Properties

Label 2268.2.l.d
Level $2268$
Weight $2$
Character orbit 2268.l
Analytic conductor $18.110$
Analytic rank $1$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.l (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.1100711784\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 756)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 + 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -2 + 3 \zeta_{6} ) q^{7} + ( -5 + 5 \zeta_{6} ) q^{13} -8 \zeta_{6} q^{19} -5 q^{25} + 7 \zeta_{6} q^{31} -11 \zeta_{6} q^{37} -5 \zeta_{6} q^{43} + ( -5 - 3 \zeta_{6} ) q^{49} + ( 13 - 13 \zeta_{6} ) q^{61} -5 \zeta_{6} q^{67} + ( 10 - 10 \zeta_{6} ) q^{73} + ( -17 + 17 \zeta_{6} ) q^{79} + ( -5 - 10 \zeta_{6} ) q^{91} -5 \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{7} + O(q^{10}) \) \( 2q - q^{7} - 5q^{13} - 8q^{19} - 10q^{25} + 7q^{31} - 11q^{37} - 5q^{43} - 13q^{49} + 13q^{61} - 5q^{67} + 10q^{73} - 17q^{79} - 20q^{91} - 5q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 −0.500000 + 2.59808i 0 0 0
541.1 0 0 0 0 0 −0.500000 2.59808i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
63.g even 3 1 inner
63.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.l.d 2
3.b odd 2 1 CM 2268.2.l.d 2
7.c even 3 1 2268.2.i.f 2
9.c even 3 1 756.2.k.b 2
9.c even 3 1 2268.2.i.f 2
9.d odd 6 1 756.2.k.b 2
9.d odd 6 1 2268.2.i.f 2
21.h odd 6 1 2268.2.i.f 2
63.g even 3 1 inner 2268.2.l.d 2
63.g even 3 1 5292.2.a.h 1
63.h even 3 1 756.2.k.b 2
63.j odd 6 1 756.2.k.b 2
63.k odd 6 1 5292.2.a.e 1
63.n odd 6 1 inner 2268.2.l.d 2
63.n odd 6 1 5292.2.a.h 1
63.s even 6 1 5292.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.k.b 2 9.c even 3 1
756.2.k.b 2 9.d odd 6 1
756.2.k.b 2 63.h even 3 1
756.2.k.b 2 63.j odd 6 1
2268.2.i.f 2 7.c even 3 1
2268.2.i.f 2 9.c even 3 1
2268.2.i.f 2 9.d odd 6 1
2268.2.i.f 2 21.h odd 6 1
2268.2.l.d 2 1.a even 1 1 trivial
2268.2.l.d 2 3.b odd 2 1 CM
2268.2.l.d 2 63.g even 3 1 inner
2268.2.l.d 2 63.n odd 6 1 inner
5292.2.a.e 1 63.k odd 6 1
5292.2.a.e 1 63.s even 6 1
5292.2.a.h 1 63.g even 3 1
5292.2.a.h 1 63.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\):

\( T_{5} \)
\( T_{13}^{2} + 5 T_{13} + 25 \)
\( T_{19}^{2} + 8 T_{19} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 + T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 25 + 5 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 64 + 8 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 49 - 7 T + T^{2} \)
$37$ \( 121 + 11 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 25 + 5 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 169 - 13 T + T^{2} \)
$67$ \( 25 + 5 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 100 - 10 T + T^{2} \)
$79$ \( 289 + 17 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 25 + 5 T + T^{2} \)
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