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Properties

Label 3060.2.a.p
Level $3060$
Weight $2$
Character orbit 3060.a
Self dual yes
Analytic conductor $24.434$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3060,2,Mod(1,3060)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3060.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3060.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-2,0,1,0,0,0,-3,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(24.4342230185\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1020)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + \beta q^{7} + (\beta - 2) q^{11} + 2 q^{13} + q^{17} + \beta q^{19} + (2 \beta - 4) q^{23} + q^{25} + ( - \beta + 2) q^{29} + 2 q^{31} - \beta q^{35} + ( - 3 \beta + 2) q^{37} + ( - \beta + 2) q^{41} + \cdots + ( - 4 \beta - 2) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + q^{7} - 3 q^{11} + 4 q^{13} + 2 q^{17} + q^{19} - 6 q^{23} + 2 q^{25} + 3 q^{29} + 4 q^{31} - q^{35} + q^{37} + 3 q^{41} + 10 q^{43} - 3 q^{47} + 3 q^{49} + 3 q^{53} + 3 q^{55} + 4 q^{61}+ \cdots - 8 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−2.37228
3.37228
0 0 0 −1.00000 0 −2.37228 0 0 0
1.2 0 0 0 −1.00000 0 3.37228 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( +1 \)
\(17\) \( -1 \)

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 3060.2.a.p 2
3.b odd 2 1 1020.2.a.j ✓ 2
12.b even 2 1 4080.2.a.bh 2
15.d odd 2 1 5100.2.a.v 2
15.e even 4 2 5100.2.g.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1020.2.a.j ✓ 2 3.b odd 2 1
3060.2.a.p 2 1.a even 1 1 trivial
4080.2.a.bh 2 12.b even 2 1
5100.2.a.v 2 15.d odd 2 1
5100.2.g.o 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3060))\):

\( T_{7}^{2} - T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} - 6 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{19}^{2} - T_{19} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T - 24 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - T - 74 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T - 8 \) Copy content Toggle raw display
$47$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$53$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$73$ \( T^{2} - 7T - 62 \) Copy content Toggle raw display
$79$ \( (T - 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 6T - 24 \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 116 \) Copy content Toggle raw display
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