# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1440 = 2^{5} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1440.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.4984578911$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{2} q^{5} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} +O(q^{10})$$ $$q -\zeta_{8}^{2} q^{5} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} + ( 4 + \zeta_{8} - \zeta_{8}^{3} ) q^{11} + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{13} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{17} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{23} - q^{25} + 6 \zeta_{8}^{2} q^{29} + ( -6 \zeta_{8} - 2 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{31} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{35} + ( -2 + 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{37} + ( -5 \zeta_{8} + 4 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{41} + ( 2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{43} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{47} + ( 1 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{49} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{53} + ( -\zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{55} + ( 8 + \zeta_{8} - \zeta_{8}^{3} ) q^{59} + ( 10 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{61} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{65} + 8 \zeta_{8}^{2} q^{67} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} + ( -2 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{73} + ( -2 \zeta_{8} + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{77} + ( -6 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{79} + ( 12 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{83} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{85} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{89} + ( 4 \zeta_{8} - 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{91} + ( -6 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 16q^{11} - 8q^{13} + 8q^{23} - 4q^{25} + 8q^{35} - 8q^{37} + 4q^{49} + 32q^{59} + 40q^{61} - 8q^{73} + 48q^{83} + 16q^{85} - 24q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$577$$ $$641$$ $$901$$ $$991$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 0.707107 + 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 0 0 1.00000i 0 0.585786i 0 0 0
1151.2 0 0 0 1.00000i 0 3.41421i 0 0 0
1151.3 0 0 0 1.00000i 0 3.41421i 0 0 0
1151.4 0 0 0 1.00000i 0 0.585786i 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
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 1440.2.h.d yes 4
3.b odd 2 1 1440.2.h.a 4
4.b odd 2 1 1440.2.h.a 4
5.b even 2 1 7200.2.h.i 4
5.c odd 4 1 7200.2.o.e 4
5.c odd 4 1 7200.2.o.m 4
8.b even 2 1 2880.2.h.a 4
8.d odd 2 1 2880.2.h.d 4
12.b even 2 1 inner 1440.2.h.d yes 4
15.d odd 2 1 7200.2.h.c 4
15.e even 4 1 7200.2.o.b 4
15.e even 4 1 7200.2.o.j 4
20.d odd 2 1 7200.2.h.c 4
20.e even 4 1 7200.2.o.b 4
20.e even 4 1 7200.2.o.j 4
24.f even 2 1 2880.2.h.a 4
24.h odd 2 1 2880.2.h.d 4
60.h even 2 1 7200.2.h.i 4
60.l odd 4 1 7200.2.o.e 4
60.l odd 4 1 7200.2.o.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.h.a 4 3.b odd 2 1
1440.2.h.a 4 4.b odd 2 1
1440.2.h.d yes 4 1.a even 1 1 trivial
1440.2.h.d yes 4 12.b even 2 1 inner
2880.2.h.a 4 8.b even 2 1
2880.2.h.a 4 24.f even 2 1
2880.2.h.d 4 8.d odd 2 1
2880.2.h.d 4 24.h odd 2 1
7200.2.h.c 4 15.d odd 2 1
7200.2.h.c 4 20.d odd 2 1
7200.2.h.i 4 5.b even 2 1
7200.2.h.i 4 60.h even 2 1
7200.2.o.b 4 15.e even 4 1
7200.2.o.b 4 20.e even 4 1
7200.2.o.e 4 5.c odd 4 1
7200.2.o.e 4 60.l odd 4 1
7200.2.o.j 4 15.e even 4 1
7200.2.o.j 4 20.e even 4 1
7200.2.o.m 4 5.c odd 4 1
7200.2.o.m 4 60.l odd 4 1

## Hecke kernels

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

 $$T_{11}^{2} - 8 T_{11} + 14$$ $$T_{23}^{2} - 4 T_{23} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$4 + 12 T^{2} + T^{4}$$
$11$ $$( 14 - 8 T + T^{2} )^{2}$$
$13$ $$( 2 + 4 T + T^{2} )^{2}$$
$17$ $$64 + 48 T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$( -4 - 4 T + T^{2} )^{2}$$
$29$ $$( 36 + T^{2} )^{2}$$
$31$ $$4624 + 152 T^{2} + T^{4}$$
$37$ $$( -46 + 4 T + T^{2} )^{2}$$
$41$ $$1156 + 132 T^{2} + T^{4}$$
$43$ $$64 + 48 T^{2} + T^{4}$$
$47$ $$( -32 + T^{2} )^{2}$$
$53$ $$64 + 48 T^{2} + T^{4}$$
$59$ $$( 62 - 16 T + T^{2} )^{2}$$
$61$ $$( 92 - 20 T + T^{2} )^{2}$$
$67$ $$( 64 + T^{2} )^{2}$$
$71$ $$( -32 + T^{2} )^{2}$$
$73$ $$( -68 + 4 T + T^{2} )^{2}$$
$79$ $$1296 + 216 T^{2} + T^{4}$$
$83$ $$( 136 - 24 T + T^{2} )^{2}$$
$89$ $$( 18 + T^{2} )^{2}$$
$97$ $$( -36 + 12 T + T^{2} )^{2}$$