# Properties

 Label 180.10.a.e Level 180 Weight 10 Character orbit 180.a Self dual yes Analytic conductor 92.706 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 180.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$92.7064505095$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{79})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 3$$ Twist minimal: no (minimal twist has level 20) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 48\sqrt{79}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 625 q^{5} + ( -190 + 23 \beta ) q^{7} +O(q^{10})$$ $$q + 625 q^{5} + ( -190 + 23 \beta ) q^{7} + ( -51360 - 10 \beta ) q^{11} + ( 89570 + 188 \beta ) q^{13} + ( -158010 - 116 \beta ) q^{17} + ( 68636 + 1720 \beta ) q^{19} + ( 332730 + 2131 \beta ) q^{23} + 390625 q^{25} + ( 3446874 + 1960 \beta ) q^{29} + ( 145916 - 8750 \beta ) q^{31} + ( -118750 + 14375 \beta ) q^{35} + ( 5630690 - 19992 \beta ) q^{37} + ( -14886726 - 24060 \beta ) q^{41} + ( -5854090 - 33205 \beta ) q^{43} + ( -31246650 - 14191 \beta ) q^{47} + ( 55968957 - 8740 \beta ) q^{49} + ( -4708890 + 70844 \beta ) q^{53} + ( -32100000 - 6250 \beta ) q^{55} + ( 46465428 + 124660 \beta ) q^{59} + ( 97836962 + 240480 \beta ) q^{61} + ( 55981250 + 117500 \beta ) q^{65} + ( -109883710 + 67713 \beta ) q^{67} + ( -155603508 + 465870 \beta ) q^{71} + ( -49612030 - 682572 \beta ) q^{73} + ( -32105280 - 1179380 \beta ) q^{77} + ( 271130888 + 23340 \beta ) q^{79} + ( 628457850 - 64319 \beta ) q^{83} + ( -98756250 - 72500 \beta ) q^{85} + ( 231145926 - 1807320 \beta ) q^{89} + ( 770018884 + 2024390 \beta ) q^{91} + ( 42897500 + 1075000 \beta ) q^{95} + ( 835858370 - 967652 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 1250q^{5} - 380q^{7} + O(q^{10})$$ $$2q + 1250q^{5} - 380q^{7} - 102720q^{11} + 179140q^{13} - 316020q^{17} + 137272q^{19} + 665460q^{23} + 781250q^{25} + 6893748q^{29} + 291832q^{31} - 237500q^{35} + 11261380q^{37} - 29773452q^{41} - 11708180q^{43} - 62493300q^{47} + 111937914q^{49} - 9417780q^{53} - 64200000q^{55} + 92930856q^{59} + 195673924q^{61} + 111962500q^{65} - 219767420q^{67} - 311207016q^{71} - 99224060q^{73} - 64210560q^{77} + 542261776q^{79} + 1256915700q^{83} - 197512500q^{85} + 462291852q^{89} + 1540037768q^{91} + 85795000q^{95} + 1671716740q^{97} + O(q^{100})$$

## 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
 −8.88819 8.88819
0 0 0 625.000 0 −10002.6 0 0 0
1.2 0 0 0 625.000 0 9622.57 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 180.10.a.e 2
3.b odd 2 1 20.10.a.b 2
12.b even 2 1 80.10.a.j 2
15.d odd 2 1 100.10.a.c 2
15.e even 4 2 100.10.c.c 4
24.f even 2 1 320.10.a.l 2
24.h odd 2 1 320.10.a.t 2
60.h even 2 1 400.10.a.l 2
60.l odd 4 2 400.10.c.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.10.a.b 2 3.b odd 2 1
80.10.a.j 2 12.b even 2 1
100.10.a.c 2 15.d odd 2 1
100.10.c.c 4 15.e even 4 2
180.10.a.e 2 1.a even 1 1 trivial
320.10.a.l 2 24.f even 2 1
320.10.a.t 2 24.h odd 2 1
400.10.a.l 2 60.h even 2 1
400.10.c.l 4 60.l odd 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(180))$$:

 $$T_{7}^{2} + 380 T_{7} - 96250364$$ $$T_{11}^{2} + 102720 T_{11} + 2619648000$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 625 T )^{2}$$
$7$ $$1 + 380 T - 15543150 T^{2} + 15334370660 T^{3} + 1628413597910449 T^{4}$$
$11$ $$1 + 102720 T + 7335543382 T^{2} + 242208386819520 T^{3} + 5559917313492231481 T^{4}$$
$13$ $$1 - 179140 T + 22798610142 T^{2} - 1899690017679220 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 + 316020 T + 259693705798 T^{2} + 37476140730581940 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$1 - 137272 T + 111610161654 T^{2} - 44295985649518888 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$
$23$ $$1 - 665460 T + 2886450615250 T^{2} - 1198595050097167980 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$1 - 6893748 T + 40195999658014 T^{2} -$$$$10\!\cdots\!12$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$31$ $$1 - 291832 T + 38964935800398 T^{2} - 7715927814392939272 T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$
$37$ $$1 - 11261380 T + 218879982937230 T^{2} -$$$$14\!\cdots\!60$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$1 + 29773452 T + 771012402449398 T^{2} +$$$$97\!\cdots\!72$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$
$43$ $$1 + 11708180 T + 838769843899386 T^{2} +$$$$58\!\cdots\!40$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 62493300 T + 3177958884734338 T^{2} +$$$$69\!\cdots\!00$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 + 9417780 T + 5708185761526990 T^{2} +$$$$31\!\cdots\!40$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$1 - 92930856 T + 16656477955483462 T^{2} -$$$$80\!\cdots\!84$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$61$ $$1 - 195673924 T + 22434263296171326 T^{2} -$$$$22\!\cdots\!84$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$
$67$ $$1 + 219767420 T + 65652945987990090 T^{2} +$$$$59\!\cdots\!40$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$1 + 311207016 T + 76405636625293726 T^{2} +$$$$14\!\cdots\!96$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$73$ $$1 + 99224060 T + 35402447061205782 T^{2} +$$$$58\!\cdots\!80$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$1 - 542261776 T + 313115996157615582 T^{2} -$$$$64\!\cdots\!44$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$
$83$ $$1 - 1256915700 T + 768086791626261130 T^{2} -$$$$23\!\cdots\!00$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$1 - 462291852 T + 159603168035249494 T^{2} -$$$$16\!\cdots\!68$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$
$97$ $$1 - 1671716740 T + 2048690578856969670 T^{2} -$$$$12\!\cdots\!80$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$