Properties

Label 4840.2.a.bc.1.2
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $1$
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] = [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: \(1\)
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 \(-0.859969\pi\)
−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 \(-0.473416\pi\)
0.0834195 + 0.996515i \(0.473416\pi\)
\(14\) 0 0
\(15\) −2.32476 −0.600251
\(16\) 0 0
\(17\) 2.64953 0.642605 0.321302 0.946977i \(-0.395879\pi\)
0.321302 + 0.946977i \(0.395879\pi\)
\(18\) 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 \(-0.318766\pi\)
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 \(-0.668762\pi\)
−0.505689 + 0.862716i \(0.668762\pi\)
\(30\) 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 \(-0.384478\pi\)
0.355009 + 0.934863i \(0.384478\pi\)
\(42\) 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 \(-0.379352\pi\)
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 \(-0.800284\pi\)
−0.809542 + 0.587063i \(0.800284\pi\)
\(62\) 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 \(-0.569109\pi\)
−0.215412 + 0.976523i \(0.569109\pi\)
\(74\) 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 \(-0.317304\pi\)
0.542960 + 0.839759i \(0.317304\pi\)
\(102\) 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 \(-0.642243\pi\)
−0.432145 + 0.901804i \(0.642243\pi\)
\(108\) 0 0
\(109\) −8.76627 −0.839656 −0.419828 0.907604i \(-0.637910\pi\)
−0.419828 + 0.907604i \(0.637910\pi\)
\(110\) 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 \(-0.666095\pi\)
−0.498443 + 0.866922i \(0.666095\pi\)
\(150\) 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 \(-0.565495\pi\)
−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 \(-0.232049\pi\)
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 \(-0.767969\pi\)
−0.745877 + 0.666084i \(0.767969\pi\)
\(174\) 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 \(-0.422878\pi\)
0.239923 + 0.970792i \(0.422878\pi\)
\(194\) 0 0
\(195\) −1.39845 −0.100145
\(196\) 0 0
\(197\) 1.66329 0.118504 0.0592522 0.998243i \(-0.481128\pi\)
0.0592522 + 0.998243i \(0.481128\pi\)
\(198\) 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 \(-0.627369\pi\)
−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 \(-0.492038\pi\)
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 \(-0.352064\pi\)
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 \(-0.477512\pi\)
0.0705892 + 0.997505i \(0.477512\pi\)
\(234\) 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 \(-0.557966\pi\)
−0.181101 + 0.983465i \(0.557966\pi\)
\(242\) 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 \(-0.662176\pi\)
−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 \(-0.751328\pi\)
−0.710050 + 0.704151i \(0.751328\pi\)
\(278\) 0 0
\(279\) −25.1459 −1.50544
\(280\) 0 0
\(281\) 6.70036 0.399710 0.199855 0.979825i \(-0.435953\pi\)
0.199855 + 0.979825i \(0.435953\pi\)
\(282\) 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 \(-0.796838\pi\)
−0.803138 + 0.595793i \(0.796838\pi\)
\(294\) 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 \(-0.564288\pi\)
−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 \(-0.743394\pi\)
−0.692280 + 0.721629i \(0.743394\pi\)
\(338\) 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 \(-0.553888\pi\)
−0.168486 + 0.985704i \(0.553888\pi\)
\(350\) 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 \(-0.591257\pi\)
−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 \(-0.895424\pi\)
−0.946516 + 0.322656i \(0.895424\pi\)
\(374\) 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 \(-0.558306\pi\)
−0.182152 + 0.983270i \(0.558306\pi\)
\(410\) 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 \(-0.766876\pi\)
−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 \(-0.775923\pi\)
−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 \(-0.554530\pi\)
−0.170473 + 0.985362i \(0.554530\pi\)
\(458\) 0 0
\(459\) 3.66785 0.171200
\(460\) 0 0
\(461\) 8.56686 0.398999 0.199499 0.979898i \(-0.436068\pi\)
0.199499 + 0.979898i \(0.436068\pi\)
\(462\) 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 \(-0.773265\pi\)
−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 \(-0.549967\pi\)
−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 \(-0.525394\pi\)
−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 \(-0.533404\pi\)
−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 \(-0.791388\pi\)
−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 \(-0.814594\pi\)
−0.835106 + 0.550089i \(0.814594\pi\)
\(542\) 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 \(-0.244203\pi\)
0.719867 + 0.694112i \(0.244203\pi\)
\(558\) 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 \(-0.550444\pi\)
−0.157812 + 0.987469i \(0.550444\pi\)
\(570\) 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 \(-0.384546\pi\)
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 \(-0.789726\pi\)
−0.789627 + 0.613587i \(0.789726\pi\)
\(594\) 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 \(-0.365955\pi\)
0.408777 + 0.912634i \(0.365955\pi\)
\(602\) 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 \(-0.361228\pi\)
0.422284 + 0.906463i \(0.361228\pi\)
\(614\) 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 \(-0.385779\pi\)
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 \(-0.259969\pi\)
0.684617 + 0.728903i \(0.259969\pi\)
\(674\) 0 0
\(675\) 1.38434 0.0532833
\(676\) 0 0
\(677\) 32.4138 1.24576 0.622882 0.782315i \(-0.285961\pi\)
0.622882 + 0.782315i \(0.285961\pi\)
\(678\) 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 \(-0.451975\pi\)
0.150305 + 0.988640i \(0.451975\pi\)
\(702\) 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 \(-0.422421\pi\)
0.241316 + 0.970447i \(0.422421\pi\)
\(734\) 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 \(-0.420433\pi\)
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 \(-0.556313\pi\)
−0.175991 + 0.984392i \(0.556313\pi\)
\(762\) 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 \(-0.590806\pi\)
−0.281421 + 0.959584i \(0.590806\pi\)
\(770\) 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 \(-0.487665\pi\)
0.0387432 + 0.999249i \(0.487665\pi\)
\(810\) 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 \(-0.579920\pi\)
−0.248447 + 0.968645i \(0.579920\pi\)
\(822\) 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 \(-0.712843\pi\)
−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 \(-0.326614\pi\)
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 \(-0.614653\pi\)
−0.352456 + 0.935828i \(0.614653\pi\)
\(854\) 0 0
\(855\) −6.25548 −0.213933
\(856\) 0 0
\(857\) 7.44802 0.254419 0.127210 0.991876i \(-0.459398\pi\)
0.127210 + 0.991876i \(0.459398\pi\)
\(858\) 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 \(-0.342645\pi\)
0.474456 + 0.880279i \(0.342645\pi\)
\(878\) 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 \(-0.354458\pi\)
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 \(-0.360537\pi\)
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 \(-0.542248\pi\)
−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 \(-0.428952\pi\)
0.221354 + 0.975193i \(0.428952\pi\)
\(938\) 0 0
\(939\) 65.4760 2.13673
\(940\) 0 0
\(941\) 11.7661 0.383564 0.191782 0.981438i \(-0.438573\pi\)
0.191782 + 0.981438i \(0.438573\pi\)
\(942\) 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 \(-0.696274\pi\)
−0.578274 + 0.815843i \(0.696274\pi\)
\(954\) 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 \(-0.493192\pi\)
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 \(-0.401691\pi\)
0.303959 + 0.952685i \(0.401691\pi\)
\(998\) 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.bc.1.2 6
4.3 odd 2 9680.2.a.db.1.5 6
11.10 odd 2 4840.2.a.bd.1.2 yes 6
44.43 even 2 9680.2.a.da.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.bc.1.2 6 1.1 even 1 trivial
4840.2.a.bd.1.2 yes 6 11.10 odd 2
9680.2.a.da.1.5 6 44.43 even 2
9680.2.a.db.1.5 6 4.3 odd 2