# Properties

 Label 720.2.t.b Level $720$ Weight $2$ Character orbit 720.t 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.t (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.74922894553$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.18939904.2 Defining polynomial: $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 240) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{4} - \beta_{5} ) q^{2} + ( -1 + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{4} -\beta_{5} q^{5} + ( \beta_{5} - \beta_{6} ) q^{7} + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{4} - \beta_{5} ) q^{2} + ( -1 + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{4} -\beta_{5} q^{5} + ( \beta_{5} - \beta_{6} ) q^{7} + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{8} + ( -\beta_{3} - \beta_{7} ) q^{10} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{11} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{13} + ( 1 + \beta_{1} + \beta_{3} + \beta_{7} ) q^{14} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{16} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{17} + ( -2 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} ) q^{19} + ( -\beta_{1} + \beta_{2} + \beta_{6} ) q^{20} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{22} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{23} -\beta_{7} q^{25} + ( 3 + \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{26} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{28} + ( -4 - 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{29} + ( 1 - \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{31} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 4 \beta_{7} ) q^{32} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{34} + ( 1 + \beta_{7} ) q^{35} + ( -2 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{37} + ( 4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} ) q^{38} + ( -1 - \beta_{1} + \beta_{3} - 2 \beta_{5} - \beta_{7} ) q^{40} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} ) q^{41} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{43} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{7} ) q^{44} + ( 2 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{46} + ( -4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{47} + 5 q^{49} + ( -1 + \beta_{2} + \beta_{6} ) q^{50} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{52} + ( 4 + 2 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{53} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{55} + ( 2 - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{56} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} ) q^{58} + ( 1 + 5 \beta_{1} + \beta_{2} - \beta_{3} - 5 \beta_{4} + 8 \beta_{5} - 2 \beta_{7} ) q^{59} + ( 1 + 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 6 \beta_{6} - 3 \beta_{7} ) q^{61} + ( -1 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{62} + ( -2 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} ) q^{64} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{65} + ( -2 - 4 \beta_{1} - 4 \beta_{4} - 2 \beta_{7} ) q^{67} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{68} + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{70} + ( 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} ) q^{71} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{73} + ( 3 - 3 \beta_{1} + \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{74} + ( -6 + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{76} + ( 2 - 2 \beta_{2} - 2 \beta_{3} ) q^{77} + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{79} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{80} + ( 12 + 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 8 \beta_{7} ) q^{82} + ( -6 + 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{6} - 2 \beta_{7} ) q^{83} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{85} + ( 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{86} + ( -4 + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{88} + ( -4 - 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} ) q^{89} + ( -2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{91} + ( -4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{92} + ( 4 + 2 \beta_{1} + 2 \beta_{2} + 8 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} ) q^{94} + ( 2 + 2 \beta_{3} - 2 \beta_{4} ) q^{95} + ( -6 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} ) q^{97} + ( 5 \beta_{4} - 5 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{4} + 12q^{8} + O(q^{10})$$ $$8q - 4q^{4} + 12q^{8} - 8q^{11} + 8q^{13} + 4q^{14} - 8q^{17} + 8q^{19} + 8q^{20} + 16q^{22} + 20q^{26} - 8q^{28} - 24q^{29} + 8q^{31} + 16q^{34} + 8q^{35} - 8q^{37} + 16q^{38} - 4q^{40} + 16q^{44} + 24q^{46} + 40q^{49} - 4q^{50} + 16q^{52} + 16q^{56} - 8q^{59} - 16q^{61} - 28q^{62} + 8q^{64} + 8q^{65} + 8q^{68} + 4q^{70} + 36q^{74} - 40q^{76} + 8q^{77} - 40q^{79} - 16q^{80} + 64q^{82} - 32q^{83} + 8q^{85} - 16q^{86} - 16q^{88} - 8q^{91} - 16q^{92} + 32q^{94} + 16q^{95} - 48q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{7} - 7 \nu^{6} + 24 \nu^{5} - 42 \nu^{4} + 59 \nu^{3} - 48 \nu^{2} + 24 \nu - 5$$ $$\beta_{2}$$ $$=$$ $$-2 \nu^{7} + 7 \nu^{6} - 24 \nu^{5} + 43 \nu^{4} - 61 \nu^{3} + 54 \nu^{2} - 29 \nu + 8$$ $$\beta_{3}$$ $$=$$ $$-3 \nu^{7} + 10 \nu^{6} - 35 \nu^{5} + 60 \nu^{4} - 87 \nu^{3} + 73 \nu^{2} - 42 \nu + 11$$ $$\beta_{4}$$ $$=$$ $$-3 \nu^{7} + 11 \nu^{6} - 38 \nu^{5} + 70 \nu^{4} - 102 \nu^{3} + 91 \nu^{2} - 53 \nu + 13$$ $$\beta_{5}$$ $$=$$ $$5 \nu^{7} - 17 \nu^{6} + 60 \nu^{5} - 105 \nu^{4} + 155 \nu^{3} - 133 \nu^{2} + 77 \nu - 19$$ $$\beta_{6}$$ $$=$$ $$-5 \nu^{7} + 18 \nu^{6} - 63 \nu^{5} + 115 \nu^{4} - 170 \nu^{3} + 152 \nu^{2} - 89 \nu + 23$$ $$\beta_{7}$$ $$=$$ $$-8 \nu^{7} + 28 \nu^{6} - 98 \nu^{5} + 175 \nu^{4} - 256 \nu^{3} + 223 \nu^{2} - 126 \nu + 31$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + 3 \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} - 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{7} - \beta_{6} + 7 \beta_{5} - \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + 3 \beta_{1} - 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$11 \beta_{7} - 15 \beta_{6} + 3 \beta_{5} + 11 \beta_{4} - \beta_{3} - 5 \beta_{2} + 9 \beta_{1} + 14$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-13 \beta_{7} - 11 \beta_{6} - 29 \beta_{5} + 17 \beta_{4} - 23 \beta_{3} + 13 \beta_{2} - 3 \beta_{1} + 18$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-67 \beta_{7} + 59 \beta_{6} - 41 \beta_{5} - 29 \beta_{4} - 15 \beta_{3} + 37 \beta_{2} - 47 \beta_{1} - 48$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-7 \beta_{7} + 113 \beta_{6} + 97 \beta_{5} - 105 \beta_{4} + 91 \beta_{3} - 31 \beta_{2} - 39 \beta_{1} - 122$$$$)/2$$

## Character values

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

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$-\beta_{7}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 0.5 + 0.0297061i 0.5 − 0.691860i 0.5 − 1.44392i 0.5 + 2.10607i 0.5 − 0.0297061i 0.5 + 0.691860i 0.5 + 1.44392i 0.5 − 2.10607i
−0.874559 1.11137i 0 −0.470294 + 1.94392i −0.707107 0.707107i 0 1.41421i 2.57172 1.17740i 0 −0.167452 + 1.40426i
181.2 −0.635665 + 1.26330i 0 −1.19186 1.60607i 0.707107 + 0.707107i 0 1.41421i 2.78658 0.484753i 0 −1.34277 + 0.443806i
181.3 0.167452 + 1.40426i 0 −1.94392 + 0.470294i −0.707107 0.707107i 0 1.41421i −0.985930 2.65103i 0 0.874559 1.11137i
181.4 1.34277 + 0.443806i 0 1.60607 + 1.19186i 0.707107 + 0.707107i 0 1.41421i 1.62764 + 2.31318i 0 0.635665 + 1.26330i
541.1 −0.874559 + 1.11137i 0 −0.470294 1.94392i −0.707107 + 0.707107i 0 1.41421i 2.57172 + 1.17740i 0 −0.167452 1.40426i
541.2 −0.635665 1.26330i 0 −1.19186 + 1.60607i 0.707107 0.707107i 0 1.41421i 2.78658 + 0.484753i 0 −1.34277 0.443806i
541.3 0.167452 1.40426i 0 −1.94392 0.470294i −0.707107 + 0.707107i 0 1.41421i −0.985930 + 2.65103i 0 0.874559 + 1.11137i
541.4 1.34277 0.443806i 0 1.60607 1.19186i 0.707107 0.707107i 0 1.41421i 1.62764 2.31318i 0 0.635665 1.26330i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 541.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.t.b 8
3.b odd 2 1 240.2.s.b 8
4.b odd 2 1 2880.2.t.b 8
12.b even 2 1 960.2.s.b 8
16.e even 4 1 inner 720.2.t.b 8
16.f odd 4 1 2880.2.t.b 8
24.f even 2 1 1920.2.s.d 8
24.h odd 2 1 1920.2.s.c 8
48.i odd 4 1 240.2.s.b 8
48.i odd 4 1 1920.2.s.c 8
48.k even 4 1 960.2.s.b 8
48.k even 4 1 1920.2.s.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.s.b 8 3.b odd 2 1
240.2.s.b 8 48.i odd 4 1
720.2.t.b 8 1.a even 1 1 trivial
720.2.t.b 8 16.e even 4 1 inner
960.2.s.b 8 12.b even 2 1
960.2.s.b 8 48.k even 4 1
1920.2.s.c 8 24.h odd 2 1
1920.2.s.c 8 48.i odd 4 1
1920.2.s.d 8 24.f even 2 1
1920.2.s.d 8 48.k even 4 1
2880.2.t.b 8 4.b odd 2 1
2880.2.t.b 8 16.f odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 8 T^{2} - 8 T^{3} + 2 T^{4} - 4 T^{5} + 2 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 1 + T^{4} )^{2}$$
$7$ $$( 2 + T^{2} )^{4}$$
$11$ $$64 - 128 T + 128 T^{2} - 16 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$13$ $$16 + 64 T + 128 T^{2} + 32 T^{3} - 8 T^{4} - 16 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$17$ $$( 8 - 12 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$19$ $$256 + 512 T + 512 T^{2} - 384 T^{3} + 224 T^{4} + 96 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$
$23$ $$200704 + 73728 T^{2} + 5248 T^{4} + 128 T^{6} + T^{8}$$
$29$ $$135424 - 11776 T + 512 T^{2} + 11392 T^{3} + 7136 T^{4} + 1888 T^{5} + 288 T^{6} + 24 T^{7} + T^{8}$$
$31$ $$( -28 + 88 T - 40 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$37$ $$414736 - 154560 T + 28800 T^{2} + 6368 T^{3} + 1016 T^{4} - 144 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$41$ $$17707264 + 1276672 T^{2} + 31072 T^{4} + 304 T^{6} + T^{8}$$
$43$ $$12544 + 28672 T + 32768 T^{2} + 18432 T^{3} + 5408 T^{4} + 256 T^{5} + T^{8}$$
$47$ $$( -224 + 416 T - 152 T^{2} + T^{4} )^{2}$$
$53$ $$73984 + 33312 T^{4} + T^{8}$$
$59$ $$7529536 + 6453888 T + 2765952 T^{2} + 482944 T^{3} + 43904 T^{4} + 784 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$61$ $$4343056 - 1667200 T + 320000 T^{2} + 27456 T^{3} + 1608 T^{4} - 416 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$67$ $$1183744 + 4224 T^{4} + T^{8}$$
$71$ $$5161984 + 699904 T^{2} + 23168 T^{4} + 272 T^{6} + T^{8}$$
$73$ $$50176 + 27136 T^{2} + 4224 T^{4} + 176 T^{6} + T^{8}$$
$79$ $$( 164 + 264 T + 120 T^{2} + 20 T^{3} + T^{4} )^{2}$$
$83$ $$2166784 + 1130496 T + 294912 T^{2} - 14336 T^{3} + 9344 T^{4} + 3328 T^{5} + 512 T^{6} + 32 T^{7} + T^{8}$$
$89$ $$118026496 + 6544640 T^{2} + 102752 T^{4} + 592 T^{6} + T^{8}$$
$97$ $$( -736 - 32 T + 120 T^{2} + 24 T^{3} + T^{4} )^{2}$$