Properties

Label 1620.1.l.a
Level $1620$
Weight $1$
Character orbit 1620.l
Analytic conductor $0.808$
Analytic rank $0$
Dimension $2$
Projective image $S_{4}$
CM/RM no
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.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.808485320465\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.0.40500.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + i q^{5} + ( -1 + i ) q^{7} +O(q^{10})\) \( q + i q^{5} + ( -1 + i ) q^{7} - q^{11} + ( -1 + i ) q^{17} -i q^{19} + ( -1 - i ) q^{23} - q^{25} + i q^{29} + q^{31} + ( -1 - i ) q^{35} - q^{41} -i q^{49} + ( 1 + i ) q^{53} -i q^{55} + i q^{59} + ( -1 + i ) q^{67} + q^{71} + ( 1 + i ) q^{73} + ( 1 - i ) q^{77} + ( -1 - i ) q^{85} -i q^{89} + q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} + O(q^{10}) \) \( 2q - 2q^{7} - 2q^{11} - 2q^{17} - 2q^{23} - 2q^{25} + 2q^{31} - 2q^{35} - 2q^{41} + 2q^{53} - 2q^{67} + 2q^{71} + 2q^{73} + 2q^{77} - 2q^{85} + 2q^{95} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
973.1
1.00000i
1.00000i
0 0 0 1.00000i 0 −1.00000 + 1.00000i 0 0 0
1297.1 0 0 0 1.00000i 0 −1.00000 1.00000i 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
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.1.l.a 2
3.b odd 2 1 1620.1.l.b yes 2
5.c odd 4 1 inner 1620.1.l.a 2
9.c even 3 2 1620.1.v.b 4
9.d odd 6 2 1620.1.v.a 4
15.e even 4 1 1620.1.l.b yes 2
45.k odd 12 2 1620.1.v.b 4
45.l even 12 2 1620.1.v.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.1.l.a 2 1.a even 1 1 trivial
1620.1.l.a 2 5.c odd 4 1 inner
1620.1.l.b yes 2 3.b odd 2 1
1620.1.l.b yes 2 15.e even 4 1
1620.1.v.a 4 9.d odd 6 2
1620.1.v.a 4 45.l even 12 2
1620.1.v.b 4 9.c even 3 2
1620.1.v.b 4 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1620, [\chi])\).

Hecke characteristic polynomials

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