Properties

Label 8030.2.a.bg.1.14
Level $8030$
Weight $2$
Character 8030.1
Self dual yes
Analytic conductor $64.120$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8030,2,Mod(1,8030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8030, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8030.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: \(64.1198728231\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 7 x^{14} - 6 x^{13} + 136 x^{12} - 149 x^{11} - 876 x^{10} + 1631 x^{9} + 2142 x^{8} - 5473 x^{7} - 1914 x^{6} + 7517 x^{5} + 392 x^{4} - 3966 x^{3} - 79 x^{2} + 491 x - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Root \(3.03129\) of defining polynomial
Character \(\chi\) \(=\) 8030.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.03129 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.03129 q^{6} +3.50775 q^{7} +1.00000 q^{8} +6.18873 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.03129 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.03129 q^{6} +3.50775 q^{7} +1.00000 q^{8} +6.18873 q^{9} -1.00000 q^{10} -1.00000 q^{11} +3.03129 q^{12} -6.77606 q^{13} +3.50775 q^{14} -3.03129 q^{15} +1.00000 q^{16} +1.63062 q^{17} +6.18873 q^{18} +7.46655 q^{19} -1.00000 q^{20} +10.6330 q^{21} -1.00000 q^{22} +3.16743 q^{23} +3.03129 q^{24} +1.00000 q^{25} -6.77606 q^{26} +9.66598 q^{27} +3.50775 q^{28} +5.19277 q^{29} -3.03129 q^{30} -3.03853 q^{31} +1.00000 q^{32} -3.03129 q^{33} +1.63062 q^{34} -3.50775 q^{35} +6.18873 q^{36} +11.5416 q^{37} +7.46655 q^{38} -20.5402 q^{39} -1.00000 q^{40} -2.42690 q^{41} +10.6330 q^{42} -10.5318 q^{43} -1.00000 q^{44} -6.18873 q^{45} +3.16743 q^{46} +2.77614 q^{47} +3.03129 q^{48} +5.30428 q^{49} +1.00000 q^{50} +4.94288 q^{51} -6.77606 q^{52} -12.7598 q^{53} +9.66598 q^{54} +1.00000 q^{55} +3.50775 q^{56} +22.6333 q^{57} +5.19277 q^{58} +4.64925 q^{59} -3.03129 q^{60} -6.19293 q^{61} -3.03853 q^{62} +21.7085 q^{63} +1.00000 q^{64} +6.77606 q^{65} -3.03129 q^{66} +6.25363 q^{67} +1.63062 q^{68} +9.60139 q^{69} -3.50775 q^{70} -6.63711 q^{71} +6.18873 q^{72} -1.00000 q^{73} +11.5416 q^{74} +3.03129 q^{75} +7.46655 q^{76} -3.50775 q^{77} -20.5402 q^{78} +10.3469 q^{79} -1.00000 q^{80} +10.7342 q^{81} -2.42690 q^{82} +9.45862 q^{83} +10.6330 q^{84} -1.63062 q^{85} -10.5318 q^{86} +15.7408 q^{87} -1.00000 q^{88} -5.49478 q^{89} -6.18873 q^{90} -23.7687 q^{91} +3.16743 q^{92} -9.21069 q^{93} +2.77614 q^{94} -7.46655 q^{95} +3.03129 q^{96} -6.93289 q^{97} +5.30428 q^{98} -6.18873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 15 q^{2} + 7 q^{3} + 15 q^{4} - 15 q^{5} + 7 q^{6} + 3 q^{7} + 15 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 15 q^{2} + 7 q^{3} + 15 q^{4} - 15 q^{5} + 7 q^{6} + 3 q^{7} + 15 q^{8} + 16 q^{9} - 15 q^{10} - 15 q^{11} + 7 q^{12} - q^{13} + 3 q^{14} - 7 q^{15} + 15 q^{16} + 2 q^{17} + 16 q^{18} + 23 q^{19} - 15 q^{20} + 20 q^{21} - 15 q^{22} + 7 q^{24} + 15 q^{25} - q^{26} + 19 q^{27} + 3 q^{28} + 23 q^{29} - 7 q^{30} + 9 q^{31} + 15 q^{32} - 7 q^{33} + 2 q^{34} - 3 q^{35} + 16 q^{36} + 11 q^{37} + 23 q^{38} + 7 q^{39} - 15 q^{40} + 27 q^{41} + 20 q^{42} + 7 q^{43} - 15 q^{44} - 16 q^{45} - 18 q^{47} + 7 q^{48} + 16 q^{49} + 15 q^{50} + 21 q^{51} - q^{52} - 19 q^{53} + 19 q^{54} + 15 q^{55} + 3 q^{56} + 11 q^{57} + 23 q^{58} + 2 q^{59} - 7 q^{60} + 31 q^{61} + 9 q^{62} + 20 q^{63} + 15 q^{64} + q^{65} - 7 q^{66} + 49 q^{67} + 2 q^{68} + 33 q^{69} - 3 q^{70} + 32 q^{71} + 16 q^{72} - 15 q^{73} + 11 q^{74} + 7 q^{75} + 23 q^{76} - 3 q^{77} + 7 q^{78} + 36 q^{79} - 15 q^{80} + 23 q^{81} + 27 q^{82} + 33 q^{83} + 20 q^{84} - 2 q^{85} + 7 q^{86} + 29 q^{87} - 15 q^{88} + 6 q^{89} - 16 q^{90} + 33 q^{91} + 20 q^{93} - 18 q^{94} - 23 q^{95} + 7 q^{96} + 30 q^{97} + 16 q^{98} - 16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 3.03129 1.75012 0.875059 0.484017i \(-0.160823\pi\)
0.875059 + 0.484017i \(0.160823\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 3.03129 1.23752
\(7\) 3.50775 1.32580 0.662902 0.748706i \(-0.269325\pi\)
0.662902 + 0.748706i \(0.269325\pi\)
\(8\) 1.00000 0.353553
\(9\) 6.18873 2.06291
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) 3.03129 0.875059
\(13\) −6.77606 −1.87934 −0.939671 0.342080i \(-0.888869\pi\)
−0.939671 + 0.342080i \(0.888869\pi\)
\(14\) 3.50775 0.937485
\(15\) −3.03129 −0.782676
\(16\) 1.00000 0.250000
\(17\) 1.63062 0.395483 0.197741 0.980254i \(-0.436639\pi\)
0.197741 + 0.980254i \(0.436639\pi\)
\(18\) 6.18873 1.45870
\(19\) 7.46655 1.71294 0.856472 0.516193i \(-0.172651\pi\)
0.856472 + 0.516193i \(0.172651\pi\)
\(20\) −1.00000 −0.223607
\(21\) 10.6330 2.32031
\(22\) −1.00000 −0.213201
\(23\) 3.16743 0.660454 0.330227 0.943902i \(-0.392875\pi\)
0.330227 + 0.943902i \(0.392875\pi\)
\(24\) 3.03129 0.618760
\(25\) 1.00000 0.200000
\(26\) −6.77606 −1.32890
\(27\) 9.66598 1.86022
\(28\) 3.50775 0.662902
\(29\) 5.19277 0.964273 0.482137 0.876096i \(-0.339861\pi\)
0.482137 + 0.876096i \(0.339861\pi\)
\(30\) −3.03129 −0.553436
\(31\) −3.03853 −0.545737 −0.272868 0.962051i \(-0.587972\pi\)
−0.272868 + 0.962051i \(0.587972\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.03129 −0.527680
\(34\) 1.63062 0.279649
\(35\) −3.50775 −0.592917
\(36\) 6.18873 1.03146
\(37\) 11.5416 1.89743 0.948714 0.316135i \(-0.102385\pi\)
0.948714 + 0.316135i \(0.102385\pi\)
\(38\) 7.46655 1.21123
\(39\) −20.5402 −3.28907
\(40\) −1.00000 −0.158114
\(41\) −2.42690 −0.379018 −0.189509 0.981879i \(-0.560690\pi\)
−0.189509 + 0.981879i \(0.560690\pi\)
\(42\) 10.6330 1.64071
\(43\) −10.5318 −1.60609 −0.803046 0.595917i \(-0.796789\pi\)
−0.803046 + 0.595917i \(0.796789\pi\)
\(44\) −1.00000 −0.150756
\(45\) −6.18873 −0.922562
\(46\) 3.16743 0.467011
\(47\) 2.77614 0.404941 0.202470 0.979288i \(-0.435103\pi\)
0.202470 + 0.979288i \(0.435103\pi\)
\(48\) 3.03129 0.437529
\(49\) 5.30428 0.757754
\(50\) 1.00000 0.141421
\(51\) 4.94288 0.692141
\(52\) −6.77606 −0.939671
\(53\) −12.7598 −1.75270 −0.876348 0.481679i \(-0.840027\pi\)
−0.876348 + 0.481679i \(0.840027\pi\)
\(54\) 9.66598 1.31537
\(55\) 1.00000 0.134840
\(56\) 3.50775 0.468742
\(57\) 22.6333 2.99785
\(58\) 5.19277 0.681844
\(59\) 4.64925 0.605281 0.302641 0.953105i \(-0.402132\pi\)
0.302641 + 0.953105i \(0.402132\pi\)
\(60\) −3.03129 −0.391338
\(61\) −6.19293 −0.792923 −0.396462 0.918051i \(-0.629762\pi\)
−0.396462 + 0.918051i \(0.629762\pi\)
\(62\) −3.03853 −0.385894
\(63\) 21.7085 2.73501
\(64\) 1.00000 0.125000
\(65\) 6.77606 0.840467
\(66\) −3.03129 −0.373126
\(67\) 6.25363 0.764003 0.382001 0.924162i \(-0.375235\pi\)
0.382001 + 0.924162i \(0.375235\pi\)
\(68\) 1.63062 0.197741
\(69\) 9.60139 1.15587
\(70\) −3.50775 −0.419256
\(71\) −6.63711 −0.787680 −0.393840 0.919179i \(-0.628854\pi\)
−0.393840 + 0.919179i \(0.628854\pi\)
\(72\) 6.18873 0.729349
\(73\) −1.00000 −0.117041
\(74\) 11.5416 1.34168
\(75\) 3.03129 0.350023
\(76\) 7.46655 0.856472
\(77\) −3.50775 −0.399745
\(78\) −20.5402 −2.32572
\(79\) 10.3469 1.16412 0.582060 0.813146i \(-0.302247\pi\)
0.582060 + 0.813146i \(0.302247\pi\)
\(80\) −1.00000 −0.111803
\(81\) 10.7342 1.19269
\(82\) −2.42690 −0.268006
\(83\) 9.45862 1.03822 0.519109 0.854708i \(-0.326264\pi\)
0.519109 + 0.854708i \(0.326264\pi\)
\(84\) 10.6330 1.16016
\(85\) −1.63062 −0.176865
\(86\) −10.5318 −1.13568
\(87\) 15.7408 1.68759
\(88\) −1.00000 −0.106600
\(89\) −5.49478 −0.582445 −0.291223 0.956655i \(-0.594062\pi\)
−0.291223 + 0.956655i \(0.594062\pi\)
\(90\) −6.18873 −0.652350
\(91\) −23.7687 −2.49164
\(92\) 3.16743 0.330227
\(93\) −9.21069 −0.955104
\(94\) 2.77614 0.286336
\(95\) −7.46655 −0.766052
\(96\) 3.03129 0.309380
\(97\) −6.93289 −0.703928 −0.351964 0.936014i \(-0.614486\pi\)
−0.351964 + 0.936014i \(0.614486\pi\)
\(98\) 5.30428 0.535813
\(99\) −6.18873 −0.621991
\(100\) 1.00000 0.100000
\(101\) −5.27322 −0.524705 −0.262353 0.964972i \(-0.584498\pi\)
−0.262353 + 0.964972i \(0.584498\pi\)
\(102\) 4.94288 0.489418
\(103\) 0.506930 0.0499493 0.0249746 0.999688i \(-0.492049\pi\)
0.0249746 + 0.999688i \(0.492049\pi\)
\(104\) −6.77606 −0.664448
\(105\) −10.6330 −1.03767
\(106\) −12.7598 −1.23934
\(107\) 1.78929 0.172977 0.0864884 0.996253i \(-0.472435\pi\)
0.0864884 + 0.996253i \(0.472435\pi\)
\(108\) 9.66598 0.930109
\(109\) 8.93951 0.856250 0.428125 0.903720i \(-0.359174\pi\)
0.428125 + 0.903720i \(0.359174\pi\)
\(110\) 1.00000 0.0953463
\(111\) 34.9860 3.32072
\(112\) 3.50775 0.331451
\(113\) −17.5669 −1.65256 −0.826278 0.563263i \(-0.809546\pi\)
−0.826278 + 0.563263i \(0.809546\pi\)
\(114\) 22.6333 2.11980
\(115\) −3.16743 −0.295364
\(116\) 5.19277 0.482137
\(117\) −41.9352 −3.87691
\(118\) 4.64925 0.427998
\(119\) 5.71979 0.524332
\(120\) −3.03129 −0.276718
\(121\) 1.00000 0.0909091
\(122\) −6.19293 −0.560681
\(123\) −7.35665 −0.663327
\(124\) −3.03853 −0.272868
\(125\) −1.00000 −0.0894427
\(126\) 21.7085 1.93395
\(127\) 8.76750 0.777990 0.388995 0.921240i \(-0.372822\pi\)
0.388995 + 0.921240i \(0.372822\pi\)
\(128\) 1.00000 0.0883883
\(129\) −31.9251 −2.81085
\(130\) 6.77606 0.594300
\(131\) 10.0699 0.879813 0.439907 0.898044i \(-0.355012\pi\)
0.439907 + 0.898044i \(0.355012\pi\)
\(132\) −3.03129 −0.263840
\(133\) 26.1908 2.27103
\(134\) 6.25363 0.540232
\(135\) −9.66598 −0.831915
\(136\) 1.63062 0.139824
\(137\) −21.8423 −1.86611 −0.933055 0.359734i \(-0.882868\pi\)
−0.933055 + 0.359734i \(0.882868\pi\)
\(138\) 9.60139 0.817325
\(139\) −14.4697 −1.22730 −0.613652 0.789576i \(-0.710300\pi\)
−0.613652 + 0.789576i \(0.710300\pi\)
\(140\) −3.50775 −0.296459
\(141\) 8.41528 0.708694
\(142\) −6.63711 −0.556974
\(143\) 6.77606 0.566643
\(144\) 6.18873 0.515728
\(145\) −5.19277 −0.431236
\(146\) −1.00000 −0.0827606
\(147\) 16.0788 1.32616
\(148\) 11.5416 0.948714
\(149\) 22.4097 1.83588 0.917938 0.396724i \(-0.129853\pi\)
0.917938 + 0.396724i \(0.129853\pi\)
\(150\) 3.03129 0.247504
\(151\) −2.42552 −0.197386 −0.0986930 0.995118i \(-0.531466\pi\)
−0.0986930 + 0.995118i \(0.531466\pi\)
\(152\) 7.46655 0.605617
\(153\) 10.0915 0.815846
\(154\) −3.50775 −0.282662
\(155\) 3.03853 0.244061
\(156\) −20.5402 −1.64453
\(157\) 12.2085 0.974344 0.487172 0.873306i \(-0.338028\pi\)
0.487172 + 0.873306i \(0.338028\pi\)
\(158\) 10.3469 0.823158
\(159\) −38.6787 −3.06742
\(160\) −1.00000 −0.0790569
\(161\) 11.1105 0.875632
\(162\) 10.7342 0.843359
\(163\) −16.8631 −1.32082 −0.660408 0.750907i \(-0.729617\pi\)
−0.660408 + 0.750907i \(0.729617\pi\)
\(164\) −2.42690 −0.189509
\(165\) 3.03129 0.235986
\(166\) 9.45862 0.734132
\(167\) 15.2813 1.18250 0.591252 0.806487i \(-0.298634\pi\)
0.591252 + 0.806487i \(0.298634\pi\)
\(168\) 10.6330 0.820354
\(169\) 32.9150 2.53193
\(170\) −1.63062 −0.125063
\(171\) 46.2085 3.53365
\(172\) −10.5318 −0.803046
\(173\) −1.38794 −0.105523 −0.0527617 0.998607i \(-0.516802\pi\)
−0.0527617 + 0.998607i \(0.516802\pi\)
\(174\) 15.7408 1.19331
\(175\) 3.50775 0.265161
\(176\) −1.00000 −0.0753778
\(177\) 14.0932 1.05931
\(178\) −5.49478 −0.411851
\(179\) −5.85001 −0.437250 −0.218625 0.975809i \(-0.570157\pi\)
−0.218625 + 0.975809i \(0.570157\pi\)
\(180\) −6.18873 −0.461281
\(181\) 25.5379 1.89822 0.949110 0.314946i \(-0.101986\pi\)
0.949110 + 0.314946i \(0.101986\pi\)
\(182\) −23.7687 −1.76185
\(183\) −18.7726 −1.38771
\(184\) 3.16743 0.233506
\(185\) −11.5416 −0.848556
\(186\) −9.21069 −0.675360
\(187\) −1.63062 −0.119243
\(188\) 2.77614 0.202470
\(189\) 33.9058 2.46628
\(190\) −7.46655 −0.541680
\(191\) 17.5333 1.26867 0.634333 0.773060i \(-0.281275\pi\)
0.634333 + 0.773060i \(0.281275\pi\)
\(192\) 3.03129 0.218765
\(193\) −6.62499 −0.476877 −0.238439 0.971158i \(-0.576636\pi\)
−0.238439 + 0.971158i \(0.576636\pi\)
\(194\) −6.93289 −0.497752
\(195\) 20.5402 1.47092
\(196\) 5.30428 0.378877
\(197\) 4.55572 0.324582 0.162291 0.986743i \(-0.448112\pi\)
0.162291 + 0.986743i \(0.448112\pi\)
\(198\) −6.18873 −0.439814
\(199\) 4.04141 0.286488 0.143244 0.989687i \(-0.454247\pi\)
0.143244 + 0.989687i \(0.454247\pi\)
\(200\) 1.00000 0.0707107
\(201\) 18.9566 1.33709
\(202\) −5.27322 −0.371023
\(203\) 18.2149 1.27844
\(204\) 4.94288 0.346071
\(205\) 2.42690 0.169502
\(206\) 0.506930 0.0353195
\(207\) 19.6023 1.36246
\(208\) −6.77606 −0.469835
\(209\) −7.46655 −0.516472
\(210\) −10.6330 −0.733747
\(211\) −17.5194 −1.20609 −0.603043 0.797708i \(-0.706045\pi\)
−0.603043 + 0.797708i \(0.706045\pi\)
\(212\) −12.7598 −0.876348
\(213\) −20.1190 −1.37853
\(214\) 1.78929 0.122313
\(215\) 10.5318 0.718266
\(216\) 9.66598 0.657686
\(217\) −10.6584 −0.723540
\(218\) 8.93951 0.605460
\(219\) −3.03129 −0.204836
\(220\) 1.00000 0.0674200
\(221\) −11.0492 −0.743247
\(222\) 34.9860 2.34811
\(223\) 8.38259 0.561340 0.280670 0.959804i \(-0.409443\pi\)
0.280670 + 0.959804i \(0.409443\pi\)
\(224\) 3.50775 0.234371
\(225\) 6.18873 0.412582
\(226\) −17.5669 −1.16853
\(227\) −0.352740 −0.0234122 −0.0117061 0.999931i \(-0.503726\pi\)
−0.0117061 + 0.999931i \(0.503726\pi\)
\(228\) 22.6333 1.49893
\(229\) −3.80347 −0.251340 −0.125670 0.992072i \(-0.540108\pi\)
−0.125670 + 0.992072i \(0.540108\pi\)
\(230\) −3.16743 −0.208854
\(231\) −10.6330 −0.699600
\(232\) 5.19277 0.340922
\(233\) −10.8696 −0.712089 −0.356045 0.934469i \(-0.615875\pi\)
−0.356045 + 0.934469i \(0.615875\pi\)
\(234\) −41.9352 −2.74139
\(235\) −2.77614 −0.181095
\(236\) 4.64925 0.302641
\(237\) 31.3646 2.03735
\(238\) 5.71979 0.370759
\(239\) 7.40131 0.478751 0.239376 0.970927i \(-0.423057\pi\)
0.239376 + 0.970927i \(0.423057\pi\)
\(240\) −3.03129 −0.195669
\(241\) −17.4401 −1.12342 −0.561708 0.827336i \(-0.689856\pi\)
−0.561708 + 0.827336i \(0.689856\pi\)
\(242\) 1.00000 0.0642824
\(243\) 3.54057 0.227128
\(244\) −6.19293 −0.396462
\(245\) −5.30428 −0.338878
\(246\) −7.35665 −0.469043
\(247\) −50.5938 −3.21921
\(248\) −3.03853 −0.192947
\(249\) 28.6718 1.81700
\(250\) −1.00000 −0.0632456
\(251\) −7.02700 −0.443540 −0.221770 0.975099i \(-0.571183\pi\)
−0.221770 + 0.975099i \(0.571183\pi\)
\(252\) 21.7085 1.36751
\(253\) −3.16743 −0.199134
\(254\) 8.76750 0.550122
\(255\) −4.94288 −0.309535
\(256\) 1.00000 0.0625000
\(257\) −21.1301 −1.31806 −0.659029 0.752118i \(-0.729032\pi\)
−0.659029 + 0.752118i \(0.729032\pi\)
\(258\) −31.9251 −1.98757
\(259\) 40.4850 2.51562
\(260\) 6.77606 0.420234
\(261\) 32.1367 1.98921
\(262\) 10.0699 0.622122
\(263\) −20.7751 −1.28105 −0.640524 0.767938i \(-0.721283\pi\)
−0.640524 + 0.767938i \(0.721283\pi\)
\(264\) −3.03129 −0.186563
\(265\) 12.7598 0.783829
\(266\) 26.1908 1.60586
\(267\) −16.6563 −1.01935
\(268\) 6.25363 0.382001
\(269\) −1.24744 −0.0760578 −0.0380289 0.999277i \(-0.512108\pi\)
−0.0380289 + 0.999277i \(0.512108\pi\)
\(270\) −9.66598 −0.588253
\(271\) 12.9392 0.785999 0.392999 0.919539i \(-0.371437\pi\)
0.392999 + 0.919539i \(0.371437\pi\)
\(272\) 1.63062 0.0988707
\(273\) −72.0499 −4.36066
\(274\) −21.8423 −1.31954
\(275\) −1.00000 −0.0603023
\(276\) 9.60139 0.577936
\(277\) 27.8940 1.67599 0.837993 0.545681i \(-0.183729\pi\)
0.837993 + 0.545681i \(0.183729\pi\)
\(278\) −14.4697 −0.867835
\(279\) −18.8047 −1.12581
\(280\) −3.50775 −0.209628
\(281\) −3.66745 −0.218782 −0.109391 0.993999i \(-0.534890\pi\)
−0.109391 + 0.993999i \(0.534890\pi\)
\(282\) 8.41528 0.501122
\(283\) −29.6089 −1.76007 −0.880034 0.474910i \(-0.842481\pi\)
−0.880034 + 0.474910i \(0.842481\pi\)
\(284\) −6.63711 −0.393840
\(285\) −22.6333 −1.34068
\(286\) 6.77606 0.400677
\(287\) −8.51295 −0.502504
\(288\) 6.18873 0.364674
\(289\) −14.3411 −0.843593
\(290\) −5.19277 −0.304930
\(291\) −21.0156 −1.23196
\(292\) −1.00000 −0.0585206
\(293\) −24.3264 −1.42116 −0.710582 0.703614i \(-0.751568\pi\)
−0.710582 + 0.703614i \(0.751568\pi\)
\(294\) 16.0788 0.937736
\(295\) −4.64925 −0.270690
\(296\) 11.5416 0.670842
\(297\) −9.66598 −0.560877
\(298\) 22.4097 1.29816
\(299\) −21.4627 −1.24122
\(300\) 3.03129 0.175012
\(301\) −36.9431 −2.12936
\(302\) −2.42552 −0.139573
\(303\) −15.9847 −0.918296
\(304\) 7.46655 0.428236
\(305\) 6.19293 0.354606
\(306\) 10.0915 0.576890
\(307\) 14.3026 0.816293 0.408147 0.912916i \(-0.366175\pi\)
0.408147 + 0.912916i \(0.366175\pi\)
\(308\) −3.50775 −0.199872
\(309\) 1.53665 0.0874171
\(310\) 3.03853 0.172577
\(311\) −0.540885 −0.0306708 −0.0153354 0.999882i \(-0.504882\pi\)
−0.0153354 + 0.999882i \(0.504882\pi\)
\(312\) −20.5402 −1.16286
\(313\) 24.8705 1.40576 0.702881 0.711307i \(-0.251897\pi\)
0.702881 + 0.711307i \(0.251897\pi\)
\(314\) 12.2085 0.688965
\(315\) −21.7085 −1.22314
\(316\) 10.3469 0.582060
\(317\) −23.8128 −1.33746 −0.668730 0.743506i \(-0.733162\pi\)
−0.668730 + 0.743506i \(0.733162\pi\)
\(318\) −38.6787 −2.16900
\(319\) −5.19277 −0.290739
\(320\) −1.00000 −0.0559017
\(321\) 5.42385 0.302730
\(322\) 11.1105 0.619165
\(323\) 12.1751 0.677440
\(324\) 10.7342 0.596345
\(325\) −6.77606 −0.375868
\(326\) −16.8631 −0.933958
\(327\) 27.0983 1.49854
\(328\) −2.42690 −0.134003
\(329\) 9.73798 0.536872
\(330\) 3.03129 0.166867
\(331\) −12.1011 −0.665137 −0.332569 0.943079i \(-0.607915\pi\)
−0.332569 + 0.943079i \(0.607915\pi\)
\(332\) 9.45862 0.519109
\(333\) 71.4279 3.91422
\(334\) 15.2813 0.836157
\(335\) −6.25363 −0.341673
\(336\) 10.6330 0.580078
\(337\) −22.6731 −1.23508 −0.617541 0.786539i \(-0.711871\pi\)
−0.617541 + 0.786539i \(0.711871\pi\)
\(338\) 32.9150 1.79034
\(339\) −53.2504 −2.89217
\(340\) −1.63062 −0.0884326
\(341\) 3.03853 0.164546
\(342\) 46.2085 2.49867
\(343\) −5.94815 −0.321170
\(344\) −10.5318 −0.567839
\(345\) −9.60139 −0.516922
\(346\) −1.38794 −0.0746163
\(347\) 22.8290 1.22552 0.612762 0.790268i \(-0.290059\pi\)
0.612762 + 0.790268i \(0.290059\pi\)
\(348\) 15.7408 0.843796
\(349\) 9.69867 0.519158 0.259579 0.965722i \(-0.416416\pi\)
0.259579 + 0.965722i \(0.416416\pi\)
\(350\) 3.50775 0.187497
\(351\) −65.4973 −3.49599
\(352\) −1.00000 −0.0533002
\(353\) 20.2399 1.07726 0.538630 0.842542i \(-0.318942\pi\)
0.538630 + 0.842542i \(0.318942\pi\)
\(354\) 14.0932 0.749047
\(355\) 6.63711 0.352261
\(356\) −5.49478 −0.291223
\(357\) 17.3384 0.917643
\(358\) −5.85001 −0.309183
\(359\) −24.6263 −1.29973 −0.649864 0.760050i \(-0.725174\pi\)
−0.649864 + 0.760050i \(0.725174\pi\)
\(360\) −6.18873 −0.326175
\(361\) 36.7494 1.93418
\(362\) 25.5379 1.34224
\(363\) 3.03129 0.159102
\(364\) −23.7687 −1.24582
\(365\) 1.00000 0.0523424
\(366\) −18.7726 −0.981258
\(367\) −31.0873 −1.62274 −0.811372 0.584530i \(-0.801279\pi\)
−0.811372 + 0.584530i \(0.801279\pi\)
\(368\) 3.16743 0.165113
\(369\) −15.0194 −0.781881
\(370\) −11.5416 −0.600020
\(371\) −44.7582 −2.32373
\(372\) −9.21069 −0.477552
\(373\) 8.77686 0.454449 0.227224 0.973842i \(-0.427035\pi\)
0.227224 + 0.973842i \(0.427035\pi\)
\(374\) −1.63062 −0.0843172
\(375\) −3.03129 −0.156535
\(376\) 2.77614 0.143168
\(377\) −35.1866 −1.81220
\(378\) 33.9058 1.74393
\(379\) −24.9480 −1.28149 −0.640747 0.767752i \(-0.721375\pi\)
−0.640747 + 0.767752i \(0.721375\pi\)
\(380\) −7.46655 −0.383026
\(381\) 26.5768 1.36157
\(382\) 17.5333 0.897082
\(383\) −26.0631 −1.33176 −0.665880 0.746059i \(-0.731944\pi\)
−0.665880 + 0.746059i \(0.731944\pi\)
\(384\) 3.03129 0.154690
\(385\) 3.50775 0.178771
\(386\) −6.62499 −0.337203
\(387\) −65.1788 −3.31322
\(388\) −6.93289 −0.351964
\(389\) 4.33744 0.219917 0.109958 0.993936i \(-0.464928\pi\)
0.109958 + 0.993936i \(0.464928\pi\)
\(390\) 20.5402 1.04009
\(391\) 5.16486 0.261198
\(392\) 5.30428 0.267907
\(393\) 30.5249 1.53978
\(394\) 4.55572 0.229514
\(395\) −10.3469 −0.520611
\(396\) −6.18873 −0.310995
\(397\) 10.0832 0.506061 0.253031 0.967458i \(-0.418573\pi\)
0.253031 + 0.967458i \(0.418573\pi\)
\(398\) 4.04141 0.202577
\(399\) 79.3918 3.97456
\(400\) 1.00000 0.0500000
\(401\) 26.7837 1.33752 0.668758 0.743480i \(-0.266826\pi\)
0.668758 + 0.743480i \(0.266826\pi\)
\(402\) 18.9566 0.945469
\(403\) 20.5893 1.02563
\(404\) −5.27322 −0.262353
\(405\) −10.7342 −0.533387
\(406\) 18.2149 0.903991
\(407\) −11.5416 −0.572096
\(408\) 4.94288 0.244709
\(409\) 7.19999 0.356017 0.178008 0.984029i \(-0.443035\pi\)
0.178008 + 0.984029i \(0.443035\pi\)
\(410\) 2.42690 0.119856
\(411\) −66.2103 −3.26591
\(412\) 0.506930 0.0249746
\(413\) 16.3084 0.802484
\(414\) 19.6023 0.963403
\(415\) −9.45862 −0.464306
\(416\) −6.77606 −0.332224
\(417\) −43.8619 −2.14793
\(418\) −7.46655 −0.365201
\(419\) 0.832557 0.0406731 0.0203365 0.999793i \(-0.493526\pi\)
0.0203365 + 0.999793i \(0.493526\pi\)
\(420\) −10.6330 −0.518837
\(421\) −32.8737 −1.60216 −0.801082 0.598555i \(-0.795742\pi\)
−0.801082 + 0.598555i \(0.795742\pi\)
\(422\) −17.5194 −0.852832
\(423\) 17.1808 0.835357
\(424\) −12.7598 −0.619671
\(425\) 1.63062 0.0790966
\(426\) −20.1190 −0.974770
\(427\) −21.7232 −1.05126
\(428\) 1.78929 0.0864884
\(429\) 20.5402 0.991692
\(430\) 10.5318 0.507891
\(431\) 10.8956 0.524821 0.262411 0.964956i \(-0.415482\pi\)
0.262411 + 0.964956i \(0.415482\pi\)
\(432\) 9.66598 0.465055
\(433\) 2.05905 0.0989515 0.0494757 0.998775i \(-0.484245\pi\)
0.0494757 + 0.998775i \(0.484245\pi\)
\(434\) −10.6584 −0.511620
\(435\) −15.7408 −0.754714
\(436\) 8.93951 0.428125
\(437\) 23.6497 1.13132
\(438\) −3.03129 −0.144841
\(439\) −17.5600 −0.838092 −0.419046 0.907965i \(-0.637635\pi\)
−0.419046 + 0.907965i \(0.637635\pi\)
\(440\) 1.00000 0.0476731
\(441\) 32.8268 1.56318
\(442\) −11.0492 −0.525555
\(443\) −30.6787 −1.45759 −0.728794 0.684733i \(-0.759919\pi\)
−0.728794 + 0.684733i \(0.759919\pi\)
\(444\) 34.9860 1.66036
\(445\) 5.49478 0.260478
\(446\) 8.38259 0.396927
\(447\) 67.9304 3.21300
\(448\) 3.50775 0.165725
\(449\) 24.7126 1.16626 0.583130 0.812379i \(-0.301828\pi\)
0.583130 + 0.812379i \(0.301828\pi\)
\(450\) 6.18873 0.291740
\(451\) 2.42690 0.114278
\(452\) −17.5669 −0.826278
\(453\) −7.35246 −0.345449
\(454\) −0.352740 −0.0165549
\(455\) 23.7687 1.11429
\(456\) 22.6333 1.05990
\(457\) −25.9039 −1.21173 −0.605867 0.795566i \(-0.707174\pi\)
−0.605867 + 0.795566i \(0.707174\pi\)
\(458\) −3.80347 −0.177724
\(459\) 15.7615 0.735684
\(460\) −3.16743 −0.147682
\(461\) 12.1699 0.566811 0.283405 0.959000i \(-0.408536\pi\)
0.283405 + 0.959000i \(0.408536\pi\)
\(462\) −10.6330 −0.494692
\(463\) −3.82987 −0.177989 −0.0889945 0.996032i \(-0.528365\pi\)
−0.0889945 + 0.996032i \(0.528365\pi\)
\(464\) 5.19277 0.241068
\(465\) 9.21069 0.427135
\(466\) −10.8696 −0.503523
\(467\) −23.8515 −1.10371 −0.551857 0.833939i \(-0.686081\pi\)
−0.551857 + 0.833939i \(0.686081\pi\)
\(468\) −41.9352 −1.93846
\(469\) 21.9362 1.01292
\(470\) −2.77614 −0.128054
\(471\) 37.0075 1.70522
\(472\) 4.64925 0.213999
\(473\) 10.5318 0.484255
\(474\) 31.3646 1.44062
\(475\) 7.46655 0.342589
\(476\) 5.71979 0.262166
\(477\) −78.9671 −3.61565
\(478\) 7.40131 0.338528
\(479\) −16.3428 −0.746721 −0.373360 0.927686i \(-0.621795\pi\)
−0.373360 + 0.927686i \(0.621795\pi\)
\(480\) −3.03129 −0.138359
\(481\) −78.2067 −3.56592
\(482\) −17.4401 −0.794375
\(483\) 33.6792 1.53246
\(484\) 1.00000 0.0454545
\(485\) 6.93289 0.314806
\(486\) 3.54057 0.160604
\(487\) −4.84862 −0.219712 −0.109856 0.993948i \(-0.535039\pi\)
−0.109856 + 0.993948i \(0.535039\pi\)
\(488\) −6.19293 −0.280341
\(489\) −51.1168 −2.31158
\(490\) −5.30428 −0.239623
\(491\) −9.77007 −0.440917 −0.220458 0.975396i \(-0.570755\pi\)
−0.220458 + 0.975396i \(0.570755\pi\)
\(492\) −7.35665 −0.331663
\(493\) 8.46742 0.381354
\(494\) −50.5938 −2.27632
\(495\) 6.18873 0.278163
\(496\) −3.03853 −0.136434
\(497\) −23.2813 −1.04431
\(498\) 28.6718 1.28482
\(499\) −9.92500 −0.444304 −0.222152 0.975012i \(-0.571308\pi\)
−0.222152 + 0.975012i \(0.571308\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 46.3222 2.06952
\(502\) −7.02700 −0.313630
\(503\) 26.0555 1.16176 0.580879 0.813990i \(-0.302709\pi\)
0.580879 + 0.813990i \(0.302709\pi\)
\(504\) 21.7085 0.966973
\(505\) 5.27322 0.234655
\(506\) −3.16743 −0.140809
\(507\) 99.7751 4.43117
\(508\) 8.76750 0.388995
\(509\) −27.9509 −1.23890 −0.619451 0.785035i \(-0.712645\pi\)
−0.619451 + 0.785035i \(0.712645\pi\)
\(510\) −4.94288 −0.218874
\(511\) −3.50775 −0.155174
\(512\) 1.00000 0.0441942
\(513\) 72.1715 3.18645
\(514\) −21.1301 −0.932007
\(515\) −0.506930 −0.0223380
\(516\) −31.9251 −1.40542
\(517\) −2.77614 −0.122094
\(518\) 40.4850 1.77881
\(519\) −4.20726 −0.184678
\(520\) 6.77606 0.297150
\(521\) 28.8879 1.26560 0.632800 0.774315i \(-0.281906\pi\)
0.632800 + 0.774315i \(0.281906\pi\)
\(522\) 32.1367 1.40658
\(523\) 19.9488 0.872299 0.436150 0.899874i \(-0.356342\pi\)
0.436150 + 0.899874i \(0.356342\pi\)
\(524\) 10.0699 0.439907
\(525\) 10.6330 0.464062
\(526\) −20.7751 −0.905837
\(527\) −4.95469 −0.215830
\(528\) −3.03129 −0.131920
\(529\) −12.9674 −0.563801
\(530\) 12.7598 0.554251
\(531\) 28.7730 1.24864
\(532\) 26.1908 1.13551
\(533\) 16.4448 0.712305
\(534\) −16.6563 −0.720788
\(535\) −1.78929 −0.0773576
\(536\) 6.25363 0.270116
\(537\) −17.7331 −0.765239
\(538\) −1.24744 −0.0537810
\(539\) −5.30428 −0.228472
\(540\) −9.66598 −0.415957
\(541\) −29.9581 −1.28800 −0.643999 0.765026i \(-0.722726\pi\)
−0.643999 + 0.765026i \(0.722726\pi\)
\(542\) 12.9392 0.555785
\(543\) 77.4129 3.32211
\(544\) 1.63062 0.0699121
\(545\) −8.93951 −0.382926
\(546\) −72.0499 −3.08345
\(547\) 28.0606 1.19979 0.599893 0.800080i \(-0.295210\pi\)
0.599893 + 0.800080i \(0.295210\pi\)
\(548\) −21.8423 −0.933055
\(549\) −38.3264 −1.63573
\(550\) −1.00000 −0.0426401
\(551\) 38.7721 1.65175
\(552\) 9.60139 0.408662
\(553\) 36.2944 1.54339
\(554\) 27.8940 1.18510
\(555\) −34.9860 −1.48507
\(556\) −14.4697 −0.613652
\(557\) −33.7799 −1.43130 −0.715649 0.698460i \(-0.753869\pi\)
−0.715649 + 0.698460i \(0.753869\pi\)
\(558\) −18.8047 −0.796065
\(559\) 71.3645 3.01840
\(560\) −3.50775 −0.148229
\(561\) −4.94288 −0.208688
\(562\) −3.66745 −0.154702
\(563\) −31.1193 −1.31152 −0.655760 0.754969i \(-0.727652\pi\)
−0.655760 + 0.754969i \(0.727652\pi\)
\(564\) 8.41528 0.354347
\(565\) 17.5669 0.739045
\(566\) −29.6089 −1.24456
\(567\) 37.6529 1.58127
\(568\) −6.63711 −0.278487
\(569\) −14.2008 −0.595330 −0.297665 0.954670i \(-0.596208\pi\)
−0.297665 + 0.954670i \(0.596208\pi\)
\(570\) −22.6333 −0.948004
\(571\) −16.1552 −0.676073 −0.338037 0.941133i \(-0.609763\pi\)
−0.338037 + 0.941133i \(0.609763\pi\)
\(572\) 6.77606 0.283321
\(573\) 53.1486 2.22031
\(574\) −8.51295 −0.355324
\(575\) 3.16743 0.132091
\(576\) 6.18873 0.257864
\(577\) 24.9530 1.03881 0.519403 0.854529i \(-0.326154\pi\)
0.519403 + 0.854529i \(0.326154\pi\)
\(578\) −14.3411 −0.596511
\(579\) −20.0823 −0.834591
\(580\) −5.19277 −0.215618
\(581\) 33.1784 1.37647
\(582\) −21.0156 −0.871125
\(583\) 12.7598 0.528458
\(584\) −1.00000 −0.0413803
\(585\) 41.9352 1.73381
\(586\) −24.3264 −1.00491
\(587\) −4.49910 −0.185698 −0.0928488 0.995680i \(-0.529597\pi\)
−0.0928488 + 0.995680i \(0.529597\pi\)
\(588\) 16.0788 0.663080
\(589\) −22.6874 −0.934817
\(590\) −4.64925 −0.191407
\(591\) 13.8097 0.568056
\(592\) 11.5416 0.474357
\(593\) 2.32311 0.0953987 0.0476993 0.998862i \(-0.484811\pi\)
0.0476993 + 0.998862i \(0.484811\pi\)
\(594\) −9.66598 −0.396600
\(595\) −5.71979 −0.234489
\(596\) 22.4097 0.917938
\(597\) 12.2507 0.501387
\(598\) −21.4627 −0.877674
\(599\) −25.2752 −1.03272 −0.516359 0.856372i \(-0.672713\pi\)
−0.516359 + 0.856372i \(0.672713\pi\)
\(600\) 3.03129 0.123752
\(601\) −16.7537 −0.683400 −0.341700 0.939809i \(-0.611003\pi\)
−0.341700 + 0.939809i \(0.611003\pi\)
\(602\) −36.9431 −1.50569
\(603\) 38.7021 1.57607
\(604\) −2.42552 −0.0986930
\(605\) −1.00000 −0.0406558
\(606\) −15.9847 −0.649333
\(607\) −7.15814 −0.290540 −0.145270 0.989392i \(-0.546405\pi\)
−0.145270 + 0.989392i \(0.546405\pi\)
\(608\) 7.46655 0.302809
\(609\) 55.2148 2.23741
\(610\) 6.19293 0.250744
\(611\) −18.8113 −0.761023
\(612\) 10.0915 0.407923
\(613\) −5.30177 −0.214137 −0.107068 0.994252i \(-0.534146\pi\)
−0.107068 + 0.994252i \(0.534146\pi\)
\(614\) 14.3026 0.577207
\(615\) 7.35665 0.296649
\(616\) −3.50775 −0.141331
\(617\) 12.8336 0.516663 0.258331 0.966056i \(-0.416827\pi\)
0.258331 + 0.966056i \(0.416827\pi\)
\(618\) 1.53665 0.0618132
\(619\) 4.98877 0.200516 0.100258 0.994961i \(-0.468033\pi\)
0.100258 + 0.994961i \(0.468033\pi\)
\(620\) 3.03853 0.122030
\(621\) 30.6163 1.22859
\(622\) −0.540885 −0.0216875
\(623\) −19.2743 −0.772208
\(624\) −20.5402 −0.822267
\(625\) 1.00000 0.0400000
\(626\) 24.8705 0.994024
\(627\) −22.6333 −0.903887
\(628\) 12.2085 0.487172
\(629\) 18.8199 0.750400
\(630\) −21.7085 −0.864887
\(631\) 25.9597 1.03344 0.516721 0.856154i \(-0.327153\pi\)
0.516721 + 0.856154i \(0.327153\pi\)
\(632\) 10.3469 0.411579
\(633\) −53.1065 −2.11079
\(634\) −23.8128 −0.945726
\(635\) −8.76750 −0.347928
\(636\) −38.6787 −1.53371
\(637\) −35.9421 −1.42408
\(638\) −5.19277 −0.205584
\(639\) −41.0753 −1.62491
\(640\) −1.00000 −0.0395285
\(641\) −42.7692 −1.68928 −0.844641 0.535333i \(-0.820186\pi\)
−0.844641 + 0.535333i \(0.820186\pi\)
\(642\) 5.42385 0.214062
\(643\) −2.69662 −0.106344 −0.0531722 0.998585i \(-0.516933\pi\)
−0.0531722 + 0.998585i \(0.516933\pi\)
\(644\) 11.1105 0.437816
\(645\) 31.9251 1.25705
\(646\) 12.1751 0.479022
\(647\) 18.3390 0.720979 0.360489 0.932763i \(-0.382610\pi\)
0.360489 + 0.932763i \(0.382610\pi\)
\(648\) 10.7342 0.421679
\(649\) −4.64925 −0.182499
\(650\) −6.77606 −0.265779
\(651\) −32.3087 −1.26628
\(652\) −16.8631 −0.660408
\(653\) 40.8513 1.59863 0.799317 0.600909i \(-0.205195\pi\)
0.799317 + 0.600909i \(0.205195\pi\)
\(654\) 27.0983 1.05963
\(655\) −10.0699 −0.393464
\(656\) −2.42690 −0.0947546
\(657\) −6.18873 −0.241445
\(658\) 9.73798 0.379626
\(659\) 22.8207 0.888967 0.444484 0.895787i \(-0.353387\pi\)
0.444484 + 0.895787i \(0.353387\pi\)
\(660\) 3.03129 0.117993
\(661\) −25.6995 −0.999593 −0.499797 0.866143i \(-0.666592\pi\)
−0.499797 + 0.866143i \(0.666592\pi\)
\(662\) −12.1011 −0.470323
\(663\) −33.4933 −1.30077
\(664\) 9.45862 0.367066
\(665\) −26.1908 −1.01563
\(666\) 71.4279 2.76777
\(667\) 16.4477 0.636858
\(668\) 15.2813 0.591252
\(669\) 25.4101 0.982411
\(670\) −6.25363 −0.241599
\(671\) 6.19293 0.239075
\(672\) 10.6330 0.410177
\(673\) 35.2433 1.35853 0.679264 0.733894i \(-0.262299\pi\)
0.679264 + 0.733894i \(0.262299\pi\)
\(674\) −22.6731 −0.873335
\(675\) 9.66598 0.372044
\(676\) 32.9150 1.26596
\(677\) 14.5548 0.559386 0.279693 0.960089i \(-0.409767\pi\)
0.279693 + 0.960089i \(0.409767\pi\)
\(678\) −53.2504 −2.04507
\(679\) −24.3188 −0.933270
\(680\) −1.63062 −0.0625313
\(681\) −1.06926 −0.0409741
\(682\) 3.03853 0.116352
\(683\) 3.17885 0.121635 0.0608176 0.998149i \(-0.480629\pi\)
0.0608176 + 0.998149i \(0.480629\pi\)
\(684\) 46.2085 1.76683
\(685\) 21.8423 0.834550
\(686\) −5.94815 −0.227102
\(687\) −11.5294 −0.439875
\(688\) −10.5318 −0.401523
\(689\) 86.4613 3.29391
\(690\) −9.60139 −0.365519
\(691\) 4.23325 0.161040 0.0805202 0.996753i \(-0.474342\pi\)
0.0805202 + 0.996753i \(0.474342\pi\)
\(692\) −1.38794 −0.0527617
\(693\) −21.7085 −0.824638
\(694\) 22.8290 0.866576
\(695\) 14.4697 0.548867
\(696\) 15.7408 0.596654
\(697\) −3.95735 −0.149895
\(698\) 9.69867 0.367100
\(699\) −32.9489 −1.24624
\(700\) 3.50775 0.132580
\(701\) −38.3654 −1.44904 −0.724521 0.689252i \(-0.757939\pi\)
−0.724521 + 0.689252i \(0.757939\pi\)
\(702\) −65.4973 −2.47204
\(703\) 86.1760 3.25019
\(704\) −1.00000 −0.0376889
\(705\) −8.41528 −0.316938
\(706\) 20.2399 0.761738
\(707\) −18.4971 −0.695656
\(708\) 14.0932 0.529656
\(709\) 46.3244 1.73975 0.869875 0.493272i \(-0.164199\pi\)
0.869875 + 0.493272i \(0.164199\pi\)
\(710\) 6.63711 0.249086
\(711\) 64.0344 2.40148
\(712\) −5.49478 −0.205926
\(713\) −9.62433 −0.360434
\(714\) 17.3384 0.648872
\(715\) −6.77606 −0.253410
\(716\) −5.85001 −0.218625
\(717\) 22.4355 0.837871
\(718\) −24.6263 −0.919047
\(719\) −37.7562 −1.40807 −0.704035 0.710166i \(-0.748620\pi\)
−0.704035 + 0.710166i \(0.748620\pi\)
\(720\) −6.18873 −0.230640
\(721\) 1.77818 0.0662229
\(722\) 36.7494 1.36767
\(723\) −52.8660 −1.96611
\(724\) 25.5379 0.949110
\(725\) 5.19277 0.192855
\(726\) 3.03129 0.112502
\(727\) 32.3561 1.20002 0.600011 0.799992i \(-0.295163\pi\)
0.600011 + 0.799992i \(0.295163\pi\)
\(728\) −23.7687 −0.880927
\(729\) −21.4701 −0.795189
\(730\) 1.00000 0.0370117
\(731\) −17.1734 −0.635182
\(732\) −18.7726 −0.693854
\(733\) −16.5130 −0.609923 −0.304961 0.952365i \(-0.598644\pi\)
−0.304961 + 0.952365i \(0.598644\pi\)
\(734\) −31.0873 −1.14745
\(735\) −16.0788 −0.593076
\(736\) 3.16743 0.116753
\(737\) −6.25363 −0.230356
\(738\) −15.0194 −0.552873
\(739\) −12.7353 −0.468474 −0.234237 0.972179i \(-0.575259\pi\)
−0.234237 + 0.972179i \(0.575259\pi\)
\(740\) −11.5416 −0.424278
\(741\) −153.365 −5.63399
\(742\) −44.7582 −1.64312
\(743\) −12.5106 −0.458969 −0.229485 0.973312i \(-0.573704\pi\)
−0.229485 + 0.973312i \(0.573704\pi\)
\(744\) −9.21069 −0.337680
\(745\) −22.4097 −0.821029
\(746\) 8.77686 0.321344
\(747\) 58.5369 2.14175
\(748\) −1.63062 −0.0596213
\(749\) 6.27636 0.229333
\(750\) −3.03129 −0.110687
\(751\) 29.5838 1.07953 0.539764 0.841817i \(-0.318514\pi\)
0.539764 + 0.841817i \(0.318514\pi\)
\(752\) 2.77614 0.101235
\(753\) −21.3009 −0.776248
\(754\) −35.1866 −1.28142
\(755\) 2.42552 0.0882737
\(756\) 33.9058 1.23314
\(757\) 10.8274 0.393528 0.196764 0.980451i \(-0.436957\pi\)
0.196764 + 0.980451i \(0.436957\pi\)
\(758\) −24.9480 −0.906153
\(759\) −9.60139 −0.348508
\(760\) −7.46655 −0.270840
\(761\) 43.3620 1.57187 0.785935 0.618309i \(-0.212182\pi\)
0.785935 + 0.618309i \(0.212182\pi\)
\(762\) 26.5768 0.962778
\(763\) 31.3575 1.13522
\(764\) 17.5333 0.634333
\(765\) −10.0915 −0.364857
\(766\) −26.0631 −0.941697
\(767\) −31.5036 −1.13753
\(768\) 3.03129 0.109382
\(769\) 24.9933 0.901282 0.450641 0.892705i \(-0.351196\pi\)
0.450641 + 0.892705i \(0.351196\pi\)
\(770\) 3.50775 0.126410
\(771\) −64.0514 −2.30675
\(772\) −6.62499 −0.238439
\(773\) −27.9458 −1.00514 −0.502570 0.864537i \(-0.667612\pi\)
−0.502570 + 0.864537i \(0.667612\pi\)
\(774\) −65.1788 −2.34280
\(775\) −3.03853 −0.109147
\(776\) −6.93289 −0.248876
\(777\) 122.722 4.40262
\(778\) 4.33744 0.155505
\(779\) −18.1206 −0.649237
\(780\) 20.5402 0.735458
\(781\) 6.63711 0.237494
\(782\) 5.16486 0.184695
\(783\) 50.1932 1.79376
\(784\) 5.30428 0.189439
\(785\) −12.2085 −0.435740
\(786\) 30.5249 1.08879
\(787\) 45.5560 1.62390 0.811949 0.583729i \(-0.198407\pi\)
0.811949 + 0.583729i \(0.198407\pi\)
\(788\) 4.55572 0.162291
\(789\) −62.9754 −2.24198
\(790\) −10.3469 −0.368127
\(791\) −61.6202 −2.19096
\(792\) −6.18873 −0.219907
\(793\) 41.9637 1.49017
\(794\) 10.0832 0.357839
\(795\) 38.6787 1.37179
\(796\) 4.04141 0.143244
\(797\) −11.2410 −0.398176 −0.199088 0.979982i \(-0.563798\pi\)
−0.199088 + 0.979982i \(0.563798\pi\)
\(798\) 79.3918 2.81044
\(799\) 4.52681 0.160147
\(800\) 1.00000 0.0353553
\(801\) −34.0057 −1.20153
\(802\) 26.7837 0.945767
\(803\) 1.00000 0.0352892
\(804\) 18.9566 0.668547
\(805\) −11.1105 −0.391594
\(806\) 20.5893 0.725227
\(807\) −3.78136 −0.133110
\(808\) −5.27322 −0.185511
\(809\) −5.48510 −0.192846 −0.0964229 0.995340i \(-0.530740\pi\)
−0.0964229 + 0.995340i \(0.530740\pi\)
\(810\) −10.7342 −0.377161
\(811\) −29.7573 −1.04492 −0.522460 0.852664i \(-0.674986\pi\)
−0.522460 + 0.852664i \(0.674986\pi\)
\(812\) 18.2149 0.639218
\(813\) 39.2224 1.37559
\(814\) −11.5416 −0.404533
\(815\) 16.8631 0.590687
\(816\) 4.94288 0.173035
\(817\) −78.6366 −2.75115
\(818\) 7.19999 0.251742
\(819\) −147.098 −5.14003
\(820\) 2.42690 0.0847511
\(821\) 30.1940 1.05378 0.526888 0.849935i \(-0.323359\pi\)
0.526888 + 0.849935i \(0.323359\pi\)
\(822\) −66.2103 −2.30935
\(823\) −9.94045 −0.346502 −0.173251 0.984878i \(-0.555427\pi\)
−0.173251 + 0.984878i \(0.555427\pi\)
\(824\) 0.506930 0.0176597
\(825\) −3.03129 −0.105536
\(826\) 16.3084 0.567442
\(827\) −12.9750 −0.451185 −0.225592 0.974222i \(-0.572432\pi\)
−0.225592 + 0.974222i \(0.572432\pi\)
\(828\) 19.6023 0.681229
\(829\) −31.1035 −1.08027 −0.540135 0.841578i \(-0.681627\pi\)
−0.540135 + 0.841578i \(0.681627\pi\)
\(830\) −9.45862 −0.328314
\(831\) 84.5548 2.93317
\(832\) −6.77606 −0.234918
\(833\) 8.64925 0.299679
\(834\) −43.8619 −1.51881
\(835\) −15.2813 −0.528832
\(836\) −7.46655 −0.258236
\(837\) −29.3704 −1.01519
\(838\) 0.832557 0.0287602
\(839\) 24.3203 0.839630 0.419815 0.907610i \(-0.362095\pi\)
0.419815 + 0.907610i \(0.362095\pi\)
\(840\) −10.6330 −0.366873
\(841\) −2.03512 −0.0701766
\(842\) −32.8737 −1.13290
\(843\) −11.1171 −0.382894
\(844\) −17.5194 −0.603043
\(845\) −32.9150 −1.13231
\(846\) 17.1808 0.590687
\(847\) 3.50775 0.120528
\(848\) −12.7598 −0.438174
\(849\) −89.7533 −3.08033
\(850\) 1.63062 0.0559297
\(851\) 36.5572 1.25316
\(852\) −20.1190 −0.689266
\(853\) 50.0979 1.71532 0.857659 0.514219i \(-0.171918\pi\)
0.857659 + 0.514219i \(0.171918\pi\)
\(854\) −21.7232 −0.743353
\(855\) −46.2085 −1.58030
\(856\) 1.78929 0.0611565
\(857\) −31.1978 −1.06570 −0.532849 0.846210i \(-0.678879\pi\)
−0.532849 + 0.846210i \(0.678879\pi\)
\(858\) 20.5402 0.701232
\(859\) −23.7018 −0.808693 −0.404347 0.914606i \(-0.632501\pi\)
−0.404347 + 0.914606i \(0.632501\pi\)
\(860\) 10.5318 0.359133
\(861\) −25.8053 −0.879441
\(862\) 10.8956 0.371105
\(863\) −16.9518 −0.577045 −0.288523 0.957473i \(-0.593164\pi\)
−0.288523 + 0.957473i \(0.593164\pi\)
\(864\) 9.66598 0.328843
\(865\) 1.38794 0.0471915
\(866\) 2.05905 0.0699692
\(867\) −43.4720 −1.47639
\(868\) −10.6584 −0.361770
\(869\) −10.3469 −0.350996
\(870\) −15.7408 −0.533663
\(871\) −42.3750 −1.43582
\(872\) 8.93951 0.302730
\(873\) −42.9058 −1.45214
\(874\) 23.6497 0.799964
\(875\) −3.50775 −0.118583
\(876\) −3.03129 −0.102418
\(877\) −2.74306 −0.0926265 −0.0463132 0.998927i \(-0.514747\pi\)
−0.0463132 + 0.998927i \(0.514747\pi\)
\(878\) −17.5600 −0.592620
\(879\) −73.7405 −2.48720
\(880\) 1.00000 0.0337100
\(881\) 53.0362 1.78684 0.893418 0.449227i \(-0.148301\pi\)
0.893418 + 0.449227i \(0.148301\pi\)
\(882\) 32.8268 1.10533
\(883\) −11.2374 −0.378168 −0.189084 0.981961i \(-0.560552\pi\)
−0.189084 + 0.981961i \(0.560552\pi\)
\(884\) −11.0492 −0.371624
\(885\) −14.0932 −0.473739
\(886\) −30.6787 −1.03067
\(887\) −12.6538 −0.424873 −0.212436 0.977175i \(-0.568140\pi\)
−0.212436 + 0.977175i \(0.568140\pi\)
\(888\) 34.9860 1.17405
\(889\) 30.7542 1.03146
\(890\) 5.49478 0.184185
\(891\) −10.7342 −0.359609
\(892\) 8.38259 0.280670
\(893\) 20.7282 0.693641
\(894\) 67.9304 2.27193
\(895\) 5.85001 0.195544
\(896\) 3.50775 0.117186
\(897\) −65.0596 −2.17228
\(898\) 24.7126 0.824670
\(899\) −15.7784 −0.526240
\(900\) 6.18873 0.206291
\(901\) −20.8064 −0.693161
\(902\) 2.42690 0.0808070
\(903\) −111.985 −3.72663
\(904\) −17.5669 −0.584267
\(905\) −25.5379 −0.848909
\(906\) −7.35246 −0.244269
\(907\) −49.8822 −1.65631 −0.828156 0.560498i \(-0.810610\pi\)
−0.828156 + 0.560498i \(0.810610\pi\)
\(908\) −0.352740 −0.0117061
\(909\) −32.6346 −1.08242
\(910\) 23.7687 0.787925
\(911\) −34.6748 −1.14883 −0.574414 0.818565i \(-0.694770\pi\)
−0.574414 + 0.818565i \(0.694770\pi\)
\(912\) 22.6333 0.749463
\(913\) −9.45862 −0.313035
\(914\) −25.9039 −0.856825
\(915\) 18.7726 0.620602
\(916\) −3.80347 −0.125670
\(917\) 35.3227 1.16646
\(918\) 15.7615 0.520207
\(919\) 2.00279 0.0660659 0.0330329 0.999454i \(-0.489483\pi\)
0.0330329 + 0.999454i \(0.489483\pi\)
\(920\) −3.16743 −0.104427
\(921\) 43.3554 1.42861
\(922\) 12.1699 0.400796
\(923\) 44.9735 1.48032
\(924\) −10.6330 −0.349800
\(925\) 11.5416 0.379486
\(926\) −3.82987 −0.125857
\(927\) 3.13725 0.103041
\(928\) 5.19277 0.170461
\(929\) −27.2884 −0.895303 −0.447652 0.894208i \(-0.647739\pi\)
−0.447652 + 0.894208i \(0.647739\pi\)
\(930\) 9.21069 0.302030
\(931\) 39.6047 1.29799
\(932\) −10.8696 −0.356045
\(933\) −1.63958 −0.0536775
\(934\) −23.8515 −0.780444
\(935\) 1.63062 0.0533269
\(936\) −41.9352 −1.37070
\(937\) −0.481367 −0.0157256 −0.00786279 0.999969i \(-0.502503\pi\)
−0.00786279 + 0.999969i \(0.502503\pi\)
\(938\) 21.9362 0.716241
\(939\) 75.3897 2.46025
\(940\) −2.77614 −0.0905475
\(941\) 26.2647 0.856204 0.428102 0.903731i \(-0.359182\pi\)
0.428102 + 0.903731i \(0.359182\pi\)
\(942\) 37.0075 1.20577
\(943\) −7.68703 −0.250324
\(944\) 4.64925 0.151320
\(945\) −33.9058 −1.10296
\(946\) 10.5318 0.342420
\(947\) −22.5959 −0.734268 −0.367134 0.930168i \(-0.619661\pi\)
−0.367134 + 0.930168i \(0.619661\pi\)
\(948\) 31.3646 1.01867
\(949\) 6.77606 0.219960
\(950\) 7.46655 0.242247
\(951\) −72.1835 −2.34071
\(952\) 5.71979 0.185379
\(953\) 4.78708 0.155069 0.0775344 0.996990i \(-0.475295\pi\)
0.0775344 + 0.996990i \(0.475295\pi\)
\(954\) −78.9671 −2.55665
\(955\) −17.5333 −0.567365
\(956\) 7.40131 0.239376
\(957\) −15.7408 −0.508828
\(958\) −16.3428 −0.528011
\(959\) −76.6171 −2.47409
\(960\) −3.03129 −0.0978345
\(961\) −21.7673 −0.702171
\(962\) −78.2067 −2.52148
\(963\) 11.0734 0.356836
\(964\) −17.4401 −0.561708
\(965\) 6.62499 0.213266
\(966\) 33.6792 1.08361
\(967\) 34.2177 1.10037 0.550184 0.835044i \(-0.314558\pi\)
0.550184 + 0.835044i \(0.314558\pi\)
\(968\) 1.00000 0.0321412
\(969\) 36.9062 1.18560
\(970\) 6.93289 0.222602
\(971\) 18.4858 0.593238 0.296619 0.954996i \(-0.404141\pi\)
0.296619 + 0.954996i \(0.404141\pi\)
\(972\) 3.54057 0.113564
\(973\) −50.7560 −1.62716
\(974\) −4.84862 −0.155360
\(975\) −20.5402 −0.657814
\(976\) −6.19293 −0.198231
\(977\) −12.8490 −0.411075 −0.205538 0.978649i \(-0.565894\pi\)
−0.205538 + 0.978649i \(0.565894\pi\)
\(978\) −51.1168 −1.63454
\(979\) 5.49478 0.175614
\(980\) −5.30428 −0.169439
\(981\) 55.3242 1.76637
\(982\) −9.77007 −0.311775
\(983\) 14.6661 0.467776 0.233888 0.972264i \(-0.424855\pi\)
0.233888 + 0.972264i \(0.424855\pi\)
\(984\) −7.35665 −0.234521
\(985\) −4.55572 −0.145157
\(986\) 8.46742 0.269658
\(987\) 29.5187 0.939589
\(988\) −50.5938 −1.60960
\(989\) −33.3588 −1.06075
\(990\) 6.18873 0.196691
\(991\) 5.74036 0.182348 0.0911742 0.995835i \(-0.470938\pi\)
0.0911742 + 0.995835i \(0.470938\pi\)
\(992\) −3.03853 −0.0964736
\(993\) −36.6820 −1.16407
\(994\) −23.2813 −0.738438
\(995\) −4.04141 −0.128121
\(996\) 28.6718 0.908502
\(997\) 36.8207 1.16612 0.583062 0.812428i \(-0.301855\pi\)
0.583062 + 0.812428i \(0.301855\pi\)
\(998\) −9.92500 −0.314170
\(999\) 111.561 3.52963
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 8030.2.a.bg.1.14 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8030.2.a.bg.1.14 15 1.1 even 1 trivial