Properties

 Label 1344.2.q.h 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 672) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 3 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 3 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{11} - q^{15} + ( 8 - 8 \zeta_{6} ) q^{17} -4 \zeta_{6} q^{19} + ( -2 + 3 \zeta_{6} ) q^{21} -4 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + q^{27} + 5 q^{29} + ( -7 + 7 \zeta_{6} ) q^{31} + \zeta_{6} q^{33} + ( 1 + 2 \zeta_{6} ) q^{35} + 8 \zeta_{6} q^{37} + 4 q^{41} -10 q^{43} + ( 1 - \zeta_{6} ) q^{45} -6 \zeta_{6} q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + 8 \zeta_{6} q^{51} + ( -1 + \zeta_{6} ) q^{53} + q^{55} + 4 q^{57} + ( 9 - 9 \zeta_{6} ) q^{59} -2 \zeta_{6} q^{61} + ( -1 - 2 \zeta_{6} ) q^{63} + ( 2 - 2 \zeta_{6} ) q^{67} + 4 q^{69} + 6 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} + 4 \zeta_{6} q^{75} + ( 2 - 3 \zeta_{6} ) q^{77} + 9 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 3 q^{83} + 8 q^{85} + ( -5 + 5 \zeta_{6} ) q^{87} + 6 \zeta_{6} q^{89} -7 \zeta_{6} q^{93} + ( 4 - 4 \zeta_{6} ) q^{95} - q^{97} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} + q^{5} + 5q^{7} - q^{9} + O(q^{10})$$ $$2q - q^{3} + q^{5} + 5q^{7} - q^{9} + q^{11} - 2q^{15} + 8q^{17} - 4q^{19} - q^{21} - 4q^{23} + 4q^{25} + 2q^{27} + 10q^{29} - 7q^{31} + q^{33} + 4q^{35} + 8q^{37} + 8q^{41} - 20q^{43} + q^{45} - 6q^{47} + 11q^{49} + 8q^{51} - q^{53} + 2q^{55} + 8q^{57} + 9q^{59} - 2q^{61} - 4q^{63} + 2q^{67} + 8q^{69} + 12q^{71} - 2q^{73} + 4q^{75} + q^{77} + 9q^{79} - q^{81} + 6q^{83} + 16q^{85} - 5q^{87} + 6q^{89} - 7q^{93} + 4q^{95} - 2q^{97} - 2q^{99} + O(q^{100})$$

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 2.50000 0.866025i 0 −0.500000 0.866025i 0
961.1 0 −0.500000 0.866025i 0 0.500000 0.866025i 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.h 2
4.b odd 2 1 1344.2.q.r 2
7.c even 3 1 inner 1344.2.q.h 2
7.c even 3 1 9408.2.a.cg 1
7.d odd 6 1 9408.2.a.bb 1
8.b even 2 1 672.2.q.g yes 2
8.d odd 2 1 672.2.q.b 2
24.f even 2 1 2016.2.s.i 2
24.h odd 2 1 2016.2.s.j 2
28.f even 6 1 9408.2.a.cp 1
28.g odd 6 1 1344.2.q.r 2
28.g odd 6 1 9408.2.a.o 1
56.j odd 6 1 4704.2.a.w 1
56.k odd 6 1 672.2.q.b 2
56.k odd 6 1 4704.2.a.bc 1
56.m even 6 1 4704.2.a.f 1
56.p even 6 1 672.2.q.g yes 2
56.p even 6 1 4704.2.a.l 1
168.s odd 6 1 2016.2.s.j 2
168.v even 6 1 2016.2.s.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.b 2 8.d odd 2 1
672.2.q.b 2 56.k odd 6 1
672.2.q.g yes 2 8.b even 2 1
672.2.q.g yes 2 56.p even 6 1
1344.2.q.h 2 1.a even 1 1 trivial
1344.2.q.h 2 7.c even 3 1 inner
1344.2.q.r 2 4.b odd 2 1
1344.2.q.r 2 28.g odd 6 1
2016.2.s.i 2 24.f even 2 1
2016.2.s.i 2 168.v even 6 1
2016.2.s.j 2 24.h odd 2 1
2016.2.s.j 2 168.s odd 6 1
4704.2.a.f 1 56.m even 6 1
4704.2.a.l 1 56.p even 6 1
4704.2.a.w 1 56.j odd 6 1
4704.2.a.bc 1 56.k odd 6 1
9408.2.a.o 1 28.g odd 6 1
9408.2.a.bb 1 7.d odd 6 1
9408.2.a.cg 1 7.c even 3 1
9408.2.a.cp 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 $$S_{2}^{\mathrm{new}}(1344, [\chi])$$:

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