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Properties

Label 1792.2.b.m
Level $1792$
Weight $2$
Character orbit 1792.b
Analytic conductor $14.309$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1792,2,Mod(897,1792)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1792.897"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1792, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,4,0,-12,0,0,0,0,0,-16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} + \beta_1) q^{5} + q^{7} + (\beta_{3} - 3) q^{9} + (2 \beta_{2} + 2 \beta_1) q^{11} + ( - \beta_{2} + \beta_1) q^{13} - 4 q^{15} + \beta_{3} q^{17} - \beta_1 q^{19} + \beta_1 q^{21}+ \cdots + (6 \beta_{2} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7} - 12 q^{9} - 16 q^{15} + 16 q^{23} - 4 q^{25} - 8 q^{31} - 32 q^{33} - 32 q^{39} + 16 q^{41} - 24 q^{47} + 4 q^{49} - 48 q^{55} + 24 q^{57} - 12 q^{63} - 8 q^{65} + 16 q^{71} - 24 q^{73} - 16 q^{79}+ \cdots + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} + 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
897.1
− 1.61803i
− 0.618034i
0.618034i
1.61803i
0 − 3.23607i 0 − 1.23607i 0 1.00000 0 −7.47214 0
897.2 0 − 1.23607i 0 − 3.23607i 0 1.00000 0 1.47214 0
897.3 0 1.23607i 0 3.23607i 0 1.00000 0 1.47214 0
897.4 0 3.23607i 0 1.23607i 0 1.00000 0 −7.47214 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.b.m 4
4.b odd 2 1 1792.2.b.k 4
8.b even 2 1 inner 1792.2.b.m 4
8.d odd 2 1 1792.2.b.k 4
16.e even 4 1 224.2.a.d yes 2
16.e even 4 1 448.2.a.i 2
16.f odd 4 1 224.2.a.c ✓ 2
16.f odd 4 1 448.2.a.j 2
48.i odd 4 1 2016.2.a.o 2
48.i odd 4 1 4032.2.a.bv 2
48.k even 4 1 2016.2.a.r 2
48.k even 4 1 4032.2.a.bw 2
80.k odd 4 1 5600.2.a.bk 2
80.q even 4 1 5600.2.a.z 2
112.j even 4 1 1568.2.a.v 2
112.j even 4 1 3136.2.a.bf 2
112.l odd 4 1 1568.2.a.k 2
112.l odd 4 1 3136.2.a.by 2
112.u odd 12 2 1568.2.i.v 4
112.v even 12 2 1568.2.i.n 4
112.w even 12 2 1568.2.i.m 4
112.x odd 12 2 1568.2.i.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.c ✓ 2 16.f odd 4 1
224.2.a.d yes 2 16.e even 4 1
448.2.a.i 2 16.e even 4 1
448.2.a.j 2 16.f odd 4 1
1568.2.a.k 2 112.l odd 4 1
1568.2.a.v 2 112.j even 4 1
1568.2.i.m 4 112.w even 12 2
1568.2.i.n 4 112.v even 12 2
1568.2.i.v 4 112.u odd 12 2
1568.2.i.w 4 112.x odd 12 2
1792.2.b.k 4 4.b odd 2 1
1792.2.b.k 4 8.d odd 2 1
1792.2.b.m 4 1.a even 1 1 trivial
1792.2.b.m 4 8.b even 2 1 inner
2016.2.a.o 2 48.i odd 4 1
2016.2.a.r 2 48.k even 4 1
3136.2.a.bf 2 112.j even 4 1
3136.2.a.by 2 112.l odd 4 1
4032.2.a.bv 2 48.i odd 4 1
4032.2.a.bw 2 48.k even 4 1
5600.2.a.z 2 80.q even 4 1
5600.2.a.bk 2 80.k odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 12T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{5}^{4} + 12T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{4} + 48T_{11}^{2} + 256 \) Copy content Toggle raw display
\( T_{23} - 4 \) Copy content Toggle raw display
\( T_{31}^{2} + 4T_{31} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$13$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$17$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$47$ \( (T^{2} + 12 T + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 108T^{2} + 1936 \) Copy content Toggle raw display
$61$ \( T^{4} + 172T^{2} + 5776 \) Copy content Toggle raw display
$67$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 8 T - 64)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T - 44)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T - 64)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 108T^{2} + 1936 \) Copy content Toggle raw display
$89$ \( (T - 6)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T + 44)^{2} \) Copy content Toggle raw display
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