Properties

Label 294.4.a.f
Level $294$
Weight $4$
Character orbit 294.a
Self dual yes
Analytic conductor $17.347$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 294.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3465615417\)
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

\(f(q)\) \(=\) \( q - 2q^{2} + 3q^{3} + 4q^{4} + 8q^{5} - 6q^{6} - 8q^{8} + 9q^{9} + O(q^{10}) \) \( q - 2q^{2} + 3q^{3} + 4q^{4} + 8q^{5} - 6q^{6} - 8q^{8} + 9q^{9} - 16q^{10} + 40q^{11} + 12q^{12} + 4q^{13} + 24q^{15} + 16q^{16} - 84q^{17} - 18q^{18} + 148q^{19} + 32q^{20} - 80q^{22} + 84q^{23} - 24q^{24} - 61q^{25} - 8q^{26} + 27q^{27} + 58q^{29} - 48q^{30} - 136q^{31} - 32q^{32} + 120q^{33} + 168q^{34} + 36q^{36} - 222q^{37} - 296q^{38} + 12q^{39} - 64q^{40} + 420q^{41} - 164q^{43} + 160q^{44} + 72q^{45} - 168q^{46} + 488q^{47} + 48q^{48} + 122q^{50} - 252q^{51} + 16q^{52} + 478q^{53} - 54q^{54} + 320q^{55} + 444q^{57} - 116q^{58} + 548q^{59} + 96q^{60} + 692q^{61} + 272q^{62} + 64q^{64} + 32q^{65} - 240q^{66} - 908q^{67} - 336q^{68} + 252q^{69} - 524q^{71} - 72q^{72} + 440q^{73} + 444q^{74} - 183q^{75} + 592q^{76} - 24q^{78} + 1216q^{79} + 128q^{80} + 81q^{81} - 840q^{82} - 684q^{83} - 672q^{85} + 328q^{86} + 174q^{87} - 320q^{88} + 604q^{89} - 144q^{90} + 336q^{92} - 408q^{93} - 976q^{94} + 1184q^{95} - 96q^{96} - 832q^{97} + 360q^{99} + O(q^{100}) \)

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
−2.00000 3.00000 4.00000 8.00000 −6.00000 0 −8.00000 9.00000 −16.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)

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 294.4.a.f yes 1
3.b odd 2 1 882.4.a.j 1
4.b odd 2 1 2352.4.a.m 1
7.b odd 2 1 294.4.a.b 1
7.c even 3 2 294.4.e.f 2
7.d odd 6 2 294.4.e.j 2
21.c even 2 1 882.4.a.q 1
21.g even 6 2 882.4.g.c 2
21.h odd 6 2 882.4.g.j 2
28.d even 2 1 2352.4.a.z 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.b 1 7.b odd 2 1
294.4.a.f yes 1 1.a even 1 1 trivial
294.4.e.f 2 7.c even 3 2
294.4.e.j 2 7.d odd 6 2
882.4.a.j 1 3.b odd 2 1
882.4.a.q 1 21.c even 2 1
882.4.g.c 2 21.g even 6 2
882.4.g.j 2 21.h odd 6 2
2352.4.a.m 1 4.b odd 2 1
2352.4.a.z 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(294))\):

\( T_{5} - 8 \)
\( T_{11} - 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( -3 + T \)
$5$ \( -8 + T \)
$7$ \( T \)
$11$ \( -40 + T \)
$13$ \( -4 + T \)
$17$ \( 84 + T \)
$19$ \( -148 + T \)
$23$ \( -84 + T \)
$29$ \( -58 + T \)
$31$ \( 136 + T \)
$37$ \( 222 + T \)
$41$ \( -420 + T \)
$43$ \( 164 + T \)
$47$ \( -488 + T \)
$53$ \( -478 + T \)
$59$ \( -548 + T \)
$61$ \( -692 + T \)
$67$ \( 908 + T \)
$71$ \( 524 + T \)
$73$ \( -440 + T \)
$79$ \( -1216 + T \)
$83$ \( 684 + T \)
$89$ \( -604 + T \)
$97$ \( 832 + T \)
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