# 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

# Learn more about

## 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

## 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

## Atkin-Lehner signs

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

## 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

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 3 T$$
$5$ $$1 - 18 T + 125 T^{2}$$
$7$ $$1 - 7 T$$
$11$ $$1 - 36 T + 1331 T^{2}$$
$13$ $$1 - 34 T + 2197 T^{2}$$
$17$ $$1 - 42 T + 4913 T^{2}$$
$19$ $$1 - 124 T + 6859 T^{2}$$
$23$ $$1 + 12167 T^{2}$$
$29$ $$1 + 102 T + 24389 T^{2}$$
$31$ $$1 + 160 T + 29791 T^{2}$$
$37$ $$1 + 398 T + 50653 T^{2}$$
$41$ $$1 + 318 T + 68921 T^{2}$$
$43$ $$1 - 268 T + 79507 T^{2}$$
$47$ $$1 - 240 T + 103823 T^{2}$$
$53$ $$1 - 498 T + 148877 T^{2}$$
$59$ $$1 - 132 T + 205379 T^{2}$$
$61$ $$1 + 398 T + 226981 T^{2}$$
$67$ $$1 + 92 T + 300763 T^{2}$$
$71$ $$1 + 720 T + 357911 T^{2}$$
$73$ $$1 + 502 T + 389017 T^{2}$$
$79$ $$1 + 1024 T + 493039 T^{2}$$
$83$ $$1 - 204 T + 571787 T^{2}$$
$89$ $$1 - 354 T + 704969 T^{2}$$
$97$ $$1 + 286 T + 912673 T^{2}$$
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