Properties

Label 294.2.a.e
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: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} + 5q^{11} - q^{12} - q^{15} + q^{16} - 4q^{17} + q^{18} + 8q^{19} + q^{20} + 5q^{22} - 4q^{23} - q^{24} - 4q^{25} - q^{27} - 5q^{29} - q^{30} + 3q^{31} + q^{32} - 5q^{33} - 4q^{34} + q^{36} - 4q^{37} + 8q^{38} + q^{40} + 2q^{43} + 5q^{44} + q^{45} - 4q^{46} - 6q^{47} - q^{48} - 4q^{50} + 4q^{51} - 9q^{53} - q^{54} + 5q^{55} - 8q^{57} - 5q^{58} - 11q^{59} - q^{60} - 6q^{61} + 3q^{62} + q^{64} - 5q^{66} - 2q^{67} - 4q^{68} + 4q^{69} + 2q^{71} + q^{72} + 10q^{73} - 4q^{74} + 4q^{75} + 8q^{76} + 3q^{79} + q^{80} + q^{81} - 7q^{83} - 4q^{85} + 2q^{86} + 5q^{87} + 5q^{88} - 6q^{89} + q^{90} - 4q^{92} - 3q^{93} - 6q^{94} + 8q^{95} - q^{96} + 7q^{97} + 5q^{99} + O(q^{100}) \)

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 1.00000 −1.00000 0 1.00000 1.00000 1.00000
\(n\): 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.e 1
3.b odd 2 1 882.2.a.c 1
4.b odd 2 1 2352.2.a.t 1
5.b even 2 1 7350.2.a.bl 1
7.b odd 2 1 294.2.a.f 1
7.c even 3 2 42.2.e.a 2
7.d odd 6 2 294.2.e.b 2
8.b even 2 1 9408.2.a.ce 1
8.d odd 2 1 9408.2.a.q 1
12.b even 2 1 7056.2.a.w 1
21.c even 2 1 882.2.a.d 1
21.g even 6 2 882.2.g.i 2
21.h odd 6 2 126.2.g.c 2
28.d even 2 1 2352.2.a.f 1
28.f even 6 2 2352.2.q.u 2
28.g odd 6 2 336.2.q.b 2
35.c odd 2 1 7350.2.a.q 1
35.j even 6 2 1050.2.i.l 2
35.l odd 12 4 1050.2.o.a 4
56.e even 2 1 9408.2.a.cr 1
56.h odd 2 1 9408.2.a.z 1
56.k odd 6 2 1344.2.q.s 2
56.p even 6 2 1344.2.q.g 2
63.g even 3 2 1134.2.h.e 2
63.h even 3 2 1134.2.e.l 2
63.j odd 6 2 1134.2.e.e 2
63.n odd 6 2 1134.2.h.l 2
84.h odd 2 1 7056.2.a.bl 1
84.n even 6 2 1008.2.s.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.a 2 7.c even 3 2
126.2.g.c 2 21.h odd 6 2
294.2.a.e 1 1.a even 1 1 trivial
294.2.a.f 1 7.b odd 2 1
294.2.e.b 2 7.d odd 6 2
336.2.q.b 2 28.g odd 6 2
882.2.a.c 1 3.b odd 2 1
882.2.a.d 1 21.c even 2 1
882.2.g.i 2 21.g even 6 2
1008.2.s.k 2 84.n even 6 2
1050.2.i.l 2 35.j even 6 2
1050.2.o.a 4 35.l odd 12 4
1134.2.e.e 2 63.j odd 6 2
1134.2.e.l 2 63.h even 3 2
1134.2.h.e 2 63.g even 3 2
1134.2.h.l 2 63.n odd 6 2
1344.2.q.g 2 56.p even 6 2
1344.2.q.s 2 56.k odd 6 2
2352.2.a.f 1 28.d even 2 1
2352.2.a.t 1 4.b odd 2 1
2352.2.q.u 2 28.f even 6 2
7056.2.a.w 1 12.b even 2 1
7056.2.a.bl 1 84.h odd 2 1
7350.2.a.q 1 35.c odd 2 1
7350.2.a.bl 1 5.b even 2 1
9408.2.a.q 1 8.d odd 2 1
9408.2.a.z 1 56.h odd 2 1
9408.2.a.ce 1 8.b even 2 1
9408.2.a.cr 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 \( T_{5} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(294))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 + T \)
$5$ \( 1 - T + 5 T^{2} \)
$7$ 1
$11$ \( 1 - 5 T + 11 T^{2} \)
$13$ \( 1 + 13 T^{2} \)
$17$ \( 1 + 4 T + 17 T^{2} \)
$19$ \( 1 - 8 T + 19 T^{2} \)
$23$ \( 1 + 4 T + 23 T^{2} \)
$29$ \( 1 + 5 T + 29 T^{2} \)
$31$ \( 1 - 3 T + 31 T^{2} \)
$37$ \( 1 + 4 T + 37 T^{2} \)
$41$ \( 1 + 41 T^{2} \)
$43$ \( 1 - 2 T + 43 T^{2} \)
$47$ \( 1 + 6 T + 47 T^{2} \)
$53$ \( 1 + 9 T + 53 T^{2} \)
$59$ \( 1 + 11 T + 59 T^{2} \)
$61$ \( 1 + 6 T + 61 T^{2} \)
$67$ \( 1 + 2 T + 67 T^{2} \)
$71$ \( 1 - 2 T + 71 T^{2} \)
$73$ \( 1 - 10 T + 73 T^{2} \)
$79$ \( 1 - 3 T + 79 T^{2} \)
$83$ \( 1 + 7 T + 83 T^{2} \)
$89$ \( 1 + 6 T + 89 T^{2} \)
$97$ \( 1 - 7 T + 97 T^{2} \)
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