Properties

Label 112.10.a.d
Level $112$
Weight $10$
Character orbit 112.a
Self dual yes
Analytic conductor $57.684$
Analytic rank $1$
Dimension $2$
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] = [112,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(57.6840136504\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11209}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 2802 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 28)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{11209}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 35) q^{3} + (19 \beta + 777) q^{5} + 2401 q^{7} + ( - 70 \beta - 7249) q^{9} + ( - 182 \beta - 31194) q^{11} + ( - 87 \beta + 61383) q^{13} + ( - 112 \beta - 185776) q^{15} + ( - 2986 \beta + 36792) q^{17}+ \cdots + (3502898 \beta + 368927966) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 70 q^{3} + 1554 q^{5} + 4802 q^{7} - 14498 q^{9} - 62388 q^{11} + 122766 q^{13} - 371552 q^{15} + 73584 q^{17} - 1171198 q^{19} + 168070 q^{21} - 2262384 q^{23} + 5394106 q^{25} - 315980 q^{27} - 1923360 q^{29}+ \cdots + 737855932 q^{99}+O(q^{100}) \) 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
53.4363
−52.4363
0 −70.8726 0 2788.58 0 2401.00 0 −14660.1 0
1.2 0 140.873 0 −1234.58 0 2401.00 0 162.080 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 112.10.a.d 2
4.b odd 2 1 28.10.a.b 2
12.b even 2 1 252.10.a.b 2
28.d even 2 1 196.10.a.b 2
28.f even 6 2 196.10.e.d 4
28.g odd 6 2 196.10.e.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.10.a.b 2 4.b odd 2 1
112.10.a.d 2 1.a even 1 1 trivial
196.10.a.b 2 28.d even 2 1
196.10.e.d 4 28.f even 6 2
196.10.e.e 4 28.g odd 6 2
252.10.a.b 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 70T_{3} - 9984 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(112))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 70T - 9984 \) Copy content Toggle raw display
$5$ \( T^{2} - 1554 T - 3442720 \) Copy content Toggle raw display
$7$ \( (T - 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 62388 T + 601778720 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 3683031768 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 98587989700 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 297234258352 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 657801801728 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 6012588512356 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 21751509340512 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 162679566869060 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 328158355074444 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 89571104447680 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 315559687006944 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 54\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 17\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 17\!\cdots\!60 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 13\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 19\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 82\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 72\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 57\!\cdots\!52 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 72\!\cdots\!36 \) Copy content Toggle raw display
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