Properties

Label 547.8.a.a
Level 547
Weight 8
Character orbit 547.a
Self dual yes
Analytic conductor 170.875
Analytic rank 1
Dimension 156
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 547.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.874608940\)
Analytic rank: \(1\)
Dimension: \(156\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 156q - 56q^{2} - 284q^{3} + 9690q^{4} - 3751q^{5} - 2322q^{6} - 2559q^{7} - 10752q^{8} + 102594q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 156q - 56q^{2} - 284q^{3} + 9690q^{4} - 3751q^{5} - 2322q^{6} - 2559q^{7} - 10752q^{8} + 102594q^{9} - 10570q^{10} - 20090q^{11} - 58311q^{12} - 63021q^{13} - 45057q^{14} - 36391q^{15} + 574338q^{16} - 232394q^{17} - 92277q^{18} - 43100q^{19} - 485568q^{20} - 231868q^{21} - 225008q^{22} - 401950q^{23} - 503569q^{24} + 2076291q^{25} - 530768q^{26} - 959873q^{27} - 617816q^{28} - 1275618q^{29} - 778474q^{30} - 485945q^{31} - 1903692q^{32} - 1050846q^{33} - 466263q^{34} - 1826209q^{35} + 5276156q^{36} - 2129902q^{37} - 2480555q^{38} - 974653q^{39} - 937648q^{40} - 2309325q^{41} - 2803500q^{42} - 1756918q^{43} - 3314520q^{44} - 7492064q^{45} - 1323786q^{46} - 6203828q^{47} - 7957494q^{48} + 15095175q^{49} - 5758152q^{50} - 1556293q^{51} - 7587898q^{52} - 13775068q^{53} - 6848423q^{54} - 4045669q^{55} - 8326655q^{56} - 9421556q^{57} - 4938892q^{58} - 7755758q^{59} - 5358502q^{60} - 11693582q^{61} - 14895366q^{62} - 9477805q^{63} + 31311690q^{64} - 15629670q^{65} - 5969892q^{66} - 9560716q^{67} - 34045735q^{68} - 17825946q^{69} - 4291177q^{70} - 13661197q^{71} - 21516953q^{72} - 17125972q^{73} - 19749599q^{74} - 21752079q^{75} - 15479244q^{76} - 55632329q^{77} - 12746879q^{78} - 9534338q^{79} - 61267539q^{80} + 58468208q^{81} - 29265046q^{82} - 38447793q^{83} - 33520873q^{84} - 22365109q^{85} - 21208733q^{86} - 27018273q^{87} - 40855385q^{88} - 62436196q^{89} - 19477679q^{90} - 20640165q^{91} - 78867734q^{92} - 77801528q^{93} + 2996793q^{94} - 30557422q^{95} - 82397286q^{96} - 56264748q^{97} - 72954494q^{98} - 43444577q^{99} + O(q^{100}) \)

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 −22.1152 83.7935 361.080 −144.371 −1853.11 55.2475 −5154.61 4834.35 3192.79
1.2 −22.0518 −47.8990 358.283 −388.461 1056.26 −154.038 −5078.15 107.313 8566.26
1.3 −21.6719 −67.1835 341.672 441.406 1455.99 −502.268 −4630.68 2326.62 −9566.11
1.4 −21.6321 −15.5597 339.946 99.8206 336.588 1402.54 −4584.82 −1944.90 −2159.32
1.5 −21.5657 −93.3798 337.078 −365.072 2013.80 −27.9939 −4508.91 6532.79 7873.02
1.6 −21.1866 −59.2391 320.872 86.1150 1255.07 −167.825 −4086.30 1322.27 −1824.48
1.7 −21.1072 70.0639 317.514 140.672 −1478.85 −431.091 −4000.10 2721.94 −2969.18
1.8 −20.7147 73.8020 301.098 73.8971 −1528.79 1490.12 −3585.68 3259.74 −1530.76
1.9 −20.7120 65.0038 300.986 −306.003 −1346.36 −1693.80 −3582.88 2038.50 6337.92
1.10 −20.6976 32.5628 300.390 −462.112 −673.972 −452.526 −3568.07 −1126.66 9564.62
1.11 −20.4450 −5.52261 289.998 429.047 112.910 1075.95 −3312.04 −2156.50 −8771.86
1.12 −20.1873 34.9985 279.528 −313.532 −706.526 1440.96 −3058.93 −962.103 6329.36
1.13 −19.9440 23.7563 269.761 −441.947 −473.794 8.48766 −2827.28 −1622.64 8814.16
1.14 −19.8798 −54.3245 267.205 −433.266 1079.96 1338.12 −2767.35 764.147 8613.23
1.15 −19.7400 −43.2832 261.668 −352.760 854.411 −1491.16 −2638.61 −313.564 6963.48
1.16 −19.6554 −60.4425 258.337 408.742 1188.02 556.317 −2561.83 1466.29 −8034.01
1.17 −19.3499 9.03641 246.418 20.2179 −174.853 193.981 −2291.37 −2105.34 −391.214
1.18 −18.7781 −27.5200 224.616 480.924 516.772 −221.639 −1814.26 −1429.65 −9030.83
1.19 −18.3687 −35.4413 209.409 105.765 651.010 751.780 −1495.38 −930.915 −1942.76
1.20 −18.2383 47.2008 204.635 260.548 −860.861 −206.356 −1397.68 40.9151 −4751.95
See next 80 embeddings (of 156 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.156
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 547.8.a.a 156
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.8.a.a 156 1.a even 1 1 trivial

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database