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Properties

Label 3724.2.a.e
Level $3724$
Weight $2$
Character orbit 3724.a
Self dual yes
Analytic conductor $29.736$
Analytic rank $1$
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] = [3724,2,Mod(1,3724)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3724.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3724, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3724.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(29.7362897127\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 532)
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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - 3 q^{5} + (\beta + 2) q^{9} + ( - \beta - 1) q^{11} + q^{13} - 3 \beta q^{15} + ( - \beta - 1) q^{17} - q^{19} + (2 \beta - 1) q^{23} + 4 q^{25} + 5 q^{27} + (\beta + 1) q^{29} + ( - 3 \beta + 1) q^{31} + \cdots + ( - 4 \beta - 7) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 6 q^{5} + 5 q^{9} - 3 q^{11} + 2 q^{13} - 3 q^{15} - 3 q^{17} - 2 q^{19} + 8 q^{25} + 10 q^{27} + 3 q^{29} - q^{31} - 12 q^{33} + 10 q^{37} + q^{39} + 3 q^{41} + 4 q^{43} - 15 q^{45} - 12 q^{47}+ \cdots - 18 q^{99}+O(q^{100}) \) 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.

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
−1.79129
2.79129
0 −1.79129 0 −3.00000 0 0 0 0.208712 0
1.2 0 2.79129 0 −3.00000 0 0 0 4.79129 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)
\(19\) \( +1 \)

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 3724.2.a.e 2
7.b odd 2 1 532.2.a.c ✓ 2
21.c even 2 1 4788.2.a.g 2
28.d even 2 1 2128.2.a.k 2
56.e even 2 1 8512.2.a.m 2
56.h odd 2 1 8512.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.2.a.c ✓ 2 7.b odd 2 1
2128.2.a.k 2 28.d even 2 1
3724.2.a.e 2 1.a even 1 1 trivial
4788.2.a.g 2 21.c even 2 1
8512.2.a.m 2 56.e even 2 1
8512.2.a.t 2 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} - 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3724))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 5 \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 3T - 3 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 3 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 21 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$31$ \( T^{2} + T - 47 \) Copy content Toggle raw display
$37$ \( (T - 5)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$43$ \( (T - 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 15 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T - 45 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T - 75 \) Copy content Toggle raw display
$61$ \( (T - 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 5T - 41 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T - 75 \) Copy content Toggle raw display
$73$ \( T^{2} + 19T + 43 \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 15T + 9 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 12 \) Copy content Toggle raw display
$97$ \( (T - 7)^{2} \) Copy content Toggle raw display
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