Properties

Label 2366.2.a.h
Level 2366
Weight 2
Character orbit 2366.a
Self dual yes
Analytic conductor 18.893
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + 3q^{3} + q^{4} + 4q^{5} - 3q^{6} + q^{7} - q^{8} + 6q^{9} + O(q^{10}) \) \( q - q^{2} + 3q^{3} + q^{4} + 4q^{5} - 3q^{6} + q^{7} - q^{8} + 6q^{9} - 4q^{10} - q^{11} + 3q^{12} - q^{14} + 12q^{15} + q^{16} - 6q^{18} + 6q^{19} + 4q^{20} + 3q^{21} + q^{22} - 7q^{23} - 3q^{24} + 11q^{25} + 9q^{27} + q^{28} - 4q^{29} - 12q^{30} - 7q^{31} - q^{32} - 3q^{33} + 4q^{35} + 6q^{36} - 9q^{37} - 6q^{38} - 4q^{40} + 3q^{41} - 3q^{42} + 4q^{43} - q^{44} + 24q^{45} + 7q^{46} - 7q^{47} + 3q^{48} + q^{49} - 11q^{50} - 9q^{54} - 4q^{55} - q^{56} + 18q^{57} + 4q^{58} + 10q^{59} + 12q^{60} + q^{61} + 7q^{62} + 6q^{63} + q^{64} + 3q^{66} - q^{67} - 21q^{69} - 4q^{70} - 16q^{71} - 6q^{72} - 5q^{73} + 9q^{74} + 33q^{75} + 6q^{76} - q^{77} + 11q^{79} + 4q^{80} + 9q^{81} - 3q^{82} + 3q^{84} - 4q^{86} - 12q^{87} + q^{88} + 6q^{89} - 24q^{90} - 7q^{92} - 21q^{93} + 7q^{94} + 24q^{95} - 3q^{96} + q^{97} - q^{98} - 6q^{99} + O(q^{100}) \)

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 3.00000 1.00000 4.00000 −3.00000 1.00000 −1.00000 6.00000 −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 2366.2.a.h 1
13.b even 2 1 182.2.a.e 1
13.d odd 4 2 2366.2.d.j 2
39.d odd 2 1 1638.2.a.j 1
52.b odd 2 1 1456.2.a.a 1
65.d even 2 1 4550.2.a.a 1
91.b odd 2 1 1274.2.a.h 1
91.r even 6 2 1274.2.f.b 2
91.s odd 6 2 1274.2.f.k 2
104.e even 2 1 5824.2.a.b 1
104.h odd 2 1 5824.2.a.bf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.a.e 1 13.b even 2 1
1274.2.a.h 1 91.b odd 2 1
1274.2.f.b 2 91.r even 6 2
1274.2.f.k 2 91.s odd 6 2
1456.2.a.a 1 52.b odd 2 1
1638.2.a.j 1 39.d odd 2 1
2366.2.a.h 1 1.a even 1 1 trivial
2366.2.d.j 2 13.d odd 4 2
4550.2.a.a 1 65.d even 2 1
5824.2.a.b 1 104.e even 2 1
5824.2.a.bf 1 104.h odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(13\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2366))\):

\( T_{3} - 3 \)
\( T_{5} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 - 3 T + 3 T^{2} \)
$5$ \( 1 - 4 T + 5 T^{2} \)
$7$ \( 1 - T \)
$11$ \( 1 + T + 11 T^{2} \)
$13$ 1
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 - 6 T + 19 T^{2} \)
$23$ \( 1 + 7 T + 23 T^{2} \)
$29$ \( 1 + 4 T + 29 T^{2} \)
$31$ \( 1 + 7 T + 31 T^{2} \)
$37$ \( 1 + 9 T + 37 T^{2} \)
$41$ \( 1 - 3 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 + 7 T + 47 T^{2} \)
$53$ \( 1 + 53 T^{2} \)
$59$ \( 1 - 10 T + 59 T^{2} \)
$61$ \( 1 - T + 61 T^{2} \)
$67$ \( 1 + T + 67 T^{2} \)
$71$ \( 1 + 16 T + 71 T^{2} \)
$73$ \( 1 + 5 T + 73 T^{2} \)
$79$ \( 1 - 11 T + 79 T^{2} \)
$83$ \( 1 + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 - T + 97 T^{2} \)
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