# Properties

 Label 1440.2.m.a Level $1440$ Weight $2$ Character orbit 1440.m Analytic conductor $11.498$ Analytic rank $0$ Dimension $4$ CM discriminant -40 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1440 = 2^{5} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1440.m (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.4984578911$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 360) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + ( -2 - \beta_{3} ) q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + ( -2 - \beta_{3} ) q^{7} + ( -\beta_{1} + 2 \beta_{2} ) q^{11} + ( -4 + \beta_{3} ) q^{13} + 2 \beta_{3} q^{19} + 2 \beta_{2} q^{23} -5 q^{25} + ( 5 \beta_{1} + 2 \beta_{2} ) q^{35} + ( 8 + \beta_{3} ) q^{37} + ( \beta_{1} + 4 \beta_{2} ) q^{41} + 2 \beta_{1} q^{47} + ( 7 + 4 \beta_{3} ) q^{49} + 4 \beta_{1} q^{53} + ( 10 - \beta_{3} ) q^{55} + ( -7 \beta_{1} + 2 \beta_{2} ) q^{59} + ( -5 \beta_{1} + 4 \beta_{2} ) q^{65} + ( -8 \beta_{1} - 2 \beta_{2} ) q^{77} + ( 7 \beta_{1} + 4 \beta_{2} ) q^{89} + ( -2 + 2 \beta_{3} ) q^{91} -10 \beta_{1} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{7} + O(q^{10})$$ $$4q - 8q^{7} - 16q^{13} - 20q^{25} + 32q^{37} + 28q^{49} + 40q^{55} - 8q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 7 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} + 7 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
719.1
 1.58114 + 0.707107i −1.58114 − 0.707107i 1.58114 − 0.707107i −1.58114 + 0.707107i
0 0 0 2.23607i 0 −5.16228 0 0 0
719.2 0 0 0 2.23607i 0 1.16228 0 0 0
719.3 0 0 0 2.23607i 0 −5.16228 0 0 0
719.4 0 0 0 2.23607i 0 1.16228 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
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
3.b odd 2 1 inner
120.m 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.m.a 4
3.b odd 2 1 inner 1440.2.m.a 4
4.b odd 2 1 360.2.m.b yes 4
5.b even 2 1 1440.2.m.b 4
5.c odd 4 2 7200.2.b.f 8
8.b even 2 1 360.2.m.a 4
8.d odd 2 1 1440.2.m.b 4
12.b even 2 1 360.2.m.b yes 4
15.d odd 2 1 1440.2.m.b 4
15.e even 4 2 7200.2.b.f 8
20.d odd 2 1 360.2.m.a 4
20.e even 4 2 1800.2.b.f 8
24.f even 2 1 1440.2.m.b 4
24.h odd 2 1 360.2.m.a 4
40.e odd 2 1 CM 1440.2.m.a 4
40.f even 2 1 360.2.m.b yes 4
40.i odd 4 2 1800.2.b.f 8
40.k even 4 2 7200.2.b.f 8
60.h even 2 1 360.2.m.a 4
60.l odd 4 2 1800.2.b.f 8
120.i odd 2 1 360.2.m.b yes 4
120.m even 2 1 inner 1440.2.m.a 4
120.q odd 4 2 7200.2.b.f 8
120.w even 4 2 1800.2.b.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.m.a 4 8.b even 2 1
360.2.m.a 4 20.d odd 2 1
360.2.m.a 4 24.h odd 2 1
360.2.m.a 4 60.h even 2 1
360.2.m.b yes 4 4.b odd 2 1
360.2.m.b yes 4 12.b even 2 1
360.2.m.b yes 4 40.f even 2 1
360.2.m.b yes 4 120.i odd 2 1
1440.2.m.a 4 1.a even 1 1 trivial
1440.2.m.a 4 3.b odd 2 1 inner
1440.2.m.a 4 40.e odd 2 1 CM
1440.2.m.a 4 120.m even 2 1 inner
1440.2.m.b 4 5.b even 2 1
1440.2.m.b 4 8.d odd 2 1
1440.2.m.b 4 15.d odd 2 1
1440.2.m.b 4 24.f even 2 1
1800.2.b.f 8 20.e even 4 2
1800.2.b.f 8 40.i odd 4 2
1800.2.b.f 8 60.l odd 4 2
1800.2.b.f 8 120.w even 4 2
7200.2.b.f 8 5.c odd 4 2
7200.2.b.f 8 15.e even 4 2
7200.2.b.f 8 40.k even 4 2
7200.2.b.f 8 120.q odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 4 T_{7} - 6$$ acting on $$S_{2}^{\mathrm{new}}(1440, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$( -6 + 4 T + T^{2} )^{2}$$
$11$ $$324 + 44 T^{2} + T^{4}$$
$13$ $$( 6 + 8 T + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( -40 + T^{2} )^{2}$$
$23$ $$( 20 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$( 54 - 16 T + T^{2} )^{2}$$
$41$ $$6084 + 164 T^{2} + T^{4}$$
$43$ $$T^{4}$$
$47$ $$( 8 + T^{2} )^{2}$$
$53$ $$( 32 + T^{2} )^{2}$$
$59$ $$6084 + 236 T^{2} + T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$324 + 356 T^{2} + T^{4}$$
$97$ $$T^{4}$$