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Properties

Label 891.2.g.b
Level $891$
Weight $2$
Character orbit 891.g
Analytic conductor $7.115$
Analytic rank $0$
Dimension $8$
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] = [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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.764411904.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 297)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (2 \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{4} - \beta_{2} q^{5} + (\beta_{4} - 2 \beta_1) q^{7} + ( - \beta_{6} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (2 \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{4} - \beta_{2} q^{5} + (\beta_{4} - 2 \beta_1) q^{7} + ( - \beta_{6} - 2 \beta_1) q^{8} + (\beta_{7} - \beta_{4} + \beta_1) q^{10} + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots - 2) q^{11}+ \cdots + (\beta_{7} + \beta_{4} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{4} - 12 q^{11} - 12 q^{14} - 4 q^{16} + 24 q^{20} - 12 q^{23} - 12 q^{25} - 8 q^{31} + 12 q^{34} - 8 q^{37} + 36 q^{47} + 8 q^{49} + 16 q^{55} + 60 q^{56} + 36 q^{58} + 36 q^{59} - 40 q^{64} - 8 q^{67} + 24 q^{70} + 48 q^{77} - 24 q^{82} - 84 q^{86} + 24 q^{88} - 108 q^{92} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 9\nu^{4} - 3\nu^{2} + 18 ) / 45 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} + 3\nu^{4} - 6\nu^{2} - 9 ) / 45 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{7} - 6\nu^{5} + 57\nu^{3} - 27\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 4\nu^{4} + 18\nu^{2} - 33 ) / 15 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 6\nu^{5} - 21\nu^{3} + 27\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 3\nu^{5} - 6\nu^{3} - 9\nu ) / 45 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 3\beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{3} + 6\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{7} + 6\beta_{6} + 12\beta_{4} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{5} - 30\beta_{3} + 6\beta_{2} - 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -24\beta_{7} + 9\beta_{6} + 9\beta_{4} - 12\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(244\) \(650\)
\(\chi(n)\) \(-1\) \(1 + \beta_{3}\)

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
−1.69185 − 0.370982i
1.27970 − 1.16721i
−1.27970 + 1.16721i
1.69185 + 0.370982i
−1.69185 + 0.370982i
1.27970 + 1.16721i
−1.27970 − 1.16721i
1.69185 − 0.370982i
−1.16721 − 2.02166i 0 −1.72474 + 2.98735i −1.22474 − 0.707107i 0 2.21650 − 1.27970i 3.38371 0 3.30136i
296.2 −0.370982 − 0.642559i 0 0.724745 − 1.25529i 1.22474 + 0.707107i 0 −2.93038 + 1.69185i −2.55940 0 − 1.04930i
296.3 0.370982 + 0.642559i 0 0.724745 − 1.25529i 1.22474 + 0.707107i 0 2.93038 − 1.69185i 2.55940 0 1.04930i
296.4 1.16721 + 2.02166i 0 −1.72474 + 2.98735i −1.22474 − 0.707107i 0 −2.21650 + 1.27970i −3.38371 0 − 3.30136i
593.1 −1.16721 + 2.02166i 0 −1.72474 − 2.98735i −1.22474 + 0.707107i 0 2.21650 + 1.27970i 3.38371 0 − 3.30136i
593.2 −0.370982 + 0.642559i 0 0.724745 + 1.25529i 1.22474 − 0.707107i 0 −2.93038 − 1.69185i −2.55940 0 1.04930i
593.3 0.370982 − 0.642559i 0 0.724745 + 1.25529i 1.22474 − 0.707107i 0 2.93038 + 1.69185i 2.55940 0 − 1.04930i
593.4 1.16721 − 2.02166i 0 −1.72474 − 2.98735i −1.22474 + 0.707107i 0 −2.21650 − 1.27970i −3.38371 0 3.30136i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 296.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
11.b odd 2 1 inner
99.g even 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.b 8
3.b odd 2 1 891.2.g.d 8
9.c even 3 1 297.2.d.b ✓ 8
9.c even 3 1 891.2.g.d 8
9.d odd 6 1 297.2.d.b ✓ 8
9.d odd 6 1 inner 891.2.g.b 8
11.b odd 2 1 inner 891.2.g.b 8
33.d even 2 1 891.2.g.d 8
36.f odd 6 1 4752.2.b.h 8
36.h even 6 1 4752.2.b.h 8
99.g even 6 1 297.2.d.b ✓ 8
99.g even 6 1 inner 891.2.g.b 8
99.h odd 6 1 297.2.d.b ✓ 8
99.h odd 6 1 891.2.g.d 8
396.k even 6 1 4752.2.b.h 8
396.o odd 6 1 4752.2.b.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.d.b ✓ 8 9.c even 3 1
297.2.d.b ✓ 8 9.d odd 6 1
297.2.d.b ✓ 8 99.g even 6 1
297.2.d.b ✓ 8 99.h odd 6 1
891.2.g.b 8 1.a even 1 1 trivial
891.2.g.b 8 9.d odd 6 1 inner
891.2.g.b 8 11.b odd 2 1 inner
891.2.g.b 8 99.g even 6 1 inner
891.2.g.d 8 3.b odd 2 1
891.2.g.d 8 9.c even 3 1
891.2.g.d 8 33.d even 2 1
891.2.g.d 8 99.h odd 6 1
4752.2.b.h 8 36.f odd 6 1
4752.2.b.h 8 36.h even 6 1
4752.2.b.h 8 396.k even 6 1
4752.2.b.h 8 396.o odd 6 1

Hecke kernels

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

\( T_{2}^{8} + 6T_{2}^{6} + 33T_{2}^{4} + 18T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{23}^{4} + 6T_{23}^{3} - 35T_{23}^{2} - 282T_{23} + 2209 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 6 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 18 T^{6} + \cdots + 5625 \) Copy content Toggle raw display
$11$ \( T^{8} + 12 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 36 T^{6} + \cdots + 90000 \) Copy content Toggle raw display
$17$ \( (T^{4} - 18 T^{2} + 27)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 36 T^{2} + 108)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} + \cdots + 2209)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 102 T^{6} + \cdots + 2518569 \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T - 5)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + 6 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$43$ \( T^{8} - 102 T^{6} + \cdots + 3515625 \) Copy content Toggle raw display
$47$ \( (T^{4} - 18 T^{3} + \cdots + 25)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 40 T^{2} + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 18 T^{3} + \cdots + 625)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 204 T^{6} + \cdots + 40297104 \) Copy content Toggle raw display
$67$ \( (T^{4} + 4 T^{3} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 160 T^{2} + 256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 216 T^{2} + 10800)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 342 T^{6} + \cdots + 796763529 \) Copy content Toggle raw display
$83$ \( T^{8} + 204 T^{6} + \cdots + 40297104 \) Copy content Toggle raw display
$89$ \( (T^{4} + 232 T^{2} + 10000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + \cdots + 2809)^{2} \) Copy content Toggle raw display
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