Properties

Label 2166.4.a.w
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: 3.3.1524.1
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.w 3
19.b odd 2 1 2166.4.a.s 3
19.d odd 6 2 114.4.e.e 6
57.f even 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.d odd 6 2
342.4.g.g 6 57.f even 6 2
2166.4.a.s 3 19.b odd 2 1
2166.4.a.w 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2166))\):

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

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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