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Properties

Label 3900.2.a.p
Level $3900$
Weight $2$
Character orbit 3900.a
Self dual yes
Analytic conductor $31.142$
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] = [3900,2,Mod(1,3900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3900, 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("3900.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3900.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,0,0,-3,0,2,0,3,0,2,0,0,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.1416567883\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 780)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( - \beta - 1) q^{7} + q^{9} + (\beta + 1) q^{11} + q^{13} + (\beta - 3) q^{17} + ( - 4 \beta + 2) q^{19} + (\beta + 1) q^{21} + (3 \beta - 3) q^{23} - q^{27} + (4 \beta - 2) q^{29} - 2 q^{31}+ \cdots + (\beta + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 3 q^{7} + 2 q^{9} + 3 q^{11} + 2 q^{13} - 5 q^{17} + 3 q^{21} - 3 q^{23} - 2 q^{27} - 4 q^{31} - 3 q^{33} + 3 q^{37} - 2 q^{39} + 5 q^{41} + 4 q^{43} - 20 q^{47} - q^{49} + 5 q^{51} + 11 q^{53}+ \cdots + 3 q^{99}+O(q^{100}) \) 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.

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
2.56155
−1.56155
0 −1.00000 0 0 0 −3.56155 0 1.00000 0
1.2 0 −1.00000 0 0 0 0.561553 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(5\) \( +1 \)
\(13\) \( -1 \)

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 3900.2.a.p 2
5.b even 2 1 780.2.a.f ✓ 2
5.c odd 4 2 3900.2.h.j 4
15.d odd 2 1 2340.2.a.k 2
20.d odd 2 1 3120.2.a.bb 2
60.h even 2 1 9360.2.a.ce 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
780.2.a.f ✓ 2 5.b even 2 1
2340.2.a.k 2 15.d odd 2 1
3120.2.a.bb 2 20.d odd 2 1
3900.2.a.p 2 1.a even 1 1 trivial
3900.2.h.j 4 5.c odd 4 2
9360.2.a.ce 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3900))\):

\( T_{7}^{2} + 3T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 5T_{17} + 2 \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} - 68 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 68 \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$41$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$47$ \( (T + 10)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 11T + 26 \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$61$ \( T^{2} + 9T - 18 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$71$ \( T^{2} + 9T - 86 \) Copy content Toggle raw display
$73$ \( (T - 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 5T - 100 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$89$ \( T^{2} + 5T - 202 \) Copy content Toggle raw display
$97$ \( T^{2} - 9T - 86 \) Copy content Toggle raw display
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