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: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\zeta_{24}^{7} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{5} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{7} + \beta_{5} + \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{3} ) / 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(1 - \beta_{1}\)

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

\( T_{2}^{8} - 6T_{2}^{6} + 35T_{2}^{4} - 6T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

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