# Properties

 Label 136.3.j.b Level $136$ Weight $3$ Character orbit 136.j Analytic conductor $3.706$ Analytic rank $0$ Dimension $64$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 136.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.70573159530$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$32$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$64 q - 8 q^{3} + 12 q^{4} + 26 q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64 q - 8 q^{3} + 12 q^{4} + 26 q^{6} - 30 q^{10} - 32 q^{11} - 14 q^{12} - 32 q^{14} - 124 q^{16} - 12 q^{17} + 84 q^{18} + 70 q^{20} - 38 q^{22} + 54 q^{24} + 160 q^{27} - 76 q^{28} + 56 q^{30} + 128 q^{33} + 54 q^{34} - 8 q^{35} - 360 q^{38} - 62 q^{40} - 88 q^{41} + 306 q^{44} - 280 q^{46} + 82 q^{48} - 164 q^{50} - 360 q^{51} - 204 q^{52} + 460 q^{54} - 328 q^{56} - 24 q^{57} + 194 q^{58} - 72 q^{62} + 204 q^{64} + 112 q^{65} - 8 q^{67} - 226 q^{68} - 36 q^{72} - 408 q^{73} - 250 q^{74} + 32 q^{75} + 508 q^{78} + 674 q^{80} + 80 q^{81} + 116 q^{82} + 1008 q^{84} - 324 q^{86} + 90 q^{88} - 8 q^{89} - 834 q^{90} + 384 q^{91} - 104 q^{92} + 154 q^{96} + 112 q^{97} - 216 q^{98} + 416 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
115.1 −1.99592 0.127650i −1.83110 1.83110i 3.96741 + 0.509561i −1.67465 + 1.67465i 3.42099 + 3.88848i −2.41883 2.41883i −7.85360 1.52349i 2.29414i 3.55625 3.12871i
115.2 −1.95592 + 0.417606i 0.749243 + 0.749243i 3.65121 1.63360i 3.60984 3.60984i −1.77834 1.15257i −3.85896 3.85896i −6.45926 + 4.71996i 7.87727i −5.55305 + 8.56803i
115.3 −1.93784 0.494748i 2.46525 + 2.46525i 3.51045 + 1.91749i −2.34604 + 2.34604i −3.55759 5.99695i 6.17762 + 6.17762i −5.85402 5.45257i 3.15495i 5.70695 3.38555i
115.4 −1.88068 0.680460i −2.79799 2.79799i 3.07395 + 2.55946i 5.84946 5.84946i 3.35822 + 7.16606i 9.27837 + 9.27837i −4.03952 6.90524i 6.65749i −14.9813 + 7.02067i
115.5 −1.76140 + 0.947342i −1.95177 1.95177i 2.20509 3.33730i −5.56104 + 5.56104i 5.28684 + 1.58886i 4.14904 + 4.14904i −0.722483 + 7.96731i 1.38120i 4.52703 15.0634i
115.6 −1.67184 + 1.09770i 2.87011 + 2.87011i 1.59010 3.67037i −0.478405 + 0.478405i −7.94889 1.64783i 5.13465 + 5.13465i 1.37059 + 7.88172i 7.47505i 0.274669 1.32496i
115.7 −1.64296 1.14047i 3.48624 + 3.48624i 1.39866 + 3.74750i 5.68473 5.68473i −1.75181 9.70371i −3.01446 3.01446i 1.97596 7.75213i 15.3077i −15.8231 + 2.85655i
115.8 −1.55587 1.25669i 0.0765415 + 0.0765415i 0.841444 + 3.91050i −1.52277 + 1.52277i −0.0228992 0.215278i −3.30548 3.30548i 3.60512 7.14165i 8.98828i 4.28288 0.455573i
115.9 −1.54442 + 1.27074i −4.05827 4.05827i 0.770451 3.92510i 3.34858 3.34858i 11.4247 + 1.11067i −7.12993 7.12993i 3.79787 + 7.04103i 23.9391i −0.916437 + 9.42676i
115.10 −1.17987 1.61490i −3.41961 3.41961i −1.21583 + 3.81074i −2.66341 + 2.66341i −1.48766 + 9.55703i −1.35907 1.35907i 7.58850 2.53271i 14.3875i 7.44363 + 1.15869i
115.11 −1.10596 + 1.66639i 1.14630 + 1.14630i −1.55370 3.68592i −4.78622 + 4.78622i −3.17793 + 0.642415i −7.03698 7.03698i 7.86051 + 1.48742i 6.37201i −2.68233 13.2691i
115.12 −0.860518 + 1.80541i 0.887294 + 0.887294i −2.51902 3.10718i 5.25581 5.25581i −2.36546 + 0.838398i −0.502888 0.502888i 7.77739 1.87408i 7.42542i 4.96617 + 14.0116i
115.13 −0.775308 + 1.84361i −2.20075 2.20075i −2.79779 2.85873i −0.154410 + 0.154410i 5.76358 2.35106i 6.51226 + 6.51226i 7.43954 2.94164i 0.686594i −0.164957 0.404387i
115.14 −0.678905 1.88125i 3.22659 + 3.22659i −3.07817 + 2.55438i −4.60664 + 4.60664i 3.87946 8.26055i −3.51075 3.51075i 6.89520 + 4.05662i 11.8217i 11.7937 + 5.53875i
115.15 −0.603225 1.90686i 0.735236 + 0.735236i −3.27224 + 2.30053i 0.444064 0.444064i 0.958480 1.84551i 8.56417 + 8.56417i 6.36070 + 4.85196i 7.91886i −1.11464 0.578897i
115.16 −0.462601 1.94576i −1.38331 1.38331i −3.57200 + 1.80023i 5.77712 5.77712i −2.05167 + 3.33151i −3.98488 3.98488i 5.15523 + 6.11748i 5.17291i −13.9134 8.56842i
115.17 0.462601 1.94576i −1.38331 1.38331i −3.57200 1.80023i −5.77712 + 5.77712i −3.33151 + 2.05167i 3.98488 + 3.98488i −5.15523 + 6.11748i 5.17291i 8.56842 + 13.9134i
115.18 0.603225 1.90686i 0.735236 + 0.735236i −3.27224 2.30053i −0.444064 + 0.444064i 1.84551 0.958480i −8.56417 8.56417i −6.36070 + 4.85196i 7.91886i 0.578897 + 1.11464i
115.19 0.678905 1.88125i 3.22659 + 3.22659i −3.07817 2.55438i 4.60664 4.60664i 8.26055 3.87946i 3.51075 + 3.51075i −6.89520 + 4.05662i 11.8217i −5.53875 11.7937i
115.20 0.775308 + 1.84361i −2.20075 2.20075i −2.79779 + 2.85873i 0.154410 0.154410i 2.35106 5.76358i −6.51226 6.51226i −7.43954 2.94164i 0.686594i 0.404387 + 0.164957i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 123.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
17.c even 4 1 inner
136.j odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.3.j.b 64
4.b odd 2 1 544.3.n.b 64
8.b even 2 1 544.3.n.b 64
8.d odd 2 1 inner 136.3.j.b 64
17.c even 4 1 inner 136.3.j.b 64
68.f odd 4 1 544.3.n.b 64
136.i even 4 1 544.3.n.b 64
136.j odd 4 1 inner 136.3.j.b 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.j.b 64 1.a even 1 1 trivial
136.3.j.b 64 8.d odd 2 1 inner
136.3.j.b 64 17.c even 4 1 inner
136.3.j.b 64 136.j odd 4 1 inner
544.3.n.b 64 4.b odd 2 1
544.3.n.b 64 8.b even 2 1
544.3.n.b 64 68.f odd 4 1
544.3.n.b 64 136.i even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$17\!\cdots\!60$$$$T_{3}^{10} -$$$$50\!\cdots\!48$$$$T_{3}^{9} +$$$$58\!\cdots\!16$$$$T_{3}^{8} -$$$$26\!\cdots\!28$$$$T_{3}^{7} +$$$$13\!\cdots\!72$$$$T_{3}^{6} -$$$$37\!\cdots\!20$$$$T_{3}^{5} +$$$$54\!\cdots\!24$$$$T_{3}^{4} -$$$$41\!\cdots\!76$$$$T_{3}^{3} +$$$$17\!\cdots\!00$$$$T_{3}^{2} -$$$$21\!\cdots\!80$$$$T_{3} + 136273199104$$">$$T_{3}^{32} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(136, [\chi])$$.