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Properties

Label 460.2.s.a
Level $460$
Weight $2$
Character orbit 460.s
Analytic conductor $3.673$
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] = [460,2,Mod(9,460)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(460, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 11, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("460.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.s (of order \(22\), degree \(10\), 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: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(12\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 20 q^{9} - 4 q^{11} + 9 q^{15} + 8 q^{19} + 14 q^{25} + 10 q^{29} - 18 q^{31} + 10 q^{35} - 60 q^{39} + 2 q^{41} - 2 q^{45} - 28 q^{49} + 24 q^{51} + 6 q^{55} - 36 q^{61} + 39 q^{65} + 118 q^{69} - 76 q^{71} + 83 q^{75} + 64 q^{79} - 160 q^{81} + 38 q^{85} - 48 q^{89} - 80 q^{91} + 21 q^{95} - 142 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −2.47974 + 2.14871i 0 1.82066 − 1.29815i 0 −0.253328 − 0.115691i 0 1.10522 − 7.68699i 0
9.2 0 −1.90027 + 1.64660i 0 −2.22785 + 0.191526i 0 −1.95670 − 0.893593i 0 0.472816 − 3.28851i 0
9.3 0 −1.79458 + 1.55501i 0 0.246165 + 2.22248i 0 1.58902 + 0.725681i 0 0.375508 − 2.61172i 0
9.4 0 −0.786010 + 0.681082i 0 2.20308 − 0.382657i 0 −3.36943 − 1.53877i 0 −0.273005 + 1.89879i 0
9.5 0 −0.410666 + 0.355844i 0 1.87331 + 1.22094i 0 2.52593 + 1.15355i 0 −0.384923 + 2.67720i 0
9.6 0 −0.391893 + 0.339577i 0 −0.773120 − 2.09816i 0 −1.88813 − 0.862281i 0 −0.388677 + 2.70331i 0
9.7 0 0.391893 − 0.339577i 0 0.150682 − 2.23099i 0 1.88813 + 0.862281i 0 −0.388677 + 2.70331i 0
9.8 0 0.410666 − 0.355844i 0 −1.45345 + 1.69926i 0 −2.52593 − 1.15355i 0 −0.384923 + 2.67720i 0
9.9 0 0.786010 − 0.681082i 0 −2.22165 + 0.253523i 0 3.36943 + 1.53877i 0 −0.273005 + 1.89879i 0
9.10 0 1.79458 − 1.55501i 0 0.389951 + 2.20180i 0 −1.58902 − 0.725681i 0 0.375508 − 2.61172i 0
9.11 0 1.90027 − 1.64660i 0 2.19157 − 0.443890i 0 1.95670 + 0.893593i 0 0.472816 − 3.28851i 0
9.12 0 2.47974 − 2.14871i 0 −2.11264 − 0.732624i 0 0.253328 + 0.115691i 0 1.10522 − 7.68699i 0
29.1 0 −0.822669 + 2.80175i 0 −0.608360 − 2.15172i 0 −0.516845 − 0.0743111i 0 −4.64928 − 2.98791i 0
29.2 0 −0.625007 + 2.12858i 0 0.203721 + 2.22677i 0 3.14354 + 0.451973i 0 −1.61646 − 1.03883i 0
29.3 0 −0.545020 + 1.85617i 0 2.03988 − 0.915906i 0 3.48084 + 0.500469i 0 −0.624547 − 0.401372i 0
29.4 0 −0.384428 + 1.30924i 0 2.22428 − 0.229290i 0 −2.93912 − 0.422582i 0 0.957431 + 0.615303i 0
29.5 0 −0.277288 + 0.944356i 0 −1.52418 − 1.63612i 0 −1.03968 − 0.149483i 0 1.70884 + 1.09821i 0
29.6 0 −0.244021 + 0.831059i 0 −1.24767 + 1.85562i 0 −3.43041 − 0.493219i 0 1.89265 + 1.21633i 0
29.7 0 0.244021 − 0.831059i 0 −2.20623 + 0.364069i 0 3.43041 + 0.493219i 0 1.89265 + 1.21633i 0
29.8 0 0.277288 − 0.944356i 0 0.855101 + 2.06611i 0 1.03968 + 0.149483i 0 1.70884 + 1.09821i 0
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.c even 11 1 inner
115.j even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.s.a ✓ 120
5.b even 2 1 inner 460.2.s.a ✓ 120
23.c even 11 1 inner 460.2.s.a ✓ 120
115.j even 22 1 inner 460.2.s.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.s.a ✓ 120 1.a even 1 1 trivial
460.2.s.a ✓ 120 5.b even 2 1 inner
460.2.s.a ✓ 120 23.c even 11 1 inner
460.2.s.a ✓ 120 115.j even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(460, [\chi])\).