# Properties

 Label 80.10.a.i 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: $$\Q(\sqrt{22})$$ Defining polynomial: $$x^{2} - 22$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}\cdot 5$$ 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 $$\beta = 40\sqrt{22}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 58 + \beta ) q^{3} -625 q^{5} + ( -5642 - 35 \beta ) q^{7} + ( 18881 + 116 \beta ) q^{9} +O(q^{10})$$ $$q + ( 58 + \beta ) q^{3} -625 q^{5} + ( -5642 - 35 \beta ) q^{7} + ( 18881 + 116 \beta ) q^{9} + ( 50704 - 106 \beta ) q^{11} + ( -10686 + 60 \beta ) q^{13} + ( -36250 - 625 \beta ) q^{15} + ( -148390 - 1804 \beta ) q^{17} + ( -137916 - 1216 \beta ) q^{19} + ( -1559236 - 7672 \beta ) q^{21} + ( 292642 - 2705 \beta ) q^{23} + 390625 q^{25} + ( 4036684 + 5926 \beta ) q^{27} + ( -4964378 + 8888 \beta ) q^{29} + ( -2565740 - 25402 \beta ) q^{31} + ( -790368 + 44556 \beta ) q^{33} + ( 3526250 + 21875 \beta ) q^{35} + ( -5503966 - 37592 \beta ) q^{37} + ( 1492212 - 7206 \beta ) q^{39} + ( -20917978 - 15076 \beta ) q^{41} + ( 11697026 + 78353 \beta ) q^{43} + ( -11800625 - 72500 \beta ) q^{45} + ( -5855874 + 292213 \beta ) q^{47} + ( 34598557 + 394940 \beta ) q^{49} + ( -72107420 - 253022 \beta ) q^{51} + ( -23192134 - 95548 \beta ) q^{53} + ( -31690000 + 66250 \beta ) q^{55} + ( -50802328 - 208444 \beta ) q^{57} + ( -89119788 - 426804 \beta ) q^{59} + ( 15912610 - 431776 \beta ) q^{61} + ( -249438602 - 1315307 \beta ) q^{63} + ( 6678750 - 37500 \beta ) q^{65} + ( -44740314 + 638803 \beta ) q^{67} + ( -78242764 + 135752 \beta ) q^{69} + ( -56159588 + 597974 \beta ) q^{71} + ( -46647262 - 531916 \beta ) q^{73} + ( 22656250 + 390625 \beta ) q^{75} + ( -155479968 - 1176588 \beta ) q^{77} + ( 95800664 + 2864292 \beta ) q^{79} + ( 71088149 + 2097164 \beta ) q^{81} + ( 8635218 - 203907 \beta ) q^{83} + ( 92743750 + 1127500 \beta ) q^{85} + ( 24923676 - 4448874 \beta ) q^{87} + ( -307533574 + 3664152 \beta ) q^{89} + ( -13629588 + 35490 \beta ) q^{91} + ( -1042963320 - 4039056 \beta ) q^{93} + ( 86197500 + 760000 \beta ) q^{95} + ( -498272734 + 4842204 \beta ) q^{97} + ( 524523024 + 3880278 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 116q^{3} - 1250q^{5} - 11284q^{7} + 37762q^{9} + O(q^{10})$$ $$2q + 116q^{3} - 1250q^{5} - 11284q^{7} + 37762q^{9} + 101408q^{11} - 21372q^{13} - 72500q^{15} - 296780q^{17} - 275832q^{19} - 3118472q^{21} + 585284q^{23} + 781250q^{25} + 8073368q^{27} - 9928756q^{29} - 5131480q^{31} - 1580736q^{33} + 7052500q^{35} - 11007932q^{37} + 2984424q^{39} - 41835956q^{41} + 23394052q^{43} - 23601250q^{45} - 11711748q^{47} + 69197114q^{49} - 144214840q^{51} - 46384268q^{53} - 63380000q^{55} - 101604656q^{57} - 178239576q^{59} + 31825220q^{61} - 498877204q^{63} + 13357500q^{65} - 89480628q^{67} - 156485528q^{69} - 112319176q^{71} - 93294524q^{73} + 45312500q^{75} - 310959936q^{77} + 191601328q^{79} + 142176298q^{81} + 17270436q^{83} + 185487500q^{85} + 49847352q^{87} - 615067148q^{89} - 27259176q^{91} - 2085926640q^{93} + 172395000q^{95} - 996545468q^{97} + 1049046048q^{99} + O(q^{100})$$

## 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
 −4.69042 4.69042
0 −129.617 0 −625.000 0 924.582 0 −2882.53 0
1.2 0 245.617 0 −625.000 0 −12208.6 0 40644.5 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.i 2
4.b odd 2 1 40.10.a.a 2
5.b even 2 1 400.10.a.n 2
5.c odd 4 2 400.10.c.k 4
8.b even 2 1 320.10.a.m 2
8.d odd 2 1 320.10.a.r 2
12.b even 2 1 360.10.a.i 2
20.d odd 2 1 200.10.a.e 2
20.e even 4 2 200.10.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.10.a.a 2 4.b odd 2 1
80.10.a.i 2 1.a even 1 1 trivial
200.10.a.e 2 20.d odd 2 1
200.10.c.c 4 20.e even 4 2
320.10.a.m 2 8.b even 2 1
320.10.a.r 2 8.d odd 2 1
360.10.a.i 2 12.b even 2 1
400.10.a.n 2 5.b even 2 1
400.10.c.k 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 116 T_{3} - 31836$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(80))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 116 T + 7530 T^{2} - 2283228 T^{3} + 387420489 T^{4}$$
$5$ $$( 1 + 625 T )^{2}$$
$7$ $$1 + 11284 T + 69419378 T^{2} + 455350101388 T^{3} + 1628413597910449 T^{4}$$
$11$ $$1 - 101408 T + 6891283798 T^{2} - 239114759448928 T^{3} + 5559917313492231481 T^{4}$$
$13$ $$1 + 21372 T + 21196469342 T^{2} + 226639360599756 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 + 296780 T + 144639901894 T^{2} + 35194509986779660 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$1 + 275832 T + 612347527414 T^{2} + 89007593053777128 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$
$23$ $$1 - 585284 T + 3430385383090 T^{2} - 1054185834311710492 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$1 + 9928756 T + 50878662529822 T^{2} +$$$$14\!\cdots\!64$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$31$ $$1 + 5131480 T + 36749057608142 T^{2} +$$$$13\!\cdots\!80$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$
$37$ $$1 + 11007932 T + 240473943386510 T^{2} +$$$$14\!\cdots\!64$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$1 + 41835956 T + 1084325213081206 T^{2} +$$$$13\!\cdots\!16$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$
$43$ $$1 - 23394052 T + 925906061281562 T^{2} -$$$$11\!\cdots\!36$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 11711748 T - 733120788879390 T^{2} +$$$$13\!\cdots\!16$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 + 46384268 T + 6816046668377422 T^{2} +$$$$15\!\cdots\!44$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$1 + 178239576 T + 18856238015031622 T^{2} +$$$$15\!\cdots\!64$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$61$ $$1 - 31825220 T + 17079149243685182 T^{2} -$$$$37\!\cdots\!20$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$
$67$ $$1 + 89480628 T + 42050726086531690 T^{2} +$$$$24\!\cdots\!16$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$1 + 112319176 T + 82264334516632606 T^{2} +$$$$51\!\cdots\!56$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$73$ $$1 + 93294524 T + 109959841455461270 T^{2} +$$$$54\!\cdots\!12$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$1 - 191601328 T - 39905777688415266 T^{2} -$$$$22\!\cdots\!32$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$
$83$ $$1 - 17270436 T + 372491529649343530 T^{2} -$$$$32\!\cdots\!08$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$1 + 615067148 T + 322694158807723094 T^{2} +$$$$21\!\cdots\!32$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$
$97$ $$1 + 996545468 T + 943405561624881990 T^{2} +$$$$75\!\cdots\!56$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$