Properties

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

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( 1 - \zeta_{12}^{2} ) q^{3} -\zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{5} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{6} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} -3 \zeta_{12}^{3} q^{8} -\zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{2} + ( 1 - \zeta_{12}^{2} ) q^{3} -\zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{5} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{6} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} -3 \zeta_{12}^{3} q^{8} -\zeta_{12}^{2} q^{9} + ( -1 + \zeta_{12}^{2} ) q^{10} + 2 \zeta_{12} q^{11} - q^{12} + 2 q^{14} + \zeta_{12} q^{15} + ( 1 - \zeta_{12}^{2} ) q^{16} -7 \zeta_{12}^{2} q^{17} -\zeta_{12}^{3} q^{18} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{19} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{20} -2 \zeta_{12}^{3} q^{21} + 2 \zeta_{12}^{2} q^{22} + ( -6 + 6 \zeta_{12}^{2} ) q^{23} -3 \zeta_{12} q^{24} + 4 q^{25} - q^{27} -2 \zeta_{12} q^{28} + ( 1 - \zeta_{12}^{2} ) q^{29} + \zeta_{12}^{2} q^{30} -4 \zeta_{12}^{3} q^{31} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{32} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{33} -7 \zeta_{12}^{3} q^{34} + 2 \zeta_{12}^{2} q^{35} + ( -1 + \zeta_{12}^{2} ) q^{36} -\zeta_{12} q^{37} + 6 q^{38} + 3 q^{40} + 9 \zeta_{12} q^{41} + ( 2 - 2 \zeta_{12}^{2} ) q^{42} + 6 \zeta_{12}^{2} q^{43} -2 \zeta_{12}^{3} q^{44} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{45} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{46} + 6 \zeta_{12}^{3} q^{47} -\zeta_{12}^{2} q^{48} + ( -3 + 3 \zeta_{12}^{2} ) q^{49} + 4 \zeta_{12} q^{50} -7 q^{51} -9 q^{53} -\zeta_{12} q^{54} + ( -2 + 2 \zeta_{12}^{2} ) q^{55} -6 \zeta_{12}^{2} q^{56} -6 \zeta_{12}^{3} q^{57} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{58} -\zeta_{12}^{3} q^{60} -\zeta_{12}^{2} q^{61} + ( 4 - 4 \zeta_{12}^{2} ) q^{62} -2 \zeta_{12} q^{63} -7 q^{64} + 2 q^{66} -2 \zeta_{12} q^{67} + ( -7 + 7 \zeta_{12}^{2} ) q^{68} + 6 \zeta_{12}^{2} q^{69} + 2 \zeta_{12}^{3} q^{70} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{71} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{72} + 11 \zeta_{12}^{3} q^{73} -\zeta_{12}^{2} q^{74} + ( 4 - 4 \zeta_{12}^{2} ) q^{75} -6 \zeta_{12} q^{76} + 4 q^{77} -4 q^{79} + \zeta_{12} q^{80} + ( -1 + \zeta_{12}^{2} ) q^{81} + 9 \zeta_{12}^{2} q^{82} + 14 \zeta_{12}^{3} q^{83} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{84} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{85} + 6 \zeta_{12}^{3} q^{86} -\zeta_{12}^{2} q^{87} + ( 6 - 6 \zeta_{12}^{2} ) q^{88} + 14 \zeta_{12} q^{89} + q^{90} + 6 q^{92} -4 \zeta_{12} q^{93} + ( -6 + 6 \zeta_{12}^{2} ) q^{94} + 6 \zeta_{12}^{2} q^{95} + 5 \zeta_{12}^{3} q^{96} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{97} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{98} -2 \zeta_{12}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 2q^{4} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - 2q^{4} - 2q^{9} - 2q^{10} - 4q^{12} + 8q^{14} + 2q^{16} - 14q^{17} + 4q^{22} - 12q^{23} + 16q^{25} - 4q^{27} + 2q^{29} + 2q^{30} + 4q^{35} - 2q^{36} + 24q^{38} + 12q^{40} + 4q^{42} + 12q^{43} - 2q^{48} - 6q^{49} - 28q^{51} - 36q^{53} - 4q^{55} - 12q^{56} - 2q^{61} + 8q^{62} - 28q^{64} + 8q^{66} - 14q^{68} + 12q^{69} - 2q^{74} + 8q^{75} + 16q^{77} - 16q^{79} - 2q^{81} + 18q^{82} - 2q^{87} + 12q^{88} + 4q^{90} + 24q^{92} - 12q^{94} + 12q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(\zeta_{12}^{2}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i 0.500000 0.866025i −0.500000 0.866025i 1.00000i −0.866025 + 0.500000i −1.73205 + 1.00000i 3.00000i −0.500000 0.866025i −0.500000 + 0.866025i
316.2 0.866025 + 0.500000i 0.500000 0.866025i −0.500000 0.866025i 1.00000i 0.866025 0.500000i 1.73205 1.00000i 3.00000i −0.500000 0.866025i −0.500000 + 0.866025i
361.1 −0.866025 + 0.500000i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000i −0.866025 0.500000i −1.73205 1.00000i 3.00000i −0.500000 + 0.866025i −0.500000 0.866025i
361.2 0.866025 0.500000i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000i 0.866025 + 0.500000i 1.73205 + 1.00000i 3.00000i −0.500000 + 0.866025i −0.500000 0.866025i
\(n\): 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.j.d 4
13.b even 2 1 inner 507.2.j.d 4
13.c even 3 1 507.2.b.b 2
13.c even 3 1 inner 507.2.j.d 4
13.d odd 4 1 39.2.e.a 2
13.d odd 4 1 507.2.e.c 2
13.e even 6 1 507.2.b.b 2
13.e even 6 1 inner 507.2.j.d 4
13.f odd 12 1 39.2.e.a 2
13.f odd 12 1 507.2.a.b 1
13.f odd 12 1 507.2.a.c 1
13.f odd 12 1 507.2.e.c 2
39.f even 4 1 117.2.g.b 2
39.h odd 6 1 1521.2.b.c 2
39.i odd 6 1 1521.2.b.c 2
39.k even 12 1 117.2.g.b 2
39.k even 12 1 1521.2.a.a 1
39.k even 12 1 1521.2.a.d 1
52.f even 4 1 624.2.q.c 2
52.l even 12 1 624.2.q.c 2
52.l even 12 1 8112.2.a.w 1
52.l even 12 1 8112.2.a.bc 1
65.f even 4 1 975.2.bb.d 4
65.g odd 4 1 975.2.i.f 2
65.k even 4 1 975.2.bb.d 4
65.o even 12 1 975.2.bb.d 4
65.s odd 12 1 975.2.i.f 2
65.t even 12 1 975.2.bb.d 4
156.l odd 4 1 1872.2.t.j 2
156.v odd 12 1 1872.2.t.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 13.d odd 4 1
39.2.e.a 2 13.f odd 12 1
117.2.g.b 2 39.f even 4 1
117.2.g.b 2 39.k even 12 1
507.2.a.b 1 13.f odd 12 1
507.2.a.c 1 13.f odd 12 1
507.2.b.b 2 13.c even 3 1
507.2.b.b 2 13.e even 6 1
507.2.e.c 2 13.d odd 4 1
507.2.e.c 2 13.f odd 12 1
507.2.j.d 4 1.a even 1 1 trivial
507.2.j.d 4 13.b even 2 1 inner
507.2.j.d 4 13.c even 3 1 inner
507.2.j.d 4 13.e even 6 1 inner
624.2.q.c 2 52.f even 4 1
624.2.q.c 2 52.l even 12 1
975.2.i.f 2 65.g odd 4 1
975.2.i.f 2 65.s odd 12 1
975.2.bb.d 4 65.f even 4 1
975.2.bb.d 4 65.k even 4 1
975.2.bb.d 4 65.o even 12 1
975.2.bb.d 4 65.t even 12 1
1521.2.a.a 1 39.k even 12 1
1521.2.a.d 1 39.k even 12 1
1521.2.b.c 2 39.h odd 6 1
1521.2.b.c 2 39.i odd 6 1
1872.2.t.j 2 156.l odd 4 1
1872.2.t.j 2 156.v odd 12 1
8112.2.a.w 1 52.l even 12 1
8112.2.a.bc 1 52.l even 12 1

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{2} + 1 \)
\( T_{5}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( 16 - 4 T^{2} + T^{4} \)
$11$ \( 16 - 4 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 49 + 7 T + T^{2} )^{2} \)
$19$ \( 1296 - 36 T^{2} + T^{4} \)
$23$ \( ( 36 + 6 T + T^{2} )^{2} \)
$29$ \( ( 1 - T + T^{2} )^{2} \)
$31$ \( ( 16 + T^{2} )^{2} \)
$37$ \( 1 - T^{2} + T^{4} \)
$41$ \( 6561 - 81 T^{2} + T^{4} \)
$43$ \( ( 36 - 6 T + T^{2} )^{2} \)
$47$ \( ( 36 + T^{2} )^{2} \)
$53$ \( ( 9 + T )^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 1 + T + T^{2} )^{2} \)
$67$ \( 16 - 4 T^{2} + T^{4} \)
$71$ \( 1296 - 36 T^{2} + T^{4} \)
$73$ \( ( 121 + T^{2} )^{2} \)
$79$ \( ( 4 + T )^{4} \)
$83$ \( ( 196 + T^{2} )^{2} \)
$89$ \( 38416 - 196 T^{2} + T^{4} \)
$97$ \( 16 - 4 T^{2} + T^{4} \)
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