Properties

Label 322.2.a.a
Level $322$
Weight $2$
Character orbit 322.a
Self dual yes
Analytic conductor $2.571$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.57118294509\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - 2q^{5} + q^{7} - q^{8} - 3q^{9} + O(q^{10}) \) \( q - q^{2} + q^{4} - 2q^{5} + q^{7} - q^{8} - 3q^{9} + 2q^{10} - 4q^{11} + 4q^{13} - q^{14} + q^{16} - 8q^{17} + 3q^{18} - 2q^{19} - 2q^{20} + 4q^{22} + q^{23} - q^{25} - 4q^{26} + q^{28} + 2q^{29} - 6q^{31} - q^{32} + 8q^{34} - 2q^{35} - 3q^{36} - 10q^{37} + 2q^{38} + 2q^{40} + 6q^{41} - 8q^{43} - 4q^{44} + 6q^{45} - q^{46} + 6q^{47} + q^{49} + q^{50} + 4q^{52} + 2q^{53} + 8q^{55} - q^{56} - 2q^{58} + 10q^{61} + 6q^{62} - 3q^{63} + q^{64} - 8q^{65} + 8q^{67} - 8q^{68} + 2q^{70} - 12q^{71} + 3q^{72} + 6q^{73} + 10q^{74} - 2q^{76} - 4q^{77} - 2q^{80} + 9q^{81} - 6q^{82} + 2q^{83} + 16q^{85} + 8q^{86} + 4q^{88} + 12q^{89} - 6q^{90} + 4q^{91} + q^{92} - 6q^{94} + 4q^{95} + 12q^{97} - q^{98} + 12q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(23\) \(-1\)

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 322.2.a.a 1
3.b odd 2 1 2898.2.a.s 1
4.b odd 2 1 2576.2.a.i 1
5.b even 2 1 8050.2.a.o 1
7.b odd 2 1 2254.2.a.d 1
23.b odd 2 1 7406.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.a.a 1 1.a even 1 1 trivial
2254.2.a.d 1 7.b odd 2 1
2576.2.a.i 1 4.b odd 2 1
2898.2.a.s 1 3.b odd 2 1
7406.2.a.d 1 23.b odd 2 1
8050.2.a.o 1 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(322))\):

\( T_{3} \)
\( T_{5} + 2 \)

Hecke characteristic polynomials

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