Properties

Label 507.2.j.c
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{4} + ( 2 - \zeta_{6} ) q^{6} + ( 4 - 2 \zeta_{6} ) q^{7} + ( 1 - 2 \zeta_{6} ) q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{4} + ( 2 - \zeta_{6} ) q^{6} + ( 4 - 2 \zeta_{6} ) q^{7} + ( 1 - 2 \zeta_{6} ) q^{8} -\zeta_{6} q^{9} + ( -2 - 2 \zeta_{6} ) q^{11} + q^{12} + 6 q^{14} + ( 5 - 5 \zeta_{6} ) q^{16} + 6 \zeta_{6} q^{17} + ( 1 - 2 \zeta_{6} ) q^{18} + ( -4 + 2 \zeta_{6} ) q^{19} + ( 2 - 4 \zeta_{6} ) q^{21} -6 \zeta_{6} q^{22} + ( -1 - \zeta_{6} ) q^{24} + 5 q^{25} - q^{27} + ( 2 + 2 \zeta_{6} ) q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} + ( -2 + 4 \zeta_{6} ) q^{31} + ( 6 - 3 \zeta_{6} ) q^{32} + ( -4 + 2 \zeta_{6} ) q^{33} + ( -6 + 12 \zeta_{6} ) q^{34} + ( 1 - \zeta_{6} ) q^{36} + ( 4 + 4 \zeta_{6} ) q^{37} -6 q^{38} + ( -4 - 4 \zeta_{6} ) q^{41} + ( 6 - 6 \zeta_{6} ) q^{42} + 4 \zeta_{6} q^{43} + ( 2 - 4 \zeta_{6} ) q^{44} + ( 2 - 4 \zeta_{6} ) q^{47} -5 \zeta_{6} q^{48} + ( 5 - 5 \zeta_{6} ) q^{49} + ( 5 + 5 \zeta_{6} ) q^{50} + 6 q^{51} + 6 q^{53} + ( -1 - \zeta_{6} ) q^{54} -6 \zeta_{6} q^{56} + ( -2 + 4 \zeta_{6} ) q^{57} + ( -12 + 6 \zeta_{6} ) q^{58} + ( -12 + 6 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} + ( -6 + 6 \zeta_{6} ) q^{62} + ( -2 - 2 \zeta_{6} ) q^{63} - q^{64} -6 q^{66} + ( -6 - 6 \zeta_{6} ) q^{67} + ( -6 + 6 \zeta_{6} ) q^{68} + ( -4 + 2 \zeta_{6} ) q^{71} + ( -2 + \zeta_{6} ) q^{72} + 12 \zeta_{6} q^{74} + ( 5 - 5 \zeta_{6} ) q^{75} + ( -2 - 2 \zeta_{6} ) q^{76} -12 q^{77} -8 q^{79} + ( -1 + \zeta_{6} ) q^{81} -12 \zeta_{6} q^{82} + ( -2 + 4 \zeta_{6} ) q^{83} + ( 4 - 2 \zeta_{6} ) q^{84} + ( -4 + 8 \zeta_{6} ) q^{86} + 6 \zeta_{6} q^{87} + ( -6 + 6 \zeta_{6} ) q^{88} + ( -4 - 4 \zeta_{6} ) q^{89} + ( 2 + 2 \zeta_{6} ) q^{93} + ( 6 - 6 \zeta_{6} ) q^{94} + ( 3 - 6 \zeta_{6} ) q^{96} + ( 16 - 8 \zeta_{6} ) q^{97} + ( 10 - 5 \zeta_{6} ) q^{98} + ( -2 + 4 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{2} + q^{3} + q^{4} + 3q^{6} + 6q^{7} - q^{9} + O(q^{10}) \) \( 2q + 3q^{2} + q^{3} + q^{4} + 3q^{6} + 6q^{7} - q^{9} - 6q^{11} + 2q^{12} + 12q^{14} + 5q^{16} + 6q^{17} - 6q^{19} - 6q^{22} - 3q^{24} + 10q^{25} - 2q^{27} + 6q^{28} - 6q^{29} + 9q^{32} - 6q^{33} + q^{36} + 12q^{37} - 12q^{38} - 12q^{41} + 6q^{42} + 4q^{43} - 5q^{48} + 5q^{49} + 15q^{50} + 12q^{51} + 12q^{53} - 3q^{54} - 6q^{56} - 18q^{58} - 18q^{59} + 2q^{61} - 6q^{62} - 6q^{63} - 2q^{64} - 12q^{66} - 18q^{67} - 6q^{68} - 6q^{71} - 3q^{72} + 12q^{74} + 5q^{75} - 6q^{76} - 24q^{77} - 16q^{79} - q^{81} - 12q^{82} + 6q^{84} + 6q^{87} - 6q^{88} - 12q^{89} + 6q^{93} + 6q^{94} + 24q^{97} + 15q^{98} + O(q^{100}) \)

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
1.50000 + 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i 0 1.50000 0.866025i 3.00000 1.73205i 1.73205i −0.500000 0.866025i 0
361.1 1.50000 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i 0 1.50000 + 0.866025i 3.00000 + 1.73205i 1.73205i −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.c 2
13.b even 2 1 507.2.j.a 2
13.c even 3 1 39.2.b.a 2
13.c even 3 1 507.2.j.a 2
13.d odd 4 2 507.2.e.e 4
13.e even 6 1 39.2.b.a 2
13.e even 6 1 inner 507.2.j.c 2
13.f odd 12 2 507.2.a.f 2
13.f odd 12 2 507.2.e.e 4
39.h odd 6 1 117.2.b.a 2
39.i odd 6 1 117.2.b.a 2
39.k even 12 2 1521.2.a.l 2
52.i odd 6 1 624.2.c.e 2
52.j odd 6 1 624.2.c.e 2
52.l even 12 2 8112.2.a.bv 2
65.l even 6 1 975.2.b.d 2
65.n even 6 1 975.2.b.d 2
65.q odd 12 2 975.2.h.f 4
65.r odd 12 2 975.2.h.f 4
91.n odd 6 1 1911.2.c.d 2
91.t odd 6 1 1911.2.c.d 2
104.n odd 6 1 2496.2.c.d 2
104.p odd 6 1 2496.2.c.d 2
104.r even 6 1 2496.2.c.k 2
104.s even 6 1 2496.2.c.k 2
156.p even 6 1 1872.2.c.e 2
156.r even 6 1 1872.2.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 13.c even 3 1
39.2.b.a 2 13.e even 6 1
117.2.b.a 2 39.h odd 6 1
117.2.b.a 2 39.i odd 6 1
507.2.a.f 2 13.f odd 12 2
507.2.e.e 4 13.d odd 4 2
507.2.e.e 4 13.f odd 12 2
507.2.j.a 2 13.b even 2 1
507.2.j.a 2 13.c even 3 1
507.2.j.c 2 1.a even 1 1 trivial
507.2.j.c 2 13.e even 6 1 inner
624.2.c.e 2 52.i odd 6 1
624.2.c.e 2 52.j odd 6 1
975.2.b.d 2 65.l even 6 1
975.2.b.d 2 65.n even 6 1
975.2.h.f 4 65.q odd 12 2
975.2.h.f 4 65.r odd 12 2
1521.2.a.l 2 39.k even 12 2
1872.2.c.e 2 156.p even 6 1
1872.2.c.e 2 156.r even 6 1
1911.2.c.d 2 91.n odd 6 1
1911.2.c.d 2 91.t odd 6 1
2496.2.c.d 2 104.n odd 6 1
2496.2.c.d 2 104.p odd 6 1
2496.2.c.k 2 104.r even 6 1
2496.2.c.k 2 104.s even 6 1
8112.2.a.bv 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 \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{2} - 3 T_{2} + 3 \)
\( T_{5} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 - 3 T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 12 - 6 T + T^{2} \)
$11$ \( 12 + 6 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 36 - 6 T + T^{2} \)
$19$ \( 12 + 6 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( 12 + T^{2} \)
$37$ \( 48 - 12 T + T^{2} \)
$41$ \( 48 + 12 T + T^{2} \)
$43$ \( 16 - 4 T + T^{2} \)
$47$ \( 12 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 108 + 18 T + T^{2} \)
$61$ \( 4 - 2 T + T^{2} \)
$67$ \( 108 + 18 T + T^{2} \)
$71$ \( 12 + 6 T + T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 12 + T^{2} \)
$89$ \( 48 + 12 T + T^{2} \)
$97$ \( 192 - 24 T + T^{2} \)
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