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Properties

Label 441.2.bb.f
Level $441$
Weight $2$
Character orbit 441.bb
Analytic conductor $3.521$
Analytic rank $0$
Dimension $72$
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(37,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 32]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.bb (of order \(21\), degree \(12\), 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: \(72\)
Relative dimension: \(6\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 14 q^{4} - 28 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 14 q^{4} - 28 q^{7} + 42 q^{13} - 26 q^{16} + 28 q^{19} + 8 q^{22} - 24 q^{25} - 28 q^{28} + 28 q^{31} + 28 q^{34} - 106 q^{37} + 70 q^{40} - 2 q^{43} + 60 q^{46} + 28 q^{49} + 70 q^{52} - 42 q^{55} + 56 q^{58} + 112 q^{61} + 38 q^{64} - 36 q^{67} - 14 q^{70} - 28 q^{73} + 56 q^{76} + 62 q^{79} - 70 q^{82} - 100 q^{85} - 262 q^{88} - 154 q^{91} - 294 q^{94} + 56 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −2.14208 + 1.46044i 0 1.72492 − 4.39502i 0.478860 + 0.147709i 0 −1.61928 − 2.09235i 1.56997 + 6.87850i 0 −1.24148 + 0.382945i
37.2 −1.14923 + 0.783534i 0 −0.0238713 + 0.0608230i −3.53905 − 1.09165i 0 −2.60134 + 0.482718i −0.639241 − 2.80070i 0 4.92254 − 1.51840i
37.3 −0.715278 + 0.487668i 0 −0.456880 + 1.16411i 2.34510 + 0.723367i 0 2.39271 − 1.12913i −0.626178 − 2.74347i 0 −2.03016 + 0.626222i
37.4 0.715278 − 0.487668i 0 −0.456880 + 1.16411i −2.34510 − 0.723367i 0 2.39271 − 1.12913i 0.626178 + 2.74347i 0 −2.03016 + 0.626222i
37.5 1.14923 − 0.783534i 0 −0.0238713 + 0.0608230i 3.53905 + 1.09165i 0 −2.60134 + 0.482718i 0.639241 + 2.80070i 0 4.92254 − 1.51840i
37.6 2.14208 − 1.46044i 0 1.72492 − 4.39502i −0.478860 − 0.147709i 0 −1.61928 − 2.09235i −1.56997 − 6.87850i 0 −1.24148 + 0.382945i
46.1 −1.00262 + 2.55465i 0 −4.05486 − 3.76236i 1.62281 + 1.10641i 0 1.40782 − 2.24010i 8.73185 − 4.20504i 0 −4.45356 + 3.03638i
46.2 −0.576178 + 1.46808i 0 −0.357164 − 0.331400i −0.810468 − 0.552568i 0 −1.54377 − 2.14867i −2.14952 + 1.03515i 0 1.27819 − 0.871452i
46.3 −0.144768 + 0.368863i 0 1.35100 + 1.25355i −2.49239 − 1.69928i 0 2.63807 − 0.201526i −1.37200 + 0.660718i 0 0.987622 − 0.673350i
46.4 0.144768 − 0.368863i 0 1.35100 + 1.25355i 2.49239 + 1.69928i 0 2.63807 − 0.201526i 1.37200 − 0.660718i 0 0.987622 − 0.673350i
46.5 0.576178 − 1.46808i 0 −0.357164 − 0.331400i 0.810468 + 0.552568i 0 −1.54377 − 2.14867i 2.14952 − 1.03515i 0 1.27819 − 0.871452i
46.6 1.00262 − 2.55465i 0 −4.05486 − 3.76236i −1.62281 − 1.10641i 0 1.40782 − 2.24010i −8.73185 + 4.20504i 0 −4.45356 + 3.03638i
100.1 −2.49548 − 0.769754i 0 3.98243 + 2.71518i −1.69843 + 0.255997i 0 −2.62887 − 0.298395i −4.59158 − 5.75765i 0 4.43546 + 0.668538i
100.2 −2.02995 − 0.626155i 0 2.07613 + 1.41548i 0.321603 − 0.0484738i 0 2.57904 + 0.590375i −0.679131 − 0.851603i 0 −0.683188 − 0.102974i
100.3 −1.12270 − 0.346307i 0 −0.511951 − 0.349042i 1.86885 − 0.281684i 0 −0.330078 − 2.62508i 1.91896 + 2.40631i 0 −2.19571 − 0.330950i
100.4 1.12270 + 0.346307i 0 −0.511951 − 0.349042i −1.86885 + 0.281684i 0 −0.330078 − 2.62508i −1.91896 − 2.40631i 0 −2.19571 − 0.330950i
100.5 2.02995 + 0.626155i 0 2.07613 + 1.41548i −0.321603 + 0.0484738i 0 2.57904 + 0.590375i 0.679131 + 0.851603i 0 −0.683188 − 0.102974i
100.6 2.49548 + 0.769754i 0 3.98243 + 2.71518i 1.69843 − 0.255997i 0 −2.62887 − 0.298395i 4.59158 + 5.75765i 0 4.43546 + 0.668538i
109.1 −1.80148 + 1.67153i 0 0.301863 − 4.02808i −1.44677 + 3.68631i 0 −0.227334 + 2.63597i 3.12479 + 3.91837i 0 −3.55545 − 9.05915i
109.2 −0.970582 + 0.900568i 0 −0.0184545 + 0.246258i −0.843767 + 2.14988i 0 −0.921217 − 2.48019i −1.85490 − 2.32597i 0 −1.11717 − 2.84651i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
49.g even 21 1 inner
147.n odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.bb.f ✓ 72
3.b odd 2 1 inner 441.2.bb.f ✓ 72
49.g even 21 1 inner 441.2.bb.f ✓ 72
147.n odd 42 1 inner 441.2.bb.f ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.bb.f ✓ 72 1.a even 1 1 trivial
441.2.bb.f ✓ 72 3.b odd 2 1 inner
441.2.bb.f ✓ 72 49.g even 21 1 inner
441.2.bb.f ✓ 72 147.n odd 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - 13 T_{2}^{70} + 79 T_{2}^{68} - 177 T_{2}^{66} + 670 T_{2}^{64} - 18148 T_{2}^{62} + \cdots + 594248265625 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display