Properties

Label 966.2.a.n
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: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−2.70156
3.70156
−1.00000 1.00000 1.00000 −2.70156 −1.00000 1.00000 −1.00000 1.00000 2.70156
1.2 −1.00000 1.00000 1.00000 3.70156 −1.00000 1.00000 −1.00000 1.00000 −3.70156
\(n\): 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.n 2
3.b odd 2 1 2898.2.a.bb 2
4.b odd 2 1 7728.2.a.bc 2
7.b odd 2 1 6762.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.n 2 1.a even 1 1 trivial
2898.2.a.bb 2 3.b odd 2 1
6762.2.a.bo 2 7.b odd 2 1
7728.2.a.bc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(966))\):

\( T_{5}^{2} - T_{5} - 10 \)
\( T_{11} + 4 \)
\( T_{13}^{2} - 5 T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -10 - T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( -4 - 5 T + T^{2} \)
$17$ \( ( -4 + T )^{2} \)
$19$ \( -40 - 2 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 2 - 7 T + T^{2} \)
$31$ \( ( 2 + T )^{2} \)
$37$ \( 46 - 15 T + T^{2} \)
$41$ \( 2 + 7 T + T^{2} \)
$43$ \( -8 - 3 T + T^{2} \)
$47$ \( -90 + 3 T + T^{2} \)
$53$ \( -32 + 6 T + T^{2} \)
$59$ \( -40 + 2 T + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( -160 - 4 T + T^{2} \)
$71$ \( -40 + 2 T + T^{2} \)
$73$ \( -16 + 10 T + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 8 - 14 T + T^{2} \)
$89$ \( -16 + 10 T + T^{2} \)
$97$ \( 80 - 19 T + T^{2} \)
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