# Properties

 Label 80.10.a.g 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: $$\Q(\sqrt{46})$$ Defining polynomial: $$x^{2} - 46$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}\cdot 3$$ 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 $$\beta = 24\sqrt{46}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -54 + \beta ) q^{3} -625 q^{5} + ( 454 + 13 \beta ) q^{7} + ( 9729 - 108 \beta ) q^{9} +O(q^{10})$$ $$q + ( -54 + \beta ) q^{3} -625 q^{5} + ( 454 + 13 \beta ) q^{7} + ( 9729 - 108 \beta ) q^{9} + ( -12560 + 134 \beta ) q^{11} + ( 73474 - 548 \beta ) q^{13} + ( 33750 - 625 \beta ) q^{15} + ( 84634 - 44 \beta ) q^{17} + ( 12740 - 3216 \beta ) q^{19} + ( 319932 - 248 \beta ) q^{21} + ( -891374 - 4033 \beta ) q^{23} + 390625 q^{25} + ( -2324052 - 4122 \beta ) q^{27} + ( 3661670 - 22088 \beta ) q^{29} + ( -5338636 + 17766 \beta ) q^{31} + ( 4228704 - 19796 \beta ) q^{33} + ( -283750 - 8125 \beta ) q^{35} + ( -2875230 + 78696 \beta ) q^{37} + ( -18487404 + 103066 \beta ) q^{39} + ( -3897882 + 78844 \beta ) q^{41} + ( -8385262 + 6001 \beta ) q^{43} + ( -6080625 + 67500 \beta ) q^{45} + ( -7696946 - 212795 \beta ) q^{47} + ( -35669667 + 11804 \beta ) q^{49} + ( -5736060 + 87010 \beta ) q^{51} + ( -29264646 - 389340 \beta ) q^{53} + ( 7850000 - 83750 \beta ) q^{55} + ( -85899096 + 186404 \beta ) q^{57} + ( -29809132 + 505884 \beta ) q^{59} + ( -94081886 - 611744 \beta ) q^{61} + ( -32783418 + 77445 \beta ) q^{63} + ( -45921250 + 342500 \beta ) q^{65} + ( 52999126 - 1153037 \beta ) q^{67} + ( -58724172 - 673592 \beta ) q^{69} + ( 84332796 - 1069994 \beta ) q^{71} + ( -195106206 + 973588 \beta ) q^{73} + ( -21093750 + 390625 \beta ) q^{75} + ( 40453792 - 102444 \beta ) q^{77} + ( -233473128 + 815332 \beta ) q^{79} + ( -175213611 + 24300 \beta ) q^{81} + ( -164767582 - 1072131 \beta ) q^{83} + ( -52896250 + 27500 \beta ) q^{85} + ( -782973828 + 4854422 \beta ) q^{87} + ( -54167430 + 6683096 \beta ) q^{89} + ( -155400308 + 706370 \beta ) q^{91} + ( 759014280 - 6298000 \beta ) q^{93} + ( -7962500 + 2010000 \beta ) q^{95} + ( 1021529314 - 1437700 \beta ) q^{97} + ( -505646352 + 2660166 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 108q^{3} - 1250q^{5} + 908q^{7} + 19458q^{9} + O(q^{10})$$ $$2q - 108q^{3} - 1250q^{5} + 908q^{7} + 19458q^{9} - 25120q^{11} + 146948q^{13} + 67500q^{15} + 169268q^{17} + 25480q^{19} + 639864q^{21} - 1782748q^{23} + 781250q^{25} - 4648104q^{27} + 7323340q^{29} - 10677272q^{31} + 8457408q^{33} - 567500q^{35} - 5750460q^{37} - 36974808q^{39} - 7795764q^{41} - 16770524q^{43} - 12161250q^{45} - 15393892q^{47} - 71339334q^{49} - 11472120q^{51} - 58529292q^{53} + 15700000q^{55} - 171798192q^{57} - 59618264q^{59} - 188163772q^{61} - 65566836q^{63} - 91842500q^{65} + 105998252q^{67} - 117448344q^{69} + 168665592q^{71} - 390212412q^{73} - 42187500q^{75} + 80907584q^{77} - 466946256q^{79} - 350427222q^{81} - 329535164q^{83} - 105792500q^{85} - 1565947656q^{87} - 108334860q^{89} - 310800616q^{91} + 1518028560q^{93} - 15925000q^{95} + 2043058628q^{97} - 1011292704q^{99} + O(q^{100})$$

## 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
 −6.78233 6.78233
0 −216.776 0 −625.000 0 −1662.09 0 27308.8 0
1.2 0 108.776 0 −625.000 0 2570.09 0 −7850.80 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.g 2
4.b odd 2 1 40.10.a.c 2
5.b even 2 1 400.10.a.r 2
5.c odd 4 2 400.10.c.m 4
8.b even 2 1 320.10.a.q 2
8.d odd 2 1 320.10.a.n 2
12.b even 2 1 360.10.a.g 2
20.d odd 2 1 200.10.a.c 2
20.e even 4 2 200.10.c.d 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 108 T_{3} - 23580$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(80))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 108 T + 15786 T^{2} + 2125764 T^{3} + 387420489 T^{4}$$
$5$ $$( 1 + 625 T )^{2}$$
$7$ $$1 - 908 T + 76435506 T^{2} - 36641075156 T^{3} + 1628413597910449 T^{4}$$
$11$ $$1 + 25120 T + 4397886806 T^{2} + 59231645997920 T^{3} + 5559917313492231481 T^{4}$$
$13$ $$1 - 146948 T + 18650572638 T^{2} - 1558309973863604 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 - 169268 T + 244287370694 T^{2} - 20073132678894196 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$1 - 25480 T + 371498689782 T^{2} - 8222082539408920 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$
$23$ $$1 + 1782748 T + 3965893132658 T^{2} + 3211001304917840324 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$1 - 7323340 T + 29495257443614 T^{2} -$$$$10\!\cdots\!60$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$31$ $$1 + 10677272 T + 73017326150862 T^{2} +$$$$28\!\cdots\!12$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$
$37$ $$1 + 5750460 T + 104099098360718 T^{2} +$$$$74\!\cdots\!20$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$1 + 7795764 T + 505248245475190 T^{2} +$$$$25\!\cdots\!04$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$
$43$ $$1 + 16770524 T + 1074543668703834 T^{2} +$$$$84\!\cdots\!32$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 15393892 T + 1097719682118050 T^{2} +$$$$17\!\cdots\!64$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 + 58529292 T + 3439533688251982 T^{2} +$$$$19\!\cdots\!36$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$1 + 59618264 T + 11433756193805126 T^{2} +$$$$51\!\cdots\!96$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$61$ $$1 + 188163772 T + 22324076261167422 T^{2} +$$$$22\!\cdots\!52$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$
$67$ $$1 - 105998252 T + 21995694557368746 T^{2} -$$$$28\!\cdots\!44$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$1 - 168665592 T + 68474091725761822 T^{2} -$$$$77\!\cdots\!52$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$73$ $$1 + 390212412 T + 130694746296409238 T^{2} +$$$$22\!\cdots\!56$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$1 + 466946256 T + 276599246367485918 T^{2} +$$$$55\!\cdots\!64$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$
$83$ $$1 + 329535164 T + 370572645121965674 T^{2} +$$$$61\!\cdots\!92$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$1 + 108334860 T - 479764388871867818 T^{2} +$$$$37\!\cdots\!40$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$
$97$ $$1 - 2043058628 T + 2509217520410601030 T^{2} -$$$$15\!\cdots\!76$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$