Properties

Label 507.2.k.j
Level $507$
Weight $2$
Character orbit 507.k
Analytic conductor $4.048$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.k (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

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

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.258819 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i −0.724745 + 1.57313i 0.866025 + 0.500000i 1.41421 + 1.41421i −1.33195 1.10721i −1.36603 + 0.366025i −2.12132 + 2.12132i −1.94949 2.28024i −1.73205 + 1.00000i
80.2 0.258819 0.965926i 1.72474 + 0.158919i 0.866025 + 0.500000i −1.41421 1.41421i 0.599900 1.62484i −1.36603 + 0.366025i 2.12132 2.12132i 2.94949 + 0.548188i −1.73205 + 1.00000i
89.1 −0.965926 0.258819i −0.724745 + 1.57313i −0.866025 0.500000i −1.41421 + 1.41421i 1.10721 1.33195i 0.366025 + 1.36603i 2.12132 + 2.12132i −1.94949 2.28024i 1.73205 1.00000i
89.2 0.965926 + 0.258819i 1.72474 + 0.158919i −0.866025 0.500000i 1.41421 1.41421i 1.62484 + 0.599900i 0.366025 + 1.36603i −2.12132 2.12132i 2.94949 + 0.548188i 1.73205 1.00000i
188.1 −0.965926 + 0.258819i −0.724745 1.57313i −0.866025 + 0.500000i −1.41421 1.41421i 1.10721 + 1.33195i 0.366025 1.36603i 2.12132 2.12132i −1.94949 + 2.28024i 1.73205 + 1.00000i
188.2 0.965926 0.258819i 1.72474 0.158919i −0.866025 + 0.500000i 1.41421 + 1.41421i 1.62484 0.599900i 0.366025 1.36603i −2.12132 + 2.12132i 2.94949 0.548188i 1.73205 + 1.00000i
488.1 −0.258819 0.965926i −0.724745 1.57313i 0.866025 0.500000i 1.41421 1.41421i −1.33195 + 1.10721i −1.36603 0.366025i −2.12132 2.12132i −1.94949 + 2.28024i −1.73205 1.00000i
488.2 0.258819 + 0.965926i 1.72474 0.158919i 0.866025 0.500000i −1.41421 + 1.41421i 0.599900 + 1.62484i −1.36603 0.366025i 2.12132 + 2.12132i 2.94949 0.548188i −1.73205 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 488.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.c even 3 1 inner
13.d odd 4 1 inner
13.f odd 12 1 inner
39.f even 4 1 inner
39.i odd 6 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.k.j 8
3.b odd 2 1 inner 507.2.k.j 8
13.b even 2 1 507.2.k.i 8
13.c even 3 1 39.2.f.a 4
13.c even 3 1 inner 507.2.k.j 8
13.d odd 4 1 507.2.k.i 8
13.d odd 4 1 inner 507.2.k.j 8
13.e even 6 1 507.2.f.a 4
13.e even 6 1 507.2.k.i 8
13.f odd 12 1 39.2.f.a 4
13.f odd 12 1 507.2.f.a 4
13.f odd 12 1 507.2.k.i 8
13.f odd 12 1 inner 507.2.k.j 8
39.d odd 2 1 507.2.k.i 8
39.f even 4 1 507.2.k.i 8
39.f even 4 1 inner 507.2.k.j 8
39.h odd 6 1 507.2.f.a 4
39.h odd 6 1 507.2.k.i 8
39.i odd 6 1 39.2.f.a 4
39.i odd 6 1 inner 507.2.k.j 8
39.k even 12 1 39.2.f.a 4
39.k even 12 1 507.2.f.a 4
39.k even 12 1 507.2.k.i 8
39.k even 12 1 inner 507.2.k.j 8
52.j odd 6 1 624.2.bf.d 4
52.l even 12 1 624.2.bf.d 4
65.n even 6 1 975.2.o.j 4
65.o even 12 1 975.2.n.c 4
65.q odd 12 1 975.2.n.c 4
65.q odd 12 1 975.2.n.d 4
65.s odd 12 1 975.2.o.j 4
65.t even 12 1 975.2.n.d 4
156.p even 6 1 624.2.bf.d 4
156.v odd 12 1 624.2.bf.d 4
195.x odd 6 1 975.2.o.j 4
195.bc odd 12 1 975.2.n.d 4
195.bh even 12 1 975.2.o.j 4
195.bl even 12 1 975.2.n.c 4
195.bl even 12 1 975.2.n.d 4
195.bn odd 12 1 975.2.n.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.f.a 4 13.c even 3 1
39.2.f.a 4 13.f odd 12 1
39.2.f.a 4 39.i odd 6 1
39.2.f.a 4 39.k even 12 1
507.2.f.a 4 13.e even 6 1
507.2.f.a 4 13.f odd 12 1
507.2.f.a 4 39.h odd 6 1
507.2.f.a 4 39.k even 12 1
507.2.k.i 8 13.b even 2 1
507.2.k.i 8 13.d odd 4 1
507.2.k.i 8 13.e even 6 1
507.2.k.i 8 13.f odd 12 1
507.2.k.i 8 39.d odd 2 1
507.2.k.i 8 39.f even 4 1
507.2.k.i 8 39.h odd 6 1
507.2.k.i 8 39.k even 12 1
507.2.k.j 8 1.a even 1 1 trivial
507.2.k.j 8 3.b odd 2 1 inner
507.2.k.j 8 13.c even 3 1 inner
507.2.k.j 8 13.d odd 4 1 inner
507.2.k.j 8 13.f odd 12 1 inner
507.2.k.j 8 39.f even 4 1 inner
507.2.k.j 8 39.i odd 6 1 inner
507.2.k.j 8 39.k even 12 1 inner
624.2.bf.d 4 52.j odd 6 1
624.2.bf.d 4 52.l even 12 1
624.2.bf.d 4 156.p even 6 1
624.2.bf.d 4 156.v odd 12 1
975.2.n.c 4 65.o even 12 1
975.2.n.c 4 65.q odd 12 1
975.2.n.c 4 195.bl even 12 1
975.2.n.c 4 195.bn odd 12 1
975.2.n.d 4 65.q odd 12 1
975.2.n.d 4 65.t even 12 1
975.2.n.d 4 195.bc odd 12 1
975.2.n.d 4 195.bl even 12 1
975.2.o.j 4 65.n even 6 1
975.2.o.j 4 65.s odd 12 1
975.2.o.j 4 195.x odd 6 1
975.2.o.j 4 195.bh even 12 1

Hecke kernels

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

\( T_{2}^{8} - T_{2}^{4} + 1 \)
\( T_{5}^{4} + 16 \)
\( T_{7}^{4} + 2 T_{7}^{3} + 2 T_{7}^{2} + 4 T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ \( ( 9 - 6 T + T^{2} - 2 T^{3} + T^{4} )^{2} \)
$5$ \( ( 16 + T^{4} )^{2} \)
$7$ \( ( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$11$ \( 65536 - 256 T^{4} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( ( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$23$ \( ( 5184 + 72 T^{2} + T^{4} )^{2} \)
$29$ \( ( 64 - 8 T^{2} + T^{4} )^{2} \)
$31$ \( ( 50 + 10 T + T^{2} )^{4} \)
$37$ \( ( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$41$ \( 256 - 16 T^{4} + T^{8} \)
$43$ \( ( 1296 - 36 T^{2} + T^{4} )^{2} \)
$47$ \( ( 256 + T^{4} )^{2} \)
$53$ \( ( 32 + T^{2} )^{4} \)
$59$ \( 65536 - 256 T^{4} + T^{8} \)
$61$ \( ( 64 + 8 T + T^{2} )^{4} \)
$67$ \( ( 2500 - 500 T + 50 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$71$ \( 65536 - 256 T^{4} + T^{8} \)
$73$ \( ( 2 - 2 T + T^{2} )^{4} \)
$79$ \( ( 10 + T )^{8} \)
$83$ \( ( 4096 + T^{4} )^{2} \)
$89$ \( 1475789056 - 38416 T^{4} + T^{8} \)
$97$ \( ( 9604 + 1372 T + 98 T^{2} + 14 T^{3} + T^{4} )^{2} \)
show more
show less