Properties

Label 731.2.bd.a
Level 731
Weight 2
Character orbit 731.bd
Analytic conductor 5.837
Analytic rank 0
Dimension 1024
CM No

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Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.bd (of order \(48\) and degree \(16\))

Newform invariants

Self dual: No
Analytic conductor: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(1024\)
Relative dimension: \(64\) over \(\Q(\zeta_{48})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{48}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1024q - 24q^{3} - 32q^{4} - 24q^{5} - 8q^{6} - 24q^{7} - 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 1024q - 24q^{3} - 32q^{4} - 24q^{5} - 8q^{6} - 24q^{7} - 8q^{9} - 8q^{10} - 48q^{11} - 24q^{12} - 16q^{13} - 8q^{14} - 8q^{15} - 8q^{17} - 48q^{18} - 24q^{19} - 24q^{20} - 32q^{21} + 40q^{24} - 24q^{25} - 24q^{26} - 24q^{28} - 24q^{29} - 24q^{30} + 24q^{31} - 24q^{34} - 192q^{35} + 8q^{36} - 24q^{37} - 16q^{38} - 8q^{40} - 32q^{41} + 24q^{43} + 32q^{44} + 72q^{46} + 48q^{47} + 48q^{48} - 8q^{49} - 144q^{52} - 8q^{53} + 144q^{54} + 72q^{55} - 8q^{56} - 24q^{57} - 128q^{58} + 96q^{59} - 112q^{60} + 24q^{61} - 192q^{62} - 24q^{63} + 192q^{64} - 136q^{66} - 8q^{68} - 96q^{69} - 24q^{71} - 432q^{72} - 24q^{73} + 88q^{74} + 144q^{76} - 24q^{77} - 496q^{78} - 40q^{79} + 264q^{80} - 120q^{81} - 16q^{83} - 48q^{86} - 32q^{87} - 24q^{89} - 112q^{90} - 24q^{91} + 184q^{92} + 168q^{93} + 72q^{95} + 40q^{96} - 160q^{97} - 432q^{98} + 8q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.52329 1.04518i 0.167312 2.55269i 3.86036 + 3.86036i 1.64362 + 1.87419i −3.09020 + 6.26630i 3.41978 3.89951i −3.61566 8.72898i −3.51390 0.462614i −2.18846 6.44700i
7.2 −2.46552 1.02125i −0.0330853 + 0.504784i 3.62163 + 3.62163i −0.699355 0.797461i 0.597085 1.21077i −0.903006 + 1.02968i −3.18811 7.69678i 2.72062 + 0.358177i 0.909865 + 2.68038i
7.3 −2.46148 1.01958i −0.197604 + 3.01486i 3.60514 + 3.60514i 2.67532 + 3.05062i 3.56029 7.21956i −0.805775 + 0.918810i −3.15911 7.62677i −6.07601 0.799922i −3.47491 10.2367i
7.4 −2.34684 0.972094i 0.0966295 1.47428i 3.14849 + 3.14849i −2.46177 2.80711i −1.65991 + 3.36597i −0.448759 + 0.511712i −2.38420 5.75597i 0.810168 + 0.106661i 3.04861 + 8.98092i
7.5 −2.32042 0.961148i −0.196000 + 2.99037i 3.04631 + 3.04631i −2.38380 2.71820i 3.32899 6.75053i −2.58084 + 2.94288i −2.21846 5.35583i −5.92959 0.780645i 2.91881 + 8.59854i
7.6 −2.19786 0.910384i −0.118993 + 1.81548i 2.58758 + 2.58758i −1.86406 2.12555i 1.91431 3.88183i 3.05317 3.48147i −1.51068 3.64712i −0.307458 0.0404776i 2.16188 + 6.36868i
7.7 −2.15398 0.892209i −0.00850084 + 0.129698i 2.42939 + 2.42939i 1.74277 + 1.98725i 0.134028 0.271782i 0.0638206 0.0727734i −1.28092 3.09243i 2.95759 + 0.389373i −1.98086 5.83541i
7.8 −2.14363 0.887919i 0.123285 1.88097i 2.39252 + 2.39252i −0.385334 0.439389i −1.93442 + 3.92262i 0.750577 0.855869i −1.22846 2.96577i −0.548508 0.0722124i 0.435870 + 1.28403i
7.9 −2.00010 0.828468i 0.00800967 0.122204i 1.89982 + 1.89982i 0.138736 + 0.158198i −0.117262 + 0.237784i −2.95893 + 3.37401i −0.568950 1.37357i 2.95946 + 0.389621i −0.146424 0.431350i
7.10 −1.91170 0.791852i −0.155821 + 2.37736i 1.61335 + 1.61335i 0.654802 + 0.746659i 2.18040 4.42142i 2.00526 2.28656i −0.223006 0.538385i −2.65325 0.349307i −0.660542 1.94589i
7.11 −1.86110 0.770892i 0.138846 2.11838i 1.45520 + 1.45520i 2.61856 + 2.98590i −1.89145 + 3.83547i −0.831524 + 0.948172i −0.0446788 0.107864i −1.49391 0.196677i −2.57160 7.57568i
7.12 −1.80829 0.749020i 0.223690 3.41284i 1.29469 + 1.29469i 0.118178 + 0.134757i −2.96079 + 6.00388i −1.90537 + 2.17266i 0.126613 + 0.305671i −8.62313 1.13526i −0.112766 0.332198i
7.13 −1.74510 0.722845i −0.119895 + 1.82924i 1.10866 + 1.10866i 0.813236 + 0.927317i 1.53149 3.10555i −2.47493 + 2.82212i 0.312353 + 0.754087i −0.357413 0.0470543i −0.748872 2.20611i
7.14 −1.73922 0.720407i 0.0730355 1.11431i 1.09167 + 1.09167i −1.79414 2.04582i −0.929779 + 1.88540i 2.46027 2.80540i 0.328608 + 0.793329i 1.73799 + 0.228811i 1.64657 + 4.85063i
7.15 −1.57134 0.650869i −0.102687 + 1.56670i 0.631253 + 0.631253i −1.97206 2.24870i 1.18107 2.39497i −0.825477 + 0.941276i 0.720690 + 1.73990i 0.530341 + 0.0698207i 1.63515 + 4.81701i
7.16 −1.50429 0.623096i −0.185410 + 2.82882i 0.460417 + 0.460417i 0.334499 + 0.381423i 2.04153 4.13982i 1.15667 1.31893i 0.840477 + 2.02909i −4.99349 0.657406i −0.265519 0.782195i
7.17 −1.41151 0.584666i 0.145113 2.21399i 0.236307 + 0.236307i 0.494155 + 0.563476i −1.49927 + 3.04022i −0.0148342 + 0.0169152i 0.973943 + 2.35131i −1.90637 0.250978i −0.368059 1.08427i
7.18 −1.39138 0.576328i −0.0198737 + 0.303214i 0.189566 + 0.189566i 2.18482 + 2.49131i 0.202402 0.410432i 1.80655 2.05998i 0.998149 + 2.40975i 2.88279 + 0.379527i −1.60410 4.72552i
7.19 −1.29704 0.537254i 0.119073 1.81670i −0.0205300 0.0205300i −2.41704 2.75611i −1.13047 + 2.29237i −3.01254 + 3.43514i 1.09011 + 2.63175i −0.311883 0.0410601i 1.65428 + 4.87336i
7.20 −1.20385 0.498652i 0.0460112 0.701995i −0.213607 0.213607i −0.570187 0.650174i −0.405442 + 0.822155i 2.09550 2.38946i 1.14794 + 2.77137i 2.48365 + 0.326979i 0.362210 + 1.06704i
See next 80 embeddings (of 1024 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 725.64
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 \(S_{2}^{\mathrm{new}}(731, [\chi])\).