Properties

Label 531.2.a.b
Level $531$
Weight $2$
Character orbit 531.a
Self dual yes
Analytic conductor $4.240$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.24005634733\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -1 + \beta ) q^{4} + ( 1 - 2 \beta ) q^{5} + ( -3 - \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( -1 + \beta ) q^{4} + ( 1 - 2 \beta ) q^{5} + ( -3 - \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + ( -2 - \beta ) q^{10} + ( -1 + 2 \beta ) q^{11} + ( -5 + 2 \beta ) q^{13} + ( -1 - 4 \beta ) q^{14} -3 \beta q^{16} + 3 \beta q^{17} -5 \beta q^{19} + ( -3 + \beta ) q^{20} + ( 2 + \beta ) q^{22} + ( 4 - \beta ) q^{23} + ( 2 - 3 \beta ) q^{26} + ( 2 - 3 \beta ) q^{28} + ( -7 - \beta ) q^{29} + ( -5 + 9 \beta ) q^{31} + ( -5 + \beta ) q^{32} + ( 3 + 3 \beta ) q^{34} + ( -1 + 7 \beta ) q^{35} + ( -2 - 3 \beta ) q^{37} + ( -5 - 5 \beta ) q^{38} + 5 q^{40} + ( -5 + 5 \beta ) q^{41} + ( -5 + 6 \beta ) q^{43} + ( 3 - \beta ) q^{44} + ( -1 + 3 \beta ) q^{46} + ( 9 - 3 \beta ) q^{47} + ( 3 + 7 \beta ) q^{49} + ( 7 - 5 \beta ) q^{52} + ( -5 + 2 \beta ) q^{53} -5 q^{55} + ( -1 + 7 \beta ) q^{56} + ( -1 - 8 \beta ) q^{58} + q^{59} + ( -5 - 3 \beta ) q^{61} + ( 9 + 4 \beta ) q^{62} + ( 1 + 2 \beta ) q^{64} + ( -9 + 8 \beta ) q^{65} + ( 7 - 6 \beta ) q^{67} + 3 q^{68} + ( 7 + 6 \beta ) q^{70} + ( 3 - 8 \beta ) q^{71} + ( -1 - 3 \beta ) q^{73} + ( -3 - 5 \beta ) q^{74} -5 q^{76} + ( 1 - 7 \beta ) q^{77} -3 q^{79} + ( 6 + 3 \beta ) q^{80} + 5 q^{82} + ( 1 - \beta ) q^{83} + ( -6 - 3 \beta ) q^{85} + ( 6 + \beta ) q^{86} -5 q^{88} + ( 7 - 11 \beta ) q^{89} + ( 13 - 3 \beta ) q^{91} + ( -5 + 4 \beta ) q^{92} + ( -3 + 6 \beta ) q^{94} + ( 10 + 5 \beta ) q^{95} + 3 q^{97} + ( 7 + 10 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - 7q^{7} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - 7q^{7} - 5q^{10} - 8q^{13} - 6q^{14} - 3q^{16} + 3q^{17} - 5q^{19} - 5q^{20} + 5q^{22} + 7q^{23} + q^{26} + q^{28} - 15q^{29} - q^{31} - 9q^{32} + 9q^{34} + 5q^{35} - 7q^{37} - 15q^{38} + 10q^{40} - 5q^{41} - 4q^{43} + 5q^{44} + q^{46} + 15q^{47} + 13q^{49} + 9q^{52} - 8q^{53} - 10q^{55} + 5q^{56} - 10q^{58} + 2q^{59} - 13q^{61} + 22q^{62} + 4q^{64} - 10q^{65} + 8q^{67} + 6q^{68} + 20q^{70} - 2q^{71} - 5q^{73} - 11q^{74} - 10q^{76} - 5q^{77} - 6q^{79} + 15q^{80} + 10q^{82} + q^{83} - 15q^{85} + 13q^{86} - 10q^{88} + 3q^{89} + 23q^{91} - 6q^{92} + 25q^{95} + 6q^{97} + 24q^{98} + O(q^{100}) \)

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
−0.618034
1.61803
−0.618034 0 −1.61803 2.23607 0 −2.38197 2.23607 0 −1.38197
1.2 1.61803 0 0.618034 −2.23607 0 −4.61803 −2.23607 0 −3.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(59\) \(-1\)

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 531.2.a.b 2
3.b odd 2 1 177.2.a.b 2
4.b odd 2 1 8496.2.a.bb 2
12.b even 2 1 2832.2.a.o 2
15.d odd 2 1 4425.2.a.t 2
21.c even 2 1 8673.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.a.b 2 3.b odd 2 1
531.2.a.b 2 1.a even 1 1 trivial
2832.2.a.o 2 12.b even 2 1
4425.2.a.t 2 15.d odd 2 1
8496.2.a.bb 2 4.b odd 2 1
8673.2.a.k 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(531))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -5 + T^{2} \)
$7$ \( 11 + 7 T + T^{2} \)
$11$ \( -5 + T^{2} \)
$13$ \( 11 + 8 T + T^{2} \)
$17$ \( -9 - 3 T + T^{2} \)
$19$ \( -25 + 5 T + T^{2} \)
$23$ \( 11 - 7 T + T^{2} \)
$29$ \( 55 + 15 T + T^{2} \)
$31$ \( -101 + T + T^{2} \)
$37$ \( 1 + 7 T + T^{2} \)
$41$ \( -25 + 5 T + T^{2} \)
$43$ \( -41 + 4 T + T^{2} \)
$47$ \( 45 - 15 T + T^{2} \)
$53$ \( 11 + 8 T + T^{2} \)
$59$ \( ( -1 + T )^{2} \)
$61$ \( 31 + 13 T + T^{2} \)
$67$ \( -29 - 8 T + T^{2} \)
$71$ \( -79 + 2 T + T^{2} \)
$73$ \( -5 + 5 T + T^{2} \)
$79$ \( ( 3 + T )^{2} \)
$83$ \( -1 - T + T^{2} \)
$89$ \( -149 - 3 T + T^{2} \)
$97$ \( ( -3 + T )^{2} \)
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