# Properties

 Label 1344.4.b.b Level $1344$ Weight $4$ Character orbit 1344.b Analytic conductor $79.299$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1344.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$79.2985670477$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-6})$$ Defining polynomial: $$x^{2} + 6$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 336) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + \beta q^{5} + ( 17 - 3 \beta ) q^{7} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + \beta q^{5} + ( 17 - 3 \beta ) q^{7} + 9 q^{9} -23 \beta q^{11} -30 \beta q^{13} -3 \beta q^{15} -21 \beta q^{17} -10 q^{19} + ( -51 + 9 \beta ) q^{21} -47 \beta q^{23} + 119 q^{25} -27 q^{27} -126 q^{29} + 8 q^{31} + 69 \beta q^{33} + ( 18 + 17 \beta ) q^{35} -244 q^{37} + 90 \beta q^{39} + 151 \beta q^{41} -66 \beta q^{43} + 9 \beta q^{45} -180 q^{47} + ( 235 - 102 \beta ) q^{49} + 63 \beta q^{51} -594 q^{53} + 138 q^{55} + 30 q^{57} + 540 q^{59} -144 \beta q^{61} + ( 153 - 27 \beta ) q^{63} + 180 q^{65} + 432 \beta q^{67} + 141 \beta q^{69} -73 \beta q^{71} + 270 \beta q^{73} -357 q^{75} + ( -414 - 391 \beta ) q^{77} + 420 \beta q^{79} + 81 q^{81} + 864 q^{83} + 126 q^{85} + 378 q^{87} -527 \beta q^{89} + ( -540 - 510 \beta ) q^{91} -24 q^{93} -10 \beta q^{95} -330 \beta q^{97} -207 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} + 34q^{7} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} + 34q^{7} + 18q^{9} - 20q^{19} - 102q^{21} + 238q^{25} - 54q^{27} - 252q^{29} + 16q^{31} + 36q^{35} - 488q^{37} - 360q^{47} + 470q^{49} - 1188q^{53} + 276q^{55} + 60q^{57} + 1080q^{59} + 306q^{63} + 360q^{65} - 714q^{75} - 828q^{77} + 162q^{81} + 1728q^{83} + 252q^{85} + 756q^{87} - 1080q^{91} - 48q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$449$$ $$577$$ $$1093$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
895.1
 − 2.44949i 2.44949i
0 −3.00000 0 2.44949i 0 17.0000 + 7.34847i 0 9.00000 0
895.2 0 −3.00000 0 2.44949i 0 17.0000 7.34847i 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.b.b 2
4.b odd 2 1 1344.4.b.c 2
7.b odd 2 1 1344.4.b.c 2
8.b even 2 1 336.4.b.d yes 2
8.d odd 2 1 336.4.b.a 2
24.f even 2 1 1008.4.b.a 2
24.h odd 2 1 1008.4.b.f 2
28.d even 2 1 inner 1344.4.b.b 2
56.e even 2 1 336.4.b.d yes 2
56.h odd 2 1 336.4.b.a 2
168.e odd 2 1 1008.4.b.f 2
168.i even 2 1 1008.4.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.4.b.a 2 8.d odd 2 1
336.4.b.a 2 56.h odd 2 1
336.4.b.d yes 2 8.b even 2 1
336.4.b.d yes 2 56.e even 2 1
1008.4.b.a 2 24.f even 2 1
1008.4.b.a 2 168.i even 2 1
1008.4.b.f 2 24.h odd 2 1
1008.4.b.f 2 168.e odd 2 1
1344.4.b.b 2 1.a even 1 1 trivial
1344.4.b.b 2 28.d even 2 1 inner
1344.4.b.c 2 4.b odd 2 1
1344.4.b.c 2 7.b odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 6$$ $$T_{19} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$6 + T^{2}$$
$7$ $$343 - 34 T + T^{2}$$
$11$ $$3174 + T^{2}$$
$13$ $$5400 + T^{2}$$
$17$ $$2646 + T^{2}$$
$19$ $$( 10 + T )^{2}$$
$23$ $$13254 + T^{2}$$
$29$ $$( 126 + T )^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$( 244 + T )^{2}$$
$41$ $$136806 + T^{2}$$
$43$ $$26136 + T^{2}$$
$47$ $$( 180 + T )^{2}$$
$53$ $$( 594 + T )^{2}$$
$59$ $$( -540 + T )^{2}$$
$61$ $$124416 + T^{2}$$
$67$ $$1119744 + T^{2}$$
$71$ $$31974 + T^{2}$$
$73$ $$437400 + T^{2}$$
$79$ $$1058400 + T^{2}$$
$83$ $$( -864 + T )^{2}$$
$89$ $$1666374 + T^{2}$$
$97$ $$653400 + T^{2}$$