Properties

Label 3264.2.l.b
Level $3264$
Weight $2$
Character orbit 3264.l
Analytic conductor $26.063$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3264,2,Mod(2209,3264)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3264, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3264.2209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3264 = 2^{6} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3264.l (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: \(26.0631712197\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - \beta_{2} q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_{2} q^{5} + q^{9} + 4 q^{11} - \beta_{2} q^{15} + ( - 2 \beta_1 - 1) q^{17} - \beta_1 q^{19} + \beta_{3} q^{23} + 3 q^{25} + q^{27} + \beta_{2} q^{29} + \beta_{3} q^{31} + 4 q^{33} - 3 \beta_{2} q^{37} - 3 \beta_1 q^{43} - \beta_{2} q^{45} + 4 \beta_{2} q^{47} + 7 q^{49} + ( - 2 \beta_1 - 1) q^{51} - 2 \beta_{3} q^{53} - 4 \beta_{2} q^{55} - \beta_1 q^{57} - 3 \beta_1 q^{59} + \beta_{2} q^{61} - \beta_1 q^{67} + \beta_{3} q^{69} - \beta_{3} q^{71} + 4 \beta_1 q^{73} + 3 q^{75} - 2 \beta_{3} q^{79} + q^{81} + \beta_1 q^{83} + (2 \beta_{3} + \beta_{2}) q^{85} + \beta_{2} q^{87} + 14 q^{89} + \beta_{3} q^{93} + \beta_{3} q^{95} + 4 \beta_1 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} + 16 q^{11} - 4 q^{17} + 12 q^{25} + 4 q^{27} + 16 q^{33} + 28 q^{49} - 4 q^{51} + 12 q^{75} + 4 q^{81} + 56 q^{89} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\zeta_{8}^{3} + 4\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + 2\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( \beta_{3} - 2\beta_{2} ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3264\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(2177\) \(2245\) \(2689\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2209.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0 1.00000 0 −2.82843 0 0 0 1.00000 0
2209.2 0 1.00000 0 −2.82843 0 0 0 1.00000 0
2209.3 0 1.00000 0 2.82843 0 0 0 1.00000 0
2209.4 0 1.00000 0 2.82843 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
68.d odd 2 1 inner
136.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3264.2.l.b yes 4
4.b odd 2 1 3264.2.l.a 4
8.b even 2 1 3264.2.l.a 4
8.d odd 2 1 inner 3264.2.l.b yes 4
17.b even 2 1 3264.2.l.a 4
68.d odd 2 1 inner 3264.2.l.b yes 4
136.e odd 2 1 3264.2.l.a 4
136.h even 2 1 inner 3264.2.l.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3264.2.l.a 4 4.b odd 2 1
3264.2.l.a 4 8.b even 2 1
3264.2.l.a 4 17.b even 2 1
3264.2.l.a 4 136.e odd 2 1
3264.2.l.b yes 4 1.a even 1 1 trivial
3264.2.l.b yes 4 8.d odd 2 1 inner
3264.2.l.b yes 4 68.d odd 2 1 inner
3264.2.l.b yes 4 136.h 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 \(S_{2}^{\mathrm{new}}(3264, [\chi])\):

\( T_{5}^{2} - 8 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T - 4)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2 T + 17)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 128)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 128)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T - 14)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
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