Properties

Label 507.2.j.f
Level $507$
Weight $2$
Character orbit 507.j
Analytic conductor $4.048$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{2} -\zeta_{24}^{4} q^{3} + ( 1 - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{4} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{5} + ( \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{6} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{8} + ( -1 + \zeta_{24}^{4} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{2} -\zeta_{24}^{4} q^{3} + ( 1 - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{4} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{5} + ( \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{6} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{8} + ( -1 + \zeta_{24}^{4} ) q^{9} + ( 2 \zeta_{24} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{7} ) q^{10} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{11} + ( -1 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{12} + ( 4 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{14} + ( -2 \zeta_{24} + 2 \zeta_{24}^{7} ) q^{15} -3 \zeta_{24}^{4} q^{16} + ( 2 + 4 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{17} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{18} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{19} + ( 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{20} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{21} + ( -2 + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{22} -4 \zeta_{24}^{4} q^{23} + ( -\zeta_{24} + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{24} -3 q^{25} + q^{27} + ( 2 \zeta_{24} - 8 \zeta_{24}^{2} + 8 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{28} -2 \zeta_{24}^{4} q^{29} + ( -4 + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{30} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{31} + ( -3 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{32} -2 \zeta_{24}^{2} q^{33} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{34} + ( 8 - 8 \zeta_{24}^{4} ) q^{35} + ( -2 \zeta_{24} + \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{36} + ( 4 \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{37} + ( 4 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{38} + ( -4 + 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{40} + ( -2 \zeta_{24} + 8 \zeta_{24}^{2} - 8 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{41} + ( 2 \zeta_{24} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{7} ) q^{42} + ( 4 - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{43} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{44} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{45} + ( 4 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{46} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{47} + ( -3 + 3 \zeta_{24}^{4} ) q^{48} + \zeta_{24}^{4} q^{49} + ( -3 \zeta_{24} + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{50} + ( -2 - 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{51} -2 q^{53} + ( \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{54} + ( -4 \zeta_{24} - 4 \zeta_{24}^{7} ) q^{55} + ( 4 - 6 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{56} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{57} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{58} + ( 2 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{59} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 8 \zeta_{24}^{6} ) q^{60} + ( -2 - 8 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 8 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{61} + ( -6 \zeta_{24} + 8 \zeta_{24}^{4} - 6 \zeta_{24}^{7} ) q^{62} + ( -2 \zeta_{24} + 2 \zeta_{24}^{7} ) q^{63} + ( 7 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{64} + ( 2 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{66} + ( 2 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{67} + 14 \zeta_{24}^{4} q^{68} + ( -4 + 4 \zeta_{24}^{4} ) q^{69} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 8 \zeta_{24}^{6} ) q^{70} -2 \zeta_{24}^{2} q^{71} + ( -3 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{72} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{73} + ( 6 - 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{74} + 3 \zeta_{24}^{4} q^{75} + ( 2 \zeta_{24} - 8 \zeta_{24}^{2} + 8 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{76} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{77} + ( -8 \zeta_{24} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} ) q^{79} + ( -6 \zeta_{24} + 6 \zeta_{24}^{7} ) q^{80} -\zeta_{24}^{4} q^{81} + ( -12 + 10 \zeta_{24}^{3} + 12 \zeta_{24}^{4} - 10 \zeta_{24}^{5} - 10 \zeta_{24}^{7} ) q^{82} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{83} + ( 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{84} + ( -16 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{85} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 12 \zeta_{24}^{6} ) q^{86} + ( -2 + 2 \zeta_{24}^{4} ) q^{87} + ( -2 \zeta_{24} + 6 \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{88} + ( -2 \zeta_{24} - 12 \zeta_{24}^{2} + 12 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{89} + ( 4 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{90} + ( -4 + 8 \zeta_{24} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} ) q^{92} + ( 2 \zeta_{24} - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{93} + ( 2 \zeta_{24} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{7} ) q^{94} + ( 8 - 8 \zeta_{24}^{4} ) q^{95} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{96} + ( 2 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{97} + ( -\zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{98} + 2 \zeta_{24}^{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} + 4q^{4} - 4q^{9} + O(q^{10}) \) \( 8q - 4q^{3} + 4q^{4} - 4q^{9} - 16q^{10} - 8q^{12} + 32q^{14} - 12q^{16} + 8q^{17} - 8q^{22} - 16q^{23} - 24q^{25} + 8q^{27} - 8q^{29} - 16q^{30} + 32q^{35} + 4q^{36} + 32q^{38} - 32q^{40} - 16q^{42} + 16q^{43} - 12q^{48} + 4q^{49} - 16q^{51} - 16q^{53} + 16q^{56} - 8q^{61} + 32q^{62} + 56q^{64} + 16q^{66} + 56q^{68} - 16q^{69} + 24q^{74} + 12q^{75} - 4q^{81} - 48q^{82} - 8q^{87} + 24q^{88} + 32q^{90} - 32q^{92} + 8q^{94} + 32q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(1 - \zeta_{24}\)

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
−0.965926 + 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 0.965926i
−2.09077 1.20711i −0.500000 + 0.866025i 1.91421 + 3.31552i 2.82843i 2.09077 1.20711i −2.44949 + 1.41421i 4.41421i −0.500000 0.866025i −3.41421 + 5.91359i
316.2 −0.358719 0.207107i −0.500000 + 0.866025i −0.914214 1.58346i 2.82843i 0.358719 0.207107i −2.44949 + 1.41421i 1.58579i −0.500000 0.866025i −0.585786 + 1.01461i
316.3 0.358719 + 0.207107i −0.500000 + 0.866025i −0.914214 1.58346i 2.82843i −0.358719 + 0.207107i 2.44949 1.41421i 1.58579i −0.500000 0.866025i −0.585786 + 1.01461i
316.4 2.09077 + 1.20711i −0.500000 + 0.866025i 1.91421 + 3.31552i 2.82843i −2.09077 + 1.20711i 2.44949 1.41421i 4.41421i −0.500000 0.866025i −3.41421 + 5.91359i
361.1 −2.09077 + 1.20711i −0.500000 0.866025i 1.91421 3.31552i 2.82843i 2.09077 + 1.20711i −2.44949 1.41421i 4.41421i −0.500000 + 0.866025i −3.41421 5.91359i
361.2 −0.358719 + 0.207107i −0.500000 0.866025i −0.914214 + 1.58346i 2.82843i 0.358719 + 0.207107i −2.44949 1.41421i 1.58579i −0.500000 + 0.866025i −0.585786 1.01461i
361.3 0.358719 0.207107i −0.500000 0.866025i −0.914214 + 1.58346i 2.82843i −0.358719 0.207107i 2.44949 + 1.41421i 1.58579i −0.500000 + 0.866025i −0.585786 1.01461i
361.4 2.09077 1.20711i −0.500000 0.866025i 1.91421 3.31552i 2.82843i −2.09077 1.20711i 2.44949 + 1.41421i 4.41421i −0.500000 + 0.866025i −3.41421 5.91359i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.4
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.f 8
13.b even 2 1 inner 507.2.j.f 8
13.c even 3 1 507.2.b.e 4
13.c even 3 1 inner 507.2.j.f 8
13.d odd 4 1 507.2.e.d 4
13.d odd 4 1 507.2.e.h 4
13.e even 6 1 507.2.b.e 4
13.e even 6 1 inner 507.2.j.f 8
13.f odd 12 1 39.2.a.b 2
13.f odd 12 1 507.2.a.h 2
13.f odd 12 1 507.2.e.d 4
13.f odd 12 1 507.2.e.h 4
39.h odd 6 1 1521.2.b.j 4
39.i odd 6 1 1521.2.b.j 4
39.k even 12 1 117.2.a.c 2
39.k even 12 1 1521.2.a.f 2
52.l even 12 1 624.2.a.k 2
52.l even 12 1 8112.2.a.bm 2
65.o even 12 1 975.2.c.h 4
65.s odd 12 1 975.2.a.l 2
65.t even 12 1 975.2.c.h 4
91.bc even 12 1 1911.2.a.h 2
104.u even 12 1 2496.2.a.bi 2
104.x odd 12 1 2496.2.a.bf 2
117.w odd 12 1 1053.2.e.m 4
117.x even 12 1 1053.2.e.e 4
117.bb odd 12 1 1053.2.e.m 4
117.bc even 12 1 1053.2.e.e 4
143.o even 12 1 4719.2.a.p 2
156.v odd 12 1 1872.2.a.w 2
195.bc odd 12 1 2925.2.c.u 4
195.bh even 12 1 2925.2.a.v 2
195.bn odd 12 1 2925.2.c.u 4
273.ca odd 12 1 5733.2.a.u 2
312.bo even 12 1 7488.2.a.cl 2
312.bq odd 12 1 7488.2.a.co 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 13.f odd 12 1
117.2.a.c 2 39.k even 12 1
507.2.a.h 2 13.f odd 12 1
507.2.b.e 4 13.c even 3 1
507.2.b.e 4 13.e even 6 1
507.2.e.d 4 13.d odd 4 1
507.2.e.d 4 13.f odd 12 1
507.2.e.h 4 13.d odd 4 1
507.2.e.h 4 13.f odd 12 1
507.2.j.f 8 1.a even 1 1 trivial
507.2.j.f 8 13.b even 2 1 inner
507.2.j.f 8 13.c even 3 1 inner
507.2.j.f 8 13.e even 6 1 inner
624.2.a.k 2 52.l even 12 1
975.2.a.l 2 65.s odd 12 1
975.2.c.h 4 65.o even 12 1
975.2.c.h 4 65.t even 12 1
1053.2.e.e 4 117.x even 12 1
1053.2.e.e 4 117.bc even 12 1
1053.2.e.m 4 117.w odd 12 1
1053.2.e.m 4 117.bb odd 12 1
1521.2.a.f 2 39.k even 12 1
1521.2.b.j 4 39.h odd 6 1
1521.2.b.j 4 39.i odd 6 1
1872.2.a.w 2 156.v odd 12 1
1911.2.a.h 2 91.bc even 12 1
2496.2.a.bf 2 104.x odd 12 1
2496.2.a.bi 2 104.u even 12 1
2925.2.a.v 2 195.bh even 12 1
2925.2.c.u 4 195.bc odd 12 1
2925.2.c.u 4 195.bn odd 12 1
4719.2.a.p 2 143.o even 12 1
5733.2.a.u 2 273.ca odd 12 1
7488.2.a.cl 2 312.bo even 12 1
7488.2.a.co 2 312.bq odd 12 1
8112.2.a.bm 2 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 \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{8} - 6 T_{2}^{6} + 35 T_{2}^{4} - 6 T_{2}^{2} + 1 \)
\( T_{5}^{2} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T^{2} + 35 T^{4} - 6 T^{6} + T^{8} \)
$3$ \( ( 1 + T + T^{2} )^{4} \)
$5$ \( ( 8 + T^{2} )^{4} \)
$7$ \( ( 64 - 8 T^{2} + T^{4} )^{2} \)
$11$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$13$ \( T^{8} \)
$17$ \( ( 784 + 112 T + 44 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$19$ \( ( 64 - 8 T^{2} + T^{4} )^{2} \)
$23$ \( ( 16 + 4 T + T^{2} )^{4} \)
$29$ \( ( 4 + 2 T + T^{2} )^{4} \)
$31$ \( ( 64 + 48 T^{2} + T^{4} )^{2} \)
$37$ \( 614656 - 56448 T^{2} + 4400 T^{4} - 72 T^{6} + T^{8} \)
$41$ \( 9834496 - 451584 T^{2} + 17600 T^{4} - 144 T^{6} + T^{8} \)
$43$ \( ( 256 + 128 T + 80 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$47$ \( ( 16 + 136 T^{2} + T^{4} )^{2} \)
$53$ \( ( 2 + T )^{8} \)
$59$ \( 614656 - 56448 T^{2} + 4400 T^{4} - 72 T^{6} + T^{8} \)
$61$ \( ( 15376 - 496 T + 140 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$67$ \( 4096 - 3072 T^{2} + 2240 T^{4} - 48 T^{6} + T^{8} \)
$71$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$73$ \( ( 16 + 136 T^{2} + T^{4} )^{2} \)
$79$ \( ( -128 + T^{2} )^{4} \)
$83$ \( ( 784 + 72 T^{2} + T^{4} )^{2} \)
$89$ \( 342102016 - 5622784 T^{2} + 73920 T^{4} - 304 T^{6} + T^{8} \)
$97$ \( 614656 - 56448 T^{2} + 4400 T^{4} - 72 T^{6} + T^{8} \)
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