# Properties

 Label 1440.2.q.j Level $1440$ Weight $2$ Character orbit 1440.q Analytic conductor $11.498$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1440 = 2^{5} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1440.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.4984578911$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.3317760000.8 Defining polynomial: $$x^{8} + 4 x^{6} + 7 x^{4} + 36 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{7} q^{3} + ( -1 + \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + ( -\beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{7} q^{3} + ( -1 + \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + ( -\beta_{2} - 2 \beta_{3} ) q^{9} + ( \beta_{4} - \beta_{5} ) q^{11} + ( -1 + 2 \beta_{2} + \beta_{3} + \beta_{6} ) q^{13} + ( -\beta_{1} - \beta_{4} - \beta_{5} ) q^{15} + ( -3 + \beta_{2} - \beta_{6} ) q^{17} + ( -3 \beta_{4} - 6 \beta_{5} ) q^{19} + ( -2 - \beta_{3} - \beta_{6} ) q^{21} + ( -\beta_{4} + \beta_{7} ) q^{23} -\beta_{3} q^{25} + ( \beta_{1} + 4 \beta_{4} + \beta_{5} + \beta_{7} ) q^{27} + ( 2 \beta_{2} - \beta_{3} + 4 \beta_{6} ) q^{29} + ( -2 \beta_{4} - 3 \beta_{5} - 4 \beta_{7} ) q^{31} + ( 1 + \beta_{2} - 2 \beta_{3} - \beta_{6} ) q^{33} + ( -\beta_{1} - \beta_{7} ) q^{35} + ( -3 - \beta_{2} + \beta_{6} ) q^{37} + ( -3 \beta_{4} + 3 \beta_{5} + 2 \beta_{7} ) q^{39} + ( 6 + 2 \beta_{2} - 6 \beta_{3} + \beta_{6} ) q^{41} + ( -\beta_{4} + \beta_{5} ) q^{43} + ( 2 + \beta_{2} + \beta_{6} ) q^{45} + ( -3 \beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{47} + ( 3 + 2 \beta_{2} - 3 \beta_{3} + \beta_{6} ) q^{49} + ( 2 \beta_{1} - 4 \beta_{4} - \beta_{5} + 4 \beta_{7} ) q^{51} -4 q^{53} + ( \beta_{4} + 2 \beta_{5} ) q^{55} + ( -3 + 6 \beta_{2} - 3 \beta_{3} + 3 \beta_{6} ) q^{57} + ( 2 \beta_{4} - 2 \beta_{7} ) q^{59} + ( -\beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{61} + ( 2 \beta_{1} - \beta_{4} - \beta_{5} + 3 \beta_{7} ) q^{63} + ( -\beta_{2} - \beta_{3} - 2 \beta_{6} ) q^{65} + ( 9 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} ) q^{67} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{6} ) q^{69} + ( 2 \beta_{1} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{71} + ( -2 + 2 \beta_{2} - 2 \beta_{6} ) q^{73} + ( \beta_{1} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{75} + ( -3 + 2 \beta_{2} + 3 \beta_{3} + \beta_{6} ) q^{77} + ( 8 \beta_{1} + 4 \beta_{4} + 4 \beta_{5} ) q^{79} + ( 1 - \beta_{3} - 4 \beta_{6} ) q^{81} + ( -9 \beta_{1} - 5 \beta_{4} - 4 \beta_{5} ) q^{83} + ( 3 - 2 \beta_{2} - 3 \beta_{3} - \beta_{6} ) q^{85} + ( -\beta_{1} + 5 \beta_{4} + 11 \beta_{5} + 3 \beta_{7} ) q^{87} + ( -9 + 2 \beta_{2} - 2 \beta_{6} ) q^{89} + ( -4 \beta_{1} + \beta_{4} + 2 \beta_{5} - 4 \beta_{7} ) q^{91} + ( -2 - \beta_{2} - 9 \beta_{3} + 2 \beta_{6} ) q^{93} + ( 6 \beta_{4} + 3 \beta_{5} ) q^{95} + ( 2 \beta_{2} - 6 \beta_{3} + 4 \beta_{6} ) q^{97} + ( 4 \beta_{1} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{5} - 8 q^{9} + O(q^{10})$$ $$8 q - 4 q^{5} - 8 q^{9} - 4 q^{13} - 24 q^{17} - 20 q^{21} - 4 q^{25} - 4 q^{29} - 24 q^{37} + 24 q^{41} + 16 q^{45} + 12 q^{49} - 32 q^{53} - 36 q^{57} - 8 q^{61} - 4 q^{65} + 4 q^{69} - 16 q^{73} - 12 q^{77} + 4 q^{81} + 12 q^{85} - 72 q^{89} - 52 q^{93} - 24 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 4 x^{6} + 7 x^{4} + 36 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} + 14 \nu^{4} - 7 \nu^{2} - 36$$$$)/63$$ $$\beta_{3}$$ $$=$$ $$($$$$4 \nu^{6} + 7 \nu^{4} + 28 \nu^{2} + 144$$$$)/63$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{7} - 7 \nu^{5} + 35 \nu^{3} - 81 \nu$$$$)/189$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{7} + 7 \nu^{5} - 35 \nu^{3} - 180 \nu$$$$)/189$$ $$\beta_{6}$$ $$=$$ $$($$$$-8 \nu^{6} - 14 \nu^{4} + 7 \nu^{2} - 162$$$$)/63$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} - 4 \nu^{5} - 7 \nu^{3} - 36 \nu$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2 \beta_{3} - 2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{7} - \beta_{5} + 3 \beta_{4} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} + 4 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-5 \beta_{7} + 7 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{6} - 7 \beta_{2} - 22$$ $$\nu^{7}$$ $$=$$ $$-21 \beta_{5} - 21 \beta_{4} - 29 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$641$$ $$901$$ $$991$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
481.1
 1.40294 + 1.01575i 0.178197 + 1.72286i −0.178197 − 1.72286i −1.40294 − 1.01575i 1.40294 − 1.01575i 0.178197 − 1.72286i −0.178197 + 1.72286i −1.40294 + 1.01575i
0 −1.40294 + 1.01575i 0 −0.500000 + 0.866025i 0 1.40294 + 2.42997i 0 0.936492 2.85008i 0
481.2 0 −0.178197 + 1.72286i 0 −0.500000 + 0.866025i 0 0.178197 + 0.308646i 0 −2.93649 0.614017i 0
481.3 0 0.178197 1.72286i 0 −0.500000 + 0.866025i 0 −0.178197 0.308646i 0 −2.93649 0.614017i 0
481.4 0 1.40294 1.01575i 0 −0.500000 + 0.866025i 0 −1.40294 2.42997i 0 0.936492 2.85008i 0
961.1 0 −1.40294 1.01575i 0 −0.500000 0.866025i 0 1.40294 2.42997i 0 0.936492 + 2.85008i 0
961.2 0 −0.178197 1.72286i 0 −0.500000 0.866025i 0 0.178197 0.308646i 0 −2.93649 + 0.614017i 0
961.3 0 0.178197 + 1.72286i 0 −0.500000 0.866025i 0 −0.178197 + 0.308646i 0 −2.93649 + 0.614017i 0
961.4 0 1.40294 + 1.01575i 0 −0.500000 0.866025i 0 −1.40294 + 2.42997i 0 0.936492 + 2.85008i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 961.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.q.j 8
3.b odd 2 1 4320.2.q.m 8
4.b odd 2 1 inner 1440.2.q.j 8
9.c even 3 1 inner 1440.2.q.j 8
9.d odd 6 1 4320.2.q.m 8
12.b even 2 1 4320.2.q.m 8
36.f odd 6 1 inner 1440.2.q.j 8
36.h even 6 1 4320.2.q.m 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.q.j 8 1.a even 1 1 trivial
1440.2.q.j 8 4.b odd 2 1 inner
1440.2.q.j 8 9.c even 3 1 inner
1440.2.q.j 8 36.f odd 6 1 inner
4320.2.q.m 8 3.b odd 2 1
4320.2.q.m 8 9.d odd 6 1
4320.2.q.m 8 12.b even 2 1
4320.2.q.m 8 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1440, [\chi])$$:

 $$T_{7}^{8} + 8 T_{7}^{6} + 63 T_{7}^{4} + 8 T_{7}^{2} + 1$$ $$T_{11}^{4} + 6 T_{11}^{2} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$81 + 36 T^{2} + 7 T^{4} + 4 T^{6} + T^{8}$$
$5$ $$( 1 + T + T^{2} )^{4}$$
$7$ $$1 + 8 T^{2} + 63 T^{4} + 8 T^{6} + T^{8}$$
$11$ $$( 36 + 6 T^{2} + T^{4} )^{2}$$
$13$ $$( 196 - 28 T + 18 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$17$ $$( -6 + 6 T + T^{2} )^{4}$$
$19$ $$( -54 + T^{2} )^{4}$$
$23$ $$1 + 8 T^{2} + 63 T^{4} + 8 T^{6} + T^{8}$$
$29$ $$( 3481 - 118 T + 63 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$31$ $$1336336 + 106352 T^{2} + 7308 T^{4} + 92 T^{6} + T^{8}$$
$37$ $$( -6 + 6 T + T^{2} )^{4}$$
$41$ $$( 441 - 252 T + 123 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$43$ $$( 36 + 6 T^{2} + T^{4} )^{2}$$
$47$ $$194481 + 21168 T^{2} + 1863 T^{4} + 48 T^{6} + T^{8}$$
$53$ $$( 4 + T )^{8}$$
$59$ $$256 + 512 T^{2} + 1008 T^{4} + 32 T^{6} + T^{8}$$
$61$ $$( 121 - 44 T + 27 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$67$ $$96059601 + 2822688 T^{2} + 73143 T^{4} + 288 T^{6} + T^{8}$$
$71$ $$( -10 + T^{2} )^{4}$$
$73$ $$( -56 + 4 T + T^{2} )^{4}$$
$79$ $$( 25600 + 160 T^{2} + T^{4} )^{2}$$
$83$ $$1632240801 + 16483608 T^{2} + 126063 T^{4} + 408 T^{6} + T^{8}$$
$89$ $$( 21 + 18 T + T^{2} )^{4}$$
$97$ $$( 576 - 288 T + 168 T^{2} + 12 T^{3} + T^{4} )^{2}$$