Properties

Label 2.28.a.b
Level 2
Weight 28
Character orbit 2.a
Self dual Yes
Analytic conductor 9.237
Analytic rank 1
Dimension 1
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 28 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(9.23711149676\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 8192q^{2} - 1016388q^{3} + 67108864q^{4} - 3341197410q^{5} - 8326250496q^{6} - 51021361384q^{7} + 549755813888q^{8} - 6592552918443q^{9} + O(q^{10}) \) \( q + 8192q^{2} - 1016388q^{3} + 67108864q^{4} - 3341197410q^{5} - 8326250496q^{6} - 51021361384q^{7} + 549755813888q^{8} - 6592552918443q^{9} - 27371089182720q^{10} - 177413845094508q^{11} - 68208644063232q^{12} - 264386643393418q^{13} - 417966992457728q^{14} + 3395952953155080q^{15} + 4503599627370496q^{16} + 76811888571465906q^{17} - 54006193507885056q^{18} - 147764402234885140q^{19} - 224223962584842240q^{20} + 51857499454360992q^{21} - 1453374219014209536q^{22} + 3583628816727527112q^{23} - 558765212165996544q^{24} + 3713019535666879975q^{25} - 2165855382678880256q^{26} + 14451157452241410840q^{27} - 3423985602213707776q^{28} - 75208679254256666970q^{29} + 27819646592246415360q^{30} - 131071363974700812448q^{31} + 36893488147419103232q^{32} + 180321303187916797104q^{33} + 629242991177448701952q^{34} + 170472440510894815440q^{35} - 442418737216594378752q^{36} - 662288183049353372674q^{37} - 1210485983108179066880q^{38} + 268719411705349334184q^{39} - 1836842701495027630080q^{40} + 638680458671096629242q^{41} + 424816635530125246464q^{42} + 6917115499338769700852q^{43} - 11906041602164404518912q^{44} + 22027020736389692832630q^{45} + 29357087266631902101504q^{46} - 30908908867583003138544q^{47} - 4577404618063843688448q^{48} - 63109183046057553744087q^{49} + 30417056036183080755200q^{50} - 78070681801375089267528q^{51} - 17742687294905387057152q^{52} - 72917877175183391659218q^{53} + 118383881848761637601280q^{54} + 592774679727911334824280q^{55} - 28049290053334694100992q^{56} + 150185965258710437674320q^{57} - 616109500450870615818240q^{58} - 428694162599071555282140q^{59} + 227898544883682634629120q^{60} - 308035309306328670786298q^{61} - 1073736613680749055574016q^{62} + 336361024895024181605112q^{63} + 302231454903657293676544q^{64} + 883367968144681832647380q^{65} + 1477192115715414401875968q^{66} - 2019363317538472735190404q^{67} + 5154758583725659766390784q^{68} - 3642357325776057826311456q^{69} + 1396510232665250328084480q^{70} + 12815663453552821957376472q^{71} - 3624294295278341150736384q^{72} - 22881210928165048507710118q^{73} - 5425464795540302828945408q^{74} - 3773868499817388804030300q^{75} - 9916301173622202915880960q^{76} + 9051895905091888301679072q^{77} + 2201349420690221745635328q^{78} + 53028364881871099261766000q^{79} - 15047415410647266345615360q^{80} + 35584171933953904647069321q^{81} + 5232070317433623586750464q^{82} + 52517451913599957686712492q^{83} + 3480097878262786019033088q^{84} - 256643683152190485030503460q^{85} + 56665010170583201389379584q^{86} + 76441199089875425228304360q^{87} - 97534292804930801818927104q^{88} - 215234163567542209160339190q^{89} + 180445353872504363684904960q^{90} + 13489366477678315864970512q^{91} + 240493258888248542015520768q^{92} + 133219361487518209362397824q^{93} - 253205781443239961710952448q^{94} + 493710038037396441415487400q^{95} - 37498098631179007495766016q^{96} - 356303023225402052577970654q^{97} - 516990427513303480271560704q^{98} + 1169610162249993034551211044q^{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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
8192.00 −1.01639e6 6.71089e7 −3.34120e9 −8.32625e9 −5.10214e10 5.49756e11 −6.59255e12 −2.73711e13
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{3} + 1016388 \) acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(2))\).