Properties

Label 1161.1.i.a
Level $1161$
Weight $1$
Character orbit 1161.i
Analytic conductor $0.579$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1161 = 3^{3} \cdot 43 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1161.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.579414479667\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.49923.1
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.4043763.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + q^{4} -2 \zeta_{6} q^{7} +O(q^{10})\) \( q + q^{4} -2 \zeta_{6} q^{7} + \zeta_{6} q^{13} + q^{16} -\zeta_{6}^{2} q^{19} -\zeta_{6} q^{25} -2 \zeta_{6} q^{28} -\zeta_{6}^{2} q^{31} + 2 \zeta_{6}^{2} q^{37} + q^{43} + 3 \zeta_{6}^{2} q^{49} + \zeta_{6} q^{52} + \zeta_{6} q^{61} + q^{64} + 2 \zeta_{6}^{2} q^{67} -2 \zeta_{6} q^{73} -\zeta_{6}^{2} q^{76} + \zeta_{6} q^{79} -2 \zeta_{6}^{2} q^{91} - q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} - 2q^{7} + O(q^{10}) \) \( 2q + 2q^{4} - 2q^{7} + q^{13} + 2q^{16} + q^{19} - q^{25} - 2q^{28} + q^{31} - 2q^{37} + 2q^{43} - 3q^{49} + q^{52} + q^{61} + 2q^{64} - 2q^{67} - 2q^{73} + q^{76} + q^{79} + 2q^{91} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1161\mathbb{Z}\right)^\times\).

\(n\) \(433\) \(947\)
\(\chi(n)\) \(-\zeta_{6}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
350.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 1.00000 0 0 −1.00000 1.73205i 0 0 0
1025.1 0 0 1.00000 0 0 −1.00000 + 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 \(\Q(\sqrt{-3}) \)
43.c even 3 1 inner
129.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1161.1.i.a 2
3.b odd 2 1 CM 1161.1.i.a 2
9.c even 3 1 3483.1.o.a 2
9.c even 3 1 3483.1.p.a 2
9.d odd 6 1 3483.1.o.a 2
9.d odd 6 1 3483.1.p.a 2
43.c even 3 1 inner 1161.1.i.a 2
129.f odd 6 1 inner 1161.1.i.a 2
387.e even 3 1 3483.1.o.a 2
387.g even 3 1 3483.1.p.a 2
387.o odd 6 1 3483.1.p.a 2
387.p odd 6 1 3483.1.o.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1161.1.i.a 2 1.a even 1 1 trivial
1161.1.i.a 2 3.b odd 2 1 CM
1161.1.i.a 2 43.c even 3 1 inner
1161.1.i.a 2 129.f odd 6 1 inner
3483.1.o.a 2 9.c even 3 1
3483.1.o.a 2 9.d odd 6 1
3483.1.o.a 2 387.e even 3 1
3483.1.o.a 2 387.p odd 6 1
3483.1.p.a 2 9.c even 3 1
3483.1.p.a 2 9.d odd 6 1
3483.1.p.a 2 387.g even 3 1
3483.1.p.a 2 387.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{1}^{\mathrm{new}}(1161, [\chi])\).