Properties

Label 3.10.a.b
Level 3
Weight 10
Character orbit 3.a
Self dual Yes
Analytic conductor 1.545
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.54510750849\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 18q^{2} \) \(\mathstrut +\mathstrut 81q^{3} \) \(\mathstrut -\mathstrut 188q^{4} \) \(\mathstrut -\mathstrut 1530q^{5} \) \(\mathstrut +\mathstrut 1458q^{6} \) \(\mathstrut +\mathstrut 9128q^{7} \) \(\mathstrut -\mathstrut 12600q^{8} \) \(\mathstrut +\mathstrut 6561q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 18q^{2} \) \(\mathstrut +\mathstrut 81q^{3} \) \(\mathstrut -\mathstrut 188q^{4} \) \(\mathstrut -\mathstrut 1530q^{5} \) \(\mathstrut +\mathstrut 1458q^{6} \) \(\mathstrut +\mathstrut 9128q^{7} \) \(\mathstrut -\mathstrut 12600q^{8} \) \(\mathstrut +\mathstrut 6561q^{9} \) \(\mathstrut -\mathstrut 27540q^{10} \) \(\mathstrut +\mathstrut 21132q^{11} \) \(\mathstrut -\mathstrut 15228q^{12} \) \(\mathstrut +\mathstrut 31214q^{13} \) \(\mathstrut +\mathstrut 164304q^{14} \) \(\mathstrut -\mathstrut 123930q^{15} \) \(\mathstrut -\mathstrut 130544q^{16} \) \(\mathstrut -\mathstrut 279342q^{17} \) \(\mathstrut +\mathstrut 118098q^{18} \) \(\mathstrut +\mathstrut 144020q^{19} \) \(\mathstrut +\mathstrut 287640q^{20} \) \(\mathstrut +\mathstrut 739368q^{21} \) \(\mathstrut +\mathstrut 380376q^{22} \) \(\mathstrut -\mathstrut 1763496q^{23} \) \(\mathstrut -\mathstrut 1020600q^{24} \) \(\mathstrut +\mathstrut 387775q^{25} \) \(\mathstrut +\mathstrut 561852q^{26} \) \(\mathstrut +\mathstrut 531441q^{27} \) \(\mathstrut -\mathstrut 1716064q^{28} \) \(\mathstrut +\mathstrut 4692510q^{29} \) \(\mathstrut -\mathstrut 2230740q^{30} \) \(\mathstrut -\mathstrut 369088q^{31} \) \(\mathstrut +\mathstrut 4101408q^{32} \) \(\mathstrut +\mathstrut 1711692q^{33} \) \(\mathstrut -\mathstrut 5028156q^{34} \) \(\mathstrut -\mathstrut 13965840q^{35} \) \(\mathstrut -\mathstrut 1233468q^{36} \) \(\mathstrut +\mathstrut 9347078q^{37} \) \(\mathstrut +\mathstrut 2592360q^{38} \) \(\mathstrut +\mathstrut 2528334q^{39} \) \(\mathstrut +\mathstrut 19278000q^{40} \) \(\mathstrut -\mathstrut 7226838q^{41} \) \(\mathstrut +\mathstrut 13308624q^{42} \) \(\mathstrut -\mathstrut 23147476q^{43} \) \(\mathstrut -\mathstrut 3972816q^{44} \) \(\mathstrut -\mathstrut 10038330q^{45} \) \(\mathstrut -\mathstrut 31742928q^{46} \) \(\mathstrut +\mathstrut 22971888q^{47} \) \(\mathstrut -\mathstrut 10574064q^{48} \) \(\mathstrut +\mathstrut 42966777q^{49} \) \(\mathstrut +\mathstrut 6979950q^{50} \) \(\mathstrut -\mathstrut 22626702q^{51} \) \(\mathstrut -\mathstrut 5868232q^{52} \) \(\mathstrut +\mathstrut 78477174q^{53} \) \(\mathstrut +\mathstrut 9565938q^{54} \) \(\mathstrut -\mathstrut 32331960q^{55} \) \(\mathstrut -\mathstrut 115012800q^{56} \) \(\mathstrut +\mathstrut 11665620q^{57} \) \(\mathstrut +\mathstrut 84465180q^{58} \) \(\mathstrut -\mathstrut 20310660q^{59} \) \(\mathstrut +\mathstrut 23298840q^{60} \) \(\mathstrut -\mathstrut 179339938q^{61} \) \(\mathstrut -\mathstrut 6643584q^{62} \) \(\mathstrut +\mathstrut 59888808q^{63} \) \(\mathstrut +\mathstrut 140663872q^{64} \) \(\mathstrut -\mathstrut 47757420q^{65} \) \(\mathstrut +\mathstrut 30810456q^{66} \) \(\mathstrut +\mathstrut 274528388q^{67} \) \(\mathstrut +\mathstrut 52516296q^{68} \) \(\mathstrut -\mathstrut 142843176q^{69} \) \(\mathstrut -\mathstrut 251385120q^{70} \) \(\mathstrut -\mathstrut 36342648q^{71} \) \(\mathstrut -\mathstrut 82668600q^{72} \) \(\mathstrut -\mathstrut 247089526q^{73} \) \(\mathstrut +\mathstrut 168247404q^{74} \) \(\mathstrut +\mathstrut 31409775q^{75} \) \(\mathstrut -\mathstrut 27075760q^{76} \) \(\mathstrut +\mathstrut 192892896q^{77} \) \(\mathstrut +\mathstrut 45510012q^{78} \) \(\mathstrut +\mathstrut 191874800q^{79} \) \(\mathstrut +\mathstrut 199732320q^{80} \) \(\mathstrut +\mathstrut 43046721q^{81} \) \(\mathstrut -\mathstrut 130083084q^{82} \) \(\mathstrut -\mathstrut 276159276q^{83} \) \(\mathstrut -\mathstrut 139001184q^{84} \) \(\mathstrut +\mathstrut 427393260q^{85} \) \(\mathstrut -\mathstrut 416654568q^{86} \) \(\mathstrut +\mathstrut 380093310q^{87} \) \(\mathstrut -\mathstrut 266263200q^{88} \) \(\mathstrut -\mathstrut 678997350q^{89} \) \(\mathstrut -\mathstrut 180689940q^{90} \) \(\mathstrut +\mathstrut 284921392q^{91} \) \(\mathstrut +\mathstrut 331537248q^{92} \) \(\mathstrut -\mathstrut 29896128q^{93} \) \(\mathstrut +\mathstrut 413493984q^{94} \) \(\mathstrut -\mathstrut 220350600q^{95} \) \(\mathstrut +\mathstrut 332214048q^{96} \) \(\mathstrut -\mathstrut 567657502q^{97} \) \(\mathstrut +\mathstrut 773401986q^{98} \) \(\mathstrut +\mathstrut 138647052q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
18.0000 81.0000 −188.000 −1530.00 1458.00 9128.00 −12600.0 6561.00 −27540.0
\(n\): 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
\(3\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut -\mathstrut 18 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\).