# Properties

 Label 1344.2.bl.j Level $1344$ Weight $2$ Character orbit 1344.bl Analytic conductor $10.732$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.7318940317$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.562828176.1 Defining polynomial: $$x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 6 x^{4} + 4 x^{3} + 4 x^{2} - 16 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{3} + ( -\beta_{2} - \beta_{5} ) q^{5} + ( \beta_{1} - \beta_{3} + \beta_{7} ) q^{7} -\beta_{1} q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{3} + ( -\beta_{2} - \beta_{5} ) q^{5} + ( \beta_{1} - \beta_{3} + \beta_{7} ) q^{7} -\beta_{1} q^{9} + ( \beta_{2} - \beta_{3} + \beta_{4} ) q^{11} + ( 1 - 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{13} -\beta_{5} q^{15} + ( -\beta_{2} - \beta_{6} + \beta_{7} ) q^{17} + ( \beta_{1} + \beta_{3} - \beta_{6} ) q^{19} + ( 1 - \beta_{3} ) q^{21} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{23} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{25} - q^{27} + ( 2 - \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{29} + ( 3 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{31} + ( \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{33} + ( 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{5} ) q^{35} + ( -\beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{39} + ( 2 - 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} ) q^{41} + ( -1 + 2 \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{43} + \beta_{2} q^{45} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{47} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{49} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{51} + ( 2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{53} + ( -2 + \beta_{3} - \beta_{4} + \beta_{6} ) q^{55} + ( 1 - \beta_{4} + \beta_{7} ) q^{57} + ( 2 - 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{59} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{61} + ( 1 - \beta_{1} - \beta_{7} ) q^{63} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{65} + ( -3 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{67} + ( \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{69} + ( -2 + 4 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{71} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{73} + ( -\beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{75} + ( -2 + 2 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{77} + ( -2 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{79} + ( -1 + \beta_{1} ) q^{81} + ( 4 \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{83} + ( 4 - 4 \beta_{2} - 2 \beta_{5} ) q^{85} + ( 2 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{87} + ( -2 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} ) q^{89} + ( 4 - \beta_{1} - \beta_{3} - 2 \beta_{4} + 3 \beta_{6} ) q^{91} + ( -3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{93} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} ) q^{95} + ( 2 - 4 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{97} + ( \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{3} + 2q^{7} - 4q^{9} + O(q^{10})$$ $$8q + 4q^{3} + 2q^{7} - 4q^{9} - 6q^{11} + 6q^{19} + 4q^{21} + 2q^{25} - 8q^{27} + 16q^{29} + 6q^{31} - 6q^{33} + 12q^{35} - 6q^{37} - 6q^{39} + 4q^{47} + 4q^{49} + 4q^{53} - 8q^{55} + 12q^{57} + 14q^{59} - 12q^{61} + 2q^{63} + 4q^{65} - 42q^{67} - 18q^{73} - 2q^{75} - 8q^{77} - 6q^{79} - 4q^{81} - 4q^{83} + 32q^{85} + 8q^{87} + 34q^{91} - 6q^{93} - 24q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 6 x^{4} + 4 x^{3} + 4 x^{2} - 16 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + \nu^{5} + 4 \nu^{4} - 6 \nu^{3} + 4 \nu - 8$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{6} + \nu^{5} + 8 \nu^{4} - 6 \nu^{3} + 16 \nu^{2} + 4 \nu - 24$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} + 4 \nu^{6} - 3 \nu^{5} + 18 \nu^{3} - 12 \nu + 40$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{7} + 4 \nu^{6} - 5 \nu^{5} - 8 \nu^{4} + 14 \nu^{3} - 16 \nu^{2} - 20 \nu + 56$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + \nu^{5} - 3 \nu^{4} + 4 \nu^{3} - 2 \nu^{2} - 8 \nu + 12$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{7} + \nu^{6} - 3 \nu^{4} + 6 \nu^{3} - 2 \nu^{2} - 4 \nu + 20$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$-9 \nu^{7} + 8 \nu^{6} - \nu^{5} - 20 \nu^{4} + 30 \nu^{3} + 8 \nu^{2} - 52 \nu + 104$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - \beta_{5} - \beta_{4} - \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - \beta_{5} - \beta_{4}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{3} - 4 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 4 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$3 \beta_{7} - 3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} + 12 \beta_{1} - 8$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-\beta_{7} - 6 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 12 \beta_{1} + 16$$$$)/2$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
703.1
 −1.33790 − 0.458297i 1.40376 + 0.171630i 0.0777157 + 1.41208i 0.856419 − 1.12541i −1.33790 + 0.458297i 1.40376 − 0.171630i 0.0777157 − 1.41208i 0.856419 + 1.12541i
0 0.500000 0.866025i 0 −2.12403 + 1.22631i 0 2.63169 + 0.272415i 0 −0.500000 0.866025i 0
703.2 0 0.500000 0.866025i 0 −0.834598 + 0.481855i 0 −1.20103 2.35744i 0 −0.500000 0.866025i 0
703.3 0 0.500000 0.866025i 0 −0.380152 + 0.219481i 0 −2.02350 + 1.70453i 0 −0.500000 0.866025i 0
703.4 0 0.500000 0.866025i 0 3.33878 1.92764i 0 1.59285 + 2.11254i 0 −0.500000 0.866025i 0
1279.1 0 0.500000 + 0.866025i 0 −2.12403 1.22631i 0 2.63169 0.272415i 0 −0.500000 + 0.866025i 0
1279.2 0 0.500000 + 0.866025i 0 −0.834598 0.481855i 0 −1.20103 + 2.35744i 0 −0.500000 + 0.866025i 0
1279.3 0 0.500000 + 0.866025i 0 −0.380152 0.219481i 0 −2.02350 1.70453i 0 −0.500000 + 0.866025i 0
1279.4 0 0.500000 + 0.866025i 0 3.33878 + 1.92764i 0 1.59285 2.11254i 0 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1279.4 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.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.2.bl.j 8
4.b odd 2 1 1344.2.bl.i 8
7.d odd 6 1 1344.2.bl.i 8
8.b even 2 1 84.2.o.a 8
8.d odd 2 1 84.2.o.b yes 8
24.f even 2 1 252.2.bf.f 8
24.h odd 2 1 252.2.bf.g 8
28.f even 6 1 inner 1344.2.bl.j 8
56.e even 2 1 588.2.o.b 8
56.h odd 2 1 588.2.o.d 8
56.j odd 6 1 84.2.o.b yes 8
56.j odd 6 1 588.2.b.a 8
56.k odd 6 1 588.2.b.a 8
56.k odd 6 1 588.2.o.d 8
56.m even 6 1 84.2.o.a 8
56.m even 6 1 588.2.b.b 8
56.p even 6 1 588.2.b.b 8
56.p even 6 1 588.2.o.b 8
168.s odd 6 1 1764.2.b.i 8
168.v even 6 1 1764.2.b.j 8
168.ba even 6 1 252.2.bf.f 8
168.ba even 6 1 1764.2.b.j 8
168.be odd 6 1 252.2.bf.g 8
168.be odd 6 1 1764.2.b.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.o.a 8 8.b even 2 1
84.2.o.a 8 56.m even 6 1
84.2.o.b yes 8 8.d odd 2 1
84.2.o.b yes 8 56.j odd 6 1
252.2.bf.f 8 24.f even 2 1
252.2.bf.f 8 168.ba even 6 1
252.2.bf.g 8 24.h odd 2 1
252.2.bf.g 8 168.be odd 6 1
588.2.b.a 8 56.j odd 6 1
588.2.b.a 8 56.k odd 6 1
588.2.b.b 8 56.m even 6 1
588.2.b.b 8 56.p even 6 1
588.2.o.b 8 56.e even 2 1
588.2.o.b 8 56.p even 6 1
588.2.o.d 8 56.h odd 2 1
588.2.o.d 8 56.k odd 6 1
1344.2.bl.i 8 4.b odd 2 1
1344.2.bl.i 8 7.d odd 6 1
1344.2.bl.j 8 1.a even 1 1 trivial
1344.2.bl.j 8 28.f even 6 1 inner
1764.2.b.i 8 168.s odd 6 1
1764.2.b.i 8 168.be odd 6 1
1764.2.b.j 8 168.v even 6 1
1764.2.b.j 8 168.ba even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1344, [\chi])$$:

 $$T_{5}^{8} - 11 T_{5}^{6} + 125 T_{5}^{4} + 264 T_{5}^{3} + 236 T_{5}^{2} + 96 T_{5} + 16$$ $$T_{11}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 - T + T^{2} )^{4}$$
$5$ $$16 + 96 T + 236 T^{2} + 264 T^{3} + 125 T^{4} - 11 T^{6} + T^{8}$$
$7$ $$2401 - 686 T - 112 T^{3} + 65 T^{4} - 16 T^{5} - 2 T^{7} + T^{8}$$
$11$ $$400 - 240 T - 212 T^{2} + 156 T^{3} + 125 T^{4} - 78 T^{5} - T^{6} + 6 T^{7} + T^{8}$$
$13$ $$256 + 1936 T^{2} + 473 T^{4} + 38 T^{6} + T^{8}$$
$17$ $$1024 + 1536 T - 128 T^{2} - 1344 T^{3} + 752 T^{4} - 28 T^{6} + T^{8}$$
$19$ $$16 + 240 T + 3628 T^{2} - 372 T^{3} + 405 T^{4} - 78 T^{5} + 43 T^{6} - 6 T^{7} + T^{8}$$
$23$ $$16384 + 12288 T - 2048 T^{2} - 3840 T^{3} + 1472 T^{4} - 40 T^{6} + T^{8}$$
$29$ $$( -512 + 352 T - 45 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$31$ $$4173849 + 551610 T + 244512 T^{2} + 1836 T^{3} + 6633 T^{4} - 36 T^{5} + 120 T^{6} - 6 T^{7} + T^{8}$$
$37$ $$355216 + 214560 T + 103972 T^{2} + 22632 T^{3} + 4605 T^{4} + 462 T^{5} + 79 T^{6} + 6 T^{7} + T^{8}$$
$41$ $$350464 + 311552 T^{2} + 14048 T^{4} + 208 T^{6} + T^{8}$$
$43$ $$1073296 + 140152 T^{2} + 6593 T^{4} + 134 T^{6} + T^{8}$$
$47$ $$4096 + 1024 T + 2048 T^{2} + 64 T^{3} + 784 T^{4} + 80 T^{5} + 44 T^{6} - 4 T^{7} + T^{8}$$
$53$ $$64 + 928 T + 12968 T^{2} + 7012 T^{3} + 3265 T^{4} + 476 T^{5} + 77 T^{6} - 4 T^{7} + T^{8}$$
$59$ $$1420864 + 548320 T + 243784 T^{2} + 20956 T^{3} + 5977 T^{4} - 542 T^{5} + 223 T^{6} - 14 T^{7} + T^{8}$$
$61$ $$1048576 + 1572864 T + 901120 T^{2} + 172032 T^{3} + 7424 T^{4} - 1344 T^{5} - 64 T^{6} + 12 T^{7} + T^{8}$$
$67$ $$4129024 - 146304 T - 337616 T^{2} + 12024 T^{3} + 30929 T^{4} + 7014 T^{5} + 755 T^{6} + 42 T^{7} + T^{8}$$
$71$ $$200704 + 173312 T^{2} + 19856 T^{4} + 280 T^{6} + T^{8}$$
$73$ $$952576 + 1124352 T + 494096 T^{2} + 61056 T^{3} - 3127 T^{4} - 954 T^{5} + 55 T^{6} + 18 T^{7} + T^{8}$$
$79$ $$241081 - 67758 T - 89888 T^{2} + 27048 T^{3} + 37649 T^{4} - 1176 T^{5} - 184 T^{6} + 6 T^{7} + T^{8}$$
$83$ $$( 196 + 304 T - 103 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$89$ $$4096 - 18432 T + 33536 T^{2} - 26496 T^{3} + 8528 T^{4} - 92 T^{6} + T^{8}$$
$97$ $$246016 + 86176 T^{2} + 8249 T^{4} + 182 T^{6} + T^{8}$$