Properties

Label 1620.1.l.b
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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

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

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

Hecke kernels

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

Hecke characteristic polynomials

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