↑

Properties

Label 441.2.h.h
Level $441$
Weight $2$
Character orbit 441.h
Analytic conductor $3.521$
Analytic rank $0$
Dimension $24$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,2,Mod(214,441)]
 
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([4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.214");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.h (of order \(3\), degree \(2\), 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: \(3.52140272914\)
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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{2} + 24 q^{4} - 24 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{2} + 24 q^{4} - 24 q^{8} - 4 q^{9} + 20 q^{11} + 4 q^{15} + 24 q^{16} - 32 q^{18} + 32 q^{23} - 12 q^{25} + 16 q^{29} - 84 q^{30} - 96 q^{32} - 4 q^{36} - 12 q^{37} + 8 q^{39} + 56 q^{44} + 24 q^{46} - 4 q^{50} + 64 q^{51} + 32 q^{53} - 12 q^{57} + 32 q^{60} + 96 q^{64} - 120 q^{65} + 24 q^{67} - 112 q^{71} + 68 q^{74} - 60 q^{78} - 24 q^{79} - 40 q^{81} + 12 q^{85} + 76 q^{86} + 16 q^{92} - 32 q^{93} - 128 q^{95} + 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1 −2.71513 −1.16958 − 1.27753i 5.37195 −0.793197 + 1.37386i 3.17555 + 3.46867i 0 −9.15528 −0.264183 + 2.98835i 2.15363 − 3.73020i
214.2 −2.71513 1.16958 + 1.27753i 5.37195 0.793197 − 1.37386i −3.17555 − 3.46867i 0 −9.15528 −0.264183 + 2.98835i −2.15363 + 3.73020i
214.3 −1.72661 −1.70981 + 0.276691i 0.981184 −1.75616 + 3.04175i 2.95217 − 0.477737i 0 1.75910 2.84688 − 0.946176i 3.03220 − 5.25192i
214.4 −1.72661 1.70981 − 0.276691i 0.981184 1.75616 − 3.04175i −2.95217 + 0.477737i 0 1.75910 2.84688 − 0.946176i −3.03220 + 5.25192i
214.5 −1.10281 −1.22001 + 1.22947i −0.783802 −0.0527330 + 0.0913363i 1.34544 − 1.35587i 0 3.07001 −0.0231690 − 2.99991i 0.0581547 − 0.100727i
214.6 −1.10281 1.22001 − 1.22947i −0.783802 0.0527330 − 0.0913363i −1.34544 + 1.35587i 0 3.07001 −0.0231690 − 2.99991i −0.0581547 + 0.100727i
214.7 0.0683740 −0.539550 + 1.64587i −1.99532 1.33190 − 2.30691i −0.0368912 + 0.112535i 0 −0.273176 −2.41777 − 1.77606i 0.0910670 − 0.157733i
214.8 0.0683740 0.539550 − 1.64587i −1.99532 −1.33190 + 2.30691i 0.0368912 − 0.112535i 0 −0.273176 −2.41777 − 1.77606i −0.0910670 + 0.157733i
214.9 1.29987 −1.47364 − 0.910162i −0.310333 1.76292 − 3.05347i −1.91554 − 1.18309i 0 −3.00314 1.34321 + 2.68250i 2.29157 − 3.96912i
214.10 1.29987 1.47364 + 0.910162i −0.310333 −1.76292 + 3.05347i 1.91554 + 1.18309i 0 −3.00314 1.34321 + 2.68250i −2.29157 + 3.96912i
214.11 2.17631 −0.507459 + 1.65605i 2.73633 −0.634145 + 1.09837i −1.10439 + 3.60407i 0 1.60248 −2.48497 − 1.68075i −1.38010 + 2.39040i
214.12 2.17631 0.507459 − 1.65605i 2.73633 0.634145 − 1.09837i 1.10439 − 3.60407i 0 1.60248 −2.48497 − 1.68075i 1.38010 − 2.39040i
373.1 −2.71513 −1.16958 + 1.27753i 5.37195 −0.793197 − 1.37386i 3.17555 − 3.46867i 0 −9.15528 −0.264183 − 2.98835i 2.15363 + 3.73020i
373.2 −2.71513 1.16958 − 1.27753i 5.37195 0.793197 + 1.37386i −3.17555 + 3.46867i 0 −9.15528 −0.264183 − 2.98835i −2.15363 − 3.73020i
373.3 −1.72661 −1.70981 − 0.276691i 0.981184 −1.75616 − 3.04175i 2.95217 + 0.477737i 0 1.75910 2.84688 + 0.946176i 3.03220 + 5.25192i
373.4 −1.72661 1.70981 + 0.276691i 0.981184 1.75616 + 3.04175i −2.95217 − 0.477737i 0 1.75910 2.84688 + 0.946176i −3.03220 − 5.25192i
373.5 −1.10281 −1.22001 − 1.22947i −0.783802 −0.0527330 − 0.0913363i 1.34544 + 1.35587i 0 3.07001 −0.0231690 + 2.99991i 0.0581547 + 0.100727i
373.6 −1.10281 1.22001 + 1.22947i −0.783802 0.0527330 + 0.0913363i −1.34544 − 1.35587i 0 3.07001 −0.0231690 + 2.99991i −0.0581547 − 0.100727i
373.7 0.0683740 −0.539550 − 1.64587i −1.99532 1.33190 + 2.30691i −0.0368912 − 0.112535i 0 −0.273176 −2.41777 + 1.77606i 0.0910670 + 0.157733i
373.8 0.0683740 0.539550 + 1.64587i −1.99532 −1.33190 − 2.30691i 0.0368912 + 0.112535i 0 −0.273176 −2.41777 + 1.77606i −0.0910670 − 0.157733i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 214.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{2}^{6} + 2T_{2}^{5} - 7T_{2}^{4} - 12T_{2}^{3} + 10T_{2}^{2} + 14T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{24} + 36 T_{5}^{22} + 831 T_{5}^{20} + 11580 T_{5}^{18} + 117495 T_{5}^{16} + 782970 T_{5}^{14} + \cdots + 2401 \) Copy content Toggle raw display