Properties

Label 668.2.a.a
Level $668$
Weight $2$
Character orbit 668.a
Self dual yes
Analytic conductor $5.334$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 668 = 2^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 668.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.33400685502\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} -3 q^{5} + ( 1 + \beta ) q^{7} + \beta q^{9} +O(q^{10})\) \( q + \beta q^{3} -3 q^{5} + ( 1 + \beta ) q^{7} + \beta q^{9} + ( 4 + \beta ) q^{13} -3 \beta q^{15} + ( 1 - \beta ) q^{17} + 2 q^{19} + ( 3 + 2 \beta ) q^{21} + ( -1 + \beta ) q^{23} + 4 q^{25} + ( 3 - 2 \beta ) q^{27} + ( 5 - 2 \beta ) q^{29} + ( 4 - 2 \beta ) q^{31} + ( -3 - 3 \beta ) q^{35} + ( 3 + 2 \beta ) q^{37} + ( 3 + 5 \beta ) q^{39} + ( -1 + 4 \beta ) q^{41} + ( 7 - 2 \beta ) q^{43} -3 \beta q^{45} + ( -1 - 2 \beta ) q^{47} + ( -3 + 3 \beta ) q^{49} -3 q^{51} + ( -2 + 2 \beta ) q^{53} + 2 \beta q^{57} -6 \beta q^{59} + ( -1 - 6 \beta ) q^{61} + ( 3 + 2 \beta ) q^{63} + ( -12 - 3 \beta ) q^{65} + ( 3 - 4 \beta ) q^{67} + 3 q^{69} -3 \beta q^{71} + ( -7 - 3 \beta ) q^{73} + 4 \beta q^{75} + ( 9 + 2 \beta ) q^{79} + ( -6 - 2 \beta ) q^{81} + ( 1 + 2 \beta ) q^{83} + ( -3 + 3 \beta ) q^{85} + ( -6 + 3 \beta ) q^{87} + ( -6 + 6 \beta ) q^{89} + ( 7 + 6 \beta ) q^{91} + ( -6 + 2 \beta ) q^{93} -6 q^{95} + ( -2 - 5 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 6q^{5} + 3q^{7} + q^{9} + O(q^{10}) \) \( 2q + q^{3} - 6q^{5} + 3q^{7} + q^{9} + 9q^{13} - 3q^{15} + q^{17} + 4q^{19} + 8q^{21} - q^{23} + 8q^{25} + 4q^{27} + 8q^{29} + 6q^{31} - 9q^{35} + 8q^{37} + 11q^{39} + 2q^{41} + 12q^{43} - 3q^{45} - 4q^{47} - 3q^{49} - 6q^{51} - 2q^{53} + 2q^{57} - 6q^{59} - 8q^{61} + 8q^{63} - 27q^{65} + 2q^{67} + 6q^{69} - 3q^{71} - 17q^{73} + 4q^{75} + 20q^{79} - 14q^{81} + 4q^{83} - 3q^{85} - 9q^{87} - 6q^{89} + 20q^{91} - 10q^{93} - 12q^{95} - 9q^{97} + O(q^{100}) \)

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
0 −1.30278 0 −3.00000 0 −0.302776 0 −1.30278 0
1.2 0 2.30278 0 −3.00000 0 3.30278 0 2.30278 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(167\) \(1\)

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 668.2.a.a 2
3.b odd 2 1 6012.2.a.a 2
4.b odd 2 1 2672.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
668.2.a.a 2 1.a even 1 1 trivial
2672.2.a.c 2 4.b odd 2 1
6012.2.a.a 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(668))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -3 - T + T^{2} \)
$5$ \( ( 3 + T )^{2} \)
$7$ \( -1 - 3 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 17 - 9 T + T^{2} \)
$17$ \( -3 - T + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( -3 + T + T^{2} \)
$29$ \( 3 - 8 T + T^{2} \)
$31$ \( -4 - 6 T + T^{2} \)
$37$ \( 3 - 8 T + T^{2} \)
$41$ \( -51 - 2 T + T^{2} \)
$43$ \( 23 - 12 T + T^{2} \)
$47$ \( -9 + 4 T + T^{2} \)
$53$ \( -12 + 2 T + T^{2} \)
$59$ \( -108 + 6 T + T^{2} \)
$61$ \( -101 + 8 T + T^{2} \)
$67$ \( -51 - 2 T + T^{2} \)
$71$ \( -27 + 3 T + T^{2} \)
$73$ \( 43 + 17 T + T^{2} \)
$79$ \( 87 - 20 T + T^{2} \)
$83$ \( -9 - 4 T + T^{2} \)
$89$ \( -108 + 6 T + T^{2} \)
$97$ \( -61 + 9 T + T^{2} \)
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