Properties

Label 2415.2.a.v.1.9
Level $2415$
Weight $2$
Character 2415.1
Self dual yes
Analytic conductor $19.284$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2415,2,Mod(1,2415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2415, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2415.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2415.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: \(19.2838720881\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 16x^{8} + 30x^{7} + 83x^{6} - 137x^{5} - 164x^{4} + 208x^{3} + 108x^{2} - 83x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(2.42079\) of defining polynomial
Character \(\chi\) \(=\) 2415.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.42079 q^{2} +1.00000 q^{3} +3.86022 q^{4} +1.00000 q^{5} +2.42079 q^{6} -1.00000 q^{7} +4.50319 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.42079 q^{2} +1.00000 q^{3} +3.86022 q^{4} +1.00000 q^{5} +2.42079 q^{6} -1.00000 q^{7} +4.50319 q^{8} +1.00000 q^{9} +2.42079 q^{10} +2.79376 q^{11} +3.86022 q^{12} -5.96954 q^{13} -2.42079 q^{14} +1.00000 q^{15} +3.18084 q^{16} +4.49734 q^{17} +2.42079 q^{18} +8.29134 q^{19} +3.86022 q^{20} -1.00000 q^{21} +6.76311 q^{22} +1.00000 q^{23} +4.50319 q^{24} +1.00000 q^{25} -14.4510 q^{26} +1.00000 q^{27} -3.86022 q^{28} -0.288553 q^{29} +2.42079 q^{30} +6.34817 q^{31} -1.30624 q^{32} +2.79376 q^{33} +10.8871 q^{34} -1.00000 q^{35} +3.86022 q^{36} -11.0997 q^{37} +20.0716 q^{38} -5.96954 q^{39} +4.50319 q^{40} -8.10472 q^{41} -2.42079 q^{42} -0.264436 q^{43} +10.7845 q^{44} +1.00000 q^{45} +2.42079 q^{46} +3.28997 q^{47} +3.18084 q^{48} +1.00000 q^{49} +2.42079 q^{50} +4.49734 q^{51} -23.0437 q^{52} -4.34703 q^{53} +2.42079 q^{54} +2.79376 q^{55} -4.50319 q^{56} +8.29134 q^{57} -0.698525 q^{58} +15.0558 q^{59} +3.86022 q^{60} -7.31629 q^{61} +15.3676 q^{62} -1.00000 q^{63} -9.52381 q^{64} -5.96954 q^{65} +6.76311 q^{66} +2.53462 q^{67} +17.3607 q^{68} +1.00000 q^{69} -2.42079 q^{70} +10.2719 q^{71} +4.50319 q^{72} -4.95971 q^{73} -26.8699 q^{74} +1.00000 q^{75} +32.0064 q^{76} -2.79376 q^{77} -14.4510 q^{78} -16.7403 q^{79} +3.18084 q^{80} +1.00000 q^{81} -19.6198 q^{82} +3.16745 q^{83} -3.86022 q^{84} +4.49734 q^{85} -0.640143 q^{86} -0.288553 q^{87} +12.5809 q^{88} +3.44448 q^{89} +2.42079 q^{90} +5.96954 q^{91} +3.86022 q^{92} +6.34817 q^{93} +7.96433 q^{94} +8.29134 q^{95} -1.30624 q^{96} -8.27346 q^{97} +2.42079 q^{98} +2.79376 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} + 10 q^{3} + 16 q^{4} + 10 q^{5} + 2 q^{6} - 10 q^{7} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{2} + 10 q^{3} + 16 q^{4} + 10 q^{5} + 2 q^{6} - 10 q^{7} + 6 q^{8} + 10 q^{9} + 2 q^{10} - q^{11} + 16 q^{12} - 2 q^{13} - 2 q^{14} + 10 q^{15} + 36 q^{16} + 8 q^{17} + 2 q^{18} + 11 q^{19} + 16 q^{20} - 10 q^{21} + 10 q^{23} + 6 q^{24} + 10 q^{25} + 15 q^{26} + 10 q^{27} - 16 q^{28} + 6 q^{29} + 2 q^{30} + 16 q^{31} - q^{32} - q^{33} + 21 q^{34} - 10 q^{35} + 16 q^{36} - 6 q^{38} - 2 q^{39} + 6 q^{40} + 23 q^{41} - 2 q^{42} + 4 q^{43} - q^{44} + 10 q^{45} + 2 q^{46} + 21 q^{47} + 36 q^{48} + 10 q^{49} + 2 q^{50} + 8 q^{51} + 10 q^{52} - q^{53} + 2 q^{54} - q^{55} - 6 q^{56} + 11 q^{57} + 8 q^{58} + 15 q^{59} + 16 q^{60} + 29 q^{61} + 12 q^{62} - 10 q^{63} + 80 q^{64} - 2 q^{65} + 32 q^{67} - 28 q^{68} + 10 q^{69} - 2 q^{70} - 4 q^{71} + 6 q^{72} + 18 q^{73} - 49 q^{74} + 10 q^{75} + 49 q^{76} + q^{77} + 15 q^{78} + 8 q^{79} + 36 q^{80} + 10 q^{81} + 6 q^{82} + 2 q^{83} - 16 q^{84} + 8 q^{85} - 14 q^{86} + 6 q^{87} - 69 q^{88} + 6 q^{89} + 2 q^{90} + 2 q^{91} + 16 q^{92} + 16 q^{93} - 2 q^{94} + 11 q^{95} - q^{96} - 2 q^{97} + 2 q^{98} - q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.42079 1.71176 0.855878 0.517178i \(-0.173017\pi\)
0.855878 + 0.517178i \(0.173017\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.86022 1.93011
\(5\) 1.00000 0.447214
\(6\) 2.42079 0.988283
\(7\) −1.00000 −0.377964
\(8\) 4.50319 1.59212
\(9\) 1.00000 0.333333
\(10\) 2.42079 0.765521
\(11\) 2.79376 0.842352 0.421176 0.906979i \(-0.361618\pi\)
0.421176 + 0.906979i \(0.361618\pi\)
\(12\) 3.86022 1.11435
\(13\) −5.96954 −1.65565 −0.827826 0.560985i \(-0.810423\pi\)
−0.827826 + 0.560985i \(0.810423\pi\)
\(14\) −2.42079 −0.646983
\(15\) 1.00000 0.258199
\(16\) 3.18084 0.795210
\(17\) 4.49734 1.09077 0.545383 0.838187i \(-0.316384\pi\)
0.545383 + 0.838187i \(0.316384\pi\)
\(18\) 2.42079 0.570585
\(19\) 8.29134 1.90216 0.951082 0.308940i \(-0.0999742\pi\)
0.951082 + 0.308940i \(0.0999742\pi\)
\(20\) 3.86022 0.863171
\(21\) −1.00000 −0.218218
\(22\) 6.76311 1.44190
\(23\) 1.00000 0.208514
\(24\) 4.50319 0.919210
\(25\) 1.00000 0.200000
\(26\) −14.4510 −2.83407
\(27\) 1.00000 0.192450
\(28\) −3.86022 −0.729512
\(29\) −0.288553 −0.0535829 −0.0267915 0.999641i \(-0.508529\pi\)
−0.0267915 + 0.999641i \(0.508529\pi\)
\(30\) 2.42079 0.441973
\(31\) 6.34817 1.14017 0.570083 0.821587i \(-0.306911\pi\)
0.570083 + 0.821587i \(0.306911\pi\)
\(32\) −1.30624 −0.230913
\(33\) 2.79376 0.486332
\(34\) 10.8871 1.86713
\(35\) −1.00000 −0.169031
\(36\) 3.86022 0.643369
\(37\) −11.0997 −1.82477 −0.912387 0.409328i \(-0.865763\pi\)
−0.912387 + 0.409328i \(0.865763\pi\)
\(38\) 20.0716 3.25604
\(39\) −5.96954 −0.955891
\(40\) 4.50319 0.712017
\(41\) −8.10472 −1.26575 −0.632873 0.774256i \(-0.718124\pi\)
−0.632873 + 0.774256i \(0.718124\pi\)
\(42\) −2.42079 −0.373536
\(43\) −0.264436 −0.0403261 −0.0201630 0.999797i \(-0.506419\pi\)
−0.0201630 + 0.999797i \(0.506419\pi\)
\(44\) 10.7845 1.62583
\(45\) 1.00000 0.149071
\(46\) 2.42079 0.356926
\(47\) 3.28997 0.479892 0.239946 0.970786i \(-0.422870\pi\)
0.239946 + 0.970786i \(0.422870\pi\)
\(48\) 3.18084 0.459115
\(49\) 1.00000 0.142857
\(50\) 2.42079 0.342351
\(51\) 4.49734 0.629754
\(52\) −23.0437 −3.19559
\(53\) −4.34703 −0.597110 −0.298555 0.954392i \(-0.596505\pi\)
−0.298555 + 0.954392i \(0.596505\pi\)
\(54\) 2.42079 0.329428
\(55\) 2.79376 0.376711
\(56\) −4.50319 −0.601764
\(57\) 8.29134 1.09821
\(58\) −0.698525 −0.0917209
\(59\) 15.0558 1.96010 0.980049 0.198757i \(-0.0636905\pi\)
0.980049 + 0.198757i \(0.0636905\pi\)
\(60\) 3.86022 0.498352
\(61\) −7.31629 −0.936755 −0.468378 0.883528i \(-0.655161\pi\)
−0.468378 + 0.883528i \(0.655161\pi\)
\(62\) 15.3676 1.95168
\(63\) −1.00000 −0.125988
\(64\) −9.52381 −1.19048
\(65\) −5.96954 −0.740430
\(66\) 6.76311 0.832482
\(67\) 2.53462 0.309654 0.154827 0.987942i \(-0.450518\pi\)
0.154827 + 0.987942i \(0.450518\pi\)
\(68\) 17.3607 2.10530
\(69\) 1.00000 0.120386
\(70\) −2.42079 −0.289340
\(71\) 10.2719 1.21905 0.609527 0.792765i \(-0.291359\pi\)
0.609527 + 0.792765i \(0.291359\pi\)
\(72\) 4.50319 0.530706
\(73\) −4.95971 −0.580491 −0.290245 0.956952i \(-0.593737\pi\)
−0.290245 + 0.956952i \(0.593737\pi\)
\(74\) −26.8699 −3.12357
\(75\) 1.00000 0.115470
\(76\) 32.0064 3.67138
\(77\) −2.79376 −0.318379
\(78\) −14.4510 −1.63625
\(79\) −16.7403 −1.88344 −0.941718 0.336404i \(-0.890789\pi\)
−0.941718 + 0.336404i \(0.890789\pi\)
\(80\) 3.18084 0.355629
\(81\) 1.00000 0.111111
\(82\) −19.6198 −2.16665
\(83\) 3.16745 0.347672 0.173836 0.984775i \(-0.444384\pi\)
0.173836 + 0.984775i \(0.444384\pi\)
\(84\) −3.86022 −0.421184
\(85\) 4.49734 0.487805
\(86\) −0.640143 −0.0690284
\(87\) −0.288553 −0.0309361
\(88\) 12.5809 1.34112
\(89\) 3.44448 0.365114 0.182557 0.983195i \(-0.441563\pi\)
0.182557 + 0.983195i \(0.441563\pi\)
\(90\) 2.42079 0.255174
\(91\) 5.96954 0.625778
\(92\) 3.86022 0.402455
\(93\) 6.34817 0.658275
\(94\) 7.96433 0.821457
\(95\) 8.29134 0.850673
\(96\) −1.30624 −0.133318
\(97\) −8.27346 −0.840042 −0.420021 0.907514i \(-0.637977\pi\)
−0.420021 + 0.907514i \(0.637977\pi\)
\(98\) 2.42079 0.244537
\(99\) 2.79376 0.280784
\(100\) 3.86022 0.386022
\(101\) −2.75695 −0.274327 −0.137164 0.990548i \(-0.543799\pi\)
−0.137164 + 0.990548i \(0.543799\pi\)
\(102\) 10.8871 1.07799
\(103\) −9.83291 −0.968865 −0.484433 0.874829i \(-0.660974\pi\)
−0.484433 + 0.874829i \(0.660974\pi\)
\(104\) −26.8820 −2.63599
\(105\) −1.00000 −0.0975900
\(106\) −10.5232 −1.02211
\(107\) −17.4587 −1.68779 −0.843896 0.536507i \(-0.819743\pi\)
−0.843896 + 0.536507i \(0.819743\pi\)
\(108\) 3.86022 0.371450
\(109\) −1.08674 −0.104091 −0.0520456 0.998645i \(-0.516574\pi\)
−0.0520456 + 0.998645i \(0.516574\pi\)
\(110\) 6.76311 0.644837
\(111\) −11.0997 −1.05353
\(112\) −3.18084 −0.300561
\(113\) −14.1873 −1.33463 −0.667315 0.744775i \(-0.732557\pi\)
−0.667315 + 0.744775i \(0.732557\pi\)
\(114\) 20.0716 1.87987
\(115\) 1.00000 0.0932505
\(116\) −1.11388 −0.103421
\(117\) −5.96954 −0.551884
\(118\) 36.4469 3.35521
\(119\) −4.49734 −0.412271
\(120\) 4.50319 0.411083
\(121\) −3.19488 −0.290444
\(122\) −17.7112 −1.60350
\(123\) −8.10472 −0.730778
\(124\) 24.5053 2.20064
\(125\) 1.00000 0.0894427
\(126\) −2.42079 −0.215661
\(127\) −21.4471 −1.90312 −0.951562 0.307457i \(-0.900522\pi\)
−0.951562 + 0.307457i \(0.900522\pi\)
\(128\) −20.4427 −1.80689
\(129\) −0.264436 −0.0232823
\(130\) −14.4510 −1.26744
\(131\) −5.79445 −0.506264 −0.253132 0.967432i \(-0.581461\pi\)
−0.253132 + 0.967432i \(0.581461\pi\)
\(132\) 10.7845 0.938673
\(133\) −8.29134 −0.718950
\(134\) 6.13579 0.530051
\(135\) 1.00000 0.0860663
\(136\) 20.2524 1.73663
\(137\) 6.13504 0.524152 0.262076 0.965047i \(-0.415593\pi\)
0.262076 + 0.965047i \(0.415593\pi\)
\(138\) 2.42079 0.206071
\(139\) −3.58042 −0.303687 −0.151843 0.988405i \(-0.548521\pi\)
−0.151843 + 0.988405i \(0.548521\pi\)
\(140\) −3.86022 −0.326248
\(141\) 3.28997 0.277066
\(142\) 24.8662 2.08672
\(143\) −16.6775 −1.39464
\(144\) 3.18084 0.265070
\(145\) −0.288553 −0.0239630
\(146\) −12.0064 −0.993658
\(147\) 1.00000 0.0824786
\(148\) −42.8471 −3.52201
\(149\) −0.559790 −0.0458598 −0.0229299 0.999737i \(-0.507299\pi\)
−0.0229299 + 0.999737i \(0.507299\pi\)
\(150\) 2.42079 0.197657
\(151\) 1.60558 0.130660 0.0653301 0.997864i \(-0.479190\pi\)
0.0653301 + 0.997864i \(0.479190\pi\)
\(152\) 37.3375 3.02847
\(153\) 4.49734 0.363589
\(154\) −6.76311 −0.544987
\(155\) 6.34817 0.509897
\(156\) −23.0437 −1.84497
\(157\) 2.55686 0.204059 0.102030 0.994781i \(-0.467466\pi\)
0.102030 + 0.994781i \(0.467466\pi\)
\(158\) −40.5248 −3.22398
\(159\) −4.34703 −0.344742
\(160\) −1.30624 −0.103267
\(161\) −1.00000 −0.0788110
\(162\) 2.42079 0.190195
\(163\) 3.77275 0.295504 0.147752 0.989024i \(-0.452796\pi\)
0.147752 + 0.989024i \(0.452796\pi\)
\(164\) −31.2860 −2.44303
\(165\) 2.79376 0.217494
\(166\) 7.66771 0.595130
\(167\) 10.5700 0.817933 0.408966 0.912549i \(-0.365889\pi\)
0.408966 + 0.912549i \(0.365889\pi\)
\(168\) −4.50319 −0.347429
\(169\) 22.6354 1.74118
\(170\) 10.8871 0.835004
\(171\) 8.29134 0.634054
\(172\) −1.02078 −0.0778337
\(173\) −5.38268 −0.409237 −0.204619 0.978842i \(-0.565595\pi\)
−0.204619 + 0.978842i \(0.565595\pi\)
\(174\) −0.698525 −0.0529551
\(175\) −1.00000 −0.0755929
\(176\) 8.88652 0.669847
\(177\) 15.0558 1.13166
\(178\) 8.33835 0.624986
\(179\) 12.0616 0.901523 0.450761 0.892645i \(-0.351153\pi\)
0.450761 + 0.892645i \(0.351153\pi\)
\(180\) 3.86022 0.287724
\(181\) −4.12989 −0.306972 −0.153486 0.988151i \(-0.549050\pi\)
−0.153486 + 0.988151i \(0.549050\pi\)
\(182\) 14.4510 1.07118
\(183\) −7.31629 −0.540836
\(184\) 4.50319 0.331980
\(185\) −11.0997 −0.816064
\(186\) 15.3676 1.12681
\(187\) 12.5645 0.918809
\(188\) 12.7000 0.926243
\(189\) −1.00000 −0.0727393
\(190\) 20.0716 1.45614
\(191\) 13.4946 0.976437 0.488218 0.872722i \(-0.337647\pi\)
0.488218 + 0.872722i \(0.337647\pi\)
\(192\) −9.52381 −0.687322
\(193\) 25.4725 1.83355 0.916776 0.399401i \(-0.130782\pi\)
0.916776 + 0.399401i \(0.130782\pi\)
\(194\) −20.0283 −1.43795
\(195\) −5.96954 −0.427488
\(196\) 3.86022 0.275730
\(197\) 8.22859 0.586263 0.293131 0.956072i \(-0.405303\pi\)
0.293131 + 0.956072i \(0.405303\pi\)
\(198\) 6.76311 0.480633
\(199\) 23.1526 1.64124 0.820621 0.571473i \(-0.193628\pi\)
0.820621 + 0.571473i \(0.193628\pi\)
\(200\) 4.50319 0.318424
\(201\) 2.53462 0.178779
\(202\) −6.67400 −0.469581
\(203\) 0.288553 0.0202524
\(204\) 17.3607 1.21549
\(205\) −8.10472 −0.566058
\(206\) −23.8034 −1.65846
\(207\) 1.00000 0.0695048
\(208\) −18.9882 −1.31659
\(209\) 23.1640 1.60229
\(210\) −2.42079 −0.167050
\(211\) −20.3171 −1.39869 −0.699344 0.714785i \(-0.746525\pi\)
−0.699344 + 0.714785i \(0.746525\pi\)
\(212\) −16.7805 −1.15249
\(213\) 10.2719 0.703821
\(214\) −42.2637 −2.88909
\(215\) −0.264436 −0.0180344
\(216\) 4.50319 0.306403
\(217\) −6.34817 −0.430942
\(218\) −2.63078 −0.178179
\(219\) −4.95971 −0.335146
\(220\) 10.7845 0.727093
\(221\) −26.8471 −1.80593
\(222\) −26.8699 −1.80339
\(223\) 8.26387 0.553390 0.276695 0.960958i \(-0.410761\pi\)
0.276695 + 0.960958i \(0.410761\pi\)
\(224\) 1.30624 0.0872769
\(225\) 1.00000 0.0666667
\(226\) −34.3445 −2.28456
\(227\) 3.84124 0.254952 0.127476 0.991842i \(-0.459312\pi\)
0.127476 + 0.991842i \(0.459312\pi\)
\(228\) 32.0064 2.11967
\(229\) 4.48106 0.296117 0.148058 0.988979i \(-0.452698\pi\)
0.148058 + 0.988979i \(0.452698\pi\)
\(230\) 2.42079 0.159622
\(231\) −2.79376 −0.183816
\(232\) −1.29941 −0.0853103
\(233\) 8.03925 0.526669 0.263334 0.964705i \(-0.415178\pi\)
0.263334 + 0.964705i \(0.415178\pi\)
\(234\) −14.4510 −0.944691
\(235\) 3.28997 0.214614
\(236\) 58.1186 3.78320
\(237\) −16.7403 −1.08740
\(238\) −10.8871 −0.705707
\(239\) −14.3411 −0.927650 −0.463825 0.885927i \(-0.653523\pi\)
−0.463825 + 0.885927i \(0.653523\pi\)
\(240\) 3.18084 0.205322
\(241\) 19.4913 1.25555 0.627774 0.778396i \(-0.283966\pi\)
0.627774 + 0.778396i \(0.283966\pi\)
\(242\) −7.73413 −0.497169
\(243\) 1.00000 0.0641500
\(244\) −28.2425 −1.80804
\(245\) 1.00000 0.0638877
\(246\) −19.6198 −1.25091
\(247\) −49.4955 −3.14932
\(248\) 28.5870 1.81528
\(249\) 3.16745 0.200729
\(250\) 2.42079 0.153104
\(251\) −12.2142 −0.770956 −0.385478 0.922717i \(-0.625963\pi\)
−0.385478 + 0.922717i \(0.625963\pi\)
\(252\) −3.86022 −0.243171
\(253\) 2.79376 0.175642
\(254\) −51.9189 −3.25768
\(255\) 4.49734 0.281635
\(256\) −30.4397 −1.90248
\(257\) −29.3237 −1.82916 −0.914581 0.404403i \(-0.867479\pi\)
−0.914581 + 0.404403i \(0.867479\pi\)
\(258\) −0.640143 −0.0398536
\(259\) 11.0997 0.689700
\(260\) −23.0437 −1.42911
\(261\) −0.288553 −0.0178610
\(262\) −14.0271 −0.866600
\(263\) 4.71668 0.290843 0.145422 0.989370i \(-0.453546\pi\)
0.145422 + 0.989370i \(0.453546\pi\)
\(264\) 12.5809 0.774298
\(265\) −4.34703 −0.267036
\(266\) −20.0716 −1.23067
\(267\) 3.44448 0.210799
\(268\) 9.78420 0.597665
\(269\) 12.4012 0.756112 0.378056 0.925783i \(-0.376593\pi\)
0.378056 + 0.925783i \(0.376593\pi\)
\(270\) 2.42079 0.147324
\(271\) 16.9980 1.03255 0.516276 0.856422i \(-0.327318\pi\)
0.516276 + 0.856422i \(0.327318\pi\)
\(272\) 14.3053 0.867388
\(273\) 5.96954 0.361293
\(274\) 14.8516 0.897220
\(275\) 2.79376 0.168470
\(276\) 3.86022 0.232358
\(277\) 12.9327 0.777051 0.388525 0.921438i \(-0.372985\pi\)
0.388525 + 0.921438i \(0.372985\pi\)
\(278\) −8.66743 −0.519838
\(279\) 6.34817 0.380055
\(280\) −4.50319 −0.269117
\(281\) 3.71629 0.221695 0.110848 0.993837i \(-0.464643\pi\)
0.110848 + 0.993837i \(0.464643\pi\)
\(282\) 7.96433 0.474269
\(283\) −18.2784 −1.08654 −0.543269 0.839559i \(-0.682813\pi\)
−0.543269 + 0.839559i \(0.682813\pi\)
\(284\) 39.6519 2.35291
\(285\) 8.29134 0.491136
\(286\) −40.3727 −2.38729
\(287\) 8.10472 0.478407
\(288\) −1.30624 −0.0769710
\(289\) 3.22610 0.189771
\(290\) −0.698525 −0.0410188
\(291\) −8.27346 −0.484999
\(292\) −19.1456 −1.12041
\(293\) −7.64958 −0.446893 −0.223447 0.974716i \(-0.571731\pi\)
−0.223447 + 0.974716i \(0.571731\pi\)
\(294\) 2.42079 0.141183
\(295\) 15.0558 0.876582
\(296\) −49.9839 −2.90526
\(297\) 2.79376 0.162111
\(298\) −1.35513 −0.0785007
\(299\) −5.96954 −0.345227
\(300\) 3.86022 0.222870
\(301\) 0.264436 0.0152418
\(302\) 3.88677 0.223658
\(303\) −2.75695 −0.158383
\(304\) 26.3734 1.51262
\(305\) −7.31629 −0.418930
\(306\) 10.8871 0.622375
\(307\) 5.37651 0.306854 0.153427 0.988160i \(-0.450969\pi\)
0.153427 + 0.988160i \(0.450969\pi\)
\(308\) −10.7845 −0.614506
\(309\) −9.83291 −0.559374
\(310\) 15.3676 0.872820
\(311\) −1.60417 −0.0909641 −0.0454820 0.998965i \(-0.514482\pi\)
−0.0454820 + 0.998965i \(0.514482\pi\)
\(312\) −26.8820 −1.52189
\(313\) 17.6416 0.997165 0.498582 0.866842i \(-0.333854\pi\)
0.498582 + 0.866842i \(0.333854\pi\)
\(314\) 6.18961 0.349300
\(315\) −1.00000 −0.0563436
\(316\) −64.6213 −3.63523
\(317\) −29.4338 −1.65317 −0.826583 0.562815i \(-0.809718\pi\)
−0.826583 + 0.562815i \(0.809718\pi\)
\(318\) −10.5232 −0.590114
\(319\) −0.806148 −0.0451356
\(320\) −9.52381 −0.532397
\(321\) −17.4587 −0.974447
\(322\) −2.42079 −0.134905
\(323\) 37.2890 2.07481
\(324\) 3.86022 0.214456
\(325\) −5.96954 −0.331130
\(326\) 9.13302 0.505831
\(327\) −1.08674 −0.0600970
\(328\) −36.4971 −2.01522
\(329\) −3.28997 −0.181382
\(330\) 6.76311 0.372297
\(331\) 21.8291 1.19984 0.599918 0.800062i \(-0.295200\pi\)
0.599918 + 0.800062i \(0.295200\pi\)
\(332\) 12.2270 0.671045
\(333\) −11.0997 −0.608258
\(334\) 25.5878 1.40010
\(335\) 2.53462 0.138481
\(336\) −3.18084 −0.173529
\(337\) −4.31057 −0.234812 −0.117406 0.993084i \(-0.537458\pi\)
−0.117406 + 0.993084i \(0.537458\pi\)
\(338\) 54.7955 2.98048
\(339\) −14.1873 −0.770549
\(340\) 17.3607 0.941517
\(341\) 17.7353 0.960420
\(342\) 20.0716 1.08535
\(343\) −1.00000 −0.0539949
\(344\) −1.19081 −0.0642039
\(345\) 1.00000 0.0538382
\(346\) −13.0303 −0.700514
\(347\) 10.9707 0.588937 0.294468 0.955661i \(-0.404857\pi\)
0.294468 + 0.955661i \(0.404857\pi\)
\(348\) −1.11388 −0.0597100
\(349\) −20.1266 −1.07735 −0.538675 0.842514i \(-0.681075\pi\)
−0.538675 + 0.842514i \(0.681075\pi\)
\(350\) −2.42079 −0.129397
\(351\) −5.96954 −0.318630
\(352\) −3.64933 −0.194510
\(353\) 31.7434 1.68953 0.844766 0.535136i \(-0.179740\pi\)
0.844766 + 0.535136i \(0.179740\pi\)
\(354\) 36.4469 1.93713
\(355\) 10.2719 0.545178
\(356\) 13.2964 0.704709
\(357\) −4.49734 −0.238025
\(358\) 29.1985 1.54319
\(359\) −19.5016 −1.02926 −0.514628 0.857413i \(-0.672070\pi\)
−0.514628 + 0.857413i \(0.672070\pi\)
\(360\) 4.50319 0.237339
\(361\) 49.7463 2.61822
\(362\) −9.99759 −0.525462
\(363\) −3.19488 −0.167688
\(364\) 23.0437 1.20782
\(365\) −4.95971 −0.259603
\(366\) −17.7112 −0.925779
\(367\) −33.4821 −1.74775 −0.873877 0.486148i \(-0.838402\pi\)
−0.873877 + 0.486148i \(0.838402\pi\)
\(368\) 3.18084 0.165813
\(369\) −8.10472 −0.421915
\(370\) −26.8699 −1.39690
\(371\) 4.34703 0.225687
\(372\) 24.5053 1.27054
\(373\) 25.9151 1.34183 0.670917 0.741532i \(-0.265901\pi\)
0.670917 + 0.741532i \(0.265901\pi\)
\(374\) 30.4160 1.57278
\(375\) 1.00000 0.0516398
\(376\) 14.8154 0.764045
\(377\) 1.72253 0.0887147
\(378\) −2.42079 −0.124512
\(379\) 14.6127 0.750606 0.375303 0.926902i \(-0.377539\pi\)
0.375303 + 0.926902i \(0.377539\pi\)
\(380\) 32.0064 1.64189
\(381\) −21.4471 −1.09877
\(382\) 32.6676 1.67142
\(383\) 36.0618 1.84267 0.921337 0.388765i \(-0.127098\pi\)
0.921337 + 0.388765i \(0.127098\pi\)
\(384\) −20.4427 −1.04321
\(385\) −2.79376 −0.142383
\(386\) 61.6636 3.13859
\(387\) −0.264436 −0.0134420
\(388\) −31.9373 −1.62137
\(389\) −15.7703 −0.799587 −0.399793 0.916605i \(-0.630918\pi\)
−0.399793 + 0.916605i \(0.630918\pi\)
\(390\) −14.4510 −0.731754
\(391\) 4.49734 0.227440
\(392\) 4.50319 0.227446
\(393\) −5.79445 −0.292292
\(394\) 19.9197 1.00354
\(395\) −16.7403 −0.842298
\(396\) 10.7845 0.541943
\(397\) 21.5785 1.08299 0.541496 0.840703i \(-0.317858\pi\)
0.541496 + 0.840703i \(0.317858\pi\)
\(398\) 56.0475 2.80941
\(399\) −8.29134 −0.415086
\(400\) 3.18084 0.159042
\(401\) −36.1057 −1.80303 −0.901515 0.432747i \(-0.857544\pi\)
−0.901515 + 0.432747i \(0.857544\pi\)
\(402\) 6.13579 0.306025
\(403\) −37.8957 −1.88772
\(404\) −10.6424 −0.529481
\(405\) 1.00000 0.0496904
\(406\) 0.698525 0.0346672
\(407\) −31.0099 −1.53710
\(408\) 20.2524 1.00264
\(409\) 15.4516 0.764034 0.382017 0.924155i \(-0.375230\pi\)
0.382017 + 0.924155i \(0.375230\pi\)
\(410\) −19.6198 −0.968954
\(411\) 6.13504 0.302619
\(412\) −37.9571 −1.87001
\(413\) −15.0558 −0.740847
\(414\) 2.42079 0.118975
\(415\) 3.16745 0.155484
\(416\) 7.79766 0.382312
\(417\) −3.58042 −0.175334
\(418\) 56.0752 2.74273
\(419\) −28.0334 −1.36952 −0.684762 0.728767i \(-0.740094\pi\)
−0.684762 + 0.728767i \(0.740094\pi\)
\(420\) −3.86022 −0.188359
\(421\) −38.2570 −1.86453 −0.932266 0.361773i \(-0.882172\pi\)
−0.932266 + 0.361773i \(0.882172\pi\)
\(422\) −49.1835 −2.39421
\(423\) 3.28997 0.159964
\(424\) −19.5755 −0.950671
\(425\) 4.49734 0.218153
\(426\) 24.8662 1.20477
\(427\) 7.31629 0.354060
\(428\) −67.3942 −3.25762
\(429\) −16.6775 −0.805197
\(430\) −0.640143 −0.0308704
\(431\) −9.32940 −0.449381 −0.224691 0.974430i \(-0.572137\pi\)
−0.224691 + 0.974430i \(0.572137\pi\)
\(432\) 3.18084 0.153038
\(433\) 19.5167 0.937913 0.468956 0.883221i \(-0.344630\pi\)
0.468956 + 0.883221i \(0.344630\pi\)
\(434\) −15.3676 −0.737668
\(435\) −0.288553 −0.0138350
\(436\) −4.19507 −0.200907
\(437\) 8.29134 0.396628
\(438\) −12.0064 −0.573689
\(439\) −9.86736 −0.470943 −0.235472 0.971881i \(-0.575663\pi\)
−0.235472 + 0.971881i \(0.575663\pi\)
\(440\) 12.5809 0.599769
\(441\) 1.00000 0.0476190
\(442\) −64.9911 −3.09131
\(443\) −2.02873 −0.0963878 −0.0481939 0.998838i \(-0.515347\pi\)
−0.0481939 + 0.998838i \(0.515347\pi\)
\(444\) −42.8471 −2.03343
\(445\) 3.44448 0.163284
\(446\) 20.0051 0.947268
\(447\) −0.559790 −0.0264771
\(448\) 9.52381 0.449958
\(449\) 32.1591 1.51768 0.758840 0.651277i \(-0.225766\pi\)
0.758840 + 0.651277i \(0.225766\pi\)
\(450\) 2.42079 0.114117
\(451\) −22.6427 −1.06620
\(452\) −54.7661 −2.57598
\(453\) 1.60558 0.0754367
\(454\) 9.29882 0.436415
\(455\) 5.96954 0.279856
\(456\) 37.3375 1.74849
\(457\) −2.49748 −0.116827 −0.0584137 0.998292i \(-0.518604\pi\)
−0.0584137 + 0.998292i \(0.518604\pi\)
\(458\) 10.8477 0.506880
\(459\) 4.49734 0.209918
\(460\) 3.86022 0.179984
\(461\) 35.4061 1.64903 0.824513 0.565843i \(-0.191449\pi\)
0.824513 + 0.565843i \(0.191449\pi\)
\(462\) −6.76311 −0.314648
\(463\) 19.0753 0.886502 0.443251 0.896397i \(-0.353825\pi\)
0.443251 + 0.896397i \(0.353825\pi\)
\(464\) −0.917840 −0.0426097
\(465\) 6.34817 0.294389
\(466\) 19.4613 0.901528
\(467\) −11.1303 −0.515049 −0.257525 0.966272i \(-0.582907\pi\)
−0.257525 + 0.966272i \(0.582907\pi\)
\(468\) −23.0437 −1.06520
\(469\) −2.53462 −0.117038
\(470\) 7.96433 0.367367
\(471\) 2.55686 0.117814
\(472\) 67.7991 3.12071
\(473\) −0.738771 −0.0339687
\(474\) −40.5248 −1.86137
\(475\) 8.29134 0.380433
\(476\) −17.3607 −0.795727
\(477\) −4.34703 −0.199037
\(478\) −34.7168 −1.58791
\(479\) 0.243772 0.0111382 0.00556912 0.999984i \(-0.498227\pi\)
0.00556912 + 0.999984i \(0.498227\pi\)
\(480\) −1.30624 −0.0596215
\(481\) 66.2599 3.02119
\(482\) 47.1844 2.14919
\(483\) −1.00000 −0.0455016
\(484\) −12.3329 −0.560588
\(485\) −8.27346 −0.375678
\(486\) 2.42079 0.109809
\(487\) −41.3260 −1.87266 −0.936329 0.351124i \(-0.885799\pi\)
−0.936329 + 0.351124i \(0.885799\pi\)
\(488\) −32.9467 −1.49143
\(489\) 3.77275 0.170610
\(490\) 2.42079 0.109360
\(491\) −26.1570 −1.18045 −0.590225 0.807239i \(-0.700961\pi\)
−0.590225 + 0.807239i \(0.700961\pi\)
\(492\) −31.2860 −1.41048
\(493\) −1.29772 −0.0584464
\(494\) −119.818 −5.39087
\(495\) 2.79376 0.125570
\(496\) 20.1925 0.906671
\(497\) −10.2719 −0.460759
\(498\) 7.66771 0.343599
\(499\) 6.49734 0.290861 0.145431 0.989368i \(-0.453543\pi\)
0.145431 + 0.989368i \(0.453543\pi\)
\(500\) 3.86022 0.172634
\(501\) 10.5700 0.472234
\(502\) −29.5681 −1.31969
\(503\) 15.8641 0.707343 0.353672 0.935370i \(-0.384933\pi\)
0.353672 + 0.935370i \(0.384933\pi\)
\(504\) −4.50319 −0.200588
\(505\) −2.75695 −0.122683
\(506\) 6.76311 0.300657
\(507\) 22.6354 1.00527
\(508\) −82.7905 −3.67324
\(509\) −26.3048 −1.16594 −0.582969 0.812494i \(-0.698109\pi\)
−0.582969 + 0.812494i \(0.698109\pi\)
\(510\) 10.8871 0.482090
\(511\) 4.95971 0.219405
\(512\) −32.8028 −1.44969
\(513\) 8.29134 0.366071
\(514\) −70.9865 −3.13108
\(515\) −9.83291 −0.433290
\(516\) −1.02078 −0.0449373
\(517\) 9.19141 0.404238
\(518\) 26.8699 1.18060
\(519\) −5.38268 −0.236273
\(520\) −26.8820 −1.17885
\(521\) −6.67223 −0.292316 −0.146158 0.989261i \(-0.546691\pi\)
−0.146158 + 0.989261i \(0.546691\pi\)
\(522\) −0.698525 −0.0305736
\(523\) 16.6224 0.726849 0.363424 0.931624i \(-0.381607\pi\)
0.363424 + 0.931624i \(0.381607\pi\)
\(524\) −22.3679 −0.977144
\(525\) −1.00000 −0.0436436
\(526\) 11.4181 0.497852
\(527\) 28.5499 1.24365
\(528\) 8.88652 0.386736
\(529\) 1.00000 0.0434783
\(530\) −10.5232 −0.457100
\(531\) 15.0558 0.653366
\(532\) −32.0064 −1.38765
\(533\) 48.3815 2.09563
\(534\) 8.33835 0.360836
\(535\) −17.4587 −0.754803
\(536\) 11.4139 0.493005
\(537\) 12.0616 0.520494
\(538\) 30.0206 1.29428
\(539\) 2.79376 0.120336
\(540\) 3.86022 0.166117
\(541\) 3.08409 0.132595 0.0662977 0.997800i \(-0.478881\pi\)
0.0662977 + 0.997800i \(0.478881\pi\)
\(542\) 41.1485 1.76748
\(543\) −4.12989 −0.177230
\(544\) −5.87462 −0.251872
\(545\) −1.08674 −0.0465510
\(546\) 14.4510 0.618445
\(547\) −2.77692 −0.118733 −0.0593663 0.998236i \(-0.518908\pi\)
−0.0593663 + 0.998236i \(0.518908\pi\)
\(548\) 23.6826 1.01167
\(549\) −7.31629 −0.312252
\(550\) 6.76311 0.288380
\(551\) −2.39249 −0.101923
\(552\) 4.50319 0.191669
\(553\) 16.7403 0.711872
\(554\) 31.3073 1.33012
\(555\) −11.0997 −0.471155
\(556\) −13.8212 −0.586149
\(557\) −8.98627 −0.380761 −0.190380 0.981710i \(-0.560972\pi\)
−0.190380 + 0.981710i \(0.560972\pi\)
\(558\) 15.3676 0.650562
\(559\) 1.57856 0.0667660
\(560\) −3.18084 −0.134415
\(561\) 12.5645 0.530474
\(562\) 8.99636 0.379489
\(563\) −17.6416 −0.743505 −0.371752 0.928332i \(-0.621243\pi\)
−0.371752 + 0.928332i \(0.621243\pi\)
\(564\) 12.7000 0.534767
\(565\) −14.1873 −0.596865
\(566\) −44.2481 −1.85989
\(567\) −1.00000 −0.0419961
\(568\) 46.2565 1.94088
\(569\) 19.0859 0.800123 0.400061 0.916488i \(-0.368989\pi\)
0.400061 + 0.916488i \(0.368989\pi\)
\(570\) 20.0716 0.840706
\(571\) −22.0717 −0.923671 −0.461836 0.886966i \(-0.652809\pi\)
−0.461836 + 0.886966i \(0.652809\pi\)
\(572\) −64.3787 −2.69181
\(573\) 13.4946 0.563746
\(574\) 19.6198 0.818915
\(575\) 1.00000 0.0417029
\(576\) −9.52381 −0.396826
\(577\) −7.03151 −0.292725 −0.146363 0.989231i \(-0.546757\pi\)
−0.146363 + 0.989231i \(0.546757\pi\)
\(578\) 7.80971 0.324841
\(579\) 25.4725 1.05860
\(580\) −1.11388 −0.0462512
\(581\) −3.16745 −0.131408
\(582\) −20.0283 −0.830199
\(583\) −12.1446 −0.502977
\(584\) −22.3345 −0.924210
\(585\) −5.96954 −0.246810
\(586\) −18.5180 −0.764972
\(587\) 40.7519 1.68201 0.841006 0.541026i \(-0.181964\pi\)
0.841006 + 0.541026i \(0.181964\pi\)
\(588\) 3.86022 0.159193
\(589\) 52.6348 2.16878
\(590\) 36.4469 1.50049
\(591\) 8.22859 0.338479
\(592\) −35.3063 −1.45108
\(593\) 47.4193 1.94728 0.973638 0.228100i \(-0.0732514\pi\)
0.973638 + 0.228100i \(0.0732514\pi\)
\(594\) 6.76311 0.277494
\(595\) −4.49734 −0.184373
\(596\) −2.16091 −0.0885143
\(597\) 23.1526 0.947571
\(598\) −14.4510 −0.590945
\(599\) −15.2730 −0.624037 −0.312018 0.950076i \(-0.601005\pi\)
−0.312018 + 0.950076i \(0.601005\pi\)
\(600\) 4.50319 0.183842
\(601\) −45.8805 −1.87150 −0.935752 0.352658i \(-0.885278\pi\)
−0.935752 + 0.352658i \(0.885278\pi\)
\(602\) 0.640143 0.0260903
\(603\) 2.53462 0.103218
\(604\) 6.19788 0.252188
\(605\) −3.19488 −0.129890
\(606\) −6.67400 −0.271113
\(607\) 24.4509 0.992431 0.496215 0.868199i \(-0.334723\pi\)
0.496215 + 0.868199i \(0.334723\pi\)
\(608\) −10.8305 −0.439234
\(609\) 0.288553 0.0116927
\(610\) −17.7112 −0.717105
\(611\) −19.6396 −0.794534
\(612\) 17.3607 0.701766
\(613\) −4.18228 −0.168921 −0.0844604 0.996427i \(-0.526917\pi\)
−0.0844604 + 0.996427i \(0.526917\pi\)
\(614\) 13.0154 0.525259
\(615\) −8.10472 −0.326814
\(616\) −12.5809 −0.506897
\(617\) −2.79324 −0.112452 −0.0562258 0.998418i \(-0.517907\pi\)
−0.0562258 + 0.998418i \(0.517907\pi\)
\(618\) −23.8034 −0.957513
\(619\) 35.0354 1.40819 0.704095 0.710106i \(-0.251353\pi\)
0.704095 + 0.710106i \(0.251353\pi\)
\(620\) 24.5053 0.984157
\(621\) 1.00000 0.0401286
\(622\) −3.88335 −0.155708
\(623\) −3.44448 −0.138000
\(624\) −18.9882 −0.760134
\(625\) 1.00000 0.0400000
\(626\) 42.7067 1.70690
\(627\) 23.1640 0.925083
\(628\) 9.87003 0.393857
\(629\) −49.9190 −1.99040
\(630\) −2.42079 −0.0964465
\(631\) −3.96520 −0.157852 −0.0789261 0.996880i \(-0.525149\pi\)
−0.0789261 + 0.996880i \(0.525149\pi\)
\(632\) −75.3850 −2.99865
\(633\) −20.3171 −0.807533
\(634\) −71.2530 −2.82982
\(635\) −21.4471 −0.851103
\(636\) −16.7805 −0.665389
\(637\) −5.96954 −0.236522
\(638\) −1.95151 −0.0772612
\(639\) 10.2719 0.406351
\(640\) −20.4427 −0.808067
\(641\) −32.2626 −1.27430 −0.637148 0.770742i \(-0.719886\pi\)
−0.637148 + 0.770742i \(0.719886\pi\)
\(642\) −42.2637 −1.66802
\(643\) 28.1370 1.10961 0.554807 0.831979i \(-0.312792\pi\)
0.554807 + 0.831979i \(0.312792\pi\)
\(644\) −3.86022 −0.152114
\(645\) −0.264436 −0.0104122
\(646\) 90.2688 3.55158
\(647\) −14.6907 −0.577549 −0.288775 0.957397i \(-0.593248\pi\)
−0.288775 + 0.957397i \(0.593248\pi\)
\(648\) 4.50319 0.176902
\(649\) 42.0623 1.65109
\(650\) −14.4510 −0.566814
\(651\) −6.34817 −0.248804
\(652\) 14.5636 0.570356
\(653\) 15.4855 0.605994 0.302997 0.952992i \(-0.402013\pi\)
0.302997 + 0.952992i \(0.402013\pi\)
\(654\) −2.63078 −0.102871
\(655\) −5.79445 −0.226408
\(656\) −25.7798 −1.00653
\(657\) −4.95971 −0.193497
\(658\) −7.96433 −0.310482
\(659\) −22.7662 −0.886846 −0.443423 0.896312i \(-0.646236\pi\)
−0.443423 + 0.896312i \(0.646236\pi\)
\(660\) 10.7845 0.419788
\(661\) −24.2449 −0.943018 −0.471509 0.881861i \(-0.656291\pi\)
−0.471509 + 0.881861i \(0.656291\pi\)
\(662\) 52.8436 2.05382
\(663\) −26.8471 −1.04265
\(664\) 14.2636 0.553535
\(665\) −8.29134 −0.321524
\(666\) −26.8699 −1.04119
\(667\) −0.288553 −0.0111728
\(668\) 40.8026 1.57870
\(669\) 8.26387 0.319500
\(670\) 6.13579 0.237046
\(671\) −20.4400 −0.789077
\(672\) 1.30624 0.0503894
\(673\) −4.32915 −0.166876 −0.0834382 0.996513i \(-0.526590\pi\)
−0.0834382 + 0.996513i \(0.526590\pi\)
\(674\) −10.4350 −0.401941
\(675\) 1.00000 0.0384900
\(676\) 87.3775 3.36067
\(677\) −14.2625 −0.548153 −0.274076 0.961708i \(-0.588372\pi\)
−0.274076 + 0.961708i \(0.588372\pi\)
\(678\) −34.3445 −1.31899
\(679\) 8.27346 0.317506
\(680\) 20.2524 0.776644
\(681\) 3.84124 0.147196
\(682\) 42.9334 1.64400
\(683\) 0.684410 0.0261882 0.0130941 0.999914i \(-0.495832\pi\)
0.0130941 + 0.999914i \(0.495832\pi\)
\(684\) 32.0064 1.22379
\(685\) 6.13504 0.234408
\(686\) −2.42079 −0.0924261
\(687\) 4.48106 0.170963
\(688\) −0.841128 −0.0320677
\(689\) 25.9498 0.988607
\(690\) 2.42079 0.0921578
\(691\) 29.0557 1.10533 0.552665 0.833403i \(-0.313611\pi\)
0.552665 + 0.833403i \(0.313611\pi\)
\(692\) −20.7783 −0.789872
\(693\) −2.79376 −0.106126
\(694\) 26.5577 1.00812
\(695\) −3.58042 −0.135813
\(696\) −1.29941 −0.0492539
\(697\) −36.4497 −1.38063
\(698\) −48.7221 −1.84416
\(699\) 8.03925 0.304072
\(700\) −3.86022 −0.145902
\(701\) 34.5747 1.30587 0.652934 0.757415i \(-0.273538\pi\)
0.652934 + 0.757415i \(0.273538\pi\)
\(702\) −14.4510 −0.545418
\(703\) −92.0311 −3.47102
\(704\) −26.6073 −1.00280
\(705\) 3.28997 0.123908
\(706\) 76.8441 2.89207
\(707\) 2.75695 0.103686
\(708\) 58.1186 2.18423
\(709\) 21.4169 0.804330 0.402165 0.915567i \(-0.368258\pi\)
0.402165 + 0.915567i \(0.368258\pi\)
\(710\) 24.8662 0.933211
\(711\) −16.7403 −0.627812
\(712\) 15.5111 0.581305
\(713\) 6.34817 0.237741
\(714\) −10.8871 −0.407440
\(715\) −16.6775 −0.623703
\(716\) 46.5602 1.74004
\(717\) −14.3411 −0.535579
\(718\) −47.2093 −1.76184
\(719\) −15.4198 −0.575062 −0.287531 0.957771i \(-0.592834\pi\)
−0.287531 + 0.957771i \(0.592834\pi\)
\(720\) 3.18084 0.118543
\(721\) 9.83291 0.366197
\(722\) 120.425 4.48176
\(723\) 19.4913 0.724891
\(724\) −15.9423 −0.592490
\(725\) −0.288553 −0.0107166
\(726\) −7.73413 −0.287040
\(727\) 30.5804 1.13416 0.567082 0.823661i \(-0.308072\pi\)
0.567082 + 0.823661i \(0.308072\pi\)
\(728\) 26.8820 0.996312
\(729\) 1.00000 0.0370370
\(730\) −12.0064 −0.444377
\(731\) −1.18926 −0.0439863
\(732\) −28.2425 −1.04387
\(733\) 25.5385 0.943286 0.471643 0.881790i \(-0.343661\pi\)
0.471643 + 0.881790i \(0.343661\pi\)
\(734\) −81.0532 −2.99173
\(735\) 1.00000 0.0368856
\(736\) −1.30624 −0.0481487
\(737\) 7.08114 0.260837
\(738\) −19.6198 −0.722216
\(739\) 45.7218 1.68190 0.840952 0.541109i \(-0.181996\pi\)
0.840952 + 0.541109i \(0.181996\pi\)
\(740\) −42.8471 −1.57509
\(741\) −49.4955 −1.81826
\(742\) 10.5232 0.386320
\(743\) −23.6223 −0.866620 −0.433310 0.901245i \(-0.642654\pi\)
−0.433310 + 0.901245i \(0.642654\pi\)
\(744\) 28.5870 1.04805
\(745\) −0.559790 −0.0205091
\(746\) 62.7350 2.29689
\(747\) 3.16745 0.115891
\(748\) 48.5018 1.77340
\(749\) 17.4587 0.637925
\(750\) 2.42079 0.0883947
\(751\) 13.9599 0.509404 0.254702 0.967020i \(-0.418023\pi\)
0.254702 + 0.967020i \(0.418023\pi\)
\(752\) 10.4649 0.381615
\(753\) −12.2142 −0.445111
\(754\) 4.16987 0.151858
\(755\) 1.60558 0.0584330
\(756\) −3.86022 −0.140395
\(757\) 30.8536 1.12139 0.560697 0.828021i \(-0.310533\pi\)
0.560697 + 0.828021i \(0.310533\pi\)
\(758\) 35.3744 1.28485
\(759\) 2.79376 0.101407
\(760\) 37.3375 1.35437
\(761\) −16.8438 −0.610587 −0.305293 0.952258i \(-0.598755\pi\)
−0.305293 + 0.952258i \(0.598755\pi\)
\(762\) −51.9189 −1.88082
\(763\) 1.08674 0.0393428
\(764\) 52.0922 1.88463
\(765\) 4.49734 0.162602
\(766\) 87.2981 3.15421
\(767\) −89.8762 −3.24524
\(768\) −30.4397 −1.09840
\(769\) −27.9453 −1.00773 −0.503866 0.863782i \(-0.668090\pi\)
−0.503866 + 0.863782i \(0.668090\pi\)
\(770\) −6.76311 −0.243726
\(771\) −29.3237 −1.05607
\(772\) 98.3295 3.53895
\(773\) −21.6878 −0.780055 −0.390028 0.920803i \(-0.627535\pi\)
−0.390028 + 0.920803i \(0.627535\pi\)
\(774\) −0.640143 −0.0230095
\(775\) 6.34817 0.228033
\(776\) −37.2570 −1.33745
\(777\) 11.0997 0.398198
\(778\) −38.1766 −1.36870
\(779\) −67.1990 −2.40765
\(780\) −23.0437 −0.825097
\(781\) 28.6974 1.02687
\(782\) 10.8871 0.389323
\(783\) −0.288553 −0.0103120
\(784\) 3.18084 0.113601
\(785\) 2.55686 0.0912582
\(786\) −14.0271 −0.500332
\(787\) 44.6257 1.59073 0.795367 0.606128i \(-0.207278\pi\)
0.795367 + 0.606128i \(0.207278\pi\)
\(788\) 31.7641 1.13155
\(789\) 4.71668 0.167918
\(790\) −40.5248 −1.44181
\(791\) 14.1873 0.504443
\(792\) 12.5809 0.447041
\(793\) 43.6749 1.55094
\(794\) 52.2369 1.85382
\(795\) −4.34703 −0.154173
\(796\) 89.3739 3.16777
\(797\) 7.74192 0.274233 0.137116 0.990555i \(-0.456217\pi\)
0.137116 + 0.990555i \(0.456217\pi\)
\(798\) −20.0716 −0.710526
\(799\) 14.7961 0.523450
\(800\) −1.30624 −0.0461826
\(801\) 3.44448 0.121705
\(802\) −87.4042 −3.08635
\(803\) −13.8563 −0.488977
\(804\) 9.78420 0.345062
\(805\) −1.00000 −0.0352454
\(806\) −91.7374 −3.23131
\(807\) 12.4012 0.436541
\(808\) −12.4151 −0.436761
\(809\) −10.4539 −0.367541 −0.183770 0.982969i \(-0.558830\pi\)
−0.183770 + 0.982969i \(0.558830\pi\)
\(810\) 2.42079 0.0850578
\(811\) −43.6119 −1.53142 −0.765710 0.643186i \(-0.777612\pi\)
−0.765710 + 0.643186i \(0.777612\pi\)
\(812\) 1.11388 0.0390894
\(813\) 16.9980 0.596145
\(814\) −75.0683 −2.63114
\(815\) 3.77275 0.132154
\(816\) 14.3053 0.500787
\(817\) −2.19253 −0.0767068
\(818\) 37.4051 1.30784
\(819\) 5.96954 0.208593
\(820\) −31.2860 −1.09255
\(821\) −7.23682 −0.252567 −0.126283 0.991994i \(-0.540305\pi\)
−0.126283 + 0.991994i \(0.540305\pi\)
\(822\) 14.8516 0.518010
\(823\) 50.2663 1.75217 0.876087 0.482153i \(-0.160145\pi\)
0.876087 + 0.482153i \(0.160145\pi\)
\(824\) −44.2795 −1.54255
\(825\) 2.79376 0.0972664
\(826\) −36.4469 −1.26815
\(827\) 46.7057 1.62412 0.812059 0.583575i \(-0.198347\pi\)
0.812059 + 0.583575i \(0.198347\pi\)
\(828\) 3.86022 0.134152
\(829\) 28.7414 0.998229 0.499115 0.866536i \(-0.333659\pi\)
0.499115 + 0.866536i \(0.333659\pi\)
\(830\) 7.66771 0.266150
\(831\) 12.9327 0.448630
\(832\) 56.8528 1.97102
\(833\) 4.49734 0.155824
\(834\) −8.66743 −0.300129
\(835\) 10.5700 0.365791
\(836\) 89.4182 3.09259
\(837\) 6.34817 0.219425
\(838\) −67.8630 −2.34429
\(839\) 10.0276 0.346193 0.173096 0.984905i \(-0.444623\pi\)
0.173096 + 0.984905i \(0.444623\pi\)
\(840\) −4.50319 −0.155375
\(841\) −28.9167 −0.997129
\(842\) −92.6121 −3.19162
\(843\) 3.71629 0.127996
\(844\) −78.4285 −2.69962
\(845\) 22.6354 0.778681
\(846\) 7.96433 0.273819
\(847\) 3.19488 0.109777
\(848\) −13.8272 −0.474828
\(849\) −18.2784 −0.627313
\(850\) 10.8871 0.373425
\(851\) −11.0997 −0.380492
\(852\) 39.6519 1.35845
\(853\) 32.9784 1.12916 0.564580 0.825379i \(-0.309038\pi\)
0.564580 + 0.825379i \(0.309038\pi\)
\(854\) 17.7112 0.606065
\(855\) 8.29134 0.283558
\(856\) −78.6196 −2.68716
\(857\) −27.1362 −0.926955 −0.463478 0.886109i \(-0.653399\pi\)
−0.463478 + 0.886109i \(0.653399\pi\)
\(858\) −40.3727 −1.37830
\(859\) 6.74621 0.230178 0.115089 0.993355i \(-0.463285\pi\)
0.115089 + 0.993355i \(0.463285\pi\)
\(860\) −1.02078 −0.0348083
\(861\) 8.10472 0.276208
\(862\) −22.5845 −0.769231
\(863\) −4.93924 −0.168134 −0.0840668 0.996460i \(-0.526791\pi\)
−0.0840668 + 0.996460i \(0.526791\pi\)
\(864\) −1.30624 −0.0444392
\(865\) −5.38268 −0.183017
\(866\) 47.2458 1.60548
\(867\) 3.22610 0.109564
\(868\) −24.5053 −0.831765
\(869\) −46.7686 −1.58651
\(870\) −0.698525 −0.0236822
\(871\) −15.1305 −0.512679
\(872\) −4.89381 −0.165725
\(873\) −8.27346 −0.280014
\(874\) 20.0716 0.678931
\(875\) −1.00000 −0.0338062
\(876\) −19.1456 −0.646869
\(877\) 5.90142 0.199277 0.0996383 0.995024i \(-0.468231\pi\)
0.0996383 + 0.995024i \(0.468231\pi\)
\(878\) −23.8868 −0.806140
\(879\) −7.64958 −0.258014
\(880\) 8.88652 0.299564
\(881\) 24.5121 0.825833 0.412917 0.910769i \(-0.364510\pi\)
0.412917 + 0.910769i \(0.364510\pi\)
\(882\) 2.42079 0.0815122
\(883\) 19.0917 0.642486 0.321243 0.946997i \(-0.395899\pi\)
0.321243 + 0.946997i \(0.395899\pi\)
\(884\) −103.636 −3.48564
\(885\) 15.0558 0.506095
\(886\) −4.91112 −0.164992
\(887\) −16.6397 −0.558705 −0.279353 0.960189i \(-0.590120\pi\)
−0.279353 + 0.960189i \(0.590120\pi\)
\(888\) −49.9839 −1.67735
\(889\) 21.4471 0.719313
\(890\) 8.33835 0.279502
\(891\) 2.79376 0.0935946
\(892\) 31.9003 1.06810
\(893\) 27.2783 0.912832
\(894\) −1.35513 −0.0453224
\(895\) 12.0616 0.403173
\(896\) 20.4427 0.682941
\(897\) −5.96954 −0.199317
\(898\) 77.8503 2.59790
\(899\) −1.83178 −0.0610934
\(900\) 3.86022 0.128674
\(901\) −19.5501 −0.651308
\(902\) −54.8132 −1.82508
\(903\) 0.264436 0.00879987
\(904\) −63.8882 −2.12489
\(905\) −4.12989 −0.137282
\(906\) 3.88677 0.129129
\(907\) 41.6157 1.38183 0.690914 0.722937i \(-0.257208\pi\)
0.690914 + 0.722937i \(0.257208\pi\)
\(908\) 14.8280 0.492085
\(909\) −2.75695 −0.0914424
\(910\) 14.4510 0.479046
\(911\) 41.6438 1.37972 0.689861 0.723942i \(-0.257672\pi\)
0.689861 + 0.723942i \(0.257672\pi\)
\(912\) 26.3734 0.873311
\(913\) 8.84910 0.292862
\(914\) −6.04588 −0.199980
\(915\) −7.31629 −0.241869
\(916\) 17.2979 0.571537
\(917\) 5.79445 0.191350
\(918\) 10.8871 0.359328
\(919\) 33.9080 1.11852 0.559261 0.828992i \(-0.311085\pi\)
0.559261 + 0.828992i \(0.311085\pi\)
\(920\) 4.50319 0.148466
\(921\) 5.37651 0.177162
\(922\) 85.7106 2.82273
\(923\) −61.3187 −2.01833
\(924\) −10.7845 −0.354785
\(925\) −11.0997 −0.364955
\(926\) 46.1772 1.51748
\(927\) −9.83291 −0.322955
\(928\) 0.376920 0.0123730
\(929\) 51.4642 1.68848 0.844242 0.535962i \(-0.180051\pi\)
0.844242 + 0.535962i \(0.180051\pi\)
\(930\) 15.3676 0.503923
\(931\) 8.29134 0.271738
\(932\) 31.0332 1.01653
\(933\) −1.60417 −0.0525181
\(934\) −26.9441 −0.881638
\(935\) 12.5645 0.410904
\(936\) −26.8820 −0.878665
\(937\) −55.7423 −1.82102 −0.910512 0.413484i \(-0.864312\pi\)
−0.910512 + 0.413484i \(0.864312\pi\)
\(938\) −6.13579 −0.200341
\(939\) 17.6416 0.575713
\(940\) 12.7000 0.414228
\(941\) 27.3939 0.893014 0.446507 0.894780i \(-0.352668\pi\)
0.446507 + 0.894780i \(0.352668\pi\)
\(942\) 6.18961 0.201668
\(943\) −8.10472 −0.263926
\(944\) 47.8901 1.55869
\(945\) −1.00000 −0.0325300
\(946\) −1.78841 −0.0581462
\(947\) −48.6525 −1.58099 −0.790497 0.612466i \(-0.790178\pi\)
−0.790497 + 0.612466i \(0.790178\pi\)
\(948\) −64.6213 −2.09880
\(949\) 29.6072 0.961090
\(950\) 20.0716 0.651208
\(951\) −29.4338 −0.954456
\(952\) −20.2524 −0.656384
\(953\) −28.4836 −0.922676 −0.461338 0.887224i \(-0.652630\pi\)
−0.461338 + 0.887224i \(0.652630\pi\)
\(954\) −10.5232 −0.340702
\(955\) 13.4946 0.436676
\(956\) −55.3598 −1.79046
\(957\) −0.806148 −0.0260591
\(958\) 0.590121 0.0190659
\(959\) −6.13504 −0.198111
\(960\) −9.52381 −0.307380
\(961\) 9.29929 0.299977
\(962\) 160.401 5.17154
\(963\) −17.4587 −0.562597
\(964\) 75.2408 2.42334
\(965\) 25.4725 0.819990
\(966\) −2.42079 −0.0778876
\(967\) 37.8183 1.21616 0.608078 0.793877i \(-0.291941\pi\)
0.608078 + 0.793877i \(0.291941\pi\)
\(968\) −14.3872 −0.462421
\(969\) 37.2890 1.19789
\(970\) −20.0283 −0.643070
\(971\) −21.5494 −0.691553 −0.345776 0.938317i \(-0.612384\pi\)
−0.345776 + 0.938317i \(0.612384\pi\)
\(972\) 3.86022 0.123817
\(973\) 3.58042 0.114783
\(974\) −100.041 −3.20553
\(975\) −5.96954 −0.191178
\(976\) −23.2720 −0.744917
\(977\) −43.8690 −1.40349 −0.701747 0.712426i \(-0.747596\pi\)
−0.701747 + 0.712426i \(0.747596\pi\)
\(978\) 9.13302 0.292042
\(979\) 9.62306 0.307554
\(980\) 3.86022 0.123310
\(981\) −1.08674 −0.0346970
\(982\) −63.3206 −2.02064
\(983\) 37.7779 1.20493 0.602464 0.798146i \(-0.294186\pi\)
0.602464 + 0.798146i \(0.294186\pi\)
\(984\) −36.4971 −1.16349
\(985\) 8.22859 0.262185
\(986\) −3.14151 −0.100046
\(987\) −3.28997 −0.104721
\(988\) −191.063 −6.07853
\(989\) −0.264436 −0.00840857
\(990\) 6.76311 0.214946
\(991\) 43.5865 1.38457 0.692285 0.721624i \(-0.256604\pi\)
0.692285 + 0.721624i \(0.256604\pi\)
\(992\) −8.29224 −0.263279
\(993\) 21.8291 0.692725
\(994\) −24.8662 −0.788707
\(995\) 23.1526 0.733986
\(996\) 12.2270 0.387428
\(997\) 3.86893 0.122530 0.0612651 0.998122i \(-0.480486\pi\)
0.0612651 + 0.998122i \(0.480486\pi\)
\(998\) 15.7287 0.497883
\(999\) −11.0997 −0.351178
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 2415.2.a.v.1.9 10
3.2 odd 2 7245.2.a.bu.1.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2415.2.a.v.1.9 10 1.1 even 1 trivial
7245.2.a.bu.1.2 10 3.2 odd 2