Properties

Label 21.6.a.a
Level 21
Weight 6
Character orbit 21.a
Self dual Yes
Analytic conductor 3.368
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 21.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(3.36806021607\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 78q^{5} \) \(\mathstrut +\mathstrut 54q^{6} \) \(\mathstrut +\mathstrut 49q^{7} \) \(\mathstrut +\mathstrut 168q^{8} \) \(\mathstrut +\mathstrut 81q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 78q^{5} \) \(\mathstrut +\mathstrut 54q^{6} \) \(\mathstrut +\mathstrut 49q^{7} \) \(\mathstrut +\mathstrut 168q^{8} \) \(\mathstrut +\mathstrut 81q^{9} \) \(\mathstrut -\mathstrut 468q^{10} \) \(\mathstrut +\mathstrut 444q^{11} \) \(\mathstrut -\mathstrut 36q^{12} \) \(\mathstrut -\mathstrut 442q^{13} \) \(\mathstrut -\mathstrut 294q^{14} \) \(\mathstrut -\mathstrut 702q^{15} \) \(\mathstrut -\mathstrut 1136q^{16} \) \(\mathstrut -\mathstrut 126q^{17} \) \(\mathstrut -\mathstrut 486q^{18} \) \(\mathstrut +\mathstrut 2684q^{19} \) \(\mathstrut +\mathstrut 312q^{20} \) \(\mathstrut -\mathstrut 441q^{21} \) \(\mathstrut -\mathstrut 2664q^{22} \) \(\mathstrut +\mathstrut 4200q^{23} \) \(\mathstrut -\mathstrut 1512q^{24} \) \(\mathstrut +\mathstrut 2959q^{25} \) \(\mathstrut +\mathstrut 2652q^{26} \) \(\mathstrut -\mathstrut 729q^{27} \) \(\mathstrut +\mathstrut 196q^{28} \) \(\mathstrut -\mathstrut 5442q^{29} \) \(\mathstrut +\mathstrut 4212q^{30} \) \(\mathstrut +\mathstrut 80q^{31} \) \(\mathstrut +\mathstrut 1440q^{32} \) \(\mathstrut -\mathstrut 3996q^{33} \) \(\mathstrut +\mathstrut 756q^{34} \) \(\mathstrut +\mathstrut 3822q^{35} \) \(\mathstrut +\mathstrut 324q^{36} \) \(\mathstrut -\mathstrut 5434q^{37} \) \(\mathstrut -\mathstrut 16104q^{38} \) \(\mathstrut +\mathstrut 3978q^{39} \) \(\mathstrut +\mathstrut 13104q^{40} \) \(\mathstrut +\mathstrut 7962q^{41} \) \(\mathstrut +\mathstrut 2646q^{42} \) \(\mathstrut -\mathstrut 11524q^{43} \) \(\mathstrut +\mathstrut 1776q^{44} \) \(\mathstrut +\mathstrut 6318q^{45} \) \(\mathstrut -\mathstrut 25200q^{46} \) \(\mathstrut -\mathstrut 13920q^{47} \) \(\mathstrut +\mathstrut 10224q^{48} \) \(\mathstrut +\mathstrut 2401q^{49} \) \(\mathstrut -\mathstrut 17754q^{50} \) \(\mathstrut +\mathstrut 1134q^{51} \) \(\mathstrut -\mathstrut 1768q^{52} \) \(\mathstrut -\mathstrut 9594q^{53} \) \(\mathstrut +\mathstrut 4374q^{54} \) \(\mathstrut +\mathstrut 34632q^{55} \) \(\mathstrut +\mathstrut 8232q^{56} \) \(\mathstrut -\mathstrut 24156q^{57} \) \(\mathstrut +\mathstrut 32652q^{58} \) \(\mathstrut +\mathstrut 27492q^{59} \) \(\mathstrut -\mathstrut 2808q^{60} \) \(\mathstrut +\mathstrut 49478q^{61} \) \(\mathstrut -\mathstrut 480q^{62} \) \(\mathstrut +\mathstrut 3969q^{63} \) \(\mathstrut +\mathstrut 27712q^{64} \) \(\mathstrut -\mathstrut 34476q^{65} \) \(\mathstrut +\mathstrut 23976q^{66} \) \(\mathstrut -\mathstrut 59356q^{67} \) \(\mathstrut -\mathstrut 504q^{68} \) \(\mathstrut -\mathstrut 37800q^{69} \) \(\mathstrut -\mathstrut 22932q^{70} \) \(\mathstrut +\mathstrut 32040q^{71} \) \(\mathstrut +\mathstrut 13608q^{72} \) \(\mathstrut -\mathstrut 61846q^{73} \) \(\mathstrut +\mathstrut 32604q^{74} \) \(\mathstrut -\mathstrut 26631q^{75} \) \(\mathstrut +\mathstrut 10736q^{76} \) \(\mathstrut +\mathstrut 21756q^{77} \) \(\mathstrut -\mathstrut 23868q^{78} \) \(\mathstrut -\mathstrut 65776q^{79} \) \(\mathstrut -\mathstrut 88608q^{80} \) \(\mathstrut +\mathstrut 6561q^{81} \) \(\mathstrut -\mathstrut 47772q^{82} \) \(\mathstrut +\mathstrut 40188q^{83} \) \(\mathstrut -\mathstrut 1764q^{84} \) \(\mathstrut -\mathstrut 9828q^{85} \) \(\mathstrut +\mathstrut 69144q^{86} \) \(\mathstrut +\mathstrut 48978q^{87} \) \(\mathstrut +\mathstrut 74592q^{88} \) \(\mathstrut -\mathstrut 7974q^{89} \) \(\mathstrut -\mathstrut 37908q^{90} \) \(\mathstrut -\mathstrut 21658q^{91} \) \(\mathstrut +\mathstrut 16800q^{92} \) \(\mathstrut -\mathstrut 720q^{93} \) \(\mathstrut +\mathstrut 83520q^{94} \) \(\mathstrut +\mathstrut 209352q^{95} \) \(\mathstrut -\mathstrut 12960q^{96} \) \(\mathstrut -\mathstrut 143662q^{97} \) \(\mathstrut -\mathstrut 14406q^{98} \) \(\mathstrut +\mathstrut 35964q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−6.00000 −9.00000 4.00000 78.0000 54.0000 49.0000 168.000 81.0000 −468.000
\(n\): 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
\(3\) \(1\)
\(7\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut +\mathstrut 6 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(21))\).