# Properties

 Label 8.20.a.a Level 8 Weight 20 Character orbit 8.a Self dual yes Analytic conductor 18.305 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$20$$ Character orbit: $$[\chi]$$ $$=$$ 8.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.3053357245$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{1453})$$ Defining polynomial: $$x^{2} - x - 363$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{7}\cdot 3\cdot 5$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 960\sqrt{1453}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -13956 - \beta ) q^{3} + ( 613310 + 44 \beta ) q^{5} + ( 44255256 + 3190 \beta ) q^{7} + ( 371593269 + 27912 \beta ) q^{9} +O(q^{10})$$ $$q + ( -13956 - \beta ) q^{3} + ( 613310 + 44 \beta ) q^{5} + ( 44255256 + 3190 \beta ) q^{7} + ( 371593269 + 27912 \beta ) q^{9} + ( -3581893804 - 223467 \beta ) q^{11} + ( -5063461802 + 71660 \beta ) q^{13} + ( -67479085560 - 1227374 \beta ) q^{15} + ( -36022539470 + 17338504 \beta ) q^{17} + ( -1560240236116 - 26137813 \beta ) q^{19} + ( -4889306864736 - 88774896 \beta ) q^{21} + ( -7379603545144 + 184291330 \beta ) q^{23} + ( -16104868999225 + 53971280 \beta ) q^{25} + ( -26341969566312 + 401128326 \beta ) q^{27} + ( -15124769622522 - 893092484 \beta ) q^{29} + ( -61694781388960 - 3864337064 \beta ) q^{31} + ( 349230172930224 + 6700599256 \beta ) q^{33} + ( 215096133585360 + 3903690164 \beta ) q^{35} + ( 1007696585087262 - 3072429508 \beta ) q^{37} + ( -25293143859288 + 4063374842 \beta ) q^{39} + ( 1270392479752122 - 47315480368 \beta ) q^{41} + ( -2816827546694732 + 16432445197 \beta ) q^{43} + ( 1872469405064790 + 33468812556 \beta ) q^{45} + ( -10974169793565168 + 13905445988 \beta ) q^{47} + ( 4186293331532393 + 282348533280 \beta ) q^{49} + ( -22714996600295880 - 205953622354 \beta ) q^{51} + ( -4709062533452338 - 810595524548 \beta ) q^{53} + ( -15363426861001640 - 294657873146 \beta ) q^{55} + ( 56775460828777296 + 1925019554344 \beta ) q^{57} + ( 49271224795203812 + 1148874921753 \beta ) q^{59} + ( 5146072688919910 - 3405518362868 \beta ) q^{61} + ( 135676101698415864 + 2420635233582 \beta ) q^{63} + ( 1116716180007380 - 178842524688 \beta ) q^{65} + ( 37876814001992252 - 6468769570097 \beta ) q^{67} + ( -143791971698754336 + 4807633743664 \beta ) q^{69} + ( 8703526283356888 - 4032598137882 \beta ) q^{71} + ( -428754127529916134 + 10723507910184 \beta ) q^{73} + ( 152487431068640100 + 15351645815545 \beta ) q^{75} + ( -1113097256235937824 - 21315830527312 \beta ) q^{77} + ( -113145960722427536 - 8826033368644 \beta ) q^{79} + ( -601404854883860151 - 11697219418248 \beta ) q^{81} + ( 383757850730492524 + 1875501918907 \beta ) q^{83} + ( 999487011407779100 + 9048886151560 \beta ) q^{85} + ( 1407007855170560232 + 27588768329226 \beta ) q^{87} + ( 3046272947217587274 + 7105196196264 \beta ) q^{89} + ( 82023827196188688 - 12981111503420 \beta ) q^{91} + ( 6035687393543352960 + 115625469454144 \beta ) q^{93} + ( -2496943855328169560 - 84681152480134 \beta ) q^{95} + ( 774074124761173538 - 186089076998104 \beta ) q^{97} + ( -9683429760739864476 - 183016652900871 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 27912q^{3} + 1226620q^{5} + 88510512q^{7} + 743186538q^{9} + O(q^{10})$$ $$2q - 27912q^{3} + 1226620q^{5} + 88510512q^{7} + 743186538q^{9} - 7163787608q^{11} - 10126923604q^{13} - 134958171120q^{15} - 72045078940q^{17} - 3120480472232q^{19} - 9778613729472q^{21} - 14759207090288q^{23} - 32209737998450q^{25} - 52683939132624q^{27} - 30249539245044q^{29} - 123389562777920q^{31} + 698460345860448q^{33} + 430192267170720q^{35} + 2015393170174524q^{37} - 50586287718576q^{39} + 2540784959504244q^{41} - 5633655093389464q^{43} + 3744938810129580q^{45} - 21948339587130336q^{47} + 8372586663064786q^{49} - 45429993200591760q^{51} - 9418125066904676q^{53} - 30726853722003280q^{55} + 113550921657554592q^{57} + 98542449590407624q^{59} + 10292145377839820q^{61} + 271352203396831728q^{63} + 2233432360014760q^{65} + 75753628003984504q^{67} - 287583943397508672q^{69} + 17407052566713776q^{71} - 857508255059832268q^{73} + 304974862137280200q^{75} - 2226194512471875648q^{77} - 226291921444855072q^{79} - 1202809709767720302q^{81} + 767515701460985048q^{83} + 1998974022815558200q^{85} + 2814015710341120464q^{87} + 6092545894435174548q^{89} + 164047654392377376q^{91} + 12071374787086705920q^{93} - 4993887710656339120q^{95} + 1548148249522347076q^{97} - 19366859521479728952q^{99} + O(q^{100})$$

## 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
 19.5591 −18.5591
0 −50549.5 0 2.22342e6 0 1.60989e8 0 1.39299e9 0
1.2 0 22637.5 0 −996804. 0 −7.24780e7 0 −6.49805e8 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 8.20.a.a 2
3.b odd 2 1 72.20.a.a 2
4.b odd 2 1 16.20.a.e 2
8.b even 2 1 64.20.a.k 2
8.d odd 2 1 64.20.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.20.a.a 2 1.a even 1 1 trivial
16.20.a.e 2 4.b odd 2 1
64.20.a.j 2 8.d odd 2 1
64.20.a.k 2 8.b even 2 1
72.20.a.a 2 3.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 27912 T_{3} - 1144314864$$ acting on $$S_{20}^{\mathrm{new}}(\Gamma_0(8))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 27912 T + 1180208070 T^{2} + 32441042066904 T^{3} + 1350851717672992089 T^{4}$$
$5$ $$1 - 1226620 T + 35930653639550 T^{2} - 23395919799804687500 T^{3} +$$$$36\!\cdots\!25$$$$T^{4}$$
$7$ $$1 - 88510512 T + 11129657221091822 T^{2} -$$$$10\!\cdots\!16$$$$T^{3} +$$$$12\!\cdots\!49$$$$T^{4}$$
$11$ $$1 + 7163787608 T + 68277596800784135798 T^{2} +$$$$43\!\cdots\!28$$$$T^{3} +$$$$37\!\cdots\!81$$$$T^{4}$$
$13$ $$1 + 10126923604 T +$$$$29\!\cdots\!58$$$$T^{2} +$$$$14\!\cdots\!08$$$$T^{3} +$$$$21\!\cdots\!29$$$$T^{4}$$
$17$ $$1 + 72045078940 T +$$$$76\!\cdots\!06$$$$T^{2} +$$$$17\!\cdots\!20$$$$T^{3} +$$$$57\!\cdots\!09$$$$T^{4}$$
$19$ $$1 + 3120480472232 T +$$$$54\!\cdots\!14$$$$T^{2} +$$$$61\!\cdots\!28$$$$T^{3} +$$$$39\!\cdots\!41$$$$T^{4}$$
$23$ $$1 + 14759207090288 T +$$$$15\!\cdots\!10$$$$T^{2} +$$$$11\!\cdots\!56$$$$T^{3} +$$$$55\!\cdots\!69$$$$T^{4}$$
$29$ $$1 + 30249539245044 T +$$$$11\!\cdots\!22$$$$T^{2} +$$$$18\!\cdots\!36$$$$T^{3} +$$$$37\!\cdots\!61$$$$T^{4}$$
$31$ $$1 + 123389562777920 T +$$$$27\!\cdots\!42$$$$T^{2} +$$$$26\!\cdots\!20$$$$T^{3} +$$$$46\!\cdots\!41$$$$T^{4}$$
$37$ $$1 - 2015393170174524 T +$$$$22\!\cdots\!90$$$$T^{2} -$$$$12\!\cdots\!52$$$$T^{3} +$$$$39\!\cdots\!29$$$$T^{4}$$
$41$ $$1 - 2540784959504244 T +$$$$74\!\cdots\!06$$$$T^{2} -$$$$11\!\cdots\!84$$$$T^{3} +$$$$19\!\cdots\!21$$$$T^{4}$$
$43$ $$1 + 5633655093389464 T +$$$$29\!\cdots\!38$$$$T^{2} +$$$$61\!\cdots\!48$$$$T^{3} +$$$$11\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 21948339587130336 T +$$$$23\!\cdots\!90$$$$T^{2} +$$$$12\!\cdots\!88$$$$T^{3} +$$$$34\!\cdots\!89$$$$T^{4}$$
$53$ $$1 + 9418125066904676 T +$$$$29\!\cdots\!78$$$$T^{2} +$$$$54\!\cdots\!92$$$$T^{3} +$$$$33\!\cdots\!89$$$$T^{4}$$
$59$ $$1 - 98542449590407624 T +$$$$95\!\cdots\!22$$$$T^{2} -$$$$43\!\cdots\!36$$$$T^{3} +$$$$19\!\cdots\!21$$$$T^{4}$$
$61$ $$1 - 10292145377839820 T +$$$$11\!\cdots\!82$$$$T^{2} -$$$$85\!\cdots\!20$$$$T^{3} +$$$$69\!\cdots\!81$$$$T^{4}$$
$67$ $$1 - 75753628003984504 T +$$$$44\!\cdots\!10$$$$T^{2} -$$$$37\!\cdots\!12$$$$T^{3} +$$$$24\!\cdots\!09$$$$T^{4}$$
$71$ $$1 - 17407052566713776 T +$$$$27\!\cdots\!06$$$$T^{2} -$$$$25\!\cdots\!56$$$$T^{3} +$$$$22\!\cdots\!61$$$$T^{4}$$
$73$ $$1 + 857508255059832268 T +$$$$53\!\cdots\!30$$$$T^{2} +$$$$21\!\cdots\!16$$$$T^{3} +$$$$64\!\cdots\!69$$$$T^{4}$$
$79$ $$1 + 226291921444855072 T +$$$$21\!\cdots\!34$$$$T^{2} +$$$$25\!\cdots\!68$$$$T^{3} +$$$$12\!\cdots\!61$$$$T^{4}$$
$83$ $$1 - 767515701460985048 T +$$$$59\!\cdots\!70$$$$T^{2} -$$$$22\!\cdots\!56$$$$T^{3} +$$$$84\!\cdots\!09$$$$T^{4}$$
$89$ $$1 - 6092545894435174548 T +$$$$31\!\cdots\!94$$$$T^{2} -$$$$66\!\cdots\!32$$$$T^{3} +$$$$11\!\cdots\!81$$$$T^{4}$$
$97$ $$1 - 1548148249522347076 T +$$$$66\!\cdots\!10$$$$T^{2} -$$$$86\!\cdots\!08$$$$T^{3} +$$$$31\!\cdots\!89$$$$T^{4}$$