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Properties

Label 1728.3.g.l
Level $1728$
Weight $3$
Character orbit 1728.g
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [1728,3,Mod(703,1728)] 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("1728.703"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.207360000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + \beta_{7} q^{7} + \beta_{3} q^{11} + ( - \beta_{6} + 1) q^{13} + (\beta_{4} + 2 \beta_1) q^{17} + \beta_{5} q^{19} + ( - \beta_{3} + \beta_{2}) q^{23} + ( - 2 \beta_{6} + 3) q^{25}+ \cdots + (8 \beta_{6} - 31) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{13} + 24 q^{25} + 56 q^{37} - 112 q^{49} - 40 q^{61} + 232 q^{73} - 416 q^{85} - 248 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{7} + 16\nu^{5} + 64\nu^{3} + 8\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} + 16\nu^{5} + 96\nu^{3} + 136\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -15\nu^{7} - 112\nu^{5} - 576\nu^{3} - 808\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -57\nu^{7} - 304\nu^{5} - 1600\nu^{3} - 152\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{6} - 48\nu^{4} - 288\nu^{2} - 120 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{6} + 216 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\nu^{6} + 80\nu^{4} + 352\nu^{2} + 152 ) / 32 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{4} + 3\beta_{2} + 7\beta_1 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} + \beta_{6} - 5\beta_{5} - 18 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{4} - 19\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{7} + 3\beta_{6} + 11\beta_{5} - 42 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{4} - 6\beta_{3} - 9\beta_{2} + 25\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -16\beta_{6} + 216 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5\beta_{4} + 32\beta_{3} + 47\beta_{2} + 131\beta_1 ) / 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
1.14412 + 1.98168i
1.14412 − 1.98168i
−0.437016 + 0.756934i
−0.437016 − 0.756934i
0.437016 − 0.756934i
0.437016 + 0.756934i
−1.14412 − 1.98168i
−1.14412 + 1.98168i
0 0 0 −7.40492 0 − 9.47802i 0 0 0
703.2 0 0 0 −7.40492 0 9.47802i 0 0 0
703.3 0 0 0 −1.08036 0 − 6.01392i 0 0 0
703.4 0 0 0 −1.08036 0 6.01392i 0 0 0
703.5 0 0 0 1.08036 0 − 6.01392i 0 0 0
703.6 0 0 0 1.08036 0 6.01392i 0 0 0
703.7 0 0 0 7.40492 0 − 9.47802i 0 0 0
703.8 0 0 0 7.40492 0 9.47802i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 703.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.g.l 8
3.b odd 2 1 inner 1728.3.g.l 8
4.b odd 2 1 inner 1728.3.g.l 8
8.b even 2 1 108.3.d.d ✓ 8
8.d odd 2 1 108.3.d.d ✓ 8
12.b even 2 1 inner 1728.3.g.l 8
24.f even 2 1 108.3.d.d ✓ 8
24.h odd 2 1 108.3.d.d ✓ 8
72.j odd 6 1 324.3.f.o 8
72.j odd 6 1 324.3.f.p 8
72.l even 6 1 324.3.f.o 8
72.l even 6 1 324.3.f.p 8
72.n even 6 1 324.3.f.o 8
72.n even 6 1 324.3.f.p 8
72.p odd 6 1 324.3.f.o 8
72.p odd 6 1 324.3.f.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.d.d ✓ 8 8.b even 2 1
108.3.d.d ✓ 8 8.d odd 2 1
108.3.d.d ✓ 8 24.f even 2 1
108.3.d.d ✓ 8 24.h odd 2 1
324.3.f.o 8 72.j odd 6 1
324.3.f.o 8 72.l even 6 1
324.3.f.o 8 72.n even 6 1
324.3.f.o 8 72.p odd 6 1
324.3.f.p 8 72.j odd 6 1
324.3.f.p 8 72.l even 6 1
324.3.f.p 8 72.n even 6 1
324.3.f.p 8 72.p odd 6 1
1728.3.g.l 8 1.a even 1 1 trivial
1728.3.g.l 8 3.b odd 2 1 inner
1728.3.g.l 8 4.b odd 2 1 inner
1728.3.g.l 8 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{4} - 56T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{4} + 126T_{7}^{2} + 3249 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 56 T^{2} + 64)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 126 T^{2} + 3249)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 360 T^{2} + 14400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T - 179)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 1016 T^{2} + 222784)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1512 T^{2} + 553536)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 896 T^{2} + 16384)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2304 T^{2} + 589824)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 14 T - 131)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 3584 T^{2} + 262144)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2880 T^{2} + 921600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 4392 T^{2} + 4562496)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 11744 T^{2} + 33547264)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 7848 T^{2} + 5875776)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 10 T - 155)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 9846 T^{2} + 7601049)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 19872 T^{2} + 90326016)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 58 T + 121)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 24030 T^{2} + 37638225)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1440 T^{2} + 230400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 1016 T^{2} + 222784)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 62 T - 10559)^{4} \) Copy content Toggle raw display
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