Properties

Label 112.10.a.e
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: \(\Q(\sqrt{193}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{193}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (11 \beta + 43) q^{3} + (95 \beta - 1119) q^{5} + 2401 q^{7} + (946 \beta + 5519) q^{9} + ( - 3326 \beta - 17658) q^{11} + ( - 10899 \beta - 13265) q^{13} + ( - 8224 \beta + 153568) q^{15} + ( - 9426 \beta - 231960) q^{17}+ \cdots + ( - 35060662 \beta - 704708930) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 86 q^{3} - 2238 q^{5} + 4802 q^{7} + 11038 q^{9} - 35316 q^{11} - 26530 q^{13} + 307136 q^{15} - 463920 q^{17} + 925426 q^{19} + 206486 q^{21} - 778128 q^{23} + 2081722 q^{25} + 2798612 q^{27} - 10003584 q^{29}+ \cdots - 1409417860 q^{99}+O(q^{100}) \) 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
−6.44622
7.44622
0 −109.817 0 −2438.78 0 2401.00 0 −7623.25 0
1.2 0 195.817 0 200.782 0 2401.00 0 18661.3 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.e 2
4.b odd 2 1 7.10.a.a 2
12.b even 2 1 63.10.a.d 2
20.d odd 2 1 175.10.a.b 2
20.e even 4 2 175.10.b.b 4
28.d even 2 1 49.10.a.b 2
28.f even 6 2 49.10.c.b 4
28.g odd 6 2 49.10.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.a.a 2 4.b odd 2 1
49.10.a.b 2 28.d even 2 1
49.10.c.b 4 28.f even 6 2
49.10.c.c 4 28.g odd 6 2
63.10.a.d 2 12.b even 2 1
112.10.a.e 2 1.a even 1 1 trivial
175.10.a.b 2 20.d odd 2 1
175.10.b.b 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 86T_{3} - 21504 \) 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
$3$ \( T^{2} - 86T - 21504 \) Copy content Toggle raw display
$5$ \( T^{2} + 2238 T - 489664 \) Copy content Toggle raw display
$7$ \( (T - 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 1823214304 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 22750162568 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 36657492732 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 213416091952 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 128613482496 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 23287739754332 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 3507668488800 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 224285819284476 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 51779041048756 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 207953886197312 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 14\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 34\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 459497424927744 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 17\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 51\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 33\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 60\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 70118242258304 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 40\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 10\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 66\!\cdots\!24 \) Copy content Toggle raw display
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