# Properties

 Label 1152.3.e.c 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: $$\Q(\sqrt{-2}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{3} ) q^{5} -2 q^{7} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{3} ) q^{5} -2 q^{7} + ( 6 \beta_{1} - 2 \beta_{3} ) q^{11} + ( 8 - \beta_{2} ) q^{13} + ( \beta_{1} + 2 \beta_{3} ) q^{17} + ( -8 - 2 \beta_{2} ) q^{19} + ( 6 \beta_{1} - 4 \beta_{3} ) q^{23} + ( -21 + 2 \beta_{2} ) q^{25} + ( -13 \beta_{1} - \beta_{3} ) q^{29} + ( -2 - 4 \beta_{2} ) q^{31} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{35} + 26 q^{37} + ( 11 \beta_{1} + 2 \beta_{3} ) q^{41} + ( -28 + 4 \beta_{2} ) q^{43} + ( 38 \beta_{1} + 4 \beta_{3} ) q^{47} -45 q^{49} + ( -21 \beta_{1} + 3 \beta_{3} ) q^{53} + ( -100 + 8 \beta_{2} ) q^{55} + 52 \beta_{1} q^{59} + ( 30 + 4 \beta_{2} ) q^{61} + ( 52 \beta_{1} - 10 \beta_{3} ) q^{65} + ( -84 - 2 \beta_{2} ) q^{67} + ( -22 \beta_{1} + 8 \beta_{3} ) q^{71} + ( -28 - 12 \beta_{2} ) q^{73} + ( -12 \beta_{1} + 4 \beta_{3} ) q^{77} + ( -38 + 4 \beta_{2} ) q^{79} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{83} + ( 86 - \beta_{2} ) q^{85} + ( 47 \beta_{1} - 12 \beta_{3} ) q^{89} + ( -16 + 2 \beta_{2} ) q^{91} + ( 80 \beta_{1} + 4 \beta_{3} ) q^{95} + ( -88 + 2 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{7} + O(q^{10})$$ $$4q - 8q^{7} + 32q^{13} - 32q^{19} - 84q^{25} - 8q^{31} + 104q^{37} - 112q^{43} - 180q^{49} - 400q^{55} + 120q^{61} - 336q^{67} - 112q^{73} - 152q^{79} + 344q^{85} - 64q^{91} - 352q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{3} - 3 \nu^{2} + 19 \nu - 9$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 2 \nu + 10$$ $$\beta_{3}$$ $$=$$ $$($$$$8 \nu^{3} - 12 \nu^{2} + 88 \nu - 42$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - 4 \beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} - 4 \beta_{1} - 18$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-8 \beta_{3} + 3 \beta_{2} + 38 \beta_{1} - 28$$$$)/4$$

## 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
 0.5 + 3.07253i 0.5 + 0.244099i 0.5 − 0.244099i 0.5 − 3.07253i
0 0 0 8.04746i 0 −2.00000 0 0 0
1025.2 0 0 0 5.21904i 0 −2.00000 0 0 0
1025.3 0 0 0 5.21904i 0 −2.00000 0 0 0
1025.4 0 0 0 8.04746i 0 −2.00000 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.c yes 4
3.b odd 2 1 inner 1152.3.e.c yes 4
4.b odd 2 1 1152.3.e.g yes 4
8.b even 2 1 1152.3.e.a 4
8.d odd 2 1 1152.3.e.e yes 4
12.b even 2 1 1152.3.e.g yes 4
16.e even 4 2 2304.3.h.l 8
16.f odd 4 2 2304.3.h.j 8
24.f even 2 1 1152.3.e.e yes 4
24.h odd 2 1 1152.3.e.a 4
48.i odd 4 2 2304.3.h.l 8
48.k even 4 2 2304.3.h.j 8

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

## Hecke kernels

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

 $$T_{5}^{4} + 92 T_{5}^{2} + 1764$$ $$T_{7} + 2$$ $$T_{13}^{2} - 16 T_{13} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$1764 + 92 T^{2} + T^{4}$$
$7$ $$( 2 + T )^{4}$$
$11$ $$10816 + 496 T^{2} + T^{4}$$
$13$ $$( -24 - 16 T + T^{2} )^{2}$$
$17$ $$30276 + 356 T^{2} + T^{4}$$
$19$ $$( -288 + 16 T + T^{2} )^{2}$$
$23$ $$399424 + 1552 T^{2} + T^{4}$$
$29$ $$86436 + 764 T^{2} + T^{4}$$
$31$ $$( -1404 + 4 T + T^{2} )^{2}$$
$37$ $$( -26 + T )^{4}$$
$41$ $$4356 + 836 T^{2} + T^{4}$$
$43$ $$( -624 + 56 T + T^{2} )^{2}$$
$47$ $$4769856 + 7184 T^{2} + T^{4}$$
$53$ $$236196 + 2556 T^{2} + T^{4}$$
$59$ $$( 5408 + T^{2} )^{2}$$
$61$ $$( -508 - 60 T + T^{2} )^{2}$$
$67$ $$( 6704 + 168 T + T^{2} )^{2}$$
$71$ $$3415104 + 7568 T^{2} + T^{4}$$
$73$ $$( -11888 + 56 T + T^{2} )^{2}$$
$79$ $$( 36 + 76 T + T^{2} )^{2}$$
$83$ $$2286144 + 3312 T^{2} + T^{4}$$
$89$ $$3678724 + 21508 T^{2} + T^{4}$$
$97$ $$( 7392 + 176 T + T^{2} )^{2}$$