Properties

Label 507.2.k.j
Level $507$
Weight $2$
Character orbit 507.k
Analytic conductor $4.048$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(80,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.80");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.k (of order \(12\), degree \(4\), not 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: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{24}^{7} q^{2} + (\zeta_{24}^{7} + \zeta_{24}^{4} - \zeta_{24}) q^{3} + \zeta_{24}^{2} q^{4} + 2 \zeta_{24}^{3} q^{5} + (\zeta_{24}^{7} - \zeta_{24}^{4} + \cdots + 1) q^{6}+ \cdots + (2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{7} q^{2} + (\zeta_{24}^{7} + \zeta_{24}^{4} - \zeta_{24}) q^{3} + \zeta_{24}^{2} q^{4} + 2 \zeta_{24}^{3} q^{5} + (\zeta_{24}^{7} - \zeta_{24}^{4} + \cdots + 1) q^{6}+ \cdots + (8 \zeta_{24}^{6} + 4 \zeta_{24}^{5} + \cdots + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 4 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 4 q^{6} - 4 q^{7} + 4 q^{9} - 8 q^{15} - 4 q^{16} + 16 q^{18} - 4 q^{19} - 8 q^{21} + 16 q^{22} - 12 q^{24} + 40 q^{27} - 4 q^{28} - 40 q^{31} + 16 q^{33} - 4 q^{37} - 48 q^{40} + 8 q^{42} + 16 q^{45} - 24 q^{46} + 4 q^{48} + 4 q^{54} - 32 q^{55} - 8 q^{57} - 8 q^{58} - 16 q^{60} - 32 q^{61} + 4 q^{63} + 32 q^{66} + 20 q^{67} + 16 q^{70} + 24 q^{72} + 8 q^{73} + 4 q^{76} - 80 q^{79} + 28 q^{81} + 4 q^{84} - 16 q^{87} - 20 q^{93} + 16 q^{94} + 40 q^{96} - 28 q^{97} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.258819 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i −0.724745 + 1.57313i 0.866025 + 0.500000i 1.41421 + 1.41421i −1.33195 1.10721i −1.36603 + 0.366025i −2.12132 + 2.12132i −1.94949 2.28024i −1.73205 + 1.00000i
80.2 0.258819 0.965926i 1.72474 + 0.158919i 0.866025 + 0.500000i −1.41421 1.41421i 0.599900 1.62484i −1.36603 + 0.366025i 2.12132 2.12132i 2.94949 + 0.548188i −1.73205 + 1.00000i
89.1 −0.965926 0.258819i −0.724745 + 1.57313i −0.866025 0.500000i −1.41421 + 1.41421i 1.10721 1.33195i 0.366025 + 1.36603i 2.12132 + 2.12132i −1.94949 2.28024i 1.73205 1.00000i
89.2 0.965926 + 0.258819i 1.72474 + 0.158919i −0.866025 0.500000i 1.41421 1.41421i 1.62484 + 0.599900i 0.366025 + 1.36603i −2.12132 2.12132i 2.94949 + 0.548188i 1.73205 1.00000i
188.1 −0.965926 + 0.258819i −0.724745 1.57313i −0.866025 + 0.500000i −1.41421 1.41421i 1.10721 + 1.33195i 0.366025 1.36603i 2.12132 2.12132i −1.94949 + 2.28024i 1.73205 + 1.00000i
188.2 0.965926 0.258819i 1.72474 0.158919i −0.866025 + 0.500000i 1.41421 + 1.41421i 1.62484 0.599900i 0.366025 1.36603i −2.12132 + 2.12132i 2.94949 0.548188i 1.73205 + 1.00000i
488.1 −0.258819 0.965926i −0.724745 1.57313i 0.866025 0.500000i 1.41421 1.41421i −1.33195 + 1.10721i −1.36603 0.366025i −2.12132 2.12132i −1.94949 + 2.28024i −1.73205 1.00000i
488.2 0.258819 + 0.965926i 1.72474 0.158919i 0.866025 0.500000i −1.41421 + 1.41421i 0.599900 + 1.62484i −1.36603 0.366025i 2.12132 + 2.12132i 2.94949 0.548188i −1.73205 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.c even 3 1 inner
13.d odd 4 1 inner
13.f odd 12 1 inner
39.f even 4 1 inner
39.i odd 6 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.k.j 8
3.b odd 2 1 inner 507.2.k.j 8
13.b even 2 1 507.2.k.i 8
13.c even 3 1 39.2.f.a 4
13.c even 3 1 inner 507.2.k.j 8
13.d odd 4 1 507.2.k.i 8
13.d odd 4 1 inner 507.2.k.j 8
13.e even 6 1 507.2.f.a 4
13.e even 6 1 507.2.k.i 8
13.f odd 12 1 39.2.f.a 4
13.f odd 12 1 507.2.f.a 4
13.f odd 12 1 507.2.k.i 8
13.f odd 12 1 inner 507.2.k.j 8
39.d odd 2 1 507.2.k.i 8
39.f even 4 1 507.2.k.i 8
39.f even 4 1 inner 507.2.k.j 8
39.h odd 6 1 507.2.f.a 4
39.h odd 6 1 507.2.k.i 8
39.i odd 6 1 39.2.f.a 4
39.i odd 6 1 inner 507.2.k.j 8
39.k even 12 1 39.2.f.a 4
39.k even 12 1 507.2.f.a 4
39.k even 12 1 507.2.k.i 8
39.k even 12 1 inner 507.2.k.j 8
52.j odd 6 1 624.2.bf.d 4
52.l even 12 1 624.2.bf.d 4
65.n even 6 1 975.2.o.j 4
65.o even 12 1 975.2.n.c 4
65.q odd 12 1 975.2.n.c 4
65.q odd 12 1 975.2.n.d 4
65.s odd 12 1 975.2.o.j 4
65.t even 12 1 975.2.n.d 4
156.p even 6 1 624.2.bf.d 4
156.v odd 12 1 624.2.bf.d 4
195.x odd 6 1 975.2.o.j 4
195.bc odd 12 1 975.2.n.d 4
195.bh even 12 1 975.2.o.j 4
195.bl even 12 1 975.2.n.c 4
195.bl even 12 1 975.2.n.d 4
195.bn odd 12 1 975.2.n.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.f.a 4 13.c even 3 1
39.2.f.a 4 13.f odd 12 1
39.2.f.a 4 39.i odd 6 1
39.2.f.a 4 39.k even 12 1
507.2.f.a 4 13.e even 6 1
507.2.f.a 4 13.f odd 12 1
507.2.f.a 4 39.h odd 6 1
507.2.f.a 4 39.k even 12 1
507.2.k.i 8 13.b even 2 1
507.2.k.i 8 13.d odd 4 1
507.2.k.i 8 13.e even 6 1
507.2.k.i 8 13.f odd 12 1
507.2.k.i 8 39.d odd 2 1
507.2.k.i 8 39.f even 4 1
507.2.k.i 8 39.h odd 6 1
507.2.k.i 8 39.k even 12 1
507.2.k.j 8 1.a even 1 1 trivial
507.2.k.j 8 3.b odd 2 1 inner
507.2.k.j 8 13.c even 3 1 inner
507.2.k.j 8 13.d odd 4 1 inner
507.2.k.j 8 13.f odd 12 1 inner
507.2.k.j 8 39.f even 4 1 inner
507.2.k.j 8 39.i odd 6 1 inner
507.2.k.j 8 39.k even 12 1 inner
624.2.bf.d 4 52.j odd 6 1
624.2.bf.d 4 52.l even 12 1
624.2.bf.d 4 156.p even 6 1
624.2.bf.d 4 156.v odd 12 1
975.2.n.c 4 65.o even 12 1
975.2.n.c 4 65.q odd 12 1
975.2.n.c 4 195.bl even 12 1
975.2.n.c 4 195.bn odd 12 1
975.2.n.d 4 65.q odd 12 1
975.2.n.d 4 65.t even 12 1
975.2.n.d 4 195.bc odd 12 1
975.2.n.d 4 195.bl even 12 1
975.2.o.j 4 65.n even 6 1
975.2.o.j 4 65.s odd 12 1
975.2.o.j 4 195.x odd 6 1
975.2.o.j 4 195.bh even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{8} - T_{2}^{4} + 1 \) Copy content Toggle raw display
\( T_{5}^{4} + 16 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 2T_{7}^{2} + 4T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{3} + T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 256 T^{4} + 65536 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 72 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 50)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 16T^{4} + 256 \) Copy content Toggle raw display
$43$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} - 256 T^{4} + 65536 \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 10 T^{3} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 256 T^{4} + 65536 \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$79$ \( (T + 10)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 4096)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 1475789056 \) Copy content Toggle raw display
$97$ \( (T^{4} + 14 T^{3} + \cdots + 9604)^{2} \) Copy content Toggle raw display
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