Properties

Label 8470.2.a.cv.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $6$
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] = [8470,2,Mod(1,8470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8470, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8470.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.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: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.4642000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 8x^{4} + 5x^{3} + 14x^{2} - 9x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.73934\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.73934 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.73934 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.50401 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.73934 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.73934 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.50401 q^{9} -1.00000 q^{10} -2.73934 q^{12} -2.65196 q^{13} -1.00000 q^{14} -2.73934 q^{15} +1.00000 q^{16} +6.67727 q^{17} -4.50401 q^{18} -5.92140 q^{19} +1.00000 q^{20} -2.73934 q^{21} -6.62208 q^{23} +2.73934 q^{24} +1.00000 q^{25} +2.65196 q^{26} -4.11999 q^{27} +1.00000 q^{28} -1.10448 q^{29} +2.73934 q^{30} -5.57898 q^{31} -1.00000 q^{32} -6.67727 q^{34} +1.00000 q^{35} +4.50401 q^{36} +4.15191 q^{37} +5.92140 q^{38} +7.26462 q^{39} -1.00000 q^{40} -9.08954 q^{41} +2.73934 q^{42} +5.51500 q^{43} +4.50401 q^{45} +6.62208 q^{46} +0.907239 q^{47} -2.73934 q^{48} +1.00000 q^{49} -1.00000 q^{50} -18.2913 q^{51} -2.65196 q^{52} -9.24121 q^{53} +4.11999 q^{54} -1.00000 q^{56} +16.2208 q^{57} +1.10448 q^{58} +1.72702 q^{59} -2.73934 q^{60} +8.93518 q^{61} +5.57898 q^{62} +4.50401 q^{63} +1.00000 q^{64} -2.65196 q^{65} -4.10699 q^{67} +6.67727 q^{68} +18.1402 q^{69} -1.00000 q^{70} -7.02166 q^{71} -4.50401 q^{72} -0.967438 q^{73} -4.15191 q^{74} -2.73934 q^{75} -5.92140 q^{76} -7.26462 q^{78} -10.7456 q^{79} +1.00000 q^{80} -2.22595 q^{81} +9.08954 q^{82} +15.9031 q^{83} -2.73934 q^{84} +6.67727 q^{85} -5.51500 q^{86} +3.02556 q^{87} +12.5040 q^{89} -4.50401 q^{90} -2.65196 q^{91} -6.62208 q^{92} +15.2827 q^{93} -0.907239 q^{94} -5.92140 q^{95} +2.73934 q^{96} +11.4362 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} - q^{9} - 6 q^{10} - q^{12} + 6 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} + 21 q^{17} + q^{18} - 3 q^{19} + 6 q^{20} - q^{21} - 10 q^{23} + q^{24} + 6 q^{25} - 6 q^{26} - 4 q^{27} + 6 q^{28} + 10 q^{29} + q^{30} - 4 q^{31} - 6 q^{32} - 21 q^{34} + 6 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} + 26 q^{39} - 6 q^{40} + 7 q^{41} + q^{42} + 19 q^{43} - q^{45} + 10 q^{46} + 10 q^{47} - q^{48} + 6 q^{49} - 6 q^{50} - 4 q^{51} + 6 q^{52} - 16 q^{53} + 4 q^{54} - 6 q^{56} + 16 q^{57} - 10 q^{58} - 3 q^{59} - q^{60} - 8 q^{61} + 4 q^{62} - q^{63} + 6 q^{64} + 6 q^{65} - 27 q^{67} + 21 q^{68} + 4 q^{69} - 6 q^{70} + 4 q^{71} + q^{72} + 13 q^{73} + 2 q^{74} - q^{75} - 3 q^{76} - 26 q^{78} + 14 q^{79} + 6 q^{80} - 14 q^{81} - 7 q^{82} + 51 q^{83} - q^{84} + 21 q^{85} - 19 q^{86} + 8 q^{87} + q^{89} + q^{90} + 6 q^{91} - 10 q^{92} + 4 q^{93} - 10 q^{94} - 3 q^{95} + q^{96} + 7 q^{97} - 6 q^{98}+O(q^{100}) \) 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
\(3\) −2.73934 −1.58156 −0.790780 0.612100i \(-0.790325\pi\)
−0.790780 + 0.612100i \(0.790325\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.73934 1.11833
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 4.50401 1.50134
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −2.73934 −0.790780
\(13\) −2.65196 −0.735520 −0.367760 0.929921i \(-0.619875\pi\)
−0.367760 + 0.929921i \(0.619875\pi\)
\(14\) −1.00000 −0.267261
\(15\) −2.73934 −0.707296
\(16\) 1.00000 0.250000
\(17\) 6.67727 1.61948 0.809738 0.586791i \(-0.199609\pi\)
0.809738 + 0.586791i \(0.199609\pi\)
\(18\) −4.50401 −1.06160
\(19\) −5.92140 −1.35846 −0.679231 0.733924i \(-0.737687\pi\)
−0.679231 + 0.733924i \(0.737687\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.73934 −0.597774
\(22\) 0 0
\(23\) −6.62208 −1.38080 −0.690400 0.723428i \(-0.742565\pi\)
−0.690400 + 0.723428i \(0.742565\pi\)
\(24\) 2.73934 0.559166
\(25\) 1.00000 0.200000
\(26\) 2.65196 0.520091
\(27\) −4.11999 −0.792892
\(28\) 1.00000 0.188982
\(29\) −1.10448 −0.205097 −0.102549 0.994728i \(-0.532700\pi\)
−0.102549 + 0.994728i \(0.532700\pi\)
\(30\) 2.73934 0.500133
\(31\) −5.57898 −1.00201 −0.501007 0.865443i \(-0.667037\pi\)
−0.501007 + 0.865443i \(0.667037\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.67727 −1.14514
\(35\) 1.00000 0.169031
\(36\) 4.50401 0.750668
\(37\) 4.15191 0.682570 0.341285 0.939960i \(-0.389138\pi\)
0.341285 + 0.939960i \(0.389138\pi\)
\(38\) 5.92140 0.960578
\(39\) 7.26462 1.16327
\(40\) −1.00000 −0.158114
\(41\) −9.08954 −1.41955 −0.709773 0.704430i \(-0.751203\pi\)
−0.709773 + 0.704430i \(0.751203\pi\)
\(42\) 2.73934 0.422690
\(43\) 5.51500 0.841029 0.420515 0.907286i \(-0.361850\pi\)
0.420515 + 0.907286i \(0.361850\pi\)
\(44\) 0 0
\(45\) 4.50401 0.671417
\(46\) 6.62208 0.976373
\(47\) 0.907239 0.132334 0.0661672 0.997809i \(-0.478923\pi\)
0.0661672 + 0.997809i \(0.478923\pi\)
\(48\) −2.73934 −0.395390
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −18.2913 −2.56130
\(52\) −2.65196 −0.367760
\(53\) −9.24121 −1.26938 −0.634689 0.772768i \(-0.718872\pi\)
−0.634689 + 0.772768i \(0.718872\pi\)
\(54\) 4.11999 0.560659
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 16.2208 2.14849
\(58\) 1.10448 0.145026
\(59\) 1.72702 0.224838 0.112419 0.993661i \(-0.464140\pi\)
0.112419 + 0.993661i \(0.464140\pi\)
\(60\) −2.73934 −0.353648
\(61\) 8.93518 1.14403 0.572016 0.820242i \(-0.306161\pi\)
0.572016 + 0.820242i \(0.306161\pi\)
\(62\) 5.57898 0.708531
\(63\) 4.50401 0.567451
\(64\) 1.00000 0.125000
\(65\) −2.65196 −0.328935
\(66\) 0 0
\(67\) −4.10699 −0.501748 −0.250874 0.968020i \(-0.580718\pi\)
−0.250874 + 0.968020i \(0.580718\pi\)
\(68\) 6.67727 0.809738
\(69\) 18.1402 2.18382
\(70\) −1.00000 −0.119523
\(71\) −7.02166 −0.833318 −0.416659 0.909063i \(-0.636799\pi\)
−0.416659 + 0.909063i \(0.636799\pi\)
\(72\) −4.50401 −0.530802
\(73\) −0.967438 −0.113230 −0.0566150 0.998396i \(-0.518031\pi\)
−0.0566150 + 0.998396i \(0.518031\pi\)
\(74\) −4.15191 −0.482650
\(75\) −2.73934 −0.316312
\(76\) −5.92140 −0.679231
\(77\) 0 0
\(78\) −7.26462 −0.822556
\(79\) −10.7456 −1.20897 −0.604487 0.796615i \(-0.706622\pi\)
−0.604487 + 0.796615i \(0.706622\pi\)
\(80\) 1.00000 0.111803
\(81\) −2.22595 −0.247328
\(82\) 9.08954 1.00377
\(83\) 15.9031 1.74559 0.872794 0.488089i \(-0.162306\pi\)
0.872794 + 0.488089i \(0.162306\pi\)
\(84\) −2.73934 −0.298887
\(85\) 6.67727 0.724252
\(86\) −5.51500 −0.594698
\(87\) 3.02556 0.324374
\(88\) 0 0
\(89\) 12.5040 1.32543 0.662713 0.748874i \(-0.269405\pi\)
0.662713 + 0.748874i \(0.269405\pi\)
\(90\) −4.50401 −0.474764
\(91\) −2.65196 −0.278000
\(92\) −6.62208 −0.690400
\(93\) 15.2827 1.58475
\(94\) −0.907239 −0.0935746
\(95\) −5.92140 −0.607523
\(96\) 2.73934 0.279583
\(97\) 11.4362 1.16117 0.580584 0.814201i \(-0.302824\pi\)
0.580584 + 0.814201i \(0.302824\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −7.67893 −0.764082 −0.382041 0.924145i \(-0.624779\pi\)
−0.382041 + 0.924145i \(0.624779\pi\)
\(102\) 18.2913 1.81111
\(103\) −1.29467 −0.127567 −0.0637836 0.997964i \(-0.520317\pi\)
−0.0637836 + 0.997964i \(0.520317\pi\)
\(104\) 2.65196 0.260046
\(105\) −2.73934 −0.267333
\(106\) 9.24121 0.897586
\(107\) 14.0391 1.35721 0.678605 0.734503i \(-0.262585\pi\)
0.678605 + 0.734503i \(0.262585\pi\)
\(108\) −4.11999 −0.396446
\(109\) 13.6269 1.30522 0.652609 0.757695i \(-0.273674\pi\)
0.652609 + 0.757695i \(0.273674\pi\)
\(110\) 0 0
\(111\) −11.3735 −1.07953
\(112\) 1.00000 0.0944911
\(113\) 12.3826 1.16486 0.582430 0.812881i \(-0.302102\pi\)
0.582430 + 0.812881i \(0.302102\pi\)
\(114\) −16.2208 −1.51921
\(115\) −6.62208 −0.617512
\(116\) −1.10448 −0.102549
\(117\) −11.9444 −1.10426
\(118\) −1.72702 −0.158985
\(119\) 6.67727 0.612105
\(120\) 2.73934 0.250067
\(121\) 0 0
\(122\) −8.93518 −0.808953
\(123\) 24.8994 2.24510
\(124\) −5.57898 −0.501007
\(125\) 1.00000 0.0894427
\(126\) −4.50401 −0.401249
\(127\) −8.64037 −0.766709 −0.383354 0.923601i \(-0.625231\pi\)
−0.383354 + 0.923601i \(0.625231\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −15.1075 −1.33014
\(130\) 2.65196 0.232592
\(131\) −6.27193 −0.547981 −0.273991 0.961732i \(-0.588344\pi\)
−0.273991 + 0.961732i \(0.588344\pi\)
\(132\) 0 0
\(133\) −5.92140 −0.513451
\(134\) 4.10699 0.354790
\(135\) −4.11999 −0.354592
\(136\) −6.67727 −0.572571
\(137\) −10.9179 −0.932783 −0.466391 0.884579i \(-0.654446\pi\)
−0.466391 + 0.884579i \(0.654446\pi\)
\(138\) −18.1402 −1.54419
\(139\) −12.5679 −1.06600 −0.532999 0.846116i \(-0.678935\pi\)
−0.532999 + 0.846116i \(0.678935\pi\)
\(140\) 1.00000 0.0845154
\(141\) −2.48524 −0.209295
\(142\) 7.02166 0.589245
\(143\) 0 0
\(144\) 4.50401 0.375334
\(145\) −1.10448 −0.0917223
\(146\) 0.967438 0.0800657
\(147\) −2.73934 −0.225937
\(148\) 4.15191 0.341285
\(149\) 18.9356 1.55126 0.775631 0.631187i \(-0.217432\pi\)
0.775631 + 0.631187i \(0.217432\pi\)
\(150\) 2.73934 0.223666
\(151\) 9.07849 0.738797 0.369398 0.929271i \(-0.379564\pi\)
0.369398 + 0.929271i \(0.379564\pi\)
\(152\) 5.92140 0.480289
\(153\) 30.0745 2.43138
\(154\) 0 0
\(155\) −5.57898 −0.448114
\(156\) 7.26462 0.581635
\(157\) 13.0301 1.03991 0.519956 0.854193i \(-0.325948\pi\)
0.519956 + 0.854193i \(0.325948\pi\)
\(158\) 10.7456 0.854874
\(159\) 25.3148 2.00760
\(160\) −1.00000 −0.0790569
\(161\) −6.62208 −0.521893
\(162\) 2.22595 0.174887
\(163\) 8.41437 0.659064 0.329532 0.944144i \(-0.393109\pi\)
0.329532 + 0.944144i \(0.393109\pi\)
\(164\) −9.08954 −0.709773
\(165\) 0 0
\(166\) −15.9031 −1.23432
\(167\) 11.2346 0.869363 0.434681 0.900584i \(-0.356861\pi\)
0.434681 + 0.900584i \(0.356861\pi\)
\(168\) 2.73934 0.211345
\(169\) −5.96713 −0.459010
\(170\) −6.67727 −0.512123
\(171\) −26.6700 −2.03951
\(172\) 5.51500 0.420515
\(173\) 0.0298132 0.00226666 0.00113333 0.999999i \(-0.499639\pi\)
0.00113333 + 0.999999i \(0.499639\pi\)
\(174\) −3.02556 −0.229367
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −4.73089 −0.355596
\(178\) −12.5040 −0.937218
\(179\) −23.6951 −1.77106 −0.885529 0.464584i \(-0.846204\pi\)
−0.885529 + 0.464584i \(0.846204\pi\)
\(180\) 4.50401 0.335709
\(181\) −5.38186 −0.400031 −0.200015 0.979793i \(-0.564099\pi\)
−0.200015 + 0.979793i \(0.564099\pi\)
\(182\) 2.65196 0.196576
\(183\) −24.4765 −1.80936
\(184\) 6.62208 0.488186
\(185\) 4.15191 0.305255
\(186\) −15.2827 −1.12058
\(187\) 0 0
\(188\) 0.907239 0.0661672
\(189\) −4.11999 −0.299685
\(190\) 5.92140 0.429584
\(191\) −12.3548 −0.893965 −0.446982 0.894543i \(-0.647501\pi\)
−0.446982 + 0.894543i \(0.647501\pi\)
\(192\) −2.73934 −0.197695
\(193\) −11.8011 −0.849464 −0.424732 0.905319i \(-0.639632\pi\)
−0.424732 + 0.905319i \(0.639632\pi\)
\(194\) −11.4362 −0.821069
\(195\) 7.26462 0.520230
\(196\) 1.00000 0.0714286
\(197\) −19.4626 −1.38665 −0.693325 0.720625i \(-0.743855\pi\)
−0.693325 + 0.720625i \(0.743855\pi\)
\(198\) 0 0
\(199\) 15.5751 1.10409 0.552043 0.833815i \(-0.313848\pi\)
0.552043 + 0.833815i \(0.313848\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 11.2504 0.793545
\(202\) 7.67893 0.540288
\(203\) −1.10448 −0.0775195
\(204\) −18.2913 −1.28065
\(205\) −9.08954 −0.634841
\(206\) 1.29467 0.0902036
\(207\) −29.8259 −2.07304
\(208\) −2.65196 −0.183880
\(209\) 0 0
\(210\) 2.73934 0.189033
\(211\) −13.8169 −0.951191 −0.475596 0.879664i \(-0.657767\pi\)
−0.475596 + 0.879664i \(0.657767\pi\)
\(212\) −9.24121 −0.634689
\(213\) 19.2347 1.31794
\(214\) −14.0391 −0.959692
\(215\) 5.51500 0.376120
\(216\) 4.11999 0.280330
\(217\) −5.57898 −0.378726
\(218\) −13.6269 −0.922929
\(219\) 2.65015 0.179080
\(220\) 0 0
\(221\) −17.7078 −1.19116
\(222\) 11.3735 0.763340
\(223\) −8.98060 −0.601385 −0.300693 0.953721i \(-0.597218\pi\)
−0.300693 + 0.953721i \(0.597218\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 4.50401 0.300267
\(226\) −12.3826 −0.823681
\(227\) 18.5359 1.23027 0.615136 0.788421i \(-0.289101\pi\)
0.615136 + 0.788421i \(0.289101\pi\)
\(228\) 16.2208 1.07425
\(229\) −26.3775 −1.74307 −0.871537 0.490329i \(-0.836876\pi\)
−0.871537 + 0.490329i \(0.836876\pi\)
\(230\) 6.62208 0.436647
\(231\) 0 0
\(232\) 1.10448 0.0725129
\(233\) −12.9030 −0.845300 −0.422650 0.906293i \(-0.638900\pi\)
−0.422650 + 0.906293i \(0.638900\pi\)
\(234\) 11.9444 0.780831
\(235\) 0.907239 0.0591817
\(236\) 1.72702 0.112419
\(237\) 29.4359 1.91207
\(238\) −6.67727 −0.432823
\(239\) −7.43785 −0.481114 −0.240557 0.970635i \(-0.577330\pi\)
−0.240557 + 0.970635i \(0.577330\pi\)
\(240\) −2.73934 −0.176824
\(241\) 29.9278 1.92782 0.963908 0.266234i \(-0.0857795\pi\)
0.963908 + 0.266234i \(0.0857795\pi\)
\(242\) 0 0
\(243\) 18.4576 1.18406
\(244\) 8.93518 0.572016
\(245\) 1.00000 0.0638877
\(246\) −24.8994 −1.58753
\(247\) 15.7033 0.999177
\(248\) 5.57898 0.354266
\(249\) −43.5640 −2.76075
\(250\) −1.00000 −0.0632456
\(251\) 2.56811 0.162098 0.0810488 0.996710i \(-0.474173\pi\)
0.0810488 + 0.996710i \(0.474173\pi\)
\(252\) 4.50401 0.283726
\(253\) 0 0
\(254\) 8.64037 0.542145
\(255\) −18.2913 −1.14545
\(256\) 1.00000 0.0625000
\(257\) −27.7714 −1.73233 −0.866166 0.499756i \(-0.833423\pi\)
−0.866166 + 0.499756i \(0.833423\pi\)
\(258\) 15.1075 0.940550
\(259\) 4.15191 0.257987
\(260\) −2.65196 −0.164467
\(261\) −4.97460 −0.307920
\(262\) 6.27193 0.387481
\(263\) 14.1621 0.873273 0.436637 0.899638i \(-0.356170\pi\)
0.436637 + 0.899638i \(0.356170\pi\)
\(264\) 0 0
\(265\) −9.24121 −0.567683
\(266\) 5.92140 0.363064
\(267\) −34.2529 −2.09624
\(268\) −4.10699 −0.250874
\(269\) 24.5685 1.49797 0.748983 0.662589i \(-0.230543\pi\)
0.748983 + 0.662589i \(0.230543\pi\)
\(270\) 4.11999 0.250734
\(271\) −10.4852 −0.636933 −0.318467 0.947934i \(-0.603168\pi\)
−0.318467 + 0.947934i \(0.603168\pi\)
\(272\) 6.67727 0.404869
\(273\) 7.26462 0.439675
\(274\) 10.9179 0.659577
\(275\) 0 0
\(276\) 18.1402 1.09191
\(277\) 30.8423 1.85313 0.926566 0.376133i \(-0.122746\pi\)
0.926566 + 0.376133i \(0.122746\pi\)
\(278\) 12.5679 0.753774
\(279\) −25.1277 −1.50436
\(280\) −1.00000 −0.0597614
\(281\) 4.44483 0.265156 0.132578 0.991173i \(-0.457674\pi\)
0.132578 + 0.991173i \(0.457674\pi\)
\(282\) 2.48524 0.147994
\(283\) 5.81252 0.345519 0.172759 0.984964i \(-0.444732\pi\)
0.172759 + 0.984964i \(0.444732\pi\)
\(284\) −7.02166 −0.416659
\(285\) 16.2208 0.960835
\(286\) 0 0
\(287\) −9.08954 −0.536538
\(288\) −4.50401 −0.265401
\(289\) 27.5860 1.62270
\(290\) 1.10448 0.0648575
\(291\) −31.3276 −1.83646
\(292\) −0.967438 −0.0566150
\(293\) −8.16127 −0.476787 −0.238393 0.971169i \(-0.576621\pi\)
−0.238393 + 0.971169i \(0.576621\pi\)
\(294\) 2.73934 0.159762
\(295\) 1.72702 0.100551
\(296\) −4.15191 −0.241325
\(297\) 0 0
\(298\) −18.9356 −1.09691
\(299\) 17.5615 1.01561
\(300\) −2.73934 −0.158156
\(301\) 5.51500 0.317879
\(302\) −9.07849 −0.522408
\(303\) 21.0352 1.20844
\(304\) −5.92140 −0.339616
\(305\) 8.93518 0.511627
\(306\) −30.0745 −1.71924
\(307\) 15.7830 0.900782 0.450391 0.892832i \(-0.351285\pi\)
0.450391 + 0.892832i \(0.351285\pi\)
\(308\) 0 0
\(309\) 3.54653 0.201755
\(310\) 5.57898 0.316865
\(311\) −6.51788 −0.369595 −0.184798 0.982777i \(-0.559163\pi\)
−0.184798 + 0.982777i \(0.559163\pi\)
\(312\) −7.26462 −0.411278
\(313\) −9.54938 −0.539763 −0.269881 0.962894i \(-0.586985\pi\)
−0.269881 + 0.962894i \(0.586985\pi\)
\(314\) −13.0301 −0.735329
\(315\) 4.50401 0.253772
\(316\) −10.7456 −0.604487
\(317\) −6.52267 −0.366350 −0.183175 0.983080i \(-0.558637\pi\)
−0.183175 + 0.983080i \(0.558637\pi\)
\(318\) −25.3148 −1.41959
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −38.4579 −2.14651
\(322\) 6.62208 0.369034
\(323\) −39.5388 −2.20000
\(324\) −2.22595 −0.123664
\(325\) −2.65196 −0.147104
\(326\) −8.41437 −0.466029
\(327\) −37.3287 −2.06428
\(328\) 9.08954 0.501886
\(329\) 0.907239 0.0500177
\(330\) 0 0
\(331\) −20.3519 −1.11864 −0.559319 0.828952i \(-0.688937\pi\)
−0.559319 + 0.828952i \(0.688937\pi\)
\(332\) 15.9031 0.872794
\(333\) 18.7002 1.02477
\(334\) −11.2346 −0.614732
\(335\) −4.10699 −0.224389
\(336\) −2.73934 −0.149443
\(337\) 20.3955 1.11101 0.555506 0.831513i \(-0.312525\pi\)
0.555506 + 0.831513i \(0.312525\pi\)
\(338\) 5.96713 0.324569
\(339\) −33.9203 −1.84230
\(340\) 6.67727 0.362126
\(341\) 0 0
\(342\) 26.6700 1.44215
\(343\) 1.00000 0.0539949
\(344\) −5.51500 −0.297349
\(345\) 18.1402 0.976634
\(346\) −0.0298132 −0.00160277
\(347\) 25.1418 1.34968 0.674841 0.737963i \(-0.264212\pi\)
0.674841 + 0.737963i \(0.264212\pi\)
\(348\) 3.02556 0.162187
\(349\) −6.48720 −0.347252 −0.173626 0.984812i \(-0.555548\pi\)
−0.173626 + 0.984812i \(0.555548\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 10.9260 0.583188
\(352\) 0 0
\(353\) −0.671105 −0.0357193 −0.0178597 0.999841i \(-0.505685\pi\)
−0.0178597 + 0.999841i \(0.505685\pi\)
\(354\) 4.73089 0.251444
\(355\) −7.02166 −0.372671
\(356\) 12.5040 0.662713
\(357\) −18.2913 −0.968081
\(358\) 23.6951 1.25233
\(359\) 33.6491 1.77593 0.887965 0.459912i \(-0.152119\pi\)
0.887965 + 0.459912i \(0.152119\pi\)
\(360\) −4.50401 −0.237382
\(361\) 16.0630 0.845422
\(362\) 5.38186 0.282864
\(363\) 0 0
\(364\) −2.65196 −0.139000
\(365\) −0.967438 −0.0506380
\(366\) 24.4765 1.27941
\(367\) −5.36793 −0.280204 −0.140102 0.990137i \(-0.544743\pi\)
−0.140102 + 0.990137i \(0.544743\pi\)
\(368\) −6.62208 −0.345200
\(369\) −40.9393 −2.13122
\(370\) −4.15191 −0.215848
\(371\) −9.24121 −0.479780
\(372\) 15.2827 0.792373
\(373\) −16.1242 −0.834878 −0.417439 0.908705i \(-0.637072\pi\)
−0.417439 + 0.908705i \(0.637072\pi\)
\(374\) 0 0
\(375\) −2.73934 −0.141459
\(376\) −0.907239 −0.0467873
\(377\) 2.92904 0.150853
\(378\) 4.11999 0.211909
\(379\) −27.3566 −1.40521 −0.702607 0.711578i \(-0.747981\pi\)
−0.702607 + 0.711578i \(0.747981\pi\)
\(380\) −5.92140 −0.303762
\(381\) 23.6689 1.21260
\(382\) 12.3548 0.632129
\(383\) 0.132352 0.00676285 0.00338142 0.999994i \(-0.498924\pi\)
0.00338142 + 0.999994i \(0.498924\pi\)
\(384\) 2.73934 0.139792
\(385\) 0 0
\(386\) 11.8011 0.600662
\(387\) 24.8396 1.26267
\(388\) 11.4362 0.580584
\(389\) 16.3662 0.829798 0.414899 0.909867i \(-0.363817\pi\)
0.414899 + 0.909867i \(0.363817\pi\)
\(390\) −7.26462 −0.367858
\(391\) −44.2175 −2.23617
\(392\) −1.00000 −0.0505076
\(393\) 17.1810 0.866666
\(394\) 19.4626 0.980509
\(395\) −10.7456 −0.540670
\(396\) 0 0
\(397\) 34.8294 1.74804 0.874018 0.485893i \(-0.161505\pi\)
0.874018 + 0.485893i \(0.161505\pi\)
\(398\) −15.5751 −0.780707
\(399\) 16.2208 0.812054
\(400\) 1.00000 0.0500000
\(401\) −11.8605 −0.592284 −0.296142 0.955144i \(-0.595700\pi\)
−0.296142 + 0.955144i \(0.595700\pi\)
\(402\) −11.2504 −0.561121
\(403\) 14.7952 0.737001
\(404\) −7.67893 −0.382041
\(405\) −2.22595 −0.110609
\(406\) 1.10448 0.0548146
\(407\) 0 0
\(408\) 18.2913 0.905557
\(409\) 7.78726 0.385055 0.192528 0.981292i \(-0.438331\pi\)
0.192528 + 0.981292i \(0.438331\pi\)
\(410\) 9.08954 0.448900
\(411\) 29.9080 1.47525
\(412\) −1.29467 −0.0637836
\(413\) 1.72702 0.0849810
\(414\) 29.8259 1.46586
\(415\) 15.9031 0.780651
\(416\) 2.65196 0.130023
\(417\) 34.4279 1.68594
\(418\) 0 0
\(419\) −9.59415 −0.468705 −0.234352 0.972152i \(-0.575297\pi\)
−0.234352 + 0.972152i \(0.575297\pi\)
\(420\) −2.73934 −0.133666
\(421\) 19.7210 0.961144 0.480572 0.876955i \(-0.340429\pi\)
0.480572 + 0.876955i \(0.340429\pi\)
\(422\) 13.8169 0.672594
\(423\) 4.08621 0.198678
\(424\) 9.24121 0.448793
\(425\) 6.67727 0.323895
\(426\) −19.2347 −0.931926
\(427\) 8.93518 0.432404
\(428\) 14.0391 0.678605
\(429\) 0 0
\(430\) −5.51500 −0.265957
\(431\) 18.5608 0.894041 0.447021 0.894524i \(-0.352485\pi\)
0.447021 + 0.894524i \(0.352485\pi\)
\(432\) −4.11999 −0.198223
\(433\) −15.6876 −0.753900 −0.376950 0.926234i \(-0.623027\pi\)
−0.376950 + 0.926234i \(0.623027\pi\)
\(434\) 5.57898 0.267800
\(435\) 3.02556 0.145064
\(436\) 13.6269 0.652609
\(437\) 39.2120 1.87577
\(438\) −2.65015 −0.126629
\(439\) 6.85571 0.327205 0.163602 0.986526i \(-0.447689\pi\)
0.163602 + 0.986526i \(0.447689\pi\)
\(440\) 0 0
\(441\) 4.50401 0.214476
\(442\) 17.7078 0.842275
\(443\) 13.6685 0.649411 0.324706 0.945815i \(-0.394735\pi\)
0.324706 + 0.945815i \(0.394735\pi\)
\(444\) −11.3735 −0.539763
\(445\) 12.5040 0.592748
\(446\) 8.98060 0.425244
\(447\) −51.8710 −2.45342
\(448\) 1.00000 0.0472456
\(449\) −36.9601 −1.74425 −0.872127 0.489280i \(-0.837259\pi\)
−0.872127 + 0.489280i \(0.837259\pi\)
\(450\) −4.50401 −0.212321
\(451\) 0 0
\(452\) 12.3826 0.582430
\(453\) −24.8691 −1.16845
\(454\) −18.5359 −0.869934
\(455\) −2.65196 −0.124326
\(456\) −16.2208 −0.759607
\(457\) −7.21944 −0.337711 −0.168856 0.985641i \(-0.554007\pi\)
−0.168856 + 0.985641i \(0.554007\pi\)
\(458\) 26.3775 1.23254
\(459\) −27.5103 −1.28407
\(460\) −6.62208 −0.308756
\(461\) 35.7004 1.66273 0.831367 0.555724i \(-0.187559\pi\)
0.831367 + 0.555724i \(0.187559\pi\)
\(462\) 0 0
\(463\) −19.9145 −0.925504 −0.462752 0.886488i \(-0.653138\pi\)
−0.462752 + 0.886488i \(0.653138\pi\)
\(464\) −1.10448 −0.0512743
\(465\) 15.2827 0.708720
\(466\) 12.9030 0.597718
\(467\) 12.8231 0.593383 0.296691 0.954973i \(-0.404117\pi\)
0.296691 + 0.954973i \(0.404117\pi\)
\(468\) −11.9444 −0.552131
\(469\) −4.10699 −0.189643
\(470\) −0.907239 −0.0418478
\(471\) −35.6938 −1.64468
\(472\) −1.72702 −0.0794924
\(473\) 0 0
\(474\) −29.4359 −1.35204
\(475\) −5.92140 −0.271693
\(476\) 6.67727 0.306052
\(477\) −41.6224 −1.90576
\(478\) 7.43785 0.340199
\(479\) 3.92060 0.179137 0.0895683 0.995981i \(-0.471451\pi\)
0.0895683 + 0.995981i \(0.471451\pi\)
\(480\) 2.73934 0.125033
\(481\) −11.0107 −0.502044
\(482\) −29.9278 −1.36317
\(483\) 18.1402 0.825406
\(484\) 0 0
\(485\) 11.4362 0.519290
\(486\) −18.4576 −0.837254
\(487\) −41.0411 −1.85975 −0.929874 0.367878i \(-0.880084\pi\)
−0.929874 + 0.367878i \(0.880084\pi\)
\(488\) −8.93518 −0.404477
\(489\) −23.0499 −1.04235
\(490\) −1.00000 −0.0451754
\(491\) −10.2674 −0.463363 −0.231681 0.972792i \(-0.574423\pi\)
−0.231681 + 0.972792i \(0.574423\pi\)
\(492\) 24.8994 1.12255
\(493\) −7.37494 −0.332150
\(494\) −15.7033 −0.706525
\(495\) 0 0
\(496\) −5.57898 −0.250504
\(497\) −7.02166 −0.314964
\(498\) 43.5640 1.95215
\(499\) 33.4423 1.49708 0.748542 0.663087i \(-0.230754\pi\)
0.748542 + 0.663087i \(0.230754\pi\)
\(500\) 1.00000 0.0447214
\(501\) −30.7756 −1.37495
\(502\) −2.56811 −0.114620
\(503\) 20.8969 0.931745 0.465872 0.884852i \(-0.345741\pi\)
0.465872 + 0.884852i \(0.345741\pi\)
\(504\) −4.50401 −0.200624
\(505\) −7.67893 −0.341708
\(506\) 0 0
\(507\) 16.3460 0.725953
\(508\) −8.64037 −0.383354
\(509\) −44.2364 −1.96074 −0.980372 0.197159i \(-0.936828\pi\)
−0.980372 + 0.197159i \(0.936828\pi\)
\(510\) 18.2913 0.809954
\(511\) −0.967438 −0.0427969
\(512\) −1.00000 −0.0441942
\(513\) 24.3961 1.07711
\(514\) 27.7714 1.22494
\(515\) −1.29467 −0.0570498
\(516\) −15.1075 −0.665070
\(517\) 0 0
\(518\) −4.15191 −0.182424
\(519\) −0.0816686 −0.00358485
\(520\) 2.65196 0.116296
\(521\) 29.2273 1.28047 0.640235 0.768179i \(-0.278837\pi\)
0.640235 + 0.768179i \(0.278837\pi\)
\(522\) 4.97460 0.217732
\(523\) 23.6402 1.03371 0.516857 0.856072i \(-0.327102\pi\)
0.516857 + 0.856072i \(0.327102\pi\)
\(524\) −6.27193 −0.273991
\(525\) −2.73934 −0.119555
\(526\) −14.1621 −0.617497
\(527\) −37.2524 −1.62274
\(528\) 0 0
\(529\) 20.8520 0.906608
\(530\) 9.24121 0.401412
\(531\) 7.77849 0.337558
\(532\) −5.92140 −0.256725
\(533\) 24.1050 1.04411
\(534\) 34.2529 1.48227
\(535\) 14.0391 0.606963
\(536\) 4.10699 0.177395
\(537\) 64.9092 2.80104
\(538\) −24.5685 −1.05922
\(539\) 0 0
\(540\) −4.11999 −0.177296
\(541\) 0.0638597 0.00274554 0.00137277 0.999999i \(-0.499563\pi\)
0.00137277 + 0.999999i \(0.499563\pi\)
\(542\) 10.4852 0.450380
\(543\) 14.7428 0.632673
\(544\) −6.67727 −0.286286
\(545\) 13.6269 0.583711
\(546\) −7.26462 −0.310897
\(547\) −17.7223 −0.757751 −0.378876 0.925448i \(-0.623689\pi\)
−0.378876 + 0.925448i \(0.623689\pi\)
\(548\) −10.9179 −0.466391
\(549\) 40.2441 1.71758
\(550\) 0 0
\(551\) 6.54009 0.278617
\(552\) −18.1402 −0.772097
\(553\) −10.7456 −0.456950
\(554\) −30.8423 −1.31036
\(555\) −11.3735 −0.482779
\(556\) −12.5679 −0.532999
\(557\) 37.7636 1.60009 0.800047 0.599937i \(-0.204808\pi\)
0.800047 + 0.599937i \(0.204808\pi\)
\(558\) 25.1277 1.06374
\(559\) −14.6255 −0.618594
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −4.44483 −0.187494
\(563\) 40.4219 1.70358 0.851791 0.523882i \(-0.175517\pi\)
0.851791 + 0.523882i \(0.175517\pi\)
\(564\) −2.48524 −0.104647
\(565\) 12.3826 0.520941
\(566\) −5.81252 −0.244319
\(567\) −2.22595 −0.0934813
\(568\) 7.02166 0.294622
\(569\) 33.7555 1.41511 0.707553 0.706660i \(-0.249799\pi\)
0.707553 + 0.706660i \(0.249799\pi\)
\(570\) −16.2208 −0.679413
\(571\) 9.90555 0.414534 0.207267 0.978284i \(-0.433543\pi\)
0.207267 + 0.978284i \(0.433543\pi\)
\(572\) 0 0
\(573\) 33.8442 1.41386
\(574\) 9.08954 0.379390
\(575\) −6.62208 −0.276160
\(576\) 4.50401 0.187667
\(577\) 25.7980 1.07398 0.536992 0.843587i \(-0.319560\pi\)
0.536992 + 0.843587i \(0.319560\pi\)
\(578\) −27.5860 −1.14742
\(579\) 32.3274 1.34348
\(580\) −1.10448 −0.0458612
\(581\) 15.9031 0.659770
\(582\) 31.3276 1.29857
\(583\) 0 0
\(584\) 0.967438 0.0400329
\(585\) −11.9444 −0.493841
\(586\) 8.16127 0.337139
\(587\) 0.761169 0.0314168 0.0157084 0.999877i \(-0.495000\pi\)
0.0157084 + 0.999877i \(0.495000\pi\)
\(588\) −2.73934 −0.112969
\(589\) 33.0354 1.36120
\(590\) −1.72702 −0.0711002
\(591\) 53.3146 2.19307
\(592\) 4.15191 0.170642
\(593\) 23.5344 0.966442 0.483221 0.875498i \(-0.339467\pi\)
0.483221 + 0.875498i \(0.339467\pi\)
\(594\) 0 0
\(595\) 6.67727 0.273741
\(596\) 18.9356 0.775631
\(597\) −42.6654 −1.74618
\(598\) −17.5615 −0.718142
\(599\) 12.2528 0.500637 0.250318 0.968164i \(-0.419465\pi\)
0.250318 + 0.968164i \(0.419465\pi\)
\(600\) 2.73934 0.111833
\(601\) −29.7034 −1.21163 −0.605813 0.795607i \(-0.707152\pi\)
−0.605813 + 0.795607i \(0.707152\pi\)
\(602\) −5.51500 −0.224775
\(603\) −18.4979 −0.753292
\(604\) 9.07849 0.369398
\(605\) 0 0
\(606\) −21.0352 −0.854498
\(607\) −38.5009 −1.56270 −0.781351 0.624092i \(-0.785469\pi\)
−0.781351 + 0.624092i \(0.785469\pi\)
\(608\) 5.92140 0.240145
\(609\) 3.02556 0.122602
\(610\) −8.93518 −0.361775
\(611\) −2.40596 −0.0973346
\(612\) 30.0745 1.21569
\(613\) −11.0299 −0.445493 −0.222746 0.974876i \(-0.571502\pi\)
−0.222746 + 0.974876i \(0.571502\pi\)
\(614\) −15.7830 −0.636949
\(615\) 24.8994 1.00404
\(616\) 0 0
\(617\) −21.3964 −0.861388 −0.430694 0.902498i \(-0.641731\pi\)
−0.430694 + 0.902498i \(0.641731\pi\)
\(618\) −3.54653 −0.142663
\(619\) −49.0198 −1.97027 −0.985136 0.171776i \(-0.945049\pi\)
−0.985136 + 0.171776i \(0.945049\pi\)
\(620\) −5.57898 −0.224057
\(621\) 27.2829 1.09482
\(622\) 6.51788 0.261343
\(623\) 12.5040 0.500964
\(624\) 7.26462 0.290817
\(625\) 1.00000 0.0400000
\(626\) 9.54938 0.381670
\(627\) 0 0
\(628\) 13.0301 0.519956
\(629\) 27.7234 1.10541
\(630\) −4.50401 −0.179444
\(631\) 16.5923 0.660527 0.330264 0.943889i \(-0.392862\pi\)
0.330264 + 0.943889i \(0.392862\pi\)
\(632\) 10.7456 0.427437
\(633\) 37.8491 1.50437
\(634\) 6.52267 0.259048
\(635\) −8.64037 −0.342883
\(636\) 25.3148 1.00380
\(637\) −2.65196 −0.105074
\(638\) 0 0
\(639\) −31.6256 −1.25109
\(640\) −1.00000 −0.0395285
\(641\) 10.8261 0.427606 0.213803 0.976877i \(-0.431415\pi\)
0.213803 + 0.976877i \(0.431415\pi\)
\(642\) 38.4579 1.51781
\(643\) −0.443804 −0.0175019 −0.00875095 0.999962i \(-0.502786\pi\)
−0.00875095 + 0.999962i \(0.502786\pi\)
\(644\) −6.62208 −0.260947
\(645\) −15.1075 −0.594856
\(646\) 39.5388 1.55563
\(647\) 31.8694 1.25292 0.626458 0.779455i \(-0.284504\pi\)
0.626458 + 0.779455i \(0.284504\pi\)
\(648\) 2.22595 0.0874437
\(649\) 0 0
\(650\) 2.65196 0.104018
\(651\) 15.2827 0.598978
\(652\) 8.41437 0.329532
\(653\) 23.5789 0.922715 0.461358 0.887214i \(-0.347363\pi\)
0.461358 + 0.887214i \(0.347363\pi\)
\(654\) 37.3287 1.45967
\(655\) −6.27193 −0.245065
\(656\) −9.08954 −0.354887
\(657\) −4.35735 −0.169996
\(658\) −0.907239 −0.0353679
\(659\) 18.1283 0.706178 0.353089 0.935590i \(-0.385131\pi\)
0.353089 + 0.935590i \(0.385131\pi\)
\(660\) 0 0
\(661\) 35.1392 1.36676 0.683378 0.730065i \(-0.260510\pi\)
0.683378 + 0.730065i \(0.260510\pi\)
\(662\) 20.3519 0.790997
\(663\) 48.5078 1.88389
\(664\) −15.9031 −0.617159
\(665\) −5.92140 −0.229622
\(666\) −18.7002 −0.724619
\(667\) 7.31398 0.283198
\(668\) 11.2346 0.434681
\(669\) 24.6009 0.951127
\(670\) 4.10699 0.158667
\(671\) 0 0
\(672\) 2.73934 0.105672
\(673\) 11.5684 0.445929 0.222965 0.974827i \(-0.428427\pi\)
0.222965 + 0.974827i \(0.428427\pi\)
\(674\) −20.3955 −0.785603
\(675\) −4.11999 −0.158578
\(676\) −5.96713 −0.229505
\(677\) 4.02406 0.154657 0.0773286 0.997006i \(-0.475361\pi\)
0.0773286 + 0.997006i \(0.475361\pi\)
\(678\) 33.9203 1.30270
\(679\) 11.4362 0.438880
\(680\) −6.67727 −0.256062
\(681\) −50.7763 −1.94575
\(682\) 0 0
\(683\) −2.85171 −0.109118 −0.0545588 0.998511i \(-0.517375\pi\)
−0.0545588 + 0.998511i \(0.517375\pi\)
\(684\) −26.6700 −1.01975
\(685\) −10.9179 −0.417153
\(686\) −1.00000 −0.0381802
\(687\) 72.2570 2.75678
\(688\) 5.51500 0.210257
\(689\) 24.5073 0.933653
\(690\) −18.1402 −0.690584
\(691\) 9.12185 0.347011 0.173506 0.984833i \(-0.444490\pi\)
0.173506 + 0.984833i \(0.444490\pi\)
\(692\) 0.0298132 0.00113333
\(693\) 0 0
\(694\) −25.1418 −0.954370
\(695\) −12.5679 −0.476729
\(696\) −3.02556 −0.114684
\(697\) −60.6933 −2.29892
\(698\) 6.48720 0.245544
\(699\) 35.3456 1.33689
\(700\) 1.00000 0.0377964
\(701\) 51.5517 1.94708 0.973540 0.228516i \(-0.0733874\pi\)
0.973540 + 0.228516i \(0.0733874\pi\)
\(702\) −10.9260 −0.412376
\(703\) −24.5851 −0.927246
\(704\) 0 0
\(705\) −2.48524 −0.0935995
\(706\) 0.671105 0.0252574
\(707\) −7.67893 −0.288796
\(708\) −4.73089 −0.177798
\(709\) 30.4453 1.14340 0.571699 0.820463i \(-0.306284\pi\)
0.571699 + 0.820463i \(0.306284\pi\)
\(710\) 7.02166 0.263518
\(711\) −48.3982 −1.81508
\(712\) −12.5040 −0.468609
\(713\) 36.9445 1.38358
\(714\) 18.2913 0.684536
\(715\) 0 0
\(716\) −23.6951 −0.885529
\(717\) 20.3748 0.760912
\(718\) −33.6491 −1.25577
\(719\) 41.3242 1.54113 0.770565 0.637361i \(-0.219974\pi\)
0.770565 + 0.637361i \(0.219974\pi\)
\(720\) 4.50401 0.167854
\(721\) −1.29467 −0.0482159
\(722\) −16.0630 −0.597803
\(723\) −81.9824 −3.04896
\(724\) −5.38186 −0.200015
\(725\) −1.10448 −0.0410195
\(726\) 0 0
\(727\) −2.01006 −0.0745492 −0.0372746 0.999305i \(-0.511868\pi\)
−0.0372746 + 0.999305i \(0.511868\pi\)
\(728\) 2.65196 0.0982880
\(729\) −43.8839 −1.62533
\(730\) 0.967438 0.0358065
\(731\) 36.8251 1.36203
\(732\) −24.4765 −0.904679
\(733\) 23.9790 0.885686 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(734\) 5.36793 0.198134
\(735\) −2.73934 −0.101042
\(736\) 6.62208 0.244093
\(737\) 0 0
\(738\) 40.9393 1.50700
\(739\) 12.9004 0.474551 0.237275 0.971442i \(-0.423746\pi\)
0.237275 + 0.971442i \(0.423746\pi\)
\(740\) 4.15191 0.152627
\(741\) −43.0167 −1.58026
\(742\) 9.24121 0.339255
\(743\) 12.6488 0.464039 0.232019 0.972711i \(-0.425467\pi\)
0.232019 + 0.972711i \(0.425467\pi\)
\(744\) −15.2827 −0.560292
\(745\) 18.9356 0.693745
\(746\) 16.1242 0.590348
\(747\) 71.6275 2.62071
\(748\) 0 0
\(749\) 14.0391 0.512977
\(750\) 2.73934 0.100027
\(751\) −7.57446 −0.276396 −0.138198 0.990405i \(-0.544131\pi\)
−0.138198 + 0.990405i \(0.544131\pi\)
\(752\) 0.907239 0.0330836
\(753\) −7.03494 −0.256367
\(754\) −2.92904 −0.106669
\(755\) 9.07849 0.330400
\(756\) −4.11999 −0.149842
\(757\) 25.1439 0.913869 0.456934 0.889500i \(-0.348947\pi\)
0.456934 + 0.889500i \(0.348947\pi\)
\(758\) 27.3566 0.993636
\(759\) 0 0
\(760\) 5.92140 0.214792
\(761\) −33.9674 −1.23132 −0.615658 0.788013i \(-0.711110\pi\)
−0.615658 + 0.788013i \(0.711110\pi\)
\(762\) −23.6689 −0.857435
\(763\) 13.6269 0.493326
\(764\) −12.3548 −0.446982
\(765\) 30.0745 1.08734
\(766\) −0.132352 −0.00478205
\(767\) −4.57997 −0.165373
\(768\) −2.73934 −0.0988476
\(769\) −22.3766 −0.806919 −0.403460 0.914997i \(-0.632192\pi\)
−0.403460 + 0.914997i \(0.632192\pi\)
\(770\) 0 0
\(771\) 76.0754 2.73979
\(772\) −11.8011 −0.424732
\(773\) 50.5235 1.81721 0.908603 0.417662i \(-0.137150\pi\)
0.908603 + 0.417662i \(0.137150\pi\)
\(774\) −24.8396 −0.892840
\(775\) −5.57898 −0.200403
\(776\) −11.4362 −0.410535
\(777\) −11.3735 −0.408022
\(778\) −16.3662 −0.586756
\(779\) 53.8228 1.92840
\(780\) 7.26462 0.260115
\(781\) 0 0
\(782\) 44.2175 1.58121
\(783\) 4.55046 0.162620
\(784\) 1.00000 0.0357143
\(785\) 13.0301 0.465063
\(786\) −17.1810 −0.612825
\(787\) 25.9303 0.924315 0.462157 0.886798i \(-0.347076\pi\)
0.462157 + 0.886798i \(0.347076\pi\)
\(788\) −19.4626 −0.693325
\(789\) −38.7949 −1.38113
\(790\) 10.7456 0.382311
\(791\) 12.3826 0.440276
\(792\) 0 0
\(793\) −23.6957 −0.841459
\(794\) −34.8294 −1.23605
\(795\) 25.3148 0.897825
\(796\) 15.5751 0.552043
\(797\) 11.5642 0.409624 0.204812 0.978801i \(-0.434342\pi\)
0.204812 + 0.978801i \(0.434342\pi\)
\(798\) −16.2208 −0.574209
\(799\) 6.05788 0.214312
\(800\) −1.00000 −0.0353553
\(801\) 56.3183 1.98991
\(802\) 11.8605 0.418808
\(803\) 0 0
\(804\) 11.2504 0.396773
\(805\) −6.62208 −0.233398
\(806\) −14.7952 −0.521139
\(807\) −67.3015 −2.36912
\(808\) 7.67893 0.270144
\(809\) −6.46814 −0.227408 −0.113704 0.993515i \(-0.536272\pi\)
−0.113704 + 0.993515i \(0.536272\pi\)
\(810\) 2.22595 0.0782120
\(811\) 35.3469 1.24120 0.620598 0.784129i \(-0.286890\pi\)
0.620598 + 0.784129i \(0.286890\pi\)
\(812\) −1.10448 −0.0387598
\(813\) 28.7227 1.00735
\(814\) 0 0
\(815\) 8.41437 0.294743
\(816\) −18.2913 −0.640325
\(817\) −32.6565 −1.14251
\(818\) −7.78726 −0.272275
\(819\) −11.9444 −0.417372
\(820\) −9.08954 −0.317420
\(821\) −30.6970 −1.07133 −0.535666 0.844430i \(-0.679939\pi\)
−0.535666 + 0.844430i \(0.679939\pi\)
\(822\) −29.9080 −1.04316
\(823\) 19.6396 0.684594 0.342297 0.939592i \(-0.388795\pi\)
0.342297 + 0.939592i \(0.388795\pi\)
\(824\) 1.29467 0.0451018
\(825\) 0 0
\(826\) −1.72702 −0.0600906
\(827\) 28.3877 0.987138 0.493569 0.869707i \(-0.335692\pi\)
0.493569 + 0.869707i \(0.335692\pi\)
\(828\) −29.8259 −1.03652
\(829\) 32.4774 1.12799 0.563994 0.825779i \(-0.309264\pi\)
0.563994 + 0.825779i \(0.309264\pi\)
\(830\) −15.9031 −0.552003
\(831\) −84.4876 −2.93084
\(832\) −2.65196 −0.0919400
\(833\) 6.67727 0.231354
\(834\) −34.4279 −1.19214
\(835\) 11.2346 0.388791
\(836\) 0 0
\(837\) 22.9853 0.794489
\(838\) 9.59415 0.331424
\(839\) 26.4066 0.911659 0.455829 0.890067i \(-0.349343\pi\)
0.455829 + 0.890067i \(0.349343\pi\)
\(840\) 2.73934 0.0945163
\(841\) −27.7801 −0.957935
\(842\) −19.7210 −0.679631
\(843\) −12.1759 −0.419361
\(844\) −13.8169 −0.475596
\(845\) −5.96713 −0.205276
\(846\) −4.08621 −0.140487
\(847\) 0 0
\(848\) −9.24121 −0.317344
\(849\) −15.9225 −0.546459
\(850\) −6.67727 −0.229029
\(851\) −27.4943 −0.942492
\(852\) 19.2347 0.658971
\(853\) −28.0478 −0.960338 −0.480169 0.877176i \(-0.659425\pi\)
−0.480169 + 0.877176i \(0.659425\pi\)
\(854\) −8.93518 −0.305756
\(855\) −26.6700 −0.912096
\(856\) −14.0391 −0.479846
\(857\) −47.9742 −1.63877 −0.819383 0.573246i \(-0.805684\pi\)
−0.819383 + 0.573246i \(0.805684\pi\)
\(858\) 0 0
\(859\) −28.1815 −0.961541 −0.480770 0.876847i \(-0.659643\pi\)
−0.480770 + 0.876847i \(0.659643\pi\)
\(860\) 5.51500 0.188060
\(861\) 24.8994 0.848568
\(862\) −18.5608 −0.632183
\(863\) 7.05526 0.240164 0.120082 0.992764i \(-0.461684\pi\)
0.120082 + 0.992764i \(0.461684\pi\)
\(864\) 4.11999 0.140165
\(865\) 0.0298132 0.00101368
\(866\) 15.6876 0.533088
\(867\) −75.5674 −2.56641
\(868\) −5.57898 −0.189363
\(869\) 0 0
\(870\) −3.02556 −0.102576
\(871\) 10.8915 0.369046
\(872\) −13.6269 −0.461464
\(873\) 51.5086 1.74330
\(874\) −39.2120 −1.32637
\(875\) 1.00000 0.0338062
\(876\) 2.65015 0.0895401
\(877\) 44.0677 1.48806 0.744030 0.668146i \(-0.232912\pi\)
0.744030 + 0.668146i \(0.232912\pi\)
\(878\) −6.85571 −0.231369
\(879\) 22.3565 0.754067
\(880\) 0 0
\(881\) 2.79232 0.0940757 0.0470379 0.998893i \(-0.485022\pi\)
0.0470379 + 0.998893i \(0.485022\pi\)
\(882\) −4.50401 −0.151658
\(883\) −14.4756 −0.487144 −0.243572 0.969883i \(-0.578319\pi\)
−0.243572 + 0.969883i \(0.578319\pi\)
\(884\) −17.7078 −0.595579
\(885\) −4.73089 −0.159027
\(886\) −13.6685 −0.459203
\(887\) 27.8169 0.933999 0.466999 0.884258i \(-0.345335\pi\)
0.466999 + 0.884258i \(0.345335\pi\)
\(888\) 11.3735 0.381670
\(889\) −8.64037 −0.289789
\(890\) −12.5040 −0.419136
\(891\) 0 0
\(892\) −8.98060 −0.300693
\(893\) −5.37213 −0.179771
\(894\) 51.8710 1.73483
\(895\) −23.6951 −0.792041
\(896\) −1.00000 −0.0334077
\(897\) −48.1069 −1.60624
\(898\) 36.9601 1.23337
\(899\) 6.16189 0.205510
\(900\) 4.50401 0.150134
\(901\) −61.7061 −2.05573
\(902\) 0 0
\(903\) −15.1075 −0.502745
\(904\) −12.3826 −0.411840
\(905\) −5.38186 −0.178899
\(906\) 24.8691 0.826221
\(907\) −29.5574 −0.981436 −0.490718 0.871319i \(-0.663265\pi\)
−0.490718 + 0.871319i \(0.663265\pi\)
\(908\) 18.5359 0.615136
\(909\) −34.5859 −1.14714
\(910\) 2.65196 0.0879115
\(911\) 48.6900 1.61317 0.806585 0.591118i \(-0.201313\pi\)
0.806585 + 0.591118i \(0.201313\pi\)
\(912\) 16.2208 0.537123
\(913\) 0 0
\(914\) 7.21944 0.238798
\(915\) −24.4765 −0.809169
\(916\) −26.3775 −0.871537
\(917\) −6.27193 −0.207117
\(918\) 27.5103 0.907974
\(919\) 41.7897 1.37852 0.689258 0.724516i \(-0.257937\pi\)
0.689258 + 0.724516i \(0.257937\pi\)
\(920\) 6.62208 0.218324
\(921\) −43.2350 −1.42464
\(922\) −35.7004 −1.17573
\(923\) 18.6211 0.612922
\(924\) 0 0
\(925\) 4.15191 0.136514
\(926\) 19.9145 0.654430
\(927\) −5.83118 −0.191521
\(928\) 1.10448 0.0362564
\(929\) 10.2472 0.336200 0.168100 0.985770i \(-0.446237\pi\)
0.168100 + 0.985770i \(0.446237\pi\)
\(930\) −15.2827 −0.501141
\(931\) −5.92140 −0.194066
\(932\) −12.9030 −0.422650
\(933\) 17.8547 0.584537
\(934\) −12.8231 −0.419585
\(935\) 0 0
\(936\) 11.9444 0.390416
\(937\) 33.3518 1.08956 0.544779 0.838580i \(-0.316614\pi\)
0.544779 + 0.838580i \(0.316614\pi\)
\(938\) 4.10699 0.134098
\(939\) 26.1590 0.853668
\(940\) 0.907239 0.0295909
\(941\) 4.77317 0.155601 0.0778005 0.996969i \(-0.475210\pi\)
0.0778005 + 0.996969i \(0.475210\pi\)
\(942\) 35.6938 1.16297
\(943\) 60.1917 1.96011
\(944\) 1.72702 0.0562096
\(945\) −4.11999 −0.134023
\(946\) 0 0
\(947\) 9.18216 0.298380 0.149190 0.988809i \(-0.452333\pi\)
0.149190 + 0.988809i \(0.452333\pi\)
\(948\) 29.4359 0.956034
\(949\) 2.56560 0.0832830
\(950\) 5.92140 0.192116
\(951\) 17.8678 0.579404
\(952\) −6.67727 −0.216412
\(953\) −49.9893 −1.61931 −0.809656 0.586905i \(-0.800346\pi\)
−0.809656 + 0.586905i \(0.800346\pi\)
\(954\) 41.6224 1.34758
\(955\) −12.3548 −0.399793
\(956\) −7.43785 −0.240557
\(957\) 0 0
\(958\) −3.92060 −0.126669
\(959\) −10.9179 −0.352559
\(960\) −2.73934 −0.0884119
\(961\) 0.125003 0.00403237
\(962\) 11.0107 0.354999
\(963\) 63.2321 2.03763
\(964\) 29.9278 0.963908
\(965\) −11.8011 −0.379892
\(966\) −18.1402 −0.583650
\(967\) −4.16857 −0.134052 −0.0670261 0.997751i \(-0.521351\pi\)
−0.0670261 + 0.997751i \(0.521351\pi\)
\(968\) 0 0
\(969\) 108.310 3.47943
\(970\) −11.4362 −0.367193
\(971\) 43.5285 1.39689 0.698447 0.715662i \(-0.253875\pi\)
0.698447 + 0.715662i \(0.253875\pi\)
\(972\) 18.4576 0.592028
\(973\) −12.5679 −0.402909
\(974\) 41.0411 1.31504
\(975\) 7.26462 0.232654
\(976\) 8.93518 0.286008
\(977\) −24.6315 −0.788033 −0.394017 0.919103i \(-0.628915\pi\)
−0.394017 + 0.919103i \(0.628915\pi\)
\(978\) 23.0499 0.737053
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 61.3755 1.95957
\(982\) 10.2674 0.327647
\(983\) 37.8291 1.20656 0.603280 0.797530i \(-0.293860\pi\)
0.603280 + 0.797530i \(0.293860\pi\)
\(984\) −24.8994 −0.793763
\(985\) −19.4626 −0.620129
\(986\) 7.37494 0.234866
\(987\) −2.48524 −0.0791061
\(988\) 15.7033 0.499588
\(989\) −36.5208 −1.16129
\(990\) 0 0
\(991\) −26.2409 −0.833570 −0.416785 0.909005i \(-0.636843\pi\)
−0.416785 + 0.909005i \(0.636843\pi\)
\(992\) 5.57898 0.177133
\(993\) 55.7507 1.76920
\(994\) 7.02166 0.222714
\(995\) 15.5751 0.493763
\(996\) −43.5640 −1.38038
\(997\) 6.27699 0.198794 0.0993971 0.995048i \(-0.468309\pi\)
0.0993971 + 0.995048i \(0.468309\pi\)
\(998\) −33.4423 −1.05860
\(999\) −17.1058 −0.541204
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 8470.2.a.cv.1.1 6
11.2 odd 10 770.2.n.g.631.1 yes 12
11.6 odd 10 770.2.n.g.421.1 12
11.10 odd 2 8470.2.a.db.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.g.421.1 12 11.6 odd 10
770.2.n.g.631.1 yes 12 11.2 odd 10
8470.2.a.cv.1.1 6 1.1 even 1 trivial
8470.2.a.db.1.1 6 11.10 odd 2