# Properties

 Label 192.2.s.a Level $192$ Weight $2$ Character orbit 192.s Analytic conductor $1.533$ Analytic rank $0$ Dimension $240$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 192.s (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.53312771881$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$30$$ over $$\Q(\zeta_{16})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10})$$ 240 * q - 8 * q^3 - 16 * q^4 - 8 * q^6 - 16 * q^7 - 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$240 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9} - 16 q^{10} - 8 q^{12} - 16 q^{13} - 8 q^{15} - 16 q^{16} - 8 q^{18} - 16 q^{19} - 8 q^{21} - 16 q^{22} - 48 q^{24} - 16 q^{25} - 8 q^{27} - 16 q^{28} - 88 q^{30} - 32 q^{31} - 16 q^{34} - 88 q^{36} - 16 q^{37} - 8 q^{39} - 16 q^{40} - 48 q^{42} - 16 q^{43} - 8 q^{45} - 16 q^{46} - 8 q^{48} - 16 q^{49} - 8 q^{51} + 32 q^{52} - 8 q^{54} - 80 q^{55} - 8 q^{57} + 128 q^{58} - 8 q^{60} - 16 q^{61} + 80 q^{64} - 24 q^{66} - 144 q^{67} - 8 q^{69} + 80 q^{70} - 8 q^{72} - 16 q^{73} - 8 q^{75} + 96 q^{76} + 16 q^{78} - 48 q^{79} - 8 q^{81} + 64 q^{82} + 104 q^{84} - 16 q^{85} - 8 q^{87} + 64 q^{88} + 136 q^{90} - 16 q^{91} - 32 q^{93} + 80 q^{94} + 128 q^{96} - 8 q^{99}+O(q^{100})$$ 240 * q - 8 * q^3 - 16 * q^4 - 8 * q^6 - 16 * q^7 - 8 * q^9 - 16 * q^10 - 8 * q^12 - 16 * q^13 - 8 * q^15 - 16 * q^16 - 8 * q^18 - 16 * q^19 - 8 * q^21 - 16 * q^22 - 48 * q^24 - 16 * q^25 - 8 * q^27 - 16 * q^28 - 88 * q^30 - 32 * q^31 - 16 * q^34 - 88 * q^36 - 16 * q^37 - 8 * q^39 - 16 * q^40 - 48 * q^42 - 16 * q^43 - 8 * q^45 - 16 * q^46 - 8 * q^48 - 16 * q^49 - 8 * q^51 + 32 * q^52 - 8 * q^54 - 80 * q^55 - 8 * q^57 + 128 * q^58 - 8 * q^60 - 16 * q^61 + 80 * q^64 - 24 * q^66 - 144 * q^67 - 8 * q^69 + 80 * q^70 - 8 * q^72 - 16 * q^73 - 8 * q^75 + 96 * q^76 + 16 * q^78 - 48 * q^79 - 8 * q^81 + 64 * q^82 + 104 * q^84 - 16 * q^85 - 8 * q^87 + 64 * q^88 + 136 * q^90 - 16 * q^91 - 32 * q^93 + 80 * q^94 + 128 * q^96 - 8 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.40912 + 0.119909i −0.169478 + 1.72374i 1.97124 0.337931i 2.90474 1.94088i 0.0321240 2.44928i 0.941226 2.27232i −2.73720 + 0.712555i −2.94255 0.584272i −3.86040 + 3.08324i
11.2 −1.38916 + 0.265037i 1.44627 + 0.953045i 1.85951 0.736355i −1.95251 + 1.30462i −2.26169 0.940613i −0.960548 + 2.31897i −2.38799 + 1.51575i 1.18341 + 2.75673i 2.36656 2.32981i
11.3 −1.35305 + 0.411422i −0.145818 1.72590i 1.66146 1.11334i 0.731423 0.488721i 0.907373 + 2.27523i 0.683144 1.64925i −1.78998 + 2.18997i −2.95747 + 0.503337i −0.788578 + 0.962186i
11.4 −1.30319 0.549270i 1.62032 0.612015i 1.39660 + 1.43161i 3.01614 2.01532i −2.44775 0.0924222i −0.924170 + 2.23114i −1.03370 2.63277i 2.25087 1.98332i −5.03756 + 0.969667i
11.5 −1.28901 0.581756i −1.06591 1.36523i 1.32312 + 1.49979i −0.780999 + 0.521847i 0.579741 + 2.37990i −1.37838 + 3.32770i −0.833011 2.70298i −0.727685 + 2.91041i 1.31031 0.218317i
11.6 −1.14703 0.827236i 1.31348 1.12905i 0.631361 + 1.89773i −2.57009 + 1.71728i −2.44060 + 0.208501i 1.31022 3.16314i 0.845681 2.69904i 0.450472 2.96599i 4.36857 + 0.156298i
11.7 −1.13891 + 0.838376i −1.71622 + 0.233636i 0.594252 1.90968i 1.07447 0.717940i 1.75875 1.70493i −0.999337 + 2.41261i 0.924224 + 2.67316i 2.89083 0.801943i −0.621830 + 1.71848i
11.8 −1.01373 0.986078i 0.252613 + 1.71353i 0.0553009 + 1.99924i −1.22014 + 0.815271i 1.43359 1.98615i 0.731141 1.76513i 1.91534 2.08122i −2.87237 + 0.865720i 2.04081 + 0.376687i
11.9 −0.814459 + 1.15614i 1.69388 + 0.361602i −0.673315 1.88325i 0.388565 0.259631i −1.79766 + 1.66386i 1.96536 4.74480i 2.72569 + 0.755387i 2.73849 + 1.22502i −0.0163008 + 0.660694i
11.10 −0.745539 1.20174i −1.61975 + 0.613518i −0.888342 + 1.79188i 1.55377 1.03819i 1.94488 + 1.48911i −0.159818 + 0.385836i 2.81567 0.268367i 2.24719 1.98749i −2.40603 1.09320i
11.11 −0.640805 + 1.26070i 0.585574 1.63006i −1.17874 1.61573i −3.57797 + 2.39072i 1.67978 + 1.78279i −0.994439 + 2.40079i 2.79229 0.450669i −2.31421 1.90904i −0.721207 6.04274i
11.12 −0.321350 1.37722i 1.56005 + 0.752482i −1.79347 + 0.885139i 1.30855 0.874343i 0.535010 2.39035i 0.168531 0.406870i 1.79536 + 2.18556i 1.86754 + 2.34783i −1.62466 1.52119i
11.13 −0.300423 + 1.38194i −1.49362 0.876991i −1.81949 0.830329i −0.191752 + 0.128125i 1.66066 1.80061i 0.553671 1.33668i 1.69408 2.26497i 1.46177 + 2.61977i −0.119453 0.303480i
11.14 −0.273585 + 1.38750i 0.865146 + 1.50051i −1.85030 0.759198i 2.09602 1.40051i −2.31864 + 0.789872i −1.76691 + 4.26571i 1.55960 2.35959i −1.50305 + 2.59632i 1.36977 + 3.29138i
11.15 −0.216199 1.39759i −1.36876 1.06137i −1.90652 + 0.604316i −2.38253 + 1.59195i −1.18743 + 2.14243i 0.537426 1.29746i 1.25677 + 2.53387i 0.746990 + 2.90551i 2.74000 + 2.98562i
11.16 0.216199 + 1.39759i 0.858398 1.50438i −1.90652 + 0.604316i 2.38253 1.59195i 2.28809 + 0.874443i 0.537426 1.29746i −1.25677 2.53387i −1.52631 2.58271i 2.74000 + 2.98562i
11.17 0.273585 1.38750i −0.225071 + 1.71737i −1.85030 0.759198i −2.09602 + 1.40051i 2.32126 + 0.782132i −1.76691 + 4.26571i −1.55960 + 2.35959i −2.89869 0.773059i 1.36977 + 3.29138i
11.18 0.300423 1.38194i 1.04431 1.38182i −1.81949 0.830329i 0.191752 0.128125i −1.59585 1.85830i 0.553671 1.33668i −1.69408 + 2.26497i −0.818831 2.88609i −0.119453 0.303480i
11.19 0.321350 + 1.37722i −1.15334 + 1.29221i −1.79347 + 0.885139i −1.30855 + 0.874343i −2.15028 1.17315i 0.168531 0.406870i −1.79536 2.18556i −0.339613 2.98072i −1.62466 1.52119i
11.20 0.640805 1.26070i −1.16480 1.28189i −1.17874 1.61573i 3.57797 2.39072i −2.36249 + 0.647019i −0.994439 + 2.40079i −2.79229 + 0.450669i −0.286494 + 2.98629i −0.721207 6.04274i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 179.30 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
64.j odd 16 1 inner
192.s even 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.2.s.a 240
3.b odd 2 1 inner 192.2.s.a 240
4.b odd 2 1 768.2.s.a 240
12.b even 2 1 768.2.s.a 240
64.i even 16 1 768.2.s.a 240
64.j odd 16 1 inner 192.2.s.a 240
192.q odd 16 1 768.2.s.a 240
192.s even 16 1 inner 192.2.s.a 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.2.s.a 240 1.a even 1 1 trivial
192.2.s.a 240 3.b odd 2 1 inner
192.2.s.a 240 64.j odd 16 1 inner
192.2.s.a 240 192.s even 16 1 inner
768.2.s.a 240 4.b odd 2 1
768.2.s.a 240 12.b even 2 1
768.2.s.a 240 64.i even 16 1
768.2.s.a 240 192.q odd 16 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(192, [\chi])$$.