Properties

Label 1502.2.a.f.1.7
Level $1502$
Weight $2$
Character 1502.1
Self dual yes
Analytic conductor $11.994$
Analytic rank $1$
Dimension $11$
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] = [1502,2,Mod(1,1502)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1502, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1502.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1502 = 2 \cdot 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1502.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: \(11.9935303836\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 20x^{9} - 7x^{8} + 134x^{7} + 70x^{6} - 354x^{5} - 193x^{4} + 341x^{3} + 163x^{2} - 72x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.434885\) of defining polynomial
Character \(\chi\) \(=\) 1502.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.565115 q^{3} +1.00000 q^{4} -0.615969 q^{5} -0.565115 q^{6} +3.05802 q^{7} +1.00000 q^{8} -2.68064 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.565115 q^{3} +1.00000 q^{4} -0.615969 q^{5} -0.565115 q^{6} +3.05802 q^{7} +1.00000 q^{8} -2.68064 q^{9} -0.615969 q^{10} -3.11632 q^{11} -0.565115 q^{12} -6.07042 q^{13} +3.05802 q^{14} +0.348093 q^{15} +1.00000 q^{16} -7.35747 q^{17} -2.68064 q^{18} +1.30846 q^{19} -0.615969 q^{20} -1.72813 q^{21} -3.11632 q^{22} -1.38346 q^{23} -0.565115 q^{24} -4.62058 q^{25} -6.07042 q^{26} +3.21022 q^{27} +3.05802 q^{28} +3.58342 q^{29} +0.348093 q^{30} +2.91808 q^{31} +1.00000 q^{32} +1.76108 q^{33} -7.35747 q^{34} -1.88364 q^{35} -2.68064 q^{36} -9.97748 q^{37} +1.30846 q^{38} +3.43049 q^{39} -0.615969 q^{40} +6.14047 q^{41} -1.72813 q^{42} -3.20281 q^{43} -3.11632 q^{44} +1.65119 q^{45} -1.38346 q^{46} +7.36719 q^{47} -0.565115 q^{48} +2.35148 q^{49} -4.62058 q^{50} +4.15782 q^{51} -6.07042 q^{52} -3.83489 q^{53} +3.21022 q^{54} +1.91956 q^{55} +3.05802 q^{56} -0.739431 q^{57} +3.58342 q^{58} -1.10698 q^{59} +0.348093 q^{60} +7.64720 q^{61} +2.91808 q^{62} -8.19746 q^{63} +1.00000 q^{64} +3.73919 q^{65} +1.76108 q^{66} -16.1681 q^{67} -7.35747 q^{68} +0.781816 q^{69} -1.88364 q^{70} +15.1235 q^{71} -2.68064 q^{72} -11.5095 q^{73} -9.97748 q^{74} +2.61116 q^{75} +1.30846 q^{76} -9.52976 q^{77} +3.43049 q^{78} +6.87816 q^{79} -0.615969 q^{80} +6.22779 q^{81} +6.14047 q^{82} +3.52486 q^{83} -1.72813 q^{84} +4.53197 q^{85} -3.20281 q^{86} -2.02504 q^{87} -3.11632 q^{88} -10.5489 q^{89} +1.65119 q^{90} -18.5635 q^{91} -1.38346 q^{92} -1.64905 q^{93} +7.36719 q^{94} -0.805970 q^{95} -0.565115 q^{96} -7.96436 q^{97} +2.35148 q^{98} +8.35375 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 11 q^{2} - 11 q^{3} + 11 q^{4} - 12 q^{5} - 11 q^{6} - 9 q^{7} + 11 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 11 q^{2} - 11 q^{3} + 11 q^{4} - 12 q^{5} - 11 q^{6} - 9 q^{7} + 11 q^{8} + 18 q^{9} - 12 q^{10} - 7 q^{11} - 11 q^{12} - 21 q^{13} - 9 q^{14} - 3 q^{15} + 11 q^{16} - 16 q^{17} + 18 q^{18} - 22 q^{19} - 12 q^{20} - q^{21} - 7 q^{22} - 2 q^{23} - 11 q^{24} + 19 q^{25} - 21 q^{26} - 44 q^{27} - 9 q^{28} + 4 q^{29} - 3 q^{30} - 28 q^{31} + 11 q^{32} - 13 q^{33} - 16 q^{34} - 11 q^{35} + 18 q^{36} - 22 q^{37} - 22 q^{38} + 9 q^{39} - 12 q^{40} - 5 q^{41} - q^{42} - 7 q^{43} - 7 q^{44} - 23 q^{45} - 2 q^{46} - 31 q^{47} - 11 q^{48} - 2 q^{49} + 19 q^{50} - 6 q^{51} - 21 q^{52} - 17 q^{53} - 44 q^{54} - 18 q^{55} - 9 q^{56} + 7 q^{57} + 4 q^{58} - 18 q^{59} - 3 q^{60} - 18 q^{61} - 28 q^{62} - 27 q^{63} + 11 q^{64} + 22 q^{65} - 13 q^{66} - 11 q^{67} - 16 q^{68} + 9 q^{69} - 11 q^{70} - 16 q^{71} + 18 q^{72} - 33 q^{73} - 22 q^{74} - 21 q^{75} - 22 q^{76} + 9 q^{78} + 9 q^{79} - 12 q^{80} + 71 q^{81} - 5 q^{82} - 18 q^{83} - q^{84} - 8 q^{85} - 7 q^{86} - 17 q^{87} - 7 q^{88} - 23 q^{90} - 22 q^{91} - 2 q^{92} + 8 q^{93} - 31 q^{94} - 23 q^{95} - 11 q^{96} - 66 q^{97} - 2 q^{98} - 5 q^{99}+O(q^{100}) \) 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
\(3\) −0.565115 −0.326270 −0.163135 0.986604i \(-0.552161\pi\)
−0.163135 + 0.986604i \(0.552161\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.615969 −0.275470 −0.137735 0.990469i \(-0.543982\pi\)
−0.137735 + 0.990469i \(0.543982\pi\)
\(6\) −0.565115 −0.230707
\(7\) 3.05802 1.15582 0.577911 0.816100i \(-0.303868\pi\)
0.577911 + 0.816100i \(0.303868\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.68064 −0.893548
\(10\) −0.615969 −0.194786
\(11\) −3.11632 −0.939606 −0.469803 0.882771i \(-0.655675\pi\)
−0.469803 + 0.882771i \(0.655675\pi\)
\(12\) −0.565115 −0.163135
\(13\) −6.07042 −1.68363 −0.841816 0.539765i \(-0.818513\pi\)
−0.841816 + 0.539765i \(0.818513\pi\)
\(14\) 3.05802 0.817290
\(15\) 0.348093 0.0898773
\(16\) 1.00000 0.250000
\(17\) −7.35747 −1.78445 −0.892224 0.451594i \(-0.850856\pi\)
−0.892224 + 0.451594i \(0.850856\pi\)
\(18\) −2.68064 −0.631834
\(19\) 1.30846 0.300181 0.150091 0.988672i \(-0.452043\pi\)
0.150091 + 0.988672i \(0.452043\pi\)
\(20\) −0.615969 −0.137735
\(21\) −1.72813 −0.377110
\(22\) −3.11632 −0.664402
\(23\) −1.38346 −0.288472 −0.144236 0.989543i \(-0.546072\pi\)
−0.144236 + 0.989543i \(0.546072\pi\)
\(24\) −0.565115 −0.115354
\(25\) −4.62058 −0.924117
\(26\) −6.07042 −1.19051
\(27\) 3.21022 0.617807
\(28\) 3.05802 0.577911
\(29\) 3.58342 0.665424 0.332712 0.943029i \(-0.392036\pi\)
0.332712 + 0.943029i \(0.392036\pi\)
\(30\) 0.348093 0.0635529
\(31\) 2.91808 0.524103 0.262051 0.965054i \(-0.415601\pi\)
0.262051 + 0.965054i \(0.415601\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.76108 0.306565
\(34\) −7.35747 −1.26179
\(35\) −1.88364 −0.318394
\(36\) −2.68064 −0.446774
\(37\) −9.97748 −1.64029 −0.820144 0.572157i \(-0.806107\pi\)
−0.820144 + 0.572157i \(0.806107\pi\)
\(38\) 1.30846 0.212260
\(39\) 3.43049 0.549318
\(40\) −0.615969 −0.0973932
\(41\) 6.14047 0.958980 0.479490 0.877547i \(-0.340822\pi\)
0.479490 + 0.877547i \(0.340822\pi\)
\(42\) −1.72813 −0.266657
\(43\) −3.20281 −0.488424 −0.244212 0.969722i \(-0.578529\pi\)
−0.244212 + 0.969722i \(0.578529\pi\)
\(44\) −3.11632 −0.469803
\(45\) 1.65119 0.246145
\(46\) −1.38346 −0.203981
\(47\) 7.36719 1.07461 0.537307 0.843387i \(-0.319442\pi\)
0.537307 + 0.843387i \(0.319442\pi\)
\(48\) −0.565115 −0.0815674
\(49\) 2.35148 0.335925
\(50\) −4.62058 −0.653449
\(51\) 4.15782 0.582211
\(52\) −6.07042 −0.841816
\(53\) −3.83489 −0.526762 −0.263381 0.964692i \(-0.584838\pi\)
−0.263381 + 0.964692i \(0.584838\pi\)
\(54\) 3.21022 0.436856
\(55\) 1.91956 0.258833
\(56\) 3.05802 0.408645
\(57\) −0.739431 −0.0979401
\(58\) 3.58342 0.470526
\(59\) −1.10698 −0.144116 −0.0720582 0.997400i \(-0.522957\pi\)
−0.0720582 + 0.997400i \(0.522957\pi\)
\(60\) 0.348093 0.0449387
\(61\) 7.64720 0.979124 0.489562 0.871969i \(-0.337157\pi\)
0.489562 + 0.871969i \(0.337157\pi\)
\(62\) 2.91808 0.370597
\(63\) −8.19746 −1.03278
\(64\) 1.00000 0.125000
\(65\) 3.73919 0.463789
\(66\) 1.76108 0.216774
\(67\) −16.1681 −1.97524 −0.987621 0.156858i \(-0.949863\pi\)
−0.987621 + 0.156858i \(0.949863\pi\)
\(68\) −7.35747 −0.892224
\(69\) 0.781816 0.0941196
\(70\) −1.88364 −0.225138
\(71\) 15.1235 1.79483 0.897417 0.441182i \(-0.145441\pi\)
0.897417 + 0.441182i \(0.145441\pi\)
\(72\) −2.68064 −0.315917
\(73\) −11.5095 −1.34708 −0.673541 0.739150i \(-0.735228\pi\)
−0.673541 + 0.739150i \(0.735228\pi\)
\(74\) −9.97748 −1.15986
\(75\) 2.61116 0.301511
\(76\) 1.30846 0.150091
\(77\) −9.52976 −1.08602
\(78\) 3.43049 0.388426
\(79\) 6.87816 0.773853 0.386927 0.922111i \(-0.373537\pi\)
0.386927 + 0.922111i \(0.373537\pi\)
\(80\) −0.615969 −0.0688674
\(81\) 6.22779 0.691976
\(82\) 6.14047 0.678102
\(83\) 3.52486 0.386904 0.193452 0.981110i \(-0.438032\pi\)
0.193452 + 0.981110i \(0.438032\pi\)
\(84\) −1.72813 −0.188555
\(85\) 4.53197 0.491561
\(86\) −3.20281 −0.345368
\(87\) −2.02504 −0.217108
\(88\) −3.11632 −0.332201
\(89\) −10.5489 −1.11818 −0.559089 0.829108i \(-0.688849\pi\)
−0.559089 + 0.829108i \(0.688849\pi\)
\(90\) 1.65119 0.174051
\(91\) −18.5635 −1.94598
\(92\) −1.38346 −0.144236
\(93\) −1.64905 −0.170999
\(94\) 7.36719 0.759867
\(95\) −0.805970 −0.0826908
\(96\) −0.565115 −0.0576769
\(97\) −7.96436 −0.808658 −0.404329 0.914614i \(-0.632495\pi\)
−0.404329 + 0.914614i \(0.632495\pi\)
\(98\) 2.35148 0.237535
\(99\) 8.35375 0.839583
\(100\) −4.62058 −0.462058
\(101\) 1.66068 0.165244 0.0826220 0.996581i \(-0.473671\pi\)
0.0826220 + 0.996581i \(0.473671\pi\)
\(102\) 4.15782 0.411685
\(103\) −14.0622 −1.38559 −0.692796 0.721134i \(-0.743621\pi\)
−0.692796 + 0.721134i \(0.743621\pi\)
\(104\) −6.07042 −0.595254
\(105\) 1.06448 0.103882
\(106\) −3.83489 −0.372477
\(107\) 12.5752 1.21570 0.607848 0.794054i \(-0.292033\pi\)
0.607848 + 0.794054i \(0.292033\pi\)
\(108\) 3.21022 0.308904
\(109\) −5.61259 −0.537589 −0.268794 0.963198i \(-0.586625\pi\)
−0.268794 + 0.963198i \(0.586625\pi\)
\(110\) 1.91956 0.183022
\(111\) 5.63843 0.535176
\(112\) 3.05802 0.288956
\(113\) 15.3358 1.44267 0.721334 0.692587i \(-0.243529\pi\)
0.721334 + 0.692587i \(0.243529\pi\)
\(114\) −0.739431 −0.0692541
\(115\) 0.852170 0.0794652
\(116\) 3.58342 0.332712
\(117\) 16.2726 1.50441
\(118\) −1.10698 −0.101906
\(119\) −22.4993 −2.06250
\(120\) 0.348093 0.0317764
\(121\) −1.28855 −0.117141
\(122\) 7.64720 0.692345
\(123\) −3.47008 −0.312886
\(124\) 2.91808 0.262051
\(125\) 5.92598 0.530035
\(126\) −8.19746 −0.730288
\(127\) −10.9479 −0.971471 −0.485736 0.874106i \(-0.661448\pi\)
−0.485736 + 0.874106i \(0.661448\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.80996 0.159358
\(130\) 3.73919 0.327948
\(131\) −14.6494 −1.27992 −0.639960 0.768408i \(-0.721049\pi\)
−0.639960 + 0.768408i \(0.721049\pi\)
\(132\) 1.76108 0.153282
\(133\) 4.00130 0.346956
\(134\) −16.1681 −1.39671
\(135\) −1.97739 −0.170187
\(136\) −7.35747 −0.630897
\(137\) 4.82347 0.412097 0.206048 0.978542i \(-0.433940\pi\)
0.206048 + 0.978542i \(0.433940\pi\)
\(138\) 0.781816 0.0665526
\(139\) −21.4686 −1.82094 −0.910470 0.413576i \(-0.864280\pi\)
−0.910470 + 0.413576i \(0.864280\pi\)
\(140\) −1.88364 −0.159197
\(141\) −4.16331 −0.350614
\(142\) 15.1235 1.26914
\(143\) 18.9174 1.58195
\(144\) −2.68064 −0.223387
\(145\) −2.20727 −0.183304
\(146\) −11.5095 −0.952531
\(147\) −1.32886 −0.109602
\(148\) −9.97748 −0.820144
\(149\) −13.8144 −1.13172 −0.565861 0.824501i \(-0.691456\pi\)
−0.565861 + 0.824501i \(0.691456\pi\)
\(150\) 2.61116 0.213201
\(151\) 8.21835 0.668800 0.334400 0.942431i \(-0.391466\pi\)
0.334400 + 0.942431i \(0.391466\pi\)
\(152\) 1.30846 0.106130
\(153\) 19.7227 1.59449
\(154\) −9.52976 −0.767930
\(155\) −1.79745 −0.144374
\(156\) 3.43049 0.274659
\(157\) −1.12390 −0.0896969 −0.0448485 0.998994i \(-0.514281\pi\)
−0.0448485 + 0.998994i \(0.514281\pi\)
\(158\) 6.87816 0.547197
\(159\) 2.16715 0.171867
\(160\) −0.615969 −0.0486966
\(161\) −4.23066 −0.333422
\(162\) 6.22779 0.489301
\(163\) 21.6481 1.69561 0.847803 0.530311i \(-0.177925\pi\)
0.847803 + 0.530311i \(0.177925\pi\)
\(164\) 6.14047 0.479490
\(165\) −1.08477 −0.0844493
\(166\) 3.52486 0.273582
\(167\) −5.38914 −0.417024 −0.208512 0.978020i \(-0.566862\pi\)
−0.208512 + 0.978020i \(0.566862\pi\)
\(168\) −1.72813 −0.133328
\(169\) 23.8500 1.83461
\(170\) 4.53197 0.347586
\(171\) −3.50752 −0.268227
\(172\) −3.20281 −0.244212
\(173\) −5.65710 −0.430101 −0.215051 0.976603i \(-0.568992\pi\)
−0.215051 + 0.976603i \(0.568992\pi\)
\(174\) −2.02504 −0.153518
\(175\) −14.1298 −1.06811
\(176\) −3.11632 −0.234901
\(177\) 0.625571 0.0470208
\(178\) −10.5489 −0.790672
\(179\) −15.0737 −1.12666 −0.563331 0.826231i \(-0.690480\pi\)
−0.563331 + 0.826231i \(0.690480\pi\)
\(180\) 1.65119 0.123073
\(181\) 26.6368 1.97990 0.989948 0.141430i \(-0.0451700\pi\)
0.989948 + 0.141430i \(0.0451700\pi\)
\(182\) −18.5635 −1.37601
\(183\) −4.32155 −0.319458
\(184\) −1.38346 −0.101990
\(185\) 6.14582 0.451849
\(186\) −1.64905 −0.120914
\(187\) 22.9282 1.67668
\(188\) 7.36719 0.537307
\(189\) 9.81691 0.714075
\(190\) −0.805970 −0.0584712
\(191\) 22.1179 1.60039 0.800196 0.599739i \(-0.204729\pi\)
0.800196 + 0.599739i \(0.204729\pi\)
\(192\) −0.565115 −0.0407837
\(193\) −4.12444 −0.296883 −0.148442 0.988921i \(-0.547426\pi\)
−0.148442 + 0.988921i \(0.547426\pi\)
\(194\) −7.96436 −0.571808
\(195\) −2.11307 −0.151320
\(196\) 2.35148 0.167963
\(197\) 26.2138 1.86765 0.933827 0.357724i \(-0.116447\pi\)
0.933827 + 0.357724i \(0.116447\pi\)
\(198\) 8.35375 0.593675
\(199\) 10.2490 0.726531 0.363265 0.931686i \(-0.381662\pi\)
0.363265 + 0.931686i \(0.381662\pi\)
\(200\) −4.62058 −0.326725
\(201\) 9.13682 0.644461
\(202\) 1.66068 0.116845
\(203\) 10.9582 0.769112
\(204\) 4.15782 0.291105
\(205\) −3.78234 −0.264170
\(206\) −14.0622 −0.979761
\(207\) 3.70857 0.257764
\(208\) −6.07042 −0.420908
\(209\) −4.07758 −0.282052
\(210\) 1.06448 0.0734558
\(211\) 0.885119 0.0609341 0.0304671 0.999536i \(-0.490301\pi\)
0.0304671 + 0.999536i \(0.490301\pi\)
\(212\) −3.83489 −0.263381
\(213\) −8.54655 −0.585600
\(214\) 12.5752 0.859626
\(215\) 1.97283 0.134546
\(216\) 3.21022 0.218428
\(217\) 8.92355 0.605770
\(218\) −5.61259 −0.380133
\(219\) 6.50419 0.439512
\(220\) 1.91956 0.129416
\(221\) 44.6629 3.00435
\(222\) 5.63843 0.378427
\(223\) −14.0423 −0.940340 −0.470170 0.882576i \(-0.655807\pi\)
−0.470170 + 0.882576i \(0.655807\pi\)
\(224\) 3.05802 0.204322
\(225\) 12.3861 0.825743
\(226\) 15.3358 1.02012
\(227\) 22.5229 1.49490 0.747449 0.664320i \(-0.231279\pi\)
0.747449 + 0.664320i \(0.231279\pi\)
\(228\) −0.739431 −0.0489700
\(229\) −17.3340 −1.14546 −0.572731 0.819743i \(-0.694116\pi\)
−0.572731 + 0.819743i \(0.694116\pi\)
\(230\) 0.852170 0.0561904
\(231\) 5.38542 0.354334
\(232\) 3.58342 0.235263
\(233\) −26.5014 −1.73617 −0.868083 0.496419i \(-0.834648\pi\)
−0.868083 + 0.496419i \(0.834648\pi\)
\(234\) 16.2726 1.06378
\(235\) −4.53796 −0.296024
\(236\) −1.10698 −0.0720582
\(237\) −3.88695 −0.252485
\(238\) −22.4993 −1.45841
\(239\) 20.0415 1.29638 0.648188 0.761480i \(-0.275527\pi\)
0.648188 + 0.761480i \(0.275527\pi\)
\(240\) 0.348093 0.0224693
\(241\) −5.59198 −0.360211 −0.180106 0.983647i \(-0.557644\pi\)
−0.180106 + 0.983647i \(0.557644\pi\)
\(242\) −1.28855 −0.0828311
\(243\) −13.1501 −0.843578
\(244\) 7.64720 0.489562
\(245\) −1.44844 −0.0925372
\(246\) −3.47008 −0.221244
\(247\) −7.94290 −0.505395
\(248\) 2.91808 0.185298
\(249\) −1.99195 −0.126235
\(250\) 5.92598 0.374792
\(251\) −15.5175 −0.979458 −0.489729 0.871875i \(-0.662904\pi\)
−0.489729 + 0.871875i \(0.662904\pi\)
\(252\) −8.19746 −0.516391
\(253\) 4.31131 0.271050
\(254\) −10.9479 −0.686934
\(255\) −2.56108 −0.160381
\(256\) 1.00000 0.0625000
\(257\) 3.79991 0.237032 0.118516 0.992952i \(-0.462186\pi\)
0.118516 + 0.992952i \(0.462186\pi\)
\(258\) 1.80996 0.112683
\(259\) −30.5113 −1.89588
\(260\) 3.73919 0.231895
\(261\) −9.60587 −0.594588
\(262\) −14.6494 −0.905040
\(263\) −3.58984 −0.221359 −0.110679 0.993856i \(-0.535303\pi\)
−0.110679 + 0.993856i \(0.535303\pi\)
\(264\) 1.76108 0.108387
\(265\) 2.36217 0.145107
\(266\) 4.00130 0.245335
\(267\) 5.96133 0.364828
\(268\) −16.1681 −0.987621
\(269\) 10.4559 0.637507 0.318753 0.947838i \(-0.396736\pi\)
0.318753 + 0.947838i \(0.396736\pi\)
\(270\) −1.97739 −0.120340
\(271\) −3.88450 −0.235966 −0.117983 0.993016i \(-0.537643\pi\)
−0.117983 + 0.993016i \(0.537643\pi\)
\(272\) −7.35747 −0.446112
\(273\) 10.4905 0.634914
\(274\) 4.82347 0.291396
\(275\) 14.3992 0.868305
\(276\) 0.781816 0.0470598
\(277\) 25.3987 1.52606 0.763029 0.646364i \(-0.223711\pi\)
0.763029 + 0.646364i \(0.223711\pi\)
\(278\) −21.4686 −1.28760
\(279\) −7.82234 −0.468311
\(280\) −1.88364 −0.112569
\(281\) 8.72731 0.520628 0.260314 0.965524i \(-0.416174\pi\)
0.260314 + 0.965524i \(0.416174\pi\)
\(282\) −4.16331 −0.247922
\(283\) −22.9032 −1.36145 −0.680725 0.732539i \(-0.738335\pi\)
−0.680725 + 0.732539i \(0.738335\pi\)
\(284\) 15.1235 0.897417
\(285\) 0.455466 0.0269795
\(286\) 18.9174 1.11861
\(287\) 18.7777 1.10841
\(288\) −2.68064 −0.157958
\(289\) 37.1323 2.18425
\(290\) −2.20727 −0.129615
\(291\) 4.50078 0.263841
\(292\) −11.5095 −0.673541
\(293\) −2.82047 −0.164774 −0.0823868 0.996600i \(-0.526254\pi\)
−0.0823868 + 0.996600i \(0.526254\pi\)
\(294\) −1.32886 −0.0775005
\(295\) 0.681864 0.0396997
\(296\) −9.97748 −0.579929
\(297\) −10.0041 −0.580495
\(298\) −13.8144 −0.800248
\(299\) 8.39820 0.485681
\(300\) 2.61116 0.150756
\(301\) −9.79425 −0.564531
\(302\) 8.21835 0.472913
\(303\) −0.938477 −0.0539141
\(304\) 1.30846 0.0750453
\(305\) −4.71044 −0.269719
\(306\) 19.7227 1.12747
\(307\) 0.877326 0.0500716 0.0250358 0.999687i \(-0.492030\pi\)
0.0250358 + 0.999687i \(0.492030\pi\)
\(308\) −9.52976 −0.543009
\(309\) 7.94678 0.452076
\(310\) −1.79745 −0.102088
\(311\) −10.6787 −0.605535 −0.302767 0.953065i \(-0.597910\pi\)
−0.302767 + 0.953065i \(0.597910\pi\)
\(312\) 3.43049 0.194213
\(313\) 27.2477 1.54013 0.770066 0.637964i \(-0.220223\pi\)
0.770066 + 0.637964i \(0.220223\pi\)
\(314\) −1.12390 −0.0634253
\(315\) 5.04938 0.284500
\(316\) 6.87816 0.386927
\(317\) −26.4081 −1.48323 −0.741615 0.670826i \(-0.765940\pi\)
−0.741615 + 0.670826i \(0.765940\pi\)
\(318\) 2.16715 0.121528
\(319\) −11.1671 −0.625236
\(320\) −0.615969 −0.0344337
\(321\) −7.10647 −0.396644
\(322\) −4.23066 −0.235765
\(323\) −9.62695 −0.535658
\(324\) 6.22779 0.345988
\(325\) 28.0489 1.55587
\(326\) 21.6481 1.19898
\(327\) 3.17176 0.175399
\(328\) 6.14047 0.339051
\(329\) 22.5290 1.24206
\(330\) −1.08477 −0.0597146
\(331\) −33.2619 −1.82824 −0.914119 0.405446i \(-0.867116\pi\)
−0.914119 + 0.405446i \(0.867116\pi\)
\(332\) 3.52486 0.193452
\(333\) 26.7461 1.46568
\(334\) −5.38914 −0.294881
\(335\) 9.95901 0.544119
\(336\) −1.72813 −0.0942774
\(337\) −22.5248 −1.22701 −0.613503 0.789692i \(-0.710240\pi\)
−0.613503 + 0.789692i \(0.710240\pi\)
\(338\) 23.8500 1.29727
\(339\) −8.66649 −0.470699
\(340\) 4.53197 0.245780
\(341\) −9.09367 −0.492450
\(342\) −3.50752 −0.189665
\(343\) −14.2153 −0.767552
\(344\) −3.20281 −0.172684
\(345\) −0.481574 −0.0259271
\(346\) −5.65710 −0.304128
\(347\) −12.8300 −0.688748 −0.344374 0.938832i \(-0.611909\pi\)
−0.344374 + 0.938832i \(0.611909\pi\)
\(348\) −2.02504 −0.108554
\(349\) −28.4649 −1.52369 −0.761847 0.647757i \(-0.775707\pi\)
−0.761847 + 0.647757i \(0.775707\pi\)
\(350\) −14.1298 −0.755271
\(351\) −19.4874 −1.04016
\(352\) −3.11632 −0.166100
\(353\) −15.6535 −0.833149 −0.416575 0.909102i \(-0.636770\pi\)
−0.416575 + 0.909102i \(0.636770\pi\)
\(354\) 0.625571 0.0332487
\(355\) −9.31563 −0.494422
\(356\) −10.5489 −0.559089
\(357\) 12.7147 0.672932
\(358\) −15.0737 −0.796671
\(359\) 30.0022 1.58345 0.791727 0.610875i \(-0.209182\pi\)
0.791727 + 0.610875i \(0.209182\pi\)
\(360\) 1.65119 0.0870255
\(361\) −17.2879 −0.909891
\(362\) 26.6368 1.40000
\(363\) 0.728180 0.0382195
\(364\) −18.5635 −0.972989
\(365\) 7.08948 0.371080
\(366\) −4.32155 −0.225891
\(367\) 26.6928 1.39335 0.696676 0.717386i \(-0.254662\pi\)
0.696676 + 0.717386i \(0.254662\pi\)
\(368\) −1.38346 −0.0721180
\(369\) −16.4604 −0.856895
\(370\) 6.14582 0.319506
\(371\) −11.7272 −0.608844
\(372\) −1.64905 −0.0854994
\(373\) 27.8906 1.44412 0.722061 0.691829i \(-0.243195\pi\)
0.722061 + 0.691829i \(0.243195\pi\)
\(374\) 22.9282 1.18559
\(375\) −3.34886 −0.172934
\(376\) 7.36719 0.379934
\(377\) −21.7528 −1.12033
\(378\) 9.81691 0.504928
\(379\) 20.0434 1.02956 0.514780 0.857322i \(-0.327874\pi\)
0.514780 + 0.857322i \(0.327874\pi\)
\(380\) −0.805970 −0.0413454
\(381\) 6.18684 0.316962
\(382\) 22.1179 1.13165
\(383\) −5.82733 −0.297763 −0.148881 0.988855i \(-0.547567\pi\)
−0.148881 + 0.988855i \(0.547567\pi\)
\(384\) −0.565115 −0.0288384
\(385\) 5.87004 0.299165
\(386\) −4.12444 −0.209928
\(387\) 8.58559 0.436430
\(388\) −7.96436 −0.404329
\(389\) −31.8482 −1.61477 −0.807384 0.590027i \(-0.799117\pi\)
−0.807384 + 0.590027i \(0.799117\pi\)
\(390\) −2.11307 −0.107000
\(391\) 10.1788 0.514763
\(392\) 2.35148 0.118768
\(393\) 8.27858 0.417599
\(394\) 26.2138 1.32063
\(395\) −4.23673 −0.213173
\(396\) 8.35375 0.419792
\(397\) −24.9264 −1.25102 −0.625509 0.780217i \(-0.715109\pi\)
−0.625509 + 0.780217i \(0.715109\pi\)
\(398\) 10.2490 0.513735
\(399\) −2.26119 −0.113201
\(400\) −4.62058 −0.231029
\(401\) −6.99992 −0.349559 −0.174780 0.984608i \(-0.555921\pi\)
−0.174780 + 0.984608i \(0.555921\pi\)
\(402\) 9.13682 0.455703
\(403\) −17.7140 −0.882396
\(404\) 1.66068 0.0826220
\(405\) −3.83612 −0.190618
\(406\) 10.9582 0.543844
\(407\) 31.0930 1.54122
\(408\) 4.15782 0.205843
\(409\) −4.72367 −0.233571 −0.116785 0.993157i \(-0.537259\pi\)
−0.116785 + 0.993157i \(0.537259\pi\)
\(410\) −3.78234 −0.186796
\(411\) −2.72582 −0.134455
\(412\) −14.0622 −0.692796
\(413\) −3.38516 −0.166573
\(414\) 3.70857 0.182266
\(415\) −2.17120 −0.106580
\(416\) −6.07042 −0.297627
\(417\) 12.1322 0.594117
\(418\) −4.07758 −0.199441
\(419\) −22.0804 −1.07870 −0.539349 0.842082i \(-0.681330\pi\)
−0.539349 + 0.842082i \(0.681330\pi\)
\(420\) 1.06448 0.0519411
\(421\) 23.1422 1.12788 0.563941 0.825815i \(-0.309285\pi\)
0.563941 + 0.825815i \(0.309285\pi\)
\(422\) 0.885119 0.0430869
\(423\) −19.7488 −0.960220
\(424\) −3.83489 −0.186239
\(425\) 33.9958 1.64904
\(426\) −8.54655 −0.414082
\(427\) 23.3853 1.13169
\(428\) 12.5752 0.607848
\(429\) −10.6905 −0.516142
\(430\) 1.97283 0.0951383
\(431\) 14.6390 0.705136 0.352568 0.935786i \(-0.385309\pi\)
0.352568 + 0.935786i \(0.385309\pi\)
\(432\) 3.21022 0.154452
\(433\) −7.87846 −0.378614 −0.189307 0.981918i \(-0.560624\pi\)
−0.189307 + 0.981918i \(0.560624\pi\)
\(434\) 8.92355 0.428344
\(435\) 1.24736 0.0598065
\(436\) −5.61259 −0.268794
\(437\) −1.81021 −0.0865939
\(438\) 6.50419 0.310782
\(439\) −0.803456 −0.0383469 −0.0191734 0.999816i \(-0.506103\pi\)
−0.0191734 + 0.999816i \(0.506103\pi\)
\(440\) 1.91956 0.0915112
\(441\) −6.30348 −0.300165
\(442\) 44.6629 2.12440
\(443\) 18.7417 0.890443 0.445222 0.895420i \(-0.353125\pi\)
0.445222 + 0.895420i \(0.353125\pi\)
\(444\) 5.63843 0.267588
\(445\) 6.49777 0.308024
\(446\) −14.0423 −0.664921
\(447\) 7.80674 0.369246
\(448\) 3.05802 0.144478
\(449\) 19.2173 0.906919 0.453460 0.891277i \(-0.350190\pi\)
0.453460 + 0.891277i \(0.350190\pi\)
\(450\) 12.3861 0.583888
\(451\) −19.1357 −0.901064
\(452\) 15.3358 0.721334
\(453\) −4.64432 −0.218209
\(454\) 22.5229 1.05705
\(455\) 11.4345 0.536058
\(456\) −0.739431 −0.0346270
\(457\) 31.7684 1.48606 0.743031 0.669257i \(-0.233388\pi\)
0.743031 + 0.669257i \(0.233388\pi\)
\(458\) −17.3340 −0.809965
\(459\) −23.6191 −1.10244
\(460\) 0.852170 0.0397326
\(461\) −20.9500 −0.975737 −0.487868 0.872917i \(-0.662225\pi\)
−0.487868 + 0.872917i \(0.662225\pi\)
\(462\) 5.38542 0.250552
\(463\) 13.6195 0.632950 0.316475 0.948601i \(-0.397501\pi\)
0.316475 + 0.948601i \(0.397501\pi\)
\(464\) 3.58342 0.166356
\(465\) 1.01576 0.0471050
\(466\) −26.5014 −1.22765
\(467\) −21.5848 −0.998827 −0.499413 0.866364i \(-0.666451\pi\)
−0.499413 + 0.866364i \(0.666451\pi\)
\(468\) 16.2726 0.752203
\(469\) −49.4422 −2.28303
\(470\) −4.53796 −0.209320
\(471\) 0.635133 0.0292654
\(472\) −1.10698 −0.0509528
\(473\) 9.98098 0.458926
\(474\) −3.88695 −0.178534
\(475\) −6.04585 −0.277403
\(476\) −22.4993 −1.03125
\(477\) 10.2800 0.470688
\(478\) 20.0415 0.916676
\(479\) 15.4702 0.706850 0.353425 0.935463i \(-0.385017\pi\)
0.353425 + 0.935463i \(0.385017\pi\)
\(480\) 0.348093 0.0158882
\(481\) 60.5675 2.76164
\(482\) −5.59198 −0.254708
\(483\) 2.39081 0.108786
\(484\) −1.28855 −0.0585705
\(485\) 4.90579 0.222761
\(486\) −13.1501 −0.596500
\(487\) 9.28893 0.420922 0.210461 0.977602i \(-0.432504\pi\)
0.210461 + 0.977602i \(0.432504\pi\)
\(488\) 7.64720 0.346173
\(489\) −12.2337 −0.553225
\(490\) −1.44844 −0.0654337
\(491\) −21.0597 −0.950411 −0.475205 0.879875i \(-0.657626\pi\)
−0.475205 + 0.879875i \(0.657626\pi\)
\(492\) −3.47008 −0.156443
\(493\) −26.3649 −1.18741
\(494\) −7.94290 −0.357368
\(495\) −5.14564 −0.231280
\(496\) 2.91808 0.131026
\(497\) 46.2481 2.07451
\(498\) −1.99195 −0.0892615
\(499\) −27.9653 −1.25190 −0.625950 0.779863i \(-0.715289\pi\)
−0.625950 + 0.779863i \(0.715289\pi\)
\(500\) 5.92598 0.265018
\(501\) 3.04549 0.136062
\(502\) −15.5175 −0.692582
\(503\) −42.8231 −1.90939 −0.954693 0.297592i \(-0.903816\pi\)
−0.954693 + 0.297592i \(0.903816\pi\)
\(504\) −8.19746 −0.365144
\(505\) −1.02293 −0.0455197
\(506\) 4.31131 0.191661
\(507\) −13.4780 −0.598579
\(508\) −10.9479 −0.485736
\(509\) 2.04492 0.0906396 0.0453198 0.998973i \(-0.485569\pi\)
0.0453198 + 0.998973i \(0.485569\pi\)
\(510\) −2.56108 −0.113407
\(511\) −35.1962 −1.55699
\(512\) 1.00000 0.0441942
\(513\) 4.20045 0.185454
\(514\) 3.79991 0.167607
\(515\) 8.66188 0.381688
\(516\) 1.80996 0.0796789
\(517\) −22.9585 −1.00971
\(518\) −30.5113 −1.34059
\(519\) 3.19692 0.140329
\(520\) 3.73919 0.163974
\(521\) 5.68194 0.248930 0.124465 0.992224i \(-0.460278\pi\)
0.124465 + 0.992224i \(0.460278\pi\)
\(522\) −9.60587 −0.420437
\(523\) 13.7368 0.600669 0.300335 0.953834i \(-0.402902\pi\)
0.300335 + 0.953834i \(0.402902\pi\)
\(524\) −14.6494 −0.639960
\(525\) 7.98498 0.348493
\(526\) −3.58984 −0.156524
\(527\) −21.4697 −0.935234
\(528\) 1.76108 0.0766412
\(529\) −21.0860 −0.916784
\(530\) 2.36217 0.102606
\(531\) 2.96742 0.128775
\(532\) 4.00130 0.173478
\(533\) −37.2752 −1.61457
\(534\) 5.96133 0.257972
\(535\) −7.74596 −0.334887
\(536\) −16.1681 −0.698354
\(537\) 8.51839 0.367596
\(538\) 10.4559 0.450785
\(539\) −7.32796 −0.315637
\(540\) −1.97739 −0.0850935
\(541\) 34.1616 1.46872 0.734361 0.678759i \(-0.237482\pi\)
0.734361 + 0.678759i \(0.237482\pi\)
\(542\) −3.88450 −0.166853
\(543\) −15.0529 −0.645980
\(544\) −7.35747 −0.315449
\(545\) 3.45718 0.148089
\(546\) 10.4905 0.448952
\(547\) −5.97905 −0.255646 −0.127823 0.991797i \(-0.540799\pi\)
−0.127823 + 0.991797i \(0.540799\pi\)
\(548\) 4.82347 0.206048
\(549\) −20.4994 −0.874894
\(550\) 14.3992 0.613985
\(551\) 4.68876 0.199748
\(552\) 0.781816 0.0332763
\(553\) 21.0335 0.894437
\(554\) 25.3987 1.07909
\(555\) −3.47310 −0.147425
\(556\) −21.4686 −0.910470
\(557\) −19.6992 −0.834681 −0.417340 0.908750i \(-0.637038\pi\)
−0.417340 + 0.908750i \(0.637038\pi\)
\(558\) −7.82234 −0.331146
\(559\) 19.4424 0.822326
\(560\) −1.88364 −0.0795985
\(561\) −12.9571 −0.547049
\(562\) 8.72731 0.368139
\(563\) −27.5569 −1.16139 −0.580693 0.814123i \(-0.697218\pi\)
−0.580693 + 0.814123i \(0.697218\pi\)
\(564\) −4.16331 −0.175307
\(565\) −9.44636 −0.397411
\(566\) −22.9032 −0.962691
\(567\) 19.0447 0.799802
\(568\) 15.1235 0.634570
\(569\) −15.7614 −0.660751 −0.330375 0.943850i \(-0.607175\pi\)
−0.330375 + 0.943850i \(0.607175\pi\)
\(570\) 0.455466 0.0190774
\(571\) −9.25289 −0.387221 −0.193611 0.981078i \(-0.562020\pi\)
−0.193611 + 0.981078i \(0.562020\pi\)
\(572\) 18.9174 0.790975
\(573\) −12.4991 −0.522159
\(574\) 18.7777 0.783765
\(575\) 6.39241 0.266582
\(576\) −2.68064 −0.111694
\(577\) −33.7348 −1.40440 −0.702198 0.711982i \(-0.747798\pi\)
−0.702198 + 0.711982i \(0.747798\pi\)
\(578\) 37.1323 1.54450
\(579\) 2.33078 0.0968641
\(580\) −2.20727 −0.0916520
\(581\) 10.7791 0.447192
\(582\) 4.50078 0.186563
\(583\) 11.9507 0.494949
\(584\) −11.5095 −0.476266
\(585\) −10.0234 −0.414418
\(586\) −2.82047 −0.116513
\(587\) −20.9968 −0.866630 −0.433315 0.901243i \(-0.642656\pi\)
−0.433315 + 0.901243i \(0.642656\pi\)
\(588\) −1.32886 −0.0548011
\(589\) 3.81819 0.157326
\(590\) 0.681864 0.0280719
\(591\) −14.8138 −0.609359
\(592\) −9.97748 −0.410072
\(593\) −8.39567 −0.344769 −0.172384 0.985030i \(-0.555147\pi\)
−0.172384 + 0.985030i \(0.555147\pi\)
\(594\) −10.0041 −0.410472
\(595\) 13.8588 0.568157
\(596\) −13.8144 −0.565861
\(597\) −5.79186 −0.237045
\(598\) 8.39820 0.343428
\(599\) −0.0503407 −0.00205686 −0.00102843 0.999999i \(-0.500327\pi\)
−0.00102843 + 0.999999i \(0.500327\pi\)
\(600\) 2.61116 0.106600
\(601\) 38.1249 1.55515 0.777573 0.628792i \(-0.216450\pi\)
0.777573 + 0.628792i \(0.216450\pi\)
\(602\) −9.79425 −0.399184
\(603\) 43.3408 1.76497
\(604\) 8.21835 0.334400
\(605\) 0.793706 0.0322688
\(606\) −0.938477 −0.0381230
\(607\) 7.64435 0.310274 0.155137 0.987893i \(-0.450418\pi\)
0.155137 + 0.987893i \(0.450418\pi\)
\(608\) 1.30846 0.0530651
\(609\) −6.19262 −0.250938
\(610\) −4.71044 −0.190720
\(611\) −44.7219 −1.80926
\(612\) 19.7227 0.797245
\(613\) −11.4483 −0.462393 −0.231196 0.972907i \(-0.574264\pi\)
−0.231196 + 0.972907i \(0.574264\pi\)
\(614\) 0.877326 0.0354060
\(615\) 2.13746 0.0861906
\(616\) −9.52976 −0.383965
\(617\) 0.885025 0.0356298 0.0178149 0.999841i \(-0.494329\pi\)
0.0178149 + 0.999841i \(0.494329\pi\)
\(618\) 7.94678 0.319666
\(619\) −17.1369 −0.688789 −0.344395 0.938825i \(-0.611916\pi\)
−0.344395 + 0.938825i \(0.611916\pi\)
\(620\) −1.79745 −0.0721872
\(621\) −4.44122 −0.178220
\(622\) −10.6787 −0.428178
\(623\) −32.2587 −1.29242
\(624\) 3.43049 0.137329
\(625\) 19.4527 0.778108
\(626\) 27.2477 1.08904
\(627\) 2.30430 0.0920250
\(628\) −1.12390 −0.0448485
\(629\) 73.4090 2.92701
\(630\) 5.04938 0.201172
\(631\) 1.37336 0.0546727 0.0273363 0.999626i \(-0.491297\pi\)
0.0273363 + 0.999626i \(0.491297\pi\)
\(632\) 6.87816 0.273598
\(633\) −0.500195 −0.0198810
\(634\) −26.4081 −1.04880
\(635\) 6.74358 0.267611
\(636\) 2.16715 0.0859333
\(637\) −14.2745 −0.565574
\(638\) −11.1671 −0.442109
\(639\) −40.5409 −1.60377
\(640\) −0.615969 −0.0243483
\(641\) 18.4117 0.727220 0.363610 0.931551i \(-0.381544\pi\)
0.363610 + 0.931551i \(0.381544\pi\)
\(642\) −7.10647 −0.280470
\(643\) 31.7565 1.25236 0.626178 0.779680i \(-0.284618\pi\)
0.626178 + 0.779680i \(0.284618\pi\)
\(644\) −4.23066 −0.166711
\(645\) −1.11488 −0.0438982
\(646\) −9.62695 −0.378767
\(647\) 6.89081 0.270906 0.135453 0.990784i \(-0.456751\pi\)
0.135453 + 0.990784i \(0.456751\pi\)
\(648\) 6.22779 0.244651
\(649\) 3.44970 0.135413
\(650\) 28.0489 1.10017
\(651\) −5.04283 −0.197644
\(652\) 21.6481 0.847803
\(653\) −1.55548 −0.0608706 −0.0304353 0.999537i \(-0.509689\pi\)
−0.0304353 + 0.999537i \(0.509689\pi\)
\(654\) 3.17176 0.124026
\(655\) 9.02354 0.352579
\(656\) 6.14047 0.239745
\(657\) 30.8528 1.20368
\(658\) 22.5290 0.878272
\(659\) 15.2618 0.594516 0.297258 0.954797i \(-0.403928\pi\)
0.297258 + 0.954797i \(0.403928\pi\)
\(660\) −1.08477 −0.0422246
\(661\) 25.6214 0.996556 0.498278 0.867017i \(-0.333966\pi\)
0.498278 + 0.867017i \(0.333966\pi\)
\(662\) −33.2619 −1.29276
\(663\) −25.2397 −0.980229
\(664\) 3.52486 0.136791
\(665\) −2.46467 −0.0955759
\(666\) 26.7461 1.03639
\(667\) −4.95753 −0.191956
\(668\) −5.38914 −0.208512
\(669\) 7.93550 0.306804
\(670\) 9.95901 0.384750
\(671\) −23.8311 −0.919990
\(672\) −1.72813 −0.0666642
\(673\) −39.7843 −1.53357 −0.766787 0.641902i \(-0.778146\pi\)
−0.766787 + 0.641902i \(0.778146\pi\)
\(674\) −22.5248 −0.867624
\(675\) −14.8331 −0.570926
\(676\) 23.8500 0.917307
\(677\) −40.3238 −1.54977 −0.774885 0.632102i \(-0.782192\pi\)
−0.774885 + 0.632102i \(0.782192\pi\)
\(678\) −8.66649 −0.332834
\(679\) −24.3552 −0.934665
\(680\) 4.53197 0.173793
\(681\) −12.7280 −0.487739
\(682\) −9.09367 −0.348215
\(683\) 1.31813 0.0504370 0.0252185 0.999682i \(-0.491972\pi\)
0.0252185 + 0.999682i \(0.491972\pi\)
\(684\) −3.50752 −0.134113
\(685\) −2.97111 −0.113520
\(686\) −14.2153 −0.542741
\(687\) 9.79571 0.373730
\(688\) −3.20281 −0.122106
\(689\) 23.2794 0.886874
\(690\) −0.481574 −0.0183332
\(691\) 10.3532 0.393855 0.196928 0.980418i \(-0.436904\pi\)
0.196928 + 0.980418i \(0.436904\pi\)
\(692\) −5.65710 −0.215051
\(693\) 25.5459 0.970409
\(694\) −12.8300 −0.487019
\(695\) 13.2240 0.501613
\(696\) −2.02504 −0.0767591
\(697\) −45.1783 −1.71125
\(698\) −28.4649 −1.07741
\(699\) 14.9764 0.566458
\(700\) −14.1298 −0.534057
\(701\) −25.2927 −0.955291 −0.477646 0.878553i \(-0.658510\pi\)
−0.477646 + 0.878553i \(0.658510\pi\)
\(702\) −19.4874 −0.735504
\(703\) −13.0551 −0.492384
\(704\) −3.11632 −0.117451
\(705\) 2.56447 0.0965835
\(706\) −15.6535 −0.589125
\(707\) 5.07840 0.190993
\(708\) 0.625571 0.0235104
\(709\) 23.3678 0.877596 0.438798 0.898586i \(-0.355404\pi\)
0.438798 + 0.898586i \(0.355404\pi\)
\(710\) −9.31563 −0.349609
\(711\) −18.4379 −0.691475
\(712\) −10.5489 −0.395336
\(713\) −4.03706 −0.151189
\(714\) 12.7147 0.475835
\(715\) −11.6525 −0.435779
\(716\) −15.0737 −0.563331
\(717\) −11.3258 −0.422968
\(718\) 30.0022 1.11967
\(719\) −40.4323 −1.50787 −0.753935 0.656949i \(-0.771847\pi\)
−0.753935 + 0.656949i \(0.771847\pi\)
\(720\) 1.65119 0.0615363
\(721\) −43.0025 −1.60150
\(722\) −17.2879 −0.643390
\(723\) 3.16012 0.117526
\(724\) 26.6368 0.989948
\(725\) −16.5575 −0.614929
\(726\) 0.728180 0.0270253
\(727\) 9.46152 0.350908 0.175454 0.984488i \(-0.443861\pi\)
0.175454 + 0.984488i \(0.443861\pi\)
\(728\) −18.5635 −0.688007
\(729\) −11.2521 −0.416743
\(730\) 7.08948 0.262393
\(731\) 23.5646 0.871567
\(732\) −4.32155 −0.159729
\(733\) 7.17204 0.264905 0.132453 0.991189i \(-0.457715\pi\)
0.132453 + 0.991189i \(0.457715\pi\)
\(734\) 26.6928 0.985249
\(735\) 0.818534 0.0301921
\(736\) −1.38346 −0.0509951
\(737\) 50.3848 1.85595
\(738\) −16.4604 −0.605916
\(739\) 17.7817 0.654111 0.327055 0.945005i \(-0.393944\pi\)
0.327055 + 0.945005i \(0.393944\pi\)
\(740\) 6.14582 0.225925
\(741\) 4.48866 0.164895
\(742\) −11.7272 −0.430518
\(743\) 33.3538 1.22363 0.611816 0.791000i \(-0.290439\pi\)
0.611816 + 0.791000i \(0.290439\pi\)
\(744\) −1.64905 −0.0604572
\(745\) 8.50925 0.311755
\(746\) 27.8906 1.02115
\(747\) −9.44889 −0.345717
\(748\) 22.9282 0.838339
\(749\) 38.4553 1.40513
\(750\) −3.34886 −0.122283
\(751\) 1.00000 0.0364905
\(752\) 7.36719 0.268654
\(753\) 8.76920 0.319567
\(754\) −21.7528 −0.792192
\(755\) −5.06225 −0.184234
\(756\) 9.81691 0.357038
\(757\) −21.3897 −0.777421 −0.388711 0.921360i \(-0.627079\pi\)
−0.388711 + 0.921360i \(0.627079\pi\)
\(758\) 20.0434 0.728009
\(759\) −2.43639 −0.0884354
\(760\) −0.805970 −0.0292356
\(761\) 38.3593 1.39052 0.695262 0.718756i \(-0.255288\pi\)
0.695262 + 0.718756i \(0.255288\pi\)
\(762\) 6.18684 0.224126
\(763\) −17.1634 −0.621357
\(764\) 22.1179 0.800196
\(765\) −12.1486 −0.439233
\(766\) −5.82733 −0.210550
\(767\) 6.71983 0.242639
\(768\) −0.565115 −0.0203918
\(769\) 1.31168 0.0473002 0.0236501 0.999720i \(-0.492471\pi\)
0.0236501 + 0.999720i \(0.492471\pi\)
\(770\) 5.87004 0.211541
\(771\) −2.14739 −0.0773363
\(772\) −4.12444 −0.148442
\(773\) −11.7641 −0.423125 −0.211563 0.977364i \(-0.567855\pi\)
−0.211563 + 0.977364i \(0.567855\pi\)
\(774\) 8.58559 0.308603
\(775\) −13.4832 −0.484332
\(776\) −7.96436 −0.285904
\(777\) 17.2424 0.618568
\(778\) −31.8482 −1.14181
\(779\) 8.03456 0.287868
\(780\) −2.11307 −0.0756601
\(781\) −47.1298 −1.68644
\(782\) 10.1788 0.363993
\(783\) 11.5036 0.411104
\(784\) 2.35148 0.0839813
\(785\) 0.692287 0.0247088
\(786\) 8.27858 0.295287
\(787\) 18.7554 0.668558 0.334279 0.942474i \(-0.391507\pi\)
0.334279 + 0.942474i \(0.391507\pi\)
\(788\) 26.2138 0.933827
\(789\) 2.02867 0.0722227
\(790\) −4.23673 −0.150736
\(791\) 46.8971 1.66747
\(792\) 8.35375 0.296837
\(793\) −46.4217 −1.64848
\(794\) −24.9264 −0.884603
\(795\) −1.33490 −0.0473440
\(796\) 10.2490 0.363265
\(797\) −4.61506 −0.163474 −0.0817369 0.996654i \(-0.526047\pi\)
−0.0817369 + 0.996654i \(0.526047\pi\)
\(798\) −2.26119 −0.0800454
\(799\) −54.2038 −1.91759
\(800\) −4.62058 −0.163362
\(801\) 28.2778 0.999146
\(802\) −6.99992 −0.247176
\(803\) 35.8672 1.26573
\(804\) 9.13682 0.322231
\(805\) 2.60595 0.0918477
\(806\) −17.7140 −0.623948
\(807\) −5.90878 −0.207999
\(808\) 1.66068 0.0584226
\(809\) 22.5980 0.794505 0.397252 0.917709i \(-0.369964\pi\)
0.397252 + 0.917709i \(0.369964\pi\)
\(810\) −3.83612 −0.134788
\(811\) −33.4298 −1.17388 −0.586940 0.809631i \(-0.699667\pi\)
−0.586940 + 0.809631i \(0.699667\pi\)
\(812\) 10.9582 0.384556
\(813\) 2.19519 0.0769887
\(814\) 31.0930 1.08981
\(815\) −13.3345 −0.467088
\(816\) 4.15782 0.145553
\(817\) −4.19075 −0.146616
\(818\) −4.72367 −0.165159
\(819\) 49.7620 1.73883
\(820\) −3.78234 −0.132085
\(821\) −38.3449 −1.33825 −0.669123 0.743151i \(-0.733330\pi\)
−0.669123 + 0.743151i \(0.733330\pi\)
\(822\) −2.72582 −0.0950738
\(823\) 1.17504 0.0409592 0.0204796 0.999790i \(-0.493481\pi\)
0.0204796 + 0.999790i \(0.493481\pi\)
\(824\) −14.0622 −0.489880
\(825\) −8.13722 −0.283302
\(826\) −3.38516 −0.117785
\(827\) 13.6131 0.473373 0.236686 0.971586i \(-0.423939\pi\)
0.236686 + 0.971586i \(0.423939\pi\)
\(828\) 3.70857 0.128882
\(829\) −11.0355 −0.383279 −0.191639 0.981465i \(-0.561380\pi\)
−0.191639 + 0.981465i \(0.561380\pi\)
\(830\) −2.17120 −0.0753635
\(831\) −14.3532 −0.497906
\(832\) −6.07042 −0.210454
\(833\) −17.3009 −0.599441
\(834\) 12.1322 0.420104
\(835\) 3.31954 0.114877
\(836\) −4.07758 −0.141026
\(837\) 9.36768 0.323794
\(838\) −22.0804 −0.762755
\(839\) 9.52844 0.328958 0.164479 0.986381i \(-0.447406\pi\)
0.164479 + 0.986381i \(0.447406\pi\)
\(840\) 1.06448 0.0367279
\(841\) −16.1591 −0.557211
\(842\) 23.1422 0.797533
\(843\) −4.93194 −0.169865
\(844\) 0.885119 0.0304671
\(845\) −14.6908 −0.505380
\(846\) −19.7488 −0.678978
\(847\) −3.94041 −0.135394
\(848\) −3.83489 −0.131691
\(849\) 12.9429 0.444200
\(850\) 33.9958 1.16605
\(851\) 13.8035 0.473177
\(852\) −8.54655 −0.292800
\(853\) −16.1384 −0.552568 −0.276284 0.961076i \(-0.589103\pi\)
−0.276284 + 0.961076i \(0.589103\pi\)
\(854\) 23.3853 0.800228
\(855\) 2.16052 0.0738882
\(856\) 12.5752 0.429813
\(857\) 29.8866 1.02091 0.510454 0.859905i \(-0.329477\pi\)
0.510454 + 0.859905i \(0.329477\pi\)
\(858\) −10.6905 −0.364968
\(859\) −47.5137 −1.62115 −0.810574 0.585636i \(-0.800845\pi\)
−0.810574 + 0.585636i \(0.800845\pi\)
\(860\) 1.97283 0.0672730
\(861\) −10.6116 −0.361641
\(862\) 14.6390 0.498606
\(863\) −21.0562 −0.716762 −0.358381 0.933575i \(-0.616671\pi\)
−0.358381 + 0.933575i \(0.616671\pi\)
\(864\) 3.21022 0.109214
\(865\) 3.48460 0.118480
\(866\) −7.87846 −0.267721
\(867\) −20.9840 −0.712655
\(868\) 8.92355 0.302885
\(869\) −21.4345 −0.727117
\(870\) 1.24736 0.0422896
\(871\) 98.1469 3.32558
\(872\) −5.61259 −0.190066
\(873\) 21.3496 0.722575
\(874\) −1.81021 −0.0612312
\(875\) 18.1217 0.612627
\(876\) 6.50419 0.219756
\(877\) −15.2397 −0.514610 −0.257305 0.966330i \(-0.582834\pi\)
−0.257305 + 0.966330i \(0.582834\pi\)
\(878\) −0.803456 −0.0271153
\(879\) 1.59389 0.0537606
\(880\) 1.91956 0.0647082
\(881\) −14.3190 −0.482419 −0.241209 0.970473i \(-0.577544\pi\)
−0.241209 + 0.970473i \(0.577544\pi\)
\(882\) −6.30348 −0.212249
\(883\) 27.2461 0.916903 0.458452 0.888719i \(-0.348404\pi\)
0.458452 + 0.888719i \(0.348404\pi\)
\(884\) 44.6629 1.50218
\(885\) −0.385332 −0.0129528
\(886\) 18.7417 0.629639
\(887\) 27.7062 0.930282 0.465141 0.885237i \(-0.346004\pi\)
0.465141 + 0.885237i \(0.346004\pi\)
\(888\) 5.63843 0.189213
\(889\) −33.4790 −1.12285
\(890\) 6.49777 0.217806
\(891\) −19.4078 −0.650185
\(892\) −14.0423 −0.470170
\(893\) 9.63967 0.322579
\(894\) 7.80674 0.261097
\(895\) 9.28493 0.310361
\(896\) 3.05802 0.102161
\(897\) −4.74595 −0.158463
\(898\) 19.2173 0.641289
\(899\) 10.4567 0.348751
\(900\) 12.3861 0.412871
\(901\) 28.2151 0.939980
\(902\) −19.1357 −0.637148
\(903\) 5.53488 0.184189
\(904\) 15.3358 0.510060
\(905\) −16.4074 −0.545401
\(906\) −4.64432 −0.154297
\(907\) −1.32457 −0.0439817 −0.0219908 0.999758i \(-0.507000\pi\)
−0.0219908 + 0.999758i \(0.507000\pi\)
\(908\) 22.5229 0.747449
\(909\) −4.45170 −0.147653
\(910\) 11.4345 0.379050
\(911\) −35.3152 −1.17004 −0.585022 0.811017i \(-0.698914\pi\)
−0.585022 + 0.811017i \(0.698914\pi\)
\(912\) −0.739431 −0.0244850
\(913\) −10.9846 −0.363537
\(914\) 31.7684 1.05080
\(915\) 2.66194 0.0880010
\(916\) −17.3340 −0.572731
\(917\) −44.7980 −1.47936
\(918\) −23.6191 −0.779546
\(919\) −32.9683 −1.08752 −0.543762 0.839239i \(-0.683001\pi\)
−0.543762 + 0.839239i \(0.683001\pi\)
\(920\) 0.852170 0.0280952
\(921\) −0.495790 −0.0163368
\(922\) −20.9500 −0.689950
\(923\) −91.8063 −3.02184
\(924\) 5.38542 0.177167
\(925\) 46.1018 1.51582
\(926\) 13.6195 0.447563
\(927\) 37.6958 1.23809
\(928\) 3.58342 0.117631
\(929\) 29.6196 0.971787 0.485893 0.874018i \(-0.338494\pi\)
0.485893 + 0.874018i \(0.338494\pi\)
\(930\) 1.01576 0.0333082
\(931\) 3.07681 0.100839
\(932\) −26.5014 −0.868083
\(933\) 6.03471 0.197567
\(934\) −21.5848 −0.706277
\(935\) −14.1231 −0.461873
\(936\) 16.2726 0.531888
\(937\) −9.17199 −0.299636 −0.149818 0.988714i \(-0.547869\pi\)
−0.149818 + 0.988714i \(0.547869\pi\)
\(938\) −49.4422 −1.61435
\(939\) −15.3981 −0.502498
\(940\) −4.53796 −0.148012
\(941\) 39.6871 1.29376 0.646882 0.762590i \(-0.276073\pi\)
0.646882 + 0.762590i \(0.276073\pi\)
\(942\) 0.635133 0.0206938
\(943\) −8.49512 −0.276639
\(944\) −1.10698 −0.0360291
\(945\) −6.04691 −0.196706
\(946\) 9.98098 0.324510
\(947\) 46.2987 1.50450 0.752252 0.658875i \(-0.228968\pi\)
0.752252 + 0.658875i \(0.228968\pi\)
\(948\) −3.88695 −0.126242
\(949\) 69.8674 2.26799
\(950\) −6.04585 −0.196153
\(951\) 14.9237 0.483933
\(952\) −22.4993 −0.729205
\(953\) −24.9769 −0.809081 −0.404541 0.914520i \(-0.632569\pi\)
−0.404541 + 0.914520i \(0.632569\pi\)
\(954\) 10.2800 0.332826
\(955\) −13.6239 −0.440859
\(956\) 20.0415 0.648188
\(957\) 6.31069 0.203996
\(958\) 15.4702 0.499819
\(959\) 14.7503 0.476311
\(960\) 0.348093 0.0112347
\(961\) −22.4848 −0.725316
\(962\) 60.5675 1.95277
\(963\) −33.7098 −1.08628
\(964\) −5.59198 −0.180106
\(965\) 2.54052 0.0817824
\(966\) 2.39081 0.0769230
\(967\) −39.6029 −1.27354 −0.636771 0.771053i \(-0.719730\pi\)
−0.636771 + 0.771053i \(0.719730\pi\)
\(968\) −1.28855 −0.0414156
\(969\) 5.44034 0.174769
\(970\) 4.90579 0.157516
\(971\) −42.4582 −1.36255 −0.681274 0.732029i \(-0.738574\pi\)
−0.681274 + 0.732029i \(0.738574\pi\)
\(972\) −13.1501 −0.421789
\(973\) −65.6512 −2.10468
\(974\) 9.28893 0.297636
\(975\) −15.8509 −0.507634
\(976\) 7.64720 0.244781
\(977\) 48.1765 1.54130 0.770652 0.637257i \(-0.219931\pi\)
0.770652 + 0.637257i \(0.219931\pi\)
\(978\) −12.2337 −0.391189
\(979\) 32.8737 1.05065
\(980\) −1.44844 −0.0462686
\(981\) 15.0454 0.480362
\(982\) −21.0597 −0.672042
\(983\) −57.1523 −1.82288 −0.911438 0.411437i \(-0.865027\pi\)
−0.911438 + 0.411437i \(0.865027\pi\)
\(984\) −3.47008 −0.110622
\(985\) −16.1469 −0.514482
\(986\) −26.3649 −0.839628
\(987\) −12.7315 −0.405248
\(988\) −7.94290 −0.252697
\(989\) 4.43097 0.140897
\(990\) −5.14564 −0.163539
\(991\) 7.28628 0.231456 0.115728 0.993281i \(-0.463080\pi\)
0.115728 + 0.993281i \(0.463080\pi\)
\(992\) 2.91808 0.0926492
\(993\) 18.7968 0.596499
\(994\) 46.2481 1.46690
\(995\) −6.31305 −0.200137
\(996\) −1.99195 −0.0631174
\(997\) −35.6704 −1.12969 −0.564847 0.825196i \(-0.691065\pi\)
−0.564847 + 0.825196i \(0.691065\pi\)
\(998\) −27.9653 −0.885227
\(999\) −32.0299 −1.01338
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 1502.2.a.f.1.7 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1502.2.a.f.1.7 11 1.1 even 1 trivial