Properties

Label 270.2.a.a
Level 270
Weight 2
Character orbit 270.a
Self dual Yes
Analytic conductor 2.156
Analytic rank 0
Dimension 1
CM No
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 270.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(2.15596085457\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + 2q^{7} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{5} + 2q^{7} - q^{8} + q^{10} + 3q^{11} - q^{13} - 2q^{14} + q^{16} + 3q^{17} + 8q^{19} - q^{20} - 3q^{22} - 3q^{23} + q^{25} + q^{26} + 2q^{28} + 9q^{29} - 7q^{31} - q^{32} - 3q^{34} - 2q^{35} + 2q^{37} - 8q^{38} + q^{40} + 12q^{41} - 7q^{43} + 3q^{44} + 3q^{46} + 3q^{47} - 3q^{49} - q^{50} - q^{52} - 12q^{53} - 3q^{55} - 2q^{56} - 9q^{58} - 12q^{59} - 10q^{61} + 7q^{62} + q^{64} + q^{65} - 4q^{67} + 3q^{68} + 2q^{70} + 2q^{73} - 2q^{74} + 8q^{76} + 6q^{77} - q^{79} - q^{80} - 12q^{82} - 18q^{83} - 3q^{85} + 7q^{86} - 3q^{88} - 2q^{91} - 3q^{92} - 3q^{94} - 8q^{95} + 14q^{97} + 3q^{98} + O(q^{100}) \)

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 0 1.00000 −1.00000 0 2.00000 −1.00000 0 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

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(270))\):

\( T_{11} - 3 \)
\( T_{13} + 1 \)