Properties

Label 112.2.j.a
Level $112$
Weight $2$
Character orbit 112.j
Analytic conductor $0.894$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,2,Mod(27,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 112.j (of order \(4\), 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: \(0.894324502638\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + 2 \beta_1 q^{4} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{2} - 1) q^{7} + ( - 2 \beta_1 + 2) q^{8} - 3 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + 2 \beta_1 q^{4} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{2} - 1) q^{7} + ( - 2 \beta_1 + 2) q^{8} - 3 \beta_1 q^{9} + 2 \beta_{2} q^{10} + (\beta_1 + 1) q^{11} + ( - \beta_{3} + \beta_{2}) q^{13} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{14} - 4 q^{16} - 2 \beta_{2} q^{17} + (3 \beta_1 - 3) q^{18} + (2 \beta_{3} + 2 \beta_{2}) q^{19} + (2 \beta_{3} - 2 \beta_{2}) q^{20} - 2 \beta_1 q^{22} + 4 q^{23} + 7 \beta_1 q^{25} + 2 \beta_{3} q^{26} + (2 \beta_{3} - 2 \beta_1) q^{28} + ( - 3 \beta_1 - 3) q^{29} + 2 \beta_{3} q^{31} + (4 \beta_1 + 4) q^{32} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{34} + (\beta_{3} + \beta_{2} + 6 \beta_1 - 6) q^{35} + 6 q^{36} + ( - 5 \beta_1 + 5) q^{37} - 4 \beta_{2} q^{38} - 4 \beta_{3} q^{40} + 2 \beta_{3} q^{41} + ( - 5 \beta_1 - 5) q^{43} + (2 \beta_1 - 2) q^{44} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{45} + ( - 4 \beta_1 - 4) q^{46} - 2 \beta_{3} q^{47} + (2 \beta_{2} - 5) q^{49} + ( - 7 \beta_1 + 7) q^{50} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{52} + (\beta_1 - 1) q^{53} - 2 \beta_{2} q^{55} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{56} + 6 \beta_1 q^{58} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{59} + (\beta_{3} - \beta_{2}) q^{61} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{62} + ( - 3 \beta_{3} + 3 \beta_1) q^{63} - 8 \beta_1 q^{64} + 12 q^{65} + ( - 5 \beta_1 + 5) q^{67} + 4 \beta_{3} q^{68} + ( - 2 \beta_{2} + 12) q^{70} + 2 q^{71} + ( - 6 \beta_1 - 6) q^{72} + 4 \beta_{3} q^{73} - 10 q^{74} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{76} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{77} + 4 \beta_1 q^{79} + (4 \beta_{3} + 4 \beta_{2}) q^{80} - 9 q^{81} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{82} + (12 \beta_1 - 12) q^{85} + 10 \beta_1 q^{86} + 4 q^{88} + 6 \beta_{3} q^{90} + (\beta_{3} - \beta_{2} + 6 \beta_1 + 6) q^{91} + 8 \beta_1 q^{92} + (2 \beta_{3} + 2 \beta_{2}) q^{94} - 24 \beta_1 q^{95} - 2 \beta_{2} q^{97} + (2 \beta_{3} - 2 \beta_{2} + 5 \beta_1 + 5) q^{98} + ( - 3 \beta_1 + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{7} + 8 q^{8} + 4 q^{11} + 4 q^{14} - 16 q^{16} - 12 q^{18} + 16 q^{23} - 12 q^{29} + 16 q^{32} - 24 q^{35} + 24 q^{36} + 20 q^{37} - 20 q^{43} - 8 q^{44} - 16 q^{46} - 20 q^{49} + 28 q^{50} - 4 q^{53} - 8 q^{56} + 48 q^{65} + 20 q^{67} + 48 q^{70} + 8 q^{71} - 24 q^{72} - 40 q^{74} - 4 q^{77} - 36 q^{81} - 48 q^{85} + 16 q^{88} + 24 q^{91} + 20 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
−1.22474 1.22474i
−1.00000 + 1.00000i 0 2.00000i −2.44949 + 2.44949i 0 −1.00000 + 2.44949i 2.00000 + 2.00000i 3.00000i 4.89898i
27.2 −1.00000 + 1.00000i 0 2.00000i 2.44949 2.44949i 0 −1.00000 2.44949i 2.00000 + 2.00000i 3.00000i 4.89898i
83.1 −1.00000 1.00000i 0 2.00000i −2.44949 2.44949i 0 −1.00000 2.44949i 2.00000 2.00000i 3.00000i 4.89898i
83.2 −1.00000 1.00000i 0 2.00000i 2.44949 + 2.44949i 0 −1.00000 + 2.44949i 2.00000 2.00000i 3.00000i 4.89898i
\(n\): 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
16.f odd 4 1 inner
112.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.2.j.a 4
4.b odd 2 1 448.2.j.b 4
7.b odd 2 1 inner 112.2.j.a 4
7.c even 3 2 784.2.w.d 8
7.d odd 6 2 784.2.w.d 8
8.b even 2 1 896.2.j.b 4
8.d odd 2 1 896.2.j.e 4
16.e even 4 1 448.2.j.b 4
16.e even 4 1 896.2.j.e 4
16.f odd 4 1 inner 112.2.j.a 4
16.f odd 4 1 896.2.j.b 4
28.d even 2 1 448.2.j.b 4
56.e even 2 1 896.2.j.e 4
56.h odd 2 1 896.2.j.b 4
112.j even 4 1 inner 112.2.j.a 4
112.j even 4 1 896.2.j.b 4
112.l odd 4 1 448.2.j.b 4
112.l odd 4 1 896.2.j.e 4
112.u odd 12 2 784.2.w.d 8
112.v even 12 2 784.2.w.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.j.a 4 1.a even 1 1 trivial
112.2.j.a 4 7.b odd 2 1 inner
112.2.j.a 4 16.f odd 4 1 inner
112.2.j.a 4 112.j even 4 1 inner
448.2.j.b 4 4.b odd 2 1
448.2.j.b 4 16.e even 4 1
448.2.j.b 4 28.d even 2 1
448.2.j.b 4 112.l odd 4 1
784.2.w.d 8 7.c even 3 2
784.2.w.d 8 7.d odd 6 2
784.2.w.d 8 112.u odd 12 2
784.2.w.d 8 112.v even 12 2
896.2.j.b 4 8.b even 2 1
896.2.j.b 4 16.f odd 4 1
896.2.j.b 4 56.h odd 2 1
896.2.j.b 4 112.j even 4 1
896.2.j.e 4 8.d odd 2 1
896.2.j.e 4 16.e even 4 1
896.2.j.e 4 56.e even 2 1
896.2.j.e 4 112.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(112, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{4} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 144 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 144 \) Copy content Toggle raw display
$17$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 2304 \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 2304 \) Copy content Toggle raw display
$61$ \( T^{4} + 144 \) Copy content Toggle raw display
$67$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$71$ \( (T - 2)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
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