# Properties

 Label 720.2.q.l Level $720$ Weight $2$ Character orbit 720.q Analytic conductor $5.749$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.856615824.2 Defining polynomial: $$x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} -\beta_{1} q^{5} + ( \beta_{2} - \beta_{4} ) q^{7} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{3} -\beta_{1} q^{5} + ( \beta_{2} - \beta_{4} ) q^{7} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{7} ) q^{9} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{11} + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{13} + \beta_{6} q^{15} + ( 2 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{17} + ( -\beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{19} + ( -1 + 4 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} ) q^{21} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + ( -1 + \beta_{1} ) q^{25} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} ) q^{27} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} ) q^{29} + ( -1 + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{31} + ( 1 + \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{33} + ( \beta_{1} + \beta_{3} + \beta_{4} ) q^{35} + ( 1 + 2 \beta_{1} + \beta_{3} + 3 \beta_{4} + \beta_{6} + \beta_{7} ) q^{37} + ( 4 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{39} + ( 1 - 3 \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{41} + ( -3 + \beta_{1} - \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{43} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} ) q^{45} + ( 2 - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{47} + ( -1 - 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{49} + ( -6 + 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{51} + ( 4 - 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{53} + ( \beta_{3} - \beta_{5} + \beta_{6} ) q^{55} + ( 9 \beta_{1} + 2 \beta_{4} + 3 \beta_{5} ) q^{57} + ( 2 + 5 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{59} + ( -4 + 6 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{61} + ( 3 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{63} + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{65} + ( -1 - 4 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{67} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{69} + ( -3 + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{71} + ( 3 - 2 \beta_{1} - 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{73} + ( -\beta_{4} - \beta_{6} ) q^{75} + ( -1 - 10 \beta_{1} - \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{77} + ( -6 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} + 4 \beta_{7} ) q^{79} + ( -3 + 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{7} ) q^{81} + ( 4 - 8 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} ) q^{83} + ( -1 - \beta_{1} - \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{85} + ( -4 - \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{87} + ( 4 + 2 \beta_{1} + \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{89} + ( -3 - 2 \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{91} + ( -2 + 6 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{93} + ( 1 + \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{95} + ( 1 - \beta_{1} + 3 \beta_{2} - 4 \beta_{4} + \beta_{6} + \beta_{7} ) q^{97} + ( 12 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{5} - q^{7} + O(q^{10})$$ $$8q - 4q^{5} - q^{7} + q^{11} - 4q^{13} - 3q^{15} + 10q^{17} - 2q^{19} + 7q^{23} - 4q^{25} + 18q^{27} - 7q^{29} - 2q^{31} - 3q^{33} + 2q^{35} + 12q^{37} + 6q^{39} - 12q^{41} - 11q^{43} - 3q^{45} + 7q^{47} - 3q^{49} - 39q^{51} + 24q^{53} - 2q^{55} + 27q^{57} + 11q^{59} - 19q^{61} + 33q^{63} - 4q^{65} - 10q^{67} - 9q^{69} - 24q^{71} + 18q^{73} + 3q^{75} - 32q^{77} - 24q^{79} - 12q^{81} + 23q^{83} - 5q^{85} - 24q^{87} + 42q^{89} - 28q^{91} + 18q^{93} + q^{95} - q^{97} + 84q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} - 7 \nu^{3} - 10 \nu + 2$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} - \nu^{3} + 6 \nu^{2} - 2 \nu + 4$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{4} + 2 \nu^{3} - 2 \nu^{2} + 10 \nu - 8$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + \nu^{5} + 9 \nu^{4} + 5 \nu^{3} + 22 \nu^{2} + 10$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 10 \nu^{5} + 4 \nu^{4} + 31 \nu^{3} + 22 \nu^{2} + 30 \nu + 10$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + 10 \nu^{5} - 9 \nu^{4} + 29 \nu^{3} - 20 \nu^{2} + 20 \nu - 6$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} - 11 \nu^{5} + 9 \nu^{4} - 32 \nu^{3} + 24 \nu^{2} - 10 \nu + 16$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{7} + 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} - 8$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 10 \beta_{2} + 2 \beta_{1} - 3$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-11 \beta_{7} - 18 \beta_{6} + 7 \beta_{5} - 11 \beta_{4} + 4 \beta_{3} + 35$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-7 \beta_{7} - 7 \beta_{5} - 11 \beta_{4} + 18 \beta_{3} + 50 \beta_{2} - 16 \beta_{1} + 17$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$19 \beta_{7} + 32 \beta_{6} - 13 \beta_{5} + 21 \beta_{4} - 4 \beta_{3} + 2 \beta_{1} - 57$$ $$\nu^{7}$$ $$=$$ $$($$$$39 \beta_{7} + 6 \beta_{6} + 45 \beta_{5} + 47 \beta_{4} - 80 \beta_{3} - 250 \beta_{2} + 128 \beta_{1} - 101$$$$)/3$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
241.1
 − 1.07834i − 2.33086i 2.06288i − 0.385731i 1.07834i 2.33086i − 2.06288i 0.385731i
0 −1.24624 + 1.20287i 0 −0.500000 0.866025i 0 0.433868 0.751481i 0 0.106223 2.99812i 0
241.2 0 −1.05903 1.37057i 0 −0.500000 0.866025i 0 1.51859 2.63027i 0 −0.756906 + 2.90295i 0
241.3 0 0.574618 1.63396i 0 −0.500000 0.866025i 0 −2.28651 + 3.96035i 0 −2.33963 1.87780i 0
241.4 0 1.73065 + 0.0696054i 0 −0.500000 0.866025i 0 −0.165947 + 0.287429i 0 2.99031 + 0.240925i 0
481.1 0 −1.24624 1.20287i 0 −0.500000 + 0.866025i 0 0.433868 + 0.751481i 0 0.106223 + 2.99812i 0
481.2 0 −1.05903 + 1.37057i 0 −0.500000 + 0.866025i 0 1.51859 + 2.63027i 0 −0.756906 2.90295i 0
481.3 0 0.574618 + 1.63396i 0 −0.500000 + 0.866025i 0 −2.28651 3.96035i 0 −2.33963 + 1.87780i 0
481.4 0 1.73065 0.0696054i 0 −0.500000 + 0.866025i 0 −0.165947 0.287429i 0 2.99031 0.240925i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 481.4 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.l 8
3.b odd 2 1 2160.2.q.l 8
4.b odd 2 1 360.2.q.e 8
9.c even 3 1 inner 720.2.q.l 8
9.c even 3 1 6480.2.a.cb 4
9.d odd 6 1 2160.2.q.l 8
9.d odd 6 1 6480.2.a.bz 4
12.b even 2 1 1080.2.q.e 8
36.f odd 6 1 360.2.q.e 8
36.f odd 6 1 3240.2.a.u 4
36.h even 6 1 1080.2.q.e 8
36.h even 6 1 3240.2.a.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.e 8 4.b odd 2 1
360.2.q.e 8 36.f odd 6 1
720.2.q.l 8 1.a even 1 1 trivial
720.2.q.l 8 9.c even 3 1 inner
1080.2.q.e 8 12.b even 2 1
1080.2.q.e 8 36.h even 6 1
2160.2.q.l 8 3.b odd 2 1
2160.2.q.l 8 9.d odd 6 1
3240.2.a.s 4 36.h even 6 1
3240.2.a.u 4 36.f odd 6 1
6480.2.a.bz 4 9.d odd 6 1
6480.2.a.cb 4 9.c even 3 1

## Hecke kernels

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

 $$T_{7}^{8} + \cdots$$ $$T_{11}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$81 - 18 T^{3} + 3 T^{4} - 6 T^{5} + T^{8}$$
$5$ $$( 1 + T + T^{2} )^{4}$$
$7$ $$16 + 28 T + 109 T^{2} - 113 T^{3} + 214 T^{4} - 29 T^{5} + 16 T^{6} + T^{7} + T^{8}$$
$11$ $$190096 + 8720 T + 18712 T^{2} + 32 T^{3} + 1348 T^{4} + 2 T^{5} + 43 T^{6} - T^{7} + T^{8}$$
$13$ $$256 - 320 T + 784 T^{2} + 352 T^{3} + 640 T^{4} - 56 T^{5} + 40 T^{6} + 4 T^{7} + T^{8}$$
$17$ $$( 172 + 40 T - 30 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$19$ $$( 1348 - 32 T - 84 T^{2} + T^{3} + T^{4} )^{2}$$
$23$ $$3844 - 2914 T + 2395 T^{2} - 727 T^{3} + 400 T^{4} - 115 T^{5} + 46 T^{6} - 7 T^{7} + T^{8}$$
$29$ $$42436 + 31930 T + 20935 T^{2} + 5209 T^{3} + 1516 T^{4} + 205 T^{5} + 64 T^{6} + 7 T^{7} + T^{8}$$
$31$ $$179776 + 8480 T + 25840 T^{2} - 2896 T^{3} + 3136 T^{4} - 160 T^{5} + 64 T^{6} + 2 T^{7} + T^{8}$$
$37$ $$( 144 + 252 T - 48 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$41$ $$5602689 + 1874664 T + 556254 T^{2} + 80568 T^{3} + 12771 T^{4} + 1224 T^{5} + 174 T^{6} + 12 T^{7} + T^{8}$$
$43$ $$44944 + 39008 T + 35128 T^{2} + 3560 T^{3} + 2272 T^{4} + 434 T^{5} + 115 T^{6} + 11 T^{7} + T^{8}$$
$47$ $$1149184 + 687152 T + 523441 T^{2} - 52297 T^{3} + 14440 T^{4} - 547 T^{5} + 154 T^{6} - 7 T^{7} + T^{8}$$
$53$ $$( 144 + 288 T - 48 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$59$ $$222784 + 273760 T + 367552 T^{2} - 27896 T^{3} + 10264 T^{4} - 434 T^{5} + 187 T^{6} - 11 T^{7} + T^{8}$$
$61$ $$196 - 350 T + 1927 T^{2} + 2857 T^{3} + 8188 T^{4} + 1717 T^{5} + 268 T^{6} + 19 T^{7} + T^{8}$$
$67$ $$2111209 - 552140 T + 231580 T^{2} - 6260 T^{3} + 5947 T^{4} + 160 T^{5} + 160 T^{6} + 10 T^{7} + T^{8}$$
$71$ $$( -72 - 36 T + 24 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$73$ $$( -1152 + 576 T - 48 T^{2} - 9 T^{3} + T^{4} )^{2}$$
$79$ $$1434288384 + 239956992 T + 35600256 T^{2} + 2578176 T^{3} + 204336 T^{4} + 9792 T^{5} + 696 T^{6} + 24 T^{7} + T^{8}$$
$83$ $$27836176 - 4785332 T + 1313317 T^{2} - 158345 T^{3} + 34786 T^{4} - 3953 T^{5} + 436 T^{6} - 23 T^{7} + T^{8}$$
$89$ $$( -7506 + 2241 T - 27 T^{2} - 21 T^{3} + T^{4} )^{2}$$
$97$ $$1459264 - 1483424 T + 1254304 T^{2} - 255464 T^{3} + 44080 T^{4} - 2666 T^{5} + 211 T^{6} + T^{7} + T^{8}$$