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Properties

Label 891.2.e.u
Level $891$
Weight $2$
Character orbit 891.e
Analytic conductor $7.115$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [891,2,Mod(298,891)]
 
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([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.298");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.e (of order \(3\), 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 \(\Q(\zeta_{3})\)
Coefficient field: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_1 - 1) q^{2} + (\beta_{7} - \beta_{5} - \beta_{2} + \cdots + 1) q^{4}+ \cdots + (2 \beta_{6} + \beta_{3} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_1 - 1) q^{2} + (\beta_{7} - \beta_{5} - \beta_{2} + \cdots + 1) q^{4}+ \cdots + (2 \beta_{6} - 2 \beta_{2} + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 2 q^{4} - 8 q^{5} + 2 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 2 q^{4} - 8 q^{5} + 2 q^{7} + 12 q^{8} + 20 q^{10} + 4 q^{11} - 4 q^{13} + 4 q^{14} - 2 q^{16} + 16 q^{17} - 16 q^{20} + 2 q^{22} - 4 q^{23} - 14 q^{25} + 16 q^{26} - 20 q^{28} - 12 q^{29} + 14 q^{31} - 4 q^{32} + 20 q^{34} + 4 q^{35} - 12 q^{38} - 12 q^{40} - 14 q^{41} + 6 q^{43} - 4 q^{44} - 56 q^{46} - 4 q^{47} + 12 q^{49} - 28 q^{50} - 26 q^{52} + 24 q^{53} - 16 q^{55} + 24 q^{56} - 24 q^{58} - 16 q^{59} - 12 q^{61} - 32 q^{62} + 8 q^{64} - 32 q^{65} - 2 q^{67} + 32 q^{68} + 14 q^{70} + 48 q^{71} + 44 q^{73} + 6 q^{74} + 6 q^{76} - 2 q^{77} - 2 q^{79} - 16 q^{80} - 4 q^{82} + 12 q^{83} - 16 q^{85} + 12 q^{86} + 6 q^{88} - 4 q^{91} + 28 q^{92} + 10 q^{94} + 12 q^{95} + 16 q^{97} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 4\nu^{4} - 3\nu^{3} - 10\nu^{2} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - \nu^{5} + 4\nu^{4} - \nu^{3} - 6\nu^{2} + 10\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} - 8\nu^{6} + 3\nu^{5} + 10\nu^{4} - 13\nu^{3} - 8\nu^{2} + 32\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} - 14\nu^{6} + 9\nu^{5} + 16\nu^{4} - 27\nu^{3} - 14\nu^{2} + 76\nu - 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 7\nu^{6} + 3\nu^{5} + 11\nu^{4} - 15\nu^{3} - 11\nu^{2} + 40\nu - 28 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 9\nu^{6} + 5\nu^{5} + 13\nu^{4} - 21\nu^{3} - 13\nu^{2} + 54\nu - 40 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} - 24\nu^{6} + 13\nu^{5} + 38\nu^{4} - 51\nu^{3} - 32\nu^{2} + 132\nu - 104 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - 2\beta_{3} + \beta_{2} - \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} - 3\beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} - 3\beta_{5} + 3\beta_{4} - 3\beta_{3} + 3\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{7} - 2\beta_{6} - 3\beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_{2} - \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{7} - 4\beta_{6} - 3\beta_{5} + 4\beta_{4} - 5\beta_{3} - 2\beta_{2} + 5\beta _1 + 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{7} - 5\beta_{6} + 9\beta_{5} - 12\beta_{3} - 3\beta_{2} + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -4\beta_{7} - 14\beta_{6} + 24\beta_{5} + 5\beta_{4} - 4\beta_{3} - 7\beta_{2} + \beta _1 + 4 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(244\) \(650\)
\(\chi(n)\) \(1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
298.1
1.20036 − 0.747754i
1.40994 + 0.109843i
−1.27597 − 0.609843i
0.665665 + 1.24775i
1.20036 + 0.747754i
1.40994 − 0.109843i
−1.27597 + 0.609843i
0.665665 − 1.24775i
−1.24775 + 2.16117i 0 −2.11378 − 3.66117i −1.70036 − 2.94511i 0 1.41342 − 2.44811i 5.55889 0 8.48652
298.2 −0.609843 + 1.05628i 0 0.256182 + 0.443720i −1.90994 − 3.30812i 0 −1.16612 + 2.01978i −3.06430 0 4.65906
298.3 0.109843 − 0.190254i 0 0.975869 + 1.69025i 0.775967 + 1.34401i 0 0.800098 − 1.38581i 0.868145 0 0.340939
298.4 0.747754 − 1.29515i 0 −0.118272 − 0.204852i −1.16567 − 2.01899i 0 −0.0473938 + 0.0820885i 2.63726 0 −3.48652
595.1 −1.24775 − 2.16117i 0 −2.11378 + 3.66117i −1.70036 + 2.94511i 0 1.41342 + 2.44811i 5.55889 0 8.48652
595.2 −0.609843 − 1.05628i 0 0.256182 − 0.443720i −1.90994 + 3.30812i 0 −1.16612 − 2.01978i −3.06430 0 4.65906
595.3 0.109843 + 0.190254i 0 0.975869 − 1.69025i 0.775967 − 1.34401i 0 0.800098 + 1.38581i 0.868145 0 0.340939
595.4 0.747754 + 1.29515i 0 −0.118272 + 0.204852i −1.16567 + 2.01899i 0 −0.0473938 − 0.0820885i 2.63726 0 −3.48652
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 298.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.e.u 8
3.b odd 2 1 891.2.e.v 8
9.c even 3 1 891.2.a.r yes 4
9.c even 3 1 inner 891.2.e.u 8
9.d odd 6 1 891.2.a.o ✓ 4
9.d odd 6 1 891.2.e.v 8
99.g even 6 1 9801.2.a.bm 4
99.h odd 6 1 9801.2.a.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
891.2.a.o ✓ 4 9.d odd 6 1
891.2.a.r yes 4 9.c even 3 1
891.2.e.u 8 1.a even 1 1 trivial
891.2.e.u 8 9.c even 3 1 inner
891.2.e.v 8 3.b odd 2 1
891.2.e.v 8 9.d odd 6 1
9801.2.a.bh 4 99.h odd 6 1
9801.2.a.bm 4 99.g even 6 1

Hecke kernels

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

\( T_{2}^{8} + 2T_{2}^{7} + 7T_{2}^{6} + 2T_{2}^{5} + 16T_{2}^{4} + 8T_{2}^{3} + 19T_{2}^{2} - 4T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{8} + 8T_{5}^{7} + 49T_{5}^{6} + 152T_{5}^{5} + 400T_{5}^{4} + 512T_{5}^{3} + 961T_{5}^{2} + 752T_{5} + 2209 \) Copy content Toggle raw display
\( T_{7}^{8} - 2T_{7}^{7} + 10T_{7}^{6} - 8T_{7}^{5} + 55T_{7}^{4} - 56T_{7}^{3} + 106T_{7}^{2} + 10T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 8 T^{7} + \cdots + 2209 \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots + 3481 \) Copy content Toggle raw display
$17$ \( (T^{4} - 8 T^{3} + \cdots - 368)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 27 T^{2} + \cdots - 27)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 4 T^{7} + \cdots + 1247689 \) Copy content Toggle raw display
$29$ \( T^{8} + 12 T^{7} + \cdots + 328329 \) Copy content Toggle raw display
$31$ \( T^{8} - 14 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$37$ \( (T^{4} - 78 T^{2} + \cdots + 213)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 14 T^{7} + \cdots + 9409 \) Copy content Toggle raw display
$43$ \( T^{8} - 6 T^{7} + \cdots + 1521 \) Copy content Toggle raw display
$47$ \( T^{8} + 4 T^{7} + \cdots + 436921 \) Copy content Toggle raw display
$53$ \( (T^{4} - 12 T^{3} + \cdots + 33)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 16 T^{7} + \cdots + 96721 \) Copy content Toggle raw display
$61$ \( T^{8} + 12 T^{7} + \cdots + 389376 \) Copy content Toggle raw display
$67$ \( T^{8} + 2 T^{7} + \cdots + 20584369 \) Copy content Toggle raw display
$71$ \( (T^{4} - 24 T^{3} + \cdots - 1371)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 22 T^{3} + \cdots - 923)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 2 T^{7} + \cdots + 361201 \) Copy content Toggle raw display
$83$ \( T^{8} - 12 T^{7} + \cdots + 10969344 \) Copy content Toggle raw display
$89$ \( (T^{4} - 225 T^{2} + \cdots - 1875)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 16 T^{7} + \cdots + 1290496 \) Copy content Toggle raw display
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