Properties

Label 3936.2.a.m.1.3
Level $3936$
Weight $2$
Character 3936.1
Self dual yes
Analytic conductor $31.429$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3936,2,Mod(1,3936)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3936, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3936.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3936 = 2^{5} \cdot 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3936.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: \(31.4291182356\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.15188.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 7x^{2} + x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.09178\) of defining polynomial
Character \(\chi\) \(=\) 3936.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.488785 q^{5} +0.956122 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.488785 q^{5} +0.956122 q^{7} +1.00000 q^{9} -2.60300 q^{11} -2.95612 q^{13} -0.488785 q^{15} +1.58057 q^{17} +3.22744 q^{19} +0.956122 q^{21} -4.40103 q^{23} -4.76109 q^{25} +1.00000 q^{27} +1.02547 q^{29} -6.26537 q^{31} -2.60300 q^{33} -0.467338 q^{35} -6.38553 q^{37} -2.95612 q^{39} -1.00000 q^{41} +0.820462 q^{43} -0.488785 q^{45} -3.15809 q^{47} -6.08583 q^{49} +1.58057 q^{51} +3.62847 q^{53} +1.27231 q^{55} +3.22744 q^{57} +9.20600 q^{59} +9.74269 q^{61} +0.956122 q^{63} +1.44491 q^{65} -12.3671 q^{67} -4.40103 q^{69} -4.40698 q^{71} -10.4489 q^{73} -4.76109 q^{75} -2.48878 q^{77} -2.27132 q^{79} +1.00000 q^{81} +15.1272 q^{83} -0.772557 q^{85} +1.02547 q^{87} -12.3223 q^{89} -2.82641 q^{91} -6.26537 q^{93} -1.57752 q^{95} +5.89080 q^{97} -2.60300 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9} + 3 q^{11} - 2 q^{13} - 4 q^{15} - 3 q^{17} - 6 q^{21} + 4 q^{27} - 13 q^{29} - 9 q^{31} + 3 q^{33} + 10 q^{35} - 7 q^{37} - 2 q^{39} - 4 q^{41} - 5 q^{43} - 4 q^{45} - 7 q^{47} - 3 q^{51} - 16 q^{53} - 16 q^{55} + 10 q^{59} - 7 q^{61} - 6 q^{63} - 2 q^{65} - 4 q^{67} - 13 q^{71} - 3 q^{73} - 12 q^{77} - 6 q^{79} + 4 q^{81} + 14 q^{83} - 16 q^{85} - 13 q^{87} - 12 q^{89} - 16 q^{91} - 9 q^{93} - 10 q^{95} - 10 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −0.488785 −0.218591 −0.109296 0.994009i \(-0.534859\pi\)
−0.109296 + 0.994009i \(0.534859\pi\)
\(6\) 0 0
\(7\) 0.956122 0.361380 0.180690 0.983540i \(-0.442167\pi\)
0.180690 + 0.983540i \(0.442167\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.60300 −0.784833 −0.392417 0.919788i \(-0.628361\pi\)
−0.392417 + 0.919788i \(0.628361\pi\)
\(12\) 0 0
\(13\) −2.95612 −0.819881 −0.409940 0.912112i \(-0.634451\pi\)
−0.409940 + 0.912112i \(0.634451\pi\)
\(14\) 0 0
\(15\) −0.488785 −0.126204
\(16\) 0 0
\(17\) 1.58057 0.383344 0.191672 0.981459i \(-0.438609\pi\)
0.191672 + 0.981459i \(0.438609\pi\)
\(18\) 0 0
\(19\) 3.22744 0.740426 0.370213 0.928947i \(-0.379285\pi\)
0.370213 + 0.928947i \(0.379285\pi\)
\(20\) 0 0
\(21\) 0.956122 0.208643
\(22\) 0 0
\(23\) −4.40103 −0.917678 −0.458839 0.888519i \(-0.651735\pi\)
−0.458839 + 0.888519i \(0.651735\pi\)
\(24\) 0 0
\(25\) −4.76109 −0.952218
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.02547 0.190426 0.0952129 0.995457i \(-0.469647\pi\)
0.0952129 + 0.995457i \(0.469647\pi\)
\(30\) 0 0
\(31\) −6.26537 −1.12529 −0.562647 0.826697i \(-0.690217\pi\)
−0.562647 + 0.826697i \(0.690217\pi\)
\(32\) 0 0
\(33\) −2.60300 −0.453124
\(34\) 0 0
\(35\) −0.467338 −0.0789945
\(36\) 0 0
\(37\) −6.38553 −1.04978 −0.524888 0.851171i \(-0.675893\pi\)
−0.524888 + 0.851171i \(0.675893\pi\)
\(38\) 0 0
\(39\) −2.95612 −0.473358
\(40\) 0 0
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) 0.820462 0.125119 0.0625596 0.998041i \(-0.480074\pi\)
0.0625596 + 0.998041i \(0.480074\pi\)
\(44\) 0 0
\(45\) −0.488785 −0.0728637
\(46\) 0 0
\(47\) −3.15809 −0.460655 −0.230327 0.973113i \(-0.573980\pi\)
−0.230327 + 0.973113i \(0.573980\pi\)
\(48\) 0 0
\(49\) −6.08583 −0.869404
\(50\) 0 0
\(51\) 1.58057 0.221324
\(52\) 0 0
\(53\) 3.62847 0.498409 0.249204 0.968451i \(-0.419831\pi\)
0.249204 + 0.968451i \(0.419831\pi\)
\(54\) 0 0
\(55\) 1.27231 0.171558
\(56\) 0 0
\(57\) 3.22744 0.427485
\(58\) 0 0
\(59\) 9.20600 1.19852 0.599259 0.800555i \(-0.295462\pi\)
0.599259 + 0.800555i \(0.295462\pi\)
\(60\) 0 0
\(61\) 9.74269 1.24742 0.623712 0.781655i \(-0.285624\pi\)
0.623712 + 0.781655i \(0.285624\pi\)
\(62\) 0 0
\(63\) 0.956122 0.120460
\(64\) 0 0
\(65\) 1.44491 0.179219
\(66\) 0 0
\(67\) −12.3671 −1.51089 −0.755443 0.655215i \(-0.772578\pi\)
−0.755443 + 0.655215i \(0.772578\pi\)
\(68\) 0 0
\(69\) −4.40103 −0.529822
\(70\) 0 0
\(71\) −4.40698 −0.523012 −0.261506 0.965202i \(-0.584219\pi\)
−0.261506 + 0.965202i \(0.584219\pi\)
\(72\) 0 0
\(73\) −10.4489 −1.22296 −0.611478 0.791262i \(-0.709425\pi\)
−0.611478 + 0.791262i \(0.709425\pi\)
\(74\) 0 0
\(75\) −4.76109 −0.549763
\(76\) 0 0
\(77\) −2.48878 −0.283623
\(78\) 0 0
\(79\) −2.27132 −0.255544 −0.127772 0.991804i \(-0.540783\pi\)
−0.127772 + 0.991804i \(0.540783\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 15.1272 1.66043 0.830215 0.557443i \(-0.188218\pi\)
0.830215 + 0.557443i \(0.188218\pi\)
\(84\) 0 0
\(85\) −0.772557 −0.0837955
\(86\) 0 0
\(87\) 1.02547 0.109942
\(88\) 0 0
\(89\) −12.3223 −1.30616 −0.653079 0.757290i \(-0.726523\pi\)
−0.653079 + 0.757290i \(0.726523\pi\)
\(90\) 0 0
\(91\) −2.82641 −0.296289
\(92\) 0 0
\(93\) −6.26537 −0.649688
\(94\) 0 0
\(95\) −1.57752 −0.161851
\(96\) 0 0
\(97\) 5.89080 0.598120 0.299060 0.954234i \(-0.403327\pi\)
0.299060 + 0.954234i \(0.403327\pi\)
\(98\) 0 0
\(99\) −2.60300 −0.261611
\(100\) 0 0
\(101\) −16.0803 −1.60005 −0.800026 0.599966i \(-0.795181\pi\)
−0.800026 + 0.599966i \(0.795181\pi\)
\(102\) 0 0
\(103\) −3.17954 −0.313289 −0.156645 0.987655i \(-0.550068\pi\)
−0.156645 + 0.987655i \(0.550068\pi\)
\(104\) 0 0
\(105\) −0.467338 −0.0456075
\(106\) 0 0
\(107\) −1.77963 −0.172043 −0.0860215 0.996293i \(-0.527415\pi\)
−0.0860215 + 0.996293i \(0.527415\pi\)
\(108\) 0 0
\(109\) −12.4334 −1.19091 −0.595454 0.803390i \(-0.703028\pi\)
−0.595454 + 0.803390i \(0.703028\pi\)
\(110\) 0 0
\(111\) −6.38553 −0.606088
\(112\) 0 0
\(113\) 4.44589 0.418234 0.209117 0.977891i \(-0.432941\pi\)
0.209117 + 0.977891i \(0.432941\pi\)
\(114\) 0 0
\(115\) 2.15115 0.200596
\(116\) 0 0
\(117\) −2.95612 −0.273294
\(118\) 0 0
\(119\) 1.51122 0.138533
\(120\) 0 0
\(121\) −4.22440 −0.384036
\(122\) 0 0
\(123\) −1.00000 −0.0901670
\(124\) 0 0
\(125\) 4.77107 0.426737
\(126\) 0 0
\(127\) −16.9981 −1.50834 −0.754168 0.656682i \(-0.771959\pi\)
−0.754168 + 0.656682i \(0.771959\pi\)
\(128\) 0 0
\(129\) 0.820462 0.0722376
\(130\) 0 0
\(131\) 0.0338981 0.00296169 0.00148084 0.999999i \(-0.499529\pi\)
0.00148084 + 0.999999i \(0.499529\pi\)
\(132\) 0 0
\(133\) 3.08583 0.267575
\(134\) 0 0
\(135\) −0.488785 −0.0420679
\(136\) 0 0
\(137\) 4.09277 0.349669 0.174834 0.984598i \(-0.444061\pi\)
0.174834 + 0.984598i \(0.444061\pi\)
\(138\) 0 0
\(139\) 11.7182 0.993924 0.496962 0.867772i \(-0.334449\pi\)
0.496962 + 0.867772i \(0.334449\pi\)
\(140\) 0 0
\(141\) −3.15809 −0.265959
\(142\) 0 0
\(143\) 7.69478 0.643470
\(144\) 0 0
\(145\) −0.501236 −0.0416254
\(146\) 0 0
\(147\) −6.08583 −0.501951
\(148\) 0 0
\(149\) −13.3386 −1.09274 −0.546371 0.837543i \(-0.683991\pi\)
−0.546371 + 0.837543i \(0.683991\pi\)
\(150\) 0 0
\(151\) 10.0529 0.818095 0.409047 0.912513i \(-0.365861\pi\)
0.409047 + 0.912513i \(0.365861\pi\)
\(152\) 0 0
\(153\) 1.58057 0.127781
\(154\) 0 0
\(155\) 3.06242 0.245979
\(156\) 0 0
\(157\) 2.17748 0.173782 0.0868909 0.996218i \(-0.472307\pi\)
0.0868909 + 0.996218i \(0.472307\pi\)
\(158\) 0 0
\(159\) 3.62847 0.287757
\(160\) 0 0
\(161\) −4.20792 −0.331631
\(162\) 0 0
\(163\) −1.52671 −0.119581 −0.0597906 0.998211i \(-0.519043\pi\)
−0.0597906 + 0.998211i \(0.519043\pi\)
\(164\) 0 0
\(165\) 1.27231 0.0990488
\(166\) 0 0
\(167\) 21.0180 1.62642 0.813212 0.581967i \(-0.197717\pi\)
0.813212 + 0.581967i \(0.197717\pi\)
\(168\) 0 0
\(169\) −4.26134 −0.327795
\(170\) 0 0
\(171\) 3.22744 0.246809
\(172\) 0 0
\(173\) −6.80497 −0.517372 −0.258686 0.965961i \(-0.583290\pi\)
−0.258686 + 0.965961i \(0.583290\pi\)
\(174\) 0 0
\(175\) −4.55218 −0.344113
\(176\) 0 0
\(177\) 9.20600 0.691965
\(178\) 0 0
\(179\) 8.48283 0.634037 0.317018 0.948419i \(-0.397318\pi\)
0.317018 + 0.948419i \(0.397318\pi\)
\(180\) 0 0
\(181\) −17.5083 −1.30138 −0.650691 0.759343i \(-0.725521\pi\)
−0.650691 + 0.759343i \(0.725521\pi\)
\(182\) 0 0
\(183\) 9.74269 0.720200
\(184\) 0 0
\(185\) 3.12115 0.229472
\(186\) 0 0
\(187\) −4.11421 −0.300861
\(188\) 0 0
\(189\) 0.956122 0.0695477
\(190\) 0 0
\(191\) 4.61849 0.334182 0.167091 0.985941i \(-0.446563\pi\)
0.167091 + 0.985941i \(0.446563\pi\)
\(192\) 0 0
\(193\) 14.5292 1.04584 0.522919 0.852382i \(-0.324843\pi\)
0.522919 + 0.852382i \(0.324843\pi\)
\(194\) 0 0
\(195\) 1.44491 0.103472
\(196\) 0 0
\(197\) −15.4744 −1.10251 −0.551253 0.834338i \(-0.685850\pi\)
−0.551253 + 0.834338i \(0.685850\pi\)
\(198\) 0 0
\(199\) −16.3457 −1.15871 −0.579357 0.815074i \(-0.696696\pi\)
−0.579357 + 0.815074i \(0.696696\pi\)
\(200\) 0 0
\(201\) −12.3671 −0.872310
\(202\) 0 0
\(203\) 0.980478 0.0688161
\(204\) 0 0
\(205\) 0.488785 0.0341382
\(206\) 0 0
\(207\) −4.40103 −0.305893
\(208\) 0 0
\(209\) −8.40103 −0.581111
\(210\) 0 0
\(211\) 20.6265 1.41999 0.709995 0.704207i \(-0.248697\pi\)
0.709995 + 0.704207i \(0.248697\pi\)
\(212\) 0 0
\(213\) −4.40698 −0.301961
\(214\) 0 0
\(215\) −0.401029 −0.0273500
\(216\) 0 0
\(217\) −5.99046 −0.406659
\(218\) 0 0
\(219\) −10.4489 −0.706074
\(220\) 0 0
\(221\) −4.67235 −0.314296
\(222\) 0 0
\(223\) −10.5875 −0.708992 −0.354496 0.935058i \(-0.615347\pi\)
−0.354496 + 0.935058i \(0.615347\pi\)
\(224\) 0 0
\(225\) −4.76109 −0.317406
\(226\) 0 0
\(227\) 16.5581 1.09900 0.549501 0.835493i \(-0.314818\pi\)
0.549501 + 0.835493i \(0.314818\pi\)
\(228\) 0 0
\(229\) −2.68480 −0.177417 −0.0887083 0.996058i \(-0.528274\pi\)
−0.0887083 + 0.996058i \(0.528274\pi\)
\(230\) 0 0
\(231\) −2.48878 −0.163750
\(232\) 0 0
\(233\) −15.7367 −1.03095 −0.515474 0.856905i \(-0.672384\pi\)
−0.515474 + 0.856905i \(0.672384\pi\)
\(234\) 0 0
\(235\) 1.54363 0.100695
\(236\) 0 0
\(237\) −2.27132 −0.147538
\(238\) 0 0
\(239\) −3.61602 −0.233901 −0.116950 0.993138i \(-0.537312\pi\)
−0.116950 + 0.993138i \(0.537312\pi\)
\(240\) 0 0
\(241\) −4.11225 −0.264893 −0.132447 0.991190i \(-0.542283\pi\)
−0.132447 + 0.991190i \(0.542283\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 2.97466 0.190044
\(246\) 0 0
\(247\) −9.54072 −0.607061
\(248\) 0 0
\(249\) 15.1272 0.958650
\(250\) 0 0
\(251\) 13.2957 0.839218 0.419609 0.907705i \(-0.362167\pi\)
0.419609 + 0.907705i \(0.362167\pi\)
\(252\) 0 0
\(253\) 11.4559 0.720224
\(254\) 0 0
\(255\) −0.772557 −0.0483794
\(256\) 0 0
\(257\) 8.53520 0.532411 0.266206 0.963916i \(-0.414230\pi\)
0.266206 + 0.963916i \(0.414230\pi\)
\(258\) 0 0
\(259\) −6.10535 −0.379368
\(260\) 0 0
\(261\) 1.02547 0.0634752
\(262\) 0 0
\(263\) 1.62346 0.100107 0.0500534 0.998747i \(-0.484061\pi\)
0.0500534 + 0.998747i \(0.484061\pi\)
\(264\) 0 0
\(265\) −1.77354 −0.108948
\(266\) 0 0
\(267\) −12.3223 −0.754111
\(268\) 0 0
\(269\) −19.1692 −1.16877 −0.584383 0.811478i \(-0.698663\pi\)
−0.584383 + 0.811478i \(0.698663\pi\)
\(270\) 0 0
\(271\) 13.2629 0.805664 0.402832 0.915274i \(-0.368026\pi\)
0.402832 + 0.915274i \(0.368026\pi\)
\(272\) 0 0
\(273\) −2.82641 −0.171062
\(274\) 0 0
\(275\) 12.3931 0.747333
\(276\) 0 0
\(277\) −9.04692 −0.543577 −0.271788 0.962357i \(-0.587615\pi\)
−0.271788 + 0.962357i \(0.587615\pi\)
\(278\) 0 0
\(279\) −6.26537 −0.375098
\(280\) 0 0
\(281\) 4.00304 0.238802 0.119401 0.992846i \(-0.461903\pi\)
0.119401 + 0.992846i \(0.461903\pi\)
\(282\) 0 0
\(283\) 28.4769 1.69278 0.846389 0.532565i \(-0.178772\pi\)
0.846389 + 0.532565i \(0.178772\pi\)
\(284\) 0 0
\(285\) −1.57752 −0.0934445
\(286\) 0 0
\(287\) −0.956122 −0.0564381
\(288\) 0 0
\(289\) −14.5018 −0.853047
\(290\) 0 0
\(291\) 5.89080 0.345325
\(292\) 0 0
\(293\) 17.4679 1.02049 0.510243 0.860030i \(-0.329555\pi\)
0.510243 + 0.860030i \(0.329555\pi\)
\(294\) 0 0
\(295\) −4.49975 −0.261985
\(296\) 0 0
\(297\) −2.60300 −0.151041
\(298\) 0 0
\(299\) 13.0100 0.752387
\(300\) 0 0
\(301\) 0.784462 0.0452156
\(302\) 0 0
\(303\) −16.0803 −0.923790
\(304\) 0 0
\(305\) −4.76207 −0.272676
\(306\) 0 0
\(307\) −21.1832 −1.20899 −0.604494 0.796609i \(-0.706625\pi\)
−0.604494 + 0.796609i \(0.706625\pi\)
\(308\) 0 0
\(309\) −3.17954 −0.180878
\(310\) 0 0
\(311\) −22.8550 −1.29599 −0.647993 0.761646i \(-0.724392\pi\)
−0.647993 + 0.761646i \(0.724392\pi\)
\(312\) 0 0
\(313\) −6.90517 −0.390304 −0.195152 0.980773i \(-0.562520\pi\)
−0.195152 + 0.980773i \(0.562520\pi\)
\(314\) 0 0
\(315\) −0.467338 −0.0263315
\(316\) 0 0
\(317\) −28.1966 −1.58368 −0.791840 0.610728i \(-0.790877\pi\)
−0.791840 + 0.610728i \(0.790877\pi\)
\(318\) 0 0
\(319\) −2.66931 −0.149452
\(320\) 0 0
\(321\) −1.77963 −0.0993291
\(322\) 0 0
\(323\) 5.10119 0.283838
\(324\) 0 0
\(325\) 14.0744 0.780705
\(326\) 0 0
\(327\) −12.4334 −0.687571
\(328\) 0 0
\(329\) −3.01952 −0.166472
\(330\) 0 0
\(331\) −5.67624 −0.311995 −0.155997 0.987757i \(-0.549859\pi\)
−0.155997 + 0.987757i \(0.549859\pi\)
\(332\) 0 0
\(333\) −6.38553 −0.349925
\(334\) 0 0
\(335\) 6.04486 0.330266
\(336\) 0 0
\(337\) −36.0116 −1.96168 −0.980838 0.194826i \(-0.937586\pi\)
−0.980838 + 0.194826i \(0.937586\pi\)
\(338\) 0 0
\(339\) 4.44589 0.241468
\(340\) 0 0
\(341\) 16.3087 0.883168
\(342\) 0 0
\(343\) −12.5117 −0.675566
\(344\) 0 0
\(345\) 2.15115 0.115814
\(346\) 0 0
\(347\) 11.2339 0.603070 0.301535 0.953455i \(-0.402501\pi\)
0.301535 + 0.953455i \(0.402501\pi\)
\(348\) 0 0
\(349\) 7.86143 0.420813 0.210406 0.977614i \(-0.432521\pi\)
0.210406 + 0.977614i \(0.432521\pi\)
\(350\) 0 0
\(351\) −2.95612 −0.157786
\(352\) 0 0
\(353\) 3.41250 0.181629 0.0908144 0.995868i \(-0.471053\pi\)
0.0908144 + 0.995868i \(0.471053\pi\)
\(354\) 0 0
\(355\) 2.15406 0.114326
\(356\) 0 0
\(357\) 1.51122 0.0799820
\(358\) 0 0
\(359\) −19.4825 −1.02825 −0.514123 0.857717i \(-0.671882\pi\)
−0.514123 + 0.857717i \(0.671882\pi\)
\(360\) 0 0
\(361\) −8.58361 −0.451769
\(362\) 0 0
\(363\) −4.22440 −0.221724
\(364\) 0 0
\(365\) 5.10728 0.267327
\(366\) 0 0
\(367\) −19.0509 −0.994447 −0.497223 0.867623i \(-0.665647\pi\)
−0.497223 + 0.867623i \(0.665647\pi\)
\(368\) 0 0
\(369\) −1.00000 −0.0520579
\(370\) 0 0
\(371\) 3.46926 0.180115
\(372\) 0 0
\(373\) 6.58155 0.340780 0.170390 0.985377i \(-0.445497\pi\)
0.170390 + 0.985377i \(0.445497\pi\)
\(374\) 0 0
\(375\) 4.77107 0.246377
\(376\) 0 0
\(377\) −3.03143 −0.156126
\(378\) 0 0
\(379\) −8.30373 −0.426534 −0.213267 0.976994i \(-0.568410\pi\)
−0.213267 + 0.976994i \(0.568410\pi\)
\(380\) 0 0
\(381\) −16.9981 −0.870838
\(382\) 0 0
\(383\) 20.4350 1.04418 0.522090 0.852891i \(-0.325153\pi\)
0.522090 + 0.852891i \(0.325153\pi\)
\(384\) 0 0
\(385\) 1.21648 0.0619975
\(386\) 0 0
\(387\) 0.820462 0.0417064
\(388\) 0 0
\(389\) 22.3152 1.13143 0.565714 0.824602i \(-0.308601\pi\)
0.565714 + 0.824602i \(0.308601\pi\)
\(390\) 0 0
\(391\) −6.95612 −0.351786
\(392\) 0 0
\(393\) 0.0338981 0.00170993
\(394\) 0 0
\(395\) 1.11019 0.0558595
\(396\) 0 0
\(397\) 37.2416 1.86910 0.934551 0.355830i \(-0.115802\pi\)
0.934551 + 0.355830i \(0.115802\pi\)
\(398\) 0 0
\(399\) 3.08583 0.154485
\(400\) 0 0
\(401\) 30.8979 1.54297 0.771483 0.636250i \(-0.219515\pi\)
0.771483 + 0.636250i \(0.219515\pi\)
\(402\) 0 0
\(403\) 18.5212 0.922606
\(404\) 0 0
\(405\) −0.488785 −0.0242879
\(406\) 0 0
\(407\) 16.6215 0.823899
\(408\) 0 0
\(409\) −8.83291 −0.436759 −0.218380 0.975864i \(-0.570077\pi\)
−0.218380 + 0.975864i \(0.570077\pi\)
\(410\) 0 0
\(411\) 4.09277 0.201881
\(412\) 0 0
\(413\) 8.80206 0.433121
\(414\) 0 0
\(415\) −7.39396 −0.362955
\(416\) 0 0
\(417\) 11.7182 0.573843
\(418\) 0 0
\(419\) 30.1068 1.47081 0.735406 0.677627i \(-0.236991\pi\)
0.735406 + 0.677627i \(0.236991\pi\)
\(420\) 0 0
\(421\) −15.9142 −0.775611 −0.387806 0.921741i \(-0.626767\pi\)
−0.387806 + 0.921741i \(0.626767\pi\)
\(422\) 0 0
\(423\) −3.15809 −0.153552
\(424\) 0 0
\(425\) −7.52522 −0.365027
\(426\) 0 0
\(427\) 9.31520 0.450794
\(428\) 0 0
\(429\) 7.69478 0.371508
\(430\) 0 0
\(431\) 15.7538 0.758833 0.379417 0.925226i \(-0.376125\pi\)
0.379417 + 0.925226i \(0.376125\pi\)
\(432\) 0 0
\(433\) 2.27782 0.109465 0.0547325 0.998501i \(-0.482569\pi\)
0.0547325 + 0.998501i \(0.482569\pi\)
\(434\) 0 0
\(435\) −0.501236 −0.0240324
\(436\) 0 0
\(437\) −14.2041 −0.679473
\(438\) 0 0
\(439\) 31.6379 1.51000 0.754998 0.655727i \(-0.227638\pi\)
0.754998 + 0.655727i \(0.227638\pi\)
\(440\) 0 0
\(441\) −6.08583 −0.289801
\(442\) 0 0
\(443\) −17.7428 −0.842987 −0.421493 0.906831i \(-0.638494\pi\)
−0.421493 + 0.906831i \(0.638494\pi\)
\(444\) 0 0
\(445\) 6.02293 0.285514
\(446\) 0 0
\(447\) −13.3386 −0.630895
\(448\) 0 0
\(449\) 0.932707 0.0440172 0.0220086 0.999758i \(-0.492994\pi\)
0.0220086 + 0.999758i \(0.492994\pi\)
\(450\) 0 0
\(451\) 2.60300 0.122570
\(452\) 0 0
\(453\) 10.0529 0.472327
\(454\) 0 0
\(455\) 1.38151 0.0647661
\(456\) 0 0
\(457\) 27.2650 1.27540 0.637701 0.770284i \(-0.279885\pi\)
0.637701 + 0.770284i \(0.279885\pi\)
\(458\) 0 0
\(459\) 1.58057 0.0737746
\(460\) 0 0
\(461\) 15.3137 0.713231 0.356615 0.934251i \(-0.383931\pi\)
0.356615 + 0.934251i \(0.383931\pi\)
\(462\) 0 0
\(463\) 6.83738 0.317760 0.158880 0.987298i \(-0.449212\pi\)
0.158880 + 0.987298i \(0.449212\pi\)
\(464\) 0 0
\(465\) 3.06242 0.142016
\(466\) 0 0
\(467\) −25.3298 −1.17212 −0.586062 0.810266i \(-0.699323\pi\)
−0.586062 + 0.810266i \(0.699323\pi\)
\(468\) 0 0
\(469\) −11.8245 −0.546004
\(470\) 0 0
\(471\) 2.17748 0.100333
\(472\) 0 0
\(473\) −2.13566 −0.0981978
\(474\) 0 0
\(475\) −15.3661 −0.705047
\(476\) 0 0
\(477\) 3.62847 0.166136
\(478\) 0 0
\(479\) −3.66224 −0.167332 −0.0836659 0.996494i \(-0.526663\pi\)
−0.0836659 + 0.996494i \(0.526663\pi\)
\(480\) 0 0
\(481\) 18.8764 0.860691
\(482\) 0 0
\(483\) −4.20792 −0.191467
\(484\) 0 0
\(485\) −2.87933 −0.130744
\(486\) 0 0
\(487\) −3.98599 −0.180623 −0.0903113 0.995914i \(-0.528786\pi\)
−0.0903113 + 0.995914i \(0.528786\pi\)
\(488\) 0 0
\(489\) −1.52671 −0.0690402
\(490\) 0 0
\(491\) −12.9805 −0.585801 −0.292900 0.956143i \(-0.594620\pi\)
−0.292900 + 0.956143i \(0.594620\pi\)
\(492\) 0 0
\(493\) 1.62083 0.0729985
\(494\) 0 0
\(495\) 1.27231 0.0571859
\(496\) 0 0
\(497\) −4.21361 −0.189006
\(498\) 0 0
\(499\) 19.0346 0.852106 0.426053 0.904698i \(-0.359904\pi\)
0.426053 + 0.904698i \(0.359904\pi\)
\(500\) 0 0
\(501\) 21.0180 0.939017
\(502\) 0 0
\(503\) 20.4804 0.913176 0.456588 0.889678i \(-0.349071\pi\)
0.456588 + 0.889678i \(0.349071\pi\)
\(504\) 0 0
\(505\) 7.85981 0.349757
\(506\) 0 0
\(507\) −4.26134 −0.189253
\(508\) 0 0
\(509\) −22.0989 −0.979514 −0.489757 0.871859i \(-0.662915\pi\)
−0.489757 + 0.871859i \(0.662915\pi\)
\(510\) 0 0
\(511\) −9.99046 −0.441952
\(512\) 0 0
\(513\) 3.22744 0.142495
\(514\) 0 0
\(515\) 1.55411 0.0684822
\(516\) 0 0
\(517\) 8.22051 0.361537
\(518\) 0 0
\(519\) −6.80497 −0.298705
\(520\) 0 0
\(521\) −30.2334 −1.32455 −0.662276 0.749260i \(-0.730409\pi\)
−0.662276 + 0.749260i \(0.730409\pi\)
\(522\) 0 0
\(523\) −35.2404 −1.54095 −0.770477 0.637468i \(-0.779982\pi\)
−0.770477 + 0.637468i \(0.779982\pi\)
\(524\) 0 0
\(525\) −4.55218 −0.198674
\(526\) 0 0
\(527\) −9.90284 −0.431374
\(528\) 0 0
\(529\) −3.63094 −0.157867
\(530\) 0 0
\(531\) 9.20600 0.399506
\(532\) 0 0
\(533\) 2.95612 0.128044
\(534\) 0 0
\(535\) 0.869854 0.0376071
\(536\) 0 0
\(537\) 8.48283 0.366061
\(538\) 0 0
\(539\) 15.8414 0.682338
\(540\) 0 0
\(541\) 36.3813 1.56415 0.782077 0.623182i \(-0.214161\pi\)
0.782077 + 0.623182i \(0.214161\pi\)
\(542\) 0 0
\(543\) −17.5083 −0.751353
\(544\) 0 0
\(545\) 6.07727 0.260322
\(546\) 0 0
\(547\) 43.1343 1.84429 0.922146 0.386842i \(-0.126434\pi\)
0.922146 + 0.386842i \(0.126434\pi\)
\(548\) 0 0
\(549\) 9.74269 0.415808
\(550\) 0 0
\(551\) 3.30966 0.140996
\(552\) 0 0
\(553\) −2.17166 −0.0923484
\(554\) 0 0
\(555\) 3.12115 0.132485
\(556\) 0 0
\(557\) 29.4236 1.24672 0.623359 0.781936i \(-0.285768\pi\)
0.623359 + 0.781936i \(0.285768\pi\)
\(558\) 0 0
\(559\) −2.42539 −0.102583
\(560\) 0 0
\(561\) −4.11421 −0.173702
\(562\) 0 0
\(563\) 33.4394 1.40930 0.704652 0.709553i \(-0.251103\pi\)
0.704652 + 0.709553i \(0.251103\pi\)
\(564\) 0 0
\(565\) −2.17308 −0.0914223
\(566\) 0 0
\(567\) 0.956122 0.0401534
\(568\) 0 0
\(569\) 19.5212 0.818373 0.409186 0.912451i \(-0.365813\pi\)
0.409186 + 0.912451i \(0.365813\pi\)
\(570\) 0 0
\(571\) −2.82499 −0.118222 −0.0591111 0.998251i \(-0.518827\pi\)
−0.0591111 + 0.998251i \(0.518827\pi\)
\(572\) 0 0
\(573\) 4.61849 0.192940
\(574\) 0 0
\(575\) 20.9537 0.873829
\(576\) 0 0
\(577\) −10.9098 −0.454180 −0.227090 0.973874i \(-0.572921\pi\)
−0.227090 + 0.973874i \(0.572921\pi\)
\(578\) 0 0
\(579\) 14.5292 0.603815
\(580\) 0 0
\(581\) 14.4635 0.600047
\(582\) 0 0
\(583\) −9.44491 −0.391168
\(584\) 0 0
\(585\) 1.44491 0.0597395
\(586\) 0 0
\(587\) 14.0030 0.577967 0.288984 0.957334i \(-0.406683\pi\)
0.288984 + 0.957334i \(0.406683\pi\)
\(588\) 0 0
\(589\) −20.2211 −0.833197
\(590\) 0 0
\(591\) −15.4744 −0.636532
\(592\) 0 0
\(593\) −8.29487 −0.340629 −0.170315 0.985390i \(-0.554478\pi\)
−0.170315 + 0.985390i \(0.554478\pi\)
\(594\) 0 0
\(595\) −0.738659 −0.0302821
\(596\) 0 0
\(597\) −16.3457 −0.668984
\(598\) 0 0
\(599\) 8.00291 0.326990 0.163495 0.986544i \(-0.447723\pi\)
0.163495 + 0.986544i \(0.447723\pi\)
\(600\) 0 0
\(601\) −3.72062 −0.151767 −0.0758837 0.997117i \(-0.524178\pi\)
−0.0758837 + 0.997117i \(0.524178\pi\)
\(602\) 0 0
\(603\) −12.3671 −0.503629
\(604\) 0 0
\(605\) 2.06482 0.0839469
\(606\) 0 0
\(607\) −26.8483 −1.08974 −0.544870 0.838520i \(-0.683421\pi\)
−0.544870 + 0.838520i \(0.683421\pi\)
\(608\) 0 0
\(609\) 0.980478 0.0397310
\(610\) 0 0
\(611\) 9.33570 0.377682
\(612\) 0 0
\(613\) 4.04344 0.163313 0.0816565 0.996661i \(-0.473979\pi\)
0.0816565 + 0.996661i \(0.473979\pi\)
\(614\) 0 0
\(615\) 0.488785 0.0197097
\(616\) 0 0
\(617\) −1.57407 −0.0633696 −0.0316848 0.999498i \(-0.510087\pi\)
−0.0316848 + 0.999498i \(0.510087\pi\)
\(618\) 0 0
\(619\) −25.2571 −1.01517 −0.507584 0.861602i \(-0.669461\pi\)
−0.507584 + 0.861602i \(0.669461\pi\)
\(620\) 0 0
\(621\) −4.40103 −0.176607
\(622\) 0 0
\(623\) −11.7816 −0.472020
\(624\) 0 0
\(625\) 21.4734 0.858937
\(626\) 0 0
\(627\) −8.40103 −0.335505
\(628\) 0 0
\(629\) −10.0928 −0.402425
\(630\) 0 0
\(631\) −11.9431 −0.475447 −0.237724 0.971333i \(-0.576401\pi\)
−0.237724 + 0.971333i \(0.576401\pi\)
\(632\) 0 0
\(633\) 20.6265 0.819832
\(634\) 0 0
\(635\) 8.30840 0.329709
\(636\) 0 0
\(637\) 17.9905 0.712808
\(638\) 0 0
\(639\) −4.40698 −0.174337
\(640\) 0 0
\(641\) −5.16615 −0.204050 −0.102025 0.994782i \(-0.532532\pi\)
−0.102025 + 0.994782i \(0.532532\pi\)
\(642\) 0 0
\(643\) −24.0280 −0.947572 −0.473786 0.880640i \(-0.657113\pi\)
−0.473786 + 0.880640i \(0.657113\pi\)
\(644\) 0 0
\(645\) −0.401029 −0.0157905
\(646\) 0 0
\(647\) 20.5437 0.807655 0.403827 0.914835i \(-0.367680\pi\)
0.403827 + 0.914835i \(0.367680\pi\)
\(648\) 0 0
\(649\) −23.9632 −0.940638
\(650\) 0 0
\(651\) −5.99046 −0.234785
\(652\) 0 0
\(653\) −26.6296 −1.04210 −0.521048 0.853527i \(-0.674459\pi\)
−0.521048 + 0.853527i \(0.674459\pi\)
\(654\) 0 0
\(655\) −0.0165688 −0.000647398 0
\(656\) 0 0
\(657\) −10.4489 −0.407652
\(658\) 0 0
\(659\) 1.43049 0.0557239 0.0278619 0.999612i \(-0.491130\pi\)
0.0278619 + 0.999612i \(0.491130\pi\)
\(660\) 0 0
\(661\) 2.69577 0.104853 0.0524266 0.998625i \(-0.483304\pi\)
0.0524266 + 0.998625i \(0.483304\pi\)
\(662\) 0 0
\(663\) −4.67235 −0.181459
\(664\) 0 0
\(665\) −1.50831 −0.0584896
\(666\) 0 0
\(667\) −4.51314 −0.174750
\(668\) 0 0
\(669\) −10.5875 −0.409337
\(670\) 0 0
\(671\) −25.3602 −0.979019
\(672\) 0 0
\(673\) −29.3956 −1.13312 −0.566559 0.824021i \(-0.691726\pi\)
−0.566559 + 0.824021i \(0.691726\pi\)
\(674\) 0 0
\(675\) −4.76109 −0.183254
\(676\) 0 0
\(677\) −16.3194 −0.627204 −0.313602 0.949555i \(-0.601536\pi\)
−0.313602 + 0.949555i \(0.601536\pi\)
\(678\) 0 0
\(679\) 5.63232 0.216149
\(680\) 0 0
\(681\) 16.5581 0.634509
\(682\) 0 0
\(683\) −24.1163 −0.922785 −0.461393 0.887196i \(-0.652650\pi\)
−0.461393 + 0.887196i \(0.652650\pi\)
\(684\) 0 0
\(685\) −2.00048 −0.0764345
\(686\) 0 0
\(687\) −2.68480 −0.102432
\(688\) 0 0
\(689\) −10.7262 −0.408636
\(690\) 0 0
\(691\) 24.7357 0.940992 0.470496 0.882402i \(-0.344075\pi\)
0.470496 + 0.882402i \(0.344075\pi\)
\(692\) 0 0
\(693\) −2.48878 −0.0945411
\(694\) 0 0
\(695\) −5.72767 −0.217263
\(696\) 0 0
\(697\) −1.58057 −0.0598683
\(698\) 0 0
\(699\) −15.7367 −0.595218
\(700\) 0 0
\(701\) 9.32868 0.352339 0.176170 0.984360i \(-0.443629\pi\)
0.176170 + 0.984360i \(0.443629\pi\)
\(702\) 0 0
\(703\) −20.6090 −0.777281
\(704\) 0 0
\(705\) 1.54363 0.0581363
\(706\) 0 0
\(707\) −15.3747 −0.578227
\(708\) 0 0
\(709\) −4.57972 −0.171995 −0.0859974 0.996295i \(-0.527408\pi\)
−0.0859974 + 0.996295i \(0.527408\pi\)
\(710\) 0 0
\(711\) −2.27132 −0.0851812
\(712\) 0 0
\(713\) 27.5741 1.03266
\(714\) 0 0
\(715\) −3.76109 −0.140657
\(716\) 0 0
\(717\) −3.61602 −0.135043
\(718\) 0 0
\(719\) 5.14564 0.191900 0.0959500 0.995386i \(-0.469411\pi\)
0.0959500 + 0.995386i \(0.469411\pi\)
\(720\) 0 0
\(721\) −3.04003 −0.113217
\(722\) 0 0
\(723\) −4.11225 −0.152936
\(724\) 0 0
\(725\) −4.88237 −0.181327
\(726\) 0 0
\(727\) −14.5542 −0.539784 −0.269892 0.962891i \(-0.586988\pi\)
−0.269892 + 0.962891i \(0.586988\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.29679 0.0479637
\(732\) 0 0
\(733\) 12.5187 0.462389 0.231194 0.972908i \(-0.425737\pi\)
0.231194 + 0.972908i \(0.425737\pi\)
\(734\) 0 0
\(735\) 2.97466 0.109722
\(736\) 0 0
\(737\) 32.1916 1.18579
\(738\) 0 0
\(739\) −0.145202 −0.00534135 −0.00267068 0.999996i \(-0.500850\pi\)
−0.00267068 + 0.999996i \(0.500850\pi\)
\(740\) 0 0
\(741\) −9.54072 −0.350487
\(742\) 0 0
\(743\) −17.8444 −0.654649 −0.327325 0.944912i \(-0.606147\pi\)
−0.327325 + 0.944912i \(0.606147\pi\)
\(744\) 0 0
\(745\) 6.51971 0.238864
\(746\) 0 0
\(747\) 15.1272 0.553477
\(748\) 0 0
\(749\) −1.70154 −0.0621730
\(750\) 0 0
\(751\) −10.6263 −0.387758 −0.193879 0.981025i \(-0.562107\pi\)
−0.193879 + 0.981025i \(0.562107\pi\)
\(752\) 0 0
\(753\) 13.2957 0.484523
\(754\) 0 0
\(755\) −4.91371 −0.178828
\(756\) 0 0
\(757\) 29.0658 1.05641 0.528207 0.849115i \(-0.322864\pi\)
0.528207 + 0.849115i \(0.322864\pi\)
\(758\) 0 0
\(759\) 11.4559 0.415822
\(760\) 0 0
\(761\) 45.2302 1.63959 0.819796 0.572656i \(-0.194087\pi\)
0.819796 + 0.572656i \(0.194087\pi\)
\(762\) 0 0
\(763\) −11.8879 −0.430370
\(764\) 0 0
\(765\) −0.772557 −0.0279318
\(766\) 0 0
\(767\) −27.2141 −0.982642
\(768\) 0 0
\(769\) 1.51426 0.0546056 0.0273028 0.999627i \(-0.491308\pi\)
0.0273028 + 0.999627i \(0.491308\pi\)
\(770\) 0 0
\(771\) 8.53520 0.307388
\(772\) 0 0
\(773\) 44.0449 1.58419 0.792093 0.610400i \(-0.208991\pi\)
0.792093 + 0.610400i \(0.208991\pi\)
\(774\) 0 0
\(775\) 29.8300 1.07152
\(776\) 0 0
\(777\) −6.10535 −0.219028
\(778\) 0 0
\(779\) −3.22744 −0.115635
\(780\) 0 0
\(781\) 11.4714 0.410478
\(782\) 0 0
\(783\) 1.02547 0.0366475
\(784\) 0 0
\(785\) −1.06432 −0.0379871
\(786\) 0 0
\(787\) 28.3855 1.01184 0.505918 0.862582i \(-0.331154\pi\)
0.505918 + 0.862582i \(0.331154\pi\)
\(788\) 0 0
\(789\) 1.62346 0.0577967
\(790\) 0 0
\(791\) 4.25082 0.151142
\(792\) 0 0
\(793\) −28.8006 −1.02274
\(794\) 0 0
\(795\) −1.77354 −0.0629010
\(796\) 0 0
\(797\) 46.5688 1.64955 0.824775 0.565461i \(-0.191302\pi\)
0.824775 + 0.565461i \(0.191302\pi\)
\(798\) 0 0
\(799\) −4.99158 −0.176589
\(800\) 0 0
\(801\) −12.3223 −0.435386
\(802\) 0 0
\(803\) 27.1986 0.959816
\(804\) 0 0
\(805\) 2.05677 0.0724915
\(806\) 0 0
\(807\) −19.1692 −0.674787
\(808\) 0 0
\(809\) 44.1687 1.55289 0.776444 0.630186i \(-0.217021\pi\)
0.776444 + 0.630186i \(0.217021\pi\)
\(810\) 0 0
\(811\) −29.6488 −1.04111 −0.520556 0.853828i \(-0.674275\pi\)
−0.520556 + 0.853828i \(0.674275\pi\)
\(812\) 0 0
\(813\) 13.2629 0.465150
\(814\) 0 0
\(815\) 0.746232 0.0261394
\(816\) 0 0
\(817\) 2.64799 0.0926416
\(818\) 0 0
\(819\) −2.82641 −0.0987629
\(820\) 0 0
\(821\) −17.5780 −0.613477 −0.306738 0.951794i \(-0.599238\pi\)
−0.306738 + 0.951794i \(0.599238\pi\)
\(822\) 0 0
\(823\) −12.3652 −0.431022 −0.215511 0.976501i \(-0.569142\pi\)
−0.215511 + 0.976501i \(0.569142\pi\)
\(824\) 0 0
\(825\) 12.3931 0.431473
\(826\) 0 0
\(827\) 7.47427 0.259906 0.129953 0.991520i \(-0.458517\pi\)
0.129953 + 0.991520i \(0.458517\pi\)
\(828\) 0 0
\(829\) 4.41598 0.153373 0.0766866 0.997055i \(-0.475566\pi\)
0.0766866 + 0.997055i \(0.475566\pi\)
\(830\) 0 0
\(831\) −9.04692 −0.313834
\(832\) 0 0
\(833\) −9.61906 −0.333281
\(834\) 0 0
\(835\) −10.2733 −0.355522
\(836\) 0 0
\(837\) −6.26537 −0.216563
\(838\) 0 0
\(839\) 53.5931 1.85024 0.925119 0.379676i \(-0.123965\pi\)
0.925119 + 0.379676i \(0.123965\pi\)
\(840\) 0 0
\(841\) −27.9484 −0.963738
\(842\) 0 0
\(843\) 4.00304 0.137872
\(844\) 0 0
\(845\) 2.08288 0.0716532
\(846\) 0 0
\(847\) −4.03904 −0.138783
\(848\) 0 0
\(849\) 28.4769 0.977326
\(850\) 0 0
\(851\) 28.1029 0.963356
\(852\) 0 0
\(853\) 44.6050 1.52725 0.763624 0.645662i \(-0.223418\pi\)
0.763624 + 0.645662i \(0.223418\pi\)
\(854\) 0 0
\(855\) −1.57752 −0.0539502
\(856\) 0 0
\(857\) 23.3486 0.797574 0.398787 0.917044i \(-0.369431\pi\)
0.398787 + 0.917044i \(0.369431\pi\)
\(858\) 0 0
\(859\) −31.9786 −1.09110 −0.545548 0.838079i \(-0.683679\pi\)
−0.545548 + 0.838079i \(0.683679\pi\)
\(860\) 0 0
\(861\) −0.956122 −0.0325846
\(862\) 0 0
\(863\) −1.28887 −0.0438738 −0.0219369 0.999759i \(-0.506983\pi\)
−0.0219369 + 0.999759i \(0.506983\pi\)
\(864\) 0 0
\(865\) 3.32616 0.113093
\(866\) 0 0
\(867\) −14.5018 −0.492507
\(868\) 0 0
\(869\) 5.91224 0.200559
\(870\) 0 0
\(871\) 36.5588 1.23875
\(872\) 0 0
\(873\) 5.89080 0.199373
\(874\) 0 0
\(875\) 4.56173 0.154214
\(876\) 0 0
\(877\) 24.2125 0.817596 0.408798 0.912625i \(-0.365948\pi\)
0.408798 + 0.912625i \(0.365948\pi\)
\(878\) 0 0
\(879\) 17.4679 0.589178
\(880\) 0 0
\(881\) 38.8979 1.31050 0.655251 0.755411i \(-0.272563\pi\)
0.655251 + 0.755411i \(0.272563\pi\)
\(882\) 0 0
\(883\) −43.8330 −1.47510 −0.737549 0.675293i \(-0.764017\pi\)
−0.737549 + 0.675293i \(0.764017\pi\)
\(884\) 0 0
\(885\) −4.49975 −0.151257
\(886\) 0 0
\(887\) 10.9358 0.367187 0.183593 0.983002i \(-0.441227\pi\)
0.183593 + 0.983002i \(0.441227\pi\)
\(888\) 0 0
\(889\) −16.2522 −0.545083
\(890\) 0 0
\(891\) −2.60300 −0.0872037
\(892\) 0 0
\(893\) −10.1926 −0.341081
\(894\) 0 0
\(895\) −4.14628 −0.138595
\(896\) 0 0
\(897\) 13.0100 0.434391
\(898\) 0 0
\(899\) −6.42497 −0.214285
\(900\) 0 0
\(901\) 5.73504 0.191062
\(902\) 0 0
\(903\) 0.784462 0.0261053
\(904\) 0 0
\(905\) 8.55779 0.284471
\(906\) 0 0
\(907\) 14.4934 0.481245 0.240622 0.970619i \(-0.422648\pi\)
0.240622 + 0.970619i \(0.422648\pi\)
\(908\) 0 0
\(909\) −16.0803 −0.533350
\(910\) 0 0
\(911\) −28.5987 −0.947518 −0.473759 0.880654i \(-0.657103\pi\)
−0.473759 + 0.880654i \(0.657103\pi\)
\(912\) 0 0
\(913\) −39.3762 −1.30316
\(914\) 0 0
\(915\) −4.76207 −0.157429
\(916\) 0 0
\(917\) 0.0324107 0.00107030
\(918\) 0 0
\(919\) 10.7297 0.353939 0.176969 0.984216i \(-0.443371\pi\)
0.176969 + 0.984216i \(0.443371\pi\)
\(920\) 0 0
\(921\) −21.1832 −0.698010
\(922\) 0 0
\(923\) 13.0276 0.428808
\(924\) 0 0
\(925\) 30.4021 0.999615
\(926\) 0 0
\(927\) −3.17954 −0.104430
\(928\) 0 0
\(929\) −47.1623 −1.54734 −0.773672 0.633586i \(-0.781582\pi\)
−0.773672 + 0.633586i \(0.781582\pi\)
\(930\) 0 0
\(931\) −19.6417 −0.643730
\(932\) 0 0
\(933\) −22.8550 −0.748238
\(934\) 0 0
\(935\) 2.01096 0.0657656
\(936\) 0 0
\(937\) 27.2063 0.888790 0.444395 0.895831i \(-0.353419\pi\)
0.444395 + 0.895831i \(0.353419\pi\)
\(938\) 0 0
\(939\) −6.90517 −0.225342
\(940\) 0 0
\(941\) 0.398995 0.0130069 0.00650344 0.999979i \(-0.497930\pi\)
0.00650344 + 0.999979i \(0.497930\pi\)
\(942\) 0 0
\(943\) 4.40103 0.143317
\(944\) 0 0
\(945\) −0.467338 −0.0152025
\(946\) 0 0
\(947\) 26.4859 0.860675 0.430338 0.902668i \(-0.358395\pi\)
0.430338 + 0.902668i \(0.358395\pi\)
\(948\) 0 0
\(949\) 30.8883 1.00268
\(950\) 0 0
\(951\) −28.1966 −0.914338
\(952\) 0 0
\(953\) −0.584594 −0.0189369 −0.00946844 0.999955i \(-0.503014\pi\)
−0.00946844 + 0.999955i \(0.503014\pi\)
\(954\) 0 0
\(955\) −2.25745 −0.0730493
\(956\) 0 0
\(957\) −2.66931 −0.0862864
\(958\) 0 0
\(959\) 3.91319 0.126363
\(960\) 0 0
\(961\) 8.25484 0.266285
\(962\) 0 0
\(963\) −1.77963 −0.0573477
\(964\) 0 0
\(965\) −7.10167 −0.228611
\(966\) 0 0
\(967\) 0.944612 0.0303767 0.0151883 0.999885i \(-0.495165\pi\)
0.0151883 + 0.999885i \(0.495165\pi\)
\(968\) 0 0
\(969\) 5.10119 0.163874
\(970\) 0 0
\(971\) 11.7149 0.375948 0.187974 0.982174i \(-0.439808\pi\)
0.187974 + 0.982174i \(0.439808\pi\)
\(972\) 0 0
\(973\) 11.2040 0.359185
\(974\) 0 0
\(975\) 14.0744 0.450740
\(976\) 0 0
\(977\) 44.7946 1.43311 0.716553 0.697532i \(-0.245719\pi\)
0.716553 + 0.697532i \(0.245719\pi\)
\(978\) 0 0
\(979\) 32.0748 1.02512
\(980\) 0 0
\(981\) −12.4334 −0.396969
\(982\) 0 0
\(983\) 17.5234 0.558909 0.279455 0.960159i \(-0.409846\pi\)
0.279455 + 0.960159i \(0.409846\pi\)
\(984\) 0 0
\(985\) 7.56365 0.240998
\(986\) 0 0
\(987\) −3.01952 −0.0961124
\(988\) 0 0
\(989\) −3.61088 −0.114819
\(990\) 0 0
\(991\) −49.6302 −1.57655 −0.788277 0.615321i \(-0.789027\pi\)
−0.788277 + 0.615321i \(0.789027\pi\)
\(992\) 0 0
\(993\) −5.67624 −0.180130
\(994\) 0 0
\(995\) 7.98952 0.253285
\(996\) 0 0
\(997\) 11.4339 0.362117 0.181058 0.983472i \(-0.442048\pi\)
0.181058 + 0.983472i \(0.442048\pi\)
\(998\) 0 0
\(999\) −6.38553 −0.202029
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 3936.2.a.m.1.3 yes 4
4.3 odd 2 3936.2.a.i.1.3 4
8.3 odd 2 7872.2.a.cj.1.2 4
8.5 even 2 7872.2.a.cf.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3936.2.a.i.1.3 4 4.3 odd 2
3936.2.a.m.1.3 yes 4 1.1 even 1 trivial
7872.2.a.cf.1.2 4 8.5 even 2
7872.2.a.cj.1.2 4 8.3 odd 2