Properties

 Label 80.10.a.e Level 80 Weight 10 Character orbit 80.a Self dual yes Analytic conductor 41.203 Analytic rank 1 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 204q^{3} + 625q^{5} - 5432q^{7} + 21933q^{9} + O(q^{10})$$ $$q + 204q^{3} + 625q^{5} - 5432q^{7} + 21933q^{9} - 73932q^{11} - 114514q^{13} + 127500q^{15} + 41682q^{17} - 1057460q^{19} - 1108128q^{21} - 1599336q^{23} + 390625q^{25} + 459000q^{27} + 2184510q^{29} + 9619648q^{31} - 15082128q^{33} - 3395000q^{35} + 4799942q^{37} - 23360856q^{39} + 9531882q^{41} + 13464484q^{43} + 13708125q^{45} - 11441952q^{47} - 10846983q^{49} + 8503128q^{51} + 53615766q^{53} - 46207500q^{55} - 215721840q^{57} - 81862620q^{59} - 104691298q^{61} - 119140056q^{63} - 71571250q^{65} - 140571092q^{67} - 326264544q^{69} - 97098792q^{71} + 171848906q^{73} + 79687500q^{75} + 401598624q^{77} + 117380080q^{79} - 338071239q^{81} - 323637636q^{83} + 26051250q^{85} + 445640040q^{87} - 894379110q^{89} + 622040048q^{91} + 1962408192q^{93} - 660912500q^{95} + 232678562q^{97} - 1621550556q^{99} + O(q^{100})$$

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
0 204.000 0 625.000 0 −5432.00 0 21933.0 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.e 1
4.b odd 2 1 10.10.a.a 1
5.b even 2 1 400.10.a.a 1
5.c odd 4 2 400.10.c.b 2
8.b even 2 1 320.10.a.a 1
8.d odd 2 1 320.10.a.j 1
12.b even 2 1 90.10.a.g 1
20.d odd 2 1 50.10.a.f 1
20.e even 4 2 50.10.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.10.a.a 1 4.b odd 2 1
50.10.a.f 1 20.d odd 2 1
50.10.b.e 2 20.e even 4 2
80.10.a.e 1 1.a even 1 1 trivial
90.10.a.g 1 12.b even 2 1
320.10.a.a 1 8.b even 2 1
320.10.a.j 1 8.d odd 2 1
400.10.a.a 1 5.b even 2 1
400.10.c.b 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 204$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(80))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 204 T + 19683 T^{2}$$
$5$ $$1 - 625 T$$
$7$ $$1 + 5432 T + 40353607 T^{2}$$
$11$ $$1 + 73932 T + 2357947691 T^{2}$$
$13$ $$1 + 114514 T + 10604499373 T^{2}$$
$17$ $$1 - 41682 T + 118587876497 T^{2}$$
$19$ $$1 + 1057460 T + 322687697779 T^{2}$$
$23$ $$1 + 1599336 T + 1801152661463 T^{2}$$
$29$ $$1 - 2184510 T + 14507145975869 T^{2}$$
$31$ $$1 - 9619648 T + 26439622160671 T^{2}$$
$37$ $$1 - 4799942 T + 129961739795077 T^{2}$$
$41$ $$1 - 9531882 T + 327381934393961 T^{2}$$
$43$ $$1 - 13464484 T + 502592611936843 T^{2}$$
$47$ $$1 + 11441952 T + 1119130473102767 T^{2}$$
$53$ $$1 - 53615766 T + 3299763591802133 T^{2}$$
$59$ $$1 + 81862620 T + 8662995818654939 T^{2}$$
$61$ $$1 + 104691298 T + 11694146092834141 T^{2}$$
$67$ $$1 + 140571092 T + 27206534396294947 T^{2}$$
$71$ $$1 + 97098792 T + 45848500718449031 T^{2}$$
$73$ $$1 - 171848906 T + 58871586708267913 T^{2}$$
$79$ $$1 - 117380080 T + 119851595982618319 T^{2}$$
$83$ $$1 + 323637636 T + 186940255267540403 T^{2}$$
$89$ $$1 + 894379110 T + 350356403707485209 T^{2}$$
$97$ $$1 - 232678562 T + 760231058654565217 T^{2}$$