Properties

Label 145.2.a.c.1.2
Level $145$
Weight $2$
Character 145.1
Self dual yes
Analytic conductor $1.158$
Analytic rank $0$
Dimension $3$
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] = [145,2,Mod(1,145)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, 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("145.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.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: \(1.15783082931\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 145.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.311108 q^{2} +2.90321 q^{3} -1.90321 q^{4} +1.00000 q^{5} +0.903212 q^{6} -0.903212 q^{7} -1.21432 q^{8} +5.42864 q^{9} +O(q^{10})\) \(q+0.311108 q^{2} +2.90321 q^{3} -1.90321 q^{4} +1.00000 q^{5} +0.903212 q^{6} -0.903212 q^{7} -1.21432 q^{8} +5.42864 q^{9} +0.311108 q^{10} -1.52543 q^{11} -5.52543 q^{12} -0.622216 q^{13} -0.280996 q^{14} +2.90321 q^{15} +3.42864 q^{16} -7.95407 q^{17} +1.68889 q^{18} -1.09679 q^{19} -1.90321 q^{20} -2.62222 q^{21} -0.474572 q^{22} +7.52543 q^{23} -3.52543 q^{24} +1.00000 q^{25} -0.193576 q^{26} +7.05086 q^{27} +1.71900 q^{28} -1.00000 q^{29} +0.903212 q^{30} -6.90321 q^{31} +3.49532 q^{32} -4.42864 q^{33} -2.47457 q^{34} -0.903212 q^{35} -10.3319 q^{36} +3.95407 q^{37} -0.341219 q^{38} -1.80642 q^{39} -1.21432 q^{40} +3.67307 q^{41} -0.815792 q^{42} -10.5161 q^{43} +2.90321 q^{44} +5.42864 q^{45} +2.34122 q^{46} +6.90321 q^{47} +9.95407 q^{48} -6.18421 q^{49} +0.311108 q^{50} -23.0923 q^{51} +1.18421 q^{52} +6.42864 q^{53} +2.19358 q^{54} -1.52543 q^{55} +1.09679 q^{56} -3.18421 q^{57} -0.311108 q^{58} -1.67307 q^{59} -5.52543 q^{60} -1.86665 q^{61} -2.14764 q^{62} -4.90321 q^{63} -5.76986 q^{64} -0.622216 q^{65} -1.37778 q^{66} +11.5254 q^{67} +15.1383 q^{68} +21.8479 q^{69} -0.280996 q^{70} +13.6731 q^{71} -6.59210 q^{72} +10.1891 q^{73} +1.23014 q^{74} +2.90321 q^{75} +2.08742 q^{76} +1.37778 q^{77} -0.561993 q^{78} +9.13828 q^{79} +3.42864 q^{80} +4.18421 q^{81} +1.14272 q^{82} +10.7096 q^{83} +4.99063 q^{84} -7.95407 q^{85} -3.27163 q^{86} -2.90321 q^{87} +1.85236 q^{88} -7.80642 q^{89} +1.68889 q^{90} +0.561993 q^{91} -14.3225 q^{92} -20.0415 q^{93} +2.14764 q^{94} -1.09679 q^{95} +10.1476 q^{96} -4.08742 q^{97} -1.92396 q^{98} -8.28100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 2 q^{3} + q^{4} + 3 q^{5} - 4 q^{6} + 4 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 2 q^{3} + q^{4} + 3 q^{5} - 4 q^{6} + 4 q^{7} + 3 q^{8} + 3 q^{9} + q^{10} + 2 q^{11} - 10 q^{12} - 2 q^{13} + 6 q^{14} + 2 q^{15} - 3 q^{16} - 4 q^{17} + 5 q^{18} - 10 q^{19} + q^{20} - 8 q^{21} - 8 q^{22} + 16 q^{23} - 4 q^{24} + 3 q^{25} - 14 q^{26} + 8 q^{27} + 12 q^{28} - 3 q^{29} - 4 q^{30} - 14 q^{31} - 3 q^{32} - 14 q^{34} + 4 q^{35} - 11 q^{36} - 8 q^{37} - 8 q^{38} + 8 q^{39} + 3 q^{40} - 2 q^{41} - 16 q^{42} + 2 q^{43} + 2 q^{44} + 3 q^{45} + 14 q^{46} + 14 q^{47} + 10 q^{48} - 5 q^{49} + q^{50} - 16 q^{51} - 10 q^{52} + 6 q^{53} + 20 q^{54} + 2 q^{55} + 10 q^{56} + 4 q^{57} - q^{58} + 8 q^{59} - 10 q^{60} - 6 q^{61} - 8 q^{63} - 11 q^{64} - 2 q^{65} - 4 q^{66} + 28 q^{67} + 12 q^{68} + 12 q^{69} + 6 q^{70} + 28 q^{71} - 13 q^{72} - 16 q^{73} + 10 q^{74} + 2 q^{75} - 14 q^{76} + 4 q^{77} + 12 q^{78} - 6 q^{79} - 3 q^{80} - q^{81} + 30 q^{82} + 12 q^{83} - 12 q^{84} - 4 q^{85} + 24 q^{86} - 2 q^{87} + 12 q^{88} - 10 q^{89} + 5 q^{90} - 12 q^{91} + 4 q^{92} - 20 q^{93} - 10 q^{95} + 24 q^{96} + 8 q^{97} + 21 q^{98} - 18 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.311108 0.219986 0.109993 0.993932i \(-0.464917\pi\)
0.109993 + 0.993932i \(0.464917\pi\)
\(3\) 2.90321 1.67617 0.838085 0.545540i \(-0.183675\pi\)
0.838085 + 0.545540i \(0.183675\pi\)
\(4\) −1.90321 −0.951606
\(5\) 1.00000 0.447214
\(6\) 0.903212 0.368735
\(7\) −0.903212 −0.341382 −0.170691 0.985325i \(-0.554600\pi\)
−0.170691 + 0.985325i \(0.554600\pi\)
\(8\) −1.21432 −0.429327
\(9\) 5.42864 1.80955
\(10\) 0.311108 0.0983809
\(11\) −1.52543 −0.459934 −0.229967 0.973198i \(-0.573862\pi\)
−0.229967 + 0.973198i \(0.573862\pi\)
\(12\) −5.52543 −1.59505
\(13\) −0.622216 −0.172572 −0.0862858 0.996270i \(-0.527500\pi\)
−0.0862858 + 0.996270i \(0.527500\pi\)
\(14\) −0.280996 −0.0750994
\(15\) 2.90321 0.749606
\(16\) 3.42864 0.857160
\(17\) −7.95407 −1.92914 −0.964572 0.263819i \(-0.915018\pi\)
−0.964572 + 0.263819i \(0.915018\pi\)
\(18\) 1.68889 0.398076
\(19\) −1.09679 −0.251620 −0.125810 0.992054i \(-0.540153\pi\)
−0.125810 + 0.992054i \(0.540153\pi\)
\(20\) −1.90321 −0.425571
\(21\) −2.62222 −0.572214
\(22\) −0.474572 −0.101179
\(23\) 7.52543 1.56916 0.784580 0.620028i \(-0.212879\pi\)
0.784580 + 0.620028i \(0.212879\pi\)
\(24\) −3.52543 −0.719625
\(25\) 1.00000 0.200000
\(26\) −0.193576 −0.0379634
\(27\) 7.05086 1.35694
\(28\) 1.71900 0.324861
\(29\) −1.00000 −0.185695
\(30\) 0.903212 0.164903
\(31\) −6.90321 −1.23985 −0.619927 0.784660i \(-0.712838\pi\)
−0.619927 + 0.784660i \(0.712838\pi\)
\(32\) 3.49532 0.617890
\(33\) −4.42864 −0.770927
\(34\) −2.47457 −0.424386
\(35\) −0.903212 −0.152671
\(36\) −10.3319 −1.72198
\(37\) 3.95407 0.650045 0.325022 0.945706i \(-0.394628\pi\)
0.325022 + 0.945706i \(0.394628\pi\)
\(38\) −0.341219 −0.0553531
\(39\) −1.80642 −0.289259
\(40\) −1.21432 −0.192001
\(41\) 3.67307 0.573637 0.286819 0.957985i \(-0.407402\pi\)
0.286819 + 0.957985i \(0.407402\pi\)
\(42\) −0.815792 −0.125879
\(43\) −10.5161 −1.60368 −0.801842 0.597536i \(-0.796146\pi\)
−0.801842 + 0.597536i \(0.796146\pi\)
\(44\) 2.90321 0.437676
\(45\) 5.42864 0.809254
\(46\) 2.34122 0.345194
\(47\) 6.90321 1.00694 0.503468 0.864014i \(-0.332057\pi\)
0.503468 + 0.864014i \(0.332057\pi\)
\(48\) 9.95407 1.43675
\(49\) −6.18421 −0.883458
\(50\) 0.311108 0.0439973
\(51\) −23.0923 −3.23357
\(52\) 1.18421 0.164220
\(53\) 6.42864 0.883042 0.441521 0.897251i \(-0.354439\pi\)
0.441521 + 0.897251i \(0.354439\pi\)
\(54\) 2.19358 0.298508
\(55\) −1.52543 −0.205689
\(56\) 1.09679 0.146564
\(57\) −3.18421 −0.421759
\(58\) −0.311108 −0.0408505
\(59\) −1.67307 −0.217815 −0.108908 0.994052i \(-0.534735\pi\)
−0.108908 + 0.994052i \(0.534735\pi\)
\(60\) −5.52543 −0.713330
\(61\) −1.86665 −0.239000 −0.119500 0.992834i \(-0.538129\pi\)
−0.119500 + 0.992834i \(0.538129\pi\)
\(62\) −2.14764 −0.272751
\(63\) −4.90321 −0.617747
\(64\) −5.76986 −0.721232
\(65\) −0.622216 −0.0771764
\(66\) −1.37778 −0.169594
\(67\) 11.5254 1.40806 0.704028 0.710173i \(-0.251383\pi\)
0.704028 + 0.710173i \(0.251383\pi\)
\(68\) 15.1383 1.83579
\(69\) 21.8479 2.63018
\(70\) −0.280996 −0.0335855
\(71\) 13.6731 1.62269 0.811347 0.584564i \(-0.198734\pi\)
0.811347 + 0.584564i \(0.198734\pi\)
\(72\) −6.59210 −0.776887
\(73\) 10.1891 1.19255 0.596274 0.802781i \(-0.296647\pi\)
0.596274 + 0.802781i \(0.296647\pi\)
\(74\) 1.23014 0.143001
\(75\) 2.90321 0.335234
\(76\) 2.08742 0.239444
\(77\) 1.37778 0.157013
\(78\) −0.561993 −0.0636331
\(79\) 9.13828 1.02814 0.514068 0.857749i \(-0.328138\pi\)
0.514068 + 0.857749i \(0.328138\pi\)
\(80\) 3.42864 0.383334
\(81\) 4.18421 0.464912
\(82\) 1.14272 0.126192
\(83\) 10.7096 1.17554 0.587768 0.809030i \(-0.300007\pi\)
0.587768 + 0.809030i \(0.300007\pi\)
\(84\) 4.99063 0.544523
\(85\) −7.95407 −0.862740
\(86\) −3.27163 −0.352789
\(87\) −2.90321 −0.311257
\(88\) 1.85236 0.197462
\(89\) −7.80642 −0.827479 −0.413740 0.910395i \(-0.635778\pi\)
−0.413740 + 0.910395i \(0.635778\pi\)
\(90\) 1.68889 0.178025
\(91\) 0.561993 0.0589128
\(92\) −14.3225 −1.49322
\(93\) −20.0415 −2.07821
\(94\) 2.14764 0.221512
\(95\) −1.09679 −0.112528
\(96\) 10.1476 1.03569
\(97\) −4.08742 −0.415015 −0.207507 0.978233i \(-0.566535\pi\)
−0.207507 + 0.978233i \(0.566535\pi\)
\(98\) −1.92396 −0.194349
\(99\) −8.28100 −0.832271
\(100\) −1.90321 −0.190321
\(101\) 13.9081 1.38391 0.691956 0.721940i \(-0.256749\pi\)
0.691956 + 0.721940i \(0.256749\pi\)
\(102\) −7.18421 −0.711343
\(103\) −12.9447 −1.27548 −0.637740 0.770252i \(-0.720130\pi\)
−0.637740 + 0.770252i \(0.720130\pi\)
\(104\) 0.755569 0.0740896
\(105\) −2.62222 −0.255902
\(106\) 2.00000 0.194257
\(107\) 11.0049 1.06389 0.531943 0.846780i \(-0.321462\pi\)
0.531943 + 0.846780i \(0.321462\pi\)
\(108\) −13.4193 −1.29127
\(109\) −18.0415 −1.72806 −0.864031 0.503439i \(-0.832068\pi\)
−0.864031 + 0.503439i \(0.832068\pi\)
\(110\) −0.474572 −0.0452487
\(111\) 11.4795 1.08959
\(112\) −3.09679 −0.292619
\(113\) −10.2810 −0.967155 −0.483577 0.875302i \(-0.660663\pi\)
−0.483577 + 0.875302i \(0.660663\pi\)
\(114\) −0.990632 −0.0927812
\(115\) 7.52543 0.701750
\(116\) 1.90321 0.176709
\(117\) −3.37778 −0.312276
\(118\) −0.520505 −0.0479164
\(119\) 7.18421 0.658575
\(120\) −3.52543 −0.321826
\(121\) −8.67307 −0.788461
\(122\) −0.580728 −0.0525767
\(123\) 10.6637 0.961514
\(124\) 13.1383 1.17985
\(125\) 1.00000 0.0894427
\(126\) −1.52543 −0.135896
\(127\) 6.22077 0.552004 0.276002 0.961157i \(-0.410990\pi\)
0.276002 + 0.961157i \(0.410990\pi\)
\(128\) −8.78568 −0.776552
\(129\) −30.5303 −2.68805
\(130\) −0.193576 −0.0169778
\(131\) −11.7605 −1.02752 −0.513759 0.857934i \(-0.671748\pi\)
−0.513759 + 0.857934i \(0.671748\pi\)
\(132\) 8.42864 0.733619
\(133\) 0.990632 0.0858987
\(134\) 3.58565 0.309753
\(135\) 7.05086 0.606841
\(136\) 9.65878 0.828234
\(137\) −3.56691 −0.304742 −0.152371 0.988323i \(-0.548691\pi\)
−0.152371 + 0.988323i \(0.548691\pi\)
\(138\) 6.79706 0.578604
\(139\) −8.56199 −0.726219 −0.363109 0.931747i \(-0.618285\pi\)
−0.363109 + 0.931747i \(0.618285\pi\)
\(140\) 1.71900 0.145282
\(141\) 20.0415 1.68780
\(142\) 4.25380 0.356971
\(143\) 0.949145 0.0793715
\(144\) 18.6128 1.55107
\(145\) −1.00000 −0.0830455
\(146\) 3.16992 0.262344
\(147\) −17.9541 −1.48083
\(148\) −7.52543 −0.618586
\(149\) −5.61285 −0.459822 −0.229911 0.973212i \(-0.573844\pi\)
−0.229911 + 0.973212i \(0.573844\pi\)
\(150\) 0.903212 0.0737469
\(151\) 10.7971 0.878652 0.439326 0.898328i \(-0.355217\pi\)
0.439326 + 0.898328i \(0.355217\pi\)
\(152\) 1.33185 0.108027
\(153\) −43.1798 −3.49088
\(154\) 0.428639 0.0345408
\(155\) −6.90321 −0.554479
\(156\) 3.43801 0.275261
\(157\) 2.28100 0.182043 0.0910217 0.995849i \(-0.470987\pi\)
0.0910217 + 0.995849i \(0.470987\pi\)
\(158\) 2.84299 0.226176
\(159\) 18.6637 1.48013
\(160\) 3.49532 0.276329
\(161\) −6.79706 −0.535683
\(162\) 1.30174 0.102274
\(163\) 16.3225 1.27848 0.639238 0.769009i \(-0.279250\pi\)
0.639238 + 0.769009i \(0.279250\pi\)
\(164\) −6.99063 −0.545877
\(165\) −4.42864 −0.344769
\(166\) 3.33185 0.258602
\(167\) −4.76986 −0.369103 −0.184551 0.982823i \(-0.559083\pi\)
−0.184551 + 0.982823i \(0.559083\pi\)
\(168\) 3.18421 0.245667
\(169\) −12.6128 −0.970219
\(170\) −2.47457 −0.189791
\(171\) −5.95407 −0.455319
\(172\) 20.0143 1.52608
\(173\) 4.23506 0.321986 0.160993 0.986956i \(-0.448530\pi\)
0.160993 + 0.986956i \(0.448530\pi\)
\(174\) −0.903212 −0.0684723
\(175\) −0.903212 −0.0682764
\(176\) −5.23014 −0.394237
\(177\) −4.85728 −0.365095
\(178\) −2.42864 −0.182034
\(179\) 9.71456 0.726100 0.363050 0.931770i \(-0.381735\pi\)
0.363050 + 0.931770i \(0.381735\pi\)
\(180\) −10.3319 −0.770091
\(181\) 0.326929 0.0243005 0.0121502 0.999926i \(-0.496132\pi\)
0.0121502 + 0.999926i \(0.496132\pi\)
\(182\) 0.174840 0.0129600
\(183\) −5.41927 −0.400604
\(184\) −9.13828 −0.673683
\(185\) 3.95407 0.290709
\(186\) −6.23506 −0.457177
\(187\) 12.1334 0.887279
\(188\) −13.1383 −0.958207
\(189\) −6.36842 −0.463234
\(190\) −0.341219 −0.0247547
\(191\) −14.9447 −1.08136 −0.540680 0.841228i \(-0.681833\pi\)
−0.540680 + 0.841228i \(0.681833\pi\)
\(192\) −16.7511 −1.20891
\(193\) −14.1476 −1.01837 −0.509185 0.860657i \(-0.670053\pi\)
−0.509185 + 0.860657i \(0.670053\pi\)
\(194\) −1.27163 −0.0912976
\(195\) −1.80642 −0.129361
\(196\) 11.7699 0.840704
\(197\) −5.70471 −0.406444 −0.203222 0.979133i \(-0.565141\pi\)
−0.203222 + 0.979133i \(0.565141\pi\)
\(198\) −2.57628 −0.183088
\(199\) −22.1432 −1.56969 −0.784845 0.619692i \(-0.787257\pi\)
−0.784845 + 0.619692i \(0.787257\pi\)
\(200\) −1.21432 −0.0858654
\(201\) 33.4608 2.36014
\(202\) 4.32693 0.304442
\(203\) 0.903212 0.0633930
\(204\) 43.9496 3.07709
\(205\) 3.67307 0.256538
\(206\) −4.02720 −0.280588
\(207\) 40.8528 2.83947
\(208\) −2.13335 −0.147921
\(209\) 1.67307 0.115729
\(210\) −0.815792 −0.0562950
\(211\) −20.8430 −1.43489 −0.717445 0.696615i \(-0.754689\pi\)
−0.717445 + 0.696615i \(0.754689\pi\)
\(212\) −12.2351 −0.840308
\(213\) 39.6958 2.71991
\(214\) 3.42372 0.234040
\(215\) −10.5161 −0.717189
\(216\) −8.56199 −0.582570
\(217\) 6.23506 0.423264
\(218\) −5.61285 −0.380150
\(219\) 29.5812 1.99891
\(220\) 2.90321 0.195735
\(221\) 4.94914 0.332916
\(222\) 3.57136 0.239694
\(223\) 9.03657 0.605133 0.302567 0.953128i \(-0.402157\pi\)
0.302567 + 0.953128i \(0.402157\pi\)
\(224\) −3.15701 −0.210937
\(225\) 5.42864 0.361909
\(226\) −3.19850 −0.212761
\(227\) 19.4050 1.28795 0.643977 0.765045i \(-0.277283\pi\)
0.643977 + 0.765045i \(0.277283\pi\)
\(228\) 6.06022 0.401348
\(229\) −25.6128 −1.69254 −0.846272 0.532751i \(-0.821158\pi\)
−0.846272 + 0.532751i \(0.821158\pi\)
\(230\) 2.34122 0.154375
\(231\) 4.00000 0.263181
\(232\) 1.21432 0.0797240
\(233\) −3.12399 −0.204659 −0.102330 0.994751i \(-0.532630\pi\)
−0.102330 + 0.994751i \(0.532630\pi\)
\(234\) −1.05086 −0.0686965
\(235\) 6.90321 0.450316
\(236\) 3.18421 0.207274
\(237\) 26.5303 1.72333
\(238\) 2.23506 0.144878
\(239\) 13.9398 0.901689 0.450845 0.892602i \(-0.351123\pi\)
0.450845 + 0.892602i \(0.351123\pi\)
\(240\) 9.95407 0.642532
\(241\) −18.4701 −1.18977 −0.594883 0.803813i \(-0.702802\pi\)
−0.594883 + 0.803813i \(0.702802\pi\)
\(242\) −2.69826 −0.173451
\(243\) −9.00492 −0.577666
\(244\) 3.55262 0.227433
\(245\) −6.18421 −0.395095
\(246\) 3.31756 0.211520
\(247\) 0.682439 0.0434225
\(248\) 8.38271 0.532302
\(249\) 31.0923 1.97040
\(250\) 0.311108 0.0196762
\(251\) −13.7921 −0.870552 −0.435276 0.900297i \(-0.643349\pi\)
−0.435276 + 0.900297i \(0.643349\pi\)
\(252\) 9.33185 0.587851
\(253\) −11.4795 −0.721710
\(254\) 1.93533 0.121433
\(255\) −23.0923 −1.44610
\(256\) 8.80642 0.550401
\(257\) −1.47949 −0.0922883 −0.0461442 0.998935i \(-0.514693\pi\)
−0.0461442 + 0.998935i \(0.514693\pi\)
\(258\) −9.49823 −0.591334
\(259\) −3.57136 −0.221914
\(260\) 1.18421 0.0734415
\(261\) −5.42864 −0.336024
\(262\) −3.65878 −0.226040
\(263\) −0.442930 −0.0273122 −0.0136561 0.999907i \(-0.504347\pi\)
−0.0136561 + 0.999907i \(0.504347\pi\)
\(264\) 5.37778 0.330980
\(265\) 6.42864 0.394908
\(266\) 0.308193 0.0188965
\(267\) −22.6637 −1.38700
\(268\) −21.9353 −1.33991
\(269\) 3.93978 0.240212 0.120106 0.992761i \(-0.461676\pi\)
0.120106 + 0.992761i \(0.461676\pi\)
\(270\) 2.19358 0.133497
\(271\) 6.20787 0.377101 0.188551 0.982063i \(-0.439621\pi\)
0.188551 + 0.982063i \(0.439621\pi\)
\(272\) −27.2716 −1.65359
\(273\) 1.63158 0.0987479
\(274\) −1.10970 −0.0670391
\(275\) −1.52543 −0.0919867
\(276\) −41.5812 −2.50289
\(277\) 5.57136 0.334751 0.167375 0.985893i \(-0.446471\pi\)
0.167375 + 0.985893i \(0.446471\pi\)
\(278\) −2.66370 −0.159758
\(279\) −37.4750 −2.24357
\(280\) 1.09679 0.0655456
\(281\) 6.69535 0.399411 0.199705 0.979856i \(-0.436001\pi\)
0.199705 + 0.979856i \(0.436001\pi\)
\(282\) 6.23506 0.371293
\(283\) 25.8020 1.53377 0.766884 0.641785i \(-0.221806\pi\)
0.766884 + 0.641785i \(0.221806\pi\)
\(284\) −26.0228 −1.54417
\(285\) −3.18421 −0.188616
\(286\) 0.295286 0.0174607
\(287\) −3.31756 −0.195829
\(288\) 18.9748 1.11810
\(289\) 46.2672 2.72160
\(290\) −0.311108 −0.0182689
\(291\) −11.8666 −0.695635
\(292\) −19.3921 −1.13484
\(293\) −18.8430 −1.10082 −0.550410 0.834895i \(-0.685528\pi\)
−0.550410 + 0.834895i \(0.685528\pi\)
\(294\) −5.58565 −0.325762
\(295\) −1.67307 −0.0974099
\(296\) −4.80150 −0.279082
\(297\) −10.7556 −0.624101
\(298\) −1.74620 −0.101155
\(299\) −4.68244 −0.270792
\(300\) −5.52543 −0.319011
\(301\) 9.49823 0.547469
\(302\) 3.35905 0.193292
\(303\) 40.3783 2.31967
\(304\) −3.76049 −0.215679
\(305\) −1.86665 −0.106884
\(306\) −13.4336 −0.767946
\(307\) −1.65878 −0.0946716 −0.0473358 0.998879i \(-0.515073\pi\)
−0.0473358 + 0.998879i \(0.515073\pi\)
\(308\) −2.62222 −0.149415
\(309\) −37.5812 −2.13792
\(310\) −2.14764 −0.121978
\(311\) 21.3002 1.20782 0.603912 0.797051i \(-0.293608\pi\)
0.603912 + 0.797051i \(0.293608\pi\)
\(312\) 2.19358 0.124187
\(313\) 8.62222 0.487356 0.243678 0.969856i \(-0.421646\pi\)
0.243678 + 0.969856i \(0.421646\pi\)
\(314\) 0.709636 0.0400471
\(315\) −4.90321 −0.276265
\(316\) −17.3921 −0.978381
\(317\) −27.5955 −1.54992 −0.774959 0.632012i \(-0.782229\pi\)
−0.774959 + 0.632012i \(0.782229\pi\)
\(318\) 5.80642 0.325608
\(319\) 1.52543 0.0854075
\(320\) −5.76986 −0.322545
\(321\) 31.9496 1.78325
\(322\) −2.11462 −0.117843
\(323\) 8.72393 0.485412
\(324\) −7.96343 −0.442413
\(325\) −0.622216 −0.0345143
\(326\) 5.07805 0.281247
\(327\) −52.3783 −2.89652
\(328\) −4.46028 −0.246278
\(329\) −6.23506 −0.343750
\(330\) −1.37778 −0.0758445
\(331\) 16.9131 0.929626 0.464813 0.885409i \(-0.346122\pi\)
0.464813 + 0.885409i \(0.346122\pi\)
\(332\) −20.3827 −1.11865
\(333\) 21.4652 1.17629
\(334\) −1.48394 −0.0811976
\(335\) 11.5254 0.629701
\(336\) −8.99063 −0.490479
\(337\) 11.9956 0.653439 0.326720 0.945121i \(-0.394057\pi\)
0.326720 + 0.945121i \(0.394057\pi\)
\(338\) −3.92396 −0.213435
\(339\) −29.8479 −1.62112
\(340\) 15.1383 0.820988
\(341\) 10.5303 0.570250
\(342\) −1.85236 −0.100164
\(343\) 11.9081 0.642979
\(344\) 12.7699 0.688505
\(345\) 21.8479 1.17625
\(346\) 1.31756 0.0708325
\(347\) 6.14764 0.330023 0.165011 0.986292i \(-0.447234\pi\)
0.165011 + 0.986292i \(0.447234\pi\)
\(348\) 5.52543 0.296194
\(349\) −7.12399 −0.381338 −0.190669 0.981654i \(-0.561066\pi\)
−0.190669 + 0.981654i \(0.561066\pi\)
\(350\) −0.280996 −0.0150199
\(351\) −4.38715 −0.234169
\(352\) −5.33185 −0.284189
\(353\) −16.9175 −0.900428 −0.450214 0.892921i \(-0.648652\pi\)
−0.450214 + 0.892921i \(0.648652\pi\)
\(354\) −1.51114 −0.0803160
\(355\) 13.6731 0.725691
\(356\) 14.8573 0.787434
\(357\) 20.8573 1.10388
\(358\) 3.02227 0.159732
\(359\) 36.7096 1.93746 0.968730 0.248116i \(-0.0798115\pi\)
0.968730 + 0.248116i \(0.0798115\pi\)
\(360\) −6.59210 −0.347434
\(361\) −17.7971 −0.936687
\(362\) 0.101710 0.00534577
\(363\) −25.1798 −1.32159
\(364\) −1.06959 −0.0560618
\(365\) 10.1891 0.533323
\(366\) −1.68598 −0.0881275
\(367\) 8.41435 0.439225 0.219613 0.975587i \(-0.429521\pi\)
0.219613 + 0.975587i \(0.429521\pi\)
\(368\) 25.8020 1.34502
\(369\) 19.9398 1.03802
\(370\) 1.23014 0.0639520
\(371\) −5.80642 −0.301455
\(372\) 38.1432 1.97763
\(373\) −8.66370 −0.448590 −0.224295 0.974521i \(-0.572008\pi\)
−0.224295 + 0.974521i \(0.572008\pi\)
\(374\) 3.77478 0.195189
\(375\) 2.90321 0.149921
\(376\) −8.38271 −0.432305
\(377\) 0.622216 0.0320457
\(378\) −1.98126 −0.101905
\(379\) −2.76986 −0.142278 −0.0711390 0.997466i \(-0.522663\pi\)
−0.0711390 + 0.997466i \(0.522663\pi\)
\(380\) 2.08742 0.107082
\(381\) 18.0602 0.925253
\(382\) −4.64941 −0.237885
\(383\) 1.67752 0.0857171 0.0428585 0.999081i \(-0.486354\pi\)
0.0428585 + 0.999081i \(0.486354\pi\)
\(384\) −25.5067 −1.30163
\(385\) 1.37778 0.0702184
\(386\) −4.40144 −0.224028
\(387\) −57.0879 −2.90194
\(388\) 7.77923 0.394930
\(389\) −5.77478 −0.292793 −0.146397 0.989226i \(-0.546768\pi\)
−0.146397 + 0.989226i \(0.546768\pi\)
\(390\) −0.561993 −0.0284576
\(391\) −59.8578 −3.02714
\(392\) 7.50961 0.379292
\(393\) −34.1432 −1.72230
\(394\) −1.77478 −0.0894122
\(395\) 9.13828 0.459797
\(396\) 15.7605 0.791994
\(397\) 29.9081 1.50105 0.750523 0.660844i \(-0.229802\pi\)
0.750523 + 0.660844i \(0.229802\pi\)
\(398\) −6.88892 −0.345310
\(399\) 2.87601 0.143981
\(400\) 3.42864 0.171432
\(401\) 8.53035 0.425985 0.212993 0.977054i \(-0.431679\pi\)
0.212993 + 0.977054i \(0.431679\pi\)
\(402\) 10.4099 0.519199
\(403\) 4.29529 0.213963
\(404\) −26.4701 −1.31694
\(405\) 4.18421 0.207915
\(406\) 0.280996 0.0139456
\(407\) −6.03164 −0.298977
\(408\) 28.0415 1.38826
\(409\) 5.09234 0.251800 0.125900 0.992043i \(-0.459818\pi\)
0.125900 + 0.992043i \(0.459818\pi\)
\(410\) 1.14272 0.0564350
\(411\) −10.3555 −0.510800
\(412\) 24.6365 1.21375
\(413\) 1.51114 0.0743582
\(414\) 12.7096 0.624645
\(415\) 10.7096 0.525715
\(416\) −2.17484 −0.106630
\(417\) −24.8573 −1.21727
\(418\) 0.520505 0.0254588
\(419\) −24.3368 −1.18893 −0.594465 0.804122i \(-0.702636\pi\)
−0.594465 + 0.804122i \(0.702636\pi\)
\(420\) 4.99063 0.243518
\(421\) 24.5018 1.19414 0.597072 0.802188i \(-0.296331\pi\)
0.597072 + 0.802188i \(0.296331\pi\)
\(422\) −6.48442 −0.315656
\(423\) 37.4750 1.82210
\(424\) −7.80642 −0.379113
\(425\) −7.95407 −0.385829
\(426\) 12.3497 0.598344
\(427\) 1.68598 0.0815902
\(428\) −20.9447 −1.01240
\(429\) 2.75557 0.133040
\(430\) −3.27163 −0.157772
\(431\) 4.26671 0.205520 0.102760 0.994706i \(-0.467233\pi\)
0.102760 + 0.994706i \(0.467233\pi\)
\(432\) 24.1748 1.16311
\(433\) 27.0049 1.29777 0.648887 0.760885i \(-0.275235\pi\)
0.648887 + 0.760885i \(0.275235\pi\)
\(434\) 1.93978 0.0931123
\(435\) −2.90321 −0.139198
\(436\) 34.3368 1.64443
\(437\) −8.25380 −0.394833
\(438\) 9.20294 0.439734
\(439\) −2.03164 −0.0969650 −0.0484825 0.998824i \(-0.515439\pi\)
−0.0484825 + 0.998824i \(0.515439\pi\)
\(440\) 1.85236 0.0883076
\(441\) −33.5718 −1.59866
\(442\) 1.53972 0.0732369
\(443\) −3.46520 −0.164637 −0.0823184 0.996606i \(-0.526232\pi\)
−0.0823184 + 0.996606i \(0.526232\pi\)
\(444\) −21.8479 −1.03686
\(445\) −7.80642 −0.370060
\(446\) 2.81135 0.133121
\(447\) −16.2953 −0.770741
\(448\) 5.21141 0.246216
\(449\) 37.3590 1.76308 0.881541 0.472107i \(-0.156506\pi\)
0.881541 + 0.472107i \(0.156506\pi\)
\(450\) 1.68889 0.0796151
\(451\) −5.60300 −0.263835
\(452\) 19.5669 0.920350
\(453\) 31.3461 1.47277
\(454\) 6.03704 0.283332
\(455\) 0.561993 0.0263466
\(456\) 3.86665 0.181072
\(457\) 13.4509 0.629207 0.314604 0.949223i \(-0.398128\pi\)
0.314604 + 0.949223i \(0.398128\pi\)
\(458\) −7.96836 −0.372337
\(459\) −56.0830 −2.61773
\(460\) −14.3225 −0.667789
\(461\) −16.2766 −0.758075 −0.379037 0.925381i \(-0.623745\pi\)
−0.379037 + 0.925381i \(0.623745\pi\)
\(462\) 1.24443 0.0578962
\(463\) −30.3926 −1.41246 −0.706231 0.707982i \(-0.749606\pi\)
−0.706231 + 0.707982i \(0.749606\pi\)
\(464\) −3.42864 −0.159171
\(465\) −20.0415 −0.929402
\(466\) −0.971896 −0.0450222
\(467\) −1.18865 −0.0550043 −0.0275022 0.999622i \(-0.508755\pi\)
−0.0275022 + 0.999622i \(0.508755\pi\)
\(468\) 6.42864 0.297164
\(469\) −10.4099 −0.480685
\(470\) 2.14764 0.0990634
\(471\) 6.62222 0.305136
\(472\) 2.03164 0.0935139
\(473\) 16.0415 0.737588
\(474\) 8.25380 0.379110
\(475\) −1.09679 −0.0503241
\(476\) −13.6731 −0.626704
\(477\) 34.8988 1.59790
\(478\) 4.33677 0.198359
\(479\) 41.0464 1.87546 0.937729 0.347367i \(-0.112924\pi\)
0.937729 + 0.347367i \(0.112924\pi\)
\(480\) 10.1476 0.463174
\(481\) −2.46028 −0.112179
\(482\) −5.74620 −0.261732
\(483\) −19.7333 −0.897896
\(484\) 16.5067 0.750304
\(485\) −4.08742 −0.185600
\(486\) −2.80150 −0.127079
\(487\) −10.1476 −0.459834 −0.229917 0.973210i \(-0.573845\pi\)
−0.229917 + 0.973210i \(0.573845\pi\)
\(488\) 2.26671 0.102609
\(489\) 47.3876 2.14294
\(490\) −1.92396 −0.0869155
\(491\) −29.2083 −1.31815 −0.659077 0.752075i \(-0.729053\pi\)
−0.659077 + 0.752075i \(0.729053\pi\)
\(492\) −20.2953 −0.914982
\(493\) 7.95407 0.358233
\(494\) 0.212312 0.00955237
\(495\) −8.28100 −0.372203
\(496\) −23.6686 −1.06275
\(497\) −12.3497 −0.553959
\(498\) 9.67307 0.433461
\(499\) 21.9813 0.984017 0.492008 0.870591i \(-0.336263\pi\)
0.492008 + 0.870591i \(0.336263\pi\)
\(500\) −1.90321 −0.0851142
\(501\) −13.8479 −0.618679
\(502\) −4.29084 −0.191510
\(503\) 5.77923 0.257683 0.128841 0.991665i \(-0.458874\pi\)
0.128841 + 0.991665i \(0.458874\pi\)
\(504\) 5.95407 0.265215
\(505\) 13.9081 0.618904
\(506\) −3.57136 −0.158766
\(507\) −36.6178 −1.62625
\(508\) −11.8394 −0.525291
\(509\) 13.6543 0.605218 0.302609 0.953115i \(-0.402142\pi\)
0.302609 + 0.953115i \(0.402142\pi\)
\(510\) −7.18421 −0.318122
\(511\) −9.20294 −0.407114
\(512\) 20.3111 0.897633
\(513\) −7.73329 −0.341433
\(514\) −0.460282 −0.0203022
\(515\) −12.9447 −0.570412
\(516\) 58.1057 2.55796
\(517\) −10.5303 −0.463124
\(518\) −1.11108 −0.0488180
\(519\) 12.2953 0.539703
\(520\) 0.755569 0.0331339
\(521\) −19.6731 −0.861893 −0.430946 0.902378i \(-0.641820\pi\)
−0.430946 + 0.902378i \(0.641820\pi\)
\(522\) −1.68889 −0.0739208
\(523\) −15.1383 −0.661951 −0.330975 0.943639i \(-0.607378\pi\)
−0.330975 + 0.943639i \(0.607378\pi\)
\(524\) 22.3827 0.977793
\(525\) −2.62222 −0.114443
\(526\) −0.137799 −0.00600832
\(527\) 54.9086 2.39186
\(528\) −15.1842 −0.660808
\(529\) 33.6321 1.46226
\(530\) 2.00000 0.0868744
\(531\) −9.08250 −0.394147
\(532\) −1.88538 −0.0817417
\(533\) −2.28544 −0.0989935
\(534\) −7.05086 −0.305120
\(535\) 11.0049 0.475784
\(536\) −13.9956 −0.604516
\(537\) 28.2034 1.21707
\(538\) 1.22570 0.0528435
\(539\) 9.43356 0.406332
\(540\) −13.4193 −0.577474
\(541\) 2.68244 0.115327 0.0576635 0.998336i \(-0.481635\pi\)
0.0576635 + 0.998336i \(0.481635\pi\)
\(542\) 1.93132 0.0829571
\(543\) 0.949145 0.0407317
\(544\) −27.8020 −1.19200
\(545\) −18.0415 −0.772812
\(546\) 0.507598 0.0217232
\(547\) −15.3635 −0.656896 −0.328448 0.944522i \(-0.606525\pi\)
−0.328448 + 0.944522i \(0.606525\pi\)
\(548\) 6.78859 0.289994
\(549\) −10.1334 −0.432481
\(550\) −0.474572 −0.0202358
\(551\) 1.09679 0.0467247
\(552\) −26.5303 −1.12921
\(553\) −8.25380 −0.350987
\(554\) 1.73329 0.0736406
\(555\) 11.4795 0.487277
\(556\) 16.2953 0.691074
\(557\) −9.87955 −0.418610 −0.209305 0.977850i \(-0.567120\pi\)
−0.209305 + 0.977850i \(0.567120\pi\)
\(558\) −11.6588 −0.493556
\(559\) 6.54326 0.276750
\(560\) −3.09679 −0.130863
\(561\) 35.2257 1.48723
\(562\) 2.08297 0.0878650
\(563\) −27.4938 −1.15872 −0.579362 0.815070i \(-0.696698\pi\)
−0.579362 + 0.815070i \(0.696698\pi\)
\(564\) −38.1432 −1.60612
\(565\) −10.2810 −0.432525
\(566\) 8.02720 0.337408
\(567\) −3.77923 −0.158713
\(568\) −16.6035 −0.696667
\(569\) 17.3590 0.727729 0.363865 0.931452i \(-0.381457\pi\)
0.363865 + 0.931452i \(0.381457\pi\)
\(570\) −0.990632 −0.0414930
\(571\) −25.4479 −1.06496 −0.532480 0.846443i \(-0.678740\pi\)
−0.532480 + 0.846443i \(0.678740\pi\)
\(572\) −1.80642 −0.0755304
\(573\) −43.3876 −1.81254
\(574\) −1.03212 −0.0430798
\(575\) 7.52543 0.313832
\(576\) −31.3225 −1.30510
\(577\) −10.6178 −0.442024 −0.221012 0.975271i \(-0.570936\pi\)
−0.221012 + 0.975271i \(0.570936\pi\)
\(578\) 14.3941 0.598715
\(579\) −41.0736 −1.70696
\(580\) 1.90321 0.0790266
\(581\) −9.67307 −0.401307
\(582\) −3.69181 −0.153030
\(583\) −9.80642 −0.406141
\(584\) −12.3729 −0.511993
\(585\) −3.37778 −0.139654
\(586\) −5.86220 −0.242165
\(587\) −8.94470 −0.369187 −0.184594 0.982815i \(-0.559097\pi\)
−0.184594 + 0.982815i \(0.559097\pi\)
\(588\) 34.1704 1.40916
\(589\) 7.57136 0.311972
\(590\) −0.520505 −0.0214289
\(591\) −16.5620 −0.681269
\(592\) 13.5571 0.557192
\(593\) 14.1619 0.581561 0.290780 0.956790i \(-0.406085\pi\)
0.290780 + 0.956790i \(0.406085\pi\)
\(594\) −3.34614 −0.137294
\(595\) 7.18421 0.294524
\(596\) 10.6824 0.437570
\(597\) −64.2864 −2.63107
\(598\) −1.45674 −0.0595707
\(599\) −22.5575 −0.921676 −0.460838 0.887484i \(-0.652451\pi\)
−0.460838 + 0.887484i \(0.652451\pi\)
\(600\) −3.52543 −0.143925
\(601\) −40.6133 −1.65665 −0.828326 0.560246i \(-0.810706\pi\)
−0.828326 + 0.560246i \(0.810706\pi\)
\(602\) 2.95497 0.120436
\(603\) 62.5674 2.54794
\(604\) −20.5491 −0.836130
\(605\) −8.67307 −0.352610
\(606\) 12.5620 0.510296
\(607\) 13.5955 0.551824 0.275912 0.961183i \(-0.411020\pi\)
0.275912 + 0.961183i \(0.411020\pi\)
\(608\) −3.83362 −0.155474
\(609\) 2.62222 0.106258
\(610\) −0.580728 −0.0235130
\(611\) −4.29529 −0.173769
\(612\) 82.1802 3.32194
\(613\) −42.0830 −1.69972 −0.849858 0.527012i \(-0.823312\pi\)
−0.849858 + 0.527012i \(0.823312\pi\)
\(614\) −0.516060 −0.0208265
\(615\) 10.6637 0.430002
\(616\) −1.67307 −0.0674099
\(617\) 33.5067 1.34893 0.674464 0.738307i \(-0.264375\pi\)
0.674464 + 0.738307i \(0.264375\pi\)
\(618\) −11.6918 −0.470313
\(619\) −14.6780 −0.589958 −0.294979 0.955504i \(-0.595313\pi\)
−0.294979 + 0.955504i \(0.595313\pi\)
\(620\) 13.1383 0.527646
\(621\) 53.0607 2.12925
\(622\) 6.62666 0.265705
\(623\) 7.05086 0.282487
\(624\) −6.19358 −0.247941
\(625\) 1.00000 0.0400000
\(626\) 2.68244 0.107212
\(627\) 4.85728 0.193981
\(628\) −4.34122 −0.173234
\(629\) −31.4509 −1.25403
\(630\) −1.52543 −0.0607745
\(631\) 11.3176 0.450545 0.225273 0.974296i \(-0.427673\pi\)
0.225273 + 0.974296i \(0.427673\pi\)
\(632\) −11.0968 −0.441407
\(633\) −60.5116 −2.40512
\(634\) −8.58517 −0.340961
\(635\) 6.22077 0.246864
\(636\) −35.5210 −1.40850
\(637\) 3.84791 0.152460
\(638\) 0.474572 0.0187885
\(639\) 74.2262 2.93634
\(640\) −8.78568 −0.347285
\(641\) 34.8988 1.37842 0.689209 0.724562i \(-0.257958\pi\)
0.689209 + 0.724562i \(0.257958\pi\)
\(642\) 9.93978 0.392292
\(643\) −41.9768 −1.65540 −0.827702 0.561168i \(-0.810352\pi\)
−0.827702 + 0.561168i \(0.810352\pi\)
\(644\) 12.9362 0.509759
\(645\) −30.5303 −1.20213
\(646\) 2.71408 0.106784
\(647\) 5.46520 0.214859 0.107430 0.994213i \(-0.465738\pi\)
0.107430 + 0.994213i \(0.465738\pi\)
\(648\) −5.08097 −0.199599
\(649\) 2.55215 0.100181
\(650\) −0.193576 −0.00759268
\(651\) 18.1017 0.709462
\(652\) −31.0651 −1.21660
\(653\) −8.76986 −0.343191 −0.171596 0.985167i \(-0.554892\pi\)
−0.171596 + 0.985167i \(0.554892\pi\)
\(654\) −16.2953 −0.637196
\(655\) −11.7605 −0.459520
\(656\) 12.5936 0.491699
\(657\) 55.3131 2.15797
\(658\) −1.93978 −0.0756204
\(659\) 3.29036 0.128174 0.0640872 0.997944i \(-0.479586\pi\)
0.0640872 + 0.997944i \(0.479586\pi\)
\(660\) 8.42864 0.328084
\(661\) 19.7560 0.768421 0.384211 0.923246i \(-0.374474\pi\)
0.384211 + 0.923246i \(0.374474\pi\)
\(662\) 5.26178 0.204505
\(663\) 14.3684 0.558023
\(664\) −13.0049 −0.504689
\(665\) 0.990632 0.0384151
\(666\) 6.67799 0.258767
\(667\) −7.52543 −0.291386
\(668\) 9.07805 0.351240
\(669\) 26.2351 1.01431
\(670\) 3.58565 0.138526
\(671\) 2.84743 0.109924
\(672\) −9.16547 −0.353566
\(673\) 44.3970 1.71138 0.855689 0.517490i \(-0.173134\pi\)
0.855689 + 0.517490i \(0.173134\pi\)
\(674\) 3.73191 0.143748
\(675\) 7.05086 0.271388
\(676\) 24.0049 0.923266
\(677\) 6.09726 0.234337 0.117168 0.993112i \(-0.462618\pi\)
0.117168 + 0.993112i \(0.462618\pi\)
\(678\) −9.28592 −0.356624
\(679\) 3.69181 0.141679
\(680\) 9.65878 0.370397
\(681\) 56.3368 2.15883
\(682\) 3.27607 0.125447
\(683\) 37.9224 1.45106 0.725531 0.688190i \(-0.241594\pi\)
0.725531 + 0.688190i \(0.241594\pi\)
\(684\) 11.3319 0.433284
\(685\) −3.56691 −0.136285
\(686\) 3.70471 0.141447
\(687\) −74.3595 −2.83699
\(688\) −36.0558 −1.37461
\(689\) −4.00000 −0.152388
\(690\) 6.79706 0.258759
\(691\) 13.3145 0.506507 0.253254 0.967400i \(-0.418499\pi\)
0.253254 + 0.967400i \(0.418499\pi\)
\(692\) −8.06022 −0.306404
\(693\) 7.47949 0.284123
\(694\) 1.91258 0.0726005
\(695\) −8.56199 −0.324775
\(696\) 3.52543 0.133631
\(697\) −29.2159 −1.10663
\(698\) −2.21633 −0.0838892
\(699\) −9.06959 −0.343043
\(700\) 1.71900 0.0649722
\(701\) −23.4893 −0.887180 −0.443590 0.896230i \(-0.646295\pi\)
−0.443590 + 0.896230i \(0.646295\pi\)
\(702\) −1.36488 −0.0515140
\(703\) −4.33677 −0.163565
\(704\) 8.80150 0.331719
\(705\) 20.0415 0.754806
\(706\) −5.26317 −0.198082
\(707\) −12.5620 −0.472442
\(708\) 9.24443 0.347427
\(709\) 11.6731 0.438391 0.219196 0.975681i \(-0.429657\pi\)
0.219196 + 0.975681i \(0.429657\pi\)
\(710\) 4.25380 0.159642
\(711\) 49.6084 1.86046
\(712\) 9.47949 0.355259
\(713\) −51.9496 −1.94553
\(714\) 6.48886 0.242840
\(715\) 0.949145 0.0354960
\(716\) −18.4889 −0.690961
\(717\) 40.4701 1.51138
\(718\) 11.4207 0.426215
\(719\) 29.5526 1.10213 0.551063 0.834463i \(-0.314222\pi\)
0.551063 + 0.834463i \(0.314222\pi\)
\(720\) 18.6128 0.693660
\(721\) 11.6918 0.435426
\(722\) −5.53680 −0.206058
\(723\) −53.6227 −1.99425
\(724\) −0.622216 −0.0231245
\(725\) −1.00000 −0.0371391
\(726\) −7.83362 −0.290733
\(727\) 3.88094 0.143936 0.0719680 0.997407i \(-0.477072\pi\)
0.0719680 + 0.997407i \(0.477072\pi\)
\(728\) −0.682439 −0.0252929
\(729\) −38.6958 −1.43318
\(730\) 3.16992 0.117324
\(731\) 83.6454 3.09374
\(732\) 10.3140 0.381217
\(733\) 14.8845 0.549771 0.274885 0.961477i \(-0.411360\pi\)
0.274885 + 0.961477i \(0.411360\pi\)
\(734\) 2.61777 0.0966236
\(735\) −17.9541 −0.662246
\(736\) 26.3037 0.969569
\(737\) −17.5812 −0.647612
\(738\) 6.20342 0.228351
\(739\) 2.24935 0.0827438 0.0413719 0.999144i \(-0.486827\pi\)
0.0413719 + 0.999144i \(0.486827\pi\)
\(740\) −7.52543 −0.276640
\(741\) 1.98126 0.0727836
\(742\) −1.80642 −0.0663159
\(743\) −3.46520 −0.127126 −0.0635630 0.997978i \(-0.520246\pi\)
−0.0635630 + 0.997978i \(0.520246\pi\)
\(744\) 24.3368 0.892229
\(745\) −5.61285 −0.205639
\(746\) −2.69535 −0.0986836
\(747\) 58.1388 2.12719
\(748\) −23.0923 −0.844340
\(749\) −9.93978 −0.363192
\(750\) 0.903212 0.0329806
\(751\) 3.16992 0.115672 0.0578360 0.998326i \(-0.481580\pi\)
0.0578360 + 0.998326i \(0.481580\pi\)
\(752\) 23.6686 0.863106
\(753\) −40.0415 −1.45919
\(754\) 0.193576 0.00704963
\(755\) 10.7971 0.392945
\(756\) 12.1204 0.440816
\(757\) −52.0785 −1.89283 −0.946413 0.322958i \(-0.895323\pi\)
−0.946413 + 0.322958i \(0.895323\pi\)
\(758\) −0.861725 −0.0312993
\(759\) −33.3274 −1.20971
\(760\) 1.33185 0.0483113
\(761\) −14.9777 −0.542942 −0.271471 0.962447i \(-0.587510\pi\)
−0.271471 + 0.962447i \(0.587510\pi\)
\(762\) 5.61868 0.203543
\(763\) 16.2953 0.589929
\(764\) 28.4429 1.02903
\(765\) −43.1798 −1.56117
\(766\) 0.521889 0.0188566
\(767\) 1.04101 0.0375887
\(768\) 25.5669 0.922567
\(769\) −1.90813 −0.0688091 −0.0344045 0.999408i \(-0.510953\pi\)
−0.0344045 + 0.999408i \(0.510953\pi\)
\(770\) 0.428639 0.0154471
\(771\) −4.29529 −0.154691
\(772\) 26.9260 0.969087
\(773\) 21.7891 0.783698 0.391849 0.920029i \(-0.371835\pi\)
0.391849 + 0.920029i \(0.371835\pi\)
\(774\) −17.7605 −0.638388
\(775\) −6.90321 −0.247971
\(776\) 4.96343 0.178177
\(777\) −10.3684 −0.371965
\(778\) −1.79658 −0.0644105
\(779\) −4.02858 −0.144339
\(780\) 3.43801 0.123100
\(781\) −20.8573 −0.746332
\(782\) −18.6222 −0.665929
\(783\) −7.05086 −0.251977
\(784\) −21.2034 −0.757265
\(785\) 2.28100 0.0814122
\(786\) −10.6222 −0.378882
\(787\) −18.1388 −0.646577 −0.323288 0.946301i \(-0.604788\pi\)
−0.323288 + 0.946301i \(0.604788\pi\)
\(788\) 10.8573 0.386775
\(789\) −1.28592 −0.0457799
\(790\) 2.84299 0.101149
\(791\) 9.28592 0.330169
\(792\) 10.0558 0.357316
\(793\) 1.16146 0.0412445
\(794\) 9.30465 0.330210
\(795\) 18.6637 0.661933
\(796\) 42.1432 1.49373
\(797\) 2.96343 0.104970 0.0524851 0.998622i \(-0.483286\pi\)
0.0524851 + 0.998622i \(0.483286\pi\)
\(798\) 0.894751 0.0316738
\(799\) −54.9086 −1.94253
\(800\) 3.49532 0.123578
\(801\) −42.3783 −1.49736
\(802\) 2.65386 0.0937110
\(803\) −15.5428 −0.548493
\(804\) −63.6829 −2.24592
\(805\) −6.79706 −0.239565
\(806\) 1.33630 0.0470691
\(807\) 11.4380 0.402637
\(808\) −16.8889 −0.594150
\(809\) −26.2953 −0.924493 −0.462247 0.886751i \(-0.652956\pi\)
−0.462247 + 0.886751i \(0.652956\pi\)
\(810\) 1.30174 0.0457385
\(811\) 24.3783 0.856037 0.428018 0.903770i \(-0.359212\pi\)
0.428018 + 0.903770i \(0.359212\pi\)
\(812\) −1.71900 −0.0603252
\(813\) 18.0228 0.632085
\(814\) −1.87649 −0.0657710
\(815\) 16.3225 0.571752
\(816\) −79.1753 −2.77169
\(817\) 11.5339 0.403520
\(818\) 1.58427 0.0553926
\(819\) 3.05086 0.106606
\(820\) −6.99063 −0.244123
\(821\) 1.52987 0.0533929 0.0266965 0.999644i \(-0.491501\pi\)
0.0266965 + 0.999644i \(0.491501\pi\)
\(822\) −3.22168 −0.112369
\(823\) 46.7195 1.62854 0.814269 0.580487i \(-0.197138\pi\)
0.814269 + 0.580487i \(0.197138\pi\)
\(824\) 15.7190 0.547597
\(825\) −4.42864 −0.154185
\(826\) 0.470127 0.0163578
\(827\) −29.6499 −1.03103 −0.515514 0.856881i \(-0.672399\pi\)
−0.515514 + 0.856881i \(0.672399\pi\)
\(828\) −77.7516 −2.70205
\(829\) −8.79706 −0.305534 −0.152767 0.988262i \(-0.548818\pi\)
−0.152767 + 0.988262i \(0.548818\pi\)
\(830\) 3.33185 0.115650
\(831\) 16.1748 0.561099
\(832\) 3.59010 0.124464
\(833\) 49.1896 1.70432
\(834\) −7.73329 −0.267782
\(835\) −4.76986 −0.165068
\(836\) −3.18421 −0.110128
\(837\) −48.6735 −1.68240
\(838\) −7.57136 −0.261548
\(839\) 11.3319 0.391219 0.195609 0.980682i \(-0.437332\pi\)
0.195609 + 0.980682i \(0.437332\pi\)
\(840\) 3.18421 0.109866
\(841\) 1.00000 0.0344828
\(842\) 7.62269 0.262695
\(843\) 19.4380 0.669481
\(844\) 39.6686 1.36545
\(845\) −12.6128 −0.433895
\(846\) 11.6588 0.400837
\(847\) 7.83362 0.269166
\(848\) 22.0415 0.756908
\(849\) 74.9086 2.57086
\(850\) −2.47457 −0.0848771
\(851\) 29.7560 1.02002
\(852\) −75.5496 −2.58829
\(853\) 54.8845 1.87921 0.939604 0.342263i \(-0.111193\pi\)
0.939604 + 0.342263i \(0.111193\pi\)
\(854\) 0.524521 0.0179487
\(855\) −5.95407 −0.203625
\(856\) −13.3635 −0.456755
\(857\) 36.4385 1.24471 0.622357 0.782733i \(-0.286175\pi\)
0.622357 + 0.782733i \(0.286175\pi\)
\(858\) 0.857279 0.0292670
\(859\) 1.72885 0.0589875 0.0294938 0.999565i \(-0.490610\pi\)
0.0294938 + 0.999565i \(0.490610\pi\)
\(860\) 20.0143 0.682482
\(861\) −9.63158 −0.328243
\(862\) 1.32741 0.0452116
\(863\) −9.40192 −0.320045 −0.160023 0.987113i \(-0.551157\pi\)
−0.160023 + 0.987113i \(0.551157\pi\)
\(864\) 24.6450 0.838439
\(865\) 4.23506 0.143996
\(866\) 8.40144 0.285493
\(867\) 134.323 4.56186
\(868\) −11.8666 −0.402780
\(869\) −13.9398 −0.472875
\(870\) −0.903212 −0.0306218
\(871\) −7.17130 −0.242990
\(872\) 21.9081 0.741903
\(873\) −22.1891 −0.750988
\(874\) −2.56782 −0.0868579
\(875\) −0.903212 −0.0305341
\(876\) −56.2993 −1.90218
\(877\) 8.91750 0.301123 0.150561 0.988601i \(-0.451892\pi\)
0.150561 + 0.988601i \(0.451892\pi\)
\(878\) −0.632060 −0.0213310
\(879\) −54.7052 −1.84516
\(880\) −5.23014 −0.176308
\(881\) −42.1245 −1.41921 −0.709605 0.704600i \(-0.751126\pi\)
−0.709605 + 0.704600i \(0.751126\pi\)
\(882\) −10.4445 −0.351683
\(883\) −38.4340 −1.29341 −0.646704 0.762741i \(-0.723853\pi\)
−0.646704 + 0.762741i \(0.723853\pi\)
\(884\) −9.41927 −0.316804
\(885\) −4.85728 −0.163276
\(886\) −1.07805 −0.0362179
\(887\) −38.6365 −1.29729 −0.648643 0.761092i \(-0.724663\pi\)
−0.648643 + 0.761092i \(0.724663\pi\)
\(888\) −13.9398 −0.467788
\(889\) −5.61868 −0.188444
\(890\) −2.42864 −0.0814082
\(891\) −6.38271 −0.213829
\(892\) −17.1985 −0.575848
\(893\) −7.57136 −0.253366
\(894\) −5.06959 −0.169552
\(895\) 9.71456 0.324722
\(896\) 7.93533 0.265101
\(897\) −13.5941 −0.453894
\(898\) 11.6227 0.387854
\(899\) 6.90321 0.230235
\(900\) −10.3319 −0.344395
\(901\) −51.1338 −1.70351
\(902\) −1.74314 −0.0580402
\(903\) 27.5754 0.917651
\(904\) 12.4844 0.415226
\(905\) 0.326929 0.0108675
\(906\) 9.75203 0.323989
\(907\) 0.534795 0.0177576 0.00887880 0.999961i \(-0.497174\pi\)
0.00887880 + 0.999961i \(0.497174\pi\)
\(908\) −36.9318 −1.22562
\(909\) 75.5022 2.50425
\(910\) 0.174840 0.00579590
\(911\) −23.6686 −0.784177 −0.392088 0.919928i \(-0.628247\pi\)
−0.392088 + 0.919928i \(0.628247\pi\)
\(912\) −10.9175 −0.361515
\(913\) −16.3368 −0.540668
\(914\) 4.18468 0.138417
\(915\) −5.41927 −0.179156
\(916\) 48.7467 1.61064
\(917\) 10.6222 0.350776
\(918\) −17.4479 −0.575865
\(919\) 35.7748 1.18010 0.590051 0.807366i \(-0.299108\pi\)
0.590051 + 0.807366i \(0.299108\pi\)
\(920\) −9.13828 −0.301280
\(921\) −4.81579 −0.158686
\(922\) −5.06376 −0.166766
\(923\) −8.50760 −0.280031
\(924\) −7.61285 −0.250444
\(925\) 3.95407 0.130009
\(926\) −9.45536 −0.310722
\(927\) −70.2721 −2.30804
\(928\) −3.49532 −0.114739
\(929\) −52.7753 −1.73150 −0.865750 0.500477i \(-0.833158\pi\)
−0.865750 + 0.500477i \(0.833158\pi\)
\(930\) −6.23506 −0.204456
\(931\) 6.78277 0.222296
\(932\) 5.94561 0.194755
\(933\) 61.8390 2.02452
\(934\) −0.369800 −0.0121002
\(935\) 12.1334 0.396803
\(936\) 4.10171 0.134069
\(937\) 42.1245 1.37615 0.688073 0.725641i \(-0.258457\pi\)
0.688073 + 0.725641i \(0.258457\pi\)
\(938\) −3.23860 −0.105744
\(939\) 25.0321 0.816892
\(940\) −13.1383 −0.428523
\(941\) −3.89829 −0.127081 −0.0635403 0.997979i \(-0.520239\pi\)
−0.0635403 + 0.997979i \(0.520239\pi\)
\(942\) 2.06022 0.0671257
\(943\) 27.6414 0.900129
\(944\) −5.73636 −0.186703
\(945\) −6.36842 −0.207165
\(946\) 4.99063 0.162259
\(947\) 9.56691 0.310883 0.155441 0.987845i \(-0.450320\pi\)
0.155441 + 0.987845i \(0.450320\pi\)
\(948\) −50.4929 −1.63993
\(949\) −6.33984 −0.205800
\(950\) −0.341219 −0.0110706
\(951\) −80.1156 −2.59793
\(952\) −8.72393 −0.282744
\(953\) 27.2070 0.881320 0.440660 0.897674i \(-0.354744\pi\)
0.440660 + 0.897674i \(0.354744\pi\)
\(954\) 10.8573 0.351517
\(955\) −14.9447 −0.483599
\(956\) −26.5303 −0.858053
\(957\) 4.42864 0.143158
\(958\) 12.7699 0.412575
\(959\) 3.22168 0.104033
\(960\) −16.7511 −0.540640
\(961\) 16.6543 0.537237
\(962\) −0.765413 −0.0246779
\(963\) 59.7418 1.92515
\(964\) 35.1526 1.13219
\(965\) −14.1476 −0.455429
\(966\) −6.13918 −0.197525
\(967\) −16.8015 −0.540300 −0.270150 0.962818i \(-0.587073\pi\)
−0.270150 + 0.962818i \(0.587073\pi\)
\(968\) 10.5319 0.338507
\(969\) 25.3274 0.813633
\(970\) −1.27163 −0.0408295
\(971\) −17.4465 −0.559884 −0.279942 0.960017i \(-0.590315\pi\)
−0.279942 + 0.960017i \(0.590315\pi\)
\(972\) 17.1383 0.549710
\(973\) 7.73329 0.247918
\(974\) −3.15701 −0.101157
\(975\) −1.80642 −0.0578519
\(976\) −6.40006 −0.204861
\(977\) 32.0513 1.02541 0.512706 0.858564i \(-0.328643\pi\)
0.512706 + 0.858564i \(0.328643\pi\)
\(978\) 14.7427 0.471418
\(979\) 11.9081 0.380586
\(980\) 11.7699 0.375974
\(981\) −97.9407 −3.12701
\(982\) −9.08694 −0.289976
\(983\) −16.5259 −0.527094 −0.263547 0.964646i \(-0.584892\pi\)
−0.263547 + 0.964646i \(0.584892\pi\)
\(984\) −12.9491 −0.412804
\(985\) −5.70471 −0.181767
\(986\) 2.47457 0.0788064
\(987\) −18.1017 −0.576184
\(988\) −1.29883 −0.0413211
\(989\) −79.1378 −2.51644
\(990\) −2.57628 −0.0818796
\(991\) −9.34920 −0.296987 −0.148494 0.988913i \(-0.547442\pi\)
−0.148494 + 0.988913i \(0.547442\pi\)
\(992\) −24.1289 −0.766094
\(993\) 49.1022 1.55821
\(994\) −3.84208 −0.121863
\(995\) −22.1432 −0.701987
\(996\) −59.1753 −1.87504
\(997\) −15.9956 −0.506584 −0.253292 0.967390i \(-0.581513\pi\)
−0.253292 + 0.967390i \(0.581513\pi\)
\(998\) 6.83854 0.216470
\(999\) 27.8796 0.882070
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 145.2.a.c.1.2 3
3.2 odd 2 1305.2.a.p.1.2 3
4.3 odd 2 2320.2.a.n.1.1 3
5.2 odd 4 725.2.b.e.349.4 6
5.3 odd 4 725.2.b.e.349.3 6
5.4 even 2 725.2.a.e.1.2 3
7.6 odd 2 7105.2.a.o.1.2 3
8.3 odd 2 9280.2.a.br.1.3 3
8.5 even 2 9280.2.a.bj.1.1 3
15.14 odd 2 6525.2.a.be.1.2 3
29.28 even 2 4205.2.a.f.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.2 3 1.1 even 1 trivial
725.2.a.e.1.2 3 5.4 even 2
725.2.b.e.349.3 6 5.3 odd 4
725.2.b.e.349.4 6 5.2 odd 4
1305.2.a.p.1.2 3 3.2 odd 2
2320.2.a.n.1.1 3 4.3 odd 2
4205.2.a.f.1.2 3 29.28 even 2
6525.2.a.be.1.2 3 15.14 odd 2
7105.2.a.o.1.2 3 7.6 odd 2
9280.2.a.bj.1.1 3 8.5 even 2
9280.2.a.br.1.3 3 8.3 odd 2