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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 $4$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} - 12 q^{4} - 24 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} - 12 q^{4} - 24 q^{8} + 8 q^{9} + 20 q^{11} + 4 q^{15} - 12 q^{16} + 4 q^{18} + 32 q^{23} - 12 q^{25} + 16 q^{29} + 48 q^{32} - 4 q^{36} + 24 q^{37} + 32 q^{39} - 112 q^{44} - 48 q^{46} - 4 q^{50} - 56 q^{51} - 64 q^{53} - 12 q^{57} - 88 q^{60} + 96 q^{64} + 60 q^{65} - 12 q^{67} - 112 q^{71} + 168 q^{72} + 68 q^{74} - 60 q^{78} + 12 q^{79} + 80 q^{81} + 12 q^{85} + 76 q^{86} + 16 q^{92} - 80 q^{93} + 64 q^{95} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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:

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 \(S_{2}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{12} - 2 T_{2}^{11} + 11 T_{2}^{10} - 10 T_{2}^{9} + 63 T_{2}^{8} - 58 T_{2}^{7} + 184 T_{2}^{6} - 74 T_{2}^{5} + 261 T_{2}^{4} - 116 T_{2}^{3} + 206 T_{2}^{2} + 14 T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{24} + 36 T_{5}^{22} + 831 T_{5}^{20} + 11580 T_{5}^{18} + 117495 T_{5}^{16} + 782970 T_{5}^{14} + 3775328 T_{5}^{12} + 10937664 T_{5}^{10} + 22667115 T_{5}^{8} + 25896660 T_{5}^{6} + 19694250 T_{5}^{4} + \cdots + 2401 \) Copy content Toggle raw display