Properties

Label 3328.2.a.m
Level $3328$
Weight $2$
Character orbit 3328.a
Self dual yes
Analytic conductor $26.574$
Analytic rank $1$
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] = [3328,2,Mod(1,3328)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3328, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3328.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3328.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: \(26.5742137927\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{3} - 2 \beta q^{5} + (\beta - 3) q^{7} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} - 2 \beta q^{5} + (\beta - 3) q^{7} + q^{9} + ( - \beta - 3) q^{11} + q^{13} + 4 \beta q^{15} + (2 \beta + 2) q^{17} + ( - \beta + 1) q^{19} + ( - 2 \beta + 6) q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 2 q^{29} + (\beta + 5) q^{31} + (2 \beta + 6) q^{33} + (6 \beta - 6) q^{35} + (4 \beta + 2) q^{37} - 2 q^{39} + ( - 4 \beta - 2) q^{41} + (2 \beta - 4) q^{43} - 2 \beta q^{45} + ( - \beta - 5) q^{47} + ( - 6 \beta + 5) q^{49} + ( - 4 \beta - 4) q^{51} + (4 \beta - 4) q^{53} + (6 \beta + 6) q^{55} + (2 \beta - 2) q^{57} + ( - 3 \beta - 5) q^{59} + (4 \beta - 4) q^{61} + (\beta - 3) q^{63} - 2 \beta q^{65} + ( - \beta + 1) q^{67} - 8 q^{69} + ( - 3 \beta - 3) q^{71} + (2 \beta + 4) q^{73} - 14 q^{75} + 6 q^{77} + ( - 2 \beta - 2) q^{79} - 11 q^{81} + (\beta - 5) q^{83} + ( - 4 \beta - 12) q^{85} - 4 q^{87} + 10 \beta q^{89} + (\beta - 3) q^{91} + ( - 2 \beta - 10) q^{93} + ( - 2 \beta + 6) q^{95} + (6 \beta - 4) q^{97} + ( - \beta - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} - 6 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} + 4 q^{17} + 2 q^{19} + 12 q^{21} + 8 q^{23} + 14 q^{25} + 8 q^{27} + 4 q^{29} + 10 q^{31} + 12 q^{33} - 12 q^{35} + 4 q^{37} - 4 q^{39} - 4 q^{41} - 8 q^{43} - 10 q^{47} + 10 q^{49} - 8 q^{51} - 8 q^{53} + 12 q^{55} - 4 q^{57} - 10 q^{59} - 8 q^{61} - 6 q^{63} + 2 q^{67} - 16 q^{69} - 6 q^{71} + 8 q^{73} - 28 q^{75} + 12 q^{77} - 4 q^{79} - 22 q^{81} - 10 q^{83} - 24 q^{85} - 8 q^{87} - 6 q^{91} - 20 q^{93} + 12 q^{95} - 8 q^{97} - 6 q^{99}+O(q^{100}) \) 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
1.73205
−1.73205
0 −2.00000 0 −3.46410 0 −1.26795 0 1.00000 0
1.2 0 −2.00000 0 3.46410 0 −4.73205 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(13\) \( -1 \)

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 3328.2.a.m 2
4.b odd 2 1 3328.2.a.bd 2
8.b even 2 1 3328.2.a.bc 2
8.d odd 2 1 3328.2.a.n 2
16.e even 4 2 416.2.b.b 4
16.f odd 4 2 104.2.b.b 4
48.i odd 4 2 3744.2.g.b 4
48.k even 4 2 936.2.g.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.b.b 4 16.f odd 4 2
416.2.b.b 4 16.e even 4 2
936.2.g.b 4 48.k even 4 2
3328.2.a.m 2 1.a even 1 1 trivial
3328.2.a.n 2 8.d odd 2 1
3328.2.a.bc 2 8.b even 2 1
3328.2.a.bd 2 4.b odd 2 1
3744.2.g.b 4 48.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3328))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{5}^{2} - 12 \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 12 \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 6 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 10T + 22 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 10T + 22 \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 32 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T - 2 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T - 32 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T - 18 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$83$ \( T^{2} + 10T + 22 \) Copy content Toggle raw display
$89$ \( T^{2} - 300 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 92 \) Copy content Toggle raw display
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