Properties

Label 507.2.a.k
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.

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

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
−1.24698
0.445042
1.80194
−1.24698 1.00000 −0.445042 2.80194 −1.24698 4.80194 3.04892 1.00000 −3.49396
1.2 0.445042 1.00000 −1.80194 −0.246980 0.445042 1.75302 −1.69202 1.00000 −0.109916
1.3 1.80194 1.00000 1.24698 1.44504 1.80194 3.44504 −1.35690 1.00000 2.60388
\(n\): 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.k yes 3
3.b odd 2 1 1521.2.a.p 3
4.b odd 2 1 8112.2.a.cf 3
13.b even 2 1 507.2.a.j 3
13.c even 3 2 507.2.e.j 6
13.d odd 4 2 507.2.b.g 6
13.e even 6 2 507.2.e.k 6
13.f odd 12 4 507.2.j.h 12
39.d odd 2 1 1521.2.a.q 3
39.f even 4 2 1521.2.b.m 6
52.b odd 2 1 8112.2.a.by 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.j 3 13.b even 2 1
507.2.a.k yes 3 1.a even 1 1 trivial
507.2.b.g 6 13.d odd 4 2
507.2.e.j 6 13.c even 3 2
507.2.e.k 6 13.e even 6 2
507.2.j.h 12 13.f odd 12 4
1521.2.a.p 3 3.b odd 2 1
1521.2.a.q 3 39.d odd 2 1
1521.2.b.m 6 39.f even 4 2
8112.2.a.by 3 52.b odd 2 1
8112.2.a.cf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(507))\):

\( T_{2}^{3} - T_{2}^{2} - 2 T_{2} + 1 \)
\( T_{5}^{3} - 4 T_{5}^{2} + 3 T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T - T^{2} + T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( 1 + 3 T - 4 T^{2} + T^{3} \)
$7$ \( -29 + 31 T - 10 T^{2} + T^{3} \)
$11$ \( -43 - 30 T + T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 7 + 14 T + 7 T^{2} + T^{3} \)
$19$ \( 113 + 10 T - 11 T^{2} + T^{3} \)
$23$ \( -83 - 43 T - 2 T^{2} + T^{3} \)
$29$ \( -43 + 5 T + 8 T^{2} + T^{3} \)
$31$ \( 197 - 23 T - 8 T^{2} + T^{3} \)
$37$ \( -91 + 63 T - 14 T^{2} + T^{3} \)
$41$ \( 1 - 2 T - T^{2} + T^{3} \)
$43$ \( 29 - 25 T + 3 T^{2} + T^{3} \)
$47$ \( -911 - 120 T + 9 T^{2} + T^{3} \)
$53$ \( 29 + 40 T + 13 T^{2} + T^{3} \)
$59$ \( -56 + 14 T^{2} + T^{3} \)
$61$ \( -223 + 12 T + 13 T^{2} + T^{3} \)
$67$ \( 97 - 22 T - 5 T^{2} + T^{3} \)
$71$ \( -461 - 79 T + 6 T^{2} + T^{3} \)
$73$ \( -167 + 101 T - 18 T^{2} + T^{3} \)
$79$ \( -169 - 22 T + 9 T^{2} + T^{3} \)
$83$ \( -43 + 55 T + 16 T^{2} + T^{3} \)
$89$ \( 1 - 8 T + 5 T^{2} + T^{3} \)
$97$ \( 1189 - 169 T - 5 T^{2} + T^{3} \)
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