# Properties

 Label 441.2.bg.a Level $441$ Weight $2$ Character orbit 441.bg Analytic conductor $3.521$ Analytic rank $0$ Dimension $216$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(17,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([21, 25]))

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

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

chi := DirichletCharacter("441.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.bg (of order $$42$$, degree $$12$$, 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: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$216$$ Relative dimension: $$18$$ over $$\Q(\zeta_{42})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$216 q - 16 q^{4} + 2 q^{7}+O(q^{10})$$ 216 * q - 16 * q^4 + 2 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$216 q - 16 q^{4} + 2 q^{7} + 12 q^{10} + 12 q^{16} - 6 q^{19} + 44 q^{22} + 26 q^{25} + 84 q^{28} - 6 q^{31} - 112 q^{34} + 60 q^{37} - 304 q^{40} + 20 q^{43} - 20 q^{46} - 86 q^{49} - 168 q^{52} - 84 q^{55} - 120 q^{58} - 2 q^{61} + 32 q^{64} + 22 q^{67} - 136 q^{70} - 6 q^{73} + 84 q^{76} + 2 q^{79} - 104 q^{82} + 96 q^{85} - 12 q^{88} + 58 q^{91} + 52 q^{94}+O(q^{100})$$ 216 * q - 16 * q^4 + 2 * q^7 + 12 * q^10 + 12 * q^16 - 6 * q^19 + 44 * q^22 + 26 * q^25 + 84 * q^28 - 6 * q^31 - 112 * q^34 + 60 * q^37 - 304 * q^40 + 20 * q^43 - 20 * q^46 - 86 * q^49 - 168 * q^52 - 84 * q^55 - 120 * q^58 - 2 * q^61 + 32 * q^64 + 22 * q^67 - 136 * q^70 - 6 * q^73 + 84 * q^76 + 2 * q^79 - 104 * q^82 + 96 * q^85 - 12 * q^88 + 58 * q^91 + 52 * q^94

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 −2.68471 + 0.201191i 0 5.18952 0.782194i −0.00530717 + 0.00492433i 0 −1.62374 2.08889i −8.52551 + 1.94589i 0 0.0132575 0.0142882i
17.2 −2.47614 + 0.185561i 0 4.11916 0.620863i 1.08753 1.00908i 0 0.513890 2.59536i −5.24274 + 1.19662i 0 −2.50562 + 2.70041i
17.3 −2.05710 + 0.154158i 0 2.23024 0.336155i −1.16105 + 1.07730i 0 1.82087 + 1.91949i −0.513717 + 0.117252i 0 2.22233 2.39510i
17.4 −1.57786 + 0.118244i 0 0.497985 0.0750592i −2.70933 + 2.51389i 0 1.38562 2.25390i 2.30834 0.526865i 0 3.97769 4.28693i
17.5 −1.51461 + 0.113504i 0 0.303503 0.0457458i 2.45092 2.27412i 0 1.78982 + 1.94847i 2.50706 0.572220i 0 −3.45407 + 3.72260i
17.6 −1.49398 + 0.111958i 0 0.241771 0.0364412i 1.96165 1.82014i 0 −2.52569 + 0.787950i 2.56409 0.585236i 0 −2.72688 + 2.93888i
17.7 −0.859142 + 0.0643838i 0 −1.24368 + 0.187455i 0.593873 0.551034i 0 −2.64451 0.0809471i 2.73633 0.624550i 0 −0.474744 + 0.511652i
17.8 −0.601956 + 0.0451104i 0 −1.61735 + 0.243776i −1.12371 + 1.04265i 0 −0.186554 2.63917i 2.13959 0.488348i 0 0.629389 0.678320i
17.9 −0.409998 + 0.0307251i 0 −1.81051 + 0.272890i −2.06043 + 1.91180i 0 −0.0705282 + 2.64481i 1.53560 0.350490i 0 0.786031 0.847140i
17.10 0.409998 0.0307251i 0 −1.81051 + 0.272890i 2.06043 1.91180i 0 −0.0705282 + 2.64481i −1.53560 + 0.350490i 0 0.786031 0.847140i
17.11 0.601956 0.0451104i 0 −1.61735 + 0.243776i 1.12371 1.04265i 0 −0.186554 2.63917i −2.13959 + 0.488348i 0 0.629389 0.678320i
17.12 0.859142 0.0643838i 0 −1.24368 + 0.187455i −0.593873 + 0.551034i 0 −2.64451 0.0809471i −2.73633 + 0.624550i 0 −0.474744 + 0.511652i
17.13 1.49398 0.111958i 0 0.241771 0.0364412i −1.96165 + 1.82014i 0 −2.52569 + 0.787950i −2.56409 + 0.585236i 0 −2.72688 + 2.93888i
17.14 1.51461 0.113504i 0 0.303503 0.0457458i −2.45092 + 2.27412i 0 1.78982 + 1.94847i −2.50706 + 0.572220i 0 −3.45407 + 3.72260i
17.15 1.57786 0.118244i 0 0.497985 0.0750592i 2.70933 2.51389i 0 1.38562 2.25390i −2.30834 + 0.526865i 0 3.97769 4.28693i
17.16 2.05710 0.154158i 0 2.23024 0.336155i 1.16105 1.07730i 0 1.82087 + 1.91949i 0.513717 0.117252i 0 2.22233 2.39510i
17.17 2.47614 0.185561i 0 4.11916 0.620863i −1.08753 + 1.00908i 0 0.513890 2.59536i 5.24274 1.19662i 0 −2.50562 + 2.70041i
17.18 2.68471 0.201191i 0 5.18952 0.782194i 0.00530717 0.00492433i 0 −1.62374 2.08889i 8.52551 1.94589i 0 0.0132575 0.0142882i
26.1 −2.68471 0.201191i 0 5.18952 + 0.782194i −0.00530717 0.00492433i 0 −1.62374 + 2.08889i −8.52551 1.94589i 0 0.0132575 + 0.0142882i
26.2 −2.47614 0.185561i 0 4.11916 + 0.620863i 1.08753 + 1.00908i 0 0.513890 + 2.59536i −5.24274 1.19662i 0 −2.50562 2.70041i
See next 80 embeddings (of 216 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.18 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
49.h odd 42 1 inner
147.o even 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.bg.a 216
3.b odd 2 1 inner 441.2.bg.a 216
49.h odd 42 1 inner 441.2.bg.a 216
147.o even 42 1 inner 441.2.bg.a 216

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.bg.a 216 1.a even 1 1 trivial
441.2.bg.a 216 3.b odd 2 1 inner
441.2.bg.a 216 49.h odd 42 1 inner
441.2.bg.a 216 147.o even 42 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(441, [\chi])$$.