# Properties

 Label 720.2.z.d Level $720$ Weight $2$ Character orbit 720.z Analytic conductor $5.749$ Analytic rank $0$ Dimension $2$ 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.z (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.74922894553$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 80) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - i + 1) q^{2} - 2 i q^{4} + ( - i + 2) q^{5} + ( - 3 i - 3) q^{7} + ( - 2 i - 2) q^{8} +O(q^{10})$$ q + (-i + 1) * q^2 - 2*i * q^4 + (-i + 2) * q^5 + (-3*i - 3) * q^7 + (-2*i - 2) * q^8 $$q + ( - i + 1) q^{2} - 2 i q^{4} + ( - i + 2) q^{5} + ( - 3 i - 3) q^{7} + ( - 2 i - 2) q^{8} + ( - 3 i + 1) q^{10} + ( - i + 1) q^{11} + 2 i q^{13} - 6 q^{14} - 4 q^{16} + ( - i - 1) q^{17} + (3 i - 3) q^{19} + ( - 4 i - 2) q^{20} - 2 i q^{22} + ( - i + 1) q^{23} + ( - 4 i + 3) q^{25} + (2 i + 2) q^{26} + (6 i - 6) q^{28} + (7 i + 7) q^{29} - 2 i q^{31} + (4 i - 4) q^{32} - 2 q^{34} + ( - 3 i - 9) q^{35} - 6 i q^{37} + 6 i q^{38} + ( - 2 i - 6) q^{40} - 4 i q^{41} - 4 i q^{43} + ( - 2 i - 2) q^{44} - 2 i q^{46} + ( - 7 i + 7) q^{47} + 11 i q^{49} + ( - 7 i - 1) q^{50} + 4 q^{52} + 8 q^{53} + ( - 3 i + 1) q^{55} + 12 i q^{56} + 14 q^{58} + ( - 3 i - 3) q^{59} + (i - 1) q^{61} + ( - 2 i - 2) q^{62} + 8 i q^{64} + (4 i + 2) q^{65} + 4 i q^{67} + (2 i - 2) q^{68} + (6 i - 12) q^{70} + (3 i + 3) q^{73} + ( - 6 i - 6) q^{74} + (6 i + 6) q^{76} - 6 q^{77} + 8 q^{79} + (4 i - 8) q^{80} + ( - 4 i - 4) q^{82} + 2 q^{83} + ( - i - 3) q^{85} + ( - 4 i - 4) q^{86} - 4 q^{88} + 6 q^{89} + ( - 6 i + 6) q^{91} + ( - 2 i - 2) q^{92} - 14 i q^{94} + (9 i - 3) q^{95} + ( - 11 i - 11) q^{97} + (11 i + 11) q^{98} +O(q^{100})$$ q + (-i + 1) * q^2 - 2*i * q^4 + (-i + 2) * q^5 + (-3*i - 3) * q^7 + (-2*i - 2) * q^8 + (-3*i + 1) * q^10 + (-i + 1) * q^11 + 2*i * q^13 - 6 * q^14 - 4 * q^16 + (-i - 1) * q^17 + (3*i - 3) * q^19 + (-4*i - 2) * q^20 - 2*i * q^22 + (-i + 1) * q^23 + (-4*i + 3) * q^25 + (2*i + 2) * q^26 + (6*i - 6) * q^28 + (7*i + 7) * q^29 - 2*i * q^31 + (4*i - 4) * q^32 - 2 * q^34 + (-3*i - 9) * q^35 - 6*i * q^37 + 6*i * q^38 + (-2*i - 6) * q^40 - 4*i * q^41 - 4*i * q^43 + (-2*i - 2) * q^44 - 2*i * q^46 + (-7*i + 7) * q^47 + 11*i * q^49 + (-7*i - 1) * q^50 + 4 * q^52 + 8 * q^53 + (-3*i + 1) * q^55 + 12*i * q^56 + 14 * q^58 + (-3*i - 3) * q^59 + (i - 1) * q^61 + (-2*i - 2) * q^62 + 8*i * q^64 + (4*i + 2) * q^65 + 4*i * q^67 + (2*i - 2) * q^68 + (6*i - 12) * q^70 + (3*i + 3) * q^73 + (-6*i - 6) * q^74 + (6*i + 6) * q^76 - 6 * q^77 + 8 * q^79 + (4*i - 8) * q^80 + (-4*i - 4) * q^82 + 2 * q^83 + (-i - 3) * q^85 + (-4*i - 4) * q^86 - 4 * q^88 + 6 * q^89 + (-6*i + 6) * q^91 + (-2*i - 2) * q^92 - 14*i * q^94 + (9*i - 3) * q^95 + (-11*i - 11) * q^97 + (11*i + 11) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 4 q^{5} - 6 q^{7} - 4 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 4 * q^5 - 6 * q^7 - 4 * q^8 $$2 q + 2 q^{2} + 4 q^{5} - 6 q^{7} - 4 q^{8} + 2 q^{10} + 2 q^{11} - 12 q^{14} - 8 q^{16} - 2 q^{17} - 6 q^{19} - 4 q^{20} + 2 q^{23} + 6 q^{25} + 4 q^{26} - 12 q^{28} + 14 q^{29} - 8 q^{32} - 4 q^{34} - 18 q^{35} - 12 q^{40} - 4 q^{44} + 14 q^{47} - 2 q^{50} + 8 q^{52} + 16 q^{53} + 2 q^{55} + 28 q^{58} - 6 q^{59} - 2 q^{61} - 4 q^{62} + 4 q^{65} - 4 q^{68} - 24 q^{70} + 6 q^{73} - 12 q^{74} + 12 q^{76} - 12 q^{77} + 16 q^{79} - 16 q^{80} - 8 q^{82} + 4 q^{83} - 6 q^{85} - 8 q^{86} - 8 q^{88} + 12 q^{89} + 12 q^{91} - 4 q^{92} - 6 q^{95} - 22 q^{97} + 22 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 4 * q^5 - 6 * q^7 - 4 * q^8 + 2 * q^10 + 2 * q^11 - 12 * q^14 - 8 * q^16 - 2 * q^17 - 6 * q^19 - 4 * q^20 + 2 * q^23 + 6 * q^25 + 4 * q^26 - 12 * q^28 + 14 * q^29 - 8 * q^32 - 4 * q^34 - 18 * q^35 - 12 * q^40 - 4 * q^44 + 14 * q^47 - 2 * q^50 + 8 * q^52 + 16 * q^53 + 2 * q^55 + 28 * q^58 - 6 * q^59 - 2 * q^61 - 4 * q^62 + 4 * q^65 - 4 * q^68 - 24 * q^70 + 6 * q^73 - 12 * q^74 + 12 * q^76 - 12 * q^77 + 16 * q^79 - 16 * q^80 - 8 * q^82 + 4 * q^83 - 6 * q^85 - 8 * q^86 - 8 * q^88 + 12 * q^89 + 12 * q^91 - 4 * q^92 - 6 * q^95 - 22 * q^97 + 22 * q^98

## Character values

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

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$i$$ $$-1$$ $$i$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 − 1.00000i 1.00000i
1.00000 + 1.00000i 0 2.00000i 2.00000 + 1.00000i 0 −3.00000 + 3.00000i −2.00000 + 2.00000i 0 1.00000 + 3.00000i
667.1 1.00000 1.00000i 0 2.00000i 2.00000 1.00000i 0 −3.00000 3.00000i −2.00000 2.00000i 0 1.00000 3.00000i
 $$n$$: 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
80.s 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.z.d 2
3.b odd 2 1 80.2.s.a yes 2
5.c odd 4 1 720.2.bd.a 2
12.b even 2 1 320.2.s.a 2
15.d odd 2 1 400.2.s.a 2
15.e even 4 1 80.2.j.a 2
15.e even 4 1 400.2.j.a 2
16.f odd 4 1 720.2.bd.a 2
24.f even 2 1 640.2.s.a 2
24.h odd 2 1 640.2.s.b 2
48.i odd 4 1 320.2.j.a 2
48.i odd 4 1 640.2.j.b 2
48.k even 4 1 80.2.j.a 2
48.k even 4 1 640.2.j.a 2
60.h even 2 1 1600.2.s.a 2
60.l odd 4 1 320.2.j.a 2
60.l odd 4 1 1600.2.j.a 2
80.s even 4 1 inner 720.2.z.d 2
120.q odd 4 1 640.2.j.b 2
120.w even 4 1 640.2.j.a 2
240.t even 4 1 400.2.j.a 2
240.z odd 4 1 80.2.s.a yes 2
240.bb even 4 1 320.2.s.a 2
240.bd odd 4 1 400.2.s.a 2
240.bd odd 4 1 640.2.s.b 2
240.bf even 4 1 640.2.s.a 2
240.bf even 4 1 1600.2.s.a 2
240.bm odd 4 1 1600.2.j.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.j.a 2 15.e even 4 1
80.2.j.a 2 48.k even 4 1
80.2.s.a yes 2 3.b odd 2 1
80.2.s.a yes 2 240.z odd 4 1
320.2.j.a 2 48.i odd 4 1
320.2.j.a 2 60.l odd 4 1
320.2.s.a 2 12.b even 2 1
320.2.s.a 2 240.bb even 4 1
400.2.j.a 2 15.e even 4 1
400.2.j.a 2 240.t even 4 1
400.2.s.a 2 15.d odd 2 1
400.2.s.a 2 240.bd odd 4 1
640.2.j.a 2 48.k even 4 1
640.2.j.a 2 120.w even 4 1
640.2.j.b 2 48.i odd 4 1
640.2.j.b 2 120.q odd 4 1
640.2.s.a 2 24.f even 2 1
640.2.s.a 2 240.bf even 4 1
640.2.s.b 2 24.h odd 2 1
640.2.s.b 2 240.bd odd 4 1
720.2.z.d 2 1.a even 1 1 trivial
720.2.z.d 2 80.s even 4 1 inner
720.2.bd.a 2 5.c odd 4 1
720.2.bd.a 2 16.f odd 4 1
1600.2.j.a 2 60.l odd 4 1
1600.2.j.a 2 240.bm odd 4 1
1600.2.s.a 2 60.h even 2 1
1600.2.s.a 2 240.bf even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 6T_{7} + 18$$ T7^2 + 6*T7 + 18 $$T_{11}^{2} - 2T_{11} + 2$$ T11^2 - 2*T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 6T + 18$$
$11$ $$T^{2} - 2T + 2$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 2T + 2$$
$19$ $$T^{2} + 6T + 18$$
$23$ $$T^{2} - 2T + 2$$
$29$ $$T^{2} - 14T + 98$$
$31$ $$T^{2} + 4$$
$37$ $$T^{2} + 36$$
$41$ $$T^{2} + 16$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} - 14T + 98$$
$53$ $$(T - 8)^{2}$$
$59$ $$T^{2} + 6T + 18$$
$61$ $$T^{2} + 2T + 2$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 6T + 18$$
$79$ $$(T - 8)^{2}$$
$83$ $$(T - 2)^{2}$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 22T + 242$$