Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1792,2,Mod(897,1792)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//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);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.b (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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - 2 \beta q^{5} + q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - 2 \beta q^{5} + q^{7} - q^{9} + 8 q^{15} - 2 q^{17} - \beta q^{19} + \beta q^{21} + 8 q^{23} - 11 q^{25} + 2 \beta q^{27} - \beta q^{29} - 4 q^{31} - 2 \beta q^{35} - 3 \beta q^{37} + 2 q^{41} - 4 \beta q^{43} + 2 \beta q^{45} + 4 q^{47} + q^{49} - 2 \beta q^{51} - 5 \beta q^{53} + 4 q^{57} - 3 \beta q^{59} - 2 \beta q^{61} - q^{63} - 6 \beta q^{67} + 8 \beta q^{69} + 14 q^{73} - 11 \beta q^{75} + 8 q^{79} - 11 q^{81} + 3 \beta q^{83} + 4 \beta q^{85} + 4 q^{87} - 10 q^{89} - 4 \beta q^{93} - 8 q^{95} - 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{7} - 2 q^{9} + 16 q^{15} - 4 q^{17} + 16 q^{23} - 22 q^{25} - 8 q^{31} + 4 q^{41} + 8 q^{47} + 2 q^{49} + 8 q^{57} - 2 q^{63} + 28 q^{73} + 16 q^{79} - 22 q^{81} + 8 q^{87} - 20 q^{89} - 16 q^{95} - 4 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
897.1
1.00000i
1.00000i
0 2.00000i 0 4.00000i 0 1.00000 0 −1.00000 0
897.2 0 2.00000i 0 4.00000i 0 1.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.b.h 2
4.b odd 2 1 1792.2.b.a 2
8.b even 2 1 inner 1792.2.b.h 2
8.d odd 2 1 1792.2.b.a 2
16.e even 4 1 112.2.a.a 1
16.e even 4 1 448.2.a.h 1
16.f odd 4 1 56.2.a.b 1
16.f odd 4 1 448.2.a.c 1
48.i odd 4 1 1008.2.a.m 1
48.i odd 4 1 4032.2.a.a 1
48.k even 4 1 504.2.a.h 1
48.k even 4 1 4032.2.a.d 1
80.i odd 4 1 2800.2.g.g 2
80.j even 4 1 1400.2.g.b 2
80.k odd 4 1 1400.2.a.a 1
80.q even 4 1 2800.2.a.bd 1
80.s even 4 1 1400.2.g.b 2
80.t odd 4 1 2800.2.g.g 2
112.j even 4 1 392.2.a.b 1
112.j even 4 1 3136.2.a.w 1
112.l odd 4 1 784.2.a.i 1
112.l odd 4 1 3136.2.a.c 1
112.u odd 12 2 392.2.i.a 2
112.v even 12 2 392.2.i.e 2
112.w even 12 2 784.2.i.j 2
112.x odd 12 2 784.2.i.b 2
176.i even 4 1 6776.2.a.h 1
208.o odd 4 1 9464.2.a.h 1
336.v odd 4 1 3528.2.a.b 1
336.y even 4 1 7056.2.a.c 1
336.br odd 12 2 3528.2.s.ba 2
336.bu even 12 2 3528.2.s.a 2
560.be even 4 1 9800.2.a.bj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 16.f odd 4 1
112.2.a.a 1 16.e even 4 1
392.2.a.b 1 112.j even 4 1
392.2.i.a 2 112.u odd 12 2
392.2.i.e 2 112.v even 12 2
448.2.a.c 1 16.f odd 4 1
448.2.a.h 1 16.e even 4 1
504.2.a.h 1 48.k even 4 1
784.2.a.i 1 112.l odd 4 1
784.2.i.b 2 112.x odd 12 2
784.2.i.j 2 112.w even 12 2
1008.2.a.m 1 48.i odd 4 1
1400.2.a.a 1 80.k odd 4 1
1400.2.g.b 2 80.j even 4 1
1400.2.g.b 2 80.s even 4 1
1792.2.b.a 2 4.b odd 2 1
1792.2.b.a 2 8.d odd 2 1
1792.2.b.h 2 1.a even 1 1 trivial
1792.2.b.h 2 8.b even 2 1 inner
2800.2.a.bd 1 80.q even 4 1
2800.2.g.g 2 80.i odd 4 1
2800.2.g.g 2 80.t odd 4 1
3136.2.a.c 1 112.l odd 4 1
3136.2.a.w 1 112.j even 4 1
3528.2.a.b 1 336.v odd 4 1
3528.2.s.a 2 336.bu even 12 2
3528.2.s.ba 2 336.br odd 12 2
4032.2.a.a 1 48.i odd 4 1
4032.2.a.d 1 48.k even 4 1
6776.2.a.h 1 176.i even 4 1
7056.2.a.c 1 336.y even 4 1
9464.2.a.h 1 208.o odd 4 1
9800.2.a.bj 1 560.be 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
\( T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{23} - 8 \) Copy content Toggle raw display
\( T_{31} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 16 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T - 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4 \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 64 \) Copy content Toggle raw display
$47$ \( (T - 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 100 \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 16 \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 14)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
show more
show less