Properties

 Label 720.2.q.g Level $720$ Weight $2$ Character orbit 720.q Analytic conductor $5.749$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.q (of order $$3$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$5.74922894553$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 360) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + \beta_{2} q^{5} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + \beta_{2} q^{5} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{9} + ( 1 + \beta_{1} - \beta_{2} ) q^{15} -2 q^{17} + ( 2 + 4 \beta_{1} - 2 \beta_{3} ) q^{19} + ( 1 - 2 \beta_{1} - 4 \beta_{2} ) q^{21} + ( -\beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{23} + ( -1 + \beta_{2} ) q^{25} + ( -5 + \beta_{3} ) q^{27} + ( -3 + 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{29} + ( -2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{31} + ( 1 + 2 \beta_{1} - \beta_{3} ) q^{35} -6 q^{37} + ( 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 2 + 4 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} ) q^{43} + ( 1 - 2 \beta_{3} ) q^{45} + ( -7 - \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{47} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{49} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{51} + ( -2 + 8 \beta_{1} - 4 \beta_{3} ) q^{53} + ( 8 - 6 \beta_{2} - 4 \beta_{3} ) q^{57} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{59} + ( 3 - 3 \beta_{2} ) q^{61} + ( -8 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{63} + ( -5 \beta_{1} - \beta_{2} - 5 \beta_{3} ) q^{67} + ( -7 - 6 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{69} + ( -8 \beta_{1} + 4 \beta_{3} ) q^{71} + ( 4 + 8 \beta_{1} - 4 \beta_{3} ) q^{73} + ( 1 + \beta_{3} ) q^{75} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{79} + ( -4 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{81} + ( -3 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{83} -2 \beta_{2} q^{85} + ( 7 - 4 \beta_{1} - 8 \beta_{2} + 5 \beta_{3} ) q^{87} + ( -7 + 8 \beta_{1} - 4 \beta_{3} ) q^{89} + ( 2 + 4 \beta_{1} - 10 \beta_{2} + 4 \beta_{3} ) q^{93} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{95} + ( -2 + 2 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$4q - 2q^{3} + 2q^{5} + 2q^{7} + 2q^{9} + 2q^{15} - 8q^{17} + 8q^{19} - 4q^{21} - 10q^{23} - 2q^{25} - 20q^{27} - 6q^{29} + 12q^{31} + 4q^{35} - 24q^{37} + 10q^{41} + 4q^{43} + 4q^{45} - 14q^{47} + 4q^{51} - 8q^{53} + 20q^{57} - 12q^{59} + 6q^{61} - 26q^{63} - 2q^{67} - 22q^{69} + 16q^{73} + 4q^{75} - 4q^{79} + 14q^{81} - 6q^{83} - 4q^{85} + 12q^{87} - 28q^{89} - 12q^{93} + 4q^{95} - 4q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

Character values

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

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\beta_{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
241.1
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
0 −1.72474 0.158919i 0 0.500000 + 0.866025i 0 −0.724745 + 1.25529i 0 2.94949 + 0.548188i 0
241.2 0 0.724745 1.57313i 0 0.500000 + 0.866025i 0 1.72474 2.98735i 0 −1.94949 2.28024i 0
481.1 0 −1.72474 + 0.158919i 0 0.500000 0.866025i 0 −0.724745 1.25529i 0 2.94949 0.548188i 0
481.2 0 0.724745 + 1.57313i 0 0.500000 0.866025i 0 1.72474 + 2.98735i 0 −1.94949 + 2.28024i 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
9.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.q.g 4
3.b odd 2 1 2160.2.q.g 4
4.b odd 2 1 360.2.q.c 4
9.c even 3 1 inner 720.2.q.g 4
9.c even 3 1 6480.2.a.bc 2
9.d odd 6 1 2160.2.q.g 4
9.d odd 6 1 6480.2.a.bl 2
12.b even 2 1 1080.2.q.c 4
36.f odd 6 1 360.2.q.c 4
36.f odd 6 1 3240.2.a.j 2
36.h even 6 1 1080.2.q.c 4
36.h even 6 1 3240.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.c 4 4.b odd 2 1
360.2.q.c 4 36.f odd 6 1
720.2.q.g 4 1.a even 1 1 trivial
720.2.q.g 4 9.c even 3 1 inner
1080.2.q.c 4 12.b even 2 1
1080.2.q.c 4 36.h even 6 1
2160.2.q.g 4 3.b odd 2 1
2160.2.q.g 4 9.d odd 6 1
3240.2.a.j 2 36.f odd 6 1
3240.2.a.o 2 36.h even 6 1
6480.2.a.bc 2 9.c even 3 1
6480.2.a.bl 2 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{4} - 2 T_{7}^{3} + 9 T_{7}^{2} + 10 T_{7} + 25$$ $$T_{11}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 6 T + T^{2} + 2 T^{3} + T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$25 + 10 T + 9 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$( 2 + T )^{4}$$
$19$ $$( -20 - 4 T + T^{2} )^{2}$$
$23$ $$361 + 190 T + 81 T^{2} + 10 T^{3} + T^{4}$$
$29$ $$225 - 90 T + 51 T^{2} + 6 T^{3} + T^{4}$$
$31$ $$144 - 144 T + 132 T^{2} - 12 T^{3} + T^{4}$$
$37$ $$( 6 + T )^{4}$$
$41$ $$1 - 10 T + 99 T^{2} - 10 T^{3} + T^{4}$$
$43$ $$8464 + 368 T + 108 T^{2} - 4 T^{3} + T^{4}$$
$47$ $$1849 + 602 T + 153 T^{2} + 14 T^{3} + T^{4}$$
$53$ $$( -92 + 4 T + T^{2} )^{2}$$
$59$ $$144 + 144 T + 132 T^{2} + 12 T^{3} + T^{4}$$
$61$ $$( 9 - 3 T + T^{2} )^{2}$$
$67$ $$22201 - 298 T + 153 T^{2} + 2 T^{3} + T^{4}$$
$71$ $$( -96 + T^{2} )^{2}$$
$73$ $$( -80 - 8 T + T^{2} )^{2}$$
$79$ $$400 - 80 T + 36 T^{2} + 4 T^{3} + T^{4}$$
$83$ $$9 + 18 T + 33 T^{2} + 6 T^{3} + T^{4}$$
$89$ $$( -47 + 14 T + T^{2} )^{2}$$
$97$ $$( 4 + 2 T + T^{2} )^{2}$$