Properties

Label 966.2.a.l
Level $966$
Weight $2$
Character orbit 966.a
Self dual yes
Analytic conductor $7.714$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + \beta q^{5} + q^{6} - q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} - q^{3} + q^{4} + \beta q^{5} + q^{6} - q^{7} - q^{8} + q^{9} -\beta q^{10} + ( 2 - 2 \beta ) q^{11} - q^{12} + ( -2 + \beta ) q^{13} + q^{14} -\beta q^{15} + q^{16} + ( 2 + 2 \beta ) q^{17} - q^{18} + ( -2 + 2 \beta ) q^{19} + \beta q^{20} + q^{21} + ( -2 + 2 \beta ) q^{22} + q^{23} + q^{24} + ( -1 + \beta ) q^{25} + ( 2 - \beta ) q^{26} - q^{27} - q^{28} + ( 6 - \beta ) q^{29} + \beta q^{30} -2 \beta q^{31} - q^{32} + ( -2 + 2 \beta ) q^{33} + ( -2 - 2 \beta ) q^{34} -\beta q^{35} + q^{36} + \beta q^{37} + ( 2 - 2 \beta ) q^{38} + ( 2 - \beta ) q^{39} -\beta q^{40} + ( 2 - \beta ) q^{41} - q^{42} + ( 2 - 3 \beta ) q^{43} + ( 2 - 2 \beta ) q^{44} + \beta q^{45} - q^{46} + ( 4 + \beta ) q^{47} - q^{48} + q^{49} + ( 1 - \beta ) q^{50} + ( -2 - 2 \beta ) q^{51} + ( -2 + \beta ) q^{52} + ( 8 - 4 \beta ) q^{53} + q^{54} -8 q^{55} + q^{56} + ( 2 - 2 \beta ) q^{57} + ( -6 + \beta ) q^{58} + ( 8 + 2 \beta ) q^{59} -\beta q^{60} + ( 4 + 2 \beta ) q^{61} + 2 \beta q^{62} - q^{63} + q^{64} + ( 4 - \beta ) q^{65} + ( 2 - 2 \beta ) q^{66} + ( 2 + 2 \beta ) q^{67} + ( 2 + 2 \beta ) q^{68} - q^{69} + \beta q^{70} + 6 \beta q^{71} - q^{72} + ( 6 - 2 \beta ) q^{73} -\beta q^{74} + ( 1 - \beta ) q^{75} + ( -2 + 2 \beta ) q^{76} + ( -2 + 2 \beta ) q^{77} + ( -2 + \beta ) q^{78} + ( 8 - 4 \beta ) q^{79} + \beta q^{80} + q^{81} + ( -2 + \beta ) q^{82} + ( -6 + 2 \beta ) q^{83} + q^{84} + ( 8 + 4 \beta ) q^{85} + ( -2 + 3 \beta ) q^{86} + ( -6 + \beta ) q^{87} + ( -2 + 2 \beta ) q^{88} + 14 q^{89} -\beta q^{90} + ( 2 - \beta ) q^{91} + q^{92} + 2 \beta q^{93} + ( -4 - \beta ) q^{94} + 8 q^{95} + q^{96} + ( -14 - \beta ) q^{97} - q^{98} + ( 2 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + q^{5} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + q^{5} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} - q^{10} + 2q^{11} - 2q^{12} - 3q^{13} + 2q^{14} - q^{15} + 2q^{16} + 6q^{17} - 2q^{18} - 2q^{19} + q^{20} + 2q^{21} - 2q^{22} + 2q^{23} + 2q^{24} - q^{25} + 3q^{26} - 2q^{27} - 2q^{28} + 11q^{29} + q^{30} - 2q^{31} - 2q^{32} - 2q^{33} - 6q^{34} - q^{35} + 2q^{36} + q^{37} + 2q^{38} + 3q^{39} - q^{40} + 3q^{41} - 2q^{42} + q^{43} + 2q^{44} + q^{45} - 2q^{46} + 9q^{47} - 2q^{48} + 2q^{49} + q^{50} - 6q^{51} - 3q^{52} + 12q^{53} + 2q^{54} - 16q^{55} + 2q^{56} + 2q^{57} - 11q^{58} + 18q^{59} - q^{60} + 10q^{61} + 2q^{62} - 2q^{63} + 2q^{64} + 7q^{65} + 2q^{66} + 6q^{67} + 6q^{68} - 2q^{69} + q^{70} + 6q^{71} - 2q^{72} + 10q^{73} - q^{74} + q^{75} - 2q^{76} - 2q^{77} - 3q^{78} + 12q^{79} + q^{80} + 2q^{81} - 3q^{82} - 10q^{83} + 2q^{84} + 20q^{85} - q^{86} - 11q^{87} - 2q^{88} + 28q^{89} - q^{90} + 3q^{91} + 2q^{92} + 2q^{93} - 9q^{94} + 16q^{95} + 2q^{96} - 29q^{97} - 2q^{98} + 2q^{99} + O(q^{100}) \)

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
−1.56155
2.56155
−1.00000 −1.00000 1.00000 −1.56155 1.00000 −1.00000 −1.00000 1.00000 1.56155
1.2 −1.00000 −1.00000 1.00000 2.56155 1.00000 −1.00000 −1.00000 1.00000 −2.56155
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(1\)
\(23\) \(-1\)

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 966.2.a.l 2
3.b odd 2 1 2898.2.a.ba 2
4.b odd 2 1 7728.2.a.bm 2
7.b odd 2 1 6762.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.l 2 1.a even 1 1 trivial
2898.2.a.ba 2 3.b odd 2 1
6762.2.a.bw 2 7.b odd 2 1
7728.2.a.bm 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(966))\):

\( T_{5}^{2} - T_{5} - 4 \)
\( T_{11}^{2} - 2 T_{11} - 16 \)
\( T_{13}^{2} + 3 T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -4 - T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -16 - 2 T + T^{2} \)
$13$ \( -2 + 3 T + T^{2} \)
$17$ \( -8 - 6 T + T^{2} \)
$19$ \( -16 + 2 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 26 - 11 T + T^{2} \)
$31$ \( -16 + 2 T + T^{2} \)
$37$ \( -4 - T + T^{2} \)
$41$ \( -2 - 3 T + T^{2} \)
$43$ \( -38 - T + T^{2} \)
$47$ \( 16 - 9 T + T^{2} \)
$53$ \( -32 - 12 T + T^{2} \)
$59$ \( 64 - 18 T + T^{2} \)
$61$ \( 8 - 10 T + T^{2} \)
$67$ \( -8 - 6 T + T^{2} \)
$71$ \( -144 - 6 T + T^{2} \)
$73$ \( 8 - 10 T + T^{2} \)
$79$ \( -32 - 12 T + T^{2} \)
$83$ \( 8 + 10 T + T^{2} \)
$89$ \( ( -14 + T )^{2} \)
$97$ \( 206 + 29 T + T^{2} \)
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