Properties

Label 7616.2.a.o.1.2
Level $7616$
Weight $2$
Character 7616.1
Self dual yes
Analytic conductor $60.814$
Analytic rank $1$
Dimension $2$
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] = [7616,2,Mod(1,7616)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7616, 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("7616.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7616.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: \(60.8140661794\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 476)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 7616.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.30278 q^{3} -2.30278 q^{5} +1.00000 q^{7} -1.30278 q^{9} +O(q^{10})\) \(q+1.30278 q^{3} -2.30278 q^{5} +1.00000 q^{7} -1.30278 q^{9} +4.00000 q^{11} -2.60555 q^{13} -3.00000 q^{15} -1.00000 q^{17} +1.39445 q^{19} +1.30278 q^{21} -4.00000 q^{23} +0.302776 q^{25} -5.60555 q^{27} +5.21110 q^{29} -3.69722 q^{31} +5.21110 q^{33} -2.30278 q^{35} +11.8167 q^{37} -3.39445 q^{39} +6.51388 q^{41} -0.697224 q^{43} +3.00000 q^{45} -4.60555 q^{47} +1.00000 q^{49} -1.30278 q^{51} -4.30278 q^{53} -9.21110 q^{55} +1.81665 q^{57} -8.00000 q^{59} +2.51388 q^{61} -1.30278 q^{63} +6.00000 q^{65} +6.30278 q^{67} -5.21110 q^{69} +10.6056 q^{71} -10.5139 q^{73} +0.394449 q^{75} +4.00000 q^{77} -4.60555 q^{79} -3.39445 q^{81} -11.2111 q^{83} +2.30278 q^{85} +6.78890 q^{87} -13.8167 q^{89} -2.60555 q^{91} -4.81665 q^{93} -3.21110 q^{95} -9.69722 q^{97} -5.21110 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - q^{5} + 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - q^{5} + 2 q^{7} + q^{9} + 8 q^{11} + 2 q^{13} - 6 q^{15} - 2 q^{17} + 10 q^{19} - q^{21} - 8 q^{23} - 3 q^{25} - 4 q^{27} - 4 q^{29} - 11 q^{31} - 4 q^{33} - q^{35} + 2 q^{37} - 14 q^{39} - 5 q^{41} - 5 q^{43} + 6 q^{45} - 2 q^{47} + 2 q^{49} + q^{51} - 5 q^{53} - 4 q^{55} - 18 q^{57} - 16 q^{59} - 13 q^{61} + q^{63} + 12 q^{65} + 9 q^{67} + 4 q^{69} + 14 q^{71} - 3 q^{73} + 8 q^{75} + 8 q^{77} - 2 q^{79} - 14 q^{81} - 8 q^{83} + q^{85} + 28 q^{87} - 6 q^{89} + 2 q^{91} + 12 q^{93} + 8 q^{95} - 23 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.30278 0.752158 0.376079 0.926588i \(-0.377272\pi\)
0.376079 + 0.926588i \(0.377272\pi\)
\(4\) 0 0
\(5\) −2.30278 −1.02983 −0.514916 0.857240i \(-0.672177\pi\)
−0.514916 + 0.857240i \(0.672177\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −1.30278 −0.434259
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −2.60555 −0.722650 −0.361325 0.932440i \(-0.617675\pi\)
−0.361325 + 0.932440i \(0.617675\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) 1.39445 0.319908 0.159954 0.987124i \(-0.448865\pi\)
0.159954 + 0.987124i \(0.448865\pi\)
\(20\) 0 0
\(21\) 1.30278 0.284289
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 0.302776 0.0605551
\(26\) 0 0
\(27\) −5.60555 −1.07879
\(28\) 0 0
\(29\) 5.21110 0.967677 0.483839 0.875157i \(-0.339242\pi\)
0.483839 + 0.875157i \(0.339242\pi\)
\(30\) 0 0
\(31\) −3.69722 −0.664041 −0.332021 0.943272i \(-0.607730\pi\)
−0.332021 + 0.943272i \(0.607730\pi\)
\(32\) 0 0
\(33\) 5.21110 0.907137
\(34\) 0 0
\(35\) −2.30278 −0.389240
\(36\) 0 0
\(37\) 11.8167 1.94265 0.971323 0.237764i \(-0.0764145\pi\)
0.971323 + 0.237764i \(0.0764145\pi\)
\(38\) 0 0
\(39\) −3.39445 −0.543547
\(40\) 0 0
\(41\) 6.51388 1.01730 0.508648 0.860974i \(-0.330145\pi\)
0.508648 + 0.860974i \(0.330145\pi\)
\(42\) 0 0
\(43\) −0.697224 −0.106326 −0.0531629 0.998586i \(-0.516930\pi\)
−0.0531629 + 0.998586i \(0.516930\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) −4.60555 −0.671789 −0.335894 0.941900i \(-0.609039\pi\)
−0.335894 + 0.941900i \(0.609039\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.30278 −0.182425
\(52\) 0 0
\(53\) −4.30278 −0.591032 −0.295516 0.955338i \(-0.595492\pi\)
−0.295516 + 0.955338i \(0.595492\pi\)
\(54\) 0 0
\(55\) −9.21110 −1.24202
\(56\) 0 0
\(57\) 1.81665 0.240622
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 2.51388 0.321869 0.160935 0.986965i \(-0.448549\pi\)
0.160935 + 0.986965i \(0.448549\pi\)
\(62\) 0 0
\(63\) −1.30278 −0.164134
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 6.30278 0.770007 0.385003 0.922915i \(-0.374200\pi\)
0.385003 + 0.922915i \(0.374200\pi\)
\(68\) 0 0
\(69\) −5.21110 −0.627343
\(70\) 0 0
\(71\) 10.6056 1.25865 0.629324 0.777143i \(-0.283332\pi\)
0.629324 + 0.777143i \(0.283332\pi\)
\(72\) 0 0
\(73\) −10.5139 −1.23056 −0.615278 0.788310i \(-0.710956\pi\)
−0.615278 + 0.788310i \(0.710956\pi\)
\(74\) 0 0
\(75\) 0.394449 0.0455470
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −4.60555 −0.518165 −0.259083 0.965855i \(-0.583420\pi\)
−0.259083 + 0.965855i \(0.583420\pi\)
\(80\) 0 0
\(81\) −3.39445 −0.377161
\(82\) 0 0
\(83\) −11.2111 −1.23058 −0.615289 0.788301i \(-0.710961\pi\)
−0.615289 + 0.788301i \(0.710961\pi\)
\(84\) 0 0
\(85\) 2.30278 0.249771
\(86\) 0 0
\(87\) 6.78890 0.727846
\(88\) 0 0
\(89\) −13.8167 −1.46456 −0.732281 0.681002i \(-0.761544\pi\)
−0.732281 + 0.681002i \(0.761544\pi\)
\(90\) 0 0
\(91\) −2.60555 −0.273136
\(92\) 0 0
\(93\) −4.81665 −0.499464
\(94\) 0 0
\(95\) −3.21110 −0.329452
\(96\) 0 0
\(97\) −9.69722 −0.984604 −0.492302 0.870424i \(-0.663844\pi\)
−0.492302 + 0.870424i \(0.663844\pi\)
\(98\) 0 0
\(99\) −5.21110 −0.523736
\(100\) 0 0
\(101\) 5.39445 0.536768 0.268384 0.963312i \(-0.413510\pi\)
0.268384 + 0.963312i \(0.413510\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) 2.60555 0.251888 0.125944 0.992037i \(-0.459804\pi\)
0.125944 + 0.992037i \(0.459804\pi\)
\(108\) 0 0
\(109\) −19.2111 −1.84009 −0.920045 0.391813i \(-0.871848\pi\)
−0.920045 + 0.391813i \(0.871848\pi\)
\(110\) 0 0
\(111\) 15.3944 1.46118
\(112\) 0 0
\(113\) −16.6056 −1.56212 −0.781059 0.624457i \(-0.785320\pi\)
−0.781059 + 0.624457i \(0.785320\pi\)
\(114\) 0 0
\(115\) 9.21110 0.858940
\(116\) 0 0
\(117\) 3.39445 0.313817
\(118\) 0 0
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 8.48612 0.765168
\(124\) 0 0
\(125\) 10.8167 0.967471
\(126\) 0 0
\(127\) 3.51388 0.311806 0.155903 0.987772i \(-0.450171\pi\)
0.155903 + 0.987772i \(0.450171\pi\)
\(128\) 0 0
\(129\) −0.908327 −0.0799737
\(130\) 0 0
\(131\) 5.21110 0.455296 0.227648 0.973743i \(-0.426896\pi\)
0.227648 + 0.973743i \(0.426896\pi\)
\(132\) 0 0
\(133\) 1.39445 0.120914
\(134\) 0 0
\(135\) 12.9083 1.11097
\(136\) 0 0
\(137\) −7.11943 −0.608254 −0.304127 0.952632i \(-0.598365\pi\)
−0.304127 + 0.952632i \(0.598365\pi\)
\(138\) 0 0
\(139\) −9.51388 −0.806957 −0.403478 0.914989i \(-0.632199\pi\)
−0.403478 + 0.914989i \(0.632199\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) −10.4222 −0.871549
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) 1.30278 0.107451
\(148\) 0 0
\(149\) 0.697224 0.0571188 0.0285594 0.999592i \(-0.490908\pi\)
0.0285594 + 0.999592i \(0.490908\pi\)
\(150\) 0 0
\(151\) −15.3028 −1.24532 −0.622661 0.782492i \(-0.713948\pi\)
−0.622661 + 0.782492i \(0.713948\pi\)
\(152\) 0 0
\(153\) 1.30278 0.105323
\(154\) 0 0
\(155\) 8.51388 0.683851
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) −5.60555 −0.444549
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 8.60555 0.674039 0.337019 0.941498i \(-0.390581\pi\)
0.337019 + 0.941498i \(0.390581\pi\)
\(164\) 0 0
\(165\) −12.0000 −0.934199
\(166\) 0 0
\(167\) 13.1194 1.01521 0.507606 0.861589i \(-0.330531\pi\)
0.507606 + 0.861589i \(0.330531\pi\)
\(168\) 0 0
\(169\) −6.21110 −0.477777
\(170\) 0 0
\(171\) −1.81665 −0.138923
\(172\) 0 0
\(173\) 11.7250 0.891434 0.445717 0.895174i \(-0.352949\pi\)
0.445717 + 0.895174i \(0.352949\pi\)
\(174\) 0 0
\(175\) 0.302776 0.0228877
\(176\) 0 0
\(177\) −10.4222 −0.783381
\(178\) 0 0
\(179\) −8.51388 −0.636357 −0.318179 0.948031i \(-0.603071\pi\)
−0.318179 + 0.948031i \(0.603071\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 3.27502 0.242096
\(184\) 0 0
\(185\) −27.2111 −2.00060
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) −5.60555 −0.407744
\(190\) 0 0
\(191\) −20.7250 −1.49961 −0.749803 0.661661i \(-0.769852\pi\)
−0.749803 + 0.661661i \(0.769852\pi\)
\(192\) 0 0
\(193\) 21.0278 1.51361 0.756806 0.653640i \(-0.226759\pi\)
0.756806 + 0.653640i \(0.226759\pi\)
\(194\) 0 0
\(195\) 7.81665 0.559762
\(196\) 0 0
\(197\) −0.605551 −0.0431437 −0.0215719 0.999767i \(-0.506867\pi\)
−0.0215719 + 0.999767i \(0.506867\pi\)
\(198\) 0 0
\(199\) −13.3028 −0.943009 −0.471504 0.881864i \(-0.656289\pi\)
−0.471504 + 0.881864i \(0.656289\pi\)
\(200\) 0 0
\(201\) 8.21110 0.579167
\(202\) 0 0
\(203\) 5.21110 0.365748
\(204\) 0 0
\(205\) −15.0000 −1.04765
\(206\) 0 0
\(207\) 5.21110 0.362197
\(208\) 0 0
\(209\) 5.57779 0.385824
\(210\) 0 0
\(211\) 27.8167 1.91498 0.957489 0.288471i \(-0.0931468\pi\)
0.957489 + 0.288471i \(0.0931468\pi\)
\(212\) 0 0
\(213\) 13.8167 0.946702
\(214\) 0 0
\(215\) 1.60555 0.109498
\(216\) 0 0
\(217\) −3.69722 −0.250984
\(218\) 0 0
\(219\) −13.6972 −0.925573
\(220\) 0 0
\(221\) 2.60555 0.175268
\(222\) 0 0
\(223\) 7.81665 0.523442 0.261721 0.965144i \(-0.415710\pi\)
0.261721 + 0.965144i \(0.415710\pi\)
\(224\) 0 0
\(225\) −0.394449 −0.0262966
\(226\) 0 0
\(227\) 12.9083 0.856756 0.428378 0.903600i \(-0.359085\pi\)
0.428378 + 0.903600i \(0.359085\pi\)
\(228\) 0 0
\(229\) −13.2111 −0.873014 −0.436507 0.899701i \(-0.643785\pi\)
−0.436507 + 0.899701i \(0.643785\pi\)
\(230\) 0 0
\(231\) 5.21110 0.342865
\(232\) 0 0
\(233\) 19.3944 1.27057 0.635286 0.772277i \(-0.280882\pi\)
0.635286 + 0.772277i \(0.280882\pi\)
\(234\) 0 0
\(235\) 10.6056 0.691830
\(236\) 0 0
\(237\) −6.00000 −0.389742
\(238\) 0 0
\(239\) 7.11943 0.460518 0.230259 0.973129i \(-0.426043\pi\)
0.230259 + 0.973129i \(0.426043\pi\)
\(240\) 0 0
\(241\) −8.48612 −0.546639 −0.273320 0.961923i \(-0.588122\pi\)
−0.273320 + 0.961923i \(0.588122\pi\)
\(242\) 0 0
\(243\) 12.3944 0.795104
\(244\) 0 0
\(245\) −2.30278 −0.147119
\(246\) 0 0
\(247\) −3.63331 −0.231182
\(248\) 0 0
\(249\) −14.6056 −0.925589
\(250\) 0 0
\(251\) −23.8167 −1.50329 −0.751647 0.659566i \(-0.770740\pi\)
−0.751647 + 0.659566i \(0.770740\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) 0 0
\(255\) 3.00000 0.187867
\(256\) 0 0
\(257\) 24.4222 1.52342 0.761708 0.647921i \(-0.224361\pi\)
0.761708 + 0.647921i \(0.224361\pi\)
\(258\) 0 0
\(259\) 11.8167 0.734251
\(260\) 0 0
\(261\) −6.78890 −0.420222
\(262\) 0 0
\(263\) −10.4222 −0.642661 −0.321330 0.946967i \(-0.604130\pi\)
−0.321330 + 0.946967i \(0.604130\pi\)
\(264\) 0 0
\(265\) 9.90833 0.608664
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 0 0
\(269\) −27.2111 −1.65909 −0.829545 0.558440i \(-0.811400\pi\)
−0.829545 + 0.558440i \(0.811400\pi\)
\(270\) 0 0
\(271\) −18.4222 −1.11907 −0.559535 0.828807i \(-0.689020\pi\)
−0.559535 + 0.828807i \(0.689020\pi\)
\(272\) 0 0
\(273\) −3.39445 −0.205441
\(274\) 0 0
\(275\) 1.21110 0.0730322
\(276\) 0 0
\(277\) −30.4222 −1.82789 −0.913947 0.405835i \(-0.866981\pi\)
−0.913947 + 0.405835i \(0.866981\pi\)
\(278\) 0 0
\(279\) 4.81665 0.288366
\(280\) 0 0
\(281\) 6.11943 0.365055 0.182527 0.983201i \(-0.441572\pi\)
0.182527 + 0.983201i \(0.441572\pi\)
\(282\) 0 0
\(283\) −19.3305 −1.14908 −0.574540 0.818476i \(-0.694819\pi\)
−0.574540 + 0.818476i \(0.694819\pi\)
\(284\) 0 0
\(285\) −4.18335 −0.247800
\(286\) 0 0
\(287\) 6.51388 0.384502
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −12.6333 −0.740578
\(292\) 0 0
\(293\) −3.21110 −0.187595 −0.0937973 0.995591i \(-0.529901\pi\)
−0.0937973 + 0.995591i \(0.529901\pi\)
\(294\) 0 0
\(295\) 18.4222 1.07258
\(296\) 0 0
\(297\) −22.4222 −1.30107
\(298\) 0 0
\(299\) 10.4222 0.602732
\(300\) 0 0
\(301\) −0.697224 −0.0401873
\(302\) 0 0
\(303\) 7.02776 0.403734
\(304\) 0 0
\(305\) −5.78890 −0.331471
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −18.2389 −1.03757
\(310\) 0 0
\(311\) −14.7250 −0.834977 −0.417489 0.908682i \(-0.637090\pi\)
−0.417489 + 0.908682i \(0.637090\pi\)
\(312\) 0 0
\(313\) −6.90833 −0.390482 −0.195241 0.980755i \(-0.562549\pi\)
−0.195241 + 0.980755i \(0.562549\pi\)
\(314\) 0 0
\(315\) 3.00000 0.169031
\(316\) 0 0
\(317\) −9.63331 −0.541060 −0.270530 0.962711i \(-0.587199\pi\)
−0.270530 + 0.962711i \(0.587199\pi\)
\(318\) 0 0
\(319\) 20.8444 1.16706
\(320\) 0 0
\(321\) 3.39445 0.189460
\(322\) 0 0
\(323\) −1.39445 −0.0775892
\(324\) 0 0
\(325\) −0.788897 −0.0437602
\(326\) 0 0
\(327\) −25.0278 −1.38404
\(328\) 0 0
\(329\) −4.60555 −0.253912
\(330\) 0 0
\(331\) −28.7250 −1.57887 −0.789434 0.613836i \(-0.789626\pi\)
−0.789434 + 0.613836i \(0.789626\pi\)
\(332\) 0 0
\(333\) −15.3944 −0.843611
\(334\) 0 0
\(335\) −14.5139 −0.792978
\(336\) 0 0
\(337\) 14.4222 0.785628 0.392814 0.919618i \(-0.371502\pi\)
0.392814 + 0.919618i \(0.371502\pi\)
\(338\) 0 0
\(339\) −21.6333 −1.17496
\(340\) 0 0
\(341\) −14.7889 −0.800864
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) 0 0
\(347\) −30.8444 −1.65581 −0.827907 0.560865i \(-0.810469\pi\)
−0.827907 + 0.560865i \(0.810469\pi\)
\(348\) 0 0
\(349\) 10.4222 0.557888 0.278944 0.960307i \(-0.410016\pi\)
0.278944 + 0.960307i \(0.410016\pi\)
\(350\) 0 0
\(351\) 14.6056 0.779587
\(352\) 0 0
\(353\) −16.0000 −0.851594 −0.425797 0.904819i \(-0.640006\pi\)
−0.425797 + 0.904819i \(0.640006\pi\)
\(354\) 0 0
\(355\) −24.4222 −1.29620
\(356\) 0 0
\(357\) −1.30278 −0.0689502
\(358\) 0 0
\(359\) 19.5139 1.02990 0.514952 0.857219i \(-0.327810\pi\)
0.514952 + 0.857219i \(0.327810\pi\)
\(360\) 0 0
\(361\) −17.0555 −0.897659
\(362\) 0 0
\(363\) 6.51388 0.341890
\(364\) 0 0
\(365\) 24.2111 1.26727
\(366\) 0 0
\(367\) 3.90833 0.204013 0.102007 0.994784i \(-0.467474\pi\)
0.102007 + 0.994784i \(0.467474\pi\)
\(368\) 0 0
\(369\) −8.48612 −0.441770
\(370\) 0 0
\(371\) −4.30278 −0.223389
\(372\) 0 0
\(373\) −24.3305 −1.25979 −0.629894 0.776681i \(-0.716902\pi\)
−0.629894 + 0.776681i \(0.716902\pi\)
\(374\) 0 0
\(375\) 14.0917 0.727691
\(376\) 0 0
\(377\) −13.5778 −0.699292
\(378\) 0 0
\(379\) 38.4222 1.97362 0.986808 0.161895i \(-0.0517605\pi\)
0.986808 + 0.161895i \(0.0517605\pi\)
\(380\) 0 0
\(381\) 4.57779 0.234528
\(382\) 0 0
\(383\) 30.4222 1.55450 0.777251 0.629191i \(-0.216614\pi\)
0.777251 + 0.629191i \(0.216614\pi\)
\(384\) 0 0
\(385\) −9.21110 −0.469441
\(386\) 0 0
\(387\) 0.908327 0.0461729
\(388\) 0 0
\(389\) −2.72498 −0.138162 −0.0690810 0.997611i \(-0.522007\pi\)
−0.0690810 + 0.997611i \(0.522007\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 6.78890 0.342455
\(394\) 0 0
\(395\) 10.6056 0.533623
\(396\) 0 0
\(397\) 22.9083 1.14974 0.574868 0.818246i \(-0.305053\pi\)
0.574868 + 0.818246i \(0.305053\pi\)
\(398\) 0 0
\(399\) 1.81665 0.0909464
\(400\) 0 0
\(401\) −4.60555 −0.229990 −0.114995 0.993366i \(-0.536685\pi\)
−0.114995 + 0.993366i \(0.536685\pi\)
\(402\) 0 0
\(403\) 9.63331 0.479869
\(404\) 0 0
\(405\) 7.81665 0.388413
\(406\) 0 0
\(407\) 47.2666 2.34292
\(408\) 0 0
\(409\) 15.8167 0.782083 0.391042 0.920373i \(-0.372115\pi\)
0.391042 + 0.920373i \(0.372115\pi\)
\(410\) 0 0
\(411\) −9.27502 −0.457503
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 25.8167 1.26729
\(416\) 0 0
\(417\) −12.3944 −0.606959
\(418\) 0 0
\(419\) 9.51388 0.464783 0.232392 0.972622i \(-0.425345\pi\)
0.232392 + 0.972622i \(0.425345\pi\)
\(420\) 0 0
\(421\) −21.9083 −1.06775 −0.533873 0.845565i \(-0.679264\pi\)
−0.533873 + 0.845565i \(0.679264\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) −0.302776 −0.0146868
\(426\) 0 0
\(427\) 2.51388 0.121655
\(428\) 0 0
\(429\) −13.5778 −0.655542
\(430\) 0 0
\(431\) −28.8444 −1.38939 −0.694693 0.719306i \(-0.744460\pi\)
−0.694693 + 0.719306i \(0.744460\pi\)
\(432\) 0 0
\(433\) −14.6056 −0.701898 −0.350949 0.936395i \(-0.614141\pi\)
−0.350949 + 0.936395i \(0.614141\pi\)
\(434\) 0 0
\(435\) −15.6333 −0.749560
\(436\) 0 0
\(437\) −5.57779 −0.266822
\(438\) 0 0
\(439\) −31.9083 −1.52290 −0.761451 0.648223i \(-0.775513\pi\)
−0.761451 + 0.648223i \(0.775513\pi\)
\(440\) 0 0
\(441\) −1.30278 −0.0620369
\(442\) 0 0
\(443\) −18.7889 −0.892687 −0.446344 0.894862i \(-0.647274\pi\)
−0.446344 + 0.894862i \(0.647274\pi\)
\(444\) 0 0
\(445\) 31.8167 1.50825
\(446\) 0 0
\(447\) 0.908327 0.0429624
\(448\) 0 0
\(449\) 31.0278 1.46429 0.732145 0.681149i \(-0.238519\pi\)
0.732145 + 0.681149i \(0.238519\pi\)
\(450\) 0 0
\(451\) 26.0555 1.22691
\(452\) 0 0
\(453\) −19.9361 −0.936679
\(454\) 0 0
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) −3.09167 −0.144622 −0.0723112 0.997382i \(-0.523037\pi\)
−0.0723112 + 0.997382i \(0.523037\pi\)
\(458\) 0 0
\(459\) 5.60555 0.261645
\(460\) 0 0
\(461\) −30.6056 −1.42544 −0.712721 0.701447i \(-0.752538\pi\)
−0.712721 + 0.701447i \(0.752538\pi\)
\(462\) 0 0
\(463\) −25.7250 −1.19554 −0.597771 0.801667i \(-0.703947\pi\)
−0.597771 + 0.801667i \(0.703947\pi\)
\(464\) 0 0
\(465\) 11.0917 0.514364
\(466\) 0 0
\(467\) −9.39445 −0.434723 −0.217362 0.976091i \(-0.569745\pi\)
−0.217362 + 0.976091i \(0.569745\pi\)
\(468\) 0 0
\(469\) 6.30278 0.291035
\(470\) 0 0
\(471\) −2.60555 −0.120057
\(472\) 0 0
\(473\) −2.78890 −0.128234
\(474\) 0 0
\(475\) 0.422205 0.0193721
\(476\) 0 0
\(477\) 5.60555 0.256661
\(478\) 0 0
\(479\) −27.9083 −1.27516 −0.637582 0.770382i \(-0.720065\pi\)
−0.637582 + 0.770382i \(0.720065\pi\)
\(480\) 0 0
\(481\) −30.7889 −1.40385
\(482\) 0 0
\(483\) −5.21110 −0.237113
\(484\) 0 0
\(485\) 22.3305 1.01398
\(486\) 0 0
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) 0 0
\(489\) 11.2111 0.506984
\(490\) 0 0
\(491\) −4.48612 −0.202456 −0.101228 0.994863i \(-0.532277\pi\)
−0.101228 + 0.994863i \(0.532277\pi\)
\(492\) 0 0
\(493\) −5.21110 −0.234696
\(494\) 0 0
\(495\) 12.0000 0.539360
\(496\) 0 0
\(497\) 10.6056 0.475724
\(498\) 0 0
\(499\) −14.7889 −0.662042 −0.331021 0.943623i \(-0.607393\pi\)
−0.331021 + 0.943623i \(0.607393\pi\)
\(500\) 0 0
\(501\) 17.0917 0.763600
\(502\) 0 0
\(503\) 5.88057 0.262202 0.131101 0.991369i \(-0.458149\pi\)
0.131101 + 0.991369i \(0.458149\pi\)
\(504\) 0 0
\(505\) −12.4222 −0.552781
\(506\) 0 0
\(507\) −8.09167 −0.359364
\(508\) 0 0
\(509\) 0.605551 0.0268406 0.0134203 0.999910i \(-0.495728\pi\)
0.0134203 + 0.999910i \(0.495728\pi\)
\(510\) 0 0
\(511\) −10.5139 −0.465107
\(512\) 0 0
\(513\) −7.81665 −0.345114
\(514\) 0 0
\(515\) 32.2389 1.42061
\(516\) 0 0
\(517\) −18.4222 −0.810208
\(518\) 0 0
\(519\) 15.2750 0.670499
\(520\) 0 0
\(521\) −18.5139 −0.811108 −0.405554 0.914071i \(-0.632921\pi\)
−0.405554 + 0.914071i \(0.632921\pi\)
\(522\) 0 0
\(523\) −34.0555 −1.48914 −0.744572 0.667542i \(-0.767346\pi\)
−0.744572 + 0.667542i \(0.767346\pi\)
\(524\) 0 0
\(525\) 0.394449 0.0172152
\(526\) 0 0
\(527\) 3.69722 0.161054
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 10.4222 0.452285
\(532\) 0 0
\(533\) −16.9722 −0.735149
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 0 0
\(537\) −11.0917 −0.478641
\(538\) 0 0
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −4.00000 −0.171973 −0.0859867 0.996296i \(-0.527404\pi\)
−0.0859867 + 0.996296i \(0.527404\pi\)
\(542\) 0 0
\(543\) 7.81665 0.335445
\(544\) 0 0
\(545\) 44.2389 1.89498
\(546\) 0 0
\(547\) 2.00000 0.0855138 0.0427569 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) 0 0
\(549\) −3.27502 −0.139774
\(550\) 0 0
\(551\) 7.26662 0.309568
\(552\) 0 0
\(553\) −4.60555 −0.195848
\(554\) 0 0
\(555\) −35.4500 −1.50477
\(556\) 0 0
\(557\) 35.2111 1.49194 0.745971 0.665978i \(-0.231986\pi\)
0.745971 + 0.665978i \(0.231986\pi\)
\(558\) 0 0
\(559\) 1.81665 0.0768363
\(560\) 0 0
\(561\) −5.21110 −0.220013
\(562\) 0 0
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 38.2389 1.60872
\(566\) 0 0
\(567\) −3.39445 −0.142553
\(568\) 0 0
\(569\) −3.11943 −0.130773 −0.0653866 0.997860i \(-0.520828\pi\)
−0.0653866 + 0.997860i \(0.520828\pi\)
\(570\) 0 0
\(571\) 19.4500 0.813956 0.406978 0.913438i \(-0.366583\pi\)
0.406978 + 0.913438i \(0.366583\pi\)
\(572\) 0 0
\(573\) −27.0000 −1.12794
\(574\) 0 0
\(575\) −1.21110 −0.0505065
\(576\) 0 0
\(577\) −29.8167 −1.24128 −0.620642 0.784094i \(-0.713128\pi\)
−0.620642 + 0.784094i \(0.713128\pi\)
\(578\) 0 0
\(579\) 27.3944 1.13847
\(580\) 0 0
\(581\) −11.2111 −0.465115
\(582\) 0 0
\(583\) −17.2111 −0.712811
\(584\) 0 0
\(585\) −7.81665 −0.323179
\(586\) 0 0
\(587\) −18.6056 −0.767933 −0.383967 0.923347i \(-0.625442\pi\)
−0.383967 + 0.923347i \(0.625442\pi\)
\(588\) 0 0
\(589\) −5.15559 −0.212432
\(590\) 0 0
\(591\) −0.788897 −0.0324509
\(592\) 0 0
\(593\) 43.8167 1.79933 0.899667 0.436576i \(-0.143809\pi\)
0.899667 + 0.436576i \(0.143809\pi\)
\(594\) 0 0
\(595\) 2.30278 0.0944046
\(596\) 0 0
\(597\) −17.3305 −0.709291
\(598\) 0 0
\(599\) 5.72498 0.233916 0.116958 0.993137i \(-0.462686\pi\)
0.116958 + 0.993137i \(0.462686\pi\)
\(600\) 0 0
\(601\) −41.2666 −1.68330 −0.841650 0.540023i \(-0.818415\pi\)
−0.841650 + 0.540023i \(0.818415\pi\)
\(602\) 0 0
\(603\) −8.21110 −0.334382
\(604\) 0 0
\(605\) −11.5139 −0.468106
\(606\) 0 0
\(607\) 39.3305 1.59638 0.798189 0.602408i \(-0.205792\pi\)
0.798189 + 0.602408i \(0.205792\pi\)
\(608\) 0 0
\(609\) 6.78890 0.275100
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) −10.7250 −0.433178 −0.216589 0.976263i \(-0.569493\pi\)
−0.216589 + 0.976263i \(0.569493\pi\)
\(614\) 0 0
\(615\) −19.5416 −0.787995
\(616\) 0 0
\(617\) −35.6333 −1.43454 −0.717271 0.696794i \(-0.754609\pi\)
−0.717271 + 0.696794i \(0.754609\pi\)
\(618\) 0 0
\(619\) 22.7889 0.915963 0.457982 0.888962i \(-0.348573\pi\)
0.457982 + 0.888962i \(0.348573\pi\)
\(620\) 0 0
\(621\) 22.4222 0.899772
\(622\) 0 0
\(623\) −13.8167 −0.553553
\(624\) 0 0
\(625\) −26.4222 −1.05689
\(626\) 0 0
\(627\) 7.26662 0.290201
\(628\) 0 0
\(629\) −11.8167 −0.471161
\(630\) 0 0
\(631\) 16.1194 0.641704 0.320852 0.947129i \(-0.396031\pi\)
0.320852 + 0.947129i \(0.396031\pi\)
\(632\) 0 0
\(633\) 36.2389 1.44037
\(634\) 0 0
\(635\) −8.09167 −0.321108
\(636\) 0 0
\(637\) −2.60555 −0.103236
\(638\) 0 0
\(639\) −13.8167 −0.546578
\(640\) 0 0
\(641\) 0.366692 0.0144835 0.00724174 0.999974i \(-0.497695\pi\)
0.00724174 + 0.999974i \(0.497695\pi\)
\(642\) 0 0
\(643\) 33.1194 1.30610 0.653051 0.757314i \(-0.273489\pi\)
0.653051 + 0.757314i \(0.273489\pi\)
\(644\) 0 0
\(645\) 2.09167 0.0823595
\(646\) 0 0
\(647\) −7.63331 −0.300096 −0.150048 0.988679i \(-0.547943\pi\)
−0.150048 + 0.988679i \(0.547943\pi\)
\(648\) 0 0
\(649\) −32.0000 −1.25611
\(650\) 0 0
\(651\) −4.81665 −0.188780
\(652\) 0 0
\(653\) −11.5778 −0.453074 −0.226537 0.974003i \(-0.572740\pi\)
−0.226537 + 0.974003i \(0.572740\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 0 0
\(657\) 13.6972 0.534380
\(658\) 0 0
\(659\) −4.11943 −0.160470 −0.0802351 0.996776i \(-0.525567\pi\)
−0.0802351 + 0.996776i \(0.525567\pi\)
\(660\) 0 0
\(661\) −31.0278 −1.20684 −0.603420 0.797424i \(-0.706196\pi\)
−0.603420 + 0.797424i \(0.706196\pi\)
\(662\) 0 0
\(663\) 3.39445 0.131829
\(664\) 0 0
\(665\) −3.21110 −0.124521
\(666\) 0 0
\(667\) −20.8444 −0.807099
\(668\) 0 0
\(669\) 10.1833 0.393711
\(670\) 0 0
\(671\) 10.0555 0.388189
\(672\) 0 0
\(673\) 29.0278 1.11894 0.559469 0.828851i \(-0.311005\pi\)
0.559469 + 0.828851i \(0.311005\pi\)
\(674\) 0 0
\(675\) −1.69722 −0.0653262
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) −9.69722 −0.372145
\(680\) 0 0
\(681\) 16.8167 0.644416
\(682\) 0 0
\(683\) 20.6056 0.788450 0.394225 0.919014i \(-0.371013\pi\)
0.394225 + 0.919014i \(0.371013\pi\)
\(684\) 0 0
\(685\) 16.3944 0.626400
\(686\) 0 0
\(687\) −17.2111 −0.656645
\(688\) 0 0
\(689\) 11.2111 0.427109
\(690\) 0 0
\(691\) 44.9361 1.70945 0.854725 0.519082i \(-0.173726\pi\)
0.854725 + 0.519082i \(0.173726\pi\)
\(692\) 0 0
\(693\) −5.21110 −0.197953
\(694\) 0 0
\(695\) 21.9083 0.831030
\(696\) 0 0
\(697\) −6.51388 −0.246731
\(698\) 0 0
\(699\) 25.2666 0.955671
\(700\) 0 0
\(701\) 40.4222 1.52673 0.763363 0.645970i \(-0.223547\pi\)
0.763363 + 0.645970i \(0.223547\pi\)
\(702\) 0 0
\(703\) 16.4777 0.621469
\(704\) 0 0
\(705\) 13.8167 0.520365
\(706\) 0 0
\(707\) 5.39445 0.202879
\(708\) 0 0
\(709\) −46.2389 −1.73654 −0.868268 0.496095i \(-0.834767\pi\)
−0.868268 + 0.496095i \(0.834767\pi\)
\(710\) 0 0
\(711\) 6.00000 0.225018
\(712\) 0 0
\(713\) 14.7889 0.553849
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) 0 0
\(717\) 9.27502 0.346382
\(718\) 0 0
\(719\) −13.3028 −0.496110 −0.248055 0.968746i \(-0.579791\pi\)
−0.248055 + 0.968746i \(0.579791\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 0 0
\(723\) −11.0555 −0.411159
\(724\) 0 0
\(725\) 1.57779 0.0585978
\(726\) 0 0
\(727\) −14.4222 −0.534890 −0.267445 0.963573i \(-0.586179\pi\)
−0.267445 + 0.963573i \(0.586179\pi\)
\(728\) 0 0
\(729\) 26.3305 0.975205
\(730\) 0 0
\(731\) 0.697224 0.0257878
\(732\) 0 0
\(733\) 34.2389 1.26464 0.632321 0.774707i \(-0.282103\pi\)
0.632321 + 0.774707i \(0.282103\pi\)
\(734\) 0 0
\(735\) −3.00000 −0.110657
\(736\) 0 0
\(737\) 25.2111 0.928663
\(738\) 0 0
\(739\) 16.3305 0.600728 0.300364 0.953825i \(-0.402892\pi\)
0.300364 + 0.953825i \(0.402892\pi\)
\(740\) 0 0
\(741\) −4.73338 −0.173885
\(742\) 0 0
\(743\) 27.3944 1.00500 0.502502 0.864576i \(-0.332413\pi\)
0.502502 + 0.864576i \(0.332413\pi\)
\(744\) 0 0
\(745\) −1.60555 −0.0588228
\(746\) 0 0
\(747\) 14.6056 0.534389
\(748\) 0 0
\(749\) 2.60555 0.0952048
\(750\) 0 0
\(751\) −10.6056 −0.387002 −0.193501 0.981100i \(-0.561984\pi\)
−0.193501 + 0.981100i \(0.561984\pi\)
\(752\) 0 0
\(753\) −31.0278 −1.13071
\(754\) 0 0
\(755\) 35.2389 1.28247
\(756\) 0 0
\(757\) 11.3028 0.410806 0.205403 0.978677i \(-0.434149\pi\)
0.205403 + 0.978677i \(0.434149\pi\)
\(758\) 0 0
\(759\) −20.8444 −0.756604
\(760\) 0 0
\(761\) 0.238859 0.00865863 0.00432931 0.999991i \(-0.498622\pi\)
0.00432931 + 0.999991i \(0.498622\pi\)
\(762\) 0 0
\(763\) −19.2111 −0.695489
\(764\) 0 0
\(765\) −3.00000 −0.108465
\(766\) 0 0
\(767\) 20.8444 0.752648
\(768\) 0 0
\(769\) 36.0555 1.30020 0.650098 0.759851i \(-0.274728\pi\)
0.650098 + 0.759851i \(0.274728\pi\)
\(770\) 0 0
\(771\) 31.8167 1.14585
\(772\) 0 0
\(773\) 19.0278 0.684381 0.342190 0.939631i \(-0.388831\pi\)
0.342190 + 0.939631i \(0.388831\pi\)
\(774\) 0 0
\(775\) −1.11943 −0.0402111
\(776\) 0 0
\(777\) 15.3944 0.552273
\(778\) 0 0
\(779\) 9.08327 0.325442
\(780\) 0 0
\(781\) 42.4222 1.51799
\(782\) 0 0
\(783\) −29.2111 −1.04392
\(784\) 0 0
\(785\) 4.60555 0.164379
\(786\) 0 0
\(787\) 21.5778 0.769165 0.384583 0.923091i \(-0.374345\pi\)
0.384583 + 0.923091i \(0.374345\pi\)
\(788\) 0 0
\(789\) −13.5778 −0.483382
\(790\) 0 0
\(791\) −16.6056 −0.590425
\(792\) 0 0
\(793\) −6.55004 −0.232599
\(794\) 0 0
\(795\) 12.9083 0.457811
\(796\) 0 0
\(797\) 24.0555 0.852090 0.426045 0.904702i \(-0.359907\pi\)
0.426045 + 0.904702i \(0.359907\pi\)
\(798\) 0 0
\(799\) 4.60555 0.162933
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) −42.0555 −1.48411
\(804\) 0 0
\(805\) 9.21110 0.324649
\(806\) 0 0
\(807\) −35.4500 −1.24790
\(808\) 0 0
\(809\) 1.81665 0.0638701 0.0319351 0.999490i \(-0.489833\pi\)
0.0319351 + 0.999490i \(0.489833\pi\)
\(810\) 0 0
\(811\) −22.5139 −0.790569 −0.395285 0.918559i \(-0.629354\pi\)
−0.395285 + 0.918559i \(0.629354\pi\)
\(812\) 0 0
\(813\) −24.0000 −0.841717
\(814\) 0 0
\(815\) −19.8167 −0.694147
\(816\) 0 0
\(817\) −0.972244 −0.0340145
\(818\) 0 0
\(819\) 3.39445 0.118612
\(820\) 0 0
\(821\) −42.2389 −1.47415 −0.737073 0.675813i \(-0.763793\pi\)
−0.737073 + 0.675813i \(0.763793\pi\)
\(822\) 0 0
\(823\) 31.3944 1.09434 0.547171 0.837021i \(-0.315705\pi\)
0.547171 + 0.837021i \(0.315705\pi\)
\(824\) 0 0
\(825\) 1.57779 0.0549318
\(826\) 0 0
\(827\) 30.7889 1.07063 0.535317 0.844651i \(-0.320192\pi\)
0.535317 + 0.844651i \(0.320192\pi\)
\(828\) 0 0
\(829\) −34.2389 −1.18916 −0.594582 0.804035i \(-0.702683\pi\)
−0.594582 + 0.804035i \(0.702683\pi\)
\(830\) 0 0
\(831\) −39.6333 −1.37486
\(832\) 0 0
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) −30.2111 −1.04550
\(836\) 0 0
\(837\) 20.7250 0.716360
\(838\) 0 0
\(839\) 10.4222 0.359814 0.179907 0.983684i \(-0.442420\pi\)
0.179907 + 0.983684i \(0.442420\pi\)
\(840\) 0 0
\(841\) −1.84441 −0.0636004
\(842\) 0 0
\(843\) 7.97224 0.274579
\(844\) 0 0
\(845\) 14.3028 0.492030
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 0 0
\(849\) −25.1833 −0.864290
\(850\) 0 0
\(851\) −47.2666 −1.62028
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 4.18335 0.143067
\(856\) 0 0
\(857\) 36.9638 1.26266 0.631330 0.775514i \(-0.282509\pi\)
0.631330 + 0.775514i \(0.282509\pi\)
\(858\) 0 0
\(859\) −47.3944 −1.61708 −0.808539 0.588443i \(-0.799741\pi\)
−0.808539 + 0.588443i \(0.799741\pi\)
\(860\) 0 0
\(861\) 8.48612 0.289206
\(862\) 0 0
\(863\) 52.3583 1.78230 0.891148 0.453712i \(-0.149900\pi\)
0.891148 + 0.453712i \(0.149900\pi\)
\(864\) 0 0
\(865\) −27.0000 −0.918028
\(866\) 0 0
\(867\) 1.30278 0.0442446
\(868\) 0 0
\(869\) −18.4222 −0.624931
\(870\) 0 0
\(871\) −16.4222 −0.556445
\(872\) 0 0
\(873\) 12.6333 0.427573
\(874\) 0 0
\(875\) 10.8167 0.365670
\(876\) 0 0
\(877\) 8.60555 0.290589 0.145294 0.989388i \(-0.453587\pi\)
0.145294 + 0.989388i \(0.453587\pi\)
\(878\) 0 0
\(879\) −4.18335 −0.141101
\(880\) 0 0
\(881\) 12.1194 0.408314 0.204157 0.978938i \(-0.434555\pi\)
0.204157 + 0.978938i \(0.434555\pi\)
\(882\) 0 0
\(883\) 22.9083 0.770927 0.385463 0.922723i \(-0.374042\pi\)
0.385463 + 0.922723i \(0.374042\pi\)
\(884\) 0 0
\(885\) 24.0000 0.806751
\(886\) 0 0
\(887\) 48.7805 1.63789 0.818944 0.573873i \(-0.194560\pi\)
0.818944 + 0.573873i \(0.194560\pi\)
\(888\) 0 0
\(889\) 3.51388 0.117852
\(890\) 0 0
\(891\) −13.5778 −0.454873
\(892\) 0 0
\(893\) −6.42221 −0.214911
\(894\) 0 0
\(895\) 19.6056 0.655341
\(896\) 0 0
\(897\) 13.5778 0.453349
\(898\) 0 0
\(899\) −19.2666 −0.642578
\(900\) 0 0
\(901\) 4.30278 0.143346
\(902\) 0 0
\(903\) −0.908327 −0.0302272
\(904\) 0 0
\(905\) −13.8167 −0.459281
\(906\) 0 0
\(907\) 12.7889 0.424648 0.212324 0.977199i \(-0.431897\pi\)
0.212324 + 0.977199i \(0.431897\pi\)
\(908\) 0 0
\(909\) −7.02776 −0.233096
\(910\) 0 0
\(911\) −5.81665 −0.192714 −0.0963572 0.995347i \(-0.530719\pi\)
−0.0963572 + 0.995347i \(0.530719\pi\)
\(912\) 0 0
\(913\) −44.8444 −1.48413
\(914\) 0 0
\(915\) −7.54163 −0.249319
\(916\) 0 0
\(917\) 5.21110 0.172086
\(918\) 0 0
\(919\) 49.3583 1.62818 0.814090 0.580739i \(-0.197236\pi\)
0.814090 + 0.580739i \(0.197236\pi\)
\(920\) 0 0
\(921\) 5.21110 0.171712
\(922\) 0 0
\(923\) −27.6333 −0.909561
\(924\) 0 0
\(925\) 3.57779 0.117637
\(926\) 0 0
\(927\) 18.2389 0.599043
\(928\) 0 0
\(929\) 46.9083 1.53901 0.769506 0.638639i \(-0.220502\pi\)
0.769506 + 0.638639i \(0.220502\pi\)
\(930\) 0 0
\(931\) 1.39445 0.0457012
\(932\) 0 0
\(933\) −19.1833 −0.628035
\(934\) 0 0
\(935\) 9.21110 0.301235
\(936\) 0 0
\(937\) −22.7889 −0.744481 −0.372240 0.928136i \(-0.621410\pi\)
−0.372240 + 0.928136i \(0.621410\pi\)
\(938\) 0 0
\(939\) −9.00000 −0.293704
\(940\) 0 0
\(941\) 37.3028 1.21604 0.608018 0.793923i \(-0.291965\pi\)
0.608018 + 0.793923i \(0.291965\pi\)
\(942\) 0 0
\(943\) −26.0555 −0.848484
\(944\) 0 0
\(945\) 12.9083 0.419908
\(946\) 0 0
\(947\) −32.8444 −1.06730 −0.533650 0.845705i \(-0.679180\pi\)
−0.533650 + 0.845705i \(0.679180\pi\)
\(948\) 0 0
\(949\) 27.3944 0.889261
\(950\) 0 0
\(951\) −12.5500 −0.406963
\(952\) 0 0
\(953\) 54.3583 1.76084 0.880419 0.474197i \(-0.157262\pi\)
0.880419 + 0.474197i \(0.157262\pi\)
\(954\) 0 0
\(955\) 47.7250 1.54434
\(956\) 0 0
\(957\) 27.1556 0.877816
\(958\) 0 0
\(959\) −7.11943 −0.229898
\(960\) 0 0
\(961\) −17.3305 −0.559049
\(962\) 0 0
\(963\) −3.39445 −0.109385
\(964\) 0 0
\(965\) −48.4222 −1.55877
\(966\) 0 0
\(967\) −8.51388 −0.273788 −0.136894 0.990586i \(-0.543712\pi\)
−0.136894 + 0.990586i \(0.543712\pi\)
\(968\) 0 0
\(969\) −1.81665 −0.0583593
\(970\) 0 0
\(971\) 17.2111 0.552331 0.276165 0.961110i \(-0.410936\pi\)
0.276165 + 0.961110i \(0.410936\pi\)
\(972\) 0 0
\(973\) −9.51388 −0.305001
\(974\) 0 0
\(975\) −1.02776 −0.0329145
\(976\) 0 0
\(977\) −12.3028 −0.393601 −0.196800 0.980444i \(-0.563055\pi\)
−0.196800 + 0.980444i \(0.563055\pi\)
\(978\) 0 0
\(979\) −55.2666 −1.76633
\(980\) 0 0
\(981\) 25.0278 0.799075
\(982\) 0 0
\(983\) 56.7805 1.81102 0.905508 0.424329i \(-0.139490\pi\)
0.905508 + 0.424329i \(0.139490\pi\)
\(984\) 0 0
\(985\) 1.39445 0.0444308
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) 2.78890 0.0886818
\(990\) 0 0
\(991\) −3.21110 −0.102004 −0.0510020 0.998699i \(-0.516241\pi\)
−0.0510020 + 0.998699i \(0.516241\pi\)
\(992\) 0 0
\(993\) −37.4222 −1.18756
\(994\) 0 0
\(995\) 30.6333 0.971141
\(996\) 0 0
\(997\) 49.3028 1.56143 0.780717 0.624884i \(-0.214854\pi\)
0.780717 + 0.624884i \(0.214854\pi\)
\(998\) 0 0
\(999\) −66.2389 −2.09570
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 7616.2.a.o.1.2 2
4.3 odd 2 7616.2.a.t.1.1 2
8.3 odd 2 476.2.a.c.1.2 2
8.5 even 2 1904.2.a.k.1.1 2
24.11 even 2 4284.2.a.l.1.1 2
56.27 even 2 3332.2.a.k.1.1 2
136.67 odd 2 8092.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.c.1.2 2 8.3 odd 2
1904.2.a.k.1.1 2 8.5 even 2
3332.2.a.k.1.1 2 56.27 even 2
4284.2.a.l.1.1 2 24.11 even 2
7616.2.a.o.1.2 2 1.1 even 1 trivial
7616.2.a.t.1.1 2 4.3 odd 2
8092.2.a.l.1.1 2 136.67 odd 2