Properties

Label 9405.2.a.w.1.5
Level $9405$
Weight $2$
Character 9405.1
Self dual yes
Analytic conductor $75.099$
Analytic rank $1$
Dimension $6$
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] = [9405,2,Mod(1,9405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9405, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9405.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9405.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: \(75.0993031010\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.131947641.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.04201\) of defining polynomial
Character \(\chi\) \(=\) 9405.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.04201 q^{2} +2.16980 q^{4} -1.00000 q^{5} -0.469142 q^{7} +0.346728 q^{8} +O(q^{10})\) \(q+2.04201 q^{2} +2.16980 q^{4} -1.00000 q^{5} -0.469142 q^{7} +0.346728 q^{8} -2.04201 q^{10} +1.00000 q^{11} +2.14036 q^{13} -0.957992 q^{14} -3.63157 q^{16} -4.47995 q^{17} -1.00000 q^{19} -2.16980 q^{20} +2.04201 q^{22} -0.200995 q^{23} +1.00000 q^{25} +4.37063 q^{26} -1.01794 q^{28} +6.92379 q^{29} +4.65688 q^{31} -8.10916 q^{32} -9.14810 q^{34} +0.469142 q^{35} -7.22421 q^{37} -2.04201 q^{38} -0.346728 q^{40} +3.34355 q^{41} +3.75162 q^{43} +2.16980 q^{44} -0.410434 q^{46} +2.69748 q^{47} -6.77991 q^{49} +2.04201 q^{50} +4.64414 q^{52} -10.4652 q^{53} -1.00000 q^{55} -0.162665 q^{56} +14.1384 q^{58} -6.05634 q^{59} -5.04325 q^{61} +9.50939 q^{62} -9.29582 q^{64} -2.14036 q^{65} +7.47221 q^{67} -9.72059 q^{68} +0.957992 q^{70} -15.9494 q^{71} -15.5740 q^{73} -14.7519 q^{74} -2.16980 q^{76} -0.469142 q^{77} +5.88178 q^{79} +3.63157 q^{80} +6.82756 q^{82} -12.9799 q^{83} +4.47995 q^{85} +7.66084 q^{86} +0.346728 q^{88} +0.794873 q^{89} -1.00413 q^{91} -0.436119 q^{92} +5.50828 q^{94} +1.00000 q^{95} +17.5493 q^{97} -13.8446 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7} + 6 q^{11} - 9 q^{13} - 18 q^{14} + 4 q^{16} + 5 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{23} + 6 q^{25} + 22 q^{26} + 10 q^{28} + 5 q^{29} - q^{31} - 15 q^{32} - 22 q^{34} - 5 q^{35} + 9 q^{37} - 25 q^{41} + 15 q^{43} + 8 q^{44} - 16 q^{46} - 24 q^{47} + 13 q^{49} - 27 q^{52} - 5 q^{53} - 6 q^{55} + 12 q^{56} + 13 q^{58} - 39 q^{59} - 11 q^{61} + 42 q^{62} - 14 q^{64} + 9 q^{65} + 24 q^{67} - 45 q^{68} + 18 q^{70} + 24 q^{71} - 26 q^{73} - q^{74} - 8 q^{76} + 5 q^{77} + 11 q^{79} - 4 q^{80} + 8 q^{82} - 39 q^{83} - 5 q^{85} - 18 q^{86} - 22 q^{89} - 26 q^{91} + 11 q^{92} - 30 q^{94} + 6 q^{95} + 22 q^{97} - 33 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.04201 1.44392 0.721959 0.691936i \(-0.243242\pi\)
0.721959 + 0.691936i \(0.243242\pi\)
\(3\) 0 0
\(4\) 2.16980 1.08490
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.469142 −0.177319 −0.0886595 0.996062i \(-0.528258\pi\)
−0.0886595 + 0.996062i \(0.528258\pi\)
\(8\) 0.346728 0.122587
\(9\) 0 0
\(10\) −2.04201 −0.645740
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.14036 0.593628 0.296814 0.954935i \(-0.404076\pi\)
0.296814 + 0.954935i \(0.404076\pi\)
\(14\) −0.957992 −0.256034
\(15\) 0 0
\(16\) −3.63157 −0.907893
\(17\) −4.47995 −1.08655 −0.543274 0.839555i \(-0.682816\pi\)
−0.543274 + 0.839555i \(0.682816\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) −2.16980 −0.485181
\(21\) 0 0
\(22\) 2.04201 0.435358
\(23\) −0.200995 −0.0419105 −0.0209552 0.999780i \(-0.506671\pi\)
−0.0209552 + 0.999780i \(0.506671\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.37063 0.857150
\(27\) 0 0
\(28\) −1.01794 −0.192373
\(29\) 6.92379 1.28571 0.642857 0.765986i \(-0.277749\pi\)
0.642857 + 0.765986i \(0.277749\pi\)
\(30\) 0 0
\(31\) 4.65688 0.836401 0.418200 0.908355i \(-0.362661\pi\)
0.418200 + 0.908355i \(0.362661\pi\)
\(32\) −8.10916 −1.43351
\(33\) 0 0
\(34\) −9.14810 −1.56889
\(35\) 0.469142 0.0792995
\(36\) 0 0
\(37\) −7.22421 −1.18765 −0.593826 0.804593i \(-0.702383\pi\)
−0.593826 + 0.804593i \(0.702383\pi\)
\(38\) −2.04201 −0.331257
\(39\) 0 0
\(40\) −0.346728 −0.0548225
\(41\) 3.34355 0.522175 0.261088 0.965315i \(-0.415919\pi\)
0.261088 + 0.965315i \(0.415919\pi\)
\(42\) 0 0
\(43\) 3.75162 0.572117 0.286058 0.958212i \(-0.407655\pi\)
0.286058 + 0.958212i \(0.407655\pi\)
\(44\) 2.16980 0.327109
\(45\) 0 0
\(46\) −0.410434 −0.0605152
\(47\) 2.69748 0.393468 0.196734 0.980457i \(-0.436966\pi\)
0.196734 + 0.980457i \(0.436966\pi\)
\(48\) 0 0
\(49\) −6.77991 −0.968558
\(50\) 2.04201 0.288784
\(51\) 0 0
\(52\) 4.64414 0.644027
\(53\) −10.4652 −1.43751 −0.718754 0.695264i \(-0.755287\pi\)
−0.718754 + 0.695264i \(0.755287\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −0.162665 −0.0217370
\(57\) 0 0
\(58\) 14.1384 1.85647
\(59\) −6.05634 −0.788468 −0.394234 0.919010i \(-0.628990\pi\)
−0.394234 + 0.919010i \(0.628990\pi\)
\(60\) 0 0
\(61\) −5.04325 −0.645722 −0.322861 0.946446i \(-0.604645\pi\)
−0.322861 + 0.946446i \(0.604645\pi\)
\(62\) 9.50939 1.20769
\(63\) 0 0
\(64\) −9.29582 −1.16198
\(65\) −2.14036 −0.265479
\(66\) 0 0
\(67\) 7.47221 0.912876 0.456438 0.889755i \(-0.349125\pi\)
0.456438 + 0.889755i \(0.349125\pi\)
\(68\) −9.72059 −1.17879
\(69\) 0 0
\(70\) 0.957992 0.114502
\(71\) −15.9494 −1.89284 −0.946421 0.322936i \(-0.895330\pi\)
−0.946421 + 0.322936i \(0.895330\pi\)
\(72\) 0 0
\(73\) −15.5740 −1.82280 −0.911400 0.411522i \(-0.864997\pi\)
−0.911400 + 0.411522i \(0.864997\pi\)
\(74\) −14.7519 −1.71487
\(75\) 0 0
\(76\) −2.16980 −0.248893
\(77\) −0.469142 −0.0534637
\(78\) 0 0
\(79\) 5.88178 0.661752 0.330876 0.943674i \(-0.392656\pi\)
0.330876 + 0.943674i \(0.392656\pi\)
\(80\) 3.63157 0.406022
\(81\) 0 0
\(82\) 6.82756 0.753978
\(83\) −12.9799 −1.42473 −0.712364 0.701810i \(-0.752376\pi\)
−0.712364 + 0.701810i \(0.752376\pi\)
\(84\) 0 0
\(85\) 4.47995 0.485919
\(86\) 7.66084 0.826090
\(87\) 0 0
\(88\) 0.346728 0.0369613
\(89\) 0.794873 0.0842564 0.0421282 0.999112i \(-0.486586\pi\)
0.0421282 + 0.999112i \(0.486586\pi\)
\(90\) 0 0
\(91\) −1.00413 −0.105262
\(92\) −0.436119 −0.0454686
\(93\) 0 0
\(94\) 5.50828 0.568136
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 17.5493 1.78186 0.890931 0.454139i \(-0.150053\pi\)
0.890931 + 0.454139i \(0.150053\pi\)
\(98\) −13.8446 −1.39852
\(99\) 0 0
\(100\) 2.16980 0.216980
\(101\) −14.0995 −1.40295 −0.701476 0.712693i \(-0.747475\pi\)
−0.701476 + 0.712693i \(0.747475\pi\)
\(102\) 0 0
\(103\) 14.7905 1.45735 0.728673 0.684861i \(-0.240137\pi\)
0.728673 + 0.684861i \(0.240137\pi\)
\(104\) 0.742122 0.0727710
\(105\) 0 0
\(106\) −21.3701 −2.07564
\(107\) −3.73310 −0.360892 −0.180446 0.983585i \(-0.557754\pi\)
−0.180446 + 0.983585i \(0.557754\pi\)
\(108\) 0 0
\(109\) 0.525483 0.0503321 0.0251661 0.999683i \(-0.491989\pi\)
0.0251661 + 0.999683i \(0.491989\pi\)
\(110\) −2.04201 −0.194698
\(111\) 0 0
\(112\) 1.70372 0.160987
\(113\) −9.63228 −0.906128 −0.453064 0.891478i \(-0.649669\pi\)
−0.453064 + 0.891478i \(0.649669\pi\)
\(114\) 0 0
\(115\) 0.200995 0.0187429
\(116\) 15.0232 1.39487
\(117\) 0 0
\(118\) −12.3671 −1.13848
\(119\) 2.10173 0.192666
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −10.2984 −0.932370
\(123\) 0 0
\(124\) 10.1045 0.907410
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −17.6992 −1.57055 −0.785276 0.619146i \(-0.787479\pi\)
−0.785276 + 0.619146i \(0.787479\pi\)
\(128\) −2.76383 −0.244290
\(129\) 0 0
\(130\) −4.37063 −0.383329
\(131\) −19.8018 −1.73009 −0.865046 0.501693i \(-0.832711\pi\)
−0.865046 + 0.501693i \(0.832711\pi\)
\(132\) 0 0
\(133\) 0.469142 0.0406798
\(134\) 15.2583 1.31812
\(135\) 0 0
\(136\) −1.55332 −0.133196
\(137\) −9.38019 −0.801403 −0.400702 0.916209i \(-0.631234\pi\)
−0.400702 + 0.916209i \(0.631234\pi\)
\(138\) 0 0
\(139\) −13.6770 −1.16007 −0.580034 0.814592i \(-0.696961\pi\)
−0.580034 + 0.814592i \(0.696961\pi\)
\(140\) 1.01794 0.0860319
\(141\) 0 0
\(142\) −32.5687 −2.73311
\(143\) 2.14036 0.178986
\(144\) 0 0
\(145\) −6.92379 −0.574989
\(146\) −31.8022 −2.63197
\(147\) 0 0
\(148\) −15.6751 −1.28848
\(149\) −9.34304 −0.765412 −0.382706 0.923870i \(-0.625008\pi\)
−0.382706 + 0.923870i \(0.625008\pi\)
\(150\) 0 0
\(151\) 7.72133 0.628353 0.314176 0.949365i \(-0.398272\pi\)
0.314176 + 0.949365i \(0.398272\pi\)
\(152\) −0.346728 −0.0281233
\(153\) 0 0
\(154\) −0.957992 −0.0771972
\(155\) −4.65688 −0.374050
\(156\) 0 0
\(157\) 2.86254 0.228455 0.114228 0.993455i \(-0.463561\pi\)
0.114228 + 0.993455i \(0.463561\pi\)
\(158\) 12.0106 0.955515
\(159\) 0 0
\(160\) 8.10916 0.641085
\(161\) 0.0942954 0.00743152
\(162\) 0 0
\(163\) 10.9042 0.854083 0.427041 0.904232i \(-0.359556\pi\)
0.427041 + 0.904232i \(0.359556\pi\)
\(164\) 7.25483 0.566507
\(165\) 0 0
\(166\) −26.5050 −2.05719
\(167\) 15.7161 1.21615 0.608074 0.793880i \(-0.291942\pi\)
0.608074 + 0.793880i \(0.291942\pi\)
\(168\) 0 0
\(169\) −8.41887 −0.647605
\(170\) 9.14810 0.701627
\(171\) 0 0
\(172\) 8.14026 0.620689
\(173\) −3.84278 −0.292161 −0.146080 0.989273i \(-0.546666\pi\)
−0.146080 + 0.989273i \(0.546666\pi\)
\(174\) 0 0
\(175\) −0.469142 −0.0354638
\(176\) −3.63157 −0.273740
\(177\) 0 0
\(178\) 1.62314 0.121659
\(179\) −19.8584 −1.48429 −0.742143 0.670242i \(-0.766190\pi\)
−0.742143 + 0.670242i \(0.766190\pi\)
\(180\) 0 0
\(181\) −15.0067 −1.11544 −0.557719 0.830030i \(-0.688323\pi\)
−0.557719 + 0.830030i \(0.688323\pi\)
\(182\) −2.05044 −0.151989
\(183\) 0 0
\(184\) −0.0696907 −0.00513767
\(185\) 7.22421 0.531134
\(186\) 0 0
\(187\) −4.47995 −0.327607
\(188\) 5.85299 0.426873
\(189\) 0 0
\(190\) 2.04201 0.148143
\(191\) −14.4281 −1.04398 −0.521990 0.852951i \(-0.674810\pi\)
−0.521990 + 0.852951i \(0.674810\pi\)
\(192\) 0 0
\(193\) 23.3554 1.68116 0.840580 0.541688i \(-0.182215\pi\)
0.840580 + 0.541688i \(0.182215\pi\)
\(194\) 35.8358 2.57286
\(195\) 0 0
\(196\) −14.7110 −1.05079
\(197\) 10.3230 0.735484 0.367742 0.929928i \(-0.380131\pi\)
0.367742 + 0.929928i \(0.380131\pi\)
\(198\) 0 0
\(199\) −10.5459 −0.747579 −0.373789 0.927514i \(-0.621942\pi\)
−0.373789 + 0.927514i \(0.621942\pi\)
\(200\) 0.346728 0.0245174
\(201\) 0 0
\(202\) −28.7913 −2.02575
\(203\) −3.24824 −0.227982
\(204\) 0 0
\(205\) −3.34355 −0.233524
\(206\) 30.2022 2.10429
\(207\) 0 0
\(208\) −7.77287 −0.538951
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 7.35896 0.506612 0.253306 0.967386i \(-0.418482\pi\)
0.253306 + 0.967386i \(0.418482\pi\)
\(212\) −22.7074 −1.55955
\(213\) 0 0
\(214\) −7.62302 −0.521099
\(215\) −3.75162 −0.255858
\(216\) 0 0
\(217\) −2.18474 −0.148310
\(218\) 1.07304 0.0726755
\(219\) 0 0
\(220\) −2.16980 −0.146288
\(221\) −9.58870 −0.645006
\(222\) 0 0
\(223\) 14.4057 0.964674 0.482337 0.875986i \(-0.339788\pi\)
0.482337 + 0.875986i \(0.339788\pi\)
\(224\) 3.80435 0.254189
\(225\) 0 0
\(226\) −19.6692 −1.30837
\(227\) 29.7139 1.97218 0.986091 0.166204i \(-0.0531510\pi\)
0.986091 + 0.166204i \(0.0531510\pi\)
\(228\) 0 0
\(229\) −11.0142 −0.727836 −0.363918 0.931431i \(-0.618561\pi\)
−0.363918 + 0.931431i \(0.618561\pi\)
\(230\) 0.410434 0.0270632
\(231\) 0 0
\(232\) 2.40067 0.157612
\(233\) −9.53690 −0.624783 −0.312391 0.949953i \(-0.601130\pi\)
−0.312391 + 0.949953i \(0.601130\pi\)
\(234\) 0 0
\(235\) −2.69748 −0.175964
\(236\) −13.1410 −0.855408
\(237\) 0 0
\(238\) 4.29176 0.278193
\(239\) −4.02979 −0.260665 −0.130333 0.991470i \(-0.541605\pi\)
−0.130333 + 0.991470i \(0.541605\pi\)
\(240\) 0 0
\(241\) 6.06607 0.390750 0.195375 0.980729i \(-0.437408\pi\)
0.195375 + 0.980729i \(0.437408\pi\)
\(242\) 2.04201 0.131265
\(243\) 0 0
\(244\) −10.9428 −0.700543
\(245\) 6.77991 0.433152
\(246\) 0 0
\(247\) −2.14036 −0.136188
\(248\) 1.61467 0.102532
\(249\) 0 0
\(250\) −2.04201 −0.129148
\(251\) 20.7498 1.30971 0.654857 0.755753i \(-0.272729\pi\)
0.654857 + 0.755753i \(0.272729\pi\)
\(252\) 0 0
\(253\) −0.200995 −0.0126365
\(254\) −36.1420 −2.26775
\(255\) 0 0
\(256\) 12.9479 0.809243
\(257\) −21.3702 −1.33304 −0.666519 0.745488i \(-0.732216\pi\)
−0.666519 + 0.745488i \(0.732216\pi\)
\(258\) 0 0
\(259\) 3.38918 0.210593
\(260\) −4.64414 −0.288017
\(261\) 0 0
\(262\) −40.4355 −2.49811
\(263\) 2.10893 0.130042 0.0650212 0.997884i \(-0.479288\pi\)
0.0650212 + 0.997884i \(0.479288\pi\)
\(264\) 0 0
\(265\) 10.4652 0.642873
\(266\) 0.957992 0.0587382
\(267\) 0 0
\(268\) 16.2132 0.990378
\(269\) −20.3100 −1.23833 −0.619163 0.785263i \(-0.712528\pi\)
−0.619163 + 0.785263i \(0.712528\pi\)
\(270\) 0 0
\(271\) −2.81864 −0.171220 −0.0856100 0.996329i \(-0.527284\pi\)
−0.0856100 + 0.996329i \(0.527284\pi\)
\(272\) 16.2693 0.986470
\(273\) 0 0
\(274\) −19.1544 −1.15716
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −16.6723 −1.00174 −0.500872 0.865522i \(-0.666987\pi\)
−0.500872 + 0.865522i \(0.666987\pi\)
\(278\) −27.9285 −1.67504
\(279\) 0 0
\(280\) 0.162665 0.00972107
\(281\) 27.9281 1.66605 0.833026 0.553234i \(-0.186607\pi\)
0.833026 + 0.553234i \(0.186607\pi\)
\(282\) 0 0
\(283\) −4.88494 −0.290380 −0.145190 0.989404i \(-0.546379\pi\)
−0.145190 + 0.989404i \(0.546379\pi\)
\(284\) −34.6069 −2.05354
\(285\) 0 0
\(286\) 4.37063 0.258441
\(287\) −1.56860 −0.0925916
\(288\) 0 0
\(289\) 3.06997 0.180587
\(290\) −14.1384 −0.830237
\(291\) 0 0
\(292\) −33.7924 −1.97755
\(293\) −0.648185 −0.0378674 −0.0189337 0.999821i \(-0.506027\pi\)
−0.0189337 + 0.999821i \(0.506027\pi\)
\(294\) 0 0
\(295\) 6.05634 0.352614
\(296\) −2.50483 −0.145591
\(297\) 0 0
\(298\) −19.0786 −1.10519
\(299\) −0.430202 −0.0248792
\(300\) 0 0
\(301\) −1.76004 −0.101447
\(302\) 15.7670 0.907290
\(303\) 0 0
\(304\) 3.63157 0.208285
\(305\) 5.04325 0.288776
\(306\) 0 0
\(307\) 11.8075 0.673891 0.336945 0.941524i \(-0.390606\pi\)
0.336945 + 0.941524i \(0.390606\pi\)
\(308\) −1.01794 −0.0580027
\(309\) 0 0
\(310\) −9.50939 −0.540097
\(311\) 14.4213 0.817756 0.408878 0.912589i \(-0.365920\pi\)
0.408878 + 0.912589i \(0.365920\pi\)
\(312\) 0 0
\(313\) −5.49337 −0.310504 −0.155252 0.987875i \(-0.549619\pi\)
−0.155252 + 0.987875i \(0.549619\pi\)
\(314\) 5.84533 0.329871
\(315\) 0 0
\(316\) 12.7623 0.717934
\(317\) −13.4742 −0.756789 −0.378395 0.925644i \(-0.623524\pi\)
−0.378395 + 0.925644i \(0.623524\pi\)
\(318\) 0 0
\(319\) 6.92379 0.387658
\(320\) 9.29582 0.519652
\(321\) 0 0
\(322\) 0.192552 0.0107305
\(323\) 4.47995 0.249271
\(324\) 0 0
\(325\) 2.14036 0.118726
\(326\) 22.2665 1.23323
\(327\) 0 0
\(328\) 1.15930 0.0640118
\(329\) −1.26550 −0.0697694
\(330\) 0 0
\(331\) 10.6406 0.584859 0.292430 0.956287i \(-0.405536\pi\)
0.292430 + 0.956287i \(0.405536\pi\)
\(332\) −28.1637 −1.54569
\(333\) 0 0
\(334\) 32.0924 1.75602
\(335\) −7.47221 −0.408250
\(336\) 0 0
\(337\) −2.99281 −0.163029 −0.0815145 0.996672i \(-0.525976\pi\)
−0.0815145 + 0.996672i \(0.525976\pi\)
\(338\) −17.1914 −0.935089
\(339\) 0 0
\(340\) 9.72059 0.527173
\(341\) 4.65688 0.252184
\(342\) 0 0
\(343\) 6.46473 0.349063
\(344\) 1.30079 0.0701340
\(345\) 0 0
\(346\) −7.84698 −0.421856
\(347\) −1.57779 −0.0847001 −0.0423500 0.999103i \(-0.513484\pi\)
−0.0423500 + 0.999103i \(0.513484\pi\)
\(348\) 0 0
\(349\) 12.9750 0.694534 0.347267 0.937766i \(-0.387110\pi\)
0.347267 + 0.937766i \(0.387110\pi\)
\(350\) −0.957992 −0.0512068
\(351\) 0 0
\(352\) −8.10916 −0.432220
\(353\) 13.5044 0.718768 0.359384 0.933190i \(-0.382987\pi\)
0.359384 + 0.933190i \(0.382987\pi\)
\(354\) 0 0
\(355\) 15.9494 0.846504
\(356\) 1.72471 0.0914096
\(357\) 0 0
\(358\) −40.5510 −2.14319
\(359\) 22.9543 1.21148 0.605741 0.795662i \(-0.292877\pi\)
0.605741 + 0.795662i \(0.292877\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −30.6438 −1.61060
\(363\) 0 0
\(364\) −2.17876 −0.114198
\(365\) 15.5740 0.815181
\(366\) 0 0
\(367\) 5.77896 0.301660 0.150830 0.988560i \(-0.451805\pi\)
0.150830 + 0.988560i \(0.451805\pi\)
\(368\) 0.729930 0.0380502
\(369\) 0 0
\(370\) 14.7519 0.766914
\(371\) 4.90967 0.254898
\(372\) 0 0
\(373\) 32.2936 1.67210 0.836051 0.548652i \(-0.184859\pi\)
0.836051 + 0.548652i \(0.184859\pi\)
\(374\) −9.14810 −0.473037
\(375\) 0 0
\(376\) 0.935292 0.0482340
\(377\) 14.8194 0.763237
\(378\) 0 0
\(379\) −12.2031 −0.626831 −0.313415 0.949616i \(-0.601473\pi\)
−0.313415 + 0.949616i \(0.601473\pi\)
\(380\) 2.16980 0.111308
\(381\) 0 0
\(382\) −29.4623 −1.50742
\(383\) −11.5424 −0.589787 −0.294894 0.955530i \(-0.595284\pi\)
−0.294894 + 0.955530i \(0.595284\pi\)
\(384\) 0 0
\(385\) 0.469142 0.0239097
\(386\) 47.6919 2.42746
\(387\) 0 0
\(388\) 38.0784 1.93314
\(389\) 4.65210 0.235871 0.117935 0.993021i \(-0.462372\pi\)
0.117935 + 0.993021i \(0.462372\pi\)
\(390\) 0 0
\(391\) 0.900450 0.0455377
\(392\) −2.35078 −0.118732
\(393\) 0 0
\(394\) 21.0797 1.06198
\(395\) −5.88178 −0.295944
\(396\) 0 0
\(397\) −23.8285 −1.19592 −0.597960 0.801526i \(-0.704022\pi\)
−0.597960 + 0.801526i \(0.704022\pi\)
\(398\) −21.5348 −1.07944
\(399\) 0 0
\(400\) −3.63157 −0.181579
\(401\) 5.11101 0.255232 0.127616 0.991824i \(-0.459268\pi\)
0.127616 + 0.991824i \(0.459268\pi\)
\(402\) 0 0
\(403\) 9.96739 0.496511
\(404\) −30.5930 −1.52206
\(405\) 0 0
\(406\) −6.63293 −0.329187
\(407\) −7.22421 −0.358091
\(408\) 0 0
\(409\) −20.0164 −0.989748 −0.494874 0.868965i \(-0.664786\pi\)
−0.494874 + 0.868965i \(0.664786\pi\)
\(410\) −6.82756 −0.337189
\(411\) 0 0
\(412\) 32.0923 1.58107
\(413\) 2.84128 0.139810
\(414\) 0 0
\(415\) 12.9799 0.637158
\(416\) −17.3565 −0.850972
\(417\) 0 0
\(418\) −2.04201 −0.0998779
\(419\) −34.3463 −1.67793 −0.838964 0.544188i \(-0.816838\pi\)
−0.838964 + 0.544188i \(0.816838\pi\)
\(420\) 0 0
\(421\) 31.9075 1.55508 0.777539 0.628835i \(-0.216468\pi\)
0.777539 + 0.628835i \(0.216468\pi\)
\(422\) 15.0271 0.731506
\(423\) 0 0
\(424\) −3.62858 −0.176220
\(425\) −4.47995 −0.217310
\(426\) 0 0
\(427\) 2.36600 0.114499
\(428\) −8.10006 −0.391531
\(429\) 0 0
\(430\) −7.66084 −0.369439
\(431\) −38.2858 −1.84416 −0.922080 0.386998i \(-0.873512\pi\)
−0.922080 + 0.386998i \(0.873512\pi\)
\(432\) 0 0
\(433\) −20.3681 −0.978827 −0.489413 0.872052i \(-0.662789\pi\)
−0.489413 + 0.872052i \(0.662789\pi\)
\(434\) −4.46126 −0.214147
\(435\) 0 0
\(436\) 1.14019 0.0546053
\(437\) 0.200995 0.00961492
\(438\) 0 0
\(439\) −24.9107 −1.18892 −0.594462 0.804123i \(-0.702635\pi\)
−0.594462 + 0.804123i \(0.702635\pi\)
\(440\) −0.346728 −0.0165296
\(441\) 0 0
\(442\) −19.5802 −0.931335
\(443\) 22.7243 1.07966 0.539832 0.841773i \(-0.318488\pi\)
0.539832 + 0.841773i \(0.318488\pi\)
\(444\) 0 0
\(445\) −0.794873 −0.0376806
\(446\) 29.4165 1.39291
\(447\) 0 0
\(448\) 4.36106 0.206041
\(449\) 41.1163 1.94040 0.970198 0.242313i \(-0.0779060\pi\)
0.970198 + 0.242313i \(0.0779060\pi\)
\(450\) 0 0
\(451\) 3.34355 0.157442
\(452\) −20.9001 −0.983057
\(453\) 0 0
\(454\) 60.6761 2.84767
\(455\) 1.00413 0.0470744
\(456\) 0 0
\(457\) 6.07340 0.284101 0.142051 0.989859i \(-0.454630\pi\)
0.142051 + 0.989859i \(0.454630\pi\)
\(458\) −22.4910 −1.05094
\(459\) 0 0
\(460\) 0.436119 0.0203342
\(461\) 6.74719 0.314248 0.157124 0.987579i \(-0.449778\pi\)
0.157124 + 0.987579i \(0.449778\pi\)
\(462\) 0 0
\(463\) 37.3772 1.73707 0.868534 0.495630i \(-0.165063\pi\)
0.868534 + 0.495630i \(0.165063\pi\)
\(464\) −25.1442 −1.16729
\(465\) 0 0
\(466\) −19.4744 −0.902135
\(467\) −28.4403 −1.31606 −0.658030 0.752992i \(-0.728610\pi\)
−0.658030 + 0.752992i \(0.728610\pi\)
\(468\) 0 0
\(469\) −3.50553 −0.161870
\(470\) −5.50828 −0.254078
\(471\) 0 0
\(472\) −2.09990 −0.0966558
\(473\) 3.75162 0.172500
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 4.56034 0.209023
\(477\) 0 0
\(478\) −8.22886 −0.376379
\(479\) −35.7235 −1.63225 −0.816124 0.577877i \(-0.803881\pi\)
−0.816124 + 0.577877i \(0.803881\pi\)
\(480\) 0 0
\(481\) −15.4624 −0.705024
\(482\) 12.3870 0.564210
\(483\) 0 0
\(484\) 2.16980 0.0986272
\(485\) −17.5493 −0.796873
\(486\) 0 0
\(487\) −9.68607 −0.438918 −0.219459 0.975622i \(-0.570429\pi\)
−0.219459 + 0.975622i \(0.570429\pi\)
\(488\) −1.74864 −0.0791570
\(489\) 0 0
\(490\) 13.8446 0.625436
\(491\) 8.75793 0.395240 0.197620 0.980279i \(-0.436679\pi\)
0.197620 + 0.980279i \(0.436679\pi\)
\(492\) 0 0
\(493\) −31.0182 −1.39699
\(494\) −4.37063 −0.196644
\(495\) 0 0
\(496\) −16.9118 −0.759363
\(497\) 7.48252 0.335637
\(498\) 0 0
\(499\) 15.3711 0.688104 0.344052 0.938951i \(-0.388200\pi\)
0.344052 + 0.938951i \(0.388200\pi\)
\(500\) −2.16980 −0.0970363
\(501\) 0 0
\(502\) 42.3712 1.89112
\(503\) −22.5117 −1.00375 −0.501874 0.864941i \(-0.667356\pi\)
−0.501874 + 0.864941i \(0.667356\pi\)
\(504\) 0 0
\(505\) 14.0995 0.627419
\(506\) −0.410434 −0.0182460
\(507\) 0 0
\(508\) −38.4037 −1.70389
\(509\) 30.6457 1.35835 0.679173 0.733978i \(-0.262339\pi\)
0.679173 + 0.733978i \(0.262339\pi\)
\(510\) 0 0
\(511\) 7.30642 0.323217
\(512\) 31.9673 1.41277
\(513\) 0 0
\(514\) −43.6381 −1.92480
\(515\) −14.7905 −0.651745
\(516\) 0 0
\(517\) 2.69748 0.118635
\(518\) 6.92073 0.304079
\(519\) 0 0
\(520\) −0.742122 −0.0325442
\(521\) −6.68929 −0.293063 −0.146532 0.989206i \(-0.546811\pi\)
−0.146532 + 0.989206i \(0.546811\pi\)
\(522\) 0 0
\(523\) 5.06543 0.221496 0.110748 0.993849i \(-0.464675\pi\)
0.110748 + 0.993849i \(0.464675\pi\)
\(524\) −42.9659 −1.87697
\(525\) 0 0
\(526\) 4.30646 0.187771
\(527\) −20.8626 −0.908790
\(528\) 0 0
\(529\) −22.9596 −0.998244
\(530\) 21.3701 0.928256
\(531\) 0 0
\(532\) 1.01794 0.0441334
\(533\) 7.15640 0.309978
\(534\) 0 0
\(535\) 3.73310 0.161396
\(536\) 2.59082 0.111907
\(537\) 0 0
\(538\) −41.4733 −1.78804
\(539\) −6.77991 −0.292031
\(540\) 0 0
\(541\) −45.1702 −1.94202 −0.971010 0.239038i \(-0.923168\pi\)
−0.971010 + 0.239038i \(0.923168\pi\)
\(542\) −5.75568 −0.247227
\(543\) 0 0
\(544\) 36.3286 1.55758
\(545\) −0.525483 −0.0225092
\(546\) 0 0
\(547\) −34.5604 −1.47769 −0.738847 0.673873i \(-0.764629\pi\)
−0.738847 + 0.673873i \(0.764629\pi\)
\(548\) −20.3531 −0.869442
\(549\) 0 0
\(550\) 2.04201 0.0870715
\(551\) −6.92379 −0.294963
\(552\) 0 0
\(553\) −2.75939 −0.117341
\(554\) −34.0450 −1.44643
\(555\) 0 0
\(556\) −29.6763 −1.25856
\(557\) −26.7387 −1.13296 −0.566478 0.824077i \(-0.691695\pi\)
−0.566478 + 0.824077i \(0.691695\pi\)
\(558\) 0 0
\(559\) 8.02981 0.339625
\(560\) −1.70372 −0.0719955
\(561\) 0 0
\(562\) 57.0294 2.40564
\(563\) 3.43066 0.144585 0.0722925 0.997383i \(-0.476968\pi\)
0.0722925 + 0.997383i \(0.476968\pi\)
\(564\) 0 0
\(565\) 9.63228 0.405233
\(566\) −9.97510 −0.419285
\(567\) 0 0
\(568\) −5.53009 −0.232037
\(569\) −6.20817 −0.260260 −0.130130 0.991497i \(-0.541539\pi\)
−0.130130 + 0.991497i \(0.541539\pi\)
\(570\) 0 0
\(571\) −4.29662 −0.179808 −0.0899040 0.995950i \(-0.528656\pi\)
−0.0899040 + 0.995950i \(0.528656\pi\)
\(572\) 4.64414 0.194181
\(573\) 0 0
\(574\) −3.20310 −0.133695
\(575\) −0.200995 −0.00838209
\(576\) 0 0
\(577\) 24.6963 1.02812 0.514060 0.857754i \(-0.328141\pi\)
0.514060 + 0.857754i \(0.328141\pi\)
\(578\) 6.26891 0.260752
\(579\) 0 0
\(580\) −15.0232 −0.623805
\(581\) 6.08941 0.252631
\(582\) 0 0
\(583\) −10.4652 −0.433425
\(584\) −5.39994 −0.223451
\(585\) 0 0
\(586\) −1.32360 −0.0546774
\(587\) −9.31384 −0.384423 −0.192212 0.981353i \(-0.561566\pi\)
−0.192212 + 0.981353i \(0.561566\pi\)
\(588\) 0 0
\(589\) −4.65688 −0.191884
\(590\) 12.3671 0.509145
\(591\) 0 0
\(592\) 26.2352 1.07826
\(593\) 25.8934 1.06331 0.531656 0.846960i \(-0.321570\pi\)
0.531656 + 0.846960i \(0.321570\pi\)
\(594\) 0 0
\(595\) −2.10173 −0.0861627
\(596\) −20.2725 −0.830394
\(597\) 0 0
\(598\) −0.878476 −0.0359236
\(599\) −24.8277 −1.01443 −0.507215 0.861819i \(-0.669325\pi\)
−0.507215 + 0.861819i \(0.669325\pi\)
\(600\) 0 0
\(601\) 9.59884 0.391545 0.195772 0.980649i \(-0.437279\pi\)
0.195772 + 0.980649i \(0.437279\pi\)
\(602\) −3.59402 −0.146481
\(603\) 0 0
\(604\) 16.7537 0.681699
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −11.9356 −0.484452 −0.242226 0.970220i \(-0.577877\pi\)
−0.242226 + 0.970220i \(0.577877\pi\)
\(608\) 8.10916 0.328870
\(609\) 0 0
\(610\) 10.2984 0.416968
\(611\) 5.77357 0.233574
\(612\) 0 0
\(613\) 9.27813 0.374740 0.187370 0.982289i \(-0.440004\pi\)
0.187370 + 0.982289i \(0.440004\pi\)
\(614\) 24.1111 0.973043
\(615\) 0 0
\(616\) −0.162665 −0.00655394
\(617\) −4.13222 −0.166357 −0.0831784 0.996535i \(-0.526507\pi\)
−0.0831784 + 0.996535i \(0.526507\pi\)
\(618\) 0 0
\(619\) −27.0790 −1.08840 −0.544198 0.838957i \(-0.683166\pi\)
−0.544198 + 0.838957i \(0.683166\pi\)
\(620\) −10.1045 −0.405806
\(621\) 0 0
\(622\) 29.4484 1.18077
\(623\) −0.372908 −0.0149403
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −11.2175 −0.448342
\(627\) 0 0
\(628\) 6.21113 0.247851
\(629\) 32.3641 1.29044
\(630\) 0 0
\(631\) −2.93731 −0.116933 −0.0584663 0.998289i \(-0.518621\pi\)
−0.0584663 + 0.998289i \(0.518621\pi\)
\(632\) 2.03938 0.0811220
\(633\) 0 0
\(634\) −27.5145 −1.09274
\(635\) 17.6992 0.702372
\(636\) 0 0
\(637\) −14.5114 −0.574963
\(638\) 14.1384 0.559746
\(639\) 0 0
\(640\) 2.76383 0.109250
\(641\) 4.32999 0.171024 0.0855122 0.996337i \(-0.472747\pi\)
0.0855122 + 0.996337i \(0.472747\pi\)
\(642\) 0 0
\(643\) −5.02416 −0.198134 −0.0990668 0.995081i \(-0.531586\pi\)
−0.0990668 + 0.995081i \(0.531586\pi\)
\(644\) 0.204602 0.00806245
\(645\) 0 0
\(646\) 9.14810 0.359927
\(647\) −10.7629 −0.423133 −0.211566 0.977364i \(-0.567856\pi\)
−0.211566 + 0.977364i \(0.567856\pi\)
\(648\) 0 0
\(649\) −6.05634 −0.237732
\(650\) 4.37063 0.171430
\(651\) 0 0
\(652\) 23.6599 0.926593
\(653\) 35.3311 1.38261 0.691307 0.722561i \(-0.257035\pi\)
0.691307 + 0.722561i \(0.257035\pi\)
\(654\) 0 0
\(655\) 19.8018 0.773721
\(656\) −12.1424 −0.474080
\(657\) 0 0
\(658\) −2.58416 −0.100741
\(659\) 18.0616 0.703580 0.351790 0.936079i \(-0.385573\pi\)
0.351790 + 0.936079i \(0.385573\pi\)
\(660\) 0 0
\(661\) 10.6516 0.414300 0.207150 0.978309i \(-0.433581\pi\)
0.207150 + 0.978309i \(0.433581\pi\)
\(662\) 21.7282 0.844489
\(663\) 0 0
\(664\) −4.50049 −0.174653
\(665\) −0.469142 −0.0181925
\(666\) 0 0
\(667\) −1.39165 −0.0538849
\(668\) 34.1007 1.31940
\(669\) 0 0
\(670\) −15.2583 −0.589480
\(671\) −5.04325 −0.194693
\(672\) 0 0
\(673\) 7.05682 0.272020 0.136010 0.990707i \(-0.456572\pi\)
0.136010 + 0.990707i \(0.456572\pi\)
\(674\) −6.11135 −0.235401
\(675\) 0 0
\(676\) −18.2672 −0.702586
\(677\) 17.0051 0.653560 0.326780 0.945100i \(-0.394036\pi\)
0.326780 + 0.945100i \(0.394036\pi\)
\(678\) 0 0
\(679\) −8.23312 −0.315958
\(680\) 1.55332 0.0595673
\(681\) 0 0
\(682\) 9.50939 0.364133
\(683\) 32.7782 1.25422 0.627112 0.778929i \(-0.284237\pi\)
0.627112 + 0.778929i \(0.284237\pi\)
\(684\) 0 0
\(685\) 9.38019 0.358399
\(686\) 13.2010 0.504018
\(687\) 0 0
\(688\) −13.6243 −0.519421
\(689\) −22.3993 −0.853346
\(690\) 0 0
\(691\) 23.4551 0.892274 0.446137 0.894965i \(-0.352799\pi\)
0.446137 + 0.894965i \(0.352799\pi\)
\(692\) −8.33805 −0.316965
\(693\) 0 0
\(694\) −3.22185 −0.122300
\(695\) 13.6770 0.518798
\(696\) 0 0
\(697\) −14.9790 −0.567369
\(698\) 26.4950 1.00285
\(699\) 0 0
\(700\) −1.01794 −0.0384746
\(701\) −6.40155 −0.241783 −0.120892 0.992666i \(-0.538575\pi\)
−0.120892 + 0.992666i \(0.538575\pi\)
\(702\) 0 0
\(703\) 7.22421 0.272466
\(704\) −9.29582 −0.350349
\(705\) 0 0
\(706\) 27.5761 1.03784
\(707\) 6.61466 0.248770
\(708\) 0 0
\(709\) 16.2437 0.610044 0.305022 0.952345i \(-0.401336\pi\)
0.305022 + 0.952345i \(0.401336\pi\)
\(710\) 32.5687 1.22228
\(711\) 0 0
\(712\) 0.275605 0.0103287
\(713\) −0.936012 −0.0350539
\(714\) 0 0
\(715\) −2.14036 −0.0800448
\(716\) −43.0887 −1.61030
\(717\) 0 0
\(718\) 46.8729 1.74928
\(719\) 2.10857 0.0786363 0.0393181 0.999227i \(-0.487481\pi\)
0.0393181 + 0.999227i \(0.487481\pi\)
\(720\) 0 0
\(721\) −6.93882 −0.258415
\(722\) 2.04201 0.0759957
\(723\) 0 0
\(724\) −32.5615 −1.21014
\(725\) 6.92379 0.257143
\(726\) 0 0
\(727\) 1.59364 0.0591048 0.0295524 0.999563i \(-0.490592\pi\)
0.0295524 + 0.999563i \(0.490592\pi\)
\(728\) −0.348160 −0.0129037
\(729\) 0 0
\(730\) 31.8022 1.17705
\(731\) −16.8071 −0.621632
\(732\) 0 0
\(733\) 10.6588 0.393692 0.196846 0.980434i \(-0.436930\pi\)
0.196846 + 0.980434i \(0.436930\pi\)
\(734\) 11.8007 0.435572
\(735\) 0 0
\(736\) 1.62990 0.0600791
\(737\) 7.47221 0.275242
\(738\) 0 0
\(739\) 3.32501 0.122313 0.0611563 0.998128i \(-0.480521\pi\)
0.0611563 + 0.998128i \(0.480521\pi\)
\(740\) 15.6751 0.576227
\(741\) 0 0
\(742\) 10.0256 0.368051
\(743\) 51.3600 1.88421 0.942107 0.335311i \(-0.108841\pi\)
0.942107 + 0.335311i \(0.108841\pi\)
\(744\) 0 0
\(745\) 9.34304 0.342302
\(746\) 65.9439 2.41438
\(747\) 0 0
\(748\) −9.72059 −0.355420
\(749\) 1.75135 0.0639930
\(750\) 0 0
\(751\) −22.1565 −0.808502 −0.404251 0.914648i \(-0.632468\pi\)
−0.404251 + 0.914648i \(0.632468\pi\)
\(752\) −9.79610 −0.357227
\(753\) 0 0
\(754\) 30.2613 1.10205
\(755\) −7.72133 −0.281008
\(756\) 0 0
\(757\) 3.94366 0.143335 0.0716674 0.997429i \(-0.477168\pi\)
0.0716674 + 0.997429i \(0.477168\pi\)
\(758\) −24.9188 −0.905092
\(759\) 0 0
\(760\) 0.346728 0.0125771
\(761\) 48.5719 1.76073 0.880366 0.474296i \(-0.157297\pi\)
0.880366 + 0.474296i \(0.157297\pi\)
\(762\) 0 0
\(763\) −0.246526 −0.00892484
\(764\) −31.3061 −1.13261
\(765\) 0 0
\(766\) −23.5696 −0.851604
\(767\) −12.9627 −0.468057
\(768\) 0 0
\(769\) −34.5865 −1.24722 −0.623611 0.781735i \(-0.714335\pi\)
−0.623611 + 0.781735i \(0.714335\pi\)
\(770\) 0.957992 0.0345236
\(771\) 0 0
\(772\) 50.6765 1.82389
\(773\) −9.98101 −0.358992 −0.179496 0.983759i \(-0.557447\pi\)
−0.179496 + 0.983759i \(0.557447\pi\)
\(774\) 0 0
\(775\) 4.65688 0.167280
\(776\) 6.08483 0.218433
\(777\) 0 0
\(778\) 9.49962 0.340578
\(779\) −3.34355 −0.119795
\(780\) 0 0
\(781\) −15.9494 −0.570713
\(782\) 1.83873 0.0657527
\(783\) 0 0
\(784\) 24.6217 0.879347
\(785\) −2.86254 −0.102168
\(786\) 0 0
\(787\) 8.47966 0.302267 0.151134 0.988513i \(-0.451708\pi\)
0.151134 + 0.988513i \(0.451708\pi\)
\(788\) 22.3988 0.797926
\(789\) 0 0
\(790\) −12.0106 −0.427319
\(791\) 4.51890 0.160674
\(792\) 0 0
\(793\) −10.7944 −0.383319
\(794\) −48.6581 −1.72681
\(795\) 0 0
\(796\) −22.8825 −0.811047
\(797\) −50.8181 −1.80007 −0.900034 0.435820i \(-0.856459\pi\)
−0.900034 + 0.435820i \(0.856459\pi\)
\(798\) 0 0
\(799\) −12.0846 −0.427522
\(800\) −8.10916 −0.286702
\(801\) 0 0
\(802\) 10.4367 0.368534
\(803\) −15.5740 −0.549595
\(804\) 0 0
\(805\) −0.0942954 −0.00332348
\(806\) 20.3535 0.716921
\(807\) 0 0
\(808\) −4.88869 −0.171983
\(809\) −26.2792 −0.923926 −0.461963 0.886899i \(-0.652855\pi\)
−0.461963 + 0.886899i \(0.652855\pi\)
\(810\) 0 0
\(811\) 44.2582 1.55412 0.777058 0.629429i \(-0.216711\pi\)
0.777058 + 0.629429i \(0.216711\pi\)
\(812\) −7.04802 −0.247337
\(813\) 0 0
\(814\) −14.7519 −0.517054
\(815\) −10.9042 −0.381957
\(816\) 0 0
\(817\) −3.75162 −0.131253
\(818\) −40.8737 −1.42912
\(819\) 0 0
\(820\) −7.25483 −0.253350
\(821\) 45.4098 1.58481 0.792406 0.609994i \(-0.208828\pi\)
0.792406 + 0.609994i \(0.208828\pi\)
\(822\) 0 0
\(823\) 20.0801 0.699950 0.349975 0.936759i \(-0.386190\pi\)
0.349975 + 0.936759i \(0.386190\pi\)
\(824\) 5.12826 0.178652
\(825\) 0 0
\(826\) 5.80192 0.201875
\(827\) −2.70217 −0.0939637 −0.0469818 0.998896i \(-0.514960\pi\)
−0.0469818 + 0.998896i \(0.514960\pi\)
\(828\) 0 0
\(829\) 43.8322 1.52236 0.761178 0.648543i \(-0.224621\pi\)
0.761178 + 0.648543i \(0.224621\pi\)
\(830\) 26.5050 0.920004
\(831\) 0 0
\(832\) −19.8964 −0.689783
\(833\) 30.3737 1.05238
\(834\) 0 0
\(835\) −15.7161 −0.543878
\(836\) −2.16980 −0.0750440
\(837\) 0 0
\(838\) −70.1354 −2.42279
\(839\) −12.7989 −0.441866 −0.220933 0.975289i \(-0.570910\pi\)
−0.220933 + 0.975289i \(0.570910\pi\)
\(840\) 0 0
\(841\) 18.9388 0.653062
\(842\) 65.1554 2.24540
\(843\) 0 0
\(844\) 15.9675 0.549623
\(845\) 8.41887 0.289618
\(846\) 0 0
\(847\) −0.469142 −0.0161199
\(848\) 38.0052 1.30510
\(849\) 0 0
\(850\) −9.14810 −0.313777
\(851\) 1.45203 0.0497750
\(852\) 0 0
\(853\) −30.8728 −1.05707 −0.528533 0.848913i \(-0.677258\pi\)
−0.528533 + 0.848913i \(0.677258\pi\)
\(854\) 4.83139 0.165327
\(855\) 0 0
\(856\) −1.29437 −0.0442406
\(857\) 18.5582 0.633935 0.316967 0.948436i \(-0.397335\pi\)
0.316967 + 0.948436i \(0.397335\pi\)
\(858\) 0 0
\(859\) −0.760628 −0.0259523 −0.0129762 0.999916i \(-0.504131\pi\)
−0.0129762 + 0.999916i \(0.504131\pi\)
\(860\) −8.14026 −0.277581
\(861\) 0 0
\(862\) −78.1799 −2.66282
\(863\) 46.7268 1.59060 0.795300 0.606216i \(-0.207313\pi\)
0.795300 + 0.606216i \(0.207313\pi\)
\(864\) 0 0
\(865\) 3.84278 0.130658
\(866\) −41.5918 −1.41335
\(867\) 0 0
\(868\) −4.74044 −0.160901
\(869\) 5.88178 0.199526
\(870\) 0 0
\(871\) 15.9932 0.541909
\(872\) 0.182200 0.00617006
\(873\) 0 0
\(874\) 0.410434 0.0138831
\(875\) 0.469142 0.0158599
\(876\) 0 0
\(877\) 38.0871 1.28611 0.643056 0.765819i \(-0.277666\pi\)
0.643056 + 0.765819i \(0.277666\pi\)
\(878\) −50.8679 −1.71671
\(879\) 0 0
\(880\) 3.63157 0.122420
\(881\) 14.4600 0.487170 0.243585 0.969880i \(-0.421677\pi\)
0.243585 + 0.969880i \(0.421677\pi\)
\(882\) 0 0
\(883\) 0.159279 0.00536017 0.00268008 0.999996i \(-0.499147\pi\)
0.00268008 + 0.999996i \(0.499147\pi\)
\(884\) −20.8055 −0.699766
\(885\) 0 0
\(886\) 46.4032 1.55895
\(887\) 9.95048 0.334104 0.167052 0.985948i \(-0.446575\pi\)
0.167052 + 0.985948i \(0.446575\pi\)
\(888\) 0 0
\(889\) 8.30345 0.278489
\(890\) −1.62314 −0.0544077
\(891\) 0 0
\(892\) 31.2574 1.04657
\(893\) −2.69748 −0.0902678
\(894\) 0 0
\(895\) 19.8584 0.663792
\(896\) 1.29663 0.0433173
\(897\) 0 0
\(898\) 83.9597 2.80177
\(899\) 32.2433 1.07537
\(900\) 0 0
\(901\) 46.8837 1.56192
\(902\) 6.82756 0.227333
\(903\) 0 0
\(904\) −3.33978 −0.111079
\(905\) 15.0067 0.498839
\(906\) 0 0
\(907\) 51.7590 1.71863 0.859315 0.511448i \(-0.170891\pi\)
0.859315 + 0.511448i \(0.170891\pi\)
\(908\) 64.4732 2.13962
\(909\) 0 0
\(910\) 2.05044 0.0679716
\(911\) 29.1062 0.964332 0.482166 0.876080i \(-0.339850\pi\)
0.482166 + 0.876080i \(0.339850\pi\)
\(912\) 0 0
\(913\) −12.9799 −0.429572
\(914\) 12.4019 0.410219
\(915\) 0 0
\(916\) −23.8985 −0.789629
\(917\) 9.28986 0.306778
\(918\) 0 0
\(919\) −12.7499 −0.420581 −0.210290 0.977639i \(-0.567441\pi\)
−0.210290 + 0.977639i \(0.567441\pi\)
\(920\) 0.0696907 0.00229764
\(921\) 0 0
\(922\) 13.7778 0.453748
\(923\) −34.1373 −1.12364
\(924\) 0 0
\(925\) −7.22421 −0.237530
\(926\) 76.3246 2.50818
\(927\) 0 0
\(928\) −56.1461 −1.84309
\(929\) −2.50821 −0.0822916 −0.0411458 0.999153i \(-0.513101\pi\)
−0.0411458 + 0.999153i \(0.513101\pi\)
\(930\) 0 0
\(931\) 6.77991 0.222202
\(932\) −20.6931 −0.677826
\(933\) 0 0
\(934\) −58.0753 −1.90028
\(935\) 4.47995 0.146510
\(936\) 0 0
\(937\) 47.9802 1.56744 0.783722 0.621112i \(-0.213319\pi\)
0.783722 + 0.621112i \(0.213319\pi\)
\(938\) −7.15832 −0.233727
\(939\) 0 0
\(940\) −5.85299 −0.190903
\(941\) 44.4858 1.45020 0.725098 0.688645i \(-0.241794\pi\)
0.725098 + 0.688645i \(0.241794\pi\)
\(942\) 0 0
\(943\) −0.672039 −0.0218846
\(944\) 21.9940 0.715845
\(945\) 0 0
\(946\) 7.66084 0.249075
\(947\) −6.70041 −0.217734 −0.108867 0.994056i \(-0.534722\pi\)
−0.108867 + 0.994056i \(0.534722\pi\)
\(948\) 0 0
\(949\) −33.3339 −1.08207
\(950\) −2.04201 −0.0662515
\(951\) 0 0
\(952\) 0.728730 0.0236183
\(953\) 57.1810 1.85227 0.926136 0.377189i \(-0.123109\pi\)
0.926136 + 0.377189i \(0.123109\pi\)
\(954\) 0 0
\(955\) 14.4281 0.466882
\(956\) −8.74382 −0.282796
\(957\) 0 0
\(958\) −72.9477 −2.35683
\(959\) 4.40064 0.142104
\(960\) 0 0
\(961\) −9.31344 −0.300434
\(962\) −31.5743 −1.01800
\(963\) 0 0
\(964\) 13.1621 0.423924
\(965\) −23.3554 −0.751837
\(966\) 0 0
\(967\) −5.24741 −0.168745 −0.0843727 0.996434i \(-0.526889\pi\)
−0.0843727 + 0.996434i \(0.526889\pi\)
\(968\) 0.346728 0.0111443
\(969\) 0 0
\(970\) −35.8358 −1.15062
\(971\) 27.4776 0.881799 0.440899 0.897557i \(-0.354660\pi\)
0.440899 + 0.897557i \(0.354660\pi\)
\(972\) 0 0
\(973\) 6.41645 0.205702
\(974\) −19.7790 −0.633761
\(975\) 0 0
\(976\) 18.3149 0.586247
\(977\) −42.6475 −1.36442 −0.682208 0.731158i \(-0.738980\pi\)
−0.682208 + 0.731158i \(0.738980\pi\)
\(978\) 0 0
\(979\) 0.794873 0.0254043
\(980\) 14.7110 0.469926
\(981\) 0 0
\(982\) 17.8838 0.570694
\(983\) 48.8786 1.55899 0.779493 0.626411i \(-0.215477\pi\)
0.779493 + 0.626411i \(0.215477\pi\)
\(984\) 0 0
\(985\) −10.3230 −0.328918
\(986\) −63.3395 −2.01714
\(987\) 0 0
\(988\) −4.64414 −0.147750
\(989\) −0.754059 −0.0239777
\(990\) 0 0
\(991\) −60.2922 −1.91524 −0.957622 0.288027i \(-0.907001\pi\)
−0.957622 + 0.288027i \(0.907001\pi\)
\(992\) −37.7634 −1.19899
\(993\) 0 0
\(994\) 15.2794 0.484632
\(995\) 10.5459 0.334327
\(996\) 0 0
\(997\) −6.50314 −0.205957 −0.102978 0.994684i \(-0.532837\pi\)
−0.102978 + 0.994684i \(0.532837\pi\)
\(998\) 31.3879 0.993565
\(999\) 0 0
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 9405.2.a.w.1.5 6
3.2 odd 2 1045.2.a.g.1.2 6
15.14 odd 2 5225.2.a.k.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.g.1.2 6 3.2 odd 2
5225.2.a.k.1.5 6 15.14 odd 2
9405.2.a.w.1.5 6 1.1 even 1 trivial