# Properties

 Label 720.3.bh.k Level $720$ Weight $3$ Character orbit 720.bh 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.bh (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.6185790339$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{5} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{7} +O(q^{10})$$ $$q + ( 1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{5} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{7} + ( 4 - 3 \beta_{1} + 3 \beta_{3} ) q^{11} + ( -8 + 8 \beta_{2} + 2 \beta_{3} ) q^{13} + ( 10 - 6 \beta_{1} + 10 \beta_{2} ) q^{17} + ( 6 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{19} + ( 14 - 14 \beta_{2} - 2 \beta_{3} ) q^{23} + ( 4 + 2 \beta_{1} + 3 \beta_{2} + 14 \beta_{3} ) q^{25} + ( 7 \beta_{1} - 18 \beta_{2} + 7 \beta_{3} ) q^{29} + ( 4 + 6 \beta_{1} - 6 \beta_{3} ) q^{31} + ( -10 + 5 \beta_{1} - 10 \beta_{2} - 5 \beta_{3} ) q^{35} + ( 16 + 18 \beta_{1} + 16 \beta_{2} ) q^{37} + ( 14 + 6 \beta_{1} - 6 \beta_{3} ) q^{41} + ( 2 - 2 \beta_{2} + 20 \beta_{3} ) q^{43} + ( 32 - 10 \beta_{1} + 32 \beta_{2} ) q^{47} + ( -4 \beta_{1} - 35 \beta_{2} - 4 \beta_{3} ) q^{49} + ( -14 + 14 \beta_{2} + 12 \beta_{3} ) q^{53} + ( 31 - 2 \beta_{1} - 3 \beta_{2} + 16 \beta_{3} ) q^{55} + ( 31 \beta_{1} + 36 \beta_{2} + 31 \beta_{3} ) q^{59} + ( 50 + 18 \beta_{1} - 18 \beta_{3} ) q^{61} + ( 28 + 14 \beta_{1} + 26 \beta_{2} - 22 \beta_{3} ) q^{65} + ( 50 + 4 \beta_{1} + 50 \beta_{2} ) q^{67} -68 q^{71} + ( 19 - 19 \beta_{2} + 48 \beta_{3} ) q^{73} + ( -22 + 14 \beta_{1} - 22 \beta_{2} ) q^{77} + ( -10 \beta_{1} - 10 \beta_{3} ) q^{79} + ( -4 + 4 \beta_{2} + 14 \beta_{3} ) q^{83} + ( 58 - 36 \beta_{1} + 16 \beta_{2} + 8 \beta_{3} ) q^{85} + ( 36 \beta_{1} + 6 \beta_{2} + 36 \beta_{3} ) q^{89} + ( 4 - 14 \beta_{1} + 14 \beta_{3} ) q^{91} + ( 36 + 18 \beta_{1} - 48 \beta_{2} - 24 \beta_{3} ) q^{95} + ( -5 + 16 \beta_{1} - 5 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{5} - 4q^{7} + O(q^{10})$$ $$4q + 4q^{5} - 4q^{7} + 16q^{11} - 32q^{13} + 40q^{17} + 56q^{23} + 16q^{25} + 16q^{31} - 40q^{35} + 64q^{37} + 56q^{41} + 8q^{43} + 128q^{47} - 56q^{53} + 124q^{55} + 200q^{61} + 112q^{65} + 200q^{67} - 272q^{71} + 76q^{73} - 88q^{77} - 16q^{83} + 232q^{85} + 16q^{91} + 144q^{95} - 20q^{97} + O(q^{100})$$

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

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
433.1
 1.22474 − 1.22474i −1.22474 + 1.22474i 1.22474 + 1.22474i −1.22474 − 1.22474i
0 0 0 −2.67423 + 4.22474i 0 1.44949 1.44949i 0 0 0
433.2 0 0 0 4.67423 + 1.77526i 0 −3.44949 + 3.44949i 0 0 0
577.1 0 0 0 −2.67423 4.22474i 0 1.44949 + 1.44949i 0 0 0
577.2 0 0 0 4.67423 1.77526i 0 −3.44949 3.44949i 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
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.3.bh.k 4
3.b odd 2 1 240.3.bg.a 4
4.b odd 2 1 45.3.g.b 4
5.c odd 4 1 inner 720.3.bh.k 4
12.b even 2 1 15.3.f.a 4
15.d odd 2 1 1200.3.bg.k 4
15.e even 4 1 240.3.bg.a 4
15.e even 4 1 1200.3.bg.k 4
20.d odd 2 1 225.3.g.a 4
20.e even 4 1 45.3.g.b 4
20.e even 4 1 225.3.g.a 4
24.f even 2 1 960.3.bg.i 4
24.h odd 2 1 960.3.bg.h 4
36.f odd 6 2 405.3.l.f 8
36.h even 6 2 405.3.l.h 8
60.h even 2 1 75.3.f.c 4
60.l odd 4 1 15.3.f.a 4
60.l odd 4 1 75.3.f.c 4
120.q odd 4 1 960.3.bg.i 4
120.w even 4 1 960.3.bg.h 4
180.v odd 12 2 405.3.l.h 8
180.x even 12 2 405.3.l.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.f.a 4 12.b even 2 1
15.3.f.a 4 60.l odd 4 1
45.3.g.b 4 4.b odd 2 1
45.3.g.b 4 20.e even 4 1
75.3.f.c 4 60.h even 2 1
75.3.f.c 4 60.l odd 4 1
225.3.g.a 4 20.d odd 2 1
225.3.g.a 4 20.e even 4 1
240.3.bg.a 4 3.b odd 2 1
240.3.bg.a 4 15.e even 4 1
405.3.l.f 8 36.f odd 6 2
405.3.l.f 8 180.x even 12 2
405.3.l.h 8 36.h even 6 2
405.3.l.h 8 180.v odd 12 2
720.3.bh.k 4 1.a even 1 1 trivial
720.3.bh.k 4 5.c odd 4 1 inner
960.3.bg.h 4 24.h odd 2 1
960.3.bg.h 4 120.w even 4 1
960.3.bg.i 4 24.f even 2 1
960.3.bg.i 4 120.q odd 4 1
1200.3.bg.k 4 15.d odd 2 1
1200.3.bg.k 4 15.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 $$S_{3}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{4} + 4 T_{7}^{3} + 8 T_{7}^{2} - 40 T_{7} + 100$$ $$T_{11}^{2} - 8 T_{11} - 38$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$625 - 100 T - 4 T^{3} + T^{4}$$
$7$ $$100 - 40 T + 8 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$( -38 - 8 T + T^{2} )^{2}$$
$13$ $$13456 + 3712 T + 512 T^{2} + 32 T^{3} + T^{4}$$
$17$ $$8464 - 3680 T + 800 T^{2} - 40 T^{3} + T^{4}$$
$19$ $$32400 + 504 T^{2} + T^{4}$$
$23$ $$144400 - 21280 T + 1568 T^{2} - 56 T^{3} + T^{4}$$
$29$ $$900 + 1236 T^{2} + T^{4}$$
$31$ $$( -200 - 8 T + T^{2} )^{2}$$
$37$ $$211600 + 29440 T + 2048 T^{2} - 64 T^{3} + T^{4}$$
$41$ $$( -20 - 28 T + T^{2} )^{2}$$
$43$ $$1420864 + 9536 T + 32 T^{2} - 8 T^{3} + T^{4}$$
$47$ $$3055504 - 223744 T + 8192 T^{2} - 128 T^{3} + T^{4}$$
$53$ $$1600 - 2240 T + 1568 T^{2} + 56 T^{3} + T^{4}$$
$59$ $$19980900 + 14124 T^{2} + T^{4}$$
$61$ $$( 556 - 100 T + T^{2} )^{2}$$
$67$ $$24522304 - 990400 T + 20000 T^{2} - 200 T^{3} + T^{4}$$
$71$ $$( 68 + T )^{4}$$
$73$ $$38316100 + 470440 T + 2888 T^{2} - 76 T^{3} + T^{4}$$
$79$ $$( 600 + T^{2} )^{2}$$
$83$ $$309136 - 8896 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$89$ $$59907600 + 15624 T^{2} + T^{4}$$
$97$ $$515524 - 14360 T + 200 T^{2} + 20 T^{3} + T^{4}$$