Properties

Label 7942.2.a.ca.1.4
Level $7942$
Weight $2$
Character 7942.1
Self dual yes
Analytic conductor $63.417$
Analytic rank $0$
Dimension $15$
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] = [7942,2,Mod(1,7942)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7942, 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("7942.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.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: \(63.4171892853\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 3 x^{14} - 33 x^{13} + 101 x^{12} + 408 x^{11} - 1314 x^{10} - 2271 x^{9} + 8292 x^{8} + 4944 x^{7} - 26254 x^{6} + 1401 x^{5} + 39318 x^{4} - 19069 x^{3} + \cdots - 3592 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.75198\) of defining polynomial
Character \(\chi\) \(=\) 7942.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.75198 q^{3} +1.00000 q^{4} +2.75002 q^{5} -1.75198 q^{6} +3.83483 q^{7} +1.00000 q^{8} +0.0694508 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.75198 q^{3} +1.00000 q^{4} +2.75002 q^{5} -1.75198 q^{6} +3.83483 q^{7} +1.00000 q^{8} +0.0694508 q^{9} +2.75002 q^{10} -1.00000 q^{11} -1.75198 q^{12} -2.99887 q^{13} +3.83483 q^{14} -4.81799 q^{15} +1.00000 q^{16} +6.94349 q^{17} +0.0694508 q^{18} +2.75002 q^{20} -6.71856 q^{21} -1.00000 q^{22} +1.05872 q^{23} -1.75198 q^{24} +2.56259 q^{25} -2.99887 q^{26} +5.13428 q^{27} +3.83483 q^{28} -6.97269 q^{29} -4.81799 q^{30} -5.40593 q^{31} +1.00000 q^{32} +1.75198 q^{33} +6.94349 q^{34} +10.5458 q^{35} +0.0694508 q^{36} +5.74381 q^{37} +5.25398 q^{39} +2.75002 q^{40} +6.29831 q^{41} -6.71856 q^{42} -5.27048 q^{43} -1.00000 q^{44} +0.190991 q^{45} +1.05872 q^{46} +9.76272 q^{47} -1.75198 q^{48} +7.70590 q^{49} +2.56259 q^{50} -12.1649 q^{51} -2.99887 q^{52} -7.56179 q^{53} +5.13428 q^{54} -2.75002 q^{55} +3.83483 q^{56} -6.97269 q^{58} +0.993331 q^{59} -4.81799 q^{60} +6.30075 q^{61} -5.40593 q^{62} +0.266332 q^{63} +1.00000 q^{64} -8.24695 q^{65} +1.75198 q^{66} +1.99535 q^{67} +6.94349 q^{68} -1.85486 q^{69} +10.5458 q^{70} +13.3663 q^{71} +0.0694508 q^{72} +8.55207 q^{73} +5.74381 q^{74} -4.48963 q^{75} -3.83483 q^{77} +5.25398 q^{78} +9.61507 q^{79} +2.75002 q^{80} -9.20353 q^{81} +6.29831 q^{82} -9.86312 q^{83} -6.71856 q^{84} +19.0947 q^{85} -5.27048 q^{86} +12.2161 q^{87} -1.00000 q^{88} +13.1889 q^{89} +0.190991 q^{90} -11.5001 q^{91} +1.05872 q^{92} +9.47110 q^{93} +9.76272 q^{94} -1.75198 q^{96} -6.09792 q^{97} +7.70590 q^{98} -0.0694508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 15 q^{2} + 3 q^{3} + 15 q^{4} + 9 q^{5} + 3 q^{6} + 15 q^{8} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 15 q^{2} + 3 q^{3} + 15 q^{4} + 9 q^{5} + 3 q^{6} + 15 q^{8} + 30 q^{9} + 9 q^{10} - 15 q^{11} + 3 q^{12} + 21 q^{15} + 15 q^{16} + 21 q^{17} + 30 q^{18} + 9 q^{20} - 9 q^{21} - 15 q^{22} + 21 q^{23} + 3 q^{24} + 24 q^{25} + 3 q^{27} - 9 q^{29} + 21 q^{30} + 18 q^{31} + 15 q^{32} - 3 q^{33} + 21 q^{34} + 18 q^{35} + 30 q^{36} - 9 q^{37} + 9 q^{40} + 15 q^{41} - 9 q^{42} + 3 q^{43} - 15 q^{44} + 54 q^{45} + 21 q^{46} + 39 q^{47} + 3 q^{48} + 33 q^{49} + 24 q^{50} + 30 q^{51} + 18 q^{53} + 3 q^{54} - 9 q^{55} - 9 q^{58} + 6 q^{59} + 21 q^{60} - 30 q^{61} + 18 q^{62} + 24 q^{63} + 15 q^{64} + 6 q^{65} - 3 q^{66} + 9 q^{67} + 21 q^{68} - 42 q^{69} + 18 q^{70} + 9 q^{71} + 30 q^{72} + 12 q^{73} - 9 q^{74} + 21 q^{75} + 12 q^{79} + 9 q^{80} + 63 q^{81} + 15 q^{82} + 30 q^{83} - 9 q^{84} - 3 q^{85} + 3 q^{86} + 9 q^{87} - 15 q^{88} - 6 q^{89} + 54 q^{90} + 96 q^{91} + 21 q^{92} + 102 q^{93} + 39 q^{94} + 3 q^{96} + 33 q^{98} - 30 q^{99}+O(q^{100}) \) 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
\(3\) −1.75198 −1.01151 −0.505754 0.862677i \(-0.668786\pi\)
−0.505754 + 0.862677i \(0.668786\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.75002 1.22985 0.614923 0.788588i \(-0.289187\pi\)
0.614923 + 0.788588i \(0.289187\pi\)
\(6\) −1.75198 −0.715245
\(7\) 3.83483 1.44943 0.724714 0.689050i \(-0.241972\pi\)
0.724714 + 0.689050i \(0.241972\pi\)
\(8\) 1.00000 0.353553
\(9\) 0.0694508 0.0231503
\(10\) 2.75002 0.869632
\(11\) −1.00000 −0.301511
\(12\) −1.75198 −0.505754
\(13\) −2.99887 −0.831737 −0.415868 0.909425i \(-0.636522\pi\)
−0.415868 + 0.909425i \(0.636522\pi\)
\(14\) 3.83483 1.02490
\(15\) −4.81799 −1.24400
\(16\) 1.00000 0.250000
\(17\) 6.94349 1.68404 0.842022 0.539444i \(-0.181365\pi\)
0.842022 + 0.539444i \(0.181365\pi\)
\(18\) 0.0694508 0.0163697
\(19\) 0 0
\(20\) 2.75002 0.614923
\(21\) −6.71856 −1.46611
\(22\) −1.00000 −0.213201
\(23\) 1.05872 0.220758 0.110379 0.993890i \(-0.464794\pi\)
0.110379 + 0.993890i \(0.464794\pi\)
\(24\) −1.75198 −0.357622
\(25\) 2.56259 0.512519
\(26\) −2.99887 −0.588127
\(27\) 5.13428 0.988092
\(28\) 3.83483 0.724714
\(29\) −6.97269 −1.29480 −0.647398 0.762152i \(-0.724143\pi\)
−0.647398 + 0.762152i \(0.724143\pi\)
\(30\) −4.81799 −0.879640
\(31\) −5.40593 −0.970933 −0.485466 0.874255i \(-0.661350\pi\)
−0.485466 + 0.874255i \(0.661350\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.75198 0.304981
\(34\) 6.94349 1.19080
\(35\) 10.5458 1.78257
\(36\) 0.0694508 0.0115751
\(37\) 5.74381 0.944277 0.472139 0.881524i \(-0.343482\pi\)
0.472139 + 0.881524i \(0.343482\pi\)
\(38\) 0 0
\(39\) 5.25398 0.841309
\(40\) 2.75002 0.434816
\(41\) 6.29831 0.983631 0.491816 0.870699i \(-0.336333\pi\)
0.491816 + 0.870699i \(0.336333\pi\)
\(42\) −6.71856 −1.03670
\(43\) −5.27048 −0.803741 −0.401870 0.915697i \(-0.631640\pi\)
−0.401870 + 0.915697i \(0.631640\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0.190991 0.0284712
\(46\) 1.05872 0.156099
\(47\) 9.76272 1.42404 0.712019 0.702160i \(-0.247781\pi\)
0.712019 + 0.702160i \(0.247781\pi\)
\(48\) −1.75198 −0.252877
\(49\) 7.70590 1.10084
\(50\) 2.56259 0.362406
\(51\) −12.1649 −1.70342
\(52\) −2.99887 −0.415868
\(53\) −7.56179 −1.03869 −0.519346 0.854564i \(-0.673824\pi\)
−0.519346 + 0.854564i \(0.673824\pi\)
\(54\) 5.13428 0.698687
\(55\) −2.75002 −0.370812
\(56\) 3.83483 0.512450
\(57\) 0 0
\(58\) −6.97269 −0.915559
\(59\) 0.993331 0.129321 0.0646603 0.997907i \(-0.479404\pi\)
0.0646603 + 0.997907i \(0.479404\pi\)
\(60\) −4.81799 −0.622000
\(61\) 6.30075 0.806729 0.403364 0.915039i \(-0.367841\pi\)
0.403364 + 0.915039i \(0.367841\pi\)
\(62\) −5.40593 −0.686553
\(63\) 0.266332 0.0335547
\(64\) 1.00000 0.125000
\(65\) −8.24695 −1.02291
\(66\) 1.75198 0.215654
\(67\) 1.99535 0.243771 0.121886 0.992544i \(-0.461106\pi\)
0.121886 + 0.992544i \(0.461106\pi\)
\(68\) 6.94349 0.842022
\(69\) −1.85486 −0.223298
\(70\) 10.5458 1.26047
\(71\) 13.3663 1.58628 0.793142 0.609037i \(-0.208444\pi\)
0.793142 + 0.609037i \(0.208444\pi\)
\(72\) 0.0694508 0.00818485
\(73\) 8.55207 1.00094 0.500472 0.865753i \(-0.333160\pi\)
0.500472 + 0.865753i \(0.333160\pi\)
\(74\) 5.74381 0.667705
\(75\) −4.48963 −0.518418
\(76\) 0 0
\(77\) −3.83483 −0.437019
\(78\) 5.25398 0.594896
\(79\) 9.61507 1.08178 0.540890 0.841093i \(-0.318087\pi\)
0.540890 + 0.841093i \(0.318087\pi\)
\(80\) 2.75002 0.307461
\(81\) −9.20353 −1.02261
\(82\) 6.29831 0.695532
\(83\) −9.86312 −1.08262 −0.541309 0.840824i \(-0.682071\pi\)
−0.541309 + 0.840824i \(0.682071\pi\)
\(84\) −6.71856 −0.733055
\(85\) 19.0947 2.07111
\(86\) −5.27048 −0.568331
\(87\) 12.2161 1.30970
\(88\) −1.00000 −0.106600
\(89\) 13.1889 1.39802 0.699009 0.715113i \(-0.253625\pi\)
0.699009 + 0.715113i \(0.253625\pi\)
\(90\) 0.190991 0.0201322
\(91\) −11.5001 −1.20554
\(92\) 1.05872 0.110379
\(93\) 9.47110 0.982107
\(94\) 9.76272 1.00695
\(95\) 0 0
\(96\) −1.75198 −0.178811
\(97\) −6.09792 −0.619150 −0.309575 0.950875i \(-0.600187\pi\)
−0.309575 + 0.950875i \(0.600187\pi\)
\(98\) 7.70590 0.778413
\(99\) −0.0694508 −0.00698007
\(100\) 2.56259 0.256259
\(101\) −17.8867 −1.77979 −0.889895 0.456166i \(-0.849222\pi\)
−0.889895 + 0.456166i \(0.849222\pi\)
\(102\) −12.1649 −1.20450
\(103\) 1.63523 0.161124 0.0805620 0.996750i \(-0.474329\pi\)
0.0805620 + 0.996750i \(0.474329\pi\)
\(104\) −2.99887 −0.294063
\(105\) −18.4762 −1.80309
\(106\) −7.56179 −0.734466
\(107\) 6.69253 0.646991 0.323496 0.946230i \(-0.395142\pi\)
0.323496 + 0.946230i \(0.395142\pi\)
\(108\) 5.13428 0.494046
\(109\) 2.64617 0.253458 0.126729 0.991937i \(-0.459552\pi\)
0.126729 + 0.991937i \(0.459552\pi\)
\(110\) −2.75002 −0.262204
\(111\) −10.0631 −0.955145
\(112\) 3.83483 0.362357
\(113\) 8.01087 0.753599 0.376800 0.926295i \(-0.377025\pi\)
0.376800 + 0.926295i \(0.377025\pi\)
\(114\) 0 0
\(115\) 2.91149 0.271498
\(116\) −6.97269 −0.647398
\(117\) −0.208274 −0.0192549
\(118\) 0.993331 0.0914435
\(119\) 26.6271 2.44090
\(120\) −4.81799 −0.439820
\(121\) 1.00000 0.0909091
\(122\) 6.30075 0.570443
\(123\) −11.0346 −0.994952
\(124\) −5.40593 −0.485466
\(125\) −6.70291 −0.599526
\(126\) 0.266332 0.0237267
\(127\) 20.7918 1.84497 0.922487 0.386027i \(-0.126153\pi\)
0.922487 + 0.386027i \(0.126153\pi\)
\(128\) 1.00000 0.0883883
\(129\) 9.23380 0.812991
\(130\) −8.24695 −0.723305
\(131\) −6.68860 −0.584386 −0.292193 0.956359i \(-0.594385\pi\)
−0.292193 + 0.956359i \(0.594385\pi\)
\(132\) 1.75198 0.152491
\(133\) 0 0
\(134\) 1.99535 0.172372
\(135\) 14.1194 1.21520
\(136\) 6.94349 0.595399
\(137\) 16.3981 1.40099 0.700494 0.713658i \(-0.252963\pi\)
0.700494 + 0.713658i \(0.252963\pi\)
\(138\) −1.85486 −0.157896
\(139\) 13.1961 1.11928 0.559640 0.828736i \(-0.310939\pi\)
0.559640 + 0.828736i \(0.310939\pi\)
\(140\) 10.5458 0.891286
\(141\) −17.1041 −1.44043
\(142\) 13.3663 1.12167
\(143\) 2.99887 0.250778
\(144\) 0.0694508 0.00578757
\(145\) −19.1750 −1.59240
\(146\) 8.55207 0.707775
\(147\) −13.5006 −1.11351
\(148\) 5.74381 0.472139
\(149\) 10.4264 0.854161 0.427080 0.904214i \(-0.359542\pi\)
0.427080 + 0.904214i \(0.359542\pi\)
\(150\) −4.48963 −0.366577
\(151\) 0.887856 0.0722527 0.0361264 0.999347i \(-0.488498\pi\)
0.0361264 + 0.999347i \(0.488498\pi\)
\(152\) 0 0
\(153\) 0.482231 0.0389860
\(154\) −3.83483 −0.309019
\(155\) −14.8664 −1.19410
\(156\) 5.25398 0.420655
\(157\) −7.69032 −0.613755 −0.306877 0.951749i \(-0.599284\pi\)
−0.306877 + 0.951749i \(0.599284\pi\)
\(158\) 9.61507 0.764934
\(159\) 13.2481 1.05065
\(160\) 2.75002 0.217408
\(161\) 4.06000 0.319973
\(162\) −9.20353 −0.723098
\(163\) 0.977244 0.0765436 0.0382718 0.999267i \(-0.487815\pi\)
0.0382718 + 0.999267i \(0.487815\pi\)
\(164\) 6.29831 0.491816
\(165\) 4.81799 0.375080
\(166\) −9.86312 −0.765526
\(167\) −14.4378 −1.11723 −0.558616 0.829427i \(-0.688667\pi\)
−0.558616 + 0.829427i \(0.688667\pi\)
\(168\) −6.71856 −0.518348
\(169\) −4.00678 −0.308214
\(170\) 19.0947 1.46450
\(171\) 0 0
\(172\) −5.27048 −0.401870
\(173\) −20.7436 −1.57711 −0.788555 0.614965i \(-0.789170\pi\)
−0.788555 + 0.614965i \(0.789170\pi\)
\(174\) 12.2161 0.926096
\(175\) 9.82711 0.742860
\(176\) −1.00000 −0.0753778
\(177\) −1.74030 −0.130809
\(178\) 13.1889 0.988547
\(179\) −19.5304 −1.45977 −0.729884 0.683571i \(-0.760426\pi\)
−0.729884 + 0.683571i \(0.760426\pi\)
\(180\) 0.190991 0.0142356
\(181\) 0.524440 0.0389813 0.0194906 0.999810i \(-0.493796\pi\)
0.0194906 + 0.999810i \(0.493796\pi\)
\(182\) −11.5001 −0.852448
\(183\) −11.0388 −0.816013
\(184\) 1.05872 0.0780497
\(185\) 15.7956 1.16131
\(186\) 9.47110 0.694455
\(187\) −6.94349 −0.507758
\(188\) 9.76272 0.712019
\(189\) 19.6891 1.43217
\(190\) 0 0
\(191\) 19.6960 1.42515 0.712576 0.701595i \(-0.247528\pi\)
0.712576 + 0.701595i \(0.247528\pi\)
\(192\) −1.75198 −0.126439
\(193\) −0.558050 −0.0401693 −0.0200846 0.999798i \(-0.506394\pi\)
−0.0200846 + 0.999798i \(0.506394\pi\)
\(194\) −6.09792 −0.437805
\(195\) 14.4485 1.03468
\(196\) 7.70590 0.550421
\(197\) −27.2349 −1.94040 −0.970202 0.242297i \(-0.922099\pi\)
−0.970202 + 0.242297i \(0.922099\pi\)
\(198\) −0.0694508 −0.00493565
\(199\) −9.66076 −0.684833 −0.342417 0.939548i \(-0.611245\pi\)
−0.342417 + 0.939548i \(0.611245\pi\)
\(200\) 2.56259 0.181203
\(201\) −3.49583 −0.246577
\(202\) −17.8867 −1.25850
\(203\) −26.7391 −1.87671
\(204\) −12.1649 −0.851712
\(205\) 17.3205 1.20971
\(206\) 1.63523 0.113932
\(207\) 0.0735287 0.00511060
\(208\) −2.99887 −0.207934
\(209\) 0 0
\(210\) −18.4762 −1.27498
\(211\) 16.7285 1.15164 0.575819 0.817577i \(-0.304683\pi\)
0.575819 + 0.817577i \(0.304683\pi\)
\(212\) −7.56179 −0.519346
\(213\) −23.4175 −1.60454
\(214\) 6.69253 0.457492
\(215\) −14.4939 −0.988477
\(216\) 5.13428 0.349343
\(217\) −20.7308 −1.40730
\(218\) 2.64617 0.179222
\(219\) −14.9831 −1.01246
\(220\) −2.75002 −0.185406
\(221\) −20.8226 −1.40068
\(222\) −10.0631 −0.675389
\(223\) −9.83783 −0.658790 −0.329395 0.944192i \(-0.606845\pi\)
−0.329395 + 0.944192i \(0.606845\pi\)
\(224\) 3.83483 0.256225
\(225\) 0.177974 0.0118650
\(226\) 8.01087 0.532875
\(227\) 0.864686 0.0573913 0.0286956 0.999588i \(-0.490865\pi\)
0.0286956 + 0.999588i \(0.490865\pi\)
\(228\) 0 0
\(229\) 8.33952 0.551091 0.275545 0.961288i \(-0.411142\pi\)
0.275545 + 0.961288i \(0.411142\pi\)
\(230\) 2.91149 0.191978
\(231\) 6.71856 0.442049
\(232\) −6.97269 −0.457780
\(233\) 19.8575 1.30091 0.650455 0.759545i \(-0.274578\pi\)
0.650455 + 0.759545i \(0.274578\pi\)
\(234\) −0.208274 −0.0136153
\(235\) 26.8476 1.75135
\(236\) 0.993331 0.0646603
\(237\) −16.8455 −1.09423
\(238\) 26.6271 1.72598
\(239\) −15.7645 −1.01972 −0.509860 0.860257i \(-0.670303\pi\)
−0.509860 + 0.860257i \(0.670303\pi\)
\(240\) −4.81799 −0.311000
\(241\) −22.0841 −1.42256 −0.711282 0.702907i \(-0.751885\pi\)
−0.711282 + 0.702907i \(0.751885\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0.721610 0.0462913
\(244\) 6.30075 0.403364
\(245\) 21.1914 1.35387
\(246\) −11.0346 −0.703537
\(247\) 0 0
\(248\) −5.40593 −0.343277
\(249\) 17.2800 1.09508
\(250\) −6.70291 −0.423929
\(251\) 7.83629 0.494623 0.247311 0.968936i \(-0.420453\pi\)
0.247311 + 0.968936i \(0.420453\pi\)
\(252\) 0.266332 0.0167773
\(253\) −1.05872 −0.0665610
\(254\) 20.7918 1.30459
\(255\) −33.4536 −2.09495
\(256\) 1.00000 0.0625000
\(257\) −20.4912 −1.27820 −0.639102 0.769122i \(-0.720694\pi\)
−0.639102 + 0.769122i \(0.720694\pi\)
\(258\) 9.23380 0.574871
\(259\) 22.0265 1.36866
\(260\) −8.24695 −0.511454
\(261\) −0.484259 −0.0299749
\(262\) −6.68860 −0.413223
\(263\) −26.8143 −1.65344 −0.826719 0.562614i \(-0.809796\pi\)
−0.826719 + 0.562614i \(0.809796\pi\)
\(264\) 1.75198 0.107827
\(265\) −20.7950 −1.27743
\(266\) 0 0
\(267\) −23.1067 −1.41411
\(268\) 1.99535 0.121886
\(269\) 10.9612 0.668316 0.334158 0.942517i \(-0.391548\pi\)
0.334158 + 0.942517i \(0.391548\pi\)
\(270\) 14.1194 0.859276
\(271\) 1.86040 0.113011 0.0565057 0.998402i \(-0.482004\pi\)
0.0565057 + 0.998402i \(0.482004\pi\)
\(272\) 6.94349 0.421011
\(273\) 20.1481 1.21942
\(274\) 16.3981 0.990648
\(275\) −2.56259 −0.154530
\(276\) −1.85486 −0.111649
\(277\) 18.7769 1.12819 0.564097 0.825709i \(-0.309225\pi\)
0.564097 + 0.825709i \(0.309225\pi\)
\(278\) 13.1961 0.791451
\(279\) −0.375446 −0.0224774
\(280\) 10.5458 0.630235
\(281\) 12.7105 0.758246 0.379123 0.925346i \(-0.376226\pi\)
0.379123 + 0.925346i \(0.376226\pi\)
\(282\) −17.1041 −1.01854
\(283\) −10.9472 −0.650741 −0.325371 0.945587i \(-0.605489\pi\)
−0.325371 + 0.945587i \(0.605489\pi\)
\(284\) 13.3663 0.793142
\(285\) 0 0
\(286\) 2.99887 0.177327
\(287\) 24.1529 1.42570
\(288\) 0.0694508 0.00409243
\(289\) 31.2120 1.83600
\(290\) −19.1750 −1.12600
\(291\) 10.6835 0.626275
\(292\) 8.55207 0.500472
\(293\) −4.29499 −0.250916 −0.125458 0.992099i \(-0.540040\pi\)
−0.125458 + 0.992099i \(0.540040\pi\)
\(294\) −13.5006 −0.787372
\(295\) 2.73168 0.159044
\(296\) 5.74381 0.333852
\(297\) −5.13428 −0.297921
\(298\) 10.4264 0.603983
\(299\) −3.17496 −0.183612
\(300\) −4.48963 −0.259209
\(301\) −20.2114 −1.16496
\(302\) 0.887856 0.0510904
\(303\) 31.3372 1.80027
\(304\) 0 0
\(305\) 17.3272 0.992152
\(306\) 0.482231 0.0275673
\(307\) 33.2043 1.89507 0.947534 0.319656i \(-0.103567\pi\)
0.947534 + 0.319656i \(0.103567\pi\)
\(308\) −3.83483 −0.218510
\(309\) −2.86490 −0.162978
\(310\) −14.8664 −0.844354
\(311\) 29.4236 1.66846 0.834230 0.551417i \(-0.185913\pi\)
0.834230 + 0.551417i \(0.185913\pi\)
\(312\) 5.25398 0.297448
\(313\) −8.73586 −0.493780 −0.246890 0.969044i \(-0.579409\pi\)
−0.246890 + 0.969044i \(0.579409\pi\)
\(314\) −7.69032 −0.433990
\(315\) 0.732417 0.0412670
\(316\) 9.61507 0.540890
\(317\) 3.01863 0.169543 0.0847715 0.996400i \(-0.472984\pi\)
0.0847715 + 0.996400i \(0.472984\pi\)
\(318\) 13.2481 0.742919
\(319\) 6.97269 0.390396
\(320\) 2.75002 0.153731
\(321\) −11.7252 −0.654437
\(322\) 4.06000 0.226255
\(323\) 0 0
\(324\) −9.20353 −0.511307
\(325\) −7.68489 −0.426281
\(326\) 0.977244 0.0541245
\(327\) −4.63606 −0.256375
\(328\) 6.29831 0.347766
\(329\) 37.4383 2.06404
\(330\) 4.81799 0.265222
\(331\) 21.0483 1.15692 0.578461 0.815710i \(-0.303654\pi\)
0.578461 + 0.815710i \(0.303654\pi\)
\(332\) −9.86312 −0.541309
\(333\) 0.398912 0.0218603
\(334\) −14.4378 −0.790002
\(335\) 5.48725 0.299801
\(336\) −6.71856 −0.366527
\(337\) 32.8578 1.78988 0.894938 0.446191i \(-0.147220\pi\)
0.894938 + 0.446191i \(0.147220\pi\)
\(338\) −4.00678 −0.217940
\(339\) −14.0349 −0.762272
\(340\) 19.0947 1.03556
\(341\) 5.40593 0.292747
\(342\) 0 0
\(343\) 2.70700 0.146164
\(344\) −5.27048 −0.284165
\(345\) −5.10089 −0.274623
\(346\) −20.7436 −1.11518
\(347\) −1.55590 −0.0835252 −0.0417626 0.999128i \(-0.513297\pi\)
−0.0417626 + 0.999128i \(0.513297\pi\)
\(348\) 12.2161 0.654849
\(349\) −7.35032 −0.393454 −0.196727 0.980458i \(-0.563031\pi\)
−0.196727 + 0.980458i \(0.563031\pi\)
\(350\) 9.82711 0.525281
\(351\) −15.3970 −0.821833
\(352\) −1.00000 −0.0533002
\(353\) −19.3609 −1.03048 −0.515240 0.857046i \(-0.672297\pi\)
−0.515240 + 0.857046i \(0.672297\pi\)
\(354\) −1.74030 −0.0924959
\(355\) 36.7575 1.95088
\(356\) 13.1889 0.699009
\(357\) −46.6502 −2.46899
\(358\) −19.5304 −1.03221
\(359\) −14.9074 −0.786785 −0.393392 0.919371i \(-0.628699\pi\)
−0.393392 + 0.919371i \(0.628699\pi\)
\(360\) 0.190991 0.0100661
\(361\) 0 0
\(362\) 0.524440 0.0275639
\(363\) −1.75198 −0.0919554
\(364\) −11.5001 −0.602772
\(365\) 23.5184 1.23101
\(366\) −11.0388 −0.577009
\(367\) −27.5633 −1.43879 −0.719397 0.694600i \(-0.755582\pi\)
−0.719397 + 0.694600i \(0.755582\pi\)
\(368\) 1.05872 0.0551894
\(369\) 0.437423 0.0227713
\(370\) 15.7956 0.821174
\(371\) −28.9982 −1.50551
\(372\) 9.47110 0.491054
\(373\) −13.6882 −0.708746 −0.354373 0.935104i \(-0.615306\pi\)
−0.354373 + 0.935104i \(0.615306\pi\)
\(374\) −6.94349 −0.359039
\(375\) 11.7434 0.606426
\(376\) 9.76272 0.503474
\(377\) 20.9102 1.07693
\(378\) 19.6891 1.01270
\(379\) −6.31554 −0.324407 −0.162204 0.986757i \(-0.551860\pi\)
−0.162204 + 0.986757i \(0.551860\pi\)
\(380\) 0 0
\(381\) −36.4269 −1.86621
\(382\) 19.6960 1.00773
\(383\) 4.01015 0.204909 0.102455 0.994738i \(-0.467330\pi\)
0.102455 + 0.994738i \(0.467330\pi\)
\(384\) −1.75198 −0.0894056
\(385\) −10.5458 −0.537466
\(386\) −0.558050 −0.0284040
\(387\) −0.366039 −0.0186068
\(388\) −6.09792 −0.309575
\(389\) −17.8884 −0.906976 −0.453488 0.891262i \(-0.649820\pi\)
−0.453488 + 0.891262i \(0.649820\pi\)
\(390\) 14.4485 0.731629
\(391\) 7.35119 0.371766
\(392\) 7.70590 0.389207
\(393\) 11.7183 0.591111
\(394\) −27.2349 −1.37207
\(395\) 26.4416 1.33042
\(396\) −0.0694508 −0.00349003
\(397\) −7.46337 −0.374576 −0.187288 0.982305i \(-0.559970\pi\)
−0.187288 + 0.982305i \(0.559970\pi\)
\(398\) −9.66076 −0.484250
\(399\) 0 0
\(400\) 2.56259 0.128130
\(401\) −16.9891 −0.848396 −0.424198 0.905569i \(-0.639444\pi\)
−0.424198 + 0.905569i \(0.639444\pi\)
\(402\) −3.49583 −0.174356
\(403\) 16.2117 0.807561
\(404\) −17.8867 −0.889895
\(405\) −25.3099 −1.25766
\(406\) −26.7391 −1.32704
\(407\) −5.74381 −0.284710
\(408\) −12.1649 −0.602252
\(409\) 3.76849 0.186340 0.0931698 0.995650i \(-0.470300\pi\)
0.0931698 + 0.995650i \(0.470300\pi\)
\(410\) 17.3205 0.855397
\(411\) −28.7293 −1.41711
\(412\) 1.63523 0.0805620
\(413\) 3.80925 0.187441
\(414\) 0.0735287 0.00361374
\(415\) −27.1237 −1.33145
\(416\) −2.99887 −0.147032
\(417\) −23.1194 −1.13216
\(418\) 0 0
\(419\) 32.5461 1.58998 0.794990 0.606622i \(-0.207476\pi\)
0.794990 + 0.606622i \(0.207476\pi\)
\(420\) −18.4762 −0.901544
\(421\) 3.97736 0.193845 0.0969223 0.995292i \(-0.469100\pi\)
0.0969223 + 0.995292i \(0.469100\pi\)
\(422\) 16.7285 0.814331
\(423\) 0.678029 0.0329669
\(424\) −7.56179 −0.367233
\(425\) 17.7933 0.863104
\(426\) −23.4175 −1.13458
\(427\) 24.1623 1.16930
\(428\) 6.69253 0.323496
\(429\) −5.25398 −0.253664
\(430\) −14.4939 −0.698959
\(431\) −4.18326 −0.201501 −0.100750 0.994912i \(-0.532124\pi\)
−0.100750 + 0.994912i \(0.532124\pi\)
\(432\) 5.13428 0.247023
\(433\) 13.7414 0.660367 0.330184 0.943917i \(-0.392889\pi\)
0.330184 + 0.943917i \(0.392889\pi\)
\(434\) −20.7308 −0.995110
\(435\) 33.5944 1.61073
\(436\) 2.64617 0.126729
\(437\) 0 0
\(438\) −14.9831 −0.715920
\(439\) 27.4116 1.30828 0.654142 0.756372i \(-0.273030\pi\)
0.654142 + 0.756372i \(0.273030\pi\)
\(440\) −2.75002 −0.131102
\(441\) 0.535181 0.0254848
\(442\) −20.8226 −0.990431
\(443\) 34.5821 1.64304 0.821521 0.570178i \(-0.193126\pi\)
0.821521 + 0.570178i \(0.193126\pi\)
\(444\) −10.0631 −0.477572
\(445\) 36.2696 1.71934
\(446\) −9.83783 −0.465835
\(447\) −18.2668 −0.863991
\(448\) 3.83483 0.181179
\(449\) 14.0064 0.661002 0.330501 0.943806i \(-0.392782\pi\)
0.330501 + 0.943806i \(0.392782\pi\)
\(450\) 0.177974 0.00838979
\(451\) −6.29831 −0.296576
\(452\) 8.01087 0.376800
\(453\) −1.55551 −0.0730843
\(454\) 0.864686 0.0405817
\(455\) −31.6256 −1.48263
\(456\) 0 0
\(457\) −34.1525 −1.59759 −0.798793 0.601606i \(-0.794528\pi\)
−0.798793 + 0.601606i \(0.794528\pi\)
\(458\) 8.33952 0.389680
\(459\) 35.6498 1.66399
\(460\) 2.91149 0.135749
\(461\) 37.5791 1.75023 0.875117 0.483912i \(-0.160784\pi\)
0.875117 + 0.483912i \(0.160784\pi\)
\(462\) 6.71856 0.312576
\(463\) 5.53335 0.257157 0.128578 0.991699i \(-0.458959\pi\)
0.128578 + 0.991699i \(0.458959\pi\)
\(464\) −6.97269 −0.323699
\(465\) 26.0457 1.20784
\(466\) 19.8575 0.919882
\(467\) 12.6215 0.584055 0.292027 0.956410i \(-0.405670\pi\)
0.292027 + 0.956410i \(0.405670\pi\)
\(468\) −0.208274 −0.00962747
\(469\) 7.65183 0.353329
\(470\) 26.8476 1.23839
\(471\) 13.4733 0.620818
\(472\) 0.993331 0.0457217
\(473\) 5.27048 0.242337
\(474\) −16.8455 −0.773738
\(475\) 0 0
\(476\) 26.6271 1.22045
\(477\) −0.525172 −0.0240460
\(478\) −15.7645 −0.721051
\(479\) −22.1665 −1.01281 −0.506407 0.862295i \(-0.669026\pi\)
−0.506407 + 0.862295i \(0.669026\pi\)
\(480\) −4.81799 −0.219910
\(481\) −17.2250 −0.785390
\(482\) −22.0841 −1.00590
\(483\) −7.11305 −0.323655
\(484\) 1.00000 0.0454545
\(485\) −16.7694 −0.761458
\(486\) 0.721610 0.0327329
\(487\) 8.56601 0.388163 0.194082 0.980985i \(-0.437827\pi\)
0.194082 + 0.980985i \(0.437827\pi\)
\(488\) 6.30075 0.285222
\(489\) −1.71212 −0.0774246
\(490\) 21.1914 0.957328
\(491\) 22.0937 0.997075 0.498537 0.866868i \(-0.333871\pi\)
0.498537 + 0.866868i \(0.333871\pi\)
\(492\) −11.0346 −0.497476
\(493\) −48.4148 −2.18049
\(494\) 0 0
\(495\) −0.190991 −0.00858440
\(496\) −5.40593 −0.242733
\(497\) 51.2573 2.29921
\(498\) 17.2800 0.774337
\(499\) −2.72781 −0.122114 −0.0610569 0.998134i \(-0.519447\pi\)
−0.0610569 + 0.998134i \(0.519447\pi\)
\(500\) −6.70291 −0.299763
\(501\) 25.2948 1.13009
\(502\) 7.83629 0.349751
\(503\) 7.28006 0.324602 0.162301 0.986741i \(-0.448108\pi\)
0.162301 + 0.986741i \(0.448108\pi\)
\(504\) 0.266332 0.0118634
\(505\) −49.1886 −2.18886
\(506\) −1.05872 −0.0470657
\(507\) 7.01981 0.311761
\(508\) 20.7918 0.922487
\(509\) −24.1341 −1.06973 −0.534863 0.844939i \(-0.679637\pi\)
−0.534863 + 0.844939i \(0.679637\pi\)
\(510\) −33.4536 −1.48135
\(511\) 32.7957 1.45080
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −20.4912 −0.903827
\(515\) 4.49691 0.198157
\(516\) 9.23380 0.406496
\(517\) −9.76272 −0.429364
\(518\) 22.0265 0.967790
\(519\) 36.3425 1.59526
\(520\) −8.24695 −0.361652
\(521\) −12.8461 −0.562800 −0.281400 0.959591i \(-0.590799\pi\)
−0.281400 + 0.959591i \(0.590799\pi\)
\(522\) −0.484259 −0.0211954
\(523\) −2.93950 −0.128535 −0.0642676 0.997933i \(-0.520471\pi\)
−0.0642676 + 0.997933i \(0.520471\pi\)
\(524\) −6.68860 −0.292193
\(525\) −17.2169 −0.751409
\(526\) −26.8143 −1.16916
\(527\) −37.5360 −1.63509
\(528\) 1.75198 0.0762454
\(529\) −21.8791 −0.951266
\(530\) −20.7950 −0.903279
\(531\) 0.0689876 0.00299381
\(532\) 0 0
\(533\) −18.8878 −0.818123
\(534\) −23.1067 −0.999924
\(535\) 18.4046 0.795699
\(536\) 1.99535 0.0861861
\(537\) 34.2169 1.47657
\(538\) 10.9612 0.472571
\(539\) −7.70590 −0.331917
\(540\) 14.1194 0.607600
\(541\) 29.2712 1.25847 0.629233 0.777217i \(-0.283369\pi\)
0.629233 + 0.777217i \(0.283369\pi\)
\(542\) 1.86040 0.0799112
\(543\) −0.918810 −0.0394299
\(544\) 6.94349 0.297700
\(545\) 7.27703 0.311714
\(546\) 20.1481 0.862258
\(547\) −28.6335 −1.22428 −0.612140 0.790750i \(-0.709691\pi\)
−0.612140 + 0.790750i \(0.709691\pi\)
\(548\) 16.3981 0.700494
\(549\) 0.437592 0.0186760
\(550\) −2.56259 −0.109269
\(551\) 0 0
\(552\) −1.85486 −0.0789479
\(553\) 36.8721 1.56796
\(554\) 18.7769 0.797753
\(555\) −27.6736 −1.17468
\(556\) 13.1961 0.559640
\(557\) 1.59398 0.0675390 0.0337695 0.999430i \(-0.489249\pi\)
0.0337695 + 0.999430i \(0.489249\pi\)
\(558\) −0.375446 −0.0158939
\(559\) 15.8055 0.668501
\(560\) 10.5458 0.445643
\(561\) 12.1649 0.513602
\(562\) 12.7105 0.536161
\(563\) 34.4090 1.45017 0.725083 0.688662i \(-0.241801\pi\)
0.725083 + 0.688662i \(0.241801\pi\)
\(564\) −17.1041 −0.720214
\(565\) 22.0300 0.926810
\(566\) −10.9472 −0.460143
\(567\) −35.2939 −1.48221
\(568\) 13.3663 0.560836
\(569\) 29.9011 1.25352 0.626760 0.779213i \(-0.284381\pi\)
0.626760 + 0.779213i \(0.284381\pi\)
\(570\) 0 0
\(571\) 9.80807 0.410455 0.205228 0.978714i \(-0.434207\pi\)
0.205228 + 0.978714i \(0.434207\pi\)
\(572\) 2.99887 0.125389
\(573\) −34.5071 −1.44155
\(574\) 24.1529 1.00812
\(575\) 2.71306 0.113143
\(576\) 0.0694508 0.00289378
\(577\) −4.11896 −0.171475 −0.0857373 0.996318i \(-0.527325\pi\)
−0.0857373 + 0.996318i \(0.527325\pi\)
\(578\) 31.2120 1.29825
\(579\) 0.977694 0.0406316
\(580\) −19.1750 −0.796200
\(581\) −37.8234 −1.56918
\(582\) 10.6835 0.442844
\(583\) 7.56179 0.313177
\(584\) 8.55207 0.353887
\(585\) −0.572757 −0.0236806
\(586\) −4.29499 −0.177424
\(587\) 18.6749 0.770795 0.385397 0.922751i \(-0.374064\pi\)
0.385397 + 0.922751i \(0.374064\pi\)
\(588\) −13.5006 −0.556756
\(589\) 0 0
\(590\) 2.73168 0.112461
\(591\) 47.7151 1.96274
\(592\) 5.74381 0.236069
\(593\) 8.40647 0.345212 0.172606 0.984991i \(-0.444781\pi\)
0.172606 + 0.984991i \(0.444781\pi\)
\(594\) −5.13428 −0.210662
\(595\) 73.2249 3.00193
\(596\) 10.4264 0.427080
\(597\) 16.9255 0.692715
\(598\) −3.17496 −0.129834
\(599\) −35.2002 −1.43824 −0.719120 0.694886i \(-0.755455\pi\)
−0.719120 + 0.694886i \(0.755455\pi\)
\(600\) −4.48963 −0.183288
\(601\) −25.9342 −1.05788 −0.528939 0.848660i \(-0.677410\pi\)
−0.528939 + 0.848660i \(0.677410\pi\)
\(602\) −20.2114 −0.823755
\(603\) 0.138579 0.00564336
\(604\) 0.887856 0.0361264
\(605\) 2.75002 0.111804
\(606\) 31.3372 1.27298
\(607\) 14.6858 0.596079 0.298039 0.954554i \(-0.403667\pi\)
0.298039 + 0.954554i \(0.403667\pi\)
\(608\) 0 0
\(609\) 46.8464 1.89831
\(610\) 17.3272 0.701557
\(611\) −29.2771 −1.18443
\(612\) 0.482231 0.0194930
\(613\) −13.9711 −0.564287 −0.282143 0.959372i \(-0.591045\pi\)
−0.282143 + 0.959372i \(0.591045\pi\)
\(614\) 33.2043 1.34002
\(615\) −30.3452 −1.22364
\(616\) −3.83483 −0.154510
\(617\) 25.2895 1.01812 0.509058 0.860732i \(-0.329994\pi\)
0.509058 + 0.860732i \(0.329994\pi\)
\(618\) −2.86490 −0.115243
\(619\) −18.7851 −0.755037 −0.377518 0.926002i \(-0.623222\pi\)
−0.377518 + 0.926002i \(0.623222\pi\)
\(620\) −14.8664 −0.597049
\(621\) 5.43575 0.218129
\(622\) 29.4236 1.17978
\(623\) 50.5770 2.02633
\(624\) 5.25398 0.210327
\(625\) −31.2461 −1.24984
\(626\) −8.73586 −0.349155
\(627\) 0 0
\(628\) −7.69032 −0.306877
\(629\) 39.8821 1.59020
\(630\) 0.732417 0.0291802
\(631\) −12.4425 −0.495327 −0.247663 0.968846i \(-0.579663\pi\)
−0.247663 + 0.968846i \(0.579663\pi\)
\(632\) 9.61507 0.382467
\(633\) −29.3081 −1.16489
\(634\) 3.01863 0.119885
\(635\) 57.1778 2.26903
\(636\) 13.2481 0.525323
\(637\) −23.1090 −0.915611
\(638\) 6.97269 0.276052
\(639\) 0.928298 0.0367229
\(640\) 2.75002 0.108704
\(641\) −12.3531 −0.487918 −0.243959 0.969786i \(-0.578446\pi\)
−0.243959 + 0.969786i \(0.578446\pi\)
\(642\) −11.7252 −0.462757
\(643\) 18.9417 0.746987 0.373494 0.927633i \(-0.378160\pi\)
0.373494 + 0.927633i \(0.378160\pi\)
\(644\) 4.06000 0.159986
\(645\) 25.3931 0.999853
\(646\) 0 0
\(647\) −8.99197 −0.353511 −0.176755 0.984255i \(-0.556560\pi\)
−0.176755 + 0.984255i \(0.556560\pi\)
\(648\) −9.20353 −0.361549
\(649\) −0.993331 −0.0389916
\(650\) −7.68489 −0.301426
\(651\) 36.3200 1.42349
\(652\) 0.977244 0.0382718
\(653\) −42.3768 −1.65833 −0.829166 0.559002i \(-0.811184\pi\)
−0.829166 + 0.559002i \(0.811184\pi\)
\(654\) −4.63606 −0.181284
\(655\) −18.3938 −0.718704
\(656\) 6.29831 0.245908
\(657\) 0.593948 0.0231721
\(658\) 37.4383 1.45950
\(659\) −29.6322 −1.15431 −0.577153 0.816636i \(-0.695837\pi\)
−0.577153 + 0.816636i \(0.695837\pi\)
\(660\) 4.81799 0.187540
\(661\) −49.1859 −1.91311 −0.956554 0.291554i \(-0.905828\pi\)
−0.956554 + 0.291554i \(0.905828\pi\)
\(662\) 21.0483 0.818067
\(663\) 36.4809 1.41680
\(664\) −9.86312 −0.382763
\(665\) 0 0
\(666\) 0.398912 0.0154575
\(667\) −7.38211 −0.285836
\(668\) −14.4378 −0.558616
\(669\) 17.2357 0.666372
\(670\) 5.48725 0.211991
\(671\) −6.30075 −0.243238
\(672\) −6.71856 −0.259174
\(673\) −31.4592 −1.21266 −0.606331 0.795212i \(-0.707359\pi\)
−0.606331 + 0.795212i \(0.707359\pi\)
\(674\) 32.8578 1.26563
\(675\) 13.1571 0.506416
\(676\) −4.00678 −0.154107
\(677\) −5.95353 −0.228813 −0.114406 0.993434i \(-0.536497\pi\)
−0.114406 + 0.993434i \(0.536497\pi\)
\(678\) −14.0349 −0.539008
\(679\) −23.3845 −0.897413
\(680\) 19.0947 0.732249
\(681\) −1.51492 −0.0580518
\(682\) 5.40593 0.207004
\(683\) 8.82659 0.337740 0.168870 0.985638i \(-0.445988\pi\)
0.168870 + 0.985638i \(0.445988\pi\)
\(684\) 0 0
\(685\) 45.0952 1.72300
\(686\) 2.70700 0.103354
\(687\) −14.6107 −0.557433
\(688\) −5.27048 −0.200935
\(689\) 22.6768 0.863918
\(690\) −5.10089 −0.194187
\(691\) 0.0845197 0.00321528 0.00160764 0.999999i \(-0.499488\pi\)
0.00160764 + 0.999999i \(0.499488\pi\)
\(692\) −20.7436 −0.788555
\(693\) −0.266332 −0.0101171
\(694\) −1.55590 −0.0590612
\(695\) 36.2896 1.37654
\(696\) 12.2161 0.463048
\(697\) 43.7323 1.65648
\(698\) −7.35032 −0.278214
\(699\) −34.7901 −1.31588
\(700\) 9.82711 0.371430
\(701\) −2.73246 −0.103204 −0.0516018 0.998668i \(-0.516433\pi\)
−0.0516018 + 0.998668i \(0.516433\pi\)
\(702\) −15.3970 −0.581124
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) −47.0367 −1.77150
\(706\) −19.3609 −0.728659
\(707\) −68.5922 −2.57968
\(708\) −1.74030 −0.0654045
\(709\) −5.26476 −0.197722 −0.0988611 0.995101i \(-0.531520\pi\)
−0.0988611 + 0.995101i \(0.531520\pi\)
\(710\) 36.7575 1.37948
\(711\) 0.667775 0.0250435
\(712\) 13.1889 0.494274
\(713\) −5.72335 −0.214341
\(714\) −46.6502 −1.74584
\(715\) 8.24695 0.308418
\(716\) −19.5304 −0.729884
\(717\) 27.6191 1.03146
\(718\) −14.9074 −0.556341
\(719\) 25.8897 0.965523 0.482762 0.875752i \(-0.339634\pi\)
0.482762 + 0.875752i \(0.339634\pi\)
\(720\) 0.190991 0.00711781
\(721\) 6.27082 0.233538
\(722\) 0 0
\(723\) 38.6911 1.43894
\(724\) 0.524440 0.0194906
\(725\) −17.8682 −0.663608
\(726\) −1.75198 −0.0650223
\(727\) 49.4783 1.83505 0.917525 0.397678i \(-0.130184\pi\)
0.917525 + 0.397678i \(0.130184\pi\)
\(728\) −11.5001 −0.426224
\(729\) 26.3463 0.975790
\(730\) 23.5184 0.870453
\(731\) −36.5955 −1.35353
\(732\) −11.0388 −0.408007
\(733\) 31.3799 1.15904 0.579522 0.814957i \(-0.303239\pi\)
0.579522 + 0.814957i \(0.303239\pi\)
\(734\) −27.5633 −1.01738
\(735\) −37.1269 −1.36945
\(736\) 1.05872 0.0390248
\(737\) −1.99535 −0.0734997
\(738\) 0.437423 0.0161018
\(739\) 33.1196 1.21832 0.609161 0.793046i \(-0.291506\pi\)
0.609161 + 0.793046i \(0.291506\pi\)
\(740\) 15.7956 0.580657
\(741\) 0 0
\(742\) −28.9982 −1.06456
\(743\) 1.63951 0.0601477 0.0300738 0.999548i \(-0.490426\pi\)
0.0300738 + 0.999548i \(0.490426\pi\)
\(744\) 9.47110 0.347227
\(745\) 28.6727 1.05049
\(746\) −13.6882 −0.501159
\(747\) −0.685001 −0.0250629
\(748\) −6.94349 −0.253879
\(749\) 25.6647 0.937768
\(750\) 11.7434 0.428808
\(751\) −28.0985 −1.02533 −0.512664 0.858589i \(-0.671342\pi\)
−0.512664 + 0.858589i \(0.671342\pi\)
\(752\) 9.76272 0.356010
\(753\) −13.7291 −0.500315
\(754\) 20.9102 0.761505
\(755\) 2.44162 0.0888597
\(756\) 19.6891 0.716084
\(757\) −24.9320 −0.906170 −0.453085 0.891467i \(-0.649677\pi\)
−0.453085 + 0.891467i \(0.649677\pi\)
\(758\) −6.31554 −0.229391
\(759\) 1.85486 0.0673270
\(760\) 0 0
\(761\) 11.5854 0.419972 0.209986 0.977704i \(-0.432658\pi\)
0.209986 + 0.977704i \(0.432658\pi\)
\(762\) −36.4269 −1.31961
\(763\) 10.1476 0.367369
\(764\) 19.6960 0.712576
\(765\) 1.32614 0.0479468
\(766\) 4.01015 0.144893
\(767\) −2.97887 −0.107561
\(768\) −1.75198 −0.0632193
\(769\) 4.99290 0.180049 0.0900243 0.995940i \(-0.471306\pi\)
0.0900243 + 0.995940i \(0.471306\pi\)
\(770\) −10.5458 −0.380046
\(771\) 35.9002 1.29291
\(772\) −0.558050 −0.0200846
\(773\) −4.30139 −0.154710 −0.0773550 0.997004i \(-0.524647\pi\)
−0.0773550 + 0.997004i \(0.524647\pi\)
\(774\) −0.366039 −0.0131570
\(775\) −13.8532 −0.497622
\(776\) −6.09792 −0.218902
\(777\) −38.5902 −1.38441
\(778\) −17.8884 −0.641329
\(779\) 0 0
\(780\) 14.4485 0.517340
\(781\) −13.3663 −0.478283
\(782\) 7.35119 0.262878
\(783\) −35.7997 −1.27938
\(784\) 7.70590 0.275211
\(785\) −21.1485 −0.754823
\(786\) 11.7183 0.417979
\(787\) 19.1746 0.683500 0.341750 0.939791i \(-0.388980\pi\)
0.341750 + 0.939791i \(0.388980\pi\)
\(788\) −27.2349 −0.970202
\(789\) 46.9782 1.67247
\(790\) 26.4416 0.940751
\(791\) 30.7203 1.09229
\(792\) −0.0694508 −0.00246783
\(793\) −18.8951 −0.670986
\(794\) −7.46337 −0.264865
\(795\) 36.4326 1.29213
\(796\) −9.66076 −0.342417
\(797\) −26.0624 −0.923179 −0.461589 0.887094i \(-0.652721\pi\)
−0.461589 + 0.887094i \(0.652721\pi\)
\(798\) 0 0
\(799\) 67.7873 2.39814
\(800\) 2.56259 0.0906014
\(801\) 0.915977 0.0323645
\(802\) −16.9891 −0.599907
\(803\) −8.55207 −0.301796
\(804\) −3.49583 −0.123288
\(805\) 11.1651 0.393517
\(806\) 16.2117 0.571032
\(807\) −19.2038 −0.676007
\(808\) −17.8867 −0.629250
\(809\) 14.0330 0.493374 0.246687 0.969095i \(-0.420658\pi\)
0.246687 + 0.969095i \(0.420658\pi\)
\(810\) −25.3099 −0.889298
\(811\) 17.0976 0.600379 0.300190 0.953880i \(-0.402950\pi\)
0.300190 + 0.953880i \(0.402950\pi\)
\(812\) −26.7391 −0.938357
\(813\) −3.25940 −0.114312
\(814\) −5.74381 −0.201321
\(815\) 2.68744 0.0941368
\(816\) −12.1649 −0.425856
\(817\) 0 0
\(818\) 3.76849 0.131762
\(819\) −0.798694 −0.0279086
\(820\) 17.3205 0.604857
\(821\) −38.3104 −1.33704 −0.668521 0.743693i \(-0.733072\pi\)
−0.668521 + 0.743693i \(0.733072\pi\)
\(822\) −28.7293 −1.00205
\(823\) −45.4170 −1.58314 −0.791568 0.611081i \(-0.790735\pi\)
−0.791568 + 0.611081i \(0.790735\pi\)
\(824\) 1.63523 0.0569659
\(825\) 4.48963 0.156309
\(826\) 3.80925 0.132541
\(827\) −38.2328 −1.32948 −0.664742 0.747073i \(-0.731459\pi\)
−0.664742 + 0.747073i \(0.731459\pi\)
\(828\) 0.0735287 0.00255530
\(829\) −9.68698 −0.336443 −0.168221 0.985749i \(-0.553802\pi\)
−0.168221 + 0.985749i \(0.553802\pi\)
\(830\) −27.1237 −0.941479
\(831\) −32.8968 −1.14118
\(832\) −2.99887 −0.103967
\(833\) 53.5058 1.85387
\(834\) −23.1194 −0.800560
\(835\) −39.7042 −1.37402
\(836\) 0 0
\(837\) −27.7555 −0.959371
\(838\) 32.5461 1.12429
\(839\) 18.0609 0.623531 0.311766 0.950159i \(-0.399080\pi\)
0.311766 + 0.950159i \(0.399080\pi\)
\(840\) −18.4762 −0.637488
\(841\) 19.6184 0.676498
\(842\) 3.97736 0.137069
\(843\) −22.2686 −0.766973
\(844\) 16.7285 0.575819
\(845\) −11.0187 −0.379055
\(846\) 0.678029 0.0233111
\(847\) 3.83483 0.131766
\(848\) −7.56179 −0.259673
\(849\) 19.1793 0.658230
\(850\) 17.7933 0.610307
\(851\) 6.08107 0.208457
\(852\) −23.4175 −0.802270
\(853\) −52.2530 −1.78911 −0.894554 0.446961i \(-0.852506\pi\)
−0.894554 + 0.446961i \(0.852506\pi\)
\(854\) 24.1623 0.826817
\(855\) 0 0
\(856\) 6.69253 0.228746
\(857\) 9.26830 0.316599 0.158299 0.987391i \(-0.449399\pi\)
0.158299 + 0.987391i \(0.449399\pi\)
\(858\) −5.25398 −0.179368
\(859\) 30.3995 1.03722 0.518609 0.855011i \(-0.326450\pi\)
0.518609 + 0.855011i \(0.326450\pi\)
\(860\) −14.4939 −0.494238
\(861\) −42.3156 −1.44211
\(862\) −4.18326 −0.142483
\(863\) 20.8169 0.708614 0.354307 0.935129i \(-0.384717\pi\)
0.354307 + 0.935129i \(0.384717\pi\)
\(864\) 5.13428 0.174672
\(865\) −57.0453 −1.93960
\(866\) 13.7414 0.466950
\(867\) −54.6830 −1.85713
\(868\) −20.7308 −0.703649
\(869\) −9.61507 −0.326169
\(870\) 33.5944 1.13896
\(871\) −5.98380 −0.202753
\(872\) 2.64617 0.0896108
\(873\) −0.423505 −0.0143335
\(874\) 0 0
\(875\) −25.7045 −0.868970
\(876\) −14.9831 −0.506232
\(877\) 38.3678 1.29559 0.647795 0.761815i \(-0.275691\pi\)
0.647795 + 0.761815i \(0.275691\pi\)
\(878\) 27.4116 0.925096
\(879\) 7.52476 0.253804
\(880\) −2.75002 −0.0927031
\(881\) −11.9200 −0.401595 −0.200797 0.979633i \(-0.564353\pi\)
−0.200797 + 0.979633i \(0.564353\pi\)
\(882\) 0.535181 0.0180205
\(883\) −8.10007 −0.272589 −0.136295 0.990668i \(-0.543519\pi\)
−0.136295 + 0.990668i \(0.543519\pi\)
\(884\) −20.8226 −0.700341
\(885\) −4.78586 −0.160875
\(886\) 34.5821 1.16181
\(887\) −16.4058 −0.550854 −0.275427 0.961322i \(-0.588819\pi\)
−0.275427 + 0.961322i \(0.588819\pi\)
\(888\) −10.0631 −0.337695
\(889\) 79.7330 2.67416
\(890\) 36.2696 1.21576
\(891\) 9.20353 0.308330
\(892\) −9.83783 −0.329395
\(893\) 0 0
\(894\) −18.2668 −0.610934
\(895\) −53.7088 −1.79529
\(896\) 3.83483 0.128113
\(897\) 5.56247 0.185726
\(898\) 14.0064 0.467399
\(899\) 37.6939 1.25716
\(900\) 0.177974 0.00593248
\(901\) −52.5052 −1.74920
\(902\) −6.29831 −0.209711
\(903\) 35.4100 1.17837
\(904\) 8.01087 0.266438
\(905\) 1.44222 0.0479409
\(906\) −1.55551 −0.0516784
\(907\) −3.06778 −0.101864 −0.0509320 0.998702i \(-0.516219\pi\)
−0.0509320 + 0.998702i \(0.516219\pi\)
\(908\) 0.864686 0.0286956
\(909\) −1.24224 −0.0412026
\(910\) −31.6256 −1.04838
\(911\) 32.5171 1.07734 0.538669 0.842517i \(-0.318927\pi\)
0.538669 + 0.842517i \(0.318927\pi\)
\(912\) 0 0
\(913\) 9.86312 0.326422
\(914\) −34.1525 −1.12966
\(915\) −30.3570 −1.00357
\(916\) 8.33952 0.275545
\(917\) −25.6496 −0.847025
\(918\) 35.6498 1.17662
\(919\) −6.71906 −0.221641 −0.110821 0.993840i \(-0.535348\pi\)
−0.110821 + 0.993840i \(0.535348\pi\)
\(920\) 2.91149 0.0959890
\(921\) −58.1734 −1.91688
\(922\) 37.5791 1.23760
\(923\) −40.0837 −1.31937
\(924\) 6.71856 0.221024
\(925\) 14.7191 0.483960
\(926\) 5.53335 0.181837
\(927\) 0.113568 0.00373006
\(928\) −6.97269 −0.228890
\(929\) 51.7217 1.69693 0.848466 0.529249i \(-0.177526\pi\)
0.848466 + 0.529249i \(0.177526\pi\)
\(930\) 26.0457 0.854072
\(931\) 0 0
\(932\) 19.8575 0.650455
\(933\) −51.5497 −1.68766
\(934\) 12.6215 0.412989
\(935\) −19.0947 −0.624464
\(936\) −0.208274 −0.00680765
\(937\) −33.3287 −1.08880 −0.544401 0.838825i \(-0.683243\pi\)
−0.544401 + 0.838825i \(0.683243\pi\)
\(938\) 7.65183 0.249841
\(939\) 15.3051 0.499463
\(940\) 26.8476 0.875674
\(941\) 32.9924 1.07552 0.537761 0.843097i \(-0.319270\pi\)
0.537761 + 0.843097i \(0.319270\pi\)
\(942\) 13.4733 0.438985
\(943\) 6.66813 0.217144
\(944\) 0.993331 0.0323302
\(945\) 54.1453 1.76135
\(946\) 5.27048 0.171358
\(947\) 6.13812 0.199462 0.0997311 0.995014i \(-0.468202\pi\)
0.0997311 + 0.995014i \(0.468202\pi\)
\(948\) −16.8455 −0.547115
\(949\) −25.6466 −0.832523
\(950\) 0 0
\(951\) −5.28859 −0.171494
\(952\) 26.6271 0.862988
\(953\) −12.0071 −0.388948 −0.194474 0.980908i \(-0.562300\pi\)
−0.194474 + 0.980908i \(0.562300\pi\)
\(954\) −0.525172 −0.0170031
\(955\) 54.1643 1.75272
\(956\) −15.7645 −0.509860
\(957\) −12.2161 −0.394889
\(958\) −22.1665 −0.716167
\(959\) 62.8841 2.03063
\(960\) −4.81799 −0.155500
\(961\) −1.77597 −0.0572894
\(962\) −17.2250 −0.555355
\(963\) 0.464802 0.0149780
\(964\) −22.0841 −0.711282
\(965\) −1.53465 −0.0494020
\(966\) −7.11305 −0.228859
\(967\) −48.3924 −1.55619 −0.778097 0.628144i \(-0.783815\pi\)
−0.778097 + 0.628144i \(0.783815\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −16.7694 −0.538432
\(971\) 29.7309 0.954109 0.477054 0.878874i \(-0.341704\pi\)
0.477054 + 0.878874i \(0.341704\pi\)
\(972\) 0.721610 0.0231456
\(973\) 50.6049 1.62232
\(974\) 8.56601 0.274473
\(975\) 13.4638 0.431187
\(976\) 6.30075 0.201682
\(977\) −28.1112 −0.899358 −0.449679 0.893190i \(-0.648462\pi\)
−0.449679 + 0.893190i \(0.648462\pi\)
\(978\) −1.71212 −0.0547474
\(979\) −13.1889 −0.421518
\(980\) 21.1914 0.676933
\(981\) 0.183779 0.00586761
\(982\) 22.0937 0.705038
\(983\) −36.4985 −1.16412 −0.582061 0.813145i \(-0.697753\pi\)
−0.582061 + 0.813145i \(0.697753\pi\)
\(984\) −11.0346 −0.351769
\(985\) −74.8964 −2.38640
\(986\) −48.4148 −1.54184
\(987\) −65.5914 −2.08780
\(988\) 0 0
\(989\) −5.57995 −0.177432
\(990\) −0.190991 −0.00607009
\(991\) −54.1787 −1.72104 −0.860521 0.509415i \(-0.829862\pi\)
−0.860521 + 0.509415i \(0.829862\pi\)
\(992\) −5.40593 −0.171638
\(993\) −36.8764 −1.17024
\(994\) 51.2573 1.62578
\(995\) −26.5673 −0.842239
\(996\) 17.2800 0.547539
\(997\) −56.3505 −1.78464 −0.892319 0.451406i \(-0.850923\pi\)
−0.892319 + 0.451406i \(0.850923\pi\)
\(998\) −2.72781 −0.0863474
\(999\) 29.4903 0.933033
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 7942.2.a.ca.1.4 15
19.4 even 9 418.2.j.d.111.4 30
19.5 even 9 418.2.j.d.177.4 yes 30
19.18 odd 2 7942.2.a.by.1.12 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.j.d.111.4 30 19.4 even 9
418.2.j.d.177.4 yes 30 19.5 even 9
7942.2.a.by.1.12 15 19.18 odd 2
7942.2.a.ca.1.4 15 1.1 even 1 trivial