Properties

Label 2268.2.l.g
Level $2268$
Weight $2$
Character orbit 2268.l
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.l (of order \(3\), 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 84)
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.

\(f(q)\) \(=\) \( q + 2 q^{5} + ( -3 + \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 2 q^{5} + ( -3 + \zeta_{6} ) q^{7} -2 q^{11} + ( 3 - 3 \zeta_{6} ) q^{13} + ( 8 - 8 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} -8 q^{23} - q^{25} + 4 \zeta_{6} q^{29} -3 \zeta_{6} q^{31} + ( -6 + 2 \zeta_{6} ) q^{35} + \zeta_{6} q^{37} + ( 6 - 6 \zeta_{6} ) q^{41} -11 \zeta_{6} q^{43} + ( 6 - 6 \zeta_{6} ) q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + ( -12 + 12 \zeta_{6} ) q^{53} -4 q^{55} + 4 \zeta_{6} q^{59} + ( 6 - 6 \zeta_{6} ) q^{61} + ( 6 - 6 \zeta_{6} ) q^{65} -13 \zeta_{6} q^{67} + 10 q^{71} + ( 11 - 11 \zeta_{6} ) q^{73} + ( 6 - 2 \zeta_{6} ) q^{77} + ( 3 - 3 \zeta_{6} ) q^{79} + 2 \zeta_{6} q^{83} + ( 16 - 16 \zeta_{6} ) q^{85} + ( -6 + 9 \zeta_{6} ) q^{91} + 2 \zeta_{6} q^{95} -10 \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 5q^{7} + O(q^{10}) \) \( 2q + 4q^{5} - 5q^{7} - 4q^{11} + 3q^{13} + 8q^{17} + q^{19} - 16q^{23} - 2q^{25} + 4q^{29} - 3q^{31} - 10q^{35} + q^{37} + 6q^{41} - 11q^{43} + 6q^{47} + 11q^{49} - 12q^{53} - 8q^{55} + 4q^{59} + 6q^{61} + 6q^{65} - 13q^{67} + 20q^{71} + 11q^{73} + 10q^{77} + 3q^{79} + 2q^{83} + 16q^{85} - 3q^{91} + 2q^{95} - 10q^{97} + O(q^{100}) \)

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 2.00000 0 −2.50000 + 0.866025i 0 0 0
541.1 0 0 0 2.00000 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.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.l.g 2
3.b odd 2 1 2268.2.l.b 2
7.c even 3 1 2268.2.i.b 2
9.c even 3 1 252.2.k.a 2
9.c even 3 1 2268.2.i.b 2
9.d odd 6 1 84.2.i.a 2
9.d odd 6 1 2268.2.i.g 2
21.h odd 6 1 2268.2.i.g 2
36.f odd 6 1 1008.2.s.c 2
36.h even 6 1 336.2.q.c 2
45.h odd 6 1 2100.2.q.b 2
45.l even 12 2 2100.2.bc.a 4
63.g even 3 1 1764.2.a.h 1
63.g even 3 1 inner 2268.2.l.g 2
63.h even 3 1 252.2.k.a 2
63.i even 6 1 588.2.i.b 2
63.j odd 6 1 84.2.i.a 2
63.k odd 6 1 1764.2.a.c 1
63.l odd 6 1 1764.2.k.j 2
63.n odd 6 1 588.2.a.a 1
63.n odd 6 1 2268.2.l.b 2
63.o even 6 1 588.2.i.b 2
63.s even 6 1 588.2.a.f 1
63.t odd 6 1 1764.2.k.j 2
72.j odd 6 1 1344.2.q.b 2
72.l even 6 1 1344.2.q.n 2
252.n even 6 1 7056.2.a.o 1
252.o even 6 1 2352.2.a.o 1
252.r odd 6 1 2352.2.q.q 2
252.s odd 6 1 2352.2.q.q 2
252.u odd 6 1 1008.2.s.c 2
252.bb even 6 1 336.2.q.c 2
252.bl odd 6 1 7056.2.a.bs 1
252.bn odd 6 1 2352.2.a.k 1
315.br odd 6 1 2100.2.q.b 2
315.bv even 12 2 2100.2.bc.a 4
504.u odd 6 1 9408.2.a.bx 1
504.y even 6 1 9408.2.a.i 1
504.bi odd 6 1 1344.2.q.b 2
504.bt even 6 1 1344.2.q.n 2
504.cy even 6 1 9408.2.a.bi 1
504.db odd 6 1 9408.2.a.cx 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a 2 9.d odd 6 1
84.2.i.a 2 63.j odd 6 1
252.2.k.a 2 9.c even 3 1
252.2.k.a 2 63.h even 3 1
336.2.q.c 2 36.h even 6 1
336.2.q.c 2 252.bb even 6 1
588.2.a.a 1 63.n odd 6 1
588.2.a.f 1 63.s even 6 1
588.2.i.b 2 63.i even 6 1
588.2.i.b 2 63.o even 6 1
1008.2.s.c 2 36.f odd 6 1
1008.2.s.c 2 252.u odd 6 1
1344.2.q.b 2 72.j odd 6 1
1344.2.q.b 2 504.bi odd 6 1
1344.2.q.n 2 72.l even 6 1
1344.2.q.n 2 504.bt even 6 1
1764.2.a.c 1 63.k odd 6 1
1764.2.a.h 1 63.g even 3 1
1764.2.k.j 2 63.l odd 6 1
1764.2.k.j 2 63.t odd 6 1
2100.2.q.b 2 45.h odd 6 1
2100.2.q.b 2 315.br odd 6 1
2100.2.bc.a 4 45.l even 12 2
2100.2.bc.a 4 315.bv even 12 2
2268.2.i.b 2 7.c even 3 1
2268.2.i.b 2 9.c even 3 1
2268.2.i.g 2 9.d odd 6 1
2268.2.i.g 2 21.h odd 6 1
2268.2.l.b 2 3.b odd 2 1
2268.2.l.b 2 63.n odd 6 1
2268.2.l.g 2 1.a even 1 1 trivial
2268.2.l.g 2 63.g even 3 1 inner
2352.2.a.k 1 252.bn odd 6 1
2352.2.a.o 1 252.o even 6 1
2352.2.q.q 2 252.r odd 6 1
2352.2.q.q 2 252.s odd 6 1
7056.2.a.o 1 252.n even 6 1
7056.2.a.bs 1 252.bl odd 6 1
9408.2.a.i 1 504.y even 6 1
9408.2.a.bi 1 504.cy even 6 1
9408.2.a.bx 1 504.u odd 6 1
9408.2.a.cx 1 504.db odd 6 1

Hecke kernels

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

\( T_{5} - 2 \)
\( T_{13}^{2} - 3 T_{13} + 9 \)
\( T_{19}^{2} - T_{19} + 1 \)