Properties

Label 819.2.j.b
Level $819$
Weight $2$
Character orbit 819.j
Analytic conductor $6.540$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(235,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("819.235");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.j (of order \(3\), degree \(2\), 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: \(2\)
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: no (minimal twist has level 91)
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + ( - 3 \zeta_{6} + 2) q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + ( - 3 \zeta_{6} + 2) q^{7} + 3 q^{8} + (3 \zeta_{6} - 3) q^{11} - q^{13} + ( - \zeta_{6} + 3) q^{14} + \zeta_{6} q^{16} + ( - 7 \zeta_{6} + 7) q^{17} + 7 \zeta_{6} q^{19} - 3 q^{22} - 6 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} - \zeta_{6} q^{26} + ( - 2 \zeta_{6} - 1) q^{28} + 5 q^{29} + ( - 5 \zeta_{6} + 5) q^{32} + 7 q^{34} - 8 \zeta_{6} q^{37} + (7 \zeta_{6} - 7) q^{38} + 2 q^{43} + 3 \zeta_{6} q^{44} + ( - 6 \zeta_{6} + 6) q^{46} + 7 \zeta_{6} q^{47} + ( - 3 \zeta_{6} - 5) q^{49} + 5 q^{50} + (\zeta_{6} - 1) q^{52} + (3 \zeta_{6} - 3) q^{53} + ( - 9 \zeta_{6} + 6) q^{56} + 5 \zeta_{6} q^{58} + (7 \zeta_{6} - 7) q^{59} + 7 \zeta_{6} q^{61} + 7 q^{64} + ( - 3 \zeta_{6} + 3) q^{67} - 7 \zeta_{6} q^{68} + 5 q^{71} + (14 \zeta_{6} - 14) q^{73} + ( - 8 \zeta_{6} + 8) q^{74} + 7 q^{76} + (6 \zeta_{6} + 3) q^{77} + 6 \zeta_{6} q^{79} + 2 \zeta_{6} q^{86} + (9 \zeta_{6} - 9) q^{88} + (3 \zeta_{6} - 2) q^{91} - 6 q^{92} + (7 \zeta_{6} - 7) q^{94} - 14 q^{97} + ( - 8 \zeta_{6} + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{4} + q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{4} + q^{7} + 6 q^{8} - 3 q^{11} - 2 q^{13} + 5 q^{14} + q^{16} + 7 q^{17} + 7 q^{19} - 6 q^{22} - 6 q^{23} + 5 q^{25} - q^{26} - 4 q^{28} + 10 q^{29} + 5 q^{32} + 14 q^{34} - 8 q^{37} - 7 q^{38} + 4 q^{43} + 3 q^{44} + 6 q^{46} + 7 q^{47} - 13 q^{49} + 10 q^{50} - q^{52} - 3 q^{53} + 3 q^{56} + 5 q^{58} - 7 q^{59} + 7 q^{61} + 14 q^{64} + 3 q^{67} - 7 q^{68} + 10 q^{71} - 14 q^{73} + 8 q^{74} + 14 q^{76} + 12 q^{77} + 6 q^{79} + 2 q^{86} - 9 q^{88} - q^{91} - 12 q^{92} - 7 q^{94} - 28 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i 0 0 0.500000 2.59808i 3.00000 0 0
352.1 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0 0.500000 + 2.59808i 3.00000 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.j.b 2
3.b odd 2 1 91.2.e.a 2
7.c even 3 1 inner 819.2.j.b 2
7.c even 3 1 5733.2.a.c 1
7.d odd 6 1 5733.2.a.d 1
12.b even 2 1 1456.2.r.g 2
21.c even 2 1 637.2.e.a 2
21.g even 6 1 637.2.a.d 1
21.g even 6 1 637.2.e.a 2
21.h odd 6 1 91.2.e.a 2
21.h odd 6 1 637.2.a.c 1
39.d odd 2 1 1183.2.e.b 2
84.n even 6 1 1456.2.r.g 2
273.w odd 6 1 1183.2.e.b 2
273.w odd 6 1 8281.2.a.f 1
273.ba even 6 1 8281.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.a 2 3.b odd 2 1
91.2.e.a 2 21.h odd 6 1
637.2.a.c 1 21.h odd 6 1
637.2.a.d 1 21.g even 6 1
637.2.e.a 2 21.c even 2 1
637.2.e.a 2 21.g even 6 1
819.2.j.b 2 1.a even 1 1 trivial
819.2.j.b 2 7.c even 3 1 inner
1183.2.e.b 2 39.d odd 2 1
1183.2.e.b 2 273.w odd 6 1
1456.2.r.g 2 12.b even 2 1
1456.2.r.g 2 84.n even 6 1
5733.2.a.c 1 7.c even 3 1
5733.2.a.d 1 7.d odd 6 1
8281.2.a.e 1 273.ba even 6 1
8281.2.a.f 1 273.w odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( (T - 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$71$ \( (T - 5)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$79$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 14)^{2} \) Copy content Toggle raw display
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