Properties

Label 547.6.a.b
Level 547
Weight 6
Character orbit 547.a
Self dual yes
Analytic conductor 87.730
Analytic rank 0
Dimension 117
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 547.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(87.7299494377\)
Analytic rank: \(0\)
Dimension: \(117\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 117q + 24q^{2} + 100q^{3} + 1962q^{4} + 749q^{5} + 306q^{6} + 393q^{7} + 1152q^{8} + 10671q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 117q + 24q^{2} + 100q^{3} + 1962q^{4} + 749q^{5} + 306q^{6} + 393q^{7} + 1152q^{8} + 10671q^{9} + 850q^{10} + 1798q^{11} + 5361q^{12} + 4419q^{13} + 3847q^{14} + 1913q^{15} + 34722q^{16} + 15252q^{17} + 2367q^{18} + 1052q^{19} + 23568q^{20} + 9212q^{21} + 9176q^{22} + 18178q^{23} + 15983q^{24} + 84312q^{25} + 21552q^{26} + 30883q^{27} + 23528q^{28} + 43620q^{29} + 23582q^{30} + 13127q^{31} + 49108q^{32} + 39222q^{33} + 32097q^{34} + 52467q^{35} + 217244q^{36} + 56152q^{37} + 76245q^{38} + 28595q^{39} + 20368q^{40} + 46679q^{41} + 78924q^{42} + 39058q^{43} + 78528q^{44} + 185770q^{45} + 41430q^{46} + 150268q^{47} + 180930q^{48} + 323802q^{49} + 91604q^{50} + 43367q^{51} + 136030q^{52} + 297398q^{53} + 116761q^{54} + 94579q^{55} + 173545q^{56} + 164740q^{57} + 87844q^{58} + 135778q^{59} + 114650q^{60} + 166976q^{61} + 229394q^{62} + 147179q^{63} + 630138q^{64} + 216626q^{65} + 82380q^{66} + 133444q^{67} + 634057q^{68} + 232986q^{69} + 30943q^{70} + 126787q^{71} + 78583q^{72} + 241702q^{73} + 242589q^{74} + 374853q^{75} + 90228q^{76} + 766693q^{77} + 82537q^{78} + 117230q^{79} + 730509q^{80} + 1051409q^{81} + 468130q^{82} + 368467q^{83} + 234191q^{84} + 261997q^{85} + 230487q^{86} + 214239q^{87} + 247415q^{88} + 494902q^{89} + 41821q^{90} + 259647q^{91} + 663682q^{92} + 767344q^{93} + 373605q^{94} + 426186q^{95} + 474162q^{96} + 733038q^{97} + 461746q^{98} + 334651q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −10.9789 22.9161 88.5367 −41.5403 −251.594 −192.368 −620.712 282.146 456.068
1.2 −10.9448 25.7259 87.7879 30.5569 −281.564 −19.7130 −610.585 418.824 −334.438
1.3 −10.9422 −27.7854 87.7311 85.5811 304.033 222.147 −609.819 529.029 −936.443
1.4 −10.7784 5.40058 84.1742 −17.6434 −58.2097 32.8655 −562.355 −213.834 190.168
1.5 −10.6726 −19.1230 81.9040 −34.6115 204.092 2.13984 −532.605 122.689 369.394
1.6 −10.1237 12.9569 70.4896 −58.7262 −131.172 −102.476 −389.658 −75.1192 594.528
1.7 −10.0439 −17.9614 68.8790 −103.266 180.402 −120.875 −370.407 79.6121 1037.19
1.8 −9.87295 20.2571 65.4751 73.8486 −199.997 −245.912 −330.498 167.350 −729.104
1.9 −9.85298 10.3729 65.0812 −1.09588 −102.204 −138.385 −325.948 −135.404 10.7977
1.10 −9.64600 29.4222 61.0453 73.3026 −283.807 119.769 −280.171 622.666 −707.077
1.11 −9.59008 16.3551 59.9697 7.17600 −156.846 253.293 −268.232 24.4881 −68.8184
1.12 −9.53545 −15.5286 58.9249 34.4651 148.072 42.6815 −256.741 −1.86228 −328.640
1.13 −9.42203 −23.2486 56.7747 38.9046 219.049 −22.9512 −233.428 297.499 −366.561
1.14 −9.17150 −10.1851 52.1165 43.6581 93.4126 205.322 −184.499 −139.264 −400.411
1.15 −9.13343 3.11578 51.4195 92.4077 −28.4577 171.805 −177.367 −233.292 −844.000
1.16 −9.01126 −22.0309 49.2028 −64.3778 198.526 52.1292 −155.019 242.362 580.125
1.17 −8.84956 5.64544 46.3147 −23.6980 −49.9597 −115.238 −126.678 −211.129 209.717
1.18 −8.60863 −25.5156 42.1085 104.634 219.655 −175.144 −87.0206 408.048 −900.759
1.19 −8.37802 −14.0231 38.1912 95.5871 117.486 24.4634 −51.8698 −46.3528 −800.830
1.20 −8.28817 23.2227 36.6938 −40.9243 −192.474 80.7858 −38.9028 296.295 339.188
See next 80 embeddings (of 117 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.117
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(547\) \(-1\)

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 547.6.a.b 117
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.6.a.b 117 1.a even 1 1 trivial

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database