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: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \beta ) q^{2} -2 \beta q^{3} + 3 q^{4} + 2 q^{5} + ( 4 + 2 \beta ) q^{6} + ( 1 - 2 \beta ) q^{8} + ( 1 + 4 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 - 2 \beta ) q^{2} -2 \beta q^{3} + 3 q^{4} + 2 q^{5} + ( 4 + 2 \beta ) q^{6} + ( 1 - 2 \beta ) q^{8} + ( 1 + 4 \beta ) q^{9} + ( 2 - 4 \beta ) q^{10} - q^{11} -6 \beta q^{12} + ( -2 + 2 \beta ) q^{13} -4 \beta q^{15} - q^{16} + ( 2 - 2 \beta ) q^{17} + ( -7 - 6 \beta ) q^{18} + ( -4 + 4 \beta ) q^{19} + 6 q^{20} + ( -1 + 2 \beta ) q^{22} -4 \beta q^{23} + ( 4 + 2 \beta ) q^{24} - q^{25} + ( -6 + 2 \beta ) q^{26} + ( -8 - 4 \beta ) q^{27} + ( 6 - 4 \beta ) q^{29} + ( 8 + 4 \beta ) q^{30} + ( 4 + 2 \beta ) q^{31} + ( -3 + 6 \beta ) q^{32} + 2 \beta q^{33} + ( 6 - 2 \beta ) q^{34} + ( 3 + 12 \beta ) q^{36} + ( -6 + 4 \beta ) q^{37} + ( -12 + 4 \beta ) q^{38} -4 q^{39} + ( 2 - 4 \beta ) q^{40} + ( 10 - 2 \beta ) q^{41} + 8 q^{43} -3 q^{44} + ( 2 + 8 \beta ) q^{45} + ( 8 + 4 \beta ) q^{46} + ( -4 - 2 \beta ) q^{47} + 2 \beta q^{48} + ( -1 + 2 \beta ) q^{50} + 4 q^{51} + ( -6 + 6 \beta ) q^{52} + ( 2 + 4 \beta ) q^{53} + 20 \beta q^{54} -2 q^{55} -8 q^{57} + ( 14 - 8 \beta ) q^{58} -2 \beta q^{59} -12 \beta q^{60} + ( 6 - 2 \beta ) q^{61} -10 \beta q^{62} -13 q^{64} + ( -4 + 4 \beta ) q^{65} + ( -4 - 2 \beta ) q^{66} + ( 12 - 4 \beta ) q^{67} + ( 6 - 6 \beta ) q^{68} + ( 8 + 8 \beta ) q^{69} + ( -8 + 4 \beta ) q^{71} + ( -7 - 6 \beta ) q^{72} + ( 2 + 2 \beta ) q^{73} + ( -14 + 8 \beta ) q^{74} + 2 \beta q^{75} + ( -12 + 12 \beta ) q^{76} + ( -4 + 8 \beta ) q^{78} + ( -4 + 8 \beta ) q^{79} -2 q^{80} + ( 5 + 12 \beta ) q^{81} + ( 14 - 18 \beta ) q^{82} + ( 4 - 12 \beta ) q^{83} + ( 4 - 4 \beta ) q^{85} + ( 8 - 16 \beta ) q^{86} + ( 8 - 4 \beta ) q^{87} + ( -1 + 2 \beta ) q^{88} -2 q^{89} + ( -14 - 12 \beta ) q^{90} -12 \beta q^{92} + ( -4 - 12 \beta ) q^{93} + 10 \beta q^{94} + ( -8 + 8 \beta ) q^{95} + ( -12 - 6 \beta ) q^{96} + ( -10 + 12 \beta ) q^{97} + ( -1 - 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 6q^{4} + 4q^{5} + 10q^{6} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 6q^{4} + 4q^{5} + 10q^{6} + 6q^{9} - 2q^{11} - 6q^{12} - 2q^{13} - 4q^{15} - 2q^{16} + 2q^{17} - 20q^{18} - 4q^{19} + 12q^{20} - 4q^{23} + 10q^{24} - 2q^{25} - 10q^{26} - 20q^{27} + 8q^{29} + 20q^{30} + 10q^{31} + 2q^{33} + 10q^{34} + 18q^{36} - 8q^{37} - 20q^{38} - 8q^{39} + 18q^{41} + 16q^{43} - 6q^{44} + 12q^{45} + 20q^{46} - 10q^{47} + 2q^{48} + 8q^{51} - 6q^{52} + 8q^{53} + 20q^{54} - 4q^{55} - 16q^{57} + 20q^{58} - 2q^{59} - 12q^{60} + 10q^{61} - 10q^{62} - 26q^{64} - 4q^{65} - 10q^{66} + 20q^{67} + 6q^{68} + 24q^{69} - 12q^{71} - 20q^{72} + 6q^{73} - 20q^{74} + 2q^{75} - 12q^{76} - 4q^{80} + 22q^{81} + 10q^{82} - 4q^{83} + 4q^{85} + 12q^{87} - 4q^{89} - 40q^{90} - 12q^{92} - 20q^{93} + 10q^{94} - 8q^{95} - 30q^{96} - 8q^{97} - 6q^{99} + O(q^{100}) \)

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
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:

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

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(539))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + 4 T^{4} \)
$3$ \( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \)
$5$ \( ( 1 - 2 T + 5 T^{2} )^{2} \)
$7$ \( \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( 1 - 2 T + 30 T^{2} - 34 T^{3} + 289 T^{4} \)
$19$ \( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 + 4 T + 30 T^{2} + 92 T^{3} + 529 T^{4} \)
$29$ \( 1 - 8 T + 54 T^{2} - 232 T^{3} + 841 T^{4} \)
$31$ \( 1 - 10 T + 82 T^{2} - 310 T^{3} + 961 T^{4} \)
$37$ \( 1 + 8 T + 70 T^{2} + 296 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 18 T + 158 T^{2} - 738 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 10 T + 114 T^{2} + 470 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 2 T + 114 T^{2} + 118 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 10 T + 142 T^{2} - 610 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 20 T + 214 T^{2} - 1340 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 12 T + 158 T^{2} + 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 6 T + 150 T^{2} - 438 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 78 T^{2} + 6241 T^{4} \)
$83$ \( 1 + 4 T - 10 T^{2} + 332 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + 2 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 8 T + 30 T^{2} + 776 T^{3} + 9409 T^{4} \)
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