Properties

Label 4840.2.a.bd.1.2
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.821728\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-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 \(-0.734178\pi\)
−0.671101 + 0.741366i \(0.734178\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.78768 1.80957 0.904786 0.425867i \(-0.140031\pi\)
0.904786 + 0.425867i \(0.140031\pi\)
\(8\) 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 \(-0.403541\pi\)
0.298418 + 0.954435i \(0.403541\pi\)
\(20\) 0 0
\(21\) −11.1302 −2.42881
\(22\) 0 0
\(23\) 5.17083 1.07819 0.539096 0.842244i \(-0.318766\pi\)
0.539096 + 0.842244i \(0.318766\pi\)
\(24\) 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 \(-0.888369\pi\)
−0.939133 + 0.343554i \(0.888369\pi\)
\(32\) 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 \(-0.585624\pi\)
−0.265765 + 0.964038i \(0.585624\pi\)
\(44\) 0 0
\(45\) 2.40452 0.358445
\(46\) 0 0
\(47\) 5.07340 0.740031 0.370016 0.929026i \(-0.379352\pi\)
0.370016 + 0.929026i \(0.379352\pi\)
\(48\) 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 \(-0.222488\pi\)
0.765507 + 0.643428i \(0.222488\pi\)
\(60\) 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 \(-0.871026\pi\)
−0.919030 + 0.394187i \(0.871026\pi\)
\(68\) 0 0
\(69\) −12.0210 −1.44715
\(70\) 0 0
\(71\) 0.346489 0.0411207 0.0205604 0.999789i \(-0.493455\pi\)
0.0205604 + 0.999789i \(0.493455\pi\)
\(72\) 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 \(-0.649657\pi\)
−0.453029 + 0.891496i \(0.649657\pi\)
\(80\) 0 0
\(81\) −10.4318 −1.15909
\(82\) 0 0
\(83\) −13.0094 −1.42796 −0.713981 0.700165i \(-0.753110\pi\)
−0.713981 + 0.700165i \(0.753110\pi\)
\(84\) 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 \(-0.813379\pi\)
−0.833001 + 0.553271i \(0.813379\pi\)
\(104\) 0 0
\(105\) −11.1302 −1.08620
\(106\) 0 0
\(107\) 8.94029 0.864291 0.432145 0.901804i \(-0.357757\pi\)
0.432145 + 0.901804i \(0.357757\pi\)
\(108\) 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 \(-0.634249\pi\)
−0.409362 + 0.912372i \(0.634249\pi\)
\(128\) 0 0
\(129\) 8.10290 0.713421
\(130\) 0 0
\(131\) 8.09882 0.707597 0.353799 0.935322i \(-0.384890\pi\)
0.353799 + 0.935322i \(0.384890\pi\)
\(132\) 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 \(-0.548941\pi\)
−0.153149 + 0.988203i \(0.548941\pi\)
\(140\) 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 \(-0.158746\pi\)
0.878198 + 0.478296i \(0.158746\pi\)
\(152\) 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 \(-0.565495\pi\)
−0.204310 + 0.978906i \(0.565495\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19.2767 −1.49168 −0.745838 0.666127i \(-0.767951\pi\)
−0.745838 + 0.666127i \(0.767951\pi\)
\(168\) 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 \(-0.587483\pi\)
−0.271388 + 0.962470i \(0.587483\pi\)
\(180\) 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 \(-0.142317\pi\)
0.901703 + 0.432355i \(0.142317\pi\)
\(192\) 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 \(-0.767242\pi\)
−0.744352 + 0.667787i \(0.767242\pi\)
\(200\) 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 \(-0.372631\pi\)
0.389548 + 0.921006i \(0.372631\pi\)
\(212\) 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 \(-0.492038\pi\)
0.0250099 + 0.999687i \(0.492038\pi\)
\(224\) 0 0
\(225\) 2.40452 0.160302
\(226\) 0 0
\(227\) −13.5057 −0.896405 −0.448203 0.893932i \(-0.647936\pi\)
−0.448203 + 0.893932i \(0.647936\pi\)
\(228\) 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 \(-0.125316\pi\)
0.923500 + 0.383600i \(0.125316\pi\)
\(240\) 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 \(-0.275590\pi\)
0.648036 + 0.761609i \(0.275590\pi\)
\(252\) 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 \(-0.337824\pi\)
0.487732 + 0.872993i \(0.337824\pi\)
\(264\) 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 \(-0.227605\pi\)
0.755065 + 0.655650i \(0.227605\pi\)
\(272\) 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 \(-0.432195\pi\)
0.211410 + 0.977398i \(0.432195\pi\)
\(284\) 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 \(-0.728967\pi\)
−0.658875 + 0.752253i \(0.728967\pi\)
\(308\) 0 0
\(309\) 39.3073 2.23611
\(310\) 0 0
\(311\) −7.07508 −0.401191 −0.200595 0.979674i \(-0.564288\pi\)
−0.200595 + 0.979674i \(0.564288\pi\)
\(312\) 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 \(-0.455344\pi\)
0.139830 + 0.990175i \(0.455344\pi\)
\(332\) 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 \(-0.659362\pi\)
−0.479997 + 0.877270i \(0.659362\pi\)
\(348\) 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 \(-0.543141\pi\)
−0.135117 + 0.990830i \(0.543141\pi\)
\(360\) 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 \(-0.591257\pi\)
−0.282782 + 0.959184i \(0.591257\pi\)
\(368\) 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 \(-0.494428\pi\)
0.0175041 + 0.999847i \(0.494428\pi\)
\(380\) 0 0
\(381\) 21.4495 1.09889
\(382\) 0 0
\(383\) 36.6876 1.87465 0.937323 0.348461i \(-0.113295\pi\)
0.937323 + 0.348461i \(0.113295\pi\)
\(384\) 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 \(-0.343094\pi\)
0.473212 + 0.880948i \(0.343094\pi\)
\(420\) 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 \(-0.233124\pi\)
0.743584 + 0.668643i \(0.233124\pi\)
\(432\) 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 \(-0.613443\pi\)
−0.348895 + 0.937162i \(0.613443\pi\)
\(440\) 0 0
\(441\) 38.2844 1.82307
\(442\) 0 0
\(443\) −32.0885 −1.52457 −0.762285 0.647241i \(-0.775923\pi\)
−0.762285 + 0.647241i \(0.775923\pi\)
\(444\) 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 \(-0.676317\pi\)
−0.526022 + 0.850471i \(0.676317\pi\)
\(464\) 0 0
\(465\) 24.3118 1.12743
\(466\) 0 0
\(467\) −31.1970 −1.44363 −0.721813 0.692089i \(-0.756691\pi\)
−0.721813 + 0.692089i \(0.756691\pi\)
\(468\) 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 \(-0.226735\pi\)
0.756856 + 0.653582i \(0.226735\pi\)
\(480\) 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 \(-0.549967\pi\)
−0.156331 + 0.987705i \(0.549967\pi\)
\(488\) 0 0
\(489\) 12.1281 0.548451
\(490\) 0 0
\(491\) 3.53178 0.159387 0.0796935 0.996819i \(-0.474606\pi\)
0.0796935 + 0.996819i \(0.474606\pi\)
\(492\) 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 \(-0.533404\pi\)
−0.104748 + 0.994499i \(0.533404\pi\)
\(500\) 0 0
\(501\) 44.8138 2.00213
\(502\) 0 0
\(503\) 35.5622 1.58564 0.792821 0.609455i \(-0.208612\pi\)
0.792821 + 0.609455i \(0.208612\pi\)
\(504\) 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 \(-0.484555\pi\)
0.0485038 + 0.998823i \(0.484555\pi\)
\(524\) 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 \(-0.585807\pi\)
−0.266319 + 0.963885i \(0.585807\pi\)
\(548\) 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 \(-0.668465\pi\)
−0.504884 + 0.863187i \(0.668465\pi\)
\(564\) 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 \(-0.408599\pi\)
0.283216 + 0.959056i \(0.408599\pi\)
\(572\) 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 \(-0.384546\pi\)
0.354809 + 0.934939i \(0.384546\pi\)
\(588\) 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 \(-0.563061\pi\)
−0.196819 + 0.980440i \(0.563061\pi\)
\(600\) 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 \(-0.615291\pi\)
−0.354331 + 0.935120i \(0.615291\pi\)
\(608\) 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 \(-0.499031\pi\)
0.00304276 + 0.999995i \(0.499031\pi\)
\(620\) 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 \(-0.441261\pi\)
0.183488 + 0.983022i \(0.441261\pi\)
\(632\) 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 \(-0.614811\pi\)
−0.352919 + 0.935654i \(0.614811\pi\)
\(644\) 0 0
\(645\) 8.10290 0.319051
\(646\) 0 0
\(647\) −22.6373 −0.889964 −0.444982 0.895539i \(-0.646790\pi\)
−0.444982 + 0.895539i \(0.646790\pi\)
\(648\) 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 \(-0.614221\pi\)
−0.351183 + 0.936307i \(0.614221\pi\)
\(660\) 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 \(-0.454987\pi\)
0.140943 + 0.990018i \(0.454987\pi\)
\(684\) 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 \(-0.504653\pi\)
−0.0146162 + 0.999893i \(0.504653\pi\)
\(692\) 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 \(-0.168800\pi\)
0.862654 + 0.505794i \(0.168800\pi\)
\(720\) 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 \(-0.352645\pi\)
0.446570 + 0.894749i \(0.352645\pi\)
\(728\) 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 \(-0.712028\pi\)
−0.617930 + 0.786233i \(0.712028\pi\)
\(740\) 0 0
\(741\) 3.63814 0.133650
\(742\) 0 0
\(743\) −27.1661 −0.996626 −0.498313 0.866997i \(-0.666047\pi\)
−0.498313 + 0.866997i \(0.666047\pi\)
\(744\) 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 \(-0.420433\pi\)
0.247371 + 0.968921i \(0.420433\pi\)
\(752\) 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 \(-0.465628\pi\)
0.107773 + 0.994175i \(0.465628\pi\)
\(788\) 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 \(-0.553863\pi\)
−0.168409 + 0.985717i \(0.553863\pi\)
\(812\) 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 \(-0.502387\pi\)
−0.00749743 + 0.999972i \(0.502387\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 35.6560 1.23988 0.619940 0.784649i \(-0.287157\pi\)
0.619940 + 0.784649i \(0.287157\pi\)
\(828\) 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 \(-0.326614\pi\)
0.518169 + 0.855278i \(0.326614\pi\)
\(840\) 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 \(-0.859579\pi\)
−0.904263 + 0.426975i \(0.859579\pi\)
\(860\) 0 0
\(861\) 50.6017 1.72450
\(862\) 0 0
\(863\) −29.1040 −0.990713 −0.495356 0.868690i \(-0.664963\pi\)
−0.495356 + 0.868690i \(0.664963\pi\)
\(864\) 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 \(-0.272666\pi\)
0.655005 + 0.755624i \(0.272666\pi\)
\(884\) 0 0
\(885\) −27.3391 −0.918993
\(886\) 0 0
\(887\) −26.2960 −0.882933 −0.441467 0.897278i \(-0.645542\pi\)
−0.441467 + 0.897278i \(0.645542\pi\)
\(888\) 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 \(-0.360537\pi\)
0.424253 + 0.905544i \(0.360537\pi\)
\(908\) 0 0
\(909\) −26.2414 −0.870373
\(910\) 0 0
\(911\) −48.0753 −1.59281 −0.796403 0.604766i \(-0.793267\pi\)
−0.796403 + 0.604766i \(0.793267\pi\)
\(912\) 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 \(-0.457752\pi\)
0.132336 + 0.991205i \(0.457752\pi\)
\(920\) 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 \(-0.459262\pi\)
0.127633 + 0.991821i \(0.459262\pi\)
\(948\) 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 \(-0.564770\pi\)
−0.202081 + 0.979369i \(0.564770\pi\)
\(968\) 0 0
\(969\) 16.0243 0.514774
\(970\) 0 0
\(971\) 1.33283 0.0427724 0.0213862 0.999771i \(-0.493192\pi\)
0.0213862 + 0.999771i \(0.493192\pi\)
\(972\) 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 \(-0.568252\pi\)
−0.212780 + 0.977100i \(0.568252\pi\)
\(984\) 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 \(-0.455910\pi\)
0.138071 + 0.990422i \(0.455910\pi\)
\(992\) 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 4840.2.a.bd.1.2 yes 6
4.3 odd 2 9680.2.a.da.1.5 6
11.10 odd 2 4840.2.a.bc.1.2 6
44.43 even 2 9680.2.a.db.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.bc.1.2 6 11.10 odd 2
4840.2.a.bd.1.2 yes 6 1.1 even 1 trivial
9680.2.a.da.1.5 6 4.3 odd 2
9680.2.a.db.1.5 6 44.43 even 2