# Properties

 Label 1344.2.q.x Level $1344$ Weight $2$ Character orbit 1344.q Analytic conductor $10.732$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( 1 + \beta_{1} - \beta_{3} ) q^{7} -\beta_{2} q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{5} + ( 1 + \beta_{1} - \beta_{3} ) q^{7} -\beta_{2} q^{9} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{11} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{15} + ( 4 - 4 \beta_{2} ) q^{17} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{19} + ( 1 + \beta_{1} - \beta_{2} ) q^{21} + 4 \beta_{2} q^{23} + ( -9 - \beta_{1} + 10 \beta_{2} + 2 \beta_{3} ) q^{25} - q^{27} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{29} + ( -1 + \beta_{2} ) q^{31} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{33} + ( -11 + 3 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{35} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{37} + ( -3 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{39} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{43} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{45} -6 \beta_{2} q^{47} + ( -4 + 3 \beta_{1} - 3 \beta_{3} ) q^{49} -4 \beta_{2} q^{51} + ( 6 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -15 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{55} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{57} + ( -2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{59} + 10 \beta_{2} q^{61} + ( -\beta_{2} + \beta_{3} ) q^{63} + ( 2 - 4 \beta_{1} - 14 \beta_{2} + 2 \beta_{3} ) q^{65} + ( -3 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{67} + 4 q^{69} -2 q^{71} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{73} + ( 1 - 2 \beta_{1} + 9 \beta_{2} + \beta_{3} ) q^{75} + ( -5 + 2 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{77} + ( -2 + 4 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{79} + ( -1 + \beta_{2} ) q^{81} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{83} + ( -4 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{85} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{87} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{89} + ( 3 - 4 \beta_{1} - 11 \beta_{2} + \beta_{3} ) q^{91} + \beta_{2} q^{93} + ( -14 + 2 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} ) q^{95} + ( 13 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{97} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - q^{5} + 6q^{7} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{3} - q^{5} + 6q^{7} - 2q^{9} + q^{11} - 10q^{13} - 2q^{15} + 8q^{17} + 5q^{19} + 3q^{21} + 8q^{23} - 19q^{25} - 4q^{27} - 6q^{29} - 2q^{31} - q^{33} - 30q^{35} + 3q^{37} - 5q^{39} + 12q^{41} - 14q^{43} - q^{45} - 12q^{47} - 10q^{49} - 8q^{51} + 11q^{53} - 58q^{55} + 10q^{57} - 5q^{59} + 20q^{61} - 3q^{63} - 26q^{65} - 7q^{67} + 16q^{69} - 8q^{71} + q^{73} + 19q^{75} - 27q^{77} + 8q^{79} - 2q^{81} + 14q^{83} - 8q^{85} - 3q^{87} + 6q^{89} - 15q^{91} + 2q^{93} - 26q^{95} + 50q^{97} - 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 5$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu + 5$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{3} + 4 \beta_{1} + 5$$

## Character values

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

 $$n$$ $$127$$ $$449$$ $$577$$ $$1093$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{2}$$ $$1$$

## 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
 −1.63746 − 1.52274i 2.13746 + 0.656712i −1.63746 + 1.52274i 2.13746 − 0.656712i
0 0.500000 0.866025i 0 −2.13746 3.70219i 0 1.50000 2.17945i 0 −0.500000 0.866025i 0
193.2 0 0.500000 0.866025i 0 1.63746 + 2.83616i 0 1.50000 + 2.17945i 0 −0.500000 0.866025i 0
961.1 0 0.500000 + 0.866025i 0 −2.13746 + 3.70219i 0 1.50000 + 2.17945i 0 −0.500000 + 0.866025i 0
961.2 0 0.500000 + 0.866025i 0 1.63746 2.83616i 0 1.50000 2.17945i 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.x 4
4.b odd 2 1 1344.2.q.w 4
7.c even 3 1 inner 1344.2.q.x 4
7.c even 3 1 9408.2.a.dp 2
7.d odd 6 1 9408.2.a.dw 2
8.b even 2 1 336.2.q.g 4
8.d odd 2 1 168.2.q.c 4
24.f even 2 1 504.2.s.i 4
24.h odd 2 1 1008.2.s.r 4
28.f even 6 1 9408.2.a.dj 2
28.g odd 6 1 1344.2.q.w 4
28.g odd 6 1 9408.2.a.ec 2
56.e even 2 1 1176.2.q.l 4
56.h odd 2 1 2352.2.q.bf 4
56.j odd 6 1 2352.2.a.ba 2
56.j odd 6 1 2352.2.q.bf 4
56.k odd 6 1 168.2.q.c 4
56.k odd 6 1 1176.2.a.k 2
56.m even 6 1 1176.2.a.n 2
56.m even 6 1 1176.2.q.l 4
56.p even 6 1 336.2.q.g 4
56.p even 6 1 2352.2.a.bf 2
168.e odd 2 1 3528.2.s.bk 4
168.s odd 6 1 1008.2.s.r 4
168.s odd 6 1 7056.2.a.cu 2
168.v even 6 1 504.2.s.i 4
168.v even 6 1 3528.2.a.bk 2
168.ba even 6 1 7056.2.a.ch 2
168.be odd 6 1 3528.2.a.bd 2
168.be odd 6 1 3528.2.s.bk 4

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

 $$T_{5}^{4} + T_{5}^{3} + 15 T_{5}^{2} - 14 T_{5} + 196$$ $$T_{11}^{4} - T_{11}^{3} + 15 T_{11}^{2} + 14 T_{11} + 196$$ $$T_{13}^{2} + 5 T_{13} - 8$$