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Properties

Label 507.2.e.f
Level $507$
Weight $2$
Character orbit 507.e
Analytic conductor $4.048$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(22,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.22");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.e (of order \(3\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - 2 \beta_1 + 2) q^{4} - 2 \beta_{3} q^{5} + (\beta_{3} - \beta_{2}) q^{7} + (\beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - 2 \beta_1 + 2) q^{4} - 2 \beta_{3} q^{5} + (\beta_{3} - \beta_{2}) q^{7} + (\beta_1 - 1) q^{9} - 2 \beta_{2} q^{11} + 2 q^{12} - 2 \beta_{2} q^{15} - 4 \beta_1 q^{16} + (2 \beta_{3} - 2 \beta_{2}) q^{19} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{20} + \beta_{3} q^{21} - 6 \beta_1 q^{23} + 7 q^{25} - q^{27} - 2 \beta_{2} q^{28} - 6 \beta_1 q^{29} - \beta_{3} q^{31} + (2 \beta_{3} - 2 \beta_{2}) q^{33} + (6 \beta_1 - 6) q^{35} + 2 \beta_1 q^{36} + 4 \beta_{2} q^{41} + (\beta_1 - 1) q^{43} - 4 \beta_{3} q^{44} + (2 \beta_{3} - 2 \beta_{2}) q^{45} + 2 \beta_{3} q^{47} + ( - 4 \beta_1 + 4) q^{48} + 4 \beta_1 q^{49} + 12 q^{53} + 12 \beta_1 q^{55} + 2 \beta_{3} q^{57} + (2 \beta_{3} - 2 \beta_{2}) q^{59} - 4 \beta_{3} q^{60} + (\beta_1 - 1) q^{61} + \beta_{2} q^{63} - 8 q^{64} - 5 \beta_{2} q^{67} + ( - 6 \beta_1 + 6) q^{69} + ( - 6 \beta_{3} + 6 \beta_{2}) q^{71} + \beta_{3} q^{73} + 7 \beta_1 q^{75} - 4 \beta_{2} q^{76} - 6 q^{77} - 11 q^{79} + 8 \beta_{2} q^{80} - \beta_1 q^{81} + 8 \beta_{3} q^{83} + (2 \beta_{3} - 2 \beta_{2}) q^{84} + ( - 6 \beta_1 + 6) q^{87} + 4 \beta_{2} q^{89} - 12 q^{92} - \beta_{2} q^{93} + (12 \beta_1 - 12) q^{95} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{97} + 2 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{4} - 2 q^{9} + 8 q^{12} - 8 q^{16} - 12 q^{23} + 28 q^{25} - 4 q^{27} - 12 q^{29} - 12 q^{35} + 4 q^{36} - 2 q^{43} + 8 q^{48} + 8 q^{49} + 48 q^{53} + 24 q^{55} - 2 q^{61} - 32 q^{64} + 12 q^{69} + 14 q^{75} - 24 q^{77} - 44 q^{79} - 2 q^{81} + 12 q^{87} - 48 q^{92} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
0 0.500000 − 0.866025i 1.00000 + 1.73205i −3.46410 0 0.866025 + 1.50000i 0 −0.500000 − 0.866025i 0
22.2 0 0.500000 − 0.866025i 1.00000 + 1.73205i 3.46410 0 −0.866025 − 1.50000i 0 −0.500000 − 0.866025i 0
484.1 0 0.500000 + 0.866025i 1.00000 − 1.73205i −3.46410 0 0.866025 − 1.50000i 0 −0.500000 + 0.866025i 0
484.2 0 0.500000 + 0.866025i 1.00000 − 1.73205i 3.46410 0 −0.866025 + 1.50000i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.e.f 4
13.b even 2 1 inner 507.2.e.f 4
13.c even 3 1 507.2.a.e 2
13.c even 3 1 inner 507.2.e.f 4
13.d odd 4 1 39.2.j.a ✓ 2
13.d odd 4 1 507.2.j.b 2
13.e even 6 1 507.2.a.e 2
13.e even 6 1 inner 507.2.e.f 4
13.f odd 12 1 39.2.j.a ✓ 2
13.f odd 12 2 507.2.b.c 2
13.f odd 12 1 507.2.j.b 2
39.f even 4 1 117.2.q.a 2
39.h odd 6 1 1521.2.a.h 2
39.i odd 6 1 1521.2.a.h 2
39.k even 12 1 117.2.q.a 2
39.k even 12 2 1521.2.b.f 2
52.f even 4 1 624.2.bv.b 2
52.i odd 6 1 8112.2.a.bu 2
52.j odd 6 1 8112.2.a.bu 2
52.l even 12 1 624.2.bv.b 2
65.f even 4 1 975.2.w.d 4
65.g odd 4 1 975.2.bc.c 2
65.k even 4 1 975.2.w.d 4
65.o even 12 1 975.2.w.d 4
65.s odd 12 1 975.2.bc.c 2
65.t even 12 1 975.2.w.d 4
156.l odd 4 1 1872.2.by.f 2
156.v odd 12 1 1872.2.by.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a ✓ 2 13.d odd 4 1
39.2.j.a ✓ 2 13.f odd 12 1
117.2.q.a 2 39.f even 4 1
117.2.q.a 2 39.k even 12 1
507.2.a.e 2 13.c even 3 1
507.2.a.e 2 13.e even 6 1
507.2.b.c 2 13.f odd 12 2
507.2.e.f 4 1.a even 1 1 trivial
507.2.e.f 4 13.b even 2 1 inner
507.2.e.f 4 13.c even 3 1 inner
507.2.e.f 4 13.e even 6 1 inner
507.2.j.b 2 13.d odd 4 1
507.2.j.b 2 13.f odd 12 1
624.2.bv.b 2 52.f even 4 1
624.2.bv.b 2 52.l even 12 1
975.2.w.d 4 65.f even 4 1
975.2.w.d 4 65.k even 4 1
975.2.w.d 4 65.o even 12 1
975.2.w.d 4 65.t even 12 1
975.2.bc.c 2 65.g odd 4 1
975.2.bc.c 2 65.s odd 12 1
1521.2.a.h 2 39.h odd 6 1
1521.2.a.h 2 39.i odd 6 1
1521.2.b.f 2 39.k even 12 2
1872.2.by.f 2 156.l odd 4 1
1872.2.by.f 2 156.v odd 12 1
8112.2.a.bu 2 52.i odd 6 1
8112.2.a.bu 2 52.j odd 6 1

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{5}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$11$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$43$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T - 12)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$71$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$73$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$79$ \( (T + 11)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$97$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
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