Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3179,2,Mod(1,3179)] 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("3179.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3179, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3179 = 11 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3179.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,4,4] 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: \(25.3844428026\)
Analytic rank: \(1\)
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 187)
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.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + \beta q^{3} + ( - 2 \beta + 2) q^{4} + ( - \beta + 2) q^{5} + ( - \beta + 3) q^{6} + 2 q^{7} + (2 \beta - 6) q^{8} + (3 \beta - 5) q^{10} - q^{11} + (2 \beta - 6) q^{12} + ( - \beta - 5) q^{13}+ \cdots + ( - 3 \beta + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 4 q^{4} + 4 q^{5} + 6 q^{6} + 4 q^{7} - 12 q^{8} - 10 q^{10} - 2 q^{11} - 12 q^{12} - 10 q^{13} - 4 q^{14} - 6 q^{15} + 16 q^{16} - 2 q^{19} + 20 q^{20} + 2 q^{22} + 4 q^{23} + 12 q^{24}+ \cdots + 6 q^{98}+O(q^{100}) \) 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
−1.73205
1.73205
−2.73205 −1.73205 5.46410 3.73205 4.73205 2.00000 −9.46410 0 −10.1962
1.2 0.732051 1.73205 −1.46410 0.267949 1.26795 2.00000 −2.53590 0 0.196152
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( +1 \)
\(17\) \( +1 \)

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 3179.2.a.k 2
17.b even 2 1 187.2.a.c ✓ 2
51.c odd 2 1 1683.2.a.r 2
68.d odd 2 1 2992.2.a.j 2
85.c even 2 1 4675.2.a.v 2
119.d odd 2 1 9163.2.a.h 2
187.b odd 2 1 2057.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.2.a.c ✓ 2 17.b even 2 1
1683.2.a.r 2 51.c odd 2 1
2057.2.a.o 2 187.b odd 2 1
2992.2.a.j 2 68.d odd 2 1
3179.2.a.k 2 1.a even 1 1 trivial
4675.2.a.v 2 85.c even 2 1
9163.2.a.h 2 119.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3179))\):

\( T_{2}^{2} + 2T_{2} - 2 \) Copy content Toggle raw display
\( T_{3}^{2} - 3 \) Copy content Toggle raw display
\( T_{5}^{2} - 4T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$3$ \( T^{2} - 3 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 1 \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 10T + 22 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 26 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 13 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12T - 12 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 24 \) Copy content Toggle raw display
$59$ \( (T - 3)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 32 \) Copy content Toggle raw display
$67$ \( (T - 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 4T + 1 \) Copy content Toggle raw display
$73$ \( T^{2} + 18T + 54 \) Copy content Toggle raw display
$79$ \( T^{2} - 6T - 18 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 188 \) Copy content Toggle raw display
$89$ \( T^{2} + 10T - 23 \) Copy content Toggle raw display
$97$ \( T^{2} + 28T + 193 \) Copy content Toggle raw display
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