Properties

Label 539.4.a.e
Level $539$
Weight $4$
Character orbit 539.a
Self dual yes
Analytic conductor $31.802$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 539.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.8020294931\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 4 \beta ) q^{3} + ( -4 + 2 \beta ) q^{4} + ( -1 - 8 \beta ) q^{5} + ( 13 + 5 \beta ) q^{6} + ( -6 - 10 \beta ) q^{8} + ( 22 + 8 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 4 \beta ) q^{3} + ( -4 + 2 \beta ) q^{4} + ( -1 - 8 \beta ) q^{5} + ( 13 + 5 \beta ) q^{6} + ( -6 - 10 \beta ) q^{8} + ( 22 + 8 \beta ) q^{9} + ( -25 - 9 \beta ) q^{10} -11 q^{11} + ( 20 - 14 \beta ) q^{12} + ( -40 + 20 \beta ) q^{13} + ( -97 - 12 \beta ) q^{15} + ( -4 - 32 \beta ) q^{16} + ( 62 - 12 \beta ) q^{17} + ( 46 + 30 \beta ) q^{18} + ( -36 - 60 \beta ) q^{19} + ( -44 + 30 \beta ) q^{20} + ( -11 - 11 \beta ) q^{22} + ( -49 - 36 \beta ) q^{23} + ( -126 - 34 \beta ) q^{24} + ( 68 + 16 \beta ) q^{25} + ( 20 - 20 \beta ) q^{26} + ( 91 - 12 \beta ) q^{27} + ( 72 - 56 \beta ) q^{29} + ( -133 - 109 \beta ) q^{30} + ( 17 - 28 \beta ) q^{31} + ( -52 + 44 \beta ) q^{32} + ( -11 - 44 \beta ) q^{33} + ( 26 + 50 \beta ) q^{34} + ( -40 + 12 \beta ) q^{36} + ( 27 - 8 \beta ) q^{37} + ( -216 - 96 \beta ) q^{38} + ( 200 - 140 \beta ) q^{39} + ( 246 + 58 \beta ) q^{40} + ( -268 + 4 \beta ) q^{41} + ( -30 - 16 \beta ) q^{43} + ( 44 - 22 \beta ) q^{44} + ( -214 - 184 \beta ) q^{45} + ( -157 - 85 \beta ) q^{46} + ( 136 + 120 \beta ) q^{47} + ( -388 - 48 \beta ) q^{48} + ( 116 + 84 \beta ) q^{50} + ( -82 + 236 \beta ) q^{51} + ( 280 - 160 \beta ) q^{52} + ( -246 - 56 \beta ) q^{53} + ( 55 + 79 \beta ) q^{54} + ( 11 + 88 \beta ) q^{55} + ( -756 - 204 \beta ) q^{57} + ( -96 + 16 \beta ) q^{58} + ( -317 + 132 \beta ) q^{59} + ( 316 - 146 \beta ) q^{60} + ( -420 - 184 \beta ) q^{61} + ( -67 - 11 \beta ) q^{62} + ( 112 + 248 \beta ) q^{64} + ( -440 + 300 \beta ) q^{65} + ( -143 - 55 \beta ) q^{66} + ( 377 - 20 \beta ) q^{67} + ( -320 + 172 \beta ) q^{68} + ( -481 - 232 \beta ) q^{69} + ( -339 + 76 \beta ) q^{71} + ( -372 - 268 \beta ) q^{72} + ( 200 + 468 \beta ) q^{73} + ( 3 + 19 \beta ) q^{74} + ( 260 + 288 \beta ) q^{75} + ( -216 + 168 \beta ) q^{76} + ( -220 + 60 \beta ) q^{78} + ( 158 + 656 \beta ) q^{79} + ( 772 + 64 \beta ) q^{80} + ( -647 + 136 \beta ) q^{81} + ( -256 - 264 \beta ) q^{82} + ( -234 - 120 \beta ) q^{83} + ( 226 - 484 \beta ) q^{85} + ( -78 - 46 \beta ) q^{86} + ( -600 + 232 \beta ) q^{87} + ( 66 + 110 \beta ) q^{88} + ( 921 + 328 \beta ) q^{89} + ( -766 - 398 \beta ) q^{90} + ( -20 + 46 \beta ) q^{92} + ( -319 + 40 \beta ) q^{93} + ( 496 + 256 \beta ) q^{94} + ( 1476 + 348 \beta ) q^{95} + ( 476 - 164 \beta ) q^{96} + ( -1097 - 144 \beta ) q^{97} + ( -242 - 88 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} - 8q^{4} - 2q^{5} + 26q^{6} - 12q^{8} + 44q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} - 8q^{4} - 2q^{5} + 26q^{6} - 12q^{8} + 44q^{9} - 50q^{10} - 22q^{11} + 40q^{12} - 80q^{13} - 194q^{15} - 8q^{16} + 124q^{17} + 92q^{18} - 72q^{19} - 88q^{20} - 22q^{22} - 98q^{23} - 252q^{24} + 136q^{25} + 40q^{26} + 182q^{27} + 144q^{29} - 266q^{30} + 34q^{31} - 104q^{32} - 22q^{33} + 52q^{34} - 80q^{36} + 54q^{37} - 432q^{38} + 400q^{39} + 492q^{40} - 536q^{41} - 60q^{43} + 88q^{44} - 428q^{45} - 314q^{46} + 272q^{47} - 776q^{48} + 232q^{50} - 164q^{51} + 560q^{52} - 492q^{53} + 110q^{54} + 22q^{55} - 1512q^{57} - 192q^{58} - 634q^{59} + 632q^{60} - 840q^{61} - 134q^{62} + 224q^{64} - 880q^{65} - 286q^{66} + 754q^{67} - 640q^{68} - 962q^{69} - 678q^{71} - 744q^{72} + 400q^{73} + 6q^{74} + 520q^{75} - 432q^{76} - 440q^{78} + 316q^{79} + 1544q^{80} - 1294q^{81} - 512q^{82} - 468q^{83} + 452q^{85} - 156q^{86} - 1200q^{87} + 132q^{88} + 1842q^{89} - 1532q^{90} - 40q^{92} - 638q^{93} + 992q^{94} + 2952q^{95} + 952q^{96} - 2194q^{97} - 484q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
−0.732051 −5.92820 −7.46410 12.8564 4.33975 0 11.3205 8.14359 −9.41154
1.2 2.73205 7.92820 −0.535898 −14.8564 21.6603 0 −23.3205 35.8564 −40.5885
\(n\): 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 539.4.a.e 2
7.b odd 2 1 11.4.a.a 2
21.c even 2 1 99.4.a.c 2
28.d even 2 1 176.4.a.i 2
35.c odd 2 1 275.4.a.b 2
35.f even 4 2 275.4.b.c 4
56.e even 2 1 704.4.a.n 2
56.h odd 2 1 704.4.a.p 2
77.b even 2 1 121.4.a.c 2
77.j odd 10 4 121.4.c.c 8
77.l even 10 4 121.4.c.f 8
84.h odd 2 1 1584.4.a.bc 2
91.b odd 2 1 1859.4.a.a 2
105.g even 2 1 2475.4.a.q 2
231.h odd 2 1 1089.4.a.v 2
308.g odd 2 1 1936.4.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 7.b odd 2 1
99.4.a.c 2 21.c even 2 1
121.4.a.c 2 77.b even 2 1
121.4.c.c 8 77.j odd 10 4
121.4.c.f 8 77.l even 10 4
176.4.a.i 2 28.d even 2 1
275.4.a.b 2 35.c odd 2 1
275.4.b.c 4 35.f even 4 2
539.4.a.e 2 1.a even 1 1 trivial
704.4.a.n 2 56.e even 2 1
704.4.a.p 2 56.h odd 2 1
1089.4.a.v 2 231.h odd 2 1
1584.4.a.bc 2 84.h odd 2 1
1859.4.a.a 2 91.b odd 2 1
1936.4.a.w 2 308.g odd 2 1
2475.4.a.q 2 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(539))\):

\( T_{2}^{2} - 2 T_{2} - 2 \)
\( T_{3}^{2} - 2 T_{3} - 47 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 - 2 T + T^{2} \)
$3$ \( -47 - 2 T + T^{2} \)
$5$ \( -191 + 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 11 + T )^{2} \)
$13$ \( 400 + 80 T + T^{2} \)
$17$ \( 3412 - 124 T + T^{2} \)
$19$ \( -9504 + 72 T + T^{2} \)
$23$ \( -1487 + 98 T + T^{2} \)
$29$ \( -4224 - 144 T + T^{2} \)
$31$ \( -2063 - 34 T + T^{2} \)
$37$ \( 537 - 54 T + T^{2} \)
$41$ \( 71776 + 536 T + T^{2} \)
$43$ \( 132 + 60 T + T^{2} \)
$47$ \( -24704 - 272 T + T^{2} \)
$53$ \( 51108 + 492 T + T^{2} \)
$59$ \( 48217 + 634 T + T^{2} \)
$61$ \( 74832 + 840 T + T^{2} \)
$67$ \( 140929 - 754 T + T^{2} \)
$71$ \( 97593 + 678 T + T^{2} \)
$73$ \( -617072 - 400 T + T^{2} \)
$79$ \( -1266044 - 316 T + T^{2} \)
$83$ \( 11556 + 468 T + T^{2} \)
$89$ \( 525489 - 1842 T + T^{2} \)
$97$ \( 1141201 + 2194 T + T^{2} \)
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