Properties

Label 112.10.a.c
Level $112$
Weight $10$
Character orbit 112.a
Self dual yes
Analytic conductor $57.684$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2305}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2305}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (5 \beta + 7) q^{3} + ( - 21 \beta - 1365) q^{5} - 2401 q^{7} + (70 \beta + 37991) q^{9} + (1050 \beta - 22470) q^{11} + (225 \beta + 50141) q^{13} + ( - 6972 \beta - 251580) q^{15} + (1590 \beta - 435204) q^{17}+ \cdots + (38317650 \beta - 684240270) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{3} - 2730 q^{5} - 4802 q^{7} + 75982 q^{9} - 44940 q^{11} + 100282 q^{13} - 503160 q^{15} - 870408 q^{17} - 508774 q^{19} - 33614 q^{21} - 79800 q^{23} + 1853210 q^{25} + 1869812 q^{27} + 2006328 q^{29}+ \cdots - 1368480540 q^{99}+O(q^{100}) \) 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
−23.5052
24.5052
0 −233.052 0 −356.781 0 −2401.00 0 34630.3 0
1.2 0 247.052 0 −2373.22 0 −2401.00 0 41351.7 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.c 2
4.b odd 2 1 14.10.a.c 2
12.b even 2 1 126.10.a.o 2
20.d odd 2 1 350.10.a.j 2
20.e even 4 2 350.10.c.j 4
28.d even 2 1 98.10.a.e 2
28.f even 6 2 98.10.c.h 4
28.g odd 6 2 98.10.c.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.c 2 4.b odd 2 1
98.10.a.e 2 28.d even 2 1
98.10.c.h 4 28.f even 6 2
98.10.c.j 4 28.g odd 6 2
112.10.a.c 2 1.a even 1 1 trivial
126.10.a.o 2 12.b even 2 1
350.10.a.j 2 20.d odd 2 1
350.10.c.j 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 14T_{3} - 57576 \) 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
$3$ \( T^{2} - 14T - 57576 \) Copy content Toggle raw display
$5$ \( T^{2} + 2730 T + 846720 \) Copy content Toggle raw display
$7$ \( (T + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 2036361600 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 2397429256 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 183575251116 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 352577596856 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 115915968000 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 31904129519604 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 2367849772544 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 60225113026444 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 527816477266884 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 271341247682336 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 494746212296124 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 85\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 83\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 68\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 76\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 50\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 45\!\cdots\!00 \) Copy content Toggle raw display
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