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Properties

Label 441.2.s.d
Level $441$
Weight $2$
Character orbit 441.s
Analytic conductor $3.521$
Analytic rank $0$
Dimension $48$
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(362,441)]
 
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([1, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.362");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.s (of order \(6\), 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: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{4} - 8 q^{9} - 40 q^{15} - 24 q^{16} + 32 q^{18} + 48 q^{25} + 48 q^{30} - 120 q^{32} - 8 q^{36} - 32 q^{39} + 96 q^{44} + 48 q^{50} + 48 q^{53} + 80 q^{57} - 72 q^{60} - 48 q^{64} - 120 q^{65} + 32 q^{72} - 88 q^{78} - 24 q^{79} + 120 q^{81} - 24 q^{85} - 144 q^{92} + 16 q^{93} - 96 q^{95} - 72 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
362.1 −2.23278 + 1.28910i −0.625134 − 1.61530i 2.32354 − 4.02449i 2.33189 3.47807 + 2.80076i 0 6.82470i −2.21841 + 2.01956i −5.20660 + 3.00603i
362.2 −2.23278 + 1.28910i 0.625134 + 1.61530i 2.32354 − 4.02449i −2.33189 −3.47807 − 2.80076i 0 6.82470i −2.21841 + 2.01956i 5.20660 − 3.00603i
362.3 −1.80506 + 1.04215i −1.73189 − 0.0239080i 1.17216 − 2.03024i 3.30465 3.15107 − 1.76173i 0 0.717672i 2.99886 + 0.0828118i −5.96509 + 3.44395i
362.4 −1.80506 + 1.04215i 1.73189 + 0.0239080i 1.17216 − 2.03024i −3.30465 −3.15107 + 1.76173i 0 0.717672i 2.99886 + 0.0828118i 5.96509 − 3.44395i
362.5 −1.61855 + 0.934468i −0.710525 + 1.57961i 0.746462 − 1.29291i 2.50573 −0.326074 − 3.22063i 0 − 0.947692i −1.99031 − 2.24470i −4.05565 + 2.34153i
362.6 −1.61855 + 0.934468i 0.710525 − 1.57961i 0.746462 − 1.29291i −2.50573 0.326074 + 3.22063i 0 − 0.947692i −1.99031 − 2.24470i 4.05565 − 2.34153i
362.7 −1.58658 + 0.916012i −0.108803 + 1.72863i 0.678156 − 1.17460i −0.645568 −1.41082 − 2.84227i 0 − 1.17925i −2.97632 − 0.376160i 1.02425 − 0.591348i
362.8 −1.58658 + 0.916012i 0.108803 − 1.72863i 0.678156 − 1.17460i 0.645568 1.41082 + 2.84227i 0 − 1.17925i −2.97632 − 0.376160i −1.02425 + 0.591348i
362.9 −0.575298 + 0.332148i −0.537154 − 1.64665i −0.779355 + 1.34988i 0.0283039 0.855956 + 0.768901i 0 − 2.36404i −2.42293 + 1.76901i −0.0162832 + 0.00940110i
362.10 −0.575298 + 0.332148i 0.537154 + 1.64665i −0.779355 + 1.34988i −0.0283039 −0.855956 − 0.768901i 0 − 2.36404i −2.42293 + 1.76901i 0.0162832 − 0.00940110i
362.11 −0.105953 + 0.0611722i −1.73002 + 0.0838860i −0.992516 + 1.71909i 0.529430 0.178170 − 0.114717i 0 − 0.487547i 2.98593 − 0.290249i −0.0560949 + 0.0323864i
362.12 −0.105953 + 0.0611722i 1.73002 − 0.0838860i −0.992516 + 1.71909i −0.529430 −0.178170 + 0.114717i 0 − 0.487547i 2.98593 − 0.290249i 0.0560949 − 0.0323864i
362.13 0.367369 − 0.212101i −1.71145 − 0.266368i −0.910027 + 1.57621i −3.60763 −0.685229 + 0.265143i 0 1.62047i 2.85810 + 0.911750i −1.32533 + 0.765180i
362.14 0.367369 − 0.212101i 1.71145 + 0.266368i −0.910027 + 1.57621i 3.60763 0.685229 − 0.265143i 0 1.62047i 2.85810 + 0.911750i 1.32533 − 0.765180i
362.15 0.850109 − 0.490811i −0.831570 + 1.51937i −0.518210 + 0.897565i −1.88120 0.0387990 + 1.69978i 0 2.98061i −1.61698 − 2.52693i −1.59922 + 0.923312i
362.16 0.850109 − 0.490811i 0.831570 − 1.51937i −0.518210 + 0.897565i 1.88120 −0.0387990 − 1.69978i 0 2.98061i −1.61698 − 2.52693i 1.59922 − 0.923312i
362.17 1.02035 − 0.589100i −1.34152 − 1.09560i −0.305921 + 0.529871i 4.33202 −2.01424 − 0.327604i 0 3.07728i 0.599340 + 2.93952i 4.42019 − 2.55200i
362.18 1.02035 − 0.589100i 1.34152 + 1.09560i −0.305921 + 0.529871i −4.33202 2.01424 + 0.327604i 0 3.07728i 0.599340 + 2.93952i −4.42019 + 2.55200i
362.19 1.28562 − 0.742253i −1.72526 + 0.153190i 0.101880 − 0.176462i −0.308431 −2.10433 + 1.47753i 0 2.66653i 2.95307 − 0.528586i −0.396525 + 0.228934i
362.20 1.28562 − 0.742253i 1.72526 − 0.153190i 0.101880 − 0.176462i 0.308431 2.10433 − 1.47753i 0 2.66653i 2.95307 − 0.528586i 0.396525 − 0.228934i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 362.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.n odd 6 1 inner
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.s.d 48
3.b odd 2 1 1323.2.s.d 48
7.b odd 2 1 inner 441.2.s.d 48
7.c even 3 1 441.2.i.d 48
7.c even 3 1 441.2.o.e ✓ 48
7.d odd 6 1 441.2.i.d 48
7.d odd 6 1 441.2.o.e ✓ 48
9.c even 3 1 1323.2.i.d 48
9.d odd 6 1 441.2.i.d 48
21.c even 2 1 1323.2.s.d 48
21.g even 6 1 1323.2.i.d 48
21.g even 6 1 1323.2.o.e 48
21.h odd 6 1 1323.2.i.d 48
21.h odd 6 1 1323.2.o.e 48
63.g even 3 1 1323.2.s.d 48
63.h even 3 1 1323.2.o.e 48
63.i even 6 1 441.2.o.e ✓ 48
63.j odd 6 1 441.2.o.e ✓ 48
63.k odd 6 1 1323.2.s.d 48
63.l odd 6 1 1323.2.i.d 48
63.n odd 6 1 inner 441.2.s.d 48
63.o even 6 1 441.2.i.d 48
63.s even 6 1 inner 441.2.s.d 48
63.t odd 6 1 1323.2.o.e 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.i.d 48 7.c even 3 1
441.2.i.d 48 7.d odd 6 1
441.2.i.d 48 9.d odd 6 1
441.2.i.d 48 63.o even 6 1
441.2.o.e ✓ 48 7.c even 3 1
441.2.o.e ✓ 48 7.d odd 6 1
441.2.o.e ✓ 48 63.i even 6 1
441.2.o.e ✓ 48 63.j odd 6 1
441.2.s.d 48 1.a even 1 1 trivial
441.2.s.d 48 7.b odd 2 1 inner
441.2.s.d 48 63.n odd 6 1 inner
441.2.s.d 48 63.s even 6 1 inner
1323.2.i.d 48 9.c even 3 1
1323.2.i.d 48 21.g even 6 1
1323.2.i.d 48 21.h odd 6 1
1323.2.i.d 48 63.l odd 6 1
1323.2.o.e 48 21.g even 6 1
1323.2.o.e 48 21.h odd 6 1
1323.2.o.e 48 63.h even 3 1
1323.2.o.e 48 63.t odd 6 1
1323.2.s.d 48 3.b odd 2 1
1323.2.s.d 48 21.c even 2 1
1323.2.s.d 48 63.g even 3 1
1323.2.s.d 48 63.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 18 T_{2}^{22} + 207 T_{2}^{20} + 12 T_{2}^{19} - 1434 T_{2}^{18} - 108 T_{2}^{17} + \cdots + 49 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display