Properties

Label 503.2.a.d
Level $503$
Weight $2$
Character orbit 503.a
Self dual yes
Analytic conductor $4.016$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 503.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.01647522167\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 3\)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( 1 + \beta_{1} - \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{2} ) q^{6} + ( \beta_{1} - \beta_{2} ) q^{7} + ( -3 + \beta_{2} ) q^{8} + \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( 1 + \beta_{1} - \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{2} ) q^{6} + ( \beta_{1} - \beta_{2} ) q^{7} + ( -3 + \beta_{2} ) q^{8} + \beta_{2} q^{9} + ( 4 - \beta_{1} ) q^{11} + 3 q^{12} + ( 2 - \beta_{1} - \beta_{2} ) q^{13} + ( -3 + 2 \beta_{2} ) q^{14} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{16} + ( 4 + 2 \beta_{2} ) q^{17} + ( 3 + \beta_{1} - \beta_{2} ) q^{18} + 4 q^{19} + ( 3 - \beta_{1} ) q^{21} + ( -\beta_{1} + 3 \beta_{2} ) q^{22} -4 q^{23} + ( -2 \beta_{1} + \beta_{2} ) q^{24} -5 q^{25} + ( -3 - 2 \beta_{1} + 2 \beta_{2} ) q^{26} + ( -2 \beta_{1} + \beta_{2} ) q^{27} + ( 6 - 3 \beta_{2} ) q^{28} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{31} -3 \beta_{1} q^{32} + ( -3 + 4 \beta_{1} - \beta_{2} ) q^{33} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{34} + ( -3 + 3 \beta_{2} ) q^{36} + ( 2 - 4 \beta_{1} ) q^{37} + 4 \beta_{2} q^{38} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{39} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{41} + ( -\beta_{1} + 2 \beta_{2} ) q^{42} + ( 1 + 2 \beta_{1} - 3 \beta_{2} ) q^{43} + ( 1 + 4 \beta_{1} - 4 \beta_{2} ) q^{44} -4 \beta_{2} q^{46} + ( 7 - 2 \beta_{1} - \beta_{2} ) q^{47} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{48} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{49} -5 \beta_{2} q^{50} + ( 6 \beta_{1} + 2 \beta_{2} ) q^{51} + ( 2 + 2 \beta_{1} - 5 \beta_{2} ) q^{52} + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{54} + ( -3 - 3 \beta_{1} + 5 \beta_{2} ) q^{56} + 4 \beta_{1} q^{57} + ( -6 - 4 \beta_{1} ) q^{58} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{59} + 2 q^{61} -6 q^{62} + ( -3 + 2 \beta_{2} ) q^{63} + ( -2 - \beta_{1} + \beta_{2} ) q^{64} + ( -3 + 3 \beta_{1} + 2 \beta_{2} ) q^{66} + ( 11 + 2 \beta_{1} - \beta_{2} ) q^{67} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{68} -4 \beta_{1} q^{69} + ( 4 - 2 \beta_{2} ) q^{71} + ( 3 + \beta_{1} - 4 \beta_{2} ) q^{72} + ( 12 - \beta_{1} + 2 \beta_{2} ) q^{73} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{74} -5 \beta_{1} q^{75} + ( 4 + 4 \beta_{1} - 4 \beta_{2} ) q^{76} + ( -3 + 5 \beta_{1} - 4 \beta_{2} ) q^{77} + ( -6 - \beta_{1} ) q^{78} + ( -4 + 5 \beta_{1} + 3 \beta_{2} ) q^{79} + ( -6 + \beta_{1} - 4 \beta_{2} ) q^{81} + ( 12 + 6 \beta_{1} - 2 \beta_{2} ) q^{82} + ( -\beta_{1} + \beta_{2} ) q^{83} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{84} + ( -9 - \beta_{1} + 6 \beta_{2} ) q^{86} + ( -6 - 2 \beta_{1} - 4 \beta_{2} ) q^{87} + ( -12 + 2 \beta_{1} + 3 \beta_{2} ) q^{88} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{91} + ( -4 - 4 \beta_{1} + 4 \beta_{2} ) q^{92} + ( 6 - 6 \beta_{1} ) q^{93} + ( -3 - 3 \beta_{1} + 6 \beta_{2} ) q^{94} + ( -9 - 3 \beta_{2} ) q^{96} + ( -6 - 5 \beta_{1} + 3 \beta_{2} ) q^{97} + ( -6 - 3 \beta_{1} ) q^{98} + ( -\beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{3} + 4q^{4} + q^{6} + q^{7} - 9q^{8} + O(q^{10}) \) \( 3q + q^{3} + 4q^{4} + q^{6} + q^{7} - 9q^{8} + 11q^{11} + 9q^{12} + 5q^{13} - 9q^{14} + 2q^{16} + 12q^{17} + 10q^{18} + 12q^{19} + 8q^{21} - q^{22} - 12q^{23} - 2q^{24} - 15q^{25} - 11q^{26} - 2q^{27} + 18q^{28} - 2q^{29} - 10q^{31} - 3q^{32} - 5q^{33} + 20q^{34} - 9q^{36} + 2q^{37} - 8q^{39} + 2q^{41} - q^{42} + 5q^{43} + 7q^{44} + 19q^{47} - 10q^{48} - 4q^{49} + 6q^{51} + 8q^{52} + 14q^{53} + 8q^{54} - 12q^{56} + 4q^{57} - 22q^{58} - 5q^{59} + 6q^{61} - 18q^{62} - 9q^{63} - 7q^{64} - 6q^{66} + 35q^{67} - 2q^{68} - 4q^{69} + 12q^{71} + 10q^{72} + 35q^{73} - 4q^{74} - 5q^{75} + 16q^{76} - 4q^{77} - 19q^{78} - 7q^{79} - 17q^{81} + 42q^{82} - q^{83} + 3q^{84} - 28q^{86} - 20q^{87} - 34q^{88} + 4q^{89} + 3q^{91} - 16q^{92} + 12q^{93} - 12q^{94} - 27q^{96} - 23q^{97} - 21q^{98} - q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 3\):

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

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
0.713538
−1.91223
2.19869
−2.49086 0.713538 4.20440 0 −1.77733 3.20440 −5.49086 −2.49086 0
1.2 0.656620 −1.91223 −1.56885 0 −1.25561 −2.56885 −2.34338 0.656620 0
1.3 1.83424 2.19869 1.36445 0 4.03293 0.364448 −1.16576 1.83424 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(503\) \(-1\)

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 503.2.a.d 3
3.b odd 2 1 4527.2.a.j 3
4.b odd 2 1 8048.2.a.m 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.2.a.d 3 1.a even 1 1 trivial
4527.2.a.j 3 3.b odd 2 1
8048.2.a.m 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(503))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 - 5 T + T^{3} \)
$3$ \( 3 - 4 T - T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 3 - 8 T - T^{2} + T^{3} \)
$11$ \( -35 + 36 T - 11 T^{2} + T^{3} \)
$13$ \( 25 - 2 T - 5 T^{2} + T^{3} \)
$17$ \( 40 + 28 T - 12 T^{2} + T^{3} \)
$19$ \( ( -4 + T )^{3} \)
$23$ \( ( 4 + T )^{3} \)
$29$ \( 72 - 40 T + 2 T^{2} + T^{3} \)
$31$ \( -72 + 10 T^{2} + T^{3} \)
$37$ \( -56 - 68 T - 2 T^{2} + T^{3} \)
$41$ \( -120 - 104 T - 2 T^{2} + T^{3} \)
$43$ \( -5 - 48 T - 5 T^{2} + T^{3} \)
$47$ \( -63 + 96 T - 19 T^{2} + T^{3} \)
$53$ \( 648 - 32 T - 14 T^{2} + T^{3} \)
$59$ \( 1 - 16 T + 5 T^{2} + T^{3} \)
$61$ \( ( -2 + T )^{3} \)
$67$ \( -1319 + 388 T - 35 T^{2} + T^{3} \)
$71$ \( -8 + 28 T - 12 T^{2} + T^{3} \)
$73$ \( -1293 + 386 T - 35 T^{2} + T^{3} \)
$79$ \( -1145 - 152 T + 7 T^{2} + T^{3} \)
$83$ \( -3 - 8 T + T^{2} + T^{3} \)
$89$ \( -168 - 92 T - 4 T^{2} + T^{3} \)
$97$ \( -1083 + 38 T + 23 T^{2} + T^{3} \)
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