Properties

Label 21.4.a.a
Level 21
Weight 4
Character orbit 21.a
Self dual Yes
Analytic conductor 1.239
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 21.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.23904011012\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut 18q^{5} \) \(\mathstrut +\mathstrut 9q^{6} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 21q^{8} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut 18q^{5} \) \(\mathstrut +\mathstrut 9q^{6} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 21q^{8} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut 54q^{10} \) \(\mathstrut -\mathstrut 36q^{11} \) \(\mathstrut -\mathstrut 3q^{12} \) \(\mathstrut -\mathstrut 34q^{13} \) \(\mathstrut -\mathstrut 21q^{14} \) \(\mathstrut +\mathstrut 54q^{15} \) \(\mathstrut -\mathstrut 71q^{16} \) \(\mathstrut +\mathstrut 42q^{17} \) \(\mathstrut -\mathstrut 27q^{18} \) \(\mathstrut -\mathstrut 124q^{19} \) \(\mathstrut -\mathstrut 18q^{20} \) \(\mathstrut -\mathstrut 21q^{21} \) \(\mathstrut +\mathstrut 108q^{22} \) \(\mathstrut -\mathstrut 63q^{24} \) \(\mathstrut +\mathstrut 199q^{25} \) \(\mathstrut +\mathstrut 102q^{26} \) \(\mathstrut -\mathstrut 27q^{27} \) \(\mathstrut +\mathstrut 7q^{28} \) \(\mathstrut +\mathstrut 102q^{29} \) \(\mathstrut -\mathstrut 162q^{30} \) \(\mathstrut -\mathstrut 160q^{31} \) \(\mathstrut +\mathstrut 45q^{32} \) \(\mathstrut +\mathstrut 108q^{33} \) \(\mathstrut -\mathstrut 126q^{34} \) \(\mathstrut -\mathstrut 126q^{35} \) \(\mathstrut +\mathstrut 9q^{36} \) \(\mathstrut +\mathstrut 398q^{37} \) \(\mathstrut +\mathstrut 372q^{38} \) \(\mathstrut +\mathstrut 102q^{39} \) \(\mathstrut -\mathstrut 378q^{40} \) \(\mathstrut -\mathstrut 318q^{41} \) \(\mathstrut +\mathstrut 63q^{42} \) \(\mathstrut -\mathstrut 268q^{43} \) \(\mathstrut -\mathstrut 36q^{44} \) \(\mathstrut -\mathstrut 162q^{45} \) \(\mathstrut +\mathstrut 240q^{47} \) \(\mathstrut +\mathstrut 213q^{48} \) \(\mathstrut +\mathstrut 49q^{49} \) \(\mathstrut -\mathstrut 597q^{50} \) \(\mathstrut -\mathstrut 126q^{51} \) \(\mathstrut -\mathstrut 34q^{52} \) \(\mathstrut -\mathstrut 498q^{53} \) \(\mathstrut +\mathstrut 81q^{54} \) \(\mathstrut +\mathstrut 648q^{55} \) \(\mathstrut +\mathstrut 147q^{56} \) \(\mathstrut +\mathstrut 372q^{57} \) \(\mathstrut -\mathstrut 306q^{58} \) \(\mathstrut -\mathstrut 132q^{59} \) \(\mathstrut +\mathstrut 54q^{60} \) \(\mathstrut +\mathstrut 398q^{61} \) \(\mathstrut +\mathstrut 480q^{62} \) \(\mathstrut +\mathstrut 63q^{63} \) \(\mathstrut +\mathstrut 433q^{64} \) \(\mathstrut +\mathstrut 612q^{65} \) \(\mathstrut -\mathstrut 324q^{66} \) \(\mathstrut +\mathstrut 92q^{67} \) \(\mathstrut +\mathstrut 42q^{68} \) \(\mathstrut +\mathstrut 378q^{70} \) \(\mathstrut -\mathstrut 720q^{71} \) \(\mathstrut +\mathstrut 189q^{72} \) \(\mathstrut -\mathstrut 502q^{73} \) \(\mathstrut -\mathstrut 1194q^{74} \) \(\mathstrut -\mathstrut 597q^{75} \) \(\mathstrut -\mathstrut 124q^{76} \) \(\mathstrut -\mathstrut 252q^{77} \) \(\mathstrut -\mathstrut 306q^{78} \) \(\mathstrut -\mathstrut 1024q^{79} \) \(\mathstrut +\mathstrut 1278q^{80} \) \(\mathstrut +\mathstrut 81q^{81} \) \(\mathstrut +\mathstrut 954q^{82} \) \(\mathstrut -\mathstrut 204q^{83} \) \(\mathstrut -\mathstrut 21q^{84} \) \(\mathstrut -\mathstrut 756q^{85} \) \(\mathstrut +\mathstrut 804q^{86} \) \(\mathstrut -\mathstrut 306q^{87} \) \(\mathstrut -\mathstrut 756q^{88} \) \(\mathstrut +\mathstrut 354q^{89} \) \(\mathstrut +\mathstrut 486q^{90} \) \(\mathstrut -\mathstrut 238q^{91} \) \(\mathstrut +\mathstrut 480q^{93} \) \(\mathstrut -\mathstrut 720q^{94} \) \(\mathstrut +\mathstrut 2232q^{95} \) \(\mathstrut -\mathstrut 135q^{96} \) \(\mathstrut -\mathstrut 286q^{97} \) \(\mathstrut -\mathstrut 147q^{98} \) \(\mathstrut -\mathstrut 324q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−3.00000 −3.00000 1.00000 −18.0000 9.00000 7.00000 21.0000 9.00000 54.0000
\(n\): 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 \(T_{2} \) \(\mathstrut +\mathstrut 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(21))\).