Properties

Label 2695.2.a.l.1.1
Level $2695$
Weight $2$
Character 2695.1
Self dual yes
Analytic conductor $21.520$
Analytic rank $0$
Dimension $4$
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] = [2695,2,Mod(1,2695)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2695, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2695.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2695 = 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2695.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: \(21.5196833447\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11348.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 385)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.723742\) of defining polynomial
Character \(\chi\) \(=\) 2695.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.03967 q^{2} +2.19994 q^{3} +2.16027 q^{4} -1.00000 q^{5} -4.48716 q^{6} -0.326891 q^{8} +1.83973 q^{9} +O(q^{10})\) \(q-2.03967 q^{2} +2.19994 q^{3} +2.16027 q^{4} -1.00000 q^{5} -4.48716 q^{6} -0.326891 q^{8} +1.83973 q^{9} +2.03967 q^{10} -1.00000 q^{11} +4.75246 q^{12} +5.64742 q^{13} -2.19994 q^{15} -3.65378 q^{16} +4.19994 q^{17} -3.75246 q^{18} -7.52683 q^{19} -2.16027 q^{20} +2.03967 q^{22} +7.79213 q^{23} -0.719140 q^{24} +1.00000 q^{25} -11.5189 q^{26} -2.55252 q^{27} +0.894968 q^{29} +4.48716 q^{30} +5.11902 q^{31} +8.10630 q^{32} -2.19994 q^{33} -8.56650 q^{34} +3.97431 q^{36} +0.287218 q^{37} +15.3523 q^{38} +12.4240 q^{39} +0.326891 q^{40} -5.43955 q^{41} +2.68710 q^{43} -2.16027 q^{44} -1.83973 q^{45} -15.8934 q^{46} -7.00636 q^{47} -8.03810 q^{48} -2.03967 q^{50} +9.23961 q^{51} +12.1999 q^{52} +13.6614 q^{53} +5.20630 q^{54} +1.00000 q^{55} -16.5586 q^{57} -1.82544 q^{58} -12.5446 q^{59} -4.75246 q^{60} +6.48716 q^{61} -10.4411 q^{62} -9.22664 q^{64} -5.64742 q^{65} +4.48716 q^{66} -11.5509 q^{67} +9.07299 q^{68} +17.1422 q^{69} +8.00000 q^{71} -0.601392 q^{72} +2.69503 q^{73} -0.585831 q^{74} +2.19994 q^{75} -16.2600 q^{76} -25.3409 q^{78} +12.1346 q^{79} +3.65378 q^{80} -11.1346 q^{81} +11.0949 q^{82} +3.12695 q^{83} -4.19994 q^{85} -5.48080 q^{86} +1.96888 q^{87} +0.326891 q^{88} +3.44748 q^{89} +3.75246 q^{90} +16.8331 q^{92} +11.2615 q^{93} +14.2907 q^{94} +7.52683 q^{95} +17.8334 q^{96} -2.34245 q^{97} -1.83973 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} + 12 q^{8} + 8 q^{9} - 2 q^{10} - 4 q^{11} + 12 q^{12} + 8 q^{13} + 2 q^{15} + 12 q^{16} + 6 q^{17} - 8 q^{18} - 6 q^{19} - 8 q^{20} - 2 q^{22} + 14 q^{23} + 6 q^{24} + 4 q^{25} + 6 q^{26} - 14 q^{27} - 4 q^{29} + 4 q^{30} - 10 q^{31} + 26 q^{32} + 2 q^{33} - 12 q^{36} - 2 q^{37} + 22 q^{38} + 16 q^{39} - 12 q^{40} + 10 q^{41} - 14 q^{43} - 8 q^{44} - 8 q^{45} - 16 q^{46} - 16 q^{47} + 18 q^{48} + 2 q^{50} + 16 q^{51} + 38 q^{52} + 2 q^{53} - 2 q^{54} + 4 q^{55} - 4 q^{57} + 8 q^{58} - 26 q^{59} - 12 q^{60} + 12 q^{61} - 50 q^{62} + 36 q^{64} - 8 q^{65} + 4 q^{66} - 10 q^{67} + 28 q^{68} + 14 q^{69} + 32 q^{71} - 10 q^{72} + 14 q^{73} - 8 q^{74} - 2 q^{75} + 6 q^{76} - 50 q^{78} + 20 q^{79} - 12 q^{80} - 16 q^{81} + 26 q^{82} + 10 q^{83} - 6 q^{85} - 20 q^{86} + 4 q^{87} - 12 q^{88} + 10 q^{89} + 8 q^{90} + 26 q^{92} + 14 q^{93} + 38 q^{94} + 6 q^{95} + 84 q^{96} + 2 q^{97} - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.03967 −1.44227 −0.721133 0.692796i \(-0.756379\pi\)
−0.721133 + 0.692796i \(0.756379\pi\)
\(3\) 2.19994 1.27014 0.635068 0.772456i \(-0.280972\pi\)
0.635068 + 0.772456i \(0.280972\pi\)
\(4\) 2.16027 1.08013
\(5\) −1.00000 −0.447214
\(6\) −4.48716 −1.83187
\(7\) 0 0
\(8\) −0.326891 −0.115573
\(9\) 1.83973 0.613245
\(10\) 2.03967 0.645001
\(11\) −1.00000 −0.301511
\(12\) 4.75246 1.37192
\(13\) 5.64742 1.56631 0.783157 0.621824i \(-0.213608\pi\)
0.783157 + 0.621824i \(0.213608\pi\)
\(14\) 0 0
\(15\) −2.19994 −0.568022
\(16\) −3.65378 −0.913445
\(17\) 4.19994 1.01863 0.509317 0.860579i \(-0.329898\pi\)
0.509317 + 0.860579i \(0.329898\pi\)
\(18\) −3.75246 −0.884462
\(19\) −7.52683 −1.72677 −0.863387 0.504543i \(-0.831661\pi\)
−0.863387 + 0.504543i \(0.831661\pi\)
\(20\) −2.16027 −0.483050
\(21\) 0 0
\(22\) 2.03967 0.434860
\(23\) 7.79213 1.62477 0.812386 0.583121i \(-0.198169\pi\)
0.812386 + 0.583121i \(0.198169\pi\)
\(24\) −0.719140 −0.146794
\(25\) 1.00000 0.200000
\(26\) −11.5189 −2.25904
\(27\) −2.55252 −0.491232
\(28\) 0 0
\(29\) 0.894968 0.166191 0.0830957 0.996542i \(-0.473519\pi\)
0.0830957 + 0.996542i \(0.473519\pi\)
\(30\) 4.48716 0.819239
\(31\) 5.11902 0.919403 0.459702 0.888073i \(-0.347956\pi\)
0.459702 + 0.888073i \(0.347956\pi\)
\(32\) 8.10630 1.43301
\(33\) −2.19994 −0.382960
\(34\) −8.56650 −1.46914
\(35\) 0 0
\(36\) 3.97431 0.662386
\(37\) 0.287218 0.0472183 0.0236092 0.999721i \(-0.492484\pi\)
0.0236092 + 0.999721i \(0.492484\pi\)
\(38\) 15.3523 2.49047
\(39\) 12.4240 1.98943
\(40\) 0.326891 0.0516860
\(41\) −5.43955 −0.849515 −0.424758 0.905307i \(-0.639641\pi\)
−0.424758 + 0.905307i \(0.639641\pi\)
\(42\) 0 0
\(43\) 2.68710 0.409778 0.204889 0.978785i \(-0.434317\pi\)
0.204889 + 0.978785i \(0.434317\pi\)
\(44\) −2.16027 −0.325672
\(45\) −1.83973 −0.274251
\(46\) −15.8934 −2.34335
\(47\) −7.00636 −1.02198 −0.510991 0.859586i \(-0.670722\pi\)
−0.510991 + 0.859586i \(0.670722\pi\)
\(48\) −8.03810 −1.16020
\(49\) 0 0
\(50\) −2.03967 −0.288453
\(51\) 9.23961 1.29380
\(52\) 12.1999 1.69183
\(53\) 13.6614 1.87654 0.938270 0.345905i \(-0.112428\pi\)
0.938270 + 0.345905i \(0.112428\pi\)
\(54\) 5.20630 0.708487
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −16.5586 −2.19324
\(58\) −1.82544 −0.239692
\(59\) −12.5446 −1.63317 −0.816583 0.577228i \(-0.804134\pi\)
−0.816583 + 0.577228i \(0.804134\pi\)
\(60\) −4.75246 −0.613539
\(61\) 6.48716 0.830595 0.415298 0.909686i \(-0.363677\pi\)
0.415298 + 0.909686i \(0.363677\pi\)
\(62\) −10.4411 −1.32602
\(63\) 0 0
\(64\) −9.22664 −1.15333
\(65\) −5.64742 −0.700477
\(66\) 4.48716 0.552331
\(67\) −11.5509 −1.41117 −0.705586 0.708624i \(-0.749316\pi\)
−0.705586 + 0.708624i \(0.749316\pi\)
\(68\) 9.07299 1.10026
\(69\) 17.1422 2.06368
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −0.601392 −0.0708748
\(73\) 2.69503 0.315429 0.157715 0.987485i \(-0.449587\pi\)
0.157715 + 0.987485i \(0.449587\pi\)
\(74\) −0.585831 −0.0681014
\(75\) 2.19994 0.254027
\(76\) −16.2600 −1.86515
\(77\) 0 0
\(78\) −25.3409 −2.86929
\(79\) 12.1346 1.36525 0.682624 0.730770i \(-0.260839\pi\)
0.682624 + 0.730770i \(0.260839\pi\)
\(80\) 3.65378 0.408505
\(81\) −11.1346 −1.23718
\(82\) 11.0949 1.22523
\(83\) 3.12695 0.343228 0.171614 0.985164i \(-0.445102\pi\)
0.171614 + 0.985164i \(0.445102\pi\)
\(84\) 0 0
\(85\) −4.19994 −0.455547
\(86\) −5.48080 −0.591010
\(87\) 1.96888 0.211086
\(88\) 0.326891 0.0348467
\(89\) 3.44748 0.365433 0.182716 0.983166i \(-0.441511\pi\)
0.182716 + 0.983166i \(0.441511\pi\)
\(90\) 3.75246 0.395544
\(91\) 0 0
\(92\) 16.8331 1.75497
\(93\) 11.2615 1.16777
\(94\) 14.2907 1.47397
\(95\) 7.52683 0.772237
\(96\) 17.8334 1.82011
\(97\) −2.34245 −0.237840 −0.118920 0.992904i \(-0.537943\pi\)
−0.118920 + 0.992904i \(0.537943\pi\)
\(98\) 0 0
\(99\) −1.83973 −0.184900
\(100\) 2.16027 0.216027
\(101\) 19.2777 1.91820 0.959102 0.283061i \(-0.0913498\pi\)
0.959102 + 0.283061i \(0.0913498\pi\)
\(102\) −18.8458 −1.86601
\(103\) −14.7585 −1.45420 −0.727100 0.686532i \(-0.759132\pi\)
−0.727100 + 0.686532i \(0.759132\pi\)
\(104\) −1.84609 −0.181024
\(105\) 0 0
\(106\) −27.8648 −2.70647
\(107\) 6.47923 0.626370 0.313185 0.949692i \(-0.398604\pi\)
0.313185 + 0.949692i \(0.398604\pi\)
\(108\) −5.51411 −0.530596
\(109\) 2.39988 0.229867 0.114933 0.993373i \(-0.463335\pi\)
0.114933 + 0.993373i \(0.463335\pi\)
\(110\) −2.03967 −0.194475
\(111\) 0.631862 0.0599737
\(112\) 0 0
\(113\) 14.3999 1.35463 0.677313 0.735695i \(-0.263144\pi\)
0.677313 + 0.735695i \(0.263144\pi\)
\(114\) 33.7741 3.16323
\(115\) −7.79213 −0.726620
\(116\) 1.93337 0.179509
\(117\) 10.3898 0.960533
\(118\) 25.5868 2.35546
\(119\) 0 0
\(120\) 0.719140 0.0656482
\(121\) 1.00000 0.0909091
\(122\) −13.2317 −1.19794
\(123\) −11.9667 −1.07900
\(124\) 11.0584 0.993078
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 6.07935 0.539455 0.269727 0.962937i \(-0.413066\pi\)
0.269727 + 0.962937i \(0.413066\pi\)
\(128\) 2.60673 0.230405
\(129\) 5.91145 0.520474
\(130\) 11.5189 1.01027
\(131\) 13.9486 1.21870 0.609349 0.792902i \(-0.291431\pi\)
0.609349 + 0.792902i \(0.291431\pi\)
\(132\) −4.75246 −0.413648
\(133\) 0 0
\(134\) 23.5601 2.03529
\(135\) 2.55252 0.219686
\(136\) −1.37292 −0.117727
\(137\) 13.0870 1.11810 0.559048 0.829135i \(-0.311167\pi\)
0.559048 + 0.829135i \(0.311167\pi\)
\(138\) −34.9645 −2.97638
\(139\) 0.863845 0.0732704 0.0366352 0.999329i \(-0.488336\pi\)
0.0366352 + 0.999329i \(0.488336\pi\)
\(140\) 0 0
\(141\) −15.4136 −1.29806
\(142\) −16.3174 −1.36932
\(143\) −5.64742 −0.472261
\(144\) −6.72199 −0.560165
\(145\) −0.894968 −0.0743231
\(146\) −5.49698 −0.454933
\(147\) 0 0
\(148\) 0.620467 0.0510021
\(149\) 7.43828 0.609368 0.304684 0.952454i \(-0.401449\pi\)
0.304684 + 0.952454i \(0.401449\pi\)
\(150\) −4.48716 −0.366375
\(151\) −4.37577 −0.356095 −0.178047 0.984022i \(-0.556978\pi\)
−0.178047 + 0.984022i \(0.556978\pi\)
\(152\) 2.46045 0.199569
\(153\) 7.72677 0.624672
\(154\) 0 0
\(155\) −5.11902 −0.411170
\(156\) 26.8391 2.14885
\(157\) −13.7522 −1.09754 −0.548771 0.835973i \(-0.684904\pi\)
−0.548771 + 0.835973i \(0.684904\pi\)
\(158\) −24.7506 −1.96905
\(159\) 30.0543 2.38346
\(160\) −8.10630 −0.640860
\(161\) 0 0
\(162\) 22.7109 1.78434
\(163\) −9.07172 −0.710552 −0.355276 0.934762i \(-0.615613\pi\)
−0.355276 + 0.934762i \(0.615613\pi\)
\(164\) −11.7509 −0.917590
\(165\) 2.19994 0.171265
\(166\) −6.37796 −0.495026
\(167\) 8.90102 0.688782 0.344391 0.938826i \(-0.388085\pi\)
0.344391 + 0.938826i \(0.388085\pi\)
\(168\) 0 0
\(169\) 18.8934 1.45334
\(170\) 8.56650 0.657021
\(171\) −13.8474 −1.05893
\(172\) 5.80484 0.442615
\(173\) 10.9204 0.830259 0.415129 0.909762i \(-0.363736\pi\)
0.415129 + 0.909762i \(0.363736\pi\)
\(174\) −4.01586 −0.304442
\(175\) 0 0
\(176\) 3.65378 0.275414
\(177\) −27.5973 −2.07434
\(178\) −7.03174 −0.527051
\(179\) −11.1203 −0.831170 −0.415585 0.909554i \(-0.636423\pi\)
−0.415585 + 0.909554i \(0.636423\pi\)
\(180\) −3.97431 −0.296228
\(181\) −14.1679 −1.05309 −0.526546 0.850147i \(-0.676513\pi\)
−0.526546 + 0.850147i \(0.676513\pi\)
\(182\) 0 0
\(183\) 14.2714 1.05497
\(184\) −2.54718 −0.187780
\(185\) −0.287218 −0.0211167
\(186\) −22.9698 −1.68423
\(187\) −4.19994 −0.307130
\(188\) −15.1356 −1.10388
\(189\) 0 0
\(190\) −15.3523 −1.11377
\(191\) −8.55857 −0.619276 −0.309638 0.950854i \(-0.600208\pi\)
−0.309638 + 0.950854i \(0.600208\pi\)
\(192\) −20.2981 −1.46489
\(193\) 4.97651 0.358217 0.179108 0.983829i \(-0.442679\pi\)
0.179108 + 0.983829i \(0.442679\pi\)
\(194\) 4.77784 0.343029
\(195\) −12.4240 −0.889700
\(196\) 0 0
\(197\) 3.07172 0.218851 0.109425 0.993995i \(-0.465099\pi\)
0.109425 + 0.993995i \(0.465099\pi\)
\(198\) 3.75246 0.266675
\(199\) −12.3126 −0.872817 −0.436409 0.899749i \(-0.643750\pi\)
−0.436409 + 0.899749i \(0.643750\pi\)
\(200\) −0.326891 −0.0231147
\(201\) −25.4114 −1.79238
\(202\) −39.3202 −2.76656
\(203\) 0 0
\(204\) 19.9600 1.39748
\(205\) 5.43955 0.379915
\(206\) 30.1025 2.09734
\(207\) 14.3354 0.996382
\(208\) −20.6345 −1.43074
\(209\) 7.52683 0.520642
\(210\) 0 0
\(211\) −2.87910 −0.198206 −0.0991028 0.995077i \(-0.531597\pi\)
−0.0991028 + 0.995077i \(0.531597\pi\)
\(212\) 29.5123 2.02691
\(213\) 17.5995 1.20590
\(214\) −13.2155 −0.903393
\(215\) −2.68710 −0.183258
\(216\) 0.834394 0.0567733
\(217\) 0 0
\(218\) −4.89497 −0.331529
\(219\) 5.92890 0.400638
\(220\) 2.16027 0.145645
\(221\) 23.7188 1.59550
\(222\) −1.28879 −0.0864980
\(223\) 11.6408 0.779523 0.389762 0.920916i \(-0.372557\pi\)
0.389762 + 0.920916i \(0.372557\pi\)
\(224\) 0 0
\(225\) 1.83973 0.122649
\(226\) −29.3710 −1.95373
\(227\) 13.2637 0.880344 0.440172 0.897913i \(-0.354917\pi\)
0.440172 + 0.897913i \(0.354917\pi\)
\(228\) −35.7709 −2.36899
\(229\) 25.6789 1.69691 0.848453 0.529270i \(-0.177534\pi\)
0.848453 + 0.529270i \(0.177534\pi\)
\(230\) 15.8934 1.04798
\(231\) 0 0
\(232\) −0.292557 −0.0192073
\(233\) 5.10284 0.334298 0.167149 0.985932i \(-0.446544\pi\)
0.167149 + 0.985932i \(0.446544\pi\)
\(234\) −21.1917 −1.38535
\(235\) 7.00636 0.457044
\(236\) −27.0996 −1.76404
\(237\) 26.6953 1.73405
\(238\) 0 0
\(239\) −12.4481 −0.805201 −0.402600 0.915376i \(-0.631894\pi\)
−0.402600 + 0.915376i \(0.631894\pi\)
\(240\) 8.03810 0.518857
\(241\) 3.77595 0.243230 0.121615 0.992577i \(-0.461193\pi\)
0.121615 + 0.992577i \(0.461193\pi\)
\(242\) −2.03967 −0.131115
\(243\) −16.8379 −1.08015
\(244\) 14.0140 0.897154
\(245\) 0 0
\(246\) 24.4081 1.55621
\(247\) −42.5072 −2.70467
\(248\) −1.67336 −0.106259
\(249\) 6.87910 0.435946
\(250\) 2.03967 0.129000
\(251\) −2.97304 −0.187657 −0.0938284 0.995588i \(-0.529911\pi\)
−0.0938284 + 0.995588i \(0.529911\pi\)
\(252\) 0 0
\(253\) −7.79213 −0.489887
\(254\) −12.3999 −0.778038
\(255\) −9.23961 −0.578607
\(256\) 13.1364 0.821025
\(257\) 14.6319 0.912710 0.456355 0.889798i \(-0.349155\pi\)
0.456355 + 0.889798i \(0.349155\pi\)
\(258\) −12.0574 −0.750663
\(259\) 0 0
\(260\) −12.1999 −0.756608
\(261\) 1.64650 0.101916
\(262\) −28.4506 −1.75769
\(263\) 10.7204 0.661049 0.330524 0.943797i \(-0.392774\pi\)
0.330524 + 0.943797i \(0.392774\pi\)
\(264\) 0.719140 0.0442600
\(265\) −13.6614 −0.839214
\(266\) 0 0
\(267\) 7.58426 0.464149
\(268\) −24.9531 −1.52425
\(269\) −1.59346 −0.0971550 −0.0485775 0.998819i \(-0.515469\pi\)
−0.0485775 + 0.998819i \(0.515469\pi\)
\(270\) −5.20630 −0.316845
\(271\) −19.9175 −1.20990 −0.604951 0.796263i \(-0.706807\pi\)
−0.604951 + 0.796263i \(0.706807\pi\)
\(272\) −15.3457 −0.930467
\(273\) 0 0
\(274\) −26.6932 −1.61259
\(275\) −1.00000 −0.0603023
\(276\) 37.0317 2.22905
\(277\) −1.40811 −0.0846054 −0.0423027 0.999105i \(-0.513469\pi\)
−0.0423027 + 0.999105i \(0.513469\pi\)
\(278\) −1.76196 −0.105675
\(279\) 9.41763 0.563819
\(280\) 0 0
\(281\) 13.4256 0.800902 0.400451 0.916318i \(-0.368853\pi\)
0.400451 + 0.916318i \(0.368853\pi\)
\(282\) 31.4386 1.87214
\(283\) −16.0921 −0.956574 −0.478287 0.878204i \(-0.658742\pi\)
−0.478287 + 0.878204i \(0.658742\pi\)
\(284\) 17.2821 1.02551
\(285\) 16.5586 0.980845
\(286\) 11.5189 0.681127
\(287\) 0 0
\(288\) 14.9134 0.878783
\(289\) 0.639491 0.0376171
\(290\) 1.82544 0.107194
\(291\) −5.15325 −0.302089
\(292\) 5.82198 0.340706
\(293\) −11.4282 −0.667640 −0.333820 0.942637i \(-0.608338\pi\)
−0.333820 + 0.942637i \(0.608338\pi\)
\(294\) 0 0
\(295\) 12.5446 0.730374
\(296\) −0.0938889 −0.00545718
\(297\) 2.55252 0.148112
\(298\) −15.1717 −0.878871
\(299\) 44.0055 2.54490
\(300\) 4.75246 0.274383
\(301\) 0 0
\(302\) 8.92514 0.513584
\(303\) 42.4098 2.43638
\(304\) 27.5014 1.57731
\(305\) −6.48716 −0.371454
\(306\) −15.7601 −0.900944
\(307\) 16.7906 0.958288 0.479144 0.877736i \(-0.340947\pi\)
0.479144 + 0.877736i \(0.340947\pi\)
\(308\) 0 0
\(309\) −32.4678 −1.84703
\(310\) 10.4411 0.593016
\(311\) −27.3663 −1.55180 −0.775899 0.630857i \(-0.782703\pi\)
−0.775899 + 0.630857i \(0.782703\pi\)
\(312\) −4.06129 −0.229925
\(313\) 14.2692 0.806541 0.403270 0.915081i \(-0.367873\pi\)
0.403270 + 0.915081i \(0.367873\pi\)
\(314\) 28.0499 1.58295
\(315\) 0 0
\(316\) 26.2139 1.47465
\(317\) 9.47379 0.532101 0.266050 0.963959i \(-0.414281\pi\)
0.266050 + 0.963959i \(0.414281\pi\)
\(318\) −61.3009 −3.43758
\(319\) −0.894968 −0.0501086
\(320\) 9.22664 0.515785
\(321\) 14.2539 0.795575
\(322\) 0 0
\(323\) −31.6122 −1.75895
\(324\) −24.0537 −1.33631
\(325\) 5.64742 0.313263
\(326\) 18.5033 1.02481
\(327\) 5.27959 0.291962
\(328\) 1.77814 0.0981814
\(329\) 0 0
\(330\) −4.48716 −0.247010
\(331\) 6.12319 0.336561 0.168280 0.985739i \(-0.446179\pi\)
0.168280 + 0.985739i \(0.446179\pi\)
\(332\) 6.75505 0.370731
\(333\) 0.528404 0.0289564
\(334\) −18.1552 −0.993407
\(335\) 11.5509 0.631095
\(336\) 0 0
\(337\) 13.2457 0.721538 0.360769 0.932655i \(-0.382514\pi\)
0.360769 + 0.932655i \(0.382514\pi\)
\(338\) −38.5363 −2.09610
\(339\) 31.6789 1.72056
\(340\) −9.07299 −0.492052
\(341\) −5.11902 −0.277210
\(342\) 28.2441 1.52727
\(343\) 0 0
\(344\) −0.878388 −0.0473595
\(345\) −17.1422 −0.922906
\(346\) −22.2739 −1.19745
\(347\) 21.0997 1.13269 0.566345 0.824168i \(-0.308357\pi\)
0.566345 + 0.824168i \(0.308357\pi\)
\(348\) 4.25330 0.228001
\(349\) 16.1666 0.865380 0.432690 0.901543i \(-0.357565\pi\)
0.432690 + 0.901543i \(0.357565\pi\)
\(350\) 0 0
\(351\) −14.4151 −0.769423
\(352\) −8.10630 −0.432067
\(353\) 17.7429 0.944362 0.472181 0.881502i \(-0.343467\pi\)
0.472181 + 0.881502i \(0.343467\pi\)
\(354\) 56.2895 2.99175
\(355\) −8.00000 −0.424596
\(356\) 7.44748 0.394716
\(357\) 0 0
\(358\) 22.6818 1.19877
\(359\) 36.5186 1.92738 0.963689 0.267028i \(-0.0860415\pi\)
0.963689 + 0.267028i \(0.0860415\pi\)
\(360\) 0.601392 0.0316962
\(361\) 37.6532 1.98175
\(362\) 28.8979 1.51884
\(363\) 2.19994 0.115467
\(364\) 0 0
\(365\) −2.69503 −0.141064
\(366\) −29.1089 −1.52155
\(367\) 35.2114 1.83802 0.919011 0.394232i \(-0.128989\pi\)
0.919011 + 0.394232i \(0.128989\pi\)
\(368\) −28.4707 −1.48414
\(369\) −10.0073 −0.520961
\(370\) 0.585831 0.0304559
\(371\) 0 0
\(372\) 24.3279 1.26134
\(373\) −33.8385 −1.75209 −0.876046 0.482228i \(-0.839828\pi\)
−0.876046 + 0.482228i \(0.839828\pi\)
\(374\) 8.56650 0.442963
\(375\) −2.19994 −0.113604
\(376\) 2.29032 0.118114
\(377\) 5.05427 0.260308
\(378\) 0 0
\(379\) 24.2735 1.24685 0.623424 0.781884i \(-0.285741\pi\)
0.623424 + 0.781884i \(0.285741\pi\)
\(380\) 16.2600 0.834118
\(381\) 13.3742 0.685181
\(382\) 17.4567 0.893162
\(383\) 4.16443 0.212793 0.106396 0.994324i \(-0.466069\pi\)
0.106396 + 0.994324i \(0.466069\pi\)
\(384\) 5.73465 0.292645
\(385\) 0 0
\(386\) −10.1504 −0.516644
\(387\) 4.94354 0.251294
\(388\) −5.06032 −0.256899
\(389\) −24.4634 −1.24034 −0.620171 0.784467i \(-0.712937\pi\)
−0.620171 + 0.784467i \(0.712937\pi\)
\(390\) 25.3409 1.28319
\(391\) 32.7265 1.65505
\(392\) 0 0
\(393\) 30.6861 1.54791
\(394\) −6.26530 −0.315641
\(395\) −12.1346 −0.610557
\(396\) −3.97431 −0.199717
\(397\) 0.683233 0.0342905 0.0171452 0.999853i \(-0.494542\pi\)
0.0171452 + 0.999853i \(0.494542\pi\)
\(398\) 25.1137 1.25884
\(399\) 0 0
\(400\) −3.65378 −0.182689
\(401\) 6.75592 0.337374 0.168687 0.985670i \(-0.446047\pi\)
0.168687 + 0.985670i \(0.446047\pi\)
\(402\) 51.8309 2.58509
\(403\) 28.9093 1.44007
\(404\) 41.6450 2.07192
\(405\) 11.1346 0.553282
\(406\) 0 0
\(407\) −0.287218 −0.0142369
\(408\) −3.02035 −0.149529
\(409\) −28.7093 −1.41959 −0.709793 0.704411i \(-0.751211\pi\)
−0.709793 + 0.704411i \(0.751211\pi\)
\(410\) −11.0949 −0.547938
\(411\) 28.7906 1.42013
\(412\) −31.8823 −1.57073
\(413\) 0 0
\(414\) −29.2396 −1.43705
\(415\) −3.12695 −0.153496
\(416\) 45.7797 2.24454
\(417\) 1.90041 0.0930633
\(418\) −15.3523 −0.750904
\(419\) −22.1508 −1.08214 −0.541068 0.840979i \(-0.681980\pi\)
−0.541068 + 0.840979i \(0.681980\pi\)
\(420\) 0 0
\(421\) 14.7572 0.719224 0.359612 0.933102i \(-0.382909\pi\)
0.359612 + 0.933102i \(0.382909\pi\)
\(422\) 5.87243 0.285865
\(423\) −12.8898 −0.626725
\(424\) −4.46579 −0.216878
\(425\) 4.19994 0.203727
\(426\) −35.8973 −1.73923
\(427\) 0 0
\(428\) 13.9969 0.676563
\(429\) −12.4240 −0.599836
\(430\) 5.48080 0.264308
\(431\) 20.5985 0.992197 0.496099 0.868266i \(-0.334765\pi\)
0.496099 + 0.868266i \(0.334765\pi\)
\(432\) 9.32634 0.448714
\(433\) −33.2631 −1.59852 −0.799262 0.600983i \(-0.794776\pi\)
−0.799262 + 0.600983i \(0.794776\pi\)
\(434\) 0 0
\(435\) −1.96888 −0.0944004
\(436\) 5.18438 0.248287
\(437\) −58.6500 −2.80561
\(438\) −12.0930 −0.577827
\(439\) −8.02798 −0.383154 −0.191577 0.981478i \(-0.561360\pi\)
−0.191577 + 0.981478i \(0.561360\pi\)
\(440\) −0.326891 −0.0155839
\(441\) 0 0
\(442\) −48.3787 −2.30114
\(443\) −33.4837 −1.59086 −0.795429 0.606046i \(-0.792755\pi\)
−0.795429 + 0.606046i \(0.792755\pi\)
\(444\) 1.36499 0.0647796
\(445\) −3.44748 −0.163426
\(446\) −23.7434 −1.12428
\(447\) 16.3638 0.773980
\(448\) 0 0
\(449\) 8.22979 0.388388 0.194194 0.980963i \(-0.437791\pi\)
0.194194 + 0.980963i \(0.437791\pi\)
\(450\) −3.75246 −0.176892
\(451\) 5.43955 0.256139
\(452\) 31.1076 1.46318
\(453\) −9.62642 −0.452289
\(454\) −27.0537 −1.26969
\(455\) 0 0
\(456\) 5.41285 0.253480
\(457\) −20.6046 −0.963843 −0.481921 0.876214i \(-0.660061\pi\)
−0.481921 + 0.876214i \(0.660061\pi\)
\(458\) −52.3765 −2.44739
\(459\) −10.7204 −0.500386
\(460\) −16.8331 −0.784846
\(461\) −29.4517 −1.37170 −0.685850 0.727743i \(-0.740570\pi\)
−0.685850 + 0.727743i \(0.740570\pi\)
\(462\) 0 0
\(463\) 24.4173 1.13477 0.567385 0.823453i \(-0.307955\pi\)
0.567385 + 0.823453i \(0.307955\pi\)
\(464\) −3.27002 −0.151807
\(465\) −11.2615 −0.522241
\(466\) −10.4081 −0.482147
\(467\) −30.5612 −1.41420 −0.707101 0.707113i \(-0.749997\pi\)
−0.707101 + 0.707113i \(0.749997\pi\)
\(468\) 22.4446 1.03750
\(469\) 0 0
\(470\) −14.2907 −0.659180
\(471\) −30.2539 −1.39403
\(472\) 4.10071 0.188751
\(473\) −2.68710 −0.123553
\(474\) −54.4498 −2.50096
\(475\) −7.52683 −0.345355
\(476\) 0 0
\(477\) 25.1334 1.15078
\(478\) 25.3901 1.16131
\(479\) −30.9524 −1.41425 −0.707126 0.707088i \(-0.750008\pi\)
−0.707126 + 0.707088i \(0.750008\pi\)
\(480\) −17.8334 −0.813978
\(481\) 1.62204 0.0739587
\(482\) −7.70170 −0.350803
\(483\) 0 0
\(484\) 2.16027 0.0981939
\(485\) 2.34245 0.106365
\(486\) 34.3437 1.55786
\(487\) 0.477034 0.0216165 0.0108082 0.999942i \(-0.496560\pi\)
0.0108082 + 0.999942i \(0.496560\pi\)
\(488\) −2.12059 −0.0959947
\(489\) −19.9572 −0.902497
\(490\) 0 0
\(491\) 18.2964 0.825706 0.412853 0.910798i \(-0.364532\pi\)
0.412853 + 0.910798i \(0.364532\pi\)
\(492\) −25.8512 −1.16546
\(493\) 3.75881 0.169288
\(494\) 86.7008 3.90085
\(495\) 1.83973 0.0826899
\(496\) −18.7038 −0.839825
\(497\) 0 0
\(498\) −14.0311 −0.628750
\(499\) −20.3840 −0.912514 −0.456257 0.889848i \(-0.650810\pi\)
−0.456257 + 0.889848i \(0.650810\pi\)
\(500\) −2.16027 −0.0966100
\(501\) 19.5817 0.874846
\(502\) 6.06404 0.270651
\(503\) 19.4005 0.865025 0.432513 0.901628i \(-0.357627\pi\)
0.432513 + 0.901628i \(0.357627\pi\)
\(504\) 0 0
\(505\) −19.2777 −0.857847
\(506\) 15.8934 0.706548
\(507\) 41.5643 1.84594
\(508\) 13.1330 0.582683
\(509\) 5.13616 0.227656 0.113828 0.993500i \(-0.463689\pi\)
0.113828 + 0.993500i \(0.463689\pi\)
\(510\) 18.8458 0.834506
\(511\) 0 0
\(512\) −32.0074 −1.41454
\(513\) 19.2124 0.848246
\(514\) −29.8442 −1.31637
\(515\) 14.7585 0.650338
\(516\) 12.7703 0.562181
\(517\) 7.00636 0.308139
\(518\) 0 0
\(519\) 24.0241 1.05454
\(520\) 1.84609 0.0809565
\(521\) 33.0445 1.44770 0.723852 0.689955i \(-0.242370\pi\)
0.723852 + 0.689955i \(0.242370\pi\)
\(522\) −3.35833 −0.146990
\(523\) −6.23138 −0.272479 −0.136240 0.990676i \(-0.543502\pi\)
−0.136240 + 0.990676i \(0.543502\pi\)
\(524\) 30.1328 1.31636
\(525\) 0 0
\(526\) −21.8661 −0.953409
\(527\) 21.4996 0.936536
\(528\) 8.03810 0.349813
\(529\) 37.7173 1.63988
\(530\) 27.8648 1.21037
\(531\) −23.0787 −1.00153
\(532\) 0 0
\(533\) −30.7195 −1.33061
\(534\) −15.4694 −0.669427
\(535\) −6.47923 −0.280121
\(536\) 3.77590 0.163094
\(537\) −24.4640 −1.05570
\(538\) 3.25014 0.140123
\(539\) 0 0
\(540\) 5.51411 0.237290
\(541\) −5.03780 −0.216592 −0.108296 0.994119i \(-0.534539\pi\)
−0.108296 + 0.994119i \(0.534539\pi\)
\(542\) 40.6252 1.74500
\(543\) −31.1685 −1.33757
\(544\) 34.0460 1.45971
\(545\) −2.39988 −0.102800
\(546\) 0 0
\(547\) 14.7686 0.631461 0.315731 0.948849i \(-0.397750\pi\)
0.315731 + 0.948849i \(0.397750\pi\)
\(548\) 28.2714 1.20769
\(549\) 11.9346 0.509358
\(550\) 2.03967 0.0869720
\(551\) −6.73628 −0.286975
\(552\) −5.60363 −0.238507
\(553\) 0 0
\(554\) 2.87209 0.122023
\(555\) −0.631862 −0.0268210
\(556\) 1.86614 0.0791418
\(557\) −14.6357 −0.620136 −0.310068 0.950714i \(-0.600352\pi\)
−0.310068 + 0.950714i \(0.600352\pi\)
\(558\) −19.2089 −0.813177
\(559\) 15.1752 0.641841
\(560\) 0 0
\(561\) −9.23961 −0.390097
\(562\) −27.3838 −1.15511
\(563\) −3.22822 −0.136053 −0.0680266 0.997684i \(-0.521670\pi\)
−0.0680266 + 0.997684i \(0.521670\pi\)
\(564\) −33.2974 −1.40207
\(565\) −14.3999 −0.605807
\(566\) 32.8225 1.37963
\(567\) 0 0
\(568\) −2.61513 −0.109728
\(569\) −3.34622 −0.140281 −0.0701404 0.997537i \(-0.522345\pi\)
−0.0701404 + 0.997537i \(0.522345\pi\)
\(570\) −33.7741 −1.41464
\(571\) −0.351656 −0.0147164 −0.00735818 0.999973i \(-0.502342\pi\)
−0.00735818 + 0.999973i \(0.502342\pi\)
\(572\) −12.1999 −0.510105
\(573\) −18.8283 −0.786565
\(574\) 0 0
\(575\) 7.79213 0.324954
\(576\) −16.9746 −0.707274
\(577\) 3.49571 0.145528 0.0727641 0.997349i \(-0.476818\pi\)
0.0727641 + 0.997349i \(0.476818\pi\)
\(578\) −1.30435 −0.0542540
\(579\) 10.9480 0.454984
\(580\) −1.93337 −0.0802788
\(581\) 0 0
\(582\) 10.5110 0.435693
\(583\) −13.6614 −0.565798
\(584\) −0.880981 −0.0364552
\(585\) −10.3898 −0.429564
\(586\) 23.3097 0.962915
\(587\) 14.6480 0.604589 0.302295 0.953215i \(-0.402247\pi\)
0.302295 + 0.953215i \(0.402247\pi\)
\(588\) 0 0
\(589\) −38.5300 −1.58760
\(590\) −25.5868 −1.05339
\(591\) 6.75759 0.277970
\(592\) −1.04943 −0.0431314
\(593\) −19.9959 −0.821135 −0.410567 0.911830i \(-0.634669\pi\)
−0.410567 + 0.911830i \(0.634669\pi\)
\(594\) −5.20630 −0.213617
\(595\) 0 0
\(596\) 16.0687 0.658198
\(597\) −27.0870 −1.10860
\(598\) −89.7567 −3.67043
\(599\) −34.6500 −1.41576 −0.707881 0.706332i \(-0.750349\pi\)
−0.707881 + 0.706332i \(0.750349\pi\)
\(600\) −0.719140 −0.0293588
\(601\) −1.21423 −0.0495295 −0.0247647 0.999693i \(-0.507884\pi\)
−0.0247647 + 0.999693i \(0.507884\pi\)
\(602\) 0 0
\(603\) −21.2507 −0.865394
\(604\) −9.45282 −0.384630
\(605\) −1.00000 −0.0406558
\(606\) −86.5021 −3.51391
\(607\) −25.8474 −1.04911 −0.524556 0.851376i \(-0.675769\pi\)
−0.524556 + 0.851376i \(0.675769\pi\)
\(608\) −61.0148 −2.47448
\(609\) 0 0
\(610\) 13.2317 0.535735
\(611\) −39.5679 −1.60074
\(612\) 16.6919 0.674729
\(613\) −34.1095 −1.37767 −0.688835 0.724918i \(-0.741878\pi\)
−0.688835 + 0.724918i \(0.741878\pi\)
\(614\) −34.2472 −1.38211
\(615\) 11.9667 0.482543
\(616\) 0 0
\(617\) −2.76644 −0.111373 −0.0556864 0.998448i \(-0.517735\pi\)
−0.0556864 + 0.998448i \(0.517735\pi\)
\(618\) 66.2238 2.66391
\(619\) 17.9121 0.719949 0.359974 0.932962i \(-0.382785\pi\)
0.359974 + 0.932962i \(0.382785\pi\)
\(620\) −11.0584 −0.444118
\(621\) −19.8895 −0.798139
\(622\) 55.8182 2.23811
\(623\) 0 0
\(624\) −45.3946 −1.81724
\(625\) 1.00000 0.0400000
\(626\) −29.1044 −1.16325
\(627\) 16.5586 0.661286
\(628\) −29.7083 −1.18549
\(629\) 1.20630 0.0480982
\(630\) 0 0
\(631\) 10.6335 0.423314 0.211657 0.977344i \(-0.432114\pi\)
0.211657 + 0.977344i \(0.432114\pi\)
\(632\) −3.96669 −0.157786
\(633\) −6.33385 −0.251748
\(634\) −19.3234 −0.767431
\(635\) −6.07935 −0.241251
\(636\) 64.9252 2.57445
\(637\) 0 0
\(638\) 1.82544 0.0722700
\(639\) 14.7179 0.582230
\(640\) −2.60673 −0.103040
\(641\) 17.5601 0.693584 0.346792 0.937942i \(-0.387271\pi\)
0.346792 + 0.937942i \(0.387271\pi\)
\(642\) −29.0733 −1.14743
\(643\) −30.0626 −1.18555 −0.592776 0.805367i \(-0.701968\pi\)
−0.592776 + 0.805367i \(0.701968\pi\)
\(644\) 0 0
\(645\) −5.91145 −0.232763
\(646\) 64.4786 2.53688
\(647\) −3.47287 −0.136532 −0.0682662 0.997667i \(-0.521747\pi\)
−0.0682662 + 0.997667i \(0.521747\pi\)
\(648\) 3.63979 0.142985
\(649\) 12.5446 0.492418
\(650\) −11.5189 −0.451808
\(651\) 0 0
\(652\) −19.5973 −0.767490
\(653\) −14.6024 −0.571437 −0.285718 0.958314i \(-0.592232\pi\)
−0.285718 + 0.958314i \(0.592232\pi\)
\(654\) −10.7686 −0.421087
\(655\) −13.9486 −0.545018
\(656\) 19.8749 0.775986
\(657\) 4.95814 0.193435
\(658\) 0 0
\(659\) 37.1876 1.44862 0.724312 0.689472i \(-0.242157\pi\)
0.724312 + 0.689472i \(0.242157\pi\)
\(660\) 4.75246 0.184989
\(661\) −34.7703 −1.35241 −0.676204 0.736714i \(-0.736376\pi\)
−0.676204 + 0.736714i \(0.736376\pi\)
\(662\) −12.4893 −0.485410
\(663\) 52.1800 2.02650
\(664\) −1.02217 −0.0396680
\(665\) 0 0
\(666\) −1.07777 −0.0417628
\(667\) 6.97371 0.270023
\(668\) 19.2286 0.743976
\(669\) 25.6090 0.990100
\(670\) −23.5601 −0.910208
\(671\) −6.48716 −0.250434
\(672\) 0 0
\(673\) −24.8299 −0.957123 −0.478562 0.878054i \(-0.658842\pi\)
−0.478562 + 0.878054i \(0.658842\pi\)
\(674\) −27.0168 −1.04065
\(675\) −2.55252 −0.0982464
\(676\) 40.8148 1.56980
\(677\) −45.4850 −1.74813 −0.874065 0.485809i \(-0.838525\pi\)
−0.874065 + 0.485809i \(0.838525\pi\)
\(678\) −64.6145 −2.48151
\(679\) 0 0
\(680\) 1.37292 0.0526492
\(681\) 29.1794 1.11816
\(682\) 10.4411 0.399811
\(683\) −40.3945 −1.54565 −0.772827 0.634617i \(-0.781158\pi\)
−0.772827 + 0.634617i \(0.781158\pi\)
\(684\) −29.9140 −1.14379
\(685\) −13.0870 −0.500028
\(686\) 0 0
\(687\) 56.4919 2.15530
\(688\) −9.81807 −0.374310
\(689\) 77.1518 2.93925
\(690\) 34.9645 1.33108
\(691\) 17.6360 0.670906 0.335453 0.942057i \(-0.391111\pi\)
0.335453 + 0.942057i \(0.391111\pi\)
\(692\) 23.5909 0.896790
\(693\) 0 0
\(694\) −43.0365 −1.63364
\(695\) −0.863845 −0.0327675
\(696\) −0.643608 −0.0243959
\(697\) −22.8458 −0.865346
\(698\) −32.9746 −1.24811
\(699\) 11.2259 0.424604
\(700\) 0 0
\(701\) −7.01815 −0.265072 −0.132536 0.991178i \(-0.542312\pi\)
−0.132536 + 0.991178i \(0.542312\pi\)
\(702\) 29.4022 1.10971
\(703\) −2.16184 −0.0815354
\(704\) 9.22664 0.347742
\(705\) 15.4136 0.580508
\(706\) −36.1898 −1.36202
\(707\) 0 0
\(708\) −59.6176 −2.24057
\(709\) −39.0733 −1.46743 −0.733714 0.679458i \(-0.762215\pi\)
−0.733714 + 0.679458i \(0.762215\pi\)
\(710\) 16.3174 0.612381
\(711\) 22.3244 0.837230
\(712\) −1.12695 −0.0422343
\(713\) 39.8881 1.49382
\(714\) 0 0
\(715\) 5.64742 0.211202
\(716\) −24.0228 −0.897774
\(717\) −27.3851 −1.02271
\(718\) −74.4860 −2.77979
\(719\) 6.82355 0.254476 0.127238 0.991872i \(-0.459389\pi\)
0.127238 + 0.991872i \(0.459389\pi\)
\(720\) 6.72199 0.250514
\(721\) 0 0
\(722\) −76.8002 −2.85821
\(723\) 8.30686 0.308935
\(724\) −30.6064 −1.13748
\(725\) 0.894968 0.0332383
\(726\) −4.48716 −0.166534
\(727\) 25.3574 0.940454 0.470227 0.882545i \(-0.344172\pi\)
0.470227 + 0.882545i \(0.344172\pi\)
\(728\) 0 0
\(729\) −3.63852 −0.134760
\(730\) 5.49698 0.203452
\(731\) 11.2856 0.417415
\(732\) 30.8299 1.13951
\(733\) −47.8560 −1.76760 −0.883801 0.467863i \(-0.845024\pi\)
−0.883801 + 0.467863i \(0.845024\pi\)
\(734\) −71.8198 −2.65092
\(735\) 0 0
\(736\) 63.1654 2.32831
\(737\) 11.5509 0.425484
\(738\) 20.4117 0.751364
\(739\) 7.35893 0.270703 0.135351 0.990798i \(-0.456784\pi\)
0.135351 + 0.990798i \(0.456784\pi\)
\(740\) −0.620467 −0.0228088
\(741\) −93.5133 −3.43530
\(742\) 0 0
\(743\) 34.4792 1.26492 0.632460 0.774593i \(-0.282045\pi\)
0.632460 + 0.774593i \(0.282045\pi\)
\(744\) −3.68129 −0.134963
\(745\) −7.43828 −0.272518
\(746\) 69.0195 2.52698
\(747\) 5.75276 0.210482
\(748\) −9.07299 −0.331741
\(749\) 0 0
\(750\) 4.48716 0.163848
\(751\) −50.3606 −1.83769 −0.918843 0.394624i \(-0.870875\pi\)
−0.918843 + 0.394624i \(0.870875\pi\)
\(752\) 25.5997 0.933525
\(753\) −6.54052 −0.238350
\(754\) −10.3091 −0.375433
\(755\) 4.37577 0.159250
\(756\) 0 0
\(757\) −15.6154 −0.567551 −0.283775 0.958891i \(-0.591587\pi\)
−0.283775 + 0.958891i \(0.591587\pi\)
\(758\) −49.5101 −1.79829
\(759\) −17.1422 −0.622223
\(760\) −2.46045 −0.0892500
\(761\) −25.3155 −0.917686 −0.458843 0.888517i \(-0.651736\pi\)
−0.458843 + 0.888517i \(0.651736\pi\)
\(762\) −27.2790 −0.988213
\(763\) 0 0
\(764\) −18.4888 −0.668901
\(765\) −7.72677 −0.279362
\(766\) −8.49408 −0.306904
\(767\) −70.8446 −2.55805
\(768\) 28.8993 1.04281
\(769\) −40.0616 −1.44466 −0.722329 0.691549i \(-0.756928\pi\)
−0.722329 + 0.691549i \(0.756928\pi\)
\(770\) 0 0
\(771\) 32.1892 1.15927
\(772\) 10.7506 0.386922
\(773\) −15.5185 −0.558162 −0.279081 0.960268i \(-0.590030\pi\)
−0.279081 + 0.960268i \(0.590030\pi\)
\(774\) −10.0832 −0.362434
\(775\) 5.11902 0.183881
\(776\) 0.765727 0.0274880
\(777\) 0 0
\(778\) 49.8973 1.78890
\(779\) 40.9426 1.46692
\(780\) −26.8391 −0.960995
\(781\) −8.00000 −0.286263
\(782\) −66.7513 −2.38702
\(783\) −2.28442 −0.0816386
\(784\) 0 0
\(785\) 13.7522 0.490835
\(786\) −62.5897 −2.23250
\(787\) 14.7204 0.524726 0.262363 0.964969i \(-0.415498\pi\)
0.262363 + 0.964969i \(0.415498\pi\)
\(788\) 6.63573 0.236388
\(789\) 23.5843 0.839622
\(790\) 24.7506 0.880586
\(791\) 0 0
\(792\) 0.601392 0.0213695
\(793\) 36.6357 1.30097
\(794\) −1.39357 −0.0494560
\(795\) −30.0543 −1.06592
\(796\) −26.5985 −0.942759
\(797\) 8.03778 0.284713 0.142356 0.989815i \(-0.454532\pi\)
0.142356 + 0.989815i \(0.454532\pi\)
\(798\) 0 0
\(799\) −29.4263 −1.04103
\(800\) 8.10630 0.286601
\(801\) 6.34245 0.224100
\(802\) −13.7799 −0.486584
\(803\) −2.69503 −0.0951055
\(804\) −54.8953 −1.93601
\(805\) 0 0
\(806\) −58.9655 −2.07697
\(807\) −3.50552 −0.123400
\(808\) −6.30171 −0.221693
\(809\) −15.5513 −0.546754 −0.273377 0.961907i \(-0.588141\pi\)
−0.273377 + 0.961907i \(0.588141\pi\)
\(810\) −22.7109 −0.797980
\(811\) 40.8052 1.43286 0.716432 0.697657i \(-0.245774\pi\)
0.716432 + 0.697657i \(0.245774\pi\)
\(812\) 0 0
\(813\) −43.8173 −1.53674
\(814\) 0.585831 0.0205334
\(815\) 9.07172 0.317768
\(816\) −33.7595 −1.18182
\(817\) −20.2253 −0.707594
\(818\) 58.5577 2.04742
\(819\) 0 0
\(820\) 11.7509 0.410359
\(821\) −26.6734 −0.930909 −0.465454 0.885072i \(-0.654109\pi\)
−0.465454 + 0.885072i \(0.654109\pi\)
\(822\) −58.7233 −2.04821
\(823\) −0.430651 −0.0150116 −0.00750578 0.999972i \(-0.502389\pi\)
−0.00750578 + 0.999972i \(0.502389\pi\)
\(824\) 4.82442 0.168067
\(825\) −2.19994 −0.0765921
\(826\) 0 0
\(827\) −49.1302 −1.70842 −0.854212 0.519924i \(-0.825960\pi\)
−0.854212 + 0.519924i \(0.825960\pi\)
\(828\) 30.9684 1.07623
\(829\) −19.4408 −0.675208 −0.337604 0.941288i \(-0.609616\pi\)
−0.337604 + 0.941288i \(0.609616\pi\)
\(830\) 6.37796 0.221382
\(831\) −3.09777 −0.107460
\(832\) −52.1068 −1.80648
\(833\) 0 0
\(834\) −3.87621 −0.134222
\(835\) −8.90102 −0.308033
\(836\) 16.2600 0.562362
\(837\) −13.0664 −0.451640
\(838\) 45.1803 1.56073
\(839\) −50.8077 −1.75408 −0.877038 0.480421i \(-0.840484\pi\)
−0.877038 + 0.480421i \(0.840484\pi\)
\(840\) 0 0
\(841\) −28.1990 −0.972380
\(842\) −30.0999 −1.03731
\(843\) 29.5354 1.01725
\(844\) −6.21963 −0.214089
\(845\) −18.8934 −0.649953
\(846\) 26.2910 0.903905
\(847\) 0 0
\(848\) −49.9158 −1.71412
\(849\) −35.4016 −1.21498
\(850\) −8.56650 −0.293829
\(851\) 2.23804 0.0767190
\(852\) 38.0196 1.30253
\(853\) 19.7547 0.676390 0.338195 0.941076i \(-0.390184\pi\)
0.338195 + 0.941076i \(0.390184\pi\)
\(854\) 0 0
\(855\) 13.8474 0.473570
\(856\) −2.11800 −0.0723918
\(857\) −9.86100 −0.336845 −0.168423 0.985715i \(-0.553867\pi\)
−0.168423 + 0.985715i \(0.553867\pi\)
\(858\) 25.3409 0.865123
\(859\) 4.47795 0.152786 0.0763929 0.997078i \(-0.475660\pi\)
0.0763929 + 0.997078i \(0.475660\pi\)
\(860\) −5.80484 −0.197944
\(861\) 0 0
\(862\) −42.0143 −1.43101
\(863\) 55.5123 1.88966 0.944830 0.327561i \(-0.106227\pi\)
0.944830 + 0.327561i \(0.106227\pi\)
\(864\) −20.6915 −0.703938
\(865\) −10.9204 −0.371303
\(866\) 67.8459 2.30550
\(867\) 1.40684 0.0477789
\(868\) 0 0
\(869\) −12.1346 −0.411637
\(870\) 4.01586 0.136151
\(871\) −65.2331 −2.21034
\(872\) −0.784499 −0.0265665
\(873\) −4.30949 −0.145854
\(874\) 119.627 4.04644
\(875\) 0 0
\(876\) 12.8080 0.432742
\(877\) 10.7024 0.361393 0.180696 0.983539i \(-0.442165\pi\)
0.180696 + 0.983539i \(0.442165\pi\)
\(878\) 16.3744 0.552611
\(879\) −25.1413 −0.847993
\(880\) −3.65378 −0.123169
\(881\) 3.64045 0.122650 0.0613249 0.998118i \(-0.480467\pi\)
0.0613249 + 0.998118i \(0.480467\pi\)
\(882\) 0 0
\(883\) −42.1889 −1.41977 −0.709884 0.704319i \(-0.751253\pi\)
−0.709884 + 0.704319i \(0.751253\pi\)
\(884\) 51.2390 1.72335
\(885\) 27.5973 0.927674
\(886\) 68.2958 2.29444
\(887\) −29.6820 −0.996624 −0.498312 0.866998i \(-0.666047\pi\)
−0.498312 + 0.866998i \(0.666047\pi\)
\(888\) −0.206550 −0.00693136
\(889\) 0 0
\(890\) 7.03174 0.235704
\(891\) 11.1346 0.373023
\(892\) 25.1471 0.841989
\(893\) 52.7357 1.76473
\(894\) −33.3767 −1.11629
\(895\) 11.1203 0.371710
\(896\) 0 0
\(897\) 96.8093 3.23237
\(898\) −16.7861 −0.560159
\(899\) 4.58136 0.152797
\(900\) 3.97431 0.132477
\(901\) 57.3771 1.91151
\(902\) −11.0949 −0.369420
\(903\) 0 0
\(904\) −4.70719 −0.156559
\(905\) 14.1679 0.470957
\(906\) 19.6348 0.652321
\(907\) −20.3971 −0.677274 −0.338637 0.940917i \(-0.609966\pi\)
−0.338637 + 0.940917i \(0.609966\pi\)
\(908\) 28.6532 0.950889
\(909\) 35.4659 1.17633
\(910\) 0 0
\(911\) −11.3589 −0.376338 −0.188169 0.982137i \(-0.560255\pi\)
−0.188169 + 0.982137i \(0.560255\pi\)
\(912\) 60.5014 2.00340
\(913\) −3.12695 −0.103487
\(914\) 42.0267 1.39012
\(915\) −14.2714 −0.471796
\(916\) 55.4732 1.83289
\(917\) 0 0
\(918\) 21.8661 0.721690
\(919\) 45.1197 1.48836 0.744181 0.667979i \(-0.232840\pi\)
0.744181 + 0.667979i \(0.232840\pi\)
\(920\) 2.54718 0.0839779
\(921\) 36.9382 1.21716
\(922\) 60.0718 1.97836
\(923\) 45.1794 1.48710
\(924\) 0 0
\(925\) 0.287218 0.00944367
\(926\) −49.8034 −1.63664
\(927\) −27.1517 −0.891780
\(928\) 7.25489 0.238153
\(929\) −5.06286 −0.166107 −0.0830536 0.996545i \(-0.526467\pi\)
−0.0830536 + 0.996545i \(0.526467\pi\)
\(930\) 22.9698 0.753211
\(931\) 0 0
\(932\) 11.0235 0.361087
\(933\) −60.2041 −1.97099
\(934\) 62.3348 2.03966
\(935\) 4.19994 0.137353
\(936\) −3.39632 −0.111012
\(937\) 9.96481 0.325536 0.162768 0.986664i \(-0.447958\pi\)
0.162768 + 0.986664i \(0.447958\pi\)
\(938\) 0 0
\(939\) 31.3913 1.02442
\(940\) 15.1356 0.493669
\(941\) 41.6118 1.35651 0.678253 0.734828i \(-0.262737\pi\)
0.678253 + 0.734828i \(0.262737\pi\)
\(942\) 61.7081 2.01056
\(943\) −42.3857 −1.38027
\(944\) 45.8352 1.49181
\(945\) 0 0
\(946\) 5.48080 0.178196
\(947\) 33.9477 1.10315 0.551576 0.834125i \(-0.314027\pi\)
0.551576 + 0.834125i \(0.314027\pi\)
\(948\) 57.6691 1.87300
\(949\) 15.2200 0.494061
\(950\) 15.3523 0.498094
\(951\) 20.8418 0.675840
\(952\) 0 0
\(953\) −52.2835 −1.69363 −0.846814 0.531889i \(-0.821482\pi\)
−0.846814 + 0.531889i \(0.821482\pi\)
\(954\) −51.2638 −1.65973
\(955\) 8.55857 0.276949
\(956\) −26.8912 −0.869724
\(957\) −1.96888 −0.0636447
\(958\) 63.1328 2.03973
\(959\) 0 0
\(960\) 20.2981 0.655117
\(961\) −4.79564 −0.154698
\(962\) −3.30843 −0.106668
\(963\) 11.9200 0.384118
\(964\) 8.15706 0.262721
\(965\) −4.97651 −0.160199
\(966\) 0 0
\(967\) −23.2044 −0.746202 −0.373101 0.927791i \(-0.621706\pi\)
−0.373101 + 0.927791i \(0.621706\pi\)
\(968\) −0.326891 −0.0105067
\(969\) −69.5450 −2.23411
\(970\) −4.77784 −0.153407
\(971\) −4.23011 −0.135751 −0.0678753 0.997694i \(-0.521622\pi\)
−0.0678753 + 0.997694i \(0.521622\pi\)
\(972\) −36.3743 −1.16670
\(973\) 0 0
\(974\) −0.972993 −0.0311767
\(975\) 12.4240 0.397886
\(976\) −23.7027 −0.758704
\(977\) −42.1067 −1.34711 −0.673557 0.739136i \(-0.735234\pi\)
−0.673557 + 0.739136i \(0.735234\pi\)
\(978\) 40.7062 1.30164
\(979\) −3.44748 −0.110182
\(980\) 0 0
\(981\) 4.41514 0.140965
\(982\) −37.3187 −1.19089
\(983\) −32.9417 −1.05068 −0.525338 0.850894i \(-0.676061\pi\)
−0.525338 + 0.850894i \(0.676061\pi\)
\(984\) 3.91180 0.124704
\(985\) −3.07172 −0.0978730
\(986\) −7.66675 −0.244159
\(987\) 0 0
\(988\) −91.8269 −2.92140
\(989\) 20.9382 0.665796
\(990\) −3.75246 −0.119261
\(991\) −34.5738 −1.09827 −0.549137 0.835732i \(-0.685043\pi\)
−0.549137 + 0.835732i \(0.685043\pi\)
\(992\) 41.4963 1.31751
\(993\) 13.4706 0.427478
\(994\) 0 0
\(995\) 12.3126 0.390336
\(996\) 14.8607 0.470879
\(997\) −17.3050 −0.548054 −0.274027 0.961722i \(-0.588356\pi\)
−0.274027 + 0.961722i \(0.588356\pi\)
\(998\) 41.5767 1.31609
\(999\) −0.733128 −0.0231951
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 2695.2.a.l.1.1 4
7.6 odd 2 385.2.a.h.1.1 4
21.20 even 2 3465.2.a.bk.1.4 4
28.27 even 2 6160.2.a.br.1.4 4
35.13 even 4 1925.2.b.p.1849.7 8
35.27 even 4 1925.2.b.p.1849.2 8
35.34 odd 2 1925.2.a.x.1.4 4
77.76 even 2 4235.2.a.r.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.a.h.1.1 4 7.6 odd 2
1925.2.a.x.1.4 4 35.34 odd 2
1925.2.b.p.1849.2 8 35.27 even 4
1925.2.b.p.1849.7 8 35.13 even 4
2695.2.a.l.1.1 4 1.1 even 1 trivial
3465.2.a.bk.1.4 4 21.20 even 2
4235.2.a.r.1.4 4 77.76 even 2
6160.2.a.br.1.4 4 28.27 even 2