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 $1$

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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: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta q^{5} + ( - 2 \beta + 2) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + \beta q^{5} + ( - 2 \beta + 2) q^{7} + q^{8} + \beta q^{10} + \beta q^{11} + (\beta - 1) q^{13} + ( - 2 \beta + 2) q^{14} + q^{16} + 6 q^{17} + 2 q^{19} + \beta q^{20} + \beta q^{22} + ( - 3 \beta + 3) q^{23} + (\beta - 2) q^{25} + (\beta - 1) q^{26} + ( - 2 \beta + 2) q^{28} + (3 \beta - 3) q^{29} + ( - \beta + 2) q^{31} + q^{32} + 6 q^{34} - 6 q^{35} + q^{37} + 2 q^{38} + \beta q^{40} + ( - 3 \beta - 3) q^{41} + (2 \beta - 4) q^{43} + \beta q^{44} + ( - 3 \beta + 3) q^{46} - 2 \beta q^{47} + ( - 4 \beta + 9) q^{49} + (\beta - 2) q^{50} + (\beta - 1) q^{52} + 6 q^{53} + (\beta + 3) q^{55} + ( - 2 \beta + 2) q^{56} + (3 \beta - 3) q^{58} + ( - 2 \beta - 6) q^{59} + (5 \beta - 4) q^{61} + ( - \beta + 2) q^{62} + q^{64} + 3 q^{65} + ( - 5 \beta + 8) q^{67} + 6 q^{68} - 6 q^{70} - 6 q^{71} + ( - \beta - 10) q^{73} + q^{74} + 2 q^{76} - 6 q^{77} + (7 \beta - 7) q^{79} + \beta q^{80} + ( - 3 \beta - 3) q^{82} + (4 \beta - 12) q^{83} + 6 \beta q^{85} + (2 \beta - 4) q^{86} + \beta q^{88} + 4 \beta q^{89} + (2 \beta - 8) q^{91} + ( - 3 \beta + 3) q^{92} - 2 \beta q^{94} + 2 \beta q^{95} + ( - 8 \beta + 2) q^{97} + ( - 4 \beta + 9) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8} + q^{10} + q^{11} - q^{13} + 2 q^{14} + 2 q^{16} + 12 q^{17} + 4 q^{19} + q^{20} + q^{22} + 3 q^{23} - 3 q^{25} - q^{26} + 2 q^{28} - 3 q^{29} + 3 q^{31} + 2 q^{32} + 12 q^{34} - 12 q^{35} + 2 q^{37} + 4 q^{38} + q^{40} - 9 q^{41} - 6 q^{43} + q^{44} + 3 q^{46} - 2 q^{47} + 14 q^{49} - 3 q^{50} - q^{52} + 12 q^{53} + 7 q^{55} + 2 q^{56} - 3 q^{58} - 14 q^{59} - 3 q^{61} + 3 q^{62} + 2 q^{64} + 6 q^{65} + 11 q^{67} + 12 q^{68} - 12 q^{70} - 12 q^{71} - 21 q^{73} + 2 q^{74} + 4 q^{76} - 12 q^{77} - 7 q^{79} + q^{80} - 9 q^{82} - 20 q^{83} + 6 q^{85} - 6 q^{86} + q^{88} + 4 q^{89} - 14 q^{91} + 3 q^{92} - 2 q^{94} + 2 q^{95} - 4 q^{97} + 14 q^{98}+O(q^{100}) \) 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
−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 \(S_{2}^{\mathrm{new}}(\Gamma_0(666))\):

\( T_{5}^{2} - T_{5} - 3 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

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