Properties

Label 2366.2.a.k
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

\(f(q)\) \(=\) \( q + q^{2} - 2q^{3} + q^{4} + 3q^{5} - 2q^{6} - q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - 2q^{3} + q^{4} + 3q^{5} - 2q^{6} - q^{7} + q^{8} + q^{9} + 3q^{10} + 6q^{11} - 2q^{12} - q^{14} - 6q^{15} + q^{16} - 3q^{17} + q^{18} + 4q^{19} + 3q^{20} + 2q^{21} + 6q^{22} - 6q^{23} - 2q^{24} + 4q^{25} + 4q^{27} - q^{28} + 3q^{29} - 6q^{30} + 10q^{31} + q^{32} - 12q^{33} - 3q^{34} - 3q^{35} + q^{36} + q^{37} + 4q^{38} + 3q^{40} + 3q^{41} + 2q^{42} - 10q^{43} + 6q^{44} + 3q^{45} - 6q^{46} - 2q^{48} + q^{49} + 4q^{50} + 6q^{51} - 9q^{53} + 4q^{54} + 18q^{55} - q^{56} - 8q^{57} + 3q^{58} + 6q^{59} - 6q^{60} - 7q^{61} + 10q^{62} - q^{63} + q^{64} - 12q^{66} - 2q^{67} - 3q^{68} + 12q^{69} - 3q^{70} + q^{72} + 7q^{73} + q^{74} - 8q^{75} + 4q^{76} - 6q^{77} + 14q^{79} + 3q^{80} - 11q^{81} + 3q^{82} + 18q^{83} + 2q^{84} - 9q^{85} - 10q^{86} - 6q^{87} + 6q^{88} + 6q^{89} + 3q^{90} - 6q^{92} - 20q^{93} + 12q^{95} - 2q^{96} - 2q^{97} + q^{98} + 6q^{99} + O(q^{100}) \)

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 −2.00000 1.00000 3.00000 −2.00000 −1.00000 1.00000 1.00000 3.00000
\(n\): 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.k 1
13.b even 2 1 2366.2.a.a 1
13.d odd 4 2 2366.2.d.a 2
13.e even 6 2 182.2.g.c 2
39.h odd 6 2 1638.2.r.m 2
52.i odd 6 2 1456.2.s.a 2
91.k even 6 2 1274.2.e.k 2
91.l odd 6 2 1274.2.e.b 2
91.p odd 6 2 1274.2.h.l 2
91.t odd 6 2 1274.2.g.c 2
91.u even 6 2 1274.2.h.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.c 2 13.e even 6 2
1274.2.e.b 2 91.l odd 6 2
1274.2.e.k 2 91.k even 6 2
1274.2.g.c 2 91.t odd 6 2
1274.2.h.e 2 91.u even 6 2
1274.2.h.l 2 91.p odd 6 2
1456.2.s.a 2 52.i odd 6 2
1638.2.r.m 2 39.h odd 6 2
2366.2.a.a 1 13.b even 2 1
2366.2.a.k 1 1.a even 1 1 trivial
2366.2.d.a 2 13.d odd 4 2

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 + 2 T + 3 T^{2} \)
$5$ \( 1 - 3 T + 5 T^{2} \)
$7$ \( 1 + T \)
$11$ \( 1 - 6 T + 11 T^{2} \)
$13$ 1
$17$ \( 1 + 3 T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 + 6 T + 23 T^{2} \)
$29$ \( 1 - 3 T + 29 T^{2} \)
$31$ \( 1 - 10 T + 31 T^{2} \)
$37$ \( 1 - T + 37 T^{2} \)
$41$ \( 1 - 3 T + 41 T^{2} \)
$43$ \( 1 + 10 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 + 9 T + 53 T^{2} \)
$59$ \( 1 - 6 T + 59 T^{2} \)
$61$ \( 1 + 7 T + 61 T^{2} \)
$67$ \( 1 + 2 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 - 7 T + 73 T^{2} \)
$79$ \( 1 - 14 T + 79 T^{2} \)
$83$ \( 1 - 18 T + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 + 2 T + 97 T^{2} \)
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