Properties

Label 294.2.a.b
Level 294
Weight 2
Character orbit 294.a
Self dual yes
Analytic conductor 2.348
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.34760181943\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + 4q^{5} + q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} + 4q^{5} + q^{6} - q^{8} + q^{9} - 4q^{10} - 4q^{11} - q^{12} + 4q^{13} - 4q^{15} + q^{16} - q^{18} + 4q^{19} + 4q^{20} + 4q^{22} + q^{24} + 11q^{25} - 4q^{26} - q^{27} + 2q^{29} + 4q^{30} + 8q^{31} - q^{32} + 4q^{33} + q^{36} - 6q^{37} - 4q^{38} - 4q^{39} - 4q^{40} + 4q^{43} - 4q^{44} + 4q^{45} - 8q^{47} - q^{48} - 11q^{50} + 4q^{52} - 10q^{53} + q^{54} - 16q^{55} - 4q^{57} - 2q^{58} + 4q^{59} - 4q^{60} - 4q^{61} - 8q^{62} + q^{64} + 16q^{65} - 4q^{66} + 4q^{67} + 8q^{71} - q^{72} - 16q^{73} + 6q^{74} - 11q^{75} + 4q^{76} + 4q^{78} - 8q^{79} + 4q^{80} + q^{81} - 12q^{83} - 4q^{86} - 2q^{87} + 4q^{88} + 8q^{89} - 4q^{90} - 8q^{93} + 8q^{94} + 16q^{95} + q^{96} + 8q^{97} - 4q^{99} + O(q^{100}) \)

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 −1.00000 1.00000 4.00000 1.00000 0 −1.00000 1.00000 −4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 294.2.a.b 1
3.b odd 2 1 882.2.a.f 1
4.b odd 2 1 2352.2.a.y 1
5.b even 2 1 7350.2.a.cj 1
7.b odd 2 1 294.2.a.c yes 1
7.c even 3 2 294.2.e.e 2
7.d odd 6 2 294.2.e.d 2
8.b even 2 1 9408.2.a.br 1
8.d odd 2 1 9408.2.a.b 1
12.b even 2 1 7056.2.a.a 1
21.c even 2 1 882.2.a.l 1
21.g even 6 2 882.2.g.a 2
21.h odd 6 2 882.2.g.f 2
28.d even 2 1 2352.2.a.b 1
28.f even 6 2 2352.2.q.y 2
28.g odd 6 2 2352.2.q.a 2
35.c odd 2 1 7350.2.a.br 1
56.e even 2 1 9408.2.a.de 1
56.h odd 2 1 9408.2.a.bo 1
84.h odd 2 1 7056.2.a.ca 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.a.b 1 1.a even 1 1 trivial
294.2.a.c yes 1 7.b odd 2 1
294.2.e.d 2 7.d odd 6 2
294.2.e.e 2 7.c even 3 2
882.2.a.f 1 3.b odd 2 1
882.2.a.l 1 21.c even 2 1
882.2.g.a 2 21.g even 6 2
882.2.g.f 2 21.h odd 6 2
2352.2.a.b 1 28.d even 2 1
2352.2.a.y 1 4.b odd 2 1
2352.2.q.a 2 28.g odd 6 2
2352.2.q.y 2 28.f even 6 2
7056.2.a.a 1 12.b even 2 1
7056.2.a.ca 1 84.h odd 2 1
7350.2.a.br 1 35.c odd 2 1
7350.2.a.cj 1 5.b even 2 1
9408.2.a.b 1 8.d odd 2 1
9408.2.a.bo 1 56.h odd 2 1
9408.2.a.br 1 8.b even 2 1
9408.2.a.de 1 56.e even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(294))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( 1 - 4 T + 5 T^{2} \)
$7$ 1
$11$ \( 1 + 4 T + 11 T^{2} \)
$13$ \( 1 - 4 T + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 - 2 T + 29 T^{2} \)
$31$ \( 1 - 8 T + 31 T^{2} \)
$37$ \( 1 + 6 T + 37 T^{2} \)
$41$ \( 1 + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 + 8 T + 47 T^{2} \)
$53$ \( 1 + 10 T + 53 T^{2} \)
$59$ \( 1 - 4 T + 59 T^{2} \)
$61$ \( 1 + 4 T + 61 T^{2} \)
$67$ \( 1 - 4 T + 67 T^{2} \)
$71$ \( 1 - 8 T + 71 T^{2} \)
$73$ \( 1 + 16 T + 73 T^{2} \)
$79$ \( 1 + 8 T + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 - 8 T + 89 T^{2} \)
$97$ \( 1 - 8 T + 97 T^{2} \)
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