# Properties

 Label 1344.2.q.j Level $1344$ Weight $2$ Character orbit 1344.q Analytic conductor $10.732$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1344.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.7318940317$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} + 4 q^{13} -3 q^{15} + 4 \zeta_{6} q^{19} + ( 2 - 3 \zeta_{6} ) q^{21} + ( -4 + 4 \zeta_{6} ) q^{25} + q^{27} -9 q^{29} + ( -1 + \zeta_{6} ) q^{31} -3 \zeta_{6} q^{33} + ( -3 - 6 \zeta_{6} ) q^{35} + 8 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{39} -10 q^{43} + ( 3 - 3 \zeta_{6} ) q^{45} -6 \zeta_{6} q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + ( -3 + 3 \zeta_{6} ) q^{53} -9 q^{55} -4 q^{57} + ( -3 + 3 \zeta_{6} ) q^{59} -10 \zeta_{6} q^{61} + ( 1 + 2 \zeta_{6} ) q^{63} + 12 \zeta_{6} q^{65} + ( 10 - 10 \zeta_{6} ) q^{67} + 6 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} -4 \zeta_{6} q^{75} + ( 6 - 9 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -9 q^{83} + ( 9 - 9 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} + ( -12 + 4 \zeta_{6} ) q^{91} -\zeta_{6} q^{93} + ( -12 + 12 \zeta_{6} ) q^{95} - q^{97} + 3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} + 3q^{5} - 5q^{7} - q^{9} + O(q^{10})$$ $$2q - q^{3} + 3q^{5} - 5q^{7} - q^{9} - 3q^{11} + 8q^{13} - 6q^{15} + 4q^{19} + q^{21} - 4q^{25} + 2q^{27} - 18q^{29} - q^{31} - 3q^{33} - 12q^{35} + 8q^{37} - 4q^{39} - 20q^{43} + 3q^{45} - 6q^{47} + 11q^{49} - 3q^{53} - 18q^{55} - 8q^{57} - 3q^{59} - 10q^{61} + 4q^{63} + 12q^{65} + 10q^{67} + 12q^{71} - 2q^{73} - 4q^{75} + 3q^{77} - q^{79} - q^{81} - 18q^{83} + 9q^{87} - 6q^{89} - 20q^{91} - q^{93} - 12q^{95} - 2q^{97} + 6q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$449$$ $$577$$ $$1093$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 + 0.866025i 0 1.50000 + 2.59808i 0 −2.50000 + 0.866025i 0 −0.500000 0.866025i 0
961.1 0 −0.500000 0.866025i 0 1.50000 2.59808i 0 −2.50000 0.866025i 0 −0.500000 + 0.866025i 0
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.2.q.j 2
4.b odd 2 1 1344.2.q.v 2
7.c even 3 1 inner 1344.2.q.j 2
7.c even 3 1 9408.2.a.bu 1
7.d odd 6 1 9408.2.a.bm 1
8.b even 2 1 336.2.q.d 2
8.d odd 2 1 42.2.e.b 2
24.f even 2 1 126.2.g.b 2
24.h odd 2 1 1008.2.s.n 2
28.f even 6 1 9408.2.a.db 1
28.g odd 6 1 1344.2.q.v 2
28.g odd 6 1 9408.2.a.d 1
40.e odd 2 1 1050.2.i.e 2
40.k even 4 2 1050.2.o.b 4
56.e even 2 1 294.2.e.f 2
56.h odd 2 1 2352.2.q.m 2
56.j odd 6 1 2352.2.a.n 1
56.j odd 6 1 2352.2.q.m 2
56.k odd 6 1 42.2.e.b 2
56.k odd 6 1 294.2.a.d 1
56.m even 6 1 294.2.a.a 1
56.m even 6 1 294.2.e.f 2
56.p even 6 1 336.2.q.d 2
56.p even 6 1 2352.2.a.m 1
72.l even 6 1 1134.2.e.p 2
72.l even 6 1 1134.2.h.a 2
72.p odd 6 1 1134.2.e.a 2
72.p odd 6 1 1134.2.h.p 2
168.e odd 2 1 882.2.g.b 2
168.s odd 6 1 1008.2.s.n 2
168.s odd 6 1 7056.2.a.g 1
168.v even 6 1 126.2.g.b 2
168.v even 6 1 882.2.a.g 1
168.ba even 6 1 7056.2.a.bz 1
168.be odd 6 1 882.2.a.k 1
168.be odd 6 1 882.2.g.b 2
280.ba even 6 1 7350.2.a.cw 1
280.bi odd 6 1 1050.2.i.e 2
280.bi odd 6 1 7350.2.a.ce 1
280.br even 12 2 1050.2.o.b 4
504.ba odd 6 1 1134.2.e.a 2
504.bt even 6 1 1134.2.h.a 2
504.ce odd 6 1 1134.2.h.p 2
504.cy even 6 1 1134.2.e.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.b 2 8.d odd 2 1
42.2.e.b 2 56.k odd 6 1
126.2.g.b 2 24.f even 2 1
126.2.g.b 2 168.v even 6 1
294.2.a.a 1 56.m even 6 1
294.2.a.d 1 56.k odd 6 1
294.2.e.f 2 56.e even 2 1
294.2.e.f 2 56.m even 6 1
336.2.q.d 2 8.b even 2 1
336.2.q.d 2 56.p even 6 1
882.2.a.g 1 168.v even 6 1
882.2.a.k 1 168.be odd 6 1
882.2.g.b 2 168.e odd 2 1
882.2.g.b 2 168.be odd 6 1
1008.2.s.n 2 24.h odd 2 1
1008.2.s.n 2 168.s odd 6 1
1050.2.i.e 2 40.e odd 2 1
1050.2.i.e 2 280.bi odd 6 1
1050.2.o.b 4 40.k even 4 2
1050.2.o.b 4 280.br even 12 2
1134.2.e.a 2 72.p odd 6 1
1134.2.e.a 2 504.ba odd 6 1
1134.2.e.p 2 72.l even 6 1
1134.2.e.p 2 504.cy even 6 1
1134.2.h.a 2 72.l even 6 1
1134.2.h.a 2 504.bt even 6 1
1134.2.h.p 2 72.p odd 6 1
1134.2.h.p 2 504.ce odd 6 1
1344.2.q.j 2 1.a even 1 1 trivial
1344.2.q.j 2 7.c even 3 1 inner
1344.2.q.v 2 4.b odd 2 1
1344.2.q.v 2 28.g odd 6 1
2352.2.a.m 1 56.p even 6 1
2352.2.a.n 1 56.j odd 6 1
2352.2.q.m 2 56.h odd 2 1
2352.2.q.m 2 56.j odd 6 1
7056.2.a.g 1 168.s odd 6 1
7056.2.a.bz 1 168.ba even 6 1
7350.2.a.ce 1 280.bi odd 6 1
7350.2.a.cw 1 280.ba even 6 1
9408.2.a.d 1 28.g odd 6 1
9408.2.a.bm 1 7.d odd 6 1
9408.2.a.bu 1 7.c even 3 1
9408.2.a.db 1 28.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1344, [\chi])$$:

 $$T_{5}^{2} - 3 T_{5} + 9$$ $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{13} - 4$$

## Hecke characteristic polynomials

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