# Properties

 Label 80.10.a.f Level $80$ Weight $10$ Character orbit 80.a Self dual yes Analytic conductor $41.203$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{1009})$$ Defining polynomial: $$x^{2} - x - 252$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 5) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{1009}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -130 - \beta ) q^{3} + 625 q^{5} + ( -850 - 107 \beta ) q^{7} + ( 1253 + 260 \beta ) q^{9} +O(q^{10})$$ $$q + ( -130 - \beta ) q^{3} + 625 q^{5} + ( -850 - 107 \beta ) q^{7} + ( 1253 + 260 \beta ) q^{9} + ( -11992 + 950 \beta ) q^{11} + ( 57510 + 676 \beta ) q^{13} + ( -81250 - 625 \beta ) q^{15} + ( 206410 - 6428 \beta ) q^{17} + ( 148260 - 1420 \beta ) q^{19} + ( 542352 + 14760 \beta ) q^{21} + ( 524610 + 9699 \beta ) q^{23} + 390625 q^{25} + ( 1346540 - 15370 \beta ) q^{27} + ( -1833490 + 53480 \beta ) q^{29} + ( -806572 + 77350 \beta ) q^{31} + ( -2275240 - 111508 \beta ) q^{33} + ( -531250 - 66875 \beta ) q^{35} + ( -10560970 - 102648 \beta ) q^{37} + ( -10204636 - 145390 \beta ) q^{39} + ( -13478638 + 77900 \beta ) q^{41} + ( -26444850 + 12899 \beta ) q^{43} + ( 783125 + 162500 \beta ) q^{45} + ( -29206090 - 261667 \beta ) q^{47} + ( 6577057 + 181900 \beta ) q^{49} + ( -889892 + 629230 \beta ) q^{51} + ( -19517570 - 568724 \beta ) q^{53} + ( -7495000 + 593750 \beta ) q^{55} + ( -13542680 + 36340 \beta ) q^{57} + ( 27497780 - 772360 \beta ) q^{59} + ( -137289858 + 346000 \beta ) q^{61} + ( -113346570 - 355071 \beta ) q^{63} + ( 35943750 + 422500 \beta ) q^{65} + ( 159290 + 1208353 \beta ) q^{67} + ( -107344464 - 1785480 \beta ) q^{69} + ( 3565468 - 3139250 \beta ) q^{71} + ( 60429090 + 4415476 \beta ) q^{73} + ( -50781250 - 390625 \beta ) q^{75} + ( -400066200 + 475644 \beta ) q^{77} + ( -3438760 + 9387820 \beta ) q^{79} + ( -137679679 - 4466020 \beta ) q^{81} + ( -701174370 - 1374201 \beta ) q^{83} + ( 129006250 - 4017500 \beta ) q^{85} + ( 22508420 - 5118910 \beta ) q^{87} + ( 415044330 + 6690840 \beta ) q^{89} + ( -340815452 - 6728170 \beta ) q^{91} + ( -207330240 - 9248928 \beta ) q^{93} + ( 92662500 - 887500 \beta ) q^{95} + ( 319197290 - 1311108 \beta ) q^{97} + ( 981866024 - 1927570 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 260q^{3} + 1250q^{5} - 1700q^{7} + 2506q^{9} + O(q^{10})$$ $$2q - 260q^{3} + 1250q^{5} - 1700q^{7} + 2506q^{9} - 23984q^{11} + 115020q^{13} - 162500q^{15} + 412820q^{17} + 296520q^{19} + 1084704q^{21} + 1049220q^{23} + 781250q^{25} + 2693080q^{27} - 3666980q^{29} - 1613144q^{31} - 4550480q^{33} - 1062500q^{35} - 21121940q^{37} - 20409272q^{39} - 26957276q^{41} - 52889700q^{43} + 1566250q^{45} - 58412180q^{47} + 13154114q^{49} - 1779784q^{51} - 39035140q^{53} - 14990000q^{55} - 27085360q^{57} + 54995560q^{59} - 274579716q^{61} - 226693140q^{63} + 71887500q^{65} + 318580q^{67} - 214688928q^{69} + 7130936q^{71} + 120858180q^{73} - 101562500q^{75} - 800132400q^{77} - 6877520q^{79} - 275359358q^{81} - 1402348740q^{83} + 258012500q^{85} + 45016840q^{87} + 830088660q^{89} - 681630904q^{91} - 414660480q^{93} + 185325000q^{95} + 638394580q^{97} + 1963732048q^{99} + O(q^{100})$$

## 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
 16.3824 −15.3824
0 −193.530 0 625.000 0 −7647.66 0 17770.7 0
1.2 0 −66.4705 0 625.000 0 5947.66 0 −15264.7 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.f 2
4.b odd 2 1 5.10.a.b 2
5.b even 2 1 400.10.a.t 2
5.c odd 4 2 400.10.c.p 4
8.b even 2 1 320.10.a.s 2
8.d odd 2 1 320.10.a.k 2
12.b even 2 1 45.10.a.f 2
20.d odd 2 1 25.10.a.b 2
20.e even 4 2 25.10.b.b 4
28.d even 2 1 245.10.a.d 2
60.h even 2 1 225.10.a.h 2
60.l odd 4 2 225.10.b.h 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 260 T_{3} + 12864$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(80))$$.