Properties

Label 819.2.et.a
Level $819$
Weight $2$
Character orbit 819.et
Analytic conductor $6.540$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(136,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.136");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.et (of order \(12\), degree \(4\), 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{12}^{3} q^{4} + ( - 2 \zeta_{12}^{2} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{12}^{3} q^{4} + ( - 2 \zeta_{12}^{2} - 1) q^{7} + (3 \zeta_{12}^{3} - 4 \zeta_{12}) q^{13} - 4 q^{16} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 3 \zeta_{12} - 5) q^{19} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{25} + ( - 6 \zeta_{12}^{3} + 4 \zeta_{12}) q^{28} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 6 \zeta_{12} - 5) q^{31} + ( - 4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} - 4) q^{37} + ( - 6 \zeta_{12}^{2} - 6) q^{43} + (8 \zeta_{12}^{2} - 3) q^{49} + ( - 8 \zeta_{12}^{2} + 2) q^{52} + (\zeta_{12}^{3} - \zeta_{12}) q^{61} - 8 \zeta_{12}^{3} q^{64} + (7 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 9 \zeta_{12} + 9) q^{67} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + 9 \zeta_{12} + 8) q^{73} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 4 \zeta_{12} - 10) q^{76} + (14 \zeta_{12}^{3} - 7 \zeta_{12}) q^{79} + ( - \zeta_{12}^{3} + 10 \zeta_{12}) q^{91} + (3 \zeta_{12}^{3} - 11 \zeta_{12}^{2} + 8 \zeta_{12} + 8) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{7} - 16 q^{16} - 16 q^{19} - 22 q^{31} - 22 q^{37} - 36 q^{43} + 4 q^{49} - 8 q^{52} + 32 q^{67} + 34 q^{73} - 28 q^{76} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(\zeta_{12}\) \(1 - \zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
0 0 2.00000i 0 0 −2.00000 + 1.73205i 0 0 0
145.1 0 0 2.00000i 0 0 −2.00000 1.73205i 0 0 0
271.1 0 0 2.00000i 0 0 −2.00000 1.73205i 0 0 0
514.1 0 0 2.00000i 0 0 −2.00000 + 1.73205i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
91.ba even 12 1 inner
273.bs odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.et.a 4
3.b odd 2 1 CM 819.2.et.a 4
7.d odd 6 1 819.2.gh.a yes 4
13.f odd 12 1 819.2.gh.a yes 4
21.g even 6 1 819.2.gh.a yes 4
39.k even 12 1 819.2.gh.a yes 4
91.ba even 12 1 inner 819.2.et.a 4
273.bs odd 12 1 inner 819.2.et.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.et.a 4 1.a even 1 1 trivial
819.2.et.a 4 3.b odd 2 1 CM
819.2.et.a 4 91.ba even 12 1 inner
819.2.et.a 4 273.bs odd 12 1 inner
819.2.gh.a yes 4 7.d odd 6 1
819.2.gh.a yes 4 13.f odd 12 1
819.2.gh.a yes 4 21.g even 6 1
819.2.gh.a yes 4 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 22T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 16 T^{3} + 113 T^{2} + \cdots + 1369 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 22 T^{3} + 137 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$37$ \( T^{4} + 22 T^{3} + 242 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 18 T + 108)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$67$ \( T^{4} - 32 T^{3} + 281 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 34 T^{3} + 338 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$79$ \( T^{4} + 147 T^{2} + 21609 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 10 T^{3} + 221 T^{2} + \cdots + 27889 \) Copy content Toggle raw display
show more
show less