Properties

Label 2268.2.i.b
Level $2268$
Weight $2$
Character orbit 2268.i
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.i (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: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q + ( -2 + 2 \zeta_{6} ) q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -2 + 2 \zeta_{6} ) q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{11} + 3 \zeta_{6} q^{13} + ( 8 - 8 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} + ( 8 - 8 \zeta_{6} ) q^{23} + \zeta_{6} q^{25} + ( 4 - 4 \zeta_{6} ) q^{29} + 3 q^{31} + ( -6 + 2 \zeta_{6} ) q^{35} + \zeta_{6} q^{37} + 6 \zeta_{6} q^{41} + ( -11 + 11 \zeta_{6} ) q^{43} -6 q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( -12 + 12 \zeta_{6} ) q^{53} -4 q^{55} -4 q^{59} -6 q^{61} -6 q^{65} + 13 q^{67} + 10 q^{71} + ( 11 - 11 \zeta_{6} ) q^{73} + ( -4 + 6 \zeta_{6} ) q^{77} -3 q^{79} + ( 2 - 2 \zeta_{6} ) q^{83} + 16 \zeta_{6} q^{85} + ( -6 + 9 \zeta_{6} ) q^{91} -2 q^{95} + ( -10 + 10 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 4 q^{7} + O(q^{10}) \) \( 2 q - 2 q^{5} + 4 q^{7} + 2 q^{11} + 3 q^{13} + 8 q^{17} + q^{19} + 8 q^{23} + q^{25} + 4 q^{29} + 6 q^{31} - 10 q^{35} + q^{37} + 6 q^{41} - 11 q^{43} - 12 q^{47} + 2 q^{49} - 12 q^{53} - 8 q^{55} - 8 q^{59} - 12 q^{61} - 12 q^{65} + 26 q^{67} + 20 q^{71} + 11 q^{73} - 2 q^{77} - 6 q^{79} + 2 q^{83} + 16 q^{85} - 3 q^{91} - 4 q^{95} - 10 q^{97} + O(q^{100}) \)

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 −1.00000 + 1.73205i 0 2.00000 + 1.73205i 0 0 0
2053.1 0 0 0 −1.00000 1.73205i 0 2.00000 1.73205i 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.b 2
3.b odd 2 1 2268.2.i.g 2
7.c even 3 1 2268.2.l.g 2
9.c even 3 1 252.2.k.a 2
9.c even 3 1 2268.2.l.g 2
9.d odd 6 1 84.2.i.a 2
9.d odd 6 1 2268.2.l.b 2
21.h odd 6 1 2268.2.l.b 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 1 252.2.k.a 2
63.h even 3 1 1764.2.a.h 1
63.h even 3 1 inner 2268.2.i.b 2
63.i even 6 1 588.2.a.f 1
63.j odd 6 1 588.2.a.a 1
63.j odd 6 1 2268.2.i.g 2
63.k odd 6 1 1764.2.k.j 2
63.l odd 6 1 1764.2.k.j 2
63.n odd 6 1 84.2.i.a 2
63.o even 6 1 588.2.i.b 2
63.s even 6 1 588.2.i.b 2
63.t odd 6 1 1764.2.a.c 1
72.j odd 6 1 1344.2.q.b 2
72.l even 6 1 1344.2.q.n 2
252.o even 6 1 336.2.q.c 2
252.r odd 6 1 2352.2.a.k 1
252.s odd 6 1 2352.2.q.q 2
252.u odd 6 1 7056.2.a.bs 1
252.bb even 6 1 2352.2.a.o 1
252.bj even 6 1 7056.2.a.o 1
252.bl odd 6 1 1008.2.s.c 2
252.bn odd 6 1 2352.2.q.q 2
315.v odd 6 1 2100.2.q.b 2
315.bx even 12 2 2100.2.bc.a 4
504.bi odd 6 1 9408.2.a.cx 1
504.bt even 6 1 9408.2.a.bi 1
504.ca even 6 1 9408.2.a.i 1
504.cm odd 6 1 9408.2.a.bx 1
504.cy even 6 1 1344.2.q.n 2
504.db odd 6 1 1344.2.q.b 2
    
        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.n odd 6 1
252.2.k.a 2 9.c even 3 1
252.2.k.a 2 63.g even 3 1
336.2.q.c 2 36.h even 6 1
336.2.q.c 2 252.o even 6 1
588.2.a.a 1 63.j odd 6 1
588.2.a.f 1 63.i even 6 1
588.2.i.b 2 63.o even 6 1
588.2.i.b 2 63.s even 6 1
1008.2.s.c 2 36.f odd 6 1
1008.2.s.c 2 252.bl odd 6 1
1344.2.q.b 2 72.j odd 6 1
1344.2.q.b 2 504.db odd 6 1
1344.2.q.n 2 72.l even 6 1
1344.2.q.n 2 504.cy even 6 1
1764.2.a.c 1 63.t odd 6 1
1764.2.a.h 1 63.h even 3 1
1764.2.k.j 2 63.k odd 6 1
1764.2.k.j 2 63.l odd 6 1
2100.2.q.b 2 45.h odd 6 1
2100.2.q.b 2 315.v odd 6 1
2100.2.bc.a 4 45.l even 12 2
2100.2.bc.a 4 315.bx even 12 2
2268.2.i.b 2 1.a even 1 1 trivial
2268.2.i.b 2 63.h even 3 1 inner
2268.2.i.g 2 3.b odd 2 1
2268.2.i.g 2 63.j odd 6 1
2268.2.l.b 2 9.d odd 6 1
2268.2.l.b 2 21.h odd 6 1
2268.2.l.g 2 7.c even 3 1
2268.2.l.g 2 9.c even 3 1
2352.2.a.k 1 252.r odd 6 1
2352.2.a.o 1 252.bb even 6 1
2352.2.q.q 2 252.s odd 6 1
2352.2.q.q 2 252.bn odd 6 1
7056.2.a.o 1 252.bj even 6 1
7056.2.a.bs 1 252.u odd 6 1
9408.2.a.i 1 504.ca even 6 1
9408.2.a.bi 1 504.bt even 6 1
9408.2.a.bx 1 504.cm odd 6 1
9408.2.a.cx 1 504.bi odd 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 + 2 T + T^{2} \)
$7$ \( 7 - 4 T + T^{2} \)
$11$ \( 4 - 2 T + T^{2} \)
$13$ \( 9 - 3 T + T^{2} \)
$17$ \( 64 - 8 T + T^{2} \)
$19$ \( 1 - T + T^{2} \)
$23$ \( 64 - 8 T + T^{2} \)
$29$ \( 16 - 4 T + T^{2} \)
$31$ \( ( -3 + T )^{2} \)
$37$ \( 1 - T + T^{2} \)
$41$ \( 36 - 6 T + T^{2} \)
$43$ \( 121 + 11 T + T^{2} \)
$47$ \( ( 6 + T )^{2} \)
$53$ \( 144 + 12 T + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( 6 + T )^{2} \)
$67$ \( ( -13 + T )^{2} \)
$71$ \( ( -10 + T )^{2} \)
$73$ \( 121 - 11 T + T^{2} \)
$79$ \( ( 3 + T )^{2} \)
$83$ \( 4 - 2 T + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 100 + 10 T + T^{2} \)
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