# Properties

 Label 441.2.f.h Level $441$ Weight $2$ Character orbit 441.f Analytic conductor $3.521$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(148,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("441.148");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.f (of order $$3$$, degree $$2$$, 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: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 4 q^{2} - 12 q^{4} - 24 q^{8} + 8 q^{9}+O(q^{10})$$ 24 * q + 4 * q^2 - 12 * q^4 - 24 * q^8 + 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 4 q^{2} - 12 q^{4} - 24 q^{8} + 8 q^{9} + 20 q^{11} + 4 q^{15} - 12 q^{16} + 4 q^{18} + 32 q^{23} - 12 q^{25} + 16 q^{29} + 48 q^{32} - 4 q^{36} + 24 q^{37} + 32 q^{39} - 112 q^{44} - 48 q^{46} - 4 q^{50} - 56 q^{51} - 64 q^{53} - 12 q^{57} - 88 q^{60} + 96 q^{64} + 60 q^{65} - 12 q^{67} - 112 q^{71} + 168 q^{72} + 68 q^{74} - 60 q^{78} + 12 q^{79} + 80 q^{81} + 12 q^{85} + 76 q^{86} + 16 q^{92} - 80 q^{93} + 64 q^{95} + 20 q^{99}+O(q^{100})$$ 24 * q + 4 * q^2 - 12 * q^4 - 24 * q^8 + 8 * q^9 + 20 * q^11 + 4 * q^15 - 12 * q^16 + 4 * q^18 + 32 * q^23 - 12 * q^25 + 16 * q^29 + 48 * q^32 - 4 * q^36 + 24 * q^37 + 32 * q^39 - 112 * q^44 - 48 * q^46 - 4 * q^50 - 56 * q^51 - 64 * q^53 - 12 * q^57 - 88 * q^60 + 96 * q^64 + 60 * q^65 - 12 * q^67 - 112 * q^71 + 168 * q^72 + 68 * q^74 - 60 * q^78 + 12 * q^79 + 80 * q^81 + 12 * q^85 + 76 * q^86 + 16 * q^92 - 80 * q^93 + 64 * q^95 + 20 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
148.1 −1.08816 1.88474i −1.68791 0.388551i −1.36816 + 2.36973i 0.634145 1.09837i 1.10439 + 3.60407i 0 1.60248 2.69806 + 1.31167i −2.76019
148.2 −1.08816 1.88474i 1.68791 + 0.388551i −1.36816 + 2.36973i −0.634145 + 1.09837i −1.10439 3.60407i 0 1.60248 2.69806 + 1.31167i 2.76019
148.3 −0.649936 1.12572i −0.0514049 1.73129i 0.155166 0.268756i 1.76292 3.05347i −1.91554 + 1.18309i 0 −3.00314 −2.99472 + 0.177994i −4.58314
148.4 −0.649936 1.12572i 0.0514049 + 1.73129i 0.155166 0.268756i −1.76292 + 3.05347i 1.91554 1.18309i 0 −3.00314 −2.99472 + 0.177994i 4.58314
148.5 −0.0341870 0.0592136i −1.69514 0.355671i 0.997662 1.72800i −1.33190 + 2.30691i 0.0368912 + 0.112535i 0 −0.273176 2.74700 + 1.20582i 0.182134
148.6 −0.0341870 0.0592136i 1.69514 + 0.355671i 0.997662 1.72800i 1.33190 2.30691i −0.0368912 0.112535i 0 −0.273176 2.74700 + 1.20582i −0.182134
148.7 0.551407 + 0.955065i −1.67475 + 0.441824i 0.391901 0.678793i 0.0527330 0.0913363i −1.34544 1.35587i 0 3.07001 2.60958 1.47989i 0.116309
148.8 0.551407 + 0.955065i 1.67475 0.441824i 0.391901 0.678793i −0.0527330 + 0.0913363i 1.34544 + 1.35587i 0 3.07001 2.60958 1.47989i −0.116309
148.9 0.863305 + 1.49529i −1.09452 + 1.34239i −0.490592 + 0.849731i 1.75616 3.04175i −2.95217 0.477737i 0 1.75910 −0.604030 2.93856i 6.06439
148.10 0.863305 + 1.49529i 1.09452 1.34239i −0.490592 + 0.849731i −1.75616 + 3.04175i 2.95217 + 0.477737i 0 1.75910 −0.604030 2.93856i −6.06439
148.11 1.35757 + 2.35137i −0.521588 1.65165i −2.68597 + 4.65224i −0.793197 + 1.37386i 3.17555 3.46867i 0 −9.15528 −2.45589 + 1.72296i −4.30727
148.12 1.35757 + 2.35137i 0.521588 + 1.65165i −2.68597 + 4.65224i 0.793197 1.37386i −3.17555 + 3.46867i 0 −9.15528 −2.45589 + 1.72296i 4.30727
295.1 −1.08816 + 1.88474i −1.68791 + 0.388551i −1.36816 2.36973i 0.634145 + 1.09837i 1.10439 3.60407i 0 1.60248 2.69806 1.31167i −2.76019
295.2 −1.08816 + 1.88474i 1.68791 0.388551i −1.36816 2.36973i −0.634145 1.09837i −1.10439 + 3.60407i 0 1.60248 2.69806 1.31167i 2.76019
295.3 −0.649936 + 1.12572i −0.0514049 + 1.73129i 0.155166 + 0.268756i 1.76292 + 3.05347i −1.91554 1.18309i 0 −3.00314 −2.99472 0.177994i −4.58314
295.4 −0.649936 + 1.12572i 0.0514049 1.73129i 0.155166 + 0.268756i −1.76292 3.05347i 1.91554 + 1.18309i 0 −3.00314 −2.99472 0.177994i 4.58314
295.5 −0.0341870 + 0.0592136i −1.69514 + 0.355671i 0.997662 + 1.72800i −1.33190 2.30691i 0.0368912 0.112535i 0 −0.273176 2.74700 1.20582i 0.182134
295.6 −0.0341870 + 0.0592136i 1.69514 0.355671i 0.997662 + 1.72800i 1.33190 + 2.30691i −0.0368912 + 0.112535i 0 −0.273176 2.74700 1.20582i −0.182134
295.7 0.551407 0.955065i −1.67475 0.441824i 0.391901 + 0.678793i 0.0527330 + 0.0913363i −1.34544 + 1.35587i 0 3.07001 2.60958 + 1.47989i 0.116309
295.8 0.551407 0.955065i 1.67475 + 0.441824i 0.391901 + 0.678793i −0.0527330 0.0913363i 1.34544 1.35587i 0 3.07001 2.60958 + 1.47989i −0.116309
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 148.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.c even 3 1 inner
63.l odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.f.h 24
3.b odd 2 1 1323.2.f.h 24
7.b odd 2 1 inner 441.2.f.h 24
7.c even 3 1 441.2.g.h 24
7.c even 3 1 441.2.h.h 24
7.d odd 6 1 441.2.g.h 24
7.d odd 6 1 441.2.h.h 24
9.c even 3 1 inner 441.2.f.h 24
9.c even 3 1 3969.2.a.bh 12
9.d odd 6 1 1323.2.f.h 24
9.d odd 6 1 3969.2.a.bi 12
21.c even 2 1 1323.2.f.h 24
21.g even 6 1 1323.2.g.h 24
21.g even 6 1 1323.2.h.h 24
21.h odd 6 1 1323.2.g.h 24
21.h odd 6 1 1323.2.h.h 24
63.g even 3 1 441.2.h.h 24
63.h even 3 1 441.2.g.h 24
63.i even 6 1 1323.2.g.h 24
63.j odd 6 1 1323.2.g.h 24
63.k odd 6 1 441.2.h.h 24
63.l odd 6 1 inner 441.2.f.h 24
63.l odd 6 1 3969.2.a.bh 12
63.n odd 6 1 1323.2.h.h 24
63.o even 6 1 1323.2.f.h 24
63.o even 6 1 3969.2.a.bi 12
63.s even 6 1 1323.2.h.h 24
63.t odd 6 1 441.2.g.h 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.h 24 1.a even 1 1 trivial
441.2.f.h 24 7.b odd 2 1 inner
441.2.f.h 24 9.c even 3 1 inner
441.2.f.h 24 63.l odd 6 1 inner
441.2.g.h 24 7.c even 3 1
441.2.g.h 24 7.d odd 6 1
441.2.g.h 24 63.h even 3 1
441.2.g.h 24 63.t odd 6 1
441.2.h.h 24 7.c even 3 1
441.2.h.h 24 7.d odd 6 1
441.2.h.h 24 63.g even 3 1
441.2.h.h 24 63.k odd 6 1
1323.2.f.h 24 3.b odd 2 1
1323.2.f.h 24 9.d odd 6 1
1323.2.f.h 24 21.c even 2 1
1323.2.f.h 24 63.o even 6 1
1323.2.g.h 24 21.g even 6 1
1323.2.g.h 24 21.h odd 6 1
1323.2.g.h 24 63.i even 6 1
1323.2.g.h 24 63.j odd 6 1
1323.2.h.h 24 21.g even 6 1
1323.2.h.h 24 21.h odd 6 1
1323.2.h.h 24 63.n odd 6 1
1323.2.h.h 24 63.s even 6 1
3969.2.a.bh 12 9.c even 3 1
3969.2.a.bh 12 63.l odd 6 1
3969.2.a.bi 12 9.d odd 6 1
3969.2.a.bi 12 63.o even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{12} - 2 T_{2}^{11} + 11 T_{2}^{10} - 10 T_{2}^{9} + 63 T_{2}^{8} - 58 T_{2}^{7} + 184 T_{2}^{6} - 74 T_{2}^{5} + 261 T_{2}^{4} - 116 T_{2}^{3} + 206 T_{2}^{2} + 14 T_{2} + 1$$ T2^12 - 2*T2^11 + 11*T2^10 - 10*T2^9 + 63*T2^8 - 58*T2^7 + 184*T2^6 - 74*T2^5 + 261*T2^4 - 116*T2^3 + 206*T2^2 + 14*T2 + 1 $$T_{5}^{24} + 36 T_{5}^{22} + 831 T_{5}^{20} + 11580 T_{5}^{18} + 117495 T_{5}^{16} + 782970 T_{5}^{14} + 3775328 T_{5}^{12} + 10937664 T_{5}^{10} + 22667115 T_{5}^{8} + 25896660 T_{5}^{6} + 19694250 T_{5}^{4} + \cdots + 2401$$ T5^24 + 36*T5^22 + 831*T5^20 + 11580*T5^18 + 117495*T5^16 + 782970*T5^14 + 3775328*T5^12 + 10937664*T5^10 + 22667115*T5^8 + 25896660*T5^6 + 19694250*T5^4 + 219030*T5^2 + 2401