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Properties

Label 441.3.t.c
Level $441$
Weight $3$
Character orbit 441.t
Analytic conductor $12.016$
Analytic rank $0$
Dimension $96$
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] = [441,3,Mod(166,441)]
 
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, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.166");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.t (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: \(12.0163796583\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 192 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 192 q^{4} - 8 q^{9} - 24 q^{11} + 104 q^{15} + 384 q^{16} - 64 q^{18} - 96 q^{23} + 240 q^{25} - 144 q^{29} - 168 q^{30} + 240 q^{32} + 136 q^{36} - 536 q^{39} - 192 q^{44} - 912 q^{50} + 648 q^{51} - 48 q^{53} - 256 q^{57} + 264 q^{60} + 768 q^{64} + 336 q^{65} + 1344 q^{71} - 1480 q^{72} - 168 q^{74} + 920 q^{78} - 48 q^{79} + 840 q^{81} - 120 q^{85} + 408 q^{86} - 1584 q^{92} - 80 q^{93} + 1344 q^{95} - 672 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
166.1 −3.80450 −2.81954 − 1.02478i 10.4742 −6.08959 + 3.51582i 10.7269 + 3.89877i 0 −24.6311 6.89965 + 5.77882i 23.1678 − 13.3759i
166.2 −3.80450 2.81954 + 1.02478i 10.4742 6.08959 − 3.51582i −10.7269 − 3.89877i 0 −24.6311 6.89965 + 5.77882i −23.1678 + 13.3759i
166.3 −3.66807 −2.99997 + 0.0127693i 9.45471 5.76370 − 3.32768i 11.0041 − 0.0468385i 0 −20.0082 8.99967 − 0.0766149i −21.1417 + 12.2061i
166.4 −3.66807 2.99997 − 0.0127693i 9.45471 −5.76370 + 3.32768i −11.0041 + 0.0468385i 0 −20.0082 8.99967 − 0.0766149i 21.1417 − 12.2061i
166.5 −3.29129 −1.93452 + 2.29296i 6.83260 −6.64948 + 3.83908i 6.36706 − 7.54678i 0 −9.32291 −1.51529 − 8.87152i 21.8854 − 12.6355i
166.6 −3.29129 1.93452 − 2.29296i 6.83260 6.64948 − 3.83908i −6.36706 + 7.54678i 0 −9.32291 −1.51529 − 8.87152i −21.8854 + 12.6355i
166.7 −2.94479 −0.220344 − 2.99190i 4.67176 −7.55585 + 4.36237i 0.648867 + 8.81049i 0 −1.97819 −8.90290 + 1.31849i 22.2503 − 12.8462i
166.8 −2.94479 0.220344 + 2.99190i 4.67176 7.55585 − 4.36237i −0.648867 − 8.81049i 0 −1.97819 −8.90290 + 1.31849i −22.2503 + 12.8462i
166.9 −2.79421 −0.935582 − 2.85038i 3.80758 0.607852 − 0.350943i 2.61421 + 7.96456i 0 0.537653 −7.24937 + 5.33354i −1.69846 + 0.980608i
166.10 −2.79421 0.935582 + 2.85038i 3.80758 −0.607852 + 0.350943i −2.61421 − 7.96456i 0 0.537653 −7.24937 + 5.33354i 1.69846 − 0.980608i
166.11 −2.36528 −1.40359 + 2.65140i 1.59456 1.57246 − 0.907857i 3.31990 − 6.27131i 0 5.68954 −5.05985 − 7.44298i −3.71930 + 2.14734i
166.12 −2.36528 1.40359 − 2.65140i 1.59456 −1.57246 + 0.907857i −3.31990 + 6.27131i 0 5.68954 −5.05985 − 7.44298i 3.71930 − 2.14734i
166.13 −2.25852 −2.85148 − 0.932222i 1.10091 4.23148 − 2.44305i 6.44014 + 2.10544i 0 6.54764 7.26192 + 5.31643i −9.55688 + 5.51767i
166.14 −2.25852 2.85148 + 0.932222i 1.10091 −4.23148 + 2.44305i −6.44014 − 2.10544i 0 6.54764 7.26192 + 5.31643i 9.55688 − 5.51767i
166.15 −1.72970 −0.787899 + 2.89469i −1.00814 −7.18312 + 4.14718i 1.36283 − 5.00694i 0 8.66258 −7.75843 − 4.56144i 12.4246 − 7.17337i
166.16 −1.72970 0.787899 − 2.89469i −1.00814 7.18312 − 4.14718i −1.36283 + 5.00694i 0 8.66258 −7.75843 − 4.56144i −12.4246 + 7.17337i
166.17 −1.24528 −2.83019 + 0.995002i −2.44928 −4.36918 + 2.52255i 3.52437 − 1.23905i 0 8.03115 7.01994 − 5.63209i 5.44084 − 3.14127i
166.18 −1.24528 2.83019 − 0.995002i −2.44928 4.36918 − 2.52255i −3.52437 + 1.23905i 0 8.03115 7.01994 − 5.63209i −5.44084 + 3.14127i
166.19 −1.21351 −0.769029 − 2.89976i −2.52740 −1.91687 + 1.10670i 0.933223 + 3.51888i 0 7.92105 −7.81719 + 4.46000i 2.32613 − 1.34299i
166.20 −1.21351 0.769029 + 2.89976i −2.52740 1.91687 − 1.10670i −0.933223 − 3.51888i 0 7.92105 −7.81719 + 4.46000i −2.32613 + 1.34299i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 166.48
Significant digits:
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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 441.3.t.c 96
7.b odd 2 1 inner 441.3.t.c 96
7.c even 3 1 441.3.k.c 96
7.c even 3 1 441.3.l.c ✓ 96
7.d odd 6 1 441.3.k.c 96
7.d odd 6 1 441.3.l.c ✓ 96
9.c even 3 1 441.3.k.c 96
63.g even 3 1 441.3.l.c ✓ 96
63.h even 3 1 inner 441.3.t.c 96
63.k odd 6 1 441.3.l.c ✓ 96
63.l odd 6 1 441.3.k.c 96
63.t odd 6 1 inner 441.3.t.c 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.3.k.c 96 7.c even 3 1
441.3.k.c 96 7.d odd 6 1
441.3.k.c 96 9.c even 3 1
441.3.k.c 96 63.l odd 6 1
441.3.l.c ✓ 96 7.c even 3 1
441.3.l.c ✓ 96 7.d odd 6 1
441.3.l.c ✓ 96 63.g even 3 1
441.3.l.c ✓ 96 63.k odd 6 1
441.3.t.c 96 1.a even 1 1 trivial
441.3.t.c 96 7.b odd 2 1 inner
441.3.t.c 96 63.h even 3 1 inner
441.3.t.c 96 63.t odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 72 T_{2}^{22} + 2232 T_{2}^{20} - 12 T_{2}^{19} - 39072 T_{2}^{18} + 684 T_{2}^{17} + \cdots - 729473 \) acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display