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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,-4,0,8,12,0,-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: \(\Q(\sqrt{3}) \)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 242)
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.

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + (7 \beta + 6) q^{5} + (11 \beta - 3) q^{7} - 8 q^{8} + ( - 14 \beta - 12) q^{10} + (6 \beta - 57) q^{13} + ( - 22 \beta + 6) q^{14} + 16 q^{16} + (9 \beta - 36) q^{17} + ( - 47 \beta - 75) q^{19}+ \cdots + (132 \beta - 58) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{4} + 12 q^{5} - 6 q^{7} - 16 q^{8} - 24 q^{10} - 114 q^{13} + 12 q^{14} + 32 q^{16} - 72 q^{17} - 150 q^{19} + 48 q^{20} + 222 q^{23} + 116 q^{25} + 228 q^{26} - 24 q^{28} + 462 q^{29}+ \cdots - 116 q^{98}+O(q^{100}) \) 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
−2.00000 0 4.00000 −6.12436 0 −22.0526 −8.00000 0 12.2487
1.2 −2.00000 0 4.00000 18.1244 0 16.0526 −8.00000 0 −36.2487
\(n\): 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.bd 2
3.b odd 2 1 242.4.a.l yes 2
11.b odd 2 1 2178.4.a.bn 2
12.b even 2 1 1936.4.a.s 2
33.d even 2 1 242.4.a.i ✓ 2
33.f even 10 4 242.4.c.t 8
33.h odd 10 4 242.4.c.p 8
132.d odd 2 1 1936.4.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
242.4.a.i ✓ 2 33.d even 2 1
242.4.a.l yes 2 3.b odd 2 1
242.4.c.p 8 33.h odd 10 4
242.4.c.t 8 33.f even 10 4
1936.4.a.r 2 132.d odd 2 1
1936.4.a.s 2 12.b even 2 1
2178.4.a.bd 2 1.a even 1 1 trivial
2178.4.a.bn 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2178))\):

\( T_{5}^{2} - 12T_{5} - 111 \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} - 354 \) Copy content Toggle raw display
\( T_{17}^{2} + 72T_{17} + 1053 \) 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
$5$ \( T^{2} - 12T - 111 \) Copy content Toggle raw display
$7$ \( T^{2} + 6T - 354 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 114T + 3141 \) Copy content Toggle raw display
$17$ \( T^{2} + 72T + 1053 \) Copy content Toggle raw display
$19$ \( T^{2} + 150T - 1002 \) Copy content Toggle raw display
$23$ \( T^{2} - 222T + 10446 \) Copy content Toggle raw display
$29$ \( T^{2} - 462T + 48069 \) Copy content Toggle raw display
$31$ \( T^{2} - 14T - 81626 \) Copy content Toggle raw display
$37$ \( T^{2} - 464T + 41917 \) Copy content Toggle raw display
$41$ \( T^{2} - 60T - 93087 \) Copy content Toggle raw display
$43$ \( T^{2} + 432T + 19008 \) Copy content Toggle raw display
$47$ \( T^{2} + 450T - 85482 \) Copy content Toggle raw display
$53$ \( T^{2} + 288T + 19413 \) Copy content Toggle raw display
$59$ \( T^{2} + 204T - 16104 \) Copy content Toggle raw display
$61$ \( T^{2} + 372T - 276456 \) Copy content Toggle raw display
$67$ \( T^{2} - 374T - 309794 \) Copy content Toggle raw display
$71$ \( T^{2} + 1092 T + 296088 \) Copy content Toggle raw display
$73$ \( T^{2} + 444T + 46932 \) Copy content Toggle raw display
$79$ \( T^{2} - 726T + 97422 \) Copy content Toggle raw display
$83$ \( T^{2} + 282T + 19854 \) Copy content Toggle raw display
$89$ \( T^{2} - 1578 T + 616173 \) Copy content Toggle raw display
$97$ \( T^{2} + 1274 T - 807719 \) Copy content Toggle raw display
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