Properties

Label 819.2.bm.b
Level $819$
Weight $2$
Character orbit 819.bm
Analytic conductor $6.540$
Analytic rank $0$
Dimension $2$
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] = [819,2,Mod(478,819)]
 
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, 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("819.478");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.bm (of order \(6\), 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
478.1
0.500000 0.866025i
0.500000 + 0.866025i
1.73205i 0 −1.00000 3.00000 + 1.73205i 0 2.50000 0.866025i 1.73205i 0 −3.00000 + 5.19615i
550.1 1.73205i 0 −1.00000 3.00000 1.73205i 0 2.50000 + 0.866025i 1.73205i 0 −3.00000 5.19615i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.bm.b 2
3.b odd 2 1 273.2.t.a 2
7.c even 3 1 819.2.do.a 2
13.e even 6 1 819.2.do.a 2
21.h odd 6 1 273.2.bl.a yes 2
39.h odd 6 1 273.2.bl.a yes 2
91.k even 6 1 inner 819.2.bm.b 2
273.bp odd 6 1 273.2.t.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.t.a 2 3.b odd 2 1
273.2.t.a 2 273.bp odd 6 1
273.2.bl.a yes 2 21.h odd 6 1
273.2.bl.a yes 2 39.h odd 6 1
819.2.bm.b 2 1.a even 1 1 trivial
819.2.bm.b 2 91.k even 6 1 inner
819.2.do.a 2 7.c even 3 1
819.2.do.a 2 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\):

\( T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{5}^{2} - 6T_{5} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$7$ \( T^{2} - 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} - 18T + 108 \) Copy content Toggle raw display
$37$ \( T^{2} + 3 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 18T + 108 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 12 \) Copy content Toggle raw display
$61$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} + 108 \) Copy content Toggle raw display
$89$ \( T^{2} + 108 \) Copy content Toggle raw display
$97$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
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