Properties

Label 222.2.a.a
Level $222$
Weight $2$
Character orbit 222.a
Self dual yes
Analytic conductor $1.773$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 222 = 2 \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 222.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.77267892487\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} - 4 q^{5} + q^{6} + 3 q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} - 4 q^{5} + q^{6} + 3 q^{7} - q^{8} + q^{9} + 4 q^{10} + 5 q^{11} - q^{12} + 3 q^{13} - 3 q^{14} + 4 q^{15} + q^{16} + 3 q^{17} - q^{18} - 7 q^{19} - 4 q^{20} - 3 q^{21} - 5 q^{22} + 9 q^{23} + q^{24} + 11 q^{25} - 3 q^{26} - q^{27} + 3 q^{28} - 4 q^{30} - 2 q^{31} - q^{32} - 5 q^{33} - 3 q^{34} - 12 q^{35} + q^{36} + q^{37} + 7 q^{38} - 3 q^{39} + 4 q^{40} + 6 q^{41} + 3 q^{42} + 4 q^{43} + 5 q^{44} - 4 q^{45} - 9 q^{46} - 10 q^{47} - q^{48} + 2 q^{49} - 11 q^{50} - 3 q^{51} + 3 q^{52} + 3 q^{53} + q^{54} - 20 q^{55} - 3 q^{56} + 7 q^{57} - 4 q^{59} + 4 q^{60} - 2 q^{61} + 2 q^{62} + 3 q^{63} + q^{64} - 12 q^{65} + 5 q^{66} + 6 q^{67} + 3 q^{68} - 9 q^{69} + 12 q^{70} - 12 q^{71} - q^{72} + 13 q^{73} - q^{74} - 11 q^{75} - 7 q^{76} + 15 q^{77} + 3 q^{78} - 6 q^{79} - 4 q^{80} + q^{81} - 6 q^{82} + 5 q^{83} - 3 q^{84} - 12 q^{85} - 4 q^{86} - 5 q^{88} + 11 q^{89} + 4 q^{90} + 9 q^{91} + 9 q^{92} + 2 q^{93} + 10 q^{94} + 28 q^{95} + q^{96} + 6 q^{97} - 2 q^{98} + 5 q^{99}+O(q^{100}) \) 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 −4.00000 1.00000 3.00000 −1.00000 1.00000 4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(37\) \(-1\)

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 222.2.a.a 1
3.b odd 2 1 666.2.a.g 1
4.b odd 2 1 1776.2.a.f 1
5.b even 2 1 5550.2.a.bh 1
8.b even 2 1 7104.2.a.bb 1
8.d odd 2 1 7104.2.a.m 1
12.b even 2 1 5328.2.a.v 1
37.b even 2 1 8214.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.a.a 1 1.a even 1 1 trivial
666.2.a.g 1 3.b odd 2 1
1776.2.a.f 1 4.b odd 2 1
5328.2.a.v 1 12.b even 2 1
5550.2.a.bh 1 5.b even 2 1
7104.2.a.m 1 8.d odd 2 1
7104.2.a.bb 1 8.b even 2 1
8214.2.a.i 1 37.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(222))\):

\( T_{5} + 4 \) Copy content Toggle raw display
\( T_{7} - 3 \) 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
$5$ \( T + 4 \) Copy content Toggle raw display
$7$ \( T - 3 \) Copy content Toggle raw display
$11$ \( T - 5 \) Copy content Toggle raw display
$13$ \( T - 3 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T + 7 \) Copy content Toggle raw display
$23$ \( T - 9 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 2 \) Copy content Toggle raw display
$37$ \( T - 1 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T + 10 \) Copy content Toggle raw display
$53$ \( T - 3 \) Copy content Toggle raw display
$59$ \( T + 4 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 6 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T - 13 \) Copy content Toggle raw display
$79$ \( T + 6 \) Copy content Toggle raw display
$83$ \( T - 5 \) Copy content Toggle raw display
$89$ \( T - 11 \) Copy content Toggle raw display
$97$ \( T - 6 \) Copy content Toggle raw display
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