Properties

Label 112.4.a.f
Level $112$
Weight $4$
Character orbit 112.a
Self dual yes
Analytic conductor $6.608$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,4,Mod(1,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 112.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.60821392064\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 2 q^{3} + 16 q^{5} + 7 q^{7} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} + 16 q^{5} + 7 q^{7} - 23 q^{9} + 8 q^{11} + 28 q^{13} + 32 q^{15} + 54 q^{17} + 110 q^{19} + 14 q^{21} - 48 q^{23} + 131 q^{25} - 100 q^{27} - 110 q^{29} - 12 q^{31} + 16 q^{33} + 112 q^{35} - 246 q^{37} + 56 q^{39} + 182 q^{41} - 128 q^{43} - 368 q^{45} - 324 q^{47} + 49 q^{49} + 108 q^{51} - 162 q^{53} + 128 q^{55} + 220 q^{57} - 810 q^{59} - 488 q^{61} - 161 q^{63} + 448 q^{65} - 244 q^{67} - 96 q^{69} + 768 q^{71} - 702 q^{73} + 262 q^{75} + 56 q^{77} - 440 q^{79} + 421 q^{81} + 1302 q^{83} + 864 q^{85} - 220 q^{87} + 730 q^{89} + 196 q^{91} - 24 q^{93} + 1760 q^{95} + 294 q^{97} - 184 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
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
0
0 2.00000 0 16.0000 0 7.00000 0 −23.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)

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 112.4.a.f 1
3.b odd 2 1 1008.4.a.c 1
4.b odd 2 1 7.4.a.a 1
7.b odd 2 1 784.4.a.g 1
8.b even 2 1 448.4.a.e 1
8.d odd 2 1 448.4.a.i 1
12.b even 2 1 63.4.a.b 1
20.d odd 2 1 175.4.a.b 1
20.e even 4 2 175.4.b.b 2
28.d even 2 1 49.4.a.b 1
28.f even 6 2 49.4.c.b 2
28.g odd 6 2 49.4.c.c 2
44.c even 2 1 847.4.a.b 1
52.b odd 2 1 1183.4.a.b 1
60.h even 2 1 1575.4.a.e 1
68.d odd 2 1 2023.4.a.a 1
84.h odd 2 1 441.4.a.i 1
84.j odd 6 2 441.4.e.e 2
84.n even 6 2 441.4.e.h 2
140.c even 2 1 1225.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 4.b odd 2 1
49.4.a.b 1 28.d even 2 1
49.4.c.b 2 28.f even 6 2
49.4.c.c 2 28.g odd 6 2
63.4.a.b 1 12.b even 2 1
112.4.a.f 1 1.a even 1 1 trivial
175.4.a.b 1 20.d odd 2 1
175.4.b.b 2 20.e even 4 2
441.4.a.i 1 84.h odd 2 1
441.4.e.e 2 84.j odd 6 2
441.4.e.h 2 84.n even 6 2
448.4.a.e 1 8.b even 2 1
448.4.a.i 1 8.d odd 2 1
784.4.a.g 1 7.b odd 2 1
847.4.a.b 1 44.c even 2 1
1008.4.a.c 1 3.b odd 2 1
1183.4.a.b 1 52.b odd 2 1
1225.4.a.j 1 140.c even 2 1
1575.4.a.e 1 60.h even 2 1
2023.4.a.a 1 68.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(112))\):

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{5} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 2 \) Copy content Toggle raw display
$5$ \( T - 16 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T - 8 \) Copy content Toggle raw display
$13$ \( T - 28 \) Copy content Toggle raw display
$17$ \( T - 54 \) Copy content Toggle raw display
$19$ \( T - 110 \) Copy content Toggle raw display
$23$ \( T + 48 \) Copy content Toggle raw display
$29$ \( T + 110 \) Copy content Toggle raw display
$31$ \( T + 12 \) Copy content Toggle raw display
$37$ \( T + 246 \) Copy content Toggle raw display
$41$ \( T - 182 \) Copy content Toggle raw display
$43$ \( T + 128 \) Copy content Toggle raw display
$47$ \( T + 324 \) Copy content Toggle raw display
$53$ \( T + 162 \) Copy content Toggle raw display
$59$ \( T + 810 \) Copy content Toggle raw display
$61$ \( T + 488 \) Copy content Toggle raw display
$67$ \( T + 244 \) Copy content Toggle raw display
$71$ \( T - 768 \) Copy content Toggle raw display
$73$ \( T + 702 \) Copy content Toggle raw display
$79$ \( T + 440 \) Copy content Toggle raw display
$83$ \( T - 1302 \) Copy content Toggle raw display
$89$ \( T - 730 \) Copy content Toggle raw display
$97$ \( T - 294 \) Copy content Toggle raw display
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