Properties

Label 2.10.a.a
Level 2
Weight 10
Character orbit 2.a
Self dual Yes
Analytic conductor 1.030
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.03007167233\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 16q^{2} \) \(\mathstrut -\mathstrut 156q^{3} \) \(\mathstrut +\mathstrut 256q^{4} \) \(\mathstrut +\mathstrut 870q^{5} \) \(\mathstrut -\mathstrut 2496q^{6} \) \(\mathstrut -\mathstrut 952q^{7} \) \(\mathstrut +\mathstrut 4096q^{8} \) \(\mathstrut +\mathstrut 4653q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 16q^{2} \) \(\mathstrut -\mathstrut 156q^{3} \) \(\mathstrut +\mathstrut 256q^{4} \) \(\mathstrut +\mathstrut 870q^{5} \) \(\mathstrut -\mathstrut 2496q^{6} \) \(\mathstrut -\mathstrut 952q^{7} \) \(\mathstrut +\mathstrut 4096q^{8} \) \(\mathstrut +\mathstrut 4653q^{9} \) \(\mathstrut +\mathstrut 13920q^{10} \) \(\mathstrut -\mathstrut 56148q^{11} \) \(\mathstrut -\mathstrut 39936q^{12} \) \(\mathstrut +\mathstrut 178094q^{13} \) \(\mathstrut -\mathstrut 15232q^{14} \) \(\mathstrut -\mathstrut 135720q^{15} \) \(\mathstrut +\mathstrut 65536q^{16} \) \(\mathstrut -\mathstrut 247662q^{17} \) \(\mathstrut +\mathstrut 74448q^{18} \) \(\mathstrut +\mathstrut 315380q^{19} \) \(\mathstrut +\mathstrut 222720q^{20} \) \(\mathstrut +\mathstrut 148512q^{21} \) \(\mathstrut -\mathstrut 898368q^{22} \) \(\mathstrut +\mathstrut 204504q^{23} \) \(\mathstrut -\mathstrut 638976q^{24} \) \(\mathstrut -\mathstrut 1196225q^{25} \) \(\mathstrut +\mathstrut 2849504q^{26} \) \(\mathstrut +\mathstrut 2344680q^{27} \) \(\mathstrut -\mathstrut 243712q^{28} \) \(\mathstrut -\mathstrut 3840450q^{29} \) \(\mathstrut -\mathstrut 2171520q^{30} \) \(\mathstrut -\mathstrut 1309408q^{31} \) \(\mathstrut +\mathstrut 1048576q^{32} \) \(\mathstrut +\mathstrut 8759088q^{33} \) \(\mathstrut -\mathstrut 3962592q^{34} \) \(\mathstrut -\mathstrut 828240q^{35} \) \(\mathstrut +\mathstrut 1191168q^{36} \) \(\mathstrut +\mathstrut 4307078q^{37} \) \(\mathstrut +\mathstrut 5046080q^{38} \) \(\mathstrut -\mathstrut 27782664q^{39} \) \(\mathstrut +\mathstrut 3563520q^{40} \) \(\mathstrut +\mathstrut 1512042q^{41} \) \(\mathstrut +\mathstrut 2376192q^{42} \) \(\mathstrut +\mathstrut 33670604q^{43} \) \(\mathstrut -\mathstrut 14373888q^{44} \) \(\mathstrut +\mathstrut 4048110q^{45} \) \(\mathstrut +\mathstrut 3272064q^{46} \) \(\mathstrut -\mathstrut 10581072q^{47} \) \(\mathstrut -\mathstrut 10223616q^{48} \) \(\mathstrut -\mathstrut 39447303q^{49} \) \(\mathstrut -\mathstrut 19139600q^{50} \) \(\mathstrut +\mathstrut 38635272q^{51} \) \(\mathstrut +\mathstrut 45592064q^{52} \) \(\mathstrut +\mathstrut 16616214q^{53} \) \(\mathstrut +\mathstrut 37514880q^{54} \) \(\mathstrut -\mathstrut 48848760q^{55} \) \(\mathstrut -\mathstrut 3899392q^{56} \) \(\mathstrut -\mathstrut 49199280q^{57} \) \(\mathstrut -\mathstrut 61447200q^{58} \) \(\mathstrut +\mathstrut 112235100q^{59} \) \(\mathstrut -\mathstrut 34744320q^{60} \) \(\mathstrut -\mathstrut 33197218q^{61} \) \(\mathstrut -\mathstrut 20950528q^{62} \) \(\mathstrut -\mathstrut 4429656q^{63} \) \(\mathstrut +\mathstrut 16777216q^{64} \) \(\mathstrut +\mathstrut 154941780q^{65} \) \(\mathstrut +\mathstrut 140145408q^{66} \) \(\mathstrut -\mathstrut 121372252q^{67} \) \(\mathstrut -\mathstrut 63401472q^{68} \) \(\mathstrut -\mathstrut 31902624q^{69} \) \(\mathstrut -\mathstrut 13251840q^{70} \) \(\mathstrut -\mathstrut 387172728q^{71} \) \(\mathstrut +\mathstrut 19058688q^{72} \) \(\mathstrut +\mathstrut 255240074q^{73} \) \(\mathstrut +\mathstrut 68913248q^{74} \) \(\mathstrut +\mathstrut 186611100q^{75} \) \(\mathstrut +\mathstrut 80737280q^{76} \) \(\mathstrut +\mathstrut 53452896q^{77} \) \(\mathstrut -\mathstrut 444522624q^{78} \) \(\mathstrut +\mathstrut 492101840q^{79} \) \(\mathstrut +\mathstrut 57016320q^{80} \) \(\mathstrut -\mathstrut 457355079q^{81} \) \(\mathstrut +\mathstrut 24192672q^{82} \) \(\mathstrut -\mathstrut 457420236q^{83} \) \(\mathstrut +\mathstrut 38019072q^{84} \) \(\mathstrut -\mathstrut 215465940q^{85} \) \(\mathstrut +\mathstrut 538729664q^{86} \) \(\mathstrut +\mathstrut 599110200q^{87} \) \(\mathstrut -\mathstrut 229982208q^{88} \) \(\mathstrut -\mathstrut 31809510q^{89} \) \(\mathstrut +\mathstrut 64769760q^{90} \) \(\mathstrut -\mathstrut 169545488q^{91} \) \(\mathstrut +\mathstrut 52353024q^{92} \) \(\mathstrut +\mathstrut 204267648q^{93} \) \(\mathstrut -\mathstrut 169297152q^{94} \) \(\mathstrut +\mathstrut 274380600q^{95} \) \(\mathstrut -\mathstrut 163577856q^{96} \) \(\mathstrut -\mathstrut 673532062q^{97} \) \(\mathstrut -\mathstrut 631156848q^{98} \) \(\mathstrut -\mathstrut 261256644q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
16.0000 −156.000 256.000 870.000 −2496.00 −952.000 4096.00 4653.00 13920.0
\(n\): 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 \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\).