Properties

Label 165.4.a.c
Level $165$
Weight $4$
Character orbit 165.a
Self dual yes
Analytic conductor $9.735$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.73531515095\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + 3 q^{3} + ( -4 + \beta ) q^{4} -5 q^{5} -3 \beta q^{6} + ( -4 + 4 \beta ) q^{7} + ( -4 + 11 \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q -\beta q^{2} + 3 q^{3} + ( -4 + \beta ) q^{4} -5 q^{5} -3 \beta q^{6} + ( -4 + 4 \beta ) q^{7} + ( -4 + 11 \beta ) q^{8} + 9 q^{9} + 5 \beta q^{10} -11 q^{11} + ( -12 + 3 \beta ) q^{12} + ( -44 - 2 \beta ) q^{13} -16 q^{14} -15 q^{15} + ( -12 - 15 \beta ) q^{16} + ( -30 + 44 \beta ) q^{17} -9 \beta q^{18} + ( -74 - 22 \beta ) q^{19} + ( 20 - 5 \beta ) q^{20} + ( -12 + 12 \beta ) q^{21} + 11 \beta q^{22} + ( -32 - 60 \beta ) q^{23} + ( -12 + 33 \beta ) q^{24} + 25 q^{25} + ( 8 + 46 \beta ) q^{26} + 27 q^{27} + ( 32 - 16 \beta ) q^{28} + ( -96 + 34 \beta ) q^{29} + 15 \beta q^{30} + ( 36 - 12 \beta ) q^{31} + ( 92 - 61 \beta ) q^{32} -33 q^{33} + ( -176 - 14 \beta ) q^{34} + ( 20 - 20 \beta ) q^{35} + ( -36 + 9 \beta ) q^{36} + ( -130 - 112 \beta ) q^{37} + ( 88 + 96 \beta ) q^{38} + ( -132 - 6 \beta ) q^{39} + ( 20 - 55 \beta ) q^{40} + ( 96 - 154 \beta ) q^{41} -48 q^{42} + ( -196 - 124 \beta ) q^{43} + ( 44 - 11 \beta ) q^{44} -45 q^{45} + ( 240 + 92 \beta ) q^{46} + ( 4 + 216 \beta ) q^{47} + ( -36 - 45 \beta ) q^{48} + ( -263 - 16 \beta ) q^{49} -25 \beta q^{50} + ( -90 + 132 \beta ) q^{51} + ( 168 - 38 \beta ) q^{52} + ( 334 - 196 \beta ) q^{53} -27 \beta q^{54} + 55 q^{55} + ( 192 - 16 \beta ) q^{56} + ( -222 - 66 \beta ) q^{57} + ( -136 + 62 \beta ) q^{58} + ( 4 + 240 \beta ) q^{59} + ( 60 - 15 \beta ) q^{60} + ( -146 + 364 \beta ) q^{61} + ( 48 - 24 \beta ) q^{62} + ( -36 + 36 \beta ) q^{63} + ( 340 + 89 \beta ) q^{64} + ( 220 + 10 \beta ) q^{65} + 33 \beta q^{66} + ( -380 + 16 \beta ) q^{67} + ( 296 - 162 \beta ) q^{68} + ( -96 - 180 \beta ) q^{69} + 80 q^{70} + ( 1008 + 44 \beta ) q^{71} + ( -36 + 99 \beta ) q^{72} + ( -272 + 58 \beta ) q^{73} + ( 448 + 242 \beta ) q^{74} + 75 q^{75} + ( 208 - 8 \beta ) q^{76} + ( 44 - 44 \beta ) q^{77} + ( 24 + 138 \beta ) q^{78} + ( 474 - 306 \beta ) q^{79} + ( 60 + 75 \beta ) q^{80} + 81 q^{81} + ( 616 + 58 \beta ) q^{82} + ( 70 - 426 \beta ) q^{83} + ( 96 - 48 \beta ) q^{84} + ( 150 - 220 \beta ) q^{85} + ( 496 + 320 \beta ) q^{86} + ( -288 + 102 \beta ) q^{87} + ( 44 - 121 \beta ) q^{88} + ( 186 - 128 \beta ) q^{89} + 45 \beta q^{90} + ( 144 - 176 \beta ) q^{91} + ( -112 + 148 \beta ) q^{92} + ( 108 - 36 \beta ) q^{93} + ( -864 - 220 \beta ) q^{94} + ( 370 + 110 \beta ) q^{95} + ( 276 - 183 \beta ) q^{96} + ( -298 + 428 \beta ) q^{97} + ( 64 + 279 \beta ) q^{98} -99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + 6q^{3} - 7q^{4} - 10q^{5} - 3q^{6} - 4q^{7} + 3q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - q^{2} + 6q^{3} - 7q^{4} - 10q^{5} - 3q^{6} - 4q^{7} + 3q^{8} + 18q^{9} + 5q^{10} - 22q^{11} - 21q^{12} - 90q^{13} - 32q^{14} - 30q^{15} - 39q^{16} - 16q^{17} - 9q^{18} - 170q^{19} + 35q^{20} - 12q^{21} + 11q^{22} - 124q^{23} + 9q^{24} + 50q^{25} + 62q^{26} + 54q^{27} + 48q^{28} - 158q^{29} + 15q^{30} + 60q^{31} + 123q^{32} - 66q^{33} - 366q^{34} + 20q^{35} - 63q^{36} - 372q^{37} + 272q^{38} - 270q^{39} - 15q^{40} + 38q^{41} - 96q^{42} - 516q^{43} + 77q^{44} - 90q^{45} + 572q^{46} + 224q^{47} - 117q^{48} - 542q^{49} - 25q^{50} - 48q^{51} + 298q^{52} + 472q^{53} - 27q^{54} + 110q^{55} + 368q^{56} - 510q^{57} - 210q^{58} + 248q^{59} + 105q^{60} + 72q^{61} + 72q^{62} - 36q^{63} + 769q^{64} + 450q^{65} + 33q^{66} - 744q^{67} + 430q^{68} - 372q^{69} + 160q^{70} + 2060q^{71} + 27q^{72} - 486q^{73} + 1138q^{74} + 150q^{75} + 408q^{76} + 44q^{77} + 186q^{78} + 642q^{79} + 195q^{80} + 162q^{81} + 1290q^{82} - 286q^{83} + 144q^{84} + 80q^{85} + 1312q^{86} - 474q^{87} - 33q^{88} + 244q^{89} + 45q^{90} + 112q^{91} - 76q^{92} + 180q^{93} - 1948q^{94} + 850q^{95} + 369q^{96} - 168q^{97} + 407q^{98} - 198q^{99} + O(q^{100}) \)

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
2.56155
−1.56155
−2.56155 3.00000 −1.43845 −5.00000 −7.68466 6.24621 24.1771 9.00000 12.8078
1.2 1.56155 3.00000 −5.56155 −5.00000 4.68466 −10.2462 −21.1771 9.00000 −7.80776
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)

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 165.4.a.c 2
3.b odd 2 1 495.4.a.d 2
5.b even 2 1 825.4.a.m 2
5.c odd 4 2 825.4.c.j 4
11.b odd 2 1 1815.4.a.n 2
15.d odd 2 1 2475.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.c 2 1.a even 1 1 trivial
495.4.a.d 2 3.b odd 2 1
825.4.a.m 2 5.b even 2 1
825.4.c.j 4 5.c odd 4 2
1815.4.a.n 2 11.b odd 2 1
2475.4.a.n 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(165))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T + T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( ( 5 + T )^{2} \)
$7$ \( -64 + 4 T + T^{2} \)
$11$ \( ( 11 + T )^{2} \)
$13$ \( 2008 + 90 T + T^{2} \)
$17$ \( -8164 + 16 T + T^{2} \)
$19$ \( 5168 + 170 T + T^{2} \)
$23$ \( -11456 + 124 T + T^{2} \)
$29$ \( 1328 + 158 T + T^{2} \)
$31$ \( 288 - 60 T + T^{2} \)
$37$ \( -18716 + 372 T + T^{2} \)
$41$ \( -100432 - 38 T + T^{2} \)
$43$ \( 1216 + 516 T + T^{2} \)
$47$ \( -185744 - 224 T + T^{2} \)
$53$ \( -107572 - 472 T + T^{2} \)
$59$ \( -229424 - 248 T + T^{2} \)
$61$ \( -561812 - 72 T + T^{2} \)
$67$ \( 137296 + 744 T + T^{2} \)
$71$ \( 1052672 - 2060 T + T^{2} \)
$73$ \( 44752 + 486 T + T^{2} \)
$79$ \( -294912 - 642 T + T^{2} \)
$83$ \( -750824 + 286 T + T^{2} \)
$89$ \( -54748 - 244 T + T^{2} \)
$97$ \( -771476 + 168 T + T^{2} \)
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