Properties

 Label 2268.1.s.d Level $2268$ Weight $1$ Character orbit 2268.s Analytic conductor $1.132$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -84 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2268 = 2^{2} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2268.s (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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 756) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.756.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

$q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} -\zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} - q^{8} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} -\zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} - q^{8} + q^{10} -\zeta_{6} q^{11} + \zeta_{6}^{2} q^{14} -\zeta_{6} q^{16} + 2 q^{17} + q^{19} + \zeta_{6} q^{20} -\zeta_{6}^{2} q^{22} + \zeta_{6}^{2} q^{23} - q^{28} + \zeta_{6}^{2} q^{31} -\zeta_{6}^{2} q^{32} + 2 \zeta_{6} q^{34} + q^{35} - q^{37} + \zeta_{6} q^{38} + \zeta_{6}^{2} q^{40} -\zeta_{6}^{2} q^{41} + q^{44} - q^{46} + \zeta_{6}^{2} q^{49} - q^{55} -\zeta_{6} q^{56} - q^{62} + q^{64} + 2 \zeta_{6}^{2} q^{68} + \zeta_{6} q^{70} + q^{71} -\zeta_{6} q^{74} + \zeta_{6}^{2} q^{76} -\zeta_{6}^{2} q^{77} - q^{80} + q^{82} -2 \zeta_{6}^{2} q^{85} + \zeta_{6} q^{88} - q^{89} -\zeta_{6} q^{92} -\zeta_{6}^{2} q^{95} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} + q^{5} + q^{7} - 2q^{8} + O(q^{10})$$ $$2q + q^{2} - q^{4} + q^{5} + q^{7} - 2q^{8} + 2q^{10} - q^{11} - q^{14} - q^{16} + 4q^{17} + 2q^{19} + q^{20} + q^{22} - q^{23} - 2q^{28} - q^{31} + q^{32} + 2q^{34} + 2q^{35} - 2q^{37} + q^{38} - q^{40} + q^{41} + 2q^{44} - 2q^{46} - q^{49} - 2q^{55} - q^{56} - 2q^{62} + 2q^{64} - 2q^{68} + q^{70} + 2q^{71} - q^{74} - q^{76} + q^{77} - 2q^{80} + 2q^{82} + 2q^{85} + q^{88} - 2q^{89} - q^{92} + q^{95} - 2q^{98} + O(q^{100})$$

Character values

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

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
755.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 + 0.866025i 0 0.500000 0.866025i −1.00000 0 1.00000
1511.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 0.866025i 0 0.500000 + 0.866025i −1.00000 0 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
84.h odd 2 1 CM by $$\Q(\sqrt{-21})$$
9.c even 3 1 inner
252.s 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.s.d 2
3.b odd 2 1 2268.1.s.a 2
4.b odd 2 1 2268.1.s.b 2
7.b odd 2 1 2268.1.s.c 2
9.c even 3 1 756.1.h.a 1
9.c even 3 1 inner 2268.1.s.d 2
9.d odd 6 1 756.1.h.d yes 1
9.d odd 6 1 2268.1.s.a 2
12.b even 2 1 2268.1.s.c 2
21.c even 2 1 2268.1.s.b 2
28.d even 2 1 2268.1.s.a 2
36.f odd 6 1 756.1.h.c yes 1
36.f odd 6 1 2268.1.s.b 2
36.h even 6 1 756.1.h.b yes 1
36.h even 6 1 2268.1.s.c 2
63.l odd 6 1 756.1.h.b yes 1
63.l odd 6 1 2268.1.s.c 2
63.o even 6 1 756.1.h.c yes 1
63.o even 6 1 2268.1.s.b 2
84.h odd 2 1 CM 2268.1.s.d 2
252.s odd 6 1 756.1.h.a 1
252.s odd 6 1 inner 2268.1.s.d 2
252.bi even 6 1 756.1.h.d yes 1
252.bi even 6 1 2268.1.s.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.1.h.a 1 9.c even 3 1
756.1.h.a 1 252.s odd 6 1
756.1.h.b yes 1 36.h even 6 1
756.1.h.b yes 1 63.l odd 6 1
756.1.h.c yes 1 36.f odd 6 1
756.1.h.c yes 1 63.o even 6 1
756.1.h.d yes 1 9.d odd 6 1
756.1.h.d yes 1 252.bi even 6 1
2268.1.s.a 2 3.b odd 2 1
2268.1.s.a 2 9.d odd 6 1
2268.1.s.a 2 28.d even 2 1
2268.1.s.a 2 252.bi even 6 1
2268.1.s.b 2 4.b odd 2 1
2268.1.s.b 2 21.c even 2 1
2268.1.s.b 2 36.f odd 6 1
2268.1.s.b 2 63.o even 6 1
2268.1.s.c 2 7.b odd 2 1
2268.1.s.c 2 12.b even 2 1
2268.1.s.c 2 36.h even 6 1
2268.1.s.c 2 63.l odd 6 1
2268.1.s.d 2 1.a even 1 1 trivial
2268.1.s.d 2 9.c even 3 1 inner
2268.1.s.d 2 84.h odd 2 1 CM
2268.1.s.d 2 252.s odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2268, [\chi])$$:

 $$T_{5}^{2} - T_{5} + 1$$ $$T_{11}^{2} + T_{11} + 1$$ $$T_{67}$$

Hecke characteristic polynomials

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