Properties

Label 2166.4.a.s
Level $2166$
Weight $4$
Character orbit 2166.a
Self dual yes
Analytic conductor $127.798$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1524.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q -2 q^{2} -3 q^{3} + 4 q^{4} + ( 1 - \beta_{2} ) q^{5} + 6 q^{6} + ( -6 - 2 \beta_{1} + \beta_{2} ) q^{7} -8 q^{8} + 9 q^{9} +O(q^{10})\) \( q -2 q^{2} -3 q^{3} + 4 q^{4} + ( 1 - \beta_{2} ) q^{5} + 6 q^{6} + ( -6 - 2 \beta_{1} + \beta_{2} ) q^{7} -8 q^{8} + 9 q^{9} + ( -2 + 2 \beta_{2} ) q^{10} + ( -18 + \beta_{1} + 2 \beta_{2} ) q^{11} -12 q^{12} + ( 25 - 2 \beta_{1} ) q^{13} + ( 12 + 4 \beta_{1} - 2 \beta_{2} ) q^{14} + ( -3 + 3 \beta_{2} ) q^{15} + 16 q^{16} + ( -17 - \beta_{1} + 3 \beta_{2} ) q^{17} -18 q^{18} + ( 4 - 4 \beta_{2} ) q^{20} + ( 18 + 6 \beta_{1} - 3 \beta_{2} ) q^{21} + ( 36 - 2 \beta_{1} - 4 \beta_{2} ) q^{22} + ( -81 + 10 \beta_{1} + 5 \beta_{2} ) q^{23} + 24 q^{24} + ( 80 + 7 \beta_{1} - 11 \beta_{2} ) q^{25} + ( -50 + 4 \beta_{1} ) q^{26} -27 q^{27} + ( -24 - 8 \beta_{1} + 4 \beta_{2} ) q^{28} + ( -3 - 13 \beta_{1} + \beta_{2} ) q^{29} + ( 6 - 6 \beta_{2} ) q^{30} + ( 33 - 3 \beta_{1} + 8 \beta_{2} ) q^{31} -32 q^{32} + ( 54 - 3 \beta_{1} - 6 \beta_{2} ) q^{33} + ( 34 + 2 \beta_{1} - 6 \beta_{2} ) q^{34} + ( -104 + 11 \beta_{1} + 18 \beta_{2} ) q^{35} + 36 q^{36} + ( 94 - 13 \beta_{1} + 23 \beta_{2} ) q^{37} + ( -75 + 6 \beta_{1} ) q^{39} + ( -8 + 8 \beta_{2} ) q^{40} + ( 4 - 26 \beta_{1} + 4 \beta_{2} ) q^{41} + ( -36 - 12 \beta_{1} + 6 \beta_{2} ) q^{42} + ( -108 - 18 \beta_{1} - 7 \beta_{2} ) q^{43} + ( -72 + 4 \beta_{1} + 8 \beta_{2} ) q^{44} + ( 9 - 9 \beta_{2} ) q^{45} + ( 162 - 20 \beta_{1} - 10 \beta_{2} ) q^{46} + ( -260 - 18 \beta_{1} + 14 \beta_{2} ) q^{47} -48 q^{48} + ( 377 - 13 \beta_{1} + 11 \beta_{2} ) q^{49} + ( -160 - 14 \beta_{1} + 22 \beta_{2} ) q^{50} + ( 51 + 3 \beta_{1} - 9 \beta_{2} ) q^{51} + ( 100 - 8 \beta_{1} ) q^{52} + ( -30 + 13 \beta_{1} - 28 \beta_{2} ) q^{53} + 54 q^{54} + ( -479 - 23 \beta_{1} + 37 \beta_{2} ) q^{55} + ( 48 + 16 \beta_{1} - 8 \beta_{2} ) q^{56} + ( 6 + 26 \beta_{1} - 2 \beta_{2} ) q^{58} + ( 315 + 24 \beta_{1} - 9 \beta_{2} ) q^{59} + ( -12 + 12 \beta_{2} ) q^{60} + ( -123 + 2 \beta_{1} - 30 \beta_{2} ) q^{61} + ( -66 + 6 \beta_{1} - 16 \beta_{2} ) q^{62} + ( -54 - 18 \beta_{1} + 9 \beta_{2} ) q^{63} + 64 q^{64} + ( 131 + 18 \beta_{1} - 23 \beta_{2} ) q^{65} + ( -108 + 6 \beta_{1} + 12 \beta_{2} ) q^{66} + ( 29 + 5 \beta_{1} - 26 \beta_{2} ) q^{67} + ( -68 - 4 \beta_{1} + 12 \beta_{2} ) q^{68} + ( 243 - 30 \beta_{1} - 15 \beta_{2} ) q^{69} + ( 208 - 22 \beta_{1} - 36 \beta_{2} ) q^{70} + ( 335 - \beta_{1} - 31 \beta_{2} ) q^{71} -72 q^{72} + ( 35 - 40 \beta_{1} - 14 \beta_{2} ) q^{73} + ( -188 + 26 \beta_{1} - 46 \beta_{2} ) q^{74} + ( -240 - 21 \beta_{1} + 33 \beta_{2} ) q^{75} + ( 11 + 16 \beta_{1} - 69 \beta_{2} ) q^{77} + ( 150 - 12 \beta_{1} ) q^{78} + ( -101 + 43 \beta_{1} - 18 \beta_{2} ) q^{79} + ( 16 - 16 \beta_{2} ) q^{80} + 81 q^{81} + ( -8 + 52 \beta_{1} - 8 \beta_{2} ) q^{82} + ( -709 - 17 \beta_{1} - 21 \beta_{2} ) q^{83} + ( 72 + 24 \beta_{1} - 12 \beta_{2} ) q^{84} + ( -576 - 12 \beta_{1} + 48 \beta_{2} ) q^{85} + ( 216 + 36 \beta_{1} + 14 \beta_{2} ) q^{86} + ( 9 + 39 \beta_{1} - 3 \beta_{2} ) q^{87} + ( 144 - 8 \beta_{1} - 16 \beta_{2} ) q^{88} + ( 380 + \beta_{1} - 24 \beta_{2} ) q^{89} + ( -18 + 18 \beta_{2} ) q^{90} + ( 436 - 62 \beta_{1} + 59 \beta_{2} ) q^{91} + ( -324 + 40 \beta_{1} + 20 \beta_{2} ) q^{92} + ( -99 + 9 \beta_{1} - 24 \beta_{2} ) q^{93} + ( 520 + 36 \beta_{1} - 28 \beta_{2} ) q^{94} + 96 q^{96} + ( 463 - 7 \beta_{1} - 7 \beta_{2} ) q^{97} + ( -754 + 26 \beta_{1} - 22 \beta_{2} ) q^{98} + ( -162 + 9 \beta_{1} + 18 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 6q^{2} - 9q^{3} + 12q^{4} + 2q^{5} + 18q^{6} - 17q^{7} - 24q^{8} + 27q^{9} + O(q^{10}) \) \( 3q - 6q^{2} - 9q^{3} + 12q^{4} + 2q^{5} + 18q^{6} - 17q^{7} - 24q^{8} + 27q^{9} - 4q^{10} - 52q^{11} - 36q^{12} + 75q^{13} + 34q^{14} - 6q^{15} + 48q^{16} - 48q^{17} - 54q^{18} + 8q^{20} + 51q^{21} + 104q^{22} - 238q^{23} + 72q^{24} + 229q^{25} - 150q^{26} - 81q^{27} - 68q^{28} - 8q^{29} + 12q^{30} + 107q^{31} - 96q^{32} + 156q^{33} + 96q^{34} - 294q^{35} + 108q^{36} + 305q^{37} - 225q^{39} - 16q^{40} + 16q^{41} - 102q^{42} - 331q^{43} - 208q^{44} + 18q^{45} + 476q^{46} - 766q^{47} - 144q^{48} + 1142q^{49} - 458q^{50} + 144q^{51} + 300q^{52} - 118q^{53} + 162q^{54} - 1400q^{55} + 136q^{56} + 16q^{58} + 936q^{59} - 24q^{60} - 399q^{61} - 214q^{62} - 153q^{63} + 192q^{64} + 370q^{65} - 312q^{66} + 61q^{67} - 192q^{68} + 714q^{69} + 588q^{70} + 974q^{71} - 216q^{72} + 91q^{73} - 610q^{74} - 687q^{75} - 36q^{77} + 450q^{78} - 321q^{79} + 32q^{80} + 243q^{81} - 32q^{82} - 2148q^{83} + 204q^{84} - 1680q^{85} + 662q^{86} + 24q^{87} + 416q^{88} + 1116q^{89} - 36q^{90} + 1367q^{91} - 952q^{92} - 321q^{93} + 1532q^{94} + 288q^{96} + 1382q^{97} - 2284q^{98} - 468q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 7 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu - 2 \)
\(\beta_{2}\)\(=\)\( 4 \nu^{2} - 2 \nu - 19 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 2\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{2} + \beta_{1} + 59\)\()/12\)

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
3.13264
−2.27307
0.140435
−2.00000 −3.00000 4.00000 −12.9884 6.00000 −25.6033 −8.00000 9.00000 25.9768
1.2 −2.00000 −3.00000 4.00000 −5.21359 6.00000 31.4905 −8.00000 9.00000 10.4272
1.3 −2.00000 −3.00000 4.00000 20.2020 6.00000 −22.8872 −8.00000 9.00000 −40.4040
\(n\): 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.s 3
19.b odd 2 1 2166.4.a.w 3
19.c even 3 2 114.4.e.e 6
57.h odd 6 2 342.4.g.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.e.e 6 19.c even 3 2
342.4.g.g 6 57.h odd 6 2
2166.4.a.s 3 1.a even 1 1 trivial
2166.4.a.w 3 19.b odd 2 1

Hecke kernels

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

\( T_{5}^{3} - 2 T_{5}^{2} - 300 T_{5} - 1368 \)
\( T_{13}^{3} - 75 T_{13}^{2} + 819 T_{13} + 13207 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T )^{3} \)
$3$ \( ( 3 + T )^{3} \)
$5$ \( -1368 - 300 T - 2 T^{2} + T^{3} \)
$7$ \( -18453 - 941 T + 17 T^{2} + T^{3} \)
$11$ \( -32688 - 888 T + 52 T^{2} + T^{3} \)
$13$ \( 13207 + 819 T - 75 T^{2} + T^{3} \)
$17$ \( 10368 - 1728 T + 48 T^{2} + T^{3} \)
$19$ \( T^{3} \)
$23$ \( -6104664 - 23052 T + 238 T^{2} + T^{3} \)
$29$ \( -306432 - 42816 T + 8 T^{2} + T^{3} \)
$31$ \( 1435247 - 14005 T - 107 T^{2} + T^{3} \)
$37$ \( 28900349 - 125173 T - 305 T^{2} + T^{3} \)
$41$ \( -7007616 - 166560 T - 16 T^{2} + T^{3} \)
$43$ \( 3121601 - 83941 T + 331 T^{2} + T^{3} \)
$47$ \( -20196504 + 91308 T + 766 T^{2} + T^{3} \)
$53$ \( -40793976 - 217980 T + 118 T^{2} + T^{3} \)
$59$ \( 31669488 + 150120 T - 936 T^{2} + T^{3} \)
$61$ \( -78172163 - 209589 T + 399 T^{2} + T^{3} \)
$67$ \( -27605943 - 188261 T - 61 T^{2} + T^{3} \)
$71$ \( 16989912 + 21420 T - 974 T^{2} + T^{3} \)
$73$ \( 167210439 - 568301 T - 91 T^{2} + T^{3} \)
$79$ \( 63732523 - 427581 T + 321 T^{2} + T^{3} \)
$83$ \( 211415616 + 1271664 T + 2148 T^{2} + T^{3} \)
$89$ \( -11038032 + 245160 T - 1116 T^{2} + T^{3} \)
$97$ \( -79278088 + 601100 T - 1382 T^{2} + T^{3} \)
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