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Properties

Label 504.2.q.d
Level $504$
Weight $2$
Character orbit 504.q
Analytic conductor $4.024$
Analytic rank $0$
Dimension $22$
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(25,504)]
 
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, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.q (of order \(3\), 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: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{3} + 3 q^{5} - 5 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{3} + 3 q^{5} - 5 q^{7} + 10 q^{9} - 3 q^{11} - 3 q^{13} - q^{15} + 7 q^{17} - q^{19} + 2 q^{23} - 10 q^{25} - 4 q^{27} + 9 q^{29} + 8 q^{31} + 29 q^{33} + 14 q^{35} + 2 q^{37} - 16 q^{39} + 16 q^{41} + q^{45} - 10 q^{47} + 15 q^{49} + 7 q^{51} + 11 q^{53} + 22 q^{55} + 7 q^{57} + 38 q^{59} + 26 q^{61} + 48 q^{63} - 26 q^{65} - 52 q^{67} - 4 q^{69} - 48 q^{71} - 35 q^{73} - 23 q^{75} + 17 q^{77} - 20 q^{79} - 38 q^{81} - 28 q^{83} - 20 q^{85} - 33 q^{87} + 6 q^{89} - 37 q^{91} + 19 q^{93} - 24 q^{95} - 29 q^{97} - 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 −1.65097 + 0.523731i 0 −0.841578 + 1.45766i 0 1.65502 − 2.06419i 0 2.45141 − 1.72933i 0
25.2 0 −1.56287 + 0.746620i 0 1.71796 − 2.97559i 0 −0.727932 + 2.54364i 0 1.88512 − 2.33374i 0
25.3 0 −1.47982 − 0.900079i 0 1.26145 − 2.18490i 0 2.63136 − 0.275550i 0 1.37972 + 2.66390i 0
25.4 0 −1.13364 − 1.30952i 0 −0.170100 + 0.294622i 0 −2.63360 + 0.253251i 0 −0.429705 + 2.96907i 0
25.5 0 −0.455209 + 1.67116i 0 0.240694 − 0.416893i 0 −1.92765 − 1.81223i 0 −2.58557 − 1.52146i 0
25.6 0 0.273951 − 1.71025i 0 −1.33401 + 2.31057i 0 0.581213 − 2.58112i 0 −2.84990 − 0.937047i 0
25.7 0 1.12528 − 1.31671i 0 −0.927957 + 1.60727i 0 0.900017 + 2.48796i 0 −0.467471 − 2.96335i 0
25.8 0 1.22753 − 1.22195i 0 1.76479 − 3.05671i 0 −2.63986 + 0.176417i 0 0.0136831 − 2.99997i 0
25.9 0 1.34477 + 1.09160i 0 0.918286 − 1.59052i 0 −0.361656 − 2.62092i 0 0.616838 + 2.93590i 0
25.10 0 1.57901 + 0.711841i 0 −1.92048 + 3.32636i 0 −2.55336 + 0.693065i 0 1.98656 + 2.24801i 0
25.11 0 1.73195 − 0.0184869i 0 0.790938 − 1.36994i 0 2.57645 + 0.601597i 0 2.99932 − 0.0640368i 0
121.1 0 −1.65097 − 0.523731i 0 −0.841578 − 1.45766i 0 1.65502 + 2.06419i 0 2.45141 + 1.72933i 0
121.2 0 −1.56287 − 0.746620i 0 1.71796 + 2.97559i 0 −0.727932 − 2.54364i 0 1.88512 + 2.33374i 0
121.3 0 −1.47982 + 0.900079i 0 1.26145 + 2.18490i 0 2.63136 + 0.275550i 0 1.37972 − 2.66390i 0
121.4 0 −1.13364 + 1.30952i 0 −0.170100 − 0.294622i 0 −2.63360 − 0.253251i 0 −0.429705 − 2.96907i 0
121.5 0 −0.455209 − 1.67116i 0 0.240694 + 0.416893i 0 −1.92765 + 1.81223i 0 −2.58557 + 1.52146i 0
121.6 0 0.273951 + 1.71025i 0 −1.33401 − 2.31057i 0 0.581213 + 2.58112i 0 −2.84990 + 0.937047i 0
121.7 0 1.12528 + 1.31671i 0 −0.927957 − 1.60727i 0 0.900017 − 2.48796i 0 −0.467471 + 2.96335i 0
121.8 0 1.22753 + 1.22195i 0 1.76479 + 3.05671i 0 −2.63986 − 0.176417i 0 0.0136831 + 2.99997i 0
121.9 0 1.34477 − 1.09160i 0 0.918286 + 1.59052i 0 −0.361656 + 2.62092i 0 0.616838 − 2.93590i 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.q.d ✓ 22
3.b odd 2 1 1512.2.q.c 22
4.b odd 2 1 1008.2.q.k 22
7.c even 3 1 504.2.t.d yes 22
9.c even 3 1 504.2.t.d yes 22
9.d odd 6 1 1512.2.t.d 22
12.b even 2 1 3024.2.q.k 22
21.h odd 6 1 1512.2.t.d 22
28.g odd 6 1 1008.2.t.k 22
36.f odd 6 1 1008.2.t.k 22
36.h even 6 1 3024.2.t.l 22
63.h even 3 1 inner 504.2.q.d ✓ 22
63.j odd 6 1 1512.2.q.c 22
84.n even 6 1 3024.2.t.l 22
252.u odd 6 1 1008.2.q.k 22
252.bb even 6 1 3024.2.q.k 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.d ✓ 22 1.a even 1 1 trivial
504.2.q.d ✓ 22 63.h even 3 1 inner
504.2.t.d yes 22 7.c even 3 1
504.2.t.d yes 22 9.c even 3 1
1008.2.q.k 22 4.b odd 2 1
1008.2.q.k 22 252.u odd 6 1
1008.2.t.k 22 28.g odd 6 1
1008.2.t.k 22 36.f odd 6 1
1512.2.q.c 22 3.b odd 2 1
1512.2.q.c 22 63.j odd 6 1
1512.2.t.d 22 9.d odd 6 1
1512.2.t.d 22 21.h odd 6 1
3024.2.q.k 22 12.b even 2 1
3024.2.q.k 22 252.bb even 6 1
3024.2.t.l 22 36.h even 6 1
3024.2.t.l 22 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{22} - 3 T_{5}^{21} + 37 T_{5}^{20} - 86 T_{5}^{19} + 790 T_{5}^{18} - 1652 T_{5}^{17} + \cdots + 217156 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display