Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{4} + ( 2 - 4 \zeta_{6} ) q^{5} + ( 2 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{4} + ( 2 - 4 \zeta_{6} ) q^{5} + ( 2 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 2 + 2 \zeta_{6} ) q^{11} -2 q^{12} + ( -2 - 2 \zeta_{6} ) q^{15} + ( -4 + 4 \zeta_{6} ) q^{16} + ( -4 + 2 \zeta_{6} ) q^{19} + ( -8 + 4 \zeta_{6} ) q^{20} + ( 1 - 2 \zeta_{6} ) q^{21} + ( 6 - 6 \zeta_{6} ) q^{23} -7 q^{25} - q^{27} + ( -2 - 2 \zeta_{6} ) q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} + ( 1 - 2 \zeta_{6} ) q^{31} + ( 4 - 2 \zeta_{6} ) q^{33} -6 \zeta_{6} q^{35} + ( -2 + 2 \zeta_{6} ) q^{36} + ( 4 + 4 \zeta_{6} ) q^{41} + \zeta_{6} q^{43} + ( 4 - 8 \zeta_{6} ) q^{44} + ( -4 + 2 \zeta_{6} ) q^{45} + ( 2 - 4 \zeta_{6} ) q^{47} + 4 \zeta_{6} q^{48} + ( -4 + 4 \zeta_{6} ) q^{49} + 12 q^{53} + ( 12 - 12 \zeta_{6} ) q^{55} + ( -2 + 4 \zeta_{6} ) q^{57} + ( 4 - 2 \zeta_{6} ) q^{59} + ( -4 + 8 \zeta_{6} ) q^{60} -\zeta_{6} q^{61} + ( -1 - \zeta_{6} ) q^{63} + 8 q^{64} + ( -5 - 5 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{69} + ( 12 - 6 \zeta_{6} ) q^{71} + ( 1 - 2 \zeta_{6} ) q^{73} + ( -7 + 7 \zeta_{6} ) q^{75} + ( 4 + 4 \zeta_{6} ) q^{76} + 6 q^{77} -11 q^{79} + ( 8 + 8 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + ( -8 + 16 \zeta_{6} ) q^{83} + ( -4 + 2 \zeta_{6} ) q^{84} + 6 \zeta_{6} q^{87} + ( -4 - 4 \zeta_{6} ) q^{89} -12 q^{92} + ( -1 - \zeta_{6} ) q^{93} + 12 \zeta_{6} q^{95} + ( 6 - 3 \zeta_{6} ) q^{97} + ( 2 - 4 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 2q^{4} + 3q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - 2q^{4} + 3q^{7} - q^{9} + 6q^{11} - 4q^{12} - 6q^{15} - 4q^{16} - 6q^{19} - 12q^{20} + 6q^{23} - 14q^{25} - 2q^{27} - 6q^{28} - 6q^{29} + 6q^{33} - 6q^{35} - 2q^{36} + 12q^{41} + q^{43} - 6q^{45} + 4q^{48} - 4q^{49} + 24q^{53} + 12q^{55} + 6q^{59} - q^{61} - 3q^{63} + 16q^{64} - 15q^{67} - 6q^{69} + 18q^{71} - 7q^{75} + 12q^{76} + 12q^{77} - 22q^{79} + 24q^{80} - q^{81} - 6q^{84} + 6q^{87} - 12q^{89} - 24q^{92} - 3q^{93} + 12q^{95} + 9q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(\zeta_{6}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i −1.00000 1.73205i 3.46410i 0 1.50000 0.866025i 0 −0.500000 0.866025i 0
361.1 0 0.500000 + 0.866025i −1.00000 + 1.73205i 3.46410i 0 1.50000 + 0.866025i 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.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.j.b 2
13.b even 2 1 39.2.j.a 2
13.c even 3 1 39.2.j.a 2
13.c even 3 1 507.2.b.c 2
13.d odd 4 2 507.2.e.f 4
13.e even 6 1 507.2.b.c 2
13.e even 6 1 inner 507.2.j.b 2
13.f odd 12 2 507.2.a.e 2
13.f odd 12 2 507.2.e.f 4
39.d odd 2 1 117.2.q.a 2
39.h odd 6 1 1521.2.b.f 2
39.i odd 6 1 117.2.q.a 2
39.i odd 6 1 1521.2.b.f 2
39.k even 12 2 1521.2.a.h 2
52.b odd 2 1 624.2.bv.b 2
52.j odd 6 1 624.2.bv.b 2
52.l even 12 2 8112.2.a.bu 2
65.d even 2 1 975.2.bc.c 2
65.h odd 4 2 975.2.w.d 4
65.n even 6 1 975.2.bc.c 2
65.q odd 12 2 975.2.w.d 4
156.h even 2 1 1872.2.by.f 2
156.p even 6 1 1872.2.by.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a 2 13.b even 2 1
39.2.j.a 2 13.c even 3 1
117.2.q.a 2 39.d odd 2 1
117.2.q.a 2 39.i odd 6 1
507.2.a.e 2 13.f odd 12 2
507.2.b.c 2 13.c even 3 1
507.2.b.c 2 13.e even 6 1
507.2.e.f 4 13.d odd 4 2
507.2.e.f 4 13.f odd 12 2
507.2.j.b 2 1.a even 1 1 trivial
507.2.j.b 2 13.e even 6 1 inner
624.2.bv.b 2 52.b odd 2 1
624.2.bv.b 2 52.j odd 6 1
975.2.w.d 4 65.h odd 4 2
975.2.w.d 4 65.q odd 12 2
975.2.bc.c 2 65.d even 2 1
975.2.bc.c 2 65.n even 6 1
1521.2.a.h 2 39.k even 12 2
1521.2.b.f 2 39.h odd 6 1
1521.2.b.f 2 39.i odd 6 1
1872.2.by.f 2 156.h even 2 1
1872.2.by.f 2 156.p even 6 1
8112.2.a.bu 2 52.l even 12 2

Hecke kernels

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

\( T_{2} \)
\( T_{5}^{2} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 12 + T^{2} \)
$7$ \( 3 - 3 T + T^{2} \)
$11$ \( 12 - 6 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( 12 + 6 T + T^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( 3 + T^{2} \)
$37$ \( T^{2} \)
$41$ \( 48 - 12 T + T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( 12 + T^{2} \)
$53$ \( ( -12 + T )^{2} \)
$59$ \( 12 - 6 T + T^{2} \)
$61$ \( 1 + T + T^{2} \)
$67$ \( 75 + 15 T + T^{2} \)
$71$ \( 108 - 18 T + T^{2} \)
$73$ \( 3 + T^{2} \)
$79$ \( ( 11 + T )^{2} \)
$83$ \( 192 + T^{2} \)
$89$ \( 48 + 12 T + T^{2} \)
$97$ \( 27 - 9 T + T^{2} \)
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