Properties

Label 3264.2.a.bj
Level $3264$
Weight $2$
Character orbit 3264.a
Self dual yes
Analytic conductor $26.063$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3264,2,Mod(1,3264)]
 
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, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3264.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3264 = 2^{6} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3264.a (trivial)

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: yes
Analytic conductor: \(26.0631712197\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 408)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta q^{5} + 4 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta q^{5} + 4 q^{7} + q^{9} + ( - \beta + 2) q^{11} - \beta q^{13} - \beta q^{15} + q^{17} + ( - \beta + 2) q^{19} - 4 q^{21} + (\beta - 2) q^{23} + (\beta + 9) q^{25} - q^{27} - 2 q^{29} - 2 \beta q^{31} + (\beta - 2) q^{33} + 4 \beta q^{35} + (2 \beta + 2) q^{37} + \beta q^{39} + (\beta + 4) q^{41} + (\beta + 6) q^{43} + \beta q^{45} + (2 \beta - 4) q^{47} + 9 q^{49} - q^{51} + 10 q^{53} + (\beta - 14) q^{55} + (\beta - 2) q^{57} - 2 \beta q^{59} + (2 \beta - 6) q^{61} + 4 q^{63} + ( - \beta - 14) q^{65} - 4 q^{67} + ( - \beta + 2) q^{69} - 4 q^{71} + 10 q^{73} + ( - \beta - 9) q^{75} + ( - 4 \beta + 8) q^{77} - 2 \beta q^{79} + q^{81} + 2 \beta q^{83} + \beta q^{85} + 2 q^{87} + ( - 2 \beta + 6) q^{89} - 4 \beta q^{91} + 2 \beta q^{93} + (\beta - 14) q^{95} + (2 \beta - 2) q^{97} + ( - \beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + q^{5} + 8 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + q^{5} + 8 q^{7} + 2 q^{9} + 3 q^{11} - q^{13} - q^{15} + 2 q^{17} + 3 q^{19} - 8 q^{21} - 3 q^{23} + 19 q^{25} - 2 q^{27} - 4 q^{29} - 2 q^{31} - 3 q^{33} + 4 q^{35} + 6 q^{37} + q^{39} + 9 q^{41} + 13 q^{43} + q^{45} - 6 q^{47} + 18 q^{49} - 2 q^{51} + 20 q^{53} - 27 q^{55} - 3 q^{57} - 2 q^{59} - 10 q^{61} + 8 q^{63} - 29 q^{65} - 8 q^{67} + 3 q^{69} - 8 q^{71} + 20 q^{73} - 19 q^{75} + 12 q^{77} - 2 q^{79} + 2 q^{81} + 2 q^{83} + q^{85} + 4 q^{87} + 10 q^{89} - 4 q^{91} + 2 q^{93} - 27 q^{95} - 2 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.27492
4.27492
0 −1.00000 0 −3.27492 0 4.00000 0 1.00000 0
1.2 0 −1.00000 0 4.27492 0 4.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(17\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3264.2.a.bj 2
3.b odd 2 1 9792.2.a.co 2
4.b odd 2 1 3264.2.a.bn 2
8.b even 2 1 408.2.a.f 2
8.d odd 2 1 816.2.a.k 2
12.b even 2 1 9792.2.a.cl 2
24.f even 2 1 2448.2.a.z 2
24.h odd 2 1 1224.2.a.k 2
136.h even 2 1 6936.2.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
408.2.a.f 2 8.b even 2 1
816.2.a.k 2 8.d odd 2 1
1224.2.a.k 2 24.h odd 2 1
2448.2.a.z 2 24.f even 2 1
3264.2.a.bj 2 1.a even 1 1 trivial
3264.2.a.bn 2 4.b odd 2 1
6936.2.a.u 2 136.h even 2 1
9792.2.a.cl 2 12.b even 2 1
9792.2.a.co 2 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3264))\):

\( T_{5}^{2} - T_{5} - 14 \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} - 12 \) Copy content Toggle raw display
\( T_{13}^{2} + T_{13} - 14 \) Copy content Toggle raw display
\( T_{19}^{2} - 3T_{19} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T - 14 \) Copy content Toggle raw display
$7$ \( (T - 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 3T - 12 \) Copy content Toggle raw display
$13$ \( T^{2} + T - 14 \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 3T - 12 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T - 12 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 56 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T - 48 \) Copy content Toggle raw display
$41$ \( T^{2} - 9T + 6 \) Copy content Toggle raw display
$43$ \( T^{2} - 13T + 28 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T - 48 \) Copy content Toggle raw display
$53$ \( (T - 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 2T - 56 \) Copy content Toggle raw display
$61$ \( T^{2} + 10T - 32 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 56 \) Copy content Toggle raw display
$83$ \( T^{2} - 2T - 56 \) Copy content Toggle raw display
$89$ \( T^{2} - 10T - 32 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T - 56 \) Copy content Toggle raw display
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