↑

Properties

Label 828.2.e.b
Level $828$
Weight $2$
Character orbit 828.e
Analytic conductor $6.612$
Analytic rank $0$
Dimension $6$
CM discriminant -23
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [828,2,Mod(91,828)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(828, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("828.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 828.e (of order \(2\), degree \(1\), 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: \(6.61161328736\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8869743.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{3} + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 92)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + \beta_{3} q^{4} + ( - \beta_{2} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + \beta_{3} q^{4} + ( - \beta_{2} + 2) q^{8} + (2 \beta_{5} + 2 \beta_{4} + \cdots + \beta_1) q^{13}+ \cdots - 7 \beta_{5} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 9 q^{8} + 30 q^{25} + 27 q^{26} - 42 q^{49} + 3 q^{52} - 15 q^{58} - 45 q^{62} - 21 q^{64} + 33 q^{82} - 39 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{3} + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + \nu^{2} + 4\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 3\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - \nu^{2} + 4\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 3\nu^{2} ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} - \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{4} - 4\beta_{3} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{5} - 3\beta_{4} + 3\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/828\mathbb{Z}\right)^\times\).

\(n\) \(415\) \(461\) \(649\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−1.07255 + 0.921756i
−1.07255 − 0.921756i
−0.261988 + 1.38973i
−0.261988 − 1.38973i
1.33454 + 0.467979i
1.33454 − 0.467979i
−1.07255 − 0.921756i 0 0.300733 + 1.97726i 0 0 0 1.50000 − 2.39792i 0 0
91.2 −1.07255 + 0.921756i 0 0.300733 − 1.97726i 0 0 0 1.50000 + 2.39792i 0 0
91.3 −0.261988 − 1.38973i 0 −1.86272 + 0.728188i 0 0 0 1.50000 + 2.39792i 0 0
91.4 −0.261988 + 1.38973i 0 −1.86272 − 0.728188i 0 0 0 1.50000 − 2.39792i 0 0
91.5 1.33454 − 0.467979i 0 1.56199 − 1.24907i 0 0 0 1.50000 − 2.39792i 0 0
91.6 1.33454 + 0.467979i 0 1.56199 + 1.24907i 0 0 0 1.50000 + 2.39792i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
4.b odd 2 1 inner
92.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 828.2.e.b 6
3.b odd 2 1 92.2.b.b ✓ 6
4.b odd 2 1 inner 828.2.e.b 6
12.b even 2 1 92.2.b.b ✓ 6
23.b odd 2 1 CM 828.2.e.b 6
24.f even 2 1 1472.2.c.c 6
24.h odd 2 1 1472.2.c.c 6
69.c even 2 1 92.2.b.b ✓ 6
92.b even 2 1 inner 828.2.e.b 6
276.h odd 2 1 92.2.b.b ✓ 6
552.b even 2 1 1472.2.c.c 6
552.h odd 2 1 1472.2.c.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.2.b.b ✓ 6 3.b odd 2 1
92.2.b.b ✓ 6 12.b even 2 1
92.2.b.b ✓ 6 69.c even 2 1
92.2.b.b ✓ 6 276.h odd 2 1
828.2.e.b 6 1.a even 1 1 trivial
828.2.e.b 6 4.b odd 2 1 inner
828.2.e.b 6 23.b odd 2 1 CM
828.2.e.b 6 92.b even 2 1 inner
1472.2.c.c 6 24.f even 2 1
1472.2.c.c 6 24.h odd 2 1
1472.2.c.c 6 552.b even 2 1
1472.2.c.c 6 552.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(828, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 3T^{3} + 8 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( (T^{3} - 39 T + 74)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{3} \) Copy content Toggle raw display
$29$ \( (T^{3} - 87 T - 282)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 186 T^{4} + \cdots + 828 \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( (T^{3} - 123 T + 426)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} \) Copy content Toggle raw display
$47$ \( T^{6} + 282 T^{4} + \cdots + 412988 \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( (T^{2} + 92)^{3} \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( T^{6} + 426 T^{4} + \cdots + 48668 \) Copy content Toggle raw display
$73$ \( (T^{3} - 219 T - 1226)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
show more
show less