Properties

Label 1425.2.a.o.1.1
Level $1425$
Weight $2$
Character 1425.1
Self dual yes
Analytic conductor $11.379$
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] = [1425,2,Mod(1,1425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1425, 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("1425.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.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: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1425.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -0.732051 q^{7} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -0.732051 q^{7} +1.73205 q^{8} +1.00000 q^{9} +1.26795 q^{11} -1.00000 q^{12} +2.73205 q^{13} +1.26795 q^{14} -5.00000 q^{16} -1.73205 q^{18} +1.00000 q^{19} +0.732051 q^{21} -2.19615 q^{22} +3.46410 q^{23} -1.73205 q^{24} -4.73205 q^{26} -1.00000 q^{27} -0.732051 q^{28} -2.19615 q^{29} -4.92820 q^{31} +5.19615 q^{32} -1.26795 q^{33} +1.00000 q^{36} -4.19615 q^{37} -1.73205 q^{38} -2.73205 q^{39} +4.73205 q^{41} -1.26795 q^{42} +6.19615 q^{43} +1.26795 q^{44} -6.00000 q^{46} -3.46410 q^{47} +5.00000 q^{48} -6.46410 q^{49} +2.73205 q^{52} +2.53590 q^{53} +1.73205 q^{54} -1.26795 q^{56} -1.00000 q^{57} +3.80385 q^{58} +9.46410 q^{59} -13.4641 q^{61} +8.53590 q^{62} -0.732051 q^{63} +1.00000 q^{64} +2.19615 q^{66} -8.00000 q^{67} -3.46410 q^{69} +16.3923 q^{71} +1.73205 q^{72} +3.07180 q^{73} +7.26795 q^{74} +1.00000 q^{76} -0.928203 q^{77} +4.73205 q^{78} +2.92820 q^{79} +1.00000 q^{81} -8.19615 q^{82} -0.928203 q^{83} +0.732051 q^{84} -10.7321 q^{86} +2.19615 q^{87} +2.19615 q^{88} +7.26795 q^{89} -2.00000 q^{91} +3.46410 q^{92} +4.92820 q^{93} +6.00000 q^{94} -5.19615 q^{96} -4.19615 q^{97} +11.1962 q^{98} +1.26795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{12} + 2 q^{13} + 6 q^{14} - 10 q^{16} + 2 q^{19} - 2 q^{21} + 6 q^{22} - 6 q^{26} - 2 q^{27} + 2 q^{28} + 6 q^{29} + 4 q^{31} - 6 q^{33} + 2 q^{36} + 2 q^{37} - 2 q^{39} + 6 q^{41} - 6 q^{42} + 2 q^{43} + 6 q^{44} - 12 q^{46} + 10 q^{48} - 6 q^{49} + 2 q^{52} + 12 q^{53} - 6 q^{56} - 2 q^{57} + 18 q^{58} + 12 q^{59} - 20 q^{61} + 24 q^{62} + 2 q^{63} + 2 q^{64} - 6 q^{66} - 16 q^{67} + 12 q^{71} + 20 q^{73} + 18 q^{74} + 2 q^{76} + 12 q^{77} + 6 q^{78} - 8 q^{79} + 2 q^{81} - 6 q^{82} + 12 q^{83} - 2 q^{84} - 18 q^{86} - 6 q^{87} - 6 q^{88} + 18 q^{89} - 4 q^{91} - 4 q^{93} + 12 q^{94} + 2 q^{97} + 12 q^{98} + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.73205 0.707107
\(7\) −0.732051 −0.276689 −0.138345 0.990384i \(-0.544178\pi\)
−0.138345 + 0.990384i \(0.544178\pi\)
\(8\) 1.73205 0.612372
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.26795 0.382301 0.191151 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.73205 0.757735 0.378867 0.925451i \(-0.376314\pi\)
0.378867 + 0.925451i \(0.376314\pi\)
\(14\) 1.26795 0.338874
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.73205 −0.408248
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.732051 0.159747
\(22\) −2.19615 −0.468221
\(23\) 3.46410 0.722315 0.361158 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) −1.73205 −0.353553
\(25\) 0 0
\(26\) −4.73205 −0.928032
\(27\) −1.00000 −0.192450
\(28\) −0.732051 −0.138345
\(29\) −2.19615 −0.407815 −0.203908 0.978990i \(-0.565364\pi\)
−0.203908 + 0.978990i \(0.565364\pi\)
\(30\) 0 0
\(31\) −4.92820 −0.885131 −0.442566 0.896736i \(-0.645932\pi\)
−0.442566 + 0.896736i \(0.645932\pi\)
\(32\) 5.19615 0.918559
\(33\) −1.26795 −0.220722
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.19615 −0.689843 −0.344922 0.938631i \(-0.612095\pi\)
−0.344922 + 0.938631i \(0.612095\pi\)
\(38\) −1.73205 −0.280976
\(39\) −2.73205 −0.437478
\(40\) 0 0
\(41\) 4.73205 0.739022 0.369511 0.929226i \(-0.379525\pi\)
0.369511 + 0.929226i \(0.379525\pi\)
\(42\) −1.26795 −0.195649
\(43\) 6.19615 0.944904 0.472452 0.881356i \(-0.343369\pi\)
0.472452 + 0.881356i \(0.343369\pi\)
\(44\) 1.26795 0.191151
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −3.46410 −0.505291 −0.252646 0.967559i \(-0.581301\pi\)
−0.252646 + 0.967559i \(0.581301\pi\)
\(48\) 5.00000 0.721688
\(49\) −6.46410 −0.923443
\(50\) 0 0
\(51\) 0 0
\(52\) 2.73205 0.378867
\(53\) 2.53590 0.348332 0.174166 0.984716i \(-0.444277\pi\)
0.174166 + 0.984716i \(0.444277\pi\)
\(54\) 1.73205 0.235702
\(55\) 0 0
\(56\) −1.26795 −0.169437
\(57\) −1.00000 −0.132453
\(58\) 3.80385 0.499470
\(59\) 9.46410 1.23212 0.616061 0.787699i \(-0.288728\pi\)
0.616061 + 0.787699i \(0.288728\pi\)
\(60\) 0 0
\(61\) −13.4641 −1.72390 −0.861951 0.506992i \(-0.830757\pi\)
−0.861951 + 0.506992i \(0.830757\pi\)
\(62\) 8.53590 1.08406
\(63\) −0.732051 −0.0922297
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.19615 0.270328
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −3.46410 −0.417029
\(70\) 0 0
\(71\) 16.3923 1.94541 0.972704 0.232048i \(-0.0745426\pi\)
0.972704 + 0.232048i \(0.0745426\pi\)
\(72\) 1.73205 0.204124
\(73\) 3.07180 0.359527 0.179763 0.983710i \(-0.442467\pi\)
0.179763 + 0.983710i \(0.442467\pi\)
\(74\) 7.26795 0.844882
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −0.928203 −0.105779
\(78\) 4.73205 0.535799
\(79\) 2.92820 0.329449 0.164724 0.986340i \(-0.447327\pi\)
0.164724 + 0.986340i \(0.447327\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −8.19615 −0.905114
\(83\) −0.928203 −0.101884 −0.0509418 0.998702i \(-0.516222\pi\)
−0.0509418 + 0.998702i \(0.516222\pi\)
\(84\) 0.732051 0.0798733
\(85\) 0 0
\(86\) −10.7321 −1.15727
\(87\) 2.19615 0.235452
\(88\) 2.19615 0.234111
\(89\) 7.26795 0.770401 0.385201 0.922833i \(-0.374132\pi\)
0.385201 + 0.922833i \(0.374132\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 3.46410 0.361158
\(93\) 4.92820 0.511031
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) −5.19615 −0.530330
\(97\) −4.19615 −0.426055 −0.213027 0.977046i \(-0.568332\pi\)
−0.213027 + 0.977046i \(0.568332\pi\)
\(98\) 11.1962 1.13098
\(99\) 1.26795 0.127434
\(100\) 0 0
\(101\) 10.3923 1.03407 0.517036 0.855963i \(-0.327035\pi\)
0.517036 + 0.855963i \(0.327035\pi\)
\(102\) 0 0
\(103\) 17.8564 1.75944 0.879722 0.475488i \(-0.157729\pi\)
0.879722 + 0.475488i \(0.157729\pi\)
\(104\) 4.73205 0.464016
\(105\) 0 0
\(106\) −4.39230 −0.426618
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.39230 0.612272 0.306136 0.951988i \(-0.400964\pi\)
0.306136 + 0.951988i \(0.400964\pi\)
\(110\) 0 0
\(111\) 4.19615 0.398281
\(112\) 3.66025 0.345861
\(113\) −5.07180 −0.477115 −0.238557 0.971128i \(-0.576674\pi\)
−0.238557 + 0.971128i \(0.576674\pi\)
\(114\) 1.73205 0.162221
\(115\) 0 0
\(116\) −2.19615 −0.203908
\(117\) 2.73205 0.252578
\(118\) −16.3923 −1.50903
\(119\) 0 0
\(120\) 0 0
\(121\) −9.39230 −0.853846
\(122\) 23.3205 2.11134
\(123\) −4.73205 −0.426675
\(124\) −4.92820 −0.442566
\(125\) 0 0
\(126\) 1.26795 0.112958
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) −12.1244 −1.07165
\(129\) −6.19615 −0.545541
\(130\) 0 0
\(131\) 15.1244 1.32142 0.660711 0.750641i \(-0.270255\pi\)
0.660711 + 0.750641i \(0.270255\pi\)
\(132\) −1.26795 −0.110361
\(133\) −0.732051 −0.0634769
\(134\) 13.8564 1.19701
\(135\) 0 0
\(136\) 0 0
\(137\) 7.85641 0.671218 0.335609 0.942001i \(-0.391058\pi\)
0.335609 + 0.942001i \(0.391058\pi\)
\(138\) 6.00000 0.510754
\(139\) 12.3923 1.05110 0.525551 0.850762i \(-0.323859\pi\)
0.525551 + 0.850762i \(0.323859\pi\)
\(140\) 0 0
\(141\) 3.46410 0.291730
\(142\) −28.3923 −2.38263
\(143\) 3.46410 0.289683
\(144\) −5.00000 −0.416667
\(145\) 0 0
\(146\) −5.32051 −0.440328
\(147\) 6.46410 0.533150
\(148\) −4.19615 −0.344922
\(149\) 7.85641 0.643622 0.321811 0.946804i \(-0.395708\pi\)
0.321811 + 0.946804i \(0.395708\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 1.73205 0.140488
\(153\) 0 0
\(154\) 1.60770 0.129552
\(155\) 0 0
\(156\) −2.73205 −0.218739
\(157\) 14.3923 1.14863 0.574315 0.818634i \(-0.305268\pi\)
0.574315 + 0.818634i \(0.305268\pi\)
\(158\) −5.07180 −0.403490
\(159\) −2.53590 −0.201110
\(160\) 0 0
\(161\) −2.53590 −0.199857
\(162\) −1.73205 −0.136083
\(163\) −12.7321 −0.997251 −0.498626 0.866817i \(-0.666162\pi\)
−0.498626 + 0.866817i \(0.666162\pi\)
\(164\) 4.73205 0.369511
\(165\) 0 0
\(166\) 1.60770 0.124781
\(167\) 3.46410 0.268060 0.134030 0.990977i \(-0.457208\pi\)
0.134030 + 0.990977i \(0.457208\pi\)
\(168\) 1.26795 0.0978244
\(169\) −5.53590 −0.425838
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 6.19615 0.472452
\(173\) 6.92820 0.526742 0.263371 0.964695i \(-0.415166\pi\)
0.263371 + 0.964695i \(0.415166\pi\)
\(174\) −3.80385 −0.288369
\(175\) 0 0
\(176\) −6.33975 −0.477876
\(177\) −9.46410 −0.711365
\(178\) −12.5885 −0.943545
\(179\) −11.3205 −0.846135 −0.423067 0.906098i \(-0.639047\pi\)
−0.423067 + 0.906098i \(0.639047\pi\)
\(180\) 0 0
\(181\) 18.3923 1.36709 0.683545 0.729909i \(-0.260437\pi\)
0.683545 + 0.729909i \(0.260437\pi\)
\(182\) 3.46410 0.256776
\(183\) 13.4641 0.995295
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) −8.53590 −0.625882
\(187\) 0 0
\(188\) −3.46410 −0.252646
\(189\) 0.732051 0.0532489
\(190\) 0 0
\(191\) 17.6603 1.27785 0.638926 0.769269i \(-0.279379\pi\)
0.638926 + 0.769269i \(0.279379\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 7.12436 0.512822 0.256411 0.966568i \(-0.417460\pi\)
0.256411 + 0.966568i \(0.417460\pi\)
\(194\) 7.26795 0.521808
\(195\) 0 0
\(196\) −6.46410 −0.461722
\(197\) 24.0000 1.70993 0.854965 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) −2.19615 −0.156074
\(199\) 19.3205 1.36959 0.684797 0.728734i \(-0.259891\pi\)
0.684797 + 0.728734i \(0.259891\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −18.0000 −1.26648
\(203\) 1.60770 0.112838
\(204\) 0 0
\(205\) 0 0
\(206\) −30.9282 −2.15487
\(207\) 3.46410 0.240772
\(208\) −13.6603 −0.947168
\(209\) 1.26795 0.0877059
\(210\) 0 0
\(211\) 14.9282 1.02770 0.513850 0.857880i \(-0.328219\pi\)
0.513850 + 0.857880i \(0.328219\pi\)
\(212\) 2.53590 0.174166
\(213\) −16.3923 −1.12318
\(214\) 0 0
\(215\) 0 0
\(216\) −1.73205 −0.117851
\(217\) 3.60770 0.244906
\(218\) −11.0718 −0.749877
\(219\) −3.07180 −0.207573
\(220\) 0 0
\(221\) 0 0
\(222\) −7.26795 −0.487793
\(223\) −9.85641 −0.660034 −0.330017 0.943975i \(-0.607054\pi\)
−0.330017 + 0.943975i \(0.607054\pi\)
\(224\) −3.80385 −0.254155
\(225\) 0 0
\(226\) 8.78461 0.584344
\(227\) 10.3923 0.689761 0.344881 0.938647i \(-0.387919\pi\)
0.344881 + 0.938647i \(0.387919\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −25.4641 −1.68272 −0.841358 0.540479i \(-0.818243\pi\)
−0.841358 + 0.540479i \(0.818243\pi\)
\(230\) 0 0
\(231\) 0.928203 0.0610713
\(232\) −3.80385 −0.249735
\(233\) −19.8564 −1.30084 −0.650418 0.759576i \(-0.725406\pi\)
−0.650418 + 0.759576i \(0.725406\pi\)
\(234\) −4.73205 −0.309344
\(235\) 0 0
\(236\) 9.46410 0.616061
\(237\) −2.92820 −0.190207
\(238\) 0 0
\(239\) −20.1962 −1.30638 −0.653190 0.757194i \(-0.726570\pi\)
−0.653190 + 0.757194i \(0.726570\pi\)
\(240\) 0 0
\(241\) −16.9282 −1.09044 −0.545221 0.838293i \(-0.683554\pi\)
−0.545221 + 0.838293i \(0.683554\pi\)
\(242\) 16.2679 1.04574
\(243\) −1.00000 −0.0641500
\(244\) −13.4641 −0.861951
\(245\) 0 0
\(246\) 8.19615 0.522568
\(247\) 2.73205 0.173836
\(248\) −8.53590 −0.542030
\(249\) 0.928203 0.0588225
\(250\) 0 0
\(251\) −10.0526 −0.634512 −0.317256 0.948340i \(-0.602761\pi\)
−0.317256 + 0.948340i \(0.602761\pi\)
\(252\) −0.732051 −0.0461149
\(253\) 4.39230 0.276142
\(254\) −6.92820 −0.434714
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 24.0000 1.49708 0.748539 0.663090i \(-0.230755\pi\)
0.748539 + 0.663090i \(0.230755\pi\)
\(258\) 10.7321 0.668148
\(259\) 3.07180 0.190872
\(260\) 0 0
\(261\) −2.19615 −0.135938
\(262\) −26.1962 −1.61840
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) −2.19615 −0.135164
\(265\) 0 0
\(266\) 1.26795 0.0777430
\(267\) −7.26795 −0.444791
\(268\) −8.00000 −0.488678
\(269\) −30.5885 −1.86501 −0.932506 0.361156i \(-0.882382\pi\)
−0.932506 + 0.361156i \(0.882382\pi\)
\(270\) 0 0
\(271\) −20.3923 −1.23874 −0.619372 0.785098i \(-0.712613\pi\)
−0.619372 + 0.785098i \(0.712613\pi\)
\(272\) 0 0
\(273\) 2.00000 0.121046
\(274\) −13.6077 −0.822071
\(275\) 0 0
\(276\) −3.46410 −0.208514
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −21.4641 −1.28733
\(279\) −4.92820 −0.295044
\(280\) 0 0
\(281\) 4.73205 0.282290 0.141145 0.989989i \(-0.454922\pi\)
0.141145 + 0.989989i \(0.454922\pi\)
\(282\) −6.00000 −0.357295
\(283\) 26.9808 1.60384 0.801920 0.597432i \(-0.203812\pi\)
0.801920 + 0.597432i \(0.203812\pi\)
\(284\) 16.3923 0.972704
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) −3.46410 −0.204479
\(288\) 5.19615 0.306186
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 4.19615 0.245983
\(292\) 3.07180 0.179763
\(293\) 27.7128 1.61900 0.809500 0.587120i \(-0.199738\pi\)
0.809500 + 0.587120i \(0.199738\pi\)
\(294\) −11.1962 −0.652973
\(295\) 0 0
\(296\) −7.26795 −0.422441
\(297\) −1.26795 −0.0735739
\(298\) −13.6077 −0.788273
\(299\) 9.46410 0.547323
\(300\) 0 0
\(301\) −4.53590 −0.261445
\(302\) −24.2487 −1.39536
\(303\) −10.3923 −0.597022
\(304\) −5.00000 −0.286770
\(305\) 0 0
\(306\) 0 0
\(307\) 11.6077 0.662486 0.331243 0.943545i \(-0.392532\pi\)
0.331243 + 0.943545i \(0.392532\pi\)
\(308\) −0.928203 −0.0528893
\(309\) −17.8564 −1.01582
\(310\) 0 0
\(311\) −26.4449 −1.49955 −0.749775 0.661692i \(-0.769838\pi\)
−0.749775 + 0.661692i \(0.769838\pi\)
\(312\) −4.73205 −0.267900
\(313\) 14.3923 0.813501 0.406751 0.913539i \(-0.366662\pi\)
0.406751 + 0.913539i \(0.366662\pi\)
\(314\) −24.9282 −1.40678
\(315\) 0 0
\(316\) 2.92820 0.164724
\(317\) −23.3205 −1.30981 −0.654905 0.755711i \(-0.727291\pi\)
−0.654905 + 0.755711i \(0.727291\pi\)
\(318\) 4.39230 0.246308
\(319\) −2.78461 −0.155908
\(320\) 0 0
\(321\) 0 0
\(322\) 4.39230 0.244774
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 22.0526 1.22138
\(327\) −6.39230 −0.353495
\(328\) 8.19615 0.452557
\(329\) 2.53590 0.139809
\(330\) 0 0
\(331\) 29.7128 1.63316 0.816582 0.577230i \(-0.195866\pi\)
0.816582 + 0.577230i \(0.195866\pi\)
\(332\) −0.928203 −0.0509418
\(333\) −4.19615 −0.229948
\(334\) −6.00000 −0.328305
\(335\) 0 0
\(336\) −3.66025 −0.199683
\(337\) 19.1244 1.04177 0.520885 0.853627i \(-0.325602\pi\)
0.520885 + 0.853627i \(0.325602\pi\)
\(338\) 9.58846 0.521543
\(339\) 5.07180 0.275462
\(340\) 0 0
\(341\) −6.24871 −0.338387
\(342\) −1.73205 −0.0936586
\(343\) 9.85641 0.532196
\(344\) 10.7321 0.578633
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) −12.9282 −0.694022 −0.347011 0.937861i \(-0.612803\pi\)
−0.347011 + 0.937861i \(0.612803\pi\)
\(348\) 2.19615 0.117726
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) −2.73205 −0.145826
\(352\) 6.58846 0.351166
\(353\) −26.7846 −1.42560 −0.712800 0.701367i \(-0.752573\pi\)
−0.712800 + 0.701367i \(0.752573\pi\)
\(354\) 16.3923 0.871241
\(355\) 0 0
\(356\) 7.26795 0.385201
\(357\) 0 0
\(358\) 19.6077 1.03630
\(359\) 17.6603 0.932073 0.466036 0.884766i \(-0.345682\pi\)
0.466036 + 0.884766i \(0.345682\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −31.8564 −1.67434
\(363\) 9.39230 0.492968
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) −23.3205 −1.21898
\(367\) −5.80385 −0.302958 −0.151479 0.988460i \(-0.548404\pi\)
−0.151479 + 0.988460i \(0.548404\pi\)
\(368\) −17.3205 −0.902894
\(369\) 4.73205 0.246341
\(370\) 0 0
\(371\) −1.85641 −0.0963798
\(372\) 4.92820 0.255515
\(373\) −4.19615 −0.217269 −0.108634 0.994082i \(-0.534648\pi\)
−0.108634 + 0.994082i \(0.534648\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −6.00000 −0.309016
\(378\) −1.26795 −0.0652163
\(379\) 7.07180 0.363254 0.181627 0.983368i \(-0.441864\pi\)
0.181627 + 0.983368i \(0.441864\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) −30.5885 −1.56504
\(383\) 30.9282 1.58036 0.790179 0.612877i \(-0.209988\pi\)
0.790179 + 0.612877i \(0.209988\pi\)
\(384\) 12.1244 0.618718
\(385\) 0 0
\(386\) −12.3397 −0.628077
\(387\) 6.19615 0.314968
\(388\) −4.19615 −0.213027
\(389\) −19.8564 −1.00676 −0.503380 0.864065i \(-0.667910\pi\)
−0.503380 + 0.864065i \(0.667910\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −11.1962 −0.565491
\(393\) −15.1244 −0.762923
\(394\) −41.5692 −2.09423
\(395\) 0 0
\(396\) 1.26795 0.0637168
\(397\) 4.92820 0.247339 0.123670 0.992323i \(-0.460534\pi\)
0.123670 + 0.992323i \(0.460534\pi\)
\(398\) −33.4641 −1.67740
\(399\) 0.732051 0.0366484
\(400\) 0 0
\(401\) 4.05256 0.202375 0.101188 0.994867i \(-0.467736\pi\)
0.101188 + 0.994867i \(0.467736\pi\)
\(402\) −13.8564 −0.691095
\(403\) −13.4641 −0.670695
\(404\) 10.3923 0.517036
\(405\) 0 0
\(406\) −2.78461 −0.138198
\(407\) −5.32051 −0.263728
\(408\) 0 0
\(409\) −5.60770 −0.277283 −0.138641 0.990343i \(-0.544274\pi\)
−0.138641 + 0.990343i \(0.544274\pi\)
\(410\) 0 0
\(411\) −7.85641 −0.387528
\(412\) 17.8564 0.879722
\(413\) −6.92820 −0.340915
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) 14.1962 0.696024
\(417\) −12.3923 −0.606854
\(418\) −2.19615 −0.107417
\(419\) 10.0526 0.491100 0.245550 0.969384i \(-0.421032\pi\)
0.245550 + 0.969384i \(0.421032\pi\)
\(420\) 0 0
\(421\) 22.7846 1.11045 0.555227 0.831699i \(-0.312631\pi\)
0.555227 + 0.831699i \(0.312631\pi\)
\(422\) −25.8564 −1.25867
\(423\) −3.46410 −0.168430
\(424\) 4.39230 0.213309
\(425\) 0 0
\(426\) 28.3923 1.37561
\(427\) 9.85641 0.476985
\(428\) 0 0
\(429\) −3.46410 −0.167248
\(430\) 0 0
\(431\) 23.3205 1.12331 0.561655 0.827372i \(-0.310165\pi\)
0.561655 + 0.827372i \(0.310165\pi\)
\(432\) 5.00000 0.240563
\(433\) −20.5885 −0.989418 −0.494709 0.869059i \(-0.664725\pi\)
−0.494709 + 0.869059i \(0.664725\pi\)
\(434\) −6.24871 −0.299948
\(435\) 0 0
\(436\) 6.39230 0.306136
\(437\) 3.46410 0.165710
\(438\) 5.32051 0.254224
\(439\) 13.0718 0.623883 0.311941 0.950101i \(-0.399021\pi\)
0.311941 + 0.950101i \(0.399021\pi\)
\(440\) 0 0
\(441\) −6.46410 −0.307814
\(442\) 0 0
\(443\) −29.3205 −1.39306 −0.696530 0.717528i \(-0.745274\pi\)
−0.696530 + 0.717528i \(0.745274\pi\)
\(444\) 4.19615 0.199141
\(445\) 0 0
\(446\) 17.0718 0.808373
\(447\) −7.85641 −0.371595
\(448\) −0.732051 −0.0345861
\(449\) 11.6603 0.550281 0.275141 0.961404i \(-0.411276\pi\)
0.275141 + 0.961404i \(0.411276\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) −5.07180 −0.238557
\(453\) −14.0000 −0.657777
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) −1.73205 −0.0811107
\(457\) −11.4641 −0.536268 −0.268134 0.963382i \(-0.586407\pi\)
−0.268134 + 0.963382i \(0.586407\pi\)
\(458\) 44.1051 2.06090
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) −1.60770 −0.0747967
\(463\) −9.51666 −0.442277 −0.221138 0.975242i \(-0.570977\pi\)
−0.221138 + 0.975242i \(0.570977\pi\)
\(464\) 10.9808 0.509769
\(465\) 0 0
\(466\) 34.3923 1.59319
\(467\) −27.4641 −1.27089 −0.635444 0.772147i \(-0.719183\pi\)
−0.635444 + 0.772147i \(0.719183\pi\)
\(468\) 2.73205 0.126289
\(469\) 5.85641 0.270424
\(470\) 0 0
\(471\) −14.3923 −0.663162
\(472\) 16.3923 0.754517
\(473\) 7.85641 0.361238
\(474\) 5.07180 0.232955
\(475\) 0 0
\(476\) 0 0
\(477\) 2.53590 0.116111
\(478\) 34.9808 1.59998
\(479\) −19.5167 −0.891739 −0.445869 0.895098i \(-0.647105\pi\)
−0.445869 + 0.895098i \(0.647105\pi\)
\(480\) 0 0
\(481\) −11.4641 −0.522718
\(482\) 29.3205 1.33551
\(483\) 2.53590 0.115387
\(484\) −9.39230 −0.426923
\(485\) 0 0
\(486\) 1.73205 0.0785674
\(487\) 11.6077 0.525995 0.262997 0.964797i \(-0.415289\pi\)
0.262997 + 0.964797i \(0.415289\pi\)
\(488\) −23.3205 −1.05567
\(489\) 12.7321 0.575763
\(490\) 0 0
\(491\) −22.0526 −0.995218 −0.497609 0.867401i \(-0.665789\pi\)
−0.497609 + 0.867401i \(0.665789\pi\)
\(492\) −4.73205 −0.213337
\(493\) 0 0
\(494\) −4.73205 −0.212905
\(495\) 0 0
\(496\) 24.6410 1.10641
\(497\) −12.0000 −0.538274
\(498\) −1.60770 −0.0720425
\(499\) 17.4641 0.781801 0.390900 0.920433i \(-0.372164\pi\)
0.390900 + 0.920433i \(0.372164\pi\)
\(500\) 0 0
\(501\) −3.46410 −0.154765
\(502\) 17.4115 0.777115
\(503\) 36.9282 1.64655 0.823274 0.567645i \(-0.192145\pi\)
0.823274 + 0.567645i \(0.192145\pi\)
\(504\) −1.26795 −0.0564789
\(505\) 0 0
\(506\) −7.60770 −0.338203
\(507\) 5.53590 0.245858
\(508\) 4.00000 0.177471
\(509\) 28.0526 1.24341 0.621704 0.783252i \(-0.286441\pi\)
0.621704 + 0.783252i \(0.286441\pi\)
\(510\) 0 0
\(511\) −2.24871 −0.0994771
\(512\) −8.66025 −0.382733
\(513\) −1.00000 −0.0441511
\(514\) −41.5692 −1.83354
\(515\) 0 0
\(516\) −6.19615 −0.272770
\(517\) −4.39230 −0.193173
\(518\) −5.32051 −0.233770
\(519\) −6.92820 −0.304114
\(520\) 0 0
\(521\) 40.7321 1.78450 0.892252 0.451538i \(-0.149125\pi\)
0.892252 + 0.451538i \(0.149125\pi\)
\(522\) 3.80385 0.166490
\(523\) −43.3205 −1.89427 −0.947137 0.320830i \(-0.896038\pi\)
−0.947137 + 0.320830i \(0.896038\pi\)
\(524\) 15.1244 0.660711
\(525\) 0 0
\(526\) 10.3923 0.453126
\(527\) 0 0
\(528\) 6.33975 0.275902
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 9.46410 0.410707
\(532\) −0.732051 −0.0317384
\(533\) 12.9282 0.559983
\(534\) 12.5885 0.544756
\(535\) 0 0
\(536\) −13.8564 −0.598506
\(537\) 11.3205 0.488516
\(538\) 52.9808 2.28416
\(539\) −8.19615 −0.353033
\(540\) 0 0
\(541\) −13.7128 −0.589560 −0.294780 0.955565i \(-0.595246\pi\)
−0.294780 + 0.955565i \(0.595246\pi\)
\(542\) 35.3205 1.51715
\(543\) −18.3923 −0.789289
\(544\) 0 0
\(545\) 0 0
\(546\) −3.46410 −0.148250
\(547\) −8.67949 −0.371108 −0.185554 0.982634i \(-0.559408\pi\)
−0.185554 + 0.982634i \(0.559408\pi\)
\(548\) 7.85641 0.335609
\(549\) −13.4641 −0.574634
\(550\) 0 0
\(551\) −2.19615 −0.0935592
\(552\) −6.00000 −0.255377
\(553\) −2.14359 −0.0911549
\(554\) 3.46410 0.147176
\(555\) 0 0
\(556\) 12.3923 0.525551
\(557\) 12.9282 0.547786 0.273893 0.961760i \(-0.411689\pi\)
0.273893 + 0.961760i \(0.411689\pi\)
\(558\) 8.53590 0.361353
\(559\) 16.9282 0.715987
\(560\) 0 0
\(561\) 0 0
\(562\) −8.19615 −0.345734
\(563\) 20.5359 0.865485 0.432742 0.901518i \(-0.357546\pi\)
0.432742 + 0.901518i \(0.357546\pi\)
\(564\) 3.46410 0.145865
\(565\) 0 0
\(566\) −46.7321 −1.96429
\(567\) −0.732051 −0.0307432
\(568\) 28.3923 1.19131
\(569\) −16.0526 −0.672958 −0.336479 0.941691i \(-0.609236\pi\)
−0.336479 + 0.941691i \(0.609236\pi\)
\(570\) 0 0
\(571\) 14.2487 0.596290 0.298145 0.954521i \(-0.403632\pi\)
0.298145 + 0.954521i \(0.403632\pi\)
\(572\) 3.46410 0.144841
\(573\) −17.6603 −0.737768
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 47.1769 1.96400 0.982000 0.188879i \(-0.0604855\pi\)
0.982000 + 0.188879i \(0.0604855\pi\)
\(578\) 29.4449 1.22474
\(579\) −7.12436 −0.296078
\(580\) 0 0
\(581\) 0.679492 0.0281901
\(582\) −7.26795 −0.301266
\(583\) 3.21539 0.133168
\(584\) 5.32051 0.220164
\(585\) 0 0
\(586\) −48.0000 −1.98286
\(587\) −3.46410 −0.142979 −0.0714894 0.997441i \(-0.522775\pi\)
−0.0714894 + 0.997441i \(0.522775\pi\)
\(588\) 6.46410 0.266575
\(589\) −4.92820 −0.203063
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) 20.9808 0.862304
\(593\) −2.78461 −0.114350 −0.0571751 0.998364i \(-0.518209\pi\)
−0.0571751 + 0.998364i \(0.518209\pi\)
\(594\) 2.19615 0.0901092
\(595\) 0 0
\(596\) 7.85641 0.321811
\(597\) −19.3205 −0.790736
\(598\) −16.3923 −0.670331
\(599\) 13.8564 0.566157 0.283079 0.959097i \(-0.408644\pi\)
0.283079 + 0.959097i \(0.408644\pi\)
\(600\) 0 0
\(601\) 15.1769 0.619079 0.309540 0.950887i \(-0.399825\pi\)
0.309540 + 0.950887i \(0.399825\pi\)
\(602\) 7.85641 0.320203
\(603\) −8.00000 −0.325785
\(604\) 14.0000 0.569652
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) 32.3923 1.31476 0.657382 0.753558i \(-0.271664\pi\)
0.657382 + 0.753558i \(0.271664\pi\)
\(608\) 5.19615 0.210732
\(609\) −1.60770 −0.0651471
\(610\) 0 0
\(611\) −9.46410 −0.382877
\(612\) 0 0
\(613\) −21.6077 −0.872727 −0.436363 0.899771i \(-0.643734\pi\)
−0.436363 + 0.899771i \(0.643734\pi\)
\(614\) −20.1051 −0.811377
\(615\) 0 0
\(616\) −1.60770 −0.0647759
\(617\) 27.7128 1.11568 0.557838 0.829950i \(-0.311631\pi\)
0.557838 + 0.829950i \(0.311631\pi\)
\(618\) 30.9282 1.24411
\(619\) 19.3205 0.776557 0.388278 0.921542i \(-0.373070\pi\)
0.388278 + 0.921542i \(0.373070\pi\)
\(620\) 0 0
\(621\) −3.46410 −0.139010
\(622\) 45.8038 1.83657
\(623\) −5.32051 −0.213162
\(624\) 13.6603 0.546848
\(625\) 0 0
\(626\) −24.9282 −0.996331
\(627\) −1.26795 −0.0506370
\(628\) 14.3923 0.574315
\(629\) 0 0
\(630\) 0 0
\(631\) −21.0718 −0.838855 −0.419427 0.907789i \(-0.637769\pi\)
−0.419427 + 0.907789i \(0.637769\pi\)
\(632\) 5.07180 0.201745
\(633\) −14.9282 −0.593343
\(634\) 40.3923 1.60418
\(635\) 0 0
\(636\) −2.53590 −0.100555
\(637\) −17.6603 −0.699725
\(638\) 4.82309 0.190948
\(639\) 16.3923 0.648470
\(640\) 0 0
\(641\) 17.4115 0.687715 0.343857 0.939022i \(-0.388266\pi\)
0.343857 + 0.939022i \(0.388266\pi\)
\(642\) 0 0
\(643\) 1.80385 0.0711368 0.0355684 0.999367i \(-0.488676\pi\)
0.0355684 + 0.999367i \(0.488676\pi\)
\(644\) −2.53590 −0.0999284
\(645\) 0 0
\(646\) 0 0
\(647\) 31.8564 1.25240 0.626202 0.779661i \(-0.284608\pi\)
0.626202 + 0.779661i \(0.284608\pi\)
\(648\) 1.73205 0.0680414
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) −3.60770 −0.141397
\(652\) −12.7321 −0.498626
\(653\) −30.9282 −1.21031 −0.605157 0.796106i \(-0.706890\pi\)
−0.605157 + 0.796106i \(0.706890\pi\)
\(654\) 11.0718 0.432942
\(655\) 0 0
\(656\) −23.6603 −0.923778
\(657\) 3.07180 0.119842
\(658\) −4.39230 −0.171230
\(659\) 18.9282 0.737338 0.368669 0.929561i \(-0.379814\pi\)
0.368669 + 0.929561i \(0.379814\pi\)
\(660\) 0 0
\(661\) −23.1769 −0.901477 −0.450739 0.892656i \(-0.648839\pi\)
−0.450739 + 0.892656i \(0.648839\pi\)
\(662\) −51.4641 −2.00021
\(663\) 0 0
\(664\) −1.60770 −0.0623907
\(665\) 0 0
\(666\) 7.26795 0.281627
\(667\) −7.60770 −0.294571
\(668\) 3.46410 0.134030
\(669\) 9.85641 0.381071
\(670\) 0 0
\(671\) −17.0718 −0.659049
\(672\) 3.80385 0.146737
\(673\) 7.12436 0.274624 0.137312 0.990528i \(-0.456154\pi\)
0.137312 + 0.990528i \(0.456154\pi\)
\(674\) −33.1244 −1.27590
\(675\) 0 0
\(676\) −5.53590 −0.212919
\(677\) −35.3205 −1.35748 −0.678739 0.734380i \(-0.737473\pi\)
−0.678739 + 0.734380i \(0.737473\pi\)
\(678\) −8.78461 −0.337371
\(679\) 3.07180 0.117885
\(680\) 0 0
\(681\) −10.3923 −0.398234
\(682\) 10.8231 0.414437
\(683\) 18.9282 0.724268 0.362134 0.932126i \(-0.382048\pi\)
0.362134 + 0.932126i \(0.382048\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) −17.0718 −0.651804
\(687\) 25.4641 0.971516
\(688\) −30.9808 −1.18113
\(689\) 6.92820 0.263944
\(690\) 0 0
\(691\) −8.39230 −0.319258 −0.159629 0.987177i \(-0.551030\pi\)
−0.159629 + 0.987177i \(0.551030\pi\)
\(692\) 6.92820 0.263371
\(693\) −0.928203 −0.0352595
\(694\) 22.3923 0.850000
\(695\) 0 0
\(696\) 3.80385 0.144184
\(697\) 0 0
\(698\) 38.1051 1.44230
\(699\) 19.8564 0.751038
\(700\) 0 0
\(701\) 21.7128 0.820082 0.410041 0.912067i \(-0.365514\pi\)
0.410041 + 0.912067i \(0.365514\pi\)
\(702\) 4.73205 0.178600
\(703\) −4.19615 −0.158261
\(704\) 1.26795 0.0477876
\(705\) 0 0
\(706\) 46.3923 1.74600
\(707\) −7.60770 −0.286117
\(708\) −9.46410 −0.355683
\(709\) 33.1769 1.24599 0.622993 0.782228i \(-0.285917\pi\)
0.622993 + 0.782228i \(0.285917\pi\)
\(710\) 0 0
\(711\) 2.92820 0.109816
\(712\) 12.5885 0.471772
\(713\) −17.0718 −0.639344
\(714\) 0 0
\(715\) 0 0
\(716\) −11.3205 −0.423067
\(717\) 20.1962 0.754239
\(718\) −30.5885 −1.14155
\(719\) 5.66025 0.211092 0.105546 0.994414i \(-0.466341\pi\)
0.105546 + 0.994414i \(0.466341\pi\)
\(720\) 0 0
\(721\) −13.0718 −0.486819
\(722\) −1.73205 −0.0644603
\(723\) 16.9282 0.629567
\(724\) 18.3923 0.683545
\(725\) 0 0
\(726\) −16.2679 −0.603760
\(727\) −8.33975 −0.309304 −0.154652 0.987969i \(-0.549426\pi\)
−0.154652 + 0.987969i \(0.549426\pi\)
\(728\) −3.46410 −0.128388
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 13.4641 0.497648
\(733\) −22.7846 −0.841569 −0.420784 0.907161i \(-0.638245\pi\)
−0.420784 + 0.907161i \(0.638245\pi\)
\(734\) 10.0526 0.371047
\(735\) 0 0
\(736\) 18.0000 0.663489
\(737\) −10.1436 −0.373644
\(738\) −8.19615 −0.301705
\(739\) 33.8564 1.24543 0.622714 0.782450i \(-0.286030\pi\)
0.622714 + 0.782450i \(0.286030\pi\)
\(740\) 0 0
\(741\) −2.73205 −0.100364
\(742\) 3.21539 0.118041
\(743\) 44.7846 1.64299 0.821494 0.570217i \(-0.193141\pi\)
0.821494 + 0.570217i \(0.193141\pi\)
\(744\) 8.53590 0.312941
\(745\) 0 0
\(746\) 7.26795 0.266099
\(747\) −0.928203 −0.0339612
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) 17.3205 0.631614
\(753\) 10.0526 0.366336
\(754\) 10.3923 0.378465
\(755\) 0 0
\(756\) 0.732051 0.0266244
\(757\) 16.2487 0.590569 0.295285 0.955409i \(-0.404585\pi\)
0.295285 + 0.955409i \(0.404585\pi\)
\(758\) −12.2487 −0.444893
\(759\) −4.39230 −0.159431
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 6.92820 0.250982
\(763\) −4.67949 −0.169409
\(764\) 17.6603 0.638926
\(765\) 0 0
\(766\) −53.5692 −1.93553
\(767\) 25.8564 0.933621
\(768\) −19.0000 −0.685603
\(769\) 48.6410 1.75404 0.877020 0.480454i \(-0.159528\pi\)
0.877020 + 0.480454i \(0.159528\pi\)
\(770\) 0 0
\(771\) −24.0000 −0.864339
\(772\) 7.12436 0.256411
\(773\) −37.1769 −1.33716 −0.668580 0.743640i \(-0.733098\pi\)
−0.668580 + 0.743640i \(0.733098\pi\)
\(774\) −10.7321 −0.385756
\(775\) 0 0
\(776\) −7.26795 −0.260904
\(777\) −3.07180 −0.110200
\(778\) 34.3923 1.23302
\(779\) 4.73205 0.169543
\(780\) 0 0
\(781\) 20.7846 0.743732
\(782\) 0 0
\(783\) 2.19615 0.0784841
\(784\) 32.3205 1.15430
\(785\) 0 0
\(786\) 26.1962 0.934386
\(787\) −43.3205 −1.54421 −0.772105 0.635495i \(-0.780796\pi\)
−0.772105 + 0.635495i \(0.780796\pi\)
\(788\) 24.0000 0.854965
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) 3.71281 0.132012
\(792\) 2.19615 0.0780369
\(793\) −36.7846 −1.30626
\(794\) −8.53590 −0.302928
\(795\) 0 0
\(796\) 19.3205 0.684797
\(797\) −3.21539 −0.113895 −0.0569475 0.998377i \(-0.518137\pi\)
−0.0569475 + 0.998377i \(0.518137\pi\)
\(798\) −1.26795 −0.0448849
\(799\) 0 0
\(800\) 0 0
\(801\) 7.26795 0.256800
\(802\) −7.01924 −0.247858
\(803\) 3.89488 0.137447
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 23.3205 0.821430
\(807\) 30.5885 1.07676
\(808\) 18.0000 0.633238
\(809\) −26.7846 −0.941697 −0.470848 0.882214i \(-0.656052\pi\)
−0.470848 + 0.882214i \(0.656052\pi\)
\(810\) 0 0
\(811\) −45.5692 −1.60015 −0.800076 0.599899i \(-0.795207\pi\)
−0.800076 + 0.599899i \(0.795207\pi\)
\(812\) 1.60770 0.0564190
\(813\) 20.3923 0.715189
\(814\) 9.21539 0.322999
\(815\) 0 0
\(816\) 0 0
\(817\) 6.19615 0.216776
\(818\) 9.71281 0.339601
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −39.4641 −1.37731 −0.688653 0.725091i \(-0.741798\pi\)
−0.688653 + 0.725091i \(0.741798\pi\)
\(822\) 13.6077 0.474623
\(823\) 38.9808 1.35878 0.679392 0.733776i \(-0.262244\pi\)
0.679392 + 0.733776i \(0.262244\pi\)
\(824\) 30.9282 1.07744
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) −29.3205 −1.01957 −0.509787 0.860301i \(-0.670276\pi\)
−0.509787 + 0.860301i \(0.670276\pi\)
\(828\) 3.46410 0.120386
\(829\) −42.1051 −1.46237 −0.731186 0.682179i \(-0.761033\pi\)
−0.731186 + 0.682179i \(0.761033\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 2.73205 0.0947168
\(833\) 0 0
\(834\) 21.4641 0.743241
\(835\) 0 0
\(836\) 1.26795 0.0438529
\(837\) 4.92820 0.170344
\(838\) −17.4115 −0.601472
\(839\) −40.3923 −1.39450 −0.697249 0.716829i \(-0.745593\pi\)
−0.697249 + 0.716829i \(0.745593\pi\)
\(840\) 0 0
\(841\) −24.1769 −0.833687
\(842\) −39.4641 −1.36002
\(843\) −4.73205 −0.162980
\(844\) 14.9282 0.513850
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) 6.87564 0.236250
\(848\) −12.6795 −0.435416
\(849\) −26.9808 −0.925977
\(850\) 0 0
\(851\) −14.5359 −0.498284
\(852\) −16.3923 −0.561591
\(853\) 35.1769 1.20443 0.602217 0.798332i \(-0.294284\pi\)
0.602217 + 0.798332i \(0.294284\pi\)
\(854\) −17.0718 −0.584185
\(855\) 0 0
\(856\) 0 0
\(857\) −6.24871 −0.213452 −0.106726 0.994288i \(-0.534037\pi\)
−0.106726 + 0.994288i \(0.534037\pi\)
\(858\) 6.00000 0.204837
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) 0 0
\(861\) 3.46410 0.118056
\(862\) −40.3923 −1.37577
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) −5.19615 −0.176777
\(865\) 0 0
\(866\) 35.6603 1.21178
\(867\) 17.0000 0.577350
\(868\) 3.60770 0.122453
\(869\) 3.71281 0.125949
\(870\) 0 0
\(871\) −21.8564 −0.740576
\(872\) 11.0718 0.374938
\(873\) −4.19615 −0.142018
\(874\) −6.00000 −0.202953
\(875\) 0 0
\(876\) −3.07180 −0.103786
\(877\) −28.8756 −0.975061 −0.487531 0.873106i \(-0.662102\pi\)
−0.487531 + 0.873106i \(0.662102\pi\)
\(878\) −22.6410 −0.764097
\(879\) −27.7128 −0.934730
\(880\) 0 0
\(881\) 15.4641 0.520999 0.260499 0.965474i \(-0.416113\pi\)
0.260499 + 0.965474i \(0.416113\pi\)
\(882\) 11.1962 0.376994
\(883\) 14.9808 0.504143 0.252071 0.967709i \(-0.418888\pi\)
0.252071 + 0.967709i \(0.418888\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 50.7846 1.70614
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 7.26795 0.243896
\(889\) −2.92820 −0.0982088
\(890\) 0 0
\(891\) 1.26795 0.0424779
\(892\) −9.85641 −0.330017
\(893\) −3.46410 −0.115922
\(894\) 13.6077 0.455109
\(895\) 0 0
\(896\) 8.87564 0.296514
\(897\) −9.46410 −0.315997
\(898\) −20.1962 −0.673954
\(899\) 10.8231 0.360970
\(900\) 0 0
\(901\) 0 0
\(902\) −10.3923 −0.346026
\(903\) 4.53590 0.150945
\(904\) −8.78461 −0.292172
\(905\) 0 0
\(906\) 24.2487 0.805609
\(907\) 11.6077 0.385427 0.192714 0.981255i \(-0.438271\pi\)
0.192714 + 0.981255i \(0.438271\pi\)
\(908\) 10.3923 0.344881
\(909\) 10.3923 0.344691
\(910\) 0 0
\(911\) −41.0718 −1.36077 −0.680385 0.732855i \(-0.738187\pi\)
−0.680385 + 0.732855i \(0.738187\pi\)
\(912\) 5.00000 0.165567
\(913\) −1.17691 −0.0389502
\(914\) 19.8564 0.656792
\(915\) 0 0
\(916\) −25.4641 −0.841358
\(917\) −11.0718 −0.365623
\(918\) 0 0
\(919\) −59.4256 −1.96027 −0.980135 0.198330i \(-0.936448\pi\)
−0.980135 + 0.198330i \(0.936448\pi\)
\(920\) 0 0
\(921\) −11.6077 −0.382487
\(922\) −10.3923 −0.342252
\(923\) 44.7846 1.47410
\(924\) 0.928203 0.0305356
\(925\) 0 0
\(926\) 16.4833 0.541676
\(927\) 17.8564 0.586481
\(928\) −11.4115 −0.374602
\(929\) −22.3923 −0.734668 −0.367334 0.930089i \(-0.619729\pi\)
−0.367334 + 0.930089i \(0.619729\pi\)
\(930\) 0 0
\(931\) −6.46410 −0.211852
\(932\) −19.8564 −0.650418
\(933\) 26.4449 0.865766
\(934\) 47.5692 1.55651
\(935\) 0 0
\(936\) 4.73205 0.154672
\(937\) −32.2487 −1.05352 −0.526760 0.850014i \(-0.676593\pi\)
−0.526760 + 0.850014i \(0.676593\pi\)
\(938\) −10.1436 −0.331200
\(939\) −14.3923 −0.469675
\(940\) 0 0
\(941\) −30.5885 −0.997155 −0.498578 0.866845i \(-0.666144\pi\)
−0.498578 + 0.866845i \(0.666144\pi\)
\(942\) 24.9282 0.812205
\(943\) 16.3923 0.533807
\(944\) −47.3205 −1.54015
\(945\) 0 0
\(946\) −13.6077 −0.442424
\(947\) −55.8564 −1.81509 −0.907545 0.419956i \(-0.862046\pi\)
−0.907545 + 0.419956i \(0.862046\pi\)
\(948\) −2.92820 −0.0951036
\(949\) 8.39230 0.272426
\(950\) 0 0
\(951\) 23.3205 0.756219
\(952\) 0 0
\(953\) −10.1436 −0.328583 −0.164292 0.986412i \(-0.552534\pi\)
−0.164292 + 0.986412i \(0.552534\pi\)
\(954\) −4.39230 −0.142206
\(955\) 0 0
\(956\) −20.1962 −0.653190
\(957\) 2.78461 0.0900136
\(958\) 33.8038 1.09215
\(959\) −5.75129 −0.185719
\(960\) 0 0
\(961\) −6.71281 −0.216542
\(962\) 19.8564 0.640196
\(963\) 0 0
\(964\) −16.9282 −0.545221
\(965\) 0 0
\(966\) −4.39230 −0.141320
\(967\) −29.1244 −0.936576 −0.468288 0.883576i \(-0.655129\pi\)
−0.468288 + 0.883576i \(0.655129\pi\)
\(968\) −16.2679 −0.522872
\(969\) 0 0
\(970\) 0 0
\(971\) 27.7128 0.889346 0.444673 0.895693i \(-0.353320\pi\)
0.444673 + 0.895693i \(0.353320\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −9.07180 −0.290828
\(974\) −20.1051 −0.644210
\(975\) 0 0
\(976\) 67.3205 2.15488
\(977\) −51.0333 −1.63270 −0.816350 0.577557i \(-0.804006\pi\)
−0.816350 + 0.577557i \(0.804006\pi\)
\(978\) −22.0526 −0.705163
\(979\) 9.21539 0.294525
\(980\) 0 0
\(981\) 6.39230 0.204091
\(982\) 38.1962 1.21889
\(983\) 6.67949 0.213043 0.106521 0.994310i \(-0.466029\pi\)
0.106521 + 0.994310i \(0.466029\pi\)
\(984\) −8.19615 −0.261284
\(985\) 0 0
\(986\) 0 0
\(987\) −2.53590 −0.0807185
\(988\) 2.73205 0.0869181
\(989\) 21.4641 0.682519
\(990\) 0 0
\(991\) 26.9282 0.855403 0.427701 0.903920i \(-0.359324\pi\)
0.427701 + 0.903920i \(0.359324\pi\)
\(992\) −25.6077 −0.813045
\(993\) −29.7128 −0.942908
\(994\) 20.7846 0.659248
\(995\) 0 0
\(996\) 0.928203 0.0294112
\(997\) 38.3923 1.21590 0.607948 0.793977i \(-0.291993\pi\)
0.607948 + 0.793977i \(0.291993\pi\)
\(998\) −30.2487 −0.957506
\(999\) 4.19615 0.132760
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 1425.2.a.o.1.1 2
3.2 odd 2 4275.2.a.t.1.2 2
5.2 odd 4 1425.2.c.k.799.2 4
5.3 odd 4 1425.2.c.k.799.3 4
5.4 even 2 285.2.a.e.1.2 2
15.14 odd 2 855.2.a.f.1.1 2
20.19 odd 2 4560.2.a.bh.1.1 2
95.94 odd 2 5415.2.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.e.1.2 2 5.4 even 2
855.2.a.f.1.1 2 15.14 odd 2
1425.2.a.o.1.1 2 1.1 even 1 trivial
1425.2.c.k.799.2 4 5.2 odd 4
1425.2.c.k.799.3 4 5.3 odd 4
4275.2.a.t.1.2 2 3.2 odd 2
4560.2.a.bh.1.1 2 20.19 odd 2
5415.2.a.r.1.1 2 95.94 odd 2