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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7605,2,Mod(1,7605)] 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("7605.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,0,4,-2,0,0,12,0,-2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(60.7262307372\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (2 \beta + 2) q^{4} - q^{5} + \beta q^{7} + (2 \beta + 6) q^{8} + ( - \beta - 1) q^{10} + 2 q^{11} + (\beta + 3) q^{14} + (4 \beta + 8) q^{16} + (\beta + 1) q^{17} + ( - 2 \beta - 4) q^{19}+ \cdots + ( - 4 \beta - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{4} - 2 q^{5} + 12 q^{8} - 2 q^{10} + 4 q^{11} + 6 q^{14} + 16 q^{16} + 2 q^{17} - 8 q^{19} - 4 q^{20} + 4 q^{22} - 4 q^{23} + 2 q^{25} + 12 q^{28} + 10 q^{29} + 2 q^{31} + 16 q^{32}+ \cdots - 8 q^{98}+O(q^{100}) \) 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
−1.73205
1.73205
−0.732051 0 −1.46410 −1.00000 0 −1.73205 2.53590 0 0.732051
1.2 2.73205 0 5.46410 −1.00000 0 1.73205 9.46410 0 −2.73205
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

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

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 7605.2.a.bj 2
3.b odd 2 1 2535.2.a.o 2
13.b even 2 1 7605.2.a.z 2
13.c even 3 2 585.2.j.c 4
39.d odd 2 1 2535.2.a.r 2
39.i odd 6 2 195.2.i.c ✓ 4
195.x odd 6 2 975.2.i.j 4
195.bl even 12 2 975.2.bb.a 4
195.bl even 12 2 975.2.bb.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.c ✓ 4 39.i odd 6 2
585.2.j.c 4 13.c even 3 2
975.2.i.j 4 195.x odd 6 2
975.2.bb.a 4 195.bl even 12 2
975.2.bb.h 4 195.bl even 12 2
2535.2.a.o 2 3.b odd 2 1
2535.2.a.r 2 39.d odd 2 1
7605.2.a.z 2 13.b even 2 1
7605.2.a.bj 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

\( T_{2}^{2} - 2T_{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 3 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 4 \) Copy content Toggle raw display
$23$ \( (T + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 10T + 22 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 11 \) Copy content Toggle raw display
$37$ \( T^{2} - 108 \) Copy content Toggle raw display
$41$ \( T^{2} - 18T + 78 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T - 11 \) Copy content Toggle raw display
$47$ \( T^{2} - 10T - 2 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 66 \) Copy content Toggle raw display
$61$ \( T^{2} + 10T - 83 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T - 11 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$73$ \( T^{2} - 3 \) Copy content Toggle raw display
$79$ \( (T + 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 8T - 32 \) Copy content Toggle raw display
$89$ \( T^{2} + 14T + 46 \) Copy content Toggle raw display
$97$ \( T^{2} - 27 \) Copy content Toggle raw display
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