# Properties

 Label 80.2.j.a Level $80$ Weight $2$ Character orbit 80.j Analytic conductor $0.639$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 80.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.638803216170$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - i + 1) q^{2} + 2 i q^{3} - 2 i q^{4} + (2 i + 1) q^{5} + (2 i + 2) q^{6} + ( - 3 i - 3) q^{7} + ( - 2 i - 2) q^{8} - q^{9} +O(q^{10})$$ q + (-i + 1) * q^2 + 2*i * q^3 - 2*i * q^4 + (2*i + 1) * q^5 + (2*i + 2) * q^6 + (-3*i - 3) * q^7 + (-2*i - 2) * q^8 - q^9 $$q + ( - i + 1) q^{2} + 2 i q^{3} - 2 i q^{4} + (2 i + 1) q^{5} + (2 i + 2) q^{6} + ( - 3 i - 3) q^{7} + ( - 2 i - 2) q^{8} - q^{9} + (i + 3) q^{10} + ( - i - 1) q^{11} + 4 q^{12} - 2 q^{13} - 6 q^{14} + (2 i - 4) q^{15} - 4 q^{16} + (i + 1) q^{17} + (i - 1) q^{18} + (3 i + 3) q^{19} + ( - 2 i + 4) q^{20} + ( - 6 i + 6) q^{21} - 2 q^{22} + (i - 1) q^{23} + ( - 4 i + 4) q^{24} + (4 i - 3) q^{25} + (2 i - 2) q^{26} + 4 i q^{27} + (6 i - 6) q^{28} + ( - 7 i + 7) q^{29} + (6 i - 2) q^{30} + 2 i q^{31} + (4 i - 4) q^{32} + ( - 2 i + 2) q^{33} + 2 q^{34} + ( - 9 i + 3) q^{35} + 2 i q^{36} - 6 q^{37} + 6 q^{38} - 4 i q^{39} + ( - 6 i + 2) q^{40} - 4 i q^{41} - 12 i q^{42} + 4 q^{43} + (2 i - 2) q^{44} + ( - 2 i - 1) q^{45} + 2 i q^{46} + ( - 7 i + 7) q^{47} - 8 i q^{48} + 11 i q^{49} + (7 i + 1) q^{50} + (2 i - 2) q^{51} + 4 i q^{52} + 8 i q^{53} + (4 i + 4) q^{54} + ( - 3 i + 1) q^{55} + 12 i q^{56} + (6 i - 6) q^{57} - 14 i q^{58} + (3 i - 3) q^{59} + (8 i + 4) q^{60} + ( - i - 1) q^{61} + (2 i + 2) q^{62} + (3 i + 3) q^{63} + 8 i q^{64} + ( - 4 i - 2) q^{65} - 4 i q^{66} + 4 q^{67} + ( - 2 i + 2) q^{68} + ( - 2 i - 2) q^{69} + ( - 12 i - 6) q^{70} + (2 i + 2) q^{72} + ( - 3 i - 3) q^{73} + (6 i - 6) q^{74} + ( - 6 i - 8) q^{75} + ( - 6 i + 6) q^{76} + 6 i q^{77} + ( - 4 i - 4) q^{78} - 8 q^{79} + ( - 8 i - 4) q^{80} - 11 q^{81} + ( - 4 i - 4) q^{82} + 2 i q^{83} + ( - 12 i - 12) q^{84} + (3 i - 1) q^{85} + ( - 4 i + 4) q^{86} + (14 i + 14) q^{87} + 4 i q^{88} + 6 q^{89} + ( - i - 3) q^{90} + (6 i + 6) q^{91} + (2 i + 2) q^{92} - 4 q^{93} - 14 i q^{94} + (9 i - 3) q^{95} + ( - 8 i - 8) q^{96} + ( - 11 i - 11) q^{97} + (11 i + 11) q^{98} + (i + 1) q^{99} +O(q^{100})$$ q + (-i + 1) * q^2 + 2*i * q^3 - 2*i * q^4 + (2*i + 1) * q^5 + (2*i + 2) * q^6 + (-3*i - 3) * q^7 + (-2*i - 2) * q^8 - q^9 + (i + 3) * q^10 + (-i - 1) * q^11 + 4 * q^12 - 2 * q^13 - 6 * q^14 + (2*i - 4) * q^15 - 4 * q^16 + (i + 1) * q^17 + (i - 1) * q^18 + (3*i + 3) * q^19 + (-2*i + 4) * q^20 + (-6*i + 6) * q^21 - 2 * q^22 + (i - 1) * q^23 + (-4*i + 4) * q^24 + (4*i - 3) * q^25 + (2*i - 2) * q^26 + 4*i * q^27 + (6*i - 6) * q^28 + (-7*i + 7) * q^29 + (6*i - 2) * q^30 + 2*i * q^31 + (4*i - 4) * q^32 + (-2*i + 2) * q^33 + 2 * q^34 + (-9*i + 3) * q^35 + 2*i * q^36 - 6 * q^37 + 6 * q^38 - 4*i * q^39 + (-6*i + 2) * q^40 - 4*i * q^41 - 12*i * q^42 + 4 * q^43 + (2*i - 2) * q^44 + (-2*i - 1) * q^45 + 2*i * q^46 + (-7*i + 7) * q^47 - 8*i * q^48 + 11*i * q^49 + (7*i + 1) * q^50 + (2*i - 2) * q^51 + 4*i * q^52 + 8*i * q^53 + (4*i + 4) * q^54 + (-3*i + 1) * q^55 + 12*i * q^56 + (6*i - 6) * q^57 - 14*i * q^58 + (3*i - 3) * q^59 + (8*i + 4) * q^60 + (-i - 1) * q^61 + (2*i + 2) * q^62 + (3*i + 3) * q^63 + 8*i * q^64 + (-4*i - 2) * q^65 - 4*i * q^66 + 4 * q^67 + (-2*i + 2) * q^68 + (-2*i - 2) * q^69 + (-12*i - 6) * q^70 + (2*i + 2) * q^72 + (-3*i - 3) * q^73 + (6*i - 6) * q^74 + (-6*i - 8) * q^75 + (-6*i + 6) * q^76 + 6*i * q^77 + (-4*i - 4) * q^78 - 8 * q^79 + (-8*i - 4) * q^80 - 11 * q^81 + (-4*i - 4) * q^82 + 2*i * q^83 + (-12*i - 12) * q^84 + (3*i - 1) * q^85 + (-4*i + 4) * q^86 + (14*i + 14) * q^87 + 4*i * q^88 + 6 * q^89 + (-i - 3) * q^90 + (6*i + 6) * q^91 + (2*i + 2) * q^92 - 4 * q^93 - 14*i * q^94 + (9*i - 3) * q^95 + (-8*i - 8) * q^96 + (-11*i - 11) * q^97 + (11*i + 11) * q^98 + (i + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{5} + 4 q^{6} - 6 q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^5 + 4 * q^6 - 6 * q^7 - 4 * q^8 - 2 * q^9 $$2 q + 2 q^{2} + 2 q^{5} + 4 q^{6} - 6 q^{7} - 4 q^{8} - 2 q^{9} + 6 q^{10} - 2 q^{11} + 8 q^{12} - 4 q^{13} - 12 q^{14} - 8 q^{15} - 8 q^{16} + 2 q^{17} - 2 q^{18} + 6 q^{19} + 8 q^{20} + 12 q^{21} - 4 q^{22} - 2 q^{23} + 8 q^{24} - 6 q^{25} - 4 q^{26} - 12 q^{28} + 14 q^{29} - 4 q^{30} - 8 q^{32} + 4 q^{33} + 4 q^{34} + 6 q^{35} - 12 q^{37} + 12 q^{38} + 4 q^{40} + 8 q^{43} - 4 q^{44} - 2 q^{45} + 14 q^{47} + 2 q^{50} - 4 q^{51} + 8 q^{54} + 2 q^{55} - 12 q^{57} - 6 q^{59} + 8 q^{60} - 2 q^{61} + 4 q^{62} + 6 q^{63} - 4 q^{65} + 8 q^{67} + 4 q^{68} - 4 q^{69} - 12 q^{70} + 4 q^{72} - 6 q^{73} - 12 q^{74} - 16 q^{75} + 12 q^{76} - 8 q^{78} - 16 q^{79} - 8 q^{80} - 22 q^{81} - 8 q^{82} - 24 q^{84} - 2 q^{85} + 8 q^{86} + 28 q^{87} + 12 q^{89} - 6 q^{90} + 12 q^{91} + 4 q^{92} - 8 q^{93} - 6 q^{95} - 16 q^{96} - 22 q^{97} + 22 q^{98} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^5 + 4 * q^6 - 6 * q^7 - 4 * q^8 - 2 * q^9 + 6 * q^10 - 2 * q^11 + 8 * q^12 - 4 * q^13 - 12 * q^14 - 8 * q^15 - 8 * q^16 + 2 * q^17 - 2 * q^18 + 6 * q^19 + 8 * q^20 + 12 * q^21 - 4 * q^22 - 2 * q^23 + 8 * q^24 - 6 * q^25 - 4 * q^26 - 12 * q^28 + 14 * q^29 - 4 * q^30 - 8 * q^32 + 4 * q^33 + 4 * q^34 + 6 * q^35 - 12 * q^37 + 12 * q^38 + 4 * q^40 + 8 * q^43 - 4 * q^44 - 2 * q^45 + 14 * q^47 + 2 * q^50 - 4 * q^51 + 8 * q^54 + 2 * q^55 - 12 * q^57 - 6 * q^59 + 8 * q^60 - 2 * q^61 + 4 * q^62 + 6 * q^63 - 4 * q^65 + 8 * q^67 + 4 * q^68 - 4 * q^69 - 12 * q^70 + 4 * q^72 - 6 * q^73 - 12 * q^74 - 16 * q^75 + 12 * q^76 - 8 * q^78 - 16 * q^79 - 8 * q^80 - 22 * q^81 - 8 * q^82 - 24 * q^84 - 2 * q^85 + 8 * q^86 + 28 * q^87 + 12 * q^89 - 6 * q^90 + 12 * q^91 + 4 * q^92 - 8 * q^93 - 6 * q^95 - 16 * q^96 - 22 * q^97 + 22 * q^98 + 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$21$$ $$31$$ $$\chi(n)$$ $$i$$ $$-i$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 − 1.00000i 1.00000i
1.00000 + 1.00000i 2.00000i 2.00000i 1.00000 2.00000i 2.00000 2.00000i −3.00000 + 3.00000i −2.00000 + 2.00000i −1.00000 3.00000 1.00000i
67.1 1.00000 1.00000i 2.00000i 2.00000i 1.00000 + 2.00000i 2.00000 + 2.00000i −3.00000 3.00000i −2.00000 2.00000i −1.00000 3.00000 + 1.00000i
 $$n$$: 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.j even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.2.j.a 2
3.b odd 2 1 720.2.bd.a 2
4.b odd 2 1 320.2.j.a 2
5.b even 2 1 400.2.j.a 2
5.c odd 4 1 80.2.s.a yes 2
5.c odd 4 1 400.2.s.a 2
8.b even 2 1 640.2.j.a 2
8.d odd 2 1 640.2.j.b 2
15.e even 4 1 720.2.z.d 2
16.e even 4 1 320.2.s.a 2
16.e even 4 1 640.2.s.a 2
16.f odd 4 1 80.2.s.a yes 2
16.f odd 4 1 640.2.s.b 2
20.d odd 2 1 1600.2.j.a 2
20.e even 4 1 320.2.s.a 2
20.e even 4 1 1600.2.s.a 2
40.i odd 4 1 640.2.s.b 2
40.k even 4 1 640.2.s.a 2
48.k even 4 1 720.2.z.d 2
80.i odd 4 1 640.2.j.b 2
80.i odd 4 1 1600.2.j.a 2
80.j even 4 1 inner 80.2.j.a 2
80.k odd 4 1 400.2.s.a 2
80.q even 4 1 1600.2.s.a 2
80.s even 4 1 400.2.j.a 2
80.s even 4 1 640.2.j.a 2
80.t odd 4 1 320.2.j.a 2
240.bd odd 4 1 720.2.bd.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.j.a 2 1.a even 1 1 trivial
80.2.j.a 2 80.j even 4 1 inner
80.2.s.a yes 2 5.c odd 4 1
80.2.s.a yes 2 16.f odd 4 1
320.2.j.a 2 4.b odd 2 1
320.2.j.a 2 80.t odd 4 1
320.2.s.a 2 16.e even 4 1
320.2.s.a 2 20.e even 4 1
400.2.j.a 2 5.b even 2 1
400.2.j.a 2 80.s even 4 1
400.2.s.a 2 5.c odd 4 1
400.2.s.a 2 80.k odd 4 1
640.2.j.a 2 8.b even 2 1
640.2.j.a 2 80.s even 4 1
640.2.j.b 2 8.d odd 2 1
640.2.j.b 2 80.i odd 4 1
640.2.s.a 2 16.e even 4 1
640.2.s.a 2 40.k even 4 1
640.2.s.b 2 16.f odd 4 1
640.2.s.b 2 40.i odd 4 1
720.2.z.d 2 15.e even 4 1
720.2.z.d 2 48.k even 4 1
720.2.bd.a 2 3.b odd 2 1
720.2.bd.a 2 240.bd odd 4 1
1600.2.j.a 2 20.d odd 2 1
1600.2.j.a 2 80.i odd 4 1
1600.2.s.a 2 20.e even 4 1
1600.2.s.a 2 80.q even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(80, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 2$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2} - 2T + 5$$
$7$ $$T^{2} + 6T + 18$$
$11$ $$T^{2} + 2T + 2$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} - 2T + 2$$
$19$ $$T^{2} - 6T + 18$$
$23$ $$T^{2} + 2T + 2$$
$29$ $$T^{2} - 14T + 98$$
$31$ $$T^{2} + 4$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} + 16$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} - 14T + 98$$
$53$ $$T^{2} + 64$$
$59$ $$T^{2} + 6T + 18$$
$61$ $$T^{2} + 2T + 2$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 6T + 18$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 4$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 22T + 242$$
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