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Properties

Label 504.2.cx.a
Level $504$
Weight $2$
Character orbit 504.cx
Analytic conductor $4.024$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,2,Mod(185,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.185");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.cx (of order \(6\), 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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{9} + 8 q^{15} - 10 q^{21} + 48 q^{25} + 18 q^{27} + 18 q^{29} + 18 q^{31} + 12 q^{33} - 4 q^{39} - 6 q^{41} - 6 q^{43} - 18 q^{45} + 18 q^{47} - 12 q^{49} + 6 q^{51} - 12 q^{53} + 4 q^{57} + 18 q^{61} - 32 q^{63} - 36 q^{65} - 12 q^{77} + 6 q^{79} + 6 q^{81} - 54 q^{87} - 18 q^{89} + 6 q^{91} + 4 q^{93} - 54 q^{95} - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
185.1 0 −1.70958 − 0.278106i 0 −0.542075 0 −2.62378 + 0.340238i 0 2.84531 + 0.950888i 0
185.2 0 −1.69924 + 0.335557i 0 −3.64567 0 1.05524 + 2.42620i 0 2.77480 − 1.14038i 0
185.3 0 −1.67230 + 0.451026i 0 −0.203178 0 −1.27132 − 2.32029i 0 2.59315 − 1.50850i 0
185.4 0 −1.46327 + 0.926729i 0 4.09884 0 −0.146661 − 2.64168i 0 1.28235 − 2.71212i 0
185.5 0 −1.45594 − 0.938214i 0 −1.81173 0 1.69266 − 2.03345i 0 1.23951 + 2.73196i 0
185.6 0 −1.14647 − 1.29831i 0 −0.0525740 0 2.44149 − 1.01937i 0 −0.371197 + 2.97695i 0
185.7 0 −1.11231 − 1.32769i 0 3.53401 0 1.30083 + 2.30388i 0 −0.525517 + 2.95361i 0
185.8 0 −1.03050 + 1.39215i 0 2.20884 0 2.16520 + 1.52049i 0 −0.876153 − 2.86921i 0
185.9 0 −0.859090 + 1.50398i 0 −4.19811 0 1.67151 − 2.05087i 0 −1.52393 − 2.58411i 0
185.10 0 −0.601162 + 1.62438i 0 −0.207028 0 −1.37075 + 2.26297i 0 −2.27721 − 1.95303i 0
185.11 0 −0.498607 − 1.65873i 0 −0.623597 0 −0.996837 + 2.45078i 0 −2.50278 + 1.65411i 0
185.12 0 −0.210634 + 1.71920i 0 −1.07485 0 −1.26701 − 2.32265i 0 −2.91127 − 0.724244i 0
185.13 0 0.149059 − 1.72562i 0 −2.22094 0 −2.45091 − 0.996507i 0 −2.95556 − 0.514438i 0
185.14 0 0.310783 − 1.70394i 0 2.76937 0 −1.21939 − 2.34800i 0 −2.80683 − 1.05911i 0
185.15 0 0.549450 − 1.64259i 0 −1.05582 0 1.79851 + 1.94045i 0 −2.39621 − 1.80504i 0
185.16 0 0.930597 + 1.46082i 0 1.36828 0 2.64451 + 0.0810554i 0 −1.26798 + 2.71887i 0
185.17 0 0.958980 + 1.44234i 0 4.11484 0 −2.54819 + 0.711830i 0 −1.16071 + 2.76636i 0
185.18 0 0.994080 + 1.41838i 0 −1.58600 0 −1.06431 + 2.42224i 0 −1.02361 + 2.81997i 0
185.19 0 1.24547 − 1.20367i 0 2.04899 0 1.41312 − 2.23676i 0 0.102371 − 2.99825i 0
185.20 0 1.51239 − 0.844203i 0 −2.84763 0 −2.64481 + 0.0704652i 0 1.57464 − 2.55353i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 185.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.cx.a yes 48
3.b odd 2 1 1512.2.cx.a 48
4.b odd 2 1 1008.2.df.e 48
7.d odd 6 1 504.2.bs.a ✓ 48
9.c even 3 1 1512.2.bs.a 48
9.d odd 6 1 504.2.bs.a ✓ 48
12.b even 2 1 3024.2.df.e 48
21.g even 6 1 1512.2.bs.a 48
28.f even 6 1 1008.2.ca.e 48
36.f odd 6 1 3024.2.ca.e 48
36.h even 6 1 1008.2.ca.e 48
63.k odd 6 1 1512.2.cx.a 48
63.s even 6 1 inner 504.2.cx.a yes 48
84.j odd 6 1 3024.2.ca.e 48
252.n even 6 1 3024.2.df.e 48
252.bn odd 6 1 1008.2.df.e 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bs.a ✓ 48 7.d odd 6 1
504.2.bs.a ✓ 48 9.d odd 6 1
504.2.cx.a yes 48 1.a even 1 1 trivial
504.2.cx.a yes 48 63.s even 6 1 inner
1008.2.ca.e 48 28.f even 6 1
1008.2.ca.e 48 36.h even 6 1
1008.2.df.e 48 4.b odd 2 1
1008.2.df.e 48 252.bn odd 6 1
1512.2.bs.a 48 9.c even 3 1
1512.2.bs.a 48 21.g even 6 1
1512.2.cx.a 48 3.b odd 2 1
1512.2.cx.a 48 63.k odd 6 1
3024.2.ca.e 48 36.f odd 6 1
3024.2.ca.e 48 84.j odd 6 1
3024.2.df.e 48 12.b even 2 1
3024.2.df.e 48 252.n even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(504, [\chi])\).