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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,9,Mod(127,288)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("288.127"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 288.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,728] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(117.325039698\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{39})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 19x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(-3.12250 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 288.127
Dual form 288.9.g.b.127.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-217.680 q^{5} -1726.24i q^{7} +5955.28i q^{11} -49121.2 q^{13} +55631.9 q^{17} -202659. i q^{19} +209565. i q^{23} -343240. q^{25} -3082.26 q^{29} +1.36035e6i q^{31} +375768. i q^{35} +1.31420e6 q^{37} -4.54489e6 q^{41} -3.48760e6i q^{43} -7.23361e6i q^{47} +2.78490e6 q^{49} +2.48557e6 q^{53} -1.29634e6i q^{55} +1.48790e7i q^{59} +4.13648e6 q^{61} +1.06927e7 q^{65} -6.04481e6i q^{67} +4.20242e7i q^{71} +9.00500e6 q^{73} +1.02802e7 q^{77} -6.81741e6i q^{79} -5.81780e7i q^{83} -1.21100e7 q^{85} -7.82499e7 q^{89} +8.47949e7i q^{91} +4.41147e7i q^{95} +6.01423e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 728 q^{5} - 12632 q^{13} + 391992 q^{17} - 791028 q^{25} + 705496 q^{29} + 4443048 q^{37} - 2953352 q^{41} + 2839044 q^{49} + 4501848 q^{53} - 40159064 q^{61} + 71183216 q^{65} - 5920824 q^{73} - 55000448 q^{77}+ \cdots + 56444872 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −217.680 −0.348288 −0.174144 − 0.984720i \(-0.555716\pi\)
−0.174144 + 0.984720i \(0.555716\pi\)
\(6\) 0 0
\(7\) − 1726.24i − 0.718967i −0.933151 − 0.359483i \(-0.882953\pi\)
0.933151 − 0.359483i \(-0.117047\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5955.28i 0.406754i 0.979101 + 0.203377i \(0.0651917\pi\)
−0.979101 + 0.203377i \(0.934808\pi\)
\(12\) 0 0
\(13\) −49121.2 −1.71987 −0.859935 − 0.510404i \(-0.829496\pi\)
−0.859935 + 0.510404i \(0.829496\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 55631.9 0.666083 0.333042 − 0.942912i \(-0.391925\pi\)
0.333042 + 0.942912i \(0.391925\pi\)
\(18\) 0 0
\(19\) − 202659.i − 1.55507i −0.628837 − 0.777537i \(-0.716469\pi\)
0.628837 − 0.777537i \(-0.283531\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 209565.i 0.748872i 0.927253 + 0.374436i \(0.122164\pi\)
−0.927253 + 0.374436i \(0.877836\pi\)
\(24\) 0 0
\(25\) −343240. −0.878696
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3082.26 −0.00435790 −0.00217895 − 0.999998i \(-0.500694\pi\)
−0.00217895 + 0.999998i \(0.500694\pi\)
\(30\) 0 0
\(31\) 1.36035e6i 1.47300i 0.676436 + 0.736502i \(0.263524\pi\)
−0.676436 + 0.736502i \(0.736476\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 375768.i 0.250407i
\(36\) 0 0
\(37\) 1.31420e6 0.701220 0.350610 − 0.936522i \(-0.385974\pi\)
0.350610 + 0.936522i \(0.385974\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.54489e6 −1.60838 −0.804189 − 0.594374i \(-0.797400\pi\)
−0.804189 + 0.594374i \(0.797400\pi\)
\(42\) 0 0
\(43\) − 3.48760e6i − 1.02013i −0.860137 − 0.510063i \(-0.829622\pi\)
0.860137 − 0.510063i \(-0.170378\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 7.23361e6i − 1.48239i −0.671288 − 0.741197i \(-0.734259\pi\)
0.671288 − 0.741197i \(-0.265741\pi\)
\(48\) 0 0
\(49\) 2.78490e6 0.483087
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.48557e6 0.315009 0.157505 − 0.987518i \(-0.449655\pi\)
0.157505 + 0.987518i \(0.449655\pi\)
\(54\) 0 0
\(55\) − 1.29634e6i − 0.141667i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.48790e7i 1.22791i 0.789340 + 0.613956i \(0.210423\pi\)
−0.789340 + 0.613956i \(0.789577\pi\)
\(60\) 0 0
\(61\) 4.13648e6 0.298752 0.149376 − 0.988780i \(-0.452273\pi\)
0.149376 + 0.988780i \(0.452273\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.06927e7 0.599009
\(66\) 0 0
\(67\) − 6.04481e6i − 0.299974i −0.988688 − 0.149987i \(-0.952077\pi\)
0.988688 − 0.149987i \(-0.0479232\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.20242e7i 1.65374i 0.562397 + 0.826868i \(0.309879\pi\)
−0.562397 + 0.826868i \(0.690121\pi\)
\(72\) 0 0
\(73\) 9.00500e6 0.317097 0.158548 − 0.987351i \(-0.449319\pi\)
0.158548 + 0.987351i \(0.449319\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.02802e7 0.292442
\(78\) 0 0
\(79\) − 6.81741e6i − 0.175029i −0.996163 − 0.0875147i \(-0.972108\pi\)
0.996163 − 0.0875147i \(-0.0278925\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 5.81780e7i − 1.22587i −0.790132 − 0.612937i \(-0.789988\pi\)
0.790132 − 0.612937i \(-0.210012\pi\)
\(84\) 0 0
\(85\) −1.21100e7 −0.231989
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.82499e7 −1.24716 −0.623582 − 0.781758i \(-0.714323\pi\)
−0.623582 + 0.781758i \(0.714323\pi\)
\(90\) 0 0
\(91\) 8.47949e7i 1.23653i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.41147e7i 0.541613i
\(96\) 0 0
\(97\) 6.01423e7 0.679350 0.339675 − 0.940543i \(-0.389683\pi\)
0.339675 + 0.940543i \(0.389683\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 288.9.g.b.127.1 4
3.2 odd 2 32.9.c.a.31.2 ✓ 4
4.3 odd 2 inner 288.9.g.b.127.2 4
12.11 even 2 32.9.c.a.31.3 yes 4
24.5 odd 2 64.9.c.f.63.3 4
24.11 even 2 64.9.c.f.63.2 4
48.5 odd 4 256.9.d.h.127.2 4
48.11 even 4 256.9.d.b.127.4 4
48.29 odd 4 256.9.d.b.127.3 4
48.35 even 4 256.9.d.h.127.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.9.c.a.31.2 ✓ 4 3.2 odd 2
32.9.c.a.31.3 yes 4 12.11 even 2
64.9.c.f.63.2 4 24.11 even 2
64.9.c.f.63.3 4 24.5 odd 2
256.9.d.b.127.3 4 48.29 odd 4
256.9.d.b.127.4 4 48.11 even 4
256.9.d.h.127.1 4 48.35 even 4
256.9.d.h.127.2 4 48.5 odd 4
288.9.g.b.127.1 4 1.1 even 1 trivial
288.9.g.b.127.2 4 4.3 odd 2 inner