Properties

Label 2166.4.a.i
Level 2166
Weight 4
Character orbit 2166.a
Self dual yes
Analytic conductor 127.798
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(127.798137072\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + 2q^{2} + 3q^{3} + 4q^{4} + 6q^{5} + 6q^{6} - 16q^{7} + 8q^{8} + 9q^{9} + O(q^{10}) \) \( q + 2q^{2} + 3q^{3} + 4q^{4} + 6q^{5} + 6q^{6} - 16q^{7} + 8q^{8} + 9q^{9} + 12q^{10} + 12q^{11} + 12q^{12} - 38q^{13} - 32q^{14} + 18q^{15} + 16q^{16} - 126q^{17} + 18q^{18} + 24q^{20} - 48q^{21} + 24q^{22} + 168q^{23} + 24q^{24} - 89q^{25} - 76q^{26} + 27q^{27} - 64q^{28} - 30q^{29} + 36q^{30} + 88q^{31} + 32q^{32} + 36q^{33} - 252q^{34} - 96q^{35} + 36q^{36} - 254q^{37} - 114q^{39} + 48q^{40} - 42q^{41} - 96q^{42} - 52q^{43} + 48q^{44} + 54q^{45} + 336q^{46} - 96q^{47} + 48q^{48} - 87q^{49} - 178q^{50} - 378q^{51} - 152q^{52} - 198q^{53} + 54q^{54} + 72q^{55} - 128q^{56} - 60q^{58} + 660q^{59} + 72q^{60} - 538q^{61} + 176q^{62} - 144q^{63} + 64q^{64} - 228q^{65} + 72q^{66} - 884q^{67} - 504q^{68} + 504q^{69} - 192q^{70} - 792q^{71} + 72q^{72} + 218q^{73} - 508q^{74} - 267q^{75} - 192q^{77} - 228q^{78} + 520q^{79} + 96q^{80} + 81q^{81} - 84q^{82} - 492q^{83} - 192q^{84} - 756q^{85} - 104q^{86} - 90q^{87} + 96q^{88} - 810q^{89} + 108q^{90} + 608q^{91} + 672q^{92} + 264q^{93} - 192q^{94} + 96q^{96} - 1154q^{97} - 174q^{98} + 108q^{99} + O(q^{100}) \)

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 −16.0000 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 2166.4.a.i 1
19.b odd 2 1 6.4.a.a 1
57.d even 2 1 18.4.a.a 1
76.d even 2 1 48.4.a.c 1
95.d odd 2 1 150.4.a.i 1
95.g even 4 2 150.4.c.d 2
133.c even 2 1 294.4.a.e 1
133.o even 6 2 294.4.e.g 2
133.r odd 6 2 294.4.e.h 2
152.b even 2 1 192.4.a.c 1
152.g odd 2 1 192.4.a.i 1
171.l even 6 2 162.4.c.c 2
171.o odd 6 2 162.4.c.f 2
209.d even 2 1 726.4.a.f 1
228.b odd 2 1 144.4.a.c 1
247.d odd 2 1 1014.4.a.g 1
247.i even 4 2 1014.4.b.d 2
285.b even 2 1 450.4.a.h 1
285.j odd 4 2 450.4.c.e 2
304.j odd 4 2 768.4.d.n 2
304.m even 4 2 768.4.d.c 2
323.c odd 2 1 1734.4.a.d 1
380.d even 2 1 1200.4.a.b 1
380.j odd 4 2 1200.4.f.j 2
399.h odd 2 1 882.4.a.n 1
399.s odd 6 2 882.4.g.f 2
399.w even 6 2 882.4.g.i 2
456.l odd 2 1 576.4.a.r 1
456.p even 2 1 576.4.a.q 1
532.b odd 2 1 2352.4.a.e 1
627.b 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 19.b odd 2 1
18.4.a.a 1 57.d even 2 1
48.4.a.c 1 76.d even 2 1
144.4.a.c 1 228.b odd 2 1
150.4.a.i 1 95.d odd 2 1
150.4.c.d 2 95.g even 4 2
162.4.c.c 2 171.l even 6 2
162.4.c.f 2 171.o odd 6 2
192.4.a.c 1 152.b even 2 1
192.4.a.i 1 152.g odd 2 1
294.4.a.e 1 133.c even 2 1
294.4.e.g 2 133.o even 6 2
294.4.e.h 2 133.r odd 6 2
450.4.a.h 1 285.b even 2 1
450.4.c.e 2 285.j odd 4 2
576.4.a.q 1 456.p even 2 1
576.4.a.r 1 456.l odd 2 1
726.4.a.f 1 209.d even 2 1
768.4.d.c 2 304.m even 4 2
768.4.d.n 2 304.j odd 4 2
882.4.a.n 1 399.h odd 2 1
882.4.g.f 2 399.s odd 6 2
882.4.g.i 2 399.w even 6 2
1014.4.a.g 1 247.d odd 2 1
1014.4.b.d 2 247.i even 4 2
1200.4.a.b 1 380.d even 2 1
1200.4.f.j 2 380.j odd 4 2
1734.4.a.d 1 323.c odd 2 1
2166.4.a.i 1 1.a even 1 1 trivial
2178.4.a.e 1 627.b odd 2 1
2352.4.a.e 1 532.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(19\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5} - 6 \)
\( T_{13} + 38 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T \)
$3$ \( 1 - 3 T \)
$5$ \( 1 - 6 T + 125 T^{2} \)
$7$ \( 1 + 16 T + 343 T^{2} \)
$11$ \( 1 - 12 T + 1331 T^{2} \)
$13$ \( 1 + 38 T + 2197 T^{2} \)
$17$ \( 1 + 126 T + 4913 T^{2} \)
$19$ 1
$23$ \( 1 - 168 T + 12167 T^{2} \)
$29$ \( 1 + 30 T + 24389 T^{2} \)
$31$ \( 1 - 88 T + 29791 T^{2} \)
$37$ \( 1 + 254 T + 50653 T^{2} \)
$41$ \( 1 + 42 T + 68921 T^{2} \)
$43$ \( 1 + 52 T + 79507 T^{2} \)
$47$ \( 1 + 96 T + 103823 T^{2} \)
$53$ \( 1 + 198 T + 148877 T^{2} \)
$59$ \( 1 - 660 T + 205379 T^{2} \)
$61$ \( 1 + 538 T + 226981 T^{2} \)
$67$ \( 1 + 884 T + 300763 T^{2} \)
$71$ \( 1 + 792 T + 357911 T^{2} \)
$73$ \( 1 - 218 T + 389017 T^{2} \)
$79$ \( 1 - 520 T + 493039 T^{2} \)
$83$ \( 1 + 492 T + 571787 T^{2} \)
$89$ \( 1 + 810 T + 704969 T^{2} \)
$97$ \( 1 + 1154 T + 912673 T^{2} \)
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