Properties

Label 2366.2.a.l
Level 2366
Weight 2
Character orbit 2366.a
Self dual yes
Analytic conductor 18.893
Analytic rank 1
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: \(1\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} - q^{7} + q^{8} - 2q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} - q^{7} + q^{8} - 2q^{9} + 2q^{10} - 5q^{11} - q^{12} - q^{14} - 2q^{15} + q^{16} + 2q^{17} - 2q^{18} + 4q^{19} + 2q^{20} + q^{21} - 5q^{22} - 9q^{23} - q^{24} - q^{25} + 5q^{27} - q^{28} - 2q^{30} - 5q^{31} + q^{32} + 5q^{33} + 2q^{34} - 2q^{35} - 2q^{36} - 3q^{37} + 4q^{38} + 2q^{40} - 5q^{41} + q^{42} - 4q^{43} - 5q^{44} - 4q^{45} - 9q^{46} - 13q^{47} - q^{48} + q^{49} - q^{50} - 2q^{51} + 14q^{53} + 5q^{54} - 10q^{55} - q^{56} - 4q^{57} + 6q^{59} - 2q^{60} - 13q^{61} - 5q^{62} + 2q^{63} + q^{64} + 5q^{66} + 3q^{67} + 2q^{68} + 9q^{69} - 2q^{70} - 2q^{72} - q^{73} - 3q^{74} + q^{75} + 4q^{76} + 5q^{77} - 15q^{79} + 2q^{80} + q^{81} - 5q^{82} + 6q^{83} + q^{84} + 4q^{85} - 4q^{86} - 5q^{88} + 6q^{89} - 4q^{90} - 9q^{92} + 5q^{93} - 13q^{94} + 8q^{95} - q^{96} - 7q^{97} + q^{98} + 10q^{99} + O(q^{100}) \)

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 2.00000 −1.00000 −1.00000 1.00000 −2.00000 2.00000
\(n\): 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.l 1
13.b even 2 1 2366.2.a.c 1
13.d odd 4 2 182.2.d.a 2
39.f even 4 2 1638.2.c.a 2
52.f even 4 2 1456.2.k.a 2
91.i even 4 2 1274.2.d.d 2
91.z odd 12 4 1274.2.n.e 4
91.bb even 12 4 1274.2.n.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.d.a 2 13.d odd 4 2
1274.2.d.d 2 91.i even 4 2
1274.2.n.b 4 91.bb even 12 4
1274.2.n.e 4 91.z odd 12 4
1456.2.k.a 2 52.f even 4 2
1638.2.c.a 2 39.f even 4 2
2366.2.a.c 1 13.b even 2 1
2366.2.a.l 1 1.a even 1 1 trivial

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2366))\):

\( T_{3} + 1 \)
\( T_{5} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 + T + 3 T^{2} \)
$5$ \( 1 - 2 T + 5 T^{2} \)
$7$ \( 1 + T \)
$11$ \( 1 + 5 T + 11 T^{2} \)
$13$ 1
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 + 9 T + 23 T^{2} \)
$29$ \( 1 + 29 T^{2} \)
$31$ \( 1 + 5 T + 31 T^{2} \)
$37$ \( 1 + 3 T + 37 T^{2} \)
$41$ \( 1 + 5 T + 41 T^{2} \)
$43$ \( 1 + 4 T + 43 T^{2} \)
$47$ \( 1 + 13 T + 47 T^{2} \)
$53$ \( 1 - 14 T + 53 T^{2} \)
$59$ \( 1 - 6 T + 59 T^{2} \)
$61$ \( 1 + 13 T + 61 T^{2} \)
$67$ \( 1 - 3 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + T + 73 T^{2} \)
$79$ \( 1 + 15 T + 79 T^{2} \)
$83$ \( 1 - 6 T + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 + 7 T + 97 T^{2} \)
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