Properties

Label 1620.1.f.d
Level $1620$
Weight $1$
Character orbit 1620.f
Self dual yes
Analytic conductor $0.808$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -20
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1620.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.808485320465\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1620.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.1620.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} - q^{14} + q^{16} + q^{20} - q^{23} + q^{25} - q^{28} - q^{29} + q^{32} - q^{35} + q^{40} - q^{41} + 2q^{43} - q^{46} - q^{47} + q^{50} - q^{56} - q^{58} - q^{61} + q^{64} - q^{67} - q^{70} + q^{80} - q^{82} - q^{83} + 2q^{86} - q^{89} - q^{92} - q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1459.1
0
1.00000 0 1.00000 1.00000 0 −1.00000 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.1.f.d 1
3.b odd 2 1 1620.1.f.a 1
4.b odd 2 1 1620.1.f.b 1
5.b even 2 1 1620.1.f.b 1
9.c even 3 2 180.1.p.a 2
9.d odd 6 2 540.1.p.b 2
12.b even 2 1 1620.1.f.c 1
15.d odd 2 1 1620.1.f.c 1
20.d odd 2 1 CM 1620.1.f.d 1
36.f odd 6 2 180.1.p.b yes 2
36.h even 6 2 540.1.p.a 2
45.h odd 6 2 540.1.p.a 2
45.j even 6 2 180.1.p.b yes 2
45.k odd 12 4 900.1.t.a 4
45.l even 12 4 2700.1.t.a 4
60.h even 2 1 1620.1.f.a 1
72.n even 6 2 2880.1.bu.b 2
72.p odd 6 2 2880.1.bu.a 2
180.n even 6 2 540.1.p.b 2
180.p odd 6 2 180.1.p.a 2
180.v odd 12 4 2700.1.t.a 4
180.x even 12 4 900.1.t.a 4
360.z odd 6 2 2880.1.bu.b 2
360.bk even 6 2 2880.1.bu.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.1.p.a 2 9.c even 3 2
180.1.p.a 2 180.p odd 6 2
180.1.p.b yes 2 36.f odd 6 2
180.1.p.b yes 2 45.j even 6 2
540.1.p.a 2 36.h even 6 2
540.1.p.a 2 45.h odd 6 2
540.1.p.b 2 9.d odd 6 2
540.1.p.b 2 180.n even 6 2
900.1.t.a 4 45.k odd 12 4
900.1.t.a 4 180.x even 12 4
1620.1.f.a 1 3.b odd 2 1
1620.1.f.a 1 60.h even 2 1
1620.1.f.b 1 4.b odd 2 1
1620.1.f.b 1 5.b even 2 1
1620.1.f.c 1 12.b even 2 1
1620.1.f.c 1 15.d odd 2 1
1620.1.f.d 1 1.a even 1 1 trivial
1620.1.f.d 1 20.d odd 2 1 CM
2700.1.t.a 4 45.l even 12 4
2700.1.t.a 4 180.v odd 12 4
2880.1.bu.a 2 72.p odd 6 2
2880.1.bu.a 2 360.bk even 6 2
2880.1.bu.b 2 72.n even 6 2
2880.1.bu.b 2 360.z odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7} + 1 \)
\( T_{23} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( -1 + T \)
$7$ \( 1 + T \)
$11$ \( T \)
$13$ \( T \)
$17$ \( T \)
$19$ \( T \)
$23$ \( 1 + T \)
$29$ \( 1 + T \)
$31$ \( T \)
$37$ \( T \)
$41$ \( 1 + T \)
$43$ \( -2 + T \)
$47$ \( 1 + T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( 1 + T \)
$67$ \( 1 + T \)
$71$ \( T \)
$73$ \( T \)
$79$ \( T \)
$83$ \( 1 + T \)
$89$ \( 1 + T \)
$97$ \( T \)
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