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Properties

Label 6336.2.d.g
Level $6336$
Weight $2$
Character orbit 6336.d
Analytic conductor $50.593$
Analytic rank $0$
Dimension $10$
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] = [6336,2,Mod(3455,6336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6336.3455");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6336.d (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: \(50.5932147207\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.5236158660608.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{8} - 4x^{7} + 3x^{6} + 8x^{5} + 6x^{4} - 16x^{3} - 16x^{2} + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 396)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} - \beta_{9} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} - \beta_{9} q^{7} - q^{11} + (\beta_{7} - 1) q^{13} + (\beta_{8} + \beta_{5} - \beta_1) q^{17} + ( - \beta_{8} + \beta_{6} - \beta_1) q^{19} + (\beta_{7} + \beta_{3} + \beta_{2} + 1) q^{23} + (\beta_{7} + \beta_{4} + \beta_{2} - 1) q^{25} + (\beta_{9} - 2 \beta_{8} + \cdots + \beta_1) q^{29}+ \cdots + ( - 4 \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{11} - 8 q^{13} + 8 q^{23} - 10 q^{25} + 16 q^{35} + 4 q^{37} - 34 q^{49} + 40 q^{59} - 20 q^{61} + 16 q^{71} + 24 q^{73} + 8 q^{83} + 36 q^{85} + 96 q^{95} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{8} - 4x^{7} + 3x^{6} + 8x^{5} + 6x^{4} - 16x^{3} - 16x^{2} + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} - 2\nu^{8} + 2\nu^{7} + 8\nu^{6} + 3\nu^{5} - 14\nu^{4} - 14\nu^{3} - 20\nu^{2} + 24\nu + 16 ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{9} - 2\nu^{8} + 6\nu^{7} - 3\nu^{5} + 2\nu^{4} + 6\nu^{3} - 4\nu^{2} - 8\nu - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + 2\nu^{7} - 4\nu^{6} - 5\nu^{5} - 8\nu^{4} + 2\nu^{3} + 16\nu^{2} + 24\nu - 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{9} - 2\nu^{8} - 2\nu^{7} + 11\nu^{5} + 2\nu^{4} - 10\nu^{3} - 12\nu^{2} + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{9} + 2\nu^{8} - 2\nu^{7} - 8\nu^{6} - 7\nu^{5} + 14\nu^{4} + 30\nu^{3} + 4\nu^{2} - 40\nu - 48 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} - 2\nu^{7} - 4\nu^{6} + 3\nu^{5} + 8\nu^{4} + 6\nu^{3} - 16\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{9} - \nu^{8} + 6\nu^{6} + 5\nu^{5} - 3\nu^{4} - 12\nu^{3} + 2\nu^{2} + 20\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{9} + 4\nu^{8} + 2\nu^{7} - 12\nu^{6} - 7\nu^{5} + 4\nu^{4} + 42\nu^{3} - 32\nu - 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7\nu^{9} + 10\nu^{8} - 2\nu^{7} - 24\nu^{6} - 11\nu^{5} + 38\nu^{4} + 46\nu^{3} - 28\nu^{2} - 88\nu - 80 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{3} - 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{9} + 2\beta_{8} + \beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} - \beta_{3} + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{9} - 2\beta_{8} + 3\beta_{7} + \beta_{6} + 4\beta_{5} - \beta_{4} + \beta_{3} + 2\beta_{2} - 2\beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{8} + 3\beta_{7} - \beta_{6} - 4\beta_{5} + 5\beta_{4} - \beta_{3} - 4\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5\beta_{7} - \beta_{6} + 10\beta_{5} - 3\beta_{4} - 3\beta_{3} + 6\beta _1 - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6\beta_{9} + 2\beta_{8} + 5\beta_{7} - 3\beta_{6} - 6\beta_{5} + 3\beta_{4} + 3\beta_{3} + 8\beta_{2} + 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 10 \beta_{9} + 6 \beta_{8} + 7 \beta_{7} - 3 \beta_{6} - 20 \beta_{5} - 5 \beta_{4} - 3 \beta_{3} + \cdots - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8\beta_{9} - 4\beta_{8} + 7\beta_{7} - 5\beta_{6} + 28\beta_{5} + 9\beta_{4} + 11\beta_{3} + 20\beta _1 + 24 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6336\mathbb{Z}\right)^\times\).

\(n\) \(1729\) \(3521\) \(4159\) \(4357\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3455.1
1.17660 + 0.784613i
−0.929398 − 1.06594i
−0.382373 + 1.36154i
−1.27340 − 0.615184i
1.40857 − 0.126175i
1.40857 + 0.126175i
−1.27340 + 0.615184i
−0.382373 − 1.36154i
−0.929398 + 1.06594i
1.17660 − 0.784613i
0 0 0 − 3.84771i 0 3.08753i 0 0 0
3455.2 0 0 0 − 3.24506i 0 1.06027i 0 0 0
3455.3 0 0 0 − 2.05483i 0 − 4.18915i 0 0 0
3455.4 0 0 0 − 0.488921i 0 − 1.61287i 0 0 0
3455.5 0 0 0 − 0.450958i 0 4.60354i 0 0 0
3455.6 0 0 0 0.450958i 0 − 4.60354i 0 0 0
3455.7 0 0 0 0.488921i 0 1.61287i 0 0 0
3455.8 0 0 0 2.05483i 0 4.18915i 0 0 0
3455.9 0 0 0 3.24506i 0 − 1.06027i 0 0 0
3455.10 0 0 0 3.84771i 0 − 3.08753i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3455.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 6336.2.d.g 10
3.b odd 2 1 6336.2.d.h 10
4.b odd 2 1 6336.2.d.h 10
8.b even 2 1 396.2.c.b yes 10
8.d odd 2 1 396.2.c.a ✓ 10
12.b even 2 1 inner 6336.2.d.g 10
24.f even 2 1 396.2.c.b yes 10
24.h odd 2 1 396.2.c.a ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
396.2.c.a ✓ 10 8.d odd 2 1
396.2.c.a ✓ 10 24.h odd 2 1
396.2.c.b yes 10 8.b even 2 1
396.2.c.b yes 10 24.f even 2 1
6336.2.d.g 10 1.a even 1 1 trivial
6336.2.d.g 10 12.b even 2 1 inner
6336.2.d.h 10 3.b odd 2 1
6336.2.d.h 10 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6336, [\chi])\):

\( T_{5}^{10} + 30T_{5}^{8} + 276T_{5}^{6} + 776T_{5}^{4} + 304T_{5}^{2} + 32 \) Copy content Toggle raw display
\( T_{23}^{5} - 4T_{23}^{4} - 58T_{23}^{3} + 352T_{23}^{2} - 404T_{23} - 336 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 30 T^{8} + \cdots + 32 \) Copy content Toggle raw display
$7$ \( T^{10} + 52 T^{8} + \cdots + 10368 \) Copy content Toggle raw display
$11$ \( (T + 1)^{10} \) Copy content Toggle raw display
$13$ \( (T^{5} + 4 T^{4} - 22 T^{3} + \cdots + 32)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 58 T^{8} + \cdots + 512 \) Copy content Toggle raw display
$19$ \( T^{10} + 152 T^{8} + \cdots + 56448 \) Copy content Toggle raw display
$23$ \( (T^{5} - 4 T^{4} + \cdots - 336)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 210 T^{8} + \cdots + 19071488 \) Copy content Toggle raw display
$31$ \( T^{10} + 216 T^{8} + \cdots + 32514048 \) Copy content Toggle raw display
$37$ \( (T^{5} - 2 T^{4} - 36 T^{3} + \cdots - 32)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 250 T^{8} + \cdots + 3527168 \) Copy content Toggle raw display
$43$ \( T^{10} + 120 T^{8} + \cdots + 10368 \) Copy content Toggle raw display
$47$ \( (T^{5} - 130 T^{3} + \cdots + 10272)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 254 T^{8} + \cdots + 3987488 \) Copy content Toggle raw display
$59$ \( (T^{5} - 20 T^{4} + \cdots + 768)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 10 T^{4} + \cdots - 1048)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 272 T^{8} + \cdots + 1179648 \) Copy content Toggle raw display
$71$ \( (T^{5} - 8 T^{4} + \cdots - 14208)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 12 T^{4} + \cdots - 448)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 9937116288 \) Copy content Toggle raw display
$83$ \( (T^{5} - 4 T^{4} + \cdots - 1152)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 589205792 \) Copy content Toggle raw display
$97$ \( (T^{5} - 12 T^{4} + \cdots + 6784)^{2} \) Copy content Toggle raw display
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