Properties

Label 6422.2.a.bm.1.9
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6422,2,Mod(1,6422)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6422, 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("6422.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.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: \(51.2799281781\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} - 22 x^{12} + 98 x^{11} + 164 x^{10} - 912 x^{9} - 374 x^{8} + 3996 x^{7} - 817 x^{6} - 8194 x^{5} + 4418 x^{4} + 6430 x^{3} - 4327 x^{2} - 922 x + 358 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 494)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(1.50159\) of defining polynomial
Character \(\chi\) \(=\) 6422.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.50159 q^{3} +1.00000 q^{4} -2.98870 q^{5} -1.50159 q^{6} -3.73146 q^{7} -1.00000 q^{8} -0.745218 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.50159 q^{3} +1.00000 q^{4} -2.98870 q^{5} -1.50159 q^{6} -3.73146 q^{7} -1.00000 q^{8} -0.745218 q^{9} +2.98870 q^{10} -3.81437 q^{11} +1.50159 q^{12} +3.73146 q^{14} -4.48781 q^{15} +1.00000 q^{16} -2.89129 q^{17} +0.745218 q^{18} -1.00000 q^{19} -2.98870 q^{20} -5.60313 q^{21} +3.81437 q^{22} -1.66948 q^{23} -1.50159 q^{24} +3.93232 q^{25} -5.62379 q^{27} -3.73146 q^{28} -8.51244 q^{29} +4.48781 q^{30} -1.71712 q^{31} -1.00000 q^{32} -5.72763 q^{33} +2.89129 q^{34} +11.1522 q^{35} -0.745218 q^{36} -5.22055 q^{37} +1.00000 q^{38} +2.98870 q^{40} -10.6438 q^{41} +5.60313 q^{42} +5.42719 q^{43} -3.81437 q^{44} +2.22723 q^{45} +1.66948 q^{46} +6.13466 q^{47} +1.50159 q^{48} +6.92379 q^{49} -3.93232 q^{50} -4.34155 q^{51} -13.2870 q^{53} +5.62379 q^{54} +11.4000 q^{55} +3.73146 q^{56} -1.50159 q^{57} +8.51244 q^{58} -5.52632 q^{59} -4.48781 q^{60} -5.99153 q^{61} +1.71712 q^{62} +2.78075 q^{63} +1.00000 q^{64} +5.72763 q^{66} +7.35059 q^{67} -2.89129 q^{68} -2.50689 q^{69} -11.1522 q^{70} +9.75998 q^{71} +0.745218 q^{72} +0.118233 q^{73} +5.22055 q^{74} +5.90474 q^{75} -1.00000 q^{76} +14.2332 q^{77} +3.04474 q^{79} -2.98870 q^{80} -6.20900 q^{81} +10.6438 q^{82} +1.22342 q^{83} -5.60313 q^{84} +8.64121 q^{85} -5.42719 q^{86} -12.7822 q^{87} +3.81437 q^{88} -14.8184 q^{89} -2.22723 q^{90} -1.66948 q^{92} -2.57842 q^{93} -6.13466 q^{94} +2.98870 q^{95} -1.50159 q^{96} -14.8790 q^{97} -6.92379 q^{98} +2.84253 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{2} + 4 q^{3} + 14 q^{4} + 2 q^{5} - 4 q^{6} - 2 q^{7} - 14 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{2} + 4 q^{3} + 14 q^{4} + 2 q^{5} - 4 q^{6} - 2 q^{7} - 14 q^{8} + 18 q^{9} - 2 q^{10} - 10 q^{11} + 4 q^{12} + 2 q^{14} + 4 q^{15} + 14 q^{16} + 2 q^{17} - 18 q^{18} - 14 q^{19} + 2 q^{20} + 18 q^{21} + 10 q^{22} + 12 q^{23} - 4 q^{24} + 44 q^{25} + 10 q^{27} - 2 q^{28} + 4 q^{29} - 4 q^{30} - 4 q^{31} - 14 q^{32} - 12 q^{33} - 2 q^{34} + 14 q^{35} + 18 q^{36} - 18 q^{37} + 14 q^{38} - 2 q^{40} - 6 q^{41} - 18 q^{42} + 28 q^{43} - 10 q^{44} - 8 q^{45} - 12 q^{46} + 20 q^{47} + 4 q^{48} + 28 q^{49} - 44 q^{50} + 10 q^{51} + 12 q^{53} - 10 q^{54} - 2 q^{55} + 2 q^{56} - 4 q^{57} - 4 q^{58} - 16 q^{59} + 4 q^{60} + 30 q^{61} + 4 q^{62} - 28 q^{63} + 14 q^{64} + 12 q^{66} - 2 q^{67} + 2 q^{68} - 42 q^{69} - 14 q^{70} - 44 q^{71} - 18 q^{72} + 36 q^{73} + 18 q^{74} + 46 q^{75} - 14 q^{76} + 68 q^{77} + 34 q^{79} + 2 q^{80} - 6 q^{81} + 6 q^{82} - 2 q^{83} + 18 q^{84} + 30 q^{85} - 28 q^{86} + 52 q^{87} + 10 q^{88} - 42 q^{89} + 8 q^{90} + 12 q^{92} + 12 q^{93} - 20 q^{94} - 2 q^{95} - 4 q^{96} - 40 q^{97} - 28 q^{98} - 12 q^{99}+O(q^{100}) \) 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
\(3\) 1.50159 0.866945 0.433473 0.901167i \(-0.357288\pi\)
0.433473 + 0.901167i \(0.357288\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.98870 −1.33659 −0.668293 0.743898i \(-0.732975\pi\)
−0.668293 + 0.743898i \(0.732975\pi\)
\(6\) −1.50159 −0.613023
\(7\) −3.73146 −1.41036 −0.705180 0.709029i \(-0.749134\pi\)
−0.705180 + 0.709029i \(0.749134\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.745218 −0.248406
\(10\) 2.98870 0.945109
\(11\) −3.81437 −1.15007 −0.575037 0.818127i \(-0.695012\pi\)
−0.575037 + 0.818127i \(0.695012\pi\)
\(12\) 1.50159 0.433473
\(13\) 0 0
\(14\) 3.73146 0.997274
\(15\) −4.48781 −1.15875
\(16\) 1.00000 0.250000
\(17\) −2.89129 −0.701242 −0.350621 0.936517i \(-0.614029\pi\)
−0.350621 + 0.936517i \(0.614029\pi\)
\(18\) 0.745218 0.175650
\(19\) −1.00000 −0.229416
\(20\) −2.98870 −0.668293
\(21\) −5.60313 −1.22270
\(22\) 3.81437 0.813225
\(23\) −1.66948 −0.348112 −0.174056 0.984736i \(-0.555687\pi\)
−0.174056 + 0.984736i \(0.555687\pi\)
\(24\) −1.50159 −0.306511
\(25\) 3.93232 0.786464
\(26\) 0 0
\(27\) −5.62379 −1.08230
\(28\) −3.73146 −0.705180
\(29\) −8.51244 −1.58072 −0.790360 0.612643i \(-0.790106\pi\)
−0.790360 + 0.612643i \(0.790106\pi\)
\(30\) 4.48781 0.819358
\(31\) −1.71712 −0.308404 −0.154202 0.988039i \(-0.549281\pi\)
−0.154202 + 0.988039i \(0.549281\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.72763 −0.997052
\(34\) 2.89129 0.495853
\(35\) 11.1522 1.88507
\(36\) −0.745218 −0.124203
\(37\) −5.22055 −0.858254 −0.429127 0.903244i \(-0.641179\pi\)
−0.429127 + 0.903244i \(0.641179\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 2.98870 0.472555
\(41\) −10.6438 −1.66229 −0.831145 0.556056i \(-0.812314\pi\)
−0.831145 + 0.556056i \(0.812314\pi\)
\(42\) 5.60313 0.864582
\(43\) 5.42719 0.827639 0.413820 0.910359i \(-0.364194\pi\)
0.413820 + 0.910359i \(0.364194\pi\)
\(44\) −3.81437 −0.575037
\(45\) 2.22723 0.332016
\(46\) 1.66948 0.246152
\(47\) 6.13466 0.894831 0.447416 0.894326i \(-0.352344\pi\)
0.447416 + 0.894326i \(0.352344\pi\)
\(48\) 1.50159 0.216736
\(49\) 6.92379 0.989113
\(50\) −3.93232 −0.556114
\(51\) −4.34155 −0.607938
\(52\) 0 0
\(53\) −13.2870 −1.82511 −0.912553 0.408959i \(-0.865892\pi\)
−0.912553 + 0.408959i \(0.865892\pi\)
\(54\) 5.62379 0.765301
\(55\) 11.4000 1.53717
\(56\) 3.73146 0.498637
\(57\) −1.50159 −0.198891
\(58\) 8.51244 1.11774
\(59\) −5.52632 −0.719465 −0.359732 0.933055i \(-0.617132\pi\)
−0.359732 + 0.933055i \(0.617132\pi\)
\(60\) −4.48781 −0.579374
\(61\) −5.99153 −0.767137 −0.383568 0.923512i \(-0.625305\pi\)
−0.383568 + 0.923512i \(0.625305\pi\)
\(62\) 1.71712 0.218075
\(63\) 2.78075 0.350342
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.72763 0.705022
\(67\) 7.35059 0.898018 0.449009 0.893527i \(-0.351777\pi\)
0.449009 + 0.893527i \(0.351777\pi\)
\(68\) −2.89129 −0.350621
\(69\) −2.50689 −0.301794
\(70\) −11.1522 −1.33294
\(71\) 9.75998 1.15830 0.579148 0.815222i \(-0.303385\pi\)
0.579148 + 0.815222i \(0.303385\pi\)
\(72\) 0.745218 0.0878248
\(73\) 0.118233 0.0138382 0.00691908 0.999976i \(-0.497798\pi\)
0.00691908 + 0.999976i \(0.497798\pi\)
\(74\) 5.22055 0.606877
\(75\) 5.90474 0.681821
\(76\) −1.00000 −0.114708
\(77\) 14.2332 1.62202
\(78\) 0 0
\(79\) 3.04474 0.342560 0.171280 0.985222i \(-0.445210\pi\)
0.171280 + 0.985222i \(0.445210\pi\)
\(80\) −2.98870 −0.334147
\(81\) −6.20900 −0.689888
\(82\) 10.6438 1.17542
\(83\) 1.22342 0.134288 0.0671439 0.997743i \(-0.478611\pi\)
0.0671439 + 0.997743i \(0.478611\pi\)
\(84\) −5.60313 −0.611352
\(85\) 8.64121 0.937271
\(86\) −5.42719 −0.585229
\(87\) −12.7822 −1.37040
\(88\) 3.81437 0.406613
\(89\) −14.8184 −1.57074 −0.785372 0.619023i \(-0.787529\pi\)
−0.785372 + 0.619023i \(0.787529\pi\)
\(90\) −2.22723 −0.234771
\(91\) 0 0
\(92\) −1.66948 −0.174056
\(93\) −2.57842 −0.267370
\(94\) −6.13466 −0.632741
\(95\) 2.98870 0.306634
\(96\) −1.50159 −0.153256
\(97\) −14.8790 −1.51073 −0.755366 0.655303i \(-0.772541\pi\)
−0.755366 + 0.655303i \(0.772541\pi\)
\(98\) −6.92379 −0.699408
\(99\) 2.84253 0.285685
\(100\) 3.93232 0.393232
\(101\) 14.4250 1.43534 0.717669 0.696384i \(-0.245209\pi\)
0.717669 + 0.696384i \(0.245209\pi\)
\(102\) 4.34155 0.429877
\(103\) 9.25772 0.912190 0.456095 0.889931i \(-0.349248\pi\)
0.456095 + 0.889931i \(0.349248\pi\)
\(104\) 0 0
\(105\) 16.7461 1.63425
\(106\) 13.2870 1.29054
\(107\) −11.6631 −1.12752 −0.563759 0.825939i \(-0.690645\pi\)
−0.563759 + 0.825939i \(0.690645\pi\)
\(108\) −5.62379 −0.541150
\(109\) −6.07064 −0.581462 −0.290731 0.956805i \(-0.593898\pi\)
−0.290731 + 0.956805i \(0.593898\pi\)
\(110\) −11.4000 −1.08695
\(111\) −7.83915 −0.744059
\(112\) −3.73146 −0.352590
\(113\) −5.29092 −0.497728 −0.248864 0.968538i \(-0.580057\pi\)
−0.248864 + 0.968538i \(0.580057\pi\)
\(114\) 1.50159 0.140637
\(115\) 4.98959 0.465281
\(116\) −8.51244 −0.790360
\(117\) 0 0
\(118\) 5.52632 0.508739
\(119\) 10.7887 0.989003
\(120\) 4.48781 0.409679
\(121\) 3.54939 0.322671
\(122\) 5.99153 0.542448
\(123\) −15.9827 −1.44111
\(124\) −1.71712 −0.154202
\(125\) 3.19098 0.285410
\(126\) −2.78075 −0.247729
\(127\) −11.2686 −0.999927 −0.499963 0.866046i \(-0.666653\pi\)
−0.499963 + 0.866046i \(0.666653\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.14944 0.717518
\(130\) 0 0
\(131\) −12.7617 −1.11499 −0.557495 0.830180i \(-0.688238\pi\)
−0.557495 + 0.830180i \(0.688238\pi\)
\(132\) −5.72763 −0.498526
\(133\) 3.73146 0.323559
\(134\) −7.35059 −0.634994
\(135\) 16.8078 1.44659
\(136\) 2.89129 0.247926
\(137\) 8.10390 0.692363 0.346182 0.938168i \(-0.387478\pi\)
0.346182 + 0.938168i \(0.387478\pi\)
\(138\) 2.50689 0.213400
\(139\) 9.90009 0.839714 0.419857 0.907590i \(-0.362080\pi\)
0.419857 + 0.907590i \(0.362080\pi\)
\(140\) 11.1522 0.942534
\(141\) 9.21176 0.775770
\(142\) −9.75998 −0.819039
\(143\) 0 0
\(144\) −0.745218 −0.0621015
\(145\) 25.4411 2.11277
\(146\) −0.118233 −0.00978506
\(147\) 10.3967 0.857506
\(148\) −5.22055 −0.429127
\(149\) 23.1692 1.89809 0.949045 0.315139i \(-0.102051\pi\)
0.949045 + 0.315139i \(0.102051\pi\)
\(150\) −5.90474 −0.482120
\(151\) 20.1390 1.63889 0.819446 0.573157i \(-0.194282\pi\)
0.819446 + 0.573157i \(0.194282\pi\)
\(152\) 1.00000 0.0811107
\(153\) 2.15464 0.174193
\(154\) −14.2332 −1.14694
\(155\) 5.13196 0.412209
\(156\) 0 0
\(157\) −13.0900 −1.04470 −0.522349 0.852732i \(-0.674944\pi\)
−0.522349 + 0.852732i \(0.674944\pi\)
\(158\) −3.04474 −0.242226
\(159\) −19.9516 −1.58227
\(160\) 2.98870 0.236277
\(161\) 6.22961 0.490962
\(162\) 6.20900 0.487825
\(163\) −5.90255 −0.462323 −0.231162 0.972915i \(-0.574253\pi\)
−0.231162 + 0.972915i \(0.574253\pi\)
\(164\) −10.6438 −0.831145
\(165\) 17.1181 1.33265
\(166\) −1.22342 −0.0949559
\(167\) −16.6567 −1.28894 −0.644468 0.764631i \(-0.722921\pi\)
−0.644468 + 0.764631i \(0.722921\pi\)
\(168\) 5.60313 0.432291
\(169\) 0 0
\(170\) −8.64121 −0.662750
\(171\) 0.745218 0.0569882
\(172\) 5.42719 0.413820
\(173\) −9.47289 −0.720211 −0.360105 0.932912i \(-0.617259\pi\)
−0.360105 + 0.932912i \(0.617259\pi\)
\(174\) 12.7822 0.969017
\(175\) −14.6733 −1.10920
\(176\) −3.81437 −0.287519
\(177\) −8.29828 −0.623737
\(178\) 14.8184 1.11068
\(179\) −5.00441 −0.374047 −0.187024 0.982355i \(-0.559884\pi\)
−0.187024 + 0.982355i \(0.559884\pi\)
\(180\) 2.22723 0.166008
\(181\) −1.97399 −0.146726 −0.0733629 0.997305i \(-0.523373\pi\)
−0.0733629 + 0.997305i \(0.523373\pi\)
\(182\) 0 0
\(183\) −8.99684 −0.665065
\(184\) 1.66948 0.123076
\(185\) 15.6027 1.14713
\(186\) 2.57842 0.189059
\(187\) 11.0285 0.806480
\(188\) 6.13466 0.447416
\(189\) 20.9850 1.52643
\(190\) −2.98870 −0.216823
\(191\) −20.8136 −1.50602 −0.753010 0.658009i \(-0.771399\pi\)
−0.753010 + 0.658009i \(0.771399\pi\)
\(192\) 1.50159 0.108368
\(193\) −21.3064 −1.53367 −0.766833 0.641847i \(-0.778169\pi\)
−0.766833 + 0.641847i \(0.778169\pi\)
\(194\) 14.8790 1.06825
\(195\) 0 0
\(196\) 6.92379 0.494556
\(197\) 8.62246 0.614325 0.307162 0.951657i \(-0.400621\pi\)
0.307162 + 0.951657i \(0.400621\pi\)
\(198\) −2.84253 −0.202010
\(199\) 4.00411 0.283844 0.141922 0.989878i \(-0.454672\pi\)
0.141922 + 0.989878i \(0.454672\pi\)
\(200\) −3.93232 −0.278057
\(201\) 11.0376 0.778532
\(202\) −14.4250 −1.01494
\(203\) 31.7638 2.22938
\(204\) −4.34155 −0.303969
\(205\) 31.8112 2.22179
\(206\) −9.25772 −0.645016
\(207\) 1.24413 0.0864730
\(208\) 0 0
\(209\) 3.81437 0.263845
\(210\) −16.7461 −1.15559
\(211\) 8.29916 0.571338 0.285669 0.958328i \(-0.407784\pi\)
0.285669 + 0.958328i \(0.407784\pi\)
\(212\) −13.2870 −0.912553
\(213\) 14.6555 1.00418
\(214\) 11.6631 0.797276
\(215\) −16.2202 −1.10621
\(216\) 5.62379 0.382651
\(217\) 6.40737 0.434961
\(218\) 6.07064 0.411155
\(219\) 0.177538 0.0119969
\(220\) 11.4000 0.768587
\(221\) 0 0
\(222\) 7.83915 0.526129
\(223\) −8.81599 −0.590362 −0.295181 0.955441i \(-0.595380\pi\)
−0.295181 + 0.955441i \(0.595380\pi\)
\(224\) 3.73146 0.249319
\(225\) −2.93043 −0.195362
\(226\) 5.29092 0.351947
\(227\) 4.18895 0.278030 0.139015 0.990290i \(-0.455606\pi\)
0.139015 + 0.990290i \(0.455606\pi\)
\(228\) −1.50159 −0.0994454
\(229\) 24.2635 1.60338 0.801689 0.597741i \(-0.203935\pi\)
0.801689 + 0.597741i \(0.203935\pi\)
\(230\) −4.98959 −0.329004
\(231\) 21.3724 1.40620
\(232\) 8.51244 0.558869
\(233\) 17.9212 1.17405 0.587027 0.809567i \(-0.300298\pi\)
0.587027 + 0.809567i \(0.300298\pi\)
\(234\) 0 0
\(235\) −18.3346 −1.19602
\(236\) −5.52632 −0.359732
\(237\) 4.57196 0.296981
\(238\) −10.7887 −0.699331
\(239\) 22.3178 1.44362 0.721808 0.692093i \(-0.243311\pi\)
0.721808 + 0.692093i \(0.243311\pi\)
\(240\) −4.48781 −0.289687
\(241\) −22.3962 −1.44267 −0.721334 0.692588i \(-0.756471\pi\)
−0.721334 + 0.692588i \(0.756471\pi\)
\(242\) −3.54939 −0.228163
\(243\) 7.54799 0.484204
\(244\) −5.99153 −0.383568
\(245\) −20.6931 −1.32203
\(246\) 15.9827 1.01902
\(247\) 0 0
\(248\) 1.71712 0.109037
\(249\) 1.83708 0.116420
\(250\) −3.19098 −0.201815
\(251\) 3.68665 0.232699 0.116350 0.993208i \(-0.462881\pi\)
0.116350 + 0.993208i \(0.462881\pi\)
\(252\) 2.78075 0.175171
\(253\) 6.36802 0.400354
\(254\) 11.2686 0.707055
\(255\) 12.9756 0.812562
\(256\) 1.00000 0.0625000
\(257\) 12.7082 0.792718 0.396359 0.918096i \(-0.370274\pi\)
0.396359 + 0.918096i \(0.370274\pi\)
\(258\) −8.14944 −0.507362
\(259\) 19.4803 1.21045
\(260\) 0 0
\(261\) 6.34362 0.392660
\(262\) 12.7617 0.788418
\(263\) 3.24916 0.200352 0.100176 0.994970i \(-0.468059\pi\)
0.100176 + 0.994970i \(0.468059\pi\)
\(264\) 5.72763 0.352511
\(265\) 39.7107 2.43941
\(266\) −3.73146 −0.228790
\(267\) −22.2512 −1.36175
\(268\) 7.35059 0.449009
\(269\) 10.9402 0.667037 0.333519 0.942744i \(-0.391764\pi\)
0.333519 + 0.942744i \(0.391764\pi\)
\(270\) −16.8078 −1.02289
\(271\) 0.968555 0.0588355 0.0294178 0.999567i \(-0.490635\pi\)
0.0294178 + 0.999567i \(0.490635\pi\)
\(272\) −2.89129 −0.175310
\(273\) 0 0
\(274\) −8.10390 −0.489575
\(275\) −14.9993 −0.904492
\(276\) −2.50689 −0.150897
\(277\) 8.02640 0.482260 0.241130 0.970493i \(-0.422482\pi\)
0.241130 + 0.970493i \(0.422482\pi\)
\(278\) −9.90009 −0.593768
\(279\) 1.27963 0.0766094
\(280\) −11.1522 −0.666472
\(281\) 4.41792 0.263551 0.131775 0.991280i \(-0.457932\pi\)
0.131775 + 0.991280i \(0.457932\pi\)
\(282\) −9.21176 −0.548552
\(283\) −15.7900 −0.938621 −0.469310 0.883033i \(-0.655497\pi\)
−0.469310 + 0.883033i \(0.655497\pi\)
\(284\) 9.75998 0.579148
\(285\) 4.48781 0.265835
\(286\) 0 0
\(287\) 39.7171 2.34442
\(288\) 0.745218 0.0439124
\(289\) −8.64042 −0.508260
\(290\) −25.4411 −1.49395
\(291\) −22.3422 −1.30972
\(292\) 0.118233 0.00691908
\(293\) −34.0058 −1.98664 −0.993320 0.115389i \(-0.963188\pi\)
−0.993320 + 0.115389i \(0.963188\pi\)
\(294\) −10.3967 −0.606349
\(295\) 16.5165 0.961627
\(296\) 5.22055 0.303438
\(297\) 21.4512 1.24473
\(298\) −23.1692 −1.34215
\(299\) 0 0
\(300\) 5.90474 0.340911
\(301\) −20.2513 −1.16727
\(302\) −20.1390 −1.15887
\(303\) 21.6604 1.24436
\(304\) −1.00000 −0.0573539
\(305\) 17.9069 1.02534
\(306\) −2.15464 −0.123173
\(307\) 29.2156 1.66743 0.833713 0.552199i \(-0.186211\pi\)
0.833713 + 0.552199i \(0.186211\pi\)
\(308\) 14.2332 0.811009
\(309\) 13.9013 0.790819
\(310\) −5.13196 −0.291476
\(311\) −13.1516 −0.745761 −0.372881 0.927879i \(-0.621630\pi\)
−0.372881 + 0.927879i \(0.621630\pi\)
\(312\) 0 0
\(313\) 22.1828 1.25385 0.626924 0.779080i \(-0.284314\pi\)
0.626924 + 0.779080i \(0.284314\pi\)
\(314\) 13.0900 0.738712
\(315\) −8.31082 −0.468262
\(316\) 3.04474 0.171280
\(317\) 25.2932 1.42061 0.710303 0.703896i \(-0.248558\pi\)
0.710303 + 0.703896i \(0.248558\pi\)
\(318\) 19.9516 1.11883
\(319\) 32.4695 1.81795
\(320\) −2.98870 −0.167073
\(321\) −17.5133 −0.977496
\(322\) −6.22961 −0.347163
\(323\) 2.89129 0.160876
\(324\) −6.20900 −0.344944
\(325\) 0 0
\(326\) 5.90255 0.326912
\(327\) −9.11563 −0.504095
\(328\) 10.6438 0.587708
\(329\) −22.8912 −1.26203
\(330\) −17.1181 −0.942323
\(331\) −27.6235 −1.51832 −0.759162 0.650901i \(-0.774391\pi\)
−0.759162 + 0.650901i \(0.774391\pi\)
\(332\) 1.22342 0.0671439
\(333\) 3.89045 0.213195
\(334\) 16.6567 0.911416
\(335\) −21.9687 −1.20028
\(336\) −5.60313 −0.305676
\(337\) −21.3984 −1.16565 −0.582824 0.812599i \(-0.698052\pi\)
−0.582824 + 0.812599i \(0.698052\pi\)
\(338\) 0 0
\(339\) −7.94481 −0.431503
\(340\) 8.64121 0.468635
\(341\) 6.54973 0.354688
\(342\) −0.745218 −0.0402968
\(343\) 0.284381 0.0153551
\(344\) −5.42719 −0.292615
\(345\) 7.49233 0.403373
\(346\) 9.47289 0.509266
\(347\) −8.49897 −0.456249 −0.228124 0.973632i \(-0.573259\pi\)
−0.228124 + 0.973632i \(0.573259\pi\)
\(348\) −12.7822 −0.685199
\(349\) 21.6315 1.15791 0.578953 0.815361i \(-0.303462\pi\)
0.578953 + 0.815361i \(0.303462\pi\)
\(350\) 14.6733 0.784320
\(351\) 0 0
\(352\) 3.81437 0.203306
\(353\) −28.9686 −1.54184 −0.770922 0.636930i \(-0.780204\pi\)
−0.770922 + 0.636930i \(0.780204\pi\)
\(354\) 8.29828 0.441048
\(355\) −29.1696 −1.54816
\(356\) −14.8184 −0.785372
\(357\) 16.2003 0.857411
\(358\) 5.00441 0.264491
\(359\) −7.42119 −0.391675 −0.195838 0.980636i \(-0.562743\pi\)
−0.195838 + 0.980636i \(0.562743\pi\)
\(360\) −2.22723 −0.117385
\(361\) 1.00000 0.0526316
\(362\) 1.97399 0.103751
\(363\) 5.32973 0.279738
\(364\) 0 0
\(365\) −0.353364 −0.0184959
\(366\) 8.99684 0.470272
\(367\) 13.6135 0.710619 0.355310 0.934749i \(-0.384375\pi\)
0.355310 + 0.934749i \(0.384375\pi\)
\(368\) −1.66948 −0.0870279
\(369\) 7.93198 0.412923
\(370\) −15.6027 −0.811144
\(371\) 49.5798 2.57405
\(372\) −2.57842 −0.133685
\(373\) −2.73685 −0.141709 −0.0708544 0.997487i \(-0.522573\pi\)
−0.0708544 + 0.997487i \(0.522573\pi\)
\(374\) −11.0285 −0.570268
\(375\) 4.79155 0.247434
\(376\) −6.13466 −0.316371
\(377\) 0 0
\(378\) −20.9850 −1.07935
\(379\) 1.62208 0.0833208 0.0416604 0.999132i \(-0.486735\pi\)
0.0416604 + 0.999132i \(0.486735\pi\)
\(380\) 2.98870 0.153317
\(381\) −16.9209 −0.866882
\(382\) 20.8136 1.06492
\(383\) −31.9773 −1.63396 −0.816981 0.576664i \(-0.804354\pi\)
−0.816981 + 0.576664i \(0.804354\pi\)
\(384\) −1.50159 −0.0766279
\(385\) −42.5386 −2.16797
\(386\) 21.3064 1.08447
\(387\) −4.04444 −0.205591
\(388\) −14.8790 −0.755366
\(389\) −4.42183 −0.224195 −0.112098 0.993697i \(-0.535757\pi\)
−0.112098 + 0.993697i \(0.535757\pi\)
\(390\) 0 0
\(391\) 4.82697 0.244110
\(392\) −6.92379 −0.349704
\(393\) −19.1628 −0.966636
\(394\) −8.62246 −0.434393
\(395\) −9.09981 −0.457861
\(396\) 2.84253 0.142843
\(397\) −0.227725 −0.0114292 −0.00571459 0.999984i \(-0.501819\pi\)
−0.00571459 + 0.999984i \(0.501819\pi\)
\(398\) −4.00411 −0.200708
\(399\) 5.60313 0.280508
\(400\) 3.93232 0.196616
\(401\) −6.80150 −0.339651 −0.169825 0.985474i \(-0.554320\pi\)
−0.169825 + 0.985474i \(0.554320\pi\)
\(402\) −11.0376 −0.550505
\(403\) 0 0
\(404\) 14.4250 0.717669
\(405\) 18.5568 0.922096
\(406\) −31.7638 −1.57641
\(407\) 19.9131 0.987056
\(408\) 4.34155 0.214939
\(409\) −16.0814 −0.795175 −0.397588 0.917564i \(-0.630153\pi\)
−0.397588 + 0.917564i \(0.630153\pi\)
\(410\) −31.8112 −1.57105
\(411\) 12.1688 0.600241
\(412\) 9.25772 0.456095
\(413\) 20.6212 1.01470
\(414\) −1.24413 −0.0611456
\(415\) −3.65644 −0.179487
\(416\) 0 0
\(417\) 14.8659 0.727986
\(418\) −3.81437 −0.186567
\(419\) −21.3995 −1.04544 −0.522718 0.852506i \(-0.675082\pi\)
−0.522718 + 0.852506i \(0.675082\pi\)
\(420\) 16.7461 0.817125
\(421\) −26.3895 −1.28615 −0.643074 0.765804i \(-0.722341\pi\)
−0.643074 + 0.765804i \(0.722341\pi\)
\(422\) −8.29916 −0.403997
\(423\) −4.57165 −0.222281
\(424\) 13.2870 0.645272
\(425\) −11.3695 −0.551501
\(426\) −14.6555 −0.710062
\(427\) 22.3571 1.08194
\(428\) −11.6631 −0.563759
\(429\) 0 0
\(430\) 16.2202 0.782210
\(431\) −22.5479 −1.08609 −0.543047 0.839702i \(-0.682729\pi\)
−0.543047 + 0.839702i \(0.682729\pi\)
\(432\) −5.62379 −0.270575
\(433\) 11.5129 0.553273 0.276636 0.960975i \(-0.410780\pi\)
0.276636 + 0.960975i \(0.410780\pi\)
\(434\) −6.40737 −0.307564
\(435\) 38.2022 1.83165
\(436\) −6.07064 −0.290731
\(437\) 1.66948 0.0798623
\(438\) −0.177538 −0.00848311
\(439\) −11.6448 −0.555778 −0.277889 0.960613i \(-0.589635\pi\)
−0.277889 + 0.960613i \(0.589635\pi\)
\(440\) −11.4000 −0.543473
\(441\) −5.15973 −0.245701
\(442\) 0 0
\(443\) 26.0168 1.23609 0.618047 0.786141i \(-0.287924\pi\)
0.618047 + 0.786141i \(0.287924\pi\)
\(444\) −7.83915 −0.372029
\(445\) 44.2877 2.09944
\(446\) 8.81599 0.417449
\(447\) 34.7906 1.64554
\(448\) −3.73146 −0.176295
\(449\) −11.7314 −0.553640 −0.276820 0.960922i \(-0.589281\pi\)
−0.276820 + 0.960922i \(0.589281\pi\)
\(450\) 2.93043 0.138142
\(451\) 40.5995 1.91176
\(452\) −5.29092 −0.248864
\(453\) 30.2406 1.42083
\(454\) −4.18895 −0.196597
\(455\) 0 0
\(456\) 1.50159 0.0703185
\(457\) 36.7678 1.71993 0.859963 0.510356i \(-0.170486\pi\)
0.859963 + 0.510356i \(0.170486\pi\)
\(458\) −24.2635 −1.13376
\(459\) 16.2600 0.758954
\(460\) 4.98959 0.232641
\(461\) 12.0022 0.558996 0.279498 0.960146i \(-0.409832\pi\)
0.279498 + 0.960146i \(0.409832\pi\)
\(462\) −21.3724 −0.994334
\(463\) −15.5101 −0.720815 −0.360407 0.932795i \(-0.617362\pi\)
−0.360407 + 0.932795i \(0.617362\pi\)
\(464\) −8.51244 −0.395180
\(465\) 7.70612 0.357363
\(466\) −17.9212 −0.830182
\(467\) −25.8302 −1.19528 −0.597639 0.801765i \(-0.703894\pi\)
−0.597639 + 0.801765i \(0.703894\pi\)
\(468\) 0 0
\(469\) −27.4284 −1.26653
\(470\) 18.3346 0.845714
\(471\) −19.6559 −0.905695
\(472\) 5.52632 0.254369
\(473\) −20.7013 −0.951847
\(474\) −4.57196 −0.209997
\(475\) −3.93232 −0.180427
\(476\) 10.7887 0.494501
\(477\) 9.90169 0.453367
\(478\) −22.3178 −1.02079
\(479\) −10.0622 −0.459753 −0.229876 0.973220i \(-0.573832\pi\)
−0.229876 + 0.973220i \(0.573832\pi\)
\(480\) 4.48781 0.204840
\(481\) 0 0
\(482\) 22.3962 1.02012
\(483\) 9.35434 0.425637
\(484\) 3.54939 0.161336
\(485\) 44.4688 2.01922
\(486\) −7.54799 −0.342384
\(487\) 5.76723 0.261338 0.130669 0.991426i \(-0.458287\pi\)
0.130669 + 0.991426i \(0.458287\pi\)
\(488\) 5.99153 0.271224
\(489\) −8.86323 −0.400809
\(490\) 20.6931 0.934820
\(491\) 19.4673 0.878546 0.439273 0.898354i \(-0.355236\pi\)
0.439273 + 0.898354i \(0.355236\pi\)
\(492\) −15.9827 −0.720557
\(493\) 24.6120 1.10847
\(494\) 0 0
\(495\) −8.49548 −0.381843
\(496\) −1.71712 −0.0771010
\(497\) −36.4190 −1.63361
\(498\) −1.83708 −0.0823215
\(499\) −4.74563 −0.212444 −0.106222 0.994342i \(-0.533875\pi\)
−0.106222 + 0.994342i \(0.533875\pi\)
\(500\) 3.19098 0.142705
\(501\) −25.0116 −1.11744
\(502\) −3.68665 −0.164543
\(503\) −30.4469 −1.35756 −0.678779 0.734342i \(-0.737491\pi\)
−0.678779 + 0.734342i \(0.737491\pi\)
\(504\) −2.78075 −0.123864
\(505\) −43.1119 −1.91845
\(506\) −6.36802 −0.283093
\(507\) 0 0
\(508\) −11.2686 −0.499963
\(509\) 6.35941 0.281876 0.140938 0.990018i \(-0.454988\pi\)
0.140938 + 0.990018i \(0.454988\pi\)
\(510\) −12.9756 −0.574568
\(511\) −0.441183 −0.0195168
\(512\) −1.00000 −0.0441942
\(513\) 5.62379 0.248297
\(514\) −12.7082 −0.560536
\(515\) −27.6685 −1.21922
\(516\) 8.14944 0.358759
\(517\) −23.3998 −1.02912
\(518\) −19.4803 −0.855914
\(519\) −14.2244 −0.624383
\(520\) 0 0
\(521\) 8.30076 0.363663 0.181832 0.983330i \(-0.441797\pi\)
0.181832 + 0.983330i \(0.441797\pi\)
\(522\) −6.34362 −0.277653
\(523\) 0.111917 0.00489377 0.00244688 0.999997i \(-0.499221\pi\)
0.00244688 + 0.999997i \(0.499221\pi\)
\(524\) −12.7617 −0.557495
\(525\) −22.0333 −0.961613
\(526\) −3.24916 −0.141670
\(527\) 4.96470 0.216266
\(528\) −5.72763 −0.249263
\(529\) −20.2128 −0.878818
\(530\) −39.7107 −1.72492
\(531\) 4.11831 0.178719
\(532\) 3.73146 0.161779
\(533\) 0 0
\(534\) 22.2512 0.962903
\(535\) 34.8576 1.50703
\(536\) −7.35059 −0.317497
\(537\) −7.51459 −0.324278
\(538\) −10.9402 −0.471667
\(539\) −26.4099 −1.13755
\(540\) 16.8078 0.723294
\(541\) −36.1474 −1.55410 −0.777049 0.629440i \(-0.783284\pi\)
−0.777049 + 0.629440i \(0.783284\pi\)
\(542\) −0.968555 −0.0416030
\(543\) −2.96414 −0.127203
\(544\) 2.89129 0.123963
\(545\) 18.1433 0.777174
\(546\) 0 0
\(547\) −5.56274 −0.237846 −0.118923 0.992904i \(-0.537944\pi\)
−0.118923 + 0.992904i \(0.537944\pi\)
\(548\) 8.10390 0.346182
\(549\) 4.46500 0.190561
\(550\) 14.9993 0.639572
\(551\) 8.51244 0.362642
\(552\) 2.50689 0.106700
\(553\) −11.3613 −0.483132
\(554\) −8.02640 −0.341009
\(555\) 23.4288 0.994499
\(556\) 9.90009 0.419857
\(557\) −12.7166 −0.538820 −0.269410 0.963026i \(-0.586829\pi\)
−0.269410 + 0.963026i \(0.586829\pi\)
\(558\) −1.27963 −0.0541711
\(559\) 0 0
\(560\) 11.1522 0.471267
\(561\) 16.5603 0.699174
\(562\) −4.41792 −0.186359
\(563\) −32.2272 −1.35821 −0.679106 0.734040i \(-0.737633\pi\)
−0.679106 + 0.734040i \(0.737633\pi\)
\(564\) 9.21176 0.387885
\(565\) 15.8130 0.665256
\(566\) 15.7900 0.663705
\(567\) 23.1686 0.972990
\(568\) −9.75998 −0.409520
\(569\) −3.69103 −0.154736 −0.0773680 0.997003i \(-0.524652\pi\)
−0.0773680 + 0.997003i \(0.524652\pi\)
\(570\) −4.48781 −0.187974
\(571\) −21.2298 −0.888440 −0.444220 0.895918i \(-0.646519\pi\)
−0.444220 + 0.895918i \(0.646519\pi\)
\(572\) 0 0
\(573\) −31.2536 −1.30564
\(574\) −39.7171 −1.65776
\(575\) −6.56495 −0.273777
\(576\) −0.745218 −0.0310507
\(577\) −16.7181 −0.695983 −0.347992 0.937498i \(-0.613136\pi\)
−0.347992 + 0.937498i \(0.613136\pi\)
\(578\) 8.64042 0.359394
\(579\) −31.9935 −1.32960
\(580\) 25.4411 1.05638
\(581\) −4.56514 −0.189394
\(582\) 22.3422 0.926113
\(583\) 50.6814 2.09901
\(584\) −0.118233 −0.00489253
\(585\) 0 0
\(586\) 34.0058 1.40477
\(587\) −25.7669 −1.06351 −0.531757 0.846897i \(-0.678468\pi\)
−0.531757 + 0.846897i \(0.678468\pi\)
\(588\) 10.3967 0.428753
\(589\) 1.71712 0.0707528
\(590\) −16.5165 −0.679973
\(591\) 12.9474 0.532586
\(592\) −5.22055 −0.214563
\(593\) 22.0804 0.906734 0.453367 0.891324i \(-0.350223\pi\)
0.453367 + 0.891324i \(0.350223\pi\)
\(594\) −21.4512 −0.880154
\(595\) −32.2443 −1.32189
\(596\) 23.1692 0.949045
\(597\) 6.01254 0.246077
\(598\) 0 0
\(599\) −24.2400 −0.990421 −0.495211 0.868773i \(-0.664909\pi\)
−0.495211 + 0.868773i \(0.664909\pi\)
\(600\) −5.90474 −0.241060
\(601\) −25.0975 −1.02375 −0.511875 0.859060i \(-0.671049\pi\)
−0.511875 + 0.859060i \(0.671049\pi\)
\(602\) 20.2513 0.825383
\(603\) −5.47779 −0.223073
\(604\) 20.1390 0.819446
\(605\) −10.6080 −0.431278
\(606\) −21.6604 −0.879895
\(607\) 39.1982 1.59101 0.795503 0.605950i \(-0.207207\pi\)
0.795503 + 0.605950i \(0.207207\pi\)
\(608\) 1.00000 0.0405554
\(609\) 47.6963 1.93275
\(610\) −17.9069 −0.725028
\(611\) 0 0
\(612\) 2.15464 0.0870963
\(613\) 31.8846 1.28781 0.643904 0.765106i \(-0.277314\pi\)
0.643904 + 0.765106i \(0.277314\pi\)
\(614\) −29.2156 −1.17905
\(615\) 47.7675 1.92617
\(616\) −14.2332 −0.573470
\(617\) −29.2084 −1.17589 −0.587944 0.808902i \(-0.700062\pi\)
−0.587944 + 0.808902i \(0.700062\pi\)
\(618\) −13.9013 −0.559193
\(619\) −34.6290 −1.39186 −0.695929 0.718111i \(-0.745007\pi\)
−0.695929 + 0.718111i \(0.745007\pi\)
\(620\) 5.13196 0.206104
\(621\) 9.38884 0.376761
\(622\) 13.1516 0.527333
\(623\) 55.2942 2.21531
\(624\) 0 0
\(625\) −29.1985 −1.16794
\(626\) −22.1828 −0.886605
\(627\) 5.72763 0.228739
\(628\) −13.0900 −0.522349
\(629\) 15.0942 0.601843
\(630\) 8.31082 0.331111
\(631\) 16.9719 0.675640 0.337820 0.941211i \(-0.390311\pi\)
0.337820 + 0.941211i \(0.390311\pi\)
\(632\) −3.04474 −0.121113
\(633\) 12.4620 0.495318
\(634\) −25.2932 −1.00452
\(635\) 33.6785 1.33649
\(636\) −19.9516 −0.791133
\(637\) 0 0
\(638\) −32.4695 −1.28548
\(639\) −7.27331 −0.287728
\(640\) 2.98870 0.118139
\(641\) −22.2794 −0.879982 −0.439991 0.898002i \(-0.645018\pi\)
−0.439991 + 0.898002i \(0.645018\pi\)
\(642\) 17.5133 0.691194
\(643\) 42.5539 1.67816 0.839080 0.544007i \(-0.183094\pi\)
0.839080 + 0.544007i \(0.183094\pi\)
\(644\) 6.22961 0.245481
\(645\) −24.3562 −0.959025
\(646\) −2.89129 −0.113756
\(647\) 37.1643 1.46108 0.730539 0.682871i \(-0.239269\pi\)
0.730539 + 0.682871i \(0.239269\pi\)
\(648\) 6.20900 0.243912
\(649\) 21.0794 0.827438
\(650\) 0 0
\(651\) 9.62126 0.377087
\(652\) −5.90255 −0.231162
\(653\) −42.4412 −1.66085 −0.830427 0.557127i \(-0.811904\pi\)
−0.830427 + 0.557127i \(0.811904\pi\)
\(654\) 9.11563 0.356449
\(655\) 38.1407 1.49028
\(656\) −10.6438 −0.415572
\(657\) −0.0881096 −0.00343748
\(658\) 22.8912 0.892392
\(659\) −32.8747 −1.28062 −0.640308 0.768118i \(-0.721193\pi\)
−0.640308 + 0.768118i \(0.721193\pi\)
\(660\) 17.1181 0.666323
\(661\) −15.2765 −0.594189 −0.297094 0.954848i \(-0.596018\pi\)
−0.297094 + 0.954848i \(0.596018\pi\)
\(662\) 27.6235 1.07362
\(663\) 0 0
\(664\) −1.22342 −0.0474779
\(665\) −11.1522 −0.432464
\(666\) −3.89045 −0.150752
\(667\) 14.2114 0.550267
\(668\) −16.6567 −0.644468
\(669\) −13.2380 −0.511812
\(670\) 21.9687 0.848725
\(671\) 22.8539 0.882264
\(672\) 5.60313 0.216146
\(673\) 6.81905 0.262855 0.131428 0.991326i \(-0.458044\pi\)
0.131428 + 0.991326i \(0.458044\pi\)
\(674\) 21.3984 0.824237
\(675\) −22.1146 −0.851190
\(676\) 0 0
\(677\) 21.0962 0.810793 0.405396 0.914141i \(-0.367134\pi\)
0.405396 + 0.914141i \(0.367134\pi\)
\(678\) 7.94481 0.305119
\(679\) 55.5203 2.13067
\(680\) −8.64121 −0.331375
\(681\) 6.29010 0.241037
\(682\) −6.54973 −0.250802
\(683\) −43.7151 −1.67271 −0.836355 0.548188i \(-0.815318\pi\)
−0.836355 + 0.548188i \(0.815318\pi\)
\(684\) 0.745218 0.0284941
\(685\) −24.2201 −0.925403
\(686\) −0.284381 −0.0108577
\(687\) 36.4339 1.39004
\(688\) 5.42719 0.206910
\(689\) 0 0
\(690\) −7.49233 −0.285228
\(691\) −43.6813 −1.66172 −0.830858 0.556485i \(-0.812150\pi\)
−0.830858 + 0.556485i \(0.812150\pi\)
\(692\) −9.47289 −0.360105
\(693\) −10.6068 −0.402919
\(694\) 8.49897 0.322617
\(695\) −29.5884 −1.12235
\(696\) 12.7822 0.484509
\(697\) 30.7745 1.16567
\(698\) −21.6315 −0.818763
\(699\) 26.9103 1.01784
\(700\) −14.6733 −0.554598
\(701\) 12.9613 0.489540 0.244770 0.969581i \(-0.421288\pi\)
0.244770 + 0.969581i \(0.421288\pi\)
\(702\) 0 0
\(703\) 5.22055 0.196897
\(704\) −3.81437 −0.143759
\(705\) −27.5312 −1.03688
\(706\) 28.9686 1.09025
\(707\) −53.8262 −2.02434
\(708\) −8.29828 −0.311868
\(709\) 3.19138 0.119855 0.0599275 0.998203i \(-0.480913\pi\)
0.0599275 + 0.998203i \(0.480913\pi\)
\(710\) 29.1696 1.09472
\(711\) −2.26899 −0.0850939
\(712\) 14.8184 0.555342
\(713\) 2.86671 0.107359
\(714\) −16.2003 −0.606281
\(715\) 0 0
\(716\) −5.00441 −0.187024
\(717\) 33.5122 1.25154
\(718\) 7.42119 0.276956
\(719\) 52.2035 1.94686 0.973431 0.228980i \(-0.0735391\pi\)
0.973431 + 0.228980i \(0.0735391\pi\)
\(720\) 2.22723 0.0830040
\(721\) −34.5448 −1.28652
\(722\) −1.00000 −0.0372161
\(723\) −33.6300 −1.25071
\(724\) −1.97399 −0.0733629
\(725\) −33.4736 −1.24318
\(726\) −5.32973 −0.197805
\(727\) −44.7559 −1.65990 −0.829952 0.557835i \(-0.811632\pi\)
−0.829952 + 0.557835i \(0.811632\pi\)
\(728\) 0 0
\(729\) 29.9610 1.10967
\(730\) 0.353364 0.0130786
\(731\) −15.6916 −0.580375
\(732\) −8.99684 −0.332533
\(733\) 12.8805 0.475753 0.237877 0.971295i \(-0.423549\pi\)
0.237877 + 0.971295i \(0.423549\pi\)
\(734\) −13.6135 −0.502484
\(735\) −31.0726 −1.14613
\(736\) 1.66948 0.0615380
\(737\) −28.0378 −1.03279
\(738\) −7.93198 −0.291980
\(739\) −31.0761 −1.14315 −0.571577 0.820549i \(-0.693668\pi\)
−0.571577 + 0.820549i \(0.693668\pi\)
\(740\) 15.6027 0.573565
\(741\) 0 0
\(742\) −49.5798 −1.82013
\(743\) −33.6944 −1.23613 −0.618064 0.786128i \(-0.712083\pi\)
−0.618064 + 0.786128i \(0.712083\pi\)
\(744\) 2.57842 0.0945294
\(745\) −69.2456 −2.53696
\(746\) 2.73685 0.100203
\(747\) −0.911715 −0.0333579
\(748\) 11.0285 0.403240
\(749\) 43.5205 1.59021
\(750\) −4.79155 −0.174963
\(751\) 1.70343 0.0621589 0.0310794 0.999517i \(-0.490106\pi\)
0.0310794 + 0.999517i \(0.490106\pi\)
\(752\) 6.13466 0.223708
\(753\) 5.53585 0.201738
\(754\) 0 0
\(755\) −60.1895 −2.19052
\(756\) 20.9850 0.763216
\(757\) 35.2712 1.28195 0.640976 0.767561i \(-0.278530\pi\)
0.640976 + 0.767561i \(0.278530\pi\)
\(758\) −1.62208 −0.0589167
\(759\) 9.56218 0.347085
\(760\) −2.98870 −0.108411
\(761\) −6.63116 −0.240379 −0.120190 0.992751i \(-0.538350\pi\)
−0.120190 + 0.992751i \(0.538350\pi\)
\(762\) 16.9209 0.612978
\(763\) 22.6523 0.820070
\(764\) −20.8136 −0.753010
\(765\) −6.43958 −0.232824
\(766\) 31.9773 1.15539
\(767\) 0 0
\(768\) 1.50159 0.0541841
\(769\) 19.7122 0.710840 0.355420 0.934707i \(-0.384338\pi\)
0.355420 + 0.934707i \(0.384338\pi\)
\(770\) 42.5386 1.53298
\(771\) 19.0826 0.687243
\(772\) −21.3064 −0.766833
\(773\) −10.8204 −0.389184 −0.194592 0.980884i \(-0.562338\pi\)
−0.194592 + 0.980884i \(0.562338\pi\)
\(774\) 4.04444 0.145374
\(775\) −6.75227 −0.242549
\(776\) 14.8790 0.534124
\(777\) 29.2515 1.04939
\(778\) 4.42183 0.158530
\(779\) 10.6438 0.381355
\(780\) 0 0
\(781\) −37.2281 −1.33213
\(782\) −4.82697 −0.172612
\(783\) 47.8722 1.71081
\(784\) 6.92379 0.247278
\(785\) 39.1221 1.39633
\(786\) 19.1628 0.683515
\(787\) 48.0124 1.71146 0.855729 0.517424i \(-0.173109\pi\)
0.855729 + 0.517424i \(0.173109\pi\)
\(788\) 8.62246 0.307162
\(789\) 4.87891 0.173694
\(790\) 9.09981 0.323757
\(791\) 19.7429 0.701975
\(792\) −2.84253 −0.101005
\(793\) 0 0
\(794\) 0.227725 0.00808165
\(795\) 59.6294 2.11484
\(796\) 4.00411 0.141922
\(797\) 22.8603 0.809755 0.404878 0.914371i \(-0.367314\pi\)
0.404878 + 0.914371i \(0.367314\pi\)
\(798\) −5.60313 −0.198349
\(799\) −17.7371 −0.627493
\(800\) −3.93232 −0.139028
\(801\) 11.0429 0.390182
\(802\) 6.80150 0.240169
\(803\) −0.450985 −0.0159149
\(804\) 11.0376 0.389266
\(805\) −18.6184 −0.656214
\(806\) 0 0
\(807\) 16.4278 0.578285
\(808\) −14.4250 −0.507469
\(809\) 16.3061 0.573294 0.286647 0.958036i \(-0.407459\pi\)
0.286647 + 0.958036i \(0.407459\pi\)
\(810\) −18.5568 −0.652020
\(811\) 0.810412 0.0284574 0.0142287 0.999899i \(-0.495471\pi\)
0.0142287 + 0.999899i \(0.495471\pi\)
\(812\) 31.7638 1.11469
\(813\) 1.45437 0.0510072
\(814\) −19.9131 −0.697954
\(815\) 17.6409 0.617935
\(816\) −4.34155 −0.151985
\(817\) −5.42719 −0.189873
\(818\) 16.0814 0.562274
\(819\) 0 0
\(820\) 31.8112 1.11090
\(821\) 6.51020 0.227208 0.113604 0.993526i \(-0.463761\pi\)
0.113604 + 0.993526i \(0.463761\pi\)
\(822\) −12.1688 −0.424434
\(823\) 22.1663 0.772670 0.386335 0.922359i \(-0.373741\pi\)
0.386335 + 0.922359i \(0.373741\pi\)
\(824\) −9.25772 −0.322508
\(825\) −22.5229 −0.784145
\(826\) −20.6212 −0.717504
\(827\) −8.74542 −0.304108 −0.152054 0.988372i \(-0.548589\pi\)
−0.152054 + 0.988372i \(0.548589\pi\)
\(828\) 1.24413 0.0432365
\(829\) −22.2016 −0.771092 −0.385546 0.922689i \(-0.625987\pi\)
−0.385546 + 0.922689i \(0.625987\pi\)
\(830\) 3.65644 0.126917
\(831\) 12.0524 0.418093
\(832\) 0 0
\(833\) −20.0187 −0.693607
\(834\) −14.8659 −0.514764
\(835\) 49.7819 1.72278
\(836\) 3.81437 0.131923
\(837\) 9.65674 0.333786
\(838\) 21.3995 0.739234
\(839\) −31.7739 −1.09696 −0.548479 0.836164i \(-0.684793\pi\)
−0.548479 + 0.836164i \(0.684793\pi\)
\(840\) −16.7461 −0.577795
\(841\) 43.4616 1.49868
\(842\) 26.3895 0.909444
\(843\) 6.63391 0.228484
\(844\) 8.29916 0.285669
\(845\) 0 0
\(846\) 4.57165 0.157177
\(847\) −13.2444 −0.455083
\(848\) −13.2870 −0.456276
\(849\) −23.7102 −0.813733
\(850\) 11.3695 0.389970
\(851\) 8.71563 0.298768
\(852\) 14.6555 0.502090
\(853\) −56.4010 −1.93113 −0.965567 0.260154i \(-0.916227\pi\)
−0.965567 + 0.260154i \(0.916227\pi\)
\(854\) −22.3571 −0.765046
\(855\) −2.22723 −0.0761697
\(856\) 11.6631 0.398638
\(857\) −26.8812 −0.918244 −0.459122 0.888373i \(-0.651836\pi\)
−0.459122 + 0.888373i \(0.651836\pi\)
\(858\) 0 0
\(859\) 21.5548 0.735440 0.367720 0.929937i \(-0.380138\pi\)
0.367720 + 0.929937i \(0.380138\pi\)
\(860\) −16.2202 −0.553106
\(861\) 59.6389 2.03249
\(862\) 22.5479 0.767984
\(863\) 28.9587 0.985767 0.492883 0.870095i \(-0.335943\pi\)
0.492883 + 0.870095i \(0.335943\pi\)
\(864\) 5.62379 0.191325
\(865\) 28.3116 0.962624
\(866\) −11.5129 −0.391223
\(867\) −12.9744 −0.440633
\(868\) 6.40737 0.217480
\(869\) −11.6137 −0.393969
\(870\) −38.2022 −1.29518
\(871\) 0 0
\(872\) 6.07064 0.205578
\(873\) 11.0881 0.375275
\(874\) −1.66948 −0.0564711
\(875\) −11.9070 −0.402530
\(876\) 0.177538 0.00599846
\(877\) −9.29627 −0.313913 −0.156956 0.987606i \(-0.550168\pi\)
−0.156956 + 0.987606i \(0.550168\pi\)
\(878\) 11.6448 0.392994
\(879\) −51.0629 −1.72231
\(880\) 11.4000 0.384294
\(881\) −16.6824 −0.562046 −0.281023 0.959701i \(-0.590674\pi\)
−0.281023 + 0.959701i \(0.590674\pi\)
\(882\) 5.15973 0.173737
\(883\) 44.8252 1.50849 0.754245 0.656594i \(-0.228003\pi\)
0.754245 + 0.656594i \(0.228003\pi\)
\(884\) 0 0
\(885\) 24.8011 0.833678
\(886\) −26.0168 −0.874050
\(887\) 51.4991 1.72917 0.864586 0.502485i \(-0.167581\pi\)
0.864586 + 0.502485i \(0.167581\pi\)
\(888\) 7.83915 0.263065
\(889\) 42.0483 1.41026
\(890\) −44.2877 −1.48453
\(891\) 23.6834 0.793423
\(892\) −8.81599 −0.295181
\(893\) −6.13466 −0.205288
\(894\) −34.7906 −1.16357
\(895\) 14.9567 0.499946
\(896\) 3.73146 0.124659
\(897\) 0 0
\(898\) 11.7314 0.391482
\(899\) 14.6169 0.487501
\(900\) −2.93043 −0.0976812
\(901\) 38.4165 1.27984
\(902\) −40.5995 −1.35182
\(903\) −30.4093 −1.01196
\(904\) 5.29092 0.175973
\(905\) 5.89967 0.196112
\(906\) −30.2406 −1.00468
\(907\) 31.7073 1.05282 0.526411 0.850230i \(-0.323537\pi\)
0.526411 + 0.850230i \(0.323537\pi\)
\(908\) 4.18895 0.139015
\(909\) −10.7497 −0.356546
\(910\) 0 0
\(911\) −46.0327 −1.52513 −0.762566 0.646911i \(-0.776061\pi\)
−0.762566 + 0.646911i \(0.776061\pi\)
\(912\) −1.50159 −0.0497227
\(913\) −4.66657 −0.154441
\(914\) −36.7678 −1.21617
\(915\) 26.8888 0.888918
\(916\) 24.2635 0.801689
\(917\) 47.6196 1.57254
\(918\) −16.2600 −0.536661
\(919\) 16.5907 0.547278 0.273639 0.961832i \(-0.411773\pi\)
0.273639 + 0.961832i \(0.411773\pi\)
\(920\) −4.98959 −0.164502
\(921\) 43.8700 1.44557
\(922\) −12.0022 −0.395270
\(923\) 0 0
\(924\) 21.3724 0.703100
\(925\) −20.5289 −0.674985
\(926\) 15.5101 0.509693
\(927\) −6.89902 −0.226593
\(928\) 8.51244 0.279434
\(929\) 48.8958 1.60422 0.802110 0.597176i \(-0.203711\pi\)
0.802110 + 0.597176i \(0.203711\pi\)
\(930\) −7.70612 −0.252693
\(931\) −6.92379 −0.226918
\(932\) 17.9212 0.587027
\(933\) −19.7484 −0.646534
\(934\) 25.8302 0.845189
\(935\) −32.9607 −1.07793
\(936\) 0 0
\(937\) 23.3908 0.764144 0.382072 0.924133i \(-0.375211\pi\)
0.382072 + 0.924133i \(0.375211\pi\)
\(938\) 27.4284 0.895570
\(939\) 33.3096 1.08702
\(940\) −18.3346 −0.598010
\(941\) −17.0539 −0.555940 −0.277970 0.960590i \(-0.589662\pi\)
−0.277970 + 0.960590i \(0.589662\pi\)
\(942\) 19.6559 0.640423
\(943\) 17.7697 0.578662
\(944\) −5.52632 −0.179866
\(945\) −62.7177 −2.04021
\(946\) 20.7013 0.673057
\(947\) 17.9107 0.582021 0.291010 0.956720i \(-0.406009\pi\)
0.291010 + 0.956720i \(0.406009\pi\)
\(948\) 4.57196 0.148490
\(949\) 0 0
\(950\) 3.93232 0.127581
\(951\) 37.9800 1.23159
\(952\) −10.7887 −0.349665
\(953\) 8.02689 0.260016 0.130008 0.991513i \(-0.458500\pi\)
0.130008 + 0.991513i \(0.458500\pi\)
\(954\) −9.90169 −0.320579
\(955\) 62.2056 2.01293
\(956\) 22.3178 0.721808
\(957\) 48.7560 1.57606
\(958\) 10.0622 0.325094
\(959\) −30.2394 −0.976481
\(960\) −4.48781 −0.144843
\(961\) −28.0515 −0.904887
\(962\) 0 0
\(963\) 8.69158 0.280082
\(964\) −22.3962 −0.721334
\(965\) 63.6783 2.04988
\(966\) −9.35434 −0.300971
\(967\) −31.4562 −1.01156 −0.505781 0.862662i \(-0.668796\pi\)
−0.505781 + 0.862662i \(0.668796\pi\)
\(968\) −3.54939 −0.114082
\(969\) 4.34155 0.139471
\(970\) −44.4688 −1.42781
\(971\) −16.7191 −0.536542 −0.268271 0.963343i \(-0.586452\pi\)
−0.268271 + 0.963343i \(0.586452\pi\)
\(972\) 7.54799 0.242102
\(973\) −36.9418 −1.18430
\(974\) −5.76723 −0.184794
\(975\) 0 0
\(976\) −5.99153 −0.191784
\(977\) 9.07353 0.290288 0.145144 0.989411i \(-0.453635\pi\)
0.145144 + 0.989411i \(0.453635\pi\)
\(978\) 8.86323 0.283415
\(979\) 56.5227 1.80647
\(980\) −20.6931 −0.661017
\(981\) 4.52395 0.144439
\(982\) −19.4673 −0.621226
\(983\) −48.1584 −1.53601 −0.768007 0.640442i \(-0.778751\pi\)
−0.768007 + 0.640442i \(0.778751\pi\)
\(984\) 15.9827 0.509511
\(985\) −25.7699 −0.821098
\(986\) −24.6120 −0.783804
\(987\) −34.3733 −1.09411
\(988\) 0 0
\(989\) −9.06061 −0.288111
\(990\) 8.49548 0.270004
\(991\) −43.3084 −1.37574 −0.687869 0.725835i \(-0.741454\pi\)
−0.687869 + 0.725835i \(0.741454\pi\)
\(992\) 1.71712 0.0545187
\(993\) −41.4793 −1.31630
\(994\) 36.4190 1.15514
\(995\) −11.9671 −0.379382
\(996\) 1.83708 0.0582101
\(997\) −57.6729 −1.82652 −0.913260 0.407377i \(-0.866444\pi\)
−0.913260 + 0.407377i \(0.866444\pi\)
\(998\) 4.74563 0.150220
\(999\) 29.3593 0.928888
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 6422.2.a.bm.1.9 14
13.2 odd 12 494.2.m.b.381.3 yes 28
13.7 odd 12 494.2.m.b.153.3 28
13.12 even 2 6422.2.a.bn.1.9 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
494.2.m.b.153.3 28 13.7 odd 12
494.2.m.b.381.3 yes 28 13.2 odd 12
6422.2.a.bm.1.9 14 1.1 even 1 trivial
6422.2.a.bn.1.9 14 13.12 even 2