Properties

 Label 1344.2.q.t 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 168) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 6 - 6 \zeta_{6} ) q^{11} + 3 q^{13} + 2 q^{15} + ( -4 + 4 \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} + ( -3 + 2 \zeta_{6} ) q^{21} -4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} - q^{27} + 4 q^{29} + ( 7 - 7 \zeta_{6} ) q^{31} -6 \zeta_{6} q^{33} + ( 2 - 6 \zeta_{6} ) q^{35} -9 \zeta_{6} q^{37} + ( 3 - 3 \zeta_{6} ) q^{39} -2 q^{41} - q^{43} + ( 2 - 2 \zeta_{6} ) q^{45} + 2 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + 4 \zeta_{6} q^{51} + ( 8 - 8 \zeta_{6} ) q^{53} + 12 q^{55} + 5 q^{57} + 10 \zeta_{6} q^{61} + ( -1 + 3 \zeta_{6} ) q^{63} + 6 \zeta_{6} q^{65} + ( 15 - 15 \zeta_{6} ) q^{67} -4 q^{69} + 6 q^{71} + ( 11 - 11 \zeta_{6} ) q^{73} -\zeta_{6} q^{75} + ( -18 + 12 \zeta_{6} ) q^{77} + \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 6 q^{83} -8 q^{85} + ( 4 - 4 \zeta_{6} ) q^{87} + 8 \zeta_{6} q^{89} + ( -6 - 3 \zeta_{6} ) q^{91} -7 \zeta_{6} q^{93} + ( -10 + 10 \zeta_{6} ) q^{95} -14 q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + 2q^{5} - 5q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} + 2q^{5} - 5q^{7} - q^{9} + 6q^{11} + 6q^{13} + 4q^{15} - 4q^{17} + 5q^{19} - 4q^{21} - 4q^{23} + q^{25} - 2q^{27} + 8q^{29} + 7q^{31} - 6q^{33} - 2q^{35} - 9q^{37} + 3q^{39} - 4q^{41} - 2q^{43} + 2q^{45} + 2q^{47} + 11q^{49} + 4q^{51} + 8q^{53} + 24q^{55} + 10q^{57} + 10q^{61} + q^{63} + 6q^{65} + 15q^{67} - 8q^{69} + 12q^{71} + 11q^{73} - q^{75} - 24q^{77} + q^{79} - q^{81} + 12q^{83} - 16q^{85} + 4q^{87} + 8q^{89} - 15q^{91} - 7q^{93} - 10q^{95} - 28q^{97} - 12q^{99} + O(q^{100})$$

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 1.00000 + 1.73205i 0 −2.50000 0.866025i 0 −0.500000 0.866025i 0
961.1 0 0.500000 + 0.866025i 0 1.00000 1.73205i 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.t 2
4.b odd 2 1 1344.2.q.i 2
7.c even 3 1 inner 1344.2.q.t 2
7.c even 3 1 9408.2.a.f 1
7.d odd 6 1 9408.2.a.cs 1
8.b even 2 1 336.2.q.a 2
8.d odd 2 1 168.2.q.b 2
24.f even 2 1 504.2.s.g 2
24.h odd 2 1 1008.2.s.m 2
28.f even 6 1 9408.2.a.bk 1
28.g odd 6 1 1344.2.q.i 2
28.g odd 6 1 9408.2.a.cd 1
56.e even 2 1 1176.2.q.e 2
56.h odd 2 1 2352.2.q.v 2
56.j odd 6 1 2352.2.a.e 1
56.j odd 6 1 2352.2.q.v 2
56.k odd 6 1 168.2.q.b 2
56.k odd 6 1 1176.2.a.d 1
56.m even 6 1 1176.2.a.e 1
56.m even 6 1 1176.2.q.e 2
56.p even 6 1 336.2.q.a 2
56.p even 6 1 2352.2.a.x 1
168.e odd 2 1 3528.2.s.d 2
168.s odd 6 1 1008.2.s.m 2
168.s odd 6 1 7056.2.a.i 1
168.v even 6 1 504.2.s.g 2
168.v even 6 1 3528.2.a.f 1
168.ba even 6 1 7056.2.a.bn 1
168.be odd 6 1 3528.2.a.y 1
168.be odd 6 1 3528.2.s.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.b 2 8.d odd 2 1
168.2.q.b 2 56.k odd 6 1
336.2.q.a 2 8.b even 2 1
336.2.q.a 2 56.p even 6 1
504.2.s.g 2 24.f even 2 1
504.2.s.g 2 168.v even 6 1
1008.2.s.m 2 24.h odd 2 1
1008.2.s.m 2 168.s odd 6 1
1176.2.a.d 1 56.k odd 6 1
1176.2.a.e 1 56.m even 6 1
1176.2.q.e 2 56.e even 2 1
1176.2.q.e 2 56.m even 6 1
1344.2.q.i 2 4.b odd 2 1
1344.2.q.i 2 28.g odd 6 1
1344.2.q.t 2 1.a even 1 1 trivial
1344.2.q.t 2 7.c even 3 1 inner
2352.2.a.e 1 56.j odd 6 1
2352.2.a.x 1 56.p even 6 1
2352.2.q.v 2 56.h odd 2 1
2352.2.q.v 2 56.j odd 6 1
3528.2.a.f 1 168.v even 6 1
3528.2.a.y 1 168.be odd 6 1
3528.2.s.d 2 168.e odd 2 1
3528.2.s.d 2 168.be odd 6 1
7056.2.a.i 1 168.s odd 6 1
7056.2.a.bn 1 168.ba even 6 1
9408.2.a.f 1 7.c even 3 1
9408.2.a.bk 1 28.f even 6 1
9408.2.a.cd 1 28.g odd 6 1
9408.2.a.cs 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 $$S_{2}^{\mathrm{new}}(1344, [\chi])$$:

 $$T_{5}^{2} - 2 T_{5} + 4$$ $$T_{11}^{2} - 6 T_{11} + 36$$ $$T_{13} - 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$4 - 2 T + T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$36 - 6 T + T^{2}$$
$13$ $$( -3 + T )^{2}$$
$17$ $$16 + 4 T + T^{2}$$
$19$ $$25 - 5 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$( -4 + T )^{2}$$
$31$ $$49 - 7 T + T^{2}$$
$37$ $$81 + 9 T + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$( 1 + T )^{2}$$
$47$ $$4 - 2 T + T^{2}$$
$53$ $$64 - 8 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$100 - 10 T + T^{2}$$
$67$ $$225 - 15 T + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$121 - 11 T + T^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$64 - 8 T + T^{2}$$
$97$ $$( 14 + T )^{2}$$