Properties

Label 6600.2.a.a
Level $6600$
Weight $2$
Character orbit 6600.a
Self dual yes
Analytic conductor $52.701$
Analytic rank $2$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(52.7012653340\)
Analytic rank: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 264)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 4q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} - 4q^{7} + q^{9} - q^{11} - 6q^{13} - 6q^{17} - 8q^{19} + 4q^{21} - q^{27} - 6q^{29} + q^{33} - 6q^{37} + 6q^{39} - 10q^{41} + 8q^{43} + 9q^{49} + 6q^{51} - 6q^{53} + 8q^{57} + 4q^{59} - 2q^{61} - 4q^{63} + 12q^{67} - 8q^{71} - 2q^{73} + 4q^{77} - 4q^{79} + q^{81} + 12q^{83} + 6q^{87} - 6q^{89} + 24q^{91} - 2q^{97} - q^{99} + O(q^{100}) \)

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
0 −1.00000 0 0 0 −4.00000 0 1.00000 0
\(n\): 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 6600.2.a.a 1
5.b even 2 1 264.2.a.b 1
5.c odd 4 2 6600.2.d.n 2
15.d odd 2 1 792.2.a.f 1
20.d odd 2 1 528.2.a.b 1
40.e odd 2 1 2112.2.a.y 1
40.f even 2 1 2112.2.a.m 1
55.d odd 2 1 2904.2.a.i 1
60.h even 2 1 1584.2.a.n 1
120.i odd 2 1 6336.2.a.v 1
120.m even 2 1 6336.2.a.o 1
165.d even 2 1 8712.2.a.r 1
220.g even 2 1 5808.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.2.a.b 1 5.b even 2 1
528.2.a.b 1 20.d odd 2 1
792.2.a.f 1 15.d odd 2 1
1584.2.a.n 1 60.h even 2 1
2112.2.a.m 1 40.f even 2 1
2112.2.a.y 1 40.e odd 2 1
2904.2.a.i 1 55.d odd 2 1
5808.2.a.f 1 220.g even 2 1
6336.2.a.o 1 120.m even 2 1
6336.2.a.v 1 120.i odd 2 1
6600.2.a.a 1 1.a even 1 1 trivial
6600.2.d.n 2 5.c odd 4 2
8712.2.a.r 1 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6600))\):

\( T_{7} + 4 \)
\( T_{13} + 6 \)
\( T_{17} + 6 \)
\( T_{19} + 8 \)