# Properties

 Label 1152.3.e.h Level $1152$ Weight $3$ Character orbit 1152.e Analytic conductor $31.390$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1152.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.3897264543$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{41}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{5} + ( 2 - 2 \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{5} + ( 2 - 2 \beta_{3} ) q^{7} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{11} + ( 8 - 3 \beta_{3} ) q^{13} + ( \beta_{1} - 6 \beta_{2} ) q^{17} + ( 8 - 2 \beta_{3} ) q^{19} + 10 \beta_{1} q^{23} + ( 11 + 2 \beta_{3} ) q^{25} + ( -3 \beta_{1} - 7 \beta_{2} ) q^{29} + ( -30 + 2 \beta_{3} ) q^{31} + ( -26 \beta_{1} + 6 \beta_{2} ) q^{35} + ( 22 + 4 \beta_{3} ) q^{37} + ( -21 \beta_{1} + 10 \beta_{2} ) q^{41} + ( 36 + 8 \beta_{3} ) q^{43} + ( 10 \beta_{1} - 8 \beta_{2} ) q^{47} + ( 51 - 8 \beta_{3} ) q^{49} + ( -27 \beta_{1} + 5 \beta_{2} ) q^{53} + ( -28 + 4 \beta_{3} ) q^{55} + ( 20 \beta_{1} - 24 \beta_{2} ) q^{59} + ( 50 + 8 \beta_{3} ) q^{61} + ( -44 \beta_{1} + 14 \beta_{2} ) q^{65} + ( 28 + 2 \beta_{3} ) q^{67} + ( 54 \beta_{1} + 12 \beta_{2} ) q^{71} + ( 68 - 4 \beta_{3} ) q^{73} + ( -52 \beta_{1} + 12 \beta_{2} ) q^{77} + ( -122 + 6 \beta_{3} ) q^{79} + ( -78 \beta_{1} - 14 \beta_{2} ) q^{83} + ( 74 - 7 \beta_{3} ) q^{85} + ( -17 \beta_{1} - 28 \beta_{2} ) q^{89} + ( 160 - 22 \beta_{3} ) q^{91} + ( -32 \beta_{1} + 12 \beta_{2} ) q^{95} + ( 40 + 26 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{7} + O(q^{10})$$ $$4q + 8q^{7} + 32q^{13} + 32q^{19} + 44q^{25} - 120q^{31} + 88q^{37} + 144q^{43} + 204q^{49} - 112q^{55} + 200q^{61} + 112q^{67} + 272q^{73} - 488q^{79} + 296q^{85} + 640q^{91} + 160q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 2$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 4 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 − 0.707107i
0 0 0 4.87832i 0 11.7980 0 0 0
1025.2 0 0 0 2.04989i 0 −7.79796 0 0 0
1025.3 0 0 0 2.04989i 0 −7.79796 0 0 0
1025.4 0 0 0 4.87832i 0 11.7980 0 0 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.3.e.h yes 4
3.b odd 2 1 inner 1152.3.e.h yes 4
4.b odd 2 1 1152.3.e.d yes 4
8.b even 2 1 1152.3.e.f yes 4
8.d odd 2 1 1152.3.e.b 4
12.b even 2 1 1152.3.e.d yes 4
16.e even 4 2 2304.3.h.i 8
16.f odd 4 2 2304.3.h.k 8
24.f even 2 1 1152.3.e.b 4
24.h odd 2 1 1152.3.e.f yes 4
48.i odd 4 2 2304.3.h.i 8
48.k even 4 2 2304.3.h.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.e.b 4 8.d odd 2 1
1152.3.e.b 4 24.f even 2 1
1152.3.e.d yes 4 4.b odd 2 1
1152.3.e.d yes 4 12.b even 2 1
1152.3.e.f yes 4 8.b even 2 1
1152.3.e.f yes 4 24.h odd 2 1
1152.3.e.h yes 4 1.a even 1 1 trivial
1152.3.e.h yes 4 3.b odd 2 1 inner
2304.3.h.i 8 16.e even 4 2
2304.3.h.i 8 48.i odd 4 2
2304.3.h.k 8 16.f odd 4 2
2304.3.h.k 8 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1152, [\chi])$$:

 $$T_{5}^{4} + 28 T_{5}^{2} + 100$$ $$T_{7}^{2} - 4 T_{7} - 92$$ $$T_{13}^{2} - 16 T_{13} - 152$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$100 + 28 T^{2} + T^{4}$$
$7$ $$( -92 - 4 T + T^{2} )^{2}$$
$11$ $$1600 + 112 T^{2} + T^{4}$$
$13$ $$( -152 - 16 T + T^{2} )^{2}$$
$17$ $$184900 + 868 T^{2} + T^{4}$$
$19$ $$( -32 - 16 T + T^{2} )^{2}$$
$23$ $$( 200 + T^{2} )^{2}$$
$29$ $$324900 + 1212 T^{2} + T^{4}$$
$31$ $$( 804 + 60 T + T^{2} )^{2}$$
$37$ $$( 100 - 44 T + T^{2} )^{2}$$
$41$ $$101124 + 4164 T^{2} + T^{4}$$
$43$ $$( -240 - 72 T + T^{2} )^{2}$$
$47$ $$322624 + 1936 T^{2} + T^{4}$$
$53$ $$1340964 + 3516 T^{2} + T^{4}$$
$59$ $$37356544 + 15424 T^{2} + T^{4}$$
$61$ $$( 964 - 100 T + T^{2} )^{2}$$
$67$ $$( 688 - 56 T + T^{2} )^{2}$$
$71$ $$16842816 + 15120 T^{2} + T^{4}$$
$73$ $$( 4240 - 136 T + T^{2} )^{2}$$
$79$ $$( 14020 + 244 T + T^{2} )^{2}$$
$83$ $$96353856 + 29040 T^{2} + T^{4}$$
$89$ $$77968900 + 19972 T^{2} + T^{4}$$
$97$ $$( -14624 - 80 T + T^{2} )^{2}$$