Properties

Label 21.6.a.b
Level 21
Weight 6
Character orbit 21.a
Self dual Yes
Analytic conductor 3.368
Analytic rank 1
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: \(1\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut 31q^{4} \) \(\mathstrut -\mathstrut 34q^{5} \) \(\mathstrut -\mathstrut 9q^{6} \) \(\mathstrut -\mathstrut 49q^{7} \) \(\mathstrut -\mathstrut 63q^{8} \) \(\mathstrut +\mathstrut 81q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut 31q^{4} \) \(\mathstrut -\mathstrut 34q^{5} \) \(\mathstrut -\mathstrut 9q^{6} \) \(\mathstrut -\mathstrut 49q^{7} \) \(\mathstrut -\mathstrut 63q^{8} \) \(\mathstrut +\mathstrut 81q^{9} \) \(\mathstrut -\mathstrut 34q^{10} \) \(\mathstrut -\mathstrut 340q^{11} \) \(\mathstrut +\mathstrut 279q^{12} \) \(\mathstrut +\mathstrut 454q^{13} \) \(\mathstrut -\mathstrut 49q^{14} \) \(\mathstrut +\mathstrut 306q^{15} \) \(\mathstrut +\mathstrut 929q^{16} \) \(\mathstrut -\mathstrut 798q^{17} \) \(\mathstrut +\mathstrut 81q^{18} \) \(\mathstrut +\mathstrut 892q^{19} \) \(\mathstrut +\mathstrut 1054q^{20} \) \(\mathstrut +\mathstrut 441q^{21} \) \(\mathstrut -\mathstrut 340q^{22} \) \(\mathstrut -\mathstrut 3192q^{23} \) \(\mathstrut +\mathstrut 567q^{24} \) \(\mathstrut -\mathstrut 1969q^{25} \) \(\mathstrut +\mathstrut 454q^{26} \) \(\mathstrut -\mathstrut 729q^{27} \) \(\mathstrut +\mathstrut 1519q^{28} \) \(\mathstrut -\mathstrut 8242q^{29} \) \(\mathstrut +\mathstrut 306q^{30} \) \(\mathstrut -\mathstrut 2496q^{31} \) \(\mathstrut +\mathstrut 2945q^{32} \) \(\mathstrut +\mathstrut 3060q^{33} \) \(\mathstrut -\mathstrut 798q^{34} \) \(\mathstrut +\mathstrut 1666q^{35} \) \(\mathstrut -\mathstrut 2511q^{36} \) \(\mathstrut +\mathstrut 9798q^{37} \) \(\mathstrut +\mathstrut 892q^{38} \) \(\mathstrut -\mathstrut 4086q^{39} \) \(\mathstrut +\mathstrut 2142q^{40} \) \(\mathstrut +\mathstrut 19834q^{41} \) \(\mathstrut +\mathstrut 441q^{42} \) \(\mathstrut -\mathstrut 17236q^{43} \) \(\mathstrut +\mathstrut 10540q^{44} \) \(\mathstrut -\mathstrut 2754q^{45} \) \(\mathstrut -\mathstrut 3192q^{46} \) \(\mathstrut +\mathstrut 8928q^{47} \) \(\mathstrut -\mathstrut 8361q^{48} \) \(\mathstrut +\mathstrut 2401q^{49} \) \(\mathstrut -\mathstrut 1969q^{50} \) \(\mathstrut +\mathstrut 7182q^{51} \) \(\mathstrut -\mathstrut 14074q^{52} \) \(\mathstrut +\mathstrut 150q^{53} \) \(\mathstrut -\mathstrut 729q^{54} \) \(\mathstrut +\mathstrut 11560q^{55} \) \(\mathstrut +\mathstrut 3087q^{56} \) \(\mathstrut -\mathstrut 8028q^{57} \) \(\mathstrut -\mathstrut 8242q^{58} \) \(\mathstrut -\mathstrut 42396q^{59} \) \(\mathstrut -\mathstrut 9486q^{60} \) \(\mathstrut +\mathstrut 14758q^{61} \) \(\mathstrut -\mathstrut 2496q^{62} \) \(\mathstrut -\mathstrut 3969q^{63} \) \(\mathstrut -\mathstrut 26783q^{64} \) \(\mathstrut -\mathstrut 15436q^{65} \) \(\mathstrut +\mathstrut 3060q^{66} \) \(\mathstrut -\mathstrut 1676q^{67} \) \(\mathstrut +\mathstrut 24738q^{68} \) \(\mathstrut +\mathstrut 28728q^{69} \) \(\mathstrut +\mathstrut 1666q^{70} \) \(\mathstrut +\mathstrut 14568q^{71} \) \(\mathstrut -\mathstrut 5103q^{72} \) \(\mathstrut +\mathstrut 78378q^{73} \) \(\mathstrut +\mathstrut 9798q^{74} \) \(\mathstrut +\mathstrut 17721q^{75} \) \(\mathstrut -\mathstrut 27652q^{76} \) \(\mathstrut +\mathstrut 16660q^{77} \) \(\mathstrut -\mathstrut 4086q^{78} \) \(\mathstrut -\mathstrut 2272q^{79} \) \(\mathstrut -\mathstrut 31586q^{80} \) \(\mathstrut +\mathstrut 6561q^{81} \) \(\mathstrut +\mathstrut 19834q^{82} \) \(\mathstrut -\mathstrut 37764q^{83} \) \(\mathstrut -\mathstrut 13671q^{84} \) \(\mathstrut +\mathstrut 27132q^{85} \) \(\mathstrut -\mathstrut 17236q^{86} \) \(\mathstrut +\mathstrut 74178q^{87} \) \(\mathstrut +\mathstrut 21420q^{88} \) \(\mathstrut -\mathstrut 117286q^{89} \) \(\mathstrut -\mathstrut 2754q^{90} \) \(\mathstrut -\mathstrut 22246q^{91} \) \(\mathstrut +\mathstrut 98952q^{92} \) \(\mathstrut +\mathstrut 22464q^{93} \) \(\mathstrut +\mathstrut 8928q^{94} \) \(\mathstrut -\mathstrut 30328q^{95} \) \(\mathstrut -\mathstrut 26505q^{96} \) \(\mathstrut +\mathstrut 10002q^{97} \) \(\mathstrut +\mathstrut 2401q^{98} \) \(\mathstrut -\mathstrut 27540q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1.00000 −9.00000 −31.0000 −34.0000 −9.00000 −49.0000 −63.0000 81.0000 −34.0000
\(n\): 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 \(T_{2} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(21))\).