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Properties

Label 126.4.t.a
Level $126$
Weight $4$
Character orbit 126.t
Analytic conductor $7.434$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [126,4,Mod(47,126)] 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("126.47"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 5])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 126.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.43424066072\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 96 q^{4} - 12 q^{7} + 18 q^{9} - 72 q^{13} - 132 q^{14} + 132 q^{15} - 384 q^{16} - 144 q^{17} - 72 q^{18} + 270 q^{21} + 48 q^{24} + 1200 q^{25} - 120 q^{26} + 450 q^{27} + 48 q^{28} + 42 q^{29}+ \cdots - 5076 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −1.73205 + 1.00000i −5.19296 + 0.182208i 2.00000 − 3.46410i 2.96004 8.81226 − 5.50855i 8.40652 + 16.5024i 8.00000i 26.9336 − 1.89240i −5.12693 + 2.96004i
47.2 −1.73205 + 1.00000i −4.77039 + 2.05995i 2.00000 − 3.46410i −18.1346 6.20260 − 8.33833i 0.905853 − 18.4981i 8.00000i 18.5132 − 19.6535i 31.4101 − 18.1346i
47.3 −1.73205 + 1.00000i −3.82633 − 3.51556i 2.00000 − 3.46410i −12.4256 10.1430 + 2.26280i −14.5199 + 11.4967i 8.00000i 2.28162 + 26.9034i 21.5217 − 12.4256i
47.4 −1.73205 + 1.00000i −3.18367 − 4.10661i 2.00000 − 3.46410i 6.95410 9.62089 + 3.92919i 3.23606 − 18.2353i 8.00000i −6.72849 + 26.1482i −12.0449 + 6.95410i
47.5 −1.73205 + 1.00000i −2.30946 + 4.65472i 2.00000 − 3.46410i 10.9611 −0.654618 − 10.3717i 16.8715 − 7.63883i 8.00000i −16.3328 − 21.4998i −18.9851 + 10.9611i
47.6 −1.73205 + 1.00000i −1.51121 + 4.97154i 2.00000 − 3.46410i 2.72245 −2.35405 − 10.1222i −16.6479 + 8.11457i 8.00000i −22.4325 − 15.0261i −4.71542 + 2.72245i
47.7 −1.73205 + 1.00000i 1.41694 − 4.99923i 2.00000 − 3.46410i 18.3227 2.54502 + 10.0759i 9.14915 + 16.1026i 8.00000i −22.9846 − 14.1672i −31.7359 + 18.3227i
47.8 −1.73205 + 1.00000i 2.80998 + 4.37081i 2.00000 − 3.46410i −7.16324 −9.23784 − 4.76049i 18.4589 − 1.50665i 8.00000i −11.2080 + 24.5638i 12.4071 − 7.16324i
47.9 −1.73205 + 1.00000i 2.86325 − 4.33610i 2.00000 − 3.46410i −6.26999 −0.623188 + 10.3736i −17.7034 − 5.43965i 8.00000i −10.6036 − 24.8307i 10.8599 − 6.26999i
47.10 −1.73205 + 1.00000i 4.56390 − 2.48412i 2.00000 − 3.46410i −16.3094 −5.42078 + 8.86652i 17.1488 + 6.99427i 8.00000i 14.6583 − 22.6745i 28.2488 − 16.3094i
47.11 −1.73205 + 1.00000i 4.81579 + 1.95145i 2.00000 − 3.46410i 3.73919 −10.2926 + 1.43579i −9.62842 + 15.8207i 8.00000i 19.3837 + 18.7955i −6.47647 + 3.73919i
47.12 −1.73205 + 1.00000i 5.19018 − 0.249057i 2.00000 − 3.46410i 14.6433 −8.74060 + 5.62156i 0.375449 − 18.5165i 8.00000i 26.8759 − 2.58530i −25.3629 + 14.6433i
47.13 1.73205 − 1.00000i −4.57477 − 2.46403i 2.00000 − 3.46410i 19.1314 −10.3878 + 0.306944i 4.28063 + 18.0188i − 8.00000i 14.8571 + 22.5448i 33.1365 − 19.1314i
47.14 1.73205 − 1.00000i −4.42087 + 2.73055i 2.00000 − 3.46410i 3.77473 −4.92662 + 9.15033i −12.9841 − 13.2066i − 8.00000i 12.0882 − 24.1428i 6.53802 − 3.77473i
47.15 1.73205 − 1.00000i −4.20058 − 3.05861i 2.00000 − 3.46410i −3.71189 −10.3342 − 1.09708i −15.6223 − 9.94703i − 8.00000i 8.28983 + 25.6959i −6.42918 + 3.71189i
47.16 1.73205 − 1.00000i −3.92150 + 3.40908i 2.00000 − 3.46410i 2.79370 −3.38317 + 9.82620i 2.52260 + 18.3477i − 8.00000i 3.75640 − 26.7374i 4.83883 − 2.79370i
47.17 1.73205 − 1.00000i −2.40296 + 4.60715i 2.00000 − 3.46410i −16.9865 0.445099 + 10.3828i 16.2157 − 8.94719i − 8.00000i −15.4516 − 22.1416i −29.4214 + 16.9865i
47.18 1.73205 − 1.00000i −0.660361 − 5.15402i 2.00000 − 3.46410i −14.9619 −6.29780 − 8.26666i −6.35399 + 17.3962i − 8.00000i −26.1278 + 6.80703i −25.9148 + 14.9619i
47.19 1.73205 − 1.00000i 0.860451 − 5.12441i 2.00000 − 3.46410i 6.32106 −3.63407 − 9.73620i 16.5946 − 8.22317i − 8.00000i −25.5192 − 8.81861i 10.9484 − 6.32106i
47.20 1.73205 − 1.00000i 1.98049 + 4.80392i 2.00000 − 3.46410i 12.8603 8.23423 + 6.34015i 7.27304 − 17.0324i − 8.00000i −19.1553 + 19.0282i 22.2747 − 12.8603i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.24
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.4.t.a yes 48
3.b odd 2 1 378.4.t.a 48
7.d odd 6 1 126.4.l.a ✓ 48
9.c even 3 1 378.4.l.a 48
9.d odd 6 1 126.4.l.a ✓ 48
21.g even 6 1 378.4.l.a 48
63.k odd 6 1 378.4.t.a 48
63.s even 6 1 inner 126.4.t.a yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.l.a ✓ 48 7.d odd 6 1
126.4.l.a ✓ 48 9.d odd 6 1
126.4.t.a yes 48 1.a even 1 1 trivial
126.4.t.a yes 48 63.s even 6 1 inner
378.4.l.a 48 9.c even 3 1
378.4.l.a 48 21.g even 6 1
378.4.t.a 48 3.b odd 2 1
378.4.t.a 48 63.k odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(126, [\chi])\).