# Properties

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

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## 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 42) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 2.00000 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.q 1
3.b odd 2 1 4032.2.a.e 1
4.b odd 2 1 1344.2.a.i 1
7.b odd 2 1 9408.2.a.n 1
8.b even 2 1 42.2.a.a 1
8.d odd 2 1 336.2.a.d 1
12.b even 2 1 4032.2.a.m 1
16.e even 4 2 5376.2.c.bc 2
16.f odd 4 2 5376.2.c.e 2
24.f even 2 1 1008.2.a.j 1
24.h odd 2 1 126.2.a.a 1
28.d even 2 1 9408.2.a.bw 1
40.e odd 2 1 8400.2.a.k 1
40.f even 2 1 1050.2.a.i 1
40.i odd 4 2 1050.2.g.a 2
56.e even 2 1 2352.2.a.l 1
56.h odd 2 1 294.2.a.g 1
56.j odd 6 2 294.2.e.a 2
56.k odd 6 2 2352.2.q.i 2
56.m even 6 2 2352.2.q.n 2
56.p even 6 2 294.2.e.c 2
72.j odd 6 2 1134.2.f.j 2
72.n even 6 2 1134.2.f.g 2
88.b odd 2 1 5082.2.a.d 1
104.e even 2 1 7098.2.a.f 1
120.i odd 2 1 3150.2.a.bo 1
120.w even 4 2 3150.2.g.r 2
168.e odd 2 1 7056.2.a.k 1
168.i even 2 1 882.2.a.b 1
168.s odd 6 2 882.2.g.h 2
168.ba even 6 2 882.2.g.j 2
280.c odd 2 1 7350.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 8.b even 2 1
126.2.a.a 1 24.h odd 2 1
294.2.a.g 1 56.h odd 2 1
294.2.e.a 2 56.j odd 6 2
294.2.e.c 2 56.p even 6 2
336.2.a.d 1 8.d odd 2 1
882.2.a.b 1 168.i even 2 1
882.2.g.h 2 168.s odd 6 2
882.2.g.j 2 168.ba even 6 2
1008.2.a.j 1 24.f even 2 1
1050.2.a.i 1 40.f even 2 1
1050.2.g.a 2 40.i odd 4 2
1134.2.f.g 2 72.n even 6 2
1134.2.f.j 2 72.j odd 6 2
1344.2.a.i 1 4.b odd 2 1
1344.2.a.q 1 1.a even 1 1 trivial
2352.2.a.l 1 56.e even 2 1
2352.2.q.i 2 56.k odd 6 2
2352.2.q.n 2 56.m even 6 2
3150.2.a.bo 1 120.i odd 2 1
3150.2.g.r 2 120.w even 4 2
4032.2.a.e 1 3.b odd 2 1
4032.2.a.m 1 12.b even 2 1
5082.2.a.d 1 88.b odd 2 1
5376.2.c.e 2 16.f odd 4 2
5376.2.c.bc 2 16.e even 4 2
7056.2.a.k 1 168.e odd 2 1
7098.2.a.f 1 104.e even 2 1
7350.2.a.f 1 280.c odd 2 1
8400.2.a.k 1 40.e odd 2 1
9408.2.a.n 1 7.b odd 2 1
9408.2.a.bw 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1344))$$:

 $$T_{5} - 2$$ $$T_{11} - 4$$ $$T_{13} + 6$$ $$T_{19} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$-2 + T$$
$7$ $$1 + T$$
$11$ $$-4 + T$$
$13$ $$6 + T$$
$17$ $$-2 + T$$
$19$ $$-4 + T$$
$23$ $$-8 + T$$
$29$ $$-2 + T$$
$31$ $$T$$
$37$ $$-10 + T$$
$41$ $$6 + T$$
$43$ $$-4 + T$$
$47$ $$T$$
$53$ $$6 + T$$
$59$ $$4 + T$$
$61$ $$6 + T$$
$67$ $$4 + T$$
$71$ $$-8 + T$$
$73$ $$-10 + T$$
$79$ $$T$$
$83$ $$-4 + T$$
$89$ $$6 + T$$
$97$ $$14 + T$$
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