Properties

Label 423.2.a.b
Level $423$
Weight $2$
Character orbit 423.a
Self dual yes
Analytic conductor $3.378$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 423 = 3^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 423.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.37767200550\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 141)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 2q^{4} + q^{5} - 3q^{7} + O(q^{10}) \) \( q - 2q^{2} + 2q^{4} + q^{5} - 3q^{7} - 2q^{10} - q^{11} - 2q^{13} + 6q^{14} - 4q^{16} - 2q^{17} + 6q^{19} + 2q^{20} + 2q^{22} - 3q^{23} - 4q^{25} + 4q^{26} - 6q^{28} - 3q^{29} + 2q^{31} + 8q^{32} + 4q^{34} - 3q^{35} - 7q^{37} - 12q^{38} - 10q^{41} - 10q^{43} - 2q^{44} + 6q^{46} + q^{47} + 2q^{49} + 8q^{50} - 4q^{52} - 4q^{53} - q^{55} + 6q^{58} - 8q^{59} - 10q^{61} - 4q^{62} - 8q^{64} - 2q^{65} + 10q^{67} - 4q^{68} + 6q^{70} + 14q^{71} - 10q^{73} + 14q^{74} + 12q^{76} + 3q^{77} + 17q^{79} - 4q^{80} + 20q^{82} - 8q^{83} - 2q^{85} + 20q^{86} - 6q^{89} + 6q^{91} - 6q^{92} - 2q^{94} + 6q^{95} + q^{97} - 4q^{98} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(47\) \(-1\)

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 423.2.a.b 1
3.b odd 2 1 141.2.a.e 1
4.b odd 2 1 6768.2.a.n 1
12.b even 2 1 2256.2.a.e 1
15.d odd 2 1 3525.2.a.c 1
21.c even 2 1 6909.2.a.k 1
24.f even 2 1 9024.2.a.bq 1
24.h odd 2 1 9024.2.a.n 1
141.c even 2 1 6627.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.2.a.e 1 3.b odd 2 1
423.2.a.b 1 1.a even 1 1 trivial
2256.2.a.e 1 12.b even 2 1
3525.2.a.c 1 15.d odd 2 1
6627.2.a.i 1 141.c even 2 1
6768.2.a.n 1 4.b odd 2 1
6909.2.a.k 1 21.c even 2 1
9024.2.a.n 1 24.h odd 2 1
9024.2.a.bq 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(423))\):

\( T_{2} + 2 \)
\( T_{5} - 1 \)
\( T_{7} + 3 \)

Hecke characteristic polynomials

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