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: \( x^{10} - 26x^{8} + 245x^{6} - 1038x^{4} + 1884x^{2} - 968 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + \beta_1 q^{3} + (\beta_{9} + 2) q^{4} + \beta_{7} q^{5} + (\beta_{4} + \beta_{3}) q^{6} + ( - \beta_{9} + \beta_{8} + 2 \beta_{5} - \beta_{2} - 1) q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + \beta_1 q^{3} + (\beta_{9} + 2) q^{4} + \beta_{7} q^{5} + (\beta_{4} + \beta_{3}) q^{6} + ( - \beta_{9} + \beta_{8} + 2 \beta_{5} - \beta_{2} - 1) q^{8} + (\beta_{2} + 2) q^{9} + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{10} + q^{11} + (\beta_{7} - \beta_{6} - 2 \beta_{3} + \beta_1) q^{12} + ( - \beta_{7} - \beta_{6} - \beta_{4}) q^{13} + (\beta_{9} - 2 \beta_{8} + 1) q^{15} + (2 \beta_{9} - \beta_{8} - \beta_{5} - \beta_{2} + 5) q^{16} + (\beta_{7} + \beta_{6} - \beta_{4}) q^{17} + (\beta_{5} + 2 \beta_{2}) q^{18} + (2 \beta_{6} - 2 \beta_{3}) q^{19} + ( - 2 \beta_{6} - \beta_{4} - 2 \beta_{3} + \beta_1) q^{20} + \beta_{5} q^{22} + ( - \beta_{9} + \beta_{8} + \beta_{5}) q^{23} + ( - 2 \beta_{7} + 4 \beta_{6} + \beta_{4} + \beta_{3} - 2 \beta_1) q^{24} + ( - 2 \beta_{9} - 2 \beta_{5} - \beta_{2} + 2) q^{25} + ( - \beta_{7} - 3 \beta_{6} - \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{26} + ( - \beta_{7} - 4 \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_1) q^{27} + ( - \beta_{9} + \beta_{8} - \beta_{5} + \beta_{2} + 1) q^{29} + ( - 3 \beta_{9} + \beta_{8} + \beta_{5} - \beta_{2} - 1) q^{30} + ( - 2 \beta_{3} + \beta_1) q^{31} + ( - 2 \beta_{9} + 5 \beta_{5} - 2 \beta_{2} - 4) q^{32} + \beta_1 q^{33} + ( - \beta_{7} + \beta_{6} - \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{34} + (\beta_{9} - 2 \beta_{5} + 2 \beta_{2}) q^{36} + (\beta_{8} - \beta_{5} + \beta_{2} + 4) q^{37} + (2 \beta_{7} + 6 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{38} + ( - \beta_{9} + \beta_{8} - 3 \beta_{5} - \beta_{2} - 1) q^{39} + ( - \beta_{7} - \beta_{6} + 2 \beta_{4} + 3 \beta_{3} - 4 \beta_1) q^{40} + (\beta_{7} - \beta_{6} + \beta_{4} - 2 \beta_1) q^{41} + (\beta_{9} + \beta_{8} + \beta_{5} + \beta_{2} - 1) q^{43} + (\beta_{9} + 2) q^{44} + (2 \beta_{7} + \beta_{6}) q^{45} + (3 \beta_{9} - \beta_{8} - \beta_{5} + \beta_{2} + 5) q^{46} + ( - \beta_{7} + 2 \beta_{6} + \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{47} + (5 \beta_{7} + 3 \beta_{6} - 2 \beta_{4} - 6 \beta_{3} + \beta_1) q^{48} + ( - 2 \beta_{8} - \beta_{5} - 6) q^{50} + ( - \beta_{9} - 3 \beta_{8} - 3 \beta_{5} - 3 \beta_{2} + 1) q^{51} + ( - \beta_{7} - 7 \beta_{6} - 3 \beta_{4} + 2 \beta_{3}) q^{52} + ( - 2 \beta_{8} - 2 \beta_{2} + 2) q^{53} + ( - 2 \beta_{7} - 8 \beta_{6} + 2 \beta_{3} + 4 \beta_1) q^{54} + \beta_{7} q^{55} + (2 \beta_{8} - 2 \beta_{5} - 2 \beta_{2}) q^{57} + (\beta_{9} - \beta_{8} - \beta_{5} + 3 \beta_{2} - 3) q^{58} + (2 \beta_{7} + 3 \beta_{6} + \beta_{4} - 2 \beta_1) q^{59} + (3 \beta_{9} + \beta_{8} - 5 \beta_{5} + \beta_{2} + 5) q^{60} + (\beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_1) q^{61} + (4 \beta_{6} + 3 \beta_{4} + \beta_{3} - 2 \beta_1) q^{62} + (3 \beta_{9} - 4 \beta_{5} + 12) q^{64} + (3 \beta_{9} - \beta_{8} + \beta_{5} + \beta_{2} - 3) q^{65} + (\beta_{4} + \beta_{3}) q^{66} + ( - \beta_{9} + \beta_{8} - 3 \beta_{5}) q^{67} + ( - \beta_{7} - 7 \beta_{6} - 3 \beta_{4} - 2 \beta_{3}) q^{68} + ( - 3 \beta_{7} + \beta_{4} + 2 \beta_{3} + \beta_1) q^{69} + ( - \beta_{9} + \beta_{8} - 3 \beta_{5} + 4) q^{71} + ( - 3 \beta_{9} + \beta_{8} - 2 \beta_{5} - \beta_{2} - 9) q^{72} + ( - 3 \beta_{7} - 5 \beta_{6} - \beta_{4}) q^{73} + (4 \beta_{5} + 2 \beta_{2} - 4) q^{74} + ( - \beta_{7} + 6 \beta_{6} - 3 \beta_{4} + 2 \beta_1) q^{75} + (6 \beta_{7} + 10 \beta_{6} + 2 \beta_{4} - 6 \beta_{3} + 2 \beta_1) q^{76} + ( - \beta_{9} - \beta_{8} - \beta_{5} - \beta_{2} - 11) q^{78} + (\beta_{9} + \beta_{8} + \beta_{5} - \beta_{2} + 1) q^{79} + (2 \beta_{7} - 2 \beta_{6} - 3 \beta_{4} - 4 \beta_{3} + 5 \beta_1) q^{80} + (2 \beta_{9} + \beta_{8} + 5 \beta_{5} + 4 \beta_{2} - 2) q^{81} + ( - \beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{82} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4}) q^{83} + ( - \beta_{9} - 3 \beta_{8} - \beta_{5} - \beta_{2} + 9) q^{85} + (\beta_{9} + \beta_{8} + \beta_{5} + \beta_{2} + 3) q^{86} + ( - 4 \beta_{7} - 4 \beta_{6} + 2 \beta_{3} + 4 \beta_1) q^{87} + ( - \beta_{9} + \beta_{8} + 2 \beta_{5} - \beta_{2} - 1) q^{88} + \beta_{7} q^{89} + ( - \beta_{7} + 3 \beta_{6} + \beta_{3}) q^{90} + ( - 3 \beta_{9} + \beta_{8} + 7 \beta_{5} - \beta_{2} - 7) q^{92} + (2 \beta_{9} + 2 \beta_{8} - 2 \beta_{5} + \beta_{2} + 5) q^{93} + (4 \beta_{7} - 2 \beta_{6} + \beta_{4} - 3 \beta_{3} + 4 \beta_1) q^{94} + (4 \beta_{9} + 2 \beta_{2} - 6) q^{95} + (10 \beta_{6} + 3 \beta_{4} + 5 \beta_{3} - 6 \beta_1) q^{96} + ( - \beta_{7} + 2 \beta_{3} - 2 \beta_1) q^{97} + (\beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9} + 10 q^{11} + 8 q^{15} + 42 q^{16} + 6 q^{18} + 2 q^{22} + 4 q^{23} + 18 q^{25} + 12 q^{29} - 4 q^{30} - 30 q^{32} - 2 q^{36} + 40 q^{37} - 16 q^{39} - 8 q^{43} + 18 q^{44} + 44 q^{46} - 62 q^{50} + 16 q^{53} - 8 q^{57} - 28 q^{58} + 36 q^{60} + 106 q^{64} - 32 q^{65} - 4 q^{67} + 36 q^{71} - 90 q^{72} - 28 q^{74} - 112 q^{78} + 8 q^{79} - 6 q^{81} + 88 q^{85} + 32 q^{86} - 6 q^{88} - 52 q^{92} + 44 q^{93} - 64 q^{95} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{9} - 160\nu^{7} + 1187\nu^{5} - 3108\nu^{3} + 1572\nu ) / 88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\nu^{9} - 324\nu^{7} + 2267\nu^{5} - 5912\nu^{3} + 3972\nu ) / 88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} - 22\nu^{6} + 157\nu^{4} - 410\nu^{2} + 252 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25\nu^{9} - 540\nu^{7} + 3749\nu^{5} - 9472\nu^{3} + 5652\nu ) / 88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -71\nu^{9} + 1516\nu^{7} - 10355\nu^{5} + 25672\nu^{3} - 14876\nu ) / 88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5\nu^{8} - 106\nu^{6} + 717\nu^{4} - 1762\nu^{2} + 1024 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5\nu^{8} + 108\nu^{6} - 749\nu^{4} + 1880\nu^{2} - 1080 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - 4\beta_{6} + \beta_{4} + 2\beta_{3} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} + \beta_{8} + 5\beta_{5} + 13\beta_{2} + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{7} - 55\beta_{6} + 18\beta_{4} + 26\beta_{3} + 58\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 34\beta_{9} + 18\beta_{8} + 80\beta_{5} + 149\beta_{2} + 277 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -151\beta_{7} - 648\beta_{6} + 229\beta_{4} + 292\beta_{3} + 541\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 434\beta_{9} + 239\beta_{8} + 979\beta_{5} + 1647\beta_{2} + 2554 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -1691\beta_{7} - 7261\beta_{6} + 2626\beta_{4} + 3166\beta_{3} + 5414\beta_1 \) 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
−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 \(S_{2}^{\mathrm{new}}(\Gamma_0(539))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} - T^{4} - 9 T^{3} + 9 T^{2} + 12 T - 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{10} - 26 T^{8} + 245 T^{6} + \cdots - 968 \) Copy content Toggle raw display
$5$ \( T^{10} - 34 T^{8} + 365 T^{6} - 1214 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T - 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - 72 T^{8} + 1696 T^{6} + \cdots - 100352 \) Copy content Toggle raw display
$17$ \( T^{10} - 136 T^{8} + 6192 T^{6} + \cdots - 430592 \) Copy content Toggle raw display
$19$ \( T^{10} - 144 T^{8} + 7312 T^{6} + \cdots - 6422528 \) Copy content Toggle raw display
$23$ \( (T^{5} - 2 T^{4} - 39 T^{3} + 126 T^{2} + \cdots - 232)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 6 T^{4} - 56 T^{3} + 256 T^{2} + \cdots + 224)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 154 T^{8} + 8469 T^{6} + \cdots - 3998792 \) Copy content Toggle raw display
$37$ \( (T^{5} - 20 T^{4} + 113 T^{3} - 34 T^{2} + \cdots - 472)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 168 T^{8} + 5680 T^{6} + \cdots - 25088 \) Copy content Toggle raw display
$43$ \( (T^{5} + 4 T^{4} - 56 T^{3} - 48 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 370 T^{8} + \cdots - 1158537248 \) Copy content Toggle raw display
$53$ \( (T^{5} - 8 T^{4} - 128 T^{3} + 864 T^{2} + \cdots - 11264)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 234 T^{8} + \cdots - 108162632 \) Copy content Toggle raw display
$61$ \( T^{10} - 144 T^{8} + 6928 T^{6} + \cdots - 3527168 \) Copy content Toggle raw display
$67$ \( (T^{5} + 2 T^{4} - 103 T^{3} - 270 T^{2} + \cdots + 4648)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 18 T^{4} + 25 T^{3} + 518 T^{2} + \cdots + 88)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} - 400 T^{8} + \cdots - 65987072 \) Copy content Toggle raw display
$79$ \( (T^{5} - 4 T^{4} - 116 T^{3} - 32 T^{2} + \cdots - 704)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} - 288 T^{8} + \cdots - 102760448 \) Copy content Toggle raw display
$89$ \( T^{10} - 34 T^{8} + 365 T^{6} - 1214 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$97$ \( T^{10} - 226 T^{8} + 12541 T^{6} + \cdots - 19208 \) Copy content Toggle raw display
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