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Properties

Label 504.2.bs.a
Level $504$
Weight $2$
Character orbit 504.bs
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(257,504)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.bs (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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{9} + 8 q^{15} + 8 q^{21} - 12 q^{23} - 24 q^{25} - 18 q^{27} + 18 q^{29} - 10 q^{39} + 6 q^{41} - 6 q^{43} + 6 q^{45} + 36 q^{47} + 6 q^{49} - 12 q^{51} + 12 q^{53} + 4 q^{57} + 46 q^{63} - 54 q^{75} - 36 q^{77} - 12 q^{79} - 24 q^{87} + 18 q^{89} + 6 q^{91} + 16 q^{93} - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1 0 −1.70597 + 0.299447i 0 1.76701 + 3.06054i 0 1.34480 + 2.27849i 0 2.82066 − 1.02170i 0
257.2 0 −1.69760 + 0.343722i 0 −0.0262870 − 0.0455305i 0 −2.10354 + 1.60471i 0 2.76371 − 1.16701i 0
257.3 0 −1.68581 − 0.397560i 0 −0.311798 − 0.540051i 0 2.62086 + 0.362103i 0 2.68389 + 1.34042i 0
257.4 0 −1.54049 + 0.791772i 0 −0.905867 − 1.56901i 0 −2.60735 + 0.449161i 0 1.74619 − 2.43943i 0
257.5 0 −1.41991 − 0.991901i 0 −1.11047 − 1.92339i 0 0.362456 − 2.62081i 0 1.03226 + 2.81681i 0
257.6 0 −1.32026 − 1.12112i 0 1.38468 + 2.39834i 0 −1.42373 − 2.23002i 0 0.486198 + 2.96034i 0
257.7 0 −1.14780 − 1.29713i 0 −0.527910 − 0.914367i 0 0.781227 + 2.52778i 0 −0.365108 + 2.97770i 0
257.8 0 −1.09564 + 1.34148i 0 −0.271038 − 0.469451i 0 1.60655 − 2.10214i 0 −0.599164 − 2.93956i 0
257.9 0 −0.559017 + 1.63936i 0 −1.82284 − 3.15725i 0 1.57353 + 2.12697i 0 −2.37500 − 1.83286i 0
257.10 0 −0.445548 + 1.67376i 0 −0.101589 − 0.175958i 0 −1.37377 − 2.26114i 0 −2.60297 − 1.49148i 0
257.11 0 −0.419673 − 1.68044i 0 1.02449 + 1.77447i 0 −2.64365 + 0.105420i 0 −2.64775 + 1.41047i 0
257.12 0 0.0250939 − 1.73187i 0 −1.42382 − 2.46612i 0 1.38343 − 2.25524i 0 −2.99874 − 0.0869188i 0
257.13 0 0.0709339 + 1.73060i 0 2.04942 + 3.54970i 0 −2.21443 − 1.44785i 0 −2.98994 + 0.245516i 0
257.14 0 0.589575 − 1.62862i 0 1.11415 + 1.92977i 0 −0.553477 + 2.58721i 0 −2.30480 − 1.92039i 0
257.15 0 0.690387 + 1.58851i 0 1.10442 + 1.91291i 0 0.234181 + 2.63537i 0 −2.04673 + 2.19337i 0
257.16 0 0.859678 − 1.50365i 0 1.29668 + 2.24592i 0 1.66807 − 2.05367i 0 −1.52191 − 2.58530i 0
257.17 0 0.872943 + 1.49598i 0 −2.09905 − 3.63567i 0 −2.61186 + 0.422135i 0 −1.47594 + 2.61182i 0
257.18 0 0.957290 − 1.44347i 0 −1.80389 − 3.12442i 0 2.05506 + 1.66636i 0 −1.16719 − 2.76363i 0
257.19 0 1.10617 + 1.33281i 0 −0.103514 − 0.179292i 0 2.64517 − 0.0556151i 0 −0.552770 + 2.94863i 0
257.20 0 1.29769 − 1.14717i 0 −0.643917 − 1.11530i 0 −2.63392 − 0.249885i 0 0.368015 − 2.97734i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.24
Significant digits:
Format:

Inner twists

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

Hecke kernels

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