Properties

Label 2695.2.a.e.1.1
Level $2695$
Weight $2$
Character 2695.1
Self dual yes
Analytic conductor $21.520$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2695,2,Mod(1,2695)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2695, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2695.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2695 = 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2695.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: \(21.5196833447\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2695.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.414214 q^{2} +1.41421 q^{3} -1.82843 q^{4} +1.00000 q^{5} -0.585786 q^{6} +1.58579 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-0.414214 q^{2} +1.41421 q^{3} -1.82843 q^{4} +1.00000 q^{5} -0.585786 q^{6} +1.58579 q^{8} -1.00000 q^{9} -0.414214 q^{10} +1.00000 q^{11} -2.58579 q^{12} -3.41421 q^{13} +1.41421 q^{15} +3.00000 q^{16} +0.585786 q^{17} +0.414214 q^{18} -1.82843 q^{20} -0.414214 q^{22} +8.82843 q^{23} +2.24264 q^{24} +1.00000 q^{25} +1.41421 q^{26} -5.65685 q^{27} -0.828427 q^{29} -0.585786 q^{30} +1.75736 q^{31} -4.41421 q^{32} +1.41421 q^{33} -0.242641 q^{34} +1.82843 q^{36} +1.17157 q^{37} -4.82843 q^{39} +1.58579 q^{40} -4.24264 q^{41} +6.00000 q^{43} -1.82843 q^{44} -1.00000 q^{45} -3.65685 q^{46} -1.41421 q^{47} +4.24264 q^{48} -0.414214 q^{50} +0.828427 q^{51} +6.24264 q^{52} +2.82843 q^{53} +2.34315 q^{54} +1.00000 q^{55} +0.343146 q^{58} +7.89949 q^{59} -2.58579 q^{60} +13.8995 q^{61} -0.727922 q^{62} -4.17157 q^{64} -3.41421 q^{65} -0.585786 q^{66} +6.48528 q^{67} -1.07107 q^{68} +12.4853 q^{69} +9.17157 q^{71} -1.58579 q^{72} +5.07107 q^{73} -0.485281 q^{74} +1.41421 q^{75} +2.00000 q^{78} +6.48528 q^{79} +3.00000 q^{80} -5.00000 q^{81} +1.75736 q^{82} -12.0000 q^{83} +0.585786 q^{85} -2.48528 q^{86} -1.17157 q^{87} +1.58579 q^{88} -0.828427 q^{89} +0.414214 q^{90} -16.1421 q^{92} +2.48528 q^{93} +0.585786 q^{94} -6.24264 q^{96} -9.31371 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{6} + 6 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{6} + 6 q^{8} - 2 q^{9} + 2 q^{10} + 2 q^{11} - 8 q^{12} - 4 q^{13} + 6 q^{16} + 4 q^{17} - 2 q^{18} + 2 q^{20} + 2 q^{22} + 12 q^{23} - 4 q^{24} + 2 q^{25} + 4 q^{29} - 4 q^{30} + 12 q^{31} - 6 q^{32} + 8 q^{34} - 2 q^{36} + 8 q^{37} - 4 q^{39} + 6 q^{40} + 12 q^{43} + 2 q^{44} - 2 q^{45} + 4 q^{46} + 2 q^{50} - 4 q^{51} + 4 q^{52} + 16 q^{54} + 2 q^{55} + 12 q^{58} - 4 q^{59} - 8 q^{60} + 8 q^{61} + 24 q^{62} - 14 q^{64} - 4 q^{65} - 4 q^{66} - 4 q^{67} + 12 q^{68} + 8 q^{69} + 24 q^{71} - 6 q^{72} - 4 q^{73} + 16 q^{74} + 4 q^{78} - 4 q^{79} + 6 q^{80} - 10 q^{81} + 12 q^{82} - 24 q^{83} + 4 q^{85} + 12 q^{86} - 8 q^{87} + 6 q^{88} + 4 q^{89} - 2 q^{90} - 4 q^{92} - 12 q^{93} + 4 q^{94} - 4 q^{96} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.414214 −0.292893 −0.146447 0.989219i \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) 1.41421 0.816497 0.408248 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) −1.82843 −0.914214
\(5\) 1.00000 0.447214
\(6\) −0.585786 −0.239146
\(7\) 0 0
\(8\) 1.58579 0.560660
\(9\) −1.00000 −0.333333
\(10\) −0.414214 −0.130986
\(11\) 1.00000 0.301511
\(12\) −2.58579 −0.746452
\(13\) −3.41421 −0.946932 −0.473466 0.880812i \(-0.656997\pi\)
−0.473466 + 0.880812i \(0.656997\pi\)
\(14\) 0 0
\(15\) 1.41421 0.365148
\(16\) 3.00000 0.750000
\(17\) 0.585786 0.142074 0.0710370 0.997474i \(-0.477369\pi\)
0.0710370 + 0.997474i \(0.477369\pi\)
\(18\) 0.414214 0.0976311
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.82843 −0.408849
\(21\) 0 0
\(22\) −0.414214 −0.0883106
\(23\) 8.82843 1.84085 0.920427 0.390914i \(-0.127841\pi\)
0.920427 + 0.390914i \(0.127841\pi\)
\(24\) 2.24264 0.457777
\(25\) 1.00000 0.200000
\(26\) 1.41421 0.277350
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) −0.828427 −0.153835 −0.0769175 0.997037i \(-0.524508\pi\)
−0.0769175 + 0.997037i \(0.524508\pi\)
\(30\) −0.585786 −0.106949
\(31\) 1.75736 0.315631 0.157816 0.987469i \(-0.449555\pi\)
0.157816 + 0.987469i \(0.449555\pi\)
\(32\) −4.41421 −0.780330
\(33\) 1.41421 0.246183
\(34\) −0.242641 −0.0416125
\(35\) 0 0
\(36\) 1.82843 0.304738
\(37\) 1.17157 0.192605 0.0963027 0.995352i \(-0.469298\pi\)
0.0963027 + 0.995352i \(0.469298\pi\)
\(38\) 0 0
\(39\) −4.82843 −0.773167
\(40\) 1.58579 0.250735
\(41\) −4.24264 −0.662589 −0.331295 0.943527i \(-0.607485\pi\)
−0.331295 + 0.943527i \(0.607485\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −1.82843 −0.275646
\(45\) −1.00000 −0.149071
\(46\) −3.65685 −0.539174
\(47\) −1.41421 −0.206284 −0.103142 0.994667i \(-0.532890\pi\)
−0.103142 + 0.994667i \(0.532890\pi\)
\(48\) 4.24264 0.612372
\(49\) 0 0
\(50\) −0.414214 −0.0585786
\(51\) 0.828427 0.116003
\(52\) 6.24264 0.865699
\(53\) 2.82843 0.388514 0.194257 0.980951i \(-0.437770\pi\)
0.194257 + 0.980951i \(0.437770\pi\)
\(54\) 2.34315 0.318862
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 0.343146 0.0450572
\(59\) 7.89949 1.02843 0.514213 0.857662i \(-0.328084\pi\)
0.514213 + 0.857662i \(0.328084\pi\)
\(60\) −2.58579 −0.333824
\(61\) 13.8995 1.77965 0.889824 0.456304i \(-0.150827\pi\)
0.889824 + 0.456304i \(0.150827\pi\)
\(62\) −0.727922 −0.0924462
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) −3.41421 −0.423481
\(66\) −0.585786 −0.0721053
\(67\) 6.48528 0.792303 0.396152 0.918185i \(-0.370345\pi\)
0.396152 + 0.918185i \(0.370345\pi\)
\(68\) −1.07107 −0.129886
\(69\) 12.4853 1.50305
\(70\) 0 0
\(71\) 9.17157 1.08847 0.544233 0.838934i \(-0.316821\pi\)
0.544233 + 0.838934i \(0.316821\pi\)
\(72\) −1.58579 −0.186887
\(73\) 5.07107 0.593524 0.296762 0.954952i \(-0.404093\pi\)
0.296762 + 0.954952i \(0.404093\pi\)
\(74\) −0.485281 −0.0564128
\(75\) 1.41421 0.163299
\(76\) 0 0
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 6.48528 0.729651 0.364826 0.931076i \(-0.381129\pi\)
0.364826 + 0.931076i \(0.381129\pi\)
\(80\) 3.00000 0.335410
\(81\) −5.00000 −0.555556
\(82\) 1.75736 0.194068
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 0.585786 0.0635375
\(86\) −2.48528 −0.267995
\(87\) −1.17157 −0.125606
\(88\) 1.58579 0.169045
\(89\) −0.828427 −0.0878131 −0.0439065 0.999036i \(-0.513980\pi\)
−0.0439065 + 0.999036i \(0.513980\pi\)
\(90\) 0.414214 0.0436619
\(91\) 0 0
\(92\) −16.1421 −1.68293
\(93\) 2.48528 0.257712
\(94\) 0.585786 0.0604193
\(95\) 0 0
\(96\) −6.24264 −0.637137
\(97\) −9.31371 −0.945664 −0.472832 0.881153i \(-0.656768\pi\)
−0.472832 + 0.881153i \(0.656768\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −1.82843 −0.182843
\(101\) −5.41421 −0.538734 −0.269367 0.963038i \(-0.586815\pi\)
−0.269367 + 0.963038i \(0.586815\pi\)
\(102\) −0.343146 −0.0339765
\(103\) 15.0711 1.48500 0.742498 0.669848i \(-0.233641\pi\)
0.742498 + 0.669848i \(0.233641\pi\)
\(104\) −5.41421 −0.530907
\(105\) 0 0
\(106\) −1.17157 −0.113793
\(107\) 19.3137 1.86713 0.933563 0.358412i \(-0.116682\pi\)
0.933563 + 0.358412i \(0.116682\pi\)
\(108\) 10.3431 0.995270
\(109\) −4.82843 −0.462479 −0.231240 0.972897i \(-0.574278\pi\)
−0.231240 + 0.972897i \(0.574278\pi\)
\(110\) −0.414214 −0.0394937
\(111\) 1.65685 0.157262
\(112\) 0 0
\(113\) −3.65685 −0.344008 −0.172004 0.985096i \(-0.555024\pi\)
−0.172004 + 0.985096i \(0.555024\pi\)
\(114\) 0 0
\(115\) 8.82843 0.823255
\(116\) 1.51472 0.140638
\(117\) 3.41421 0.315644
\(118\) −3.27208 −0.301219
\(119\) 0 0
\(120\) 2.24264 0.204724
\(121\) 1.00000 0.0909091
\(122\) −5.75736 −0.521247
\(123\) −6.00000 −0.541002
\(124\) −3.21320 −0.288554
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −7.31371 −0.648987 −0.324493 0.945888i \(-0.605194\pi\)
−0.324493 + 0.945888i \(0.605194\pi\)
\(128\) 10.5563 0.933058
\(129\) 8.48528 0.747087
\(130\) 1.41421 0.124035
\(131\) −11.3137 −0.988483 −0.494242 0.869325i \(-0.664554\pi\)
−0.494242 + 0.869325i \(0.664554\pi\)
\(132\) −2.58579 −0.225064
\(133\) 0 0
\(134\) −2.68629 −0.232060
\(135\) −5.65685 −0.486864
\(136\) 0.928932 0.0796553
\(137\) 8.48528 0.724947 0.362473 0.931994i \(-0.381932\pi\)
0.362473 + 0.931994i \(0.381932\pi\)
\(138\) −5.17157 −0.440234
\(139\) 12.4853 1.05899 0.529494 0.848314i \(-0.322382\pi\)
0.529494 + 0.848314i \(0.322382\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) −3.79899 −0.318804
\(143\) −3.41421 −0.285511
\(144\) −3.00000 −0.250000
\(145\) −0.828427 −0.0687971
\(146\) −2.10051 −0.173839
\(147\) 0 0
\(148\) −2.14214 −0.176082
\(149\) 11.6569 0.954967 0.477483 0.878641i \(-0.341549\pi\)
0.477483 + 0.878641i \(0.341549\pi\)
\(150\) −0.585786 −0.0478293
\(151\) 14.4853 1.17880 0.589398 0.807843i \(-0.299365\pi\)
0.589398 + 0.807843i \(0.299365\pi\)
\(152\) 0 0
\(153\) −0.585786 −0.0473580
\(154\) 0 0
\(155\) 1.75736 0.141154
\(156\) 8.82843 0.706840
\(157\) −6.48528 −0.517582 −0.258791 0.965933i \(-0.583324\pi\)
−0.258791 + 0.965933i \(0.583324\pi\)
\(158\) −2.68629 −0.213710
\(159\) 4.00000 0.317221
\(160\) −4.41421 −0.348974
\(161\) 0 0
\(162\) 2.07107 0.162718
\(163\) −6.48528 −0.507966 −0.253983 0.967209i \(-0.581741\pi\)
−0.253983 + 0.967209i \(0.581741\pi\)
\(164\) 7.75736 0.605748
\(165\) 1.41421 0.110096
\(166\) 4.97056 0.385790
\(167\) −20.4853 −1.58520 −0.792599 0.609743i \(-0.791273\pi\)
−0.792599 + 0.609743i \(0.791273\pi\)
\(168\) 0 0
\(169\) −1.34315 −0.103319
\(170\) −0.242641 −0.0186097
\(171\) 0 0
\(172\) −10.9706 −0.836498
\(173\) −2.92893 −0.222683 −0.111341 0.993782i \(-0.535515\pi\)
−0.111341 + 0.993782i \(0.535515\pi\)
\(174\) 0.485281 0.0367891
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 11.1716 0.839707
\(178\) 0.343146 0.0257199
\(179\) 3.31371 0.247678 0.123839 0.992302i \(-0.460479\pi\)
0.123839 + 0.992302i \(0.460479\pi\)
\(180\) 1.82843 0.136283
\(181\) 25.3137 1.88155 0.940777 0.339027i \(-0.110098\pi\)
0.940777 + 0.339027i \(0.110098\pi\)
\(182\) 0 0
\(183\) 19.6569 1.45308
\(184\) 14.0000 1.03209
\(185\) 1.17157 0.0861358
\(186\) −1.02944 −0.0754820
\(187\) 0.585786 0.0428369
\(188\) 2.58579 0.188588
\(189\) 0 0
\(190\) 0 0
\(191\) 26.6274 1.92669 0.963346 0.268261i \(-0.0864491\pi\)
0.963346 + 0.268261i \(0.0864491\pi\)
\(192\) −5.89949 −0.425759
\(193\) 13.6569 0.983042 0.491521 0.870866i \(-0.336441\pi\)
0.491521 + 0.870866i \(0.336441\pi\)
\(194\) 3.85786 0.276979
\(195\) −4.82843 −0.345771
\(196\) 0 0
\(197\) −18.6274 −1.32715 −0.663574 0.748110i \(-0.730961\pi\)
−0.663574 + 0.748110i \(0.730961\pi\)
\(198\) 0.414214 0.0294369
\(199\) −19.8995 −1.41064 −0.705319 0.708890i \(-0.749196\pi\)
−0.705319 + 0.708890i \(0.749196\pi\)
\(200\) 1.58579 0.112132
\(201\) 9.17157 0.646913
\(202\) 2.24264 0.157792
\(203\) 0 0
\(204\) −1.51472 −0.106052
\(205\) −4.24264 −0.296319
\(206\) −6.24264 −0.434945
\(207\) −8.82843 −0.613618
\(208\) −10.2426 −0.710199
\(209\) 0 0
\(210\) 0 0
\(211\) 12.9706 0.892930 0.446465 0.894801i \(-0.352683\pi\)
0.446465 + 0.894801i \(0.352683\pi\)
\(212\) −5.17157 −0.355185
\(213\) 12.9706 0.888728
\(214\) −8.00000 −0.546869
\(215\) 6.00000 0.409197
\(216\) −8.97056 −0.610369
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 7.17157 0.484610
\(220\) −1.82843 −0.123273
\(221\) −2.00000 −0.134535
\(222\) −0.686292 −0.0460609
\(223\) 14.3848 0.963276 0.481638 0.876370i \(-0.340042\pi\)
0.481638 + 0.876370i \(0.340042\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 1.51472 0.100758
\(227\) 3.31371 0.219939 0.109969 0.993935i \(-0.464925\pi\)
0.109969 + 0.993935i \(0.464925\pi\)
\(228\) 0 0
\(229\) −16.1421 −1.06670 −0.533351 0.845894i \(-0.679068\pi\)
−0.533351 + 0.845894i \(0.679068\pi\)
\(230\) −3.65685 −0.241126
\(231\) 0 0
\(232\) −1.31371 −0.0862492
\(233\) −1.85786 −0.121713 −0.0608564 0.998147i \(-0.519383\pi\)
−0.0608564 + 0.998147i \(0.519383\pi\)
\(234\) −1.41421 −0.0924500
\(235\) −1.41421 −0.0922531
\(236\) −14.4437 −0.940202
\(237\) 9.17157 0.595758
\(238\) 0 0
\(239\) 18.3431 1.18652 0.593260 0.805011i \(-0.297841\pi\)
0.593260 + 0.805011i \(0.297841\pi\)
\(240\) 4.24264 0.273861
\(241\) −21.2132 −1.36646 −0.683231 0.730202i \(-0.739426\pi\)
−0.683231 + 0.730202i \(0.739426\pi\)
\(242\) −0.414214 −0.0266267
\(243\) 9.89949 0.635053
\(244\) −25.4142 −1.62698
\(245\) 0 0
\(246\) 2.48528 0.158456
\(247\) 0 0
\(248\) 2.78680 0.176962
\(249\) −16.9706 −1.07547
\(250\) −0.414214 −0.0261972
\(251\) 18.7279 1.18210 0.591048 0.806636i \(-0.298714\pi\)
0.591048 + 0.806636i \(0.298714\pi\)
\(252\) 0 0
\(253\) 8.82843 0.555038
\(254\) 3.02944 0.190084
\(255\) 0.828427 0.0518781
\(256\) 3.97056 0.248160
\(257\) 12.1421 0.757406 0.378703 0.925518i \(-0.376370\pi\)
0.378703 + 0.925518i \(0.376370\pi\)
\(258\) −3.51472 −0.218817
\(259\) 0 0
\(260\) 6.24264 0.387152
\(261\) 0.828427 0.0512784
\(262\) 4.68629 0.289520
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 2.24264 0.138025
\(265\) 2.82843 0.173749
\(266\) 0 0
\(267\) −1.17157 −0.0716991
\(268\) −11.8579 −0.724334
\(269\) 17.3137 1.05564 0.527818 0.849358i \(-0.323010\pi\)
0.527818 + 0.849358i \(0.323010\pi\)
\(270\) 2.34315 0.142599
\(271\) 23.7990 1.44569 0.722843 0.691012i \(-0.242835\pi\)
0.722843 + 0.691012i \(0.242835\pi\)
\(272\) 1.75736 0.106556
\(273\) 0 0
\(274\) −3.51472 −0.212332
\(275\) 1.00000 0.0603023
\(276\) −22.8284 −1.37411
\(277\) −2.34315 −0.140786 −0.0703930 0.997519i \(-0.522425\pi\)
−0.0703930 + 0.997519i \(0.522425\pi\)
\(278\) −5.17157 −0.310170
\(279\) −1.75736 −0.105210
\(280\) 0 0
\(281\) −32.8284 −1.95838 −0.979190 0.202946i \(-0.934948\pi\)
−0.979190 + 0.202946i \(0.934948\pi\)
\(282\) 0.828427 0.0493321
\(283\) 13.6569 0.811816 0.405908 0.913914i \(-0.366955\pi\)
0.405908 + 0.913914i \(0.366955\pi\)
\(284\) −16.7696 −0.995090
\(285\) 0 0
\(286\) 1.41421 0.0836242
\(287\) 0 0
\(288\) 4.41421 0.260110
\(289\) −16.6569 −0.979815
\(290\) 0.343146 0.0201502
\(291\) −13.1716 −0.772131
\(292\) −9.27208 −0.542607
\(293\) −12.3848 −0.723526 −0.361763 0.932270i \(-0.617825\pi\)
−0.361763 + 0.932270i \(0.617825\pi\)
\(294\) 0 0
\(295\) 7.89949 0.459926
\(296\) 1.85786 0.107986
\(297\) −5.65685 −0.328244
\(298\) −4.82843 −0.279703
\(299\) −30.1421 −1.74316
\(300\) −2.58579 −0.149290
\(301\) 0 0
\(302\) −6.00000 −0.345261
\(303\) −7.65685 −0.439875
\(304\) 0 0
\(305\) 13.8995 0.795883
\(306\) 0.242641 0.0138708
\(307\) 6.14214 0.350550 0.175275 0.984519i \(-0.443919\pi\)
0.175275 + 0.984519i \(0.443919\pi\)
\(308\) 0 0
\(309\) 21.3137 1.21249
\(310\) −0.727922 −0.0413432
\(311\) −20.5858 −1.16731 −0.583656 0.812001i \(-0.698378\pi\)
−0.583656 + 0.812001i \(0.698378\pi\)
\(312\) −7.65685 −0.433484
\(313\) 19.6569 1.11107 0.555536 0.831493i \(-0.312513\pi\)
0.555536 + 0.831493i \(0.312513\pi\)
\(314\) 2.68629 0.151596
\(315\) 0 0
\(316\) −11.8579 −0.667057
\(317\) −24.6274 −1.38321 −0.691607 0.722274i \(-0.743097\pi\)
−0.691607 + 0.722274i \(0.743097\pi\)
\(318\) −1.65685 −0.0929118
\(319\) −0.828427 −0.0463830
\(320\) −4.17157 −0.233198
\(321\) 27.3137 1.52450
\(322\) 0 0
\(323\) 0 0
\(324\) 9.14214 0.507896
\(325\) −3.41421 −0.189386
\(326\) 2.68629 0.148780
\(327\) −6.82843 −0.377613
\(328\) −6.72792 −0.371487
\(329\) 0 0
\(330\) −0.585786 −0.0322465
\(331\) −25.6569 −1.41023 −0.705114 0.709094i \(-0.749104\pi\)
−0.705114 + 0.709094i \(0.749104\pi\)
\(332\) 21.9411 1.20418
\(333\) −1.17157 −0.0642018
\(334\) 8.48528 0.464294
\(335\) 6.48528 0.354329
\(336\) 0 0
\(337\) 20.4853 1.11590 0.557952 0.829873i \(-0.311587\pi\)
0.557952 + 0.829873i \(0.311587\pi\)
\(338\) 0.556349 0.0302614
\(339\) −5.17157 −0.280881
\(340\) −1.07107 −0.0580868
\(341\) 1.75736 0.0951663
\(342\) 0 0
\(343\) 0 0
\(344\) 9.51472 0.512999
\(345\) 12.4853 0.672185
\(346\) 1.21320 0.0652222
\(347\) −10.9706 −0.588931 −0.294465 0.955662i \(-0.595142\pi\)
−0.294465 + 0.955662i \(0.595142\pi\)
\(348\) 2.14214 0.114831
\(349\) −16.7279 −0.895425 −0.447713 0.894178i \(-0.647761\pi\)
−0.447713 + 0.894178i \(0.647761\pi\)
\(350\) 0 0
\(351\) 19.3137 1.03089
\(352\) −4.41421 −0.235278
\(353\) 20.3431 1.08276 0.541378 0.840779i \(-0.317903\pi\)
0.541378 + 0.840779i \(0.317903\pi\)
\(354\) −4.62742 −0.245944
\(355\) 9.17157 0.486777
\(356\) 1.51472 0.0802799
\(357\) 0 0
\(358\) −1.37258 −0.0725433
\(359\) −27.1716 −1.43406 −0.717030 0.697042i \(-0.754499\pi\)
−0.717030 + 0.697042i \(0.754499\pi\)
\(360\) −1.58579 −0.0835783
\(361\) −19.0000 −1.00000
\(362\) −10.4853 −0.551094
\(363\) 1.41421 0.0742270
\(364\) 0 0
\(365\) 5.07107 0.265432
\(366\) −8.14214 −0.425596
\(367\) −29.6985 −1.55025 −0.775124 0.631809i \(-0.782313\pi\)
−0.775124 + 0.631809i \(0.782313\pi\)
\(368\) 26.4853 1.38064
\(369\) 4.24264 0.220863
\(370\) −0.485281 −0.0252286
\(371\) 0 0
\(372\) −4.54416 −0.235604
\(373\) −19.3137 −1.00003 −0.500013 0.866018i \(-0.666671\pi\)
−0.500013 + 0.866018i \(0.666671\pi\)
\(374\) −0.242641 −0.0125467
\(375\) 1.41421 0.0730297
\(376\) −2.24264 −0.115655
\(377\) 2.82843 0.145671
\(378\) 0 0
\(379\) −29.1716 −1.49844 −0.749222 0.662319i \(-0.769572\pi\)
−0.749222 + 0.662319i \(0.769572\pi\)
\(380\) 0 0
\(381\) −10.3431 −0.529895
\(382\) −11.0294 −0.564315
\(383\) 5.89949 0.301450 0.150725 0.988576i \(-0.451839\pi\)
0.150725 + 0.988576i \(0.451839\pi\)
\(384\) 14.9289 0.761839
\(385\) 0 0
\(386\) −5.65685 −0.287926
\(387\) −6.00000 −0.304997
\(388\) 17.0294 0.864539
\(389\) 38.2843 1.94109 0.970545 0.240921i \(-0.0774494\pi\)
0.970545 + 0.240921i \(0.0774494\pi\)
\(390\) 2.00000 0.101274
\(391\) 5.17157 0.261538
\(392\) 0 0
\(393\) −16.0000 −0.807093
\(394\) 7.71573 0.388713
\(395\) 6.48528 0.326310
\(396\) 1.82843 0.0918819
\(397\) 31.1716 1.56446 0.782228 0.622992i \(-0.214083\pi\)
0.782228 + 0.622992i \(0.214083\pi\)
\(398\) 8.24264 0.413166
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) 5.31371 0.265354 0.132677 0.991159i \(-0.457643\pi\)
0.132677 + 0.991159i \(0.457643\pi\)
\(402\) −3.79899 −0.189476
\(403\) −6.00000 −0.298881
\(404\) 9.89949 0.492518
\(405\) −5.00000 −0.248452
\(406\) 0 0
\(407\) 1.17157 0.0580727
\(408\) 1.31371 0.0650383
\(409\) −21.2132 −1.04893 −0.524463 0.851433i \(-0.675734\pi\)
−0.524463 + 0.851433i \(0.675734\pi\)
\(410\) 1.75736 0.0867898
\(411\) 12.0000 0.591916
\(412\) −27.5563 −1.35760
\(413\) 0 0
\(414\) 3.65685 0.179725
\(415\) −12.0000 −0.589057
\(416\) 15.0711 0.738920
\(417\) 17.6569 0.864660
\(418\) 0 0
\(419\) 4.10051 0.200323 0.100161 0.994971i \(-0.468064\pi\)
0.100161 + 0.994971i \(0.468064\pi\)
\(420\) 0 0
\(421\) −28.9706 −1.41194 −0.705969 0.708242i \(-0.749488\pi\)
−0.705969 + 0.708242i \(0.749488\pi\)
\(422\) −5.37258 −0.261533
\(423\) 1.41421 0.0687614
\(424\) 4.48528 0.217825
\(425\) 0.585786 0.0284148
\(426\) −5.37258 −0.260302
\(427\) 0 0
\(428\) −35.3137 −1.70695
\(429\) −4.82843 −0.233119
\(430\) −2.48528 −0.119851
\(431\) −25.1127 −1.20964 −0.604818 0.796364i \(-0.706754\pi\)
−0.604818 + 0.796364i \(0.706754\pi\)
\(432\) −16.9706 −0.816497
\(433\) −0.142136 −0.00683060 −0.00341530 0.999994i \(-0.501087\pi\)
−0.00341530 + 0.999994i \(0.501087\pi\)
\(434\) 0 0
\(435\) −1.17157 −0.0561726
\(436\) 8.82843 0.422805
\(437\) 0 0
\(438\) −2.97056 −0.141939
\(439\) −33.4558 −1.59676 −0.798380 0.602154i \(-0.794309\pi\)
−0.798380 + 0.602154i \(0.794309\pi\)
\(440\) 1.58579 0.0755994
\(441\) 0 0
\(442\) 0.828427 0.0394043
\(443\) −28.6274 −1.36013 −0.680065 0.733152i \(-0.738048\pi\)
−0.680065 + 0.733152i \(0.738048\pi\)
\(444\) −3.02944 −0.143771
\(445\) −0.828427 −0.0392712
\(446\) −5.95837 −0.282137
\(447\) 16.4853 0.779727
\(448\) 0 0
\(449\) 18.3431 0.865667 0.432833 0.901474i \(-0.357514\pi\)
0.432833 + 0.901474i \(0.357514\pi\)
\(450\) 0.414214 0.0195262
\(451\) −4.24264 −0.199778
\(452\) 6.68629 0.314497
\(453\) 20.4853 0.962482
\(454\) −1.37258 −0.0644185
\(455\) 0 0
\(456\) 0 0
\(457\) 32.9706 1.54230 0.771149 0.636655i \(-0.219682\pi\)
0.771149 + 0.636655i \(0.219682\pi\)
\(458\) 6.68629 0.312430
\(459\) −3.31371 −0.154671
\(460\) −16.1421 −0.752631
\(461\) −6.58579 −0.306731 −0.153365 0.988170i \(-0.549011\pi\)
−0.153365 + 0.988170i \(0.549011\pi\)
\(462\) 0 0
\(463\) 40.1421 1.86556 0.932782 0.360442i \(-0.117374\pi\)
0.932782 + 0.360442i \(0.117374\pi\)
\(464\) −2.48528 −0.115376
\(465\) 2.48528 0.115252
\(466\) 0.769553 0.0356488
\(467\) −12.0416 −0.557220 −0.278610 0.960404i \(-0.589874\pi\)
−0.278610 + 0.960404i \(0.589874\pi\)
\(468\) −6.24264 −0.288566
\(469\) 0 0
\(470\) 0.585786 0.0270203
\(471\) −9.17157 −0.422604
\(472\) 12.5269 0.576598
\(473\) 6.00000 0.275880
\(474\) −3.79899 −0.174493
\(475\) 0 0
\(476\) 0 0
\(477\) −2.82843 −0.129505
\(478\) −7.59798 −0.347524
\(479\) 4.97056 0.227111 0.113555 0.993532i \(-0.463776\pi\)
0.113555 + 0.993532i \(0.463776\pi\)
\(480\) −6.24264 −0.284936
\(481\) −4.00000 −0.182384
\(482\) 8.78680 0.400228
\(483\) 0 0
\(484\) −1.82843 −0.0831103
\(485\) −9.31371 −0.422914
\(486\) −4.10051 −0.186003
\(487\) 15.4558 0.700371 0.350186 0.936680i \(-0.386119\pi\)
0.350186 + 0.936680i \(0.386119\pi\)
\(488\) 22.0416 0.997778
\(489\) −9.17157 −0.414753
\(490\) 0 0
\(491\) 20.1421 0.909002 0.454501 0.890746i \(-0.349818\pi\)
0.454501 + 0.890746i \(0.349818\pi\)
\(492\) 10.9706 0.494591
\(493\) −0.485281 −0.0218560
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 5.27208 0.236723
\(497\) 0 0
\(498\) 7.02944 0.314997
\(499\) 6.82843 0.305682 0.152841 0.988251i \(-0.451158\pi\)
0.152841 + 0.988251i \(0.451158\pi\)
\(500\) −1.82843 −0.0817697
\(501\) −28.9706 −1.29431
\(502\) −7.75736 −0.346228
\(503\) −18.3431 −0.817880 −0.408940 0.912561i \(-0.634102\pi\)
−0.408940 + 0.912561i \(0.634102\pi\)
\(504\) 0 0
\(505\) −5.41421 −0.240929
\(506\) −3.65685 −0.162567
\(507\) −1.89949 −0.0843595
\(508\) 13.3726 0.593312
\(509\) −12.8284 −0.568610 −0.284305 0.958734i \(-0.591763\pi\)
−0.284305 + 0.958734i \(0.591763\pi\)
\(510\) −0.343146 −0.0151947
\(511\) 0 0
\(512\) −22.7574 −1.00574
\(513\) 0 0
\(514\) −5.02944 −0.221839
\(515\) 15.0711 0.664111
\(516\) −15.5147 −0.682997
\(517\) −1.41421 −0.0621970
\(518\) 0 0
\(519\) −4.14214 −0.181820
\(520\) −5.41421 −0.237429
\(521\) −2.68629 −0.117689 −0.0588443 0.998267i \(-0.518742\pi\)
−0.0588443 + 0.998267i \(0.518742\pi\)
\(522\) −0.343146 −0.0150191
\(523\) −1.17157 −0.0512293 −0.0256147 0.999672i \(-0.508154\pi\)
−0.0256147 + 0.999672i \(0.508154\pi\)
\(524\) 20.6863 0.903685
\(525\) 0 0
\(526\) −4.97056 −0.216727
\(527\) 1.02944 0.0448430
\(528\) 4.24264 0.184637
\(529\) 54.9411 2.38874
\(530\) −1.17157 −0.0508899
\(531\) −7.89949 −0.342809
\(532\) 0 0
\(533\) 14.4853 0.627427
\(534\) 0.485281 0.0210002
\(535\) 19.3137 0.835004
\(536\) 10.2843 0.444213
\(537\) 4.68629 0.202228
\(538\) −7.17157 −0.309188
\(539\) 0 0
\(540\) 10.3431 0.445098
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −9.85786 −0.423432
\(543\) 35.7990 1.53628
\(544\) −2.58579 −0.110865
\(545\) −4.82843 −0.206827
\(546\) 0 0
\(547\) 3.02944 0.129529 0.0647647 0.997901i \(-0.479370\pi\)
0.0647647 + 0.997901i \(0.479370\pi\)
\(548\) −15.5147 −0.662756
\(549\) −13.8995 −0.593216
\(550\) −0.414214 −0.0176621
\(551\) 0 0
\(552\) 19.7990 0.842701
\(553\) 0 0
\(554\) 0.970563 0.0412353
\(555\) 1.65685 0.0703295
\(556\) −22.8284 −0.968141
\(557\) 17.6569 0.748145 0.374072 0.927399i \(-0.377961\pi\)
0.374072 + 0.927399i \(0.377961\pi\)
\(558\) 0.727922 0.0308154
\(559\) −20.4853 −0.866435
\(560\) 0 0
\(561\) 0.828427 0.0349762
\(562\) 13.5980 0.573596
\(563\) 19.7990 0.834428 0.417214 0.908808i \(-0.363007\pi\)
0.417214 + 0.908808i \(0.363007\pi\)
\(564\) 3.65685 0.153981
\(565\) −3.65685 −0.153845
\(566\) −5.65685 −0.237775
\(567\) 0 0
\(568\) 14.5442 0.610259
\(569\) −12.1421 −0.509025 −0.254512 0.967070i \(-0.581915\pi\)
−0.254512 + 0.967070i \(0.581915\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 6.24264 0.261018
\(573\) 37.6569 1.57314
\(574\) 0 0
\(575\) 8.82843 0.368171
\(576\) 4.17157 0.173816
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 6.89949 0.286981
\(579\) 19.3137 0.802650
\(580\) 1.51472 0.0628953
\(581\) 0 0
\(582\) 5.45584 0.226152
\(583\) 2.82843 0.117141
\(584\) 8.04163 0.332765
\(585\) 3.41421 0.141160
\(586\) 5.12994 0.211916
\(587\) 27.5563 1.13737 0.568686 0.822555i \(-0.307452\pi\)
0.568686 + 0.822555i \(0.307452\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −3.27208 −0.134709
\(591\) −26.3431 −1.08361
\(592\) 3.51472 0.144454
\(593\) −18.7279 −0.769064 −0.384532 0.923112i \(-0.625637\pi\)
−0.384532 + 0.923112i \(0.625637\pi\)
\(594\) 2.34315 0.0961404
\(595\) 0 0
\(596\) −21.3137 −0.873044
\(597\) −28.1421 −1.15178
\(598\) 12.4853 0.510561
\(599\) −9.85786 −0.402781 −0.201391 0.979511i \(-0.564546\pi\)
−0.201391 + 0.979511i \(0.564546\pi\)
\(600\) 2.24264 0.0915554
\(601\) −19.2721 −0.786124 −0.393062 0.919512i \(-0.628584\pi\)
−0.393062 + 0.919512i \(0.628584\pi\)
\(602\) 0 0
\(603\) −6.48528 −0.264101
\(604\) −26.4853 −1.07767
\(605\) 1.00000 0.0406558
\(606\) 3.17157 0.128836
\(607\) −25.1716 −1.02168 −0.510841 0.859675i \(-0.670666\pi\)
−0.510841 + 0.859675i \(0.670666\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −5.75736 −0.233109
\(611\) 4.82843 0.195337
\(612\) 1.07107 0.0432954
\(613\) −20.4853 −0.827393 −0.413696 0.910415i \(-0.635762\pi\)
−0.413696 + 0.910415i \(0.635762\pi\)
\(614\) −2.54416 −0.102674
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) −2.82843 −0.113868 −0.0569341 0.998378i \(-0.518132\pi\)
−0.0569341 + 0.998378i \(0.518132\pi\)
\(618\) −8.82843 −0.355131
\(619\) −12.1005 −0.486360 −0.243180 0.969981i \(-0.578191\pi\)
−0.243180 + 0.969981i \(0.578191\pi\)
\(620\) −3.21320 −0.129045
\(621\) −49.9411 −2.00407
\(622\) 8.52691 0.341898
\(623\) 0 0
\(624\) −14.4853 −0.579875
\(625\) 1.00000 0.0400000
\(626\) −8.14214 −0.325425
\(627\) 0 0
\(628\) 11.8579 0.473180
\(629\) 0.686292 0.0273642
\(630\) 0 0
\(631\) 7.02944 0.279837 0.139919 0.990163i \(-0.455316\pi\)
0.139919 + 0.990163i \(0.455316\pi\)
\(632\) 10.2843 0.409086
\(633\) 18.3431 0.729075
\(634\) 10.2010 0.405134
\(635\) −7.31371 −0.290236
\(636\) −7.31371 −0.290007
\(637\) 0 0
\(638\) 0.343146 0.0135853
\(639\) −9.17157 −0.362822
\(640\) 10.5563 0.417276
\(641\) 36.0000 1.42191 0.710957 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) −11.3137 −0.446516
\(643\) −16.2426 −0.640547 −0.320274 0.947325i \(-0.603775\pi\)
−0.320274 + 0.947325i \(0.603775\pi\)
\(644\) 0 0
\(645\) 8.48528 0.334108
\(646\) 0 0
\(647\) −30.5858 −1.20245 −0.601226 0.799079i \(-0.705321\pi\)
−0.601226 + 0.799079i \(0.705321\pi\)
\(648\) −7.92893 −0.311478
\(649\) 7.89949 0.310082
\(650\) 1.41421 0.0554700
\(651\) 0 0
\(652\) 11.8579 0.464390
\(653\) −6.68629 −0.261655 −0.130827 0.991405i \(-0.541763\pi\)
−0.130827 + 0.991405i \(0.541763\pi\)
\(654\) 2.82843 0.110600
\(655\) −11.3137 −0.442063
\(656\) −12.7279 −0.496942
\(657\) −5.07107 −0.197841
\(658\) 0 0
\(659\) −43.4558 −1.69280 −0.846400 0.532548i \(-0.821235\pi\)
−0.846400 + 0.532548i \(0.821235\pi\)
\(660\) −2.58579 −0.100652
\(661\) −17.7990 −0.692300 −0.346150 0.938179i \(-0.612511\pi\)
−0.346150 + 0.938179i \(0.612511\pi\)
\(662\) 10.6274 0.413046
\(663\) −2.82843 −0.109847
\(664\) −19.0294 −0.738485
\(665\) 0 0
\(666\) 0.485281 0.0188043
\(667\) −7.31371 −0.283188
\(668\) 37.4558 1.44921
\(669\) 20.3431 0.786511
\(670\) −2.68629 −0.103780
\(671\) 13.8995 0.536584
\(672\) 0 0
\(673\) 11.5147 0.443860 0.221930 0.975063i \(-0.428764\pi\)
0.221930 + 0.975063i \(0.428764\pi\)
\(674\) −8.48528 −0.326841
\(675\) −5.65685 −0.217732
\(676\) 2.45584 0.0944555
\(677\) −10.4437 −0.401382 −0.200691 0.979655i \(-0.564319\pi\)
−0.200691 + 0.979655i \(0.564319\pi\)
\(678\) 2.14214 0.0822682
\(679\) 0 0
\(680\) 0.928932 0.0356229
\(681\) 4.68629 0.179579
\(682\) −0.727922 −0.0278736
\(683\) 17.3137 0.662491 0.331245 0.943545i \(-0.392531\pi\)
0.331245 + 0.943545i \(0.392531\pi\)
\(684\) 0 0
\(685\) 8.48528 0.324206
\(686\) 0 0
\(687\) −22.8284 −0.870959
\(688\) 18.0000 0.686244
\(689\) −9.65685 −0.367897
\(690\) −5.17157 −0.196878
\(691\) −24.5858 −0.935287 −0.467644 0.883917i \(-0.654897\pi\)
−0.467644 + 0.883917i \(0.654897\pi\)
\(692\) 5.35534 0.203579
\(693\) 0 0
\(694\) 4.54416 0.172494
\(695\) 12.4853 0.473594
\(696\) −1.85786 −0.0704222
\(697\) −2.48528 −0.0941367
\(698\) 6.92893 0.262264
\(699\) −2.62742 −0.0993780
\(700\) 0 0
\(701\) −50.7696 −1.91754 −0.958770 0.284184i \(-0.908277\pi\)
−0.958770 + 0.284184i \(0.908277\pi\)
\(702\) −8.00000 −0.301941
\(703\) 0 0
\(704\) −4.17157 −0.157222
\(705\) −2.00000 −0.0753244
\(706\) −8.42641 −0.317132
\(707\) 0 0
\(708\) −20.4264 −0.767671
\(709\) −32.6274 −1.22535 −0.612674 0.790336i \(-0.709906\pi\)
−0.612674 + 0.790336i \(0.709906\pi\)
\(710\) −3.79899 −0.142574
\(711\) −6.48528 −0.243217
\(712\) −1.31371 −0.0492333
\(713\) 15.5147 0.581031
\(714\) 0 0
\(715\) −3.41421 −0.127684
\(716\) −6.05887 −0.226431
\(717\) 25.9411 0.968789
\(718\) 11.2548 0.420027
\(719\) −36.3848 −1.35692 −0.678462 0.734636i \(-0.737353\pi\)
−0.678462 + 0.734636i \(0.737353\pi\)
\(720\) −3.00000 −0.111803
\(721\) 0 0
\(722\) 7.87006 0.292893
\(723\) −30.0000 −1.11571
\(724\) −46.2843 −1.72014
\(725\) −0.828427 −0.0307670
\(726\) −0.585786 −0.0217406
\(727\) 45.6985 1.69486 0.847431 0.530905i \(-0.178148\pi\)
0.847431 + 0.530905i \(0.178148\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) −2.10051 −0.0777432
\(731\) 3.51472 0.129997
\(732\) −35.9411 −1.32842
\(733\) −50.7279 −1.87368 −0.936839 0.349760i \(-0.886263\pi\)
−0.936839 + 0.349760i \(0.886263\pi\)
\(734\) 12.3015 0.454057
\(735\) 0 0
\(736\) −38.9706 −1.43647
\(737\) 6.48528 0.238888
\(738\) −1.75736 −0.0646893
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) −2.14214 −0.0787465
\(741\) 0 0
\(742\) 0 0
\(743\) −15.3137 −0.561805 −0.280903 0.959736i \(-0.590634\pi\)
−0.280903 + 0.959736i \(0.590634\pi\)
\(744\) 3.94113 0.144489
\(745\) 11.6569 0.427074
\(746\) 8.00000 0.292901
\(747\) 12.0000 0.439057
\(748\) −1.07107 −0.0391621
\(749\) 0 0
\(750\) −0.585786 −0.0213899
\(751\) 33.4558 1.22082 0.610411 0.792085i \(-0.291004\pi\)
0.610411 + 0.792085i \(0.291004\pi\)
\(752\) −4.24264 −0.154713
\(753\) 26.4853 0.965177
\(754\) −1.17157 −0.0426662
\(755\) 14.4853 0.527173
\(756\) 0 0
\(757\) 24.6274 0.895099 0.447549 0.894259i \(-0.352297\pi\)
0.447549 + 0.894259i \(0.352297\pi\)
\(758\) 12.0833 0.438884
\(759\) 12.4853 0.453187
\(760\) 0 0
\(761\) 2.78680 0.101021 0.0505106 0.998724i \(-0.483915\pi\)
0.0505106 + 0.998724i \(0.483915\pi\)
\(762\) 4.28427 0.155203
\(763\) 0 0
\(764\) −48.6863 −1.76141
\(765\) −0.585786 −0.0211792
\(766\) −2.44365 −0.0882927
\(767\) −26.9706 −0.973851
\(768\) 5.61522 0.202622
\(769\) 0.242641 0.00874985 0.00437492 0.999990i \(-0.498607\pi\)
0.00437492 + 0.999990i \(0.498607\pi\)
\(770\) 0 0
\(771\) 17.1716 0.618419
\(772\) −24.9706 −0.898710
\(773\) 13.3137 0.478861 0.239430 0.970914i \(-0.423039\pi\)
0.239430 + 0.970914i \(0.423039\pi\)
\(774\) 2.48528 0.0893316
\(775\) 1.75736 0.0631262
\(776\) −14.7696 −0.530196
\(777\) 0 0
\(778\) −15.8579 −0.568532
\(779\) 0 0
\(780\) 8.82843 0.316108
\(781\) 9.17157 0.328185
\(782\) −2.14214 −0.0766026
\(783\) 4.68629 0.167474
\(784\) 0 0
\(785\) −6.48528 −0.231470
\(786\) 6.62742 0.236392
\(787\) −45.4558 −1.62033 −0.810163 0.586205i \(-0.800621\pi\)
−0.810163 + 0.586205i \(0.800621\pi\)
\(788\) 34.0589 1.21330
\(789\) 16.9706 0.604168
\(790\) −2.68629 −0.0955740
\(791\) 0 0
\(792\) −1.58579 −0.0563485
\(793\) −47.4558 −1.68521
\(794\) −12.9117 −0.458219
\(795\) 4.00000 0.141865
\(796\) 36.3848 1.28962
\(797\) 40.1421 1.42191 0.710954 0.703239i \(-0.248264\pi\)
0.710954 + 0.703239i \(0.248264\pi\)
\(798\) 0 0
\(799\) −0.828427 −0.0293076
\(800\) −4.41421 −0.156066
\(801\) 0.828427 0.0292710
\(802\) −2.20101 −0.0777204
\(803\) 5.07107 0.178954
\(804\) −16.7696 −0.591417
\(805\) 0 0
\(806\) 2.48528 0.0875403
\(807\) 24.4853 0.861923
\(808\) −8.58579 −0.302047
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 2.07107 0.0727699
\(811\) 41.4558 1.45571 0.727856 0.685730i \(-0.240517\pi\)
0.727856 + 0.685730i \(0.240517\pi\)
\(812\) 0 0
\(813\) 33.6569 1.18040
\(814\) −0.485281 −0.0170091
\(815\) −6.48528 −0.227169
\(816\) 2.48528 0.0870023
\(817\) 0 0
\(818\) 8.78680 0.307223
\(819\) 0 0
\(820\) 7.75736 0.270899
\(821\) 12.3431 0.430779 0.215389 0.976528i \(-0.430898\pi\)
0.215389 + 0.976528i \(0.430898\pi\)
\(822\) −4.97056 −0.173368
\(823\) −35.9411 −1.25283 −0.626414 0.779490i \(-0.715478\pi\)
−0.626414 + 0.779490i \(0.715478\pi\)
\(824\) 23.8995 0.832578
\(825\) 1.41421 0.0492366
\(826\) 0 0
\(827\) 40.6274 1.41275 0.706377 0.707836i \(-0.250328\pi\)
0.706377 + 0.707836i \(0.250328\pi\)
\(828\) 16.1421 0.560978
\(829\) −1.02944 −0.0357538 −0.0178769 0.999840i \(-0.505691\pi\)
−0.0178769 + 0.999840i \(0.505691\pi\)
\(830\) 4.97056 0.172531
\(831\) −3.31371 −0.114951
\(832\) 14.2426 0.493775
\(833\) 0 0
\(834\) −7.31371 −0.253253
\(835\) −20.4853 −0.708922
\(836\) 0 0
\(837\) −9.94113 −0.343616
\(838\) −1.69848 −0.0586732
\(839\) 32.5858 1.12499 0.562493 0.826802i \(-0.309842\pi\)
0.562493 + 0.826802i \(0.309842\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) 12.0000 0.413547
\(843\) −46.4264 −1.59901
\(844\) −23.7157 −0.816329
\(845\) −1.34315 −0.0462056
\(846\) −0.585786 −0.0201398
\(847\) 0 0
\(848\) 8.48528 0.291386
\(849\) 19.3137 0.662845
\(850\) −0.242641 −0.00832251
\(851\) 10.3431 0.354558
\(852\) −23.7157 −0.812487
\(853\) −52.6690 −1.80335 −0.901677 0.432410i \(-0.857663\pi\)
−0.901677 + 0.432410i \(0.857663\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 30.6274 1.04682
\(857\) 30.0416 1.02620 0.513101 0.858328i \(-0.328497\pi\)
0.513101 + 0.858328i \(0.328497\pi\)
\(858\) 2.00000 0.0682789
\(859\) 46.0416 1.57092 0.785460 0.618912i \(-0.212426\pi\)
0.785460 + 0.618912i \(0.212426\pi\)
\(860\) −10.9706 −0.374093
\(861\) 0 0
\(862\) 10.4020 0.354294
\(863\) 35.6569 1.21377 0.606887 0.794788i \(-0.292418\pi\)
0.606887 + 0.794788i \(0.292418\pi\)
\(864\) 24.9706 0.849516
\(865\) −2.92893 −0.0995867
\(866\) 0.0588745 0.00200064
\(867\) −23.5563 −0.800016
\(868\) 0 0
\(869\) 6.48528 0.219998
\(870\) 0.485281 0.0164526
\(871\) −22.1421 −0.750258
\(872\) −7.65685 −0.259294
\(873\) 9.31371 0.315221
\(874\) 0 0
\(875\) 0 0
\(876\) −13.1127 −0.443037
\(877\) 20.4853 0.691739 0.345869 0.938283i \(-0.387584\pi\)
0.345869 + 0.938283i \(0.387584\pi\)
\(878\) 13.8579 0.467680
\(879\) −17.5147 −0.590757
\(880\) 3.00000 0.101130
\(881\) 16.6274 0.560192 0.280096 0.959972i \(-0.409634\pi\)
0.280096 + 0.959972i \(0.409634\pi\)
\(882\) 0 0
\(883\) 21.3137 0.717263 0.358632 0.933479i \(-0.383243\pi\)
0.358632 + 0.933479i \(0.383243\pi\)
\(884\) 3.65685 0.122993
\(885\) 11.1716 0.375528
\(886\) 11.8579 0.398373
\(887\) 22.8284 0.766504 0.383252 0.923644i \(-0.374804\pi\)
0.383252 + 0.923644i \(0.374804\pi\)
\(888\) 2.62742 0.0881703
\(889\) 0 0
\(890\) 0.343146 0.0115023
\(891\) −5.00000 −0.167506
\(892\) −26.3015 −0.880640
\(893\) 0 0
\(894\) −6.82843 −0.228377
\(895\) 3.31371 0.110765
\(896\) 0 0
\(897\) −42.6274 −1.42329
\(898\) −7.59798 −0.253548
\(899\) −1.45584 −0.0485551
\(900\) 1.82843 0.0609476
\(901\) 1.65685 0.0551978
\(902\) 1.75736 0.0585137
\(903\) 0 0
\(904\) −5.79899 −0.192872
\(905\) 25.3137 0.841456
\(906\) −8.48528 −0.281905
\(907\) −48.4264 −1.60797 −0.803986 0.594648i \(-0.797291\pi\)
−0.803986 + 0.594648i \(0.797291\pi\)
\(908\) −6.05887 −0.201071
\(909\) 5.41421 0.179578
\(910\) 0 0
\(911\) −20.4853 −0.678708 −0.339354 0.940659i \(-0.610208\pi\)
−0.339354 + 0.940659i \(0.610208\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) −13.6569 −0.451729
\(915\) 19.6569 0.649836
\(916\) 29.5147 0.975194
\(917\) 0 0
\(918\) 1.37258 0.0453020
\(919\) −41.9411 −1.38351 −0.691755 0.722132i \(-0.743162\pi\)
−0.691755 + 0.722132i \(0.743162\pi\)
\(920\) 14.0000 0.461566
\(921\) 8.68629 0.286223
\(922\) 2.72792 0.0898393
\(923\) −31.3137 −1.03070
\(924\) 0 0
\(925\) 1.17157 0.0385211
\(926\) −16.6274 −0.546411
\(927\) −15.0711 −0.494999
\(928\) 3.65685 0.120042
\(929\) −6.68629 −0.219370 −0.109685 0.993966i \(-0.534984\pi\)
−0.109685 + 0.993966i \(0.534984\pi\)
\(930\) −1.02944 −0.0337566
\(931\) 0 0
\(932\) 3.39697 0.111271
\(933\) −29.1127 −0.953107
\(934\) 4.98781 0.163206
\(935\) 0.585786 0.0191573
\(936\) 5.41421 0.176969
\(937\) 17.2721 0.564254 0.282127 0.959377i \(-0.408960\pi\)
0.282127 + 0.959377i \(0.408960\pi\)
\(938\) 0 0
\(939\) 27.7990 0.907186
\(940\) 2.58579 0.0843391
\(941\) 5.61522 0.183051 0.0915255 0.995803i \(-0.470826\pi\)
0.0915255 + 0.995803i \(0.470826\pi\)
\(942\) 3.79899 0.123778
\(943\) −37.4558 −1.21973
\(944\) 23.6985 0.771320
\(945\) 0 0
\(946\) −2.48528 −0.0808035
\(947\) −25.5980 −0.831823 −0.415911 0.909405i \(-0.636537\pi\)
−0.415911 + 0.909405i \(0.636537\pi\)
\(948\) −16.7696 −0.544650
\(949\) −17.3137 −0.562027
\(950\) 0 0
\(951\) −34.8284 −1.12939
\(952\) 0 0
\(953\) 41.2548 1.33638 0.668188 0.743993i \(-0.267070\pi\)
0.668188 + 0.743993i \(0.267070\pi\)
\(954\) 1.17157 0.0379311
\(955\) 26.6274 0.861643
\(956\) −33.5391 −1.08473
\(957\) −1.17157 −0.0378716
\(958\) −2.05887 −0.0665192
\(959\) 0 0
\(960\) −5.89949 −0.190405
\(961\) −27.9117 −0.900377
\(962\) 1.65685 0.0534191
\(963\) −19.3137 −0.622376
\(964\) 38.7868 1.24924
\(965\) 13.6569 0.439630
\(966\) 0 0
\(967\) −5.02944 −0.161736 −0.0808679 0.996725i \(-0.525769\pi\)
−0.0808679 + 0.996725i \(0.525769\pi\)
\(968\) 1.58579 0.0509691
\(969\) 0 0
\(970\) 3.85786 0.123869
\(971\) 12.5858 0.403897 0.201949 0.979396i \(-0.435273\pi\)
0.201949 + 0.979396i \(0.435273\pi\)
\(972\) −18.1005 −0.580574
\(973\) 0 0
\(974\) −6.40202 −0.205134
\(975\) −4.82843 −0.154633
\(976\) 41.6985 1.33474
\(977\) −9.31371 −0.297972 −0.148986 0.988839i \(-0.547601\pi\)
−0.148986 + 0.988839i \(0.547601\pi\)
\(978\) 3.79899 0.121478
\(979\) −0.828427 −0.0264766
\(980\) 0 0
\(981\) 4.82843 0.154160
\(982\) −8.34315 −0.266240
\(983\) −14.1005 −0.449736 −0.224868 0.974389i \(-0.572195\pi\)
−0.224868 + 0.974389i \(0.572195\pi\)
\(984\) −9.51472 −0.303318
\(985\) −18.6274 −0.593519
\(986\) 0.201010 0.00640147
\(987\) 0 0
\(988\) 0 0
\(989\) 52.9706 1.68437
\(990\) 0.414214 0.0131646
\(991\) −37.4558 −1.18982 −0.594912 0.803791i \(-0.702813\pi\)
−0.594912 + 0.803791i \(0.702813\pi\)
\(992\) −7.75736 −0.246296
\(993\) −36.2843 −1.15145
\(994\) 0 0
\(995\) −19.8995 −0.630856
\(996\) 31.0294 0.983205
\(997\) −30.2426 −0.957794 −0.478897 0.877871i \(-0.658963\pi\)
−0.478897 + 0.877871i \(0.658963\pi\)
\(998\) −2.82843 −0.0895323
\(999\) −6.62742 −0.209682
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 2695.2.a.e.1.1 2
7.6 odd 2 385.2.a.d.1.1 2
21.20 even 2 3465.2.a.u.1.2 2
28.27 even 2 6160.2.a.y.1.2 2
35.13 even 4 1925.2.b.j.1849.3 4
35.27 even 4 1925.2.b.j.1849.2 4
35.34 odd 2 1925.2.a.n.1.2 2
77.76 even 2 4235.2.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.a.d.1.1 2 7.6 odd 2
1925.2.a.n.1.2 2 35.34 odd 2
1925.2.b.j.1849.2 4 35.27 even 4
1925.2.b.j.1849.3 4 35.13 even 4
2695.2.a.e.1.1 2 1.1 even 1 trivial
3465.2.a.u.1.2 2 21.20 even 2
4235.2.a.h.1.2 2 77.76 even 2
6160.2.a.y.1.2 2 28.27 even 2