Properties

Label 507.2.b.f
Level $507$
Weight $2$
Character orbit 507.b
Analytic conductor $4.048$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{2} - q^{3} + ( -3 - \beta_{2} - \beta_{4} ) q^{4} + ( \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{5} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{6} + ( \beta_{1} - \beta_{5} ) q^{7} + ( -3 \beta_{3} - 5 \beta_{5} ) q^{8} + q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{2} - q^{3} + ( -3 - \beta_{2} - \beta_{4} ) q^{4} + ( \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{5} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{6} + ( \beta_{1} - \beta_{5} ) q^{7} + ( -3 \beta_{3} - 5 \beta_{5} ) q^{8} + q^{9} + ( 1 + 2 \beta_{2} - 4 \beta_{4} ) q^{10} + ( 3 \beta_{1} - \beta_{3} - 3 \beta_{5} ) q^{11} + ( 3 + \beta_{2} + \beta_{4} ) q^{12} + ( -3 + 2 \beta_{2} + 2 \beta_{4} ) q^{14} + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{15} + ( 6 \beta_{2} + 5 \beta_{4} ) q^{16} + ( -1 + 3 \beta_{2} + \beta_{4} ) q^{17} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{18} + ( \beta_{3} - 2 \beta_{5} ) q^{19} + ( -5 \beta_{1} + 7 \beta_{3} - \beta_{5} ) q^{20} + ( -\beta_{1} + \beta_{5} ) q^{21} + ( -7 + 7 \beta_{2} + 5 \beta_{4} ) q^{22} + ( 1 - 4 \beta_{2} + \beta_{4} ) q^{23} + ( 3 \beta_{3} + 5 \beta_{5} ) q^{24} + ( -6 + 5 \beta_{2} + 2 \beta_{4} ) q^{25} - q^{27} + ( -3 \beta_{1} + \beta_{3} + 3 \beta_{5} ) q^{28} + ( -1 - \beta_{2} + 2 \beta_{4} ) q^{29} + ( -1 - 2 \beta_{2} + 4 \beta_{4} ) q^{30} + ( 5 \beta_{1} - 3 \beta_{3} - 8 \beta_{5} ) q^{31} + ( -7 \beta_{1} + 7 \beta_{3} + 12 \beta_{5} ) q^{32} + ( -3 \beta_{1} + \beta_{3} + 3 \beta_{5} ) q^{33} + ( -6 \beta_{1} + 7 \beta_{3} + 7 \beta_{5} ) q^{34} + ( 3 - \beta_{2} - 4 \beta_{4} ) q^{35} + ( -3 - \beta_{2} - \beta_{4} ) q^{36} + ( \beta_{3} - 7 \beta_{5} ) q^{37} + ( -2 + \beta_{2} + 5 \beta_{4} ) q^{38} + ( 3 - 7 \beta_{2} + \beta_{4} ) q^{40} + ( -3 \beta_{1} + \beta_{3} + 5 \beta_{5} ) q^{41} + ( 3 - 2 \beta_{2} - 2 \beta_{4} ) q^{42} + ( 3 + 4 \beta_{2} + 2 \beta_{4} ) q^{43} + ( -10 \beta_{1} + 7 \beta_{3} + 11 \beta_{5} ) q^{44} + ( \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{45} + ( 10 \beta_{1} - 12 \beta_{3} - 5 \beta_{5} ) q^{46} + ( \beta_{3} - 2 \beta_{5} ) q^{47} + ( -6 \beta_{2} - 5 \beta_{4} ) q^{48} + ( 4 + \beta_{2} + 2 \beta_{4} ) q^{49} + ( -14 \beta_{1} + 7 \beta_{3} + 8 \beta_{5} ) q^{50} + ( 1 - 3 \beta_{2} - \beta_{4} ) q^{51} + ( -4 - \beta_{2} - 4 \beta_{4} ) q^{53} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{54} + ( 5 - 2 \beta_{2} - 10 \beta_{4} ) q^{55} + ( 1 - 3 \beta_{2} - \beta_{4} ) q^{56} + ( -\beta_{3} + 2 \beta_{5} ) q^{57} + ( 3 \beta_{1} - 6 \beta_{3} + \beta_{5} ) q^{58} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{59} + ( 5 \beta_{1} - 7 \beta_{3} + \beta_{5} ) q^{60} + ( -3 + \beta_{2} - 5 \beta_{4} ) q^{61} + ( -9 + 16 \beta_{2} + 13 \beta_{4} ) q^{62} + ( \beta_{1} - \beta_{5} ) q^{63} + ( 7 - 14 \beta_{2} - 7 \beta_{4} ) q^{64} + ( 7 - 7 \beta_{2} - 5 \beta_{4} ) q^{66} + ( 7 \beta_{1} - 4 \beta_{3} ) q^{67} + ( 2 - 14 \beta_{2} - 5 \beta_{4} ) q^{68} + ( -1 + 4 \beta_{2} - \beta_{4} ) q^{69} + ( \beta_{1} + 4 \beta_{3} - 7 \beta_{5} ) q^{70} + ( 5 \beta_{1} - 7 \beta_{3} - 4 \beta_{5} ) q^{71} + ( -3 \beta_{3} - 5 \beta_{5} ) q^{72} + ( 9 \beta_{1} - 6 \beta_{3} - 7 \beta_{5} ) q^{73} + ( -2 + 6 \beta_{2} + 15 \beta_{4} ) q^{74} + ( 6 - 5 \beta_{2} - 2 \beta_{4} ) q^{75} + ( \beta_{1} - 2 \beta_{3} + 6 \beta_{5} ) q^{76} + ( -7 + 2 \beta_{2} + 4 \beta_{4} ) q^{77} + 3 \beta_{4} q^{79} + ( 8 \beta_{1} - 5 \beta_{3} - 11 \beta_{5} ) q^{80} + q^{81} + ( 7 - 9 \beta_{2} - 9 \beta_{4} ) q^{82} + ( 2 \beta_{1} - \beta_{3} + 3 \beta_{5} ) q^{83} + ( 3 \beta_{1} - \beta_{3} - 3 \beta_{5} ) q^{84} + ( -3 \beta_{1} + \beta_{3} - 5 \beta_{5} ) q^{85} + ( -3 \beta_{1} + 13 \beta_{3} + 15 \beta_{5} ) q^{86} + ( 1 + \beta_{2} - 2 \beta_{4} ) q^{87} + ( 2 - 14 \beta_{2} - 5 \beta_{4} ) q^{88} + ( -\beta_{1} - 6 \beta_{3} - 2 \beta_{5} ) q^{89} + ( 1 + 2 \beta_{2} - 4 \beta_{4} ) q^{90} + ( -4 + 19 \beta_{2} ) q^{92} + ( -5 \beta_{1} + 3 \beta_{3} + 8 \beta_{5} ) q^{93} + ( -2 + \beta_{2} + 5 \beta_{4} ) q^{94} + ( 10 - 5 \beta_{2} - 4 \beta_{4} ) q^{95} + ( 7 \beta_{1} - 7 \beta_{3} - 12 \beta_{5} ) q^{96} + ( -2 \beta_{1} + 12 \beta_{3} + 3 \beta_{5} ) q^{97} + ( 4 \beta_{1} + 5 \beta_{3} + 10 \beta_{5} ) q^{98} + ( 3 \beta_{1} - \beta_{3} - 3 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} - 22q^{4} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} - 22q^{4} + 6q^{9} + 2q^{10} + 22q^{12} - 10q^{14} + 22q^{16} + 2q^{17} - 18q^{22} - 22q^{25} - 6q^{27} - 4q^{29} - 2q^{30} + 8q^{35} - 22q^{36} + 6q^{40} + 10q^{42} + 30q^{43} - 22q^{48} + 30q^{49} - 2q^{51} - 34q^{53} + 6q^{55} - 2q^{56} - 26q^{61} + 4q^{62} + 18q^{66} - 26q^{68} + 30q^{74} + 22q^{75} - 30q^{77} + 6q^{79} + 6q^{81} + 6q^{82} + 4q^{87} - 26q^{88} + 2q^{90} + 14q^{92} + 42q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 5 x^{4} + 6 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 3 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} + 3 \nu^{2} + 1 \)
\(\beta_{5}\)\(=\)\( \nu^{5} + 4 \nu^{3} + 3 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} - 3 \beta_{2} + 5\)
\(\nu^{5}\)\(=\)\(\beta_{5} - 4 \beta_{3} + 9 \beta_{1}\)

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
0.445042i
1.80194i
1.24698i
1.24698i
1.80194i
0.445042i
2.69202i −1.00000 −5.24698 1.04892i 2.69202i 0.554958i 8.74094i 1.00000 2.82371
337.2 2.35690i −1.00000 −3.55496 3.69202i 2.35690i 0.801938i 3.66487i 1.00000 −8.70171
337.3 2.04892i −1.00000 −2.19806 3.35690i 2.04892i 2.24698i 0.405813i 1.00000 6.87800
337.4 2.04892i −1.00000 −2.19806 3.35690i 2.04892i 2.24698i 0.405813i 1.00000 6.87800
337.5 2.35690i −1.00000 −3.55496 3.69202i 2.35690i 0.801938i 3.66487i 1.00000 −8.70171
337.6 2.69202i −1.00000 −5.24698 1.04892i 2.69202i 0.554958i 8.74094i 1.00000 2.82371
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.b.f 6
3.b odd 2 1 1521.2.b.k 6
13.b even 2 1 inner 507.2.b.f 6
13.c even 3 2 507.2.j.i 12
13.d odd 4 1 507.2.a.i 3
13.d odd 4 1 507.2.a.l yes 3
13.e even 6 2 507.2.j.i 12
13.f odd 12 2 507.2.e.i 6
13.f odd 12 2 507.2.e.l 6
39.d odd 2 1 1521.2.b.k 6
39.f even 4 1 1521.2.a.n 3
39.f even 4 1 1521.2.a.s 3
52.f even 4 1 8112.2.a.cg 3
52.f even 4 1 8112.2.a.cp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.i 3 13.d odd 4 1
507.2.a.l yes 3 13.d odd 4 1
507.2.b.f 6 1.a even 1 1 trivial
507.2.b.f 6 13.b even 2 1 inner
507.2.e.i 6 13.f odd 12 2
507.2.e.l 6 13.f odd 12 2
507.2.j.i 12 13.c even 3 2
507.2.j.i 12 13.e even 6 2
1521.2.a.n 3 39.f even 4 1
1521.2.a.s 3 39.f even 4 1
1521.2.b.k 6 3.b odd 2 1
1521.2.b.k 6 39.d odd 2 1
8112.2.a.cg 3 52.f even 4 1
8112.2.a.cp 3 52.f even 4 1

Hecke kernels

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

\( T_{2}^{6} + 17 T_{2}^{4} + 94 T_{2}^{2} + 169 \)
\( T_{5}^{6} + 26 T_{5}^{4} + 181 T_{5}^{2} + 169 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 169 + 94 T^{2} + 17 T^{4} + T^{6} \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( 169 + 181 T^{2} + 26 T^{4} + T^{6} \)
$7$ \( 1 + 5 T^{2} + 6 T^{4} + T^{6} \)
$11$ \( 1681 + 474 T^{2} + 41 T^{4} + T^{6} \)
$13$ \( T^{6} \)
$17$ \( ( -13 - 16 T - T^{2} + T^{3} )^{2} \)
$19$ \( 49 + 98 T^{2} + 21 T^{4} + T^{6} \)
$23$ \( ( -91 - 49 T + T^{3} )^{2} \)
$29$ \( ( -29 - 15 T + 2 T^{2} + T^{3} )^{2} \)
$31$ \( 38809 + 7985 T^{2} + 174 T^{4} + T^{6} \)
$37$ \( 142129 + 8693 T^{2} + 166 T^{4} + T^{6} \)
$41$ \( 841 + 1214 T^{2} + 73 T^{4} + T^{6} \)
$43$ \( ( -41 + 47 T - 15 T^{2} + T^{3} )^{2} \)
$47$ \( 49 + 98 T^{2} + 21 T^{4} + T^{6} \)
$53$ \( ( -41 + 66 T + 17 T^{2} + T^{3} )^{2} \)
$59$ \( 10816 + 1504 T^{2} + 68 T^{4} + T^{6} \)
$61$ \( ( -167 - 16 T + 13 T^{2} + T^{3} )^{2} \)
$67$ \( 1681 + 1214 T^{2} + 213 T^{4} + T^{6} \)
$71$ \( 41209 + 8281 T^{2} + 182 T^{4} + T^{6} \)
$73$ \( 851929 + 29301 T^{2} + 306 T^{4} + T^{6} \)
$79$ \( ( 27 - 18 T - 3 T^{2} + T^{3} )^{2} \)
$83$ \( 1849 + 649 T^{2} + 62 T^{4} + T^{6} \)
$89$ \( 12769 + 10226 T^{2} + 201 T^{4} + T^{6} \)
$97$ \( 2679769 + 95331 T^{2} + 587 T^{4} + T^{6} \)
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