# Properties

 Label 720.3.l.c Level $720$ Weight $3$ Character orbit 720.l Analytic conductor $19.619$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 720.l (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.6185790339$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4x^{2} + 9$$ x^4 - 4*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 180) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{5} + ( - \beta_{3} + 4) q^{7}+O(q^{10})$$ q - b2 * q^5 + (-b3 + 4) * q^7 $$q - \beta_{2} q^{5} + ( - \beta_{3} + 4) q^{7} + (6 \beta_{2} - \beta_1) q^{11} + (\beta_{3} + 2) q^{13} - 4 \beta_1 q^{17} + ( - 2 \beta_{3} - 8) q^{19} + (6 \beta_{2} - 2 \beta_1) q^{23} - 5 q^{25} + ( - 6 \beta_{2} + 8 \beta_1) q^{29} + ( - 2 \beta_{3} - 2) q^{31} + ( - 4 \beta_{2} + 5 \beta_1) q^{35} + ( - 3 \beta_{3} - 34) q^{37} + (24 \beta_{2} - 3 \beta_1) q^{41} + (2 \beta_{3} - 20) q^{43} + (12 \beta_{2} + 14 \beta_1) q^{47} + ( - 8 \beta_{3} + 57) q^{49} + (24 \beta_{2} + 10 \beta_1) q^{53} + ( - \beta_{3} + 30) q^{55} + ( - 18 \beta_{2} - 17 \beta_1) q^{59} + ( - 6 \beta_{3} - 10) q^{61} + ( - 2 \beta_{2} - 5 \beta_1) q^{65} + 76 q^{67} + (12 \beta_{2} - 12 \beta_1) q^{71} + ( - 6 \beta_{3} + 38) q^{73} + (42 \beta_{2} - 34 \beta_1) q^{77} + ( - 6 \beta_{3} - 50) q^{79} + (24 \beta_{2} - 2 \beta_1) q^{83} - 4 \beta_{3} q^{85} + (12 \beta_{2} + 21 \beta_1) q^{89} + (2 \beta_{3} - 82) q^{91} + (8 \beta_{2} + 10 \beta_1) q^{95} + (2 \beta_{3} - 106) q^{97}+O(q^{100})$$ q - b2 * q^5 + (-b3 + 4) * q^7 + (6*b2 - b1) * q^11 + (b3 + 2) * q^13 - 4*b1 * q^17 + (-2*b3 - 8) * q^19 + (6*b2 - 2*b1) * q^23 - 5 * q^25 + (-6*b2 + 8*b1) * q^29 + (-2*b3 - 2) * q^31 + (-4*b2 + 5*b1) * q^35 + (-3*b3 - 34) * q^37 + (24*b2 - 3*b1) * q^41 + (2*b3 - 20) * q^43 + (12*b2 + 14*b1) * q^47 + (-8*b3 + 57) * q^49 + (24*b2 + 10*b1) * q^53 + (-b3 + 30) * q^55 + (-18*b2 - 17*b1) * q^59 + (-6*b3 - 10) * q^61 + (-2*b2 - 5*b1) * q^65 + 76 * q^67 + (12*b2 - 12*b1) * q^71 + (-6*b3 + 38) * q^73 + (42*b2 - 34*b1) * q^77 + (-6*b3 - 50) * q^79 + (24*b2 - 2*b1) * q^83 - 4*b3 * q^85 + (12*b2 + 21*b1) * q^89 + (2*b3 - 82) * q^91 + (8*b2 + 10*b1) * q^95 + (2*b3 - 106) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 16 q^{7}+O(q^{10})$$ 4 * q + 16 * q^7 $$4 q + 16 q^{7} + 8 q^{13} - 32 q^{19} - 20 q^{25} - 8 q^{31} - 136 q^{37} - 80 q^{43} + 228 q^{49} + 120 q^{55} - 40 q^{61} + 304 q^{67} + 152 q^{73} - 200 q^{79} - 328 q^{91} - 424 q^{97}+O(q^{100})$$ 4 * q + 16 * q^7 + 8 * q^13 - 32 * q^19 - 20 * q^25 - 8 * q^31 - 136 * q^37 - 80 * q^43 + 228 * q^49 + 120 * q^55 - 40 * q^61 + 304 * q^67 + 152 * q^73 - 200 * q^79 - 328 * q^91 - 424 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu$$ v^3 - v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 7\nu$$ -v^3 + 7*v
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 6$$ (b3 + b1) / 6 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$( \beta_{3} + 7\beta_1 ) / 6$$ (b3 + 7*b1) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 1.58114 + 0.707107i −1.58114 − 0.707107i 1.58114 − 0.707107i −1.58114 + 0.707107i
0 0 0 2.23607i 0 −5.48683 0 0 0
161.2 0 0 0 2.23607i 0 13.4868 0 0 0
161.3 0 0 0 2.23607i 0 −5.48683 0 0 0
161.4 0 0 0 2.23607i 0 13.4868 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.3.l.c 4
3.b odd 2 1 inner 720.3.l.c 4
4.b odd 2 1 180.3.g.a 4
5.b even 2 1 3600.3.l.n 4
5.c odd 4 2 3600.3.c.k 8
8.b even 2 1 2880.3.l.f 4
8.d odd 2 1 2880.3.l.b 4
12.b even 2 1 180.3.g.a 4
15.d odd 2 1 3600.3.l.n 4
15.e even 4 2 3600.3.c.k 8
20.d odd 2 1 900.3.g.d 4
20.e even 4 2 900.3.b.b 8
24.f even 2 1 2880.3.l.b 4
24.h odd 2 1 2880.3.l.f 4
36.f odd 6 2 1620.3.o.f 8
36.h even 6 2 1620.3.o.f 8
60.h even 2 1 900.3.g.d 4
60.l odd 4 2 900.3.b.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.g.a 4 4.b odd 2 1
180.3.g.a 4 12.b even 2 1
720.3.l.c 4 1.a even 1 1 trivial
720.3.l.c 4 3.b odd 2 1 inner
900.3.b.b 8 20.e even 4 2
900.3.b.b 8 60.l odd 4 2
900.3.g.d 4 20.d odd 2 1
900.3.g.d 4 60.h even 2 1
1620.3.o.f 8 36.f odd 6 2
1620.3.o.f 8 36.h even 6 2
2880.3.l.b 4 8.d odd 2 1
2880.3.l.b 4 24.f even 2 1
2880.3.l.f 4 8.b even 2 1
2880.3.l.f 4 24.h odd 2 1
3600.3.c.k 8 5.c odd 4 2
3600.3.c.k 8 15.e even 4 2
3600.3.l.n 4 5.b even 2 1
3600.3.l.n 4 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 8T_{7} - 74$$ acting on $$S_{3}^{\mathrm{new}}(720, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$(T^{2} - 8 T - 74)^{2}$$
$11$ $$T^{4} + 396 T^{2} + 26244$$
$13$ $$(T^{2} - 4 T - 86)^{2}$$
$17$ $$(T^{2} + 288)^{2}$$
$19$ $$(T^{2} + 16 T - 296)^{2}$$
$23$ $$T^{4} + 504 T^{2} + 11664$$
$29$ $$T^{4} + 2664 T^{2} + 944784$$
$31$ $$(T^{2} + 4 T - 356)^{2}$$
$37$ $$(T^{2} + 68 T + 346)^{2}$$
$41$ $$T^{4} + 6084 T^{2} + \cdots + 7387524$$
$43$ $$(T^{2} + 40 T + 40)^{2}$$
$47$ $$T^{4} + 8496 T^{2} + \cdots + 7884864$$
$53$ $$T^{4} + 9360 T^{2} + \cdots + 1166400$$
$59$ $$T^{4} + 13644 T^{2} + \cdots + 12830724$$
$61$ $$(T^{2} + 20 T - 3140)^{2}$$
$67$ $$(T - 76)^{4}$$
$71$ $$T^{4} + 6624 T^{2} + \cdots + 3504384$$
$73$ $$(T^{2} - 76 T - 1796)^{2}$$
$79$ $$(T^{2} + 100 T - 740)^{2}$$
$83$ $$T^{4} + 5904 T^{2} + \cdots + 7884864$$
$89$ $$T^{4} + 17316 T^{2} + \cdots + 52099524$$
$97$ $$(T^{2} + 212 T + 10876)^{2}$$