# Properties

 Label 720.4.f.j Level $720$ Weight $4$ Character orbit 720.f Analytic conductor $42.481$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 720.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$42.4813752041$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{41})$$ Defining polynomial: $$x^{4} + 21 x^{2} + 100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 15) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} - \beta_{3} ) q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{7} +O(q^{10})$$ $$q + ( -1 - \beta_{2} - \beta_{3} ) q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{7} + ( -20 - \beta_{2} - 2 \beta_{3} ) q^{11} + ( -\beta_{1} - 4 \beta_{2} ) q^{13} + ( -8 \beta_{1} - 3 \beta_{2} ) q^{17} + ( 32 - 4 \beta_{2} - 8 \beta_{3} ) q^{19} + ( -\beta_{1} + 3 \beta_{2} ) q^{23} + ( 61 + 5 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} ) q^{25} + ( -166 + 7 \beta_{2} + 14 \beta_{3} ) q^{29} + ( -28 + 2 \beta_{2} + 4 \beta_{3} ) q^{31} + ( -80 - 15 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{35} + ( 33 \beta_{1} + 6 \beta_{2} ) q^{37} + ( 202 + 2 \beta_{2} + 4 \beta_{3} ) q^{41} + ( 41 \beta_{1} - 7 \beta_{2} ) q^{43} + ( -27 \beta_{1} + 19 \beta_{2} ) q^{47} + ( -41 + 6 \beta_{2} + 12 \beta_{3} ) q^{49} + ( -14 \beta_{1} + 27 \beta_{2} ) q^{53} + ( 204 + 5 \beta_{1} + 14 \beta_{2} + 24 \beta_{3} ) q^{55} + ( 92 + \beta_{2} + 2 \beta_{3} ) q^{59} + ( 154 + 16 \beta_{2} + 32 \beta_{3} ) q^{61} + ( -248 + 30 \beta_{1} - 13 \beta_{2} + 22 \beta_{3} ) q^{65} + ( -53 \beta_{1} - 23 \beta_{2} ) q^{67} + ( -48 + 30 \beta_{2} + 60 \beta_{3} ) q^{71} + ( -96 \beta_{1} - 42 \beta_{2} ) q^{73} + ( -66 \beta_{1} - 12 \beta_{2} ) q^{77} + ( 140 + 50 \beta_{2} + 100 \beta_{3} ) q^{79} + ( 119 \beta_{1} + 15 \beta_{2} ) q^{83} + ( -128 + 95 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} ) q^{85} + ( 582 - 24 \beta_{2} - 48 \beta_{3} ) q^{89} + ( -528 + 42 \beta_{2} + 84 \beta_{3} ) q^{91} + ( 704 + 20 \beta_{1} - 56 \beta_{2} - 16 \beta_{3} ) q^{95} + ( -122 \beta_{1} - 2 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{5} + O(q^{10})$$ $$4 q - 6 q^{5} - 84 q^{11} + 112 q^{19} + 256 q^{25} - 636 q^{29} - 104 q^{31} - 300 q^{35} + 816 q^{41} - 140 q^{49} + 864 q^{55} + 372 q^{59} + 680 q^{61} - 948 q^{65} - 72 q^{71} + 760 q^{79} - 508 q^{85} + 2232 q^{89} - 1944 q^{91} + 2784 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 21 x^{2} + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{3} + 32 \nu$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$4 \nu^{3} + 34 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{3} + 15 \nu^{2} - 17 \nu + 160$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + 2 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{3} + \beta_{2} - 64$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$16 \beta_{2} - 17 \beta_{1}$$$$)/6$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 2.70156i − 2.70156i − 3.70156i 3.70156i
0 0 0 −11.1047 1.29844i 0 16.2094i 0 0 0
289.2 0 0 0 −11.1047 + 1.29844i 0 16.2094i 0 0 0
289.3 0 0 0 8.10469 7.70156i 0 22.2094i 0 0 0
289.4 0 0 0 8.10469 + 7.70156i 0 22.2094i 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.4.f.j 4
3.b odd 2 1 240.4.f.f 4
4.b odd 2 1 45.4.b.b 4
5.b even 2 1 inner 720.4.f.j 4
12.b even 2 1 15.4.b.a 4
15.d odd 2 1 240.4.f.f 4
15.e even 4 1 1200.4.a.bn 2
15.e even 4 1 1200.4.a.bt 2
20.d odd 2 1 45.4.b.b 4
20.e even 4 1 225.4.a.i 2
20.e even 4 1 225.4.a.o 2
24.f even 2 1 960.4.f.q 4
24.h odd 2 1 960.4.f.p 4
60.h even 2 1 15.4.b.a 4
60.l odd 4 1 75.4.a.c 2
60.l odd 4 1 75.4.a.f 2
120.i odd 2 1 960.4.f.p 4
120.m even 2 1 960.4.f.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 12.b even 2 1
15.4.b.a 4 60.h even 2 1
45.4.b.b 4 4.b odd 2 1
45.4.b.b 4 20.d odd 2 1
75.4.a.c 2 60.l odd 4 1
75.4.a.f 2 60.l odd 4 1
225.4.a.i 2 20.e even 4 1
225.4.a.o 2 20.e even 4 1
240.4.f.f 4 3.b odd 2 1
240.4.f.f 4 15.d odd 2 1
720.4.f.j 4 1.a even 1 1 trivial
720.4.f.j 4 5.b even 2 1 inner
960.4.f.p 4 24.h odd 2 1
960.4.f.p 4 120.i odd 2 1
960.4.f.q 4 24.f even 2 1
960.4.f.q 4 120.m even 2 1
1200.4.a.bn 2 15.e even 4 1
1200.4.a.bt 2 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 $$S_{4}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{4} + 756 T_{7}^{2} + 129600$$ $$T_{11}^{2} + 42 T_{11} + 72$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$15625 + 750 T - 110 T^{2} + 6 T^{3} + T^{4}$$
$7$ $$129600 + 756 T^{2} + T^{4}$$
$11$ $$( 72 + 42 T + T^{2} )^{2}$$
$13$ $$1327104 + 3780 T^{2} + T^{4}$$
$17$ $$2483776 + 7252 T^{2} + T^{4}$$
$19$ $$( -5120 - 56 T + T^{2} )^{2}$$
$23$ $$6400 + 2464 T^{2} + T^{4}$$
$29$ $$( 7200 + 318 T + T^{2} )^{2}$$
$31$ $$( -800 + 52 T + T^{2} )^{2}$$
$37$ $$41990400 + 106596 T^{2} + T^{4}$$
$41$ $$( 40140 - 408 T + T^{2} )^{2}$$
$43$ $$8256266496 + 196128 T^{2} + T^{4}$$
$47$ $$6186766336 + 189712 T^{2} + T^{4}$$
$53$ $$813390400 + 218644 T^{2} + T^{4}$$
$59$ $$( 8280 - 186 T + T^{2} )^{2}$$
$61$ $$( -65564 - 340 T + T^{2} )^{2}$$
$67$ $$9419867136 + 341712 T^{2} + T^{4}$$
$71$ $$( -331776 + 36 T + T^{2} )^{2}$$
$73$ $$104976000000 + 1126224 T^{2} + T^{4}$$
$79$ $$( -886400 - 380 T + T^{2} )^{2}$$
$83$ $$40558737664 + 1371040 T^{2} + T^{4}$$
$89$ $$( 98820 - 1116 T + T^{2} )^{2}$$
$97$ $$196199387136 + 1475712 T^{2} + T^{4}$$