Properties

Label 2.16.a.a
Level 2
Weight 16
Character orbit 2.a
Self dual Yes
Analytic conductor 2.854
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 16 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(2.853870102\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 128q^{2} \) \(\mathstrut +\mathstrut 6252q^{3} \) \(\mathstrut +\mathstrut 16384q^{4} \) \(\mathstrut +\mathstrut 90510q^{5} \) \(\mathstrut -\mathstrut 800256q^{6} \) \(\mathstrut +\mathstrut 56q^{7} \) \(\mathstrut -\mathstrut 2097152q^{8} \) \(\mathstrut +\mathstrut 24738597q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 128q^{2} \) \(\mathstrut +\mathstrut 6252q^{3} \) \(\mathstrut +\mathstrut 16384q^{4} \) \(\mathstrut +\mathstrut 90510q^{5} \) \(\mathstrut -\mathstrut 800256q^{6} \) \(\mathstrut +\mathstrut 56q^{7} \) \(\mathstrut -\mathstrut 2097152q^{8} \) \(\mathstrut +\mathstrut 24738597q^{9} \) \(\mathstrut -\mathstrut 11585280q^{10} \) \(\mathstrut -\mathstrut 95889948q^{11} \) \(\mathstrut +\mathstrut 102432768q^{12} \) \(\mathstrut -\mathstrut 59782138q^{13} \) \(\mathstrut -\mathstrut 7168q^{14} \) \(\mathstrut +\mathstrut 565868520q^{15} \) \(\mathstrut +\mathstrut 268435456q^{16} \) \(\mathstrut -\mathstrut 1355814414q^{17} \) \(\mathstrut -\mathstrut 3166540416q^{18} \) \(\mathstrut +\mathstrut 3783593180q^{19} \) \(\mathstrut +\mathstrut 1482915840q^{20} \) \(\mathstrut +\mathstrut 350112q^{21} \) \(\mathstrut +\mathstrut 12273913344q^{22} \) \(\mathstrut -\mathstrut 11608845528q^{23} \) \(\mathstrut -\mathstrut 13111394304q^{24} \) \(\mathstrut -\mathstrut 22325518025q^{25} \) \(\mathstrut +\mathstrut 7652113664q^{26} \) \(\mathstrut +\mathstrut 64956341880q^{27} \) \(\mathstrut +\mathstrut 917504q^{28} \) \(\mathstrut -\mathstrut 28959105930q^{29} \) \(\mathstrut -\mathstrut 72431170560q^{30} \) \(\mathstrut +\mathstrut 253685353952q^{31} \) \(\mathstrut -\mathstrut 34359738368q^{32} \) \(\mathstrut -\mathstrut 599503954896q^{33} \) \(\mathstrut +\mathstrut 173544244992q^{34} \) \(\mathstrut +\mathstrut 5068560q^{35} \) \(\mathstrut +\mathstrut 405317173248q^{36} \) \(\mathstrut +\mathstrut 817641294446q^{37} \) \(\mathstrut -\mathstrut 484299927040q^{38} \) \(\mathstrut -\mathstrut 373757926776q^{39} \) \(\mathstrut -\mathstrut 189813227520q^{40} \) \(\mathstrut -\mathstrut 682333284198q^{41} \) \(\mathstrut -\mathstrut 44814336q^{42} \) \(\mathstrut +\mathstrut 366945604292q^{43} \) \(\mathstrut -\mathstrut 1571060908032q^{44} \) \(\mathstrut +\mathstrut 2239090414470q^{45} \) \(\mathstrut +\mathstrut 1485932227584q^{46} \) \(\mathstrut +\mathstrut 695741581776q^{47} \) \(\mathstrut +\mathstrut 1678258470912q^{48} \) \(\mathstrut -\mathstrut 4747561506807q^{49} \) \(\mathstrut +\mathstrut 2857666307200q^{50} \) \(\mathstrut -\mathstrut 8476551716328q^{51} \) \(\mathstrut -\mathstrut 979470548992q^{52} \) \(\mathstrut +\mathstrut 12993372468702q^{53} \) \(\mathstrut -\mathstrut 8314411760640q^{54} \) \(\mathstrut -\mathstrut 8678999193480q^{55} \) \(\mathstrut -\mathstrut 117440512q^{56} \) \(\mathstrut +\mathstrut 23655024561360q^{57} \) \(\mathstrut +\mathstrut 3706765559040q^{58} \) \(\mathstrut +\mathstrut 9209035340340q^{59} \) \(\mathstrut +\mathstrut 9271189831680q^{60} \) \(\mathstrut -\mathstrut 42338641200298q^{61} \) \(\mathstrut -\mathstrut 32471725305856q^{62} \) \(\mathstrut +\mathstrut 1385361432q^{63} \) \(\mathstrut +\mathstrut 4398046511104q^{64} \) \(\mathstrut -\mathstrut 5410881310380q^{65} \) \(\mathstrut +\mathstrut 76736506226688q^{66} \) \(\mathstrut +\mathstrut 30029787950636q^{67} \) \(\mathstrut -\mathstrut 22213663358976q^{68} \) \(\mathstrut -\mathstrut 72578502241056q^{69} \) \(\mathstrut -\mathstrut 648775680q^{70} \) \(\mathstrut +\mathstrut 115328696975352q^{71} \) \(\mathstrut -\mathstrut 51880598175744q^{72} \) \(\mathstrut +\mathstrut 43787346432122q^{73} \) \(\mathstrut -\mathstrut 104658085689088q^{74} \) \(\mathstrut -\mathstrut 139579138692300q^{75} \) \(\mathstrut +\mathstrut 61990390661120q^{76} \) \(\mathstrut -\mathstrut 5369837088q^{77} \) \(\mathstrut +\mathstrut 47841014627328q^{78} \) \(\mathstrut +\mathstrut 79603813043120q^{79} \) \(\mathstrut +\mathstrut 24296093122560q^{80} \) \(\mathstrut +\mathstrut 51135221770281q^{81} \) \(\mathstrut +\mathstrut 87338660377344q^{82} \) \(\mathstrut -\mathstrut 3417068864868q^{83} \) \(\mathstrut +\mathstrut 5736235008q^{84} \) \(\mathstrut -\mathstrut 122714762611140q^{85} \) \(\mathstrut -\mathstrut 46969037349376q^{86} \) \(\mathstrut -\mathstrut 181052330274360q^{87} \) \(\mathstrut +\mathstrut 201095796228096q^{88} \) \(\mathstrut -\mathstrut 377306179184790q^{89} \) \(\mathstrut -\mathstrut 286603573052160q^{90} \) \(\mathstrut -\mathstrut 3347799728q^{91} \) \(\mathstrut -\mathstrut 190199325130752q^{92} \) \(\mathstrut +\mathstrut 1586040832907904q^{93} \) \(\mathstrut -\mathstrut 89054922467328q^{94} \) \(\mathstrut +\mathstrut 342453018721800q^{95} \) \(\mathstrut -\mathstrut 214817084276736q^{96} \) \(\mathstrut -\mathstrut 166982186657374q^{97} \) \(\mathstrut +\mathstrut 607687872871296q^{98} \) \(\mathstrut -\mathstrut 2372182779922956q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−128.000 6252.00 16384.0 90510.0 −800256. 56.0000 −2.09715e6 2.47386e7 −1.15853e7
\(n\): 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

There are no other newforms in \(S_{16}^{\mathrm{new}}(\Gamma_0(2))\).