Properties

Label 891.2.e.i
Level $891$
Weight $2$
Character orbit 891.e
Analytic conductor $7.115$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(244\) \(650\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0 0.500000 + 0.866025i 0.500000 + 0.866025i 0 1.00000 1.73205i 3.00000 0 1.00000
595.1 0.500000 + 0.866025i 0 0.500000 0.866025i 0.500000 0.866025i 0 1.00000 + 1.73205i 3.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
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.i 2
3.b odd 2 1 891.2.e.d 2
9.c even 3 1 891.2.a.b 1
9.c even 3 1 inner 891.2.e.i 2
9.d odd 6 1 891.2.a.f yes 1
9.d odd 6 1 891.2.e.d 2
99.g even 6 1 9801.2.a.d 1
99.h odd 6 1 9801.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
891.2.a.b 1 9.c even 3 1
891.2.a.f yes 1 9.d odd 6 1
891.2.e.d 2 3.b odd 2 1
891.2.e.d 2 9.d odd 6 1
891.2.e.i 2 1.a even 1 1 trivial
891.2.e.i 2 9.c even 3 1 inner
9801.2.a.d 1 99.g even 6 1
9801.2.a.j 1 99.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 \(S_{2}^{\mathrm{new}}(891, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T - 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$37$ \( (T + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$61$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 11)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$89$ \( (T + 15)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
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