Properties

Label 8030.2.a.bg.1.9
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 1.9
Root \(1.35747\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.35747 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.35747 q^{6} +0.608019 q^{7} +1.00000 q^{8} -1.15727 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.35747 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.35747 q^{6} +0.608019 q^{7} +1.00000 q^{8} -1.15727 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.35747 q^{12} -1.27461 q^{13} +0.608019 q^{14} -1.35747 q^{15} +1.00000 q^{16} +5.40721 q^{17} -1.15727 q^{18} -5.57690 q^{19} -1.00000 q^{20} +0.825369 q^{21} -1.00000 q^{22} +6.46978 q^{23} +1.35747 q^{24} +1.00000 q^{25} -1.27461 q^{26} -5.64338 q^{27} +0.608019 q^{28} +5.96812 q^{29} -1.35747 q^{30} +3.61424 q^{31} +1.00000 q^{32} -1.35747 q^{33} +5.40721 q^{34} -0.608019 q^{35} -1.15727 q^{36} +3.57853 q^{37} -5.57690 q^{38} -1.73025 q^{39} -1.00000 q^{40} +4.81019 q^{41} +0.825369 q^{42} +4.63923 q^{43} -1.00000 q^{44} +1.15727 q^{45} +6.46978 q^{46} -3.42434 q^{47} +1.35747 q^{48} -6.63031 q^{49} +1.00000 q^{50} +7.34014 q^{51} -1.27461 q^{52} +6.33551 q^{53} -5.64338 q^{54} +1.00000 q^{55} +0.608019 q^{56} -7.57049 q^{57} +5.96812 q^{58} +8.43549 q^{59} -1.35747 q^{60} -11.5083 q^{61} +3.61424 q^{62} -0.703640 q^{63} +1.00000 q^{64} +1.27461 q^{65} -1.35747 q^{66} +3.72409 q^{67} +5.40721 q^{68} +8.78255 q^{69} -0.608019 q^{70} -10.2186 q^{71} -1.15727 q^{72} -1.00000 q^{73} +3.57853 q^{74} +1.35747 q^{75} -5.57690 q^{76} -0.608019 q^{77} -1.73025 q^{78} +1.90817 q^{79} -1.00000 q^{80} -4.18893 q^{81} +4.81019 q^{82} +1.16567 q^{83} +0.825369 q^{84} -5.40721 q^{85} +4.63923 q^{86} +8.10156 q^{87} -1.00000 q^{88} +14.1616 q^{89} +1.15727 q^{90} -0.774989 q^{91} +6.46978 q^{92} +4.90624 q^{93} -3.42434 q^{94} +5.57690 q^{95} +1.35747 q^{96} -4.84035 q^{97} -6.63031 q^{98} +1.15727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 15 q^{2} + 7 q^{3} + 15 q^{4} - 15 q^{5} + 7 q^{6} + 3 q^{7} + 15 q^{8} + 16 q^{9}+O(q^{10}) \) 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
\(3\) 1.35747 0.783737 0.391869 0.920021i \(-0.371829\pi\)
0.391869 + 0.920021i \(0.371829\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.35747 0.554186
\(7\) 0.608019 0.229809 0.114905 0.993377i \(-0.463344\pi\)
0.114905 + 0.993377i \(0.463344\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.15727 −0.385756
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) 1.35747 0.391869
\(13\) −1.27461 −0.353514 −0.176757 0.984254i \(-0.556561\pi\)
−0.176757 + 0.984254i \(0.556561\pi\)
\(14\) 0.608019 0.162500
\(15\) −1.35747 −0.350498
\(16\) 1.00000 0.250000
\(17\) 5.40721 1.31144 0.655721 0.755004i \(-0.272365\pi\)
0.655721 + 0.755004i \(0.272365\pi\)
\(18\) −1.15727 −0.272771
\(19\) −5.57690 −1.27943 −0.639715 0.768612i \(-0.720948\pi\)
−0.639715 + 0.768612i \(0.720948\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0.825369 0.180110
\(22\) −1.00000 −0.213201
\(23\) 6.46978 1.34904 0.674521 0.738255i \(-0.264350\pi\)
0.674521 + 0.738255i \(0.264350\pi\)
\(24\) 1.35747 0.277093
\(25\) 1.00000 0.200000
\(26\) −1.27461 −0.249972
\(27\) −5.64338 −1.08607
\(28\) 0.608019 0.114905
\(29\) 5.96812 1.10825 0.554126 0.832433i \(-0.313053\pi\)
0.554126 + 0.832433i \(0.313053\pi\)
\(30\) −1.35747 −0.247839
\(31\) 3.61424 0.649137 0.324569 0.945862i \(-0.394781\pi\)
0.324569 + 0.945862i \(0.394781\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.35747 −0.236306
\(34\) 5.40721 0.927329
\(35\) −0.608019 −0.102774
\(36\) −1.15727 −0.192878
\(37\) 3.57853 0.588306 0.294153 0.955758i \(-0.404962\pi\)
0.294153 + 0.955758i \(0.404962\pi\)
\(38\) −5.57690 −0.904693
\(39\) −1.73025 −0.277062
\(40\) −1.00000 −0.158114
\(41\) 4.81019 0.751226 0.375613 0.926777i \(-0.377432\pi\)
0.375613 + 0.926777i \(0.377432\pi\)
\(42\) 0.825369 0.127357
\(43\) 4.63923 0.707476 0.353738 0.935344i \(-0.384910\pi\)
0.353738 + 0.935344i \(0.384910\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.15727 0.172515
\(46\) 6.46978 0.953917
\(47\) −3.42434 −0.499491 −0.249746 0.968312i \(-0.580347\pi\)
−0.249746 + 0.968312i \(0.580347\pi\)
\(48\) 1.35747 0.195934
\(49\) −6.63031 −0.947188
\(50\) 1.00000 0.141421
\(51\) 7.34014 1.02783
\(52\) −1.27461 −0.176757
\(53\) 6.33551 0.870250 0.435125 0.900370i \(-0.356704\pi\)
0.435125 + 0.900370i \(0.356704\pi\)
\(54\) −5.64338 −0.767966
\(55\) 1.00000 0.134840
\(56\) 0.608019 0.0812499
\(57\) −7.57049 −1.00274
\(58\) 5.96812 0.783653
\(59\) 8.43549 1.09821 0.549104 0.835754i \(-0.314969\pi\)
0.549104 + 0.835754i \(0.314969\pi\)
\(60\) −1.35747 −0.175249
\(61\) −11.5083 −1.47348 −0.736740 0.676176i \(-0.763636\pi\)
−0.736740 + 0.676176i \(0.763636\pi\)
\(62\) 3.61424 0.459009
\(63\) −0.703640 −0.0886503
\(64\) 1.00000 0.125000
\(65\) 1.27461 0.158096
\(66\) −1.35747 −0.167093
\(67\) 3.72409 0.454970 0.227485 0.973782i \(-0.426950\pi\)
0.227485 + 0.973782i \(0.426950\pi\)
\(68\) 5.40721 0.655721
\(69\) 8.78255 1.05730
\(70\) −0.608019 −0.0726721
\(71\) −10.2186 −1.21272 −0.606362 0.795189i \(-0.707372\pi\)
−0.606362 + 0.795189i \(0.707372\pi\)
\(72\) −1.15727 −0.136385
\(73\) −1.00000 −0.117041
\(74\) 3.57853 0.415995
\(75\) 1.35747 0.156747
\(76\) −5.57690 −0.639715
\(77\) −0.608019 −0.0692902
\(78\) −1.73025 −0.195913
\(79\) 1.90817 0.214685 0.107343 0.994222i \(-0.465766\pi\)
0.107343 + 0.994222i \(0.465766\pi\)
\(80\) −1.00000 −0.111803
\(81\) −4.18893 −0.465436
\(82\) 4.81019 0.531197
\(83\) 1.16567 0.127949 0.0639745 0.997952i \(-0.479622\pi\)
0.0639745 + 0.997952i \(0.479622\pi\)
\(84\) 0.825369 0.0900551
\(85\) −5.40721 −0.586494
\(86\) 4.63923 0.500261
\(87\) 8.10156 0.868579
\(88\) −1.00000 −0.106600
\(89\) 14.1616 1.50112 0.750561 0.660801i \(-0.229783\pi\)
0.750561 + 0.660801i \(0.229783\pi\)
\(90\) 1.15727 0.121987
\(91\) −0.774989 −0.0812409
\(92\) 6.46978 0.674521
\(93\) 4.90624 0.508753
\(94\) −3.42434 −0.353193
\(95\) 5.57690 0.572178
\(96\) 1.35747 0.138546
\(97\) −4.84035 −0.491464 −0.245732 0.969338i \(-0.579028\pi\)
−0.245732 + 0.969338i \(0.579028\pi\)
\(98\) −6.63031 −0.669763
\(99\) 1.15727 0.116310
\(100\) 1.00000 0.100000
\(101\) 3.27870 0.326243 0.163122 0.986606i \(-0.447844\pi\)
0.163122 + 0.986606i \(0.447844\pi\)
\(102\) 7.34014 0.726782
\(103\) 6.53669 0.644080 0.322040 0.946726i \(-0.395632\pi\)
0.322040 + 0.946726i \(0.395632\pi\)
\(104\) −1.27461 −0.124986
\(105\) −0.825369 −0.0805477
\(106\) 6.33551 0.615359
\(107\) 13.5564 1.31055 0.655275 0.755390i \(-0.272553\pi\)
0.655275 + 0.755390i \(0.272553\pi\)
\(108\) −5.64338 −0.543034
\(109\) −2.59741 −0.248786 −0.124393 0.992233i \(-0.539698\pi\)
−0.124393 + 0.992233i \(0.539698\pi\)
\(110\) 1.00000 0.0953463
\(111\) 4.85775 0.461078
\(112\) 0.608019 0.0574524
\(113\) 19.6270 1.84636 0.923178 0.384373i \(-0.125582\pi\)
0.923178 + 0.384373i \(0.125582\pi\)
\(114\) −7.57049 −0.709042
\(115\) −6.46978 −0.603310
\(116\) 5.96812 0.554126
\(117\) 1.47507 0.136370
\(118\) 8.43549 0.776550
\(119\) 3.28768 0.301382
\(120\) −1.35747 −0.123920
\(121\) 1.00000 0.0909091
\(122\) −11.5083 −1.04191
\(123\) 6.52971 0.588764
\(124\) 3.61424 0.324569
\(125\) −1.00000 −0.0894427
\(126\) −0.703640 −0.0626853
\(127\) 4.68487 0.415715 0.207858 0.978159i \(-0.433351\pi\)
0.207858 + 0.978159i \(0.433351\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.29763 0.554476
\(130\) 1.27461 0.111791
\(131\) 2.46553 0.215414 0.107707 0.994183i \(-0.465649\pi\)
0.107707 + 0.994183i \(0.465649\pi\)
\(132\) −1.35747 −0.118153
\(133\) −3.39086 −0.294025
\(134\) 3.72409 0.321713
\(135\) 5.64338 0.485705
\(136\) 5.40721 0.463664
\(137\) −8.44166 −0.721220 −0.360610 0.932717i \(-0.617431\pi\)
−0.360610 + 0.932717i \(0.617431\pi\)
\(138\) 8.78255 0.747620
\(139\) 20.5520 1.74320 0.871600 0.490218i \(-0.163083\pi\)
0.871600 + 0.490218i \(0.163083\pi\)
\(140\) −0.608019 −0.0513870
\(141\) −4.64845 −0.391470
\(142\) −10.2186 −0.857525
\(143\) 1.27461 0.106589
\(144\) −1.15727 −0.0964390
\(145\) −5.96812 −0.495625
\(146\) −1.00000 −0.0827606
\(147\) −9.00047 −0.742346
\(148\) 3.57853 0.294153
\(149\) −1.05695 −0.0865883 −0.0432942 0.999062i \(-0.513785\pi\)
−0.0432942 + 0.999062i \(0.513785\pi\)
\(150\) 1.35747 0.110837
\(151\) 10.2067 0.830607 0.415304 0.909683i \(-0.363675\pi\)
0.415304 + 0.909683i \(0.363675\pi\)
\(152\) −5.57690 −0.452347
\(153\) −6.25759 −0.505896
\(154\) −0.608019 −0.0489955
\(155\) −3.61424 −0.290303
\(156\) −1.73025 −0.138531
\(157\) 7.28511 0.581415 0.290707 0.956812i \(-0.406109\pi\)
0.290707 + 0.956812i \(0.406109\pi\)
\(158\) 1.90817 0.151806
\(159\) 8.60029 0.682047
\(160\) −1.00000 −0.0790569
\(161\) 3.93375 0.310023
\(162\) −4.18893 −0.329113
\(163\) 3.70131 0.289909 0.144955 0.989438i \(-0.453696\pi\)
0.144955 + 0.989438i \(0.453696\pi\)
\(164\) 4.81019 0.375613
\(165\) 1.35747 0.105679
\(166\) 1.16567 0.0904736
\(167\) 14.8922 1.15239 0.576197 0.817311i \(-0.304536\pi\)
0.576197 + 0.817311i \(0.304536\pi\)
\(168\) 0.825369 0.0636786
\(169\) −11.3754 −0.875028
\(170\) −5.40721 −0.414714
\(171\) 6.45397 0.493547
\(172\) 4.63923 0.353738
\(173\) 4.99951 0.380106 0.190053 0.981774i \(-0.439134\pi\)
0.190053 + 0.981774i \(0.439134\pi\)
\(174\) 8.10156 0.614178
\(175\) 0.608019 0.0459619
\(176\) −1.00000 −0.0753778
\(177\) 11.4510 0.860706
\(178\) 14.1616 1.06145
\(179\) −20.3087 −1.51794 −0.758970 0.651126i \(-0.774297\pi\)
−0.758970 + 0.651126i \(0.774297\pi\)
\(180\) 1.15727 0.0862576
\(181\) −12.2620 −0.911425 −0.455713 0.890127i \(-0.650615\pi\)
−0.455713 + 0.890127i \(0.650615\pi\)
\(182\) −0.774989 −0.0574460
\(183\) −15.6221 −1.15482
\(184\) 6.46978 0.476959
\(185\) −3.57853 −0.263099
\(186\) 4.90624 0.359743
\(187\) −5.40721 −0.395414
\(188\) −3.42434 −0.249746
\(189\) −3.43128 −0.249589
\(190\) 5.57690 0.404591
\(191\) −18.4536 −1.33526 −0.667628 0.744495i \(-0.732690\pi\)
−0.667628 + 0.744495i \(0.732690\pi\)
\(192\) 1.35747 0.0979672
\(193\) 18.2662 1.31483 0.657415 0.753529i \(-0.271650\pi\)
0.657415 + 0.753529i \(0.271650\pi\)
\(194\) −4.84035 −0.347517
\(195\) 1.73025 0.123906
\(196\) −6.63031 −0.473594
\(197\) 12.1870 0.868288 0.434144 0.900843i \(-0.357051\pi\)
0.434144 + 0.900843i \(0.357051\pi\)
\(198\) 1.15727 0.0822434
\(199\) 18.6779 1.32404 0.662021 0.749485i \(-0.269699\pi\)
0.662021 + 0.749485i \(0.269699\pi\)
\(200\) 1.00000 0.0707107
\(201\) 5.05535 0.356577
\(202\) 3.27870 0.230689
\(203\) 3.62873 0.254687
\(204\) 7.34014 0.513913
\(205\) −4.81019 −0.335959
\(206\) 6.53669 0.455433
\(207\) −7.48727 −0.520401
\(208\) −1.27461 −0.0883786
\(209\) 5.57690 0.385762
\(210\) −0.825369 −0.0569559
\(211\) −5.62682 −0.387366 −0.193683 0.981064i \(-0.562043\pi\)
−0.193683 + 0.981064i \(0.562043\pi\)
\(212\) 6.33551 0.435125
\(213\) −13.8715 −0.950457
\(214\) 13.5564 0.926699
\(215\) −4.63923 −0.316393
\(216\) −5.64338 −0.383983
\(217\) 2.19753 0.149178
\(218\) −2.59741 −0.175919
\(219\) −1.35747 −0.0917295
\(220\) 1.00000 0.0674200
\(221\) −6.89210 −0.463613
\(222\) 4.85775 0.326031
\(223\) 15.6774 1.04983 0.524917 0.851153i \(-0.324096\pi\)
0.524917 + 0.851153i \(0.324096\pi\)
\(224\) 0.608019 0.0406250
\(225\) −1.15727 −0.0771512
\(226\) 19.6270 1.30557
\(227\) 15.5410 1.03149 0.515746 0.856741i \(-0.327515\pi\)
0.515746 + 0.856741i \(0.327515\pi\)
\(228\) −7.57049 −0.501368
\(229\) 10.7970 0.713486 0.356743 0.934203i \(-0.383887\pi\)
0.356743 + 0.934203i \(0.383887\pi\)
\(230\) −6.46978 −0.426605
\(231\) −0.825369 −0.0543053
\(232\) 5.96812 0.391826
\(233\) −11.7711 −0.771150 −0.385575 0.922676i \(-0.625997\pi\)
−0.385575 + 0.922676i \(0.625997\pi\)
\(234\) 1.47507 0.0964283
\(235\) 3.42434 0.223379
\(236\) 8.43549 0.549104
\(237\) 2.59028 0.168257
\(238\) 3.28768 0.213109
\(239\) −26.9077 −1.74052 −0.870259 0.492595i \(-0.836048\pi\)
−0.870259 + 0.492595i \(0.836048\pi\)
\(240\) −1.35747 −0.0876245
\(241\) −28.4825 −1.83472 −0.917360 0.398059i \(-0.869684\pi\)
−0.917360 + 0.398059i \(0.869684\pi\)
\(242\) 1.00000 0.0642824
\(243\) 11.2438 0.721289
\(244\) −11.5083 −0.736740
\(245\) 6.63031 0.423595
\(246\) 6.52971 0.416319
\(247\) 7.10840 0.452297
\(248\) 3.61424 0.229505
\(249\) 1.58237 0.100278
\(250\) −1.00000 −0.0632456
\(251\) 11.7900 0.744176 0.372088 0.928197i \(-0.378642\pi\)
0.372088 + 0.928197i \(0.378642\pi\)
\(252\) −0.703640 −0.0443252
\(253\) −6.46978 −0.406752
\(254\) 4.68487 0.293955
\(255\) −7.34014 −0.459657
\(256\) 1.00000 0.0625000
\(257\) 24.6820 1.53962 0.769810 0.638273i \(-0.220351\pi\)
0.769810 + 0.638273i \(0.220351\pi\)
\(258\) 6.29763 0.392074
\(259\) 2.17581 0.135198
\(260\) 1.27461 0.0790482
\(261\) −6.90671 −0.427515
\(262\) 2.46553 0.152321
\(263\) −2.79278 −0.172210 −0.0861051 0.996286i \(-0.527442\pi\)
−0.0861051 + 0.996286i \(0.527442\pi\)
\(264\) −1.35747 −0.0835467
\(265\) −6.33551 −0.389187
\(266\) −3.39086 −0.207907
\(267\) 19.2239 1.17648
\(268\) 3.72409 0.227485
\(269\) 6.34426 0.386817 0.193408 0.981118i \(-0.438046\pi\)
0.193408 + 0.981118i \(0.438046\pi\)
\(270\) 5.64338 0.343445
\(271\) −20.0684 −1.21907 −0.609534 0.792760i \(-0.708644\pi\)
−0.609534 + 0.792760i \(0.708644\pi\)
\(272\) 5.40721 0.327860
\(273\) −1.05203 −0.0636715
\(274\) −8.44166 −0.509979
\(275\) −1.00000 −0.0603023
\(276\) 8.78255 0.528648
\(277\) −10.7700 −0.647105 −0.323552 0.946210i \(-0.604877\pi\)
−0.323552 + 0.946210i \(0.604877\pi\)
\(278\) 20.5520 1.23263
\(279\) −4.18265 −0.250409
\(280\) −0.608019 −0.0363361
\(281\) −18.2462 −1.08848 −0.544239 0.838930i \(-0.683181\pi\)
−0.544239 + 0.838930i \(0.683181\pi\)
\(282\) −4.64845 −0.276811
\(283\) −1.64455 −0.0977581 −0.0488790 0.998805i \(-0.515565\pi\)
−0.0488790 + 0.998805i \(0.515565\pi\)
\(284\) −10.2186 −0.606362
\(285\) 7.57049 0.448437
\(286\) 1.27461 0.0753695
\(287\) 2.92469 0.172639
\(288\) −1.15727 −0.0681927
\(289\) 12.2379 0.719878
\(290\) −5.96812 −0.350460
\(291\) −6.57065 −0.385178
\(292\) −1.00000 −0.0585206
\(293\) 14.0979 0.823609 0.411805 0.911272i \(-0.364899\pi\)
0.411805 + 0.911272i \(0.364899\pi\)
\(294\) −9.00047 −0.524918
\(295\) −8.43549 −0.491133
\(296\) 3.57853 0.207998
\(297\) 5.64338 0.327462
\(298\) −1.05695 −0.0612272
\(299\) −8.24647 −0.476906
\(300\) 1.35747 0.0783737
\(301\) 2.82074 0.162585
\(302\) 10.2067 0.587328
\(303\) 4.45075 0.255689
\(304\) −5.57690 −0.319857
\(305\) 11.5083 0.658961
\(306\) −6.25759 −0.357723
\(307\) −10.1607 −0.579902 −0.289951 0.957041i \(-0.593639\pi\)
−0.289951 + 0.957041i \(0.593639\pi\)
\(308\) −0.608019 −0.0346451
\(309\) 8.87338 0.504789
\(310\) −3.61424 −0.205275
\(311\) −8.66290 −0.491228 −0.245614 0.969368i \(-0.578990\pi\)
−0.245614 + 0.969368i \(0.578990\pi\)
\(312\) −1.73025 −0.0979563
\(313\) 4.68241 0.264666 0.132333 0.991205i \(-0.457753\pi\)
0.132333 + 0.991205i \(0.457753\pi\)
\(314\) 7.28511 0.411122
\(315\) 0.703640 0.0396456
\(316\) 1.90817 0.107343
\(317\) −14.1539 −0.794963 −0.397482 0.917610i \(-0.630116\pi\)
−0.397482 + 0.917610i \(0.630116\pi\)
\(318\) 8.60029 0.482280
\(319\) −5.96812 −0.334151
\(320\) −1.00000 −0.0559017
\(321\) 18.4025 1.02713
\(322\) 3.93375 0.219219
\(323\) −30.1555 −1.67790
\(324\) −4.18893 −0.232718
\(325\) −1.27461 −0.0707029
\(326\) 3.70131 0.204997
\(327\) −3.52591 −0.194983
\(328\) 4.81019 0.265599
\(329\) −2.08206 −0.114788
\(330\) 1.35747 0.0747264
\(331\) −4.41687 −0.242773 −0.121387 0.992605i \(-0.538734\pi\)
−0.121387 + 0.992605i \(0.538734\pi\)
\(332\) 1.16567 0.0639745
\(333\) −4.14132 −0.226943
\(334\) 14.8922 0.814866
\(335\) −3.72409 −0.203469
\(336\) 0.825369 0.0450276
\(337\) −20.8916 −1.13804 −0.569020 0.822324i \(-0.692677\pi\)
−0.569020 + 0.822324i \(0.692677\pi\)
\(338\) −11.3754 −0.618738
\(339\) 26.6432 1.44706
\(340\) −5.40721 −0.293247
\(341\) −3.61424 −0.195722
\(342\) 6.45397 0.348991
\(343\) −8.28748 −0.447482
\(344\) 4.63923 0.250131
\(345\) −8.78255 −0.472837
\(346\) 4.99951 0.268776
\(347\) 32.1610 1.72649 0.863247 0.504781i \(-0.168427\pi\)
0.863247 + 0.504781i \(0.168427\pi\)
\(348\) 8.10156 0.434289
\(349\) 8.52683 0.456431 0.228215 0.973611i \(-0.426711\pi\)
0.228215 + 0.973611i \(0.426711\pi\)
\(350\) 0.608019 0.0325000
\(351\) 7.19313 0.383941
\(352\) −1.00000 −0.0533002
\(353\) −12.2726 −0.653206 −0.326603 0.945162i \(-0.605904\pi\)
−0.326603 + 0.945162i \(0.605904\pi\)
\(354\) 11.4510 0.608611
\(355\) 10.2186 0.542346
\(356\) 14.1616 0.750561
\(357\) 4.46294 0.236204
\(358\) −20.3087 −1.07335
\(359\) −5.96357 −0.314745 −0.157373 0.987539i \(-0.550302\pi\)
−0.157373 + 0.987539i \(0.550302\pi\)
\(360\) 1.15727 0.0609934
\(361\) 12.1019 0.636940
\(362\) −12.2620 −0.644475
\(363\) 1.35747 0.0712488
\(364\) −0.774989 −0.0406205
\(365\) 1.00000 0.0523424
\(366\) −15.6221 −0.816582
\(367\) −15.5688 −0.812684 −0.406342 0.913721i \(-0.633196\pi\)
−0.406342 + 0.913721i \(0.633196\pi\)
\(368\) 6.46978 0.337261
\(369\) −5.56668 −0.289790
\(370\) −3.57853 −0.186039
\(371\) 3.85211 0.199992
\(372\) 4.90624 0.254377
\(373\) −6.58501 −0.340959 −0.170479 0.985361i \(-0.554532\pi\)
−0.170479 + 0.985361i \(0.554532\pi\)
\(374\) −5.40721 −0.279600
\(375\) −1.35747 −0.0700996
\(376\) −3.42434 −0.176597
\(377\) −7.60705 −0.391783
\(378\) −3.43128 −0.176486
\(379\) −18.6144 −0.956159 −0.478080 0.878316i \(-0.658667\pi\)
−0.478080 + 0.878316i \(0.658667\pi\)
\(380\) 5.57690 0.286089
\(381\) 6.35959 0.325811
\(382\) −18.4536 −0.944168
\(383\) 0.874732 0.0446967 0.0223484 0.999750i \(-0.492886\pi\)
0.0223484 + 0.999750i \(0.492886\pi\)
\(384\) 1.35747 0.0692732
\(385\) 0.608019 0.0309875
\(386\) 18.2662 0.929725
\(387\) −5.36884 −0.272913
\(388\) −4.84035 −0.245732
\(389\) −2.34683 −0.118989 −0.0594946 0.998229i \(-0.518949\pi\)
−0.0594946 + 0.998229i \(0.518949\pi\)
\(390\) 1.73025 0.0876148
\(391\) 34.9835 1.76919
\(392\) −6.63031 −0.334881
\(393\) 3.34689 0.168828
\(394\) 12.1870 0.613972
\(395\) −1.90817 −0.0960102
\(396\) 1.15727 0.0581549
\(397\) −10.1299 −0.508405 −0.254203 0.967151i \(-0.581813\pi\)
−0.254203 + 0.967151i \(0.581813\pi\)
\(398\) 18.6779 0.936239
\(399\) −4.60300 −0.230438
\(400\) 1.00000 0.0500000
\(401\) 25.3431 1.26558 0.632788 0.774325i \(-0.281911\pi\)
0.632788 + 0.774325i \(0.281911\pi\)
\(402\) 5.05535 0.252138
\(403\) −4.60677 −0.229479
\(404\) 3.27870 0.163122
\(405\) 4.18893 0.208150
\(406\) 3.62873 0.180091
\(407\) −3.57853 −0.177381
\(408\) 7.34014 0.363391
\(409\) −27.1863 −1.34427 −0.672137 0.740426i \(-0.734624\pi\)
−0.672137 + 0.740426i \(0.734624\pi\)
\(410\) −4.81019 −0.237559
\(411\) −11.4593 −0.565247
\(412\) 6.53669 0.322040
\(413\) 5.12894 0.252379
\(414\) −7.48727 −0.367979
\(415\) −1.16567 −0.0572205
\(416\) −1.27461 −0.0624931
\(417\) 27.8988 1.36621
\(418\) 5.57690 0.272775
\(419\) −13.2670 −0.648136 −0.324068 0.946034i \(-0.605051\pi\)
−0.324068 + 0.946034i \(0.605051\pi\)
\(420\) −0.825369 −0.0402739
\(421\) −32.3904 −1.57861 −0.789305 0.614002i \(-0.789559\pi\)
−0.789305 + 0.614002i \(0.789559\pi\)
\(422\) −5.62682 −0.273909
\(423\) 3.96288 0.192682
\(424\) 6.33551 0.307680
\(425\) 5.40721 0.262288
\(426\) −13.8715 −0.672074
\(427\) −6.99723 −0.338620
\(428\) 13.5564 0.655275
\(429\) 1.73025 0.0835374
\(430\) −4.63923 −0.223724
\(431\) 7.86742 0.378960 0.189480 0.981885i \(-0.439320\pi\)
0.189480 + 0.981885i \(0.439320\pi\)
\(432\) −5.64338 −0.271517
\(433\) 34.1919 1.64316 0.821578 0.570096i \(-0.193094\pi\)
0.821578 + 0.570096i \(0.193094\pi\)
\(434\) 2.19753 0.105485
\(435\) −8.10156 −0.388440
\(436\) −2.59741 −0.124393
\(437\) −36.0813 −1.72600
\(438\) −1.35747 −0.0648626
\(439\) 12.0906 0.577052 0.288526 0.957472i \(-0.406835\pi\)
0.288526 + 0.957472i \(0.406835\pi\)
\(440\) 1.00000 0.0476731
\(441\) 7.67305 0.365383
\(442\) −6.89210 −0.327824
\(443\) 23.9911 1.13985 0.569926 0.821696i \(-0.306972\pi\)
0.569926 + 0.821696i \(0.306972\pi\)
\(444\) 4.85775 0.230539
\(445\) −14.1616 −0.671322
\(446\) 15.6774 0.742345
\(447\) −1.43477 −0.0678625
\(448\) 0.608019 0.0287262
\(449\) −23.0951 −1.08993 −0.544964 0.838460i \(-0.683457\pi\)
−0.544964 + 0.838460i \(0.683457\pi\)
\(450\) −1.15727 −0.0545541
\(451\) −4.81019 −0.226503
\(452\) 19.6270 0.923178
\(453\) 13.8553 0.650978
\(454\) 15.5410 0.729375
\(455\) 0.774989 0.0363320
\(456\) −7.57049 −0.354521
\(457\) −24.6058 −1.15101 −0.575505 0.817798i \(-0.695194\pi\)
−0.575505 + 0.817798i \(0.695194\pi\)
\(458\) 10.7970 0.504511
\(459\) −30.5149 −1.42431
\(460\) −6.46978 −0.301655
\(461\) 20.7783 0.967740 0.483870 0.875140i \(-0.339231\pi\)
0.483870 + 0.875140i \(0.339231\pi\)
\(462\) −0.825369 −0.0383996
\(463\) 18.1764 0.844729 0.422364 0.906426i \(-0.361200\pi\)
0.422364 + 0.906426i \(0.361200\pi\)
\(464\) 5.96812 0.277063
\(465\) −4.90624 −0.227521
\(466\) −11.7711 −0.545285
\(467\) −5.88706 −0.272421 −0.136210 0.990680i \(-0.543492\pi\)
−0.136210 + 0.990680i \(0.543492\pi\)
\(468\) 1.47507 0.0681851
\(469\) 2.26432 0.104556
\(470\) 3.42434 0.157953
\(471\) 9.88933 0.455676
\(472\) 8.43549 0.388275
\(473\) −4.63923 −0.213312
\(474\) 2.59028 0.118976
\(475\) −5.57690 −0.255886
\(476\) 3.28768 0.150691
\(477\) −7.33188 −0.335704
\(478\) −26.9077 −1.23073
\(479\) −8.58515 −0.392266 −0.196133 0.980577i \(-0.562838\pi\)
−0.196133 + 0.980577i \(0.562838\pi\)
\(480\) −1.35747 −0.0619599
\(481\) −4.56124 −0.207975
\(482\) −28.4825 −1.29734
\(483\) 5.33995 0.242976
\(484\) 1.00000 0.0454545
\(485\) 4.84035 0.219789
\(486\) 11.2438 0.510028
\(487\) −21.4531 −0.972132 −0.486066 0.873922i \(-0.661568\pi\)
−0.486066 + 0.873922i \(0.661568\pi\)
\(488\) −11.5083 −0.520954
\(489\) 5.02443 0.227213
\(490\) 6.63031 0.299527
\(491\) −21.0332 −0.949215 −0.474608 0.880197i \(-0.657410\pi\)
−0.474608 + 0.880197i \(0.657410\pi\)
\(492\) 6.52971 0.294382
\(493\) 32.2709 1.45341
\(494\) 7.10840 0.319822
\(495\) −1.15727 −0.0520153
\(496\) 3.61424 0.162284
\(497\) −6.21309 −0.278695
\(498\) 1.58237 0.0709075
\(499\) −21.7241 −0.972505 −0.486253 0.873818i \(-0.661637\pi\)
−0.486253 + 0.873818i \(0.661637\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 20.2158 0.903175
\(502\) 11.7900 0.526212
\(503\) −38.2583 −1.70585 −0.852926 0.522031i \(-0.825174\pi\)
−0.852926 + 0.522031i \(0.825174\pi\)
\(504\) −0.703640 −0.0313426
\(505\) −3.27870 −0.145900
\(506\) −6.46978 −0.287617
\(507\) −15.4417 −0.685792
\(508\) 4.68487 0.207858
\(509\) 43.3695 1.92232 0.961161 0.275989i \(-0.0890054\pi\)
0.961161 + 0.275989i \(0.0890054\pi\)
\(510\) −7.34014 −0.325027
\(511\) −0.608019 −0.0268972
\(512\) 1.00000 0.0441942
\(513\) 31.4726 1.38955
\(514\) 24.6820 1.08868
\(515\) −6.53669 −0.288041
\(516\) 6.29763 0.277238
\(517\) 3.42434 0.150602
\(518\) 2.17581 0.0955997
\(519\) 6.78671 0.297903
\(520\) 1.27461 0.0558955
\(521\) 0.164537 0.00720849 0.00360424 0.999994i \(-0.498853\pi\)
0.00360424 + 0.999994i \(0.498853\pi\)
\(522\) −6.90671 −0.302299
\(523\) 4.84807 0.211991 0.105996 0.994367i \(-0.466197\pi\)
0.105996 + 0.994367i \(0.466197\pi\)
\(524\) 2.46553 0.107707
\(525\) 0.825369 0.0360220
\(526\) −2.79278 −0.121771
\(527\) 19.5430 0.851305
\(528\) −1.35747 −0.0590764
\(529\) 18.8581 0.819916
\(530\) −6.33551 −0.275197
\(531\) −9.76213 −0.423640
\(532\) −3.39086 −0.147012
\(533\) −6.13114 −0.265569
\(534\) 19.2239 0.831900
\(535\) −13.5564 −0.586096
\(536\) 3.72409 0.160856
\(537\) −27.5684 −1.18967
\(538\) 6.34426 0.273521
\(539\) 6.63031 0.285588
\(540\) 5.64338 0.242852
\(541\) 1.68555 0.0724673 0.0362336 0.999343i \(-0.488464\pi\)
0.0362336 + 0.999343i \(0.488464\pi\)
\(542\) −20.0684 −0.862012
\(543\) −16.6453 −0.714318
\(544\) 5.40721 0.231832
\(545\) 2.59741 0.111261
\(546\) −1.05203 −0.0450226
\(547\) 0.160774 0.00687418 0.00343709 0.999994i \(-0.498906\pi\)
0.00343709 + 0.999994i \(0.498906\pi\)
\(548\) −8.44166 −0.360610
\(549\) 13.3181 0.568404
\(550\) −1.00000 −0.0426401
\(551\) −33.2836 −1.41793
\(552\) 8.78255 0.373810
\(553\) 1.16020 0.0493367
\(554\) −10.7700 −0.457572
\(555\) −4.85775 −0.206200
\(556\) 20.5520 0.871600
\(557\) −22.6746 −0.960753 −0.480376 0.877062i \(-0.659500\pi\)
−0.480376 + 0.877062i \(0.659500\pi\)
\(558\) −4.18265 −0.177066
\(559\) −5.91323 −0.250103
\(560\) −0.608019 −0.0256935
\(561\) −7.34014 −0.309901
\(562\) −18.2462 −0.769670
\(563\) −7.53055 −0.317375 −0.158687 0.987329i \(-0.550726\pi\)
−0.158687 + 0.987329i \(0.550726\pi\)
\(564\) −4.64845 −0.195735
\(565\) −19.6270 −0.825715
\(566\) −1.64455 −0.0691254
\(567\) −2.54695 −0.106962
\(568\) −10.2186 −0.428763
\(569\) 39.3562 1.64990 0.824948 0.565208i \(-0.191204\pi\)
0.824948 + 0.565208i \(0.191204\pi\)
\(570\) 7.57049 0.317093
\(571\) 12.9036 0.540000 0.270000 0.962860i \(-0.412976\pi\)
0.270000 + 0.962860i \(0.412976\pi\)
\(572\) 1.27461 0.0532943
\(573\) −25.0503 −1.04649
\(574\) 2.92469 0.122074
\(575\) 6.46978 0.269809
\(576\) −1.15727 −0.0482195
\(577\) 3.42560 0.142609 0.0713047 0.997455i \(-0.477284\pi\)
0.0713047 + 0.997455i \(0.477284\pi\)
\(578\) 12.2379 0.509031
\(579\) 24.7959 1.03048
\(580\) −5.96812 −0.247813
\(581\) 0.708750 0.0294039
\(582\) −6.57065 −0.272362
\(583\) −6.33551 −0.262390
\(584\) −1.00000 −0.0413803
\(585\) −1.47507 −0.0609866
\(586\) 14.0979 0.582380
\(587\) 11.8855 0.490565 0.245283 0.969452i \(-0.421119\pi\)
0.245283 + 0.969452i \(0.421119\pi\)
\(588\) −9.00047 −0.371173
\(589\) −20.1563 −0.830525
\(590\) −8.43549 −0.347284
\(591\) 16.5435 0.680510
\(592\) 3.57853 0.147077
\(593\) −45.6647 −1.87523 −0.937613 0.347682i \(-0.886969\pi\)
−0.937613 + 0.347682i \(0.886969\pi\)
\(594\) 5.64338 0.231551
\(595\) −3.28768 −0.134782
\(596\) −1.05695 −0.0432942
\(597\) 25.3548 1.03770
\(598\) −8.24647 −0.337223
\(599\) −24.0849 −0.984081 −0.492040 0.870572i \(-0.663749\pi\)
−0.492040 + 0.870572i \(0.663749\pi\)
\(600\) 1.35747 0.0554186
\(601\) −4.20692 −0.171604 −0.0858019 0.996312i \(-0.527345\pi\)
−0.0858019 + 0.996312i \(0.527345\pi\)
\(602\) 2.82074 0.114965
\(603\) −4.30977 −0.175507
\(604\) 10.2067 0.415304
\(605\) −1.00000 −0.0406558
\(606\) 4.45075 0.180799
\(607\) 20.4287 0.829174 0.414587 0.910010i \(-0.363926\pi\)
0.414587 + 0.910010i \(0.363926\pi\)
\(608\) −5.57690 −0.226173
\(609\) 4.92590 0.199608
\(610\) 11.5083 0.465956
\(611\) 4.36471 0.176577
\(612\) −6.25759 −0.252948
\(613\) 45.2149 1.82621 0.913105 0.407724i \(-0.133677\pi\)
0.913105 + 0.407724i \(0.133677\pi\)
\(614\) −10.1607 −0.410053
\(615\) −6.52971 −0.263303
\(616\) −0.608019 −0.0244978
\(617\) −26.7362 −1.07636 −0.538180 0.842830i \(-0.680888\pi\)
−0.538180 + 0.842830i \(0.680888\pi\)
\(618\) 8.87338 0.356940
\(619\) −18.2248 −0.732516 −0.366258 0.930513i \(-0.619361\pi\)
−0.366258 + 0.930513i \(0.619361\pi\)
\(620\) −3.61424 −0.145152
\(621\) −36.5114 −1.46515
\(622\) −8.66290 −0.347351
\(623\) 8.61049 0.344972
\(624\) −1.73025 −0.0692656
\(625\) 1.00000 0.0400000
\(626\) 4.68241 0.187147
\(627\) 7.57049 0.302336
\(628\) 7.28511 0.290707
\(629\) 19.3499 0.771529
\(630\) 0.703640 0.0280337
\(631\) 5.40052 0.214991 0.107496 0.994206i \(-0.465717\pi\)
0.107496 + 0.994206i \(0.465717\pi\)
\(632\) 1.90817 0.0759028
\(633\) −7.63825 −0.303593
\(634\) −14.1539 −0.562124
\(635\) −4.68487 −0.185913
\(636\) 8.60029 0.341023
\(637\) 8.45109 0.334844
\(638\) −5.96812 −0.236280
\(639\) 11.8256 0.467815
\(640\) −1.00000 −0.0395285
\(641\) 3.87647 0.153111 0.0765557 0.997065i \(-0.475608\pi\)
0.0765557 + 0.997065i \(0.475608\pi\)
\(642\) 18.4025 0.726289
\(643\) 30.2728 1.19384 0.596922 0.802299i \(-0.296390\pi\)
0.596922 + 0.802299i \(0.296390\pi\)
\(644\) 3.93375 0.155011
\(645\) −6.29763 −0.247969
\(646\) −30.1555 −1.18645
\(647\) −31.9670 −1.25675 −0.628376 0.777910i \(-0.716280\pi\)
−0.628376 + 0.777910i \(0.716280\pi\)
\(648\) −4.18893 −0.164557
\(649\) −8.43549 −0.331122
\(650\) −1.27461 −0.0499945
\(651\) 2.98308 0.116916
\(652\) 3.70131 0.144955
\(653\) 0.256173 0.0100248 0.00501241 0.999987i \(-0.498404\pi\)
0.00501241 + 0.999987i \(0.498404\pi\)
\(654\) −3.52591 −0.137874
\(655\) −2.46553 −0.0963363
\(656\) 4.81019 0.187807
\(657\) 1.15727 0.0451493
\(658\) −2.08206 −0.0811672
\(659\) −8.16400 −0.318024 −0.159012 0.987277i \(-0.550831\pi\)
−0.159012 + 0.987277i \(0.550831\pi\)
\(660\) 1.35747 0.0528396
\(661\) 1.39542 0.0542756 0.0271378 0.999632i \(-0.491361\pi\)
0.0271378 + 0.999632i \(0.491361\pi\)
\(662\) −4.41687 −0.171667
\(663\) −9.35584 −0.363351
\(664\) 1.16567 0.0452368
\(665\) 3.39086 0.131492
\(666\) −4.14132 −0.160473
\(667\) 38.6124 1.49508
\(668\) 14.8922 0.576197
\(669\) 21.2816 0.822794
\(670\) −3.72409 −0.143874
\(671\) 11.5083 0.444271
\(672\) 0.825369 0.0318393
\(673\) −12.5908 −0.485339 −0.242670 0.970109i \(-0.578023\pi\)
−0.242670 + 0.970109i \(0.578023\pi\)
\(674\) −20.8916 −0.804715
\(675\) −5.64338 −0.217214
\(676\) −11.3754 −0.437514
\(677\) 3.86811 0.148663 0.0743317 0.997234i \(-0.476318\pi\)
0.0743317 + 0.997234i \(0.476318\pi\)
\(678\) 26.6432 1.02322
\(679\) −2.94303 −0.112943
\(680\) −5.40721 −0.207357
\(681\) 21.0965 0.808419
\(682\) −3.61424 −0.138397
\(683\) −31.9618 −1.22298 −0.611492 0.791251i \(-0.709430\pi\)
−0.611492 + 0.791251i \(0.709430\pi\)
\(684\) 6.45397 0.246774
\(685\) 8.44166 0.322539
\(686\) −8.28748 −0.316418
\(687\) 14.6566 0.559186
\(688\) 4.63923 0.176869
\(689\) −8.07533 −0.307646
\(690\) −8.78255 −0.334346
\(691\) 36.2225 1.37797 0.688984 0.724777i \(-0.258057\pi\)
0.688984 + 0.724777i \(0.258057\pi\)
\(692\) 4.99951 0.190053
\(693\) 0.703640 0.0267291
\(694\) 32.1610 1.22082
\(695\) −20.5520 −0.779583
\(696\) 8.10156 0.307089
\(697\) 26.0097 0.985189
\(698\) 8.52683 0.322745
\(699\) −15.9789 −0.604379
\(700\) 0.608019 0.0229809
\(701\) −33.0030 −1.24651 −0.623253 0.782020i \(-0.714189\pi\)
−0.623253 + 0.782020i \(0.714189\pi\)
\(702\) 7.19313 0.271487
\(703\) −19.9571 −0.752697
\(704\) −1.00000 −0.0376889
\(705\) 4.64845 0.175071
\(706\) −12.2726 −0.461887
\(707\) 1.99351 0.0749737
\(708\) 11.4510 0.430353
\(709\) −0.716856 −0.0269221 −0.0134610 0.999909i \(-0.504285\pi\)
−0.0134610 + 0.999909i \(0.504285\pi\)
\(710\) 10.2186 0.383497
\(711\) −2.20826 −0.0828162
\(712\) 14.1616 0.530727
\(713\) 23.3834 0.875714
\(714\) 4.46294 0.167021
\(715\) −1.27461 −0.0476679
\(716\) −20.3087 −0.758970
\(717\) −36.5265 −1.36411
\(718\) −5.96357 −0.222558
\(719\) −3.29324 −0.122817 −0.0614086 0.998113i \(-0.519559\pi\)
−0.0614086 + 0.998113i \(0.519559\pi\)
\(720\) 1.15727 0.0431288
\(721\) 3.97443 0.148016
\(722\) 12.1019 0.450384
\(723\) −38.6642 −1.43794
\(724\) −12.2620 −0.455713
\(725\) 5.96812 0.221650
\(726\) 1.35747 0.0503805
\(727\) −38.4438 −1.42580 −0.712901 0.701265i \(-0.752619\pi\)
−0.712901 + 0.701265i \(0.752619\pi\)
\(728\) −0.774989 −0.0287230
\(729\) 27.8299 1.03074
\(730\) 1.00000 0.0370117
\(731\) 25.0853 0.927814
\(732\) −15.6221 −0.577411
\(733\) 49.9850 1.84624 0.923119 0.384515i \(-0.125631\pi\)
0.923119 + 0.384515i \(0.125631\pi\)
\(734\) −15.5688 −0.574655
\(735\) 9.00047 0.331987
\(736\) 6.46978 0.238479
\(737\) −3.72409 −0.137179
\(738\) −5.56668 −0.204912
\(739\) −8.94290 −0.328970 −0.164485 0.986380i \(-0.552596\pi\)
−0.164485 + 0.986380i \(0.552596\pi\)
\(740\) −3.57853 −0.131549
\(741\) 9.64946 0.354482
\(742\) 3.85211 0.141415
\(743\) −47.5663 −1.74504 −0.872520 0.488578i \(-0.837516\pi\)
−0.872520 + 0.488578i \(0.837516\pi\)
\(744\) 4.90624 0.179871
\(745\) 1.05695 0.0387235
\(746\) −6.58501 −0.241094
\(747\) −1.34899 −0.0493571
\(748\) −5.40721 −0.197707
\(749\) 8.24257 0.301177
\(750\) −1.35747 −0.0495679
\(751\) −45.0276 −1.64308 −0.821540 0.570151i \(-0.806885\pi\)
−0.821540 + 0.570151i \(0.806885\pi\)
\(752\) −3.42434 −0.124873
\(753\) 16.0046 0.583239
\(754\) −7.60705 −0.277032
\(755\) −10.2067 −0.371459
\(756\) −3.43128 −0.124794
\(757\) 15.4036 0.559853 0.279926 0.960021i \(-0.409690\pi\)
0.279926 + 0.960021i \(0.409690\pi\)
\(758\) −18.6144 −0.676107
\(759\) −8.78255 −0.318786
\(760\) 5.57690 0.202296
\(761\) −28.9539 −1.04958 −0.524790 0.851232i \(-0.675856\pi\)
−0.524790 + 0.851232i \(0.675856\pi\)
\(762\) 6.35959 0.230383
\(763\) −1.57927 −0.0571735
\(764\) −18.4536 −0.667628
\(765\) 6.25759 0.226244
\(766\) 0.874732 0.0316053
\(767\) −10.7520 −0.388232
\(768\) 1.35747 0.0489836
\(769\) 4.98586 0.179795 0.0898973 0.995951i \(-0.471346\pi\)
0.0898973 + 0.995951i \(0.471346\pi\)
\(770\) 0.608019 0.0219115
\(771\) 33.5051 1.20666
\(772\) 18.2662 0.657415
\(773\) −8.43430 −0.303361 −0.151680 0.988430i \(-0.548468\pi\)
−0.151680 + 0.988430i \(0.548468\pi\)
\(774\) −5.36884 −0.192979
\(775\) 3.61424 0.129827
\(776\) −4.84035 −0.173759
\(777\) 2.95361 0.105960
\(778\) −2.34683 −0.0841380
\(779\) −26.8260 −0.961141
\(780\) 1.73025 0.0619530
\(781\) 10.2186 0.365650
\(782\) 34.9835 1.25101
\(783\) −33.6804 −1.20364
\(784\) −6.63031 −0.236797
\(785\) −7.28511 −0.260017
\(786\) 3.34689 0.119380
\(787\) −11.9511 −0.426012 −0.213006 0.977051i \(-0.568325\pi\)
−0.213006 + 0.977051i \(0.568325\pi\)
\(788\) 12.1870 0.434144
\(789\) −3.79112 −0.134968
\(790\) −1.90817 −0.0678895
\(791\) 11.9336 0.424310
\(792\) 1.15727 0.0411217
\(793\) 14.6686 0.520896
\(794\) −10.1299 −0.359497
\(795\) −8.60029 −0.305021
\(796\) 18.6779 0.662021
\(797\) 9.14504 0.323934 0.161967 0.986796i \(-0.448216\pi\)
0.161967 + 0.986796i \(0.448216\pi\)
\(798\) −4.60300 −0.162944
\(799\) −18.5161 −0.655053
\(800\) 1.00000 0.0353553
\(801\) −16.3887 −0.579066
\(802\) 25.3431 0.894897
\(803\) 1.00000 0.0352892
\(804\) 5.05535 0.178289
\(805\) −3.93375 −0.138646
\(806\) −4.60677 −0.162266
\(807\) 8.61217 0.303163
\(808\) 3.27870 0.115344
\(809\) 39.5390 1.39012 0.695058 0.718953i \(-0.255379\pi\)
0.695058 + 0.718953i \(0.255379\pi\)
\(810\) 4.18893 0.147184
\(811\) 19.4631 0.683443 0.341721 0.939801i \(-0.388990\pi\)
0.341721 + 0.939801i \(0.388990\pi\)
\(812\) 3.62873 0.127343
\(813\) −27.2423 −0.955430
\(814\) −3.57853 −0.125427
\(815\) −3.70131 −0.129651
\(816\) 7.34014 0.256956
\(817\) −25.8726 −0.905166
\(818\) −27.1863 −0.950546
\(819\) 0.896870 0.0313392
\(820\) −4.81019 −0.167979
\(821\) 41.6945 1.45515 0.727574 0.686029i \(-0.240648\pi\)
0.727574 + 0.686029i \(0.240648\pi\)
\(822\) −11.4593 −0.399690
\(823\) −2.79126 −0.0972970 −0.0486485 0.998816i \(-0.515491\pi\)
−0.0486485 + 0.998816i \(0.515491\pi\)
\(824\) 6.53669 0.227717
\(825\) −1.35747 −0.0472611
\(826\) 5.12894 0.178459
\(827\) 32.9618 1.14619 0.573097 0.819488i \(-0.305742\pi\)
0.573097 + 0.819488i \(0.305742\pi\)
\(828\) −7.48727 −0.260201
\(829\) 27.4543 0.953528 0.476764 0.879031i \(-0.341810\pi\)
0.476764 + 0.879031i \(0.341810\pi\)
\(830\) −1.16567 −0.0404610
\(831\) −14.6199 −0.507160
\(832\) −1.27461 −0.0441893
\(833\) −35.8515 −1.24218
\(834\) 27.8988 0.966057
\(835\) −14.8922 −0.515367
\(836\) 5.57690 0.192881
\(837\) −20.3965 −0.705008
\(838\) −13.2670 −0.458302
\(839\) 11.2227 0.387452 0.193726 0.981056i \(-0.437943\pi\)
0.193726 + 0.981056i \(0.437943\pi\)
\(840\) −0.825369 −0.0284779
\(841\) 6.61847 0.228223
\(842\) −32.3904 −1.11625
\(843\) −24.7687 −0.853080
\(844\) −5.62682 −0.193683
\(845\) 11.3754 0.391324
\(846\) 3.96288 0.136246
\(847\) 0.608019 0.0208918
\(848\) 6.33551 0.217562
\(849\) −2.23243 −0.0766167
\(850\) 5.40721 0.185466
\(851\) 23.1523 0.793650
\(852\) −13.8715 −0.475228
\(853\) 7.82241 0.267834 0.133917 0.990993i \(-0.457244\pi\)
0.133917 + 0.990993i \(0.457244\pi\)
\(854\) −6.99723 −0.239440
\(855\) −6.45397 −0.220721
\(856\) 13.5564 0.463350
\(857\) −7.54594 −0.257764 −0.128882 0.991660i \(-0.541139\pi\)
−0.128882 + 0.991660i \(0.541139\pi\)
\(858\) 1.73025 0.0590699
\(859\) 21.5268 0.734486 0.367243 0.930125i \(-0.380302\pi\)
0.367243 + 0.930125i \(0.380302\pi\)
\(860\) −4.63923 −0.158197
\(861\) 3.97018 0.135303
\(862\) 7.86742 0.267965
\(863\) −2.00484 −0.0682455 −0.0341228 0.999418i \(-0.510864\pi\)
−0.0341228 + 0.999418i \(0.510864\pi\)
\(864\) −5.64338 −0.191992
\(865\) −4.99951 −0.169989
\(866\) 34.1919 1.16189
\(867\) 16.6126 0.564195
\(868\) 2.19753 0.0745889
\(869\) −1.90817 −0.0647301
\(870\) −8.10156 −0.274669
\(871\) −4.74678 −0.160838
\(872\) −2.59741 −0.0879593
\(873\) 5.60159 0.189585
\(874\) −36.0813 −1.22047
\(875\) −0.608019 −0.0205548
\(876\) −1.35747 −0.0458648
\(877\) −6.29155 −0.212451 −0.106225 0.994342i \(-0.533876\pi\)
−0.106225 + 0.994342i \(0.533876\pi\)
\(878\) 12.0906 0.408037
\(879\) 19.1375 0.645493
\(880\) 1.00000 0.0337100
\(881\) −18.7614 −0.632087 −0.316043 0.948745i \(-0.602355\pi\)
−0.316043 + 0.948745i \(0.602355\pi\)
\(882\) 7.67305 0.258365
\(883\) 20.2339 0.680924 0.340462 0.940258i \(-0.389416\pi\)
0.340462 + 0.940258i \(0.389416\pi\)
\(884\) −6.89210 −0.231807
\(885\) −11.4510 −0.384920
\(886\) 23.9911 0.805998
\(887\) −16.1549 −0.542429 −0.271215 0.962519i \(-0.587425\pi\)
−0.271215 + 0.962519i \(0.587425\pi\)
\(888\) 4.85775 0.163016
\(889\) 2.84849 0.0955353
\(890\) −14.1616 −0.474696
\(891\) 4.18893 0.140334
\(892\) 15.6774 0.524917
\(893\) 19.0972 0.639064
\(894\) −1.43477 −0.0479860
\(895\) 20.3087 0.678843
\(896\) 0.608019 0.0203125
\(897\) −11.1944 −0.373769
\(898\) −23.0951 −0.770695
\(899\) 21.5702 0.719408
\(900\) −1.15727 −0.0385756
\(901\) 34.2574 1.14128
\(902\) −4.81019 −0.160162
\(903\) 3.82908 0.127424
\(904\) 19.6270 0.652785
\(905\) 12.2620 0.407602
\(906\) 13.8553 0.460311
\(907\) −1.37058 −0.0455094 −0.0227547 0.999741i \(-0.507244\pi\)
−0.0227547 + 0.999741i \(0.507244\pi\)
\(908\) 15.5410 0.515746
\(909\) −3.79434 −0.125850
\(910\) 0.774989 0.0256906
\(911\) −39.1208 −1.29613 −0.648064 0.761586i \(-0.724421\pi\)
−0.648064 + 0.761586i \(0.724421\pi\)
\(912\) −7.57049 −0.250684
\(913\) −1.16567 −0.0385781
\(914\) −24.6058 −0.813887
\(915\) 15.6221 0.516452
\(916\) 10.7970 0.356743
\(917\) 1.49909 0.0495043
\(918\) −30.5149 −1.00714
\(919\) −45.9131 −1.51453 −0.757266 0.653106i \(-0.773466\pi\)
−0.757266 + 0.653106i \(0.773466\pi\)
\(920\) −6.46978 −0.213302
\(921\) −13.7929 −0.454491
\(922\) 20.7783 0.684295
\(923\) 13.0248 0.428715
\(924\) −0.825369 −0.0271526
\(925\) 3.57853 0.117661
\(926\) 18.1764 0.597314
\(927\) −7.56470 −0.248458
\(928\) 5.96812 0.195913
\(929\) −48.6547 −1.59631 −0.798155 0.602452i \(-0.794190\pi\)
−0.798155 + 0.602452i \(0.794190\pi\)
\(930\) −4.90624 −0.160882
\(931\) 36.9766 1.21186
\(932\) −11.7711 −0.385575
\(933\) −11.7597 −0.384994
\(934\) −5.88706 −0.192630
\(935\) 5.40721 0.176835
\(936\) 1.47507 0.0482142
\(937\) −12.3082 −0.402090 −0.201045 0.979582i \(-0.564434\pi\)
−0.201045 + 0.979582i \(0.564434\pi\)
\(938\) 2.26432 0.0739326
\(939\) 6.35625 0.207428
\(940\) 3.42434 0.111690
\(941\) −22.9424 −0.747900 −0.373950 0.927449i \(-0.621997\pi\)
−0.373950 + 0.927449i \(0.621997\pi\)
\(942\) 9.88933 0.322212
\(943\) 31.1209 1.01344
\(944\) 8.43549 0.274552
\(945\) 3.43128 0.111620
\(946\) −4.63923 −0.150834
\(947\) 59.1141 1.92095 0.960475 0.278368i \(-0.0897934\pi\)
0.960475 + 0.278368i \(0.0897934\pi\)
\(948\) 2.59028 0.0841285
\(949\) 1.27461 0.0413757
\(950\) −5.57690 −0.180939
\(951\) −19.2136 −0.623043
\(952\) 3.28768 0.106554
\(953\) −1.33413 −0.0432166 −0.0216083 0.999767i \(-0.506879\pi\)
−0.0216083 + 0.999767i \(0.506879\pi\)
\(954\) −7.33188 −0.237378
\(955\) 18.4536 0.597144
\(956\) −26.9077 −0.870259
\(957\) −8.10156 −0.261886
\(958\) −8.58515 −0.277374
\(959\) −5.13269 −0.165743
\(960\) −1.35747 −0.0438122
\(961\) −17.9372 −0.578621
\(962\) −4.56124 −0.147060
\(963\) −15.6884 −0.505553
\(964\) −28.4825 −0.917360
\(965\) −18.2662 −0.588010
\(966\) 5.33995 0.171810
\(967\) 50.0088 1.60817 0.804087 0.594512i \(-0.202655\pi\)
0.804087 + 0.594512i \(0.202655\pi\)
\(968\) 1.00000 0.0321412
\(969\) −40.9353 −1.31503
\(970\) 4.84035 0.155414
\(971\) −0.494504 −0.0158694 −0.00793469 0.999969i \(-0.502526\pi\)
−0.00793469 + 0.999969i \(0.502526\pi\)
\(972\) 11.2438 0.360644
\(973\) 12.4960 0.400604
\(974\) −21.4531 −0.687401
\(975\) −1.73025 −0.0554125
\(976\) −11.5083 −0.368370
\(977\) −54.3441 −1.73862 −0.869310 0.494267i \(-0.835437\pi\)
−0.869310 + 0.494267i \(0.835437\pi\)
\(978\) 5.02443 0.160664
\(979\) −14.1616 −0.452605
\(980\) 6.63031 0.211798
\(981\) 3.00589 0.0959708
\(982\) −21.0332 −0.671197
\(983\) −32.8799 −1.04871 −0.524353 0.851501i \(-0.675693\pi\)
−0.524353 + 0.851501i \(0.675693\pi\)
\(984\) 6.52971 0.208159
\(985\) −12.1870 −0.388310
\(986\) 32.2709 1.02771
\(987\) −2.82634 −0.0899634
\(988\) 7.10840 0.226148
\(989\) 30.0148 0.954416
\(990\) −1.15727 −0.0367804
\(991\) −9.97742 −0.316943 −0.158472 0.987364i \(-0.550657\pi\)
−0.158472 + 0.987364i \(0.550657\pi\)
\(992\) 3.61424 0.114752
\(993\) −5.99578 −0.190270
\(994\) −6.21309 −0.197067
\(995\) −18.6779 −0.592130
\(996\) 1.58237 0.0501392
\(997\) −4.82352 −0.152762 −0.0763812 0.997079i \(-0.524337\pi\)
−0.0763812 + 0.997079i \(0.524337\pi\)
\(998\) −21.7241 −0.687665
\(999\) −20.1950 −0.638941
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.1.9 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8030.2.a.bg.1.9 15 1.1 even 1 trivial