Properties

Label 966.2.a.o
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: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

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

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
3.37228
−2.37228
1.00000 −1.00000 1.00000 −4.37228 −1.00000 −1.00000 1.00000 1.00000 −4.37228
1.2 1.00000 −1.00000 1.00000 1.37228 −1.00000 −1.00000 1.00000 1.00000 1.37228
\(n\): 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.o 2
3.b odd 2 1 2898.2.a.x 2
4.b odd 2 1 7728.2.a.bh 2
7.b odd 2 1 6762.2.a.cd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.o 2 1.a even 1 1 trivial
2898.2.a.x 2 3.b odd 2 1
6762.2.a.cd 2 7.b odd 2 1
7728.2.a.bh 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(966))\):

\( T_{5}^{2} + 3 T_{5} - 6 \)
\( T_{11} - 4 \)
\( T_{13}^{2} + 3 T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -6 + 3 T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( -6 + 3 T + T^{2} \)
$17$ \( -32 - 2 T + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( 34 - 13 T + T^{2} \)
$31$ \( -32 - 2 T + T^{2} \)
$37$ \( 22 - 11 T + T^{2} \)
$41$ \( -74 - T + T^{2} \)
$43$ \( -68 + 5 T + T^{2} \)
$47$ \( 48 - 15 T + T^{2} \)
$53$ \( -32 + 2 T + T^{2} \)
$59$ \( -24 - 6 T + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( 48 - 18 T + T^{2} \)
$73$ \( 16 + 14 T + T^{2} \)
$79$ \( -128 + 4 T + T^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( -116 + 8 T + T^{2} \)
$97$ \( 34 + 13 T + T^{2} \)
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