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 $1$

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: \(0\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1 + 1) q^{2} - q^{3} + ( - \beta_1 + 4) q^{4} + ( - \beta_{2} + 2 \beta_1 + 1) q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} + (\beta_{2} + 1) q^{7} + ( - 3 \beta_1 + 5) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1 + 1) q^{2} - q^{3} + ( - \beta_1 + 4) q^{4} + ( - \beta_{2} + 2 \beta_1 + 1) q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} + (\beta_{2} + 1) q^{7} + ( - 3 \beta_1 + 5) q^{8} + q^{9} + ( - 6 \beta_{2} + 2 \beta_1 - 3) q^{10} + (3 \beta_{2} - \beta_1 + 3) q^{11} + (\beta_1 - 4) q^{12} + ( - 2 \beta_1 - 1) q^{14} + (\beta_{2} - 2 \beta_1 - 1) q^{15} + (\beta_{2} - 6 \beta_1 + 6) q^{16} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{17} + ( - \beta_{2} - \beta_1 + 1) q^{18} + ( - \beta_1 - 2) q^{19} + ( - 5 \beta_{2} + 7 \beta_1 + 1) q^{20} + ( - \beta_{2} - 1) q^{21} + (2 \beta_{2} - 7 \beta_1) q^{22} + (5 \beta_{2} - 4 \beta_1 + 3) q^{23} + (3 \beta_1 - 5) q^{24} + ( - 3 \beta_{2} + 5 \beta_1 + 1) q^{25} - q^{27} + (3 \beta_{2} - \beta_1 + 3) q^{28} + ( - 3 \beta_{2} + \beta_1 - 2) q^{29} + (6 \beta_{2} - 2 \beta_1 + 3) q^{30} + ( - 5 \beta_{2} + 3 \beta_1 - 8) q^{31} + (7 \beta_{2} - 7 \beta_1 + 12) q^{32} + ( - 3 \beta_{2} + \beta_1 - 3) q^{33} + ( - 6 \beta_{2} + 7 \beta_1 - 7) q^{34} + (3 \beta_{2} + \beta_1 + 2) q^{35} + ( - \beta_1 + 4) q^{36} + (\beta_1 + 7) q^{37} + (4 \beta_{2} + \beta_1 + 1) q^{38} + ( - 8 \beta_{2} + 7 \beta_1 - 4) q^{40} + (3 \beta_{2} - \beta_1 + 5) q^{41} + (2 \beta_1 + 1) q^{42} + ( - 2 \beta_{2} + 4 \beta_1 - 7) q^{43} + (10 \beta_{2} - 7 \beta_1 + 11) q^{44} + ( - \beta_{2} + 2 \beta_1 + 1) q^{45} + (10 \beta_{2} - 12 \beta_1 + 5) q^{46} + (\beta_1 + 2) q^{47} + ( - \beta_{2} + 6 \beta_1 - 6) q^{48} + (\beta_{2} + \beta_1 - 5) q^{49} + ( - 14 \beta_{2} + 7 \beta_1 - 8) q^{50} + (2 \beta_{2} - 3 \beta_1 + 2) q^{51} + (3 \beta_{2} + \beta_1 - 5) q^{53} + (\beta_{2} + \beta_1 - 1) q^{54} + (8 \beta_{2} + 2 \beta_1 + 3) q^{55} + (2 \beta_{2} - 3 \beta_1 + 2) q^{56} + (\beta_1 + 2) q^{57} + ( - 3 \beta_{2} + 6 \beta_1 + 1) q^{58} + (2 \beta_{2} - 4 \beta_1) q^{59} + (5 \beta_{2} - 7 \beta_1 - 1) q^{60} + (6 \beta_{2} - \beta_1 - 2) q^{61} + ( - 3 \beta_{2} + 16 \beta_1 - 7) q^{62} + (\beta_{2} + 1) q^{63} + (7 \beta_{2} - 14 \beta_1 + 7) q^{64} + ( - 2 \beta_{2} + 7 \beta_1) q^{66} + ( - 7 \beta_{2} + 4 \beta_1) q^{67} + ( - 9 \beta_{2} + 14 \beta_1 - 12) q^{68} + ( - 5 \beta_{2} + 4 \beta_1 - 3) q^{69} + ( - \beta_{2} - 4 \beta_1 - 7) q^{70} + ( - 5 \beta_{2} + 7 \beta_1 - 4) q^{71} + ( - 3 \beta_1 + 5) q^{72} + (9 \beta_{2} - 6 \beta_1 + 7) q^{73} + ( - 9 \beta_{2} - 6 \beta_1 + 4) q^{74} + (3 \beta_{2} - 5 \beta_1 - 1) q^{75} + (\beta_{2} - 2 \beta_1 - 6) q^{76} + (2 \beta_{2} + 2 \beta_1 + 5) q^{77} - 3 \beta_{2} q^{79} + ( - 8 \beta_{2} + 5 \beta_1 - 11) q^{80} + q^{81} + ( - 9 \beta_1 + 2) q^{82} + ( - 2 \beta_{2} + \beta_1 + 3) q^{83} + ( - 3 \beta_{2} + \beta_1 - 3) q^{84} + ( - 3 \beta_{2} + \beta_1 + 5) q^{85} + ( - 3 \beta_{2} + 13 \beta_1 - 15) q^{86} + (3 \beta_{2} - \beta_1 + 2) q^{87} + (9 \beta_{2} - 14 \beta_1 + 12) q^{88} + ( - \beta_{2} - 6 \beta_1 + 2) q^{89} + ( - 6 \beta_{2} + 2 \beta_1 - 3) q^{90} + (19 \beta_{2} - 19 \beta_1 + 15) q^{92} + (5 \beta_{2} - 3 \beta_1 + 8) q^{93} + ( - 4 \beta_{2} - \beta_1 - 1) q^{94} + (\beta_{2} - 5 \beta_1 - 5) q^{95} + ( - 7 \beta_{2} + 7 \beta_1 - 12) q^{96} + (2 \beta_{2} - 12 \beta_1 + 3) q^{97} + (4 \beta_{2} + 5 \beta_1 - 10) q^{98} + (3 \beta_{2} - \beta_1 + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 11 q^{4} + 6 q^{5} - 3 q^{6} + 2 q^{7} + 12 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 11 q^{4} + 6 q^{5} - 3 q^{6} + 2 q^{7} + 12 q^{8} + 3 q^{9} - q^{10} + 5 q^{11} - 11 q^{12} - 5 q^{14} - 6 q^{15} + 11 q^{16} - q^{17} + 3 q^{18} - 7 q^{19} + 15 q^{20} - 2 q^{21} - 9 q^{22} - 12 q^{24} + 11 q^{25} - 3 q^{27} + 5 q^{28} - 2 q^{29} + q^{30} - 16 q^{31} + 22 q^{32} - 5 q^{33} - 8 q^{34} + 4 q^{35} + 11 q^{36} + 22 q^{37} + 3 q^{40} + 11 q^{41} + 5 q^{42} - 15 q^{43} + 16 q^{44} + 6 q^{45} - 7 q^{46} + 7 q^{47} - 11 q^{48} - 15 q^{49} - 3 q^{50} + q^{51} - 17 q^{53} - 3 q^{54} + 3 q^{55} + q^{56} + 7 q^{57} + 12 q^{58} - 6 q^{59} - 15 q^{60} - 13 q^{61} - 2 q^{62} + 2 q^{63} + 9 q^{66} + 11 q^{67} - 13 q^{68} - 24 q^{70} + 12 q^{72} + 6 q^{73} + 15 q^{74} - 11 q^{75} - 21 q^{76} + 15 q^{77} + 3 q^{79} - 20 q^{80} + 3 q^{81} - 3 q^{82} + 12 q^{83} - 5 q^{84} + 19 q^{85} - 29 q^{86} + 2 q^{87} + 13 q^{88} + q^{89} - q^{90} + 7 q^{92} + 16 q^{93} - 21 q^{95} - 22 q^{96} - 5 q^{97} - 29 q^{98} + 5 q^{99}+O(q^{100}) \) 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
−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 \(S_{2}^{\mathrm{new}}(\Gamma_0(507))\):

\( T_{2}^{3} - 3T_{2}^{2} - 4T_{2} + 13 \) Copy content Toggle raw display
\( T_{5}^{3} - 6T_{5}^{2} + 5T_{5} + 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

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