Properties

Label 5929.2.a.m.1.2
Level $5929$
Weight $2$
Character 5929.1
Self dual yes
Analytic conductor $47.343$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5929,2,Mod(1,5929)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5929, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5929.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5929 = 7^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5929.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: \(47.3433033584\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 5929.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.23607 q^{2} -3.23607 q^{3} +3.00000 q^{4} +2.00000 q^{5} -7.23607 q^{6} +2.23607 q^{8} +7.47214 q^{9} +O(q^{10})\) \(q+2.23607 q^{2} -3.23607 q^{3} +3.00000 q^{4} +2.00000 q^{5} -7.23607 q^{6} +2.23607 q^{8} +7.47214 q^{9} +4.47214 q^{10} -9.70820 q^{12} -1.23607 q^{13} -6.47214 q^{15} -1.00000 q^{16} +1.23607 q^{17} +16.7082 q^{18} -2.47214 q^{19} +6.00000 q^{20} -6.47214 q^{23} -7.23607 q^{24} -1.00000 q^{25} -2.76393 q^{26} -14.4721 q^{27} +0.472136 q^{29} -14.4721 q^{30} +7.23607 q^{31} -6.70820 q^{32} +2.76393 q^{34} +22.4164 q^{36} +0.472136 q^{37} -5.52786 q^{38} +4.00000 q^{39} +4.47214 q^{40} -6.76393 q^{41} -8.00000 q^{43} +14.9443 q^{45} -14.4721 q^{46} -7.23607 q^{47} +3.23607 q^{48} -2.23607 q^{50} -4.00000 q^{51} -3.70820 q^{52} +8.47214 q^{53} -32.3607 q^{54} +8.00000 q^{57} +1.05573 q^{58} -3.23607 q^{59} -19.4164 q^{60} -2.76393 q^{61} +16.1803 q^{62} -13.0000 q^{64} -2.47214 q^{65} +5.52786 q^{67} +3.70820 q^{68} +20.9443 q^{69} -1.52786 q^{71} +16.7082 q^{72} -5.23607 q^{73} +1.05573 q^{74} +3.23607 q^{75} -7.41641 q^{76} +8.94427 q^{78} -8.94427 q^{79} -2.00000 q^{80} +24.4164 q^{81} -15.1246 q^{82} +15.4164 q^{83} +2.47214 q^{85} -17.8885 q^{86} -1.52786 q^{87} -2.00000 q^{89} +33.4164 q^{90} -19.4164 q^{92} -23.4164 q^{93} -16.1803 q^{94} -4.94427 q^{95} +21.7082 q^{96} +9.41641 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 6 q^{4} + 4 q^{5} - 10 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 6 q^{4} + 4 q^{5} - 10 q^{6} + 6 q^{9} - 6 q^{12} + 2 q^{13} - 4 q^{15} - 2 q^{16} - 2 q^{17} + 20 q^{18} + 4 q^{19} + 12 q^{20} - 4 q^{23} - 10 q^{24} - 2 q^{25} - 10 q^{26} - 20 q^{27} - 8 q^{29} - 20 q^{30} + 10 q^{31} + 10 q^{34} + 18 q^{36} - 8 q^{37} - 20 q^{38} + 8 q^{39} - 18 q^{41} - 16 q^{43} + 12 q^{45} - 20 q^{46} - 10 q^{47} + 2 q^{48} - 8 q^{51} + 6 q^{52} + 8 q^{53} - 20 q^{54} + 16 q^{57} + 20 q^{58} - 2 q^{59} - 12 q^{60} - 10 q^{61} + 10 q^{62} - 26 q^{64} + 4 q^{65} + 20 q^{67} - 6 q^{68} + 24 q^{69} - 12 q^{71} + 20 q^{72} - 6 q^{73} + 20 q^{74} + 2 q^{75} + 12 q^{76} - 4 q^{80} + 22 q^{81} + 10 q^{82} + 4 q^{83} - 4 q^{85} - 12 q^{87} - 4 q^{89} + 40 q^{90} - 12 q^{92} - 20 q^{93} - 10 q^{94} + 8 q^{95} + 30 q^{96} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607 1.58114 0.790569 0.612372i \(-0.209785\pi\)
0.790569 + 0.612372i \(0.209785\pi\)
\(3\) −3.23607 −1.86834 −0.934172 0.356822i \(-0.883860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 3.00000 1.50000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −7.23607 −2.95411
\(7\) 0 0
\(8\) 2.23607 0.790569
\(9\) 7.47214 2.49071
\(10\) 4.47214 1.41421
\(11\) 0 0
\(12\) −9.70820 −2.80252
\(13\) −1.23607 −0.342824 −0.171412 0.985199i \(-0.554833\pi\)
−0.171412 + 0.985199i \(0.554833\pi\)
\(14\) 0 0
\(15\) −6.47214 −1.67110
\(16\) −1.00000 −0.250000
\(17\) 1.23607 0.299791 0.149895 0.988702i \(-0.452106\pi\)
0.149895 + 0.988702i \(0.452106\pi\)
\(18\) 16.7082 3.93816
\(19\) −2.47214 −0.567147 −0.283573 0.958951i \(-0.591520\pi\)
−0.283573 + 0.958951i \(0.591520\pi\)
\(20\) 6.00000 1.34164
\(21\) 0 0
\(22\) 0 0
\(23\) −6.47214 −1.34953 −0.674767 0.738031i \(-0.735756\pi\)
−0.674767 + 0.738031i \(0.735756\pi\)
\(24\) −7.23607 −1.47706
\(25\) −1.00000 −0.200000
\(26\) −2.76393 −0.542052
\(27\) −14.4721 −2.78516
\(28\) 0 0
\(29\) 0.472136 0.0876734 0.0438367 0.999039i \(-0.486042\pi\)
0.0438367 + 0.999039i \(0.486042\pi\)
\(30\) −14.4721 −2.64224
\(31\) 7.23607 1.29964 0.649818 0.760090i \(-0.274845\pi\)
0.649818 + 0.760090i \(0.274845\pi\)
\(32\) −6.70820 −1.18585
\(33\) 0 0
\(34\) 2.76393 0.474010
\(35\) 0 0
\(36\) 22.4164 3.73607
\(37\) 0.472136 0.0776187 0.0388093 0.999247i \(-0.487644\pi\)
0.0388093 + 0.999247i \(0.487644\pi\)
\(38\) −5.52786 −0.896738
\(39\) 4.00000 0.640513
\(40\) 4.47214 0.707107
\(41\) −6.76393 −1.05635 −0.528174 0.849136i \(-0.677123\pi\)
−0.528174 + 0.849136i \(0.677123\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 14.9443 2.22776
\(46\) −14.4721 −2.13380
\(47\) −7.23607 −1.05549 −0.527744 0.849403i \(-0.676962\pi\)
−0.527744 + 0.849403i \(0.676962\pi\)
\(48\) 3.23607 0.467086
\(49\) 0 0
\(50\) −2.23607 −0.316228
\(51\) −4.00000 −0.560112
\(52\) −3.70820 −0.514235
\(53\) 8.47214 1.16374 0.581869 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) −32.3607 −4.40373
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 1.05573 0.138624
\(59\) −3.23607 −0.421300 −0.210650 0.977562i \(-0.567558\pi\)
−0.210650 + 0.977562i \(0.567558\pi\)
\(60\) −19.4164 −2.50665
\(61\) −2.76393 −0.353885 −0.176943 0.984221i \(-0.556621\pi\)
−0.176943 + 0.984221i \(0.556621\pi\)
\(62\) 16.1803 2.05491
\(63\) 0 0
\(64\) −13.0000 −1.62500
\(65\) −2.47214 −0.306631
\(66\) 0 0
\(67\) 5.52786 0.675336 0.337668 0.941265i \(-0.390362\pi\)
0.337668 + 0.941265i \(0.390362\pi\)
\(68\) 3.70820 0.449686
\(69\) 20.9443 2.52139
\(70\) 0 0
\(71\) −1.52786 −0.181324 −0.0906621 0.995882i \(-0.528898\pi\)
−0.0906621 + 0.995882i \(0.528898\pi\)
\(72\) 16.7082 1.96908
\(73\) −5.23607 −0.612835 −0.306418 0.951897i \(-0.599130\pi\)
−0.306418 + 0.951897i \(0.599130\pi\)
\(74\) 1.05573 0.122726
\(75\) 3.23607 0.373669
\(76\) −7.41641 −0.850720
\(77\) 0 0
\(78\) 8.94427 1.01274
\(79\) −8.94427 −1.00631 −0.503155 0.864196i \(-0.667827\pi\)
−0.503155 + 0.864196i \(0.667827\pi\)
\(80\) −2.00000 −0.223607
\(81\) 24.4164 2.71293
\(82\) −15.1246 −1.67023
\(83\) 15.4164 1.69217 0.846085 0.533048i \(-0.178953\pi\)
0.846085 + 0.533048i \(0.178953\pi\)
\(84\) 0 0
\(85\) 2.47214 0.268141
\(86\) −17.8885 −1.92897
\(87\) −1.52786 −0.163804
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 33.4164 3.52240
\(91\) 0 0
\(92\) −19.4164 −2.02430
\(93\) −23.4164 −2.42817
\(94\) −16.1803 −1.66887
\(95\) −4.94427 −0.507272
\(96\) 21.7082 2.21558
\(97\) 9.41641 0.956091 0.478046 0.878335i \(-0.341345\pi\)
0.478046 + 0.878335i \(0.341345\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −3.00000 −0.300000
\(101\) −9.23607 −0.919023 −0.459512 0.888172i \(-0.651976\pi\)
−0.459512 + 0.888172i \(0.651976\pi\)
\(102\) −8.94427 −0.885615
\(103\) 5.70820 0.562446 0.281223 0.959642i \(-0.409260\pi\)
0.281223 + 0.959642i \(0.409260\pi\)
\(104\) −2.76393 −0.271026
\(105\) 0 0
\(106\) 18.9443 1.84003
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −43.4164 −4.17775
\(109\) −4.47214 −0.428353 −0.214176 0.976795i \(-0.568707\pi\)
−0.214176 + 0.976795i \(0.568707\pi\)
\(110\) 0 0
\(111\) −1.52786 −0.145018
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 17.8885 1.67542
\(115\) −12.9443 −1.20706
\(116\) 1.41641 0.131510
\(117\) −9.23607 −0.853875
\(118\) −7.23607 −0.666134
\(119\) 0 0
\(120\) −14.4721 −1.32112
\(121\) 0 0
\(122\) −6.18034 −0.559542
\(123\) 21.8885 1.97362
\(124\) 21.7082 1.94945
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −20.9443 −1.85850 −0.929252 0.369447i \(-0.879547\pi\)
−0.929252 + 0.369447i \(0.879547\pi\)
\(128\) −15.6525 −1.38350
\(129\) 25.8885 2.27936
\(130\) −5.52786 −0.484826
\(131\) −13.8885 −1.21345 −0.606724 0.794913i \(-0.707517\pi\)
−0.606724 + 0.794913i \(0.707517\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 12.3607 1.06780
\(135\) −28.9443 −2.49113
\(136\) 2.76393 0.237005
\(137\) 7.52786 0.643149 0.321574 0.946884i \(-0.395788\pi\)
0.321574 + 0.946884i \(0.395788\pi\)
\(138\) 46.8328 3.98667
\(139\) −10.4721 −0.888235 −0.444117 0.895969i \(-0.646483\pi\)
−0.444117 + 0.895969i \(0.646483\pi\)
\(140\) 0 0
\(141\) 23.4164 1.97202
\(142\) −3.41641 −0.286699
\(143\) 0 0
\(144\) −7.47214 −0.622678
\(145\) 0.944272 0.0784175
\(146\) −11.7082 −0.968978
\(147\) 0 0
\(148\) 1.41641 0.116428
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 7.23607 0.590822
\(151\) 8.94427 0.727875 0.363937 0.931423i \(-0.381432\pi\)
0.363937 + 0.931423i \(0.381432\pi\)
\(152\) −5.52786 −0.448369
\(153\) 9.23607 0.746692
\(154\) 0 0
\(155\) 14.4721 1.16243
\(156\) 12.0000 0.960769
\(157\) 6.94427 0.554213 0.277107 0.960839i \(-0.410624\pi\)
0.277107 + 0.960839i \(0.410624\pi\)
\(158\) −20.0000 −1.59111
\(159\) −27.4164 −2.17426
\(160\) −13.4164 −1.06066
\(161\) 0 0
\(162\) 54.5967 4.28953
\(163\) 23.4164 1.83411 0.917057 0.398755i \(-0.130558\pi\)
0.917057 + 0.398755i \(0.130558\pi\)
\(164\) −20.2918 −1.58452
\(165\) 0 0
\(166\) 34.4721 2.67556
\(167\) −12.9443 −1.00166 −0.500829 0.865546i \(-0.666971\pi\)
−0.500829 + 0.865546i \(0.666971\pi\)
\(168\) 0 0
\(169\) −11.4721 −0.882472
\(170\) 5.52786 0.423968
\(171\) −18.4721 −1.41260
\(172\) −24.0000 −1.82998
\(173\) −17.2361 −1.31043 −0.655217 0.755441i \(-0.727423\pi\)
−0.655217 + 0.755441i \(0.727423\pi\)
\(174\) −3.41641 −0.258997
\(175\) 0 0
\(176\) 0 0
\(177\) 10.4721 0.787134
\(178\) −4.47214 −0.335201
\(179\) 8.94427 0.668526 0.334263 0.942480i \(-0.391513\pi\)
0.334263 + 0.942480i \(0.391513\pi\)
\(180\) 44.8328 3.34164
\(181\) −1.41641 −0.105281 −0.0526404 0.998614i \(-0.516764\pi\)
−0.0526404 + 0.998614i \(0.516764\pi\)
\(182\) 0 0
\(183\) 8.94427 0.661180
\(184\) −14.4721 −1.06690
\(185\) 0.944272 0.0694243
\(186\) −52.3607 −3.83927
\(187\) 0 0
\(188\) −21.7082 −1.58323
\(189\) 0 0
\(190\) −11.0557 −0.802067
\(191\) −20.9443 −1.51547 −0.757737 0.652560i \(-0.773695\pi\)
−0.757737 + 0.652560i \(0.773695\pi\)
\(192\) 42.0689 3.03606
\(193\) 23.8885 1.71954 0.859768 0.510686i \(-0.170608\pi\)
0.859768 + 0.510686i \(0.170608\pi\)
\(194\) 21.0557 1.51171
\(195\) 8.00000 0.572892
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) −20.1803 −1.43055 −0.715273 0.698845i \(-0.753698\pi\)
−0.715273 + 0.698845i \(0.753698\pi\)
\(200\) −2.23607 −0.158114
\(201\) −17.8885 −1.26176
\(202\) −20.6525 −1.45310
\(203\) 0 0
\(204\) −12.0000 −0.840168
\(205\) −13.5279 −0.944827
\(206\) 12.7639 0.889305
\(207\) −48.3607 −3.36130
\(208\) 1.23607 0.0857059
\(209\) 0 0
\(210\) 0 0
\(211\) −21.8885 −1.50687 −0.753435 0.657523i \(-0.771604\pi\)
−0.753435 + 0.657523i \(0.771604\pi\)
\(212\) 25.4164 1.74561
\(213\) 4.94427 0.338776
\(214\) 8.94427 0.611418
\(215\) −16.0000 −1.09119
\(216\) −32.3607 −2.20187
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 16.9443 1.14499
\(220\) 0 0
\(221\) −1.52786 −0.102775
\(222\) −3.41641 −0.229294
\(223\) −12.1803 −0.815656 −0.407828 0.913059i \(-0.633714\pi\)
−0.407828 + 0.913059i \(0.633714\pi\)
\(224\) 0 0
\(225\) −7.47214 −0.498142
\(226\) 4.47214 0.297482
\(227\) −29.8885 −1.98377 −0.991886 0.127129i \(-0.959424\pi\)
−0.991886 + 0.127129i \(0.959424\pi\)
\(228\) 24.0000 1.58944
\(229\) −4.47214 −0.295527 −0.147764 0.989023i \(-0.547207\pi\)
−0.147764 + 0.989023i \(0.547207\pi\)
\(230\) −28.9443 −1.90853
\(231\) 0 0
\(232\) 1.05573 0.0693119
\(233\) 17.4164 1.14099 0.570493 0.821302i \(-0.306752\pi\)
0.570493 + 0.821302i \(0.306752\pi\)
\(234\) −20.6525 −1.35009
\(235\) −14.4721 −0.944058
\(236\) −9.70820 −0.631950
\(237\) 28.9443 1.88013
\(238\) 0 0
\(239\) −25.8885 −1.67459 −0.837295 0.546751i \(-0.815864\pi\)
−0.837295 + 0.546751i \(0.815864\pi\)
\(240\) 6.47214 0.417775
\(241\) 27.1246 1.74725 0.873625 0.486600i \(-0.161763\pi\)
0.873625 + 0.486600i \(0.161763\pi\)
\(242\) 0 0
\(243\) −35.5967 −2.28353
\(244\) −8.29180 −0.530828
\(245\) 0 0
\(246\) 48.9443 3.12057
\(247\) 3.05573 0.194431
\(248\) 16.1803 1.02745
\(249\) −49.8885 −3.16156
\(250\) −26.8328 −1.69706
\(251\) −17.7082 −1.11773 −0.558866 0.829258i \(-0.688763\pi\)
−0.558866 + 0.829258i \(0.688763\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −46.8328 −2.93855
\(255\) −8.00000 −0.500979
\(256\) −9.00000 −0.562500
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 57.8885 3.60398
\(259\) 0 0
\(260\) −7.41641 −0.459946
\(261\) 3.52786 0.218369
\(262\) −31.0557 −1.91863
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 16.9443 1.04088
\(266\) 0 0
\(267\) 6.47214 0.396088
\(268\) 16.5836 1.01300
\(269\) 13.4164 0.818013 0.409006 0.912532i \(-0.365875\pi\)
0.409006 + 0.912532i \(0.365875\pi\)
\(270\) −64.7214 −3.93882
\(271\) −1.52786 −0.0928111 −0.0464056 0.998923i \(-0.514777\pi\)
−0.0464056 + 0.998923i \(0.514777\pi\)
\(272\) −1.23607 −0.0749476
\(273\) 0 0
\(274\) 16.8328 1.01691
\(275\) 0 0
\(276\) 62.8328 3.78209
\(277\) −15.8885 −0.954650 −0.477325 0.878727i \(-0.658394\pi\)
−0.477325 + 0.878727i \(0.658394\pi\)
\(278\) −23.4164 −1.40442
\(279\) 54.0689 3.23702
\(280\) 0 0
\(281\) 12.4721 0.744025 0.372013 0.928228i \(-0.378668\pi\)
0.372013 + 0.928228i \(0.378668\pi\)
\(282\) 52.3607 3.11803
\(283\) −5.88854 −0.350038 −0.175019 0.984565i \(-0.555999\pi\)
−0.175019 + 0.984565i \(0.555999\pi\)
\(284\) −4.58359 −0.271986
\(285\) 16.0000 0.947758
\(286\) 0 0
\(287\) 0 0
\(288\) −50.1246 −2.95362
\(289\) −15.4721 −0.910126
\(290\) 2.11146 0.123989
\(291\) −30.4721 −1.78631
\(292\) −15.7082 −0.919253
\(293\) 15.1246 0.883589 0.441795 0.897116i \(-0.354342\pi\)
0.441795 + 0.897116i \(0.354342\pi\)
\(294\) 0 0
\(295\) −6.47214 −0.376822
\(296\) 1.05573 0.0613629
\(297\) 0 0
\(298\) −31.3050 −1.81345
\(299\) 8.00000 0.462652
\(300\) 9.70820 0.560503
\(301\) 0 0
\(302\) 20.0000 1.15087
\(303\) 29.8885 1.71705
\(304\) 2.47214 0.141787
\(305\) −5.52786 −0.316525
\(306\) 20.6525 1.18062
\(307\) 8.94427 0.510477 0.255238 0.966878i \(-0.417846\pi\)
0.255238 + 0.966878i \(0.417846\pi\)
\(308\) 0 0
\(309\) −18.4721 −1.05084
\(310\) 32.3607 1.83796
\(311\) 21.7082 1.23096 0.615480 0.788153i \(-0.288962\pi\)
0.615480 + 0.788153i \(0.288962\pi\)
\(312\) 8.94427 0.506370
\(313\) 2.94427 0.166420 0.0832100 0.996532i \(-0.473483\pi\)
0.0832100 + 0.996532i \(0.473483\pi\)
\(314\) 15.5279 0.876288
\(315\) 0 0
\(316\) −26.8328 −1.50946
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) −61.3050 −3.43781
\(319\) 0 0
\(320\) −26.0000 −1.45344
\(321\) −12.9443 −0.722479
\(322\) 0 0
\(323\) −3.05573 −0.170025
\(324\) 73.2492 4.06940
\(325\) 1.23607 0.0685647
\(326\) 52.3607 2.89999
\(327\) 14.4721 0.800311
\(328\) −15.1246 −0.835117
\(329\) 0 0
\(330\) 0 0
\(331\) 21.8885 1.20310 0.601552 0.798834i \(-0.294549\pi\)
0.601552 + 0.798834i \(0.294549\pi\)
\(332\) 46.2492 2.53826
\(333\) 3.52786 0.193326
\(334\) −28.9443 −1.58376
\(335\) 11.0557 0.604039
\(336\) 0 0
\(337\) 20.4721 1.11519 0.557594 0.830114i \(-0.311725\pi\)
0.557594 + 0.830114i \(0.311725\pi\)
\(338\) −25.6525 −1.39531
\(339\) −6.47214 −0.351518
\(340\) 7.41641 0.402211
\(341\) 0 0
\(342\) −41.3050 −2.23352
\(343\) 0 0
\(344\) −17.8885 −0.964486
\(345\) 41.8885 2.25520
\(346\) −38.5410 −2.07198
\(347\) −3.05573 −0.164040 −0.0820200 0.996631i \(-0.526137\pi\)
−0.0820200 + 0.996631i \(0.526137\pi\)
\(348\) −4.58359 −0.245706
\(349\) −2.76393 −0.147950 −0.0739749 0.997260i \(-0.523568\pi\)
−0.0739749 + 0.997260i \(0.523568\pi\)
\(350\) 0 0
\(351\) 17.8885 0.954820
\(352\) 0 0
\(353\) 15.8885 0.845662 0.422831 0.906209i \(-0.361036\pi\)
0.422831 + 0.906209i \(0.361036\pi\)
\(354\) 23.4164 1.24457
\(355\) −3.05573 −0.162181
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) 7.05573 0.372387 0.186194 0.982513i \(-0.440385\pi\)
0.186194 + 0.982513i \(0.440385\pi\)
\(360\) 33.4164 1.76120
\(361\) −12.8885 −0.678344
\(362\) −3.16718 −0.166464
\(363\) 0 0
\(364\) 0 0
\(365\) −10.4721 −0.548137
\(366\) 20.0000 1.04542
\(367\) −17.1246 −0.893897 −0.446949 0.894560i \(-0.647489\pi\)
−0.446949 + 0.894560i \(0.647489\pi\)
\(368\) 6.47214 0.337383
\(369\) −50.5410 −2.63106
\(370\) 2.11146 0.109769
\(371\) 0 0
\(372\) −70.2492 −3.64225
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 38.8328 2.00532
\(376\) −16.1803 −0.834437
\(377\) −0.583592 −0.0300565
\(378\) 0 0
\(379\) −25.3050 −1.29983 −0.649914 0.760008i \(-0.725195\pi\)
−0.649914 + 0.760008i \(0.725195\pi\)
\(380\) −14.8328 −0.760907
\(381\) 67.7771 3.47233
\(382\) −46.8328 −2.39618
\(383\) −26.6525 −1.36188 −0.680939 0.732340i \(-0.738428\pi\)
−0.680939 + 0.732340i \(0.738428\pi\)
\(384\) 50.6525 2.58485
\(385\) 0 0
\(386\) 53.4164 2.71882
\(387\) −59.7771 −3.03864
\(388\) 28.2492 1.43414
\(389\) −19.8885 −1.00839 −0.504195 0.863590i \(-0.668211\pi\)
−0.504195 + 0.863590i \(0.668211\pi\)
\(390\) 17.8885 0.905822
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) 44.9443 2.26714
\(394\) 4.47214 0.225303
\(395\) −17.8885 −0.900070
\(396\) 0 0
\(397\) 0.111456 0.00559383 0.00279691 0.999996i \(-0.499110\pi\)
0.00279691 + 0.999996i \(0.499110\pi\)
\(398\) −45.1246 −2.26189
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 5.05573 0.252471 0.126236 0.992000i \(-0.459710\pi\)
0.126236 + 0.992000i \(0.459710\pi\)
\(402\) −40.0000 −1.99502
\(403\) −8.94427 −0.445546
\(404\) −27.7082 −1.37853
\(405\) 48.8328 2.42652
\(406\) 0 0
\(407\) 0 0
\(408\) −8.94427 −0.442807
\(409\) −31.1246 −1.53901 −0.769507 0.638639i \(-0.779498\pi\)
−0.769507 + 0.638639i \(0.779498\pi\)
\(410\) −30.2492 −1.49390
\(411\) −24.3607 −1.20162
\(412\) 17.1246 0.843669
\(413\) 0 0
\(414\) −108.138 −5.31468
\(415\) 30.8328 1.51352
\(416\) 8.29180 0.406539
\(417\) 33.8885 1.65953
\(418\) 0 0
\(419\) 6.65248 0.324995 0.162497 0.986709i \(-0.448045\pi\)
0.162497 + 0.986709i \(0.448045\pi\)
\(420\) 0 0
\(421\) −22.3607 −1.08979 −0.544896 0.838503i \(-0.683431\pi\)
−0.544896 + 0.838503i \(0.683431\pi\)
\(422\) −48.9443 −2.38257
\(423\) −54.0689 −2.62892
\(424\) 18.9443 0.920015
\(425\) −1.23607 −0.0599581
\(426\) 11.0557 0.535652
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) −35.7771 −1.72532
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 14.4721 0.696291
\(433\) −0.472136 −0.0226894 −0.0113447 0.999936i \(-0.503611\pi\)
−0.0113447 + 0.999936i \(0.503611\pi\)
\(434\) 0 0
\(435\) −3.05573 −0.146511
\(436\) −13.4164 −0.642529
\(437\) 16.0000 0.765384
\(438\) 37.8885 1.81038
\(439\) 1.52786 0.0729210 0.0364605 0.999335i \(-0.488392\pi\)
0.0364605 + 0.999335i \(0.488392\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.41641 −0.162502
\(443\) −7.05573 −0.335228 −0.167614 0.985853i \(-0.553606\pi\)
−0.167614 + 0.985853i \(0.553606\pi\)
\(444\) −4.58359 −0.217528
\(445\) −4.00000 −0.189618
\(446\) −27.2361 −1.28967
\(447\) 45.3050 2.14285
\(448\) 0 0
\(449\) −19.5279 −0.921577 −0.460788 0.887510i \(-0.652433\pi\)
−0.460788 + 0.887510i \(0.652433\pi\)
\(450\) −16.7082 −0.787632
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) −28.9443 −1.35992
\(454\) −66.8328 −3.13662
\(455\) 0 0
\(456\) 17.8885 0.837708
\(457\) −24.8328 −1.16163 −0.580815 0.814036i \(-0.697266\pi\)
−0.580815 + 0.814036i \(0.697266\pi\)
\(458\) −10.0000 −0.467269
\(459\) −17.8885 −0.834966
\(460\) −38.8328 −1.81059
\(461\) 10.1803 0.474146 0.237073 0.971492i \(-0.423812\pi\)
0.237073 + 0.971492i \(0.423812\pi\)
\(462\) 0 0
\(463\) −14.4721 −0.672577 −0.336289 0.941759i \(-0.609172\pi\)
−0.336289 + 0.941759i \(0.609172\pi\)
\(464\) −0.472136 −0.0219184
\(465\) −46.8328 −2.17182
\(466\) 38.9443 1.80406
\(467\) 34.0689 1.57652 0.788260 0.615342i \(-0.210982\pi\)
0.788260 + 0.615342i \(0.210982\pi\)
\(468\) −27.7082 −1.28081
\(469\) 0 0
\(470\) −32.3607 −1.49269
\(471\) −22.4721 −1.03546
\(472\) −7.23607 −0.333067
\(473\) 0 0
\(474\) 64.7214 2.97275
\(475\) 2.47214 0.113429
\(476\) 0 0
\(477\) 63.3050 2.89853
\(478\) −57.8885 −2.64776
\(479\) 22.4721 1.02678 0.513389 0.858156i \(-0.328390\pi\)
0.513389 + 0.858156i \(0.328390\pi\)
\(480\) 43.4164 1.98168
\(481\) −0.583592 −0.0266095
\(482\) 60.6525 2.76264
\(483\) 0 0
\(484\) 0 0
\(485\) 18.8328 0.855154
\(486\) −79.5967 −3.61058
\(487\) 8.36068 0.378859 0.189429 0.981894i \(-0.439336\pi\)
0.189429 + 0.981894i \(0.439336\pi\)
\(488\) −6.18034 −0.279771
\(489\) −75.7771 −3.42676
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 65.6656 2.96044
\(493\) 0.583592 0.0262837
\(494\) 6.83282 0.307423
\(495\) 0 0
\(496\) −7.23607 −0.324909
\(497\) 0 0
\(498\) −111.554 −4.99886
\(499\) 10.4721 0.468797 0.234399 0.972141i \(-0.424688\pi\)
0.234399 + 0.972141i \(0.424688\pi\)
\(500\) −36.0000 −1.60997
\(501\) 41.8885 1.87144
\(502\) −39.5967 −1.76729
\(503\) −3.41641 −0.152330 −0.0761650 0.997095i \(-0.524268\pi\)
−0.0761650 + 0.997095i \(0.524268\pi\)
\(504\) 0 0
\(505\) −18.4721 −0.821999
\(506\) 0 0
\(507\) 37.1246 1.64876
\(508\) −62.8328 −2.78776
\(509\) −31.5279 −1.39745 −0.698724 0.715391i \(-0.746248\pi\)
−0.698724 + 0.715391i \(0.746248\pi\)
\(510\) −17.8885 −0.792118
\(511\) 0 0
\(512\) 11.1803 0.494106
\(513\) 35.7771 1.57960
\(514\) 13.4164 0.591772
\(515\) 11.4164 0.503067
\(516\) 77.6656 3.41904
\(517\) 0 0
\(518\) 0 0
\(519\) 55.7771 2.44834
\(520\) −5.52786 −0.242413
\(521\) 14.3607 0.629153 0.314576 0.949232i \(-0.398138\pi\)
0.314576 + 0.949232i \(0.398138\pi\)
\(522\) 7.88854 0.345272
\(523\) 44.0000 1.92399 0.961993 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) −41.6656 −1.82017
\(525\) 0 0
\(526\) 0 0
\(527\) 8.94427 0.389619
\(528\) 0 0
\(529\) 18.8885 0.821241
\(530\) 37.8885 1.64577
\(531\) −24.1803 −1.04934
\(532\) 0 0
\(533\) 8.36068 0.362141
\(534\) 14.4721 0.626271
\(535\) 8.00000 0.345870
\(536\) 12.3607 0.533900
\(537\) −28.9443 −1.24904
\(538\) 30.0000 1.29339
\(539\) 0 0
\(540\) −86.8328 −3.73669
\(541\) 32.8328 1.41159 0.705797 0.708415i \(-0.250589\pi\)
0.705797 + 0.708415i \(0.250589\pi\)
\(542\) −3.41641 −0.146747
\(543\) 4.58359 0.196701
\(544\) −8.29180 −0.355508
\(545\) −8.94427 −0.383131
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 22.5836 0.964723
\(549\) −20.6525 −0.881426
\(550\) 0 0
\(551\) −1.16718 −0.0497237
\(552\) 46.8328 1.99334
\(553\) 0 0
\(554\) −35.5279 −1.50943
\(555\) −3.05573 −0.129708
\(556\) −31.4164 −1.33235
\(557\) 21.0557 0.892160 0.446080 0.894993i \(-0.352820\pi\)
0.446080 + 0.894993i \(0.352820\pi\)
\(558\) 120.902 5.11818
\(559\) 9.88854 0.418241
\(560\) 0 0
\(561\) 0 0
\(562\) 27.8885 1.17641
\(563\) 39.4164 1.66120 0.830602 0.556867i \(-0.187997\pi\)
0.830602 + 0.556867i \(0.187997\pi\)
\(564\) 70.2492 2.95803
\(565\) 4.00000 0.168281
\(566\) −13.1672 −0.553458
\(567\) 0 0
\(568\) −3.41641 −0.143349
\(569\) −16.4721 −0.690548 −0.345274 0.938502i \(-0.612214\pi\)
−0.345274 + 0.938502i \(0.612214\pi\)
\(570\) 35.7771 1.49854
\(571\) 32.9443 1.37867 0.689337 0.724440i \(-0.257902\pi\)
0.689337 + 0.724440i \(0.257902\pi\)
\(572\) 0 0
\(573\) 67.7771 2.83143
\(574\) 0 0
\(575\) 6.47214 0.269907
\(576\) −97.1378 −4.04741
\(577\) 28.4721 1.18531 0.592655 0.805456i \(-0.298080\pi\)
0.592655 + 0.805456i \(0.298080\pi\)
\(578\) −34.5967 −1.43903
\(579\) −77.3050 −3.21268
\(580\) 2.83282 0.117626
\(581\) 0 0
\(582\) −68.1378 −2.82440
\(583\) 0 0
\(584\) −11.7082 −0.484489
\(585\) −18.4721 −0.763729
\(586\) 33.8197 1.39708
\(587\) 13.1246 0.541711 0.270855 0.962620i \(-0.412693\pi\)
0.270855 + 0.962620i \(0.412693\pi\)
\(588\) 0 0
\(589\) −17.8885 −0.737085
\(590\) −14.4721 −0.595808
\(591\) −6.47214 −0.266228
\(592\) −0.472136 −0.0194047
\(593\) −32.2918 −1.32607 −0.663033 0.748591i \(-0.730731\pi\)
−0.663033 + 0.748591i \(0.730731\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −42.0000 −1.72039
\(597\) 65.3050 2.67275
\(598\) 17.8885 0.731517
\(599\) 3.41641 0.139591 0.0697953 0.997561i \(-0.477765\pi\)
0.0697953 + 0.997561i \(0.477765\pi\)
\(600\) 7.23607 0.295411
\(601\) 3.12461 0.127456 0.0637278 0.997967i \(-0.479701\pi\)
0.0637278 + 0.997967i \(0.479701\pi\)
\(602\) 0 0
\(603\) 41.3050 1.68207
\(604\) 26.8328 1.09181
\(605\) 0 0
\(606\) 66.8328 2.71490
\(607\) −4.94427 −0.200682 −0.100341 0.994953i \(-0.531993\pi\)
−0.100341 + 0.994953i \(0.531993\pi\)
\(608\) 16.5836 0.672553
\(609\) 0 0
\(610\) −12.3607 −0.500469
\(611\) 8.94427 0.361847
\(612\) 27.7082 1.12004
\(613\) 47.3050 1.91063 0.955315 0.295591i \(-0.0955166\pi\)
0.955315 + 0.295591i \(0.0955166\pi\)
\(614\) 20.0000 0.807134
\(615\) 43.7771 1.76526
\(616\) 0 0
\(617\) 33.4164 1.34529 0.672647 0.739964i \(-0.265157\pi\)
0.672647 + 0.739964i \(0.265157\pi\)
\(618\) −41.3050 −1.66153
\(619\) 29.1246 1.17062 0.585308 0.810811i \(-0.300973\pi\)
0.585308 + 0.810811i \(0.300973\pi\)
\(620\) 43.4164 1.74364
\(621\) 93.6656 3.75867
\(622\) 48.5410 1.94632
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) −19.0000 −0.760000
\(626\) 6.58359 0.263133
\(627\) 0 0
\(628\) 20.8328 0.831320
\(629\) 0.583592 0.0232693
\(630\) 0 0
\(631\) −24.0000 −0.955425 −0.477712 0.878516i \(-0.658534\pi\)
−0.477712 + 0.878516i \(0.658534\pi\)
\(632\) −20.0000 −0.795557
\(633\) 70.8328 2.81535
\(634\) 31.3050 1.24328
\(635\) −41.8885 −1.66230
\(636\) −82.2492 −3.26139
\(637\) 0 0
\(638\) 0 0
\(639\) −11.4164 −0.451626
\(640\) −31.3050 −1.23744
\(641\) −24.4721 −0.966591 −0.483296 0.875457i \(-0.660560\pi\)
−0.483296 + 0.875457i \(0.660560\pi\)
\(642\) −28.9443 −1.14234
\(643\) 29.1246 1.14856 0.574281 0.818658i \(-0.305282\pi\)
0.574281 + 0.818658i \(0.305282\pi\)
\(644\) 0 0
\(645\) 51.7771 2.03872
\(646\) −6.83282 −0.268834
\(647\) 22.0689 0.867617 0.433809 0.901005i \(-0.357169\pi\)
0.433809 + 0.901005i \(0.357169\pi\)
\(648\) 54.5967 2.14476
\(649\) 0 0
\(650\) 2.76393 0.108410
\(651\) 0 0
\(652\) 70.2492 2.75117
\(653\) 42.9443 1.68054 0.840270 0.542169i \(-0.182397\pi\)
0.840270 + 0.542169i \(0.182397\pi\)
\(654\) 32.3607 1.26540
\(655\) −27.7771 −1.08534
\(656\) 6.76393 0.264087
\(657\) −39.1246 −1.52640
\(658\) 0 0
\(659\) −17.8885 −0.696839 −0.348419 0.937339i \(-0.613281\pi\)
−0.348419 + 0.937339i \(0.613281\pi\)
\(660\) 0 0
\(661\) −12.8328 −0.499139 −0.249569 0.968357i \(-0.580289\pi\)
−0.249569 + 0.968357i \(0.580289\pi\)
\(662\) 48.9443 1.90227
\(663\) 4.94427 0.192020
\(664\) 34.4721 1.33778
\(665\) 0 0
\(666\) 7.88854 0.305675
\(667\) −3.05573 −0.118318
\(668\) −38.8328 −1.50249
\(669\) 39.4164 1.52393
\(670\) 24.7214 0.955069
\(671\) 0 0
\(672\) 0 0
\(673\) −5.41641 −0.208787 −0.104394 0.994536i \(-0.533290\pi\)
−0.104394 + 0.994536i \(0.533290\pi\)
\(674\) 45.7771 1.76327
\(675\) 14.4721 0.557033
\(676\) −34.4164 −1.32371
\(677\) 3.70820 0.142518 0.0712589 0.997458i \(-0.477298\pi\)
0.0712589 + 0.997458i \(0.477298\pi\)
\(678\) −14.4721 −0.555799
\(679\) 0 0
\(680\) 5.52786 0.211984
\(681\) 96.7214 3.70637
\(682\) 0 0
\(683\) 29.8885 1.14365 0.571827 0.820374i \(-0.306235\pi\)
0.571827 + 0.820374i \(0.306235\pi\)
\(684\) −55.4164 −2.11890
\(685\) 15.0557 0.575250
\(686\) 0 0
\(687\) 14.4721 0.552146
\(688\) 8.00000 0.304997
\(689\) −10.4721 −0.398957
\(690\) 93.6656 3.56579
\(691\) 48.5410 1.84659 0.923294 0.384095i \(-0.125486\pi\)
0.923294 + 0.384095i \(0.125486\pi\)
\(692\) −51.7082 −1.96565
\(693\) 0 0
\(694\) −6.83282 −0.259370
\(695\) −20.9443 −0.794462
\(696\) −3.41641 −0.129499
\(697\) −8.36068 −0.316683
\(698\) −6.18034 −0.233929
\(699\) −56.3607 −2.13176
\(700\) 0 0
\(701\) 24.4721 0.924300 0.462150 0.886802i \(-0.347078\pi\)
0.462150 + 0.886802i \(0.347078\pi\)
\(702\) 40.0000 1.50970
\(703\) −1.16718 −0.0440212
\(704\) 0 0
\(705\) 46.8328 1.76383
\(706\) 35.5279 1.33711
\(707\) 0 0
\(708\) 31.4164 1.18070
\(709\) 2.94427 0.110574 0.0552872 0.998470i \(-0.482393\pi\)
0.0552872 + 0.998470i \(0.482393\pi\)
\(710\) −6.83282 −0.256431
\(711\) −66.8328 −2.50643
\(712\) −4.47214 −0.167600
\(713\) −46.8328 −1.75390
\(714\) 0 0
\(715\) 0 0
\(716\) 26.8328 1.00279
\(717\) 83.7771 3.12871
\(718\) 15.7771 0.588796
\(719\) −33.4853 −1.24879 −0.624395 0.781108i \(-0.714655\pi\)
−0.624395 + 0.781108i \(0.714655\pi\)
\(720\) −14.9443 −0.556940
\(721\) 0 0
\(722\) −28.8197 −1.07256
\(723\) −87.7771 −3.26447
\(724\) −4.24922 −0.157921
\(725\) −0.472136 −0.0175347
\(726\) 0 0
\(727\) 51.0132 1.89197 0.945987 0.324206i \(-0.105097\pi\)
0.945987 + 0.324206i \(0.105097\pi\)
\(728\) 0 0
\(729\) 41.9443 1.55349
\(730\) −23.4164 −0.866680
\(731\) −9.88854 −0.365741
\(732\) 26.8328 0.991769
\(733\) 13.2361 0.488885 0.244443 0.969664i \(-0.421395\pi\)
0.244443 + 0.969664i \(0.421395\pi\)
\(734\) −38.2918 −1.41338
\(735\) 0 0
\(736\) 43.4164 1.60035
\(737\) 0 0
\(738\) −113.013 −4.16007
\(739\) −7.05573 −0.259549 −0.129775 0.991544i \(-0.541425\pi\)
−0.129775 + 0.991544i \(0.541425\pi\)
\(740\) 2.83282 0.104136
\(741\) −9.88854 −0.363265
\(742\) 0 0
\(743\) 33.8885 1.24325 0.621625 0.783315i \(-0.286473\pi\)
0.621625 + 0.783315i \(0.286473\pi\)
\(744\) −52.3607 −1.91964
\(745\) −28.0000 −1.02584
\(746\) −13.4164 −0.491210
\(747\) 115.193 4.21471
\(748\) 0 0
\(749\) 0 0
\(750\) 86.8328 3.17069
\(751\) 38.4721 1.40387 0.701934 0.712242i \(-0.252320\pi\)
0.701934 + 0.712242i \(0.252320\pi\)
\(752\) 7.23607 0.263872
\(753\) 57.3050 2.08831
\(754\) −1.30495 −0.0475235
\(755\) 17.8885 0.651031
\(756\) 0 0
\(757\) −19.8885 −0.722861 −0.361431 0.932399i \(-0.617712\pi\)
−0.361431 + 0.932399i \(0.617712\pi\)
\(758\) −56.5836 −2.05521
\(759\) 0 0
\(760\) −11.0557 −0.401033
\(761\) 17.5967 0.637882 0.318941 0.947775i \(-0.396673\pi\)
0.318941 + 0.947775i \(0.396673\pi\)
\(762\) 151.554 5.49023
\(763\) 0 0
\(764\) −62.8328 −2.27321
\(765\) 18.4721 0.667861
\(766\) −59.5967 −2.15332
\(767\) 4.00000 0.144432
\(768\) 29.1246 1.05094
\(769\) 31.7082 1.14343 0.571714 0.820453i \(-0.306279\pi\)
0.571714 + 0.820453i \(0.306279\pi\)
\(770\) 0 0
\(771\) −19.4164 −0.699265
\(772\) 71.6656 2.57930
\(773\) −6.36068 −0.228778 −0.114389 0.993436i \(-0.536491\pi\)
−0.114389 + 0.993436i \(0.536491\pi\)
\(774\) −133.666 −4.80451
\(775\) −7.23607 −0.259927
\(776\) 21.0557 0.755857
\(777\) 0 0
\(778\) −44.4721 −1.59440
\(779\) 16.7214 0.599105
\(780\) 24.0000 0.859338
\(781\) 0 0
\(782\) −17.8885 −0.639693
\(783\) −6.83282 −0.244185
\(784\) 0 0
\(785\) 13.8885 0.495703
\(786\) 100.498 3.58466
\(787\) −16.5836 −0.591141 −0.295571 0.955321i \(-0.595510\pi\)
−0.295571 + 0.955321i \(0.595510\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −40.0000 −1.42314
\(791\) 0 0
\(792\) 0 0
\(793\) 3.41641 0.121320
\(794\) 0.249224 0.00884461
\(795\) −54.8328 −1.94472
\(796\) −60.5410 −2.14582
\(797\) −2.94427 −0.104291 −0.0521457 0.998639i \(-0.516606\pi\)
−0.0521457 + 0.998639i \(0.516606\pi\)
\(798\) 0 0
\(799\) −8.94427 −0.316426
\(800\) 6.70820 0.237171
\(801\) −14.9443 −0.528030
\(802\) 11.3050 0.399192
\(803\) 0 0
\(804\) −53.6656 −1.89264
\(805\) 0 0
\(806\) −20.0000 −0.704470
\(807\) −43.4164 −1.52833
\(808\) −20.6525 −0.726552
\(809\) −38.9443 −1.36921 −0.684604 0.728915i \(-0.740025\pi\)
−0.684604 + 0.728915i \(0.740025\pi\)
\(810\) 109.193 3.83667
\(811\) 18.8328 0.661310 0.330655 0.943752i \(-0.392730\pi\)
0.330655 + 0.943752i \(0.392730\pi\)
\(812\) 0 0
\(813\) 4.94427 0.173403
\(814\) 0 0
\(815\) 46.8328 1.64048
\(816\) 4.00000 0.140028
\(817\) 19.7771 0.691913
\(818\) −69.5967 −2.43339
\(819\) 0 0
\(820\) −40.5836 −1.41724
\(821\) 8.83282 0.308267 0.154134 0.988050i \(-0.450741\pi\)
0.154134 + 0.988050i \(0.450741\pi\)
\(822\) −54.4721 −1.89993
\(823\) −49.8885 −1.73901 −0.869503 0.493928i \(-0.835561\pi\)
−0.869503 + 0.493928i \(0.835561\pi\)
\(824\) 12.7639 0.444653
\(825\) 0 0
\(826\) 0 0
\(827\) −4.94427 −0.171929 −0.0859646 0.996298i \(-0.527397\pi\)
−0.0859646 + 0.996298i \(0.527397\pi\)
\(828\) −145.082 −5.04195
\(829\) 16.8328 0.584628 0.292314 0.956322i \(-0.405575\pi\)
0.292314 + 0.956322i \(0.405575\pi\)
\(830\) 68.9443 2.39309
\(831\) 51.4164 1.78362
\(832\) 16.0689 0.557088
\(833\) 0 0
\(834\) 75.7771 2.62395
\(835\) −25.8885 −0.895910
\(836\) 0 0
\(837\) −104.721 −3.61970
\(838\) 14.8754 0.513862
\(839\) 14.0689 0.485712 0.242856 0.970062i \(-0.421916\pi\)
0.242856 + 0.970062i \(0.421916\pi\)
\(840\) 0 0
\(841\) −28.7771 −0.992313
\(842\) −50.0000 −1.72311
\(843\) −40.3607 −1.39010
\(844\) −65.6656 −2.26030
\(845\) −22.9443 −0.789307
\(846\) −120.902 −4.15669
\(847\) 0 0
\(848\) −8.47214 −0.290934
\(849\) 19.0557 0.653991
\(850\) −2.76393 −0.0948021
\(851\) −3.05573 −0.104749
\(852\) 14.8328 0.508164
\(853\) 0.652476 0.0223403 0.0111702 0.999938i \(-0.496444\pi\)
0.0111702 + 0.999938i \(0.496444\pi\)
\(854\) 0 0
\(855\) −36.9443 −1.26347
\(856\) 8.94427 0.305709
\(857\) 10.7639 0.367689 0.183844 0.982955i \(-0.441146\pi\)
0.183844 + 0.982955i \(0.441146\pi\)
\(858\) 0 0
\(859\) −40.5410 −1.38324 −0.691621 0.722261i \(-0.743103\pi\)
−0.691621 + 0.722261i \(0.743103\pi\)
\(860\) −48.0000 −1.63679
\(861\) 0 0
\(862\) −26.8328 −0.913929
\(863\) −20.9443 −0.712951 −0.356476 0.934305i \(-0.616022\pi\)
−0.356476 + 0.934305i \(0.616022\pi\)
\(864\) 97.0820 3.30280
\(865\) −34.4721 −1.17209
\(866\) −1.05573 −0.0358751
\(867\) 50.0689 1.70043
\(868\) 0 0
\(869\) 0 0
\(870\) −6.83282 −0.231654
\(871\) −6.83282 −0.231521
\(872\) −10.0000 −0.338643
\(873\) 70.3607 2.38135
\(874\) 35.7771 1.21018
\(875\) 0 0
\(876\) 50.8328 1.71748
\(877\) −41.4164 −1.39853 −0.699266 0.714861i \(-0.746490\pi\)
−0.699266 + 0.714861i \(0.746490\pi\)
\(878\) 3.41641 0.115298
\(879\) −48.9443 −1.65085
\(880\) 0 0
\(881\) −29.4164 −0.991064 −0.495532 0.868590i \(-0.665027\pi\)
−0.495532 + 0.868590i \(0.665027\pi\)
\(882\) 0 0
\(883\) 8.94427 0.300999 0.150499 0.988610i \(-0.451912\pi\)
0.150499 + 0.988610i \(0.451912\pi\)
\(884\) −4.58359 −0.154163
\(885\) 20.9443 0.704034
\(886\) −15.7771 −0.530042
\(887\) 40.3607 1.35518 0.677589 0.735440i \(-0.263025\pi\)
0.677589 + 0.735440i \(0.263025\pi\)
\(888\) −3.41641 −0.114647
\(889\) 0 0
\(890\) −8.94427 −0.299813
\(891\) 0 0
\(892\) −36.5410 −1.22348
\(893\) 17.8885 0.598617
\(894\) 101.305 3.38814
\(895\) 17.8885 0.597948
\(896\) 0 0
\(897\) −25.8885 −0.864393
\(898\) −43.6656 −1.45714
\(899\) 3.41641 0.113944
\(900\) −22.4164 −0.747214
\(901\) 10.4721 0.348877
\(902\) 0 0
\(903\) 0 0
\(904\) 4.47214 0.148741
\(905\) −2.83282 −0.0941660
\(906\) −64.7214 −2.15022
\(907\) 13.5279 0.449185 0.224593 0.974453i \(-0.427895\pi\)
0.224593 + 0.974453i \(0.427895\pi\)
\(908\) −89.6656 −2.97566
\(909\) −69.0132 −2.28902
\(910\) 0 0
\(911\) 33.5279 1.11083 0.555414 0.831574i \(-0.312560\pi\)
0.555414 + 0.831574i \(0.312560\pi\)
\(912\) −8.00000 −0.264906
\(913\) 0 0
\(914\) −55.5279 −1.83670
\(915\) 17.8885 0.591377
\(916\) −13.4164 −0.443291
\(917\) 0 0
\(918\) −40.0000 −1.32020
\(919\) −6.11146 −0.201598 −0.100799 0.994907i \(-0.532140\pi\)
−0.100799 + 0.994907i \(0.532140\pi\)
\(920\) −28.9443 −0.954264
\(921\) −28.9443 −0.953746
\(922\) 22.7639 0.749690
\(923\) 1.88854 0.0621622
\(924\) 0 0
\(925\) −0.472136 −0.0155237
\(926\) −32.3607 −1.06344
\(927\) 42.6525 1.40089
\(928\) −3.16718 −0.103968
\(929\) −28.2492 −0.926827 −0.463413 0.886142i \(-0.653376\pi\)
−0.463413 + 0.886142i \(0.653376\pi\)
\(930\) −104.721 −3.43395
\(931\) 0 0
\(932\) 52.2492 1.71148
\(933\) −70.2492 −2.29986
\(934\) 76.1803 2.49270
\(935\) 0 0
\(936\) −20.6525 −0.675047
\(937\) 20.6525 0.674687 0.337343 0.941382i \(-0.390472\pi\)
0.337343 + 0.941382i \(0.390472\pi\)
\(938\) 0 0
\(939\) −9.52786 −0.310930
\(940\) −43.4164 −1.41609
\(941\) −41.5967 −1.35602 −0.678008 0.735055i \(-0.737156\pi\)
−0.678008 + 0.735055i \(0.737156\pi\)
\(942\) −50.2492 −1.63721
\(943\) 43.7771 1.42558
\(944\) 3.23607 0.105325
\(945\) 0 0
\(946\) 0 0
\(947\) −58.8328 −1.91181 −0.955905 0.293677i \(-0.905121\pi\)
−0.955905 + 0.293677i \(0.905121\pi\)
\(948\) 86.8328 2.82020
\(949\) 6.47214 0.210094
\(950\) 5.52786 0.179348
\(951\) −45.3050 −1.46911
\(952\) 0 0
\(953\) −5.05573 −0.163771 −0.0818855 0.996642i \(-0.526094\pi\)
−0.0818855 + 0.996642i \(0.526094\pi\)
\(954\) 141.554 4.58299
\(955\) −41.8885 −1.35548
\(956\) −77.6656 −2.51189
\(957\) 0 0
\(958\) 50.2492 1.62348
\(959\) 0 0
\(960\) 84.1378 2.71553
\(961\) 21.3607 0.689054
\(962\) −1.30495 −0.0420733
\(963\) 29.8885 0.963145
\(964\) 81.3738 2.62087
\(965\) 47.7771 1.53800
\(966\) 0 0
\(967\) −21.8885 −0.703888 −0.351944 0.936021i \(-0.614479\pi\)
−0.351944 + 0.936021i \(0.614479\pi\)
\(968\) 0 0
\(969\) 9.88854 0.317666
\(970\) 42.1115 1.35212
\(971\) 29.1246 0.934653 0.467327 0.884085i \(-0.345217\pi\)
0.467327 + 0.884085i \(0.345217\pi\)
\(972\) −106.790 −3.42530
\(973\) 0 0
\(974\) 18.6950 0.599028
\(975\) −4.00000 −0.128103
\(976\) 2.76393 0.0884713
\(977\) 5.05573 0.161747 0.0808735 0.996724i \(-0.474229\pi\)
0.0808735 + 0.996724i \(0.474229\pi\)
\(978\) −169.443 −5.41818
\(979\) 0 0
\(980\) 0 0
\(981\) −33.4164 −1.06690
\(982\) 0 0
\(983\) −44.1803 −1.40913 −0.704567 0.709637i \(-0.748859\pi\)
−0.704567 + 0.709637i \(0.748859\pi\)
\(984\) 48.9443 1.56029
\(985\) 4.00000 0.127451
\(986\) 1.30495 0.0415581
\(987\) 0 0
\(988\) 9.16718 0.291647
\(989\) 51.7771 1.64642
\(990\) 0 0
\(991\) −26.2492 −0.833834 −0.416917 0.908945i \(-0.636889\pi\)
−0.416917 + 0.908945i \(0.636889\pi\)
\(992\) −48.5410 −1.54118
\(993\) −70.8328 −2.24781
\(994\) 0 0
\(995\) −40.3607 −1.27952
\(996\) −149.666 −4.74234
\(997\) 32.6525 1.03411 0.517057 0.855951i \(-0.327027\pi\)
0.517057 + 0.855951i \(0.327027\pi\)
\(998\) 23.4164 0.741233
\(999\) −6.83282 −0.216181
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 5929.2.a.m.1.2 2
7.6 odd 2 847.2.a.f.1.2 2
11.10 odd 2 539.2.a.f.1.1 2
21.20 even 2 7623.2.a.bl.1.1 2
33.32 even 2 4851.2.a.y.1.2 2
44.43 even 2 8624.2.a.ce.1.2 2
77.6 even 10 847.2.f.n.729.1 4
77.10 even 6 539.2.e.i.177.2 4
77.13 even 10 847.2.f.n.323.1 4
77.20 odd 10 847.2.f.b.323.1 4
77.27 odd 10 847.2.f.b.729.1 4
77.32 odd 6 539.2.e.j.177.2 4
77.41 even 10 847.2.f.a.372.1 4
77.48 odd 10 847.2.f.m.148.1 4
77.54 even 6 539.2.e.i.67.2 4
77.62 even 10 847.2.f.a.148.1 4
77.65 odd 6 539.2.e.j.67.2 4
77.69 odd 10 847.2.f.m.372.1 4
77.76 even 2 77.2.a.d.1.1 2
231.230 odd 2 693.2.a.h.1.2 2
308.307 odd 2 1232.2.a.m.1.1 2
385.153 odd 4 1925.2.b.h.1849.4 4
385.307 odd 4 1925.2.b.h.1849.1 4
385.384 even 2 1925.2.a.r.1.2 2
616.307 odd 2 4928.2.a.bv.1.2 2
616.461 even 2 4928.2.a.bm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.d.1.1 2 77.76 even 2
539.2.a.f.1.1 2 11.10 odd 2
539.2.e.i.67.2 4 77.54 even 6
539.2.e.i.177.2 4 77.10 even 6
539.2.e.j.67.2 4 77.65 odd 6
539.2.e.j.177.2 4 77.32 odd 6
693.2.a.h.1.2 2 231.230 odd 2
847.2.a.f.1.2 2 7.6 odd 2
847.2.f.a.148.1 4 77.62 even 10
847.2.f.a.372.1 4 77.41 even 10
847.2.f.b.323.1 4 77.20 odd 10
847.2.f.b.729.1 4 77.27 odd 10
847.2.f.m.148.1 4 77.48 odd 10
847.2.f.m.372.1 4 77.69 odd 10
847.2.f.n.323.1 4 77.13 even 10
847.2.f.n.729.1 4 77.6 even 10
1232.2.a.m.1.1 2 308.307 odd 2
1925.2.a.r.1.2 2 385.384 even 2
1925.2.b.h.1849.1 4 385.307 odd 4
1925.2.b.h.1849.4 4 385.153 odd 4
4851.2.a.y.1.2 2 33.32 even 2
4928.2.a.bm.1.1 2 616.461 even 2
4928.2.a.bv.1.2 2 616.307 odd 2
5929.2.a.m.1.2 2 1.1 even 1 trivial
7623.2.a.bl.1.1 2 21.20 even 2
8624.2.a.ce.1.2 2 44.43 even 2