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Properties

Label 891.2.e.o
Level $891$
Weight $2$
Character orbit 891.e
Analytic conductor $7.115$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{3} - \beta_1 + 1) q^{5} + (\beta_{2} - 2 \beta_1 + 1) q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{3} - \beta_1 + 1) q^{5} + (\beta_{2} - 2 \beta_1 + 1) q^{7} - 3 q^{8} + ( - \beta_{3} + 4) q^{10} + (\beta_{2} + 1) q^{11} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{3} - 6 \beta_{2} - \beta_1 + 1) q^{14} + (2 \beta_{2} - \beta_1 + 2) q^{16} + ( - 2 \beta_{3} + 2) q^{17} + ( - \beta_{3} - 3) q^{19} + (3 \beta_{2} + 2 \beta_1 + 3) q^{20} + (\beta_{3} + \beta_1 - 1) q^{22} + ( - 2 \beta_{3} + 3 \beta_{2} + \cdots + 2) q^{23}+ \cdots + (6 \beta_{3} - 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 3 q^{4} + q^{5} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - 3 q^{4} + q^{5} - 12 q^{8} + 14 q^{10} + 2 q^{11} + 9 q^{13} + 13 q^{14} + 3 q^{16} + 4 q^{17} - 14 q^{19} + 8 q^{20} - q^{22} - 4 q^{23} + 3 q^{25} + 22 q^{26} + 26 q^{28} + 3 q^{29} - 15 q^{31} - 7 q^{32} + 14 q^{34} - 26 q^{35} + 16 q^{37} + 3 q^{38} - 3 q^{40} + 13 q^{41} + 16 q^{43} - 6 q^{44} + 22 q^{46} - q^{47} - 12 q^{49} + 5 q^{50} + 20 q^{52} - 18 q^{53} + 2 q^{55} + 18 q^{58} + 20 q^{59} - 8 q^{61} - 2 q^{62} - 8 q^{64} - 11 q^{65} - 13 q^{67} - 16 q^{68} - 13 q^{70} - 6 q^{71} - 12 q^{73} - 9 q^{74} + 4 q^{76} - q^{79} - 10 q^{80} + 26 q^{82} + 12 q^{83} + 14 q^{85} - 21 q^{86} - 6 q^{88} + 6 q^{89} - 26 q^{91} + 7 q^{92} + 7 q^{94} + 3 q^{95} + 8 q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 7 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 7 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
298.1
−0.651388 + 1.12824i
1.15139 − 1.99426i
−0.651388 − 1.12824i
1.15139 + 1.99426i
−0.651388 + 1.12824i 0 0.151388 + 0.262211i −0.651388 − 1.12824i 0 1.80278 − 3.12250i −3.00000 0 1.69722
298.2 1.15139 − 1.99426i 0 −1.65139 − 2.86029i 1.15139 + 1.99426i 0 −1.80278 + 3.12250i −3.00000 0 5.30278
595.1 −0.651388 − 1.12824i 0 0.151388 − 0.262211i −0.651388 + 1.12824i 0 1.80278 + 3.12250i −3.00000 0 1.69722
595.2 1.15139 + 1.99426i 0 −1.65139 + 2.86029i 1.15139 − 1.99426i 0 −1.80278 − 3.12250i −3.00000 0 5.30278
\(n\): 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.o 4
3.b odd 2 1 891.2.e.n 4
9.c even 3 1 891.2.a.i ✓ 2
9.c even 3 1 inner 891.2.e.o 4
9.d odd 6 1 891.2.a.j yes 2
9.d odd 6 1 891.2.e.n 4
99.g even 6 1 9801.2.a.o 2
99.h odd 6 1 9801.2.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
891.2.a.i ✓ 2 9.c even 3 1
891.2.a.j yes 2 9.d odd 6 1
891.2.e.n 4 3.b odd 2 1
891.2.e.n 4 9.d odd 6 1
891.2.e.o 4 1.a even 1 1 trivial
891.2.e.o 4 9.c even 3 1 inner
9801.2.a.o 2 99.g even 6 1
9801.2.a.ba 2 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 \(S_{2}^{\mathrm{new}}(891, [\chi])\):

\( T_{2}^{4} - T_{2}^{3} + 4T_{2}^{2} + 3T_{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{4} - T_{5}^{3} + 4T_{5}^{2} + 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{4} + 13T_{7}^{2} + 169 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + 4 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{3} + 4 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 9 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T - 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 7 T + 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{4} - 3 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$31$ \( T^{4} + 15 T^{3} + \cdots + 2809 \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 13 T^{3} + \cdots + 1521 \) Copy content Toggle raw display
$43$ \( T^{4} - 16 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$47$ \( T^{4} + T^{3} + 4 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$53$ \( (T^{2} + 9 T - 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 20 T^{3} + \cdots + 7569 \) Copy content Toggle raw display
$61$ \( T^{4} + 8 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$67$ \( T^{4} + 13 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$71$ \( (T^{2} + 3 T - 27)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T - 43)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + T^{3} + \cdots + 69169 \) Copy content Toggle raw display
$83$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 3 T - 27)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
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