Properties

Label 2166.4.a.u
Level $2166$
Weight $4$
Character orbit 2166.a
Self dual yes
Analytic conductor $127.798$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.14457.1
Defining polynomial: \(x^{3} - x^{2} - 32 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} -3 q^{3} + 4 q^{4} + ( 3 - \beta_{1} ) q^{5} -6 q^{6} + ( 2 - \beta_{1} ) q^{7} + 8 q^{8} + 9 q^{9} +O(q^{10})\) \( q + 2 q^{2} -3 q^{3} + 4 q^{4} + ( 3 - \beta_{1} ) q^{5} -6 q^{6} + ( 2 - \beta_{1} ) q^{7} + 8 q^{8} + 9 q^{9} + ( 6 - 2 \beta_{1} ) q^{10} + ( -15 - \beta_{2} ) q^{11} -12 q^{12} + ( -1 + 6 \beta_{1} ) q^{13} + ( 4 - 2 \beta_{1} ) q^{14} + ( -9 + 3 \beta_{1} ) q^{15} + 16 q^{16} + ( -30 - 7 \beta_{1} + \beta_{2} ) q^{17} + 18 q^{18} + ( 12 - 4 \beta_{1} ) q^{20} + ( -6 + 3 \beta_{1} ) q^{21} + ( -30 - 2 \beta_{2} ) q^{22} + ( 3 + 9 \beta_{1} + 2 \beta_{2} ) q^{23} -24 q^{24} + ( -29 - 5 \beta_{1} + \beta_{2} ) q^{25} + ( -2 + 12 \beta_{1} ) q^{26} -27 q^{27} + ( 8 - 4 \beta_{1} ) q^{28} + ( 36 + 15 \beta_{1} + \beta_{2} ) q^{29} + ( -18 + 6 \beta_{1} ) q^{30} + ( -28 + 24 \beta_{1} + \beta_{2} ) q^{31} + 32 q^{32} + ( 45 + 3 \beta_{2} ) q^{33} + ( -60 - 14 \beta_{1} + 2 \beta_{2} ) q^{34} + ( 93 - 4 \beta_{1} + \beta_{2} ) q^{35} + 36 q^{36} + ( 77 - 17 \beta_{1} + 3 \beta_{2} ) q^{37} + ( 3 - 18 \beta_{1} ) q^{39} + ( 24 - 8 \beta_{1} ) q^{40} + ( -222 + 20 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -12 + 6 \beta_{1} ) q^{42} + ( -28 + 21 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -60 - 4 \beta_{2} ) q^{44} + ( 27 - 9 \beta_{1} ) q^{45} + ( 6 + 18 \beta_{1} + 4 \beta_{2} ) q^{46} + ( 102 - 10 \beta_{1} - 6 \beta_{2} ) q^{47} -48 q^{48} + ( -252 - 3 \beta_{1} + \beta_{2} ) q^{49} + ( -58 - 10 \beta_{1} + 2 \beta_{2} ) q^{50} + ( 90 + 21 \beta_{1} - 3 \beta_{2} ) q^{51} + ( -4 + 24 \beta_{1} ) q^{52} + ( -435 + 18 \beta_{1} - \beta_{2} ) q^{53} -54 q^{54} + ( -66 + 55 \beta_{1} - 5 \beta_{2} ) q^{55} + ( 16 - 8 \beta_{1} ) q^{56} + ( 72 + 30 \beta_{1} + 2 \beta_{2} ) q^{58} + ( 81 - 9 \beta_{1} ) q^{59} + ( -36 + 12 \beta_{1} ) q^{60} + ( -157 - 32 \beta_{1} - 4 \beta_{2} ) q^{61} + ( -56 + 48 \beta_{1} + 2 \beta_{2} ) q^{62} + ( 18 - 9 \beta_{1} ) q^{63} + 64 q^{64} + ( -525 + 13 \beta_{1} - 6 \beta_{2} ) q^{65} + ( 90 + 6 \beta_{2} ) q^{66} + ( -244 - 12 \beta_{1} - \beta_{2} ) q^{67} + ( -120 - 28 \beta_{1} + 4 \beta_{2} ) q^{68} + ( -9 - 27 \beta_{1} - 6 \beta_{2} ) q^{69} + ( 186 - 8 \beta_{1} + 2 \beta_{2} ) q^{70} + ( -18 + 19 \beta_{1} - 11 \beta_{2} ) q^{71} + 72 q^{72} + ( -217 - 66 \beta_{1} - 4 \beta_{2} ) q^{73} + ( 154 - 34 \beta_{1} + 6 \beta_{2} ) q^{74} + ( 87 + 15 \beta_{1} - 3 \beta_{2} ) q^{75} + ( -51 + 55 \beta_{1} - 4 \beta_{2} ) q^{77} + ( 6 - 36 \beta_{1} ) q^{78} + ( -172 - 30 \beta_{1} + 3 \beta_{2} ) q^{79} + ( 48 - 16 \beta_{1} ) q^{80} + 81 q^{81} + ( -444 + 40 \beta_{1} + 4 \beta_{2} ) q^{82} + ( 846 + 37 \beta_{1} + 5 \beta_{2} ) q^{83} + ( -24 + 12 \beta_{1} ) q^{84} + ( 540 - 24 \beta_{1} + 12 \beta_{2} ) q^{85} + ( -56 + 42 \beta_{1} - 4 \beta_{2} ) q^{86} + ( -108 - 45 \beta_{1} - 3 \beta_{2} ) q^{87} + ( -120 - 8 \beta_{2} ) q^{88} + ( 543 + 34 \beta_{1} + 11 \beta_{2} ) q^{89} + ( 54 - 18 \beta_{1} ) q^{90} + ( -524 + 7 \beta_{1} - 6 \beta_{2} ) q^{91} + ( 12 + 36 \beta_{1} + 8 \beta_{2} ) q^{92} + ( 84 - 72 \beta_{1} - 3 \beta_{2} ) q^{93} + ( 204 - 20 \beta_{1} - 12 \beta_{2} ) q^{94} -96 q^{96} + ( 320 - 13 \beta_{1} - \beta_{2} ) q^{97} + ( -504 - 6 \beta_{1} + 2 \beta_{2} ) q^{98} + ( -135 - 9 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 6q^{2} - 9q^{3} + 12q^{4} + 10q^{5} - 18q^{6} + 7q^{7} + 24q^{8} + 27q^{9} + O(q^{10}) \) \( 3q + 6q^{2} - 9q^{3} + 12q^{4} + 10q^{5} - 18q^{6} + 7q^{7} + 24q^{8} + 27q^{9} + 20q^{10} - 44q^{11} - 36q^{12} - 9q^{13} + 14q^{14} - 30q^{15} + 48q^{16} - 84q^{17} + 54q^{18} + 40q^{20} - 21q^{21} - 88q^{22} - 2q^{23} - 72q^{24} - 83q^{25} - 18q^{26} - 81q^{27} + 28q^{28} + 92q^{29} - 60q^{30} - 109q^{31} + 96q^{32} + 132q^{33} - 168q^{34} + 282q^{35} + 108q^{36} + 245q^{37} + 27q^{39} + 80q^{40} - 688q^{41} - 42q^{42} - 103q^{43} - 176q^{44} + 90q^{45} - 4q^{46} + 322q^{47} - 144q^{48} - 754q^{49} - 166q^{50} + 252q^{51} - 36q^{52} - 1322q^{53} - 162q^{54} - 248q^{55} + 56q^{56} + 184q^{58} + 252q^{59} - 120q^{60} - 435q^{61} - 218q^{62} + 63q^{63} + 192q^{64} - 1582q^{65} + 264q^{66} - 719q^{67} - 336q^{68} + 6q^{69} + 564q^{70} - 62q^{71} + 216q^{72} - 581q^{73} + 490q^{74} + 249q^{75} - 204q^{77} + 54q^{78} - 489q^{79} + 160q^{80} + 243q^{81} - 1376q^{82} + 2496q^{83} - 84q^{84} + 1632q^{85} - 206q^{86} - 276q^{87} - 352q^{88} + 1584q^{89} + 180q^{90} - 1573q^{91} - 8q^{92} + 327q^{93} + 644q^{94} - 288q^{96} + 974q^{97} - 1508q^{98} - 396q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 32 x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 4 \nu^{2} - 6 \nu - 85 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 3 \beta_{1} + 88\)\()/4\)

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
6.22121
−0.0940524
−5.12716
2.00000 −3.00000 4.00000 −8.44242 −6.00000 −9.44242 8.00000 9.00000 −16.8848
1.2 2.00000 −3.00000 4.00000 4.18810 −6.00000 3.18810 8.00000 9.00000 8.37621
1.3 2.00000 −3.00000 4.00000 14.2543 −6.00000 13.2543 8.00000 9.00000 28.5086
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 2166.4.a.u 3
19.b odd 2 1 2166.4.a.t 3
19.c even 3 2 114.4.e.d 6
57.h odd 6 2 342.4.g.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.e.d 6 19.c even 3 2
342.4.g.h 6 57.h odd 6 2
2166.4.a.t 3 19.b odd 2 1
2166.4.a.u 3 1.a even 1 1 trivial

Hecke kernels

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

\( T_{5}^{3} - 10 T_{5}^{2} - 96 T_{5} + 504 \)
\( T_{13}^{3} + 9 T_{13}^{2} - 4629 T_{13} - 37685 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{3} \)
$3$ \( ( 3 + T )^{3} \)
$5$ \( 504 - 96 T - 10 T^{2} + T^{3} \)
$7$ \( 399 - 113 T - 7 T^{2} + T^{3} \)
$11$ \( -217224 - 4740 T + 44 T^{2} + T^{3} \)
$13$ \( -37685 - 4629 T + 9 T^{2} + T^{3} \)
$17$ \( -820800 - 10080 T + 84 T^{2} + T^{3} \)
$19$ \( T^{3} \)
$23$ \( 106776 - 30192 T + 2 T^{2} + T^{3} \)
$29$ \( -1302336 - 30144 T - 92 T^{2} + T^{3} \)
$31$ \( -9721957 - 73489 T + 109 T^{2} + T^{3} \)
$37$ \( -1317799 - 71005 T - 245 T^{2} + T^{3} \)
$41$ \( -10279872 + 88560 T + 688 T^{2} + T^{3} \)
$43$ \( 6249329 - 79297 T + 103 T^{2} + T^{3} \)
$47$ \( -11695896 - 166164 T - 322 T^{2} + T^{3} \)
$53$ \( 66978360 + 533448 T + 1322 T^{2} + T^{3} \)
$59$ \( 367416 + 10692 T - 252 T^{2} + T^{3} \)
$61$ \( -73871 - 142557 T + 435 T^{2} + T^{3} \)
$67$ \( 9614433 + 149527 T + 719 T^{2} + T^{3} \)
$71$ \( -111044664 - 718212 T + 62 T^{2} + T^{3} \)
$73$ \( 70657335 - 510269 T + 581 T^{2} + T^{3} \)
$79$ \( -51818777 - 94281 T + 489 T^{2} + T^{3} \)
$83$ \( -372278592 + 1783728 T - 2496 T^{2} + T^{3} \)
$89$ \( 395518680 + 73116 T - 1584 T^{2} + T^{3} \)
$97$ \( -24200200 + 290300 T - 974 T^{2} + T^{3} \)
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