Properties

Label 731.2.be.a
Level 731
Weight 2
Character orbit 731.be
Analytic conductor 5.837
Analytic rank 0
Dimension 1536
CM No

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.be (of order \(56\) and degree \(24\))

Newform invariants

Self dual: No
Analytic conductor: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(1536\)
Relative dimension: \(64\) over \(\Q(\zeta_{56})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{56}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1536q - 20q^{2} - 20q^{3} - 28q^{5} - 48q^{6} - 48q^{7} - 36q^{8} - 44q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 1536q - 20q^{2} - 20q^{3} - 28q^{5} - 48q^{6} - 48q^{7} - 36q^{8} - 44q^{9} - 20q^{10} - 20q^{11} - 20q^{12} - 52q^{14} - 20q^{15} + 192q^{16} - 20q^{17} - 64q^{18} - 20q^{19} - 52q^{20} - 20q^{22} - 16q^{23} - 12q^{25} - 4q^{26} - 44q^{27} + 28q^{28} - 20q^{29} - 4q^{31} - 36q^{32} + 32q^{33} - 52q^{34} - 8q^{35} - 120q^{36} - 16q^{37} - 44q^{39} - 108q^{40} - 36q^{41} - 48q^{42} - 76q^{43} + 48q^{44} - 100q^{45} - 100q^{46} - 132q^{48} - 88q^{49} - 320q^{50} - 20q^{51} - 40q^{52} - 24q^{53} + 60q^{54} + 88q^{56} + 52q^{57} + 36q^{58} - 44q^{59} + 36q^{60} + 16q^{61} + 84q^{62} - 44q^{63} - 60q^{65} + 76q^{66} + 120q^{67} - 52q^{68} - 88q^{69} - 4q^{70} - 68q^{71} - 40q^{73} - 252q^{74} + 120q^{75} - 208q^{76} - 100q^{77} + 204q^{78} + 80q^{79} + 32q^{80} - 140q^{82} - 96q^{83} - 104q^{84} + 192q^{85} - 24q^{86} - 88q^{87} + 128q^{88} - 52q^{90} + 200q^{91} + 592q^{92} - 400q^{93} - 188q^{94} - 84q^{95} + 88q^{96} - 60q^{97} + 112q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −2.34206 1.47161i 0.145057 + 0.102923i 2.45183 + 5.09127i 0.480201 + 1.66682i −0.188268 0.454520i −3.24749 + 1.34515i 1.13066 10.0349i −0.980389 2.80179i 1.32825 4.61045i
59.2 −2.26390 1.42250i −1.53996 1.09266i 2.23396 + 4.63887i −0.815247 2.82979i 1.93201 + 4.66428i 1.66054 0.687817i 0.942615 8.36594i 0.186741 + 0.533676i −2.17974 + 7.56604i
59.3 −2.18953 1.37577i −2.34049 1.66066i 2.03352 + 4.22265i 0.552015 + 1.91609i 2.83987 + 6.85605i 0.0348702 0.0144437i 0.777898 6.90404i 1.72924 + 4.94188i 1.42745 4.95478i
59.4 −2.11046 1.32609i 2.28054 + 1.61813i 1.82776 + 3.79537i 0.228454 + 0.792980i −2.66720 6.43919i 0.870869 0.360726i 0.617458 5.48010i 1.59168 + 4.54877i 0.569420 1.97650i
59.5 −2.07661 1.30482i 0.254844 + 0.180822i 1.74198 + 3.61727i 0.889546 + 3.08768i −0.293272 0.708022i 4.10843 1.70177i 0.553273 4.91043i −0.958588 2.73949i 2.18163 7.57261i
59.6 −2.04157 1.28281i 0.345756 + 0.245327i 1.65467 + 3.43595i −0.431255 1.49692i −0.391180 0.944392i −2.11387 + 0.875595i 0.489608 4.34539i −0.931475 2.66200i −1.03982 + 3.60929i
59.7 −2.02398 1.27175i 2.04231 + 1.44909i 1.61138 + 3.34606i 0.563014 + 1.95427i −2.29070 5.53025i 0.0630526 0.0261172i 0.458687 4.07096i 1.08031 + 3.08734i 1.34581 4.67141i
59.8 −1.94177 1.22010i −1.28164 0.909373i 1.41408 + 2.93637i 0.0278087 + 0.0965260i 1.37913 + 3.32952i 2.68860 1.11366i 0.323299 2.86936i −0.175193 0.500673i 0.0637730 0.221361i
59.9 −1.84294 1.15800i 0.271878 + 0.192908i 1.18771 + 2.46630i −1.04595 3.63057i −0.277669 0.670353i 0.626004 0.259300i 0.179699 1.59487i −0.954133 2.72675i −2.27656 + 7.90213i
59.10 −1.77214 1.11351i −0.156933 0.111350i 1.03281 + 2.14466i −0.475124 1.64919i 0.154118 + 0.372073i −2.24185 + 0.928604i 0.0891399 0.791139i −0.978608 2.79670i −0.994406 + 3.45166i
59.11 −1.67502 1.05249i −2.57521 1.82721i 0.830209 + 1.72395i −0.911231 3.16295i 2.39042 + 5.77099i −3.94144 + 1.63260i −0.0191731 + 0.170166i 2.30217 + 6.57923i −1.80263 + 6.25708i
59.12 −1.60800 1.01037i −1.46926 1.04250i 0.697037 + 1.44741i 0.311704 + 1.08195i 1.30926 + 3.16083i −4.49816 + 1.86320i −0.0836683 + 0.742577i 0.0810942 + 0.231754i 0.591951 2.05471i
59.13 −1.57989 0.992711i 2.03132 + 1.44130i 0.642812 + 1.33481i −0.907639 3.15049i −1.77847 4.29360i 4.62744 1.91675i −0.108315 + 0.961322i 1.05808 + 3.02382i −1.69355 + 5.87845i
59.14 −1.49819 0.941376i 1.42757 + 1.01291i 0.490620 + 1.01878i 1.15114 + 3.99569i −1.18524 2.86142i −3.49084 + 1.44595i −0.172204 + 1.52835i 0.0211196 + 0.0603563i 2.03682 7.06995i
59.15 −1.45286 0.912890i 1.32843 + 0.942568i 0.409656 + 0.850659i −0.163059 0.565989i −1.06955 2.58212i 0.972866 0.402974i −0.202842 + 1.80028i −0.114558 0.327389i −0.279785 + 0.971156i
59.16 −1.41877 0.891475i −1.74699 1.23955i 0.350426 + 0.727667i 0.415357 + 1.44174i 1.37355 + 3.31604i −0.456128 + 0.188934i −0.223695 + 1.98535i 0.524634 + 1.49932i 0.695974 2.41578i
59.17 −1.34805 0.847036i 2.65624 + 1.88470i 0.232002 + 0.481757i −0.0257895 0.0895172i −1.98433 4.79060i −1.60590 + 0.665187i −0.261197 + 2.31819i 2.51266 + 7.18077i −0.0410588 + 0.142518i
59.18 −1.33188 0.836875i −0.817319 0.579919i 0.205773 + 0.427293i 1.09966 + 3.81700i 0.603251 + 1.45638i 0.267978 0.111000i −0.268710 + 2.38487i −0.659132 1.88369i 1.72974 6.00406i
59.19 −1.23009 0.772914i −1.98496 1.40841i 0.0479473 + 0.0995635i −0.994433 3.45175i 1.35310 + 3.26666i 2.78266 1.15262i −0.307340 + 2.72772i 0.965629 + 2.75961i −1.44467 + 5.01456i
59.20 −1.03459 0.650076i −1.12412 0.797604i −0.219989 0.456812i −0.0749620 0.260199i 0.644498 + 1.55596i 2.47802 1.02643i −0.342977 + 3.04401i −0.363369 1.03845i −0.0915941 + 0.317930i
See next 80 embeddings (of 1536 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 729.64
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(731, [\chi])\).