Properties

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

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + \beta_{2} q^{4} + ( - \beta_{2} + \beta_1 - 2) q^{5} - \beta_1 q^{6} + ( - \beta_{2} + \beta_1 - 4) q^{7} + ( - \beta_{2} + 2 \beta_1 - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + \beta_{2} q^{4} + ( - \beta_{2} + \beta_1 - 2) q^{5} - \beta_1 q^{6} + ( - \beta_{2} + \beta_1 - 4) q^{7} + ( - \beta_{2} + 2 \beta_1 - 1) q^{8} + q^{9} + (2 \beta_1 - 1) q^{10} + (4 \beta_{2} - \beta_1 + 2) q^{11} + \beta_{2} q^{12} + (4 \beta_1 - 1) q^{14} + ( - \beta_{2} + \beta_1 - 2) q^{15} + ( - 3 \beta_{2} + \beta_1 - 3) q^{16} + (\beta_{2} - 2) q^{17} - \beta_1 q^{18} + (\beta_{2} - 4 \beta_1 - 2) q^{19} - \beta_1 q^{20} + ( - \beta_{2} + \beta_1 - 4) q^{21} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{22} + ( - 5 \beta_{2} + 3 \beta_1 - 2) q^{23} + ( - \beta_{2} + 2 \beta_1 - 1) q^{24} + (2 \beta_{2} - 3 \beta_1) q^{25} + q^{27} + ( - 2 \beta_{2} - \beta_1) q^{28} + (2 \beta_{2} - 3 \beta_1 - 1) q^{29} + (2 \beta_1 - 1) q^{30} + (2 \beta_{2} + 3 \beta_1 - 3) q^{31} + (4 \beta_{2} - \beta_1 + 3) q^{32} + (4 \beta_{2} - \beta_1 + 2) q^{33} + ( - \beta_{2} + 2 \beta_1 - 1) q^{34} + (4 \beta_{2} - 5 \beta_1 + 9) q^{35} + \beta_{2} q^{36} + ( - \beta_{2} - 5) q^{37} + (3 \beta_{2} + 2 \beta_1 + 7) q^{38} + (\beta_{2} - 4 \beta_1 + 4) q^{40} - \beta_1 q^{41} + (4 \beta_1 - 1) q^{42} + ( - 2 \beta_{2} - 2 \beta_1 - 1) q^{43} + ( - 3 \beta_{2} + 4 \beta_1 + 3) q^{44} + ( - \beta_{2} + \beta_1 - 2) q^{45} + (2 \beta_{2} + 2 \beta_1 - 1) q^{46} + ( - 3 \beta_{2} - 6 \beta_1 + 4) q^{47} + ( - 3 \beta_{2} + \beta_1 - 3) q^{48} + (6 \beta_{2} - 7 \beta_1 + 10) q^{49} + (\beta_{2} + 4) q^{50} + (\beta_{2} - 2) q^{51} + ( - 2 \beta_{2} + 3 \beta_1 - 6) q^{53} - \beta_1 q^{54} + ( - 2 \beta_{2} - 5) q^{55} + (3 \beta_{2} - 8 \beta_1 + 6) q^{56} + (\beta_{2} - 4 \beta_1 - 2) q^{57} + (\beta_{2} + \beta_1 + 4) q^{58} + ( - 6 \beta_{2} + 2 \beta_1 + 2) q^{59} - \beta_1 q^{60} + (3 \beta_{2} + 2 \beta_1 - 4) q^{61} + ( - 5 \beta_{2} + 3 \beta_1 - 8) q^{62} + ( - \beta_{2} + \beta_1 - 4) q^{63} + (3 \beta_{2} - 5 \beta_1 + 4) q^{64} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{66} + (3 \beta_{2} + \beta_1 - 1) q^{67} + ( - 3 \beta_{2} + \beta_1 + 1) q^{68} + ( - 5 \beta_{2} + 3 \beta_1 - 2) q^{69} + (\beta_{2} - 9 \beta_1 + 6) q^{70} + ( - 2 \beta_{2} + 7 \beta_1 - 1) q^{71} + ( - \beta_{2} + 2 \beta_1 - 1) q^{72} + (\beta_{2} + \beta_1 - 6) q^{73} + (\beta_{2} + 5 \beta_1 + 1) q^{74} + (2 \beta_{2} - 3 \beta_1) q^{75} + ( - 7 \beta_{2} + \beta_1 - 3) q^{76} + ( - 10 \beta_{2} + 2 \beta_1 - 9) q^{77} + ( - 5 \beta_{2} + \beta_1 - 5) q^{79} + (3 \beta_{2} - 2 \beta_1 + 7) q^{80} + q^{81} + (\beta_{2} + 2) q^{82} + ( - 3 \beta_{2} + 4 \beta_1 + 3) q^{83} + ( - 2 \beta_{2} - \beta_1) q^{84} + (2 \beta_{2} - 3 \beta_1 + 4) q^{85} + (4 \beta_{2} + \beta_1 + 6) q^{86} + (2 \beta_{2} - 3 \beta_1 - 1) q^{87} + (5 \beta_{2} + \beta_1 - 1) q^{88} + (\beta_{2} - 3 \beta_1 + 3) q^{89} + (2 \beta_1 - 1) q^{90} + (6 \beta_{2} - 5 \beta_1 - 2) q^{92} + (2 \beta_{2} + 3 \beta_1 - 3) q^{93} + (9 \beta_{2} - 4 \beta_1 + 15) q^{94} + (2 \beta_{2} + 5 \beta_1) q^{95} + (4 \beta_{2} - \beta_1 + 3) q^{96} + ( - 10 \beta_{2} + 6 \beta_1 - 7) q^{97} + (\beta_{2} - 10 \beta_1 + 8) q^{98} + (4 \beta_{2} - \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{3} - q^{4} - 4 q^{5} - q^{6} - 10 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 3 q^{3} - q^{4} - 4 q^{5} - q^{6} - 10 q^{7} + 3 q^{9} - q^{10} + q^{11} - q^{12} + q^{14} - 4 q^{15} - 5 q^{16} - 7 q^{17} - q^{18} - 11 q^{19} - q^{20} - 10 q^{21} - 5 q^{22} + 2 q^{23} - 5 q^{25} + 3 q^{27} + q^{28} - 8 q^{29} - q^{30} - 8 q^{31} + 4 q^{32} + q^{33} + 18 q^{35} - q^{36} - 14 q^{37} + 20 q^{38} + 7 q^{40} - q^{41} + q^{42} - 3 q^{43} + 16 q^{44} - 4 q^{45} - 3 q^{46} + 9 q^{47} - 5 q^{48} + 17 q^{49} + 11 q^{50} - 7 q^{51} - 13 q^{53} - q^{54} - 13 q^{55} + 7 q^{56} - 11 q^{57} + 12 q^{58} + 14 q^{59} - q^{60} - 13 q^{61} - 16 q^{62} - 10 q^{63} + 4 q^{64} - 5 q^{66} - 5 q^{67} + 7 q^{68} + 2 q^{69} + 8 q^{70} + 6 q^{71} - 18 q^{73} + 7 q^{74} - 5 q^{75} - q^{76} - 15 q^{77} - 9 q^{79} + 16 q^{80} + 3 q^{81} + 5 q^{82} + 16 q^{83} + q^{84} + 7 q^{85} + 15 q^{86} - 8 q^{87} - 7 q^{88} + 5 q^{89} - q^{90} - 17 q^{92} - 8 q^{93} + 32 q^{94} + 3 q^{95} + 4 q^{96} - 5 q^{97} + 13 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

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

\( T_{2}^{3} + T_{2}^{2} - 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{3} + 4T_{5}^{2} + 3T_{5} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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