Properties

Label 138.3.b.a.91.8
Level $138$
Weight $3$
Character 138.91
Analytic conductor $3.760$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [138,3,Mod(91,138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(138, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("138.91");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 138.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(3.76022764817\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1358954496.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 91.8
Root \(1.98289i\) of defining polynomial
Character \(\chi\) \(=\) 138.91
Dual form 138.3.b.a.91.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +4.92225i q^{5} +2.44949 q^{6} +5.13851i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +4.92225i q^{5} +2.44949 q^{6} +5.13851i q^{7} +2.82843 q^{8} +3.00000 q^{9} +6.96111i q^{10} -7.71925i q^{11} +3.46410 q^{12} +0.944060 q^{13} +7.26695i q^{14} +8.52559i q^{15} +4.00000 q^{16} -27.6556i q^{17} +4.24264 q^{18} -9.08137i q^{19} +9.84450i q^{20} +8.90017i q^{21} -10.9167i q^{22} +(-20.0652 + 11.2423i) q^{23} +4.89898 q^{24} +0.771456 q^{25} +1.33510 q^{26} +5.19615 q^{27} +10.2770i q^{28} -14.4550 q^{29} +12.0570i q^{30} -0.830916 q^{31} +5.65685 q^{32} -13.3701i q^{33} -39.1110i q^{34} -25.2930 q^{35} +6.00000 q^{36} +23.8369i q^{37} -12.8430i q^{38} +1.63516 q^{39} +13.9222i q^{40} -26.8431 q^{41} +12.5867i q^{42} -26.7736i q^{43} -15.4385i q^{44} +14.7667i q^{45} +(-28.3764 + 15.8990i) q^{46} -38.2280 q^{47} +6.92820 q^{48} +22.5957 q^{49} +1.09100 q^{50} -47.9010i q^{51} +1.88812 q^{52} -92.3727i q^{53} +7.34847 q^{54} +37.9961 q^{55} +14.5339i q^{56} -15.7294i q^{57} -20.4425 q^{58} -47.8457 q^{59} +17.0512i q^{60} +121.824i q^{61} -1.17509 q^{62} +15.4155i q^{63} +8.00000 q^{64} +4.64690i q^{65} -18.9082i q^{66} +8.10477i q^{67} -55.3113i q^{68} +(-34.7539 + 19.4722i) q^{69} -35.7698 q^{70} -23.8139 q^{71} +8.48528 q^{72} +98.1035 q^{73} +33.7105i q^{74} +1.33620 q^{75} -18.1627i q^{76} +39.6654 q^{77} +2.31246 q^{78} -70.0030i q^{79} +19.6890i q^{80} +9.00000 q^{81} -37.9618 q^{82} +50.7347i q^{83} +17.8003i q^{84} +136.128 q^{85} -37.8636i q^{86} -25.0368 q^{87} -21.8333i q^{88} +87.6727i q^{89} +20.8833i q^{90} +4.85106i q^{91} +(-40.1304 + 22.4845i) q^{92} -1.43919 q^{93} -54.0626 q^{94} +44.7008 q^{95} +9.79796 q^{96} +36.7344i q^{97} +31.9551 q^{98} -23.1577i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{4} + 24 q^{9} + 16 q^{13} + 32 q^{16} + 16 q^{23} + 72 q^{25} + 32 q^{26} - 144 q^{29} - 128 q^{31} - 112 q^{35} + 48 q^{36} + 48 q^{39} - 16 q^{41} - 80 q^{46} - 112 q^{47} + 40 q^{49} - 160 q^{50} + 32 q^{52} - 64 q^{55} + 128 q^{58} + 80 q^{59} - 96 q^{62} + 64 q^{64} - 72 q^{69} - 144 q^{70} + 32 q^{71} + 64 q^{73} + 48 q^{75} + 224 q^{77} - 144 q^{78} + 72 q^{81} + 48 q^{85} + 96 q^{87} + 32 q^{92} + 192 q^{93} - 16 q^{94} + 112 q^{95} + 224 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 0.707107
\(3\) 1.73205 0.577350
\(4\) 2.00000 0.500000
\(5\) 4.92225i 0.984450i 0.870468 + 0.492225i \(0.163816\pi\)
−0.870468 + 0.492225i \(0.836184\pi\)
\(6\) 2.44949 0.408248
\(7\) 5.13851i 0.734073i 0.930206 + 0.367037i \(0.119628\pi\)
−0.930206 + 0.367037i \(0.880372\pi\)
\(8\) 2.82843 0.353553
\(9\) 3.00000 0.333333
\(10\) 6.96111i 0.696111i
\(11\) 7.71925i 0.701750i −0.936422 0.350875i \(-0.885884\pi\)
0.936422 0.350875i \(-0.114116\pi\)
\(12\) 3.46410 0.288675
\(13\) 0.944060 0.0726200 0.0363100 0.999341i \(-0.488440\pi\)
0.0363100 + 0.999341i \(0.488440\pi\)
\(14\) 7.26695i 0.519068i
\(15\) 8.52559i 0.568372i
\(16\) 4.00000 0.250000
\(17\) 27.6556i 1.62680i −0.581703 0.813401i \(-0.697613\pi\)
0.581703 0.813401i \(-0.302387\pi\)
\(18\) 4.24264 0.235702
\(19\) 9.08137i 0.477967i −0.971024 0.238983i \(-0.923186\pi\)
0.971024 0.238983i \(-0.0768141\pi\)
\(20\) 9.84450i 0.492225i
\(21\) 8.90017i 0.423817i
\(22\) 10.9167i 0.496212i
\(23\) −20.0652 + 11.2423i −0.872399 + 0.488794i
\(24\) 4.89898 0.204124
\(25\) 0.771456 0.0308582
\(26\) 1.33510 0.0513501
\(27\) 5.19615 0.192450
\(28\) 10.2770i 0.367037i
\(29\) −14.4550 −0.498449 −0.249225 0.968446i \(-0.580176\pi\)
−0.249225 + 0.968446i \(0.580176\pi\)
\(30\) 12.0570i 0.401900i
\(31\) −0.830916 −0.0268037 −0.0134019 0.999910i \(-0.504266\pi\)
−0.0134019 + 0.999910i \(0.504266\pi\)
\(32\) 5.65685 0.176777
\(33\) 13.3701i 0.405155i
\(34\) 39.1110i 1.15032i
\(35\) −25.2930 −0.722658
\(36\) 6.00000 0.166667
\(37\) 23.8369i 0.644241i 0.946699 + 0.322120i \(0.104396\pi\)
−0.946699 + 0.322120i \(0.895604\pi\)
\(38\) 12.8430i 0.337973i
\(39\) 1.63516 0.0419272
\(40\) 13.9222i 0.348056i
\(41\) −26.8431 −0.654709 −0.327354 0.944902i \(-0.606157\pi\)
−0.327354 + 0.944902i \(0.606157\pi\)
\(42\) 12.5867i 0.299684i
\(43\) 26.7736i 0.622642i −0.950305 0.311321i \(-0.899229\pi\)
0.950305 0.311321i \(-0.100771\pi\)
\(44\) 15.4385i 0.350875i
\(45\) 14.7667i 0.328150i
\(46\) −28.3764 + 15.8990i −0.616879 + 0.345630i
\(47\) −38.2280 −0.813363 −0.406681 0.913570i \(-0.633314\pi\)
−0.406681 + 0.913570i \(0.633314\pi\)
\(48\) 6.92820 0.144338
\(49\) 22.5957 0.461136
\(50\) 1.09100 0.0218201
\(51\) 47.9010i 0.939235i
\(52\) 1.88812 0.0363100
\(53\) 92.3727i 1.74288i −0.490502 0.871440i \(-0.663186\pi\)
0.490502 0.871440i \(-0.336814\pi\)
\(54\) 7.34847 0.136083
\(55\) 37.9961 0.690837
\(56\) 14.5339i 0.259534i
\(57\) 15.7294i 0.275954i
\(58\) −20.4425 −0.352457
\(59\) −47.8457 −0.810944 −0.405472 0.914108i \(-0.632893\pi\)
−0.405472 + 0.914108i \(0.632893\pi\)
\(60\) 17.0512i 0.284186i
\(61\) 121.824i 1.99711i 0.0537398 + 0.998555i \(0.482886\pi\)
−0.0537398 + 0.998555i \(0.517114\pi\)
\(62\) −1.17509 −0.0189531
\(63\) 15.4155i 0.244691i
\(64\) 8.00000 0.125000
\(65\) 4.64690i 0.0714907i
\(66\) 18.9082i 0.286488i
\(67\) 8.10477i 0.120967i 0.998169 + 0.0604834i \(0.0192642\pi\)
−0.998169 + 0.0604834i \(0.980736\pi\)
\(68\) 55.3113i 0.813401i
\(69\) −34.7539 + 19.4722i −0.503680 + 0.282205i
\(70\) −35.7698 −0.510997
\(71\) −23.8139 −0.335407 −0.167704 0.985837i \(-0.553635\pi\)
−0.167704 + 0.985837i \(0.553635\pi\)
\(72\) 8.48528 0.117851
\(73\) 98.1035 1.34388 0.671942 0.740604i \(-0.265460\pi\)
0.671942 + 0.740604i \(0.265460\pi\)
\(74\) 33.7105i 0.455547i
\(75\) 1.33620 0.0178160
\(76\) 18.1627i 0.238983i
\(77\) 39.6654 0.515136
\(78\) 2.31246 0.0296470
\(79\) 70.0030i 0.886114i −0.896493 0.443057i \(-0.853894\pi\)
0.896493 0.443057i \(-0.146106\pi\)
\(80\) 19.6890i 0.246112i
\(81\) 9.00000 0.111111
\(82\) −37.9618 −0.462949
\(83\) 50.7347i 0.611261i 0.952150 + 0.305631i \(0.0988672\pi\)
−0.952150 + 0.305631i \(0.901133\pi\)
\(84\) 17.8003i 0.211909i
\(85\) 136.128 1.60151
\(86\) 37.8636i 0.440274i
\(87\) −25.0368 −0.287780
\(88\) 21.8333i 0.248106i
\(89\) 87.6727i 0.985086i 0.870288 + 0.492543i \(0.163933\pi\)
−0.870288 + 0.492543i \(0.836067\pi\)
\(90\) 20.8833i 0.232037i
\(91\) 4.85106i 0.0533084i
\(92\) −40.1304 + 22.4845i −0.436200 + 0.244397i
\(93\) −1.43919 −0.0154751
\(94\) −54.0626 −0.575134
\(95\) 44.7008 0.470534
\(96\) 9.79796 0.102062
\(97\) 36.7344i 0.378705i 0.981909 + 0.189353i \(0.0606390\pi\)
−0.981909 + 0.189353i \(0.939361\pi\)
\(98\) 31.9551 0.326073
\(99\) 23.1577i 0.233917i
\(100\) 1.54291 0.0154291
\(101\) 189.428 1.87553 0.937764 0.347273i \(-0.112892\pi\)
0.937764 + 0.347273i \(0.112892\pi\)
\(102\) 67.7422i 0.664139i
\(103\) 85.5225i 0.830316i 0.909749 + 0.415158i \(0.136274\pi\)
−0.909749 + 0.415158i \(0.863726\pi\)
\(104\) 2.67020 0.0256750
\(105\) −43.8088 −0.417227
\(106\) 130.635i 1.23240i
\(107\) 138.422i 1.29366i −0.762634 0.646830i \(-0.776094\pi\)
0.762634 0.646830i \(-0.223906\pi\)
\(108\) 10.3923 0.0962250
\(109\) 49.7589i 0.456503i 0.973602 + 0.228252i \(0.0733009\pi\)
−0.973602 + 0.228252i \(0.926699\pi\)
\(110\) 53.7345 0.488496
\(111\) 41.2867i 0.371952i
\(112\) 20.5541i 0.183518i
\(113\) 189.178i 1.67414i 0.547096 + 0.837070i \(0.315733\pi\)
−0.547096 + 0.837070i \(0.684267\pi\)
\(114\) 22.2447i 0.195129i
\(115\) −55.3373 98.7658i −0.481193 0.858833i
\(116\) −28.9101 −0.249225
\(117\) 2.83218 0.0242067
\(118\) −67.6640 −0.573424
\(119\) 142.109 1.19419
\(120\) 24.1140i 0.200950i
\(121\) 61.4133 0.507548
\(122\) 172.285i 1.41217i
\(123\) −46.4935 −0.377996
\(124\) −1.66183 −0.0134019
\(125\) 126.854i 1.01483i
\(126\) 21.8009i 0.173023i
\(127\) −32.2016 −0.253556 −0.126778 0.991931i \(-0.540464\pi\)
−0.126778 + 0.991931i \(0.540464\pi\)
\(128\) 11.3137 0.0883883
\(129\) 46.3733i 0.359483i
\(130\) 6.57170i 0.0505516i
\(131\) 88.6957 0.677066 0.338533 0.940954i \(-0.390069\pi\)
0.338533 + 0.940954i \(0.390069\pi\)
\(132\) 26.7402i 0.202578i
\(133\) 46.6647 0.350863
\(134\) 11.4619i 0.0855364i
\(135\) 25.5768i 0.189457i
\(136\) 78.2220i 0.575162i
\(137\) 88.3772i 0.645089i −0.946554 0.322544i \(-0.895462\pi\)
0.946554 0.322544i \(-0.104538\pi\)
\(138\) −49.1495 + 27.5378i −0.356155 + 0.199549i
\(139\) −183.605 −1.32090 −0.660449 0.750871i \(-0.729634\pi\)
−0.660449 + 0.750871i \(0.729634\pi\)
\(140\) −50.5861 −0.361329
\(141\) −66.2129 −0.469595
\(142\) −33.6780 −0.237169
\(143\) 7.28743i 0.0509610i
\(144\) 12.0000 0.0833333
\(145\) 71.1513i 0.490698i
\(146\) 138.739 0.950269
\(147\) 39.1369 0.266237
\(148\) 47.6738i 0.322120i
\(149\) 86.5480i 0.580859i −0.956896 0.290430i \(-0.906202\pi\)
0.956896 0.290430i \(-0.0937982\pi\)
\(150\) 1.88967 0.0125978
\(151\) −263.765 −1.74679 −0.873393 0.487016i \(-0.838085\pi\)
−0.873393 + 0.487016i \(0.838085\pi\)
\(152\) 25.6860i 0.168987i
\(153\) 82.9669i 0.542268i
\(154\) 56.0954 0.364256
\(155\) 4.08998i 0.0263869i
\(156\) 3.27032 0.0209636
\(157\) 83.4996i 0.531845i −0.963994 0.265922i \(-0.914324\pi\)
0.963994 0.265922i \(-0.0856765\pi\)
\(158\) 98.9992i 0.626577i
\(159\) 159.994i 1.00625i
\(160\) 27.8445i 0.174028i
\(161\) −57.7685 103.105i −0.358811 0.640405i
\(162\) 12.7279 0.0785674
\(163\) −178.643 −1.09597 −0.547986 0.836487i \(-0.684605\pi\)
−0.547986 + 0.836487i \(0.684605\pi\)
\(164\) −53.6861 −0.327354
\(165\) 65.8111 0.398855
\(166\) 71.7497i 0.432227i
\(167\) −125.596 −0.752072 −0.376036 0.926605i \(-0.622713\pi\)
−0.376036 + 0.926605i \(0.622713\pi\)
\(168\) 25.1735i 0.149842i
\(169\) −168.109 −0.994726
\(170\) 192.514 1.13244
\(171\) 27.2441i 0.159322i
\(172\) 53.5472i 0.311321i
\(173\) 145.986 0.843849 0.421925 0.906631i \(-0.361355\pi\)
0.421925 + 0.906631i \(0.361355\pi\)
\(174\) −35.4074 −0.203491
\(175\) 3.96414i 0.0226522i
\(176\) 30.8770i 0.175437i
\(177\) −82.8712 −0.468199
\(178\) 123.988i 0.696561i
\(179\) −152.764 −0.853428 −0.426714 0.904387i \(-0.640329\pi\)
−0.426714 + 0.904387i \(0.640329\pi\)
\(180\) 29.5335i 0.164075i
\(181\) 133.516i 0.737658i −0.929497 0.368829i \(-0.879759\pi\)
0.929497 0.368829i \(-0.120241\pi\)
\(182\) 6.86044i 0.0376947i
\(183\) 211.005i 1.15303i
\(184\) −56.7529 + 31.7979i −0.308440 + 0.172815i
\(185\) −117.331 −0.634223
\(186\) −2.03532 −0.0109426
\(187\) −213.481 −1.14161
\(188\) −76.4561 −0.406681
\(189\) 26.7005i 0.141272i
\(190\) 63.2164 0.332718
\(191\) 271.502i 1.42148i 0.703456 + 0.710739i \(0.251639\pi\)
−0.703456 + 0.710739i \(0.748361\pi\)
\(192\) 13.8564 0.0721688
\(193\) 242.088 1.25434 0.627172 0.778881i \(-0.284212\pi\)
0.627172 + 0.778881i \(0.284212\pi\)
\(194\) 51.9503i 0.267785i
\(195\) 8.04866i 0.0412752i
\(196\) 45.1914 0.230568
\(197\) −251.316 −1.27571 −0.637857 0.770155i \(-0.720179\pi\)
−0.637857 + 0.770155i \(0.720179\pi\)
\(198\) 32.7500i 0.165404i
\(199\) 176.314i 0.885999i −0.896522 0.443000i \(-0.853914\pi\)
0.896522 0.443000i \(-0.146086\pi\)
\(200\) 2.18201 0.0109100
\(201\) 14.0379i 0.0698402i
\(202\) 267.892 1.32620
\(203\) 74.2773i 0.365898i
\(204\) 95.8020i 0.469617i
\(205\) 132.128i 0.644528i
\(206\) 120.947i 0.587122i
\(207\) −60.1955 + 33.7268i −0.290800 + 0.162931i
\(208\) 3.77624 0.0181550
\(209\) −70.1013 −0.335413
\(210\) −61.9551 −0.295024
\(211\) 215.285 1.02031 0.510155 0.860082i \(-0.329588\pi\)
0.510155 + 0.860082i \(0.329588\pi\)
\(212\) 184.745i 0.871440i
\(213\) −41.2469 −0.193647
\(214\) 195.758i 0.914756i
\(215\) 131.786 0.612960
\(216\) 14.6969 0.0680414
\(217\) 4.26967i 0.0196759i
\(218\) 70.3697i 0.322797i
\(219\) 169.920 0.775892
\(220\) 75.9921 0.345419
\(221\) 26.1086i 0.118138i
\(222\) 58.3882i 0.263010i
\(223\) 237.892 1.06678 0.533391 0.845869i \(-0.320918\pi\)
0.533391 + 0.845869i \(0.320918\pi\)
\(224\) 29.0678i 0.129767i
\(225\) 2.31437 0.0102861
\(226\) 267.538i 1.18380i
\(227\) 1.80104i 0.00793410i −0.999992 0.00396705i \(-0.998737\pi\)
0.999992 0.00396705i \(-0.00126275\pi\)
\(228\) 31.4588i 0.137977i
\(229\) 230.994i 1.00871i 0.863497 + 0.504353i \(0.168269\pi\)
−0.863497 + 0.504353i \(0.831731\pi\)
\(230\) −78.2587 139.676i −0.340255 0.607287i
\(231\) 68.7026 0.297414
\(232\) −40.8850 −0.176228
\(233\) −50.0293 −0.214718 −0.107359 0.994220i \(-0.534239\pi\)
−0.107359 + 0.994220i \(0.534239\pi\)
\(234\) 4.00531 0.0171167
\(235\) 188.168i 0.800715i
\(236\) −95.6914 −0.405472
\(237\) 121.249i 0.511598i
\(238\) 200.972 0.844422
\(239\) 405.258 1.69564 0.847820 0.530284i \(-0.177915\pi\)
0.847820 + 0.530284i \(0.177915\pi\)
\(240\) 34.1023i 0.142093i
\(241\) 348.532i 1.44619i 0.690748 + 0.723095i \(0.257281\pi\)
−0.690748 + 0.723095i \(0.742719\pi\)
\(242\) 86.8515 0.358890
\(243\) 15.5885 0.0641500
\(244\) 243.647i 0.998555i
\(245\) 111.222i 0.453966i
\(246\) −65.7518 −0.267284
\(247\) 8.57335i 0.0347099i
\(248\) −2.35019 −0.00947655
\(249\) 87.8750i 0.352912i
\(250\) 179.398i 0.717592i
\(251\) 5.76047i 0.0229501i −0.999934 0.0114750i \(-0.996347\pi\)
0.999934 0.0114750i \(-0.00365270\pi\)
\(252\) 30.8311i 0.122346i
\(253\) 86.7818 + 154.888i 0.343011 + 0.612206i
\(254\) −45.5399 −0.179291
\(255\) 235.781 0.924630
\(256\) 16.0000 0.0625000
\(257\) 26.3305 0.102453 0.0512267 0.998687i \(-0.483687\pi\)
0.0512267 + 0.998687i \(0.483687\pi\)
\(258\) 65.5817i 0.254193i
\(259\) −122.486 −0.472920
\(260\) 9.29379i 0.0357454i
\(261\) −43.3651 −0.166150
\(262\) 125.435 0.478758
\(263\) 61.0826i 0.232253i −0.993234 0.116127i \(-0.962952\pi\)
0.993234 0.116127i \(-0.0370478\pi\)
\(264\) 37.8164i 0.143244i
\(265\) 454.681 1.71578
\(266\) 65.9939 0.248097
\(267\) 151.853i 0.568740i
\(268\) 16.2095i 0.0604834i
\(269\) −8.00773 −0.0297685 −0.0148843 0.999889i \(-0.504738\pi\)
−0.0148843 + 0.999889i \(0.504738\pi\)
\(270\) 36.1710i 0.133967i
\(271\) 347.677 1.28294 0.641470 0.767148i \(-0.278325\pi\)
0.641470 + 0.767148i \(0.278325\pi\)
\(272\) 110.623i 0.406701i
\(273\) 8.40229i 0.0307776i
\(274\) 124.984i 0.456147i
\(275\) 5.95506i 0.0216547i
\(276\) −69.5078 + 38.9444i −0.251840 + 0.141103i
\(277\) −320.160 −1.15581 −0.577907 0.816103i \(-0.696130\pi\)
−0.577907 + 0.816103i \(0.696130\pi\)
\(278\) −259.657 −0.934017
\(279\) −2.49275 −0.00893458
\(280\) −71.5395 −0.255498
\(281\) 77.6750i 0.276424i 0.990403 + 0.138212i \(0.0441355\pi\)
−0.990403 + 0.138212i \(0.955865\pi\)
\(282\) −93.6392 −0.332054
\(283\) 449.812i 1.58944i 0.606974 + 0.794721i \(0.292383\pi\)
−0.606974 + 0.794721i \(0.707617\pi\)
\(284\) −47.6278 −0.167704
\(285\) 77.4240 0.271663
\(286\) 10.3060i 0.0360349i
\(287\) 137.933i 0.480604i
\(288\) 16.9706 0.0589256
\(289\) −475.835 −1.64649
\(290\) 100.623i 0.346976i
\(291\) 63.6259i 0.218646i
\(292\) 196.207 0.671942
\(293\) 437.459i 1.49304i −0.665366 0.746518i \(-0.731724\pi\)
0.665366 0.746518i \(-0.268276\pi\)
\(294\) 55.3479 0.188258
\(295\) 235.508i 0.798334i
\(296\) 67.4209i 0.227773i
\(297\) 40.1104i 0.135052i
\(298\) 122.397i 0.410729i
\(299\) −18.9427 + 10.6134i −0.0633536 + 0.0354962i
\(300\) 2.67240 0.00890800
\(301\) 137.577 0.457065
\(302\) −373.020 −1.23516
\(303\) 328.100 1.08284
\(304\) 36.3255i 0.119492i
\(305\) −599.647 −1.96605
\(306\) 117.333i 0.383441i
\(307\) −465.359 −1.51583 −0.757914 0.652354i \(-0.773781\pi\)
−0.757914 + 0.652354i \(0.773781\pi\)
\(308\) 79.3309 0.257568
\(309\) 148.129i 0.479383i
\(310\) 5.78410i 0.0186584i
\(311\) 596.299 1.91736 0.958679 0.284489i \(-0.0918238\pi\)
0.958679 + 0.284489i \(0.0918238\pi\)
\(312\) 4.62493 0.0148235
\(313\) 411.536i 1.31481i −0.753536 0.657406i \(-0.771654\pi\)
0.753536 0.657406i \(-0.228346\pi\)
\(314\) 118.086i 0.376071i
\(315\) −75.8791 −0.240886
\(316\) 140.006i 0.443057i
\(317\) −348.576 −1.09961 −0.549804 0.835293i \(-0.685298\pi\)
−0.549804 + 0.835293i \(0.685298\pi\)
\(318\) 226.266i 0.711528i
\(319\) 111.582i 0.349786i
\(320\) 39.3780i 0.123056i
\(321\) 239.753i 0.746895i
\(322\) −81.6970 145.813i −0.253718 0.452835i
\(323\) −251.151 −0.777558
\(324\) 18.0000 0.0555556
\(325\) 0.728300 0.00224092
\(326\) −252.640 −0.774969
\(327\) 86.1849i 0.263562i
\(328\) −75.9236 −0.231474
\(329\) 196.435i 0.597068i
\(330\) 93.0709 0.282033
\(331\) −62.3646 −0.188413 −0.0942064 0.995553i \(-0.530031\pi\)
−0.0942064 + 0.995553i \(0.530031\pi\)
\(332\) 101.469i 0.305631i
\(333\) 71.5107i 0.214747i
\(334\) −177.620 −0.531795
\(335\) −39.8937 −0.119086
\(336\) 35.6007i 0.105954i
\(337\) 106.990i 0.317477i 0.987321 + 0.158738i \(0.0507427\pi\)
−0.987321 + 0.158738i \(0.949257\pi\)
\(338\) −237.742 −0.703378
\(339\) 327.666i 0.966565i
\(340\) 272.256 0.800753
\(341\) 6.41404i 0.0188095i
\(342\) 38.5290i 0.112658i
\(343\) 367.895i 1.07258i
\(344\) 75.7272i 0.220137i
\(345\) −95.8469 171.067i −0.277817 0.495848i
\(346\) 206.455 0.596692
\(347\) −319.649 −0.921177 −0.460589 0.887614i \(-0.652362\pi\)
−0.460589 + 0.887614i \(0.652362\pi\)
\(348\) −50.0737 −0.143890
\(349\) −156.934 −0.449667 −0.224834 0.974397i \(-0.572184\pi\)
−0.224834 + 0.974397i \(0.572184\pi\)
\(350\) 5.60613i 0.0160175i
\(351\) 4.90548 0.0139757
\(352\) 43.6666i 0.124053i
\(353\) 157.767 0.446933 0.223466 0.974712i \(-0.428263\pi\)
0.223466 + 0.974712i \(0.428263\pi\)
\(354\) −117.198 −0.331066
\(355\) 117.218i 0.330192i
\(356\) 175.345i 0.492543i
\(357\) 246.140 0.689467
\(358\) −216.040 −0.603465
\(359\) 149.911i 0.417580i 0.977961 + 0.208790i \(0.0669525\pi\)
−0.977961 + 0.208790i \(0.933047\pi\)
\(360\) 41.7667i 0.116019i
\(361\) 278.529 0.771548
\(362\) 188.820i 0.521603i
\(363\) 106.371 0.293033
\(364\) 9.70212i 0.0266542i
\(365\) 482.890i 1.32299i
\(366\) 298.406i 0.815317i
\(367\) 643.183i 1.75254i −0.481818 0.876271i \(-0.660023\pi\)
0.481818 0.876271i \(-0.339977\pi\)
\(368\) −80.2607 + 44.9691i −0.218100 + 0.122199i
\(369\) −80.5292 −0.218236
\(370\) −165.931 −0.448463
\(371\) 474.658 1.27940
\(372\) −2.87838 −0.00773757
\(373\) 240.542i 0.644884i −0.946589 0.322442i \(-0.895496\pi\)
0.946589 0.322442i \(-0.104504\pi\)
\(374\) −301.907 −0.807239
\(375\) 219.717i 0.585911i
\(376\) −108.125 −0.287567
\(377\) −13.6464 −0.0361974
\(378\) 37.7602i 0.0998947i
\(379\) 690.895i 1.82294i −0.411364 0.911471i \(-0.634948\pi\)
0.411364 0.911471i \(-0.365052\pi\)
\(380\) 89.4015 0.235267
\(381\) −55.7747 −0.146390
\(382\) 383.962i 1.00514i
\(383\) 249.591i 0.651674i 0.945426 + 0.325837i \(0.105646\pi\)
−0.945426 + 0.325837i \(0.894354\pi\)
\(384\) 19.5959 0.0510310
\(385\) 195.243i 0.507125i
\(386\) 342.365 0.886955
\(387\) 80.3208i 0.207547i
\(388\) 73.4689i 0.189353i
\(389\) 515.323i 1.32474i −0.749177 0.662369i \(-0.769551\pi\)
0.749177 0.662369i \(-0.230449\pi\)
\(390\) 11.3825i 0.0291860i
\(391\) 310.912 + 554.915i 0.795172 + 1.41922i
\(392\) 63.9103 0.163036
\(393\) 153.625 0.390904
\(394\) −355.414 −0.902066
\(395\) 344.572 0.872335
\(396\) 46.3155i 0.116958i
\(397\) −219.200 −0.552140 −0.276070 0.961138i \(-0.589032\pi\)
−0.276070 + 0.961138i \(0.589032\pi\)
\(398\) 249.345i 0.626496i
\(399\) 80.8257 0.202571
\(400\) 3.08582 0.00771456
\(401\) 110.690i 0.276034i −0.990430 0.138017i \(-0.955927\pi\)
0.990430 0.138017i \(-0.0440729\pi\)
\(402\) 19.8525i 0.0493844i
\(403\) −0.784434 −0.00194649
\(404\) 378.857 0.937764
\(405\) 44.3002i 0.109383i
\(406\) 105.044i 0.258729i
\(407\) 184.003 0.452096
\(408\) 135.484i 0.332070i
\(409\) 632.891 1.54741 0.773706 0.633545i \(-0.218401\pi\)
0.773706 + 0.633545i \(0.218401\pi\)
\(410\) 186.858i 0.455750i
\(411\) 153.074i 0.372442i
\(412\) 171.045i 0.415158i
\(413\) 245.856i 0.595292i
\(414\) −85.1293 + 47.6969i −0.205626 + 0.115210i
\(415\) −249.729 −0.601756
\(416\) 5.34041 0.0128375
\(417\) −318.013 −0.762621
\(418\) −99.1382 −0.237173
\(419\) 655.862i 1.56530i 0.622461 + 0.782651i \(0.286133\pi\)
−0.622461 + 0.782651i \(0.713867\pi\)
\(420\) −87.6177 −0.208614
\(421\) 257.748i 0.612228i −0.951995 0.306114i \(-0.900971\pi\)
0.951995 0.306114i \(-0.0990288\pi\)
\(422\) 304.460 0.721468
\(423\) −114.684 −0.271121
\(424\) 261.269i 0.616201i
\(425\) 21.3351i 0.0502003i
\(426\) −58.3319 −0.136929
\(427\) −625.993 −1.46603
\(428\) 276.843i 0.646830i
\(429\) 12.6222i 0.0294224i
\(430\) 186.374 0.433428
\(431\) 703.840i 1.63304i 0.577317 + 0.816520i \(0.304100\pi\)
−0.577317 + 0.816520i \(0.695900\pi\)
\(432\) 20.7846 0.0481125
\(433\) 621.176i 1.43459i 0.696772 + 0.717293i \(0.254619\pi\)
−0.696772 + 0.717293i \(0.745381\pi\)
\(434\) 6.03823i 0.0139130i
\(435\) 123.238i 0.283305i
\(436\) 99.5177i 0.228252i
\(437\) 102.095 + 182.219i 0.233627 + 0.416978i
\(438\) 240.304 0.548638
\(439\) −292.224 −0.665659 −0.332830 0.942987i \(-0.608003\pi\)
−0.332830 + 0.942987i \(0.608003\pi\)
\(440\) 107.469 0.244248
\(441\) 67.7871 0.153712
\(442\) 36.9231i 0.0835364i
\(443\) −541.206 −1.22168 −0.610842 0.791753i \(-0.709169\pi\)
−0.610842 + 0.791753i \(0.709169\pi\)
\(444\) 82.5734i 0.185976i
\(445\) −431.547 −0.969768
\(446\) 336.430 0.754328
\(447\) 149.906i 0.335359i
\(448\) 41.1081i 0.0917592i
\(449\) 554.320 1.23457 0.617283 0.786741i \(-0.288233\pi\)
0.617283 + 0.786741i \(0.288233\pi\)
\(450\) 3.27301 0.00727335
\(451\) 207.208i 0.459442i
\(452\) 378.356i 0.837070i
\(453\) −456.854 −1.00851
\(454\) 2.54706i 0.00561026i
\(455\) −23.8781 −0.0524794
\(456\) 44.4894i 0.0975645i
\(457\) 740.778i 1.62096i −0.585767 0.810480i \(-0.699207\pi\)
0.585767 0.810480i \(-0.300793\pi\)
\(458\) 326.674i 0.713263i
\(459\) 143.703i 0.313078i
\(460\) −110.675 197.532i −0.240597 0.429417i
\(461\) 74.1009 0.160739 0.0803697 0.996765i \(-0.474390\pi\)
0.0803697 + 0.996765i \(0.474390\pi\)
\(462\) 97.1601 0.210303
\(463\) 704.351 1.52128 0.760638 0.649176i \(-0.224886\pi\)
0.760638 + 0.649176i \(0.224886\pi\)
\(464\) −57.8201 −0.124612
\(465\) 7.08405i 0.0152345i
\(466\) −70.7520 −0.151828
\(467\) 584.754i 1.25215i −0.779763 0.626075i \(-0.784660\pi\)
0.779763 0.626075i \(-0.215340\pi\)
\(468\) 5.66436 0.0121033
\(469\) −41.6465 −0.0887984
\(470\) 266.110i 0.566191i
\(471\) 144.626i 0.307061i
\(472\) −135.328 −0.286712
\(473\) −206.672 −0.436939
\(474\) 171.472i 0.361755i
\(475\) 7.00587i 0.0147492i
\(476\) 284.218 0.597096
\(477\) 277.118i 0.580960i
\(478\) 573.122 1.19900
\(479\) 586.758i 1.22496i 0.790485 + 0.612482i \(0.209829\pi\)
−0.790485 + 0.612482i \(0.790171\pi\)
\(480\) 48.2280i 0.100475i
\(481\) 22.5035i 0.0467847i
\(482\) 492.899i 1.02261i
\(483\) −100.058 178.583i −0.207160 0.369738i
\(484\) 122.827 0.253774
\(485\) −180.816 −0.372817
\(486\) 22.0454 0.0453609
\(487\) −143.981 −0.295650 −0.147825 0.989014i \(-0.547227\pi\)
−0.147825 + 0.989014i \(0.547227\pi\)
\(488\) 344.569i 0.706085i
\(489\) −309.420 −0.632760
\(490\) 157.291i 0.321002i
\(491\) 672.591 1.36984 0.684920 0.728618i \(-0.259837\pi\)
0.684920 + 0.728618i \(0.259837\pi\)
\(492\) −92.9871 −0.188998
\(493\) 399.763i 0.810879i
\(494\) 12.1245i 0.0245436i
\(495\) 113.988 0.230279
\(496\) −3.32366 −0.00670094
\(497\) 122.368i 0.246213i
\(498\) 124.274i 0.249546i
\(499\) 110.196 0.220834 0.110417 0.993885i \(-0.464781\pi\)
0.110417 + 0.993885i \(0.464781\pi\)
\(500\) 253.707i 0.507414i
\(501\) −217.539 −0.434209
\(502\) 8.14653i 0.0162281i
\(503\) 697.316i 1.38631i −0.720786 0.693157i \(-0.756219\pi\)
0.720786 0.693157i \(-0.243781\pi\)
\(504\) 43.6017i 0.0865114i
\(505\) 932.414i 1.84636i
\(506\) 122.728 + 219.045i 0.242546 + 0.432895i
\(507\) −291.173 −0.574306
\(508\) −64.4031 −0.126778
\(509\) 702.717 1.38058 0.690292 0.723531i \(-0.257482\pi\)
0.690292 + 0.723531i \(0.257482\pi\)
\(510\) 333.444 0.653812
\(511\) 504.106i 0.986509i
\(512\) 22.6274 0.0441942
\(513\) 47.1882i 0.0919847i
\(514\) 37.2370 0.0724454
\(515\) −420.963 −0.817405
\(516\) 92.7465i 0.179741i
\(517\) 295.092i 0.570777i
\(518\) −173.222 −0.334405
\(519\) 252.855 0.487197
\(520\) 13.1434i 0.0252758i
\(521\) 130.216i 0.249935i 0.992161 + 0.124968i \(0.0398827\pi\)
−0.992161 + 0.124968i \(0.960117\pi\)
\(522\) −61.3275 −0.117486
\(523\) 966.051i 1.84713i −0.383436 0.923567i \(-0.625259\pi\)
0.383436 0.923567i \(-0.374741\pi\)
\(524\) 177.391 0.338533
\(525\) 6.86608i 0.0130783i
\(526\) 86.3838i 0.164228i
\(527\) 22.9795i 0.0436044i
\(528\) 53.4805i 0.101289i
\(529\) 276.223 451.156i 0.522160 0.852847i
\(530\) 643.017 1.21324
\(531\) −143.537 −0.270315
\(532\) 93.3294 0.175431
\(533\) −25.3414 −0.0475449
\(534\) 214.753i 0.402160i
\(535\) 681.346 1.27354
\(536\) 22.9237i 0.0427682i
\(537\) −264.594 −0.492727
\(538\) −11.3246 −0.0210495
\(539\) 174.422i 0.323602i
\(540\) 51.1535i 0.0947287i
\(541\) 250.233 0.462539 0.231269 0.972890i \(-0.425712\pi\)
0.231269 + 0.972890i \(0.425712\pi\)
\(542\) 491.689 0.907175
\(543\) 231.257i 0.425887i
\(544\) 156.444i 0.287581i
\(545\) −244.926 −0.449405
\(546\) 11.8826i 0.0217631i
\(547\) 226.518 0.414109 0.207055 0.978329i \(-0.433612\pi\)
0.207055 + 0.978329i \(0.433612\pi\)
\(548\) 176.754i 0.322544i
\(549\) 365.471i 0.665703i
\(550\) 8.42172i 0.0153122i
\(551\) 131.271i 0.238242i
\(552\) −98.2989 + 55.0756i −0.178078 + 0.0997747i
\(553\) 359.711 0.650473
\(554\) −452.775 −0.817284
\(555\) −203.224 −0.366169
\(556\) −367.210 −0.660449
\(557\) 693.109i 1.24436i −0.782874 0.622181i \(-0.786247\pi\)
0.782874 0.622181i \(-0.213753\pi\)
\(558\) −3.52528 −0.00631770
\(559\) 25.2759i 0.0452162i
\(560\) −101.172 −0.180665
\(561\) −369.759 −0.659108
\(562\) 109.849i 0.195461i
\(563\) 232.265i 0.412549i −0.978494 0.206275i \(-0.933866\pi\)
0.978494 0.206275i \(-0.0661340\pi\)
\(564\) −132.426 −0.234798
\(565\) −931.180 −1.64811
\(566\) 636.131i 1.12391i
\(567\) 46.2466i 0.0815637i
\(568\) −67.3559 −0.118584
\(569\) 285.831i 0.502340i 0.967943 + 0.251170i \(0.0808153\pi\)
−0.967943 + 0.251170i \(0.919185\pi\)
\(570\) 109.494 0.192095
\(571\) 572.347i 1.00236i −0.865344 0.501179i \(-0.832900\pi\)
0.865344 0.501179i \(-0.167100\pi\)
\(572\) 14.5749i 0.0254805i
\(573\) 470.255i 0.820690i
\(574\) 195.067i 0.339838i
\(575\) −15.4794 + 8.67291i −0.0269207 + 0.0150833i
\(576\) 24.0000 0.0416667
\(577\) 157.233 0.272501 0.136251 0.990674i \(-0.456495\pi\)
0.136251 + 0.990674i \(0.456495\pi\)
\(578\) −672.932 −1.16424
\(579\) 419.310 0.724196
\(580\) 142.303i 0.245349i
\(581\) −260.701 −0.448710
\(582\) 89.9806i 0.154606i
\(583\) −713.047 −1.22307
\(584\) 277.479 0.475135
\(585\) 13.9407i 0.0238302i
\(586\) 618.661i 1.05574i
\(587\) −212.520 −0.362044 −0.181022 0.983479i \(-0.557941\pi\)
−0.181022 + 0.983479i \(0.557941\pi\)
\(588\) 78.2738 0.133119
\(589\) 7.54585i 0.0128113i
\(590\) 333.059i 0.564507i
\(591\) −435.292 −0.736534
\(592\) 95.3476i 0.161060i
\(593\) −937.090 −1.58025 −0.790127 0.612944i \(-0.789985\pi\)
−0.790127 + 0.612944i \(0.789985\pi\)
\(594\) 56.7246i 0.0954960i
\(595\) 699.495i 1.17562i
\(596\) 173.096i 0.290430i
\(597\) 305.385i 0.511532i
\(598\) −26.7891 + 15.0096i −0.0447978 + 0.0250996i
\(599\) −318.493 −0.531708 −0.265854 0.964013i \(-0.585654\pi\)
−0.265854 + 0.964013i \(0.585654\pi\)
\(600\) 3.77935 0.00629891
\(601\) 799.628 1.33050 0.665248 0.746623i \(-0.268326\pi\)
0.665248 + 0.746623i \(0.268326\pi\)
\(602\) 194.563 0.323194
\(603\) 24.3143i 0.0403222i
\(604\) −527.529 −0.873393
\(605\) 302.291i 0.499655i
\(606\) 464.003 0.765681
\(607\) −185.298 −0.305268 −0.152634 0.988283i \(-0.548776\pi\)
−0.152634 + 0.988283i \(0.548776\pi\)
\(608\) 51.3720i 0.0844934i
\(609\) 128.652i 0.211251i
\(610\) −848.029 −1.39021
\(611\) −36.0896 −0.0590664
\(612\) 165.934i 0.271134i
\(613\) 399.963i 0.652469i −0.945289 0.326234i \(-0.894220\pi\)
0.945289 0.326234i \(-0.105780\pi\)
\(614\) −658.117 −1.07185
\(615\) 228.853i 0.372118i
\(616\) 112.191 0.182128
\(617\) 308.392i 0.499826i 0.968268 + 0.249913i \(0.0804020\pi\)
−0.968268 + 0.249913i \(0.919598\pi\)
\(618\) 209.487i 0.338975i
\(619\) 659.561i 1.06553i 0.846264 + 0.532763i \(0.178846\pi\)
−0.846264 + 0.532763i \(0.821154\pi\)
\(620\) 8.17995i 0.0131935i
\(621\) −104.262 + 58.4165i −0.167893 + 0.0940685i
\(622\) 843.294 1.35578
\(623\) −450.507 −0.723125
\(624\) 6.54064 0.0104818
\(625\) −605.118 −0.968190
\(626\) 582.000i 0.929713i
\(627\) −121.419 −0.193651
\(628\) 166.999i 0.265922i
\(629\) 659.225 1.04805
\(630\) −107.309 −0.170332
\(631\) 102.663i 0.162699i 0.996686 + 0.0813497i \(0.0259231\pi\)
−0.996686 + 0.0813497i \(0.974077\pi\)
\(632\) 197.998i 0.313289i
\(633\) 372.885 0.589076
\(634\) −492.961 −0.777541
\(635\) 158.504i 0.249613i
\(636\) 319.988i 0.503126i
\(637\) 21.3317 0.0334877
\(638\) 157.801i 0.247336i
\(639\) −71.4417 −0.111802
\(640\) 55.6889i 0.0870139i
\(641\) 721.856i 1.12614i −0.826409 0.563070i \(-0.809620\pi\)
0.826409 0.563070i \(-0.190380\pi\)
\(642\) 339.062i 0.528135i
\(643\) 177.237i 0.275641i −0.990457 0.137821i \(-0.955990\pi\)
0.990457 0.137821i \(-0.0440097\pi\)
\(644\) −115.537 206.210i −0.179405 0.320202i
\(645\) 228.261 0.353893
\(646\) −355.181 −0.549816
\(647\) 568.014 0.877920 0.438960 0.898507i \(-0.355347\pi\)
0.438960 + 0.898507i \(0.355347\pi\)
\(648\) 25.4558 0.0392837
\(649\) 369.333i 0.569079i
\(650\) 1.02997 0.00158457
\(651\) 7.39529i 0.0113599i
\(652\) −357.287 −0.547986
\(653\) −550.970 −0.843752 −0.421876 0.906653i \(-0.638628\pi\)
−0.421876 + 0.906653i \(0.638628\pi\)
\(654\) 121.884i 0.186367i
\(655\) 436.582i 0.666538i
\(656\) −107.372 −0.163677
\(657\) 294.311 0.447961
\(658\) 277.801i 0.422191i
\(659\) 663.157i 1.00631i −0.864197 0.503154i \(-0.832173\pi\)
0.864197 0.503154i \(-0.167827\pi\)
\(660\) 131.622 0.199428
\(661\) 893.783i 1.35217i −0.736825 0.676084i \(-0.763676\pi\)
0.736825 0.676084i \(-0.236324\pi\)
\(662\) −88.1969 −0.133228
\(663\) 45.2214i 0.0682072i
\(664\) 143.499i 0.216113i
\(665\) 229.695i 0.345407i
\(666\) 101.131i 0.151849i
\(667\) 290.043 162.507i 0.434847 0.243639i
\(668\) −251.192 −0.376036
\(669\) 412.041 0.615906
\(670\) −56.4182 −0.0842063
\(671\) 940.387 1.40147
\(672\) 50.3469i 0.0749210i
\(673\) 1201.94 1.78594 0.892969 0.450118i \(-0.148618\pi\)
0.892969 + 0.450118i \(0.148618\pi\)
\(674\) 151.306i 0.224490i
\(675\) 4.00860 0.00593867
\(676\) −336.218 −0.497363
\(677\) 338.932i 0.500639i 0.968163 + 0.250319i \(0.0805356\pi\)
−0.968163 + 0.250319i \(0.919464\pi\)
\(678\) 463.389i 0.683465i
\(679\) −188.760 −0.277998
\(680\) 385.028 0.566218
\(681\) 3.11949i 0.00458076i
\(682\) 9.07083i 0.0133003i
\(683\) −951.288 −1.39281 −0.696404 0.717650i \(-0.745218\pi\)
−0.696404 + 0.717650i \(0.745218\pi\)
\(684\) 54.4882i 0.0796611i
\(685\) 435.015 0.635058
\(686\) 520.283i 0.758429i
\(687\) 400.093i 0.582377i
\(688\) 107.094i 0.155661i
\(689\) 87.2053i 0.126568i
\(690\) −135.548 241.926i −0.196446 0.350617i
\(691\) −407.739 −0.590071 −0.295036 0.955486i \(-0.595332\pi\)
−0.295036 + 0.955486i \(0.595332\pi\)
\(692\) 291.972 0.421925
\(693\) 118.996 0.171712
\(694\) −452.051 −0.651371
\(695\) 903.749i 1.30036i
\(696\) −70.8149 −0.101746
\(697\) 742.362i 1.06508i
\(698\) −221.938 −0.317963
\(699\) −86.6532 −0.123967
\(700\) 7.92827i 0.0113261i
\(701\) 1016.66i 1.45029i 0.688594 + 0.725147i \(0.258228\pi\)
−0.688594 + 0.725147i \(0.741772\pi\)
\(702\) 6.93739 0.00988233
\(703\) 216.472 0.307926
\(704\) 61.7540i 0.0877187i
\(705\) 325.917i 0.462293i
\(706\) 223.117 0.316029
\(707\) 973.380i 1.37678i
\(708\) −165.742 −0.234099
\(709\) 121.061i 0.170749i −0.996349 0.0853744i \(-0.972791\pi\)
0.996349 0.0853744i \(-0.0272086\pi\)
\(710\) 165.771i 0.233481i
\(711\) 210.009i 0.295371i
\(712\) 247.976i 0.348281i
\(713\) 16.6725 9.34138i 0.0233836 0.0131015i
\(714\) 348.094 0.487527
\(715\) 35.8705 0.0501686
\(716\) −305.527 −0.426714
\(717\) 701.928 0.978979
\(718\) 212.006i 0.295274i
\(719\) 160.481 0.223200 0.111600 0.993753i \(-0.464402\pi\)
0.111600 + 0.993753i \(0.464402\pi\)
\(720\) 59.0670i 0.0820375i
\(721\) −439.459 −0.609513
\(722\) 393.899 0.545567
\(723\) 603.675i 0.834959i
\(724\) 267.032i 0.368829i
\(725\) −11.1514 −0.0153813
\(726\) 150.431 0.207205
\(727\) 328.928i 0.452446i −0.974076 0.226223i \(-0.927362\pi\)
0.974076 0.226223i \(-0.0726378\pi\)
\(728\) 13.7209i 0.0188474i
\(729\) 27.0000 0.0370370
\(730\) 682.910i 0.935493i
\(731\) −740.441 −1.01292
\(732\) 422.010i 0.576516i
\(733\) 528.441i 0.720929i 0.932773 + 0.360465i \(0.117382\pi\)
−0.932773 + 0.360465i \(0.882618\pi\)
\(734\) 909.598i 1.23923i
\(735\) 192.641i 0.262097i
\(736\) −113.506 + 63.5959i −0.154220 + 0.0864074i
\(737\) 62.5627 0.0848883
\(738\) −113.885 −0.154316
\(739\) 816.397 1.10473 0.552366 0.833602i \(-0.313725\pi\)
0.552366 + 0.833602i \(0.313725\pi\)
\(740\) −234.662 −0.317111
\(741\) 14.8495i 0.0200398i
\(742\) 671.268 0.904674
\(743\) 501.473i 0.674930i 0.941338 + 0.337465i \(0.109570\pi\)
−0.941338 + 0.337465i \(0.890430\pi\)
\(744\) −4.07064 −0.00547129
\(745\) 426.011 0.571827
\(746\) 340.177i 0.456002i
\(747\) 152.204i 0.203754i
\(748\) −426.961 −0.570804
\(749\) 711.281 0.949641
\(750\) 310.726i 0.414302i
\(751\) 706.401i 0.940614i 0.882503 + 0.470307i \(0.155857\pi\)
−0.882503 + 0.470307i \(0.844143\pi\)
\(752\) −152.912 −0.203341
\(753\) 9.97742i 0.0132502i
\(754\) −19.2989 −0.0255954
\(755\) 1298.32i 1.71962i
\(756\) 53.4010i 0.0706362i
\(757\) 1154.74i 1.52542i −0.646740 0.762711i \(-0.723868\pi\)
0.646740 0.762711i \(-0.276132\pi\)
\(758\) 977.073i 1.28901i
\(759\) 150.311 + 268.274i 0.198038 + 0.353457i
\(760\) 126.433 0.166359
\(761\) 834.803 1.09698 0.548491 0.836157i \(-0.315203\pi\)
0.548491 + 0.836157i \(0.315203\pi\)
\(762\) −78.8774 −0.103514
\(763\) −255.687 −0.335107
\(764\) 543.004i 0.710739i
\(765\) 408.384 0.533835
\(766\) 352.975i 0.460803i
\(767\) −45.1692 −0.0588907
\(768\) 27.7128 0.0360844
\(769\) 154.214i 0.200538i 0.994960 + 0.100269i \(0.0319703\pi\)
−0.994960 + 0.100269i \(0.968030\pi\)
\(770\) 276.116i 0.358592i
\(771\) 45.6058 0.0591515
\(772\) 484.177 0.627172
\(773\) 487.995i 0.631301i 0.948876 + 0.315650i \(0.102223\pi\)
−0.948876 + 0.315650i \(0.897777\pi\)
\(774\) 113.591i 0.146758i
\(775\) −0.641015 −0.000827116
\(776\) 103.901i 0.133893i
\(777\) −212.152 −0.273040
\(778\) 728.777i 0.936732i
\(779\) 243.772i 0.312929i
\(780\) 16.0973i 0.0206376i
\(781\) 183.825i 0.235372i
\(782\) 439.696 + 784.769i 0.562271 + 1.00354i
\(783\) −75.1105 −0.0959266
\(784\) 90.3827 0.115284
\(785\) 411.006 0.523574
\(786\) 217.259 0.276411
\(787\) 193.960i 0.246455i −0.992378 0.123228i \(-0.960675\pi\)
0.992378 0.123228i \(-0.0393246\pi\)
\(788\) −502.631 −0.637857
\(789\) 105.798i 0.134091i
\(790\) 487.299 0.616834
\(791\) −972.092 −1.22894
\(792\) 65.5000i 0.0827020i
\(793\) 115.009i 0.145030i
\(794\) −309.995 −0.390422
\(795\) 787.531 0.990605
\(796\) 352.628i 0.443000i
\(797\) 916.068i 1.14939i 0.818366 + 0.574697i \(0.194880\pi\)
−0.818366 + 0.574697i \(0.805120\pi\)
\(798\) 114.305 0.143239
\(799\) 1057.22i 1.32318i
\(800\) 4.36401 0.00545502
\(801\) 263.018i 0.328362i
\(802\) 156.539i 0.195186i
\(803\) 757.285i 0.943070i
\(804\) 28.0757i 0.0349201i
\(805\) 507.509 284.351i 0.630447 0.353231i
\(806\) −1.10936 −0.00137637
\(807\) −13.8698 −0.0171869
\(808\) 535.784 0.663099
\(809\) −81.4166 −0.100639 −0.0503193 0.998733i \(-0.516024\pi\)
−0.0503193 + 0.998733i \(0.516024\pi\)
\(810\) 62.6500i 0.0773457i
\(811\) 1503.38 1.85374 0.926870 0.375382i \(-0.122489\pi\)
0.926870 + 0.375382i \(0.122489\pi\)
\(812\) 148.555i 0.182949i
\(813\) 602.193 0.740705
\(814\) 260.219 0.319680
\(815\) 879.328i 1.07893i
\(816\) 191.604i 0.234809i
\(817\) −243.141 −0.297602
\(818\) 895.044 1.09419
\(819\) 14.5532i 0.0177695i
\(820\) 264.256i 0.322264i
\(821\) −136.557 −0.166330 −0.0831651 0.996536i \(-0.526503\pi\)
−0.0831651 + 0.996536i \(0.526503\pi\)
\(822\) 216.479i 0.263356i
\(823\) −134.188 −0.163048 −0.0815239 0.996671i \(-0.525979\pi\)
−0.0815239 + 0.996671i \(0.525979\pi\)
\(824\) 241.894i 0.293561i
\(825\) 10.3145i 0.0125024i
\(826\) 347.692i 0.420935i
\(827\) 1190.70i 1.43979i 0.694085 + 0.719893i \(0.255809\pi\)
−0.694085 + 0.719893i \(0.744191\pi\)
\(828\) −120.391 + 67.4536i −0.145400 + 0.0814657i
\(829\) 167.849 0.202471 0.101236 0.994862i \(-0.467720\pi\)
0.101236 + 0.994862i \(0.467720\pi\)
\(830\) −353.170 −0.425506
\(831\) −554.534 −0.667309
\(832\) 7.55248 0.00907750
\(833\) 624.898i 0.750178i
\(834\) −449.738 −0.539255
\(835\) 618.215i 0.740378i
\(836\) −140.203 −0.167706
\(837\) −4.31757 −0.00515838
\(838\) 927.528i 1.10684i
\(839\) 620.476i 0.739542i 0.929123 + 0.369771i \(0.120564\pi\)
−0.929123 + 0.369771i \(0.879436\pi\)
\(840\) −123.910 −0.147512
\(841\) −632.052 −0.751548
\(842\) 364.511i 0.432910i
\(843\) 134.537i 0.159593i
\(844\) 430.571 0.510155
\(845\) 827.473i 0.979258i
\(846\) −162.188 −0.191711
\(847\) 315.573i 0.372577i
\(848\) 369.491i 0.435720i
\(849\) 779.098i 0.917665i
\(850\) 30.1724i 0.0354969i
\(851\) −267.981 478.292i −0.314901 0.562035i
\(852\) −82.4938 −0.0968237
\(853\) 639.619 0.749846 0.374923 0.927056i \(-0.377669\pi\)
0.374923 + 0.927056i \(0.377669\pi\)
\(854\) −885.287 −1.03664
\(855\) 134.102 0.156845
\(856\) 391.516i 0.457378i
\(857\) 351.336 0.409960 0.204980 0.978766i \(-0.434287\pi\)
0.204980 + 0.978766i \(0.434287\pi\)
\(858\) 17.8505i 0.0208048i
\(859\) −1548.57 −1.80276 −0.901382 0.433024i \(-0.857446\pi\)
−0.901382 + 0.433024i \(0.857446\pi\)
\(860\) 263.573 0.306480
\(861\) 238.908i 0.277477i
\(862\) 995.381i 1.15473i
\(863\) −1502.94 −1.74153 −0.870763 0.491703i \(-0.836375\pi\)
−0.870763 + 0.491703i \(0.836375\pi\)
\(864\) 29.3939 0.0340207
\(865\) 718.579i 0.830727i
\(866\) 878.475i 1.01441i
\(867\) −824.170 −0.950600
\(868\) 8.53934i 0.00983795i
\(869\) −540.370 −0.621830
\(870\) 174.284i 0.200327i
\(871\) 7.65138i 0.00878460i
\(872\) 140.739i 0.161398i
\(873\) 110.203i 0.126235i
\(874\) 144.384 + 257.697i 0.165200 + 0.294848i
\(875\) −651.839 −0.744958
\(876\) 339.841 0.387946
\(877\) −474.173 −0.540676 −0.270338 0.962765i \(-0.587135\pi\)
−0.270338 + 0.962765i \(0.587135\pi\)
\(878\) −413.268 −0.470692
\(879\) 757.702i 0.862004i
\(880\) 151.984 0.172709
\(881\) 727.978i 0.826309i −0.910661 0.413154i \(-0.864427\pi\)
0.910661 0.413154i \(-0.135573\pi\)
\(882\) 95.8654 0.108691
\(883\) 134.373 0.152178 0.0760890 0.997101i \(-0.475757\pi\)
0.0760890 + 0.997101i \(0.475757\pi\)
\(884\) 52.2172i 0.0590692i
\(885\) 407.913i 0.460918i
\(886\) −765.381 −0.863861
\(887\) 510.194 0.575190 0.287595 0.957752i \(-0.407144\pi\)
0.287595 + 0.957752i \(0.407144\pi\)
\(888\) 116.776i 0.131505i
\(889\) 165.468i 0.186128i
\(890\) −610.299 −0.685729
\(891\) 69.4732i 0.0779722i
\(892\) 475.784 0.533391
\(893\) 347.163i 0.388760i
\(894\) 211.998i 0.237135i
\(895\) 751.941i 0.840157i
\(896\) 58.1356i 0.0648835i
\(897\) −32.8098 + 18.3829i −0.0365772 + 0.0204938i
\(898\) 783.927 0.872970
\(899\) 12.0109 0.0133603
\(900\) 4.62873 0.00514304
\(901\) −2554.63 −2.83532
\(902\) 293.037i 0.324874i
\(903\) 238.290 0.263887
\(904\) 535.076i 0.591898i
\(905\) 657.199 0.726187
\(906\) −646.089 −0.713122
\(907\) 594.907i 0.655907i −0.944694 0.327953i \(-0.893641\pi\)
0.944694 0.327953i \(-0.106359\pi\)
\(908\) 3.60208i 0.00396705i
\(909\) 568.285 0.625176
\(910\) −33.7688 −0.0371086
\(911\) 1025.46i 1.12564i −0.826579 0.562821i \(-0.809716\pi\)
0.826579 0.562821i \(-0.190284\pi\)
\(912\) 62.9176i 0.0689886i
\(913\) 391.633 0.428952
\(914\) 1047.62i 1.14619i
\(915\) −1038.62 −1.13510
\(916\) 461.987i 0.504353i
\(917\) 455.764i 0.497016i
\(918\) 203.227i 0.221380i
\(919\) 1333.78i 1.45133i −0.688046 0.725667i \(-0.741531\pi\)
0.688046 0.725667i \(-0.258469\pi\)
\(920\) −156.517 279.352i −0.170128 0.303643i
\(921\) −806.026 −0.875164
\(922\) 104.794 0.113660
\(923\) −22.4817 −0.0243573
\(924\) 137.405 0.148707
\(925\) 18.3891i 0.0198801i
\(926\) 996.103 1.07570
\(927\) 256.568i 0.276772i
\(928\) −81.7700 −0.0881142
\(929\) −1146.22 −1.23382 −0.616911 0.787033i \(-0.711616\pi\)
−0.616911 + 0.787033i \(0.711616\pi\)
\(930\) 10.0184i 0.0107724i
\(931\) 205.200i 0.220408i
\(932\) −100.059 −0.107359
\(933\) 1032.82 1.10699
\(934\) 826.968i 0.885404i
\(935\) 1050.81i 1.12386i
\(936\) 8.01061 0.00855834
\(937\) 1095.51i 1.16916i 0.811334 + 0.584582i \(0.198742\pi\)
−0.811334 + 0.584582i \(0.801258\pi\)
\(938\) −58.8970 −0.0627900
\(939\) 712.802i 0.759107i
\(940\) 376.336i 0.400357i
\(941\) 422.732i 0.449237i 0.974447 + 0.224618i \(0.0721135\pi\)
−0.974447 + 0.224618i \(0.927886\pi\)
\(942\) 204.531i 0.217125i
\(943\) 538.611 301.777i 0.571167 0.320018i
\(944\) −191.383 −0.202736
\(945\) −131.427 −0.139076
\(946\) −292.278 −0.308962
\(947\) 823.217 0.869289 0.434645 0.900602i \(-0.356874\pi\)
0.434645 + 0.900602i \(0.356874\pi\)
\(948\) 242.498i 0.255799i
\(949\) 92.6156 0.0975928
\(950\) 9.90780i 0.0104293i
\(951\) −603.751 −0.634859
\(952\) 401.945 0.422211
\(953\) 397.062i 0.416644i 0.978060 + 0.208322i \(0.0668002\pi\)
−0.978060 + 0.208322i \(0.933200\pi\)
\(954\) 391.904i 0.410801i
\(955\) −1336.40 −1.39937
\(956\) 810.516 0.847820
\(957\) 193.266i 0.201949i
\(958\) 829.801i 0.866180i
\(959\) 454.127 0.473543
\(960\) 68.2047i 0.0710466i
\(961\) −960.310 −0.999282
\(962\) 31.8247i 0.0330818i
\(963\) 415.265i 0.431220i
\(964\) 697.064i 0.723095i
\(965\) 1191.62i 1.23484i
\(966\) −141.503 252.555i −0.146484 0.261444i
\(967\) −981.927 −1.01544 −0.507718 0.861523i \(-0.669511\pi\)
−0.507718 + 0.861523i \(0.669511\pi\)
\(968\) 173.703 0.179445
\(969\) −435.006 −0.448923
\(970\) −255.713 −0.263621
\(971\) 985.395i 1.01483i 0.861703 + 0.507413i \(0.169398\pi\)
−0.861703 + 0.507413i \(0.830602\pi\)
\(972\) 31.1769 0.0320750
\(973\) 943.456i 0.969637i
\(974\) −203.620 −0.209056
\(975\) 1.26145 0.00129380
\(976\) 487.295i 0.499277i
\(977\) 1772.59i 1.81432i −0.420783 0.907161i \(-0.638245\pi\)
0.420783 0.907161i \(-0.361755\pi\)
\(978\) −437.585 −0.447429
\(979\) 676.767 0.691284
\(980\) 222.443i 0.226983i
\(981\) 149.277i 0.152168i
\(982\) 951.188 0.968623
\(983\) 1328.52i 1.35149i −0.737135 0.675746i \(-0.763822\pi\)
0.737135 0.675746i \(-0.236178\pi\)
\(984\) −131.504 −0.133642
\(985\) 1237.04i 1.25588i
\(986\) 565.350i 0.573378i
\(987\) 340.236i 0.344717i
\(988\) 17.1467i 0.0173550i
\(989\) 300.996 + 537.217i 0.304344 + 0.543192i
\(990\) 161.204 0.162832
\(991\) −274.330 −0.276821 −0.138411 0.990375i \(-0.544199\pi\)
−0.138411 + 0.990375i \(0.544199\pi\)
\(992\) −4.70037 −0.00473828
\(993\) −108.019 −0.108780
\(994\) 173.055i 0.174099i
\(995\) 867.861 0.872222
\(996\) 175.750i 0.176456i
\(997\) −1081.51 −1.08476 −0.542380 0.840133i \(-0.682477\pi\)
−0.542380 + 0.840133i \(0.682477\pi\)
\(998\) 155.841 0.156153
\(999\) 123.860i 0.123984i
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 138.3.b.a.91.8 yes 8
3.2 odd 2 414.3.b.c.91.2 8
4.3 odd 2 1104.3.c.c.1057.4 8
23.22 odd 2 inner 138.3.b.a.91.7 8
69.68 even 2 414.3.b.c.91.3 8
92.91 even 2 1104.3.c.c.1057.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.3.b.a.91.7 8 23.22 odd 2 inner
138.3.b.a.91.8 yes 8 1.1 even 1 trivial
414.3.b.c.91.2 8 3.2 odd 2
414.3.b.c.91.3 8 69.68 even 2
1104.3.c.c.1057.1 8 92.91 even 2
1104.3.c.c.1057.4 8 4.3 odd 2