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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,0,2,0,0,-4,6,0,0,0,0,4] 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: \(66.4754596827\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2775)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - 2 q^{7} + (\beta + 3) q^{8} - 2 \beta q^{11} + (2 \beta + 2) q^{13} + ( - 2 \beta - 2) q^{14} + 3 q^{16} + (4 \beta - 2) q^{17} + ( - 3 \beta + 3) q^{19} + ( - 2 \beta - 4) q^{22} + \cdots + ( - 3 \beta - 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 6 q^{8} + 4 q^{13} - 4 q^{14} + 6 q^{16} - 4 q^{17} + 6 q^{19} - 8 q^{22} + 6 q^{23} + 12 q^{26} - 4 q^{28} + 4 q^{29} + 12 q^{31} - 6 q^{32} + 12 q^{34} - 2 q^{37} - 6 q^{38}+ \cdots - 6 q^{98}+O(q^{100}) \) 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.

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
−1.41421
1.41421
−0.414214 0 −1.82843 0 0 −2.00000 1.58579 0 0
1.2 2.41421 0 3.82843 0 0 −2.00000 4.41421 0 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)
\(37\) \( +1 \)

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 8325.2.a.bn 2
3.b odd 2 1 2775.2.a.k ✓ 2
5.b even 2 1 8325.2.a.bg 2
15.d odd 2 1 2775.2.a.p yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2775.2.a.k ✓ 2 3.b odd 2 1
2775.2.a.p yes 2 15.d odd 2 1
8325.2.a.bg 2 5.b even 2 1
8325.2.a.bn 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8325))\):

\( T_{2}^{2} - 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 8 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 8 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T - 9 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 7 \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$31$ \( (T - 6)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 10T - 7 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T - 9 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T - 23 \) Copy content Toggle raw display
$59$ \( T^{2} + 18T + 79 \) Copy content Toggle raw display
$61$ \( (T + 12)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T - 112 \) Copy content Toggle raw display
$73$ \( T^{2} - 6T - 63 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T - 25 \) Copy content Toggle raw display
$83$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$89$ \( T^{2} - 20T + 92 \) Copy content Toggle raw display
$97$ \( T^{2} - 72 \) Copy content Toggle raw display
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