Properties

Label 6026.2.a.f.1.14
Level $6026$
Weight $2$
Character 6026.1
Self dual yes
Analytic conductor $48.118$
Analytic rank $1$
Dimension $20$
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] = [6026,2,Mod(1,6026)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6026, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6026.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6026 = 2 \cdot 23 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6026.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: \(48.1178522580\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 5 x^{19} - 17 x^{18} + 115 x^{17} + 78 x^{16} - 1083 x^{15} + 248 x^{14} + 5359 x^{13} - 3723 x^{12} - 14776 x^{11} + 14837 x^{10} + 21886 x^{9} - 28084 x^{8} - 14682 x^{7} + \cdots - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Root \(-0.590088\) of defining polynomial
Character \(\chi\) \(=\) 6026.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.590088 q^{3} +1.00000 q^{4} +0.936829 q^{5} +0.590088 q^{6} +1.40799 q^{7} +1.00000 q^{8} -2.65180 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.590088 q^{3} +1.00000 q^{4} +0.936829 q^{5} +0.590088 q^{6} +1.40799 q^{7} +1.00000 q^{8} -2.65180 q^{9} +0.936829 q^{10} -3.44267 q^{11} +0.590088 q^{12} +2.00043 q^{13} +1.40799 q^{14} +0.552812 q^{15} +1.00000 q^{16} +0.141401 q^{17} -2.65180 q^{18} -6.42748 q^{19} +0.936829 q^{20} +0.830839 q^{21} -3.44267 q^{22} +1.00000 q^{23} +0.590088 q^{24} -4.12235 q^{25} +2.00043 q^{26} -3.33506 q^{27} +1.40799 q^{28} -8.45079 q^{29} +0.552812 q^{30} -6.58167 q^{31} +1.00000 q^{32} -2.03148 q^{33} +0.141401 q^{34} +1.31905 q^{35} -2.65180 q^{36} -2.90009 q^{37} -6.42748 q^{38} +1.18043 q^{39} +0.936829 q^{40} +3.03496 q^{41} +0.830839 q^{42} -0.862799 q^{43} -3.44267 q^{44} -2.48428 q^{45} +1.00000 q^{46} -4.76503 q^{47} +0.590088 q^{48} -5.01756 q^{49} -4.12235 q^{50} +0.0834389 q^{51} +2.00043 q^{52} -5.92958 q^{53} -3.33506 q^{54} -3.22519 q^{55} +1.40799 q^{56} -3.79278 q^{57} -8.45079 q^{58} -3.96755 q^{59} +0.552812 q^{60} +1.48572 q^{61} -6.58167 q^{62} -3.73370 q^{63} +1.00000 q^{64} +1.87406 q^{65} -2.03148 q^{66} +5.37393 q^{67} +0.141401 q^{68} +0.590088 q^{69} +1.31905 q^{70} +14.5679 q^{71} -2.65180 q^{72} +10.7599 q^{73} -2.90009 q^{74} -2.43255 q^{75} -6.42748 q^{76} -4.84724 q^{77} +1.18043 q^{78} -6.15727 q^{79} +0.936829 q^{80} +5.98741 q^{81} +3.03496 q^{82} +10.3262 q^{83} +0.830839 q^{84} +0.132468 q^{85} -0.862799 q^{86} -4.98671 q^{87} -3.44267 q^{88} -2.22548 q^{89} -2.48428 q^{90} +2.81659 q^{91} +1.00000 q^{92} -3.88377 q^{93} -4.76503 q^{94} -6.02145 q^{95} +0.590088 q^{96} -2.07398 q^{97} -5.01756 q^{98} +9.12925 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{2} - 5 q^{3} + 20 q^{4} - 6 q^{5} - 5 q^{6} - 12 q^{7} + 20 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{2} - 5 q^{3} + 20 q^{4} - 6 q^{5} - 5 q^{6} - 12 q^{7} + 20 q^{8} - q^{9} - 6 q^{10} - 3 q^{11} - 5 q^{12} - 13 q^{13} - 12 q^{14} - 10 q^{15} + 20 q^{16} - 14 q^{17} - q^{18} - 21 q^{19} - 6 q^{20} - 8 q^{21} - 3 q^{22} + 20 q^{23} - 5 q^{24} - 14 q^{25} - 13 q^{26} - 5 q^{27} - 12 q^{28} - 27 q^{29} - 10 q^{30} - 27 q^{31} + 20 q^{32} - 12 q^{33} - 14 q^{34} - 23 q^{35} - q^{36} - 19 q^{37} - 21 q^{38} - 35 q^{39} - 6 q^{40} - 17 q^{41} - 8 q^{42} - 27 q^{43} - 3 q^{44} + 4 q^{45} + 20 q^{46} - 28 q^{47} - 5 q^{48} - 10 q^{49} - 14 q^{50} + 6 q^{51} - 13 q^{52} - 47 q^{53} - 5 q^{54} - 4 q^{55} - 12 q^{56} - 16 q^{57} - 27 q^{58} - 16 q^{59} - 10 q^{60} - 9 q^{61} - 27 q^{62} - 9 q^{63} + 20 q^{64} + 9 q^{65} - 12 q^{66} - 8 q^{67} - 14 q^{68} - 5 q^{69} - 23 q^{70} - 30 q^{71} - q^{72} - 26 q^{73} - 19 q^{74} - 18 q^{75} - 21 q^{76} - 50 q^{77} - 35 q^{78} - 35 q^{79} - 6 q^{80} - 60 q^{81} - 17 q^{82} + 2 q^{83} - 8 q^{84} - 62 q^{85} - 27 q^{86} + q^{87} - 3 q^{88} - 25 q^{89} + 4 q^{90} + 22 q^{91} + 20 q^{92} - 21 q^{93} - 28 q^{94} - 14 q^{95} - 5 q^{96} + 2 q^{97} - 10 q^{98} - 5 q^{99}+O(q^{100}) \) 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
\(3\) 0.590088 0.340688 0.170344 0.985385i \(-0.445512\pi\)
0.170344 + 0.985385i \(0.445512\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.936829 0.418963 0.209481 0.977813i \(-0.432822\pi\)
0.209481 + 0.977813i \(0.432822\pi\)
\(6\) 0.590088 0.240903
\(7\) 1.40799 0.532171 0.266085 0.963949i \(-0.414270\pi\)
0.266085 + 0.963949i \(0.414270\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.65180 −0.883932
\(10\) 0.936829 0.296251
\(11\) −3.44267 −1.03800 −0.519001 0.854773i \(-0.673696\pi\)
−0.519001 + 0.854773i \(0.673696\pi\)
\(12\) 0.590088 0.170344
\(13\) 2.00043 0.554820 0.277410 0.960752i \(-0.410524\pi\)
0.277410 + 0.960752i \(0.410524\pi\)
\(14\) 1.40799 0.376301
\(15\) 0.552812 0.142735
\(16\) 1.00000 0.250000
\(17\) 0.141401 0.0342947 0.0171473 0.999853i \(-0.494542\pi\)
0.0171473 + 0.999853i \(0.494542\pi\)
\(18\) −2.65180 −0.625034
\(19\) −6.42748 −1.47456 −0.737282 0.675585i \(-0.763891\pi\)
−0.737282 + 0.675585i \(0.763891\pi\)
\(20\) 0.936829 0.209481
\(21\) 0.830839 0.181304
\(22\) −3.44267 −0.733979
\(23\) 1.00000 0.208514
\(24\) 0.590088 0.120451
\(25\) −4.12235 −0.824470
\(26\) 2.00043 0.392317
\(27\) −3.33506 −0.641832
\(28\) 1.40799 0.266085
\(29\) −8.45079 −1.56927 −0.784636 0.619956i \(-0.787150\pi\)
−0.784636 + 0.619956i \(0.787150\pi\)
\(30\) 0.552812 0.100929
\(31\) −6.58167 −1.18210 −0.591051 0.806634i \(-0.701287\pi\)
−0.591051 + 0.806634i \(0.701287\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.03148 −0.353635
\(34\) 0.141401 0.0242500
\(35\) 1.31905 0.222960
\(36\) −2.65180 −0.441966
\(37\) −2.90009 −0.476772 −0.238386 0.971170i \(-0.576618\pi\)
−0.238386 + 0.971170i \(0.576618\pi\)
\(38\) −6.42748 −1.04267
\(39\) 1.18043 0.189020
\(40\) 0.936829 0.148126
\(41\) 3.03496 0.473981 0.236990 0.971512i \(-0.423839\pi\)
0.236990 + 0.971512i \(0.423839\pi\)
\(42\) 0.830839 0.128201
\(43\) −0.862799 −0.131576 −0.0657878 0.997834i \(-0.520956\pi\)
−0.0657878 + 0.997834i \(0.520956\pi\)
\(44\) −3.44267 −0.519001
\(45\) −2.48428 −0.370335
\(46\) 1.00000 0.147442
\(47\) −4.76503 −0.695052 −0.347526 0.937670i \(-0.612978\pi\)
−0.347526 + 0.937670i \(0.612978\pi\)
\(48\) 0.590088 0.0851719
\(49\) −5.01756 −0.716795
\(50\) −4.12235 −0.582988
\(51\) 0.0834389 0.0116838
\(52\) 2.00043 0.277410
\(53\) −5.92958 −0.814490 −0.407245 0.913319i \(-0.633510\pi\)
−0.407245 + 0.913319i \(0.633510\pi\)
\(54\) −3.33506 −0.453844
\(55\) −3.22519 −0.434885
\(56\) 1.40799 0.188151
\(57\) −3.79278 −0.502366
\(58\) −8.45079 −1.10964
\(59\) −3.96755 −0.516531 −0.258266 0.966074i \(-0.583151\pi\)
−0.258266 + 0.966074i \(0.583151\pi\)
\(60\) 0.552812 0.0713677
\(61\) 1.48572 0.190226 0.0951132 0.995466i \(-0.469679\pi\)
0.0951132 + 0.995466i \(0.469679\pi\)
\(62\) −6.58167 −0.835873
\(63\) −3.73370 −0.470403
\(64\) 1.00000 0.125000
\(65\) 1.87406 0.232449
\(66\) −2.03148 −0.250058
\(67\) 5.37393 0.656530 0.328265 0.944586i \(-0.393536\pi\)
0.328265 + 0.944586i \(0.393536\pi\)
\(68\) 0.141401 0.0171473
\(69\) 0.590088 0.0710383
\(70\) 1.31905 0.157656
\(71\) 14.5679 1.72889 0.864443 0.502731i \(-0.167671\pi\)
0.864443 + 0.502731i \(0.167671\pi\)
\(72\) −2.65180 −0.312517
\(73\) 10.7599 1.25935 0.629677 0.776857i \(-0.283187\pi\)
0.629677 + 0.776857i \(0.283187\pi\)
\(74\) −2.90009 −0.337129
\(75\) −2.43255 −0.280887
\(76\) −6.42748 −0.737282
\(77\) −4.84724 −0.552395
\(78\) 1.18043 0.133658
\(79\) −6.15727 −0.692747 −0.346374 0.938097i \(-0.612587\pi\)
−0.346374 + 0.938097i \(0.612587\pi\)
\(80\) 0.936829 0.104741
\(81\) 5.98741 0.665268
\(82\) 3.03496 0.335155
\(83\) 10.3262 1.13345 0.566724 0.823908i \(-0.308211\pi\)
0.566724 + 0.823908i \(0.308211\pi\)
\(84\) 0.830839 0.0906520
\(85\) 0.132468 0.0143682
\(86\) −0.862799 −0.0930380
\(87\) −4.98671 −0.534632
\(88\) −3.44267 −0.366989
\(89\) −2.22548 −0.235901 −0.117950 0.993019i \(-0.537632\pi\)
−0.117950 + 0.993019i \(0.537632\pi\)
\(90\) −2.48428 −0.261866
\(91\) 2.81659 0.295259
\(92\) 1.00000 0.104257
\(93\) −3.88377 −0.402728
\(94\) −4.76503 −0.491476
\(95\) −6.02145 −0.617788
\(96\) 0.590088 0.0602256
\(97\) −2.07398 −0.210581 −0.105291 0.994441i \(-0.533577\pi\)
−0.105291 + 0.994441i \(0.533577\pi\)
\(98\) −5.01756 −0.506850
\(99\) 9.12925 0.917524
\(100\) −4.12235 −0.412235
\(101\) −3.99612 −0.397629 −0.198815 0.980037i \(-0.563709\pi\)
−0.198815 + 0.980037i \(0.563709\pi\)
\(102\) 0.0834389 0.00826168
\(103\) −9.91039 −0.976499 −0.488250 0.872704i \(-0.662364\pi\)
−0.488250 + 0.872704i \(0.662364\pi\)
\(104\) 2.00043 0.196158
\(105\) 0.778354 0.0759596
\(106\) −5.92958 −0.575931
\(107\) −1.48043 −0.143118 −0.0715592 0.997436i \(-0.522797\pi\)
−0.0715592 + 0.997436i \(0.522797\pi\)
\(108\) −3.33506 −0.320916
\(109\) 12.9418 1.23960 0.619798 0.784761i \(-0.287215\pi\)
0.619798 + 0.784761i \(0.287215\pi\)
\(110\) −3.22519 −0.307510
\(111\) −1.71131 −0.162430
\(112\) 1.40799 0.133043
\(113\) −13.0599 −1.22857 −0.614286 0.789083i \(-0.710556\pi\)
−0.614286 + 0.789083i \(0.710556\pi\)
\(114\) −3.79278 −0.355226
\(115\) 0.936829 0.0873598
\(116\) −8.45079 −0.784636
\(117\) −5.30474 −0.490423
\(118\) −3.96755 −0.365243
\(119\) 0.199091 0.0182506
\(120\) 0.552812 0.0504646
\(121\) 0.851951 0.0774501
\(122\) 1.48572 0.134510
\(123\) 1.79089 0.161479
\(124\) −6.58167 −0.591051
\(125\) −8.54609 −0.764385
\(126\) −3.73370 −0.332625
\(127\) 15.9305 1.41360 0.706800 0.707413i \(-0.250138\pi\)
0.706800 + 0.707413i \(0.250138\pi\)
\(128\) 1.00000 0.0883883
\(129\) −0.509128 −0.0448262
\(130\) 1.87406 0.164366
\(131\) −1.00000 −0.0873704
\(132\) −2.03148 −0.176817
\(133\) −9.04983 −0.784720
\(134\) 5.37393 0.464237
\(135\) −3.12438 −0.268904
\(136\) 0.141401 0.0121250
\(137\) 2.71237 0.231734 0.115867 0.993265i \(-0.463035\pi\)
0.115867 + 0.993265i \(0.463035\pi\)
\(138\) 0.590088 0.0502317
\(139\) −6.50910 −0.552095 −0.276047 0.961144i \(-0.589025\pi\)
−0.276047 + 0.961144i \(0.589025\pi\)
\(140\) 1.31905 0.111480
\(141\) −2.81179 −0.236796
\(142\) 14.5679 1.22251
\(143\) −6.88682 −0.575905
\(144\) −2.65180 −0.220983
\(145\) −7.91695 −0.657467
\(146\) 10.7599 0.890498
\(147\) −2.96080 −0.244203
\(148\) −2.90009 −0.238386
\(149\) −6.06260 −0.496668 −0.248334 0.968674i \(-0.579883\pi\)
−0.248334 + 0.968674i \(0.579883\pi\)
\(150\) −2.43255 −0.198617
\(151\) 10.8693 0.884531 0.442266 0.896884i \(-0.354175\pi\)
0.442266 + 0.896884i \(0.354175\pi\)
\(152\) −6.42748 −0.521337
\(153\) −0.374966 −0.0303142
\(154\) −4.84724 −0.390602
\(155\) −6.16590 −0.495257
\(156\) 1.18043 0.0945102
\(157\) 22.1495 1.76772 0.883862 0.467747i \(-0.154934\pi\)
0.883862 + 0.467747i \(0.154934\pi\)
\(158\) −6.15727 −0.489846
\(159\) −3.49897 −0.277487
\(160\) 0.936829 0.0740629
\(161\) 1.40799 0.110965
\(162\) 5.98741 0.470415
\(163\) −0.433839 −0.0339809 −0.0169904 0.999856i \(-0.505408\pi\)
−0.0169904 + 0.999856i \(0.505408\pi\)
\(164\) 3.03496 0.236990
\(165\) −1.90315 −0.148160
\(166\) 10.3262 0.801468
\(167\) 23.0084 1.78044 0.890220 0.455531i \(-0.150551\pi\)
0.890220 + 0.455531i \(0.150551\pi\)
\(168\) 0.830839 0.0641006
\(169\) −8.99827 −0.692175
\(170\) 0.132468 0.0101599
\(171\) 17.0444 1.30341
\(172\) −0.862799 −0.0657878
\(173\) 3.71260 0.282264 0.141132 0.989991i \(-0.454926\pi\)
0.141132 + 0.989991i \(0.454926\pi\)
\(174\) −4.98671 −0.378042
\(175\) −5.80423 −0.438759
\(176\) −3.44267 −0.259501
\(177\) −2.34121 −0.175976
\(178\) −2.22548 −0.166807
\(179\) 2.42372 0.181157 0.0905787 0.995889i \(-0.471128\pi\)
0.0905787 + 0.995889i \(0.471128\pi\)
\(180\) −2.48428 −0.185167
\(181\) −14.2979 −1.06276 −0.531378 0.847135i \(-0.678326\pi\)
−0.531378 + 0.847135i \(0.678326\pi\)
\(182\) 2.81659 0.208780
\(183\) 0.876704 0.0648078
\(184\) 1.00000 0.0737210
\(185\) −2.71689 −0.199750
\(186\) −3.88377 −0.284771
\(187\) −0.486795 −0.0355980
\(188\) −4.76503 −0.347526
\(189\) −4.69573 −0.341564
\(190\) −6.02145 −0.436842
\(191\) −20.4296 −1.47824 −0.739118 0.673576i \(-0.764757\pi\)
−0.739118 + 0.673576i \(0.764757\pi\)
\(192\) 0.590088 0.0425860
\(193\) −10.8576 −0.781548 −0.390774 0.920487i \(-0.627793\pi\)
−0.390774 + 0.920487i \(0.627793\pi\)
\(194\) −2.07398 −0.148903
\(195\) 1.10586 0.0791925
\(196\) −5.01756 −0.358397
\(197\) 14.0192 0.998829 0.499415 0.866363i \(-0.333548\pi\)
0.499415 + 0.866363i \(0.333548\pi\)
\(198\) 9.12925 0.648787
\(199\) 7.76683 0.550576 0.275288 0.961362i \(-0.411227\pi\)
0.275288 + 0.961362i \(0.411227\pi\)
\(200\) −4.12235 −0.291494
\(201\) 3.17110 0.223672
\(202\) −3.99612 −0.281166
\(203\) −11.8986 −0.835121
\(204\) 0.0834389 0.00584189
\(205\) 2.84324 0.198580
\(206\) −9.91039 −0.690489
\(207\) −2.65180 −0.184313
\(208\) 2.00043 0.138705
\(209\) 22.1277 1.53060
\(210\) 0.778354 0.0537116
\(211\) −11.8639 −0.816746 −0.408373 0.912815i \(-0.633904\pi\)
−0.408373 + 0.912815i \(0.633904\pi\)
\(212\) −5.92958 −0.407245
\(213\) 8.59632 0.589010
\(214\) −1.48043 −0.101200
\(215\) −0.808296 −0.0551253
\(216\) −3.33506 −0.226922
\(217\) −9.26693 −0.629080
\(218\) 12.9418 0.876527
\(219\) 6.34931 0.429046
\(220\) −3.22519 −0.217442
\(221\) 0.282862 0.0190274
\(222\) −1.71131 −0.114856
\(223\) −29.2901 −1.96141 −0.980706 0.195487i \(-0.937371\pi\)
−0.980706 + 0.195487i \(0.937371\pi\)
\(224\) 1.40799 0.0940753
\(225\) 10.9316 0.728775
\(226\) −13.0599 −0.868732
\(227\) 14.0155 0.930244 0.465122 0.885247i \(-0.346010\pi\)
0.465122 + 0.885247i \(0.346010\pi\)
\(228\) −3.79278 −0.251183
\(229\) 16.2255 1.07221 0.536105 0.844152i \(-0.319895\pi\)
0.536105 + 0.844152i \(0.319895\pi\)
\(230\) 0.936829 0.0617727
\(231\) −2.86030 −0.188194
\(232\) −8.45079 −0.554822
\(233\) −24.5063 −1.60546 −0.802729 0.596344i \(-0.796620\pi\)
−0.802729 + 0.596344i \(0.796620\pi\)
\(234\) −5.30474 −0.346781
\(235\) −4.46402 −0.291201
\(236\) −3.96755 −0.258266
\(237\) −3.63334 −0.236010
\(238\) 0.199091 0.0129051
\(239\) −11.1756 −0.722892 −0.361446 0.932393i \(-0.617717\pi\)
−0.361446 + 0.932393i \(0.617717\pi\)
\(240\) 0.552812 0.0356839
\(241\) 12.1844 0.784868 0.392434 0.919780i \(-0.371633\pi\)
0.392434 + 0.919780i \(0.371633\pi\)
\(242\) 0.851951 0.0547655
\(243\) 13.5383 0.868481
\(244\) 1.48572 0.0951132
\(245\) −4.70060 −0.300310
\(246\) 1.79089 0.114183
\(247\) −12.8577 −0.818118
\(248\) −6.58167 −0.417936
\(249\) 6.09337 0.386152
\(250\) −8.54609 −0.540502
\(251\) −16.4845 −1.04049 −0.520247 0.854016i \(-0.674160\pi\)
−0.520247 + 0.854016i \(0.674160\pi\)
\(252\) −3.73370 −0.235201
\(253\) −3.44267 −0.216439
\(254\) 15.9305 0.999566
\(255\) 0.0781680 0.00489507
\(256\) 1.00000 0.0625000
\(257\) −16.5385 −1.03164 −0.515821 0.856696i \(-0.672513\pi\)
−0.515821 + 0.856696i \(0.672513\pi\)
\(258\) −0.509128 −0.0316969
\(259\) −4.08330 −0.253724
\(260\) 1.87406 0.116224
\(261\) 22.4098 1.38713
\(262\) −1.00000 −0.0617802
\(263\) 27.6647 1.70588 0.852940 0.522009i \(-0.174817\pi\)
0.852940 + 0.522009i \(0.174817\pi\)
\(264\) −2.03148 −0.125029
\(265\) −5.55500 −0.341241
\(266\) −9.04983 −0.554881
\(267\) −1.31323 −0.0803685
\(268\) 5.37393 0.328265
\(269\) −22.9212 −1.39753 −0.698766 0.715350i \(-0.746267\pi\)
−0.698766 + 0.715350i \(0.746267\pi\)
\(270\) −3.12438 −0.190144
\(271\) −4.22389 −0.256583 −0.128291 0.991737i \(-0.540949\pi\)
−0.128291 + 0.991737i \(0.540949\pi\)
\(272\) 0.141401 0.00857367
\(273\) 1.66204 0.100591
\(274\) 2.71237 0.163860
\(275\) 14.1919 0.855802
\(276\) 0.590088 0.0355191
\(277\) 10.7621 0.646632 0.323316 0.946291i \(-0.395202\pi\)
0.323316 + 0.946291i \(0.395202\pi\)
\(278\) −6.50910 −0.390390
\(279\) 17.4532 1.04490
\(280\) 1.31905 0.0788282
\(281\) −5.81258 −0.346749 −0.173375 0.984856i \(-0.555467\pi\)
−0.173375 + 0.984856i \(0.555467\pi\)
\(282\) −2.81179 −0.167440
\(283\) 14.4146 0.856858 0.428429 0.903575i \(-0.359067\pi\)
0.428429 + 0.903575i \(0.359067\pi\)
\(284\) 14.5679 0.864443
\(285\) −3.55319 −0.210473
\(286\) −6.88682 −0.407226
\(287\) 4.27319 0.252239
\(288\) −2.65180 −0.156259
\(289\) −16.9800 −0.998824
\(290\) −7.91695 −0.464899
\(291\) −1.22383 −0.0717424
\(292\) 10.7599 0.629677
\(293\) −3.66943 −0.214370 −0.107185 0.994239i \(-0.534184\pi\)
−0.107185 + 0.994239i \(0.534184\pi\)
\(294\) −2.96080 −0.172678
\(295\) −3.71692 −0.216407
\(296\) −2.90009 −0.168564
\(297\) 11.4815 0.666224
\(298\) −6.06260 −0.351197
\(299\) 2.00043 0.115688
\(300\) −2.43255 −0.140443
\(301\) −1.21481 −0.0700207
\(302\) 10.8693 0.625458
\(303\) −2.35807 −0.135467
\(304\) −6.42748 −0.368641
\(305\) 1.39186 0.0796978
\(306\) −0.374966 −0.0214354
\(307\) 5.54957 0.316731 0.158365 0.987381i \(-0.449378\pi\)
0.158365 + 0.987381i \(0.449378\pi\)
\(308\) −4.84724 −0.276197
\(309\) −5.84800 −0.332681
\(310\) −6.16590 −0.350200
\(311\) 24.3526 1.38091 0.690456 0.723375i \(-0.257410\pi\)
0.690456 + 0.723375i \(0.257410\pi\)
\(312\) 1.18043 0.0668288
\(313\) −2.81619 −0.159181 −0.0795903 0.996828i \(-0.525361\pi\)
−0.0795903 + 0.996828i \(0.525361\pi\)
\(314\) 22.1495 1.24997
\(315\) −3.49784 −0.197081
\(316\) −6.15727 −0.346374
\(317\) 5.22230 0.293313 0.146657 0.989187i \(-0.453149\pi\)
0.146657 + 0.989187i \(0.453149\pi\)
\(318\) −3.49897 −0.196213
\(319\) 29.0933 1.62891
\(320\) 0.936829 0.0523704
\(321\) −0.873583 −0.0487587
\(322\) 1.40799 0.0784643
\(323\) −0.908849 −0.0505697
\(324\) 5.98741 0.332634
\(325\) −8.24648 −0.457432
\(326\) −0.433839 −0.0240281
\(327\) 7.63678 0.422315
\(328\) 3.03496 0.167577
\(329\) −6.70912 −0.369886
\(330\) −1.90315 −0.104765
\(331\) 27.1414 1.49183 0.745913 0.666044i \(-0.232014\pi\)
0.745913 + 0.666044i \(0.232014\pi\)
\(332\) 10.3262 0.566724
\(333\) 7.69045 0.421434
\(334\) 23.0084 1.25896
\(335\) 5.03446 0.275062
\(336\) 0.830839 0.0453260
\(337\) −32.4158 −1.76580 −0.882900 0.469562i \(-0.844412\pi\)
−0.882900 + 0.469562i \(0.844412\pi\)
\(338\) −8.99827 −0.489441
\(339\) −7.70650 −0.418560
\(340\) 0.132468 0.00718410
\(341\) 22.6585 1.22703
\(342\) 17.0444 0.921653
\(343\) −16.9206 −0.913627
\(344\) −0.862799 −0.0465190
\(345\) 0.552812 0.0297624
\(346\) 3.71260 0.199590
\(347\) −18.9938 −1.01964 −0.509820 0.860281i \(-0.670288\pi\)
−0.509820 + 0.860281i \(0.670288\pi\)
\(348\) −4.98671 −0.267316
\(349\) 14.2188 0.761113 0.380556 0.924758i \(-0.375733\pi\)
0.380556 + 0.924758i \(0.375733\pi\)
\(350\) −5.80423 −0.310249
\(351\) −6.67156 −0.356101
\(352\) −3.44267 −0.183495
\(353\) −2.04492 −0.108840 −0.0544201 0.998518i \(-0.517331\pi\)
−0.0544201 + 0.998518i \(0.517331\pi\)
\(354\) −2.34121 −0.124434
\(355\) 13.6476 0.724339
\(356\) −2.22548 −0.117950
\(357\) 0.117481 0.00621776
\(358\) 2.42372 0.128098
\(359\) 27.8198 1.46828 0.734138 0.679001i \(-0.237587\pi\)
0.734138 + 0.679001i \(0.237587\pi\)
\(360\) −2.48428 −0.130933
\(361\) 22.3125 1.17434
\(362\) −14.2979 −0.751482
\(363\) 0.502726 0.0263863
\(364\) 2.81659 0.147629
\(365\) 10.0802 0.527622
\(366\) 0.876704 0.0458260
\(367\) −13.4702 −0.703139 −0.351570 0.936162i \(-0.614352\pi\)
−0.351570 + 0.936162i \(0.614352\pi\)
\(368\) 1.00000 0.0521286
\(369\) −8.04808 −0.418967
\(370\) −2.71689 −0.141244
\(371\) −8.34879 −0.433448
\(372\) −3.88377 −0.201364
\(373\) 23.3405 1.20852 0.604262 0.796786i \(-0.293468\pi\)
0.604262 + 0.796786i \(0.293468\pi\)
\(374\) −0.486795 −0.0251716
\(375\) −5.04295 −0.260417
\(376\) −4.76503 −0.245738
\(377\) −16.9052 −0.870664
\(378\) −4.69573 −0.241522
\(379\) −7.47226 −0.383824 −0.191912 0.981412i \(-0.561469\pi\)
−0.191912 + 0.981412i \(0.561469\pi\)
\(380\) −6.02145 −0.308894
\(381\) 9.40038 0.481596
\(382\) −20.4296 −1.04527
\(383\) 4.45858 0.227823 0.113911 0.993491i \(-0.463662\pi\)
0.113911 + 0.993491i \(0.463662\pi\)
\(384\) 0.590088 0.0301128
\(385\) −4.54104 −0.231433
\(386\) −10.8576 −0.552638
\(387\) 2.28797 0.116304
\(388\) −2.07398 −0.105291
\(389\) −19.1214 −0.969491 −0.484746 0.874655i \(-0.661088\pi\)
−0.484746 + 0.874655i \(0.661088\pi\)
\(390\) 1.10586 0.0559976
\(391\) 0.141401 0.00715094
\(392\) −5.01756 −0.253425
\(393\) −0.590088 −0.0297660
\(394\) 14.0192 0.706279
\(395\) −5.76832 −0.290235
\(396\) 9.12925 0.458762
\(397\) −26.9060 −1.35037 −0.675187 0.737647i \(-0.735937\pi\)
−0.675187 + 0.737647i \(0.735937\pi\)
\(398\) 7.76683 0.389316
\(399\) −5.34020 −0.267344
\(400\) −4.12235 −0.206118
\(401\) 24.0408 1.20054 0.600270 0.799798i \(-0.295060\pi\)
0.600270 + 0.799798i \(0.295060\pi\)
\(402\) 3.17110 0.158160
\(403\) −13.1662 −0.655854
\(404\) −3.99612 −0.198815
\(405\) 5.60918 0.278722
\(406\) −11.8986 −0.590519
\(407\) 9.98405 0.494891
\(408\) 0.0834389 0.00413084
\(409\) −6.20525 −0.306830 −0.153415 0.988162i \(-0.549027\pi\)
−0.153415 + 0.988162i \(0.549027\pi\)
\(410\) 2.84324 0.140417
\(411\) 1.60054 0.0789488
\(412\) −9.91039 −0.488250
\(413\) −5.58628 −0.274883
\(414\) −2.65180 −0.130329
\(415\) 9.67389 0.474872
\(416\) 2.00043 0.0980792
\(417\) −3.84095 −0.188092
\(418\) 22.1277 1.08230
\(419\) −31.9967 −1.56314 −0.781571 0.623817i \(-0.785581\pi\)
−0.781571 + 0.623817i \(0.785581\pi\)
\(420\) 0.778354 0.0379798
\(421\) 3.43322 0.167325 0.0836625 0.996494i \(-0.473338\pi\)
0.0836625 + 0.996494i \(0.473338\pi\)
\(422\) −11.8639 −0.577526
\(423\) 12.6359 0.614378
\(424\) −5.92958 −0.287966
\(425\) −0.582903 −0.0282749
\(426\) 8.59632 0.416493
\(427\) 2.09188 0.101233
\(428\) −1.48043 −0.0715592
\(429\) −4.06383 −0.196204
\(430\) −0.808296 −0.0389795
\(431\) 25.7714 1.24136 0.620682 0.784062i \(-0.286856\pi\)
0.620682 + 0.784062i \(0.286856\pi\)
\(432\) −3.33506 −0.160458
\(433\) −27.2835 −1.31116 −0.655581 0.755125i \(-0.727576\pi\)
−0.655581 + 0.755125i \(0.727576\pi\)
\(434\) −9.26693 −0.444827
\(435\) −4.67170 −0.223991
\(436\) 12.9418 0.619798
\(437\) −6.42748 −0.307468
\(438\) 6.34931 0.303382
\(439\) −0.346172 −0.0165219 −0.00826094 0.999966i \(-0.502630\pi\)
−0.00826094 + 0.999966i \(0.502630\pi\)
\(440\) −3.22519 −0.153755
\(441\) 13.3055 0.633598
\(442\) 0.282862 0.0134544
\(443\) 25.3185 1.20292 0.601459 0.798904i \(-0.294586\pi\)
0.601459 + 0.798904i \(0.294586\pi\)
\(444\) −1.71131 −0.0812152
\(445\) −2.08490 −0.0988336
\(446\) −29.2901 −1.38693
\(447\) −3.57747 −0.169209
\(448\) 1.40799 0.0665213
\(449\) 23.8631 1.12617 0.563084 0.826400i \(-0.309615\pi\)
0.563084 + 0.826400i \(0.309615\pi\)
\(450\) 10.9316 0.515322
\(451\) −10.4483 −0.491993
\(452\) −13.0599 −0.614286
\(453\) 6.41385 0.301349
\(454\) 14.0155 0.657782
\(455\) 2.63866 0.123702
\(456\) −3.79278 −0.177613
\(457\) 15.1988 0.710968 0.355484 0.934682i \(-0.384316\pi\)
0.355484 + 0.934682i \(0.384316\pi\)
\(458\) 16.2255 0.758166
\(459\) −0.471579 −0.0220114
\(460\) 0.936829 0.0436799
\(461\) 21.3324 0.993550 0.496775 0.867879i \(-0.334517\pi\)
0.496775 + 0.867879i \(0.334517\pi\)
\(462\) −2.86030 −0.133073
\(463\) −41.1982 −1.91464 −0.957320 0.289029i \(-0.906668\pi\)
−0.957320 + 0.289029i \(0.906668\pi\)
\(464\) −8.45079 −0.392318
\(465\) −3.63843 −0.168728
\(466\) −24.5063 −1.13523
\(467\) 6.68526 0.309357 0.154679 0.987965i \(-0.450566\pi\)
0.154679 + 0.987965i \(0.450566\pi\)
\(468\) −5.30474 −0.245212
\(469\) 7.56645 0.349386
\(470\) −4.46402 −0.205910
\(471\) 13.0702 0.602242
\(472\) −3.96755 −0.182621
\(473\) 2.97033 0.136576
\(474\) −3.63334 −0.166885
\(475\) 26.4963 1.21573
\(476\) 0.199091 0.00912531
\(477\) 15.7240 0.719954
\(478\) −11.1756 −0.511162
\(479\) 25.4479 1.16274 0.581371 0.813639i \(-0.302516\pi\)
0.581371 + 0.813639i \(0.302516\pi\)
\(480\) 0.552812 0.0252323
\(481\) −5.80144 −0.264523
\(482\) 12.1844 0.554986
\(483\) 0.830839 0.0378045
\(484\) 0.851951 0.0387251
\(485\) −1.94297 −0.0882257
\(486\) 13.5383 0.614109
\(487\) −34.7009 −1.57245 −0.786224 0.617941i \(-0.787967\pi\)
−0.786224 + 0.617941i \(0.787967\pi\)
\(488\) 1.48572 0.0672552
\(489\) −0.256003 −0.0115769
\(490\) −4.70060 −0.212351
\(491\) −21.7959 −0.983634 −0.491817 0.870699i \(-0.663667\pi\)
−0.491817 + 0.870699i \(0.663667\pi\)
\(492\) 1.79089 0.0807397
\(493\) −1.19495 −0.0538177
\(494\) −12.8577 −0.578497
\(495\) 8.55255 0.384408
\(496\) −6.58167 −0.295526
\(497\) 20.5114 0.920062
\(498\) 6.09337 0.273050
\(499\) −29.3583 −1.31426 −0.657128 0.753779i \(-0.728229\pi\)
−0.657128 + 0.753779i \(0.728229\pi\)
\(500\) −8.54609 −0.382193
\(501\) 13.5770 0.606574
\(502\) −16.4845 −0.735740
\(503\) −28.0748 −1.25179 −0.625896 0.779906i \(-0.715267\pi\)
−0.625896 + 0.779906i \(0.715267\pi\)
\(504\) −3.73370 −0.166312
\(505\) −3.74369 −0.166592
\(506\) −3.44267 −0.153045
\(507\) −5.30978 −0.235815
\(508\) 15.9305 0.706800
\(509\) −29.7702 −1.31954 −0.659771 0.751466i \(-0.729347\pi\)
−0.659771 + 0.751466i \(0.729347\pi\)
\(510\) 0.0781680 0.00346134
\(511\) 15.1499 0.670191
\(512\) 1.00000 0.0441942
\(513\) 21.4360 0.946423
\(514\) −16.5385 −0.729481
\(515\) −9.28434 −0.409117
\(516\) −0.509128 −0.0224131
\(517\) 16.4044 0.721466
\(518\) −4.08330 −0.179410
\(519\) 2.19076 0.0961637
\(520\) 1.87406 0.0821831
\(521\) −5.61415 −0.245961 −0.122980 0.992409i \(-0.539245\pi\)
−0.122980 + 0.992409i \(0.539245\pi\)
\(522\) 22.4098 0.980849
\(523\) −23.2859 −1.01822 −0.509110 0.860701i \(-0.670025\pi\)
−0.509110 + 0.860701i \(0.670025\pi\)
\(524\) −1.00000 −0.0436852
\(525\) −3.42501 −0.149480
\(526\) 27.6647 1.20624
\(527\) −0.930652 −0.0405398
\(528\) −2.03148 −0.0884087
\(529\) 1.00000 0.0434783
\(530\) −5.55500 −0.241294
\(531\) 10.5211 0.456578
\(532\) −9.04983 −0.392360
\(533\) 6.07122 0.262974
\(534\) −1.31323 −0.0568291
\(535\) −1.38691 −0.0599613
\(536\) 5.37393 0.232119
\(537\) 1.43021 0.0617181
\(538\) −22.9212 −0.988205
\(539\) 17.2738 0.744035
\(540\) −3.12438 −0.134452
\(541\) 23.1219 0.994087 0.497043 0.867726i \(-0.334419\pi\)
0.497043 + 0.867726i \(0.334419\pi\)
\(542\) −4.22389 −0.181432
\(543\) −8.43703 −0.362068
\(544\) 0.141401 0.00606250
\(545\) 12.1242 0.519345
\(546\) 1.66204 0.0711286
\(547\) 2.84341 0.121575 0.0607876 0.998151i \(-0.480639\pi\)
0.0607876 + 0.998151i \(0.480639\pi\)
\(548\) 2.71237 0.115867
\(549\) −3.93982 −0.168147
\(550\) 14.1919 0.605144
\(551\) 54.3173 2.31399
\(552\) 0.590088 0.0251158
\(553\) −8.66939 −0.368660
\(554\) 10.7621 0.457238
\(555\) −1.60321 −0.0680523
\(556\) −6.50910 −0.276047
\(557\) 37.7432 1.59923 0.799616 0.600512i \(-0.205037\pi\)
0.799616 + 0.600512i \(0.205037\pi\)
\(558\) 17.4532 0.738855
\(559\) −1.72597 −0.0730008
\(560\) 1.31905 0.0557399
\(561\) −0.287252 −0.0121278
\(562\) −5.81258 −0.245189
\(563\) −37.6061 −1.58491 −0.792454 0.609932i \(-0.791197\pi\)
−0.792454 + 0.609932i \(0.791197\pi\)
\(564\) −2.81179 −0.118398
\(565\) −12.2349 −0.514726
\(566\) 14.4146 0.605890
\(567\) 8.43022 0.354036
\(568\) 14.5679 0.611254
\(569\) −1.58471 −0.0664344 −0.0332172 0.999448i \(-0.510575\pi\)
−0.0332172 + 0.999448i \(0.510575\pi\)
\(570\) −3.55319 −0.148827
\(571\) −17.4302 −0.729433 −0.364716 0.931119i \(-0.618834\pi\)
−0.364716 + 0.931119i \(0.618834\pi\)
\(572\) −6.88682 −0.287952
\(573\) −12.0553 −0.503617
\(574\) 4.27319 0.178360
\(575\) −4.12235 −0.171914
\(576\) −2.65180 −0.110491
\(577\) −34.6626 −1.44302 −0.721512 0.692402i \(-0.756553\pi\)
−0.721512 + 0.692402i \(0.756553\pi\)
\(578\) −16.9800 −0.706275
\(579\) −6.40695 −0.266264
\(580\) −7.91695 −0.328733
\(581\) 14.5392 0.603187
\(582\) −1.22383 −0.0507295
\(583\) 20.4136 0.845443
\(584\) 10.7599 0.445249
\(585\) −4.96963 −0.205469
\(586\) −3.66943 −0.151583
\(587\) 29.2368 1.20673 0.603366 0.797465i \(-0.293826\pi\)
0.603366 + 0.797465i \(0.293826\pi\)
\(588\) −2.96080 −0.122102
\(589\) 42.3035 1.74309
\(590\) −3.71692 −0.153023
\(591\) 8.27259 0.340289
\(592\) −2.90009 −0.119193
\(593\) −22.5904 −0.927676 −0.463838 0.885920i \(-0.653528\pi\)
−0.463838 + 0.885920i \(0.653528\pi\)
\(594\) 11.4815 0.471091
\(595\) 0.186514 0.00764633
\(596\) −6.06260 −0.248334
\(597\) 4.58312 0.187574
\(598\) 2.00043 0.0818037
\(599\) 25.3413 1.03542 0.517710 0.855556i \(-0.326785\pi\)
0.517710 + 0.855556i \(0.326785\pi\)
\(600\) −2.43255 −0.0993085
\(601\) −28.1890 −1.14985 −0.574926 0.818205i \(-0.694969\pi\)
−0.574926 + 0.818205i \(0.694969\pi\)
\(602\) −1.21481 −0.0495121
\(603\) −14.2506 −0.580328
\(604\) 10.8693 0.442266
\(605\) 0.798133 0.0324487
\(606\) −2.35807 −0.0957899
\(607\) −30.5821 −1.24129 −0.620645 0.784092i \(-0.713129\pi\)
−0.620645 + 0.784092i \(0.713129\pi\)
\(608\) −6.42748 −0.260669
\(609\) −7.02125 −0.284515
\(610\) 1.39186 0.0563549
\(611\) −9.53213 −0.385629
\(612\) −0.374966 −0.0151571
\(613\) −29.1459 −1.17719 −0.588596 0.808428i \(-0.700319\pi\)
−0.588596 + 0.808428i \(0.700319\pi\)
\(614\) 5.54957 0.223963
\(615\) 1.67776 0.0676538
\(616\) −4.84724 −0.195301
\(617\) −4.72723 −0.190311 −0.0951555 0.995462i \(-0.530335\pi\)
−0.0951555 + 0.995462i \(0.530335\pi\)
\(618\) −5.84800 −0.235241
\(619\) −7.98997 −0.321144 −0.160572 0.987024i \(-0.551334\pi\)
−0.160572 + 0.987024i \(0.551334\pi\)
\(620\) −6.16590 −0.247629
\(621\) −3.33506 −0.133831
\(622\) 24.3526 0.976452
\(623\) −3.13346 −0.125539
\(624\) 1.18043 0.0472551
\(625\) 12.6055 0.504221
\(626\) −2.81619 −0.112558
\(627\) 13.0573 0.521457
\(628\) 22.1495 0.883862
\(629\) −0.410075 −0.0163508
\(630\) −3.49784 −0.139357
\(631\) −30.6118 −1.21864 −0.609318 0.792926i \(-0.708557\pi\)
−0.609318 + 0.792926i \(0.708557\pi\)
\(632\) −6.15727 −0.244923
\(633\) −7.00076 −0.278255
\(634\) 5.22230 0.207404
\(635\) 14.9241 0.592246
\(636\) −3.49897 −0.138743
\(637\) −10.0373 −0.397692
\(638\) 29.0933 1.15181
\(639\) −38.6310 −1.52822
\(640\) 0.936829 0.0370314
\(641\) −34.3486 −1.35669 −0.678343 0.734745i \(-0.737302\pi\)
−0.678343 + 0.734745i \(0.737302\pi\)
\(642\) −0.873583 −0.0344776
\(643\) −30.5130 −1.20331 −0.601657 0.798754i \(-0.705493\pi\)
−0.601657 + 0.798754i \(0.705493\pi\)
\(644\) 1.40799 0.0554826
\(645\) −0.476966 −0.0187805
\(646\) −0.908849 −0.0357582
\(647\) −28.0890 −1.10429 −0.552147 0.833747i \(-0.686191\pi\)
−0.552147 + 0.833747i \(0.686191\pi\)
\(648\) 5.98741 0.235208
\(649\) 13.6590 0.536161
\(650\) −8.24648 −0.323454
\(651\) −5.46831 −0.214320
\(652\) −0.433839 −0.0169904
\(653\) 6.21190 0.243090 0.121545 0.992586i \(-0.461215\pi\)
0.121545 + 0.992586i \(0.461215\pi\)
\(654\) 7.63678 0.298622
\(655\) −0.936829 −0.0366050
\(656\) 3.03496 0.118495
\(657\) −28.5331 −1.11318
\(658\) −6.70912 −0.261549
\(659\) 32.4953 1.26584 0.632919 0.774218i \(-0.281857\pi\)
0.632919 + 0.774218i \(0.281857\pi\)
\(660\) −1.90315 −0.0740799
\(661\) −31.0967 −1.20952 −0.604761 0.796407i \(-0.706731\pi\)
−0.604761 + 0.796407i \(0.706731\pi\)
\(662\) 27.1414 1.05488
\(663\) 0.166914 0.00648239
\(664\) 10.3262 0.400734
\(665\) −8.47815 −0.328769
\(666\) 7.69045 0.297999
\(667\) −8.45079 −0.327216
\(668\) 23.0084 0.890220
\(669\) −17.2838 −0.668229
\(670\) 5.03446 0.194498
\(671\) −5.11483 −0.197456
\(672\) 0.830839 0.0320503
\(673\) −46.4303 −1.78976 −0.894879 0.446309i \(-0.852738\pi\)
−0.894879 + 0.446309i \(0.852738\pi\)
\(674\) −32.4158 −1.24861
\(675\) 13.7483 0.529172
\(676\) −8.99827 −0.346087
\(677\) −34.2065 −1.31466 −0.657331 0.753602i \(-0.728315\pi\)
−0.657331 + 0.753602i \(0.728315\pi\)
\(678\) −7.70650 −0.295966
\(679\) −2.92015 −0.112065
\(680\) 0.132468 0.00507993
\(681\) 8.27041 0.316923
\(682\) 22.6585 0.867638
\(683\) 30.5656 1.16956 0.584779 0.811193i \(-0.301181\pi\)
0.584779 + 0.811193i \(0.301181\pi\)
\(684\) 17.0444 0.651707
\(685\) 2.54103 0.0970878
\(686\) −16.9206 −0.646032
\(687\) 9.57446 0.365288
\(688\) −0.862799 −0.0328939
\(689\) −11.8617 −0.451895
\(690\) 0.552812 0.0210452
\(691\) 25.5364 0.971449 0.485725 0.874112i \(-0.338556\pi\)
0.485725 + 0.874112i \(0.338556\pi\)
\(692\) 3.71260 0.141132
\(693\) 12.8539 0.488279
\(694\) −18.9938 −0.720994
\(695\) −6.09792 −0.231307
\(696\) −4.98671 −0.189021
\(697\) 0.429145 0.0162550
\(698\) 14.2188 0.538188
\(699\) −14.4609 −0.546960
\(700\) −5.80423 −0.219379
\(701\) 43.9053 1.65828 0.829140 0.559041i \(-0.188831\pi\)
0.829140 + 0.559041i \(0.188831\pi\)
\(702\) −6.67156 −0.251802
\(703\) 18.6403 0.703032
\(704\) −3.44267 −0.129750
\(705\) −2.63417 −0.0992085
\(706\) −2.04492 −0.0769616
\(707\) −5.62651 −0.211607
\(708\) −2.34121 −0.0879879
\(709\) 43.4728 1.63265 0.816327 0.577590i \(-0.196007\pi\)
0.816327 + 0.577590i \(0.196007\pi\)
\(710\) 13.6476 0.512185
\(711\) 16.3278 0.612342
\(712\) −2.22548 −0.0834035
\(713\) −6.58167 −0.246485
\(714\) 0.117481 0.00439662
\(715\) −6.45178 −0.241283
\(716\) 2.42372 0.0905787
\(717\) −6.59461 −0.246280
\(718\) 27.8198 1.03823
\(719\) −13.5046 −0.503638 −0.251819 0.967774i \(-0.581029\pi\)
−0.251819 + 0.967774i \(0.581029\pi\)
\(720\) −2.48428 −0.0925837
\(721\) −13.9537 −0.519664
\(722\) 22.3125 0.830385
\(723\) 7.18989 0.267395
\(724\) −14.2979 −0.531378
\(725\) 34.8371 1.29382
\(726\) 0.502726 0.0186579
\(727\) −23.0798 −0.855983 −0.427991 0.903783i \(-0.640779\pi\)
−0.427991 + 0.903783i \(0.640779\pi\)
\(728\) 2.81659 0.104390
\(729\) −9.97345 −0.369387
\(730\) 10.0802 0.373085
\(731\) −0.122000 −0.00451235
\(732\) 0.876704 0.0324039
\(733\) 24.7561 0.914389 0.457195 0.889367i \(-0.348854\pi\)
0.457195 + 0.889367i \(0.348854\pi\)
\(734\) −13.4702 −0.497194
\(735\) −2.77377 −0.102312
\(736\) 1.00000 0.0368605
\(737\) −18.5007 −0.681480
\(738\) −8.04808 −0.296254
\(739\) −27.0300 −0.994315 −0.497157 0.867660i \(-0.665623\pi\)
−0.497157 + 0.867660i \(0.665623\pi\)
\(740\) −2.71689 −0.0998749
\(741\) −7.58720 −0.278723
\(742\) −8.34879 −0.306494
\(743\) 23.6428 0.867369 0.433685 0.901065i \(-0.357213\pi\)
0.433685 + 0.901065i \(0.357213\pi\)
\(744\) −3.88377 −0.142386
\(745\) −5.67963 −0.208085
\(746\) 23.3405 0.854555
\(747\) −27.3830 −1.00189
\(748\) −0.486795 −0.0177990
\(749\) −2.08443 −0.0761634
\(750\) −5.04295 −0.184142
\(751\) 43.9382 1.60333 0.801664 0.597775i \(-0.203948\pi\)
0.801664 + 0.597775i \(0.203948\pi\)
\(752\) −4.76503 −0.173763
\(753\) −9.72732 −0.354483
\(754\) −16.9052 −0.615652
\(755\) 10.1827 0.370586
\(756\) −4.69573 −0.170782
\(757\) −21.4475 −0.779522 −0.389761 0.920916i \(-0.627442\pi\)
−0.389761 + 0.920916i \(0.627442\pi\)
\(758\) −7.47226 −0.271405
\(759\) −2.03148 −0.0737380
\(760\) −6.02145 −0.218421
\(761\) 22.5640 0.817945 0.408972 0.912547i \(-0.365887\pi\)
0.408972 + 0.912547i \(0.365887\pi\)
\(762\) 9.40038 0.340540
\(763\) 18.2219 0.659677
\(764\) −20.4296 −0.739118
\(765\) −0.351279 −0.0127005
\(766\) 4.45858 0.161095
\(767\) −7.93682 −0.286582
\(768\) 0.590088 0.0212930
\(769\) 44.8392 1.61694 0.808471 0.588536i \(-0.200296\pi\)
0.808471 + 0.588536i \(0.200296\pi\)
\(770\) −4.54104 −0.163648
\(771\) −9.75916 −0.351468
\(772\) −10.8576 −0.390774
\(773\) 44.4563 1.59898 0.799490 0.600679i \(-0.205103\pi\)
0.799490 + 0.600679i \(0.205103\pi\)
\(774\) 2.28797 0.0822393
\(775\) 27.1319 0.974608
\(776\) −2.07398 −0.0744517
\(777\) −2.40951 −0.0864407
\(778\) −19.1214 −0.685534
\(779\) −19.5071 −0.698915
\(780\) 1.10586 0.0395962
\(781\) −50.1523 −1.79459
\(782\) 0.141401 0.00505648
\(783\) 28.1839 1.00721
\(784\) −5.01756 −0.179199
\(785\) 20.7503 0.740611
\(786\) −0.590088 −0.0210478
\(787\) 52.7211 1.87930 0.939652 0.342133i \(-0.111149\pi\)
0.939652 + 0.342133i \(0.111149\pi\)
\(788\) 14.0192 0.499415
\(789\) 16.3246 0.581172
\(790\) −5.76832 −0.205227
\(791\) −18.3882 −0.653810
\(792\) 9.12925 0.324394
\(793\) 2.97207 0.105541
\(794\) −26.9060 −0.954858
\(795\) −3.27794 −0.116257
\(796\) 7.76683 0.275288
\(797\) 11.2097 0.397068 0.198534 0.980094i \(-0.436382\pi\)
0.198534 + 0.980094i \(0.436382\pi\)
\(798\) −5.34020 −0.189041
\(799\) −0.673779 −0.0238366
\(800\) −4.12235 −0.145747
\(801\) 5.90153 0.208520
\(802\) 24.0408 0.848910
\(803\) −37.0428 −1.30721
\(804\) 3.17110 0.111836
\(805\) 1.31905 0.0464903
\(806\) −13.1662 −0.463759
\(807\) −13.5256 −0.476122
\(808\) −3.99612 −0.140583
\(809\) 22.7509 0.799878 0.399939 0.916542i \(-0.369031\pi\)
0.399939 + 0.916542i \(0.369031\pi\)
\(810\) 5.60918 0.197087
\(811\) −34.2471 −1.20258 −0.601289 0.799031i \(-0.705346\pi\)
−0.601289 + 0.799031i \(0.705346\pi\)
\(812\) −11.8986 −0.417560
\(813\) −2.49247 −0.0874146
\(814\) 9.98405 0.349941
\(815\) −0.406433 −0.0142367
\(816\) 0.0834389 0.00292094
\(817\) 5.54562 0.194017
\(818\) −6.20525 −0.216961
\(819\) −7.46902 −0.260989
\(820\) 2.84324 0.0992901
\(821\) 0.758459 0.0264704 0.0132352 0.999912i \(-0.495787\pi\)
0.0132352 + 0.999912i \(0.495787\pi\)
\(822\) 1.60054 0.0558252
\(823\) 42.7619 1.49059 0.745293 0.666737i \(-0.232310\pi\)
0.745293 + 0.666737i \(0.232310\pi\)
\(824\) −9.91039 −0.345245
\(825\) 8.37446 0.291561
\(826\) −5.58628 −0.194371
\(827\) −36.5280 −1.27020 −0.635102 0.772428i \(-0.719042\pi\)
−0.635102 + 0.772428i \(0.719042\pi\)
\(828\) −2.65180 −0.0921563
\(829\) 27.7964 0.965408 0.482704 0.875783i \(-0.339655\pi\)
0.482704 + 0.875783i \(0.339655\pi\)
\(830\) 9.67389 0.335786
\(831\) 6.35059 0.220299
\(832\) 2.00043 0.0693525
\(833\) −0.709486 −0.0245822
\(834\) −3.84095 −0.133001
\(835\) 21.5549 0.745938
\(836\) 22.1277 0.765301
\(837\) 21.9502 0.758712
\(838\) −31.9967 −1.10531
\(839\) −14.0685 −0.485700 −0.242850 0.970064i \(-0.578082\pi\)
−0.242850 + 0.970064i \(0.578082\pi\)
\(840\) 0.778354 0.0268558
\(841\) 42.4159 1.46262
\(842\) 3.43322 0.118317
\(843\) −3.42993 −0.118133
\(844\) −11.8639 −0.408373
\(845\) −8.42985 −0.289996
\(846\) 12.6359 0.434431
\(847\) 1.19954 0.0412167
\(848\) −5.92958 −0.203623
\(849\) 8.50588 0.291921
\(850\) −0.582903 −0.0199934
\(851\) −2.90009 −0.0994139
\(852\) 8.59632 0.294505
\(853\) −4.51815 −0.154698 −0.0773492 0.997004i \(-0.524646\pi\)
−0.0773492 + 0.997004i \(0.524646\pi\)
\(854\) 2.09188 0.0715825
\(855\) 15.9677 0.546082
\(856\) −1.48043 −0.0506000
\(857\) −38.4586 −1.31372 −0.656861 0.754012i \(-0.728116\pi\)
−0.656861 + 0.754012i \(0.728116\pi\)
\(858\) −4.06383 −0.138737
\(859\) 7.79124 0.265834 0.132917 0.991127i \(-0.457566\pi\)
0.132917 + 0.991127i \(0.457566\pi\)
\(860\) −0.808296 −0.0275627
\(861\) 2.52156 0.0859345
\(862\) 25.7714 0.877777
\(863\) 45.7539 1.55748 0.778740 0.627347i \(-0.215859\pi\)
0.778740 + 0.627347i \(0.215859\pi\)
\(864\) −3.33506 −0.113461
\(865\) 3.47807 0.118258
\(866\) −27.2835 −0.927132
\(867\) −10.0197 −0.340287
\(868\) −9.26693 −0.314540
\(869\) 21.1974 0.719074
\(870\) −4.67170 −0.158385
\(871\) 10.7502 0.364256
\(872\) 12.9418 0.438263
\(873\) 5.49978 0.186139
\(874\) −6.42748 −0.217413
\(875\) −12.0328 −0.406783
\(876\) 6.34931 0.214523
\(877\) 17.3672 0.586449 0.293225 0.956044i \(-0.405272\pi\)
0.293225 + 0.956044i \(0.405272\pi\)
\(878\) −0.346172 −0.0116827
\(879\) −2.16529 −0.0730334
\(880\) −3.22519 −0.108721
\(881\) 12.9803 0.437319 0.218659 0.975801i \(-0.429832\pi\)
0.218659 + 0.975801i \(0.429832\pi\)
\(882\) 13.3055 0.448021
\(883\) −21.2687 −0.715748 −0.357874 0.933770i \(-0.616498\pi\)
−0.357874 + 0.933770i \(0.616498\pi\)
\(884\) 0.282862 0.00951369
\(885\) −2.19331 −0.0737273
\(886\) 25.3185 0.850591
\(887\) 2.61349 0.0877524 0.0438762 0.999037i \(-0.486029\pi\)
0.0438762 + 0.999037i \(0.486029\pi\)
\(888\) −1.71131 −0.0574278
\(889\) 22.4299 0.752276
\(890\) −2.08490 −0.0698859
\(891\) −20.6126 −0.690550
\(892\) −29.2901 −0.980706
\(893\) 30.6272 1.02490
\(894\) −3.57747 −0.119649
\(895\) 2.27061 0.0758983
\(896\) 1.40799 0.0470377
\(897\) 1.18043 0.0394135
\(898\) 23.8631 0.796321
\(899\) 55.6203 1.85504
\(900\) 10.9316 0.364388
\(901\) −0.838446 −0.0279327
\(902\) −10.4483 −0.347892
\(903\) −0.716847 −0.0238552
\(904\) −13.0599 −0.434366
\(905\) −13.3947 −0.445255
\(906\) 6.41385 0.213086
\(907\) 35.6302 1.18308 0.591540 0.806275i \(-0.298520\pi\)
0.591540 + 0.806275i \(0.298520\pi\)
\(908\) 14.0155 0.465122
\(909\) 10.5969 0.351477
\(910\) 2.63866 0.0874709
\(911\) 3.27725 0.108580 0.0542900 0.998525i \(-0.482710\pi\)
0.0542900 + 0.998525i \(0.482710\pi\)
\(912\) −3.79278 −0.125592
\(913\) −35.5497 −1.17652
\(914\) 15.1988 0.502730
\(915\) 0.821322 0.0271521
\(916\) 16.2255 0.536105
\(917\) −1.40799 −0.0464960
\(918\) −0.471579 −0.0155644
\(919\) 18.4871 0.609832 0.304916 0.952379i \(-0.401372\pi\)
0.304916 + 0.952379i \(0.401372\pi\)
\(920\) 0.936829 0.0308864
\(921\) 3.27474 0.107906
\(922\) 21.3324 0.702546
\(923\) 29.1420 0.959221
\(924\) −2.86030 −0.0940970
\(925\) 11.9552 0.393084
\(926\) −41.1982 −1.35386
\(927\) 26.2803 0.863159
\(928\) −8.45079 −0.277411
\(929\) 53.7010 1.76187 0.880937 0.473234i \(-0.156913\pi\)
0.880937 + 0.473234i \(0.156913\pi\)
\(930\) −3.63843 −0.119309
\(931\) 32.2503 1.05696
\(932\) −24.5063 −0.802729
\(933\) 14.3702 0.470459
\(934\) 6.68526 0.218749
\(935\) −0.456044 −0.0149142
\(936\) −5.30474 −0.173391
\(937\) 0.0532100 0.00173829 0.000869147 1.00000i \(-0.499723\pi\)
0.000869147 1.00000i \(0.499723\pi\)
\(938\) 7.56645 0.247053
\(939\) −1.66180 −0.0542309
\(940\) −4.46402 −0.145600
\(941\) −51.3211 −1.67302 −0.836510 0.547952i \(-0.815408\pi\)
−0.836510 + 0.547952i \(0.815408\pi\)
\(942\) 13.0702 0.425849
\(943\) 3.03496 0.0988318
\(944\) −3.96755 −0.129133
\(945\) −4.39910 −0.143103
\(946\) 2.97033 0.0965738
\(947\) −4.21479 −0.136962 −0.0684812 0.997652i \(-0.521815\pi\)
−0.0684812 + 0.997652i \(0.521815\pi\)
\(948\) −3.63334 −0.118005
\(949\) 21.5245 0.698715
\(950\) 26.4963 0.859654
\(951\) 3.08162 0.0999282
\(952\) 0.199091 0.00645257
\(953\) −39.7049 −1.28617 −0.643083 0.765796i \(-0.722345\pi\)
−0.643083 + 0.765796i \(0.722345\pi\)
\(954\) 15.7240 0.509084
\(955\) −19.1391 −0.619326
\(956\) −11.1756 −0.361446
\(957\) 17.1676 0.554949
\(958\) 25.4479 0.822183
\(959\) 3.81899 0.123322
\(960\) 0.552812 0.0178419
\(961\) 12.3184 0.397366
\(962\) −5.80144 −0.187046
\(963\) 3.92579 0.126507
\(964\) 12.1844 0.392434
\(965\) −10.1717 −0.327439
\(966\) 0.830839 0.0267318
\(967\) −24.0568 −0.773614 −0.386807 0.922161i \(-0.626422\pi\)
−0.386807 + 0.922161i \(0.626422\pi\)
\(968\) 0.851951 0.0273827
\(969\) −0.536301 −0.0172285
\(970\) −1.94297 −0.0623850
\(971\) 24.9454 0.800535 0.400267 0.916398i \(-0.368917\pi\)
0.400267 + 0.916398i \(0.368917\pi\)
\(972\) 13.5383 0.434240
\(973\) −9.16476 −0.293809
\(974\) −34.7009 −1.11189
\(975\) −4.86615 −0.155842
\(976\) 1.48572 0.0475566
\(977\) −9.20105 −0.294368 −0.147184 0.989109i \(-0.547021\pi\)
−0.147184 + 0.989109i \(0.547021\pi\)
\(978\) −0.256003 −0.00818608
\(979\) 7.66159 0.244866
\(980\) −4.70060 −0.150155
\(981\) −34.3189 −1.09572
\(982\) −21.7959 −0.695534
\(983\) −8.26546 −0.263627 −0.131814 0.991275i \(-0.542080\pi\)
−0.131814 + 0.991275i \(0.542080\pi\)
\(984\) 1.79089 0.0570916
\(985\) 13.1336 0.418472
\(986\) −1.19495 −0.0380549
\(987\) −3.95898 −0.126016
\(988\) −12.8577 −0.409059
\(989\) −0.862799 −0.0274354
\(990\) 8.55255 0.271818
\(991\) 14.8110 0.470487 0.235243 0.971937i \(-0.424411\pi\)
0.235243 + 0.971937i \(0.424411\pi\)
\(992\) −6.58167 −0.208968
\(993\) 16.0158 0.508246
\(994\) 20.5114 0.650582
\(995\) 7.27619 0.230671
\(996\) 6.09337 0.193076
\(997\) −19.4318 −0.615412 −0.307706 0.951481i \(-0.599561\pi\)
−0.307706 + 0.951481i \(0.599561\pi\)
\(998\) −29.3583 −0.929320
\(999\) 9.67198 0.306008
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 6026.2.a.f.1.14 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6026.2.a.f.1.14 20 1.1 even 1 trivial