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Properties

Label 2112.4.a.bi
Level $2112$
Weight $4$
Character orbit 2112.a
Self dual yes
Analytic conductor $124.612$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2112,4,Mod(1,2112)] 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("2112.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2112.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,6,0,-10,0,2] 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: \(124.612033932\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 66)
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 \(\beta = \sqrt{97}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( - \beta - 5) q^{5} + (3 \beta + 1) q^{7} + 9 q^{9} - 11 q^{11} + ( - 7 \beta - 7) q^{13} + ( - 3 \beta - 15) q^{15} + ( - 2 \beta - 40) q^{17} + ( - 10 \beta - 30) q^{19} + (9 \beta + 3) q^{21}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 10 q^{5} + 2 q^{7} + 18 q^{9} - 22 q^{11} - 14 q^{13} - 30 q^{15} - 80 q^{17} - 60 q^{19} + 6 q^{21} + 202 q^{23} - 6 q^{25} + 54 q^{27} + 336 q^{29} - 128 q^{31} - 66 q^{33} - 592 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) \) 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
5.42443
−4.42443
0 3.00000 0 −14.8489 0 30.5466 0 9.00000 0
1.2 0 3.00000 0 4.84886 0 −28.5466 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
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 2112.4.a.bi 2
4.b odd 2 1 2112.4.a.bb 2
8.b even 2 1 528.4.a.n 2
8.d odd 2 1 66.4.a.c ✓ 2
24.f even 2 1 198.4.a.h 2
24.h odd 2 1 1584.4.a.ba 2
40.e odd 2 1 1650.4.a.s 2
40.k even 4 2 1650.4.c.u 4
88.g even 2 1 726.4.a.o 2
264.p odd 2 1 2178.4.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.4.a.c ✓ 2 8.d odd 2 1
198.4.a.h 2 24.f even 2 1
528.4.a.n 2 8.b even 2 1
726.4.a.o 2 88.g even 2 1
1584.4.a.ba 2 24.h odd 2 1
1650.4.a.s 2 40.e odd 2 1
1650.4.c.u 4 40.k even 4 2
2112.4.a.bb 2 4.b odd 2 1
2112.4.a.bi 2 1.a even 1 1 trivial
2178.4.a.bf 2 264.p odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2112))\):

\( T_{5}^{2} + 10T_{5} - 72 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 872 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 10T - 72 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 872 \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 14T - 4704 \) Copy content Toggle raw display
$17$ \( T^{2} + 80T + 1212 \) Copy content Toggle raw display
$19$ \( T^{2} + 60T - 8800 \) Copy content Toggle raw display
$23$ \( T^{2} - 202T + 7776 \) Copy content Toggle raw display
$29$ \( T^{2} - 336T + 24732 \) Copy content Toggle raw display
$31$ \( T^{2} + 128T - 20736 \) Copy content Toggle raw display
$37$ \( T^{2} + 188T - 90492 \) Copy content Toggle raw display
$41$ \( T^{2} + 132T - 51516 \) Copy content Toggle raw display
$43$ \( T^{2} - 480T + 51392 \) Copy content Toggle raw display
$47$ \( T^{2} - 390T + 37152 \) Copy content Toggle raw display
$53$ \( T^{2} + 610T - 25800 \) Copy content Toggle raw display
$59$ \( T^{2} - 372T + 3168 \) Copy content Toggle raw display
$61$ \( T^{2} + 1050 T + 270872 \) Copy content Toggle raw display
$67$ \( T^{2} - 408T + 16784 \) Copy content Toggle raw display
$71$ \( T^{2} + 942T + 200016 \) Copy content Toggle raw display
$73$ \( T^{2} - 740T - 260412 \) Copy content Toggle raw display
$79$ \( T^{2} - 1646 T + 606616 \) Copy content Toggle raw display
$83$ \( T^{2} + 352T - 231312 \) Copy content Toggle raw display
$89$ \( T^{2} - 2036 T + 1011492 \) Copy content Toggle raw display
$97$ \( T^{2} - 336T - 447076 \) Copy content Toggle raw display
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