# Properties

 Label 80.10.a.k Level $80$ Weight $10$ Character orbit 80.a Self dual yes Analytic conductor $41.203$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.7117.1 Defining polynomial: $$x^{3} - x^{2} - 19 x - 22$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}\cdot 3^{3}\cdot 5$$ Twist minimal: no (minimal twist has level 40) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -28 - \beta_{1} ) q^{3} + 625 q^{5} + ( 1840 - 17 \beta_{1} + \beta_{2} ) q^{7} + ( 15693 + 76 \beta_{1} + 4 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -28 - \beta_{1} ) q^{3} + 625 q^{5} + ( 1840 - 17 \beta_{1} + \beta_{2} ) q^{7} + ( 15693 + 76 \beta_{1} + 4 \beta_{2} ) q^{9} + ( -1852 - 292 \beta_{1} - \beta_{2} ) q^{11} + ( -27698 - 28 \beta_{1} - 16 \beta_{2} ) q^{13} + ( -17500 - 625 \beta_{1} ) q^{15} + ( 122354 - 1596 \beta_{1} ) q^{17} + ( -496372 + 378 \beta_{1} - 81 \beta_{2} ) q^{19} + ( 524576 - 5248 \beta_{1} + 60 \beta_{2} ) q^{21} + ( 166640 + 1417 \beta_{1} + 319 \beta_{2} ) q^{23} + 390625 q^{25} + ( -2565144 - 16554 \beta_{1} - 336 \beta_{2} ) q^{27} + ( 1744894 - 6128 \beta_{1} - 380 \beta_{2} ) q^{29} + ( -4236304 - 6022 \beta_{1} - 238 \beta_{2} ) q^{31} + ( 10164688 + 20092 \beta_{1} + 1176 \beta_{2} ) q^{33} + ( 1150000 - 10625 \beta_{1} + 625 \beta_{2} ) q^{35} + ( 7241478 + 50680 \beta_{1} - 1544 \beta_{2} ) q^{37} + ( 1935608 + 96626 \beta_{1} + 240 \beta_{2} ) q^{39} + ( 9146826 + 3188 \beta_{1} - 556 \beta_{2} ) q^{41} + ( 7739420 + 162615 \beta_{1} - 1134 \beta_{2} ) q^{43} + ( 9808125 + 47500 \beta_{1} + 2500 \beta_{2} ) q^{45} + ( 9567176 - 115901 \beta_{1} - 2081 \beta_{2} ) q^{47} + ( 9031641 - 218524 \beta_{1} + 1292 \beta_{2} ) q^{49} + ( 51782920 - 45746 \beta_{1} + 6384 \beta_{2} ) q^{51} + ( -15209994 - 175444 \beta_{1} - 5104 \beta_{2} ) q^{53} + ( -1157500 - 182500 \beta_{1} - 625 \beta_{2} ) q^{55} + ( 1792048 + 820372 \beta_{1} - 864 \beta_{2} ) q^{57} + ( 89573956 - 183246 \beta_{1} - 5775 \beta_{2} ) q^{59} + ( 51990046 - 362224 \beta_{1} + 12800 \beta_{2} ) q^{61} + ( 129915888 - 191501 \beta_{1} + 829 \beta_{2} ) q^{63} + ( -17311250 - 17500 \beta_{1} - 10000 \beta_{2} ) q^{65} + ( 175638868 - 277043 \beta_{1} + 4270 \beta_{2} ) q^{67} + ( -57500576 - 1582112 \beta_{1} - 8220 \beta_{2} ) q^{69} + ( 79964808 + 772366 \beta_{1} - 12320 \beta_{2} ) q^{71} + ( 66120810 + 460180 \beta_{1} - 4832 \beta_{2} ) q^{73} + ( -10937500 - 390625 \beta_{1} ) q^{75} + ( 128606336 - 1549988 \beta_{1} + 25888 \beta_{2} ) q^{77} + ( 137946576 - 298676 \beta_{1} + 29392 \beta_{2} ) q^{79} + ( 339595929 + 3283092 \beta_{1} - 9828 \beta_{2} ) q^{81} + ( -123983276 + 2169743 \beta_{1} - 21868 \beta_{2} ) q^{83} + ( 76471250 - 997500 \beta_{1} ) q^{85} + ( 167670584 + 154370 \beta_{1} + 27552 \beta_{2} ) q^{87} + ( 251642202 + 2366280 \beta_{1} + 20184 \beta_{2} ) q^{89} + ( -614129888 + 1749570 \beta_{1} + 7470 \beta_{2} ) q^{91} + ( 329777920 + 5530672 \beta_{1} + 25992 \beta_{2} ) q^{93} + ( -310232500 + 236250 \beta_{1} - 50625 \beta_{2} ) q^{95} + ( -301150334 - 1338708 \beta_{1} - 18648 \beta_{2} ) q^{97} + ( -957255180 - 10349092 \beta_{1} - 70093 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 84q^{3} + 1875q^{5} + 5520q^{7} + 47079q^{9} + O(q^{10})$$ $$3q - 84q^{3} + 1875q^{5} + 5520q^{7} + 47079q^{9} - 5556q^{11} - 83094q^{13} - 52500q^{15} + 367062q^{17} - 1489116q^{19} + 1573728q^{21} + 499920q^{23} + 1171875q^{25} - 7695432q^{27} + 5234682q^{29} - 12708912q^{31} + 30494064q^{33} + 3450000q^{35} + 21724434q^{37} + 5806824q^{39} + 27440478q^{41} + 23218260q^{43} + 29424375q^{45} + 28701528q^{47} + 27094923q^{49} + 155348760q^{51} - 45629982q^{53} - 3472500q^{55} + 5376144q^{57} + 268721868q^{59} + 155970138q^{61} + 389747664q^{63} - 51933750q^{65} + 526916604q^{67} - 172501728q^{69} + 239894424q^{71} + 198362430q^{73} - 32812500q^{75} + 385819008q^{77} + 413839728q^{79} + 1018787787q^{81} - 371949828q^{83} + 229413750q^{85} + 503011752q^{87} + 754926606q^{89} - 1842389664q^{91} + 989333760q^{93} - 930697500q^{95} - 903451002q^{97} - 2871765540q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 19 x - 22$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-12 \nu^{2} + 84 \nu + 128$$ $$\beta_{2}$$ $$=$$ $$1272 \nu^{2} - 3144 \nu - 15488$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 106 \beta_{1} + 1920$$$$)/5760$$ $$\nu^{2}$$ $$=$$ $$($$$$7 \beta_{2} + 262 \beta_{1} + 74880$$$$)/5760$$

## 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
 5.33466 −1.41013 −2.92453
0 −262.608 0 625.000 0 1790.86 0 49280.0 0
1.2 0 −13.6876 0 625.000 0 −6441.91 0 −19495.7 0
1.3 0 192.296 0 625.000 0 10171.1 0 17294.6 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.k 3
4.b odd 2 1 40.10.a.d 3
5.b even 2 1 400.10.a.x 3
5.c odd 4 2 400.10.c.r 6
8.b even 2 1 320.10.a.v 3
8.d odd 2 1 320.10.a.u 3
12.b even 2 1 360.10.a.j 3
20.d odd 2 1 200.10.a.f 3
20.e even 4 2 200.10.c.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.10.a.d 3 4.b odd 2 1
80.10.a.k 3 1.a even 1 1 trivial
200.10.a.f 3 20.d odd 2 1
200.10.c.f 6 20.e even 4 2
320.10.a.u 3 8.d odd 2 1
320.10.a.v 3 8.b even 2 1
360.10.a.j 3 12.b even 2 1
400.10.a.x 3 5.b even 2 1
400.10.c.r 6 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 84 T_{3}^{2} - 49536 T_{3} - 691200$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(80))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 84 T + 9513 T^{2} + 2615544 T^{3} + 187244379 T^{4} + 32543321076 T^{5} + 7625597484987 T^{6}$$
$5$ $$( 1 - 625 T )^{3}$$
$7$ $$1 - 5520 T + 62218149 T^{2} - 328164567136 T^{3} + 2510726733013443 T^{4} - 8988843060465678480 T^{5} +$$$$65\!\cdots\!43$$$$T^{6}$$
$11$ $$1 + 5556 T + 2594856177 T^{2} + 72877014191416 T^{3} + 6118535131034237307 T^{4} +$$$$30\!\cdots\!36$$$$T^{5} +$$$$13\!\cdots\!71$$$$T^{6}$$
$13$ $$1 + 83094 T + 20077033299 T^{2} + 754871996338436 T^{3} +$$$$21\!\cdots\!27$$$$T^{4} +$$$$93\!\cdots\!26$$$$T^{5} +$$$$11\!\cdots\!17$$$$T^{6}$$
$17$ $$1 - 367062 T + 268505189631 T^{2} - 69711192496917428 T^{3} +$$$$31\!\cdots\!07$$$$T^{4} -$$$$51\!\cdots\!58$$$$T^{5} +$$$$16\!\cdots\!73$$$$T^{6}$$
$19$ $$1 + 1489116 T + 1342580105481 T^{2} + 830560424045529832 T^{3} +$$$$43\!\cdots\!99$$$$T^{4} +$$$$15\!\cdots\!56$$$$T^{5} +$$$$33\!\cdots\!39$$$$T^{6}$$
$23$ $$1 - 499920 T - 191246545323 T^{2} + 4150866877865471776 T^{3} -$$$$34\!\cdots\!49$$$$T^{4} -$$$$16\!\cdots\!80$$$$T^{5} +$$$$58\!\cdots\!47$$$$T^{6}$$
$29$ $$1 - 5234682 T + 42736985914563 T^{2} -$$$$14\!\cdots\!08$$$$T^{3} +$$$$61\!\cdots\!47$$$$T^{4} -$$$$11\!\cdots\!02$$$$T^{5} +$$$$30\!\cdots\!09$$$$T^{6}$$
$31$ $$1 + 12708912 T + 128130963574173 T^{2} +$$$$72\!\cdots\!04$$$$T^{3} +$$$$33\!\cdots\!83$$$$T^{4} +$$$$88\!\cdots\!92$$$$T^{5} +$$$$18\!\cdots\!11$$$$T^{6}$$
$37$ $$1 - 21724434 T + 286542561812475 T^{2} -$$$$28\!\cdots\!32$$$$T^{3} +$$$$37\!\cdots\!75$$$$T^{4} -$$$$36\!\cdots\!86$$$$T^{5} +$$$$21\!\cdots\!33$$$$T^{6}$$
$41$ $$1 - 27440478 T + 1215792148252263 T^{2} -$$$$18\!\cdots\!72$$$$T^{3} +$$$$39\!\cdots\!43$$$$T^{4} -$$$$29\!\cdots\!38$$$$T^{5} +$$$$35\!\cdots\!81$$$$T^{6}$$
$43$ $$1 - 23218260 T + 251755624778721 T^{2} -$$$$25\!\cdots\!68$$$$T^{3} +$$$$12\!\cdots\!03$$$$T^{4} -$$$$58\!\cdots\!40$$$$T^{5} +$$$$12\!\cdots\!07$$$$T^{6}$$
$47$ $$1 - 28701528 T + 2689800471894429 T^{2} -$$$$45\!\cdots\!28$$$$T^{3} +$$$$30\!\cdots\!43$$$$T^{4} -$$$$35\!\cdots\!92$$$$T^{5} +$$$$14\!\cdots\!63$$$$T^{6}$$
$53$ $$1 + 45629982 T + 7541283402476907 T^{2} +$$$$30\!\cdots\!96$$$$T^{3} +$$$$24\!\cdots\!31$$$$T^{4} +$$$$49\!\cdots\!98$$$$T^{5} +$$$$35\!\cdots\!37$$$$T^{6}$$
$59$ $$1 - 268721868 T + 46457702853914817 T^{2} -$$$$50\!\cdots\!40$$$$T^{3} +$$$$40\!\cdots\!63$$$$T^{4} -$$$$20\!\cdots\!28$$$$T^{5} +$$$$65\!\cdots\!19$$$$T^{6}$$
$61$ $$1 - 155970138 T + 27601780143557283 T^{2} -$$$$34\!\cdots\!16$$$$T^{3} +$$$$32\!\cdots\!03$$$$T^{4} -$$$$21\!\cdots\!78$$$$T^{5} +$$$$15\!\cdots\!21$$$$T^{6}$$
$67$ $$1 - 526916604 T + 169230786613422441 T^{2} -$$$$33\!\cdots\!48$$$$T^{3} +$$$$46\!\cdots\!27$$$$T^{4} -$$$$39\!\cdots\!36$$$$T^{5} +$$$$20\!\cdots\!23$$$$T^{6}$$
$71$ $$1 - 239894424 T + 117827037562982037 T^{2} -$$$$16\!\cdots\!88$$$$T^{3} +$$$$54\!\cdots\!47$$$$T^{4} -$$$$50\!\cdots\!64$$$$T^{5} +$$$$96\!\cdots\!91$$$$T^{6}$$
$73$ $$1 - 198362430 T + 177547240540777287 T^{2} -$$$$22\!\cdots\!56$$$$T^{3} +$$$$10\!\cdots\!31$$$$T^{4} -$$$$68\!\cdots\!70$$$$T^{5} +$$$$20\!\cdots\!97$$$$T^{6}$$
$79$ $$1 - 413839728 T + 365148310414214253 T^{2} -$$$$92\!\cdots\!64$$$$T^{3} +$$$$43\!\cdots\!07$$$$T^{4} -$$$$59\!\cdots\!08$$$$T^{5} +$$$$17\!\cdots\!59$$$$T^{6}$$
$83$ $$1 + 371949828 T + 338245334081122329 T^{2} +$$$$14\!\cdots\!04$$$$T^{3} +$$$$63\!\cdots\!87$$$$T^{4} +$$$$12\!\cdots\!52$$$$T^{5} +$$$$65\!\cdots\!27$$$$T^{6}$$
$89$ $$1 - 754926606 T + 926540806511526711 T^{2} -$$$$52\!\cdots\!12$$$$T^{3} +$$$$32\!\cdots\!99$$$$T^{4} -$$$$92\!\cdots\!86$$$$T^{5} +$$$$43\!\cdots\!29$$$$T^{6}$$
$97$ $$1 + 903451002 T + 2439888940908067119 T^{2} +$$$$13\!\cdots\!72$$$$T^{3} +$$$$18\!\cdots\!23$$$$T^{4} +$$$$52\!\cdots\!78$$$$T^{5} +$$$$43\!\cdots\!13$$$$T^{6}$$