Properties

Label 547.6.a.a
Level 547
Weight 6
Character orbit 547.a
Self dual yes
Analytic conductor 87.730
Analytic rank 1
Dimension 111
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: \(1\)
Dimension: \(111\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 111q - 28q^{2} - 98q^{3} + 1722q^{4} - 801q^{5} - 414q^{6} - 587q^{7} - 1344q^{8} + 8241q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 111q - 28q^{2} - 98q^{3} + 1722q^{4} - 801q^{5} - 414q^{6} - 587q^{7} - 1344q^{8} + 8241q^{9} - 950q^{10} - 1832q^{11} - 4143q^{12} - 4369q^{13} - 4777q^{14} - 3487q^{15} + 26274q^{16} - 13648q^{17} - 10269q^{18} - 5446q^{19} - 26032q^{20} - 8428q^{21} - 8248q^{22} - 24142q^{23} - 18577q^{24} + 58062q^{25} - 17656q^{26} - 33269q^{27} - 23512q^{28} - 33752q^{29} - 12418q^{30} - 13781q^{31} - 44076q^{32} - 39186q^{33} - 7207q^{34} - 30833q^{35} + 120044q^{36} - 61582q^{37} - 91259q^{38} - 20077q^{39} - 66032q^{40} - 54181q^{41} - 69252q^{42} - 38600q^{43} - 95712q^{44} - 190880q^{45} - 9354q^{46} - 83886q^{47} - 173886q^{48} + 194148q^{49} - 70896q^{50} - 60673q^{51} - 145186q^{52} - 286874q^{53} - 116519q^{54} - 74821q^{55} - 240407q^{56} - 95180q^{57} - 66900q^{58} - 135740q^{59} - 144550q^{60} - 227450q^{61} - 308766q^{62} - 249721q^{63} + 347514q^{64} - 290374q^{65} - 178980q^{66} - 91006q^{67} - 521943q^{68} - 414510q^{69} - 165057q^{70} - 236165q^{71} - 527945q^{72} - 184618q^{73} - 206443q^{74} - 243897q^{75} - 221676q^{76} - 751131q^{77} - 306839q^{78} - 107446q^{79} - 856691q^{80} + 382187q^{81} - 244614q^{82} - 499547q^{83} - 330289q^{84} - 287103q^{85} - 272441q^{86} - 391281q^{87} - 588937q^{88} - 740774q^{89} - 687179q^{90} - 237213q^{91} - 1367678q^{92} - 754880q^{93} - 32851q^{94} - 295814q^{95} - 816078q^{96} - 320770q^{97} - 661922q^{98} - 547439q^{99} + O(q^{100}) \)

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 −11.1760 15.2016 92.9024 82.7786 −169.892 77.1869 −680.644 −11.9127 −925.132
1.2 −11.1575 −27.7524 92.4888 −19.1317 309.646 −173.776 −674.902 527.195 213.461
1.3 −10.8972 −3.04130 86.7487 −42.8787 33.1417 139.007 −596.607 −233.750 467.257
1.4 −10.7598 −8.37500 83.7726 78.2630 90.1130 −21.6310 −557.061 −172.859 −842.092
1.5 −10.6929 24.2490 82.3384 −108.003 −259.292 200.220 −538.264 345.014 1154.86
1.6 −10.6698 −9.29200 81.8438 43.9375 99.1434 −209.001 −531.821 −156.659 −468.803
1.7 −10.3059 −5.69318 74.2112 −45.6121 58.6732 125.560 −435.024 −210.588 470.073
1.8 −10.2910 1.80530 73.9047 −105.302 −18.5783 −122.352 −431.241 −239.741 1083.66
1.9 −10.1611 −23.3655 71.2488 −39.6822 237.421 190.155 −398.813 302.949 403.217
1.10 −9.96663 11.0844 67.3337 58.1385 −110.474 77.9760 −352.158 −120.137 −579.445
1.11 −9.83916 24.1089 64.8091 −28.9694 −237.211 28.9470 −322.814 338.239 285.035
1.12 −9.31382 28.8189 54.7473 21.1798 −268.414 16.1468 −211.864 587.530 −197.265
1.13 −9.26272 1.07541 53.7981 53.2230 −9.96122 −134.918 −201.909 −241.843 −492.990
1.14 −8.96480 −25.8952 48.3677 54.0730 232.145 −81.1746 −146.733 427.559 −484.754
1.15 −8.79797 −13.8260 45.4043 −43.4417 121.641 −186.463 −117.931 −51.8414 382.199
1.16 −8.49621 −28.8934 40.1856 −77.1552 245.484 144.097 −69.5467 591.826 655.527
1.17 −8.46822 15.0261 39.7108 103.484 −127.245 −74.7851 −65.2966 −17.2156 −876.328
1.18 −8.25035 −30.3390 36.0683 −26.4241 250.308 −114.994 −33.5649 677.458 218.008
1.19 −8.11392 15.5937 33.8357 −82.1739 −126.526 −140.403 −14.8948 0.163009 666.752
1.20 −8.10730 18.6684 33.7283 −66.0394 −151.350 35.8802 −14.0116 105.509 535.401
See next 80 embeddings (of 111 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.111
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.a 111
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.6.a.a 111 1.a even 1 1 trivial

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database