Properties

Label 3648.2.a.bs
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: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 456)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} + (\beta - 2) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta q^{5} + (\beta - 2) q^{7} + q^{9} + (\beta - 2) q^{11} - 6 q^{13} + \beta q^{15} + \beta q^{17} - q^{19} + (\beta - 2) q^{21} + 4 q^{23} + (\beta + 5) q^{25} + q^{27} - 2 q^{29} + ( - 2 \beta + 4) q^{31} + (\beta - 2) q^{33} + ( - \beta + 10) q^{35} + ( - 2 \beta - 2) q^{37} - 6 q^{39} + (2 \beta + 2) q^{41} + (3 \beta - 2) q^{43} + \beta q^{45} + (\beta + 2) q^{47} + ( - 3 \beta + 7) q^{49} + \beta q^{51} + 6 q^{53} + ( - \beta + 10) q^{55} - q^{57} + 4 q^{59} + ( - \beta - 4) q^{61} + (\beta - 2) q^{63} - 6 \beta q^{65} + 12 q^{67} + 4 q^{69} + ( - \beta + 4) q^{73} + (\beta + 5) q^{75} + ( - 3 \beta + 14) q^{77} + 4 \beta q^{79} + q^{81} - 4 \beta q^{83} + (\beta + 10) q^{85} - 2 q^{87} + (4 \beta - 2) q^{89} + ( - 6 \beta + 12) q^{91} + ( - 2 \beta + 4) q^{93} - \beta q^{95} - 6 q^{97} + (\beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + q^{5} - 3 q^{7} + 2 q^{9} - 3 q^{11} - 12 q^{13} + q^{15} + q^{17} - 2 q^{19} - 3 q^{21} + 8 q^{23} + 11 q^{25} + 2 q^{27} - 4 q^{29} + 6 q^{31} - 3 q^{33} + 19 q^{35} - 6 q^{37} - 12 q^{39} + 6 q^{41} - q^{43} + q^{45} + 5 q^{47} + 11 q^{49} + q^{51} + 12 q^{53} + 19 q^{55} - 2 q^{57} + 8 q^{59} - 9 q^{61} - 3 q^{63} - 6 q^{65} + 24 q^{67} + 8 q^{69} + 7 q^{73} + 11 q^{75} + 25 q^{77} + 4 q^{79} + 2 q^{81} - 4 q^{83} + 21 q^{85} - 4 q^{87} + 18 q^{91} + 6 q^{93} - q^{95} - 12 q^{97} - 3 q^{99}+O(q^{100}) \) 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
−2.70156
3.70156
0 1.00000 0 −2.70156 0 −4.70156 0 1.00000 0
1.2 0 1.00000 0 3.70156 0 1.70156 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.bs 2
4.b odd 2 1 3648.2.a.bn 2
8.b even 2 1 456.2.a.e 2
8.d odd 2 1 912.2.a.o 2
24.f even 2 1 2736.2.a.bb 2
24.h odd 2 1 1368.2.a.l 2
152.g odd 2 1 8664.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.a.e 2 8.b even 2 1
912.2.a.o 2 8.d odd 2 1
1368.2.a.l 2 24.h odd 2 1
2736.2.a.bb 2 24.f even 2 1
3648.2.a.bn 2 4.b odd 2 1
3648.2.a.bs 2 1.a even 1 1 trivial
8664.2.a.v 2 152.g odd 2 1

Hecke kernels

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

\( T_{5}^{2} - T_{5} - 10 \) Copy content Toggle raw display
\( T_{7}^{2} + 3T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} - 8 \) Copy content Toggle raw display
\( T_{23} - 4 \) Copy content Toggle raw display
\( T_{31}^{2} - 6T_{31} - 32 \) 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
$5$ \( T^{2} - T - 10 \) Copy content Toggle raw display
$7$ \( T^{2} + 3T - 8 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T - 8 \) Copy content Toggle raw display
$13$ \( (T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - T - 10 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 6T - 32 \) Copy content Toggle raw display
$37$ \( T^{2} + 6T - 32 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 32 \) Copy content Toggle raw display
$43$ \( T^{2} + T - 92 \) Copy content Toggle raw display
$47$ \( T^{2} - 5T - 4 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 9T + 10 \) Copy content Toggle raw display
$67$ \( (T - 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 7T + 2 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 160 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 160 \) Copy content Toggle raw display
$89$ \( T^{2} - 164 \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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