# Properties

 Label 8.12.a.b Level 8 Weight 12 Character orbit 8.a Self dual yes Analytic conductor 6.147 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$8 = 2^{3}$$ Weight: $$k$$ = $$12$$ Character orbit: $$[\chi]$$ = 8.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.14674544448$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{109})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 64\sqrt{109}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 28 + \beta ) q^{3} + ( 3934 - 12 \beta ) q^{5} + ( 45528 + 42 \beta ) q^{7} + ( 270101 + 56 \beta ) q^{9} +O(q^{10})$$ $$q + ( 28 + \beta ) q^{3} + ( 3934 - 12 \beta ) q^{5} + ( 45528 + 42 \beta ) q^{7} + ( 270101 + 56 \beta ) q^{9} + ( 79540 - 693 \beta ) q^{11} + ( 525238 + 948 \beta ) q^{13} + ( -5247416 + 3598 \beta ) q^{15} + ( 715442 - 10824 \beta ) q^{17} + ( -10933300 - 4587 \beta ) q^{19} + ( 20026272 + 46704 \beta ) q^{21} + ( 17903368 - 24738 \beta ) q^{23} + ( 30939047 - 94416 \beta ) q^{25} + ( 27604696 + 94522 \beta ) q^{27} + ( -114413850 + 59556 \beta ) q^{29} + ( 32361056 - 28632 \beta ) q^{31} + ( -307172432 + 60136 \beta ) q^{33} + ( -45910704 - 381108 \beta ) q^{35} + ( 37779390 - 81564 \beta ) q^{37} + ( 437954536 + 551782 \beta ) q^{39} + ( 600607098 + 647088 \beta ) q^{41} + ( -22759916 + 324051 \beta ) q^{43} + ( 762553526 - 3020908 \beta ) q^{45} + ( -614539632 - 816420 \beta ) q^{47} + ( 883034537 + 3824352 \beta ) q^{49} + ( -4812493960 + 412370 \beta ) q^{51} + ( -1904274962 + 3514980 \beta ) q^{53} + ( 4025704984 - 3680742 \beta ) q^{55} + ( -2354062768 - 11061736 \beta ) q^{57} + ( -3006463292 + 4307463 \beta ) q^{59} + ( 4894896454 + 4588692 \beta ) q^{61} + ( 13347241656 + 13893810 \beta ) q^{63} + ( -3012688172 - 2573424 \beta ) q^{65} + ( 7351547612 - 21317583 \beta ) q^{67} + ( -10543332128 + 17210704 \beta ) q^{69} + ( 2159995544 - 20099142 \beta ) q^{71} + ( 5527819738 - 3259368 \beta ) q^{73} + ( -41287051708 + 28295399 \beta ) q^{75} + ( -9373484064 - 28210224 \beta ) q^{77} + ( 25978811632 + 41373444 \beta ) q^{79} + ( -4873980151 + 20331080 \beta ) q^{81} + ( -54113987956 - 13552251 \beta ) q^{83} + ( 60804864860 - 51166920 \beta ) q^{85} + ( 23386022184 - 112746282 \beta ) q^{87} + ( 35594145930 + 90758808 \beta ) q^{89} + ( 41689446288 + 65220540 \beta ) q^{91} + ( -11877047680 + 31559360 \beta ) q^{93} + ( -18436437784 + 113154342 \beta ) q^{95} + ( -849903838 - 64199400 \beta ) q^{97} + ( 4157458628 - 182725753 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 56q^{3} + 7868q^{5} + 91056q^{7} + 540202q^{9} + O(q^{10})$$ $$2q + 56q^{3} + 7868q^{5} + 91056q^{7} + 540202q^{9} + 159080q^{11} + 1050476q^{13} - 10494832q^{15} + 1430884q^{17} - 21866600q^{19} + 40052544q^{21} + 35806736q^{23} + 61878094q^{25} + 55209392q^{27} - 228827700q^{29} + 64722112q^{31} - 614344864q^{33} - 91821408q^{35} + 75558780q^{37} + 875909072q^{39} + 1201214196q^{41} - 45519832q^{43} + 1525107052q^{45} - 1229079264q^{47} + 1766069074q^{49} - 9624987920q^{51} - 3808549924q^{53} + 8051409968q^{55} - 4708125536q^{57} - 6012926584q^{59} + 9789792908q^{61} + 26694483312q^{63} - 6025376344q^{65} + 14703095224q^{67} - 21086664256q^{69} + 4319991088q^{71} + 11055639476q^{73} - 82574103416q^{75} - 18746968128q^{77} + 51957623264q^{79} - 9747960302q^{81} - 108227975912q^{83} + 121609729720q^{85} + 46772044368q^{87} + 71188291860q^{89} + 83378892576q^{91} - 23754095360q^{93} - 36872875568q^{95} - 1699807676q^{97} + 8314917256q^{99} + O(q^{100})$$

## 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
 −4.72015 5.72015
0 −640.180 0 11952.2 0 17464.5 0 232683. 0
1.2 0 696.180 0 −4084.16 0 73591.5 0 307519. 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 8.12.a.b 2
3.b odd 2 1 72.12.a.e 2
4.b odd 2 1 16.12.a.d 2
5.b even 2 1 200.12.a.d 2
5.c odd 4 2 200.12.c.c 4
8.b even 2 1 64.12.a.h 2
8.d odd 2 1 64.12.a.k 2
12.b even 2 1 144.12.a.p 2
16.e even 4 2 256.12.b.h 4
16.f odd 4 2 256.12.b.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.12.a.b 2 1.a even 1 1 trivial
16.12.a.d 2 4.b odd 2 1
64.12.a.h 2 8.b even 2 1
64.12.a.k 2 8.d odd 2 1
72.12.a.e 2 3.b odd 2 1
144.12.a.p 2 12.b even 2 1
200.12.a.d 2 5.b even 2 1
200.12.c.c 4 5.c odd 4 2
256.12.b.h 4 16.e even 4 2
256.12.b.k 4 16.f odd 4 2

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 56 T_{3} - 445680$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(8))$$.