Properties

 Label 1344.4.b.c Level $1344$ Weight $4$ Character orbit 1344.b Analytic conductor $79.299$ Analytic rank $1$ 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: $$1$$ 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.c 2
4.b odd 2 1 1344.4.b.b 2
7.b odd 2 1 1344.4.b.b 2
8.b even 2 1 336.4.b.a 2
8.d odd 2 1 336.4.b.d yes 2
24.f even 2 1 1008.4.b.f 2
24.h odd 2 1 1008.4.b.a 2
28.d even 2 1 inner 1344.4.b.c 2
56.e even 2 1 336.4.b.a 2
56.h odd 2 1 336.4.b.d yes 2
168.e odd 2 1 1008.4.b.a 2
168.i even 2 1 1008.4.b.f 2

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

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

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