Properties

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

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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: no (minimal twist has level 222)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 0 0 −1.00000 −1.00000 0 0
\(n\): 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.a 1
3.b odd 2 1 222.2.a.e 1
4.b odd 2 1 5328.2.a.l 1
12.b even 2 1 1776.2.a.c 1
15.d odd 2 1 5550.2.a.h 1
24.f even 2 1 7104.2.a.u 1
24.h odd 2 1 7104.2.a.g 1
111.d odd 2 1 8214.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.a.e 1 3.b odd 2 1
666.2.a.a 1 1.a even 1 1 trivial
1776.2.a.c 1 12.b even 2 1
5328.2.a.l 1 4.b odd 2 1
5550.2.a.h 1 15.d odd 2 1
7104.2.a.g 1 24.h odd 2 1
7104.2.a.u 1 24.f even 2 1
8214.2.a.d 1 111.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(666))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T + 3 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T + 7 \) Copy content Toggle raw display
$23$ \( T + 3 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 2 \) Copy content Toggle raw display
$37$ \( T - 1 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 6 \) Copy content Toggle raw display
$53$ \( T + 9 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T - 2 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T - 5 \) Copy content Toggle raw display
$79$ \( T - 2 \) Copy content Toggle raw display
$83$ \( T + 3 \) Copy content Toggle raw display
$89$ \( T - 3 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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