Properties

Label 306.2.a.f
Level $306$
Weight $2$
Character orbit 306.a
Self dual yes
Analytic conductor $2.443$
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] = [306,2,Mod(1,306)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(306, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("306.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 306 = 2 \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 306.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: \(2.44342230185\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta q^{5} + ( - \beta + 2) q^{7} + q^{8} + \beta q^{10} - 2 \beta q^{11} + (2 \beta + 2) q^{13} + ( - \beta + 2) q^{14} + q^{16} - q^{17} + ( - 2 \beta + 2) q^{19} + \beta q^{20} + \cdots + ( - 4 \beta + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 4 q^{13} + 4 q^{14} + 2 q^{16} - 2 q^{17} + 4 q^{19} - 12 q^{23} + 2 q^{25} + 4 q^{26} + 4 q^{28} + 4 q^{31} + 2 q^{32} - 2 q^{34} - 12 q^{35} - 8 q^{37} + 4 q^{38}+ \cdots + 6 q^{98}+O(q^{100}) \) 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
−2.44949
2.44949
1.00000 0 1.00000 −2.44949 0 4.44949 1.00000 0 −2.44949
1.2 1.00000 0 1.00000 2.44949 0 −0.449490 1.00000 0 2.44949
\(n\): 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 306.2.a.f yes 2
3.b odd 2 1 306.2.a.e 2
4.b odd 2 1 2448.2.a.w 2
5.b even 2 1 7650.2.a.cq 2
8.b even 2 1 9792.2.a.ct 2
8.d odd 2 1 9792.2.a.cp 2
12.b even 2 1 2448.2.a.x 2
15.d odd 2 1 7650.2.a.cz 2
17.b even 2 1 5202.2.a.z 2
24.f even 2 1 9792.2.a.cq 2
24.h odd 2 1 9792.2.a.cu 2
51.c odd 2 1 5202.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
306.2.a.e 2 3.b odd 2 1
306.2.a.f yes 2 1.a even 1 1 trivial
2448.2.a.w 2 4.b odd 2 1
2448.2.a.x 2 12.b even 2 1
5202.2.a.q 2 51.c odd 2 1
5202.2.a.z 2 17.b even 2 1
7650.2.a.cq 2 5.b even 2 1
7650.2.a.cz 2 15.d odd 2 1
9792.2.a.cp 2 8.d odd 2 1
9792.2.a.cq 2 24.f even 2 1
9792.2.a.ct 2 8.b even 2 1
9792.2.a.cu 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(306))\):

\( T_{5}^{2} - 6 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} - 2 \) Copy content Toggle raw display
\( T_{23}^{2} + 12T_{23} + 30 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 6 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T - 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 24 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 20 \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 20 \) Copy content Toggle raw display
$23$ \( T^{2} + 12T + 30 \) Copy content Toggle raw display
$29$ \( T^{2} - 6 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 50 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 10 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 24 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 12 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 12 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 10 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 20 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T - 18 \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 50 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T + 12 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T - 60 \) Copy content Toggle raw display
$97$ \( T^{2} + 20T + 76 \) Copy content Toggle raw display
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