# Properties

 Label 704.2.be.a Level $704$ Weight $2$ Character orbit 704.be Analytic conductor $5.621$ Analytic rank $0$ Dimension $176$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,2,Mod(49,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([0, 5, 8]))

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

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

chi := DirichletCharacter("704.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$704 = 2^{6} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 704.be (of order $$20$$, degree $$8$$, not 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: $$5.62146830230$$ Analytic rank: $$0$$ Dimension: $$176$$ Relative dimension: $$22$$ over $$\Q(\zeta_{20})$$ Twist minimal: no (minimal twist has level 176) Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$176 q + 6 q^{3} - 6 q^{5}+O(q^{10})$$ 176 * q + 6 * q^3 - 6 * q^5 $$\operatorname{Tr}(f)(q) =$$ $$176 q + 6 q^{3} - 6 q^{5} + 12 q^{11} - 6 q^{13} + 12 q^{15} - 12 q^{17} + 6 q^{19} - 28 q^{21} + 18 q^{27} - 22 q^{29} + 12 q^{31} - 16 q^{33} + 26 q^{35} + 18 q^{37} + 40 q^{43} - 24 q^{45} + 12 q^{47} + 8 q^{49} - 6 q^{51} - 30 q^{53} - 10 q^{59} - 6 q^{61} + 28 q^{63} - 32 q^{65} - 24 q^{67} + 12 q^{69} + 46 q^{75} - 14 q^{77} + 52 q^{79} - 54 q^{83} + 14 q^{85} + 122 q^{91} + 6 q^{93} - 52 q^{95} - 12 q^{97} - 92 q^{99}+O(q^{100})$$ 176 * q + 6 * q^3 - 6 * q^5 + 12 * q^11 - 6 * q^13 + 12 * q^15 - 12 * q^17 + 6 * q^19 - 28 * q^21 + 18 * q^27 - 22 * q^29 + 12 * q^31 - 16 * q^33 + 26 * q^35 + 18 * q^37 + 40 * q^43 - 24 * q^45 + 12 * q^47 + 8 * q^49 - 6 * q^51 - 30 * q^53 - 10 * q^59 - 6 * q^61 + 28 * q^63 - 32 * q^65 - 24 * q^67 + 12 * q^69 + 46 * q^75 - 14 * q^77 + 52 * q^79 - 54 * q^83 + 14 * q^85 + 122 * q^91 + 6 * q^93 - 52 * q^95 - 12 * q^97 - 92 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1 0 −3.10742 + 0.492167i 0 −1.27797 + 0.651156i 0 −2.30623 3.17425i 0 6.56066 2.13169i 0
49.2 0 −2.68580 + 0.425388i 0 0.102082 0.0520132i 0 2.36506 + 3.25523i 0 4.17938 1.35796i 0
49.3 0 −2.39469 + 0.379282i 0 2.15797 1.09954i 0 0.678287 + 0.933582i 0 2.73753 0.889476i 0
49.4 0 −2.35196 + 0.372513i 0 −3.24121 + 1.65148i 0 −0.349519 0.481071i 0 2.53977 0.825220i 0
49.5 0 −2.09875 + 0.332409i 0 −1.30316 + 0.663991i 0 0.801284 + 1.10287i 0 1.44108 0.468236i 0
49.6 0 −1.33374 + 0.211244i 0 0.141837 0.0722696i 0 −1.45574 2.00365i 0 −1.11893 + 0.363561i 0
49.7 0 −1.22266 + 0.193650i 0 2.51383 1.28086i 0 −2.24956 3.09626i 0 −1.39578 + 0.453517i 0
49.8 0 −0.901291 + 0.142750i 0 3.45027 1.75800i 0 1.81367 + 2.49631i 0 −2.06122 + 0.669732i 0
49.9 0 −0.662536 + 0.104935i 0 −0.264470 + 0.134754i 0 0.696271 + 0.958335i 0 −2.42523 + 0.788004i 0
49.10 0 −0.562832 + 0.0891439i 0 1.83674 0.935866i 0 −0.677755 0.932849i 0 −2.54434 + 0.826705i 0
49.11 0 −0.310032 + 0.0491042i 0 −3.81933 + 1.94605i 0 −1.09214 1.50320i 0 −2.75946 + 0.896603i 0
49.12 0 0.417657 0.0661504i 0 −0.736011 + 0.375016i 0 −1.33653 1.83957i 0 −2.68311 + 0.871795i 0
49.13 0 0.757844 0.120031i 0 −0.814347 + 0.414930i 0 2.91240 + 4.00857i 0 −2.29325 + 0.745122i 0
49.14 0 0.925602 0.146601i 0 −1.96526 + 1.00135i 0 −0.432076 0.594702i 0 −2.01792 + 0.655663i 0
49.15 0 1.17460 0.186039i 0 2.66815 1.35949i 0 −2.54995 3.50970i 0 −1.50808 + 0.490006i 0
49.16 0 1.67818 0.265797i 0 −1.78176 + 0.907854i 0 2.03350 + 2.79887i 0 −0.107544 + 0.0349431i 0
49.17 0 1.93030 0.305729i 0 1.24626 0.635003i 0 1.09820 + 1.51155i 0 0.779411 0.253246i 0
49.18 0 1.99895 0.316603i 0 −2.43455 + 1.24047i 0 −2.40520 3.31048i 0 1.04239 0.338694i 0
49.19 0 2.20058 0.348538i 0 2.16222 1.10171i 0 0.296642 + 0.408292i 0 1.86792 0.606923i 0
49.20 0 2.54744 0.403475i 0 2.98748 1.52220i 0 1.07945 + 1.48574i 0 3.47349 1.12860i 0
See next 80 embeddings (of 176 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
16.e even 4 1 inner
176.w even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 704.2.be.a 176
4.b odd 2 1 176.2.w.a 176
11.c even 5 1 inner 704.2.be.a 176
16.e even 4 1 inner 704.2.be.a 176
16.f odd 4 1 176.2.w.a 176
44.h odd 10 1 176.2.w.a 176
176.v odd 20 1 176.2.w.a 176
176.w even 20 1 inner 704.2.be.a 176

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.w.a 176 4.b odd 2 1
176.2.w.a 176 16.f odd 4 1
176.2.w.a 176 44.h odd 10 1
176.2.w.a 176 176.v odd 20 1
704.2.be.a 176 1.a even 1 1 trivial
704.2.be.a 176 11.c even 5 1 inner
704.2.be.a 176 16.e even 4 1 inner
704.2.be.a 176 176.w even 20 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(704, [\chi])$$.