Properties

Label 1620.2.d.a
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: \(x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + ( -1 - 2 i ) q^{5} + 2 i q^{7} +O(q^{10})\) \( q + ( -1 - 2 i ) q^{5} + 2 i q^{7} + q^{11} -2 i q^{17} + 3 q^{19} -4 i q^{23} + ( -3 + 4 i ) q^{25} + 3 q^{29} + q^{31} + ( 4 - 2 i ) q^{35} -10 i q^{37} + q^{41} + 10 i q^{43} -10 i q^{47} + 3 q^{49} -6 i q^{53} + ( -1 - 2 i ) q^{55} + 13 q^{59} -10 q^{61} -8 i q^{67} + 9 q^{71} -14 i q^{73} + 2 i q^{77} -10 i q^{83} + ( -4 + 2 i ) q^{85} -13 q^{89} + ( -3 - 6 i ) q^{95} + 10 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{5} + 2q^{11} + 6q^{19} - 6q^{25} + 6q^{29} + 2q^{31} + 8q^{35} + 2q^{41} + 6q^{49} - 2q^{55} + 26q^{59} - 20q^{61} + 18q^{71} - 8q^{85} - 26q^{89} - 6q^{95} + O(q^{100}) \)

Character values

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

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

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

Hecke kernels

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

\( T_{7}^{2} + 4 \)
\( T_{11} - 1 \)

Hecke characteristic polynomials

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