# Properties

 Label 384.2.v.a Level $384$ Weight $2$ Character orbit 384.v Analytic conductor $3.066$ Analytic rank $0$ Dimension $512$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,2,Mod(13,384)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(32))

chi = DirichletCharacter(H, H._module([0, 15, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("384.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$3.06625543762$$ Analytic rank: $$0$$ Dimension: $$512$$ Relative dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$512 q+O(q^{10})$$ 512 * q $$\operatorname{Tr}(f)(q) =$$ $$512 q - 96 q^{50} - 96 q^{52} - 32 q^{54} - 224 q^{56} - 192 q^{60} - 192 q^{62} - 192 q^{64} - 192 q^{66} - 192 q^{68} - 192 q^{70} - 224 q^{74} - 32 q^{76} - 96 q^{78} - 96 q^{80}+O(q^{100})$$ 512 * q - 96 * q^50 - 96 * q^52 - 32 * q^54 - 224 * q^56 - 192 * q^60 - 192 * q^62 - 192 * q^64 - 192 * q^66 - 192 * q^68 - 192 * q^70 - 224 * q^74 - 32 * q^76 - 96 * q^78 - 96 * q^80

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −1.40051 0.196427i 0.290285 + 0.956940i 1.92283 + 0.550195i 0.0502132 + 0.509823i −0.218576 1.39722i −1.66106 + 2.48595i −2.58487 1.14825i −0.831470 + 0.555570i 0.0298193 0.723874i
13.2 −1.39658 0.222610i −0.290285 0.956940i 1.90089 + 0.621788i 0.0409738 + 0.416014i 0.192382 + 1.40107i −0.338624 + 0.506787i −2.51633 1.29154i −0.831470 + 0.555570i 0.0353857 0.590120i
13.3 −1.34449 + 0.438569i −0.290285 0.956940i 1.61531 1.17931i 0.414961 + 4.21317i 0.809970 + 1.15929i −0.841223 + 1.25898i −1.65457 + 2.29399i −0.831470 + 0.555570i −2.40568 5.48258i
13.4 −1.30780 0.538203i 0.290285 + 0.956940i 1.42068 + 1.40772i 0.403677 + 4.09860i 0.135394 1.40772i 2.40587 3.60064i −1.10032 2.60563i −0.831470 + 0.555570i 1.67795 5.57741i
13.5 −1.26580 + 0.630679i 0.290285 + 0.956940i 1.20449 1.59662i −0.0580215 0.589102i −0.970964 1.02822i 0.672130 1.00591i −0.517679 + 2.78065i −0.831470 + 0.555570i 0.444978 + 0.709091i
13.6 −1.21456 + 0.724467i −0.290285 0.956940i 0.950294 1.75981i −0.290371 2.94818i 1.04584 + 0.951956i 1.76246 2.63771i 0.120741 + 2.82585i −0.831470 + 0.555570i 2.48853 + 3.37037i
13.7 −1.06403 0.931580i 0.290285 + 0.956940i 0.264317 + 1.98246i −0.231940 2.35493i 0.582595 1.28864i 0.430118 0.643717i 1.56558 2.35563i −0.831470 + 0.555570i −1.94701 + 2.72179i
13.8 −1.06073 0.935332i −0.290285 0.956940i 0.250309 + 1.98427i −0.385901 3.91812i −0.587142 + 1.28657i −2.38112 + 3.56360i 1.59044 2.33891i −0.831470 + 0.555570i −3.25540 + 4.51703i
13.9 −0.922626 + 1.07180i −0.290285 0.956940i −0.297523 1.97775i −0.0692954 0.703568i 1.29348 + 0.571770i −2.09840 + 3.14048i 2.39426 + 1.50583i −0.831470 + 0.555570i 0.818020 + 0.574859i
13.10 −0.870630 + 1.11445i 0.290285 + 0.956940i −0.484007 1.94055i −0.328555 3.33588i −1.31919 0.509633i −1.53196 + 2.29274i 2.58404 + 1.15010i −0.831470 + 0.555570i 4.00372 + 2.53816i
13.11 −0.612256 1.27481i −0.290285 0.956940i −1.25028 + 1.56102i 0.189500 + 1.92402i −1.04219 + 0.955951i 0.476963 0.713825i 2.75550 + 0.638130i −0.831470 + 0.555570i 2.33674 1.41957i
13.12 −0.607361 + 1.27715i −0.290285 0.956940i −1.26223 1.55138i 0.269039 + 2.73160i 1.39846 + 0.210471i 0.857384 1.28317i 2.74797 0.669804i −0.831470 + 0.555570i −3.65206 1.31546i
13.13 −0.542361 1.30608i 0.290285 + 0.956940i −1.41169 + 1.41673i 0.155407 + 1.57788i 1.09240 0.898142i 0.0321569 0.0481261i 2.61601 + 1.07540i −0.831470 + 0.555570i 1.97655 1.05875i
13.14 −0.283082 + 1.38559i 0.290285 + 0.956940i −1.83973 0.784471i −0.166665 1.69218i −1.40810 + 0.131324i 1.76235 2.63755i 1.60775 2.32704i −0.831470 + 0.555570i 2.39185 + 0.248095i
13.15 −0.229611 1.39545i 0.290285 + 0.956940i −1.89456 + 0.640821i −0.159671 1.62116i 1.26871 0.624802i −2.66520 + 3.98875i 1.32924 + 2.49662i −0.831470 + 0.555570i −2.22559 + 0.595050i
13.16 −0.162120 1.40489i −0.290285 0.956940i −1.94743 + 0.455521i −0.344842 3.50124i −1.29734 + 0.562957i 1.89335 2.83360i 0.955675 + 2.66208i −0.831470 + 0.555570i −4.86295 + 1.05208i
13.17 −0.156266 + 1.40555i −0.290285 0.956940i −1.95116 0.439279i −0.103867 1.05457i 1.39039 0.258474i −0.0783117 + 0.117202i 0.922330 2.67382i −0.831470 + 0.555570i 1.49849 + 0.0188036i
13.18 0.147414 1.40651i −0.290285 0.956940i −1.95654 0.414679i 0.138503 + 1.40624i −1.38874 + 0.267222i −1.77662 + 2.65890i −0.871671 + 2.69076i −0.831470 + 0.555570i 1.99831 + 0.0124945i
13.19 0.246397 + 1.39258i 0.290285 + 0.956940i −1.87858 + 0.686257i 0.267357 + 2.71452i −1.26109 + 0.640033i −0.343422 + 0.513967i −1.41855 2.44698i −0.831470 + 0.555570i −3.71432 + 1.04117i
13.20 0.320440 1.37743i 0.290285 + 0.956940i −1.79464 0.882769i −0.362380 3.67931i 1.41114 0.0932050i 1.08924 1.63016i −1.79103 + 2.18911i −0.831470 + 0.555570i −5.18411 0.679844i
See next 80 embeddings (of 512 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
128.k 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.v.a 512
128.k even 32 1 inner 384.2.v.a 512

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.v.a 512 1.a even 1 1 trivial
384.2.v.a 512 128.k even 32 1 inner

## Hecke kernels

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