Properties

Label 2166.4.a.p
Level $2166$
Weight $4$
Character orbit 2166.a
Self dual yes
Analytic conductor $127.798$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( -4 - 5 \beta ) q^{5} + 6 q^{6} + ( -12 + 7 \beta ) q^{7} + 8 q^{8} + 9 q^{9} +O(q^{10})\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( -4 - 5 \beta ) q^{5} + 6 q^{6} + ( -12 + 7 \beta ) q^{7} + 8 q^{8} + 9 q^{9} + ( -8 - 10 \beta ) q^{10} + ( 3 + 8 \beta ) q^{11} + 12 q^{12} + ( 12 - 5 \beta ) q^{13} + ( -24 + 14 \beta ) q^{14} + ( -12 - 15 \beta ) q^{15} + 16 q^{16} + ( 11 - 34 \beta ) q^{17} + 18 q^{18} + ( -16 - 20 \beta ) q^{20} + ( -36 + 21 \beta ) q^{21} + ( 6 + 16 \beta ) q^{22} + ( -35 + 16 \beta ) q^{23} + 24 q^{24} + ( -84 + 65 \beta ) q^{25} + ( 24 - 10 \beta ) q^{26} + 27 q^{27} + ( -48 + 28 \beta ) q^{28} + ( -22 - 86 \beta ) q^{29} + ( -24 - 30 \beta ) q^{30} + ( -200 + 161 \beta ) q^{31} + 32 q^{32} + ( 9 + 24 \beta ) q^{33} + ( 22 - 68 \beta ) q^{34} + ( 13 - 3 \beta ) q^{35} + 36 q^{36} + ( -154 + 160 \beta ) q^{37} + ( 36 - 15 \beta ) q^{39} + ( -32 - 40 \beta ) q^{40} + ( 128 + 75 \beta ) q^{41} + ( -72 + 42 \beta ) q^{42} + ( -159 + 94 \beta ) q^{43} + ( 12 + 32 \beta ) q^{44} + ( -36 - 45 \beta ) q^{45} + ( -70 + 32 \beta ) q^{46} + ( 176 - 19 \beta ) q^{47} + 48 q^{48} + ( -150 - 119 \beta ) q^{49} + ( -168 + 130 \beta ) q^{50} + ( 33 - 102 \beta ) q^{51} + ( 48 - 20 \beta ) q^{52} + ( -217 - 332 \beta ) q^{53} + 54 q^{54} + ( -52 - 87 \beta ) q^{55} + ( -96 + 56 \beta ) q^{56} + ( -44 - 172 \beta ) q^{58} + ( -18 - 424 \beta ) q^{59} + ( -48 - 60 \beta ) q^{60} + ( 372 - 250 \beta ) q^{61} + ( -400 + 322 \beta ) q^{62} + ( -108 + 63 \beta ) q^{63} + 64 q^{64} + ( -23 - 15 \beta ) q^{65} + ( 18 + 48 \beta ) q^{66} + ( -131 + 129 \beta ) q^{67} + ( 44 - 136 \beta ) q^{68} + ( -105 + 48 \beta ) q^{69} + ( 26 - 6 \beta ) q^{70} + ( -352 + 245 \beta ) q^{71} + 72 q^{72} + ( 161 + 74 \beta ) q^{73} + ( -308 + 320 \beta ) q^{74} + ( -252 + 195 \beta ) q^{75} + ( 20 - 19 \beta ) q^{77} + ( 72 - 30 \beta ) q^{78} + ( 375 + 250 \beta ) q^{79} + ( -64 - 80 \beta ) q^{80} + 81 q^{81} + ( 256 + 150 \beta ) q^{82} + ( 69 - 212 \beta ) q^{83} + ( -144 + 84 \beta ) q^{84} + ( 126 + 251 \beta ) q^{85} + ( -318 + 188 \beta ) q^{86} + ( -66 - 258 \beta ) q^{87} + ( 24 + 64 \beta ) q^{88} + ( -404 - 372 \beta ) q^{89} + ( -72 - 90 \beta ) q^{90} + ( -179 + 109 \beta ) q^{91} + ( -140 + 64 \beta ) q^{92} + ( -600 + 483 \beta ) q^{93} + ( 352 - 38 \beta ) q^{94} + 96 q^{96} + ( -1030 + 122 \beta ) q^{97} + ( -300 - 238 \beta ) q^{98} + ( 27 + 72 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 13 q^{5} + 12 q^{6} - 17 q^{7} + 16 q^{8} + 18 q^{9} + O(q^{10}) \) \( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 13 q^{5} + 12 q^{6} - 17 q^{7} + 16 q^{8} + 18 q^{9} - 26 q^{10} + 14 q^{11} + 24 q^{12} + 19 q^{13} - 34 q^{14} - 39 q^{15} + 32 q^{16} - 12 q^{17} + 36 q^{18} - 52 q^{20} - 51 q^{21} + 28 q^{22} - 54 q^{23} + 48 q^{24} - 103 q^{25} + 38 q^{26} + 54 q^{27} - 68 q^{28} - 130 q^{29} - 78 q^{30} - 239 q^{31} + 64 q^{32} + 42 q^{33} - 24 q^{34} + 23 q^{35} + 72 q^{36} - 148 q^{37} + 57 q^{39} - 104 q^{40} + 331 q^{41} - 102 q^{42} - 224 q^{43} + 56 q^{44} - 117 q^{45} - 108 q^{46} + 333 q^{47} + 96 q^{48} - 419 q^{49} - 206 q^{50} - 36 q^{51} + 76 q^{52} - 766 q^{53} + 108 q^{54} - 191 q^{55} - 136 q^{56} - 260 q^{58} - 460 q^{59} - 156 q^{60} + 494 q^{61} - 478 q^{62} - 153 q^{63} + 128 q^{64} - 61 q^{65} + 84 q^{66} - 133 q^{67} - 48 q^{68} - 162 q^{69} + 46 q^{70} - 459 q^{71} + 144 q^{72} + 396 q^{73} - 296 q^{74} - 309 q^{75} + 21 q^{77} + 114 q^{78} + 1000 q^{79} - 208 q^{80} + 162 q^{81} + 662 q^{82} - 74 q^{83} - 204 q^{84} + 503 q^{85} - 448 q^{86} - 390 q^{87} + 112 q^{88} - 1180 q^{89} - 234 q^{90} - 249 q^{91} - 216 q^{92} - 717 q^{93} + 666 q^{94} + 192 q^{96} - 1938 q^{97} - 838 q^{98} + 126 q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
2.00000 3.00000 4.00000 −12.0902 6.00000 −0.673762 8.00000 9.00000 −24.1803
1.2 2.00000 3.00000 4.00000 −0.909830 6.00000 −16.3262 8.00000 9.00000 −1.81966
\(n\): 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.p yes 2
19.b odd 2 1 2166.4.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2166.4.a.j 2 19.b odd 2 1
2166.4.a.p yes 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5}^{2} + 13 T_{5} + 11 \)
\( T_{13}^{2} - 19 T_{13} + 59 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( 11 + 13 T + T^{2} \)
$7$ \( 11 + 17 T + T^{2} \)
$11$ \( -31 - 14 T + T^{2} \)
$13$ \( 59 - 19 T + T^{2} \)
$17$ \( -1409 + 12 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( 409 + 54 T + T^{2} \)
$29$ \( -5020 + 130 T + T^{2} \)
$31$ \( -18121 + 239 T + T^{2} \)
$37$ \( -26524 + 148 T + T^{2} \)
$41$ \( 20359 - 331 T + T^{2} \)
$43$ \( 1499 + 224 T + T^{2} \)
$47$ \( 27271 - 333 T + T^{2} \)
$53$ \( 8909 + 766 T + T^{2} \)
$59$ \( -171820 + 460 T + T^{2} \)
$61$ \( -17116 - 494 T + T^{2} \)
$67$ \( -16379 + 133 T + T^{2} \)
$71$ \( -22361 + 459 T + T^{2} \)
$73$ \( 32359 - 396 T + T^{2} \)
$79$ \( 171875 - 1000 T + T^{2} \)
$83$ \( -54811 + 74 T + T^{2} \)
$89$ \( 175120 + 1180 T + T^{2} \)
$97$ \( 920356 + 1938 T + T^{2} \)
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