Properties

Label 2268.2.i.a
Level $2268$
Weight $2$
Character orbit 2268.i
Analytic conductor $18.110$
Analytic rank $1$
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.i (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 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.

\(f(q)\) \(=\) \( q + ( -3 + 3 \zeta_{6} ) q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -3 + 3 \zeta_{6} ) q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{11} -2 \zeta_{6} q^{13} + ( -3 + 3 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} + ( -3 + 3 \zeta_{6} ) q^{23} -4 \zeta_{6} q^{25} + ( 6 - 6 \zeta_{6} ) q^{29} -7 q^{31} + ( 6 + 3 \zeta_{6} ) q^{35} + \zeta_{6} q^{37} -6 \zeta_{6} q^{41} + ( 4 - 4 \zeta_{6} ) q^{43} -9 q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + ( -3 + 3 \zeta_{6} ) q^{53} -9 q^{55} + 9 q^{59} - q^{61} + 6 q^{65} -7 q^{67} + ( 1 - \zeta_{6} ) q^{73} + ( 9 - 6 \zeta_{6} ) q^{77} -13 q^{79} + ( -12 + 12 \zeta_{6} ) q^{83} -9 \zeta_{6} q^{85} -15 \zeta_{6} q^{89} + ( -6 + 4 \zeta_{6} ) q^{91} -3 q^{95} + ( 10 - 10 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{5} - q^{7} + O(q^{10}) \) \( 2q - 3q^{5} - q^{7} + 3q^{11} - 2q^{13} - 3q^{17} + q^{19} - 3q^{23} - 4q^{25} + 6q^{29} - 14q^{31} + 15q^{35} + q^{37} - 6q^{41} + 4q^{43} - 18q^{47} - 13q^{49} - 3q^{53} - 18q^{55} + 18q^{59} - 2q^{61} + 12q^{65} - 14q^{67} + q^{73} + 12q^{77} - 26q^{79} - 12q^{83} - 9q^{85} - 15q^{89} - 8q^{91} - 6q^{95} + 10q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −1.50000 + 2.59808i 0 −0.500000 2.59808i 0 0 0
2053.1 0 0 0 −1.50000 2.59808i 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
63.h 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.i.a 2
3.b odd 2 1 2268.2.i.h 2
7.c even 3 1 2268.2.l.h 2
9.c even 3 1 28.2.e.a 2
9.c even 3 1 2268.2.l.h 2
9.d odd 6 1 252.2.k.c 2
9.d odd 6 1 2268.2.l.a 2
21.h odd 6 1 2268.2.l.a 2
36.f odd 6 1 112.2.i.b 2
36.h even 6 1 1008.2.s.p 2
45.j even 6 1 700.2.i.c 2
45.k odd 12 2 700.2.r.b 4
63.g even 3 1 28.2.e.a 2
63.h even 3 1 196.2.a.b 1
63.h even 3 1 inner 2268.2.i.a 2
63.i even 6 1 1764.2.a.j 1
63.j odd 6 1 1764.2.a.a 1
63.j odd 6 1 2268.2.i.h 2
63.k odd 6 1 196.2.e.a 2
63.l odd 6 1 196.2.e.a 2
63.n odd 6 1 252.2.k.c 2
63.o even 6 1 1764.2.k.b 2
63.s even 6 1 1764.2.k.b 2
63.t odd 6 1 196.2.a.a 1
72.n even 6 1 448.2.i.e 2
72.p odd 6 1 448.2.i.c 2
252.n even 6 1 784.2.i.d 2
252.o even 6 1 1008.2.s.p 2
252.r odd 6 1 7056.2.a.bw 1
252.u odd 6 1 784.2.a.d 1
252.bb even 6 1 7056.2.a.f 1
252.bi even 6 1 784.2.i.d 2
252.bj even 6 1 784.2.a.g 1
252.bl odd 6 1 112.2.i.b 2
315.q odd 6 1 4900.2.a.n 1
315.r even 6 1 4900.2.a.g 1
315.bo even 6 1 700.2.i.c 2
315.bs even 12 2 4900.2.e.h 2
315.bt odd 12 2 4900.2.e.i 2
315.ch odd 12 2 700.2.r.b 4
504.w even 6 1 448.2.i.e 2
504.ba odd 6 1 448.2.i.c 2
504.bf even 6 1 3136.2.a.k 1
504.bp odd 6 1 3136.2.a.v 1
504.ce odd 6 1 3136.2.a.s 1
504.cq even 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.c even 3 1
28.2.e.a 2 63.g even 3 1
112.2.i.b 2 36.f odd 6 1
112.2.i.b 2 252.bl odd 6 1
196.2.a.a 1 63.t odd 6 1
196.2.a.b 1 63.h even 3 1
196.2.e.a 2 63.k odd 6 1
196.2.e.a 2 63.l odd 6 1
252.2.k.c 2 9.d odd 6 1
252.2.k.c 2 63.n odd 6 1
448.2.i.c 2 72.p odd 6 1
448.2.i.c 2 504.ba odd 6 1
448.2.i.e 2 72.n even 6 1
448.2.i.e 2 504.w even 6 1
700.2.i.c 2 45.j even 6 1
700.2.i.c 2 315.bo even 6 1
700.2.r.b 4 45.k odd 12 2
700.2.r.b 4 315.ch odd 12 2
784.2.a.d 1 252.u odd 6 1
784.2.a.g 1 252.bj even 6 1
784.2.i.d 2 252.n even 6 1
784.2.i.d 2 252.bi even 6 1
1008.2.s.p 2 36.h even 6 1
1008.2.s.p 2 252.o even 6 1
1764.2.a.a 1 63.j odd 6 1
1764.2.a.j 1 63.i even 6 1
1764.2.k.b 2 63.o even 6 1
1764.2.k.b 2 63.s even 6 1
2268.2.i.a 2 1.a even 1 1 trivial
2268.2.i.a 2 63.h even 3 1 inner
2268.2.i.h 2 3.b odd 2 1
2268.2.i.h 2 63.j odd 6 1
2268.2.l.a 2 9.d odd 6 1
2268.2.l.a 2 21.h odd 6 1
2268.2.l.h 2 7.c even 3 1
2268.2.l.h 2 9.c even 3 1
3136.2.a.h 1 504.cq even 6 1
3136.2.a.k 1 504.bf even 6 1
3136.2.a.s 1 504.ce odd 6 1
3136.2.a.v 1 504.bp odd 6 1
4900.2.a.g 1 315.r even 6 1
4900.2.a.n 1 315.q odd 6 1
4900.2.e.h 2 315.bs even 12 2
4900.2.e.i 2 315.bt odd 12 2
7056.2.a.f 1 252.bb even 6 1
7056.2.a.bw 1 252.r odd 6 1

Hecke kernels

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

\( T_{5}^{2} + 3 T_{5} + 9 \)
\( T_{13}^{2} + 2 T_{13} + 4 \)
\( T_{19}^{2} - T_{19} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( 1 + T + 7 T^{2} \)
$11$ \( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \)
$29$ \( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + 7 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \)
$41$ \( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 + 9 T + 47 T^{2} )^{2} \)
$53$ \( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \)
$59$ \( ( 1 - 9 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + T + 61 T^{2} )^{2} \)
$67$ \( ( 1 + 7 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 + 13 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 10 T + 3 T^{2} - 970 T^{3} + 9409 T^{4} \)
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