Properties

Label 294.4.a.e
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 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 3q^{3} + 4q^{4} - 6q^{5} - 6q^{6} - 8q^{8} + 9q^{9} + O(q^{10}) \) \( q - 2q^{2} + 3q^{3} + 4q^{4} - 6q^{5} - 6q^{6} - 8q^{8} + 9q^{9} + 12q^{10} + 12q^{11} + 12q^{12} - 38q^{13} - 18q^{15} + 16q^{16} + 126q^{17} - 18q^{18} - 20q^{19} - 24q^{20} - 24q^{22} + 168q^{23} - 24q^{24} - 89q^{25} + 76q^{26} + 27q^{27} + 30q^{29} + 36q^{30} + 88q^{31} - 32q^{32} + 36q^{33} - 252q^{34} + 36q^{36} + 254q^{37} + 40q^{38} - 114q^{39} + 48q^{40} - 42q^{41} - 52q^{43} + 48q^{44} - 54q^{45} - 336q^{46} + 96q^{47} + 48q^{48} + 178q^{50} + 378q^{51} - 152q^{52} + 198q^{53} - 54q^{54} - 72q^{55} - 60q^{57} - 60q^{58} + 660q^{59} - 72q^{60} + 538q^{61} - 176q^{62} + 64q^{64} + 228q^{65} - 72q^{66} + 884q^{67} + 504q^{68} + 504q^{69} + 792q^{71} - 72q^{72} - 218q^{73} - 508q^{74} - 267q^{75} - 80q^{76} + 228q^{78} - 520q^{79} - 96q^{80} + 81q^{81} + 84q^{82} + 492q^{83} - 756q^{85} + 104q^{86} + 90q^{87} - 96q^{88} - 810q^{89} + 108q^{90} + 672q^{92} + 264q^{93} - 192q^{94} + 120q^{95} - 96q^{96} - 1154q^{97} + 108q^{99} + O(q^{100}) \)

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 −6.00000 −6.00000 0 −8.00000 9.00000 12.0000
\(n\): 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 294.4.a.e 1
3.b odd 2 1 882.4.a.n 1
4.b odd 2 1 2352.4.a.e 1
7.b odd 2 1 6.4.a.a 1
7.c even 3 2 294.4.e.g 2
7.d odd 6 2 294.4.e.h 2
21.c even 2 1 18.4.a.a 1
21.g even 6 2 882.4.g.i 2
21.h odd 6 2 882.4.g.f 2
28.d even 2 1 48.4.a.c 1
35.c odd 2 1 150.4.a.i 1
35.f even 4 2 150.4.c.d 2
56.e even 2 1 192.4.a.c 1
56.h odd 2 1 192.4.a.i 1
63.l odd 6 2 162.4.c.f 2
63.o even 6 2 162.4.c.c 2
77.b even 2 1 726.4.a.f 1
84.h odd 2 1 144.4.a.c 1
91.b odd 2 1 1014.4.a.g 1
91.i even 4 2 1014.4.b.d 2
105.g even 2 1 450.4.a.h 1
105.k odd 4 2 450.4.c.e 2
112.j even 4 2 768.4.d.c 2
112.l odd 4 2 768.4.d.n 2
119.d odd 2 1 1734.4.a.d 1
133.c even 2 1 2166.4.a.i 1
140.c even 2 1 1200.4.a.b 1
140.j odd 4 2 1200.4.f.j 2
168.e odd 2 1 576.4.a.r 1
168.i even 2 1 576.4.a.q 1
231.h odd 2 1 2178.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 7.b odd 2 1
18.4.a.a 1 21.c even 2 1
48.4.a.c 1 28.d even 2 1
144.4.a.c 1 84.h odd 2 1
150.4.a.i 1 35.c odd 2 1
150.4.c.d 2 35.f even 4 2
162.4.c.c 2 63.o even 6 2
162.4.c.f 2 63.l odd 6 2
192.4.a.c 1 56.e even 2 1
192.4.a.i 1 56.h odd 2 1
294.4.a.e 1 1.a even 1 1 trivial
294.4.e.g 2 7.c even 3 2
294.4.e.h 2 7.d odd 6 2
450.4.a.h 1 105.g even 2 1
450.4.c.e 2 105.k odd 4 2
576.4.a.q 1 168.i even 2 1
576.4.a.r 1 168.e odd 2 1
726.4.a.f 1 77.b even 2 1
768.4.d.c 2 112.j even 4 2
768.4.d.n 2 112.l odd 4 2
882.4.a.n 1 3.b odd 2 1
882.4.g.f 2 21.h odd 6 2
882.4.g.i 2 21.g even 6 2
1014.4.a.g 1 91.b odd 2 1
1014.4.b.d 2 91.i even 4 2
1200.4.a.b 1 140.c even 2 1
1200.4.f.j 2 140.j odd 4 2
1734.4.a.d 1 119.d odd 2 1
2166.4.a.i 1 133.c even 2 1
2178.4.a.e 1 231.h odd 2 1
2352.4.a.e 1 4.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

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

\( T_{5} + 6 \)
\( T_{11} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T \)
$3$ \( 1 - 3 T \)
$5$ \( 1 + 6 T + 125 T^{2} \)
$7$ 1
$11$ \( 1 - 12 T + 1331 T^{2} \)
$13$ \( 1 + 38 T + 2197 T^{2} \)
$17$ \( 1 - 126 T + 4913 T^{2} \)
$19$ \( 1 + 20 T + 6859 T^{2} \)
$23$ \( 1 - 168 T + 12167 T^{2} \)
$29$ \( 1 - 30 T + 24389 T^{2} \)
$31$ \( 1 - 88 T + 29791 T^{2} \)
$37$ \( 1 - 254 T + 50653 T^{2} \)
$41$ \( 1 + 42 T + 68921 T^{2} \)
$43$ \( 1 + 52 T + 79507 T^{2} \)
$47$ \( 1 - 96 T + 103823 T^{2} \)
$53$ \( 1 - 198 T + 148877 T^{2} \)
$59$ \( 1 - 660 T + 205379 T^{2} \)
$61$ \( 1 - 538 T + 226981 T^{2} \)
$67$ \( 1 - 884 T + 300763 T^{2} \)
$71$ \( 1 - 792 T + 357911 T^{2} \)
$73$ \( 1 + 218 T + 389017 T^{2} \)
$79$ \( 1 + 520 T + 493039 T^{2} \)
$83$ \( 1 - 492 T + 571787 T^{2} \)
$89$ \( 1 + 810 T + 704969 T^{2} \)
$97$ \( 1 + 1154 T + 912673 T^{2} \)
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