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Properties

Label 1728.3.g.n
Level $1728$
Weight $3$
Character orbit 1728.g
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
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] = [1728,3,Mod(703,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
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

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: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{5} + \beta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{5} + \beta_{6} q^{7} + (\beta_{6} + \beta_{5} + \beta_{4}) q^{11} + ( - \beta_1 + 2) q^{13} + (\beta_1 - 6) q^{17} + (\beta_{6} - \beta_{5} - 3 \beta_{3}) q^{19} + (3 \beta_{6} - \beta_{5} + \cdots - \beta_{3}) q^{23}+ \cdots + ( - 2 \beta_{7} + 4 \beta_{2} + \cdots - 17) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} + 16 q^{13} - 48 q^{17} + 48 q^{25} - 32 q^{29} + 96 q^{37} + 128 q^{41} + 168 q^{53} - 32 q^{61} + 112 q^{65} + 24 q^{73} - 440 q^{77} - 144 q^{85} - 624 q^{89} - 136 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 7\nu^{5} + 6\nu^{4} + 9\nu^{3} - 16\nu^{2} + 8\nu + 20 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{7} - 12\nu^{6} + \nu^{5} + 18\nu^{4} - 15\nu^{3} - 20\nu^{2} + 52\nu - 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{7} - 21\nu^{6} + 9\nu^{5} + 33\nu^{4} - 45\nu^{3} - 33\nu^{2} + 120\nu - 90 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -4\nu^{7} + 11\nu^{6} - 6\nu^{5} - 17\nu^{4} + 24\nu^{3} + 15\nu^{2} - 62\nu + 48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19\nu^{7} - 56\nu^{6} + 39\nu^{5} + 74\nu^{4} - 129\nu^{3} - 48\nu^{2} + 332\nu - 288 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 45\nu^{7} - 114\nu^{6} + 57\nu^{5} + 172\nu^{4} - 235\nu^{3} - 178\nu^{2} + 644\nu - 484 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -11\nu^{7} + 33\nu^{6} - 19\nu^{5} - 45\nu^{4} + 75\nu^{3} + 41\nu^{2} - 184\nu + 146 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{2} + \beta _1 + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 3\beta_{6} + 3\beta_{5} - 3\beta_{4} + 3\beta_{3} + 4\beta_{2} + 18 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 3\beta_{6} + 3\beta_{5} - 7\beta_{3} + 4\beta_{2} + \beta _1 - 6 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} - 3\beta_{6} - \beta_{5} - 19\beta_{4} - \beta_{3} - 6 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{7} + 9\beta_{6} - \beta_{5} - 4\beta_{4} - 13\beta_{3} - 6\beta_{2} - 5\beta _1 + 42 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{7} - 3\beta_{4} + 6\beta_{3} - 14\beta_{2} + \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4\beta_{7} + 3\beta_{6} + 2\beta_{5} + 11\beta_{4} + 13\beta_{3} + \beta_{2} - 6\beta _1 + 48 ) / 12 \) 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.

comment: embeddings in the coefficient field
 
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
1.20036 + 0.747754i
1.20036 − 0.747754i
−1.27597 − 0.609843i
−1.27597 + 0.609843i
1.40994 − 0.109843i
1.40994 + 0.109843i
0.665665 + 1.24775i
0.665665 − 1.24775i
0 0 0 −4.18059 0 − 2.58429i 0 0 0
703.2 0 0 0 −4.18059 0 2.58429i 0 0 0
703.3 0 0 0 −3.22512 0 − 6.57221i 0 0 0
703.4 0 0 0 −3.22512 0 6.57221i 0 0 0
703.5 0 0 0 1.76102 0 − 12.0363i 0 0 0
703.6 0 0 0 1.76102 0 12.0363i 0 0 0
703.7 0 0 0 9.64469 0 − 1.12019i 0 0 0
703.8 0 0 0 9.64469 0 1.12019i 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
4.b odd 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.n 8
3.b odd 2 1 1728.3.g.k 8
4.b odd 2 1 inner 1728.3.g.n 8
8.b even 2 1 864.3.g.a ✓ 8
8.d odd 2 1 864.3.g.a ✓ 8
12.b even 2 1 1728.3.g.k 8
24.f even 2 1 864.3.g.c yes 8
24.h odd 2 1 864.3.g.c yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.3.g.a ✓ 8 8.b even 2 1
864.3.g.a ✓ 8 8.d odd 2 1
864.3.g.c yes 8 24.f even 2 1
864.3.g.c yes 8 24.h odd 2 1
1728.3.g.k 8 3.b odd 2 1
1728.3.g.k 8 12.b even 2 1
1728.3.g.n 8 1.a even 1 1 trivial
1728.3.g.n 8 4.b odd 2 1 inner

Hecke kernels

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

\( T_{5}^{4} - 4T_{5}^{3} - 54T_{5}^{2} - 28T_{5} + 229 \) Copy content Toggle raw display
\( T_{7}^{8} + 196T_{7}^{6} + 7758T_{7}^{4} + 51220T_{7}^{2} + 52441 \) 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
$5$ \( (T^{4} - 4 T^{3} + \cdots + 229)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 196 T^{6} + \cdots + 52441 \) Copy content Toggle raw display
$11$ \( T^{8} + 604 T^{6} + \cdots + 269517889 \) Copy content Toggle raw display
$13$ \( (T^{4} - 8 T^{3} + \cdots + 15616)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 24 T^{3} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 1200 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 8962787584 \) Copy content Toggle raw display
$29$ \( (T^{4} + 16 T^{3} + \cdots - 2288)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 3220 T^{6} + \cdots + 116920969 \) Copy content Toggle raw display
$37$ \( (T^{4} - 48 T^{3} + \cdots - 354096)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 64 T^{3} + \cdots - 3217904)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 947180525824 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 1218851328256 \) Copy content Toggle raw display
$53$ \( (T^{4} - 84 T^{3} + \cdots - 1618803)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 157628225880064 \) Copy content Toggle raw display
$61$ \( (T^{4} + 16 T^{3} + \cdots + 3739648)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 54344260884736 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 55832696848384 \) Copy content Toggle raw display
$73$ \( (T^{4} - 12 T^{3} + \cdots - 7709463)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 122448233690689 \) Copy content Toggle raw display
$89$ \( (T^{4} + 312 T^{3} + \cdots - 3170736)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 68 T^{3} + \cdots - 19173311)^{2} \) Copy content Toggle raw display
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