi\)
−0.521335 + 0.853352i \(0.674566\pi\)
\(80\) −1.89319e15 −0.126162
\(81\) −1.44948e16 −0.869140
\(82\) −1.72309e16 −0.930875
\(83\) −2.12860e16 −1.03736 −0.518682 0.854968i \(-0.673577\pi\)
−0.518682 + 0.854968i \(0.673577\pi\)
\(84\) 9.34150e15 0.411190
\(85\) 7.84163e15 0.312137
\(86\) −1.65506e16 −0.596455
\(87\) 1.71780e16 0.561123
\(88\) −2.15630e16 −0.639154
\(89\) −9.39599e15 −0.253004 −0.126502 0.991966i \(-0.540375\pi\)
−0.126502 + 0.991966i \(0.540375\pi\)
\(90\) 9.42634e15 0.230824
\(91\) 2.38492e16 0.531646
\(92\) −3.09530e15 −0.0628790
\(93\) 1.20163e17 2.22672
\(94\) 2.46609e16 0.417274
\(95\) 3.27261e16 0.506109
\(96\) −1.06130e17 −1.50152
\(97\) 1.65321e16 0.214175 0.107088 0.994250i \(-0.465847\pi\)
0.107088 + 0.994250i \(0.465847\pi\)
\(98\) −6.04363e15 −0.0717589
\(99\) −7.46750e16 −0.813347
\(100\) 6.21174e16 0.621174
\(101\) 5.36191e16 0.492706 0.246353 0.969180i \(-0.420768\pi\)
0.246353 + 0.969180i \(0.420768\pi\)
\(102\) 6.56289e16 0.554618
\(103\) −2.06651e16 −0.160739 −0.0803695 0.996765i \(-0.525610\pi\)
−0.0803695 + 0.996765i \(0.525610\pi\)
\(104\) −1.72342e17 −1.23483
\(105\) 3.42480e16 0.226218
\(106\) −9.70530e16 −0.591438
\(107\) 4.13751e16 0.232797 0.116398 0.993203i \(-0.462865\pi\)
0.116398 + 0.993203i \(0.462865\pi\)
\(108\) −2.45128e16 −0.127436
\(109\) 4.14022e17 1.99021 0.995103 0.0988449i \(-0.0315148\pi\)
0.995103 + 0.0988449i \(0.0315148\pi\)
\(110\) −3.38206e16 −0.150433
\(111\) −4.46603e17 −1.83940
\(112\) −3.03762e16 −0.115926
\(113\) −2.51260e17 −0.889113 −0.444556 0.895751i \(-0.646639\pi\)
−0.444556 + 0.895751i \(0.646639\pi\)
\(114\) 2.73895e17 0.899274
\(115\) −1.13480e16 −0.0345931
\(116\) −1.01811e17 −0.288337
\(117\) −5.96841e17 −1.57137
\(118\) 2.47608e17 0.606411
\(119\) 1.25819e17 0.286811
\(120\) −2.47487e17 −0.525426
\(121\) −2.37522e17 −0.469924
\(122\) −6.52595e16 −0.120388
\(123\) 1.56670e18 2.69645
\(124\) −7.12184e17 −1.14422
\(125\) 5.01853e17 0.753081
\(126\) 1.51245e17 0.212096
\(127\) 3.58876e17 0.470558 0.235279 0.971928i \(-0.424400\pi\)
0.235279 + 0.971928i \(0.424400\pi\)
\(128\) 7.54612e17 0.925633
\(129\) 1.50484e18 1.72774
\(130\) −2.70312e17 −0.290634
\(131\) −1.68323e18 −1.69565 −0.847825 0.530276i \(-0.822088\pi\)
−0.847825 + 0.530276i \(0.822088\pi\)
\(132\) 8.38763e17 0.792065
\(133\) 5.25090e17 0.465044
\(134\) −6.59136e17 −0.547752
\(135\) −8.98695e16 −0.0701093
\(136\) −9.09206e17 −0.666164
\(137\) −2.82456e18 −1.94458 −0.972290 0.233778i \(-0.924891\pi\)
−0.972290 + 0.233778i \(0.924891\pi\)
\(138\) −9.49751e16 −0.0614664
\(139\) −4.10036e17 −0.249572 −0.124786 0.992184i \(-0.539824\pi\)
−0.124786 + 0.992184i \(0.539824\pi\)
\(140\) −2.02982e17 −0.116243
\(141\) −2.24225e18 −1.20871
\(142\) 1.49328e18 0.758038
\(143\) 2.14140e18 1.02410
\(144\) 7.60181e17 0.342638
\(145\) −3.73261e17 −0.158630
\(146\) −3.07826e17 −0.123397
\(147\) 5.49508e17 0.207863
\(148\) 2.64694e18 0.945189
\(149\) −5.35731e18 −1.80661 −0.903304 0.429001i \(-0.858866\pi\)
−0.903304 + 0.429001i \(0.858866\pi\)
\(150\) 1.90599e18 0.607218
\(151\) 2.59085e18 0.780077 0.390039 0.920798i \(-0.372462\pi\)
0.390039 + 0.920798i \(0.372462\pi\)
\(152\) −3.79446e18 −1.08014
\(153\) −3.14869e18 −0.847719
\(154\) −5.42650e17 −0.138227
\(155\) −2.61102e18 −0.629494
\(156\) 6.70382e18 1.53025
\(157\) −1.02610e18 −0.221842 −0.110921 0.993829i \(-0.535380\pi\)
−0.110921 + 0.993829i \(0.535380\pi\)
\(158\) 2.55686e18 0.523746
\(159\) 8.82441e18 1.71321
\(160\) 2.30610e18 0.424481
\(161\) −1.82079e17 −0.0317863
\(162\) 2.63598e18 0.436580
\(163\) 4.62746e18 0.727358 0.363679 0.931524i \(-0.381521\pi\)
0.363679 + 0.931524i \(0.381521\pi\)
\(164\) −9.28554e18 −1.38559
\(165\) 3.07509e18 0.435757
\(166\) 3.87100e18 0.521080
\(167\) −8.36584e18 −1.07009 −0.535044 0.844824i \(-0.679705\pi\)
−0.535044 + 0.844824i \(0.679705\pi\)
\(168\) −3.97092e18 −0.482794
\(169\) 8.46471e18 0.978532
\(170\) −1.42605e18 −0.156790
\(171\) −1.31407e19 −1.37452
\(172\) −8.91893e18 −0.887812
\(173\) 1.05223e19 0.997058 0.498529 0.866873i \(-0.333874\pi\)
0.498529 + 0.866873i \(0.333874\pi\)
\(174\) −3.12393e18 −0.281859
\(175\) 3.65402e18 0.314013
\(176\) −2.72744e18 −0.223305
\(177\) −2.25134e19 −1.75658
\(178\) 1.70872e18 0.127087
\(179\) −2.28203e18 −0.161835 −0.0809173 0.996721i \(-0.525785\pi\)
−0.0809173 + 0.996721i \(0.525785\pi\)
\(180\) 5.07973e18 0.343578
\(181\) 1.33314e19 0.860220 0.430110 0.902777i \(-0.358475\pi\)
0.430110 + 0.902777i \(0.358475\pi\)
\(182\) −4.33714e18 −0.267052
\(183\) 5.93362e18 0.348727
\(184\) 1.31576e18 0.0738287
\(185\) 9.70425e18 0.519999
\(186\) −2.18524e19 −1.11851
\(187\) 1.12971e19 0.552477
\(188\) 1.32894e19 0.621105
\(189\) −1.44195e18 −0.0644208
\(190\) −5.95146e18 −0.254225
\(191\) −1.82726e19 −0.746478 −0.373239 0.927735i \(-0.621753\pi\)
−0.373239 + 0.927735i \(0.621753\pi\)
\(192\) 7.88046e18 0.307958
\(193\) −2.95687e19 −1.10559 −0.552796 0.833317i \(-0.686439\pi\)
−0.552796 + 0.833317i \(0.686439\pi\)
\(194\) −3.00648e18 −0.107583
\(195\) 2.45777e19 0.841874
\(196\) −3.25683e18 −0.106812
\(197\) 3.08412e19 0.968653 0.484326 0.874887i \(-0.339065\pi\)
0.484326 + 0.874887i \(0.339065\pi\)
\(198\) 1.35801e19 0.408555
\(199\) 3.29269e19 0.949074 0.474537 0.880235i \(-0.342615\pi\)
0.474537 + 0.880235i \(0.342615\pi\)
\(200\) −2.64051e19 −0.729345
\(201\) 5.99310e19 1.58666
\(202\) −9.75098e18 −0.247492
\(203\) −5.98895e18 −0.145759
\(204\) 3.53666e19 0.825537
\(205\) −3.40428e19 −0.762286
\(206\) 3.75809e18 0.0807412
\(207\) 4.55663e18 0.0939498
\(208\) −2.17991e19 −0.431420
\(209\) 4.71472e19 0.895803
\(210\) −6.22822e18 −0.113632
\(211\) −4.31546e18 −0.0756181 −0.0378091 0.999285i \(-0.512038\pi\)
−0.0378091 + 0.999285i \(0.512038\pi\)
\(212\) −5.23006e19 −0.880344
\(213\) −1.35774e20 −2.19579
\(214\) −7.52433e18 −0.116937
\(215\) −3.26988e19 −0.488432
\(216\) 1.04200e19 0.149627
\(217\) −4.18938e19 −0.578418
\(218\) −7.52926e19 −0.999705
\(219\) 2.79886e19 0.357443
\(220\) −1.82255e19 −0.223917
\(221\) 9.02923e19 1.06737
\(222\) 8.12177e19 0.923954
\(223\) −8.31952e19 −0.910976 −0.455488 0.890242i \(-0.650535\pi\)
−0.455488 + 0.890242i \(0.650535\pi\)
\(224\) 3.70013e19 0.390039
\(225\) −9.14439e19 −0.928118
\(226\) 4.56933e19 0.446612
\(227\) 3.19266e19 0.300561 0.150281 0.988643i \(-0.451982\pi\)
0.150281 + 0.988643i \(0.451982\pi\)
\(228\) 1.47598e20 1.33855
\(229\) −5.56717e19 −0.486444 −0.243222 0.969971i \(-0.578204\pi\)
−0.243222 + 0.969971i \(0.578204\pi\)
\(230\) 2.06372e18 0.0173765
\(231\) 4.93397e19 0.400401
\(232\) 4.32781e19 0.338548
\(233\) 5.46463e19 0.412131 0.206065 0.978538i \(-0.433934\pi\)
0.206065 + 0.978538i \(0.433934\pi\)
\(234\) 1.08539e20 0.789319
\(235\) 4.87220e19 0.341703
\(236\) 1.33433e20 0.902631
\(237\) −2.32478e20 −1.51713
\(238\) −2.28810e19 −0.144069
\(239\) 2.48148e20 1.50775 0.753875 0.657018i \(-0.228182\pi\)
0.753875 + 0.657018i \(0.228182\pi\)
\(240\) −3.13040e19 −0.183571
\(241\) −1.79431e20 −1.01567 −0.507834 0.861455i \(-0.669554\pi\)
−0.507834 + 0.861455i \(0.669554\pi\)
\(242\) 4.31949e19 0.236049
\(243\) −2.71974e20 −1.43507
\(244\) −3.51675e19 −0.179196
\(245\) −1.19403e19 −0.0587628
\(246\) −2.84914e20 −1.35446
\(247\) 3.76825e20 1.73067
\(248\) 3.02738e20 1.34347
\(249\) −3.51965e20 −1.50940
\(250\) −9.12653e19 −0.378282
\(251\) 3.77682e20 1.51321 0.756606 0.653871i \(-0.226856\pi\)
0.756606 + 0.653871i \(0.226856\pi\)
\(252\) 8.15041e19 0.315700
\(253\) −1.63487e19 −0.0612291
\(254\) −6.52640e19 −0.236367
\(255\) 1.29662e20 0.454172
\(256\) −1.99699e20 −0.676606
\(257\) −5.38973e20 −1.76659 −0.883294 0.468819i \(-0.844680\pi\)
−0.883294 + 0.468819i \(0.844680\pi\)
\(258\) −2.73665e20 −0.867865
\(259\) 1.55704e20 0.477807
\(260\) −1.45668e20 −0.432603
\(261\) 1.49877e20 0.430815
\(262\) 3.06106e20 0.851746
\(263\) 3.86743e20 1.04183 0.520917 0.853607i \(-0.325590\pi\)
0.520917 + 0.853607i \(0.325590\pi\)
\(264\) −3.56544e20 −0.929994
\(265\) −1.91746e20 −0.484324
\(266\) −9.54910e19 −0.233598
\(267\) −1.55363e20 −0.368131
\(268\) −3.55200e20 −0.815319
\(269\) −6.51556e19 −0.144896 −0.0724481 0.997372i \(-0.523081\pi\)
−0.0724481 + 0.997372i \(0.523081\pi\)
\(270\) 1.63434e19 0.0352168
\(271\) −1.10328e20 −0.230381 −0.115191 0.993343i \(-0.536748\pi\)
−0.115191 + 0.993343i \(0.536748\pi\)
\(272\) −1.15003e20 −0.232742
\(273\) 3.94348e20 0.773566
\(274\) 5.13665e20 0.976787
\(275\) 3.28090e20 0.604875
\(276\) −5.11809e19 −0.0914915
\(277\) 2.03040e20 0.351969 0.175984 0.984393i \(-0.443689\pi\)
0.175984 + 0.984393i \(0.443689\pi\)
\(278\) 7.45676e19 0.125363
\(279\) 1.04842e21 1.70961
\(280\) 8.62842e19 0.136486
\(281\) −2.83182e20 −0.434572 −0.217286 0.976108i \(-0.569720\pi\)
−0.217286 + 0.976108i \(0.569720\pi\)
\(282\) 4.07769e20 0.607151
\(283\) −4.43477e20 −0.640746 −0.320373 0.947291i \(-0.603808\pi\)
−0.320373 + 0.947291i \(0.603808\pi\)
\(284\) 8.04711e20 1.12832
\(285\) 5.41128e20 0.736409
\(286\) −3.89427e20 −0.514416
\(287\) −5.46216e20 −0.700436
\(288\) −9.25978e20 −1.15283
\(289\) −3.50895e20 −0.424176
\(290\) 6.78799e19 0.0796817
\(291\) 2.73360e20 0.311633
\(292\) −1.65883e20 −0.183674
\(293\) −6.05067e20 −0.650772 −0.325386 0.945581i \(-0.605494\pi\)
−0.325386 + 0.945581i \(0.605494\pi\)
\(294\) −9.99316e19 −0.104412
\(295\) 4.89194e20 0.496585
\(296\) −1.12517e21 −1.10978
\(297\) −1.29471e20 −0.124092
\(298\) 9.74263e20 0.907482
\(299\) −1.30667e20 −0.118293
\(300\) 1.02711e21 0.903833
\(301\) −5.24650e20 −0.448802
\(302\) −4.71162e20 −0.391843
\(303\) 8.86594e20 0.716906
\(304\) −4.79952e20 −0.377374
\(305\) −1.28932e20 −0.0985852
\(306\) 5.72609e20 0.425820
\(307\) −1.90711e21 −1.37943 −0.689713 0.724083i \(-0.742263\pi\)
−0.689713 + 0.724083i \(0.742263\pi\)
\(308\) −2.92427e20 −0.205749
\(309\) −3.41698e20 −0.233882
\(310\) 4.74832e20 0.316203
\(311\) 3.83279e20 0.248343 0.124171 0.992261i \(-0.460373\pi\)
0.124171 + 0.992261i \(0.460373\pi\)
\(312\) −2.84968e21 −1.79673
\(313\) 3.72919e20 0.228817 0.114408 0.993434i \(-0.463503\pi\)
0.114408 + 0.993434i \(0.463503\pi\)
\(314\) 1.86603e20 0.111434
\(315\) 2.98812e20 0.173684
\(316\) 1.37786e21 0.779586
\(317\) −1.31935e21 −0.726700 −0.363350 0.931653i \(-0.618367\pi\)
−0.363350 + 0.931653i \(0.618367\pi\)
\(318\) −1.60478e21 −0.860565
\(319\) −5.37742e20 −0.280771
\(320\) −1.71235e20 −0.0870597
\(321\) 6.84138e20 0.338728
\(322\) 3.31123e19 0.0159667
\(323\) 1.98797e21 0.933659
\(324\) 1.42049e21 0.649840
\(325\) 2.62226e21 1.16861
\(326\) −8.41534e20 −0.365361
\(327\) 6.84587e21 2.89583
\(328\) 3.94713e21 1.62687
\(329\) 7.81743e20 0.313978
\(330\) −5.59225e20 −0.218886
\(331\) −4.78390e21 −1.82492 −0.912460 0.409165i \(-0.865820\pi\)
−0.912460 + 0.409165i \(0.865820\pi\)
\(332\) 2.08603e21 0.775618
\(333\) −3.89659e21 −1.41224
\(334\) 1.52138e21 0.537519
\(335\) −1.30224e21 −0.448550
\(336\) −5.02271e20 −0.168677
\(337\) 2.40928e21 0.788919 0.394459 0.918913i \(-0.370932\pi\)
0.394459 + 0.918913i \(0.370932\pi\)
\(338\) −1.53936e21 −0.491529
\(339\) −4.15460e21 −1.29369
\(340\) −7.68482e20 −0.233379
\(341\) −3.76160e21 −1.11419
\(342\) 2.38972e21 0.690438
\(343\) −1.91581e20 −0.0539949
\(344\) 3.79129e21 1.04241
\(345\) −1.87640e20 −0.0503343
\(346\) −1.91356e21 −0.500835
\(347\) −4.59537e21 −1.17360 −0.586800 0.809732i \(-0.699612\pi\)
−0.586800 + 0.809732i \(0.699612\pi\)
\(348\) −1.68345e21 −0.419542
\(349\) −3.89744e21 −0.947901 −0.473950 0.880552i \(-0.657172\pi\)
−0.473950 + 0.880552i \(0.657172\pi\)
\(350\) −6.64507e20 −0.157733
\(351\) −1.03480e21 −0.239743
\(352\) 3.32230e21 0.751323
\(353\) −4.92306e21 −1.08680 −0.543401 0.839473i \(-0.682864\pi\)
−0.543401 + 0.839473i \(0.682864\pi\)
\(354\) 4.09420e21 0.882352
\(355\) 2.95025e21 0.620751
\(356\) 9.20809e20 0.189167
\(357\) 2.08042e21 0.417321
\(358\) 4.15003e20 0.0812915
\(359\) 2.24060e21 0.428609 0.214305 0.976767i \(-0.431251\pi\)
0.214305 + 0.976767i \(0.431251\pi\)
\(360\) −2.15931e21 −0.403408
\(361\) 2.81618e21 0.513865
\(362\) −2.42441e21 −0.432099
\(363\) −3.92743e21 −0.683758
\(364\) −2.33723e21 −0.397502
\(365\) −6.08165e20 −0.101049
\(366\) −1.07907e21 −0.175170
\(367\) 9.77120e21 1.54984 0.774919 0.632061i \(-0.217791\pi\)
0.774919 + 0.632061i \(0.217791\pi\)
\(368\) 1.66427e20 0.0257940
\(369\) 1.36694e22 2.07026
\(370\) −1.76478e21 −0.261202
\(371\) −3.07655e21 −0.445027
\(372\) −1.17760e22 −1.66488
\(373\) −2.05508e21 −0.283991 −0.141995 0.989867i \(-0.545352\pi\)
−0.141995 + 0.989867i \(0.545352\pi\)
\(374\) −2.05446e21 −0.277516
\(375\) 8.29816e21 1.09576
\(376\) −5.64912e21 −0.729263
\(377\) −4.29790e21 −0.542444
\(378\) 2.62229e20 0.0323593
\(379\) 5.45735e21 0.658489 0.329245 0.944245i \(-0.393206\pi\)
0.329245 + 0.944245i \(0.393206\pi\)
\(380\) −3.20717e21 −0.378409
\(381\) 5.93404e21 0.684681
\(382\) 3.32299e21 0.374965
\(383\) −5.20172e21 −0.574060 −0.287030 0.957922i \(-0.592668\pi\)
−0.287030 + 0.957922i \(0.592668\pi\)
\(384\) 1.24775e22 1.34683
\(385\) −1.07210e21 −0.113193
\(386\) 5.37725e21 0.555352
\(387\) 1.31297e22 1.32651
\(388\) −1.62015e21 −0.160135
\(389\) 1.16024e22 1.12196 0.560981 0.827829i \(-0.310424\pi\)
0.560981 + 0.827829i \(0.310424\pi\)
\(390\) −4.46961e21 −0.422884
\(391\) −6.89344e20 −0.0638166
\(392\) 1.38443e21 0.125412
\(393\) −2.78322e22 −2.46724
\(394\) −5.60867e21 −0.486566
\(395\) 5.05153e21 0.428892
\(396\) 7.31817e21 0.608125
\(397\) −8.51122e21 −0.692265 −0.346133 0.938186i \(-0.612505\pi\)
−0.346133 + 0.938186i \(0.612505\pi\)
\(398\) −5.98798e21 −0.476732
\(399\) 8.68238e21 0.676658
\(400\) −3.33991e21 −0.254815
\(401\) 1.27994e22 0.956014 0.478007 0.878356i \(-0.341359\pi\)
0.478007 + 0.878356i \(0.341359\pi\)
\(402\) −1.08988e22 −0.797001
\(403\) −3.00646e22 −2.15260
\(404\) −5.25468e21 −0.368387
\(405\) 5.20785e21 0.357512
\(406\) 1.08913e21 0.0732165
\(407\) 1.39805e22 0.920388
\(408\) −1.50338e22 −0.969295
\(409\) 8.57747e21 0.541641 0.270820 0.962630i \(-0.412705\pi\)
0.270820 + 0.962630i \(0.412705\pi\)
\(410\) 6.19091e21 0.382906
\(411\) −4.67042e22 −2.82944
\(412\) 2.02518e21 0.120182
\(413\) 7.84909e21 0.456293
\(414\) −8.28653e20 −0.0471922
\(415\) 7.64787e21 0.426709
\(416\) 2.65535e22 1.45154
\(417\) −6.77996e21 −0.363137
\(418\) −8.57403e21 −0.449973
\(419\) −1.00651e22 −0.517606 −0.258803 0.965930i \(-0.583328\pi\)
−0.258803 + 0.965930i \(0.583328\pi\)
\(420\) −3.35631e21 −0.169139
\(421\) 2.06716e22 1.02088 0.510441 0.859913i \(-0.329482\pi\)
0.510441 + 0.859913i \(0.329482\pi\)
\(422\) 7.84794e20 0.0379839
\(423\) −1.95636e22 −0.928015
\(424\) 2.22321e22 1.03365
\(425\) 1.38340e22 0.630436
\(426\) 2.46915e22 1.10297
\(427\) −2.06871e21 −0.0905862
\(428\) −4.05476e21 −0.174058
\(429\) 3.54081e22 1.49010
\(430\) 5.94648e21 0.245346
\(431\) 5.85800e20 0.0236969 0.0118485 0.999930i \(-0.496228\pi\)
0.0118485 + 0.999930i \(0.496228\pi\)
\(432\) 1.31800e21 0.0522762
\(433\) −2.65233e21 −0.103153 −0.0515763 0.998669i \(-0.516425\pi\)
−0.0515763 + 0.998669i \(0.516425\pi\)
\(434\) 7.61866e21 0.290547
\(435\) −6.17188e21 −0.230812
\(436\) −4.05742e22 −1.48804
\(437\) −2.87690e21 −0.103474
\(438\) −5.08991e21 −0.179548
\(439\) 6.42495e21 0.222291 0.111145 0.993804i \(-0.464548\pi\)
0.111145 + 0.993804i \(0.464548\pi\)
\(440\) 7.74736e21 0.262909
\(441\) 4.79443e21 0.159591
\(442\) −1.64203e22 −0.536155
\(443\) 1.53379e22 0.491285 0.245643 0.969360i \(-0.421001\pi\)
0.245643 + 0.969360i \(0.421001\pi\)
\(444\) 4.37672e22 1.37529
\(445\) 3.37589e21 0.104071
\(446\) 1.51296e22 0.457595
\(447\) −8.85834e22 −2.62868
\(448\) −2.74746e21 −0.0799959
\(449\) 2.38064e22 0.680142 0.340071 0.940400i \(-0.389549\pi\)
0.340071 + 0.940400i \(0.389549\pi\)
\(450\) 1.66297e22 0.466205
\(451\) −4.90441e22 −1.34923
\(452\) 2.46235e22 0.664774
\(453\) 4.28398e22 1.13504
\(454\) −5.80606e21 −0.150976
\(455\) −8.56880e21 −0.218687
\(456\) −6.27416e22 −1.57165
\(457\) 2.08677e22 0.513082 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(458\) 1.01243e22 0.244347
\(459\) −5.45918e21 −0.129336
\(460\) 1.11211e21 0.0258646
\(461\) −5.38735e22 −1.23003 −0.615017 0.788513i \(-0.710851\pi\)
−0.615017 + 0.788513i \(0.710851\pi\)
\(462\) −8.97275e21 −0.201126
\(463\) 9.60815e21 0.211447 0.105723 0.994396i \(-0.466284\pi\)
0.105723 + 0.994396i \(0.466284\pi\)
\(464\) 5.47414e21 0.118280
\(465\) −4.31734e22 −0.915938
\(466\) −9.93778e21 −0.207018
\(467\) −4.38850e22 −0.897683 −0.448841 0.893611i \(-0.648163\pi\)
−0.448841 + 0.893611i \(0.648163\pi\)
\(468\) 5.84905e22 1.17489
\(469\) −2.08944e22 −0.412156
\(470\) −8.86042e21 −0.171642
\(471\) −1.69666e22 −0.322788
\(472\) −5.67200e22 −1.05981
\(473\) −4.71078e22 −0.864516
\(474\) 4.22777e22 0.762071
\(475\) 5.77345e22 1.02221
\(476\) −1.23303e22 −0.214444
\(477\) 7.69925e22 1.31535
\(478\) −4.51274e22 −0.757362
\(479\) 2.82508e21 0.0465778 0.0232889 0.999729i \(-0.492586\pi\)
0.0232889 + 0.999729i \(0.492586\pi\)
\(480\) 3.81315e22 0.617636
\(481\) 1.11739e23 1.77817
\(482\) 3.26307e22 0.510183
\(483\) −3.01068e21 −0.0462503
\(484\) 2.32772e22 0.351354
\(485\) −5.93984e21 −0.0880988
\(486\) 4.94603e22 0.720855
\(487\) 2.34513e22 0.335870 0.167935 0.985798i \(-0.446290\pi\)
0.167935 + 0.985798i \(0.446290\pi\)
\(488\) 1.49491e22 0.210401
\(489\) 7.65153e22 1.05833
\(490\) 2.17142e21 0.0295173
\(491\) 3.75824e22 0.502103 0.251051 0.967974i \(-0.419224\pi\)
0.251051 + 0.967974i \(0.419224\pi\)
\(492\) −1.53537e23 −2.01609
\(493\) −2.26740e22 −0.292637
\(494\) −6.85280e22 −0.869339
\(495\) 2.68300e22 0.334562
\(496\) 3.82925e22 0.469375
\(497\) 4.73366e22 0.570385
\(498\) 6.40072e22 0.758192
\(499\) 2.82679e21 0.0329184 0.0164592 0.999865i \(-0.494761\pi\)
0.0164592 + 0.999865i \(0.494761\pi\)
\(500\) −4.91817e22 −0.563065
\(501\) −1.38330e23 −1.55702
\(502\) −6.86840e22 −0.760105
\(503\) −4.74984e22 −0.516834 −0.258417 0.966033i \(-0.583201\pi\)
−0.258417 + 0.966033i \(0.583201\pi\)
\(504\) −3.46461e22 −0.370676
\(505\) −1.92648e22 −0.202669
\(506\) 2.97311e21 0.0307561
\(507\) 1.39964e23 1.42380
\(508\) −3.51700e22 −0.351828
\(509\) 3.19613e22 0.314430 0.157215 0.987564i \(-0.449748\pi\)
0.157215 + 0.987564i \(0.449748\pi\)
\(510\) −2.35798e22 −0.228136
\(511\) −9.75799e21 −0.0928501
\(512\) −6.25919e22 −0.585765
\(513\) −2.27833e22 −0.209710
\(514\) 9.80158e22 0.887380
\(515\) 7.42478e21 0.0661183
\(516\) −1.47475e23 −1.29180
\(517\) 7.01918e22 0.604807
\(518\) −2.83159e22 −0.240009
\(519\) 1.73987e23 1.45076
\(520\) 6.19209e22 0.507936
\(521\) −1.75671e23 −1.41768 −0.708841 0.705369i \(-0.750782\pi\)
−0.708841 + 0.705369i \(0.750782\pi\)
\(522\) −2.72561e22 −0.216404
\(523\) 2.05125e23 1.60234 0.801168 0.598439i \(-0.204212\pi\)
0.801168 + 0.598439i \(0.204212\pi\)
\(524\) 1.64956e23 1.26781
\(525\) 6.04193e22 0.456901
\(526\) −7.03318e22 −0.523326
\(527\) −1.58608e23 −1.16128
\(528\) −4.50984e22 −0.324917
\(529\) −1.40052e23 −0.992927
\(530\) 3.48702e22 0.243282
\(531\) −1.96428e23 −1.34865
\(532\) −5.14589e22 −0.347705
\(533\) −3.91986e23 −2.60669
\(534\) 2.82538e22 0.184917
\(535\) −1.48657e22 −0.0957585
\(536\) 1.50990e23 0.957297
\(537\) −3.77335e22 −0.235476
\(538\) 1.18490e22 0.0727832
\(539\) −1.72019e22 −0.104009
\(540\) 8.80723e21 0.0524195
\(541\) 1.41206e23 0.827325 0.413663 0.910430i \(-0.364249\pi\)
0.413663 + 0.910430i \(0.364249\pi\)
\(542\) 2.00639e22 0.115723
\(543\) 2.20436e23 1.25165
\(544\) 1.40085e23 0.783073
\(545\) −1.48754e23 −0.818651
\(546\) −7.17148e22 −0.388572
\(547\) 3.33679e23 1.78007 0.890033 0.455896i \(-0.150681\pi\)
0.890033 + 0.455896i \(0.150681\pi\)
\(548\) 2.76807e23 1.45393
\(549\) 5.17706e22 0.267743
\(550\) −5.96654e22 −0.303836
\(551\) −9.46272e22 −0.474490
\(552\) 2.17562e22 0.107424
\(553\) 8.10516e22 0.394092
\(554\) −3.69242e22 −0.176798
\(555\) 1.60460e23 0.756619
\(556\) 4.01836e22 0.186600
\(557\) 2.77830e23 1.27061 0.635303 0.772263i \(-0.280875\pi\)
0.635303 + 0.772263i \(0.280875\pi\)
\(558\) −1.90661e23 −0.858760
\(559\) −3.76509e23 −1.67023
\(560\) 1.09139e22 0.0476849
\(561\) 1.86798e23 0.803875
\(562\) 5.14985e22 0.218291
\(563\) −2.71407e22 −0.113318 −0.0566590 0.998394i \(-0.518045\pi\)
−0.0566590 + 0.998394i \(0.518045\pi\)
\(564\) 2.19741e23 0.903731
\(565\) 9.02754e22 0.365727
\(566\) 8.06491e22 0.321855
\(567\) 8.35597e22 0.328504
\(568\) −3.42069e23 −1.32481
\(569\) −1.86156e23 −0.710270 −0.355135 0.934815i \(-0.615565\pi\)
−0.355135 + 0.934815i \(0.615565\pi\)
\(570\) −9.84077e22 −0.369907
\(571\) −5.54352e22 −0.205295 −0.102648 0.994718i \(-0.532731\pi\)
−0.102648 + 0.994718i \(0.532731\pi\)
\(572\) −2.09857e23 −0.765698
\(573\) −3.02138e23 −1.08615
\(574\) 9.93329e22 0.351838
\(575\) −2.00199e22 −0.0698691
\(576\) 6.87567e22 0.236442
\(577\) −1.70022e23 −0.576117 −0.288059 0.957613i \(-0.593010\pi\)
−0.288059 + 0.957613i \(0.593010\pi\)
\(578\) 6.38126e22 0.213069
\(579\) −4.88919e23 −1.60868
\(580\) 3.65796e22 0.118605
\(581\) 1.22710e23 0.392086
\(582\) −4.97122e22 −0.156537
\(583\) −2.76240e23 −0.857244
\(584\) 7.05143e22 0.215659
\(585\) 2.14439e23 0.646367
\(586\) 1.10035e23 0.326891
\(587\) 3.25116e23 0.951951 0.475975 0.879459i \(-0.342095\pi\)
0.475975 + 0.879459i \(0.342095\pi\)
\(588\) −5.38519e22 −0.155415
\(589\) −6.61934e23 −1.88293
\(590\) −8.89631e22 −0.249441
\(591\) 5.09960e23 1.40943
\(592\) −1.42320e23 −0.387731
\(593\) −3.66434e23 −0.984081 −0.492041 0.870572i \(-0.663749\pi\)
−0.492041 + 0.870572i \(0.663749\pi\)
\(594\) 2.35452e22 0.0623330
\(595\) −4.52055e22 −0.117977
\(596\) 5.25018e23 1.35077
\(597\) 5.44448e23 1.38094
\(598\) 2.37626e22 0.0594202
\(599\) −6.01696e23 −1.48337 −0.741684 0.670750i \(-0.765973\pi\)
−0.741684 + 0.670750i \(0.765973\pi\)
\(600\) −4.36609e23 −1.06122
\(601\) 3.50996e23 0.841142 0.420571 0.907260i \(-0.361830\pi\)
0.420571 + 0.907260i \(0.361830\pi\)
\(602\) 9.54111e22 0.225439
\(603\) 5.22895e23 1.21820
\(604\) −2.53904e23 −0.583250
\(605\) 8.53393e22 0.193298
\(606\) −1.61233e23 −0.360111
\(607\) −2.25258e21 −0.00496107 −0.00248054 0.999997i \(-0.500790\pi\)
−0.00248054 + 0.999997i \(0.500790\pi\)
\(608\) 5.84630e23 1.26970
\(609\) −9.90276e22 −0.212085
\(610\) 2.34471e22 0.0495206
\(611\) 5.61009e23 1.16847
\(612\) 3.08572e23 0.633825
\(613\) −2.91522e23 −0.590550 −0.295275 0.955412i \(-0.595411\pi\)
−0.295275 + 0.955412i \(0.595411\pi\)
\(614\) 3.46820e23 0.692903
\(615\) −5.62900e23 −1.10916
\(616\) 1.24306e23 0.241577
\(617\) −9.48194e23 −1.81750 −0.908748 0.417346i \(-0.862960\pi\)
−0.908748 + 0.417346i \(0.862960\pi\)
\(618\) 6.21401e22 0.117482
\(619\) 3.06667e23 0.571870 0.285935 0.958249i \(-0.407696\pi\)
0.285935 + 0.958249i \(0.407696\pi\)
\(620\) 2.55881e23 0.470661
\(621\) 7.90028e21 0.0143339
\(622\) −6.97018e22 −0.124746
\(623\) 5.41660e22 0.0956265
\(624\) −3.60449e23 −0.627733
\(625\) 3.03278e23 0.521028
\(626\) −6.78178e22 −0.114937
\(627\) 7.79581e23 1.30343
\(628\) 1.00558e23 0.165867
\(629\) 5.89491e23 0.959283
\(630\) −5.43410e22 −0.0872434
\(631\) 7.21382e23 1.14266 0.571329 0.820721i \(-0.306428\pi\)
0.571329 + 0.820721i \(0.306428\pi\)
\(632\) −5.85705e23 −0.915342
\(633\) −7.13563e22 −0.110027
\(634\) 2.39932e23 0.365031
\(635\) −1.28941e23 −0.193559
\(636\) −8.64793e23 −1.28093
\(637\) −1.37486e23 −0.200943
\(638\) 9.77918e22 0.141035
\(639\) −1.18463e24 −1.68587
\(640\) −2.71125e23 −0.380750
\(641\) 6.65756e22 0.0922618 0.0461309 0.998935i \(-0.485311\pi\)
0.0461309 + 0.998935i \(0.485311\pi\)
\(642\) −1.24415e23 −0.170147
\(643\) 5.21871e23 0.704320 0.352160 0.935940i \(-0.385447\pi\)
0.352160 + 0.935940i \(0.385447\pi\)
\(644\) 1.78438e22 0.0237660
\(645\) −5.40675e23 −0.710688
\(646\) −3.61525e23 −0.468989
\(647\) 3.24456e22 0.0415403 0.0207702 0.999784i \(-0.493388\pi\)
0.0207702 + 0.999784i \(0.493388\pi\)
\(648\) −6.03829e23 −0.763003
\(649\) 7.04761e23 0.878946
\(650\) −4.76876e23 −0.587005
\(651\) −6.92715e23 −0.841621
\(652\) −4.53492e23 −0.543832
\(653\) 6.71134e23 0.794414 0.397207 0.917729i \(-0.369979\pi\)
0.397207 + 0.917729i \(0.369979\pi\)
\(654\) −1.24497e24 −1.45461
\(655\) 6.04767e23 0.697488
\(656\) 4.99262e23 0.568390
\(657\) 2.44199e23 0.274434
\(658\) −1.42165e23 −0.157715
\(659\) 1.18160e24 1.29403 0.647013 0.762479i \(-0.276018\pi\)
0.647013 + 0.762479i \(0.276018\pi\)
\(660\) −3.01359e23 −0.325808
\(661\) −1.60861e24 −1.71687 −0.858435 0.512923i \(-0.828563\pi\)
−0.858435 + 0.512923i \(0.828563\pi\)
\(662\) 8.69983e23 0.916681
\(663\) 1.49299e24 1.55307
\(664\) −8.86739e23 −0.910683
\(665\) −1.88660e23 −0.191291
\(666\) 7.08621e23 0.709386
\(667\) 3.28127e22 0.0324319
\(668\) 8.19854e23 0.800086
\(669\) −1.37564e24 −1.32551
\(670\) 2.36821e23 0.225312
\(671\) −1.85747e23 −0.174494
\(672\) 6.11818e23 0.567523
\(673\) 6.49909e23 0.595284 0.297642 0.954678i \(-0.403800\pi\)
0.297642 + 0.954678i \(0.403800\pi\)
\(674\) −4.38143e23 −0.396284
\(675\) −1.58545e23 −0.141603
\(676\) −8.29543e23 −0.731631
\(677\) 8.20394e23 0.714527 0.357264 0.934004i \(-0.383710\pi\)
0.357264 + 0.934004i \(0.383710\pi\)
\(678\) 7.55541e23 0.649839
\(679\) −9.53045e22 −0.0809506
\(680\) 3.26669e23 0.274020
\(681\) 5.27908e23 0.437328
\(682\) 6.84071e23 0.559672
\(683\) −1.13385e23 −0.0916180 −0.0458090 0.998950i \(-0.514587\pi\)
−0.0458090 + 0.998950i \(0.514587\pi\)
\(684\) 1.28779e24 1.02770
\(685\) 1.01484e24 0.799883
\(686\) 3.48403e22 0.0271223
\(687\) −9.20534e23 −0.707795
\(688\) 4.79551e23 0.364194
\(689\) −2.20785e24 −1.65618
\(690\) 3.41236e22 0.0252836
\(691\) −1.72294e24 −1.26097 −0.630486 0.776201i \(-0.717144\pi\)
−0.630486 + 0.776201i \(0.717144\pi\)
\(692\) −1.03119e24 −0.745483
\(693\) 4.30487e23 0.307416
\(694\) 8.35698e23 0.589513
\(695\) 1.47322e23 0.102659
\(696\) 7.15605e23 0.492600
\(697\) −2.06795e24 −1.40625
\(698\) 7.08774e23 0.476142
\(699\) 9.03578e23 0.599666
\(700\) −3.58094e23 −0.234782
\(701\) −3.05383e23 −0.197807 −0.0989034 0.995097i \(-0.531533\pi\)
−0.0989034 + 0.995097i \(0.531533\pi\)
\(702\) 1.88185e23 0.120426
\(703\) 2.46017e24 1.55541
\(704\) −2.46691e23 −0.154094
\(705\) 8.05621e23 0.497191
\(706\) 8.95291e23 0.545914
\(707\) −3.09103e23 −0.186225
\(708\) 2.20631e24 1.31336
\(709\) −1.59973e24 −0.940925 −0.470462 0.882420i \(-0.655913\pi\)
−0.470462 + 0.882420i \(0.655913\pi\)
\(710\) −5.36522e23 −0.311811
\(711\) −2.02836e24 −1.16481
\(712\) −3.91421e23 −0.222108
\(713\) 2.29531e23 0.128700
\(714\) −3.78338e23 −0.209626
\(715\) −7.69383e23 −0.421251
\(716\) 2.23640e23 0.121001
\(717\) 4.10314e24 2.19384
\(718\) −4.07468e23 −0.215296
\(719\) −2.58321e24 −1.34885 −0.674425 0.738343i \(-0.735609\pi\)
−0.674425 + 0.738343i \(0.735609\pi\)
\(720\) −2.73126e23 −0.140941
\(721\) 1.19130e23 0.0607536
\(722\) −5.12140e23 −0.258121
\(723\) −2.96690e24 −1.47784
\(724\) −1.30648e24 −0.643171
\(725\) −6.58496e23 −0.320391
\(726\) 7.14229e23 0.343460
\(727\) −2.76273e24 −1.31309 −0.656547 0.754285i \(-0.727984\pi\)
−0.656547 + 0.754285i \(0.727984\pi\)
\(728\) 9.93518e23 0.466723
\(729\) −2.62524e24 −1.21895
\(730\) 1.10599e23 0.0507582
\(731\) −1.98631e24 −0.901050
\(732\) −5.81496e23 −0.260737
\(733\) 2.01898e24 0.894845 0.447423 0.894323i \(-0.352342\pi\)
0.447423 + 0.894323i \(0.352342\pi\)
\(734\) −1.77696e24 −0.778503
\(735\) −1.97433e23 −0.0855022
\(736\) −2.02725e23 −0.0867853
\(737\) −1.87609e24 −0.793925
\(738\) −2.48586e24 −1.03992
\(739\) 4.01532e24 1.66051 0.830256 0.557382i \(-0.188194\pi\)
0.830256 + 0.557382i \(0.188194\pi\)
\(740\) −9.51019e23 −0.388794
\(741\) 6.23081e24 2.51820
\(742\) 5.59491e23 0.223543
\(743\) 3.85515e24 1.52278 0.761388 0.648296i \(-0.224518\pi\)
0.761388 + 0.648296i \(0.224518\pi\)
\(744\) 5.00578e24 1.95480
\(745\) 1.92483e24 0.743130
\(746\) 3.73730e23 0.142652
\(747\) −3.07088e24 −1.15888
\(748\) −1.10712e24 −0.413077
\(749\) −2.38519e23 −0.0879888
\(750\) −1.50908e24 −0.550415
\(751\) 4.15814e24 1.49954 0.749772 0.661696i \(-0.230163\pi\)
0.749772 + 0.661696i \(0.230163\pi\)
\(752\) −7.14543e23 −0.254787
\(753\) 6.24499e24 2.20178
\(754\) 7.81602e23 0.272477
\(755\) −9.30867e23 −0.320877
\(756\) 1.41312e23 0.0481662
\(757\) −9.26353e23 −0.312221 −0.156110 0.987740i \(-0.549896\pi\)
−0.156110 + 0.987740i \(0.549896\pi\)
\(758\) −9.92455e23 −0.330767
\(759\) −2.70326e23 −0.0890907
\(760\) 1.36331e24 0.444304
\(761\) −1.38871e24 −0.447551 −0.223776 0.974641i \(-0.571838\pi\)
−0.223776 + 0.974641i \(0.571838\pi\)
\(762\) −1.07914e24 −0.343924
\(763\) −2.38675e24 −0.752227
\(764\) 1.79072e24 0.558128
\(765\) 1.13129e24 0.348701
\(766\) 9.45967e23 0.288358
\(767\) 5.63281e24 1.69811
\(768\) −3.30203e24 −0.984489
\(769\) −2.57456e23 −0.0759151 −0.0379576 0.999279i \(-0.512085\pi\)
−0.0379576 + 0.999279i \(0.512085\pi\)
\(770\) 1.94969e23 0.0568584
\(771\) −8.91194e24 −2.57046
\(772\) 2.89773e24 0.826631
\(773\) −8.82985e23 −0.249131 −0.124565 0.992211i \(-0.539754\pi\)
−0.124565 + 0.992211i \(0.539754\pi\)
\(774\) −2.38772e24 −0.666323
\(775\) −4.60629e24 −1.27141
\(776\) 6.88700e23 0.188021
\(777\) 2.57458e24 0.695228
\(778\) −2.10998e24 −0.563575
\(779\) −8.63036e24 −2.28014
\(780\) −2.40862e24 −0.629454
\(781\) 4.25030e24 1.09872
\(782\) 1.25362e23 0.0320559
\(783\) 2.59856e23 0.0657292
\(784\) 1.75113e23 0.0438158
\(785\) 3.68668e23 0.0912523
\(786\) 5.06147e24 1.23932
\(787\) 5.12110e24 1.24045 0.620223 0.784426i \(-0.287042\pi\)
0.620223 + 0.784426i \(0.287042\pi\)
\(788\) −3.02244e24 −0.724244
\(789\) 6.39481e24 1.51591
\(790\) −9.18654e23 −0.215438
\(791\) 1.44846e24 0.336053
\(792\) −3.11083e24 −0.714024
\(793\) −1.48458e24 −0.337118
\(794\) 1.54782e24 0.347734
\(795\) −3.17052e24 −0.704710
\(796\) −3.22685e24 −0.709606
\(797\) −3.84853e24 −0.837335 −0.418667 0.908140i \(-0.637503\pi\)
−0.418667 + 0.908140i \(0.637503\pi\)
\(798\) −1.57895e24 −0.339894
\(799\) 2.95965e24 0.630366
\(800\) 4.06835e24 0.857341
\(801\) −1.35554e24 −0.282641
\(802\) −2.32766e24 −0.480218
\(803\) −8.76159e23 −0.178855
\(804\) −5.87325e24 −1.18632
\(805\) 6.54192e22 0.0130750
\(806\) 5.46744e24 1.08128
\(807\) −1.07735e24 −0.210830
\(808\) 2.23368e24 0.432538
\(809\) −6.88172e23 −0.131867 −0.0659333 0.997824i \(-0.521002\pi\)
−0.0659333 + 0.997824i \(0.521002\pi\)
\(810\) −9.47081e23 −0.179583
\(811\) 4.38304e24 0.822429 0.411215 0.911539i \(-0.365105\pi\)
0.411215 + 0.911539i \(0.365105\pi\)
\(812\) 5.86919e23 0.108981
\(813\) −1.82428e24 −0.335214
\(814\) −2.54245e24 −0.462322
\(815\) −1.66260e24 −0.299191
\(816\) −1.90158e24 −0.338648
\(817\) −8.28962e24 −1.46099
\(818\) −1.55987e24 −0.272073
\(819\) 3.44067e24 0.593922
\(820\) 3.33620e24 0.569948
\(821\) 6.99923e24 1.18341 0.591703 0.806156i \(-0.298456\pi\)
0.591703 + 0.806156i \(0.298456\pi\)
\(822\) 8.49347e24 1.42126
\(823\) −4.65710e24 −0.771288 −0.385644 0.922648i \(-0.626021\pi\)
−0.385644 + 0.922648i \(0.626021\pi\)
\(824\) −8.60873e23 −0.141110
\(825\) 5.42499e24 0.880116
\(826\) −1.42741e24 −0.229202
\(827\) 3.76035e24 0.597629 0.298815 0.954311i \(-0.403409\pi\)
0.298815 + 0.954311i \(0.403409\pi\)
\(828\) −4.46551e23 −0.0702446
\(829\) 8.04278e24 1.25225 0.626127 0.779721i \(-0.284639\pi\)
0.626127 + 0.779721i \(0.284639\pi\)
\(830\) −1.39081e24 −0.214341
\(831\) 3.35728e24 0.512128
\(832\) −1.97168e24 −0.297706
\(833\) −7.25320e23 −0.108404
\(834\) 1.23298e24 0.182408
\(835\) 3.00577e24 0.440170
\(836\) −4.62044e24 −0.669776
\(837\) 1.81774e24 0.260835
\(838\) 1.83041e24 0.260000
\(839\) 9.27061e24 1.30356 0.651781 0.758407i \(-0.274022\pi\)
0.651781 + 0.758407i \(0.274022\pi\)
\(840\) 1.42671e24 0.198592
\(841\) −6.17787e24 −0.851281
\(842\) −3.75926e24 −0.512802
\(843\) −4.68243e24 −0.632319
\(844\) 4.22916e23 0.0565383
\(845\) −3.04129e24 −0.402509
\(846\) 3.55776e24 0.466153
\(847\) 1.36927e24 0.177615
\(848\) 2.81209e24 0.361131
\(849\) −7.33290e24 −0.932311
\(850\) −2.51580e24 −0.316676
\(851\) −8.53084e23 −0.106314
\(852\) 1.33059e25 1.64176
\(853\) −1.02012e25 −1.24619 −0.623094 0.782147i \(-0.714124\pi\)
−0.623094 + 0.782147i \(0.714124\pi\)
\(854\) 3.76208e23 0.0455026
\(855\) 4.72132e24 0.565394
\(856\) 1.72361e24 0.204368
\(857\) −1.52446e25 −1.78970 −0.894848 0.446370i \(-0.852716\pi\)
−0.894848 + 0.446370i \(0.852716\pi\)
\(858\) −6.43919e24 −0.748496
\(859\) 7.61607e24 0.876575 0.438287 0.898835i \(-0.355585\pi\)
0.438287 + 0.898835i \(0.355585\pi\)
\(860\) 3.20449e24 0.365192
\(861\) −9.03170e24 −1.01916
\(862\) −1.06532e23 −0.0119033
\(863\) −1.50221e24 −0.166203 −0.0831016 0.996541i \(-0.526483\pi\)
−0.0831016 + 0.996541i \(0.526483\pi\)
\(864\) −1.60546e24 −0.175886
\(865\) −3.78058e24 −0.410130
\(866\) 4.82344e23 0.0518149
\(867\) −5.80207e24 −0.617192
\(868\) 4.10560e24 0.432473
\(869\) 7.27753e24 0.759130
\(870\) 1.12240e24 0.115940
\(871\) −1.49946e25 −1.53385
\(872\) 1.72474e25 1.74717
\(873\) 2.38505e24 0.239263
\(874\) 5.23183e23 0.0519764
\(875\) −2.89308e24 −0.284638
\(876\) −2.74289e24 −0.267253
\(877\) −1.14810e25 −1.10786 −0.553928 0.832564i \(-0.686872\pi\)
−0.553928 + 0.832564i \(0.686872\pi\)
\(878\) −1.16842e24 −0.111659
\(879\) −1.00048e25 −0.946899
\(880\) 9.79944e23 0.0918541
\(881\) 7.47390e24 0.693828 0.346914 0.937897i \(-0.387230\pi\)
0.346914 + 0.937897i \(0.387230\pi\)
\(882\) −8.71899e23 −0.0801647
\(883\) −6.25058e24 −0.569186 −0.284593 0.958648i \(-0.591858\pi\)
−0.284593 + 0.958648i \(0.591858\pi\)
\(884\) −8.84867e24 −0.798056
\(885\) 8.08884e24 0.722551
\(886\) −2.78929e24 −0.246779
\(887\) 1.91184e25 1.67533 0.837664 0.546186i \(-0.183921\pi\)
0.837664 + 0.546186i \(0.183921\pi\)
\(888\) −1.86047e25 −1.61478
\(889\) −2.06885e24 −0.177854
\(890\) −6.13928e23 −0.0522760
\(891\) 7.50273e24 0.632789
\(892\) 8.15315e24 0.681121
\(893\) 1.23518e25 1.02210
\(894\) 1.61095e25 1.32042
\(895\) 8.19913e23 0.0665690
\(896\) −4.35019e24 −0.349856
\(897\) −2.16058e24 −0.172121
\(898\) −4.32935e24 −0.341644
\(899\) 7.54974e24 0.590167
\(900\) 8.96152e24 0.693938
\(901\) −1.16477e25 −0.893471
\(902\) 8.91899e24 0.677736
\(903\) −8.67512e24 −0.653024
\(904\) −1.04671e25 −0.780537
\(905\) −4.78986e24 −0.353842
\(906\) −7.79069e24 −0.570146
\(907\) 2.92316e24 0.211929 0.105964 0.994370i \(-0.466207\pi\)
0.105964 + 0.994370i \(0.466207\pi\)
\(908\) −3.12881e24 −0.224724
\(909\) 7.73549e24 0.550421
\(910\) 1.55829e24 0.109849
\(911\) 1.06255e25 0.742068 0.371034 0.928619i \(-0.379003\pi\)
0.371034 + 0.928619i \(0.379003\pi\)
\(912\) −7.93603e24 −0.549095
\(913\) 1.10180e25 0.755266
\(914\) −3.79493e24 −0.257727
\(915\) −2.13189e24 −0.143445
\(916\) 5.45584e24 0.363706
\(917\) 9.70346e24 0.640896
\(918\) 9.92788e23 0.0649671
\(919\) 2.50761e25 1.62584 0.812920 0.582376i \(-0.197877\pi\)
0.812920 + 0.582376i \(0.197877\pi\)
\(920\) −4.72740e23 −0.0303687
\(921\) −3.15341e25 −2.00712
\(922\) 9.79725e24 0.617862
\(923\) 3.39706e25 2.12270
\(924\) −4.83530e24 −0.299372
\(925\) 1.71200e25 1.05026
\(926\) −1.74731e24 −0.106212
\(927\) −2.98130e24 −0.179568
\(928\) −6.66805e24 −0.397962
\(929\) −2.11336e24 −0.124980 −0.0624898 0.998046i \(-0.519904\pi\)
−0.0624898 + 0.998046i \(0.519904\pi\)
\(930\) 7.85137e24 0.460087
\(931\) −3.02704e24 −0.175770
\(932\) −5.35534e24 −0.308143
\(933\) 6.33753e24 0.361349
\(934\) 7.98078e24 0.450917
\(935\) −4.05895e24 −0.227256
\(936\) −2.48634e25 −1.37948
\(937\) −2.92602e25 −1.60876 −0.804379 0.594117i \(-0.797502\pi\)
−0.804379 + 0.594117i \(0.797502\pi\)
\(938\) 3.79979e24 0.207031
\(939\) 6.16623e24 0.332937
\(940\) −4.77477e24 −0.255485
\(941\) −2.89238e25 −1.53371 −0.766856 0.641819i \(-0.778180\pi\)
−0.766856 + 0.641819i \(0.778180\pi\)
\(942\) 3.08549e24 0.162141
\(943\) 2.99265e24 0.155850
\(944\) −7.17438e24 −0.370273
\(945\) 5.18080e23 0.0264988
\(946\) 8.56686e24 0.434257
\(947\) −3.39090e25 −1.70349 −0.851746 0.523954i \(-0.824456\pi\)
−0.851746 + 0.523954i \(0.824456\pi\)
\(948\) 2.27829e25 1.13433
\(949\) −7.00271e24 −0.345544
\(950\) −1.04994e25 −0.513468
\(951\) −2.18155e25 −1.05738
\(952\) 5.24139e24 0.251786
\(953\) −2.63092e24 −0.125262 −0.0626308 0.998037i \(-0.519949\pi\)
−0.0626308 + 0.998037i \(0.519949\pi\)
\(954\) −1.40016e25 −0.660718
\(955\) 6.56517e24 0.307056
\(956\) −2.43186e25 −1.12732
\(957\) −8.89158e24 −0.408533
\(958\) −5.13760e23 −0.0233966
\(959\) 1.62830e25 0.734982
\(960\) −2.83138e24 −0.126675
\(961\) 3.02616e25 1.34197
\(962\) −2.03205e25 −0.893197
\(963\) 5.96908e24 0.260066
\(964\) 1.75843e25 0.759397
\(965\) 1.06237e25 0.454774
\(966\) 5.47513e23 0.0232321
\(967\) −4.60012e25 −1.93484 −0.967419 0.253181i \(-0.918523\pi\)
−0.967419 + 0.253181i \(0.918523\pi\)
\(968\) −9.89475e24 −0.412538
\(969\) 3.28712e25 1.35851
\(970\) 1.08020e24 0.0442531
\(971\) −7.14898e24 −0.290322 −0.145161 0.989408i \(-0.546370\pi\)
−0.145161 + 0.989408i \(0.546370\pi\)
\(972\) 2.66535e25 1.07298
\(973\) 2.36377e24 0.0943293
\(974\) −4.26477e24 −0.168712
\(975\) 4.33592e25 1.70037
\(976\) 1.89088e24 0.0735089
\(977\) 3.25867e25 1.25585 0.627923 0.778276i \(-0.283905\pi\)
0.627923 + 0.778276i \(0.283905\pi\)
\(978\) −1.39148e25 −0.531614
\(979\) 4.86350e24 0.184203
\(980\) 1.17015e24 0.0439359
\(981\) 5.97299e25 2.22334
\(982\) −6.83461e24 −0.252212
\(983\) −3.29176e25 −1.20427 −0.602135 0.798395i \(-0.705683\pi\)
−0.602135 + 0.798395i \(0.705683\pi\)
\(984\) 6.52659e25 2.36716
\(985\) −1.10809e25 −0.398445
\(986\) 4.12341e24 0.146995
\(987\) 1.29261e25 0.456850
\(988\) −3.69289e25 −1.29399
\(989\) 2.87449e24 0.0998603
\(990\) −4.87921e24 −0.168055
\(991\) 1.54947e25 0.529124 0.264562 0.964369i \(-0.414773\pi\)
0.264562 + 0.964369i \(0.414773\pi\)
\(992\) −4.66442e25 −1.57924
\(993\) −7.91020e25 −2.65533
\(994\) −8.60847e24 −0.286511
\(995\) −1.18303e25 −0.390392
\(996\) 3.44927e25 1.12855
\(997\) 4.99514e25 1.62046 0.810231 0.586110i \(-0.199342\pi\)
0.810231 + 0.586110i \(0.199342\pi\)
\(998\) −5.14069e23 −0.0165353
\(999\) −6.75590e24 −0.215465
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 7.18.a.a.1.2 4
3.2 odd 2 63.18.a.b.1.3 4
4.3 odd 2 112.18.a.f.1.1 4
7.6 odd 2 49.18.a.c.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.18.a.a.1.2 4 1.1 even 1 trivial
49.18.a.c.1.2 4 7.6 odd 2
63.18.a.b.1.3 4 3.2 odd 2
112.18.a.f.1.1 4 4.3 odd 2