Properties

Label 507.2.a.l
Level $507$
Weight $2$
Character orbit 507.a
Self dual yes
Analytic conductor $4.048$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.80194
0.445042
−1.24698
−2.04892 −1.00000 2.19806 3.35690 2.04892 2.24698 −0.405813 1.00000 −6.87800
1.2 2.35690 −1.00000 3.55496 3.69202 −2.35690 −0.801938 3.66487 1.00000 8.70171
1.3 2.69202 −1.00000 5.24698 −1.04892 −2.69202 0.554958 8.74094 1.00000 −2.82371
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(507))\):

\( T_{2}^{3} - 3 T_{2}^{2} - 4 T_{2} + 13 \)
\( T_{5}^{3} - 6 T_{5}^{2} + 5 T_{5} + 13 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 13 - 4 T - 3 T^{2} + T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 13 + 5 T - 6 T^{2} + T^{3} \)
$7$ \( 1 - T - 2 T^{2} + T^{3} \)
$11$ \( 41 - 8 T - 5 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 13 - 16 T + T^{2} + T^{3} \)
$19$ \( 7 + 14 T + 7 T^{2} + T^{3} \)
$23$ \( 91 - 49 T + T^{3} \)
$29$ \( -29 - 15 T + 2 T^{2} + T^{3} \)
$31$ \( -197 + 41 T + 16 T^{2} + T^{3} \)
$37$ \( -377 + 159 T - 22 T^{2} + T^{3} \)
$41$ \( 29 + 24 T - 11 T^{2} + T^{3} \)
$43$ \( 41 + 47 T + 15 T^{2} + T^{3} \)
$47$ \( -7 + 14 T - 7 T^{2} + T^{3} \)
$53$ \( -41 + 66 T + 17 T^{2} + T^{3} \)
$59$ \( -104 - 16 T + 6 T^{2} + T^{3} \)
$61$ \( -167 - 16 T + 13 T^{2} + T^{3} \)
$67$ \( -41 - 46 T - 11 T^{2} + T^{3} \)
$71$ \( 203 - 91 T + T^{3} \)
$73$ \( 923 - 135 T - 6 T^{2} + T^{3} \)
$79$ \( 27 - 18 T - 3 T^{2} + T^{3} \)
$83$ \( -43 + 41 T - 12 T^{2} + T^{3} \)
$89$ \( 113 - 100 T - T^{2} + T^{3} \)
$97$ \( -1637 - 281 T + 5 T^{2} + T^{3} \)
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