# Properties

 Label 1440.2.bi.d Level $1440$ Weight $2$ Character orbit 1440.bi Analytic conductor $11.498$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1440 = 2^{5} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1440.bi (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.4984578911$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 5 x^{12} + 28 x^{8} + 80 x^{4} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{49}]$$ Coefficient ring index: $$2^{18}$$ Twist minimal: no (minimal twist has level 360) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{9} q^{5} -\beta_{10} q^{7} +O(q^{10})$$ $$q -\beta_{9} q^{5} -\beta_{10} q^{7} -\beta_{15} q^{11} + \beta_{7} q^{13} + \beta_{13} q^{17} -4 \beta_{4} q^{19} + ( -2 \beta_{5} + 2 \beta_{9} ) q^{23} + ( -1 - 2 \beta_{4} - \beta_{11} ) q^{25} + ( 2 \beta_{1} - \beta_{2} + \beta_{5} - 2 \beta_{9} ) q^{29} + ( \beta_{6} + \beta_{7} ) q^{31} + ( -\beta_{3} - \beta_{13} + \beta_{15} ) q^{35} + \beta_{6} q^{37} + ( \beta_{3} - \beta_{13} ) q^{41} + ( 2 - 2 \beta_{4} - 2 \beta_{11} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{9} ) q^{47} + ( -5 \beta_{4} - \beta_{11} - \beta_{12} ) q^{49} + ( -\beta_{1} - \beta_{2} ) q^{53} + ( \beta_{6} + \beta_{7} - \beta_{8} ) q^{55} -\beta_{14} q^{59} + ( \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{10} ) q^{61} + ( \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{65} + ( -2 - 2 \beta_{4} - 2 \beta_{12} ) q^{67} + ( 5 - 5 \beta_{4} ) q^{73} + ( -3 \beta_{1} + 3 \beta_{2} + \beta_{5} + \beta_{9} ) q^{77} + ( -\beta_{6} + \beta_{7} ) q^{79} + ( 2 \beta_{3} + \beta_{14} + \beta_{15} ) q^{83} + ( \beta_{7} + 2 \beta_{8} - 2 \beta_{10} ) q^{85} + ( -\beta_{3} - \beta_{13} - 4 \beta_{14} ) q^{89} + ( 4 - 2 \beta_{11} + 2 \beta_{12} ) q^{91} -4 \beta_{2} q^{95} + ( -3 - 3 \beta_{4} + 2 \beta_{12} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + O(q^{10})$$ $$16 q - 24 q^{25} + 16 q^{43} - 16 q^{67} + 80 q^{73} + 32 q^{91} - 64 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 5 x^{12} + 28 x^{8} + 80 x^{4} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{15} - 4 \nu^{13} + 11 \nu^{11} - 36 \nu^{9} - 44 \nu^{7} + 64 \nu^{5} + 144 \nu^{3} + 256 \nu$$$$)/640$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{15} - 4 \nu^{13} - 11 \nu^{11} - 36 \nu^{9} + 44 \nu^{7} + 64 \nu^{5} - 144 \nu^{3} + 256 \nu$$$$)/640$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{13} + 27 \nu^{9} + 112 \nu^{5} + 288 \nu$$$$)/80$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{14} - 7 \nu^{10} - 12 \nu^{6} - 48 \nu^{2}$$$$)/320$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{15} - 4 \nu^{13} + 11 \nu^{11} + 4 \nu^{9} + 36 \nu^{7} - 56 \nu^{5} + 224 \nu^{3} + 96 \nu$$$$)/320$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{12} + 10 \nu^{10} + 7 \nu^{8} + 10 \nu^{6} + 92 \nu^{4} + 80 \nu^{2} + 128$$$$)/80$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{12} - 10 \nu^{10} + 7 \nu^{8} - 10 \nu^{6} + 92 \nu^{4} - 80 \nu^{2} + 128$$$$)/80$$ $$\beta_{8}$$ $$=$$ $$($$$$-5 \nu^{14} - 4 \nu^{12} - 25 \nu^{10} + 44 \nu^{8} - 140 \nu^{6} + 144 \nu^{4} - 80 \nu^{2} + 576$$$$)/320$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{15} + 4 \nu^{13} + 11 \nu^{11} - 4 \nu^{9} + 36 \nu^{7} + 56 \nu^{5} + 224 \nu^{3} - 96 \nu$$$$)/320$$ $$\beta_{10}$$ $$=$$ $$($$$$5 \nu^{14} - 4 \nu^{12} + 25 \nu^{10} + 44 \nu^{8} + 140 \nu^{6} + 144 \nu^{4} + 80 \nu^{2} + 576$$$$)/320$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{14} + 4 \nu^{12} + 5 \nu^{10} + 20 \nu^{8} + 28 \nu^{6} + 48 \nu^{4} + 144 \nu^{2} + 192$$$$)/64$$ $$\beta_{12}$$ $$=$$ $$($$$$\nu^{14} - 4 \nu^{12} + 5 \nu^{10} - 20 \nu^{8} + 28 \nu^{6} - 48 \nu^{4} + 144 \nu^{2} - 192$$$$)/64$$ $$\beta_{13}$$ $$=$$ $$($$$$-\nu^{15} - 9 \nu^{11} - 24 \nu^{7} + 24 \nu^{3}$$$$)/80$$ $$\beta_{14}$$ $$=$$ $$($$$$11 \nu^{15} + 12 \nu^{13} + 39 \nu^{11} + 28 \nu^{9} + 164 \nu^{7} + 48 \nu^{5} + 496 \nu^{3} + 832 \nu$$$$)/640$$ $$\beta_{15}$$ $$=$$ $$($$$$-11 \nu^{15} + 12 \nu^{13} - 39 \nu^{11} + 28 \nu^{9} - 164 \nu^{7} + 48 \nu^{5} - 496 \nu^{3} + 832 \nu$$$$)/640$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} + \beta_{14} - \beta_{9} + \beta_{5} + \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{12} + \beta_{11} - \beta_{10} + \beta_{8}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} + 2 \beta_{13} + \beta_{9} + \beta_{5} - \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 4$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-3 \beta_{15} - 3 \beta_{14} + 3 \beta_{9} - 3 \beta_{5} + 2 \beta_{3} + 3 \beta_{2} + 3 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{12} + \beta_{11} + 5 \beta_{10} - 5 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 20 \beta_{4}$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$\beta_{15} - \beta_{14} - 2 \beta_{13} + 7 \beta_{9} + 7 \beta_{5} + 17 \beta_{2} - 17 \beta_{1}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-5 \beta_{12} + 5 \beta_{11} + 7 \beta_{10} + 7 \beta_{8} - 6 \beta_{7} - 6 \beta_{6} - 36$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-5 \beta_{15} - 5 \beta_{14} - 11 \beta_{9} + 11 \beta_{5} + 6 \beta_{3} - 19 \beta_{2} - 19 \beta_{1}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$-9 \beta_{12} - 9 \beta_{11} + 3 \beta_{10} - 3 \beta_{8} - 18 \beta_{7} + 18 \beta_{6} - 20 \beta_{4}$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$7 \beta_{15} - 7 \beta_{14} - 30 \beta_{13} + \beta_{9} + \beta_{5} - 41 \beta_{2} + 41 \beta_{1}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$-19 \beta_{12} + 19 \beta_{11} - 47 \beta_{10} - 47 \beta_{8} + 6 \beta_{7} + 6 \beta_{6} + 36$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$61 \beta_{15} + 61 \beta_{14} + 83 \beta_{9} - 83 \beta_{5} - 22 \beta_{3} - 37 \beta_{2} - 37 \beta_{1}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$\beta_{12} + \beta_{11} - 11 \beta_{10} + 11 \beta_{8} + 34 \beta_{7} - 34 \beta_{6} - 460 \beta_{4}$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$-111 \beta_{15} + 111 \beta_{14} + 46 \beta_{13} - 153 \beta_{9} - 153 \beta_{5} - 63 \beta_{2} + 63 \beta_{1}$$$$)/4$$

## Character values

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

 $$n$$ $$577$$ $$641$$ $$901$$ $$991$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
847.1
 −0.512386 − 1.31813i −1.31813 − 0.512386i 0.788026 − 1.17431i −1.17431 + 0.788026i −0.788026 + 1.17431i 1.17431 − 0.788026i 0.512386 + 1.31813i 1.31813 + 0.512386i −0.512386 + 1.31813i −1.31813 + 0.512386i 0.788026 + 1.17431i −1.17431 − 0.788026i −0.788026 − 1.17431i 1.17431 + 0.788026i 0.512386 − 1.31813i 1.31813 − 0.512386i
0 0 0 −1.83051 + 1.28422i 0 −2.94984 + 2.94984i 0 0 0
847.2 0 0 0 −1.83051 + 1.28422i 0 2.94984 2.94984i 0 0 0
847.3 0 0 0 −0.386289 2.20245i 0 −1.51606 + 1.51606i 0 0 0
847.4 0 0 0 −0.386289 2.20245i 0 1.51606 1.51606i 0 0 0
847.5 0 0 0 0.386289 + 2.20245i 0 −1.51606 + 1.51606i 0 0 0
847.6 0 0 0 0.386289 + 2.20245i 0 1.51606 1.51606i 0 0 0
847.7 0 0 0 1.83051 1.28422i 0 −2.94984 + 2.94984i 0 0 0
847.8 0 0 0 1.83051 1.28422i 0 2.94984 2.94984i 0 0 0
1423.1 0 0 0 −1.83051 1.28422i 0 −2.94984 2.94984i 0 0 0
1423.2 0 0 0 −1.83051 1.28422i 0 2.94984 + 2.94984i 0 0 0
1423.3 0 0 0 −0.386289 + 2.20245i 0 −1.51606 1.51606i 0 0 0
1423.4 0 0 0 −0.386289 + 2.20245i 0 1.51606 + 1.51606i 0 0 0
1423.5 0 0 0 0.386289 2.20245i 0 −1.51606 1.51606i 0 0 0
1423.6 0 0 0 0.386289 2.20245i 0 1.51606 + 1.51606i 0 0 0
1423.7 0 0 0 1.83051 + 1.28422i 0 −2.94984 2.94984i 0 0 0
1423.8 0 0 0 1.83051 + 1.28422i 0 2.94984 + 2.94984i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1423.8 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
5.c odd 4 1 inner
8.d odd 2 1 inner
15.e even 4 1 inner
24.f even 2 1 inner
40.k even 4 1 inner
120.q odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.bi.d 16
3.b odd 2 1 inner 1440.2.bi.d 16
4.b odd 2 1 360.2.w.d 16
5.c odd 4 1 inner 1440.2.bi.d 16
8.b even 2 1 360.2.w.d 16
8.d odd 2 1 inner 1440.2.bi.d 16
12.b even 2 1 360.2.w.d 16
15.e even 4 1 inner 1440.2.bi.d 16
20.e even 4 1 360.2.w.d 16
24.f even 2 1 inner 1440.2.bi.d 16
24.h odd 2 1 360.2.w.d 16
40.i odd 4 1 360.2.w.d 16
40.k even 4 1 inner 1440.2.bi.d 16
60.l odd 4 1 360.2.w.d 16
120.q odd 4 1 inner 1440.2.bi.d 16
120.w even 4 1 360.2.w.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.w.d 16 4.b odd 2 1
360.2.w.d 16 8.b even 2 1
360.2.w.d 16 12.b even 2 1
360.2.w.d 16 20.e even 4 1
360.2.w.d 16 24.h odd 2 1
360.2.w.d 16 40.i odd 4 1
360.2.w.d 16 60.l odd 4 1
360.2.w.d 16 120.w even 4 1
1440.2.bi.d 16 1.a even 1 1 trivial
1440.2.bi.d 16 3.b odd 2 1 inner
1440.2.bi.d 16 5.c odd 4 1 inner
1440.2.bi.d 16 8.d odd 2 1 inner
1440.2.bi.d 16 15.e even 4 1 inner
1440.2.bi.d 16 24.f even 2 1 inner
1440.2.bi.d 16 40.k even 4 1 inner
1440.2.bi.d 16 120.q odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 324 T_{7}^{4} + 6400$$ acting on $$S_{2}^{\mathrm{new}}(1440, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$( 625 + 150 T^{2} + 18 T^{4} + 6 T^{6} + T^{8} )^{2}$$
$7$ $$( 6400 + 324 T^{4} + T^{8} )^{2}$$
$11$ $$( 40 - 18 T^{2} + T^{4} )^{4}$$
$13$ $$( 102400 + 804 T^{4} + T^{8} )^{2}$$
$17$ $$( 25600 + 1796 T^{4} + T^{8} )^{2}$$
$19$ $$( 16 + T^{2} )^{8}$$
$23$ $$( 16384 + 2880 T^{4} + T^{8} )^{2}$$
$29$ $$( 800 - 58 T^{2} + T^{4} )^{4}$$
$31$ $$( 1280 + 76 T^{2} + T^{4} )^{4}$$
$37$ $$( 102400 + 804 T^{4} + T^{8} )^{2}$$
$41$ $$( 640 - 92 T^{2} + T^{4} )^{4}$$
$43$ $$( 6400 + 320 T + 8 T^{2} - 4 T^{3} + T^{4} )^{4}$$
$47$ $$( 262144 + 11520 T^{4} + T^{8} )^{2}$$
$53$ $$( 16384 + 420 T^{4} + T^{8} )^{2}$$
$59$ $$( 40 + 18 T^{2} + T^{4} )^{4}$$
$61$ $$( 8000 + 220 T^{2} + T^{4} )^{4}$$
$67$ $$( 6400 - 320 T + 8 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$71$ $$T^{16}$$
$73$ $$( 50 - 10 T + T^{2} )^{8}$$
$79$ $$( 1280 - 76 T^{2} + T^{4} )^{4}$$
$83$ $$( 104857600 + 21136 T^{4} + T^{8} )^{2}$$
$89$ $$( 2560 + 348 T^{2} + T^{4} )^{4}$$
$97$ $$( 2500 - 800 T + 128 T^{2} + 16 T^{3} + T^{4} )^{4}$$