Properties

Label 294.4.a.g
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: no (minimal twist has level 42)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 3q^{3} + 4q^{4} + 15q^{5} - 6q^{6} - 8q^{8} + 9q^{9} + O(q^{10}) \) \( q - 2q^{2} + 3q^{3} + 4q^{4} + 15q^{5} - 6q^{6} - 8q^{8} + 9q^{9} - 30q^{10} - 9q^{11} + 12q^{12} + 88q^{13} + 45q^{15} + 16q^{16} + 84q^{17} - 18q^{18} - 104q^{19} + 60q^{20} + 18q^{22} - 84q^{23} - 24q^{24} + 100q^{25} - 176q^{26} + 27q^{27} + 51q^{29} - 90q^{30} - 185q^{31} - 32q^{32} - 27q^{33} - 168q^{34} + 36q^{36} + 44q^{37} + 208q^{38} + 264q^{39} - 120q^{40} + 168q^{41} + 326q^{43} - 36q^{44} + 135q^{45} + 168q^{46} + 138q^{47} + 48q^{48} - 200q^{50} + 252q^{51} + 352q^{52} + 639q^{53} - 54q^{54} - 135q^{55} - 312q^{57} - 102q^{58} - 159q^{59} + 180q^{60} - 722q^{61} + 370q^{62} + 64q^{64} + 1320q^{65} + 54q^{66} - 166q^{67} + 336q^{68} - 252q^{69} + 1086q^{71} - 72q^{72} - 218q^{73} - 88q^{74} + 300q^{75} - 416q^{76} - 528q^{78} - 583q^{79} + 240q^{80} + 81q^{81} - 336q^{82} + 597q^{83} + 1260q^{85} - 652q^{86} + 153q^{87} + 72q^{88} + 1038q^{89} - 270q^{90} - 336q^{92} - 555q^{93} - 276q^{94} - 1560q^{95} - 96q^{96} + 169q^{97} - 81q^{99} + O(q^{100}) \)

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 15.0000 −6.00000 0 −8.00000 9.00000 −30.0000
\(n\): 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.g 1
3.b odd 2 1 882.4.a.h 1
4.b odd 2 1 2352.4.a.q 1
7.b odd 2 1 294.4.a.a 1
7.c even 3 2 294.4.e.e 2
7.d odd 6 2 42.4.e.b 2
21.c even 2 1 882.4.a.r 1
21.g even 6 2 126.4.g.a 2
21.h odd 6 2 882.4.g.l 2
28.d even 2 1 2352.4.a.u 1
28.f even 6 2 336.4.q.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 7.d odd 6 2
126.4.g.a 2 21.g even 6 2
294.4.a.a 1 7.b odd 2 1
294.4.a.g 1 1.a even 1 1 trivial
294.4.e.e 2 7.c even 3 2
336.4.q.d 2 28.f even 6 2
882.4.a.h 1 3.b odd 2 1
882.4.a.r 1 21.c even 2 1
882.4.g.l 2 21.h odd 6 2
2352.4.a.q 1 4.b odd 2 1
2352.4.a.u 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(294))\):

\( T_{5} - 15 \)
\( T_{11} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( -3 + T \)
$5$ \( -15 + T \)
$7$ \( T \)
$11$ \( 9 + T \)
$13$ \( -88 + T \)
$17$ \( -84 + T \)
$19$ \( 104 + T \)
$23$ \( 84 + T \)
$29$ \( -51 + T \)
$31$ \( 185 + T \)
$37$ \( -44 + T \)
$41$ \( -168 + T \)
$43$ \( -326 + T \)
$47$ \( -138 + T \)
$53$ \( -639 + T \)
$59$ \( 159 + T \)
$61$ \( 722 + T \)
$67$ \( 166 + T \)
$71$ \( -1086 + T \)
$73$ \( 218 + T \)
$79$ \( 583 + T \)
$83$ \( -597 + T \)
$89$ \( -1038 + T \)
$97$ \( -169 + T \)
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