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Properties

Label 441.2.w.a
Level $441$
Weight $2$
Character orbit 441.w
Analytic conductor $3.521$
Analytic rank $0$
Dimension $120$
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(62,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.62");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.w (of order \(14\), degree \(6\), 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: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 24 q^{4} - 32 q^{16} - 44 q^{22} - 4 q^{25} - 56 q^{28} + 112 q^{34} - 76 q^{37} + 28 q^{40} + 8 q^{43} - 40 q^{46} - 84 q^{49} - 140 q^{52} + 12 q^{58} - 84 q^{61} + 24 q^{64} + 16 q^{67} + 112 q^{70} - 84 q^{76} - 24 q^{79} + 140 q^{82} - 96 q^{85} - 24 q^{88} - 112 q^{91} - 112 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
62.1 −2.63522 + 0.601472i 0 4.78069 − 2.30226i 2.19013 + 2.74633i 0 −1.96904 + 1.76717i −6.98686 + 5.57183i 0 −7.42332 − 5.91990i
62.2 −2.41221 + 0.550570i 0 3.71367 − 1.78841i −0.289860 − 0.363473i 0 1.34601 + 2.27777i −4.10462 + 3.27333i 0 0.899319 + 0.717183i
62.3 −2.38347 + 0.544011i 0 3.58304 − 1.72550i −1.76892 − 2.21816i 0 1.05543 − 2.42612i −3.77859 + 3.01333i 0 5.42288 + 4.32461i
62.4 −2.08702 + 0.476349i 0 2.32682 − 1.12054i 2.25096 + 2.82261i 0 0.212729 − 2.63719i −0.975036 + 0.777565i 0 −6.04235 − 4.81862i
62.5 −1.79728 + 0.410217i 0 1.26000 − 0.606784i −2.31994 − 2.90911i 0 −1.91410 − 1.82654i 0.866954 − 0.691373i 0 5.36294 + 4.27680i
62.6 −1.61687 + 0.369040i 0 0.676145 − 0.325614i −1.24446 − 1.56050i 0 0.123672 + 2.64286i 1.62019 − 1.29205i 0 2.58801 + 2.06387i
62.7 −0.880699 + 0.201014i 0 −1.06671 + 0.513702i 2.70158 + 3.38767i 0 2.62942 + 0.293490i 2.24872 − 1.79330i 0 −3.06025 − 2.44047i
62.8 −0.843773 + 0.192586i 0 −1.12707 + 0.542770i −0.0384932 − 0.0482690i 0 −2.43436 + 1.03628i 2.19977 − 1.75426i 0 0.0417755 + 0.0333148i
62.9 −0.514347 + 0.117396i 0 −1.55117 + 0.747002i 1.31830 + 1.65310i 0 1.93236 + 1.80720i 1.53509 − 1.22420i 0 −0.872134 − 0.695503i
62.10 −0.138369 + 0.0315817i 0 −1.78379 + 0.859028i −0.192202 − 0.241014i 0 0.709889 − 2.54874i 0.441617 − 0.352178i 0 0.0342063 + 0.0272786i
62.11 0.138369 − 0.0315817i 0 −1.78379 + 0.859028i 0.192202 + 0.241014i 0 0.709889 − 2.54874i −0.441617 + 0.352178i 0 0.0342063 + 0.0272786i
62.12 0.514347 − 0.117396i 0 −1.55117 + 0.747002i −1.31830 − 1.65310i 0 1.93236 + 1.80720i −1.53509 + 1.22420i 0 −0.872134 − 0.695503i
62.13 0.843773 − 0.192586i 0 −1.12707 + 0.542770i 0.0384932 + 0.0482690i 0 −2.43436 + 1.03628i −2.19977 + 1.75426i 0 0.0417755 + 0.0333148i
62.14 0.880699 − 0.201014i 0 −1.06671 + 0.513702i −2.70158 − 3.38767i 0 2.62942 + 0.293490i −2.24872 + 1.79330i 0 −3.06025 − 2.44047i
62.15 1.61687 − 0.369040i 0 0.676145 − 0.325614i 1.24446 + 1.56050i 0 0.123672 + 2.64286i −1.62019 + 1.29205i 0 2.58801 + 2.06387i
62.16 1.79728 − 0.410217i 0 1.26000 − 0.606784i 2.31994 + 2.90911i 0 −1.91410 − 1.82654i −0.866954 + 0.691373i 0 5.36294 + 4.27680i
62.17 2.08702 − 0.476349i 0 2.32682 − 1.12054i −2.25096 − 2.82261i 0 0.212729 − 2.63719i 0.975036 − 0.777565i 0 −6.04235 − 4.81862i
62.18 2.38347 − 0.544011i 0 3.58304 − 1.72550i 1.76892 + 2.21816i 0 1.05543 − 2.42612i 3.77859 − 3.01333i 0 5.42288 + 4.32461i
62.19 2.41221 − 0.550570i 0 3.71367 − 1.78841i 0.289860 + 0.363473i 0 1.34601 + 2.27777i 4.10462 − 3.27333i 0 0.899319 + 0.717183i
62.20 2.63522 − 0.601472i 0 4.78069 − 2.30226i −2.19013 − 2.74633i 0 −1.96904 + 1.76717i 6.98686 − 5.57183i 0 −7.42332 − 5.91990i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 62.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
49.f odd 14 1 inner
147.k even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.w.a ✓ 120
3.b odd 2 1 inner 441.2.w.a ✓ 120
49.f odd 14 1 inner 441.2.w.a ✓ 120
147.k even 14 1 inner 441.2.w.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.w.a ✓ 120 1.a even 1 1 trivial
441.2.w.a ✓ 120 3.b odd 2 1 inner
441.2.w.a ✓ 120 49.f odd 14 1 inner
441.2.w.a ✓ 120 147.k even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(441, [\chi])\).