Properties

 Label 1344.2.q.p 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( 2 - 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( 2 - 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{11} -5 q^{13} + ( 2 - 2 \zeta_{6} ) q^{17} + 3 \zeta_{6} q^{19} + ( -1 - 2 \zeta_{6} ) q^{21} -2 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} - q^{27} -8 q^{29} + ( 1 - \zeta_{6} ) q^{31} -2 \zeta_{6} q^{33} -5 \zeta_{6} q^{37} + ( -5 + 5 \zeta_{6} ) q^{39} + 2 q^{41} -7 q^{43} + 8 \zeta_{6} q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} -2 \zeta_{6} q^{51} + ( -2 + 2 \zeta_{6} ) q^{53} + 3 q^{57} + ( 10 - 10 \zeta_{6} ) q^{59} -2 \zeta_{6} q^{61} + ( -3 + \zeta_{6} ) q^{63} + ( -11 + 11 \zeta_{6} ) q^{67} -2 q^{69} + 12 q^{71} + ( 3 - 3 \zeta_{6} ) q^{73} -5 \zeta_{6} q^{75} + ( -2 - 4 \zeta_{6} ) q^{77} -17 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 16 q^{83} + ( -8 + 8 \zeta_{6} ) q^{87} -12 \zeta_{6} q^{89} + ( -10 + 15 \zeta_{6} ) q^{91} -\zeta_{6} q^{93} -14 q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} + q^{7} - q^{9} + 2q^{11} - 10q^{13} + 2q^{17} + 3q^{19} - 4q^{21} - 2q^{23} + 5q^{25} - 2q^{27} - 16q^{29} + q^{31} - 2q^{33} - 5q^{37} - 5q^{39} + 4q^{41} - 14q^{43} + 8q^{47} - 13q^{49} - 2q^{51} - 2q^{53} + 6q^{57} + 10q^{59} - 2q^{61} - 5q^{63} - 11q^{67} - 4q^{69} + 24q^{71} + 3q^{73} - 5q^{75} - 8q^{77} - 17q^{79} - q^{81} + 32q^{83} - 8q^{87} - 12q^{89} - 5q^{91} - q^{93} - 28q^{97} - 4q^{99} + O(q^{100})$$

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 0 0.500000 2.59808i 0 −0.500000 0.866025i 0
961.1 0 0.500000 + 0.866025i 0 0 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.p 2
4.b odd 2 1 1344.2.q.f 2
7.c even 3 1 inner 1344.2.q.p 2
7.c even 3 1 9408.2.a.s 1
7.d odd 6 1 9408.2.a.cj 1
8.b even 2 1 672.2.q.c 2
8.d odd 2 1 672.2.q.i yes 2
24.f even 2 1 2016.2.s.f 2
24.h odd 2 1 2016.2.s.g 2
28.f even 6 1 9408.2.a.y 1
28.g odd 6 1 1344.2.q.f 2
28.g odd 6 1 9408.2.a.cm 1
56.j odd 6 1 4704.2.a.i 1
56.k odd 6 1 672.2.q.i yes 2
56.k odd 6 1 4704.2.a.h 1
56.m even 6 1 4704.2.a.x 1
56.p even 6 1 672.2.q.c 2
56.p even 6 1 4704.2.a.bb 1
168.s odd 6 1 2016.2.s.g 2
168.v even 6 1 2016.2.s.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.c 2 8.b even 2 1
672.2.q.c 2 56.p even 6 1
672.2.q.i yes 2 8.d odd 2 1
672.2.q.i yes 2 56.k odd 6 1
1344.2.q.f 2 4.b odd 2 1
1344.2.q.f 2 28.g odd 6 1
1344.2.q.p 2 1.a even 1 1 trivial
1344.2.q.p 2 7.c even 3 1 inner
2016.2.s.f 2 24.f even 2 1
2016.2.s.f 2 168.v even 6 1
2016.2.s.g 2 24.h odd 2 1
2016.2.s.g 2 168.s odd 6 1
4704.2.a.h 1 56.k odd 6 1
4704.2.a.i 1 56.j odd 6 1
4704.2.a.x 1 56.m even 6 1
4704.2.a.bb 1 56.p even 6 1
9408.2.a.s 1 7.c even 3 1
9408.2.a.y 1 28.f even 6 1
9408.2.a.cj 1 7.d odd 6 1
9408.2.a.cm 1 28.g odd 6 1

Hecke kernels

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

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