Properties

Label 666.2.a.k
Level $666$
Weight $2$
Character orbit 666.a
Self dual yes
Analytic conductor $5.318$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + 2 q^{5} + ( 1 - \beta ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + 2 q^{5} + ( 1 - \beta ) q^{7} + q^{8} + 2 q^{10} + ( 1 - \beta ) q^{11} + ( -1 + 3 \beta ) q^{13} + ( 1 - \beta ) q^{14} + q^{16} + ( 1 + \beta ) q^{17} + ( -1 - \beta ) q^{19} + 2 q^{20} + ( 1 - \beta ) q^{22} + ( 1 + 3 \beta ) q^{23} - q^{25} + ( -1 + 3 \beta ) q^{26} + ( 1 - \beta ) q^{28} + ( 4 - 2 \beta ) q^{29} + ( -4 - 2 \beta ) q^{31} + q^{32} + ( 1 + \beta ) q^{34} + ( 2 - 2 \beta ) q^{35} - q^{37} + ( -1 - \beta ) q^{38} + 2 q^{40} + ( 6 + 2 \beta ) q^{41} + ( -4 + 2 \beta ) q^{43} + ( 1 - \beta ) q^{44} + ( 1 + 3 \beta ) q^{46} -4 \beta q^{47} + ( -2 - \beta ) q^{49} - q^{50} + ( -1 + 3 \beta ) q^{52} + ( 1 - \beta ) q^{53} + ( 2 - 2 \beta ) q^{55} + ( 1 - \beta ) q^{56} + ( 4 - 2 \beta ) q^{58} + ( 6 - 2 \beta ) q^{59} + ( -2 - 4 \beta ) q^{61} + ( -4 - 2 \beta ) q^{62} + q^{64} + ( -2 + 6 \beta ) q^{65} + ( -8 + 4 \beta ) q^{67} + ( 1 + \beta ) q^{68} + ( 2 - 2 \beta ) q^{70} + ( -8 + 4 \beta ) q^{71} + ( 1 + \beta ) q^{73} - q^{74} + ( -1 - \beta ) q^{76} + ( 5 - \beta ) q^{77} -6 q^{79} + 2 q^{80} + ( 6 + 2 \beta ) q^{82} + ( -3 + 7 \beta ) q^{83} + ( 2 + 2 \beta ) q^{85} + ( -4 + 2 \beta ) q^{86} + ( 1 - \beta ) q^{88} + ( 5 - 7 \beta ) q^{89} + ( -13 + \beta ) q^{91} + ( 1 + 3 \beta ) q^{92} -4 \beta q^{94} + ( -2 - 2 \beta ) q^{95} + ( 8 - 2 \beta ) q^{97} + ( -2 - \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 4q^{5} + q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 4q^{5} + q^{7} + 2q^{8} + 4q^{10} + q^{11} + q^{13} + q^{14} + 2q^{16} + 3q^{17} - 3q^{19} + 4q^{20} + q^{22} + 5q^{23} - 2q^{25} + q^{26} + q^{28} + 6q^{29} - 10q^{31} + 2q^{32} + 3q^{34} + 2q^{35} - 2q^{37} - 3q^{38} + 4q^{40} + 14q^{41} - 6q^{43} + q^{44} + 5q^{46} - 4q^{47} - 5q^{49} - 2q^{50} + q^{52} + q^{53} + 2q^{55} + q^{56} + 6q^{58} + 10q^{59} - 8q^{61} - 10q^{62} + 2q^{64} + 2q^{65} - 12q^{67} + 3q^{68} + 2q^{70} - 12q^{71} + 3q^{73} - 2q^{74} - 3q^{76} + 9q^{77} - 12q^{79} + 4q^{80} + 14q^{82} + q^{83} + 6q^{85} - 6q^{86} + q^{88} + 3q^{89} - 25q^{91} + 5q^{92} - 4q^{94} - 6q^{95} + 14q^{97} - 5q^{98} + O(q^{100}) \)

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
2.56155
−1.56155
1.00000 0 1.00000 2.00000 0 −1.56155 1.00000 0 2.00000
1.2 1.00000 0 1.00000 2.00000 0 2.56155 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.k yes 2
3.b odd 2 1 666.2.a.h 2
4.b odd 2 1 5328.2.a.bh 2
12.b even 2 1 5328.2.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.a.h 2 3.b odd 2 1
666.2.a.k yes 2 1.a even 1 1 trivial
5328.2.a.y 2 12.b even 2 1
5328.2.a.bh 2 4.b odd 2 1

Hecke kernels

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

\( T_{5} - 2 \)
\( T_{7}^{2} - T_{7} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -2 + T )^{2} \)
$7$ \( -4 - T + T^{2} \)
$11$ \( -4 - T + T^{2} \)
$13$ \( -38 - T + T^{2} \)
$17$ \( -2 - 3 T + T^{2} \)
$19$ \( -2 + 3 T + T^{2} \)
$23$ \( -32 - 5 T + T^{2} \)
$29$ \( -8 - 6 T + T^{2} \)
$31$ \( 8 + 10 T + T^{2} \)
$37$ \( ( 1 + T )^{2} \)
$41$ \( 32 - 14 T + T^{2} \)
$43$ \( -8 + 6 T + T^{2} \)
$47$ \( -64 + 4 T + T^{2} \)
$53$ \( -4 - T + T^{2} \)
$59$ \( 8 - 10 T + T^{2} \)
$61$ \( -52 + 8 T + T^{2} \)
$67$ \( -32 + 12 T + T^{2} \)
$71$ \( -32 + 12 T + T^{2} \)
$73$ \( -2 - 3 T + T^{2} \)
$79$ \( ( 6 + T )^{2} \)
$83$ \( -208 - T + T^{2} \)
$89$ \( -206 - 3 T + T^{2} \)
$97$ \( 32 - 14 T + T^{2} \)
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