# Properties

 Label 1344.2.q.e 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: $$x^{2} - x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} + ( - \zeta_{6} - 2) q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^3 + (-z - 2) * q^7 - z * q^9 $$q + (\zeta_{6} - 1) q^{3} + ( - \zeta_{6} - 2) q^{7} - \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{11} - q^{13} + ( - 2 \zeta_{6} + 2) q^{17} + 5 \zeta_{6} q^{19} + ( - 2 \zeta_{6} + 3) q^{21} - 6 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} + q^{27} + 8 q^{29} + ( - 3 \zeta_{6} + 3) q^{31} - 2 \zeta_{6} q^{33} - 9 \zeta_{6} q^{37} + ( - \zeta_{6} + 1) q^{39} + 2 q^{41} - q^{43} - 8 \zeta_{6} q^{47} + (5 \zeta_{6} + 3) q^{49} + 2 \zeta_{6} q^{51} + ( - 6 \zeta_{6} + 6) q^{53} - 5 q^{57} + ( - 6 \zeta_{6} + 6) q^{59} - 2 \zeta_{6} q^{61} + (3 \zeta_{6} - 1) q^{63} + (5 \zeta_{6} - 5) q^{67} + 6 q^{69} + 4 q^{71} + ( - 11 \zeta_{6} + 11) q^{73} + 5 \zeta_{6} q^{75} + ( - 4 \zeta_{6} + 6) q^{77} + 5 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + (8 \zeta_{6} - 8) q^{87} - 12 \zeta_{6} q^{89} + (\zeta_{6} + 2) q^{91} + 3 \zeta_{6} q^{93} + 18 q^{97} + 2 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 + (-z - 2) * q^7 - z * q^9 + (2*z - 2) * q^11 - q^13 + (-2*z + 2) * q^17 + 5*z * q^19 + (-2*z + 3) * q^21 - 6*z * q^23 + (-5*z + 5) * q^25 + q^27 + 8 * q^29 + (-3*z + 3) * q^31 - 2*z * q^33 - 9*z * q^37 + (-z + 1) * q^39 + 2 * q^41 - q^43 - 8*z * q^47 + (5*z + 3) * q^49 + 2*z * q^51 + (-6*z + 6) * q^53 - 5 * q^57 + (-6*z + 6) * q^59 - 2*z * q^61 + (3*z - 1) * q^63 + (5*z - 5) * q^67 + 6 * q^69 + 4 * q^71 + (-11*z + 11) * q^73 + 5*z * q^75 + (-4*z + 6) * q^77 + 5*z * q^79 + (z - 1) * q^81 + (8*z - 8) * q^87 - 12*z * q^89 + (z + 2) * q^91 + 3*z * q^93 + 18 * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 5 q^{7} - q^{9}+O(q^{10})$$ 2 * q - q^3 - 5 * q^7 - q^9 $$2 q - q^{3} - 5 q^{7} - q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{17} + 5 q^{19} + 4 q^{21} - 6 q^{23} + 5 q^{25} + 2 q^{27} + 16 q^{29} + 3 q^{31} - 2 q^{33} - 9 q^{37} + q^{39} + 4 q^{41} - 2 q^{43} - 8 q^{47} + 11 q^{49} + 2 q^{51} + 6 q^{53} - 10 q^{57} + 6 q^{59} - 2 q^{61} + q^{63} - 5 q^{67} + 12 q^{69} + 8 q^{71} + 11 q^{73} + 5 q^{75} + 8 q^{77} + 5 q^{79} - q^{81} - 8 q^{87} - 12 q^{89} + 5 q^{91} + 3 q^{93} + 36 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q - q^3 - 5 * q^7 - q^9 - 2 * q^11 - 2 * q^13 + 2 * q^17 + 5 * q^19 + 4 * q^21 - 6 * q^23 + 5 * q^25 + 2 * q^27 + 16 * q^29 + 3 * q^31 - 2 * q^33 - 9 * q^37 + q^39 + 4 * q^41 - 2 * q^43 - 8 * q^47 + 11 * q^49 + 2 * q^51 + 6 * q^53 - 10 * q^57 + 6 * q^59 - 2 * q^61 + q^63 - 5 * q^67 + 12 * q^69 + 8 * q^71 + 11 * q^73 + 5 * q^75 + 8 * q^77 + 5 * q^79 - q^81 - 8 * q^87 - 12 * q^89 + 5 * q^91 + 3 * q^93 + 36 * q^97 + 4 * q^99

## 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 −2.50000 0.866025i 0 −0.500000 0.866025i 0
961.1 0 −0.500000 0.866025i 0 0 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.e 2
4.b odd 2 1 1344.2.q.q 2
7.c even 3 1 inner 1344.2.q.e 2
7.c even 3 1 9408.2.a.cn 1
7.d odd 6 1 9408.2.a.x 1
8.b even 2 1 672.2.q.h yes 2
8.d odd 2 1 672.2.q.d 2
24.f even 2 1 2016.2.s.h 2
24.h odd 2 1 2016.2.s.e 2
28.f even 6 1 9408.2.a.ci 1
28.g odd 6 1 1344.2.q.q 2
28.g odd 6 1 9408.2.a.t 1
56.j odd 6 1 4704.2.a.y 1
56.k odd 6 1 672.2.q.d 2
56.k odd 6 1 4704.2.a.ba 1
56.m even 6 1 4704.2.a.j 1
56.p even 6 1 672.2.q.h yes 2
56.p even 6 1 4704.2.a.g 1
168.s odd 6 1 2016.2.s.e 2
168.v even 6 1 2016.2.s.h 2

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

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{11}^{2} + 2T_{11} + 4$$ T11^2 + 2*T11 + 4 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 5T + 7$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2} - 5T + 25$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$(T - 8)^{2}$$
$31$ $$T^{2} - 3T + 9$$
$37$ $$T^{2} + 9T + 81$$
$41$ $$(T - 2)^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} + 8T + 64$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} - 6T + 36$$
$61$ $$T^{2} + 2T + 4$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$(T - 4)^{2}$$
$73$ $$T^{2} - 11T + 121$$
$79$ $$T^{2} - 5T + 25$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 12T + 144$$
$97$ $$(T - 18)^{2}$$