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Properties

Label 112.3.r.b
Level $112$
Weight $3$
Character orbit 112.r
Analytic conductor $3.052$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,3,Mod(79,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.79");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 112.r (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(3.05177896084\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.259470000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 14x^{4} - x^{3} + 176x^{2} - 91x + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 1) q^{3} + ( - \beta_{3} + \beta_1) q^{5} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - 1) q^{7} + (4 \beta_{5} - 2 \beta_{4} - \beta_{3} + \cdots + 7) q^{9} + (\beta_{5} + \beta_{3} - 3 \beta_{2} + \cdots + 7) q^{11}+ \cdots + (40 \beta_{5} - 40 \beta_{4} + \cdots + 49) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} - q^{5} - 14 q^{7} + 14 q^{9} + 33 q^{11} + 28 q^{13} - 5 q^{17} - 63 q^{19} + 29 q^{21} + 33 q^{23} - 32 q^{25} - 100 q^{29} + 69 q^{31} - 71 q^{33} - 189 q^{35} + 15 q^{37} + 246 q^{39}+ \cdots + 124 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 14x^{4} - x^{3} + 176x^{2} - 91x + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{5} + 28\nu^{4} - 392\nu^{3} + 352\nu^{2} - 182\nu + 175 ) / 2373 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 26\nu^{5} - 25\nu^{4} + 350\nu^{3} + 170\nu^{2} + 4400\nu + 98 ) / 2373 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -26\nu^{5} + 25\nu^{4} - 350\nu^{3} - 170\nu^{2} + 346\nu - 98 ) / 2373 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 78\nu^{5} + 38\nu^{4} + 1050\nu^{3} + 1301\nu^{2} + 13878\nu + 8204 ) / 2373 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -91\nu^{5} + 144\nu^{4} - 1225\nu^{3} + 987\nu^{2} - 15061\nu + 13104 ) / 2373 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{5} - \beta_{4} + 10\beta_{2} - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} - 13\beta _1 - 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -14\beta_{5} + 28\beta_{4} - 3\beta_{3} - 136\beta_{2} - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -80\beta_{5} + 40\beta_{4} - 175\beta_{3} - 379\beta_{2} + 175\beta _1 + 299 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.264167 + 0.457551i
−1.70587 − 2.95466i
1.94170 + 3.36313i
0.264167 − 0.457551i
−1.70587 + 2.95466i
1.94170 − 3.36313i
0 −4.86043 + 2.80617i 0 0.0283339 − 0.0490758i 0 −2.52833 − 6.52744i 0 11.2492 − 19.4842i 0
79.2 0 0.819997 − 0.473425i 0 −3.91174 + 6.77534i 0 1.41174 + 6.85616i 0 −4.05174 + 7.01781i 0
79.3 0 2.54043 − 1.46672i 0 3.38341 − 5.86024i 0 −5.88341 − 3.79282i 0 −0.197460 + 0.342011i 0
95.1 0 −4.86043 − 2.80617i 0 0.0283339 + 0.0490758i 0 −2.52833 + 6.52744i 0 11.2492 + 19.4842i 0
95.2 0 0.819997 + 0.473425i 0 −3.91174 − 6.77534i 0 1.41174 − 6.85616i 0 −4.05174 − 7.01781i 0
95.3 0 2.54043 + 1.46672i 0 3.38341 + 5.86024i 0 −5.88341 + 3.79282i 0 −0.197460 − 0.342011i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.3.r.b ✓ 6
3.b odd 2 1 1008.3.cd.j 6
4.b odd 2 1 112.3.r.c yes 6
7.b odd 2 1 784.3.r.q 6
7.c even 3 1 112.3.r.c yes 6
7.c even 3 1 784.3.d.l 6
7.d odd 6 1 784.3.d.k 6
7.d odd 6 1 784.3.r.p 6
8.b even 2 1 448.3.r.e 6
8.d odd 2 1 448.3.r.d 6
12.b even 2 1 1008.3.cd.k 6
21.h odd 6 1 1008.3.cd.k 6
28.d even 2 1 784.3.r.p 6
28.f even 6 1 784.3.d.k 6
28.f even 6 1 784.3.r.q 6
28.g odd 6 1 inner 112.3.r.b ✓ 6
28.g odd 6 1 784.3.d.l 6
56.k odd 6 1 448.3.r.e 6
56.p even 6 1 448.3.r.d 6
84.n even 6 1 1008.3.cd.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.3.r.b ✓ 6 1.a even 1 1 trivial
112.3.r.b ✓ 6 28.g odd 6 1 inner
112.3.r.c yes 6 4.b odd 2 1
112.3.r.c yes 6 7.c even 3 1
448.3.r.d 6 8.d odd 2 1
448.3.r.d 6 56.p even 6 1
448.3.r.e 6 8.b even 2 1
448.3.r.e 6 56.k odd 6 1
784.3.d.k 6 7.d odd 6 1
784.3.d.k 6 28.f even 6 1
784.3.d.l 6 7.c even 3 1
784.3.d.l 6 28.g odd 6 1
784.3.r.p 6 7.d odd 6 1
784.3.r.p 6 28.d even 2 1
784.3.r.q 6 7.b odd 2 1
784.3.r.q 6 28.f even 6 1
1008.3.cd.j 6 3.b odd 2 1
1008.3.cd.j 6 84.n even 6 1
1008.3.cd.k 6 12.b even 2 1
1008.3.cd.k 6 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 3T_{3}^{5} - 16T_{3}^{4} - 57T_{3}^{3} + 388T_{3}^{2} - 513T_{3} + 243 \) acting on \(S_{3}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 3 T^{5} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( T^{6} + T^{5} + 54 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} + 14 T^{5} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{6} - 33 T^{5} + \cdots + 3251043 \) Copy content Toggle raw display
$13$ \( (T^{3} - 14 T^{2} + \cdots + 728)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 5 T^{5} + \cdots + 540225 \) Copy content Toggle raw display
$19$ \( T^{6} + 63 T^{5} + \cdots + 324723 \) Copy content Toggle raw display
$23$ \( T^{6} - 33 T^{5} + \cdots + 231317883 \) Copy content Toggle raw display
$29$ \( (T^{3} + 50 T^{2} + \cdots + 1560)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 69 T^{5} + \cdots + 3479787 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 1995855625 \) Copy content Toggle raw display
$41$ \( (T^{3} - 62 T^{2} + \cdots + 216)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 10673651712 \) Copy content Toggle raw display
$47$ \( T^{6} + 171 T^{5} + \cdots + 5250987 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 17908060041 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 1821240963 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 5424764409 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 7174510227 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 33942454272 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 20655150961 \) Copy content Toggle raw display
$79$ \( T^{6} - 201 T^{5} + \cdots + 2187 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 72559411200 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 14363303409 \) Copy content Toggle raw display
$97$ \( (T^{3} - 62 T^{2} + \cdots - 195304)^{2} \) Copy content Toggle raw display
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