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Properties

Label 1782.2.e.e
Level $1782$
Weight $2$
Character orbit 1782.e
Analytic conductor $14.229$
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] = [1782,2,Mod(595,1782)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1782, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1782.595");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1782 = 2 \cdot 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1782.e (of order \(3\), degree \(2\), 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: \(14.2293416402\)
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 66)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - 2 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + q^{8} + 2 q^{10} + ( - \zeta_{6} + 1) q^{11} + 6 \zeta_{6} q^{13} + 4 \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} + \cdots + 9 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - 2 q^{5} + 4 q^{7} + 2 q^{8} + 4 q^{10} + q^{11} + 6 q^{13} + 4 q^{14} - q^{16} + 4 q^{17} + 8 q^{19} - 2 q^{20} + q^{22} - 4 q^{23} + q^{25} - 12 q^{26} - 8 q^{28} - 6 q^{29}+ \cdots + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1782\mathbb{Z}\right)^\times\).

\(n\) \(1135\) \(1541\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
595.1
0.500000 − 0.866025i
0.500000 + 0.866025i
−0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.00000 + 1.73205i 0 2.00000 + 3.46410i 1.00000 0 2.00000
1189.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.00000 − 1.73205i 0 2.00000 − 3.46410i 1.00000 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1782.2.e.e 2
3.b odd 2 1 1782.2.e.v 2
9.c even 3 1 66.2.a.b ✓ 1
9.c even 3 1 inner 1782.2.e.e 2
9.d odd 6 1 198.2.a.a 1
9.d odd 6 1 1782.2.e.v 2
36.f odd 6 1 528.2.a.j 1
36.h even 6 1 1584.2.a.f 1
45.h odd 6 1 4950.2.a.bu 1
45.j even 6 1 1650.2.a.k 1
45.k odd 12 2 1650.2.c.e 2
45.l even 12 2 4950.2.c.p 2
63.l odd 6 1 3234.2.a.t 1
63.o even 6 1 9702.2.a.x 1
72.j odd 6 1 6336.2.a.bw 1
72.l even 6 1 6336.2.a.cj 1
72.n even 6 1 2112.2.a.r 1
72.p odd 6 1 2112.2.a.e 1
99.g even 6 1 2178.2.a.g 1
99.h odd 6 1 726.2.a.c 1
99.m even 15 4 726.2.e.g 4
99.o odd 30 4 726.2.e.o 4
396.k even 6 1 5808.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.b ✓ 1 9.c even 3 1
198.2.a.a 1 9.d odd 6 1
528.2.a.j 1 36.f odd 6 1
726.2.a.c 1 99.h odd 6 1
726.2.e.g 4 99.m even 15 4
726.2.e.o 4 99.o odd 30 4
1584.2.a.f 1 36.h even 6 1
1650.2.a.k 1 45.j even 6 1
1650.2.c.e 2 45.k odd 12 2
1782.2.e.e 2 1.a even 1 1 trivial
1782.2.e.e 2 9.c even 3 1 inner
1782.2.e.v 2 3.b odd 2 1
1782.2.e.v 2 9.d odd 6 1
2112.2.a.e 1 72.p odd 6 1
2112.2.a.r 1 72.n even 6 1
2178.2.a.g 1 99.g even 6 1
3234.2.a.t 1 63.l odd 6 1
4950.2.a.bu 1 45.h odd 6 1
4950.2.c.p 2 45.l even 12 2
5808.2.a.bc 1 396.k even 6 1
6336.2.a.bw 1 72.j odd 6 1
6336.2.a.cj 1 72.l even 6 1
9702.2.a.x 1 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1782, [\chi])\):

\( T_{5}^{2} + 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 16 \) Copy content Toggle raw display
\( T_{13}^{2} - 6T_{13} + 36 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display
\( T_{23}^{2} + 4T_{23} + 16 \) 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
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
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