Properties

Label 1275.2.a.n.1.2
Level $1275$
Weight $2$
Character 1275.1
Self dual yes
Analytic conductor $10.181$
Analytic rank $0$
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] = [1275,2,Mod(1,1275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1275.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1275.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: \(10.1809262577\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 1275.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} +2.56155 q^{6} +6.56155 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} +2.56155 q^{6} +6.56155 q^{8} +1.00000 q^{9} +1.56155 q^{11} +4.56155 q^{12} -0.438447 q^{13} +7.68466 q^{16} -1.00000 q^{17} +2.56155 q^{18} -4.68466 q^{19} +4.00000 q^{22} +2.43845 q^{23} +6.56155 q^{24} -1.12311 q^{26} +1.00000 q^{27} -8.24621 q^{29} +3.12311 q^{31} +6.56155 q^{32} +1.56155 q^{33} -2.56155 q^{34} +4.56155 q^{36} +5.12311 q^{37} -12.0000 q^{38} -0.438447 q^{39} -3.56155 q^{41} -4.68466 q^{43} +7.12311 q^{44} +6.24621 q^{46} +11.1231 q^{47} +7.68466 q^{48} -7.00000 q^{49} -1.00000 q^{51} -2.00000 q^{52} -12.2462 q^{53} +2.56155 q^{54} -4.68466 q^{57} -21.1231 q^{58} +7.12311 q^{59} +9.12311 q^{61} +8.00000 q^{62} +1.43845 q^{64} +4.00000 q^{66} -4.00000 q^{67} -4.56155 q^{68} +2.43845 q^{69} -6.24621 q^{71} +6.56155 q^{72} +12.2462 q^{73} +13.1231 q^{74} -21.3693 q^{76} -1.12311 q^{78} -9.36932 q^{79} +1.00000 q^{81} -9.12311 q^{82} +0.876894 q^{83} -12.0000 q^{86} -8.24621 q^{87} +10.2462 q^{88} -1.12311 q^{89} +11.1231 q^{92} +3.12311 q^{93} +28.4924 q^{94} +6.56155 q^{96} +2.87689 q^{97} -17.9309 q^{98} +1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} + 5 q^{4} + q^{6} + 9 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 2 q^{3} + 5 q^{4} + q^{6} + 9 q^{8} + 2 q^{9} - q^{11} + 5 q^{12} - 5 q^{13} + 3 q^{16} - 2 q^{17} + q^{18} + 3 q^{19} + 8 q^{22} + 9 q^{23} + 9 q^{24} + 6 q^{26} + 2 q^{27} - 2 q^{31} + 9 q^{32} - q^{33} - q^{34} + 5 q^{36} + 2 q^{37} - 24 q^{38} - 5 q^{39} - 3 q^{41} + 3 q^{43} + 6 q^{44} - 4 q^{46} + 14 q^{47} + 3 q^{48} - 14 q^{49} - 2 q^{51} - 4 q^{52} - 8 q^{53} + q^{54} + 3 q^{57} - 34 q^{58} + 6 q^{59} + 10 q^{61} + 16 q^{62} + 7 q^{64} + 8 q^{66} - 8 q^{67} - 5 q^{68} + 9 q^{69} + 4 q^{71} + 9 q^{72} + 8 q^{73} + 18 q^{74} - 18 q^{76} + 6 q^{78} + 6 q^{79} + 2 q^{81} - 10 q^{82} + 10 q^{83} - 24 q^{86} + 4 q^{88} + 6 q^{89} + 14 q^{92} - 2 q^{93} + 24 q^{94} + 9 q^{96} + 14 q^{97} - 7 q^{98} - q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56155 1.81129 0.905646 0.424035i \(-0.139387\pi\)
0.905646 + 0.424035i \(0.139387\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.56155 2.28078
\(5\) 0 0
\(6\) 2.56155 1.04575
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 6.56155 2.31986
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.56155 0.470826 0.235413 0.971895i \(-0.424356\pi\)
0.235413 + 0.971895i \(0.424356\pi\)
\(12\) 4.56155 1.31681
\(13\) −0.438447 −0.121603 −0.0608017 0.998150i \(-0.519366\pi\)
−0.0608017 + 0.998150i \(0.519366\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 7.68466 1.92116
\(17\) −1.00000 −0.242536
\(18\) 2.56155 0.603764
\(19\) −4.68466 −1.07473 −0.537367 0.843348i \(-0.680581\pi\)
−0.537367 + 0.843348i \(0.680581\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 2.43845 0.508451 0.254226 0.967145i \(-0.418179\pi\)
0.254226 + 0.967145i \(0.418179\pi\)
\(24\) 6.56155 1.33937
\(25\) 0 0
\(26\) −1.12311 −0.220259
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.24621 −1.53128 −0.765641 0.643268i \(-0.777578\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) 3.12311 0.560926 0.280463 0.959865i \(-0.409512\pi\)
0.280463 + 0.959865i \(0.409512\pi\)
\(32\) 6.56155 1.15993
\(33\) 1.56155 0.271831
\(34\) −2.56155 −0.439303
\(35\) 0 0
\(36\) 4.56155 0.760259
\(37\) 5.12311 0.842233 0.421117 0.907006i \(-0.361638\pi\)
0.421117 + 0.907006i \(0.361638\pi\)
\(38\) −12.0000 −1.94666
\(39\) −0.438447 −0.0702077
\(40\) 0 0
\(41\) −3.56155 −0.556221 −0.278111 0.960549i \(-0.589708\pi\)
−0.278111 + 0.960549i \(0.589708\pi\)
\(42\) 0 0
\(43\) −4.68466 −0.714404 −0.357202 0.934027i \(-0.616269\pi\)
−0.357202 + 0.934027i \(0.616269\pi\)
\(44\) 7.12311 1.07385
\(45\) 0 0
\(46\) 6.24621 0.920954
\(47\) 11.1231 1.62247 0.811236 0.584719i \(-0.198795\pi\)
0.811236 + 0.584719i \(0.198795\pi\)
\(48\) 7.68466 1.10918
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) −2.00000 −0.277350
\(53\) −12.2462 −1.68215 −0.841073 0.540921i \(-0.818076\pi\)
−0.841073 + 0.540921i \(0.818076\pi\)
\(54\) 2.56155 0.348583
\(55\) 0 0
\(56\) 0 0
\(57\) −4.68466 −0.620498
\(58\) −21.1231 −2.77360
\(59\) 7.12311 0.927349 0.463675 0.886006i \(-0.346531\pi\)
0.463675 + 0.886006i \(0.346531\pi\)
\(60\) 0 0
\(61\) 9.12311 1.16809 0.584047 0.811720i \(-0.301468\pi\)
0.584047 + 0.811720i \(0.301468\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.43845 0.179806
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −4.56155 −0.553170
\(69\) 2.43845 0.293555
\(70\) 0 0
\(71\) −6.24621 −0.741289 −0.370644 0.928775i \(-0.620863\pi\)
−0.370644 + 0.928775i \(0.620863\pi\)
\(72\) 6.56155 0.773286
\(73\) 12.2462 1.43331 0.716655 0.697428i \(-0.245672\pi\)
0.716655 + 0.697428i \(0.245672\pi\)
\(74\) 13.1231 1.52553
\(75\) 0 0
\(76\) −21.3693 −2.45123
\(77\) 0 0
\(78\) −1.12311 −0.127167
\(79\) −9.36932 −1.05413 −0.527065 0.849825i \(-0.676708\pi\)
−0.527065 + 0.849825i \(0.676708\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −9.12311 −1.00748
\(83\) 0.876894 0.0962517 0.0481258 0.998841i \(-0.484675\pi\)
0.0481258 + 0.998841i \(0.484675\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) −8.24621 −0.884087
\(88\) 10.2462 1.09225
\(89\) −1.12311 −0.119049 −0.0595245 0.998227i \(-0.518958\pi\)
−0.0595245 + 0.998227i \(0.518958\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 11.1231 1.15966
\(93\) 3.12311 0.323851
\(94\) 28.4924 2.93877
\(95\) 0 0
\(96\) 6.56155 0.669686
\(97\) 2.87689 0.292104 0.146052 0.989277i \(-0.453343\pi\)
0.146052 + 0.989277i \(0.453343\pi\)
\(98\) −17.9309 −1.81129
\(99\) 1.56155 0.156942
\(100\) 0 0
\(101\) 10.8769 1.08229 0.541146 0.840929i \(-0.317991\pi\)
0.541146 + 0.840929i \(0.317991\pi\)
\(102\) −2.56155 −0.253632
\(103\) −16.6847 −1.64399 −0.821994 0.569496i \(-0.807138\pi\)
−0.821994 + 0.569496i \(0.807138\pi\)
\(104\) −2.87689 −0.282103
\(105\) 0 0
\(106\) −31.3693 −3.04686
\(107\) 4.68466 0.452883 0.226442 0.974025i \(-0.427291\pi\)
0.226442 + 0.974025i \(0.427291\pi\)
\(108\) 4.56155 0.438936
\(109\) −6.87689 −0.658687 −0.329344 0.944210i \(-0.606827\pi\)
−0.329344 + 0.944210i \(0.606827\pi\)
\(110\) 0 0
\(111\) 5.12311 0.486264
\(112\) 0 0
\(113\) 0.438447 0.0412456 0.0206228 0.999787i \(-0.493435\pi\)
0.0206228 + 0.999787i \(0.493435\pi\)
\(114\) −12.0000 −1.12390
\(115\) 0 0
\(116\) −37.6155 −3.49251
\(117\) −0.438447 −0.0405345
\(118\) 18.2462 1.67970
\(119\) 0 0
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 23.3693 2.11576
\(123\) −3.56155 −0.321134
\(124\) 14.2462 1.27935
\(125\) 0 0
\(126\) 0 0
\(127\) −19.8078 −1.75765 −0.878827 0.477140i \(-0.841674\pi\)
−0.878827 + 0.477140i \(0.841674\pi\)
\(128\) −9.43845 −0.834249
\(129\) −4.68466 −0.412461
\(130\) 0 0
\(131\) 14.4384 1.26149 0.630746 0.775989i \(-0.282749\pi\)
0.630746 + 0.775989i \(0.282749\pi\)
\(132\) 7.12311 0.619987
\(133\) 0 0
\(134\) −10.2462 −0.885138
\(135\) 0 0
\(136\) −6.56155 −0.562649
\(137\) −0.246211 −0.0210352 −0.0105176 0.999945i \(-0.503348\pi\)
−0.0105176 + 0.999945i \(0.503348\pi\)
\(138\) 6.24621 0.531713
\(139\) −0.876894 −0.0743772 −0.0371886 0.999308i \(-0.511840\pi\)
−0.0371886 + 0.999308i \(0.511840\pi\)
\(140\) 0 0
\(141\) 11.1231 0.936734
\(142\) −16.0000 −1.34269
\(143\) −0.684658 −0.0572540
\(144\) 7.68466 0.640388
\(145\) 0 0
\(146\) 31.3693 2.59614
\(147\) −7.00000 −0.577350
\(148\) 23.3693 1.92095
\(149\) 12.2462 1.00325 0.501624 0.865086i \(-0.332736\pi\)
0.501624 + 0.865086i \(0.332736\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −30.7386 −2.49323
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −6.68466 −0.533494 −0.266747 0.963767i \(-0.585949\pi\)
−0.266747 + 0.963767i \(0.585949\pi\)
\(158\) −24.0000 −1.90934
\(159\) −12.2462 −0.971188
\(160\) 0 0
\(161\) 0 0
\(162\) 2.56155 0.201255
\(163\) 15.1231 1.18453 0.592267 0.805742i \(-0.298233\pi\)
0.592267 + 0.805742i \(0.298233\pi\)
\(164\) −16.2462 −1.26862
\(165\) 0 0
\(166\) 2.24621 0.174340
\(167\) −19.8078 −1.53277 −0.766385 0.642381i \(-0.777947\pi\)
−0.766385 + 0.642381i \(0.777947\pi\)
\(168\) 0 0
\(169\) −12.8078 −0.985213
\(170\) 0 0
\(171\) −4.68466 −0.358245
\(172\) −21.3693 −1.62940
\(173\) −1.80776 −0.137442 −0.0687209 0.997636i \(-0.521892\pi\)
−0.0687209 + 0.997636i \(0.521892\pi\)
\(174\) −21.1231 −1.60134
\(175\) 0 0
\(176\) 12.0000 0.904534
\(177\) 7.12311 0.535405
\(178\) −2.87689 −0.215632
\(179\) −0.876894 −0.0655422 −0.0327711 0.999463i \(-0.510433\pi\)
−0.0327711 + 0.999463i \(0.510433\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 9.12311 0.674399
\(184\) 16.0000 1.17954
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) −1.56155 −0.114192
\(188\) 50.7386 3.70050
\(189\) 0 0
\(190\) 0 0
\(191\) −4.87689 −0.352880 −0.176440 0.984311i \(-0.556458\pi\)
−0.176440 + 0.984311i \(0.556458\pi\)
\(192\) 1.43845 0.103811
\(193\) 7.75379 0.558130 0.279065 0.960272i \(-0.409976\pi\)
0.279065 + 0.960272i \(0.409976\pi\)
\(194\) 7.36932 0.529086
\(195\) 0 0
\(196\) −31.9309 −2.28078
\(197\) 8.93087 0.636298 0.318149 0.948041i \(-0.396939\pi\)
0.318149 + 0.948041i \(0.396939\pi\)
\(198\) 4.00000 0.284268
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 27.8617 1.96035
\(203\) 0 0
\(204\) −4.56155 −0.319373
\(205\) 0 0
\(206\) −42.7386 −2.97774
\(207\) 2.43845 0.169484
\(208\) −3.36932 −0.233620
\(209\) −7.31534 −0.506013
\(210\) 0 0
\(211\) 13.3693 0.920382 0.460191 0.887820i \(-0.347781\pi\)
0.460191 + 0.887820i \(0.347781\pi\)
\(212\) −55.8617 −3.83660
\(213\) −6.24621 −0.427983
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 6.56155 0.446457
\(217\) 0 0
\(218\) −17.6155 −1.19307
\(219\) 12.2462 0.827522
\(220\) 0 0
\(221\) 0.438447 0.0294931
\(222\) 13.1231 0.880765
\(223\) 14.9309 0.999845 0.499922 0.866070i \(-0.333362\pi\)
0.499922 + 0.866070i \(0.333362\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.12311 0.0747079
\(227\) 14.0540 0.932795 0.466398 0.884575i \(-0.345552\pi\)
0.466398 + 0.884575i \(0.345552\pi\)
\(228\) −21.3693 −1.41522
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −54.1080 −3.55236
\(233\) 3.56155 0.233325 0.116663 0.993172i \(-0.462780\pi\)
0.116663 + 0.993172i \(0.462780\pi\)
\(234\) −1.12311 −0.0734197
\(235\) 0 0
\(236\) 32.4924 2.11508
\(237\) −9.36932 −0.608603
\(238\) 0 0
\(239\) −6.24621 −0.404034 −0.202017 0.979382i \(-0.564750\pi\)
−0.202017 + 0.979382i \(0.564750\pi\)
\(240\) 0 0
\(241\) 3.36932 0.217037 0.108518 0.994094i \(-0.465389\pi\)
0.108518 + 0.994094i \(0.465389\pi\)
\(242\) −21.9309 −1.40977
\(243\) 1.00000 0.0641500
\(244\) 41.6155 2.66416
\(245\) 0 0
\(246\) −9.12311 −0.581668
\(247\) 2.05398 0.130691
\(248\) 20.4924 1.30127
\(249\) 0.876894 0.0555709
\(250\) 0 0
\(251\) 8.49242 0.536037 0.268018 0.963414i \(-0.413631\pi\)
0.268018 + 0.963414i \(0.413631\pi\)
\(252\) 0 0
\(253\) 3.80776 0.239392
\(254\) −50.7386 −3.18363
\(255\) 0 0
\(256\) −27.0540 −1.69087
\(257\) 15.3693 0.958712 0.479356 0.877621i \(-0.340870\pi\)
0.479356 + 0.877621i \(0.340870\pi\)
\(258\) −12.0000 −0.747087
\(259\) 0 0
\(260\) 0 0
\(261\) −8.24621 −0.510428
\(262\) 36.9848 2.28493
\(263\) 20.4924 1.26362 0.631808 0.775125i \(-0.282313\pi\)
0.631808 + 0.775125i \(0.282313\pi\)
\(264\) 10.2462 0.630611
\(265\) 0 0
\(266\) 0 0
\(267\) −1.12311 −0.0687329
\(268\) −18.2462 −1.11456
\(269\) 16.4384 1.00227 0.501135 0.865369i \(-0.332916\pi\)
0.501135 + 0.865369i \(0.332916\pi\)
\(270\) 0 0
\(271\) 19.8078 1.20324 0.601618 0.798784i \(-0.294523\pi\)
0.601618 + 0.798784i \(0.294523\pi\)
\(272\) −7.68466 −0.465951
\(273\) 0 0
\(274\) −0.630683 −0.0381010
\(275\) 0 0
\(276\) 11.1231 0.669532
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) −2.24621 −0.134719
\(279\) 3.12311 0.186975
\(280\) 0 0
\(281\) −10.8769 −0.648861 −0.324431 0.945910i \(-0.605173\pi\)
−0.324431 + 0.945910i \(0.605173\pi\)
\(282\) 28.4924 1.69670
\(283\) −21.3693 −1.27027 −0.635137 0.772399i \(-0.719056\pi\)
−0.635137 + 0.772399i \(0.719056\pi\)
\(284\) −28.4924 −1.69071
\(285\) 0 0
\(286\) −1.75379 −0.103704
\(287\) 0 0
\(288\) 6.56155 0.386643
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 2.87689 0.168647
\(292\) 55.8617 3.26906
\(293\) −1.12311 −0.0656125 −0.0328063 0.999462i \(-0.510444\pi\)
−0.0328063 + 0.999462i \(0.510444\pi\)
\(294\) −17.9309 −1.04575
\(295\) 0 0
\(296\) 33.6155 1.95386
\(297\) 1.56155 0.0906105
\(298\) 31.3693 1.81718
\(299\) −1.06913 −0.0618294
\(300\) 0 0
\(301\) 0 0
\(302\) 20.4924 1.17921
\(303\) 10.8769 0.624861
\(304\) −36.0000 −2.06474
\(305\) 0 0
\(306\) −2.56155 −0.146434
\(307\) −32.4924 −1.85444 −0.927220 0.374516i \(-0.877809\pi\)
−0.927220 + 0.374516i \(0.877809\pi\)
\(308\) 0 0
\(309\) −16.6847 −0.949157
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) −2.87689 −0.162872
\(313\) −33.6155 −1.90006 −0.950031 0.312156i \(-0.898949\pi\)
−0.950031 + 0.312156i \(0.898949\pi\)
\(314\) −17.1231 −0.966313
\(315\) 0 0
\(316\) −42.7386 −2.40424
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −31.3693 −1.75910
\(319\) −12.8769 −0.720968
\(320\) 0 0
\(321\) 4.68466 0.261472
\(322\) 0 0
\(323\) 4.68466 0.260661
\(324\) 4.56155 0.253420
\(325\) 0 0
\(326\) 38.7386 2.14553
\(327\) −6.87689 −0.380293
\(328\) −23.3693 −1.29035
\(329\) 0 0
\(330\) 0 0
\(331\) −34.9309 −1.91997 −0.959987 0.280044i \(-0.909651\pi\)
−0.959987 + 0.280044i \(0.909651\pi\)
\(332\) 4.00000 0.219529
\(333\) 5.12311 0.280744
\(334\) −50.7386 −2.77629
\(335\) 0 0
\(336\) 0 0
\(337\) 16.7386 0.911811 0.455906 0.890028i \(-0.349315\pi\)
0.455906 + 0.890028i \(0.349315\pi\)
\(338\) −32.8078 −1.78451
\(339\) 0.438447 0.0238132
\(340\) 0 0
\(341\) 4.87689 0.264099
\(342\) −12.0000 −0.648886
\(343\) 0 0
\(344\) −30.7386 −1.65732
\(345\) 0 0
\(346\) −4.63068 −0.248947
\(347\) 8.49242 0.455897 0.227949 0.973673i \(-0.426798\pi\)
0.227949 + 0.973673i \(0.426798\pi\)
\(348\) −37.6155 −2.01640
\(349\) 11.5616 0.618876 0.309438 0.950920i \(-0.399859\pi\)
0.309438 + 0.950920i \(0.399859\pi\)
\(350\) 0 0
\(351\) −0.438447 −0.0234026
\(352\) 10.2462 0.546125
\(353\) 10.4924 0.558455 0.279228 0.960225i \(-0.409922\pi\)
0.279228 + 0.960225i \(0.409922\pi\)
\(354\) 18.2462 0.969775
\(355\) 0 0
\(356\) −5.12311 −0.271524
\(357\) 0 0
\(358\) −2.24621 −0.118716
\(359\) 14.2462 0.751886 0.375943 0.926643i \(-0.377319\pi\)
0.375943 + 0.926643i \(0.377319\pi\)
\(360\) 0 0
\(361\) 2.94602 0.155054
\(362\) 15.3693 0.807793
\(363\) −8.56155 −0.449365
\(364\) 0 0
\(365\) 0 0
\(366\) 23.3693 1.22153
\(367\) 1.75379 0.0915470 0.0457735 0.998952i \(-0.485425\pi\)
0.0457735 + 0.998952i \(0.485425\pi\)
\(368\) 18.7386 0.976819
\(369\) −3.56155 −0.185407
\(370\) 0 0
\(371\) 0 0
\(372\) 14.2462 0.738632
\(373\) 0.246211 0.0127483 0.00637417 0.999980i \(-0.497971\pi\)
0.00637417 + 0.999980i \(0.497971\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 72.9848 3.76391
\(377\) 3.61553 0.186209
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) −19.8078 −1.01478
\(382\) −12.4924 −0.639168
\(383\) −6.24621 −0.319166 −0.159583 0.987184i \(-0.551015\pi\)
−0.159583 + 0.987184i \(0.551015\pi\)
\(384\) −9.43845 −0.481654
\(385\) 0 0
\(386\) 19.8617 1.01094
\(387\) −4.68466 −0.238135
\(388\) 13.1231 0.666225
\(389\) 35.8617 1.81826 0.909131 0.416510i \(-0.136747\pi\)
0.909131 + 0.416510i \(0.136747\pi\)
\(390\) 0 0
\(391\) −2.43845 −0.123318
\(392\) −45.9309 −2.31986
\(393\) 14.4384 0.728323
\(394\) 22.8769 1.15252
\(395\) 0 0
\(396\) 7.12311 0.357950
\(397\) 19.3693 0.972118 0.486059 0.873926i \(-0.338434\pi\)
0.486059 + 0.873926i \(0.338434\pi\)
\(398\) 40.9848 2.05438
\(399\) 0 0
\(400\) 0 0
\(401\) 39.1771 1.95641 0.978205 0.207641i \(-0.0665787\pi\)
0.978205 + 0.207641i \(0.0665787\pi\)
\(402\) −10.2462 −0.511035
\(403\) −1.36932 −0.0682105
\(404\) 49.6155 2.46846
\(405\) 0 0
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) −6.56155 −0.324845
\(409\) −14.6847 −0.726110 −0.363055 0.931768i \(-0.618266\pi\)
−0.363055 + 0.931768i \(0.618266\pi\)
\(410\) 0 0
\(411\) −0.246211 −0.0121447
\(412\) −76.1080 −3.74957
\(413\) 0 0
\(414\) 6.24621 0.306985
\(415\) 0 0
\(416\) −2.87689 −0.141051
\(417\) −0.876894 −0.0429417
\(418\) −18.7386 −0.916537
\(419\) −0.492423 −0.0240564 −0.0120282 0.999928i \(-0.503829\pi\)
−0.0120282 + 0.999928i \(0.503829\pi\)
\(420\) 0 0
\(421\) 24.4384 1.19106 0.595529 0.803334i \(-0.296943\pi\)
0.595529 + 0.803334i \(0.296943\pi\)
\(422\) 34.2462 1.66708
\(423\) 11.1231 0.540824
\(424\) −80.3542 −3.90234
\(425\) 0 0
\(426\) −16.0000 −0.775203
\(427\) 0 0
\(428\) 21.3693 1.03292
\(429\) −0.684658 −0.0330556
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 7.68466 0.369728
\(433\) −26.6847 −1.28238 −0.641191 0.767381i \(-0.721560\pi\)
−0.641191 + 0.767381i \(0.721560\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −31.3693 −1.50232
\(437\) −11.4233 −0.546450
\(438\) 31.3693 1.49888
\(439\) −22.2462 −1.06175 −0.530877 0.847449i \(-0.678137\pi\)
−0.530877 + 0.847449i \(0.678137\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 1.12311 0.0534207
\(443\) 31.1231 1.47870 0.739352 0.673319i \(-0.235132\pi\)
0.739352 + 0.673319i \(0.235132\pi\)
\(444\) 23.3693 1.10906
\(445\) 0 0
\(446\) 38.2462 1.81101
\(447\) 12.2462 0.579226
\(448\) 0 0
\(449\) 36.7386 1.73380 0.866902 0.498479i \(-0.166108\pi\)
0.866902 + 0.498479i \(0.166108\pi\)
\(450\) 0 0
\(451\) −5.56155 −0.261883
\(452\) 2.00000 0.0940721
\(453\) 8.00000 0.375873
\(454\) 36.0000 1.68956
\(455\) 0 0
\(456\) −30.7386 −1.43947
\(457\) −13.8078 −0.645900 −0.322950 0.946416i \(-0.604675\pi\)
−0.322950 + 0.946416i \(0.604675\pi\)
\(458\) 15.3693 0.718161
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) −8.24621 −0.384064 −0.192032 0.981389i \(-0.561508\pi\)
−0.192032 + 0.981389i \(0.561508\pi\)
\(462\) 0 0
\(463\) 40.9848 1.90473 0.952364 0.304965i \(-0.0986447\pi\)
0.952364 + 0.304965i \(0.0986447\pi\)
\(464\) −63.3693 −2.94185
\(465\) 0 0
\(466\) 9.12311 0.422620
\(467\) −21.3693 −0.988854 −0.494427 0.869219i \(-0.664622\pi\)
−0.494427 + 0.869219i \(0.664622\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −6.68466 −0.308013
\(472\) 46.7386 2.15132
\(473\) −7.31534 −0.336360
\(474\) −24.0000 −1.10236
\(475\) 0 0
\(476\) 0 0
\(477\) −12.2462 −0.560715
\(478\) −16.0000 −0.731823
\(479\) 24.3002 1.11030 0.555152 0.831749i \(-0.312660\pi\)
0.555152 + 0.831749i \(0.312660\pi\)
\(480\) 0 0
\(481\) −2.24621 −0.102418
\(482\) 8.63068 0.393117
\(483\) 0 0
\(484\) −39.0540 −1.77518
\(485\) 0 0
\(486\) 2.56155 0.116194
\(487\) −17.3693 −0.787079 −0.393539 0.919308i \(-0.628750\pi\)
−0.393539 + 0.919308i \(0.628750\pi\)
\(488\) 59.8617 2.70981
\(489\) 15.1231 0.683890
\(490\) 0 0
\(491\) −21.3693 −0.964384 −0.482192 0.876066i \(-0.660159\pi\)
−0.482192 + 0.876066i \(0.660159\pi\)
\(492\) −16.2462 −0.732436
\(493\) 8.24621 0.371391
\(494\) 5.26137 0.236720
\(495\) 0 0
\(496\) 24.0000 1.07763
\(497\) 0 0
\(498\) 2.24621 0.100655
\(499\) 13.3693 0.598493 0.299246 0.954176i \(-0.403265\pi\)
0.299246 + 0.954176i \(0.403265\pi\)
\(500\) 0 0
\(501\) −19.8078 −0.884946
\(502\) 21.7538 0.970919
\(503\) 29.5616 1.31808 0.659042 0.752106i \(-0.270962\pi\)
0.659042 + 0.752106i \(0.270962\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 9.75379 0.433609
\(507\) −12.8078 −0.568813
\(508\) −90.3542 −4.00882
\(509\) 25.1231 1.11356 0.556781 0.830659i \(-0.312036\pi\)
0.556781 + 0.830659i \(0.312036\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −50.4233 −2.22842
\(513\) −4.68466 −0.206833
\(514\) 39.3693 1.73651
\(515\) 0 0
\(516\) −21.3693 −0.940732
\(517\) 17.3693 0.763902
\(518\) 0 0
\(519\) −1.80776 −0.0793520
\(520\) 0 0
\(521\) −35.5616 −1.55798 −0.778990 0.627036i \(-0.784268\pi\)
−0.778990 + 0.627036i \(0.784268\pi\)
\(522\) −21.1231 −0.924533
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 65.8617 2.87718
\(525\) 0 0
\(526\) 52.4924 2.28878
\(527\) −3.12311 −0.136045
\(528\) 12.0000 0.522233
\(529\) −17.0540 −0.741477
\(530\) 0 0
\(531\) 7.12311 0.309116
\(532\) 0 0
\(533\) 1.56155 0.0676384
\(534\) −2.87689 −0.124495
\(535\) 0 0
\(536\) −26.2462 −1.13366
\(537\) −0.876894 −0.0378408
\(538\) 42.1080 1.81540
\(539\) −10.9309 −0.470826
\(540\) 0 0
\(541\) 34.1080 1.46642 0.733208 0.680005i \(-0.238022\pi\)
0.733208 + 0.680005i \(0.238022\pi\)
\(542\) 50.7386 2.17941
\(543\) 6.00000 0.257485
\(544\) −6.56155 −0.281324
\(545\) 0 0
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −1.12311 −0.0479767
\(549\) 9.12311 0.389365
\(550\) 0 0
\(551\) 38.6307 1.64572
\(552\) 16.0000 0.681005
\(553\) 0 0
\(554\) −15.3693 −0.652980
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −26.4924 −1.12252 −0.561260 0.827640i \(-0.689683\pi\)
−0.561260 + 0.827640i \(0.689683\pi\)
\(558\) 8.00000 0.338667
\(559\) 2.05398 0.0868739
\(560\) 0 0
\(561\) −1.56155 −0.0659288
\(562\) −27.8617 −1.17528
\(563\) −31.1231 −1.31168 −0.655841 0.754899i \(-0.727686\pi\)
−0.655841 + 0.754899i \(0.727686\pi\)
\(564\) 50.7386 2.13648
\(565\) 0 0
\(566\) −54.7386 −2.30084
\(567\) 0 0
\(568\) −40.9848 −1.71969
\(569\) 21.1231 0.885527 0.442763 0.896639i \(-0.353998\pi\)
0.442763 + 0.896639i \(0.353998\pi\)
\(570\) 0 0
\(571\) −30.7386 −1.28637 −0.643186 0.765710i \(-0.722388\pi\)
−0.643186 + 0.765710i \(0.722388\pi\)
\(572\) −3.12311 −0.130584
\(573\) −4.87689 −0.203735
\(574\) 0 0
\(575\) 0 0
\(576\) 1.43845 0.0599353
\(577\) 3.94602 0.164275 0.0821376 0.996621i \(-0.473825\pi\)
0.0821376 + 0.996621i \(0.473825\pi\)
\(578\) 2.56155 0.106547
\(579\) 7.75379 0.322236
\(580\) 0 0
\(581\) 0 0
\(582\) 7.36932 0.305468
\(583\) −19.1231 −0.791998
\(584\) 80.3542 3.32508
\(585\) 0 0
\(586\) −2.87689 −0.118843
\(587\) 28.9848 1.19633 0.598166 0.801372i \(-0.295896\pi\)
0.598166 + 0.801372i \(0.295896\pi\)
\(588\) −31.9309 −1.31681
\(589\) −14.6307 −0.602847
\(590\) 0 0
\(591\) 8.93087 0.367367
\(592\) 39.3693 1.61807
\(593\) −27.7538 −1.13971 −0.569856 0.821745i \(-0.693001\pi\)
−0.569856 + 0.821745i \(0.693001\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 55.8617 2.28819
\(597\) 16.0000 0.654836
\(598\) −2.73863 −0.111991
\(599\) −0.384472 −0.0157091 −0.00785455 0.999969i \(-0.502500\pi\)
−0.00785455 + 0.999969i \(0.502500\pi\)
\(600\) 0 0
\(601\) −30.9848 −1.26390 −0.631949 0.775010i \(-0.717745\pi\)
−0.631949 + 0.775010i \(0.717745\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 36.4924 1.48486
\(605\) 0 0
\(606\) 27.8617 1.13181
\(607\) 9.36932 0.380289 0.190144 0.981756i \(-0.439104\pi\)
0.190144 + 0.981756i \(0.439104\pi\)
\(608\) −30.7386 −1.24662
\(609\) 0 0
\(610\) 0 0
\(611\) −4.87689 −0.197298
\(612\) −4.56155 −0.184390
\(613\) −14.6847 −0.593108 −0.296554 0.955016i \(-0.595837\pi\)
−0.296554 + 0.955016i \(0.595837\pi\)
\(614\) −83.2311 −3.35893
\(615\) 0 0
\(616\) 0 0
\(617\) 44.2462 1.78129 0.890643 0.454704i \(-0.150255\pi\)
0.890643 + 0.454704i \(0.150255\pi\)
\(618\) −42.7386 −1.71920
\(619\) 5.36932 0.215811 0.107906 0.994161i \(-0.465586\pi\)
0.107906 + 0.994161i \(0.465586\pi\)
\(620\) 0 0
\(621\) 2.43845 0.0978515
\(622\) 0 0
\(623\) 0 0
\(624\) −3.36932 −0.134881
\(625\) 0 0
\(626\) −86.1080 −3.44157
\(627\) −7.31534 −0.292147
\(628\) −30.4924 −1.21678
\(629\) −5.12311 −0.204272
\(630\) 0 0
\(631\) −0.684658 −0.0272558 −0.0136279 0.999907i \(-0.504338\pi\)
−0.0136279 + 0.999907i \(0.504338\pi\)
\(632\) −61.4773 −2.44543
\(633\) 13.3693 0.531383
\(634\) 46.1080 1.83118
\(635\) 0 0
\(636\) −55.8617 −2.21506
\(637\) 3.06913 0.121603
\(638\) −32.9848 −1.30588
\(639\) −6.24621 −0.247096
\(640\) 0 0
\(641\) −28.9309 −1.14270 −0.571350 0.820706i \(-0.693580\pi\)
−0.571350 + 0.820706i \(0.693580\pi\)
\(642\) 12.0000 0.473602
\(643\) 13.7538 0.542396 0.271198 0.962524i \(-0.412580\pi\)
0.271198 + 0.962524i \(0.412580\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 9.36932 0.368346 0.184173 0.982894i \(-0.441039\pi\)
0.184173 + 0.982894i \(0.441039\pi\)
\(648\) 6.56155 0.257762
\(649\) 11.1231 0.436620
\(650\) 0 0
\(651\) 0 0
\(652\) 68.9848 2.70166
\(653\) 32.9309 1.28868 0.644342 0.764737i \(-0.277131\pi\)
0.644342 + 0.764737i \(0.277131\pi\)
\(654\) −17.6155 −0.688822
\(655\) 0 0
\(656\) −27.3693 −1.06859
\(657\) 12.2462 0.477770
\(658\) 0 0
\(659\) −9.86174 −0.384159 −0.192079 0.981379i \(-0.561523\pi\)
−0.192079 + 0.981379i \(0.561523\pi\)
\(660\) 0 0
\(661\) 13.3153 0.517907 0.258953 0.965890i \(-0.416622\pi\)
0.258953 + 0.965890i \(0.416622\pi\)
\(662\) −89.4773 −3.47763
\(663\) 0.438447 0.0170279
\(664\) 5.75379 0.223290
\(665\) 0 0
\(666\) 13.1231 0.508510
\(667\) −20.1080 −0.778583
\(668\) −90.3542 −3.49591
\(669\) 14.9309 0.577261
\(670\) 0 0
\(671\) 14.2462 0.549969
\(672\) 0 0
\(673\) 0.738634 0.0284722 0.0142361 0.999899i \(-0.495468\pi\)
0.0142361 + 0.999899i \(0.495468\pi\)
\(674\) 42.8769 1.65156
\(675\) 0 0
\(676\) −58.4233 −2.24705
\(677\) 1.31534 0.0505527 0.0252763 0.999681i \(-0.491953\pi\)
0.0252763 + 0.999681i \(0.491953\pi\)
\(678\) 1.12311 0.0431326
\(679\) 0 0
\(680\) 0 0
\(681\) 14.0540 0.538550
\(682\) 12.4924 0.478360
\(683\) 9.56155 0.365863 0.182931 0.983126i \(-0.441441\pi\)
0.182931 + 0.983126i \(0.441441\pi\)
\(684\) −21.3693 −0.817076
\(685\) 0 0
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) −36.0000 −1.37249
\(689\) 5.36932 0.204555
\(690\) 0 0
\(691\) −28.9848 −1.10264 −0.551318 0.834295i \(-0.685875\pi\)
−0.551318 + 0.834295i \(0.685875\pi\)
\(692\) −8.24621 −0.313474
\(693\) 0 0
\(694\) 21.7538 0.825763
\(695\) 0 0
\(696\) −54.1080 −2.05096
\(697\) 3.56155 0.134903
\(698\) 29.6155 1.12096
\(699\) 3.56155 0.134710
\(700\) 0 0
\(701\) 15.3693 0.580491 0.290246 0.956952i \(-0.406263\pi\)
0.290246 + 0.956952i \(0.406263\pi\)
\(702\) −1.12311 −0.0423889
\(703\) −24.0000 −0.905177
\(704\) 2.24621 0.0846573
\(705\) 0 0
\(706\) 26.8769 1.01153
\(707\) 0 0
\(708\) 32.4924 1.22114
\(709\) −44.7386 −1.68019 −0.840097 0.542436i \(-0.817502\pi\)
−0.840097 + 0.542436i \(0.817502\pi\)
\(710\) 0 0
\(711\) −9.36932 −0.351377
\(712\) −7.36932 −0.276177
\(713\) 7.61553 0.285204
\(714\) 0 0
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) −6.24621 −0.233269
\(718\) 36.4924 1.36189
\(719\) −11.8078 −0.440355 −0.220178 0.975460i \(-0.570664\pi\)
−0.220178 + 0.975460i \(0.570664\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 7.54640 0.280848
\(723\) 3.36932 0.125306
\(724\) 27.3693 1.01717
\(725\) 0 0
\(726\) −21.9309 −0.813931
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.68466 0.173268
\(732\) 41.6155 1.53815
\(733\) 11.7538 0.434136 0.217068 0.976156i \(-0.430351\pi\)
0.217068 + 0.976156i \(0.430351\pi\)
\(734\) 4.49242 0.165818
\(735\) 0 0
\(736\) 16.0000 0.589768
\(737\) −6.24621 −0.230082
\(738\) −9.12311 −0.335826
\(739\) −20.6847 −0.760897 −0.380449 0.924802i \(-0.624230\pi\)
−0.380449 + 0.924802i \(0.624230\pi\)
\(740\) 0 0
\(741\) 2.05398 0.0754547
\(742\) 0 0
\(743\) −28.4924 −1.04529 −0.522643 0.852552i \(-0.675054\pi\)
−0.522643 + 0.852552i \(0.675054\pi\)
\(744\) 20.4924 0.751289
\(745\) 0 0
\(746\) 0.630683 0.0230909
\(747\) 0.876894 0.0320839
\(748\) −7.12311 −0.260447
\(749\) 0 0
\(750\) 0 0
\(751\) −25.3693 −0.925740 −0.462870 0.886426i \(-0.653180\pi\)
−0.462870 + 0.886426i \(0.653180\pi\)
\(752\) 85.4773 3.11704
\(753\) 8.49242 0.309481
\(754\) 9.26137 0.337279
\(755\) 0 0
\(756\) 0 0
\(757\) −16.0540 −0.583492 −0.291746 0.956496i \(-0.594236\pi\)
−0.291746 + 0.956496i \(0.594236\pi\)
\(758\) −30.7386 −1.11648
\(759\) 3.80776 0.138213
\(760\) 0 0
\(761\) −15.7538 −0.571074 −0.285537 0.958368i \(-0.592172\pi\)
−0.285537 + 0.958368i \(0.592172\pi\)
\(762\) −50.7386 −1.83807
\(763\) 0 0
\(764\) −22.2462 −0.804840
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) −3.12311 −0.112769
\(768\) −27.0540 −0.976226
\(769\) 40.5464 1.46214 0.731070 0.682302i \(-0.239021\pi\)
0.731070 + 0.682302i \(0.239021\pi\)
\(770\) 0 0
\(771\) 15.3693 0.553512
\(772\) 35.3693 1.27297
\(773\) 8.63068 0.310424 0.155212 0.987881i \(-0.450394\pi\)
0.155212 + 0.987881i \(0.450394\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) 18.8769 0.677641
\(777\) 0 0
\(778\) 91.8617 3.29340
\(779\) 16.6847 0.597790
\(780\) 0 0
\(781\) −9.75379 −0.349018
\(782\) −6.24621 −0.223364
\(783\) −8.24621 −0.294696
\(784\) −53.7926 −1.92116
\(785\) 0 0
\(786\) 36.9848 1.31921
\(787\) 10.2462 0.365238 0.182619 0.983184i \(-0.441543\pi\)
0.182619 + 0.983184i \(0.441543\pi\)
\(788\) 40.7386 1.45125
\(789\) 20.4924 0.729550
\(790\) 0 0
\(791\) 0 0
\(792\) 10.2462 0.364083
\(793\) −4.00000 −0.142044
\(794\) 49.6155 1.76079
\(795\) 0 0
\(796\) 72.9848 2.58688
\(797\) 9.61553 0.340599 0.170300 0.985392i \(-0.445526\pi\)
0.170300 + 0.985392i \(0.445526\pi\)
\(798\) 0 0
\(799\) −11.1231 −0.393507
\(800\) 0 0
\(801\) −1.12311 −0.0396830
\(802\) 100.354 3.54363
\(803\) 19.1231 0.674840
\(804\) −18.2462 −0.643494
\(805\) 0 0
\(806\) −3.50758 −0.123549
\(807\) 16.4384 0.578661
\(808\) 71.3693 2.51076
\(809\) 15.9460 0.560632 0.280316 0.959908i \(-0.409561\pi\)
0.280316 + 0.959908i \(0.409561\pi\)
\(810\) 0 0
\(811\) −45.3693 −1.59313 −0.796566 0.604551i \(-0.793352\pi\)
−0.796566 + 0.604551i \(0.793352\pi\)
\(812\) 0 0
\(813\) 19.8078 0.694689
\(814\) 20.4924 0.718259
\(815\) 0 0
\(816\) −7.68466 −0.269017
\(817\) 21.9460 0.767794
\(818\) −37.6155 −1.31520
\(819\) 0 0
\(820\) 0 0
\(821\) −12.4384 −0.434105 −0.217052 0.976160i \(-0.569644\pi\)
−0.217052 + 0.976160i \(0.569644\pi\)
\(822\) −0.630683 −0.0219976
\(823\) −3.50758 −0.122266 −0.0611332 0.998130i \(-0.519471\pi\)
−0.0611332 + 0.998130i \(0.519471\pi\)
\(824\) −109.477 −3.81382
\(825\) 0 0
\(826\) 0 0
\(827\) −47.4233 −1.64907 −0.824535 0.565811i \(-0.808563\pi\)
−0.824535 + 0.565811i \(0.808563\pi\)
\(828\) 11.1231 0.386555
\(829\) 17.5076 0.608063 0.304032 0.952662i \(-0.401667\pi\)
0.304032 + 0.952662i \(0.401667\pi\)
\(830\) 0 0
\(831\) −6.00000 −0.208138
\(832\) −0.630683 −0.0218650
\(833\) 7.00000 0.242536
\(834\) −2.24621 −0.0777799
\(835\) 0 0
\(836\) −33.3693 −1.15410
\(837\) 3.12311 0.107950
\(838\) −1.26137 −0.0435732
\(839\) −26.0540 −0.899483 −0.449742 0.893159i \(-0.648484\pi\)
−0.449742 + 0.893159i \(0.648484\pi\)
\(840\) 0 0
\(841\) 39.0000 1.34483
\(842\) 62.6004 2.15735
\(843\) −10.8769 −0.374620
\(844\) 60.9848 2.09918
\(845\) 0 0
\(846\) 28.4924 0.979590
\(847\) 0 0
\(848\) −94.1080 −3.23168
\(849\) −21.3693 −0.733393
\(850\) 0 0
\(851\) 12.4924 0.428235
\(852\) −28.4924 −0.976134
\(853\) 28.7386 0.983992 0.491996 0.870597i \(-0.336267\pi\)
0.491996 + 0.870597i \(0.336267\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 30.7386 1.05062
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) −1.75379 −0.0598734
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −61.4773 −2.09392
\(863\) 9.75379 0.332023 0.166011 0.986124i \(-0.446911\pi\)
0.166011 + 0.986124i \(0.446911\pi\)
\(864\) 6.56155 0.223229
\(865\) 0 0
\(866\) −68.3542 −2.32277
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −14.6307 −0.496312
\(870\) 0 0
\(871\) 1.75379 0.0594249
\(872\) −45.1231 −1.52806
\(873\) 2.87689 0.0973681
\(874\) −29.2614 −0.989780
\(875\) 0 0
\(876\) 55.8617 1.88739
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) −56.9848 −1.92315
\(879\) −1.12311 −0.0378814
\(880\) 0 0
\(881\) 40.2462 1.35593 0.677965 0.735095i \(-0.262862\pi\)
0.677965 + 0.735095i \(0.262862\pi\)
\(882\) −17.9309 −0.603764
\(883\) 23.4233 0.788257 0.394128 0.919055i \(-0.371047\pi\)
0.394128 + 0.919055i \(0.371047\pi\)
\(884\) 2.00000 0.0672673
\(885\) 0 0
\(886\) 79.7235 2.67836
\(887\) −18.4384 −0.619102 −0.309551 0.950883i \(-0.600179\pi\)
−0.309551 + 0.950883i \(0.600179\pi\)
\(888\) 33.6155 1.12806
\(889\) 0 0
\(890\) 0 0
\(891\) 1.56155 0.0523140
\(892\) 68.1080 2.28042
\(893\) −52.1080 −1.74373
\(894\) 31.3693 1.04915
\(895\) 0 0
\(896\) 0 0
\(897\) −1.06913 −0.0356972
\(898\) 94.1080 3.14042
\(899\) −25.7538 −0.858937
\(900\) 0 0
\(901\) 12.2462 0.407980
\(902\) −14.2462 −0.474347
\(903\) 0 0
\(904\) 2.87689 0.0956841
\(905\) 0 0
\(906\) 20.4924 0.680815
\(907\) 9.86174 0.327454 0.163727 0.986506i \(-0.447648\pi\)
0.163727 + 0.986506i \(0.447648\pi\)
\(908\) 64.1080 2.12750
\(909\) 10.8769 0.360764
\(910\) 0 0
\(911\) 24.3002 0.805101 0.402551 0.915398i \(-0.368124\pi\)
0.402551 + 0.915398i \(0.368124\pi\)
\(912\) −36.0000 −1.19208
\(913\) 1.36932 0.0453178
\(914\) −35.3693 −1.16991
\(915\) 0 0
\(916\) 27.3693 0.904308
\(917\) 0 0
\(918\) −2.56155 −0.0845438
\(919\) −16.6847 −0.550376 −0.275188 0.961390i \(-0.588740\pi\)
−0.275188 + 0.961390i \(0.588740\pi\)
\(920\) 0 0
\(921\) −32.4924 −1.07066
\(922\) −21.1231 −0.695652
\(923\) 2.73863 0.0901432
\(924\) 0 0
\(925\) 0 0
\(926\) 104.985 3.45002
\(927\) −16.6847 −0.547996
\(928\) −54.1080 −1.77618
\(929\) 3.06913 0.100695 0.0503474 0.998732i \(-0.483967\pi\)
0.0503474 + 0.998732i \(0.483967\pi\)
\(930\) 0 0
\(931\) 32.7926 1.07473
\(932\) 16.2462 0.532162
\(933\) 0 0
\(934\) −54.7386 −1.79110
\(935\) 0 0
\(936\) −2.87689 −0.0940342
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) −33.6155 −1.09700
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) −17.1231 −0.557901
\(943\) −8.68466 −0.282811
\(944\) 54.7386 1.78159
\(945\) 0 0
\(946\) −18.7386 −0.609246
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −42.7386 −1.38809
\(949\) −5.36932 −0.174295
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −36.3542 −1.17763 −0.588813 0.808269i \(-0.700405\pi\)
−0.588813 + 0.808269i \(0.700405\pi\)
\(954\) −31.3693 −1.01562
\(955\) 0 0
\(956\) −28.4924 −0.921511
\(957\) −12.8769 −0.416251
\(958\) 62.2462 2.01108
\(959\) 0 0
\(960\) 0 0
\(961\) −21.2462 −0.685362
\(962\) −5.75379 −0.185510
\(963\) 4.68466 0.150961
\(964\) 15.3693 0.495012
\(965\) 0 0
\(966\) 0 0
\(967\) −42.4384 −1.36473 −0.682364 0.731012i \(-0.739048\pi\)
−0.682364 + 0.731012i \(0.739048\pi\)
\(968\) −56.1771 −1.80560
\(969\) 4.68466 0.150493
\(970\) 0 0
\(971\) −43.6155 −1.39969 −0.699844 0.714295i \(-0.746747\pi\)
−0.699844 + 0.714295i \(0.746747\pi\)
\(972\) 4.56155 0.146312
\(973\) 0 0
\(974\) −44.4924 −1.42563
\(975\) 0 0
\(976\) 70.1080 2.24410
\(977\) −8.24621 −0.263820 −0.131910 0.991262i \(-0.542111\pi\)
−0.131910 + 0.991262i \(0.542111\pi\)
\(978\) 38.7386 1.23872
\(979\) −1.75379 −0.0560513
\(980\) 0 0
\(981\) −6.87689 −0.219562
\(982\) −54.7386 −1.74678
\(983\) −30.9309 −0.986542 −0.493271 0.869876i \(-0.664199\pi\)
−0.493271 + 0.869876i \(0.664199\pi\)
\(984\) −23.3693 −0.744987
\(985\) 0 0
\(986\) 21.1231 0.672697
\(987\) 0 0
\(988\) 9.36932 0.298078
\(989\) −11.4233 −0.363240
\(990\) 0 0
\(991\) 42.7386 1.35764 0.678819 0.734306i \(-0.262492\pi\)
0.678819 + 0.734306i \(0.262492\pi\)
\(992\) 20.4924 0.650635
\(993\) −34.9309 −1.10850
\(994\) 0 0
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 34.2462 1.08404
\(999\) 5.12311 0.162088
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 1275.2.a.n.1.2 2
3.2 odd 2 3825.2.a.s.1.1 2
5.2 odd 4 1275.2.b.d.1174.4 4
5.3 odd 4 1275.2.b.d.1174.1 4
5.4 even 2 51.2.a.b.1.1 2
15.14 odd 2 153.2.a.e.1.2 2
20.19 odd 2 816.2.a.m.1.2 2
35.34 odd 2 2499.2.a.o.1.1 2
40.19 odd 2 3264.2.a.bg.1.1 2
40.29 even 2 3264.2.a.bl.1.1 2
55.54 odd 2 6171.2.a.p.1.2 2
60.59 even 2 2448.2.a.v.1.1 2
65.64 even 2 8619.2.a.q.1.2 2
85.4 even 4 867.2.d.c.577.4 4
85.9 even 8 867.2.e.f.829.3 8
85.14 odd 16 867.2.h.j.757.2 16
85.19 even 8 867.2.e.f.616.1 8
85.24 odd 16 867.2.h.j.712.4 16
85.29 odd 16 867.2.h.j.688.3 16
85.39 odd 16 867.2.h.j.688.4 16
85.44 odd 16 867.2.h.j.712.3 16
85.49 even 8 867.2.e.f.616.2 8
85.54 odd 16 867.2.h.j.757.1 16
85.59 even 8 867.2.e.f.829.4 8
85.64 even 4 867.2.d.c.577.3 4
85.74 odd 16 867.2.h.j.733.1 16
85.79 odd 16 867.2.h.j.733.2 16
85.84 even 2 867.2.a.f.1.1 2
105.104 even 2 7497.2.a.v.1.2 2
120.29 odd 2 9792.2.a.cy.1.2 2
120.59 even 2 9792.2.a.cz.1.2 2
255.254 odd 2 2601.2.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
51.2.a.b.1.1 2 5.4 even 2
153.2.a.e.1.2 2 15.14 odd 2
816.2.a.m.1.2 2 20.19 odd 2
867.2.a.f.1.1 2 85.84 even 2
867.2.d.c.577.3 4 85.64 even 4
867.2.d.c.577.4 4 85.4 even 4
867.2.e.f.616.1 8 85.19 even 8
867.2.e.f.616.2 8 85.49 even 8
867.2.e.f.829.3 8 85.9 even 8
867.2.e.f.829.4 8 85.59 even 8
867.2.h.j.688.3 16 85.29 odd 16
867.2.h.j.688.4 16 85.39 odd 16
867.2.h.j.712.3 16 85.44 odd 16
867.2.h.j.712.4 16 85.24 odd 16
867.2.h.j.733.1 16 85.74 odd 16
867.2.h.j.733.2 16 85.79 odd 16
867.2.h.j.757.1 16 85.54 odd 16
867.2.h.j.757.2 16 85.14 odd 16
1275.2.a.n.1.2 2 1.1 even 1 trivial
1275.2.b.d.1174.1 4 5.3 odd 4
1275.2.b.d.1174.4 4 5.2 odd 4
2448.2.a.v.1.1 2 60.59 even 2
2499.2.a.o.1.1 2 35.34 odd 2
2601.2.a.t.1.2 2 255.254 odd 2
3264.2.a.bg.1.1 2 40.19 odd 2
3264.2.a.bl.1.1 2 40.29 even 2
3825.2.a.s.1.1 2 3.2 odd 2
6171.2.a.p.1.2 2 55.54 odd 2
7497.2.a.v.1.2 2 105.104 even 2
8619.2.a.q.1.2 2 65.64 even 2
9792.2.a.cy.1.2 2 120.29 odd 2
9792.2.a.cz.1.2 2 120.59 even 2