Properties

Label 21.6.a.d
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 10q^{2} \) \(\mathstrut +\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 68q^{4} \) \(\mathstrut -\mathstrut 106q^{5} \) \(\mathstrut +\mathstrut 90q^{6} \) \(\mathstrut -\mathstrut 49q^{7} \) \(\mathstrut +\mathstrut 360q^{8} \) \(\mathstrut +\mathstrut 81q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 10q^{2} \) \(\mathstrut +\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 68q^{4} \) \(\mathstrut -\mathstrut 106q^{5} \) \(\mathstrut +\mathstrut 90q^{6} \) \(\mathstrut -\mathstrut 49q^{7} \) \(\mathstrut +\mathstrut 360q^{8} \) \(\mathstrut +\mathstrut 81q^{9} \) \(\mathstrut -\mathstrut 1060q^{10} \) \(\mathstrut +\mathstrut 92q^{11} \) \(\mathstrut +\mathstrut 612q^{12} \) \(\mathstrut +\mathstrut 670q^{13} \) \(\mathstrut -\mathstrut 490q^{14} \) \(\mathstrut -\mathstrut 954q^{15} \) \(\mathstrut +\mathstrut 1424q^{16} \) \(\mathstrut -\mathstrut 222q^{17} \) \(\mathstrut +\mathstrut 810q^{18} \) \(\mathstrut -\mathstrut 908q^{19} \) \(\mathstrut -\mathstrut 7208q^{20} \) \(\mathstrut -\mathstrut 441q^{21} \) \(\mathstrut +\mathstrut 920q^{22} \) \(\mathstrut -\mathstrut 1176q^{23} \) \(\mathstrut +\mathstrut 3240q^{24} \) \(\mathstrut +\mathstrut 8111q^{25} \) \(\mathstrut +\mathstrut 6700q^{26} \) \(\mathstrut +\mathstrut 729q^{27} \) \(\mathstrut -\mathstrut 3332q^{28} \) \(\mathstrut +\mathstrut 1118q^{29} \) \(\mathstrut -\mathstrut 9540q^{30} \) \(\mathstrut +\mathstrut 3696q^{31} \) \(\mathstrut +\mathstrut 2720q^{32} \) \(\mathstrut +\mathstrut 828q^{33} \) \(\mathstrut -\mathstrut 2220q^{34} \) \(\mathstrut +\mathstrut 5194q^{35} \) \(\mathstrut +\mathstrut 5508q^{36} \) \(\mathstrut +\mathstrut 4182q^{37} \) \(\mathstrut -\mathstrut 9080q^{38} \) \(\mathstrut +\mathstrut 6030q^{39} \) \(\mathstrut -\mathstrut 38160q^{40} \) \(\mathstrut -\mathstrut 6662q^{41} \) \(\mathstrut -\mathstrut 4410q^{42} \) \(\mathstrut -\mathstrut 3700q^{43} \) \(\mathstrut +\mathstrut 6256q^{44} \) \(\mathstrut -\mathstrut 8586q^{45} \) \(\mathstrut -\mathstrut 11760q^{46} \) \(\mathstrut -\mathstrut 7056q^{47} \) \(\mathstrut +\mathstrut 12816q^{48} \) \(\mathstrut +\mathstrut 2401q^{49} \) \(\mathstrut +\mathstrut 81110q^{50} \) \(\mathstrut -\mathstrut 1998q^{51} \) \(\mathstrut +\mathstrut 45560q^{52} \) \(\mathstrut -\mathstrut 37578q^{53} \) \(\mathstrut +\mathstrut 7290q^{54} \) \(\mathstrut -\mathstrut 9752q^{55} \) \(\mathstrut -\mathstrut 17640q^{56} \) \(\mathstrut -\mathstrut 8172q^{57} \) \(\mathstrut +\mathstrut 11180q^{58} \) \(\mathstrut +\mathstrut 32700q^{59} \) \(\mathstrut -\mathstrut 64872q^{60} \) \(\mathstrut -\mathstrut 10802q^{61} \) \(\mathstrut +\mathstrut 36960q^{62} \) \(\mathstrut -\mathstrut 3969q^{63} \) \(\mathstrut -\mathstrut 18368q^{64} \) \(\mathstrut -\mathstrut 71020q^{65} \) \(\mathstrut +\mathstrut 8280q^{66} \) \(\mathstrut +\mathstrut 64996q^{67} \) \(\mathstrut -\mathstrut 15096q^{68} \) \(\mathstrut -\mathstrut 10584q^{69} \) \(\mathstrut +\mathstrut 51940q^{70} \) \(\mathstrut -\mathstrut 61320q^{71} \) \(\mathstrut +\mathstrut 29160q^{72} \) \(\mathstrut +\mathstrut 38922q^{73} \) \(\mathstrut +\mathstrut 41820q^{74} \) \(\mathstrut +\mathstrut 72999q^{75} \) \(\mathstrut -\mathstrut 61744q^{76} \) \(\mathstrut -\mathstrut 4508q^{77} \) \(\mathstrut +\mathstrut 60300q^{78} \) \(\mathstrut -\mathstrut 88096q^{79} \) \(\mathstrut -\mathstrut 150944q^{80} \) \(\mathstrut +\mathstrut 6561q^{81} \) \(\mathstrut -\mathstrut 66620q^{82} \) \(\mathstrut +\mathstrut 71892q^{83} \) \(\mathstrut -\mathstrut 29988q^{84} \) \(\mathstrut +\mathstrut 23532q^{85} \) \(\mathstrut -\mathstrut 37000q^{86} \) \(\mathstrut +\mathstrut 10062q^{87} \) \(\mathstrut +\mathstrut 33120q^{88} \) \(\mathstrut +\mathstrut 111818q^{89} \) \(\mathstrut -\mathstrut 85860q^{90} \) \(\mathstrut -\mathstrut 32830q^{91} \) \(\mathstrut -\mathstrut 79968q^{92} \) \(\mathstrut +\mathstrut 33264q^{93} \) \(\mathstrut -\mathstrut 70560q^{94} \) \(\mathstrut +\mathstrut 96248q^{95} \) \(\mathstrut +\mathstrut 24480q^{96} \) \(\mathstrut -\mathstrut 150846q^{97} \) \(\mathstrut +\mathstrut 24010q^{98} \) \(\mathstrut +\mathstrut 7452q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
10.0000 9.00000 68.0000 −106.000 90.0000 −49.0000 360.000 81.0000 −1060.00
\(n\): 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 \(T_{2} \) \(\mathstrut -\mathstrut 10 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(21))\).