Properties

Label 330.2.a.d
Level $330$
Weight $2$
Character orbit 330.a
Self dual yes
Analytic conductor $2.635$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.63506326670\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} + q^{11} - q^{12} + 6q^{13} - q^{15} + q^{16} + 2q^{17} + q^{18} - 4q^{19} + q^{20} + q^{22} - q^{24} + q^{25} + 6q^{26} - q^{27} - 10q^{29} - q^{30} + q^{32} - q^{33} + 2q^{34} + q^{36} + 6q^{37} - 4q^{38} - 6q^{39} + q^{40} + 2q^{41} + 4q^{43} + q^{44} + q^{45} - 8q^{47} - q^{48} - 7q^{49} + q^{50} - 2q^{51} + 6q^{52} - 10q^{53} - q^{54} + q^{55} + 4q^{57} - 10q^{58} - 4q^{59} - q^{60} - 2q^{61} + q^{64} + 6q^{65} - q^{66} - 4q^{67} + 2q^{68} - 8q^{71} + q^{72} + 2q^{73} + 6q^{74} - q^{75} - 4q^{76} - 6q^{78} - 8q^{79} + q^{80} + q^{81} + 2q^{82} - 12q^{83} + 2q^{85} + 4q^{86} + 10q^{87} + q^{88} - 6q^{89} + q^{90} - 8q^{94} - 4q^{95} - q^{96} + 18q^{97} - 7q^{98} + q^{99} + O(q^{100}) \)

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 0 1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 330.2.a.d 1
3.b odd 2 1 990.2.a.b 1
4.b odd 2 1 2640.2.a.t 1
5.b even 2 1 1650.2.a.h 1
5.c odd 4 2 1650.2.c.g 2
11.b odd 2 1 3630.2.a.f 1
12.b even 2 1 7920.2.a.m 1
15.d odd 2 1 4950.2.a.bg 1
15.e even 4 2 4950.2.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.a.d 1 1.a even 1 1 trivial
990.2.a.b 1 3.b odd 2 1
1650.2.a.h 1 5.b even 2 1
1650.2.c.g 2 5.c odd 4 2
2640.2.a.t 1 4.b odd 2 1
3630.2.a.f 1 11.b odd 2 1
4950.2.a.bg 1 15.d odd 2 1
4950.2.c.j 2 15.e even 4 2
7920.2.a.m 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(330))\):

\( T_{7} \)
\( T_{13} - 6 \)