Properties

 Label 1344.4.a.ba Level $1344$ Weight $4$ Character orbit 1344.a Self dual yes Analytic conductor $79.299$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1344.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$79.2985670477$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} + 18q^{5} + 7q^{7} + 9q^{9} + O(q^{10})$$ $$q + 3q^{3} + 18q^{5} + 7q^{7} + 9q^{9} + 36q^{11} + 34q^{13} + 54q^{15} + 42q^{17} + 124q^{19} + 21q^{21} + 199q^{25} + 27q^{27} - 102q^{29} - 160q^{31} + 108q^{33} + 126q^{35} - 398q^{37} + 102q^{39} - 318q^{41} + 268q^{43} + 162q^{45} + 240q^{47} + 49q^{49} + 126q^{51} + 498q^{53} + 648q^{55} + 372q^{57} + 132q^{59} - 398q^{61} + 63q^{63} + 612q^{65} - 92q^{67} - 720q^{71} - 502q^{73} + 597q^{75} + 252q^{77} - 1024q^{79} + 81q^{81} + 204q^{83} + 756q^{85} - 306q^{87} + 354q^{89} + 238q^{91} - 480q^{93} + 2232q^{95} - 286q^{97} + 324q^{99} + O(q^{100})$$

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 3.00000 0 18.0000 0 7.00000 0 9.00000 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 1344.4.a.ba 1
4.b odd 2 1 1344.4.a.n 1
8.b even 2 1 21.4.a.a 1
8.d odd 2 1 336.4.a.f 1
24.f even 2 1 1008.4.a.v 1
24.h odd 2 1 63.4.a.c 1
40.f even 2 1 525.4.a.g 1
40.i odd 4 2 525.4.d.c 2
56.e even 2 1 2352.4.a.r 1
56.h odd 2 1 147.4.a.c 1
56.j odd 6 2 147.4.e.g 2
56.p even 6 2 147.4.e.i 2
120.i odd 2 1 1575.4.a.b 1
168.i even 2 1 441.4.a.j 1
168.s odd 6 2 441.4.e.b 2
168.ba even 6 2 441.4.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.a 1 8.b even 2 1
63.4.a.c 1 24.h odd 2 1
147.4.a.c 1 56.h odd 2 1
147.4.e.g 2 56.j odd 6 2
147.4.e.i 2 56.p even 6 2
336.4.a.f 1 8.d odd 2 1
441.4.a.j 1 168.i even 2 1
441.4.e.b 2 168.s odd 6 2
441.4.e.d 2 168.ba even 6 2
525.4.a.g 1 40.f even 2 1
525.4.d.c 2 40.i odd 4 2
1008.4.a.v 1 24.f even 2 1
1344.4.a.n 1 4.b odd 2 1
1344.4.a.ba 1 1.a even 1 1 trivial
1575.4.a.b 1 120.i odd 2 1
2352.4.a.r 1 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1344))$$:

 $$T_{5} - 18$$ $$T_{11} - 36$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$-18 + T$$
$7$ $$-7 + T$$
$11$ $$-36 + T$$
$13$ $$-34 + T$$
$17$ $$-42 + T$$
$19$ $$-124 + T$$
$23$ $$T$$
$29$ $$102 + T$$
$31$ $$160 + T$$
$37$ $$398 + T$$
$41$ $$318 + T$$
$43$ $$-268 + T$$
$47$ $$-240 + T$$
$53$ $$-498 + T$$
$59$ $$-132 + T$$
$61$ $$398 + T$$
$67$ $$92 + T$$
$71$ $$720 + T$$
$73$ $$502 + T$$
$79$ $$1024 + T$$
$83$ $$-204 + T$$
$89$ $$-354 + T$$
$97$ $$286 + T$$
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