Properties

Label 539.2.a.f
Level $539$
Weight $2$
Character orbit 539.a
Self dual yes
Analytic conductor $4.304$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q - \beta q^{2} + ( - \beta - 1) q^{3} + 3 q^{4} + 2 q^{5} + (\beta + 5) q^{6} - \beta q^{8} + (2 \beta + 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + ( - \beta - 1) q^{3} + 3 q^{4} + 2 q^{5} + (\beta + 5) q^{6} - \beta q^{8} + (2 \beta + 3) q^{9} - 2 \beta q^{10} - q^{11} + ( - 3 \beta - 3) q^{12} + (\beta - 1) q^{13} + ( - 2 \beta - 2) q^{15} - q^{16} + ( - \beta + 1) q^{17} + ( - 3 \beta - 10) q^{18} + (2 \beta - 2) q^{19} + 6 q^{20} + \beta q^{22} + ( - 2 \beta - 2) q^{23} + (\beta + 5) q^{24} - q^{25} + (\beta - 5) q^{26} + ( - 2 \beta - 10) q^{27} + ( - 2 \beta + 4) q^{29} + (2 \beta + 10) q^{30} + (\beta + 5) q^{31} + 3 \beta q^{32} + (\beta + 1) q^{33} + ( - \beta + 5) q^{34} + (6 \beta + 9) q^{36} + (2 \beta - 4) q^{37} + (2 \beta - 10) q^{38} - 4 q^{39} - 2 \beta q^{40} + ( - \beta + 9) q^{41} + 8 q^{43} - 3 q^{44} + (4 \beta + 6) q^{45} + (2 \beta + 10) q^{46} + ( - \beta - 5) q^{47} + (\beta + 1) q^{48} + \beta q^{50} + 4 q^{51} + (3 \beta - 3) q^{52} + (2 \beta + 4) q^{53} + (10 \beta + 10) q^{54} - 2 q^{55} - 8 q^{57} + ( - 4 \beta + 10) q^{58} + ( - \beta - 1) q^{59} + ( - 6 \beta - 6) q^{60} + ( - \beta + 5) q^{61} + ( - 5 \beta - 5) q^{62} - 13 q^{64} + (2 \beta - 2) q^{65} + ( - \beta - 5) q^{66} + ( - 2 \beta + 10) q^{67} + ( - 3 \beta + 3) q^{68} + (4 \beta + 12) q^{69} + (2 \beta - 6) q^{71} + ( - 3 \beta - 10) q^{72} + (\beta + 3) q^{73} + (4 \beta - 10) q^{74} + (\beta + 1) q^{75} + (6 \beta - 6) q^{76} + 4 \beta q^{78} + 4 \beta q^{79} - 2 q^{80} + (6 \beta + 11) q^{81} + ( - 9 \beta + 5) q^{82} + ( - 6 \beta - 2) q^{83} + ( - 2 \beta + 2) q^{85} - 8 \beta q^{86} + ( - 2 \beta + 6) q^{87} + \beta q^{88} - 2 q^{89} + ( - 6 \beta - 20) q^{90} + ( - 6 \beta - 6) q^{92} + ( - 6 \beta - 10) q^{93} + (5 \beta + 5) q^{94} + (4 \beta - 4) q^{95} + ( - 3 \beta - 15) q^{96} + (6 \beta - 4) q^{97} + ( - 2 \beta - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 6 q^{4} + 4 q^{5} + 10 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 6 q^{4} + 4 q^{5} + 10 q^{6} + 6 q^{9} - 2 q^{11} - 6 q^{12} - 2 q^{13} - 4 q^{15} - 2 q^{16} + 2 q^{17} - 20 q^{18} - 4 q^{19} + 12 q^{20} - 4 q^{23} + 10 q^{24} - 2 q^{25} - 10 q^{26} - 20 q^{27} + 8 q^{29} + 20 q^{30} + 10 q^{31} + 2 q^{33} + 10 q^{34} + 18 q^{36} - 8 q^{37} - 20 q^{38} - 8 q^{39} + 18 q^{41} + 16 q^{43} - 6 q^{44} + 12 q^{45} + 20 q^{46} - 10 q^{47} + 2 q^{48} + 8 q^{51} - 6 q^{52} + 8 q^{53} + 20 q^{54} - 4 q^{55} - 16 q^{57} + 20 q^{58} - 2 q^{59} - 12 q^{60} + 10 q^{61} - 10 q^{62} - 26 q^{64} - 4 q^{65} - 10 q^{66} + 20 q^{67} + 6 q^{68} + 24 q^{69} - 12 q^{71} - 20 q^{72} + 6 q^{73} - 20 q^{74} + 2 q^{75} - 12 q^{76} - 4 q^{80} + 22 q^{81} + 10 q^{82} - 4 q^{83} + 4 q^{85} + 12 q^{87} - 4 q^{89} - 40 q^{90} - 12 q^{92} - 20 q^{93} + 10 q^{94} - 8 q^{95} - 30 q^{96} - 8 q^{97} - 6 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−2.23607 −3.23607 3.00000 2.00000 7.23607 0 −2.23607 7.47214 −4.47214
1.2 2.23607 1.23607 3.00000 2.00000 2.76393 0 2.23607 −1.47214 4.47214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(11\) \(1\)

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 539.2.a.f 2
3.b odd 2 1 4851.2.a.y 2
4.b odd 2 1 8624.2.a.ce 2
7.b odd 2 1 77.2.a.d 2
7.c even 3 2 539.2.e.j 4
7.d odd 6 2 539.2.e.i 4
11.b odd 2 1 5929.2.a.m 2
21.c even 2 1 693.2.a.h 2
28.d even 2 1 1232.2.a.m 2
35.c odd 2 1 1925.2.a.r 2
35.f even 4 2 1925.2.b.h 4
56.e even 2 1 4928.2.a.bv 2
56.h odd 2 1 4928.2.a.bm 2
77.b even 2 1 847.2.a.f 2
77.j odd 10 2 847.2.f.a 4
77.j odd 10 2 847.2.f.n 4
77.l even 10 2 847.2.f.b 4
77.l even 10 2 847.2.f.m 4
231.h odd 2 1 7623.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 7.b odd 2 1
539.2.a.f 2 1.a even 1 1 trivial
539.2.e.i 4 7.d odd 6 2
539.2.e.j 4 7.c even 3 2
693.2.a.h 2 21.c even 2 1
847.2.a.f 2 77.b even 2 1
847.2.f.a 4 77.j odd 10 2
847.2.f.b 4 77.l even 10 2
847.2.f.m 4 77.l even 10 2
847.2.f.n 4 77.j odd 10 2
1232.2.a.m 2 28.d even 2 1
1925.2.a.r 2 35.c odd 2 1
1925.2.b.h 4 35.f even 4 2
4851.2.a.y 2 3.b odd 2 1
4928.2.a.bm 2 56.h odd 2 1
4928.2.a.bv 2 56.e even 2 1
5929.2.a.m 2 11.b odd 2 1
7623.2.a.bl 2 231.h odd 2 1
8624.2.a.ce 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(539))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$5$ \( (T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$31$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 18T + 76 \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$67$ \( T^{2} - 20T + 80 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$73$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 80 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 176 \) Copy content Toggle raw display
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 164 \) Copy content Toggle raw display
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