Properties

Label 2268.1.m.a
Level $2268$
Weight $1$
Character orbit 2268.m
Analytic conductor $1.132$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2268.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.13187944865\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
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)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.588.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.15431472.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{7} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{7} -\zeta_{6}^{2} q^{13} + \zeta_{6} q^{19} + q^{25} + \zeta_{6} q^{31} + \zeta_{6} q^{37} + \zeta_{6} q^{43} -\zeta_{6} q^{49} + 2 \zeta_{6}^{2} q^{61} + \zeta_{6} q^{67} -\zeta_{6}^{2} q^{73} -\zeta_{6}^{2} q^{79} + \zeta_{6} q^{91} -2 \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{7} + O(q^{10}) \) \( 2q - q^{7} + q^{13} + q^{19} + 2q^{25} + q^{31} + q^{37} + q^{43} - q^{49} - 2q^{61} + q^{67} + q^{73} + q^{79} + q^{91} - 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(-\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 −0.500000 + 0.866025i 0 0 0
1241.1 0 0 0 0 0 −0.500000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
63.g even 3 1 inner
63.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.1.m.a 2
3.b odd 2 1 CM 2268.1.m.a 2
7.c even 3 1 2268.1.bh.b 2
9.c even 3 1 84.1.p.a 2
9.c even 3 1 2268.1.bh.b 2
9.d odd 6 1 84.1.p.a 2
9.d odd 6 1 2268.1.bh.b 2
21.h odd 6 1 2268.1.bh.b 2
36.f odd 6 1 336.1.bn.a 2
36.h even 6 1 336.1.bn.a 2
45.h odd 6 1 2100.1.bn.c 2
45.j even 6 1 2100.1.bn.c 2
45.k odd 12 2 2100.1.bh.a 4
45.l even 12 2 2100.1.bh.a 4
63.g even 3 1 588.1.c.b 1
63.g even 3 1 inner 2268.1.m.a 2
63.h even 3 1 84.1.p.a 2
63.i even 6 1 588.1.p.a 2
63.j odd 6 1 84.1.p.a 2
63.k odd 6 1 588.1.c.a 1
63.l odd 6 1 588.1.p.a 2
63.n odd 6 1 588.1.c.b 1
63.n odd 6 1 inner 2268.1.m.a 2
63.o even 6 1 588.1.p.a 2
63.s even 6 1 588.1.c.a 1
63.t odd 6 1 588.1.p.a 2
72.j odd 6 1 1344.1.bn.b 2
72.l even 6 1 1344.1.bn.a 2
72.n even 6 1 1344.1.bn.b 2
72.p odd 6 1 1344.1.bn.a 2
252.n even 6 1 2352.1.d.b 1
252.o even 6 1 2352.1.d.a 1
252.r odd 6 1 2352.1.bn.a 2
252.s odd 6 1 2352.1.bn.a 2
252.u odd 6 1 336.1.bn.a 2
252.bb even 6 1 336.1.bn.a 2
252.bi even 6 1 2352.1.bn.a 2
252.bj even 6 1 2352.1.bn.a 2
252.bl odd 6 1 2352.1.d.a 1
252.bn odd 6 1 2352.1.d.b 1
315.r even 6 1 2100.1.bn.c 2
315.br odd 6 1 2100.1.bn.c 2
315.bt odd 12 2 2100.1.bh.a 4
315.bv even 12 2 2100.1.bh.a 4
504.bi odd 6 1 1344.1.bn.b 2
504.bt even 6 1 1344.1.bn.a 2
504.ce odd 6 1 1344.1.bn.a 2
504.cq even 6 1 1344.1.bn.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.1.p.a 2 9.c even 3 1
84.1.p.a 2 9.d odd 6 1
84.1.p.a 2 63.h even 3 1
84.1.p.a 2 63.j odd 6 1
336.1.bn.a 2 36.f odd 6 1
336.1.bn.a 2 36.h even 6 1
336.1.bn.a 2 252.u odd 6 1
336.1.bn.a 2 252.bb even 6 1
588.1.c.a 1 63.k odd 6 1
588.1.c.a 1 63.s even 6 1
588.1.c.b 1 63.g even 3 1
588.1.c.b 1 63.n odd 6 1
588.1.p.a 2 63.i even 6 1
588.1.p.a 2 63.l odd 6 1
588.1.p.a 2 63.o even 6 1
588.1.p.a 2 63.t odd 6 1
1344.1.bn.a 2 72.l even 6 1
1344.1.bn.a 2 72.p odd 6 1
1344.1.bn.a 2 504.bt even 6 1
1344.1.bn.a 2 504.ce odd 6 1
1344.1.bn.b 2 72.j odd 6 1
1344.1.bn.b 2 72.n even 6 1
1344.1.bn.b 2 504.bi odd 6 1
1344.1.bn.b 2 504.cq even 6 1
2100.1.bh.a 4 45.k odd 12 2
2100.1.bh.a 4 45.l even 12 2
2100.1.bh.a 4 315.bt odd 12 2
2100.1.bh.a 4 315.bv even 12 2
2100.1.bn.c 2 45.h odd 6 1
2100.1.bn.c 2 45.j even 6 1
2100.1.bn.c 2 315.r even 6 1
2100.1.bn.c 2 315.br odd 6 1
2268.1.m.a 2 1.a even 1 1 trivial
2268.1.m.a 2 3.b odd 2 1 CM
2268.1.m.a 2 63.g even 3 1 inner
2268.1.m.a 2 63.n odd 6 1 inner
2268.1.bh.b 2 7.c even 3 1
2268.1.bh.b 2 9.c even 3 1
2268.1.bh.b 2 9.d odd 6 1
2268.1.bh.b 2 21.h odd 6 1
2352.1.d.a 1 252.o even 6 1
2352.1.d.a 1 252.bl odd 6 1
2352.1.d.b 1 252.n even 6 1
2352.1.d.b 1 252.bn odd 6 1
2352.1.bn.a 2 252.r odd 6 1
2352.1.bn.a 2 252.s odd 6 1
2352.1.bn.a 2 252.bi even 6 1
2352.1.bn.a 2 252.bj even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} - T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2268, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 1 - T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 1 - T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 1 - T + T^{2} \)
$37$ \( 1 - T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 4 + 2 T + T^{2} \)
$67$ \( 1 - T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 1 - T + T^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 4 + 2 T + T^{2} \)
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