Properties

Label 507.2.b.b.337.2
Level $507$
Weight $2$
Character 507.337
Analytic conductor $4.048$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(337,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.b (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: \(4.04841538248\)
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 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.2.b.b.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +3.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +3.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.00000i q^{11} -1.00000 q^{12} +2.00000 q^{14} +1.00000i q^{15} -1.00000 q^{16} +7.00000 q^{17} +1.00000i q^{18} -6.00000i q^{19} -1.00000i q^{20} +2.00000i q^{21} -2.00000 q^{22} +6.00000 q^{23} -3.00000i q^{24} +4.00000 q^{25} -1.00000 q^{27} -2.00000i q^{28} -1.00000 q^{29} -1.00000 q^{30} +4.00000i q^{31} +5.00000i q^{32} -2.00000i q^{33} +7.00000i q^{34} -2.00000 q^{35} +1.00000 q^{36} -1.00000i q^{37} +6.00000 q^{38} +3.00000 q^{40} +9.00000i q^{41} -2.00000 q^{42} -6.00000 q^{43} +2.00000i q^{44} -1.00000i q^{45} +6.00000i q^{46} -6.00000i q^{47} +1.00000 q^{48} +3.00000 q^{49} +4.00000i q^{50} -7.00000 q^{51} -9.00000 q^{53} -1.00000i q^{54} +2.00000 q^{55} +6.00000 q^{56} +6.00000i q^{57} -1.00000i q^{58} +1.00000i q^{60} +1.00000 q^{61} -4.00000 q^{62} -2.00000i q^{63} -7.00000 q^{64} +2.00000 q^{66} -2.00000i q^{67} +7.00000 q^{68} -6.00000 q^{69} -2.00000i q^{70} +6.00000i q^{71} +3.00000i q^{72} -11.0000i q^{73} +1.00000 q^{74} -4.00000 q^{75} -6.00000i q^{76} +4.00000 q^{77} -4.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} -9.00000 q^{82} -14.0000i q^{83} +2.00000i q^{84} -7.00000i q^{85} -6.00000i q^{86} +1.00000 q^{87} -6.00000 q^{88} +14.0000i q^{89} +1.00000 q^{90} +6.00000 q^{92} -4.00000i q^{93} +6.00000 q^{94} -6.00000 q^{95} -5.00000i q^{96} -2.00000i q^{97} +3.00000i q^{98} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{9} + 2 q^{10} - 2 q^{12} + 4 q^{14} - 2 q^{16} + 14 q^{17} - 4 q^{22} + 12 q^{23} + 8 q^{25} - 2 q^{27} - 2 q^{29} - 2 q^{30} - 4 q^{35} + 2 q^{36} + 12 q^{38} + 6 q^{40} - 4 q^{42} - 12 q^{43} + 2 q^{48} + 6 q^{49} - 14 q^{51} - 18 q^{53} + 4 q^{55} + 12 q^{56} + 2 q^{61} - 8 q^{62} - 14 q^{64} + 4 q^{66} + 14 q^{68} - 12 q^{69} + 2 q^{74} - 8 q^{75} + 8 q^{77} - 8 q^{79} + 2 q^{81} - 18 q^{82} + 2 q^{87} - 12 q^{88} + 2 q^{90} + 12 q^{92} + 12 q^{94} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) − 1.00000i − 0.447214i −0.974679 0.223607i \(-0.928217\pi\)
0.974679 0.223607i \(-0.0717831\pi\)
\(6\) − 1.00000i − 0.408248i
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 3.00000i 1.06066i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) 1.00000i 0.258199i
\(16\) −1.00000 −0.250000
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 1.00000i 0.235702i
\(19\) − 6.00000i − 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) 2.00000i 0.436436i
\(22\) −2.00000 −0.426401
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) − 3.00000i − 0.612372i
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) − 2.00000i − 0.377964i
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) −1.00000 −0.182574
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 5.00000i 0.883883i
\(33\) − 2.00000i − 0.348155i
\(34\) 7.00000i 1.20049i
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) − 1.00000i − 0.164399i −0.996616 0.0821995i \(-0.973806\pi\)
0.996616 0.0821995i \(-0.0261945\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 9.00000i 1.40556i 0.711405 + 0.702782i \(0.248059\pi\)
−0.711405 + 0.702782i \(0.751941\pi\)
\(42\) −2.00000 −0.308607
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 2.00000i 0.301511i
\(45\) − 1.00000i − 0.149071i
\(46\) 6.00000i 0.884652i
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) 1.00000 0.144338
\(49\) 3.00000 0.428571
\(50\) 4.00000i 0.565685i
\(51\) −7.00000 −0.980196
\(52\) 0 0
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) 2.00000 0.269680
\(56\) 6.00000 0.801784
\(57\) 6.00000i 0.794719i
\(58\) − 1.00000i − 0.131306i
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) −4.00000 −0.508001
\(63\) − 2.00000i − 0.251976i
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) − 2.00000i − 0.244339i −0.992509 0.122169i \(-0.961015\pi\)
0.992509 0.122169i \(-0.0389851\pi\)
\(68\) 7.00000 0.848875
\(69\) −6.00000 −0.722315
\(70\) − 2.00000i − 0.239046i
\(71\) 6.00000i 0.712069i 0.934473 + 0.356034i \(0.115871\pi\)
−0.934473 + 0.356034i \(0.884129\pi\)
\(72\) 3.00000i 0.353553i
\(73\) − 11.0000i − 1.28745i −0.765256 0.643726i \(-0.777388\pi\)
0.765256 0.643726i \(-0.222612\pi\)
\(74\) 1.00000 0.116248
\(75\) −4.00000 −0.461880
\(76\) − 6.00000i − 0.688247i
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) −9.00000 −0.993884
\(83\) − 14.0000i − 1.53670i −0.640030 0.768350i \(-0.721078\pi\)
0.640030 0.768350i \(-0.278922\pi\)
\(84\) 2.00000i 0.218218i
\(85\) − 7.00000i − 0.759257i
\(86\) − 6.00000i − 0.646997i
\(87\) 1.00000 0.107211
\(88\) −6.00000 −0.639602
\(89\) 14.0000i 1.48400i 0.670402 + 0.741999i \(0.266122\pi\)
−0.670402 + 0.741999i \(0.733878\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) − 4.00000i − 0.414781i
\(94\) 6.00000 0.618853
\(95\) −6.00000 −0.615587
\(96\) − 5.00000i − 0.510310i
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 2.00000i 0.201008i
\(100\) 4.00000 0.400000
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) − 7.00000i − 0.693103i
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) − 9.00000i − 0.874157i
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −1.00000 −0.0962250
\(109\) − 2.00000i − 0.191565i −0.995402 0.0957826i \(-0.969465\pi\)
0.995402 0.0957826i \(-0.0305354\pi\)
\(110\) 2.00000i 0.190693i
\(111\) 1.00000i 0.0949158i
\(112\) 2.00000i 0.188982i
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) −6.00000 −0.561951
\(115\) − 6.00000i − 0.559503i
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) 0 0
\(119\) − 14.0000i − 1.28338i
\(120\) −3.00000 −0.273861
\(121\) 7.00000 0.636364
\(122\) 1.00000i 0.0905357i
\(123\) − 9.00000i − 0.811503i
\(124\) 4.00000i 0.359211i
\(125\) − 9.00000i − 0.804984i
\(126\) 2.00000 0.178174
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 3.00000i 0.265165i
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) − 2.00000i − 0.174078i
\(133\) −12.0000 −1.04053
\(134\) 2.00000 0.172774
\(135\) 1.00000i 0.0860663i
\(136\) 21.0000i 1.80074i
\(137\) 3.00000i 0.256307i 0.991754 + 0.128154i \(0.0409051\pi\)
−0.991754 + 0.128154i \(0.959095\pi\)
\(138\) − 6.00000i − 0.510754i
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) −2.00000 −0.169031
\(141\) 6.00000i 0.505291i
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 1.00000i 0.0830455i
\(146\) 11.0000 0.910366
\(147\) −3.00000 −0.247436
\(148\) − 1.00000i − 0.0821995i
\(149\) 3.00000i 0.245770i 0.992421 + 0.122885i \(0.0392146\pi\)
−0.992421 + 0.122885i \(0.960785\pi\)
\(150\) − 4.00000i − 0.326599i
\(151\) 2.00000i 0.162758i 0.996683 + 0.0813788i \(0.0259324\pi\)
−0.996683 + 0.0813788i \(0.974068\pi\)
\(152\) 18.0000 1.45999
\(153\) 7.00000 0.565916
\(154\) 4.00000i 0.322329i
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −3.00000 −0.239426 −0.119713 0.992809i \(-0.538197\pi\)
−0.119713 + 0.992809i \(0.538197\pi\)
\(158\) − 4.00000i − 0.318223i
\(159\) 9.00000 0.713746
\(160\) 5.00000 0.395285
\(161\) − 12.0000i − 0.945732i
\(162\) 1.00000i 0.0785674i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 9.00000i 0.702782i
\(165\) −2.00000 −0.155700
\(166\) 14.0000 1.08661
\(167\) − 16.0000i − 1.23812i −0.785345 0.619059i \(-0.787514\pi\)
0.785345 0.619059i \(-0.212486\pi\)
\(168\) −6.00000 −0.462910
\(169\) 0 0
\(170\) 7.00000 0.536875
\(171\) − 6.00000i − 0.458831i
\(172\) −6.00000 −0.457496
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 1.00000i 0.0758098i
\(175\) − 8.00000i − 0.604743i
\(176\) − 2.00000i − 0.150756i
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) 18.0000i 1.32698i
\(185\) −1.00000 −0.0735215
\(186\) 4.00000 0.293294
\(187\) 14.0000i 1.02378i
\(188\) − 6.00000i − 0.437595i
\(189\) 2.00000i 0.145479i
\(190\) − 6.00000i − 0.435286i
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 7.00000 0.505181
\(193\) 9.00000i 0.647834i 0.946085 + 0.323917i \(0.105000\pi\)
−0.946085 + 0.323917i \(0.895000\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) −2.00000 −0.142134
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 12.0000i 0.848528i
\(201\) 2.00000i 0.141069i
\(202\) − 3.00000i − 0.211079i
\(203\) 2.00000i 0.140372i
\(204\) −7.00000 −0.490098
\(205\) 9.00000 0.628587
\(206\) − 6.00000i − 0.418040i
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 2.00000i 0.138013i
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −9.00000 −0.618123
\(213\) − 6.00000i − 0.411113i
\(214\) − 6.00000i − 0.410152i
\(215\) 6.00000i 0.409197i
\(216\) − 3.00000i − 0.204124i
\(217\) 8.00000 0.543075
\(218\) 2.00000 0.135457
\(219\) 11.0000i 0.743311i
\(220\) 2.00000 0.134840
\(221\) 0 0
\(222\) −1.00000 −0.0671156
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 10.0000 0.668153
\(225\) 4.00000 0.266667
\(226\) − 15.0000i − 0.997785i
\(227\) − 14.0000i − 0.929213i −0.885517 0.464606i \(-0.846196\pi\)
0.885517 0.464606i \(-0.153804\pi\)
\(228\) 6.00000i 0.397360i
\(229\) 22.0000i 1.45380i 0.686743 + 0.726900i \(0.259040\pi\)
−0.686743 + 0.726900i \(0.740960\pi\)
\(230\) 6.00000 0.395628
\(231\) −4.00000 −0.263181
\(232\) − 3.00000i − 0.196960i
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 14.0000 0.907485
\(239\) 30.0000i 1.94054i 0.242028 + 0.970269i \(0.422188\pi\)
−0.242028 + 0.970269i \(0.577812\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) − 7.00000i − 0.450910i −0.974254 0.225455i \(-0.927613\pi\)
0.974254 0.225455i \(-0.0723868\pi\)
\(242\) 7.00000i 0.449977i
\(243\) −1.00000 −0.0641500
\(244\) 1.00000 0.0640184
\(245\) − 3.00000i − 0.191663i
\(246\) 9.00000 0.573819
\(247\) 0 0
\(248\) −12.0000 −0.762001
\(249\) 14.0000i 0.887214i
\(250\) 9.00000 0.569210
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) 12.0000i 0.754434i
\(254\) − 20.0000i − 1.25491i
\(255\) 7.00000i 0.438357i
\(256\) −17.0000 −1.06250
\(257\) 7.00000 0.436648 0.218324 0.975876i \(-0.429941\pi\)
0.218324 + 0.975876i \(0.429941\pi\)
\(258\) 6.00000i 0.373544i
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) − 8.00000i − 0.494242i
\(263\) −30.0000 −1.84988 −0.924940 0.380114i \(-0.875885\pi\)
−0.924940 + 0.380114i \(0.875885\pi\)
\(264\) 6.00000 0.369274
\(265\) 9.00000i 0.552866i
\(266\) − 12.0000i − 0.735767i
\(267\) − 14.0000i − 0.856786i
\(268\) − 2.00000i − 0.122169i
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −7.00000 −0.424437
\(273\) 0 0
\(274\) −3.00000 −0.181237
\(275\) 8.00000i 0.482418i
\(276\) −6.00000 −0.361158
\(277\) 31.0000 1.86261 0.931305 0.364241i \(-0.118672\pi\)
0.931305 + 0.364241i \(0.118672\pi\)
\(278\) 12.0000i 0.719712i
\(279\) 4.00000i 0.239474i
\(280\) − 6.00000i − 0.358569i
\(281\) 19.0000i 1.13344i 0.823909 + 0.566722i \(0.191789\pi\)
−0.823909 + 0.566722i \(0.808211\pi\)
\(282\) −6.00000 −0.357295
\(283\) 18.0000 1.06999 0.534994 0.844856i \(-0.320314\pi\)
0.534994 + 0.844856i \(0.320314\pi\)
\(284\) 6.00000i 0.356034i
\(285\) 6.00000 0.355409
\(286\) 0 0
\(287\) 18.0000 1.06251
\(288\) 5.00000i 0.294628i
\(289\) 32.0000 1.88235
\(290\) −1.00000 −0.0587220
\(291\) 2.00000i 0.117242i
\(292\) − 11.0000i − 0.643726i
\(293\) 9.00000i 0.525786i 0.964825 + 0.262893i \(0.0846766\pi\)
−0.964825 + 0.262893i \(0.915323\pi\)
\(294\) − 3.00000i − 0.174964i
\(295\) 0 0
\(296\) 3.00000 0.174371
\(297\) − 2.00000i − 0.116052i
\(298\) −3.00000 −0.173785
\(299\) 0 0
\(300\) −4.00000 −0.230940
\(301\) 12.0000i 0.691669i
\(302\) −2.00000 −0.115087
\(303\) 3.00000 0.172345
\(304\) 6.00000i 0.344124i
\(305\) − 1.00000i − 0.0572598i
\(306\) 7.00000i 0.400163i
\(307\) − 14.0000i − 0.799022i −0.916728 0.399511i \(-0.869180\pi\)
0.916728 0.399511i \(-0.130820\pi\)
\(308\) 4.00000 0.227921
\(309\) 6.00000 0.341328
\(310\) 4.00000i 0.227185i
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) − 3.00000i − 0.169300i
\(315\) −2.00000 −0.112687
\(316\) −4.00000 −0.225018
\(317\) − 25.0000i − 1.40414i −0.712108 0.702070i \(-0.752259\pi\)
0.712108 0.702070i \(-0.247741\pi\)
\(318\) 9.00000i 0.504695i
\(319\) − 2.00000i − 0.111979i
\(320\) 7.00000i 0.391312i
\(321\) 6.00000 0.334887
\(322\) 12.0000 0.668734
\(323\) − 42.0000i − 2.33694i
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 2.00000i 0.110600i
\(328\) −27.0000 −1.49083
\(329\) −12.0000 −0.661581
\(330\) − 2.00000i − 0.110096i
\(331\) − 4.00000i − 0.219860i −0.993939 0.109930i \(-0.964937\pi\)
0.993939 0.109930i \(-0.0350627\pi\)
\(332\) − 14.0000i − 0.768350i
\(333\) − 1.00000i − 0.0547997i
\(334\) 16.0000 0.875481
\(335\) −2.00000 −0.109272
\(336\) − 2.00000i − 0.109109i
\(337\) 33.0000 1.79762 0.898812 0.438334i \(-0.144431\pi\)
0.898812 + 0.438334i \(0.144431\pi\)
\(338\) 0 0
\(339\) 15.0000 0.814688
\(340\) − 7.00000i − 0.379628i
\(341\) −8.00000 −0.433224
\(342\) 6.00000 0.324443
\(343\) − 20.0000i − 1.07990i
\(344\) − 18.0000i − 0.970495i
\(345\) 6.00000i 0.323029i
\(346\) 6.00000i 0.322562i
\(347\) 18.0000 0.966291 0.483145 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 1.00000 0.0536056
\(349\) 26.0000i 1.39175i 0.718164 + 0.695874i \(0.244983\pi\)
−0.718164 + 0.695874i \(0.755017\pi\)
\(350\) 8.00000 0.427618
\(351\) 0 0
\(352\) −10.0000 −0.533002
\(353\) − 11.0000i − 0.585471i −0.956193 0.292735i \(-0.905434\pi\)
0.956193 0.292735i \(-0.0945655\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 14.0000i 0.741999i
\(357\) 14.0000i 0.740959i
\(358\) 2.00000i 0.105703i
\(359\) 18.0000i 0.950004i 0.879985 + 0.475002i \(0.157553\pi\)
−0.879985 + 0.475002i \(0.842447\pi\)
\(360\) 3.00000 0.158114
\(361\) −17.0000 −0.894737
\(362\) 7.00000i 0.367912i
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) −11.0000 −0.575766
\(366\) − 1.00000i − 0.0522708i
\(367\) 10.0000 0.521996 0.260998 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) −6.00000 −0.312772
\(369\) 9.00000i 0.468521i
\(370\) − 1.00000i − 0.0519875i
\(371\) 18.0000i 0.934513i
\(372\) − 4.00000i − 0.207390i
\(373\) −11.0000 −0.569558 −0.284779 0.958593i \(-0.591920\pi\)
−0.284779 + 0.958593i \(0.591920\pi\)
\(374\) −14.0000 −0.723923
\(375\) 9.00000i 0.464758i
\(376\) 18.0000 0.928279
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) − 36.0000i − 1.84920i −0.380945 0.924598i \(-0.624401\pi\)
0.380945 0.924598i \(-0.375599\pi\)
\(380\) −6.00000 −0.307794
\(381\) 20.0000 1.02463
\(382\) − 4.00000i − 0.204658i
\(383\) 8.00000i 0.408781i 0.978889 + 0.204390i \(0.0655212\pi\)
−0.978889 + 0.204390i \(0.934479\pi\)
\(384\) − 3.00000i − 0.153093i
\(385\) − 4.00000i − 0.203859i
\(386\) −9.00000 −0.458088
\(387\) −6.00000 −0.304997
\(388\) − 2.00000i − 0.101535i
\(389\) −19.0000 −0.963338 −0.481669 0.876353i \(-0.659969\pi\)
−0.481669 + 0.876353i \(0.659969\pi\)
\(390\) 0 0
\(391\) 42.0000 2.12403
\(392\) 9.00000i 0.454569i
\(393\) 8.00000 0.403547
\(394\) −6.00000 −0.302276
\(395\) 4.00000i 0.201262i
\(396\) 2.00000i 0.100504i
\(397\) 34.0000i 1.70641i 0.521575 + 0.853206i \(0.325345\pi\)
−0.521575 + 0.853206i \(0.674655\pi\)
\(398\) − 14.0000i − 0.701757i
\(399\) 12.0000 0.600751
\(400\) −4.00000 −0.200000
\(401\) − 1.00000i − 0.0499376i −0.999688 0.0249688i \(-0.992051\pi\)
0.999688 0.0249688i \(-0.00794864\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 0 0
\(404\) −3.00000 −0.149256
\(405\) − 1.00000i − 0.0496904i
\(406\) −2.00000 −0.0992583
\(407\) 2.00000 0.0991363
\(408\) − 21.0000i − 1.03965i
\(409\) 7.00000i 0.346128i 0.984911 + 0.173064i \(0.0553667\pi\)
−0.984911 + 0.173064i \(0.944633\pi\)
\(410\) 9.00000i 0.444478i
\(411\) − 3.00000i − 0.147979i
\(412\) −6.00000 −0.295599
\(413\) 0 0
\(414\) 6.00000i 0.294884i
\(415\) −14.0000 −0.687233
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 12.0000i 0.586939i
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 2.00000 0.0975900
\(421\) − 19.0000i − 0.926003i −0.886357 0.463002i \(-0.846772\pi\)
0.886357 0.463002i \(-0.153228\pi\)
\(422\) − 8.00000i − 0.389434i
\(423\) − 6.00000i − 0.291730i
\(424\) − 27.0000i − 1.31124i
\(425\) 28.0000 1.35820
\(426\) 6.00000 0.290701
\(427\) − 2.00000i − 0.0967868i
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) − 30.0000i − 1.44505i −0.691345 0.722525i \(-0.742982\pi\)
0.691345 0.722525i \(-0.257018\pi\)
\(432\) 1.00000 0.0481125
\(433\) −19.0000 −0.913082 −0.456541 0.889702i \(-0.650912\pi\)
−0.456541 + 0.889702i \(0.650912\pi\)
\(434\) 8.00000i 0.384012i
\(435\) − 1.00000i − 0.0479463i
\(436\) − 2.00000i − 0.0957826i
\(437\) − 36.0000i − 1.72211i
\(438\) −11.0000 −0.525600
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 6.00000i 0.286039i
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 1.00000i 0.0474579i
\(445\) 14.0000 0.663664
\(446\) −16.0000 −0.757622
\(447\) − 3.00000i − 0.141895i
\(448\) 14.0000i 0.661438i
\(449\) − 34.0000i − 1.60456i −0.596948 0.802280i \(-0.703620\pi\)
0.596948 0.802280i \(-0.296380\pi\)
\(450\) 4.00000i 0.188562i
\(451\) −18.0000 −0.847587
\(452\) −15.0000 −0.705541
\(453\) − 2.00000i − 0.0939682i
\(454\) 14.0000 0.657053
\(455\) 0 0
\(456\) −18.0000 −0.842927
\(457\) − 13.0000i − 0.608114i −0.952654 0.304057i \(-0.901659\pi\)
0.952654 0.304057i \(-0.0983414\pi\)
\(458\) −22.0000 −1.02799
\(459\) −7.00000 −0.326732
\(460\) − 6.00000i − 0.279751i
\(461\) 19.0000i 0.884918i 0.896789 + 0.442459i \(0.145894\pi\)
−0.896789 + 0.442459i \(0.854106\pi\)
\(462\) − 4.00000i − 0.186097i
\(463\) 26.0000i 1.20832i 0.796862 + 0.604161i \(0.206492\pi\)
−0.796862 + 0.604161i \(0.793508\pi\)
\(464\) 1.00000 0.0464238
\(465\) −4.00000 −0.185496
\(466\) − 10.0000i − 0.463241i
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) − 6.00000i − 0.276759i
\(471\) 3.00000 0.138233
\(472\) 0 0
\(473\) − 12.0000i − 0.551761i
\(474\) 4.00000i 0.183726i
\(475\) − 24.0000i − 1.10120i
\(476\) − 14.0000i − 0.641689i
\(477\) −9.00000 −0.412082
\(478\) −30.0000 −1.37217
\(479\) − 24.0000i − 1.09659i −0.836286 0.548294i \(-0.815277\pi\)
0.836286 0.548294i \(-0.184723\pi\)
\(480\) −5.00000 −0.228218
\(481\) 0 0
\(482\) 7.00000 0.318841
\(483\) 12.0000i 0.546019i
\(484\) 7.00000 0.318182
\(485\) −2.00000 −0.0908153
\(486\) − 1.00000i − 0.0453609i
\(487\) 18.0000i 0.815658i 0.913058 + 0.407829i \(0.133714\pi\)
−0.913058 + 0.407829i \(0.866286\pi\)
\(488\) 3.00000i 0.135804i
\(489\) − 4.00000i − 0.180886i
\(490\) 3.00000 0.135526
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) − 9.00000i − 0.405751i
\(493\) −7.00000 −0.315264
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) − 4.00000i − 0.179605i
\(497\) 12.0000 0.538274
\(498\) −14.0000 −0.627355
\(499\) 24.0000i 1.07439i 0.843459 + 0.537194i \(0.180516\pi\)
−0.843459 + 0.537194i \(0.819484\pi\)
\(500\) − 9.00000i − 0.402492i
\(501\) 16.0000i 0.714827i
\(502\) − 12.0000i − 0.535586i
\(503\) −2.00000 −0.0891756 −0.0445878 0.999005i \(-0.514197\pi\)
−0.0445878 + 0.999005i \(0.514197\pi\)
\(504\) 6.00000 0.267261
\(505\) 3.00000i 0.133498i
\(506\) −12.0000 −0.533465
\(507\) 0 0
\(508\) −20.0000 −0.887357
\(509\) 7.00000i 0.310270i 0.987893 + 0.155135i \(0.0495812\pi\)
−0.987893 + 0.155135i \(0.950419\pi\)
\(510\) −7.00000 −0.309965
\(511\) −22.0000 −0.973223
\(512\) − 11.0000i − 0.486136i
\(513\) 6.00000i 0.264906i
\(514\) 7.00000i 0.308757i
\(515\) 6.00000i 0.264392i
\(516\) 6.00000 0.264135
\(517\) 12.0000 0.527759
\(518\) − 2.00000i − 0.0878750i
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) − 1.00000i − 0.0437688i
\(523\) 14.0000 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) −8.00000 −0.349482
\(525\) 8.00000i 0.349149i
\(526\) − 30.0000i − 1.30806i
\(527\) 28.0000i 1.21970i
\(528\) 2.00000i 0.0870388i
\(529\) 13.0000 0.565217
\(530\) −9.00000 −0.390935
\(531\) 0 0
\(532\) −12.0000 −0.520266
\(533\) 0 0
\(534\) 14.0000 0.605839
\(535\) 6.00000i 0.259403i
\(536\) 6.00000 0.259161
\(537\) −2.00000 −0.0863064
\(538\) − 14.0000i − 0.603583i
\(539\) 6.00000i 0.258438i
\(540\) 1.00000i 0.0430331i
\(541\) − 45.0000i − 1.93470i −0.253442 0.967351i \(-0.581563\pi\)
0.253442 0.967351i \(-0.418437\pi\)
\(542\) 0 0
\(543\) −7.00000 −0.300399
\(544\) 35.0000i 1.50061i
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) 3.00000i 0.128154i
\(549\) 1.00000 0.0426790
\(550\) −8.00000 −0.341121
\(551\) 6.00000i 0.255609i
\(552\) − 18.0000i − 0.766131i
\(553\) 8.00000i 0.340195i
\(554\) 31.0000i 1.31706i
\(555\) 1.00000 0.0424476
\(556\) 12.0000 0.508913
\(557\) 9.00000i 0.381342i 0.981654 + 0.190671i \(0.0610664\pi\)
−0.981654 + 0.190671i \(0.938934\pi\)
\(558\) −4.00000 −0.169334
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) − 14.0000i − 0.591080i
\(562\) −19.0000 −0.801467
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 6.00000i 0.252646i
\(565\) 15.0000i 0.631055i
\(566\) 18.0000i 0.756596i
\(567\) − 2.00000i − 0.0839921i
\(568\) −18.0000 −0.755263
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 6.00000i 0.251312i
\(571\) −26.0000 −1.08807 −0.544033 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) 0 0
\(573\) 4.00000 0.167102
\(574\) 18.0000i 0.751305i
\(575\) 24.0000 1.00087
\(576\) −7.00000 −0.291667
\(577\) 11.0000i 0.457936i 0.973434 + 0.228968i \(0.0735351\pi\)
−0.973434 + 0.228968i \(0.926465\pi\)
\(578\) 32.0000i 1.33102i
\(579\) − 9.00000i − 0.374027i
\(580\) 1.00000i 0.0415227i
\(581\) −28.0000 −1.16164
\(582\) −2.00000 −0.0829027
\(583\) − 18.0000i − 0.745484i
\(584\) 33.0000 1.36555
\(585\) 0 0
\(586\) −9.00000 −0.371787
\(587\) 16.0000i 0.660391i 0.943913 + 0.330195i \(0.107115\pi\)
−0.943913 + 0.330195i \(0.892885\pi\)
\(588\) −3.00000 −0.123718
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) − 6.00000i − 0.246807i
\(592\) 1.00000i 0.0410997i
\(593\) − 13.0000i − 0.533846i −0.963718 0.266923i \(-0.913993\pi\)
0.963718 0.266923i \(-0.0860069\pi\)
\(594\) 2.00000 0.0820610
\(595\) −14.0000 −0.573944
\(596\) 3.00000i 0.122885i
\(597\) 14.0000 0.572982
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) − 12.0000i − 0.489898i
\(601\) −5.00000 −0.203954 −0.101977 0.994787i \(-0.532517\pi\)
−0.101977 + 0.994787i \(0.532517\pi\)
\(602\) −12.0000 −0.489083
\(603\) − 2.00000i − 0.0814463i
\(604\) 2.00000i 0.0813788i
\(605\) − 7.00000i − 0.284590i
\(606\) 3.00000i 0.121867i
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 30.0000 1.21666
\(609\) − 2.00000i − 0.0810441i
\(610\) 1.00000 0.0404888
\(611\) 0 0
\(612\) 7.00000 0.282958
\(613\) − 23.0000i − 0.928961i −0.885583 0.464481i \(-0.846241\pi\)
0.885583 0.464481i \(-0.153759\pi\)
\(614\) 14.0000 0.564994
\(615\) −9.00000 −0.362915
\(616\) 12.0000i 0.483494i
\(617\) 13.0000i 0.523360i 0.965155 + 0.261680i \(0.0842766\pi\)
−0.965155 + 0.261680i \(0.915723\pi\)
\(618\) 6.00000i 0.241355i
\(619\) 24.0000i 0.964641i 0.875995 + 0.482321i \(0.160206\pi\)
−0.875995 + 0.482321i \(0.839794\pi\)
\(620\) 4.00000 0.160644
\(621\) −6.00000 −0.240772
\(622\) − 18.0000i − 0.721734i
\(623\) 28.0000 1.12180
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 6.00000i 0.239808i
\(627\) −12.0000 −0.479234
\(628\) −3.00000 −0.119713
\(629\) − 7.00000i − 0.279108i
\(630\) − 2.00000i − 0.0796819i
\(631\) − 20.0000i − 0.796187i −0.917345 0.398094i \(-0.869672\pi\)
0.917345 0.398094i \(-0.130328\pi\)
\(632\) − 12.0000i − 0.477334i
\(633\) 8.00000 0.317971
\(634\) 25.0000 0.992877
\(635\) 20.0000i 0.793676i
\(636\) 9.00000 0.356873
\(637\) 0 0
\(638\) 2.00000 0.0791808
\(639\) 6.00000i 0.237356i
\(640\) 3.00000 0.118585
\(641\) 31.0000 1.22443 0.612213 0.790693i \(-0.290279\pi\)
0.612213 + 0.790693i \(0.290279\pi\)
\(642\) 6.00000i 0.236801i
\(643\) − 16.0000i − 0.630978i −0.948929 0.315489i \(-0.897831\pi\)
0.948929 0.315489i \(-0.102169\pi\)
\(644\) − 12.0000i − 0.472866i
\(645\) − 6.00000i − 0.236250i
\(646\) 42.0000 1.65247
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 3.00000i 0.117851i
\(649\) 0 0
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) 4.00000i 0.156652i
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 8.00000i 0.312586i
\(656\) − 9.00000i − 0.351391i
\(657\) − 11.0000i − 0.429151i
\(658\) − 12.0000i − 0.467809i
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) −2.00000 −0.0778499
\(661\) − 45.0000i − 1.75030i −0.483854 0.875149i \(-0.660764\pi\)
0.483854 0.875149i \(-0.339236\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 42.0000 1.62992
\(665\) 12.0000i 0.465340i
\(666\) 1.00000 0.0387492
\(667\) −6.00000 −0.232321
\(668\) − 16.0000i − 0.619059i
\(669\) − 16.0000i − 0.618596i
\(670\) − 2.00000i − 0.0772667i
\(671\) 2.00000i 0.0772091i
\(672\) −10.0000 −0.385758
\(673\) 29.0000 1.11787 0.558934 0.829212i \(-0.311211\pi\)
0.558934 + 0.829212i \(0.311211\pi\)
\(674\) 33.0000i 1.27111i
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) 15.0000i 0.576072i
\(679\) −4.00000 −0.153506
\(680\) 21.0000 0.805313
\(681\) 14.0000i 0.536481i
\(682\) − 8.00000i − 0.306336i
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) − 6.00000i − 0.229416i
\(685\) 3.00000 0.114624
\(686\) 20.0000 0.763604
\(687\) − 22.0000i − 0.839352i
\(688\) 6.00000 0.228748
\(689\) 0 0
\(690\) −6.00000 −0.228416
\(691\) 42.0000i 1.59776i 0.601494 + 0.798878i \(0.294573\pi\)
−0.601494 + 0.798878i \(0.705427\pi\)
\(692\) 6.00000 0.228086
\(693\) 4.00000 0.151947
\(694\) 18.0000i 0.683271i
\(695\) − 12.0000i − 0.455186i
\(696\) 3.00000i 0.113715i
\(697\) 63.0000i 2.38630i
\(698\) −26.0000 −0.984115
\(699\) 10.0000 0.378235
\(700\) − 8.00000i − 0.302372i
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) −6.00000 −0.226294
\(704\) − 14.0000i − 0.527645i
\(705\) 6.00000 0.225973
\(706\) 11.0000 0.413990
\(707\) 6.00000i 0.225653i
\(708\) 0 0
\(709\) 11.0000i 0.413114i 0.978435 + 0.206557i \(0.0662258\pi\)
−0.978435 + 0.206557i \(0.933774\pi\)
\(710\) 6.00000i 0.225176i
\(711\) −4.00000 −0.150012
\(712\) −42.0000 −1.57402
\(713\) 24.0000i 0.898807i
\(714\) −14.0000 −0.523937
\(715\) 0 0
\(716\) 2.00000 0.0747435
\(717\) − 30.0000i − 1.12037i
\(718\) −18.0000 −0.671754
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) 12.0000i 0.446903i
\(722\) − 17.0000i − 0.632674i
\(723\) 7.00000i 0.260333i
\(724\) 7.00000 0.260153
\(725\) −4.00000 −0.148556
\(726\) − 7.00000i − 0.259794i
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 11.0000i − 0.407128i
\(731\) −42.0000 −1.55343
\(732\) −1.00000 −0.0369611
\(733\) − 15.0000i − 0.554038i −0.960864 0.277019i \(-0.910654\pi\)
0.960864 0.277019i \(-0.0893464\pi\)
\(734\) 10.0000i 0.369107i
\(735\) 3.00000i 0.110657i
\(736\) 30.0000i 1.10581i
\(737\) 4.00000 0.147342
\(738\) −9.00000 −0.331295
\(739\) 16.0000i 0.588570i 0.955718 + 0.294285i \(0.0950814\pi\)
−0.955718 + 0.294285i \(0.904919\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) −18.0000 −0.660801
\(743\) − 36.0000i − 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 12.0000 0.439941
\(745\) 3.00000 0.109911
\(746\) − 11.0000i − 0.402739i
\(747\) − 14.0000i − 0.512233i
\(748\) 14.0000i 0.511891i
\(749\) 12.0000i 0.438470i
\(750\) −9.00000 −0.328634
\(751\) 34.0000 1.24068 0.620339 0.784334i \(-0.286995\pi\)
0.620339 + 0.784334i \(0.286995\pi\)
\(752\) 6.00000i 0.218797i
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 2.00000 0.0727875
\(756\) 2.00000i 0.0727393i
\(757\) −50.0000 −1.81728 −0.908640 0.417579i \(-0.862879\pi\)
−0.908640 + 0.417579i \(0.862879\pi\)
\(758\) 36.0000 1.30758
\(759\) − 12.0000i − 0.435572i
\(760\) − 18.0000i − 0.652929i
\(761\) − 50.0000i − 1.81250i −0.422744 0.906249i \(-0.638933\pi\)
0.422744 0.906249i \(-0.361067\pi\)
\(762\) 20.0000i 0.724524i
\(763\) −4.00000 −0.144810
\(764\) −4.00000 −0.144715
\(765\) − 7.00000i − 0.253086i
\(766\) −8.00000 −0.289052
\(767\) 0 0
\(768\) 17.0000 0.613435
\(769\) 30.0000i 1.08183i 0.841078 + 0.540914i \(0.181921\pi\)
−0.841078 + 0.540914i \(0.818079\pi\)
\(770\) 4.00000 0.144150
\(771\) −7.00000 −0.252099
\(772\) 9.00000i 0.323917i
\(773\) − 14.0000i − 0.503545i −0.967786 0.251773i \(-0.918987\pi\)
0.967786 0.251773i \(-0.0810135\pi\)
\(774\) − 6.00000i − 0.215666i
\(775\) 16.0000i 0.574737i
\(776\) 6.00000 0.215387
\(777\) 2.00000 0.0717496
\(778\) − 19.0000i − 0.681183i
\(779\) 54.0000 1.93475
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 42.0000i 1.50192i
\(783\) 1.00000 0.0357371
\(784\) −3.00000 −0.107143
\(785\) 3.00000i 0.107075i
\(786\) 8.00000i 0.285351i
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 30.0000 1.06803
\(790\) −4.00000 −0.142314
\(791\) 30.0000i 1.06668i
\(792\) −6.00000 −0.213201
\(793\) 0 0
\(794\) −34.0000 −1.20661
\(795\) − 9.00000i − 0.319197i
\(796\) −14.0000 −0.496217
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 12.0000i 0.424795i
\(799\) − 42.0000i − 1.48585i
\(800\) 20.0000i 0.707107i
\(801\) 14.0000i 0.494666i
\(802\) 1.00000 0.0353112
\(803\) 22.0000 0.776363
\(804\) 2.00000i 0.0705346i
\(805\) −12.0000 −0.422944
\(806\) 0 0
\(807\) 14.0000 0.492823
\(808\) − 9.00000i − 0.316619i
\(809\) 33.0000 1.16022 0.580109 0.814539i \(-0.303010\pi\)
0.580109 + 0.814539i \(0.303010\pi\)
\(810\) 1.00000 0.0351364
\(811\) − 28.0000i − 0.983213i −0.870817 0.491606i \(-0.836410\pi\)
0.870817 0.491606i \(-0.163590\pi\)
\(812\) 2.00000i 0.0701862i
\(813\) 0 0
\(814\) 2.00000i 0.0701000i
\(815\) 4.00000 0.140114
\(816\) 7.00000 0.245049
\(817\) 36.0000i 1.25948i
\(818\) −7.00000 −0.244749
\(819\) 0 0
\(820\) 9.00000 0.314294
\(821\) 50.0000i 1.74501i 0.488603 + 0.872506i \(0.337507\pi\)
−0.488603 + 0.872506i \(0.662493\pi\)
\(822\) 3.00000 0.104637
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) − 18.0000i − 0.627060i
\(825\) − 8.00000i − 0.278524i
\(826\) 0 0
\(827\) − 16.0000i − 0.556375i −0.960527 0.278187i \(-0.910266\pi\)
0.960527 0.278187i \(-0.0897336\pi\)
\(828\) 6.00000 0.208514
\(829\) −17.0000 −0.590434 −0.295217 0.955430i \(-0.595392\pi\)
−0.295217 + 0.955430i \(0.595392\pi\)
\(830\) − 14.0000i − 0.485947i
\(831\) −31.0000 −1.07538
\(832\) 0 0
\(833\) 21.0000 0.727607
\(834\) − 12.0000i − 0.415526i
\(835\) −16.0000 −0.553703
\(836\) 12.0000 0.415029
\(837\) − 4.00000i − 0.138260i
\(838\) 16.0000i 0.552711i
\(839\) − 12.0000i − 0.414286i −0.978311 0.207143i \(-0.933583\pi\)
0.978311 0.207143i \(-0.0664165\pi\)
\(840\) 6.00000i 0.207020i
\(841\) −28.0000 −0.965517
\(842\) 19.0000 0.654783
\(843\) − 19.0000i − 0.654395i
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) − 14.0000i − 0.481046i
\(848\) 9.00000 0.309061
\(849\) −18.0000 −0.617758
\(850\) 28.0000i 0.960392i
\(851\) − 6.00000i − 0.205677i
\(852\) − 6.00000i − 0.205557i
\(853\) − 21.0000i − 0.719026i −0.933140 0.359513i \(-0.882943\pi\)
0.933140 0.359513i \(-0.117057\pi\)
\(854\) 2.00000 0.0684386
\(855\) −6.00000 −0.205196
\(856\) − 18.0000i − 0.615227i
\(857\) 31.0000 1.05894 0.529470 0.848329i \(-0.322391\pi\)
0.529470 + 0.848329i \(0.322391\pi\)
\(858\) 0 0
\(859\) 34.0000 1.16007 0.580033 0.814593i \(-0.303040\pi\)
0.580033 + 0.814593i \(0.303040\pi\)
\(860\) 6.00000i 0.204598i
\(861\) −18.0000 −0.613438
\(862\) 30.0000 1.02180
\(863\) 10.0000i 0.340404i 0.985409 + 0.170202i \(0.0544420\pi\)
−0.985409 + 0.170202i \(0.945558\pi\)
\(864\) − 5.00000i − 0.170103i
\(865\) − 6.00000i − 0.204006i
\(866\) − 19.0000i − 0.645646i
\(867\) −32.0000 −1.08678
\(868\) 8.00000 0.271538
\(869\) − 8.00000i − 0.271381i
\(870\) 1.00000 0.0339032
\(871\) 0 0
\(872\) 6.00000 0.203186
\(873\) − 2.00000i − 0.0676897i
\(874\) 36.0000 1.21772
\(875\) −18.0000 −0.608511
\(876\) 11.0000i 0.371656i
\(877\) 17.0000i 0.574049i 0.957923 + 0.287025i \(0.0926662\pi\)
−0.957923 + 0.287025i \(0.907334\pi\)
\(878\) − 14.0000i − 0.472477i
\(879\) − 9.00000i − 0.303562i
\(880\) −2.00000 −0.0674200
\(881\) −37.0000 −1.24656 −0.623281 0.781998i \(-0.714201\pi\)
−0.623281 + 0.781998i \(0.714201\pi\)
\(882\) 3.00000i 0.101015i
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) − 4.00000i − 0.134383i
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −3.00000 −0.100673
\(889\) 40.0000i 1.34156i
\(890\) 14.0000i 0.469281i
\(891\) 2.00000i 0.0670025i
\(892\) 16.0000i 0.535720i
\(893\) −36.0000 −1.20469
\(894\) 3.00000 0.100335
\(895\) − 2.00000i − 0.0668526i
\(896\) 6.00000 0.200446
\(897\) 0 0
\(898\) 34.0000 1.13459
\(899\) − 4.00000i − 0.133407i
\(900\) 4.00000 0.133333
\(901\) −63.0000 −2.09883
\(902\) − 18.0000i − 0.599334i
\(903\) − 12.0000i − 0.399335i
\(904\) − 45.0000i − 1.49668i
\(905\) − 7.00000i − 0.232688i
\(906\) 2.00000 0.0664455
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) − 14.0000i − 0.464606i
\(909\) −3.00000 −0.0995037
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) − 6.00000i − 0.198680i
\(913\) 28.0000 0.926665
\(914\) 13.0000 0.430002
\(915\) 1.00000i 0.0330590i
\(916\) 22.0000i 0.726900i
\(917\) 16.0000i 0.528367i
\(918\) − 7.00000i − 0.231034i
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 18.0000 0.593442
\(921\) 14.0000i 0.461316i
\(922\) −19.0000 −0.625732
\(923\) 0 0
\(924\) −4.00000 −0.131590
\(925\) − 4.00000i − 0.131519i
\(926\) −26.0000 −0.854413
\(927\) −6.00000 −0.197066
\(928\) − 5.00000i − 0.164133i
\(929\) − 27.0000i − 0.885841i −0.896561 0.442921i \(-0.853942\pi\)
0.896561 0.442921i \(-0.146058\pi\)
\(930\) − 4.00000i − 0.131165i
\(931\) − 18.0000i − 0.589926i
\(932\) −10.0000 −0.327561
\(933\) 18.0000 0.589294
\(934\) − 6.00000i − 0.196326i
\(935\) 14.0000 0.457849
\(936\) 0 0
\(937\) −49.0000 −1.60076 −0.800380 0.599493i \(-0.795369\pi\)
−0.800380 + 0.599493i \(0.795369\pi\)
\(938\) − 4.00000i − 0.130605i
\(939\) −6.00000 −0.195803
\(940\) −6.00000 −0.195698
\(941\) − 38.0000i − 1.23876i −0.785090 0.619382i \(-0.787383\pi\)
0.785090 0.619382i \(-0.212617\pi\)
\(942\) 3.00000i 0.0977453i
\(943\) 54.0000i 1.75848i
\(944\) 0 0
\(945\) 2.00000 0.0650600
\(946\) 12.0000 0.390154
\(947\) 48.0000i 1.55979i 0.625910 + 0.779895i \(0.284728\pi\)
−0.625910 + 0.779895i \(0.715272\pi\)
\(948\) 4.00000 0.129914
\(949\) 0 0
\(950\) 24.0000 0.778663
\(951\) 25.0000i 0.810681i
\(952\) 42.0000 1.36123
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) − 9.00000i − 0.291386i
\(955\) 4.00000i 0.129437i
\(956\) 30.0000i 0.970269i
\(957\) 2.00000i 0.0646508i
\(958\) 24.0000 0.775405
\(959\) 6.00000 0.193750
\(960\) − 7.00000i − 0.225924i
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) − 7.00000i − 0.225455i
\(965\) 9.00000 0.289720
\(966\) −12.0000 −0.386094
\(967\) 2.00000i 0.0643157i 0.999483 + 0.0321578i \(0.0102379\pi\)
−0.999483 + 0.0321578i \(0.989762\pi\)
\(968\) 21.0000i 0.674966i
\(969\) 42.0000i 1.34923i
\(970\) − 2.00000i − 0.0642161i
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) −1.00000 −0.0320750
\(973\) − 24.0000i − 0.769405i
\(974\) −18.0000 −0.576757
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) 33.0000i 1.05576i 0.849318 + 0.527882i \(0.177014\pi\)
−0.849318 + 0.527882i \(0.822986\pi\)
\(978\) 4.00000 0.127906
\(979\) −28.0000 −0.894884
\(980\) − 3.00000i − 0.0958315i
\(981\) − 2.00000i − 0.0638551i
\(982\) − 6.00000i − 0.191468i
\(983\) − 4.00000i − 0.127580i −0.997963 0.0637901i \(-0.979681\pi\)
0.997963 0.0637901i \(-0.0203188\pi\)
\(984\) 27.0000 0.860729
\(985\) 6.00000 0.191176
\(986\) − 7.00000i − 0.222925i
\(987\) 12.0000 0.381964
\(988\) 0 0
\(989\) −36.0000 −1.14473
\(990\) 2.00000i 0.0635642i
\(991\) −2.00000 −0.0635321 −0.0317660 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) −20.0000 −0.635001
\(993\) 4.00000i 0.126936i
\(994\) 12.0000i 0.380617i
\(995\) 14.0000i 0.443830i
\(996\) 14.0000i 0.443607i
\(997\) −35.0000 −1.10846 −0.554231 0.832363i \(-0.686987\pi\)
−0.554231 + 0.832363i \(0.686987\pi\)
\(998\) −24.0000 −0.759707
\(999\) 1.00000i 0.0316386i
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 507.2.b.b.337.2 2
3.2 odd 2 1521.2.b.c.1351.1 2
13.2 odd 12 39.2.e.a.22.1 yes 2
13.3 even 3 507.2.j.d.316.1 4
13.4 even 6 507.2.j.d.361.1 4
13.5 odd 4 507.2.a.c.1.1 1
13.6 odd 12 39.2.e.a.16.1 2
13.7 odd 12 507.2.e.c.484.1 2
13.8 odd 4 507.2.a.b.1.1 1
13.9 even 3 507.2.j.d.361.2 4
13.10 even 6 507.2.j.d.316.2 4
13.11 odd 12 507.2.e.c.22.1 2
13.12 even 2 inner 507.2.b.b.337.1 2
39.2 even 12 117.2.g.b.100.1 2
39.5 even 4 1521.2.a.a.1.1 1
39.8 even 4 1521.2.a.d.1.1 1
39.32 even 12 117.2.g.b.55.1 2
39.38 odd 2 1521.2.b.c.1351.2 2
52.15 even 12 624.2.q.c.529.1 2
52.19 even 12 624.2.q.c.289.1 2
52.31 even 4 8112.2.a.w.1.1 1
52.47 even 4 8112.2.a.bc.1.1 1
65.2 even 12 975.2.bb.d.724.1 4
65.19 odd 12 975.2.i.f.601.1 2
65.28 even 12 975.2.bb.d.724.2 4
65.32 even 12 975.2.bb.d.874.2 4
65.54 odd 12 975.2.i.f.451.1 2
65.58 even 12 975.2.bb.d.874.1 4
156.71 odd 12 1872.2.t.j.289.1 2
156.119 odd 12 1872.2.t.j.1153.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.e.a.16.1 2 13.6 odd 12
39.2.e.a.22.1 yes 2 13.2 odd 12
117.2.g.b.55.1 2 39.32 even 12
117.2.g.b.100.1 2 39.2 even 12
507.2.a.b.1.1 1 13.8 odd 4
507.2.a.c.1.1 1 13.5 odd 4
507.2.b.b.337.1 2 13.12 even 2 inner
507.2.b.b.337.2 2 1.1 even 1 trivial
507.2.e.c.22.1 2 13.11 odd 12
507.2.e.c.484.1 2 13.7 odd 12
507.2.j.d.316.1 4 13.3 even 3
507.2.j.d.316.2 4 13.10 even 6
507.2.j.d.361.1 4 13.4 even 6
507.2.j.d.361.2 4 13.9 even 3
624.2.q.c.289.1 2 52.19 even 12
624.2.q.c.529.1 2 52.15 even 12
975.2.i.f.451.1 2 65.54 odd 12
975.2.i.f.601.1 2 65.19 odd 12
975.2.bb.d.724.1 4 65.2 even 12
975.2.bb.d.724.2 4 65.28 even 12
975.2.bb.d.874.1 4 65.58 even 12
975.2.bb.d.874.2 4 65.32 even 12
1521.2.a.a.1.1 1 39.5 even 4
1521.2.a.d.1.1 1 39.8 even 4
1521.2.b.c.1351.1 2 3.2 odd 2
1521.2.b.c.1351.2 2 39.38 odd 2
1872.2.t.j.289.1 2 156.71 odd 12
1872.2.t.j.1153.1 2 156.119 odd 12
8112.2.a.w.1.1 1 52.31 even 4
8112.2.a.bc.1.1 1 52.47 even 4