Properties

Label 322.2.a.b
Level $322$
Weight $2$
Character orbit 322.a
Self dual yes
Analytic conductor $2.571$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: \(0\)
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

\(f(q)\) \(=\) \( q - q^{2} + 2q^{3} + q^{4} - 2q^{6} + q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + 2q^{3} + q^{4} - 2q^{6} + q^{7} - q^{8} + q^{9} + 4q^{11} + 2q^{12} - q^{14} + q^{16} + 6q^{17} - q^{18} - 6q^{19} + 2q^{21} - 4q^{22} - q^{23} - 2q^{24} - 5q^{25} - 4q^{27} + q^{28} + 10q^{29} + 4q^{31} - q^{32} + 8q^{33} - 6q^{34} + q^{36} - 2q^{37} + 6q^{38} - 10q^{41} - 2q^{42} - 4q^{43} + 4q^{44} + q^{46} + 12q^{47} + 2q^{48} + q^{49} + 5q^{50} + 12q^{51} - 6q^{53} + 4q^{54} - q^{56} - 12q^{57} - 10q^{58} - 2q^{59} - 4q^{62} + q^{63} + q^{64} - 8q^{66} + 6q^{68} - 2q^{69} - 8q^{71} - q^{72} - 6q^{73} + 2q^{74} - 10q^{75} - 6q^{76} + 4q^{77} - 8q^{79} - 11q^{81} + 10q^{82} - 14q^{83} + 2q^{84} + 4q^{86} + 20q^{87} - 4q^{88} - 14q^{89} - q^{92} + 8q^{93} - 12q^{94} - 2q^{96} - 2q^{97} - q^{98} + 4q^{99} + O(q^{100}) \)

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 0 −2.00000 1.00000 −1.00000 1.00000 0
\(n\): 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.b 1
3.b odd 2 1 2898.2.a.p 1
4.b odd 2 1 2576.2.a.c 1
5.b even 2 1 8050.2.a.n 1
7.b odd 2 1 2254.2.a.a 1
23.b odd 2 1 7406.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.a.b 1 1.a even 1 1 trivial
2254.2.a.a 1 7.b odd 2 1
2576.2.a.c 1 4.b odd 2 1
2898.2.a.p 1 3.b odd 2 1
7406.2.a.e 1 23.b odd 2 1
8050.2.a.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(322))\):

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

Hecke characteristic polynomials

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