Show commands: Magma / Pari/GP / SageMath

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 = [2,4,0,8,8,0,-36] 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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{70}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 70 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 198)
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 \(\beta = 2\sqrt{70}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + (\beta + 4) q^{5} + ( - \beta - 18) q^{7} + 8 q^{8} + (2 \beta + 8) q^{10} + ( - \beta - 30) q^{13} + ( - 2 \beta - 36) q^{14} + 16 q^{16} + (4 \beta + 46) q^{17} + (4 \beta - 22) q^{19}+ \cdots + (72 \beta + 522) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} - 36 q^{7} + 16 q^{8} + 16 q^{10} - 60 q^{13} - 72 q^{14} + 32 q^{16} + 92 q^{17} - 44 q^{19} + 32 q^{20} - 24 q^{23} + 342 q^{25} - 120 q^{26} - 144 q^{28} - 100 q^{29}+ \cdots + 1044 q^{98}+O(q^{100}) \) 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
−8.36660
8.36660
2.00000 0 4.00000 −12.7332 0 −1.26680 8.00000 0 −25.4664
1.2 2.00000 0 4.00000 20.7332 0 −34.7332 8.00000 0 41.4664
\(n\): 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.bm 2
3.b odd 2 1 2178.4.a.w 2
11.b odd 2 1 198.4.a.i ✓ 2
33.d even 2 1 198.4.a.j yes 2
44.c even 2 1 1584.4.a.bh 2
132.d odd 2 1 1584.4.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
198.4.a.i ✓ 2 11.b odd 2 1
198.4.a.j yes 2 33.d even 2 1
1584.4.a.bb 2 132.d odd 2 1
1584.4.a.bh 2 44.c even 2 1
2178.4.a.w 2 3.b odd 2 1
2178.4.a.bm 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2178))\):

\( T_{5}^{2} - 8T_{5} - 264 \) Copy content Toggle raw display
\( T_{7}^{2} + 36T_{7} + 44 \) Copy content Toggle raw display
\( T_{17}^{2} - 92T_{17} - 2364 \) 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
$5$ \( T^{2} - 8T - 264 \) Copy content Toggle raw display
$7$ \( T^{2} + 36T + 44 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 60T + 620 \) Copy content Toggle raw display
$17$ \( T^{2} - 92T - 2364 \) Copy content Toggle raw display
$19$ \( T^{2} + 44T - 3996 \) Copy content Toggle raw display
$23$ \( T^{2} + 24T - 2376 \) Copy content Toggle raw display
$29$ \( T^{2} + 100T + 1380 \) Copy content Toggle raw display
$31$ \( T^{2} - 264T + 16304 \) Copy content Toggle raw display
$37$ \( T^{2} - 36T - 54556 \) Copy content Toggle raw display
$41$ \( T^{2} + 188T - 46044 \) Copy content Toggle raw display
$43$ \( T^{2} - 12T - 189244 \) Copy content Toggle raw display
$47$ \( T^{2} - 448T + 49896 \) Copy content Toggle raw display
$53$ \( T^{2} - 688T + 104616 \) Copy content Toggle raw display
$59$ \( T^{2} - 24T - 90576 \) Copy content Toggle raw display
$61$ \( T^{2} + 84T - 267316 \) Copy content Toggle raw display
$67$ \( T^{2} + 352T - 81024 \) Copy content Toggle raw display
$71$ \( T^{2} + 304T - 125016 \) Copy content Toggle raw display
$73$ \( T^{2} - 1156 T + 329604 \) Copy content Toggle raw display
$79$ \( T^{2} - 1932 T + 852236 \) Copy content Toggle raw display
$83$ \( T^{2} - 968T + 44976 \) Copy content Toggle raw display
$89$ \( T^{2} - 1312 T + 143616 \) Copy content Toggle raw display
$97$ \( T^{2} - 84T - 217756 \) Copy content Toggle raw display
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