Properties

Label 539.2.a.l
Level $539$
Weight $2$
Character orbit 539.a
Self dual yes
Analytic conductor $4.304$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 26 x^{8} + 245 x^{6} - 1038 x^{4} + 1884 x^{2} - 968\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + \beta_{1} q^{3} + ( 2 + \beta_{9} ) q^{4} + \beta_{7} q^{5} + ( \beta_{3} + \beta_{4} ) q^{6} + ( -1 - \beta_{2} + 2 \beta_{5} + \beta_{8} - \beta_{9} ) q^{8} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{2} + \beta_{1} q^{3} + ( 2 + \beta_{9} ) q^{4} + \beta_{7} q^{5} + ( \beta_{3} + \beta_{4} ) q^{6} + ( -1 - \beta_{2} + 2 \beta_{5} + \beta_{8} - \beta_{9} ) q^{8} + ( 2 + \beta_{2} ) q^{9} + ( \beta_{3} + \beta_{6} - \beta_{7} ) q^{10} + q^{11} + ( \beta_{1} - 2 \beta_{3} - \beta_{6} + \beta_{7} ) q^{12} + ( -\beta_{4} - \beta_{6} - \beta_{7} ) q^{13} + ( 1 - 2 \beta_{8} + \beta_{9} ) q^{15} + ( 5 - \beta_{2} - \beta_{5} - \beta_{8} + 2 \beta_{9} ) q^{16} + ( -\beta_{4} + \beta_{6} + \beta_{7} ) q^{17} + ( 2 \beta_{2} + \beta_{5} ) q^{18} + ( -2 \beta_{3} + 2 \beta_{6} ) q^{19} + ( \beta_{1} - 2 \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{20} + \beta_{5} q^{22} + ( \beta_{5} + \beta_{8} - \beta_{9} ) q^{23} + ( -2 \beta_{1} + \beta_{3} + \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{24} + ( 2 - \beta_{2} - 2 \beta_{5} - 2 \beta_{9} ) q^{25} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{26} + ( \beta_{1} + 2 \beta_{3} + \beta_{4} - 4 \beta_{6} - \beta_{7} ) q^{27} + ( 1 + \beta_{2} - \beta_{5} + \beta_{8} - \beta_{9} ) q^{29} + ( -1 - \beta_{2} + \beta_{5} + \beta_{8} - 3 \beta_{9} ) q^{30} + ( \beta_{1} - 2 \beta_{3} ) q^{31} + ( -4 - 2 \beta_{2} + 5 \beta_{5} - 2 \beta_{9} ) q^{32} + \beta_{1} q^{33} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{34} + ( 2 \beta_{2} - 2 \beta_{5} + \beta_{9} ) q^{36} + ( 4 + \beta_{2} - \beta_{5} + \beta_{8} ) q^{37} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 6 \beta_{6} + 2 \beta_{7} ) q^{38} + ( -1 - \beta_{2} - 3 \beta_{5} + \beta_{8} - \beta_{9} ) q^{39} + ( -4 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{40} + ( -2 \beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{41} + ( -1 + \beta_{2} + \beta_{5} + \beta_{8} + \beta_{9} ) q^{43} + ( 2 + \beta_{9} ) q^{44} + ( \beta_{6} + 2 \beta_{7} ) q^{45} + ( 5 + \beta_{2} - \beta_{5} - \beta_{8} + 3 \beta_{9} ) q^{46} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{47} + ( \beta_{1} - 6 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} + 5 \beta_{7} ) q^{48} + ( -6 - \beta_{5} - 2 \beta_{8} ) q^{50} + ( 1 - 3 \beta_{2} - 3 \beta_{5} - 3 \beta_{8} - \beta_{9} ) q^{51} + ( 2 \beta_{3} - 3 \beta_{4} - 7 \beta_{6} - \beta_{7} ) q^{52} + ( 2 - 2 \beta_{2} - 2 \beta_{8} ) q^{53} + ( 4 \beta_{1} + 2 \beta_{3} - 8 \beta_{6} - 2 \beta_{7} ) q^{54} + \beta_{7} q^{55} + ( -2 \beta_{2} - 2 \beta_{5} + 2 \beta_{8} ) q^{57} + ( -3 + 3 \beta_{2} - \beta_{5} - \beta_{8} + \beta_{9} ) q^{58} + ( -2 \beta_{1} + \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{59} + ( 5 + \beta_{2} - 5 \beta_{5} + \beta_{8} + 3 \beta_{9} ) q^{60} + ( -2 \beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{61} + ( -2 \beta_{1} + \beta_{3} + 3 \beta_{4} + 4 \beta_{6} ) q^{62} + ( 12 - 4 \beta_{5} + 3 \beta_{9} ) q^{64} + ( -3 + \beta_{2} + \beta_{5} - \beta_{8} + 3 \beta_{9} ) q^{65} + ( \beta_{3} + \beta_{4} ) q^{66} + ( -3 \beta_{5} + \beta_{8} - \beta_{9} ) q^{67} + ( -2 \beta_{3} - 3 \beta_{4} - 7 \beta_{6} - \beta_{7} ) q^{68} + ( \beta_{1} + 2 \beta_{3} + \beta_{4} - 3 \beta_{7} ) q^{69} + ( 4 - 3 \beta_{5} + \beta_{8} - \beta_{9} ) q^{71} + ( -9 - \beta_{2} - 2 \beta_{5} + \beta_{8} - 3 \beta_{9} ) q^{72} + ( -\beta_{4} - 5 \beta_{6} - 3 \beta_{7} ) q^{73} + ( -4 + 2 \beta_{2} + 4 \beta_{5} ) q^{74} + ( 2 \beta_{1} - 3 \beta_{4} + 6 \beta_{6} - \beta_{7} ) q^{75} + ( 2 \beta_{1} - 6 \beta_{3} + 2 \beta_{4} + 10 \beta_{6} + 6 \beta_{7} ) q^{76} + ( -11 - \beta_{2} - \beta_{5} - \beta_{8} - \beta_{9} ) q^{78} + ( 1 - \beta_{2} + \beta_{5} + \beta_{8} + \beta_{9} ) q^{79} + ( 5 \beta_{1} - 4 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{80} + ( -2 + 4 \beta_{2} + 5 \beta_{5} + \beta_{8} + 2 \beta_{9} ) q^{81} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{82} + ( -2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{83} + ( 9 - \beta_{2} - \beta_{5} - 3 \beta_{8} - \beta_{9} ) q^{85} + ( 3 + \beta_{2} + \beta_{5} + \beta_{8} + \beta_{9} ) q^{86} + ( 4 \beta_{1} + 2 \beta_{3} - 4 \beta_{6} - 4 \beta_{7} ) q^{87} + ( -1 - \beta_{2} + 2 \beta_{5} + \beta_{8} - \beta_{9} ) q^{88} + \beta_{7} q^{89} + ( \beta_{3} + 3 \beta_{6} - \beta_{7} ) q^{90} + ( -7 - \beta_{2} + 7 \beta_{5} + \beta_{8} - 3 \beta_{9} ) q^{92} + ( 5 + \beta_{2} - 2 \beta_{5} + 2 \beta_{8} + 2 \beta_{9} ) q^{93} + ( 4 \beta_{1} - 3 \beta_{3} + \beta_{4} - 2 \beta_{6} + 4 \beta_{7} ) q^{94} + ( -6 + 2 \beta_{2} + 4 \beta_{9} ) q^{95} + ( -6 \beta_{1} + 5 \beta_{3} + 3 \beta_{4} + 10 \beta_{6} ) q^{96} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{7} ) q^{97} + ( 2 + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9} + O(q^{10}) \) \( 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9} + 10 q^{11} + 8 q^{15} + 42 q^{16} + 6 q^{18} + 2 q^{22} + 4 q^{23} + 18 q^{25} + 12 q^{29} - 4 q^{30} - 30 q^{32} - 2 q^{36} + 40 q^{37} - 16 q^{39} - 8 q^{43} + 18 q^{44} + 44 q^{46} - 62 q^{50} + 16 q^{53} - 8 q^{57} - 28 q^{58} + 36 q^{60} + 106 q^{64} - 32 q^{65} - 4 q^{67} + 36 q^{71} - 90 q^{72} - 28 q^{74} - 112 q^{78} + 8 q^{79} - 6 q^{81} + 88 q^{85} + 32 q^{86} - 6 q^{88} - 52 q^{92} + 44 q^{93} - 64 q^{95} + 22 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 26 x^{8} + 245 x^{6} - 1038 x^{4} + 1884 x^{2} - 968\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{9} - 160 \nu^{7} + 1187 \nu^{5} - 3108 \nu^{3} + 1572 \nu \)\()/88\)
\(\beta_{4}\)\(=\)\((\)\( 15 \nu^{9} - 324 \nu^{7} + 2267 \nu^{5} - 5912 \nu^{3} + 3972 \nu \)\()/88\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{8} - 22 \nu^{6} + 157 \nu^{4} - 410 \nu^{2} + 252 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( 25 \nu^{9} - 540 \nu^{7} + 3749 \nu^{5} - 9472 \nu^{3} + 5652 \nu \)\()/88\)
\(\beta_{7}\)\(=\)\((\)\( -71 \nu^{9} + 1516 \nu^{7} - 10355 \nu^{5} + 25672 \nu^{3} - 14876 \nu \)\()/88\)
\(\beta_{8}\)\(=\)\((\)\( 5 \nu^{8} - 106 \nu^{6} + 717 \nu^{4} - 1762 \nu^{2} + 1024 \)\()/4\)
\(\beta_{9}\)\(=\)\((\)\( -5 \nu^{8} + 108 \nu^{6} - 749 \nu^{4} + 1880 \nu^{2} - 1080 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(-\beta_{7} - 4 \beta_{6} + \beta_{4} + 2 \beta_{3} + 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{9} + \beta_{8} + 5 \beta_{5} + 13 \beta_{2} + 34\)
\(\nu^{5}\)\(=\)\(-13 \beta_{7} - 55 \beta_{6} + 18 \beta_{4} + 26 \beta_{3} + 58 \beta_{1}\)
\(\nu^{6}\)\(=\)\(34 \beta_{9} + 18 \beta_{8} + 80 \beta_{5} + 149 \beta_{2} + 277\)
\(\nu^{7}\)\(=\)\(-151 \beta_{7} - 648 \beta_{6} + 229 \beta_{4} + 292 \beta_{3} + 541 \beta_{1}\)
\(\nu^{8}\)\(=\)\(434 \beta_{9} + 239 \beta_{8} + 979 \beta_{5} + 1647 \beta_{2} + 2554\)
\(\nu^{9}\)\(=\)\(-1691 \beta_{7} - 7261 \beta_{6} + 2626 \beta_{4} + 3166 \beta_{3} + 5414 \beta_{1}\)

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
−2.10267
2.10267
−2.15293
2.15293
−2.32267
2.32267
−3.27614
3.27614
−0.903205
0.903205
−2.74816 −2.10267 5.55241 −2.44342 5.77848 0 −9.76260 1.42122 6.71492
1.2 −2.74816 2.10267 5.55241 2.44342 −5.77848 0 −9.76260 1.42122 −6.71492
1.3 −1.14898 −2.15293 −0.679834 3.87589 2.47369 0 3.07909 1.63513 −4.45334
1.4 −1.14898 2.15293 −0.679834 −3.87589 −2.47369 0 3.07909 1.63513 4.45334
1.5 0.566092 −2.32267 −1.67954 −3.58219 −1.31484 0 −2.08296 2.39479 −2.02785
1.6 0.566092 2.32267 −1.67954 3.58219 1.31484 0 −2.08296 2.39479 2.02785
1.7 1.70296 −3.27614 0.900071 0.246676 −5.57913 0 −1.87313 7.73309 0.420079
1.8 1.70296 3.27614 0.900071 −0.246676 5.57913 0 −1.87313 7.73309 −0.420079
1.9 2.62810 −0.903205 4.90690 0.337987 −2.37371 0 7.63960 −2.18422 0.888264
1.10 2.62810 0.903205 4.90690 −0.337987 2.37371 0 7.63960 −2.18422 −0.888264
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.a.l 10
3.b odd 2 1 4851.2.a.cg 10
4.b odd 2 1 8624.2.a.df 10
7.b odd 2 1 inner 539.2.a.l 10
7.c even 3 2 539.2.e.o 20
7.d odd 6 2 539.2.e.o 20
11.b odd 2 1 5929.2.a.bv 10
21.c even 2 1 4851.2.a.cg 10
28.d even 2 1 8624.2.a.df 10
77.b even 2 1 5929.2.a.bv 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.2.a.l 10 1.a even 1 1 trivial
539.2.a.l 10 7.b odd 2 1 inner
539.2.e.o 20 7.c even 3 2
539.2.e.o 20 7.d odd 6 2
4851.2.a.cg 10 3.b odd 2 1
4851.2.a.cg 10 21.c even 2 1
5929.2.a.bv 10 11.b odd 2 1
5929.2.a.bv 10 77.b even 2 1
8624.2.a.df 10 4.b odd 2 1
8624.2.a.df 10 28.d even 2 1

Hecke kernels

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

\( T_{2}^{5} - T_{2}^{4} - 9 T_{2}^{3} + 9 T_{2}^{2} + 12 T_{2} - 8 \)
\( T_{3}^{10} - 26 T_{3}^{8} + 245 T_{3}^{6} - 1038 T_{3}^{4} + 1884 T_{3}^{2} - 968 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -8 + 12 T + 9 T^{2} - 9 T^{3} - T^{4} + T^{5} )^{2} \)
$3$ \( -968 + 1884 T^{2} - 1038 T^{4} + 245 T^{6} - 26 T^{8} + T^{10} \)
$5$ \( -8 + 204 T^{2} - 1214 T^{4} + 365 T^{6} - 34 T^{8} + T^{10} \)
$7$ \( T^{10} \)
$11$ \( ( -1 + T )^{10} \)
$13$ \( -100352 + 69120 T^{2} - 16672 T^{4} + 1696 T^{6} - 72 T^{8} + T^{10} \)
$17$ \( -430592 + 705536 T^{2} - 110752 T^{4} + 6192 T^{6} - 136 T^{8} + T^{10} \)
$19$ \( -6422528 + 1736704 T^{2} - 167424 T^{4} + 7312 T^{6} - 144 T^{8} + T^{10} \)
$23$ \( ( -232 + 28 T + 126 T^{2} - 39 T^{3} - 2 T^{4} + T^{5} )^{2} \)
$29$ \( ( 224 + 528 T + 256 T^{2} - 56 T^{3} - 6 T^{4} + T^{5} )^{2} \)
$31$ \( -3998792 + 1812028 T^{2} - 199230 T^{4} + 8469 T^{6} - 154 T^{8} + T^{10} \)
$37$ \( ( -472 - 756 T - 34 T^{2} + 113 T^{3} - 20 T^{4} + T^{5} )^{2} \)
$41$ \( -25088 + 97280 T^{2} - 51616 T^{4} + 5680 T^{6} - 168 T^{8} + T^{10} \)
$43$ \( ( 256 + 256 T - 48 T^{2} - 56 T^{3} + 4 T^{4} + T^{5} )^{2} \)
$47$ \( -1158537248 + 108191568 T^{2} - 3514992 T^{4} + 52488 T^{6} - 370 T^{8} + T^{10} \)
$53$ \( ( -11264 + 1792 T + 864 T^{2} - 128 T^{3} - 8 T^{4} + T^{5} )^{2} \)
$59$ \( -108162632 + 15918364 T^{2} - 838702 T^{4} + 20469 T^{6} - 234 T^{8} + T^{10} \)
$61$ \( -3527168 + 1196544 T^{2} - 139552 T^{4} + 6928 T^{6} - 144 T^{8} + T^{10} \)
$67$ \( ( 4648 + 1580 T - 270 T^{2} - 103 T^{3} + 2 T^{4} + T^{5} )^{2} \)
$71$ \( ( 88 - 436 T + 518 T^{2} + 25 T^{3} - 18 T^{4} + T^{5} )^{2} \)
$73$ \( -65987072 + 19299840 T^{2} - 1721376 T^{4} + 46800 T^{6} - 400 T^{8} + T^{10} \)
$79$ \( ( -704 + 928 T - 32 T^{2} - 116 T^{3} - 4 T^{4} + T^{5} )^{2} \)
$83$ \( -102760448 + 17694720 T^{2} - 1067008 T^{4} + 27136 T^{6} - 288 T^{8} + T^{10} \)
$89$ \( -8 + 204 T^{2} - 1214 T^{4} + 365 T^{6} - 34 T^{8} + T^{10} \)
$97$ \( -19208 + 718732 T^{2} - 212606 T^{4} + 12541 T^{6} - 226 T^{8} + T^{10} \)
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