# Properties

 Label 192.3.t.a Level $192$ Weight $3$ Character orbit 192.t Analytic conductor $5.232$ Analytic rank $0$ Dimension $256$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.23162107572$$ Analytic rank: $$0$$ Dimension: $$256$$ Relative dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$256q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$256q + 144q^{22} + 400q^{26} + 240q^{28} - 80q^{32} - 240q^{34} - 560q^{38} - 720q^{40} - 208q^{44} - 624q^{50} + 384q^{51} - 528q^{52} - 144q^{54} + 512q^{55} - 784q^{56} + 512q^{59} - 288q^{60} - 96q^{62} + 96q^{64} + 288q^{66} - 128q^{67} + 480q^{68} + 672q^{70} - 1024q^{71} + 1232q^{74} - 768q^{75} + 208q^{76} + 720q^{78} - 512q^{79} + 816q^{80} + 1040q^{82} + 560q^{88} + 96q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.99032 0.196561i 1.44015 0.962276i 3.92273 + 0.782437i −1.54570 7.77077i −3.05550 + 1.63216i −11.5402 + 4.78010i −7.65368 2.32835i 1.14805 2.77164i 1.54901 + 15.7701i
19.2 −1.97540 0.312752i −1.44015 + 0.962276i 3.80437 + 1.23562i −0.279220 1.40373i 3.14581 1.45047i −3.62990 + 1.50355i −7.12870 3.63066i 1.14805 2.77164i 0.112549 + 2.86026i
19.3 −1.92244 + 0.551559i −1.44015 + 0.962276i 3.39156 2.12068i 1.64535 + 8.27175i 2.23785 2.64425i 8.64107 3.57925i −5.35040 + 5.94754i 1.14805 2.77164i −7.72546 14.9945i
19.4 −1.84535 + 0.771151i 1.44015 0.962276i 2.81065 2.84609i −0.118194 0.594202i −1.91552 + 2.88631i 4.91350 2.03524i −2.99188 + 7.41948i 1.14805 2.77164i 0.676329 + 1.00537i
19.5 −1.77755 0.916684i 1.44015 0.962276i 2.31938 + 3.25891i 0.937462 + 4.71294i −3.44204 + 0.390335i 6.22692 2.57928i −1.13543 7.91901i 1.14805 2.77164i 2.65389 9.23685i
19.6 −1.75449 + 0.960092i −1.44015 + 0.962276i 2.15645 3.36894i −0.982422 4.93897i 1.60285 3.07097i 2.14513 0.888543i −0.548972 + 7.98114i 1.14805 2.77164i 6.46551 + 7.72214i
19.7 −1.70771 1.04102i −1.44015 + 0.962276i 1.83257 + 3.55552i 1.29633 + 6.51710i 3.46110 0.144072i −5.83647 + 2.41754i 0.571859 7.97953i 1.14805 2.77164i 4.57065 12.4788i
19.8 −1.36103 1.46546i 1.44015 0.962276i −0.295169 + 3.98909i 0.242100 + 1.21712i −3.37027 0.800794i −4.94543 + 2.04846i 6.24761 4.99674i 1.14805 2.77164i 1.45414 2.01133i
19.9 −1.27767 + 1.53868i −1.44015 + 0.962276i −0.735101 3.93187i 0.251501 + 1.26438i 0.359399 3.44541i −9.71768 + 4.02520i 6.98913 + 3.89256i 1.14805 2.77164i −2.26682 1.22849i
19.10 −1.27046 + 1.54465i 1.44015 0.962276i −0.771866 3.92482i 0.951985 + 4.78595i −0.343273 + 3.44705i −6.47911 + 2.68373i 7.04309 + 3.79407i 1.14805 2.77164i −8.60206 4.60988i
19.11 −1.26075 1.55258i −1.44015 + 0.962276i −0.821030 + 3.91483i −1.08391 5.44919i 3.30968 + 1.02276i −3.97032 + 1.64456i 7.11321 3.66090i 1.14805 2.77164i −7.09379 + 8.55292i
19.12 −1.08433 1.68055i 1.44015 0.962276i −1.64847 + 3.64452i −1.65925 8.34160i −3.17874 1.37681i 7.15690 2.96448i 7.91227 1.18154i 1.14805 2.77164i −12.2193 + 11.8335i
19.13 −0.760241 + 1.84987i −1.44015 + 0.962276i −2.84407 2.81270i −0.194779 0.979220i −0.685230 3.39565i 6.99962 2.89934i 7.36532 3.12284i 1.14805 2.77164i 1.95951 + 0.384127i
19.14 −0.441777 1.95060i −1.44015 + 0.962276i −3.60967 + 1.72346i 0.109322 + 0.549600i 2.51324 + 2.38404i 6.48355 2.68558i 4.95644 + 6.27962i 1.14805 2.77164i 1.02375 0.456044i
19.15 −0.399279 1.95974i 1.44015 0.962276i −3.68115 + 1.56497i 0.758038 + 3.81092i −2.46083 2.43810i −10.7874 + 4.46830i 4.53673 + 6.58924i 1.14805 2.77164i 7.16573 3.00718i
19.16 −0.358174 + 1.96767i 1.44015 0.962276i −3.74342 1.40954i −0.809137 4.06781i 1.37761 + 3.17839i −6.21464 + 2.57419i 4.11429 6.86095i 1.14805 2.77164i 8.29390 0.135127i
19.17 0.143105 + 1.99487i 1.44015 0.962276i −3.95904 + 0.570954i −0.196739 0.989075i 2.12571 + 2.73521i 12.0254 4.98107i −1.70554 7.81608i 1.14805 2.77164i 1.94492 0.534012i
19.18 0.195531 + 1.99042i −1.44015 + 0.962276i −3.92354 + 0.778375i 1.68587 + 8.47543i −2.19693 2.67834i −3.67509 + 1.52227i −2.31646 7.65728i 1.14805 2.77164i −16.5400 + 5.01279i
19.19 0.377294 1.96409i 1.44015 0.962276i −3.71530 1.48208i 1.80347 + 9.06666i −1.34664 3.19164i 9.72216 4.02705i −4.31270 + 6.73800i 1.14805 2.77164i 18.4882 0.121382i
19.20 0.679497 1.88103i −1.44015 + 0.962276i −3.07657 2.55631i 0.892221 + 4.48550i 0.831497 + 3.36283i −2.40197 + 0.994929i −6.89902 + 4.05012i 1.14805 2.77164i 9.04363 + 1.36958i
See next 80 embeddings (of 256 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 187.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
64.j odd 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.3.t.a 256
64.j odd 16 1 inner 192.3.t.a 256

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.3.t.a 256 1.a even 1 1 trivial
192.3.t.a 256 64.j odd 16 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(192, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database