Properties

Label 2268.2.l.a
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 28)
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.

\(f(q)\) \(=\) \( q -3 q^{5} + ( 2 + \zeta_{6} ) q^{7} +O(q^{10})\) \( q -3 q^{5} + ( 2 + \zeta_{6} ) q^{7} + 3 q^{11} + ( -2 + 2 \zeta_{6} ) q^{13} + ( 3 - 3 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} -3 q^{23} + 4 q^{25} -6 \zeta_{6} q^{29} + 7 \zeta_{6} q^{31} + ( -6 - 3 \zeta_{6} ) q^{35} + \zeta_{6} q^{37} + ( 6 - 6 \zeta_{6} ) q^{41} + 4 \zeta_{6} q^{43} + ( -9 + 9 \zeta_{6} ) q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + ( 3 - 3 \zeta_{6} ) q^{53} -9 q^{55} + 9 \zeta_{6} q^{59} + ( 1 - \zeta_{6} ) q^{61} + ( 6 - 6 \zeta_{6} ) q^{65} + 7 \zeta_{6} q^{67} + ( 1 - \zeta_{6} ) q^{73} + ( 6 + 3 \zeta_{6} ) q^{77} + ( 13 - 13 \zeta_{6} ) q^{79} + 12 \zeta_{6} q^{83} + ( -9 + 9 \zeta_{6} ) q^{85} + 15 \zeta_{6} q^{89} + ( -6 + 4 \zeta_{6} ) q^{91} -3 \zeta_{6} q^{95} + 10 \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{5} + 5q^{7} + O(q^{10}) \) \( 2q - 6q^{5} + 5q^{7} + 6q^{11} - 2q^{13} + 3q^{17} + q^{19} - 6q^{23} + 8q^{25} - 6q^{29} + 7q^{31} - 15q^{35} + q^{37} + 6q^{41} + 4q^{43} - 9q^{47} + 11q^{49} + 3q^{53} - 18q^{55} + 9q^{59} + q^{61} + 6q^{65} + 7q^{67} + q^{73} + 15q^{77} + 13q^{79} + 12q^{83} - 9q^{85} + 15q^{89} - 8q^{91} - 3q^{95} + 10q^{97} + O(q^{100}) \)

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 −3.00000 0 2.50000 + 0.866025i 0 0 0
541.1 0 0 0 −3.00000 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.a 2
3.b odd 2 1 2268.2.l.h 2
7.c even 3 1 2268.2.i.h 2
9.c even 3 1 252.2.k.c 2
9.c even 3 1 2268.2.i.h 2
9.d odd 6 1 28.2.e.a 2
9.d odd 6 1 2268.2.i.a 2
21.h odd 6 1 2268.2.i.a 2
36.f odd 6 1 1008.2.s.p 2
36.h even 6 1 112.2.i.b 2
45.h odd 6 1 700.2.i.c 2
45.l even 12 2 700.2.r.b 4
63.g even 3 1 1764.2.a.a 1
63.g even 3 1 inner 2268.2.l.a 2
63.h even 3 1 252.2.k.c 2
63.i even 6 1 196.2.e.a 2
63.j odd 6 1 28.2.e.a 2
63.k odd 6 1 1764.2.a.j 1
63.l odd 6 1 1764.2.k.b 2
63.n odd 6 1 196.2.a.b 1
63.n odd 6 1 2268.2.l.h 2
63.o even 6 1 196.2.e.a 2
63.s even 6 1 196.2.a.a 1
63.t odd 6 1 1764.2.k.b 2
72.j odd 6 1 448.2.i.e 2
72.l even 6 1 448.2.i.c 2
252.n even 6 1 7056.2.a.bw 1
252.o even 6 1 784.2.a.d 1
252.r odd 6 1 784.2.i.d 2
252.s odd 6 1 784.2.i.d 2
252.u odd 6 1 1008.2.s.p 2
252.bb even 6 1 112.2.i.b 2
252.bl odd 6 1 7056.2.a.f 1
252.bn odd 6 1 784.2.a.g 1
315.u even 6 1 4900.2.a.n 1
315.v odd 6 1 4900.2.a.g 1
315.br odd 6 1 700.2.i.c 2
315.bv even 12 2 700.2.r.b 4
315.bw odd 12 2 4900.2.e.h 2
315.bx even 12 2 4900.2.e.i 2
504.u odd 6 1 3136.2.a.k 1
504.y even 6 1 3136.2.a.v 1
504.bi odd 6 1 448.2.i.e 2
504.bt even 6 1 448.2.i.c 2
504.cy even 6 1 3136.2.a.s 1
504.db odd 6 1 3136.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.e.a 2 9.d odd 6 1
28.2.e.a 2 63.j odd 6 1
112.2.i.b 2 36.h even 6 1
112.2.i.b 2 252.bb even 6 1
196.2.a.a 1 63.s even 6 1
196.2.a.b 1 63.n odd 6 1
196.2.e.a 2 63.i even 6 1
196.2.e.a 2 63.o even 6 1
252.2.k.c 2 9.c even 3 1
252.2.k.c 2 63.h even 3 1
448.2.i.c 2 72.l even 6 1
448.2.i.c 2 504.bt even 6 1
448.2.i.e 2 72.j odd 6 1
448.2.i.e 2 504.bi odd 6 1
700.2.i.c 2 45.h odd 6 1
700.2.i.c 2 315.br odd 6 1
700.2.r.b 4 45.l even 12 2
700.2.r.b 4 315.bv even 12 2
784.2.a.d 1 252.o even 6 1
784.2.a.g 1 252.bn odd 6 1
784.2.i.d 2 252.r odd 6 1
784.2.i.d 2 252.s odd 6 1
1008.2.s.p 2 36.f odd 6 1
1008.2.s.p 2 252.u odd 6 1
1764.2.a.a 1 63.g even 3 1
1764.2.a.j 1 63.k odd 6 1
1764.2.k.b 2 63.l odd 6 1
1764.2.k.b 2 63.t odd 6 1
2268.2.i.a 2 9.d odd 6 1
2268.2.i.a 2 21.h odd 6 1
2268.2.i.h 2 7.c even 3 1
2268.2.i.h 2 9.c even 3 1
2268.2.l.a 2 1.a even 1 1 trivial
2268.2.l.a 2 63.g even 3 1 inner
2268.2.l.h 2 3.b odd 2 1
2268.2.l.h 2 63.n odd 6 1
3136.2.a.h 1 504.db odd 6 1
3136.2.a.k 1 504.u odd 6 1
3136.2.a.s 1 504.cy even 6 1
3136.2.a.v 1 504.y even 6 1
4900.2.a.g 1 315.v odd 6 1
4900.2.a.n 1 315.u even 6 1
4900.2.e.h 2 315.bw odd 12 2
4900.2.e.i 2 315.bx even 12 2
7056.2.a.f 1 252.bl odd 6 1
7056.2.a.bw 1 252.n even 6 1

Hecke kernels

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

\( T_{5} + 3 \)
\( T_{13}^{2} + 2 T_{13} + 4 \)
\( T_{19}^{2} - T_{19} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 3 T + 5 T^{2} )^{2} \)
$7$ \( 1 - 5 T + 7 T^{2} \)
$11$ \( ( 1 - 3 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \)
$17$ \( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( ( 1 + 3 T + 23 T^{2} )^{2} \)
$29$ \( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \)
$37$ \( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \)
$41$ \( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \)
$67$ \( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \)
$83$ \( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 10 T + 3 T^{2} - 970 T^{3} + 9409 T^{4} \)
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