Properties

Label 507.2.f.a
Level $507$
Weight $2$
Character orbit 507.f
Analytic conductor $4.048$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(239,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.f (of order \(4\), 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: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) 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: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} + ( - \zeta_{8}^{3} - \zeta_{8} - 1) q^{3} - \zeta_{8}^{2} q^{4} - 2 \zeta_{8} q^{5} + ( - \zeta_{8}^{2} - \zeta_{8} + 1) q^{6} + ( - \zeta_{8}^{2} - 1) q^{7} - 3 \zeta_{8}^{3} q^{8} + (2 \zeta_{8}^{3} + 2 \zeta_{8} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8} q^{2} + ( - \zeta_{8}^{3} - \zeta_{8} - 1) q^{3} - \zeta_{8}^{2} q^{4} - 2 \zeta_{8} q^{5} + ( - \zeta_{8}^{2} - \zeta_{8} + 1) q^{6} + ( - \zeta_{8}^{2} - 1) q^{7} - 3 \zeta_{8}^{3} q^{8} + (2 \zeta_{8}^{3} + 2 \zeta_{8} - 1) q^{9} - 2 \zeta_{8}^{2} q^{10} + 4 \zeta_{8}^{3} q^{11} + (\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8}) q^{12} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{14} + (2 \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{15} + q^{16} + (2 \zeta_{8}^{2} - \zeta_{8} - 2) q^{18} + (\zeta_{8}^{2} - 1) q^{19} + 2 \zeta_{8}^{3} q^{20} + (2 \zeta_{8}^{3} + \zeta_{8}^{2} + 1) q^{21} - 4 q^{22} + (6 \zeta_{8}^{3} - 6 \zeta_{8}) q^{23} + (3 \zeta_{8}^{3} - 3 \zeta_{8}^{2} - 3) q^{24} - \zeta_{8}^{2} q^{25} + ( - \zeta_{8}^{3} - \zeta_{8} + 5) q^{27} + (\zeta_{8}^{2} - 1) q^{28} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{29} + (2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2 \zeta_{8}) q^{30} + ( - 5 \zeta_{8}^{2} + 5) q^{31} - 5 \zeta_{8} q^{32} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 4) q^{33} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{35} + ( - 2 \zeta_{8}^{3} + \cdots + 2 \zeta_{8}) q^{36} + \cdots + ( - 4 \zeta_{8}^{3} - 8 \zeta_{8}^{2} - 8) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{6} - 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 4 q^{6} - 4 q^{7} - 4 q^{9} - 8 q^{15} + 4 q^{16} - 8 q^{18} - 4 q^{19} + 4 q^{21} - 16 q^{22} - 12 q^{24} + 20 q^{27} - 4 q^{28} + 20 q^{31} + 16 q^{33} - 4 q^{37} - 24 q^{40} - 8 q^{42} + 16 q^{45} - 24 q^{46} - 4 q^{48} + 4 q^{54} + 32 q^{55} + 4 q^{57} - 8 q^{58} + 8 q^{60} + 32 q^{61} + 4 q^{63} + 16 q^{66} + 20 q^{67} - 8 q^{70} + 24 q^{72} - 4 q^{73} + 4 q^{76} - 40 q^{79} - 28 q^{81} + 4 q^{84} + 16 q^{87} - 20 q^{93} - 16 q^{94} - 20 q^{96} - 28 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(-\zeta_{8}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i −1.00000 + 1.41421i 1.00000i 1.41421 + 1.41421i 1.70711 0.292893i −1.00000 1.00000i −2.12132 + 2.12132i −1.00000 2.82843i 2.00000i
239.2 0.707107 + 0.707107i −1.00000 1.41421i 1.00000i −1.41421 1.41421i 0.292893 1.70711i −1.00000 1.00000i 2.12132 2.12132i −1.00000 + 2.82843i 2.00000i
437.1 −0.707107 + 0.707107i −1.00000 1.41421i 1.00000i 1.41421 1.41421i 1.70711 + 0.292893i −1.00000 + 1.00000i −2.12132 2.12132i −1.00000 + 2.82843i 2.00000i
437.2 0.707107 0.707107i −1.00000 + 1.41421i 1.00000i −1.41421 + 1.41421i 0.292893 + 1.70711i −1.00000 + 1.00000i 2.12132 + 2.12132i −1.00000 2.82843i 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

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

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 2 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 256 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 16 \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 256 \) Copy content Toggle raw display
$53$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 256 \) Copy content Toggle raw display
$61$ \( (T - 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 256 \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T + 10)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 4096 \) Copy content Toggle raw display
$89$ \( T^{4} + 38416 \) Copy content Toggle raw display
$97$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
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