Properties

Label 666.2.a.e
Level $666$
Weight $2$
Character orbit 666.a
Self dual yes
Analytic conductor $5.318$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.31803677462\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - 2q^{5} - 3q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} - 2q^{5} - 3q^{7} + q^{8} - 2q^{10} - 5q^{11} - 3q^{13} - 3q^{14} + q^{16} - 3q^{17} + 5q^{19} - 2q^{20} - 5q^{22} - 3q^{23} - q^{25} - 3q^{26} - 3q^{28} + 4q^{31} + q^{32} - 3q^{34} + 6q^{35} + q^{37} + 5q^{38} - 2q^{40} - 6q^{41} + 4q^{43} - 5q^{44} - 3q^{46} + 4q^{47} + 2q^{49} - q^{50} - 3q^{52} + 3q^{53} + 10q^{55} - 3q^{56} - 14q^{59} - 14q^{61} + 4q^{62} + q^{64} + 6q^{65} + 12q^{67} - 3q^{68} + 6q^{70} - 12q^{71} + 13q^{73} + q^{74} + 5q^{76} + 15q^{77} + 6q^{79} - 2q^{80} - 6q^{82} + 7q^{83} + 6q^{85} + 4q^{86} - 5q^{88} + q^{89} + 9q^{91} - 3q^{92} + 4q^{94} - 10q^{95} - 12q^{97} + 2q^{98} + O(q^{100}) \)

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 −2.00000 0 −3.00000 1.00000 0 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(37\) \(-1\)

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 666.2.a.e yes 1
3.b odd 2 1 666.2.a.c 1
4.b odd 2 1 5328.2.a.f 1
12.b even 2 1 5328.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.a.c 1 3.b odd 2 1
666.2.a.e yes 1 1.a even 1 1 trivial
5328.2.a.f 1 4.b odd 2 1
5328.2.a.s 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(666))\):

\( T_{5} + 2 \)
\( T_{7} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( 2 + T \)
$7$ \( 3 + T \)
$11$ \( 5 + T \)
$13$ \( 3 + T \)
$17$ \( 3 + T \)
$19$ \( -5 + T \)
$23$ \( 3 + T \)
$29$ \( T \)
$31$ \( -4 + T \)
$37$ \( -1 + T \)
$41$ \( 6 + T \)
$43$ \( -4 + T \)
$47$ \( -4 + T \)
$53$ \( -3 + T \)
$59$ \( 14 + T \)
$61$ \( 14 + T \)
$67$ \( -12 + T \)
$71$ \( 12 + T \)
$73$ \( -13 + T \)
$79$ \( -6 + T \)
$83$ \( -7 + T \)
$89$ \( -1 + T \)
$97$ \( 12 + T \)
show more
show less