Properties

Label 1425.2.c.k.799.3
Level $1425$
Weight $2$
Character 1425.799
Analytic conductor $11.379$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1425,2,Mod(799,1425)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1425.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 285)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.3
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1425.799
Dual form 1425.2.c.k.799.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.73205 q^{6} +0.732051i q^{7} +1.73205i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.73205 q^{6} +0.732051i q^{7} +1.73205i q^{8} -1.00000 q^{9} +1.26795 q^{11} +1.00000i q^{12} +2.73205i q^{13} -1.26795 q^{14} -5.00000 q^{16} -1.73205i q^{18} -1.00000 q^{19} +0.732051 q^{21} +2.19615i q^{22} +3.46410i q^{23} +1.73205 q^{24} -4.73205 q^{26} +1.00000i q^{27} -0.732051i q^{28} +2.19615 q^{29} -4.92820 q^{31} -5.19615i q^{32} -1.26795i q^{33} +1.00000 q^{36} +4.19615i q^{37} -1.73205i q^{38} +2.73205 q^{39} +4.73205 q^{41} +1.26795i q^{42} +6.19615i q^{43} -1.26795 q^{44} -6.00000 q^{46} +3.46410i q^{47} +5.00000i q^{48} +6.46410 q^{49} -2.73205i q^{52} +2.53590i q^{53} -1.73205 q^{54} -1.26795 q^{56} +1.00000i q^{57} +3.80385i q^{58} -9.46410 q^{59} -13.4641 q^{61} -8.53590i q^{62} -0.732051i q^{63} -1.00000 q^{64} +2.19615 q^{66} +8.00000i q^{67} +3.46410 q^{69} +16.3923 q^{71} -1.73205i q^{72} +3.07180i q^{73} -7.26795 q^{74} +1.00000 q^{76} +0.928203i q^{77} +4.73205i q^{78} -2.92820 q^{79} +1.00000 q^{81} +8.19615i q^{82} -0.928203i q^{83} -0.732051 q^{84} -10.7321 q^{86} -2.19615i q^{87} +2.19615i q^{88} -7.26795 q^{89} -2.00000 q^{91} -3.46410i q^{92} +4.92820i q^{93} -6.00000 q^{94} -5.19615 q^{96} +4.19615i q^{97} +11.1962i q^{98} -1.26795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{9} + 12 q^{11} - 12 q^{14} - 20 q^{16} - 4 q^{19} - 4 q^{21} - 12 q^{26} - 12 q^{29} + 8 q^{31} + 4 q^{36} + 4 q^{39} + 12 q^{41} - 12 q^{44} - 24 q^{46} + 12 q^{49} - 12 q^{56} - 24 q^{59} - 40 q^{61} - 4 q^{64} - 12 q^{66} + 24 q^{71} - 36 q^{74} + 4 q^{76} + 16 q^{79} + 4 q^{81} + 4 q^{84} - 36 q^{86} - 36 q^{89} - 8 q^{91} - 24 q^{94} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205i 1.22474i 0.790569 + 0.612372i \(0.209785\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.73205 0.707107
\(7\) 0.732051i 0.276689i 0.990384 + 0.138345i \(0.0441781\pi\)
−0.990384 + 0.138345i \(0.955822\pi\)
\(8\) 1.73205i 0.612372i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.26795 0.382301 0.191151 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 2.73205i 0.757735i 0.925451 + 0.378867i \(0.123686\pi\)
−0.925451 + 0.378867i \(0.876314\pi\)
\(14\) −1.26795 −0.338874
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) − 1.73205i − 0.408248i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.732051 0.159747
\(22\) 2.19615i 0.468221i
\(23\) 3.46410i 0.722315i 0.932505 + 0.361158i \(0.117618\pi\)
−0.932505 + 0.361158i \(0.882382\pi\)
\(24\) 1.73205 0.353553
\(25\) 0 0
\(26\) −4.73205 −0.928032
\(27\) 1.00000i 0.192450i
\(28\) − 0.732051i − 0.138345i
\(29\) 2.19615 0.407815 0.203908 0.978990i \(-0.434636\pi\)
0.203908 + 0.978990i \(0.434636\pi\)
\(30\) 0 0
\(31\) −4.92820 −0.885131 −0.442566 0.896736i \(-0.645932\pi\)
−0.442566 + 0.896736i \(0.645932\pi\)
\(32\) − 5.19615i − 0.918559i
\(33\) − 1.26795i − 0.220722i
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.19615i 0.689843i 0.938631 + 0.344922i \(0.112095\pi\)
−0.938631 + 0.344922i \(0.887905\pi\)
\(38\) − 1.73205i − 0.280976i
\(39\) 2.73205 0.437478
\(40\) 0 0
\(41\) 4.73205 0.739022 0.369511 0.929226i \(-0.379525\pi\)
0.369511 + 0.929226i \(0.379525\pi\)
\(42\) 1.26795i 0.195649i
\(43\) 6.19615i 0.944904i 0.881356 + 0.472452i \(0.156631\pi\)
−0.881356 + 0.472452i \(0.843369\pi\)
\(44\) −1.26795 −0.191151
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 3.46410i 0.505291i 0.967559 + 0.252646i \(0.0813007\pi\)
−0.967559 + 0.252646i \(0.918699\pi\)
\(48\) 5.00000i 0.721688i
\(49\) 6.46410 0.923443
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.73205i − 0.378867i
\(53\) 2.53590i 0.348332i 0.984716 + 0.174166i \(0.0557230\pi\)
−0.984716 + 0.174166i \(0.944277\pi\)
\(54\) −1.73205 −0.235702
\(55\) 0 0
\(56\) −1.26795 −0.169437
\(57\) 1.00000i 0.132453i
\(58\) 3.80385i 0.499470i
\(59\) −9.46410 −1.23212 −0.616061 0.787699i \(-0.711272\pi\)
−0.616061 + 0.787699i \(0.711272\pi\)
\(60\) 0 0
\(61\) −13.4641 −1.72390 −0.861951 0.506992i \(-0.830757\pi\)
−0.861951 + 0.506992i \(0.830757\pi\)
\(62\) − 8.53590i − 1.08406i
\(63\) − 0.732051i − 0.0922297i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.19615 0.270328
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 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 \(0.0745426\pi\)
\(72\) − 1.73205i − 0.204124i
\(73\) 3.07180i 0.359527i 0.983710 + 0.179763i \(0.0575332\pi\)
−0.983710 + 0.179763i \(0.942467\pi\)
\(74\) −7.26795 −0.844882
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0.928203i 0.105779i
\(78\) 4.73205i 0.535799i
\(79\) −2.92820 −0.329449 −0.164724 0.986340i \(-0.552673\pi\)
−0.164724 + 0.986340i \(0.552673\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 8.19615i 0.905114i
\(83\) − 0.928203i − 0.101884i −0.998702 0.0509418i \(-0.983778\pi\)
0.998702 0.0509418i \(-0.0162223\pi\)
\(84\) −0.732051 −0.0798733
\(85\) 0 0
\(86\) −10.7321 −1.15727
\(87\) − 2.19615i − 0.235452i
\(88\) 2.19615i 0.234111i
\(89\) −7.26795 −0.770401 −0.385201 0.922833i \(-0.625868\pi\)
−0.385201 + 0.922833i \(0.625868\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) − 3.46410i − 0.361158i
\(93\) 4.92820i 0.511031i
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) −5.19615 −0.530330
\(97\) 4.19615i 0.426055i 0.977046 + 0.213027i \(0.0683323\pi\)
−0.977046 + 0.213027i \(0.931668\pi\)
\(98\) 11.1962i 1.13098i
\(99\) −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
\(103\) 17.8564i 1.75944i 0.475488 + 0.879722i \(0.342271\pi\)
−0.475488 + 0.879722i \(0.657729\pi\)
\(104\) −4.73205 −0.464016
\(105\) 0 0
\(106\) −4.39230 −0.426618
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −6.39230 −0.612272 −0.306136 0.951988i \(-0.599036\pi\)
−0.306136 + 0.951988i \(0.599036\pi\)
\(110\) 0 0
\(111\) 4.19615 0.398281
\(112\) − 3.66025i − 0.345861i
\(113\) − 5.07180i − 0.477115i −0.971128 0.238557i \(-0.923326\pi\)
0.971128 0.238557i \(-0.0766745\pi\)
\(114\) −1.73205 −0.162221
\(115\) 0 0
\(116\) −2.19615 −0.203908
\(117\) − 2.73205i − 0.252578i
\(118\) − 16.3923i − 1.50903i
\(119\) 0 0
\(120\) 0 0
\(121\) −9.39230 −0.853846
\(122\) − 23.3205i − 2.11134i
\(123\) − 4.73205i − 0.426675i
\(124\) 4.92820 0.442566
\(125\) 0 0
\(126\) 1.26795 0.112958
\(127\) − 4.00000i − 0.354943i −0.984126 0.177471i \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) − 12.1244i − 1.07165i
\(129\) 6.19615 0.545541
\(130\) 0 0
\(131\) 15.1244 1.32142 0.660711 0.750641i \(-0.270255\pi\)
0.660711 + 0.750641i \(0.270255\pi\)
\(132\) 1.26795i 0.110361i
\(133\) − 0.732051i − 0.0634769i
\(134\) −13.8564 −1.19701
\(135\) 0 0
\(136\) 0 0
\(137\) − 7.85641i − 0.671218i −0.942001 0.335609i \(-0.891058\pi\)
0.942001 0.335609i \(-0.108942\pi\)
\(138\) 6.00000i 0.510754i
\(139\) −12.3923 −1.05110 −0.525551 0.850762i \(-0.676141\pi\)
−0.525551 + 0.850762i \(0.676141\pi\)
\(140\) 0 0
\(141\) 3.46410 0.291730
\(142\) 28.3923i 2.38263i
\(143\) 3.46410i 0.289683i
\(144\) 5.00000 0.416667
\(145\) 0 0
\(146\) −5.32051 −0.440328
\(147\) − 6.46410i − 0.533150i
\(148\) − 4.19615i − 0.344922i
\(149\) −7.85641 −0.643622 −0.321811 0.946804i \(-0.604292\pi\)
−0.321811 + 0.946804i \(0.604292\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) − 1.73205i − 0.140488i
\(153\) 0 0
\(154\) −1.60770 −0.129552
\(155\) 0 0
\(156\) −2.73205 −0.218739
\(157\) − 14.3923i − 1.14863i −0.818634 0.574315i \(-0.805268\pi\)
0.818634 0.574315i \(-0.194732\pi\)
\(158\) − 5.07180i − 0.403490i
\(159\) 2.53590 0.201110
\(160\) 0 0
\(161\) −2.53590 −0.199857
\(162\) 1.73205i 0.136083i
\(163\) − 12.7321i − 0.997251i −0.866817 0.498626i \(-0.833838\pi\)
0.866817 0.498626i \(-0.166162\pi\)
\(164\) −4.73205 −0.369511
\(165\) 0 0
\(166\) 1.60770 0.124781
\(167\) − 3.46410i − 0.268060i −0.990977 0.134030i \(-0.957208\pi\)
0.990977 0.134030i \(-0.0427919\pi\)
\(168\) 1.26795i 0.0978244i
\(169\) 5.53590 0.425838
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) − 6.19615i − 0.472452i
\(173\) 6.92820i 0.526742i 0.964695 + 0.263371i \(0.0848343\pi\)
−0.964695 + 0.263371i \(0.915166\pi\)
\(174\) 3.80385 0.288369
\(175\) 0 0
\(176\) −6.33975 −0.477876
\(177\) 9.46410i 0.711365i
\(178\) − 12.5885i − 0.943545i
\(179\) 11.3205 0.846135 0.423067 0.906098i \(-0.360953\pi\)
0.423067 + 0.906098i \(0.360953\pi\)
\(180\) 0 0
\(181\) 18.3923 1.36709 0.683545 0.729909i \(-0.260437\pi\)
0.683545 + 0.729909i \(0.260437\pi\)
\(182\) − 3.46410i − 0.256776i
\(183\) 13.4641i 0.995295i
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) −8.53590 −0.625882
\(187\) 0 0
\(188\) − 3.46410i − 0.252646i
\(189\) −0.732051 −0.0532489
\(190\) 0 0
\(191\) 17.6603 1.27785 0.638926 0.769269i \(-0.279379\pi\)
0.638926 + 0.769269i \(0.279379\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 7.12436i 0.512822i 0.966568 + 0.256411i \(0.0825401\pi\)
−0.966568 + 0.256411i \(0.917460\pi\)
\(194\) −7.26795 −0.521808
\(195\) 0 0
\(196\) −6.46410 −0.461722
\(197\) − 24.0000i − 1.70993i −0.518686 0.854965i \(-0.673579\pi\)
0.518686 0.854965i \(-0.326421\pi\)
\(198\) − 2.19615i − 0.156074i
\(199\) −19.3205 −1.36959 −0.684797 0.728734i \(-0.740109\pi\)
−0.684797 + 0.728734i \(0.740109\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 18.0000i 1.26648i
\(203\) 1.60770i 0.112838i
\(204\) 0 0
\(205\) 0 0
\(206\) −30.9282 −2.15487
\(207\) − 3.46410i − 0.240772i
\(208\) − 13.6603i − 0.947168i
\(209\) −1.26795 −0.0877059
\(210\) 0 0
\(211\) 14.9282 1.02770 0.513850 0.857880i \(-0.328219\pi\)
0.513850 + 0.857880i \(0.328219\pi\)
\(212\) − 2.53590i − 0.174166i
\(213\) − 16.3923i − 1.12318i
\(214\) 0 0
\(215\) 0 0
\(216\) −1.73205 −0.117851
\(217\) − 3.60770i − 0.244906i
\(218\) − 11.0718i − 0.749877i
\(219\) 3.07180 0.207573
\(220\) 0 0
\(221\) 0 0
\(222\) 7.26795i 0.487793i
\(223\) − 9.85641i − 0.660034i −0.943975 0.330017i \(-0.892946\pi\)
0.943975 0.330017i \(-0.107054\pi\)
\(224\) 3.80385 0.254155
\(225\) 0 0
\(226\) 8.78461 0.584344
\(227\) − 10.3923i − 0.689761i −0.938647 0.344881i \(-0.887919\pi\)
0.938647 0.344881i \(-0.112081\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) 25.4641 1.68272 0.841358 0.540479i \(-0.181757\pi\)
0.841358 + 0.540479i \(0.181757\pi\)
\(230\) 0 0
\(231\) 0.928203 0.0610713
\(232\) 3.80385i 0.249735i
\(233\) − 19.8564i − 1.30084i −0.759576 0.650418i \(-0.774594\pi\)
0.759576 0.650418i \(-0.225406\pi\)
\(234\) 4.73205 0.309344
\(235\) 0 0
\(236\) 9.46410 0.616061
\(237\) 2.92820i 0.190207i
\(238\) 0 0
\(239\) 20.1962 1.30638 0.653190 0.757194i \(-0.273430\pi\)
0.653190 + 0.757194i \(0.273430\pi\)
\(240\) 0 0
\(241\) −16.9282 −1.09044 −0.545221 0.838293i \(-0.683554\pi\)
−0.545221 + 0.838293i \(0.683554\pi\)
\(242\) − 16.2679i − 1.04574i
\(243\) − 1.00000i − 0.0641500i
\(244\) 13.4641 0.861951
\(245\) 0 0
\(246\) 8.19615 0.522568
\(247\) − 2.73205i − 0.173836i
\(248\) − 8.53590i − 0.542030i
\(249\) −0.928203 −0.0588225
\(250\) 0 0
\(251\) −10.0526 −0.634512 −0.317256 0.948340i \(-0.602761\pi\)
−0.317256 + 0.948340i \(0.602761\pi\)
\(252\) 0.732051i 0.0461149i
\(253\) 4.39230i 0.276142i
\(254\) 6.92820 0.434714
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) − 24.0000i − 1.49708i −0.663090 0.748539i \(-0.730755\pi\)
0.663090 0.748539i \(-0.269245\pi\)
\(258\) 10.7321i 0.668148i
\(259\) −3.07180 −0.190872
\(260\) 0 0
\(261\) −2.19615 −0.135938
\(262\) 26.1962i 1.61840i
\(263\) − 6.00000i − 0.369976i −0.982741 0.184988i \(-0.940775\pi\)
0.982741 0.184988i \(-0.0592246\pi\)
\(264\) 2.19615 0.135164
\(265\) 0 0
\(266\) 1.26795 0.0777430
\(267\) 7.26795i 0.444791i
\(268\) − 8.00000i − 0.488678i
\(269\) 30.5885 1.86501 0.932506 0.361156i \(-0.117618\pi\)
0.932506 + 0.361156i \(0.117618\pi\)
\(270\) 0 0
\(271\) −20.3923 −1.23874 −0.619372 0.785098i \(-0.712613\pi\)
−0.619372 + 0.785098i \(0.712613\pi\)
\(272\) 0 0
\(273\) 2.00000i 0.121046i
\(274\) 13.6077 0.822071
\(275\) 0 0
\(276\) −3.46410 −0.208514
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) − 21.4641i − 1.28733i
\(279\) 4.92820 0.295044
\(280\) 0 0
\(281\) 4.73205 0.282290 0.141145 0.989989i \(-0.454922\pi\)
0.141145 + 0.989989i \(0.454922\pi\)
\(282\) 6.00000i 0.357295i
\(283\) 26.9808i 1.60384i 0.597432 + 0.801920i \(0.296188\pi\)
−0.597432 + 0.801920i \(0.703812\pi\)
\(284\) −16.3923 −0.972704
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 3.46410i 0.204479i
\(288\) 5.19615i 0.306186i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 4.19615 0.245983
\(292\) − 3.07180i − 0.179763i
\(293\) 27.7128i 1.61900i 0.587120 + 0.809500i \(0.300262\pi\)
−0.587120 + 0.809500i \(0.699738\pi\)
\(294\) 11.1962 0.652973
\(295\) 0 0
\(296\) −7.26795 −0.422441
\(297\) 1.26795i 0.0735739i
\(298\) − 13.6077i − 0.788273i
\(299\) −9.46410 −0.547323
\(300\) 0 0
\(301\) −4.53590 −0.261445
\(302\) 24.2487i 1.39536i
\(303\) − 10.3923i − 0.597022i
\(304\) 5.00000 0.286770
\(305\) 0 0
\(306\) 0 0
\(307\) − 11.6077i − 0.662486i −0.943545 0.331243i \(-0.892532\pi\)
0.943545 0.331243i \(-0.107468\pi\)
\(308\) − 0.928203i − 0.0528893i
\(309\) 17.8564 1.01582
\(310\) 0 0
\(311\) −26.4449 −1.49955 −0.749775 0.661692i \(-0.769838\pi\)
−0.749775 + 0.661692i \(0.769838\pi\)
\(312\) 4.73205i 0.267900i
\(313\) 14.3923i 0.813501i 0.913539 + 0.406751i \(0.133338\pi\)
−0.913539 + 0.406751i \(0.866662\pi\)
\(314\) 24.9282 1.40678
\(315\) 0 0
\(316\) 2.92820 0.164724
\(317\) 23.3205i 1.30981i 0.755711 + 0.654905i \(0.227291\pi\)
−0.755711 + 0.654905i \(0.772709\pi\)
\(318\) 4.39230i 0.246308i
\(319\) 2.78461 0.155908
\(320\) 0 0
\(321\) 0 0
\(322\) − 4.39230i − 0.244774i
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 22.0526 1.22138
\(327\) 6.39230i 0.353495i
\(328\) 8.19615i 0.452557i
\(329\) −2.53590 −0.139809
\(330\) 0 0
\(331\) 29.7128 1.63316 0.816582 0.577230i \(-0.195866\pi\)
0.816582 + 0.577230i \(0.195866\pi\)
\(332\) 0.928203i 0.0509418i
\(333\) − 4.19615i − 0.229948i
\(334\) 6.00000 0.328305
\(335\) 0 0
\(336\) −3.66025 −0.199683
\(337\) − 19.1244i − 1.04177i −0.853627 0.520885i \(-0.825602\pi\)
0.853627 0.520885i \(-0.174398\pi\)
\(338\) 9.58846i 0.521543i
\(339\) −5.07180 −0.275462
\(340\) 0 0
\(341\) −6.24871 −0.338387
\(342\) 1.73205i 0.0936586i
\(343\) 9.85641i 0.532196i
\(344\) −10.7321 −0.578633
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) 12.9282i 0.694022i 0.937861 + 0.347011i \(0.112803\pi\)
−0.937861 + 0.347011i \(0.887197\pi\)
\(348\) 2.19615i 0.117726i
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) −2.73205 −0.145826
\(352\) − 6.58846i − 0.351166i
\(353\) − 26.7846i − 1.42560i −0.701367 0.712800i \(-0.747427\pi\)
0.701367 0.712800i \(-0.252573\pi\)
\(354\) −16.3923 −0.871241
\(355\) 0 0
\(356\) 7.26795 0.385201
\(357\) 0 0
\(358\) 19.6077i 1.03630i
\(359\) −17.6603 −0.932073 −0.466036 0.884766i \(-0.654318\pi\)
−0.466036 + 0.884766i \(0.654318\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 31.8564i 1.67434i
\(363\) 9.39230i 0.492968i
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) −23.3205 −1.21898
\(367\) 5.80385i 0.302958i 0.988460 + 0.151479i \(0.0484037\pi\)
−0.988460 + 0.151479i \(0.951596\pi\)
\(368\) − 17.3205i − 0.902894i
\(369\) −4.73205 −0.246341
\(370\) 0 0
\(371\) −1.85641 −0.0963798
\(372\) − 4.92820i − 0.255515i
\(373\) − 4.19615i − 0.217269i −0.994082 0.108634i \(-0.965352\pi\)
0.994082 0.108634i \(-0.0346477\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) 6.00000i 0.309016i
\(378\) − 1.26795i − 0.0652163i
\(379\) −7.07180 −0.363254 −0.181627 0.983368i \(-0.558136\pi\)
−0.181627 + 0.983368i \(0.558136\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 30.5885i 1.56504i
\(383\) 30.9282i 1.58036i 0.612877 + 0.790179i \(0.290012\pi\)
−0.612877 + 0.790179i \(0.709988\pi\)
\(384\) −12.1244 −0.618718
\(385\) 0 0
\(386\) −12.3397 −0.628077
\(387\) − 6.19615i − 0.314968i
\(388\) − 4.19615i − 0.213027i
\(389\) 19.8564 1.00676 0.503380 0.864065i \(-0.332090\pi\)
0.503380 + 0.864065i \(0.332090\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 11.1962i 0.565491i
\(393\) − 15.1244i − 0.762923i
\(394\) 41.5692 2.09423
\(395\) 0 0
\(396\) 1.26795 0.0637168
\(397\) − 4.92820i − 0.247339i −0.992323 0.123670i \(-0.960534\pi\)
0.992323 0.123670i \(-0.0394663\pi\)
\(398\) − 33.4641i − 1.67740i
\(399\) −0.732051 −0.0366484
\(400\) 0 0
\(401\) 4.05256 0.202375 0.101188 0.994867i \(-0.467736\pi\)
0.101188 + 0.994867i \(0.467736\pi\)
\(402\) 13.8564i 0.691095i
\(403\) − 13.4641i − 0.670695i
\(404\) −10.3923 −0.517036
\(405\) 0 0
\(406\) −2.78461 −0.138198
\(407\) 5.32051i 0.263728i
\(408\) 0 0
\(409\) 5.60770 0.277283 0.138641 0.990343i \(-0.455726\pi\)
0.138641 + 0.990343i \(0.455726\pi\)
\(410\) 0 0
\(411\) −7.85641 −0.387528
\(412\) − 17.8564i − 0.879722i
\(413\) − 6.92820i − 0.340915i
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 14.1962 0.696024
\(417\) 12.3923i 0.606854i
\(418\) − 2.19615i − 0.107417i
\(419\) −10.0526 −0.491100 −0.245550 0.969384i \(-0.578968\pi\)
−0.245550 + 0.969384i \(0.578968\pi\)
\(420\) 0 0
\(421\) 22.7846 1.11045 0.555227 0.831699i \(-0.312631\pi\)
0.555227 + 0.831699i \(0.312631\pi\)
\(422\) 25.8564i 1.25867i
\(423\) − 3.46410i − 0.168430i
\(424\) −4.39230 −0.213309
\(425\) 0 0
\(426\) 28.3923 1.37561
\(427\) − 9.85641i − 0.476985i
\(428\) 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 \(0.310165\pi\)
\(432\) − 5.00000i − 0.240563i
\(433\) − 20.5885i − 0.989418i −0.869059 0.494709i \(-0.835275\pi\)
0.869059 0.494709i \(-0.164725\pi\)
\(434\) 6.24871 0.299948
\(435\) 0 0
\(436\) 6.39230 0.306136
\(437\) − 3.46410i − 0.165710i
\(438\) 5.32051i 0.254224i
\(439\) −13.0718 −0.623883 −0.311941 0.950101i \(-0.600979\pi\)
−0.311941 + 0.950101i \(0.600979\pi\)
\(440\) 0 0
\(441\) −6.46410 −0.307814
\(442\) 0 0
\(443\) − 29.3205i − 1.39306i −0.717528 0.696530i \(-0.754726\pi\)
0.717528 0.696530i \(-0.245274\pi\)
\(444\) −4.19615 −0.199141
\(445\) 0 0
\(446\) 17.0718 0.808373
\(447\) 7.85641i 0.371595i
\(448\) − 0.732051i − 0.0345861i
\(449\) −11.6603 −0.550281 −0.275141 0.961404i \(-0.588724\pi\)
−0.275141 + 0.961404i \(0.588724\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 5.07180i 0.238557i
\(453\) − 14.0000i − 0.657777i
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) −1.73205 −0.0811107
\(457\) 11.4641i 0.536268i 0.963382 + 0.268134i \(0.0864070\pi\)
−0.963382 + 0.268134i \(0.913593\pi\)
\(458\) 44.1051i 2.06090i
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 1.60770i 0.0747967i
\(463\) − 9.51666i − 0.442277i −0.975242 0.221138i \(-0.929023\pi\)
0.975242 0.221138i \(-0.0709772\pi\)
\(464\) −10.9808 −0.509769
\(465\) 0 0
\(466\) 34.3923 1.59319
\(467\) 27.4641i 1.27089i 0.772147 + 0.635444i \(0.219183\pi\)
−0.772147 + 0.635444i \(0.780817\pi\)
\(468\) 2.73205i 0.126289i
\(469\) −5.85641 −0.270424
\(470\) 0 0
\(471\) −14.3923 −0.663162
\(472\) − 16.3923i − 0.754517i
\(473\) 7.85641i 0.361238i
\(474\) −5.07180 −0.232955
\(475\) 0 0
\(476\) 0 0
\(477\) − 2.53590i − 0.116111i
\(478\) 34.9808i 1.59998i
\(479\) 19.5167 0.891739 0.445869 0.895098i \(-0.352895\pi\)
0.445869 + 0.895098i \(0.352895\pi\)
\(480\) 0 0
\(481\) −11.4641 −0.522718
\(482\) − 29.3205i − 1.33551i
\(483\) 2.53590i 0.115387i
\(484\) 9.39230 0.426923
\(485\) 0 0
\(486\) 1.73205 0.0785674
\(487\) − 11.6077i − 0.525995i −0.964797 0.262997i \(-0.915289\pi\)
0.964797 0.262997i \(-0.0847111\pi\)
\(488\) − 23.3205i − 1.05567i
\(489\) −12.7321 −0.575763
\(490\) 0 0
\(491\) −22.0526 −0.995218 −0.497609 0.867401i \(-0.665789\pi\)
−0.497609 + 0.867401i \(0.665789\pi\)
\(492\) 4.73205i 0.213337i
\(493\) 0 0
\(494\) 4.73205 0.212905
\(495\) 0 0
\(496\) 24.6410 1.10641
\(497\) 12.0000i 0.538274i
\(498\) − 1.60770i − 0.0720425i
\(499\) −17.4641 −0.781801 −0.390900 0.920433i \(-0.627836\pi\)
−0.390900 + 0.920433i \(0.627836\pi\)
\(500\) 0 0
\(501\) −3.46410 −0.154765
\(502\) − 17.4115i − 0.777115i
\(503\) 36.9282i 1.64655i 0.567645 + 0.823274i \(0.307855\pi\)
−0.567645 + 0.823274i \(0.692145\pi\)
\(504\) 1.26795 0.0564789
\(505\) 0 0
\(506\) −7.60770 −0.338203
\(507\) − 5.53590i − 0.245858i
\(508\) 4.00000i 0.177471i
\(509\) −28.0526 −1.24341 −0.621704 0.783252i \(-0.713559\pi\)
−0.621704 + 0.783252i \(0.713559\pi\)
\(510\) 0 0
\(511\) −2.24871 −0.0994771
\(512\) 8.66025i 0.382733i
\(513\) − 1.00000i − 0.0441511i
\(514\) 41.5692 1.83354
\(515\) 0 0
\(516\) −6.19615 −0.272770
\(517\) 4.39230i 0.193173i
\(518\) − 5.32051i − 0.233770i
\(519\) 6.92820 0.304114
\(520\) 0 0
\(521\) 40.7321 1.78450 0.892252 0.451538i \(-0.149125\pi\)
0.892252 + 0.451538i \(0.149125\pi\)
\(522\) − 3.80385i − 0.166490i
\(523\) − 43.3205i − 1.89427i −0.320830 0.947137i \(-0.603962\pi\)
0.320830 0.947137i \(-0.396038\pi\)
\(524\) −15.1244 −0.660711
\(525\) 0 0
\(526\) 10.3923 0.453126
\(527\) 0 0
\(528\) 6.33975i 0.275902i
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) 9.46410 0.410707
\(532\) 0.732051i 0.0317384i
\(533\) 12.9282i 0.559983i
\(534\) −12.5885 −0.544756
\(535\) 0 0
\(536\) −13.8564 −0.598506
\(537\) − 11.3205i − 0.488516i
\(538\) 52.9808i 2.28416i
\(539\) 8.19615 0.353033
\(540\) 0 0
\(541\) −13.7128 −0.589560 −0.294780 0.955565i \(-0.595246\pi\)
−0.294780 + 0.955565i \(0.595246\pi\)
\(542\) − 35.3205i − 1.51715i
\(543\) − 18.3923i − 0.789289i
\(544\) 0 0
\(545\) 0 0
\(546\) −3.46410 −0.148250
\(547\) 8.67949i 0.371108i 0.982634 + 0.185554i \(0.0594080\pi\)
−0.982634 + 0.185554i \(0.940592\pi\)
\(548\) 7.85641i 0.335609i
\(549\) 13.4641 0.574634
\(550\) 0 0
\(551\) −2.19615 −0.0935592
\(552\) 6.00000i 0.255377i
\(553\) − 2.14359i − 0.0911549i
\(554\) −3.46410 −0.147176
\(555\) 0 0
\(556\) 12.3923 0.525551
\(557\) − 12.9282i − 0.547786i −0.961760 0.273893i \(-0.911689\pi\)
0.961760 0.273893i \(-0.0883113\pi\)
\(558\) 8.53590i 0.361353i
\(559\) −16.9282 −0.715987
\(560\) 0 0
\(561\) 0 0
\(562\) 8.19615i 0.345734i
\(563\) 20.5359i 0.865485i 0.901518 + 0.432742i \(0.142454\pi\)
−0.901518 + 0.432742i \(0.857546\pi\)
\(564\) −3.46410 −0.145865
\(565\) 0 0
\(566\) −46.7321 −1.96429
\(567\) 0.732051i 0.0307432i
\(568\) 28.3923i 1.19131i
\(569\) 16.0526 0.672958 0.336479 0.941691i \(-0.390764\pi\)
0.336479 + 0.941691i \(0.390764\pi\)
\(570\) 0 0
\(571\) 14.2487 0.596290 0.298145 0.954521i \(-0.403632\pi\)
0.298145 + 0.954521i \(0.403632\pi\)
\(572\) − 3.46410i − 0.144841i
\(573\) − 17.6603i − 0.737768i
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 47.1769i − 1.96400i −0.188879 0.982000i \(-0.560485\pi\)
0.188879 0.982000i \(-0.439515\pi\)
\(578\) 29.4449i 1.22474i
\(579\) 7.12436 0.296078
\(580\) 0 0
\(581\) 0.679492 0.0281901
\(582\) 7.26795i 0.301266i
\(583\) 3.21539i 0.133168i
\(584\) −5.32051 −0.220164
\(585\) 0 0
\(586\) −48.0000 −1.98286
\(587\) 3.46410i 0.142979i 0.997441 + 0.0714894i \(0.0227752\pi\)
−0.997441 + 0.0714894i \(0.977225\pi\)
\(588\) 6.46410i 0.266575i
\(589\) 4.92820 0.203063
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) − 20.9808i − 0.862304i
\(593\) − 2.78461i − 0.114350i −0.998364 0.0571751i \(-0.981791\pi\)
0.998364 0.0571751i \(-0.0182093\pi\)
\(594\) −2.19615 −0.0901092
\(595\) 0 0
\(596\) 7.85641 0.321811
\(597\) 19.3205i 0.790736i
\(598\) − 16.3923i − 0.670331i
\(599\) −13.8564 −0.566157 −0.283079 0.959097i \(-0.591356\pi\)
−0.283079 + 0.959097i \(0.591356\pi\)
\(600\) 0 0
\(601\) 15.1769 0.619079 0.309540 0.950887i \(-0.399825\pi\)
0.309540 + 0.950887i \(0.399825\pi\)
\(602\) − 7.85641i − 0.320203i
\(603\) − 8.00000i − 0.325785i
\(604\) −14.0000 −0.569652
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) − 32.3923i − 1.31476i −0.753558 0.657382i \(-0.771664\pi\)
0.753558 0.657382i \(-0.228336\pi\)
\(608\) 5.19615i 0.210732i
\(609\) 1.60770 0.0651471
\(610\) 0 0
\(611\) −9.46410 −0.382877
\(612\) 0 0
\(613\) − 21.6077i − 0.872727i −0.899771 0.436363i \(-0.856266\pi\)
0.899771 0.436363i \(-0.143734\pi\)
\(614\) 20.1051 0.811377
\(615\) 0 0
\(616\) −1.60770 −0.0647759
\(617\) − 27.7128i − 1.11568i −0.829950 0.557838i \(-0.811631\pi\)
0.829950 0.557838i \(-0.188369\pi\)
\(618\) 30.9282i 1.24411i
\(619\) −19.3205 −0.776557 −0.388278 0.921542i \(-0.626930\pi\)
−0.388278 + 0.921542i \(0.626930\pi\)
\(620\) 0 0
\(621\) −3.46410 −0.139010
\(622\) − 45.8038i − 1.83657i
\(623\) − 5.32051i − 0.213162i
\(624\) −13.6603 −0.546848
\(625\) 0 0
\(626\) −24.9282 −0.996331
\(627\) 1.26795i 0.0506370i
\(628\) 14.3923i 0.574315i
\(629\) 0 0
\(630\) 0 0
\(631\) −21.0718 −0.838855 −0.419427 0.907789i \(-0.637769\pi\)
−0.419427 + 0.907789i \(0.637769\pi\)
\(632\) − 5.07180i − 0.201745i
\(633\) − 14.9282i − 0.593343i
\(634\) −40.3923 −1.60418
\(635\) 0 0
\(636\) −2.53590 −0.100555
\(637\) 17.6603i 0.699725i
\(638\) 4.82309i 0.190948i
\(639\) −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
\(643\) 1.80385i 0.0711368i 0.999367 + 0.0355684i \(0.0113242\pi\)
−0.999367 + 0.0355684i \(0.988676\pi\)
\(644\) 2.53590 0.0999284
\(645\) 0 0
\(646\) 0 0
\(647\) − 31.8564i − 1.25240i −0.779661 0.626202i \(-0.784608\pi\)
0.779661 0.626202i \(-0.215392\pi\)
\(648\) 1.73205i 0.0680414i
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) −3.60770 −0.141397
\(652\) 12.7321i 0.498626i
\(653\) − 30.9282i − 1.21031i −0.796106 0.605157i \(-0.793110\pi\)
0.796106 0.605157i \(-0.206890\pi\)
\(654\) −11.0718 −0.432942
\(655\) 0 0
\(656\) −23.6603 −0.923778
\(657\) − 3.07180i − 0.119842i
\(658\) − 4.39230i − 0.171230i
\(659\) −18.9282 −0.737338 −0.368669 0.929561i \(-0.620186\pi\)
−0.368669 + 0.929561i \(0.620186\pi\)
\(660\) 0 0
\(661\) −23.1769 −0.901477 −0.450739 0.892656i \(-0.648839\pi\)
−0.450739 + 0.892656i \(0.648839\pi\)
\(662\) 51.4641i 2.00021i
\(663\) 0 0
\(664\) 1.60770 0.0623907
\(665\) 0 0
\(666\) 7.26795 0.281627
\(667\) 7.60770i 0.294571i
\(668\) 3.46410i 0.134030i
\(669\) −9.85641 −0.381071
\(670\) 0 0
\(671\) −17.0718 −0.659049
\(672\) − 3.80385i − 0.146737i
\(673\) 7.12436i 0.274624i 0.990528 + 0.137312i \(0.0438463\pi\)
−0.990528 + 0.137312i \(0.956154\pi\)
\(674\) 33.1244 1.27590
\(675\) 0 0
\(676\) −5.53590 −0.212919
\(677\) 35.3205i 1.35748i 0.734380 + 0.678739i \(0.237473\pi\)
−0.734380 + 0.678739i \(0.762527\pi\)
\(678\) − 8.78461i − 0.337371i
\(679\) −3.07180 −0.117885
\(680\) 0 0
\(681\) −10.3923 −0.398234
\(682\) − 10.8231i − 0.414437i
\(683\) 18.9282i 0.724268i 0.932126 + 0.362134i \(0.117952\pi\)
−0.932126 + 0.362134i \(0.882048\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) −17.0718 −0.651804
\(687\) − 25.4641i − 0.971516i
\(688\) − 30.9808i − 1.18113i
\(689\) −6.92820 −0.263944
\(690\) 0 0
\(691\) −8.39230 −0.319258 −0.159629 0.987177i \(-0.551030\pi\)
−0.159629 + 0.987177i \(0.551030\pi\)
\(692\) − 6.92820i − 0.263371i
\(693\) − 0.928203i − 0.0352595i
\(694\) −22.3923 −0.850000
\(695\) 0 0
\(696\) 3.80385 0.144184
\(697\) 0 0
\(698\) 38.1051i 1.44230i
\(699\) −19.8564 −0.751038
\(700\) 0 0
\(701\) 21.7128 0.820082 0.410041 0.912067i \(-0.365514\pi\)
0.410041 + 0.912067i \(0.365514\pi\)
\(702\) − 4.73205i − 0.178600i
\(703\) − 4.19615i − 0.158261i
\(704\) −1.26795 −0.0477876
\(705\) 0 0
\(706\) 46.3923 1.74600
\(707\) 7.60770i 0.286117i
\(708\) − 9.46410i − 0.355683i
\(709\) −33.1769 −1.24599 −0.622993 0.782228i \(-0.714083\pi\)
−0.622993 + 0.782228i \(0.714083\pi\)
\(710\) 0 0
\(711\) 2.92820 0.109816
\(712\) − 12.5885i − 0.471772i
\(713\) − 17.0718i − 0.639344i
\(714\) 0 0
\(715\) 0 0
\(716\) −11.3205 −0.423067
\(717\) − 20.1962i − 0.754239i
\(718\) − 30.5885i − 1.14155i
\(719\) −5.66025 −0.211092 −0.105546 0.994414i \(-0.533659\pi\)
−0.105546 + 0.994414i \(0.533659\pi\)
\(720\) 0 0
\(721\) −13.0718 −0.486819
\(722\) 1.73205i 0.0644603i
\(723\) 16.9282i 0.629567i
\(724\) −18.3923 −0.683545
\(725\) 0 0
\(726\) −16.2679 −0.603760
\(727\) 8.33975i 0.309304i 0.987969 + 0.154652i \(0.0494256\pi\)
−0.987969 + 0.154652i \(0.950574\pi\)
\(728\) − 3.46410i − 0.128388i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) − 13.4641i − 0.497648i
\(733\) − 22.7846i − 0.841569i −0.907161 0.420784i \(-0.861755\pi\)
0.907161 0.420784i \(-0.138245\pi\)
\(734\) −10.0526 −0.371047
\(735\) 0 0
\(736\) 18.0000 0.663489
\(737\) 10.1436i 0.373644i
\(738\) − 8.19615i − 0.301705i
\(739\) −33.8564 −1.24543 −0.622714 0.782450i \(-0.713970\pi\)
−0.622714 + 0.782450i \(0.713970\pi\)
\(740\) 0 0
\(741\) −2.73205 −0.100364
\(742\) − 3.21539i − 0.118041i
\(743\) 44.7846i 1.64299i 0.570217 + 0.821494i \(0.306859\pi\)
−0.570217 + 0.821494i \(0.693141\pi\)
\(744\) −8.53590 −0.312941
\(745\) 0 0
\(746\) 7.26795 0.266099
\(747\) 0.928203i 0.0339612i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) − 17.3205i − 0.631614i
\(753\) 10.0526i 0.366336i
\(754\) −10.3923 −0.378465
\(755\) 0 0
\(756\) 0.732051 0.0266244
\(757\) − 16.2487i − 0.590569i −0.955409 0.295285i \(-0.904585\pi\)
0.955409 0.295285i \(-0.0954145\pi\)
\(758\) − 12.2487i − 0.444893i
\(759\) 4.39230 0.159431
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) − 6.92820i − 0.250982i
\(763\) − 4.67949i − 0.169409i
\(764\) −17.6603 −0.638926
\(765\) 0 0
\(766\) −53.5692 −1.93553
\(767\) − 25.8564i − 0.933621i
\(768\) − 19.0000i − 0.685603i
\(769\) −48.6410 −1.75404 −0.877020 0.480454i \(-0.840472\pi\)
−0.877020 + 0.480454i \(0.840472\pi\)
\(770\) 0 0
\(771\) −24.0000 −0.864339
\(772\) − 7.12436i − 0.256411i
\(773\) − 37.1769i − 1.33716i −0.743640 0.668580i \(-0.766902\pi\)
0.743640 0.668580i \(-0.233098\pi\)
\(774\) 10.7321 0.385756
\(775\) 0 0
\(776\) −7.26795 −0.260904
\(777\) 3.07180i 0.110200i
\(778\) 34.3923i 1.23302i
\(779\) −4.73205 −0.169543
\(780\) 0 0
\(781\) 20.7846 0.743732
\(782\) 0 0
\(783\) 2.19615i 0.0784841i
\(784\) −32.3205 −1.15430
\(785\) 0 0
\(786\) 26.1962 0.934386
\(787\) 43.3205i 1.54421i 0.635495 + 0.772105i \(0.280796\pi\)
−0.635495 + 0.772105i \(0.719204\pi\)
\(788\) 24.0000i 0.854965i
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) 3.71281 0.132012
\(792\) − 2.19615i − 0.0780369i
\(793\) − 36.7846i − 1.30626i
\(794\) 8.53590 0.302928
\(795\) 0 0
\(796\) 19.3205 0.684797
\(797\) 3.21539i 0.113895i 0.998377 + 0.0569475i \(0.0181368\pi\)
−0.998377 + 0.0569475i \(0.981863\pi\)
\(798\) − 1.26795i − 0.0448849i
\(799\) 0 0
\(800\) 0 0
\(801\) 7.26795 0.256800
\(802\) 7.01924i 0.247858i
\(803\) 3.89488i 0.137447i
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 23.3205 0.821430
\(807\) − 30.5885i − 1.07676i
\(808\) 18.0000i 0.633238i
\(809\) 26.7846 0.941697 0.470848 0.882214i \(-0.343948\pi\)
0.470848 + 0.882214i \(0.343948\pi\)
\(810\) 0 0
\(811\) −45.5692 −1.60015 −0.800076 0.599899i \(-0.795207\pi\)
−0.800076 + 0.599899i \(0.795207\pi\)
\(812\) − 1.60770i − 0.0564190i
\(813\) 20.3923i 0.715189i
\(814\) −9.21539 −0.322999
\(815\) 0 0
\(816\) 0 0
\(817\) − 6.19615i − 0.216776i
\(818\) 9.71281i 0.339601i
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −39.4641 −1.37731 −0.688653 0.725091i \(-0.741798\pi\)
−0.688653 + 0.725091i \(0.741798\pi\)
\(822\) − 13.6077i − 0.474623i
\(823\) 38.9808i 1.35878i 0.733776 + 0.679392i \(0.237756\pi\)
−0.733776 + 0.679392i \(0.762244\pi\)
\(824\) −30.9282 −1.07744
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 29.3205i 1.01957i 0.860301 + 0.509787i \(0.170276\pi\)
−0.860301 + 0.509787i \(0.829724\pi\)
\(828\) 3.46410i 0.120386i
\(829\) 42.1051 1.46237 0.731186 0.682179i \(-0.238967\pi\)
0.731186 + 0.682179i \(0.238967\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) − 2.73205i − 0.0947168i
\(833\) 0 0
\(834\) −21.4641 −0.743241
\(835\) 0 0
\(836\) 1.26795 0.0438529
\(837\) − 4.92820i − 0.170344i
\(838\) − 17.4115i − 0.601472i
\(839\) 40.3923 1.39450 0.697249 0.716829i \(-0.254407\pi\)
0.697249 + 0.716829i \(0.254407\pi\)
\(840\) 0 0
\(841\) −24.1769 −0.833687
\(842\) 39.4641i 1.36002i
\(843\) − 4.73205i − 0.162980i
\(844\) −14.9282 −0.513850
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) − 6.87564i − 0.236250i
\(848\) − 12.6795i − 0.435416i
\(849\) 26.9808 0.925977
\(850\) 0 0
\(851\) −14.5359 −0.498284
\(852\) 16.3923i 0.561591i
\(853\) 35.1769i 1.20443i 0.798332 + 0.602217i \(0.205716\pi\)
−0.798332 + 0.602217i \(0.794284\pi\)
\(854\) 17.0718 0.584185
\(855\) 0 0
\(856\) 0 0
\(857\) 6.24871i 0.213452i 0.994288 + 0.106726i \(0.0340368\pi\)
−0.994288 + 0.106726i \(0.965963\pi\)
\(858\) 6.00000i 0.204837i
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 0 0
\(861\) 3.46410 0.118056
\(862\) 40.3923i 1.37577i
\(863\) 12.0000i 0.408485i 0.978920 + 0.204242i \(0.0654731\pi\)
−0.978920 + 0.204242i \(0.934527\pi\)
\(864\) 5.19615 0.176777
\(865\) 0 0
\(866\) 35.6603 1.21178
\(867\) − 17.0000i − 0.577350i
\(868\) 3.60770i 0.122453i
\(869\) −3.71281 −0.125949
\(870\) 0 0
\(871\) −21.8564 −0.740576
\(872\) − 11.0718i − 0.374938i
\(873\) − 4.19615i − 0.142018i
\(874\) 6.00000 0.202953
\(875\) 0 0
\(876\) −3.07180 −0.103786
\(877\) 28.8756i 0.975061i 0.873106 + 0.487531i \(0.162102\pi\)
−0.873106 + 0.487531i \(0.837898\pi\)
\(878\) − 22.6410i − 0.764097i
\(879\) 27.7128 0.934730
\(880\) 0 0
\(881\) 15.4641 0.520999 0.260499 0.965474i \(-0.416113\pi\)
0.260499 + 0.965474i \(0.416113\pi\)
\(882\) − 11.1962i − 0.376994i
\(883\) 14.9808i 0.504143i 0.967709 + 0.252071i \(0.0811118\pi\)
−0.967709 + 0.252071i \(0.918888\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 50.7846 1.70614
\(887\) − 24.0000i − 0.805841i −0.915235 0.402921i \(-0.867995\pi\)
0.915235 0.402921i \(-0.132005\pi\)
\(888\) 7.26795i 0.243896i
\(889\) 2.92820 0.0982088
\(890\) 0 0
\(891\) 1.26795 0.0424779
\(892\) 9.85641i 0.330017i
\(893\) − 3.46410i − 0.115922i
\(894\) −13.6077 −0.455109
\(895\) 0 0
\(896\) 8.87564 0.296514
\(897\) 9.46410i 0.315997i
\(898\) − 20.1962i − 0.673954i
\(899\) −10.8231 −0.360970
\(900\) 0 0
\(901\) 0 0
\(902\) 10.3923i 0.346026i
\(903\) 4.53590i 0.150945i
\(904\) 8.78461 0.292172
\(905\) 0 0
\(906\) 24.2487 0.805609
\(907\) − 11.6077i − 0.385427i −0.981255 0.192714i \(-0.938271\pi\)
0.981255 0.192714i \(-0.0617288\pi\)
\(908\) 10.3923i 0.344881i
\(909\) −10.3923 −0.344691
\(910\) 0 0
\(911\) −41.0718 −1.36077 −0.680385 0.732855i \(-0.738187\pi\)
−0.680385 + 0.732855i \(0.738187\pi\)
\(912\) − 5.00000i − 0.165567i
\(913\) − 1.17691i − 0.0389502i
\(914\) −19.8564 −0.656792
\(915\) 0 0
\(916\) −25.4641 −0.841358
\(917\) 11.0718i 0.365623i
\(918\) 0 0
\(919\) 59.4256 1.96027 0.980135 0.198330i \(-0.0635518\pi\)
0.980135 + 0.198330i \(0.0635518\pi\)
\(920\) 0 0
\(921\) −11.6077 −0.382487
\(922\) 10.3923i 0.342252i
\(923\) 44.7846i 1.47410i
\(924\) −0.928203 −0.0305356
\(925\) 0 0
\(926\) 16.4833 0.541676
\(927\) − 17.8564i − 0.586481i
\(928\) − 11.4115i − 0.374602i
\(929\) 22.3923 0.734668 0.367334 0.930089i \(-0.380271\pi\)
0.367334 + 0.930089i \(0.380271\pi\)
\(930\) 0 0
\(931\) −6.46410 −0.211852
\(932\) 19.8564i 0.650418i
\(933\) 26.4449i 0.865766i
\(934\) −47.5692 −1.55651
\(935\) 0 0
\(936\) 4.73205 0.154672
\(937\) 32.2487i 1.05352i 0.850014 + 0.526760i \(0.176593\pi\)
−0.850014 + 0.526760i \(0.823407\pi\)
\(938\) − 10.1436i − 0.331200i
\(939\) 14.3923 0.469675
\(940\) 0 0
\(941\) −30.5885 −0.997155 −0.498578 0.866845i \(-0.666144\pi\)
−0.498578 + 0.866845i \(0.666144\pi\)
\(942\) − 24.9282i − 0.812205i
\(943\) 16.3923i 0.533807i
\(944\) 47.3205 1.54015
\(945\) 0 0
\(946\) −13.6077 −0.442424
\(947\) 55.8564i 1.81509i 0.419956 + 0.907545i \(0.362046\pi\)
−0.419956 + 0.907545i \(0.637954\pi\)
\(948\) − 2.92820i − 0.0951036i
\(949\) −8.39230 −0.272426
\(950\) 0 0
\(951\) 23.3205 0.756219
\(952\) 0 0
\(953\) − 10.1436i − 0.328583i −0.986412 0.164292i \(-0.947466\pi\)
0.986412 0.164292i \(-0.0525338\pi\)
\(954\) 4.39230 0.142206
\(955\) 0 0
\(956\) −20.1962 −0.653190
\(957\) − 2.78461i − 0.0900136i
\(958\) 33.8038i 1.09215i
\(959\) 5.75129 0.185719
\(960\) 0 0
\(961\) −6.71281 −0.216542
\(962\) − 19.8564i − 0.640196i
\(963\) 0 0
\(964\) 16.9282 0.545221
\(965\) 0 0
\(966\) −4.39230 −0.141320
\(967\) 29.1244i 0.936576i 0.883576 + 0.468288i \(0.155129\pi\)
−0.883576 + 0.468288i \(0.844871\pi\)
\(968\) − 16.2679i − 0.522872i
\(969\) 0 0
\(970\) 0 0
\(971\) 27.7128 0.889346 0.444673 0.895693i \(-0.353320\pi\)
0.444673 + 0.895693i \(0.353320\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 9.07180i − 0.290828i
\(974\) 20.1051 0.644210
\(975\) 0 0
\(976\) 67.3205 2.15488
\(977\) 51.0333i 1.63270i 0.577557 + 0.816350i \(0.304006\pi\)
−0.577557 + 0.816350i \(0.695994\pi\)
\(978\) − 22.0526i − 0.705163i
\(979\) −9.21539 −0.294525
\(980\) 0 0
\(981\) 6.39230 0.204091
\(982\) − 38.1962i − 1.21889i
\(983\) 6.67949i 0.213043i 0.994310 + 0.106521i \(0.0339713\pi\)
−0.994310 + 0.106521i \(0.966029\pi\)
\(984\) 8.19615 0.261284
\(985\) 0 0
\(986\) 0 0
\(987\) 2.53590i 0.0807185i
\(988\) 2.73205i 0.0869181i
\(989\) −21.4641 −0.682519
\(990\) 0 0
\(991\) 26.9282 0.855403 0.427701 0.903920i \(-0.359324\pi\)
0.427701 + 0.903920i \(0.359324\pi\)
\(992\) 25.6077i 0.813045i
\(993\) − 29.7128i − 0.942908i
\(994\) −20.7846 −0.659248
\(995\) 0 0
\(996\) 0.928203 0.0294112
\(997\) − 38.3923i − 1.21590i −0.793977 0.607948i \(-0.791993\pi\)
0.793977 0.607948i \(-0.208007\pi\)
\(998\) − 30.2487i − 0.957506i
\(999\) −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.c.k.799.3 4
5.2 odd 4 1425.2.a.o.1.1 2
5.3 odd 4 285.2.a.e.1.2 2
5.4 even 2 inner 1425.2.c.k.799.2 4
15.2 even 4 4275.2.a.t.1.2 2
15.8 even 4 855.2.a.f.1.1 2
20.3 even 4 4560.2.a.bh.1.1 2
95.18 even 4 5415.2.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.e.1.2 2 5.3 odd 4
855.2.a.f.1.1 2 15.8 even 4
1425.2.a.o.1.1 2 5.2 odd 4
1425.2.c.k.799.2 4 5.4 even 2 inner
1425.2.c.k.799.3 4 1.1 even 1 trivial
4275.2.a.t.1.2 2 15.2 even 4
4560.2.a.bh.1.1 2 20.3 even 4
5415.2.a.r.1.1 2 95.18 even 4