Properties

Label 891.2.a.q
Level $891$
Weight $2$
Character orbit 891.a
Self dual yes
Analytic conductor $7.115$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [891,2,Mod(1,891)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(891, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.a (trivial)

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: yes
Analytic conductor: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.22545.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{3} + 3) q^{4} + ( - \beta_{3} - \beta_1 + 1) q^{5} + \beta_1 q^{7} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{3} + 3) q^{4} + ( - \beta_{3} - \beta_1 + 1) q^{5} + \beta_1 q^{7} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{8} + ( - 3 \beta_1 + 1) q^{10} + q^{11} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{13} + (\beta_{3} + 2 \beta_1) q^{14} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 + 5) q^{16} + (\beta_{2} - 2 \beta_1 - 1) q^{17} + (\beta_{3} + 2 \beta_1 + 2) q^{19} + ( - \beta_{3} + \beta_{2} - 4 \beta_1 - 2) q^{20} + \beta_{2} q^{22} + (\beta_{3} - \beta_1 + 4) q^{23} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{25} + (3 \beta_{3} + \beta_1 + 6) q^{26} + (\beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{28} + (\beta_{3} - \beta_{2} - 1) q^{29} + (\beta_{3} - \beta_{2}) q^{31} + ( - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 10) q^{32} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 + 5) q^{34} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{35} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{37} + (\beta_{3} + 3 \beta_{2} + 5 \beta_1 - 1) q^{38} + ( - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 4) q^{40} + ( - \beta_{2} + 3 \beta_1 - 1) q^{41} + ( - \beta_{2} - 5) q^{43} + (\beta_{3} + 3) q^{44} + ( - 2 \beta_{3} + 5 \beta_{2} - \beta_1 - 1) q^{46} + ( - \beta_{2} + 2 \beta_1 - 2) q^{47} + (\beta_{2} - 4) q^{49} + (2 \beta_{3} + \beta_{2} - 4 \beta_1 + 7) q^{50} + (7 \beta_{2} + 3 \beta_1 - 5) q^{52} + ( - \beta_{3} - 3 \beta_1 - 1) q^{53} + ( - \beta_{3} - \beta_1 + 1) q^{55} + (\beta_{3} + 3 \beta_1 + 4) q^{56} + ( - 2 \beta_{3} + \beta_1 - 6) q^{58} + (\beta_{3} - \beta_1 + 1) q^{59} + ( - \beta_1 + 4) q^{61} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 6) q^{62} + (2 \beta_{3} - 7 \beta_{2} + \beta_1 + 6) q^{64} + ( - 4 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 5) q^{65} + ( - 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{67} + ( - 4 \beta_{3} + 2 \beta_{2} - 5 \beta_1 - 2) q^{68} + ( - 3 \beta_{2} + \beta_1 - 9) q^{70} + ( - 2 \beta_{2} + 3 \beta_1 - 1) q^{71} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 6) q^{73} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 - 8) q^{74} + (5 \beta_{3} + 7 \beta_1 + 10) q^{76} + \beta_1 q^{77} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{79} + ( - 2 \beta_{3} - 9) q^{80} + (2 \beta_{3} - \beta_{2} + 6 \beta_1 - 5) q^{82} + (3 \beta_{3} - \beta_{2} + 2 \beta_1 + 5) q^{83} + (3 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 4) q^{85} + ( - \beta_{3} - 5 \beta_{2} - 5) q^{86} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{88} + ( - 2 \beta_{2} - 4 \beta_1) q^{89} + (3 \beta_1 + 4) q^{91} + (4 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 19) q^{92} + (\beta_{3} - 2 \beta_{2} + 4 \beta_1 - 5) q^{94} + ( - 2 \beta_{3} - 3 \beta_{2} - \beta_1 - 7) q^{95} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 7) q^{97} + (\beta_{3} - 4 \beta_{2} + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 11 q^{4} + 4 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 11 q^{4} + 4 q^{5} + q^{7} + q^{10} + 4 q^{11} + 7 q^{13} + q^{14} + 17 q^{16} - 5 q^{17} + 9 q^{19} - 10 q^{20} + q^{22} + 14 q^{23} + 14 q^{25} + 22 q^{26} - q^{28} - 6 q^{29} - 2 q^{31} - 34 q^{32} + 16 q^{34} - 8 q^{35} + 3 q^{37} + 3 q^{38} + 12 q^{40} - 2 q^{41} - 21 q^{43} + 11 q^{44} + 2 q^{46} - 7 q^{47} - 15 q^{49} + 23 q^{50} - 10 q^{52} - 6 q^{53} + 4 q^{55} + 18 q^{56} - 21 q^{58} + 2 q^{59} + 15 q^{61} - 20 q^{62} + 16 q^{64} + 19 q^{65} + 14 q^{67} - 7 q^{68} - 38 q^{70} - 3 q^{71} + 22 q^{73} - 36 q^{74} + 42 q^{76} + q^{77} + 11 q^{79} - 34 q^{80} - 17 q^{82} + 18 q^{83} + 13 q^{85} - 24 q^{86} - 6 q^{89} + 19 q^{91} + 67 q^{92} - 19 q^{94} - 30 q^{95} + 26 q^{97} + 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.519120
1.45106
−2.27060
2.33866
−2.73051 0 5.45571 −0.936586 0 −0.519120 −9.43585 0 2.55736
1.2 −0.894434 0 −1.19999 3.74893 0 1.45106 2.86218 0 −3.35317
1.3 2.15561 0 2.64667 3.62393 0 −2.27060 1.39396 0 7.81179
1.4 2.46934 0 4.09762 −2.43628 0 2.33866 5.17972 0 −6.01598
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.a.q 4
3.b odd 2 1 891.2.a.p 4
9.c even 3 2 99.2.e.e 8
9.d odd 6 2 297.2.e.e 8
11.b odd 2 1 9801.2.a.bi 4
33.d even 2 1 9801.2.a.bl 4
99.h odd 6 2 1089.2.e.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.e.e 8 9.c even 3 2
297.2.e.e 8 9.d odd 6 2
891.2.a.p 4 3.b odd 2 1
891.2.a.q 4 1.a even 1 1 trivial
1089.2.e.i 8 99.h odd 6 2
9801.2.a.bi 4 11.b odd 2 1
9801.2.a.bl 4 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(891))\):

\( T_{2}^{4} - T_{2}^{3} - 9T_{2}^{2} + 8T_{2} + 13 \) Copy content Toggle raw display
\( T_{5}^{4} - 4T_{5}^{3} - 9T_{5}^{2} + 29T_{5} + 31 \) Copy content Toggle raw display
\( T_{7}^{4} - T_{7}^{3} - 6T_{7}^{2} + 5T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 9 T^{2} + 8 T + 13 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} - 9 T^{2} + 29 T + 31 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} - 6 T^{2} + 5 T + 4 \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 7 T^{3} - 15 T^{2} + 155 T - 158 \) Copy content Toggle raw display
$17$ \( T^{4} + 5 T^{3} - 24 T^{2} - 169 T - 236 \) Copy content Toggle raw display
$19$ \( T^{4} - 9 T^{3} + 81 T - 54 \) Copy content Toggle raw display
$23$ \( T^{4} - 14 T^{3} + 51 T^{2} + \cdots - 188 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} - 9 T^{2} - 105 T - 150 \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} - 21 T^{2} - 73 T - 59 \) Copy content Toggle raw display
$37$ \( T^{4} - 3 T^{3} - 81 T^{2} + 144 T + 57 \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} - 63 T^{2} + 161 T - 26 \) Copy content Toggle raw display
$43$ \( T^{4} + 21 T^{3} + 156 T^{2} + \cdots + 498 \) Copy content Toggle raw display
$47$ \( T^{4} + 7 T^{3} - 15 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} - 45 T^{2} - 165 T - 123 \) Copy content Toggle raw display
$59$ \( T^{4} - 2 T^{3} - 21 T^{2} + 49 T + 13 \) Copy content Toggle raw display
$61$ \( T^{4} - 15 T^{3} + 78 T^{2} + \cdots + 120 \) Copy content Toggle raw display
$67$ \( T^{4} - 14 T^{3} - 21 T^{2} + \cdots - 2171 \) Copy content Toggle raw display
$71$ \( T^{4} + 3 T^{3} - 87 T^{2} + 270 T - 195 \) Copy content Toggle raw display
$73$ \( T^{4} - 22 T^{3} + 129 T^{2} + \cdots - 1028 \) Copy content Toggle raw display
$79$ \( T^{4} - 11 T^{3} - 24 T^{2} + 25 T + 40 \) Copy content Toggle raw display
$83$ \( T^{4} - 18 T^{3} - 3 T^{2} + \cdots + 1548 \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} - 132 T^{2} + \cdots + 2064 \) Copy content Toggle raw display
$97$ \( T^{4} - 26 T^{3} + 156 T^{2} + \cdots - 1229 \) Copy content Toggle raw display
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