Properties

Label 3648.2.a.bo
Level $3648$
Weight $2$
Character orbit 3648.a
Self dual yes
Analytic conductor $29.129$
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] = [3648,2,Mod(1,3648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3648, 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("3648.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3648.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: \(29.1294266574\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1824)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + (\beta + 1) q^{5} + (\beta - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + (\beta + 1) q^{5} + (\beta - 1) q^{7} + q^{9} + (\beta + 1) q^{11} + ( - 2 \beta + 4) q^{13} + ( - \beta - 1) q^{15} + (\beta + 1) q^{17} - q^{19} + ( - \beta + 1) q^{21} - 2 \beta q^{23} + 3 \beta q^{25} - q^{27} + 4 q^{29} + (2 \beta - 2) q^{31} + ( - \beta - 1) q^{33} + (\beta + 3) q^{35} + (2 \beta + 4) q^{37} + (2 \beta - 4) q^{39} - 4 q^{41} + (3 \beta + 1) q^{43} + (\beta + 1) q^{45} + (\beta - 7) q^{47} + ( - \beta - 2) q^{49} + ( - \beta - 1) q^{51} + (2 \beta + 2) q^{53} + (3 \beta + 5) q^{55} + q^{57} + ( - 2 \beta + 2) q^{59} + (\beta + 9) q^{61} + (\beta - 1) q^{63} - 4 q^{65} + ( - 2 \beta - 10) q^{67} + 2 \beta q^{69} + ( - 6 \beta + 2) q^{71} + (\beta + 1) q^{73} - 3 \beta q^{75} + (\beta + 3) q^{77} + ( - 2 \beta + 2) q^{79} + q^{81} + ( - 6 \beta + 8) q^{83} + (3 \beta + 5) q^{85} - 4 q^{87} + (2 \beta + 2) q^{89} + (4 \beta - 12) q^{91} + ( - 2 \beta + 2) q^{93} + ( - \beta - 1) q^{95} + ( - 6 \beta - 4) q^{97} + (\beta + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 3 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 3 q^{5} - q^{7} + 2 q^{9} + 3 q^{11} + 6 q^{13} - 3 q^{15} + 3 q^{17} - 2 q^{19} + q^{21} - 2 q^{23} + 3 q^{25} - 2 q^{27} + 8 q^{29} - 2 q^{31} - 3 q^{33} + 7 q^{35} + 10 q^{37} - 6 q^{39} - 8 q^{41} + 5 q^{43} + 3 q^{45} - 13 q^{47} - 5 q^{49} - 3 q^{51} + 6 q^{53} + 13 q^{55} + 2 q^{57} + 2 q^{59} + 19 q^{61} - q^{63} - 8 q^{65} - 22 q^{67} + 2 q^{69} - 2 q^{71} + 3 q^{73} - 3 q^{75} + 7 q^{77} + 2 q^{79} + 2 q^{81} + 10 q^{83} + 13 q^{85} - 8 q^{87} + 6 q^{89} - 20 q^{91} + 2 q^{93} - 3 q^{95} - 14 q^{97} + 3 q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
0 −1.00000 0 −0.561553 0 −2.56155 0 1.00000 0
1.2 0 −1.00000 0 3.56155 0 1.56155 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(1\)

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 3648.2.a.bo 2
4.b odd 2 1 3648.2.a.bv 2
8.b even 2 1 1824.2.a.p yes 2
8.d odd 2 1 1824.2.a.n 2
24.f even 2 1 5472.2.a.bk 2
24.h odd 2 1 5472.2.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1824.2.a.n 2 8.d odd 2 1
1824.2.a.p yes 2 8.b even 2 1
3648.2.a.bo 2 1.a even 1 1 trivial
3648.2.a.bv 2 4.b odd 2 1
5472.2.a.bj 2 24.h odd 2 1
5472.2.a.bk 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3648))\):

\( T_{5}^{2} - 3T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} - 2 \) Copy content Toggle raw display
\( T_{23}^{2} + 2T_{23} - 16 \) Copy content Toggle raw display
\( T_{31}^{2} + 2T_{31} - 16 \) 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
$5$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$29$ \( (T - 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$41$ \( (T + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$47$ \( T^{2} + 13T + 38 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$59$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 19T + 86 \) Copy content Toggle raw display
$67$ \( T^{2} + 22T + 104 \) Copy content Toggle raw display
$71$ \( T^{2} + 2T - 152 \) Copy content Toggle raw display
$73$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$83$ \( T^{2} - 10T - 128 \) Copy content Toggle raw display
$89$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$97$ \( T^{2} + 14T - 104 \) Copy content Toggle raw display
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