# Properties

 Label 384.2.w.a Level $384$ Weight $2$ Character orbit 384.w Analytic conductor $3.066$ Analytic rank $0$ Dimension $992$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 384.w (of order $$32$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.06625543762$$ Analytic rank: $$0$$ Dimension: $$992$$ Relative dimension: $$62$$ over $$\Q(\zeta_{32})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{32}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$992q - 16q^{3} - 32q^{4} - 16q^{6} - 32q^{7} - 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$992q - 16q^{3} - 32q^{4} - 16q^{6} - 32q^{7} - 16q^{9} - 32q^{10} - 16q^{12} - 32q^{13} - 16q^{15} - 32q^{16} - 16q^{18} - 32q^{19} - 16q^{21} - 32q^{22} - 16q^{24} - 32q^{25} - 16q^{27} - 32q^{28} - 16q^{30} - 32q^{31} - 16q^{33} - 32q^{34} - 16q^{36} - 32q^{37} - 16q^{39} - 32q^{40} - 16q^{42} - 32q^{43} - 16q^{45} - 32q^{46} - 16q^{48} - 32q^{49} - 16q^{51} - 128q^{52} - 16q^{54} - 32q^{55} - 16q^{57} - 320q^{58} - 16q^{60} - 32q^{61} - 32q^{63} - 224q^{64} - 16q^{66} - 32q^{67} - 16q^{69} - 224q^{70} - 16q^{72} - 32q^{73} - 16q^{75} - 256q^{76} - 16q^{78} - 32q^{79} - 16q^{81} - 32q^{82} - 16q^{84} - 32q^{85} - 16q^{87} - 32q^{88} - 16q^{90} - 32q^{91} - 16q^{93} - 32q^{94} - 16q^{96} - 32q^{97} - 16q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.41413 0.0157861i 1.41096 + 1.00458i 1.99950 + 0.0446470i −3.04456 1.62735i −1.97942 1.44288i −1.52522 0.303386i −2.82684 0.0947007i 0.981620 + 2.83486i 4.27970 + 2.34934i
11.2 −1.41347 + 0.0458784i −0.204040 1.71999i 1.99579 0.129695i −0.561681 0.300225i 0.367315 + 2.42179i −0.946225 0.188216i −2.81504 + 0.274884i −2.91674 + 0.701894i 0.807693 + 0.398589i
11.3 −1.41282 0.0626704i 1.71388 0.250263i 1.99214 + 0.177085i 1.34559 + 0.719232i −2.43709 + 0.246168i 2.49948 + 0.497176i −2.80345 0.375038i 2.87474 0.857839i −1.85601 1.10048i
11.4 −1.39869 0.208968i −0.339181 + 1.69852i 1.91266 + 0.584564i −0.321371 0.171776i 0.829345 2.30482i −1.36158 0.270834i −2.55307 1.21731i −2.76991 1.15221i 0.413602 + 0.307418i
11.5 −1.37752 + 0.320040i −1.61000 0.638681i 1.79515 0.881727i −0.665070 0.355487i 2.42221 + 0.364535i 0.505023 + 0.100455i −2.19067 + 1.78912i 2.18417 + 2.05655i 1.02992 + 0.276843i
11.6 −1.30952 + 0.533994i 1.15242 + 1.29303i 1.42970 1.39855i 3.25884 + 1.74189i −2.19959 1.07787i −3.48323 0.692857i −1.12541 + 2.59489i −0.343871 + 2.98023i −5.19768 0.540839i
11.7 −1.30056 0.555477i −1.67960 + 0.423001i 1.38289 + 1.44486i 3.35351 + 1.79249i 2.41939 + 0.382846i 3.20524 + 0.637561i −0.995942 2.64728i 2.64214 1.42095i −3.36574 4.19402i
11.8 −1.29156 0.576082i −1.73162 + 0.0387632i 1.33626 + 1.48809i −0.861594 0.460532i 2.25882 + 0.947488i −4.16278 0.828028i −0.868599 2.69175i 2.99699 0.134246i 0.847497 + 1.09115i
11.9 −1.27198 + 0.618111i −1.15591 + 1.28991i 1.23588 1.57245i 1.21911 + 0.651630i 0.672992 2.35522i 1.31145 + 0.260863i −0.600063 + 2.76404i −0.327733 2.98204i −1.95347 0.0753137i
11.10 −1.24002 + 0.679964i 1.09320 1.34347i 1.07530 1.68634i 0.708082 + 0.378478i −0.442086 + 2.40927i −4.90117 0.974904i −0.186740 + 2.82226i −0.609807 2.93737i −1.13539 + 0.0121507i
11.11 −1.15643 0.814046i −0.824531 1.52320i 0.674657 + 1.88277i −0.611958 0.327098i −0.286446 + 2.43268i 2.75895 + 0.548789i 0.752472 2.72650i −1.64030 + 2.51186i 0.441413 + 0.876428i
11.12 −1.12499 + 0.856964i 0.434523 + 1.67666i 0.531225 1.92816i −2.13158 1.13935i −1.92567 1.51386i 1.86189 + 0.370353i 1.05474 + 2.62441i −2.62238 + 1.45710i 3.37440 0.544921i
11.13 −1.11732 0.866940i 1.27976 1.16714i 0.496829 + 1.93731i −2.95304 1.57843i −2.44175 + 0.194595i −1.29330 0.257253i 1.12441 2.59532i 0.275578 2.98732i 1.93109 + 4.32373i
11.14 −1.11401 0.871200i 0.593110 + 1.62734i 0.482021 + 1.94104i 3.29266 + 1.75996i 0.757006 2.32958i −0.370976 0.0737916i 1.15406 2.58227i −2.29644 + 1.93038i −2.13477 4.82918i
11.15 −1.10845 + 0.878258i −0.117171 1.72808i 0.457325 1.94701i 3.15748 + 1.68771i 1.64758 + 1.81259i 4.43843 + 0.882860i 1.20306 + 2.55982i −2.97254 + 0.404962i −4.98216 + 0.902342i
11.16 −1.07112 + 0.923423i 1.49221 0.879386i 0.294582 1.97819i −1.88241 1.00617i −0.786282 + 2.31986i 2.66028 + 0.529163i 1.51117 + 2.39089i 1.45336 2.62445i 2.94540 0.660535i
11.17 −1.02490 0.974463i 1.27873 + 1.16826i 0.100842 + 1.99746i −1.18082 0.631161i −0.172143 2.44343i 4.72705 + 0.940268i 1.84309 2.14546i 0.270316 + 2.98780i 0.595179 + 1.79754i
11.18 −0.917171 + 1.07647i −0.843817 1.51260i −0.317594 1.97462i −3.45356 1.84597i 2.40220 + 0.478969i −1.46677 0.291758i 2.41692 + 1.46919i −1.57594 + 2.55272i 5.15464 2.02460i
11.19 −0.845116 1.13392i −0.765283 + 1.55382i −0.571557 + 1.91659i −0.974296 0.520772i 2.40866 0.445384i −1.27617 0.253846i 2.65630 0.971642i −1.82869 2.37822i 0.232879 + 1.54489i
11.20 −0.813493 + 1.15682i −1.73071 0.0681515i −0.676460 1.88213i 1.66341 + 0.889109i 1.48676 1.94668i −2.12405 0.422500i 2.72758 + 0.748555i 2.99071 + 0.235901i −2.38171 + 1.20098i
See next 80 embeddings (of 992 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 371.62 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
128.l odd 32 1 inner
384.w even 32 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.2.w.a 992
3.b odd 2 1 inner 384.2.w.a 992
128.l odd 32 1 inner 384.2.w.a 992
384.w even 32 1 inner 384.2.w.a 992

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.w.a 992 1.a even 1 1 trivial
384.2.w.a 992 3.b odd 2 1 inner
384.2.w.a 992 128.l odd 32 1 inner
384.2.w.a 992 384.w even 32 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(384, [\chi])$$.