Properties

Label 3850.2.c.o.1849.1
Level $3850$
Weight $2$
Character 3850.1849
Analytic conductor $30.742$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3850,2,Mod(1849,3850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3850, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("3850.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(30.7424047782\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3850.1849
Dual form 3850.2.c.o.1849.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{12} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -2.00000 q^{19} -2.00000 q^{21} +1.00000i q^{22} +6.00000i q^{23} -2.00000 q^{24} -2.00000 q^{26} +4.00000i q^{27} -1.00000i q^{28} +8.00000 q^{31} -1.00000i q^{32} -2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} +2.00000i q^{38} +4.00000 q^{39} +12.0000 q^{41} +2.00000i q^{42} +4.00000i q^{43} +1.00000 q^{44} +6.00000 q^{46} +12.0000i q^{47} +2.00000i q^{48} -1.00000 q^{49} +12.0000 q^{51} +2.00000i q^{52} +4.00000 q^{54} -1.00000 q^{56} -4.00000i q^{57} +2.00000 q^{61} -8.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} +8.00000i q^{67} +6.00000i q^{68} -12.0000 q^{69} +12.0000 q^{71} -1.00000i q^{72} -2.00000i q^{73} -4.00000 q^{74} +2.00000 q^{76} -1.00000i q^{77} -4.00000i q^{78} -14.0000 q^{79} -11.0000 q^{81} -12.0000i q^{82} -12.0000i q^{83} +2.00000 q^{84} +4.00000 q^{86} -1.00000i q^{88} -6.00000 q^{89} +2.00000 q^{91} -6.00000i q^{92} +16.0000i q^{93} +12.0000 q^{94} +2.00000 q^{96} +8.00000i q^{97} +1.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} - 2 q^{11} + 2 q^{14} + 2 q^{16} - 4 q^{19} - 4 q^{21} - 4 q^{24} - 4 q^{26} + 16 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 24 q^{41} + 2 q^{44} + 12 q^{46} - 2 q^{49} + 24 q^{51} + 8 q^{54} - 2 q^{56} + 4 q^{61} - 2 q^{64} - 4 q^{66} - 24 q^{69} + 24 q^{71} - 8 q^{74} + 4 q^{76} - 28 q^{79} - 22 q^{81} + 4 q^{84} + 8 q^{86} - 12 q^{89} + 4 q^{91} + 24 q^{94} + 4 q^{96} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times\).

\(n\) \(1751\) \(2201\) \(2927\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 1.00000i 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) − 2.00000i − 0.577350i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 1.00000i 0.213201i
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) −2.00000 −0.408248
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 4.00000i 0.769800i
\(28\) − 1.00000i − 0.188982i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 2.00000i − 0.348155i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 4.00000i − 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) 2.00000i 0.324443i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 2.00000i 0.308607i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 2.00000i 0.288675i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 2.00000i 0.277350i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) − 4.00000i − 0.529813i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) − 8.00000i − 1.01600i
\(63\) − 1.00000i − 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 6.00000i 0.727607i
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) − 1.00000i − 0.113961i
\(78\) − 4.00000i − 0.452911i
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) − 12.0000i − 1.32518i
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) − 1.00000i − 0.106600i
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) − 6.00000i − 0.625543i
\(93\) 16.0000i 1.65912i
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) − 12.0000i − 1.18818i
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) − 4.00000i − 0.384900i
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 1.00000i 0.0944911i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 2.00000i − 0.181071i
\(123\) 24.0000i 2.16401i
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 2.00000i 0.174078i
\(133\) − 2.00000i − 0.173422i
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 12.0000i 1.02151i
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) − 12.0000i − 1.00702i
\(143\) 2.00000i 0.167248i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) − 2.00000i − 0.164957i
\(148\) 4.00000i 0.328798i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) − 2.00000i − 0.162221i
\(153\) 6.00000i 0.485071i
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 14.0000i 1.11378i
\(159\) 0 0
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 11.0000i 0.864242i
\(163\) − 8.00000i − 0.626608i −0.949653 0.313304i \(-0.898564\pi\)
0.949653 0.313304i \(-0.101436\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) − 4.00000i − 0.304997i
\(173\) 18.0000i 1.36851i 0.729241 + 0.684257i \(0.239873\pi\)
−0.729241 + 0.684257i \(0.760127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 6.00000i 0.449719i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) − 2.00000i − 0.148250i
\(183\) 4.00000i 0.295689i
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 16.0000 1.17318
\(187\) 6.00000i 0.438763i
\(188\) − 12.0000i − 0.875190i
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) − 2.00000i − 0.144338i
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) − 1.00000i − 0.0710669i
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 6.00000i 0.422159i
\(203\) 0 0
\(204\) −12.0000 −0.840168
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) − 6.00000i − 0.417029i
\(208\) − 2.00000i − 0.138675i
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 24.0000i 1.64445i
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) 8.00000i 0.543075i
\(218\) − 16.0000i − 1.08366i
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) − 8.00000i − 0.536925i
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 24.0000i 1.59294i 0.604681 + 0.796468i \(0.293301\pi\)
−0.604681 + 0.796468i \(0.706699\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) − 28.0000i − 1.81880i
\(238\) − 6.00000i − 0.388922i
\(239\) −18.0000 −1.16432 −0.582162 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) − 1.00000i − 0.0642824i
\(243\) − 10.0000i − 0.641500i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 24.0000 1.53018
\(247\) 4.00000i 0.254514i
\(248\) 8.00000i 0.508001i
\(249\) 24.0000 1.52094
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 1.00000i 0.0629941i
\(253\) − 6.00000i − 0.377217i
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 24.0000i 1.49708i 0.663090 + 0.748539i \(0.269245\pi\)
−0.663090 + 0.748539i \(0.730755\pi\)
\(258\) 8.00000i 0.498058i
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 18.0000i 1.11204i
\(263\) − 24.0000i − 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) − 12.0000i − 0.734388i
\(268\) − 8.00000i − 0.488678i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) 4.00000i 0.242091i
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) 14.0000i 0.839664i
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 24.0000i 1.42918i
\(283\) 16.0000i 0.951101i 0.879688 + 0.475551i \(0.157751\pi\)
−0.879688 + 0.475551i \(0.842249\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 12.0000i 0.708338i
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) 2.00000i 0.117041i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) − 4.00000i − 0.232104i
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) − 14.0000i − 0.805609i
\(303\) − 12.0000i − 0.689382i
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 4.00000i − 0.228292i −0.993464 0.114146i \(-0.963587\pi\)
0.993464 0.114146i \(-0.0364132\pi\)
\(308\) 1.00000i 0.0569803i
\(309\) −32.0000 −1.82042
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 4.00000i 0.226455i
\(313\) − 8.00000i − 0.452187i −0.974106 0.226093i \(-0.927405\pi\)
0.974106 0.226093i \(-0.0725954\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) − 12.0000i − 0.673987i −0.941507 0.336994i \(-0.890590\pi\)
0.941507 0.336994i \(-0.109410\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 6.00000i 0.334367i
\(323\) 12.0000i 0.667698i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) −8.00000 −0.443079
\(327\) 32.0000i 1.76960i
\(328\) 12.0000i 0.662589i
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 12.0000i 0.658586i
\(333\) 4.00000i 0.219199i
\(334\) 0 0
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) − 2.00000i − 0.108148i
\(343\) − 1.00000i − 0.0539949i
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) 1.00000i 0.0533002i
\(353\) 24.0000i 1.27739i 0.769460 + 0.638696i \(0.220526\pi\)
−0.769460 + 0.638696i \(0.779474\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 12.0000i 0.635107i
\(358\) − 12.0000i − 0.634220i
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) − 2.00000i − 0.105118i
\(363\) 2.00000i 0.104973i
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) 6.00000i 0.312772i
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) − 16.0000i − 0.829561i
\(373\) − 14.0000i − 0.724893i −0.932005 0.362446i \(-0.881942\pi\)
0.932005 0.362446i \(-0.118058\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 4.00000i 0.205738i
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 24.0000i 1.22795i
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) − 4.00000i − 0.203331i
\(388\) − 8.00000i − 0.406138i
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) − 1.00000i − 0.0505076i
\(393\) − 36.0000i − 1.81596i
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) 38.0000i 1.90717i 0.301131 + 0.953583i \(0.402636\pi\)
−0.301131 + 0.953583i \(0.597364\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 16.0000i 0.798007i
\(403\) − 16.0000i − 0.797017i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) 4.00000i 0.198273i
\(408\) 12.0000i 0.594089i
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) − 16.0000i − 0.788263i
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) − 28.0000i − 1.37117i
\(418\) − 2.00000i − 0.0978232i
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 4.00000i 0.194717i
\(423\) − 12.0000i − 0.583460i
\(424\) 0 0
\(425\) 0 0
\(426\) 24.0000 1.16280
\(427\) 2.00000i 0.0967868i
\(428\) − 12.0000i − 0.580042i
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 4.00000i 0.192450i
\(433\) 4.00000i 0.192228i 0.995370 + 0.0961139i \(0.0306413\pi\)
−0.995370 + 0.0961139i \(0.969359\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) − 12.0000i − 0.574038i
\(438\) − 4.00000i − 0.191127i
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 12.0000i 0.570782i
\(443\) − 24.0000i − 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) − 1.00000i − 0.0472456i
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) − 6.00000i − 0.282216i
\(453\) 28.0000i 1.31555i
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) 24.0000 1.12022
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) − 2.00000i − 0.0930484i
\(463\) 22.0000i 1.02243i 0.859454 + 0.511213i \(0.170804\pi\)
−0.859454 + 0.511213i \(0.829196\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) − 6.00000i − 0.277647i −0.990317 0.138823i \(-0.955668\pi\)
0.990317 0.138823i \(-0.0443321\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −28.0000 −1.29017
\(472\) 0 0
\(473\) − 4.00000i − 0.183920i
\(474\) −28.0000 −1.28608
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) 0 0
\(478\) 18.0000i 0.823301i
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) − 20.0000i − 0.910975i
\(483\) − 12.0000i − 0.546019i
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 2.00000i 0.0905357i
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) − 24.0000i − 1.08200i
\(493\) 0 0
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 12.0000i 0.538274i
\(498\) − 24.0000i − 1.07547i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) 18.0000i 0.799408i
\(508\) − 8.00000i − 0.354943i
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) − 1.00000i − 0.0441942i
\(513\) − 8.00000i − 0.353209i
\(514\) 24.0000 1.05859
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) − 12.0000i − 0.527759i
\(518\) − 4.00000i − 0.175750i
\(519\) −36.0000 −1.58022
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) − 44.0000i − 1.92399i −0.273075 0.961993i \(-0.588041\pi\)
0.273075 0.961993i \(-0.411959\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) − 48.0000i − 2.09091i
\(528\) − 2.00000i − 0.0870388i
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 2.00000i 0.0867110i
\(533\) − 24.0000i − 1.03956i
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) 24.0000i 1.03568i
\(538\) − 18.0000i − 0.776035i
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) − 20.0000i − 0.859074i
\(543\) 4.00000i 0.171656i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 4.00000 0.171184
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) 6.00000i 0.256307i
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) − 12.0000i − 0.510754i
\(553\) − 14.0000i − 0.595341i
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) 6.00000i 0.254228i 0.991888 + 0.127114i \(0.0405714\pi\)
−0.991888 + 0.127114i \(0.959429\pi\)
\(558\) 8.00000i 0.338667i
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) − 6.00000i − 0.253095i
\(563\) 24.0000i 1.01148i 0.862686 + 0.505740i \(0.168780\pi\)
−0.862686 + 0.505740i \(0.831220\pi\)
\(564\) 24.0000 1.01058
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) − 11.0000i − 0.461957i
\(568\) 12.0000i 0.503509i
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) − 2.00000i − 0.0836242i
\(573\) − 48.0000i − 2.00523i
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 8.00000i 0.333044i 0.986038 + 0.166522i \(0.0532537\pi\)
−0.986038 + 0.166522i \(0.946746\pi\)
\(578\) 19.0000i 0.790296i
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 16.0000i 0.663221i
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) − 30.0000i − 1.23823i −0.785299 0.619116i \(-0.787491\pi\)
0.785299 0.619116i \(-0.212509\pi\)
\(588\) 2.00000i 0.0824786i
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) − 4.00000i − 0.164399i
\(593\) − 18.0000i − 0.739171i −0.929197 0.369586i \(-0.879500\pi\)
0.929197 0.369586i \(-0.120500\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 0 0
\(597\) 32.0000i 1.30967i
\(598\) − 12.0000i − 0.490716i
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 32.0000 1.30531 0.652654 0.757656i \(-0.273656\pi\)
0.652654 + 0.757656i \(0.273656\pi\)
\(602\) 4.00000i 0.163028i
\(603\) − 8.00000i − 0.325785i
\(604\) −14.0000 −0.569652
\(605\) 0 0
\(606\) −12.0000 −0.487467
\(607\) − 40.0000i − 1.62355i −0.583970 0.811775i \(-0.698502\pi\)
0.583970 0.811775i \(-0.301498\pi\)
\(608\) 2.00000i 0.0811107i
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) − 6.00000i − 0.242536i
\(613\) − 14.0000i − 0.565455i −0.959200 0.282727i \(-0.908761\pi\)
0.959200 0.282727i \(-0.0912392\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 32.0000i 1.28723i
\(619\) 16.0000 0.643094 0.321547 0.946894i \(-0.395797\pi\)
0.321547 + 0.946894i \(0.395797\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) − 24.0000i − 0.962312i
\(623\) − 6.00000i − 0.240385i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) 4.00000i 0.159745i
\(628\) − 14.0000i − 0.558661i
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) − 14.0000i − 0.556890i
\(633\) − 8.00000i − 0.317971i
\(634\) −12.0000 −0.476581
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 24.0000i 0.947204i
\(643\) 34.0000i 1.34083i 0.741987 + 0.670415i \(0.233884\pi\)
−0.741987 + 0.670415i \(0.766116\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) − 11.0000i − 0.432121i
\(649\) 0 0
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) 8.00000i 0.313304i
\(653\) − 24.0000i − 0.939193i −0.882881 0.469596i \(-0.844399\pi\)
0.882881 0.469596i \(-0.155601\pi\)
\(654\) 32.0000 1.25130
\(655\) 0 0
\(656\) 12.0000 0.468521
\(657\) 2.00000i 0.0780274i
\(658\) 12.0000i 0.467809i
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 28.0000i 1.08825i
\(663\) − 24.0000i − 0.932083i
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 0 0
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 2.00000i 0.0771517i
\(673\) 10.0000i 0.385472i 0.981251 + 0.192736i \(0.0617360\pi\)
−0.981251 + 0.192736i \(0.938264\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 42.0000i − 1.61419i −0.590421 0.807096i \(-0.701038\pi\)
0.590421 0.807096i \(-0.298962\pi\)
\(678\) 12.0000i 0.460857i
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) −48.0000 −1.83936
\(682\) 8.00000i 0.306336i
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 20.0000i 0.763048i
\(688\) 4.00000i 0.152499i
\(689\) 0 0
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) − 18.0000i − 0.684257i
\(693\) 1.00000i 0.0379869i
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) − 72.0000i − 2.72719i
\(698\) 14.0000i 0.529908i
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) − 8.00000i − 0.301941i
\(703\) 8.00000i 0.301726i
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) − 6.00000i − 0.225653i
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) − 6.00000i − 0.224860i
\(713\) 48.0000i 1.79761i
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) − 36.0000i − 1.34444i
\(718\) 30.0000i 1.11959i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 15.0000i 0.558242i
\(723\) 40.0000i 1.48762i
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) 20.0000i 0.741759i 0.928681 + 0.370879i \(0.120944\pi\)
−0.928681 + 0.370879i \(0.879056\pi\)
\(728\) 2.00000i 0.0741249i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) − 4.00000i − 0.147844i
\(733\) − 14.0000i − 0.517102i −0.965998 0.258551i \(-0.916755\pi\)
0.965998 0.258551i \(-0.0832450\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) − 8.00000i − 0.294684i
\(738\) 12.0000i 0.441726i
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) −16.0000 −0.586588
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) 12.0000i 0.439057i
\(748\) − 6.00000i − 0.219382i
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 12.0000i 0.437595i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) − 16.0000i − 0.581530i −0.956795 0.290765i \(-0.906090\pi\)
0.956795 0.290765i \(-0.0939098\pi\)
\(758\) − 28.0000i − 1.01701i
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 16.0000i 0.579619i
\(763\) 16.0000i 0.579239i
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) 2.00000i 0.0721688i
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) −48.0000 −1.72868
\(772\) 2.00000i 0.0719816i
\(773\) − 42.0000i − 1.51064i −0.655359 0.755318i \(-0.727483\pi\)
0.655359 0.755318i \(-0.272517\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −8.00000 −0.287183
\(777\) 8.00000i 0.286998i
\(778\) − 6.00000i − 0.215110i
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) − 36.0000i − 1.28736i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) −36.0000 −1.28408
\(787\) − 16.0000i − 0.570338i −0.958477 0.285169i \(-0.907950\pi\)
0.958477 0.285169i \(-0.0920498\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 48.0000 1.70885
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 1.00000i 0.0355335i
\(793\) − 4.00000i − 0.142044i
\(794\) 38.0000 1.34857
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) − 42.0000i − 1.48772i −0.668338 0.743858i \(-0.732994\pi\)
0.668338 0.743858i \(-0.267006\pi\)
\(798\) − 4.00000i − 0.141598i
\(799\) 72.0000 2.54718
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) − 30.0000i − 1.05934i
\(803\) 2.00000i 0.0705785i
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 36.0000i 1.26726i
\(808\) − 6.00000i − 0.211079i
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 0 0
\(813\) 40.0000i 1.40286i
\(814\) 4.00000 0.140200
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) − 8.00000i − 0.279885i
\(818\) − 4.00000i − 0.139857i
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) − 12.0000i − 0.418548i
\(823\) − 50.0000i − 1.74289i −0.490493 0.871445i \(-0.663183\pi\)
0.490493 0.871445i \(-0.336817\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 6.00000i 0.208514i
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 44.0000 1.52634
\(832\) 2.00000i 0.0693375i
\(833\) 6.00000i 0.207888i
\(834\) −28.0000 −0.969561
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) 32.0000i 1.10608i
\(838\) 0 0
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 10.0000i 0.344623i
\(843\) 12.0000i 0.413302i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) 1.00000i 0.0343604i
\(848\) 0 0
\(849\) −32.0000 −1.09824
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) − 24.0000i − 0.822226i
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 42.0000i 1.43469i 0.696717 + 0.717346i \(0.254643\pi\)
−0.696717 + 0.717346i \(0.745357\pi\)
\(858\) 4.00000i 0.136558i
\(859\) 16.0000 0.545913 0.272956 0.962026i \(-0.411998\pi\)
0.272956 + 0.962026i \(0.411998\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) − 30.0000i − 1.02180i
\(863\) − 18.0000i − 0.612727i −0.951915 0.306364i \(-0.900888\pi\)
0.951915 0.306364i \(-0.0991123\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) 4.00000 0.135926
\(867\) − 38.0000i − 1.29055i
\(868\) − 8.00000i − 0.271538i
\(869\) 14.0000 0.474917
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 16.0000i 0.541828i
\(873\) − 8.00000i − 0.270759i
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) − 10.0000i − 0.337676i −0.985644 0.168838i \(-0.945999\pi\)
0.985644 0.168838i \(-0.0540015\pi\)
\(878\) − 28.0000i − 0.944954i
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) − 1.00000i − 0.0336718i
\(883\) − 20.0000i − 0.673054i −0.941674 0.336527i \(-0.890748\pi\)
0.941674 0.336527i \(-0.109252\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 48.0000i 1.61168i 0.592132 + 0.805841i \(0.298286\pi\)
−0.592132 + 0.805841i \(0.701714\pi\)
\(888\) 8.00000i 0.268462i
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 11.0000 0.368514
\(892\) 8.00000i 0.267860i
\(893\) − 24.0000i − 0.803129i
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 24.0000i 0.801337i
\(898\) − 18.0000i − 0.600668i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 12.0000i 0.399556i
\(903\) − 8.00000i − 0.266223i
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 28.0000 0.930238
\(907\) 20.0000i 0.664089i 0.943264 + 0.332045i \(0.107738\pi\)
−0.943264 + 0.332045i \(0.892262\pi\)
\(908\) − 24.0000i − 0.796468i
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) 12.0000i 0.397142i
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) − 18.0000i − 0.594412i
\(918\) − 24.0000i − 0.792118i
\(919\) 34.0000 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) 18.0000i 0.592798i
\(923\) − 24.0000i − 0.789970i
\(924\) −2.00000 −0.0657952
\(925\) 0 0
\(926\) 22.0000 0.722965
\(927\) − 16.0000i − 0.525509i
\(928\) 0 0
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 6.00000i 0.196537i
\(933\) 48.0000i 1.57145i
\(934\) −6.00000 −0.196326
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 38.0000i 1.24141i 0.784046 + 0.620703i \(0.213153\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(938\) 8.00000i 0.261209i
\(939\) 16.0000 0.522140
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 28.0000i 0.912289i
\(943\) 72.0000i 2.34464i
\(944\) 0 0
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) − 12.0000i − 0.389948i −0.980808 0.194974i \(-0.937538\pi\)
0.980808 0.194974i \(-0.0624622\pi\)
\(948\) 28.0000i 0.909398i
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 24.0000 0.778253
\(952\) 6.00000i 0.194461i
\(953\) 30.0000i 0.971795i 0.874016 + 0.485898i \(0.161507\pi\)
−0.874016 + 0.485898i \(0.838493\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 18.0000 0.582162
\(957\) 0 0
\(958\) 24.0000i 0.775405i
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 8.00000i 0.257930i
\(963\) − 12.0000i − 0.386695i
\(964\) −20.0000 −0.644157
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) 8.00000i 0.257263i 0.991692 + 0.128631i \(0.0410584\pi\)
−0.991692 + 0.128631i \(0.958942\pi\)
\(968\) 1.00000i 0.0321412i
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 10.0000i 0.320750i
\(973\) − 14.0000i − 0.448819i
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) − 16.0000i − 0.511624i
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) 36.0000i 1.14881i
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −24.0000 −0.765092
\(985\) 0 0
\(986\) 0 0
\(987\) − 24.0000i − 0.763928i
\(988\) − 4.00000i − 0.127257i
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) − 8.00000i − 0.254000i
\(993\) − 56.0000i − 1.77711i
\(994\) 12.0000 0.380617
\(995\) 0 0
\(996\) −24.0000 −0.760469
\(997\) 38.0000i 1.20347i 0.798695 + 0.601736i \(0.205524\pi\)
−0.798695 + 0.601736i \(0.794476\pi\)
\(998\) − 4.00000i − 0.126618i
\(999\) 16.0000 0.506218
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 3850.2.c.o.1849.1 2
5.2 odd 4 3850.2.a.ba.1.1 1
5.3 odd 4 770.2.a.a.1.1 1
5.4 even 2 inner 3850.2.c.o.1849.2 2
15.8 even 4 6930.2.a.bm.1.1 1
20.3 even 4 6160.2.a.k.1.1 1
35.13 even 4 5390.2.a.r.1.1 1
55.43 even 4 8470.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.a.1.1 1 5.3 odd 4
3850.2.a.ba.1.1 1 5.2 odd 4
3850.2.c.o.1849.1 2 1.1 even 1 trivial
3850.2.c.o.1849.2 2 5.4 even 2 inner
5390.2.a.r.1.1 1 35.13 even 4
6160.2.a.k.1.1 1 20.3 even 4
6930.2.a.bm.1.1 1 15.8 even 4
8470.2.a.r.1.1 1 55.43 even 4