# Properties

 Label 1440.2.o.a Level $1440$ Weight $2$ Character orbit 1440.o Analytic conductor $11.498$ Analytic rank $0$ Dimension $12$ 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.o (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.4984578911$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.426337261060096.1 Defining polynomial: $$x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{10}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + \beta_{9} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + \beta_{9} q^{7} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{11} + ( \beta_{7} - \beta_{8} ) q^{13} + \beta_{6} q^{17} -2 \beta_{7} q^{19} + ( \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{11} ) q^{23} + ( -1 + \beta_{1} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{25} + ( \beta_{2} + \beta_{3} + 2 \beta_{11} ) q^{29} + ( \beta_{1} + 2 \beta_{7} - \beta_{8} + \beta_{10} ) q^{31} + ( 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{11} ) q^{35} + ( \beta_{1} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{37} + ( -\beta_{5} + 2 \beta_{11} ) q^{41} + ( -4 - 2 \beta_{1} + 2 \beta_{9} + 2 \beta_{10} ) q^{43} + ( 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{47} + ( -1 + \beta_{1} - 2 \beta_{9} - \beta_{10} ) q^{49} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} ) q^{53} + ( 4 + \beta_{1} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{55} + ( -\beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{59} + ( -2 + 2 \beta_{1} - 2 \beta_{10} ) q^{61} + ( \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{65} + ( 4 - 2 \beta_{1} + 2 \beta_{10} ) q^{67} + ( -2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{6} ) q^{71} + ( \beta_{1} - 4 \beta_{8} + \beta_{10} ) q^{73} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{77} + ( 3 \beta_{1} + 2 \beta_{7} + \beta_{8} + 3 \beta_{10} ) q^{79} + ( 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{5} + 2 \beta_{11} ) q^{83} + ( 2 - \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} ) q^{85} + ( \beta_{5} + 2 \beta_{11} ) q^{89} + ( -\beta_{1} - 4 \beta_{7} + 3 \beta_{8} - \beta_{10} ) q^{91} + ( -2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{11} ) q^{95} + ( \beta_{1} + 4 \beta_{7} + 2 \beta_{8} + \beta_{10} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + O(q^{10})$$ $$12q - 8q^{25} - 64q^{43} - 4q^{49} + 48q^{55} - 8q^{61} + 32q^{67} + 20q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{11} + 12 \nu^{9} + 16 \nu^{8} + 11 \nu^{7} - 28 \nu^{6} - 44 \nu^{5} + 52 \nu^{4} + 100 \nu^{3} + 56 \nu^{2} - 96 \nu - 288$$$$)/80$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{11} + \nu^{10} + 22 \nu^{9} + 2 \nu^{8} - 19 \nu^{7} - 67 \nu^{6} - 22 \nu^{5} + 110 \nu^{4} + 172 \nu^{3} - 40 \nu^{2} - 272 \nu - 352$$$$)/160$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{11} + 5 \nu^{10} + 8 \nu^{9} - 6 \nu^{8} + 4 \nu^{7} - 27 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} + 60 \nu^{3} + 104 \nu^{2} + 16 \nu - 32$$$$)/160$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{11} + 4 \nu^{10} + \nu^{9} + 2 \nu^{8} - 7 \nu^{7} - 20 \nu^{6} + \nu^{5} + 18 \nu^{4} + 18 \nu^{3} - 16 \nu^{2} - 72 \nu$$$$)/80$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{10} - \nu^{9} + 2 \nu^{7} + 3 \nu^{6} - 5 \nu^{5} - 4 \nu^{4} - 2 \nu^{3} + 8 \nu^{2} + 24 \nu$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{11} - 8 \nu^{10} + 18 \nu^{9} + 16 \nu^{8} + 29 \nu^{7} - 60 \nu^{6} - 102 \nu^{5} + 4 \nu^{4} + 24 \nu^{3} + 352 \nu^{2} + 64 \nu - 320$$$$)/160$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{11} - 12 \nu^{10} - 20 \nu^{9} - 12 \nu^{8} + 35 \nu^{7} + 48 \nu^{6} + 36 \nu^{5} - 136 \nu^{4} - 164 \nu^{3} - 48 \nu^{2} + 432 \nu + 448$$$$)/160$$ $$\beta_{8}$$ $$=$$ $$($$$$-3 \nu^{11} - \nu^{10} + 4 \nu^{8} + 5 \nu^{7} - \nu^{6} - 12 \nu^{5} - 8 \nu^{4} - 12 \nu^{3} + 36 \nu^{2} + 16 \nu + 64$$$$)/40$$ $$\beta_{9}$$ $$=$$ $$($$$$-7 \nu^{11} - 12 \nu^{10} + 12 \nu^{9} + 44 \nu^{8} + 21 \nu^{7} - 40 \nu^{6} - 108 \nu^{5} - 24 \nu^{4} + 196 \nu^{3} + 208 \nu^{2} + 176 \nu - 320$$$$)/160$$ $$\beta_{10}$$ $$=$$ $$($$$$2 \nu^{11} + \nu^{10} + 2 \nu^{9} - 3 \nu^{8} - 4 \nu^{7} + 3 \nu^{6} - 2 \nu^{5} + 5 \nu^{4} + 2 \nu^{3} - 30 \nu^{2} - 12 \nu - 72$$$$)/20$$ $$\beta_{11}$$ $$=$$ $$($$$$-11 \nu^{11} - 22 \nu^{10} - 30 \nu^{9} + 28 \nu^{8} + 85 \nu^{7} + 38 \nu^{6} - 94 \nu^{5} - 256 \nu^{4} - 64 \nu^{3} + 352 \nu^{2} + 352 \nu + 448$$$$)/160$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{11} + \beta_{9} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{10} - \beta_{8} - 2 \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{11} + \beta_{9} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} + 4$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{11} - 3 \beta_{10} - 2 \beta_{9} + 6 \beta_{5} - \beta_{3} - \beta_{2} + 3 \beta_{1} + 2$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{11} - 4 \beta_{10} + \beta_{9} - 4 \beta_{8} + \beta_{7} - 7 \beta_{5} - 7 \beta_{4} + \beta_{3} - \beta_{2} - 8$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{10} + 5 \beta_{8} - 6 \beta_{7} - 5 \beta_{6} - 14 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$3 \beta_{11} - 9 \beta_{9} - 2 \beta_{8} + 9 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} + 16 \beta_{1} + 4$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-5 \beta_{11} - 5 \beta_{10} + 10 \beta_{9} - 14 \beta_{5} - 15 \beta_{3} - 15 \beta_{2} + 5 \beta_{1} - 2$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-3 \beta_{11} + 12 \beta_{10} + 7 \beta_{9} + 16 \beta_{8} + 7 \beta_{7} + 4 \beta_{6} - 33 \beta_{5} - 33 \beta_{4} + 7 \beta_{3} + \beta_{2} + 32$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$15 \beta_{10} + 11 \beta_{8} - 2 \beta_{7} - 11 \beta_{6} + 38 \beta_{4} + 37 \beta_{3} - 37 \beta_{2} + 15 \beta_{1}$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$\beta_{11} + \beta_{9} - 50 \beta_{8} - \beta_{7} + 22 \beta_{6} - 7 \beta_{5} + 7 \beta_{4} - 21 \beta_{3} - 23 \beta_{2} + 8 \beta_{1} + 100$$$$)/4$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1439.1
 −0.0912546 + 1.41127i −0.0912546 − 1.41127i −0.760198 + 1.19252i −0.760198 − 1.19252i −1.35818 + 0.394157i −1.35818 − 0.394157i −0.394157 − 1.35818i −0.394157 + 1.35818i 1.19252 + 0.760198i 1.19252 − 0.760198i 1.41127 + 0.0912546i 1.41127 − 0.0912546i
0 0 0 −2.20963 0.342849i 0 2.64002 0 0 0
1439.2 0 0 0 −2.20963 + 0.342849i 0 2.64002 0 0 0
1439.3 0 0 0 −1.24561 1.85700i 0 0.864641 0 0 0
1439.4 0 0 0 −1.24561 + 1.85700i 0 0.864641 0 0 0
1439.5 0 0 0 −0.256912 2.22126i 0 −3.50466 0 0 0
1439.6 0 0 0 −0.256912 + 2.22126i 0 −3.50466 0 0 0
1439.7 0 0 0 0.256912 2.22126i 0 −3.50466 0 0 0
1439.8 0 0 0 0.256912 + 2.22126i 0 −3.50466 0 0 0
1439.9 0 0 0 1.24561 1.85700i 0 0.864641 0 0 0
1439.10 0 0 0 1.24561 + 1.85700i 0 0.864641 0 0 0
1439.11 0 0 0 2.20963 0.342849i 0 2.64002 0 0 0
1439.12 0 0 0 2.20963 + 0.342849i 0 2.64002 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1439.12 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
20.d odd 2 1 inner
60.h 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.o.a 12
3.b odd 2 1 inner 1440.2.o.a 12
4.b odd 2 1 1440.2.o.b yes 12
5.b even 2 1 1440.2.o.b yes 12
5.c odd 4 1 7200.2.h.l 12
5.c odd 4 1 7200.2.h.m 12
8.b even 2 1 2880.2.o.f 12
8.d odd 2 1 2880.2.o.e 12
12.b even 2 1 1440.2.o.b yes 12
15.d odd 2 1 1440.2.o.b yes 12
15.e even 4 1 7200.2.h.l 12
15.e even 4 1 7200.2.h.m 12
20.d odd 2 1 inner 1440.2.o.a 12
20.e even 4 1 7200.2.h.l 12
20.e even 4 1 7200.2.h.m 12
24.f even 2 1 2880.2.o.e 12
24.h odd 2 1 2880.2.o.f 12
40.e odd 2 1 2880.2.o.f 12
40.f even 2 1 2880.2.o.e 12
60.h even 2 1 inner 1440.2.o.a 12
60.l odd 4 1 7200.2.h.l 12
60.l odd 4 1 7200.2.h.m 12
120.i odd 2 1 2880.2.o.e 12
120.m even 2 1 2880.2.o.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.o.a 12 1.a even 1 1 trivial
1440.2.o.a 12 3.b odd 2 1 inner
1440.2.o.a 12 20.d odd 2 1 inner
1440.2.o.a 12 60.h even 2 1 inner
1440.2.o.b yes 12 4.b odd 2 1
1440.2.o.b yes 12 5.b even 2 1
1440.2.o.b yes 12 12.b even 2 1
1440.2.o.b yes 12 15.d odd 2 1
2880.2.o.e 12 8.d odd 2 1
2880.2.o.e 12 24.f even 2 1
2880.2.o.e 12 40.f even 2 1
2880.2.o.e 12 120.i odd 2 1
2880.2.o.f 12 8.b even 2 1
2880.2.o.f 12 24.h odd 2 1
2880.2.o.f 12 40.e odd 2 1
2880.2.o.f 12 120.m even 2 1
7200.2.h.l 12 5.c odd 4 1
7200.2.h.l 12 15.e even 4 1
7200.2.h.l 12 20.e even 4 1
7200.2.h.l 12 60.l odd 4 1
7200.2.h.m 12 5.c odd 4 1
7200.2.h.m 12 15.e even 4 1
7200.2.h.m 12 20.e even 4 1
7200.2.h.m 12 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{3} - 10 T_{7} + 8$$ acting on $$S_{2}^{\mathrm{new}}(1440, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$15625 + 2500 T^{2} - 425 T^{4} - 152 T^{6} - 17 T^{8} + 4 T^{10} + T^{12}$$
$7$ $$( 8 - 10 T + T^{3} )^{4}$$
$11$ $$( -512 + 228 T^{2} - 28 T^{4} + T^{6} )^{2}$$
$13$ $$( 16 + 52 T^{2} + 32 T^{4} + T^{6} )^{2}$$
$17$ $$( -128 + 288 T^{2} - 34 T^{4} + T^{6} )^{2}$$
$19$ $$( 4096 + 1600 T^{2} + 80 T^{4} + T^{6} )^{2}$$
$23$ $$( 512 + 528 T^{2} + 88 T^{4} + T^{6} )^{2}$$
$29$ $$( 3200 + 2144 T^{2} + 98 T^{4} + T^{6} )^{2}$$
$31$ $$( 6400 + 1424 T^{2} + 88 T^{4} + T^{6} )^{2}$$
$37$ $$( 64 + 132 T^{2} + 44 T^{4} + T^{6} )^{2}$$
$41$ $$( 13448 + 2828 T^{2} + 102 T^{4} + T^{6} )^{2}$$
$43$ $$( -512 + 8 T + 16 T^{2} + T^{3} )^{4}$$
$47$ $$( 51200 + 5696 T^{2} + 176 T^{4} + T^{6} )^{2}$$
$53$ $$( -991232 + 30464 T^{2} - 306 T^{4} + T^{6} )^{2}$$
$59$ $$( -359552 + 15652 T^{2} - 220 T^{4} + T^{6} )^{2}$$
$61$ $$( -328 - 100 T + 2 T^{2} + T^{3} )^{4}$$
$67$ $$( 512 - 80 T - 8 T^{2} + T^{3} )^{4}$$
$71$ $$( -204800 + 15616 T^{2} - 256 T^{4} + T^{6} )^{2}$$
$73$ $$( 215296 + 15872 T^{2} + 276 T^{4} + T^{6} )^{2}$$
$79$ $$( 215296 + 29584 T^{2} + 344 T^{4} + T^{6} )^{2}$$
$83$ $$( 204800 + 15616 T^{2} + 256 T^{4} + T^{6} )^{2}$$
$89$ $$( 200 + 1356 T^{2} + 118 T^{4} + T^{6} )^{2}$$
$97$ $$( 719104 + 24704 T^{2} + 276 T^{4} + T^{6} )^{2}$$