Properties

Label 547.8.a.b
Level 547
Weight 8
Character orbit 547.a
Self dual yes
Analytic conductor 170.875
Analytic rank 0
Dimension 162
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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: \(0\)
Dimension: \(162\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 162q + 48q^{2} + 310q^{3} + 10650q^{4} + 3999q^{5} + 1998q^{6} + 4301q^{7} + 9216q^{8} + 124464q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 162q + 48q^{2} + 310q^{3} + 10650q^{4} + 3999q^{5} + 1998q^{6} + 4301q^{7} + 9216q^{8} + 124464q^{9} + 7430q^{10} + 19840q^{11} + 55737q^{12} + 51223q^{13} + 75679q^{14} + 44609q^{15} + 709506q^{16} + 258906q^{17} + 135171q^{18} + 80362q^{19} + 506432q^{20} + 138572q^{21} + 158320q^{22} + 571410q^{23} + 325871q^{24} + 2732541q^{25} + 488640q^{26} + 772231q^{27} + 699304q^{28} + 968170q^{29} + 301526q^{30} + 348203q^{31} + 1078196q^{32} + 1536618q^{33} + 870073q^{34} + 1089291q^{35} + 8775356q^{36} + 2226256q^{37} + 3884597q^{38} + 923555q^{39} + 2518352q^{40} + 1825935q^{41} + 3419892q^{42} + 1582376q^{43} + 4352040q^{44} + 9457186q^{45} + 1012278q^{46} + 4801410q^{47} + 9073674q^{48} + 21448221q^{49} + 2366848q^{50} + 3749747q^{51} + 7035334q^{52} + 17191348q^{53} + 5748697q^{54} + 5271331q^{55} + 14854657q^{56} + 5393884q^{57} + 4036260q^{58} + 8263804q^{59} + 10193498q^{60} + 12366404q^{61} + 18470554q^{62} + 15526895q^{63} + 49399626q^{64} + 17325330q^{65} + 11279868q^{66} + 5477434q^{67} + 44562265q^{68} + 26851278q^{69} + 9428823q^{70} + 12108395q^{71} + 22153063q^{72} + 13995388q^{73} + 13478769q^{74} + 24654171q^{75} + 8225460q^{76} + 61240119q^{77} + 17624449q^{78} + 8215066q^{79} + 65708461q^{80} + 112675190q^{81} + 29179962q^{82} + 33597369q^{83} + 13895447q^{84} + 24308391q^{85} + 22043075q^{86} + 25661967q^{87} + 32743591q^{88} + 47538968q^{89} + 46132321q^{90} + 24574095q^{91} + 108017386q^{92} + 63765304q^{93} + 41203657q^{94} + 38032578q^{95} + 41465754q^{96} + 45954628q^{97} + 37164970q^{98} + 43882333q^{99} + O(q^{100}) \)

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 −22.5767 33.0813 381.706 −358.894 −746.864 574.689 −5727.83 −1092.63 8102.63
1.2 −22.4516 −37.2542 376.074 77.8611 836.417 1130.45 −5569.67 −799.124 −1748.11
1.3 −22.2173 16.7054 365.608 173.020 −371.148 −1471.69 −5279.02 −1907.93 −3844.03
1.4 −21.7039 18.5452 343.061 347.311 −402.503 −741.863 −4667.67 −1843.08 −7538.02
1.5 −20.8479 45.1465 306.633 322.346 −941.209 −5.40800 −3724.13 −148.790 −6720.22
1.6 −20.8479 65.8328 306.633 502.171 −1372.47 1563.55 −3724.12 2146.95 −10469.2
1.7 −20.8460 −48.0992 306.555 −148.257 1002.67 −895.682 −3722.16 126.530 3090.56
1.8 −20.7350 −17.8303 301.938 −317.707 369.710 −851.936 −3606.60 −1869.08 6587.63
1.9 −20.7005 −6.00300 300.512 −17.8219 124.265 −716.757 −3571.08 −2150.96 368.922
1.10 −20.3517 46.7393 286.191 64.7634 −951.224 714.717 −3219.46 −2.43807 −1318.04
1.11 −20.3261 92.1539 285.149 506.323 −1873.13 −1193.46 −3194.21 6305.34 −10291.6
1.12 −19.9380 −73.5654 269.525 155.076 1466.75 −1751.54 −2821.73 3224.86 −3091.91
1.13 −19.8867 −90.0590 267.483 141.094 1790.98 585.768 −2773.86 5923.62 −2805.90
1.14 −19.7278 −72.6529 261.187 49.2471 1433.28 1221.88 −2627.48 3091.44 −971.538
1.15 −19.2376 56.5363 242.086 −95.4839 −1087.62 −118.574 −2194.74 1009.35 1836.88
1.16 −19.0596 −63.9317 235.270 −257.662 1218.51 712.325 −2044.53 1900.26 4910.95
1.17 −18.8770 6.74768 228.342 −192.693 −127.376 1188.09 −1894.17 −2141.47 3637.46
1.18 −18.4008 82.6740 210.589 −485.077 −1521.27 −163.489 −1519.70 4647.99 8925.80
1.19 −18.3148 82.2716 207.430 −333.251 −1506.78 748.313 −1454.75 4581.61 6103.42
1.20 −17.8711 9.30587 191.378 516.703 −166.306 −1632.07 −1132.63 −2100.40 −9234.07
See next 80 embeddings (of 162 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.162
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.b 162
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.8.a.b 162 1.a even 1 1 trivial

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database