Properties

Label 891.2.j.a
Level $891$
Weight $2$
Character orbit 891.j
Analytic conductor $7.115$
Analytic rank $0$
Dimension $6$
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(100,891)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(891, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.100");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.j (of order \(9\), degree \(6\), 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: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 297)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{18}^{3} - \zeta_{18}^{2}) q^{2} + (\zeta_{18}^{4} + \zeta_{18}^{3} - 1) q^{4} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{5} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3}) q^{7} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{18}^{3} - \zeta_{18}^{2}) q^{2} + (\zeta_{18}^{4} + \zeta_{18}^{3} - 1) q^{4} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{5} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3}) q^{7} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{8} + (\zeta_{18}^{5} + \zeta_{18}^{4} - 3 \zeta_{18}^{2} + 2 \zeta_{18}) q^{10} - \zeta_{18} q^{11} + ( - 3 \zeta_{18}^{4} + \zeta_{18}^{3} - 3 \zeta_{18}^{2} + \zeta_{18} - 3) q^{13} + \zeta_{18}^{2} q^{14} + (\zeta_{18}^{5} - \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 2 \zeta_{18} - 2) q^{16} + ( - 4 \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} + 4) q^{17} + (2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 2 \zeta_{18}) q^{19} + (3 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{20} + ( - \zeta_{18}^{4} + \zeta_{18}^{3}) q^{22} + ( - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2) q^{23} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} - 1) q^{25} + ( - 4 \zeta_{18}^{5} + \zeta_{18}^{4} + 3 \zeta_{18}^{2} + 3 \zeta_{18} - 4) q^{26} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{28} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 6 \zeta_{18}^{3} - 6 \zeta_{18}^{2} - 2 \zeta_{18} - 2) q^{29} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 2 \zeta_{18} - 4) q^{31} + ( - 3 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + \zeta_{18} + 1) q^{32} + (5 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{3} - 4 \zeta_{18}^{2} + 4) q^{34} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3} - \zeta_{18}^{2} - 2 \zeta_{18}) q^{35} + (4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18}) q^{37} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 5 \zeta_{18} + 1) q^{38} + ( - 4 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} + \zeta_{18} - 4) q^{40} + ( - \zeta_{18}^{4} + 4 \zeta_{18}^{3} - \zeta_{18}^{2} + 4 \zeta_{18} - 1) q^{41} + (6 \zeta_{18}^{5} - 6 \zeta_{18}^{3} - 4 \zeta_{18}^{2} + \zeta_{18} + 2) q^{43} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}) q^{44} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2 \zeta_{18}) q^{46} + ( - 3 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2) q^{47} + ( - 4 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 3 \zeta_{18} - 3) q^{49} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 3 \zeta_{18} - 1) q^{50} + ( - 5 \zeta_{18}^{5} - \zeta_{18}^{4} - 6 \zeta_{18}^{3} + 6 \zeta_{18}^{2} + \zeta_{18} + 5) q^{52} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{2} + 2 \zeta_{18} + 7) q^{53} + (\zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{55} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} - 2 \zeta_{18} - 1) q^{56} + ( - 10 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 4) q^{58} + ( - 3 \zeta_{18}^{5} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{59} + ( - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{61} + ( - 3 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 6 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - 3 \zeta_{18}) q^{62} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - \zeta_{18}^{3} + 4 \zeta_{18}^{2} - \zeta_{18} + 1) q^{64} + ( - 4 \zeta_{18}^{5} + 4 \zeta_{18}^{3} - 5 \zeta_{18}^{2} - 3 \zeta_{18} - 9) q^{65} + (\zeta_{18}^{4} + 7 \zeta_{18}^{3} - \zeta_{18}^{2} + 7 \zeta_{18} + 1) q^{67} + ( - \zeta_{18}^{4} + 5 \zeta_{18}^{3} - \zeta_{18}^{2} + 5 \zeta_{18} - 1) q^{68} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + 2 \zeta_{18}^{2} - \zeta_{18} + 1) q^{70} + (6 \zeta_{18}^{5} + 6 \zeta_{18}^{4} - 7 \zeta_{18}^{3} - \zeta_{18}^{2} - 5 \zeta_{18} + 7) q^{71} + ( - 7 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - \zeta_{18}^{3} + 5 \zeta_{18}^{2} - 7 \zeta_{18}) q^{73} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{3} - 4 \zeta_{18}^{2} + 4) q^{74} + ( - 4 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 4 \zeta_{18}^{2} + 5 \zeta_{18} + 5) q^{76} + (\zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 1) q^{77} + ( - 5 \zeta_{18}^{5} + 7 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 7 \zeta_{18} + 5) q^{79} + ( - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 2 \zeta_{18} - 11) q^{80} + ( - 5 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 5) q^{82} + ( - 9 \zeta_{18}^{5} + \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - \zeta_{18} + 9) q^{83} + (4 \zeta_{18}^{5} - 7 \zeta_{18}^{4} - 7 \zeta_{18}^{3} + \zeta_{18} + 6) q^{85} + (8 \zeta_{18}^{5} - \zeta_{18}^{4} - 5 \zeta_{18}^{3} - 8 \zeta_{18}^{2} + 6 \zeta_{18} + 6) q^{86} + (\zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{2} + 1) q^{88} + ( - 6 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 6 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 6 \zeta_{18}) q^{89} + (7 \zeta_{18}^{5} + 7 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 5 \zeta_{18} - 4) q^{91} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 4 \zeta_{18}) q^{92} + ( - 4 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 5 \zeta_{18}^{2} + 2 \zeta_{18} - 4) q^{94} + ( - 11 \zeta_{18}^{4} + \zeta_{18}^{2} - 11) q^{95} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} - 4 \zeta_{18}^{2} - 9 \zeta_{18} - 6) q^{97} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 7 \zeta_{18}^{2} - 6 \zeta_{18} + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 3 q^{7} - 3 q^{8} - 15 q^{13} - 15 q^{16} + 12 q^{17} - 12 q^{19} - 3 q^{20} + 3 q^{22} + 6 q^{23} - 9 q^{25} - 24 q^{26} + 12 q^{28} + 6 q^{29} - 18 q^{31} - 3 q^{32} + 27 q^{34} - 6 q^{35} - 3 q^{38} - 21 q^{40} + 6 q^{41} - 6 q^{43} + 6 q^{46} - 15 q^{47} - 15 q^{49} - 12 q^{50} + 12 q^{52} + 42 q^{53} + 6 q^{55} - 3 q^{56} - 12 q^{58} + 6 q^{59} + 6 q^{61} - 18 q^{62} + 3 q^{64} - 42 q^{65} + 27 q^{67} + 9 q^{68} + 9 q^{70} + 21 q^{71} - 3 q^{73} + 18 q^{74} + 21 q^{76} - 3 q^{77} + 18 q^{79} - 66 q^{80} - 30 q^{82} + 39 q^{83} + 15 q^{85} + 21 q^{86} + 6 q^{88} + 18 q^{89} - 12 q^{91} + 6 q^{92} - 18 q^{94} - 66 q^{95} - 30 q^{97} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(244\) \(650\)
\(\chi(n)\) \(1\) \(-\zeta_{18}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
100.1
−0.173648 + 0.984808i
0.939693 + 0.342020i
−0.766044 0.642788i
−0.766044 + 0.642788i
0.939693 0.342020i
−0.173648 0.984808i
1.43969 0.524005i 0 0.266044 0.223238i 0.560307 + 3.17766i 0 −0.500000 0.419550i −1.26604 + 2.19285i 0 2.47178 + 4.28125i
199.1 −0.266044 + 0.223238i 0 −0.326352 + 1.85083i 2.26604 0.824773i 0 −0.500000 2.83564i −0.673648 1.16679i 0 −0.418748 + 0.725293i
397.1 0.326352 1.85083i 0 −1.43969 0.524005i 1.67365 1.40436i 0 −0.500000 + 0.181985i 0.439693 0.761570i 0 −2.05303 3.55596i
496.1 0.326352 + 1.85083i 0 −1.43969 + 0.524005i 1.67365 + 1.40436i 0 −0.500000 0.181985i 0.439693 + 0.761570i 0 −2.05303 + 3.55596i
694.1 −0.266044 0.223238i 0 −0.326352 1.85083i 2.26604 + 0.824773i 0 −0.500000 + 2.83564i −0.673648 + 1.16679i 0 −0.418748 0.725293i
793.1 1.43969 + 0.524005i 0 0.266044 + 0.223238i 0.560307 3.17766i 0 −0.500000 + 0.419550i −1.26604 2.19285i 0 2.47178 4.28125i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 100.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.j.a 6
3.b odd 2 1 297.2.j.a 6
27.e even 9 1 inner 891.2.j.a 6
27.e even 9 1 8019.2.a.b 3
27.f odd 18 1 297.2.j.a 6
27.f odd 18 1 8019.2.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.j.a 6 3.b odd 2 1
297.2.j.a 6 27.f odd 18 1
891.2.j.a 6 1.a even 1 1 trivial
891.2.j.a 6 27.e even 9 1 inner
8019.2.a.a 3 27.f odd 18 1
8019.2.a.b 3 27.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 3T_{2}^{5} + 6T_{2}^{4} - 8T_{2}^{3} + 3T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 9 T^{5} + 45 T^{4} - 152 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{5} + 12 T^{4} + 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$13$ \( T^{6} + 15 T^{5} + 132 T^{4} + \cdots + 5329 \) Copy content Toggle raw display
$17$ \( T^{6} - 12 T^{5} + 99 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$19$ \( T^{6} + 12 T^{5} + 117 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} + 24 T^{4} - 64 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{6} - 6 T^{5} + 36 T^{4} + \cdots + 207936 \) Copy content Toggle raw display
$31$ \( T^{6} + 18 T^{5} + 126 T^{4} + \cdots + 1369 \) Copy content Toggle raw display
$37$ \( T^{6} + 36 T^{4} - 144 T^{3} + \cdots + 5184 \) Copy content Toggle raw display
$41$ \( T^{6} - 6 T^{5} + 66 T^{4} - 422 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$43$ \( T^{6} + 6 T^{5} + 126 T^{4} + \cdots + 45369 \) Copy content Toggle raw display
$47$ \( T^{6} + 15 T^{5} + 156 T^{4} + \cdots + 289 \) Copy content Toggle raw display
$53$ \( (T^{3} - 21 T^{2} + 135 T - 267)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 6 T^{5} - 12 T^{4} + 53 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$61$ \( T^{6} - 6 T^{5} + 36 T^{4} - 192 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$67$ \( T^{6} - 27 T^{5} + 432 T^{4} + \cdots + 683929 \) Copy content Toggle raw display
$71$ \( T^{6} - 21 T^{5} + 387 T^{4} + \cdots + 356409 \) Copy content Toggle raw display
$73$ \( T^{6} + 3 T^{5} + 123 T^{4} + \cdots + 72361 \) Copy content Toggle raw display
$79$ \( T^{6} - 18 T^{5} + 234 T^{4} + \cdots + 72361 \) Copy content Toggle raw display
$83$ \( T^{6} - 39 T^{5} + 693 T^{4} + \cdots + 1172889 \) Copy content Toggle raw display
$89$ \( T^{6} - 18 T^{5} + 405 T^{4} + \cdots + 1896129 \) Copy content Toggle raw display
$97$ \( T^{6} + 30 T^{5} + 330 T^{4} + \cdots + 26569 \) Copy content Toggle raw display
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