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: \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{5} - \beta_{4} + 3 \beta_{2} + 2) q^{4} + ( - 2 \beta_{7} - \beta_{6}) q^{5} + (\beta_{6} - \beta_1) q^{6} + ( - \beta_{7} + \beta_{6} - \beta_{3} - \beta_1) q^{7} + ( - 4 \beta_{7} + \beta_{6}) q^{8} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{5} - \beta_{4} + 3 \beta_{2} + 2) q^{4} + ( - 2 \beta_{7} - \beta_{6}) q^{5} + (\beta_{6} - \beta_1) q^{6} + ( - \beta_{7} + \beta_{6} - \beta_{3} - \beta_1) q^{7} + ( - 4 \beta_{7} + \beta_{6}) q^{8} + ( - \beta_{2} - 1) q^{9} + (\beta_{5} - 3 \beta_{2}) q^{10} + 2 \beta_{3} q^{11} + (\beta_{4} - 2) q^{12} + (2 \beta_{4} - 4) q^{14} + ( - 2 \beta_{3} + \beta_1) q^{15} + (3 \beta_{5} + 3 \beta_{2}) q^{16} + (\beta_{5} - \beta_{4} + \beta_{2}) q^{17} - \beta_{6} q^{18} + ( - 4 \beta_{7} - 2 \beta_{6} - 4 \beta_{3} + 2 \beta_1) q^{19} + ( - \beta_{6} + \beta_1) q^{20} + (\beta_{7} - \beta_{6}) q^{21} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{22} - 2 \beta_{2} q^{23} + ( - 4 \beta_{3} - \beta_1) q^{24} + ( - 3 \beta_{4} - 3) q^{25} + q^{27} + ( - 6 \beta_{3} - 4 \beta_1) q^{28} + ( - 3 \beta_{5} - \beta_{2}) q^{29} + ( - \beta_{5} + \beta_{4} + 3 \beta_{2} + 4) q^{30} + (\beta_{7} + \beta_{6}) q^{31} + ( - 4 \beta_{7} + \beta_{6} - 4 \beta_{3} - \beta_1) q^{32} + ( - 2 \beta_{7} - 2 \beta_{3}) q^{33} + ( - 4 \beta_{7} + \beta_{6}) q^{34} + (2 \beta_{2} + 2) q^{35} + ( - \beta_{5} - 3 \beta_{2}) q^{36} + ( - 6 \beta_{3} + \beta_1) q^{37} + (2 \beta_{4} + 8) q^{38} + ( - 3 \beta_{4} - 4) q^{40} + \beta_1 q^{41} + ( - 2 \beta_{5} - 6 \beta_{2}) q^{42} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} + 3) q^{43} + ( - 4 \beta_{7} + 2 \beta_{6}) q^{44} + (2 \beta_{7} + \beta_{6} + 2 \beta_{3} - \beta_1) q^{45} + ( - 2 \beta_{6} + 2 \beta_1) q^{46} + ( - 2 \beta_{7} - 4 \beta_{6}) q^{47} + ( - 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{2}) q^{48} + ( - 3 \beta_{5} - \beta_{2}) q^{49} + 12 \beta_{3} q^{50} + \beta_{4} q^{51} + ( - 3 \beta_{4} + 4) q^{53} + \beta_1 q^{54} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{55} + ( - 6 \beta_{5} + 6 \beta_{4} - 14 \beta_{2} - 8) q^{56} + (4 \beta_{7} + 2 \beta_{6}) q^{57} + (12 \beta_{7} - \beta_{6} + 12 \beta_{3} + \beta_1) q^{58} + ( - 6 \beta_{7} + 2 \beta_{6} - 6 \beta_{3} - 2 \beta_1) q^{59} + \beta_{6} q^{60} + ( - 2 \beta_{5} + 2 \beta_{4} - 9 \beta_{2} - 7) q^{61} + 4 \beta_{2} q^{62} + (\beta_{3} + \beta_1) q^{63} + ( - \beta_{4} - 4) q^{64} + 2 \beta_{4} q^{66} + (3 \beta_{3} - \beta_1) q^{67} + (3 \beta_{5} + 7 \beta_{2}) q^{68} + (2 \beta_{2} + 2) q^{69} + 2 \beta_{6} q^{70} + ( - 14 \beta_{7} - 14 \beta_{3}) q^{71} + (4 \beta_{7} - \beta_{6} + 4 \beta_{3} + \beta_1) q^{72} + (7 \beta_{7} + 2 \beta_{6}) q^{73} + ( - 5 \beta_{5} + 5 \beta_{4} - \beta_{2} + 4) q^{74} + 3 \beta_{5} q^{75} + 2 \beta_1 q^{76} + (2 \beta_{4} - 2) q^{77} + ( - \beta_{4} + 7) q^{79} + (12 \beta_{3} - 3 \beta_1) q^{80} + \beta_{2} q^{81} + (\beta_{5} - \beta_{4} + 5 \beta_{2} + 4) q^{82} + ( - 4 \beta_{7} + 2 \beta_{6}) q^{83} + (6 \beta_{7} - 4 \beta_{6} + 6 \beta_{3} + 4 \beta_1) q^{84} + (4 \beta_{7} + \beta_{6} + 4 \beta_{3} - \beta_1) q^{85} + (4 \beta_{7} + 2 \beta_{6}) q^{86} + (3 \beta_{5} - 3 \beta_{4} + \beta_{2} - 2) q^{87} + (2 \beta_{5} + 10 \beta_{2}) q^{88} + ( - 8 \beta_{3} - 2 \beta_1) q^{89} + ( - \beta_{4} - 4) q^{90} + ( - 2 \beta_{4} + 4) q^{92} + (\beta_{3} - \beta_1) q^{93} + ( - 2 \beta_{5} - 18 \beta_{2}) q^{94} + (6 \beta_{5} - 6 \beta_{4} - 10 \beta_{2} - 16) q^{95} + (4 \beta_{7} - \beta_{6}) q^{96} + (7 \beta_{7} + \beta_{6} + 7 \beta_{3} - \beta_1) q^{97} + (12 \beta_{7} - \beta_{6} + 12 \beta_{3} + \beta_1) q^{98} + 2 \beta_{7} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 10 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 10 q^{4} - 4 q^{9} + 14 q^{10} - 20 q^{12} - 40 q^{14} - 6 q^{16} + 2 q^{17} + 4 q^{22} + 8 q^{23} - 12 q^{25} + 8 q^{27} - 2 q^{29} + 14 q^{30} + 8 q^{35} + 10 q^{36} + 56 q^{38} - 20 q^{40} + 20 q^{42} + 10 q^{43} - 6 q^{48} - 2 q^{49} - 4 q^{51} + 44 q^{53} - 12 q^{55} - 44 q^{56} - 32 q^{61} - 16 q^{62} - 28 q^{64} - 8 q^{66} - 22 q^{68} + 8 q^{69} + 6 q^{74} + 6 q^{75} - 24 q^{77} + 60 q^{79} - 4 q^{81} + 18 q^{82} - 2 q^{87} - 36 q^{88} - 28 q^{90} + 40 q^{92} + 68 q^{94} - 52 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 1296 ) / 1040 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 181\nu ) / 260 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 116 ) / 65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} - 65\nu^{5} + 585\nu^{3} - 256\nu ) / 1040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} - 65\nu^{5} + 377\nu^{3} - 256\nu ) / 832 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + 5\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{7} + 5\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{5} + 29\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -36\beta_{7} + 29\beta_{6} - 36\beta_{3} - 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 65\beta_{4} - 116 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -260\beta_{3} - 181\beta_1 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{8} - 9T_{2}^{6} + 65T_{2}^{4} - 144T_{2}^{2} + 256 \) Copy content Toggle raw display
\( T_{5}^{4} + 13T_{5}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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