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Properties

Label 464.4.k.c
Level $464$
Weight $4$
Character orbit 464.k
Analytic conductor $27.377$
Analytic rank $0$
Dimension $60$
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] = [464,4,Mod(191,464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(464, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("464.191");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 464.k (of order \(4\), degree \(2\), 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: \(27.3768862427\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 20 q^{17} - 304 q^{21} - 1156 q^{25} + 576 q^{29} + 244 q^{37} - 500 q^{41} + 1760 q^{45} - 796 q^{49} - 2272 q^{53} - 2236 q^{61} + 3008 q^{65} + 1296 q^{69} - 1660 q^{73} + 3568 q^{77} - 6940 q^{81} - 2352 q^{85} - 2292 q^{89} - 1228 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1 0 −7.26513 − 7.26513i 0 8.45399i 0 − 10.2200i 0 78.5643i 0
191.2 0 −6.67098 − 6.67098i 0 − 16.7758i 0 − 23.1088i 0 62.0040i 0
191.3 0 −5.86557 − 5.86557i 0 − 1.96751i 0 28.5268i 0 41.8098i 0
191.4 0 −5.17566 − 5.17566i 0 14.6001i 0 − 17.3780i 0 26.5749i 0
191.5 0 −4.92620 − 4.92620i 0 − 16.9233i 0 16.6731i 0 21.5349i 0
191.6 0 −4.56580 − 4.56580i 0 − 8.30996i 0 − 12.0895i 0 14.6931i 0
191.7 0 −4.35326 − 4.35326i 0 3.47594i 0 8.50378i 0 10.9017i 0
191.8 0 −4.17221 − 4.17221i 0 8.00723i 0 32.0912i 0 7.81472i 0
191.9 0 −3.82060 − 3.82060i 0 8.04907i 0 − 25.3813i 0 2.19396i 0
191.10 0 −2.45795 − 2.45795i 0 12.8277i 0 7.22386i 0 − 14.9170i 0
191.11 0 −2.20790 − 2.20790i 0 − 5.95138i 0 − 23.8484i 0 − 17.2503i 0
191.12 0 −1.61617 − 1.61617i 0 − 19.5678i 0 5.46577i 0 − 21.7760i 0
191.13 0 −1.02079 − 1.02079i 0 21.0668i 0 16.3031i 0 − 24.9160i 0
191.14 0 −0.796713 − 0.796713i 0 1.33216i 0 − 7.34842i 0 − 25.7305i 0
191.15 0 −0.499153 − 0.499153i 0 − 8.31724i 0 − 21.1814i 0 − 26.5017i 0
191.16 0 0.499153 + 0.499153i 0 − 8.31724i 0 21.1814i 0 − 26.5017i 0
191.17 0 0.796713 + 0.796713i 0 1.33216i 0 7.34842i 0 − 25.7305i 0
191.18 0 1.02079 + 1.02079i 0 21.0668i 0 − 16.3031i 0 − 24.9160i 0
191.19 0 1.61617 + 1.61617i 0 − 19.5678i 0 − 5.46577i 0 − 21.7760i 0
191.20 0 2.20790 + 2.20790i 0 − 5.95138i 0 23.8484i 0 − 17.2503i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
29.c odd 4 1 inner
116.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 464.4.k.c ✓ 60
4.b odd 2 1 inner 464.4.k.c ✓ 60
29.c odd 4 1 inner 464.4.k.c ✓ 60
116.e even 4 1 inner 464.4.k.c ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
464.4.k.c ✓ 60 1.a even 1 1 trivial
464.4.k.c ✓ 60 4.b odd 2 1 inner
464.4.k.c ✓ 60 29.c odd 4 1 inner
464.4.k.c ✓ 60 116.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} + 34540 T_{3}^{56} + 481277666 T_{3}^{52} + 3565956037700 T_{3}^{48} + \cdots + 47\!\cdots\!56 \) acting on \(S_{4}^{\mathrm{new}}(464, [\chi])\). Copy content Toggle raw display