Properties

Label 547.2.e.a
Level 547
Weight 2
Character orbit 547.e
Analytic conductor 4.368
Analytic rank 0
Dimension 264
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 547.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.36781699056\)
Analytic rank: \(0\)
Dimension: \(264\)
Relative dimension: \(44\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 264q - 2q^{2} - q^{3} - 44q^{4} - 3q^{5} - 10q^{6} - 18q^{7} - q^{8} - 35q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 264q - 2q^{2} - q^{3} - 44q^{4} - 3q^{5} - 10q^{6} - 18q^{7} - q^{8} - 35q^{9} + 13q^{10} - 4q^{11} + 31q^{12} + 7q^{13} - 44q^{14} + 7q^{15} - 76q^{16} + 13q^{17} - 22q^{18} + 11q^{19} + 30q^{20} + 42q^{21} - 28q^{22} + 21q^{23} + 85q^{24} - 37q^{25} - 150q^{26} - 46q^{27} + 34q^{28} + 25q^{29} + 10q^{30} - 25q^{31} + 44q^{32} - 18q^{33} + 41q^{34} - 3q^{35} + 21q^{36} - 29q^{37} - 18q^{38} - 11q^{39} + 90q^{40} - 42q^{41} + 11q^{42} - 43q^{43} - 22q^{44} - 26q^{45} + 78q^{46} - 6q^{47} + 31q^{48} - 70q^{49} + 6q^{50} - 54q^{51} + 87q^{52} + 6q^{53} - 20q^{54} - 5q^{55} - 127q^{56} + 61q^{57} + 10q^{58} - 16q^{59} + 30q^{60} + 27q^{61} + 26q^{62} - 29q^{63} - 57q^{64} - 52q^{65} + 39q^{66} - 2q^{67} - 8q^{68} - 55q^{69} - 34q^{70} - 3q^{71} - 82q^{72} - 8q^{73} + 7q^{74} - 136q^{75} + 125q^{76} - 11q^{77} + 10q^{78} - 15q^{79} - q^{80} - 79q^{81} - 75q^{82} - 72q^{83} - 96q^{84} - 3q^{85} + 121q^{86} + 33q^{87} - 20q^{88} - 60q^{89} - 31q^{90} + 51q^{91} + 104q^{92} + 44q^{93} - 58q^{94} + 95q^{95} - 162q^{96} - 45q^{97} + 103q^{98} - 234q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −0.595191 + 2.60770i −0.495101 0.238428i −4.64393 2.23640i 1.45928 + 1.82988i 0.916430 1.14917i −0.911665 + 0.439035i 5.26051 6.59647i −1.68219 2.10940i −5.64035 + 2.71625i
9.2 −0.580540 + 2.54351i 1.37424 + 0.661801i −4.33048 2.08545i −1.73005 2.16941i −2.48110 + 3.11120i −3.62207 + 1.74430i 4.56510 5.72446i −0.419904 0.526543i 6.52229 3.14097i
9.3 −0.569327 + 2.49438i 2.72887 + 1.31416i −4.09588 1.97247i 0.314234 + 0.394037i −4.83163 + 6.05868i −1.02655 + 0.494361i 4.06156 5.09304i 3.84928 + 4.82684i −1.16178 + 0.559484i
9.4 −0.531576 + 2.32899i −1.23305 0.593806i −3.33966 1.60830i −1.80655 2.26534i 2.03842 2.55610i 0.803964 0.387169i 2.54210 3.18770i −0.702662 0.881110i 6.23627 3.00323i
9.5 −0.512100 + 2.24366i −1.76126 0.848179i −2.96982 1.43019i 1.48262 + 1.85914i 2.80497 3.51732i 1.07224 0.516363i 1.85996 2.33231i 0.512167 + 0.642236i −4.93053 + 2.37442i
9.6 −0.475423 + 2.08296i 0.520852 + 0.250829i −2.31077 1.11281i −1.27883 1.60361i −0.770092 + 0.965665i 1.59880 0.769940i 0.752322 0.943382i −1.66210 2.08421i 3.94824 1.90137i
9.7 −0.458352 + 2.00817i −2.82736 1.36159i −2.02074 0.973135i 0.518676 + 0.650399i 4.03023 5.05375i −4.55321 + 2.19271i 0.311883 0.391089i 4.26960 + 5.35391i −1.54385 + 0.743479i
9.8 −0.455062 + 1.99376i 1.77328 + 0.853969i −1.96604 0.946797i 2.51219 + 3.15019i −2.50956 + 3.14689i 2.12663 1.02413i 0.232245 0.291225i 0.544806 + 0.683165i −7.42391 + 3.57517i
9.9 −0.412323 + 1.80651i 1.82260 + 0.877718i −1.29151 0.621960i 0.884143 + 1.10868i −2.33710 + 2.93063i −0.937096 + 0.451282i −0.654513 + 0.820733i 0.681012 + 0.853963i −2.36739 + 1.14007i
9.10 −0.387202 + 1.69644i −2.32711 1.12068i −0.926057 0.445966i 0.458209 + 0.574576i 2.80223 3.51389i 4.08231 1.96594i −1.05471 + 1.32256i 2.28906 + 2.87040i −1.15216 + 0.554849i
9.11 −0.379656 + 1.66338i 2.89065 + 1.39206i −0.820767 0.395261i −2.52981 3.17229i −3.41299 + 4.27975i 3.93562 1.89529i −1.15847 + 1.45267i 4.54754 + 5.70244i 6.23719 3.00367i
9.12 −0.363899 + 1.59434i 0.165866 + 0.0798767i −0.607575 0.292593i 1.15116 + 1.44351i −0.187709 + 0.235380i −3.58469 + 1.72630i −1.35165 + 1.69492i −1.84934 2.31900i −2.72036 + 1.31006i
9.13 −0.319463 + 1.39966i −0.881702 0.424606i −0.0550501 0.0265107i −0.988555 1.23961i 0.875974 1.09844i −1.79033 + 0.862177i −1.73554 + 2.17630i −1.27336 1.59674i 2.05084 0.987630i
9.14 −0.264233 + 1.15768i −2.41010 1.16064i 0.531537 + 0.255975i −2.54037 3.18553i 1.98048 2.48344i −1.02886 + 0.495474i −1.91751 + 2.40448i 2.59103 + 3.24905i 4.35906 2.09921i
9.15 −0.246653 + 1.08066i 1.76480 + 0.849885i 0.694955 + 0.334673i −2.52009 3.16010i −1.35373 + 1.69752i −4.70478 + 2.26570i −1.91529 + 2.40170i 0.521760 + 0.654266i 4.03657 1.94391i
9.16 −0.228467 + 1.00098i 0.653373 + 0.314648i 0.852178 + 0.410387i −0.0133888 0.0167890i −0.464230 + 0.582126i 1.59577 0.768480i −1.88578 + 2.36470i −1.54258 1.93433i 0.0198643 0.00956615i
9.17 −0.217875 + 0.954575i −0.359677 0.173211i 0.938195 + 0.451811i −0.0979486 0.122824i 0.243708 0.305600i 3.84057 1.84952i −1.85664 + 2.32816i −1.77110 2.22089i 0.138585 0.0667390i
9.18 −0.183655 + 0.804646i −2.23554 1.07658i 1.18821 + 0.572213i 2.30110 + 2.88549i 1.27683 1.60110i 0.712284 0.343018i −1.70783 + 2.14155i 1.96815 + 2.46798i −2.74440 + 1.32163i
9.19 −0.173579 + 0.760500i 2.53318 + 1.21992i 1.25371 + 0.603753i 0.424209 + 0.531942i −1.36745 + 1.71473i −0.719571 + 0.346527i −1.64949 + 2.06839i 3.05835 + 3.83505i −0.478176 + 0.230277i
9.20 −0.102615 + 0.449585i 1.96249 + 0.945083i 1.61034 + 0.775499i 0.923383 + 1.15789i −0.626276 + 0.785325i 3.50362 1.68725i −1.08894 + 1.36549i 1.08770 + 1.36393i −0.615321 + 0.296323i
See next 80 embeddings (of 264 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 544.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
547.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 547.2.e.a 264
547.e even 7 1 inner 547.2.e.a 264
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.2.e.a 264 1.a even 1 1 trivial
547.2.e.a 264 547.e even 7 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(547, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database