Properties

Label 731.2.n.a
Level 731
Weight 2
Character orbit 731.n
Analytic conductor 5.837
Analytic rank 0
Dimension 256
CM No

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Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.n (of order \(12\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(64\) over \(\Q(\zeta_{12})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256q - 6q^{3} - 264q^{4} + 2q^{5} - 2q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 256q - 6q^{3} - 264q^{4} + 2q^{5} - 2q^{6} + 2q^{10} + 4q^{11} + 8q^{12} - 8q^{13} - 6q^{14} + 248q^{16} - 2q^{17} + 16q^{18} - 14q^{20} - 16q^{21} - 4q^{22} + 8q^{23} + 12q^{24} - 12q^{27} - 14q^{28} + 2q^{29} + 8q^{30} - 24q^{31} + 20q^{33} + 16q^{34} + 40q^{35} + 18q^{37} + 8q^{38} + 36q^{39} - 10q^{40} + 8q^{41} - 80q^{44} - 4q^{45} + 2q^{46} + 24q^{47} + 24q^{48} + 92q^{50} - 20q^{51} + 4q^{52} - 88q^{54} - 80q^{55} + 60q^{56} - 44q^{57} + 34q^{58} - 8q^{61} + 24q^{62} - 26q^{63} - 200q^{64} - 8q^{65} + 44q^{67} - 58q^{68} + 40q^{69} - 26q^{71} - 48q^{72} + 36q^{73} + 90q^{74} - 156q^{75} - 24q^{78} + 22q^{79} + 30q^{80} + 132q^{81} + 156q^{82} - 160q^{84} - 28q^{85} + 52q^{86} + 28q^{88} - 20q^{89} + 28q^{90} + 34q^{91} - 70q^{92} + 40q^{95} - 16q^{96} - 92q^{98} + 30q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1 2.76205i 1.78973 + 0.479556i −5.62893 −4.22937 1.13326i 1.32456 4.94332i −0.560575 + 0.150206i 10.0233i 0.375075 + 0.216550i −3.13011 + 11.6817i
208.2 2.67881i −0.393175 0.105351i −5.17600 3.35902 + 0.900047i −0.282214 + 1.05324i −1.82716 + 0.489587i 8.50788i −2.45459 1.41716i 2.41105 8.99816i
208.3 2.67476i −2.11059 0.565531i −5.15436 2.13934 + 0.573235i −1.51266 + 5.64533i 0.737688 0.197663i 8.43716i 1.53669 + 0.887206i 1.53327 5.72224i
208.4 2.61030i 2.69586 + 0.722353i −4.81365 1.96090 + 0.525421i 1.88556 7.03699i −1.40725 + 0.377070i 7.34447i 4.14778 + 2.39472i 1.37151 5.11853i
208.5 2.53263i −2.42847 0.650707i −4.41419 −2.11001 0.565376i −1.64800 + 6.15041i −4.08395 + 1.09429i 6.11424i 2.87598 + 1.66045i −1.43189 + 5.34387i
208.6 2.49716i 0.186349 + 0.0499322i −4.23579 0.0534817 + 0.0143304i 0.124688 0.465344i 4.60333 1.23346i 5.58312i −2.56584 1.48139i 0.0357852 0.133552i
208.7 2.28700i 1.18102 + 0.316454i −3.23035 −0.126627 0.0339295i 0.723730 2.70100i −4.66179 + 1.24912i 2.81381i −1.30340 0.752519i −0.0775966 + 0.289595i
208.8 2.26939i 1.08136 + 0.289751i −3.15015 1.08055 + 0.289533i 0.657559 2.45404i −0.970179 + 0.259959i 2.61014i −1.51268 0.873348i 0.657064 2.45220i
208.9 2.24302i −2.58532 0.692735i −3.03116 −2.60177 0.697143i −1.55382 + 5.79894i 2.01210 0.539141i 2.31291i 3.60594 + 2.08189i −1.56371 + 5.83584i
208.10 2.21617i −0.0692057 0.0185436i −2.91143 −1.25645 0.336665i −0.0410959 + 0.153372i 0.0334492 0.00896268i 2.01988i −2.59363 1.49743i −0.746107 + 2.78451i
208.11 2.16713i −3.18659 0.853843i −2.69647 2.45506 + 0.657832i −1.85039 + 6.90576i 2.85934 0.766159i 1.50935i 6.82721 + 3.94169i 1.42561 5.32045i
208.12 2.13900i 3.06744 + 0.821917i −2.57533 −0.243980 0.0653743i 1.75808 6.56125i 3.12655 0.837757i 1.23062i 6.13555 + 3.54236i −0.139836 + 0.521874i
208.13 1.83727i 1.97172 + 0.528320i −1.37557 −3.10860 0.832946i 0.970668 3.62258i −0.484923 + 0.129935i 1.14725i 1.01047 + 0.583396i −1.53035 + 5.71134i
208.14 1.81631i −1.48008 0.396586i −1.29900 −3.73299 1.00025i −0.720326 + 2.68829i 4.01589 1.07605i 1.27324i −0.564718 0.326040i −1.81677 + 6.78028i
208.15 1.79720i −1.99454 0.534434i −1.22993 0.473839 + 0.126965i −0.960485 + 3.58458i −3.30462 + 0.885471i 1.38398i 1.09448 + 0.631896i 0.228181 0.851582i
208.16 1.73334i 2.03902 + 0.546353i −1.00446 3.75820 + 1.00701i 0.947015 3.53431i 0.573555 0.153684i 1.72560i 1.26102 + 0.728048i 1.74548 6.51423i
208.17 1.62839i 0.474953 + 0.127263i −0.651670 2.37962 + 0.637618i 0.207235 0.773411i 4.72094 1.26497i 2.19561i −2.38869 1.37911i 1.03829 3.87497i
208.18 1.62403i −0.958554 0.256844i −0.637473 3.71240 + 0.994733i −0.417122 + 1.55672i −0.920765 + 0.246718i 2.21278i −1.74522 1.00760i 1.61548 6.02904i
208.19 1.44677i −2.41714 0.647670i −0.0931316 0.138660 + 0.0371537i −0.937027 + 3.49703i 0.626167 0.167781i 2.75879i 2.82501 + 1.63102i 0.0537528 0.200608i
208.20 1.24128i 2.55087 + 0.683505i 0.459214 1.75642 + 0.470632i 0.848423 3.16636i −1.74616 + 0.467882i 3.05258i 3.44170 + 1.98707i 0.584188 2.18022i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 608.64
Significant digits:
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Inner twists

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

Hecke kernels

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