Properties

Label 891.2.e.q
Level $891$
Weight $2$
Character orbit 891.e
Analytic conductor $7.115$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 297)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4}) q^{2} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{5} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3}) q^{7} + ( - \beta_{4} + \beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4}) q^{2} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{5} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3}) q^{7} + ( - \beta_{4} + \beta_{2} + 1) q^{8} + (2 \beta_{4} - \beta_{2} - 1) q^{10} + \beta_{3} q^{11} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{13} + (2 \beta_{5} + 4 \beta_{3} - \beta_{2} - \beta_1 - 4) q^{14} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{16} + (\beta_{4} + \beta_{2} + 1) q^{17} + (2 \beta_{4} - \beta_{2} + 1) q^{19} + (2 \beta_{5} - 2 \beta_{4} + 7 \beta_{3} - \beta_1) q^{20} - \beta_{5} q^{22} + ( - 2 \beta_{5} + 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{23} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{25} + ( - \beta_{2} - 1) q^{26} + ( - 3 \beta_{4} + 3 \beta_{2} + 5) q^{28} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{29} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{31} + ( - \beta_{5} + 6 \beta_{3} - 6) q^{32} + ( - 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3}) q^{34} + (2 \beta_{4} - 3 \beta_{2} - 3) q^{35} + (4 \beta_{4} + 3 \beta_{2} + 2) q^{37} + (9 \beta_{3} - 3 \beta_1) q^{38} + ( - 4 \beta_{5} - 7 \beta_{3} + \beta_{2} + \beta_1 + 7) q^{40} + ( - 3 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{41} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \beta_1) q^{43} + ( - \beta_{2} - 2) q^{44} + ( - \beta_{4} - 3 \beta_{2} - 9) q^{46} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_1) q^{47} + (4 \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{49} + (\beta_{5} + 8 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 8) q^{50} + (2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + \beta_1) q^{52} + ( - 2 \beta_{4} + 2 \beta_{2} - 4) q^{53} + (\beta_{2} + 1) q^{55} + ( - 4 \beta_{5} + 4 \beta_{4} - 7 \beta_{3} + 4 \beta_1) q^{56} + (4 \beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{58} + (4 \beta_{5} + 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{59} + ( - 2 \beta_{5} + 2 \beta_{4} + 7 \beta_{3} - 3 \beta_1) q^{61} + ( - \beta_{2} - 1) q^{62} + ( - 2 \beta_{4} - \beta_{2} - 6) q^{64} + ( - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 2 \beta_1) q^{65} + (4 \beta_{5} + 3 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{67} + ( - \beta_{5} + 6 \beta_{3} - 6) q^{68} + (6 \beta_{5} - 6 \beta_{4} + 11 \beta_{3} - 5 \beta_1) q^{70} + ( - 2 \beta_{4} + 2 \beta_{2} - 4) q^{71} + ( - 2 \beta_{4} - 2 \beta_{2}) q^{73} + ( - 5 \beta_{5} + 5 \beta_{4} + 13 \beta_{3} - \beta_1) q^{74} + ( - 8 \beta_{5} - 5 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{76} + ( - \beta_{5} - 2 \beta_{3} + 2) q^{77} + (\beta_{5} - \beta_{4} - 7 \beta_{3} + 5 \beta_1) q^{79} + (4 \beta_{4} - 3 \beta_{2} - 3) q^{80} + ( - 4 \beta_{4} - \beta_{2} - 10) q^{82} + ( - 3 \beta_{3} - 3 \beta_1) q^{83} + ( - 5 \beta_{3} - \beta_{2} - \beta_1 + 5) q^{85} + (4 \beta_{5} - 11 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 11) q^{86} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_1) q^{88} + ( - 4 \beta_{4} - 2 \beta_{2} + 4) q^{89} + ( - 3 \beta_{2} + 1) q^{91} + (8 \beta_{5} - 8 \beta_{4} + 3 \beta_{3}) q^{92} + (\beta_{5} - 6 \beta_{3} + 6) q^{94} + (6 \beta_{5} + 6 \beta_{3} - 6) q^{95} + (6 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} + 3 \beta_1) q^{97} + ( - 2 \beta_{4} + 5 \beta_{2} + 17) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 5 q^{4} + 2 q^{5} - 7 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} - 5 q^{4} + 2 q^{5} - 7 q^{7} + 6 q^{8} - 8 q^{10} + 3 q^{11} - 4 q^{13} - 13 q^{14} - 5 q^{16} + 2 q^{17} + 4 q^{19} + 22 q^{20} + q^{22} - 5 q^{23} - 5 q^{25} - 4 q^{26} + 30 q^{28} - 3 q^{29} - 4 q^{31} - 17 q^{32} + 7 q^{34} - 16 q^{35} - 2 q^{37} + 24 q^{38} + 24 q^{40} - q^{41} - 5 q^{43} - 10 q^{44} - 46 q^{46} + 7 q^{47} - 6 q^{49} - 23 q^{50} + 12 q^{52} - 24 q^{53} + 4 q^{55} - 21 q^{56} + 3 q^{58} - 11 q^{59} + 16 q^{61} - 4 q^{62} - 30 q^{64} - 16 q^{65} - 12 q^{67} - 17 q^{68} + 34 q^{70} - 24 q^{71} + 8 q^{73} + 33 q^{74} + 22 q^{76} + 7 q^{77} - 15 q^{79} - 20 q^{80} - 50 q^{82} - 12 q^{83} + 16 q^{85} + 27 q^{86} + 3 q^{88} + 36 q^{89} + 12 q^{91} + 17 q^{92} + 17 q^{94} - 24 q^{95} + 3 q^{97} + 96 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - \nu^{3} + 9\nu^{2} - 21\nu - 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - \nu^{3} - 18\nu^{2} + 33\nu - 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} - \nu^{4} + 7\nu^{3} + 9\nu^{2} + 12\nu + 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} + 3\nu - 72 ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 7\beta_{4} - 11\beta_{3} + \beta_{2} - \beta _1 + 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} - 7\beta_{3} + 8\beta_{2} + 10\beta _1 + 20 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{5} - 2\beta_{4} - 20\beta_{3} + \beta_{2} - 10\beta _1 + 43 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
298.1
0.403374 1.68443i
−1.62241 + 0.606458i
1.71903 + 0.211943i
0.403374 + 1.68443i
−1.62241 0.606458i
1.71903 0.211943i
−1.25707 + 2.17731i 0 −2.16044 3.74200i 1.66044 + 2.87597i 0 −2.25707 + 3.90936i 5.83502 0 −8.34916
298.2 −0.285997 + 0.495361i 0 0.836412 + 1.44871i −1.33641 2.31473i 0 −1.28600 + 2.22741i −2.10083 0 1.52884
298.3 1.04307 1.80664i 0 −1.17597 2.03684i 0.675970 + 1.17081i 0 0.0430651 0.0745909i −0.734191 0 2.82032
595.1 −1.25707 2.17731i 0 −2.16044 + 3.74200i 1.66044 2.87597i 0 −2.25707 3.90936i 5.83502 0 −8.34916
595.2 −0.285997 0.495361i 0 0.836412 1.44871i −1.33641 + 2.31473i 0 −1.28600 2.22741i −2.10083 0 1.52884
595.3 1.04307 + 1.80664i 0 −1.17597 + 2.03684i 0.675970 1.17081i 0 0.0430651 + 0.0745909i −0.734191 0 2.82032
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 298.3
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.q 6
3.b odd 2 1 891.2.e.t 6
9.c even 3 1 297.2.a.h yes 3
9.c even 3 1 inner 891.2.e.q 6
9.d odd 6 1 297.2.a.g 3
9.d odd 6 1 891.2.e.t 6
36.f odd 6 1 4752.2.a.bg 3
36.h even 6 1 4752.2.a.bo 3
45.h odd 6 1 7425.2.a.bn 3
45.j even 6 1 7425.2.a.bm 3
99.g even 6 1 3267.2.a.w 3
99.h odd 6 1 3267.2.a.t 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.a.g 3 9.d odd 6 1
297.2.a.h yes 3 9.c even 3 1
891.2.e.q 6 1.a even 1 1 trivial
891.2.e.q 6 9.c even 3 1 inner
891.2.e.t 6 3.b odd 2 1
891.2.e.t 6 9.d odd 6 1
3267.2.a.t 3 99.h odd 6 1
3267.2.a.w 3 99.g even 6 1
4752.2.a.bg 3 36.f odd 6 1
4752.2.a.bo 3 36.h even 6 1
7425.2.a.bm 3 45.j even 6 1
7425.2.a.bn 3 45.h odd 6 1

Hecke kernels

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

\( T_{2}^{6} + T_{2}^{5} + 6T_{2}^{4} + T_{2}^{3} + 28T_{2}^{2} + 15T_{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{6} - 2T_{5}^{5} + 12T_{5}^{4} - 8T_{5}^{3} + 88T_{5}^{2} - 96T_{5} + 144 \) Copy content Toggle raw display
\( T_{7}^{6} + 7T_{7}^{5} + 38T_{7}^{4} + 79T_{7}^{3} + 128T_{7}^{2} - 11T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{5} + 6 T^{4} + T^{3} + 28 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + 12 T^{4} - 8 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{6} + 7 T^{5} + 38 T^{4} + 79 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{6} + 4 T^{5} + 20 T^{4} - 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( (T^{3} - T^{2} - 11 T + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 2 T^{2} - 36 T + 108)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 5 T^{5} + 54 T^{4} + \cdots + 19881 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} + 42 T^{4} + \cdots + 13689 \) Copy content Toggle raw display
$31$ \( T^{6} + 4 T^{5} + 20 T^{4} - 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( (T^{3} + T^{2} - 129 T - 141)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + T^{5} + 66 T^{4} - 179 T^{3} + \cdots + 3249 \) Copy content Toggle raw display
$43$ \( T^{6} + 5 T^{5} + 64 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$47$ \( T^{6} - 7 T^{5} + 78 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$53$ \( (T^{3} + 12 T^{2} - 24 T - 432)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 11 T^{5} + 162 T^{4} + \cdots + 269361 \) Copy content Toggle raw display
$61$ \( T^{6} - 16 T^{5} + 256 T^{4} + \cdots + 318096 \) Copy content Toggle raw display
$67$ \( T^{6} + 12 T^{5} + 204 T^{4} + \cdots + 15376 \) Copy content Toggle raw display
$71$ \( (T^{3} + 12 T^{2} - 24 T - 432)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 4 T^{2} - 40 T + 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 15 T^{5} + 372 T^{4} + \cdots + 4338889 \) Copy content Toggle raw display
$83$ \( T^{6} + 12 T^{5} + 180 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$89$ \( (T^{3} - 18 T^{2} + 12 T + 648)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 3 T^{5} + 222 T^{4} + \cdots + 1408969 \) Copy content Toggle raw display
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