Properties

Label 1792.2.e.c
Level $1792$
Weight $2$
Character orbit 1792.e
Analytic conductor $14.309$
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] = [1792,2,Mod(895,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.895");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.e (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: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 112)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{2} + 2) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{2} + 2) q^{7} - q^{9} - \beta_{3} q^{11} + \beta_{3} q^{13} + 4 \beta_{2} q^{15} + \beta_1 q^{19} + ( - \beta_{3} + 2 \beta_1) q^{21} - 2 \beta_{2} q^{23} + 7 q^{25} + 2 \beta_1 q^{27} + 3 \beta_1 q^{29} - 8 q^{31} - 4 \beta_{2} q^{33} + (2 \beta_{3} + 3 \beta_1) q^{35} + \beta_1 q^{37} + 4 \beta_{2} q^{39} - 4 \beta_{2} q^{41} + 3 \beta_{3} q^{43} - \beta_{3} q^{45} + (4 \beta_{2} + 1) q^{49} - 3 \beta_1 q^{53} - 12 q^{55} - 4 q^{57} - 3 \beta_1 q^{59} + \beta_{3} q^{61} + ( - \beta_{2} - 2) q^{63} + 12 q^{65} + \beta_{3} q^{67} + 2 \beta_{3} q^{69} + 2 \beta_{2} q^{71} + 4 \beta_{2} q^{73} + 7 \beta_1 q^{75} + ( - 2 \beta_{3} - 3 \beta_1) q^{77} - 2 \beta_{2} q^{79} - 11 q^{81} - 3 \beta_1 q^{83} - 12 q^{87} - 4 \beta_{2} q^{89} + (2 \beta_{3} + 3 \beta_1) q^{91} - 8 \beta_1 q^{93} + 4 \beta_{2} q^{95} - 8 \beta_{2} q^{97} + \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{7} - 4 q^{9} + 28 q^{25} - 32 q^{31} + 4 q^{49} - 48 q^{55} - 16 q^{57} - 8 q^{63} + 48 q^{65} - 44 q^{81} - 48 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{12}^{3} + 4\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
895.1
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0 2.00000i 0 −3.46410 0 2.00000 + 1.73205i 0 −1.00000 0
895.2 0 2.00000i 0 3.46410 0 2.00000 1.73205i 0 −1.00000 0
895.3 0 2.00000i 0 −3.46410 0 2.00000 1.73205i 0 −1.00000 0
895.4 0 2.00000i 0 3.46410 0 2.00000 + 1.73205i 0 −1.00000 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
8.b even 2 1 inner
28.d even 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.e.c 4
4.b odd 2 1 1792.2.e.a 4
7.b odd 2 1 1792.2.e.a 4
8.b even 2 1 inner 1792.2.e.c 4
8.d odd 2 1 1792.2.e.a 4
16.e even 4 1 112.2.f.b yes 2
16.e even 4 1 448.2.f.a 2
16.f odd 4 1 112.2.f.a 2
16.f odd 4 1 448.2.f.c 2
28.d even 2 1 inner 1792.2.e.c 4
48.i odd 4 1 1008.2.b.b 2
48.i odd 4 1 4032.2.b.b 2
48.k even 4 1 1008.2.b.g 2
48.k even 4 1 4032.2.b.h 2
56.e even 2 1 inner 1792.2.e.c 4
56.h odd 2 1 1792.2.e.a 4
80.i odd 4 1 2800.2.e.b 4
80.j even 4 1 2800.2.e.c 4
80.k odd 4 1 2800.2.k.e 2
80.q even 4 1 2800.2.k.b 2
80.s even 4 1 2800.2.e.c 4
80.t odd 4 1 2800.2.e.b 4
112.j even 4 1 112.2.f.b yes 2
112.j even 4 1 448.2.f.a 2
112.l odd 4 1 112.2.f.a 2
112.l odd 4 1 448.2.f.c 2
112.u odd 12 1 784.2.p.e 2
112.u odd 12 1 784.2.p.f 2
112.v even 12 1 784.2.p.a 2
112.v even 12 1 784.2.p.b 2
112.w even 12 1 784.2.p.a 2
112.w even 12 1 784.2.p.b 2
112.x odd 12 1 784.2.p.e 2
112.x odd 12 1 784.2.p.f 2
336.v odd 4 1 1008.2.b.b 2
336.v odd 4 1 4032.2.b.b 2
336.y even 4 1 1008.2.b.g 2
336.y even 4 1 4032.2.b.h 2
560.r even 4 1 2800.2.e.c 4
560.u odd 4 1 2800.2.e.b 4
560.be even 4 1 2800.2.k.b 2
560.bf odd 4 1 2800.2.k.e 2
560.bm odd 4 1 2800.2.e.b 4
560.bn even 4 1 2800.2.e.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.f.a 2 16.f odd 4 1
112.2.f.a 2 112.l odd 4 1
112.2.f.b yes 2 16.e even 4 1
112.2.f.b yes 2 112.j even 4 1
448.2.f.a 2 16.e even 4 1
448.2.f.a 2 112.j even 4 1
448.2.f.c 2 16.f odd 4 1
448.2.f.c 2 112.l odd 4 1
784.2.p.a 2 112.v even 12 1
784.2.p.a 2 112.w even 12 1
784.2.p.b 2 112.v even 12 1
784.2.p.b 2 112.w even 12 1
784.2.p.e 2 112.u odd 12 1
784.2.p.e 2 112.x odd 12 1
784.2.p.f 2 112.u odd 12 1
784.2.p.f 2 112.x odd 12 1
1008.2.b.b 2 48.i odd 4 1
1008.2.b.b 2 336.v odd 4 1
1008.2.b.g 2 48.k even 4 1
1008.2.b.g 2 336.y even 4 1
1792.2.e.a 4 4.b odd 2 1
1792.2.e.a 4 7.b odd 2 1
1792.2.e.a 4 8.d odd 2 1
1792.2.e.a 4 56.h odd 2 1
1792.2.e.c 4 1.a even 1 1 trivial
1792.2.e.c 4 8.b even 2 1 inner
1792.2.e.c 4 28.d even 2 1 inner
1792.2.e.c 4 56.e even 2 1 inner
2800.2.e.b 4 80.i odd 4 1
2800.2.e.b 4 80.t odd 4 1
2800.2.e.b 4 560.u odd 4 1
2800.2.e.b 4 560.bm odd 4 1
2800.2.e.c 4 80.j even 4 1
2800.2.e.c 4 80.s even 4 1
2800.2.e.c 4 560.r even 4 1
2800.2.e.c 4 560.bn even 4 1
2800.2.k.b 2 80.q even 4 1
2800.2.k.b 2 560.be even 4 1
2800.2.k.e 2 80.k odd 4 1
2800.2.k.e 2 560.bf odd 4 1
4032.2.b.b 2 48.i odd 4 1
4032.2.b.b 2 336.v odd 4 1
4032.2.b.h 2 48.k even 4 1
4032.2.b.h 2 336.y even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display
\( T_{31} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
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