Properties

Label 165.4.a.d
Level $165$
Weight $4$
Character orbit 165.a
Self dual yes
Analytic conductor $9.735$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.23612.1
Defining polynomial: \(x^{3} - x^{2} - 20 x + 26\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} ) q^{2} -3 q^{3} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} -5 q^{5} + ( 3 + 3 \beta_{1} ) q^{6} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{7} + ( -15 - 3 \beta_{1} - 4 \beta_{2} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{1} ) q^{2} -3 q^{3} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} -5 q^{5} + ( 3 + 3 \beta_{1} ) q^{6} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{7} + ( -15 - 3 \beta_{1} - 4 \beta_{2} ) q^{8} + 9 q^{9} + ( 5 + 5 \beta_{1} ) q^{10} + 11 q^{11} + ( -21 - 3 \beta_{1} - 3 \beta_{2} ) q^{12} + ( -4 + 12 \beta_{1} + 2 \beta_{2} ) q^{13} + ( -22 + 10 \beta_{1} + 4 \beta_{2} ) q^{14} + 15 q^{15} + ( 9 + 23 \beta_{1} + 7 \beta_{2} ) q^{16} + ( -76 + 10 \beta_{1} + 6 \beta_{2} ) q^{17} + ( -9 - 9 \beta_{1} ) q^{18} + ( 48 + 2 \beta_{1} + 4 \beta_{2} ) q^{19} + ( -35 - 5 \beta_{1} - 5 \beta_{2} ) q^{20} + ( 6 - 6 \beta_{1} + 6 \beta_{2} ) q^{21} + ( -11 - 11 \beta_{1} ) q^{22} + ( -60 - 20 \beta_{1} - 8 \beta_{2} ) q^{23} + ( 45 + 9 \beta_{1} + 12 \beta_{2} ) q^{24} + 25 q^{25} + ( -168 - 4 \beta_{1} - 18 \beta_{2} ) q^{26} -27 q^{27} + ( -110 - 10 \beta_{1} - 6 \beta_{2} ) q^{28} + ( 34 - 34 \beta_{1} + 20 \beta_{2} ) q^{29} + ( -15 - 15 \beta_{1} ) q^{30} + ( -24 + 4 \beta_{1} - 16 \beta_{2} ) q^{31} + ( -225 - 13 \beta_{1} - 12 \beta_{2} ) q^{32} -33 q^{33} + ( -76 + 52 \beta_{1} - 28 \beta_{2} ) q^{34} + ( 10 - 10 \beta_{1} + 10 \beta_{2} ) q^{35} + ( 63 + 9 \beta_{1} + 9 \beta_{2} ) q^{36} + ( -122 - 24 \beta_{1} - 4 \beta_{2} ) q^{37} + ( -84 - 64 \beta_{1} - 14 \beta_{2} ) q^{38} + ( 12 - 36 \beta_{1} - 6 \beta_{2} ) q^{39} + ( 75 + 15 \beta_{1} + 20 \beta_{2} ) q^{40} + ( -66 + 2 \beta_{1} - 20 \beta_{2} ) q^{41} + ( 66 - 30 \beta_{1} - 12 \beta_{2} ) q^{42} + ( -150 - 74 \beta_{1} + 14 \beta_{2} ) q^{43} + ( 77 + 11 \beta_{1} + 11 \beta_{2} ) q^{44} -45 q^{45} + ( 356 + 92 \beta_{1} + 44 \beta_{2} ) q^{46} + ( -36 + 48 \beta_{1} + 28 \beta_{2} ) q^{47} + ( -27 - 69 \beta_{1} - 21 \beta_{2} ) q^{48} + ( -51 - 4 \beta_{1} - 20 \beta_{2} ) q^{49} + ( -25 - 25 \beta_{1} ) q^{50} + ( 228 - 30 \beta_{1} - 18 \beta_{2} ) q^{51} + ( 292 + 144 \beta_{1} + 42 \beta_{2} ) q^{52} + ( -42 - 32 \beta_{1} + 52 \beta_{2} ) q^{53} + ( 27 + 27 \beta_{1} ) q^{54} -55 q^{55} + ( 438 + 54 \beta_{1} - 4 \beta_{2} ) q^{56} + ( -144 - 6 \beta_{1} - 12 \beta_{2} ) q^{57} + ( 402 - 114 \beta_{1} - 26 \beta_{2} ) q^{58} + ( -308 - 120 \beta_{1} - 4 \beta_{2} ) q^{59} + ( 105 + 15 \beta_{1} + 15 \beta_{2} ) q^{60} + ( 218 - 12 \beta_{1} - 44 \beta_{2} ) q^{61} + ( 88 \beta_{1} + 44 \beta_{2} ) q^{62} + ( -18 + 18 \beta_{1} - 18 \beta_{2} ) q^{63} + ( 359 + 89 \beta_{1} - 7 \beta_{2} ) q^{64} + ( 20 - 60 \beta_{1} - 10 \beta_{2} ) q^{65} + ( 33 + 33 \beta_{1} ) q^{66} + ( -128 + 148 \beta_{1} - 28 \beta_{2} ) q^{67} + ( 12 + 108 \beta_{1} - 16 \beta_{2} ) q^{68} + ( 180 + 60 \beta_{1} + 24 \beta_{2} ) q^{69} + ( 110 - 50 \beta_{1} - 20 \beta_{2} ) q^{70} + ( -216 + 104 \beta_{1} - 16 \beta_{2} ) q^{71} + ( -135 - 27 \beta_{1} - 36 \beta_{2} ) q^{72} + ( 356 - 168 \beta_{1} + 14 \beta_{2} ) q^{73} + ( 466 + 138 \beta_{1} + 36 \beta_{2} ) q^{74} -75 q^{75} + ( 624 + 124 \beta_{1} + 74 \beta_{2} ) q^{76} + ( -22 + 22 \beta_{1} - 22 \beta_{2} ) q^{77} + ( 504 + 12 \beta_{1} + 54 \beta_{2} ) q^{78} + ( -524 - 14 \beta_{1} - 52 \beta_{2} ) q^{79} + ( -45 - 115 \beta_{1} - 35 \beta_{2} ) q^{80} + 81 q^{81} + ( 78 + 146 \beta_{1} + 58 \beta_{2} ) q^{82} + ( -582 + 164 \beta_{1} + 70 \beta_{2} ) q^{83} + ( 330 + 30 \beta_{1} + 18 \beta_{2} ) q^{84} + ( 380 - 50 \beta_{1} - 30 \beta_{2} ) q^{85} + ( 1158 + 94 \beta_{1} + 32 \beta_{2} ) q^{86} + ( -102 + 102 \beta_{1} - 60 \beta_{2} ) q^{87} + ( -165 - 33 \beta_{1} - 44 \beta_{2} ) q^{88} + ( -730 + 68 \beta_{1} ) q^{89} + ( 45 + 45 \beta_{1} ) q^{90} + ( 56 - 176 \beta_{1} + 4 \beta_{2} ) q^{91} + ( -1252 - 372 \beta_{1} - 160 \beta_{2} ) q^{92} + ( 72 - 12 \beta_{1} + 48 \beta_{2} ) q^{93} + ( -692 - 76 \beta_{1} - 132 \beta_{2} ) q^{94} + ( -240 - 10 \beta_{1} - 20 \beta_{2} ) q^{95} + ( 675 + 39 \beta_{1} + 36 \beta_{2} ) q^{96} + ( 286 - 240 \beta_{1} + 64 \beta_{2} ) q^{97} + ( 147 + 131 \beta_{1} + 64 \beta_{2} ) q^{98} + 99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 4q^{2} - 9q^{3} + 22q^{4} - 15q^{5} + 12q^{6} - 4q^{7} - 48q^{8} + 27q^{9} + O(q^{10}) \) \( 3q - 4q^{2} - 9q^{3} + 22q^{4} - 15q^{5} + 12q^{6} - 4q^{7} - 48q^{8} + 27q^{9} + 20q^{10} + 33q^{11} - 66q^{12} - 56q^{14} + 45q^{15} + 50q^{16} - 218q^{17} - 36q^{18} + 146q^{19} - 110q^{20} + 12q^{21} - 44q^{22} - 200q^{23} + 144q^{24} + 75q^{25} - 508q^{26} - 81q^{27} - 340q^{28} + 68q^{29} - 60q^{30} - 68q^{31} - 688q^{32} - 99q^{33} - 176q^{34} + 20q^{35} + 198q^{36} - 390q^{37} - 316q^{38} + 240q^{40} - 196q^{41} + 168q^{42} - 524q^{43} + 242q^{44} - 135q^{45} + 1160q^{46} - 60q^{47} - 150q^{48} - 157q^{49} - 100q^{50} + 654q^{51} + 1020q^{52} - 158q^{53} + 108q^{54} - 165q^{55} + 1368q^{56} - 438q^{57} + 1092q^{58} - 1044q^{59} + 330q^{60} + 642q^{61} + 88q^{62} - 36q^{63} + 1166q^{64} + 132q^{66} - 236q^{67} + 144q^{68} + 600q^{69} + 280q^{70} - 544q^{71} - 432q^{72} + 900q^{73} + 1536q^{74} - 225q^{75} + 1996q^{76} - 44q^{77} + 1524q^{78} - 1586q^{79} - 250q^{80} + 243q^{81} + 380q^{82} - 1582q^{83} + 1020q^{84} + 1090q^{85} + 3568q^{86} - 204q^{87} - 528q^{88} - 2122q^{89} + 180q^{90} - 8q^{91} - 4128q^{92} + 204q^{93} - 2152q^{94} - 730q^{95} + 2064q^{96} + 618q^{97} + 572q^{98} + 297q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 20 x + 26\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 14 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - \beta_{1} + 14\)

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
4.26150
1.32906
−4.59056
−5.26150 −3.00000 19.6833 −5.00000 15.7845 −10.3207 −61.4719 9.00000 26.3075
1.2 −2.32906 −3.00000 −2.57547 −5.00000 6.98719 22.4672 24.6309 9.00000 11.6453
1.3 3.59056 −3.00000 4.89212 −5.00000 −10.7717 −16.1465 −11.1590 9.00000 −17.9528
\(n\): 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.d 3
3.b odd 2 1 495.4.a.l 3
5.b even 2 1 825.4.a.s 3
5.c odd 4 2 825.4.c.l 6
11.b odd 2 1 1815.4.a.s 3
15.d odd 2 1 2475.4.a.s 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.d 3 1.a even 1 1 trivial
495.4.a.l 3 3.b odd 2 1
825.4.a.s 3 5.b even 2 1
825.4.c.l 6 5.c odd 4 2
1815.4.a.s 3 11.b odd 2 1
2475.4.a.s 3 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 4 T_{2}^{2} - 15 T_{2} - 44 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(165))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -44 - 15 T + 4 T^{2} + T^{3} \)
$3$ \( ( 3 + T )^{3} \)
$5$ \( ( 5 + T )^{3} \)
$7$ \( -3744 - 428 T + 4 T^{2} + T^{3} \)
$11$ \( ( -11 + T )^{3} \)
$13$ \( -34144 - 3560 T + T^{3} \)
$17$ \( -235104 + 9680 T + 218 T^{2} + T^{3} \)
$19$ \( -30960 + 5376 T - 146 T^{2} + T^{3} \)
$23$ \( 1664 - 2672 T + 200 T^{2} + T^{3} \)
$29$ \( 3163056 - 54364 T - 68 T^{2} + T^{3} \)
$31$ \( -1812096 - 23232 T + 68 T^{2} + T^{3} \)
$37$ \( 618952 + 36460 T + 390 T^{2} + T^{3} \)
$41$ \( -4364208 - 26076 T + 196 T^{2} + T^{3} \)
$43$ \( -31273920 - 28668 T + 524 T^{2} + T^{3} \)
$47$ \( -20966976 - 135920 T + 60 T^{2} + T^{3} \)
$53$ \( 39574952 - 260852 T + 158 T^{2} + T^{3} \)
$59$ \( -84227264 + 64144 T + 1044 T^{2} + T^{3} \)
$61$ \( 22757384 - 60548 T - 642 T^{2} + T^{3} \)
$67$ \( 87537664 - 462208 T + 236 T^{2} + T^{3} \)
$71$ \( 6553600 - 129728 T + 544 T^{2} + T^{3} \)
$73$ \( -5609344 - 299576 T - 900 T^{2} + T^{3} \)
$79$ \( -14694992 + 562208 T + 1586 T^{2} + T^{3} \)
$83$ \( -924645384 - 307644 T + 1582 T^{2} + T^{3} \)
$89$ \( 293444632 + 1406940 T + 2122 T^{2} + T^{3} \)
$97$ \( -223543736 - 1291700 T - 618 T^{2} + T^{3} \)
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