# Properties

 Label 1344.2.h.e Level 1344 Weight 2 Character orbit 1344.h Analytic conductor 10.732 Analytic rank 0 Dimension 4 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.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.7318940317$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 672) 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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{8} - \zeta_{8}^{2} ) q^{3} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} + \zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{8} - \zeta_{8}^{2} ) q^{3} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} + \zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{9} + 2 q^{11} + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{13} + ( 3 - 2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{15} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{17} + ( -3 \zeta_{8} + 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{19} + ( 1 + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{21} + 4 q^{23} + ( -1 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{25} + ( 1 + \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{27} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{29} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{31} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{33} + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{35} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{37} + ( -1 - 2 \zeta_{8} + 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{39} + 10 \zeta_{8}^{2} q^{41} + ( -6 \zeta_{8} + 2 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{43} + ( 2 + 3 \zeta_{8} - 4 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{45} + ( 4 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{47} - q^{49} + ( -4 + 4 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{51} + ( -8 \zeta_{8} + 2 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{53} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{55} + ( 5 - 6 \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{57} + ( -2 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{59} + ( -10 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{61} + ( 1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{63} + ( 4 \zeta_{8} - 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{65} + ( 8 \zeta_{8} + 2 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{67} + ( 4 + 4 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{69} + ( 6 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{71} + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{73} + ( 3 - \zeta_{8} + 5 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{75} + 2 \zeta_{8}^{2} q^{77} + 4 \zeta_{8}^{2} q^{79} + ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} + ( 10 - \zeta_{8} + \zeta_{8}^{3} ) q^{83} + ( 8 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{85} + ( 4 - 4 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{87} + ( 4 \zeta_{8} - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{89} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{91} + ( 2 + 4 \zeta_{8} + 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{93} + ( -10 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{95} + ( -6 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{97} + ( 4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} + O(q^{10})$$ $$4q + 4q^{3} + 8q^{11} - 8q^{13} + 12q^{15} + 4q^{21} + 16q^{23} - 4q^{25} + 4q^{27} + 8q^{33} - 8q^{35} - 8q^{37} - 4q^{39} + 8q^{45} + 16q^{47} - 4q^{49} - 16q^{51} + 20q^{57} - 8q^{59} - 40q^{61} + 4q^{63} + 16q^{69} + 24q^{71} - 24q^{73} + 12q^{75} + 28q^{81} + 40q^{83} + 32q^{85} + 16q^{87} + 8q^{93} - 40q^{95} - 24q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
575.1
 −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i
0 0.292893 1.70711i 0 3.41421i 0 1.00000i 0 −2.82843 1.00000i 0
575.2 0 0.292893 + 1.70711i 0 3.41421i 0 1.00000i 0 −2.82843 + 1.00000i 0
575.3 0 1.70711 0.292893i 0 0.585786i 0 1.00000i 0 2.82843 1.00000i 0
575.4 0 1.70711 + 0.292893i 0 0.585786i 0 1.00000i 0 2.82843 + 1.00000i 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
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.2.h.e 4
3.b odd 2 1 1344.2.h.a 4
4.b odd 2 1 1344.2.h.a 4
8.b even 2 1 672.2.h.a 4
8.d odd 2 1 672.2.h.d yes 4
12.b even 2 1 inner 1344.2.h.e 4
24.f even 2 1 672.2.h.a 4
24.h odd 2 1 672.2.h.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.h.a 4 8.b even 2 1
672.2.h.a 4 24.f even 2 1
672.2.h.d yes 4 8.d odd 2 1
672.2.h.d yes 4 24.h odd 2 1
1344.2.h.a 4 3.b odd 2 1
1344.2.h.a 4 4.b odd 2 1
1344.2.h.e 4 1.a even 1 1 trivial
1344.2.h.e 4 12.b 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 $$S_{2}^{\mathrm{new}}(1344, [\chi])$$:

 $$T_{5}^{4} + 12 T_{5}^{2} + 4$$ $$T_{11} - 2$$ $$T_{13}^{2} + 4 T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 4 T + 8 T^{2} - 12 T^{3} + 9 T^{4}$$
$5$ $$1 - 8 T^{2} + 34 T^{4} - 200 T^{6} + 625 T^{8}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 1 - 2 T + 11 T^{2} )^{4}$$
$13$ $$( 1 + 4 T + 28 T^{2} + 52 T^{3} + 169 T^{4} )^{2}$$
$17$ $$1 - 44 T^{2} + 934 T^{4} - 12716 T^{6} + 83521 T^{8}$$
$19$ $$1 - 32 T^{2} + 690 T^{4} - 11552 T^{6} + 130321 T^{8}$$
$23$ $$( 1 - 4 T + 23 T^{2} )^{4}$$
$29$ $$1 - 92 T^{2} + 3670 T^{4} - 77372 T^{6} + 707281 T^{8}$$
$31$ $$1 - 76 T^{2} + 2854 T^{4} - 73036 T^{6} + 923521 T^{8}$$
$37$ $$( 1 + 4 T + 70 T^{2} + 148 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 8 T + 41 T^{2} )^{2}( 1 + 8 T + 41 T^{2} )^{2}$$
$43$ $$1 - 20 T^{2} + 2646 T^{4} - 36980 T^{6} + 3418801 T^{8}$$
$47$ $$( 1 - 8 T + 78 T^{2} - 376 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$1 + 52 T^{2} + 4246 T^{4} + 146068 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 + 4 T + 104 T^{2} + 236 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 20 T + 204 T^{2} + 1220 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 - 4 T^{2} + 6934 T^{4} - 17956 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 12 T + 146 T^{2} - 852 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 12 T + 174 T^{2} + 876 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 142 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 20 T + 264 T^{2} - 1660 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$1 - 284 T^{2} + 35494 T^{4} - 2249564 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 + 12 T + 102 T^{2} + 1164 T^{3} + 9409 T^{4} )^{2}$$