↑

Properties

Label 112.5.c.c
Level $112$
Weight $5$
Character orbit 112.c
Analytic conductor $11.577$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,5,Mod(97,112)]
 
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, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.97");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.c (of order \(2\), degree \(1\), not 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: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.1308672.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 72x^{2} + 1278 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} + \beta_1) q^{5} + (3 \beta_{3} + \beta_{2} - \beta_1 + 19) q^{7} + (2 \beta_{3} - 63) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} + \beta_1) q^{5} + (3 \beta_{3} + \beta_{2} - \beta_1 + 19) q^{7} + (2 \beta_{3} - 63) q^{9} + (12 \beta_{3} - 90) q^{11} + (\beta_{2} - \beta_1) q^{13} + ( - 46 \beta_{3} - 96) q^{15} + ( - 2 \beta_{2} - 20 \beta_1) q^{17} + (10 \beta_{2} - 19 \beta_1) q^{19} + ( - 50 \beta_{3} + 9 \beta_{2} + \cdots + 192) q^{21}+ \cdots + ( - 936 \beta_{3} + 7398) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 76 q^{7} - 252 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 76 q^{7} - 252 q^{9} - 360 q^{11} - 384 q^{15} + 768 q^{21} + 792 q^{23} - 2300 q^{25} + 1224 q^{29} - 4032 q^{35} - 3896 q^{37} + 768 q^{39} - 3688 q^{43} - 1532 q^{49} + 11136 q^{51} + 5832 q^{53} + 12864 q^{57} - 3060 q^{63} - 4032 q^{65} + 1048 q^{67} + 21528 q^{71} + 3528 q^{77} - 12776 q^{79} - 29628 q^{81} + 16512 q^{85} - 5568 q^{91} + 38016 q^{93} - 36864 q^{95} + 29592 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 72x^{2} + 1278 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{3} + 144\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 72 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{2} - 72\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
− 6.34371i
− 5.63537i
5.63537i
6.34371i
0 − 12.6874i 0 23.1980i 0 −6.45584 + 48.5729i 0 −79.9706 0
97.2 0 − 11.2707i 0 − 43.1492i 0 44.4558 − 20.6077i 0 −46.0294 0
97.3 0 11.2707i 0 43.1492i 0 44.4558 + 20.6077i 0 −46.0294 0
97.4 0 12.6874i 0 − 23.1980i 0 −6.45584 − 48.5729i 0 −79.9706 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.5.c.c 4
3.b odd 2 1 1008.5.f.h 4
4.b odd 2 1 14.5.b.a ✓ 4
7.b odd 2 1 inner 112.5.c.c 4
8.b even 2 1 448.5.c.f 4
8.d odd 2 1 448.5.c.e 4
12.b even 2 1 126.5.c.a 4
20.d odd 2 1 350.5.b.a 4
20.e even 4 2 350.5.d.a 8
21.c even 2 1 1008.5.f.h 4
28.d even 2 1 14.5.b.a ✓ 4
28.f even 6 2 98.5.d.d 8
28.g odd 6 2 98.5.d.d 8
56.e even 2 1 448.5.c.e 4
56.h odd 2 1 448.5.c.f 4
84.h odd 2 1 126.5.c.a 4
140.c even 2 1 350.5.b.a 4
140.j odd 4 2 350.5.d.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.5.b.a ✓ 4 4.b odd 2 1
14.5.b.a ✓ 4 28.d even 2 1
98.5.d.d 8 28.f even 6 2
98.5.d.d 8 28.g odd 6 2
112.5.c.c 4 1.a even 1 1 trivial
112.5.c.c 4 7.b odd 2 1 inner
126.5.c.a 4 12.b even 2 1
126.5.c.a 4 84.h odd 2 1
350.5.b.a 4 20.d odd 2 1
350.5.b.a 4 140.c even 2 1
350.5.d.a 8 20.e even 4 2
350.5.d.a 8 140.j odd 4 2
448.5.c.e 4 8.d odd 2 1
448.5.c.e 4 56.e even 2 1
448.5.c.f 4 8.b even 2 1
448.5.c.f 4 56.h odd 2 1
1008.5.f.h 4 3.b odd 2 1
1008.5.f.h 4 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 288T_{3}^{2} + 20448 \) acting on \(S_{5}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 288 T^{2} + 20448 \) Copy content Toggle raw display
$5$ \( T^{4} + 2400 T^{2} + 1001952 \) Copy content Toggle raw display
$7$ \( T^{4} - 76 T^{3} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{2} + 180 T - 2268)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 2784 T^{2} + 1001952 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 2769149952 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 3940759008 \) Copy content Toggle raw display
$23$ \( (T^{2} - 396 T - 34524)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 612 T - 322236)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 137379151872 \) Copy content Toggle raw display
$37$ \( (T^{2} + 1948 T + 855364)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 2977977687552 \) Copy content Toggle raw display
$43$ \( (T^{2} + 1844 T - 186716)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 299010613248 \) Copy content Toggle raw display
$53$ \( (T^{2} - 2916 T - 6394428)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 8158764780000 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 1641231749088 \) Copy content Toggle raw display
$67$ \( (T^{2} - 524 T - 594908)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 10764 T + 19321092)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 5086562360832 \) Copy content Toggle raw display
$79$ \( (T^{2} + 6388 T - 4378364)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 123161511902688 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 79956812938752 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 58\!\cdots\!52 \) Copy content Toggle raw display
show more
show less