# Properties

 Label 135.2.q.a Level $135$ Weight $2$ Character orbit 135.q Analytic conductor $1.078$ Analytic rank $0$ Dimension $192$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 135.q (of order $$36$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.07798042729$$ Analytic rank: $$0$$ Dimension: $$192$$ Relative dimension: $$16$$ over $$\Q(\zeta_{36})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$192 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 36 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10})$$ 192 * q - 12 * q^2 - 12 * q^3 - 12 * q^5 - 36 * q^6 - 12 * q^7 - 18 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$192 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 36 q^{6} - 12 q^{7} - 18 q^{8} - 6 q^{10} - 36 q^{11} - 12 q^{12} - 12 q^{13} - 12 q^{15} - 24 q^{16} - 18 q^{17} - 54 q^{18} + 36 q^{20} - 24 q^{21} - 12 q^{22} - 36 q^{23} - 30 q^{25} - 36 q^{27} - 24 q^{28} + 60 q^{30} - 24 q^{31} - 48 q^{32} - 6 q^{33} + 36 q^{35} + 12 q^{36} - 6 q^{37} + 12 q^{38} - 36 q^{40} + 24 q^{41} - 24 q^{42} - 12 q^{43} + 18 q^{45} - 12 q^{46} - 6 q^{47} + 12 q^{48} + 36 q^{50} + 144 q^{51} + 12 q^{52} - 24 q^{55} + 180 q^{56} - 12 q^{57} - 12 q^{58} - 36 q^{60} - 60 q^{61} - 18 q^{62} - 54 q^{63} - 84 q^{65} + 72 q^{66} + 24 q^{67} - 60 q^{68} - 12 q^{70} - 36 q^{71} + 180 q^{72} - 6 q^{73} - 60 q^{75} - 72 q^{76} + 132 q^{77} + 78 q^{78} + 12 q^{81} - 24 q^{82} + 48 q^{83} - 12 q^{85} + 12 q^{86} + 144 q^{87} - 48 q^{88} + 48 q^{90} - 12 q^{91} + 258 q^{92} + 180 q^{93} + 18 q^{95} - 12 q^{96} + 24 q^{97} + 324 q^{98}+O(q^{100})$$ 192 * q - 12 * q^2 - 12 * q^3 - 12 * q^5 - 36 * q^6 - 12 * q^7 - 18 * q^8 - 6 * q^10 - 36 * q^11 - 12 * q^12 - 12 * q^13 - 12 * q^15 - 24 * q^16 - 18 * q^17 - 54 * q^18 + 36 * q^20 - 24 * q^21 - 12 * q^22 - 36 * q^23 - 30 * q^25 - 36 * q^27 - 24 * q^28 + 60 * q^30 - 24 * q^31 - 48 * q^32 - 6 * q^33 + 36 * q^35 + 12 * q^36 - 6 * q^37 + 12 * q^38 - 36 * q^40 + 24 * q^41 - 24 * q^42 - 12 * q^43 + 18 * q^45 - 12 * q^46 - 6 * q^47 + 12 * q^48 + 36 * q^50 + 144 * q^51 + 12 * q^52 - 24 * q^55 + 180 * q^56 - 12 * q^57 - 12 * q^58 - 36 * q^60 - 60 * q^61 - 18 * q^62 - 54 * q^63 - 84 * q^65 + 72 * q^66 + 24 * q^67 - 60 * q^68 - 12 * q^70 - 36 * q^71 + 180 * q^72 - 6 * q^73 - 60 * q^75 - 72 * q^76 + 132 * q^77 + 78 * q^78 + 12 * q^81 - 24 * q^82 + 48 * q^83 - 12 * q^85 + 12 * q^86 + 144 * q^87 - 48 * q^88 + 48 * q^90 - 12 * q^91 + 258 * q^92 + 180 * q^93 + 18 * q^95 - 12 * q^96 + 24 * q^97 + 324 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −1.52658 2.18018i 1.25829 + 1.19025i −1.73870 + 4.77703i −1.77382 + 1.36146i 0.674088 4.56031i 2.78918 + 1.30062i 7.92739 2.12414i 0.166591 + 2.99537i 5.67609 + 1.78885i
2.2 −1.48477 2.12047i 0.685393 1.59067i −1.60780 + 4.41741i 1.13563 1.92622i −4.39061 + 0.908422i −2.68394 1.25154i 6.75336 1.80956i −2.06047 2.18047i −5.77064 + 0.451915i
2.3 −1.16608 1.66533i −0.967587 + 1.43658i −0.729546 + 2.00441i 1.90549 1.17008i 3.52066 0.0638164i 2.43427 + 1.13512i 0.261271 0.0700073i −1.12755 2.78004i −4.17052 1.80887i
2.4 −0.996232 1.42277i −1.44830 0.949962i −0.347747 + 0.955427i −1.86340 1.23602i 0.0912678 + 3.00698i 0.920112 + 0.429055i −1.64960 + 0.442010i 1.19514 + 2.75166i 0.0978025 + 3.88255i
2.5 −0.856769 1.22359i −0.960742 + 1.44117i −0.0790861 + 0.217287i −1.10572 + 1.94355i 2.58654 0.0591935i −4.36642 2.03610i −2.55204 + 0.683816i −1.15395 2.76919i 3.32546 0.312216i
2.6 −0.761943 1.08817i 1.60100 0.660916i 0.0804885 0.221140i 1.62191 + 1.53928i −1.93906 1.23857i 1.77434 + 0.827388i −2.86825 + 0.768546i 2.12638 2.11625i 0.439192 2.93776i
2.7 −0.565301 0.807333i 1.55815 + 0.756415i 0.351818 0.966613i −0.542991 2.16914i −0.270145 1.68555i −0.590505 0.275357i −2.88324 + 0.772562i 1.85567 + 2.35722i −1.44426 + 1.66459i
2.8 −0.264318 0.377486i −0.938886 1.45550i 0.611409 1.67983i 2.04357 + 0.907638i −0.301267 + 0.739132i −3.26325 1.52168i −1.68596 + 0.451753i −1.23699 + 2.73311i −0.197534 1.01132i
2.9 0.0649826 + 0.0928047i 0.728671 1.57132i 0.679650 1.86732i −2.22708 + 0.200340i 0.193177 0.0344841i 0.739528 + 0.344848i 0.436329 0.116914i −1.93808 2.28995i −0.163314 0.193665i
2.10 0.184358 + 0.263290i −1.70366 + 0.312334i 0.648706 1.78231i −0.0853346 + 2.23444i −0.396317 0.390975i 4.46037 + 2.07991i 1.20979 0.324162i 2.80490 1.06422i −0.604038 + 0.389469i
2.11 0.343150 + 0.490069i 0.150104 + 1.72553i 0.561625 1.54305i 2.04155 0.912186i −0.794122 + 0.665679i −0.136472 0.0636379i 2.10468 0.563947i −2.95494 + 0.518020i 1.14759 + 0.687482i
2.12 0.606324 + 0.865920i 1.47671 + 0.905161i 0.301851 0.829330i −1.02123 + 1.98924i 0.111569 + 1.82754i −2.61975 1.22161i 2.94330 0.788655i 1.36137 + 2.67333i −2.34172 + 0.321816i
2.13 0.866263 + 1.23715i −0.938839 1.45553i −0.0960923 + 0.264011i 0.525423 2.17346i 0.987435 2.42236i 1.45360 + 0.677826i 2.50778 0.671958i −1.23716 + 2.73303i 3.14405 1.23276i
2.14 1.15565 + 1.65044i −0.352907 + 1.69572i −0.704386 + 1.93528i −2.13942 0.650308i −3.20652 + 1.37721i 1.32708 + 0.618828i −0.115769 + 0.0310202i −2.75091 1.19686i −1.39912 4.28251i
2.15 1.36466 + 1.94894i −1.63096 + 0.583078i −1.25203 + 3.43992i 2.13659 + 0.659531i −3.36209 2.38294i −2.19758 1.02475i −3.81650 + 1.02263i 2.32004 1.90195i 1.63034 + 5.06413i
2.16 1.43892 + 2.05499i 1.64201 0.551185i −1.46845 + 4.03452i −0.953539 2.02256i 3.49539 + 2.58120i −2.44940 1.14217i −5.55747 + 1.48912i 2.39239 1.81010i 2.78428 4.86981i
23.1 −1.15365 2.47401i −1.57848 0.713012i −3.50423 + 4.17618i −1.08960 1.95263i 0.0570200 + 4.72775i −0.00587402 + 0.0671403i 9.10104 + 2.43862i 1.98323 + 2.25096i −3.57380 + 4.94833i
23.2 −0.849158 1.82102i 1.18657 1.26176i −1.30949 + 1.56058i −1.94472 + 1.10365i −3.30529 1.08934i 0.278616 3.18459i 0.0721951 + 0.0193446i −0.184096 2.99435i 3.66116 + 2.60421i
23.3 −0.846935 1.81626i 1.73205 0.00445475i −1.29592 + 1.54442i 1.48888 1.66830i −1.47502 3.14207i −0.193523 + 2.21198i 0.0311508 + 0.00834683i 2.99996 0.0154316i −4.29105 1.29125i
23.4 −0.768168 1.64734i −1.46825 + 0.918833i −0.838077 + 0.998782i 0.104070 + 2.23364i 2.64149 + 1.71288i −0.393694 + 4.49995i −1.22229 0.327512i 1.31149 2.69815i 3.59963 1.88725i
See next 80 embeddings (of 192 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 128.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
27.f odd 18 1 inner
135.q even 36 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.2.q.a 192
3.b odd 2 1 405.2.r.a 192
5.b even 2 1 675.2.ba.b 192
5.c odd 4 1 inner 135.2.q.a 192
5.c odd 4 1 675.2.ba.b 192
15.e even 4 1 405.2.r.a 192
27.e even 9 1 405.2.r.a 192
27.f odd 18 1 inner 135.2.q.a 192
135.n odd 18 1 675.2.ba.b 192
135.q even 36 1 inner 135.2.q.a 192
135.q even 36 1 675.2.ba.b 192
135.r odd 36 1 405.2.r.a 192

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.q.a 192 1.a even 1 1 trivial
135.2.q.a 192 5.c odd 4 1 inner
135.2.q.a 192 27.f odd 18 1 inner
135.2.q.a 192 135.q even 36 1 inner
405.2.r.a 192 3.b odd 2 1
405.2.r.a 192 15.e even 4 1
405.2.r.a 192 27.e even 9 1
405.2.r.a 192 135.r odd 36 1
675.2.ba.b 192 5.b even 2 1
675.2.ba.b 192 5.c odd 4 1
675.2.ba.b 192 135.n odd 18 1
675.2.ba.b 192 135.q even 36 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(135, [\chi])$$.