# Properties

 Label 1344.4.b.d 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: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( 8 - 16 \zeta_{6} ) q^{5} + ( 7 + 14 \zeta_{6} ) q^{7} + 9 q^{9} +O(q^{10})$$ $$q + 3 q^{3} + ( 8 - 16 \zeta_{6} ) q^{5} + ( 7 + 14 \zeta_{6} ) q^{7} + 9 q^{9} + ( -2 + 4 \zeta_{6} ) q^{11} + ( 8 - 16 \zeta_{6} ) q^{13} + ( 24 - 48 \zeta_{6} ) q^{15} + ( -44 + 88 \zeta_{6} ) q^{17} -52 q^{19} + ( 21 + 42 \zeta_{6} ) q^{21} + ( 66 - 132 \zeta_{6} ) q^{23} -67 q^{25} + 27 q^{27} + 246 q^{29} + 116 q^{31} + ( -6 + 12 \zeta_{6} ) q^{33} + ( 280 - 224 \zeta_{6} ) q^{35} + 314 q^{37} + ( 24 - 48 \zeta_{6} ) q^{39} + ( -156 + 312 \zeta_{6} ) q^{41} + ( 218 - 436 \zeta_{6} ) q^{43} + ( 72 - 144 \zeta_{6} ) q^{45} -192 q^{47} + ( -147 + 392 \zeta_{6} ) q^{49} + ( -132 + 264 \zeta_{6} ) q^{51} + 150 q^{53} + 48 q^{55} -156 q^{57} + 204 q^{59} + ( 336 - 672 \zeta_{6} ) q^{61} + ( 63 + 126 \zeta_{6} ) q^{63} -192 q^{65} + ( -294 + 588 \zeta_{6} ) q^{67} + ( 198 - 396 \zeta_{6} ) q^{69} + ( -470 + 940 \zeta_{6} ) q^{71} + ( -72 + 144 \zeta_{6} ) q^{73} -201 q^{75} + ( -70 + 56 \zeta_{6} ) q^{77} + ( 794 - 1588 \zeta_{6} ) q^{79} + 81 q^{81} + 252 q^{83} + 1056 q^{85} + 738 q^{87} + ( 124 - 248 \zeta_{6} ) q^{89} + ( 280 - 224 \zeta_{6} ) q^{91} + 348 q^{93} + ( -416 + 832 \zeta_{6} ) q^{95} + ( 832 - 1664 \zeta_{6} ) q^{97} + ( -18 + 36 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} + 28q^{7} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} + 28q^{7} + 18q^{9} - 104q^{19} + 84q^{21} - 134q^{25} + 54q^{27} + 492q^{29} + 232q^{31} + 336q^{35} + 628q^{37} - 384q^{47} + 98q^{49} + 300q^{53} + 96q^{55} - 312q^{57} + 408q^{59} + 252q^{63} - 384q^{65} - 402q^{75} - 84q^{77} + 162q^{81} + 504q^{83} + 2112q^{85} + 1476q^{87} + 336q^{91} + 696q^{93} + O(q^{100})$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 3.00000 0 13.8564i 0 14.0000 + 12.1244i 0 9.00000 0
895.2 0 3.00000 0 13.8564i 0 14.0000 12.1244i 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.d 2
4.b odd 2 1 1344.4.b.a 2
7.b odd 2 1 1344.4.b.a 2
8.b even 2 1 336.4.b.b 2
8.d odd 2 1 336.4.b.c yes 2
24.f even 2 1 1008.4.b.b 2
24.h odd 2 1 1008.4.b.e 2
28.d even 2 1 inner 1344.4.b.d 2
56.e even 2 1 336.4.b.b 2
56.h odd 2 1 336.4.b.c yes 2
168.e odd 2 1 1008.4.b.e 2
168.i even 2 1 1008.4.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.4.b.b 2 8.b even 2 1
336.4.b.b 2 56.e even 2 1
336.4.b.c yes 2 8.d odd 2 1
336.4.b.c yes 2 56.h odd 2 1
1008.4.b.b 2 24.f even 2 1
1008.4.b.b 2 168.i even 2 1
1008.4.b.e 2 24.h odd 2 1
1008.4.b.e 2 168.e odd 2 1
1344.4.b.a 2 4.b odd 2 1
1344.4.b.a 2 7.b odd 2 1
1344.4.b.d 2 1.a even 1 1 trivial
1344.4.b.d 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 $$S_{4}^{\mathrm{new}}(1344, [\chi])$$:

 $$T_{5}^{2} + 192$$ $$T_{19} + 52$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$192 + T^{2}$$
$7$ $$343 - 28 T + T^{2}$$
$11$ $$12 + T^{2}$$
$13$ $$192 + T^{2}$$
$17$ $$5808 + T^{2}$$
$19$ $$( 52 + T )^{2}$$
$23$ $$13068 + T^{2}$$
$29$ $$( -246 + T )^{2}$$
$31$ $$( -116 + T )^{2}$$
$37$ $$( -314 + T )^{2}$$
$41$ $$73008 + T^{2}$$
$43$ $$142572 + T^{2}$$
$47$ $$( 192 + T )^{2}$$
$53$ $$( -150 + T )^{2}$$
$59$ $$( -204 + T )^{2}$$
$61$ $$338688 + T^{2}$$
$67$ $$259308 + T^{2}$$
$71$ $$662700 + T^{2}$$
$73$ $$15552 + T^{2}$$
$79$ $$1891308 + T^{2}$$
$83$ $$( -252 + T )^{2}$$
$89$ $$46128 + T^{2}$$
$97$ $$2076672 + T^{2}$$