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Properties

Label 126.4.g.d
Level $126$
Weight $4$
Character orbit 126.g
Analytic conductor $7.434$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [126,4,Mod(37,126)] 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("126.37"); 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([0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 126.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,0,-4,-9] 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: \(7.43424066072\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} + (9 \zeta_{6} - 9) q^{5} + ( - 14 \zeta_{6} - 7) q^{7} - 8 q^{8} + 18 \zeta_{6} q^{10} - 57 \zeta_{6} q^{11} - 70 q^{13} + (14 \zeta_{6} - 42) q^{14} + \cdots + (294 \zeta_{6} + 490) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{4} - 9 q^{5} - 28 q^{7} - 16 q^{8} + 18 q^{10} - 57 q^{11} - 140 q^{13} - 70 q^{14} - 16 q^{16} + 51 q^{17} - 5 q^{19} + 72 q^{20} - 228 q^{22} + 69 q^{23} + 44 q^{25} - 140 q^{26}+ \cdots + 1274 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 + 0.866025i
0.500000 − 0.866025i
1.00000 − 1.73205i 0 −2.00000 − 3.46410i −4.50000 + 7.79423i 0 −14.0000 − 12.1244i −8.00000 0 9.00000 + 15.5885i
109.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −4.50000 − 7.79423i 0 −14.0000 + 12.1244i −8.00000 0 9.00000 − 15.5885i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.4.g.d 2
3.b odd 2 1 14.4.c.a ✓ 2
7.b odd 2 1 882.4.g.u 2
7.c even 3 1 inner 126.4.g.d 2
7.c even 3 1 882.4.a.f 1
7.d odd 6 1 882.4.a.c 1
7.d odd 6 1 882.4.g.u 2
12.b even 2 1 112.4.i.a 2
15.d odd 2 1 350.4.e.e 2
15.e even 4 2 350.4.j.b 4
21.c even 2 1 98.4.c.a 2
21.g even 6 1 98.4.a.f 1
21.g even 6 1 98.4.c.a 2
21.h odd 6 1 14.4.c.a ✓ 2
21.h odd 6 1 98.4.a.d 1
24.f even 2 1 448.4.i.e 2
24.h odd 2 1 448.4.i.b 2
84.j odd 6 1 784.4.a.c 1
84.n even 6 1 112.4.i.a 2
84.n even 6 1 784.4.a.p 1
105.o odd 6 1 350.4.e.e 2
105.o odd 6 1 2450.4.a.q 1
105.p even 6 1 2450.4.a.d 1
105.x even 12 2 350.4.j.b 4
168.s odd 6 1 448.4.i.b 2
168.v even 6 1 448.4.i.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a ✓ 2 3.b odd 2 1
14.4.c.a ✓ 2 21.h odd 6 1
98.4.a.d 1 21.h odd 6 1
98.4.a.f 1 21.g even 6 1
98.4.c.a 2 21.c even 2 1
98.4.c.a 2 21.g even 6 1
112.4.i.a 2 12.b even 2 1
112.4.i.a 2 84.n even 6 1
126.4.g.d 2 1.a even 1 1 trivial
126.4.g.d 2 7.c even 3 1 inner
350.4.e.e 2 15.d odd 2 1
350.4.e.e 2 105.o odd 6 1
350.4.j.b 4 15.e even 4 2
350.4.j.b 4 105.x even 12 2
448.4.i.b 2 24.h odd 2 1
448.4.i.b 2 168.s odd 6 1
448.4.i.e 2 24.f even 2 1
448.4.i.e 2 168.v even 6 1
784.4.a.c 1 84.j odd 6 1
784.4.a.p 1 84.n even 6 1
882.4.a.c 1 7.d odd 6 1
882.4.a.f 1 7.c even 3 1
882.4.g.u 2 7.b odd 2 1
882.4.g.u 2 7.d odd 6 1
2450.4.a.d 1 105.p even 6 1
2450.4.a.q 1 105.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 9T_{5} + 81 \) acting on \(S_{4}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$7$ \( T^{2} + 28T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} + 57T + 3249 \) Copy content Toggle raw display
$13$ \( (T + 70)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 51T + 2601 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} - 69T + 4761 \) Copy content Toggle raw display
$29$ \( (T + 114)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 23T + 529 \) Copy content Toggle raw display
$37$ \( T^{2} - 253T + 64009 \) Copy content Toggle raw display
$41$ \( (T - 42)^{2} \) Copy content Toggle raw display
$43$ \( (T + 124)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 201T + 40401 \) Copy content Toggle raw display
$53$ \( T^{2} + 393T + 154449 \) Copy content Toggle raw display
$59$ \( T^{2} - 219T + 47961 \) Copy content Toggle raw display
$61$ \( T^{2} - 709T + 502681 \) Copy content Toggle raw display
$67$ \( T^{2} + 419T + 175561 \) Copy content Toggle raw display
$71$ \( (T - 96)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 313T + 97969 \) Copy content Toggle raw display
$79$ \( T^{2} + 461T + 212521 \) Copy content Toggle raw display
$83$ \( (T - 588)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 1017 T + 1034289 \) Copy content Toggle raw display
$97$ \( (T + 1834)^{2} \) Copy content Toggle raw display
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