Properties

Label 30.2.a.a
Level 30
Weight 2
Character orbit 30.a
Self dual Yes
Analytic conductor 0.240
Analytic rank 0
Dimension 1
CM No
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 30.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.239551206064\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + 2q^{13} + 4q^{14} - q^{15} + q^{16} + 6q^{17} - q^{18} - 4q^{19} - q^{20} - 4q^{21} - q^{24} + q^{25} - 2q^{26} + q^{27} - 4q^{28} - 6q^{29} + q^{30} + 8q^{31} - q^{32} - 6q^{34} + 4q^{35} + q^{36} + 2q^{37} + 4q^{38} + 2q^{39} + q^{40} - 6q^{41} + 4q^{42} - 4q^{43} - q^{45} + q^{48} + 9q^{49} - q^{50} + 6q^{51} + 2q^{52} - 6q^{53} - q^{54} + 4q^{56} - 4q^{57} + 6q^{58} - q^{60} - 10q^{61} - 8q^{62} - 4q^{63} + q^{64} - 2q^{65} - 4q^{67} + 6q^{68} - 4q^{70} - q^{72} + 2q^{73} - 2q^{74} + q^{75} - 4q^{76} - 2q^{78} + 8q^{79} - q^{80} + q^{81} + 6q^{82} + 12q^{83} - 4q^{84} - 6q^{85} + 4q^{86} - 6q^{87} + 18q^{89} + q^{90} - 8q^{91} + 8q^{93} + 4q^{95} - q^{96} + 2q^{97} - 9q^{98} + O(q^{100}) \)

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 1.00000 1.00000 −1.00000 −1.00000 −4.00000 −1.00000 1.00000 1.00000
\(n\): 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
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(\Gamma_0(30))\).