Properties

Label 3.14.a.a
Level 3
Weight 14
Character orbit 3.a
Self dual Yes
Analytic conductor 3.217
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 14 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(3.21692786856\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 12q^{2} \) \(\mathstrut -\mathstrut 729q^{3} \) \(\mathstrut -\mathstrut 8048q^{4} \) \(\mathstrut -\mathstrut 30210q^{5} \) \(\mathstrut +\mathstrut 8748q^{6} \) \(\mathstrut +\mathstrut 235088q^{7} \) \(\mathstrut +\mathstrut 194880q^{8} \) \(\mathstrut +\mathstrut 531441q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 12q^{2} \) \(\mathstrut -\mathstrut 729q^{3} \) \(\mathstrut -\mathstrut 8048q^{4} \) \(\mathstrut -\mathstrut 30210q^{5} \) \(\mathstrut +\mathstrut 8748q^{6} \) \(\mathstrut +\mathstrut 235088q^{7} \) \(\mathstrut +\mathstrut 194880q^{8} \) \(\mathstrut +\mathstrut 531441q^{9} \) \(\mathstrut +\mathstrut 362520q^{10} \) \(\mathstrut -\mathstrut 11182908q^{11} \) \(\mathstrut +\mathstrut 5866992q^{12} \) \(\mathstrut +\mathstrut 8049614q^{13} \) \(\mathstrut -\mathstrut 2821056q^{14} \) \(\mathstrut +\mathstrut 22023090q^{15} \) \(\mathstrut +\mathstrut 63590656q^{16} \) \(\mathstrut -\mathstrut 117494622q^{17} \) \(\mathstrut -\mathstrut 6377292q^{18} \) \(\mathstrut -\mathstrut 214061380q^{19} \) \(\mathstrut +\mathstrut 243130080q^{20} \) \(\mathstrut -\mathstrut 171379152q^{21} \) \(\mathstrut +\mathstrut 134194896q^{22} \) \(\mathstrut +\mathstrut 830555544q^{23} \) \(\mathstrut -\mathstrut 142067520q^{24} \) \(\mathstrut -\mathstrut 308059025q^{25} \) \(\mathstrut -\mathstrut 96595368q^{26} \) \(\mathstrut -\mathstrut 387420489q^{27} \) \(\mathstrut -\mathstrut 1891988224q^{28} \) \(\mathstrut -\mathstrut 1252400250q^{29} \) \(\mathstrut -\mathstrut 264277080q^{30} \) \(\mathstrut +\mathstrut 6159350552q^{31} \) \(\mathstrut -\mathstrut 2359544832q^{32} \) \(\mathstrut +\mathstrut 8152339932q^{33} \) \(\mathstrut +\mathstrut 1409935464q^{34} \) \(\mathstrut -\mathstrut 7102008480q^{35} \) \(\mathstrut -\mathstrut 4277037168q^{36} \) \(\mathstrut -\mathstrut 5498191402q^{37} \) \(\mathstrut +\mathstrut 2568736560q^{38} \) \(\mathstrut -\mathstrut 5868168606q^{39} \) \(\mathstrut -\mathstrut 5887324800q^{40} \) \(\mathstrut -\mathstrut 4678687878q^{41} \) \(\mathstrut +\mathstrut 2056549824q^{42} \) \(\mathstrut +\mathstrut 7115013764q^{43} \) \(\mathstrut +\mathstrut 90000043584q^{44} \) \(\mathstrut -\mathstrut 16054832610q^{45} \) \(\mathstrut -\mathstrut 9966666528q^{46} \) \(\mathstrut -\mathstrut 29528776992q^{47} \) \(\mathstrut -\mathstrut 46357588224q^{48} \) \(\mathstrut -\mathstrut 41622642663q^{49} \) \(\mathstrut +\mathstrut 3696708300q^{50} \) \(\mathstrut +\mathstrut 85653579438q^{51} \) \(\mathstrut -\mathstrut 64783293472q^{52} \) \(\mathstrut -\mathstrut 204125042466q^{53} \) \(\mathstrut +\mathstrut 4649045868q^{54} \) \(\mathstrut +\mathstrut 337835650680q^{55} \) \(\mathstrut +\mathstrut 45813949440q^{56} \) \(\mathstrut +\mathstrut 156050746020q^{57} \) \(\mathstrut +\mathstrut 15028803000q^{58} \) \(\mathstrut -\mathstrut 29909821020q^{59} \) \(\mathstrut -\mathstrut 177241828320q^{60} \) \(\mathstrut -\mathstrut 134392006738q^{61} \) \(\mathstrut -\mathstrut 73912206624q^{62} \) \(\mathstrut +\mathstrut 124935401808q^{63} \) \(\mathstrut -\mathstrut 492620115968q^{64} \) \(\mathstrut -\mathstrut 243178838940q^{65} \) \(\mathstrut -\mathstrut 97828079184q^{66} \) \(\mathstrut +\mathstrut 348518801948q^{67} \) \(\mathstrut +\mathstrut 945596717856q^{68} \) \(\mathstrut -\mathstrut 605474991576q^{69} \) \(\mathstrut +\mathstrut 85224101760q^{70} \) \(\mathstrut +\mathstrut 1314335409192q^{71} \) \(\mathstrut +\mathstrut 103567222080q^{72} \) \(\mathstrut -\mathstrut 1178875922326q^{73} \) \(\mathstrut +\mathstrut 65978296824q^{74} \) \(\mathstrut +\mathstrut 224575029225q^{75} \) \(\mathstrut +\mathstrut 1722765986240q^{76} \) \(\mathstrut -\mathstrut 2628967475904q^{77} \) \(\mathstrut +\mathstrut 70418023272q^{78} \) \(\mathstrut -\mathstrut 1072420659640q^{79} \) \(\mathstrut -\mathstrut 1921073717760q^{80} \) \(\mathstrut +\mathstrut 282429536481q^{81} \) \(\mathstrut +\mathstrut 56144254536q^{82} \) \(\mathstrut +\mathstrut 1124025139644q^{83} \) \(\mathstrut +\mathstrut 1379259415296q^{84} \) \(\mathstrut +\mathstrut 3549512530620q^{85} \) \(\mathstrut -\mathstrut 85380165168q^{86} \) \(\mathstrut +\mathstrut 912999782250q^{87} \) \(\mathstrut -\mathstrut 2179325111040q^{88} \) \(\mathstrut +\mathstrut 2235610909530q^{89} \) \(\mathstrut +\mathstrut 192657991320q^{90} \) \(\mathstrut +\mathstrut 1892367656032q^{91} \) \(\mathstrut -\mathstrut 6684311018112q^{92} \) \(\mathstrut -\mathstrut 4490166552408q^{93} \) \(\mathstrut +\mathstrut 354345323904q^{94} \) \(\mathstrut +\mathstrut 6466794289800q^{95} \) \(\mathstrut +\mathstrut 1720108182528q^{96} \) \(\mathstrut -\mathstrut 14215257165502q^{97} \) \(\mathstrut +\mathstrut 499471711956q^{98} \) \(\mathstrut -\mathstrut 5943055810428q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−12.0000 −729.000 −8048.00 −30210.0 8748.00 235088. 194880. 531441. 362520.
\(n\): 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 \(T_{2} \) \(\mathstrut +\mathstrut 12 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(3))\).