Properties

Label 1620.2.i.l
Level $1620$
Weight $2$
Character orbit 1620.i
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.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + ( - 3 \zeta_{6} + 3) q^{11} + 4 \zeta_{6} q^{13} + 5 q^{19} - 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + (9 \zeta_{6} - 9) q^{29} - 5 \zeta_{6} q^{31} + 4 q^{35} + 2 q^{37} - 9 \zeta_{6} q^{41} + ( - 10 \zeta_{6} + 10) q^{43} + (6 \zeta_{6} - 6) q^{47} - 9 \zeta_{6} q^{49} + 12 q^{53} + 3 q^{55} + 9 \zeta_{6} q^{59} + ( - 10 \zeta_{6} + 10) q^{61} + (4 \zeta_{6} - 4) q^{65} - 2 \zeta_{6} q^{67} - 3 q^{71} - 4 q^{73} - 12 \zeta_{6} q^{77} + ( - 4 \zeta_{6} + 4) q^{79} + ( - 6 \zeta_{6} + 6) q^{83} + 9 q^{89} + 16 q^{91} + 5 \zeta_{6} q^{95} + (2 \zeta_{6} - 2) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{5} + 4 q^{7} + 3 q^{11} + 4 q^{13} + 10 q^{19} - 6 q^{23} - q^{25} - 9 q^{29} - 5 q^{31} + 8 q^{35} + 4 q^{37} - 9 q^{41} + 10 q^{43} - 6 q^{47} - 9 q^{49} + 24 q^{53} + 6 q^{55} + 9 q^{59} + 10 q^{61} - 4 q^{65} - 2 q^{67} - 6 q^{71} - 8 q^{73} - 12 q^{77} + 4 q^{79} + 6 q^{83} + 18 q^{89} + 32 q^{91} + 5 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
541.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0.500000 + 0.866025i 0 2.00000 3.46410i 0 0 0
1081.1 0 0 0 0.500000 0.866025i 0 2.00000 + 3.46410i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.i.l 2
3.b odd 2 1 1620.2.i.e 2
9.c even 3 1 1620.2.a.a 1
9.c even 3 1 inner 1620.2.i.l 2
9.d odd 6 1 1620.2.a.d yes 1
9.d odd 6 1 1620.2.i.e 2
36.f odd 6 1 6480.2.a.m 1
36.h even 6 1 6480.2.a.y 1
45.h odd 6 1 8100.2.a.n 1
45.j even 6 1 8100.2.a.m 1
45.k odd 12 2 8100.2.d.b 2
45.l even 12 2 8100.2.d.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.a 1 9.c even 3 1
1620.2.a.d yes 1 9.d odd 6 1
1620.2.i.e 2 3.b odd 2 1
1620.2.i.e 2 9.d odd 6 1
1620.2.i.l 2 1.a even 1 1 trivial
1620.2.i.l 2 9.c even 3 1 inner
6480.2.a.m 1 36.f odd 6 1
6480.2.a.y 1 36.h even 6 1
8100.2.a.m 1 45.j even 6 1
8100.2.a.n 1 45.h odd 6 1
8100.2.d.b 2 45.k odd 12 2
8100.2.d.g 2 45.l even 12 2

Hecke kernels

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

\( T_{7}^{2} - 4T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{17} \) 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
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( (T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$71$ \( (T + 3)^{2} \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$89$ \( (T - 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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