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Properties

Label 112.2.f.b
Level $112$
Weight $2$
Character orbit 112.f
Analytic conductor $0.894$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,2,Mod(111,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.111"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 112.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.894324502638\)
Analytic rank: \(0\)
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 q^{3} + 2 \beta q^{5} + ( - \beta - 2) q^{7} + q^{9} - 2 \beta q^{11} - 2 \beta q^{13} + 4 \beta q^{15} + 2 q^{19} + ( - 2 \beta - 4) q^{21} + 2 \beta q^{23} - 7 q^{25} - 4 q^{27} + 6 q^{29} - 8 q^{31} + \cdots - 2 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} - 4 q^{7} + 2 q^{9} + 4 q^{19} - 8 q^{21} - 14 q^{25} - 8 q^{27} + 12 q^{29} - 16 q^{31} + 12 q^{35} - 4 q^{37} + 2 q^{49} + 12 q^{53} + 24 q^{55} + 8 q^{57} + 12 q^{59} - 4 q^{63} + 24 q^{65}+ \cdots - 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 2.00000 0 − 3.46410i 0 −2.00000 + 1.73205i 0 1.00000 0
111.2 0 2.00000 0 3.46410i 0 −2.00000 − 1.73205i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.2.f.b yes 2
3.b odd 2 1 1008.2.b.b 2
4.b odd 2 1 112.2.f.a ✓ 2
5.b even 2 1 2800.2.k.b 2
5.c odd 4 2 2800.2.e.b 4
7.b odd 2 1 112.2.f.a ✓ 2
7.c even 3 1 784.2.p.a 2
7.c even 3 1 784.2.p.b 2
7.d odd 6 1 784.2.p.e 2
7.d odd 6 1 784.2.p.f 2
8.b even 2 1 448.2.f.a 2
8.d odd 2 1 448.2.f.c 2
12.b even 2 1 1008.2.b.g 2
16.e even 4 2 1792.2.e.c 4
16.f odd 4 2 1792.2.e.a 4
20.d odd 2 1 2800.2.k.e 2
20.e even 4 2 2800.2.e.c 4
21.c even 2 1 1008.2.b.g 2
24.f even 2 1 4032.2.b.h 2
24.h odd 2 1 4032.2.b.b 2
28.d even 2 1 inner 112.2.f.b yes 2
28.f even 6 1 784.2.p.a 2
28.f even 6 1 784.2.p.b 2
28.g odd 6 1 784.2.p.e 2
28.g odd 6 1 784.2.p.f 2
35.c odd 2 1 2800.2.k.e 2
35.f even 4 2 2800.2.e.c 4
56.e even 2 1 448.2.f.a 2
56.h odd 2 1 448.2.f.c 2
84.h odd 2 1 1008.2.b.b 2
112.j even 4 2 1792.2.e.c 4
112.l odd 4 2 1792.2.e.a 4
140.c even 2 1 2800.2.k.b 2
140.j odd 4 2 2800.2.e.b 4
168.e odd 2 1 4032.2.b.b 2
168.i even 2 1 4032.2.b.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.f.a ✓ 2 4.b odd 2 1
112.2.f.a ✓ 2 7.b odd 2 1
112.2.f.b yes 2 1.a even 1 1 trivial
112.2.f.b yes 2 28.d even 2 1 inner
448.2.f.a 2 8.b even 2 1
448.2.f.a 2 56.e even 2 1
448.2.f.c 2 8.d odd 2 1
448.2.f.c 2 56.h odd 2 1
784.2.p.a 2 7.c even 3 1
784.2.p.a 2 28.f even 6 1
784.2.p.b 2 7.c even 3 1
784.2.p.b 2 28.f even 6 1
784.2.p.e 2 7.d odd 6 1
784.2.p.e 2 28.g odd 6 1
784.2.p.f 2 7.d odd 6 1
784.2.p.f 2 28.g odd 6 1
1008.2.b.b 2 3.b odd 2 1
1008.2.b.b 2 84.h odd 2 1
1008.2.b.g 2 12.b even 2 1
1008.2.b.g 2 21.c even 2 1
1792.2.e.a 4 16.f odd 4 2
1792.2.e.a 4 112.l odd 4 2
1792.2.e.c 4 16.e even 4 2
1792.2.e.c 4 112.j even 4 2
2800.2.e.b 4 5.c odd 4 2
2800.2.e.b 4 140.j odd 4 2
2800.2.e.c 4 20.e even 4 2
2800.2.e.c 4 35.f even 4 2
2800.2.k.b 2 5.b even 2 1
2800.2.k.b 2 140.c even 2 1
2800.2.k.e 2 20.d odd 2 1
2800.2.k.e 2 35.c odd 2 1
4032.2.b.b 2 24.h odd 2 1
4032.2.b.b 2 168.e odd 2 1
4032.2.b.h 2 24.f even 2 1
4032.2.b.h 2 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 12 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 12 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 12 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 48 \) Copy content Toggle raw display
$43$ \( T^{2} + 108 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 12 \) Copy content Toggle raw display
$67$ \( T^{2} + 12 \) Copy content Toggle raw display
$71$ \( T^{2} + 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 48 \) Copy content Toggle raw display
$79$ \( T^{2} + 12 \) Copy content Toggle raw display
$83$ \( (T + 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 192 \) Copy content Toggle raw display
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