# Properties

 Label 1344.2.q.s 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -5 + 5 \zeta_{6} ) q^{11} + q^{15} + ( 4 - 4 \zeta_{6} ) q^{17} -8 \zeta_{6} q^{19} + ( -2 - \zeta_{6} ) q^{21} -4 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} - q^{27} + 5 q^{29} + ( 3 - 3 \zeta_{6} ) q^{31} + 5 \zeta_{6} q^{33} + ( 3 - 2 \zeta_{6} ) q^{35} -4 \zeta_{6} q^{37} + 2 q^{43} + ( 1 - \zeta_{6} ) q^{45} -6 \zeta_{6} q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} -4 \zeta_{6} q^{51} + ( -9 + 9 \zeta_{6} ) q^{53} -5 q^{55} -8 q^{57} + ( 11 - 11 \zeta_{6} ) q^{59} -6 \zeta_{6} q^{61} + ( -3 + 2 \zeta_{6} ) q^{63} + ( 2 - 2 \zeta_{6} ) q^{67} -4 q^{69} -2 q^{71} + ( -10 + 10 \zeta_{6} ) q^{73} -4 \zeta_{6} q^{75} + ( 10 + 5 \zeta_{6} ) q^{77} + 3 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -7 q^{83} + 4 q^{85} + ( 5 - 5 \zeta_{6} ) q^{87} + 6 \zeta_{6} q^{89} -3 \zeta_{6} q^{93} + ( 8 - 8 \zeta_{6} ) q^{95} + 7 q^{97} + 5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + q^{5} - q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} + q^{5} - q^{7} - q^{9} - 5q^{11} + 2q^{15} + 4q^{17} - 8q^{19} - 5q^{21} - 4q^{23} + 4q^{25} - 2q^{27} + 10q^{29} + 3q^{31} + 5q^{33} + 4q^{35} - 4q^{37} + 4q^{43} + q^{45} - 6q^{47} - 13q^{49} - 4q^{51} - 9q^{53} - 10q^{55} - 16q^{57} + 11q^{59} - 6q^{61} - 4q^{63} + 2q^{67} - 8q^{69} - 4q^{71} - 10q^{73} - 4q^{75} + 25q^{77} + 3q^{79} - q^{81} - 14q^{83} + 8q^{85} + 5q^{87} + 6q^{89} - 3q^{93} + 8q^{95} + 14q^{97} + 10q^{99} + O(q^{100})$$

## 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 0.500000 + 0.866025i 0 −0.500000 2.59808i 0 −0.500000 0.866025i 0
961.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 −0.500000 + 2.59808i 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.s 2
4.b odd 2 1 1344.2.q.g 2
7.c even 3 1 inner 1344.2.q.s 2
7.c even 3 1 9408.2.a.q 1
7.d odd 6 1 9408.2.a.cr 1
8.b even 2 1 336.2.q.b 2
8.d odd 2 1 42.2.e.a 2
24.f even 2 1 126.2.g.c 2
24.h odd 2 1 1008.2.s.k 2
28.f even 6 1 9408.2.a.z 1
28.g odd 6 1 1344.2.q.g 2
28.g odd 6 1 9408.2.a.ce 1
40.e odd 2 1 1050.2.i.l 2
40.k even 4 2 1050.2.o.a 4
56.e even 2 1 294.2.e.b 2
56.h odd 2 1 2352.2.q.u 2
56.j odd 6 1 2352.2.a.f 1
56.j odd 6 1 2352.2.q.u 2
56.k odd 6 1 42.2.e.a 2
56.k odd 6 1 294.2.a.e 1
56.m even 6 1 294.2.a.f 1
56.m even 6 1 294.2.e.b 2
56.p even 6 1 336.2.q.b 2
56.p even 6 1 2352.2.a.t 1
72.l even 6 1 1134.2.e.e 2
72.l even 6 1 1134.2.h.l 2
72.p odd 6 1 1134.2.e.l 2
72.p odd 6 1 1134.2.h.e 2
168.e odd 2 1 882.2.g.i 2
168.s odd 6 1 1008.2.s.k 2
168.s odd 6 1 7056.2.a.w 1
168.v even 6 1 126.2.g.c 2
168.v even 6 1 882.2.a.c 1
168.ba even 6 1 7056.2.a.bl 1
168.be odd 6 1 882.2.a.d 1
168.be odd 6 1 882.2.g.i 2
280.ba even 6 1 7350.2.a.q 1
280.bi odd 6 1 1050.2.i.l 2
280.bi odd 6 1 7350.2.a.bl 1
280.br even 12 2 1050.2.o.a 4
504.ba odd 6 1 1134.2.e.l 2
504.bt even 6 1 1134.2.h.l 2
504.ce odd 6 1 1134.2.h.e 2
504.cy even 6 1 1134.2.e.e 2

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

## Hecke kernels

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

 $$T_{5}^{2} - T_{5} + 1$$ $$T_{11}^{2} + 5 T_{11} + 25$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$25 + 5 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$16 - 4 T + T^{2}$$
$19$ $$64 + 8 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$( -5 + T )^{2}$$
$31$ $$9 - 3 T + T^{2}$$
$37$ $$16 + 4 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$81 + 9 T + T^{2}$$
$59$ $$121 - 11 T + T^{2}$$
$61$ $$36 + 6 T + T^{2}$$
$67$ $$4 - 2 T + T^{2}$$
$71$ $$( 2 + T )^{2}$$
$73$ $$100 + 10 T + T^{2}$$
$79$ $$9 - 3 T + T^{2}$$
$83$ $$( 7 + T )^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$( -7 + T )^{2}$$