Properties

Label 66.2.a.a
Level $66$
Weight $2$
Character orbit 66.a
Self dual yes
Analytic conductor $0.527$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 66 = 2 \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 66.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.527012653340\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{6} + 2 q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} - q^{6} + 2 q^{7} - q^{8} + q^{9} - q^{11} + q^{12} - 4 q^{13} - 2 q^{14} + q^{16} - 6 q^{17} - q^{18} - 4 q^{19} + 2 q^{21} + q^{22} + 6 q^{23} - q^{24} - 5 q^{25} + 4 q^{26} + q^{27} + 2 q^{28} + 6 q^{29} + 8 q^{31} - q^{32} - q^{33} + 6 q^{34} + q^{36} - 10 q^{37} + 4 q^{38} - 4 q^{39} + 6 q^{41} - 2 q^{42} + 8 q^{43} - q^{44} - 6 q^{46} - 6 q^{47} + q^{48} - 3 q^{49} + 5 q^{50} - 6 q^{51} - 4 q^{52} - q^{54} - 2 q^{56} - 4 q^{57} - 6 q^{58} + 8 q^{61} - 8 q^{62} + 2 q^{63} + q^{64} + q^{66} - 4 q^{67} - 6 q^{68} + 6 q^{69} + 6 q^{71} - q^{72} + 2 q^{73} + 10 q^{74} - 5 q^{75} - 4 q^{76} - 2 q^{77} + 4 q^{78} + 14 q^{79} + q^{81} - 6 q^{82} - 12 q^{83} + 2 q^{84} - 8 q^{86} + 6 q^{87} + q^{88} - 6 q^{89} - 8 q^{91} + 6 q^{92} + 8 q^{93} + 6 q^{94} - q^{96} + 14 q^{97} + 3 q^{98} - q^{99}+O(q^{100}) \) 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
0
−1.00000 1.00000 1.00000 0 −1.00000 2.00000 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(11\) \(1\)

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 66.2.a.a 1
3.b odd 2 1 198.2.a.e 1
4.b odd 2 1 528.2.a.d 1
5.b even 2 1 1650.2.a.m 1
5.c odd 4 2 1650.2.c.d 2
7.b odd 2 1 3234.2.a.d 1
8.b even 2 1 2112.2.a.i 1
8.d odd 2 1 2112.2.a.v 1
9.c even 3 2 1782.2.e.s 2
9.d odd 6 2 1782.2.e.f 2
11.b odd 2 1 726.2.a.i 1
11.c even 5 4 726.2.e.k 4
11.d odd 10 4 726.2.e.b 4
12.b even 2 1 1584.2.a.h 1
15.d odd 2 1 4950.2.a.g 1
15.e even 4 2 4950.2.c.r 2
21.c even 2 1 9702.2.a.bu 1
24.f even 2 1 6336.2.a.bf 1
24.h odd 2 1 6336.2.a.bj 1
33.d even 2 1 2178.2.a.b 1
44.c even 2 1 5808.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.a 1 1.a even 1 1 trivial
198.2.a.e 1 3.b odd 2 1
528.2.a.d 1 4.b odd 2 1
726.2.a.i 1 11.b odd 2 1
726.2.e.b 4 11.d odd 10 4
726.2.e.k 4 11.c even 5 4
1584.2.a.h 1 12.b even 2 1
1650.2.a.m 1 5.b even 2 1
1650.2.c.d 2 5.c odd 4 2
1782.2.e.f 2 9.d odd 6 2
1782.2.e.s 2 9.c even 3 2
2112.2.a.i 1 8.b even 2 1
2112.2.a.v 1 8.d odd 2 1
2178.2.a.b 1 33.d even 2 1
3234.2.a.d 1 7.b odd 2 1
4950.2.a.g 1 15.d odd 2 1
4950.2.c.r 2 15.e even 4 2
5808.2.a.l 1 44.c even 2 1
6336.2.a.bf 1 24.f even 2 1
6336.2.a.bj 1 24.h odd 2 1
9702.2.a.bu 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(66))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T + 6 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T - 6 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T - 14 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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