# Properties

 Label 80.10.a.d Level 80 Weight 10 Character orbit 80.a Self dual yes Analytic conductor 41.203 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 80.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$41.2028668931$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 5) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 114q^{3} - 625q^{5} - 4242q^{7} - 6687q^{9} + O(q^{10})$$ $$q + 114q^{3} - 625q^{5} - 4242q^{7} - 6687q^{9} + 46208q^{11} - 115934q^{13} - 71250q^{15} + 494842q^{17} + 1008740q^{19} - 483588q^{21} + 532554q^{23} + 390625q^{25} - 3006180q^{27} + 4196390q^{29} + 3365028q^{31} + 5267712q^{33} + 2651250q^{35} - 14931358q^{37} - 13216476q^{39} + 11056262q^{41} + 6396794q^{43} + 4179375q^{45} + 35559158q^{47} - 22359043q^{49} + 56411988q^{51} + 39738586q^{53} - 28880000q^{55} + 114996360q^{57} + 85185620q^{59} + 45748642q^{61} + 28366254q^{63} + 72458750q^{65} + 45286158q^{67} + 60711156q^{69} + 189967468q^{71} + 412170946q^{73} + 44531250q^{75} - 196014336q^{77} - 95040840q^{79} - 211084299q^{81} - 261706326q^{83} - 309276250q^{85} + 478388460q^{87} - 19938630q^{89} + 491792028q^{91} + 383613192q^{93} - 630462500q^{95} - 19503358q^{97} - 308992896q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 114.000 0 −625.000 0 −4242.00 0 −6687.00 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.10.a.d 1
4.b odd 2 1 5.10.a.a 1
5.b even 2 1 400.10.a.c 1
5.c odd 4 2 400.10.c.e 2
8.b even 2 1 320.10.a.c 1
8.d odd 2 1 320.10.a.h 1
12.b even 2 1 45.10.a.c 1
20.d odd 2 1 25.10.a.a 1
20.e even 4 2 25.10.b.a 2
28.d even 2 1 245.10.a.a 1
60.h even 2 1 225.10.a.b 1
60.l odd 4 2 225.10.b.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.a.a 1 4.b odd 2 1
25.10.a.a 1 20.d odd 2 1
25.10.b.a 2 20.e even 4 2
45.10.a.c 1 12.b even 2 1
80.10.a.d 1 1.a even 1 1 trivial
225.10.a.b 1 60.h even 2 1
225.10.b.d 2 60.l odd 4 2
245.10.a.a 1 28.d even 2 1
320.10.a.c 1 8.b even 2 1
320.10.a.h 1 8.d odd 2 1
400.10.a.c 1 5.b even 2 1
400.10.c.e 2 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 114$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(80))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 114 T + 19683 T^{2}$$
$5$ $$1 + 625 T$$
$7$ $$1 + 4242 T + 40353607 T^{2}$$
$11$ $$1 - 46208 T + 2357947691 T^{2}$$
$13$ $$1 + 115934 T + 10604499373 T^{2}$$
$17$ $$1 - 494842 T + 118587876497 T^{2}$$
$19$ $$1 - 1008740 T + 322687697779 T^{2}$$
$23$ $$1 - 532554 T + 1801152661463 T^{2}$$
$29$ $$1 - 4196390 T + 14507145975869 T^{2}$$
$31$ $$1 - 3365028 T + 26439622160671 T^{2}$$
$37$ $$1 + 14931358 T + 129961739795077 T^{2}$$
$41$ $$1 - 11056262 T + 327381934393961 T^{2}$$
$43$ $$1 - 6396794 T + 502592611936843 T^{2}$$
$47$ $$1 - 35559158 T + 1119130473102767 T^{2}$$
$53$ $$1 - 39738586 T + 3299763591802133 T^{2}$$
$59$ $$1 - 85185620 T + 8662995818654939 T^{2}$$
$61$ $$1 - 45748642 T + 11694146092834141 T^{2}$$
$67$ $$1 - 45286158 T + 27206534396294947 T^{2}$$
$71$ $$1 - 189967468 T + 45848500718449031 T^{2}$$
$73$ $$1 - 412170946 T + 58871586708267913 T^{2}$$
$79$ $$1 + 95040840 T + 119851595982618319 T^{2}$$
$83$ $$1 + 261706326 T + 186940255267540403 T^{2}$$
$89$ $$1 + 19938630 T + 350356403707485209 T^{2}$$
$97$ $$1 + 19503358 T + 760231058654565217 T^{2}$$