Properties

Label 9680.2.a.da.1.5
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.22733568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4840)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.821728\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.32476 q^{3} +1.00000 q^{5} -4.78768 q^{7} +2.40452 q^{9} +O(q^{10})\) \(q+2.32476 q^{3} +1.00000 q^{5} -4.78768 q^{7} +2.40452 q^{9} -0.601547 q^{13} +2.32476 q^{15} -2.64953 q^{17} -2.60155 q^{19} -11.1302 q^{21} -5.17083 q^{23} +1.00000 q^{25} -1.38434 q^{27} +5.44643 q^{29} +10.4577 q^{31} -4.78768 q^{35} +9.39816 q^{37} -1.39845 q^{39} -4.54633 q^{41} +3.48547 q^{43} +2.40452 q^{45} -5.07340 q^{47} +15.9218 q^{49} -6.15952 q^{51} +3.76714 q^{53} -6.04798 q^{57} -11.7599 q^{59} +12.6454 q^{61} -11.5121 q^{63} -0.601547 q^{65} +15.0452 q^{67} -12.0210 q^{69} -0.346489 q^{71} +3.68096 q^{73} +2.32476 q^{75} +8.05322 q^{79} -10.4318 q^{81} +13.0094 q^{83} -2.64953 q^{85} +12.6617 q^{87} +0.462692 q^{89} +2.88001 q^{91} +24.3118 q^{93} -2.60155 q^{95} +7.97011 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{15} + 8 q^{17} - 12 q^{19} - 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} + 16 q^{29} + 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} + 32 q^{41} + 4 q^{43} + 4 q^{45} + 6 q^{47} + 16 q^{49} - 40 q^{51} + 8 q^{53} - 16 q^{57} - 4 q^{59} + 16 q^{61} - 28 q^{63} + 2 q^{67} + 8 q^{69} + 28 q^{71} + 16 q^{73} + 2 q^{75} - 10 q^{81} - 12 q^{83} + 8 q^{85} + 24 q^{87} + 18 q^{89} + 24 q^{91} + 20 q^{93} - 12 q^{95}+O(q^{100}) \) 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
\(3\) 2.32476 1.34220 0.671101 0.741366i \(-0.265822\pi\)
0.671101 + 0.741366i \(0.265822\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.78768 −1.80957 −0.904786 0.425867i \(-0.859969\pi\)
−0.904786 + 0.425867i \(0.859969\pi\)
\(8\) 0 0
\(9\) 2.40452 0.801508
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −0.601547 −0.166839 −0.0834195 0.996515i \(-0.526584\pi\)
−0.0834195 + 0.996515i \(0.526584\pi\)
\(14\) 0 0
\(15\) 2.32476 0.600251
\(16\) 0 0
\(17\) −2.64953 −0.642605 −0.321302 0.946977i \(-0.604121\pi\)
−0.321302 + 0.946977i \(0.604121\pi\)
\(18\) 0 0
\(19\) −2.60155 −0.596836 −0.298418 0.954435i \(-0.596459\pi\)
−0.298418 + 0.954435i \(0.596459\pi\)
\(20\) 0 0
\(21\) −11.1302 −2.42881
\(22\) 0 0
\(23\) −5.17083 −1.07819 −0.539096 0.842244i \(-0.681234\pi\)
−0.539096 + 0.842244i \(0.681234\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.38434 −0.266417
\(28\) 0 0
\(29\) 5.44643 1.01138 0.505689 0.862716i \(-0.331238\pi\)
0.505689 + 0.862716i \(0.331238\pi\)
\(30\) 0 0
\(31\) 10.4577 1.87827 0.939133 0.343554i \(-0.111631\pi\)
0.939133 + 0.343554i \(0.111631\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.78768 −0.809265
\(36\) 0 0
\(37\) 9.39816 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(38\) 0 0
\(39\) −1.39845 −0.223932
\(40\) 0 0
\(41\) −4.54633 −0.710018 −0.355009 0.934863i \(-0.615522\pi\)
−0.355009 + 0.934863i \(0.615522\pi\)
\(42\) 0 0
\(43\) 3.48547 0.531530 0.265765 0.964038i \(-0.414376\pi\)
0.265765 + 0.964038i \(0.414376\pi\)
\(44\) 0 0
\(45\) 2.40452 0.358445
\(46\) 0 0
\(47\) −5.07340 −0.740031 −0.370016 0.929026i \(-0.620648\pi\)
−0.370016 + 0.929026i \(0.620648\pi\)
\(48\) 0 0
\(49\) 15.9218 2.27455
\(50\) 0 0
\(51\) −6.15952 −0.862506
\(52\) 0 0
\(53\) 3.76714 0.517456 0.258728 0.965950i \(-0.416697\pi\)
0.258728 + 0.965950i \(0.416697\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.04798 −0.801074
\(58\) 0 0
\(59\) −11.7599 −1.53101 −0.765507 0.643428i \(-0.777512\pi\)
−0.765507 + 0.643428i \(0.777512\pi\)
\(60\) 0 0
\(61\) 12.6454 1.61908 0.809542 0.587063i \(-0.199716\pi\)
0.809542 + 0.587063i \(0.199716\pi\)
\(62\) 0 0
\(63\) −11.5121 −1.45039
\(64\) 0 0
\(65\) −0.601547 −0.0746127
\(66\) 0 0
\(67\) 15.0452 1.83806 0.919030 0.394187i \(-0.128974\pi\)
0.919030 + 0.394187i \(0.128974\pi\)
\(68\) 0 0
\(69\) −12.0210 −1.44715
\(70\) 0 0
\(71\) −0.346489 −0.0411207 −0.0205604 0.999789i \(-0.506545\pi\)
−0.0205604 + 0.999789i \(0.506545\pi\)
\(72\) 0 0
\(73\) 3.68096 0.430823 0.215412 0.976523i \(-0.430891\pi\)
0.215412 + 0.976523i \(0.430891\pi\)
\(74\) 0 0
\(75\) 2.32476 0.268441
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.05322 0.906058 0.453029 0.891496i \(-0.350343\pi\)
0.453029 + 0.891496i \(0.350343\pi\)
\(80\) 0 0
\(81\) −10.4318 −1.15909
\(82\) 0 0
\(83\) 13.0094 1.42796 0.713981 0.700165i \(-0.246890\pi\)
0.713981 + 0.700165i \(0.246890\pi\)
\(84\) 0 0
\(85\) −2.64953 −0.287381
\(86\) 0 0
\(87\) 12.6617 1.35747
\(88\) 0 0
\(89\) 0.462692 0.0490453 0.0245226 0.999699i \(-0.492193\pi\)
0.0245226 + 0.999699i \(0.492193\pi\)
\(90\) 0 0
\(91\) 2.88001 0.301907
\(92\) 0 0
\(93\) 24.3118 2.52101
\(94\) 0 0
\(95\) −2.60155 −0.266913
\(96\) 0 0
\(97\) 7.97011 0.809242 0.404621 0.914484i \(-0.367403\pi\)
0.404621 + 0.914484i \(0.367403\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.9134 −1.08592 −0.542960 0.839759i \(-0.682696\pi\)
−0.542960 + 0.839759i \(0.682696\pi\)
\(102\) 0 0
\(103\) 16.9081 1.66600 0.833001 0.553271i \(-0.186621\pi\)
0.833001 + 0.553271i \(0.186621\pi\)
\(104\) 0 0
\(105\) −11.1302 −1.08620
\(106\) 0 0
\(107\) −8.94029 −0.864291 −0.432145 0.901804i \(-0.642243\pi\)
−0.432145 + 0.901804i \(0.642243\pi\)
\(108\) 0 0
\(109\) 8.76627 0.839656 0.419828 0.907604i \(-0.362090\pi\)
0.419828 + 0.907604i \(0.362090\pi\)
\(110\) 0 0
\(111\) 21.8485 2.07377
\(112\) 0 0
\(113\) −2.49000 −0.234240 −0.117120 0.993118i \(-0.537366\pi\)
−0.117120 + 0.993118i \(0.537366\pi\)
\(114\) 0 0
\(115\) −5.17083 −0.482182
\(116\) 0 0
\(117\) −1.44643 −0.133723
\(118\) 0 0
\(119\) 12.6851 1.16284
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −10.5692 −0.952988
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 9.22655 0.818724 0.409362 0.912372i \(-0.365751\pi\)
0.409362 + 0.912372i \(0.365751\pi\)
\(128\) 0 0
\(129\) 8.10290 0.713421
\(130\) 0 0
\(131\) −8.09882 −0.707597 −0.353799 0.935322i \(-0.615110\pi\)
−0.353799 + 0.935322i \(0.615110\pi\)
\(132\) 0 0
\(133\) 12.4554 1.08002
\(134\) 0 0
\(135\) −1.38434 −0.119145
\(136\) 0 0
\(137\) 6.57195 0.561480 0.280740 0.959784i \(-0.409420\pi\)
0.280740 + 0.959784i \(0.409420\pi\)
\(138\) 0 0
\(139\) 3.61119 0.306297 0.153149 0.988203i \(-0.451059\pi\)
0.153149 + 0.988203i \(0.451059\pi\)
\(140\) 0 0
\(141\) −11.7945 −0.993272
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 5.44643 0.452302
\(146\) 0 0
\(147\) 37.0145 3.05291
\(148\) 0 0
\(149\) 12.1685 0.996886 0.498443 0.866922i \(-0.333905\pi\)
0.498443 + 0.866922i \(0.333905\pi\)
\(150\) 0 0
\(151\) −21.5830 −1.75640 −0.878198 0.478296i \(-0.841254\pi\)
−0.878198 + 0.478296i \(0.841254\pi\)
\(152\) 0 0
\(153\) −6.37085 −0.515053
\(154\) 0 0
\(155\) 10.4577 0.839986
\(156\) 0 0
\(157\) 7.68096 0.613007 0.306504 0.951870i \(-0.400841\pi\)
0.306504 + 0.951870i \(0.400841\pi\)
\(158\) 0 0
\(159\) 8.75771 0.694531
\(160\) 0 0
\(161\) 24.7562 1.95107
\(162\) 0 0
\(163\) 5.21691 0.408620 0.204310 0.978906i \(-0.434505\pi\)
0.204310 + 0.978906i \(0.434505\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.2767 1.49168 0.745838 0.666127i \(-0.232049\pi\)
0.745838 + 0.666127i \(0.232049\pi\)
\(168\) 0 0
\(169\) −12.6381 −0.972165
\(170\) 0 0
\(171\) −6.25548 −0.478369
\(172\) 0 0
\(173\) 19.6210 1.49175 0.745877 0.666084i \(-0.232031\pi\)
0.745877 + 0.666084i \(0.232031\pi\)
\(174\) 0 0
\(175\) −4.78768 −0.361914
\(176\) 0 0
\(177\) −27.3391 −2.05493
\(178\) 0 0
\(179\) 7.26184 0.542775 0.271388 0.962470i \(-0.412517\pi\)
0.271388 + 0.962470i \(0.412517\pi\)
\(180\) 0 0
\(181\) −8.78809 −0.653214 −0.326607 0.945160i \(-0.605905\pi\)
−0.326607 + 0.945160i \(0.605905\pi\)
\(182\) 0 0
\(183\) 29.3977 2.17314
\(184\) 0 0
\(185\) 9.39816 0.690967
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 6.62778 0.482100
\(190\) 0 0
\(191\) −24.9236 −1.80341 −0.901703 0.432355i \(-0.857683\pi\)
−0.901703 + 0.432355i \(0.857683\pi\)
\(192\) 0 0
\(193\) −6.66623 −0.479845 −0.239923 0.970792i \(-0.577122\pi\)
−0.239923 + 0.970792i \(0.577122\pi\)
\(194\) 0 0
\(195\) −1.39845 −0.100145
\(196\) 0 0
\(197\) −1.66329 −0.118504 −0.0592522 0.998243i \(-0.518872\pi\)
−0.0592522 + 0.998243i \(0.518872\pi\)
\(198\) 0 0
\(199\) 21.0008 1.48870 0.744352 0.667787i \(-0.232758\pi\)
0.744352 + 0.667787i \(0.232758\pi\)
\(200\) 0 0
\(201\) 34.9765 2.46705
\(202\) 0 0
\(203\) −26.0758 −1.83016
\(204\) 0 0
\(205\) −4.54633 −0.317530
\(206\) 0 0
\(207\) −12.4334 −0.864180
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −11.3170 −0.779095 −0.389548 0.921006i \(-0.627369\pi\)
−0.389548 + 0.921006i \(0.627369\pi\)
\(212\) 0 0
\(213\) −0.805506 −0.0551924
\(214\) 0 0
\(215\) 3.48547 0.237707
\(216\) 0 0
\(217\) −50.0683 −3.39886
\(218\) 0 0
\(219\) 8.55735 0.578252
\(220\) 0 0
\(221\) 1.59381 0.107212
\(222\) 0 0
\(223\) −0.746956 −0.0500198 −0.0250099 0.999687i \(-0.507962\pi\)
−0.0250099 + 0.999687i \(0.507962\pi\)
\(224\) 0 0
\(225\) 2.40452 0.160302
\(226\) 0 0
\(227\) 13.5057 0.896405 0.448203 0.893932i \(-0.352064\pi\)
0.448203 + 0.893932i \(0.352064\pi\)
\(228\) 0 0
\(229\) 2.36737 0.156440 0.0782201 0.996936i \(-0.475076\pi\)
0.0782201 + 0.996936i \(0.475076\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.15499 −0.141178 −0.0705892 0.997505i \(-0.522488\pi\)
−0.0705892 + 0.997505i \(0.522488\pi\)
\(234\) 0 0
\(235\) −5.07340 −0.330952
\(236\) 0 0
\(237\) 18.7218 1.21611
\(238\) 0 0
\(239\) −28.5539 −1.84700 −0.923500 0.383600i \(-0.874684\pi\)
−0.923500 + 0.383600i \(0.874684\pi\)
\(240\) 0 0
\(241\) 5.62289 0.362202 0.181101 0.983465i \(-0.442034\pi\)
0.181101 + 0.983465i \(0.442034\pi\)
\(242\) 0 0
\(243\) −20.0985 −1.28932
\(244\) 0 0
\(245\) 15.9218 1.01721
\(246\) 0 0
\(247\) 1.56495 0.0995755
\(248\) 0 0
\(249\) 30.2437 1.91661
\(250\) 0 0
\(251\) −20.5337 −1.29607 −0.648036 0.761609i \(-0.724410\pi\)
−0.648036 + 0.761609i \(0.724410\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −6.15952 −0.385724
\(256\) 0 0
\(257\) −15.7592 −0.983030 −0.491515 0.870869i \(-0.663557\pi\)
−0.491515 + 0.870869i \(0.663557\pi\)
\(258\) 0 0
\(259\) −44.9954 −2.79588
\(260\) 0 0
\(261\) 13.0961 0.810627
\(262\) 0 0
\(263\) −15.8194 −0.975464 −0.487732 0.872993i \(-0.662176\pi\)
−0.487732 + 0.872993i \(0.662176\pi\)
\(264\) 0 0
\(265\) 3.76714 0.231413
\(266\) 0 0
\(267\) 1.07565 0.0658287
\(268\) 0 0
\(269\) 18.1701 1.10785 0.553924 0.832567i \(-0.313130\pi\)
0.553924 + 0.832567i \(0.313130\pi\)
\(270\) 0 0
\(271\) −24.8599 −1.51013 −0.755065 0.655650i \(-0.772395\pi\)
−0.755065 + 0.655650i \(0.772395\pi\)
\(272\) 0 0
\(273\) 6.69534 0.405221
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 23.6352 1.42010 0.710050 0.704151i \(-0.248672\pi\)
0.710050 + 0.704151i \(0.248672\pi\)
\(278\) 0 0
\(279\) 25.1459 1.50544
\(280\) 0 0
\(281\) −6.70036 −0.399710 −0.199855 0.979825i \(-0.564047\pi\)
−0.199855 + 0.979825i \(0.564047\pi\)
\(282\) 0 0
\(283\) −7.11292 −0.422819 −0.211410 0.977398i \(-0.567805\pi\)
−0.211410 + 0.977398i \(0.567805\pi\)
\(284\) 0 0
\(285\) −6.04798 −0.358251
\(286\) 0 0
\(287\) 21.7664 1.28483
\(288\) 0 0
\(289\) −9.98001 −0.587059
\(290\) 0 0
\(291\) 18.5286 1.08617
\(292\) 0 0
\(293\) 27.4950 1.60628 0.803138 0.595793i \(-0.203162\pi\)
0.803138 + 0.595793i \(0.203162\pi\)
\(294\) 0 0
\(295\) −11.7599 −0.684690
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.11049 0.179885
\(300\) 0 0
\(301\) −16.6873 −0.961841
\(302\) 0 0
\(303\) −25.3710 −1.45752
\(304\) 0 0
\(305\) 12.6454 0.724076
\(306\) 0 0
\(307\) 23.0888 1.31775 0.658875 0.752253i \(-0.271033\pi\)
0.658875 + 0.752253i \(0.271033\pi\)
\(308\) 0 0
\(309\) 39.3073 2.23611
\(310\) 0 0
\(311\) 7.07508 0.401191 0.200595 0.979674i \(-0.435712\pi\)
0.200595 + 0.979674i \(0.435712\pi\)
\(312\) 0 0
\(313\) −28.1646 −1.59196 −0.795979 0.605325i \(-0.793043\pi\)
−0.795979 + 0.605325i \(0.793043\pi\)
\(314\) 0 0
\(315\) −11.5121 −0.648632
\(316\) 0 0
\(317\) 23.4408 1.31657 0.658284 0.752770i \(-0.271283\pi\)
0.658284 + 0.752770i \(0.271283\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −20.7841 −1.16005
\(322\) 0 0
\(323\) 6.89287 0.383529
\(324\) 0 0
\(325\) −0.601547 −0.0333678
\(326\) 0 0
\(327\) 20.3795 1.12699
\(328\) 0 0
\(329\) 24.2898 1.33914
\(330\) 0 0
\(331\) −5.08798 −0.279661 −0.139830 0.990175i \(-0.544656\pi\)
−0.139830 + 0.990175i \(0.544656\pi\)
\(332\) 0 0
\(333\) 22.5981 1.23837
\(334\) 0 0
\(335\) 15.0452 0.822006
\(336\) 0 0
\(337\) 25.4172 1.38456 0.692280 0.721629i \(-0.256606\pi\)
0.692280 + 0.721629i \(0.256606\pi\)
\(338\) 0 0
\(339\) −5.78867 −0.314398
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −42.7149 −2.30639
\(344\) 0 0
\(345\) −12.0210 −0.647186
\(346\) 0 0
\(347\) 17.8827 0.959995 0.479997 0.877270i \(-0.340638\pi\)
0.479997 + 0.877270i \(0.340638\pi\)
\(348\) 0 0
\(349\) 6.29515 0.336971 0.168486 0.985704i \(-0.446112\pi\)
0.168486 + 0.985704i \(0.446112\pi\)
\(350\) 0 0
\(351\) 0.832746 0.0444487
\(352\) 0 0
\(353\) 3.17636 0.169060 0.0845302 0.996421i \(-0.473061\pi\)
0.0845302 + 0.996421i \(0.473061\pi\)
\(354\) 0 0
\(355\) −0.346489 −0.0183898
\(356\) 0 0
\(357\) 29.4898 1.56077
\(358\) 0 0
\(359\) 5.12020 0.270234 0.135117 0.990830i \(-0.456859\pi\)
0.135117 + 0.990830i \(0.456859\pi\)
\(360\) 0 0
\(361\) −12.2320 −0.643787
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.68096 0.192670
\(366\) 0 0
\(367\) 10.8346 0.565564 0.282782 0.959184i \(-0.408743\pi\)
0.282782 + 0.959184i \(0.408743\pi\)
\(368\) 0 0
\(369\) −10.9318 −0.569085
\(370\) 0 0
\(371\) −18.0358 −0.936374
\(372\) 0 0
\(373\) 36.5605 1.89303 0.946516 0.322656i \(-0.104576\pi\)
0.946516 + 0.322656i \(0.104576\pi\)
\(374\) 0 0
\(375\) 2.32476 0.120050
\(376\) 0 0
\(377\) −3.27628 −0.168737
\(378\) 0 0
\(379\) −0.681538 −0.0350083 −0.0175041 0.999847i \(-0.505572\pi\)
−0.0175041 + 0.999847i \(0.505572\pi\)
\(380\) 0 0
\(381\) 21.4495 1.09889
\(382\) 0 0
\(383\) −36.6876 −1.87465 −0.937323 0.348461i \(-0.886705\pi\)
−0.937323 + 0.348461i \(0.886705\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.38090 0.426025
\(388\) 0 0
\(389\) 36.9171 1.87177 0.935887 0.352301i \(-0.114601\pi\)
0.935887 + 0.352301i \(0.114601\pi\)
\(390\) 0 0
\(391\) 13.7002 0.692851
\(392\) 0 0
\(393\) −18.8278 −0.949739
\(394\) 0 0
\(395\) 8.05322 0.405201
\(396\) 0 0
\(397\) −29.6085 −1.48601 −0.743003 0.669288i \(-0.766599\pi\)
−0.743003 + 0.669288i \(0.766599\pi\)
\(398\) 0 0
\(399\) 28.9558 1.44960
\(400\) 0 0
\(401\) −17.1919 −0.858523 −0.429262 0.903180i \(-0.641226\pi\)
−0.429262 + 0.903180i \(0.641226\pi\)
\(402\) 0 0
\(403\) −6.29082 −0.313368
\(404\) 0 0
\(405\) −10.4318 −0.518362
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 7.36760 0.364304 0.182152 0.983270i \(-0.441694\pi\)
0.182152 + 0.983270i \(0.441694\pi\)
\(410\) 0 0
\(411\) 15.2782 0.753619
\(412\) 0 0
\(413\) 56.3028 2.77048
\(414\) 0 0
\(415\) 13.0094 0.638604
\(416\) 0 0
\(417\) 8.39516 0.411113
\(418\) 0 0
\(419\) −19.3728 −0.946425 −0.473212 0.880948i \(-0.656906\pi\)
−0.473212 + 0.880948i \(0.656906\pi\)
\(420\) 0 0
\(421\) 17.2101 0.838769 0.419384 0.907809i \(-0.362246\pi\)
0.419384 + 0.907809i \(0.362246\pi\)
\(422\) 0 0
\(423\) −12.1991 −0.593141
\(424\) 0 0
\(425\) −2.64953 −0.128521
\(426\) 0 0
\(427\) −60.5423 −2.92985
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −30.8744 −1.48717 −0.743584 0.668643i \(-0.766876\pi\)
−0.743584 + 0.668643i \(0.766876\pi\)
\(432\) 0 0
\(433\) 2.48737 0.119536 0.0597678 0.998212i \(-0.480964\pi\)
0.0597678 + 0.998212i \(0.480964\pi\)
\(434\) 0 0
\(435\) 12.6617 0.607080
\(436\) 0 0
\(437\) 13.4522 0.643504
\(438\) 0 0
\(439\) 14.6203 0.697790 0.348895 0.937162i \(-0.386557\pi\)
0.348895 + 0.937162i \(0.386557\pi\)
\(440\) 0 0
\(441\) 38.2844 1.82307
\(442\) 0 0
\(443\) 32.0885 1.52457 0.762285 0.647241i \(-0.224077\pi\)
0.762285 + 0.647241i \(0.224077\pi\)
\(444\) 0 0
\(445\) 0.462692 0.0219337
\(446\) 0 0
\(447\) 28.2890 1.33802
\(448\) 0 0
\(449\) 11.2208 0.529544 0.264772 0.964311i \(-0.414703\pi\)
0.264772 + 0.964311i \(0.414703\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −50.1753 −2.35744
\(454\) 0 0
\(455\) 2.88001 0.135017
\(456\) 0 0
\(457\) 7.28861 0.340947 0.170473 0.985362i \(-0.445470\pi\)
0.170473 + 0.985362i \(0.445470\pi\)
\(458\) 0 0
\(459\) 3.66785 0.171200
\(460\) 0 0
\(461\) −8.56686 −0.398999 −0.199499 0.979898i \(-0.563932\pi\)
−0.199499 + 0.979898i \(0.563932\pi\)
\(462\) 0 0
\(463\) 22.6373 1.05204 0.526022 0.850471i \(-0.323683\pi\)
0.526022 + 0.850471i \(0.323683\pi\)
\(464\) 0 0
\(465\) 24.3118 1.12743
\(466\) 0 0
\(467\) 31.1970 1.44363 0.721813 0.692089i \(-0.243309\pi\)
0.721813 + 0.692089i \(0.243309\pi\)
\(468\) 0 0
\(469\) −72.0314 −3.32610
\(470\) 0 0
\(471\) 17.8564 0.822780
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −2.60155 −0.119367
\(476\) 0 0
\(477\) 9.05817 0.414745
\(478\) 0 0
\(479\) −33.1292 −1.51371 −0.756856 0.653582i \(-0.773265\pi\)
−0.756856 + 0.653582i \(0.773265\pi\)
\(480\) 0 0
\(481\) −5.65343 −0.257774
\(482\) 0 0
\(483\) 57.5524 2.61873
\(484\) 0 0
\(485\) 7.97011 0.361904
\(486\) 0 0
\(487\) 6.89985 0.312662 0.156331 0.987705i \(-0.450033\pi\)
0.156331 + 0.987705i \(0.450033\pi\)
\(488\) 0 0
\(489\) 12.1281 0.548451
\(490\) 0 0
\(491\) −3.53178 −0.159387 −0.0796935 0.996819i \(-0.525394\pi\)
−0.0796935 + 0.996819i \(0.525394\pi\)
\(492\) 0 0
\(493\) −14.4305 −0.649916
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.65888 0.0744109
\(498\) 0 0
\(499\) 4.67978 0.209496 0.104748 0.994499i \(-0.466596\pi\)
0.104748 + 0.994499i \(0.466596\pi\)
\(500\) 0 0
\(501\) 44.8138 2.00213
\(502\) 0 0
\(503\) −35.5622 −1.58564 −0.792821 0.609455i \(-0.791388\pi\)
−0.792821 + 0.609455i \(0.791388\pi\)
\(504\) 0 0
\(505\) −10.9134 −0.485638
\(506\) 0 0
\(507\) −29.3807 −1.30484
\(508\) 0 0
\(509\) −10.7168 −0.475012 −0.237506 0.971386i \(-0.576330\pi\)
−0.237506 + 0.971386i \(0.576330\pi\)
\(510\) 0 0
\(511\) −17.6232 −0.779606
\(512\) 0 0
\(513\) 3.60143 0.159007
\(514\) 0 0
\(515\) 16.9081 0.745059
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 45.6141 2.00224
\(520\) 0 0
\(521\) −2.19519 −0.0961729 −0.0480865 0.998843i \(-0.515312\pi\)
−0.0480865 + 0.998843i \(0.515312\pi\)
\(522\) 0 0
\(523\) −2.21848 −0.0970075 −0.0485038 0.998823i \(-0.515445\pi\)
−0.0485038 + 0.998823i \(0.515445\pi\)
\(524\) 0 0
\(525\) −11.1302 −0.485762
\(526\) 0 0
\(527\) −27.7081 −1.20698
\(528\) 0 0
\(529\) 3.73746 0.162498
\(530\) 0 0
\(531\) −28.2771 −1.22712
\(532\) 0 0
\(533\) 2.73483 0.118459
\(534\) 0 0
\(535\) −8.94029 −0.386523
\(536\) 0 0
\(537\) 16.8821 0.728514
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 38.8481 1.67021 0.835106 0.550089i \(-0.185406\pi\)
0.835106 + 0.550089i \(0.185406\pi\)
\(542\) 0 0
\(543\) −20.4302 −0.876745
\(544\) 0 0
\(545\) 8.76627 0.375506
\(546\) 0 0
\(547\) 12.4573 0.532637 0.266319 0.963885i \(-0.414193\pi\)
0.266319 + 0.963885i \(0.414193\pi\)
\(548\) 0 0
\(549\) 30.4063 1.29771
\(550\) 0 0
\(551\) −14.1691 −0.603626
\(552\) 0 0
\(553\) −38.5562 −1.63958
\(554\) 0 0
\(555\) 21.8485 0.927417
\(556\) 0 0
\(557\) −33.9789 −1.43973 −0.719867 0.694112i \(-0.755797\pi\)
−0.719867 + 0.694112i \(0.755797\pi\)
\(558\) 0 0
\(559\) −2.09667 −0.0886799
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 23.9594 1.00977 0.504884 0.863187i \(-0.331535\pi\)
0.504884 + 0.863187i \(0.331535\pi\)
\(564\) 0 0
\(565\) −2.49000 −0.104755
\(566\) 0 0
\(567\) 49.9443 2.09746
\(568\) 0 0
\(569\) 7.52879 0.315623 0.157812 0.987469i \(-0.449556\pi\)
0.157812 + 0.987469i \(0.449556\pi\)
\(570\) 0 0
\(571\) −13.5353 −0.566433 −0.283216 0.959056i \(-0.591401\pi\)
−0.283216 + 0.959056i \(0.591401\pi\)
\(572\) 0 0
\(573\) −57.9414 −2.42054
\(574\) 0 0
\(575\) −5.17083 −0.215638
\(576\) 0 0
\(577\) 27.3683 1.13936 0.569678 0.821868i \(-0.307068\pi\)
0.569678 + 0.821868i \(0.307068\pi\)
\(578\) 0 0
\(579\) −15.4974 −0.644050
\(580\) 0 0
\(581\) −62.2846 −2.58400
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −1.44643 −0.0598027
\(586\) 0 0
\(587\) −17.1927 −0.709618 −0.354809 0.934939i \(-0.615454\pi\)
−0.354809 + 0.934939i \(0.615454\pi\)
\(588\) 0 0
\(589\) −27.2063 −1.12102
\(590\) 0 0
\(591\) −3.86675 −0.159057
\(592\) 0 0
\(593\) 38.4574 1.57925 0.789627 0.613587i \(-0.210274\pi\)
0.789627 + 0.613587i \(0.210274\pi\)
\(594\) 0 0
\(595\) 12.6851 0.520037
\(596\) 0 0
\(597\) 48.8218 1.99814
\(598\) 0 0
\(599\) 9.63410 0.393639 0.196819 0.980440i \(-0.436939\pi\)
0.196819 + 0.980440i \(0.436939\pi\)
\(600\) 0 0
\(601\) −20.0426 −0.817555 −0.408777 0.912634i \(-0.634045\pi\)
−0.408777 + 0.912634i \(0.634045\pi\)
\(602\) 0 0
\(603\) 36.1765 1.47322
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 17.4596 0.708662 0.354331 0.935120i \(-0.384709\pi\)
0.354331 + 0.935120i \(0.384709\pi\)
\(608\) 0 0
\(609\) −60.6200 −2.45644
\(610\) 0 0
\(611\) 3.05189 0.123466
\(612\) 0 0
\(613\) −20.9105 −0.844569 −0.422284 0.906463i \(-0.638772\pi\)
−0.422284 + 0.906463i \(0.638772\pi\)
\(614\) 0 0
\(615\) −10.5692 −0.426189
\(616\) 0 0
\(617\) −41.2590 −1.66103 −0.830513 0.557000i \(-0.811952\pi\)
−0.830513 + 0.557000i \(0.811952\pi\)
\(618\) 0 0
\(619\) −0.151406 −0.00608551 −0.00304276 0.999995i \(-0.500969\pi\)
−0.00304276 + 0.999995i \(0.500969\pi\)
\(620\) 0 0
\(621\) 7.15819 0.287248
\(622\) 0 0
\(623\) −2.21522 −0.0887510
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −24.9007 −0.992855
\(630\) 0 0
\(631\) −9.21832 −0.366976 −0.183488 0.983022i \(-0.558739\pi\)
−0.183488 + 0.983022i \(0.558739\pi\)
\(632\) 0 0
\(633\) −26.3094 −1.04570
\(634\) 0 0
\(635\) 9.22655 0.366145
\(636\) 0 0
\(637\) −9.57773 −0.379483
\(638\) 0 0
\(639\) −0.833142 −0.0329586
\(640\) 0 0
\(641\) −19.8665 −0.784678 −0.392339 0.919821i \(-0.628334\pi\)
−0.392339 + 0.919821i \(0.628334\pi\)
\(642\) 0 0
\(643\) 17.8982 0.705838 0.352919 0.935654i \(-0.385189\pi\)
0.352919 + 0.935654i \(0.385189\pi\)
\(644\) 0 0
\(645\) 8.10290 0.319051
\(646\) 0 0
\(647\) 22.6373 0.889964 0.444982 0.895539i \(-0.353210\pi\)
0.444982 + 0.895539i \(0.353210\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −116.397 −4.56195
\(652\) 0 0
\(653\) 11.1498 0.436325 0.218163 0.975912i \(-0.429994\pi\)
0.218163 + 0.975912i \(0.429994\pi\)
\(654\) 0 0
\(655\) −8.09882 −0.316447
\(656\) 0 0
\(657\) 8.85095 0.345308
\(658\) 0 0
\(659\) 18.0304 0.702366 0.351183 0.936307i \(-0.385779\pi\)
0.351183 + 0.936307i \(0.385779\pi\)
\(660\) 0 0
\(661\) −16.1373 −0.627668 −0.313834 0.949478i \(-0.601613\pi\)
−0.313834 + 0.949478i \(0.601613\pi\)
\(662\) 0 0
\(663\) 3.70524 0.143900
\(664\) 0 0
\(665\) 12.4554 0.482998
\(666\) 0 0
\(667\) −28.1626 −1.09046
\(668\) 0 0
\(669\) −1.73649 −0.0671368
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −35.5210 −1.36923 −0.684617 0.728903i \(-0.740031\pi\)
−0.684617 + 0.728903i \(0.740031\pi\)
\(674\) 0 0
\(675\) −1.38434 −0.0532833
\(676\) 0 0
\(677\) −32.4138 −1.24576 −0.622882 0.782315i \(-0.714039\pi\)
−0.622882 + 0.782315i \(0.714039\pi\)
\(678\) 0 0
\(679\) −38.1583 −1.46438
\(680\) 0 0
\(681\) 31.3976 1.20316
\(682\) 0 0
\(683\) −7.36687 −0.281885 −0.140943 0.990018i \(-0.545013\pi\)
−0.140943 + 0.990018i \(0.545013\pi\)
\(684\) 0 0
\(685\) 6.57195 0.251101
\(686\) 0 0
\(687\) 5.50357 0.209974
\(688\) 0 0
\(689\) −2.26611 −0.0863319
\(690\) 0 0
\(691\) 0.768429 0.0292324 0.0146162 0.999893i \(-0.495347\pi\)
0.0146162 + 0.999893i \(0.495347\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.61119 0.136980
\(696\) 0 0
\(697\) 12.0456 0.456261
\(698\) 0 0
\(699\) −5.00985 −0.189490
\(700\) 0 0
\(701\) −7.95905 −0.300609 −0.150305 0.988640i \(-0.548025\pi\)
−0.150305 + 0.988640i \(0.548025\pi\)
\(702\) 0 0
\(703\) −24.4498 −0.922140
\(704\) 0 0
\(705\) −11.7945 −0.444205
\(706\) 0 0
\(707\) 52.2496 1.96505
\(708\) 0 0
\(709\) 15.3307 0.575755 0.287877 0.957667i \(-0.407050\pi\)
0.287877 + 0.957667i \(0.407050\pi\)
\(710\) 0 0
\(711\) 19.3642 0.726212
\(712\) 0 0
\(713\) −54.0752 −2.02513
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −66.3811 −2.47905
\(718\) 0 0
\(719\) −46.2627 −1.72531 −0.862654 0.505794i \(-0.831200\pi\)
−0.862654 + 0.505794i \(0.831200\pi\)
\(720\) 0 0
\(721\) −80.9504 −3.01475
\(722\) 0 0
\(723\) 13.0719 0.486148
\(724\) 0 0
\(725\) 5.44643 0.202275
\(726\) 0 0
\(727\) −24.0817 −0.893140 −0.446570 0.894749i \(-0.647355\pi\)
−0.446570 + 0.894749i \(0.647355\pi\)
\(728\) 0 0
\(729\) −15.4288 −0.571437
\(730\) 0 0
\(731\) −9.23485 −0.341563
\(732\) 0 0
\(733\) −13.0668 −0.482632 −0.241316 0.970447i \(-0.577579\pi\)
−0.241316 + 0.970447i \(0.577579\pi\)
\(734\) 0 0
\(735\) 37.0145 1.36530
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 33.5963 1.23586 0.617930 0.786233i \(-0.287972\pi\)
0.617930 + 0.786233i \(0.287972\pi\)
\(740\) 0 0
\(741\) 3.63814 0.133650
\(742\) 0 0
\(743\) 27.1661 0.996626 0.498313 0.866997i \(-0.333953\pi\)
0.498313 + 0.866997i \(0.333953\pi\)
\(744\) 0 0
\(745\) 12.1685 0.445821
\(746\) 0 0
\(747\) 31.2813 1.14452
\(748\) 0 0
\(749\) 42.8032 1.56400
\(750\) 0 0
\(751\) −13.5581 −0.494742 −0.247371 0.968921i \(-0.579567\pi\)
−0.247371 + 0.968921i \(0.579567\pi\)
\(752\) 0 0
\(753\) −47.7359 −1.73959
\(754\) 0 0
\(755\) −21.5830 −0.785485
\(756\) 0 0
\(757\) 32.1498 1.16850 0.584252 0.811572i \(-0.301388\pi\)
0.584252 + 0.811572i \(0.301388\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.70988 0.351983 0.175991 0.984392i \(-0.443687\pi\)
0.175991 + 0.984392i \(0.443687\pi\)
\(762\) 0 0
\(763\) −41.9701 −1.51942
\(764\) 0 0
\(765\) −6.37085 −0.230339
\(766\) 0 0
\(767\) 7.07415 0.255433
\(768\) 0 0
\(769\) 15.6081 0.562841 0.281421 0.959584i \(-0.409194\pi\)
0.281421 + 0.959584i \(0.409194\pi\)
\(770\) 0 0
\(771\) −36.6363 −1.31943
\(772\) 0 0
\(773\) −21.4874 −0.772849 −0.386425 0.922321i \(-0.626290\pi\)
−0.386425 + 0.922321i \(0.626290\pi\)
\(774\) 0 0
\(775\) 10.4577 0.375653
\(776\) 0 0
\(777\) −104.604 −3.75263
\(778\) 0 0
\(779\) 11.8275 0.423764
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −7.53972 −0.269448
\(784\) 0 0
\(785\) 7.68096 0.274145
\(786\) 0 0
\(787\) −6.04684 −0.215547 −0.107773 0.994175i \(-0.534372\pi\)
−0.107773 + 0.994175i \(0.534372\pi\)
\(788\) 0 0
\(789\) −36.7763 −1.30927
\(790\) 0 0
\(791\) 11.9213 0.423874
\(792\) 0 0
\(793\) −7.60682 −0.270126
\(794\) 0 0
\(795\) 8.75771 0.310604
\(796\) 0 0
\(797\) 32.0918 1.13675 0.568376 0.822769i \(-0.307572\pi\)
0.568376 + 0.822769i \(0.307572\pi\)
\(798\) 0 0
\(799\) 13.4421 0.475547
\(800\) 0 0
\(801\) 1.11255 0.0393102
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 24.7562 0.872543
\(806\) 0 0
\(807\) 42.2411 1.48696
\(808\) 0 0
\(809\) −2.20394 −0.0774865 −0.0387432 0.999249i \(-0.512335\pi\)
−0.0387432 + 0.999249i \(0.512335\pi\)
\(810\) 0 0
\(811\) 9.59190 0.336817 0.168409 0.985717i \(-0.446137\pi\)
0.168409 + 0.985717i \(0.446137\pi\)
\(812\) 0 0
\(813\) −57.7933 −2.02690
\(814\) 0 0
\(815\) 5.21691 0.182741
\(816\) 0 0
\(817\) −9.06762 −0.317236
\(818\) 0 0
\(819\) 6.92505 0.241981
\(820\) 0 0
\(821\) 14.2376 0.496894 0.248447 0.968645i \(-0.420080\pi\)
0.248447 + 0.968645i \(0.420080\pi\)
\(822\) 0 0
\(823\) 0.430172 0.0149949 0.00749743 0.999972i \(-0.497613\pi\)
0.00749743 + 0.999972i \(0.497613\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −35.6560 −1.23988 −0.619940 0.784649i \(-0.712843\pi\)
−0.619940 + 0.784649i \(0.712843\pi\)
\(828\) 0 0
\(829\) 10.9221 0.379339 0.189669 0.981848i \(-0.439258\pi\)
0.189669 + 0.981848i \(0.439258\pi\)
\(830\) 0 0
\(831\) 54.9462 1.90606
\(832\) 0 0
\(833\) −42.1853 −1.46164
\(834\) 0 0
\(835\) 19.2767 0.667098
\(836\) 0 0
\(837\) −14.4771 −0.500401
\(838\) 0 0
\(839\) −30.0181 −1.03634 −0.518169 0.855278i \(-0.673386\pi\)
−0.518169 + 0.855278i \(0.673386\pi\)
\(840\) 0 0
\(841\) 0.663634 0.0228839
\(842\) 0 0
\(843\) −15.5768 −0.536492
\(844\) 0 0
\(845\) −12.6381 −0.434765
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −16.5359 −0.567509
\(850\) 0 0
\(851\) −48.5963 −1.66586
\(852\) 0 0
\(853\) 20.5878 0.704912 0.352456 0.935828i \(-0.385347\pi\)
0.352456 + 0.935828i \(0.385347\pi\)
\(854\) 0 0
\(855\) −6.25548 −0.213933
\(856\) 0 0
\(857\) −7.44802 −0.254419 −0.127210 0.991876i \(-0.540602\pi\)
−0.127210 + 0.991876i \(0.540602\pi\)
\(858\) 0 0
\(859\) 53.0056 1.80853 0.904263 0.426975i \(-0.140421\pi\)
0.904263 + 0.426975i \(0.140421\pi\)
\(860\) 0 0
\(861\) 50.6017 1.72450
\(862\) 0 0
\(863\) 29.1040 0.990713 0.495356 0.868690i \(-0.335037\pi\)
0.495356 + 0.868690i \(0.335037\pi\)
\(864\) 0 0
\(865\) 19.6210 0.667132
\(866\) 0 0
\(867\) −23.2012 −0.787953
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −9.05037 −0.306660
\(872\) 0 0
\(873\) 19.1643 0.648614
\(874\) 0 0
\(875\) −4.78768 −0.161853
\(876\) 0 0
\(877\) −28.1013 −0.948912 −0.474456 0.880279i \(-0.657355\pi\)
−0.474456 + 0.880279i \(0.657355\pi\)
\(878\) 0 0
\(879\) 63.9194 2.15595
\(880\) 0 0
\(881\) 29.5508 0.995591 0.497796 0.867294i \(-0.334143\pi\)
0.497796 + 0.867294i \(0.334143\pi\)
\(882\) 0 0
\(883\) −38.9274 −1.31001 −0.655005 0.755624i \(-0.727334\pi\)
−0.655005 + 0.755624i \(0.727334\pi\)
\(884\) 0 0
\(885\) −27.3391 −0.918993
\(886\) 0 0
\(887\) 26.2960 0.882933 0.441467 0.897278i \(-0.354458\pi\)
0.441467 + 0.897278i \(0.354458\pi\)
\(888\) 0 0
\(889\) −44.1737 −1.48154
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 13.1987 0.441677
\(894\) 0 0
\(895\) 7.26184 0.242736
\(896\) 0 0
\(897\) 7.23116 0.241441
\(898\) 0 0
\(899\) 56.9574 1.89964
\(900\) 0 0
\(901\) −9.98113 −0.332520
\(902\) 0 0
\(903\) −38.7941 −1.29099
\(904\) 0 0
\(905\) −8.78809 −0.292126
\(906\) 0 0
\(907\) −25.5540 −0.848505 −0.424253 0.905544i \(-0.639463\pi\)
−0.424253 + 0.905544i \(0.639463\pi\)
\(908\) 0 0
\(909\) −26.2414 −0.870373
\(910\) 0 0
\(911\) 48.0753 1.59281 0.796403 0.604766i \(-0.206733\pi\)
0.796403 + 0.604766i \(0.206733\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 29.3977 0.971857
\(916\) 0 0
\(917\) 38.7745 1.28045
\(918\) 0 0
\(919\) −8.02354 −0.264672 −0.132336 0.991205i \(-0.542248\pi\)
−0.132336 + 0.991205i \(0.542248\pi\)
\(920\) 0 0
\(921\) 53.6761 1.76869
\(922\) 0 0
\(923\) 0.208430 0.00686054
\(924\) 0 0
\(925\) 9.39816 0.309010
\(926\) 0 0
\(927\) 40.6559 1.33531
\(928\) 0 0
\(929\) 47.7554 1.56680 0.783402 0.621516i \(-0.213483\pi\)
0.783402 + 0.621516i \(0.213483\pi\)
\(930\) 0 0
\(931\) −41.4214 −1.35753
\(932\) 0 0
\(933\) 16.4479 0.538480
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −13.5515 −0.442709 −0.221354 0.975193i \(-0.571048\pi\)
−0.221354 + 0.975193i \(0.571048\pi\)
\(938\) 0 0
\(939\) −65.4760 −2.13673
\(940\) 0 0
\(941\) −11.7661 −0.383564 −0.191782 0.981438i \(-0.561427\pi\)
−0.191782 + 0.981438i \(0.561427\pi\)
\(942\) 0 0
\(943\) 23.5083 0.765536
\(944\) 0 0
\(945\) 6.62778 0.215602
\(946\) 0 0
\(947\) −7.85540 −0.255266 −0.127633 0.991821i \(-0.540738\pi\)
−0.127633 + 0.991821i \(0.540738\pi\)
\(948\) 0 0
\(949\) −2.21427 −0.0718782
\(950\) 0 0
\(951\) 54.4944 1.76710
\(952\) 0 0
\(953\) 35.7034 1.15655 0.578274 0.815843i \(-0.303726\pi\)
0.578274 + 0.815843i \(0.303726\pi\)
\(954\) 0 0
\(955\) −24.9236 −0.806508
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −31.4644 −1.01604
\(960\) 0 0
\(961\) 78.3643 2.52788
\(962\) 0 0
\(963\) −21.4971 −0.692736
\(964\) 0 0
\(965\) −6.66623 −0.214593
\(966\) 0 0
\(967\) 12.5681 0.404162 0.202081 0.979369i \(-0.435230\pi\)
0.202081 + 0.979369i \(0.435230\pi\)
\(968\) 0 0
\(969\) 16.0243 0.514774
\(970\) 0 0
\(971\) −1.33283 −0.0427724 −0.0213862 0.999771i \(-0.506808\pi\)
−0.0213862 + 0.999771i \(0.506808\pi\)
\(972\) 0 0
\(973\) −17.2892 −0.554267
\(974\) 0 0
\(975\) −1.39845 −0.0447864
\(976\) 0 0
\(977\) −44.0415 −1.40901 −0.704506 0.709698i \(-0.748831\pi\)
−0.704506 + 0.709698i \(0.748831\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 21.0787 0.672991
\(982\) 0 0
\(983\) 13.3425 0.425560 0.212780 0.977100i \(-0.431748\pi\)
0.212780 + 0.977100i \(0.431748\pi\)
\(984\) 0 0
\(985\) −1.66329 −0.0529968
\(986\) 0 0
\(987\) 56.4680 1.79740
\(988\) 0 0
\(989\) −18.0228 −0.573091
\(990\) 0 0
\(991\) −8.69302 −0.276143 −0.138071 0.990422i \(-0.544090\pi\)
−0.138071 + 0.990422i \(0.544090\pi\)
\(992\) 0 0
\(993\) −11.8283 −0.375361
\(994\) 0 0
\(995\) 21.0008 0.665769
\(996\) 0 0
\(997\) −19.1952 −0.607918 −0.303959 0.952685i \(-0.598309\pi\)
−0.303959 + 0.952685i \(0.598309\pi\)
\(998\) 0 0
\(999\) −13.0103 −0.411626
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 9680.2.a.da.1.5 6
4.3 odd 2 4840.2.a.bd.1.2 yes 6
11.10 odd 2 9680.2.a.db.1.5 6
44.43 even 2 4840.2.a.bc.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.bc.1.2 6 44.43 even 2
4840.2.a.bd.1.2 yes 6 4.3 odd 2
9680.2.a.da.1.5 6 1.1 even 1 trivial
9680.2.a.db.1.5 6 11.10 odd 2