Properties

Label 731.2.u.a
Level 731
Weight 2
Character orbit 731.u
Analytic conductor 5.837
Analytic rank 0
Dimension 348
CM No

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Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.u (of order \(21\) and degree \(12\))

Newform invariants

Self dual: No
Analytic conductor: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(348\)
Relative dimension: \(29\) over \(\Q(\zeta_{21})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 348q - 2q^{2} + 5q^{3} - 54q^{4} + 3q^{5} + 8q^{6} - 71q^{7} - 26q^{8} + 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 348q - 2q^{2} + 5q^{3} - 54q^{4} + 3q^{5} + 8q^{6} - 71q^{7} - 26q^{8} + 16q^{9} - 4q^{10} + 8q^{12} - 9q^{13} - 19q^{14} - 2q^{15} - 30q^{16} + 29q^{17} + 180q^{18} + 4q^{19} - 21q^{20} + 26q^{21} - 53q^{22} + 51q^{23} + 28q^{24} + 10q^{25} - 6q^{26} - 28q^{27} + 25q^{28} - 94q^{29} - 30q^{30} - 52q^{31} - 23q^{32} - 5q^{33} + q^{34} + 12q^{35} - 129q^{36} - 82q^{37} - 29q^{38} - 123q^{39} + 97q^{40} - 27q^{41} + 56q^{42} - 21q^{43} - 122q^{44} - 184q^{45} + 26q^{46} + 114q^{47} + 34q^{48} - 195q^{49} - 39q^{50} - 10q^{51} - 49q^{52} + 10q^{53} + 10q^{54} - 22q^{55} + 158q^{56} - 72q^{57} - 18q^{58} + q^{59} - 92q^{60} + 98q^{61} - 45q^{62} - 97q^{63} - 112q^{64} - 105q^{65} + 546q^{66} - 11q^{67} + 20q^{68} - 5q^{69} - 80q^{70} + 69q^{71} - 143q^{72} + 42q^{73} + 77q^{74} + 35q^{75} - 208q^{76} + 95q^{77} - 124q^{78} + 10q^{79} + 62q^{80} - 85q^{81} - 212q^{82} + 47q^{83} + 50q^{84} + 50q^{85} + 132q^{86} - 78q^{87} - 64q^{88} + 33q^{89} + 461q^{90} + 2q^{91} - 4q^{92} - 140q^{93} + 16q^{94} + 108q^{95} - 178q^{96} + 41q^{97} - 12q^{98} + 162q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
52.1 −1.63866 2.05481i −2.28610 0.344574i −1.09201 + 4.78443i −3.10060 2.11395i 3.03810 + 5.26215i 0.378768 0.656045i 6.88469 3.31549i 2.24080 + 0.691196i 0.737047 + 9.83521i
52.2 −1.62919 2.04294i −1.11228 0.167649i −1.07429 + 4.70679i 3.54342 + 2.41587i 1.46961 + 2.54545i 1.14433 1.98204i 6.65741 3.20604i −1.65766 0.511320i −0.837442 11.1749i
52.3 −1.49691 1.87707i −2.23244 0.336487i −0.837601 + 3.66977i 0.525293 + 0.358138i 2.71016 + 4.69414i −0.291720 + 0.505274i 3.81602 1.83770i 2.00386 + 0.618108i −0.114067 1.52211i
52.4 −1.45147 1.82009i 1.04795 + 0.157954i −0.760906 + 3.33375i 0.648967 + 0.442458i −1.23359 2.13663i 0.330221 0.571959i 2.97727 1.43378i −1.79346 0.553209i −0.136644 1.82339i
52.5 −1.27132 1.59418i 0.488205 + 0.0735850i −0.480123 + 2.10356i −1.73427 1.18240i −0.503355 0.871836i 1.29743 2.24721i 0.289629 0.139478i −2.63379 0.812416i 0.319838 + 4.26794i
52.6 −0.951366 1.19298i 0.415637 + 0.0626473i −0.0730511 + 0.320058i 2.78277 + 1.89726i −0.320687 0.555446i −1.33438 + 2.31122i −2.29821 + 1.10676i −2.69789 0.832188i −0.384048 5.12476i
52.7 −0.914320 1.14652i −0.681161 0.102669i −0.0334880 + 0.146720i −0.399887 0.272639i 0.505088 + 0.874838i −2.43725 + 4.22144i −2.44363 + 1.17679i −2.41328 0.744398i 0.0530392 + 0.707758i
52.8 −0.898774 1.12703i 2.02130 + 0.304663i −0.0173535 + 0.0760309i −1.87413 1.27776i −1.47333 2.55189i −1.91386 + 3.31491i −2.49625 + 1.20213i 1.12613 + 0.347365i 0.244349 + 3.26062i
52.9 −0.760722 0.953916i 2.86326 + 0.431567i 0.113785 0.498526i −0.172885 0.117871i −1.76647 3.05961i 0.654433 1.13351i −2.76066 + 1.32946i 5.14527 + 1.58711i 0.0190785 + 0.254585i
52.10 −0.623125 0.781374i −1.01366 0.152785i 0.222781 0.976068i 1.41491 + 0.964669i 0.512256 + 0.887254i 1.69493 2.93571i −2.70238 + 1.30140i −1.86255 0.574521i −0.127898 1.70668i
52.11 −0.592982 0.743576i −3.02986 0.456678i 0.243764 1.06800i 2.13428 + 1.45513i 1.45708 + 2.52373i −2.14683 + 3.71842i −2.65246 + 1.27736i 6.10478 + 1.88307i −0.183592 2.44986i
52.12 −0.250802 0.314495i −2.19641 0.331055i 0.409036 1.79210i −1.91395 1.30491i 0.446747 + 0.773789i −0.720517 + 1.24797i −1.39103 + 0.669886i 1.84788 + 0.569996i 0.0696339 + 0.929200i
52.13 −0.140426 0.176089i 1.06730 + 0.160869i 0.433754 1.90040i 3.39049 + 2.31160i −0.121549 0.210530i 0.225543 0.390652i −0.801395 + 0.385931i −1.75348 0.540876i −0.0690673 0.921639i
52.14 −0.0413957 0.0519085i 2.72264 + 0.410371i 0.444061 1.94556i 1.77635 + 1.21110i −0.0914035 0.158316i 1.41437 2.44976i −0.239010 + 0.115101i 4.37762 + 1.35032i −0.0106670 0.142342i
52.15 0.0261406 + 0.0327793i −0.733041 0.110488i 0.444651 1.94814i −0.752885 0.513308i −0.0155404 0.0269167i 1.97546 3.42159i 0.151031 0.0727325i −2.34158 0.722281i −0.00285499 0.0380972i
52.16 0.205456 + 0.257634i −1.99291 0.300383i 0.420879 1.84399i 0.980625 + 0.668579i −0.332067 0.575157i 0.142364 0.246582i 1.15533 0.556379i 1.01474 + 0.313006i 0.0292269 + 0.390006i
52.17 0.226429 + 0.283933i 0.883362 + 0.133145i 0.415694 1.82127i −1.90936 1.30178i 0.162214 + 0.280964i −0.768024 + 1.33026i 1.26564 0.609502i −2.10412 0.649034i −0.0627162 0.836890i
52.18 0.427167 + 0.535651i −3.38252 0.509832i 0.340592 1.49223i −0.790363 0.538860i −1.17181 2.02963i 0.823766 1.42680i 2.17935 1.04952i 8.31477 + 2.56477i −0.0489762 0.653542i
52.19 0.607502 + 0.761783i 2.59131 + 0.390577i 0.233787 1.02429i −0.368169 0.251013i 1.27669 + 2.21129i 0.0206336 0.0357384i 2.67804 1.28968i 3.69561 + 1.13994i −0.0324456 0.432956i
52.20 0.615504 + 0.771817i 1.80469 + 0.272014i 0.228185 0.999743i 1.07898 + 0.735639i 0.900850 + 1.56032i −2.48864 + 4.31044i 2.69092 1.29588i 0.316206 + 0.0975368i 0.0963399 + 1.28557i
See next 80 embeddings (of 348 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 698.29
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{348} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\).