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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2178,4,Mod(1,2178)] 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("2178.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2178, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2178.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-4,0,8,-6,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(128.506159993\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{273}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 68 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 726)
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 \(\beta = \sqrt{273}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + ( - \beta - 3) q^{5} + ( - \beta - 3) q^{7} - 8 q^{8} + (2 \beta + 6) q^{10} + ( - \beta - 57) q^{13} + (2 \beta + 6) q^{14} + 16 q^{16} + (6 \beta - 12) q^{17} + ( - 4 \beta - 30) q^{19}+ \cdots + ( - 12 \beta + 122) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{4} - 6 q^{5} - 6 q^{7} - 16 q^{8} + 12 q^{10} - 114 q^{13} + 12 q^{14} + 32 q^{16} - 24 q^{17} - 60 q^{19} - 24 q^{20} + 222 q^{23} + 314 q^{25} + 228 q^{26} - 24 q^{28} - 12 q^{29}+ \cdots + 244 q^{98}+O(q^{100}) \) 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
8.76136
−7.76136
−2.00000 0 4.00000 −19.5227 0 −19.5227 −8.00000 0 39.0454
1.2 −2.00000 0 4.00000 13.5227 0 13.5227 −8.00000 0 −27.0454
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(11\) \( +1 \)

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 2178.4.a.x 2
3.b odd 2 1 726.4.a.s yes 2
11.b odd 2 1 2178.4.a.bg 2
33.d even 2 1 726.4.a.m ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
726.4.a.m ✓ 2 33.d even 2 1
726.4.a.s yes 2 3.b odd 2 1
2178.4.a.x 2 1.a even 1 1 trivial
2178.4.a.bg 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2178))\):

\( T_{5}^{2} + 6T_{5} - 264 \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} - 264 \) Copy content Toggle raw display
\( T_{17}^{2} + 24T_{17} - 9684 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 6T - 264 \) Copy content Toggle raw display
$7$ \( T^{2} + 6T - 264 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 114T + 2976 \) Copy content Toggle raw display
$17$ \( T^{2} + 24T - 9684 \) Copy content Toggle raw display
$19$ \( T^{2} + 60T - 3468 \) Copy content Toggle raw display
$23$ \( T^{2} - 222T + 5496 \) Copy content Toggle raw display
$29$ \( T^{2} + 12T - 39276 \) Copy content Toggle raw display
$31$ \( T^{2} + 316T + 15136 \) Copy content Toggle raw display
$37$ \( T^{2} + 40T - 88052 \) Copy content Toggle raw display
$41$ \( T^{2} - 192T - 79236 \) Copy content Toggle raw display
$43$ \( T^{2} - 288T + 10908 \) Copy content Toggle raw display
$47$ \( T^{2} + 54T - 1728 \) Copy content Toggle raw display
$53$ \( T^{2} - 306T - 38016 \) Copy content Toggle raw display
$59$ \( T^{2} - 444T - 4224 \) Copy content Toggle raw display
$61$ \( T^{2} - 618T - 75144 \) Copy content Toggle raw display
$67$ \( (T + 524)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 294T + 14784 \) Copy content Toggle raw display
$73$ \( T^{2} + 984T - 194736 \) Copy content Toggle raw display
$79$ \( T^{2} + 1722 T + 695184 \) Copy content Toggle raw display
$83$ \( T^{2} + 2604 T + 1685376 \) Copy content Toggle raw display
$89$ \( T^{2} - 2100 T + 1098132 \) Copy content Toggle raw display
$97$ \( T^{2} - 184 T - 2202836 \) Copy content Toggle raw display
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