Properties

Label 342.4.a.c
Level $342$
Weight $4$
Character orbit 342.a
Self dual yes
Analytic conductor $20.179$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 342.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.1786532220\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} + 4q^{4} + 7q^{5} - 15q^{7} + 8q^{8} + O(q^{10}) \) \( q + 2q^{2} + 4q^{4} + 7q^{5} - 15q^{7} + 8q^{8} + 14q^{10} + 49q^{11} + 14q^{13} - 30q^{14} + 16q^{16} + 33q^{17} - 19q^{19} + 28q^{20} + 98q^{22} + 148q^{23} - 76q^{25} + 28q^{26} - 60q^{28} + 278q^{29} + 94q^{31} + 32q^{32} + 66q^{34} - 105q^{35} + 160q^{37} - 38q^{38} + 56q^{40} - 400q^{41} + 73q^{43} + 196q^{44} + 296q^{46} - 173q^{47} - 118q^{49} - 152q^{50} + 56q^{52} - 170q^{53} + 343q^{55} - 120q^{56} + 556q^{58} + 12q^{59} + 419q^{61} + 188q^{62} + 64q^{64} + 98q^{65} + 444q^{67} + 132q^{68} - 210q^{70} + 952q^{71} - 27q^{73} + 320q^{74} - 76q^{76} - 735q^{77} - 556q^{79} + 112q^{80} - 800q^{82} + 276q^{83} + 231q^{85} + 146q^{86} + 392q^{88} - 1386q^{89} - 210q^{91} + 592q^{92} - 346q^{94} - 133q^{95} + 130q^{97} - 236q^{98} + O(q^{100}) \)

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 4.00000 7.00000 0 −15.0000 8.00000 0 14.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(19\) \(1\)

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 342.4.a.c 1
3.b odd 2 1 114.4.a.b 1
12.b even 2 1 912.4.a.b 1
57.d even 2 1 2166.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.a.b 1 3.b odd 2 1
342.4.a.c 1 1.a even 1 1 trivial
912.4.a.b 1 12.b even 2 1
2166.4.a.d 1 57.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 7 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(342))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( T \)
$5$ \( -7 + T \)
$7$ \( 15 + T \)
$11$ \( -49 + T \)
$13$ \( -14 + T \)
$17$ \( -33 + T \)
$19$ \( 19 + T \)
$23$ \( -148 + T \)
$29$ \( -278 + T \)
$31$ \( -94 + T \)
$37$ \( -160 + T \)
$41$ \( 400 + T \)
$43$ \( -73 + T \)
$47$ \( 173 + T \)
$53$ \( 170 + T \)
$59$ \( -12 + T \)
$61$ \( -419 + T \)
$67$ \( -444 + T \)
$71$ \( -952 + T \)
$73$ \( 27 + T \)
$79$ \( 556 + T \)
$83$ \( -276 + T \)
$89$ \( 1386 + T \)
$97$ \( -130 + T \)
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