Properties

Label 112.4.a.e
Level $112$
Weight $4$
Character orbit 112.a
Self dual yes
Analytic conductor $6.608$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{3} - 12 q^{5} - 7 q^{7} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} - 12 q^{5} - 7 q^{7} - 23 q^{9} - 48 q^{11} + 56 q^{13} - 24 q^{15} - 114 q^{17} - 2 q^{19} - 14 q^{21} + 120 q^{23} + 19 q^{25} - 100 q^{27} - 54 q^{29} - 236 q^{31} - 96 q^{33} + 84 q^{35} + 146 q^{37} + 112 q^{39} + 126 q^{41} + 376 q^{43} + 276 q^{45} + 12 q^{47} + 49 q^{49} - 228 q^{51} + 174 q^{53} + 576 q^{55} - 4 q^{57} - 138 q^{59} + 380 q^{61} + 161 q^{63} - 672 q^{65} + 484 q^{67} + 240 q^{69} - 576 q^{71} - 1150 q^{73} + 38 q^{75} + 336 q^{77} - 776 q^{79} + 421 q^{81} - 378 q^{83} + 1368 q^{85} - 108 q^{87} - 390 q^{89} - 392 q^{91} - 472 q^{93} + 24 q^{95} - 1330 q^{97} + 1104 q^{99}+O(q^{100}) \) 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 −12.0000 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.e 1
3.b odd 2 1 1008.4.a.r 1
4.b odd 2 1 14.4.a.b 1
7.b odd 2 1 784.4.a.h 1
8.b even 2 1 448.4.a.g 1
8.d odd 2 1 448.4.a.k 1
12.b even 2 1 126.4.a.d 1
20.d odd 2 1 350.4.a.f 1
20.e even 4 2 350.4.c.g 2
28.d even 2 1 98.4.a.e 1
28.f even 6 2 98.4.c.b 2
28.g odd 6 2 98.4.c.c 2
44.c even 2 1 1694.4.a.b 1
52.b odd 2 1 2366.4.a.c 1
84.h odd 2 1 882.4.a.b 1
84.j odd 6 2 882.4.g.v 2
84.n even 6 2 882.4.g.p 2
140.c even 2 1 2450.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.b 1 4.b odd 2 1
98.4.a.e 1 28.d even 2 1
98.4.c.b 2 28.f even 6 2
98.4.c.c 2 28.g odd 6 2
112.4.a.e 1 1.a even 1 1 trivial
126.4.a.d 1 12.b even 2 1
350.4.a.f 1 20.d odd 2 1
350.4.c.g 2 20.e even 4 2
448.4.a.g 1 8.b even 2 1
448.4.a.k 1 8.d odd 2 1
784.4.a.h 1 7.b odd 2 1
882.4.a.b 1 84.h odd 2 1
882.4.g.p 2 84.n even 6 2
882.4.g.v 2 84.j odd 6 2
1008.4.a.r 1 3.b odd 2 1
1694.4.a.b 1 44.c even 2 1
2366.4.a.c 1 52.b odd 2 1
2450.4.a.i 1 140.c even 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
\( T_{5} + 12 \) 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
$5$ \( T + 12 \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T + 48 \) Copy content Toggle raw display
$13$ \( T - 56 \) Copy content Toggle raw display
$17$ \( T + 114 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T - 120 \) Copy content Toggle raw display
$29$ \( T + 54 \) Copy content Toggle raw display
$31$ \( T + 236 \) Copy content Toggle raw display
$37$ \( T - 146 \) Copy content Toggle raw display
$41$ \( T - 126 \) Copy content Toggle raw display
$43$ \( T - 376 \) Copy content Toggle raw display
$47$ \( T - 12 \) Copy content Toggle raw display
$53$ \( T - 174 \) Copy content Toggle raw display
$59$ \( T + 138 \) Copy content Toggle raw display
$61$ \( T - 380 \) Copy content Toggle raw display
$67$ \( T - 484 \) Copy content Toggle raw display
$71$ \( T + 576 \) Copy content Toggle raw display
$73$ \( T + 1150 \) Copy content Toggle raw display
$79$ \( T + 776 \) Copy content Toggle raw display
$83$ \( T + 378 \) Copy content Toggle raw display
$89$ \( T + 390 \) Copy content Toggle raw display
$97$ \( T + 1330 \) Copy content Toggle raw display
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