Properties

Label 666.2.a.j
Level $666$
Weight $2$
Character orbit 666.a
Self dual yes
Analytic conductor $5.318$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta q^{5} + ( 2 - 2 \beta ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + \beta q^{5} + ( 2 - 2 \beta ) q^{7} + q^{8} + \beta q^{10} + \beta q^{11} + ( -1 + \beta ) q^{13} + ( 2 - 2 \beta ) q^{14} + q^{16} + 6 q^{17} + 2 q^{19} + \beta q^{20} + \beta q^{22} + ( 3 - 3 \beta ) q^{23} + ( -2 + \beta ) q^{25} + ( -1 + \beta ) q^{26} + ( 2 - 2 \beta ) q^{28} + ( -3 + 3 \beta ) q^{29} + ( 2 - \beta ) q^{31} + q^{32} + 6 q^{34} -6 q^{35} + q^{37} + 2 q^{38} + \beta q^{40} + ( -3 - 3 \beta ) q^{41} + ( -4 + 2 \beta ) q^{43} + \beta q^{44} + ( 3 - 3 \beta ) q^{46} -2 \beta q^{47} + ( 9 - 4 \beta ) q^{49} + ( -2 + \beta ) q^{50} + ( -1 + \beta ) q^{52} + 6 q^{53} + ( 3 + \beta ) q^{55} + ( 2 - 2 \beta ) q^{56} + ( -3 + 3 \beta ) q^{58} + ( -6 - 2 \beta ) q^{59} + ( -4 + 5 \beta ) q^{61} + ( 2 - \beta ) q^{62} + q^{64} + 3 q^{65} + ( 8 - 5 \beta ) q^{67} + 6 q^{68} -6 q^{70} -6 q^{71} + ( -10 - \beta ) q^{73} + q^{74} + 2 q^{76} -6 q^{77} + ( -7 + 7 \beta ) q^{79} + \beta q^{80} + ( -3 - 3 \beta ) q^{82} + ( -12 + 4 \beta ) q^{83} + 6 \beta q^{85} + ( -4 + 2 \beta ) q^{86} + \beta q^{88} + 4 \beta q^{89} + ( -8 + 2 \beta ) q^{91} + ( 3 - 3 \beta ) q^{92} -2 \beta q^{94} + 2 \beta q^{95} + ( 2 - 8 \beta ) q^{97} + ( 9 - 4 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8} + q^{10} + q^{11} - q^{13} + 2 q^{14} + 2 q^{16} + 12 q^{17} + 4 q^{19} + q^{20} + q^{22} + 3 q^{23} - 3 q^{25} - q^{26} + 2 q^{28} - 3 q^{29} + 3 q^{31} + 2 q^{32} + 12 q^{34} - 12 q^{35} + 2 q^{37} + 4 q^{38} + q^{40} - 9 q^{41} - 6 q^{43} + q^{44} + 3 q^{46} - 2 q^{47} + 14 q^{49} - 3 q^{50} - q^{52} + 12 q^{53} + 7 q^{55} + 2 q^{56} - 3 q^{58} - 14 q^{59} - 3 q^{61} + 3 q^{62} + 2 q^{64} + 6 q^{65} + 11 q^{67} + 12 q^{68} - 12 q^{70} - 12 q^{71} - 21 q^{73} + 2 q^{74} + 4 q^{76} - 12 q^{77} - 7 q^{79} + q^{80} - 9 q^{82} - 20 q^{83} + 6 q^{85} - 6 q^{86} + q^{88} + 4 q^{89} - 14 q^{91} + 3 q^{92} - 2 q^{94} + 2 q^{95} - 4 q^{97} + 14 q^{98} + O(q^{100}) \)

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
−1.30278
2.30278
1.00000 0 1.00000 −1.30278 0 4.60555 1.00000 0 −1.30278
1.2 1.00000 0 1.00000 2.30278 0 −2.60555 1.00000 0 2.30278
\(n\): 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.j 2
3.b odd 2 1 74.2.a.a 2
4.b odd 2 1 5328.2.a.bf 2
12.b even 2 1 592.2.a.f 2
15.d odd 2 1 1850.2.a.u 2
15.e even 4 2 1850.2.b.i 4
21.c even 2 1 3626.2.a.a 2
24.f even 2 1 2368.2.a.ba 2
24.h odd 2 1 2368.2.a.s 2
33.d even 2 1 8954.2.a.p 2
111.d odd 2 1 2738.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 3.b odd 2 1
592.2.a.f 2 12.b even 2 1
666.2.a.j 2 1.a even 1 1 trivial
1850.2.a.u 2 15.d odd 2 1
1850.2.b.i 4 15.e even 4 2
2368.2.a.s 2 24.h odd 2 1
2368.2.a.ba 2 24.f even 2 1
2738.2.a.l 2 111.d odd 2 1
3626.2.a.a 2 21.c even 2 1
5328.2.a.bf 2 4.b odd 2 1
8954.2.a.p 2 33.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(666))\):

\( T_{5}^{2} - T_{5} - 3 \)
\( T_{7}^{2} - 2 T_{7} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -3 - T + T^{2} \)
$7$ \( -12 - 2 T + T^{2} \)
$11$ \( -3 - T + T^{2} \)
$13$ \( -3 + T + T^{2} \)
$17$ \( ( -6 + T )^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( -27 - 3 T + T^{2} \)
$29$ \( -27 + 3 T + T^{2} \)
$31$ \( -1 - 3 T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( -9 + 9 T + T^{2} \)
$43$ \( -4 + 6 T + T^{2} \)
$47$ \( -12 + 2 T + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 36 + 14 T + T^{2} \)
$61$ \( -79 + 3 T + T^{2} \)
$67$ \( -51 - 11 T + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( 107 + 21 T + T^{2} \)
$79$ \( -147 + 7 T + T^{2} \)
$83$ \( 48 + 20 T + T^{2} \)
$89$ \( -48 - 4 T + T^{2} \)
$97$ \( -204 + 4 T + T^{2} \)
show more
show less