# Properties

 Label 1440.2.h.a 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: $$x^{4} + 1$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7}+O(q^{10})$$ q + b1 * q^5 + (-b2 + 2*b1) * q^7 $$q + \beta_1 q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7} + ( - \beta_{3} - 4) q^{11} + (\beta_{3} - 2) q^{13} + ( - 2 \beta_{2} - 4 \beta_1) q^{17} + (2 \beta_{3} - 2) q^{23} - q^{25} - 6 \beta_1 q^{29} + ( - 6 \beta_{2} - 2 \beta_1) q^{31} + (\beta_{3} - 2) q^{35} + (5 \beta_{3} - 2) q^{37} + (5 \beta_{2} - 4 \beta_1) q^{41} + (2 \beta_{2} - 4 \beta_1) q^{43} - 4 \beta_{3} q^{47} + (4 \beta_{3} + 1) q^{49} + ( - 2 \beta_{2} - 4 \beta_1) q^{53} + ( - \beta_{2} - 4 \beta_1) q^{55} + ( - \beta_{3} - 8) q^{59} + ( - 2 \beta_{3} + 10) q^{61} + (\beta_{2} - 2 \beta_1) q^{65} + 8 \beta_1 q^{67} + 4 \beta_{3} q^{71} + (6 \beta_{3} - 2) q^{73} + (2 \beta_{2} - 6 \beta_1) q^{77} + ( - 6 \beta_{2} + 6 \beta_1) q^{79} + ( - 2 \beta_{3} - 12) q^{83} + (2 \beta_{3} + 4) q^{85} - 3 \beta_{2} q^{89} + (4 \beta_{2} - 6 \beta_1) q^{91} + ( - 6 \beta_{3} - 6) q^{97}+O(q^{100})$$ q + b1 * q^5 + (-b2 + 2*b1) * q^7 + (-b3 - 4) * q^11 + (b3 - 2) * q^13 + (-2*b2 - 4*b1) * q^17 + (2*b3 - 2) * q^23 - q^25 - 6*b1 * q^29 + (-6*b2 - 2*b1) * q^31 + (b3 - 2) * q^35 + (5*b3 - 2) * q^37 + (5*b2 - 4*b1) * q^41 + (2*b2 - 4*b1) * q^43 - 4*b3 * q^47 + (4*b3 + 1) * q^49 + (-2*b2 - 4*b1) * q^53 + (-b2 - 4*b1) * q^55 + (-b3 - 8) * q^59 + (-2*b3 + 10) * q^61 + (b2 - 2*b1) * q^65 + 8*b1 * q^67 + 4*b3 * q^71 + (6*b3 - 2) * q^73 + (2*b2 - 6*b1) * q^77 + (-6*b2 + 6*b1) * q^79 + (-2*b3 - 12) * q^83 + (2*b3 + 4) * q^85 - 3*b2 * q^89 + (4*b2 - 6*b1) * q^91 + (-6*b3 - 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 16 q^{11} - 8 q^{13} - 8 q^{23} - 4 q^{25} - 8 q^{35} - 8 q^{37} + 4 q^{49} - 32 q^{59} + 40 q^{61} - 8 q^{73} - 48 q^{83} + 16 q^{85} - 24 q^{97}+O(q^{100})$$ 4 * q - 16 * q^11 - 8 * q^13 - 8 * q^23 - 4 * q^25 - 8 * q^35 - 8 * q^37 + 4 * q^49 - 32 * q^59 + 40 * q^61 - 8 * q^73 - 48 * q^83 + 16 * q^85 - 24 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 1.00000i 0 3.41421i 0 0 0
1151.2 0 0 0 1.00000i 0 0.585786i 0 0 0
1151.3 0 0 0 1.00000i 0 0.585786i 0 0 0
1151.4 0 0 0 1.00000i 0 3.41421i 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.a 4
3.b odd 2 1 1440.2.h.d yes 4
4.b odd 2 1 1440.2.h.d yes 4
5.b even 2 1 7200.2.h.c 4
5.c odd 4 1 7200.2.o.b 4
5.c odd 4 1 7200.2.o.j 4
8.b even 2 1 2880.2.h.d 4
8.d odd 2 1 2880.2.h.a 4
12.b even 2 1 inner 1440.2.h.a 4
15.d odd 2 1 7200.2.h.i 4
15.e even 4 1 7200.2.o.e 4
15.e even 4 1 7200.2.o.m 4
20.d odd 2 1 7200.2.h.i 4
20.e even 4 1 7200.2.o.e 4
20.e even 4 1 7200.2.o.m 4
24.f even 2 1 2880.2.h.d 4
24.h odd 2 1 2880.2.h.a 4
60.h even 2 1 7200.2.h.c 4
60.l odd 4 1 7200.2.o.b 4
60.l odd 4 1 7200.2.o.j 4

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

## Hecke kernels

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

 $$T_{11}^{2} + 8T_{11} + 14$$ T11^2 + 8*T11 + 14 $$T_{23}^{2} + 4T_{23} - 4$$ T23^2 + 4*T23 - 4

## Hecke characteristic polynomials

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