Properties

Label 507.2.e.k
Level $507$
Weight $2$
Character orbit 507.e
Analytic conductor $4.048$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.64827.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots + \beta_1) q^{4}+ \cdots - \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots + \beta_1) q^{4}+ \cdots + ( - 4 \beta_{3} + 3 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 3 q^{3} + q^{4} - 8 q^{5} + q^{6} + 10 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - 3 q^{3} + q^{4} - 8 q^{5} + q^{6} + 10 q^{7} - 3 q^{9} + q^{10} - q^{11} - 2 q^{12} + 2 q^{14} + 4 q^{15} + 5 q^{16} + 7 q^{17} - 2 q^{18} + 11 q^{19} + q^{20} - 20 q^{21} + 5 q^{22} - 2 q^{23} - 10 q^{25} + 6 q^{27} - q^{28} + 8 q^{29} + q^{30} - 16 q^{31} - 4 q^{32} - q^{33} - 18 q^{35} + q^{36} + 14 q^{37} + 40 q^{38} + 14 q^{40} + q^{41} - q^{42} + 3 q^{43} + 32 q^{44} + 4 q^{45} + 3 q^{46} + 18 q^{47} + 5 q^{48} - 17 q^{49} - 11 q^{50} - 14 q^{51} - 26 q^{53} + q^{54} + 13 q^{55} - 7 q^{56} - 22 q^{57} - 12 q^{58} - 14 q^{59} - 2 q^{60} + 13 q^{61} + 16 q^{62} + 10 q^{63} + 8 q^{64} - 10 q^{66} + 5 q^{67} - 7 q^{68} - 2 q^{69} + 16 q^{70} - 6 q^{71} - 36 q^{73} - 7 q^{74} + 5 q^{75} + q^{76} - 30 q^{77} - 18 q^{79} - 16 q^{80} - 3 q^{81} - 5 q^{82} + 32 q^{83} - q^{84} - 7 q^{85} + 30 q^{86} + 8 q^{87} + 7 q^{88} - 5 q^{89} - 2 q^{90} - 34 q^{92} + 8 q^{93} - 32 q^{94} - 3 q^{95} + 8 q^{96} + 5 q^{97} - 13 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-\beta_{5}\)

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
−0.623490 + 1.07992i
0.222521 0.385418i
0.900969 1.56052i
−0.623490 1.07992i
0.222521 + 0.385418i
0.900969 + 1.56052i
−0.623490 + 1.07992i −0.500000 + 0.866025i 0.222521 + 0.385418i −2.80194 −0.623490 1.07992i 2.40097 + 4.15860i −3.04892 −0.500000 0.866025i 1.74698 3.02586i
22.2 0.222521 0.385418i −0.500000 + 0.866025i 0.900969 + 1.56052i 0.246980 0.222521 + 0.385418i 0.876510 + 1.51816i 1.69202 −0.500000 0.866025i 0.0549581 0.0951903i
22.3 0.900969 1.56052i −0.500000 + 0.866025i −0.623490 1.07992i −1.44504 0.900969 + 1.56052i 1.72252 + 2.98349i 1.35690 −0.500000 0.866025i −1.30194 + 2.25502i
484.1 −0.623490 1.07992i −0.500000 0.866025i 0.222521 0.385418i −2.80194 −0.623490 + 1.07992i 2.40097 4.15860i −3.04892 −0.500000 + 0.866025i 1.74698 + 3.02586i
484.2 0.222521 + 0.385418i −0.500000 0.866025i 0.900969 1.56052i 0.246980 0.222521 0.385418i 0.876510 1.51816i 1.69202 −0.500000 + 0.866025i 0.0549581 + 0.0951903i
484.3 0.900969 + 1.56052i −0.500000 0.866025i −0.623490 + 1.07992i −1.44504 0.900969 1.56052i 1.72252 2.98349i 1.35690 −0.500000 + 0.866025i −1.30194 2.25502i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.e.k 6
13.b even 2 1 507.2.e.j 6
13.c even 3 1 507.2.a.j 3
13.c even 3 1 inner 507.2.e.k 6
13.d odd 4 2 507.2.j.h 12
13.e even 6 1 507.2.a.k yes 3
13.e even 6 1 507.2.e.j 6
13.f odd 12 2 507.2.b.g 6
13.f odd 12 2 507.2.j.h 12
39.h odd 6 1 1521.2.a.p 3
39.i odd 6 1 1521.2.a.q 3
39.k even 12 2 1521.2.b.m 6
52.i odd 6 1 8112.2.a.cf 3
52.j odd 6 1 8112.2.a.by 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.j 3 13.c even 3 1
507.2.a.k yes 3 13.e even 6 1
507.2.b.g 6 13.f odd 12 2
507.2.e.j 6 13.b even 2 1
507.2.e.j 6 13.e even 6 1
507.2.e.k 6 1.a even 1 1 trivial
507.2.e.k 6 13.c even 3 1 inner
507.2.j.h 12 13.d odd 4 2
507.2.j.h 12 13.f odd 12 2
1521.2.a.p 3 39.h odd 6 1
1521.2.a.q 3 39.i odd 6 1
1521.2.b.m 6 39.k even 12 2
8112.2.a.by 3 52.j odd 6 1
8112.2.a.cf 3 52.i odd 6 1

Hecke kernels

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

\( T_{2}^{6} - T_{2}^{5} + 3T_{2}^{4} + 5T_{2}^{2} - 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{3} + 4T_{5}^{2} + 3T_{5} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{3} + 4 T^{2} + 3 T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} - 10 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$11$ \( T^{6} + T^{5} + \cdots + 1849 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 7 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$19$ \( T^{6} - 11 T^{5} + \cdots + 12769 \) Copy content Toggle raw display
$23$ \( T^{6} + 2 T^{5} + \cdots + 6889 \) Copy content Toggle raw display
$29$ \( T^{6} - 8 T^{5} + \cdots + 1849 \) Copy content Toggle raw display
$31$ \( (T^{3} + 8 T^{2} + \cdots - 197)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 14 T^{5} + \cdots + 8281 \) Copy content Toggle raw display
$41$ \( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$47$ \( (T^{3} - 9 T^{2} + \cdots + 911)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 13 T^{2} + \cdots + 29)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 14 T^{5} + \cdots + 3136 \) Copy content Toggle raw display
$61$ \( T^{6} - 13 T^{5} + \cdots + 49729 \) Copy content Toggle raw display
$67$ \( T^{6} - 5 T^{5} + \cdots + 9409 \) Copy content Toggle raw display
$71$ \( T^{6} + 6 T^{5} + \cdots + 212521 \) Copy content Toggle raw display
$73$ \( (T^{3} + 18 T^{2} + \cdots + 167)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 9 T^{2} + \cdots - 169)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 16 T^{2} + \cdots + 43)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{6} - 5 T^{5} + \cdots + 1413721 \) Copy content Toggle raw display
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