Properties

 Label 1344.2.a.p Level $1344$ Weight $2$ Character orbit 1344.a Self dual yes Analytic conductor $10.732$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1344.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$10.7318940317$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 672) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 1.00000 0 0 0 −1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.2.a.p 1
3.b odd 2 1 4032.2.a.q 1
4.b odd 2 1 1344.2.a.e 1
7.b odd 2 1 9408.2.a.w 1
8.b even 2 1 672.2.a.c 1
8.d odd 2 1 672.2.a.g yes 1
12.b even 2 1 4032.2.a.x 1
16.e even 4 2 5376.2.c.y 2
16.f odd 4 2 5376.2.c.j 2
24.f even 2 1 2016.2.a.i 1
24.h odd 2 1 2016.2.a.d 1
28.d even 2 1 9408.2.a.ch 1
56.e even 2 1 4704.2.a.k 1
56.h odd 2 1 4704.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.c 1 8.b even 2 1
672.2.a.g yes 1 8.d odd 2 1
1344.2.a.e 1 4.b odd 2 1
1344.2.a.p 1 1.a even 1 1 trivial
2016.2.a.d 1 24.h odd 2 1
2016.2.a.i 1 24.f even 2 1
4032.2.a.q 1 3.b odd 2 1
4032.2.a.x 1 12.b even 2 1
4704.2.a.k 1 56.e even 2 1
4704.2.a.z 1 56.h odd 2 1
5376.2.c.j 2 16.f odd 4 2
5376.2.c.y 2 16.e even 4 2
9408.2.a.w 1 7.b odd 2 1
9408.2.a.ch 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1344))$$:

 $$T_{5}$$ $$T_{11} - 2$$ $$T_{13} - 2$$ $$T_{19} - 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$-2 + T$$
$13$ $$-2 + T$$
$17$ $$-4 + T$$
$19$ $$-4 + T$$
$23$ $$6 + T$$
$29$ $$-2 + T$$
$31$ $$T$$
$37$ $$-6 + T$$
$41$ $$-8 + T$$
$43$ $$-8 + T$$
$47$ $$4 + T$$
$53$ $$-6 + T$$
$59$ $$T$$
$61$ $$-14 + T$$
$67$ $$4 + T$$
$71$ $$2 + T$$
$73$ $$2 + T$$
$79$ $$-4 + T$$
$83$ $$12 + T$$
$89$ $$T$$
$97$ $$-6 + T$$