Properties

Label 5415.2.a.a.1.1
Level $5415$
Weight $2$
Character 5415.1
Self dual yes
Analytic conductor $43.239$
Analytic rank $2$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5415,2,Mod(1,5415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5415, 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("5415.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5415.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: \(43.2389926945\)
Analytic rank: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5415.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -2.00000 q^{7} +1.00000 q^{9} +2.00000 q^{10} -3.00000 q^{11} -2.00000 q^{12} -6.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} -4.00000 q^{16} +6.00000 q^{17} -2.00000 q^{18} -2.00000 q^{20} +2.00000 q^{21} +6.00000 q^{22} -8.00000 q^{23} +1.00000 q^{25} +12.0000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -7.00000 q^{29} -2.00000 q^{30} -9.00000 q^{31} +8.00000 q^{32} +3.00000 q^{33} -12.0000 q^{34} +2.00000 q^{35} +2.00000 q^{36} +2.00000 q^{37} +6.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} +10.0000 q^{43} -6.00000 q^{44} -1.00000 q^{45} +16.0000 q^{46} +4.00000 q^{47} +4.00000 q^{48} -3.00000 q^{49} -2.00000 q^{50} -6.00000 q^{51} -12.0000 q^{52} -14.0000 q^{53} +2.00000 q^{54} +3.00000 q^{55} +14.0000 q^{58} -3.00000 q^{59} +2.00000 q^{60} -7.00000 q^{61} +18.0000 q^{62} -2.00000 q^{63} -8.00000 q^{64} +6.00000 q^{65} -6.00000 q^{66} -4.00000 q^{67} +12.0000 q^{68} +8.00000 q^{69} -4.00000 q^{70} -7.00000 q^{71} +2.00000 q^{73} -4.00000 q^{74} -1.00000 q^{75} +6.00000 q^{77} -12.0000 q^{78} -5.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} -6.00000 q^{83} +4.00000 q^{84} -6.00000 q^{85} -20.0000 q^{86} +7.00000 q^{87} -3.00000 q^{89} +2.00000 q^{90} +12.0000 q^{91} -16.0000 q^{92} +9.00000 q^{93} -8.00000 q^{94} -8.00000 q^{96} +12.0000 q^{97} +6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.00000 1.00000
\(5\) −1.00000 −0.447214
\(6\) 2.00000 0.816497
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −2.00000 −0.577350
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 4.00000 1.06904
\(15\) 1.00000 0.258199
\(16\) −4.00000 −1.00000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −2.00000 −0.471405
\(19\) 0 0
\(20\) −2.00000 −0.447214
\(21\) 2.00000 0.436436
\(22\) 6.00000 1.27920
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 12.0000 2.35339
\(27\) −1.00000 −0.192450
\(28\) −4.00000 −0.755929
\(29\) −7.00000 −1.29987 −0.649934 0.759991i \(-0.725203\pi\)
−0.649934 + 0.759991i \(0.725203\pi\)
\(30\) −2.00000 −0.365148
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 8.00000 1.41421
\(33\) 3.00000 0.522233
\(34\) −12.0000 −2.05798
\(35\) 2.00000 0.338062
\(36\) 2.00000 0.333333
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −4.00000 −0.617213
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) −6.00000 −0.904534
\(45\) −1.00000 −0.149071
\(46\) 16.0000 2.35907
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 4.00000 0.577350
\(49\) −3.00000 −0.428571
\(50\) −2.00000 −0.282843
\(51\) −6.00000 −0.840168
\(52\) −12.0000 −1.66410
\(53\) −14.0000 −1.92305 −0.961524 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 2.00000 0.272166
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 14.0000 1.83829
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 2.00000 0.258199
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 18.0000 2.28600
\(63\) −2.00000 −0.251976
\(64\) −8.00000 −1.00000
\(65\) 6.00000 0.744208
\(66\) −6.00000 −0.738549
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 12.0000 1.45521
\(69\) 8.00000 0.963087
\(70\) −4.00000 −0.478091
\(71\) −7.00000 −0.830747 −0.415374 0.909651i \(-0.636349\pi\)
−0.415374 + 0.909651i \(0.636349\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −4.00000 −0.464991
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) −12.0000 −1.35873
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 4.00000 0.436436
\(85\) −6.00000 −0.650791
\(86\) −20.0000 −2.15666
\(87\) 7.00000 0.750479
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 2.00000 0.210819
\(91\) 12.0000 1.25794
\(92\) −16.0000 −1.66812
\(93\) 9.00000 0.933257
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 6.00000 0.606092
\(99\) −3.00000 −0.301511
\(100\) 2.00000 0.200000
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 12.0000 1.18818
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 28.0000 2.71960
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −2.00000 −0.192450
\(109\) −3.00000 −0.287348 −0.143674 0.989625i \(-0.545892\pi\)
−0.143674 + 0.989625i \(0.545892\pi\)
\(110\) −6.00000 −0.572078
\(111\) −2.00000 −0.189832
\(112\) 8.00000 0.755929
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) −14.0000 −1.29987
\(117\) −6.00000 −0.554700
\(118\) 6.00000 0.552345
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 14.0000 1.26750
\(123\) 6.00000 0.541002
\(124\) −18.0000 −1.61645
\(125\) −1.00000 −0.0894427
\(126\) 4.00000 0.356348
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) −10.0000 −0.880451
\(130\) −12.0000 −1.05247
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 6.00000 0.522233
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −16.0000 −1.36201
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 4.00000 0.338062
\(141\) −4.00000 −0.336861
\(142\) 14.0000 1.17485
\(143\) 18.0000 1.50524
\(144\) −4.00000 −0.333333
\(145\) 7.00000 0.581318
\(146\) −4.00000 −0.331042
\(147\) 3.00000 0.247436
\(148\) 4.00000 0.328798
\(149\) −3.00000 −0.245770 −0.122885 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(150\) 2.00000 0.163299
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) −12.0000 −0.966988
\(155\) 9.00000 0.722897
\(156\) 12.0000 0.960769
\(157\) −20.0000 −1.59617 −0.798087 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) 10.0000 0.795557
\(159\) 14.0000 1.11027
\(160\) −8.00000 −0.632456
\(161\) 16.0000 1.26098
\(162\) −2.00000 −0.157135
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) −12.0000 −0.937043
\(165\) −3.00000 −0.233550
\(166\) 12.0000 0.931381
\(167\) −22.0000 −1.70241 −0.851206 0.524832i \(-0.824128\pi\)
−0.851206 + 0.524832i \(0.824128\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 12.0000 0.920358
\(171\) 0 0
\(172\) 20.0000 1.52499
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −14.0000 −1.06134
\(175\) −2.00000 −0.151186
\(176\) 12.0000 0.904534
\(177\) 3.00000 0.225494
\(178\) 6.00000 0.449719
\(179\) −1.00000 −0.0747435 −0.0373718 0.999301i \(-0.511899\pi\)
−0.0373718 + 0.999301i \(0.511899\pi\)
\(180\) −2.00000 −0.149071
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −24.0000 −1.77900
\(183\) 7.00000 0.517455
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) −18.0000 −1.31982
\(187\) −18.0000 −1.31629
\(188\) 8.00000 0.583460
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −5.00000 −0.361787 −0.180894 0.983503i \(-0.557899\pi\)
−0.180894 + 0.983503i \(0.557899\pi\)
\(192\) 8.00000 0.577350
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) −24.0000 −1.72310
\(195\) −6.00000 −0.429669
\(196\) −6.00000 −0.428571
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 6.00000 0.426401
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 6.00000 0.422159
\(203\) 14.0000 0.982607
\(204\) −12.0000 −0.840168
\(205\) 6.00000 0.419058
\(206\) −16.0000 −1.11477
\(207\) −8.00000 −0.556038
\(208\) 24.0000 1.66410
\(209\) 0 0
\(210\) 4.00000 0.276026
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) −28.0000 −1.92305
\(213\) 7.00000 0.479632
\(214\) 24.0000 1.64061
\(215\) −10.0000 −0.681994
\(216\) 0 0
\(217\) 18.0000 1.22192
\(218\) 6.00000 0.406371
\(219\) −2.00000 −0.135147
\(220\) 6.00000 0.404520
\(221\) −36.0000 −2.42162
\(222\) 4.00000 0.268462
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −16.0000 −1.06904
\(225\) 1.00000 0.0666667
\(226\) 12.0000 0.798228
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 0 0
\(229\) 29.0000 1.91637 0.958187 0.286143i \(-0.0923732\pi\)
0.958187 + 0.286143i \(0.0923732\pi\)
\(230\) −16.0000 −1.05501
\(231\) −6.00000 −0.394771
\(232\) 0 0
\(233\) 16.0000 1.04819 0.524097 0.851658i \(-0.324403\pi\)
0.524097 + 0.851658i \(0.324403\pi\)
\(234\) 12.0000 0.784465
\(235\) −4.00000 −0.260931
\(236\) −6.00000 −0.390567
\(237\) 5.00000 0.324785
\(238\) 24.0000 1.55569
\(239\) −9.00000 −0.582162 −0.291081 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) −4.00000 −0.258199
\(241\) −21.0000 −1.35273 −0.676364 0.736567i \(-0.736446\pi\)
−0.676364 + 0.736567i \(0.736446\pi\)
\(242\) 4.00000 0.257130
\(243\) −1.00000 −0.0641500
\(244\) −14.0000 −0.896258
\(245\) 3.00000 0.191663
\(246\) −12.0000 −0.765092
\(247\) 0 0
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 2.00000 0.126491
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) −4.00000 −0.251976
\(253\) 24.0000 1.50887
\(254\) −16.0000 −1.00393
\(255\) 6.00000 0.375735
\(256\) 16.0000 1.00000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 20.0000 1.24515
\(259\) −4.00000 −0.248548
\(260\) 12.0000 0.744208
\(261\) −7.00000 −0.433289
\(262\) 16.0000 0.988483
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 14.0000 0.860013
\(266\) 0 0
\(267\) 3.00000 0.183597
\(268\) −8.00000 −0.488678
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) −2.00000 −0.121716
\(271\) −5.00000 −0.303728 −0.151864 0.988401i \(-0.548528\pi\)
−0.151864 + 0.988401i \(0.548528\pi\)
\(272\) −24.0000 −1.45521
\(273\) −12.0000 −0.726273
\(274\) −4.00000 −0.241649
\(275\) −3.00000 −0.180907
\(276\) 16.0000 0.963087
\(277\) 16.0000 0.961347 0.480673 0.876900i \(-0.340392\pi\)
0.480673 + 0.876900i \(0.340392\pi\)
\(278\) 8.00000 0.479808
\(279\) −9.00000 −0.538816
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 8.00000 0.476393
\(283\) −26.0000 −1.54554 −0.772770 0.634686i \(-0.781129\pi\)
−0.772770 + 0.634686i \(0.781129\pi\)
\(284\) −14.0000 −0.830747
\(285\) 0 0
\(286\) −36.0000 −2.12872
\(287\) 12.0000 0.708338
\(288\) 8.00000 0.471405
\(289\) 19.0000 1.11765
\(290\) −14.0000 −0.822108
\(291\) −12.0000 −0.703452
\(292\) 4.00000 0.234082
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) −6.00000 −0.349927
\(295\) 3.00000 0.174667
\(296\) 0 0
\(297\) 3.00000 0.174078
\(298\) 6.00000 0.347571
\(299\) 48.0000 2.77591
\(300\) −2.00000 −0.115470
\(301\) −20.0000 −1.15278
\(302\) 10.0000 0.575435
\(303\) 3.00000 0.172345
\(304\) 0 0
\(305\) 7.00000 0.400819
\(306\) −12.0000 −0.685994
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 12.0000 0.683763
\(309\) −8.00000 −0.455104
\(310\) −18.0000 −1.02233
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 40.0000 2.25733
\(315\) 2.00000 0.112687
\(316\) −10.0000 −0.562544
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) −28.0000 −1.57016
\(319\) 21.0000 1.17577
\(320\) 8.00000 0.447214
\(321\) 12.0000 0.669775
\(322\) −32.0000 −1.78329
\(323\) 0 0
\(324\) 2.00000 0.111111
\(325\) −6.00000 −0.332820
\(326\) 20.0000 1.10770
\(327\) 3.00000 0.165900
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 6.00000 0.330289
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −12.0000 −0.658586
\(333\) 2.00000 0.109599
\(334\) 44.0000 2.40757
\(335\) 4.00000 0.218543
\(336\) −8.00000 −0.436436
\(337\) 12.0000 0.653682 0.326841 0.945079i \(-0.394016\pi\)
0.326841 + 0.945079i \(0.394016\pi\)
\(338\) −46.0000 −2.50207
\(339\) 6.00000 0.325875
\(340\) −12.0000 −0.650791
\(341\) 27.0000 1.46213
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) −8.00000 −0.430706
\(346\) 12.0000 0.645124
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) 14.0000 0.750479
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 4.00000 0.213809
\(351\) 6.00000 0.320256
\(352\) −24.0000 −1.27920
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −6.00000 −0.318896
\(355\) 7.00000 0.371521
\(356\) −6.00000 −0.317999
\(357\) 12.0000 0.635107
\(358\) 2.00000 0.105703
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) −28.0000 −1.47165
\(363\) 2.00000 0.104973
\(364\) 24.0000 1.25794
\(365\) −2.00000 −0.104685
\(366\) −14.0000 −0.731792
\(367\) −20.0000 −1.04399 −0.521996 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 32.0000 1.66812
\(369\) −6.00000 −0.312348
\(370\) 4.00000 0.207950
\(371\) 28.0000 1.45369
\(372\) 18.0000 0.933257
\(373\) 24.0000 1.24267 0.621336 0.783544i \(-0.286590\pi\)
0.621336 + 0.783544i \(0.286590\pi\)
\(374\) 36.0000 1.86152
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 42.0000 2.16311
\(378\) −4.00000 −0.205738
\(379\) 25.0000 1.28416 0.642082 0.766636i \(-0.278071\pi\)
0.642082 + 0.766636i \(0.278071\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 10.0000 0.511645
\(383\) 34.0000 1.73732 0.868659 0.495410i \(-0.164982\pi\)
0.868659 + 0.495410i \(0.164982\pi\)
\(384\) 0 0
\(385\) −6.00000 −0.305788
\(386\) −36.0000 −1.83235
\(387\) 10.0000 0.508329
\(388\) 24.0000 1.21842
\(389\) 37.0000 1.87597 0.937987 0.346670i \(-0.112688\pi\)
0.937987 + 0.346670i \(0.112688\pi\)
\(390\) 12.0000 0.607644
\(391\) −48.0000 −2.42746
\(392\) 0 0
\(393\) 8.00000 0.403547
\(394\) 36.0000 1.81365
\(395\) 5.00000 0.251577
\(396\) −6.00000 −0.301511
\(397\) 12.0000 0.602263 0.301131 0.953583i \(-0.402636\pi\)
0.301131 + 0.953583i \(0.402636\pi\)
\(398\) 22.0000 1.10276
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 23.0000 1.14857 0.574283 0.818657i \(-0.305281\pi\)
0.574283 + 0.818657i \(0.305281\pi\)
\(402\) −8.00000 −0.399004
\(403\) 54.0000 2.68993
\(404\) −6.00000 −0.298511
\(405\) −1.00000 −0.0496904
\(406\) −28.0000 −1.38962
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) −11.0000 −0.543915 −0.271957 0.962309i \(-0.587671\pi\)
−0.271957 + 0.962309i \(0.587671\pi\)
\(410\) −12.0000 −0.592638
\(411\) −2.00000 −0.0986527
\(412\) 16.0000 0.788263
\(413\) 6.00000 0.295241
\(414\) 16.0000 0.786357
\(415\) 6.00000 0.294528
\(416\) −48.0000 −2.35339
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) −4.00000 −0.195180
\(421\) −9.00000 −0.438633 −0.219317 0.975654i \(-0.570383\pi\)
−0.219317 + 0.975654i \(0.570383\pi\)
\(422\) 26.0000 1.26566
\(423\) 4.00000 0.194487
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) −14.0000 −0.678302
\(427\) 14.0000 0.677507
\(428\) −24.0000 −1.16008
\(429\) −18.0000 −0.869048
\(430\) 20.0000 0.964486
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) 4.00000 0.192450
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) −36.0000 −1.72806
\(435\) −7.00000 −0.335624
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) 9.00000 0.429547 0.214773 0.976664i \(-0.431099\pi\)
0.214773 + 0.976664i \(0.431099\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 72.0000 3.42469
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) −4.00000 −0.189832
\(445\) 3.00000 0.142214
\(446\) 16.0000 0.757622
\(447\) 3.00000 0.141895
\(448\) 16.0000 0.755929
\(449\) −13.0000 −0.613508 −0.306754 0.951789i \(-0.599243\pi\)
−0.306754 + 0.951789i \(0.599243\pi\)
\(450\) −2.00000 −0.0942809
\(451\) 18.0000 0.847587
\(452\) −12.0000 −0.564433
\(453\) 5.00000 0.234920
\(454\) 28.0000 1.31411
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) 12.0000 0.561336 0.280668 0.959805i \(-0.409444\pi\)
0.280668 + 0.959805i \(0.409444\pi\)
\(458\) −58.0000 −2.71016
\(459\) −6.00000 −0.280056
\(460\) 16.0000 0.746004
\(461\) 33.0000 1.53696 0.768482 0.639872i \(-0.221013\pi\)
0.768482 + 0.639872i \(0.221013\pi\)
\(462\) 12.0000 0.558291
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) 28.0000 1.29987
\(465\) −9.00000 −0.417365
\(466\) −32.0000 −1.48237
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) −12.0000 −0.554700
\(469\) 8.00000 0.369406
\(470\) 8.00000 0.369012
\(471\) 20.0000 0.921551
\(472\) 0 0
\(473\) −30.0000 −1.37940
\(474\) −10.0000 −0.459315
\(475\) 0 0
\(476\) −24.0000 −1.10004
\(477\) −14.0000 −0.641016
\(478\) 18.0000 0.823301
\(479\) −15.0000 −0.685367 −0.342684 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(480\) 8.00000 0.365148
\(481\) −12.0000 −0.547153
\(482\) 42.0000 1.91305
\(483\) −16.0000 −0.728025
\(484\) −4.00000 −0.181818
\(485\) −12.0000 −0.544892
\(486\) 2.00000 0.0907218
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) 0 0
\(489\) 10.0000 0.452216
\(490\) −6.00000 −0.271052
\(491\) −5.00000 −0.225647 −0.112823 0.993615i \(-0.535989\pi\)
−0.112823 + 0.993615i \(0.535989\pi\)
\(492\) 12.0000 0.541002
\(493\) −42.0000 −1.89158
\(494\) 0 0
\(495\) 3.00000 0.134840
\(496\) 36.0000 1.61645
\(497\) 14.0000 0.627986
\(498\) −12.0000 −0.537733
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) −2.00000 −0.0894427
\(501\) 22.0000 0.982888
\(502\) −6.00000 −0.267793
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 3.00000 0.133498
\(506\) −48.0000 −2.13386
\(507\) −23.0000 −1.02147
\(508\) 16.0000 0.709885
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) −12.0000 −0.531369
\(511\) −4.00000 −0.176950
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) 4.00000 0.176432
\(515\) −8.00000 −0.352522
\(516\) −20.0000 −0.880451
\(517\) −12.0000 −0.527759
\(518\) 8.00000 0.351500
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 7.00000 0.306676 0.153338 0.988174i \(-0.450998\pi\)
0.153338 + 0.988174i \(0.450998\pi\)
\(522\) 14.0000 0.612763
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) −16.0000 −0.698963
\(525\) 2.00000 0.0872872
\(526\) −24.0000 −1.04645
\(527\) −54.0000 −2.35228
\(528\) −12.0000 −0.522233
\(529\) 41.0000 1.78261
\(530\) −28.0000 −1.21624
\(531\) −3.00000 −0.130189
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) −6.00000 −0.259645
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) 1.00000 0.0431532
\(538\) 42.0000 1.81075
\(539\) 9.00000 0.387657
\(540\) 2.00000 0.0860663
\(541\) −9.00000 −0.386940 −0.193470 0.981106i \(-0.561974\pi\)
−0.193470 + 0.981106i \(0.561974\pi\)
\(542\) 10.0000 0.429537
\(543\) −14.0000 −0.600798
\(544\) 48.0000 2.05798
\(545\) 3.00000 0.128506
\(546\) 24.0000 1.02711
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 4.00000 0.170872
\(549\) −7.00000 −0.298753
\(550\) 6.00000 0.255841
\(551\) 0 0
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) −32.0000 −1.35955
\(555\) 2.00000 0.0848953
\(556\) −8.00000 −0.339276
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 18.0000 0.762001
\(559\) −60.0000 −2.53773
\(560\) −8.00000 −0.338062
\(561\) 18.0000 0.759961
\(562\) 28.0000 1.18111
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) −8.00000 −0.336861
\(565\) 6.00000 0.252422
\(566\) 52.0000 2.18572
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) 11.0000 0.461144 0.230572 0.973055i \(-0.425940\pi\)
0.230572 + 0.973055i \(0.425940\pi\)
\(570\) 0 0
\(571\) −15.0000 −0.627730 −0.313865 0.949468i \(-0.601624\pi\)
−0.313865 + 0.949468i \(0.601624\pi\)
\(572\) 36.0000 1.50524
\(573\) 5.00000 0.208878
\(574\) −24.0000 −1.00174
\(575\) −8.00000 −0.333623
\(576\) −8.00000 −0.333333
\(577\) 32.0000 1.33218 0.666089 0.745873i \(-0.267967\pi\)
0.666089 + 0.745873i \(0.267967\pi\)
\(578\) −38.0000 −1.58059
\(579\) −18.0000 −0.748054
\(580\) 14.0000 0.581318
\(581\) 12.0000 0.497844
\(582\) 24.0000 0.994832
\(583\) 42.0000 1.73946
\(584\) 0 0
\(585\) 6.00000 0.248069
\(586\) −24.0000 −0.991431
\(587\) 22.0000 0.908037 0.454019 0.890992i \(-0.349990\pi\)
0.454019 + 0.890992i \(0.349990\pi\)
\(588\) 6.00000 0.247436
\(589\) 0 0
\(590\) −6.00000 −0.247016
\(591\) 18.0000 0.740421
\(592\) −8.00000 −0.328798
\(593\) −10.0000 −0.410651 −0.205325 0.978694i \(-0.565825\pi\)
−0.205325 + 0.978694i \(0.565825\pi\)
\(594\) −6.00000 −0.246183
\(595\) 12.0000 0.491952
\(596\) −6.00000 −0.245770
\(597\) 11.0000 0.450200
\(598\) −96.0000 −3.92573
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) −1.00000 −0.0407909 −0.0203954 0.999792i \(-0.506493\pi\)
−0.0203954 + 0.999792i \(0.506493\pi\)
\(602\) 40.0000 1.63028
\(603\) −4.00000 −0.162893
\(604\) −10.0000 −0.406894
\(605\) 2.00000 0.0813116
\(606\) −6.00000 −0.243733
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) −14.0000 −0.567309
\(610\) −14.0000 −0.566843
\(611\) −24.0000 −0.970936
\(612\) 12.0000 0.485071
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 32.0000 1.29141
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 28.0000 1.12724 0.563619 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) 16.0000 0.643614
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) 18.0000 0.722897
\(621\) 8.00000 0.321029
\(622\) 16.0000 0.641542
\(623\) 6.00000 0.240385
\(624\) −24.0000 −0.960769
\(625\) 1.00000 0.0400000
\(626\) 16.0000 0.639489
\(627\) 0 0
\(628\) −40.0000 −1.59617
\(629\) 12.0000 0.478471
\(630\) −4.00000 −0.159364
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 0 0
\(633\) 13.0000 0.516704
\(634\) 24.0000 0.953162
\(635\) −8.00000 −0.317470
\(636\) 28.0000 1.11027
\(637\) 18.0000 0.713186
\(638\) −42.0000 −1.66280
\(639\) −7.00000 −0.276916
\(640\) 0 0
\(641\) −19.0000 −0.750455 −0.375227 0.926933i \(-0.622435\pi\)
−0.375227 + 0.926933i \(0.622435\pi\)
\(642\) −24.0000 −0.947204
\(643\) 32.0000 1.26196 0.630978 0.775800i \(-0.282654\pi\)
0.630978 + 0.775800i \(0.282654\pi\)
\(644\) 32.0000 1.26098
\(645\) 10.0000 0.393750
\(646\) 0 0
\(647\) 2.00000 0.0786281 0.0393141 0.999227i \(-0.487483\pi\)
0.0393141 + 0.999227i \(0.487483\pi\)
\(648\) 0 0
\(649\) 9.00000 0.353281
\(650\) 12.0000 0.470679
\(651\) −18.0000 −0.705476
\(652\) −20.0000 −0.783260
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −6.00000 −0.234619
\(655\) 8.00000 0.312586
\(656\) 24.0000 0.937043
\(657\) 2.00000 0.0780274
\(658\) 16.0000 0.623745
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) −6.00000 −0.233550
\(661\) −17.0000 −0.661223 −0.330612 0.943767i \(-0.607255\pi\)
−0.330612 + 0.943767i \(0.607255\pi\)
\(662\) −16.0000 −0.621858
\(663\) 36.0000 1.39812
\(664\) 0 0
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 56.0000 2.16833
\(668\) −44.0000 −1.70241
\(669\) 8.00000 0.309298
\(670\) −8.00000 −0.309067
\(671\) 21.0000 0.810696
\(672\) 16.0000 0.617213
\(673\) 36.0000 1.38770 0.693849 0.720121i \(-0.255914\pi\)
0.693849 + 0.720121i \(0.255914\pi\)
\(674\) −24.0000 −0.924445
\(675\) −1.00000 −0.0384900
\(676\) 46.0000 1.76923
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) −12.0000 −0.460857
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 14.0000 0.536481
\(682\) −54.0000 −2.06777
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) −2.00000 −0.0764161
\(686\) −40.0000 −1.52721
\(687\) −29.0000 −1.10642
\(688\) −40.0000 −1.52499
\(689\) 84.0000 3.20015
\(690\) 16.0000 0.609110
\(691\) −1.00000 −0.0380418 −0.0190209 0.999819i \(-0.506055\pi\)
−0.0190209 + 0.999819i \(0.506055\pi\)
\(692\) −12.0000 −0.456172
\(693\) 6.00000 0.227921
\(694\) −4.00000 −0.151838
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 4.00000 0.151402
\(699\) −16.0000 −0.605176
\(700\) −4.00000 −0.151186
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) −12.0000 −0.452911
\(703\) 0 0
\(704\) 24.0000 0.904534
\(705\) 4.00000 0.150649
\(706\) −4.00000 −0.150542
\(707\) 6.00000 0.225653
\(708\) 6.00000 0.225494
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) −14.0000 −0.525411
\(711\) −5.00000 −0.187515
\(712\) 0 0
\(713\) 72.0000 2.69642
\(714\) −24.0000 −0.898177
\(715\) −18.0000 −0.673162
\(716\) −2.00000 −0.0747435
\(717\) 9.00000 0.336111
\(718\) 64.0000 2.38846
\(719\) 33.0000 1.23069 0.615346 0.788257i \(-0.289016\pi\)
0.615346 + 0.788257i \(0.289016\pi\)
\(720\) 4.00000 0.149071
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) 21.0000 0.780998
\(724\) 28.0000 1.04061
\(725\) −7.00000 −0.259973
\(726\) −4.00000 −0.148454
\(727\) 14.0000 0.519231 0.259616 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 4.00000 0.148047
\(731\) 60.0000 2.21918
\(732\) 14.0000 0.517455
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 40.0000 1.47643
\(735\) −3.00000 −0.110657
\(736\) −64.0000 −2.35907
\(737\) 12.0000 0.442026
\(738\) 12.0000 0.441726
\(739\) −29.0000 −1.06678 −0.533391 0.845869i \(-0.679083\pi\)
−0.533391 + 0.845869i \(0.679083\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) −56.0000 −2.05582
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 0 0
\(745\) 3.00000 0.109911
\(746\) −48.0000 −1.75740
\(747\) −6.00000 −0.219529
\(748\) −36.0000 −1.31629
\(749\) 24.0000 0.876941
\(750\) −2.00000 −0.0730297
\(751\) −29.0000 −1.05823 −0.529113 0.848552i \(-0.677475\pi\)
−0.529113 + 0.848552i \(0.677475\pi\)
\(752\) −16.0000 −0.583460
\(753\) −3.00000 −0.109326
\(754\) −84.0000 −3.05910
\(755\) 5.00000 0.181969
\(756\) 4.00000 0.145479
\(757\) 46.0000 1.67190 0.835949 0.548807i \(-0.184918\pi\)
0.835949 + 0.548807i \(0.184918\pi\)
\(758\) −50.0000 −1.81608
\(759\) −24.0000 −0.871145
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 16.0000 0.579619
\(763\) 6.00000 0.217215
\(764\) −10.0000 −0.361787
\(765\) −6.00000 −0.216930
\(766\) −68.0000 −2.45694
\(767\) 18.0000 0.649942
\(768\) −16.0000 −0.577350
\(769\) 13.0000 0.468792 0.234396 0.972141i \(-0.424689\pi\)
0.234396 + 0.972141i \(0.424689\pi\)
\(770\) 12.0000 0.432450
\(771\) 2.00000 0.0720282
\(772\) 36.0000 1.29567
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) −20.0000 −0.718885
\(775\) −9.00000 −0.323290
\(776\) 0 0
\(777\) 4.00000 0.143499
\(778\) −74.0000 −2.65303
\(779\) 0 0
\(780\) −12.0000 −0.429669
\(781\) 21.0000 0.751439
\(782\) 96.0000 3.43295
\(783\) 7.00000 0.250160
\(784\) 12.0000 0.428571
\(785\) 20.0000 0.713831
\(786\) −16.0000 −0.570701
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −36.0000 −1.28245
\(789\) −12.0000 −0.427211
\(790\) −10.0000 −0.355784
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 42.0000 1.49146
\(794\) −24.0000 −0.851728
\(795\) −14.0000 −0.496529
\(796\) −22.0000 −0.779769
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 8.00000 0.282843
\(801\) −3.00000 −0.106000
\(802\) −46.0000 −1.62432
\(803\) −6.00000 −0.211735
\(804\) 8.00000 0.282138
\(805\) −16.0000 −0.563926
\(806\) −108.000 −3.80414
\(807\) 21.0000 0.739235
\(808\) 0 0
\(809\) −37.0000 −1.30085 −0.650425 0.759570i \(-0.725409\pi\)
−0.650425 + 0.759570i \(0.725409\pi\)
\(810\) 2.00000 0.0702728
\(811\) 7.00000 0.245803 0.122902 0.992419i \(-0.460780\pi\)
0.122902 + 0.992419i \(0.460780\pi\)
\(812\) 28.0000 0.982607
\(813\) 5.00000 0.175358
\(814\) 12.0000 0.420600
\(815\) 10.0000 0.350285
\(816\) 24.0000 0.840168
\(817\) 0 0
\(818\) 22.0000 0.769212
\(819\) 12.0000 0.419314
\(820\) 12.0000 0.419058
\(821\) 15.0000 0.523504 0.261752 0.965135i \(-0.415700\pi\)
0.261752 + 0.965135i \(0.415700\pi\)
\(822\) 4.00000 0.139516
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) 3.00000 0.104447
\(826\) −12.0000 −0.417533
\(827\) −48.0000 −1.66912 −0.834562 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) −16.0000 −0.556038
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) −12.0000 −0.416526
\(831\) −16.0000 −0.555034
\(832\) 48.0000 1.66410
\(833\) −18.0000 −0.623663
\(834\) −8.00000 −0.277017
\(835\) 22.0000 0.761341
\(836\) 0 0
\(837\) 9.00000 0.311086
\(838\) 18.0000 0.621800
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 18.0000 0.620321
\(843\) 14.0000 0.482186
\(844\) −26.0000 −0.894957
\(845\) −23.0000 −0.791224
\(846\) −8.00000 −0.275046
\(847\) 4.00000 0.137442
\(848\) 56.0000 1.92305
\(849\) 26.0000 0.892318
\(850\) −12.0000 −0.411597
\(851\) −16.0000 −0.548473
\(852\) 14.0000 0.479632
\(853\) −34.0000 −1.16414 −0.582069 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) −28.0000 −0.958140
\(855\) 0 0
\(856\) 0 0
\(857\) 26.0000 0.888143 0.444072 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) 36.0000 1.22902
\(859\) 5.00000 0.170598 0.0852989 0.996355i \(-0.472815\pi\)
0.0852989 + 0.996355i \(0.472815\pi\)
\(860\) −20.0000 −0.681994
\(861\) −12.0000 −0.408959
\(862\) −42.0000 −1.43053
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) −8.00000 −0.272166
\(865\) 6.00000 0.204006
\(866\) −20.0000 −0.679628
\(867\) −19.0000 −0.645274
\(868\) 36.0000 1.22192
\(869\) 15.0000 0.508840
\(870\) 14.0000 0.474644
\(871\) 24.0000 0.813209
\(872\) 0 0
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) −4.00000 −0.135147
\(877\) −12.0000 −0.405211 −0.202606 0.979260i \(-0.564941\pi\)
−0.202606 + 0.979260i \(0.564941\pi\)
\(878\) −18.0000 −0.607471
\(879\) −12.0000 −0.404750
\(880\) −12.0000 −0.404520
\(881\) 3.00000 0.101073 0.0505363 0.998722i \(-0.483907\pi\)
0.0505363 + 0.998722i \(0.483907\pi\)
\(882\) 6.00000 0.202031
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −72.0000 −2.42162
\(885\) −3.00000 −0.100844
\(886\) 0 0
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) −6.00000 −0.201120
\(891\) −3.00000 −0.100504
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 1.00000 0.0334263
\(896\) 0 0
\(897\) −48.0000 −1.60267
\(898\) 26.0000 0.867631
\(899\) 63.0000 2.10117
\(900\) 2.00000 0.0666667
\(901\) −84.0000 −2.79845
\(902\) −36.0000 −1.19867
\(903\) 20.0000 0.665558
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) −10.0000 −0.332228
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) −28.0000 −0.929213
\(909\) −3.00000 −0.0995037
\(910\) 24.0000 0.795592
\(911\) 45.0000 1.49092 0.745458 0.666552i \(-0.232231\pi\)
0.745458 + 0.666552i \(0.232231\pi\)
\(912\) 0 0
\(913\) 18.0000 0.595713
\(914\) −24.0000 −0.793849
\(915\) −7.00000 −0.231413
\(916\) 58.0000 1.91637
\(917\) 16.0000 0.528367
\(918\) 12.0000 0.396059
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) 16.0000 0.527218
\(922\) −66.0000 −2.17359
\(923\) 42.0000 1.38245
\(924\) −12.0000 −0.394771
\(925\) 2.00000 0.0657596
\(926\) 52.0000 1.70883
\(927\) 8.00000 0.262754
\(928\) −56.0000 −1.83829
\(929\) −15.0000 −0.492134 −0.246067 0.969253i \(-0.579138\pi\)
−0.246067 + 0.969253i \(0.579138\pi\)
\(930\) 18.0000 0.590243
\(931\) 0 0
\(932\) 32.0000 1.04819
\(933\) 8.00000 0.261908
\(934\) 36.0000 1.17796
\(935\) 18.0000 0.588663
\(936\) 0 0
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) −16.0000 −0.522419
\(939\) 8.00000 0.261070
\(940\) −8.00000 −0.260931
\(941\) −17.0000 −0.554184 −0.277092 0.960843i \(-0.589371\pi\)
−0.277092 + 0.960843i \(0.589371\pi\)
\(942\) −40.0000 −1.30327
\(943\) 48.0000 1.56310
\(944\) 12.0000 0.390567
\(945\) −2.00000 −0.0650600
\(946\) 60.0000 1.95077
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 10.0000 0.324785
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 28.0000 0.906533
\(955\) 5.00000 0.161796
\(956\) −18.0000 −0.582162
\(957\) −21.0000 −0.678834
\(958\) 30.0000 0.969256
\(959\) −4.00000 −0.129167
\(960\) −8.00000 −0.258199
\(961\) 50.0000 1.61290
\(962\) 24.0000 0.773791
\(963\) −12.0000 −0.386695
\(964\) −42.0000 −1.35273
\(965\) −18.0000 −0.579441
\(966\) 32.0000 1.02958
\(967\) 40.0000 1.28631 0.643157 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 24.0000 0.770594
\(971\) 32.0000 1.02693 0.513464 0.858111i \(-0.328362\pi\)
0.513464 + 0.858111i \(0.328362\pi\)
\(972\) −2.00000 −0.0641500
\(973\) 8.00000 0.256468
\(974\) −44.0000 −1.40985
\(975\) 6.00000 0.192154
\(976\) 28.0000 0.896258
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −20.0000 −0.639529
\(979\) 9.00000 0.287641
\(980\) 6.00000 0.191663
\(981\) −3.00000 −0.0957826
\(982\) 10.0000 0.319113
\(983\) −26.0000 −0.829271 −0.414636 0.909988i \(-0.636091\pi\)
−0.414636 + 0.909988i \(0.636091\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 84.0000 2.67510
\(987\) 8.00000 0.254643
\(988\) 0 0
\(989\) −80.0000 −2.54385
\(990\) −6.00000 −0.190693
\(991\) −56.0000 −1.77890 −0.889449 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) −72.0000 −2.28600
\(993\) −8.00000 −0.253872
\(994\) −28.0000 −0.888106
\(995\) 11.0000 0.348723
\(996\) 12.0000 0.380235
\(997\) −40.0000 −1.26681 −0.633406 0.773819i \(-0.718344\pi\)
−0.633406 + 0.773819i \(0.718344\pi\)
\(998\) −24.0000 −0.759707
\(999\) −2.00000 −0.0632772
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 5415.2.a.a.1.1 1
19.8 odd 6 285.2.i.a.121.1 yes 2
19.12 odd 6 285.2.i.a.106.1 2
19.18 odd 2 5415.2.a.k.1.1 1
57.8 even 6 855.2.k.d.406.1 2
57.50 even 6 855.2.k.d.676.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.i.a.106.1 2 19.12 odd 6
285.2.i.a.121.1 yes 2 19.8 odd 6
855.2.k.d.406.1 2 57.8 even 6
855.2.k.d.676.1 2 57.50 even 6
5415.2.a.a.1.1 1 1.1 even 1 trivial
5415.2.a.k.1.1 1 19.18 odd 2