Properties

Label 10.12.a.b
Level 10
Weight 12
Character orbit 10.a
Self dual Yes
Analytic conductor 7.683
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 10.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(7.6834318056\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 32q^{2} \) \(\mathstrut +\mathstrut 738q^{3} \) \(\mathstrut +\mathstrut 1024q^{4} \) \(\mathstrut -\mathstrut 3125q^{5} \) \(\mathstrut -\mathstrut 23616q^{6} \) \(\mathstrut +\mathstrut 25574q^{7} \) \(\mathstrut -\mathstrut 32768q^{8} \) \(\mathstrut +\mathstrut 367497q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 32q^{2} \) \(\mathstrut +\mathstrut 738q^{3} \) \(\mathstrut +\mathstrut 1024q^{4} \) \(\mathstrut -\mathstrut 3125q^{5} \) \(\mathstrut -\mathstrut 23616q^{6} \) \(\mathstrut +\mathstrut 25574q^{7} \) \(\mathstrut -\mathstrut 32768q^{8} \) \(\mathstrut +\mathstrut 367497q^{9} \) \(\mathstrut +\mathstrut 100000q^{10} \) \(\mathstrut +\mathstrut 769152q^{11} \) \(\mathstrut +\mathstrut 755712q^{12} \) \(\mathstrut -\mathstrut 918982q^{13} \) \(\mathstrut -\mathstrut 818368q^{14} \) \(\mathstrut -\mathstrut 2306250q^{15} \) \(\mathstrut +\mathstrut 1048576q^{16} \) \(\mathstrut +\mathstrut 10312794q^{17} \) \(\mathstrut -\mathstrut 11759904q^{18} \) \(\mathstrut -\mathstrut 5521660q^{19} \) \(\mathstrut -\mathstrut 3200000q^{20} \) \(\mathstrut +\mathstrut 18873612q^{21} \) \(\mathstrut -\mathstrut 24612864q^{22} \) \(\mathstrut -\mathstrut 39973422q^{23} \) \(\mathstrut -\mathstrut 24182784q^{24} \) \(\mathstrut +\mathstrut 9765625q^{25} \) \(\mathstrut +\mathstrut 29407424q^{26} \) \(\mathstrut +\mathstrut 140478300q^{27} \) \(\mathstrut +\mathstrut 26187776q^{28} \) \(\mathstrut -\mathstrut 15269010q^{29} \) \(\mathstrut +\mathstrut 73800000q^{30} \) \(\mathstrut -\mathstrut 241583788q^{31} \) \(\mathstrut -\mathstrut 33554432q^{32} \) \(\mathstrut +\mathstrut 567634176q^{33} \) \(\mathstrut -\mathstrut 330009408q^{34} \) \(\mathstrut -\mathstrut 79918750q^{35} \) \(\mathstrut +\mathstrut 376316928q^{36} \) \(\mathstrut -\mathstrut 25751446q^{37} \) \(\mathstrut +\mathstrut 176693120q^{38} \) \(\mathstrut -\mathstrut 678208716q^{39} \) \(\mathstrut +\mathstrut 102400000q^{40} \) \(\mathstrut -\mathstrut 1217700138q^{41} \) \(\mathstrut -\mathstrut 603955584q^{42} \) \(\mathstrut -\mathstrut 683436262q^{43} \) \(\mathstrut +\mathstrut 787611648q^{44} \) \(\mathstrut -\mathstrut 1148428125q^{45} \) \(\mathstrut +\mathstrut 1279149504q^{46} \) \(\mathstrut +\mathstrut 1537395294q^{47} \) \(\mathstrut +\mathstrut 773849088q^{48} \) \(\mathstrut -\mathstrut 1323297267q^{49} \) \(\mathstrut -\mathstrut 312500000q^{50} \) \(\mathstrut +\mathstrut 7610841972q^{51} \) \(\mathstrut -\mathstrut 941037568q^{52} \) \(\mathstrut +\mathstrut 3572891298q^{53} \) \(\mathstrut -\mathstrut 4495305600q^{54} \) \(\mathstrut -\mathstrut 2403600000q^{55} \) \(\mathstrut -\mathstrut 838008832q^{56} \) \(\mathstrut -\mathstrut 4074985080q^{57} \) \(\mathstrut +\mathstrut 488608320q^{58} \) \(\mathstrut -\mathstrut 1069039020q^{59} \) \(\mathstrut -\mathstrut 2361600000q^{60} \) \(\mathstrut -\mathstrut 2091535078q^{61} \) \(\mathstrut +\mathstrut 7730681216q^{62} \) \(\mathstrut +\mathstrut 9398368278q^{63} \) \(\mathstrut +\mathstrut 1073741824q^{64} \) \(\mathstrut +\mathstrut 2871818750q^{65} \) \(\mathstrut -\mathstrut 18164293632q^{66} \) \(\mathstrut -\mathstrut 1462369186q^{67} \) \(\mathstrut +\mathstrut 10560301056q^{68} \) \(\mathstrut -\mathstrut 29500385436q^{69} \) \(\mathstrut +\mathstrut 2557400000q^{70} \) \(\mathstrut +\mathstrut 9660178332q^{71} \) \(\mathstrut -\mathstrut 12042141696q^{72} \) \(\mathstrut -\mathstrut 5603447662q^{73} \) \(\mathstrut +\mathstrut 824046272q^{74} \) \(\mathstrut +\mathstrut 7207031250q^{75} \) \(\mathstrut -\mathstrut 5654179840q^{76} \) \(\mathstrut +\mathstrut 19670293248q^{77} \) \(\mathstrut +\mathstrut 21702678912q^{78} \) \(\mathstrut +\mathstrut 5026936280q^{79} \) \(\mathstrut -\mathstrut 3276800000q^{80} \) \(\mathstrut +\mathstrut 38571994341q^{81} \) \(\mathstrut +\mathstrut 38966404416q^{82} \) \(\mathstrut -\mathstrut 38405955462q^{83} \) \(\mathstrut +\mathstrut 19326578688q^{84} \) \(\mathstrut -\mathstrut 32227481250q^{85} \) \(\mathstrut +\mathstrut 21869960384q^{86} \) \(\mathstrut -\mathstrut 11268529380q^{87} \) \(\mathstrut -\mathstrut 25203572736q^{88} \) \(\mathstrut +\mathstrut 35558583210q^{89} \) \(\mathstrut +\mathstrut 36749700000q^{90} \) \(\mathstrut -\mathstrut 23502045668q^{91} \) \(\mathstrut -\mathstrut 40932784128q^{92} \) \(\mathstrut -\mathstrut 178288835544q^{93} \) \(\mathstrut -\mathstrut 49196649408q^{94} \) \(\mathstrut +\mathstrut 17255187500q^{95} \) \(\mathstrut -\mathstrut 24763170816q^{96} \) \(\mathstrut +\mathstrut 10572232514q^{97} \) \(\mathstrut +\mathstrut 42345512544q^{98} \) \(\mathstrut +\mathstrut 282661052544q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−32.0000 738.000 1024.00 −3125.00 −23616.0 25574.0 −32768.0 367497. 100000.
\(n\): 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
\(2\) \(1\)
\(5\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) \(\mathstrut -\mathstrut 738 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(10))\).