↑

Properties

Label 738.2.u.c
Level $738$
Weight $2$
Character orbit 738.u
Analytic conductor $5.893$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [738,2,Mod(289,738)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(738, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("738.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 738 = 2 \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 738.u (of order \(20\), degree \(8\), 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: \(5.89295966917\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{4} + 4 q^{10} - 4 q^{11} + 2 q^{13} - 6 q^{16} - 10 q^{17} - 8 q^{19} - 10 q^{20} + 4 q^{22} + 4 q^{23} + 6 q^{25} + 8 q^{26} + 14 q^{29} + 24 q^{31} + 20 q^{34} + 56 q^{37} - 8 q^{38} + 16 q^{40} - 4 q^{41} - 20 q^{43} + 4 q^{44} + 20 q^{46} + 12 q^{47} + 40 q^{49} + 8 q^{52} - 26 q^{53} - 4 q^{55} - 14 q^{58} + 8 q^{59} + 40 q^{61} + 6 q^{64} + 12 q^{65} + 8 q^{67} + 10 q^{68} - 60 q^{70} - 48 q^{71} + 10 q^{74} + 8 q^{76} + 20 q^{77} + 28 q^{79} - 10 q^{80} - 2 q^{82} + 80 q^{83} - 30 q^{85} + 8 q^{86} + 16 q^{88} - 58 q^{89} - 4 q^{92} - 8 q^{94} + 68 q^{95} - 86 q^{97} - 2 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 0.951057 − 0.309017i 0 0.809017 − 0.587785i −1.28802 − 1.77281i 0 −1.13802 − 2.23348i 0.587785 − 0.809017i 0 −1.77281 − 1.28802i
289.2 0.951057 − 0.309017i 0 0.809017 − 0.587785i 0.299981 + 0.412889i 0 −0.410194 − 0.805051i 0.587785 − 0.809017i 0 0.412889 + 0.299981i
289.3 0.951057 − 0.309017i 0 0.809017 − 0.587785i 1.04558 + 1.43911i 0 1.54821 + 3.03854i 0.587785 − 0.809017i 0 1.43911 + 1.04558i
307.1 −0.951057 − 0.309017i 0 0.809017 + 0.587785i −2.42150 + 3.33291i 0 −4.09923 − 2.08866i −0.587785 − 0.809017i 0 3.33291 − 2.42150i
307.2 −0.951057 − 0.309017i 0 0.809017 + 0.587785i −0.718242 + 0.988576i 0 3.30863 + 1.68583i −0.587785 − 0.809017i 0 0.988576 − 0.718242i
307.3 −0.951057 − 0.309017i 0 0.809017 + 0.587785i 0.846136 − 1.16461i 0 0.790595 + 0.402828i −0.587785 − 0.809017i 0 −1.16461 + 0.846136i
361.1 −0.587785 + 0.809017i 0 −0.309017 − 0.951057i −1.42143 + 0.461849i 0 0.476878 − 3.01089i 0.951057 + 0.309017i 0 0.461849 − 1.42143i
361.2 −0.587785 + 0.809017i 0 −0.309017 − 0.951057i 1.26355 − 0.410553i 0 −0.447363 + 2.82454i 0.951057 + 0.309017i 0 −0.410553 + 1.26355i
361.3 −0.587785 + 0.809017i 0 −0.309017 − 0.951057i 3.17802 − 1.03260i 0 −0.0295159 + 0.186356i 0.951057 + 0.309017i 0 −1.03260 + 3.17802i
415.1 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i −1.42143 − 0.461849i 0 0.476878 + 3.01089i 0.951057 − 0.309017i 0 0.461849 + 1.42143i
415.2 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i 1.26355 + 0.410553i 0 −0.447363 − 2.82454i 0.951057 − 0.309017i 0 −0.410553 − 1.26355i
415.3 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i 3.17802 + 1.03260i 0 −0.0295159 − 0.186356i 0.951057 − 0.309017i 0 −1.03260 − 3.17802i
487.1 0.587785 + 0.809017i 0 −0.309017 + 0.951057i −3.30848 − 1.07499i 0 3.25475 − 0.515502i −0.951057 + 0.309017i 0 −1.07499 − 3.30848i
487.2 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 0.210253 + 0.0683152i 0 −4.01959 + 0.636641i −0.951057 + 0.309017i 0 0.0683152 + 0.210253i
487.3 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 2.31414 + 0.751911i 0 0.764846 − 0.121140i −0.951057 + 0.309017i 0 0.751911 + 2.31414i
541.1 0.587785 − 0.809017i 0 −0.309017 − 0.951057i −3.30848 + 1.07499i 0 3.25475 + 0.515502i −0.951057 − 0.309017i 0 −1.07499 + 3.30848i
541.2 0.587785 − 0.809017i 0 −0.309017 − 0.951057i 0.210253 − 0.0683152i 0 −4.01959 − 0.636641i −0.951057 − 0.309017i 0 0.0683152 − 0.210253i
541.3 0.587785 − 0.809017i 0 −0.309017 − 0.951057i 2.31414 − 0.751911i 0 0.764846 + 0.121140i −0.951057 − 0.309017i 0 0.751911 − 2.31414i
595.1 0.951057 + 0.309017i 0 0.809017 + 0.587785i −1.28802 + 1.77281i 0 −1.13802 + 2.23348i 0.587785 + 0.809017i 0 −1.77281 + 1.28802i
595.2 0.951057 + 0.309017i 0 0.809017 + 0.587785i 0.299981 − 0.412889i 0 −0.410194 + 0.805051i 0.587785 + 0.809017i 0 0.412889 − 0.299981i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.g even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 738.2.u.c ✓ 24
3.b odd 2 1 738.2.u.d yes 24
41.g even 20 1 inner 738.2.u.c ✓ 24
123.m odd 20 1 738.2.u.d yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
738.2.u.c ✓ 24 1.a even 1 1 trivial
738.2.u.c ✓ 24 41.g even 20 1 inner
738.2.u.d yes 24 3.b odd 2 1
738.2.u.d yes 24 123.m odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 18 T_{5}^{22} - 50 T_{5}^{21} + 246 T_{5}^{20} + 900 T_{5}^{19} - 2245 T_{5}^{18} + \cdots + 32041 \) acting on \(S_{2}^{\mathrm{new}}(738, [\chi])\). Copy content Toggle raw display