Properties

Label 507.2.j.g
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: 8.0.1731891456.1
Defining polynomial: \(x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{3} + ( 2 + 3 \beta_{2} - \beta_{4} + \beta_{5} ) q^{4} + ( -\beta_{6} - 2 \beta_{7} ) q^{5} + ( -\beta_{1} + \beta_{6} ) q^{6} + ( -\beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{7} + ( \beta_{6} - 4 \beta_{7} ) q^{8} + ( -1 - \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{3} + ( 2 + 3 \beta_{2} - \beta_{4} + \beta_{5} ) q^{4} + ( -\beta_{6} - 2 \beta_{7} ) q^{5} + ( -\beta_{1} + \beta_{6} ) q^{6} + ( -\beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{7} + ( \beta_{6} - 4 \beta_{7} ) q^{8} + ( -1 - \beta_{2} ) q^{9} + ( -3 \beta_{2} + \beta_{5} ) q^{10} + 2 \beta_{3} q^{11} + ( -2 + \beta_{4} ) q^{12} + ( -4 + 2 \beta_{4} ) q^{14} + ( \beta_{1} - 2 \beta_{3} ) q^{15} + ( 3 \beta_{2} + 3 \beta_{5} ) q^{16} + ( \beta_{2} - \beta_{4} + \beta_{5} ) q^{17} -\beta_{6} q^{18} + ( 2 \beta_{1} - 4 \beta_{3} - 2 \beta_{6} - 4 \beta_{7} ) q^{19} + ( \beta_{1} - \beta_{6} ) q^{20} + ( -\beta_{6} + \beta_{7} ) q^{21} + ( 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{22} -2 \beta_{2} q^{23} + ( -\beta_{1} - 4 \beta_{3} ) q^{24} + ( -3 - 3 \beta_{4} ) q^{25} + q^{27} + ( -4 \beta_{1} - 6 \beta_{3} ) q^{28} + ( -\beta_{2} - 3 \beta_{5} ) q^{29} + ( 4 + 3 \beta_{2} + \beta_{4} - \beta_{5} ) q^{30} + ( \beta_{6} + \beta_{7} ) q^{31} + ( -\beta_{1} - 4 \beta_{3} + \beta_{6} - 4 \beta_{7} ) q^{32} + ( -2 \beta_{3} - 2 \beta_{7} ) q^{33} + ( \beta_{6} - 4 \beta_{7} ) q^{34} + ( 2 + 2 \beta_{2} ) q^{35} + ( -3 \beta_{2} - \beta_{5} ) q^{36} + ( \beta_{1} - 6 \beta_{3} ) q^{37} + ( 8 + 2 \beta_{4} ) q^{38} + ( -4 - 3 \beta_{4} ) q^{40} + \beta_{1} q^{41} + ( -6 \beta_{2} - 2 \beta_{5} ) q^{42} + ( 3 + 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{43} + ( 2 \beta_{6} - 4 \beta_{7} ) q^{44} + ( -\beta_{1} + 2 \beta_{3} + \beta_{6} + 2 \beta_{7} ) q^{45} + ( 2 \beta_{1} - 2 \beta_{6} ) q^{46} + ( -4 \beta_{6} - 2 \beta_{7} ) q^{47} + ( -3 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} ) q^{48} + ( -\beta_{2} - 3 \beta_{5} ) q^{49} + 12 \beta_{3} q^{50} + \beta_{4} q^{51} + ( 4 - 3 \beta_{4} ) q^{53} + \beta_{1} q^{54} + ( 2 \beta_{2} - 2 \beta_{5} ) q^{55} + ( -8 - 14 \beta_{2} + 6 \beta_{4} - 6 \beta_{5} ) q^{56} + ( 2 \beta_{6} + 4 \beta_{7} ) q^{57} + ( \beta_{1} + 12 \beta_{3} - \beta_{6} + 12 \beta_{7} ) q^{58} + ( -2 \beta_{1} - 6 \beta_{3} + 2 \beta_{6} - 6 \beta_{7} ) q^{59} + \beta_{6} q^{60} + ( -7 - 9 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{61} + 4 \beta_{2} q^{62} + ( \beta_{1} + \beta_{3} ) q^{63} + ( -4 - \beta_{4} ) q^{64} + 2 \beta_{4} q^{66} + ( -\beta_{1} + 3 \beta_{3} ) q^{67} + ( 7 \beta_{2} + 3 \beta_{5} ) q^{68} + ( 2 + 2 \beta_{2} ) q^{69} + 2 \beta_{6} q^{70} + ( -14 \beta_{3} - 14 \beta_{7} ) q^{71} + ( \beta_{1} + 4 \beta_{3} - \beta_{6} + 4 \beta_{7} ) q^{72} + ( 2 \beta_{6} + 7 \beta_{7} ) q^{73} + ( 4 - \beta_{2} + 5 \beta_{4} - 5 \beta_{5} ) q^{74} + 3 \beta_{5} q^{75} + 2 \beta_{1} q^{76} + ( -2 + 2 \beta_{4} ) q^{77} + ( 7 - \beta_{4} ) q^{79} + ( -3 \beta_{1} + 12 \beta_{3} ) q^{80} + \beta_{2} q^{81} + ( 4 + 5 \beta_{2} - \beta_{4} + \beta_{5} ) q^{82} + ( 2 \beta_{6} - 4 \beta_{7} ) q^{83} + ( 4 \beta_{1} + 6 \beta_{3} - 4 \beta_{6} + 6 \beta_{7} ) q^{84} + ( -\beta_{1} + 4 \beta_{3} + \beta_{6} + 4 \beta_{7} ) q^{85} + ( 2 \beta_{6} + 4 \beta_{7} ) q^{86} + ( -2 + \beta_{2} - 3 \beta_{4} + 3 \beta_{5} ) q^{87} + ( 10 \beta_{2} + 2 \beta_{5} ) q^{88} + ( -2 \beta_{1} - 8 \beta_{3} ) q^{89} + ( -4 - \beta_{4} ) q^{90} + ( 4 - 2 \beta_{4} ) q^{92} + ( -\beta_{1} + \beta_{3} ) q^{93} + ( -18 \beta_{2} - 2 \beta_{5} ) q^{94} + ( -16 - 10 \beta_{2} - 6 \beta_{4} + 6 \beta_{5} ) q^{95} + ( -\beta_{6} + 4 \beta_{7} ) q^{96} + ( -\beta_{1} + 7 \beta_{3} + \beta_{6} + 7 \beta_{7} ) q^{97} + ( \beta_{1} + 12 \beta_{3} - \beta_{6} + 12 \beta_{7} ) q^{98} + 2 \beta_{7} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} + 10q^{4} - 4q^{9} + O(q^{10}) \) \( 8q - 4q^{3} + 10q^{4} - 4q^{9} + 14q^{10} - 20q^{12} - 40q^{14} - 6q^{16} + 2q^{17} + 4q^{22} + 8q^{23} - 12q^{25} + 8q^{27} - 2q^{29} + 14q^{30} + 8q^{35} + 10q^{36} + 56q^{38} - 20q^{40} + 20q^{42} + 10q^{43} - 6q^{48} - 2q^{49} - 4q^{51} + 44q^{53} - 12q^{55} - 44q^{56} - 32q^{61} - 16q^{62} - 28q^{64} - 8q^{66} - 22q^{68} + 8q^{69} + 6q^{74} + 6q^{75} - 24q^{77} + 60q^{79} - 4q^{81} + 18q^{82} - 2q^{87} - 36q^{88} - 28q^{90} + 40q^{92} + 68q^{94} - 52q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 9 \nu^{6} - 65 \nu^{4} + 585 \nu^{2} - 1296 \)\()/1040\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 181 \nu \)\()/260\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 116 \)\()/65\)
\(\beta_{5}\)\(=\)\((\)\( -29 \nu^{6} + 325 \nu^{4} - 1885 \nu^{2} + 4176 \)\()/1040\)
\(\beta_{6}\)\(=\)\((\)\( 9 \nu^{7} - 65 \nu^{5} + 585 \nu^{3} - 256 \nu \)\()/1040\)
\(\beta_{7}\)\(=\)\((\)\( 9 \nu^{7} - 65 \nu^{5} + 377 \nu^{3} - 256 \nu \)\()/832\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + 5 \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-4 \beta_{7} + 5 \beta_{6}\)
\(\nu^{4}\)\(=\)\(9 \beta_{5} + 29 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-36 \beta_{7} + 29 \beta_{6} - 36 \beta_{3} - 29 \beta_{1}\)
\(\nu^{6}\)\(=\)\(65 \beta_{4} - 116\)
\(\nu^{7}\)\(=\)\(-260 \beta_{3} - 181 \beta_{1}\)

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(1 + \beta_{2}\)

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
−2.21837 1.28078i
−1.35234 0.780776i
1.35234 + 0.780776i
2.21837 + 1.28078i
−2.21837 + 1.28078i
−1.35234 + 0.780776i
1.35234 0.780776i
2.21837 1.28078i
−2.21837 1.28078i −0.500000 + 0.866025i 2.28078 + 3.95042i 0.561553i 2.21837 1.28078i 3.08440 1.78078i 6.56155i −0.500000 0.866025i 0.719224 1.24573i
316.2 −1.35234 0.780776i −0.500000 + 0.866025i 0.219224 + 0.379706i 3.56155i 1.35234 0.780776i 0.486319 0.280776i 2.43845i −0.500000 0.866025i 2.78078 4.81645i
316.3 1.35234 + 0.780776i −0.500000 + 0.866025i 0.219224 + 0.379706i 3.56155i −1.35234 + 0.780776i −0.486319 + 0.280776i 2.43845i −0.500000 0.866025i 2.78078 4.81645i
316.4 2.21837 + 1.28078i −0.500000 + 0.866025i 2.28078 + 3.95042i 0.561553i −2.21837 + 1.28078i −3.08440 + 1.78078i 6.56155i −0.500000 0.866025i 0.719224 1.24573i
361.1 −2.21837 + 1.28078i −0.500000 0.866025i 2.28078 3.95042i 0.561553i 2.21837 + 1.28078i 3.08440 + 1.78078i 6.56155i −0.500000 + 0.866025i 0.719224 + 1.24573i
361.2 −1.35234 + 0.780776i −0.500000 0.866025i 0.219224 0.379706i 3.56155i 1.35234 + 0.780776i 0.486319 + 0.280776i 2.43845i −0.500000 + 0.866025i 2.78078 + 4.81645i
361.3 1.35234 0.780776i −0.500000 0.866025i 0.219224 0.379706i 3.56155i −1.35234 0.780776i −0.486319 0.280776i 2.43845i −0.500000 + 0.866025i 2.78078 + 4.81645i
361.4 2.21837 1.28078i −0.500000 0.866025i 2.28078 3.95042i 0.561553i −2.21837 1.28078i −3.08440 1.78078i 6.56155i −0.500000 + 0.866025i 0.719224 + 1.24573i
\(n\): 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.g 8
13.b even 2 1 inner 507.2.j.g 8
13.c even 3 1 507.2.b.d 4
13.c even 3 1 inner 507.2.j.g 8
13.d odd 4 1 39.2.e.b 4
13.d odd 4 1 507.2.e.g 4
13.e even 6 1 507.2.b.d 4
13.e even 6 1 inner 507.2.j.g 8
13.f odd 12 1 39.2.e.b 4
13.f odd 12 1 507.2.a.d 2
13.f odd 12 1 507.2.a.g 2
13.f odd 12 1 507.2.e.g 4
39.f even 4 1 117.2.g.c 4
39.h odd 6 1 1521.2.b.h 4
39.i odd 6 1 1521.2.b.h 4
39.k even 12 1 117.2.g.c 4
39.k even 12 1 1521.2.a.g 2
39.k even 12 1 1521.2.a.m 2
52.f even 4 1 624.2.q.h 4
52.l even 12 1 624.2.q.h 4
52.l even 12 1 8112.2.a.bk 2
52.l even 12 1 8112.2.a.bo 2
65.f even 4 1 975.2.bb.i 8
65.g odd 4 1 975.2.i.k 4
65.k even 4 1 975.2.bb.i 8
65.o even 12 1 975.2.bb.i 8
65.s odd 12 1 975.2.i.k 4
65.t even 12 1 975.2.bb.i 8
156.l odd 4 1 1872.2.t.r 4
156.v odd 12 1 1872.2.t.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 13.d odd 4 1
39.2.e.b 4 13.f odd 12 1
117.2.g.c 4 39.f even 4 1
117.2.g.c 4 39.k even 12 1
507.2.a.d 2 13.f odd 12 1
507.2.a.g 2 13.f odd 12 1
507.2.b.d 4 13.c even 3 1
507.2.b.d 4 13.e even 6 1
507.2.e.g 4 13.d odd 4 1
507.2.e.g 4 13.f odd 12 1
507.2.j.g 8 1.a even 1 1 trivial
507.2.j.g 8 13.b even 2 1 inner
507.2.j.g 8 13.c even 3 1 inner
507.2.j.g 8 13.e even 6 1 inner
624.2.q.h 4 52.f even 4 1
624.2.q.h 4 52.l even 12 1
975.2.i.k 4 65.g odd 4 1
975.2.i.k 4 65.s odd 12 1
975.2.bb.i 8 65.f even 4 1
975.2.bb.i 8 65.k even 4 1
975.2.bb.i 8 65.o even 12 1
975.2.bb.i 8 65.t even 12 1
1521.2.a.g 2 39.k even 12 1
1521.2.a.m 2 39.k even 12 1
1521.2.b.h 4 39.h odd 6 1
1521.2.b.h 4 39.i odd 6 1
1872.2.t.r 4 156.l odd 4 1
1872.2.t.r 4 156.v odd 12 1
8112.2.a.bk 2 52.l even 12 1
8112.2.a.bo 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 \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{8} - 9 T_{2}^{6} + 65 T_{2}^{4} - 144 T_{2}^{2} + 256 \)
\( T_{5}^{4} + 13 T_{5}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 - 144 T^{2} + 65 T^{4} - 9 T^{6} + T^{8} \)
$3$ \( ( 1 + T + T^{2} )^{4} \)
$5$ \( ( 4 + 13 T^{2} + T^{4} )^{2} \)
$7$ \( 16 - 52 T^{2} + 165 T^{4} - 13 T^{6} + T^{8} \)
$11$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$13$ \( T^{8} \)
$17$ \( ( 16 + 4 T + 5 T^{2} - T^{3} + T^{4} )^{2} \)
$19$ \( 4096 - 3328 T^{2} + 2640 T^{4} - 52 T^{6} + T^{8} \)
$23$ \( ( 4 - 2 T + T^{2} )^{4} \)
$29$ \( ( 1444 - 38 T + 39 T^{2} + T^{3} + T^{4} )^{2} \)
$31$ \( ( 16 + 9 T^{2} + T^{4} )^{2} \)
$37$ \( 456976 - 46644 T^{2} + 4085 T^{4} - 69 T^{6} + T^{8} \)
$41$ \( 256 - 144 T^{2} + 65 T^{4} - 9 T^{6} + T^{8} \)
$43$ \( ( 4 - 10 T + 23 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$47$ \( ( 68 + T^{2} )^{4} \)
$53$ \( ( -8 - 11 T + T^{2} )^{4} \)
$59$ \( 1048576 - 135168 T^{2} + 16400 T^{4} - 132 T^{6} + T^{8} \)
$61$ \( ( 2209 + 752 T + 209 T^{2} + 16 T^{3} + T^{4} )^{2} \)
$67$ \( 16 - 84 T^{2} + 437 T^{4} - 21 T^{6} + T^{8} \)
$71$ \( ( 38416 - 196 T^{2} + T^{4} )^{2} \)
$73$ \( ( 361 + 106 T^{2} + T^{4} )^{2} \)
$79$ \( ( 52 - 15 T + T^{2} )^{4} \)
$83$ \( ( 64 + 84 T^{2} + T^{4} )^{2} \)
$89$ \( 16777216 - 802816 T^{2} + 34320 T^{4} - 196 T^{6} + T^{8} \)
$97$ \( 2085136 - 134292 T^{2} + 7205 T^{4} - 93 T^{6} + T^{8} \)
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