Properties

Label 2366.4.a.h
Level $2366$
Weight $4$
Character orbit 2366.a
Self dual yes
Analytic conductor $139.599$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2366.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(139.598519074\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} + 8q^{3} + 4q^{4} + 14q^{5} + 16q^{6} + 7q^{7} + 8q^{8} + 37q^{9} + O(q^{10}) \) \( q + 2q^{2} + 8q^{3} + 4q^{4} + 14q^{5} + 16q^{6} + 7q^{7} + 8q^{8} + 37q^{9} + 28q^{10} + 28q^{11} + 32q^{12} + 14q^{14} + 112q^{15} + 16q^{16} + 74q^{17} + 74q^{18} - 80q^{19} + 56q^{20} + 56q^{21} + 56q^{22} - 112q^{23} + 64q^{24} + 71q^{25} + 80q^{27} + 28q^{28} + 190q^{29} + 224q^{30} - 72q^{31} + 32q^{32} + 224q^{33} + 148q^{34} + 98q^{35} + 148q^{36} + 346q^{37} - 160q^{38} + 112q^{40} - 162q^{41} + 112q^{42} - 412q^{43} + 112q^{44} + 518q^{45} - 224q^{46} - 24q^{47} + 128q^{48} + 49q^{49} + 142q^{50} + 592q^{51} + 318q^{53} + 160q^{54} + 392q^{55} + 56q^{56} - 640q^{57} + 380q^{58} + 200q^{59} + 448q^{60} - 198q^{61} - 144q^{62} + 259q^{63} + 64q^{64} + 448q^{66} + 716q^{67} + 296q^{68} - 896q^{69} + 196q^{70} - 392q^{71} + 296q^{72} - 538q^{73} + 692q^{74} + 568q^{75} - 320q^{76} + 196q^{77} + 240q^{79} + 224q^{80} - 359q^{81} - 324q^{82} + 1072q^{83} + 224q^{84} + 1036q^{85} - 824q^{86} + 1520q^{87} + 224q^{88} - 810q^{89} + 1036q^{90} - 448q^{92} - 576q^{93} - 48q^{94} - 1120q^{95} + 256q^{96} - 1354q^{97} + 98q^{98} + 1036q^{99} + O(q^{100}) \)

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 8.00000 4.00000 14.0000 16.0000 7.00000 8.00000 37.0000 28.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 2366.4.a.h 1
13.b even 2 1 14.4.a.a 1
39.d odd 2 1 126.4.a.h 1
52.b odd 2 1 112.4.a.a 1
65.d even 2 1 350.4.a.l 1
65.h odd 4 2 350.4.c.b 2
91.b odd 2 1 98.4.a.a 1
91.r even 6 2 98.4.c.d 2
91.s odd 6 2 98.4.c.f 2
104.e even 2 1 448.4.a.b 1
104.h odd 2 1 448.4.a.o 1
143.d odd 2 1 1694.4.a.g 1
156.h even 2 1 1008.4.a.s 1
273.g even 2 1 882.4.a.i 1
273.w odd 6 2 882.4.g.b 2
273.ba even 6 2 882.4.g.k 2
364.h even 2 1 784.4.a.s 1
455.h odd 2 1 2450.4.a.bo 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 13.b even 2 1
98.4.a.a 1 91.b odd 2 1
98.4.c.d 2 91.r even 6 2
98.4.c.f 2 91.s odd 6 2
112.4.a.a 1 52.b odd 2 1
126.4.a.h 1 39.d odd 2 1
350.4.a.l 1 65.d even 2 1
350.4.c.b 2 65.h odd 4 2
448.4.a.b 1 104.e even 2 1
448.4.a.o 1 104.h odd 2 1
784.4.a.s 1 364.h even 2 1
882.4.a.i 1 273.g even 2 1
882.4.g.b 2 273.w odd 6 2
882.4.g.k 2 273.ba even 6 2
1008.4.a.s 1 156.h even 2 1
1694.4.a.g 1 143.d odd 2 1
2366.4.a.h 1 1.a even 1 1 trivial
2450.4.a.bo 1 455.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2366))\):

\( T_{3} - 8 \)
\( T_{5} - 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( -8 + T \)
$5$ \( -14 + T \)
$7$ \( -7 + T \)
$11$ \( -28 + T \)
$13$ \( T \)
$17$ \( -74 + T \)
$19$ \( 80 + T \)
$23$ \( 112 + T \)
$29$ \( -190 + T \)
$31$ \( 72 + T \)
$37$ \( -346 + T \)
$41$ \( 162 + T \)
$43$ \( 412 + T \)
$47$ \( 24 + T \)
$53$ \( -318 + T \)
$59$ \( -200 + T \)
$61$ \( 198 + T \)
$67$ \( -716 + T \)
$71$ \( 392 + T \)
$73$ \( 538 + T \)
$79$ \( -240 + T \)
$83$ \( -1072 + T \)
$89$ \( 810 + T \)
$97$ \( 1354 + T \)
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