Properties

Label 8112.2.a.cq
Level $8112$
Weight $2$
Character orbit 8112.a
Self dual yes
Analytic conductor $64.775$
Analytic rank $0$
Dimension $4$
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] = [8112,2,Mod(1,8112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8112, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8112.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8112.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: \(64.7746461197\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.25488.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 9x^{2} - 6x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 312)
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 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.

\(f(q)\) \(=\) \( q - q^{3} + ( - \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + ( - \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{7} + q^{9} + ( - \beta_{3} + \beta_{2} + 2) q^{11} + (\beta_{2} - \beta_1 + 1) q^{15} + (\beta_{2} + \beta_1 + 3) q^{17} + ( - \beta_{3} - \beta_{2} + 2 \beta_1) q^{19} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{21} + ( - \beta_{3} - \beta_{2}) q^{23} + ( - \beta_{3} + 4 \beta_{2} - 3 \beta_1) q^{25} - q^{27} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{29} + ( - \beta_1 + 5) q^{31} + (\beta_{3} - \beta_{2} - 2) q^{33} + ( - \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 2) q^{35} + ( - \beta_{3} + \beta_1 - 3) q^{37} + (\beta_{3} + 3 \beta_1 - 1) q^{41} + ( - \beta_{2} + 3 \beta_1 + 2) q^{43} + ( - \beta_{2} + \beta_1 - 1) q^{45} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1 + 4) q^{47} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 5) q^{49} + ( - \beta_{2} - \beta_1 - 3) q^{51} + (\beta_{3} - 2 \beta_{2} - 3 \beta_1 + 1) q^{53} + (\beta_{3} - 7 \beta_{2} + 4 \beta_1 - 2) q^{55} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{57} + ( - 4 \beta_{2} + 2 \beta_1 + 2) q^{59} + ( - \beta_{3} - 5 \beta_{2} - 1) q^{61} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{63} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 3) q^{67} + (\beta_{3} + \beta_{2}) q^{69} + (3 \beta_{3} - \beta_{2} - 2 \beta_1) q^{71} + (\beta_{2} + 2 \beta_1 - 2) q^{73} + (\beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{75} + (2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 6) q^{77} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{79} + q^{81} + (\beta_{3} - \beta_{2} + 2 \beta_1 + 8) q^{83} + (\beta_{3} - 4 \beta_{2} + 3 \beta_1 - 1) q^{85} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{87} + (2 \beta_{3} - 2 \beta_{2} - 8) q^{89} + (\beta_1 - 5) q^{93} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 8) q^{95} + ( - 6 \beta_{2} + \beta_1 + 3) q^{97} + ( - \beta_{3} + \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 4 q^{5} + 2 q^{7} + 4 q^{9} + 10 q^{11} + 4 q^{15} + 12 q^{17} + 2 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 4 q^{27} - 6 q^{29} + 20 q^{31} - 10 q^{33} + 10 q^{35} - 10 q^{37} - 6 q^{41} + 8 q^{43} - 4 q^{45} + 14 q^{47} + 18 q^{49} - 12 q^{51} + 2 q^{53} - 10 q^{55} - 2 q^{57} + 8 q^{59} - 2 q^{61} + 2 q^{63} - 10 q^{67} - 2 q^{69} - 6 q^{71} - 8 q^{73} - 2 q^{75} - 28 q^{77} + 6 q^{79} + 4 q^{81} + 30 q^{83} - 6 q^{85} + 6 q^{87} - 36 q^{89} - 20 q^{93} + 34 q^{95} + 12 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 9x^{2} - 6x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 6\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 7\beta _1 + 5 \) 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
−1.47535
−2.28657
3.20740
0.554520
0 −1.00000 0 −4.20740 0 −3.55539 0 1.00000 0
1.2 0 −1.00000 0 −1.55452 0 2.96046 0 1.00000 0
1.3 0 −1.00000 0 0.475353 0 4.55539 0 1.00000 0
1.4 0 −1.00000 0 1.28657 0 −1.96046 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(13\) \(-1\)

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 8112.2.a.cq 4
4.b odd 2 1 4056.2.a.bd 4
13.b even 2 1 8112.2.a.cs 4
13.f odd 12 2 624.2.bv.g 8
39.k even 12 2 1872.2.by.m 8
52.b odd 2 1 4056.2.a.be 4
52.f even 4 2 4056.2.c.p 8
52.l even 12 2 312.2.bf.b 8
156.v odd 12 2 936.2.bi.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.bf.b 8 52.l even 12 2
624.2.bv.g 8 13.f odd 12 2
936.2.bi.c 8 156.v odd 12 2
1872.2.by.m 8 39.k even 12 2
4056.2.a.bd 4 4.b odd 2 1
4056.2.a.be 4 52.b odd 2 1
4056.2.c.p 8 52.f even 4 2
8112.2.a.cq 4 1.a even 1 1 trivial
8112.2.a.cs 4 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8112))\):

\( T_{5}^{4} + 4T_{5}^{3} - 3T_{5}^{2} - 8T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 2T_{7}^{3} - 21T_{7}^{2} + 22T_{7} + 94 \) Copy content Toggle raw display
\( T_{11}^{4} - 10T_{11}^{3} + 18T_{11}^{2} + 56T_{11} - 104 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} - 3 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} - 21 T^{2} + 22 T + 94 \) Copy content Toggle raw display
$11$ \( T^{4} - 10 T^{3} + 18 T^{2} + \cdots - 104 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 12 T^{3} + 33 T^{2} - 48 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} - 54 T^{2} - 8 T + 376 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} - 30 T^{2} - 32 T - 8 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} - 15 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$31$ \( T^{4} - 20 T^{3} + 141 T^{2} + \cdots + 376 \) Copy content Toggle raw display
$37$ \( T^{4} + 10 T^{3} + 9 T^{2} - 80 T - 128 \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} - 87 T^{2} + \cdots - 1776 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} - 45 T^{2} + 190 T + 694 \) Copy content Toggle raw display
$47$ \( T^{4} - 14 T^{3} + 18 T^{2} + \cdots - 1448 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} - 147 T^{2} + \cdots + 208 \) Copy content Toggle raw display
$59$ \( T^{4} - 8 T^{3} - 60 T^{2} + 448 T - 512 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} - 198 T^{2} + \cdots + 6481 \) Copy content Toggle raw display
$67$ \( T^{4} + 10 T^{3} - 45 T^{2} + \cdots + 214 \) Copy content Toggle raw display
$71$ \( T^{4} + 6 T^{3} - 198 T^{2} + \cdots - 1128 \) Copy content Toggle raw display
$73$ \( T^{4} + 8 T^{3} - 30 T^{2} - 280 T - 431 \) Copy content Toggle raw display
$79$ \( T^{4} - 6 T^{3} - 63 T^{2} + 108 T + 216 \) Copy content Toggle raw display
$83$ \( T^{4} - 30 T^{3} + 294 T^{2} + \cdots + 1248 \) Copy content Toggle raw display
$89$ \( T^{4} + 36 T^{3} + 408 T^{2} + \cdots - 768 \) Copy content Toggle raw display
$97$ \( T^{4} - 12 T^{3} - 135 T^{2} + \cdots + 5748 \) Copy content Toggle raw display
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