Properties

Label 221.2.a.b
Level $221$
Weight $2$
Character orbit 221.a
Self dual yes
Analytic conductor $1.765$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 221 = 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 221.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.76469388467\)
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

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

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
0
1.00000 2.00000 −1.00000 2.00000 2.00000 2.00000 −3.00000 1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(1\)
\(17\) \(-1\)

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 221.2.a.b 1
3.b odd 2 1 1989.2.a.a 1
4.b odd 2 1 3536.2.a.c 1
5.b even 2 1 5525.2.a.c 1
13.b even 2 1 2873.2.a.a 1
17.b even 2 1 3757.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
221.2.a.b 1 1.a even 1 1 trivial
1989.2.a.a 1 3.b odd 2 1
2873.2.a.a 1 13.b even 2 1
3536.2.a.c 1 4.b odd 2 1
3757.2.a.d 1 17.b even 2 1
5525.2.a.c 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(221))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -2 + T \)
$5$ \( -2 + T \)
$7$ \( -2 + T \)
$11$ \( 6 + T \)
$13$ \( 1 + T \)
$17$ \( -1 + T \)
$19$ \( -4 + T \)
$23$ \( -6 + T \)
$29$ \( 6 + T \)
$31$ \( 2 + T \)
$37$ \( -2 + T \)
$41$ \( 6 + T \)
$43$ \( T \)
$47$ \( 4 + T \)
$53$ \( -14 + T \)
$59$ \( -4 + T \)
$61$ \( -2 + T \)
$67$ \( T \)
$71$ \( 10 + T \)
$73$ \( -10 + T \)
$79$ \( -14 + T \)
$83$ \( -12 + T \)
$89$ \( 18 + T \)
$97$ \( -2 + T \)
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