Properties

Label 2268.2.x.g
Level $2268$
Weight $2$
Character orbit 2268.x
Analytic conductor $18.110$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.x (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 756)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 3 \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} + ( 3 + 3 \zeta_{6} ) q^{11} + ( 4 - 2 \zeta_{6} ) q^{13} + 6 q^{17} + ( -1 + 2 \zeta_{6} ) q^{19} + ( 6 - 3 \zeta_{6} ) q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + ( -6 - 6 \zeta_{6} ) q^{29} + ( 6 - 3 \zeta_{6} ) q^{31} + ( -3 - 6 \zeta_{6} ) q^{35} + q^{37} -3 \zeta_{6} q^{41} + ( -10 + 10 \zeta_{6} ) q^{43} + ( -6 + 6 \zeta_{6} ) q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + ( -9 + 18 \zeta_{6} ) q^{55} + 6 \zeta_{6} q^{59} + ( 8 + 8 \zeta_{6} ) q^{61} + ( 6 + 6 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} + ( -3 + 6 \zeta_{6} ) q^{71} + ( 2 - 4 \zeta_{6} ) q^{73} + ( -12 - 3 \zeta_{6} ) q^{77} + ( -14 + 14 \zeta_{6} ) q^{79} + ( 6 - 6 \zeta_{6} ) q^{83} + 18 \zeta_{6} q^{85} -9 q^{89} + ( -10 + 8 \zeta_{6} ) q^{91} + ( -6 + 3 \zeta_{6} ) q^{95} + ( 4 + 4 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{5} - 5q^{7} + O(q^{10}) \) \( 2q + 3q^{5} - 5q^{7} + 9q^{11} + 6q^{13} + 12q^{17} + 9q^{23} - 4q^{25} - 18q^{29} + 9q^{31} - 12q^{35} + 2q^{37} - 3q^{41} - 10q^{43} - 6q^{47} + 11q^{49} + 6q^{59} + 24q^{61} + 18q^{65} - 2q^{67} - 27q^{77} - 14q^{79} + 6q^{83} + 18q^{85} - 18q^{89} - 12q^{91} - 9q^{95} + 12q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 1.50000 2.59808i 0 −2.50000 0.866025i 0 0 0
1889.1 0 0 0 1.50000 + 2.59808i 0 −2.50000 + 0.866025i 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
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.x.g 2
3.b odd 2 1 2268.2.x.a 2
7.b odd 2 1 2268.2.x.b 2
9.c even 3 1 756.2.f.a 2
9.c even 3 1 2268.2.x.h 2
9.d odd 6 1 756.2.f.c yes 2
9.d odd 6 1 2268.2.x.b 2
21.c even 2 1 2268.2.x.h 2
36.f odd 6 1 3024.2.k.a 2
36.h even 6 1 3024.2.k.d 2
63.l odd 6 1 756.2.f.c yes 2
63.l odd 6 1 2268.2.x.a 2
63.o even 6 1 756.2.f.a 2
63.o even 6 1 inner 2268.2.x.g 2
252.s odd 6 1 3024.2.k.a 2
252.bi even 6 1 3024.2.k.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.f.a 2 9.c even 3 1
756.2.f.a 2 63.o even 6 1
756.2.f.c yes 2 9.d odd 6 1
756.2.f.c yes 2 63.l odd 6 1
2268.2.x.a 2 3.b odd 2 1
2268.2.x.a 2 63.l odd 6 1
2268.2.x.b 2 7.b odd 2 1
2268.2.x.b 2 9.d odd 6 1
2268.2.x.g 2 1.a even 1 1 trivial
2268.2.x.g 2 63.o even 6 1 inner
2268.2.x.h 2 9.c even 3 1
2268.2.x.h 2 21.c even 2 1
3024.2.k.a 2 36.f odd 6 1
3024.2.k.a 2 252.s odd 6 1
3024.2.k.d 2 36.h even 6 1
3024.2.k.d 2 252.bi even 6 1

Hecke kernels

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

\( T_{5}^{2} - 3 T_{5} + 9 \)
\( T_{11}^{2} - 9 T_{11} + 27 \)
\( T_{13}^{2} - 6 T_{13} + 12 \)