Properties

Label 539.4.a.g
Level $539$
Weight $4$
Character orbit 539.a
Self dual yes
Analytic conductor $31.802$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.522072.1
Defining polynomial: \( x^{4} - x^{3} - 12x^{2} + 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_1 - 4) q^{3} + ( - 2 \beta_{3} - \beta_{2} + 6) q^{4} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{5} + ( - 3 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 5) q^{6} + (5 \beta_{2} - 4 \beta_1 - 4) q^{8} + (6 \beta_{3} + 4 \beta_1 + 21) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_1 - 4) q^{3} + ( - 2 \beta_{3} - \beta_{2} + 6) q^{4} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{5} + ( - 3 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 5) q^{6} + (5 \beta_{2} - 4 \beta_1 - 4) q^{8} + (6 \beta_{3} + 4 \beta_1 + 21) q^{9} + ( - 7 \beta_{3} - 9 \beta_{2} + 6 \beta_1 - 1) q^{10} - 11 q^{11} + (13 \beta_{3} + 11 \beta_{2} - 2 \beta_1 - 13) q^{12} + (10 \beta_{3} + 10 \beta_{2} + 3 \beta_1 - 8) q^{13} + (4 \beta_{3} - 8 \beta_{2} + 2 \beta_1 + 68) q^{15} + ( - 6 \beta_{3} - 5 \beta_{2} + 8 \beta_1 + 2) q^{16} + ( - 9 \beta_{3} - 13 \beta_{2} + 5 \beta_1 - 5) q^{17} + (18 \beta_{3} + 7 \beta_{2} + 4 \beta_1 - 10) q^{18} + (7 \beta_{3} + 15 \beta_{2} + 4 \beta_1 - 55) q^{19} + (21 \beta_{3} + 27 \beta_{2} - 10 \beta_1 - 37) q^{20} - 11 \beta_{2} q^{22} + ( - 6 \beta_{3} - 2 \beta_{2} + 18 \beta_1 + 10) q^{23} + (9 \beta_{3} - 25 \beta_{2} + 14 \beta_1 + 119) q^{24} + (2 \beta_{3} - 26 \beta_{2} - 4 \beta_1 + 5) q^{25} + ( - \beta_{3} - 45 \beta_{2} + 14 \beta_1 + 105) q^{26} + ( - 54 \beta_{3} - 18 \beta_{2} - 12 \beta_1 - 122) q^{27} + ( - 8 \beta_{3} + 4 \beta_{2} + 14 \beta_1 - 90) q^{29} + (26 \beta_{3} + 66 \beta_{2} + 4 \beta_1 - 122) q^{30} + (25 \beta_{3} - 7 \beta_{2} + 9 \beta_1 + 15) q^{31} + (28 \beta_{3} - 7 \beta_{2} + 4 \beta_1 + 32) q^{32} + (11 \beta_1 + 44) q^{33} + (32 \beta_{3} + 40 \beta_{2} - 28 \beta_1 - 112) q^{34} + ( - 32 \beta_{3} - 67 \beta_{2} - 4 \beta_1 - 140) q^{36} + ( - 32 \beta_{3} + 14 \beta_{2} - 16 \beta_1 + 20) q^{37} + ( - 11 \beta_{3} - 87 \beta_{2} + 6 \beta_1 + 195) q^{38} + ( - 98 \beta_{3} - 80 \beta_{2} - 2 \beta_1 - 144) q^{39} + ( - 7 \beta_{3} - 65 \beta_{2} + 14 \beta_1 + 231) q^{40} + ( - 23 \beta_{3} - 51 \beta_{2} + 23 \beta_1 - 27) q^{41} + ( - 74 \beta_{2} + 2 \beta_1 + 66) q^{43} + (22 \beta_{3} + 11 \beta_{2} - 66) q^{44} + ( - 35 \beta_{3} + \beta_{2} - 42 \beta_1 - 227) q^{45} + (52 \beta_{3} + 48 \beta_{2} - 48 \beta_1 + 92) q^{46} + ( - \beta_{3} + 15 \beta_{2} + 39 \beta_1 + 25) q^{47} + ( - 3 \beta_{3} + 43 \beta_{2} + 6 \beta_1 - 221) q^{48} + (42 \beta_{3} + 21 \beta_{2} + 12 \beta_1 - 394) q^{50} + (54 \beta_{3} + 92 \beta_{2} + 6 \beta_1 - 48) q^{51} + (51 \beta_{3} + 87 \beta_{2} - 54 \beta_1 - 491) q^{52} + (10 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 150) q^{53} + ( - 54 \beta_{3} + 46 \beta_{2} - 84 \beta_1 - 42) q^{54} + ( - 11 \beta_{3} - 11 \beta_{2} + 22 \beta_1 + 33) q^{55} + ( - 104 \beta_{3} - 96 \beta_{2} + 64 \beta_1 - 4) q^{57} + (26 \beta_{3} - 56 \beta_{2} - 44 \beta_1 + 166) q^{58} + (24 \beta_{3} - 124 \beta_{2} - 3 \beta_1 - 4) q^{59} + ( - 126 \beta_{3} - 198 \beta_{2} + 28 \beta_1 + 270) q^{60} + ( - 120 \beta_{3} - 24 \beta_{2} - 29 \beta_1 + 14) q^{61} + (66 \beta_{3} - 44 \beta_{2} + 32 \beta_1 - 178) q^{62} + (102 \beta_{3} - \beta_{2} - 16 \beta_1 - 234) q^{64} + ( - 142 \beta_{3} - 146 \beta_{2} + 54 \beta_1 - 54) q^{65} + (33 \beta_{3} + 55 \beta_{2} - 22 \beta_1 + 55) q^{66} + (124 \beta_{3} + 120 \beta_{2} - 62 \beta_1 + 364) q^{67} + ( - 60 \beta_{3} - 172 \beta_{2} + 80 \beta_1 + 300) q^{68} + ( - 72 \beta_{3} + 28 \beta_{2} + 4 \beta_1 - 588) q^{69} + (96 \beta_{3} + 92 \beta_{2} - 26 \beta_1 + 480) q^{71} + ( - 54 \beta_{3} - 37 \beta_{2} - 88 \beta_1 - 718) q^{72} + ( - 61 \beta_{3} - 217 \beta_{2} + 47 \beta_1 - 449) q^{73} + ( - 108 \beta_{3} + 86 \beta_{2} - 32 \beta_1 + 276) q^{74} + (92 \beta_{3} + 124 \beta_{2} - 63 \beta_1 + 232) q^{75} + (125 \beta_{3} + 201 \beta_{2} - 66 \beta_1 - 693) q^{76} + (56 \beta_{3} + 228 \beta_{2} - 192 \beta_1 - 640) q^{78} + ( - 114 \beta_{3} + 28 \beta_{2} - 6 \beta_1 - 320) q^{79} + ( - 3 \beta_{3} + 115 \beta_{2} + 38 \beta_1 - 509) q^{80} + (234 \beta_{3} + 252 \beta_{2} + 140 \beta_1 + 557) q^{81} + (148 \beta_{3} + 116 \beta_{2} - 92 \beta_1 - 484) q^{82} + (43 \beta_{3} - 25 \beta_{2} + 76 \beta_1 - 87) q^{83} + (82 \beta_{3} + 186 \beta_{2} - 2 \beta_1 - 342) q^{85} + (154 \beta_{3} + 142 \beta_{2} - 4 \beta_1 - 1026) q^{86} + ( - 56 \beta_{3} + 4 \beta_{2} + 122 \beta_1 - 84) q^{87} + ( - 55 \beta_{2} + 44 \beta_1 + 44) q^{88} + (104 \beta_{3} - 44 \beta_{2} + 48 \beta_1 + 770) q^{89} + ( - 163 \beta_{3} - 165 \beta_{2} + 14 \beta_1 - 21) q^{90} + ( - 140 \beta_{3} - 144 \beta_{2} + 56 \beta_1 + 92) q^{92} + ( - 158 \beta_{3} - 40 \beta_{2} - 104 \beta_1 - 388) q^{93} + (86 \beta_{3} + 52 \beta_{2} - 80 \beta_1 + 410) q^{94} + ( - 220 \beta_{3} - 216 \beta_{2} + 196 \beta_1 - 4) q^{95} + ( - 143 \beta_{3} - 49 \beta_{2} - 130 \beta_1 - 305) q^{96} + ( - 32 \beta_{3} + 176 \beta_{2} - 154 \beta_1 + 174) q^{97} + ( - 66 \beta_{3} - 44 \beta_1 - 231) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 14 q^{3} + 26 q^{4} - 10 q^{5} - 14 q^{6} - 18 q^{8} + 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 14 q^{3} + 26 q^{4} - 10 q^{5} - 14 q^{6} - 18 q^{8} + 76 q^{9} + 2 q^{10} - 44 q^{11} - 70 q^{12} - 58 q^{13} + 284 q^{15} + 2 q^{16} - 4 q^{17} - 62 q^{18} - 258 q^{19} - 182 q^{20} + 22 q^{22} + 8 q^{23} + 498 q^{24} + 80 q^{25} + 482 q^{26} - 428 q^{27} - 396 q^{29} - 628 q^{30} + 56 q^{31} + 134 q^{32} + 154 q^{33} - 472 q^{34} - 418 q^{36} + 84 q^{37} + 942 q^{38} - 412 q^{39} + 1026 q^{40} - 52 q^{41} + 408 q^{43} - 286 q^{44} - 826 q^{45} + 368 q^{46} - 8 q^{47} - 982 q^{48} - 1642 q^{50} - 388 q^{51} - 2030 q^{52} + 624 q^{53} - 92 q^{54} + 110 q^{55} + 48 q^{57} + 864 q^{58} + 238 q^{59} + 1420 q^{60} + 162 q^{61} - 688 q^{62} - 902 q^{64} - 32 q^{65} + 154 q^{66} + 1340 q^{67} + 1384 q^{68} - 2416 q^{69} + 1788 q^{71} - 2622 q^{72} - 1456 q^{73} + 996 q^{74} + 806 q^{75} - 3042 q^{76} - 2632 q^{78} - 1324 q^{79} - 2342 q^{80} + 1444 q^{81} - 1984 q^{82} - 450 q^{83} - 1736 q^{85} - 4380 q^{86} - 588 q^{87} + 198 q^{88} + 3072 q^{89} + 218 q^{90} + 544 q^{92} - 1264 q^{93} + 1696 q^{94} + 24 q^{95} - 862 q^{96} + 652 q^{97} - 836 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 12x^{2} + 5x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - 12\nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 11\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 13\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta_{2} + 2\beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{3} + 6\beta_{2} + \beta _1 + 9 \) 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
−3.20317
0.555307
3.79597
−0.148103
−4.89098 −6.57251 15.9217 −15.5514 32.1460 0 −38.7449 16.1978 76.0614
1.2 −3.24550 5.49244 2.53327 16.0955 −17.8257 0 17.7423 3.16692 −52.2379
1.3 1.53253 −10.1459 −5.65135 −8.69995 −15.5490 0 −20.9211 75.9402 −13.3330
1.4 4.60395 −2.77399 13.1964 −1.84418 −12.7713 0 23.9238 −19.3050 −8.49053
\(n\): 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.g 4
7.b odd 2 1 77.4.a.d 4
21.c even 2 1 693.4.a.l 4
28.d even 2 1 1232.4.a.s 4
35.c odd 2 1 1925.4.a.p 4
77.b even 2 1 847.4.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.a.d 4 7.b odd 2 1
539.4.a.g 4 1.a even 1 1 trivial
693.4.a.l 4 21.c even 2 1
847.4.a.d 4 77.b even 2 1
1232.4.a.s 4 28.d even 2 1
1925.4.a.p 4 35.c odd 2 1

Hecke kernels

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

\( T_{2}^{4} + 2T_{2}^{3} - 27T_{2}^{2} - 40T_{2} + 112 \) Copy content Toggle raw display
\( T_{3}^{4} + 14T_{3}^{3} + 6T_{3}^{2} - 436T_{3} - 1016 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} - 27 T^{2} - 40 T + 112 \) Copy content Toggle raw display
$3$ \( T^{4} + 14 T^{3} + 6 T^{2} + \cdots - 1016 \) Copy content Toggle raw display
$5$ \( T^{4} + 10 T^{3} - 240 T^{2} + \cdots - 4016 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T + 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 58 T^{3} - 4926 T^{2} + \cdots - 4947656 \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} - 6186 T^{2} + \cdots - 2705024 \) Copy content Toggle raw display
$19$ \( T^{4} + 258 T^{3} + \cdots - 14423904 \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} - 22392 T^{2} + \cdots - 17449856 \) Copy content Toggle raw display
$29$ \( T^{4} + 396 T^{3} + \cdots + 22336464 \) Copy content Toggle raw display
$31$ \( T^{4} - 56 T^{3} - 31890 T^{2} + \cdots - 11250248 \) Copy content Toggle raw display
$37$ \( T^{4} - 84 T^{3} - 63516 T^{2} + \cdots + 11157312 \) Copy content Toggle raw display
$41$ \( T^{4} + 52 T^{3} + \cdots - 659233664 \) Copy content Toggle raw display
$43$ \( T^{4} - 408 T^{3} + \cdots + 1210397376 \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} - 121650 T^{2} + \cdots - 318931592 \) Copy content Toggle raw display
$53$ \( T^{4} - 624 T^{3} + \cdots + 403923072 \) Copy content Toggle raw display
$59$ \( T^{4} - 238 T^{3} + \cdots + 17599820728 \) Copy content Toggle raw display
$61$ \( T^{4} - 162 T^{3} + \cdots + 6668930664 \) Copy content Toggle raw display
$67$ \( T^{4} - 1340 T^{3} + \cdots - 140865466496 \) Copy content Toggle raw display
$71$ \( T^{4} - 1788 T^{3} + \cdots - 72982082688 \) Copy content Toggle raw display
$73$ \( T^{4} + 1456 T^{3} + \cdots - 322052228384 \) Copy content Toggle raw display
$79$ \( T^{4} + 1324 T^{3} + \cdots - 59537293568 \) Copy content Toggle raw display
$83$ \( T^{4} + 450 T^{3} + \cdots + 21951092064 \) Copy content Toggle raw display
$89$ \( T^{4} - 3072 T^{3} + \cdots - 109303561968 \) Copy content Toggle raw display
$97$ \( T^{4} - 652 T^{3} + \cdots + 868634650768 \) Copy content Toggle raw display
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