Properties

Label 1620.2.d.b
Level $1620$
Weight $2$
Character orbit 1620.d
Analytic conductor $12.936$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{5} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{5} + \beta q^{7} - q^{11} + \beta q^{17} + 3 q^{19} + 2 \beta q^{23} + (2 \beta - 3) q^{25} - 3 q^{29} + q^{31} + (\beta - 4) q^{35} - 5 \beta q^{37} - q^{41} + 5 \beta q^{43} + 5 \beta q^{47} + 3 q^{49} + 3 \beta q^{53} + ( - \beta - 1) q^{55} - 13 q^{59} - 10 q^{61} - 4 \beta q^{67} - 9 q^{71} - 7 \beta q^{73} - \beta q^{77} + 5 \beta q^{83} + (\beta - 4) q^{85} + 13 q^{89} + (3 \beta + 3) q^{95} + 5 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{11} + 6 q^{19} - 6 q^{25} - 6 q^{29} + 2 q^{31} - 8 q^{35} - 2 q^{41} + 6 q^{49} - 2 q^{55} - 26 q^{59} - 20 q^{61} - 18 q^{71} - 8 q^{85} + 26 q^{89} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.00000i
1.00000i
0 0 0 1.00000 2.00000i 0 2.00000i 0 0 0
649.2 0 0 0 1.00000 + 2.00000i 0 2.00000i 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.d.b yes 2
3.b odd 2 1 1620.2.d.a 2
5.b even 2 1 inner 1620.2.d.b yes 2
5.c odd 4 1 8100.2.a.c 1
5.c odd 4 1 8100.2.a.k 1
9.c even 3 2 1620.2.r.b 4
9.d odd 6 2 1620.2.r.e 4
15.d odd 2 1 1620.2.d.a 2
15.e even 4 1 8100.2.a.d 1
15.e even 4 1 8100.2.a.l 1
45.h odd 6 2 1620.2.r.e 4
45.j even 6 2 1620.2.r.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.d.a 2 3.b odd 2 1
1620.2.d.a 2 15.d odd 2 1
1620.2.d.b yes 2 1.a even 1 1 trivial
1620.2.d.b yes 2 5.b even 2 1 inner
1620.2.r.b 4 9.c even 3 2
1620.2.r.b 4 45.j even 6 2
1620.2.r.e 4 9.d odd 6 2
1620.2.r.e 4 45.h odd 6 2
8100.2.a.c 1 5.c odd 4 1
8100.2.a.d 1 15.e even 4 1
8100.2.a.k 1 5.c odd 4 1
8100.2.a.l 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} + 1 \) 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
$5$ \( T^{2} - 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T - 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T + 3)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( (T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 100 \) Copy content Toggle raw display
$47$ \( T^{2} + 100 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T + 13)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T + 9)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 196 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 100 \) Copy content Toggle raw display
$89$ \( (T - 13)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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