Properties

Label 891.2.g.a
Level $891$
Weight $2$
Character orbit 891.g
Analytic conductor $7.115$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [891,2,Mod(296,891)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(891, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.296");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.g (of order \(6\), degree \(2\), 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: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} + 2) q^{4} + \beta_1 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} + 2) q^{4} + \beta_1 q^{5} + \beta_{3} q^{11} - 4 \beta_{2} q^{16} + 2 \beta_{3} q^{20} + \beta_1 q^{23} + 6 \beta_{2} q^{25} + (5 \beta_{2} - 5) q^{31} - 7 q^{37} + (2 \beta_{3} - 2 \beta_1) q^{44} + 2 \beta_{3} q^{47} + (7 \beta_{2} - 7) q^{49} + (4 \beta_{3} - 4 \beta_1) q^{53} + 11 q^{55} + \beta_1 q^{59} - 8 q^{64} + ( - 13 \beta_{2} + 13) q^{67} + ( - 5 \beta_{3} + 5 \beta_1) q^{71} + (4 \beta_{3} - 4 \beta_1) q^{80} + ( - 5 \beta_{3} + 5 \beta_1) q^{89} + 2 \beta_{3} q^{92} - 17 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 8 q^{16} + 12 q^{25} - 10 q^{31} - 28 q^{37} - 14 q^{49} + 44 q^{55} - 32 q^{64} + 26 q^{67} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} + 10\nu - 9 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{3} + 2\nu^{2} + 10\nu + 15 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(650\)
\(\chi(n)\) \(-1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
296.1
−1.18614 1.26217i
1.68614 + 0.396143i
−1.18614 + 1.26217i
1.68614 0.396143i
0 0 1.00000 1.73205i −2.87228 1.65831i 0 0 0 0 0
296.2 0 0 1.00000 1.73205i 2.87228 + 1.65831i 0 0 0 0 0
593.1 0 0 1.00000 + 1.73205i −2.87228 + 1.65831i 0 0 0 0 0
593.2 0 0 1.00000 + 1.73205i 2.87228 1.65831i 0 0 0 0 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
33.d even 2 1 inner
99.g even 6 1 inner
99.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.g.a 4
3.b odd 2 1 inner 891.2.g.a 4
9.c even 3 1 33.2.d.a 2
9.c even 3 1 inner 891.2.g.a 4
9.d odd 6 1 33.2.d.a 2
9.d odd 6 1 inner 891.2.g.a 4
11.b odd 2 1 CM 891.2.g.a 4
33.d even 2 1 inner 891.2.g.a 4
36.f odd 6 1 528.2.b.a 2
36.h even 6 1 528.2.b.a 2
45.h odd 6 1 825.2.f.a 2
45.j even 6 1 825.2.f.a 2
45.k odd 12 2 825.2.d.a 4
45.l even 12 2 825.2.d.a 4
72.j odd 6 1 2112.2.b.e 2
72.l even 6 1 2112.2.b.f 2
72.n even 6 1 2112.2.b.e 2
72.p odd 6 1 2112.2.b.f 2
99.g even 6 1 33.2.d.a 2
99.g even 6 1 inner 891.2.g.a 4
99.h odd 6 1 33.2.d.a 2
99.h odd 6 1 inner 891.2.g.a 4
99.m even 15 4 363.2.f.c 8
99.n odd 30 4 363.2.f.c 8
99.o odd 30 4 363.2.f.c 8
99.p even 30 4 363.2.f.c 8
396.k even 6 1 528.2.b.a 2
396.o odd 6 1 528.2.b.a 2
495.o odd 6 1 825.2.f.a 2
495.r even 6 1 825.2.f.a 2
495.bd odd 12 2 825.2.d.a 4
495.bf even 12 2 825.2.d.a 4
792.s odd 6 1 2112.2.b.f 2
792.w even 6 1 2112.2.b.e 2
792.z even 6 1 2112.2.b.f 2
792.be odd 6 1 2112.2.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.d.a 2 9.c even 3 1
33.2.d.a 2 9.d odd 6 1
33.2.d.a 2 99.g even 6 1
33.2.d.a 2 99.h odd 6 1
363.2.f.c 8 99.m even 15 4
363.2.f.c 8 99.n odd 30 4
363.2.f.c 8 99.o odd 30 4
363.2.f.c 8 99.p even 30 4
528.2.b.a 2 36.f odd 6 1
528.2.b.a 2 36.h even 6 1
528.2.b.a 2 396.k even 6 1
528.2.b.a 2 396.o odd 6 1
825.2.d.a 4 45.k odd 12 2
825.2.d.a 4 45.l even 12 2
825.2.d.a 4 495.bd odd 12 2
825.2.d.a 4 495.bf even 12 2
825.2.f.a 2 45.h odd 6 1
825.2.f.a 2 45.j even 6 1
825.2.f.a 2 495.o odd 6 1
825.2.f.a 2 495.r even 6 1
891.2.g.a 4 1.a even 1 1 trivial
891.2.g.a 4 3.b odd 2 1 inner
891.2.g.a 4 9.c even 3 1 inner
891.2.g.a 4 9.d odd 6 1 inner
891.2.g.a 4 11.b odd 2 1 CM
891.2.g.a 4 33.d even 2 1 inner
891.2.g.a 4 99.g even 6 1 inner
891.2.g.a 4 99.h odd 6 1 inner
2112.2.b.e 2 72.j odd 6 1
2112.2.b.e 2 72.n even 6 1
2112.2.b.e 2 792.w even 6 1
2112.2.b.e 2 792.be odd 6 1
2112.2.b.f 2 72.l even 6 1
2112.2.b.f 2 72.p odd 6 1
2112.2.b.f 2 792.s odd 6 1
2112.2.b.f 2 792.z even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{23}^{4} - 11T_{23}^{2} + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 11T^{2} + 121 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 11T^{2} + 121 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 11T^{2} + 121 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$37$ \( (T + 7)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 44T^{2} + 1936 \) Copy content Toggle raw display
$53$ \( (T^{2} + 176)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 11T^{2} + 121 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 13 T + 169)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 275)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 275)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 17 T + 289)^{2} \) Copy content Toggle raw display
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